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John Carter (2012) – Review
January 16, 2013 Jason Rugaard	1 Comment 2012, andrew stanton, avatar, Battlefield Earth, battleship, Dune, finding nemo, Friday Night Lights, john carter, Lynn Collins, Mark Strong, planet of the apes, star wars, taylor kitsch
What went wrong at the Disney movie factory to produce a movie this expensive only to come out with a product so bland? On paper everything lines up, the director of Pixar's Finding Nemo steps behind the camera for his first foray in to live-action cinema, a story that influenced Star Wars and numerous artists, plus a hefty budget to afford the best special effects in the industry. So it's a bit stunning to gaze at the screen and realize that some of the most gifted craftsmen in Hollywood have created the Dune of our generation.
This overlong and self important work lacks humor or any sense of self depreciation. The writing is on the level of George Lucas and the acting is passable, especially when considering the actors have to mouth nearly unpronounceable names and places. There are too many scenes that just fall flat, stopping any sort of momentum sustained to that point.
Front, and center is Taylor Kitsch an actor that I very much admired as Tim Riggins on television's Friday Night Lights. That is a well written show full of originality, wit and gravitas, all which John Carter is lacking in every department. I do not want to harp on Kitsch too harshly, but his two attempts at action lead in this and Battleship have not shown the promise that was glimpsed in his previous work.
The ironic part of the problem here is that the central themes, and even character archetypes have been borrowed heavily in some of the biggest sci-fi blockbuster of the last century (Star Wars, Planet of the Apes, Avatar). So even though Princess of Mars is the original text it's the last to the screen making the proceedings feel redundant and uninvolved. Even the so called wow moments fall flat and the CGI ranges from laughable to adequate, sometimes within the same scene.
Is John Carter the equivalent of Battlefield Earth? No. Its much more handsome visually than anything in the grungy looking Hubbard adaption. Regardless of the financial loss this film generated, it is not the worst movie of all time, it's not even the worst movie of the year. That is about all the praise I can think to bestow on John Carter.
Director: Andrew Stanton
Stars: Taylor Kitsch, Lynn Collins, Mark Strong
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June 14, 2013 June 14, 2013 Jason Rugaard 0
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One thought on "John Carter (2012) – Review"
I understand the "hate' if you will, but I really enjoyed the film (to an extent). Visually, I thought it was one of the best looking films of the year, with the CGI aliens blending in flawlessly to the backdrops and scenery. The problem? The disgusting script that never knows where to go. Too much time is spent on explaining the setting and the people instead of just going on the adventure we so desired.
It doesn't help that Disney butchered the marketing on the film, flip-flopping titles and never knowing how to sell the movie. Taylor Kitsch was a worthwhile risk, I think, but like you said, he doesn't bring any of the charm or wit that made him good in Friday Night Lights.
While I won't defend the film to its death, I actually look forward to a sequel (that will never happen). Now that the world is created, and somewhat explained, I want a worthwhile adventure. Give Kitsch a witty sidekick, ease up his eyebrows and brooding stares, and I'll definitely go back to Mars for another trip.
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Tony Stark, a.k.a. Iron Man, is in a bind, but NASA wants to help.
In the early moments of the "Avengers: Endgame" trailer, we see Stark aboard a ship, "adrift in space with zero promise of rescue."
He tells us food and water ran out a few days ago and oxygen will run out soon.
NASA, however, was not willing to sit idly by.
In a tweet, NASA reached out to Marvel, writing, "Hey @Marvel, we heard about Tony Stark. As we know, the first thing you should do is listen in mission control for '@Avengers, we have a problem.' But if he can't communicate, then we recommend ground teams use all resources to scan the skies for your missing man."
The National Aeronautics and Space Administration's message did not fall on deaf ears.
Marvel Studies answered back, "Appreciate any help you can provide."
And Robert Downey Jr. followed suit, writing, "Always good to know @NASA has your back."
"Failure is not an option," NASA responded with a thumbs-up emoji.
Needless to say, the exchange had the internet going wild.


text:
\section{Introduction}

The study of open quantum systems is one of the fundamental problems
of modern physics~\cite{rhbook,bpbook}.
An open quantum system consists of a region of space where
quantum processes take place (and which can be studied by the
experimenter) in contact with a decohering and dissipative
environment that is typically beyond the control of the experimenter.
Various instances of concrete open quantum systems can be found
in different areas of physics such as, for example, quantum optics,
atomic and mesoscopic physics, biophysics or, at even shorter
distances, nuclear physics.
The interdisciplinary character of the theory of open quantum systems
calls for a variety of different approaches.
Here,~we are concerned in particular with a formalism that adopts
non-Hermitian Hamiltonian operators, a~theoretical approach that
is routinely called non-Hermitian quantum mechanics~\cite{moyeseyev}.
The description of open quantum systems in terms of non-Hermitian
Hamiltonians~\cite{rotter-bird} can be rigorously derived, in~the case
of a localised quantum subsystem coupled to a continuum of scattering
states, by means of the
Feshbach projection formalism~\cite{fesh,fesh2,rotter}.
{Such an approach has been successfully employed to illustrate
the complexities of exceptional points, which do occur
when resonances coalesce in a non-avoided
crossing~\cite{rotter-bird}.
When one uses the full non-Hermitian Hamiltonian,
left and right
eigenvectors~\cite{eleuch-rotter,eleuch-rotter2,eleuch-rotter3}
must be distinguished.
From this perspective, the occurrence of exceptional points
may create problems for defining the density matrix.
On the other side, one can always use a Hermitian basis
(which, for~example, but not necessarily,
arises from the Hermitian part of the full
non-Hermitian operators)
to represent non-Hermitian operators and the density matrix.
From such a vantage point, the coalescence of the eigenvalues
of the non-Hermitian Hamiltonian appears to be a foregoer of
such major problems.}
It is worth mentioning that non-Hermitian Hamiltonians also appear
in parity-time (PT) symmetric generalisations of
quantum mechanics~\cite{bender,mostafazadeh}.
Such {theories} have recently found concrete applications
in lossy optical waveguides~\cite{optics,optics2}
and photonic lattices~\cite{lattice,lattice2}.

However, we are interested here in open systems that can be
effectively described by non-Hermitian Hamiltonians that are
not necessarily PT-symmetric (and which, for such a reason, will
be called general in the rest of this paper).
For such Hamiltonians, it has been shown how to define
a proper statistical mechanics~\cite{ks} in order to study the
behaviour of non equilibrium averages (e.g., the
purity of quantum states~\cite{kostya-purity}) and to provide the
definition of correlation functions~\cite{sk-corr}.

In order to try to build
possible measures of quantum
information~{\cite{mahler,mahler2,qinfo}}
for systems with
general non-Hermitian Hamiltonians, one can start
by defining an entropy functional~\cite{vonneumann,ohya}.
To~this end, a~non-Hermitian generalisation
of the von Neumann entropy has been introduced in \cite{sk-S}.
Nevertheless,~entropies of the von Neumann form cannot be used
when quantum theory is formulated by means of the Wigner function~\cite{manfredi}.
Since the (partial) Wigner representation is particularly useful
in order to derive a mixed quantum-classical description of
non-Hermitian systems~\cite{nhps}, it becomes interesting to
study the properties of the
so-called linear entropy~\cite{manfredi,zurek,pattanayak}
and its generalisation to the case of open quantum systems
described by general non-Hermitian Hamiltonians.
{To this end, we~present in this paper,
for the first time to our
knowledge, a generalisation of the entropy for
systems with non-Hermitian Hamiltonians that must be adopted
when there is an embedding of the quantum subsystem in phase space.
We associate the term ``linear'' to such an
entropy as it arises from its first appearance
in the literature~\cite{manfredi,zurek,pattanayak}.}

This paper is organised as follows. In Section~\ref{sec:s-nh},
we summarise the results of the
density-matrix approach~\cite{ks,kostya-purity,sk-corr,sk-S}
to non-Hermitian dynamics that are useful for the study and
generalisation of the linear entropy~\cite{manfredi,zurek,pattanayak}.
In particular, we introduce the equations of motion
for the density matrices~\cite{ks} and the von Neumann-like entropies
studied in \cite{sk-S}.
In Section~\ref{sec:qlinS}, we study the linear entropy
and its non-Hermitian generalisation, along the lines
followed in \cite{sk-S}.
Analytical solutions are given in the
{basic} case of
a constant decay operator.
{It is worth noting that even basic models with
constant decay operators become interesting when one adds
the additional level of complexity provided by the classical-like
environment represented by means of the partially Wigner-transformed
Hermitian part of the Hamiltonian. In order to fix the ideas,
one can think of a light-emitting quantum dot coupled to
an energy-absorbing optical guide in a classical environment,
which introduces thermal fluctuations or some other type of noise.
It is not even difficult to imagine how models like these one can be
made more and more complex within our approach.}
In Section~\ref{sec:qcnh}, we briefly recall how to formulate
the dynamics of a non-Hermitian system that is embedded in
a classical bath of degrees of freedom.
In Section~\ref{sec:qcentropy}, we study the behaviour of
the linear entropy and its non-Hermitian generalisation
in a quantum-classical set-up. Once again,
analytical solutions are provided for the case
of a constant decay operator.
Finally, our conclusions are presented in Section~\ref{sec:concl}.

\section{Quantum Dynamics with Non-Hermitian Hamiltonians}
\label{sec:s-nh}

Let us consider a non-Hermitian Hamiltonian composed of two terms:
\begin{equation}
\hat{\cal H}=
\hat H -i \hat\Gamma \;.
\label{eq:tot-NH-H}
\end{equation}

Both operators on the right-hand side,
$\hat H$ and $\hat\Gamma$, are Hermitian;
$\hat\Gamma$ is often called the decay rate~operator.
The quantum states $|\Psi\rangle$ and $\langle\Psi|$
evolve according to the Schr\"odinger equations
\begin{eqnarray}
\partial_t|{\Psi}\rangle&=&-\frac{i}{\hbar}\hat{\cal H}|\Psi\rangle=
-\frac{i}{\hbar}\hat H|\Psi\rangle-\frac{1}{\hbar}\hat\Gamma|\Psi\rangle
\;, \\
\partial_t \langle {\Psi}|&=&\frac{i}{\hbar}\langle\Psi |\hat{\cal H}^\dagger
=\frac{i}{\hbar}\langle\Psi|\hat H-\frac{1}{\hbar}\langle\Psi|\hat\Gamma
.
\end{eqnarray}

On conceptual grounds, we can expect that the open quantum system
dynamics produces statistical mixtures. Indeed,
we have shown that the purity is not conserved \cite{ks,kostya-purity}.
Defining the non-normalised density matrix as
\begin{equation}
\hat{\Omega} =\sum_k{\cal P}_k|\Psi^k \rangle\langle\Psi^k |
\;,
\end{equation}
{where
$(|\Psi^k \rangle,\langle\Psi^k |)$  are the eigenstates of any good
\emph{Hermitian} operator that can cover the Hilbert space
of the system and
${\cal P}_k$ is their} probability of occurrence,
the equation of motion can be written as
\begin{equation}
\partial_t {\hat{\Omega}} =-\frac{i}{\hbar}\left[\hat H, \hat{\Omega}\right]_- -
\frac{1}{\hbar}
\left[\hat{\Gamma},\hat{\Omega} \right]_+
\;,
\label{eq:dotOmega}
\end{equation}
with $[~,~]_-$ and $[~,~]_+$ denoting the commutator
and anticommutator, respectively.~Equation (\ref{eq:dotOmega}) effectively describes the subsystem
(with Hamiltonian $\hat{H}$) coupled to the environment (represented
by $\hat{\Gamma}$). \linebreak
{It~is worth remarking again and explicitly that, in our
approach~\cite{ks,sk-corr,sk-S,nhps}, we use
Hermitian basis sets to represent the equations of motion.
This situation is commonly found when, for example,
the~non-Hermitian creation and destruction operators, $\hat{a}$
and $\hat{a^\dag}$, respectively, are represented in the
basis of the Hermitian number operator.
It should be evident that, because of this, we do not need to worry about
the left/right eigenvectors of the full
non-Hermitian Hamiltonian~\cite{leftright1,leftright2}.
}

Non-Hermitian dynamics do not conserve the probability.
This can be easily seen by
taking the trace of both sides of Equation~(\ref{eq:dotOmega}):
\begin{equation}
\partial_t{\rm Tr}\,\hat{\Omega}
=-\frac{2}{\hbar}{\rm Tr}\left(\hat{\Gamma} \,\hat{\Omega} \right)
\;.
\label{eq:dotTrOmega}
\end{equation}

However, we can define a normalised density matrix~\cite{ks} as
\begin{equation}
\hat{\rho} =\frac{\hat{\Omega} }{{\rm Tr}\,\hat{\Omega} } \;.
\label{eq:rho}
\end{equation}

The density matrix in Equation (\ref{eq:rho})
can be used in the calculation of statistical averages: \linebreak
$\langle\chi\rangle_t={\rm Tr} \left( \hat\chi\hat{\rho}(t) \right),$
where $\hat\chi$ is an arbitrary operator.
The normalised density matrix $\hat\rho$
obeys the equation~\cite{ks}:
\begin{equation}
\partial_t {\hat{\rho}} =
-\frac{i}{\hbar}\left[\hat H, \hat{\rho}\right]_-
-\frac{1}{\hbar}\left[\hat{\Gamma},\hat{\rho}\right]_+
+\frac{2}{\hbar}\hat{\rho} \, {\rm Tr} (\hat{\Gamma}\hat{\rho})\;.
\label{eq:dotrho}
\end{equation}

Similarly to Equation (\ref{eq:dotOmega}),
Equation (\ref{eq:dotrho}) effectively describes the evolution of
the subsystem coupled to the environment;
the role of the third term on the right-hand side is to
conserve the probability during the dynamics.
Equation~(\ref{eq:dotrho}) is nonlinear.
This property was also noted when considering
operator averages in \cite{emg-korsch}).
Within the Feshbach--Fano projection formalism,
the nonlinearity of the non-Hermitian approach
has been suggested in~\cite{zno02} as well.
While the density operator $\hat{\rho}$
is bounded and useful in the calculation of
of the statistical averages,
the gain or loss of probability of open systems
are properly described
by the non-normalised density operator $\hat{\Omega}$.
Hence, it turns out that both $\hat{\Omega}$ and $\hat{\rho}$
are useful in the non-Hermitian formalism~\cite{sk-corr,sk-S}.

The normalised density matrix $\hat\rho$ allows us to
define~\cite{sk-S}
the von Neumann entropy of a non-Hermitian system as
\begin{equation}
S_{\rm vN} \equiv -k_B \left\langle   \ln\hat{\rho} \right\rangle
= -k_B{\rm Tr}\left( \hat{\rho} \ln\hat{\rho} \right)
\; .
\label{eq:SVNnH}
\end{equation}

The rate of the von Neumann entropy production is~\cite{sk-S}:
\begin{eqnarray}
\partial_t S_{\rm vN}
=
\frac{2k_B}{\hbar}{\rm Tr}\left(\hat{\Gamma}\hat{\rho} \ln\hat{\rho} \right)
+
\frac{2}{\hbar}{\rm Tr}\left( \hat{\Gamma}\hat{\rho} \right)
S_{\rm vN}
. \label{eq:dotSVNnH}
\end{eqnarray}

However, the gain or loss of information in a non-Hermitian system
are more properly represented by introducing
another entropy, given by
the statistical average of the logarithm
of the \emph{non-normalised} density operator~\cite{sk-S}:
\begin{equation}
S_{\rm NH} \equiv -k_B \langle   \ln\hat{\Omega} \rangle
= -k_B{\rm Tr} ( \hat{\rho} \ln\hat{\Omega} )
= - k_B \frac{ {\rm Tr}( \hat{\Omega} \ln\hat{\Omega} ) }{
{\rm Tr}\, \hat{\Omega} } \; .
\label{eq:SVNnH2}
\end{equation}

The rate of change of $S_{\rm NH}$ is~\cite{sk-S}
\begin{equation}
\partial_t S_{\rm NH}
=\frac{2k_{\rm B}}{\hbar}{\rm Tr}\left(\hat{\Gamma}\hat{\rho}\ln\hat{\Omega}\right)
+\frac{2}{\hbar} {\rm Tr}\left(\hat\Gamma \rho\right) S_{\rm NH}
+2\frac{k_B}{\hbar}{\rm Tr}\left(\hat{\Gamma}\hat{\rho}\right)\;,
\end{equation}
while the difference between the two entropies reads
\begin{equation}
S_{\rm vN} - S_{\rm NH} = k_{\rm B}
\ln\left( {\rm Tr}\, \hat\Omega\right)\; .
\label{e:diffentr}
\end{equation}

The fact that the $S_{\rm NH}$ entropy captures the
expected physical behaviour of the flow of information
out of an open system can be seen by considering
the models where $\hat H$ is an arbitrary self-adjoint
operator while $\hat \Gamma$ is proportional to the
identity operator:
\begin{equation}
\hat\Gamma = \frac{1}{2} \hbar \gamma_0 \hat I \;,
\label{eq:Gammamodgau}
\end{equation}
where the parameter $\gamma_0$ is assumed to be real-valued.
For such models, after imposing the initial conditions
${\rm Tr}\,\hat{\Omega} (0) = 1$),
we obtain~\cite{sk-S}:
\begin{eqnarray}
{\rm Tr}\,\hat{\Omega} (t) &=& \exp{(-\gamma_0 t)}\;,\label{eq:trOmega}\\
S_{\rm vN} (t) &=& S_{\rm vN}^{(0)} = \text{const}\;,\\
S_{\rm NH} (t) &=& S_{\rm vN}^{(0)} + k_{\rm B} \gamma_0 t\;.
\end{eqnarray}

One can then see that, for positive values of $\gamma_0$,
the $S_{\rm NH}$ entropy diverges
at large times, as
{a good entropy functional}
of an open system is expected to do.
On the contrary, the von Neumann entropy $S_{\rm vN}$
is always constant.

\section{Non-Hermitian Dynamics and Quantum Linear Entropy}
\label{sec:qlinS}

The quantum linear entropy is
\begin{equation}
S_{\rm lin}=1-{\rm Tr}\left[\hat{\rho}^2(t)\right].
\label{eq:Slin}
\end{equation}

The entropy production is
\begin{equation}
\dot{S}_{\rm lin}
=-2{\rm Tr}\left[\hat{\rho}(t)\frac{\partial\hat{\rho}(t)}{\partial t}
\right]\;.
\label{eq:dotSlin-ini}
\end{equation}

Substituting Equation~(\ref{eq:dotrho}) in Equation~(\ref{eq:dotSlin-ini})
and using the following identities
\begin{eqnarray}
{\rm Tr}\left[\hat{\rho}\hat{H}\hat{\rho}-\hat{\rho}\hat{\rho}\hat{H}\right]
&=&
{\rm Tr}\left[\hat{\rho}^2\hat{H}-\hat{\rho}^2\hat{H}\right]=0\;,
\\
{\rm Tr}\left[\hat{\rho}\hat{\Gamma}\hat{\rho}+\hat{\rho}\hat{\rho}\hat{\Gamma}
\right]
&=&
2{\rm Tr}\left[\hat{\Gamma}\hat{\rho}^2\right] \;,
\end{eqnarray}
we obtain
\begin{equation}
\dot{S}_{\rm lin}
=\frac{4}{\hbar}{\rm Tr}\left[\hat{\Gamma}\hat{\rho}^2(t)\right]
-\frac{4}{\hbar}{\rm Tr}\left[\hat{\Gamma}\hat{\rho}(t)\right]
{\rm Tr}\left[\hat{\rho}^2(t)\right]\;.
\label{eq:dotSlin}
\end{equation}

Analogously with the entropy of Equation~(\ref{eq:SVNnH2}),
we can also introduce a linear entropy involving the
non-normalised density matrix as
\begin{equation}
S_{\rm lin}^{\rm NH}=1-{\rm Tr}\left[\hat{\rho}(t)\hat{\Omega}(t)\right]
\;.
\label{eq:SlinNH}
\end{equation}

The rate of production of $S_{\rm lin}^{\rm NH}$ is
\begin{eqnarray}
\dot{S}_{\rm lin}^{\rm NH}
&=&
-\frac{2}{{\rm Tr}\left[\hat{\Omega}(t)\right]}
{\rm Tr}\left[\hat{\Omega}(t)\partial_t\hat{\Omega}(t)\right]
-
\frac{2{\rm Tr}\left(\hat{\Omega}^2(t)\right)}
{\hbar\left[{\rm Tr}\left(\hat{\Omega}(t)\right)\right]^2}
{\rm Tr}\left(\hat{\Gamma}\hat{\Omega}(t)\right)
\;.\label{eq:dotSNHlin-ini}
\end{eqnarray}

Using Equation~(\ref{eq:dotOmega}), together with the identity
\begin{eqnarray}
{\rm Tr}\left[\hat{\Omega}[\hat{\Gamma},\hat{\Omega}]_+\right]
&=&
2{\rm Tr}\left[\hat{\Gamma}\hat{\Omega}^2\right]\;,
\end{eqnarray}
in Equation (\ref{eq:dotSNHlin-ini}), we obtain
\begin{eqnarray}
\dot{S}_{\rm lin}^{\rm NH}
&=&
\frac{4{\rm Tr}\left[\hat{\Gamma}\hat{\Omega}^2(t)\right]}
{\hbar{\rm Tr}\left[\hat{\Omega}(t)\right]}
-
\frac{2{\rm Tr}\left(\hat{\Omega}^2(t)\right)
{\rm Tr}\left(\hat{\Gamma}\hat{\Omega}(t)\right)
}
{\hbar\left[{\rm Tr}\left(\hat{\Omega}(t)\right)\right]^2}
\;.
\label{eq:dotSlinNH}
\end{eqnarray}

\subsection*{Linear Entropy Production and Constant Decay Operator}

Let us consider Equations (\ref{eq:dotSlin}) and~(\ref{eq:dotSlinNH})
in the case of a decay operator defined by Equation~(\ref{eq:Gammamodgau}).
In~such a case, the temporal dependence of ${\rm Tr}(\hat\Omega(t))$
is given, when choosing ${\rm Tr}\hat\Omega(0)=1$,
by Equation~(\ref{eq:trOmega}).
Using Equation~(\ref{eq:dotOmega}), we easily obtain
\begin{eqnarray}
\partial_t {\rm Tr} \hat \Omega^2(t)
&=& -2{\gamma_0}{\rm Tr}\hat\Omega^2(t),
\\
{\rm Tr} \hat \Omega^2(t)
&=&
{\rm Tr} \hat \Omega^2(0)\exp[-2{\gamma_0} t]
\;.
\end{eqnarray}

Hence, the calculation of
\begin{equation}
\partial_t {\rm Tr} \hat \rho^2(t) = 2 {\rm Tr}\left[\hat\rho(t)
\partial_t\hat\rho(t)\right] \;
\label{eq:dotTrrho2-ini}
\end{equation}
can proceed upon considering the identities
\begin{eqnarray}
-\frac{2}{\hbar}{\rm Tr}
\left\{\hat\rho(t)\left[\hat\Gamma,\hat\rho(t)\right]_+\right\}
&=&-2{\gamma_0}{\rm Tr}\left[\hat\rho^2(t)\right]
\;,\\
\frac{4}{\hbar}{\rm Tr}\left\{\hat\rho^2(t)
{\rm Tr}\left[\hat\Gamma\hat\rho(t)\right]\right\}
&=&
2{\gamma_0} {\rm Tr}\left[\hat\rho^2(t)\right] \;.
\end{eqnarray}

Therefore, Equation~(\ref{eq:dotTrrho2-ini}) is found to give

\begin{equation}
\partial_t{\rm Tr}\hat\rho^2(t)
=-2{\gamma_0} {\rm Tr}\left[\hat\rho^2(t)\right]
+2{\gamma_0} {\rm Tr}\left[\hat\rho^2(t)\right]=0\;.
\end{equation}

Given the above result, we can choose

\begin{equation}
{\rm Tr}\hat\rho^2(t)={\rm const.}={\rm Tr}\hat\rho^2(0) \;.
\end{equation}

Finally, Equation~(\ref{eq:dotSlin}) becomes
\begin{eqnarray}
\dot{S}_{\rm lin}
&=&
2{\gamma_0}{\rm Tr}[\hat\rho^2(0)]
-2{\gamma_0}{\rm Tr}[\hat\rho^2(t)]=0\;.
\label{eq:dotSlingamma-const}
\end{eqnarray}

Equation~(\ref{eq:dotSlingamma-const}) shows that $S_{\rm lin}$
is identically constant and is thus not suitable
to describe the information flow or the evolution
of the entanglement in systems with non-Hermitian Hamiltonians.

Let us now consider Equation~(\ref{eq:dotSlinNH}): it becomes
\begin{eqnarray}
\dot{S}_{\rm lin}^{\rm NH}
&=&
2{\gamma_0}\frac{{\rm Tr}\hat\Omega^2(t)}{{\rm Tr}\hat\Omega(t)}
-{\gamma_0}\frac{{\rm Tr}\hat\Omega^2(t)}{{\rm Tr}\hat\Omega(t)}
\nonumber\\
&=&
{\gamma_0}\left[{\rm Tr}\hat\Omega^2(0)\right]e^{-\gamma t}\;.
\end{eqnarray}

Integrating between $0$ and $t$, we obtain
\begin{equation}
S_{\rm lin}^{\rm NH}=\left[1-e^{-{\gamma_0}t}\right]
{\rm Tr} \hat\Omega^2(0)
\;.\label{eq:SlinNH-t}
\end{equation}

Equation~(\ref{eq:SlinNH-t}) describes the increase of the linear entropy
$S_{\rm lin}^{\rm NH}$ from the value of $0$ at $t=0$
to the plateau value of ${\rm Tr} \hat\Omega^2(0)$ at $t=\infty$.
Because of the choice of the initial condition ${\rm Tr} \hat\Omega(0)=1$,
the~quantity ${\rm Tr} \hat\Omega^2(0)$ is the purity of the
non-Hermitian system. Hence, Equation~(\ref{eq:SlinNH-t}) monitors the loss
of the initial purity of the system.

\section{Non-Hermitian Dynamics in a Classical Environment}
\label{sec:qcnh}

One particular class of open quantum systems is obtained when
a quantum subsystem is embedded in a classical environment.~In \cite{nhps}, an equation of motion for a quantum subsystem
embedded in~a~classical bath, described in terms of its phase space
coordinates, has been derived.
To~this end, we consider a~total~Hamiltonian

\begin{equation}
\hat{\cal H}(\hat{r},\hat{p},\hat{R},\hat{P})
=\hat{H}(\hat{r},\hat{p},\hat{R},\hat{P})
-i\hat{\Gamma}(\hat{r},\hat{p}) \;,
\label{eq:calH}
\end{equation}
where $(\hat{r},\hat{p})$ are $n$ light degrees of freedom
with mass $m$, and $(\hat{R},\hat{P})$ are $N$ heavy degrees of
freedom of mass $M$. The small expansion parameter $\mu=\sqrt{m/M}
<<1$ can be used to obtain the classical limit for the
$(\hat{R},\hat{P})$ degrees of freedom, after taking
a partial Wigner transform over the $2N$ heavy coordinates.
Using a multidimensional notation and denoting the phase space point
$(R,P)$ with $X$, the~partial Wigner transform
of the density matrix is defined as

\begin{equation}
\hat{\Omega}_{\rm W}(X,t)
=\frac{1}{(2\pi\hbar)^N}\int dZe^{P\cdot Z/\hbar}
\langle R-Z/2|\hat{\Omega}(t)|R+Z/2\rangle
\;,
\end{equation}
while the partial Wigner transform of an arbitrary
operator $\hat{\chi}$ is defined as

\begin{equation}
\hat{\chi}_{\rm W}(X)
=\int dZe^{P\cdot Z/\hbar}
\langle R-Z/2|\hat{\chi}|R+Z/2\rangle
\;.
\end{equation}

In \cite{nhps}, it was shown that, upon taking the
partial Wigner transform of Equation~(\ref{eq:dotOmega}),
with the $\hat{H}$ and $\hat{\Gamma}$ of Equation~(\ref{eq:calH}),
and performing a linear expansion in $\mu$, one obtains
the equation of motion
\begin{eqnarray}
\frac{\partial}{\partial t}\hat{\Omega}_{\rm W}(X,t)
&=&-\frac{i}{\hbar}
\left[\hat{H}_{\rm W},\hat{\Omega}_{\rm W}(X,t)\right]_-
+
\frac{1}{2}
{\cal B}_{ab} \left(\partial_a\hat{H}_{\rm W}\right)
\left(\partial_b\hat{\Omega}_{\rm W}(X,t)\right)
\nonumber\\
&-&
\frac{1}{2}
{\cal B}_{ab}\left(\partial_a\hat{\Omega}_{\rm W}(X,t)\right)
\left(\partial_b\hat{H}_{\rm W}\right)
-
\frac{1}{\hbar}\left[\hat{\Gamma},\hat{\Omega}_{\rm W}(X,t)\right]_+
\;, \label{eq:qcdotOm}
\end{eqnarray}
where ${\cal B}_{ab}=-{\cal B}_{ba}^{\rm T}$
is the symplectic matrix~\cite{mccauley}
and $\partial_a=(\partial/\partial X_a)$ is
the gradient operator in phase space.
The Einstein convention of summing over repeated indices
is used throughout this paper.
One~can note that ${\cal B}_{ab} (\partial_a\hat{H}_{\rm W})
(\partial_b\hat{\Omega}_{\rm W})$
is the Poisson bracket between $ \hat{H}_{\rm W}$ and
$\hat{\Omega}_{\rm W}$.

Equation~(\ref{eq:qcdotOm}) describes the evolution of the
non-normalised density matrix, $\hat{\Omega}_{\rm W}(X,t)$,
when a quantum subsystem with probability sinks or sources
(represented by the decay operator $\hat{\Gamma}$)
is embedded in a classical environment (with phase space coordinates
$X$). The classical bath produces both statistical noise and
 decoherence in addition to those eventually represented by the
decay operator.
As a consequence of Equation~(\ref{eq:qcdotOm}), the trace of
$\hat{\Omega}_{\rm W}(X,t)$ is not a conserved quantity:
\begin{eqnarray}
\frac{d}{dt}{\rm Tr}^\prime\int dX \hat{\Omega}_{\rm W}(X,t)
&=&
\frac{d}{dt}\tilde{\rm T}{\rm r}\left[ \hat{\Omega}_{\rm W}(X,t)\right]
\nonumber\\
&=&
\tilde{\rm T}{\rm r}\left[ \frac{\partial }{\partial t}\hat{\Omega}_{\rm W}(X,t)\right]
\neq 0\;,
\end{eqnarray}
where we have denoted with the symbol ${\rm Tr}^\prime$ a partial trace
over the quantal degrees of freedom, with the symbol $\int dX$ the phase
space integral, and with the symbol $\tilde{\rm T}{\rm r}$ both the
partial trace and the phase space integral.

Using the cyclic invariance of the trace, we can easily see that
\begin{eqnarray}
\tilde{\rm T}{\rm r}\left\{
\left[\hat{H}_{\rm W},\hat{\Omega}_{\rm W}\right]_-
\right\}
&=&
\tilde{\rm T}{\rm r}\left\{
\hat{H}_{\rm W}\hat{\Omega}_{\rm W}-\hat{H}_{\rm W}\hat{\Omega}_{\rm W}
\right\} =0\;,
\\
\tilde{\rm T}{\rm r}\left\{
\hat{H}_{\rm W}\overleftarrow{\nabla}_a
{\cal B}_{ab}
\overrightarrow{\nabla}_b \hat{\Omega}_{\rm W}
-
\hat{\Omega}_{\rm W} \overleftarrow{\nabla}_a
{\cal B}_{ab}
\overrightarrow{\nabla}_b \hat{H}_{\rm W}
\right\}
&=& 0 \;,
\end{eqnarray}
where, in the last identity, we have also performed an
integration by parts and exploited
the fact that ${\cal B}_{ab}$
are constants.
If we also use the identity
\begin{eqnarray}
\tilde{\rm T}{\rm r}\left\{
\left[\hat{\Gamma},\hat{\Omega}_{\rm W}\right]_+
\right\}
&=&
2{\rm Tr}^\prime\left[\hat{\Gamma}\hat{\Omega}_{\rm S}\right]
\;,
\end{eqnarray}
where $\hat{\Omega}_{\rm S}=\int dX \hat{\Omega}_{\rm W}(X)$,
we can then find
\begin{equation}
\frac{d}{dt}\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}(X,t)\right]
=
-\frac{2}{\hbar}
{\rm Tr}^\prime\left[\hat{\Gamma}\hat{\Omega}_{\rm S}(t)\right]
\;.
\label{eq:dotqcTrace}
\end{equation}

Equation~(\ref{eq:dotqcTrace}) is analogous to Equation~(\ref{eq:dotTrOmega})
and shows that the probability is not conserved
for the quantum-classical system because of the action of
the decay operator.
We can introduce a normalised density matrix as

\begin{equation}
\hat{\rho}_{\rm W}(X,t)=\frac{\hat{\Omega}_{\rm W}(X,t)}
{\tilde{\rm T}{\rm r}\left[ \hat{\Omega}_{\rm W}(X,t)\right]} \;,
\label{eq:rhoW}
\end{equation}
and, using Equations~(\ref{eq:qcdotOm}) and~(\ref{eq:dotqcTrace}),
find its equation of motion:
\begin{equation}
\begin{array}{ll}
\frac{\partial}{\partial t}\hat{\rho}_{\rm W}(X,t)
=-\frac{i}{\hbar}
\left[\hat{H}_{\rm W},\hat{\rho}_{\rm W}(X,t)\right]_-
+
\frac{1}{2} \hat{H}_{\rm W}\overleftarrow{\nabla} \cdot
\mbox{\boldmath${\cal B}$} \cdot
\overrightarrow{\nabla} \hat{\rho}_{\rm W}(X,t)\vspace{5pt} \\
~~~~~~~~~~~~~~~~~~~~~~~-\frac{1}{2} \hat{\rho}_{\rm W}(X,t) \overleftarrow{\nabla} \cdot
\mbox{\boldmath${\cal B}$} \cdot
\overrightarrow{\nabla} \hat{H}_{\rm W} \vspace{5pt} \\
~~~~~~~~~~~~~~~~~~~~~~~-\frac{1}{\hbar}\left[\hat{\Gamma},\hat{\rho}_{\rm W}(X,t)\right]_+
+\frac{2}{\hbar}\hat{\rho}_{\rm W}(X,t)
\tilde{\rm T}{\rm r}\left[\hat{\Gamma}\hat{\rho}_{\rm W}(X,t)\right]
\;.
\label{eq:qcdotrhoW}
\end{array}
\end{equation}

At variance with Equation~(\ref{eq:qcdotOm}), Equation~(\ref{eq:qcdotrhoW})
is nonlinear and allows one to define averages of the dynamical
variables of the quantum-classical system with a non-Hermitian
Hamiltonian that has a probabilistic meaning.

\section{Entropy Production and Quantum-Classical Non-Hermitian
Hamiltonians}
\label{sec:qcentropy}

As noted in~\cite{manfredi}, when considering the definition of
the entropy for a quantum system in terms of the Wigner function,
the typical choice in terms of the von Neumann definition,
found in Equation~(\ref{eq:SVNnH}) when the Wigner function
$f_{\rm W}(x,X,t)$ replaces the density matrix $\hat\rho$,
cannot work: $f_{\rm W}(x,X,t)$ can be negative in general.
What one can do~\cite{manfredi} is start from the linear
entropy~\cite{zurek,pattanayak}, $S_{\rm lin}=1-{\rm Tr}(\hat{\rho}^2)$,
and perform the Wigner transform in order to obtain:
\begin{equation}
S_{\rm lin}=1-(2\pi\hbar)^{n+N}\int dxdX f_{\rm W}^2(x,X,t)\;,
\label{eq:s2}
\end{equation}
where $f_{\rm W}(x,X,t)$ is the Wigner function, obtained
by transforming $\hat{\rho}$ over all the coordinates.

In a mixed quantum-classical framework, the natural extension
of Equation~(\ref{eq:s2}) is given by
\begin{equation}
S_{\rm lin,W}=1-(2\pi\hbar)^{N}{\rm Tr}^\prime\int dX\hat{\rho}_{\rm W}^2(X,t)
=1-(2\pi\hbar)^{N}\tilde{\rm T}{\rm r}\left[\hat{\rho}_{\rm W}^2(X,t)\right].
\label{eq:s2W}
\end{equation}

When considering the non-Hermitian dynamics of the quantum subsystem
embedded in the classical environment, given by Equation~(\ref{eq:qcdotrhoW}),
we obtain the linear entropy production
\begin{equation}
\dot{S}_{\rm lin,W}=
-2(2\pi\hbar)^{N}\tilde{\rm T}{\rm r}\left[\hat{\rho}_{\rm W}
\frac{\partial\hat{\rho}_{\rm W}}{\partial t}\right].
\label{eq:s2pW}
\end{equation}

{We have obtained
Equation~(\ref{eq:s2pW}) by using the identities
\begin{eqnarray}
\tilde{\rm T}{\rm r}\left\{\hat{\rho}_{\rm W}
\left[\hat{H}_{\rm W},\hat{\rho}_{\rm W}\right]_-
\right\}&=&0\;,
\\
\tilde{\rm T}{\rm r}\left\{\hat{\rho}_{\rm W}
\left[{\cal B}_{ab}(\nabla_a\hat{H}_{\rm W})(\nabla_b\hat{\rho}_{\rm W})
-
{\cal B}_{ab}(\nabla_a\hat{\rho}_{\rm W})(\nabla_b\hat{H}_{\rm W})
\right]
\right\}
&=&0\;,
\end{eqnarray}
 together with}
${\cal B}_{ab} (\nabla_{ab}^2\hat{\rho}_{\rm W}) \hat{\rho}_{\rm W}
\hat{H}_{\rm W}=0$ and ${\cal B}_{ab} \hat{\rho}_{\rm W}
(\nabla_{ab}^2\hat{\rho}_{\rm W}) \hat{H}_{\rm W}=0$,
which follow from taking the trace of an antisymmetric matrix, ${\cal B}_{ab}$,
and a symmetric one, $\nabla_{ab}^2\hat{\rho}_{\rm W}$.
Noting that we also have
\begin{eqnarray}
\tilde{\rm T}{\rm r}\left\{\hat{\rho}_{\rm W}
\left[\hat{\Gamma},\hat{\rho}_{\rm W}\right]_+
\right\}
&=&
2\tilde{\rm T}{\rm r}\left\{
\hat{\Gamma}\hat{\rho}_{\rm W}^2
\right\}\;,
\end{eqnarray}
we finally obtain the entropy production
\begin{eqnarray}
\dot{S}_{\rm lin,W}
&=&
\frac{4}{\hbar}(2\pi\hbar)^N
\left\{
{\rm Tr}^\prime\left[\hat{\Gamma}\hat{\rho}_{\rm S}^2(t)\right]
-{\rm Tr}^\prime\left[\hat{\Gamma}\hat{\rho}_{\rm S}(t)\right]
{\rm Tr}^\prime\left[\hat{\rho}_{\rm S}^2(t)\right]
\right\}\;.
\label{eq:SlinW}
\end{eqnarray}

Within the quantum-classical framework,
we can also introduce a non-Hermitian linear entropy~as
\begin{eqnarray}
S_{{\rm lin},W}^{\rm NH}&=&
1-(2\pi\hbar)^N
\tilde{\rm T}{\rm r}\left[\hat{\rho}_{\rm W}(X,t)\hat{\Omega}_{\rm W}(X,t)\right].
\label{eq:s2WNH}
\end{eqnarray}

The entropy production is given by
\begin{eqnarray}
\dot{S}_{{\rm lin},W}^{\rm NH}
&=&
-2\frac{(2\pi\hbar)^N}{Z_{\rm W}}
\tilde{\rm T}{\rm r}\left[
\hat{\Omega}_{\rm W}
\frac{\partial\hat{\Omega}_{\rm W}}{\partial t}
\right]
-
\frac{2(2\pi\hbar)^N}{\hbar}
\tilde{\rm Tr}\left[\hat{\Gamma}\hat{\Omega}_{\rm W}\right]
\tilde{\rm Tr}\left[\hat{\rho}_{\rm W}^2\right],
\end{eqnarray}
where we have defined
\begin{eqnarray}
Z_{\rm W}&=&\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}(X,t)\right].
\end{eqnarray}

In the following, we will use
\begin{eqnarray}
\dot{Z}_{\rm W}&=&
-\frac{2}{\hbar}\tilde{\rm T}{\rm r}\left[\hat{\Gamma}\hat{\Omega}_{\rm W}(X,t)\right].
\end{eqnarray}

In order to calculate $\tilde{\rm T}{\rm r}
[\hat{\Omega}_{\rm W} \partial\hat{\Omega}_{\rm W}/\partial t]$,
we are led to consider the following identities:
\begin{eqnarray}
\tilde{\rm T}{\rm r}\left[
{\cal B}_{ab}\hat{\Omega}_{\rm W}(\nabla_a\hat{H}_W)(\nabla_b\hat{\rho}_W)
\right.
-
\left.
{\cal B}_{ab}\hat{\Omega}_{\rm W}(\nabla_a\hat{\rho}_W)(\nabla_b\hat{H}_W)
\right]
&=&0
\;, \\
\tilde{\rm T}{\rm r}\left\{
\hat{\Omega}_{\rm W}\left[\hat{\Gamma},\hat{\Omega}_{\rm W}\right]_+
\right\}
&=&
2\tilde{\rm Tr}^\prime\left[\hat{\Gamma}\hat{\Omega}_{\rm S}^2\right].
\end{eqnarray}

Finally, we obtain
\begin{eqnarray}
\dot{S}_{\rm lin,W}^{\rm NH}
&=&
\frac{4(2\pi\hbar)^N}{\hbar}
{\rm Tr}^\prime\left[
\hat{\Gamma}\hat{\rho}_{\rm S}\hat{\Omega}_{\rm S}
\right]
-
\frac{2(2\pi\hbar)^N}{\hbar}
{\rm Tr}^\prime\left[\hat{\Gamma}\hat{\Omega}_{\rm S}\right]
{\rm Tr}^\prime\left[\hat{\rho}_{\rm S}^2\right].
\label{eq:SlinWNH}
\end{eqnarray}

\subsection*{Quantum-Classical Linear Entropy Production
and Constant Decay Operator}

When the decay operator $\hat\Gamma$ is given by Equation~(\ref{eq:Gammamodgau}),
Equation~(\ref{eq:qcdotOm}) becomes
\begin{eqnarray}
\frac{\partial}{\partial t}\hat{\Omega}_{\rm W}(X,t)
=&-&\frac{i}{\hbar}
\left[\hat{H}_{\rm W},\hat{\Omega}_{\rm W}(X,t)\right]_-
+
\frac{1}{2} \hat{H}_{\rm W}\overleftarrow{\nabla} \cdot
\mbox{\boldmath${\cal B}$} \cdot
\overrightarrow{\nabla} \hat{\Omega}_{\rm W}(X,t)
\nonumber\\
&-&
\frac{1}{2} \hat{\Omega}_{\rm W}(X,t) \overleftarrow{\nabla} \cdot
\mbox{\boldmath${\cal B}$} \cdot
\overrightarrow{\nabla} \hat{H}_{\rm W}
-
\gamma_0\hat{\Omega}_{\rm W}(X,t) \;,
\label{eq:qcdotOm-Gamconst}
\end{eqnarray}
and Equation~(\ref{eq:dotqcTrace}) becomes
\begin{equation}
\frac{d}{dt}\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}(X,t)\right]
=
-\gamma_0
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}(X,t)\right]
\;.
\label{eq:dotqcTrace-Gamconst}
\end{equation}

Upon choosing the initial condition
$\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}(X,0)\right]=1$,
Equation~(\ref{eq:dotqcTrace-Gamconst}) has the solution
\begin{equation}
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}(X,t)\right]
=
\exp\left[-\gamma_0 t\right]\;,
\end{equation}
which is analogous to Equation~(\ref{eq:trOmega}).
Equation~(\ref{eq:qcdotrhoW}) becomes
\begin{equation}
\begin{array}{ll}
\frac{\partial}{\partial t}\hat{\rho}_{\rm W}(X,t)
=-\frac{i}{\hbar}
\left[\hat{H}_{\rm W},\hat{\rho}_{\rm W}(X,t)\right]_-
+
\frac{1}{2} \hat{H}_{\rm W}\overleftarrow{\nabla} \cdot
\mbox{\boldmath${\cal B}$} \cdot \vspace{4pt}
\overrightarrow{\nabla} \hat{\rho}_{\rm W}(X,t) \\
~~~~~~~~~~~~~~~~~~~~~~~-\frac{1}{2} \hat{\rho}_{\rm W}(X,t) \overleftarrow{\nabla} \cdot
\mbox{\boldmath${\cal B}$} \cdot
\overrightarrow{\nabla} \hat{H}_{\rm W}
\;. \\
\label{eq:qcdotrhoW-Gamconst}
\end{array}
\end{equation}

 \vspace{-5pt}

Equation~(\ref{eq:qcdotrhoW-Gamconst}) shows that, in the case
considered, the normalised density matrix $\hat{\rho}_{\rm W}(X,t)$
is not influenced by $\hat\Gamma$, so this
evolves according to the unitary quantum-classical dynamics
{that were first derived in \cite{kapra-cicco}.}

We also have that
Equations~(\ref{eq:SlinW}) and~(\ref{eq:SlinWNH}) become
\begin{eqnarray}
\dot{S}_{\rm lin,W}
&=&
2\gamma_0(2\pi\hbar)^N
\left\{
\tilde{\rm T}{\rm r}^\prime\left[\hat{\rho}_{\rm W}^2(X,t)\right]
-
\tilde{\rm T}{\rm r}\left[\hat{\rho}_{\rm W}^2(X,t)\right]
\right\}=0\;, \label{eq:SlinW-Gamconst} \\
\dot{S}_{\rm lin,W}^{\rm NH}
&=&
\frac{4(2\pi\hbar)^N}{\hbar}
\tilde{\rm T}{\rm r}\left[
\hat{\Gamma}\hat{\rho}_{\rm W}(X,t)\hat{\Omega}_{\rm W}(X,t)
\right]
-
\frac{2(2\pi\hbar)^N}{\hbar}
\tilde{\rm T}{\rm r}\left[\hat{\Gamma}\hat{\Omega}_{\rm W}(X,t)\right]
\tilde{\rm T}{\rm r}\left[\hat{\rho}_{\rm W}^2(X,t)\right]
\nonumber\\
&=&
(2\pi\hbar)^N\gamma_0e^{\gamma_0 t}
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}^2(X,t)\right]\;.
\label{eq:dotSlinWNH-Gamconst}
\end{eqnarray}

In order to evaluate Equation~(\ref{eq:dotSlinWNH-Gamconst}), we need
to calculate $\tilde{\rm T}{\rm r}[\hat{\Omega}_{\rm W}^2(X,t)]$.
From Equation~(\ref{eq:qcdotOm-Gamconst}), we get
\begin{eqnarray}
\frac{\partial}{\partial t}
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}^2(X,t)\right]
&=&
-2\gamma_0
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}^2(X,t)
\right]\;,
\\
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}^2(X,t)\right]
&=&
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}^2(X,0)\right]
\exp\left[-2\gamma_0 t\right]\;.\label{eq:TrOmW-t}
\end{eqnarray}

Upon substituting Equation~(\ref{eq:TrOmW-t}) into Equation~(\ref{eq:dotSlinWNH-Gamconst})
and integrating, we finally obtain
\begin{equation}
S_{\rm lin,W}^{\rm NH}=(2\pi\hbar)^N
\tilde{\rm T}{\rm r}\left[\hat{\Omega}_{\rm W}^2(X,0)\right]
\left(1-e^{-\gamma_0t}\right)\;.
\label{eq:SlinWNH-t}
\end{equation}

Analogously to the pure quantum case, the rate of production
of the quantum-classical entropy in Equation~(\ref{eq:SlinWNH-t})
monitors the flow of information associated with the decay of the purity
of the quantum-classical non-Hermitian system (for positive
$\gamma_0$).

\section{Conclusions}
\label{sec:concl}

In this paper, we have shown that it is possible to define
meaningful entropy functionals for open quantum systems
described by non-Hermitian Hamiltonians.
In particular, a non-Hermitian generalisation of
the von Neumann entropy, which is able to signal
the loss of information of the quantum subsystem,
requires both the normalised and the non-normalised density matrix:
this entropy can be defined as the normalised average of the logarithm
of the non-normalised density matrix~\cite{sk-S}.

Motivated by the Wigner representation of quantum mechanics,
we have also introduced the non-Hermitian generalisation of
the linear entropy, defined as one minus
the normalised average of the square of the non-normalised
density matrix. Through the analytical solution of the
 {basic}
case of a constant decay operator, we have shown
that the non-Hermitian linear entropy is able to describe
the loss of purity of the quantum subsystem.
This is true both for pure non-Hermitian subsystems
as well as for non-Hermitian subsystems embedded in
a classical environment.
 {It is worth repeating that even basic models with
constant decay operators are interesting when one adds
the additional level of complexity provided by the classical-like
environment represented by means of the partially Wigner-transformed
Hermitian part of the Hamiltonian, as in the case of
a light-emitting quantum dot coupled to
an energy-absorbing optical guide in a classical environment.}

The results obtained so far~\cite{ks,kostya-purity,sk-corr,sk-S}
show that the correct description of the dynamics and of the
information flow of systems
described by non-Hermitian Hamiltonians needs the use
of both the normalised and non-normalised density matrix.
In this way, reasonable entropy functionals
can be introduced.
On conceptual grounds, one might have expected that the
foundation of the non-Hermitian theory on the \emph{normalised}
density matrix alone would hide the interesting effects
arising from the coupling to the probability sinks or sources.~As a matter of fact, the density matrix $\hat\rho$ is constrained
to be normalised in order to be able to define correctly
(normalised) statistical averages. However, such a procedure
inevitably masks the flow of information:
it is as if one would like to study the motion of a body by
choosing the frame of reference that moves together with the
body itself.
On~the contrary, the flow of information in systems
modelled with non-Hermitian Hamiltonians
can be \emph{solely}
captured through the use of the \emph{non-normalised}
density matrix.

 {We} hope that
the results discussed in this paper
may be a first step toward
{a rigorous analysis of the quantum information flow in}
 systems
with non-Hermitian Hamiltonians, after removing the constraints
of PT-symmetry~\cite{gardas}.

\acknowledgments{Our research has been self-funded through the salaries
received from the University of Messina. We are very grateful to
Professor Ignazio Licata who encouraged and supported us through the
waiver of the publications costs in the journal \emph{Entropy}.}

redpajama_set_name: RedPajamaArXiv


text:
#include "tensorflow/compiler/xla/service/while_util.h"

#include "tensorflow/compiler/xla/service/hlo_matchers.h"
#include "tensorflow/compiler/xla/test.h"
#include "tensorflow/compiler/xla/tools/parser/hlo_parser.h"

namespace xla {
namespace {

namespace op = ::xla::testing::opcode_matchers;

StatusOr<std::unique_ptr<HloModule>> GetParsedModule(
    HloComputation** entry_computation, HloInstruction** param0,
    HloInstruction** param1, HloInstruction** param2) {
  const char* const hlo_string = R"(
HloModule ModuleWithWhile

while_body {
  ROOT p_body = (f32[32,32]{1,0}, f32[32,32]{1,0}) parameter(0)
}

while_condition {
  p_cond = f32[32,32]{1,0} parameter(0)
  ROOT result = pred[] constant(true)
}

ENTRY entry {
  p_entry_0 = f32[32,32]{1,0} parameter(0)
  p_entry_1 = s32[32,32]{1,0} parameter(1)
  p_entry_2 = s64[32,32]{1,0} parameter(2)
  while_init = (f32[32,32]{1,0}, f32[32,32]{1,0}) tuple(p_entry_0, p_entry_0)
  ROOT while = (f32[32,32]{1,0}, f32[32,32]{1,0}) while(while_init), condition=while_condition, body=while_body
}
)";

  TF_ASSIGN_OR_RETURN(std::unique_ptr<HloModule> module,
                      tools::Parse(hlo_string));

  *entry_computation = module->entry_computation();
  *param0 = (*entry_computation)->parameter_instruction(0);
  *param1 = (*entry_computation)->parameter_instruction(1);
  *param2 = (*entry_computation)->parameter_instruction(2);

  return std::move(module);
}

TEST(WhileUtil, MakeZeroInstructionsLiveOp) {
  HloInstruction *param0, *param1, *param2;
  HloComputation* entry_computation;

  TF_ASSERT_OK_AND_ASSIGN(
      std::unique_ptr<HloModule> module,
      GetParsedModule(&entry_computation, &param0, &param1, &param2));

  HloInstruction* while_instr = entry_computation->root_instruction();
  ASSERT_EQ(while_instr->opcode(), HloOpcode::kWhile);

  TF_ASSERT_OK_AND_ASSIGN(
      WhileUtil::MakeInstructionsLiveInResult make_live_in_result,
      WhileUtil::MakeInstructionsLiveIn(while_instr, /*instructions=*/{}));

  HloInstruction* new_while_instr = make_live_in_result.new_while_instr;

  EXPECT_THAT(
      entry_computation->root_instruction(),
      op::Tuple(op::GetTupleElement(::testing::Eq(new_while_instr), 0),
                op::GetTupleElement(::testing::Eq(new_while_instr), 1)));

  auto param_reconstructed =
      op::Tuple(op::GetTupleElement(op::Parameter(0), 0),
                op::GetTupleElement(op::Parameter(0), 1));

  EXPECT_THAT(new_while_instr->while_body()->root_instruction(),
              op::Tuple(op::GetTupleElement(param_reconstructed, 0),
                        op::GetTupleElement(param_reconstructed, 1)));
}

TEST(WhileUtilTest, MakeTwoInstructionsLive) {
  HloInstruction *param0, *param1, *param2;
  HloComputation* entry_computation;

  TF_ASSERT_OK_AND_ASSIGN(
      std::unique_ptr<HloModule> module,
      GetParsedModule(&entry_computation, &param0, &param1, &param2));

  HloInstruction* while_instr = entry_computation->root_instruction();
  ASSERT_EQ(while_instr->opcode(), HloOpcode::kWhile);

  TF_ASSERT_OK_AND_ASSIGN(
      WhileUtil::MakeInstructionsLiveInResult make_live_in_result,
      WhileUtil::MakeInstructionsLiveIn(while_instr,
                                        /*instructions=*/{param0, param1}));

  HloInstruction* new_while_instr = make_live_in_result.new_while_instr;

  XLA_VLOG_LINES(3, module->ToString());

  EXPECT_THAT(
      entry_computation->root_instruction(),
      op::Tuple(op::GetTupleElement(::testing::Eq(new_while_instr), 0),
                op::GetTupleElement(::testing::Eq(new_while_instr), 1)));

  auto first_half_param_reconstructed =
      op::Tuple(op::GetTupleElement(op::Parameter(0), 0),
                op::GetTupleElement(op::Parameter(0), 1));

  EXPECT_THAT(new_while_instr->while_body()->root_instruction(),
              op::Tuple(op::GetTupleElement(first_half_param_reconstructed, 0),
                        op::GetTupleElement(first_half_param_reconstructed, 1),
                        op::GetTupleElement(op::Parameter(0), 2),
                        op::GetTupleElement(op::Parameter(0), 3)));
}

TEST(WhileUtilTest, GetInvariantGTEsForWhileBody) {
  const char* const hlo_string = R"(
HloModule ModuleWithWhile

body {
  param.b = (s32[], s32[]) parameter(0)
  gte.0 = s32[] get-tuple-element(param.b), index=0
  gte.1 = s32[] get-tuple-element(param.b), index=1
  add = s32[] add(gte.0, gte.1)
  ROOT tuple = (s32[], s32[]) tuple(gte.0, add)
}

cond {
  param.c = (s32[], s32[]) parameter(0)
  ROOT constant = pred[] constant(true)
}

ENTRY main {
  init = (s32[], s32[]) parameter(0)
  ROOT while = (s32[], s32[]) while(init), condition=cond, body=body
}
)";

  TF_ASSERT_OK_AND_ASSIGN(std::unique_ptr<HloModule> module,
                          tools::Parse(hlo_string));

  HloComputation* while_body = module->GetComputationWithName("body");

  ASSERT_NE(while_body, nullptr)
      << "Expected exactly one while_body computation";

  std::vector<HloInstruction*> gte_list =
      WhileUtil::GetInvariantGTEsForWhileBody(*while_body);

  ASSERT_EQ(gte_list.size(), 1);
  EXPECT_EQ((*gte_list.begin())->name(), "gte.0");
}
}  // namespace
}  // namespace xla

redpajama_set_name: RedPajamaGithub


text:
EMBA in Switzerland: Executive education in banking & finance in Zurich & Geneva. MBA in wealth management degree in English for international students / American students. Undergraduate & graduate study abroad programs in Europe.
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School Search» Business Schools / MBA Programs» Switzerland» EMBA in Switzerland, executive education
Executive Banking & Finance Education in Switzerland
About Programs Contact
Swiss Finance Institute:
Based in both Zurich and Geneva, Swiss Finance Institute is a private foundation created by Switzerland's banking and finance community in co-operation with leading Swiss universities to support and advance research and executive education in banking and finance. As one of the major European providers of advanced executive education in banking and finance and delivering world-class research, the institute offers a wide variety of high-quality international and national executive education programmes including an executive MBA programme in Asset and Wealth Management taught in English, additional executive programmes taught in English and German, a PhD programme in finance and more.
Email: Contact us here
( or request a callback ).
Zurich's closest international airport is Zurich Airport, approximately 10 km from the city centre, and the nearest major airport to Geneva is Geneva International Airport, approximately 6 km from the city.
Established in 2006, Swiss Finance Institute belongs to the most important knowledge pools of the Swiss financial centre, emerging from the integration of three institutes: The International Center FAME for Financial Asset Management and Engineering in Geneva, the Swiss Banking School in Zurich and the Stiftung Banking and Finance at the University of Zurich.
The Swiss Finance Institute programmes combine state-of-the-art knowledge and skills in management with practical experience and insights into the functioning of the financial sector. Every Swiss Finance Institute course has an intense and pragmatic curriculum. Students are exposed to the latest developments within the most important business areas and technology of the sector. The faculty has over 50 academic professors and lecturers at its disposal.
Since the Swiss Finance Institute was founded in 2006, it has become a leading research and teaching organization in the area of banking and finance. It aims to further strengthen this position by achieving top-3 status in Europe and top-10 in the world. To reach this objective, a total of 50 faculties at our partner universities are being supported with appropriate incentive, selected fundamental and applied research projects are being financed on an ongoing basis, and our PhD program in finance, which has already grown into one of the most recognized programs in the world, will continue to be strengthened.
Moreover, we want to make the Swiss Finance Institute, and thus Switzerland, the benchmark for high level executive education in banking and finance.
Our Activities:
The Swiss Finance Institute is active in both research and executive education. Excellence at both levels will increase Switzerland's attractiveness for outstanding researchers, teachers, students, as well as participants in executive education programs, and sustain the ambition of becoming a world leader in banking and finance education.
Executive Education Programmes in Banking & Finance:
- Master programme: Dual Degree Executive MBA (Master of Business Administration) in Asset and Wealth Management (in English)
- Senior Management programme in Banking (in English)
- Advanced Executive programme (in German)
- Diploma of Advanced Studies (DAS) in Banking (in German)
- Financial Asset Management and Engineering programme (FAME)
- International Private Banking & Wealth Management Retreat
- PhD programme in Finance
- Private Banking – Cross Border Activities (Workshop / Seminars)
Co-operation Partners:
* Carnegie Mellon University / University of Lausanne
* Rochester-Bern Executive Programmes / Universität Bern
* Singapore Management University / Wealth Management Institute
Quality Faculty & Research:
The Swiss Finance Institute is a research organization with a faculty of more than 50 full time researchers engaged in high level research in the field of banking and finance and related disciplines. The Swiss Finance Institute Research center aims at increasing Switzerland's attractiveness for outstanding researchers, teachers, students and participants in executive education programs. Over the medium term, Swiss Finance Institute wants to achieve a leading position within Europe in major research areas relevant for banking and finance.
The Swiss Finance Institute has set up an Executive Education Advisory Board comprising leading specialists and experienced practitioners from the fields of education in banking and finance to oversee the strategic orientation of the courses offered and to ensure that highest standards of quality are upheld.
See Also: - Courses / Programs
Address: Walchestrasse 9 CH-8006, Zürich, Switzerland
Related Schools & Programs. Inquire to our admissions team today:
View this School
Lausanne Hotel School
Courses / Programs
Located in Lausanne, a beautiful city which lies on Lake Geneva in Western Switzerland, The Ecole hôtelière de Lausanne (EHL) is a leading hospitality school which is part of the University of Applied Sciences of Western Switzerland. EHL is an international, bilingual school that offers hotel management and hospitality programmes in both English and French.
We provide the solid management education of a top business school while basing our programmes on our own unique educational philosophy: Hospitality management as both a science and an art. We are accredited by the New England Association of Schools and Colleges (NEASC).
Founded a hundred and fifteen years ago the school has never ceased to grow and innovate. During this time, the hospitality industry too has greatly evolved. It is now the fastest-growing economic sector in the world and is reaching out into many new business areas.
In response to these rapid changes, EHL continually evaluates and updates its programmes to correspond to new industry needs. With more than a century of experience, EHL is steeped in rich traditions yet a visit to the school reveals a young, dynamic institution with ultra-modern facilities.
- 4-year Bachelor of Science in Hospitality
- 2-year Hotel Management Diploma
- 14-month Master in Hospitality
* Programmes are offered in English and French.
* All EHL programmes strike a careful balance between the science and art of hospitality management.
* Students learn the hard financial and business aspects of hotel management as well as practical and professional experience.
Glion Institute of Higher Education, Online MBA & Hospitality Postgraduate Programs
Established in 1962, Glion Institute of Higher Education is a private university in Montreaux, Switzerland, and is a global leader in international hospitality education.
Glion has been ranked among the Top 3 Hospitality Management Schools in the world for an international career.
We provide an accredited online MBA in International Hospitality & Service Industries Management degree program taught in English that is available to students and working professionals around the world.
In addition, we provide Postgraduate Certificate & Diploma programs that are also taught entirely via the internet. We offer two start dates in 2014.
Our MBA offers a management curriculum that balances hospitality and tourism applied knowledge with a core of management and leadership theory.
It is designed to equip working professionals with the expertise to lead and motivate individuals, teams and organizations within a diverse and multicultural global business environment.
The program is intensive and relevant, yet we offer an MBA program that fits into your work and life, in an online environment which is 'asynchronous'.
There are no fixed lecture times, the classroom is always open and you don't need to synchronize your study with anyone else.
Online Master of Business Administration (MBA) in International Hospitality & Service Industries Management degree with the following specializations:
- General Management for Service Industries
- Marketing & Innovation
- Asset & Revenue Management
- 'Self Designed' option: Any 3 courses from the other specializations
Online Postgraduate Programs:
- Postgraduate Certificate in International Hospitality & Services Administration
- Postgraduate Diploma in International Hospitality & Services Management
Business & Hotel Management School (BHMS)
Located in Luzern in north-central Switzerland, Business & Hotel Management School (BHMS) is an accredited hospitality management school. We offer high-quality undergraduate diploma and degree courses in the fields of hospitality management and culinary management as well as an MBA in Hospitality and Global Management to students from Switzerland and around the world.
BHMS is a member of the Bénédict Switzerland chain of schools which is a world-renowned learning organisation, with specialisation in hotel management being offered exclusively at BHMS in Luzern. Here, a selected group of approximately 200 students per year have the opportunity to prepare themselves for the hospitality opportunities available in the world's fastest growing market and career segment - the Service Sector.
Switzerland is perfectly situated in the centre of Europe, bordered by France, Germany, Austria, Italy and Liechtenstein, and is an internationally recognized destination for students seeking top quality education in Business & Hospitality Management. Luzern is conveniently located less than a 1 hour drive from Zurich and approximately 2.5 hours drive from Geneva.
Culinary Management Programs
- Diploma in Culinary Management (6-12 months)
- Advanced Diploma in Culinary Management (6-12 months)
- Associate Degree in Culinary Entrepreneurship (12 months)
Hospitality Management Programs
- Diploma in Hospitality Management (1 yr)
- Higher Diploma in Hospitality Management (2 yrs)
- Bachelor of Arts Degree in Hotel & Hospitality Management (3 yrs)
- Post Graduate Diploma in Hospitality Management (1 yr)
- MBA in Global Management & Hospitality Management (2 yrs)
IMI International Management Institute Switzerland
IMI International Management Institute in Switzerland is a private hospitality and business management school founded in 1991. We offer a family atmosphere and personalised learning and careers support.
Our undergraduate degrees in hospitality, business and marketing management are validated by Manchester Metropolitan University and our culinary arts degree by Oxford Brookes University.
We have an MSc in international hospitality and events management, which provides work experience for those requiring it, and postgraduate diplomas offering specialisation in hotel and events or culinary.
Our Diplom HF in International Hospitality Management offers three specialist internships alongside three study semesters with a strong focus on language skills.
The combination of quality academic teaching, Swiss practical training, excellent industry contacts and breath-taking scenery are a recipe for success. Our graduates are found in Senior Management positions all around the globe. Their success shows that, to reach the top, it pays to start at the top!
- BA (Hons) Degrees in Hotel, Tourism, Events, Business, Marketing or Culinary Arts
- MSc in International Hospitality Management
- Postgraduate Diplomas in International Hotel and Events or Culinary Arts
EU Business School - Geneva & Montreux
• Established in 1973, EU Business School (EU) is an international, triple-accredited business school that provides business administration, management, and finance programs taught in English at undergraduate and graduate level.
• EU campuses in Switzerland are located in Geneva & Montreux.
• Taught by international faculty who are experts in their fields, our practical and hands-on Bachelor and Masters programs include management theory and case studies.
• We offer high quality teaching, a competitive learning environment and a network of campuses in different countries. This broad network allows students to participate in inter-campus exchanges in: Spain, Switzerland, Germany, UK, USA, Russia, Kazakhstan, Malaysia & Taiwan.
• We combine the U.S. university model with top European management practices to provide our students with the necessary tools to be successful in the international business world.
• Geneva is a truly international city making it an ideal place to study international management. The city is host to the United Nations, and many multinationals are located in the region due to the excellent logistical network and central location of Geneva at the heart of Europe, one hour by air from major European cities.
• A one hour drive from Geneva, Montreux is located on the shores of Lake Geneva crowned by the Alps.
"EU not only met my expectations, it even surpassed them. I feel I made the right choice and I would recommend it to anyone."
- Antonella, Argentina
• BBA - Bachelor of Business Administration
• Business Bridging Foundation
• English Foundation Program
• Business Management
• Communication & PR
• Finance
• International Relations
• International Business
• Digital Business, Design & Innovation
MBA Degree Specializations
• Design Management
• Digital Business
• Entrepreneurship
• Global Banking & Finance
• Blockchain Management...
Doctorate Program
• Doctor of Business Administration (DBA)
Online Business Programs
• BBA / MBA
• Masters Programs
• Executive BBA
All degrees are taught in English
BSL - Business School Lausanne
Business School Lausanne (BSL), a leading innovator in business education with entrepreneurial spirit, is committed to practical and pragmatic learning. The school ranked among the Top 3 Swiss business schools in a recent employer choice study, conducted by QS (Quacquarelli Symonds Ltd 2010).
BSL takes a pragmatic approach to learning by applying theory to practice and is backed by a multidisciplinary faculty of business professionals. BSL attracts students from around the world, creating a multicultural environment of more than 40 nationalities. Studying at BSL is an interactive and engaging learning experience. Established in 1987, BSL is a co-founder of the World Business School Council of Sustainable Business.
- Accelerated 2 - year BBA with an internship option (3 years)
- Master of Business Administration (MBA)
- Executive Master of Business Administration (EMBA)
- Doctorate of Business Administration (DBA)
- MSc in Finance with CFA preparation
- Master in International Business (MIB)
- MSc in Global Marketing
- MSc in Management
- MSc in Luxury Management
- Diploma in Sustainable Business - executive 1-year program with a joint degree by BSL and University of St. Gallen
You are here: School & College Search> Business Schools in Switzerland> EMBA / banking education in Zurich & Geneva
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redpajama_set_name: RedPajamaCommonCrawl


text:
Burnet, Duckworth & Palmer LLP welcomes associate Erin Moch
Burnet, Duckworth & Palmer LLP is pleased to welcome Erin Moch who has joined its business law group as an associate.
Erin was called to the Bar in 2018 and has experience advising on a range of business law matters such as M&A, dispositions, amalgamations and financings for a range of clients in the agribusiness, energy, financial services and manufacturing sectors. She also advises clients on tax law, including tax litigation and the tax aspects of M&A, joint ventures and company reorganizations and restructurings.
Commenting on her arrival at BD&P, Erin said, "I am looking forward to joining BD&P, a top tier Calgary firm, and being part of a team of talented people, experiencing the firm culture from within, while delivering high quality work with precision."
Burnet, Duckworth & Palmer LLP
https://www.bdplaw.com/
The world is changing, and it's happening faster than ever. Businesses and people are adapting, but nobody can do it alone. Maybe it's easier for us to keep up because we've always been nimble and brought creative thinking to the way we practise law. BD&P is a full-service boutique law firm headquartered in Calgary, Canada. Our approximately 120 lawyers are bright, deeply talented legal minds who work on a broad spectrum of corporate and litigation matters, sitting across the table from national and international firms. Our clients live a variety of sectors, including energy, renewables, agribusiness, technology and life sciences. We are not just legal advisors, we are true partners. We've been called unconventional, and we think that makes us better partners to our clients for now — and for the future.
Erin Moch


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text:
Gassour
Vassiliou
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Ahmed Zaki
Poster Tours
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Home / The Singer - المغنواتي
The Singer - المغنواتي
Sayed Issa directed this musical that endures as an almost-forgotten film from the waning years of the Egyptian musical genre. The story is set in the 1940s, during the battles for Egypt's liberation from British rule. Ali El-Haggar plays a singer who loves the daughter of a village leader. Her family disapproves and another woman (Soheir El-Morshedi) creates a love triangle by trying to win his heart.
This is a very rare poster by artists Khalil and Anis, and demonstrates an advanced talent for visual balance and artistry. Ali El-Haggar is dressed in a suit and traditional tarboosh cap, as he plays the oud. Soheir El-Morshedi, the troupe's dancer, is elegant and exotic in a traditional green dance costume. The other performers in the troupe are de-emphasized, using a black & white cross-hatching illustration technique. The overall look is complex, densely arranged and very attractive, with a palette of gold, green and red colors that makes this poster stand out as a décor object.
Scarcity Index
Dimensions 27 x 39 in
Condition Very good vintage condition. Minimal signs of wear along the edges and fold lines.
Artist Khalil
Printer Arab Cinema Press
Medium Stone lithography
Wherever you are located, we will mail your poster via express mail. For FREE!
Rolled in a sturdy, artistic tube, your poster is guaranteed ultimate protection. Thinking of a gift? It will never fail to impress.
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Your poster arrives with a Certificate of Authenticity. Each poster has a unique code that can be tracked and verified through its certificate.
We use 256-bit SSL certificates to ensure highest standards of payment security.
If you enjoy vintage Egyptian movie posters as much as we do, sign up to learn about our latest poster releases & special offers.


Guests that are camping on EXPO grounds and are not associated with an event that has contracted for all, or a portion of the grounds, will be required to camp in Area 1 (see map), or the Southern Oregon RV Park. Individuals traveling with equine, or other livestock that require boarding horses in the EXPO stables will be allowed to camp in Area 1 camp spaces, if they are available and not reserved by another party. Guests can make reservations by calling the Jackson County Parks office at (541) 774-8183 in advance of arrival, or guests can arrive on a first-come, first-serve basis. If arriving on a first-come, first-serve basis, please be aware that spaces may be fully occupied due to large events and as such, a camp space in Area 1 may not be available. Guests are strongly encouraged to make reservations in advance. When making a reservation, you may pay in advance. If arriving on a first-come, first-serve basis, guests will be required to pay immediately upon arrival at the Southern Oregon RV Park located at Gate 5 of the EXPO. Cash/checks will be accepted, no credit cards processed on site. If guests require a stable rental, they must contact the EXPO directly at (541) 774-8270. Please direct onsite questions to Camp Hosts at the Southern Oregon RV Park, or by calling the Parks office at (541) 774-8183. ALL guests not related to a contracted event traveling without livestock will be required to camp at the Southern Oregon RV Park.
In an effort to streamline event coordination for EXPO event planners, guests that are camping at the EXPO and are associated with a contracted facility event, such as an equine show, or dog show will be required to coordinate and pay for camping services directly with the EXPO. In most cases this will be done by the event coordinator directly and the guest will reserve camping with the event coordinator. Please check with your event planner for details regarding camping on EXPO grounds during your specific event. Coordinators will reserve camping spaces in one, or multiple camping areas as shown in the map. Areas 1-3 have a variety of camp utilities available (details on map), while areas 4 and 5 are primarily for dry camping during large events. If needed, temporary utilities can be set up in Areas 4 and 5 for large events, but the event will be required to reserve a minimum of 120 camping nights for setup to occur. For example, the event coordinator could reserve 60 spaces for 2 nights, or 40 spaces for 3 nights. Setting up temporary utilities is not a simple task, hence this minimum number of nights is required to make this task feasible. Event coordinators must be very clear in identifying camping needs while negotiating facility contract with EXPO. Coordinators will be required to give EXPO staff a firm count of campers arriving for their event 30 days prior to event. This will give staff time to properly plan and coordinate camping services. Once the contract is executed with EXPO staff, coordinators can direct all operational questions regarding camping to the Jackson County Parks staff at (541) 774-8183.


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TO REACH SPLIT ROCK PICNIC AREA of Joshua Tree National Park, take the marked road that leads north off the Park Boulevard at a point 1.2 miles past Jumbo Rocks Campground. This road ends at the Split Rock Parking Lot. Picnic tables and a restroom are available, and on just about any day you'll find rock climbers on the surrounding large granite formations.
Because the Split Rock area is quite extensive, I am making seven 3D Galleries which you can link to above or under each numbered heading. 3D images require red-cyan glasses to view properly. All photographs in this Main Index will be 2D.
Note: Split Rock proper—the large boulder immediately north of the parking area—has been declared off limits to climbing activities.
Click images to enlarge. 2016p (4K) resolution. Or Windows users, right-click and choose SAVE LINK AS.
The West Tiers are the rocks that form the eastern side of the small valley east of the Split Dome. They face Split Dome's east face and the road. Several 'tiers' of rock characterize the northern part of this formation, with numerous crack systems. A side hike takes you to the hilltop or summit of West Tiers where some beautiful scenery provides views of 29 Palms and Belle Mountain, Belle and White Tank Campgrounds, and all the way south to The Pope's Hat rock pile at Live Oak picnic area.
The northern end of West Towers (see above) contains even more interesting formations, some of which are favorites with rock climbers, others just attract photographers looking for some fascinating subjects to frame and capture in digital images. Future Games Rock is a large wall that, because it is north-facing, gets shade most of the day.
Beyond and to the left of Frigid Tower lies the aforementioned steep north-facing face of Future Games Rock, about one-third mile hike northeast of the parking lot. More than a dozen climbing routes are recorded.
From the base of the West Tiers (el. 4,275ft) to the sloping summit area is approximate 100ft elevation gain, which provides the viewer a scenic 360-degree vista. From up top can be seen the 29 Palms desert floor to the north, Belle Mountain to the east, and Live Oak Picnic Area with its dominant Pope's Hat rock pile plainly visible. West lies the main Split Rock complex with the 2-mile loop trail starting from the parking circle.
As you climb the West Tiers slope you will pass on your right a massive rock pile with one extremely large boulder stuck in it; numerous pinnacle-pointed funky-looking rocks jut up elsewhere out of the desert like they just grew there willy-nilly. Lying southeast of the parking lot, this feature has been given the rather colorless name: SE Boulders.
The 2-mile Split Rock Loop hike passes through a landscape of uncanny sculpted rock gardens highlighted by my favorite feature, Tulip Rock, as well as The Tooth and Face Rock—three geologic formations with likenesses to their given names.
Since it was well past the noon hour, I headed down the Loop clockwise starting about 2 P.M. on a beautiful late-November day. (Note the proximity to the picnic area. The obvious trail begins west of the parking loop.) The trail is well marked by cairns and easy to follow. I gave myself two to two-and-one-half hours since 3D picture taking is a tedious process. The trail can be done in much less time if you stroll along.
With the sun getting low in the sky, I'd have it at my back much of the time casting its welcoming golden-hour glow and late-autumn deep shadows that give Joshua Tree's formations their spectacular appearance this time of year.
Several formations named by the rock climbers were passed within minutes, the first being the Grand Canyon. West of this lies a very large boulder, Rubicon Formation. Opposite Rubicon takes you to The Snake Pit, To Cling or Fling Corridor, and Screamer Rocks with its darkly sinister Brit Corridor. (Claustrophobic types may elect to bypass Brit Corridor.) East of Rubicon, For Peter Block contains three climbs between 55ft and 90ft (large image below).
← Once you exit the Rubicon Formation, the clockwise route passes a few hundred yards south of monolithic Tulip Rock. Its most photogenic side faces southeast, with direct sun illuminating it and enhancing photo shoots nicely. In this gallery you will find several distant shots of Tulip Rock as well as a couple taken close up. Immediately west of Tulip Rock is 'The Tooth'—a largish boulder that resembles a molar.
Further south and on the west side the trail passes along the base of the massive Isles in the Sky and Isles Corridor. This scenic loop trail meanders nicely in, around, and through the formations. You will enjoy doing it!
We begin the second half of our journey. Leaving the "Snake Pit" area, the Split Rock Loop Trail swings around the southern base of towering Isles in the Sky, where a signpost points the way to a side trip (further south) and Face Rock. Cattlemen used this area long ago, indicated by an old cement watering trough and some iron pipes. It is on the left side of the trail as you approach Face Rock.
From Face Rock, you backtrack and the trail continues along the western side of Isles in the Sky. This is a long formation but soon returns you to the Tulip Rock corridor after passing through an opening. Before you, a huge balanced rock that has a hole at its base. Several hundred yards beyond, Rubicon Formation looks much smaller than when you stood beside it earlier. Tulip Rock and The Tooth, a small molar-shaped rock, are the most notable features.
As an aside, there are several large and inviting pine trees which offer shade here; they can be found along the northern base of Isles in the Sky. Once you leave the rocky confines of Tulip Rock corridor, the trail descends a long slope into the open desert. Visible once more, Split Rock and Split Dome's massive formations half a mile east and the end of your hike. The Split Rock Loop Trail makes a final, leisurely swing behind a photogenic rock pile before heading across a sandy wash. You have before you Frigid Tower, its pyramid-shaped bulk protecting the surrounding granite massif like a medieval guardhouse.


text:
Inga-Maria Elisabeth (Lill) Thiel, född 13 oktober 1913 i Stockholm, död 18 november 1996 i Alvesta, var en svensk författare och målare.

Hon var dotter till bankdirektören Ernest Thiel och hans andra hustru Signe Maria Thiel. Carin Thiel och Signe Henschen var hennes halvsystrar. Thiel studerade konst på egen hand men fick stor vägledning av sin fars vänner i konstnärskretsarna. Separat ställde hon ut på Gummesons konsthall 1928 som senare följdes av medverkan i flera samlingsutställningar. Tillsammans med tecknaren Folke Karlsson gav hon 1945 ut barnboken Så långt fjädern räcker  där Thiel svarade för texten och fotografierna. Lill Thiel är begravd på Borgholms kyrkogård på Öland.

Källor
Svenskt konstnärslexikon del V, sid 423, Allhems Förlag, Malmö.
 Inga-Maria Elisabet Thiel på Gravar.se

Noter

Svenska målare under 1900-talet
Konstnärer från Stockholm
Födda 1913
Avlidna 1996
Kvinnor
Lill

redpajama_set_name: RedPajamaWikipedia


All prints in the "On Paper" section are printed on Torchon by Hahnemuhle. This is an acid free cotton rag watercolor paper with a parchment or pastel paper surface. Especially suitable for reproductions of oil, watercolor and pastel paintings. The printed side of this paper has been specially coated for excellent image sharpness and optimum color graduation. The coating also offers a very high level of water resistance.


Hello all - I'd greatly appreciate some help in making a destination decision.
I plan to go with the whole family (6 of us, the youngest being 19, a 23 year old and 2x25 year olds, so no young'uns) to either Central America, Caribbean or Mexico. We would like to have a beach vacation at an all inclusive but have at least a couple of days off of the resort seeing nature. Rainforests, rivers, animals, birds - that type of stuff. We're looking for a fairly luxurious resort, but not over the top...ie: 4 star is good. We will be going for 1 week only, and will be going at the end of December - leaving anywhere between December 26th and January 1st.
I've heard about Costa Rica, Honduras (Roatan), Nicaragua for Central America, and I'm not that familiar with Mexico. 7 days on the beach would be too boring for us as the family likes adventure, but we would also like to have some downtime at a beautiful beach.
I just want to point out that you are traveling in the very highest of the high seasons in the Caribbean and anywhere on the coast of Mexico, esp the Caribbean coast. The high Christmas season extends from about Dec. 20 until Jan. 10. Rates will be at their highest of the entire year at that time. Just want to point that out in case you didn't understand that. Rates can be 50%, 75%, to 100% higher than any other time of year (aside from Semaana Santa) then.
I am not at all familiar with Central America, but hopefully rates do not jump as high there as they do in Caribbean and Caribbean of MX then.
You do not mention ancient sites so perhaps The Yucatan is not for you.
In any event if you are staying at AI it does not really matter all that much which beach you choose IMO.
Have you thought about Belize? There are a number of AIs on Ambergris Caye. Although the beaches have sea grass, the snorkeling and fishing are great. Many offer day excursions to Mayan ruins in the rain forest, birdwatching, cave exploring, river trips, etc.
Your descriptions sound a lot like Costa Rica. But I'm not sure about the availability of "all inclusives" there?
ֿCancun/ Playa Del Carmen in Mexico is nice. There isn't really that much animal stuff there from what I know of, but there is a lot to do there. (From Water Stuff to ATVs, Ziplines, horseback riding. There is also ancient sites. ruins there that sound good but haven't been.) The beaches are also nice. I know that Playa Del Carmen is in the Caribbean, but not sure about Cancun. Also as all inclusive, there are A LOT there!! Playa Del Carmen is good because it is close to some of the wilderness/ forest type activities, has a nice beach, and is close to the airport. (Cancun Airport) In Cancun, there is an area called the Hotel Zone, with a bunch of hotels. I've bean to Cancun a bunch of times and it is nice. Another is is Cabo San Lucas of Mexico but haven't been there. There aren't as many hotels, though. I don't know much about The Caribbean, but I've bean to Punta Cana in the Dominican Republic before and there is nice beaches, a lot to do (not as much as Cancun), nice beaches, and lots of hotels. (Like Cancun/Playa Del Carmen, a lot All Inclusive.


text:
Haiti's First Electric Powered Moto!
DigitalKap recently announced the launch of its sister company, So-Mob! So-Mob, with its slogan "Mobility Fueled by Solar", is a first of its kind in Haiti, electric powered motorcycle. These motorcycles have been designed and optimized to be charged by solar energy. DigitalKap offers charging station solar kits for purchase and use at home along with the motorcycles.
This endeavor has been in the works for over a year, but in light of the current fuel crisis in Haiti, it couldn't have come to fruition at a more important time. Before So-Mob there was no other option to gasoline or diesel fuel dependence when it came to transportation. We are proud to be pioneering a green transportation alternative in Haiti and to show Haitians that there are other choices when it comes to getting around.
You can learn a little more about So-Mob at so-mob.com, following on social media (@ridesomob) and by viewing our launch video on Facebook.
#solar energy in Haiti
#electric motorcyle
#electric vehicle


text:
Турпа́н білокрилий (Melanitta fusca) — водоплавний птах, що гніздиться на крайній півночі Європи та Азії на захід від Єнісею. Невелика ізольована популяція цього виду є в східній частині Туреччини. В Україні рідкісний під час сезонних міграцій та на зимівлі. Східносибірський, або американський, вид Melanitta deglandi часто також вважають конспецифічним із турпаном білокрилим і розглядають як його підвид (M. f. deglandi).

Опис
Маса тіла 0,8—2,0 кг, довжина тіла 51—58 см, розмах крил 90—99 см. Дорослий самець у шлюбному вбранні чорний, з зеленкувато-фіолетовим металічним полиском; смуга під оком і «дзеркальце» білі; здуття основи верхньої щелепи чорне, решта дзьоба жовтогаряча; ноги червоні; райдужна оболонка ока білувата; у позашлюбному вбранні з'являються окремі бурі пера. Доросла самка темно-бура; перед і за оком білуваті плями; дзьоб темно-бурий; ноги червонуваті; райдужна оболонка ока коричнева. Молодий птах подібний до дорослої самки, але світліший; груди і черево білуваті.

Від синьги відрізняється білим «дзеркальцем» та червоним забарвленням ніг; дорослий самець від дорослого самця синьги — також білуватою райдужною оболонкою ока та білою смугою під оком.

Гніздування

Гніздо будують на землі поблизу озер, річок або узбережжя. У тайзі воно розташоване не далі 100 м від відкритого водоймища. У кладці зазвичай від 7 до 9 яєць. Яйця овальної форми біло-кремового кольору. Виводки нерідко об'єднуються, і одна самка водить каченят двох або трьох різних вікових груп, тому число пташенят буває вдвічі більшим, ніж в одному виводку. Розвиток пташенят йде досить повільно й починають літати не раніше середини вересня.

Живлення
Живиться молюсками, дрібною рибою, рідше комахами, їх личинками та рослинною їжею (наприклад, рдестом). За кормом пірнає на глибину до 10 м, залишаючись під водою до 1 хвилини.

Охорона
Турпан перебуває під охороною відповідно до Угоди AEWA.

Посилання

Крехові
Птахи України
Птахи Європи
Птахи Азії
Птахи, описані 1758


So, when the media decides not to cover serious people making a serious effort, why would anyone be surprised that some candidates resort to tactics of outrage or stunt? Too often Libertarians have been ignored – and some find a way to reach the public.
Such was my thought when my friend and colleague U.S. Senate hopeful Brian Ellison, in an effort to bring to light a subject that matters to him, advocated his plan "Arm the Homeless". For a media that hasn't given his candidacy any attention, this caught the imagination of media between political scandals – whether a long-serving Congressman whose #metoo was really an admission of past crimes – to the many-sided lineup of silliness coming from Washington these days – Brian's suggestion simply provided a brief distraction from all that is wrong with politics.
Politics has long been full of proposals designed to garner attention – with no real expectation that public policy would be formed. Perhaps the most well known is Jonathan Swift's "A Modest Proposal". And to some, arming our homeless population may seem as ridiculous as feeding the babies of poor people to the rich.
As most people go about their day-to-day lives, but for turning a blind eye when confronted with a homeless person, most Americans don't give them much thought. Perhaps the thought of all the unclaimed Doe's resident in the coroner's office is a fact too few Americans choose to give a damn about – but, let's at least give credit to Brian for raising a few eyebrows and maybe a little awareness about this population.
Maybe, just maybe, a more serious interview of Brian's background would reveal his service to the country in the military. Concern for the reality that so many veterans are homeless and living in despair ought to be somewhere on the agenda of a serious media. That so many are left defenseless is a metaphor, stupid!
As a candidate for Governor, I'm not prepared to endorse providing shotguns to the homeless. But, please stop acting like Libertarians are the only folks tossing out unusual solutions to complicated and intractable problems.
Remember Midnight Basketball? Did that solve the gang problems plaguing our cities? Not even the most ardent supporter of Bill Clinton's agenda would argue that this was even marginally effective. And these days we talk about investing billions to build a border wall. Why not just go straight to building pyramids to distract the people? By comparison, Brian's rationale seems quite grounded: At least by taking a homeless person through a program of training we acknowledge their existence – and maybe by investing our time and resources we can remove some of the stink of our ongoing neglect.
If homelessness and despair were a simple problem to solve, some crafty politician would have done so long ago. But before you can marshal appropriate resources, you must admit there is a problem. Let's consider Brian's suggestion at least as the beginning of our public 12-step program.
The question for our fellow citizens lay in Step No. 8. Are we a good enough people to make amends with those we've wronged? Put your outrage in your pocket – and either lead, follow, or get the hell out of the way.
Some media outlets have finally begun to understand the liberty movement and give candidates an opportunity to offer serious ideas. If a major party candidate for the United States Senate can't get a fair hearing of his ideas, then the result is a candidate finding the circus and trying to play ringmaster.
Will the media play a constructive role in facilitating debate with a serious Libertarian candidate? Or, will they try to find the unusual, attention-getting byline to present their own idea of the crazy Libertarian. Unless you're willing to dig deeper, spare us the outrage.


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