text:
\section{Introduction}

In physics of real materials edge effects often play an important role. One of the principle examples is the quantum Hall state, for which the sample edges are the loci of all transport in the system, while the bulk of the material is insulating. Another interesting example is the Josephson effect, namely a current across an interface of two superconductors. These phenomena have motivated the authors to study the holographic description of edge or interface physics. (For earlier results in this direction see~\cite{QHall1,QHall2,QHall3,QHall4,Gorsky:2001iq,QHall5,QHall6,QHall7,QHall8,QHall9,QHall10,QHall11,QHall12,QHall13,JJ1,JJ2,JJ3,JJ4,JJ5,JJ6,JJ7,JJ8} and references therein.)

In~\cite{taka} Takayanagi proposed an extension of the AdS/CFT correspondence~\cite{maldacena}, to the case in which the CFT is defined in a space with a boundary, a Boundary Conformal Field Theory (BCFT), \emph{cf.}~\cite{cardy}. The correspondence was dubbed AdS/BCFT. The idea behind the proposal was an appropriate extension of the CFT boundary inside the bulk of the $AdS$ space. The extension (boundary profile in the bulk space, which we henceforth label $Q$) should be dynamical, \emph{e.g.} governed by a variational principle. As proposed in~\cite{taka}, the dynamical feature can be attained via supplementing the variational principle with Neumann boundary conditions on $Q$.

The Neumann boundary conditions used in~\cite{taka} and most of other works imply that the renormalized stress-energy tensor on $Q$ vanishes. This condition is quite restrictive. In particular it appears to be difficult to extend the known solutions of Einstein equations in empty AdS to the case of non-zero temperature. The exercise can be done in the case of $AdS_3$~\cite{taka2,taka3}, but higher dimensional generalizations are still unknown. However, given a boundary condition, one can build a solution perturbatively, as demonstrated in~\cite{taka3}.

In this paper we explore an alternative path. Instead of modifying the bulk geometry we modify the boundary condition itself. This can be done by tuning the energy-momentum tensor residing on $Q$, so as to allow the bulk solution of interest. A modified energy momentum tensor naturally appears when considering some extra degrees of freedom (matter) living on $Q$, which are then expected to be thermally excited by interaction with the black hole thermal radiation. This is very close in spirit to the approach of the fluid/gravity correspondence~\cite{fluidgravity}, which states the equivalence of the certain bulk gravity dynamics and the equations of motion of the dual field theory in the hydrodynamical regime. The connection is clearer if noticed that the gravity definition of the boundary stress-energy tensor, used in fluid/gravity correspondence is equivalent to the Neumann boundary condition for the metric. Therefore one can adopt the fluid/gravity framework in order to study the AdS/BCFT problem.

Within this framework we describe a family of boundary stress-energy tensors $T_{ab}$ residing on $Q$, consistent with the simple $AdS_4$ Schwarzschild black hole in the bulk. Each of the $T_{ab}$ corresponds to a hypersurface in the bulk that bounds a subspace of the black hole solution. We consider the hypersurfaces that preserve all but one spatial translation symmetries. From the point of view of the AdS/BCFT correspondence this is an AdS/CFT problem in a half-space or an infinite strip.

It is well-known in the fluid/gravity correspondence that constant radius slices of the $AdS$ (black hole) space, lead to the stress-energy tensor of a conformal fluid. In this work we are rather interested in those hypersurfaces that extend along the radial direction, from the boundary into the bulk. As a result we find that the corresponding $T_{ab}$ do not generally take the simple fluid-like form. Specifically, in the local rest frame, we find that the stress tensor is not proportional to the unit matrix. This suggests a natural condition to discriminate between different profiles of $Q$, which is to look for energy momentum tensors of the fluid-like form, in local thermal equilibrium with the black hole radiation. We show that there is a unique geometry of the hypersurface that yields a fluid-like $T_{ab}$. Remarkably, the equation for the hypersurface is integrable with the solution given by an elliptic integral.

We further study properties of the special fluid-like solution. The fluid on the boundary extension is subject to a  curved metric, or equivalently to an external field. As a result, the thermodynamic quantities, except entropy density, are coordinate dependent. Nevertheless at every spatial position the fluid has the same equation of state as the well-known conformal fluid defined at the corresponding energy scale. The total entropy of the boundary fluid is consistent with the Bekenstein-Hawking formula. It is proportional to the area of the horizon swept by the boundary hypersurface. Although our work is aiming at the study of edge and interface physics, the solution described in this article might be of interest for the fluid/gravity program itself, as the one exploring all energy scales of the dual field theory.

We also analyze the thermodynamics of the BCFT. The thermodynamics consists of the ``bulk'' and ``boundary'' contributions. The tension of the boundary hypersurface sets a characteristic scale for the corresponding thermodynamical quantities. Computing the boundary contribution to the total action we find that its associated entropy is not equal to the contribution coming from the Bekenstein-Hawking formula alone, as \emph{e.g.} in the 3-dimensional case in~\cite{taka}.

The paper is organized as follows. In section~\ref{sec:AdS-BCFT} we review the AdS/BCFT construction of Takayanagi. In section~\ref{sec:fluid} we describe a family of hypersurfaces and stress-energy tensors consistent with the $AdS_4$ Schwarzschild black hole geometry. We find that the fluid-like condition uniquely defines the geometry of the profile $Q$, and we discuss properties of the boundary fluid. In section~\ref{sec:blackhole} we compute the free energy of the BCFT and derive the boundary contribution to the entropy. We conclude in section~\ref{sec:conclusions}, where we speculate on the dual field theory interpretation and possible applications of our results. In the appendices we review earlier results on AdS/BCFT and fluid/gravity duality. Appendix~\ref{sec:examples} reviews some known solutions of AdS/BCFT, including the BTZ black hole. Appendix~\ref{sec:conformalfluid} overviews the basic example of the conformal fluid in the fluid/gravity framework.

\section{AdS/BCFT}
\label{sec:AdS-BCFT}

The AdS/BCFT correspondence proposed by Takayanagi in \cite{taka} is a suggestion for the gravity dual of a $d$-dimensional CFT defined in a space $M$ with a boundary $P$.\footnote{See the original work of Takayanagi as well as more detailed reviews~\cite{taka2,taka3}. Also see other examples of the AdS/CFT correspondence for the CFT's with boundaries~\cite{AdSCFTb}.} Such CFT's are called boundary CFT's, or BCFT's, if the boundary preserves the $SO(2,d-1)$ subgroup of the $d$-dimensional $SO(2,d)$ conformal group \cite{cardy}.

\begin{figure}
 \centering
 \begin{minipage}{0.45\linewidth}
 \includegraphics[width=\linewidth]{./AdSBCFT.pdf}
 \end{minipage}
 \hfill{
\begin{minipage}{0.45\linewidth}
 \includegraphics[width=\linewidth]{./profile0.pdf}
\end{minipage}
}
\caption{Left: $N$ is the subspace of the bulk of $AdS_{d+1}$, bounded by $Q$. It ``encodes'' physics of $M$. $P$ is the common boundary of $Q$ and $M$. Right: solutions to the AdS/BCFT problem in empty AdS-space are hyperplanes foliating $AdS_{d+1}$ in $AdS_d$ slices.}
\label{fig:AdS/BCFT}
\end{figure}

It is natural to think that the boundary $P$ must be extended to the bulk to cut out a region of the bulk that ``encodes'' the physics of the BCFT. This extension is labelled by $Q$, figure~\ref{fig:AdS/BCFT} (left), while $N$ will label the part of the bulk bounded by $Q$ and $M$: $\partial N=M\cup Q$ and $P=\partial M=\partial Q$. It is also natural to demand that the profile of the boundary $Q$ is determined dynamically. Dynamics can be introduced through the specification of the boundary conditions of the variational problem. For it to be non-trivial one has to choose Neumann boundary conditions for the boundary $Q$ as opposed to the Dirichlet boundary conditions on $M$. Consider the following (4-dimensional) action
\begin{multline}
\label{action}
I =  \frac{1}{2\kappa}\int_N {\rm d}^4x\, \sqrt{-g}(R-2\Lambda)+\frac{1}{\kappa}\int_Q {\rm d}^3x\, \sqrt{-h}(K-\Sigma) +\frac{1}{\kappa}\int_M{\rm d}^3x\, \sqrt{-\gamma}(K^{(\gamma)}-\Sigma^{(\gamma)})+ \\
+ \int_Q{\rm d}^3x\, \mathcal{L_{\rm mat}}  + \Delta I\,.
\end{multline}
Here $\kappa=8\pi G$ is the gravitational coupling constant, $g_{\mu\nu}$ is the bulk metric, $h_{ab}$ and $\gamma_{ij}$ are induced metrics on $Q$ and $M$, $K$ and $K^{(\gamma)}$ are corresponding traces of the extrinsic curvature, $\Sigma$ and $\Sigma^{(\gamma)}$ are tensions of $Q$ and $M$ respectively. $\mathcal{L}_{\rm mat}$ is a Lagrangian of possible matter fields on $Q$. $\Delta I$ is the part of the action that contains possible counter-terms and contact terms, localized on $P$. They do not affect the bulk dynamics and will be introduced later.

The variation of action~(\ref{action}) on-shell amounts to the surface terms
\begin{multline}
\delta I = \frac{1}{2\kappa}\int_Q \sqrt{-h}(K_{ab}-\Sigma h_{ab})\delta h^{ab}+\frac{1}{2\kappa}\int_M \sqrt{-\gamma}(K^{(\gamma)}_{ij}-\Sigma^{(\gamma)}\gamma_{ij})\delta\gamma^{ij} -
\\  - \frac{1}{2}\int_Q{\rm d}^3x\,\sqrt{-h}T_{ab}\,\delta h^{ab} + \delta(\Delta I)\,,
\end{multline}
where $T_{ab}$ is the matter stress-energy tensor on $Q$. Typically, in the variational principle, one imposes the Dirichlet boundary conditions at the boundary. For the AdS/CFT correspondence this means, in particular, that we fix the boundary metric on $M$, $\delta\gamma_{ij}=0$\,. However one can alternatively choose the Neumann boundary conditions. Let us do this for the induced metric on $Q$, that is let us impose
\begin{equation}\bea\ds
\label{Neumann-metric}
K_{ab} - (K-\Sigma) h_{ab} = 8\pi G T_{ab}\,.
\ea\end{equation}
This is the dynamical equation for the induced metric $h_{ab}$, or equivalently, for the profile of the hypersurface $Q$. The solution of the AdS/BCFT then requires a solution of the equations of motion, derived from action~(\ref{action}), which additionally satisfies the Dirichlet boundary conditions on $M$ and Neumann boundary conditions~(\ref{Neumann-metric}) on $Q$.\footnote{Although the AdS/BCFT is formulated in a bottom-up fashion one can embed it in a top-down string theory construction~\cite{taka2}.}

In this work we are interested in studying the holographic models of boundary CFT's at finite temperature. For this we would like to find a solution of the AdS/BCFT problem, which has an asymptotically $AdS_4$  black hole geometry. For simplicity, we will consider the boundary $P$ specified by the condition $y={\rm const}$, where $y$ is one of the coordinates on $M$, figure~\ref{fig:AdS/BCFT} (right), that is we consider the AdS/BCFT problem on a half of Minkowski space.

To solve equation~(\ref{Neumann-metric}) one needs to specify an energy-momentum tensor $T_{ab}$ for the matter fields on $Q$. In the simplest case~\cite{taka} one assumes $T_{ab}=0$.\footnote{Notice that here, as opposed to~\cite{taka}, we do not consider the vacuum term $-\Sigma h_{ab}/{8\pi G}$ to be part of $T_{ab}$.} With this assumption one can solve the problem for the empty AdS case, and for the BTZ black hole in three dimensions. These two cases are reviewed in detail in appendix~\ref{sec:examples}. On the other hand it is easily checked that the AdS-Schwarzschild (or other plane-symmetric) solution does not satisfy~(\ref{Neumann-metric}) with $T_{ab}=0$ for a number of dimensions greater than three, except for $\Sigma=0$. One possibility, explored in~\cite{taka3}, is to construct a solution as a perturbative expansion over the  vacuum AdS solution. This has however the disadvantage of working with complicated metric from the start.

Another possibility, which we explore in this article, is to allow matter on $Q$. Specifically, the objective of this work will be to analyze possible theories on $Q$, which would be consistent with the Neumann boundary conditions and the $AdS_4$-Schwarzschild black hole geometry:
\begin{equation}\bea\ds
\label{ansatz-metric}
{\rm d} s^2 = \frac{L^2}{z^2}\left(- f(z){\rm d} t^2 + {\rm d} x^2 + {\rm d} y^2 + \frac{{\rm d} z^2}{f(z)} \right),
\ea\end{equation}
where the function $f(z)$ is given by
\begin{equation}\bea\ds
f(z)=1-\left(\frac{z}{z_h}\right)^3 \,.
\ea\end{equation}
This solution describes a black hole with the horizon radius $z_h$ and the Hawking temperature
\begin{equation}\bea\ds
\label{HawkingT}
T_H=\frac{3}{4\pi z_h}\,.
\ea\end{equation}

\section{Boundary fluid from AdS/BCFT}
\label{sec:fluid}

As explained in the previous section, there are two approaches to the AdS/BCFT problem. In the first, one fixes the boundary stress (-energy-momentum) tensor, \emph{e.g.} $T_{ab}=0$, or in a more general form, and finds a solution to the bulk Einstein equations, satisfying~(\ref{Neumann-metric}). In the other, one fixes the bulk metric and finds an appropriate matter content on $Q$, parameterized by stress-energy tensors $T_{ab}$, which supports the metric of interest. Put this way the AdS/BCFT problem is in fact equivalent to a holographic duality, between gravity in the bulk and the matter theory on the boundary $Q$, analogous to the AdS/CFT correspondence itself.\footnote{AdS/CFT correspondence with Neumann boundary conditions was previously discussed in~\cite{Compere:2008us}. AdS/BCFT may be considered as an extension of that approach to the case of arbitrary boundaries.} In a particular regime it is equivalent to the fluid/gravity correspondence~\cite{fluidgravity}, which we shall discuss momentarily.

The fluid/gravity correspondence is based on the observed equivalence of the Einstein equations with a negative cosmological constant in the bulk of a space-time and fluid-like equations on a time-like hypersurface, which is considered as a boundary of that space-time. The equations of the fluid are nothing but the statement of the covariant conservation of the stress-energy tensor seen by an observer placed on the hypersurface. In the long-wave hydrodynamical regime this is equivalent to the equations of relativistic hydrodynamics. The correspondence is a duality of two different descriptions of the same physics. By duality, one can reconstruct the bulk metric from a given solution to the hydrodynamical equations and vice-versa.

Given a hypersurface $Q$ the stress-energy tensor $T_{ab}$ residing on it is defined through the variation of the action with respect to the induced metric on $Q$. Naive unrenormalized action may lead to a diverging $T_{ab}$, or physical quantities computed from it. The divergences can be cancelled by subtracting an appropriately chosen vacuum solution (AdS in our case), or equivalently by adding appropriate boundary terms as specified by the procedure of holographic renormalization~\cite{skenderis,vijay}. More specifically the renormalization procedure leads to the following form of $T_{ab}$:
\begin{equation}\bea\ds
\label{Tab}
T_{ab} = -\frac{L^{d-2}}{\kappa z^{d-2}}\left(K_{ab}-K h_{ab}+\Sigma h_{ab} -\kappa T_{ab}^{(R)} - \kappa T_{ab}^{(ct)}\right).
\ea\end{equation}
Notice that this is the ``intrinsic'' stress-energy tensor, which is defined with respect to the intrinsic hypersurface metric,
\begin{equation}\bea\ds
\label{intr-metric}
\hat{h}_{ab}=h_{ab}\cdot \frac{z^2}{L^2}\,.
\ea\end{equation}
In the expression above we have written explicitly the part coming from the Gibbons-Hawking and tension-like terms in~(\ref{action}), while $T_{ab}^{(R)}$ and $T_{ab}^{(ct)}$ are additional possible contributions from the intrinsic curvature and counter-terms respectively.

As follows from the discussion, the fluid/gravity correspondence, within the limit of its validity, is equivalent to AdS/BCFT. The Brown-York type procedure of extracting the dual stress-energy tensor~(\ref{Tab}) can be mathematically formalized as the variational principle for~(\ref{action}) supplemented with Neumann boundary conditions~(\ref{Neumann-metric}). The $\mathcal{L}_{\rm mat}$-term in the action~(\ref{action}) plays a role of a source term with respect to the boundary (bulk) metric. In what follows we identify the stress-energy tensor of the matter fields with the $T_{ab}$ in the left hand side of~(\ref{Tab}). From this perspective, finding a finite temperature solution to the AdS/BCFT problem~(\ref{Neumann-metric}) translates into finding a dual fluid theory living on $Q$ supporting the black hole solution~(\ref{ansatz-metric}).

Considering the problem in a half-space $y<0$, let us parameterize a generic hypersurface $Q$ by the function $y(z)$, figure~\ref{fig:AdS/BCFT} (right), and restrict to the first three terms in~(\ref{Tab}). We then find the boundary stress-energy tensor in the following form
\begin{equation}\bea\ds
\label{Tab-gen}
T_{ab} = \left(
\begin{array}{ccc}
- \hat{h}_{tt}\, \epsilon(z) && \\
& \hat{h}_{xx}p_x(z) & \\
&&  \hat{h}_{zz} p_z(z)
\end{array}
\right),
\ea\end{equation}
where $\hat{h}_{ab}$ is the intrinsic metric~(\ref{intr-metric}) on $Q$ and the following functions have been introduced
\begin{eqnarray}
\label{epsilon}\epsilon(z) & = & \frac{L^2}{2\kappa z^3}\left(2\Sigma L+\frac{(zf'-4f)y'(z)-4f^2y'(z)^3+2zfy''(z)}{(1+fy'(z)^2)^{3/2}}\right) ,\\
p_x(z) & = & \frac{L^2}{2\kappa z^3}\left(-2\Sigma L+\frac{2(2f-zf')y'(z)+f(4f-zf')y'(z)^3-2zfy''(z)}{(1+fy'(z)^2)^{3/2}}\right), \\
p_z(z) & = & \frac{L^2}{2\kappa z^3}\left(-2\Sigma L+\frac{y'(z)(4 f-z f')}{\sqrt{1+fy'(z)^2}}\right).
\end{eqnarray}

Close to the boundary $z\to 0$ the bulk metric approaches that of the empty $AdS$-space. Therefore we expect to recover solution~(\ref{QprofileT}) of appendix~\ref{sec:emptyAdS} in this limit. For this reason we may set $\Sigma L =2\cos\theta$, where $\theta$ is the angle that the profile $Q$ makes with the $y$-axis, figure~\ref{fig:AdS/BCFT} (right). With this condition~(\ref{Tab-gen}) describes a generic theory on the boundary hypersurface $Q$, which supports the Schwarzschild black-hole solution in the bulk.

Interestingly generic $T_{ab}$ does not describe a fluid-like system ($p_x\neq p_z$). One can ask whether the above class of stress-energy tensors can at all describe a fluid. Apparently this is a restriction on the profile $y(x)$. Demanding $p_x=p_z$ one arrives at a particularly simple equation
\begin{equation}\bea\ds
\label{Yeqn0}
 \frac{2f y''(z) + f'y'(z)}{\sqrt{1+fy'(z)^2}} =0\,,
\ea\end{equation}
which can be easily integrated to get
\begin{equation}\bea\ds
\label{Yeqn1}
f{y'}^2 = {\rm const}\,.
\ea\end{equation}
So, the general solution, which yields a fluid-like theory on $Q$, is provided by the profile
\begin{equation}\bea\ds
\label{profileQ1}
y=y_0 + \int\limits_0^z \frac{\cot\theta{\rm d} q}{\sqrt{f(q)}}\,,
\ea\end{equation}
parameterized by the ``contact" angle $\theta$. For $f=1$ the result~(\ref{QprofileT}) of Takayanagi is reproduced, while for the black hole solution $f=1-z^3/z_h^3$ the profiles are shown on figure~\ref{fig_profile1} (left) for different values of $\theta$. In the limit $\theta\to 0$ we recover the well-known results of the conformal fluid (see appendix~\ref{sec:conformalfluid}). In particular, $p_x=p_z=p$ in this limit.

\begin{figure}
 \begin{minipage}{0.45\linewidth}
 \includegraphics[width=\linewidth]{./profile1.pdf}
 \end{minipage}
 \hfill{
\begin{minipage}{0.45\linewidth}
 \includegraphics[width=\linewidth]{./2profile.pdf}
\end{minipage}
}

 \caption{Profiles of the brane $Q$ described by (\ref{profileQ1}) for different angles $\theta$ (left). AdS/BCFT model of an infinite stripe (right). $N'$ is a part of the bulk bounded by the dashed lines. Areas of the horizon ``swept" by the branches of the hypersurface $Q$ are colored with red.}
 \label{fig_profile1}
\end{figure}

More interesting is the behaviour close to the horizon. The profile is regular and has the expansion
\begin{equation}\bea\ds
y= {\rm const} + \frac{2\cot\theta}{\sqrt{3}}\sqrt{1-z}+ O\left((1-z)^{3/2}\right)\,,
\ea\end{equation}
that is it reaches the horizon at a finite value $y_h$. This value gives a characteristic distance scale $\Delta y=y_h-y_0$ set by angle $\theta$. The solution can be extended by adding a second branch of the surface $Q$ as shown in figure~\ref{fig_profile1} (right). This way one gets a configuration, similar to the one found in the BTZ case~\cite{taka}, \emph{cf.} figure~\ref{fig:BTZ} from appendix~\ref{sec:BTZ}. Notice that the walls of the configuration in figure~\ref{fig_profile1} (right) must satisfy $y'(z)>0$ ($y'(z)<0$) for the right (left) branch of the boundary $Q$. Otherwise one will end up with a theory with negative energy density and temperature (null/weak energy condition).

The renormalized energy density and pressure on the hypersurface $Q$ are given by
\begin{eqnarray}
\label{edensity} \epsilon & = &  \frac{2L^2\cos\theta}{\kappa z^3}\left(1-\sqrt{f}\right), \\
\label{pressure} p & = &  \frac{L^2\cos\theta}{2\kappa z^3\sqrt{f}}\left(4f-zf'-4\sqrt{f}\right).
\end{eqnarray}
For a given value of the coordinate $z$ these are the results~(\ref{ezconst}) and~(\ref{pzconst}) for the conformal fluid renormalized by a factor $\cos\theta$, see appendix~\ref{sec:conformalfluid}. Put differently, the two fluids have the same equation of state. The dependence of $\epsilon$ and $p$ on the position on $Q$ (equivalently on coordinate $y$) is demonstrated on figure~\ref{fig:density-pressure} (left).

\begin{figure}[htb]
 \begin{minipage}{0.45\linewidth}
 \includegraphics[width=\linewidth]{./endens.pdf}
 \end{minipage}
 \hfill{
\begin{minipage}{0.45\linewidth}
 \includegraphics[width=\linewidth]{./sound.pdf}
\end{minipage}
}
\caption{Renormalized energy density (blue) and pressure (magenta) in the effective theory on the surface $Q$ (left). Dependence of the square of speed of sound on the position on $Q$ (right).}
\label{fig:density-pressure}
 \end{figure}

As hypersurface $Q$ has a non-flat induced metric, the fluid is subject to a gravitational force, or, in other words, it is subject to an external field. As a result equilibrium thermodynamical quantities depend on the coordinates. The dependence on the $z$-coordinate is interpreted as a renormalization of physical quantities. In this respect our fluid is made of ``dressed" filaments of the same fluid~\cite{Brattan:2011my}. Close to the boundary the system is conformal $\epsilon=2p$. At the opposite end the system is strongly non-conformal: while the energy density is finite, the pressure diverges as $y\to y(z_h)$. Similar behavior is well known in the models of fluid dynamics~\cite{Bludman:1968zz}, see~\cite{Brattan:2011my} for the discussion in the context of fluid/gravity correspondence.

Assuming the local temperature on the surface $Q$ is given by
\begin{equation}\bea\ds
\label{localT}
T=\frac{T_H}{\sqrt{f(z)}}\,,
\ea\end{equation}
the local entropy density is
\begin{equation}\bea\ds
\label{Yentropy}
s=\,\frac{\epsilon+p}{T}= \frac{L^2\cos\theta}{4Gz_h^2}\,.
\ea\end{equation}
We notice that the entropy density is constant over the surface. This is consistent with the general observation that the entropy of the fluid on a hypersurface in the bulk is proportional to the area of the horizon swept by the hypersurface. Indeed, if this entropy density is integrated with respect to the intrinsic metric~(\ref{intr-metric}) on $Q$, we find that the total entropy is

\begin{equation}\bea\ds
\label{Sfluid}
S_Q= \frac{L^2(y(z_h)-y_0)}{4Gz_h^2}\,\Delta x\,
\ea\end{equation}
for each branch of the configuration, consistent with the Bekenstein-Hawking entropy.

Although we have concentrated in four space-time dimensions the results of this section can be easily generalized to the case of the $AdS_{d+1}$-Schwarzschild black hole with $d>3$. There is a unique hypersurface $Q$ satisfying the condition $p_z=p_i$, where $i$ label the spatial coordinates transverse to $y$. The profile of the hypersurface stays defined by equation~(\ref{Yeqn1}) with function $f(z)$ appropriately modified.

\section{Black hole thermodynamics}
\label{sec:blackhole}

Let us derive the equilibrium thermodynamics of the BCFT from the gravity configuration described by figure~\ref{fig_profile1} (right). The free energy of the system is computed from the total Euclidean action. We split the latter in two parts: one, which we call the ``bulk'' contribution, will be independent from the tension parameter (angle $\theta$); and the other, ``boundary" contribution, will be $\theta$-dependent. The bulk part of the total (renormalized) action reads
\begin{equation}\bea\ds
\label{Sbulk}
I_{\rm bulk}= -\frac{1}{16\pi G}\int_{N'} {\rm d}^4x\, \sqrt{g_E}(R-2\Lambda) - \frac{1}{8\pi G}\int_M \sqrt{\gamma}(K^{(\gamma)}-\Sigma^{(\gamma)})= -\frac{L^2}{12Gz_h^2}\,V\,,
\ea\end{equation}
where the first integral is taken over the subspace $N'$ of the bulk space $N$, which excludes the two wedges underneath the hypersurface $Q$, figure~\ref{fig_profile1} (right). $V$ stands for the volume of the BCFT. In this calculation the tension of the boundary $M$ must be selected to be $\Sigma^{(\gamma)}=2/L$. The renormalization procedure here is equivalent to subtracting the action of the vacuum AdS solution. In both ways one derives the standard AdS/CFT result.

The $\theta$-dependent part consists of two pieces: the contribution from the bulk of the wedges\footnote{In this calculation one may find useful formulae for the integrals presented in appendix~\ref{sec:appendix}.}
\begin{multline}
\label{SbulkQ}
2\times\frac{6\beta L^2}{16\pi G }\int {\rm d} x\, \int\limits_{\epsilon}^{z_h} \frac{{\rm d} z}{z^4}\int\limits_{y_0}^{y(z)} {\rm d} y = \frac{z_h L^2}{G } \,\cot\theta\int {\rm d} x\, \int\limits_{\epsilon}^{z_h} \frac{{\rm d} z}{z^4}\int\limits_0^z\frac{{\rm d} q}{\sqrt{f(q)}} =
\\ = -\frac{z_h L^2}{3G } \, \Delta x \left(\frac{y(z_h)-y_0}{z_h^3}-\frac{y(\epsilon)-y_0}{\epsilon^3}-\frac{y(z_h)-y_0}{4z_h^3}-\frac{\cot\theta}{2\epsilon^2}+O(\epsilon)\right)
\end{multline}
and the boundary action part
\begin{multline}
\label{SQ}
2\times -\frac{1}{8\pi G}\int {\rm d}^3x\, \sqrt{h_E}(K-\Sigma) = \frac{z_hL^2}{6G}\, \cot\theta\int{\rm d} x \int\limits_\epsilon^{z_h}\frac{{\rm d} z}{z^3}\,\left(4+\frac{zf'(z)-6f}{\sqrt{f(z)}}\right)=
\\ = -\frac{L^2\Delta x}{3Gz_h}\,\cot\theta\left(1-\frac{z_h^2}{\epsilon^2}\right) - \frac{L^2\Delta x}{2Gz_h^2}\left(y(z_h)-y(\epsilon)\right) - \frac{z_hL^2}{G}\,\Delta x\left(-\frac{3(y(z_h)-y_0)}{4z_h^3}+\frac{\cot\theta}{2\epsilon^2}+O(\epsilon)\right).
\end{multline}

The bulk action possessed cubic divergences, which were cancelled by the Gibbons-Hawking term on the boundary $M$. The boundary contribution has quadratic divergences. Adding~(\ref{SbulkQ}) and~(\ref{SQ}) do not remove the divergence completely, only up to the $(y(\epsilon)-y_0)/\epsilon^3$ term. The remaining divergence is the UV divergence of the Takayanagi's vacuum solution~(\ref{QprofileT}) reviewed in appendix~\ref{sec:emptyAdS}. We also observe a similar (linearly-divergent) term in the lower-dimensional example, as reviewed in the appendix~\ref{sec:BTZ}, equation~(\ref{Ibdry1}).

The remaining divergence can be removed by a counter-term at the boundary $P=Q\cap M$ of the CFT:
\begin{equation}\bea\ds
-\,\frac{1}{4\pi G}\int_P{\rm d}^2x\,\sqrt{\sigma}\cot\theta\,,
\ea\end{equation}
where $\sigma_{ij}$ is the induced metric on $P$. Adding extra counter-terms on $P$ does not affect the dynamics discussed before.\footnote{This counter-term would be taken into account automatically should one renormalize the action via the subtraction of the vacuum (empty AdS) solution.} One should also remember that since the boundary $P$ between $Q$ and $M$ is not smooth one has to add an extra counter-term
\begin{equation}\bea\ds
\,\frac{1}{8\pi G}\int_P{\rm d}^2x\,\sqrt{\sigma}\,\theta\,
\ea\end{equation}
to cancel the divergency associated with the discontinuity of the normal vector across $P$~\cite{Hayward:1993my}. Indeed, the extrinsic curvature $K$ depends on the second derivative of the profile $y(z)$. Since the profile function is not smooth, the second derivative contains a delta-function with support on $P$. The above (infinite) counter-term merely cancels the contribution of this delta-function. The importance of the second counter-term was stressed in~\cite{taka3}. Specifically the stress-energy tensor defined on $P$ needs this term to correctly compute the Weyl anomaly in the AdS$_3$/CFT$_2$ case.

After all the divergencies are taken care of one arrives at
\begin{equation}\bea\ds
\label{Sbdry}
2I_{\rm bdry}= - \frac{L^2\cot\theta}{3Gz_h}\,\Delta x \,.
\ea\end{equation}
Now the free energy of the configuration illustrated by figure~\ref{fig_profile1} (right) reads
\begin{equation}\bea\ds
\label{freee}
F=T_HI_E = T_HI_{\rm bulk} +2T_HI_{\rm bdry}= -\frac{L^2V}{16\pi G z_h^3} - \frac{L^2\cot\theta}{4\pi G z_h^2}\,\Delta x\,.
\ea\end{equation}
From the free energy one can compute the entropy
\begin{equation}\bea\ds
\label{Snew}
S\equiv S_{\rm bulk} + 2S_{\rm bdry} = -\frac{\partial F}{\partial T_H}= \frac{L^2V}{4Gz_h^2}+ 2\,\frac{L^2\cot\theta}{3G z_h}\,\Delta x\,,
\ea\end{equation}
and the internal energy of the configuration,
\begin{equation}\bea\ds
E\equiv E_{\rm bulk} + 2E_{\rm bdry}=\frac{\partial I}{\partial \beta}= \frac{L^2V}{8\pi G z_h^3} + \frac{L^2\cot\theta}{4\pi G z_h^2}\,\Delta x\,.
\ea\end{equation}
The above expressions satisfy the thermodynamical relation $F=E-T_HS$.

We note that as far as the boundary contribution is concerned the above entropy is not the one of the Bekenstein-Hawking formula~(\ref{Sfluid}). While it is the case for the AdS/BCFT of the BTZ black hole, appendix~\ref{sec:BTZ}, here we find that $S_Q\simeq 1.05S_{\rm bdry}$. We are not aware of the reason, why the two entropies should be the same, or approximately the same. It would be interesting to derive the relation between them. For example, one can notice that the above boundary free energy can be calculated from the fluid data as follows
\begin{equation}\bea\ds
2F_{\rm bdry}= - \int_Q p\sqrt{f}\,{\rm d} V\,,
\ea\end{equation}
where the volume integration should be performed with respect to the intrinsic metric~(\ref{intr-metric}).

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

In this work we have studied the AdS/BCFT problem at finite temperature. We looked for the solutions of the AdS/BCFT boundary conditions consistent with the planar $AdS$-Schwarzschild geometry in the bulk. Such solutions can be found if certain matter field fields are introduced on the AdS/BCFT boundary $Q$. The solutions can be classified by the stress-energy tensor of the matter theory. As the main result of our investigation we have noticed that among a continuum of the profiles of $Q$ (or stress-energy tensors $T_{ab}$) there is a single profile, for which gravity yields a fluid-like stress-energy tensor, in local thermodynamic equilibrium with the black hole radiation. The profile function in this case happens to be integrable, provided by an elliptic integral.

From the perspective of this special solution, the AdS/BCFT problem turns out to be equivalent to the problem of the fluid/gravity correspondence. Indeed, one can understand the fluid/gravity correspondence as a problem of finding a dual description of the bulk gravity theory in terms of a given fluid living on a boundary of the bulk space. The stress-energy tensor of the fluid in this case acts as a source for the boundary metric.

Having noticed the equivalence with fluid-gravity correspondence we have analyzed the matter theory on $Q$ with ideal-fluid-like $T_{ab}$. We have found that this special solution yields the same ``conformal'' fluid of the standard fluid/gravity calculation in the $AdS$-Schwarzschild geometry, albeit with thermodynamical quantities dependent on position (inhomogeneous fluid). We have computed the local quantities describing this boundary fluid and shown that they satisfy the first law of thermodynamics, describing a fluid in local thermal equilibrium with the Hawking radiation of the black hole. In particular, we have found that the entropy density does not depend on the coordinates and is consistent with the Bekenstein-Hawking formula, which tells that the total entropy is proportional to the area of the horizon swept by the hypersurface $Q$.

The coordinate dependence reflects the fact the fluid is subject to an external force with thermodynamical quantities renormalized according to the position. Our fluid explores then all energy scales of the boundary field theory, in particular that of the boundary edge $P$. This is a novel feature which we have not found in the literature and that we would like to understand better in future work. It might be interesting for the fluid/gravity program itself.

Finally we have computed the free energy of the BCFT, its entropy and internal energy. They satisfy the thermodynamic relation $F=E-TS$. Interestingly, although the boundary entropy does not exactly coincide with the Bekenstein-Hawking entropy, proportional to the horizon area swept by the profile $Q$, the difference between them is rather small. We remark that there is no \emph{a priori} reason for this to be the case, as the boundary entropy of $P$ is not directly related with the black hole entropy. The latter correctly reproduce the entropy of the fluid living on $Q$. It is nevertheless interesting to better understand the relation between the two.

We believe that the most interesting direction to develop from our results is the extension of this AdS/BCFT framework to a model of interface physics\footnote{Also see the recent developments on the holographic constructions of boundary/defect CFTs~\cite{Defects1,Defects2,Defects3,Defects4,Defects5,Defects6}.}. Indeed in the Takayanagi's construction there is no part of the bulk space beyond $N$, the part of interest. It seems quite natural to extend this correspondence to an interface problem, where the bulk space exists on both sides of the boundary $Q$, and the interface separates two distinct phases, such as holographic superconductors~\cite{Gubser:2008px,Hartnoll:2008vx} with different order parameters, simulating a holographic Josephson junction, or phases at different filling fraction, in the quantum Hall effect, \emph{cf.}~\cite{QHall7,QHall9,Defects4}. Indeed, an interesting feature of the solution concerning these problems is that the geometry of the profile $Q$ defines a finite ``width'' associated with the interface $P$. We expect this width to be related to some potential barrier of the boundary interface, such as a penetration length or escape energy from $P$. The volume associated to this width become parameter-dependent, which indicates that such a system is a candidate for a model of a gapped system with a gapless boundary.

\vspace{0.5cm}
\paragraph{Acknowledgements.} The authors would like to thank I.~Gordeli, Y.~Oz and P.~Sodano for useful conversations and T.~Takayanagi for correspondence. DM gratefully acknowledges the hospitality of the Freiburg Institute for Advanced Studies, where the part of this work was done. The work was supported by the Brazilian Ministry of Science, Technology and Innovations and the graduate program of the DFTE-UFRN. The work of DM was also supported by the Science without Borders program of the CNPq-Brazil, MIT-IIP exchange program, by the Russian RFBR grant 14-02-00627 and the grant for support of Scientific Schools NSh~1500.2014.2.

\begin{appendix}

\section{Previous examples of AdS/BCFT}
\label{sec:examples}

In this appendix we review the solutions to~(\ref{Neumann-metric}) found in~\cite{taka} for the empty AdS case and for the $AdS_3$ black hole. The first case describes the asymptotic behavior of solution~(\ref{profileQ1}) and it is responsible for the divergence we discuss in section~\ref{sec:blackhole}. The second solution is a lower dimensional  analog of solution~(\ref{profileQ1}).

\subsection{Solution in empty $AdS$}
\label{sec:emptyAdS}

Let us recall how the AdS/BCFT construction works in the case of empty $AdS$ space, \emph{i.e.} the $f(z)=1$ case of metric~(\ref{ansatz-metric}). We are looking for the solution of the AdS/BCFT equations for the boundary CFT defined on a half-space $y<0$. Although we review the 4-dimensional case, $d+1$-dimensional generalization is straightforward~\cite{taka}.

In the situation with no extra matter on the boundary $Q$, equation~(\ref{Neumann-metric}) becomes
\begin{equation}\bea\ds
\label{TakayanagiEq}
K_{ab}-(K-\Sigma)h_{ab}=0\,.
\ea\end{equation}
In empty $AdS$ space ($T=0$) this equation can be solved with tension $\Sigma\neq 0$.  Let us parameterize $Q$ as $y=y(z)$. The unit normal vector $n^\mu$ to $Q$ for a metric of the form~(\ref{ansatz-metric}) is given by the following expression.
\begin{equation}\bea\ds
\label{normalQ}
(n^t,n^x,n^y,n^z)=\left(0,0,\,\frac{z}{L\sqrt{1+f(z)y'(z)^2}}\,,\,-\frac{z f(z)y'(z)}{L\sqrt{1+f(z)y'(z)^2}}\right).
\ea\end{equation}
The sign of the normal vector is fixed by the requirement that the vector is pointing outside of the space $N$ (figure~{\ref{fig:AdS/BCFT}}). The above sign corresponds to the situation $M: \{(t,x,y)|y\leq0\}$. In this convention the extrinsic curvature is computed as
 \begin{equation}\bea\ds
 \label{ExtCurv}
 K_{\mu\nu}= {h_\mu}^{\alpha}{h_\nu}^{\beta}\nabla_\alpha n_\beta\,,
 \ea\end{equation}
 where ${h_\mu}^{\nu}$ is the projector onto $Q$:
 \begin{equation}\bea\ds
 \label{projector}
 {h_\mu}^\nu = {\delta_\mu}^\nu - n_\mu n^\nu\,.
 \ea\end{equation}
The 3-dimensional tensors $K_{ab}$, $h_{ab}$ can be obtained via the projector
 \begin{equation}\bea\ds
 h^{\mu}_a = \frac{\partial x^\mu}{\partial\tau^a}\,, \qquad e.g. \qquad K_{ab}= h^{\mu}_a h^{\nu}_b K_{\mu\nu} = h^{\mu}_a h^{\nu}_b \nabla_\mu n_\nu\,,
 \ea\end{equation}
where $\tau^a=\{t,x,z\}$ is a parametrization of $Q$.

One can readily find that~(\ref{TakayanagiEq}) is solved by
\begin{equation}\bea\ds
\label{Qtension}
y'(z)=\, \frac{L\Sigma}{\sqrt{4-L^2\Sigma^2}}\,.
\ea\end{equation}
Introducing $L\Sigma=2\cos\theta$ one obtains
\begin{equation}\bea\ds
\label{QprofileT}
Q\,:\quad y=z\cot\theta\,,
\ea\end{equation}
where $\theta$ is the angle between the positive direction of the $y$-axis and the hyperplane $Q$, or more generally, the tangential to $Q$ hyperplane at $z=0$, as shown on figure~\ref{fig:AdS/BCFT}.

Let us notice that the collection of half-hyperplanes~(\ref{QprofileT}) parameterized by the angle $\theta$ describes the foliation of the $AdS_4$ space in $AdS_3$ planes.  Indeed, the induced metric on $Q$ is just
\begin{equation}\bea\ds
{\rm d} s^2 = \frac{L^2}{z^2}\left(-{\rm d} t^2 + {\rm d} x^2 +\frac{{\rm d} z^2}{\sin^2\theta}\right).
\ea\end{equation}
This way the embedding of $SO(2,2)$ into $SO(2,3)$ is realized geometrically.

For $T\neq 0$ there are no solutions to~(\ref{TakayanagiEq}) except for the tensionless brane $\Sigma=0$, which corresponds to
\begin{equation}\bea\ds
\label{Qzerotension}
Q\,:\quad y={\rm const}\,, \qquad \text{or} \qquad \theta=\frac{\pi}{2}\,.
\ea\end{equation}
However, in the $d=2$ case black-hole solutions of the AdS/BCFT problem with $\Sigma\neq 0$ do exist. Let us now review this case.

\subsection{BTZ black hole}
\label{sec:BTZ}

The problem of finding a boundary surface $Q$ for a black hole geometry was solved by Takayanagi in the $AdS_3/CFT_2$ case~\cite{taka}. The BTZ black hole~\cite{btz},
\begin{equation}\bea\ds
{\rm d} s^2 = \frac{L^2}{z^2}\left(-f(z){\rm d} t^2 + {\rm d} y^2 +\frac{{\rm d} z^2}{f(z)} \right), \qquad f(z)=1-\frac{z^2}{z_h^2}\,,
\ea\end{equation}
has a temperature given by
\begin{equation}\bea\ds
T_{BTZ}=\frac{1}{2\pi z_h}\,.
\ea\end{equation}
The solution to equation~(\ref{TakayanagiEq}) for the profile $y(z)$ in this case is
\begin{equation}\bea\ds
y(z)=y_0 + z_h\,{\rm arcsinh}\left(\frac{z}{z_h}\cot\theta'\right)\,, \qquad \cos\theta'=L\Sigma\,,
\ea\end{equation}
where $\theta'$ has the same meaning as $\theta$ in the previous section (figure~\ref{fig:BTZ}). For $z\to 0$, one reproduces the result of empty $AdS$~(\ref{QprofileT}). For $z\to z_h$ the profile enters the horizon at a finite angle, $y=y_0+\Delta y+\cos\theta'(z-z_h)$. Angle $\theta$ (tension), sets a characteristic distance scale $\Delta y=z_h{\rm arcsinh}(\cot\theta')$. It is natural to associate this scale with the ``width" of the boundary in the CFT.

\begin{figure}
 \centering
 \includegraphics[width=0.5\linewidth]{./profileBTZ.pdf}

 \caption{Profile of the boundary $Q$ in the case of the BTZ black hole.}
 \label{fig:BTZ}
\end{figure}

We also notice that the profile of $Q$ is open in the direction of the horizon, that is $\theta\leq\pi/2$ for the right branch in figure~\ref{fig:BTZ} and $\theta\geq\pi/2$ for the left branch. This is a consequence of a ``null (weak) energy condition" on $Q$. One has to demand this in order for the temperature or energy density on $Q$ to be non-negative.

In~\cite{taka} the boundary entropy associated to this solution was computed. One first finds the full Euclidean action
\begin{multline}
I_{E}= -\frac{1}{2\kappa}\int_N {\rm d}^3x\, \sqrt{g_E}(R-2\Lambda) - \frac{1}{\kappa}\int_M {\rm d}^2x\, \sqrt{\gamma_E}(K^{(\gamma)}-\Sigma^{(\gamma)})  -\frac{1}{\kappa}\int_Q {\rm d}^2x\, \sqrt{h_E}(K-\Sigma)+\ldots=
\\ = I_{\rm bulk} + 2 I_{\rm bdry}+\ldots\,,
\end{multline}
where the ``bulk'' part, which does not depend on $\Sigma$, was separated from the $\Sigma$-dependent contribution of $Q$, and ellipses stand for the counter-terms necessary to remove the divergencies. The bulk part is given by
\begin{multline}
I_{\rm bulk}= -\frac{\beta}{2\kappa}\int\limits_{\epsilon}^{z_h}{\rm d} z\int\limits_{-\Delta y/2}^{\Delta y/2}{\rm d} y\,\sqrt{g_E}(R-2\Lambda)  - \frac{\beta}{\kappa}\int\limits_{-\Delta y/2}^{\Delta y/2}{\rm d} y\, \sqrt{\gamma_E}(K^{(\gamma)}-\Sigma^{(\gamma)})=
\\ = \frac{Lz_h}{2G}\Delta y\int\limits_{\epsilon}^{z_h}\frac{{\rm d} z}{z^3}  - \frac{Lz_h}{4G}\,\Delta y \,\frac{\sqrt{f(\epsilon)}}{\epsilon^2} = - \frac{L}{4G}\,\Delta y\left(\frac{1}{z_h}-\frac{z_h}{\epsilon^2}\right)- \frac{L}{4G}\,\Delta y \left(\frac{z_h}{\epsilon^2}-\frac{1}{2z_h}\right)+O(\epsilon)\,,
\end{multline}
where $\epsilon$ is an IR-regulator and $\Delta y=2y$ is now the length of the boundary interval. $\Sigma$-dependent part for one of the branches of $Q$ can be computed as follows.
\begin{eqnarray}
I_{\rm bdry} & = &  -\frac{\beta}{2\kappa}\int\limits_{\epsilon}^{z_h}{\rm d} z\int\limits_{y_0}^{y(z)}{\rm d} y\,\sqrt{g_E}(R-2\Lambda) - \frac{\beta}{\kappa}\int\limits_{\epsilon}^{z_h} {\rm d} z \,\sqrt{h_E}(K-\Sigma)= \nonumber
\\  & = &   \frac{Lz_h}{2G}\int\limits_{\epsilon}^{z_h}\frac{{\rm d} z}{z^3}\,y(z)-\frac{Lz_h}{4 G}\,\cos\theta'\,\int\limits_\epsilon^{z_h}\frac{{\rm d} z}{z^2}\, \frac{1}{\sqrt{1-f(z)\cos^2\theta'}} \nonumber
\\ & = & \frac{Lz_h}{4G\epsilon}\,\cot\theta' - \frac{L}{4G}\,{\rm arcsinh}({\cot\theta'}) + O(\epsilon)\,.
\label{Ibdry1}
\end{eqnarray}
Removing the divergent parts one arrives at the following expression for the entropy\footnote{The $O(\epsilon^{-2})$ divergence has been removed by the Gibbons-Hawking term at $M$, while one needs an extra counter-term to remove the $O(\epsilon^{-1})$ piece. We discuss the corresponding term in section~\ref{sec:blackhole} for the higher dimensional example.}
\begin{equation}\bea\ds
\label{BTZentropy}
S = -\frac{\partial F}{\partial T}= \frac{L}{4G}\frac{\Delta y}{z_h}+ \frac{L}{2G}\,{\rm arcsinh}({\cot\theta'})\,.
\ea\end{equation}

We notice that the boundary contribution to the entropy is nothing but the area of the extra part of the horizon swept by the hypersurface $Q$. Strictly speaking this contribution must be corrected in the following way
\begin{equation}\bea\ds
\label{BTZentropy-b}
S_Q =  \frac{L}{4G}\,{\rm arcsinh}(|\cot\theta'|)\,.
\ea\end{equation}
The non-analytic form of the entropy at $\theta'=\pi/2$ is related to the fact that going through the point $\theta'=\pi/2$ we should also change the direction of the normal to the surface $Q$ in order to satisfy the null (weak) energy condition. For the same reason the contribution of the second branch of $Q$ contributes the same amount to the entropy, rather than cancels the total result.

\section{Conformal fluids for constant $z$ slices}
\label{sec:conformalfluid}

In this appendix we review the simplest and most paradigmatic example of the fluid/gravity correspondence, which is given when considering $Q$ defined by the equation $z=z_0$ in the black hole geometry (\ref{ansatz-metric}). We choose the normal vector, which points towards the boundary of AdS space:
\begin{equation}\bea\ds
\label{normalQz}
(n^t,n^x,n^y,n^z)=\left(0,0,0,-\,\frac{z \sqrt{f(z)}}{L}\right).
\ea\end{equation}
This choice is dictated by the requirement that the theory on $Q$ has positive energy density and temperature. This is then seen to be a degenerate limit of the AdS/BCFT where the part of the space with $z<z_0$ has been excised.

Here we only need the first three terms in (\ref{Tab}). $T_{ab}$ takes the form of the energy-momentum tensor of an ideal fluid
\begin{equation}\bea\ds
\label{IdealFluid}
T_{ab}= (\epsilon+p)u_a u_b + p g_{ab}\,,
\ea\end{equation}
evaluated in the co-moving frame
\begin{equation}\bea\ds
(u^t,u^x,u^y) = \left(\frac{z}{L\sqrt{f(z)}},0,0\right).
\ea\end{equation}
The energy density and the pressure are thus given by
\begin{eqnarray}
\label{ezconst}
\epsilon & = &  u^a u^a T_{ab} = \frac{L^{2}}{\kappa z^3}\left(L\Sigma-2\sqrt{f}\right)  \\
\label{pzconst}
p & = & \frac{\epsilon + {T^a}_a}{2}= -\frac{L^2 \left(z f'+2 L \Sigma \sqrt{f}-4 f\right)}{2\kappa z^3 \sqrt{f(z)}}\,,
\end{eqnarray}
and we get the following equation of state
\begin{equation}\bea\ds
\frac{\epsilon}{p}=\frac{4f-2L\Sigma\sqrt{f}}{zf'-4f+2L\Sigma\sqrt{f}}\,.
\ea\end{equation}
In the empty $AdS$-space, $f(z)=1$, and general $\Sigma$, $T_{ab}$ describes an energy dominated universe. The hydrodynamical quantities diverge as $z\to 0$. For a finite temperature case this is also the leading order result for general $\Sigma$ in the $z\to 0$ asymptotics. The choice $\Sigma=2/L$ removes the UV divergences and is called holographic renormalization. In fluid/gravity correspondence this means subtracting non-hydrodynamic degrees of freedom from the energy-momentum tensor. The resulting equation of state at the $AdS$ boundary is that of a conformal incompressible fluid
\begin{equation}\bea\ds
\epsilon = 2 p\,, \qquad \text{with} \qquad \epsilon= \frac{L^2}{\kappa z_h^3}
\ea\end{equation}

One can also find the local temperature and local entropy density of the thermodynamical system~(\ref{ezconst})-(\ref{pzconst}). The local temperature $T$ should be given by the red-shifted Hawking temperature $T_H$~(\ref{HawkingT})
\begin{equation}\bea\ds
T=\frac{T_H}{\sqrt{f(z)}}\,.
\ea\end{equation}
From the thermodynamic relation
\begin{equation}\bea\ds
\label{ensemble}
\epsilon +p = Ts\,,
\ea\end{equation}
one can find that the entropy density is precisely the Bekenstein-Hawking entropy density
\begin{equation}\bea\ds
s=\frac{L^2}{4Gz_h^2}\,.
\ea\end{equation}
The consistency of the first law of thermodynamics can be verified locally through the relations
\begin{equation}\bea\ds
\label{1stlaw}
\frac{ds}{s}=\frac{d\epsilon}{\epsilon+p}\,, \qquad \frac{dT}{T}=\frac{dp}{\epsilon + p}\,.
\ea\end{equation}

These results are well-known in the study of fluid-gravity correspondence, \emph{e.g.} see reviews~\cite{Rangamani:2009xk,Hubeny:2010wp,f/g-review} and references therein. Changing the position of the surface along the $z$ axis one can study the renormalization of the physical quantities in the hydrodynamical theory~\cite{Brattan:2011my}. It is noticed that the speed of sound, see figure~\ref{fig:density-pressure} (right), given by
\begin{equation}\bea\ds
v_s^2 = \frac{\partial p}{\partial\epsilon}=\frac{\partial p/\partial z_h}{\partial\epsilon/\partial z_h}= \frac{3-f(z)}{4f(z)}\,,
\ea\end{equation}
becomes superluminal for $z>z_s$,
\begin{equation}\bea\ds
z_s^3 = \frac25\,z_h^3\,.
\ea\end{equation}
In~\cite{Brattan:2011my} it was suggested that this indicates that the gravity solution must be modified beyond this point.

\section{Useful formulae}
\label{sec:appendix}
\begin{eqnarray}
\int\limits_0^1\frac{{\rm d} \zeta}{\sqrt{1-\zeta^3}} & = & \frac{\sqrt{\pi }\, \Gamma \left(\frac{4}{3}\right)}{\Gamma \left(\frac{5}{6}\right)}\,, \\
\int\limits_0^1{\rm d} \zeta\,\frac{1-\sqrt{1-\zeta^3}}{\zeta^3\sqrt{1-\zeta^3}} & = & \frac12 + \frac14 \int\limits_0^1\frac{{\rm d} \zeta}{\sqrt{1-\zeta^3}}\,, \\
\int\limits_0^1{\rm d} \zeta\,\frac{\sqrt{1-\zeta^3}-1}{\zeta^3} & = & \frac12 - \frac34 \int\limits_0^1\frac{{\rm d} \zeta}{\sqrt{1-\zeta^3}}\,.
\end{eqnarray}

\end{appendix}

\addcontentsline{toc}{section}{Bibliography}

redpajama_set_name: RedPajamaArXiv


text:
Осман Абібулла огли Абібуллаєв (;  28 вересня 1913, Бахчисарай, Сімферопольський повіт, Таврійська губернія, Російська імперія — 9 вересня  1970) — азербайджанський радянський археолог, кандидат історичних наук, старший науковий співробітник Інституту історії Академії наук Азербайджанської РСР, дослідник пам'яток Нахічеванської культури епохи бронзи і раннього заліза тощо, що здійснив значний вклад у дослідження археології Азербайджану і Кавказу.

У роботах Абібуллаєва отримали подальший розвиток проблеми куро-аракської культури, збагачені новим археологічним матеріалом. Абібуллаєв близько десяти років займався дослідженням пам'ятника Кюльтепе I поблизу Нахічевані, де вперше в Закавказзі під куро-аракським шаром були відкриті більш давні культурні нашарування. Вчений присвятив спеціальну монографію даного пам'ятника, а також написав серію публікацій.

Біографія

Ранні роки
Осман Абібулла огли Абібуллаєв народився 28 вересня 1913 року в місті Бахчисарай Таврійської губернії в сім'ї вчителя. Батьків рано втратив і ріс у дитячому будинку, де здобув початкову освіту.

У 1930 році закінчив середню школу в Сімферополі. Потім працював учителем, фрезерувальником. Незабаром переїхав до Нахічевані, де був призначений на посаду технічного секретаря обкому партії, а потім бібліотекаря обласного парткабінету. У 1938 році Абібуллаєв закінчив учительські курси і був направлений учителем у село Баскал. Педагогічну діяльність Абібуллаєва перервала німецько-радянська війна. Осман Абібуллаєв брав участь у боях за Крим і на Північному Кавказі, тричі був важко поранений, контужений. За участь у радянсько-німецькій війні Осман Абібуллаєв був нагороджений медалями «За оборону Кавказу» та «За перемогу над Німеччиною».

Початок роботи в Інституті історії
Після демобілізації з інвалідності наприкінці 1943 року Абібуллаєв у 1944 році вступив на сходознавчий факультет Азербайджанського державного університету, який успішно закінчив у 1948 році.

Наприкінці 1947 року Абібуллаєв почав працювати в Інституті історії Академії наук Азербайджанської РСР. Спочатку обіймав посаду лаборанта у відділі археології, а потім став молодшим науковим співробітником. Брав участь у роботі низки великих археологічних експедицій Академії наук Азербайджанської РСР і Академії наук СРСР. Зокрема, брав участь у Гобустанській, Мінгячевірській, Мільській (на городищі Оренкала) та інших експедиціях. Пізніше Абібуллаєв очолив загін, а незабаром і експедицію з дослідження давніх пам'яток на території Нахічеванської АРСР. Так, під керівництвом Олександра Ієссена він очолював загін експедиції з дослідження пам'яток енеоліту та доби бронзи на території Нахічеванської АРСР і в Ходжали (Нагірно-Карабаська АО).

Вивчення пам'яток Нахічевані
У 1951 році Абібуллаєв став вести систематичні ремонтні роботи на багатошаровому поселенні Кюльтепе I поблизу міста Нахічевань. У 1959 році Абібуллаєв захистив кандидатську дисертацію, в основу якої лягли перші підсумки його розкопок на Кюльтепе I. Того ж року була опублікована книга Османа Абібуллаєва «Археологічні розкопки в Кюльтепі», в якій були узагальнені результати його перших розкопок.

Розкопки, які Абібуллаєв вів на пагорбі Кюльтепе, зафіксували послідовні нашарування різних культурних шарів. Археологом, серед інших, були також встановлені культурні нашарування епохи енеоліту в Кюльтепі. Це були перші дані найдавнішою осіло-землеробської культури Закавказзя.

У середині 1960-х років Осман Абібуллаєв приступив до написання монографії «Енеоліт та епоха бронзи на території Нахічеванської АРСР», опублікованій в 1982 році. Робота, яка за задумом автора, повинна була охопити доби енеоліту, ранньої та середньої бронзи Нахічевані, була написана на основі археологічних матеріалів, більшість яких, було здобуто самим Абібуллаєвим. Крім місцевих матеріалів Абібуллаєв досліджував і величезну кількість порівняльного матеріалу з інших пам'яток Кавказу і суміжних областей Сходу. Дана робота Абібуллаєва, як зазначає історик , являє собою цінний внесок у радянську археологію. Ісмайлов зазначає, що ознайомлення з цією працею Абібуллаєва переконує в тому, що «глибоке та ґрунтовне знання автором досліджуваних питань дозволило йому на високому науковому рівні» виконати завдання висвітлення культурно-історичного розвитку Нахічеванського краю на великому відрізку часу. Ця монографія Абібуллаєва була представлена до захисту на здобуття наукового ступеня доктора історичних наук.

Наприкінці 1960-их років Осман Абібуллаєв брав участь у роботі Нахічеванської археологічної експедиції. У 1969 році поблизу могильника Кизил-Ванк Абібуллаєвим були відкриті залишки стародавнього поселення, що належало до цього некрополя. У цей же період Абібуллаєв брав участь у розкопках багатошарового пам'ятника Кюльтепе II.

Крім присвяченій Кюльтепе монографії, Абібуллаєв за час своєї роботи в Інституті історії Азербайджанської РСР опублікував ряд статей на сторінках періодичних наукових видань центральних інститутів Академії наук СРСР і Академії наук Азербайджанської РСР, а також статті у Великій радянській енциклопедії і в багатотомній історії СРСР, був одним із авторів колективної праці «Археологія Азербайджану», де були узагальнені його дослідження доби енеоліту, ранньої і середньої бронзи.

Осман Абібуллаєв займався підготовкою молодих кадрів, якими були успішно захищені кандидатські дисертації на тему давньої археології Азербайджану. Абібуллаєв виступав також на наукових сесіях, проведених Інститутом археології АН СРСР та Інститутами історії АН Азербайджанської, Вірменської та Грузинської РСР.

Помер Осман Абібуллаєв 9 вересня 1970 року. За словами історика і археолога Гудрата Ісмайлова, смерть Абібуллаєва, який «так багато зробив для археології Кавказу», стала важкою втратою для археології країни. На думку археологів В. Алієва та М. В. Мінкевич-Мустафаєвої, Осман Абібуллаєв вніс великий внесок у розвиток археологічної науки в Азербайджані.

Деякі роботи

  (разом з В. Алієвим)
  (разом з В. Алієвим)
  (разом з В. Алієвим)

Примітки

Література

Дослідники доісторичної доби
Радянські педагоги
Азербайджанські педагоги
Азербайджанські археологи
Археологи XX століття
Радянські археологи
Персоналії за алфавітом

redpajama_set_name: RedPajamaWikipedia


Two NFC Forum members announced significant industry achievements this month securing competitive market advantages through the NFC Forum Certification Program.
FIME announced that it is the only testing laboratory in Japan allowed to provide NFC Forum accredited testing and certification services. Identiv's news was equally significant as it unveiled a new NFC Forum certified NFC tag making Identiv the first tag manufacturer to achieve certification in the NFC Forum Certification Program.
The NFC Forum Certification Program allows tag/inlay, NFC reader and handset manufacturers to test and verify the performance and interoperability of all the key components in the NFC eco-system. Certification helps ensure that NFC tags, readers and handsets provide consistent, compelling and connected user experiences.
FIME is the only testing laboratory in Japan accredited to provide NFC Forum testing and certification services. Japanese tag, terminal and mobile device manufacturers can now benefit from local end-to-end testing support. Japan is an important market for NFC technology and FIME is now the only test lab capable of providing this NFC Forum accredited service.
With the release of its Tamper Detection Label NTAG 424 DNA, Identiv is the first tag manufacturer to achieve certification in the NFC Forum Certification Program. NFC Forum Certification indicates that the label has demonstrated its conformance to the applicable certification requirements for Near Field Communication (NFC) tags for performance and interoperability. Identiv's Tamper Detection Label utilizing NXP® Semiconductors N.V.'s NTAG® 424 DNA, strengthens its radio-frequency identification (RFID) offerings to provide enhanced security for the Internet of Things (IoT).
Find out more about NFC Forum Certification and why it should be part of your product strategy, click here.


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Q: OneNote Links with a # won't open in IE 11 I'm using OneNote 2010. I am creating links that contain a hash #, for example http://a.com/index.html#MainMenu. When clicked, the address sent to IE 11 is changed to http://a.com/index.html%20-%20MainMenu. I have tried changing the # to %23 but the link fails to load entirely then as my intranet apparently doesn't support encoding. I've seen other posts that state there's a way for the support for encoding to be disabled server-side.
Has anyone heard of this and know of a workaround? Prior to being forced to upgrade to IE 11 this functioned properly when clicking links in OneNote.

redpajama_set_name: RedPajamaStackExchange


Learn flamenco guitar from Gerardo Núñez. Gerardo highlights an enormous amount of techniques, exercises, scales and falsetas. He also demonstrates how to play seven compositions from his early and current repertoire.


PoE Door Controller is an ethernet power sourcing device, which allows network communication with entry points installed in the access control system. Borer PoE Controller incorporates PoE technology for power delivery to equipment located at the door, such as electric door locks, biometric readers, exit request devices, etc.
PoE Ethernet Door Controller is an ethernet power sourcing device, which allows network communication with entry points installed in the access control system. Borer PoE Controller incorporates PoE technology for power delivery to equipment located at the door, such as electric door locks, biometric readers, exit request devices, etc.
The PoE Door Controller/Ethernet Lock Manager has a small footprint and is designed to be mounted on the secure side of the door.
Manages power and data supplied in a CAT5 cable from a standard PoE switch or power injector.
Monitors door status (Door Open, Lock Status and Request to Exit button) and redirects to central database.
Delivers power to electric door locks, magnetic locks, strikes, turnstiles.
Power over CAT5e/6 Cable technology powers equipment where it is inconvenient, expensive or difficult to provide a mains power outlet to the door.
Borer ELM supports IEEE 802.3af (PoE) and IEEE 802.3at (PoE+) standards, which allows a single cable to deliver both data and power to doors over data cable, providing protection against short circuit, polarity reversal and accidental connection of non-PoE compliant equipment.
The Biometric enabled version of the Lock Manager holds a library of up to 64,000 biometric templates. The template library held by the Lock Manager is referenced by biometric reader heads every time a person types in a unique PIN or presents their ID card. The template corresponding to the PIN or card is sent to the reader head where it is compared to the live finger to verify that the presenter of the card or PIN is the true owner.


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Q: Is Sao Feng's map magical? The map that Sao Feng's hands to Barbossa and co to go rescue Sparrow from Davy Jones' locker contains many information hidden to strange places, some of them out of this world.
But the map itself seems to be quite a feat of craftmaship: you can fold it yet the 'wheels' inside can be turned with no difficulty and show little wear and tear, and it has some inscriptions in English and Spanish despite being a seemingly old Chinese map. It doesn't seem that the way the wheels are positioned matters much when the map is folded as well.
Plus, the map changes in between movies: for example, Florida wasn't on the map on At World's End but shows up, along with Ponce de Leon's trip, on On Stranger Tides.
So, was it ever established in-universe that the map was magical as an explanation to the way it works and the changes between movies?


Mr. Makarov has been with Study Medicine Europe for the last 2 years. As a Country Manager in Ukraine he aims at dealing and managing other City Managers across the country. Sergiy provides and is responsible for professional consultation, mentoring and advice to our prospective international students who relocate to different Universities in Ukraine to assure a students`steady transition and adaptation process. He is a Master`s Degree holder from Kiev Slavic University in International Relations, European Studies as well as a qualified English translator and interpreter.
Miss Tarashchanska has been with Study Medicine Europe for the past 2 years. Svitlana has been helping students relocate to Ukraine by arranging their various needs and necessities, consultation, mentoring and advice as well as providing ongoing support, in case of emergency, for the duration of a student's academic life. She has a Master`s Degree in linguistics as a qualified English translator and interpreter.


A project we created alongside award winning photographer Michael Corridore (Photoplay films) for Ward6 depicting a specific course of medical treatment using a glitch sequence as the hero device.
The glitch represents a need for a total "reset" to the immune system of a patient who is suffering from Multiple Sclerosis (MS), and the treatment has the desired impact to remove or "reset" the glitch in the immune system.
The effect was designed and created from the ground up to facilitate the flexibility the project required. The final effect is a result of layering a series of highly customisable elements that were rendered in 3D over the base imagery. Each element could be manipulated in shape, speed and scale independently to reveal specific sections of distortion and finer details.
The glitch effects were also translated to a host of other video assets, including text based key selling messages, which accompany the hero motion pieces when displayed at various medical conferences.


Vetruria is specialised in the sale and distribution of glass containers from the best companies in the sector: glass bottles for oil, wine, spirits, jars for food and accessories such as synthetic corks and twist off caps.
Professionalism, competence and passion, these are our principal characteristics.
Our company will try to satisfy every requirement of the customer in the choice of a container in terms of shape, colour and characteristics.
Our response is a distribution which is precise, punctual and convenient.
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Using a benchmarking tool to test the limits of your apache server will help you prepare for upgrades.
cPanel adds in an exclusion for all packages named httpd* (any files starting with httpd) to prevent accidentally overwriting apache.
Once the entry has been removed, save the config file and exit.
Run the following command to install ab.
Make sure to go back into the /etc/yum.conf and add httpd* back to the exclude list, otherwise you risk breaking your cpanel installation.
Once you have the program installed, you can run a stress test against a domain name.
WARNING: Do not use this against a site you do not control as it will be considered an attack.
Here is an example of a stress test that will simulate 100 users simultaneously making accessing a website.
You can check the manual for ab to find out more information about it's options.


Looking for a Unique experience to celebrate finishing school?
Keen to meet other like-minded people and explore the real world?
Would you rather an Adrenalin Rush rather than an Epic Hangover?
Want to leave with Awesome Memories rather than hazy regrets?
JOIN Us on a Schoolies Radventure in 2017. Grab a few like-minded mates, or sign up solo and become part of a team with other keen beans! With options to design & create a trip or book onto one of our pre-existing programs – whatever you choose, you'll be living out your dream Radventure before you know it!
With white-water paddling, bush exploration, sea kayaking and sailing adventures on offer around Australia, we will tailor an experience to match your interests and budget. You will team up with a Journey guide, an experienced Outdoor guide and Wilderness first aider who will accompany you to ensure your trip runs smoothly and is everything you wanted it to be.
Keen to do something Rad for schoolies?


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Extraterrestrial civilization was responsible for the emergence of life on Earth 3.8 billion years ago
Part of the scientific community believes that an extraterrestrial civilization was responsible for the emergence of life on Earth 3.8 billion years ago.
Could there be planets on which advanced extraterrestrial civilizations existed? For several years now, there has been a question that has plagued the minds of many people. Could there be an extraterrestrial civilization that is much more advanced than ours?
This is due to the fact that many scientists are wondering if there are solar systems in the universe that are more than 10 billion years old. Considering that ours is only 4,600,000 years old. Francis Crick, the Nobel laureate in medicine who is credited with discovering the structure of DNA, also wrote about this.
If solar systems have existed much longer than ours, then there is a possibility that technologically advanced alien civilizations exist. Then there is the possibility that technologically advanced alien civilizations exist. Hence, more advanced than us by thousands of years. They may have existed even before the birth of Earth.
Recent discoveries of exoplanets found far from our solar system support this theory.
A planet with an advanced alien civilization
Proof of this is the planet PSR B1620. It is an exoplanet located about 12,400 light years from Earth in the constellation of Scorpius. It has the unofficial names "Methuselah" and "planet Genesis.
Francis Crick proposed a new theory to the scientific community, which he called "infectious. It includes an alien civilization that may have given rise to primitive life on Earth.
Obviously, this raises questions: how was it brought to Earth? One theory is that they did so by means of a spaceship specially designed for this mission. In this way, they were able to bring in these microorganisms responsible for the origin of life.
Apparently, the space journey was carried out with all possible safety precautions. Until the ship reached our planet, which was its goal.
The origin of terrestrial life – space
Subsequent discoveries in the field of genetics have shown that: "All life on Earth, from bacteria to man, originated from an ancestor cell." The more important question is: What is the origin of this ancestral cell?
Manfred Eigen, a Nobel laureate in chemistry, and a group from the Max Planck Society, the German Society for the Advancement of Science, were able to sequence "reverse" RNA containing DNA genetic information that can be used in protein synthesis.
According to Eigen, the "primary gene" could have appeared at this very time. It can be assumed that if the human genetic code is more than 3.8 billion years old, then it must have extraterrestrial origin.
Our scientific community has refused for years to acknowledge the possibility of extraterrestrial life. But now they are considering the possibility that our life is of extraterrestrial origin. And an extraterrestrial civilization is responsible for life on Earth.
Already 4 billion years ago there was life on Earth
Japanese geologists have found deposits of organic rocks in the northeast of Canada, indicating that life should have existed on…
NASA showed how life on Earth changed for 20 years
NASA published a video in which in just a few minutes it was shown how the life on the planet…
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Signs & Symptoms of Earring Allergies With 14 Karat Gold
Alexis Jenkins
Alexis Jenkins writes to motivate others in areas of health including nutrition, fitness training and improving lifestyle choices. She graduated with a Bachelor of Science in health science from Brigham Young University-Idaho.
Allergic reactions to 14 karat gold earrings and other metal jewelry are fairly common. Signs and symptoms can appear soon after exposure to new jewelry, or may develop over time as the metal on your earrings wear down. Reactions vary from localized, mild irritation at the site of the piercing to widespread and painful. If you have concerns about metal allergies or reactions to jewelry that you wear, consider speaking with your allergist or doctor.
14 karat gold can often cause allergic reactions in and around the earlobes. (Image: Design Pics/Design Pics/Getty Images)
Nickel is a metal component used in the gold alloy of most gold jewelry. It is used primarily because it is an inexpensive, hard metal that has a lightening and whitening effect and is often used in expensive jewewlry. 14 karat gold is 58.3 percent pure 24 karat gold, while the remaining percentage is composed of metal alloys including silver, nickel, copper and zinc, according to the Amazon Precious Metals Guide.
Allergic Contact Dermatitis
Allergic contact dermatitis occurs when the skin around the earring, usually the earlobe and parts of the neck, reacts to the nickel and other metal elements found in the earring. The skin usually becomes swollen and a rash appears between 24 and 48 hours after exposure to the irritant. Inflammation may vary from mild irritation to open sores, depending on the severity of your reaction.
While you may not experience an allergic reaction after initially wearing new jewelry, the gold molecules may begin to wear away over time, exposing your earlobes to nickel and other metal alloys. Once you develop a sensitivity to the metal allergen, even brief contact can cause your contact dermatitis to reappear. In some cases, an allergic reaction that begins in the earlobes may initiate allergic reactions on other jewelry-adorned areas of the body that had previously never experienced an allergic response, according to the American Academy of Dermatology.
Treat your reaction by removing your earrings and washing the area with plenty of water to remove any remaining irritants. You can also apply an over-the-counter topical cortisone cream such as hydrocortisone cream. If your symptoms do not improve in five to seven days, you may need to speak with a doctor.
To prevent the wear of your gold earrings, consider getting them gold-plated with 24K gold, suggests dermatologist Audrey Kunin, M.D., of ShareCare, a healthcare information website. You can also try storing jewelry in airtight, dry pouch or bag that is free from solvents, detergents or adhesives. Polish your earrings regularly and wipe them clean after wearing them. If allergic reactions continue to occur, you may need to consider switching your earring and other jewelry metal to stainless steel, titanium or other hypoallergenic metal.
Roman Research, Inc: Excerpts from the Academy of Dermatology News Releases: Ear Piercing Can Cause Metal Allergies
ShareCare; Can Pure Gold Jewelry Cause a Nickel Allergy Reaction?; Audrey Kunin, MD
Jewellery Catalogue: Gold Allergy
University of Maryland Medical Center: Contact Dermatitis
Asthma and Allergy Foundation of America: Contact Dermatitis
Allergic Reaction to Platinum on the Ring Finger
Allergies to Metal in Pierced Ears
Allergic Reactions to Silver Jewelry
How to Stop Pierced Ears From Seeping
How to Change Pierced Earrings
How to Remove Discoloration on Jewelry


Bobby Joe Jones was born on December 6, 1954 to Carl and Bessie in Houston Texas. He passed away on April 5, 2018. He is survived by his wife of 12 years Zona Fenley Jones. Son; Joshua Fenley and wife Jessica. Daughter's; Doranda Jordan and husband Lee; Crystal Rodgers; Nicke Behm and husband Rusty; Donia Behm and Husband Travis. Brother's; Carl Clinton Jones; Claude Jones; Robert Jones and wife Melissa. Sister's; Margaret Evans; Frances Lirette, Deborah Cohn and husband Larry. Grandchildren; Brooklyn; Kylia; Mary; Amelia; Grant; and Matt.
He is preceded in death by parents; Carl Cavanaugh Jones and Bessie Lee; Brother Jimmy Jones; Sister's; Jodean Fain; Billie Jean; and Patricia Cupp. He was loved by many family and friends.
Dad... I am sitting here at home after your service and it seems like its all still a dream. I can not believe you are gone. You were the sunshine in everyone's life, you shaped me in to the person I am today and I am truly grateful, Along the road you taught me valuable lessons and the importance of family and friends, Looking through all of your photos.. every single one of them had your wonderful smile..
When I close my eyes for the final time I better see that smile of yours. Thank you for being such a wonderful husband to my mother and an amazing father to us, and a loving poppy to the grand kids.
--Until we see you again.
Bobby it's the after we had your service.
I'm thinking about you and a warmth came over me.
I know it's you telling me everything is ok .
You are my heart and soul.I love you always and forever.
Its been two days since we got to say goodbye for a final time. Im sitting in class thinking about you and my eyes keep tearing up. Im gonna miss you. i saw a news article today and i thought "gotta tell boby about that". then i remembered i cant. goodbye pop. i love you.


Abstract: In this document we show how to handle counterparty risk for Interest Rate Swaps (IRS). First we establish a general formula, showing that counterparty risk adds one level of optionality to the contract. Then we introduce the default probabilities using a deterministic intensity model where the default time is modeled as the first jump of a time inhomogeneous Poisson process.
We consider Credit Default Swaps as liquid sources of market default probabilities. We then apply the general formula to a single IRS. The IRS price under counterparty risk turns out to be the sum of swaption prices with different maturities, each weighted with the probability of defaulting around that maturity. Then we consider a portfolio of IRS's in presence of a netting agreement. The related option cannot be valued as a standard swaption, and we resort both to Monte Carlo simulation and to two analytical approximations, investigating them by Monte Carlo simulation under several configurations of market inputs. We find a good agreement between the formula and the simulations in most cases.
The approximated formula is well suited to risk management, where the computational time under each risk factors scenario is crucial and an analytical approximation contains it.
Keywords: Interest Rate Swap, Counterparty Risk Pricing, Netting Agreements, Libor Model.


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Jerzy Egon Kwiatkowski (ur. 1928 w Królewskiej Hucie, zm. 17 maja 2005 w Rudzie Śląskiej) – polski rzeźbiarz z Górnego Śląska.

Życiorys
Ukończył Państwową Wyższą Szkołę Sztuk Plastycznych we Wrocławiu w 1952 roku. Jest znany z wykonania licznych rzeźb o tematyce świeckiej i religijnej, jego prace zdobią m.in. place i skwery w Katowicach, Chorzowie, Wodzisławiu Śląskim i Dąbrowie Górniczej.

Jego kariera jako rzeźbiarza rozpoczęła się w latach sześćdziesiątych XX wieku, gdy wykonał wiele prac w kościołach. Nawiązał kontakt wraz z architektem Mieczysławem Królem z którym wspólnie stworzyli projekt wystroju kościoła w Łagiewnikach Wielkich (1969). Ta sam dwójka w 1973 roku dokonała wykończenia wnętrza i drzwi w katedrze katowickiej: projekt rzeźb ołtarza ze scenami biblijnymi oraz monumentalnej rzeźby Chrystusa Króla w prezbiterium, kaplicy Najświętszego Sakramentu oraz kaplicy chrzcielnej (1980–1986).

Jednym z największych dzieł Kwiatkowskiego jest Pomnik Powstańców Śląskich w Wodzisławiu Śląskim. Zaprojektował także wnętrze kościoła Krzyża Świętego w Czułowie. Pomógł przy realizacji Drogi Krzyżowej w kościele parafialnym w Roju oraz przy wystroju wnętrza kościoła w Drogomyślu. Rzeźbiarz jest także autorem zamkniętej przestrzeni nad ołtarzem z wizerunkiem Chrystusa Króla w archikatedrze w Katowicach, w której umieścił formę rzeźbiarską. Zaprojektował też główne drzwi katedry (1968). Po przeniesieniu z Krakowa do Katowic artysta zaprojektował wystrój kaplicy seminaryjnej w Wyższym Śląskim Seminarium Duchownym w Katowicach.

Był członkiem Związku Polskich Artystów Plastyków.

Jedno z ostatnich przedsięwzięć śląskiego rzeźbiarza jest rzeźba bł. ks. Emila Szramka, która mieści się w kościele Niepokalanego Poczęcia Najświętszej Maryi Panny w Katowicach. Zmarł w Rudzie Śląskiej, pogrzeb odbył się 18 maja 2005 roku w Rudzie Śląskiej-Kochłowicach.

Wybrane dzieła
 Chrystus Król, archikatedra w Katowicach
 drzwi katedry katowickiej (projekt, wyk. Stefan Gaida)
 Pomnik Powstańców Śląskich w Wodzisławiu Śląskim
 rzeźba w prezbiterium kościoła Mariackiego w Katowicach
 prace w kościele św. Jacka w Ochojcu,
 rzeźba w kościele Wniebowzięcia Najświętszej Maryi Panny w Katowicach,
 wystrój kościoła św. Apostołów Piotra i Pawła w Katowicach,
 Narodzenia Najświętszej Maryi Panny w Golasowicach,
 NSPJ w Mysłowicach,
 NMP Matki Kościoła w Jastrzębiu Zdroju (górny kościół),
 Wszystkich Świętych w Pszczynie,
 Plac Grunwaldzki w Katowicach, rzeźba rzeźba Rodzina z 1963 roku, pierwotną okładzinę zaprojektowała Lidia Kwiatkowska, żona artysty
 płaskorzeźby Pałacu Ślubów w Katowicach
 płaskorzeźby na fasadzie Galerii Sztuki Współczesnej BWA w Katowicach (1972).
 rzeźby plenerowe Dwie góralki i Rodzina w Parku Śląskim (zob. rzeźby w Parku Śląskim)
 tablice pamiątkowe poświęcone Mieczysławowi Dzieńdzielowi, Władysławowi Kiełbasie, Ludwikowi Musiołowi, Cyprianowi Norwidowi, Konstantemu Prusowi, Rudolfowi Virchowowi, Rafałowi Wojaczkowi

Bibliografia
 L. Makówka, Sztuka sakralna na Górnym Śląsku w II połowie XX wieku. Malarstwo i rzeźba, Katowice 2008, s. 234-239.

Przypisy

Urodzeni w 1928
Ludzie urodzeni w Chorzowie
Polscy architekci
Śląscy rzeźbiarze
Zmarli w 2005
Członkowie Związku Polskich Artystów Plastyków