text:
falling in reverse meaning
Lead singer Ronnie Radke, guitarits Jacky Vincent & Derek Jones, drummer Ryan Seaman, and bassist Ronnie Ficarro. Falling in Reverse Fans Also Like: One Direction song meanings Beatles song meanings Yes song meanings David Bowie song meanings My Chemical Romance song meanings Submit Your Interpretation Falling is typically seen as a bad thing, regressing. The fine line between genius and insanity, self-seriousness and self-deprecation, implosion and explosion: that is the phantom zone where FALLING IN REVERSE thrives. The fights are getting old I think this is one of Ronnie's hidden, analytical, meanings that let's out just enough to show what he's really talking about but hides enough to not say it blatantly. So it sounds like it's about a girl..but really it's not. 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falling in reverse meaning 2021


NOTE: As of 2014, Herb is still on the air at WOSN-FM in Vero Beach, FL.
At the very end of the above WDGY interview with Tony Andreason, Bill Diehl references the new Music Survey that has Freddy Cannon's Abigail Beecher at No. 13. Here is a link to the WDGY Music Survey on Oldiesloon from Sunday, February 22, 1964 which shows the song in that position.
Includes a newscast and commercials for Vern Donnay's Brookdale Estates and Dayton's Top 10 Club.
Sam Babcock counting down the "Tall Tiger Top 50." Included are many commercials and jingles. You will also hear the influence The Beatles had on the station's programming. #1 29 MB - #2 27 MB (provided by Phil Kitchen) Phil says that the aircheck was recorded on a Wollensak tape recorder with an audio cable clipped to the speaker terminals of a Setchell Carlson tube table radio.
Includes commercials for Minnesota Dragways, Mennen Dry Deodorant and Schells Beer.
Includes a newscast with Mark Anderson and commercials for Pannekoeken Huis, WDGY Incredible Contest, Holiday Village Stores, Dayton's, Levi's for Feet, Goodman Jewelers, 1977 Subaru, Northwestern Bell, Winn Stephens Datsun North, Dolomite ski boots and WCCO-TV Scene Tonight.
Includes commercials for Y-11 Album Special, Coca-Cola, The Amazing Rhythm Aces at O'Shaughnessy Auditorium, Y-11 Count The Winners weekend, 7-11 fight against Muscular Dystrophy, Supertramp "Even in the Quietest Moments" album discounted with Butterfinger or Baby Ruth wrapper, and Black Sunday (movie).
Jay Philpott writes, "Hal worked at KQRS from about May 1977 to February 1981. On his last show, he hinted that he would be back on Tuesday "somewhere over the radio." That turned out to be a six-month stint at WDGY. He debuted on Stereo 101 during the Minnesota State Fair exactly six months after leaving KQRS. Apparently, KQ was OK with him working at WDGY, but held him to the six month non compete before he could go to KDWB-FM. He stayed at 101 for about three years and went back to KQ in '84 as KDWB-FM started sliding back to CHR. He left KQ about '85 or '86 and worked at KJJO Hot Rockin' 104 for a couple of years there before heading up to Duluth where he worked for KKCB, B-105. He's out of radio now and in the automotive business."
WDGY (740 AM) - "The WDGY Classic Weekly Countdown" with Larry Ravenswood featuring the top 10 songs from the WDGY 30 Star Survey from Sept 4, 1970. 13 MB. The actual WDGY survey featured can be found at Oldiesloon here.
A video and photos of Johnny Canton, who was inducted into the MN Broadcasting Hall of Fame in 2006, can be found on the Pavek Museum website.
The following WDGY-AM North Country Radio images are scanned from a children's book called "Careers with a Radio Station" (written in 1983 by Mark Lerner with photographs by Milton J. Blumenfeld). Click on photos to enlarge.


text:
\section{Introduction}

The Gaussian isoperimetric inequality says that the halfspace has the smallest Gaussian perimeter among all sets with prescribed Gaussian measure, \cite{Borell}.
In the Euclidean setting, an increasing interest has been devoted to the study of non local approximations of the perimeter and their isoperimetric shapes, since the pioneering work of Caffarelli, Roquejoffre and Savin, \cite{CRS}.

The aim of this paper is to provide an analogous non local approximation of the Gaussian perimeter, showing the Gamma convergence to the local one. Moreover, we study the isoperimetric properties of this non local functional and observe that, in contrast with the local setting, an halfspace is a volume constrained critical point if and only if it has Gaussian measure $\frac 12$. In particular, we deduce that Ehrhard symmetrization can in general increase the considered non local Gaussian perimeter.

We remark that the non local approximation of the Gaussian perimeter we study is different from the one recently proposed in \cite{NPS}.
The non local functional we introduce has the advantage of having a more explicit formulation, while has the drawback that the isoperimetric shapes and the Ehrhard symmetrization are not preserved.

Inspired by  \cite{ADPM}, for a measurable set $E\subset\R{n}$, $n\ge 1$, $0<s<1$, and a
connected, open set $\Omega\Subset\R{n}$ with Lipschitz boundary (or
simply $\Omega=(a,b)\Subset\R{}$ if $n=1$), we define the Gaussian, nonlocal
functional
$$\M{J}^\gamma_s(E, \Omega):=\M{J}^{1, \, \gamma}_s(E,\Omega)+\M{J}^{2, \, \gamma}_s(E, \Omega),$$
where
\begin{equation*}
\begin{split}
\M{J}^{1, \, \gamma}_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap \Omega}\frac{\gamma(x, \, y)}{|x-y|^{n+s}} \, dxdy,\\
\M{J}^{2, \, \gamma}_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap
\Omega^c}\frac{\gamma(x, \, y)}{|x-y|^{n+s}} \, dxdy+ \int_{E\cap
\Omega^c}\int_{E^c\cap \Omega}\frac{\gamma(x, \, y)}{|x-y|^{n+s}} \, dxdy,
\end{split}
\end{equation*}
and
\[
\gamma \fromto{\R{n} \times \R{n}}{\R{+}},\ \gamma(x, \, y) = \exp\left(- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)\right).
\]
When $\Omega$ coincides with the whole space, we just write $\M{J}^{\gamma}_s(E)$.

In \cite{ADPM}, Amborsio, De Philippis and Martinazzi have studied the Euclidean version of it, namely $\M{J}_s = \M{J}_s^1 + \M{J}_s^2$, and
\begin{equation*}
\begin{split}
\M{J}^1_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap \Omega}\frac{1}{|x-y|^{n+s}} \, dxdy,\\
\M{J}^2_s(E, \Omega)&:=\int_{E\cap \Omega}\int_{E^c\cap
\Omega^c}\frac{1}{|x-y|^{n+s}} \, dxdy+ \int_{E\cap
\Omega^c}\int_{E^c\cap \Omega}\frac{1}{|x-y|^{n+s}} \, dxdy.
\end{split}
\end{equation*}
The authors point out that $\M{J}_s(E,\Omega)$ can be thought of as a fractional perimeter of $E$ in $\Omega$, and they show the $\Gamma$-convergence of $(1-s)\M{J}_s(\cdot,\Omega)$ to $\omega_{n-1}P(\cdot,\Omega)$  as $s\to 1^-$, where $\omega_{n-1}$ is the volume of the unit ball in $\R{n-1}$, $P(E, \, \Omega):= \mathcal{H}^{n-1}(\M{F} E \cap \Omega)$ is the Euclidean perimeter, $\mathcal{H}^\alpha$ denotes the classical $\alpha$-Hausdorff measure and $\M{F} E$ the reduced boundary of $E$.
Moreover, they prove the convergence of any sequence $\{E_i\}$ of local minimizers for
$\M{J}_{s_i}(\cdot, \Omega)$ to a local minimizer for $P(\cdot,\Omega)$, see \cite[Theorem 3]{ADPM}.

The first aim of this paper is to generalize \cite[Theorem 3]{ADPM} to the Gaussian case, thus building a relation between the functional $\M{J}^{\gamma}_s$ and the Gaussian perimeter
\[
P^\gamma(E, \, \Omega):= \int_{\M{F} E \cap \Omega} e^{-\frac{1}{2} \abs{x}^2} \, d\mathcal{H}^{n-1}(x).
\]
The second goal is to investigate whether the halfspaces are volume constrained critical points of $\M{J}^{\gamma}_s$. This turns out to be true if and only if the boundary hyperplane passes through the origin.

The paper is divided in four Sections. In Section 2 we prove the $\Gamma-$convergence of the functional $\M{J}^{\gamma}_s$ to $P^\gamma$.
In Section 3 we compute the first and second variation of $\M{J}^{\gamma}_s$
(for the local framework see \cite{BBJ} or \cite{L}).
In Section 4 we prove that halfspaces are volume constrained stationary points for $\M{J}^{\gamma}_s$ if and only if their Gaussian volume is $\frac 12$.

\section{The Gamma-convergence }
In this section we extend \cite[Theorem 3]{ADPM} to the Gaussian case. Namely, we show:
\begin{trm}[Convergence of local minimizers]\label{trm4}
Assume that $s_i\uparrow 1$, $E_i$ are local minimizers of
$\M{J}^{\gamma}_{s_i}(\cdot, \Omega)$, and $\chi_{E_i}\to\chi_E$ in
$L^1_{\loc}(\R{n})$. Then
\begin{equation}\label{Genergybou}
\limsup_{i\to \infty}(1-s_i)\M{J}^{\gamma}_{s_i}(E_i,\Omega')< + \infty \qquad \forall\Omega'\Subset\Omega,
\end{equation}
$E$ is a local minimizer of $P^{\gamma}(\cdot,\Omega)$ and
$(1-s_i)\M{J}^{\gamma}_{s_i}(E_i,\Omega')\to \omega_{n-1}P(E,\Omega')$ whenever
$\Omega'\Subset\Omega$ and $P(E,\partial\Omega')=0$.
\end{trm}
The proof of Theorem \ref{trm4} is almost identical to the Euclidean one for \cite[Theorem 3]{ADPM}. We limit our study to the parts which differ from it.  In particular we will prove the following two propositions. Let $\omega_k$ denote the volume of the unit ball in $\R{k}$ for $k\ge 1$, and set $\omega_0:=1$.
\begin{prop}\label{GammaLiminfProp}
For every measurable set $E\subset\R{n}$ we have
\begin{equation}\label{Gliminf}
\Gamma-\liminf_{s\uparrow 1}(1-s)\M{J}^{1, \, \gamma}_{s}(E,\Omega) \geq \omega_{n-1}P^\gamma(E,\Omega)
\end{equation}
w.r.t. the $L^1_{\rm
loc}$ convergence of the corresponding characteristic functions in
$\R{n}$, i.e.
$$\liminf_{i\to \infty}(1-s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i,\Omega)\geq \omega_{n-1}P^\gamma(E,\Omega)\qquad \text{whenever } \chi_{E_i}\to \chi_E \text{ in } L^1_{\loc}(\R{n}),\;s_i\uparrow 1.$$
\end{prop}

\begin{prop}\label{GammaLimsupProp}
For every measurable set $E\subset\R{n}$ we have
\begin{equation}\label{Glimsup}
\Gamma-\limsup_{s\uparrow 1}(1-s)\M{J}^\gamma_{s}(E,\Omega) \leq \omega_{n-1}P^\gamma(E,\Omega)
\end{equation}
w.r.t. the $L^1_{\rm
loc}$ convergence of the corresponding characteristic functions in
$\R{n}$. Inequality \eqref{Glimsup} means that for every measurable set $E$ and sequence $s_i\uparrow 1$ there exists a sequence $E_i$ with $\chi_{E_i}\to \chi_E$ in $L^1_{\loc}(\R{n})$ such that
$$\limsup_{i\to\infty}(1-s_i)\M{J}^\gamma_{s_i}(E_i,\Omega)\leq \omega_{n-1} P^\gamma(E,\Omega).$$
\end{prop}

In these two propositions lurk the main differences between the Gaussian case and the Euclidean case. Once we have proved Proposition \ref{GammaLiminfProp} and Proposition \ref{GammaLimsupProp}, we are done: the proof of Theorem \ref{trm4} is completely identical to the proof of \cite[Theorem 3]{ADPM}, with the only forethought of adding a $\gamma$-superscript in every considered functional, and remembering the simple inequality $\gamma(x, \, y) \le 1$.

\medskip

We will use the following notation: we write $x\in\R{n}$ as $(x',x_n)$ with $x'\in\R{n-1}$
and $x_n\in\R{}$; we denote by $H$ the halfspace $\{x:\ x_n\leq 0\}$ and
by $Q=(-1/2,1/2)^n$ the canonical unit cube; we denote by $B_r(x)$ the ball of radius $r$ centered at $x$ and,
unless otherwise specified, $B_r:= B_r(0)$;  for every $h\in \R{n}$ and function $u$ defined on
$U\subset \R{n}$ we set $\tau_h u(x):= u(x+h)$ for all $x\in U-h$. For the definition and basic properties of the perimeter $P(E,\Omega)$ in the
sense of De Giorgi, we refer to \cite{AFP,giu}.

\subsection{Proof of proposition \ref{GammaLiminfProp}}
We denote by ${\mathcal C}$ the family of all $n$-cubes in $\R{n}$
$$
{\mathcal C}:=\left\{R(x+rQ):\ x\in\R{n},\,\,r>0,\,\,R\in
SO(n)\right\}.
$$

Let $s_i\uparrow 1$ and sets $E_i\subset\R{n}$ with $\chi_{E_i}\to \chi_E$
in $L^1_{\loc}(\R{n})$ as $i\to \infty$ be given. We  need to show the inequality
\begin{equation}\label{liminf}
\liminf_{i\to\infty}(1-s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i,\Omega)\ge \omega_{n-1} P^{\gamma}(E,\Omega).
\end{equation}
We can assume that the left-hand side of \eqref{liminf} is finite,
otherwise the inequality is trivial.  We choose an arbitrary $\Omega' \Subset \Omega$, and find a positive constant $c_0=c_0(\Omega')$ so that $c_0 \le \gamma(x, \, y),\ \forall \, x, \, y \in \Omega'$. Then we easily obtain the inequality
\[
c_0 \limsup_i \M{J}^1_{s_i} (1-s_i) (E_i, \, \Omega'_i) \le \lim_i (1- s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i, \, \Omega') < +\infty .
\]
By \cite[Theorem 1]{ADPM} and the arbitrariness of $\Omega'$, we conclude that $E$ has locally finite perimeter.  We shall denote by $\mu$ its perimeter measure, i.e.
$\mu(A)=|D\chi_E|(A)$ for any Borel set $A\subset\Omega$, and we
shall use the following property of sets of finite perimeter: for
$\mu$-a.e. $x_0\in\Omega$ there exists $R_{x_0}\in SO(n)$ such that
$(E-x_0)/r$ locally converge in measure to $R_{x_0}H$ as $r\to 0$.
In addition,
\begin{equation}\label{densitycube}
\lim_{r\to 0}\frac{\mu(x_0+rR_{x_0}Q)}{r^{n-1}}=1,\quad \text{for  $\mu$-a.e. } x_0.
\end{equation}
Indeed this property holds for every $x_0\in \M{F} E$, see \cite[Theorem 3.59(b)]{AFP}.

Now, given a cube $C\in{\mathcal C}$ contained in $\Omega$, we set
$$\alpha_i(C):=(1-s_i)\M{J}^{1, \, \gamma}_{s_i}(E_i,C), \qquad \mbox{and} \qquad \alpha (C):=\liminf_{i\to\infty}\alpha_i(C).$$
Moreover, we define $C_r(x_0):=x_0+rR_{x_0}Q$, where $R_{x_0}$ is as in
\eqref{densitycube}, and the measure
\[
\nu(E) = \int_{E} e^{-\frac{1}{2} \abs{x}^2} \, d\mu(x),\qquad \mbox{for every } E \mbox{ Borel set}.
\]
We claim that  for $\mu$-a.e. $x_0\in \R{n}$ it holds
\begin{equation}\label{eq1}
\omega_{n-1} \le \liminf_{r\to 0}\frac{\alpha(C_r(x_0))}{\nu( C_r(x_0))}.
\end{equation}
If the claim is true, then we observe that for all $\varepsilon>0$ the family
$$\M{A}:=\Big\{C_r(x_0)\subset\Omega\;:\; \omega_{n-1} \nu(C_r(x_0)) \le (1+\varepsilon) \alpha(C_r(x_0))   \Big\}$$
is a fine covering of $\mu$-almost all of $\Omega$. By a suitable
variant of Vitali's theorem (see \cite{M}), we can extract a
countable subfamily of disjoint cubes $\{C_j\subset\Omega:j\in J\}$ such that
$\nu\big(\Omega\setminus\bigcup\limits_{j\in J} C_j\big)=0$, whence
\begin{equation*}
\begin{split}
\omega_{n-1} P^\gamma(E,\Omega)&=\omega_{n-1} \nu\Big(\bigcup_{j\in J}C_j\Big)=\omega_{n-1}\sum_{j\in J}\nu(C_j)\leq (1+\varepsilon)\sum_{j\in J}\alpha(C_j)\le (1+\varepsilon)\liminf_{i\to\infty}\sum_{j\in J}\alpha_i(C_j)\\
&\leq (1+\varepsilon)\liminf_{i\to\infty}(1-s_i)\M{J}^{1,\gamma}_{s_i}(E_i,\Omega).
\end{split}
\end{equation*}
Since $\varepsilon>0$ is arbitrary, we get the $\Gamma-\liminf$ estimate.

We now prove the inequality in \eqref{eq1} at any point $x_0$ such
that $(E-x_0)/r$ converges locally in measure as $r\to 0$ to
$R_{x_0}H$ and \eqref{densitycube} holds. Because of
\eqref{densitycube} and the continuity of the exponential, we know that
\[
\lim_{r \to 0} \Intm_{C_r(x_0)} e^{- \frac{1}{2} \abs{x}^2 } \, d\mu(x) = e^{-\frac{1}{2} \abs{x_0}^2}.
\]
Thus we just need to show the inequality
\begin{equation}\label{eq11}
\liminf_{r\to 0}\frac{\alpha(C_r(x_0))}{r^{n-1}}\geq \omega_{n-1} e^{-\frac{1}{2} \abs{x_0}^2}.
\end{equation}
Since from now on $x_0$ is fixed, we assume $R_{x_0}=I$, so that the limit
hyperplane is $H$ and the cubes $C_r(x_0)$ are the standard ones
$x_0+rQ$. Let us choose a sequence $r_k\to 0$ such that
$$\liminf_{r\to 0}\frac{\alpha(C_r(x_0))}{r^{n-1}}=
\lim_{k\to \infty} \frac{\alpha(C_{r_k}(x_0))}{r_k^{n-1}}.$$ For $k>0$
we can choose $i(k)$ large enough that the following conditions hold:
\begin{equation*}
\left\{
\begin{aligned}
&\alpha_{i(k)}(C_{r_k}(x_0))\leq \alpha(C_{r_k}(x_0))+r_k^n,\\
&r_k^{1-s_{i(k)}}\ge 1-\frac{1}{k},\\
&\Intm_{C_{r_k}(x_0)}|\chi_{E_{i(k)}}-\chi_E|dx<\frac{1}{k}.
\end{aligned}
\right.
\end{equation*}
We observe that, although $\M{J}^{1, \, \gamma}_s$ does not enjoy the nice scaling properties of $\M{J}^{1}_s$, it still satisfies the equality
\[
\M{J}^{1, \, \gamma}_s (E, \, C_r(x_0))= r^{n - s}\M{J}^{1, \, \gamma_{x_0, \, r}}_s((E- x_0)/r, \, Q),
\]
where we have set
\[
\gamma_{x_0, \, r}(x, \, y) = \exp \left(- \frac{1}{4} \left( \abs{x_0 + rx}^2 + \abs{x_0 + r y}^2  \right)\right).
\]
In particular, for $r$ sufficiently small, and thus for every $r_k$ with $k$ sufficiently big, the following inequality holds:
\[
\abs{\gamma_{x_0, \, r}(x, \, y) - e^{- \frac{1}{2} \abs{x_0}^2}} \le 4 r.
\]
Then we infer
\begin{equation*}
\begin{split}
\frac{\alpha(C_{r_k}(x_0))}{r_k^{n-1}}&\ge \frac{\alpha_{i(k)}(C_{r_k}(x_0))}{r_k^{n-1}}-r_k=\frac{(1-s_{i(k)})\M{J}^{1, \, \gamma_{x_0, \, r_k}}_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q)r_k^{n-s_{i(k)}}}{r_k^{n-1}}-r_k\\
&\ge
\Big(1-\frac{1}{k}\Big)(1-s_{i(k)})\M{J}^{1, \, \gamma_{x_0, \, r_k}}_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q)-r_k \\
&\ge
\Big(1-\frac{1}{k}\Big)(1-s_{i(k)})\M{J}^{1}_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q)(e^{- \frac{1}{2} \abs{x_0^2}} - r_k)-r_k,
\end{split}
\end{equation*}
i.e.
$$\lim_{k\to\infty} \frac{\alpha(C_{r_k}(x_0))}{r_k^{n-1}}\ge e^{- \frac{1}{2} \abs{x_0}^2}
\liminf_{k\to\infty}(1-s_{i(k)})\M{J}^1_{s_{i(k)}}((E_{{i(k)}}-x_0)/r_k,Q).$$
Since we have
$$\lim_{k\to\infty}\int_{Q}|\chi_{(E_{{i(k)}}-x_0)/r_k}-\chi_{(E-x_0)/r_k}|dx=0,$$
and
$$\lim_{k\to\infty}\int_{Q}|\chi_{(E-x_0)/r_k}-\chi_H|dx=0,$$
it follows that $(E_{{i(k)}}-x_0)/r_k\to H$ in $L^1(Q)$.
If we define
\begin{equation}\label{defGamman}
\Gamma_n:=\inf\Big\{\liminf_{s\uparrow 1}(1-s)\M{J}^1_s(E_s,Q)\;\Big|\;
\chi_{E_s}\to \chi_H\text{ in }L^1(Q)\Big\},
\end{equation}
it has been proved in \cite[Lemmata 7, 11, 12]{ADPM} that $\Gamma_n=\omega_{n-1}$.
Hence we conclude the claimed inequality \eqref{eq11}.

\subsection{Proof of proposition \ref{GammaLimsupProp}}
As in \cite{ADPM}, it is enough to prove the $\Gamma-\limsup$ inequality for the
collection $\M{B}$ of polyhedra $\Pi$  of finite perimeter which
satisfy $P(\Pi,\partial\Omega)=0$. $\M{B}$ is dense in
energy, i.e. such that for every set $E$ of finite perimeter there
exists $E_k\in\M{B}$ with $\chi_{E_k}\to \chi_E$ in
$L^1_{\loc}(\R{n})$ as $k\to\infty$ and
$\limsup_kP^\gamma(E_k,\Omega)=P^\gamma(E,\Omega)$. We recall that a polyhedron $\Pi$ is in the class $\M{B}$ if and only if
$$\lim_{\delta\to 0}P(\Pi,\Omega^+_\delta\cup \Omega^-_\delta)=0,\quad \mbox{
or equivalently} \quad \lim_{\delta\to 0}P^\gamma(\Pi,\Omega^+_\delta\cup \Omega^-_\delta)=0, $$
where we have set
\begin{equation}\label{omegadelta}
\begin{split}
\Omega^+_\delta:=\{x\in \Omega^c\;|\;d(x,\Omega)<\delta\}, \qquad \Omega^-_\delta:=\{x\in \Omega\;|\;d(x,\Omega^c)<\delta\}.
\end{split}
\end{equation}

We are going to prove that for a polyhedron $\Pi\subset\R{n}$ there holds
\begin{equation}\label{poly}
\limsup_{s\uparrow 1}(1-s)\M{J}^\gamma_s(\Pi,\Omega)\leq
\Gamma_n^* P^\gamma(\Pi,\Omega)+ 2\Gamma_n^* \lim_{\delta\to 0} P^\gamma(\Pi,\Omega^+_\delta\cup \Omega^-_\delta),
\end{equation}
where
\begin{equation}\label{gamman*}
\Gamma_n^*:= \limsup_{s\uparrow 1}(1-s)\M{J}^1_s(H,Q).
\end{equation}
Again, as in \cite[Lemmata 7, 11, 12]{ADPM} we have the equality $\Gamma_n^* = \omega_{n-1}$. We shall divide the proof into two main steps.

 \emph{Step 1.} We first estimate $\M{J}_s^{1, \, \gamma}(\Pi, \Omega)$. For a fixed $\varepsilon>0$ set
\begin{equation*}
(\partial\Pi)_\varepsilon:=\{x\in\Omega\;|\; d(x,\partial\Pi)<\varepsilon\},\quad (\partial\Pi)_\varepsilon^-:=(\partial\Pi)_\varepsilon \cap \Pi.
\end{equation*}
We can find $N_\varepsilon$ disjoint cubes $Q^\varepsilon_i\subset\Omega$, $1\leq
i\le N_\varepsilon$, of side length $\varepsilon$ satisfying the following
properties:
\begin{itemize}
\item[(i)]
if $\tilde Q_i^\varepsilon$ denotes the dilation of $Q_i^\varepsilon$ by a factor
$(1+\varepsilon)$, then each cube $\tilde Q_i^\varepsilon$ intersects exactly one
face $\Sigma$ of $\partial\Pi$, its barycenter belongs to $\Sigma$ and
each of its sides is either parallel or orthogonal to $\Sigma$;
\item[(ii)]  $\M{H}^{n-1}\left(((\partial\Pi)\cap \Omega)\setminus \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon
\right)=|P(\Pi,\Omega)-N_\varepsilon\ve^{n-1} |\to 0$ as $\varepsilon \to 0$.
\end{itemize}
Property (ii), combined with the continuity of the exponential and the property of measures, easily implies
\begin{equation}\label{expdecay}
\abs{P^\gamma(\Pi, \Omega) - \varepsilon^{n-1}   \sum_{i=1}^{N_\varepsilon}  e^{-\frac{1}{2} \abs{x_i^\varepsilon}^2 } } \to 0 \mbox{ as } \varepsilon \to 0,
\end{equation}
where we have set by $x_i^\varepsilon$ the center of the cubes $Q^\epsilon_i$.
For $x\in\R{n}$ set
$$I_s(x):=\int_{\Pi^c\cap \Omega}\frac{e^{-\frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy.$$
We consider several cases.

\medskip

\noindent\emph{Case 1:} $x\in (\Pi\cap \Omega)\setminus(\partial\Pi)_\varepsilon^-$. Then for $y\in \Pi^c\cap \Omega$ we have $|x-y|\geq \varepsilon$, hence
$$I_s(x)\leq \int_{(B_\varepsilon(x))^c} \frac{1}{|x-y|^{n+s}}dy=n\omega_n\int_\varepsilon^\infty \frac{1}{\rho^{s+1}}d\rho=\frac{n\omega_{n}}{s\varepsilon^s},$$
since $n\omega_n=\M{H}^{n-1}(S^{n-1})$. Therefore
\begin{equation}\label{case1}
\int_{(\Pi\cap \Omega)\setminus(\partial\Pi)_\varepsilon^-}  I_s(x) e^{- \frac{1}{4} \abs{x}^2} \, dx\leq \frac{n\omega_{n} }{s\varepsilon^s} \int_{\Pi\cap \Omega} e^{- \frac{1}{4} \abs{x}^2} \, dx.
\end{equation}

\noindent\emph{Case 2:} $x\in
(\partial\Pi)_\varepsilon^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon$. Then
\begin{equation}\label{case2.0}
I_s(x)\leq\int_{(B_{d(x,\Pi^c \cap
\Omega)}(x))^c}\frac{1}{|x-y|^{n+s}}dy=
n\omega_{n}\int_{d(x,\Pi^c\cap\Omega)}^\infty\frac{1}{\rho^{s-1}}d\rho=\frac{n\omega_{n}}{s[d(x,\Pi^c\cap\Omega)]^s}.
\end{equation}
Now write $(\partial\Pi)\cap \Omega=\bigcup_{j=1}^J\Sigma_j$, where each $\Sigma_j$ is the intersection of a face of $\partial\Pi$ with $\Omega$, and define
$$(\partial\Pi)^-_{\varepsilon,j}:=\{x\in (\partial\Pi)^-_\varepsilon:\dist(x,\Pi^c\cap\Omega)=\dist(x,\Sigma_j)\}.$$
Clearly $(\partial\Pi)^-_\varepsilon=\bigcup_{j=1}^J (\partial\Pi)^-_{\varepsilon,j}$. Moreover we have
$$(\partial\Pi)^-_{\varepsilon,j}\subset\{x+t\nu:x\in \Sigma_{\varepsilon,j},\, t\in (0,\varepsilon),\, \nu\text{ is the interior unit normal to }\Sigma_{\varepsilon,j}\}, $$
and $\Sigma_{\varepsilon,j}$ is the set of points $x$ belonging to the same hyperplane as $\Sigma_j$ and with $\dist(x,\Sigma_j)\le \varepsilon$. Clearly $\M{H}^{n-1}(\Sigma_{\varepsilon,j})\le \M{H}^{n-1}(\Sigma_j)+C\varepsilon$ as $\varepsilon\to 0$. Then from \eqref{case2.0} we infer
\begin{equation}\label{case2}
\begin{split}
\int_{(\partial\Pi)_\varepsilon^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}I_s(x) e^{- \frac{1}{4} \abs{x}^2} \, dx&\leq \frac{n\omega_{n}}{s}\sum_{j=1}^J\int_{(\partial\Pi)_{\varepsilon,j}^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}\frac{1}{[d(x,\Pi^c)]^s} \, dx\\
&\leq \frac{n\omega_{n}}{s}\sum_{j=1}^J\int_{(\partial\Pi)_{\varepsilon,j}^-\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}\frac{1}{[d(x,\Sigma_{\varepsilon,j})]^s} \, dx\\
&\leq\frac{n\omega_{n}}{s}\sum_{j=1}^J\int_{(\Sigma_{\varepsilon,j})\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}\bigg(\int_0^\varepsilon \frac{dt}{t^s}\bigg)\, d\M{H}^{n-1}\\
&= \frac{n\omega_{n}\varepsilon^{1-s}}{s(1-s)}\M{H}^{n-1}\left(\bigg(\bigcup_{j=1}^J\Sigma_{\varepsilon,j}\bigg)\setminus\bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon\right)=\frac{\varepsilon^{1-s}o(1)}{s(1-s)},
\end{split}
\end{equation}
with error $o(1)\to 0$ as $\varepsilon\to 0$ and independent of $s$.

\medskip

\noindent\emph{Case 3:} $x\in \Pi\cap \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon$. In this case we write
\begin{equation*}
\begin{split}
I_s(x)&=\int_{(\Pi^c\cap \Omega)\cap \{y:|x-y|\geq \varepsilon^2\}}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy +\int_{(\Pi^c\cap \Omega)\cap \{y:|x-y|< \varepsilon^2\}}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy\\
&=:I_s^1(x)+I_s^2(x).
\end{split}
\end{equation*}
Then, similar to the case 1,
$$I_s^1(x)\leq n\omega_{n}\int_{\varepsilon^2}^\infty\frac{1}{\rho^{s+1}} \, d\rho=\frac{n\omega_{n}}{s\varepsilon^{2s}},$$
hence (since all cubes are contained in $\Omega$)
\begin{equation}\label{case3.1}
\int_{\Pi\cap \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}I_s^1(x) e^{-\frac{1}{4} \abs{x}^2 } \, dx\leq
\frac{n\omega_{n}}{s\varepsilon^{2s}} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx.
\end{equation}
As for $I_s^2(x)$ observe that if $x\in Q_i^\varepsilon$ and $|x-y|\leq
\varepsilon^2$, then $y\in \tilde Q_i^\varepsilon$, where $\tilde Q_i^\varepsilon$ is the
cube obtained by dilating $Q_i^\varepsilon$ by a factor $1+\varepsilon$ (hence the
side length of $\tilde Q_i^\varepsilon$ is $\varepsilon+\varepsilon^2$). Then
\begin{equation}\label{case3.2}
\begin{split}
\int_{\Pi\cap \bigcup_{i=1}^{N_\varepsilon}Q_i^\varepsilon}I_s^2(x) e^{- \frac{1}{4} \abs{x}^2} \, dx
&\le\sum_{i=1}^{N_\varepsilon}\int_{\Pi\cap Q_i^\varepsilon}\int_{\Pi^c\cap \tilde Q_i^\varepsilon}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx \\
&\le\sum_{i=1}^{N_\varepsilon}\int_{\Pi\cap \tilde Q_i^\varepsilon}\int_{\Pi^c\cap \tilde Q_i^\varepsilon}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx \\
&\le \left(\sum_{i=1}^{N_\varepsilon} e^{- \frac{1}{2} \abs{x^\varepsilon_i}^2} \right) \M{J}^1_s(H,(\varepsilon+\varepsilon^2) Q)(1 + \varepsilon^2) \\
&= \underbrace{\left( \varepsilon^{n-1 } \sum_{i=1}^{N_\varepsilon} e^{- \frac{1}{2} \abs{x^\varepsilon_i}^2} \right)}_{= P^\gamma(\Pi, \, \Omega) + o(1)} \varepsilon^{1-s}(1+\varepsilon)^{n-s}\M{J}^1_s(H,Q)(1 + \varepsilon^2),
\end{split}
\end{equation}
where in the last identity we used the scaling property
\begin{equation}\label{scale}
\M{J}^i_s(\lambda
E,\lambda\Omega)=\lambda^{n-s}\M{J}^i_s(E,\Omega)\qquad \text{for
}\lambda>0, \; i=1,2.
\end{equation}
Keeping $\varepsilon>0$ fixed, letting $s$ go to $1$ and putting \eqref{case1}-\eqref{case3.2} together, we infer
\begin{equation*}
\begin{split}
\limsup_{s\uparrow 1}(1-s)\M{J}^{1, \, \gamma}_s(\Pi,\Omega) \leq o(1)+\Gamma_n^*P^{\gamma}(\Pi,\Omega) = o(1) + \omega_{n-1} P^{\gamma}(\Pi, \, \Omega),
\end{split}
\end{equation*}
with error $o(1)\to 0$ as $\varepsilon\to 0$ uniformly in $s$. Since $\varepsilon>0$
is arbitrary, we conclude
\begin{equation}\label{limsupJ1}
\limsup_{s\uparrow 1}(1-s)\M{J}^{1, \, \gamma}_s(\Pi,\Omega) \leq \omega_{n-1} P^{\gamma}(\Pi,\Omega).
\end{equation}

\medskip

\noindent\emph{Step 2.} It now remains to estimate $\M{J}_s^{2,\gamma}$. Let us start by considering the term
$$\int_{\Pi\cap \Omega}\int_{\Pi^c\cap \Omega^c}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx.$$

\noindent\emph{Case 1:} $x\in \Pi\cap (\Omega\setminus\Omega_\delta^-)$. Then for
$y\in \Pi^c\cap\Omega^c$ we have $|x-y|\ge \delta$, whence
$$I(x):=\int_{\Pi^c\cap \Omega^c}\frac{e^{ - \frac{1}{4} \abs{y}^2 }}{|x-y|^{n+s}} \, dy\leq
n\omega_{n}\int_\delta^\infty\frac{d\rho}{\rho^{1+s}}=\frac{n\omega_{n}}{s\delta^s}.$$

\noindent\emph{Case 2:} $x\in \Pi\cap \Omega_\delta^-$. In this
case, using the same argument of case $1$ for $y\in
\Pi^c\cap(\Omega^c\setminus\Omega_\delta^+)$, we have
\begin{equation*}
\begin{split}
I(x)=\int_{\Pi^c\cap\Omega_\delta^+}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy +
\int_{\Pi^c\cap(\Omega^c\setminus\Omega_\delta^+)}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy\le
\int_{\Pi^c\cap\Omega_\delta^+}\frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}} \, dy +\frac{n\omega_{n}}{s\delta^s}.
\end{split}
\end{equation*}
Therefore
\begin{equation*}
\begin{split}
\int_{\Pi\cap \Omega}\int_{\Pi^c\cap \Omega^c}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}
{|x-y|^{n+s}} \, dy dx &\le\frac{2n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx+
\int_{\Pi\cap\Omega_\delta^-}\int_{\Pi^c\cap\Omega_\delta^+}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dy dx\\
&\le\frac{2n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx + \int_{\Pi\cap(\Omega_\delta^-\cup \Omega_\delta^+)}
\int_{\Pi^c\cap(\Omega_\delta^-\cup \Omega_\delta^+)}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx.
\end{split}
\end{equation*}
An obvious similar estimate can be obtained by swapping $\Pi$ and
$\Pi^c$, finally yielding
\begin{equation*}
\begin{split}
\M{J}_s^{2,\gamma}(\Pi,\Omega)&\le \frac{4n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx  + 2 \int_{\Pi\cap(\Omega_\delta^-\cup \Omega_\delta^+)}\int_{\Pi^c\cap(\Omega_\delta^-\cup \Omega_\delta^+)}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}} \, dydx\\
&=
\frac{4n\omega_{n}}{s\delta^s} \int_\Omega e^{- \frac{1}{4} \abs{x}^2} \, dx   +2\M{J}_s^{1, \, \gamma}(\Pi,\Omega_\delta^-\cup
\Omega_\delta^+).
\end{split}
\end{equation*}
Using inequality \eqref{limsupJ1} applied with the open set $ \Omega_\delta^-\cup \Omega_\delta^+ $, we get
$$\limsup_{s\uparrow 1}(1-s)\M{J}_s^{2, \, \gamma}(\Pi,\Omega)\le 2 \omega_{n-1} P^\gamma(\Pi,\Omega_\delta^-\cup \Omega_\delta^+).$$
Since $\delta>0$ is arbitrary, letting $\delta$ go to zero, we conclude the proof of the Proposition.

\section{First and second variation}
In this section we calculate the first and second variation of $\M{J}^\gamma_{s}(E)$. A similar analysis has been done in \cite{FFMMM} in order to prove the local minimality of the ball for a functional involving nonlocal terms.

First, we fix some notation. Given a vector field  $X \in C_{c}^{2}(\mathbb{R}^n , \mathbb{R}^{n})$, the {\it associated flow} is defined as the  solution of the Cauchy problem
\begin{eqnarray} \label{flusso}
\begin{cases}
\displaystyle\frac{\partial}{\partial t}\Phi(x,t)=X(\Phi (x,t)) \vspace{5pt}
\\
\Phi(x,0)=x.
\end{cases}
\end{eqnarray}
In the following, we shall always write $\Phi_t$ to denote the map $\Phi(\cdot,t)$.
Note that for any given $X$ there exists $\delta>0$ such that, for $t\in[-\delta,\delta]$, the map $\Phi_t$ is a diffeomorphism coinciding with the identity map outside a compact set.

If $E\subset \mathbb{R}^n$ is  measurable, we set $E_t:=\Phi_t(E)$. Denoting by $J\Phi_t$ the $n$-dimensional Jacobian of $\Phi_t$, the first and second derivatives of $J\Phi_t$ are given by
\begin{equation}\label{utili}
\frac{\partial}{\partial t} \restrict{t=0} J\Phi_t =\text{div}X,\qquad\frac{\partial^2}{\partial t^2} \restrict{t=0}   J\Phi_t=\text{div} ((\text{div}X)X).
\end{equation}

Finally, given a sufficiently smooth  bounded open set $E\subset \R{n}$ and a vector field $X$, we recall that the {\it first variation of $\M{J}^\gamma_{s}(E)$} along the vector field $X$ is defined by
$$
\delta \M{J}^\gamma_{s}(E)[X]:=\frac{d}{dt}\restrict{t=0}\M{J}^\gamma_{s}(E_t),
$$
where $\Phi_t$ is the flow associated with $X$.
The {\it second variation of $\M{J}^\gamma_{s}(E)$} along the vector field $X$ is defined by
$$
\delta^2 \M{J}^\gamma_{s}(E)[X]= \frac{d^2}{dt^2}\restrict{t=0} \M{J}^\gamma_s (E_t).
$$
If $X$ is a vector field such that $X:=\phi \nu_E$ on $\partial E$, where $\nu_E$ denotes the exterior normal to $E$, using the area formula and the divergence theorem, the first  variation of the Gaussian volume can be computed as
\begin{eqnarray} \label{vol1}
\nonumber
\frac{d}{dt} \restrict{t=0}\gamma (E_t)=\frac{d}{dt} \restrict{t=0}\int_{E} J\Phi_t (x)e^{-\frac{|\Phi_t |^2}{2}}dx
= \int_E \left( \diver X - \langle X,  x\rangle \right) e^{-\frac{|x|^2}{2}}dx \\=
\int_E  \diver \left(X e^{-\frac{|x|^2}{2}}\right)dx= \int_{\partial E}\phi(x) e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}_x.
\end{eqnarray}
If $E$ is a set of class $C^2$, given a smooth function $\phi: \partial E\rightarrow \mathbb{R}$, it can be extended in a neighborhood $U$ of $\partial E$ so that
\begin{equation} \label{hyp}
\frac{\partial}{\partial \nu} \phi +\phi(H-\langle x, \nu_E\rangle)=0 \quad \text{on}\,\, \partial E.
\end{equation}
The second variation of the Gaussian volume along the vector field $X$ such that $X=\phi \nu_E$ on $\partial E$ and $\phi$ satisfies \eqref{hyp}, can be calculated using the divergence theorem and reads as
\begin{eqnarray*} \label{vol2}
\begin{split}
\frac{d^2}{dt^2} \restrict{t=0}\gamma (E_t)&=\int_E\diver \left(  \diver \left(X e^{-\frac{|x|^2}{2}} \right) X\right)dx=
\int_{\partial E}\phi\left(\frac{\partial}{\partial \nu} \phi +\phi(H-\langle x, \nu_E\rangle )\right) e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}_x=0
\end{split}
\end{eqnarray*}
Thus, we say that a vector field preserves the Gaussian volume of $E$ if it satisfies
\begin{eqnarray} \label{tecnico}
\int_{\partial E}\phi(x) e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}_x= 0  \quad\text{ and }\quad \frac{\partial \phi}{\partial \nu}
+\phi(H-\langle x, \nu_E\rangle)=0 \; \text{ for  }\, x\in \partial E.
\end{eqnarray}
We note that without this assumptions the expression of the second variation of the Gaussian perimeter even in the local framework is quite complicated, see \cite[Eq. (17)]{BBJ}.

In order to compute the first and second variation of $\M{J}^\gamma_s$, due to the singularity of the Kernel in the integrand, we need to pass through approximations. Thus, given $\delta \in [0,1)$, let $\eta_\delta \in C^\infty_c([0,+\infty),[0,1])$ be such that $\eta_\delta = 1$ on $[0,\delta]\cup [1/\delta, \infty]$, $\eta_\delta=0$
on $[2\delta, 2/\delta]$, $|\eta'|\leq 2/\delta$ on $[0,\infty)$, and $\eta \downarrow 0$ for every $s \in (0,1)$ as $\delta \rightarrow 0^+$. Then we define
$$
K_{\delta} (z):=(1- \eta_\delta(|z|))\frac{1}{|z|^{n+s}}.
$$
Now we show the following theorem.
\begin{trm} \label{variations}
Let $E$ be an open set of class $C^2$ and $X \in C^2_c(\R n,\R n)$ a vector field such that $X = \phi \nu_E$ on $\partial E$. Then the first variation of $\M{J}^\gamma_s (E)$ along a vector field $X$ is given by
\begin{equation}\label{varprima}
\partial \M{J}^\gamma_s (E)[X]=\int_{\partial E}H^{*}_{\partial E}(x)(X(x), \nu_E(x))d\mathcal{H} ^{n-1}_x,
\end{equation}
while the second variation reads as
\begin{equation}\label{varseconda}
\begin{split}
\partial ^2  \M{J}^\gamma_s (E)[X]&=\int_ {\partial E}\int_{\partial E}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}
\left( \abs{\phi (x)-\phi(y) }^2 -\phi ^2 |\nu_E(x)-\nu_E(y)|\right) d\mathcal{H}^{n-1}_xd\mathcal{H}^{n-1}_y
\\
&\qquad  +\int_{\partial E}H^*_\delta \left( \phi \left(H-\langle x,\nu_E \rangle \right) +\frac{\partial \phi}{\partial \nu}\right)\phi d\mathcal{H} ^{n-1}_x \\
&\qquad -\int_{\partial E}\phi^2(x)\int_{\R n}\frac{\langle y- x, \nu_E\rangle }{2} \frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}dy \, d\mathcal{H}^{n-1}_x.
\end{split}
\end{equation}
Moreover, if $X$ is volume preserving, then
\begin{equation}\label{varseconda1}
\begin{split}
\partial ^2  \M{J}^\gamma_s (E)[X]&=\int_ {\partial E}\int_{\partial E}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}
\left( \abs{\phi (x)-\phi(y) }^2 -\phi ^2 |\nu_E(x)-\nu_E(y)|\right) d\mathcal{H}^{n-1}_xd\mathcal{H}^{n-1}_y
\\
&\qquad -\int_{\partial E}\phi^2(x)\int_{\R n}\frac{\langle y-x,\nu_E(x)\rangle}{2} \frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}dy \, d\mathcal{H}^{n-1}_x.
\end{split}
\end{equation}
\end{trm}
\begin{proof}
Let us call $\M{J}_s^\delta$ the integral associated to the regularized kernel, namely
$$
\M{J}_s^\delta(E)= \int_{E^c}\int_{E}\frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}+\delta}dxdy.
$$
By the definition of $\Phi_t$, the implicit function theorem gives the existence of $\varepsilon>0$ such that the map $\Phi_t$ is a diffeomorphism for all $t \in [-\delta, \delta]$.
Using the area formula, we compute
$$
\M{J}_s^\delta(E_t)= \int_{E^c}\int_{E}\frac{e^{- \frac{1}{4} \left(\abs{\Phi(x,t)}^2 + \abs{\Phi(y,t)}^2\right)}}{|\Phi(x,t)-\Phi(y,t)|^{n+s}+\delta}J\Phi(x,t)J\Phi(y,t)dxdy
$$
We use the first equation in $\eqref{utili}$ to compute the first variation of $\M{J}_s^\delta$
\begin{equation*}
\begin{split}
\frac{d}{dt}_{|_{t=0}}J_s^\delta (E_t)&= \int_{E^c}\int_{E}e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}D_z\left(K_{\delta} (x-y)\right)(X(x)-X(y))dxdy
\\
& \, \, \, \, + \int_{E^c}\int_{E}K_{\delta} (x-y)\left( D_x(e^{- \frac{1}{4}\left(\abs{x}^2 + \abs{y}^2\right)})X(x)+ D_y(e^{- \frac{1}{4}\left(\abs{x}^2 + \abs{y}^2\right)})X(y)\right)dxdy
\\
& \, \, \, \, + \int_{E^c}\int_{E}K_{\delta} (x-y)e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}\left(\diver X(x)+\diver X(y)\right)dxdy\\
&  =
 \int_{E^c}\int_{E}\diver_x\left( e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}K_{\delta} (x-y)X(x)\right)dxdy\\
& \, \, \, \, +\int_{E^c}\int_{E}\diver_y\left(  e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}K_{\delta} (x-y)X(y)\right)dxdy
\\
&  =\int_{\partial E}\int_{\R n}\left(\chi_{E^c}(y)-\chi_E (y)    \right) K_{\delta} (x-y)e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}\langle X(x),\nu_E(x)\rangle dyd\mathcal{H}^{n-1}_x.
\end{split}
\end{equation*}

Now we compute the second variation of $\M{J}_s^\delta$.
\begin{equation}
\begin{split}
\delta^2 \M{J}_s^\delta[X]&=\frac{d^2}{dt^2}_{|_{t=0}} \int_{E^c}\int_E K_\delta(\Phi(x,t)-\Phi(y,t)) e^{- \frac{1}{4} \left(\abs{\Phi(x,t)}^2 + \abs{\Phi(y,t)}^2\right)}\diver X(x)\diver X(y)dxdy\\
&= \int_{E^c}\int_{E} D^2_{xx} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right)[X(x),X(x)]
dxdy\\
& \quad+  \int_{E^c}\int_{E}\left \langle D_{x} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),DX(x)X(x)\right \rangle dxdy
\nonumber
\\
& \quad+2\int_{E^c}\int_{E} D^2_{yx} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right)[X(x),X(y)]dxdy
\\
& \quad+\int_{E^c}\int_{E} D^2_{yy} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right)[X(y),X(y)]dxdy\\
& \quad+\int_{E^c}\int_{E}\left \langle D_{y} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),DX(y)X(y)\right \rangle dxdy\\
& \quad +2\int_{E^c}\int_{E}\left \langle D_{x} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),X(x)\right \rangle
(\diver_x X(x)+\diver_y X(y))dxdy
\\
& \quad +2\int_{E^c}\int_{E}\left \langle D_{y} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}} \right),X(y)\right \rangle
(\diver_x X(x)+\diver_y X(y))dxdy
\\
& \quad +\int_{E^c}\int_{E} K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}
\big( \diver [X(x)\diver(X(x))]+\diver [X(y)\diver(X(y))]\big)dxdy
\\
& \quad +2\int_{E^c}\int_{E} K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\diver_x X(x)\diver_y X(y)dxdy
\end{split}
\end{equation}

We now use the divergence theorem and exploit the symmetry of $K_\delta$ in order to simplify the above expression as follows
\begin{equation}
\begin{split}
\delta^2 \M{J}_s^\delta&= \int_{E^c}\int_{E}\diver_x \left[X(x) \diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)\right]dxdy
\\
& \quad + \int_{E^c}\int_{E}\diver_y \left[X(y) \diver_y \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(y)\right)\right]dxdy
\\ & \quad + \int_{E^c}\int_{E}\diver_x \left[X(x) \diver_y \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(y)\right)\right]dxdy
\\& \quad + \int_{E^c}\int_{E}\diver_y \left[X(y) \diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)\right]dxdy\\
&=I_1+I_2+I_3+I_4.
\end{split}
\end{equation}
 Using Fubini and the divergence theorems we have
$$
I_1=\int_{\partial E} \langle X(x),\nu_E(x) \rangle \int_{E^c}\diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)dyd\mathcal{H}^{n-1}_x
$$
and
\begin{equation}
\begin{split}
I_3&=\int_{\partial E} \langle X(x),\nu_E(x) \rangle \int_{E^c}\diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(y)\right)dyd\mathcal{H}^{n-1}_x
\\
&=-\int_{\partial E}\int_{\partial E}(K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\langle X(y),\nu_E(y)\rangle \langle X(x),\nu_E(x)\rangle d\mathcal{H}_x^{n-1}d\mathcal{H}_y^{n-1}.
\end{split}
\end{equation}
We remark that $I_1$ (resp. $I_3$) has the same expression of $I_2$ (resp. $I_4$) exchanging $x$ and $y$. Using this observation and the symmetry of $K_\delta$, we compute
$$
I_1+I_2=\int_{\partial E}\langle X(x),\nu_E(x)\rangle  \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\diver_x \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)dyd\mathcal{H}^{n-1}_x \\,
$$
and
$$I_3+I_4=-2\int_{\partial E}\int_{\partial E}(K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\langle X(y),\nu_E(y)\rangle \langle X(x),\nu_E(x)\rangle  d\mathcal{H}_x^{n-1}d\mathcal{H}_y^{n-1}.
$$
Next we write $\diver_x X(x)= \diver_{\nu(x)} X(x)+ \diver_{\tau (x)} X(x)$, where $\diver_{\nu(x)}X(x):= \langle DX[\nu_E(x)], \nu_E(x)\rangle$. Using  Fubini's theorem and the divergence theorem on manifolds, we get
\begin{equation}
\begin{split}
I_1+I_2&= \int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\langle X(x),\nu_E(x)\rangle  \diver_{\tau (x)}\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy\\
&\quad+
\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\langle X(x),\nu_E(x)\rangle   \diver_{\nu(x)}  \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy
\\
&=
\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}H(x)\langle X(x),\nu_E(x)\rangle ^2 \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy\\
&\quad+
\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\langle X(x),\nu_E(x)\rangle   \diver_{\nu(x)} \left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy \\
&=
\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}H(x)\phi^2 (x)\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}X(x)\right)d\mathcal{H}^{n-1}_xdy\\
&\quad +
\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\phi^2(x)\frac{\partial}{\partial \nu (x)}\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\right)d\mathcal{H}^{n-1}_xdy\\
&\quad+
\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\int_{\partial E}\phi(x)\frac{\partial \phi}{\partial \nu(x)}K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}d\mathcal{H}^{n-1}_xdy,
\end{split}
\end{equation}
where we used that $X=\phi \nu_E$ and then $\langle D_\tau f,X \rangle =0$ for every $f \in C^1(\partial E)$.
Regarding the second addend of the above expression, using again Fubini's theorem  and the fact that $D_x K_\delta =-D_y K_\delta$, we get
\begin{equation}
\begin{split}
&\int_{E}\frac{\partial}{\partial \nu (x)}\left( K_\delta (x-y) {e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}\right)dy\\
&=
-\int_E \left[\left \langle D_y K_\delta (x-y) , \nu_E(x)\right \rangle +\frac{\langle x,\nu_E(x)\rangle }{2} K_\delta (x-y) \right] e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}dy\\
&=-\int_{\partial E}   K_\delta (x-y) e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}\left \langle \nu_E(x),\nu_E(y)\right \rangle d\mathcal{H}^{n-1}_y
-\int_E \frac{\langle x+y,\nu_E(x)\rangle}{2} K_\delta(x-y) e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}dy.
\end{split}
\end{equation}
Finally, thanks to the identity $|\nu_E(x)-\nu_E(y)|^2=2- 2\langle \nu_E(x),\nu_E(y)\rangle$, after some elementary calculations we deduce
\begin{equation}
\begin{split}
\partial ^2  \M{J}^\delta_s (E)[X]&=\int_ {\partial E}\int_{\partial E}e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}K_\delta(x-y)
\left( \abs{\phi (x)-\phi(y) }^2 -\phi ^2 |\nu_E(x)-\nu_E(y)|\right) d\mathcal{H}^{n-1}d\mathcal{H}^{n-1}\\
&\quad +
\int_{\partial E}H^*_\delta \left( \phi \left(H-\langle x,\nu_E\rangle \right) +\frac{\partial \phi}{\partial \nu}\right)\phi d\mathcal{H} ^{n-1}\\
&\quad -\int_{\partial E}\phi^2(x)\int_{\R n}\left(\chi_{E^c} (y)-\chi_E (y)\right)\frac{\langle y-x,\nu_E(x)\rangle }{2} K_\delta(x-y) e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}dy d\mathcal{H}^{n-1}_x.
\end{split}
\end{equation}
At this point we just need to show that the first and second variation of $\M{ J}_s^\delta$ converge, respectively, to the first and second variation of $\M{J}^\gamma_s$ as $\delta$ goes to $0$. The proof of this fact is exactly the same as in \cite{FFMMM}.
\end{proof}

Note that, since
$$
\frac{d}{dt}\restrict{t=0}\int_{E_t} e^{-\frac{x^2}{2}}dx=
\int_{\partial E}e^{-\frac{|x|^2}{2}}\langle X(x),\nu_E(x)\rangle d\mathcal{H}^{n-1},
$$
we have that the flow $\Phi$ associated to $X$ preserves the Gaussian volume if
$$
\int_{\partial E}e^{-\frac{|x|^2}{2}}\langle X(x),\nu_E(x)\rangle d\mathcal{H}^{n-1}=0.
$$
Thus, the Euler-Lagrange equation for the problem
\begin{equation}\label{minimiz}
\min_{|E|=m} \M{J}^\gamma_s(E)
\end{equation}
is
$$
\int_{\partial E} \phi(x) \int_{\R n}\left(\chi_{E^c}(y)-\chi_E (y)    \right) \frac{e^{- \frac{1}{4} \left(\abs{x}^2 + \abs{y}^2\right)}}{|x-y|^{n+s}}dyd\mathcal{H}^{n-1}_x
= \lambda \int_{\partial E} \phi(x) e^{-\frac{|x|^2}{2}}d\mathcal{H}_x^{n-1}.
$$
Moreover, if $E$ is a set of class $C^2$, then thanks to the fundamental lemma of the calculus of variations the above equation can be rewritten as
\begin{equation}\label{minimizza}
\int_{\R n}\left(\chi_{E^c}(y)-\chi_E (y)    \right) \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy= \lambda e^{-\frac{\abs{x}^2 }{4}}, \qquad \forall x \in \partial E.
\end{equation}
$E$ is said to be stationary with respect to the non local Gaussian isoperimetric problem, or equivalently a volume constrained critical point, if it satisfies equation \eqref{minimizza}.

\section{Volume constrained stationary shapes}
In this section we prove that, as opposed to the local setting, the only halfspaces which are stationary with respect to the non local Gaussian isoperimetric problem are the ones generated by hyperplanes passing through the origin.
\begin{trm}
We fix $a \in \mathbb R$ and $\omega \in \mathbb S^{n-1}$. If $H_{\omega,a}:= \{x \in \R n: \langle x, \omega\rangle <a\}$ is stationary with respect to the non local Gaussian isoperimetric problem, then $a=0$, or equivalently,
$$\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{H_{\omega, a}}e^{-\frac{|x|^2}{2}}dx= \frac 12.$$
\end{trm}
\begin{proof}
Up to rotation, we can assume $\omega=e_n$.
We start observing that, for every $x \in \partial E$, it holds $\langle x, e_n\rangle=a$. This implies that, with the change of coordinate $z=y-x$, if $\langle y, e_n\rangle<a$, then $\langle z, e_n\rangle<0$ and then we can write
\begin{equation}\label{minimizza1}
\begin{split}
\int_{E} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy&= \int_{\{ y_n<a\}} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy=\int_{\{ z_n<0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2+2\langle z, x\rangle)}}{|z|^{n+s}}dy\\
&=\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2-2\langle z, x\rangle)}}{|z|^{n+s}}dy, \qquad \forall x \in \partial E.
\end{split}
\end{equation}
Analogously, we compute
\begin{equation}\label{minimizza2}
\int_{E^c} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy= \int_{\{ y_n>a\}} \frac{e^{- \frac{1}{4} \abs{y}^2}}{|x-y|^{n+s}}dy=\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2+2\langle z, x\rangle)}}{|z|^{n+s}}dz \qquad \forall x \in \partial E.
\end{equation}
Plugging equations \eqref{minimizza1}, \eqref{minimizza2} in equation \eqref{minimizza}, we get
\begin{equation}\label{minimizza3}
\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} (\abs{z}^2+\abs{x}^2})}{|z|^{n+s}}\left (e^{ \frac{\langle z, x\rangle}{2}}- e^{- \frac{\langle z, x\rangle}{2}}\right)dz= \lambda e^{-\frac{\abs{x}^2 }{4}}, \qquad \forall x \in \partial E,
\end{equation}
which in turn reads
\begin{equation}\label{minimizza4}
2\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\sinh\left(\frac{\langle z, x\rangle}{2}\right)dz= \lambda , \qquad \forall x \in \partial E.
\end{equation}
We remark that the integral in \eqref{minimizza4} is well defined, since $\lim_{x\to 0}\frac{\sinh(x)}{x}=1$.

We split $x=(x',x_n)$ and we observe that
\begin{equation}\label{minimizza5}
\begin{split}
\sinh\left(\frac{\langle z, x\rangle}{2}\right)&=\sinh\left(\frac{\langle z', x'\rangle+ z_nx_n}{2}\right)\\
&=\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_nx_n}{2}\right)+\cosh\left(\frac{\langle z', x'\rangle}{2}\right)\sinh\left(\frac{ z_nx_n}{2}\right).
\end{split}
\end{equation}
Plugging \eqref{minimizza5} in \eqref{minimizza4}, we deduce the following equation for every $x \in \partial E$
\begin{equation}\label{minimizza45}
\begin{split}
A+B:=\int_{\{ z_n>0\}}& \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_na}{2}\right)dz\\
&+\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\cosh\left(\frac{\langle z', x'\rangle}{2}\right)\sinh\left(\frac{ z_na}{2}\right)dz= \frac \lambda 2.
\end{split}
\end{equation}
Since $\frac{e^{- \frac{1}{4} (\abs{z'}^2+\abs{z_n}^2})}{(\abs{z'}^2+\abs{z_n}^2)^{\frac{n+s}{2}}}\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_na}{2}\right)$ is odd in $z'$, we deduce
$$A=\int_0^\infty\int_{\mathbb R^{n-1}} \frac{e^{- \frac{1}{4}( \abs{z'}^2+\abs{z_n}^2)}}{(\abs{z'}^2+\abs{z_n}^2)^{\frac{n+s}{2}}}\sinh\left(\frac{\langle z', x'\rangle}{2}\right)\cosh\left(\frac{ z_na}{2}\right)dz'dz_n=0.$$
Plugging this information in \eqref{minimizza5} and taking the partial derivative in $x_j$, for every $j=1,\dots,n-1$, we deduce
$$
\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\frac{\partial}{ \partial x_j}\left(\cosh\left(\frac{\langle z', x'\rangle}{2}\right)\right)\sinh\left(\frac{ z_na}{2}\right)dz= 0.
$$
Assuming without loss of generality that $j=n-1$ and denoting $x'=(x'',x_{n-1})$, we obtain
\begin{equation}\label{minimizza6}
\begin{split}
C&+D:=\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\frac{\partial}{ \partial x_j}\left(\cosh\left(\frac{\langle z'', x''\rangle}{2}\right)\cosh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)\right) \sinh\left(\frac{ z_na}{2}\right)dz\\
& +\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\frac{\partial}{ \partial x_j}\left(\sinh\left(\frac{\langle z'', x''\rangle}{2}\right)\sinh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)\right)\sinh\left(\frac{ z_na}{2}\right)dz= 0.
\end{split}
\end{equation}
Since $\frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\cosh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)z_{n-1}$ is odd in $z_{n-1}$, we deduce
$$D=\int_0^\infty \int_{\mathbb R^{n-2}} \int_{\mathbb R} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\sinh\left(\frac{\langle z'', x''\rangle}{2}\right)\cosh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)z_{n-1}\sinh\left(\frac{ z_na}{2}\right)dz=0.$$
Plugging this information in \eqref{minimizza6}, we get that for every $x \in \partial E$ it holds
\begin{equation}\label{minimizza7}
\begin{split}
\int_{\{ z_n>0\}} \frac{e^{- \frac{1}{4} \abs{z}^2}}{|z|^{n+s}}\cosh\left(\frac{\langle z'', x''\rangle}{2}\right)\sinh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)\sinh\left(\frac{ z_na}{2}\right)z_{n-1}dz=0.
\end{split}
\end{equation}
We denote
$$C(z_n,x):=\int_{\mathbb R^{n-1}} \frac{e^{- \frac{1}{4} \abs{(z',z_n)}^2}}{|(z',z_n)|^{n+s}}\cosh\left(\frac{\langle z'', x''\rangle}{2}\right)\sinh\left(\frac{\langle z_{n-1}, x_{n-1}\rangle}{2}\right)z_{n-1}dz''dz_{n-1},$$
and we observe that if $x_{n-1}\neq 0$, then $C(z_n,x)\neq 0$ since the integrand is even in the variables $z''$ and $z_{n-1}$. Equation \eqref{minimizza7} then reads
\begin{equation}\label{minimizza8}
\int_0^\infty C(z_n,x)\sinh\left(\frac{ z_na}{2}\right)dz_n=0, \qquad \forall x \in \partial E,
\end{equation}
and since for every $z_n>0$
$$\sinh\left(\frac{ z_na}{2}\right)\left\{ \begin{array}{ccc}>0 & \text{if } a>0\\ =0 & \text{if } a=0\\ <0 & \text{if } a<0 \end{array}\right.,$$
equation \eqref{minimizza8} can hold if and only if $a=0$.
\end{proof}

\section{Acknowledgements}
The first author has been partially supported by the NSF DMS Grant No.~1906451.
The second author was partially supported by the Academy of Finland grant 314227.

redpajama_set_name: RedPajamaArXiv


text:
import os
import subprocess
import sys

from django.core.exceptions import SuspiciousOperation

from pymongo import MongoClient

import djcelery
djcelery.setup_loader()

CURRENT_FILE = os.path.abspath(__file__)
PROJECT_ROOT = os.path.realpath(
    os.path.join(os.path.dirname(CURRENT_FILE), '..//'))
PRINT_EXCEPTION = False

TEMPLATED_EMAIL_TEMPLATE_DIR = 'templated_email/'

ADMINS = (
    # ('Your Name', 'your_email@example.com'),
)
MANAGERS = ADMINS

DATABASES = {
    'default': {
        'ENGINE': 'django.db.backends.postgresql_psycopg2',
        'NAME': 'FormhubDjangoDB',
        'USER': 'formhubDjangoApp',
        'HOST': 'localhost',
        'PORT': '',                      # Set to empty string for default.
    }
}

# Local time zone for this installation. Choices can be found here:
# http://en.wikipedia.org/wiki/List_of_tz_zones_by_name
# although not all choices may be available on all operating systems.
# On Unix systems, a value of None will cause Django to use the same
# timezone as the operating system.
# If running in a Windows environment this must be set to the same as your
# system time zone.
TIME_ZONE = 'America/New_York'

# Language code for this installation. All choices can be found here:
# http://www.i18nguy.com/unicode/language-identifiers.html
LANGUAGE_CODE = 'en-us'

ugettext = lambda s: s

LANGUAGES = (
    ('fr', u'Français'),
    ('en', u'English'),
    ('es', u'Español'),
    ('it', u'Italiano'),
    ('km', u'ភាសាខ្មែរ'),
    ('ne', u'नेपाली'),
    ('nl', u'Nederlands'),
    ('zh', u'中文'),
)

SITE_ID = 1

# If you set this to False, Django will make some optimizations so as not
# to load the internationalization machinery.
USE_I18N = True

# If you set this to False, Django will not format dates, numbers and
# calendars according to the current locale
USE_L10N = True

# URL that handles the media served from MEDIA_ROOT. Make sure to use a
# trailing slash.
# Examples: "http://media.lawrence.com/media/", "http://example.com/media/"
MEDIA_URL = ''

# Absolute path to the directory static files should be collected to.
# Don't put anything in this directory yourself; store your static files
# in apps' "static/" subdirectories and in STATICFILES_DIRS.
# Example: "/home/media/media.lawrence.com/static/"
STATIC_ROOT = os.path.join(PROJECT_ROOT, 'static')

# URL prefix for static files.
# Example: "http://media.lawrence.com/static/"
STATIC_URL = '/static/'

#ENKETO URL
ENKETO_URL = 'https://enketo.formhub.org/'
ENKETO_API_SURVEY_PATH = '/api_v1/survey'
ENKETO_API_INSTANCE_PATH = '/api_v1/instance'
ENKETO_PREVIEW_URL = ENKETO_URL + 'webform/preview'
ENKETO_API_TOKEN = ''
ENKETO_API_INSTANCE_IFRAME_URL = ENKETO_URL + "api_v1/instance/iframe"

# Login URLs
LOGIN_URL = '/accounts/login/'
LOGIN_REDIRECT_URL = '/login_redirect/'

# URL prefix for admin static files -- CSS, JavaScript and images.
# Make sure to use a trailing slash.
# Examples: "http://foo.com/static/admin/", "/static/admin/".
ADMIN_MEDIA_PREFIX = '/static/admin/'

# Additional locations of static files
STATICFILES_DIRS = (
    # Put strings here, like "/home/html/static" or "C:/www/django/static".
    # Always use forward slashes, even on Windows.
    # Don't forget to use absolute paths, not relative paths.
)

# List of finder classes that know how to find static files in
# various locations.
STATICFILES_FINDERS = (
    'django.contrib.staticfiles.finders.FileSystemFinder',
    'django.contrib.staticfiles.finders.AppDirectoriesFinder',
    # 'django.contrib.staticfiles.finders.DefaultStorageFinder',
)

# List of callables that know how to import templates from various sources.
TEMPLATE_LOADERS = (
    'django.template.loaders.filesystem.Loader',
    'django.template.loaders.app_directories.Loader',
    # 'django.template.loaders.eggs.Loader',
)

MIDDLEWARE_CLASSES = (
    'django.middleware.common.CommonMiddleware',
    'django.contrib.sessions.middleware.SessionMiddleware',
    # 'django.middleware.locale.LocaleMiddleware',
    'utils.middleware.LocaleMiddlewareWithTweaks',
    'django.middleware.csrf.CsrfViewMiddleware',
    'corsheaders.middleware.CorsMiddleware',
    'django.contrib.auth.middleware.AuthenticationMiddleware',
    'django.contrib.messages.middleware.MessageMiddleware',
    'django.middleware.transaction.TransactionMiddleware',
    'utils.middleware.HTTPResponseNotAllowedMiddleware',
)

LOCALE_PATHS = (os.path.join(PROJECT_ROOT, 'formhub', 'locale'), )

ROOT_URLCONF = 'formhub.urls'
USE_TZ = True

TEMPLATE_DIRS = (
    os.path.join(PROJECT_ROOT, 'templates'),
    # Put strings here, like "/home/html/django_templates"
    # or "C:/www/django/templates".
    # Always use forward slashes, even on Windows.
    # Don't forget to use absolute paths, not relative paths.
)

# needed by guardian
ANONYMOUS_USER_ID = -1

INSTALLED_APPS = (
    'django.contrib.auth',
    'django.contrib.contenttypes',
    'django.contrib.sessions',
    'django.contrib.sites',
    'django.contrib.messages',
    'django.contrib.staticfiles',
    'django.contrib.humanize',
    # Uncomment the next line to enable the admin:
    'django.contrib.admin',
    # Uncomment the next line to enable admin documentation:
    'django.contrib.admindocs',
    'registration',
    'south',
    'django_nose',
    'django_digest',
    'corsheaders',
    'oauth2_provider',
    'rest_framework',
    'rest_framework.authtoken',
    'taggit',
    'odk_logger',
    'odk_viewer',
    'main',
    'restservice',
    'staff',
    'api',
    'guardian',
    'djcelery',
    'stats',
    'sms_support',
)

OAUTH2_PROVIDER = {
    # this is the list of available scopes
    'SCOPES': {
        'read': 'Read scope',
        'write': 'Write scope',
        'groups': 'Access to your groups'}
}

REST_FRAMEWORK = {
    # Use hyperlinked styles by default.
    # Only used if the `serializer_class` attribute is not set on a view.
    'DEFAULT_MODEL_SERIALIZER_CLASS':
    'rest_framework.serializers.HyperlinkedModelSerializer',

    # Use Django's standard `django.contrib.auth` permissions,
    # or allow read-only access for unauthenticated users.
    'DEFAULT_PERMISSION_CLASSES': [
        'rest_framework.permissions.IsAuthenticatedOrReadOnly',
        'rest_framework.permissions.DjangoModelPermissions'
    ],
    'DEFAULT_AUTHENTICATION_CLASSES': (
        'oauth2_provider.ext.rest_framework.OAuth2Authentication',
        #'rest_framework.authentication.BasicAuthentication',
        'rest_framework.authentication.SessionAuthentication',
        'rest_framework.authentication.TokenAuthentication',
    )
}

SWAGGER_SETTINGS = {
    "exclude_namespaces": [],    # List URL namespaces to ignore
    "api_version": '1.0',  # Specify your API's version (optional)
    "enabled_methods": [         # Methods to enable in UI
        'get',
        'post',
        'put',
        'patch',
        'delete'
    ],
}

CORS_ORIGIN_ALLOW_ALL = False
CORS_ALLOW_CREDENTIALS = True
CORS_ORIGIN_WHITELIST = (
    'dev.formhub.org',
)

USE_THOUSAND_SEPARATOR = True

COMPRESS = True

# extra data stored with users
AUTH_PROFILE_MODULE = 'main.UserProfile'

# case insensitive usernames
AUTHENTICATION_BACKENDS = (
    'main.backends.ModelBackend',
    'guardian.backends.ObjectPermissionBackend',
)

# Settings for Django Registration
ACCOUNT_ACTIVATION_DAYS = 1

def skip_suspicious_operations(record):
    """Prevent django from sending 500 error
    email notifications for SuspiciousOperation
    events, since they are not true server errors,
    especially when related to the ALLOWED_HOSTS
    configuration

    background and more information:
    http://www.tiwoc.de/blog/2013/03/django-prevent-email-notification-on-suspiciousoperation/

    """
    if record.exc_info:
        exc_value = record.exc_info[1]
        if isinstance(exc_value, SuspiciousOperation):
            return False
    return True

# A sample logging configuration. The only tangible logging
# performed by this configuration is to send an email to
# the site admins on every HTTP 500 error.
# See http://docs.djangoproject.com/en/dev/topics/logging for
# more details on how to customize your logging configuration.
LOGGING = {
    'version': 1,
    'disable_existing_loggers': False,
    'formatters': {
        'verbose': {
            'format': '%(levelname)s %(asctime)s %(module)s' +
                      ' %(process)d %(thread)d %(message)s'
        },
        'simple': {
            'format': '%(levelname)s %(message)s'
        },
    },
    'filters': {
        'require_debug_false': {
            '()': 'django.utils.log.RequireDebugFalse'
        },
        # Define filter for suspicious urls
        'skip_suspicious_operations': {
            '()': 'django.utils.log.CallbackFilter',
            'callback': skip_suspicious_operations,
        },
    },
    'handlers': {
        'mail_admins': {
            'level': 'ERROR',
            'filters': ['require_debug_false', 'skip_suspicious_operations'],
            'class': 'django.utils.log.AdminEmailHandler'
        },
        'console': {
            'level': 'DEBUG',
            'class': 'logging.StreamHandler',
            'formatter': 'verbose'
        },
        'audit': {
            'level': 'DEBUG',
            'class': 'utils.log.AuditLogHandler',
            'formatter': 'verbose',
            'model': 'main.models.audit.AuditLog'
        },
    },
    'loggers': {
        'django.request': {
            'handlers': ['mail_admins'],
            'level': 'DEBUG',
            'propagate': True,
        },
        'console_logger': {
            'handlers': ['console'],
            'level': 'DEBUG',
            'propagate': True
        },
        'audit_logger': {
            'handlers': ['audit'],
            'level': 'DEBUG',
            'propagate': True
        }
    }
}

MONGO_DATABASE = {
    'HOST': 'localhost',
    'PORT': 27017,
    'NAME': 'formhub',
    'USER': '',
    'PASSWORD': ''
}

GOOGLE_STEP2_URI = 'http://formhub.org/gwelcome'
GOOGLE_CLIENT_ID = '617113120802.apps.googleusercontent.com'
GOOGLE_CLIENT_SECRET = '9reM29qpGFPyI8TBuB54Z4fk'

THUMB_CONF = {
    'large': {'size': 1280, 'suffix': '-large'},
    'medium': {'size': 640, 'suffix': '-medium'},
    'small': {'size': 240, 'suffix': '-small'},
}
# order of thumbnails from largest to smallest
THUMB_ORDER = ['large', 'medium', 'small']
IMG_FILE_TYPE = 'jpg'

# celery
BROKER_BACKEND = "rabbitmq"
BROKER_URL = 'amqp://guest:guest@localhost:5672/'
CELERY_RESULT_BACKEND = "amqp"  # telling Celery to report results to RabbitMQ
CELERY_ALWAYS_EAGER = False

# auto add crowdform to new registration
AUTO_ADD_CROWDFORM = False
DEFAULT_CROWDFORM = {'xform_username': 'bob', 'xform_id_string': 'transport'}

# duration to keep zip exports before deletion (in seconds)
ZIP_EXPORT_COUNTDOWN = 3600  # 1 hour

# default content length for submission requests
DEFAULT_CONTENT_LENGTH = 10000000

TEST_RUNNER = 'django_nose.NoseTestSuiteRunner'
NOSE_ARGS = ['--with-fixture-bundling']

# re-captcha in registrations
REGISTRATION_REQUIRE_CAPTCHA = False
RECAPTCHA_USE_SSL = False
RECAPTCHA_PRIVATE_KEY = ''
RECAPTCHA_PUBLIC_KEY = '6Ld52OMSAAAAAJJ4W-0TFDTgbznnWWFf0XuOSaB6'

# MongoDB
if MONGO_DATABASE.get('USER') and MONGO_DATABASE.get('PASSWORD'):
    MONGO_CONNECTION_URL = (
        "mongodb://%(USER)s:%(PASSWORD)s@%(HOST)s:%(PORT)s") % MONGO_DATABASE
else:
    MONGO_CONNECTION_URL = "mongodb://%(HOST)s:%(PORT)s" % MONGO_DATABASE

MONGO_CONNECTION = MongoClient(
    MONGO_CONNECTION_URL, safe=True, j=True, tz_aware=True)
MONGO_DB = MONGO_CONNECTION[MONGO_DATABASE['NAME']]

# this is rather pointless, currently, given the state of the tests
# however, some functions require its being defined
TESTING_MODE = False

redpajama_set_name: RedPajamaGithub


text:
Q: django 'load staticfiles' failed to load HTML5 static file I was trying to use Django's template language to load static files into my Django template, but it seemed something went wrong. Could anyone help? Thanks in advance!
The static part I was struggling to load was this, part of bootstrap framework:

I tried two methods in my destination template, all of which failed:
(1)

(2)

With this HTML5 tag, how to get it around?

redpajama_set_name: RedPajamaStackExchange


How to find a particular barricade/cache?
How to obtain a sports drink?
How can I find the inventory?
How do I change my name and/or avatar?
How to start a new game?
How do I increase my Toughness and Endurance?
How to unlock Industrial Zone/Suburbs/etc?
How to collect (earn) stars?
If you want to change your game avatar, please open the 'Shop' menu by pressing the icon in the top middle of the game screen.
then press the 'Edit character' icon on the right.
In the character customization menu you may change your character's name, gender, skintone, haircut etc.


Does anyone know for sure what the maximum level is that a player can reach? I suspect its level 78 but does anyone know for sure?
Its level 78 but you can still get more xp for when the level cap will be raised you will automaticly level up.


Online Journal of Sudhir Parasuram (Lakkaraju): Microsoft Starter Edition (aka Demo Edition) to be launched in India ... Y is that bad ?
Microsoft Starter Edition (aka Demo Edition) to be launched in India ... Y is that bad ?
Yeah yeah, it's been doing the press rounds lately. M$'s starter edition - one which can run only 3 applications at a time [ when has even a full fledged one been able to run more ? ;-) ] - is going for commercial launch in beginning of 2005. Here's a press report on that.
For instance, the OS has severely limited multitasking to the point where Windows can only run three applications at a time; reverts back to a single default desktop and removes personalization features that allow multiple user accounts on a single PC; and fails to address security concerns by not bundling an antivirus solution, a reliable patching system or materials educating users about risks to their security.
The software also lacks the ability to grow with the users as they become more computer literate.


My 5 year old son loves this necklace! It makes him look like a mini surfer dude...he can put it on and take it off by himself, and I feel that it is safe for him to wear.
Absolutely beautiful! I had bought this to help my 4 year old daughter who is getting her back teeth through. We had previously passed on her amber pumpkin necklace as we thought the teething was over, nope. We have had a different girl since we go it she has been a delight! Has finally after months of being all bones has started to eat again and is actually plumping out! She's sleeping at night and no more tantrums and emotional mealy downs seem to be a thing of the past. Not only has it helped so much physically, it's stunning, she loves it! Will not take it off! Which resulted in buying more jewellery which she chose online for her and her little brother! Thank you amber pumpkin you've really helped our family with your amazing product!
Very pretty. Super quick delivery and I wish they did adult size in that necklace. My daughter adores it.


During the D-Day period I was a Red Cross VAD working as an Operating Room Assistant [ORA] at the Royal Naval Hospital, Haslar, Gosport, Hampshire. The theatre was one of two built in the vast cellars with the main steam pipes running under the ceilings and was always hot and smelly. So we used to put perfume on our masks to try and counter this. Have you ever smelt gas gangrene?
The theatre usually had only one table but two more were added and as each table had its own team consisting of surgeon, sister, two VADs, anaesthetist and his assistant we were pretty crowded.
Casualties started to come in from 7th June and x-rays were taken of their wounds so that the surgeons would know where to probe for the bullets, etc. We were classed as a base hospital so after surgery the wounds were plastered over with a little note on the outside to show what had been done and then most of the casualties were transferred to other hospitals up country.
I shall never forget the wonderful feeling of "all pulling together". Amongst the theatre staff everybody was so good-tempered and helpful.
…And strawberries! Whenever I see one now, I think of D-Day. The twenty-four hours were split into three watches: 4am to 12noon, 12noon to 8pm and 8pm to 4am and I can well remember going on duty at 4am having just demolished a dish of lovely luscious strawberries!


Brian, Golda and Roadie Rob react and discuss Takeover Chicago. Asuka, Bobby Roode and AOP continue on their undefeated streaks. Hideo doesn't impress, Dunne vs Bate put on the possible match of the year and #DIY swerves the world with an emotional classic tag team break-up. Follow Marked Out Pro Wrestling Podcast on Twitter @smartmarkshow or Instagram @markedoutwrestling Comment please!
‹ E154: Shane McMahon, Show Me the Money!


text:
import sqlalchemy as sa

def upgrade(migrate_engine):
    meta = sa.MetaData(bind=migrate_engine)
    event = sa.Table('event', meta, autoload=True)
    raw = sa.Column('raw', sa.Text)
    event.create_column(raw)


text:
Bob Looby 5:00am - 11:00am
KBHB ON DEMAND
LOCAL Weather Sports Obituaries Business Agriculture News Markets
Contact Us Airstaff Programming Schedule Jobs Privacy Policy Terms of Service Contest Rules EEO Public Inspection File
KBHB Store
Town & Country Market
Closings, Cancellations and Delays
Monument Health-Wind Power
Monument Health to get 20 percent of power from wind
News Release - February 13, 2020
RAPID CITY, S.D. – Monument Health has committed to receiving 20 percent of its electrical power from a wind power generating facility operated by Black Hills Energy.
The program, called "Renewable Ready," gives subscribers the option of having up to 100 percent of their electricity needs supplied by Wyoming wind energy, according to Black Hills Energy. As a Renewable Ready subscriber, Monument Health will be powered by 20 percent wind energy for the next 15 years, starting in late 2020.
"By choosing low-cost renewable energy resources to power Monument Health, we're able to advance our goals and sustainability objectives while also supporting the expansion of affordable, renewable energy development in our region," said Dave Ellenbecker, Vice President of Facilities Management and Plant Operations.
Black Hills Energy plans to build a utility-scale wind power generating facility in 2020 to supply the energy for its Renewable Ready program. Located near Cheyenne, the Corriedale Wind Energy Project will produce 52 megawatts (MW) of energy when operational.
According to Black Hills Energy, Renewable Ready is now fully subscribed. The program was designed for and open to large commercial and industrial customers and governmental agencies in the company's electric service territories in South Dakota and Wyoming who use a minimum of 300,000 kilowatt-hours of energy per year.
kbhbradio.com
Address: 1612 Junction Avenue, Suite #1
Sturgis, SD 57785
BIG 81 ON FACEBOOK

redpajama_set_name: RedPajamaCommonCrawl


So what's the difference between that noisy corded drill out in your garage and the sleek, new, cordless one sitting next to it on the workbench?
The obvious difference is mobility; pop the battery in your cordless drill, and you're free to roam until your heart's desire (or until your battery runs out of juice). Your corded drill, on the other hand, is of course reliant upon that cumbersome extension cord that most DIY'ers have a love/hate relationship with.
Makita vs. DeWalt – Who Makes A Better Set of Power Tools?


Jodi is our amazing Children's Pastor – She loves to love people! She has a heart for the hurting and anyone in need of a friend. An ideal mom, Jodi likes to make sure every child is OK & feels welcome and loved. She will take the time to answer all parents' questions. You can be sure your kids are in good hands!
Kristine is a super mom! She is the most kindhearted, caring and loving person and loves the children with all her heart. Kristine is exceptionally organized & handles every detail. She helps Pastor Jodi direct the Anchor Kids Team and invests in the lives of every volunteer. Her gift of hospitality will have you feeling right at home.
We love Robin! Robin is very approachable and she'll make sure to find the answers to all your questions as well as taking care of your kids! The youth love her!
In February of 2014 we opened our brand new Children's facility! The rooms are designated by age groups & each room has its own underwater theme.
Anchor Kids is offered during all weekend services for infants to 6th grade. Come to the Anchor Kids check-in desk to sign up your kids!
Anchor Sidekicks, Special Needs Ministry is aimed at meeting the spiritual needs of individuals with disabilities. We celebrate each child's uniqueness and provide a safe and fun environment so those with special needs can participate in Sunday classes.
The purpose of Anchor Moms is to give women the opportunity to connect and encourage each other through the ups and downs of life, build spiritual foundations through weekly Bible based studies and rejuvenate their soul with social activities and guest speakers. Anchor Moms is completely free with the occasional cost of a study guide.


Send your affection with a spray of gorgeous fresh lavender flowers featuring roses, delphinium, and Liatris.
This beautiful standing spray conveys deep emotion with hues of violet and lavender coming together to create an astonishing design. Featuring premium lavender roses, blue delphinium, and purple Liatris, amongst mums, statice and arrow-shaped TI leaves.


In October 2015, Ping Ming Health clinic director Ping Wang gave a seminar to traditional Chinese medicine students at China Capital Medical University (CCMU) in Beijing, China.
Ping introduced the history and development of Chinese medicine in Australia over the past 150 years.
She also shared her extensive experience of practicing and teaching Chinese medicine, and the most common health issues successfully helped by Chinese medicine in Australia.
Students from CCMU were very interested to learn about the Chinese medicine profession in Australia and many were interested in visiting Ping Ming Health for clinical observation and experience.
Ping also held meetings with the head of the School of Traditional Chinese Medicine to further cooperation between our clinic and the university in Beijing.
China Capital Medical University ranks among the top academic medical universities in China and is one of the key municipal universities in Beijing. The university and the affiliated hospitals have a staff of about 20,000. Among the staff, there are 6 members of the Chinese Academy of Sciences or the Chinese Academy of Engineering, over 1,000 professors, and over 2,000 associate professors. The university has over 9,000 enrolled students.
Ping graduated from and lectured at Capital Medical University prior to coming to Australia and founding Ping Ming Health in 1992.
Ping's meeting coincides with the recent signing of the China Australia Free Trade Agreement (ChAFTA), which provides significant opportunities for Australian healthcare providers to assist China with its growing healthcare challenges.
George Institute China relations director Professor John Knight said the trade agreement positioned "Australian healthcare to be not just for our country, but for the region" (AFR).
Australia was the first western country to introduce national registration of the Chinese medicine profession in 2012 and has internationally recognised tertiary education and research programs for Chinese medicine.
In the near future, Ping Ming Health will continue to develop our ability to train new Chinese medicine graduates into highly skilled, experienced and caring practitioners.
We look forward to offering Chinese graduates the experience of learning from our Australian trained practitioners, as well as giving Australian practitioners and health professionals the opportunity to visit hospitals in China.


text:
Q: How to make Mobile version of website I am looking for something that will help me learn how to set up my site so that when people go to it there is the regular version and the mobile version. Problem I am having is that most of the sites I see on google are pay sites that help you set that up. Can I just make two versions of my site and load them both on my hosting? Also how does it know when to load the mobile versus the regular? These are the types of things I am looking to learn, any suggestions of places to start?

A: Please please please make sure you do the following:

*

*Only do a mobile version if it adds value.

*If a user requests a deep link from a mobile device, don't redirect to the mobile home page

*Allow the user to choose to view the full version

*Make sure tablets such as the iPad uses the full version by default

*Don't serve WML to the hi-res smartphones such as the iPhone

*If your full sized website is unusable on a phone, consider tweaking it to be more suitable (don't print content text too wide).

Modern iPhones and iPads are perfectly capable of handling most full sized websites, there is little need for a mobile version unless it actually improves the user experience.  I hate to get redirected to a baby interface that doesn't provide the information I need.
Good examples of mobile versions include:

*

*Google website

*Gmail website

Mediocre examples include:

*

*Any blogpress site

Bad examples include

*

*anything that uses WML

Here's Apple's advice on providing mobile versions to the iPad:

Note that the Safari on iPad user
  agent string contains the word
  "Mobile", but does not contain the
  word "iPhone". If you are currently
  serving mobile content to any browser
  that self-identifies as "Mobile", you
  should modify your user agent string
  checks to look for iPad and avoid
  sending it the wrong version of your
  site. The version numbers in this
  string are subject to change over time
  as new versions of Safari on iPad
  become available, so any code that
  checks the user agent string should
  not rely on version numbers.

A: Something to look into would be the @mobile css media type, which is used to load a different styles when loading in a mobile device. I am not sure which devices support it, but I imagine it would be most of the popular models. As far as your suggestion, you can certainly host two separate sites, but I would defiantly go through Alex's suggestions before you go through the trouble.
http://www.w3.org/TR/css-mobile/

A: Alex had some good suggestions, but if you really want to serve up a specific version based on mobile or non-mobile you can take a look at the http header.  In some cases the User Agent will give it away, but not always.  Check out this link for details: http://detectmobilebrowsers.mobi/


text:
Search Records by Author: Joshi, S. T.
Found 229 items on 10 pages.
"On 'The Book.'"
Joshi, S. T.
in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 347-351.) (Reprinted from: Nyctalops 3(): 4: 9-13. April 1983.)
'Reality' and Knowledge: Some Notes on the Aesthetic Thought of H. P. Lovecraft
Lovecraft Studies 1(3): 17-27. Fall 1980. (Reprinted in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 105-114.)
21st-Century Horror: Weird Fiction at the Turn of the Millenium
Seattle, WA : Sarnath Press, 2018. 346 p.
Subjects: HORROR GENRE
A Dreamer and a Visionary: H. P. Lovecraft in His Time
Liverpool, UK: Liverpool University Press, 2001. 422 p.
A Literary Tutelage: Robert Bloch and H. P. Lovecraft
Studies in Weird Fiction No. 16: 13-25. Winter 1995. (Reprinted in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 548-565.)
A Look at Lovecraft's Fantastic Poetry
Aklo p. 20-30. Summer 1991. (Reprinted in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 453-464.)
Subjects: LOVECRAFT, H. P.; POETRY
A Look at Lovecraft's Letters,
Crypt of Cthulhu 6(4): 3-12. 1987. (No. 46) (Reprinted in: Van Hise, James, ed. The Fantastic Worlds of H. P. Lovecraft. Yucca Valley, CA: James Van Hise, 1999. p. 132-141.) (Reprinted in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 440-452.)
in: Van Hise, James, ed. The Fantastic Worlds of H. P. Lovecraft. Yucca Valley, CA: James Van Hise, 1999. p. 132-141. (Reprinted from Crypt of Cthulhu No. 46, 1987.)
A Master of Symbol and Metaphor,
in: Nolan, William F. Nolan on Bradbury: Sixty Years of Writing about the Master of Science Fiction. New York: Hippocampus Press, 2013. p. 253-256.
A RARY Tutelage: Robert Bloch and H. P. Lovecraft,
in: Szumskyj, Benjamin, ed. The Man Who Collected Psychos: Critical Essays on Robert Bloch. Jefferson, NC: McFarland, 2009. p. 23-40.
A Style Sheet for Lovecraftian Studies
Lovecraft Studies 1(1): 27-29. Fall 1979.
A Subtler Magick: The Writings and Philosophy of H. P. Lovecraft
San Bernardino, CA: Borgo Press, 1996. 312pp. 2nd Edition, rev. and expanded.
A Subtler Magick: The Writings and Philosophy of H. P. Lovecraft. 3rd. Edition
Berkeley Heights, NJ: Wildside Press, 1999. 312pp.
A Weird Writer in Our Midst: Early Criticism of H. P. Lovecraft, ed. by S. T. Joshi.
New York: Hippocampus, 2010. 134 p.
Against Religion: The Atheist Writings of H. P:. Lovecraft.
New York: Sporting Gentlemen, 2010. 190 p.
Algernon Blackwood and the Ghost Story
In: Joshi, S. T. The Development of the Weird Tale. Seattle: Sarnath Press, 2019. Kindle Edition, unpaged.
Subjects: BLACKWOOD, ALGERNON
Ambrose Bierce: An Annotated Bibliography of Primary Sources
Joshi, S. T.; Schultz, David E.
Westport, CT: Greenwood, 1999. 247pp.
Subjects: BIERCE, AMBROSE
An Epicure in the Terrible: A Centennial Anthology of Essays in Honor of H. P. Lovecraft, ed. by S. T. Joshi and David E. Schultz.
Rutherford, NJ: Fairleigh Dickinson University Press, 1991. 347pp.
An Index to the Fiction and Poetry of H. P. Lovecraft
West Warwick, RI: Necronomicon Press, 1992. 42pp.
Subjects: INDEXES; LOVECRAFT, H. P.
An Index to the Selected Letters of H. P. Lovecraft
Anne Rice: The Philosophy of Vampirism
Interzone No. 75: 47-50, 61. September 1993.
Subjects: RICE, ANNE; VAMPIRES
Arthur Machen: Philosophy and Fiction
Studies in Weird Fiction No. 2: 3-26. Summer 1987. Also in: Schweitzer, Darrell, ed. Discovering Classic Horror Fiction I. Mercer Island, WA: Starmont, 1992. pp.1-33.
Subjects: MACHEN, ARTHUR
Atheism and Anticlericalism in the Films of Guillermo del Toro
Subjects: ATHEISM; DEL TORO, GUILLERMO; MOTION PICTURES
Autobiography in Lovecraft
Lovecraft Studies 1(1): 7-19. Fall 1979. (Reprinted in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 219-231.)
Barbarism and Civilisation: Robert E. Howard and H. P. Lovecraft in Their Correspondence
Studies in the Fantastic No. 1: 95-124. Summer 2008. (Reprinted in: Joshi, S. T., ed. Lovecraft and a World in Transition: Collected Essays on H. P. Lovecraft. New York: Hippocampus, 2014. p. 73-96.)
Subjects: HOWARD, ROBERT E.; LOVECRAFT, H. P.
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text:
China is Trying to Match New US Air Force Bomber
This new bomber, which will likely be operated by the People's Liberation Army Air Force, is being called the H-20
As the U.S. continues to develop the B-21 Raider, a long-range, stealth strategic bomber, peers and rivals around the world are working to stay competitive. While Russia works on the PAK DA, Communist China is trying to counter the future backbone of the United States Air Force's strategic bomber force with a design of their own.
This new bomber, which will likely be operated by the People's Liberation Army Air Force, is being called the H-20. Details remain sparse, but reports state that it will have a top speed of 600 miles per hour and a maximum range of 5,282 miles. Although we're not certain about the ordnance it'll carry, it's likely that it'll carry a variety of dumb bombs, smart bombs, and missiles, just like the B-2.
Currently, the People's Liberation Army Air Force leans heavily on the H-6 Badger as the primary airframe in their strategic bomber force. This plane is a far cry from the original Soviet Tu-16 on which it's based. The current version, according to MilitaryFactory.com, has a range of 3,728 miles and a top speed of 652 miles per hour. Unlike the up-and-coming H-20, the H-6 doesn't have stealth technology, and while its range allows it to operate against naval units in the South China Sea, it doesn't have the reach to hit American bases in Guam or Okinawa.
The Chinese Communists used the H-6 to send a message in December 2016, when one of these bombers flew along the so-called "nine-dash line," which delineates Chinese claims in the South China Sea. About 180 of these bombers were built and the People's Liberation Army Air Force has roughly 120 H-6s in service, with another 30 in service with the People's Liberation Army Navy Air Force.
A People's Liberation Army Air Force H-6K Badger bomber. (Japanese Ministry of Defense photo)
The H-20 is slated to enter service in 2025, coincidentally around the time that the B-21 Raider is set to deploy.
More From We Are the Mighty
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More Weapons and Technology - WARRIOR MAVEN (CLICK HERE) --
Kris Osborn can be reached at Krisosborn.ko@gmail.com


text:
Händelser

Astronomi och rymdfart
1 april - USA skjuter ut världens första värdersatellit, TIROS-1, från Cape Canaveral AFS i Florida, USA, klockan 06.40 EST. TIRROS står för Television Infra-Red Observation Satellite.  Samma kväll introduceras satellitväderprognoser till världen via TV för första gången. Bilderna tas från en höjd på 450 engelska mil .
15 maj - Den sovjetiska satelliten Sputnik 4 skjuts upp.
12 augusti - Första experimeniella Project Echo-kommunikationssatelliten går i omloppsbana.
19 augusti - Den sovjetiska satelliten Sputnik 5 skjuts upp.
Okänt datum - Astronomer upptäcker vad som först antas var en ljusstark stjärna som sänder ut radiovågor, men visar sig vara en rödförskjutning från en kvasar .

Teknik
Okänt datum - Theodore Maiman konstruerar den första användbara lasern.

Pristagare
Copleymedaljen: Harold Jeffreys
Darwinmedaljen: Edred John Henry Corner
Ingenjörsvetenskapsakademien utdelar Stora guldmedaljen till Gudmund Borelius
Nobelpriset:
Fysik: Donald A. Glaser
Kemi: Willard Frank Libby
Fysiologi/medicin: Frank Macfarlane Burnet, Peter Medawar
Penrosemedaljen: Walter Hermann Bucher
Wollastonmedaljen: Cecil Edgar Tilley

Födda
18 oktober – Craig Mello, amerikansk utvecklingsbiolog, nobelpristagare.

Avlidna
24 april – Max von Laue, tysk fysiker, nobelpristagare.

Källor

Fotnoter

Externa länkar

redpajama_set_name: RedPajamaWikipedia


SJR Summer club details are released.
This summer we have more clubs than ever, Sign-up Letters have gone out today but if you are worried it has gone missing, they can be downloaded from this page. We have also responded to requests to have more clubs for our younger pupils starting with Archery and Fencing for Years 1 - 3. We are lucky to enjoy a wonderful partnership with many of our local providers.
You will see the clubs that start in the next couple of weeks below.
We also have some old favourites and new clubs returning soon. Letters will be distributed in school shortly and made available in school.