text:
\section{Introduction}\label{sec:introduction}

The study of minimal surfaces in product spaces $M^2\times \mathbb{R}$ was initiated by Rosenberg and Meeks~\cite{Rosenberg2002,MR2005} and has been very active since then. Among that spaces, there are three homogeneous Riemannian manifolds: $\mathbb{R}^3$, where the classical theory of minimal surfaces has been developed, and $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$, where many authors have been actively working. Giving a complete list of references in the subject is far from being possible so we will only mention a few of them: Nelli and Rosenberg~\cite{NR2002} proved a Jenkins-Serrin-type theorem in $\mathbb{H}^2\times \mathbb{R}$, Hauswirth~\cite{Hauswirth2006} constructed minimal examples or Riemann type, Sá Earp and Tobiana~\cite{ST2004} investigated the screw motion invariant surfaces in $\mathbb{H}^2\times \mathbb{R}$, Daniel~\cite{Daniel2009} and Hauwirth, Sá Earp and Tobiana~\cite{HST2008} showed, independently, the existence of an associated family of minimal immersions for simply connected minimal surfaces in $\mathbb{S}^2\times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$, Urbano and the author~\cite{TU2013} tackled a general study of minimal surfaces in $\mathbb{S}^2\times \mathbb{S}^2$ with applications to $\mathbb{S}^2\times \mathbb{R}$, and very recently Manzano, Plehnert and the author~\cite{MPT2013} constructed orientable and non-orientable even Euler characteristic embedded minimal surfaces in the quotient $\mathbb{S}^2\times \mathbb{S}^1$ and Martín, Mazzeo and Rogríguez~\cite{MMR2014} constructed the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2\times \mathbb{R}$ with finite total curvature and positive genus.

In this paper we are going to show a geometric relation between maximal surfaces in anti-De Sitter space-time $\mathbf{H}^3_1$ and minimal immersions in $\mathbb{H}^2\times \mathbb{R}$. It is well-known that the Gauss map of a spacelike maximal immersion in $\mathbf{H}^3_1$ is always a minimal Lagrangian immersion in $\mathbb{H}^2\times \mathbb{H}^2$ (see~\cite{Torralbo2007} and also~\cite{CU2007} where an analogous case for Lagrangian minimal immersions in $\mathbb{S}^2\times \mathbb{S}^2$ is studied). We are going to get new minimal immersions in $\mathbb{H}^2\times\mathbb{H}^2$ by pairing different components of the Gauss map of two suitable maximal immersions in anti-De Sitter space-time (see Theorem~\ref{thm:Gauss-map-pair}). This construction is in the same spirit as in~\cite{TU2013}. From that, and under an appropriate choice of one element in the pair, we will establish a conformal correspondence between maximal immersions in anti-De Sitter space-time and minimal immersions in $\mathbb{H}^2\times\mathbb{R}$ (see Corollary~\ref{cor:Gauss-map-H2xR}). We will also show that this result admits a local converse, i.e.\ that roughly speaking every minimal surface in $\mathbb{H}^2\times \mathbb{R}$ is locally the Gauss map of a maximal immersion in anti-De Sitter space (see Theorem~\ref{thm:local-correspondence}).

Finally, we will illustrate this geometric correspondence by showing new examples of maximal surfaces in anti-De Sitter space in Proposition~\ref{prop:examples-AdS} and computing their Gauss map (in the sense of Corollary~\ref{cor:Gauss-map-H2xR}). This will provide us with two $1$-parameter families of minimal examples in $\mathbb{H}^2\times\mathbb{R}$. We want to point out that, although the constructed maximal immersions in $\mathbf{H}^3_1$ are non-complete (see Proposition~\ref{prop:examples-AdS}), their corresponding minimal immersions in $\mathbb{H}^2\times \mathbb{R}$ induce complete metrics in the surface. Moreover, they are invariant by screw motions and they were first described in~\cite{ST2004}.

The structure of the paper is the following: Section~\ref{sec:preliminaries} introduces both anti-De Sitter space-time $\mathbf{H}^3_1$ and the Riemannian products $\mathbb{H}^2\times\mathbb{H}^2$ and $\mathbb{H}^2\times \mathbb{R}$, where $\mathbb{H}^2$ stands for the hyperbolic plane. In Section~\ref{sec:maximal-AdS}, we will briefly present some basic facts about maximal surfaces in anti-De Sitter space as well as some examples. Section~\ref{sec:Gauss-map-of-a-pair} contains the main theorems that are illustrated in Section~\ref{sec:examples}. Finally, Section~\ref{sec:appendix} contains an analysis of the solutions to the sinh-Gordon equation that only depend on one variable.

\section{Preliminaries}\label{sec:preliminaries}

\subsection{The hyperbolic plane and the product manifold $\mathbb{H}^2 \times \mathbb{H}^2$}\label{subsec:hyperbolic-plane}

Let $\mathbb{H}^2$ be the hyperbolic plane and $\prodesc{\,}{\,}$ its metric. Although for all computations we will use the hyperboloid model of $\mathbb{H}^2$, i.e.\ $\mathbb{H}^2 = \{p \in \mathbb{R}^3:\, \prodesc{p}{p} = -1,\, p_1 > 0\}$, where $\prodesc{x}{y} = -x_1y_1 + x_2y_2 + x_3y_3$, the Poincaré disc model is also considered in Figures~\ref{fig:positive-energy} and~\ref{fig:negative-energy}.

We endow $\mathbb{H}^2 \times \mathbb{H}^2$ with the product metric, also denoted by $\prodesc{\,}{\,}$. So $\mathbb{H}^2 \times \mathbb{H}^2$ is an Einstein manifold with constant scalar curvature $-4$.

We will consider $\mathbb{H}^2 \times \mathbb{R}$ as the totally geodesic submanifold of $\mathbb{H}^2 \times \mathbb{H}^2$ given by the image of the map $i:\mathbb{H}^2 \times \mathbb{R} \rightarrow \mathbb{H}^2 \times \mathbb{H}^2$ defined by $i(p,t) = [p, (\sinh(t), 0, \cosh(t))]$.

\subsection{The anti-De Sitter $3$-space}\label{subsec:anti-De-Sitter}
The anti-De Sitter $3$-space, that it is usually denoted by $\mathbf{H}^3_1$, is a Lorentz manifold of dimension $3$ and constant curvature $-1$. It is defined as a quadric in a vector space. More precisely, let $\mathbb{R}^4_2$ be the euclidean $4$-space endow with the metric $\vprodesc{u}{v} = u_1v_1 + u_2v_2 - u_3v_3 - u_4v_4$. Then $\mathbf{H}^3_1 = \{p \in \mathbb{R}^4_2:\, \vprodesc{p}{p} = -1\}$. Moreover, the map $\pi: \mathbf{H}^3_1 \subset \mathbb{R}^4_2 \equiv \mathbb{C}^2_1 \rightarrow \mathbb{H}^2(-2)$, where $\mathbb{H}^2(c)$ stands for the hyperbolic plane of constant curvature $c < 0$, given by
\[
\pi(z, w) = \left(z\bar{w}, \frac{1}{2}(\abs{z}^2 + \abs{w}^2) \right),
\]
is a semi-Riemannian submersion with totally geodesic fibers generated by the unit temporal vector field $\xi_{(z,w)} = (iz, iw)$. The fiber of a point $(z_0,w_0) \in \mathbf{H}^3_1$ is the circle $(z_0e^{it}, w_0e^{it})$.

\subsection{The Gauss map}\label{subsec:definition-gauss-map}
Let $\phi: \Sigma \rightarrow \mathbb{R}^4_2$ be a spacelike immersion of an oriented surface $\Sigma$. Its Gauss map assigns to each point of the surface its oriented tangent plane. In this particular case, the image of the Gauss map is contained in $G^+_*(2,4)$, the Grassmann manifold of oriented spacelike planes of $\mathbb{R}^4_2$. It is well-known that $G^+_*(2,4)$ is diffeomorphic to $\mathbb{H}^2 \times \mathbb{H}^2$ (see, for example, \cite[Section 1]{Palmer1991}). To understand the construction in Section~\ref{sec:Gauss-map-of-a-pair} a suitable identification between $\mathbb{H}^2 \times \mathbb{H}^2$ and $G^+_*(2,4)$ is needed.

Let $\Lambda^2\mathbb{R}^4_2 = \mathrm{span}\{v \wedge w:\, v, w\in \mathbb{R}^4_2\} \equiv \mathbb{R}^6$ be the linear space generated by the $2$-vectors in $\mathbb{R}^4$ endowed with the index $2$ metric
\[
g(v \wedge w, v' \wedge, w') = \vprodesc{v}{w'}\vprodesc{w}{v'} - \vprodesc{v}{v'}\vprodesc{w}{w'}.
\]
The star operator $\star: \Lambda^2 \mathbb{R}^4_2\rightarrow \Lambda^2\mathbb{R}^4_2$ defined by $\alpha\wedge (\star \beta) = g(\alpha, \beta)\Omega$ for all $\alpha, \beta \in \Lambda^2 \mathbb{R}^4_2$, where $\Omega$ is the orientation form of $\mathbb{R}^4$, is an linear automorphism of $\Lambda^2 \mathbb{R}^4_2$ with eigenvalues $\pm 1$. Consider $\Lambda^2_\pm \mathbb{R}^4_2$ the eigenspaces of $\star$ associated to the eigenvalues $\pm 1$. Observe that we can decompose $\Lambda^2 \mathbb{R}^4_2 = \Lambda^2_+ \mathbb{R}^4_2 \oplus \Lambda^2_-\mathbb{R}^4_2$. Let $\{e_1, e_2, e_3, e_4\}$ be an oriented orthonormal frame of $\mathbb{R}^4_2$, i.e.\ $\abs{e_1}^2 = \abs{e_2}^2 = -\abs{e_3}^2 = -\abs{e_4}^2 = 1$ and $\vprodesc{e_i}{e_j} = 0$, $i \neq j$. The frame $\{E^\pm_j:\, j = 1, 2, 3\}$ given by:
\[
\begin{split}
E^\pm_1 &= \tfrac{1}{\sqrt{2}}(e_1 \wedge e_2 \pm e_4 \wedge e_3),\,
E^\pm_2 = \tfrac{1}{\sqrt{2}}(e_1 \wedge e_3 \pm e_4 \wedge e_2),\,
E^\pm_3 = \tfrac{1}{\sqrt{2}}(e_1 \wedge e_4 \pm e_2 \wedge e_3),
\end{split}
\]
is an orthonormal oriented reference in $\Lambda^2_\pm \mathbb{R}^4_2$, i.e.\ $g(E^\pm_i, E^\pm_j) = \epsilon_i\delta_{ij}$, where $\epsilon_1 = -1$ and $\epsilon_2 = \epsilon_3 = 1$. Hence each $\Lambda^2_\pm \mathbb{R}^4_2$ is isometric to the Lorentz-Minkowski $3$-space. We denote by $\mathbb{H}^2_\pm$ the hyperbolic plane in the $3$-space $\Lambda^2_\pm \mathbb{R}^4_2$.

Finally, if $\{v, w\}$ is an oriented orthonormal frame of a plane $P \in G^+_*(2,4)$, then the map $G^+_*(2,4) \rightarrow \mathbb{H}^2_+ \times \mathbb{H}^2_-$ given by
\[
P \mapsto \frac{1}{\sqrt{2}} [ v \wedge w + \star(v \wedge w), v\wedge w - \star(v \wedge w)]
\]
is a diffeomorphism.

\section{Maximal surfaces in anti-De Sitter space}
\label{sec:maximal-AdS}

Let $\phi: \Sigma \rightarrow \mathbf{H}^3_1$ be a spacelike maximal immersion, i.e.\ with zero mean curvature, of an oriented surface $\Sigma$ and $N$ a unit normal vector field to $\phi$. Given a conformal parameter $z = x+iy$ on $\Sigma$, it is well-known (see for instance~\cite{Palmer1990}) that the $2$-differential $\Theta_\phi(z) = \theta(z) \mathrm{d} z \otimes \mathrm{d} z= \vprodesc{\phi_z}{N_z}\mathrm{d} z\otimes \mathrm{d} z$ is holomorphic, where $N$ is the (timelike) unit normal vector field to $\phi$ such that $\{\phi_x, \phi_y, \phi, N\}$ is a positively oriented frame in $\mathbb{R}^4_2$ (we are using subscripts to indicate derivatives). The associated conformal factor $e^{2v}$ satisfies $v_{z\bar{z}} + e^{-2v}\abs{\Theta_\phi}^2 - \frac{1}{4}e^{2v} = 0$.  Moreover, the Frenet equations of the immersion are given by
\begin{equation}\label{eq:frenet-maxima-AdS}
\phi_{zz} = 2v_z\phi_z + \theta N, \quad \phi_{z\bar{z}} = \tfrac{1}{2}e^{2v}\phi, \quad N_z = 2e^{-2v}\theta \phi_{\bar{z}}.
\end{equation}

Conversely, we get the following result (see~\cite[Proposition 2.1]{Palmer1990} and also~\cite[Lemma~3.3]{Perdomo2009}):
\begin{quote}
\itshape
For any solution $v:D \rightarrow \mathbb{R}$, $D \subset \mathbb{C}$, to the equation $v_{z\bar{z}} - \frac{1}{2}\sinh(2v) = 0$ there exists a $1$-parameter family $\phi_t:\mathbb{C} \rightarrow \mathbf{H}^3_1$ of maximal immersions whose induced metric is $e^{2v}\abs{\mathrm{d} z}^2$ and whose Hopf differential is $\Theta_{\phi_t}(z) = \frac{i}{2}e^{it}\mathrm{d} z \otimes \mathrm{d} z$.
\end{quote}

In this section we are going to present some examples of spacelike maximal surfaces in $\mathbf{H}^3_1$ that will be useful in the sequel. The first simple example is the totally geodesic embedding of the hyperbolic plane $\mathbb{H}^2$ into $\mathbf{H}^3_1$ given by
$
\mathbf{B} = \{(z, w) \in \mathbb{R}^4_2 \equiv \mathbb{C}^2:\, \mathrm{Im}(w) = 0\}
$
, up to isometries of $\mathbf{H}^3_1$.

The second example, that will play an important role in the following section (see Corollary~\ref{cor:Gauss-map-H2xR}), is the so-called \emph{hyperbolic cylinder}
\[
\mathbf{C} = \{(z, w) \in \mathbb{R}^4_2 \equiv \mathbb{C}^2:\, \Re(z)^2 - \Re(w)^2 = \Im(z)^2 - \Im(w)^2 = -\tfrac{1}{2}\}.
\]
It is a complete spacelike maximal surface with vanishing Gauss curvature, constant principal curvatures $\lambda_1 = -\lambda_2 = 1$ and the norm of the second fundamental form is $\abs{\sigma}^2 = 2$. It was characterized by Ishihara~\cite{Ishihara88} as the only complete maximal surface in $\mathbf{H}^3_1$, up to rigid motions, with $\abs{\sigma}^2 = 2$. We can parametrize the hyperbolic cylinder $\mathbf{C}$ by
\begin{equation}\label{eq:immersion-hyperbolic-cylinder}
\psi_t(x, y) = \frac{1}{\sqrt{2}} (\sinh a_t(x, y), \sinh b_t(x, y), \cosh a_t(x, y), \cosh b_t(x, y) ),
\end{equation}
where
\[
\begin{split}
a_t(x, y) &= (x+y)\cos \tfrac{t}{2} + (x-y)\sin \tfrac{t}{2}, \\
b_t(x, y) &= (y-x)\cos \tfrac{t}{2} + (x+y)\sin \tfrac{t}{2}.
\end{split}
\]
Then $\psi_t(\mathbb{R}^2) = \mathbf{C}$, $z = x+iy$ is a conformal parameter with conformal factor $e^{2u(x, y)}$ where $u(x, y) = 0$, and the associated Hopf differential is $\Theta_{\psi_t} = \frac{i}{2}e^{it}\mathrm{d} z \otimes \mathrm{d} z$.

Next we are going to show examples invariant under a $1$-parameter group of isometries of $\mathbf{H}^3_1$. In that case, the equation for its conformal factor $v_{z\bar{z}} - \frac{1}{2}\sinh(2v) = 0$ becomes an ordinary differential equation
\begin{equation}\label{eq:sinh-Gordon-ordinary}
v''(x) -2\sinh(2v(x)) = 0, \text{ with energy } E =  \frac{1}{2}v'(x)^2 - \cosh( 2v(x) ).
\end{equation}
It is possible to integrate explicitly the Frenet system~\eqref{eq:frenet-maxima-AdS} for some values of $E$ obtaining the following result (see also Section~\ref{sec:appendix} where we get all the solutions to the previous equation in terms of Jacobi elliptic functions).

\begin{proposition}\label{prop:examples-AdS}
Let $v: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a solution to~\eqref{eq:sinh-Gordon-ordinary} with energy $E$. Then the map $\phi_E: \Sigma_v = (I \times \mathbb{R}, e^{2v}g_0) \rightarrow \mathbf{H}^3_1$ given by:
\[
\begin{split}
\phi_E&(x, y) = \frac{1}{\sqrt{2E}}\left(e^{v(x)} \cos(\sqrt{2E} y), -e^{v(x)} \sin(\sqrt{2E} y), \right.\\
& \left.-\sqrt{2E + e^{2v(x)}} \cos(\sqrt{2E} G(x)), -\sqrt{2E + e^{2v(x)}} \sin(\sqrt{2E} G(x))\right), \, E > 0, \\
\phi_E&(x,y) = \frac{1}{\sqrt{-2E}}\left( \sqrt{-2E - e^{2v(x)}} \sinh(\sqrt{-2E}G(x)), e^{v(x)}\sinh(\sqrt{-2E}y)\right. \\
	&\left. e^{v(x)}\cosh(\sqrt{-2E}y), \sqrt{-2E - e^{2v(x)}} \cosh(\sqrt{-2E}G(x))\right),\, E < 0,
\end{split}
\]
is an isometric maximal immersion with associated Hopf differential $\Theta(z) = \frac{1}{2}\mathrm{d} z\otimes \mathrm{d} z$, where $G(x) = \int_0^x \frac{\mathrm{d} t}{2E + e^{2v(t)}}$ and $g_0$ is the Euclidean metric in $\mathbb{R}^2$.

Moreover, all the surfaces $\Sigma_v$, except the one associated to the trivial solution $v(x) = 0$ which is the hyperbolic cylinder, are not complete.
\end{proposition}

\begin{remark}\label{rmk:examples-maximal-AdS}
The obtained examples in Proposition~\ref{prop:examples-AdS} are invariant by the following $1$-parameter group of isometries depending on the sign of $E$:

\begin{center}
\begin{tabular}{cc}
	$E > 0$ &  $E < 0$ \\
	$\begin{pmatrix}
			\cos \theta 	&	-\sin \theta 	&	0 	& 	0 \\
			\sin \theta 	&	\cos \theta 	&	0 	&	0 \\
			0 	&	0 	&	1 	&	0 \\
			0 	& 	0 	& 	0 	& 	1
		\end{pmatrix}$
	&
	$\begin{pmatrix}
			1 	&	 0 	& 	0 	& 	0 	\\
			0 	&	\cosh \theta 	&	\sinh \theta 	& 	0 \\
			0 	&	\sinh \theta 	&	\cosh \theta 	&	0 \\
			0 	& 	0 	& 	0 	& 	1
		\end{pmatrix}$
\end{tabular}
\end{center}

Besides, when $E < 0$ the immersion $\phi_E$ is only defined for $2E < -e^{2v(x)}$. A deep analysis of the solutions (see Section~\ref{sec:appendix}) shows that there is always a non-empty interval $I' \subseteq I$ where this happens, being $I$ the maximal interval of definition of $v$ (see also the proof of Proposition~\ref{prop:examples-AdS}).
\end{remark}

\begin{remark}
It is not surprising that the obtained examples are not complete. In~\cite{Perdomo2009}, the author pointed out the difficulties of getting complete maximal immersions in anti-De Sitter space-time and he also provided with complete examples looking for radial solutions of the sinh-Gordon equation. Palmer in~\cite[Theorem I]{Palmer1990} also provide complete minimal examples in $\mathbf{H}^3_1$ in terms of holomorphic quadratic differentials.
\end{remark}

\begin{proof}
Taking equation~\eqref{eq:sinh-Gordon-ordinary} into account, it is straightforward to check that $\phi_E$ is a maximal isometric immersion. We will now show that the metric $e^{2v}g_0$ is non-complete except for the trivial solution $v = 0$., by finding a divergent curve $\gamma$ in $\Sigma$ with finite length. Notice that we can consider the solutions to~\eqref{eq:sinh-Gordon-ordinary} given in Lemma~\ref{lm:solutions-sinh-Gordon-ordinary} for $a_0 = 0$, which are always defined in an interval $I = ]0, \ell[$. Observe that if $v$ is a solution then $-v$ is also a solution (see Section~\ref{sec:appendix}) so we have to deal with both cases.

If $E > -1$, thanks to the symmetries of the solutions (see Remark~\ref{rmk:properties-solutions-sinh-Gordon}.(1)), $v$ and $-v$ are symmetric so, considering the one given in Lemma~\ref{lm:solutions-sinh-Gordon-ordinary} we have that $e^{v(x)} \leq 1$ in $]0,\ell/2[$. Hence the curve $\gamma:]0, \tfrac{\ell}{2}[ \rightarrow \Sigma$, given by $\gamma(t) = \tfrac{\ell}{2} - t$, diverges in $\Sigma$ but has finite length.

If $E = -1$ we have three different solutions: (1) $v(x) = 0$, which produces the hyperbolic cylinder which is complete; (2) $v(x) = \log \tanh(x)$, in this case $\Sigma = (\mathbb{R}^+\times \mathbb{R}, \tanh^2(x)g_0)$ and so the curve $\gamma(t) = (a-t,0)$, $t\in ]0,a[$, $a\in \mathbb{R}$ arbitrary, diverges in $\Sigma$ but has finite length; and (3) $v(x) = \log \cotanh(x)$. In this case the immersion is only defined when $\cotanh^2(x) < 2$ (see Remark~\ref{rmk:examples-maximal-AdS}), i.e. $\Sigma = (]0, \arccotanh(\sqrt{2})[ \times \mathbb{R}, \cotanh^2(x)g_0)$. Hence the curve $\gamma(t) = (t,0)$, $t\in]\tfrac{1}{2}, \arccotanh(\sqrt{2})[$ diverges in $\Sigma$ but has finite length.

Finally, if $E < -1$ we have two different types of solutions, namely $v_1(x) = -\log(\tfrac{1}{\lambda}\sn_\mu(\lambda x))$ and $v_2(x) = -v_1(x)$ (see Lemma~\ref{lm:solutions-sinh-Gordon-ordinary}). In the first case the immersion is only defined when $2E+e^{2v_1(x)} < 0$, i.e.\ for $x \in J = ]c, \ell - c[$ where $c = \tfrac{1}{\lambda}\arcsn_\mu(\tfrac{\lambda}{\sqrt{-2E}})$ (see Lemma~\ref{lm:solutions-sinh-Gordon-ordinary} for the definition of $\lambda$ and $\mu$). But $e^{v_1(x)}\leq \sqrt{-2E}$ for $x \in ]c, \ell -c[$ and so $\Sigma$ is also incomplete in this case.

In the second case, $2E +e^{2v_2(x)} < 0$ so $\Sigma = (]0, \ell[ \times \mathbb{R}, e^{2v_2(x)}g_0)$, but $e^{v_2(x)}\leq \tfrac{1}{\lambda}$ so the curve $\gamma(t) = (\ell/2 - t, 0)$, $t \in ]0, \ell/2[$, diverges in $\Sigma$ and has finite length.
\end{proof}

\section{The Gauss map of a pair of maximal surfaces in $\mathbf{H}^3_1$}
\label{sec:Gauss-map-of-a-pair}

Let $\phi: \Sigma \rightarrow \mathbf{H}^3_1 \subset \mathbb{R}^4_2$ be a spacelike immersion of an oriented surface $\Sigma$. The \emph{Gauss map} of $\phi: \Sigma \rightarrow \mathbb{R}^4_2$ is the map $\nu_\phi = (\nu^+_\phi, \nu^-_\phi): \Sigma \rightarrow \mathbb{H}^2_+ \times \mathbb{H}^2_-$ defined by
\[
\nu^\pm_\phi(p) = \frac{1}{\sqrt{2}}[e_1 \wedge e_2 \pm N(p) \wedge \phi(p)],
\]
where $\{e_1, e_2\}$ is an oriented orthonormal basis in $T_p \Sigma$ and $N$ is the unit (timelike) normal vector field to the immersion $\phi$ such that $\{e_1, e_2, \phi(p), N_p\}$ is oriented in $\mathbb{R}^4_2$ (see Section~\ref{subsec:definition-gauss-map}).

If $\phi$ is maximal then its Gauss map is a Lagrangian minimal immersion in $\mathbb{H}^2\times\mathbb{H}^2$ (see~\cite{Torralbo2007}). For instance:
\begin{itemize}
	\item The Gauss map of the totally geodesic embedding of the hyperbolic plane in $\mathbf{H}^3_1$ given in Section~\ref{sec:maximal-AdS} is the diagonal map $\nu: \mathbb{H}^2\to \mathbb{H}^2\times \mathbb{H}^2$, $\nu(p) = (p,p)$.
	\item The Gauss map of the hyperbolic cylinder is the product of two geodesics of $\mathbb{H}^2$.
\end{itemize}

\begin{theorem}\label{thm:Gauss-map-pair}
Let $\Sigma$ be a Riemann surface and $\phi$, $\psi: \Sigma \rightarrow \mathbf{H}^3_1$ two conformal spacelike maximal immersions with the same Hopf differentials $\Theta_\phi = \Theta_\psi$. Then
\[
	\nu_{\{\phi, \psi\}}: (\nu^+_\phi, \nu^-_\psi): \Sigma \rightarrow \mathbb{H}^2 \times \mathbb{H}^2
\]
is a conformal minimal immersion. Moreover, the induced metric by $\nu_{\{\phi, \psi\}}$ is
\[
	g = \frac{1}{2}\Bigl[ (2 + \abs{\sigma_\phi}^2) g_\phi + (2 + \abs{\sigma_\psi}^2)g_\psi\Bigr],
\]
where $g_\phi$ and $g_\psi$ are the induced metrics on $\Sigma$ by $\phi$ and $\psi$, respectively. Here $|\sigma_\phi|$ and $|\sigma_\psi|$ are the lengths of the second fundamental forms of $\phi$ and $\psi$ in $\mathbf{H}^3_1$, computed with respect to $g_\phi$ and $g_\psi$, respectively.
\end{theorem}

\begin{remark}~
\begin{enumerate}
	\item If $\phi = \psi$, then $\nu_{\{\phi, \psi\}} = \nu_\phi$ is the Gauss map of $\phi$.
	\item Given a maximal immersion $\phi: \Sigma \rightarrow \mathbf{H}^3_1$, its \emph{polar} immersion (possibly branched) is $N: \Sigma \rightarrow \mathbf{H}^3_1$, where $N$ is a unit normal vector field to $\phi$. $N$ is also a maximal conformal immersion with the same Hopf differential as $\phi$. Nevertheless, $\nu_{\{\phi, N\}}$ is congruent to $\nu_\phi$, the Gauss map of $\phi$.
	\item Given $A \in \mathcal{O}_2(4)$, then it is easy to check that $\nu_{\{A\phi, \psi\}}$ is congruent to $\nu_{\{\phi, \psi\}}$.
\end{enumerate}

\end{remark}

\begin{proof}
	The immersion $\phi: \Sigma \rightarrow  \mathbb{R}^4_2$ has parallel mean curvature vector because it is contained in $\mathbf{H}^3_1$ as a maximal surface. From~\cite[Theorem 3.2]{Palmer1991} we deduce that each $\nu^\pm_\phi$ is a harmonic map. Analogously $\nu_\psi^\pm:\Sigma \rightarrow \mathbb{H}^2$ are also harmonic maps. Hence $\nu_{\{\phi, \psi\}} = (\nu_\phi^+, \nu_\psi^-):\Sigma \rightarrow \mathbb{H}^2\times \mathbb{H}^2$ is a harmonic map.

	It remains to check that $\nu$ is conformal (and so minimal). Let $z = x+iy$ a conformal parameter over $\Sigma$ and $N_\phi$, $N_\psi$ the temporal unit normal vector field to $\phi$ and $\psi$ respectively such that $\{\phi_x, \phi_y, \phi, N_\phi\}$ and $\{\psi_x, \psi_y, \psi, N_\psi\}$ are oriented references in $\mathbb{R}^4_2$. Since $\phi$ and $\psi$ are conformal immersions the induced metrics by $\phi$ and $\psi$ in $\Sigma$ are given by $g_\phi = e^{2u}\abs{\mathrm{d} z}^2$ and $g_\psi = e^{2w}\abs{\mathrm{d} z}^2$ for certain functions $u$ and $w$. Moreover, $\Theta_\phi = \Theta_\psi = \theta \mathrm{d} z \otimes \mathrm{d} z$ for some function $\theta(z)$ by hypothesis.

	We can express the component of the Gauss map $\nu_{\{\phi, \psi\}}$ as
	\[
		\begin{split}
			v_\phi^+(z) = \tfrac{1}{\sqrt{2}}\bigl( -2ie^{-2u} \phi_z \wedge \phi_{\bar{z}}  - \phi \wedge N_\phi \bigr), \\
			v_\psi^-(z) = \tfrac{1}{\sqrt{2}}\bigl( -2ie^{-2w} \psi_z \wedge \psi_{\bar{z}}  + \psi \wedge N_\psi \bigr), \\
		\end{split}
	\]
	Taking the Frenet equations~\eqref{eq:frenet-maxima-AdS} of $\phi$ and $\psi$ and Section~\ref{subsec:definition-gauss-map} into account, we easily get that
	\[
	\begin{split}
		(\nu_\phi^+)_z &= \tfrac{1}{2}e^u(-i + 2\theta e^{-2u})E^+_2(z) + \tfrac{i}{2}e^u(i + 2\theta e^{-2u})E^+_3(z), \\
		%
		(\nu_\psi^-)_z &= \tfrac{1}{2}e^w(-i - 2\theta e^{-2w})E^-_2(z) + \tfrac{i}{2}e^w(-i + 2\theta e^{-2w})E^-_3(z). \\
		%
	\end{split}
	\]
	Then we deduce from the previous equations that:
	\begin{align*}
		\vprodesc{(\nu_\phi^+)_z}{(\nu_\phi^+)_z} = -2i\theta,
		& & \vprodesc{(\nu_\psi^-)_z}{(\nu_\psi^-)_z} = 2i\theta, \\
		%
		|\nu_\phi^+|^2 = \tfrac{1}{2}\bigl(e^{2u} + 4e^{-2u}\abs{\theta}^2 \bigr),
		& & |\nu_\psi^-|^2 = \tfrac{1}{2}\bigl(e^{2w} + 4e^{-2w}\abs{\theta}^2 \bigr).
	\end{align*}
	Finally, $\vprodesc{\nu_z}{\nu_z} = 0$ and so $\nu = \nu_{\{\phi,\psi\}}$ is a conformal map. Moreover, from $8\abs{\theta}^2 = e^{4u}\abs{\sigma_\phi}^2 = e^{4w}\abs{\sigma_\psi}^2$ and
	\[
		\abs{\nu_z}^2 = \tfrac{1}{2}\bigl(e^{2u} + e^{2w} + 4\abs{\theta}^2(e^{-2u} + e^{-2w}) \bigr),
	\]
	we get the expression of the induced metric on $\Sigma$ by $\nu$.
\end{proof}

Now, let $\phi: \Sigma \rightarrow \mathbf{H}^3_1$ be a conformal maximal immersion. There is no loss of generality in assuming that locally the Hopf differential $\Theta(z) = \tfrac{i}{2}e^{it}\mathrm{d} z \otimes \mathrm{d} z$ (observe that either $\Theta = 0$ and so the immersion is totally geodesic or the zeroes of $\Theta$ are isolated and we can locally normalize $\Theta$ away from the zeroes). Then $\phi$ and $\psi_t$, the immersion of the hyperbolic cylinder given in Section~\ref{sec:maximal-AdS}, are two conformal maximal immersions with the same Hopf differentials. Then, thanks to the previous theorem, $\hat{\nu}_\phi = \nu_{\{\phi, \psi_t\}}: \Sigma \rightarrow \mathbb{H}^2\times\mathbb{H}^2$ is a minimal immersion that we call the \emph{modified Gauss map} of $\phi$. Now, the Gauss map of the hyperbolic cylinder $\psi_t$ is the product of two geodesics in $\mathbb{H}^2\times\mathbb{H}^2$ so its second component can be viewed as a map from $\Sigma$ to $\mathbb{R}$. Hence the \emph{modified Gauss map of a maximal immersion $\phi:\Sigma \rightarrow \mathbf{H}^3_1$ is a conformal minimal immersion $\hat{\nu}_\phi: \Sigma \rightarrow \mathbb{H}^2\times \mathbb{R}$}. We get the following result:

\begin{corollary}\label{cor:Gauss-map-H2xR}
	Let $u:D \subseteq \mathbb{C} \rightarrow \mathbb{R}$ be a solution of $u_{z\bar{z}} - \tfrac{1}{2}\sinh(2u) = 0$. Then the $1$-parameter family of minimal immersions $\Phi_t: (D, 4\cosh^2 u \abs{\mathrm{d} z}^2) \rightarrow \mathbb{H}^2\times \mathbb{R}$ with Hopf differential $\Theta = e^{it}\mathrm{d} z\otimes \mathrm{d} z$ (see~\cite[Corollary~10]{HST2008}) associated to $u$ is given by:
	\[
		\Phi_t(z) = \bigl( \nu_{\phi_t}^+(z), 2\,\mathrm{Im}(ze^{it/2})\bigr),
	\]
	where $\phi_t: (D, e^{2u}\abs{\mathrm{d} z}^2) \rightarrow \mathbf{H}^3_1$ is the $1$-parameter family of immersions associated to $u$ with Hopf differential $\Theta_{\phi_t} = \tfrac{i}{2}e^{it}\mathrm{d} z \otimes \mathrm{d} z$.
\end{corollary}

\begin{proof}
	Let $\psi_t = (\mathbb{C}, \abs{\mathrm{d} z}^2) \rightarrow \mathbf{H}^3_1$ the immersion of the hyperbolic cylinder given in equation~\eqref{eq:immersion-hyperbolic-cylinder}. Then $\phi_t, \psi_t:D \subseteq \mathbb{C} \rightarrow \mathbf{H}^3_1$ are conformal maximal immersions with the same Hopf differentials. Hence, applying the previous theorem we get that $\nu_t = \nu_{\{\phi_t, \psi_t\}}$ is a conformal minimal immersion in $\mathbb{H}^2\times \mathbb{H}^2$ with induced metric $4\cosh^2 u\abs{\mathrm{d} z}^2$.

	Furthermore, a straightforward computation shows that
	\[
	\nu_\psi^-(z) = \cosh[2\, \mathrm{Im}(ze^{it/2})] E_1^- + \sinh[2\, \mathrm{Im}(z e^{it/2}) ]E_3^-,
	\]
	where $\{E_1^-, E_2^-, E_3^-\}$ is the orthonormal reference in $\Lambda^2_-\mathbb{R}^4_2$ associated with the canonical base of $\mathbb{R}^4_2$ (see Section~\ref{subsec:definition-gauss-map}). Hence, as we have mentioned above, $\nu_\psi^-(D)$ is contained in a geodesic of $\mathbb{H}^2$. Considering $\mathbb{R}$ embedded in $\mathbb{H}^2$ as such geodesic we get the result. Finally, it is easy to check that $\Theta_{\nu_t} = -2i \Theta_{\phi_t} = \Theta$.
\end{proof}

The next result is a local converse of Theorem~\ref{thm:Gauss-map-pair} in the special case of surfaces immersed in $\mathbb{H}^2\times \mathbb{R}$.

\begin{theorem}\label{thm:local-correspondence}
	Let $\phi: \Sigma \rightarrow \mathbb{H}^2\times \mathbb{R}$ an isometric minimal immersion of a simply connected Riemann surface $\Sigma$ satisfying $\nu^2 < 1$, where $\nu =\prodesc{N}{\partial_t}$. Then, there exists a conformal maximal immersion $\psi:\Sigma \rightarrow \mathbf{H}^3_1$ such that $\phi = \hat{\nu}_{\psi}$, up to an ambient isometry.
\end{theorem}

\begin{proof}
Let $w$ a conformal parameter over $\Sigma$. Then $\Upsilon(w) = \prodesc{\phi_w}{\partial_t}\mathrm{d} w$ is a holomorphic  1-form without zeroes (note that $\abs{\Upsilon}^2 = \tfrac{1}{4}(1-\nu^2) > 0$ by assumption). Then we can always find another conformal parameter $z$ such that $\Upsilon = \mathrm{d} z$. The conformal factor induced by $\phi$ in this new parameter is $4\cosh^2 u$, where $u = \tanh(\nu)$ satisfies $u_{z\bar{z}} - \tfrac{1}{2}\sinh(2u) = 0$. Moreover, the fundamental data of the immersion $\phi$ can be expressed in terms of $u$ (see~\cite[Theorem 2.3]{FM2010}).

Let $\psi: \Sigma \rightarrow \mathbf{H}^3_1$ be the maximal conformal immersion associated to $u$ with Hopf differential $\Theta_\psi(z) = \frac{i}{2}\mathrm{d} z\otimes \mathrm{d} z$ (see Section~\ref{sec:maximal-AdS}). Then, $\hat{\nu}_\psi: \Sigma \rightarrow \mathbb{H}^2\times \mathbb{R}$ is a minimal immersion with the same fundamental data as $\phi$ and so both immersions differ in an ambient isometry.
\end{proof}

\begin{remark}
It is possible to get a similar result for minimal immersion of $\mathbb{H}^2\times \mathbb{H}^2$ without complex points as in~\cite[Theorem 3]{TU2013}, that is, every minimal immersion in $\mathbb{H}^2\times\mathbb{H}^2$ without complex points is locally congruent to the Gauss map of the pair of two maximal immersion in the anti-De Sitter space-time.
\end{remark}

\section{Examples}\label{sec:examples}

In this section we are going to use Corollary~\ref{cor:Gauss-map-H2xR} to compute the minimal immersions associated to the maximal immersions in $\mathbf{H}^3_1$ given by Proposition~\ref{prop:examples-AdS}. As we shall see, the obtained examples are invariant by $1$-parameter groups of isometries of $\mathbb{H}^2\times\mathbb{R}$, namely, \emph{elliptic} and \emph{hyperbolic} screw motions (see figures~\ref{fig:positive-energy} and~\ref{fig:negative-energy}). Moreover,  although the considered maximal immersions in $\mathbf{H}^3_1$ are not complete (see Remark~\ref{rmk:examples-maximal-AdS}), their Gauss maps, in the sense of Corollary~\ref{cor:Gauss-map-H2xR}, are complete immersions.

{\itshape
	Let $v: I \subseteq \mathbb{R} \rightarrow \mathbb{R}$ be a solution of $v''(x) - 2\sinh(2v) = 0$ with energy $E$ (cf.\ equation~\eqref{eq:sinh-Gordon-ordinary}). Then, the map $\Phi_E: \Sigma = (I \times \mathbb{R}, 4\cosh^2(v)g_0) \rightarrow \mathbb{H}^2\times\mathbb{R}$ given by:
	\[
	\begin{split}
		\Phi_E&(x,y) = \tfrac{1}{\sqrt{2E}}\left(v'(x), \tfrac{\sqrt{2E}e^{-v(x)}\cos \sqrt{2E}\bigl(y-G(x) \bigr) - v'(x)e^{v(x)}\sin \sqrt{2E}\bigl(y-G(x)\bigr)}{\sqrt{2E + e^{2v(x)}}}, \right. \\
		&\left. \tfrac{\sqrt{2E}e^{-v(x)}\sin \sqrt{2E}\bigl(y-G(x) \bigr) + v'(x)e^{v(x)}\cos \sqrt{2E}\bigl(y-G(x)\bigr)}{\sqrt{2E + e^{2v(x)}}}, 2\sqrt{E}(y-x) \right),\quad E > 0, \\
		%
		\Phi_E&(x,y) = \tfrac{1}{\sqrt{-2E}}\left( \tfrac{\sqrt{-2E}e^{-v(x)} \cosh\sqrt{-2E}\bigl(y-G(x)\bigr) - v'(x)e^{v(x)}\sinh \sqrt{-2E}\bigl( y-G(x)\bigr)}{\sqrt{-2E-e^{2v(x)}}}, v'(x), \right. \\
		&\left. \tfrac{-\sqrt{-2E}e^{-v(x)} \sinh\sqrt{-2E}\bigl(y-G(x)\bigr) + v'(x)e^{v(x)}\cosh \sqrt{-2E}\bigl( y-G(x)\bigr)}{\sqrt{-2E-e^{2v(x)}}}, 2\sqrt{-E}(y-x)\right), \quad E < 0,
	\end{split}
	\]
	is an isometric minimal immersion with associated Hopf differential $\Theta = -i \mathrm{d} z\otimes \mathrm{d} z$, where $G(x) = \int_0^x \frac{\mathrm{d} t}{2E + e^{2v(t)}}$  and $g_0$ stands for the Euclidean metric in $\mathbb{R}^2$.
}

\begin{figure}[htbp]
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth]{gfront-4}	&  \includegraphics[width=0.33\textwidth]{gfront-1} & \includegraphics[width=0.33\textwidth]{gfront-01} \\
\includegraphics[width=0.33\textwidth]{gtop-4}	&  \includegraphics[width=0.33\textwidth]{gtop-1} & \includegraphics[width=0.33\textwidth]{gtop-01}
\end{tabular}
\caption{From left to right, typical solutions for positive energy $E = 4, 1, 0.1$ in $\mathbb{H}^2\times \mathbb{R}$ being $\mathbb{H}^2$ the disc model. Below each surface the top view has been drawn. The boundary of $\mathbb{H}^2$ is drawn to help the visualization.} \label{fig:positive-energy}
\end{figure}

\begin{figure}[htbp]
\begin{tabular}{ccc}
\reflectbox{\includegraphics[width=0.33\textwidth]{gfront-n05}}	&  \includegraphics[width=0.33\textwidth]{gfront-n1} & \includegraphics[width=0.33\textwidth]{gfront-n6} \\
\includegraphics[width=0.33\textwidth]{gtop-n05}	&  \includegraphics[width=0.33\textwidth]{gtop-n1} & \includegraphics[width=0.33\textwidth]{gtop-n6}
\end{tabular}
\caption{From left to right, typical solutions for negative energy $E = -0.5, -1, -6$ in $\mathbb{H}^2\times \mathbb{R}$ being $\mathbb{H}^2$ the disc model. Below each surface the top view has been drawn.} \label{fig:negative-energy}
\end{figure}

This can be checked directly from the definition or, taking into account Corollary~\ref{cor:Gauss-map-H2xR}, computing the first component of the Gauss map of the maximal immersion $\phi_E$ given in Proposition~\ref{prop:examples-AdS}.

These examples are invariant by the $1$-parameter group of isometries $A_\theta\times \tau_\theta$ of $\mathbb{H}^2\times \mathbb{R}$, where $A_\theta$ is an isometry of $\mathbb{H}^2$ given by:

\begin{center}
\begin{tabular}{ccc}
	&	$E > 0$ &  $E < 0$ \\
	$A_\theta$ & $\begin{pmatrix}
		1 	& 	0 	& 	0 	\\
		0 	&	\cos \theta 	&	-\sin \theta 	\\
		0 	&	\sin \theta 	&	\cos \theta
		\end{pmatrix}$
	&
	$\begin{pmatrix}
			\cosh\theta 	&	 0 	& 	\sinh\theta	\\
			0 	&	1 	&	0	\\
			\sinh \theta 	&	0 	& 	\cosh \theta
		\end{pmatrix}$
\end{tabular}
\end{center}
and $\tau_\theta:\mathbb{R} \to \mathbb{R}$ is given by $\tau_\theta(t) = t + \tfrac{\theta}{\sqrt{\abs{E}}}$.

The complete classification of constant mean curvature surfaces (in particular the minimal ones) invariant by a $1$-parameter group of $\mathbb{H}^2\times \mathbb{R}$ can be found in~\cite{Onnis2008} and the references therein.

\section{Appendix}\label{sec:appendix}

In this section we will exhibit explicit solutions for the equation $\Delta v - 2\sinh(2v) = 0$. We will restring ourselves to the simplest case, that is, when the function only depends on one variable, i.e.\ $v = v(x)$. In that case it is easy to find a first integral of the equation, namely the energy $E = (v')^2/2 - \cosh(2v)$ is constant for every solution $v$ (cf.~\eqref{eq:sinh-Gordon-ordinary}). Moreover, if $v$ is a solution then $u(x) = -v(x)$ and $w(x) = v(-x)$ are also solutions with the same energy of $v$. Hence, we only need to consider initial conditions $v(0) = v_0 \geq 0$ and $v'(0) = \sqrt{2(E + \cosh(2v_0))} \geq 0$ (note that $E + \cosh (2v(x)) \geq 0$ by the definition of $E$). Thus we are interested in solving the following initial value problem:
\begin{equation}\label{eq:edo-problem}
\begin{split}
v''(x) -2 \sinh(2v(x)) &= 0, \\
v(0) = v_0 \geq 0, \quad  v'(0) &= \sqrt{2(E + \cosh(2v_0))}.
\end{split}
\end{equation}

It is possible to obtain all the solutions to that problem in terms of Elliptic Jacobi functions (see for instance~\cite{BF1971} for further details). Let
\[
F(\varphi, \mu) = \int_0^\varphi \tfrac{\mathrm{d} \theta}{\sqrt{1-\mu\sin^2 \theta}}, \quad 0 \leq \mu \leq 1,
\]
be the \emph{elliptic integral of the first kind with modulus $\mu$}. Then denoting the inverse of $\varphi \mapsto F(\varphi, \mu)$ by $\varphi = \am_\mu(x)$, the elementary Jacobi elliptic functions are given by:
\begin{align*}
\sn_\mu(x) &= \sin \am_\mu(x), &  \dn_\mu(x) &= \sqrt{1 - \mu\sin^2 \am_\mu(x)} \\
\cn_\mu(x) &= \cos \am_\mu(x), & \tn_\mu(x) &= \sn_\mu(x)/\cn_\mu(x)
\end{align*}
The basic properties of these functions are:
\begin{gather*}
	\sn_\mu(x)^2 + \cn_\mu(x)^2 = 1, \quad \mu \sn_\mu(x)^2 + \dn_\mu(x)^2 = 1 \\
	\sn_\mu\bigl(x + 2K(\mu)\bigr) = -\sn_\mu(x), \quad \cn_\mu\bigl(x + 2K(\mu) \bigr) = -\cn_\mu(x), \\
	\dn_\mu(x+2K(\mu)) = \dn_\mu(x), \quad \tn_\mu\bigl( x + 2K(\mu) \bigr) = \tn_\mu(x),
\end{gather*}
where $K(\mu) = F(\tfrac{\pi}{2},\mu)$ is the complete elliptic integral of the first kind. Moreover, the derivaties of the Jacobi elliptic functions are:
\begin{align*}
	\sn_\mu'(x) &= \cn_\mu(x)\dn_\mu(x), & \cn_\mu'(x) &= -\sn_\mu(x) \dn_\mu(x), \\
	\am_m'(x) &= \dn_\mu(x), & \dn_\mu'(x) &= -\mu \sn_\mu(x) \cn_\mu(x).
\end{align*}

\begin{lemma}\label{lm:solutions-sinh-Gordon-ordinary}
The solution $v:I \rightarrow \mathbb{R}$ of the initial value problem~\eqref{eq:edo-problem} and its maximal definition interval $I$ are given, in terms of the energy $E$, by:
\begin{align*}
	[E > 1]\quad v(x) &= \log\bigl(\lambda \tn_\mu(\lambda^{-1}x + a_0))\bigr), & \mu &= 1-\lambda^4,\\
	I &= ]-a_0, \lambda K(\mu)-a_0[, & a_0 &= \arctn_\mu(\lambda^{-1}e^{v_0}),\\
	[\abs{E}\leq 1]\quad  v(x) &= \log \bigl( \tn_\mu(x+a_0)\dn_\mu(x+a_0) \bigr),& \mu &= \tfrac{1-E}{2},\\
	I &= ]-a_0, K(\mu)-a_0[, & a_0 &= \tfrac{1}{2}\arccn_\mu(\tanh(v_0)),\\
	[E < -1]\quad v(x) &= -\log\bigl( \lambda^{-1} \sn_\mu(\lambda x + a_0) \bigr), & \mu &= \lambda^{-4},\\
	I &= ]-a_0, 2\lambda^{-1} K(\mu) - a_0[, & a_0 &= \arcsn_\mu(\lambda e^{-v_0})
\end{align*}
where $\lambda^2 = |E - \sqrt{E^2 - 1}|$ for $\abs{E} > 1$.
\end{lemma}

\begin{proof}
It is a direct computation taking into account the aforementioned properties of the Jacobi elliptic functions.
\end{proof}

\begin{remark}\label{rmk:properties-solutions-sinh-Gordon}

In the special cases $E = 1$ and $E = -1$ we get solutions in terms of elementary functions, namely, $v_1(x) = \log \tan(x)$, $v_{-1}(x) = \log \cotanh(x)$ as well as the constant solution $v(x) = 0$ (also with $E = -1$).

On the one hand, the solutions of the sinh-Gordon equation with energy $E > -1$ are symmetric with respect to the middle point of the maximal interval of definition. On the other hand, the solutions $v$ with energy $E < -1$ never vanish and are symmetric with respect to the vertical line passing through the middle point of the maximal interval of definition.
\end{remark}


A lawsuit recently filed in Madison County Circuit Court claims an Alton nursing home resident died as a result of the poor quality of care she allegedly received at the facility. In the case, the estate of an elderly woman claims Eldercare of Alton allowed the woman to develop a severe infection that ultimately led to her death. The senior citizen was reportedly a patient at the facility for less than one month in 2011. The woman allegedly died at Saint Anthony's Health Center in Alton more than three weeks after she was taken by ambulance from the skilled nursing facility.
In the complaint, the woman's family claims she suffered septic shock, bedsores, dehydration, and other injuries while in the care of the nursing home. The lawsuit also claims an excessive amount of a blood thinning medication was administered to the woman while she resided at the long-term care facility. In addition, the woman's family accuses her physician of failing to provide the senior citizen with appropriate health care and refusing to speak with her family regarding her condition.
An investigation report issued by the Illinois Department of Public Health states the Alton nursing home failed to sufficiently assess and monitor the elderly woman's care. The report also claims the skilled nursing facility failed to adequately prevent mistreatment of individuals in its care as required by state law. According to the report, neglect on the part of the long-term care facility resulted in a number of serious health conditions for the woman including septic shock, heart attack, low blood pressure, low blood sugar, a serious urinary tract infection, and brain dysfunction. Additionally, the report accuses the nursing home of providing the elderly woman with an incorrect dose of more than one prescription medication.
All Illinois nursing homes must be licensed by the state. In addition, skilled nursing facilities that receive federal Medicaid and Medicare insurance program payments are considered certified facilities. A certified nursing home is subject to both state and federal laws and regulations. Despite that the Illinois Nursing Home Care Act does not require long-term care facilities operating in the state to become certified, most nursing homes in Illinois are in fact certified.
Federal law requires certified nursing homes to ensure that all patients maintain the same level of health they enjoyed when they entered the facility. If a resident's condition deteriorates, the skilled nursing facility must work to restore the resident to his or her original condition. A common sign of neglect in Illinois nursing homes is the presence of bedsores on a patient's body. Bedsores, or pressure ulcers, are normally caused by remaining in the same position for too long. If provided with proper care, no nursing home patient should suffer from bedsores.
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text:
Don't evict people from land, says President
President Edgar Lungu says he will not allow the developer who has acquired huge tracts of land covering three chiefdoms in Lufwanyama district on the Copperbelt province to displace local people.
The President said he was shocked to learn that the land, which could be bigger than Lusaka, has been given to one developer who was now threatening to evict people in Lufwanyama.
The Head of State said this when he met Lamba chiefs from Copperbelt at State House today.
President Lungu has described the development as a big scandal which government has to deal with to protect the people from eviction.
He has since assured the people of Lufwanyama district that government will support them and ensure that they are not evicted.
He has also clarified that Government had no intentions of abolishing chiefdoms through the proposed land policy which was at draft stage.
The President said chiefs should not be misled by politicians and some civil society organisations who alleged that the draft land policy was meant to abolish chiefdoms. He said government was consulting with stakeholders and would follow what will come out of the discussions.
And speaking on behalf of other traditional leaders, senior chief Chiwala of Masaiti district said the House of Chiefs rejected the land policy in its draft form when the Zambia Land Alliance presented it because their submissions were not considered. The traditional leader said the Zambia Land Alliance was scheduled to meet the house of chiefs on January 20, 2016 to get their resolutions after consultations with their subjects.
Meanwhile, Copperbelt Minister Mwenya Musenge, who was also in the meeting, told the Head of State that the upgrading of the feeder roads by the Zambia National Service (ZNS) is progressing well following the President's directive.
– ZNBC
Veteran coach backs Katongo, Chansa
Police need 11,000 houses, says Kanganja

redpajama_set_name: RedPajamaCommonCrawl


This field includes studies that examine the common assumption of a harmony of interests between creators (e.g. authors, performers) and investors (e.g. publishers, producers), a simplifying hypothesis that facilitates analytical solutions, which however finds weak empirical support. Papers included under this category also examine collecting societies and relate to the area of contract theory.
3. Harmony of interest assumption between authors and publishers (creators and producers/investors) is a fundamental issue defined within the Copyright Evidence wiki.
Kretschmer (2012) Kretschmer, M. (2012). Does Copyright Law Matter? An Empirical Analysis of Creators' Earnings. An Empirical Analysis of Creators' Earnings (May 21, 2012).
Mandel (2013) Mandel, Gregory N., The Public Psychology of Intellectual Property (March 27, 2013). Florida Law Review, Vol. 66, 2013 (Forthcoming); Temple University Legal Studies Research Paper No. 2013-23.
Mustonen (2005) Mustonen, Mikko. When does a firm support substitute open source programming?. Journal of Economics & Management Strategy 14.1 (2005): 121-139.
Piolatto and Schuett (2012) Piolatto, Amedeo, and Florian Schuett. Music piracy: A case of "the rich get richer and the poor get poorer". Information Economics and Policy 24.1 (2012): 30-39.
Towse (1999) Towse, R. (1999). Copyright and economic incentives: an application to performers' rights in the music industry. Kyklos, 52(3), 369-390.
Towse (2013) Towse, Ruth. The economic effects of digitization on the administration of musical copyrights. Review of Economic Research on Copyright Issues 10.2 (2013): 55-67.
Travis (2010) Travis, H., 2010. Estimating the economic impact of mass digitization projects on copyright holders: Evidence from the Google Book search litigation. Journal of the Copyright Society of the USA, 57, pp.41-62.


text:
Nietzsche: The Superman and Will to Power
Rejecting Positivist Optimism for Decandentism
The Artistic and Social Movement of Futurism
The Conquest of African Territories by Europe
The Decolonization of Asia and Influence on Colonial Powers
The 20th century, which started January 1, 1901, and ended December 31, 2000, was dominated by significant events. Events such as the Spanish flu, the two World Wars, nuclear power, imperialism, nationalism, nuclear power and weapons, space exploration, and decolonization.
These events, which also included technological advances, the Cold War, and post-Cold War conflicts, reshaped the globe's structure, politically and socially. One of the big changes that occurred was the massive transformation of the relationship humanity had with the natural world.
Nevertheless, at the beginning of the century, these changes were not apparent; they were not even anywhere in sight. There were still issues such as the tension between colonial politics and imperialists in the great European forces.
Both systems had contradictions that threatened the progress of humanity and the progressiveness of different countries in Europe. In this article, we will examine the culture of the early 20th century, including the rejection of rationalism, among other things.
The Growing Dissatisfaction with the Positivist Culture
There was a deepening dissatisfaction with the positivist culture, which was prominent in the early twentieth century. This revolt was particularly common among the younger generations and it led to the conscious rejection of rationalism in any form.
A series of events and attitudes of people led to this discovery of the rejection of positivist optimism. These include the increasingly bitter tensions between the great European powers, which were created by imperialism and colonial politics.
Also, there were inherent contradictions in the capitalist system, which were often the roots of heated and violent social conflicts. Additionally, there was the success of the Marxist ideology that supports the maturation of solid class consciousness in the proletariat. In general, there was a change in the social and political climate, thus signifying that people's dissatisfaction with positivism was deepening.
The Failure of Positivism in the Early Twentieth Century
Positivism, developed by August Comte in the nineteenth century, has to do with scientific facts. The concept posits the notion that knowledge backed by scientific findings was the only reliable form of knowledge.
Positivism is a subject under epistemological philosophy, projecting an objective rather than a subjective view. The concept does not bother with how things ought to be; thus, every decision made must rely on scientific methods.
For the younger generation, positivist optimism failed and pessimism and faith in humanity's unstoppable progress took over. This pessimism is one that often led to the expectation of some calamity or catastrophe, of which traces already existed.
Additionally, philosophy and art had now converged their interests on the centrality of man being a subject. They rose against the cognitive domain of science and the accentuated transformation of advancing technology.
The Prevalence of Irrationalism
With the rejection of positivist optimism came the prevalence of irrationalism as the world synthetically passed from rationalism. Problems such as interiority, destiny, and man's function in a world where he is pushed by hidden solicitations are returned to the forefront.
Irrationalism is a phenomenon that concerned the whole culture of the early twentieth century, and it was concretized in Decadentism. Furthermore, irrationalism found precedents in the philosophical currents that arose in the nineteenth century, and in the last decades.
Meanwhile, in reality, the prevalence of irrationalism was closely related to the crisis those ideals were going through. These ideals had been the pride of the primacy and expansion of the bourgeoisie, which was now beginning to lose momentum. The ideals were already losing confidence in their own myths for the same reasons it rejected the positivist culture.
Meanwhile, in industrial society, the contrast between society and man, especially intellectuals, was becoming more and more serious. Writers felt marginalized and the poet sought his own function, looking in vain to create a new relationship with the public.
However, rather than offer solutions to these contradictions, writers took refuge in activism and irrationalism, thus avoiding commitment. Instead, they side with the less evolved bourgeoisie, searching for authoritarian solutions before finally reaffirming bloodbath purifying and war world hygiene.
Decadentism Takes Over Romantic Subjectivism
As irrationalism faltered, decadentism resumed and exasperated romantic subjectivism, discovering the presence of the unconscious in man. Finally, research appeared, offering a deeper reality that escaped reason transcended physical appearance and could only be approached through artistic intuition.
Thus, poetry became understood as the revelation of a noumenal reality that contrasted with the phenomenal. It became viewed as an immediate illumination of the unconscious such that it was free from any logical, stylistic, and metric constraint. The poet learned to express the deepest reality of things by relying on words-music, synesthesia, and symbols, thus becoming a seer.
Meanwhile, Decandentism, which was born from an age of strong spiritual crisis, created awareness of what precarious conditions humanity was in. Subsequently, man was discovered to be lonely, despite facing a hostile and incomprehensible society, and unable to communicate with other men. As a result, the existential anguish that, at the time, flowed into the existentialist philosophical current was born.
However, thanks to art, the decadent poet could create his art and feel the master of his fate. He feels that it is his job to reveal the unknown and unravel the mystery, thus developing various mythical attitudes.
Decadentism is characterized by the heterogeneity of attitudes, in which we find refuge in faith and the celebration of heroism. One of the most prominent poets of the time, D'Annunzio, collected greater consensus among other poets, expressing the aspirations of the Italian middle and petty bourgeoisie then.
D'Annunzio's ideals inspired the founders of the Florentine magazines, Papini and Prezzolini, which is not at all surprising. In their pages, the Florentine magazines, Il Leonardo, La voice, Lacerba, found expression of these ideals. As such, they ignited a significant debate about the purposes of the Italian intervention in World War I.
Thus, it is safe to conclude that stating the triumph of D'Annunzio's ideals primarily favors Italy's involvement in World War I. We could also say these ideals led to the victory of fascism which the poet himself propitiated.
Positivist optimism was rejected at a time when the world was in a profound spiritual crisis. Man became increasingly dissatisfied with positivism and instead preferred irrationalism, which eventually brought about decadentism. Meanwhile, the heterogeneity of attitudes in decadentism brought a refuge in faith and celebration of heroism in D'Annunzio, who created the superman myth.


"The term "blue revolution" refers to the intense growth in aquaculture as a widespread and highly productive activity that has developed all over the world in the last 30 years.
The growth of this activity has been fuelled by the development of more efficient technologies in breeding and fattening, based on the implementation of animal enhancement programmes, the use of new species, the improvement of food products, new cultivation techniques... "
Aquaculture was and is still seen as the activity that should grow to cater for this growing demand. It is thus necessary to turn this sector into an industry that directs its efforts towards sustainability, as is the case today with agriculture.
In the year 2014 aquaculture sector to the supply of fish for human consumption exceeded for the first time that of fish caught in the natural environment.
We work with national cooperatives and farmers who share our idea of sustainability. We consider that sustainability is, today more than ever, a very important factor for the people and activities that we associate so closely with the natural environment.
In the rural development areas where we work, we are carrying out a project to provide economic support for the establishment of drip irrigation to optimize the water resources used, also increasing yields per hectare.
We provide training courses to farmers to teach them best farming practices and also implement a quality standard movement in all our products. In this way, we are able to control first-hand the use of the seeds that we consider most appropriate and promote the responsible use of pesticides that cause little harm to the land and produce no consumer waste.
The digitalization of the primary sector, especially in rural areas, is another one of the projects on which our strategy focuses.
The digitization of this sector improves the value chain, generating "table-field, field-table" feedback through an "integrated" process that allows us to study consumers' needs and thus always drive "continuous improvement" at source, i.e. in the countryside or rivers.
We believe the most important thing is to raise awareness of the need for collaboration that, through a paradigm shift, allows us to realize that small producers can do great things, working together in the company's value chain. And that is one of the great benefits that digitization brings in this field.
The digitalization project we have launched in our rural areas of influence has focused for two years on overcoming the barriers that arise due to farmers' low level of training and the complication of telecommunications in areas far from cities.


The 3300 Spyder line is a special variant of the 'cult' model of the Tonino Lamborghini Swiss Watches collection: the Spyder. Designed in 2008 by the Centro Stile Tonino Lamborghini, the Spyder model was created as a homage to the shield that encloses the famous 'Miura raging bull' in the brand's logo. The shield is the inspiration behind the design of the case, the bezel and many other details for all the Spyder models. The 3300 line is enriched by a bezel with inserts in carbon fiber, material largely used in the luxury sports car sector. The 3305 model is a quartz chronograph with Ronda RL 5040.D movement. The case is in polished/satin stainless steel with black-PVD treatment; the bezel is in polished/satin stainless steel with carbon fiber insert and red anodized aluminium shield of the Tonino Lamborghini brand. The dial has a matt red ground with a central part surface worked with a black bee nest pattern, date at 4H and Tonino Lamborghini brand's shield at 12H; the hands are red/nickel coloured with SPL. The special buttons are in red aluminium at 2H and stainless steel with red insert at 4H. The letter 'L' of the family name is engraved on the crown. The black leather strap with PU carbon fiber pattern and red stitching is enriched by a deployante buckle with black-PVD treatment, engraved Tonino Lamborghini name and 3D shield.
Dimensions: 47,50 mm without crown x 54,00 mm with lugs. Case: Polished/satin stainless steel with black-PVD treatment. Bezel: polished/satin stainless steel, polished ss screw, carbon fiber insert, anodized red aluminum shield. Glass: S apphire coated mineral glass. Crown: Polished stainless steel, letter 'L' of the family name engraved. Pusher: Special pusher in red aluminium 2H. Stainless steel, red insert 4H. Dial: Ground in matt red, central part worked surface in black bee nest pattern. Second component black. Black bague, red numbers and white lines. 11 index nickel colour with SPL red. Date at 4H. Tonino Lamborghini shield at 12H. Hands: Red/nickel colour with SPL. Strap: Black leather strap with PU carbon fiber pattern, red stitching. PVD-Black deployante buckle with Tonino Lamborghini engraved and 3D shield. Movement: RL 5040.D quartz chronograph. Waterproof: Guarantee 100 meters/330 ft.


Roman has a more than 10 years experiences in ICT market with main focus on networks, ICT infrastructure technologies, information & cyber security and business intelligence. In few last years he has been focusing on Flowmon technologies that bring new dimensions in the field of Artificial Intelligence and Machine Learning based security for sophisticated threats real-time detection and respond. As a Principal Consultant, Roman brings new visions to enterprise clients when connecting IT trends with their business and legislative requirements.


text:
Manulea crassifolia är en flenörtsväxtart. Manulea crassifolia ingår i släktet Manulea och familjen flenörtsväxter.

Underarter

Arten delas in i följande underarter:

 M. c. crassifolia
 M. c. thodeana

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redpajama_set_name: RedPajamaWikipedia


Graduate Record Examinations. GRE is a standardised test that is an admission requirement for most graduate schools (without MBA school) in the United States . Created and administered by Educational Testing Service (ETS) .


Hearing the bark of a dog coming from the old house was not uncommon for neighbors. The abandoned home had, for some reason, become a place where people dumped dogs. Once the barking began, someone in the community typically worked to contact animal control to help the dog. This time was no exception; the loud barking continued day and night until someone did something. When animal control arrived, they found a St. Bernard that was around five years old that had been locked in the dark basement with no food or water and no stairs to get out.
Detroit Animal Control, along with the Detroit Land Bank Authority arrived quickly at the home, but soon realized it was going to be difficult to get the dog out. Detroit Dog Rescue (DDR) also pitched in to help, and the groups worked together to get the dog out of the basement and straight to the vet for medical care. He was very dehydrated and sick and was approximately 40 pounds underweight for his large breed. He was rescued just in time.
Stranger still, the dog had a collar on and was very sweet and well-behaved, despite being in such a dire condition. They began to slowly feed the dog so his system would not be overwhelmed, and they administered fluids to help with dehydration. He had been chewing on wood scraps and was very hungry. The dog, who is named Bentley, was posted to the DDR Facebook page and that is when something amazing happened: his family recognized him. Bentley had been stolen from his family's backyard and they had desperately trying to locate him for over a month. Bentley is now slowly healing and getting back to his old self, and now he's back with the family who loves him and missed him so much while he was missing.


text:
You are here: Home / Association, Organization, Not-For-Profit, Philanthropy / Steven M. Bernstein, Partner at Fisher Phillips, Installed as Chair of the Greater Tampa Chamber of Commerce
Steven M. Bernstein, Partner at Fisher Phillips, Installed as Chair of the Greater Tampa Chamber of Commerce
December 17, 2017 by Post
Steven M. Bernstein, partner in the Tampa office of Fisher Phillips, was installed as Chair of the Greater Tampa Chamber of Commerce. He takes on this role for a one-year term, effective December 14, 2017.
"I have been involved with the Chamber for a number of years, and admire the important work this group does in support of businesses throughout our region," said Bernstein. "I hope to fulfill the Chamber's vision to remain a vibrant business hub for the community and be a catalyst for positive change on behalf of our members. As we move forward with these ideals, we will be steadfast in our goal to always be inclusive and to be a strong advocate for businesses in the Greater Tampa Bay region."
The mission of the Greater Tampa Chamber of Commerce is to serve its members and enhance the community by building business success. The group has a long history under different names, with a goal of an engaged and inclusive community driven by business prosperity.
"Steve has a diverse and successful track record of leading several areas of our organization over the last decade," said president and CEO Bob Rohrlack. "Couple that with his commitment to the Tampa business community at large, and I can't think of a better Chairman to lead us in year two of our 10-year vision plan, Vision 2026."
Bernstein, who is past president of the Academy of Florida Management Attorneys, maintains a traditional labor practice in which he represents employers throughout the United States in both state and federal courts, as well as before the National Labor Relations Board (NLRB), the U.S. Department of Labor (DOL), the Equal Employment Opportunity Commission (EEOC), and other state and federal agencies. He was recently named Labor and Employment Lawyer of the Year for the Tampa region by Best Lawyers in America.
Previously, Bernstein worked for the International Brotherhood of Teamsters and with the Minority Staff of the U.S. Senate Labor Committee.
Bernstein says that he hopes to build strong relationships not only in Tampa but throughout the entire Bay area by working cooperatively with other business organizations.
"I'm a strong believer in collaboration, and I believe that uniting our efforts will give us greater strength and accomplishment," added Bernstein. "I look forward to working with the many dedicated people – both volunteers and staff – that make the Chamber such a vital part of the Tampa Bay business community."
About Fisher Phillips (www.fisherphillips.com)
Employers often must take a stand: in court, with employees and unions, or with competitors. Fisher Phillips has the experience and resolve to back up management. That's why some of the savviest employers come to the firm to handle their toughest labor and employment cases.
Fisher Phillips has 350 attorneys in 32 offices. In addition to Tampa, the offices are in Atlanta, Baltimore, Boston, Charlotte, Chicago, Cleveland, Columbia, Columbus, Dallas, Denver, Fort Lauderdale, Gulfport, Houston, Irvine, Kansas City, Las Vegas, Los Angeles, Louisville, Memphis, New Jersey, New Orleans, Orlando, Philadelphia, Phoenix, Portland, Sacramento, San Diego, San Francisco, Seattle, and Washington, D.C.
Filed Under: Association, Organization, Not-For-Profit, Philanthropy, Legal


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Home » Football News » Spanish La Liga
Toni Kroos – I always save energy for second half
luka 01 Jan 2020 49 views
A documentary about Toni Kroos has shown different sides of the player. The movie was premiered a few days ago, and we saw a different side of the German player.
Toni Kroos as an emotional person
As you may know, Toni Kroos is the most stable player in Germany and Real Madrid. German player never shows the emotion and expression on his face is always the same. Even if a player scores a goal, his face stays the same.
Toni's wife talked about his emotional side. Jessy noted that Toni is a compassionate person, but it is not visible on the pitch. Kroos wife mentioned that the player needs time to open up in every relationship. It does not matter whether you get the functional expression on him or not; it will take time for Toni to show the real emotional side. Jessy said that Toni is one of the most emotional persons on the earth. Media could show his emotionless face, but his true friends know the real personality.
Taking care of his boots
The German player is addicted to boots. Toni notes that from early childhood, he was the one taking care of boots. Kroos believes that cleaning the boots is more than just a ritual. Los Blancos player noted that it's essential to have clean boots while playing on a big stage.
Advice from grandparents
Real Madrid's player said that he is always saving energy for the second half. Toni recalled the moment when his grandpa told about the second half and energy saving. Since then, Toni is trying hard to save energy for the second half.
The bad scenario at Bayern Munich
Toni Kroos said that he was not happy in Munich. Toni's wife, Jessy, noted that the player was not a happy man at Bayern Munich. Bayern's authorities recall the moment when they gave the player the most significant wage possible and how Toni said that he would pass on this opportunity. It was a life-changing moment for Toni Kroos as he signed for Real Madrid, and the rest is history.
Quique Setien
Quique Setien – I did not like the performance from Barcelona
The new coach of Barcelona is very clear about the current team situation. After a very struggling win against Ibiza, Quique said that he is not happy with the performance. Quique Setien was not impressed with his squad's performance Before the cup match, Setien noted that Pique and Messi would get a rest. Journalists asked
Valverde and Casemiro in Copa Del Rey – Zidane wants the cup
It's interesting to see Real Madrid's main squad in Copa Del Rey. Copa of Spain was always a tournament for testing bench squad strength and length. Zinedine Zidane is very serious about Copa Del Rey French coach understands that each trophy counts. If we take a look at SuperCopa of Spain, Zidane was very serious
Barcelona once more displayed Master Class
Antoine Griezmann scored two goals without Messi. Ibiza loses at home. Quique Setien wins his second match with Barcelona Quique Setien wins again Ladies and gentlemen, football supporters, today we saw another success of new coach Quique setien. On Copa Del Rey, the Ibiza team hosted Barcelona and lost the match. Flawless perform shown
Quique Setien – We may sign a forward in a near future
Barcelona's new coach, Quique Setien, met with the press. It's believed that Barcelona may struggle against FC Ibiza without Lionel Messi. Are you signing a new striker in January? Spanish journalists asked Quique about a new striker. Spanish coach noted that signing a new footballer could a new challenge for him. At the same time,
Kylian Mbappe – I've to focus on PSG, we will see what happens in summer
It's not a surprise that Kylian MBappe is one of the biggest fans of Real Madrid. When he was a child, he had posters of Real Madrid in his room. Kylian's dream to play for Zidane and Ronaldo French talent is also a big fan of Cristiano Ronaldo. When he joined Monaco and journalists asked
Van De Beek doesn't confirm his Possible Transfer to Real Madrid
The Ajax star Donny van de Beek has been one of the most wanted midfielders in Europe since last season when his stunning performances helped the club reach the semi-finals of the Champion's League. Was the Transfer-story Rumour? A week ago, some rumors assured football fans that Real Madrid agreed £47m deal with Ajax
Ronaldinho – Lionel Messi has everything and does not need my skills
As you know, Ronaldinho and Lionel Messi were teammates. The Brazilian magician was playing for Barcelona when Lionel started his professional sporting career. Ronaldinho met with journalists on Wednesday evening and talked about relationship with Lionel Messi. The early stage of Lionel Messi Barcelona's former player said that Messi was a real magician on
Real Madrid V Sevilla
Real Madrid will face Sevilla without Gareth Bale and Sergio Ramos
Real Madrid will face Sevilla in La Liga on Saturday evening. The new champions of Supercopa are back in Spain with new injured players. Sergio Ramos has an injury and can't play against Sevilla As you may know, Sergio Ramos is the former player of Sevilla. He started a professional career from Sevilla as he


text:
MSNBC Apologizes After Labeling Segregationist Democratic Congressmen Republicans
by Virginia Carter
in World Media
California Senator Kamala Harris accused Biden of "coddling the reputations of segregationists".
"I have to tell Vice President Biden, as someone I respect, that he is wrong for using his relationships with Eastland and Talmadge as examples of how to bring our country together", said Booker, who is African American. "I never thought I'd say it". As he put it Tuesday: "Today, you look at the other side and you're the enemy". Not the opposition, the enemy.
Others, however, have spoken out against Biden, including some of his 2020 competitors. Sen. Cory Booker called on Biden to apologize, saying on CNN Wednesday that Biden's comments resonated with him on a personal level, recalling the "indignity" of "watch (ing) another man call my father "boy" or being called that himself. So if he didn't say 'Boy, ' I think it would have landed differently.
"Cory shared directly what he said publicly - including helping Vice President Biden understand why the word "boy" is painful to so many", said Sabrina Singh, a Booker campaign spokeswoman. "So maybe the best way to beat him is to go make your own relationships". This is Biden responding. "He's asking you to apologize". "Apologize for what? Cory should apologize". "There's not a racist bone in my body", he said.
Actually, the controversy has thrown the spotlight on a particularly sordid and reactionary episode in Biden's early career in the Senate.
First came the "touching" scandal, in which a former candidate for political office in Texas accused Biden of kissing the back of her neck without permission and making her feel uncomfortable.
Biden reportedly called Booker this week to discuss the disagreement but did not apologize.
Biden supporters have repeatedly pointed to his efforts on civil rights issues to cast him as a champion of equality. African-Americans make up 28 percent of the state's population but more than half of the Democratic primary votes.
Biden was talking about the breakdown between Republicans and Democrats, but he may as well have been talking about the Democratic Party itself, which is headed in a risky and discouraging direction when a history of cooperating is so quickly attacked as a commitment to complicity.
Biden had raised Eastland and Talmadge specifically because they were so far from his moral compass, but the anger flew at Biden from Booker and others for mentioning the segregationists, let alone working with them decades ago.
A second anonymous adviser to Biden's campaign also spoke to The Washington Post, brushing off the wave of criticism as a passing matter. "In 1997, Mr. Biden effusively praised him, including glowing words for Thurmond's life before public office".
Goldberg's comments come after Biden has become embroiled in a feud with Sen. That was inartful.' I didn't mean in it that way and I think we have to - sometimes people say things in a conversation and you get what I'm trying to say, but people turn it into something else.
He continued, "Not a joke". Quoted in Politico, Thompson was quoted as saying, "If he (Biden) was able to work with Eastland, he's a great person". He went on to warn, however, "when we have income inequality as large as we have in the United States today, it brews and ferments political discord and basic revolution".
When Biden was waxing on about the old Senate days, he lamented that things are different now.
"You impress not occupy to agree with of us to work with them", said Mr Clyburn, who's Dwelling of Representatives majority whip. No one's standard of living would change.
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Andy Murray and Feliciano Lopez conclude dream week with title — ATP Queen's
Both singles semi-finals were absorbing in their own way, while the doubles that followed was filled with tension and excitement. The three time Grand Slam singles champion said he felt "as well as I did there, not ideal in terms of my movement".
Andy Murray says winning at Queen's with a metal hip is 'mental'
I've been rejected a couple of times so far". 'I don't anticipate it would be much longer than that'. "I'm very happy to be in the final".
Hartson labels Arsenal's bid for Tierney as 'an embarrassment'
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Trump won't say if he'll ask Federal Bureau of Investigation to probe Khashoggi killing
The onus is on Saudi Arabia, which is liable for the killing irrespective of whether it was intended or not, she argues. As a journalist, she said Khashoggi was "extraordinary" and that he was "fighting for democracy".
2018 Cheltenham Festival: An Early Look At The Arkle
He will take some beating if aimed at this race; Mullins has trained the Arkle winner in two of the last three years. With the right schooling over fences, Charli Parcs could feature prominently if aimed at the Arkle.
The Continuity of the Multimedia Exhibition of Santiago Ribeiro in Times Square in New York in the International Press
As Ribeiro continues planning the project's next phase, he is testing innovative exhibition spaces such as Times Square. He conceived the International Surrealism Now movement in Coimbra with a major exhibit organized by Bissaya Barreto.
Are Mobile Games the Right Way to Fight Stress?
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General concerns of beginners in the casino
In general, there are three main concerns related to fraud in the casino. - I will not receive a prize Is this possible? Yes. Some of the online casinos can be fakes that having worked for just a week and collecting bets will simply disappear.
Stenson Named European Golfer of the Year
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I had planned on writing up this post so long ago that I almost decided to just not post it… I mean it's been over a month since I shared with you my pieces for the fall 10×10 challenge and nearly as long since I completed my participation.
Never heard of the challenge? All you do is: choose 10 items to dress with for 10 days and then create 10 different outfits over those 10 days — fun right? Tops, jackets, bottoms and shoes all count toward the 10 piece total. Accessories like: bags, hats, scarves and jewelry are fair game. Although since it's all constructed anyway, feel free to change up the rules however you see fit!
If you want a more detailed overview check out Lee's amazing guide or Caroline's quick start method.
Be content with my 'right now' closet. Period.
There is a difference between wanting more and being consumed by that — allowing that sense of urgency to cause you to move at a rate that also causes you to miss out on more meaningful experiences. I know that now. And while that wishlist of mine hasn't gone anywhere, I also don't feel consumed by it anymore — and I'm feeling pretty content with that. Even a month later! Every time I feel that sense of urgency rise up in me, I feel content now to quiet it with that quote and the sentiments I've been left with from this particular 10 x 10 challenge.
Full disclosure: I officially caved on my #fall10x10 on the LAST DAY and reached for two of the warmest sweaters in my closet (this Madewell sweater I found secondhand and this Everlane cashmere sweater) when getting dressed. Because 37 degrees. 37 DEGREES. (And to think that now we're in the 20's…) I mean, whaaat?! 10 days and only 12 pieces worn? I decided to call it win anyway.
Did any of you participate in the fall 10 x 10 challenge? What reasons did you join? Or didn't you? I hope to back next week reviewing two Everlane Cashmere sweaters that I've been wearing since Spring — pretty exciting stuff!
Laura, your looks are so, so good! I feel like you are really finding your style lately, and it's envy-worthy! Love that long cardigan.
PS – You're a babe.
I love your blog! So glad you posted this!
Love this! Looking forward to seeing some more outfit/capsule posts!


text:
from argparse import ArgumentParser
from shutil import rmtree
from subprocess import Popen
from sys import argv
from sys import exit
from tempfile import mkdtemp

DEFAULT_PORT = 8080
DEFAULT_HOSTNAME = 'localhost'

def run_server(srcdir, objdir, js_file, hostname=DEFAULT_HOSTNAME,
        port=DEFAULT_PORT):

    dist_dir = '%s/dist' % objdir
    head_dir = '%s/services/common/tests/unit' % srcdir

    head_paths = [
        'head_global.js',
        'head_helpers.js',
        'head_http.js',
    ]

    head_paths = ['"%s/%s"' % (head_dir, path) for path in head_paths]

    args = [
        '%s/bin/xpcshell' % dist_dir,
        '-g', '%s/bin' % dist_dir,
        '-a', '%s/bin' % dist_dir,
        '-r', '%s/bin/components/httpd.manifest' % dist_dir,
        '-m',
        '-n',
        '-s',
        '-f', '%s/testing/xpcshell/head.js' % srcdir,
        '-e', 'const _SERVER_ADDR = "%s";' % hostname,
        '-e', 'const _TESTING_MODULES_DIR = "%s/_tests/modules";' % objdir,
        '-e', 'const SERVER_PORT = "%s";' % port,
        '-e', 'const INCLUDE_FILES = [%s];' % ', '.join(head_paths),
        '-e', '_register_protocol_handlers();',
        '-e', 'for each (let name in INCLUDE_FILES) load(name);',
        '-e', '_fakeIdleService.activate();',
        '-f', '%s/services/common/tests/%s' % (srcdir, js_file)
    ]

    profile_dir = mkdtemp()
    print 'Created profile directory: %s' % profile_dir

    try:
        env = {'XPCSHELL_TEST_PROFILE_DIR': profile_dir}
        proc = Popen(args, env=env)

        return proc.wait()

    finally:
        print 'Removing profile directory %s' % profile_dir
        rmtree(profile_dir)

if __name__ == '__main__':
    parser = ArgumentParser(description="Run a standalone JS server.")
    parser.add_argument('srcdir',
        help="Root directory of Firefox source code.")
    parser.add_argument('objdir',
        help="Root directory object directory created during build.")
    parser.add_argument('js_file',
        help="JS file (in this directory) to execute.")
    parser.add_argument('--port', default=DEFAULT_PORT, type=int,
        help="Port to run server on.")
    parser.add_argument('--address', default=DEFAULT_HOSTNAME,
        help="Hostname to bind server to.")

    args = parser.parse_args()

    exit(run_server(args.srcdir, args.objdir, args.js_file, args.address,
        args.port))

redpajama_set_name: RedPajamaGithub