text:
\section{Introduction}
\noindent
We consider the existence problem for embedded compact spacelike hypersurfaces $\Sigma$ in de Sitter space $S^{n+1}_1$  satisfying a prescribed curvature equation of the form
\begin{equation}
  \label{ActualEquation}
   H_k^\frac 1 k(\lambda[A]) = \psi.
\end{equation}
\noindent
Here $1\leq k\leq n$ is fixed, $A$ is the second fundamental form of $\Sigma $, $\lambda[A]=(\lambda_1, \ldots, \lambda_n)$ are the eigenvalues of the shape operator $A^i_j$, and $H_k$ is the $k$-th normalised symmetric polynomial in $\lambda$, that is
\begin{equation}
    S_{k}(\lambda) :=  \sum_{1\leq i_1<\cdots<i_k\leq n}\lambda_{i_1}\cdots\lambda_{i_k}, \qquad\qquad H_k:= \begin{pmatrix} n \\ k \end{pmatrix}^{-1}S_k.
\end{equation}
\noindent
Furthermore, we will choose our prescription function  $\psi$ to depend on both position in $S^{n+1}_1$ and the tilt, $\tau$, which is a measure of how spacelike $\Sigma$ is at the position, see \eqref{Tiltdefinition} below. We note that, as every compact embedded spacelike surface in $S^{n+1}_1$ may be written as a graph, (\ref{ActualEquation}) may be rewritten as a fully nonlinear elliptic second order partial differential equation in the graph function. Our main result is that, assuming some natural structural assumptions on $\psi$, the prescribed curvature equation (\ref{ActualEquation}) has a smooth spacelike solution $\Sigma$.

\noindent
The existence of solutions of such equations was studied in \cite{CNS3} by L. Caffarelli, L. Nirenberg and J. Spruck. In \cite{CNS4}, they proved the existence of starshaped hypersurfaces in Euclidean space
with prescribed $k$-symmetric curvature using an priori $C^{2,\alpha}$ estimate needed to carry out
the continuity method.
Curvature estimates for starshaped hypersurfaces with given $k$-symmetric curvature have
been established for various ambient Riemannian manifolds. For hypersurfaces in the sphere, the lower order and the curvature estimate are given in \cite{Oliker1} by M. Barbosa, L. Herbert and V. Oliker. These were used to prove the existence result by Y. Li and V. Oliker in \cite{YYLi:elli} via a degree theory argument. The curvature estimate and the existence result for prescribed curvature hypersurfaces in the hyperbolic space was proved by Q. Jin and Y. Li in \cite{YYLi:hyp} using similar arguments of W. Sheng, J. Urbas and X. Wang in \cite{sheng2004}.

\noindent
For spacelike hypersurfaces in Lorentz manifolds, less is known. The case of prescribed mean curvature was studied by C. Gerhardt \cite{Gerhardt1983, Gerhardt1997} and R. Bartnik and L. Simon \cite{bartnik1982}.  C. Gerhardt \cite{CG:Lorentz} also studied more general curvature functionals, but was forced to only consider a comparatively restrictive class of curvature functionals due to a problematic sign on a term in the curvature estimates (in particular all of these functionals are zero on the boundary of the positive cone,
 $\Gamma^+$). Urbas \cite{Urbas:cespacelike} obtained curvature estimates for prescribed symmetric curvature under the assumption that the mean curvature was in $L^p$ for sufficiently large $p$. Y. Huang \cite{Huang:01} noted that over compact domains Minkowski space, if the prescription function is additionally required to depend on the tilt and satisfies certain structural assumptions, then the problematic term in the curvature estimates may be cancelled  and curvature estimates may be obtained. D. Ballesteros-Ch\'avez \cite{Ballesteros2019} extended this result to compact domains in de Sitter space.

 \noindent
We will prove the following:
\begin{thm}\label{FullTheorem}
Suppose that $\psi:S^{n+1}_1\times\bb{R}_+\rightarrow\bb{R}$ is a smooth positive function satisfying the structural conditions \ref{BarriersE}--\ref{convex} and let $1\leq k\leq n$. Then there exists a smooth embedded $k$-admissable spacelike hypersurface $\Sigma\subset S^{n+1}_1$ satisfying (\ref{ActualEquation}).
\end{thm}
\noindent
In Section \ref{Preliminaries} we will collect all required definitions and some preliminary calculations. In Sections \ref{sec:c0-estimate} and \ref{sectiontilt} we will prove the required $C^0$ and $C^1$ estimates respectively. In Section \ref{sec:curvests} we extend earlier results \cite{Ballesteros2019,Huang:01} to give the curvature estimates (and therefore $C^2$ estimates). In Section \ref{sec:exists} we prove existence of a solution via regularity result of Evans and Krylov  \cite{Evans, Krylov} and the degree theory of Y. Li \cite{YYLi:1989}.

\noindent
We now state our structural assumptions on the prescription $\psi$:
\begin{assump*}
We impose the following structural assumptions on $\psi:S^{n+1}_1\times[1.\infty)\rightarrow \mathbb{R}$:
\begin{enumerate}[label=\textbf{\Alph*)}]
\item \label{BarriersE}(Barrier conditions) There exist constants $0<R_1<R_2<\infty$ such that
  \begin{flalign*}
       \tanh(r)>\psi(Y(r, \xi), \cosh(r))&\qquad\quad \mbox{for all } \xi\in\mathbb{S}^n, r< R_1,\\
      \tanh(r)<\psi(Y(r, \xi), \cosh(r))&\quad\qquad\mbox{for all } \xi\in\mathbb{S}^n, r > R_2.
      \end{flalign*}
      where $Y$ is as in equation (\ref{eq:37}).
\item (Differential inequality)  For all $x\in S^{n+1}_1$ and $\tau\in [1,\infty)$,\label{differentialineq}
\[\psi_\tau(x,\tau) \tau\geq\psi(x,\tau)\]
\item (Asymptotics) $\frac{\psi(x,\tau)}{\tau} \rightarrow \infty$ as $\tau \rightarrow \infty$ for all $x\in S^{n+1}_1$.\label{asymptotics}
\item ($C^1$ bound) Taking coordinates $(x^1, \ldots, x^{n+1})=(r, \xi_1, \ldots, \xi_n)$ on $S^{n+1}_1$ as in (\ref{eq:37}) we have that there exists a uniform $C>0$ such that for all $x\in S^{n+1}_1$ and $\tau\in\bb{R}$,\label{estimates}
\[\left|\pard{\psi(x,\tau)}{x^i}\right|\leq C \psi(x,\tau)\ .\]
\item (Convexity in $\tau$) $\psi_{\tau\tau}(x,\tau)\geq 0$ for all $x\in S^{n+1}_1$ and $\tau\in [1,\infty)$.\label{convex}
\end{enumerate}
\end{assump*}
\begin{rem}
Assumption \ref{BarriersE} is simply to ensure the existence of suitable barriers. The relevance of the rate $\tanh(r)$ is that this is curvature of the natural foliation of totally umbillic hypersurfaces in de Sitter space.
\end{rem}
\begin{rem} We note that condition \ref{differentialineq} already implies that $\psi(x,\tau)\geq\tau\psi(x,1)$, so \ref{asymptotics} may be considered as the smallest possible increase in growth on top of this assumption.
\end{rem}
\begin{rem} Condition \ref{estimates} above is used to estimate the space derivative of $\psi$ with respect to a multiple of $\psi$. This is a necessary condition in our tilt estimates as the derivative may be vastly larger with respect to $\tau$, for example if in local coordinates $\psi(x,\tau)=\tau^2+x^1e^\tau$ then at $x^1=0$, $\pard{\psi(x,\tau)}{x^1}$ cannot be estimated by $\psi$.
\end{rem}
\begin{rem}
There are an abundance of functions $\psi$ which satisfy the structure conditions \ref{BarriersE}--\ref{convex} (see also Lemma \ref{BarriersExist}). Our model function is $\psi(x,\tau) = \Psi(x)\tau^p$ where $p>1$ and $\Psi$ is a smooth bounded function satisfying the conditions of Lemma \ref{BarriersExist}.
\end{rem}

\section{Preliminaries}
\label{Preliminaries}

\label{sec:geom-form-hypers}

\subsection{Subspace geometry in {L}orentzian manifolds}
To avoid confusion with signs, we now collect some geometric formulae for hypersurfaces in Lorentzian manifolds
\noindent
Let $\{\partial_1,...,\partial_n,N\}$ be a basis for a Lorentzian
manifold $(\ov{M},\bar{g})$ and $M$ a Lorentzian  (not necessarily spacelike) hypersurface with induced metric $g$ such that
$\{\partial_i\}$ span $TM$, and let $N$ be the unit normal field to $M$
and put $\epsilon = \bar{g}(N,N)$. When the induced metric is positive
definite, then we say that $M$ is a spacelike hypersurface, then $g$
can be represented by the matrix
$g_{ij} = g(\partial_i,\partial_j)$ with inverse denoted by $g^{ij}$.
\noindent
The \textit{Gauss formula} for $X,Y\in T\Sigma$ reads
\begin{equation*}
  \label{eq:4}
  D_{X}Y = \nabla_{X}Y +\epsilon\, h(X,Y)N,
\end{equation*}
here $D$ is the connection on $\ov{M}$, $\nabla$ is the induced connection on $M$ and the \textit{second fundamental form} $h$ is
the normal projection of $D$. In coordinate basis we write
\begin{equation*}
  \label{eq:5}
h_{ij}= h(\partial_i,\partial_j).
\end{equation*}
The \textit{shape operator} obtained by raising an index with the inverse of the metric
\begin{equation*}
  \label{eq:6}
  h^i_j = g^{ik}h_{kj}.
\end{equation*}
The \textit{principal curvatures} of the hypersurface $\Sigma$ are the
eigenvalues of the symmetric matrix $(h^i_j)$. The tangential projection of the
covariant derivative of the normal vector field $N$ on $\Sigma$, $\nabla_j N = (D_{\partial_j}N)^{\top}$,
is related to the second fundamental form by the \textit{Weingarten equation}
      \begin{equation}
        \label{eq:7}
        \nabla_{j} N = -h^i_j\partial_i = -g^{ik}h_{kj}\partial_i.
      \end{equation}
      The \emph{curvature tensor} is defined for $X,Y,Z \in T\Sigma$ as
      \begin{equation*}
        \label{eq:8}
        R(X,Y)Z = \nabla_{Y}\nabla_{X}Z - \nabla_{X}\nabla_{Y}Z + \nabla_{[X,Y]}Z.
      \end{equation*}
      Contracting with the metric
      \begin{equation*}
        \label{eq:11}
        R_{ijkl} = g\left(R(\partial_i,\partial_j)\partial_k,\partial_l\right) = g_{lm}R^{m}_{ijk}.
        \end{equation*}
  To relate curvature and second fundamental form, we have the \emph{Codazzi equations}
 \begin{equation}
  \label{GeneralCodazzi}
  \ip{\ov{R}_{ijk}}{N} = \nabla_j h_{ik}-\nabla_i h_{jk}.
\end{equation}
and the \textit{the Gauss equation},
\noindent
\begin{equation}
  \label{eq:15}
  \ov{R}_{ijkl} = R_{ijkl} - \epsilon \left(h_{ik}h_{jl} - h_{il}h_{jk} \right).
\end{equation}

\noindent
Note that if $T$ is a symmetric tensor, then the following \textit{Ricci identity} holds
      \begin{equation}
        \label{eq:17}
{\nabla_{k}}\nabla_{l}T_{ij} - {\nabla_{l}}\nabla_{k}T_{ij} =  R_{kljr}T_{ir} + R_{klir}T_{rj}.
      \end{equation}

\subsection{The geometry of de {S}itter space}
We now consider manifolds $\Sigma\subset S_1^{n+1}\subset\mathbb{R}^{n+2}_1$ where
\begin{itemize}
    \item  $\mathbb{R}^{n+2}_{1}=(\mathbb{R}^{n+2},\bar{g})$ is \emph{Minkowski space} with metric \[\bar{\bar{g}} = -dx_1^2 + dx_2^2 + \cdots + dx_{n+2}^2\] and
covariant derivative $\bar D$.
\item $S^{n+1}_1$ is \emph{de Sitter space}, defined by
\[{S}^{n+1}_{1} = \left\{ x\in\mathbb{R}^{n+2}_{1} : -x_1^2 + x_2^2 + \cdots + x_{n+2}^2 = 1\right\}\] with the
induced Lorentzian metric $\ov{g}$, covariant derivative $D$, unit normal $N$  and second fundamental form $h$.
\item $\Sigma \subset S^{n+1}_1$ is a embedded spacelike hypersurface of $S^{n+1}_1$ with induced Riemannian metric $g$, covariant derivative $\nabla$, unit normal $\nu$ and second fundamental form $A$.
\end{itemize}
Let $\bb{S}^n$ be the standard round sphere. Then de Sitter space may be parametrised by $Y:\bb{S}^n\times \bb{R}\rightarrow S^{n+1}_1$ given by
\begin{equation}
             \label{eq:37}
             Y(r,\xi) = \sinh(r)E_1 + \cosh(r) \xi
           \end{equation}
and in these coordinates,
the induced metric is
\begin{equation*}
  \label{eq:18}
  \ov{g} = -dr^2 + \cosh^2(r)\sigma,
\end{equation*}
where $\sigma$ is the round metric on $\mathbb{S}^n$.
\noindent
We note that as $|Y|^2=1$ we have that $Y=N$ is a unit normal to $S^{n+1}_1$ and (as $0=\ip{Y_{\alpha\beta}}{Y}+\ip{Y_\alpha}{Y_\beta}$)
\begin{equation}h_{\alpha\beta}=-\ov{g}_{\alpha\beta} \ ,\label{hisg}\end{equation}
(where $1\leq \alpha, \beta\leq n+1$) and so using the Gauss equation we have
\[R^{S^{n+1}_1}_{\alpha \beta\gamma\delta} = \ov{g}_{\alpha\gamma}\ov{g}_{\beta\delta}-\ov{g}_{\alpha\delta}\ov{g}_{\beta\gamma}\ ,\]
where we used that Minkowski space is flat. This implies that on $\Sigma$ (which has a timelike normal) we have that
\begin{equation}
\nabla_ih_{jk}=\nabla_jh_{ik}\label{CM}
\end{equation}
and
\begin{equation}R_{ijkl}=A_{il}A_{jk}-A_{ik}A_{jl}+{g}_{ik}{g}_{jl}-{g}_{il}{g}_{jk}\ .\label{GE}\end{equation}
We define the \emph{tilt function} on $\Sigma$ to be the function
\begin{equation}\tau = \langle \nu , E_1 \rangle ,\label{Tiltdefinition}\end{equation}
where $\nu$ is a unit normal to $\Sigma$ which has been chosen so that $\tau$ is positive.
\noindent
We now represent $\Sigma$ as a graph, that is we take $u:\bb{S}^n\rightarrow\bb{R}$ so that $\Sigma$ is parametrised by $X:\bb{S}^n\rightarrow S^{n+1}_1$ given by $X(\xi)=Y(u(\xi),\xi)$. We will use $\tilde{\nabla}$ to denote the standard covariant derivative for the metric $\sigma$ on $\bb{S}^n$, and our indices $\partial_{i},\partial_j,...$, etc., take values from $1$ to $n$,
except for the vector field $\partial_{r}$ which will be considered separately.
The tangent space of the hypersurface at a point $Y\in\Sigma$
is spanned by the tangent vectors
$Y_j = u_{j}\partial_{r} + \partial_{j}$, the covariant derivative
 $\nabla$ corresponding to the induced metric
on $\Sigma$  is given by
\begin{equation*}
  \label{eq:19}
  g_{ij} = -u_{i} u_{j} + \cosh^2(u)\sigma_{ij} .
\end{equation*}
We write
\[\tilde{\tau}=\frac{\cosh^2(u)}{\sqrt{\cosh^2(u)-|\tilde{\nabla} u|^2}}\ ,\]
where $\tilde{\nabla}u = \sigma^{ij}u_j\partial_i$ and $|\tilde{\nabla}u| := \sigma^{ij}u_iu_j$ (we will see shortly that $\tilde{\tau}=\tau$). $\Sigma$ is spacelike at a point if $g_{ij}$ is invertable, which is equivalent to $\tilde{\tau}$ being finite at that point.
Since $\Sigma$ is spacelike, we calculate the inverse of $g_{ij}$ to be
  \begin{equation}
    \label{eq:ginverse}
    \begin{split}
      g^{ij}
    &=\cosh^{-2}(u) \sigma^{ij} +
    \frac{ \sigma^{il}u_{l} \sigma^{jm}u_{m}}{\cosh^4(u) - \cosh^{2}(u)|\tilde{\nabla} u|^2}=\cosh^{-2}(u)\left[\sigma^{ij} +\tilde{\tau}^2 \frac{\sigma^{il}u_{l} \sigma^{jm}u_{m}}{\cosh^4(u)}\right]
    \end{split}
  \end{equation}
 A unit normal vector
  to $\Sigma$ at the point $Y$ can be obtained by solving the equation $\ov{g}(Y_{\alpha},\hat{n})=0$, and then we get
\begin{equation*}
  \label{eq:21}
  \nu =- \frac{ \cosh^2(u)\partial_{r}+ \tilde{\nabla} u }{
    \sqrt{\cosh^4(u) - \cosh^2(u)|\tilde{\nabla} u|^2}}=\cosh^{-3}(u)\tilde{\tau}(\cosh^2(u)\partial_r+\tilde \nabla u).
\end{equation*}
We note that
           \begin{equation*}
             \label{eq:38}
             \cosh(u)\partial_r =
             E_1 + \sinh(u)Y.
           \end{equation*}
and so we see that
\[\tau=\ip{E_1}{\nu}=\tilde{\tau}\ ,\]
as claimed.
\noindent
The second fundamental form is the projection of the second derivatives of the
parametrisation $D_{Y_{\alpha}}Y_{\beta}$ on the normal direction. Writing $\tilde{\Gamma}$ for the Christoffel symbols of the metric $\sigma$, we have
  \begin{equation*}
    \label{eq:23}
    D_{\partial_r}\partial_r = 0; \quad
    D_{\partial_r}{\partial_j} = \tanh(r)\partial_j;\quad
    D_{\partial_i}{\partial_j}= \cosh(r)\sinh(r)\sigma_{ij}\partial_{r}+ \tilde\Gamma_{ij}^k \partial_k ,
  \end{equation*}
and using these identities we compute
  \begin{equation*}
    \label{eq:24}
    \begin{split}
    D_{Y_i}Y_{j}& = D_{u_i\partial_{r}+\partial_i}\left(u_j\partial_{r}+\partial_j\right)\\
    &=
    u_ju_jD_{\partial_{r}}\partial_{r} +
    u_iD_{\partial_{r}}\partial_{j} +
    u_{ij} \partial_{r} +
    u_{j}D_{\partial_{i}}\partial_{r} +
    D_{\partial_{i}}\partial_{j}.
    \end{split}
  \end{equation*}
It follows that  $ A_{ij}=  \ov{g}(D_{Y_{i}}Y_{j}, \nu)$ is given explicitly by
 \begin{equation}
  \label{eq:Secondff}
  A_{ij} = \cosh^{-1}(u)\tau
  \left(\tilde{\nabla}^2_{ij}u - 2\tanh(u)u_{i}u_{j} + \sinh(u)\cosh(u)\sigma_{ij}\right).
\end{equation}

\noindent
Finally, we define a notion of partial derivatives for $\psi$ on $S^{n+1}_1$. Suppose we have a function $f:S^{n+1}_1\times \bb{R}\rightarrow \bb{R}$, then we define the partial derivative on $S^{n+1}_1$ and $\Sigma$ by
\begin{align*}
D^x f =\pard{\psi}{x^\alpha}\ov{g}^{\alpha\beta}\partial_{\beta}, \qquad
\nabla^x \psi = (D^x f)^\top\ .
\end{align*}
Similarly, we may define second derivatives of $f$ in the usual tensorial way.

\subsection{Curvature functionals and admissability}
Throughout this section we fix $1\leq k\leq n$. As described above we will consider functions
\[F[A] := H_k^\frac 1 k(\lambda[A]),\]
where $\lambda[A]=(\lambda_1,\ldots, \lambda_n)$ are the eigenvalues of the symmetric matrix $A$ and we will define $f:\bb{R}^n\rightarrow\bb{R}$ by $f=H_k^\frac{1}{k}$. We define the \emph{admissable cone} of $H_k$ to be
$\Gamma_{k}$ which is defined to be the connected component of $H_k^{-1}(\{x\in\bb{R}|x>0\})$ which contains the positive cone
\[\Gamma^{+}=\{\lambda\in\mathbb{R}^n\,|\, \lambda_i>0, \forall i = 1,2,\dots,n\}.\]
We have that for all $\lambda\in\Gamma_k$, $f_{\lambda_i}(\lambda) >0$ and $f$ is concave in $\Gamma_k$.
Since $f \in C^2(\Gamma_k)\cap C^{0}(\ov{\Gamma_k})$ it follows that $F[A]$ is elliptic and concave if the eigenvalues of $A$ lie in $\Gamma_k$.
\noindent
A spacelike hypersurface $\Sigma\subset S^{n+1}_1$ will be called \emph{$k$-admissable} if for all $p\in\Sigma$ the eigenvalues of the shape operator $A_i^j=A_{ik}g^{kj}$ are in $\Gamma_k$.
\noindent
A function $u\in C^2(\bb{S}^n)$ will be called \emph{$k$-admissable} if the graph of $u$ is a spacelike admissable hypersurface. We note that this implies that $u$ is positive, as if there is a negative minimum of $u$ at $p\in\Sigma$, the shape operator is negative definite so $\lambda[A]\notin \Gamma_k$.
\noindent
As is standard, and we will write
\[F^{ij} := \pard{F}{A_{ij}}, \qquad\qquad F^{ij,kl} := \frac{\partial^2F}{\partial A_{ij}\partial A_{kl}}\ .\]

\section{A priori $C^0$ estimate}
\label{sec:c0-estimate}
\noindent
Considering $\Sigma$ graphically, any solution $u\in C^2\left(\mathbb{S}^n\right)$ of \eqref{ActualEquation}, the barrier conditions \eqref{barrier} will ensure that $R_1 \leq u(\xi) \leq R_2$ for all $\xi\in\mathbb{S}^n$. The proof follows maximum principle arguments similar to \cite[Lemma 3.1]{Oliker1}.
\begin{lem}\label{BarrierConditions}
  Let $1\leq k\leq n$ and let $\psi: (0,\infty)\times \mathbb{S}^n\times[1, \infty)\to \mathbb{R} $ is a continuous positive function such that there exist constants $0<R_1<R_2<\infty$ such that
  \begin{equation}
    \label{barrier}
    \begin{split}
       \tanh(r)>\psi(r, \xi, \cosh(r)),\quad \mbox{for } \xi\in\mathbb{S}^n, r< R_1,\\
      \tanh(r)<\psi(r, \xi, \cosh(r)),\quad\mbox{for } \xi\in\mathbb{S}^n, r > R_2.
      \end{split}
    \end{equation}
    Then $u\in C^2\left(\mathbb{S}^n\right)$ is a solution of \eqref{ActualEquation} then for all $\xi\in\mathbb{S}^n$, $u$ satisfies
    \begin{equation}
      \label{eq:25}
      R_1 \leq u(\xi) \leq R_2.
    \end{equation}
\end{lem}
  \begin{proof}
    Suppose there is a point $\xi_0\in\mathbb{S}^n$ where the maximum of $u$ is attained, say $R_2<r_0 = u(\xi_0)$. Then at the maximum we have $\tilde{\nabla} u \dot{=} 0$ and $\tilde{\nabla}^2 u \dot{\leq} 0$. Substituting these into the equations of the previous section Then the inverse of the  metric, the tilt and the second fundamental form at $\xi_0$ are respectively
      \begin{equation*}
        \label{eq:26}
        g^{ij} \dot{=}\frac{1}{\cosh^2(r_0)}\sigma^{ij}, \qquad \tau \dot{ =}  \cosh u(u)=\cosh(r_0), \qquad
        A_{ij}\dot{\leq} \sinh(r_0)\cosh(r_0)\sigma_{ij}.
      \end{equation*}
    At $\xi_0$, the shape operator therefore satisfies
      \begin{equation}
        \label{eq:28}
        A^i_j \leq \tanh(r_0) \delta^i_j,
      \end{equation}
      and so $\lambda_i\leq\tanh (r_0)$ for $1\leq i \leq n$. Substituting into \eqref{eq:44},
      \begin{equation}
        \label{eq:29}
       \psi(r_0,\xi_0, \cosh(r_0)) \dot{=} F[A^i_j] \leq \tanh(r_0) < \psi(r_0,\xi_0, \cosh(r_0)),
      \end{equation}
     which is a contradiction, and so  $u \leq R_2$. An analogous argument at the minimum completes the proof.
  \end{proof}
\noindent
Observe that, we may impose a few fairly mild assumptions on $\psi_r$ to ensure that barriers exist.
\begin{lem}
 Let $\psi: [0,\infty)\times \mathbb{S}^n\times[1, \infty)\to \mathbb{R}$ be a uniformly bounded in $C^2$ which is positive for $r>0$. We give conditions for upper and lower barriers:\label{BarriersExist}
\begin{description}[leftmargin=*]
 \item[Lower Barriers]
 If for all $\xi\in \mathbb{S}^n$
\[\psi(0,\xi, 1)=0 \qquad\text{ and }\qquad \psi_r(0,\xi,1)<1\]
 then a lower barrier exists, that is there exists an $0<R_1$ such that for all $\xi\in\mathbb{S}^n$ and $r<R_1$,
 \[\tanh(r)>\psi(r, \xi, \cosh(r))\ .\]
\item[Upper Barriers]  Suppose $\psi$ satisfies the structural condition $\psi_\tau\tau>\psi$ and let $\gamma(r)$ be any smooth monotonic function s.t. $\gamma(r)\rightarrow\infty$ as $r\rightarrow\infty$. Suppose additionally that for all $\xi\in \mathbb{S}^n$ and $r>\tilde{R}$,
\[ \psi_r(r,\xi,\cosh(r))>\left[\gamma'(r)-\tanh(r)\right]\psi\ ,\]
then an upper barrier exists, that is there exists an $0<R_2$ such that for all $\xi\in\mathbb{S}^n$ and $r>R_2$,
 \[\tanh(r)<\psi(r, \xi, \cosh(r))\ .\]
\end{description}
\end{lem}
 \begin{proof}
 We fix $\xi$ and consider the function
 \[\Psi(r,\xi) = \frac{\psi(r, \xi, \cosh (r))}{\tanh(r)}\ .\]
 \noindent
 Finding a lower barrier is equivalent to showing that for all $\xi\in\mathbb{S}^n$ there exists a $R_1>0$ such that for all $0<r<R_1$, $\Psi(r,\xi)<1$. By the assumptions, $\Psi$ is continuous up to $r=0$, and so by compactness, there exists a $\delta>0$ such that for all $\xi\in \mathbb{S}^n$, $\Psi(0,\xi)<1-\delta$. By continuity and compactness there exists an $R_1$ such that $\Psi(r,\xi)<1$ for all $0<r<R_1$ as claimed.
 \noindent
  Finding an upper barrier is equivalent to showing that for all $\xi\in\mathbb{S}^n$ there exists a $R_2>0$ such that for all $r>R_2$, $\Psi(r,\xi)>1$. We calculate that
  \begin{flalign*}
  \frac{d}{dr}\Psi(r,\xi) &= -\frac{1}{\sinh^2(r)}\psi(r,\xi,\cosh(r)) +\coth(r)(\psi_r+\psi_\tau \sinh(r))\\
  &\geq\left[1-\frac{1}{\sinh^2(r)}\right]\psi +\coth(r)\psi_r\\
  &\geq \frac{\gamma'(r)}{\tanh(r)}\Psi
  \end{flalign*}
 for $r>\hat{R}(\tilde{R})$ sufficiently large depending on $\tilde{R}$. Since $\tanh(r)<1$, we seee that \[\frac{d}{dr}\Psi(r)>\gamma'(r)\Psi\ ,\]
 which implies $\Psi(\xi,r)>\Psi(\hat{R}, \xi)e^{\gamma(r)}$. Since $\mathbb{S}^n$ is compact, and $\gamma(r)\rightarrow\infty$ as $r \rightarrow\infty$, this implies that the claimed upper barrier conditions are met.
 \end{proof}

 \begin{rem}
 A function $\psi$ satisfying lower barrier conditions in Lemma \ref{BarriersExist} allows the solution to \eqref{eq:44} given by $u\equiv0$. As noted earlier, this solution is inadmissable, as we require strictly positive $u$ for the shape operator to be in the admissable cone everywhere.
 \end{rem}

\section{Tilt estimate}
\label{sectiontilt}
\noindent
We now demonstrate a strict spacelikeness estimate by estimating the tilt function $\tau$. The height function is defined by
           \begin{equation}
             \eta := -\langle Y, E_1\rangle, \label{etadef}
           \end{equation}
and we note that in terms of the graph function, $\eta=\sinh(u)$.
We now demonstrate the following identities.
           \begin{prop}
             \label{prop:1}
             The tilt and height functions satisfy the following identities:
             \begin{enumerate}
             \item\label{p:1}{$\nabla^2_{ij}\eta = \tau A_{ij} + \eta g_{ij}$.}
             \item\label{p:2}{$\nabla_j \tau = A_{j}^i\nabla_i\eta$,}
             \item\label{p:3}{$\nabla_j\nabla_i \tau =  \nabla_{k}A_{ij}\nabla^k\eta + \tau  A^2_{ij} +  A_{ij} \eta$,}
             \end{enumerate}
             where $A^2_{ij}:=A_{kj}A^k_i$.
             \end{prop}

    \begin{proof}
           Using (\ref{hisg}) and the Gauss formula we have that
           \begin{equation*}
             \begin{split}
             \label{eq:42}
             \nabla^2_{ij}\eta & = Y_{j}(Y_{i}\eta) - (\nabla_{Y_{i}}Y_{j}) \eta\\
             & = Y_{j}(Y_{i}\eta) - (\bar{D}_{Y_{i}}Y_{j}-h_{ij}N + A_{ij}\hat{n}) \eta\\
             & = \bar{D}^2_{Y_{j}Y_{i}}\eta + (h_{ij}N - A_{ij}\nu) \eta\\
             & =\tau A_{ij} + \eta g_{ij},
             \end{split}
           \end{equation*}
where we used that $X(\eta) = -\ip{X}{E_1}$ and $\bar{D}^2\eta=0$.
 \noindent
               Using the Weingarten equation (\ref{eq:7}) we obtain
           \begin{multline*}
             \label{eq:41}
             \nabla_j \tau = \langle \nabla_j \nu, E_1\rangle

             = -g^{ik}A_{kj}\langle Y_i, E_1\rangle
             = -g^{ik}A_{kj}\nabla_i\langle Y, E_1\rangle
             = g^{ik}A_{kj}\nabla_i\eta.
           \end{multline*}
           and from this we have
           \begin{equation*}
             \begin{split}
             \label{eq:43}
             \nabla^2_{ij}\tau &= \nabla_{j}(g^{mn}A_{ni}\nabla_{m}\eta)\\
             & = g^{mn}\nabla_{j}A_{ni}\nabla_{m}\eta + g^{mn}A_{ni}\nabla_{mj}\eta\\
             & = g^{mn}\nabla_{n}A_{ij}\nabla_{m}\eta + \tau  A_{mj}g^{mn}A_{ni} +  A_{ij} \eta
           \end{split}
         \end{equation*}
         where we used (\ref{CM}) on the third line.
               \end{proof}

\begin{prop}\label{TiltEstimate}
Suppose that $\psi:S^{n+1}\times\bb{R}\rightarrow\bb{R}$ is smooth and positive and satisfies assumptions \ref{asymptotics} and \ref{estimates} above. Suppose that $u\in C^3(\bb{S}^n)$ satisfies (\ref{ActualEquation}) so that there exist $R_1, R_2 >0$ such that,
\[0<R_1<u(\xi)<R_2\]
for all $\xi\in\mathbb{S}^n$. Then there exists a constant $C_\tau$, depending only on $n$, $k$, $R_1$,$R_2$ and $\psi$ such that
\[\tau<C_\tau\ .\]
\end{prop}

\begin{proof}
We have that
\begin{flalign*}F^{ij}\nabla_j\nabla_i \tau &=  \nabla^k F\nabla_{k}\eta + \tau  F^{ij}A_{mj}A^m_{i} +  F \eta\\
&=\ip{\nabla^x \psi}{\nabla \eta} + \psi_\tau \ip{\nabla \tau}{\nabla \eta}+\tau F^{ij}A^2_{ij} +\psi \eta
\end{flalign*}
Due to structural assumption \ref{estimates} on $\psi$, we have that
\[|\ip{\nabla^x \psi}{\nabla\eta}|\leq C\tau^2\psi .\]
Furthermore, following \cite[equation 3.8]{Urbas:cespacelike} and using Newton's inequalities \cite{NewtonInequalities} (which are valid for $S_k$ \emph{outside} $\Gamma_k$),
\begin{equation}F^{ij}A^2_{ij} \geq  H_k^\frac 1 k H_1\geq H_k^\frac{2}{k}=\psi^2\ .\label{UrbasMagicInequality}\end{equation}
Substituting this into the above equation we see that at a maximum of $\tau$, as $\nabla \tau=0$ and $\nabla_i\nabla_j\tau\leq 0$, we have that
\begin{flalign*}
0&\geq - C\tau^2\psi +\psi \eta +\tau\psi^2
\end{flalign*}
which implies, by using the $C^0$ estimates that
\[\psi \leq C \tau\ .\]
The $C^0$ estimates imply that the solution stays in a compact region of $S^{n+1}_1$ and so, due to structural assumption \ref{asymptotics} on $\psi$, there exists a uniform $\tau_0$ such that for all $x$ in this region and for all $\tau>\tau_0$, $\tau<\frac{\psi}{2C}$. This yields a contradiction to $\tau>\tau_0$ and proves the Lemma.
\end{proof}

\section{A priori $C^2$ estimate}
\label{sec:curvests}
\noindent
From \cite{Ballesteros2019} we have the following curvature estimates over domains $\Omega \subset S^n$.
               \begin{thm}[Ballesteros-Ch\'avez 2019]
                 \label{thm:01}
                 Let $\Omega\subset\mathbb{S}^n$ be a domain in the round sphere, and let $u\in C^{4}(\Omega) \cap C^2(\bar{\Omega})$ an admissible solution of
                 the boundary value problem
                 \begin{equation*}
                   \label{eq:44}
                   \left\{
                     \begin{array}{rcll}
                      F(A) =  H_{k}^{\frac{1}{k}}(\lambda(A)) &=& \psi(Y,\tau) & \mbox{in}\quad \Omega\\
                       u  &=& \varphi & \mbox{on}\quad \partial\Omega
                     \end{array}
                     \right.,
                   \end{equation*}
                   where $A$ is the second fundamental form of a spacelike surface $\Sigma$ in de Sitter space given by
                   (\ref{eq:Secondff}) and $\psi$ is a smooth positive function satisfying \ref{differentialineq} and \ref{convex}. Assume additionally that there exists a $R_1,R_2, C_\tau>0$ such that
     \[R_1<u(\xi)<R_2, \qquad \tau(\xi)<C_\tau .\]
                   Then
                   \begin{equation*}
                     \label{eq:46}
                     \sup_{\Omega}|A| \leq C,
                   \end{equation*}
                   where $C$ depends on $n$, $|\varphi|_{C^1(\bar{\Omega})}$,$R_1$, $R_2$, $C_\tau$, $|\psi|_{C^2([R_1,R_2]\times\Omega\times[1,C_{\tau}])}$ and $\sup_{\partial\Omega}|A|$.
                 \end{thm}
\noindent
We will now extend this result to all of $\bb{S}^n$, or equivalently obtain estimates on all of $\Sigma$.
\begin{thm}\label{CurvatureEstimate}
Suppose that $\psi$ is a smooth positive function which satisfies \ref{differentialineq} and \ref{convex}. Suppose $u\in C^4(\bb{S}^n)$ is a solution of (\ref{ActualEquation}) such that there exist constants $0<R_1<R_2$ and $C_\tau>0$ such that for all $\xi\in \bb{S}^2$,
\[R_1<u(\xi)<R_2, \qquad \tau(\xi)<C_\tau .\]
Then there exists a constant $C_A=C_A(k,n,R_1,R_2,C_\tau,|\psi|_{C^2([R_1,R_2]\times\bb{S}^n\times[1,C_\tau))})$ so that
\[|A|<C_A\ .\]
\end{thm}
\begin{proof}
Suppose first that $k=1$. In this case (\ref{eq:ginverse}) and (\ref{eq:Secondff}) imply that if we write (\ref{ActualEquation}) in terms of the graph function $u$, we obtain a quasilinear equation which is uniformly elliptic if we have a uniform bound on $\tau$. Therefore if we have the assumed bounds, we see that this equation is uniform ellipticity with uniform $C^1$ estimates on the solution. We may therefore apply De Giorgi--Nash--Moser estimates and Schauder estimates to imply uniform $C^2$ estimates. In this case, the theorem is therefore proven.

\noindent
Suppose now that $k\geq 2$. We begin by proving a Simon's-type identity for the second fundamental form. At an arbitrary point $p\in\Sigma$, we choose coordinates so that $A_{ij}$ is diagonal and $g_{ij}=\delta_{ij}$. In these coordinates $F^{ij}$ is also diagonal.
\noindent
The Codazzi equation (\ref{CM}) and the Ricci identity (\ref{eq:17}) imply that
\begin{equation*}

             \begin{split}
               \nabla_{i}\nabla_{j}A_{kk} =& \nabla_{i}\nabla_{k}A_{kj} \\
               =&  \nabla_{k}\nabla_{i} A_{kj} + R_{ikkr}A^r_{j} + R_{ikjr}A_{k}^r\\
               =& \nabla_{k}\nabla_{k} A_{ij} + R_{ikkr}A_{j}^r + R_{ikjr}A_{k}^r.

               \end{split}
           \end{equation*}
where we only sum over indices where one is raised and one is lowered. Choosing $i=j$ in the above, and applying the Gauss equation (\ref{GE}) we obtain
\begin{align*}
\nabla_{j}\nabla_{j}A_{kk} =& \nabla_{k}\nabla_{k} A_{jj} + R_{jkkr}A_{j}^r + R_{jkjr}A_{k}^r\\
&=\nabla_{k}\nabla_{k} A_{jj} + (A_{jr}A_{kk} - A_{jk}A_{kr}+g_{jk}g_{kr}-g_{jr}g_{kk})A_{j}^r\\
&\qquad +(A_{jr}A_{kj}-A_{jj}A_{kr}+g_{jj}g_{kr}-g_{jr}g_{kj})A_{k}^r\\
&=\nabla_{k}\nabla_{k} A_{jj} + A^2_{jj}A_{kk} - A_{jk}A^2_{kj}+g_{jk}A_{jk}-A_{jj}g_{kk}\\
&\qquad +A^2_{jk}A_{kj}-A_{jj}A^2_{kk}+g_{jj}A_{kk}-A_{kj}g_{kj}\\
&=\nabla_{k}\nabla_{k} A_{jj} + A^2_{jj}A_{kk} -A_{jj}g_{kk}-A_{jj}A^2_{kk}+g_{jj}A_{kk}\ .
\end{align*}

\noindent
We therefore see that
\begin{flalign*}
F^{ij}\nabla_i\nabla_j A_{kk}&=\sum_{j=1}^nF^{jj}\nabla_j\nabla_j A_kk\\
&=F^{ij}\nabla_k\nabla_kA_{ij}+A_{kk}F^{ij}A^2_{ij}-F^{ij}A_{ij}A^2_{kk}+A_{kk}\text{tr}{F^{ij}}-F^{ij}A_{ij}\\
&=F^{ij}\nabla_k\nabla_kA_{ij}+A_{kk}F^{ij}A^2_{ij}-\psi A^2_{kk}+A_{kk}\text{tr}{F^{ij}}-\psi
\end{flalign*}
and so writing $H=nH_1 = \sum_{k=1}^n A_{kk}$ we have that
\begin{equation}
    F^{ij}\nabla_i\nabla_j H  = F^{ij}\Delta A_{ij}+H F^{ij}A^2_{ij} + H \text{tr} F^{ij} -\psi\left(n+|A|^2\right)
\end{equation}
where $\Delta$ is the Laplace--Beltrami operator. To estimate the first term on the right hand side we note that by differentiating (\ref{ActualEquation}) twice gives
\[\Delta \psi = F^{ij}\Delta A_{ij} + F^{ij,kl}\nabla_t A_{ij}\nabla^tA_{kl}\leq F^{ij}\Delta A_{ij}\]
where we used the well known concavity of $F$.
\noindent
We have that
\begin{flalign*}\nabla_l\nabla_k \psi &= \nabla^x_l\nabla^x_k \psi  +\nabla^x_k \psi_\tau \nabla_l \tau+\nabla^x_l \psi_\tau \nabla_k \tau +\psi_\tau \nabla_l\nabla_k \tau +\psi_{\tau \tau}\nabla_l \tau \nabla_k\tau\\
&=D_l^xD_k^x \psi - A_{lk}D^x_{\hat{n}}\psi+\nabla^x_l \psi_\tau \nabla_k \tau +\psi_\tau \nabla_l\nabla_k \tau+\psi_{\tau \tau}\nabla_l \tau \nabla_k\tau\ .
\end{flalign*}
Since $H^2 = 2S_2+|A|^2$, so if $\l[A]\in \Gamma_k$, $|A|<H$ and so
\begin{flalign*}\Delta \psi&=g^{kl}D_l^xD_k^x \psi - H D^x_{\hat{n}}\phi+2\ip{\nabla^x \psi_\tau}{\nabla \tau} +\psi_\tau \Delta \tau+\psi_{\tau \tau}|\nabla \tau|^2\\
&=\psi_\tau\left[\ip{\nabla \eta}{\nabla H}+\tau |A|^2+H\eta\right] +\psi_{\tau \tau}|\nabla \tau|^2- H D^x_{\hat{n}}\psi+2\ip{\nabla^x \psi_\tau}{\nabla \tau} +g^{kl}D_l^xD_k^x \psi\\
&\geq\psi_\tau\ip{\nabla \eta}{\nabla H}+\psi_\tau \tau |A|^2 - C_1 H-C_2\ ,
\end{flalign*}
where used structural assumption \ref{convex} of $\psi$ and we estimated using the bounds on $\tau$ and $u$ in compactness arguments to estimate derivatives of $\psi$.
\noindent
Overall we have that
\begin{flalign*}
F^{ij}\nabla_i\nabla_j H & \geq H F^{ij}A^2_{ij} + H \text{tr} F^{ij}+(\psi_\tau\tau-\psi)|A|^2\\
&\qquad+\psi_\tau\ip{\nabla \eta}{\nabla H} -C_1 H -C_2-n\psi \ .
\end{flalign*}
Using (\ref{UrbasMagicInequality}) and the structural assumption \ref{differentialineq} on $\psi$ we have that
\begin{flalign*}
F^{ij}\nabla_i\nabla_j H &\geq \frac 1 n H^2\psi +H\text{tr}F^{ij}+\psi_\tau\ip{\nabla \eta}{\nabla H} -C_1 H -C_2-n\psi\ .
\end{flalign*}
Using the bounds on $u$, there exists a small constant $\delta>0$ such that $\psi>\delta$, and so at a maximum point of $H$,
\begin{flalign*}
0&\geq F^{ij}\nabla_i\nabla_j H\\
&\geq \frac \delta n H^2-C_1H-C_3 ,
\end{flalign*}
and so $H$ is bounded. This implies a uniform bound on $H$ and a uniform bound on $|A|$ now follows as $k\geq 2$.
\end{proof}

 \section{Proof of existence}
 \label{sec:exists}
\noindent
We now prove existence of solutions to \eqref{ActualEquation}, following the proof of V. Oliker and Y. Li \cite{YYLi:elli}. Throughout this section $\Sigma$ will be considered in graphical coordinates with graph function $u$. Consider for $0<\alpha<1$, the subset of functions in $C^{4,\alpha}(\mathbb{S}^n)$ which are $k$-admissible, denoted by $C^{4,\alpha}_{ad}(\mathbb{S}^n)$
The idea is to consider a one parameter family of prescription functions $\psi_t$ where
\[\psi_t(\xi,u,\tau) := t\psi(\xi,u,\tau) +(1-t)\Psi(\xi,u,\tau) , \]
where $\Psi(\xi,u,\tau)$ is to be chosen shortly. We define  $\Phi:C^{4,\alpha}_{ad}(S^n)\times[0,1]\rightarrow C^{2,\alpha}$, by
\begin{equation}\Phi(u,t):=H_k^\frac{1}{k}(u_t)-\psi_t(\xi,u_t,\tau(u_t))\label{oneparamfamily}\end{equation}
for all $t\in[0,1]$. We will apply degree theory to ensure that there exists at least one solution to $\Phi(u_t,t)=0$ for all $t\in[0,1]$. As in \cite{YYLi:elli}, to be able to apply the beautiful degree theory of Y. Li \cite{YYLi:1989}, we need to verify the following three steps:
\begin{description
\item[Step 1a)] Show that there exists a unique solution to
\[H_k^\frac 1 k(u_0) = \Psi(x,u_0, \tau(u_0))\ .\]
\item[Step 1b)] Show that at $u_0$ the the linearisation of $\Phi$ is invertible.
\item[Step 2] Define a suitable set of admissable functions and show that all solutions of (\ref{oneparamfamily}) stay in this set.
\item[Step 3] Verify that we may apply degree theory to ensure that the degree of $\Phi(\cdot,1)$ is not zero, and so a solution exists as claimed.
\end{description}

We choose $\Psi$ to be
\[\Psi(\xi, u, \tau) = \tau^p u \tanh(u)\ .\]
for some $p>1$.
\begin{proof}[Proof of Step 1a)]
We may easily verify that a solution exists to $\Phi(u_0,0)=0$, by considering constant functions. The hypersurfaces corresponding to $u_0=\lambda$ are totally umbillic with principal curvatures $\tanh(\lambda)$, and so $H_k^\frac 1 k=\tanh(\lambda)$. We may see that on such a hypersurface, $\tau = \cosh(\lambda)$ and so if $\lambda$ satisfies $\lambda \cosh^p(\lambda) = 1$ then $u_0$ is a solution. Clearly such a $\lambda$ exists as, writing the continuous function $\varphi(x):=x\cosh^p(x)$, $\varphi(0)=0$ and $\varphi(1)>1$.

\noindent
Suppose that there exists another $u\in C^{4,\alpha}_a(S^n)$ satisfying $\Phi(u,0)=0$. Suppose furthermore that $\max u = u(\xi_0)=\lambda_0>\lambda$. As in the proof of the $C^0$ estimates we have that at $\xi_0$,
\[\psi(\xi_0,u(\xi_0),\tau(\xi_0)) = \cosh^p(\lambda_0)\lambda_0\tanh(\lambda_0) =H_k^\frac 1 k|_{\xi_0}\leq\tanh(\xi_0)\]
which is a contradiction as $x\cosh^p(x)$ is a monotonically increasing function. Therefore $\max u \leq \lambda$. An identical argument implies $\min u\geq \lambda$, implying that $u(\xi)=u_0$.
\end{proof}

\begin{proof}[Proof of step 1b)]
Considering $\tau$, $A_i^k$ as algebraic functions of $\xi$, $u$, $\tilde \nabla u$, $\tilde \nabla^2 u$, which we will write with the variables $x$, $r$, $p^i$, $z_{ij}$ respectively then $A_i^k=A_i^k(x,r,p,z)$ and $\tau = \tau(x,r,p)$. Then the linearisation of the above in direction $v$ is given by
\begin{flalign}
\nonumber 0&=\frac{d}{ds}\left(\left[H^\frac{1}{k}_k-\psi_t\right](u+sv) \right)\\
&=F^{i}_k\pard{A^k_i}{z_{ij}}\tilde\nabla_{ij}v + \left[F^{i}_k\pard{A^k_i}{p_{k}}-\psi_\tau\pard{\tau}{p_k}\right]\tilde\nabla_kv +\left[F^{i}_k\pard{A^k_i}{r}-\psi_r - \psi_\tau\pard{\tau}{r}\right]v\ , \label{genlin}
\end{flalign}
From equations (\ref{eq:ginverse}) and (\ref{eq:Secondff}) we have that
\begin{align*}
A_i^k &=\frac {\tau}{\cosh^3(u)}\left(\sigma^{ij} + \tau^2\frac{u_l\sigma^{li}u_m\sigma^{mj}}{\cosh^4(u)}\right)\left(\tilde\nabla_{ij} u - 2\tanh(u)u_iu_j+\sinh(u)\cosh(u)\sigma_{ij}\right)
\end{align*}
and so we see that
\begin{align*}
\pard{A^k_i}{r}\big|_{(x,u,\tilde \nabla u, \tilde \nabla^2 u)}&=\frac{\pard{\tau}{r}}{\tau}A_i^k-3\tanh(u)A_i^k+\tau\frac{\cosh^2(u)+\sinh^2(u)}{\cosh^3(u)}\delta_i^k \\&\qquad\qquad+ u_lu_m\left[\text{ bounded terms }\right]\ ,
\end{align*}
and
\[\pard{\tau}{r}\big|_{(x,u,\tilde \nabla u, \tilde \nabla^2 u)}=\tanh(u)\tau +|\tilde \nabla u|^2\left[\text{ bounded terms }\right]\ .\]
\noindent
We are interested in (\ref{genlin}) when $t=0$, that is when $u=\l>0$ and $\psi = \Psi = \tau^pu\tanh(u)$. In this case, we have that
\[\tau = \cosh(u), \qquad A_i^k = \tanh(u) \delta_i^k,\qquad H_k^\frac 1 k = \tanh(u),\qquad F_k^i = \frac 1 n \delta_k^i\ .\]
At such a $u$, the linearisation becomes
\[0=a^{ij}\tilde \nabla_{ij} v +b^i \tilde\nabla_i v +c v\ ,\]
where $a$ is elliptic, $b$ is bounded, and
\begin{flalign*}
c&=\frac{\pard{\tau}{r}}{\tau}H_k^\frac 1 k-3\tanh(u)H_k^\frac 1 k+\tau \frac{\cosh^2(u)+\sinh^2(u)}{\cosh^3(u)}\text{tr}F^{ij}-\psi_r - \psi_\tau \pard \tau z\\
&=\cosh^{-2}(u)-\cosh^p(u)\tanh(u)-u\cosh^{p-2}(u) - p\cosh(u)^{p-1}u\tanh(u) \sinh(u) .
\end{flalign*}
We recall that $u=\lambda$ was chosen so that $\lambda\cosh^p\lambda=1$, which implies that $u\cosh^{p-2}u=\cosh^{-2}(u)$, and so we see that $c< 0$. The strong maximum principle now implies that the only solution $v\in C^{4,\alpha}(S^n)$ of $\Phi_u(\cdot,0)(v)=0$ is $v=0$. This implies that $\text{ker}(\Phi_u)=\{0\}$ and so, the standard theory of second order elliptic equations imply that $\Phi_u$ is invertable, as required.
\end{proof}

\begin{proof}[Proof of Step 2]
By assumption, we have that $\psi(\xi, z, \cosh(z))<\tanh(z)$ for $z<R_1$ and $\psi(\xi, z, \cosh(z))>\tanh(z)$ for $z>R_2$. Similarly we see directly that there exists $R^\Psi_1, R^\Psi_2>0$ such that
$\Psi(\xi, z, \cosh(z))<\tanh(z)$ for $z<R^\Psi_1$ and $\psi(\xi, z, \cosh(z))>\tanh(z)$ for $z>R^\Psi_2$. Setting $\overline{R}_1=\min\{R_1,R^\Psi_1\}$ and $\overline{R}_2=\max\{R_2,R^\Psi_2\}$, then for all $t\in[0,1]$, $\psi_t(\xi, z, \cosh(z))<\tanh(z)$ for $z<\overline{R}_1$ and $\psi_t(\xi, z, \cosh(z))>\tanh(z)$ for $z>\overline{R}_2$. Lemma \ref{BarrierConditions}, therefore yields uniform $C^0$ estimates
\[0<\overline{R}_2\leq u_t\leq \overline{R}_2<\infty . \]
Proposition \ref{TiltEstimate}, in addition to giving a $C^1$ estimate,  implies uniform spacelikeness, and so we may apply Theorem \ref{CurvatureEstimate} to yield $|\lambda_i|<C_A$, which implies uniform $C^2$ estimates in this situation. Uniform parabolicity of the equation now follows, and so due to the classical regularity theory for uniformly elliptic equations and the Evans-Krylov theorem \cite{Evans, Krylov} we obtain
\begin{equation}
  \label{eq:102}
  \|u_t\|_{C^{4,\alpha}(\mathbb{S}^n)} < C,
\end{equation}
for any admissible solution $u_t\in C^{4,\alpha}_{ad}(\mathbb{S}^n)$, where the constant $C$ depends on
$k, n, R_1, R_2$ and $\|\psi\|_{C^{2,\alpha}(\mathbb{S}^n)}$.
\noindent
Due to compactness and the above estimates, there exists a constant $\delta>0$ such that $\delta<\psi_t(\xi, u(\xi), \tau(u))$ for all $\xi\in S^n$. We define the bounded open set $V:=\{\lambda\in\Gamma_k : H_k^\frac 1 k(\lambda)\geq \delta , |\lambda|<\sqrt{n}C_A \}\subset\Gamma_k$ and we define the bounded set
\begin{align*}
  \label{eq:104}
  \mathcal{B} &=  \Big\{u \in C^{4,\alpha}(\mathbb{S}^n) \Big|\\
  &\qquad\qquad\frac 1 2 \overline{R}_1 < u < 2 \overline{R}_2 ,  \| u \|_{C^{4,\alpha}(\mathbb{S}^n)} < C
\mbox{ and } \lambda(A[u(\xi)])\in V \ \forall \ \xi\in\mathbb{S}^n\Big\}.
\end{align*}
\noindent
Clearly the arguments of the previous paragraphs imply that any admissable solution $u_t\in C^{4,\alpha}_{ad}(S^n)$ is contined in $\mathcal{B}$, and $\partial \mathcal{B}\cap \Phi^{-1}(\cdot,t)=\emptyset$ for all $t\in[0,1]$.
\end{proof}

\begin{proof}[Proof of Step 3]
This step now follows exactly as in \cite{YYLi:elli}. Due to Step 2 and \cite[Definition 2.2, Proposition 2.2]{YYLi:1989}, the degree is defined and constant for $t\in[0,1]$. By \cite[Proposition 2.3, Proposition 2.2]{YYLi:1989}, $\text{deg}(\Phi(\cdot,0), \mathcal{B}, 0)=\text{deg}(\Phi_u(\cdot,0), B_1, 0)$ , where $B_1$ is the open unit ball in $C^{4, \alpha}(S^n)$. However, Step 1 and \cite[Proposition 2.4]{YYLi:1989} imply that $\text{deg}(\Phi(\cdot,0), \mathcal{B}, 0)=\pm 1=\text{deg}(\Phi(\cdot,1), \mathcal{B}, 1)$, and we conclude that a $k$-admissable solution $u_1$ to $\Phi(u_1,1)=0$ exists. Standard elliptic estimates imply that $u_1$ is smooth and therefore the proof of Theorem \ref{FullTheorem} is complete.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
package org.pentaho.hadoop.shim.common;

import org.apache.hadoop.util.VersionInfo;
import org.junit.Test;
import org.pentaho.hadoop.shim.api.ConfigurationException;
import org.pentaho.hadoop.shim.api.internal.Configuration;
import org.pentaho.hadoop.shim.api.internal.DistributedCacheUtil;
import org.pentaho.hadoop.shim.spi.HadoopShim;

import java.util.ArrayList;
import java.util.List;

import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertNull;
import static org.junit.Assert.assertTrue;
import static org.junit.Assert.fail;
import static org.junit.Assert.assertNotNull;

public class CommonHadoopShimTest {

  @Test
  public void getHadoopVersion() {
    CommonHadoopShim shim = new CommonHadoopShim();
    assertEquals( VersionInfo.getVersion(), shim.getHadoopVersion() );
  }

  @Test( timeout = 10000 )
  public void getNamenodeConnectionInfo() {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration c = shim.createConfiguration();
    c.set( "fs.default.name", "localhost:54310" );
    String[] info = shim.getNamenodeConnectionInfo( c );

    assertEquals( 2, info.length );
    assertEquals( "localhost", info[ 0 ] );
    assertEquals( "54310", info[ 1 ] );
  }

  @Test
  public void getNamenodeConnectionInfo_local() {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration c = shim.createConfiguration();
    c.set( "fs.default.name", "file:///" );
    String[] info = shim.getNamenodeConnectionInfo( c );

    assertEquals( 2, info.length );
    assertNull( info[ 0 ] );
    assertNull( info[ 1 ] );
  }

  @Test( timeout = 10000 )
  public void getJobtrackerConnectionInfo() {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration c = shim.createConfiguration();
    c.set( "mapred.job.tracker", "anotherhost:54311" );
    String[] info = shim.getJobtrackerConnectionInfo( c );

    assertEquals( 2, info.length );
    assertEquals( "anotherhost", info[ 0 ] );
    assertEquals( "54311", info[ 1 ] );
  }

  @Test
  public void getJobtrackerConnectionInfo_local() {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration c = shim.createConfiguration();
    c.set( "mapred.job.tracker", "local" );
    String[] info = shim.getJobtrackerConnectionInfo( c );

    assertEquals( 2, info.length );
    assertNull( info[ 0 ] );
    assertNull( info[ 1 ] );
  }

  @Test
  public void configureConnectionInformation() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( "namenodeHost", "namenodePort", "jobtrackerHost", "jobtrackerPort", conf,
      logMessages );
    assertEquals( "hdfs://namenodeHost:namenodePort", conf.get( "fs.default.name" ) );
    assertEquals( "jobtrackerHost:jobtrackerPort", conf.get( "mapred.job.tracker" ) );
    assertEquals( 0, logMessages.size() );
  }

  @Test( expected = Exception.class )
  public void configureConnectionInformation_null_namenodeHost() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( null, "namenodePort", "jobtrackerHost", "jobtrackerPort", conf, logMessages );
  }

  @Test( expected = Exception.class )
  public void configureConnectionInformation_empty_namenodeHost() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( "", "namenodePort", "jobtrackerHost", "jobtrackerPort", conf, logMessages );
  }

  @Test
  public void configureConnectionInformation_null_namenodePort() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();

    shim.configureConnectionInformation( "namenodeHost", null, "jobtrackerHost", "jobtrackerPort", conf, logMessages );
    assertEquals( "hdfs://namenodeHost", conf.get( "fs.default.name" ) );
    assertEquals( "jobtrackerHost:jobtrackerPort", conf.get( "mapred.job.tracker" ) );
    assertEquals( 1, logMessages.size() );
    String message = logMessages.get( 0 );
    assertTrue( "Unexpected message: " + message, message.contains( "HA?" ) );

    logMessages.clear();
    shim.configureConnectionInformation( "namenodeHost", "   \t   ", "jobtrackerHost", "jobtrackerPort", conf,
      logMessages );
    assertEquals( "hdfs://namenodeHost", conf.get( "fs.default.name" ) );
    assertEquals( "jobtrackerHost:jobtrackerPort", conf.get( "mapred.job.tracker" ) );
    assertEquals( 1, logMessages.size() );
    message = logMessages.get( 0 );
    assertTrue( "Unexpected message: " + message, message.contains( "HA?" ) );

    logMessages.clear();
    shim.configureConnectionInformation( "namenodeHost", "   -1   ", "jobtrackerHost", "jobtrackerPort", conf,
      logMessages );
    assertEquals( "hdfs://namenodeHost", conf.get( "fs.default.name" ) );
    assertEquals( "jobtrackerHost:jobtrackerPort", conf.get( "mapred.job.tracker" ) );
    assertEquals( 1, logMessages.size() );
    message = logMessages.get( 0 );
    assertTrue( "Unexpected message: " + message, message.contains( "HA?" ) );
  }

  @Test( expected = Exception.class )
  public void configureConnectionInformation_null_jobtrackerHost() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( "namenodeHost", "namenodePort", null, "jobtrackerPort", conf, logMessages );
  }

  @Test( expected = Exception.class )
  public void configureConnectionInformation_empty_jobtrackerHost() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( "namenodeHost", "namenodePort", "", "jobtrackerPort", conf, logMessages );
  }

  @Test
  public void configureConnectionInformation_null_jobtrackerPort() throws Exception {
    CommonHadoopShim shim = new CommonHadoopShim();
    Configuration conf = new ConfigurationProxy();
    List<String> logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( "namenodeHost", "namenodePort", "jobtrackerHost", null, conf, logMessages );
    assertEquals( "hdfs://namenodeHost:namenodePort", conf.get( "fs.default.name" ) );
    assertEquals( conf.get( "mapred.job.tracker" ), "jobtrackerHost:" + shim.getDefaultJobtrackerPort() );
    assertEquals( 1, logMessages.size() );
    String message = logMessages.get( 0 );
    assertTrue( "Unexpected message: " + message, message.contains( "using default" ) );

    logMessages = new ArrayList<String>();
    shim.configureConnectionInformation( "namenodeHost", "namenodePort", "jobtrackerHost", "", conf, logMessages );
    assertEquals( "hdfs://namenodeHost:namenodePort", conf.get( "fs.default.name" ) );
    assertEquals( conf.get( "mapred.job.tracker" ), "jobtrackerHost:" + shim.getDefaultJobtrackerPort() );
    assertEquals( 1, logMessages.size() );
    message = logMessages.get( 0 );
    assertTrue( "Unexpected message: " + message, message.contains( "using default" ) );
  }

  @Test
  public void getDistributedCacheUtil() {
    HadoopShim shim = new CommonHadoopShim();
    try {
      DistributedCacheUtil dcUtil = shim.getDistributedCacheUtil();
      assertNotNull( dcUtil );
    } catch ( ConfigurationException ex ) {
      fail( "Caught exception " + ex.getMessage() );
    }
  }

  @Test
  public void setDistributedCacheUtil_null() {
    CommonHadoopShim shim = new CommonHadoopShim();
    try {
      shim.setDistributedCacheUtil( null );
      fail( "expected exception" );
    } catch ( NullPointerException ex ) {
      assertNotNull( ex );
    }
  }

  /*  BAD-215 enable after decisionod enabling functionality
  @Test( expected = SQLException.class )
  public void getJdbcDriver_notSupportedShim_acceptsUrl() throws SQLException {
    CommonHadoopShim shim = new CommonHadoopShim();
    Driver notSupportedShimDriver = shim.getJdbcDriver( "hive2Simba" );
    notSupportedShimDriver.acceptsURL( "" );
  }

  @Test( expected = SQLException.class )
  public void getJdbcDriver_notSupportedShim_connect() throws SQLException {
    CommonHadoopShim shim = new CommonHadoopShim();
    Driver notSupportedShimDriver = shim.getJdbcDriver( "hive2Simba" );
    notSupportedShimDriver.connect( "", null );
  }

  @Test( expected = SQLException.class )
  public void getJdbcDriver_supportedShim_acceptsUrl() {
    CommonHadoopShim shim = new CommonHadoopShim() {
      {
        JDBC_POSSIBLE_DRIVER_MAP.put( "hive2Simba", SupportedDriver.class.getCanonicalName() );
      }
    };
    Driver notSupportedShimDriver = shim.getJdbcDriver( "hive2Simba" );
    try {
      notSupportedShimDriver.acceptsURL( "" );
    } catch ( SQLException e ) {
      fail( "In supported shim driver should be added to the Classes map" + e.getMessage() );
    }
  }

  @Test( expected = SQLException.class )
  public void getJdbcDriver_supportedShim_connect() {
    CommonHadoopShim shim = new CommonHadoopShim() {
      {
        JDBC_POSSIBLE_DRIVER_MAP.put( "hive2Simba", SupportedDriver.class.getCanonicalName() );
      }
    };
    Driver notSupportedShimDriver = shim.getJdbcDriver( "hive2Simba" );
    try {
      notSupportedShimDriver.connect( "", null );
    } catch ( SQLException e ) {
      fail( "In supported shim driver should be added to the Classes map" + e.getMessage() );
    }
  }

  private static class SupportedDriver implements Driver {
    @Override public Connection connect( String url, Properties info ) throws SQLException {
      return null;
    }

    @Override public boolean acceptsURL( String url ) throws SQLException {
      return false;
    }

    @Override public DriverPropertyInfo[] getPropertyInfo( String url, Properties info ) throws SQLException {
      return new DriverPropertyInfo[ 0 ];
    }

    @Override public int getMajorVersion() {
      return 0;
    }

    @Override public int getMinorVersion() {
      return 0;
    }

    @Override public boolean jdbcCompliant() {
      return false;
    }

    @Override public Logger getParentLogger() throws SQLFeatureNotSupportedException {
      return null;
    }
  } */
}

redpajama_set_name: RedPajamaGithub


text:
Q: Explicit formula for IFS fractal dimesnion Is there an explicit formula for finding the box counting dimension of an arbitrary IFS fractal, such as the IFS fern or any other random IFS fractal? If not, is there at least a summation, or recurrence relation that could find the fractal dimension? An example of how this would work would also be appreciated. If there is really no formula, is it just because we don't know it or because it cannot be expressed?

A: No - there is no such formula as discussed in this MathOverflow post.
The most important work in this area has been done by Ken Falconer.  His first paper on the subject "The Hausdorff dimension of self-affine fractals" Math. Proc. Cambr. Phil. Soc. 103 (1988) 339-350 contains an algorithm that yields an upper bound on the box-counting dimension which almost always yields the Hausdorff dimension as well.  Using these ideas, I estimate the dimension of the fern to be about 1.84.  I'm not certain, though, as the technique can fail in special cases.  Later papers by Falconer and others explore situations where the technique is guaranteed to work.

A: For graph-directed IFS of similarities satisfying an open set condition with every cycle contracting, it's possible to compute the Hausdorff dimension:
Hausdorff Dimension in Graph Directed Constructions
R Daniel Mauldin, S C Williams
Transactions of AMS, vol309 no2, October 1988, 811-829
A regular IFS of can be considered as a graph-directed IFS with one node, so every similarity must be contracting.
However, similarities are not very general (you can have IFS with all sorts of transformation functions), box-dimension is sometimes different from Hausdorff dimension, and I don't know if any algorithm exists for verifying that the IFS passes the open set condition.
I have a Javascript implementation of the algorithm described in the paper (view page source), with some more information in two blog posts.

redpajama_set_name: RedPajamaStackExchange


Longtime teacher, bestselling author, and original education Hacker Mark Barnes is taking old school to task and giving you right now solutions for your biggest problems, in brief, no-nonsense episodes of the Hack Learning Podcast.
Founder and publisher at Times 10 Books, Mark brings Hacks that help you solve problems with what-you-can-do-tomorrow tips and strategies.
Yet another great Hack Learning product!
139 — Are teachers doing enough for Generation Alpha?
119 – What if You Knew Every Student's Quit Point?
117 – What if teachers could extinguish shame?
113 – No, Mr. President, We Will NOT Arm our Teachers!
99 – What's Your Parent Mantra?
98 – What Is Your Inspirational Story?
76 – Hacking PD: What Do You Want?
45 – Hacking Homework Again. Your assignment is. . .
26 – Teach One Lesson — to the World!
Tell us about your review in a quick email here, or tweet us here, and we may feature it on this page.


text:
>RR-2682
The tanKer Aerial Refueling Model for Analysis (KARMA)
by Dahlia Anne Goldfeld, Richard Mason
Air Warfare,
Fighter Aircraft,
Modeling and Simulation,
Tanker Aircraft,
This report describes the detailed algorithms contained within the tanKer Air Refueling Model for Analysis (KARMA) and explains how to use the model. KARMA was developed as part of a RAND Project AIR FORCE project to evaluate the capability of the Mobility Air Forces to support combat operations in an anti-access/area denial environment in the 2030 time frame. KARMA enables an analyst to simulate the extent to which a tanker force can aerially refuel combat aircraft in a denied environment. It also assesses the impact that missile attacks have on the sustainment of combat sorties. In this application, the authors used the U.S. Air Force Synthetic Theater Operations Research Model to generate an air tasking order for the combat aircraft in a scenario as an input to KARMA. The authors also created KARMA to be compatible with the RAND Combat Operations in Denied Environments (CODE) family of models and used CODE outputs to provide expected levels of air base damage given specific ballistic and cruise missile attacks. KARMA is a flexible, fast-running model, allowing users to explore a variety of scenarios, basing structures, concepts of operations, enemy attack patterns, and tanker aircraft capabilities.
Receiver Aircraft Demand
Tanker Allocation
Air Base Attacks
Future Work and Conclusions
Input Files and Data
Output Files
The research reported here was commissioned by the Air Force Air Mobility Command, Strategic Plans and Programs (AF AMC/A5/8), and conducted by the Force Modernization and Employment Program within RAND Project AIR FORCE.
This report is part of the RAND Corporation research report series. RAND reports present research findings and objective analysis that address the challenges facing the public and private sectors. All RAND reports undergo rigorous peer review to ensure high standards for research quality and objectivity.
Paperback ISBN/EAN: 9781977401502
DOI: https://doi.org/10.7249/RR2682
Document Number: RR-2682-AF
Series: Research Reports
Tanker Aircraft
Goldfeld, Dahlia Anne and Richard Mason, The tanKer Aerial Refueling Model for Analysis (KARMA). Santa Monica, CA: RAND Corporation, 2019. https://www.rand.org/pubs/research_reports/RR2682.html. Also available in print form.
Goldfeld, Dahlia Anne and Richard Mason, The tanKer Aerial Refueling Model for Analysis (KARMA), Santa Monica, Calif.: RAND Corporation, RR-2682-AF, 2019. As of January 14, 2020: https://www.rand.org/pubs/research_reports/RR2682.html

redpajama_set_name: RedPajamaCommonCrawl


I hope you find great value in these Quotes by Milton Caniff from my large collection of motivational and inspirational quotes and sayings.
stores to get their stuff in the hands of readers.
There is too much illustrating of the news these days.
what the cartoonists are saying or how they feel about a certain issue.
Thank you for visiting: Milton Caniff Quotes and Sayings.


Dvoretskii, N.; Anikanova, L.; Malysheva, Z.; Koshel', G.
The kinetic parameters of dissolution in hydrochloric acid of Fe2O3 (hematite) samples of varied thermal and chemical prehistory are calculated in terms of Delmon's model of instantaneous nucleation on the surface of a solid reagent.
Dvoretskii, N., Anikanova, L., Malysheva, Z., & Koshel', G. (2004). Kinetic Parameters of Dissolution of Iron(III) Oxides of Varied Thermal and Chemical Prehistory in Hydrochloric Acid. Russian Journal of Applied Chemistry, 75(8), 1207-1210.
Dvoretskii, N., L. Anikanova, Z. Malysheva, and G. Koshel'. "Kinetic Parameters of Dissolution of Iron(III) Oxides of Varied Thermal and Chemical Prehistory in Hydrochloric Acid." Russian Journal of Applied Chemistry 75.8 (2004): 1207-1210.


text:
Anti-Human Rights, Phenomenon of Terrorism, Syria, United States
Syria crisis and America's funeral
Written by ORIENTAL REVIEW on 28/06/2013
More in Anti-Human Rights:
Out Of Kilter: National Security And Press Freedoms In Australia 10/07/2019
The European Union Approves The Nomination Of Four Senior Civil Servants 10/07/2019
Facebook, Funny Money And Libra 03/07/2019
America, at least publicly, seems destined to put its military might against Syria, notwithstanding the fact that America will have to arm and train terrorists it has been fighting for a decade, notwithstanding America will be risking a world war and notwithstanding the simple fact that every authority advises against this policy and the American people want no part of it.
American Policy Heads to the "Other Side of the Mirror"
When American CIA and Special Forces personnel received the orders to arm and train Syrian rebels, a week ago, they were flabbergasted. Many had been "on the ground" for months or longer, had seen the 'Free Syrian Army' purged and overrun with Jihadists, that and supplanted in its combat role by al-Qaeda in Iraq partner, al-Nusra.
Not all Jihadists come to Jordan and Turkey and, eventually enter Syria (and Iraq) as terrorists. Most come for the $3,000 cash payment to their families, for the $200 per month, for the food and, frankly, for the free cigarettes.
Most don't arrive as hardcore Jihadists or terrorists, in fact few do. Most, if not all, however, those who survive, will leave, US trained and supplied but al-Qaeda led, "blooded," radicalized and ready to move against Iraq, against Jordan or to set up terror cells across Europe and North America.
Insanity or Beyond Insanity?
What is America thinking, has it, as Paul Craig Roberts and so many others, even Zbigniew Brzezinski have observed, "lost their minds?"
To all but the mentally challenged, the Global War on Terror is an artificial construct, 90% false flag terror, 90% bank robbery, 90% "Hollywoodism."
You say the total is more than 100%? Welcome to the "new math."
The terror "game" was supposed to replace the Cold War but better, helpless enemies, the chance to use new hi-tech weapons on defenseless civilians, a change to use a controlled press to manage the narrative and the lesson of World War II, a "slam bang" take-off, propelled by 9/11, long admitted, long proven to be the "New Pearl Harbor."
The basis of America's program of encirclement of Russia and China that began with the Truman Doctrine in 1947 was the "domino theory."
If one nation fell to communism, such as Cuba or Nicaragua or Greece, we could name dozens, perhaps even mention Vietnam; the entire world would fall, "like dominos."
Domino Theory in Reverse
The site of a bomb attack in Damascus, Syria
It is now clear that the "Arab Spring" revolutions, purported to be an "instantaneous uprising" of "freedom loving" Muslims, a "grass roots" movement based on social media, whistleblowing and democratic reform has taken on a sinister side.
Tunisia's could be believed, initially at least. In Libya, Gaddafi's flamboyance and alienated many but when the turn came for the UN to authorize use of force to bring about regime change, a move both Russia and China backed, more should have seen something very wrong.
When Gaddafi was brutally murdered, the outline of what would continue was now unquestionable. Freedom was the last thing these revolutions would bring; this was clearly as "softening up" in preparation for what many now see a reestablishment of the Ottoman Empire.
Syria and Iraq, both beleaguered by terrorist groups, Western backed, air support from the international psychopaths, Turkey and Israel, are going to be the dominos that bring the United States to its knees.
A Warning Anyone Could See
Egypt's revolution has been the greatest disaster so far. A nation long under the thumb of Mubarak's corrupt dictatorship, his long partnership with Israel and their Arab cohorts, the general uprising of the Egyptian people held great promise.
It would and well should have immediately opened Gaza, removed walls, fences, ended the naval blockade and put a friendly air force in Gaza skies.
Nothing of the kind happened, nothing of the kind is even spoken of.
While Mubarak awaits trial and execution for his secret complicity with Israel, Morsi calls on Egyptians to march on Syria in a jihad on behalf of Israel, something even Mubarak would never have contemplated.
You see, the Muslim Brotherhood is but another CIA "construct" from the Cold War days, one of several such as the Mujahedeen, ideological powerhouses to feed the wars on the periphery of the Soviet Union, Bosnia and Kosovo, Chechnya and Afghanistan.
All that was needed was to reach out and "draw water from the well." Revolution for freedom was derailed and reactionary totalitarianism again reared its head.
This was proof, undeniable. A Shiite majority, long oppressed, long denied political expression, rose against a police state.
Rows of tanks from Saudi and the Persian Gulf States flowed in, painting a clear picture: Revolution is OK, so long as the new regime is weak, corrupt, compliant and servile.
Bahrain's government, on each above count, had already gone as far as possible. There was no hope for improvement.
It had also become abundantly clear that a third force was managing the entire movement and that things were not as they seemed.
Nexus Control, "Strange Bedfellows"
Simply put, if America "wins" and Syria falls, America loses, in fact loses more than even the most imagine and America's greatest critics are aware of.
Vis-à-vis Israel, the nomination of Hagel, the appointment of Dempsey and Rice were strong indications that the United States, at least the White House, was willing to steer a genuinely pro-American course, finally after years of unmistakable Zionist control and the ensuing disasters, incident after incident of false flag terrorism, espionage and economic warfare on America waged from Tel Aviv on an unprecedented scale and gross interference in US elections, bribery of public officials on a massive scale and orchestration of a "ham handed" and almost infantile program of psychological warfare using press and entertainment industry assets.
Then, in an inexplicable and puzzling way, particularly after President Obama's May 23, 2013 speech calling for an end to new US military involvement, the president backed a report citing WMD use by the Assad government and authorized military aid to groups he knows are terrorists bent on killing Americans.
Al-Nusra insurgents
Al-Nusra is, in fact, made of up Saddam's former Baathist security personnel credit for the killing of 5,000 American soldiers in Iraq.
Not only is this group, an al-Qaeda affiliate, still killing more and more every day in Iraq, they have already begun ethnically cleansing Christians in rebel occupied Syria and are militarily operating with air and artillery support from both Israel and Turkey.
In fact, there is no more determined enemy on earth set on destroying the United States than al-Nusra, a terrorist organization the US is now planning on handing, not only Syria to but Iraq as well.
Jordan would soon follow and, based on the flow of billions in oil revenue to al-Qaeda, now partnered with Egypt, Turkey, Saudi Arabia and Israel, the nuclear threat continually parroted by the blitherings of DEBKA and the Times of Israel would become a reality.
The Syrian conflict that the US has agreed to bolster despite clear evidence that the Syrian people choose their own imperfect government over what is now obvious, a decade of sectarian violence followed by broad corruption and Western dominance, makes America's bankrupt policies clear.
In fact, calling them "American policies" is, in itself, an absurdity. What nation would spend a decade creating an enemy where there was none, the purpose of the "war on terror," quell that same conflict with military brutality and billions in bribes and then try to unleash another, guaranteed to be worse, perhaps endlessly so?
What has been set in motion is a plan, one tied to re-establishment of feudal rule, one likely to be built on the ashes of sectarian wars, ethnic cleansing and brutal suppression of human aspirations.
A similar plan is being put in motion for the Middle East as well.
Source: Press TV
Jabhat Al-Nusra
Independent Moscow-based Internet journal focusing on the past, present and future political and security issues in Eurasia and beyond, working for you since 2010.
A world war for sure will be started. Can Russia and China prevent it? The Shias are now turned into warriors and its about their survival with Iran leading the crescent pack from Iran to Iraq to Syria to Hezbollah. Haven't the sunni Iraqis under Saddam learnt their lesson when they took on Iran? have not the Israelis learnt their lessons from Hezbollah in 2006?
It's very obvious that the west is gathering a bunch of spoilt oil rich Arabs to lead them into war against a very strong Iran. The Saudis, Qataris, Emiratis are not warriors. They're a bunch of monkeys who like to show off their wealth and buy the goodies from Harrods and suppress the poor workers from third world countries. That's not Iranians.They build their country with their own hands.
Anyone should realize that the west has opened Qatar's doors to the Afghani taliban so as to put together a Sunni pack against Iran. After all the west knows that the Taliban are warriors and now need them to start the last great game. The end game of the west – accept it or not- is Pakistan's denuclearization.
Pingback: Sunday 30/6 Reads « Here comes the pain
Oriental Review should not post anything written by this compulsive liar who claims Fidel Castro works for the CIA and that he personally insured Obama's election victory. Keeping on lying Doff!
Andre Fomine
I'm afraid I didn't quite catch you, Dean. Do you have some substantial objections regarding the text above? Please express'em calmly with argumentation.
It is true he claims that Fidel Castro is a CIA agent:
http://www.veteranstoday.com/2012/11/01/romney-leaks-drugs-blood-diamonds-and-a-cuban-mistress/
http://www.veteranstoday.com/2012/11/05/vt-hacked-over-romneys-lifting-of-cuba-embargo-document/
If you want to find out what type of a liar that guy is please listen to this: http://www.youtube.com/watch?v=E0Plve7o1O4
Duff also says George Soros and Obama re great men. He claims that Russia is trying to destroy the US,


"With over 30 000 solar home systems produced, at Specialized Solar Systems we are only in the beginning stages of an incredible journey – both humbling and exciting – to bring sustainable, renewable, affordable, off-grid solar energy into the estimated 600 million non-electrified homes in Africa [source: World Bank]: fast-tracking job creation, infrastructure, communication, water purification, education, entertainment, possibilities, hope …and a better standard of living for all."
Built on the backbone of entrepreneurship and micro-economies the data has been carefully evaluated over many years and strategic partnerships formed.
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Above: A short video of our dedicated factory staff in action.
In this video, our factory staff show some of the detail involved when building our immensely popular managed off-grid solar home system for DCGO Power.Today solar electrification project. With over 30 000 units produced by Specialized Solar Systems over the years, these economical and reliable homegrown DC micro-grids are used to solar-power small households with lighting, entertainment, and refrigeration throughout Africa.
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Based on a pay-for-energy-usage business model and using solar power generation for energy delivery, a percentage of recaptured revenue supports product expansion, employment and sustainability.
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The core architectural design is to securely assist with the management function by providing an integrated framework for remote system monitoring, GPS system allocation, client and system management, billing options, vendor management and automated system switching in a simple and intuitive user interface that is accessible anywhere where there is an internet connection. The framework allows for the creation of independent utilities or ecosystems where administrator users are created then assigned to their own unique environment.
Vendors (P.O.S) are set up and managed by these administrators. Using a pay-for-energy-business model, the vendor business model is based on a wallet fund system. For a vendor to dispense energy, funds must be available in their account thus eliminating risk for the independent utility. Adequate training is provided in all the framework functions in 3 modules.
The cloud-based framework is constantly monitored and support is always available. The solution is now used extensively in many off-grid solar electrification projects where remote administrators and their ground operations rely on real-time data to fulfill their operational criteria.
" Our passion is using renewable clean energy to empower indigent communities; fast-tracking equality acceleration and socio-economic transformation. "
SSS-EPOWER is a centralised on-line solar system management framework that is accessible with any modern internet browser. It is built on dedicated and reliable cloud services and allows the creation of system management clusters for solar energy service providers (utilities) for efficient management of macro and micro solar energy solutions. The use of solar energy production as a primary mechanism for energy distribution for large-scale and diverse demographic coverage is the culmination of many years of product development leading to the formation of an integrated solar energy supply solution.
The management framework supports automated billing services that can be set up and then allocated to a solar system . The payment packages work primarily with upfront payments for energy usage and once allocated, perform an automated pay-for-energy-usage function. All payment packages include an optional switching function that allows the solar system to be changed to varied states - ON, WARNING and OFF - all dependent on the user account balance and the allocated payment package. The user can also be notified of the different status changes via SMS or email.
A vendor outlet can be established anywhere where there is an internet connection. With the vendor application, the point of sale outlet can receive payment and print invoices or mini statements for the end-user as a proof of payment. The receipt shows the last 3 transactions, the payment value , account balance and the current system status.All vendor transactions happen in real-time. For example: If a unit is OFF and a payment is processed that results in a system status change to ON, the transaction will update the systems status immediately if criteria are met. The vendor module is created and managed within the main framework.
Having designed and manufactured over 30 000 DC microgrids units installed throughout Africa ranging from simple unmanaged solar systems to sophisticated remotely managed and monitored systems, Specialized Solar Systems has a proven track record in solar energy product offerings and services that can fulfill almost any individuals or energy service provider's energy requirements.
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Q: GLPK Simplex behavior with meth=GLP_PRIMAL vs meth=GLP_DUALP I have a linear program with no objective function. So I just want to test its feasibility. I am using GLPK api for simplex to do that. When I run simplex with the default method (meth=GLP_PRIMAL), the solver fails to converge in 100000 iterations (that is the limit I have set). However, when I use the method GLP_DUALP, after a few iterations I get the message "Warning: dual degeneracy; switching to primal simplex" and it goes on to converge in a reasonable number of iterations.
So my question is if it ultimately uses the primal simplex in both cases, why does it not converge in the first case. What might be going on.
Thanks in advance.

A: It's hard to say what exactly is happening without detailed information about the problem, but basically the primal simplex in the case of dual degeneracy is some sort of "warm-started".
While using the dual algorithm the optimization process is generating a dual problem, which will start with an optimal solution and then tries to find a feasible solution which still contains the optimal condition. Primal simplex instead will start the other way around with a feasible solution and then search for an optimal solution.
When the strong duality theorem is true both optimal solution should be the same. In your Problem you get a "dual degeneracy" warning, which means that in the dual problem there is a equation which turns to be 0. So the variable inside this equation has no influence to the objective function (no matter if X is 100 or just 1), this is plausible since you dont have an objective function. GLPK is then switching to primal simplex, because for the dual problem there exists alternative optimal solutions. With the already derived informations the primal simplex could be faster. I don't know what GLPK is exactly doing, but normally one can use the feasible solutions of the dual problem as a lower bound to the primal problem.
What stalls your primal approach is perhaps the same issue. The Problem degenerated and the simplex algorithm is getting stuck at one variable which has no influence to the objective, therefor it's hard to find an optimal value for this variable.


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WENDY RANKIN CARMICKLE
Wendy Rankin Carmickle (49) was born on January 4, 1972, and passed away on August 29, 2021. She was born in Bristol, Tennessee, to Jim and Alyce Rankin, and grew up in Bristol, Virginia. A 1990 graduate of John Battle High School, she attended Virginia Tech and completed training as a paralegal in Atlanta, Georgia, in 1994. At the time of her death, she was pursuing additional training in paralegal studies through Liberty University online. She lived most of her life in southwestern Virginia (Bristol, Roanoke, Chilhowie, and Marion). Along with Bill and her children, she lived in Bluffton, South Carolina from 2018 until her death.
Wendy was preceded in death by her mother, Alyce Mumpower Rankin, and her grandparents Lena Rush Rankin, George Washington Rankin, Helen Mary Mumpower, and Troy Bolden Mumpower Sr. She is survived by her father, Jim Rankin; her sister Debbie Rankin; her husband, William "Bill" Randolph Carmickle; her children Andrew Christopher Snodgrass (25), Caroline Alena Snodgrass (22), Emma Katheryn Snodgrass (20), and Bentley Reese Carmickle (10); and her stepdaughter Elisbaeth Renei Carmickle (29).
Wendy spent countless hours going the extra mile to make other people's lives more special. She created scrapbooks to preserve the special moments in her family's lives. She was successful in numerous home businesses where she brightened birthdays with balloon bouquets; assisted parents in dressing their children in matching outfits; and boosted women's confidence in their appearance using a fine line of skin care products. Through her latest business, Sweet Tees, she spent immeasurable energy and creativity personalizing bridal party attire, matching t-shirts for family reunions and vacations, and one-of-a-kind items to match her clients' unique personalities.
Her love for life was best seen through Wendy's love for her family, who meant everything to her. She put her career on hold as she dedicated herself to being a full-time mother. The greatest joy of her life was watching her children grow, supporting them in their studies and favorite activities, encouraging them to follow their dreams, and celebrating their accomplishments. The beach was her truest happy place, especially when she was there surrounded by her husband and her children.
The family will celebrate Wendy's life at a private memorial service. They would like to thank everyone for their acts of kindness and love during this difficult time.
Linda Toomer says:
Wendy was a very important part of our team at Burr & Forman and she will be truly missed. She wasn't here very long but made a huge impact on all of the staff. I personally treasure our hugs and look forward to seeing you again. My heart goes out to the family and friends of such a good person. We will miss her.
Penny McCarty says:
Bill, Andrew, Caroline, Emma, and Bentley,
I am so sorry for your loss my heart breaks for you all.
May you gather strength from everything yWendy was and taught you to be.
Please let me know if you need anything.
Susan Bennett Stanley says:
Debbie and Jim,
My heart breaks for you. So sorry for the loss of Wendy. My prayers are for you, her children and her husband.
Travis and Jessica Johnson says:
Our thoughts and prayers go out to you all. We're so heartbroken and sorry for your loss. Wendy was such a beautiful person inside and out. Sending our love, hugs and prayers for the days, weeks, and months ahead.
Bob and Ann Gowin says:
We only met Wendy once but she was so sweet, kind and accepting of us that she became instant family.
Our hearts go out to her father, Bill and their children. Rest In Peace Wendy!
Debbie Ringler-Boggs says:
Wendy was kind, giving, a fellow Hokie and such a sweet soul. Working together in Roanoke, VA, we became not just co-workers but friends. She was a great confidante, a wonderful mother and family meant the world to her. My heart and prayers are with the family during this difficult time.
Kim Ellison says:
I can't even say enough about the person Wendy was. She dedicated her life to her family…and friends. She was a hero in my book! She lit up the room with her smile and fun/loving personality. She will be missed by so many! My heart goes out to you -Bill, Andrew, Caroline, Emma, and Bentley -she loved you all with everything in her. Please know we are here…anytime. Carol and Emma …you have a room at our house anytime…love y'all dearly!! Until we meet again Wendy…much love


It's been 139 days since I found out about my husband's multiple affairs throughout our marriage.
Some days I feel filled to overflowing with love for him. Some days I look at him and my heart is heavy with the reality of it all and I wonder what on earth I'm doing still with this man, because clearly no marriage can survive this. Sometime I believe that love always wins, and sometimes I get online and shop for an apartment for myself because I just need to run away and be alone for a while. Sometimes I just can't stop kissing him and sometimes I want to throat punch him in his sleep.
I think all of these feelings are healthy.
Most days the possibility of hope, the hope for hope, is all I'm hanging by. Today I have hope; today is a good day. Tomorrow will likely be less hopeful, and that's okay. Everything is as it should be. It's absolutely terrible, it's a nightmare, its heavy and usually unbearable, and it takes more strength than anyone would ever have on their own. I wouldn't wish a moment of this on anyone- not even the women who took my husband. No one should ever have to endure this, but it happened and I can't change it. I can only choose what I can choose. I can decide each day if I love this man enough that I am willing to endure this with him, because leaving would be SO much easier. Repairing a relationship after so many lies and so much damage has been done, repairing it with 2 completely broken people who are working hard just to repair themselves individually, sometimes it doesn't even seem possible.
If this is true, and I believe that it is, then we were perfect today. Perfectly imperfect and that is the best any of us can be. Today we chose what our hearts told us was best, and that was to stay together. Anytime we follow our hearts we are living perfectly. If my heart told me to let go of this marriage I would say the same. My instincts have yet to ever lead me in the wrong direction and I intend to always follow them. As long as I do that there is hope in at least one of her forms.


"Roots" and images of root vegetables are abundant in their metaphorical revelations.
bring about terms such as Eros, Thanatos, conception, birth and waning.
weaves the tale of a gripping journey into the Italian traditions of mural, mosaic and grotesque.


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Sister Wives, Surrogates and Sex Workers: Outlaws by Choice? (Gender in Law) (Hardcover)
By Angela Campbell
Did she choose that?' Or, more normatively, 'Why would she choose that?' This book critiques and offers an alternative to these questions, which have traditionally framed law and policy discussions circulating around controversial genderized practices. It examines the simplicity and incompleteness of choice-based rhetoric and of presumptions that women's conduct is shaped, in an absolute way, either by choice or by coercion. This book develops an analytical framework that aims to discern the meaning and value that women may ascribe to morally ambiguous practices. An analysis of law's approach to polygamy, surrogacy and sex work, particularly in Canada, the United Kingdom and Australia, provides a basis for evaluating the choice-coercion binary and for contemplating alternate modes for assessing, from a law and policy standpoint, the palatability of social practices that appear pernicious to women. Weaving together interdisciplinary research, an innovative analytical framework for assessing choices ostensibly harmful to women, and a critique of the legal rules governing such choices, this book bears relevance for students, scholars, practicing jurists and policymakers seeking a richer understanding of conduct that moves women to the margins of law and society.
Angela Campbell is a professor at McGill University's Faculty of Law and a member of McGill's Institute of Comparative Law and Centre for Human Rights and Legal Pluralism. She teaches in the areas of family law, criminal law, successions, and health law. Her research in these areas aims to interrogate the interface between law and controversial or unorthodox practices associated with marriage, kinship, sexuality and reproduction. Professor Campbell received her B.A., LL.B. and B.C.L. degrees from McGill University, and her LL.M. degree from Harvard Law School, where she was a Frank Knox Fellow. Prior to becoming a law professor, she served as a law clerk to The Honourable Mr. Justice Frank Iacobucci at the Supreme Court of Canada. Sister Wives, Surrogates and Sex Workers: Outlaws by Choice is her first book.
Family Law - Marriage
Publication Date: December 16th, 2013
Series: Gender in Law


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Florida woman arrested on suspicion that she killed her young son over 30 years ago
A Florida woman was arrested on suspicion that she killed her young son who went missing in Nevada more than 30 years ago, authorities said Monday,Amy Elizabeth Fleming, 60, was picked up by U.S. Marshals and officers in Boca Raton, Florida, on Jan. 29 before she was turned over to the North Las Vegas police over the weekend.Read More
Saudi Arabia regrets EU inclusion in dirty-money blacklist – statement
How Ariana Grande is saying 'thank u, next' to the traditional music business model


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US ship arrives in DPRK
WFP announced an agreement with the Democratic People's Republic of Korea that paves the way for a rapid expansion of its food assistance to more than 5 million hungry people throughout the country.
AP feeding time 1st of July 07:30 GMT
Reuters (similar VNR) 1st of July 07:15 GMT
Food Prices, Logistics
WFP Marks World Food Day With Focus on Bold Partnerships that Spur Change (For the Media)
WFP Distributing Food in Worst Affected Areas of Haiti (For the Media)
El Nino's Hidden Impact on People living with HIV-AIDS (For the Media)
Urgently Needed Food Reaches Besieged Towns in Syria (For the Media)
El Nino Doubling Hunger Rates in southern Madagascar (For the Media)
WFP Emergency Operation to Assist around 80,000 South Sudanese Refugees into Uganda (For the Media)
WFP Delivers Life Saving Food to People Stranded at Syria-Jordan Border (For the Media)
Over 60 Million Affected Globally by El Niño, More at Risk from La Nina Impact (For the Media)


Following in a tradition that has been established over the past few years, the Western Wayne School District invites the community to attend to its annual homecoming game this Friday, September 28 and accept the challenge of helping others. Along with coming to cheer on the students, they ask that spectators bring donations of canned goods and other nonperishable food items for a drive that will benefit those in need in Wayne County.
The Western Wayne Homecoming Game against Lakeland will begin at 7 p.m. with the presentation of the Homecoming Court about a half hour before kick-off. The ten seniors on this year's court feel extremely honored and excited to be a part of this ensemble.
Members of the court include: Jake Kunz, Maya Black, Nicholas McGlone, Samantha Atcavage, Dylan Walck, Gianna DeBastiani, Bryce Urian, Allison Mattern, RJ Clemens, and Paige Barillo.
The part of the homecoming game that senior Jake Kunz looks forward to the most is the support from his fellow Western Wayne students and community members in the stands while he plays.
"I look forward to seeing all of the kids in the stands having a good time," Jake said.
Jake plays both football and basketball for Western Wayne along with being a member of the National Honor Society and Future Business Leaders of America.
He very much enjoys his time on the football team because he gets to spend time with his friends while working hard on the game.
Jake aspires to study business marketing in college.
Fellow homecoming court member Maya Black also likes playing sports for Western Wayne.
Maya participates in track and field.
"I really enjoy winning with my teammates and bonding through track," Maya, who does the long, triple, and high jump events, explained.
Maya additionally is part of the Technology Student Association, NHS, and FBLA.
Next year, Maya wishes to attend Temple University to study broadcast journalism.
Maya is also involved in fashion classes at Western Wayne and looks forward to dressing up for the spirit week leading up to the game along with getting to wear her gown on Friday night.
Like Jake and Maya, homecoming court member Nick McGlone participates in FBLA as well.
Nick feels most proud of placing in the top five of last year's FBLA regional conference. This allowed him to go to the FBLA state conference in Hersey where his event was advertising.
Nick, also a member of Students Against Destructive Decisions, looks forward to continuing his academic efforts at college next year and plans to soak in all of the excitement from the student section at the game.
"Everyone will be very hyped up," Nick explained.
Likewise senior Sam Atcavage looks forward to feeling all of the positive energy at the game.
"I look forward to making a memory with my friends who are on the court," Sam said.
Sam, who wishes to study nursing in college, also likes to bond with her friends in the various sports and clubs she participates in at Western Wayne.
She is part of the tennis team, indoor track club, and outdoor track, along with being a member of NHS and FBLA.
Sam has many great memories from her time on the tennis team.
"We have a small team and put in a lot of hard work while playing a fun sport," Sam explained.
Fellow homecoming court member Dylan Walck agrees that some of his favorite high school memories come from the time he has spent in Western Wayne sports programs.
Dylan, who plays multiple positions on the football team, has a passion for the sport.
Dylan greatly looks forward to the energy of the crowd at the homecoming game.
Dylan's friend senior Gianna DeBastiani likewise looks forward to experiencing the excitement of homecoming with all of her classmates on the court.
"I look forward to hanging out with my friends," she said.
Gianna plays on the volleyball and track teams at Western Wayne along with being a member of NHS, FBLA, and Pennsylvania Junior Academy of Science.
She has especially enjoyed her time on the volleyball team working with all of her teammates to have good seasons.
She plans to study business administration.
Homecoming court member Bryce Urian, like his classmates, has had a lot of fun in the many activities he participates in at Western Wayne.
Bryce performs with the Western Wayne Band along with being a member of NHS and the Video Game Club.
He wishes to study chemistry in college and really looks forward to adding to his high school experiences the honor of being on the homecoming court.
Fellow band member Allison Mattern also looks forward to her time Friday night at homecoming.
Allison is a member of NHS, FBLA, PJAS, the Scholastic Team, and the tennis team.
She especially loves her experiences with the tennis team.
Like his friend Allison, homecoming court member RJ Clemens likes to have a good time while participating in a variety of activities at school.
RJ is in NHS, PJAS, TSA, Political Science Club, Scholastic Team, Spanish Club, Envirothon, Band, Drama Club, musical theater, and the Reading Team.
RJ, who wishes to study science in college, is glad to participate in all of his academic clubs along performing a variety of lead roles over the years in Western Wayne musicals.
He looks forward to getting to dress up for all of spirit week, especially for the homecoming game.
Finally, homecoming court member Paige Barillo agrees with RJ that dressing up is one of her favorite parts of homecoming along with getting to spend time with friends and being recognized by her peers at the game.
Paige, who plans to study nursing in college, is involved in a variety of clubs and sports including NHS, PJAS, FBLA, Wayne Memorial Community Health Board, track and field, indoor track club, and the cross country team.
"I love the cross country team because it's such a small team and we are so close with each other," Paige said.
Paige and all of the members of the homecoming court look forward to bonding with each other and making memories during homecoming at Friday's game. The court along with the rest of the Western Wayne community invite the public to join them on Friday, Sept. 28 at 6:30 p.m. for the homecoming festivities followed by the game.
Bring some canned goods to donate and contribute to the positive atmosphere of Western Wayne Homecoming.


Project logistics business in Samudera Indonesia initially started by one division in 1970. In 1980, the division was established into an independent subsidiary for freight forwarding service especially for project cargo..
Services provided for the customers including planning, operational design, inland and seaway transportation, stevedoring, and customs formalities.
This subsidiary has been dealing with various infrastructure development and maintenance projects in Indonesia such as power plant projects, petrochemicals, oil and gas and other large-scale projects. This subsidiary has been certified ISO 9001 - 2008 to ensure service quality.


Sign up for Broadjam today to follow warren niles boulanger, and be notified when they upload new stuff or update their news!
Baptized Warren Niles Baker, ( the surname Boulanger - pronounced "Boo-Lawn-Jay" - was translated to English by French-Canadian ancestry ) Warren grew up in the south Louisiana city of Baton Rouge. Though attending music school on a percussion scholarship, he had a deep desire to write and so majored in music theory and composition. It was there that he took up piano as a second instrument. His instrumental and vocal music has been extremely well received wherever he has performed, be it for concert, banquet, coffeehouse or seminar.