text:
\section{Introduction}

This short note serves primarily as a commentary on the very recent topic of pointwise estimates for the operator that restricts the Fourier transform to a hypersurface. We will concentrate exclusively on the case of the two-dimensional unit sphere $\mathbb{S}^2$ in three-dimensional Euclidean space $\mathbb{R}^3$. This both simplifies the exposition and enables the formulation of more general results.

If $A,B\colon X\to[0,\infty)$ are two functions (or functional expressions) such that,  for each $x\in X$,  $A(x)\leq CB(x)$ for some unimportant constant $0\leq C<\infty$, then we will write $A(x)\lesssim B(x)$ or $A(x) = O(B(x))$. If the constant $C$ depends on a set of parameters $P$, we may emphasize it notationally by writing $A(x)\lesssim_P B(x)$ or $A(x) = O_P(B(x))$.

The {\it Fourier transform} $f\mapsto\widehat{f}$ is one of the most fundamental integral operators. For a function $f\in\textup{L}^1(\mathbb{R}^3)$, the definition of $\widehat{f}$ that we use is
\[ \widehat{f}(y) := \int_{\mathbb{R}^3} f(x) e^{-2\pi i x\cdot y} \textup{d}x, \]
for each $y\in\mathbb{R}^3$.
Here $x\cdot y$ denotes the standard scalar product of vectors $x,y\in\mathbb{R}^3$, and integration is performed with respect to  Lebesgue measure.
The map $f\mapsto\widehat{f}$ is then extended by continuity to bounded linear operators $\textup{L}^p(\mathbb{R}^3)\to\textup{L}^{p'}(\mathbb{R}^3)$ for each $p\in(1,2]$ and $p'=p/(p-1)$.

Classical Fourier restriction theory seeks for {\it a priori} $\textup{L}^p$-estimates for the operator $f\mapsto \widehat{f}|_S$, where $S$ is a hypersurface in the Euclidean space.
In the case of $\mathbb{S}^2\subset\mathbb{R}^3$, the endpoint Tomas--Stein inequality \cite{S93:habook, T75:restr} reads as follows:
\begin{equation}\label{eq:restriction}
\big\| \widehat{f}\,\big\vert_{\mathbb{S}^2} \big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)} \lesssim \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)}.
\end{equation}
Here $\sigma$ denotes the standard surface measure on $\mathbb{S}^2$.
It is well-known that $4/3$ is the largest exponent that can appear on the right-hand side of \eqref{eq:restriction} provided that we stick to the $\textup{L}^2$-norm on the left-hand side. Estimate \eqref{eq:restriction} enables the {\it Fourier restriction operator}
\begin{equation}\label{eq:fourrestrop}
\mathcal{R}\colon\textup{L}^{4/3}(\mathbb{R}^3)\to\textup{L}^2(\mathbb{S}^2,\sigma)
\end{equation}
to be defined via an approximation of identity argument as follows.
The set of complex-valued Schwartz functions on $\mathbb{R}^3$ will be denoted by $\mathcal{S}(\mathbb{R}^3)$.
Fix $\chi\in\mathcal{S}(\mathbb{R}^3)$ such that $\int_{\mathbb{R}^3}\chi(x)\,\textup{d}x=1$,
and write $\chi_{\varepsilon}$ for the $\textup{L}^1$-normalized dilate of $\chi$, defined as
\[ \chi_{\varepsilon}(x):=\varepsilon^{-3}\chi(\varepsilon^{-1}x), \]
for $x\in\mathbb{R}^3$ and $\varepsilon>0$.
Given $f\in\textup{L}^{4/3}(\mathbb{R}^3)$, then, thanks to \eqref{eq:restriction}, $\mathcal{R}f$ can be defined as the limit
\[ \lim_{\varepsilon\to0^+}(\widehat{f}\ast\chi_{\varepsilon})\big\vert_{\mathbb{S}^2} \]
in the norm of the space $\textup{L}^2(\mathbb{S}^2,\sigma)$.

Maximal restriction theorems were recently inaugurated by M\"{u}ller, Ricci, and Wright \cite{MRW16:maxrestr}.
In that work, the authors considered general $\textup{C}^2$ planar curves with nonnegative signed curvature equipped with affine arclength measure, and established a maximal restriction theorem in the full range of exponents where the usual restriction estimate is known to hold.
Shortly thereafter, Vitturi \cite{V17:maxrestr} provided an elementary argument which leads to a partial generalization to higher-dimensional spheres. In $\mathbb{R}^3$, Vitturi's result covers the full Tomas--Stein range, whose endpoint estimate amounts to
\begin{equation}\label{eq:maxrestriction}
\Big\| \sup_{\varepsilon>0} \big| (\widehat{f}\ast\chi_{\varepsilon})(\omega) \big| \Big\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)} \lesssim_{\chi} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)}.
\end{equation}
Here and in what follows, we write a variable in the subscript of the letter denoting a function space whenever we need to emphasize with respect to which variable the corresponding (semi)norm is taken.
An easy consequence of \eqref{eq:maxrestriction} and of obvious convergence properties in the dense class of Schwartz functions is the fact that the limit
\[ \lim_{\varepsilon\to0^+}(\widehat{f}\ast\chi_{\varepsilon})(\omega) \]
exists for each $f\in\textup{L}^{4/3}(\mathbb{R}^3)$ and for $\sigma$-almost every $\omega\in\mathbb{S}^2$.
This enables us to recover the operator \eqref{eq:fourrestrop} also in the pointwise sense, and not only in the $\textup{L}^2$-norm, which was the main motivation behind the paper \cite{MRW16:maxrestr}.

For the elegant proof of \eqref{eq:maxrestriction}, Vitturi \cite{V17:maxrestr} used the following equivalent, non-oscillatory reformulation of the ordinary restriction estimate \eqref{eq:restriction}:
\begin{equation}\label{eq:restriction2}
\Big| \int_{(\mathbb{S}^2)^2} g(\omega) \overline{g(\omega')} h(\omega-\omega') \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega') \Big|
\lesssim \|g\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}^2 \|h\|_{\textup{L}^{2}(\mathbb{R}^3)}.
\end{equation}
The proof of the equivalence between \eqref{eq:restriction} and \eqref{eq:restriction2} amounts to passing to the adjoint operator (i.e.\@ to a Fourier extension estimate) and expanding out the $\textup{L}^4$-norm using Plancherel's identity. These steps make the choice of exponents $4/3$ and $2$ into the most convenient one. The advantage of the expanded adjoint formulation \eqref{eq:restriction2} is that one can easily insert the iterated maximal function operator in $h$, and simply invoke its boundedness on $\textup{L}^2(\mathbb{R}^3)$. We refer the reader to \cite{V17:maxrestr} for details. In some sense, we will be following a similar step below.

It is also clear from the proof in \cite{V17:maxrestr} that the function $\chi$ does not need to be smooth. One can, for instance, take $\chi$ to be the $\textup{L}^1$-normalized indicator function of the unit ball in $\mathbb{R}^3$, in which case $(\widehat{f}\ast\chi_{\varepsilon})(\omega)$ become the usual Hardy--Littlewood averages of $\widehat{f}$ over Euclidean balls $\textup{B}(\omega,\varepsilon)$,
\[ \frac{1}{|\textup{B}(\omega,\varepsilon)|} \int_{\textup{B}(\omega,\varepsilon)} \widehat{f}(y) \textup{d}y. \]
Moreover, Ramos \cite{Ra18} concluded that, for each $f\in\textup{L}^{p}(\mathbb{R}^3)$ and $1\leq p\leq 4/3$, almost every point on the sphere is a Lebesgue point of $\widehat{f}$, i.e.\@ for $\sigma$-a.e.\@ $\omega\in\mathbb{S}^2$ we have that
\[ \lim_{\varepsilon\to0^+} \frac{1}{|B(\omega,\varepsilon)|} \int_{B(\omega,\varepsilon)} \big|\widehat{f}(y) - (\mathcal{R}f)(y) \big| \,\textup{d}y = 0. \]
Prior to \cite{Ra18} this had been confirmed by Vitturi \cite{V17:maxrestr} for functions $f\in\textup{L}^{p}(\mathbb{R}^3)$, $1\leq p\leq 8/7$, who repeated the two-dimensional argument of M\"{u}ller, Ricci, and Wright \cite{MRW16:maxrestr}.

Variational estimates establish convergence in the whole $\textup{L}^p$-space in an explicit and quantitative manner, without the need for pre-existing convergence results on a dense subspace.
They were introduced by L\'{e}pingle \cite{L76:var} in the context of probability theory. Various modifications were then defined and used by Bourgain for numerous problems in harmonic analysis and ergodic theory; see for instance \cite{B89:pt}.
Variational estimates for various averages and truncations of integral operators have been extensively studied; the papers \cite{AJS98,BMSW18,CJRW00,JSW08:var,MSZ20:var} are just a sample from the available literature.

Recall that, given a function $a\colon(0,\infty)\to\mathbb{C}$ and an exponent $\varrho\in[1,\infty)$, the \emph{$\varrho$-variation seminorm} of $a$ is defined as
\[ \|a\|_{\widetilde{\textup{V}}^{\varrho}} := \sup_{\substack{m\in\mathbb{N}\\ 0<\varepsilon_0<\varepsilon_1<\cdots<\varepsilon_m}}
\Big( \sum_{j=1}^{m} |a(\varepsilon_{j})-a(\varepsilon_{j-1})|^{\varrho} \Big)^{1/\varrho}. \]
In other to turn it into a \emph{$\varrho$-variation norm}, we simply add the term $|a(\varepsilon_0)|^{\varrho}$ as follows:
\[ \|a\|_{\textup{V}^{\varrho}} := \sup_{\substack{m\in\mathbb{N}\\ 0<\varepsilon_0<\varepsilon_1<\cdots<\varepsilon_m}}
\Big( |a(\varepsilon_0)|^{\varrho} + \sum_{j=1}^{m} |a(\varepsilon_{j})-a(\varepsilon_{j-1})|^{\varrho} \Big)^{1/\varrho}. \]
Since this is not the standard convention in the literature,  we make a distinction as we are going to use both quantities. Clearly, $\|a\|_{\textup{V}^{\varrho}}$ controls both $\sup_{\varepsilon>0}|a(\varepsilon)|$ and the number of ``jumps'' of $a(\varepsilon)$ as $\varepsilon\to0^+$ (and even as $\varepsilon\to\infty$, which is not the interesting case here).
Given a function of two variables $b\colon(0,\infty)\times(0,\infty)\to\mathbb{C}$, we can define the \emph{biparameter $\varrho$-variation seminorm} as
\[ \|b\|_{\widetilde{\textup{W}}^{\varrho}} := \sup_{\substack{m,n\in\mathbb{N}\\ 0<\varepsilon_0<\varepsilon_1<\cdots<\varepsilon_m\\ 0<\eta_0<\eta_1<\cdots<\eta_n}}
\Big( \sum_{\substack{1\leq j\leq m\\ 1\leq k\leq n}}  \big| b(\varepsilon_{j},\eta_{k})-b(\varepsilon_{j},\eta_{k-1})-b(\varepsilon_{j-1},\eta_{k})+b(\varepsilon_{j-1},\eta_{k-1}) \big|^{\varrho} \Big)^{1/\varrho}. \]

In the first preprint of this paper, the authors established the following variational generalization of estimate \eqref{eq:maxrestriction},
\begin{equation}\label{eq:varrest}
\Big\| \big\| (\widehat{f}\ast\chi_{\varepsilon})(\omega) \big\|_{\textup{V}^{\varrho}_{\varepsilon}} \Big\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)} \lesssim_{\chi,\varrho} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)},
\end{equation}
when $\varrho\in(2,\infty)$, $f\in\textup{L}^{4/3}(\mathbb{R}^3)$, and $\chi$ is either a Schwartz function or the indicator function of the unit ball.
This result inspired several subsequent papers, such as \cite{Kov19:var} and \cite{Ra19}, which we are about to discuss.
In light of the more recent research, we now take the opportunity to both generalize \eqref{eq:varrest} and complete our short survey of maximal and variational Fourier restriction theories, with papers that appeared in the meantime.

Instead of the averaging function $\chi$ one can, more generally, consider an averaging measure $\mu$. In what follows, $\mu$ will be a finite complex measure defined on the Borel sets in $\mathbb{R}^3$; its dilates $\mu_{\varepsilon}$ are now defined as
\[ \mu_{\varepsilon}(E):=\mu(\varepsilon^{-1}E), \]
for every Borel set $E\subseteq\mathbb{R}^3$ and $\varepsilon>0$. For reasons of elegance, one can additionally assume that $\mu$ is {\it normalized} by $\mu(\mathbb{R}^3)=1$ and that it is {\it even}, i.e., centrally symmetric with respect to the origin, which means that
\[ \mu(-E)=\mu(E), \]
for each Borel set $E\subseteq \mathbb{R}^3$.
The Fourier transform of such a measure $\mu$ is the function $\widehat{\mu}$ defined as
\[ \widehat{\mu}(y) := \int_{\mathbb{R}^3} e^{-2\pi i x\cdot y} \textup{d}\mu(x), \]
for each $y\in\mathbb{R}^3$.

In \cite{Kov19:var} one of the present authors showed analogues of \eqref{eq:maxrestriction} and \eqref{eq:varrest} when $\chi$ is replaced by a measure $\mu$ whose Fourier transform $\widehat{\mu}$ is $\textup{C}^\infty$ and satisfies the decay condition
\begin{equation}\label{eq:decaycond}
|\nabla\widehat{\mu}(x)| \lesssim (1+|x|)^{-1-\delta},
\end{equation}
for some $\delta>0$.
Here and in what follows, $|v|$ denotes the Euclidean norm of a vector $v\in\mathbb{R}^3$.
It is interesting to make the following observation, which applies to the cases of two or three dimensions only, as things improve in higher dimensions. If one takes $\mu$ to be the normalized spherical measure, i.e.\@ $\mu=\sigma/\sigma(\mathbb{S}^2)$, then the decay of $|\nabla\widehat{\mu}(x)|$ as $|x|\to\infty$ is only $O(|x|^{-1})$; see \cite{AS92:hmf}. Consequently, the results from \cite{Kov19:var} do not apply. This was one of the sources of motivation for Ramos \cite{Ra19}, who reused Vitturi's argument from \cite{V17:maxrestr} to conclude that the maximal estimate
\[ \Big\| \sup_{\varepsilon>0} \big| (\widehat{f}\ast\mu_{\varepsilon})(\omega) \big| \Big\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)} \lesssim_{\mu} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)} \]
holds as soon as the maximal operator $h\mapsto\sup_{\varepsilon>0}(h\ast\mu_{\varepsilon})$ is bounded on $\textup{L}^2(\mathbb{R}^3)$. Relating this to the work of Rubio de Francia \cite{RdF86:max}, he further deduced the following sufficient condition in terms of the decays of $\widehat{\mu}$ and $\nabla\widehat{\mu}$:
\[ |\widehat{\mu}(x)| \lesssim (1+|x|)^{-\alpha} \text{ and } |\nabla\widehat{\mu}(x)| \lesssim (1+|x|)^{-\beta}, \text{ with } \alpha+\beta>1. \]
This condition includes the spherical measure as it satisfies $|\widehat{\mu}(x)|=O(|x|^{-1})$ and $|\nabla\widehat{\mu}(x)|=O(|x|^{-1})$ as $|x|\to\infty$.

The main result of this note is a variational estimate which generalizes \eqref{eq:varrest} slightly beyond the previously covered cases of the averaging functions $\chi$ or measures $\mu$.

\begin{theorem}\label{thm:maintheorem}
Suppose that $\varrho\in(1,\infty)$, and that $\mu$ is a normalized, even complex measure defined on the Borel subsets of $\mathbb{R}^3$, and satisfying \emph{any} of the following three conditions:
\begin{itemize}
\item[(a)] $-x\cdot\nabla\widehat{\mu}(x)\geq 0$ for each $x\in\mathbb{R}^3$, and
\begin{equation}\label{eq:condvariation}
\Big\| \big\| (h\ast\mu_{\varepsilon})(x) \big\|_{\widetilde{\textup{V}}^{\varrho}_{\varepsilon}} \Big\|_{\textup{L}_x^{2}(\mathbb{R}^3)}
\lesssim_{\mu,\varrho} \|h\|_{\textup{L}^{2}(\mathbb{R}^3)}
\end{equation}
holds for each $h\in\mathcal{S}(\mathbb{R}^3);$
\item[(b)] $\varrho\in(2,\infty)$, while $\widehat{\mu}$ is $\textup{C}^2$ and satisfies the decay condition \eqref{eq:decaycond}$;$
\item[(c)] the inequality
\begin{equation}\label{eq:conddouble}
\Big\| \big\| (h\ast\mu_{\varepsilon}\ast\overline{\mu}_{\eta})(x) \big\|_{\widetilde{\textup{W}}^{\varrho}_{\varepsilon,\eta}} \Big\|_{\textup{L}_x^{2}(\mathbb{R}^3)}
\lesssim_{\mu,\varrho} \|h\|_{\textup{L}^{2}(\mathbb{R}^3)}
\end{equation}
holds for each $h\in\mathcal{S}(\mathbb{R}^3)$.
\end{itemize}
Then, for each function $f\in\mathcal{S}(\mathbb{R}^3)$, the following estimate holds:
\begin{equation}\label{eq:varrestmu}
\Big\| \big\| (\widehat{f}\ast\mu_{\varepsilon})(\omega) \big\|_{\textup{V}^{\varrho}_{\varepsilon}} \Big\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)} \lesssim_{\mu,\varrho} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)}.
\end{equation}
\end{theorem}

Let us immediately clarify one minor technical issue. Theorem~\ref{thm:maintheorem} claims estimate \eqref{eq:varrestmu} for Schwartz functions $f$ only, but it immediately extends to all $f\in\textup{L}^{4/3}(\mathbb{R}^3)$ whenever $\mu$ is absolutely continuous with respect to Lebesgue measure. Otherwise, we could run into measurability issues on the left-hand side of \eqref{eq:varrestmu} for singular measures $\mu$.

Condition (a) in Theorem \ref{thm:maintheorem} is quite restrictive, but it is satisfied at least when $\varrho>2$ and the Radon--Nikodym density of $\mu$ is a radial Gaussian function. Indeed, if $\textup{d}\mu(x)=\alpha^3 e^{-\pi\alpha^2|x|^2}\,\textup{d}x$ for some $\alpha\in(0,\infty)$, then
\begin{equation}\label{eq:forGauss}
-x\cdot\nabla\widehat{\mu}(x) = 2\pi\alpha^{-2}|x|^2 e^{-\pi\alpha^{-2}|x|^2}
\end{equation}
is nonnegative, and \eqref{eq:condvariation} is a standard estimate by Bourgain \cite[Lemma 3.28]{B89:pt}. In fact, Bourgain formulated \eqref{eq:condvariation} for one-dimensional Schwartz averaging functions in \cite{B89:pt}, but the proof carries over to higher dimensions. Alternatively, one can invoke more general results from the subsequent literature, such as the work of Jones, Seeger, and Wright \cite{JSW08:var}, which covered higher-dimensional convolutions, more general dilation structures, and both strong and weak-type variational estimates in a range of $\textup{L}^p$-spaces.

Theorem~\ref{thm:maintheorem} under condition (b) was covered by the paper \cite{Kov19:var}, up to minor technicalities, such as the fact that here we do not need $\widehat{\mu}$ to be smoother than $\textup{C}^2$. However, \cite{Kov19:var} was concerned with more general surfaces and more general measures $\sigma$ on them, while here we are able to give a more direct proof that is specific to the sphere and to the stated choice of Lebesgue space exponents. In fact, as we have already noted, the present proof predates \cite{Kov19:var}.

Condition (c) above is somewhat artificial and difficult to verify, but we include it since the proof that it implies \eqref{eq:varrestmu} will be the most straightforward.

Maximal restriction estimates have found a nice application in the very recent work of Bilz \cite{B20}, who used them to show that there exists a compact set $E\subset\mathbb{R}^3$ of full Hausdorff dimension that does not allow any nontrivial a priori Fourier restriction estimates for any nontrivial Borel measure on $E$. We do not discuss the details here, but rather refer an interested reader to \cite{B20}.

The remainder of this paper is devoted to the proof of Theorem~\ref{thm:maintheorem}.

\section{Proof of Theorem~\ref{thm:maintheorem}}

We need to establish \eqref{eq:varrestmu} assuming any one of the three conditions from the statement of Theorem~\ref{thm:maintheorem}.
Let us start with condition (a).

\begin{proof}[Proof of Theorem~\ref{thm:maintheorem} assuming condition (a)]
Start by observing that
\[ \sup_{\varepsilon_0>0} \big|\widehat{f}\ast\mu_{\varepsilon_0}\big| \leq \big|\widehat{f}\ast\mu\big| + \sup_{\varepsilon_0>0} \big|\widehat{f}\ast(\mu_{\varepsilon_0}-\mu_1)\big|, \]
that $\widehat{f}\ast\mu = (f\widehat{\mu})\widehat{}\,$, and that the ordinary restriction estimate \eqref{eq:restriction} applies to $f\widehat{\mu}$ and yields
\[ \big\| \widehat{f}\ast\mu \big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)} \lesssim \|f\widehat{\mu}\|_{\textup{L}^{4/3}(\mathbb{R}^3)} \lesssim_{\mu} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)}. \]
Thus, inequality \eqref{eq:varrestmu} reduces to two applications of
\begin{equation}\label{eq:varrest0}
\Big\| \big\| (\widehat{f}\ast\mu_{\varepsilon})(\omega) \big\|_{\widetilde{\textup{V}}^{\varrho}_{\varepsilon}} \Big\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)} \lesssim_{\mu,\varrho} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)},
\end{equation}
which we proceed to establish.
The desired estimate \eqref{eq:varrest0} unfolds as
\begin{equation}\label{eq:varrest1}
\bigg\| \sup_{\substack{m\in\mathbb{N}\\ 0<\varepsilon_0<\varepsilon_1<\cdots<\varepsilon_m}} \Big( \sum_{j=1}^{m} \big| \big(\widehat{f} \ast (\mu_{\varepsilon_{j-1}} - \mu_{\varepsilon_{j}}) \big) (\omega) \big|^\varrho\Big)^{1/\varrho} \bigg\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)}
\lesssim_{\mu,\varrho} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)}.
\end{equation}
The numbers $\varepsilon_j$ in the above supremum can be restricted to a fixed interval $[\varepsilon_\textup{min},\varepsilon_\textup{max}]$ with $0<\varepsilon_\textup{min}<\varepsilon_\textup{max}$, but the estimate needs to be established with a constant independent of $\varepsilon_\textup{min}$ and $\varepsilon_\textup{max}$. Afterwards one simply applies the monotone convergence theorem letting $\varepsilon_\textup{min}\to0^+$ and $\varepsilon_\textup{max}\to\infty$. Moreover, by only increasing the left-hand side of \eqref{eq:varrest1}, we can also achieve $\varepsilon_0=\varepsilon_\textup{min}$ and $\varepsilon_m=\varepsilon_\textup{max}$.

Next, by continuity one may further restrict attention to rational numbers in the interval $[\varepsilon_\textup{min},\varepsilon_\textup{max}]$, and by yet another application of the monotone convergence theorem one may consider only finitely many values in that interval.
In this way, no generality is lost in assuming that the supremum in \eqref{eq:varrest1} is achieved for some $m\in\mathbb{N}$ and for some measurable functions $\varepsilon_k\colon\mathbb{S}^2\to[\varepsilon_\textup{min},\varepsilon_\textup{max}]$, $k\in\{0,1,\ldots,m\}$, such that $\varepsilon_0(\omega)\equiv\varepsilon_{\textup{min}}$, $\varepsilon_m(\omega)\equiv\varepsilon_{\textup{max}}$.
Estimate \eqref{eq:varrest1} then becomes
\[ \Big\| \Big( \sum_{j=1}^{m} \big| \big(\widehat{f} \ast (\mu_{\varepsilon_{j-1}(\omega)} - \mu_{\varepsilon_{j}(\omega)}) \big) (\omega) \big|^\varrho\Big)^{1/\varrho} \Big\|_{\textup{L}^2_{\omega}(\mathbb{S}^2,\sigma)} \lesssim_{\mu,\varrho} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)}. \]
Once again, the implicit constant needs to be independent of $m$ and the functions $\varepsilon_k$. The reduction we just performed is an instance of the Kolmogorov--Seliverstov--Plessner linearization method used in \cite{MRW16:maxrestr}.

Dualizing the mixed $\textup{L}_\omega^2(\ell_j^\varrho)$-norm, see \cite{BP61}, we turn the latter estimate into
\begin{equation}\label{eq:varrest2}
\big| \Lambda(f,\mathbf{g}) \big| \lesssim_{\mu,\varrho} \|f\|_{\textup{L}^{4/3}(\mathbb{R}^3)} \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)},
\end{equation}
where the bilinear form $\Lambda$ is defined via
\[ \Lambda(f,\mathbf{g}) := \int_{\mathbb{S}^2} \sum_{j=1}^{m} \big(\widehat{f} \ast (\mu_{\varepsilon_{j-1}(\omega)} - \mu_{\varepsilon_{j}(\omega)}) \big) (\omega) \overline{g_j(\omega)} \,\textup{d}\sigma(\omega). \]
Here, $\varrho'=\varrho/(\varrho-1)$ denotes the exponent conjugate to $\varrho$ as usual, and $g_j\colon\mathbb{S}^2\to\mathbb{C}$ are arbitrary measurable functions, $j\in\{1,2,\ldots,m\}$, gathered in a single vector-valued function $\mathbf{g}=(g_j)_{j=1}^{m}$.
By elementary properties of the Fourier transform, $\Lambda$ can be rewritten as
\[ \Lambda(f,\mathbf{g}) = \int_{\mathbb{R}^3} f(x) \overline{\mathcal{E}(\mathbf{g})(x)} \textup{d}x, \]
where $\mathcal{E}$ is a certain extension-type operator given by
\begin{equation}\label{eq:defofE}
\mathcal{E}(\mathbf{g})(x) := \int_{\mathbb{S}^2} \sum_{j=1}^{m} \Big( \widehat{\mu}\big(\varepsilon_{j-1}(\omega)x\big) - \widehat{\mu}\big(\varepsilon_{j}(\omega)x\big) \Big) g_j(\omega) e^{2\pi i x\cdot\omega} \,\textup{d}\sigma(\omega).
\end{equation}
By H\"{o}lder's inequality, \eqref{eq:varrest2} is in turn equivalent to
\begin{equation}\label{eq:varrest3}
\| \mathcal{E}(\mathbf{g}) \|_{\textup{L}^{4}(\mathbb{R}^3)}
\lesssim_{\mu,\varrho} \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}.
\end{equation}

If we denote
\begin{equation}\label{eq:defofvartheta}
\vartheta(x) := - x \cdot (\nabla\widehat{\mu})(x),
\end{equation}
then we also have
\[ - t \frac{\textup{d}}{\textup{d}t} \widehat{\mu}(tx) = -(tx) \cdot (\nabla\widehat{\mu})(tx) = \vartheta(tx), \]
for any $x\in\mathbb{R}^3$, which in turn implies
\begin{equation}\label{eq:fundthmcalc}
\widehat{\mu}(ax) - \widehat{\mu}(bx) = \int_{a}^{b} \vartheta(t x) \frac{\textup{d}t}{t},
\end{equation}
for any $0<a<b$.
Substituting this into the definition of $\mathcal{E}$, yields
\[ \mathcal{E}(\mathbf{g})(x) = \int_{\varepsilon_{\textup{min}}}^{\varepsilon_{\textup{max}}} \vartheta(tx) \int_{\mathbb{S}^2}
g_{j(t,\omega)}(\omega) e^{2\pi i x\cdot\omega} \,\textup{d}\sigma(\omega) \frac{\textup{d}t}{t}, \]
where, for each $t\in[\varepsilon_{\textup{min}},\varepsilon_{\textup{max}})$ and each $\omega\in\mathbb{S}^2$, we denote by $j(t,\omega)$ the unique index $j\in\{1,2,\ldots,m\}$ such that $t\in[\varepsilon_{j-1}(\omega),\varepsilon_{j}(\omega))$.
Since $\vartheta(x)\geq 0$, by the standing assumption in (a), we can apply the Cauchy--Schwarz inequality in the variable $t$ to estimate
\begin{equation}\label{eq:auxest1}
| \mathcal{E}(\mathbf{g})(x) |^2 \lesssim_{\mu} \mathcal{A}(x) \mathcal{B}(\mathbf{g})(x),
\end{equation}
where
\[ \mathcal{A}(x) := \int_{\varepsilon_{\textup{min}}}^{\varepsilon_{\textup{max}}} \vartheta(tx) \frac{\textup{d}t}{t} \]
and
\begin{equation}\label{eq:calbdef}
\mathcal{B}(\mathbf{g})(x) := \int_{\varepsilon_{\textup{min}}}^{\varepsilon_{\textup{max}}} \vartheta(tx) \Big| \int_{\mathbb{S}^2} g_{j(t,\omega)}(\omega) e^{2\pi i x\cdot\omega} \,\textup{d}\sigma(\omega) \Big|^2 \frac{\textup{d}t}{t}.
\end{equation}
By \eqref{eq:fundthmcalc},
\begin{equation}\label{eq:auxest2}
\mathcal{A}(x) = \widehat{\mu}\big(\varepsilon_{\textup{min}}x\big) - \widehat{\mu}\big(\varepsilon_{\textup{max}}x\big) \lesssim 1.
\end{equation}
From \eqref{eq:auxest1} and \eqref{eq:auxest2}, we see that \eqref{eq:varrest3} will be established once we prove
\begin{equation}\label{eq:varrest9}
\| \mathcal{B}(\mathbf{g}) \|_{\textup{L}^{2}(\mathbb{R}^3)}
\lesssim_{\mu,\varrho} \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}^2 .
\end{equation}

Expanding out the square in the definition of $\mathcal{B}(\mathbf{g})(x)$ yields
\begin{equation}\label{eq:calb}
\mathcal{B}(\mathbf{g})(x)
= \int_{(\mathbb{S}^2)^2} \int_{\varepsilon_{\textup{min}}}^{\varepsilon_{\textup{max}}} g_{j(t,\omega)}(\omega) \overline{g_{j(t,\omega')}(\omega')} \,e^{2\pi i x\cdot(\omega-\omega')} \,\vartheta(tx) \frac{\textup{d}t}{t} \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega').
\end{equation}
For fixed $\omega,\omega'\in\mathbb{S}^2$, consider
\[ J(\omega,\omega') := \big\{ (j,j')\in\{1,2,\ldots,m\}^2 \,:\, [\varepsilon_{j-1}(\omega),\varepsilon_{j}(\omega)) \cap [\varepsilon_{j'-1}(\omega'),\varepsilon_{j'}(\omega')) \neq\emptyset \big\}. \]
The intersection of two half-open intervals is either the empty set or again a half-open interval.
For each pair $(j,j')\in J(\omega,\omega')$, it follows that there exist unique real numbers $a(j,j',\omega,\omega')$ and $b(j,j',\omega,\omega')$, such that
\begin{equation}\label{eq:intervals}
[\varepsilon_{j-1}(\omega), \varepsilon_{j}(\omega)) \cap [\varepsilon_{j'-1}(\omega'), \varepsilon_{j'}(\omega'))
= [a(j,j',\omega,\omega'), b(j,j',\omega,\omega')).
\end{equation}
Clearly the intervals \eqref{eq:intervals} constitute a finite partition of $[\varepsilon_{\textup{min}},\varepsilon_{\textup{max}})$. Using \eqref{eq:fundthmcalc}, we can rewrite \eqref{eq:calb} as
\begin{align*}
\mathcal{B}(\mathbf{g})(x) & = \int_{(\mathbb{S}^2)^2} \sum_{(j,j')\in J(\omega,\omega')} g_{j}(\omega) \overline{g_{j'}(\omega')}
\,e^{2\pi i x\cdot(\omega-\omega')}
\left(\int_{a(j,j',\omega,\omega')}^{b(j,j',\omega,\omega')} \vartheta(tx) \frac{\textup{d}t}{t} \right)\,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega') \\
& = \int_{(\mathbb{S}^2)^2} \sum_{(j,j')\in J(\omega,\omega')} g_{j}(\omega) \overline{g_{j'}(\omega')}
\,e^{2\pi i x\cdot(\omega-\omega')} \\
& \qquad\qquad \Big( \widehat{\mu}\big(a(j,j',\omega,\omega')x\big) - \widehat{\mu}\big(b(j,j',\omega,\omega')x\big) \Big) \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega').
\end{align*}
Taking $h\in\mathcal{S}(\mathbb{R}^3)$ and dualizing with $\widehat{h}$ leads to the form
\[ \Theta(\mathbf{g},h) := \int_{\mathbb{R}^3} \mathcal{B}(\mathbf{g})(x) \widehat{h}(x) \,\textup{d}x. \]
By Plancherel's identity we have
\[ \| \mathcal{B}(\mathbf{g}) \|_{\textup{L}^{2}(\mathbb{R}^3)}
= \sup \big\{ |\Theta(\mathbf{g},h)| \,:\, h\in\mathcal{S}(\mathbb{R}^3),\ \|h\|_{\textup{L}^2(\mathbb{R}^3)}=1 \big\}, \]
so \eqref{eq:varrest9} will follow from
\begin{equation}\label{eq:varrest4}
| \Theta(\mathbf{g},h) | \lesssim_{\mu,\varrho} \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}^2 \|h\|_{\textup{L}^{2}(\mathbb{R}^3)},
\end{equation}
which we proceed to establish.

Using basic properties of the Fourier transform, the form $\Theta$ can be rewritten as
\begin{align*}
\Theta(\mathbf{g},h)
& = \int_{(\mathbb{S}^2)^2} \sum_{(j,j')\in J(\omega,\omega')} g_{j}(\omega) \overline{g_{j'}(\omega')} \\
& \qquad\qquad \big(h\ast\mu_{a(j,j',\omega,\omega')} - h\ast\mu_{b(j,j',\omega,\omega')}\big)(\omega-\omega') \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega')
\end{align*}
for any Schwartz function $h$.
Applying H\"{o}lder's inequality to the sum in $(j,j')$, and recalling the definition of the $\varrho$-variation yields
\begin{align*}
|\Theta(\mathbf{g},h)| \leq \int_{(\mathbb{S}^2)^2}
 \Big(\sum_{j=1}^{m}|g_{j}(\omega)|^{\varrho'}\Big)^{1/\varrho'} \Big(\sum_{j=1}^{m}|g_{j}(\omega')|^{\varrho'}\Big)^{1/\varrho'}
 \| (h \ast \mu_\varepsilon)(\omega-\omega') \|_{\widetilde{\textup{V}}^\varrho_\varepsilon} \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega').
\end{align*}
By the usual Tomas--Stein  restriction theorem in the formulation \eqref{eq:restriction2}, applied with $g$ replaced by $\big(\sum_{j=1}^{m}|g_{j}|^{\varrho'}\big)^{1/\varrho'}$ and with $h$ replaced by $\|h\ast\mu_\varepsilon\|_{\widetilde{\textup{V}}^\varrho_\varepsilon}$, we obtain
\[ |\Theta(\mathbf{g},h)| \lesssim \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}^2
\big\| \|h \ast \mu_\varepsilon\|_{\widetilde{\textup{V}}^\varrho_\varepsilon} \big\|_{\textup{L}^{2}(\mathbb{R}^3)}. \]
Invoking assumption \eqref{eq:condvariation} completes the proof of estimate \eqref{eq:varrest4}, and therefore also that of \eqref{eq:varrestmu}.
\end{proof}

Next, we will impose condition (b) and reduce the proof to the previous one by replacing $\mu$ with a superposition of ``nicer'' measures.

\begin{proof}[Proof of Theorem~\ref{thm:maintheorem} assuming condition (b)]
We can repeat the same steps as before, reducing \eqref{eq:varrestmu} to \eqref{eq:varrest9}, where $\mathcal{B}$ is as in \eqref{eq:calbdef} and $\vartheta$ is defined by \eqref{eq:defofvartheta}.

We have already noted that \eqref{eq:condvariation} is satisfied for measures $\mu$ with Gaussian densities, i.e., when $\textup{d}\mu(x)=\alpha^3 e^{-\pi \alpha^2|x|^2}\,\textup{d}x$ for some $\alpha\in(0,\infty)$, and that in this case \eqref{eq:forGauss} equals $\psi(x/\alpha)$,
where
\[ \psi(x) := 2\pi|x|^2 e^{-\pi|x|^2}. \]
Therefore, the previous proof of \eqref{eq:varrest9} specialized to this measure yields
\begin{equation}\label{eq:varrest8}
\bigg\| \int_{\varepsilon_{\textup{min}}}^{\varepsilon_{\textup{max}}} \psi\Big(\frac{tx}{\alpha}\Big) \Big| \int_{\mathbb{S}^2} g_{j(t,\omega)}(\omega) e^{2\pi i x\cdot\omega} \,\textup{d}\sigma(\omega) \Big|^2 \frac{\textup{d}t}{t} \bigg\|_{\textup{L}^{2}_{x}(\mathbb{R}^3)}
\lesssim_{\varrho} \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}^2 .
\end{equation}
Also note that the $\textup{L}^1$-normalization of the above Gaussians guarantees that the left-hand side of \eqref{eq:condvariation} does not depend on the parameter $\alpha$. Consequently, the previous proof makes the constant in the bound \eqref{eq:varrest8} independent of $\alpha$ as well.
In this way, estimate \eqref{eq:varrest9} for a general measure $\mu$ satisfying condition (b) will be a consequence of \eqref{eq:varrest8} and Minkowski's inequality for integrals if we can only dominate $\vartheta(x)$ pointwise as follows:
\begin{equation}\label{eq:gausslemma3}
|\vartheta(x)| \lesssim_{\mu,\delta} \int_{1}^{\infty} \psi\Big(\frac{x}{\alpha}\Big) \frac{\textup{d}\alpha}{\alpha^{1+\delta}},
\end{equation}
for each $x\in\mathbb{R}^3$.

Denote by $\Psi(x)$ the right-hand side of \eqref{eq:gausslemma3}, and observe that $\Psi(0)=0$ and $\Psi(x)>0$ for each $x\neq 0$.
By continuity and compactness it suffices to show that the ratio $|\vartheta(x)|/\Psi(x)$ remains bounded as $|x|\to\infty$ or $|x|\to0^+$.

Substituting $r=\pi\alpha^{-2}|x|^2$, we may rewrite $\Psi$ as
\begin{align}
\Psi(x) & = 2\pi |x|^2 \int_{1}^{\infty} e^{-\pi\alpha^{-2} |x|^2} \frac{\textup{d}\alpha}{\alpha^{3+\delta}} \label{eq:gausspsi1} \\
& = \int_{0}^{\pi|x|^2} e^{-r} \Big(\frac{\sqrt{r}}{\sqrt{\pi}|x|}\Big)^{\delta} \,\textup{d}r. \label{eq:gausspsi2}
\end{align}
From \eqref{eq:gausspsi2} we see that $\lim_{|x|\to\infty} \Psi(x)/|x|^{-\delta} \in (0,\infty)$, while decay condition \eqref{eq:decaycond} gives
$|\vartheta(x)|=O(|x|^{-\delta})$ as $|x|\to\infty$.

On the other hand, using Taylor's formula for the function $x\mapsto e^{-\pi\alpha^{-2} |x|^2}$ and substituting into \eqref{eq:gausspsi1}, we easily obtain
\[ \Psi(x) = 2\pi |x|^2 \Big(\frac{1}{2+\delta} + O_{\delta}(|x|^2)\Big) \]
on a neighborhood of the origin. Moreover, $\widehat{\mu}$ is $\textup{C}^2$ and even, since $\mu$ is even, and so we have that $(\nabla\widehat{\mu})(0)=0$. Taylor's formula then yields
\[ \vartheta(x) = O_{\mu}(|x|^2) \]
on a neighborhood of the origin.
It follows that $|\vartheta(x)|/\Psi(x) = O_{\mu,\delta}(1)$ for sufficiently small nonzero $|x|$, and this completes the proof of \eqref{eq:gausslemma3}.
\end{proof}

The Gaussian domination trick which we have just used can be attributed to Stein, see \cite[Chapter~V, \S 3.1]{S70:sing}. It was generalized and used in a slightly different context by Durcik \cite{D15:L4entangled}.

\begin{proof}[Proof of Theorem~\ref{thm:maintheorem} assuming condition (c)]
We can repeat the same steps as before that reduce \eqref{eq:varrestmu} to \eqref{eq:varrest3}.
This time we define the form $\Theta$ differently, via
\[ \Theta(\mathbf{g},h) := \int_{\mathbb{R}^3} |\mathcal{E}(\mathbf{g})(x)|^2 \,\widehat{h}(x) \,\textup{d}x, \]
where $\mathbf{g}$ is as before and $h\in\mathcal{S}(\mathbb{R}^3)$.
Again, by duality, we only need to establish \eqref{eq:varrest4}.

Squaring out \eqref{eq:defofE} and inserting that into the above definition of $\Theta$, we obtain
\begin{align*}
\Theta(\mathbf{g},h)
& = \int_{\mathbb{R}^3} \int_{(\mathbb{S}^2)^2} \sum_{\substack{1\leq j\leq m\\ 1\leq k\leq m}} g_{j}(\omega) \overline{g_{k}(\omega')}
\,\Big( \widehat{\mu}\big(\varepsilon_{j-1}(\omega)x\big) - \widehat{\mu}\big(\varepsilon_{j}(\omega)x\big) \Big) \\
& \qquad\qquad\quad \Big( \overline{\widehat{\mu}\big(\varepsilon_{k-1}(\omega')x\big)} - \overline{\widehat{\mu}\big(\varepsilon_{k}(\omega')x\big)} \Big)
\,\widehat{h}(x) \,e^{2\pi i x\cdot(\omega-\omega')} \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega') \,\textup{d}x,
\end{align*}
i.e.,
\begin{align*}
\Theta(\mathbf{g},h)
& = \int_{(\mathbb{S}^2)^2} \sum_{\substack{1\leq j\leq m\\ 1\leq k\leq m}} g_{j}(\omega) \overline{g_{k}(\omega')} \\
& \qquad\qquad \big(h \ast ( \mu_{\varepsilon_{j-1}(\omega)} - \mu_{\varepsilon_{j}(\omega)} ) \ast \overline{( \mu_{\varepsilon_{k-1}(\omega)} - \mu_{\varepsilon_{k}(\omega)} )} \,\big) (\omega-\omega') \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega').
\end{align*}
Applying H\"{o}lder's inequality to the sum in $(j,k)$, and recalling the definition of the biparameter $\varrho$-variation seminorm yields
\begin{align*}
|\Theta(\mathbf{g},h)| \leq \int_{(\mathbb{S}^2)^2}
& \Big(\sum_{j=1}^{m}|g_{j}(\omega)|^{\varrho'}\Big)^{1/\varrho'} \Big(\sum_{k=1}^{m}|g_{k}(\omega')|^{\varrho'}\Big)^{1/\varrho'} \\
& \| (h \ast \mu_\varepsilon \ast \overline{\mu}_\eta)(\omega-\omega') \|_{\widetilde{\textup{W}}^\varrho_{\varepsilon,\eta}} \,\textup{d}\sigma(\omega) \,\textup{d}\sigma(\omega').
\end{align*}
Using \eqref{eq:restriction2} we obtain
\[ |\Theta(\mathbf{g},h)| \lesssim \Big\| \Big( \sum_{j=1}^{m} |g_j|^{\varrho'} \Big)^{1/\varrho'} \Big\|_{\textup{L}^2(\mathbb{S}^2,\sigma)}^2
\big\| \| h \ast \mu_\varepsilon \ast \overline{\mu}_\eta \|_{\widetilde{\textup{W}}^\varrho_{\varepsilon,\eta}} \big\|_{\textup{L}^{2}(\mathbb{R}^3)}, \]
and it remains to invoke the assumption \eqref{eq:conddouble}. This proves \eqref{eq:varrest4} and, thus, also completes the proof of the theorem assuming condition (c).
\end{proof}

\section*{Acknowledgments}
The authors are grateful to Polona Durcik and Jim Wright for useful discussions surrounding the problem.

V.K. was supported in part by the Croatian Science Foundation under the project UIP-2017-05-4129 (MUNHANAP).
V.K. also acknowledges partial support of the DAAD--MZO bilateral grant \emph{Multilinear singular integrals and applications} and that of the \emph{Fulbright Scholar Program}.
D.O.S. acknowledges support by
the EPSRC New Investigator Award \emph{Sharp Fourier Restriction Theory}, grant no.\@ EP/T001364/1, and the College Early Career Travel Fund of the University of Birmingham.
This work was started during a pleasant visit of the second author to the University of Zagreb, whose hospitality is greatly appreciated.

redpajama_set_name: RedPajamaArXiv


text:
Nørre Herlev Kirke er en kirke i Nørre Herlev Sogn, Hillerød Kommune.

Kirkens orgel er fra 1963 og blev udvidet og omdisponeret i 2005.

Eksterne kilder og henvisninger
 Nørre Herlev Kirke hos KortTilKirken.dk
 Nørre Herlev Kirke hos danmarkskirker.natmus.dk (Danmarks Kirker, Nationalmuseet)

Kirker i Hillerød Kommune
Kirker i Helsingør Stift
Kirker i Frederiksborg Amt

redpajama_set_name: RedPajamaWikipedia


text:
Campillos-Paravientos é um município da Espanha, na província de Cuenca, comunidade autônoma de Castela-Mancha. Tem  de área e em  tinha  habitantes (). Limita com os municípios de Alcalá de la Vega, Boniches, Fuentelespino de Moya e San Martín de Boniches.

Municípios de Cuenca (província)
Municípios por nome de Castela-Mancha
Municípios da Espanha por nome
Localidades de Cuenca (província)
Localidades de Castela-Mancha
Localidades da Espanha


Please Note: This proposal does not require funding for just an idea.
The founders have already spent US$150,000 for product architecture, video analytics, design, Android mobile app as well as website development to build the MVP (minimum viable product).
NEM community members can try out the MVP at https://mvp.prover.io37.
At the moment the service operates on the basis of Ropsten Testnet. We want to move our technology to NEM, because Ethereum has extremely expensive transactions and works too slowly.
PROVER is a blockсhain service for confirming the authenticity of the events, facts, or phenomena. This is achieved through the confirming the authenticity of the video on which this event was recorded. If the video is real - the fact is true. If the video being faked - no. It can be taken into account to make legal and financial decisions based on it.
It guarantees people and companies the protection from false information that criminals create for their criminal purposes.
Everyone can record a video at any time. It is fast, simple, cheap and convenient! Many companies use video content in their business processes - it gives them competitive advantages. But technologies allow to edit and fake videos, which seriously undermines the credibility of that format.
1:The video was recorded on a real camera, and not mounted in advance and emulated via a virtual camera as fake.
2:The video does not contain traces of editing and modification.
3:The video was recorded in a strictly defined time interval. Not earlier and no later than specific timestamps, objectively stored and verified in the blockchain.
The service will make money by providing technology for implementation in third-party solutions. As analogue of business model of CAPTCHA services. Payment is expected to be taken at each act of video verification.
We are seeking 600,000 XEMs to complete the development project for NEM blockchain and create good self-funded business by the end of 2018.
PROVER will work on the basis of NEM blockchain and utilize XEMs. Therefore, this platform will not only lead to growing demand for XEMs, it will also increase transactions on the NEM blockchain, thereby contributing to the NEM community. Hence, the success of this project will contribute to the ROI of every NEM supporter because of the increased visibility of the NEM blockchain as a business-focused smart asset blockchain as well as growth in XEM value.
Furthermore, the code that runs PROVER will be open sourced in order to contribute to the knowledge base on which more applications can be built on NEM. Our GitHub - https://github.com/proverproject.
Please support this proposal so that we can all benefit together as a community.
We live in the world of rapidly evolving technology that dictates new consumption patterns and communication culture bringing faster, cheaper and more convenient ways for people to interact with each other and get services from vendors. On the other hand, this new environment leaves concerning possibilities for exploiting these new forms of communication against its adopters threatening to undermine the trust between the members of the new digital society. New security solutions are required to address these concerns and eliminate the risks of fraud and disinformation in the market.
We introduce PROVER – an open source decentralized Blockchain platform for verification of authenticity of user-generated video files. The purpose of PROVER is to eliminate forgery of video materials and confirm their authenticity using video analysis algorithms and Blockchain.
Growing affordability of smartphone devices contributes to their massive adoption worldwide. The average price of an android smartphone in 2016 was $208 compared to $380 5 years ago, which translates into 11.4% annual decrease in price. This resulted in YoY increase in a number of smartphones users by 12% in 2016 and 25% in 2015 respectively reaching 1.48 bln units in January – August 2017. The estimated total number of smartphone users globally is of around 2.9 bln. There is hardly any country in the world that does not have a significant smartphone users' community.
These fundamentals shape the behavioral patterns of younger generations and significantly influence those of older generations. In 2016 an average US citizen preferred to spend almost 13% of his lifetime looking at smartphone screen. The vast majority of that time is dedicated to social networking and direct communication. People value their time more than ever and often prefer texting and video messaging instead of calling or meeting in person.
It is only natural that many projects these habits and patterns on other spheres of life and business and request a similar functionality from their counterparts. Many businesses are facing a sharp need to pursue digitalization in order to retain clients. Many large corporations are allocating increasingly high budgets in 2016 to develop new client-oriented products and solutions that would fit in the new digital paradigm (bring examples). Others are falling behind and lose their market share to new ventures oriented on digital audience from day one.
This tendency supports creation of a vast amount of user generated video content, for example – a trend that has become a phenomenon in recent years shaking media, news, education and entertainment industries all over the globe. Moreover, such video content is also required and actively relied on in financial, insurance, judicial, medical and other industries.
However, the authenticity of such digital video recordings, when it comes to correct depiction of the events and facts of commercial and legal nature, can often be questioned due to numerous ways of manipulating, editing and faking the video files. There are multiple ways of compromising the authenticity of the user generated video recordings not limited to using a virtual camera (emulator), including) changing the date of the recording, artificial alteration of the video.
In this regard, there is a fundamental need for an independent, decentralized service that will objectively guarantee the authenticity of the created video content and protect it from a possible forgery and unfair editing. PROVER platform relies on Blockchain to guarantee the authenticity of the video recording referencing to a specific time of the recording and its authenticity. We expect PROVER solution to make a serious positive impact on the development of the digital economy and help thousands of businesses from dozens of different areas as well as make the lives of their clients easier.
PROVER is an open source decentralized Blockchain platform for verification of authenticity of user-generated video files.
The purpose of PROVER is to eliminate forgery of video materials and confirm their authenticity using video analysis algorithms and Blockchain.
To confirm the authenticity of the video recording PROVER relies on its original open source technology, which allows a user to record a video file on top of the swype code randomly generated by the PROVER smart contract creating a unique digital signature of the file.
PROVER relies on the immutability of Blockchain to store and timestamp the digital signature of the video file that can later be used as a reference to prove the authenticity of the video recording when the user presents the video file to PROVER client.
To eliminate forgery (editing, CGI, video emulation) of the presented video file PROVER uses its original open source video analysis algorithm that detects any manipulations with the original unedited video content.
PROVER team has been working on the solution of this problem since October 2016, MVP is in development since August 2017. MVP of the project is available at https://play.google.com/store/apps/details?id=io.prover.provermvp and https://mvp.prover.io45.
1:NEM's Smart Asset System allows us to define and launch our own PROOF token. PROOF Tokens owners have the right to access PROVER and to obtain services that will be provided by PROVER.
2:NEM blocks complete every 60 seconds. This is important for the PROVER. We must store the information about the swype code creation in the blockchain, and the time of the user's result waiting depends on the transaction confirmation time.
3:We are going to create our own Namespace. It lets us create a unique place for our project data on the NEM blockchain. This makes our assets unique, easy to use, and more trustworthy.
4:We will use Messages to store information in NEM blockchain.
5:Transaction fees are kept low. For each video confirmation we need two transactions, one for creating and saving the swype code, and the other for storing the hash of the recorded video file.
1:User follows the app instructions and moves the smartphone up, down, left, right or diagonal in a certain pattern before, during or after the recording (maintaining the integrity of the recording).
2:If user follows the instructions correctly SWYPE ID creates a unique video sequence with embedded smartphone camera movements and hashes the file creating a unique digital number.
3:SWYPE ID addresses PROVER smart contract, which places the hash of the video file to Blockchain, thus putting a timestamp on the file.
4:User sends the video file to PROVER client (bank, insurance company, news media, etc.) to provide access to its services.
5:PROVER client uses PROVER platform to confirm the authenticity of the recording and to hash the received video file to confirm the hash was properly placed on the Blockchain.
1:User launches the app and inputs the text information which he wants to be saved in blockchain and to be associated with video.
2:Then user sends entered text to the blockchain and gets the hash of the transaction and block, which contains this text in blockchain. The hash of the transaction and block are displayed as QR code.
3:QR code appears on the screen of the app and user can capture it while filming the video by any kind of digital cameras.
4:User can check the video file with that QR code by uploading it using frontend. Video analytics (on a backend) finds and recognizes QR-code, searches for a block in the blockchain, then retrieves the stored information and detects the block time. If found, - the video is confirmed.
If system confirms the file as truthful - it generates the certificate that can be used as a key to access the actual blocks containing this information (hash, e.t.c).
・Prover Swype-ID application v.1 release (Android and iOS).
・Prover Clapperboard application v.1 release (Android and iOS).
・PROVER technology (Swype-ID or Clapperboard).
・Algorithm for smartphone movements tracking based on accelerometer and gyroscope data.
・Algorithm of manipulation detection during the input of the swype code on the generated flat image, for example from the TV screen or poster.
・Algorithm for confirming the continuity of video recording.
・Algorithm for detection frame falsification (frame insertion and replacement).
・Algorithm for detecting mounted video (replacing an object in a current background).
・Algorithm for detection the deletion of objects.
・Prover Swype-ID application v.2 release (Android and iOS).
・Prover Clapperboard application v.2 release (Android and iOS).
Our technology can put a reliable barrier in the way of scammers and open widely the doors to online and crypto-economy for whole branches of banking and insurance, legal and other conservative spheres.
It should also note the global benefits of our project for the development of the blockchain community. Our system could be a driver for the popularization of blockchain technology and crypto-currency among the population of our planet. It will bring millions of new users from all over the world to blockchain economy, increasing the overall volume and popularity of the blockchain economy!
1. Why PROVER platform needs decentralization?
Decentralization is essential for PROVER platform users to guarantee transparent process of video verification and eliminate trust issues to any particular black box protocol. User generated video content needs an authenticity industry standard that will be used by third party developers to create applications without the risk of being fraudulent or compromised.
2. Will the entire PROVER algorithm be in open source?
Yes, we are convinced that this is the only way to gain trust in PROVER protocol and allow third party developers contribute to developing the platform and its features going forward to reach massive adoption.
3. Does PROVER get access to the user videos?
Neither PROVER nor SWYPE ID app gets access to the user videos. Video files are stored on user's smartphones and are transferred to recipients directly.
4. How does the recipient confirm the authenticity of the video file?
The recipient runs PROVER algorithm when opening the video file to confirm its authenticity and hash on the Blockchain.
5. Video authenticity check on the recipient side requires processing power to perform certain calculations, can the calculations be performed in a decentralized manner as well?
We believe that in most cases processing will happen on either the recipient or the developer's server. PROVER Platform does not aim to solve the problem of distributed computing; however, we are closely following the developments in this field.


text:
Home Miscellaneous How many battalions are in the REME?
How many battalions are in the REME?
Where are 2 Battalion REME based?
What does the REME cap badge mean?
What Colour is the REME beret?
How many trades are in the REME?
When did the REME cap badge change?
What weapons do the REME use?
What happened to the 1st Close Support Battalion REME?
What is the role of a REME in the Army?
How many REME Battalions are there in Germany?
Additionally, the REME consist of eight Regular Battalions and three Reserve Battalions. If variety is what you want – REME is the place for you.
Leuchars Station
2 Close Support Battalion REME is a battalion of the Royal Electrical and Mechanical Engineers of the British Army….
2 Close Support Battalion REME
Size Battalion 278 personnel
Part of 101 Logistic Brigade
Garrison/HQ Leuchars Station, Fife
Engagements Operation Telic Operation Herrick
What does Remy mean in the Army?
The Corps of the Royal Electrical and Mechanical Engineers (REME) provides engineering support and is responsible for maintaining and repairing the Army's equipment. They will be found wherever the army is located at home or overseas.
The horse stands on a globe and above it is a scroll inscribed 'REME' surmounted by a crown. Behind the horse there is a lightning flash. This symbolises electrical engineering while the globe stands for the world-wide role of the unit. The chained horse symbolises power under control.
Blue Beret
The Blue Beret represents a huge achievement within the Aviation trades of REME and is awarded to students who have successfully completed basic trade training to progress in their career within the Aviation field of the Corps.
Does REME fight?
The role of the Royal Electrical and Mechanical Engineers (REME) is to do just that; maintain, repair and manufacture equipment to keep battle winning aircraft, tanks and weapons in fighting order. The REME exists to 'keep the punch in the Army's fist'.
8 trades
All 8 trades of the REME offer incredible opportunities for women to work on the cutting edge of engineering. Here you'll find out more about the role of an Aircraft Technician, from a Lance Corporal exploring engineering opportunities within REME.
Shortly after the accession of Queen Elizabeth II in 1952 the crown of the REME badge, in common with other Regiments and Corps, was changed to the King Edward's crown (also known as the Queen's Crown). This is the badge we still wear today.
What Colour is REME blazer?
Make your blazer stand out with this hand embroidered REME Cap badge with added Gold Wire – only the best material for our customers….REME Blazer Badge Gold Wired.
Colour Black, Navy
In total, the Museum's weapons collection numbers more than 800 objects and includes firearms, edged weapons such as swords and knives, polearms, artillery and ammunition.
Shortly after returning, the battalion was redesignated as part of the 2003 Defence White Paper to become 1st Close Support Battalion REME. As part of this reform, one brigade, 4th Armoured Brigade, was selected to move to Catterick Garrison in 2008, and the battalion followed, settling at Megiddo Lines, where they remain today.
How are REME Battalions formed?
Under this paper, new 'REME battalions' were to be formed by grouping previously independent REME workshops and companies within the two deployable divisions. At first all independent REME units would be grouped into these battalions, but this was later reduced to just three based in Germany.
At first all independent REME units would be grouped into these battalions, but this was later reduced to just three based in Germany. In September 1993, 1st Battalion REME was formed through the grouping of the following independent units: 12 Armoured Workshop HQ, 4 Close Support Company, and 12 General Support Company.
Previous Article How many numbers of unrooted trees are expected for 5 taxa?
Next Article What is acclimatization in plant tissue culture?

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Memory allocator for a light tree based data structure for frequent allocation and deallocation I have a binary tree based data structure that does all operations in O(log N). It allocates / deallocates memory via new and delete. In my typical usage, the data structure is in use for a 40 minutes run and allocates/de-allocates small chunks tree nodes half a billion times. The overall memory footprint of this data structure is low as de-allocation happens soon enough. This has been identified as a performance bottleneck.
I am thinking of overloading operator new and delete or modifying my existing code to be able to use allocators and create a custom allocator for such purpose. I am thinking of allocating contiguous large amount of memory upfront and re-using chunks of memory from it. Does C++11 standard have such a pool based memory allocator?
What memory allocation/de-allocation strategy would be best suited for such usage?

A: There is no out-of-the-box feature in C++11 for memory pooling. But you can use Boost.Pool to this aim. From the Boost.Pool Documentation:

When should I use Pool?
Pools are generally used when there is a lot of allocation and deallocation of small objects. Another common
usage is the situation above, where many objects may be dropped out of
memory.
In general, use Pools when you need a more efficient way to do unusual memory control.

A: Use C++17 std::pmr::unsynchronized_pool_resource or the synchronized version.

redpajama_set_name: RedPajamaStackExchange


text:
Centre de recherche sur les civilisations de l'Asie orientale - CRCAO
Présentation de l'unité
Équipe Chine
Équipe Japon
Equipe Tibet (TBACT)
Recherche par équipe
Programmes transversaux
Dunhuang Academy
Mission archéologique franco-chinoise dans le Nord-Est de la Chine (Mafnec)
TibArmy - L'armée tibétaine des Dalaï-lamas (1642-1959)
Encyclopaedia of Manuscript Cultures in Asia and Africa
Mission archéologique franco-indienne au Ladakh (Mafil)
Histoire sociale des sociétés tibétaines
Hosei University
Royal University of Bhutan
Revue d'études tibétaines
EMSCAT
Bourses doctorales
Accueil > Coopérations internationales
par Moretti Costantino - publié le 7 mars 2016 , mis à jour le 8 mars 2016 à 13h02min
Cooperation agreement between the CRCAO and the Dunhuang Academy
On Monday the 23rd of October 2013 the CRCAO and the Dunhuang Academy signed a cooperation agreement, through the Research and International Relations Division of the EPHE. The agreement was concluded following the joint drafting of a scientific cooperation plan.
The CRCAO and the Dunhuang Academy have collaborated on an informal basis for at least a decade due to the important role played by this field of research in the work of the CRCAO. Due to the regularity with which the two institutions have collaborated since 2009, an agreement was established to formalize that collaboration to promote the research work of the two institutions and to further foster scientific exchange.
Since 2009, joint activities of the two Institutes contribute to animate scientific life of the CRCAO through various regular exchanges of researchers and shared activities. Several Chinese researchers were invited in Paris by the CRCAO for a period of one month each. Their contribution to the French Research Team's activities includes giving lectures, conducting round table discussions, taking part in seminars led by members of the Research Team, etc. Similarly, every year, various members of the CRCAO (professors, research fellows, doctoral candidates, post-docs, etc.) working on Dunhuang, Turfan and Central Asian studies, have been hosted by the Academy to conduct research. The Academy notably helped them to facilitate the access to the archeological sites of the region and the primary sources found in such sites. In 2011 the cooperation between the two institutions gave life to a Meeting organized by the CRCAO in cooperation with the EFEO (French School of Asian Studies), the BnF (National Library of France) and Guimet Museum : Franco-Chinese Meeting on Dunhuang Studies — Research News and Recent Publications (14th-16th June, Paris, Collège de France and Maison de l'Asie).
Visiting Scholars : http://silkroad.hypotheses.org/category/visiting-scholars-and-lectures-topic
Dunhuang Academy : http://en.dha.ac.cn


text:
Who was/is Hanns Dieter Hüsch ? - CDs, Vinyl LPs, DVD and more
Hanns Dieter Hüsch
* May 6, 1925 - December 6, 2005
Hanns Dieter Hüsch * May 6, 1925 - December 6, 2005 After studying theater studies, the man from Moers lands in 1947 in Mainz at the student cabaret "Die Toleranten". In 1949, he released his first own programs. In 1956 he founded "Die Arche Nova". Since then, Hüsch has been travelling in the German-speaking world as a cabaret artist and small artist, writing books, producing records (for example, in the student year 1968 "Der Lästerer" at Polydor) and appearing on radio and television. His work has earned him many honours (including the Mainz Gutenberg Plaque and the State Prize of the State of North Rhine-Westphalia, 1994).
Various Artists -
Symphony of the Stars Symphony of the Stars,
1920-90s (8-CD & 1-DVD with 192-page book)
Read more at: https://www.bear-family.de/various-sinfonie-der-sterne-sinfonie-der-sterne-20-90er-8-cd-und-1-dvd-mit-192-seitigem-buch.html
Copyright © Bear Family Records
"He is - a bush" (Thomas Rothschild)
Hanns Dieter Hüsch (1925-2005) was not really a songwriter in the narrower sense. "But also any other job title - such as cabaret artist, humorist, solo entertainer - would be too narrow", Thomas Rothschild writes in his portrait book 'Liedermacher'. But Hüsch "is literally unique within the Federal Republican scene". Therefore the last 'word' should be granted to him at this point. With a commentary on the Deutschfolkwelle in the seventies. Which is just as unique - a Hüsch. This recording was made on 6 May 1978 and was the 23rd episode of 'Hanns Dieter Hüsch's Gesellschaftsabend' at Saarländischer Rundfunk. With a selection of highlights, it was entirely dedicated to the artist's 30th stage anniversary. However, the premiere of the folklore number had already taken place two and a half years earlier, in November 1975 at the 11th 'Gesellschaftsabend'. Hüsch's own traumatic experiences at the Waldeck Festival in 1968 are clearly echoed in the satirical introduction.
www.huesch.info (Fanpage)
Various - songwriter in Germany
Vol.1, For whom we sing (3-CD)
/various-songwriter-in-Germany-vol.1-for-who-who-we-sing-3-cd.html
Copyright © Bear Family Records®. Copying, also of extracts, or any other form of reproduction, including the adaptation into electronic data bases and copying onto any data mediums, in English or in any other language is permissible only and exclusively with the written consent of Bear Family Records® GmbH.
More information about Hanns Dieter Hüsch on Wikipedia.org


Professional photos are a well-known component to a successful Agent's marketing plan and success.
But we also recognize that there are other ingredients that make up a successful, 360-degree marketing strategy.
Professional & affordable photography packages designed for every property type. All interior photos utilize our proprietary CLEARpic processing technology designed to ensure clear, high-dynamic-range situations.
Use Twilight photography to differentiate yourself from the competition.
Video is one of the fastest growing segments in Real Estate Marketing today. We can help you create high-impact property videos for every budget and marketing plan.
Use Zillow Walkthru videos to drive your listing rank to the top of the page, or have us shoot and develop dynamic videos which can be incorporated into a number of mediums.
We offer quality templates, or we can develop a custom, personalized template, specific to your needs.
Create dramatic Virtual Tours and Single Property websites to professionally showcase your properties.
Our tours can use photos, videos, and interactive 3D panoramas & floorplans. All can be customized to your taste with hundreds of available options.
Convert your Virtual Tour into a Hi-Definition Video for use on YouTube, Facebook, & other social sites.
is integrally linked to yours.
That's why we're constantly adding new services & products which are designed to help you market your home or property.
We can make for you.
Let's face it -- everyone has a camera or a phone, so we can all take photos easily. The challenge is taking good ones.
Over the past six years, we have continually honed our processing technology & workflow to provide the best possible photos at an affordable price.
Happy customers are good ones. So we try to do our absolute best, whenever & where-ever we can.
Our online scheduler lets you lock in shoot appointments in real-time -- no waiting for verbal confirmation, Saves the back & forth with your clients.
Agent schedules are always tight. Listing "go-live" deadlines have very little wiggle-room. We get it! That's why we guarantee "next day" turnaround on all photos by noon.
If you need them even faster, no problem. For a small rush fee, we will process & deliver your photos the same day we shoot.
Because we're small, we are flexible -- no big company rules. If we need to "tweak" something, we can. Need to reach us before & after regular business hours, give us a call. If it's urgent, we will respond ASAP, because like all real estate agents, we enjoy no lives, as well.
New Services on the Way!
We understand that you, the Agent, need EVERY advantage to help differentiate yourself and your team.
We're focused on the newest emerging trends & technologies, so you don't have to.
If you're not completely satisfied with our services, we guarantee we will get it right!.
See our Terms of Service for the details.


Our firm is committed to excellence in serving all individuals including those with disabilities. We work to create and promote equality for people with disabilities through awareness and open communication. In doing so, we take the following measures into consideration.
Our services are accessible to disabled individuals. Assisted devices such as wheelchairs, walkers, and oxygen tanks are accommodated in our workplace. Our premises are wheelchair accessible. Our staff is trained not to touch or handle any assistive device without permission. We take caution not to move the assistive device or equipment such as canes, walkers out of the customers reach. We make our customers aware of any accessibility features in the area that are appropriate to their needs.
We communicate with individuals with disabilities in ways that take into account their disability.
Speak directly to the individual so they can read our lips if they are able.
Offer pen and paper for written communication if they are unable to read lips.
Offer to read out loud to individuals.
Post policies printed in large, plain format.
Post on our website in simple formatting which is easy to read.
We communicate with an accompanied intervener or support person. We would take care to speak with the customer who is deafblind and not to the intervener.
We speak slowly and clearly, communicate by pen and paper, or their choice of communication method. We are patient while allowing the individual to finish their sentences. When possible we ask questions that may be answered with a "yes", or "no".
We take our time and are patient when communicating with someone with a learning disability. We take into account the customer`s disability when communicating with them.
We do not make assumptions about what they can and cannot do. We use plain language when communicating with them and provide information one piece at a time.
We demonstrate the same respect and consideration we show everyone else. When communicating, we are confident, calm and reassuring, and if in a crisis situation ask the individual to tell us the best way we can help.
For any disabled persons attempting to access our services, we individualize accommodation as required. As technology continues to change and improve, we will provide the best solution available at that time.
We are aware that a service animal is not a pet and will refrain from touching or addressing the animal.
If we are unaware as to whether or not the animal is a pet or a service animal, we ask the individual.
We welcome any person who may support our customer by means of communication, assisting with mobility, personal care or medical needs.
A person with a disability who is accompanied by a support person is welcome to have that person accompany them anywhere on our premises.
In the reception area, we ensure there is a chair available beside the candidate who requires the assistance. Additionally, we ensure the consultant has a second guest chair in their office to accommodate seating for the support person. We will take care to direct our communication to our customer and not to the support person.
Should we notice that our customer is having difficulty accessing our services, we ask how we can be of assistance.
Should a temporary disruption occur we will notify individuals promptly. This notice will be clearly posted and will include information about the reason for the disruption, its anticipated length of time, and a description of alternative facilities or services, if available.
Telephone, email, poster on main door, elevators and in reception.
Send email to staff so they can communicate to those affected.
We train all permanent staff members on our accessibility policies including team members in the positions of First Impressions Specialist, Placement Consultant, Recruiter, Corporate Service, Accounting, all Management and Principles of the firm and Temporary Employees.
This Accessible Customer Service Plan will be provided to new staff members during their initial training period, which spans approximately two weeks from their start date.
Our customer service training will be incorporated into our Initial Training Guide. A training log of who was trained when and on what will be maintained.
Our plan related to the customer service standard.
Video "Service-Ability: Transforming Ontario`s Customer"
What to do if a person with a disability is having difficulty in accessing our services.
Health and Safety Handbook for Internal Employees and New Candidates.
Our customer service plan is in a format, which takes into account people with disabilities.
All feedback will be directed to Accessibility Ambassador. Those offering feedback can expect to hear back from our firm within seven business days.
Complaints will be addressed according to our organization`s regular complaint management procedures.
Any policy of ours that does not respect and promote the dignity and independence of people with disabilities will be modified or removed.
An individualized plan for emergency preparedness for employees with disabilities will be made if a disability is brought to our attention.
This 2014-21 accessibility plan outlines the policies and actions that Turtle Island Staffing will put in place to improve opportunities for people with disabilities.
Turtle Island Staffing is committed to treating all people in a way that allows them to maintain their dignity and independence. We believe in integration and equal opportunity. We are committed to meeting the needs of people with disabilities in a timely manner and will do so by preventing and removing barriers to accessibility and meeting accessibility requirements under the Accessibility for Ontarians with Disabilities Act.
Turtle Island Staffing will continue to provide employees with disabilities with individualized emergency response information when necessary.
Turtle Island Staffing will continue to provide training to employees and staff members on Ontario's accessibility laws and on the Human Rights Code as it relates to people with disabilities. Training has been integrated into the company's existing training schedule.
Turtle Island Staffing is committed to meeting the communication needs of people with disabilities. We have, and will continue to, consult with people with disabilities to determine their information and communication requirements.
When building and updating the company's websites, we referred to WCAG 2.0 to ensure appropriate measures to comply were fulfilled.
We took into account (and will continue to) all feedback about our website in order to accommodate the needs of people with disabilities, to the best of our ability.
Those who wish to provide feedback on the way Turtle Island Staffing provides services to people with disabilities can email or telephone our firm. Contact information is posted on our website and in our reception areas.
As of January 1, 2016 those who require publicly available information can make a request via the contact information posted on our website. We will take into account their disability when making the necessary provisions.
Turtle Island Staffing will take the following steps to make all websites and content confirm with WCAG2.0 Level AA by January 1, 2021.
When building or significantly changing the company's website, we will refer to the required standards of WCAG 2.0 by January 1, 2021.
We will take into account any feedback about our website to accommodate the needs of people with disabilities, to the best of our ability.
Turtle Island Staffing is committed to fair and accessible employment practices. We will continue take the following steps to notify candidates and staff, when requested the company will accommodate people with disabilities during the recruitment and assessment processes and when people are hired.
The company's Accessible Customer Service Plan welcoming Assistive Devices, Service Animals, and Support Persons is posted on our website and in reception areas.
During the recruitment and assessment process we communicate with candidates with disabilities in ways that take into account their disability and request that our clients make accommodations accordingly.
Turtle Island Staffing will take the following steps to develop and put in place a process for developing individual accommodation plans and return to work policies for employees that have been absent due to a disability.
An individualized plan for emergency preparedness for employees with disabilities will continue to be created should a disability be brought to our attention.
An Individualized return to work plan will be created for an employee who has been absent due to a disability.
We will take the following steps to ensure the accessibility needs of employees are taken into account if the company uses performance management, career development and redeployment processes.
To identify and locate opportunities for performance management, career development and redeployment processes that are available to those employees with disabilities, are comparable to those employees without.
Turtle Island Staffing will meet the Accessibility Standards for the Design of Public Spaces when building or making major modifications to public spaces.
In the event of a service disruption, we will notify our candidates and employees of the service disruption and of alternatives, if available.


text:
Home News Politics Buhari ignores Saraki, Dogara, others in new peace talk
Buhari ignores Saraki, Dogara, others in new peace talk
President Muhammadu Buhari has called the bluff of the new Peoples' Democratic Party (nPDP) faction of the All Progressives Congress (APC), National Daily has gathered.
The bloc which has the Senate President, Bukola Saraki, Speaker of the House, Yakubu Dogara and several others pulled out of the PDP to team up with the APC for the 2015 elections had demanded to meet with the president to discuss some pressing issues.
A presidential source confirmed to National Daily that President Buhari has vowed never to sit down for a discussion with the aggrieved members.
The source said the president made his stance known to governors elected on the platform of the APC when he met them recently.
"The president told the governors as well as national leaders of the party pointedly to resolve the issues," the source said.
He said the issue was a party matter and he was not ready to sit with any faction.
"It is a party matter. I am not ready to sit down with any faction. If they have problems, they should go to the party. I will not interfere," Buhari was quoted to have said.
"Governors as party leaders in the states should deal with all issues. Where there is need, the party leadership can come in. I will not get involved."
Buhari was said to have commended the earlier meeting with Vice-President Yemi Osinbajo and gave his nod that the discussion should continue.
The source also added that opinions were divided on the issue among the governors that attended.
"Some hardline governors asked the President to ignore the nPDP while the majority felt the party and the VP should continue to talk to them," the source said.
Garba Shehu, senior special assistant to the president on media and publicity, did not respond to inquiries as of the time of filing this report.
Last week, Kawu Baraje, leader of the nPDP, said the group would take a decision if the ruling party calls its bluff.
He had led the bloc to meetings with the national working committee of the APC and Osinbajo.
But follow-up meeting did not hold in protest of alleged harassment of Senate President Bukola Saraki and Yakubu Dogara, speaker of the House of Representatives, who are members of nPDP.
Baraje had kicked against the withdrawal of some of the security aides of the national assembly leaders, last month.
The security aides were later reinstated.
There are reports that the nPDP would pull out of the ruling party before 2019 elections.
Previous articleSickle cell patients get succor as M&B commences production of drug
Next article ISIS attacks: FG intensifies screening of passengers, luggage at airports
Tension in Presidency over Supreme Court's ghost votes for APC in Imo


The 2019 Toyota Prius now comes with optional all-wheel-drive. AWD is a key feature if you live in a climate that has rough weather. Buying a Toyota Prius is also doing a favor for the environment. Why? It is one of the world's most efficient mass produced vehicles. Prius owners report on Fuelly that they get about 45 MPG in real-world driving. This is useful when fuel prices are high.
The Toyota Prius has a "funky" interior, notes Car and Driver Magazine. It has sleek headlights, and a bold large navigation device with a bright screen. Both the headlights and taillights use LED's, which will give the new Prius a long illumination life. If you pay more money, you can get amazing gadgets like a Heads Up Display (HUD) and lane guidance system. The Toyota Prius is packed full of modern amenities that will make the owner happy. All 2019 Toyota Prius vehicles come with standard Safety Sense-P features. These features include pedestrian detection and adaptive cruise control.
The hybrid engine of the 2019 Toyota Prius is a 1.8L gas internal combustion engine coupled to a nickel hydride rechargeable battery. Combined, the engine outputs 121 horsepower. Car and Driver emphasizes how the nickel hydride battery will have good performance in cold weather environments. Nickel is a silvery metal discovered in 1751 that is highly resistant to corrosion. Hydride is a name for hydrogen when hydrogen is connected to a metal in a substance. Hydrogen is a low density gas that has no color. Hydrogen was discovered in 1766, and it is one of the two elements inside water molecules. Nickel hydride batteries are environmentally friendly, have high energy capacity, and are very recyclable. The Toyota Prius was designed so that its power-plant components are effective and add value to the driver's experience. Nickel hydride batteries are also used for digital cameras because of their beneficial qualities.
The NHTSA has revealed the Toyota Prius safety ratings for front, side, rollover, and overall. The Toyota Prius gets 4/5 stars for front crashes. It scores in at 5/5 stars for side crashes. Finally, the score for rollover on the Prius is 4/5 stars. The Toyota Prius is safest when dealing with side crashes. This is a useful thing to consider if your commute involves a lot of busy intersections in a dense urban setting. The overall Toyota Prius rating from U.S. News and World Report NHTSA data is 5/5 stars.


text:
WrestleMania X And XII Are My Favourites Because… (Pt. 1)
Lee Herbert
I would like to begin with an apology for what has now become a slightly misleading title. I was asked to write about my favourite WrestleMania and immediately chose X and XII as, from memory, they were my joint favourites. As I hadn't watched either in full since they were originally broadcast I decided to revisit both which is so convenient to do now with the fantastic WWE Network. While I watched both I quickly realised as much as we complain about things in the present, we can also sometimes reflect on the past with rose-tinted specs. The great memories of the past were maybe not all we make them out to be and it is amazing how childhood memories can be distorted over time as we grow older (just look listen to Hulk Hogan for evidence of how more recent memories can become distorted!).
Regardless the standout WrestleMania memories I had were of my heroes as a kid, "Mr Wrestlemania" himself, "The Heartbreak Kid" Shawn Michaels and "The best there is, the best there was and the best there ever will be", one of the greatest technical wrestlers in WWE history Bret "The Hitman" Hart. The reason these WrestleMania's became so lodged in my brain were their amazing, and timeless performances where they pushed themselves to their physical and mental limits.
Thankfully this is something that I hadn't exaggerated in my mind (phew!) but whether or not I would still count these as my favourite WrestleMania events all these years later you will have to read the rest of this article to find out!
WrestleMania X
A landmark WrestleMania in that it was the first not to feature Hulk Hogan and it was also the last televised WWE appearance for the soon to be inducted in the Hall Of Fame "The Macho Man" Randy Savage.
Even back in 1994 WWE couldn't make a firm decision on who they were putting in the WrestleMania main event. It was decided that in front of over 18,000 people in Madison Square Garden there would be two title matches. This came about after Bret "The Hitman" Hart and Lex Luger simultaneously eliminated each other at the Royal Rumble. A good old fashioned coin toss was used to decide who would take on the champion Yokozuna first, "The Hitman" lost and therefore took on his brother Owen Hart in the opening match.
This first match was exactly as you would expect from an encounter between Bret and Owen, a 5/5 and what must have been on the short-list for match of the year. An old fashioned tale of face against heel, excellent technical wrestling straight out of Stampede, the match did what a match at WrestleMania should and actually told a story and the 30 mins or so went by in a flash. Ok there was one surprise as Bret didn't win but then he was suffering with an ongoing knee injury (kayfabe) that looked certain to cost him his title opportunity later.
How can you follow that? Well in the sort of move you would imagine now 21 years later we go from the magnificent to the ridiculous and we are treated to a "skit" with Howard Finkel wearing a wig and then a 6 minute "comedy" match between a pair of clowns (and that isn't just me throwing insults, it was Doink and Dink) against the man that was to incredibly headline XI Bam Bam Bigelow and his girlfriend (kayfabe) Luna Vachon. A Bigelow diving headbutt (a move that makes me shudder when I see it these days) wrapped things up quickly to put everyone out of their misery. If that wasn't a lame enough attempt to bring humour to the occasion the camera then bizarrely panned to a Bill Clinton lookalike.
Next was a "falls count anywhere" with a weird stipulation that you only won after a pin if your opponent didn't return to the ring in 60 seconds. At the time I probably thought this was "extreme" and I was a fan of both Savage and Crush as a kid. However watching this now the strange gimmick didn't do anything to help the quality of match and the hardcore element, fighting backstage etc, is pretty tame in comparison to what was being shown on Raw just a few years later. The fact Crush had two haircuts on one head couldn't help distract me from the fact Savage was massively over selling some lame shots from Mr Fuji's flag. Over in ten mins after Randy tied up Crush, I'd probably give it 2/5 just because it was Savage's last 'Mania.
We then get taken back to the Bill Clinton lookalike but this time for an interview! Absolutely bonkers. Back to the ring and in a match straight out of a 2015 edition of Raw, Alundra Blayze defended the newly reinstated Women's Title against Leilani Kai in a match that was given less airtime than Finkel's wig modelling earlier. The 2015 Hall Of Fame inductee wrapped things up in 3 minutes. Some things never change I guess (apart from hash-tags).
The Tag Team Titles were on the line next as The Quebecers defended their title against Men On A Mission with a future "King Of The Ring" Mabel looking slightly slimmer than in his later days as Viscera and Big Daddy V. It was actually a better match than the horror show I expected and went at a decent pace but ended with The Quebecers running away and keeping hold of the belts with a countout defeat.
The first of the nights title matches saw a now babyface Lex Luger dressed in the stars and stripes oiled up and ready to take on Yokozuna. The twist in the tale was Mr Perfect was to be the special guest referee on his return to WWE. He had actually feuded with Luger previously and they had fought at WrestleMania IX. Just as you could predict how Owen vs Bret would turn out, you can guess how this one played out. If you had a match like this in Stu's "Dungeon" you probably wouldn't get out alive. Luger couldn't even be bothered to look interested during the many rest holds. The match did end with some excitment when Luger had Yokozuna beat after a running forearm smash but Perfect was busy getting the battered bodies of Jim Cornette and Mr Fuji, who Luger had also taken care of, out of the ring. In his frustration at the lack of a count Luger grabbed Perfect who then disqualified Luger and turned heel in the process.
I'm not even going to give the next "match" any credibility by writing about it, but there were more wig jokes and Eathquake beat Adam Bomb in about twenty seconds. Of course then we had to hear more from the fake Bill Clinton, this time with Ted Dibiase. Flogging a dead horse springs to mind.
The next match was a ladder match, an Intercontinental Title match and the second 5/5 match of the evening. Shawn Michaels, accompanied by bodyguard Diesel against Razor Ramon in a coming together of "The Kliq". They may have been men with big egos but they knew how to deliver on"The Grandest Stage of Them All". It wasn't long before Kevin "Diesel" Nash was sent to the back after taking out Ramon despite being told a few minutes earlier that there were "no rules" by the announce team.
What then follows is about fifteen minutes or so of WWE at its finest. Even if you have never seen the match in full you will have seen most of the iconic moments that are used in almost every Wrestlemania (or just WWE for that matter) montage. The match ended similarly to Crush vs Savage earlier in the evening with Michaels left tied up for Razor Ramon to climb the ladder and collect the belts. There may have been much dangerous matches in the years since this, more insane bumps but at the time this was a game changer. If you haven't seen it, I implore you to get on the WWE Network right now and play it back.
The next match, an eight man tag, was cancelled due to time constraints (they could have shaved ten mins off Yokozuna vs Luger with no complaints) so in order to justify its exclusion we next see a kayfabe argument between the heel team unable to decide who would be captain.
Yet still before the main event we welcome Burt Reynolds as ring announcer and the legendary Roddy Piper as guest referree. As previously Yokozuna tries his best to stink the place out but Bret does all he can to make a match of it and for that alone he deserved the win and his WrestleMania moment. After a ten minute bout Bret "The Hitman Hart" wins by pinfall and is then joined in the ring by all the roster babyfaces who hold him aloft in the centre of the ring. A classic WrestleMania moment.
In part two I will look at WrestleMania XII, which was the best of the two and whether either can stands tall with the more modern WrestleMania events. Ok I'm sorry I may have slightly misled you again!
Follow me on twitter if you like to tweet @LeeHWWENews
Bret HartShawn MichaelsYokozunaWresltemaniaLex LugerOwen HartRazor Ramon
Best of SummerSlam Part 1: Let's Get This Party Started!
Triple H - The True Mr. WrestleMania? Part II: A Degenerate's Rise
Triple H - The True 'Mr. WrestleMania'? - Part One: Obscure Rise
Impact Wrestling Viewership Report (01.03.19)
Raw Live Coverage 7-30-18
Zelina Vega to Make In-Ring Debut on This Week's Smackdown
Smackdown Live Coverage and Results (7/17/18)


text:
Q: PHP GET Data Fowarding I have set up a website in a way that leaves the user on a confirmation page to submit form data with the GET data present in the URL of the confirmation page (It's nothing sensitive). I'd like it so that upon pushing a button, the GET data on that page's URL is sent to another URL, is this possible?
I could probably just copy and paste the form and set it to be hidden and set the values to the individual values but that isn't what I want to do because it'd create extra work in the future adding more form inputs.
Edit:
Here is the code for the form:
<form action="confirmrequest.php">

Please choose a method of contacting you in the event more detail from you is required

            <select name="socialmediatype" required>
                <option>-Choose an option below-</option>
                <option value="Instagram">Instagram</option>
                <option value="Snapchat">Snapchat</option>
                <option value="Whatsapp">Whatsapp</option>
                <option value="Skype">Skype</option>
            </select>

            Platform Username (or phone for Whatsapp):

            <input type="text" name="contactpin" required>

Please enter your email address to receive confirmation email & download link

            Email Address

            <input type="email" name="email" required>

Which type of art style are you requesting?

            <input type="radio" name="style" value="Lineart" required> Lineart - $5

            <input type="radio" name="style" value="Coloured"> Coloured - $6

            <input type="radio" name="style" value="Background & Shading"> Background/Shading - $7

Please include a full description of what you would like

            <textarea name="description" cols="50" rows="10"></textarea>

        <input type="submit" class="btn btn-primary" value="Submit Request">
    </form>


Jesus was always telling religious people to stop it! If people who claimed to love God made a public show of their faith, Jesus saw it as a real turn-off. I imagine he felt this way because he knew irreligious people felt that way too. They were tired of the religious crowd always behaving in ways that communicated 1) we are really good, and 2) you non-religious people are not. Even fasting and praying had become a kind of public show to prove the holiness of the faithful.
It's an odd thing, but the Christian season of Lent, designed to help us draw near to God, can in a way actually become something that takes us further from the heart of God if we let it. Because we are a people who are trying to become Christ-like, and trying to set ourselves apart for God's purposes, we can begin to think of the effort we put into spiritual disciplines such as prayer and fasting as kind of heroic. And in turn we can look at others who are going about their typical lives, and for a lot of us the judgment begins to flow very easily, almost effortlessly.
So Jesus offered a strategy to his disciples to help keep them from this kind of folly. Keep your practices to yourself. This does not mean we keep our faith to ourselves. But don't turn your private disciplines into public ones. Keep them low-key, and let people see the outward workings of your faith take the form of offering grace, encouragement, compassion and service.
May you draw near to God as you practice spiritual disciplines over the season of Lent. May your practices of prayer, fasting, solitude, meditation, study, or whatever you chose to do help you to walk more faithfully and courageously in the way of Jesus. And may the things you do in private, when seen by God, pay off for you in ways beyond imagination, just as Jesus has promised.


I did not plan to make a blog post out of this but I got asked about my outfit a lot so I feel I would like to share it with all of you. Last weekend I went with my friend Tamas to Melrose Avenue in Los Angeles. He wanted to get an earring pierced and I had to be the emotional support for this adventure so we decided to make a day out of it.
On the way we found some very colorful places and of course we had to make some fun pictures, it was a perfect day, blue sky, sun, warm weather and we had a lot of fun. Sometimes it is good to see different places, it is a colorful world and age is just a number….


text:
Air Drop for Ship Stuck in Antarctica
WELLINGTON, New Zealand — A New Zealand air force cargo plane flew to Antarctica on Wednesday to parachute sea pumps and hull patches to a leaking Russian fishing vessel that had been stuck in the frigid waters after striking sea ice last week.
The vessel Sparta, with 32 crew aboard, hit underwater ice Friday that tore a 30-centimeter hole in its hull and caused it to list at 13 degrees. Rescue ships, hampered by heavy sea ice, were still several days away from the Ross Sea shelf area of northern Antarctica where the stricken ship sits immobilized.
Maritime New Zealand, which is coordinating rescue attempts, said Wednesday that this second air drop of vital pumps and patches will help the crew in its fight to keep the ship afloat.
Rescue Coordination Center spokesman Ross Henderson confirmed the successful drop of three pallets of equipment by a New Zealand Defense Force C-130 Hercules plane onto the ice near the stricken vessel mid-afternoon Wednesday.
"The crew will now recover the equipment and focus on making more permanent repairs to the hole in the side of the ship," he said in a statement.
Search and rescue mission coordinator John Dickson said the crew's efforts over the past few days meant the vessel was now back on an even keel and "the crew only needs to resume pumping occasionally to keep ahead of the water ingress."
Weather conditions in the area were reasonably good, with occasional snow showers and clouds, but were forecast to worsen Thursday, Dickson said.
The crew is made up of 15 Russians, 16 Indonesians and one Ukrainian, the agency said.
Several nearby vessels have been tasked by the rescue center with trying to reach the Sparta to assist its crew.
Dickson said the Norwegian vessel Seljavaer was heading away from the stranded vessel in an attempt to navigate a path around heavy ice, while the Sparta's sister ship, the Chiyo Maru No. 3, is still some 160 kilometers away.
The ice-strengthened polar research vessel Araon is estimated to arrive in the area on Dec. 26, he said.


When you establish a new venture, the initial thrill of starting something new is enough to inspire you to better your trade. However, after a while in the business the risk of monotony setting in is high, which gives rise to stagnation and hinders the growth of your organization. This is the reason why most small businesses that start-off with a bang tend to lose focus after a few months of success. You can counter stagnation in your organization by being an active member of trade associations and attending business events and trade fairs, which can bring many benefits to your business. Since there are plenty of business events organized all through the year, deciding on which ones to go might get confusing. Here are some ways to help you judge if the event is worth attending or not. 1. Identify all the upcoming business events: Even though attending trade fairs and events is the best way of networking with prospective clients and partners, for a businessperson being away from office every few days is not a possibility. Hence, it is best that you begin by identifying all the upcoming business events and trade fairs organized around the world. Social media and other business connections will come in handy while doing this. Once you jot down all the events, quickly categorize them by type: business conferences, trade fairs, seminars and sales opportunity. 2. Importance of the event: Once you have the list of all the upcoming events, you will need to decide which ones are worth attending. There are two main questions you need to ask yourself before making the decision – a. Does the event have the ability to bring in business? b. Will it help me improve my skill set? 3. Cost: Another important factor to consider while making the decision on attending an event is to calculate the cost. Since business events and trade fairs are held all over the world, attending all of them might not fit in your budget. Calculate the flight rates, accommodation and the event entry (if any), and decide whether the event is important enough to spend the requisite amount. 4. Time: In any business, time is equivalent to money. If it is a day or a weekend event it might not be a problem, but many events go on for a week. Think about how much time you are willing to be away from your office and if the event is worth the time spent away from work. Consider all these factors before you decide which business events to attend. These events are definitely one of the best ways networking with other entrepreneurs and small business owners who share the same concerns and challenges as you and also open up opportunities for collaboration. Meeting like-minded people and discussing business will give you a broader perspective on things and will help you make better decisions. Hence, make sure you attend at least a few events in a year to help your business grow.