text:
\section{Introduction}

Quantum chromodynamics (QCD) tells us that quarks and gluons are confined
inside hadrons in vacuum or ground state. No free quarks and gluons
are detected in any hadronic collisions. Finite temperature theory
of QCD predicts that the deconfinement of quarks and gluons can be
reached at high temperatures and densities and a new state of matter,
the quark gluon plasma (QGP), can be created \citep{Lee:1974ma}.
According to lattice QCD calculations the confinement-deconfinement
phase transition takes place at temperature of about 170 MeV in three
flavor case \citep{Karsch:2000ps}. In early 1970s Lee and Greiner
and his collaborators proposed that the deconfinement could be realized
by smashing two heavy ions together at ultrarelativistic energies
\citep{Hofmann:1974}. This is the only way of generating QGP in laboratories.
The minimum energy density for deconfinement to take place is about
an order of magnitude higher than normal nuclear matter density. The
current energy frontier of heavy ion collisions is the Relativistic
Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL),
which has been running since the summer of 2000. A lot of evidences
imply that the QGP at RHIC is near perfect fluid (strongly coupled
QGP or sQGP), which is contrary to the conventional concept of QGP
as a weakly interacting gas of quarks and gluons, see e.g. \citet{Shuryak:2004cy,gyulassy:2005}.

While perturbation can be used to deal with weak coupling problems,
there are a lot of difficulties in describing a strongly coupled system
due to its non-linear feature and failure of perturbative methods.
In recent years a promising method to deal with strong coupling problems
in gauge theory is through string/gauge duality or precisely AdS/CFT
duality, where AdS and CFT are abbreviations for anti-de Sitter space
and conformal field theory respectively. The duality was first proposed
by Maldacena and many others \citep{Maldacena:1997re,Gubser:1998bc,Witten:1998zw},
where a strongly coupled conformal gauge field theory corresponds
to weekly coupled closed strings at 't Hooft limit. So properties
of a weekly coupled string can provide information for strongly coupled
gauge theory.

Policastro, Son and Starinets first used AdS/CFT duality to calculate
transport coefficients of strongly coupled dense matter \citep{Policastro:2001yc}.
Kovtun, and Son and Starinets derived a lower bound (called KSS bound)
for the ratio of shear viscosity to entropy density, $\eta/s\gtrsim1/(4\pi)$,
consistent to the RHIC data \citet{Kovtun:2004de}. These works open
an avenue to study properties of strongly coupled matter followed
by a lot of developments in hydrodynamic properties of sQGP \citet{Buchel:2003tz,Maeda:2006by,Natsuume:2007ty,Baier:2007ix}.
Recently the jet quenching effect has been investigated with AdS/CFT
techniques \citet{Liu:2006ug,Herzog:2006gh,Gubser:2006bz}. The sceening
length of the heavy quark potential can also be described from the
AdS/CFT duality \citet{Liu:2006nn,Peeters:2006iu,Hou:2007uk,Li:2008py},
which is closely related to the $J/\psi$ suppression in hot medium.

The KSS bound is valid for any conformal fluids. Altough it is debatable
that the sQGP is a conformal fluid, the ratio $\eta/s$ is fixed to
be the KSS bound in all periods of the sQGP evolution in some hydrodynamic
simulations \citep{Song:2007ux,Chaudhuri:2008sj}. However the sQGP
formed in ultra-relativistic heavy ion collisions has particular geometric
configurations such as boost invariance \citep{Bjorken:1982qr} etc..
Janik and Peschanski first considered a proper time dependent $AdS_{5}$
metric with boost invariance through holographic renormalization in
late time of fluid evolution \citet{Janik:2005zt,Janik:2006ft,Benincasa:2007tp}.
The method has been extended to sQGP with shear viscosity \citep{Nakamura:2006ih,Sin:2006pv},
where the shear viscosity appears in energy-momentum tensor in CFT
and finally enter in the $AdS_{5}$ metric. The early time behavior
of sQGP has also been addressed following similar method and can then
join the late time one to provide a global picture for time evolution
in heavy ion collision \citet{Kovchegov:2007pq,Albacete:2008vs}.
The exact solution to the gravity dual of 1+1 dimensional sQGP with
Bjorken scaling has been found by Kajantie, Louko and Tahkokallio,
which is helpful to understand what happens in 1+4 dimension in the
real world \citet{Kajantie:2007bn}. The proper time dependent $AdS_{5}$
metric without shear viscosity does not necessarily mean that the
${\cal N}=4$ super Yang-Mills (SYM) plasma is non-viscous, noting
that the shear viscosity with AdS/CFT duality is obtained by a D3
black AdS metric in Ref. \citep{Son:2002sd,Son:2007vk}. If we treat
the proper time dependent $AdS_{5}$ metric without viscosity as an
extension of standard AdS metric we can build up a connection between
the sQGP as an ideal fluid to the SYM plasma, then the ratio $\eta/s$
may not be zero.

In this paper, we investigate the 2+1-dimensional expansion (also
including 1+1-dimensional expansion) of the sQGP in late time and
obtain the proper time dependent $AdS_{5}$ metric through holographic
renormalization. The metric dual to the sQGP fluid can be obtained
by solving the Einstein equation with the stress tensor of sQGP as
boundary value of the metric if transverse expansion is small and
can be treated as a perturbation. The metric has off-diagonal elements
proportional to transverse rapidity. We compute with the metric the
shear viscosity in 1+1 and 2+1 dimension through Kubo formula with
AdS/CFT duality. In 1+1 dimension with transverse expansion turned
off, the shear viscosity is found to be vanishing if only the leading
terms in proper time are considered in the Einstein equation or the
equation of motion for perturbations to the metric. Only including
the next-to-leading terms can one get the ratio $\eta/s=1/(4\pi)$,
indicating the shear viscosity is a higher order effect in late time
behavior. In 2+1 dimension the result is unchanged in the leading
order of transverse rapidity.

The structure of the paper is as follows. In Sec. \ref{sec:AdS5},
we obtain the proper time dependent $AdS_{5}$ metric corresponding
to the sQGP in 2+1 dimension. In Sec. \ref{sec:Boost-geometry} we
derive the effective action for perturbations in the AdS metric. We
compute the ratio $\eta/s$ in Sec. \ref{sec:ratio1+1} and \ref{sec:ratio2+1}
for 1+1 and 2+1 dimensional cases respectively. We make summary and
conclusions in Sec. \ref{sec:Conclusion}. In the paper we take the
signature $(-,+,+,+,+)$ for the metric.

\section{2+1 hydrodynamics with proper time dependent $AdS_{5}$ metric\label{sec:AdS5}}

\subsection{2+1 hydrodynamics for ideal fluids}

In order to describe the hydrodynamic evolution of an ultrarelativistic
system with transverse expansion, e.g. in heavy ion collisions, we
use cylindrical coordinates, \begin{eqnarray}
x^{\mu} & = & (\tau,y,r,\theta),\label{eq:dim4}\end{eqnarray}
where $y$ is the rapidity, $\tau$ the transverse proper time, $r$
the radius in transverse plane, and $\theta$ the azimuthal angle
around longitudinal direction. Note that the full proper time $\tau_{f}$
is related to $\tau$ by $\tau_{f}^{2}=\tau^{2}-r^{2}$. Hereafter
the proper time means the transverse one if not explicitly stated.
The line element is given by \begin{equation}
ds^{2}=-d\tau^{2}+\tau^{2}dy^{2}+dr^{2}+r^{2}d\theta^{2},\label{eq:metric-dim4}\end{equation}
which gives the metric $\widetilde{g}_{\mu\nu}^{(0)}=\mathrm{diag}(-1,\tau^{2},1,r^{2})$.
The velocity four-vector can be parametrized as \begin{eqnarray}
u^{\mu} & = & \frac{dX^{\mu}}{d\tau_{f}}=\frac{1}{\sqrt{1-R^{2}(\tau,r)/\tau^{2}}}\left[1,0,\frac{R(\tau,r)}{\tau},0\right]\nonumber \\
 & \equiv & \left[\cosh\alpha(\tau,r),0,\sinh\alpha(\tau,r),0\right],\end{eqnarray}
where $R$ is a function of $(\tau,r)$ and $\alpha$ is the radial
rapidity in transverse direction. The radial velocity is defined by
$u_{r}\equiv\sinh\alpha$. One can verify that the velocity satisfies
$u^{\mu}u_{\mu}=-1$.

\begin{figure}[t]
\includegraphics[scale=0.42]{ur_r1} \includegraphics[scale=0.42]{ur_tau2}

\caption{\label{fig:radial-velocity}{\small The radial velocity $u_{r}=\sinh\alpha$
as functions of $r$ and $\tau$.}}

\end{figure}

\begin{figure}
\includegraphics[scale=0.45]{rho_tau3}

\caption{\label{fig:density-time}{\small The radial velocity $u_{r}=\sinh\alpha$
as functions of $r$ and $\tau$.}}

\end{figure}

For ideal fluid with the equation of state $\rho=3P$ where $\rho$
and $P$ are energy density and pressure respectively, the energy-momentum
tensor is \begin{eqnarray}
T_{\mu\nu} & = & \left(\rho+P\right)u_{\mu}u_{\mu}+P\widetilde{g}_{\mu\nu}^{(0)}(x)\nonumber \\
 & = & \left(\begin{array}{cccc}
-\frac{\rho}{3}+\frac{4}{3}\rho\cosh^{2}\alpha & 0 & -\frac{4}{3}\rho\sinh\alpha\cosh\alpha & 0\\
0 & \frac{\rho}{3}\tau^{2} & 0 & 0\\
-\frac{4}{3}\rho\sinh\alpha\cosh\alpha & 0 & \frac{\rho}{3}+\frac{4}{3}\rho\sinh^{2}\alpha & 0\\
0 & 0 & 0 & \frac{\rho}{3}r^{2}\end{array}\right).\label{eq:EMT01}\end{eqnarray}
One can verify that $T_{\mu\nu}$ satisfies conformal invariance $T_{\mu}^{\mu}=0$.
The energy-momentum conservation equations read \begin{equation}
\nabla_{\mu}T^{\mu\nu}\equiv\partial_{\mu}T^{\mu\nu}+\Gamma_{\mu\sigma}^{\mu}T^{\sigma\nu}+\Gamma_{\mu\sigma}^{\nu}T^{\mu\sigma}=0,\end{equation}
where $\Gamma_{\mu\sigma}^{\mu}$ are Christoffel symbols for the
metric in (\ref{eq:metric-dim4}). The above equations lead to \begin{eqnarray}
\partial_{\tau}\ln\rho & = & -\frac{2}{\tau}\frac{2\cosh^{2}\alpha}{2\cosh^{2}\alpha+1}\left(1+\frac{R}{\tau}+\partial_{\tau}R\right),\nonumber \\
\partial_{\tau}\ln R & = & \frac{1}{\tau}\frac{R/\tau+2+2(1-\partial_{r}R)\cosh^{2}\alpha}{2\cosh^{2}\alpha+1}.\label{eq:evolution}\end{eqnarray}
Given the initial condition $R(\tau_{0},r)=\xi\tau_{0}r$ where $\xi$
is set to 0.05 $\mathrm{fm}^{-1}$, Eq. (\ref{eq:evolution}) can
be solved numerically whose results are shown in Fig. (\ref{fig:radial-velocity})(\ref{fig:density-time})
and are comparable with those in Ref. \citep{Kolb:2003dz}. The results
show that the radial velocity is linearly proportional to the radial
distance, i.e. $u_{r}\sim\xi r$, where $\xi$ is a small number.
One can see in the figure that $u_{r}$ rises sharply in a very short
early time and gradually falls with increasing $\tau$. If the expansion
in transverse direction is negligible compared to that in longitudinal
direction the energy density damps in the same way as in the case
of 1+1 dimension, i.e. $\rho\sim\tau^{-4/3}$.

\subsection{General setup of holographic model for 2+1 hydrodynamics\label{sub:General-setup-of}}

In this section we briefly introduce the general idea of the holographic
model for relativistic hydrodynamics. The metric in $AdS$ space can
in general be written as \citep{fefferman1985}, \begin{eqnarray}
ds^{2} & = & g_{MN}dX^{M}dX^{N}=\frac{1}{z^{2}}\left[\widetilde{g}_{\mu\nu}(x,z)dx^{\mu}dx^{\nu}+dz^{2}\right],\label{eq:gen-metric01}\end{eqnarray}
where $X^{M}=(x^{\mu},z)$ and $\widetilde{g}_{\mu\nu}(x,z)$ is the
metric tensor in the 4-dimensional space (\ref{eq:dim4}) and generally
depends on $z$. Normally one assumes the following form of $\widetilde{g}_{\mu\nu}(x,z)$,
\begin{equation}
\widetilde{g}_{\mu\nu}(x,z)dx^{\mu}dx^{\nu}=-Ad\tau^{2}+B\tau^{2}dy^{2}+C\left(dr^{2}+r^{2}d\theta^{2}\right)+2Dd\tau dr,\end{equation}
where we have introduced the off-diagonal term proportional to $d\tau dr$
to account for transverse expansion or radial flow implied by energy-momentum
tensor in (\ref{eq:EMT01}). Note that the off-diagonal term related
to transverse expansion is absent in the metric (\ref{eq:metric-dim4}).
One can write the coefficients $A$, $B$, $C$ and $D$ in exponential
forms, \begin{equation}
A=e^{w},\; B=e^{b},\; C=e^{c},\; D=e^{d}.\end{equation}
In general these coefficients are functions of $x$ and $z$. A standard
way of holographic renormalization \citep{Karch:2005ms,Skenderis:2002wp}
is to expand the metric $\widetilde{g}_{\mu\nu}(x,z)$ or these coefficients
in the fifth coordinate $z$, \begin{equation}
\widetilde{g}_{\mu\nu}(x,z)=\sum_{n=0}z^{2n}\widetilde{g}_{\mu\nu}^{(2n)}(x),\end{equation}
where $\widetilde{g}_{\mu\nu}^{(0)}(x)$ is just the metric (\ref{eq:metric-dim4})
in 4-dimensional space. The second order term $\widetilde{g}_{\mu\nu}^{(2)}(x)$
can be proved to be vanishing. The fourth order term $\widetilde{g}_{\mu\nu}^{(4)}(x)$
is given by the energy-momentum tensor, \begin{equation}
\widetilde{g}_{\mu\nu}^{(4)}(x)\propto\left\langle T_{\mu\nu}\right\rangle ,\end{equation}
where we set factor $\frac{N^{2}}{2\pi^{2}}=1$ ($N$ is the number
of colors) and we will finally restore it. Given by these boundary
conditions one can solve the $AdS$ metric (\ref{eq:gen-metric01})
from the Einstein equation with the cosmological constant $\Lambda=-6$
which we will discuss about in Sec. (\ref{sec:Boost-geometry}), \begin{equation}
R_{MN}-\frac{1}{2}g_{MN}R+6g_{MN}=0,\label{eq:einstein01}\end{equation}
where $R_{MN}$ and $R$ are the curvature tensor and scalar in $AdS$
space with the metric $g_{MN}$ given in (\ref{eq:gen-metric01}).

\subsection{Holographic model in late time}

The introduction of the $r$ dependence of the metric (\ref{eq:metric01})
makes solving the Einstein equation a formidable task even with the
method of Janik and Peschanski \citep{Janik:2005zt,Nakamura:2006ih,Sin:2006pv}.
So the simplification is necessary. In this section we consider a
simplified problem with perturbation in transverse direction based
on the Janik-Peschanski method.

Supposing $\alpha$ is small in Eq. (\ref{eq:EMT01}) we can make
an expansion of $T_{\mu\nu}$ in $\alpha$,

\begin{eqnarray}
T_{\mu\nu} & = & \left(\begin{array}{cccc}
\rho & 0 & 0 & 0\\
0 & \tau^{2}\frac{1}{3}\rho & 0 & 0\\
0 & 0 & \frac{1}{3}\rho & 0\\
0 & 0 & 0 & r^{2}\frac{1}{3}\rho\end{array}\right)+\alpha\left(\begin{array}{cccc}
0 & 0 & -\frac{4}{3}\rho & 0\\
0 & 0 & 0 & 0\\
-\frac{4}{3}\rho & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array}\right)\nonumber \\
 &  & +\alpha^{2}\left(\begin{array}{cccc}
\frac{4}{3}\rho & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & \frac{4}{3}\rho & 0\\
0 & 0 & 0 & 0\end{array}\right)+O\left(\alpha^{3}\right).\label{eq:expansion01}\end{eqnarray}
In this expansion, we see that off-diagonal elements are proportional
to odd power of $\alpha$, while modifications of diagonal elements
are proportional to even power of $\alpha$. In leading order the
transverse velocity vanishes and we reproduce the results in Ref.
\citep{Janik:2005zt,Nakamura:2006ih}. In next-to-leading order $O(\alpha)$
the transverse velocity does not vanish while the energy density and
pressure do not change, whose modifications are in $O(\alpha^{2})$
or higher.

If we assume that there is no linear terms of $\alpha$ in diagonal
components of the AdS metric $g_{MN}$, we get the following result
by solving the Einstein equation (\ref{eq:einstein01}), \begin{equation}
g_{MN}=\frac{1}{z^{2}}\left(\begin{array}{ccccc}
-\frac{(1-a)^{2}}{1+a} & 0 & \frac{4}{3}\alpha\frac{(1-a)^{2}}{1+a} & 0 & 0\\
0 & \tau^{2}(1+a) & 0 & 0 & 0\\
\frac{4}{3}\alpha\frac{(1-a)^{2}}{1+a} & 0 & 1+a & 0 & 0\\
0 & 0 & 0 & r^{2}(1+a) & 0\\
0 & 0 & 0 & 0 & 1\end{array}\right),\label{eq:solution01}\end{equation}
with $a=\frac{\rho_{0}z^{4}}{3\tau^{4/3}}$ and $\rho_{0}$ is a constant
energy density.

In the case without off-diagonal components in the metric (\ref{eq:solution01}),
i.e. $\alpha=0$, the line element in central rapidity region (not
for non-central rapidity region) has the form, \begin{equation}
ds^{2}=-\frac{(1-z^{4}/z_{H}^{4})^{2}}{1+z^{4}/z_{H}^{4}}\frac{d\tau^{2}}{z^{2}}+\frac{1+z^{4}/z_{H}^{4}}{z^{2}}[\tau^{2}dy^{2}+dr^{2}+r^{2}d\theta^{2}]+\frac{dz^{2}}{z^{2}},\label{eq:static-metric}\end{equation}
where the black hole locates at \begin{equation}
z_{H}=\left(\frac{\rho_{0}}{3}\right)^{-1/4}\tau^{1/3},\label{eq:bh-zh}\end{equation}
which depends on $\tau$. See Appendix (\ref{sec:app-b}) for the
derivation of the metric (\ref{eq:static-metric}). Changing variables,
\begin{equation}
z\rightarrow\widetilde{z}=\frac{z}{\sqrt{1+\frac{z^{4}}{z_{H}^{4}}}},\qquad z_{H}\rightarrow\widetilde{z}_{H}=\frac{z_{H}}{\sqrt{2}},\label{eq:trans01}\end{equation}
we can verify that the line element (\ref{eq:static-metric}) is the
standard one in D3 black $AdS_{5}$ space if $z_{0}$ is constant,
see Sec. \ref{sec:Boost-geometry}, \begin{equation}
ds^{2}=-\frac{1-\widetilde{z}^{4}/\widetilde{z}_{H}^{4}}{\widetilde{z}^{2}}d\tau^{2}+\frac{d\mathbf{x}^{2}}{\widetilde{z}^{2}}+\frac{1}{1-\widetilde{z}^{4}/\widetilde{z}_{H}^{4}}\frac{d\widetilde{z}^{2}}{\widetilde{z}^{2}},\label{eq:ads-metric01}\end{equation}
where $d\mathbf{x}^{2}\equiv\tau^{2}dy^{2}+dr^{2}+r^{2}d\theta^{2}$
and $\widetilde{z}_{H}$ is the location of the black hole horizon,
the cosmology constant for $AdS_{d+1}$ is \citep{Skenderis:2002wp}
\begin{equation}
\Lambda=-\frac{d(d-1)}{2}=-6,\label{eq:cosmolog}\end{equation}
and the Hawking temperature is determined by the behavior of the metric
near the horizon, \begin{equation}
T_{H}=\frac{1}{\pi\widetilde{z}_{H}}=\frac{\sqrt{2}}{\pi z_{H}}.\label{Hawking temp}\end{equation}

Hereafter we call the proper time dependent $AdS_{5}$ metric the
$AdS^{\#}$ one in (\ref{eq:solution01}) with transverse expansion
or in (\ref{eq:static-metric}) without transverse expansion.

\section{Effective action for perturbations in $AdS$ metric\label{sec:Boost-geometry}}

The low energy limit of type IIB string theory in $AdS_{5}\times S^{5}$
space can be approximated by a five-dimensional theory (see, e.g.
\citet{Son:2007vk}) whose action is \begin{equation}
I_{5D}\approx\frac{N^{2}}{8\pi^{2}R^{3}}\int d^{5}x(\mathcal{R}_{5D}-2\Lambda+\cdots),\label{eq:action01}\end{equation}
where $\mathcal{R}_{5D}$ is the scalar curvature in $AdS_{5}$, $\Lambda$
the cosmological constant, and $R$ the radius of $S^{5}$. The line
element in $AdS_{5}$ space without black hole is \begin{equation}
ds^{2}=\frac{r^{2}}{R^{2}}(-dt^{2}+d\mathbf{x}^{2})+\frac{R^{2}}{r^{2}}dr^{2}.\label{eq:metric01}\end{equation}
Using the variable $z=R^{2}/r$, the above line element can be written
as \begin{equation}
ds^{2}=\frac{R^{2}}{z^{2}}(-dt^{2}+d\mathbf{x}^{2}+dz^{2}).\label{eq:metric02}\end{equation}
The line element with a black hole has the form \begin{eqnarray}
ds^{2} & = & \frac{R^{2}}{z^{2}}\left[-(1-z^{4}/z_{H}^{4})dt^{2}+d\mathbf{x}^{2}+\frac{1}{1-z^{4}/z_{H}^{4}}dz^{2}\right]\nonumber \\
 & = & \frac{\pi^{2}T^{2}R^{2}}{u}[-(1-u^{2})dt^{2}+d\mathbf{x}^{2}]+\frac{R^{2}}{4u^{2}(1-u^{2})}du^{2},\label{eq:metric03}\end{eqnarray}
where $u=z^{2}/z_{H}^{2}$ and $z_{H}=1/(\pi T)$.

The AdS/CFT correspondence can be expressed by \begin{equation}
Z_{4D}[J]=\left.e^{iS_{5D}[\phi]}\right|_{J(x)=\phi(x,z=0)},\label{eq:duality}\end{equation}
where $Z_{4D}[J]$ is the partition function of CFT in 4-dimensional
Minkowski space and $S_{5D}[\phi]$ the action in $AdS_{5}$ given
by (\ref{eq:action01}) for the classical bulk field $\phi(x,z)$,
$J(x)$ is the source in 4-dimension taking the value of the bulk
field on the boundary $z=0$ in $AdS_{5}$. The partition function
$Z_{4D}[J]$ is a functional of $J(x)$, \begin{equation}
Z_{4D}[J]=\int[d\varphi]\mathrm{exp}\left\{ iS_{CFT}[\varphi]+i\int d^{4}xJO[\varphi]\right\} ,\end{equation}
where $\varphi$ denotes the CFT fields, $O[\varphi]$ is the operator
coupling to $J(x)$, and $S_{CFT}[\varphi]$ is the CFT action. Following
the AdS/CFT correspondence the Green functions of operators $O$ in
CFT can be evaluated in terms of $S_{5D}[\phi]$ in $AdS_{5}$, for
example, a two-point Green function is \begin{equation}
\left\langle TO(x_{1})O(x_{2})\right\rangle _{4D}=\left.\frac{\delta^{2}S_{5D}[\phi]}{i\delta\phi(x_{1})\delta\phi(x_{2})}\right|_{J(x)=\phi(x,z=0)}.\label{eq:green-f}\end{equation}

We note that the stress tensor is coupled with the metric, \begin{equation}
T_{\mu\nu}=-\frac{2}{\sqrt{-\widetilde{g}}}\frac{\delta S_{CFT}}{\delta\widetilde{g}_{\mu\nu}},\end{equation}
where $T_{\mu\nu}$ can be regarded as an operator in CFT and the
metric $\widetilde{g}_{\mu\nu}$ in 4-dimension as its source. In
order to obtain the shear viscosity $\eta$ through the Kubo formula,
\begin{equation}
\eta=-\lim_{\omega\rightarrow0}\frac{1}{\omega}\mathsf{\mathrm{Im}}G_{12,12}^{R}(\omega,\mathbf{0}),\label{eq:kubo01}\end{equation}
we need to know the Green function of $T_{12}$ at low energy limit,
\begin{equation}
G_{12,12}^{R}(\omega,\mathbf{k})=-i\int d^{4}xe^{ik\cdot x}\Theta(t)\left\langle [T_{12}(x),T_{12}(0)]\right\rangle .\label{eq:kubo02}\end{equation}
To obtain $G_{12,12}^{R}(\omega,\mathbf{k})$ from the gravitational
dual we should know the action $S_{5D}[g_{\mu\nu}]$ in $AdS_{5}$
as a functional of $g_{\mu\nu}$ with the boundary value $g_{\mu\nu}(z=0)\rightarrow\widetilde{g}_{\mu\nu}$.
For this purpose we consider a variation in the $AdS_{5}$ metric,
$g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu}$. We can choose for simplicity
$h_{z\mu}=h_{\mu z}=0$ and that $h_{\mu\nu}$ depend on $t$, $x^{3}$
and $z$ (or $u$). The line element in (\ref{eq:metric03}) then
becomes \begin{equation}
ds^{2}=\frac{\pi^{2}T^{2}R^{2}}{u}[-(1-u^{2})dt^{2}+d\mathbf{x}^{2}]+\frac{R^{2}}{4u^{2}(1-u^{2})}du^{2}+h_{\mu\nu}dx^{\mu}dx^{\nu}.\label{eq:metric04}\end{equation}
We can expand the action (\ref{eq:action01}) in terms of $h_{\mu\nu}$,
\begin{eqnarray}
I_{5D} & \approx & \frac{N^{2}}{8\pi^{2}}\int d^{5}x\sqrt{-g}(\mathcal{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\mathcal{R}+\Lambda g_{\mu\nu})h^{\mu\nu}\nonumber \\
 & = & \frac{N^{2}}{8\pi^{2}}\int d^{5}x\sqrt{-g}(\mathcal{R}_{\mu\nu}^{(0)}-\frac{1}{2}g_{\mu\nu}^{(0)}\mathcal{R}_{(0)}+\Lambda g_{\mu\nu}^{(0)})h^{\mu\nu}\nonumber \\
 &  & +\frac{N^{2}}{8\pi^{2}}\int d^{5}x\sqrt{-g}(\delta\mathcal{R}_{\mu\nu}-\frac{1}{2}h_{\mu\nu}\mathcal{R}_{(0)}-\frac{1}{2}g_{\mu\nu}^{(0)}\delta\mathcal{R}+\Lambda h_{\mu\nu})h^{\mu\nu}\nonumber \\
 & = & \frac{N^{2}}{8\pi^{2}}\int d^{5}x\sqrt{-g}(\delta\mathcal{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}^{(0)}\delta\mathcal{R}+4h_{\mu\nu})h^{\mu\nu},\label{eq:ation02}\end{eqnarray}
where we have used the fact that the first term of the second equality
is vanishing from the Einstein equation for the unperturbed metric
$g_{\mu\nu}^{(0)}$. We have also used $\mathcal{R}_{0}=-20$ and
$\Lambda=-6$. Here we have chosen the $S^{5}$ radius $R=1$. We
consider the components $h_{12}=h_{21}$ which can be verified to
decouple from others. For the metric (\ref{eq:metric04}), the action
(\ref{eq:ation02}) can be simplified as \begin{eqnarray}
I_{5D} & \approx & \frac{N^{2}}{8\pi^{2}}\int d^{5}x\sqrt{-g}(\delta R_{12}+4h_{12})h^{12}=\frac{N^{2}}{8\pi^{2}}\int d^{5}x\sqrt{-g}\left(-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\cdots\right),\label{eq:action03}\end{eqnarray}
with $\phi=h_{2}^{1}=g^{11}h_{12}$ and $h^{12}=-g^{11}g^{22}h_{12}$.
The retarded two-point Green functions of the CFT stress tensor can
then be derived from the quadratic terms in $I_{5D}$ as above by
using Eq. (\ref{eq:green-f}). The shear viscosity is then obtained
through the Kubo formula (\ref{eq:kubo01}).

\section{Shear viscosity and entropy density in $AdS_{5}^{\#}$ }

In this section we will calculate the shear viscosity in $AdS_{5}^{\#}$
following the procedure of AdS/CFT duality presented in the previous
section. Also we will compute the entropy density in $AdS_{5}^{\#}$.
We consider fluid evolutions in 1+1/2+1 dimension without/with transverse
expansion or radial flow.

\subsection{1+1 dimension without transverse expansion\label{sec:ratio1+1} }

Following Janik and Peschanski's solution \citep{Janik:2005zt} in
late time, the line element in $AdS_{5}^{\#}$ in 1+1 dimension can
be written as\begin{equation}
ds^{2}=\frac{1}{z^{2}}\left\{ -\frac{(1-a)^{2}}{1+a}d\tau^{2}+(1+a)[\tau^{2}dy^{2}+dx_{1}^{2}+dx_{2}^{2}]\right\} +\frac{dz^{2}}{z^{2}},\label{eq:dim11-metric}\end{equation}
where $a=\frac{\rho_{0}z^{4}}{3\tau^{4/3}}$. Note that the derivative
of $a$ is non-zero, i.e. $\partial_{\tau,z}a\neq0$ in computing
the Ricci tensor and the scalar curvature. Finally we will keep a
constant $a$ while taking the limit $\tau\rightarrow\infty$ when
doing power counting in $\tau$. In adopting the above measure we
can verify after a lenghy but straightforward algebra that the Ricci
tensor $\mathcal{R}_{\mu\nu}=-4g_{\mu\nu}$ and $\mathcal{R}_{5D}=-20$
upto higher order terms in negative powers of $\tau$. The determinant
of the metric is $\sqrt{-g}=\frac{1}{z^{5}}(1-a^{2})\tau$. Taking
a perturbation in the metric $g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu}$
with $h_{12}=h_{12}(\tau,y,z)$ and $h_{\mu\nu}=0$ for all other
indices $\mu,\nu$, we obtain the quadratic terms of the effective
action up to higher order contributions in negative powers of $\tau$,
\begin{equation}
I_{5D}\approx-\frac{N^{2}}{16\pi^{2}}\int d^{5}x\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,\label{eq:action01-1}\end{equation}
where $\phi\equiv h_{2}^{1}=g^{11}h_{12}$, $d^{5}x=d\tau dydx_{1}dx_{2}dz$
and $\mu=\tau,y,1,2,z$. The equation of motion for $\phi$ is then
\begin{equation}
\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)=0.\label{eq:EOM001}\end{equation}
We assume a factorized form for $\phi$, \begin{equation}
\phi(\tau,y,z)=\phi_{0}(y,\tau)f(z)=\int\frac{d\omega dp_{3}}{(2\pi)^{2}}\exp(-i\omega\tau\cosh y+ip_{3}\tau\sinh y)\phi_{0}(\omega,p_{3})f_{p}(z)\label{eq:factorization}\end{equation}
where $p_{3}$ is the momentum along the third axis (we do not distinguish
subscript or superscript for this mometum). Inserting the above into
Eq. (\ref{eq:EOM001}) and taking the limit $\tau\rightarrow\infty$
while keeping $v$ or $a$ constant, the leading order contribution
of $O(\tau^{0})$ in Eq. (\ref{eq:EMT01}) gives \begin{eqnarray}
0 & = & (1-a^{2})\frac{d^{2}f_{p}}{dz^{2}}+f_{p}\frac{(1+a)^{2}}{1-a}\omega^{2}\cosh^{2}y-f_{p}(1-a)(-\omega\sinh y+p_{3}\cosh y)^{2}.\end{eqnarray}
We consider the central rapidity $y=0$ and static limit $p_{3}=0$,
the above equation becomes \begin{equation}
\frac{d^{2}f_{p}}{dz^{2}}+\frac{(1+a)}{(1-a)^{2}}\omega^{2}f_{p}=0.\label{eq:eom-f}\end{equation}
The reason for choosing central rapidity is that the metric (\ref{eq:dim11-metric})
can only be treated as an extension of the standard AdS one at central
rapidity, see Appendix \ref{sec:app-b}. Considering the solution
near the horizon at $a_{H}=1$ and changing variables, \begin{eqnarray}
z & \rightarrow & z'=\left(\frac{\rho_{0}}{3}\right)^{1/4}\tau^{-1/3}z,\nonumber \\
\omega' & \rightarrow & \frac{1}{2\sqrt{2}}\left(\frac{\rho_{0}}{3}\right)^{-1/4}\tau^{1/3}\omega,\label{eq:change-var}\end{eqnarray}
Eq. (\ref{eq:eom-f}) can be rewritten\begin{equation}
\frac{d^{2}f_{p}}{d{z'}^{2}}+\frac{1}{(1-z')^{2}}{\omega'}^{2}f_{p}=0.\label{eq:eom-f1}\end{equation}
The solution to Eq. (\ref{eq:eom-f1}) can be found, \begin{equation}
f_{p}(z)=(1-z')^{1/2(1\pm\sqrt{1-4{\omega'}^{2}})}\approx(1-z')^{1/2\pm i\omega'},\label{eq:solution-f}\end{equation}
where we have approximated $\sqrt{1-4{\omega'}^{2}}\approx i2\omega'$
at the limit $\tau\rightarrow\infty$, since $\omega'\sim\tau^{1/3}\gg1$.
Using the incoming wave solution corresponding to the positive sign
in (\ref{eq:solution-f}) and substituting it back into the action
(\ref{eq:action01-1}), the boundary term at $z=z_{H}$ or the black
hole horizon at $a_{H}=1$, the retarded Green function $G_{12,12}^{R}(\omega)$
at the static limt can be obtained, which is vanishing due to the
presence of the real part $\frac{1}{2}$ in the exponent of the solution
(\ref{eq:solution-f}). This can be seen from the fact that there
is a factor $(1-z')$ which is zero at the boundary, see Eq. (\ref{eq:sol1})
in Appendix (\ref{sec:app-a}). Therefore we have shown that the shear
viscosity is absent from the scaling solution in 1+1 dimension in
the leading order.

Picking up the next-to-leading order contribution of $O(1/\tau^{1/3})$
in Eq. (\ref{eq:EOM001}), Eq. (\ref{eq:eom-f}) becomes \begin{equation}
\frac{d^{2}f_{p}}{dz^{2}}-\frac{3+5a^{2}}{z(1-a^{2})}\frac{df_{p}}{dz}+\frac{(1+a)}{(1-a)^{2}}\omega^{2}f_{p}=0.\label{eq:eom-f2}\end{equation}
Changing variables as in Eq. (\ref{eq:change-var}), the above equation
near the horizon can be rewritten in the form \begin{equation}
\frac{d^{2}f_{p}}{d{z'}^{2}}-\frac{1}{1-z'}\frac{df_{p}}{dz'}+\frac{1}{(1-z')^{2}}{\omega'}^{2}f_{p}=0,\label{eq:eom-f3}\end{equation}
whose solutions are \begin{equation}
f_{\pm p}(z)=(1-z')^{\pm i\omega'}.\label{eq:solution-f1}\end{equation}
As derived in Eq. (\ref{eq:sol2}) in Appendix \ref{sec:app-a}, the
retarded Green function at static limit is obtained, \begin{equation}
G_{12,12}^{R}(\omega,0)=-2F(\omega)=-i\frac{\sqrt{2}}{3^{3/4}\times4\pi^{2}}N^{2}\rho_{0}^{3/4}\tau^{-1}\omega.\end{equation}
The shear viscosity per unit transverse area can be obtained from
the Kubo formula (\ref{eq:kubo01}), \begin{equation}
\eta=\frac{\rho_{0}^{3/4}\sqrt{N}}{6^{3/4}\sqrt{\pi}}\frac{1}{\tau}.\label{eq:shear01}\end{equation}
where we have recovered the factor $\rho_{0}\rightarrow\rho_{0}\left(\frac{N^{2}}{2\pi^{2}}\right)^{-1}$.

The Hawking temperature is $T=\frac{\sqrt{2}}{\pi z_{H}}$ from Eq.
(\ref{Hawking temp}) where $z_{H}$ is the horizon of the black hole
given in Eq. (\ref{eq:bh-zh}). Then the initial energy density is
a constant due to $T\sim\tau^{-1/3}$, \begin{equation}
T=\frac{\sqrt{2}}{\pi z_{H}}=\frac{\sqrt{2}}{\pi}\left(\frac{2\pi^{2}\rho_{0}}{3N^{2}}\right)^{1/4}\tau^{-1/3}.\end{equation}
The entropy per unit rapidity and unit transverse area is given by
\citep{Janik:2005zt,Sin:2006pv},\begin{equation}
S=\left(\frac{N^{2}}{2\pi}\right)^{1/4}\left(\frac{\pi}{3}\right)^{3/4}2\sqrt{2}\rho_{0}^{3/4}=\frac{N^{2}}{2}\pi^{2}T^{3}\tau.\end{equation}
The entropy density is obtained, \begin{equation}
s=\frac{S}{\tau}=\frac{1}{2}N_{c}^{2}\pi^{2}T^{3},\label{eq:entropy01}\end{equation}
where we see that the entropy density has an asymptotic behavior $s\sim\tau^{-1}$.
From Eq. (\ref{eq:shear01}) and (\ref{eq:entropy01}), we get the
well-known value, \begin{equation}
\frac{\eta}{s}=\frac{1}{4\pi}.\end{equation}

\subsection{2+1 dimension with transverse expansion \label{sec:ratio2+1}}

The metric in 2+1 dimension with radial flow is given in (\ref{eq:solution01}),
in order to calculate the shear viscosity we have to introduce a perturbation
to the background metric. It is then convenient to explicitly use
rectangular transverse coordinates $(x_{1},x_{2})$ instead of cylindrical
ones $(r,\theta)$. The corresponding line element is \begin{equation}
ds^{2}=\frac{1}{z^{2}}\left\{ -\frac{(1-a)^{2}}{1+a}dt^{2}+(1+a)(dx_{3}dx_{3}+dx_{i}dx_{i})+\frac{8}{3}\alpha\frac{(1-a)^{2}}{1+a}\frac{1}{r}x_{i}dx_{i}dt+\frac{dz^{2}}{z^{2}}\right\} ,\label{eq:back-tr}\end{equation}
where $r=\sqrt{x_{i}x_{i}}$ and the summation over $i=1,2$ is implied.
We keep in mind that $\alpha$ is small. The metric determinant has
the same form as in the 1+1 dimensional case in the leading order,
\begin{equation}
\sqrt{-g}=\frac{1}{z^{5}}(1-a^{2})+O(\alpha^{2}).\end{equation}
Now we consider the perturbation $h_{12}(t,x_{3},z)$ to the background
metric (\ref{eq:back-tr}). Denoting $\phi=h_{2}^{1}(t,x_{3},z)$,
we find the equation of motion for $\phi$, \begin{equation}
0=\delta\mathcal{R}_{2}^{1}+4\phi=-\frac{1}{2\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi).\label{eq:eom-h12}\end{equation}
It can be verified that the entangled term $\sim\alpha\phi$ appear
in $O(t^{-1/3})$, while leading terms are of $O(t^{2/3})$. So the
transverse part is decoupled from $\phi$ in the leading order. The
shear viscosity and entropy density per unit transverse area are the
same as in the 1+1 dimensional case, Eqs. (\ref{eq:shear01}) and
(\ref{eq:entropy01}).

We can also consider the perturbation along $x_{2}$ axis (or equivalently
$x_{1}$ axis), i.e. $\phi=h_{3}^{1}(t,z,x_{2})$. To the leading
order $O(\tau^{2/3})$ we find the equation of motion for $\phi$,
\begin{equation}
0=\delta\mathcal{R}_{3}^{1}+4\phi=-\frac{1}{2\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi)+t^{2/3}\frac{2}{3}\alpha\frac{1}{1+a}\frac{(x^{2})^{2}}{r^{3}}\partial_{t}\phi,\label{eq:eom-h13-01}\end{equation}
whose explicit form reads, \begin{equation}
-\partial_{z}^{2}\phi+\frac{1+a}{(1-a)^{2}}\partial_{t}^{2}\phi-\frac{4}{3}\alpha\frac{1}{1+a}\frac{(x^{1})^{2}}{r^{3}}\partial_{t}\phi=0.\label{eq:eom-h13-02}\end{equation}
We see that the transverse part enters the equation of motion in the
leading order. We assume that the solution has the factorization form
$\phi(t,x_{2},z)=\phi_{0}(p)f_{p}(z)e^{-i\omega t+ip_{2}x_{2}}$,
then we derive from Eq. (\ref{eq:eom-h13-02}) the differential equation
for $f_{p}(z)$,

\begin{equation}
\frac{d^{2}f_{p}}{dz^{2}}-\frac{3+5a}{z(1-a^{2})}\frac{df_{p}}{dz}+\frac{1+a}{(1-a)^{2}}\omega^{2}f_{p}-\frac{i}{1+a}\frac{2\alpha}{3r}\omega f_{p}=0,\end{equation}
where we can expand the equation near the horzion $z\rightarrow z_{H}$
in Eq. (\ref{eq:bh-zh}),

\begin{equation}
\frac{d^{2}f_{p}}{d{z'}^{2}}-\frac{1}{1-z'}\frac{df_{p}}{dz'}+\frac{1}{(1-z')^{2}}{\omega'}^{2}f_{p}-i\alpha\frac{2\sqrt{2}z_{H}}{3r}\omega'f_{p}=0.\label{eq:EOM-21}\end{equation}
The solution is a linear combination of Bessel functions, \begin{eqnarray*}
f_{p1} & = & J_{i\omega'}\left[i^{3/2}(1-z')z_{H}\sqrt{\frac{\alpha\omega}{3r}}\right],\\
f_{p2} & = & Y_{i\omega'}\left[i^{3/2}(1-z')z_{H}\sqrt{\frac{\alpha\omega}{3r}}\right].\end{eqnarray*}
Since $\alpha$ is small, we can make expansion in $\alpha$ for the
solutions $f_{p1}$ and $f_{p2}$. One can verify that the solution
is a linear combination of $f_{p}$ and $f_{-p}$, \begin{eqnarray}
f_{p} & = & (1-z')^{i\omega'}z_{H}^{i\omega'}\left(\frac{\alpha\omega}{3r}\right)^{i\omega'}\left[\frac{(-1)^{i3\omega'/4}2^{-i\omega'}}{\Gamma(1+i\omega')}+\frac{i(-1)^{i3\omega'/4}2^{-2-i\omega'}}{(1+i\omega')\Gamma(1+i\omega')}(1-z')^{2}z_{H}^{2}\frac{\alpha\omega}{3r}\right],\nonumber \\
f_{-p} & = & f_{p}(i\omega'\rightarrow-i\omega').\label{eq:sol21}\end{eqnarray}
One sees that $f_{-p}=f_{p}^{*}$. Following Eq. (\ref{eq:sol3})
in Appendix \ref{sec:app-a} and steps in previous section, we get
the same value as in the case of 1+1 dimension, i.e. $\eta/s=1/(4\pi)$.

\section{Summary and discussions \label{sec:Conclusion}}

We derive a time dependent metric dual to sQGP fluid in 2+1 dimension
with radial flow in late time by holographic renormalization. It is
difficult to obtain the exact solution to the Einstein equation with
this metric, especially when the metric has off-diagonal components
for radial flows. If transverse expansion is small and can be treated
as a perturbation, the late time asymptotic solution, the metric in
(\ref{eq:solution01}) with off-diagonal elements, can be found by
using $v=z^{4}/\tau^{4/3}$ as a scaling and expansion parameter.
With this metric we calculate the ratio $\eta/s$ of shear viscosity
$\eta$ to entropy density $s$ for sQGP in SYM field theory with
the KSS method. As a first attempt we consider 1+1 dimension with
only longitudinal flow whose metric is diagonal. If we include only
the leading order terms in the equation of motion for perturbations
to the metric, the shear viscosity is vanishing, consistent to the
assumption that the fluid is ideal. We reproduce KSS bound $1/(4\pi)$
for $\eta/s$ if we pick up the next-to-leading order term in the
equation of motion, indicating that the shear viscosity is a higher
order effect. Our derivation is based on the Janik and Peschanski's
method and is valid in late time, $\tau\rightarrow\infty$. For intermidiate
stage of hydrodynamic evolution the ratio is not necessarily $1/(4\pi)$
for an expanding fluid, so our result is not trivial or obvious. We
further show that the ratio for fluids in 2+1 dimension in late time
with transverse flow is the same as in 1+1 dimension in the leading
order of transverse rapidity. We remember that the mean free path
is $l_{\mathrm{mfp}}\sim1/(n\sigma v)$, where $n\sim T^{3}$ is the
particle number density, $\sigma\sim g^{2}T^{-2}$ ($g$ is the coupling
constant) the typical scattering cross section and $v\sim1$ the typical
velocity. The shear viscosity can then be estimated as $\eta\sim\rho l_{\mathrm{mfp}}\sim g^{-2}T^{3}$,
where $\rho$ is the energy density. In comparsion with our result
in Eq. (\ref{eq:shear01}), we have $\eta\sim\tau^{-1}$, $T\sim\tau^{-1/3}$
and $g$ does not change with the proper time. This implies that if
the coupling is as strong at the beginning as in late time of fluid
expansion or hydrodynamic evolution does not influence the strength
of the interaction.

In 1+1 dimension one can introduce the shear viscosity of the next-to-leading
order in the stress tensor in late time solution \citet{Janik:2006ft,Nakamura:2006ih,Sin:2006pv}.
So an interesting attempt is to calculate the shear viscosity with
the metric dual to the stress tensor with shear terms. We found that
the correction to the shear viscosity is also of the next-to-leading
order, $\eta/s=1/(4\pi)+O(\tau^{-1-s})$.

The same analysis can also be applied to hydrodynamic behaviors of
sQGP in early time, which is important to understand the initial state
of QGP \citet{Kovchegov:2007pq}. However early time behaviors of
fluids show anisotropic feature and therefore more complicated. We
will reserve it for a future investigation.

\begin{acknowledgments}
We thank D. Rischke, H.-c. Ren and P.-f. Zhuang for helpful discussions.
Q.W. is supported in part by '100 talents' project of Chinese Academy
of Sciences (CAS), by National Natural Science Foundation of China
(NSFC) under the grants 10675109 and 10735040.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Always ready to assist you
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redpajama_set_name: RedPajamaCommonCrawl


Well, it seems to me that many artists out there are questioning the upcoming deadline for entries to Studio Visit Magazine. I've had quite a few hits to my blog searching for Studio Visit or Studio Visit scam. So, here's something that I can discuss without getting into trouble for a change. Yes, you should enter! Studio Visit is the sister publication to New American Paintings. Both of which are published by The Open Studios Press. I'm sure many of you are unsure of Studio Visit because you've never heard of it. True, it's new and fairly unknown, but I've noticed a remarkable increase in popularity over the past few years, so they are building a following. Studio Visit just started a matter of years ago- 3 or 4, if memory serves. They are lacking the legions of fans of New American Paintings, but that's not to say they won't build their own impressive fan base. Good things come to those who wait… or those who with good PR efforts. Regardless, the magazine is a good-looking publication. I've been in it before, so I can vouch for their slick presentation. The jurors aren't too shabby, either. As with New American Paintings (or NAP, for short), they've managed to get prestigious museum directors and curators to jury their exhibitions-in-print.
I won't say that getting into Studio Visit will be the overnight success you are looking for… then again, I wouldn't say that about anything. However, it will get your work in front of galleries. The magazine is shipped out to galleries and museums all over the place, as is NAP. I can say that I've had a gallery contact me as a result and it helped grease the wheels to get me representation with another gallery. It jogs people's memories and gets them thinking about your work. And when your work is in front of a gallery over and over again, your name starts to stick. So I leave you with this: if you'd like to have your work noticed by important people, in a high-quality publication, then the upcoming deadline is February 5th.


If you have diabetes, managing your HbA1C levels is essential. During the festive season it can be a bigger challenge to stick to your meal plan and keep your blood sugar levels steady.
Here are some simple planning strategies and nutrition swaps to help you enjoy the festivities and yet stay on track.
Before you get to any event, have a plan in place for what you will eat. For example, if you know that there will only be snacks, plan to select those that are lower in calories and won't spike your blood sugar, such as vegetables and lean proteins. Be cautious of dips or sauces that can add hidden calories, salt or sugar. You can also plan ahead by looking at online restaurant menus to plan your meal ahead of time. If the event is at a house, ask the host what he or she is planning to serve, and offer to bring your own healthy dish that complements the meal.
Skipping a meal can negatively impact your blood glucose level, says Varma . If the holiday schedule has you busy, keep diabetes friendly options close like Ensure Diabetes Care. Made by Abbott, Ensure Diabetes Care has blends of carbohydrates that are slowly digested and absorbed to help minimize blood sugar spikes. With fewer than 230 calories per serve it's a smart and portion controlled choice.
Staying hydrated and drinking water is important for our overall health and well-being, but recent research shows it may also help with the calories we consume. Using the National Health and Nutrition Examination Survey data, researchers found that drinking more water is associated with eating fewer calories, as well as less sugar, salt and cholesterol. While more controlled research on this topic should be done to confirm this finding, choosing water may be helpful to limit other sugar-sweetened drinks.
Foods rich in carbohydrates can provide a lot of good nutrients. But compared to fats and protein, carbs have the most adverse effect on blood sugar levels. This is why it's important to choose your carbs wisely, and go with low glycemic foods – carbohydrates that are slowly digested and don't impact your blood sugar levels. In contrast, foods with a high glycemic index and glycemic load are digested rapidly and cause spikes in blood sugars. Limit refined and processed carbohydrates and choose fiber-rich whole foods such as apples, carrots, beans and oats.
Most traditional holiday recipes can be made healthier with simple substitutions.
Did you know that it takes 20 minutes for your brain to catch up with your stomach and let you know that you are "full"? Research has shown that when meals are consumed slowly, people eat significantly fewer calories and consume more water. To help slow down your eating – try to chew slowly, place your utensil down or take a sip of water in between bites.
If dessert is important, then plan ahead for it. Eat fewer carbs throughout the day, so you can enjoy a mithai. Remember to be mindful of portion sizes and skip the extra helpings.
Although most holiday traditions revolve around meal time, consider adding a physical activity into it. Perhaps go for a family walk after your meal.
Lastly, if a food setback happens, don't give up. "A lot of us can tend to overindulge despite our best efforts," says Varma . "Instead of feeling guilt about it, acknowledge your feelings and focus on getting back on track at your next meal". With the above tips, it is possible to enjoy the holidays and stay on track with your health. Just make a plan… and be merry!


Anguilla is a special and unusual island, in so many ways.
Its dry and sunny weather, warm and genuinely friendly people, and pristine, white-sand beaches are the obvious and well-known.
It is also famous for its luxurious Anguilla villas for rent, five-star resorts and high-end beach-side dining. For an idea of luxury Anguilla villa living, see the video below.
These "cachet factors" include the little bit of extra travel it takes to reach Anguilla, its people's passion for freedom (including not paying taxes), the scarcity of land, and its special status and stability as a separate British dependency since 1980.
Still relatively unknown, celebrities and the business elite may go to St. Barths to be seen, but they come to Anguilla to disappear. Its popularity has grown as the likes of Brad Pitt and Jay-Z vacation on Anguilla's sandy shores (leaving the paparazzi and their bodyguards behind in St. Barths!).
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A high-end luxury Anguilla villa, or economical living, everything feels more flexible when staying in your own villa. We always felt more "on our own" and enjoyed that independence.
"So What's Your Favorite Anguilla Villa?"
That's usually the second question people ask me when we talk about Anguilla (after the beaches of course!).
Have you fallen in love with your villa? Share it with everyone!
Visitors to this site would love to hear your favorites, too!
"I didn't even know I needed that," is a phrase you find yourself saying a lot at Champagne Shores. Plush outdoor and indoor living areas and five luxurious suites lay out over the villa's 7,000 square feet. It is all air conditioned (central air) and it all gazes out to one of Anguilla's most panoramic views of the Caribbean Sea and St. Martin. Beyond the villa itself, service is outstanding. It is the only villa in Anguilla at this price point that includes a personal service attendant throughout the day. Champagne Shores provides exceptional value in every way.
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One of Anguilla's premier ultra-luxury villas, Nevaeh sits right on the secluded shores of Long Bay beach. Peaceful, with a zen-like ambiance, a Balinese design, eight plush bedrooms and some of the most unique amenities, Nevaeh is heaven.
Bird of Paradise is an award-winning luxury Anguilla villa. Built with passion and love, the Dill family tells their story of how they constructed this most unique Balinese-inspired villa, and how they found and fell in love with Anguilla.
Sheriva's exclusive property features three ultra-luxurious villas. Each villa boasts 14,000-21,000 feet, with multiple pools, bedrooms, dining and living spaces. With terrific views of the Caribbean Sea, and just a 2 minute drive from Maunday's Bay, the location is divine, too!
Exclusivity Villa is, as the name suggests, truly exclusive. Located in the remote Eastern End, on Captain's Bay, Exclusivity stands alone. This luxury villa is for those seeking total privacy and tranquility.
A five-bedroom villa that blends a classic Caribbean feel with beach house chic and comfort, Beach Escape Villa is true to its name. This hideaway sits directly on one of Anguilla's favorite hidden gem beaches. It is a "best" if you are seeking true Caribbean beachfront villa living!
With expansive views of St. Martin, an infinity edge pool, 3 master bedrooms (with ensuite bathrooms and private patios/balconies), Dropsey Bay at our finger tips, signing privileges at CuisinArt, and personalized, caring five-star service, this is Anguilla luxury for a relatively more affordable price.
The Globus family tells how they came to build Tequila Sunrise, a villa full of warm family memories.
My sister and my Dad's high-class, luxury Anguilla villa getaway!
My Mom and Dad's beach-side Anguilla villa experience!
With sweeping views of Anguilla's Southern shore, clean lines, modern Italian furnishings and a large outdoor living space that maximizes the view, Rum Punch impresses from the start.
A Caribbean dream home designed for luxurious comfort, tucked away on Blackgarden Bay.
This luxury beachfront Anguilla Villa is a top pick for families with children and multi-generational family gatherings. A warm and inviting Tuscan villa with amenities that delight kids from 1-99. Multiple outdoor spaces provide the opportunity to enjoy the sunsets and sounds of crashing waves in a private moment, or all together!
Set on the south shore of Anguilla, overlooking St. Martin, Three Dolphins Villa is one luxury villa that exudes a certain warmth and true, tropical feel. Owned by Chelle, here she shares her story of how she came to fall in love with the island and find her dream home, Three Dolphins.
Perched on the edge of North Hill overlooking Sandy Ground, this villa is world-famous for its endless views and sensational sunsets. Featured on HGTV, Spyglass Hill is both luxurious and comfortable. With its warm tones and use of wood throughout the house, Spyglass Hill truly feels like "home." From Montserrat, Ian "Sugar George" and Janine Edwards share their story of perserverence and how their project of passion, Spyglass Hill, came to be.
One of Anguilla's coolest villas, Sandcastle Villa: The Beach House sits directly on the sands of Limestone Bay. It is one of the few villas in Anguilla that is on the beach and available at a reasonable cost. With all modern furnishings and plenty of activities included (including Jeep and Dingy!), Sandcastle Villa: The Beach House is a high-value option.
Contemporary and chic, Beaches Edge is made up of two five-bedroom, modern villas on the island's south cast. Facing due south to St. Martin, the views are panoramic, some of the most stunning on the island. Each villa features a kitchen (with every superior appliances of every kind), living room, dining room and media center. The villas can be rented together or separately.
Modern and newly built, Kiki Villa offers all of the conveniences of home just a short drive from Shoal Bay East. It features luxurious furnishings and many Anguillian accents throughout the villa, in addition to flat-screen TVs, washing machine and dryer, a foosball table and outdoor living spaces with hammock, BBQ and a pool.
Perched overlooking windswept Savannah Bay, Wesley House is one of Anguilla's most unique villas. Whimsical in its open-air Caribbean design, and being one of the few villa rentals that overlooks the vast and scenic east end of Anguilla, Wesley House is a personal favorite. Wesley House's official website here.
A charming Anguila villa enclave, Twin Palms is made up of two villas, Coconut Palm and Beach Palm. Both have a laid-back, Caribbean beach cottage feel, with all of the amenities you could need to feel right at home in Anguilla, with Meads Bay just a 2 minute walk away (literally across the street from the villa).
Our second favorite Anguilla villa, our "happy disaster"... which ended up in one of our best stays ever at an Anguilla villa, perched spectacularly above Shoal Bay East, at a surprising do-able cost.
With a stunning view of charming Sandy Ground, and vistas out to Sandy Island and the open ocean, Topaz Villa has one of the best, central locations. Inside there are modern furnishings and plenty of amenities including a pretty infinity-edge pool.
Owned by Mary Pat & Raoul of Hibernia Restaurant, Villa Hibernia is an exotic beauty. The three bedroom villa is a far-eastern dream, complete with a pool and set on the sea rocks overlooking idyllic Island Harbour.
Tucked into the hill surrounding Long Bay, Morgan Hill Villa offers comfortable, clean and private accommodations with endless views of the Atlantic ocean.
Our very first of all of the villas in Anguilla that ever hosted us, of course has very fond memories.
After many vacations spent in Anguilla, we return to our "first" Anguilla beach, this time trying a different villa rental. Cheap and clean and right on the same beach as "Ella's"!
Belonging to the famous "Sugar George" and Janine Edwards, Little Butterfly is a charming, affordable vacation rental in the Little Harbour area. With modern, luxurious fixtures and finishings, it's an excellent choice for a long-term stay.
With Susan & Ambrose as your kind and accommodating hosts, a stay in their home (whether to rent a room, private studio, or their full villa) is infused with Caribbean charm and a true warm spirit. With sea views, a full kitchen and living/dining area it is the complete package.
Spacious and bright Anguilla condos with sweeping views of the Caribbean Sea, Ocean Terrace Condos offers super high-value for long-term stays.
These villa rentals (complete with private pools) rent as both vacation units and long-term. For the space and design, this is one of the best accommodations deals to be had.
Not a full-size Anguilla villa, this one bedroom Anguilla rental is perfect for a couple traveling alone and seeking affordable accommodations. Complete with flatscreen TV, kitchen, terrace with view of St. Martin and even pool access, this gem of a rental is just 30 seconds from Rendezvous Bay.
Visit the link above for more reviews of Anguilla villas written by readers of this site.
Are You An Anguilla Villa Owner?
E-mail me (nori@anguilla-beaches.com) with a summary. Please attach a couple of photos of your villa and yourelf.
If you have a story that others will find fascintating, I'll reply with details on how to submit your story.
These Anguilla villas, while no longer available, remain in our memories.
Cozy beach cottage meets Caribbean cool at Moondance Villa! Set on a peaceful ridge with a vivid view of St. Martin and with excellent amenities, Moondance Villa offers exceptional value.
Little Harbour Villa Our second stay on Anguilla was in a spectacular, private villa, gazing out at the St. Martin scenery. We loved it there so much we stayed there twice, before the family moved back in!
C-Breeze Inn Villa A friend told us about this economical little cottage on a lake. It's an ideal villa for two travelers on a budget.
The Thai House Exotic and mysterious, Nick Douglas' 300 year old Thai House on Shoal Bay East is not only beautiful, but makes for one of the most intriguing Anguilla villa stories.
Picture-perfect, The Boat House sits directly over calm waters in the Corito Bay area.
Airbnb has reached the shores of Anguilla! Today, more and more villas are listed on Airbnb. It begs the question: is there an advantage to booking via Airbnb vs. with an agent or with the villa directly? More on Airbnb Anguilla villas here.
Book early. If your week or two in the sun and surf is already booked, my best suggestion is to contact Jackie Pascher, our favorite Anguilla real estate agent.
Jackie knows all of the Anguilla villas, and seems to know every Anguilla villa owner on the island. We depended on her to find us our villa before we built our own home and moved here.
Here is an important lesson that we learned, the hard way, about why it's so important to have someone in Anguilla, like Jackie, who can take care of you, in the event of problems.
My dad originally asked several people about her. She has a great reputation. And since first meeting her, we've come to know her very well. We like her, and more importantly, we trust her completely.
She will take care of all your needs and set you up with the most perfect Anguilla villa!
Remember, summer rates are much cheaper, and the August boat races are fantastic. So don't eliminate the idea of a summer vacation here.


Where does the User Experience (UX) start?
I recently came across two examples of bad UX and in both cases, these occurred without me even using the product.
Now the likelihood of someone remembering the URL from the poster is zero. They would have to write it down or photograph it on their phone. Before the user has even accessed the site, they have been presented with an unnecessarily complex URL. Why didn't the Web Team create a more user-friendly web address of, for example, http://helpdesk. Easy to remember and helps to get the UX journey off to a positive start.
If no links are working, it may be that you are using a Pop Up Blocker. You will need to allow ***** to use Pop Ups and we suggest that you add www.*****.com to your list of trusted sites.
If you are having trouble accessing content, it may be that you need to activate Flash in your browser.
Before I have accessed the website, I have given expectations that there may be issues with it. Not a great way for me to start my UX experience. Remember this website was designed for children to do their homework. Why not just remove the requirements for pop-ups and Flash?
User Experience begins before the user starts to use the website. Don't get your users off to a bad start.


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Photo credit: Unchained at Last | Susan Landmann
There is no publicly available government data on child marriage in the United States.
Internationally there is growing recognition that child marriage is a human rights violation and a severe impediment to social and economic development. As a leading donor for international development, the United States can play an important role in the global movement to end child marriage.
The Trump administration's 2018 budget has proposed complete elimination of all funding for reproductive health and family planning, which could have a detrimental impact on efforts internationally to address child marriage.
Child marriage is driven by gender inequality and the belief that women and girls are somehow inferior to men and boys.
There is limited information on child marriage in the United States.
In February 2017, a study by Girls Not Brides member Unchained at Last who are based in the US revealed that over 248,000 children had been married there between 2000 and 2010, mostly to adult men.
In the U.S federal system, state legislatures set the minimum age of marriage for each state. The minimum age in most states is 18, but exceptions in every state allow those younger than 18 to marry. Laws in 23 states do not set a minimum age below which a child cannot marry, meaning that those states' laws are weaker than child marriage laws in countries like Afghanistan, Honduras and Malawi.
The United States has committed to eliminate child, early and forced marriage by 2030 in line with target 5.3 of the Sustainable Development Goals.
The United States co-sponsored the 2015 Human Rights Council resolution to end child, early and forced marriage, recognising that it is a violation of human rights.
The United States co-sponsored the 2013 and 2014 UN General Assembly resolutions on child, early and forced marriage, and the 2013 Human Rights Council resolution on child, early and forced marriage. In 2014, the United States signed a joint statement at the Human Rights Council calling for a resolution on child marriage.
The United States signed, but is one of few countries not yet to have ratified, the Convention on the Rights of the Child in 1995, which sets a minimum age of marriage of 18. In 1980 the United States signed, but has not yet ratified, the Convention on the Elimination of All Forms of Discrimination Against Women (CEDAW) in 1981, which obligates states to ensure free and full consent to marriage.
In March 2016, the U.S. State Department adopted the Global Strategy To Empower Adolescent Girls which includes specific provisions on ending child marriage and addressing the needs of married girls globally due to the ongoing advocacy of Girls Not Brides USA.
This meets the requirement of the Violence Against Women Reauthorization Act of 2013 that the U.S. Secretary of State must "establish and implement a multi-year, multi-sectoral strategy to end child marriage."
The government is also addressing child marriage through:
Let Girls Learn, an initiative that addresses barriers to education, including child marriage.
Developmental assistance programmes at the United States Agency for International Development (USAID) which address child marriage in order to promote gender equality and women's empowerment.
The USAID Vision for Action on Ending Child Marriage and Meeting the Needs of Married Children is a strong step forward in addressing the root causes of this harmful practice. USAID's Resource Guide outlines how the government and other countries can address child, early and forced marriage through multi-sectoral and sector-specific approaches.
In the US federal system, state legislatures set the minimum age of marriage for each state. The minimum age of marriage in most US states is 18, but exceptions in every state allow those younger than 18 to marry.
In May 2018, Delaware's governor signed a bill prohibiting all marriages before the age of 18, becoming the first US state to prohibit all forms of child marriage. New Jersey followed in June 2018, passing legislation to raise the legal age to 18.
Girls Not Brides members are building coalitions in multiple states to address the legal loopholes that allow for the marriage of minors through parental or judicial consent.
Human Rights Watch, Delaware ends child marriage; 49 to go and counting, [website], 2018, https://www.hrw.org/news/2018/05/10/delaware-ends-child-marriage-49-go-and-counting (accessed May 2018)
International Women's Health Coalition, Trump's 2018 Budget Wages War on Women, [website], 2017,
https://iwhc.org/2017/05/trumps-2018-budget-wages-war-women/ (accessed June 2018)
United States Agency for International Development, CHILD, EARLY, AND FORCED MARRIAGE RESOURCE GUIDE, 2015, https://www.usaid.gov/sites/default/files/documents/1865/USAID_CEFM_Resource-Guide.PDF (accessed June 2018)
United States Agency for International Development, ENDING CHILD MARRIAGE & MEETING THE NEEDS OF MARRIED CHILDREN: THE USAID VISION FOR ACTION, 2012, https://pdf.usaid.gov/pdf_docs/PDACU300.pdf (accessed June 2018)
United States State Department, Global Strategy to Empower Adolescent Girls, 2016, http://www.state.gov/documents/organization/254904.pdf (accessed May 2018)
National Partnership in United States
Girls Not Brides USA is the official Girls Not Brides National Partnership in the United States.
View their profile
Members In United States
A Sisters Project
Advocates for Youth
AHA Foundation
AIDS-Free World
Akili Dada
Association for Community Empowerment Solutions (ACES)
Camfed
Center on Gender Equity and Health at the University of California, San Diego
Child Marriage and Youth Empowerment Group, Innovations for Youth (i4Y), University of California, Berkeley
ChildFund International
Educate the Children (ETC)
EngenderHealth
Enhance Worldwide
Every Mother Counts
Expanding Lives
Fistula Foundation
Free the Slaves
Freedom United
Friends of UNFPA
Girl Up Campaign
Girls Rights Project
Global Fund for Children (GFC)
Global Rights
GreeneWorks
Hekima Place
Human Rights Center, University of California, Berkeley, School of Law
Ibis Reproductive Health
International Action Network for Gender Equity and Law (IANGEL)
International Center for Research on Women (ICRW)
International Rescue Committee (IRC)
International Women's Health Coalition (IWHC)
Kakenya Center for Excellence (KCE)
Landesa
Management Sciences for Health
Men's Resources International
Milaan Foundation
National Council of Jewish Women
Parliamentarians for Global Action
Pathfinder International
Population Council
Population Institute
Population Media Center (PMC)
Promundo
Redner Consulting LLC
Sahar Education
School Girls Unite
Sexuality Information and Education Council of the U.S. (SIECUS)
Shadhika Foundation
Tahirih Justice Center
The Hunger Project
The Paz Foundation
Tostan
U.S. Fund for UNICEF
U.S. National Committee for UN Women (USNC)
Unchained At last
Women Deliver
Women for Afghan Women (WAW)
Women's Refugee Commission
WomenOne
World Pulse
Yellitaare
Resources related to United States
Girls Not Brides USA welcomes the bipartisan re-introduction of the Keeping Girls in School Act in the House and Senate
Girls Not Brides USA welcomes the House introduction of the Keeping Girls in School Act
Falling through the cracks: How laws allow child marriage to happen in today's America
Fact sheet and brief
Child Marriage In The United States: A Serious Problem With a Simple First-Step Solution
Despite progress, child marriage is still legal in all 50 states (New York Times)
Efforts to bar child marriage in California runs into opposition (San Francisco Chronicle)
Child marriage is sharply curtailed by New York legislature (New York Times)
US governor rejects measure to ban child marriage (Thomson Reuters Foundation)
From Baltimore to Bangladesh, let's end child marriage (Thomson Reuters Foundation)
Former child bride in Texas aims to change marriage laws (Houston Chronicle)
It's legal for 14-year-olds to marry. Should it be? (New York Times)
New Jersey just became the first state to completely ban child marriage (PBS News Hour)
Connecticut, other states look to stop child marriages (CBS St Louis)
NY Senate passes bill to end child marriage (Auburnpub)
New Hampshire law allows 13 year olds to get married (NBC4i)
Why does the United States still let 12-year-olds get married? (The Washington Post)
Married in middle school: Bonita Springs girl a "bride" at 13 (Knox News)
How I came to report on forced and child marriage – in the U.S. (PBS Newshour)
America's child marriage problem (The New York Times)
The incredible true story of a 15-year-old American escaping forced marriage (Epoch Times)
Advocates seize on White House Africa Summit to call for an end to child marriage (TIME)


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Quick Answer: When Was The Last Person Killed By The Death Penalty?
What happens to the bodies of executed inmates?
Has anyone survived an execution?
Can you watch an execution?
Why do people stay on death row so long?
Who has been on death row the longest?
Who is next to be executed in Texas?
Has anyone been executed in 2019?
Who was the last person killed by lethal injection?
Has anyone innocent been put to death?
When was the most recent death penalty execution?
Is Dustin Honken still alive?
Does Canada have a death sentence?
What countries have the death penalty?
Is electric chair still used in USA?
Did China execute Schellenberg?
Who is on death row now?
How many executions have there been in China in 2019?
A prison cemetery is a graveyard reserved for the dead bodies of prisoners.
Generally, the remains of inmates who are not claimed by family or friends are interred in prison cemeteries and include convicts executed for capital crimes..
He was 17 when he survived the first attempt to execute him, as the chair malfunctioned….Willie FrancisDiedMay 9, 1947 (aged 18) LouisianaKnown forFirst known incident of a failed execution by electrocution in the United States2 more rows
In the United States, an execution chamber will usually contain a lethal injection table. In most cases, a witness room is located adjacent to an execution chamber, where witnesses may watch the execution through glass windows.
In the United States, prisoners may wait many years before execution can be carried out due to the complex and time-consuming appeals procedures mandated in the jurisdiction. … Nearly a quarter of inmates on death row in the U.S. die of natural causes while awaiting execution.
David Lee PowellLongest Time on Death Row Prior to Execution:NameTime on Death Row1.David Lee Powell11,575 days (31 years)2.Lester Bower11,346 days (31 years)
Scheduled ExecutionsScheduled ExecutionLinkLast Name02/10/2021Inmate InformationBusby, Jr.03/04/2021Inmate InformationIbarra04/20/2021Inmate InformationGonzales05/19/2021Inmate InformationJones2 more rows
A total of twenty-two death row inmates, all men, were executed in the United States in 2019, twenty by lethal injection and two, in Tennessee, by electrocution. …
Gary Ray BowlesAndrée Borrel, a member of Britain's Special Operations Executive, was executed by the Nazis with an injection of phenol. Gary Ray Bowles, also known as the I-95 Killer, was executed in 2019 for the murders of six men in 1994….Executions.NameBrandon BernardBorn1980Died2020Age40NationalityAmerican63 more columns
A variety of individuals are claimed to have been innocent victims of the death penalty. … Newly available DNA evidence has allowed the exoneration and release of more than 20 death row inmates since 1992 in the United States, but DNA evidence is available in only a fraction of capital cases.
As of December 17, 2019, 2,656 convicts are still on death row. The Trump administration's Department of Justice announced its plans to resume executions for federal crimes in 2019. On July 14, 2020, Daniel Lewis Lee became the first inmate executed by the federal government since 2003.
A total of seventeen death row inmates, all men, were executed in the United States in 2020, sixteen by lethal injection and one by electrocution.
Honken was sentenced to death and executed by the federal government on July 17, 2020, the first defendant from Iowa to be executed since 1963….1993 Iowa murders.Dustin HonkenDiedJuly 17, 2020 (aged 52) USP Terre Haute, Terre Haute, Indiana, U.S.Cause of deathExecution by lethal injectionConviction(s)Murder x 57 more rows
The death penalty was de facto abolished in Canada in Jan 1963 and de jure in Sep 1999. … In 1976, Bill C-84 was enacted, abolishing the death penalty for murder, treason, and piracy.
Although most nations have abolished capital punishment, over 60% of the world's population live in countries where the death penalty is retained, such as China, India, the United States, Indonesia, Pakistan, Bangladesh, Nigeria, Egypt, Saudi Arabia, Iran, as well as in Japan and Taiwan.
As of 2015, the only places in the world that still reserve the electric chair as an option for execution are the U.S. states of Alabama, Florida, South Carolina, Kentucky, Tennessee, and Virginia. … Electrocution is also authorized in Kentucky in the event that lethal injection is found unconstitutional by a court.
China has arrested two Canadian citizens for jeopardising national security….This article's lead section may be too long for the length of the article.Robert Lloyd SchellenbergCriminal chargeDrug smugglingPenaltyexecution (pending appeal)3 more rows
List of federal death row inmatesNameCrimeJurijus KadamovasRansom-related kidnapping and murder of five people.Iouri MikhelRansom-related kidnapping and murder of five people.Alfonso Rodriguez Jr.Rape and murder of Dru Sjodin in Grand Forks, North Dakota.10 more rows
Number of executions worldwide in 2019Number of executionsChina **1,000Iran*251Saudi Arabia184Iraq*1009 more rows•Dec 2, 2020
Who Declared Themselves As National Assembly?
Why did the Third Estate declared themselves national assembly?
Quick Answer: What Do Deer Do When They Get Shot?
What smells are deer attracted to? Liquid Doe Urine
Question: What Does The Bible Say About The Will Of God?
How do you know when God wants you to be with someone?
Question: What City Has No Homeless?
What are the 3 types of homelessness? Homelessness
Quick Answer: Does The Seller Have To Be Out Of The House At Closing?
Do you tip your realtor? You should not tip your Realtor


text:
Q: How to match two tables and insert it into another table in laravel 5.4? I need to match two tables and insert it into the disbursment table, so this is my code...
    public function store(Request $request)
{
    //Approved Request
    $approvedRequest= DB::table('request')
    ->where('users_MemId',Auth::user()->MemId)
    ->where('requestStatus','Approved')
    ->join('requestdetails','request.requestId','=','requestdetails.request_requestId')
    ->join('itemattribute','requestdetails.RequestDetailsId','=','itemattribute.RequestDetailsId')
    ->join('exudeinventory', 'itemattribute.AttrName', '=', 'exudeinventory.ItemName')
    ->select('itemattribute.AttrName', 'exudeinventory.InventoryId', 'request.requestId')
    ->get();
    // dd($approvedRequest);

store function.

I've tried matching the tables and I can't figure out on how to insert the values in another table or if I am doing it right.

I've been looking for solutions but still I can't figure it out, Hope that someone would help me and explain it to me, I am new to laravel T,T

A: Maybe you can do this:
        public function store(Request $request)
        {
            //Approved Request
            $approvedRequest= DB::table('request')
            ->where('users_MemId',Auth::user()->MemId)
            ->where('requestStatus','Approved')
            ->join('requestdetails','request.requestId','=','requestdetails.request_requestId')
            ->join('itemattribute','requestdetails.RequestDetailsId','=','itemattribute.RequestDetailsId')
            ->join('exudeinventory', 'itemattribute.AttrName', '=', 'exudeinventory.ItemName')
            ->select('itemattribute.AttrName', 'exudeinventory.InventoryId', 'request.requestId')
            ->get()
            ->each (function ($request, $key) {
                $dis = new Disbursment;
                $dis->DisbursmentId = rand(1, 999999);
                $dis->ItemImage = 'default.jpg';
                $dis->ItemTypeId = $request->AttrName;
                $dis->exudeinventory_InventoryId = $request->InventoryId;
                $dis->request_requestId = $request->requestId;
                $dis->save();
            });
        }

redpajama_set_name: RedPajamaStackExchange


Featured Resource: Spring is the time to look into summer camps!
While CSFP does not fund summer camp or extra-curricular activities for your children, we know that out of school time is important. Check out our listing of free and low-cost summer learning opportunities today!
CSFP provides partial, need-based scholarships to be used in grades K-8. All families who are selected from CSFP's random scholarship lottery must contribute tuition to their child's school.
As the tuition rates at CSFP's more than 170 partner schools vary greatly, we understand that sometimes families need additional support. If you are researching for additional scholarships for your children, use the links below as a starting point.
Let us help simplify the process of finding a great school for your child!
Find schools by type, focus, and more.
You can estimate the amount of a partial, needs-based, CSFP scholarship your family may receive.
Join us each fall for a free, open-to-the-public, Philly K-8 School Fair.
Join CSFP for special events just for our families.
All CSFP families with children in grades 3-5 are invited to participate in our new book club!
Free academic support for CSFP 8th graders and alumni.
Find free and low-cost out-of-school-time activities for your child.
Find opportunities for your child to continue their education in the summertime!
While CSFP is the largest and most diverse provider of K-8th grade scholarships in PA, other opportunities do exist.
CSFP funds children through 8th grade, but there are many oppotunities to secure funding for high school!
Advice on keeping children in school and families invested in their children's education.
Are you a CSFP partner school or interested in becoming one?
Find resources on maintaining and increasing quality Pre-K at your school.
A quality Pre-K experience can dramatically impact your child's future. Learn what to look for and how to afford great Pre-K programs.
Find ways to get involved in the movement for quality Pre-K in Pennsylvania.
Find support and activities to help your former CSFP child succeed in high school.
Learn about colleges and the admissions process, and get help getting into college.
Have you received CSFP's support in the past? We would love to get to know you!


Agreement and Limited License ----------------------------- I agree and understand that by entering my Social Security number, PACSES Case ID, and my email address to access this Internet site, I am certifying that I am the individual named above. I agree and understand that a record of this agreement will be maintained by the Department of Human Services and that it is a crime to use [claim] the identity of another person to obtain child support information. I agree and understand that the information provided by this Internet site does not constitute an official record of the activities associated with my child support case. I agree and understand that the Department of Human Services has no liability if the information provided on this Internet site is incorrect or out of date. I agree and understand that all case specific inquiries related to my child support cases(s) should be directed to the Domestic Relations Section that manages my case(s). This statement is made to the penalties for unsworn falsification to authorities pursuant to 18 Pa. C.S. §4904. Click "I Agree" below to complete registration for this website. If you click "I Disagree", your registration will not be complete and you will be sent back to the home page.


Company Description: Druva is the global leader in Cloud Data Protection and Management, delivering the industry's first data management-as-a-service solution that aggregates data from endpoints, servers and cloud applications and leverages the public cloud. Druva's award-winning solutions is trusted by over 4,000 global organizations, and protects over 40 PB of data.
The past year has represented another year of incredible growth for Druva. As the industry's fastest growing data protection provider, Druva is trusted by over 4,000 global organizations, and protects over 40 PB of data. A customer-first attitude & delivering "WOW" in every interaction with our customers is a part of Druva's culture. There has been a consistent focus to ensure we maintain this culture and all our processes and systems are designed to ensure customer experience is enhanced.
Innovations in Customer Success has been Druva's key focus; implement the best technology and fully integrated solutions for a seamless customer experience. Our contact center systems are designed to ensure our customers are able to reach our Tech Support teams without any hurdles. Multiple channels such as chat, email, community, and phone are available for our customers to instantly reach out to our Tech Support Teams.
Sunil Wadhwa joined Druva as VP - Customer Success and Operations, he own's customer support, professional services, customer education and operations functions at Druva.
-Implemented Cloud-based Omni-channel contact center for Phone, Email, Chat and Online support tickets. Providing in-a-minute response times for all media interactions.
-Streamlined contact center implementation for enhanced customer and agent experience. Customers communicate via the channel of their choice, and agents manage the context-based communication through a single interface.
-200+ KnowledgeBase articles published by Support Engineers focusing on troubleshooting and FAQs as part of offering self-services to our customers.
-99.9% achievement of in-minute Service Level Agreement for all customers interactions .
-Druva Status is a real-time, 24/7 health check of Druva services.
-Systemized approach to ensure customers have all resources and support necessary to be successful at every stage of the customer lifecycle.
-Beyond tools and philosophy, the Druva team is the most critical component to the overall success.


Ne day i received a strange request i from Ginger Houghton who has been the inspiration and driving force behind much of our work in Lithuania. She said a member of the Knights of Lithuania had heard the name "Rugute", a program of some sort that was doing good things in Lithuania.
In life, as in the magazine business, it seems it is all too easy to overlook the many good things in life, and focus much more easily on the negatives.


Secret treasures of our homeland.
We have slept in many beds and ate lots of breakfasts before selecting these 14 addresses in the cities of art, some better known, others less. From Florence to Palermo, here is the list. Our criteria for selection? Elegance a few steps away from where it's all happening.
In the street of antique shops, of art galleries and bistrots, Margutta 54 is the address where you can find four rooms decorated is a classic way with elements of design set in a historical building. To greet customers there is a buttler. The rooms: if you have a family choose a suite or the studio for four.


Hooperman is a U.S. television series starring John Ritter. It ran for two seasons on ABC, from 1987 to 1989. A comedy-drama, the series was created by Steven Bochco and Terry Louise Fisher who were the team responsible for creating L.A. Law.
Ritter played San Francisco plainclothes police officer Harry Hooperman. In the first episode, Hooperman inherits the rundown apartment building he lives in when his elderly landlady is killed. He also inherits her temperamental pet Jack Russell terrier, Bijoux. Due to the demands of his job as a police officer, he hires Susan Smith (played by Debrah Farentino) to be the building manager, and the pair become romantically involved.
Also starring on the show was Alix Elias as the cheerful and bubbly police dispatcher, Betty Bushkin; Barbara Bosson was Hooperman's divorced superior, Capt. Celeste "C.Z." Stern; Harry's partner, Inspector Clarence McNeil (Felton Perry); Clarence Felder as redneck inspector Boris "Bobo" Pritzger; Joseph Gian as Rick Silardi, a gay cop, and Sydney Walsh as officer Maureen "Mo" DeMott, his patrol partner intent on "saving" him from being gay by making unwanted passes. Dan Lauria (later of the series, The Wonder Years) played Celeste's former husband, Lou Stern.
Like Bochco's other hit series' Hill Street Blues and L.A. Law, the theme music was composed by Mike Post who would also go on to compose the theme for Bochco's 90s hit series NYPD Blue.


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Matt Rogers (born March 5, 1990) is an American comedian, actor, writer, podcaster, and television host. He is best known for co-hosting the pop culture podcast Las Culturistas with Bowen Yang since 2016.

Early life and education
Rogers was born to Richard and Katrina Rogers, and raised on Long Island, New York.

Rogers attended Islip High School and was named prom king at his senior prom. Rogers got a 31 on the ACT and a 2010 on his SAT. After graduating, he earned a BFA in dramatic writing from New York University. While studying there, Rogers became a member of the improvisational group Hammerkatz and started studying at the Upright Citizens Brigade in 2009. It was while at NYU that Rogers first met Yang.

Career
While studying at UCB, Rogers performed in several shows, including Characters Welcome and Amazing Welcome; he also performed in the Maude team and served as the artistic director of the musical sketch comedy group Pop Roulette. In 2016, Rogers was recognized as a "Comic to Watch" by Comedy Central.

Since 2016, Rogers has co-hosted the podcast Las Culturistas with fellow NYU alumnus Bowen Yang. Guests on the show have included Alan Cumming, Bianca Del Rio, Katie Couric, Michelle Yeoh, Trixie Mattel, Sean Hayes and Will Ferrell.

In 2020, Rogers hosted two television competition series. Gayme Show, co-hosted with Dave Mizzoni, was based on a popular comedy night in which straight men were quizzed on queer culture; the show aired for one season on the streaming platform Quibi. After initially being renewed for a second series, the show's current status remains in limbo following the closure of Quibi in October 2020. Also in 2020, Rogers became the host of Haute Dog, which aired on HBO Max and saw dog groomers compete for a cash prize.

As an actor, Rogers has made guest appearances on multiple television series, including Shrill, Awkwafina Is Nora from Queens, and Search Party. In 2021, it was announced that Rogers would have a starring role on the comedy series I Love That for You, in addition to a supporting role in the film Fire Island, a gay retelling of Pride and Prejudice.

As a voice actor, Rogers serves as the voice of several recurring characters on the comedy series Our Cartoon President. Since 2021, he has voiced the character of Twink on the Netflix comedy Q-Force, on which he also serves as a writer.

In December 2022 Rogers released his musical comedy special, Have You Heard of Christmas? on Showtime. The special follows Rogers's journey to entering the Christmas canon as the "Pop Prince of Christmas". The TV special features original songs and guest appearances by Bowen Yang, Josh Sharp, and Aaron Jackson.

Rogers has been cited in academic work for his insight into the entertainment industry.

Personal life
After an extended period of living in Brooklyn, Rogers resides in Los Angeles .

Rogers is openly gay, having come out while a student at NYU. Rogers is also famously red-green color blind.

Filmography

Film

Television

Writer

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References

External links
 Official website

1990 births
21st-century American comedians
21st-century American male actors
21st-century American male writers
American gay actors
American gay writers
American podcasters
Comedians from New York (state)
Comedians from New York City
Gay comedians
LGBT people from New York (state)
Living people
Male actors from New York City
New York University alumni
People from Long Island
Writers from Brooklyn
Writers from Los Angeles
American LGBT comedians

redpajama_set_name: RedPajamaWikipedia


Ex-Brazil President Lula sentenced to nearly 10 years in prison for corruption – Royal Times of Nigeria.
Former Brazilian leader Luiz Inacio Lula da Silva, 71, was convicted on corruption charges on Wednesday and sentenced to nearly 10 years in prison.
Judge Sergio Moro found Lula guilty of accepting $1.2 million worth of bribes from the engineering firm OAS SA. Prosecutors said the company spent the same amount refurbishing a beach apartment for Lula in return for his helping in winning contracts with state oil company Petroleo Brasileiro.
Moro said he did not order Silva's immediate arrest because the conviction of a president is such a serious matter that he felt an appeal should be heard first.
Lula was president between 2003 and 2010. He is one of the country's most popular politicians and a top contender to win next year's presidential election, the ruling is a serious blow to his chances of a political comeback. His handpicked successor, Dilma Rousseff, was impeached last year over accusations of budget mishandling.


Volunteers are still needed to wrap gifts as well. The Giving tree location is open from 2pm to 7pm during the week.
This year approximately 517 children, 209 families, are being served. Please help make a child's Christmas more merry!


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Concord Adex
Roadmap Satellite Hybrid Terrain
Concord Adex has continued to be at the front line of Canada's most extensive skyline-defining communities. The company works hand-in-hand with many large-scale master-planned communities and expands its network all the way to the U.K. The real estate company boasts of over 150 completed mixed and residential buildings across the country. As of the time of writing, the company has more than 50 ongoing projects at different stages of development.
In the past 25 years, Concord Adex has diversified across other industries, including telecommunications, technology and software, wind and hydroelectric power generation, and green energy projects in solar. The company has also expanded into hospitality and bought the Westin Bayshore and the Sundal Hotel, only a few distances away from the Blackcomb gondolas and Whistler.
Concord Adex embraces timeless designs, quality materials, and liveable floorplans that fit into contemporary life to build communities that last a lifetime. The company Concord Pacific place was the first to meet the LEED Gold standards for community development and it has become a source of inspiration for other real estate developers around the globe.
Concord Adex uses the Concord Tech Green Approach to build communities using a minimal amount of energy. The company focuses on creating green communities by using automated intelligent building systems and offsetting energy usage. The Concord green energy projects have hydro, solar, and wind medium at the various stages of community developments. Also, the company aims at leading North America with E.V. parking infrastructures. Concord Adex holds several developments and design awards, including the Best Master Planned Community in B.C. in 2000.
Concord Adex recognizes that building communities extends beyond bricks and mortars so it actively participates in community events and supports charitable organizations. The company infuses over 2 million dollars annually into charitable programs and community events. The company also extends its community involvement to youths training and employment opportunities.
Richmond: Richmond finds its base in one of the largest world-class communities in Canada. It is surrounded by meandering walkways and beautiful water garden landscaping. The community is an 18-storied building with many amenities close by. The children's play area in the community makes it a perfect place to raise a family. The security features also allow Richmond to give residents tranquil peace of mind.
The Arc: If you are looking for a community with a solid build and architectural response in Vancouver, the Arch is a perfect fit. The 28 and 30-storied buildings have many amenities that are guaranteed to exceed your expectation. The community has luxurious yet timeless designs that redefine the future of Vancouverism. For people who want the world at their feet, the Arch offers everything that makes living easy.
Opus: Opus is located in New York, Ontario, where you can enjoy life to the fullest. Whether it's a first kiss or a first date, the space perfects a moment and gives you a memory to hold onto for a lifetime. The community hangs every moment around your heart, like a diamond, thanks to the variety of amenities for residents.


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Recognized by patients and peers as one of the top plastic surgeons in New York City, Robert Tornambe, MD, offers a variety of precision surgical techniques at Madison Plastic Surgery in Manhattan. Dr. Tornambe has been named one of the "Best Doctors" in New York Magazine and has appeared on numerous television and radio shows, including Discovery Health, NBC News, and the popular Extreme Makeover show on ABC.
Speaking throughout North America and Europe on the latest advances in aesthetic medicine, Dr. Tornambe is also a frequent guest lecturer. He regularly authors articles and blog posts that appear in regarded medical journals and popular media outlets, such as Oprah.com and Huffington Post. Dr. Tornambe also published The Beauty Quotient Formula, a book that discusses lifestyle choices, confidence, and contentment that contributes to one's overall attractiveness.
Dr. Tornambe completed his first residency in general surgery and his fellowship training in hand surgery at Cabrini Medical Center in New York City. He completed his second residency in plastic surgery at University of Texas Health Science Center in Houston. Dr. Tornambe also underwent fellowship training in Nashville, Tennessee, in cosmetic and reconstructive breast surgery with the renowned plastic surgeon, G. Patrick Maxwell, MD.
In addition to his fellowship in the American College of Surgeons, Dr. Tornambe is also a diplomat of the American Board of Plastic Surgery. He values a less-is-more approach in cosmetic surgery and tailors each procedure to the patient's individual anatomical needs and personal desires. Dr. Tornambe also focuses on the entire patient, and not just the procedure, and believes in building long-term, honest relationships with his patients.
Park Plaza Dermatology: Pinkas E. Lebovits MD, PC
Midtown Manhattan Office: 212-757-7010
200 West 58th Street, Suite 1C, New York, NY 10019
Brooklyn Office: 212-757-7010
158 Noble Street, Brooklyn, NY 11222
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286 Fort Washington Ave, New York, NY 10032
601 West 182nd Street, New York, NY 10033
West Harlem Office: 646-412-5911
5 Hamilton Place, Floor 2, New York, NY 10031
Bronx Office #1: 347-918-5500
3224 Grand Concourse Suite BA, Bronx NY 10458
Park Plaza Dermatology: Dr. Pinkas E. Lebovits MD, PC, New York, NY
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Address: 601 West 182nd Street, New York, NY 10033
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