text:
\section{Introduction}
Let $X$ be a random vector in $\R^d$ with density $f:\R^d\longrightarrow  [0,\infty)$, a relation denoted by $X\sim f$. Its entropy is defined to be
$$\E(X)=-\int_{\R^d}f\lg f$$
provided $\int_{\R^d}f\lg_+ f < \infty$. We then say that $X$ has finite entropy.

We shall say that a random vector $X$ on $\R^d$, or a probability density $f$, is
 \emph{isotropic} if it is centered and has a  covariance matrix equal to the identity:
 $$\mathbb E [X] = \int_{\R^d} x \, f(x)\, dx = 0$$
and
$$ \mathbb E [X_i X_j] = \int_{\R^d} x_i x_j \, f(x) \, dx = \delta_{i,j}, \quad i,j=1\ldots , d .$$
This normalization can be realized by an affine transformation.

 Among random vectors with a given covariance matrix, the corresponding Gaussian has the largest entropy.
 The gap between the entropy of a random vector $X$ and that of a Gaussian with same covariance matrix is a strong measure of how close $X$ is to being Gaussian.
 For instance, if $X$ has  mean zero and is isotropic with density $f$, and if $G$ is a standard (normal) Gaussian vector with density $g$, then the Pinsker-Csisz\'ar-Kullback inequality (see \cite{P, C} or \cite{BL}) implies that
 $$\frac{1}{2}\bigl{(}\int_{\R^d}|f-g|\bigl{)}^2\leq \E(G)-\E(X).$$

The celebrated Shannon-Stam inequality (see \cite{SW, S}) says that if $X$ and $Y$ are independent identically distributed (\emph{iid} in short) random vectors, then the normalized sum $(X+Y)/\sqrt{2}$ has entropy at least as large as that of $X$ and $Y$:
\begin{equation*}\label{ST}
 \E\Big(\frac{X+Y}{\sqrt{2}}\Big)\ge\E(X).
 \end{equation*}
Moreover,  this inequality is strict if $X$ is not itself a Gaussian random vector. A challenging problem is to quantify this phenomenon, i.e. for fixed covariance matrix (say the identity), bound $ \E(\frac{X+Y}{\sqrt{2}})-\E(X)$ from below by a non-negative (and positive outside zero) function of $\E(G)-\E(X)$. The first result in this direction was obtained by Carlen and Soffer~\cite{CS} who proved, under technical assumptions, a non-explicit bound based on a compactness argument.  Extra assumptions cannot be avoided if one aims at universal entropic estimates: it is  easy  to construct (by taking a carefully chosen double bumped Gaussian) a random variable $X$ for which, the convolution does not greatly modify the entropy, $\E(\frac{X+Y}{\sqrt{2}}) \simeq \E(X)$, but with $\E(X) \ll \E(G)$.
A surprisingly neat result holds in the case where $X\in \R$ is a random variable with variance $1$ and with a density $f$ that satisfies a Poincar\'e (or spectral gap) inequality in the sense that for some positive $c$ and
any smooth function $s$ with $\int_{\R}fs=0$ $$  c\int_{\R}f \, s^2\leq \int_{\R}f\, \big(s'\big)^2.$$
Indeed, we then have
 \begin{equation}\label{bbn}
 \E\Big(\frac{X+Y}{\sqrt{2}}\Big)-\E(X)\geq \frac{c}{2+2c}\big(\E(G)-\E(X)\big)
 \end{equation}
for $Y$  an independent copy of $X$. This result was proved by Ball, Barthe and Naor in~\cite{BBN} using a variational formula for the Fisher information of a marginal density and spectral analysis to get an information jump in the presence of a spectral gap (see \cite[Theorem 2]{BBN}) and then  using a relation between the Fisher information and entropy provided by the adjoint Ornstein-Uhlenbeck semigroup. In the paper~\cite{BJ}, Barron and Johnson obtained  a result similar to~\eqref{bbn} under the same hypothesis, but their method is different (at least in details) to that of \cite{BBN}. In their paper, Barron and Johnson give an upper bound for the $L^2$ distance of the score function of $(X+Y)/\sqrt{2}$ to the space of additive functions of $X$ and $Y$, using $L^2$-orthogonal projections; they eventually use the Poincar\'e inequality to get the result (see \cite[Proposition 2.1 and 3.1]{BJ} for details).

 The aim of the present paper is to investigate similar results for random vectors, and incidentally to give a new approach to~\eqref{bbn}. A random vector $X\in \R^d$ with density $f$ is said to satisfy a Poincar\'e or spectral gap inequality with constant $c>0$ if for any smooth function $s$ with $\int_{\R^d}fs=0$
 \begin{equation}\label{spectralgap}
  c\int_{\R^d}f s^2\leq \int_{\R^d}f\, |\nabla s|^2.
 \end{equation}
That largest constant $c$ in this inequality is indeed the spectral gap for the operator $-L$ on $L^2(f)$ where $Ls:= \Delta s - \nabla (\log(f)\cdot \nabla s$ for suitable functions; the Poincar\'e constant ${\rm c_p}(f)$  refers rather to the inverse of the spectral gap, i.e. to the smallest constant in the inequality
  \begin{equation}\label{spectralgap2}
  \int_{\R^d}f s^2\leq  {\rm c_p}(f)\int_{\R^d}f\, |\nabla s|^2.
 \end{equation}

 A simplistic adaptation of the argument used~\cite{BBN} in higher dimensions leads to an inequality of the form~\eqref{bbn} for random vectors  but with an extra dependance in $d$, the dimension. In the present paper we prove the result without the extra dependence for log-concave random vectors, i.e. those having a density $f$ such that $-\log(f)$ is convex on $\R^n$. It is well known that such random vectors have finite entropy and have a positive spectral gap (see below). This family is central in many high-dimensional problems.

\begin{theorem}\label{main}
Let $X$ be an isotropic  log-concave random vector in $\R^d$. Assume its density $f$ satisfies a Poincar\'e inequality~\eqref{spectralgap} with constant $c>0$.
Then, if $Y$ is an independent copy of $X$, we have
\begin{equation}
\label{vectorjump}
\E\left(\frac{X+Y}{\sqrt{2}}\right)-\E(X)\geq \frac{c}{4(1+c)}(\E(G)-\E(X)).
\end{equation}
\end{theorem}
Since $c\le 1$ when $f$ is isotropic,  the constant $\frac{c}{4(1+c)}$ may be replaced by $\frac{c}{8}$. The log-concavity assumption will be crucial on the proof of inequality~\eqref{marg1bis} below, we do not know whether it holds without this assumption.

As mentioned above, we need to develop a method different to the one of~\cite{BBN}. Our alternative approach relies on the study of \emph{second} derivatives of the entropy  along the heat semi-group (or rather along the Ornstein-Uhlenbeck semi-group). Note that it also gives an alternative proof of the one dimensional case~\eqref{bbn}, up to a numerical (nonessential) constant.

Let us make some general comments on  log-concave random vectors. If $X$ and $Y$ are \emph{i.i.d} random vectors with density $f$, the normalized sum $\frac{X+Y}{\sqrt{2}}$ has density
\begin{equation}\label{convol}
u\longto \int_{\R^d}f(\frac{u+v}{\sqrt{2}})f(\frac{u-v}{\sqrt{2}})dv
\end{equation}
which is a marginal of the joint density on $\R^{2d}$ of the pair $(X,Y)$. It is a consequence of the Brunn-Minkowski inequality (in its functional form due to Pr\'ekopa~\cite{Prekopa}) that log-concave random vectors have log-concave marginals and hence that if $X$ and $Y$ are log-concave, then so is $\frac{X+Y}{\sqrt{2}}$. It is also well known that a log-concave density verifies a Poincar\'e inequality for some constant $c>0$. It was proven by Kannan, Lovasz and Simonovitz~\cite{KLS} and independently by Bobkov~\cite{Bo} that if $X$ is an isotropic log-concave random vector, then it satifies a Poincar\'e inequality~\eqref{spectralgap} with consant $\frac{C}{n}$ for some numerical constant $C$ (thus independent of $n$ and $X$). Actually, slightly better depedence in $n$ is known.The Kannan-Lovasz-Simonovitz (KLS) open conjecture states that there exists a universal constant $c>0$ such that for every $n$, every every isotropic log-concave random vector in $\R^n$ verifies a Poincar\'e inequality with constant $c$. It was noticed some time ago by the first named author, as part of a general program of understanding information theory (and entropy) in the context of convexity in high dimensions,  that using~\eqref{vectorjump} we can prove that the KLS conjecture implies the celebrated hyperplane (or slicing) conjecture.  We shall return to this in the last section.

The organization of the paper is as follows. In the next section, we recall some standard facts about the Ornstein-Uhlenbeck semi-group and about the first derivative of entropy (Fisher information) and the second derivative (information production). Next we investigate how information production behaves under convolution and state a general inequality relating information production of a random vector to the information of a marginal. The subsequent section contains the proof of Theorem~\ref{main}. The final section discusses the connections between entropy jump and the isotropic constant of  log-concave distribution.

\section{Classical results on Ornstein-Uhlenbeck semi-group and Fisher information}
\label{sect:OU}

For any random vector $X$ with smooth enough density $f$ -we require that $\sqrt{f} \in H^1(\R^n)$, but later the density will have even smoother behavior- its Fisher information is defined by
$$J(X):=J(f):=\int_{\R^d}\frac{|\nabla f|^2}{f}.$$

Among random vectors with  given covariance matrix, the Gaussian has the smallest Fisher information, as shown by the following straightforward computation:
if $X$ is an isotropic mean-zero  log-concave random with density $f$, and $G$ is a standard Gaussian with density $g: x\to (1/\sqrt{2\pi})^de^{-|x|^2/2}$, then
$$J(G)=\int_{\R^d}\frac{|\nabla g|^2}{g}=\frac{1}{\sqrt{2\pi}^d}\int_{\R^d}\sum_{i=1}^dx_i^2e^{-|x|^2/2}dx=d.$$
and, by integration by parts,
$$
0\leq\int_{\R^d}\Bigl{|}\frac{\nabla f}{f}+x\Bigl{|}^2 f(x)dx=J(X)-2\int_{\R^d} \textrm{div}(x) f  + d =  J(X)-J(G).
$$

The Fisher information appears as the derivative of the entropy  along the Ornstein-Uhlenbeck semi-group, a property central in the works by Bakry and  \'Emery \cite{BE} and also in Barron's work \cite{B} on the convergence of entropy in the central limit theorem. The Ornstein-Uhlenbeck semi-group can be constructed in several (equivalent) ways and we choose the following. If $X$ is a random vector with density $f$ and $G$ is a standard Gaussian, independent of $X$, we consider the random vector  $X_t=e^{-t}X+\sqrt{1-e^{-2t}}G$, whose law is the Ornstein-Uhlenbeck evolute at time $t$ of the law of $X$. More precisely, the density $f_t$ of $X_t$ is the solution of the  Fokker-Planck equation with generator $L$ :
 \begin{equation}\label{f-p}
 f_0 = f ; \qquad \frac{\partial}{\partial t} f_t(x) =L(f_t)(x) := \Delta_x f_t+\d_x(xf_t),
 \end{equation}
for all $t>0$ and $x\in \R^n$.

It is indeed well known, that starting with a (continuous, say) density $f$, the density $f_t$ is, for $t>0$, strictly positive, $C^\infty$-smooth on $\R^n$ and that $f_t$ and its derivatives decay  exponentially fast to zero at $\infty$ ; in particular $f_t$ has a finite Fisher information and it is readily checked that
\begin{align*}
\frac{\partial}{\partial_t}\E(f_t)&=-\int_{\R^d}(Lf_t)\log(f_t) = -\int_{\R^d} f_t \Delta \log(f_t) -d\int_{\R^d}f_t\\
&=J(f_t)-d.
\end{align*}
Hence, we have the classical expression of the entropy gap as the integral of the information gap
\begin{equation}\label{EFI}
\E(G)-\E(X)=\int_0^{\infty}(J(f_t)-d)dt.
\end{equation}
We refer to Carlen and Soffer \cite{CS} for details and precise justifications.

Let us mention for further reference some other nice stability properties of the Ornstein-Uhlenbeck semi-group.
It can only improve the spectral gap: if $X\sim f$ is isotropic and satisfies a Poincar\'e inequality~\eqref{spectralgap} with constant $c>0$ (actually, $c\in ]0,1]$), then $f_t$ satisfies a Poincar\'e inequality with the same constant $c>0$. This follows easily from Fubini's theorem, H\"older's inequality and the fact that the Gaussian density has spectral gap of size $1$ (see~\cite{BBN}).
Next,  it is again a consequence of Pr\'ekopa's theorem that if $X$ (or $f$) is log-concave, then so is $X_t$ (or $f_t$). Finally, it is also classical that the operation of taking marginals commutes with the Ornstein-Uhlenbeck semi-group in the following sense. Let $X$ and $Y$ be two independent random vectors and $X_t=e^{-t}X+\sqrt{1-e^{-2t}}G_1$ and $Y_t=e^{-t}Y+\sqrt{1-e^{-2t}}G_2$ their independent evolutes along the Ornstein-Uhlenbeck semi-group, where each $G_i$ $(i=1,2$) is a  standard Gaussian vector independent of all the other vectors. Then
\begin{equation}\label{OUmarg}
\frac{X_t+Y_t}{\sqrt2 }= e^{-t}\, \frac{X+Y}{\sqrt2} \;+\; \sqrt{1-e^{-2t}}\, G
\end{equation}
where $G=\frac{G_1+G_2}{\sqrt2}$ is a standard Gaussian vector.

Throughout the rest of the section, $X$ will be an isotropic  log-concave random vector with mean zero,  and density $f$. The density $f_t$ of $X_t=e^{-t}X+\sqrt{1-e^{-2t}}G$, the evolute of $X$ along the Ornstein-Uhlenbeck semi-group satisfies~\eqref{f-p} and takes the form $f_t=e^{-\varphi_t}$ with  $\varphi_t:=\log(f_t)$ convex on $\R^d$. Its  Fisher information will be denoted by
\begin{equation}\label{defJt}
J(t):=J(f_t)=\int_{\R^d}\frac{|\nabla f_t|^2}{f_t}=-\int_{\R^d} f_t\Delta\lg f_t=\tr\int_{\R^d}f_t\hess\varphi_t.
\end{equation}
We will  work with the derivative $\partial_t J(t)$ of the Fisher information along the Ornstein-Uhlenbeck semi-group. The following result is classical in the context of Bakry-Emery's $\Gamma_2$ calculus, although it is not usually written in this form which for us will prove useful. We include a proof for completeness.

\begin{lemma}~\label{lem:dtJ}
With the previous notation we have
\begin{equation}\label{derJ}
\partial_t J(t)=2J(t)-2\tr\int_{\R^d} f_t\, \big(\hess\varphi_t\big)^2.
\end{equation}
\end{lemma}

\begin{proof}
Denoting $\partial_j$ the partial derivative (in space) with respect to $x_j$ we have
\begin{align*}
\partial_t J(t)&=\sum_{i=1}^d\int_{\R^d}2\frac{\partial_if_t}{f_t}\partial_i\partial_tf_t-\int_{\R^d}\big(\frac{\partial_if_t}{f_t}\big)^2\partial_tf_t\\
&=\sum_{i=1}^d\int_{\R^d}-2\partial_i\varphi_t\partial_i\Big[\sum_{j=1}^d\partial_j\big((-\partial_j\varphi_t+x_j) f_t\big) \Big]\\
&\ \ \ \ -\int_{\R^d}\sum_{j=1}^d\partial_j\big((-\partial_j\varphi_t+x_j) f_t \big) (\partial_i\varphi_t)^2.
\end{align*}
where we used that $f_t$ follows~\eqref{f-p} and $Lg = \sum_j \partial_j \big(\partial_j g + x_j g \big)$.
Let $A$ and $B$ be the first and the second terms in the above sum, respectively. Then, we have, by integration by parts and~\eqref{defJt},
\begin{align*}
A&=2\sum_{i,j=1}^d\int_{\R^d}(\partial_{ij}\varphi_t)\partial_i\big((-\partial_j\varphi_t+x_j)f_t\big)\\
&=2J(t)-2\tr\int_{\R^d}f_t\, \big(\hess\varphi_t\big)^2+2\sum_{i,j=1}^d\int_{\R^d}f_t\, \big(\partial_{ij}\varphi_t\big)\big(\partial_i\varphi_t\big)\big(\partial_j\varphi_t-x_j\big)
\end{align*}
and
\begin{align*}
B&=-\sum_{i=1}^d\int_{\R^d}\sum_{j=1}^d\partial_j(f_t(-\partial_j\varphi_t+x_j)) (\partial_i\varphi_t)^2\\
&=2\sum_{i,j=1}^d\int_{\R^d}f_t(-\partial_j\varphi_t+x_j)(\partial_i\varphi_t)(\partial_{ij}\varphi_t)\\
&=-2\sum_{i,j=1}^d\int_{\R^d}f_t (\partial_{ij}\varphi_t) (\partial_i\varphi_t) (\partial_j\varphi_t-x_j).
\end{align*}
Taking the sum of $A$ and $B$, one gets the result of this lemma.
\end{proof}

Note that the formula in the previous lemma can be rewritten in the following equivalent form, which is more standard:
\begin{eqnarray}
\partial_t \big(J(t) - d \big) &=& -2 \big(J(t)-d\big) -2\tr\int_{\R^d} f_t\, \big(\hess\varphi_t- \textrm{Id}\big)^2\nonumber\\
&\le&  -2 \big(J(t)-d\big). \label{Gamma2}
\end{eqnarray}

The next lemma will allow us to control the tails of the entropy production.

\begin{lemma}\label{lem:intJ}
With the previous notation we have
$$\E(G)-\E(X)\leq 2\int_0^{\infty}e^{-2t}(J(t)-d)dt.$$
\end{lemma}
\begin{proof}
Integration of inequality ~\eqref{Gamma2} leads to the following classical Gaussian Log-Sobolev inequality
$$J(t)-d \geq 2(\E(G)-\E(X_t))\text{ }\forall t > 0.$$
By integration by parts, we get
$$\int_0^{\infty}e^{-2t}(J(t)-d)dt\geq \E(G)-\E(X)-\int_0^{\infty}e^{-2t}(J(t)-d)dt,$$
or equivalentely
$$\E(G)-\E(X)\leq 2\int_0^{\infty}e^{-2t}(J(t)-d)dt.$$
\end{proof}

\section{A result for the information production of marginals}

As we saw in ~\eqref{derJ}, the information production $ \partial_t J(t)$ along the Ornstein-Uhlenbeck semi-group is given by quantities of the form
$$\tr\int \big(\hess\log f\big)^2 f  .$$
For our argument, we need to analyze how such quantities can be estimated for marginal densities. Assume $Z$ is a random vector with density $\omega:\R^N\to \R_+$ ($N\ge 1$) and consider the projection $P_E Z$ of $Z$ onto a subspace $E\subset \R^N$. It has a density on $E\simeq \R^{\textrm{dim}(E)}$ which we denote by $h$. A useful observation due to Carlen~\cite{Ca} for Fisher information is that
$$J(h) \le \int \frac{|P_E\nabla \omega|^2}{\omega}.$$
The next result provides an analogue for  information production. However, we are able to establish it only in the case of log-concave densities: here is where the restriction in  our main theorem comes from. Using it, we can then state the central inequality that will be used in the proof of the main theorem.

\begin{lemma}\label{margentprod}
Let $N\ge 1$ and  $\omega=e^{-\phi}:\R^N \to \R_+$ be a smooth positive function. Given a subspace $E\subset \R^N$ define the marginal function on $E$ by
$$\forall x \in E, \qquad h(x) := e^{-\psi(x)} := \int_{E^\perp} \omega(x+y) \, dy=  \int_{E^\perp}e^{-\phi(x+y)} \, dy.$$
Denote by $P_E$ the orthogonal projection onto $E$.
Then, for every $x\in E$ we have,
$$h(x) \hess \psi(x) \le
 \int_{E^\perp}  \omega(x+y)\, P_E \hess \phi (x+y) P_E  \,dy
$$
 in the operator sense (for symmetric operators on $E$) and if $\hess \psi(x) \ge 0$ , then
\begin{equation}\label{marg1}
\tr\left[ \big(\hess \psi(x)\big)^2  h(x)  \right] \le\int_{E^\perp}   \tr\big[\big(P_E \hess \phi (x+y) P_E \big)^2\big]  \omega(x+y) \, dy.
\end{equation}
Therefore, if $\hess\psi\ge 0$ we have
\begin{equation}\label{marg1bis}
 \int_E  \tr\left[\big(\hess \log h\big)^2\right]  h \le \int_{\R^N}   \tr\left[\big(P_E (\hess \log \omega) P_E \big)^2\right]  \omega .
\end{equation}
\end{lemma}

\begin{proof}
We start with the observation that for $x\in E$,
\begin{equation}\label{CS1}
\frac{\nabla h(x) \otimes \nabla h(x)}{h(x)} \le  \int_{E^\perp} \frac{P_E \nabla \omega(x+y) \otimes P_E\nabla \omega(x+y)}{\omega(x+y)} \, dy
\end{equation}
in the symmetric operator sense (on $E$). Indeed, we have
$$\nabla h(x) = \int_{E^\perp} P_E \nabla \omega(x+y) \, dy$$
and for any $v \in E$ we have, using the Cauchy-Schwartz inequality
$$
(\nabla h(x) , v)^2  \le \int_{E^\perp} \frac{( \nabla \omega(x+y) , v )^2}{\omega(x+y)} \, dy \int_{E^\perp} \omega(x+y) \, dy
$$
as claimed.
Next, observe that
$$\hess h(x) = \int_{E_\perp} P_E \hess \omega (x+y) P_E  \, dy $$
and
$$h(x) \hess \psi(x) = h(x) \hess (-\log h)(x) = \frac{\nabla h(x) \otimes \nabla h(x)}{h(x)} - \hess h (x) .$$
Thus~\eqref{CS1} leads to the inequality
\begin{align*}
h(x) \hess \psi(x)  &\le \int_{E^\perp} \Big(\frac{P_E \nabla \omega(x+y) \otimes P_E\nabla \omega(x+y)}{\omega(x+y)}  -
 P_E \hess \omega (x+y) P_E  \Big) \, dy \\
 &= \int_{E^\perp}  \omega(x+y)\, P_E \hess \phi (x+y) P_E dy
 \end{align*}
 in the operator sense on $E$, as wanted.

 Using that  that for symmetric operators $A\ge B \Rightarrow \text{Tr}(AH) \ge \text{Tr}(BH)$ whenever $H\ge 0$, we deduce that, when $\hess \psi (x) \ge 0$,
$$ \tr\left[ \big(\hess \psi(x)\big)^2  h(x)  \right] \le\int_{E^\perp}   \tr\big[P_E \hess \phi (x+y) P_E \hess \psi(x) \big]  \omega(x+y) \, dy. $$
By the Cauchy-Schwartz inequality  (in vectorial form, for the Hilbert-Schmidt scalar product) we then have
\begin{align*}
 \tr\left[ \big(\hess \psi(x)\big)^2  h(x)  \right]  & \le  \sqrt{
 \int_{E^\perp}   \tr\big[\big(P_E \hess \phi (x+y) P_E  \big)^2 \big] \omega(x+y) \, dy. }\\
 &\qquad \times
\sqrt{ \int_{E^\perp}   \tr\big[\big(\hess \psi(x) \big)^2\big]  \omega(x+y) \, dy. }
\end{align*}
Noting that  the second integral equals $ \tr\left[ \big(\hess \psi(x)\big)^2  h(x)  \right]  $, we arrive to inequality~\eqref{marg1}. Integration over $E$ then gives~\eqref{marg1bis}.
\end{proof}

For our argument, we will need the following  useful observation.

\begin{theorem}\label{K2}
Let $X$ be a log-concave random vector in $\R^d$ with smooth density $f=e^{-\varphi}$ where $\varphi$ is a convex function on $\R^d$, and let $Y$ be an independent copy of $X$. Denote by  $h=e^{-\psi}$ the density on $\R^d$ of the random vector
 $\frac{X+Y}{\sqrt{2}}$ and put
$$K=\tr\Big[\int_{\R^d} \big(\hess \varphi\big)^2\,  f \Big], \quad K_2=\tr\Big[\int_{\R^d} \big(\hess\psi\big)^2 h\Big],$$
and
$$ M=\tr\Big[\Big(\int_{\R^d} (\hess \varphi )\, f \Big)^2\Big].$$
Then, we have
$$K_2\leq \frac{K+M}{2}.$$
\end{theorem}
\begin{proof}
As mentioned earlier, we know by Pr\'{e}kopa's theorem that $h$ is log-concave, i.e. $\hess \psi \ge 0$.

We denote $\omega(x,y)=f(x)f(y)$ the density of $(X,Y)$ on $\R^d\times \R^d=\R^{2d}$. For $i=1, \ldots d$ we set $e_i=(0,...,\frac{1}{\sqrt{2}},0,\ldots, 0,\frac{1}{\sqrt{2}},0, \ldots)$ where the $i$-th and $(d+i)$-th coordinates are equal to $\frac{1}{\sqrt{2}}$ and the others are zero. Let $E$ be the vector subspace of $\R^{2d}$ spanned by the orthogonal family $\{e_1,\cdots,e_d\}$.We can assume that the density $h=e^{-\psi}$ of  the random vector $\frac{X+Y}{\sqrt 2}$ is defined on $E$ by identification of $\R^d$ and $E$ through the orthonormal basis $\{e_i\}$ of $E$. Then the Lemma~\ref{margentprod} gives
\begin{align*}
K_2&\leq
 \int_{\R^{2d}}\omega(x,y)\tr\left[ \Big[ \big(\hess(-\log\omega)(x,y)e_i,e_j\big)\Big]_{i,j}^2 \right]\, dx dy \\
&=\int_{\R^{2d}}f(x)f(y)\sum_{i,j=1}^d \big(\hess(-\log\omega)(x,y)e_i,e_j\big)^2\, dxdy.
\end{align*}
Direct computation gives
$$ \big(\hess(-\log\omega)(x,y)e_i,e_j\big)^2=\frac{1}{4}(\partial_{ji}\varphi(x)+\partial_{ji}\varphi(y))^2$$
and hence
\begin{align*}
K_2&\leq \sum_{i,j=1}^d\frac{1}{4}\int_{\R^{2d}}f(x)f(y)\big((\partial_{ji}\varphi(x))^2+2\partial_{ji}\varphi(x)\partial_{ji}\varphi(y)+(\partial_{ji}\varphi(y))^2\big)\, dxdy\\
&=\frac{1}{2}\sum_{i,j=1}^d\int_{\R^{d}}f(\partial_{ji}\varphi)^2+\frac{1}{2}\sum_{i,j=1}^d\biggl(\int_{\R^d}f\partial_{ji}\varphi\biggl)^2\\
&=\frac{1}{2}(K+M).
\end{align*}
\end{proof}

\section{Proof of Theorem 1.1}
We go back to the situation and the notation of Section~\S\ref{sect:OU}. $X$ is an isotropic  log-concave random vector with mean zero,  and density $f$, and $X_t$ is its evolute along the  Ornstein-Uhlenbeck semi-group.
The (log-concave) density of $X_t$ is denoted by $f_t=e^{-\varphi_t}$ and we set
$$
J(t):=J(X_t)=\tr\int_{\R^d}f_t\hess\varphi_t
$$
and
$$
K(t):=\tr\int_{\R^d}f_t(\hess\varphi_t)^2=-\frac{1}{2}e^{2t}\partial_t(e^{-2t}J(t))
$$
where we used Lemma~\ref{lem:dtJ} for the last equality.

We now consider $Z_t$, the Ornstein-Uhlenbeck evolute of $\frac{X+Y}{\sqrt 2}$ where $Y$ is an independent copy of $X$. As mentioned earlier~\eqref{OUmarg}, $Z_t = \frac{X_t+Y_t}{\sqrt 2}$ where $Y_t$ is an Ornstein-Uhlenbeck evolute of $Y$ independent of $X_t$. Denote by $h_t=e^{-\psi_t}$ the smooth (log-concave) density of $Z_t$ and set accordingly
$$
J_2(t):=J(Z_t)=\tr\int_{\R^d}h_t\hess\psi_t
$$
and
$$
K_2(t):=\tr\int_{\R^d}h_t(\hess\psi_t)^2=-\frac{1}{2}e^{2t}\partial_t(e^{-2t}J_2(t))
$$

Theorem~\ref{K2} applied to $X_t$ and $Z_t=\frac{X_t+Y_t}{\sqrt 2}$ then gives that
$$K_2(t)\leq \frac{K(t)+M(t)}{2}=K(t)-\frac{K(t)-M(t)}{2}$$
where,
$$M(t):=\tr\Big[\Big(\int_{\R^d} (\hess \varphi_t )\, f_t \Big)^2\Big].$$
This can be rewritten as
\begin{equation}\label{5}
\partial_t(e^{-2t}(J_2(t)-J(t)))\geq e^{-2t}(K(t)-M(t))
\end{equation}

We next claim that
\begin{equation}\label{spectK}
K(t)-M(t)\geq \frac{c}{1+c}(K(t)-J(t))
\end{equation}
To prove this, remember, as recalled in Section~\S\ref{sect:OU}, that $f_t$ verifies Poincar\'e's inequality with the  same (or better) constant $c$ as $f$. We apply the Poincar\'e inequality~\eqref{spectralgap} to the density $f_t=e^{-\varphi_t}$ and to the functions
$$s_i(x)=\partial_i\varphi_t (x)-\sum_{j=1}^dx_j\int_{\R^d}\big(\partial_{ij}\varphi_t \big) f_t$$
which verify that $\int s_i \, f_t=0$, for $i=1, \ldots, d$. After summing the inequalities $\int_{\R^d}|\nabla s_i|^2f_t \ge c\int_{\R^d}s_i^2 f_t$ we find
$$\tr\int_{\R^d}f_t \, \big(\hess\varphi_t\big)^2-\tr\Big(\int_{\R^d}f_t\hess\varphi_t\Big)^2\geq c\Big(\tr\Big(\int_{\R^d}f_t\hess\varphi_t\Big)^2-\tr\int_{\R^d}f_t\hess\varphi_t\Big).$$
This rewrites as $K(t)-M(t)\geq c(M(t)-J(t))$, which is equivalent to the desired inequality~\eqref{spectK}.

Substituting  inequality~\eqref{spectK} in~\eqref{5}, we find
$$\partial_t(e^{-2t}(J_2(t)-J(t)))\geq \frac{c}{1+c}e^{-2t}(K(t)-J(t)).$$
Integrating this inequality from $t$ to $\infty$, we obtain
$$J(t)-J_2(t)\geq \frac{c}{1+c}e^{2t}\int_t^{\infty}e^{-2s}(K(s)-J(s))ds.$$
Hence, using~\eqref{EFI},
\begin{align*}
\E\left(\frac{X+Y}{\sqrt{2}}\right)-\E(X)&=\int_0^{\infty}(J(t)-J_2(t))dt\\
&\geq\frac{c}{1+c}\int_0^{\infty}e^{2t}\int_t^{\infty}e^{-2s}(K(s)-J(s))dsdt\\
&=\frac{c}{2(1+c)}\int_0^{\infty}(1-e^{-2t})(K(t)-J(t))dt\\
&=\frac{c}{2(1+c)}\int_0^{\infty}(1-e^{-2t})(-\frac{1}{2}\partial_t(J(t)-d))dt\\
&=\frac{c}{2(1+c)}\int_0^{\infty}e^{-2t}(J(t)-d)dt.
\end{align*}
Applying Lemma~\ref{lem:intJ} we get
$$\E\left(\frac{X+Y}{\sqrt{2}}\right)-\E(X)\geq \frac{c}{4(1+c)}(\E(G)-\E(X)).$$
This ends the proof of Theorem~\ref{main}.

\section{Links with the Isotropic constant}

The isotropic constant of an isotropic log-concave random vector $X\sim f$ on $\R^d$ is defined by
$$L_X:= L_f := f(0)^{1/d}$$
This quantity  appears in several high-dimensional problems and a challenging open problem  in asymptotic convex geometry raised by Bourgain and known as the Slicing or Hyperplane conjecture,  is whether it is universally bounded (independently of $f$ and $d$). The best known bound is $L_f \le c\,  d^{1/4}$ for some universal constant $c>0$ (\cite{K1}).  See~\cite{Ba1,MP,K1,EK} for background, equivalent formulations and related results.

\begin{theorem}\label{L}
Let $X$ be an isotropic log-concave random vector in $\R^d$.
Assume that it satisfies an entropy jump with constant $\kappa \in (0,1)$:
$$
\E\left(\frac{X+Y}{\sqrt{2}}\right)-\E(X)\geq \kappa\,  (\E(G)-\E(X)).
$$
for $Y$  an independent copy of $X$. By Theorem~\ref{main}, this  holds with $\kappa=c/8$ if $f$ satisfies a spectral gap inequality~\eqref{spectralgap} with constant $c>0$. Then we have
\begin{equation}\label{Lbound}
\displaystyle L_X \le  e^{2/\kappa}.
\end{equation}
\end{theorem}
Note that the bound also reads as
$$L_X \le e^{16\,  {\rm c_p}(X)}$$
in terms of  ${\rm c_p}(X):={\rm c_p}(f)$, the Poincar\'e constant ~\eqref{spectralgap2} of $X\sim f$.

As a consequence, we see that the KLS conjecture (asserting that isotropic log-concave distributions satisfy a Poincar\'e inequality~\eqref{spectralgap}  with some  universal constant) implies the Hyperplance conjecture. In this direction, a better result is  known; indeed,   Eldan and Klartag~\cite{EK} recently proved that the \emph{variance conjecture} implies as well the   Hyperplane conjecture. The variance conjecture asserts that  inequality~\eqref{spectralgap} for the particular function  $s(x)=|x|^2 - \int |y|^2\, f(y)\, dy$ holds with a universal constant for every log-concave isotropic distribution $f$ on every dimension $d$.  However, it is worth noting  that unlike the Eldan-Klartag result, our estimate above holds at the level of an individual distribution $X$.

Theorem~\ref{L} was presented by the first named author in 2003 at a conference in Kiel and then expanded in a series of lectures in 2006 at the conference \emph{Phenomena in High Dimensions} at the I.H.P., as part of a more general program proposing a probabilistic viewpoint on the geometry of convex bodies in high dimensions. A similar program was also recently and independently proposed by Bobkov and Madiman (see e.g.~\cite{BoMa1, BoMa2}).

Let us now explain the short and simple argument allowing us to pass  from the entropy jump to a bound on the isotropic constant. It relies on a classical rigidity property  of  isotropic log-concave distributions
$X\sim f$ in $\R^d$, namely that  up to non-essential linear terms in $d$, we have $\log f(0)\simeq -\E(X)$.
The following bound
\begin{equation}\label{fm}
-\log f(0)\leq \E(X)\leq -\log f(0)+d
\end{equation}
is for instance implicit in~\cite{FM} and the easy proof is as  follows. Write $f=e^{-\varphi}$ with $\varphi$ convex . For the lower bound use that $\int_{\R^d} x f(x)\, dx = 0$ together with  Jensen's inequality to get
$$-\log f(0)= \varphi(0) \le  \int_{\R^d} \varphi(x) \, f(x)\, dx = \E (X).$$
The upper bound combines the convexity of $f$ and an integration by parts as follows:
$$\E(X) -\varphi(0)=  \int_{\R^d} f(x) \big(\varphi(x) - \varphi(0)\big)\, dx \le \int_{\R^d} f(x) \nabla\varphi(x) \cdot x \, dx = d.$$

Let us mention that in the definition of $L_X$ and in the entropic bounds above
, we can replace, up to numerical constants,  $f(0)$ by
$||f||_{\infty}:=\sup_{\R^d}|f|$, since it is known (see~\cite{Fr}) that $||f||_{\infty}\leq e^df(0)$ for an isotropic log-concave distribution $f$ on $\R^d$.

To finish the proof of~\eqref{Lbound}, assume first that $X\sim f$ is symmetric, which means that $f$ is even. If $h$ denotes the density of $\frac{X+Y}{\sqrt{2}}$, then
$h(x)=2^{\frac{d}{2}}\int_{\R^d} f(x-y)f(y)\, dy$.
It then follows from the log-concavity of $f$ that
$$h(0)=2^{\frac{d}{2}}\int_{\R^d} f(y)^2\, dy\geq 2^{\frac{d}{2}}\int_{\R^d} f(2y)f(0)\, dy=2^{-\frac{d}{2}}f(0).$$
Hence, using~\eqref{fm}  we have that
\begin{equation}\label{jumpS}
\E\left(\frac{X+Y}{\sqrt{2}}\right)\leq \frac{d}{2}\log 2\, -\log f(0) +d\le -\log f(0) +  \frac32 d.
\end{equation}
Let us now go back to the general case where $X\sim f$ is not necessarily symmetric, and consider $Y, X', Y'$ i.i.d. copies of $X$.  Then $\frac{X-X'}{\sqrt 2}, \frac{Y-Y'}{\sqrt 2}$ are symmetric log-concave isotropic random vectors in $\R^d$, independent  and identically distributed according to the  density
$g(x)=2^{d/2}\int_{\R^d} f(x+y)f(y).$
It follows from the argument above that $g(0)\geq 2^{-d/2} f(0)$.
Thus, by the Shannon-Stam inequality
$\frac12\E(Z)+ \frac12 \E(U)\le \E\big(\frac{Z+U}{\sqrt{2}}\big)$ for $Z= (X+Y)/\sqrt2$ and
$U=-(X'+Y')/\sqrt 2$ two independent random vectors, and the bound~\eqref{jumpS} obtained in the symmetric case, one gets
\begin{align*}
\E\left(\frac{X+Y}{\sqrt{2}}\right)&\leq \E\left(\frac{X+Y - X' - Y'}{\sqrt{4}}\right) =  \E\left(\frac{\frac{X-X'}{\sqrt 2} + \frac{Y-Y'}{\sqrt 2}}{\sqrt{2}}\right)
\\
&\leq -\log g(0) +\frac32 d \\
&\leq -\log f(0)+2d.
\end{align*}
On the other hand, the assumption on the Entropy jump implies
$$(1-\kappa)\E(X)\leq \E\left(\frac{X+Y}{\sqrt{2}}\right)-\kappa\E(G) \le  \E\left(\frac{X+Y}{\sqrt{2}}\right)$$
since $\E(G)=\frac{d}{2}\log 2\pi e \ge 0$. Therefore, using again~\eqref{fm} we get that
$$(1-\kappa)\big(-\log f(0) \big) \leq -\log f(0)+ 2d. $$
This implies
$$\kappa \log f(0)\leq 2d.$$
and the desired bound~\eqref{Lbound}.

\subsection*{Acknowledgements}
The second author would like to thank his PhD advisor Dario Cordero-Erausquin for all his help and advice.

redpajama_set_name: RedPajamaArXiv


text:
Highlands and Islands Labour MSP Rhoda Grant has raised with the First Minister why the air ambulance was not initially called when a Caithness mother went into labour at 30 weeks with twins.
At First Minister's Questions today, Mrs Grant told Nicola Sturgeon the woman had bravely shared her experience of giving birth under the current maternity provisions in Caithness.
After going to Caithness General Hospital, the mother was informed that she would have to go to Inverness by road ambulance, over a 100 miles away and two and half hour drive.
Mrs Grant told the First Minister: "Half way into that journey they had to stop at a community hospital at Golspie when the first twin was born breech.
"The air ambulance was then tasked but because it would take two hours to arrive the first twin would be sent by road to Inverness.
"The helicopter could not land, another air ambulance was tasked but this would take too long therefore a second ambulance resumed the journey to Inverness where the second twin was born.
"Thankfully after prolonged stay in hospital all are now doing well.
"However, it begs the question why was the air ambulance or emergency retrieval team not tasked initially airlifting the mum from Caithness.
"Will the First Minister investigate this and will she make sure that the air ambulance treats situations like this as a priority?"
Nicola Sturgeon promised to investigate and conveyed her good wishes to the family. She said she could not answer immediately as to why the air ambulance was not initially tasked, and did not have information in the chamber, and but she asked the Health Secretary, Jeane Freeman, to look into this.
Posting from a Caithness mum on Caithness Health Action Team's Facebook site
Caithness Health Action Team - CHAT
14 February at 21:30 ·
A Mums recent story -
What's it going to take for NHS Highland to change things? Where is this going to end?
While at my work in November 2018 my waters broke and I started labour. I never expected labour as I was only 30 weeks pregnant with my twins. A few hours later we made our way through to Caithness General Hospital. I was taken good care of by the midwives in Wick while they decided on what to do next.
I was in Wick for a couple of hours and was informed I would have to go to Inverness by ambulance. I was put in the ambulance with two midwives. The ambulance crew took an incubator as well. There was no panic and everyone seemed very calm. My partner followed in the car.
We made our way to Inverness. Contractions came stronger and closer so the midwife checked me to which they weren't concerned. About 10/15mins later I had an urge to push. We stopped outside Golspie for the midwives to do a second check. It was decided we must stop at Golspie hospital. The midwives informed my partner and we made our way into Golspie.
Minutes later we arrived and I was wheeled into a very small room with a number of staff inside. Shortly after arrival about 10.30pm twin 1 was born breech. He was being looked after by doctors and nurses very soon after birth.
Thankfully twin 2 was in no rush to appear.
A TV with a video link to Inverness was wheeled into the room with a number of staff on the screen. They used this link to consult with Inverness to decide on what to do next.
I was informed a helicopter was being sent out but wouldn't be arriving till sometime later. So they sent up two nurses and a doctor by police car from Inverness to look after twin one.
We were told that a helicopter was coming but then it couldn't land due to frost on the propellers so it had to go back down the line. They would have been able to send another helicopter but this would have been many hours later.
There was a lot of confusion, questions and lack of decisions being made on what was the best option going forward. It was eventually decided we should make our way to Inverness by ambulance.
About 2am we left for Inverness by convoy, 2 ambulances, one with me and two midwives and other with twin one and two nurses, a police car and two other cars with Drs and my partner.
When we arrived in Inverness to our surprise the Inverness midwives took over and the Wick midwives involvement was finished.
I was induced and twin 2 was born head down. He was taken to SCBU along with his brother.
We spent 6 weeks in Inverness until the twins grew strong enough. I was provided with accommodation for the full 6 weeks.
We now have many appointments in Raigmore due to no-one being able to do checks up here. Why is this?
I am expected to drive two newborns to Inverness for an appointment lasting a couple of minutes.
We all have nothing but the greatest of praise for all staff involved but often think what if something went wrong while on the A9 or even in Golspie hospital.
All parties involved did the best they could with the situation. They made us feel very comfortable and safe in their hands.
My delivery should never have happened the way it did. It could had gone so wrong.
We would like to know,
Why weren't we flown to Inverness?
Why could specialists not come up to Wick?
Why are Wick and Golspie staff expected to work like that?
Golspie wasn't prepared for such an emergency but they did what they could and they did amazing. This shouldn't be the case.
We know of another birth which took place in Golspie in 2017 and wonder if anything has been learned or changed from that.
We dread to think what could have happened if the road was closed by an accident or bad weather.
There is an investigation into our incident but we are unsure if the results will be made public.
With respect to the new hospital at Aviemore Dr Boyd Peters said "We have been on this journey since 2012 and the land purchase is a significant step towards securing much needed modern healthcare facilities for the population of Badenoch and Strathspey" After seven years it finally reaches the planning stage so that must mean yet another two or even three years until its on the ground and working. In Caithness redesign of health services has been going on and on for several years and recently we have the early stages of a "hub" idea being promoted.

redpajama_set_name: RedPajamaCommonCrawl


Best Home For Sale in Miami: BLUE ONE Realty Wins Outstanding Achievement in Website Development from Interactive Media Awards.
BLUE ONE Realty Wins Outstanding Achievement in Website Development from Interactive Media Awards.
DORAL, FL — The South Florida Real Estate company BLUE ONE Realty, today announced that it has been awarded Outstanding Achievement in Website Development by the Interactive Media Awards™ for its work on the BLUE ONE Realty Website. The honor recognizes that the project met and surpassed the basic standards of excellence that comprise the web's most professional work. The site was honored specifically for excellence in Real Estate.
Natalia Arias, Broker / Owner of BLUE ONE Realty, LLC, said, "It's an honor to have our work recognized by the Interactive Media Awards. We feel strongly that our website is the perfect examples of how the Internet can be used to help home buyers find their dream home."
At BLUE ONE Realty, we offer each of our clients the kind of personalized, professional service we would expect to receive if we were buying or selling our own home. We know that your real estate transaction may be the most important step in your entire financial lifetime. We attend to every detail along the way with that significance in mind. Regardless of whether you are a first time home buyer or seller or a seasoned professional investor, BLUE ONE Realty treats every transaction like it is our own. Your success means our success, and we stake our reputation on that conviction.


Incoterms are a series of international trade terms developed by the International Chamber of Commerce (ICC) that define both the roles of buyers and sellers in a commercial transaction and when transfer of risk takes place. To keep up with the rapid expansion of world trade and globalization, the Incoterms rules are revised about once a decade. Since the last revision in 2000, much has changed in global trade and the 2010 revision will take into account issues such as developments in cargo security and the need to replace paper documents with electronic ones. Incoterms 2010 will come into effect on January 1, 2011.
1.The following terms from Incoterms 2000 have been deleted from the list: DES (Delivered Ex Ship), DEQ (Delivered Ex Quay), DDU (Delivered Duty Unpaid) and DAF (Delivered at Frontier).
2.They have been replaced by two new terms: DAP (Delivered at Place) and DAT (Delivered at Terminal). Both terms may be used for both sea and air transport.
Delivered at Place means that the seller delivers when the goods are placed at the disposal of the buyer on the arriving means of transport ready for unloading at the named place of destination.
Delivered at Terminal means that the seller delivers when the goods are unloaded from the arriving means of transport and placed at the disposal of the seller at a named port or place of destination. Terminal includes any place, whether covered or not, and includes quay, warehouse, container yard, road, rail or cargo terminal.
If you need any further information please contact one of the Customs brokers or consultants at your local Winstar office.


Do you know what a temporary injunction is? When can you request an injunction in your Florida inheritance or divorce lawsuit? Did you know that an injunction can sometimes be used to stop a trustee from acting without Court permission? Are you entitled to an injunction from the probate court in West Palm Beach? You may want to read a recent Fourth DCA opinion, Olson v. Olson.
A November 14, 2018 Fourth DCA opinion, Olson v. Olson, is a family law opinion dealing with a Florida temporary injunction. Although this is not a Florida trust and estates case, Olson v. Olson reminds us of important rules regarding temporary injunctions in Florida litigation.
What is an unjust enrichment action? When can a West Palm Beach probate lawyer bring an unjust enrichment claim? What should Florida trust attorneys know about unjust enrichment claims? Are these claims common in probate or guardianship lawsuits? To learn more about unjust enrichment claims, you may want to read a November 21, 2018 Fourth DCA opinion, Southern Specialties Inc. v. Farmhouse Tomatoes Inc.
What is a Reasonable Fee to Pay a Florida Probate Lawyer for a Florida Estate?
What should I expect to pay a West Palm Beach probate lawyer for administering my mom's estate? How much does it cost to have an experienced trust and estates attorney run an Orlando trust? How much should a probate lawyer charge me per hour? Do I have to pay my trust lawyer hourly or will trust lawyers take my case on a contingency?
If someone is the subject of a guardianship in Orlando, can they get married? Can a Ward get married? Is the right to marry taken away if you are deemed incompetent by a probate court? Can the court take away a Ward's right to marry? Can a Ward's right to marry be "subject to court approval"? What happens if a Ward marries without getting permission from the court? What is Florida Statute, Section 744.3215?
Pursuant to Florida guardianship law, who is an interested person? Can I participate in a Florida guardianship proceeding for my spouse, mom, dad, or aunt? Can I participate in a guardianship proceeding of just anyone? Who has the ability to go into a probate court West Palm Beach and have a say in someone's guardianship proceeding? What do Palm Beach courts consider to be frivolous claims? Anyone who is involved in a guardianship matter in Florida should read St. Pete v. Osorio-Khor.


Preheat oven to 400 degrees and line 18 muffin cups with baking cups. In large bowl combine flour, sugar, baking powder, salt and cinnamon. In a small bowl combine milk, oil, vanilla, and egg. Add liquid ingredients all at once to dry, stirring gently just until moistened. Gently fold rhubarb in and then spoon batter into cups, filling 2/3 full. Bake 20-25 minutes or until light golden brown. Immediately remove from pan and sprinkle with powdered sugar. Serve warm.


text:
Professor Erica Charters
Collecting COVID: Oral Histories
Georgina Ferry interviews Erica Charters, Professor of the Global History of Medicine, 6 December 2021.
Topics discussed include (00:00:30) EC's early studies in the history of disease, doctorate at Oxford on the history of medicine (under Professor Mark Harrison, with particular interest in disease during the Seven Years' War), lecture post at Oxford from 2009; (00:01:40) interest in disease and the human experience; (00:02:47) long-term perspectives on disease and historical perspectives including social, cultural, political; (00:05:15) historic pandemics and Charles Rosenberg article "What is an Epidemic?" written during the AIDS epidemic; (00:07:50) tools used to explore the narrative of an epidemic; (00:10:12) first memory of COVID-19 in early 2020; (00:11:42) EC's research work and collaboration with other historians on COVID-19 and the relevance of history in understanding the pandemic; (00:14:23) Centaurus journal contributions [published by the European Society for the History of Science], work of Margaret Pelling (particularly relating to cholera in the 19th century) and work of anthropologists in East Africa [Wenzel, Geissler and Prince]; (00:16:05) COVID-19 specific research, online workshops 'How Epidemics End' (in collaboration with Kristin Heitman); (00:19:01) examples of epidemics ending (including Historian Sam Cohn's perspective, mathematical modelling, anthropologist perspectives); (00:23:10) concepts of excess deaths in disease and warfare; (00:27:20) grant applications for projects; (00:29:00) colleagues involved in policy work in terms of social responses to disease and public health restrictions; (00:31:15) working with health and medical practitioners and interdisciplinary research; (00:32:49) EC's personal experience of COVID-19 and the impact on the academic and research community; (00:34:31) impact on the well-being of students and colleagues, including remote teaching; (00:36:30) positives of online working, including broadening audiences internationally, online reading groups; (00:38:00) competing theories and frameworks in medical science; (00:39:41) EC's personal reaction to the threat from COVID-19; (00:41:40) working long hours and returning to the office; (00:43:32) maintaining productiveness and wellbeing through taking on the project; (00:44:57) how future historians might frame the global response to COVID-19; (00:47:55) future research interests owing to the pandemic, including how these events are measured; (00:50:30) changes in approach to work as a result of COVID-19.
Professor Andy Pollard
Georgina Ferry interviews Sir Andrew Pollard, Professor of Paediatric Infection and Immunity and Director of the Oxford Vaccine Group, 3 December 2021.
Professor Peter Horby
Georgina Ferry interviews Sir Peter Horby, Professor of Emerging Infectious Diseases, 6 December 2021.
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Erica Charters
Georgina Ferry
epidemics and society
diseases and history
covid-19 (disease)
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Welcome to QUARTO! We are an artist-duo, Anna af Sillén de Mesquita and Leandro Zappala, collaborating since 2003. Feel free to read more about US and check out our PERFORMANCES or Send us a mail.
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Mom to Hudson + Hawkins, Wife to my soulmate, Daughter to the most inspiring woman. Vegetarian and lover of all things living. Poet at heart. A tried-and-true California girl with deep Midwest roots. Writer + Blogger.


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For a county that had for the previous four years been either champions or separated by just a score from whoever else lifted the All-Ireland, 2018 for Tipperary was a kind of appalling vista, albeit in the rear-view mirror.
This year's captain Séamus Callanan acknowledges that with the excitement of the new round-robin format and the blazing weather it was a great championship for the neutral, but far from that for Tipperary.
Completely unexpectedly Tipp crashed out at the first phase, two draws all they had to show for four matches. It appeared as if their world just spun out of control, and the margins were so tight at the end that had a late, late defeat by Clare in Thurles been reversed they would have survived.
The problems began on the first day when Tipperary lost to a Limerick side already beginning to bristle with the intent that would close a 45-year gap back to their previous All-Ireland. Had that effectively derailed Tipp at the outset?
"To be honest you don't really have much time to reflect on it because you're in the middle of four games in four weeks. When each game is over you have to just reset and go again, so you don't get time to reflect too much on the games.
Nonetheless the return of 2010 All-Ireland winning manager Liam Sheedy and a full league campaign under his belt has Callanan in optimistic mood.


text:
package dockerclient

import (
	"fmt"
	"io"
	"time"

	"github.com/docker/docker/pkg/units"
)

type ContainerConfig struct {
	Hostname        string
	Domainname      string
	User            string
	AttachStdin     bool
	AttachStdout    bool
	AttachStderr    bool
	ExposedPorts    map[string]struct{}
	Tty             bool
	OpenStdin       bool
	StdinOnce       bool
	Env             []string
	Cmd             []string
	Image           string
	Volumes         map[string]struct{}
	VolumeDriver    string
	WorkingDir      string
	Entrypoint      []string
	NetworkDisabled bool
	MacAddress      string
	OnBuild         []string
	Labels          map[string]string

	// FIXME: The following fields have been removed since API v1.18
	Memory     int64
	MemorySwap int64
	CpuShares  int64
	Cpuset     string
	PortSpecs  []string

	// This is used only by the create command
	HostConfig HostConfig
}

type HostConfig struct {
	Binds           []string
	ContainerIDFile string
	LxcConf         []map[string]string
	Memory          int64
	MemorySwap      int64
	CpuShares       int64
	CpuPeriod       int64
	CpusetCpus      string
	CpusetMems      string
	CpuQuota        int64
	BlkioWeight     int64
	OomKillDisable  bool
	Privileged      bool
	PortBindings    map[string][]PortBinding
	Links           []string
	PublishAllPorts bool
	Dns             []string
	DnsSearch       []string
	ExtraHosts      []string
	VolumesFrom     []string
	Devices         []DeviceMapping
	NetworkMode     string
	IpcMode         string
	PidMode         string
	UTSMode         string
	CapAdd          []string
	CapDrop         []string
	RestartPolicy   RestartPolicy
	SecurityOpt     []string
	ReadonlyRootfs  bool
	Ulimits         []Ulimit
	LogConfig       LogConfig
	CgroupParent    string
}

type DeviceMapping struct {
	PathOnHost        string `json:"PathOnHost"`
	PathInContainer   string `json:"PathInContainer"`
	CgroupPermissions string `json:"CgroupPermissions"`
}

type ExecConfig struct {
	AttachStdin  bool
	AttachStdout bool
	AttachStderr bool
	Tty          bool
	Cmd          []string
	Container    string
	Detach       bool
}

type LogOptions struct {
	Follow     bool
	Stdout     bool
	Stderr     bool
	Timestamps bool
	Tail       int64
}

type MonitorEventsFilters struct {
	Event     string `json:",omitempty"`
	Image     string `json:",omitempty"`
	Container string `json:",omitempty"`
}

type MonitorEventsOptions struct {
	Since   int
	Until   int
	Filters *MonitorEventsFilters `json:",omitempty"`
}

type RestartPolicy struct {
	Name              string
	MaximumRetryCount int64
}

type PortBinding struct {
	HostIp   string
	HostPort string
}

type State struct {
	Running    bool
	Paused     bool
	Restarting bool
	OOMKilled  bool
	Dead       bool
	Pid        int
	ExitCode   int
	Error      string // contains last known error when starting the container
	StartedAt  time.Time
	FinishedAt time.Time
	Ghost      bool
}

// String returns a human-readable description of the state
// Stoken from docker/docker/daemon/state.go
func (s *State) String() string {
	if s.Running {
		if s.Paused {
			return fmt.Sprintf("Up %s (Paused)", units.HumanDuration(time.Now().UTC().Sub(s.StartedAt)))
		}
		if s.Restarting {
			return fmt.Sprintf("Restarting (%d) %s ago", s.ExitCode, units.HumanDuration(time.Now().UTC().Sub(s.FinishedAt)))
		}

		return fmt.Sprintf("Up %s", units.HumanDuration(time.Now().UTC().Sub(s.StartedAt)))
	}

	if s.Dead {
		return "Dead"
	}

	if s.FinishedAt.IsZero() {
		return ""
	}

	return fmt.Sprintf("Exited (%d) %s ago", s.ExitCode, units.HumanDuration(time.Now().UTC().Sub(s.FinishedAt)))
}

// StateString returns a single string to describe state
// Stoken from docker/docker/daemon/state.go
func (s *State) StateString() string {
	if s.Running {
		if s.Paused {
			return "paused"
		}
		if s.Restarting {
			return "restarting"
		}
		return "running"
	}

	if s.Dead {
		return "dead"
	}

	return "exited"
}

type ImageInfo struct {
	Architecture    string
	Author          string
	Comment         string
	Config          *ContainerConfig
	Container       string
	ContainerConfig *ContainerConfig
	Created         time.Time
	DockerVersion   string
	Id              string
	Os              string
	Parent          string
	Size            int64
	VirtualSize     int64
}

type ContainerInfo struct {
	Id      string
	Created string
	Path    string
	Name    string
	Args    []string
	ExecIDs []string
	Config  *ContainerConfig
	State   *State
	Image   string
	Node    struct {
		ID     string
		IP     string
		Addr   string
		Name   string
		Cpus   int
		Memory int
		Labels map[string]string
	}
	NetworkSettings struct {
		IPAddress   string `json:"IpAddress"`
		IPPrefixLen int    `json:"IpPrefixLen"`
		Gateway     string
		Bridge      string
		Ports       map[string][]PortBinding
	}
	SysInitPath    string
	ResolvConfPath string
	Volumes        map[string]string
	HostConfig     *HostConfig
}

type ContainerChanges struct {
	Path string
	Kind int
}

type Port struct {
	IP          string
	PrivatePort int
	PublicPort  int
	Type        string
}

type Container struct {
	Id         string
	Names      []string
	Image      string
	Command    string
	Created    int64
	Status     string
	Ports      []Port
	SizeRw     int64
	SizeRootFs int64
	Labels     map[string]string
}

type Event struct {
	Id     string
	Status string
	From   string
	Time   int64
}

type Version struct {
	ApiVersion    string
	Arch          string
	GitCommit     string
	GoVersion     string
	KernelVersion string
	Os            string
	Version       string
}

type RespContainersCreate struct {
	Id       string
	Warnings []string
}

type Image struct {
	Created     int64
	Id          string
	Labels      map[string]string
	ParentId    string
	RepoDigests []string
	RepoTags    []string
	Size        int64
	VirtualSize int64
}

// Info is the struct returned by /info
// The API is currently in flux, so Debug, MemoryLimit, SwapLimit, and
// IPv4Forwarding are interfaces because in docker 1.6.1 they are 0 or 1 but in
// master they are bools.
type Info struct {
	ID                 string
	Containers         int64
	Driver             string
	DriverStatus       [][]string
	ExecutionDriver    string
	Images             int64
	KernelVersion      string
	OperatingSystem    string
	NCPU               int64
	MemTotal           int64
	Name               string
	Labels             []string
	Debug              interface{}
	NFd                int64
	NGoroutines        int64
	SystemTime         string
	NEventsListener    int64
	InitPath           string
	InitSha1           string
	IndexServerAddress string
	MemoryLimit        interface{}
	SwapLimit          interface{}
	IPv4Forwarding     interface{}
	BridgeNfIptables   bool
	BridgeNfIp6tables  bool
	DockerRootDir      string
	HttpProxy          string
	HttpsProxy         string
	NoProxy            string
}

type ImageDelete struct {
	Deleted  string
	Untagged string
}

type EventOrError struct {
	Event
	Error error
}

type WaitResult struct {
	ExitCode int
	Error    error
}

type decodingResult struct {
	result interface{}
	err    error
}

// The following are types for the API stats endpoint
type ThrottlingData struct {
	// Number of periods with throttling active
	Periods uint64 `json:"periods"`
	// Number of periods when the container hit its throttling limit.
	ThrottledPeriods uint64 `json:"throttled_periods"`
	// Aggregate time the container was throttled for in nanoseconds.
	ThrottledTime uint64 `json:"throttled_time"`
}

type CpuUsage struct {
	// Total CPU time consumed.
	// Units: nanoseconds.
	TotalUsage uint64 `json:"total_usage"`
	// Total CPU time consumed per core.
	// Units: nanoseconds.
	PercpuUsage []uint64 `json:"percpu_usage"`
	// Time spent by tasks of the cgroup in kernel mode.
	// Units: nanoseconds.
	UsageInKernelmode uint64 `json:"usage_in_kernelmode"`
	// Time spent by tasks of the cgroup in user mode.
	// Units: nanoseconds.
	UsageInUsermode uint64 `json:"usage_in_usermode"`
}

type CpuStats struct {
	CpuUsage       CpuUsage       `json:"cpu_usage"`
	SystemUsage    uint64         `json:"system_cpu_usage"`
	ThrottlingData ThrottlingData `json:"throttling_data,omitempty"`
}

type NetworkStats struct {
	RxBytes   uint64 `json:"rx_bytes"`
	RxPackets uint64 `json:"rx_packets"`
	RxErrors  uint64 `json:"rx_errors"`
	RxDropped uint64 `json:"rx_dropped"`
	TxBytes   uint64 `json:"tx_bytes"`
	TxPackets uint64 `json:"tx_packets"`
	TxErrors  uint64 `json:"tx_errors"`
	TxDropped uint64 `json:"tx_dropped"`
}

type MemoryStats struct {
	Usage    uint64            `json:"usage"`
	MaxUsage uint64            `json:"max_usage"`
	Stats    map[string]uint64 `json:"stats"`
	Failcnt  uint64            `json:"failcnt"`
	Limit    uint64            `json:"limit"`
}

type BlkioStatEntry struct {
	Major uint64 `json:"major"`
	Minor uint64 `json:"minor"`
	Op    string `json:"op"`
	Value uint64 `json:"value"`
}

type BlkioStats struct {
	// number of bytes tranferred to and from the block device
	IoServiceBytesRecursive []BlkioStatEntry `json:"io_service_bytes_recursive"`
	IoServicedRecursive     []BlkioStatEntry `json:"io_serviced_recursive"`
	IoQueuedRecursive       []BlkioStatEntry `json:"io_queue_recursive"`
	IoServiceTimeRecursive  []BlkioStatEntry `json:"io_service_time_recursive"`
	IoWaitTimeRecursive     []BlkioStatEntry `json:"io_wait_time_recursive"`
	IoMergedRecursive       []BlkioStatEntry `json:"io_merged_recursive"`
	IoTimeRecursive         []BlkioStatEntry `json:"io_time_recursive"`
	SectorsRecursive        []BlkioStatEntry `json:"sectors_recursive"`
}

type Stats struct {
	Read         time.Time    `json:"read"`
	NetworkStats NetworkStats `json:"network,omitempty"`
	CpuStats     CpuStats     `json:"cpu_stats,omitempty"`
	MemoryStats  MemoryStats  `json:"memory_stats,omitempty"`
	BlkioStats   BlkioStats   `json:"blkio_stats,omitempty"`
}

type Ulimit struct {
	Name string `json:"name"`
	Soft uint64 `json:"soft"`
	Hard uint64 `json:"hard"`
}

type LogConfig struct {
	Type   string            `json:"type"`
	Config map[string]string `json:"config"`
}

type BuildImage struct {
	Config         *ConfigFile
	DockerfileName string
	Context        io.Reader
	RemoteURL      string
	RepoName       string
	SuppressOutput bool
	NoCache        bool
	Remove         bool
	ForceRemove    bool
	Pull           bool
	Memory         int64
	MemorySwap     int64
	CpuShares      int64
	CpuPeriod      int64
	CpuQuota       int64
	CpuSetCpus     string
	CpuSetMems     string
	CgroupParent   string
}

type Volume struct {
	Name       string // Name is the name of the volume
	Driver     string // Driver is the Driver name used to create the volume
	Mountpoint string // Mountpoint is the location on disk of the volume
}

type VolumesListResponse struct {
	Volumes []*Volume // Volumes is the list of volumes being returned
}

type VolumeCreateRequest struct {
	Name       string            // Name is the requested name of the volume
	Driver     string            // Driver is the name of the driver that should be used to create the volume
	DriverOpts map[string]string // DriverOpts holds the driver specific options to use for when creating the volume.
}

// IPAM represents IP Address Management
type IPAM struct {
	Driver string       `json:"driver"`
	Config []IPAMConfig `json:"config"`
}

// IPAMConfig represents IPAM configurations
type IPAMConfig struct {
	Subnet     string            `json:"subnet,omitempty"`
	IPRange    string            `json:"ip_range,omitempty"`
	Gateway    string            `json:"gateway,omitempty"`
	AuxAddress map[string]string `json:"auxiliary_address,omitempty"`
}

// NetworkResource is the body of the "get network" http response message
type NetworkResource struct {
	Name       string                      `json:"name"`
	ID         string                      `json:"id"`
	Scope      string                      `json:"scope"`
	Driver     string                      `json:"driver"`
	IPAM       IPAM                        `json:"ipam"`
	Containers map[string]EndpointResource `json:"containers"`
}

//EndpointResource contains network resources allocated and usd for a container in a network
type EndpointResource struct {
	EndpointID  string `json:"endpoint"`
	MacAddress  string `json:"mac_address"`
	IPv4Address string `json:"ipv4_address"`
	IPv6Address string `json:"ipv6_address"`
}

// NetworkCreate is the expected body of the "create network" http request message
type NetworkCreate struct {
	Name           string `json:"name"`
	CheckDuplicate bool   `json:"check_duplicate"`
	Driver         string `json:"driver"`
	IPAM           IPAM   `json:"ipam"`
}

// NetworkCreateResponse is the response message sent by the server for network create call
type NetworkCreateResponse struct {
	ID      string `json:"id"`
	Warning string `json:"warning"`
}

// NetworkConnect represents the data to be used to connect a container to the network
type NetworkConnect struct {
	Container string `json:"container"`
}

// NetworkDisconnect represents the data to be used to disconnect a container from the network
type NetworkDisconnect struct {
	Container string `json:"container"`
}

redpajama_set_name: RedPajamaGithub


text:
Q: Is there any default function in C++ to search for a word in a string? Is there any default function in C++ to search for a word in a string?

A: Check out string::find()

A: That would be strstr if you want a plain function to work with char*, std::string::find if you work with std::string.

redpajama_set_name: RedPajamaStackExchange


text:
An Anti-government protester clashes with police in Lima, 20 January 2023. (Photo by Lucas Aguayo Araos/ dpa/ picture-alliance/ Newscom/ MaxPPP).
By Ivan Fernandes |
Catholic bishops in Peru are offering to mediate for peace after Pope Francis decried the ongoing civil unrest against the Latin American country's new president saw dozens of people killed.
"It is time to raise our heads and move forward towards meeting and reconciling with justice. The country must not continue in anxiety, fear and uncertainty. Let us take on the commitment to rebuild Peru and let us also ask civil society to assume its responsibilities," the Peruvian Episcopal Conference said.
International media reported that 60 people have died in the continuous unrest since Congress on December 7 overwhelmingly voted to dismiss Pedro Castillo as the president and swore in his vice-president, Dina Boluarte, as the new leader. Boluarte is the first woman to lead Peru and will govern until July 2026, when Castillo's term would have come to an end.
Protests over Castillo's dismissal began in the capital Lima and soon spread elsewhere with demonstrations, riots, strikes and blockades of roads, railroads and airports paralyzing much of the country. Last week the government was forced to indefinitely close the Inca citadel of Machu Picchu, Peru's most famous tourist site, leaving hundreds of foreigners and Peruvians stranded. Over the weekend, anti-riot police raided San Marcos University – the oldest in the Americas – to violently quell demonstrators who have been demanding fresh elections and calling for Boluarte to stand down, something she has so far refused to do.
"No to violence... no to death"
"No to violence from wherever it comes! no to death!" Pope Francis said following his January 22 Sunday Angelus address to the faithful gathered in St. Peter's Square, asking them to pray that violence in Peru might cease. "Violence quenches the hope for a just solution to problems. I encourage all the parties involved to undertake the path of dialogue as brothers of the same nation, in full respect for human rights and the rule of law," the Latin American pope said.
The bishops in Peru said they will not tire calling for dialogue amid the "harsh political and social confrontation" taking place and have offered their services "to mediate and build bridges of encounter", because "the Church must be truly in solidarity with all humanity and its history".
The death of so many "is a deep wound in the heart of our people", as well as for the suffering of all the wounded, both civilians and police, said the bishops. "This obliges us to resolutely change course: we want peace!" the bishops wrote, stressing such violence requires dialogue, listening but at the same time cannot go unpunished. "It is time for the authorities and all the political actors to commit themselves in a responsible manner, in order to find a consensual solution to this serious crisis", they said.
Peru, a divided country
A tightly fought presidential election last June saw socialist leader Castillo marginally win the presidential election after Peruvians voted to select their fifth president in three years. Castillo, 51, a former primary school teacher and leader of the Free Peru party has galvanized rural and poorer voters who are fed up with politicians after years of corruption and a rift between Congress and the president played out with swift changes in leadership.
However, Castillo form the onset faced attacks and allegations of corruption from various far-right opposition groups – who until now sought unsuccessfully to impeach him. His 18 months in office saw constant ministerial changes and he had five prime ministers.
The bishops had repeatedly stressed the need to overcome the crisis and called on the people, who are predominantly Catholic, to unite for the common good and for politics to be orientated towards care for those most in need, the pursuit of dialogue, justice, reconciliation and service to others.
Visiting bishops to the Holy Land praise local Christian communities