text:
\section{Introduction}

In 1993, Almgren-Taylor-Wang \cite{ATW93} proposed an implicit time discretization for mean curvature flow, which comes as a family of variational problems. Given an open subset $E_0\subset\mathbb{R}^n$ and a time-step size $h>0$, the sets $E_1,E_2,\ldots$ are successively obtained by solving
\begin{equation}\label{MM}
E_{k} \in \arg\min_E \Big\{ P(E)+ \frac1h \int_{E \Delta E_{k-1}} d_{E_{k-1}} \Big\},
\end{equation}
where $P(E) = \sup \{ \int_E \diver \xi \colon \|\xi\|_\infty \leq 1\}$ denotes the De Giorgi  perimeter of a subset of $\mathbb{R}^n$, $d_E$ the distance function to the boundary of $E$ and $E\Delta E_{k-1}$ the symmetric difference of $E$ and $E_{k-1}$.

\medskip

At the very heart of their idea lies the gradient-flow structure of mean curvature flow: trajectories in state space follow the steepest descent of the area functional with respect to an $L^2$-type metric.
In fact, this scheme inspired Ennio De Giorgi \cite{de1993new} to define his minimizing movements for general gradient flows in metric spaces, see \cite{ambrosio2006gradient}. Given a metric $\dist $ and an energy functional $E$, each time step of his abstract scheme is a minimization problem of the form
\[
x_{k} \in \arg \min_x \Big\{ E(x) + \frac1{2h} {\dist}^2(x,x_{k-1})\Big\}.
\]
In the smooth finite dimensional case when $\dist$ is the induced distance of a Riemannian metric, the Euler-Lagrange equation of the scheme boils down to the implicit Euler scheme.

\medskip

In case of mean curvature flow, the metric tensor ($L^2$-metric on normal velocities) is completely degenerate in the sense that the induced distance vanishes identically \cite{michor2004riemannian}. This  explains the use of the proxy $2\int_{E_{k+1}\Delta E_{k}} d_{E_k}$ for the squared distance in the minimizing movements scheme \eqref{MM}.

\medskip

The initial motivation of \cite{ATW93} was to define a generalized mean curvature flow through singularities as limits of the scheme \eqref{MM}. The convergence analysis as $h\downarrow0$ has a long history: Compactness of the approximate solutions was already established in \cite{ATW93}, together with the consistency of the scheme, in the sense that the approximations converge to the smooth mean curvature flow as long as the latter exists. In \cite{chambolle2004algorithm}, Chambolle simplified the proof and, furthermore,   proved  convergence to the viscosity solution (see \cite{evans1995motion}), provided the latter is unique. More precisely,  setting  $E_h(t) = E_k$, $t\in [kh, (k+1)h)$ to be the piecewise constant in  time interpolation of the sets $E_k$ obtained from \eqref{MM}, then the result reads as follows, see \cite{BarlesSonerSouganidis} for the notion of viscosity solution in this context.
\begin{thm}[{Convergence to viscosity solution \cite[Theorem 4]{chambolle2004algorithm}}]\label{thm Chambolle}
	Suppose $T<\infty$ and $E_0$ is a bounded  set in $\mathbb{R}^n$ with $\mathcal{L}^n(\partial E_0)=0$ such that the viscosity solution $\mathbf{1}_{E(t)}$ starting from $\mathbf{1}_{E_0}$ is unique, then $E_h\to E$ in $L^1$, i.e., $\int_0^T|E_h(t) \Delta E(t)|\,dt \to 0$ as $h\downarrow 0$.
\end{thm}

\medskip

Only shortly after \cite{ATW93}, Luckhaus-Sturzenhecker \cite{LucStu95} published a conditional convergence result which does not rely on the comparison principle but is purely based on the gradient-flow structure of mean curvature flow. In particular they showed that, \emph{conditioned on the convergence of the perimeters}, the scheme converges to a \(BV\) solution of mean curvature flow, according to the following definition.

\begin{defn}
A set of finite perimeter  \(E\subset \mathbb{R}_+\times \mathbb{R}^n\) is a \(BV\) solution of mean curvature flow if there exists \(V\in L^2(0,T;L^2(\mathcal{H}^{n-1}\lcorner \partial^\ast E(t)))\) such that
\begin{align}
&\int_0^T \int_{\partial^\ast E(t)} \left(\diver \xi - \nu \cdot D\xi \, \nu\right) d\mathcal{H}^{n-1}\,dt
= -\int_0^T \int_{\partial^\ast E(t)} V \, \xi \cdot \nu\, d\mathcal{H}^{n-1}\,dt,
\\\label{BV sol}
&\int_0^T\int_{E(t)}\partial_t \psi(t,x)\,dx\,dt+\int_{E(0)}\psi(0,x)\,dx=-\int_0^T \int_{\partial^\ast E(t)} \psi(t,x) V d\mathcal{H}^{n-1}(x)\,dt
\end{align}
for all \(\xi\in C^1_{c}([0,T)\times \mathbb{R}^n;\mathbb{R}^n)\) and \(\psi \in C^1_{c}([0,T)\times \mathbb{R}^n;\mathbb{R})\). Here \(E(t)\) is the time slice of \(E\),  \(\partial^\ast\) denotes the reduced boundary, and $\nu$ the (measure theoretic) exterior normal.
\end{defn}

The main result in \cite{LucStu95} is the following conditional convergence result:

\begin{thm}[{Conditional convergence \cite[Theorem 2.3]{LucStu95}}]\label{thm LS} Let $n\leq 7$ and let  \(E_h\) be the (time) piecewise constant approximation built by the Almgren-Taylor-Wang scheme. Then there exists a set \(E\subset \mathbb{R}_+\times \mathbb{R}^n\) and a subsequence \(\{h_j\}\) such that $E_{h_j}(t) \to E$ in $L^1$. Moreover, if
\begin{equation}\label{conv_ass}
\lim_{h_j \downarrow 0} \int_0^T P(E_{h_j}(t)) \,dt = \int_0^T P(E(t))\,dt,
\end{equation}
then $E$ is a ${BV}$ solution of mean curvature flow.
\end{thm}
We also  refer the reader to the  work of Mugnai-Seis-Spadaro \cite{mugnai2015global} where the proof of \cite{LucStu95}  is revisited in the case of volume-preserving mean curvature flow.

\medskip
 To the best of our knowledge, the only two cases in which  assumption \eqref{conv_ass} has been shown to be satisfied \emph{a-priori} is in the graphical case \cite{logaritsch2016obstacle}, in which no singularities occur, cf.\ \cite{ecker1989mean}, and in the convex case \cite{caselles2006anisotropic}, in which no singularities appear until the solution disappears in a round point \cite{huisken1984flow}.

 The main result of the present paper is to show that for a relevant class of initial data \eqref{conv_ass} holds true. The class of sets we will work with  is the class of {\em strictly mean convex sets}. Recall that a set is said to be strictly mean convex if $H>0$. Note that then, at least locally, $E$ solves a one-sided variational problem, called $\delta$-outward minimization, see Definition \ref{def H>0} below.

More precisely, our main theorem reads as follows.
\begin{thm}\label{thm main}
Let  $E_0\subset \mathbb{R}^n$ be a compact set with \(C^2\) boundary and let \(n\le 7\). Assume that \(E_0\)  is strictly mean convex in the sense that $H_{\partial E_0}>0$, then \eqref{conv_ass} holds.
\end{thm}

It is easy to construct strictly mean convex sets such that the mean curvature flow starting from them develops singularities in finite time. Hence our result is the first one establishing the validity of \eqref{conv_ass} under the possible development of singularities. Note also that, to the best of our knowledge, there are no examples of initial data for which \eqref{conv_ass} does not hold.

Let us also remark that a similar question was raised by Ilmanen for the approximation of the mean curvature flow via the Allen-Cahn equation \cite[Section 13, Question 4]{ilmanen1993convergence}.

Along the way we establish the following natural properties of the minimizing movements scheme \eqref{MM} for mean convex sets, which mirror Huisken's results for mean curvature flow \cite{huisken1984flow}:
\begin{itemize}
\item The sets $E_k$ are nested in the sense that $E_{k+1} \subset E_k$ for all $k\geq 1$.
\item The scheme preserves $\delta$-outward minimality and moreover, if $n\leq 7$, the minimum of the mean curvature of \(\partial E_k\),  $\min H_{\partial E_k}$ is increasing in $k$.
\end{itemize}
While Huisken's proofs are based on the maximum principle, our proofs are solely of variational nature.

Inspired by the work of Evans-Spruck \cite{evans1995motion} on mean curvature flow, we introduce the arrival time $u_h$ of the scheme. As the name suggests, the arrival time $u(x)$ of the mean curvature flow starting from $E_0\subset \mathbb{R}^n$ at a point $x\in E_0$ is the first time $t>0$ at which the flow reaches $x$, i.e., the super level set $\{u>t\}$ is equal to $ E(t) $. Similarly, as the sets $E_k$ obtained by the scheme are nested, one may also define the arrival time $u_h$ of the scheme so that $E_h(t) = \{u_h > t\}$. As one would expect,  \(u_h\) converges to \(u\), see Proposition \ref{thm:convergence}. By the  coarea formula, the proof of Theorem \ref{thm main} then boils down to the convergence of the total variation of the functions $u_h$.
This can be obtained by using a compensated compactness argument in line with the one in  \cite{evans1995motion}, together with some duality formulation of the obstacle problem established in \cite{scheven2017dual}.
However, we also present a much simpler direct proof which is self-contained and again based on the variational principle for $u_h$.
\medskip

As an immediate consequence of our main theorem, the convergence result of Luckhaus-Sturzenhecker becomes unconditional in the case of mean convex initial data:
\begin{cor}\label{c:conv}
Suppose $n\leq 7$ and $E_0$ is strictly mean convex, then any $L^1$-limit of the approximations $E_h(t)$ is a ${BV}$ solution of mean curvature flow.
\end{cor}

The paper is organized as follows. In Section \ref{sec:basic}, we establish some basic properties of strictly mean convex, so $\delta$-outward minimizing, sets and of the minimization scheme when applied to such sets.
In Section \ref{sec:arrival time} we define the arrival time of the scheme and prove that it solves an obstacle problem. In Section \ref{sec:conv} we show it converges to the arrival time of the discrete evolution and eventually in Section  \ref{sec:CC uh}, we prove Theorem \ref{thm main}.

\medskip

 \subsection*{Acknowledgements}

G.~D.~P.\ is supported by the MIUR SIR-grant ``Geometric Variational Problems" (RBSI14RVEZ). The authors would like to thank the referee for the careful reading and helpful comments which highly improved the quality of the manuscript.

\section{Basic properties of the scheme and mean convexity}\label{sec:basic}

We recall the definition and derive some first properties for the implicit time discretization scheme \eqref{MM} when the initial set is mean convex. The basis of our analysis is Lemma \ref{la stay mean convex}, which states that the scheme preserves mean convexity and that $\min H_{\partial E(t)}$ is non-decreasing in $t$.

  \medskip

Let us state  the minimization problem \eqref{MM} in a more precise language:  Given initial conditions $E_0\subset \mathbb{R}^n$, obtain $E_k$ for $k\in \mathbb{N}$ by successively minimizing $\mathcal{F}_h(E,E_{k-1})$:
  \begin{equation}\label{minimizing movements}
   E_k \in \arg\min \mathcal{F}_h(\,\cdot\,,E_{k-1}),
  \end{equation}
  where the functional $\mathcal{F}_h$ is given by
  \[
    \mathcal{F}_h(E,F) : = P(E) + \frac1h \int_{E\Delta F} d_{F}.
  \]
  Here and throughout the paper $d_F (x) := \textup{dist}(x,\partial F)$ denotes the distance function to the  boundary of $F$. We will always work with  the representative of \(F\) for which \(\overline { \partial^\ast F}=\partial F\), \( \partial^\ast F\) being the reduced boundary of \(F\), see \cite[Remark 15.3]{Maggi12}.

 We denote by $E_h$ the piecewise constant interpolation of the sets $E_0,E_1,E_2,\ldots $, i.e.,
  \[E_h(t)=E_k\quad \text{for }t\in[kh,(k+1)h).\]

  \begin{rem}\label{rem F tilde}
  It  is easy to see that the metric term $\int_{E\Delta F} d_F$ can be rewritten as
	\[
	\int_{E\Delta F} d_{F} = \int_{E} sd_{F} -\int_{F} sd_{F},
	\]
	where $sd_{F} := d_{F} - d_{\mathbb{R}^n\setminus F} $ denotes the signed distance function to the  boundary $\partial F$. Therefore the minimization of $\mathcal{F}_h(\,\cdot\,,F)$ is equivalent to minimizing
	\[
	 	P(E) + \frac1h \int_{E} sd_{F}.
	\]
	Testing \eqref{minimizing movements} with $E_{k-1}$ and summing over $k$ implies the following a priori estimate for the implicit time discretization
  \begin{equation}\label{a priori}
  	\sup_{N\geq1}  \Big\{ P(E_N) + \sum_{k=1}^N \frac1h \int_{E_k \Delta E_{k-1}} d_{E_{k-1}} \Big\} \leq P(E_0),
  \end{equation}
  which underlies Luckhaus-Sturzenhecker's compactness and conditional convergence Theorem \ref{thm LS}.
	\end{rem}

\begin{rem}\label{example ball}

In the radially symmetric case $E_0=B_{r_0}$, a Steiner symmetrization argument shows that the
minimizers are radially symmetric. Therefore, the minimization problem \eqref{minimizing movements} reduces to finding radii $r_0 > r_1 >r_2>\ldots$ so that each $r_k$ minimizes the function
  $$
    r^{n-1} + \frac1h \int_r^{r_{k-1}}  \rho^{n-1} (r_{k-1}-\rho) d\rho.
  $$
  The Euler-Lagrange equation is
  $$
    r_k^2 -r_{k-1}\,r_k +(n-1)h=0 \textrm{  (or equivalently $\frac{r_k-r_{k-1}}{h} = - \frac{n-1}{r_k}$)},
  $$
   so that for sufficiently small $h$ the optimal radius is explicitly given by
  \[
    r_k= \frac12 \Big( r_{k-1} + \sqrt{ r_{k-1}^2 -4 (n-1) h}\Big).
  \]
  Note that for fixed $h$, after $O(r_0^{2}h^{-1})$ steps we have $r_k = 0$. Note also that, as one can easily see by induction
  \[
  r_k \ge \sqrt{r_0^2-2k(n-1)h}.
  \]
  \end{rem}

It is a well known fact in the study of mean curvature flow that mean-convexity of the initial condition (i.e. \(H_{\partial E_0}\ge 0 \)) is preserved, \cite{huisken1984flow} and that in this setting much stronger results can be obtained, see for instance  \cite{HK17,White00,White03} for an incomplete list and \cite{MetzgerSchulze08} where a  problem similar to ours is studied.

Here, as in \cite{HI01}, we introduce the variational analog of mean convexity:

  \begin{defn}\label{def H>0}
  Let $\Omega \subset \mathbb{R}^n$. A set $E\subset \Omega$ is called \emph{outward minimizing in $\Omega$} if
  \begin{equation}\label{H>0_Omega}
  P(E) \leq P(F)\quad \text{for all }F \text{ with } E\subset F \subset \Omega.
  \end{equation}
  If $E$ is outward minimizing in $\Omega = E+B_\delta = \{x\in \mathbb{R}^n \colon \dist(x,E) < \delta \} $ the $\delta$-neighborhood of $E$, then $E$ is called \emph{$\delta$-outward minimizing}, and \eqref{H>0_Omega} simply reads
  \begin{equation}\label{H>0}
  P(E) \leq P(F)\quad \text{for all }F\supset E \text{ with } \sup_{x\in F} \dist(x,E) < \delta.
  \end{equation}
  \end{defn}

\begin{rem}\label{rmk:H}
  Outward minimality as defined  above is the variational formulation of the pointwise inequality $H\geq0$.
  It is easy to see that in our case of a smooth and strictly mean convex set $E_0$ there exists $\delta>0$ such that $E_0$ is \emph{$\delta$-outward minimizing}, see for instance \cite[Lemma 5.12]{DLMV18}.
  Note carefully that $P(E)$ denotes the perimeter in $\mathbb{R}^n$, not the one relative to $\Omega$.

 Each iteration of the scheme does not move further than \(O(\sqrt{h})\) in Hausdorff distance, see  \cite[Lemma 2.1,(1)]{LucStu95}, i.e., there exists a universal constant $C=C(n)$ such that
 \begin{equation}\label{eq:LS sqrt h}
 	\sup_{x\in \partial E_k} d_{E_{k-1}}(x) \leq C \sqrt{h}.
 \end{equation}

  \end{rem}
  \medskip

 Let us now recall a few basic properties of $\delta$-outward minimizing sets which will be useful in the sequel.  They are well known to experts, but for the sake of completeness we report here their simple proof, see also \cite[Section 3]{spadaro2011mean} and \cite[Section 1]{De-PhilippisPaolini09a}.

\begin{lem}\label{la H>0 variant}
  $E$ is outward minimizing in $\Omega$ if and only if
    \begin{equation}\label{H>0 variant}
      P(E\cap G) \leq P(G)\quad \text{for all }G\subset \Omega.
    \end{equation}
  \end{lem}

  \begin{proof}
   We employ the basic inequality
  \begin{equation} \label{perimeter inequality}
   P(E\cap F) + P(E\cup F) \leq P(E) + P(F).
  \end{equation}
   Given any set $G\subset \Omega$, the outward minimizing property  \eqref{H>0_Omega} of $E$ tested with $F= E\cup G$ yields
    \[
    P(E) \overset{\eqref{H>0_Omega}}{\leq} P(E\cup G) \leq P(E) + P(G) - P(E \cap G),
    \]
    which simplifies to \eqref{H>0 variant}.

Vice versa, if $F\supset E$, we can apply \eqref{H>0 variant} with \(G=F\) to obtain \eqref{H>0_Omega}.

  \end{proof}

  A  direct consequence of this characterization is that outward minimality is stable under  $L^1$-convergence.
  \begin{cor}\label{cor H>0 stable}
    Let $E_h\to E$ in $L^1$ for some sequence $\{E_h\}_h$ of outward minimizing sets in $ \Omega$. Then $E$ is outward minimizing in $\Omega$.
  \end{cor}

  \begin{proof}
    By Lemma \ref{la H>0 variant} it is enough to show  \eqref{H>0 variant} instead of \eqref{H>0_Omega} for $E$, which in turn follows immediately from \eqref{H>0 variant} for $E_h$ and the lower semi-continuity of the perimeter.
  \end{proof}

  If $\Sigma(t)$ is a smooth mean curvature flow then the scalar mean curvature $H$ of $\Sigma(t)$ solves
  \[
  \partial_t H - \Delta H = \left|A\right|^2 H,
  \]
  where $A$ denotes the second fundamental form of $\Sigma(t)$ and $\Delta$ the Laplace-Beltrami operator on $\Sigma(t)$,  cf.\ \cite[Corollary 3.5]{huisken1984flow}.  In particular, if $H\geq 0$ at $t=0$, by the maximum principle $H\geq0$ for $t\geq0$ and $\min H(t)$ is non-decreasing in $t$.
  By the strong maximum principle we even have $H>0$ for $t>0$.

  It is well known and easy to see that $\delta$-outward minimality is preserved by the implicit time discretization \eqref{minimizing movements}, see for instance \cite{spadaro2011mean}. We report  the simple proof of this fact in the next lemma where we also establish the monotonicity of $\min H_{\partial E_h(t)}$.

  \begin{lem}\label{la stay mean convex}
  Let $E_0\subset\subset \Omega$ be outward minimizing in $\Omega$. Then there exists $h_0>0$ such that for all $0<h<h_0$ the implicit time discretizations $E_h$ are non-increasing in $t$, i.e.,
  \begin{equation}\label{nested}
    E_h(t) \subset E_h(s) \quad \text{for all } 0\leq s \leq t,
  \end{equation}
  $E_h(t)$ is outward minimzing in $\Omega$ for all $t\geq0$, and $E_h(t)$ solves the Euler-Lagrange equation
    \begin{equation}\label{Euler Lagrange equation}
    H_{\partial E_h(t)}(x) = \frac{d_{E_{h}(t-h)}(x)}{h}\geq0,\qquad x\in \partial^\ast E_h(t).
    \end{equation}
    Furthermore, if \(n\le 7\), $ \min H_{\partial E_h(t)} $ is non-decreasing in $t$.
  \end{lem}

  Note that by classical regularity for minimizers of \eqref{MM}, see e.g.\ \cite{Maggi12},  \(\partial^\ast E_h(t)\) is a \(C^2\)-manifold relatively open in \(\partial E_h(t)\) and \(\partial E_h(t)\setminus \partial^\ast E_h(t)\) has Hausdorff dimension at most \(n-8\). In particular \eqref{Euler Lagrange equation} makes sense.

We also believe that the restriction \(n\le 7\) needed to show the monotonicity of $ \min H_{\partial E_h(t)} $ can be actually avoided. It seems however that this would require some version of the maximum principle for singular hypersurfaces in the spirit of \cite{Simon87}. Since however in Theorem \ref{thm LS} this restriction does not seem to be easily avoidable, we decided to restrict ourselves to this case.

  By Remark \ref{rmk:H}, if $E_0$ is a bounded open set of class $C^2$ with $H_{\partial E_0}>0$, there exists $\delta>0$ such that $E_0$ is outward minimizing in its $\delta$-neighborhood $\Omega=E_0+B_\delta$.
  The smallness condition on $h$  can be dropped if $E_0$ is outward minimizing in $\mathbb{R}^n$.

    \begin{proof}
    	Let $h>0$ be such that $h< h_0:= \frac1{C^2} \min_{x\in\partial \Omega}\dist^2(x,E_0)$ with $C$ from \eqref{eq:LS sqrt h}.

    	Let $k\geq 1$ and assume that $E_{k-1}$ is outward minimizing in $\Omega$.
    We first prove $E_k \subset E_{k-1}$ and then the outward minimality of $E_k$ in $\Omega$.

     Since by assumption $E_{k-1}$ is outward minimizing in $\Omega$, by \eqref{eq:LS sqrt h} and our choice of $h_0$, we may employ the characterization \eqref{H>0 variant}:
    \[
      P(E_{k-1}\cap E_k) \leq P(E_k).
    \]
    We want to use $E_{k-1} \cap E_k$ as a competitor for the minimization of $\mathcal{F}_{h}(\,\cdot\,,E_{k-1})$.  Since
    \[
      (E_{k-1}\cap E_k ) \Delta E_{k-1} = E_{k-1} \setminus E_k \subset E_k \Delta E_{k-1}
    \]
    we have
    \[
      \frac1h\int_{(E_{k-1}\cap E_k ) \Delta E_{k-1}} d_{E_{k-1}} \leq \frac1h \int_{E_k \Delta E_{k-1}} d_{E_{k-1}}
    \]
    with strict inequality if $\mathcal{L}^n(E_k \setminus E_{k-1})>0$. Hence
    \[
      \mathcal{F}_{h}(E_{k-1}\cap E_k,E_{k-1}) \leq \mathcal{F}_{h}(E_k,E_{k-1})
    \]
    with strict inequality if $\mathcal{L}^n(E_k \setminus E_{k-1}) >0$, which proves $E_k \subset E_{k-1}$ (up to Lebesgue null sets).

    Let $F$ be such that $ E_k \subset F \subset \Omega$; we want to verify $P(E_k) \leq P(F)$.  Using the outward minimality of the predecessor  $E_{k-1}$ we have
    \[
    P(F\cap E_{k-1}) \overset{\eqref{H>0 variant}}{\leq} P(F)
    \]
    and hence it is enough to prove the inequality \eqref{H>0_Omega} for sets $F$ with $E_k \subset F\subset E_{k-1}$. Using these inclusions we have
    \[
    F\Delta E_{k-1}  = E_{k-1} \setminus F \subset E_{k-1} \setminus E_k = E_k \Delta E_{k-1}
    \]
    and therefore
    \[
    \frac1h \int_{F\Delta E_{k-1}} d_{E_{k-1}} \leq  \frac1h\int_{E_k \Delta E_{k-1}} d_{E_{k-1}}.
    \]
   Now the minimality $\mathcal{F}_{h} (E_k,E_{k-1}) \leq \mathcal{F}_{h}(F,E_{k-1})$ implies $P(E_k) \leq P(F)$ and hence $E_k$ is indeed outward minimizing in $\Omega$.

    Since \eqref{Euler Lagrange equation} is classical, we now  turn to the proof of the monotonicity of $\inf H_{\partial E_h(t)}$. Fix $k \in \mathbb{N}$ and let
    \[
     x_0\in \arg\min H_{\partial E_k}.
    \]
    Since $H_{\partial E_h(t)} = \frac1h d_{E_{k-1}}$ is Lipschitz continuous and $\partial E_k$ is compact, at least one such $x_0$ exists.
    We shift $E_{k-1}$ by $h\, H_{\partial E_k}(x_0) = d_{E_{k-1}}(x_0)$ in the fixed direction $\nu_{\partial E_k}(x_0)$, i.e.,
    \[
      F_{k-1} := E_{k-1} + h\, H_{\partial E_k}(x_0)\,\nu_{\partial E_k}(x_0).
    \]
    By definition of $x_0$ we have $E_k \subset F_{k-1}$ and $x_0 \in \partial E_k \cap \partial F_{k-1}$ and, since \(n \le 7\), both boundaries are smooth in a neighborhood of \(x_0\).  Thus
    \[
     H_{\partial E_k}(x_0) \geq H_{\partial F_{k-1}}(x_0) \geq \min H_{\partial F_{k-1}} = \min H_{\partial E_{k-1}},
    \]
    which is precisely our claim.
  \end{proof}

  By Corollary \ref{cor H>0 stable},  limits of outward minimizing sets are  outward minimizing. From this we can easily infer the monotonicity of the perimeters.

  \begin{cor}\label{cor monotonicity Per}
    Let $E_0\subset\subset \Omega$ be outward minimizing in $\Omega$ and let $E(t)$ be an $L^1$-limit of the implicit time discretizations $E_h(t)$. Then $E(t)$ is outward minimizing in $\Omega$  for a.e.\ $t$ and $P(E(t))$ is non-increasing in $t$.
  \end{cor}

  \begin{proof}
    The outward minimizing  property of $E(t)$ is an immediate consequence of Lemma \ref{la stay mean convex} and Corollary \ref{cor H>0 stable}.
    Since by Lemma \ref{la stay mean convex} we have $E(t) \subset E(s)$ for $t\geq s$ we can use the mean convexity \eqref{H>0} of $E(t)$ to conclude
    $ P(E(t)) \leq P(E(s))$ for $t\geq s$.
  \end{proof}

  \medskip

  The basic inequality \eqref{perimeter inequality}
  and the observation that we have the analogous equality for the distance term in $\mathcal{F}$ yield the general inequality
  \begin{equation}\label{general F inequality}
   \mathcal{F}_h(E\cap F,E_{k-1})+ \mathcal{F}_h(E\cup F,E_{k-1}) \leq \mathcal{F}_h(E,E_{k-1}) + \mathcal{F}_h( F,E_{k-1}).
  \end{equation}
  Therefore, if $E$ and $F$ are minimizers, so are $E\cap F$ and $E\cup F$.
  In our setting, where $E_{k-1}$ is outward minimizing , this implies the outward minimality  of all these sets and we have equality in \eqref{perimeter inequality}.

The following general lemma is a comparison result which holds independently of the initial conditions $E_0$ being mean convex and revisits Chambolle's ideas \cite{chambolle2004algorithm}.

\begin{lem}[Comparison principle, \cite{chambolle2004algorithm}]\label{lem_comparison}
	Let \(n\le 7\) and let  $E_0 , F_0 \subset \mathbb{R}^n$ be two bounded open sets of finite perimeter such that $E_0$ is properly contained in $F_0$ in the sense that $E_0\subset \subset F_0$.
	Let $E$ and $F$ be minimizers of $\mathcal{F}_h(\,\cdot\, , E_0)$ and $\mathcal{F}_h(\,\cdot\, , F_0)$, respectively, then $E$ is properly contained in $F$, i.e., $E\subset \subset F$.
\end{lem}

\begin{proof}
	The proof consists of two steps. First we prove the inclusion $E\subset F$, second we prove
	$\min_{x\in \partial E} d(x,\partial F) >0$.

	Inasmuch as $E_0\subset \subset F_0$, the boundaries have a definite distance
	$\min_{x\in \partial E_0} d(x,\partial F_0) >0$, which implies the strict inequality
	\begin{equation}\label{f<g}
		 sd_{F_0} <sd_{E_0} \quad \text{in } \mathbb{R}^n.
	\end{equation}
	Probing the minimality of $E$ and $F$ for the modified functionals in Remark \ref{rem F tilde}  with $E\cap F$ and $E\cup F$, respectively, yields
	\[
		P(E) +\frac1h \int_E sd_{E_0} \leq P(E\cap F) + \frac1h \int_{E\cap F} sd_{E_0}
	\]
	and
	\[
		P(F) +\frac1h \int_F sd_{F_0} \leq P(E\cup F) + \frac1h \int_{E\cup F} sd_{F_0}.
	\]
	Summing these two inequalities and using the general inequality for the perimeter of intersections and unions of sets \eqref{perimeter inequality} we obtain
	\[
		\int_E sd_{E_0} + \int_F sd_{F_0}  \leq   \int_{E\cap F} sd_{E_0} +\int_{E\cup F} sd_{F_0}.
	\]
	Rearranging the terms and using the obvious identities $\chi_{E\cap F} = \chi_E \chi_F$ and $\chi_{E\cup F} = \chi_E + \chi_F - \chi_E\chi_F$ along the way, we obtain
	\[
	0\leq \int \left( sd_{E_0} - sd_{F_0}\right) \chi_E \left( 1 - \chi_F\right)
	= \int_{E\setminus F}  \left( sd_{E_0} - sd_{F_0}\right).
	\]
	Since by \eqref{f<g} the integrand is strictly negative, this means that $\mathcal{L}^n(E\setminus F)=0$
	and hence $E\subset F$.

	Now assume for a contradiction $\partial E \cap \partial F \neq \emptyset$. Let $x_0 \in \partial E \cap \partial F$ be a point in the intersection. Since $E \subset F$ we have $H_{\partial E} \geq H_{\partial F}$ at that point $x_0$ and therefore
	\[
		\frac1h sd_{E_0} = -H_{\partial E} \leq - H_{\partial F} = \frac1h sd_{F_0},
	\]
	a contradiction to \eqref{f<g} (note that as in the proof of Lemma \ref{la stay mean convex} we have used the restriction \(n\le 7\) to ensure smoothness of the boundaries at the touching point).
	\end{proof}

  \section{The arrival time for the implicit time discretization}\label{sec:arrival time}

Since by Lemma \ref{la stay mean convex} the sets $E_h(t)$ are nested, we can define the (discrete) arrival time \(u_h\) for the scheme. In this section we show that, up to subsequences, \(u_h\) converges uniformly to some continuous function  \(u\). In the next section we will identify \(u\) as the arrival  time for the limiting evolution starting from \(E_0\).

\begin{defn}\label{dfn:at}
	Let $E_0$ be outward minimizing in the sense of Definition \ref{def H>0}, let $E_k$, $k\geq1$, be given by \eqref{minimizing movements} and let $E_h$ denote their piecewise constant interpolation in time. We define the \emph{arrival time} $u_h\colon \mathbb{R}^n \to [0,\infty)$ by
	\begin{equation}\label{def uh}
		u_h(x) := h\sum_{k\geq 0} \chi_{E_k}(x) = \int_0^\infty \chi_{E_h(t)}(x)\,dt \quad (x\in \mathbb{R}^n).
	\end{equation}
\end{defn}

Let us first note that $u_h \in {BV}(\mathbb{R}^n)$ since the a priori estimate \eqref{a priori} implies
	\begin{equation}\label{a priori uh}
	\int_{\mathbb{R}^n}\left| Du_h\right| = \int_0^{T_h} P(E_h(t))\,dt \leq T_h P(E_0),
	\end{equation}
	where $T_h$ denotes the extinction time of $(E_h(t))_{t\geq0}$. Note that  the extinction time is finite: If $R>0$ is sufficiently large such that $E_0\subset B_R$, then by Lemma \ref{lem_comparison} we have $E_h(t) \subset B_{r_h(t)}$, where $r_h$ is given in Remark \ref{example ball} and satisfies $r_h(t)=0$ for $t$ larger than $O(R^2)$.

	\medskip

The following lemma  states that for our mean convex initial condition, the arrival time solves a (one-sided) variational problem.
\begin{lem}\label{lem uh outward minimizing}
 	 Let $E_0\subset\subset \Omega$ be outward minimizing in $\Omega$ in the sense of Definition \ref{def H>0}. Then there exists $h_0>0$ such that for $0<h<h_0$, the arrival time $u_h$ is outward minimizing in $\Omega$ in the sense that
 	 \begin{equation}\label{eq:uh outward minimizing}
 	 	\int_{\mathbb{R}^n} \left| D u_h \right| \leq \int_{\mathbb{R}^n} \left|D v\right| \quad \text{for all } v\in {BV}(\mathbb{R}^n) \text{ s.t. } v\geq u_h \text{ and } v=0 \text{ in }\mathbb{R}^n\setminus \Omega.
 	 \end{equation}
  \end{lem}

	Again, the smallness condition on $h$ can be dropped in case of $\Omega=\mathbb{R}^n$.

\begin{proof}
	Given $v \in {BV}(\mathbb{R}^n)$ with $v\geq u_h$ and $v=0$ in $\mathbb{R}^n\setminus \Omega$ we employ the coarea formula, cf.\ \cite[Theorem 3.40]{ambrosio2000functions}, to manipulate the total variation of $v$:
	\[
		 \int_{\mathbb{R}^n} \left|D v\right| = \int_0^\infty P(\{ x\in \mathbb{R}^n \colon v(x)> t \}) \,dt.
	\]
	Since $v\geq u_h$ and $v=0$ in $\mathbb{R}^n\setminus \Omega$ imply
	\[
		E_h(t) = \{ x\in \mathbb{R}^n \colon u_h(x)> t \} \subset \{ x\in \mathbb{R}^n \colon v(x)> t \} \subset \Omega,
	\]
	the super level sets of $v$ are admissible for \eqref{H>0_Omega} and we obtain
	\[
		\int_{\mathbb{R}^n} \left|D u_h\right| = \int_0^\infty P(E_h(t) ) \,dt
		\leq \int_0^\infty P(\{ x\in \mathbb{R}^n \colon v(x)> t \}) \,dt =  \int_{\mathbb{R}^n} \left|D v\right| . \qedhere
	\]
\end{proof}

The next lemma states that we have a uniform estimate on the modulus of continuity of $u_h$ except for fluctuations on scales below $ h$; and hence after passing to a subsequence, we obtain uniform convergence to a continuous function.

\begin{lem}\label{lm:conv}
	Let \(n\le 7\) and let  $E_0$ be a  bounded open set of class $C^2$ with $ H_{\partial E_0} >0$. Then there exists a subsequence $h_j\downarrow0$ and a continuous function $u\colon \mathbb{R}^n\to [0,\infty) $ with $\supp u \subset \bar{E_0}$ such that
	\begin{align}\label{uniform convergence}
 u_{h_j} &\to u & & \text{uniformly} \\
	 D u_{h_j} &\rightharpoonup  Du & &  \text{as measures}  \label{gradient weak convergence}
	\end{align}
\end{lem}

\begin{proof}
	Let $H_0 := \min H_{\partial E_0} >0$, which by Lemma \ref{la stay mean convex} implies
	$\min H_{\partial E_k} \geq H_0$ for all $k\geq0$.

	We claim that we have a uniform bound on the modulus of continuity up to fluctuations on scales below $ h$, i.e.,
	\begin{equation}\label{mod of cont}
		\left| u_h(x)-u_h(y) \right| \leq \frac{1}{H_0} |x-y| + h \quad \text{ for all } x,y\in \mathbb{R}^n.
	\end{equation}
	In order to prove \eqref{mod of cont} let $x,y\in E_0$ be given.
	Without loss of generality  we may assume $x \in E_n$ and $y\in E_{m}$ with $-1 \leq m < n$, where we have set  $E_{-1} := \mathbb{R}^n \setminus E_0$.
	Since  the sets $E_k$, $k\geq0$ are nested, the segment $[x,y] $ intersects the intermediate boundaries non-trivially: There are points $z_k$, $k=m+1,\dots,n$, such that $z_k \in \partial E_k \cap [x,y]$. Using the Euler-Lagrange equation \eqref{Euler Lagrange equation} along these points we obtain
	\[
		\left| x-y \right| \geq\left| z_{n} - z_{m+1} \right|
		=  \sum_{k=m+2}^{n} \left| z_k - z_{k-1} \right|
		\geq \sum_{k=m+2}^{n} d( z_k, \partial E_{k-1})
		\geq (m-n-1) h  H_0.
	\]
	Since $|u(x)-u(y)| = (m-n)h$, this is precisely our claim \eqref{mod of cont}.
	Therefore, by Arzel\`{a}-Ascoli, we obtain the compactness \eqref{uniform convergence}. The weak convergence of the gradients \eqref{gradient weak convergence} follows immediately from the uniform bound \eqref{a priori uh}.
\end{proof}

\section{Convergence to the continuous arrival time}\label{sec:conv}

Let \(E_0\) be an outward minimizing set such that \(H_{\partial E_0}>0\). According to the  previous section the  arrival times \(u_h\) of the discrete  scheme converge, up to subsequences, to a limiting function \(u\). In this section we identify this function as the arrival time of the limiting equation. We start by recalling the following

\begin{thm}[Evans-Spruck \cite{evans1995motion}]\label{thm:es}
Let \(E_0\) be a bounded open set of class $C^2$ with \(H_{\partial E_0}>0\). Then there exists a unique continuous viscosity solution \(u\) of
\begin{equation}\label{e:at}
\begin{cases}
 |Du|\diver\Bigg(\dfrac{Du}{|Du|}\Bigg)=-1\qquad &\textrm{in \(E_0\)}\\
u=0&\textrm{on \(\partial E_0\).}
\end{cases}
\end{equation}
Moreover,   for all \(t\in[0, \sup u]\) the set \(\{u\ge t\}\) is the evolution of  \(\overline{E_0}=\{u\ge 0\}\) via mean curvature flow.
\end{thm}

Here  a solution of \eqref{e:at} is understood in the viscosity sense, that is for all \(x\in E_0\) and all  \(\varphi \in C^2(E_0)\) such that \(u-\varphi\) has a minimum at \(x\) (resp. a maximum) then
\begin{align}
\Delta \varphi(x) -\frac{D^2 \varphi(x)[D\varphi(x),D\varphi(x)]}{|D\varphi(x)|^2}\le -1 \quad(\ge -1) \qquad&\textrm{if \(D\varphi(x)\ne 0\)} \label{e:sup1}
\\
\exists \eta\in \mathbb S^{n-1} \textrm{ such that } \Delta \varphi(x)-D^2\varphi(x)[\eta, \eta] \le -1\quad (\ge -1)\qquad&\textrm{if \(D\varphi(x)= 0\).}\label{e:sup2}
\end{align}

The following proposition is the elliptic analog of \cite{chambolle2004algorithm}, see also \cite[Section 7]{chambolle2015nonlocal}, and shows that the discrete arrival time converges to the (unique) viscosity solution of  \eqref{e:at}.

\begin{prop}\label{thm:convergence}

Let \(E_0\) be as in Theorem \ref{thm:es} and let \(u_h\) be as in Definition \ref{dfn:at}. Then every limit point of \(u\) of \(u_h\) is a viscosity solution of \eqref{e:at}. In particular  the whole sequence \(u_h\) converges to \(u\).
\end{prop}

\begin{proof}
Let \(u\) be such that (up to subsequences) \(u_{h}\to u\) uniformly.  Let \(x\in E_0\)  and  \(\varphi \in C^{2}(E_0)\) be such that \(u-\varphi\) has a minimum at \(x\). By changing coordinates we may assume without loss of generality that \(x=0\), moreover, by replacing \(\varphi\) by \(\varphi-C|x|^4\) we may assume that the minimum is global and strict:
\begin{equation}\label{minstrict}
u(x)-\varphi(x)>u(0)-\varphi(0) \qquad \textrm{for all \(x\in E_0\setminus\{0\}\).}
\end{equation}
By classical arguments we can find a sequence of points \(x_h\) such that \(x_h \to 0\) and
\[
(u_h)_*(x)-\varphi(x)\ge (u_h)_*(x_h)-\varphi(x_h)
\]
where \((u_h)_*\) is the lower semicontinuous envelop of \(u_h\), namely
\[
(u_h)_*=\sum_{k=1}^{T_h/h} h \chi_{\Int(E_k)}.
\]
Here \(T_h\) is the extinction time of the scheme. Note in particular that \( (u_h)_*\to u\) uniformly. For simplicity, from now on we assume that the sets \(E_k\) are open and that \(u_h\) is already lower-semicontinuous  (observe  that by the regularity theory for almost minimizers of the perimeter \(|\overline{E_k} \setminus \Int(E_k)|=0\) which allows us to choose such a representative). We also let \(k_h\in \mathbb N\) be the unique integer such that \(u_h(x_h)=k_h h\). In particular \(x_h \in E_{k_h}\).

 We now distinguish two cases.

\medskip
\noindent
{\em Case 1: \(D\varphi(0)\ne 0\).} Since \(x_h\to 0\) we have \(D\varphi(x_h)\ne 0\) if \(h\) is sufficiently small.  Hence, \(u_h\) cannot be flat constant in a neighborhood of \(x_h\), so \(x_h \notin \Int(E_{k_h}\setminus E_{k_h+1})\) and thus, since \(E_{k_h}\) is open,
\[
x_h \in \partial E_{k_h+1}.
\]
In particular
\[
U:=\big\{ \varphi>\varphi (x_h)\} \subset E_{k_h+1} \qquad\textrm{and}\qquad x_h \in \partial U\cap\partial E_{k_h+1}.
\]
Since both \(\partial U\) and \(\partial E_{k_h+1}\) are smooth in a neighborhood of \(x_h\), the comparison principle and the Euler-Lagrange equation \eqref{Euler Lagrange equation} yield
 \begin{equation}\label{e:dis}
 \begin{split}
 \diver\Bigg(\frac{D\varphi(x_h)}{|D\varphi(x_h)|}\Bigg)&=-H_{\partial U}(x_h)\le -H_{\partial E_{k_h+1}}(x_h)
 \\
 &= -\frac{\dist (x_h, \partial E_{k_h})}{h}\le -\frac{\dist (x_h, \partial \{\varphi> \varphi(x_h)-h\})}{h},
 \end{split}
 \end{equation}
 where in the last inequality we have used that
 \[
 E_{k_h}^c=\{u\le u(y_h)-h\}\subset \{\varphi\le \varphi(x_h)-h\}.
 \]
Moreover, by Taylor expansion, one easily verifies
\begin{equation}\label{e:taylor}
\frac{\dist (x_h, \partial \{\varphi> \varphi(x_h)-h\})|D\varphi(x_h)|}{h}\to 1\qquad\textrm{as \(h\to 0\)}.
\end{equation}
Combining \eqref{e:dis} and \eqref{e:taylor} we conclude the validity of \eqref{e:sup1}.

\medskip
\noindent
{\em Case 2: \(D\varphi(0)=0\).} This time we can not assume a priori that \(D\varphi (x_h)\neq 0\). To overcome this difficulty we exploit Jensen's $\inf$-convolution (on a fixed scale of order \(h\)). To this aim let us define
\[
v_h(x):=\inf_{y\in \overline{E_0}} \Big\{u_h(y)+\frac{ |x-y|^4}{2c^4_nh}\Big\} \quad \text{for \(x\in E_0\)},
\]
where \(c_n\) is a constant that will be fixed later in dependence only on the dimension \(n\). We also let \(z_h\) be a minimum point of \(v_h-\varphi\), namely
\[
v_h(x)-\varphi(x)\ge v_h(z_h)-\varphi(z_h) \quad \text{for all } x\in \bar{E_0}
\]
and let \(y_h\in \overline{E_0}\) be such that
\[
v_h(z_h) =u_h(y_h)+\frac{|z_h-y_h|^4}{2c^4_nh}.
\]
Note that the existence of \(y_h\) is ensured by the lower semicontinuity of \(u_h\).

We now divide the proof in some  steps:

\medskip
\noindent
{\em Step 1: \(|z_h-y_h| \to 0\).}  Indeed, since \(v_h\le u_h\le 2 \|u\|_{\infty}\) we obtain
\[
|z_h-y_h|^4\le 8c^4_n h \|u\|_{\infty}\to 0 \quad\textrm{as \(h\to 0\)}.
\]

\medskip
\noindent
{\em Step 2: \(z_h \to 0\).} Indeed, by \(v_h\le u_h\) and the definition of $z_h$,
\[
u_h(x_h)-\varphi(x_h) \ge v_h(z_h)-\varphi(z_h)\ge u_h(y_h)-\varphi(y_h)+\varphi(y_h)-\varphi(z_h).
\]
If we let \(\bar z\in \overline {E_0} \) be an accumulation point  of \(z_h\) (and hence of \(y_h\)) we deduce from the above inequality and the uniform convergence of \(u_h\) to \(u\)  that
\[
u(0)-\varphi(0)\ge u(\bar z)-\varphi (\bar z)
\]
which in view of \eqref{minstrict} forces \(\bar z=0\).

\medskip
\noindent
{\em Step 3: \(z_h \ne y_h\).} Let us assume by contradiction that \(z_h=y_h\). By the very definition of \(v_h\) this means that
\begin{equation}\label{e:bound}
u_h(z_h)=v_h(z_h) \le u(y)+\frac{|y-z_h|^4}{c^4_n h} \quad \text{for all } y\in \bar{E_0}.
\end{equation}
Let also  \(j_h\in \mathbb{N}\) be such that \(u_h(z_h)=j_h h\). Note that since \(u>0\) in \(E_0\) and \(u_h(z_h)\to u(0)>0\) we may assume that \(j_h \gg 1\). In particular
\[
z_h\in E_{j_{h}}\setminus E_{j_{h}+1}.
\]
We now note that \eqref{e:bound} implies
\begin{equation}\label{e:inc}
F_0:=B\big(z_h,c_n \sqrt{h}\big)\subset\subset \big\{u_h\ge (j_h -1)h\big\}=E_{j_h-1}.
\end{equation}
If we let \(F_1\) and \(F_2\) be minimizers of \eqref{minimizing movements} starting from \(F_0\) and \(F_1\), respectively, Remark \ref{example ball} ensures that
\[
F_2=B\big(z_h,r_h\big) \qquad\textrm{with} \qquad  r_h\ge \sqrt{c_n-4(n-1)}>0
\]
provided \(c_n\) is chosen sufficiently large. However, by  Lemma \ref{lem_comparison} and \eqref{e:inc}
\[
z_h \in F_2\subset\subset E_{j_h+1},
\]
a contradiction.

\medskip
\noindent
{\em Step 4: Conclusion.} By the very definitions of \(v_h\), \(y_h\) and \(z_h\) we have
\begin{equation}\label{e:key}
u_h(y_h)+\frac{|z_h-y_h|^4}{2c^4_n h}-\varphi(z_h)\le  u_h(y)+\frac{|x-y|^4}{2c^4_n h}-\varphi(x) \quad \text{for all } x,y\in \bar{E_0}.
\end{equation}
In particular, the optimality condition in the \(x\)-variable implies
\[
D \varphi(z_h)=\frac{2|z_h-y_h|^2(z_h-y_h)}{c_n^4 h} \ne 0.
\]
Moreover, if we set
 \[
 \psi_h(x):=\varphi(x+(z_h-y_h))+\frac{|z_h-y_h|^4}{2c^4_n h},
\]
the function \(u-\psi_h\) has a minimum at \(y_h\)  with \(D \psi_h(z_h)\ne 0\). By the very same arguments of Case 1 we obtain that
\[
\Delta \varphi(z_h) -\frac{D^2 \varphi(z_h)[D\varphi(z_h),D\varphi(z_h)]}{|D\varphi(z_h)|^2}\le -1+o(1),
\]
which gives \eqref{e:sup2} with \(\eta\) being any limiting point of the sequence \(\frac{D\varphi(z_h)}{|D\varphi(z_h)|}\).

Since the case in which \(u-\varphi\) has a maximum at some \(x\in E_0\) can be treated analogously, this completes the proof.
\end{proof}

\section{Compensated compactness for the arrival time and proof of Theorem~\ref{thm main}}\label{sec:CC uh}

In this section we establish the convergence of the total variations of the arrival times $u_h$ and prove Theorem \ref{thm main}.  Our proof is elementary and only uses the variational principle for $u_h$ established in Lemma \ref{lem uh outward minimizing}. We also state a second proof which seems more robust and might be applicable to similar problems. This second proof based on the compensated compactness argument of  Evans-Spruck \cite{evans1995motion} together with the dual problem of the variational principle for $u_h$ viewed as an obstacle problem for \(BV\) functions established in \cite{scheven2017dual}.

\begin{prop}\label{prop CC uh}
	Let $E_0$ be strictly mean convex in the sense of Definition \ref{def H>0} and let $u_{h_j}$ defined by \eqref{def uh} satisfy \eqref{uniform convergence} and \eqref{gradient weak convergence}.
	Then,
	\[
		\left| Du_{h_j}\right| \rightharpoonup  \left| Du \right| \quad \text{as measures.}
	\]
	In particular it holds
	\[
		\int_{\mathbb{R}^n} \left| Du_{h_j}\right| \to \int_{\mathbb{R}^n}  \left| Du \right|.
	\]
\end{prop}

While the compensated compactness argument of Evans-Spruck is based on the curious estimate
\begin{equation}\label{EvansSpruck}
\sup_{\varepsilon>0} \int_{\mathbb{R}^n} |H_\varepsilon(x)| \,dx < \infty,
\end{equation}
which miraculously holds true for the elliptic regularizations $u_\varepsilon$ of the level set formulation, this estimate is very intuitive in our situation:

Informally, the Euler-Lagrange equation of the minimization problem in Lemma \ref{lem uh outward minimizing} reads
	\[
		- \diver \left(  \frac{Du_h}{|Du_h|}\right) \geq 0.
	\]
	This means that these distributions are in fact measures, for which it should be reasonable to get appropriate bounds. This resembles the $L^1$-bound \eqref{EvansSpruck} and would allow us to pass to the limit in
	\begin{equation}\label{heuristics}
	\int \zeta \left| Du_h\right| = \int  \zeta Du_h \cdot  \frac{Du_h}{|Du_h|} = -\int \zeta u_h
	\diver \left(  \frac{Du_h}{|Du_h|}\right) - \int u_h \,\frac{Du_h}{|Du_h|} \cdot D\zeta.
	\end{equation}
	This argument can be made rigorous, see Remark \ref{rmk_oldproof} below. Let us first show a simpler direct proof.

	\begin{proof}[Proof of Proposition \ref{prop CC uh}]
		By lower semi-continuity, we only need to prove the inequality
		\begin{equation}\label{eq:proof of prop CC uh}
			\limsup_{h\downarrow0} \int \left| Du_h\right| \leq \int \left| Du\right|.
		\end{equation}
		Since $u_h\to u$ uniformly, for any $\varepsilon>0$ there exists $h_0>0$ such that
		\[
			u_h<u+\varepsilon \quad \text{whenever } 0<h<h_0.
		\]
		Multiplying $u+\varepsilon$ with a cutoff $\eta\in C_0^\infty(\Omega)$ of $E_0$ in $\Omega$, $v=(u+\varepsilon)\eta$ is an admissible competitor for \eqref{eq:uh outward minimizing}, so that
		\[
			 \int \left| Du_h\right|
			 \stackrel{\eqref{eq:uh outward minimizing}}{\leq}
			 \int \left|Dv\right|
			 = \int \eta \left| D u\right| + \int (u+\varepsilon)\left|D\eta \right|
			 \leq   \int \left| D u\right| + \varepsilon \int \left|D\eta \right|,
		\]
		where we have used $\eta \leq 1$ for the first, and $u+\varepsilon = \varepsilon $ on $\supp \eta \subset  \Omega \setminus E_0$ for the second right-hand side term. Passing first to the limit $h\downarrow0$ and then $\varepsilon\downarrow0$ yields \eqref{eq:proof of prop CC uh}.
		\end{proof}

\begin{rem}\label{rmk_oldproof}
	For the alternative proof of Proposition \ref{prop CC uh}, which makes the compensated compactness argument \eqref{heuristics} rigorous, we interpret the minimization problem in Lemma \ref{lem uh outward minimizing} as an obstacle problem in a $\delta$-neighborhood $\Omega$ of $ E_0$ with homogeneous Dirichlet boundary conditions. Here the obstacle is of class BV and happens to be our minimizer $u_h$ itself. This allows us to use the general theory for dual formulations of obstacle problems: By \cite[Theorem 3.6, Remark 3.8]{scheven2017dual} the dual problem reads
	\[
		\max  \llbracket\sigma, Du_h^+ \rrbracket(\bar{\Omega}).
	\]
	where the maximum runs over all measurable vector fields $\sigma\colon \Omega\to \mathbb{R}^n$ with $|\sigma|\leq 1$ a.e. \ in $\Omega$ and $\diver \sigma \leq 0$ distributionally in $\Omega$.
	Note that this implies that $\diver \sigma$ is a measure on $\bar{\Omega}$ and
	\begin{equation}\label{diver sigma bounded}
		(-\diver \sigma)(\bar \Omega) \leq \mathcal{H}^{n-1}(\partial \Omega).
	\end{equation}
	 Here
	 \[
	 u_h^+(x)=\aplimsup_{y\to x} u_h(y)
	 \]
	  denotes the largest representative of $u_h$, see \cite{scheven2017dual}, and the measure $\llbracket\sigma, Du_h^+ \rrbracket$ is defined as
	\begin{equation}
		\llbracket\sigma, Du_h^+ \rrbracket (\zeta) := -\int_{\Omega} \zeta u_h^+ \diver(\sigma)\, dx
		- \int_{\Omega} u_h \left( \sigma \cdot D \zeta\right) dx,
	\end{equation}
	for test functions $\zeta\in C^1(\bar{\Omega})$.
	This yields a vector field $\sigma_h$ for any $h>0$ with the above mentioned properties and such that
	\begin{equation}\label{duality equation uh}
		\int \left| Du_h\right|
		=\llbracket\sigma, Du_h^+ \rrbracket(\bar{\Omega}) = \llbracket\sigma, Du_h^+ \rrbracket(\mathbb{R}^n).
	\end{equation}
	Here we used the fact that $u_h $ vanishes away from $E_0 \subset\subset \Omega$.

	Since $|\sigma_h|\leq 1 $, we may assume that there exists a measurable vector field $\sigma$ with $|\sigma|\leq1$ such that
	\begin{equation}
		\sigma_{h_j} \overset{\ast}{\rightharpoonup} \sigma\quad \text{in }L^\infty.
	\end{equation}
	Moreover, by \eqref{diver sigma bounded}, there exists a subsequence, which we do not relabel, and  a measure $\mu$ such that
	\begin{equation}\label{diver sigma convergence}
		\diver \sigma_{h_j} \rightharpoonup \mu \quad \text{as measures}.
	\end{equation}
	In particular
 	\begin{equation}
 	\diver \sigma = \mu \quad \text{  in }\Omega.
 	\end{equation}

	Now we can make the idea of the aforementioned compensated compactness argument rigorous.
	By \eqref{duality equation uh} we have
	\begin{equation}\label{Du rigorous}
		\int \zeta \left| Du_h\right|
		= -\int \zeta u_h^+ \diver(\sigma_h)\,dx
		- \int u_h \left( \sigma_h \cdot D \zeta\right) dx,
	\end{equation}
	which is precisely the analogue of \eqref{heuristics} with the important difference that we can give a meaning to (and have precise estimates for) all products appearing on the right.
	Along the subsequence $h_j\downarrow0$, on the one hand, since $u=\lim u_{h_j}$ is continuous, we have
	\[
		\lim_{h_j\downarrow0}-\int \zeta u \diver(\sigma_{h_j})\,dx
		\stackrel{\eqref{diver sigma convergence}}{=} - \int \zeta u \,d\mu.
	\]
	On the other hand, by the uniform convergence \eqref{uniform convergence}, we have
	\[
		\left| -\int \zeta\, \big( u_{h_j}^+ -u\big) \diver(\sigma_{h_j})\,dx \right|
		\stackrel{\eqref{diver sigma bounded}}{\leq} \|\zeta\|_\infty \| u_{h_j}^+ - u\|_\infty \mathcal{H}^{n-1}(\partial \Omega) \to 0.
	\]
	Therefore, we can pass to the limit in the first right-hand side product of \eqref{Du rigorous}:
	\begin{equation}
		\lim_{h_j\downarrow0} -\int \zeta u_{h_j}^+ \diver(\sigma_{h_j})\,dx
		= - \int \zeta u \,d\mu = \int D(\zeta u) \cdot \sigma \,dx.
	\end{equation}
	Since $\supp u_h \subset \Omega$ is equibounded, the convergence $u_{h_j} \to u$ is strong in $L^1$ and hence we may pass to the limit in the second right-hand side product of \eqref{Du rigorous}. Therefore, for any non-negative test function $\zeta\in C^1(\mathbb{R}^n)$ we obtain
	\[
			\lim_{h_j\downarrow0}\int \zeta \,| Du_{h_j}|
			= \int D(\zeta u) \cdot \sigma \,dx
			-\int u \left( \sigma \cdot D \zeta\right) dx
			= \int \zeta\, \sigma \cdot Du
			\leq \int \zeta \left| Du\right|,
	\]
	where we used the pointwise bound $\left| \sigma \right| \leq 1$ a.e.\ in the last inequality.
	The lower semicontinuity of the total variation implies
	\[
	\int \zeta \left| Du\right|\leq \liminf_{h_j\downarrow0}\int \zeta \left| Du_{h_j}\right|
	\]
	for all non-negative test function $\zeta\in C^1(\mathbb{R}^n)$.
	Therefore
	\[
	\lim_{h_j\downarrow0}\int \zeta \left| Du_{h_j}\right| = \int \zeta \left| Du\right|
	\]
	holds for all non-negative test functions $\zeta \in C^1(\mathbb{R}^n)$. By linearity and continuity in $\zeta$ the convergence holds for all continuous test functions $\zeta\in C( \mathbb{R}^n)$ without restriction on the sign, which proves $\left| Du_{h_j}\right| \rightharpoonup  \left| Du \right| $ as measures.
\end{rem}

We are now ready to prove Theorem \ref{thm main}:

\begin{proof}[Proof of Theorem \ref{thm main}]
	 Passing to a subsequence, we may assume $E_{h_j} \to E $ in $L^1$. By Proposition \ref{thm:convergence}  \(u_{h_j}\) converges  to the arrival time of the limiting evolution \(u\). By the co-area formula and Proposition \ref{prop CC uh}
	 \[
\lim_{h\to 0} \int_0^\infty P(E_{h}(t)) \,dt = \lim_{h \to0} \int_{\mathbb{R}^n} 	|Du_{h}| = \int_{\mathbb{R}^n} |Du|=\int_0^\infty P(E(t))\,dt,
	 \]
	which proves \eqref{conv_ass}.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Givry-en-Argonne
French Version : http://www.cartesfrance.fr/carte-france-ville/51272_Givry-en-Argonne.html
> Champagne-Ardenne
> Marne
> Givry-en-Argonne
Town of Givry-en-Argonne
The village of Givry-en-Argonne is a small village located north east of France. The town of Givry-en-Argonne is located in the department of Marne of the french region Champagne-Ardenne.	The town of Givry-en-Argonne is located in the township of Givry-en-Argonne part of the district of Sainte-Menehould. The area code for Givry-en-Argonne is 51272 (also known as code INSEE), and the Givry-en-Argonne zip code is 51330.
Geography and map of Givry-en-Argonne :
The altitude of the city hall of Givry-en-Argonne is approximately 170 meters. The Givry-en-Argonne surface is 7.66 km ². The latitude and longitude of Givry-en-Argonne are 48.95 degrees North and 4.889 degrees East. Nearby cities and towns of Givry-en-Argonne are : Remicourt (51330) at 0.95 km, Le Châtelier (51330) at 1.78 km, La Neuville-aux-Bois (51330) at 2.05 km, Le Vieil-Dampierre (51330) at 3.52 km, Saint-Mard-sur-le-Mont (51330) at 4.44 km, Épense (51330) at 4.69 km, Noirlieu (51330) at 5.61 km, Sivry-Ante (51800) at 5.88 km.
(The distances to these nearby towns of Givry-en-Argonne are calculated as the crow flies)
Population and housing of Givry-en-Argonne :
The population of Givry-en-Argonne was 491 in 1999, 471 in 2006 and 468 in 2007. The population density of Givry-en-Argonne is 61.10 inhabitants per km². The number of housing of Givry-en-Argonne was 296 in 2007. These homes of Givry-en-Argonne consist of 238 main residences, 23 second or occasional homes and 35 vacant homes.
I like Givry-en-Argonne !
Administrative Information of Givry-en-Argonne
Region : Champagne-Ardenne (21)
Departement : Marne (51)
District : Sainte-Menehould
Township : Givry-en-Argonne
Nearby towns of Givry-en-Argonne
Here are maps and information of the close municipalities near Givry-en-Argonne.
See all the towns of Marne (Distances are calculated as the crow flies)
Remicourt (51) at 0.95 km Le Châtelier (51) at 1.78 km
La Neuville-aux-Bois (51) at 2.05 km Le Vieil-Dampierre (51) at 3.52 km
Saint-Mard-sur-le-Mont (51) at 4.44 km Épense (51) at 4.69 km
Noirlieu (51) at 5.61 km Sivry-Ante (51) at 5.88 km
Sommeilles (55) at 7.06 km Braux-Saint-Remy (51) at 8.06 km
Contault (51) at 8.29 km Dommartin-Varimont (51) at 8.30 km
Belval-en-Argonne (51) at 8.43 km Les Charmontois (51) at 8.50 km
Villers-en-Argonne (51) at 8.67 km Le Chemin (51) at 8.69 km
Charmont (51) at 8.88 km Possesse (51) at 9.03 km
Nettancourt (55) at 9.25 km Somme-Yèvre (51) at 9.48 km
Geography of Givry-en-Argonne
Area : 7.66 km² (766 hectares)
Decimal geographical coordinates : Latitude: 48.95 degrees (48.95° North)
Population and housings of Givry-en-Argonne
Inhabitants in 1999 : 491 inhabitants
Population data : Population Givry-en-Argonne
Variation of population density : -0.60 %/year
Housings data : Housing Givry-en-Argonne
Hotel Givry-en-Argonne
Search hotels near Givry-en-Argonne : Book your hotel now !
Find all hotels of Givry-en-Argonne :
Guest accommodation : Le Néerléain
Located in Noirlieu, Le Néerléain features accommodation with barbecue facilities, free WiFi, a shared kitchen, and a shared lounge. This holiday home provides accommodation wit...
Address : 1 Rue Du Marronnier 51330 Noirlieu
Distance Guest accommodation - Givry-en-Argonne : 5.67 km
Bed and Breakfast : Les Aneries
Located between Reims and Verdun amongst the fields and countryside, just 8 km from the A4 motorway, Les Aneries offers B&B accommodation in a renovated typical Argonne house. G...
Contact : 24, Grande Rue 51800 Braux-Saint-Rémy
Distance Bed and Breakfast - Givry-en-Argonne : 8.25 km
Guest accommodation : Nuits Insolites d'Argonne
Offering self-catering wooden caravans in the countryside, Nuits Insolites d'Argonne is 18 km from Grande Romanie Golf Club. You can make grilled meals on the barbecue and parki...
Contact : 13 rue Saint Nicolas 51330 Varimont
Guest accommodation : Maison De Vacances - Nettancourt
Maison De Vacances - Nettancourt is set in Revigny-sur-Ornain and offers a tennis court. Guests benefit from a terrace and a barbecue. The holiday home has 3 bedrooms, a TV and...
Contact : 55800 Revigny-sur-Ornain
Bed and Breakfast : Château de Nettancourt
Situated in Nettancourt, Château de Nettancourt provides free WiFi, and guests can enjoy a seasonal outdoor swimming pool, a garden and a terrace. Guests at the bed and breakfa...
Address : 7 Rue Du Chée 55800 Nettancourt
Guest accommodation : Le Gîte de la Faïencerie
Le Gîte de la Faïencerie is set in Passavant-en-Argonne. This property offers access to free WiFi. The holiday home includes a kitchen with a dishwasher and an oven, as well as...
Contact : La Faïencerie Grande Rue 51800 Passavant-en-Argonne
Distance Guest accommodation - Givry-en-Argonne : 10.66 km
Guest accommodation : La Grange aux Orchidées
Located in Bussy-le-Repos, La Grange aux Orchidées offers a self-catering house with a garden next to a vineyard. Chalons-sur-Marne is only a 35-minute drive away. The cottage ...
Contact : 26 rue Royale 51330 Bussy-le-Repos
Guest accommodation : Gîte De la chee
Boasting accommodation with a terrace, Gîte De la chee is situated in Laheycourt. Featuring garden views, this holiday home also comes with free WiFi. This holiday home feature...
Price : from 250.00 €
Contact : 33 rue du général porson 55800 Laheycourt
Bed and Breakfast : Chambres d'hôtes La Maison Forte
Located in a 17th-century building, this B&B features rooms with private bathroom and a view of the landscape. A breakfast is served daily with supplement and free private parki...
Address : 6 palce Henriot du Coudray 55800 Revigny-sur-Ornain
Distance Bed and Breakfast - Givry-en-Argonne : 15.28 km
Bed and Breakfast : Demeure D'Argonne
Located in Sainte-Menehould in the Champagne - Ardenne region, Demeure D'Argonne features accommodation with free WiFi. There is a fully equipped private bathroom with shower a...
Rate : from 79.00 € to 129.00 €
Contact : 4 Place du Général Leclerc 51800 Sainte-Menehould
Find another hotel near Givry-en-Argonne :
Givry-en-Argonne hotels list
Map of Givry-en-Argonne
Map of Givry-en-Argonne :
At right you can find the localization of Givry-en-Argonne on the map of France. Below, this is the satellite map of Givry-en-Argonne. A road map, and maps templates of Givry-en-Argonne are available here : "road map of Givry-en-Argonne". This map show Givry-en-Argonne seen by the satellite of Google Map. To see the streets of Givry-en-Argonne or move on another zone, use the buttons "zoom" and "map" on top of this dynamic map.
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To search hotels, housings, tourist information office, administrations or other services, use the Google search integrated to the map on the following page : "map Givry-en-Argonne".
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Photos Givry-en-Argonne
Most popular and closest pictures of Givry-en-Argonne town hall :
Psyllobora vigintiduopunctata in copula
Araschnia levana
Brenthis daphne
Anisosticta novemdecimpunctata
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Other photos of Givry-en-Argonne and nearby towns can be found here: photos Givry-en-Argonne
Givry-en-Argonne weather forecast
Find next hours and 7 days weather forecast for Givry-en-Argonne here : weather Givry-en-Argonne (with english metrics)
This is the last weather forecast for Givry-en-Argonne collected by the nearest observation station of Sainte-Menehould.
The latest weather data for Givry-en-Argonne were collected Monday, 17 January 2022 at 17:25 from the nearest observation station of Sainte-Menehould.
Overcast throughout the day.
Temperature Max: 6°C Temperature Min: 1°C
Wind speed: 2 km/h East
Pressure: 1028 mbars
Wind speed: 4 km/h North-East
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Make a link to this page of Givry-en-Argonne with the following code :
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redpajama_set_name: RedPajamaCommonCrawl


"up to one in four women will lose a baby during pregnancy or birth" (Tommy's, 2017) Given the statistics early pregnancy and infant loss is still a subject rarely talked about. It's often a silent loss, known only by those who knew about the pregnancy in those first few weeks or much a wider affair at the later stages. It's when excitement suddenly turns to grief and people experience difficulties with the loss. It is a loss that is so often seen as out of the order of life's events.
Having trained and volunteered with Cruse Bereavement in Cornwall my bereavement knowledge was fairly comprehensive. However, reflecting on this experience I identified the gap in my knowledge and was rapidly becoming aware of just how common this type of loss is. This training was an opportunity to explore as fully as possible how this type of loss may impact on people's lives. While never wanting to lose sight of the individuals experience or understanding the difficulties they are experiencing in the moment, it provided me an opportunity to explore how to create an environment that allows the person or couple to explore their own feelings around the loss. It also provided me with an opportunity to consider the complexities around pregnancy, stillborn and early infant loss and what can be done during that difficult time.
Chantal Lockey created online learning material which detailed a mother's first-hand experience of losing their child. As a result of the loss of her 6 week old daughter, Chantal wanted to share her experience, raise awareness and help break down barriers in order to help others. A decade on, and the Foundation for Infant Loss Training is well established and recognised by many.
The training material provided an overview of infant loss, the initial emotional and physical reactions to pregnancy and infant loss, a parent's perspective of loss, when a baby dies (what to expect as a professional i.e. those in attendance), a baby that dies in hospital, a baby's funeral care, memory boxes, relationships after pregnancy and infant loss, including the difference in how men and women may grieve, returning to work and life after loss, including subsequent children, family inclusion and milestones.
I originally completed my training back in 2014 and this year Chantal has changed the delivery format of the training and additional information has been added, bringing me more up to date. This covers termination, subsequent pregnancies after loss, grief theory, when a twin dies, infertility, relationship issues after loss, supporting children after loss, pre and post-natal depression and post-traumatic stress disorder. I'm really pleased to see this additional information as these aspects are really important especially the post-traumatic stress. This is an area of considerable importance and has not always been recognised.
Sadly pregnancy and early infant loss is difficult to talk about; people just don't know what to say. On the whole within society we have moved more towards recognising the importance of acknowledging such a loss but there is still much work to be done.
Recently Coronation Street featured a story following the loss of a baby to character Michelle Connor, played by Kym Marsh who herself tragically lost her own son at a similar stage of pregnancy over eight years ago (Digital Spy, 2017). The storyline followed the impact such a loss can have on people. This was challenging also for Simon Gregson, who plays onscreen husband Steve McDonald. He and his wife Emma have lost 11 babies, including one at 21 weeks and 4 days (Hello, 2017). The programme faced the tragic loss of a baby and followed the parents, close family and friends grief and hi-lighted the differences between people and how they grieve, in particular men and women.
If you feel you would benefit from meeting others who have experienced a pregnancy or early infant loss and bereavement please contact me to discuss attending a Support Group. Individual and couple counselling is also available if you prefer.


The AK4181A is a 4-wire touch screen controller that incorporates a 12-bit 125kHz sampling SAR A/D converter. The AK4181A can detect the pressed screen location by performing two A/D conversions. In addition to location, the AK4181A also measures touch screen pressure. On-chip VREF can be utilized for two analog auxiliary inputs and battery monitoring, with the ability to measure voltages from 0V to 5V. The AK4181A also has an on-chip temperature sensor.


text:
Can This Sexologist Help Her "Married at First Sight" Clients in Bed?
Studio@Gawker
Filed to: fyi networkFiled to: fyi network
fyi network
asked and answered
Sexologist Dr. Logan Levkoff helps newlyweds — who met at the altar — navigate the kinks in their newly-minted sex life on the new FYI Network show, Married at First Sight. We asked her a few probing questions about satisfaction, sex, and how to find it all in one relationship. Here's what she said. (Marrieds: feel free to use her advice the next time you're "just too tired.")
Q. Can you explain your role on Married at First Sight?
My role on Married at First Sight is to get an understanding of who the singles are as sexual beings and, once they're married, to help them communicate about sexuality and intimacy. Now, that doesn't mean I focus on how they have sex — rather, I'm looking to identify the role that sex plays in their lives, the relationship models that they had growing up, what they typically find attractive, any deal breakers, how much they're willing to experiment, and what their sexual and reproductive values are. We don't talk about it very often, but political or religious values can make or break a relationship.
Q. How did you get involved with the show?
I got a call this past winter and was asked to participate in a show that had to do with arranged marriages. I said no immediately. But after some prompting from the people in my life, I took the call and listened to the description. I was intrigued but cautious. "Does anyone get voted off an island? Does anyone win a pile of money?" "Absolutely not," the producer told me. "This is not one of those shows." In an effort to show me what this experiment was really about, I was sent the entire season of the original show that aired in Denmark. For eight straight hours, I watched the season with Danish subtitles on my iPhone. And I was hooked! I rooted for these couples. The show was thoughtful, provocative, and incredibly hopeful. I was in.
Q. What makes this show an extreme social experiment?
I think when someone agrees to marry a complete stranger, that's pretty extreme. And let's be clear: this is a real marriage — not just one made for television. This experiment seeks to determine whether social science can play a role in marital success. If we can offer a new way to find a connection (and a long-lasting one), imagine the possibilities!
Q. What qualities do you look at in two people to judge whether you think they'd make a good match?
My role is to explore an individual's sexuality (as a whole) and how he or she may complement someone else. My perspective alone is not enough to make a match. That's why there are four experts. Each of us offers a unique perspective. Together, we can create what we believe is a great match.
Q. What are some of the ways that you think people sabotage their relationships or sell themselves short when dating?
It seems as if we have a short attention span when it comes to modern relationships. The grass is always greener — or so we think. And sure, online dating can add to this feeling because there are thousands of people to choose from. Add to this pop culture, where we watch celebrities jump in and out of relationships on the cover of tabloids — and yes, I know this is rarely representative of their real lives — and it just seems like we don't know what's worth fighting for anymore. We tend to jump ship at the first obstacle because we don't know how to work through something or we assume that something better (or easier) is out there.
I don't believe there are rules in love, dating, and sex. I think that if you want something, you need to be authentic and up-0front with your needs. We still buy into these old ideas of how a man or woman is "supposed" to act in order to attract a partner. If we aren't being genuine, we'll never be satisfied.
Q. Is there any way to predict two people's sexual compatibility before you've seen them together?
Can you know for sure whether two people are going to have great orgasms together? No. But I can get a sense of whether a couple will be compatible with respect to what they want sexually and whether their values about sex are aligned. While it's difficult to predict chemistry prior to seeing a couple together, from a completely superficial perspective, you can tell if someone would find the other attractive based on interviews and past history. But it is important to remember that while chemistry is sometimes instantaneous, it doesn't have much weight if you don't communicate your physical and emotional needs to each other. From my meetings with the men and women [on the show] and the detailed reports that they filled out, I got a good sense of how aware they were of their own sexuality, how conscious they were of how others viewed them, their assertiveness in a relationship, and how communicative they are. I would never have matched couples who were polar opposites in these areas — especially not in an experience like this one.
Q. How much of a role do you think sexual compatibility plays in making a marriage last?
Sex and physical intimacy are important in a marriage — in any romantic relationship, for that matter. While sexual incompatibility poses certain problems, these can be managed if partners are willing to work on their issues. If there isn't any compromise, a relationship can be severely challenged. On the other hand, great lust and sexual connection isn't enough to sustain a marriage if the emotional connection is lacking.
Q. Do you think a marriage can exist without both parties being sexually satisfied?
There are always ebbs and flows when it comes to sex. I would love to tell everyone that it's possible to have sexual experiences with a long-term partner with the same intensity and novelty as when you met, but that's just not the case. Relationships evolve and yes, sex changes. But that doesn't make it bad, just different. Sex plays different roles in the life of a relationship, but if someone is unsatisfied (and a partner is unwilling to work at changing that), then a traditional monogamous marriage may end.
Q. What are your thoughts about the fact that more couples these days are exploring non-monogamy in their marriages?
Monogamy is definitely not for everyone. Many people choose it because that's the way our society is structured, but not all do. Non-monogamy (in any relationship — not just marriage) isn't a panacea for everyone. It is highly structured and negotiated; it's not simply a legitimate "hall pass." Can it work? Yes. Is it for everyone? No.
Q. Do you have any advice for people who are anxious about discussing sex and sexuality with their partners?
Acknowledge the anxiety. We are so keen to look sexually confident and smooth that we wind up sabotaging ourselves. Tackle the discomfort head-on: "You know, I'm not used to talking about sex, but there are things that I was wondering..." (or something like that). If we don't talk about sex and sexuality, then our relationships progress with inaccurate expectations and assumptions about one another and unsurprisingly, unmet needs.
Can communication heal all sexual hang-ups? Leave your own expertise in the comments, and watch Dr. Logan Levkoff in action on Married at First Sight, premiering Tuesday, July 8 at 9p ET/10p PT on FYI Network.
This post is a sponsored collaboration between FYI Network and Studio@Gawker.


text:
Ziauddin Yousafzai on being a feminist father: I raise my children to fight for equality
Ziauddin Yousafzai, the father of Malala, stood in direct opposition of the Taliban when they banned girls' education and has rebelled against inequality all his life. Here, he talks about his journey and how we can all instigate social change.
Ziauddin Yousafzai and his children.
I come from Pakistan, a land so beautiful that for years, since moving to the UK, I have returned there in my dreams. It is a land known for its warm hospitality and rich culture. But it is also a land where men and women are not equal. Throughout my childhood and into my adolescence, I was viewed as being superior to my sisters. The women in my life, my mothers and my sisters, served me. I come from a family in which my gender made me special. This was not unusual. But I never wanted to be special for this reason. As a child, I accepted this status unchallenged, but as I grew up, I gained the confidence to question it, challenge it and finally got myself free of it.
By the time Malala was born I was a feminist in the way I lived my life, but I did not have the the terminology to describe this change. It is a valuable term that I only came to know of after living in the UK. For more than forty years, I had no idea what the word 'feminist' meant. When it was explained to me, I said, 'Oh, I have been a feminist for most of my life, almost from the beginning!'. I was living that word instinctively, but not hearing it.
My wife and I have always wanted to raise our children to be the kind of people who will continue to fight for equality for their own generation and for generations to follow.
When I married Malala's mother, Toor Pekai, and then when Malala arrived, I felt so privileged to be sharing my life with these precious females. The love I felt for them was enough to move mountains. I tell people now that when I applied the basic principle of gender equality to my own family, it changed my life. It changed my wife's life. It changed my daughter's life and it changed my sons' lives.
Even though Malala was born into a patriarchal society, I was not worried that it would limit her in this world. As I looked at her lying in her cradle, I believed that she could do anything. I had faith in her and that was enough. I needed faith in my own position as her father too, and pledged to her and myself that as long as I was beside her, supporting her, believing in her, nothing could stand in her way. I look back and I see myself resolved and determined that these social norms we lived with in Pakistan, traditions with deep winding roots across hundreds of years of patriarchy, unconscious, unspoken, natural, would not cut her down. I was ready to be Malala's shield whenever and wherever she needed me.
Whenever anybody asks me how Malala became who she is, I have often used the response 'Ask me not what I did but what I did not do. I did not clip her wings.' In Pakistan, I occasionally came across families who kept a bird, perhaps a dove, in their courtyard, and this bird was no longer able to fly. It would waddle around the dusty floor, lifting its head and moving it from one side to the other, but the vital ingredient in its life was gone. Somebody, no doubt a father or a brother, had taken some scissors to its primary feathers and clipped them so short that flight was no longer possible. It was an act of ownership over that poor defenceless creature, for entertainment or the desire to have an obedient pet, forced to live against its primal instinct of flight. When I say for Malala that 'I did not clip her wings' what I mean is that when she was small, I broke the scissors used by society to clip girls' wings. I did not let those scissors near Malala. I wanted her to fly high in the sky, not scratch around in a dusty courtyard, grounded by social norms, and I would stand by her, protecting her, until she had the confidence and strength to fly high herself, no longer in need of protection.
I tell people now that when I applied the basic principle of gender equality to my own family, it changed my life.
For girls to grow up feeling truly equal, I say the same thing to all fathers, brothers, men and boys: 'The world will not come to you, inside your home, and introduce your daughter or your sister to you, when she is still a child, as the next great woman, the next great scientist or politician, the next Malala. It is up to you as a father or a brother, and a mother, to be the first to accept and encourage potential in the girl you love. It is up to you to recognise her and to believe in her, that she can grow up to be these things. If you do not say that your children are the best and can work towards their dreams, then who will? Some fathers feel embarrassed or ashamed to believe in the empowerment of women. There are many men who are prepared to think differently about their daughters' futures, but some men encourage their daughters' freedom without owning the change in themselves. These fathers believe in equality, and yet they do not speak about it. But it needs to be shouted about because misogyny is still everywhere. Sometimes it is in jokes, sometimes it is in subtle casual comments but it all comes from the same place: a place where women are not seen as equal.
The Yousafzai family
My attempt at being a different kind of father was not confined only to raising my daughter Malala. After her birth, we had two boys, Khushal, now 18 years old, and Atal now 14. I believe that boys and men everywhere should understand the ethos of equality; that gender alone does not bring privilege. In the case of patriarchal societies, where there is an ingrained cultural bias towards men, there is an even greater urgency for men to contribute to the fight for equality. This is how I wanted to raise my sons. When boys are aware of what girls face, when they are taught from infancy that it is their responsibility as they grow into young men to take measures to make the lives of girls and women easier, it is not patronising to girls and women, but rather I feel it offers much-needed support.
When I became a father to the boys, I defined my family – to myself and to the society in which we lived – as one that believed first and foremost in equality. We did not write on the walls of our house 'All women and men are equal, all have freedom of speech,' but families are like small institutions and our lives together echoed those values of equality. I was determined to raise a family in which all three children saw no preference between genders.
All around my boys were other boys and men who were schooled in the old ways, but I knew that if my sons saw me treating their mother and sister with respect and love, they would think that that way was the normal way. If they saw their mother's voice was as valid as mine, and that there were no limits on their sister's future, then surely this would set them on a different road themselves.
Let Her Fly
Ziauddin Yousafzai
Louise Carpenter
In this intimate and extraordinary memoir, Ziauddin Yousafzai, the father of Malala, gives a moving account of fatherhood and his lifelong fight for equality – proving there are many faces of feminism.
"Whenever anybody has asked me how Malala became who she is, I have often used the phrase. 'Ask me not what I did but what I did not do. I did not clip her wings'"
For over twenty years, Ziauddin Yousafzai has been fighting for equality – first for Malala, his daughter – and then for all girls throughout the world living in patriarchal societies. Taught as a young boy in Pakistan to believe that he was inherently better than his sisters, Ziauddin rebelled against inequality at a young age. And when he had a daughter himself he vowed that Malala would have an education, something usually only given to boys, and he founded a school that Malala could attend.
Then in 2012, Malala was shot for standing up to the Taliban by continuing to go to her father's school, and Ziauddin almost lost the very person for whom his fight for equality began.
Let Her Fly is Ziauddin's journey from a stammering boy growing up in a tiny village high in the mountains of Pakistan, through to being an activist for equality and the father of the youngest ever recipient of the Nobel Peace Prize, and now one of the most influential and inspiring young women on the planet.
Told through intimate portraits of each of Ziauddin's closest relationships – as a son to a traditional father; as a father to Malala and her brothers, educated and growing up in the West; as a husband to a wife finally learning to read and write; as a brother to five sisters still living in the patriarchy – Let Her Fly looks at what it means to love, to have courage and fight for what is inherently right. Personal in its detail and universal in its themes, this landmark book shows why we must all keep fighting for the rights of girls and women everywhere.
The best celebrity books of 2019
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Investing in the Griffin Institutional Access Real Estate Fund (the "Fund") involves risks, including the risk that you may receive little or no return on your investment or that you may lose part or all of your investment. The Fund is a newly formed entity with no significant operating history upon which prospective investors may evaluate the Fund's potential performance. The ability of the Fund to achieve its investment objective depends, in part, on the ability of the Advisor to allocate effectively the assets of the Fund among the various securities and investments in which the Fund invests. There can be no assurance that the actual allocations will be effective in achieving the Fund's investment objective or delivering positive returns.
The Fund's investments may be negatively affected by the broad investment environment and capital markets in which the Fund invests, including the real estate market, the debt market and/or the equity securities market. The value of the Fund's investments will increase or decrease based on changes in the prices of the investments it holds. This will cause the value of the Fund's shares to increase or decrease. The Fund is "non-diversified" under the Investment Company Act of 1940 since changes in the financial condition or market value of a single issuer may cause a greater fluctuation in the Fund's net asset value than in a "diversified" fund. The Fund is not intended to be a complete investment program.
There is no guarantee that shareholders will be able to sell all of the shares they desire in a quarterly repurchase offer. Limited liquidity is provided to shareholders only through the Fund's quarterly repurchase offers for no less than 5% of the Fund's shares outstanding at net asset value. Quarterly repurchases by the Fund of its shares typically will be funded from available cash or sales of portfolio securities. The sale of securities to fund repurchases could reduce the market price of those securities, which in turn would reduce the Fund's net asset value. Currently, no secondary market exists for the Fund's shares, and the Fund expects that no secondary market will develop.
The Fund's investment in Private Investment Funds will require it to bear a pro rata share of the vehicles' expenses, including management and performance fees. Also, once an investment is made in a Private Investment Fund, neither the Advisor nor any Sub-Advisor will be able to exercise control over investment decisions made by the Private Investment Fund.
This is a closed end interval fund. Limited liquidity is provided to shareholders only through the fund's quarterly repurchase offers for no less than 5% of the fund's shares outstanding at net asset value.
Please refer to the Prospectus and Statement of Additional Information for more complete details about this investment.


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C3–C4intermediacy in grasses:organelle enrichment and distribution, glycine decarboxylase expression, and the rise of C2photosynthesis
Khoshravesh, Roxana and Stinson, Corey S. and Stata, Matt and Busch, Florian A. and Sage, Rowan F. and Ludwig, Martha and Sage, Tammy L. (2016) C3–C4intermediacy in grasses:organelle enrichment and distribution, glycine decarboxylase expression, and the rise of C2photosynthesis. Journal of Experimental Botany, 67 (10). pp. 3065-3078. ISSN 0022-0957
Official URL: https://doi.org/10.1093/jxb/erw150
Photorespiratory glycine shuttling and decarboxylation in bundle sheath (BS) cells exhibited by C2 species is proposed to be the evolutionary bridge to C4 photosynthesis in eudicots. To evaluate this in grasses, we compare anatomy, cellular localization of glycine decarboxylase (GDC), and photosynthetic physiology of a suspected C2 grass, Homolepis aturensis, with these traits in known C2 grasses, Neurachne minor and Steinchisma hians, and C3 S. laxum that is sister to S. hians. We also use publicly available genome and RNA-sequencing data to examine the evolution of GDC subunits and enhance our understanding of the evolution of BS-specific GDC expression in C2 and C4 grasses. Our results confirm the identity of H. aturensis as a C2 species; GDC is confined predominantly to the organelle-enriched BS cells in H. aturensis and S. hians and to mestome sheath cells of N. minor. Phylogenetic analyses and data obtained from immunodetection of the P-subunit of GDC are consistent with the hypothesis that the BS dominant levels of GDC in C2 and C4 species are due to changes in expression of a single GLDP gene in M and BS cells. All BS mitochondria and peroxisomes and most chloroplasts in H. aturensis and S. hians are situated centripetally in a pattern identical to C2 eudicots. In S. laxum, which has C3-like gas exchange patterns, mitochondria and peroxisomes are positioned centripetally as they are in S. hians. This subcellular phenotype, also present in eudicots, is posited to initiate a facilitation cascade leading to C2 and C4 photosynthesis.


The 2019 model is strutting its way into the new year with a lighter and more rigid structure, an upgraded emergency braking system and all the connectivity the entire family needs. While the interior of the vehicle was designed to be more spacious, there were also some changes under the hood with the offer of all-wheel drive and the SR and Platinum models coming with the option of a Turbo Four engine.
We wanted to share some of the other highlights of the brand new 2019 Nissan Altima.
Intelligent Mobility allows for the driver to speak commands while driving or there is the option to use a touchscreen while parked. The system is compatible with either an Android or iPhone by just plugging in. Android Auto allows the driver to use Google Maps or Google Play with voice commands. For the iPhone, Apple Carplay enables all the features of the phone by simply speaking what you need to Siri or toggling through screens.
The Door to Door Navigation system is adaptable with a mobile phone and brings the maps to the vehicle screen. The features of the navigation system include real time traffic, voice recognition, Bluetooth and will even send walking directions to your phone if you need to park a distance from your destination.
Remote Access allows you to lock and unlock your car with the tap of the screen on your mobile phone or by talking commands while you're walking. The system can sync with Amazon Alexa, Google Assistant and a smartwatch too. With this feature, you will never lose your car in a parking lot or ramp again. You can use the car finder feature or access and turn on the horn and lights from your phone as well.
This vehicle is certainly not lacking USB ports. There are two regular ports along with two USB type-C plug-ins too. There is plenty of access for everyone in the vehicle to stay charged up.
The 2019 Nissan Altima is outfitted with Sirius XM satellite radio. To enhance the listening experience in the vehicle, the Bose Premium Audio System is an available option to car buyers. Once the music starts playing, the speaker lights flash and come alive. The lights continue to dance with the music. There are nine speakers strategically located throughout the front and rear of the car and the noise cancellation system adds to the listening pleasure.
Everybody in the car can stay comfortable with Nissan's Intelligent Climate Control. Each of the front seats of the car has its own temperature controls and vents. The front seats are heated and the controls have been enhanced to get warm and cozy much faster than previous models. If you forget your gloves in the middle of winter, there's no need to worry because the steering wheel is heated too.
There is plenty of space to load up and transport luggage, long or bulky materials or whatever you need to transport. The rear seats feature a 60/40 split, allowing the back seats to fold down for more storage room. The spacious trunk measures 15.4 cubic feet and can add to the amount of cargo space. Aside from moving things, the vehicle is great for long road trips too. When it's time to take a break from the open road, both the driver and passenger can take a rest with the reclining seats.
This all new model of the Altima was designed with keeping the safety of drivers and their passengers in mind. The manufacturer did this with Nissan Intelligent Mobility. This feature monitors all sides of the vehicle and can sense when there is a car getting too close. The system can enable automatic emergency braking, a rear cross-traffic alert, rear automatic braking, warnings when leaving your lane, blind spot warnings and turn on high beam assistance.
ProPILOT Assist alleviates stress on the drive, especially during busy traffic times. This feature is able to keep a safe distance from the car in front of it in heavy traffic. It also senses when traffic slows suddenly and reduces the speed or even brings the car to a complete stop, if needed. ProPILOT keeps the vehicle centered in its own lane when driving on straight roads or lightly curved roads.
Passengers are kept safe with 10 air bags throughout the vehicle. There are front and rear seat air bags as well as knee-mounted bags on the front passenger and driver seats. The rollover sensor can activate the roof-mounted air bags.
The advanced braking system senses and adjusts to the passenger load in the vehicle. It also applies maximum braking in the front and rear when danger is detected.
The backseat is equipped with Nissan's LATCH system to ensure the utmost safety when it comes to children. The system is compatible with certain designs of child safety seats.
With the introduction of the world's first VC-turbo engine, the vehicle gets approximately 25 mpg in the city and 34 mpg on the highway. This model is also the manufacturer's first all-wheel drive sedan. The features list of the 2019 Nissan Altima could go on and on.
Port City Nissan serves customers throughout the Portsmouth and greater Seacoast region and all of New Hampshire, as well as Maine and Massachusetts. We are ready to introduce the fully redesigned 2019 Nissan Altima to you. Come into Port City Nissan today to test drive this cutting edge vehicle and to explore the plethora of additional highlights and options.


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Doug Coster to Serve as PMPA President
The secret to longevity starts with having great people, and our employees are key in what makes our business go forward.
Precision Machined Products Association
PMPA announced Doug Coster as its new president for the upcoming term. Mr. Coster is the president and owner of Nolte Precise Manufacturing, a fourth-generation, family-owned contract manufacturing business located in Cincinnati, Ohio.
Mr. Coster earned a bachelor's degree in accounting from John Carroll University and his certification to be a certified public accountant in 1998. After college, Mr. Coster spent three years working as a senior auditor for Deloitte & Touche. Mr. Coster soon felt the urge to return to Nolte and carry on the legacy his ancestors started a century ago.
So, in 2001, Mr. Coster returned to Nolte to work for the company full-time as its vice president. "I knew I had a calling and passion for manufacturing that was coupled with the personal belief that the health of manufacturing is key to the health of our country," Mr. Coster says. "In my mid-20s, I felt the industry was going through changes and challenges, and I wanted to have an impact."
Prior to his return in 2001, Mr. Coster had spent many of his early years working on Nolte's shop floor. From quality, to shipping, to production and expediting, Mr. Coster spent many summers working for his father, who owned the company at the time. After serving as vice president for more than seven years, Mr. Coster became president of Nolte in 2008. In 2011, Mr. Coster and his father completed
the transition of ownership.
Nolte Precise Manufacturing was started by Mr. Coster's great-grandfather, Louis G. Nolte, in 1916. Mr. Nolte was originally a screw machine operator at Cincinnati Milling Machine. For more than 100 years, Nolte has been on the leading edge of innovation by incorporating the most efficient business practices and using the latest technology and equipment. This helps Nolte provide total cost solutions for the aerospace, automotive, pneumatic, industrial equipment, energy and defense industries.
With so many changes in markets and the economy, it's no question that maintaining a successful business over the course of a century is no easy task. However, Mr. Coster believes it's the company's employees that allow the company to thrive during times of major change.
"The secret to longevity starts with having great people, and our employees are key in what makes our business go forward," Mr. Coster says. "When my grandfather passed away at a young age, my dad took over the business in his early 20s. For that transition to be a success, the employees really had to rally around my dad and make it known that they were there for him. That remains true today. I'm blessed to have a wonderful team of people passionate about our vision and working hard to continue Nolte's success."
Another major factor in Nolte's success has been its longstanding involvement with PMPA. A member since 1943, Nolte has been quite active within the organization, serving on numerous committees and boards throughout the years. Mr. Coster's father, Bob Coster, who is a recipient of the PMPA Merit Award, recently attended his 90th PMPA meeting. Mr. Coster and his father choose to be so heavily active in PMPA because of the value that Nolte receives from its membership. As president, Mr. Coster looks forward to incorporating his financial skill set and knowledge as he works with members to learn new trends and combat new challenges within the industry.
"As PMPA president, I think it's exciting to help the organization's members adapt and thrive," Mr. Coster says. "When you walk through some of what PMPA is doing, there's a lot of effort happening now from the standpoint of technology leadership, regulatory procedures, connecting members, engaging the current workforce and developing the future workforce. I think it's proof that the association wants to help move the industry and members forward."
Mr. Coster currently lives in Cincinnati, Ohio. In his spare time, he enjoys running, hiking, reading, coaching and spending time with his family.
PMPA President Doug Coster can be contacted at: dcoster@nolteprecise.com.
Postprocessors: An Integral Part Of Machine Tools
Why Material Performance can be Different


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As we did last year, below we present a list of the Library of America series "backlist titles" (i.e., volumes published prior to 2012) selling the most copies in 2013.
There are some noteworthy movements in this year's list. Perhaps not surprisingly, the debut volume of the LOA edition of Kurt Vonnegut's collected fiction tops the list in its first year of eligibility. An end-of-the-year Ben Stiller remake of The Secret Life of Walter Mitty more than tripled sales of the LOA James Thurber edition. Likewise, a screen adaptation of On the Road motivated readers to snap up the Kerouac volume containing the novel; its sales increased by over 30% in 2013. An encore broadcast in August of the American Masters program James Baldwin: The Price of the Ticket fueled a similar increase in sales of the collection of Baldwin's essays.
Just as impressive was the 50% increase in sales for the Ulysses S. Grant volume, probably helped along by commemorations of the 150th anniversary of the Civil War. Somewhat mysterious are the unexpected 30% increases in sales for such stalwart evergreens as John Muir and John Dos Passos's U.S.A.—not that we're complaining. At #15, sales of the Dos Passos volume were virtually identical to those of Shirley Jackson and Walt Whitman, which also enjoyed modest lifts.
Following each entry are the year the volume was originally published and last year's ranking:
Kurt Vonnegut: Novels & Stories 1963–1973 [2011, —]
The Philip K. Dick Collection [three volumes, 2007–2009; ranked #1 last year]
Flannery O'Connor: Collected Works [1988; #3]
Jack Kerouac: Road Novels 1957–1960 [2007; #7]
Raymond Carver: Collected Stories [2009; #4]
H. P. Lovecraft: Tales [2005; #5] *
Ulysses S. Grant: Memoirs & Selected Letters [1990; #14] *
Alexis de Tocqueville: Democracy in America [2004; #6] *
American Earth: Environmental Writing Since Thoreau [2008; #2]
Dashiell Hammett: Complete Novels [1999; #8]
John Muir: Nature Writings [1997; #13] *
James Thurber: Writings and Drawings [1996; —] *
James Baldwin: Collected Essays [1998; —]
The Collected Plays of Tennessee Williams [two volumes, 2000; #11]
John Dos Passos: U.S.A. [1988; —]
(tie) Shirley Jackson: Novels & Stories [2010; —]
(tie) Walt Whitman: Poetry & Prose [1982; —]
* Titles indicated by a red asterisk were available this past year as e-books, which contribute a relatively small—but rapidly growing—source of sales for LOA backlist titles. For a complete list of LOA e-books currently available, visit www.loa.org/ebooks.
This past year also marks a milestone of sorts: with the publication of volume #246 (Susan Sontag: Essays of the 1960s & 70s), the LOA series now contains over a quarter of a million pages.
Stephen Dunne February 5, 2014 at 8:06 PM
Thank you so much for this annual update, It is pleasing to see certain authors' work get new readership, regardless of the catalyst.
Lovely to see Muir doing so well too.
Forthcoming from The Library of America (Fall 2014...


Yea so if you catch an AC out at sea with out a DD you may get the ftrs too if they can't land within 1 space.
LHTR fixes some issues with box rules. You can find the rules, and a discussion about them, on Larry's site.


The Portfolio Summary report provides an overview of your customer revenue and insight into customer segments.
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This graph tracks the MRR count of your accounts, depending on the reporting period. You can also have it track the total number of accounts.
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La stagione 2006-07 dei Tulsa 66ers fu la 6ª nella NBA D-League per la franchigia.

I Tulsa 66ers arrivarono quarti nella Eastern Division con un record di 21-29, non qualificandosi per i play-off.

Roster

Staff tecnico
Allenatore: Joey Meyer
Vice-allenatore: Roy Rogers
Preparatore atletico: John Joslin

Collegamenti esterni

Stagione 2006-2007 delle squadre di pallacanestro
2006-2007

redpajama_set_name: RedPajamaWikipedia


Realtek High Definition Audio is a MEDIA device. The developer of this driver was Realtek. The hardware id of this driver is HDAUDIO/FUNC_01&VEN_10EC&DEV_0272&SUBSYS_17AA3658.
You can download from the link below the driver installer file for the Realtek Realtek High Definition Audio driver. The archive contains version 6.0.1.6782 released on 2012-11-20 of the driver.
This driver was installed by many users and received an average rating of 3.8 stars out of 77839 votes.