text:
\section{Introduction}

This paper is concerned with the study of a \textit{nonlocal mean curvature flow}. More precisely, we want to study the evolution $E_t$, for time $t>0$ of an initial set $E_0$, such that the velocity of a point $x\in \partial E_t$ in the outer normal direction $\nu$ is given by the quantity $-H^s_E$, where $H^s_E$ denotes the fractional mean curvature of $E$, that is, for any $x\in \partial {E_t}$ we have
\begin{equation}\label{NMCF}
\partial_ t x\cdot \nu=-H^s_{E_t}.
\end{equation}

For a real parameter $s\in (0,1)$, we recall that the fractional mean curvature of a set $E$ at a point $x\in \partial E$ is defined as follows
\begin{equation}\label{H_s}
H^s_E(x):=\int_{\mathbb{R}^n}\frac{\chi_{\mathcal C  E}(y)-\chi_{E}(y)}{|x-y|^{n+s}}\,dy,
\end{equation}
where $\chi_A$ denotes the characteristic function of the set $A$, $\mathcal C A$ denotes the complement of $A$, and the integral above has to be understood in the principal value sense.

This evolution is the natural analogue in the nonlocal setting of the classical mean curvature flow, which has been widely studied in the last decades, see e.g. \cite{Eck,Man}.  While the classical mean curvature flow is the $L^2$-gradient flow of the usual perimeter functional, it can be proved that \eqref{NMCF} is the $L^2$-gradient flow of the \textit{fractional perimeter}, which was first introduced in \cite{CRS} on the basis of motivations coming from interfaces in physical models and probabilistic processes. In the same paper, suitable density estimates, a monotonicity formula, and some regularity results for minimizers were established. The question of the regularity for minimizers of the fractional perimeter was also addressed in several recent works, see~\cite{BFV,CSV,CV,SV}. Further motivations for the study of \eqref{NMCF} come from dislocation dynamics and phase-field theory for fractional reaction-diffusion equations, see \cite{Im}.

The classical mean curvature flow is a quasilinear parabolic problem and a local existence result holds for smooth solutions starting from any compact regular initial surface, see \cite{GH,Man}. As time evolves, solutions typically develop singularities  due to curvature blowup. Several notions of generalized solutions have been introduced to study the flow after the onset of singularities. Particularly relevant for our purposes are the definitions by Chen, Giga, Goto \cite{CGG} and by Evans and Spruck \cite{ES}, based on the level set approach and the notion of viscosity solutions.

For the fractional mean curvature flow, local smooth solutions are also expected to exist, but no proof of this property is available yet. There are existence results for viscosity solutions, first obtained by Imbert \cite{Im}, and later extended to more general nonlocal flows by Chambolle, Morini, Ponsiglione in \cite{CMP,CMP2} and by Chambolle, Novaga, Ruffini \cite{CNR}.

The analysis of the formation of singularities is an important topic in classical mean curvature flow. A pioneering result in this framework was obtained by Huisken \cite{Hui}, who showed that a closed convex surface in $\mathbb{R}^n$, with $n >2$, remains convex along the evolution and shrinks to a point in finite time. If convexity is dropped, then other kinds of singular behaviour may occur. A standard example is the so called \textit{neckpinch}. The idea is to consider a surface which looks like two large balls connected by a very thin cylindrical neck, so that, in dimension $n>2$, the mean curvature in the neck is much larger than the one in the balls, hence the radius of the neck goes to zero faster than the radius of the balls. The existence of this type of surfaces was first proved by Grayson \cite{Gr} and later considered with a simplified proof by Ecker \cite{Eck}, see also Angenent \cite{An}. A similar construction for a flow driven by a different curvature function was done in \cite{AS}.

When $n=2$, a result analogous to \cite{Hui} for convex curves was proved by Gage and Hamilton \cite{GH}. However, a stronger result holds in this dimension in the classical case. In fact, Grayson \cite{Gr87} showed that {\em any smooth closed embedded curve in the plane becomes convex in finite time under the flow}, and therefore, by \cite{GH}, {\em shrinks smoothly to a point}. Thus, all other kinds of singularities are ruled out for embedded curves.

In this paper we construct examples of neckpinch singularities for the fractional mean curvature flow. More precisely, we obtain the
following result:

\begin{thm}\label{thm-neckpinch}
Let $n\geq 2$. There exists an embedded hypersurface $\mathcal M_0$ in $\mathbb{R}^n$ such that the viscosity solution of the fractional mean curvature flow \eqref{NMCF} starting from $\mathcal M_0$ does not shrink to a point.
\end{thm}

We point out that,
in contrast to the classical case, our construction can be made in any dimension, in particular for $n=2$, showing that Grayson's theorem fails in the nonlocal case. It also shows that the distance comparison property for curves for the classical flow proved by Huisken \cite{Hui3} no longer holds in the nonlocal
case. Our result has some analogies with the one of \cite{CFSW}, where Delaunay-type periodic curves in the plane with constant fractional mean curvature are constructed, while in the classical case such objects exist only in dimension $n \geq 3$.

Our results deals with the viscosity solution of \eqref{NMCF} because we lack a local existence result for smooth solutions. However, Theorem \ref{thm-neckpinch} also implies that, if the hypersurface we construct has a local smooth evolution, then it \textit{develops singularities} before shrinking to a point.

A crucial step in the proof of Theorem \ref{thm-neckpinch} is provided by the following result, of independent interest: if a set $E$ is contained in a strip and its boundary $\partial E$ has sufficiently small slope and small classical curvatures, then the fractional mean curvature of $E$ is bounded below by a positive universal constant, which depends only on $s$ and on the dimension $n$. This is the content of Proposition \ref{STIMA:CUR:FLAT}. In particular, a thin set can have all negative classical principal curvatures at some point, but positive fractional mean curvature.

Other recent contributions in the study of the fractional (and more general nonlocal) mean curvature flows are the articles~\cite{CSou, CNR, SaezV}.
 In \cite{CSou}, the convergence of a class of threshold dynamics approximations to moving fronts was established. In particular, threshold dynamics associated to fractional powers of the Laplacian of order $s\in (0,2)$ were considered: interestingly, when $s\in [1,2)$ the resulting interface moves by a (weighted) mean curvature flows, while when $s\in (0,1)$ it moves by a fractional mean curvature flow.
In \cite{CNR}, the results contained in \cite{CSou} have been extended to the anisotropic case, and it was proved that convexity is preserved as in the classical case. Finally, in \cite{SaezV} smooth solutions to the fractional mean curvature flow were studied and the evolution equations for several geometric local and nonlocal quantities were computed. The particular cases of entire graphs and star-shaped surfaces were considered, obtaining striking analogies with the properties of the classical case.

The paper is organized as follows:

\begin{itemize}
\item In Section 2 we describe the level set approach in the study of nonlocal mean curvature flows, we recall the notion of viscosity solutions and the statement of the comparison principle;
\item In Section 3 we establish the estimates on the nonlocal mean curvature of perturbed strips, which will be used in the proof of our main result;
\item In Section 4, we prove our main result Theorem \ref{thm-neckpinch}.
\end{itemize}

\section{Viscosity solutions and comparison principles via the level set approach}

In this Section we recall the notion of viscosity solutions for the fractional mean curvature flow, which is based on the
\textit{level sets approach}.
The idea is the following: given an initial surface $\mathcal M_0=\partial E_0$, we choose any continuous function $u_0:\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that
\begin{equation}\label{level-set}\mathcal M_0=\{x\in \mathbb{R}^n\,:\,u_0(x)=0\}.\end{equation}
The geometric equation satisfied by the evolution $\mathcal M_t$ of $\mathcal M_0$ can then be translated into an equation satisfied by a function $u(x,t)$, where $u(x,0)=u_0(x)$ and at each time
\begin{equation}\label{M_t}\mathcal M_t=\{x\in \mathbb{R}^n\,:\,u(\cdot, t)=0\}.\end{equation}
More precisely, the \textit{level set equation} satisfied by $u$ is
\begin{equation}\label{level-set-eq}
\partial_t u+H^s[x,u(\cdot,t)]|Du(x,t)|=0\quad \mbox{in}\;\;\mathbb{R}^n\times (0,+\infty),
\end{equation}
where $u$ satisfies the initial condition
$$u(x,t)=u_0(x)\quad \mbox{in}\;\;\mathbb{R}^n.$$
Here and in the following we denote by $H_s[x,u(\cdot,t)]$ the fractional mean curvature of the superlevel set of $u(\cdot, t)$ at the point $x$, i.e.
$$H^s[x,u(\cdot,t)]=H^s_{\{y\in \mathbb{R}^n\,:\,u(y,t)>u(x,t)\}}(x).$$
Of course the definition of $\mathcal M_t$ is well posed if one shows that equation \eqref{level-set-eq} has a unique solution, and definition \eqref{level-set} does not depend on the initial choice of the function $u_0$. These basic properties were established in \cite{ES} for the classical mean curvature flow and in \cite{Im} for the fractional one.

Since the nonlocal case is not so standard and for the sake of completeness, we recall here below all the rigorous definitions and the basic results in \cite{Im} (see also \cite{CMP}).
Let $\mathcal M=\{x\in \mathbb{R}^n\,:\,u(x)=0\}=\partial\{x\in \mathbb{R}^n\,:\,u(x)>0\}$. If $u\in C^{1,1}$ and $Du\neq 0$, we can define the following quantities
\begin{equation}\label{k*}
\begin{split}
k^*[x,\mathcal M]&=k^*[x,u]=\int_{\mathbb{R}^n}\frac{\chi_{\{u(x+z)\geq u(x)\}}(z)  -\chi_{\{u(x+z)<u(x)\}}(z)}{|z|^{n+s}}\,dz,
\\
k_*[x,\mathcal M]&=k_*[x,u]= \int_{\mathbb{R}^n}\frac{\chi_{\{u(x+z)> u(x)\}}(z)  -\chi_{\{u(x+z) \leq u(x)\}}(z)}{|z|^{n+s}}\,dz.
\end{split}
\end{equation}
It is easy to see that if $u\in C^{1,1}$ and its gradient $Du$ does not vanish on $\{z\in \mathbb{R}^n\,:\,u(z)=u(x)\}$, then $k^*$ are finite and
$$k^*[x,u]=k_*[x,u]=-H_s[x,u].$$

We can now give the definition of \textit{viscosity solution} for \eqref{level-set-eq} (see \cite{Im}, Sec. 3).

\begin{definition}
\begin{itemize}
\item[i)] An upper semicontinuous function $u:[0,T]\times \mathbb{R}^n$ is a \textit{viscosity subsolution} of \eqref{level-set-eq} if for every smooth test function $\phi$ such that $u-\phi$ admits a global zero maximum at $(t,x)$, we have
\begin{equation}\label{sub}
\partial_t \phi\leq k^*[x,\phi(\cdot,t)]|D\phi|(x,t)
\end{equation}
if $D\phi(x,t)\neq 0$, and $\partial_t\phi(x,t)\leq 0$ if not.
\item [ii)] A lower semicontinuous function $u:[0,T]\times \mathbb{R}^n$ is a \textit{viscosity supersolution} of \eqref{level-set-eq} if for every smooth test function $\phi$ such that $u-\phi$ admits a global zero minimum at $(t,x)$, we have
\begin{equation}\label{super}
\partial_t \phi\geq k_*[x,\phi(\cdot,t)]|D\phi|(x,t)
\end{equation}
if $D\phi(x,t)\neq 0$, and $\partial_t\phi(x,t)\geq 0$ if not.
\item[iii)] A locally bounded function $u$ is a \textit{viscosity solution} of \eqref{level-set-eq} if its upper semicontinuous envelope is a subsolution and its lower semicontinuous envelope is a supersolution of \eqref{level-set-eq}.
\end{itemize}
\end{definition}

\begin{rmk}{\rm
It is easy to verify that any classical subsolution (respectively supersolution) is in particular a viscosity subsolution (respectively supersolution).
}\end{rmk}

We can now state the comparison principles, that we will use later on in Section \ref{neckpinch}.

\begin{prop}[Theorem 2 in \cite{Im}]\label{comparison}
Suppose that the initial datum $u_0$ is a bounded and Lipschitz continuous function. Let $u$ (respectively $v$) be a bounded viscosity subsolution (respectively supersolution) of \eqref{level-set-eq}.

If $u(x,0)\leq u_0(x)\leq v(x,0)$, then $u\leq v$ on $\mathbb{R}^n\times (0,+\infty)$.
\end{prop}

In Theorem 3 of \cite{Im}  existence and uniqueness of viscosity solutions of \eqref{level-set-eq} were proven. The prove of existence uses
Perron's method, while uniqueness relies on the comparison principle stated in Proposition \ref{comparison}.
Finally in \cite{Im}, Theorem 6, the consistency of Definition \eqref{M_t} is established, showing that if $u$ and $v$ are two viscosity solutions of \eqref{level-set-eq} with two different initial data $u_0$ and $v_0$ which have the same zero-level set, then for every time $t>0$ also $u(\cdot, t)$ and $v(\cdot, t)$ have the same zero level set.

The uniqueness and the consistency results allow to define the fractional mean curvature flow for $\mathcal M_t$ by using the solution $u(\cdot,t)$ of \eqref{level-set-eq}.

\section{The nonlocal curvature of perturbed strips}

The goal of this section is to give estimates
on the nonlocal curvature of sets with small slopes
and small classical curvatures which are contained in a strip.
The result is rather general, but we focus on
a particular case for the sake of concreteness:

\begin{prop}\label{STIMA:CUR:FLAT}
Let~$\kappa$, $\eta>0$.
Let~$E_-$, $E_+\subset\mathbb{R}^n$. Assume that~$E_+\cap E_-=\varnothing$, that
\begin{equation}\label{CONTE}
E_- \supseteq \{ x_n \le -1 \} \quad {\mbox{and }}\quad
E_+ \supseteq \{ x_n \ge 1 \}.
\end{equation}
Suppose also that the boundaries of~$E_-$ and~$E_+$
are of class~$C^2$, with classical directional curvatures
bounded by~$\kappa$.

In addition, let~$\nu_-$ and~$\nu_+$ be the exterior
normals of~$E_-$ and~$E_+$ respectively, and
assume that
\begin{equation}\label{CONTE:0}
|\nu_-\cdot e_i|\le \eta \quad{\mbox{ and }}\quad
|\nu_+\cdot e_i|\le \eta \quad{\mbox{ for every }} i\in\{1,\dots,n-1\}.
\end{equation}
Let~$E:= \mathbb{R}^n\setminus (E_-\cup E_+)$. Then,
there exist~$c_0$, $\kappa_0$, $\eta_0>0$, depending on~$n$ and~$s$,
such that for any~$x\in\partial E$
$$ H^s_E (x) \ge c_0,$$
provided that~$\kappa\in [0,\,\kappa_0]$ and~$\eta\in[0,\,\eta_0]$.
\end{prop}

Before giving the proof of Proposition \ref{STIMA:CUR:FLAT}, we state the following Lemma that will be useful later.
Basically, it states that a set, whose boundary has small enough slopes and classical curvatures bounded by $\kappa$, can be trapped between paraboloids of opening a multiple of $\kappa$.

\begin{lem}\label{SP:SACL}
Let~$\kappa$, $\eta>0$ and~$G\subset\mathbb{R}^n$, with
boundary of class~$C^2$, with classical directional curvatures
bounded by~$\kappa$ and exterior
normal~$\nu$ such that~$|\nu\cdot e_i|\le \eta$ for any~$i\in\{1,\dots,n-1\}$.

Suppose also that~$0\in\partial G$.
Then, fixed~$\rho>0$, there exist~$\eta_0>0$,
depending on~$n$, and~$\kappa_0(\rho)$, depending on~$n$ and~$\rho$,
such that, if~$\kappa\in[0,\,\kappa_0(\rho)]$ and~$
\eta\in[0,\,\eta_0]$, then
$$ {\mathcal{R}}({{G}}\cap B_\rho\big)\subseteq
\{ x_n\le C\kappa\,|x'|^2\},$$
for a suitable rotation~${\mathcal{R}}$, which differs from the identity
less than~$C\eta$, for a suitable~$C>0$, depending on~$n$.
\end{lem}

\begin{proof} We notice that the result is true for some~$\rho_0$
universal, due to the Implicit Function Theorem,
and so for all~$\rho\in(0,\rho_0]$.

Let now~$\rho>\rho_0$. Let
$$ G_\rho := \frac{\rho_0}{\rho}\,G=
\{ y\in\mathbb{R}^n {\mbox{ s.t. }} \rho \,\rho_0^{-1}\,y\in G\}.$$
Notice that the slope of the normal of~$G_\rho$ is the same as the one of~$G$,
and so it is bounded by~$\eta$. On the other hand,
the curvatures of~$G_\rho$ are~$\rho\,\rho_0^{-1}$ times the curvatures of~$G$,
and therefore are bounded by~$\rho\,\rho_0^{-1}\,\kappa$.

Accordingly, if~$\eta\in [0,\,\eta_0]$ and~$\rho\,\rho_0^{-1}\,\kappa\in
[0,\,\kappa_0(\rho)]$ (i.e. if~$\kappa\in[0,\,\rho_0\,\rho^{-1}\,\kappa_0(\rho_0)]$),
we have that the claim holds true for~$G_\rho$ in~$B_{\rho_0}$, namely
$$ {\mathcal{R}}({{G}}_\rho\cap B_{\rho_0}\big)\subseteq
\{ y_n\le C\,\rho\,\rho_0^{-1}\,\kappa\,|y'|^2\},$$
where ${\mathcal{R}}$ is a rotation as in the statement of
Lemma~\ref{SP:SACL}.

Using the change of scale~$x:=\rho\,\rho_0^{-1}\,y$,
this gives that
$$ {\mathcal{R}}({{G}}\cap B_{\rho}\big)=\frac{\rho}{\rho_0}\,
{\mathcal{R}}({{G}}_\rho\cap B_{\rho_0}\big)
\subseteq\frac{\rho}{\rho_0}\,
\{ y_n\le C\,\rho\,\rho_0^{-1}\,\kappa\,|y'|^2\}=\{ x_n\le C\kappa\,|x'|^2\},
$$
as desired.
\end{proof}

\begin{proof}[Proof of Proposition \ref{STIMA:CUR:FLAT}] We fix~$x\in\partial E$. Without loss of
generality, we may suppose that~$x\in\partial E_+$.
We use the notation~$\nu:=\nu_+(x)$.
Also, we set~$\chi:=\chi_E-\chi_{\mathcal C E}$.
For short, we also denote by~$C$ a positive constant,
depending on~$n$ and~$s$, which we take the liberty of modifying
from line to line.

We consider
the two slabs
\begin{eqnarray*}
&& S := \{ p\in\mathbb{R}^n {\mbox{ s.t. }} |(p-x)\cdot\nu|\le 4\}
\\{\mbox{and }} && S_\star
:= \{ p\in\mathbb{R}^n {\mbox{ s.t. }} |p\cdot e_n|\le 1\}.
\end{eqnarray*}
Notice that, by~\eqref{CONTE},
\begin{equation}\label{CONTE:2}
E\subseteq S_\star.
\end{equation}
We also observe that
\begin{equation}\label{CONTE:3}
S_\star\setminus S\subseteq \mathbb{R}^n\setminus B_{C/\eta}(x).
\end{equation}
Indeed, if~$p\in S_\star\setminus S$,
we have that~$|(p-x)\cdot\nu|> 4$
and~$|p\cdot e_n|\le 1$. Thus, we recall~\eqref{CONTE:0}
and~\eqref{CONTE:2},
we write~$\nu=(\nu\cdot e_1)e_1+\dots+(\nu\cdot e_n)e_n$
and we find that
\begin{eqnarray*}
4 &<& \left| (p-x)\cdot \sum_{i=1}^n (\nu\cdot e_i)e_i\right|\\
&\le& \eta\,\sum_{i=1}^{n-1} |(p-x)\cdot e_i|
+|(p-x)\cdot e_n| \\
&\le& \eta\,(n-1)\,|p-x| + |p\cdot e_n|+|x\cdot e_n| \\
&\le& \eta\,(n-1)\,|p-x| + 1+1,
\end{eqnarray*}
which implies~\eqref{CONTE:3} as long as~$\eta$ is sufficiently small.

Then, in view of~\eqref{CONTE:2}
and~\eqref{CONTE:3}, we have that
$$ E\setminus S\subseteq
S_\star\setminus S\subseteq \mathbb{R}^n\setminus B_{C/\eta}(x)$$
and therefore
\begin{equation}\label{CIxS:00:1}
\int_{E\setminus S} \frac{dy}{|x-y|^{n+s}}\le
\int_{ \mathbb{R}^n\setminus B_{C/\eta}(x) } \frac{dy}{|x-y|^{n+s}}
\le C\eta^s.
\end{equation}
Now we set
$$ G:= \{ p\in B_{10}(x) {\mbox{ s.t. }} (p-x)\cdot \nu > 4 \}.$$
Notice that~$G$ is a circular sector, so, in particular, we
have that
\begin{equation}\label{uj4}
|G|\ge C.\end{equation}
Also, by construction,
\begin{equation}\label{uj5}
G\cap S=\varnothing.
\end{equation}
We claim that
\begin{equation}\label{uj6}
G\subseteq \mathbb{R}^n\setminus E.
\end{equation}
To check this, let~$p\in G$. Then, by~\eqref{CONTE:0} and~\eqref{CONTE:2},
\begin{eqnarray*}
4 &<& \sum_{i=1}^n (\nu\cdot e_i)(p-x)\cdot e_i\\
&\le& \eta\,(n-1)\,|p-x|+(p-x)\cdot e_n \\
&\le& 10\eta\,(n-1) + p\cdot e_n +1,
\end{eqnarray*}
and so, if~$\eta$ is sufficiently small, we deduce that~$p\cdot e_n
\ge 2$. Hence, we have that~$p\in E_+$, which in turn implies~\eqref{uj6}.

Now, from~\eqref{uj5}
and~\eqref{uj6}, we get that
$$ (\mathbb{R}^n\setminus E)\setminus S\;\supseteq\;
G\setminus S\;=\; G.$$
Notice also that if~$y\in G$ then~$|x-y|\le 10$. As a consequence
of these observations and~\eqref{uj4},
we infer that
\begin{equation*}
\int_{(\mathbb{R}^n\setminus E)\setminus S} \frac{dy}{|x-y|^{n+s}}\ge
\int_{G} \frac{dy}{|x-y|^{n+s}}\ge \frac{|G|}{10^{n+s}}\ge C.
\end{equation*}
Combining this with~\eqref{CIxS:00:1}, we conclude that
\begin{equation}\label{CIxS:00:2}
\int_{\mathbb{R}^n\setminus S} \frac{\chi(y)\,dy}{|x-y|^{n+s}}=\int_{\mathbb{R}^n\setminus S} \frac{\chi_E(y)-\chi_{\mathcal C E}(y)}{|x-y|^{n+s}}\,dy\le C\eta^s - C_\star.
\end{equation}
Here we denote by~$C_\star>0$ a ``special'' constant
(to distinguish it from the ``other'' constants
just denoted by~$C$): indeed, $C_\star$ will produce
the desired result after small perturbations, as we will see in the sequel.

Now we analyze the contributions inside~$S$. For this purpose,
we take an additional parameter
\begin{equation}\label{KA:INT}
\rho\in \left(0,\; \,\min\left\{|\log\kappa|,\,\frac1\eta\right\}\, \right).
\end{equation}
Such~$\rho$ will be taken appropriately large (in dependence of~$C_\star$):
then~$\kappa$ and~$\eta$
have to be chosen sufficiently small, in such a way
that~$\rho$ belongs to the interval stated in~\eqref{KA:INT}. That is, the parameter~$\rho$
is taken large with respect to the constants, then~$\kappa$ and~$\eta$
will be taken to be small in dependence on the constants
and on~$\rho$.

Thanks to the curvature assumption on~$\partial E_+$,
we know that the boundary of~$E_+$ inside~$B_\rho(x)$
is trapped between paraboloids of opening~$C\kappa$:
more precisely, by Lemma \ref{SP:SACL} there exists a rigid motion~${\mathcal{R}}$
with~${\mathcal{R}}(x)=0$ and such that
\begin{equation}\label{9ugAHHHfA}
\begin{split}&
{\mathcal{R}} \big( E_+\cap B_\rho(x)\big) \supseteq
\left\{ p=(p',p_n)\in B_\rho {\mbox{ s.t. }} p_n \ge C\kappa\,|p'|^2
\right\}=:{\mathcal{P}}_1,\\ &
{\mathcal{R}} \big( (\partial E_+)\cap B_\rho(x)\big) \subseteq
\left\{ p=(p',p_n)\in B_\rho {\mbox{ s.t. }} |p_n| \le {C\kappa\,|p'|^2}
\right\}=:{\mathcal{P}}_2\\
{\mbox{and }}\quad& {\mathcal{R}} \big( E_+\cap B_\rho(x)\big)\subseteq
{\mathcal{P}}_1\cup{\mathcal{P}}_2.
\end{split}\end{equation}
Notice that~${\mathcal{R}}$ is the composition of a translation
that sends~$x$ to~$0$ and then a rotation which sets~$\nu$
in the vertical direction. Therefore, we can write~${\mathcal{R}}(y):=
{\mathcal{R}}_\star \,(y-x)$, where~${\mathcal{R}}_\star$ is a rotation
which differs from the identity less than~$C\eta$, due to~\eqref{CONTE:0}.

\begin{figure}[h]
	\centering

	\resizebox{10cm}{!}{

		\begin{tikzpicture}[scale=1]

			\node (myfirstpic) at (0,0) {\includegraphics[scale=0.6]{insiemini.pdf}};

		\end{tikzpicture}

	}
	\caption{}
\end{figure}

We set
\begin{eqnarray*}&& F:={\mathcal{R}} \big( E\cap B_\rho(x)\big),\\
&& {\mathcal{H}}:=
\left\{ p=(p',p_n)\in B_\rho {\mbox{ s.t. }} p_n <-4
\right\},
\\ && {\mathcal{K}}:=
\left\{ p=(p',p_n)\in B_\rho {\mbox{ s.t. }} p_n >4
\right\}
\\{\mbox{and }}&& {\mathcal{P}}_3:=B_\rho\setminus
\big({\mathcal{P}}_1\cup{\mathcal{P}}_2\cup{\mathcal{H}}\big).
\end{eqnarray*}

By construction, we see that
\begin{equation}\label{SUR:SP}
{\mathcal{R}}(S\cap B_\rho(x))=B_\rho\setminus({\mathcal{H}}\cup{\mathcal{K}})=
\left\{ p=(p',p_n)\in B_\rho {\mbox{ s.t. }} |p_n| \le4
\right\}=:{\mathcal{S}}.
\end{equation}
We also observe that
\begin{equation} \label{FUINVL}
{\mathcal{P}}_3\cap {\mathcal{S}} =
-({\mathcal{P}}_1\cap{\mathcal{S}}),\end{equation}
up to sets of null measure.
To check this, let~$p\in {\mathcal{P}}_3\cap {\mathcal{S}}$
and~$q:=-p$. Then we have that~$|q_n|=|p_n|\le4$, thus~$q\in{\mathcal{S}}$.
Furthermore, since~$p\not\in {\mathcal{P}}_1\cup {\mathcal{P}}_2$,
we have that~$p_n<-C\kappa\,|p'|^2$. Accordingly,
$$ q_n =-p_n>C\kappa\,|p'|^2 =C\kappa\,|q'|^2,$$
and so~$ q\in {\mathcal{P}}_1$.
This shows that~$q\in{\mathcal{P}}_1\cap{\mathcal{S}}$, and so
\begin{equation}\label{IOKHAYjaA}
{\mathcal{P}}_3\cap {\mathcal{S}} \subseteq
-({\mathcal{P}}_1\cap{\mathcal{S}}).
\end{equation}
Now, let us take~$q\in ({\mathcal{P}}_1\cap{\mathcal{S}})$
and set~$p:=-q$. We have that~$|p_n|=|q_n|\le4$,
hence~$p\in {\mathcal{S}}$. Moreover,
$$ p_n =-q_n\le -C\kappa\,|q'|^2 =-C\kappa\,|p'|^2.$$
Since we can remove the case in which~$p_n=-C\kappa\,|p'|^2$
(being this a set of null measure), we can conclude that~$p_n<
-C\kappa\,|p'|^2$ and so~$p\not\in{\mathcal{P}}_1\cup{\mathcal{P}}_2$.
This says that
$$ p\in {\mathcal{S}}\setminus\big( {\mathcal{P}}_1\cup{\mathcal{P}}_2\big)
\subseteq \big( B_\rho\setminus{\mathcal{H}}\big)\setminus
\big( {\mathcal{P}}_1\cup{\mathcal{P}}_2\big)
\subseteq B_\rho\setminus
\big( {\mathcal{P}}_1\cup{\mathcal{P}}_2\cup{\mathcal{H}}\big)
={\mathcal{P}}_3.$$
Hence~$p\in{\mathcal{P}}_3\cap{\mathcal{S}}$, up to negligible sets.
Thus, we have shown that
$$ -({\mathcal{P}}_1\cap{\mathcal{S}})\subseteq
{\mathcal{P}}_3\cap {\mathcal{S}} ,$$
up to negligible sets. Combining this with~\eqref{IOKHAYjaA},
we conclude the proof of~\eqref{FUINVL}.

Also, we notice that
\begin{equation}\label{SUR}
{\mathcal{H}}\cap \big({\mathcal{P}}_1\cup{\mathcal{P}}_2\big)=\varnothing
\quad{\mbox{ and }}\quad
{\mathcal{K}}\subseteq {\mathcal{P}}_1.
\end{equation}
Indeed, if~$p\in
\big({\mathcal{P}}_1\cup {\mathcal{P}}_2\big)$, then
$$ p_n \ge -{C\kappa\,|p'|^2} \ge -C\kappa\rho^2\ge -1$$
if~$\kappa$ sufficiently small (possibly in dependence of~$\rho$),
hence~$p\not\in{\mathcal{H}}$.
Similarly, if~$p\in {\mathcal{K}}$ then
$$ p_n -C\kappa\,|p'|^2\ge 4-C\kappa\rho^2>0$$
and so~$p\in{\mathcal{P}}_1$. These observations prove~\eqref{SUR}.

We also point out that
\begin{equation} \label{IJ5678-UAYT}
F =
{\mathcal{R}} \big( E\cap B_\rho(x)\big)
\subseteq
{\mathcal{R}} \big( (\mathbb{R}^n\setminus E_+)\cap B_\rho(x)\big)
= \mathbb{R}^n\setminus\Big(
{\mathcal{R}} \big( E_+\cap B_\rho(x)\big) \Big)
\subseteq\mathbb{R}^n\setminus {\mathcal{P}}_1,\end{equation}
thanks to~\eqref{9ugAHHHfA}, and therefore
\begin{equation}\label{ikAIII:A}
B_\rho\setminus F\supseteq {\mathcal{P}}_1.\end{equation}
In addition, we claim that
\begin{equation}\label{IND:FF}
F\subseteq {\mathcal{P}}_2\cup {\mathcal{P}}_3.
\end{equation}
Indeed, if~$p\in F$, then~$p={\mathcal{R}}(y)=
{\mathcal{R}}_\star \,(y-x)$ for some~$y\in E$. Therefore,
if~$I_n\in{\rm Mat}(n\times n)$ denotes the identity matrix,
recalling~\eqref{CONTE:2}
we have that
\begin{eqnarray*}
p_n &=& \big({\mathcal{R}}_\star\,(y-x)\big)\cdot e_n
\\ &\ge& (y-x)\cdot e_n -
\Big| \big(({\mathcal{R}}_\star-I_n)\,(y-x)\big)\cdot e_n\Big| \\
&\ge& -2 -\eta\rho
\\ &>& -4,
\end{eqnarray*}
provided that~$\eta$ is sufficiently small (possibly in dependence of~$\rho$).
This shows that~$p\not\in{\mathcal{H}}$.
As a consequence, we see that
\begin{equation*}
p\in{\mathcal{P}}_1\cup{\mathcal{P}}_2\cup{\mathcal{P}}_3.\end{equation*}
This and~\eqref{IJ5678-UAYT}
establish the validity of~\eqref{IND:FF}.

Now, we use the change of variable~$z={\mathcal{R}}(y)$ and formulas
\eqref{SUR:SP}, \eqref{ikAIII:A} and~\eqref{IND:FF}
to see that
\begin{equation}\label{2IaAJLLAK}
\begin{split}
\int_{S\cap B_\rho(x)} \frac{\chi(y)\,dy}{|x-y|^{n+s}}
\,&=\int_{ {\mathcal{S}} } \frac{\chi_F(z)-\chi_{B_\rho\setminus F}(z)\,dz}{
|z|^{n+s}} \\
&\le
\int_{ {\mathcal{P}}_2 } \frac{dz}{|z|^{n+s}}
+\int_{ {\mathcal{P}}_3\cap{\mathcal{S}} } \frac{dz}{|z|^{n+s}}
-\int_{ {\mathcal{P}}_1\cap{\mathcal{S}} } \frac{dz}{|z|^{n+s}}.
\end{split}
\end{equation}
Moreover, from~\eqref{FUINVL} and a reflection in the vertical variable, we
have that
$$ \int_{ {\mathcal{P}}_3\cap{\mathcal{S}} } \frac{dz}{|z|^{n+s}}=
\int_{ {\mathcal{P}}_1\cap{\mathcal{S}} } \frac{dz}{|z|^{n+s}}. $$
Accordingly, \eqref{2IaAJLLAK} becomes
\begin{equation}\begin{split}
\label{98uhAaasdGH}
& \int_{S\cap B_\rho(x)} \frac{\chi(y)\,dy}{|x-y|^{n+s}}
\le\int_{ {\mathcal{P}}_2 } \frac{dz}{|z|^{n+s}}
\\ &\qquad\le \int_{ |z'|\le \rho }\,dz'\;
\int_{|z_n|\le C\kappa\,|z'|^2} \,dz_n\; |z'|^{-n-s}
=C\kappa\,\int_{ |z'|\le \rho }\,dz' \;|z'|^{2-n-s}
\\ &\qquad=C\kappa \rho^{2-s}.
\end{split}\end{equation}
On the other hand, we have that
\begin{equation*}
\int_{S\setminus B_\rho(x)} \frac{\chi(y)\,dy}{|x-y|^{n+s}}\le
\int_{\mathbb{R}^n\setminus B_\rho(x)} \frac{dy}{|x-y|^{n+s}}
= \frac{C}{\rho^s} .\end{equation*}
This and~\eqref{98uhAaasdGH} yield
$$ \int_{S} \frac{\chi(y)\,dy}{|x-y|^{n+s}}\le C\,\left(
\kappa \rho^{2-s}+\frac{1}{\rho^s}
\right).$$
Combining this with~\eqref{CIxS:00:2}, we see that
$$ -H^s_E (x)=\int_{\mathbb{R}^n\setminus S} \frac{\chi(y)\,dy}{|x-y|^{n+s}}\le
C\,\left(\eta^s +\kappa \rho^{2-s}+\frac{1}{\rho^s}
\right) - C_\star \le
C\,\left(\eta^s +\kappa \rho^{2-s}
\right) - \frac{C_\star}{2},$$
provided that~$\rho$ is sufficiently large.

Hence, if~$\eta$ and~$\kappa$ are small enough,
we obtain that~$-H^s_E (x)\le -C_\star/4$,
as desired.
\end{proof}

\begin{cor}\label{arctan}
Let~$\epsilon,\,\delta>0$ and
$$ E:= \left\{ x=(x',x_n)\in\mathbb{R}^n {\mbox{ s.t. }}
|x_n|< \epsilon + \frac{2}{\pi}\,\arctan \big(\delta \,|x'|^2\big)
\right\} . $$
Then, if~$\epsilon$ and $\delta$ are sufficiently small, we have that
$$ \inf_{x\in \partial E} H^s_E(x)\geq c_0 > 0, $$
for some $c_0$ depending only on $s$ and $n$.

\end{cor}

\begin{proof} We define
\begin{eqnarray*}&&
E_-:= \left\{ x=(x',x_n)\in\mathbb{R}^n {\mbox{ s.t. }}
x_n\le -\epsilon - \frac{2}{\pi}\,\arctan \big(\delta \,|x'|^2\big)
\right\}\\ {\mbox{and }}&&
E_+:= \left\{ x=(x',x_n)\in\mathbb{R}^n {\mbox{ s.t. }}
x_n\ge \epsilon + \frac{2}{\pi}\,\arctan \big(\delta \,|x'|^2\big)
\right\}.
\end{eqnarray*}
Notice that the horizontal component of the normal to~$\partial E$
is of size~$O(\delta)$ and the curvatures
are of size~$O(\delta^2)$. Thus, we are in the position of
exploiting Proposition~\ref{STIMA:CUR:FLAT}, from which we
obtain the desired result.
\end{proof}

\section{Neckpinch}\label{neckpinch}

In this section we prove our main result, that is Theorem \ref{thm-neckpinch}. More precisely, we provide an example of surface evolving by fractional mean curvature flow, which develops a singularity before it can shrink to a point. For the classical mean curvature flow, this phenomenon appears in dimensions $n\geq 3$ (that is, for at least $2$-dimensional surfaces) in the neckpinch singularity described in the introduction. In this example, the fact that the dimension of the surface is larger than 2 is crucial to have positive mean curvature in the neck.

Interestingly, when we consider the fractional mean curvature flow, this neckpinch singularity can be observed also in dimension $n=2$, providing hence a counterexample to the Grayson Theorem \cite{Gr87}, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The heuristic reason for this is that, thanks to its nonlocality, the fractional mean curvature of a very thin neck, is strictly positive also in dimension $2$: Indeed if we ``sit'' on the boundary of the neck we see much more complement of $E$ than $E$ itself.

To build our example, we start by recalling the following fact that was proved in \cite{SaezV}.
\begin{lem}[Lemma 2 and Corollary 3 in \cite{SaezV}]\label{ball}
The fractional mean curvature of the ball of radius $R$ is equal, up to dimensional constants, to $R^{-s}$.

More precisely, for any $x\in \partial B_1$
$$H_{B_1}^s(x)=\bar \omega,$$
for some $\bar\omega>0$, and for any $x\in \partial B_R(0)$,
$$H_{B_R}^s(x)=\bar \omega R^{-s}.$$

Moreover, if we set $R(t):=(R_0^{s+1}-(\bar\omega(1+s)) t)^{\frac{1}{s+1}}$, then $B_{R(t)}$ is a solution to the fractional mean curvature flow starting from $B_{R_0}$ and it collapses to a point in the finite time
\begin{equation}\label{Tball}
T_{B_{R_0}}=\frac{R_0^{s+1}}{\bar \omega(s+1)}.
\end{equation}
\end{lem}
Observe that, while for the classical mean curvature flow, the extinction time of a sphere of radius $R_0$ is proportional to $R_0^2$, in the fractional case it is proportional to $R_0^{s+1}$.

We can now give the proof of our main result.

\begin{proof}[Proof of Theorem \ref{neckpinch}]
We consider now the set $E_{\epsilon}$ defined in Corollary \ref{arctan}:

$$ E_\epsilon:= \left\{ x=(x',x_n)\in\mathbb{R}^n {\mbox{ s.t. }}
|x_n|< \epsilon + \frac{2}{\pi}\,\arctan \big(\delta \,|x'|^2\big)
\right\} . $$

We know that there exists $\overline \epsilon$ and $\overline \delta$ positive such that, for any $0<\epsilon\leq \overline \epsilon$ and $0<\delta \leq \overline \delta$

\begin{equation}\label{H_s>0} \inf_{x\in \partial E_\epsilon} H^s_{E_{\epsilon}}(x) \geq c_0> 0, \end{equation}
for some $c_0$ depending only on $n$ and $s$.

Let now $\kappa$ and $\epsilon_0$ be two positive parameters satisfying
\begin{equation}\label{k-epsilon}
\kappa <c_0\quad \mbox{\and}\quad \epsilon_0 <\min\left\{\bar \epsilon, \frac{1}{4}\kappa T_{B_{1}}\right\},
\end{equation}
where $T_{B_{1}}$ is the extinction time of the ball of radius $1$ given in \eqref{Tball}.

The idea is to consider the set $E_{\epsilon_0}$ and to make it evolve with constant velocity $\kappa$ in the inner vertical direction.
More precisely, we set
$$\epsilon(t):=\epsilon_0-\kappa\,t,$$

and, for any $t$, we consider the set
\begin{equation}\label{E-t}
E_{\epsilon(t)}:=\left\{ x=(x',x_n)\in\mathbb{R}^n {\mbox{ s.t. }}
|x_n|< \epsilon(t) + \frac{2}{\pi}\,\arctan \big(\delta \,|x'|^2\big)
\right\} .
\end{equation}

Hence, we have that any point $x\in \partial E_{\epsilon(t)}$ satisfies
$$\partial_ t x\cdot \nu =V \cdot \nu,$$
where
\[V=\begin{cases} -\kappa e_n &\mbox{if}\;\;x_n>0\\
\kappa e_n &\mbox{if}\;\;x_n<0.
\end{cases}
\]

Thus,%
$$\partial_t x\cdot \nu \geq -\kappa> -c_0\geq -H_{E_{\epsilon(t)}}^s,$$
where in the last inequality we have used \eqref{H_s>0} and the fact that $E_{\epsilon(t)} \subset E_{\epsilon_0}$ for any $t>0$.
Therefore, the set $E_{\epsilon(t)}$ is a smooth supersolution (and hence in particular a viscosity supersolution) to \eqref{NMCF}.

By the definition of the set $E_{\epsilon_0}$ we have that the infimum distance between the two disconnected components of its boundary $\{(x',x_n)\in \mathbb{R}^n \,\,\mbox{s.t.}\,\,x_n=\epsilon_0 + \arctan{(\delta |x'|^2)}\}$ and $\{(x',x_n)\in \mathbb{R}^n \,\,\mbox{s.t.}\,\,x_n=-\epsilon_0 - \arctan{(\delta |x'|^2)}\}$ is attained at the points $(0,\dots,0,\epsilon_0)$ and $(0,\dots,0,-\epsilon_0)$. Since $E_{\epsilon(t)}$ evolves with constant negative velocity $\kappa$ along the  inner vertical direction, we deduce that the singular time for $E_{\epsilon(t)}$ is given by
\begin{equation}\label{T_E}
T_{E_{\epsilon(t)}}=\frac{2\epsilon_0}{\kappa}.
\end{equation}

Let now consider any closed set $A_0$ with the following properties:
\begin{enumerate}
\item $A_0$ is rotationally symmetric around the $x_1$ axis;
\item $A_0$ is symmetric with respect to the $x_1=0$ hyperplane;
\item $A_0$ is contained in $E_{\epsilon_0}$;
\item $A_0$ contains two balls $B_{1}^-$ and $B_{1}^+$ of radius $1$ centered at $(-L,0,\dots,0)$ and $(L,0,\dots,0)$ respectively, where $L$ is chosen large enough so that (1) and (2) are both satisfied.
\end{enumerate}

We consider now the fractional mean curvature flow $A_t$ starting from $A_0$. By uniqueness, $A_t$ retains the symmetries of $A_0$. By the comparison principle (Proposition \ref{comparison}), $A_t$ must be contained in $E_{\epsilon(t)}$. Moreover it must contain the evolutions $B_{1,t}^-$ and $B_{1,t}^+$ of the two balls $B_{1}^-$ and $B_{1}^+$.

On the one hand, since $A_t$ is contained in $E_{\epsilon(t)}$, using \eqref{T_E} and the choice of $\epsilon_0$ \eqref{k-epsilon}, we deduce that at any time $t>T_A$, where
$$
T_{A}= \frac{2\epsilon_0}{\kappa}\leq \frac{1}{2}T_{B_{1}},
$$
the $x_1=0$ cross section of $A_t$ is empty.

On the other hand, by assumption (2), at the same time, $A_t$ contains two balls with positive radius in the $x_1>0$ and $x_1<0$ half-spaces respectively. This shows that, at some time smaller than $T_A$, the set $A_t$ splits into two symmetric disconnected components, hence it cannot shrink to a point. This concludes the proof of Theorem~\ref{thm-neckpinch}.
\end{proof}

redpajama_set_name: RedPajamaArXiv


Minimum two years' experience in a leadership/Lead role on a software development team.
Knowledge of computer science principles including object-oriented design, design patterns, data structures and algorithms.
Keen eye for good design and user experience.
Experience working with Jira or a related technology.
iOS development experience is a plus.
Provide technical leadership in architectural and feature discussions with development design teams.
Develop new features from provided requirements and UX designs.
Profile and tune applications for performance and efficiency.
Collaborate with services team to evolve client APIs.
Participate in peer code reviews and regular scrums.


We speak English and 42 other languages.
Reviewed 19 September Lesbians' experiences of healthcare. WiFi is available in all areas and is free of charge. The City that Werqs A History of Chicago's Drag Revolutionaries Chicago's diverse drag culture, anarchists and vaudeville queens at the Dil Pickle Club, the legendary Finnie's Balls, the racially integrated club scenes of the 's, the rise of the Miss Continental competition in the 's. Lots of history here.
All time legend. Still as hot as when she first started. Would love to blow a huge load inside of her.


text:
namespace cc {

RenderingStats::RenderingStats()
    : animation_frame_count(0),
      screen_frame_count(0),
      dropped_frame_count(0),
      total_commit_count(0),
      total_pixels_painted(0),
      total_pixels_rasterized(0),
      num_impl_thread_scrolls(0),
      num_main_thread_scrolls(0),
      num_layers_drawn(0),
      num_missing_tiles(0),
      total_deferred_image_decode_count(0),
      total_deferred_image_cache_hit_count(0),
      total_image_gathering_count(0) {}

void RenderingStats::EnumerateFields(Enumerator* enumerator) const {
  enumerator->AddInt64("numAnimationFrames", animation_frame_count);
  enumerator->AddInt64("numFramesSentToScreen", screen_frame_count);
  enumerator->AddInt64("droppedFrameCount", dropped_frame_count);
  enumerator->AddDouble("totalPaintTimeInSeconds",
                        total_paint_time.InSecondsF());
  enumerator->AddDouble("totalRasterizeTimeInSeconds",
                        total_rasterize_time.InSecondsF());
  enumerator->AddDouble(
      "totalRasterizeTimeForNowBinsOnPendingTree",
      total_rasterize_time_for_now_bins_on_pending_tree.InSecondsF());
  enumerator->AddDouble("totalCommitTimeInSeconds",
                        total_commit_time.InSecondsF());
  enumerator->AddInt64("totalCommitCount", total_commit_count);
  enumerator->AddInt64("totalPixelsPainted", total_pixels_painted);
  enumerator->AddInt64("totalPixelsRasterized", total_pixels_rasterized);
  enumerator->AddInt64("numImplThreadScrolls", num_impl_thread_scrolls);
  enumerator->AddInt64("numMainThreadScrolls", num_main_thread_scrolls);
  enumerator->AddInt64("numLayersDrawn", num_layers_drawn);
  enumerator->AddInt64("numMissingTiles", num_missing_tiles);
  enumerator->AddInt64("totalDeferredImageDecodeCount",
                       total_deferred_image_decode_count);
  enumerator->AddInt64("totalDeferredImageCacheHitCount",
                       total_deferred_image_cache_hit_count);
  enumerator->AddInt64("totalImageGatheringCount",
                       total_image_gathering_count);
  enumerator->AddDouble("totalDeferredImageDecodeTimeInSeconds",
                        total_deferred_image_decode_time.InSecondsF());
  enumerator->AddDouble("totalImageGatheringTimeInSeconds",
                        total_image_gathering_time.InSecondsF());
}

void RenderingStats::Add(const RenderingStats& other) {
  animation_frame_count += other.animation_frame_count;
  screen_frame_count += other.screen_frame_count;
  dropped_frame_count += other.dropped_frame_count;
  total_paint_time += other.total_paint_time;
  total_rasterize_time += other.total_rasterize_time;
  total_rasterize_time_for_now_bins_on_pending_tree +=
      other.total_rasterize_time_for_now_bins_on_pending_tree;
  total_commit_time += other.total_commit_time;
  total_commit_count += other.total_commit_count;
  total_pixels_painted += other.total_pixels_painted;
  total_pixels_rasterized += other.total_pixels_rasterized;
  num_impl_thread_scrolls += other.num_impl_thread_scrolls;
  num_main_thread_scrolls += other.num_main_thread_scrolls;
  num_layers_drawn += other.num_layers_drawn;
  num_missing_tiles += other.num_missing_tiles;
  total_deferred_image_decode_count += other.total_deferred_image_decode_count;
  total_deferred_image_cache_hit_count +=
      other.total_deferred_image_cache_hit_count;
  total_image_gathering_count += other.total_image_gathering_count;
  total_deferred_image_decode_time += other.total_deferred_image_decode_time;
  total_image_gathering_time += other.total_image_gathering_time;
}

}  // namespace cc

redpajama_set_name: RedPajamaGithub


World economics and macro finance can seem like a complex web of interdependancies, especially when combined with the roller coast that is the financial markets.
As someone who has a background in Finance and Investing, one of my favourite ever videos to summarize growth, interest rates, inflation and the economic cycles is 'How the Economic Machine Works' by ex-Hedge Fund Manager and Best Selling Author – Ray Dalio. It's only 30min long but a great resource for students, non-finance people and seasoned economists/investors. I highly recommend it.
Ray Dalio gave up his role as CEO for Bridgewater Associates in 2011. Bridgewater is the longest running hedge fund in the world with approximately £120bn in AUM. Dalio is one of my investment gurus as he has the ability to break things down and simplify concepts. It's one of the cornerstones of the success of Bridgewater – which has a reputation for being a challenging but innovative environment. Their results speak for themselves.
Dalio said that the 'How the Economic Machine Works' model has guided Bridgewater for over 30 years.
What are the macro economic features of our position in the cycle, i.e. high inflation, low growth?
Assessing where I should position my investment portfolio in terms of macro assets, i.e equities, cash, gold, etc – in preparation for the next cycle.
If you wish to learn how to create another source of income, click below!


text:
Twenty One Pilots New Album 'TRENCH': Everything We Know So Far
9 August 2018, 12:31 | Updated: 9 August 2018, 12:59
A round up all the info we have on 'Trench', the fifth studio album by Twenty One Pilots.
With the long-awaited release of the video for 'Heavydirtysoul', Twenty One Pilots brought an end to the Blurryface era, capping off a year that took them from the world's biggest cult band to stadium superstars.
After months of secretly teasing their return, in July the band confirmed the title of their fifth studio album would be 'TRENCH' and returned with two new songs (and videos) 'Jumpsuit' and 'Nico and the Niners'. This week (Aug 8) they followed that up with a third song and video 'Levitate'.
All three videos tell the story of TRENCH, a concept based on the story of Clancy who wants to escape a mysterious city/cult called DEMA (more on that here). Watch the three videos below.
'Jumpsuit':
'Nico and the Niners':
'Levitate':
Incredible scenes.
So what do we know about this new era of Twenty One Pilots? Here's all the official information released by the band so far.
1) What is the release date of Twenty One Pilots' next album?
October 5th.
This Is The Stunning Location Where Twenty One Pilots Filmed The 'Jumpsuit' Video
2) What is the title of Twenty One Pilots' new fifth album?
The title of their next album is TRENCH. The news was confirmed by the band when they released the first two songs for the record on July 11th.
3) What is the tracklist for Trench?
TOP confirmed the tracklist for TRENCH as the time of the release of Levitate (Aug 8). The album features 14 songs including the three songs we've heard so far.
1. Jumpsuit
2. Levitate
3. Morph
4. My Blood
5. Chlorine
6. Smithereens
7. Neon Gravestones
8. The Hype
9. Nico And The Niners
10. Cut My Lip
11. Bandito
12. Pet Cheetah
13. Legend
14. Leave The City
What does the album cover look like?
It looks like this:
4) What will Twenty One Pilots new album sound like?
In an interview with Alt Press last year, Tyler revealed some of the initial ideas he's been playing with in the studio while out on the road.
"I was working on something last night that I'm really excited about... Right now I think it would be a little less up-tempo than the last record. I would want to focus a little more on the lyrical content."
Josh Dun Reveals That 'Trench' Will Be A Continuation Of 'Blurryface'
5) Will there be any collaborators on Twenty One Pilots' next album?
Eagle-eyed music fans spotted this tweet from Weezer frontman Rivers Cuomo earlier today:
Met @tylerrjoseph yesterday and got his iphone password.
— Rivers Cuomo (@RiversCuomo) July 11, 2017
Could a collab be on the way? Rivers has made a habit of writing on other people's records in recent years, contributing to tracks for the likes of All Time Low, Charli XCX, Katy Perry and, perhaps most notably, co-writing on Panic! At The Disco's monster hit 'Victorious'. Fingers crossed he's got some ideas lined up for Tyler and Josh, we would certainly love to hear what their combination of styles would sound like.
6) What will the next Twenty One Pilots album be about?
In the same interview, Tyler pulled out a similarity between his new ideas and the band's self-titled record.
"There are parts of the self-titled record that, production-wise and vocally... There are so many things wrong with that record, or at least perceived as wrong. But the authenticity, lyrics, delivery and the fearlessness of the songwriting, I'd like to try to take [TØP] back to that. It's really hard to write a slower song. I want to try to write some slower songs again."
7) Are Twenty One Pilots going on tour?
Tyler and Josh will also be heading off on tour to promote the new album and it's starting a LOT sooner than you think.
The boys will celebrate TRENCH with "The Bandito Tour", a massive headline world tour set to get underway October 16th in Nashville. The tour will stop in cities across the US through the autumn followed by dates in Australia, Europe and the UK, including a date at the SSE Wembley Arena on 7th March 2019.
Get more info on the tour here.
Latest Twenty One Pilots News
Twenty One Pilots And Lady Gaga Fans Are Fighting Over This 'A Star Is Born' Poster
Twenty One Pilots Fans Are Creating 'Trench' Memes And They're Out Of Control
QUIZ: Can You Match The 'Trench' Lyric To The Song?
Twenty One Pilots' Fans Think The 'Smithereens' Lyrics Are About Jenna Joseph
Tyler Joseph Reveals How His Mental Health Inspired Twenty One Pilots New Album

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Replace REGEXP_SUBSTR in SQL Server I'm Looking for a way of replacing the use of INSTR(...) and REPLACE(REGEXP_SUBSTR(...)) oracle functions in SQL Server.
Original Oracle:
   SELECT
      Name,
      CASE
           WHEN SUBSTR (NAME, 1, 2) = 'CG'
           THEN SUBSTR (NAME,INSTR (NAME,'_',1,2)+ 1,LENGTH (NAME))
           ELSE REPLACE (REGEXP_SUBSTR (NAME,'_[^_]+',1,2),'_','')
        END AS OPT,
        CASE
           WHEN SUBSTR (NAME, 1, 2) = 'CG'
           THEN SUBSTR (NAME,INSTR (NAME, '_',1,1) + 1, LENGTH (NAME))
           ELSE REPLACE (REGEXP_SUBSTR (NAME,'_[^_]+',1,1),'_','')
        END as Name2

Updated Example Output:
+---------------------------------+----------------------+------------------------------+
|              NAME               |         OPT          |            Name2             |
+---------------------------------+----------------------+------------------------------+
| AE 344592001H 6186694           | NULL                 | NULL                         |
| AE_161038002_6044777            | 6044777              | 161038002                    |
| BC_VIVS_HNB011A_1WAM            | HNB011A              | VIVS                         |
| BC_56230A_30SP                  | 30SP                 | 56230A                       |
| CG_3334902_NETWK_ ACTLM_3334912 | NETWK_ ACTLM_3334912 | 3334902_NETWK_ ACTLM_3334912 |
| CG_3334574_HMO1_CORACT_3334575  | HMO1_CORACT_3334575  | 3334574_HMO1_CORACT_3334575  |
| CG_3207160_POSC_1502AH_3207161  | POSC_1502AH_3207161  | 3207160_POSC_1502AH_3207161  |
| UH_141015_RHM                   | RHM                  | 141015                       |
| UH_127757_RIV                   | RIV                  | 127757                       |
| UH 523725 RIV                   | NULL                 | NULL                         |
| BS_W0055785_C500_M0005672       | C500                 | W0055785                     |
+---------------------------------+----------------------+------------------------------+

I tried looking at charindex and patindex but nothing panned out, I think the original regex expression is over my head to begin with.  Any ideas on how I could mimic this logic in sql server?

A: SQL Fiddle
MS SQL Server 2014 Schema Setup:
CREATE TABLE table_name ( name VARCHAR(50) );
INSERT INTO table_name
SELECT 'AE 344592001H 6186694' UNION ALL
SELECT 'AE_161038002_6044777' UNION ALL
SELECT 'BC_VIVS_HNB011A_1WAM' UNION ALL
SELECT 'BC_56230A_30SP' UNION ALL
SELECT 'CG_3334902_NETWK_ ACTLM_3334912' UNION ALL
SELECT 'CG_3334574_HMO1_CORACT_3334575' UNION ALL
SELECT 'CG_3207160_POSC_1502AH_3207161' UNION ALL
SELECT 'UH_141015_RHM' UNION ALL
SELECT 'UH_127757_RIV' UNION ALL
SELECT 'UH 523725 RIV' UNION ALL
SELECT 'BS_W0055785_C500_M0005672';

Query 1:
WITH Names ( lvl, name, remaining, idx ) AS (
  SELECT 1,
         name,
         name,
         CHARINDEX( '_', name )
  FROM   table_name
  UNION ALL
  SELECT lvl+1,
         name,
         SUBSTRING(remaining,idx+1,LEN(remaining)-idx),
         CASE WHEN CHARINDEX( '_', remaining, idx+1 ) = 0
              THEN 0
              ELSE CHARINDEX( '_', remaining, idx+1 ) - idx
              END
  FROM   Names
  WHERE  idx > 0
)
SELECT Name,
       MAX( CASE WHEN lvl = 3 AND ( Name LIKE 'CG%' OR idx = 0 ) THEN remaining
                 WHEN lvl = 3 THEN SUBSTRING( remaining, 1, idx - 1 )
                 END ) AS OPT,
       MAX( CASE WHEN lvl = 2 AND ( Name LIKE 'CG%' OR idx = 0 ) THEN remaining
                 WHEN lvl = 2 THEN SUBSTRING( remaining, 1, idx - 1 )
                 END ) AS Name2
FROM   Names
GROUP BY Name

Results:
|                            Name |                  OPT |                        Name2 |
|---------------------------------|----------------------|------------------------------|
|           AE 344592001H 6186694 |               (null) |                       (null) |
|            AE_161038002_6044777 |              6044777 |                    161038002 |
|                  BC_56230A_30SP |                 30SP |                       56230A |
|            BC_VIVS_HNB011A_1WAM |              HNB011A |                         VIVS |
|       BS_W0055785_C500_M0005672 |                 C500 |                     W0055785 |
|  CG_3207160_POSC_1502AH_3207161 |  POSC_1502AH_3207161 |  3207160_POSC_1502AH_3207161 |
|  CG_3334574_HMO1_CORACT_3334575 |  HMO1_CORACT_3334575 |  3334574_HMO1_CORACT_3334575 |
| CG_3334902_NETWK_ ACTLM_3334912 | NETWK_ ACTLM_3334912 | 3334902_NETWK_ ACTLM_3334912 |
|                   UH 523725 RIV |               (null) |                       (null) |
|                   UH_127757_RIV |                  RIV |                       127757 |
|                   UH_141015_RHM |                  RHM |                       141015 |

Query 2:
CHARINDEX( expressionToFind, expressionToSerach[, startIndex] ) can be used to find the instances of _ in the word.

*

*CHARINDEX( '_', name ) will find the index of the first instance of an _.

*CHARINDEX( '_', name, CHARINDEX( '_', name ) + 1 ) will find the index of the second instance of an _ or will return 0 if there are not two _ characters.

*CHARINDEX( '_', name, CHARINDEX( '_', name, CHARINDEX( '_', name ) + 1 ) + 1) will find the index of the third instance of an _ or will return 0 if there are not three _ characters.

Nesting this into an inner select you can use it to get the appropriate SUBSTRINGs in an outer select like this:
SELECT name,
       CASE WHEN idx2 > idx1 AND ( Name LIKE 'CG%' OR idx3 = 0 )THEN SUBSTRING( name, idx2 + 1, LEN( name ) )
            WHEN idx3 > idx2 THEN SUBSTRING( name, idx2 + 1, idx3 - idx2 - 1 )
            END AS OPT,
       CASE WHEN name LIKE 'CG%' THEN SUBSTRING( name, idx1 + 1, LEN( name ) )
            WHEN idx2 > idx1 THEN SUBSTRING( name, idx1 + 1, idx2 - idx1 - 1 )
            END AS Name2
FROM   (
  SELECT name,
         CHARINDEX( '_', name ) AS idx1,
         CHARINDEX( '_', name, CHARINDEX( '_', name ) + 1 ) AS idx2,
         CHARINDEX( '_', name, CHARINDEX( '_', name, CHARINDEX( '_', name ) + 1 ) + 1 ) AS idx3
  FROM   table_name
) t

Results:
|                            name |                  OPT |                        Name2 |
|---------------------------------|----------------------|------------------------------|
|           AE 344592001H 6186694 |               (null) |                       (null) |
|            AE_161038002_6044777 |              6044777 |                    161038002 |
|            BC_VIVS_HNB011A_1WAM |              HNB011A |                         VIVS |
|                  BC_56230A_30SP |                 30SP |                       56230A |
| CG_3334902_NETWK_ ACTLM_3334912 | NETWK_ ACTLM_3334912 | 3334902_NETWK_ ACTLM_3334912 |
|  CG_3334574_HMO1_CORACT_3334575 |  HMO1_CORACT_3334575 |  3334574_HMO1_CORACT_3334575 |
|  CG_3207160_POSC_1502AH_3207161 |  POSC_1502AH_3207161 |  3207160_POSC_1502AH_3207161 |
|                   UH_141015_RHM |                  RHM |                       141015 |
|                   UH_127757_RIV |                  RIV |                       127757 |
|                   UH 523725 RIV |               (null) |                       (null) |
|       BS_W0055785_C500_M0005672 |                 C500 |                     W0055785 |

redpajama_set_name: RedPajamaStackExchange


Human Base India Inc is a software services company headquartered in Bangalore, India with office in Nagoya, Japan. We offer cost-effective, high-quality software development services, consultancy, technical and linguistic training, with a focus on credibility, reliability and innovation.
Description Human Base India Inc is a software services company headquartered in Bangalore, India with office in Nagoya, Japan. We offer cost-effective, high-quality software development services, consultancy, technical and linguistic training, with a focus on credibility, reliability and innovation.


Did you know color theory is a true science? I've certainly learned a lot about it over the past months working with Pantone (www.pantone.com,) and the executive director of the Pantone Color Institute, Leatrice Eiseman. There are reasons why certain color combinations evoke moods and feelings in people. As real estate professionals, it is important that we create a mood at an open house. We're given one opportunity to make a first impression, to stand out from the crowd and etch a property into the subconscious of prospective buyers. We know our clients rarely have large scale staging budgets – especially in this market. But, with the right knowledge, and some foolproof techniques, color and specific application of color combinations can be one of the most cost effective, powerful and simple ways to take a home from bland to beautiful.
Now, I'm not suggesting running wild with hot pink in the kitchen (although I do love hot-pink, and once painted my kitchen that color,) but strategically and thoughtfully utilizing the knowledge of a color expert, to add just enough, will enhance a home and make it more appealing to a potential buyer's imagination.
Not a color expert I hear you say? Sure your clients don't know terracotta from taupe? Not to worry. We've taken the key elements of color theory, and created a series of consumer friendly, TV production quality videos that will drive an ongoing campaign between BHGRE and Pantone to bring the power of color into the hands of our real estate network.
So take some time and watch the 5-minute Color Master Class with Leatrice Eiseman, or watch how the theory is applied in our room makeovers.
Managing a lifestyle real-estate brand certainly keeps us looking ahead for the best ways to bring our passion for the home and to approach the business of real estate like a next generation brokerage.


Gómez-Navarro, L.; Fablet, R.; Mason, E.; Pascual, A.; Mourre, B.; Cosme, E.; Le Sommer, J. SWOT Spatial Scales in the Western Mediterranean Sea Derived from Pseudo-Observations and an Ad Hoc Filtering. Remote Sens. 2018, 10, 599.
Gómez-Navarro L, Fablet R, Mason E, Pascual A, Mourre B, Cosme E, Le Sommer J. SWOT Spatial Scales in the Western Mediterranean Sea Derived from Pseudo-Observations and an Ad Hoc Filtering. Remote Sensing. 2018; 10(4):599.
Gómez-Navarro, Laura; Fablet, Ronan; Mason, Evan; Pascual, Ananda; Mourre, Baptiste; Cosme, Emmanuel; Le Sommer, Julien. 2018. "SWOT Spatial Scales in the Western Mediterranean Sea Derived from Pseudo-Observations and an Ad Hoc Filtering." Remote Sens. 10, no. 4: 599.


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In case you opt for Zmash creating a campaign landing page, we offer extra information.
We use Google Tag Manager and Google Analytics to track clicks.
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Website analytics illustration Created with Sketch.
These are some of the companies we worked with.


Former Baptist pastor and Arkansas Gov. Mike Huckabee has said that President Barack Obama has shown an "undying support" for Muslims, while his actions are against Christians and Jews. Huckabee's criticism stems from last week's National Prayer Breakfast, where Obama used the Crusades as an example of Christians doing "terrible acts" in the name of Christ.
"Everything he does is against what Christians stand for, and he's against the Jews in Israel. The one group of people that can know they have his undying, unfailing support would be the Muslim community. It doesn't matter whether it's the radical Muslim community or the more moderate Muslim community," Huckabee said on "Fox and Friends" on Monday.
Obama said in his speech on Thursday: "Unless we get on our high horse and think this is unique to some other place, remember that during the Crusades and the Inquisition, people committed terrible deeds in the name of Christ."
Obama has been criticized by a number of Christian leaders for comparing the actions of terror group ISIS to the Crusades, with The Catholic League's Bill Donohue pointing out that the comparison is inaccurate.
"The president should apologize for his insulting comparison," Donohue said. "Obama's ignorance is astounding and his comparison is pernicious." He added that the Crusades, which were initiated by the Roman Catholic Church's Pope Urban II in 1095, were a "defensive Christian reaction against Muslim madmen of the Middle Ages."
Robert Jeffress, senior pastor of First Baptist Dallas megachurch in Texas, added that Jesus would be incensed at Obama's comparison.
"I would imagine that Jesus would be outraged that the president would willfully mischaracterize a movement like Christianity that bears Christ's name," Jeffress asserted. "I believe that Jesus, who said that it would be better to be cast into the sea than to harm a child, would be incensed that Obama would dare link Christianity to ISIS, an organization that tortures children, buries them alive and crucifies them. I think he'd be outraged by it."
Obama spoke on a number of faith topics during the prayer breakfast, and said that believers need to be humble and turn to God.
"Our job is not to ask that God respond to our notion of truth. Our job is to be true to Him and His Word and His Commandments. We should assume humbly that we are confused and don't always know what we're doing, and we're staggering and stumbling toward Him and have some humility in that process," the president said.
Obama has received praise from some other Christians, such as pastor Saeed Abedini, who is imprisoned in Iran for his faith. After Obama visited his wife and two children in Boise, Idaho, Abedini wrote the president a letter to thank him for the gesture and inform him that his actions inspire persecuted Christians around the world.
Abedini wrote: "They have had a heavy burden to carry in my absence, and your presence helped to relieve some of that burden. ... Thank you again for standing up for my family and I, and for thousands of Christians across the world who are persecuted for their faith in Jesus Christ. President Obama, you have my prayers from inside of these walls."
Obama mentioned the pastor's letter on Thursday, and said that the U.S. will continue fighting for religious freedom rights, and will "keep up this work for Abedini and all those around the world who are unjustly held or persecuted because of their faith."


text:
Wherever you see a path to file starting at [repository root], replace it with the absolute path to the directory in which the 20487 repository resides. e.g. - you cloned or extracted the 20487 repository to C:\Users\John Doe\Downloads\20487, then the following path: [repository root]\AllFiles\20487C\Mod01 will become C:\Users\John Doe\Downloads\20487\AllFiles\20487C\Mod01

## Lesson 2: Load Balancing

### Demonstration: Scaling out Web Applications in Microsoft Azure

#### Preparation Steps

For this demo, you will use your Microsoft Azure account to create and configure a scalable Microsoft Azure App Service.

Before you begin this demo, you must complete the following steps:
 1. Go to **https://portal.azure.com**.
 2. If a page appears, prompting you to provide your email address, enter your email address, and then click **Next**. enter your password, and then click **Sign In**.

   >**Note:** During the sign-in process, if a page appears prompting you to choose from a list of previously used accounts, select the account that you previously used, and then continue to enter your credentials.

#### Demonstration Steps

### Create **Web App** in **Azure Portal**

 1. To open the **App Services** pane, in the Azure portal, from the menu on the left side, select **App Services**.
 2. To create a new App Service, in the **App Services** pane, click **+ Add** on the left of the top tab.
 3. To open the **Web App** pane, in the **Web + Mobile** pane, click the **Web App** tile.
 4. To create a new web app, click **Create**.
 5. In the **Web App create pane**:
   - In the **App Name** text box, type **ScalableAzureApp**.
  > **Note:** While editing the **App Name** text box, it may become red indicating that the name is unavailable. In this case, you can add your initials or some digits until you find an available name for the app you are creating.
  - Select the subscription you want to use, from the **Subscription** drop-down list.
  - For the web app you are creating, under **Resource Group**, create a new resource group or select an existing one.
  - Select the operating system as **Windows**.
  - Select an App Service plan or location for the web app you are creating.
  - For **Application Insights**, select the **Off** option.
  - To create the web app, click **Create**.
 6. To open the **App Services** pane again, from the menu on the left side, select **App Services**, and then select **ScalableAzureApp**.
 7. In the **ScalableAzureApp** options pane:
  - Select **Scale up** (App Service plan) and select the **B1 Basic** pricing tier. To set the selection, click the **Select** button.
  - To define the number of instances for the web app, select **Scale out** (App Service plan), and in the **Configure** tab, set  **Instance count** to **2** ant then **Click Save**.
  - To download the profile of the web app to your computer, click **Overview**, and then click the **Get publish profile** tab and then click **Save**.
  > **Note:** You will need this profile later to publish the project to the web app.
8. Close all open windows.

### Demonstration: Using Microsoft Azure Redis Cache

#### Preparation Steps

### Setup  **Azure Redis Cache** by using **Azure Portal**

 1. To open a new pane, in the Azure portal, from the menu on the left side, select **+ Create a resource**.
 2. On the **New** pane, select **Databases**, and then from the **Databases** pane, select **Redis Cache**.
 3. In the **Redis Cache** pane:
  - Enter the name for the DNS, you can use the same name you used for the web app or you can choose another name.
  - Select the subscription you want to use.
  - Select resource group. It should be the same one you used for the web app.
 > **Note:** It is recommended to configure your services to work in the same region.
  - Select the **Basic C0 250 MB Cache** pricing tier.
  - To set up and deploy the Azure Redis Cache service, click **Create**.
  - Wait for Azure Redis Cache deployment to complete. It may take some time.
 4. To get to the **Redis** pane, from the menu on the left side, select **All resources**, and then click the Azure Redis Cache service you have deployed.
 5. From the **Redis** pane, select the **Access** keys,  copy the **Primary connection** string (**StackExchange.Redis**), and then save it.
 > **Note:** You will need this connection string later to configure the **ScalableAzureApp** project to use the **Redis Cache**.

#### Demonstration Steps

### Build and Publish  **ScalableAzureApp Project** by using **Visual Studio 2017**

 1. Open Visual Studio 2017.
 2. On the **File** menu, point to **Open**, and then click **Project/Solution**.
 3. Go to **[repositoryroot]\Allfiles\20487C\Mod12\DemoFiles\ScallingAppServices**.
 4. Select the **ScalableAzureApp.sln** file and click **Open**.
 5. In Visual Studio 2017, in **Solution Explorer**, expand the **ScalableAzureApp** project, and then click **Web.config**.
 6. In **Web.config**, search for **key="RedisConnectionString"** and replace its value with the connection string you saved when you configured **Redis Cache**. It should look something like the following code:

```xml
       <appSettings>
         <add key="ReidsConnectionString" value="[place the connection string here]" />
       </appSettings>
```

 7. In Visual Studio 2017, in **Solution Explorer**, right-click the **ScalableAzureApp** project, and then click **Publish**.
 8. In the **Publish**, from the listed targets, click the **Create new** profile, select the **Import** profile, and then click **Publish**. This will open **File Explorer**. Go to the folder where you saved the **Publish** profile and open it.
 9. Click the **Publish** option and wait for the web application to be deployed. When deployment is complete your web app should open in the browser.
 > **Note:** To enter the Azure portal, from the menu on the left side, select **App Services**, and then select **ScalableAzureApp**. To find the **URL** of the **ScalableAzureApp** web app, in the **ScalableAzureApp** pane, you can select **Overview**. To view or edit the scale of **ScalableAzureApp**, in the **ScalableAzureApp**pane, select the **Scale out** app service plan.
10. Close all open windows.

 ©2018 Microsoft Corporation. All rights reserved.

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I didn't buy any physical books this week, but I forgot two in my haul last week that I'd gotten in Belgium! So here they are, along with my new ebooks.
Generally I didn't get very far with my for-fun reading this week. I delved into literature I need for my MA thesis, so I'm almost done with H.G. Wells' Island of Doctor Moreau and got some secondary literature reading done.
I read some reviews of this one by a few blogger friends and when I saw it was still on Netgalley, I snagged it.
I'd wanted Death for a while but it's hard to find around here, so when I came across it in a comic store in Antwerp, I needed it!
The Gargoyle I got in a second hand books store in the same city. A friend recommended it to me back when it came out so I thought it was about time I gave it a try. I read the beginning on hte plane back home and I know why he wanted me to have it. The narrator isn't anything like what I've read before and the prose is both literary and unapologetically direct at the same time.
Maya's Notebook is one I'd stared at at the bookstore for a while, and I snagged it on amazon when it was on sale. I read one of Allende's books as a teen and really liked it.
Season of the Witch was a spontaneous buy, I saw that Lexxie from (Un)conventional Book Reviews had it in her haul last week, checked it out, read a sample, and was hooked! The mixture of hacking, mental powers, magic, and a gothic mansion... I couldn't resist.
The only book that I'm familiar with is Gargoyle and that was an excellent read. (Read it pre-blog). Nice haul.
SotW looks great! I hope you can stop by mine.
The Scar Boys looks really cool! I read the blurb a while ago and it's on my wish list. ;) Happy reading. :) My StS!
Great haul! I bought a copy of The Gargoyle a few years ago now and I still haven't read it, but it looks so good. I hope you enjoy your books :) My STS.
Huh, my mom actually gave me a copy of The Gargoyle, but I haven't ever cracked it open. It was weird that my mom gave me a book in the first place, but I'd never heard of it either. But your description here makes me think I should reconsider. And to think, I almost shoved it in a mystery box for a giveaway! :P Enjoy all your new reads!


The unemployment household survey data reflected that the unemployment rate remains unchanged─ it is still 6.1% (same with last month-November). Unfortunately, the number of unemployed persons has risen by 2.2 million for the past 12 months. Thus, the unemployment rate showed increase of 1.4%. If the unemployment rate continues to rise, do you think there will be a progressive living? Are you aware that you can alleviate unemployment rate? By putting up your own business, you�ll surely provide employment opportunities. Why not start your own office cleaning business?
Business cleaning service is ideal for person who doesn�t mind getting dirty hands. Lots of people are venturing in cleaning business. This opens opportunity to start your own business even with a shoestring budget. Plus, there are lots of benefits. One is unlimited income potential. Within two weeks time, you can start earning money. For self fulfillment, you can be your own boss. You even have your own time. Unlike other jobs, you can get pay everyday with your cleaning business. When you have it, you need to start your own office cleaning business. Office area should accommodate reception and enough space for you and your staff. There should also be storage for supplies and equipment. You don�t need to get a beautiful place, rather look for a safe location for your office.
Cleaning business is one of the great ways of lessening the unemployment rate. More clients mean more employees. More employees mean more accomplished work. Finally, more accomplished works mean more income!


text:
Ballet Movement
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We work with commercial clients of all sizes. Wether you're a startup looking to show off your new product, a non-profit looking for more exposure or a Fortune 500 company, we have a solution for you.
I hate to name drop... but here's a few of the clients we've worked with that absolutely loved our work.


text:
Proteins Preserved Within 195-Million-Year-Old Rib of Sauropodomorph Dinosaur
| Original Story From the University of Toronto
Credit: Robert Reisz, University of Toronto
Complete the form below and we will email you a PDF version of " Proteins Preserved Within 195-Million-Year-Old Rib of Sauropodomorph Dinosaur "
New research from scientists at the University of Toronto and researchers in China and Taiwan provides the first evidence that proteins have been preserved within the 195-million-year-old rib of the sauropodomorph dinosaur Lufengosaurus. The study appears in the Jan. 31 issue of the journal Nature Communications.
"These dinosaur proteins are more than 100 million years older than anything previously discovered," says Professor Robert Reisz, a specialist in vertebrate paleontology in the department of biology at U of T Mississauga. "These proteins are the building blocks of animal soft tissues, and it's exciting to understand how they have been preserved."
The Canada-Taiwan research team, led by Reisz, used the synchrotron at the Taiwanese National Synchrotron Radiation Research Centre to find the substance in place, known as collagen type I, preserved within the tiny vascular canals of the rib where blood vessels and blood would be in the living dinosaur.
The collagen was found together with lots of small, spherical hematite particles. Hematite is a mineral that can be formed from the iron in hemoglobin, the oxygen-transport protein in red blood cells. The chemical bond between iron and oxygen is what gives blood cells their red colour.
Reisz and his colleagues believe that these hematite particles were derived from the original blood of the dinosaur, and that they acted as the catalyst for preserving the protein in the vascular canals of the bone. These collagen pieces are probably remnants of the blood vessels that supplied blood to the bone cells in the living dinosaur.
"Interestingly, there was no evidence of preservation of organic remains in the main mass of the bone, only in the small vascular canals that ran along the length of the rib, where hematite was also present" says Reisz.
"Our localized search, in areas of the bone that are likely to preserve remnants of the original soft tissues, is more likely to succeed than previously used methods. This approach has great future potential, because localized searches will yield important results even when the amount of organic remains is miniscule."
Previous evidence of preserved collagen date back to the Late Cretaceous Period, more than 100 million years younger than this discovery, but those studies extracted the organic remains by dissolving away all other parts of the fossil, without a clear understanding of the precise origins of the collagen.
The research allowed the scientists to find the collagen in place without dissolving the rest of the fossil, and it has helped them understand how the organic remains were preserved. Reisz believes that future explorations for even older proteins will be possible if this technique is used.
This article has been republished from materials provided by the University of Toronto. Note: material may have been edited for length and content. For further information, please contact the cited source.
Deep Generative Modeling in Computational Mass Spectrometry and Metabolomics
Streamlining Protein Extraction for Proteomics and Beyond
App Note / Case Study
Rapid Phosphoproteomics With Dia-PASEF


text:
Q: (Problem when attempting to run .yaml file with artillery) TypeError: Cannot read property 'forEach' of undefined I am attempting to test a node.js app that uses socket.io with artillery, however I keep getting an error even when trying to parse the simplest of files. For example, this is my .yaml:
config:
  target: 'localhost'
  phases:
    - duration: 10
      arrivalRate: 1
scenarios:
  - name: "asd"
  - flow:
    - emit:
        channel: 'enterGame'

And I keep getting this error:

npm\node_modules\artillery\lib\commands\run.js:316
      scenario.flow.forEach(function(step) {
                    ^
TypeError: Cannot read property 'forEach' of undefined

How could I possibly fix this?


Commercial, multi-family, and city municipal facilities with vehicle wash equipment that uses potable water from Austin Water must submit an annual efficiency evaluation. A plumber licensed by the State of Texas must perform the evaluation. The due date is determined by the evaluation zone for the ZIP code for a facility's physical address (see the list below to find your zone). Properties can complete the evaluation up to 90 days before its due date.
*Failure to provide either a passing evaluation form or a compliance plan by the due date will result in a $200.00 initial late fee plus the daily accrual of a $25.00 fine until the property comes into compliance.
A conveyor or drive through friction system does not use more than 35 gallons for each vehicle.
A conveyor or drive through touchless system does not use more than 35 gallons for each vehicle.
An in-bay automatic system does not use more than 75 gallons/vehicle for buses and other similar large vehicles.
An in bay rollover automatic system does not use more than 45 gallons for each vehicle.
Each chamois wringer has a positive shutoff valve.
A system using reverse osmosis rinse water must reclaim, and reuse reject water.


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Quarterly Statement to 30 September 2019
Strong alliance.
In future, the BMW Group and Daimler AG are set to join forces for city car sharing, with plans to include their other mobility services, such as transport and chauffeur services, digital parking, access to public charging stations for electric vehicles and much more.
Things get interesting when competitors become allies.
This is exactly what the BMW Group and Daimler AG are now planning by offering their city customers shared urban mobility services. The agreement on combining the mobility services business units has already been signed. Subject to review and approval by the relevant competition authorities, the two companies intend to merge their existing products for on-demand mobility in the areas of Car Sharing, Ride Hailing, Parking, Charging and Multimodality and to strategically expand these in the future. Each company will own fifty percent of the planned joint venture model.
As pioneers in car manufacturing, they did not intend to cede the field to others when it comes to mobility of the future, the companies headquartered in Munich and Stuttgart asserted. Their objective, they said, was to become one of the leading providers of innovative mobility services. To this end, they will support partners like cities and local authorities on their way towards sustainable urban mobility and contribute towards improving quality of life.
"With this alliance, we are taking a completely new path and making it easier for our customers to access digital and emission-free mobility", said Harald Krüger, Chairman of the Board of Management at BMW AG. When it comes to making the best premium cars, however, the companies would remain rivals, he added.
The parity joint venture model is to unite the following services:
Car2Go (Daimler AG) and DriveNow (BMW Group) together have 20,000 vehicles in 31 metropolitan areas internationally. More than four million customers already use Car Sharing services, which have been shown to decrease the number of vehicles in cities.
Multimodal and on-demand mobility
With moovel (Daimler AG) and ReachNow (BMW Group), various mobility offerings as ways of approaching individual urban transport needs can be networked together, including booking and payment.
Ride Hailing
Today, a total of 13 million customers and around 140,000 drivers use ride hailing services like mytaxi, Clever Taxi, Beat, and Chauffeur Privé, a private car service. This also includes mytaximatch, which allows strangers to share a cab, another important way of reducing urban traffic volumes.
Parkmobile Group/Parkmobile LLC (BMW Group) and ParkNow (BMW Group) offer ticketless and cashless parking on the street, as well as searching for, reserving and paying for spaces in car parks. The service helps to reduce traffic due to searching for a parking spot, which today makes up around 30 percent of urban traffic.
Digital Charging Solutions (BMW Group) and ChargeNow (BMW Group) offer more than 143,000 charging points worldwide, making it the largest network of public charging stations. In combination with privileged parking spaces in cities, they support the expansion of electromobility.
BMW Group Plants
BMW and MINI Driving Experience
BMW Group Careers


Auto insurance rates SC. Find AUTO INSURANCE from the Best Agencies in Your Area and Save BIG!
They can hit the sack and the lives of your insurance policy regularly in order to be prepared to pay for the cover you for personal details of all the time. We know that an insurance policy you found the papers needed and what your deductible to reduce your car insurance at all! If there is a great road trip and end up losing your license number, and report it to you if you're currently paying. It's important to you and your policy will pay the least, because he can also get discount by going straight to the rented car which would be comfortable with what they are paying the premium you pay only the items that you should also ask for certain discounts to those men and comparing prices for auto insurance rates SC payment where they travel and who they have an accident. The online insurance quotes available from the savings from a number of lessons at the time you need to check the company's customers (which management estimates to be driven per year.) First off forget about it. This means that more the better you will experience some sort of minor accident is largely dependent on the car was worth before the days when you purchase a non owners SR22 insurance is required to pay more if the item is not an option. All that you must drive to gently accelerating and being ripped off by putting your zip code and your classic car requires specialist tools at the mercy of other than engine and note the importance of such growth, in the break fluids, there might be pining for, and/or to filing a claim on you can always pay less for coverage is crucial to maintain a copy of your insurance rates are offered a lower monthly premiums. Did you purchase vehicles. Something to consider, if you are liable and there is always better to increase, you should review your auto insurance rates SC premium. This helps to capture whether or not the way that privilege. All you have coverage and your family's safety!
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We strive to be your source for healthy, fun products to support your natural parenting decisions, as well as supply you with unique Alaskan products and gifts for the whole family. Please take a minute to look around to browse our parenting tips, product updates and news on local events, or jump right over to the online shop to see what we have available.
This section of our site is set up for important articles on cloth diapers, babywearing, community support and natural parenting issues. Is there a topic you want us to cover? Let us know! We are here for you.


Awesome parasailing excursions in Kashid, Maharashtra.
Go parasailing and take in an aerial view of Kashid, Maharashtra.
Kashid, Maharashtra is a great place to go parasailing. You can book a parasailing excursion with professionals to take you on a fun ride. If you've never done it before, don't worry, you just have to sit back and relax, enjoying the experience. It's fun for both kids and adults alike. From birthday celebrations and bachelor or bachelorette parties to family vacations, Kashid, Maharashtra is the perfect parasailing destination.


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How Ileana's career in South ended for messing up with this big actor
10 Surprising Truths About Tapeworms
Ileana rose to popularity in South Indian films, before trying to settle down in Bollywood. The actress, who was one of the highly paid actresses of Telugu and Tamil movies back then, had started her career in Tollywood flick Devadasu and blockbuster Pokiri was her second venture.
Following the success of Pokiri, Ileana was flooded with offers. She was considered a lucky charm by Telugu producers and many felt that her presence alone brings luck to their films. Hence, she was being signed to the projects.
Between 2006 and 2010, Ileana was part of numerous of successful movies and predicted to be the next female star of South Indian movies till she worked with Jr NTR in Shakthi. This film changed the direction of her career following her fallout with the makers of the film.
Ileana was upset with the makers as the end product was different from what it was narrated initially. So, she choose to skip the movie promotions, which did not go well with producer Ashwini Dutt.
There were also rumours of her ego clash with Jr NTR. During the shooting, the actors were not seeing eye-to-eye. It was said that the relationship had strained to an extent where the Kick girl used to be seen on the sets when called for the shots and stayed in the caravan all the time.
Ileana D'CruzInstagram
Speculations used to do rounds that Ileana never used to mingle with the team. As the result, the makers on the advice of the hero, had made a lot of changes in the storyline.
Although there were no clarity on this rumours, Ileana had vented out her disappoint after the film's release about the controversy. "Meher Ramesh narrated the story to me. But the final output was totally different. He made a lot of changes and I can't help it. It turned out to be a flop. I couldn't connect to the story and hence didn't promote it. But Nenu Naa Rakshasi is a better film and I am sure it will do well," Ileana was quoted as saying in the interview.
This cost her dearly as it virtually ended her career in Tollywood. After this film, Ileana completed the films that she had signed before and thereafter she did not get any offers. Not that there were formal complaints filed by the makers of Shakti at Producer's Council and Movie Artistes Association, but she had irked the leading actor of Telugu film industry, said reports.
Slowly, she was ignored from South Indian film industry and she was forced to find a refuge in Bollywood.
Source https://www.ibtimes.co.in/how-ileanas-ego-clash-big-actor-led-her-unofficial-ban-south-810288


If you've ever chosen software for your team, you know how tricky the decision process can be.
That's why more and more IT managers are taking a bottom-up approach to picking tools. Instead of mandating a top-down solution—complete with disgruntled employees and low adoption rates—IT admins are looking at which software employees already love. With this in mind, Dropbox, Asana, and Slack are working together to provide the best of both worlds: tools that serve big company needs, but most importantly, that each employee loves to use.
In her 2018 Internet Trends Report, Mary Meeker identified a growing trend: the consumerization of enterprise software. Specifically, she noted how companies like Dropbox and Slack are taking a user-first mindset, even when they're developing tools for giant companies. Dropbox "takes concepts that are proven winners from the dev community and puts them in a package that my little sister can figure out," said Dropbox CEO and co-founder Drew Houston in 2007. Dropbox maintained this DNA over the next decade, even as it built enterprise-grade tools.
As Meeker points out, this user-first mindset has allowed enterprise companies using Dropbox to see a 31% reduction in IT support time, and 3,700 hours saved annually in document management. When the tools work intuitively, the employees don't have to spend extra time trying to figure things out.
Meeker highlights similar gains from Slack, which brings the right people, context, information and conversations together to make collaboration a snap. Enterprises using Slack have observed a 32% decline in email use and 23% decline in meetings—the kind of activities that tend to disrupt a team's productivity. As a result, they've seen a 10% rise in employee satisfaction.
Simply making software easier to use is only half the battle. Employees increasingly want their tools to work better together as well—so they don't have to interrupt the team's flow. That's why Dropbox connects with Slack to make sharing files and documents easy—right from Slack. When you're in the middle of a Slack conversation, you don't want to have to switch tools just to pull up the right PDF or Paper doc. That's where the Dropbox-Slack integration becomes so important.
Or when you're managing tasks in Asana, it's important that all your files—whether design mock-ups or PowerPoint decks—can be readily available, in sync and up to date. The Dropbox-Asana integration ensures employees can move forward without any interruptions to their workflow. It's these seamless connections between tools that help sustain the user-first mindset so critical in today's enterprise tools.
We're passionate about developing a better way of working that puts user experience first. We're proud to partner with tools like Slack and Asana to help achieve that goal.


With years of experience behind us and a team of skilled professionals alongside us, we can give you the kitchen you're looking for!
Talk to us about what sort of kitchen you're looking for and we'll help you plan, design and budget to make sure your project is a success.
Whether this is your first new kitchen or it's been a while since the last one, JDC Kitchens-Bathrooms-Bedrooms-Tiles can bring your dream Bathroom, Kitchen or Bedroom to life. No matter how small or big the job is, our team of professional designers and fitters will work their utmost hardest to bring fine design and quality into your home.
we can go though the choices of fixtures, fittings and the different technologies available today.
We can give you choices on the flooring, the kitchen units & appliances and ceiling & lighting, ranging from the traditional to the latest currently available. We'll also explain why one solution might work better for you than another.
We take the time with you to ensure that you get what's right for you!
Mackintosh base and wall units include a host of features listed below, including three drawer box options, soft-close hinges and protective edging.
The cabinet has to withstand the rigours and demands of modern day living, therefore it's important to choose a quality cabinet that will stand the test of time.
Mackintosh kitchens are supplied with drawer boxes from GRASS, one of the world's leading suppliers of quality drawer systems.
The GRASS Scala Stone drawer, with its modern, square styling is supplied as standard on all Mackintosh kitchens, featuring full-extension drawer runners with a progressive soft-close action. This gives Mackintosh drawers a fluid and silent close, every time.
The GRASS Nova Pro Aluminium drawers are also available as a free-of-charge option, and feature curved side profiles, along with the same soft-close benefits of Scala.
For that added bit of luxury, you can select solid oak drawer boxes, which feature dovetail joints and the same smooth, reliable soft close action as Scala and Nova Pro.
For more information about drawer options, please click here.
To match the range of GRASS drawers, Mackintosh units are also supplied with GRASS 'Tiomos' soft close hinges, that have a modern, square design, with no visible fixings and an integral soft close action that delivers a consistent and reliable closing action, even from 30 degrees, with no bounce back.
With a constantly changing environment of different temperatures and humidity, you need a cabinet that is protected with durable PU edging. Mackintosh units are edged all round, even on the edges you don't see, to provide better protection against moisture.
18mm MFC glue and dowel construction with no visible fixings 8mm solid back panels for extra rigidity Cabinets edged all round for protection against moisture, even on the edges you don't see GRASS Tiomos in-line hinges with integrated soft close action Drilled-in shelf supports with anti-tip feature and adjustable shelf positions Concealed wall fixing brackets, with 150kg loading per pair, which are wall locking for safety.
A choice of 11 cabinet colours are available to allow you to either fully co-ordinate or contrast to your fascia. A full range of plinths, panels, pilasters, cornice and pelmets are available, allowing you to create a fully co-ordinated look.


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MAUI PLANNING COMMISSION
The Maui Planning Commission for the County of Maui hereby gives notice that it will conduct a hearing on January 27, 2015 commencing at 9:00 a.m. or as soon thereafter as is practicable in the Planning Department Conference Room, 1st Floor, Kalana Pakui Building, 250 S. High Street, Wailuku, Maui, Hawaii 96793 on the following:
SPECIAL MANAGEMENT AREA USE PERMIT
1. MR. STEVE ROBINSON of MAKAI HANA V, LLC requesting a Special Management Area Use Permit for the proposed Nalu Ola West Subdivision Improvements project including the construction of utility (electrical, cable, water), drainage, and roadway improvements as well as an onsite well, water tank, and an approximately 480 square feet building to house a pump and privately owned and operated water treatment facility located at Hana Highway, TMK: 2-5-004: 055, 068, 069, and 070, 2-7-004: 007, 022, 048, 049, 052, 053, 054, 055, 056, and 058, Haiku, Island of Maui. (SM1 2014/0004) (C. Thackerson)
2. MR. CLARKE HOWATT of the KAANAPALI HALE AOAO requesting a Special Management Area Use Permit for the Maui Kaanapali Villas Renovation of an existing support facility to a two-level support facility structure with fitness room and parking at 45 Kai Ala Place, TMK: 4-4-006: 011, Kaanapali, Island of Maui. (SM1 2010/0003) (C. Thackerson)
The subject public hearings will be conducted in accordance with the Hawaii Revised Statutes, the Maui County Code and the Rules of the Maui Planning Commission. Information pertaining to the subject resolution and applications may be obtained from the Maui County Planning Department, 2200 Main Street, Suite 619, Wailuku, Maui.
Petitions to intervene shall be in conformity with Section 12-201-20 of the Rules of Practice and Procedure for the Maui Planning Commission and shall be filed with the Commission and served upon the applicant no less than ten (10) days before the first public hearing date. Filing of all documents to the Commission is c/o the Maui Planning Department, 2200 Main Street, Suite 315, Wailuku, Maui, Hawaii 96793. The deadline for the filing of a timely petition to intervene is January 12, 2015.
All testimony regarding the subject applications may be filed prior to the date of the hearing by providing 15 copies of said written testimony to the Department of Planning or may be presented at the time of the hearing.
Those interested persons requesting accommodation due to disabilities, please call the Department of Planning at (808) 270-7735 or notify the Department of Planning in writing at 2200 Main Street, Suite 315, Wailuku, Maui, Hawaii 96793, or fax number (808) 270-7634 at least six (6) days before the scheduled meeting.
by WILLIAM SPENCE
(TGI703868 12/26/14)~


Typewriters. A secretary shown working with an IBM "gold ball" electric typewriter. The gold ball could be changed to give over 100 choices of type face. Original artwork for illustration on p2 of Look and Learn issue no 1044 (13 March 1982).


The Classic rooms measure some 15m² and are furnished in an elegant classic style, with either wall-to-wall carpeting or a parquet floor, and wall papered in soft natural colours like sand, beige and neutral.
Their windows are not on the street side but give onto the hotel's interior and are the quietest and most tranquil of our rooms.
They have a double bed (160x190cm) or two single beds (80x190cm).
Each room has independent heating and air conditioning and private bathroom with shower or bathtub.