text:
\section*{Introduction}
A communication protocol with $l$ possible inputs and $k$ possible outputs  can be described by a \emph{transition matrix} $A=(a_{ij})\in [0,1]^{k\times l}$, where $a_{ij}$ is the conditional probability of output $i$  if the input is $j$. This is a \emph{stochastic} matri
: for all $j$, we have $\sum_{i=1}^ka_{ij}=1$. A \emph{communication channel} can be described  by the set of transition matrices that it affords.  Channel Q \emph{can be simulated} by channel C  if all transition matrices afforded by Q are convex combinations of transition matrices afforded by C.  Such convex combinations occur naturally in information theory; they correspond to the sender and receiver having access to (unlimited)  shared randomness. The relation `can be simulated by' is obviously reflexive and transitive.  Two channels are \emph{equivalent} if each can be simulated by the other.

The \emph{classical channel with $n$ states} affords stochastic 0-1 matrices with at most $n$ nonzero rows.   The \emph{quantum channel of level $n$} affords channel matrices of the form $(\operatorname{tr} E_i\rho_j)$, where $\rho_1, \dots, \rho_l\in M_n(\mathbb{C})$ are \emph{density matrices},
and $E_1, \dots, E_k\in M_n(\mathbb{C})$ is a \emph{positive operator valued measure (POVM)}.
  It is easy to see that the classical channel with $n$ states can be simulated by the quantum channel of level $n$. By \cite[Theorem 3]{FW} of Weiner and the present author, the converse also holds.  The present paper is about variants of this theorem   for  general probabilistic channels (Section~\ref{GPT}) and for noisy quantum channels (Section~\ref{quantum}).  In Section~\ref{noiselesssimul}, we discuss noiseless classical simulations of noisy channels.  Section~\ref{future} contains an open problem tentatively linking classical simulations of quantum channels to the more traditional way of comparing  efficiency of classical and quantum communication, involving von Neumann entropy, mutual information and Holevo's inequality. The reader who is  interested in  quantum information theory but not in general probabilistic theory can safely skip Section~\ref{GPT}.

\bigskip\noindent
\bf Notations and terminology.
\rm The set $\{1,\dots, k\}$ is denoted by  $[k]$.  For a real number $a$, we write  $a_+=\max (a, 0)$. The indicator of an event $A$ is written $\mathbb 1(A)$. A \emph{convex body} is a  convex compact set with nonempty interior.

 A matrix is \it stochastic
\rm if all entries are nonnegative reals and each column sums to 1.  The set of $n$-square matrices with complex entries is written $M_n(\mathbb{C})$. The identity matrix is $\bf 1$.
A complex  matrix $A$  is \it psdh \rm if it is positive semi-definite Hermitian, written $A\ge 0$. A \emph{positive operator valued measure (POVM)} is a sequence $E_1$, \dots, $E_k$ of psdh matrices summing to $\bf 1$.
A \it density matrix \rm is a psdh matrix with trace 1.

For $0\le\delta\le 1$, the \emph{$\delta$-noisy classical channel with $n$ states} affords transition matrices of the form $EX\in [0,1]^{k\times l}$, where  $E\in \{0,1\}^{k\times n}$ is a stochastic 0-1 matrix and $X$
  is an $n\times l$ stochastic matrix with each column containing $n-1$ entries equal to $\delta/n$. Here  $E$ can be interpreted as  a classical decoding map $[n]\to[k]$, and the columns of $X$ can be interpreted as extremal $\delta$-noisy classical states. The presence of noise impedes the exact transmission of pure states, the pure state chosen by the sender is transmitted unchanged with probability $1-(n-1)\delta/n$ but turns into each one  of the other $n-1$ pure states with probability $\delta/n$.

Note that the 0-noisy classical channel with $n$ states is  the same as the classical channel with $n$ states defined previously.

Following the  terminology of \cite{A, DC}, the \emph{signalling dimension} $\signdim \mathrm Q$ of a  channel $\mathrm Q$ is the smallest positive integer $n$ such that  $\mathrm Q$ can be simulated by the (noiseless) classical channel with $n$ states.

\section{General probabilistic theory}\label{GPT}
Let $S$ be  a convex body in a finite dimensional real affine space. Let $E$ be the cone of \emph{effects}, i.e., affine linear functions $e:S\to[0,\infty)$.  A \emph{partition of unity} is   a  sequence $e_1, \dots, e_k\in E$ of effects such that $e_1+\dots +e_k=1$  (the constant 1 function). The \emph{channel with state space $S$}  affords transition matrices of the form $(e_i(x_j))\in[0,1]^{k\times l}$, where $x_1, \dots, x_l\in S$, and $e_1$, \dots, $e_k$ is a partition of unity.

\subsection{Signalling dimension vs.\ information storability}
Following terminology introduced in \cite{A}, the \emph{signalling dimension} $\signdim S$ of $S$ is the signalling dimension of the channel with state space $S$, i.e., the smallest positive integer $n$ such that the channel with state space $S$ can be simulated by the classical channel with $n$ states.  By \cite[Theorem 3]{FW} mentioned in the Introduction, the signalling dimension of the set of $n$-square density matrices is $n$.

Calculating, or even efficiently estimating the signalling dimension of a given convex body seems to be a   difficult problem, and strong general theorems are yet to be searched for. In this section, we  start with weak  general results and work our way towards deeper results for   special cases.

The \emph{affine dimension} $\affdim S$ of $S$ is the minimal dimension of an affine space containing $S$.  Adding 1, we get the  \emph{linear dimension} $\lindim S$ of $S$, i.e., the dimension of the vector space of affine linear functions on $S$.  For example, the affine dimension of the set of $n$-square density matrices is $n^2-1$, while its linear dimension is $n^2$.

A partition of unity is \emph{extremal} if it cannot be written as a convex combination of two partitions of unity in a nontrivial way. The nonzero effects appearing in an extremal partition of unity need not lie on  extremal rays of the cone $E$ of effects. When they do, a characterization of extremal partitions of unity  is given in \cite[Theorem 2]{A}. We now give a necessary condition of extremality for a general partition of unity. Although  this
is implicitly contained  in the paper cited above (see the proof given there), we include  a proof.

\begin{Pro}\label{rang}  The nonzero effects in an extremal partition of unity are linearly independent. Thus, their number is $\le$ the linear dimension of $S$.
\end{Pro}

\begin{proof} Let $e_1$, \dots, $e_k$ be an extremal partition of unity. If $\lambda_1 e_1+\dots +\lambda_k e_k=0$ and $|\epsilon|\le 1/\max \{|\lambda_i|:  \lambda_i\ne 0\}$, then $(1\pm\epsilon\lambda_1)e_1$, \dots, $(1\pm\epsilon\lambda_k)e_k$ is also a partition of unity, which must coincide with $e_1$, \dots, $e_k$ because of extremality. Thus $\lambda_ie_i=0$ for all $i$.
\end{proof}

Consider the   transition matrix $A=(a_{ij})\in [0,1]^{k\times l}$ of some communication protocol, where $a_{ij}$ is the conditional probability of output $i\in[k]$ if the input was $j\in[l]$. Let us try to guess the input based on the output, using a  function $G:[k]\to [l]$.  If input $j$ occurs with  probability $q_j$, then the probability of success will be $$\sum_{j=1}^l q_j\sum _{i=1}^ka_{ij}\mathbb 1(G(i)=j)=\sum_{i=1}^kq_{G(i)}a_{i,G(i)}.$$  Choosing the best possible guessing function $G$, the probability of success is  $$\sum_{i=1}^k\max_j q_ja_{ij}.$$  Without any communication, the probability of successfully guessing the input, with the optimal  strategy, is $\max_j q_j$.  The ratio $\sum_{i=1}^k\max_j q_ja_{ij}/\max_j q_j$ is maximized when $q_j=1/l$ for all $j$, in which case it simplifies to  $\sum_{i=1}^k\max_j a_{ij}$.  Motivated by these considerations, and following \cite{MK} by Matsumoto and Kimura, the \emph{information storability} $\infstor S$ of $S$ is defined to be the maximum of $\sum_{i=1}^k\max_j a_{ij}$
over all transition matrices $(a_{ij})$  afforded by $S$, or, equivalently, the maximum of   $\sum_{i=1}^k\max_S e_{i}$  over all partitions of unity $e_1$, \dots, $e_k$.
When taking these maxima, it suffices to consider extremal partitions of unity. Then  Proposition~\ref{rang} and a  simple compactness argument shows  that these maxima are attained.

As a simple example, let $S=[0,1]$. Then we can choose the partition of unity $1=x+(1-x)$ to show that  $$\infstor S\ge \max_{0\le x\le 1}x+\max_{0\le x\le 1}(1-x)=1+1=2.$$ On the other hand, as any  affine  linear function on $S$ takes its maximum at 0 or 1, we have $$\sum_{i=1}^k\max_S e_i=\sum_{e_i(0)\ge e_i(1)} e_i(0)+\sum_{e_i(0)< e_i(1)} e_i(1)\le 1+1=2$$ for any partition of unity $e_1$, \dots, $e_n$ on $S$,  whence $\infstor S=2$.
This is the easiest special case of \cite[Theorems 1 and 4]{MK},  cited below in relation to  Theorem~\ref{signdim} and Proposition~\ref{MK}.

By \cite[Theorem 4]{MK},  $\infstor S\le \lindim S$. We  refine this inequality as follows.

\begin{Th}\label{signdim}
\begin{enumerate}
\item $\infstor S\le\signdim S \le \lindim S$.

\item
If  $\infstor S\le \affdim S$, then $\signdim S\le \affdim S$.

\end{enumerate}
\end{Th}

This theorem is closely related to \cite[Theorem 1(i)]{DC}.

\begin{proof}
(1)  Let $n=\signdim S$. Any transition matrix afforded by $S$ is a convex combination of transition matrices afforded by the classical channel with $n$ states. Such a matrix has $\le n$ nonzero rows and therefore sum of row-maxima $\le n$. This property is preserved  when taking convex combinations. This proves the first inequality.

Any transition matrix afforded by $S$ is a convex combination of transition matrices of the form  $(e_i(x_j))$, where $e_1$, \dots, $e_k$ is an \emph{extremal}  partition of unity, and $x_j\in S$. By Proposition~\ref{rang}, such a matrix has $\le\lindim S$ nonzero rows, and therefore is a convex combination  of matrices afforded by the classical channel with $\lindim S$ states. This proves the second inequality.

(2)
Let $\infstor S\le \affdim S=n$. Any transition matrix afforded by $S$ is a convex combination of matrices of the form $A=(a_{ij})\in [0,1]^{k\times l}$, where  $a_{ij}=e_i(x_j)$, $e_1$, \dots, $e_k$ is an \emph{extremal} partition of unity, and $x_j\in S$. We shall show that such an $A$ is always a convex combination of transition matrices afforded by the classical channel with $n$ states.  Using Proposition~\ref{rang}, we may assume that $k=n+1$.  Set $m_i=\max_S e_i\in [0,1]$ for each $i\in [k]$.
Note that
$\sum_{i=1}^k(1-m_i)\ge n+1-\infstor S\ge 1$.   Choose a probability distribution $p_1$, \dots, $p_k$ such that  $p_i\le 1-m_i$ for all $i$. Then $$p_i\le 1-a_{ij}=\sum_{i'\ne i} a_{i'j}$$  for all $i$ and $j$, and  $$\sum_{i\in T}p_i\le 1=\sum_{i=1}^k a_{ij}$$ for all $T\subseteq [k]$.

For any fixed $j$, put  supply $a_{ij}$ and demand $p_i$  at each node $i$  of the complete (but loopless)  graph on  $k$ nodes. Then, for the total supply  at the neighbors of any  subset $T\subseteq [k]$, we have $$\sum_{i\in N(T)}a_{ij}\ge \sum_{i\in T}p_i.$$  By the Supply--Demand Theorem~\cite[2.1.5.\ Corollary]{LoPlu}, the demands can be met: there exist stochastic column vectors $b_j(1)$, \dots, $b_j(k)$  such that  the $i$-th entry of $b_j(i)$ is zero for all $i$, and $\sum_{i=1}^kp_ib_j(i)$  is the $j$-th column of $A$. Now let $B(i)$ be the matrix with columns  $b_1(i)$, \dots, $b_l(i)$.  Then the $i$-th row of $B(i)$ is zero, so $B(i)$ has $\le k-1=n$ nonzero rows, so $B(i)$ is a convex  combination of transition matrices afforded by the classical channel with $n$ states.  Then so is $A$, since  $$A=\sum_{i=1}^k p_iB(i).$$
\end{proof}

For the remainder of this section, assume that $S$ is not just a point.  A \emph{chord} \rm  of $S$ is a segment $AB$ whose endpoints $A$ and $B$ belong to the boundary  of $S$.  We write $AOB$ for a chord $AB$ with a distinguished point $O$ on the chord. The convex body $S$ is \emph{centrally symmetric} if  there exists a point $O\in S$ such that for any chord $AOB$ of $S$, we have $|OA|=|OB|$ for the lengths of the segments $OA$ and $OB$.
The \emph{Minkowski measure of asymmetry}  $\asymm S$ of $S$ is  the smallest real number $m\ge 1$  such that  there exists a point $O\in S$ such that for any chord $AOB$ of $S$, we have $|OB|\le m|OA|$.

 By \cite[Theorem 1]{MK}  of Matsumoto and Kimura, the information storability  is related to the Minkowski measure of asymmetry  as follows.

\begin{Pro}\label{MK}  $\infstor S = \asymm S +1$
\end{Pro}
Although this is a known statement, we include the sketch of a geometric proof for the convenience of the reader.
\begin{proof}
$\le$:
There exists a point $O\in S$ such that for any chord $AOB$ of $S$, we have $|OB|\le (\asymm S)|OA|$.
 Let $n=\asymm S +1$.  Then $e(x)\le ne(O)$ for all  $e\in E$ and $x\in S$, whence  $$\sum_{i=1}^k\max_S e_{i}\le n\sum_{i=1}^ke_i(O)=n$$   for all partitions of unity $e_1$, \dots, $e_k$.

$\ge$: Let $n=\infstor S$. Then   $\sum_{i=1}^k\max_S e_{i}\le n$   for all partitions of unity $e_1$, \dots, $e_k$.  When $k$ is the linear dimension of $S$, this tells us that for any simplex $\Delta$ containing $S$, there exists a point each of whose  barycentric coordinates with respect to $\Delta$  is at least $1/n$ times  the maximum value of that barycentric coordinate on $S$.
  Using Helly's theorem, we see that  there exists a point $O$ that divides  the distance between any two parallel supporting hyperplanes of $S$  in a ratio at least as equitable as $1:(n-1)$.  Then, for any chord $AOB$  of $S$ with $|AO|\le |OB|$, considering the supporting hyperplane  of $S$ at $A$  and the parallel supporting hyperplane, we get that $|OB|\le (n-1)|OA|$.
\end{proof}

\begin{Cor} For the regular octahedron, we have $\asymm =1$, $\infstor =2$,  $\signdim=\affdim =3$, and $\lindim =4$.
\end{Cor}

\begin{proof}  The regular octahedron is centrally symmetric, which means that  $\asymm =1$.
By Proposition~\ref{MK}, we have $\infstor=\asymm +1 =2$. Obviously, $\affdim =3$ and $\lindim=\affdim +1=4$.

By Theorem~\ref{signdim}(2), we have $\signdim\le 3$.  To prove the converse inequality, let $$X=\begin{pmatrix} 1 & -1 &&&&\\&&1&-1&&\\&&&&1&-1\end{pmatrix}$$  be the matrix whose columns are the vertices of the octahedron (the entries not shown are zero).  Let $$V=\begin{pmatrix} 1&1&1\\1&-1&-1\\-1&1&-1\\-1&-1&1\end{pmatrix},$$ then $$VX=\begin{pmatrix} 1&-1&1&-1&1&-1\\1&-1&-1&1&-1&1\\-1&1&1&-1&-1&1\\-1&1&-1&1&1&-1\end{pmatrix}.$$ Adding 1 to each entry and dividing by 4, we get the stochastic matrix
$$A=\frac12\begin{pmatrix} 1&0&1&0&1&0\\1&0&0&1&0&1\\0&1&1&0&0&1\\0&1&0&1&1&0\end{pmatrix},$$
which is therefore a  transition matrix afforded by the octahedron.  Since any two rows of $A$ have an 1/2 at the same position, we have $$\sum_{1\le i<i'\le 4}\max_{1\le j\le 6}(a_{ij}+a_{i'j})= {4\choose 2} =6.$$ On the other hand, any $4\times 6$ transition matrix afforded by the classical channel with 2  states has at least $4-2=2$ zero rows, so the sum above would be
$\le  {4\choose 2}-{{4-2}\choose 2}=5$  --- note that this is a special case of  \cite[inequality (3.6)]{FW}.   This inequality  is preserved under convex combinations.  Therefore, the octahedron cannot be simulated by the classical 2-state channel, hence its signalling dimension is (at least) 3.
\end{proof}

\subsection{Noisy balls}
If an  origin is chosen in $S$, and $0\le \delta \le 1$, then the \emph{$\delta$-noisy channel with state space $S$} affords the transition matrices $(e_i(x_j))$, where $e_1$, \dots, $e_k$  is a partition of unity and  $x_j\in (1-\delta)S$ for all $j$.  This is analogous to the partial depolarization channel in quantum information theory, cf.\ Subsection~\ref{depol}. Note that $e_i\ge 0$ is required on all of $S$.

It is easy to see that if $S'=f(S)$ is an affine image of $S$, then $S'$ can be simulated by $S$. If, in addition,    $O'=f(O)$, then $\delta$-noisy $S'$ can be simulated by $\delta$-noisy $S$. In particular, a  classical bit can be simulated by $S$ unless $S$ is just a point, and a $\delta$-noisy classical bit can be simulated by any $\delta$-noisy $S\ne\{O\}$  that   is symmetric with respect to $O$.

\begin{Th}\label{ball} Let $n$ be an even positive integer. Put $$S=\{x\in \mathbb{R}^d: \|x\|_{n/(n-1)}\le 1\},$$  the unit ball of the  $n/(n-1)$-norm. Let $0\le \delta \le 1$.
\begin{enumerate}
\item The $\delta$-noisy channel with state space $S$ can be simulated by the $\delta$-noisy classical channel with $n$ states.
\item The signalling dimension of $S$ is $\le n$.

\item For an ellipsoid of arbitrary affine dimension $\ge 1$, the signalling dimension  is $2$. A $\delta$-noisy ellipsoid can be simulated by a $\delta$-noisy classical bit.
\end{enumerate}
\end{Th}

The proof below is similar to that of \cite[Theorem 3]{FW}.  However, the mixed discriminant used there (and used in Section~\ref{quantum} of the present paper) must be replaced by a  different $n$-linear symmetric function $\{\cdot, \dots, \cdot\}$.

To introduce $\{\cdot, \dots, \cdot\}$,  we can think of an affine linear function $e:S\to \mathbb{R}$ as a formal sum of a number and a vector: $e=c+v\in\mathbb{R}\oplus \mathbb{R}^d=\mathbb{R}^{d+1}$, meaning that $e(x)=c+vx$ for $x\in S$, where $vx$ is the usual inner product. For an effect $e\in E$, the condition $e\ge 0$ translates to $\|v\|_n\le c$ because $$(n/(n-1))^{-1}+n^{-1}=1.$$  Given $e_1, \dots, e_n\in \mathbb{R}^{d+1}$, where $e_i=c_i+v_i$, we define $$\{e_1,\dots, e_n\}=c_1\cdots c_n-v_1\cdots v_n,$$ where $v_1\cdots v_n$ means that we take the coordinatewise product and then add up the coordinates (which is an $n$-linear generalization of the usual inner product). For $n=2$, $\{\cdot,\cdot\}$ is the Lorentzian indefinite symmetric bilinear product well known from the special theory of relativity.  For general $n$,   $\{\cdot,\dots,\cdot\}$ is symmetric,  multilinear and $\{1, \dots, 1\}=1$.  When $e_1, \dots, e_n\in E$, we have
$\{e_1,\dots, e_n\}\ge 0$ by repeated application of H\"older's inequality.  Further,  if $0\le e\le 1$ holds pointwise on $S$, then writing $e=c+v$ and $a=\|v\|_n$, we have $0\le a\le\min (c, 1-c)$ and therefore  \begin{align*}\{e,\dots, e\}=c^n-v^n\overset * =c^n-a^n=\\= (c-a)(c^{n-1}+c^{n-2}a+\dots +ca^{n-2}+a^{n-1})\le \\\le (c-a)(c+(1-c))^{n-1}=c-a=\min_{x\in S} e(x).\end{align*}  Note that the equality marked by a * holds because $n$ is even.

We are now ready to start the proof of Theorem~\ref{ball}.

\begin{proof}(1)
Let  $A\in[0,1]^{k\times l}$ be  a $\delta$-noisy transition matrix afforded by $S$, i.e., $$a_{ij}=e_i((1-\delta)x_j),$$ where $x_1, \dots, x_l\in S$, $e_i\in E$, and $e_1+\dots +e_k=1$. We shall prove that $A$ is a convex combination  of $\delta$-noisy $n$-state classical transition matrices.

If $e_i=c_i+v_i$ as before, then $c_1+\dots+c_k=1$, $v_1+\dots+v_k=0$, and  $$a_{ij}=c_i+(1-\delta)v_ix_j=\delta c_i+(1-\delta)e_i(x_j),$$ so $A=\delta C+(1-\delta) A'$, where $C$ is the matrix with entries $c_{ij}=c_i$  not depending  on $j$, and $A'$ is the matrix with entries $a'_{ij}=e_i(x_j)$.

For $I=(i_1,\dots, i_n)\in [k]^n$, put
\[p_I=\{e_{i_1},\dots, e_{i_n}\}.\]
We have $p_I\ge 0$ for all $I$. Thus, we get a  measure $P$
on $[k]^n$ defined by $$P(T)=\sum_{I\in T}p_I.$$
Using the multilinearity of the bracket and the assumption that $e_1$, \dots, $e_k$ is a partition of unity, we see that
\[P([k]^n)=\{ 1,\dots,  1\}=\rm 1,\]
so $P$ is  a probability measure.

Let $D(I)$ be the matrix with entries $d(I)_{ij}=m(i,I)/n$  not depending on $j$, where $m(i,I)$ is the number of occurrences of $i$ in the sequence $I$. Then $\int D\mathrm d P=C$ because $$\int d_{ij}\mathrm d P=\sum_{I\in[k]^n}p_Im(i,I)/n=\{e_i, 1, \dots, 1\}=c_i=c_{ij}.$$

For any $R\subseteq [k]$, we may put $e_R=\sum_{i\in R} e_i$, and then  we have
\[P(R^n)=\{e_R, \dots, e_R\}\le\min_{x\in S} e_R(x)\le e_R(x_j)\] for all $j$ since $0\le e_R\le 1$.
The right hand side here is $A'_j(R)$, where $A'_j$ is the probability measure on $[k]$ given by the numbers $e_i(x_j)$. So we have
\[A'_j(R)\ge P(R^n)\qquad\textrm{ for all }R\subseteq [k].\]

Let us connect $I\in [k]^n$ to $i\in [k]$ by an edge if $i$ occurs in $I$. This gives us a bipartite graph. The neighborhood of any set $T\subseteq [k]^n$ is the set $R\subseteq [k]$ of indices occurring in some element of $T$. We always have $T\subseteq R^n$, whence
$$A'_j(R)\ge P(R^n)\ge P(T).$$ Thus, by the Supply--Demand Theorem~\cite[2.1.5.\ Corollary]{LoPlu}, and using the fact that both $A'_j$ and $P$ are probability measures, there exists a probability measure $P_j$ on $[k]^n\times [k]$ which is supported on the edges of the graph and has marginals $P$ and $A'_j$. Whenever $p_I\ne 0$, let $B'(I)$ be the $k\times l$ stochastic matrix whose $j$-th column is given by the conditional distribution  $P_j|I$ on $[k]$.  We have  $A'=\int B'\mathrm dP$.

 Now $B(I)=\delta D(I)+(1-\delta) B'(I)$  is a convex combination of $\delta$-noisy  $n$-state classical transition matrices,  and, in turn,  $A=\int B\mathrm d P$ is a convex combination of the $B(I)$, as desired.

\bigskip

(2)  Set $\delta=0$ in (1).

\bigskip

(3) The signalling dimension of an ellipsoid is the same as that of the Euclidean unit ball. This is $\le 2$ by (2), and is $\ge 2$ because the unit ball is not a point. The noisy claim follows from (1).
\end{proof}

\section{Noisy quantum channels}\label{quantum}
Let $$K\subseteq \Delta_n=\{(\xi_1,\dots, \xi_n): \xi_i\ge0 \; \textrm{ for all } \; i, \;  \xi_1+\dots +\xi_n=1\}$$
  be  a convex set of probability distributions that is invariant under all permutations of the $n$ coordinates.  The \emph{$K$-noisy classical channel}  affords  transition matrices of the form $EX\in [0,1]^{k\times l}$, where  $X\in K^l$  is an $n\times l$ matrix  with all columns in $K$, and $E$ is a  $k\times n$ stochastic 0-1 matrix. A density matrix is  \emph{$K$-noisy} if the sequence of its eigenvalues is in $K$. The \emph{$K$-noisy quantum channel} affords transition matrices of the form $(\operatorname{tr} E_i\rho_j)$, where $E_1$, \dots, $E_k$ is a POVM and $\rho_j$ is a $K$-noisy density matrix for $j=1, \dots, l$.

It is easy to see that  the $K$-noisy classical channel can be simulated by  the $K$-noisy quantum channel. Our goal is to prove the converse, which is a far-reaching generalization of \cite[Theorem 3]{FW}  mentioned in the Introduction.

In fact, we may generalize further. Let $K_j\subseteq \Delta_n$   ($j=1, \dots, l$)  be   convex sets, each of them  invariant under all permutations of the $n$ coordinates.  The \emph{$(K_1, \dots, K_l)$-noisy classical channel}  affords  transition matrices of the form $EX\in [0,1]^{k\times l}$, where  $X\in K_1\times\dots\times K_l$  is an $n\times l$ matrix  with $j$-th column in $K_j$, and $E$ is a  $k\times n$ stochastic 0-1 matrix.  The \emph{$(K_1, \dots, K_l)$-noisy quantum channel} affords transition matrices of the form $(\operatorname{tr} E_i\rho_j)$, where $E_1$, \dots, $E_k$ is a POVM and $\rho_j$ is a $K_j$-noisy density matrix for $j=1, \dots, l$.

It is easy to see that  the $(K_1, \dots, K_l)$-noisy classical channel can be simulated by  the $(K_1, \dots, K_l)$-noisy quantum channel. We shall prove the converse.

As in \cite{FW}, our  main tool is the \emph{mixed discriminant}, the unique symmetric  $n$-linear function $D$ on $M_n(\mathbb{C})$ such that  $D(E, \dots, E)=\det E$ for
all  $E\in M_n(\mathbb{C})$. Explicitly, if  $E_i=\left[e_i^1, \dots, e_i^n\right]$  are the columns, then \begin{equation}\label{*} D(E_1, \dots, E_n)=\frac1{n!}\sum_ {\pi\in \mathfrak S_n} \det\left[e_{\pi(1)}^1, \dots, e_{\pi(n)}^n\right].\end{equation}

We shall need the
following inequalities.

\begin{Lemma}\label{L1}  For $\lambda_1, \dots, \lambda_n\in [0,1]$ and $r=1,2, \dots, n$, we have
\begin{equation}\label{lambda}  \sum_{Q\subseteq[n]}(r-|Q|)_+\prod_{m\notin Q}\lambda_m\prod_{m\in Q}(1-\lambda_m)\le \lambda_1+\dots+\lambda_r,\end{equation} where $a_+=\max (a, 0)$.
\end{Lemma}

\begin{proof}We have $$(r-|Q|)_+\le \left|[r]\setminus Q\right|=\sum_{s=1}^r{\mathbb {1}} (s\notin Q)$$ for all $Q$. Thus, the left hand side of \eqref{lambda} is $$\le\sum_{s=1}^r\sum_{Q\subseteq [n]\setminus\{s\}}\prod_{m\notin Q}\lambda_m\prod_{m\in Q}(1-\lambda_m)=\sum_{s=1}^r\lambda_j\prod_{m\ne s}(\lambda_m+(1-\lambda_m))= \lambda_1+\dots+\lambda_r.$$
\end{proof}

\begin{Lemma}\label{L2}
For an $n$-square Hermitian matrix $0\le E\le\bf 1$ with eigenvalues $\lambda_1$,  \dots, $\lambda_n$, and $r=1,2, \dots, n$, we have  $$\sum_{q=0}^{r-1}  (r-q){n\choose q}  D(\underbrace{E, \dots, E}_{n-q}, \underbrace{{\bf 1}-E, \dots,{\bf 1}- E}_{q})\le\lambda_1+\dots+\lambda_r. $$
\end{Lemma}

\begin{proof}
  Since the spectrum and the mixed discriminant are both invariant under unitary conjugation, we may assume  that $E$ is a diagonal matrix. Then \eqref{*} reduces Lemma~\ref{L2} to Lemma~\ref{L1}.
\end{proof}

By Bapat's  \cite[Lemma 2(vi)]{B}, if  $E_1$, \dots, $E_n$ are all positive semidefinite Hermitian matrices, then  \begin{equation}\label{**}D(E_1, \dots, E_n)\ge 0.\end{equation}
Given a POVM $E_1, \dots, E_k\in M_n(\mathbb{C})$, we define  \begin{equation}\label {p}  p_I=D(E_{i_1}, \dots, E_{i_n})\end{equation} for all $I=(i_1, \dots, i_n)\in [k]^n$. By multilinearity and \eqref{**}, this defines  a probability distribution on $[k]^n$.

\begin{Lemma}\label{L3}
If  $E_1, \dots, E_k\in M_n(\mathbb{C})$ is  a POVM, $u_1$, \dots,  $u_k$  are real numbers, and $\lambda_1$, \dots, $\lambda_n$ are the eigenvalues of $E=\sum_{i=1}^k u_iE_i$,  then \begin{equation}\label{***}
\sum_{I\in[k]^n}  p_I\min\left\{\sum_{m\in S}u_{i_m}: S\subseteq [n], |S|=r\right\}\le \lambda_1+\dots+\lambda_r\end{equation} for all $r=1,2,\dots, n$.
\end{Lemma}

\begin{proof}
We may assume that all $u_i\ge 0$ because adding $u$ to all $u_i$ adds $ru$ to both sides of \eqref{***}.  We may assume $u_1\ge\dots \ge u_k$. Put $u_{k+1}=0$.  Write  $E=\sum_{i=1}^kv_iF_i$, where $v_i =u_i-u_{i+1}$ and $F_i=E_1+\dots+E_i$.

Let $\sigma_i$ be the sum of the $r$ smallest eigenvalues of $F_i$. Then \begin{equation}\label{1} \sum_{i=1}^kv_i\sigma_i\le\lambda_1+\dots+\lambda _r.\end{equation}
As $0\le F_i\le\bf 1$, we have \begin{equation}\label{2}  \sum_{q=0}^{r-1}   (r-q)\binom{n}{q} D(\underbrace{F_i, \dots, F_i}_{n-q}, \underbrace{{\bf 1}-F_i, \dots,{\bf 1}- F_i}_{q})\le  \sigma_i\end{equation}  for all $i$, by Lemma~\ref{L2}.

On the other hand, since  $u_i=v_i+\dots +v_k$, we have $$\min\left\{\sum_{m\in S}u_{i_m}: S\subseteq [n], |S|=r\right\}=\sum_{i=1}^kv_i\left(r-|\{m\in [n]: i_m>i\}|\right)_+.$$ It remains to check that  \begin{align*}\sum_{I\in[k]^n}  p_I\left(r-|\{m\in [n]: i_m>i\}|\right)_+=\\=\sum_{q=0}^{r-1}  (r-q){n\choose q} D(\underbrace{F_i, \dots, F_i}_{n-q}, \underbrace{{\bf 1}-F_i, \dots,{\bf 1}- F_i}_{q})\end{align*} for all $i\in [k]$. This follows from \begin{align*}\sum\left(p_I:I\in [k]^n, |\{m\in [n]:i_m>i\}|=q\right)=\\={n\choose q} D(\underbrace{F_i, \dots, F_i}_{n-q}, \underbrace{{\bf 1}-F_i,\dots,{\bf 1}- F_i}_{q}),\end{align*} which is clear from the definitions of $p_I$ and $F_i$, and from the symmetry and multilinearity of $D$.
\end{proof}

We are ready for the main result of this paper.

\begin{Th}\label{main} The $(K_1, \dots, K_l)$-noisy quantum channel can be simulated by  the $(K_1, \dots, K_l)$-noisy classical channel.
In particular, the $K$-noisy quantum channel can be simulated by the $K$-noisy classical channel.
\end{Th}

\begin{proof}
It suffices to prove that  for any  POVM $E_1$, \dots, $E_k$, and any $K$-noisy density matrix $\rho$, there exist points $x_I=(x_{I,1}, \dots, x_{I,n})\in K$  for each  $I=(i_1, \dots, i_n)\in [k]^n$  such that \begin{equation}\label{fo}
\operatorname{tr} E_i\rho=\sum_{I\in [k]^n}p_I\sum(x_{I,m}:m\in [n], i_m=i)
\end{equation} for each $i\in [k]$.  Here the $p_I$ are defined as in \eqref{p}.

Let the eigenvalues of $\rho$ be $0\le\mu_1\le\dots\le \mu_n$; we have $\mu_1+\dots+\mu_n=1$.   Since $\rho$ is $K$-noisy, we have $\mu=(\mu_1, \dots, \mu_n)\in K$.   Since $K$ is convex and invariant with respect to permutations, any  convex combination of permutations of $\mu$ is in $K$. Thus, if $x\in [0,1]^n$ is a stochastic  vector, and any $r$ distinct coordinates of $x$ sum to $\ge  \mu_1+\dots +\mu_r$ for each $r=1,2, \dots, n$, then $x\in K$. If we require
\begin{itemize}
\item these $2^n$ inequalities for each $x_I$, together with

\item $x_{I,m}\ge 0$  for all $I$ and $m$, and

\item \eqref{fo} for all $i$,

\end{itemize}
then
each $x_I$ will be  a stochastic vector  since
setting $r=n$  yields  $$x_{I,1}+\dots+x_{I,n}\ge \mu_1+\dots+\mu_n=1,$$ while summing \eqref{fo} for  $i=1,2,\dots, k$  yields $$1=\sum_{I\in[k]^n} p_I(x_{I,1}+\dots+x_{I,n}).$$
Therefore, it suffices to prove that the system of $(2^n+n)k^n$ inequalities and $k$ equations above has  a solution.
By the well-known Farkas Lemma, this is equivalent to saying that  a linear combination of the  inequalities and equations in the system cannot lead to the contradictory inequality $0\ge 1$. That is, it suffices to prove that if  nonnegative numbers $w_{I,H}$  $(I\in [k]^n, H\subseteq [n])$  and real numbers $u_1$, \dots, $u_k$ satisfy
\begin{equation}\label{start}\sum(w_{I,H}:H\subseteq [n], H\ni m)\le p_I u_{i_m}\end{equation} for all  $I\in [k]^n$ and all $m\in [n]$, then \begin{equation}\label{cel}\sum_{I\in[k]^n}\sum_{H\subseteq [n]}w_{I,H}(\mu_1+\dots +\mu_{|H|})\le\sum_{i=1}^ku_i\operatorname{tr} E_i\rho.\end{equation}

Let $\lambda_1\le\dots \le\lambda_n$  be the eigenvalues of $u_1E_1+\dots+u_kE_k$.   By von Neumann's inequality, the  right hand side of \eqref{cel} is  $$\ge \lambda_1\mu_n+\dots+\lambda_n\mu_1.$$

The coefficient of any $\mu_t$  on the left hand side  of \eqref{cel} is  $$\sum_{I\in[k]^n}\sum_{|H|\ge t}w_{I,H},$$ so it suffices to prove that  $$\sum_{t=n-r+1}^n\sum_{I\in[k]^n}\sum_{|H|\ge t}w_{I,H}\le\lambda_1+\dots+\lambda_r$$ for $r=1,\dots, n$. In view of Lemma~\ref{L3}, this follows if  $$\sum_{t=n-r+1}^n\sum_{|H|\ge t}w_{I,H}\le p_I\sum_{m\in S}u_{i_m}$$  for all $I\in [k]^n$ and all  $S\subseteq [n]$ with $|S|=r$. This follows from \eqref{start}  and the fact that  $$\sum_{n-r<t\le|H|}1=(|H|+r-n)_+\le |S\cap H|=\sum_{m\in S\cap H} 1$$ for all $H,S\subseteq [n]$ with $|S|=r$.
\end{proof}

\section{Simulation of a noisy channel by a noiseless one}\label{noiselesssimul}
Given the $K$-noisy channel, we might try to determine its signalling dimension, i.e., simulate it by a noiseless classical channel with as few states as possible.  In view of Theorem~\ref{main}, it makes no difference whether the given channel is classical or quantum
\begin{Th}\label{noiseless} The $K$-noisy classical (or, equivalently, quantum) channel can be simulated by the noiseless d-state classical (or, equivalently, level $d$ quantum) channel  if and only if we have \begin{equation}\label{binom} \mu_1+\dots+\mu_r\ge\binom rd\bigg/\binom nd\end{equation}  for all $\mu=(\mu_1\le \dots\le\mu _n)\in K$ and all integers $d\le r\le n$.
\end{Th}

\begin{proof}  `Only if':  Let $\mu=(\mu_1\le \dots\le\mu _n)\in K$. Let $A=(a_{ij})$ be an $n\times n!$  stochastic matrix whose  columns are the $n!$ permutations of $\mu$.  Then $A$ is a transition matrix afforded by  the $K$-noisy channel, thus also by the noiseless $d$-state channel.  By \cite[Section 3]{FW}, we then have $$\binom nr  (\mu_1+\dots+\mu_r)=\sum_{|S|=r}\min_{j\in[l]}\sum_{i\in S}a_{ij}
\ge \binom{n-d}{n-r}$$ for all $d\le r\le n$, which is equivalent to \eqref{binom}.

`If': Let $S$ be a uniform random $d$-element subset of $[n]$.  It suffices to prove that, for any $\mu\in K$, there is a random element $m$ of $S$ whose distribution is given by $\mathbb P(m=r)=\mu_r$ for all $r=1, \dots, n$. We may assume $\mu_1\le \dots\le\mu _n$. Let $\nu_r=\mathbb P(\max S=r)$, then $$\nu_1+\dots+\nu_r=\mathbb P(\max S\le r)=\binom rd\bigg/\binom nd \le \mu_1+\dots+\mu_r,$$ so $\mu$ is a convex combination of the permutations of $\nu$. But $\nu$ is the distribution of the greatest element of $S$, so each permutation of $\nu$ is the  distribution of an element of $S$, thus $\mu$ is the distribution of a random element of $S$, as claimed.
\end{proof}

For $0\le\delta\le 1$,  the \emph{$\delta$-noisy  quantum channel of level $n$} affords transition matrices of the form $(\operatorname{tr} E_i\rho_j)$, where $E_1, \dots, E_k\in M_n(\mathbb{C})$ is a POVM and $\rho_1, \dots, \rho_l\in M_n(\mathbb{C})$ are density matrices with all eigenvalues $\ge\delta/n$.
This channel is equivalent to the  $\delta$-noisy classical channel with $n$ states. This is a special case of Theorem~\ref{main}. Alternatively, it can be shown by  combining ideas  from the proofs of Theorem~\ref{ball}(1) and \cite[Theorem 3]{FW}.
\begin{Cor}\label{cor} Let $0\le\delta\le 1$. The signalling dimension of the $\delta$-noisy $n$-state classical (or, equivalently, $n$-level quantum) channel is $\lceil (1-\delta)n +\delta\rceil$.
\end{Cor}

\begin{proof}
The $\delta$-noisy $n$-state classical  channel can
be simulated by the noiseless $d$-state classical  channel  if and only if we have \begin{equation}\label{convex}r\delta/n\ge\binom rd\bigg/\binom nd\end{equation}  for all integers $d\le r\le n-1$  --- note that both sides of \eqref{binom} are 1 for $r=n$. In inequality~\eqref{convex}, the left hand side is linear in $r$, while the right hand side is convex for $r=0, 1, \dots$. Also, the inequality holds for $r=0,1,\dots, d-1$.  Therefore, it holds  for all integers $d\le r\le n-1$ if and only if it holds for $r=n-1$, i.e., $(n-1)\delta/n\ge (n-d)/n$, or, equivalently, $d\ge (1-\delta)n +\delta$.
\end{proof}

\subsection{Partial replacer quantum channels}\label{depol}
The usual mathematical model for a  noisy quantum channel is given in terms of a completely positive trace-preserving map $\mathcal N:M_m(\mathbb{C})\to M_n(\mathbb{C})$. Let $\ran \mathcal N$ stand for the set of density matrices $\mathcal N(\sigma)\in M_n(\mathbb{C})$, where $\sigma\in M_m(\mathbb{C})$ is a density matrix. The channel affords transition matrices of the form $(\operatorname{tr} E_i\rho_j)\in [0,1]^{k\times l}$, where $E_1$, \dots, $E_k$ is a POVM in $M_n(\mathbb{C})$ and each $\rho_j$ is contained in  $\ran \mathcal N$. The \emph{signalling dimension} $\signdim\mathcal N$ of $\mathcal N$ is the signalling dimension of this channel, i.e., the smallest $d$ such that the channel can be simulated by the noiseless classical channel with $d$ states. If the spectrum of every $\rho\in\ran\mathcal N$ is contained in a  given permutation-invariant set $K\subseteq\Delta_n$, then  every transition matrix afforded by $\mathcal N$ is also afforded by the $K$-noisy quantum channel, so we can  can use Theorem~\ref{main}  to show that $\mathcal N$ can be simulated by the $K$-noisy classical channel. Then Theorem~\ref{noiseless} can be used to give an upper bound on the signalling dimension of $\mathcal N$.

An important special case is given by \emph{partial replacer channels}. Let $m\le n$. We embed $M_m(\mathbb{C})$ into $M_n(\mathbb{C})$  as the set of matrices that are zero outside of the upper  left $m$-square block.  We fix a density matrix $\rho\in M_n(\mathbb{C})$. The \emph{replacer channel} $\mathcal N_\rho: M_m(\mathbb{C})\to M_n(\mathbb{C})$ is given by $\mathcal N_\rho(X)=(\operatorname{tr} X)\rho$. Given $0\le\delta\le 1$, the  \emph{partial replacer channel}  $\mathcal  N_\rho(\delta): M_m(\mathbb{C})\to M_n(\mathbb{C})$ is given by $\mathcal N_\rho(\delta)(X)=(1-\delta)X+\delta(\operatorname{tr} X)\rho$.  In \cite[Theorem 3]{DC} by Doolittle and Chitambar, it is shown that \begin{equation}\label{dcbounds}\lceil(1-\delta) m+\delta\rceil\le\signdim \mathcal N_\rho(\delta)\le \min\{m, \lceil(1-\delta) m+1\rceil\},\end{equation} and the upper bound is tight for the \emph{partial erasure channel}  given by the  \emph{erasure flag} $\rho$ which has entry 1 at position $(m+1, m+1)$  and zero elsewhere. Note that the difference between the upper and the lower bound in \eqref{dcbounds} is at most 1.

We shall now prove that the lower bound is tight if $m=n$ and $\rho$ is sufficiently mixed, in particular, if $\rho=\mathbf 1/n$ is the \emph{maximally mixed state}, yielding the \emph{partial depolarization channel}  $$\mathcal N(\delta)(X)=(1-\delta)X+(\delta/n)(\operatorname{tr} X)\mathbf 1.$$

From now on, we let $m=n$.
 Let $d=\lceil (1-\delta)n+\delta\rceil$ stand for the lower bound in  \eqref{dcbounds}.
Let $\mu_1\le\dots\le\mu_n$ stand for the eigenvalues of a  fixed density matrix $\rho$.

\begin{Pro} \begin{enumerate}
\item If $\delta(\mu_1+\dots+\mu_r)\ge\binom rd /\binom nd$ holds for $r=d, \dots, n-1$, then $\signdim \mathcal N_\rho(\delta)=d$.
\item
The partial depolarization channel is equivalent to the $\delta$-noisy classical channel with $n$ states.
\item The signalling dimension of the partial depolarization channel is $d$.
\end{enumerate}
\end{Pro}

\begin{proof} (1)
The eigenvalues $\mu_1'\le\dots\le\mu_n'$ of $\mathcal N_\rho(\delta) (\sigma)=(1-\delta)\sigma+\delta\rho\ge \delta\rho$ satisfy $\mu_1'+\dots+\mu_r'\ge\delta(\mu_1+\dots+\mu_r)$ for any density matrix $\sigma\in M_m(\mathbb{C})$ and any $r=1, \dots, n$. Thus, $\mu_1'+\dots+\mu_r'\ge \binom rd /\binom nd$ for $r=1, \dots, n-1$, but also, trivially, for $r=n$. The claim  now follows from Theorem~\ref{noiseless} together with the first inequality in \eqref{dcbounds}.

(2)  The range $\ran \mathcal N(\delta) $  is the set of density matrices with all eigenvalues $\ge\delta/n$, so the  claim follows from Theorem~\ref{main}.

(3)  follows from (2) together with Corollary~\ref{cor}.
\end{proof}

\section{Future research}\label{future} It is well known that quantum communication can outperform classical communication if entanglement is used cleverly. On the other hand, in certain scenarios not involving entanglement, it can be proved that passing from classical to  quantum cannot increase efficiency.

 A fundamental result in this direction is the Holevo bound \cite{H}
   which we now recall.  For any stochastic matrix $A=(a_{ij})\in[0,1]^{k\times l}$  and input probabilities $q_j\ge 0$ $(j=1, \dots, l)$ summing to 1,  we define the \emph{mutual information} $$\Info(A, q)=H(j)+H(i)-H(i,j).$$ Here $H$ stands for the Shannon entropy of a random variable, and
 the joint distribution of the random pair $(i,j)$ is given by the probabilities $q_ja_{ij}$. Now,   for any density matrices $\rho_j\in M_n(\mathbb{C})$  and any POVM $E_1, \dots, E_k\in M_n(\mathbb{C})$, the
 Holevo inequality  reads \begin{equation}\label{holevo}\Info(A,q)\le\chi,\end{equation}  where $a_{ij}=\operatorname{tr} E_i\rho_j$ and the  Holevo quantity $\chi$ is defined by $$\chi=S\left(\sum_{j=1}^l q_j\rho_j\right)-\sum_{j=1}^l q_jS(\rho_j),$$ where $S$ is von Neumann entropy, i.e., the Shannon entropy of the spectrum.  If all $\rho_j$ with $q_j>0$ commute, then a  POVM $E_1$, \dots $E_k$ can be found so that equality holds in \eqref{holevo}. Otherwise, the inequality is strict for any POVM.

 Another result in the above mentioned direction is \cite[Theorem 3]{FW}: the $n$-level quantum channel can be simulated by the $n$-state classical channel.

  It would be nice to unify these two results. Let  a probability distribution $q_1$, \dots, $q_l$
   be given.

    Can every quantum transition matrix $A=(a_{ij})=(\operatorname{tr} E_i\rho_j)\in[0,1]^{k\times l}$, where $E_1, \dots, E_k\in M_n(\mathbb{C})$ is a POVM, and $\rho_1, \dots, \rho_l\in M_n(\mathbb{C})$ are density matrices, be written as a convex combination $A=\su
    p_IA_I$ of stochastic matrices $A_I$, each  with $\le n$ nonzero rows, and each satisfying $\Info(A_I, q)\le\chi $ ?
      Can the proof of Theorem~\ref{main} be modified to yield this result and thus, maybe,  a new proof  of Holevo's inequality?

\bigskip

\noindent
{\bf Acknowledgement.}
I am grateful to Mih\'aly Weiner for useful conversations.

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From Nashville to LA to India via Palette's Virtual Studio Nashville (VSN)
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From Nashville to LA to India via Virtual Studio Nashville (VSN)
— Jeff Silverman
Two years in the making, Aeone delivers a powerful yet, inspiring body of work in her 6th studio album including the title track video release of – Love Is The Healer. Jeff Silverman and Aeone produced and arranged the album. Jeff also mixed, mastered and engineered the project and co-wrote on a number of tracks from Palette Music Studio Productions (MSP) while connecting with Aeone via Palette's Virtual Studio Nashville (VSN) in Nashville, TN. Aeone recorded from The Chateau, her studio in Los Angeles, CA.
From Aeone
Love Is The Healer was created by Aeone and longtime collaborator, veteran producer and songwriter Jeff Silverman, 2000 miles apart in Los Angeles, CA and Nashville, TN. During the entire creative process, including recording, mixing and mastering, Jeff and I used Palette's VSN virtual studio set up (Virtual Studio Nashville) to communicate. Files were sent back and forth via the internet until the mix/ mastering stage. This process was a continual "passing of the torch." For the creative process, files were constantly being traded off to each other as we were working on the arrangements. When it came to the mix, Jeff would take the torch and paint his picture. Then send it off to me to listen and comment. Then we both would hook up via VSN, streaming audio directly out of Jeff's Pro Tools session, screen sharing and talking on the phone while making our final mix /mastering tweaks.
In addition, files were transferred to India and back while collaborating with Ricky Kej, 2015 Grammy Award winner for Best New Age Album.
I am so proud of the work we have done together, each song feels seamless – where one person's ideas end and the other's begin. It was not only a great collaboration, but also a tribute to Jeff and his unique ability to tune into where I wanted to go as an artist and expand on that, rather than change it for the sake of making his mark. That being said, his imprint and input are absolutely apparent throughout the album.
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New AEONE Album Produced by Jeff Silverman and Aeone
Posted:August 1, 2015
AEONE - LOVE IS THE HEALER
Jeff Silverman and Palette Records are proud to announce LOVE IS THE HEALER, AEONE'S new album and video. Now available on iTunes, Amazon, CD Baby and https://paletterecords.net/store/
CLICK HERE to stream samples of the songs and read more about the project!
This beautifully crafted 10-song album brings Aeone's signature ethereal voice and sound with a touch of the ancient Celt together, in a cinematic blend of otherworldly pop and organic rhythms. Hypnotic and uplifting, the 10 songs follow a path of grace and beauty deep into the heart of love. Aeone weaves her story in a spellbinding tapestry of voice and sound from the romantic poetry of love to the dark ancient longings of the soul - taking us from multi layered vocals and ambient grooves in rich soundscapes with ethnic instruments, down to the single purity of her clear and haunting voice. LOVE IS THE HEALER is an emotional and ambitious body of work that leaves its mark well after the last song ends.
Artist and composer Aeone delivers a powerful yet, inspirational body of work in her 6th studio album – LOVE IS THE HEALER. Jeff Silverman and Aeone produced and arranged the album. Jeff also mixed, mastered and engineered the project and co-wrote on a number of tracks from Palette Music Studio Productions (MSP), Mount Juliet / Nashville, TN • www.palettemusic.net. Aeone recorded from The Chateau, her studio in Los Angeles, CA. www.aeone.com.
Aeone was honored to have Ricky Kej, 2015 Best New Age Album, Grammy award winner, co-write and co-arrange on Boat of Angels with her.
Known for her rich haunting style, Aeone's voice and music has been heard in countless Film, TV and CD projects and in over seventy movie trailers including "The Hobbit," "300," "The Last Samurai," "Beowulf," "Tomb Raider," and "Something's Gotta Give." Aeone was also the voice of Emmy nominated "Mists of Avalon" and has released 5 critically acclaimed albums.
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Ten Acres Lane Sports Complex is also the proud home of the National Taekwondo Centre and multi-medal winning Team GB Taekwondo.
When Gym membership is included it will be with Sports Direct Fitness, Manchester.
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I am in the Nijmegen area but customer of PC Amsterdam.
Keep us posted on your delivery and post some pictures when you get the car. You won't be disappointed.
I will do so. Have a nice Sunday (drive)!
My mother was from Gemert in Noord Brabandt; my son lived in Delft for 10 years as a teacher. I have many, many relatives in Holland. I always enjoy it there, but the roundabouts with trams, bikes and cars takes a little getting used to!
I am from Berlicum originally and worked in Uden, so I do know Gemert as well. But 6 years ago I moved to North Limburg. Grtz Rik.
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text:
En matemàtiques, específicament en càlcul i anàlisi complexa, la derivada logarítmica d'una funció f es defineix per la fórmula on  és la derivada de f.  Intuïtivament, aquest és el canvi relatiu infinitesimal en f; és a dir, el canvi absolut infinitesimal en f, és a dir  escalada pel valor actual de f.

Quan f és una funció f ( x ) d'una variable real x, i pren valors reals estrictament positius, això és igual a la derivada de ln( f ), o al logaritme natural de f. Això es desprèn directament de la regla de la cadena:  Moltes propietats del logaritme real també s'apliquen a la derivada logarítmica, fins i tot quan la funció no pren valors en els reals positius. Per exemple, com que el logaritme d'un producte és la suma dels logaritmes dels factors, tenim Així, per a les funcions de valor positiu-real, la derivada logarítmica d'un producte és la suma de les derivades logarítmiques dels factors. Però també podem utilitzar la llei de Leibniz per obtenir la derivada d'un producteExemples:

 El creixement exponencial i la decadència exponencial són processos amb derivada logarítmica constant.
 En finances matemàtiques, el símbol grec λ és la derivada logarítmica del preu del derivat respecte al preu subjacent.
 En l'anàlisi numèrica, el nombre de condició és el canvi relatiu infinitesimal en la sortida per a un canvi relatiu en l'entrada i, per tant, és una relació de derivades logarítmiques.

Referències

Anàlisi complexa

redpajama_set_name: RedPajamaWikipedia


Our rich, lathering shampoo is formulated using a high level of Aloe and Cannabis Sativa (Hemp) seed oil, which provide essential fatty acids that are optimal for hair and scalp health. Omega 3 & 6 properties also moisturize, without weighing down hair. Suitable for all ages, our aloe conditioning shampoo is formulated using vegetable based cleansers and is ultra mild. Plant extracts of Aloe, Chamomile, Calendula, Red Clover, Willow Bark and Elder Flower are used to nourish and revitalize your hair. Panthenol (Pro Vitamin B 5) deeply penetrates the hair shaft and locks in moisture. Acts as a "2-in-1", adding volume and conditioning the hair. No product buildup even from daily shampooing.
A balanced pH level helps ensure hair proteins are kept strong and are resistant to breakage.
Wet hair thoroughly, lather with shampoo, rinse well. For optimal results, leave shampoo on your hair/scalp for a few minutes. Rinse thoroughly. Avoid contact with eyes. (A conditioner after use is not recommended for fine or medium hair, as the shampoo is a "2 in 1" that conditions while cleansing).
Order online from an innovative Canadian company that creates skin and body care using botanical, natural ingredients to make unscented skin care that is not merely safe for your skin, but a comfort to put on. Loads of typical fragrance free products leave the skin appearing oily or sticky. Wisp's creative skin and body care leaves all types of skin feeling tremendous.
Fragrance, as well as Natural skin care Fragrance, can damage the skin barrier after a while; keeping in mind that fragrance and colorants are clearly two of the ingredients to blame. Allowing your skin to relax from skin care manufactured with fragrance, colorants and harsh elements will help your skin to attain stability. You don't have to give up on effectiveness to experience better and healthier skincare.
It doesn't matter if you are purchasing a wonderful aloe conditioning shampoo or a spectacular hemp and olive body butter, Wisp ships everywhere in Canada and the USA, with orders leaving within within a day or two on week days.
My sister got me hooked on this last month. It costs 1/3 of the salon product that I was buying and seems to give me a fuller body feel. My hair is silky with a noticeable shine. I am very impressed.
I have been using some of these products for a few weeks now and love them!!! The shampoo is amazing and I love that I don't need conditioner and my hair still looks and feels great AND my scalp appreciates all the goodness too! The facial cleanser and baby face lotion are quickly becoming must have at all times for me!


We are excited to announce a successful event collaboration with our strategic partners at AdvaMed 2015! We hosted more than 250 1-on-1 meetings over the course of three days. The partnering event consisted of meetings between over 40 US companies from AdvaMed's network and 15 Chinese healthcare companies or investment institutions from TPP's global network.
As one of the most significant events in the medical device and diagnostics industries, AdvaMed 2015 was able to offer an extended opportunity for US companies to engage with Chinese groups looking to invest, license, and distribute products and technologies to a market with tremendous potential.
"Building partnerships is a key focus of our conference, especially amongst international parties, and we look forward to working with The Pharma Partners team to help foster growth for U.S.-based MedTech companies and investors in China," said Ray Briscuso, Managing Partner and Producer of AdvaMed 2015.


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Best General Surgery Hospitals in Dubai
Choose from 6 best General Surgery hospitals in Dubai
Top Hospitals For General Surgery In Dubai
Al Zahra Hospital
Canadian Specialist Hospital, Dubai
Dr. Sulaiman Al-Habib Hospital
Thumbay Hospital
Dubai, United Arab Emirates - 4 / 5 - (1 Review)
Al Zahra Private Hospital Dubai is a world-class medical facility providing tertiary care in multiple specialties and has attained national and international accreditations. The 187 bedded hospital has state-of-the-art infrastructure and is equipped with top-notch technology, offering basic to complex medical interventions. The hospital has a team of accomplished medical professionals having international reputation along with highly trained support staff team constituting specialized nurses, paramedical team and others. This premier healthcare service provider aims to deliver first-rate quality clinical care to all the patients on a global level.
Bariatric Surgery, Cardiology, Colorectal medicine, Ear Nose & Throat, Endocrinology, Gastroenterology, General Medicine, General Surgery, Hematology, Nephrology, Neurology, Neurosurgery, Nuclear Medicine, Obstetrics & Gynecology, Oncology, Ophthalmology, Orthopedics, Pediatric Cardiology, Plastic Surgery, Pulmonology, Radiation Therapy, Reproductive medicine, Spinal Surgery, Urology, Vascular surgery
Guinness world record for the most blood cholesterol readings in 8 hours.
UAE's Distinguished Hospital Award
Dubai, United Arab Emirates - 4 / 5 - (2 Reviews)
Canadian Specialist Hospital is premier medical facility providing comprehensive tertiary healthcare services and is a referral facility from other healthcare centres in the Middle East. With a capacity of 200 beds, this hospital has more than 300 specialized centre and is one of the largest hospitals in Dubai with world-class infrastructure and advanced technology. The hospital houses a panel of renowned doctors with reliable experience in their field of medical expertise along with a trained support staff to provide immaculate medical care to all the patients.
Canadian Specialist Hospital won Best Hospital for Medical Tourism Award at the DHA Health Awards 2018
Mr. Smith from Zimbabwe Traveled to India for Cardiac & Hernia Surgery
I approached Dr. Surbhi to have her opinion on whether the repair of my umbilical hernia with absorbable mesh was a viable option. It was not this simple as I am a heart patient and was also looking f
Zulekha Hospital Dubai is a reputed multi-specialty hospital, with a capacity of 179 beds encompassing 32 departments and specialties operated under the highly qualified and accomplished set of doctors. This hospital, established in 2004, is a part of Zulekha Healthcare Group which was founded under the esteemed leadership of Dr. Zulekha Daud in 1964 when she started her practice in Sharjah, UAE. Dr. Zulekha has been the winner of the International Women's Day Excellence Award by 'Dhwani', the New Global Indian Excellence Award and the Lifetime Achievement – Healthcare Award for her services and contribution to the healthcare sector.
Bariatric Surgery, Cardiology, Colorectal medicine, Ear Nose & Throat, Endocrinology, Gastroenterology, General Medicine, General Surgery, Nephrology, Neurology, Neurosurgery, Nuclear Medicine, Obstetrics & Gynecology, Oncology, Ophthalmology, Orthopedics, Pediatric Cardiology, Plastic Surgery, Pulmonology, Radiation Therapy, Reproductive medicine, Spinal Surgery, Urology, Vascular surgery
Dubai Quality Awards 2015
The Dubai Chamber CSR Label Award 2014
Unilever MENA Sustainable Living Plan Award 2014
Business Excellence Department of Dubai Economic Development awards the organizations with the "Dubai Quality Award" which has three levels of honors for implementing the EFQM Excellence Model successfully
Dubai Quality Appreciation Award (DQAP) and Dubai Quality Award (DQA) in the year 2007, 2011 and 2014.
Zulekha Hospitals honored by Medigo For Excellent Patient Satisfaction Score
Dubai, United Arab Emirates - 3.8 / 5
Dr. Sulaiman Al-Habib Hospital is a 200 bedded state-of-the-art medical facility. The hospital is a part of Dr. Sulaiman Al-Habib Medical Group (HMG), one of the largest private healthcare providers in the Saudi Arabia. The hospital is a multi-disciplinary health centre offering a wide range of medical services and world-class facilities in combination with international standard of quality and clinical excellence. Sulaiman Al-Habib Hospital ensures that all the patients, domestic and international, receive utmost care and reliable medical treatment. The hospital is equipped with the latest medical equipment and technology for diagnostics, therapeutic use, and critical care.
Bariatric Surgery, Cardiology, Dentistry, Ear Nose & Throat, Endocrinology, Gastroenterology, General Surgery, Nephrology, Neurology, Neurosurgery, Obstetrics & Gynecology, Ophthalmology, Orthopedics, Urology, Vascular surgery
Saudi German Hospital is a multi-specialty hospital offering comprehensive medical facilities, established in 1988 by one of the largest group of healthcare providers - Saudi German Hospital Group (SHG). With a capacity of over 300 beds, the tertiary care hospital offers excellent medical services with outpatient, inpatient and critical care units in a multitude of specialties and departments. The hospital is a world-class internationally accredited medical center operates with advanced curative techniques, state-of-the art centres for excellence, providing high standard clinical care.
HIMSS by Dubai Health Authority
Best Hospital Award 2014 – 9th Edition DHA ( WHTC)
Best Hospital in Medial Tourism – Socrates Awards Ceremony (ITALY)
Asian Patient Safety Awards 2014 (Infection Prevention and Practice )
Best Hospital Award 2015 – Gulf Business Magazine
Best Hospital Award 2016 – 11the Edition DHA ( WHTC)
Best Urgent Care Hospital (Winners Review 2017)
The Most Outstanding Medical Tourism executive of the World
Dubai, United Arab Emirates - 4 / 5
Thumbay Hospital in Dubai is top-class multi-specialty hospital in Dubai providing multidisciplinary medical services, and is owned by Thumbay Group which is one of the largest healthcare providers in the area. The state-of-the-art facility has a capacity of 150 beds and offers excellent outpatient, inpatient and critical care services to the all the patients. The hospital has made its mark in the sector based on international standard quality clinical care offered by the medical experts, nurses, and the support staff at the hospital.
Bariatric Surgery, Cardiology, Colorectal medicine, Dentistry, Ear Nose & Throat, Gastroenterology, General Medicine, General Surgery, Hematology, Nephrology, Neurology, Neurosurgery, Nuclear Medicine, Obstetrics & Gynecology, Oncology, Ophthalmology, Orthopedics, Pediatric Cardiology, Plastic Surgery, Pulmonology, Radiation Therapy, Reproductive medicine, Spinal Surgery, Urology, Vascular surgery
Dubai Quality Appreciation Award
Dubai Human Development & Appreciation Award
Sheikh Khalifa Excellence Award
Best General Surgeon in Dubai
Dr. Murtaza Pithawala
Specialist General Surgeon - Advanced Laparoscopy and Bariatric Surgery
Dr. Anwar Kamel Bahat Mohamed Oraby
Consultant General Surgery
Dr. Ritu Khare
Specialist - General Surgery
Dr. Uday Shankar
Dr. James Zachariah Pulimuttil


This wonderful articulated gold-tone necklace by Sarah Coventry is called Night Owl and dates to 1975. It is composed of 3 "hinged" segments; head, body and tail. The body is made of of open-work "squiggle" designs and the eyes are opaque black plastic cabochons.
Chain available upon request at no extra charge.


This publication ... serves as a useful 'wake-up call' to bat surveyors and researchers, identifying which interventions are lacking scientific evidence. This is very helpful in directing future bat research.
This book brings together scientific evidence and experience relevant to the practical conservation of bats. The authors worked with an international group of bat experts and conservationists to develop a global list of interventions that could benefit bats.
For each intervention, the book summarises studies captured by the Conservation Evidence project, where that intervention has been tested and its effects on bats quantified. The result is a thorough guide to what is known, or not known, about the effectiveness of bat conservation actions throughout the world.
Bat Conservation is the fifth in a series of Synopses that will cover different species groups and habitats, gradually building into a comprehensive summary of evidence on the effects of conservation interventions for all biodiversity throughout the world.
By making evidence accessible in this way, we hope to enable a change in the practice of conservation, so it can become more evidence-based. We also aim to highlight where there are gaps in knowledge.
Evidence from all around the world is included. If there appears to be a bias towards evidence from northern European or North American temperate environments, this reflects a current bias in the published research that is available to us. Conservation interventions are grouped primarily according to the relevant direct threats, as defined in the International Union for the Conservation of Nature (IUCN)'s Unified Classification of Direct Threats (www.iucnredlist.org/technical-documents/classification-schemes).
Anna Berthinussen is a post-doctorate researcher at the University of Leeds, currently working on a Defra-funded study of the interactions between bats and roads. She holds degrees in bat ecology and conservation (PhD) and Zoology (BSc), both from the University of Leeds. She has published several scientific papers and contributed to book chapters on bats, and has a keen interest in wildlife conservation.
research interests include bat conservation, urban ecology, citizen science and applied ecology and its translation into policy and practice.
John Altringham is Professor of Animal Ecology & Conservation at the University of Leeds. He works primarily on the ecology and conservation of bats, but has broad interests in conservation. In the past he has studied animals as diverse as tunas and tarantulas. He is the author of numerous scientific papers and reports, and three books on bats, the most recent being Bats, from evolution to conservation, published by OUP in 2011. He is a scientific advisor to the National Trust and other conservation organisations.


Get reliable power and long life at a lower total cost with the new 313FL GC. A 70-hp engine lets you tackle big jobs without paying a huge fuel bill. And the maintenance-free emissions package saves time and cuts costs. It's a tough little excavator, loaded with features you don't always find in this size class.
Get more details and talk to your Cat dealer today.


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Q: Hide the extra row at the bottom in a DataGridview in a Winform Application I have binded a datagridview to a datatable. But the gridview displays an extra empty  row at the bottom ? How should I hide it ? Thanks in Advance

A: The extra row is to manually add a line to your datagridview. If you don't need it you can disable it by disallowing the user to add rows:
this.dataGridView.AllowUserToAddRows = false;

A: You can try this:
CurrencyManager cm = myGrid.BindingContext[myGrid.DataSource, myGrid.DataMember]
                     as CurrencyManager;
DataView dv = cm.List as DataView;
dv.AllowNew = false;

redpajama_set_name: RedPajamaStackExchange


DUBAI (Reuters) - The United Arab Emirates cabinet approved steps on Sunday that would allow for 100 percent ownership of UAE-based businesses by foreign investors by year-end, the state news agency WAM said.
The decision is part of a wider change to the system that would grant residency visas of up to 10 years to investors and specialists in scientific, technical, medical and research fields, the agency said.
The cabinet meeting was chaired by Sheikh Mohammed bin Rashid Al Maktoum, vice president and prime minister of the UAE and ruler of Dubai. He directed the economy minister to implement the resolution and submit a study following up its impact in the third quarter of this year, WAM said.
At present, foreigners generally cannot own more than 49 percent of any UAE firm unless it is incorporated in a special "free zone".


I need support paying my gas bill to keep my cooking gas on. Being able to cook and feed myself is vital to my healing!
I am living on my own and receiving Social Security Income. I have many health problems and I am taking many medications. Paying for my medical needs has made me fall behind on my bills. In addition I will be getting surgery soon.
My goals are to recuperate after my surgery and I hope to be healthy once again so I can do the things I enjoy. The first step would be to pay my bills. I need food to stay healthy and gas in order to cook my food to eat.
I need support paying my gas bill to keep my cooking gas on. I have tried applying for LIHEAP but was not covered because it is not for heat.
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Thank you for helping Woiciech fund this need.
Woiciech has been working with CCL for a few months. He is reliable he shows up when he needs to and calls when he can't make it. He is a very friendly and resourceful person. I believe that paying his bill will help him keep positive and help him recover after his surgery with fewer worries.


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What's on in Thurrock
Wednesday, 18 March 2020 - 7:30pm to Sunday, 22 March 2020 - 2:30pm
Consider yourself at home with Lionel Bart's classic musical based on Charles Dickens's novel, 'Oliver Twist'.
The Tony and Olivier Award-winning show is one of the few musicals to win an Academy Award for Best Picture and is widely hailed as a true theatrical masterpiece by actors and audience members alike.
The streets of Victorian England come to life in one of the world's best loved musicals.
This is an amateur production by TOPS on Stage. Performances will be on:
Wednesday 18 March at 7:30pm
Thursday 19 March at 7:30pm
Friday 20 March at 7:30pm
Saturday 21 March at 2:30pm
Sunday 22 March at 2:30pm
If we don't have your favourite seat it may still be available directly from TOPS on Stage - buy tickets at www.ticketsource.co.uk/topsonstage or phone 01375 675 591.
Thameside Theatre, Orsett Road, Grays, RM17 5DX


The caramelized sugar offers a rich topping for the individual custards --a delicious lightened version of the classic Spanish dessert. Be careful not to stir too vigorously when dissolving the sugar in water, or large sugar crystals will form. Prepare this dessert up to one day ahead.
Coat 12 (6-ounce) custard cups with cooking spray.
Combine sugar and 1/4 cup water in a small, heavy saucepan; cook over medium-high heat, stirring gently as needed to dissolve sugar evenly. Continue cooking 4 minutes or until golden, stirring constantly. Immediately pour about 1 tablespoon sugar mixture into each of 12 prepared custard cups, tipping quickly until caramelized to coat the bottom of prepared cups.
Combine fat-free milk, vanilla, cinnamon, eggs, and condensed milk in a medium bowl, stirring with a whisk until well blended. Pour 1/2 cup milk mixture into each custard cup. Place cups in a jelly-roll pan; add hot water to pan to a depth of 1/4 inch. Bake at 350° for 35 minutes or until a knife inserted in the center comes out clean. Remove cups from pan; cool completely on a wire rack. Cover and chill at least 3 hours.
Loosen edges of custards with a knife or rubber spatula. Place a dessert plate, upside down, on top of each custard cup; invert custard onto each of 12 plates. Drizzle any remaining caramelized syrup over custards.


As we approach the execution of Cal Coburn Brown, opponents of the death penalty argue we should cease executions because it is cheaper than continuing them ["State prepares for first execution since 2001," Local News, March 9].
There are a number of problems with this argument. First, the reason the death penalty is so expensive is because of the frivolous legal challenges mounted by these very same opponents. Their tactics are akin to extortion. If they really wanted to lower the cost of justice, they would help expedite the executions, not inhibit them.
Second, by basing their argument on costs, they imply they would favor executions if a way could be found to lower the costs of execution to a price less than life in prison.
than wave their Bible at us, they are tossing out tortured, amoral arguments in the hopes of duping less-emotional thinkers.


Nov 8 (Reuters) - Canada's main stock index remained flat on Thursday, as robust earnings by a handful of companies helped offset losses in energy shares.
* At 9:40 a.m. ET (1440 GMT), the Toronto Stock Exchange's S&P/TSX Composite index was up 8.46 points, or 0.06 percent, at 15,377.89.
* Seven of the index's 11 major sectors were higher, led by gains in the consumer discretionary sector, which rose 1.5 percent.
* The financial sector rose 0.3 percent after two of Canada's biggest insurers, Manulife Financial Corp and Sun Life Financial, posted third-quarter earnings that surpassed market expectations, helped by sales growth in Asia.
* But, the energy sector dropped 0.4 percent after oil prices edged lower, as investors focused on the pace of growth in global crude supply, which is increasing more quickly than many had expected.
* The materials sector, which includes precious and base metals miners, lost 0.2 percent as copper and gold prices fell on a firming dollar.
* The most heavily traded shares by volume were Bombardier B , Aurora Cannabis and Trevali Mining.


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L'Anguille (Eel) est un super-vilain appartenant à l'univers de Marvel Comics. Deux personnes ont porté le costume de l'Anguille : le premier est apparu dans Strange Tales #112 et le deuxième dans Power Man and Iron Fist #92.

Origine

Leopold Stryke
Le premier, Leopold Stryke, gérait un zoo marin le jour et volait les banques la nuit. Il fut vaincu plusieurs fois par la Torche Humaine et les X-Men.
Il fut recruté par le Comte Nefaria, puis s'associa avec l'Homme-plante, le Porc-épic et l'Épouvantail.

Il fut par la suite recruté pour faire partie de l'Escouade des serpents.

Il fut tué par le Gladiateur à Las Vegas.

Edward Lavell
Edward Lavell prit possession du costume et attaqua Iron Fist et Power Man pour libérer Hammerhead de prison. On le vit ensuite travailler pour le compte du Fantôme, mais ils furent vaincus par Spider-Man.

Il fit ensuite partie des Maîtres du Mal, qui furent battus par les Thunderbolts.

On le revit dans un tournoi de combattants à Madripoor. Il fut vaincu et blessé par le Crapaud.

Durant Civil War
Remis sur pied et rentré aux États-Unis, il fut appréhendé par les Thunderbolts et forcé à rejoindre leur armée. Il fit partie des super-vilains empoisonnés par le Punisher lors de la veillée pour l'Homme aux échasses, au Bar Sans Nom. Il survécut pourtant à l'explosion du bar.

Récemment, on le revit au sein d'une nouvelle Escouade des serpents, dirigée par Sin, la fille de Crâne Rouge.

Pouvoirs

Le costume de l'Anguille génère de l'électricité et isole totalement celui qui le porte. La puissance est telle que le tissu brille dans le noir.
Un champ électrique entoure le porteur, aussi personne ne peut l'approcher sans être détecté.
Les décharges peuvent assommer leur victime
Le costume graisseux de l'Anguille et son agilité naturelle font de lui un adversaire difficile à attraper.

Ennemi de Luke Cage
Ennemi d'Iron Fist
Personnage créé par Stan Lee


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Sen. Duckworth Owns Trump: I Won't Be Lectured On Military Needs By a 'Five-Deferment Draft Dodger'
Addressing the White House response to the federal government shutdown, Democratic Sen. Tammy Duckworth tore into President Trump on the Senate floor Saturday, calling him a "five-deferment draft dodger" and slamming him for his provocative comments toward North Korean leader Kim Jong Un.
"Does he even know that there are service members who are in harm's way right now, watching him, looking for their commander in chief to show leadership, rather than [trying] to deflect blame?" the Illinois Senator said. "Or that his own Pentagon says that the short-term funding plans he seems intent on pushing is actually harmful to not just the military, but to our national security?"
"I spent my entire adult life looking out for the well-being, the training, the equipping of the troops for whom I was responsible," she continued. "Sadly, this is something that the current occupant of the Oval Office does not seem to care to do — and I will not be lectured about what our military needs by a five-deferment draft dodger."
"And I have a message for cadet bone spurs: If you cared about our military, you'd stop baiting Kim Jong Un into a war that could put 85,000 American troops, and millions of innocent civilians, in danger."
The federal government shut down Friday at midnight after the Senate failed to pass a short-term funding bill.
Trump has blamed Democrats for the shutdown, tweeting Saturday that they decided to "play shutdown politics. The White House has also blasted Democrats, accusing them of putting "unlawful immigrants" ahead of the military and other funding needs.
"We will not negotiate the status of unlawful immigrants while Democrats hold our lawful citizens hostage over their reckless demands," White House press secretary Sarah Huckabee Sanders said. "When Democrats start paying our armed forces and first responders, we will reopen negotiations on immigration reform."
But Democrats are firm and have countered that Trump's demands on an immigration compromise led to the shutdown. Senate Minority Leader Charles Schumer (D-N.Y.) said he thought he had come to a deal with Trump on Friday by offering funding for Trump's U.S.-Mexico border wall, but the president ultimately walked away from that deal.
"Negotiating with this White House is like negotiating with Jell-O," Schumer said Saturday. "It's next to impossible. As soon as you take one step forward, the hard-right forces the president three steps back."
Watch Sen. Tammy Duckworth's epic response below:
https://youtu.be/_F_o6J25nrc
In this article:donald trump, government shutdown, Tammy Duckworth


The interior and exterior were designed for comfort and entertaining. The top level encompasses the main living space with an elegant dining area, showcase-worthy front and back kitchen, butler's pantry, four seasons room and 125 windows to ensure full enjoyment of the spectacular view. The finished lower level includes a guest bedroom suite, home office and theater room with surround sound and wall-to-wall projection screen.
Oak cabinets with vertically matched grain make up the kitchen, dining hutch and entertainment area built-ins. Paired with hardwood floors and a neutral color scheme, the mixed materials work together to reflect the home's naturally beautiful location. The front kitchen features open shelving to display china and glassware. A back kitchen provides additional utility space, which keeps the front kitchen uncluttered and primed as a gathering space.


Because that's what living is...the highs and lows; the falling and rising. But the rising part is our choice. And not always the easy one. If we keep rising though, friends, we''ll continue to get stronger, and it will get easier. Then, it's our strength that will shape our world around us.
As I do my last minute prep work, before heading to Chile for the Atacama Crossing, I've had to remind myself of those words a few times.
I had a mild knee sprain about a month back that has altered my training and I'm not in the shape I'd like to be. BUT I've recently had blood work and a CT scan that indicate my health is good. I have to look at the bigger picture here. So despite not being as fit as I'd like, and not having done any altitude training, I'm alive and going to freaking Chile! How awesome is that!? I don't care where I finish in the standings, I'm just going to do my best to make sure I finish.
At the end of my life, one of the things I want to be remembered for is being the kinda dude who always made the effort to rise up.


Site is offline.	CANDUNDAR.COM.TR has #1 004 301 rank on the internet. This rank shows site's popularity. Lower rank means more visitors that site gets. This website is estimated to get 66 unique visitors per day. These unique visitors make 125 pageviews. We estimate that this website earns at least $0.48 USD per day with advertising revenues so it can be valued at least $499 USD. This site has a /10 PageRank. It has backward links from domains, backward links from .edu domains and links from .gov domains. IP address of this site is . We detected that the average page load time of this website is 0.00 seconds.
CANDUNDAR.COM.TR is ranked #1 004 301 on the internet. This rank shows site's popularity. Lower rank means more visitors that site gets. Most of users come to this website from Turkey. This website is ranked #20348 in Turkey. It has 66 visitors per day, and has 125 pageviews per day. Click on the tabs below to get more info.
We found that the main page of candundar.com.tr has external links and internal links.


In the rest of the French press, the revelations only added to a growing sentiment that Macron — who currently enjoys an approval rating of just 36 percent — has gone too far in cultivating what some see as a kinglike persona.
This latest makeup scandal was quickly compared to the revelations from last year that the administration of former French president François Hollande, a Socialist, paid about $11,000 a month on the president's haircuts. By comparison, Macron actually spends more or less the same on makeup than his last two predecessors, according to Le Point: Hollande spent slightly more per quarter, while Nicolas Sarkozy spent slightly less.
But for a new president floundering in the polls, the optics are less than ideal. In France, the average disposable annual income per capita is roughly 25,000 euros ($29,700), according to the OCED. That is slightly less than Macron's makeup expenditure.
But most of all, the makeup news comes at the end of the August recess, right before the French parliament convenes to discuss one of the most controversial issues here: labor reform — and, in particular, the potential cuts in government spending that labor reform would mean.
Macron, a self-styled political centrist, ran on promises to reform France's famously rigid labor market to stimulate growth. He has repeatedly made clear his view that slashing public spending is an absolute necessity to achieve that goal, which has lost him a considerable amount of support in the eyes of the public. In July, for instance, Macron insisted that military spending would have to temporarily decrease by approximately $1 billion. Pierre de Villiers, France's top general, resigned in protest.
Jaunta Swirlbush says: Don't these hoes know that beauty ain't cheap? No one just wakes up looking gorgeous. It takes a lot of work!


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HMWhite and Patricia McCobb collaborated on the landscape master plan and site design for the rehabilitation of one of New York City's first public parks. Commissioned by a private / public partnership -- the City Parks Foundation and neighborhood business sponsors, the landscape transformation secured the Park's future as an internationally recognized public park.
Thorough research and urban analysis led to a contemporary schematic plan grounded in park history. A complete tree inventory was conducted to chart present and future conditions, which established a preservation and landscape management program. The park's design anticipates future losses and proposes new planting design to reinforce a simplified visual and spatial organization.
Anchored by a new water basin and a restored historic fountain, the redesign reinforced the Park's central oval at its core. A permeable, stabilized crushed stone walking surface replaced trampled lawn and created a new gathering area, while promoting the health and growth conditions of numerous mature trees. A new architectural order developed a hierarchy of entrances, pathways and plaza nodes to minimize visual clutter and highlight the Park's historic landscape characteristics.
Environmental Signage: Calori & Vanden-Eynden, Ltd.


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Looking for the v10 manual? Visit our new user's guide!
New! Responsive Designs
Search Descriptions All MSx ML8 Version
Select your edition:
DotFeed
Plugin Documentation
New: Developer Resources
Server Configuration Info
E-commerce Basics
3rd Party Resources
PCI PA-DSS Guidelines
Object Hierarchy Overview
This article applies to: ML8
SEO stands for Search Engine Optimization, which is the study of ways to increase search engine traffic to a site in order to increase that site's search engine rank (how high it appears in search results). Most people find sites through search engines, so high search engine rank is vitally important to e-commerce sites.
SEO experts generally focus their efforts on increasing ranking on Google and Yahoo search results, as those 2 search engines account for the vast majority of Internet searches - around 75% of all searches between them (with Google making up around 50% on its own).
When a person performs a search on one of these (or other) search engines, the search is not actually looking through live websites in real-time. Instead, the search engine checks its internal indexes, which contain a list of keywords they have found on the Internet, and where they found them.
How are Sites Indexed?
How, then, are these lists created? Search engines use programs called web crawlers (also called bots, worms, spiders, etc) to browse the web in a methodical, automated manner. Each crawler starts with a list of URLs to visit, called "seeds'. As the crawler visits these seed sites, it identifies all of the links leading from those pages to other pages, and adds them to the list of sites to visit. Those new URLs are then visited and the new link there are added to the list, and so on.
The sites that these crawlers visit are indexed, according to complex internal algorithms. Each search engine has its own proprietary algorithm that is not known precisely by the public, but there are some methods of increasing page rank that are well-established and reliable. Information on some of those concepts and how AspDotNetStorefront software assists with them is found below.
Note: There are many SEO concepts that have no bearing on the software being used, and the concepts discussed below can be much more complex than described herein. This guide covers these ideas at a basic level and only as they relate to the software. Store admins may wish to work with an SEO firm to increase their success in this very complex area. AspDotNetStorefront recommends Vertical Measures.
A page's URL has a relatively minor effect on how well that page will rank, but it can have an effect on whether or not the page gets indexed at all. There are 2 basic types of URLs - static, and dynamic. Static URLs are simpler, usually shorter, and do not change. Dynamic URLs are often very long and can contain special characters/sequences like %, ?, and 'cgi'. Dynamic URLs usually represent pages whose contents are created programmatically or by reading from a database, rather than simple flat HTML files. The contents of these pages can change frequently and even vary within a single visit based on customer input, page referrer, etc. Dynamic URLs are usually worse for SEO for several reasons:
- It is difficult to accurately index a page whose contents may be different 5 minutes later, or can vary by visitor.
- Search engines only want to list pages in their index that are unique. This means they often cut off URLs after a specific number of variable strings (eg ?=, &, etc). For example, the following 3 URLs may be cut off before the question mark:
http://www.someforumsite.com/thread.php?threadid1234
So you are left with:
http://www.someforumsite.com/thread.php
Now you have duplicate URLs which will be thrown out, and your pages aren't indexed. Dynamic URLs also don't generally contain any keywords. This can hurt traffic, as visitors often try to get an idea of whether or not a page in search results contains what they want by looking at the URL.
How Does AspDotNetStorefront Help?
While most pages on an AspDotNetStorefront site are dynamically generated, the application uses a set of URL rewrite rules to make those URLs appear static. Each page gets a unique URL that contains the product name. This helps increase the chances that the page will be indexed, and puts human-readable keywords right in the URL For example, instead of:
http://www.yourstore.com/showproduct.aspx?productid=4
Customers (and web crawlers) will see:
http://www.yourstore.com/p-4-your-product-name.aspx
Meta tags are information inserted into the 'head' section of web pages. With the exception of the Title tag (see below), these tags contain information that site visitors don't see and probably aren't interested in - such as what character set to use, whether a page is self-rated as containing adult content, etc.
Description and Keywords
These tags can also be used for a page description and to list page keywords. Historically these 2 tags were very important in boosting page rank, but because of overuse, search engines do not rely on these tags very much anymore. That said, they are still important as the contents of those tags are often displayed on search engine results pages:
The title tag isn't really a meta tag, but it functions like one. The contents of that tag are seen by customers at the top of the browser and are used for the default name given to a page when adding it to the browser bookmarks. That tile is also displayed on search engine results pages:
Every page in the AspDotNetStorefront software can be given a title, meta description, and keywords. You have the option of either setting up unique meta information for every product, entity, and topic page, or of setting default meta values that will be used across the site.
To set that default information, simply populate the following AppConfigs:
- SE_MetaDescription
- SE_MetaKeywords
- SE_MetaTitle
If that is all you do, then every page will use the values you entered above (with the exception of titles, which are taken from product/entity names by default). If, however, you'd like to customize the meta information for each page, those tags can be set individually as shown below:
The software has rules for what title information to set if none is entered by the store admin (see below for how to customize what is used instead). The defaults are:
- Product pages use the store name (set in the StoreName AppConfig) plus the product name
- Entity pages use the store name plus the entity name
- Topic pages use the topic name
Each product can have its own meta information, which is entered on the 'Search Engine' tab in Product Manager:
Like products, entities get their own meta information from the data in the 'Search Engine' tab:
Topic pages can have their own meta information as well:
Image Alt Text
Alt text is text that can be set on images which will be displayed when the images cannot be viewed for whatever reason - a broken link, missing file, browser error, etc. Alt text can also tell search engines what an image is about, since web crawlers cannot see images.
An added non-SEO benefit is that alt text is often used by software designed to assist handicapped visitors (such as software that can read a website's content aloud), so it benefits potential customers.
New in version 8.0!
Every product and entity image can be given its own unique alt text. Simply add the text on the 'Search Engine' tab in Product Manager:
When a search engine finds multiple copies of the same content, it will attempt to pick the "best" page from the bunch and then down-rank the other copies. If a site has too much duplicate content, the entire site may be penalized in search rank.
Many CMS/ecommerce applications have a tendency to create multiple URLs that contain the same content, especially on sites with a lot of dynamically generated content. AspDotNetStorefront's URL rewrite rules ensure that each product, entity and topic page has a single unique URL regardless of how the page is reached.
Site Depth and Unreachable Pages
Site Depth
Site depth is a measure of how many 'clicks' it can take to get from your home page to any one page on your site. Search engines do not always 'dig deep'. If a page is more than 2 or 3 links from your home page, search engines may not index that page.
Unreachable Pages
Large sites can sometimes end up with pages that are not linked to from anywhere else on the site by mistake. Because there are no links for web crawlers to go through to reach them, those pages will never be indexed, and your site visitors may miss out on important information.
The sitemap feature can prevent both of these issues from becoming a problem on your site. The sitemap contains links to every page on the site - products, entities, topics, etc. The default storefront skin places a link to the sitemap in the footer of every page on the site (and this is easily done in custom skins as well). With a link to the sitemap on every page, a web crawler is never more than 2 clicks away from any page and no page is unreachable.
The AspDotNetStorefront software also has the ability to generate XML sitemaps. These XML files can be submitted to the larger search engines (through special pages they maintain) to keep them updated of changes on your site. These sitemaps can help get your site indexed more frequently, and ensure that search engines are aware of all the pages on your site.
Please see the pages on the built-in sitemap and Google/Yahoo Sitemaps within the relevant product manuals on this site for more information about sitemaps and how to submit them to search engines.
Page rank is a complex algorithm developed by Stanford University and used by Google to assign a numeric value to pages within a given set of hyperlinks. This value is used to determine the relative importance of the various pages within the set. To quote Google:
"PageRank relies on the uniquely democratic nature of the web by using its vast link structure as an indicator of an individual page's value. In essence, Google interprets a link from page A to page B as a vote, by page A, for page B. But, Google looks at more than the sheer volume of votes, or links a page receives; it also analyzes the page that casts the vote. Votes cast by pages that are themselves "important" weigh more heavily and help to make other pages "important"."
In other words, the more links to your site from other high-quality sites, the higher your page rank will be and the closer to the top of search results your site will be.
While much of page rank is determined by factors outside of the software's control, the affiliate features can be used to work with other sites to increase traffic to your site and reward referrers. Bear in mind that while generally speaking, the more inbound links you have the better, Google does judge the 'quality' of those links. Using linking services or 'spamming' links to your site inappropriately can actually damage your page rank.
Please see the pages affiliate tracking within the relevant product manuals on this site for more information about affiliate features.
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About one year ago, Twentieth Century Fox sued the company releasing Zoom, claiming that this film infringed upon the copyrights to the X-Men franchise. Fox claimed that Zoom was too similar - both stories were about groups of young superheroes brought together to train as a team. There was even a room that simulated dangerous conditions where they could train in realistic environments. Well, Fox and Marvel have little to worry about - Zoom is a piece of junk. The aim of the film, based on the novel by Jason Lethcoe, is more along the lines of Sky High, both in terms of story and quality. Zoom aims more towards a younger audience, and doesn't focus on action, special effect, or story. It's more of a family friendly movie with an ooey-gooey moral. After all, it does star Tim Allen (The Shaggy Dog, Christmas with the Kranks), who does seem to gravitate towards these projects.
Allen is Jack Shepherd, also known as Captain Zoom. Shepherd was a member of a supergroup formed by the government years ago, until something went terribly wrong. He went into retirement, until Marsha Holloway (Courtney Cox, Barnyard, The Longest Yard), tracked him down in order to help train a new generation of superheroes. Shepherd is initially reluctant, swayed only by the promise of a lot of money. What he doesn't know is that seven days from now, a danger from his past is coming, and that he is training the new group, composed of kids, to combat and hopefully defeat this danger.
Zoom is a bit too obvious, which is a problem. Pete Hewitt (Garfield, Thunderpants), who directed and adapted the story with Adam Rifkin (Night at the Golden Eagle, Small Soldiers) and David Berenbaum (The Haunted Mansion, Elf), throw together a thoroughly predictable story. It's not a surprise that the four kids, Dylan West (Michael Cassidy, The Girl from Monday), Summer Jones (Kate Mara, Brokeback Mountain, Tadpole), Tucker Williams (Spencer Breslin, The Shaggy Dog, The Princess Diaries 2), and Cindy Collins (Ryan Newman, Monster House) will come together as a team, convince Zoom to don his uniform again, and save the day. And it's pretty obvious what Hewitt is foreshadowing when he says that Zoom's brother died years ago.
In addition to being predictable, Zoom is also boring. It's hard to figure out what audience this film was going for. It is obviously too simplistic for older viewers, and probably too not engaging enough for younger ones. Things move pretty slowly, causing the filmmakers to rely on slapstick humor or lame jokes. The first time they show Newman throwing heavy objects is kinda funny. The tenth time isn't. Allen often mugs for the camera, and Cox seems to fall down a lot. The film takes place over the course of a week, and feels like an extended montage, with pauses for boring conversation. Worst part? The superpowers aren't that interesting, and the ending plays out with no logical sense whatsoever.
Haro Rates It: Pretty Bad.
1 hour, 23 minutes, Rated PG for brief rude humor, language, and mild violence.


My husband Neal and I were expecting a grand adventure as part of our honeymoon, but we had no idea what lay in store for us and how grateful we would be for US Rider. On Sunday, September 21st, the day after our wedding, my new husband and I left central North Carolina with our quarter horse gelding Max and molly mule Sally in tow. We were headed to Tennessee's Big South Fork Recreation Area where we had rented a secluded cabin outside the park and couldn't wait to spend a week exploring the area from horseback. The drive had gone smoothly until without warning the truck just shut off and wouldn't start again. The last major town we had gone through was Knoxville one hour southeast with our final destination still one hour west of us. My husband is a talented mechanic, but whatever was wrong with our truck wasn't something that was going to be fixed on the side of the road. Our secluded honeymoon location was suddenly presenting a really big problem in the middle of nowhere on a Sunday evening. How were we going to find a service station in an area we had no familiarity with? If we did find one, how would we know they could work on a diesel truck if they weren't going to open until the next day? Even if we did find one how were we going to get there and what were we going to do with the horses? Where could we take them? Where would we stay? Just at the moment when real panic should have been setting in I was picking up the phone and calling US Rider.
Working with the representative on the phone we were able to locate a service shop in Knoxville equipped to work on our vehicle as well as a wrecker service to come tow both our truck and trailer. Clinton Hwy Wrecker Service out of the Knoxville area was able to easily find us on a country state road in a surprisingly reasonable amount of time. They brought with them a roll back wrecker that took our truck to a dealership service shop in Knoxville and a wheel lift wrecker to tow me, my husband, and our two mounts the remaining hour to our cabin rental location. The driving skills of the Clinton Hwy Wrecker driver were greatly appreciated as he navigated the Big South Fork River gorge with our trailer and most precious cargo inside. We also experienced wonderful hospitality and genuine care as our driver stopped at a small convenience store to allow us to purchase a few supplies and remained friendly as we picked through the back roads of Tennessee to our secluded cabin site. I couldn't believe we were able to even find such service that day much less that it was covered fully by my US Rider membership.
It was an experience we could only laugh at as we dined on left over wedding cake and convenience store snacks on our first full day as a married couple. We were able to arrange for a ride into the nearest town with a rental agency the following day which provided us the transportation we needed to continue to enjoy our honeymoon week though not quite in the fashion we had planned. If it hadn't been for US Rider and the services provided through them I'm not sure that could have happened, but I know for certain we would not be able to tell this story with a smile.
Are you a member and have a story to share? You could win a free 1-year membership to USRider by entering our USRider Member Story contest!