text:
Thor (Chris Hemsworth) is imprisoned on the planet Sakaar, and must race against time to return to Asgard and stop Ragnarök, the destruction of his world, at the hands of the powerful and ruthless villain Hela (Cate Blanchett).
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Skurge
(voice)
Rachel House
Hogun
Volstagg
Eric Pearson, Craig Kyle, Christopher L. Yost, Stan Lee (based on the Marvel comics by), Larry Lieber (based on the Marvel comics by), Jack Kirby (based on the Marvel comics by)
27 October 2017 | ctowyi
| This Thor is Thor-some!
Frankly, I didn't like the first two Thor movies, the game plan was so clear from the get go. Same old "save the world" routines, brooding hero with a burden, a love connection born out of necessity and bloated battle scenes. It is the same blueprint for many superhero movies, but Thor has the God element that made the stories feel even more preposterous and ridiculous. It is hard to be vested in his quests because God don't die, hurt as much and can always employ one last deus ex machina that solves the unsolvable problem. But Thor: Ragnarok throws that over-used game plan out from Bifrost into an unknown territory, making it this year's best superhero movie. I know I know… Justice League has not descended, but looking at the trailer it doesn't take a genius to see that it will be doing the above-mentioned routine.
New Zealand director Taika Waititi seems like an unusual choice to helm Thor's third standalone movie. He has the indie root in him and he makes superbly smart comedies with oddles of heart, just watch What We Do in the Shadows (2014) and Hunt for the Wilderpeople (2016) to see his outstanding oeuvre. Apparently, the Marvel-Disney studios have given him full reins and he went ahead and make a comedy. But how do you inject an indie comedic vibe into a superhero franchise that has not been done before?
Essentially, Waititi throws out all the cumbersome weights that hampered Thor and tells a simpler but no less cool story about a superhero trying to get back to his home to clean house. In between he gives Thor ample opportunities to be a beer-swigging beefcake, negotiate some treacherously outlandish situations and strip him off the one thing that makes him Thor, the Mjolnir. At one point, Odin even asks a defeated Thor "What are you, the God of Hammers? " that had me laughing till my tears rolled down.
The tone is light but no less serious, populated by so many colourful characters in a vivid world trapped in impossible situations. At first I thought the introduction of Hulk in the early trailer is a misstep, but Mark Ruffalo's casting would have been close to impossible to keep under wraps. By giving Planet Hulk comic fans a look-in would have been a better move. The repartee between Thor and Hulk/Banner is so hilarious. No less funny is also the dialogue between Thor and Loki, gone is that "he ain't heavy, he is my brother" vibe and this is a Thor that will give Loki a smackdown if he is out of step.
Cate Blanchett plays Thor's estranged sister with evil relish and you don't cast Jeff Goldblum without allowing him to be Goldblum. Waititi even voices a gladiator-mentor character named Korg that practically steals all the scenes he is in. Chris Hemsworth hams it up just to a safe degree and nails all the one-liners with an A+. In fact, the whole cast seems to be having fun.
Thor: Ragnarok embraces its preposterousness and swims in a pool of ridiculousness with the deadpan and self-mocking humour hitting all the bullseyes. In the midst of it all, it still manages to tell a super duper cool story of a superhero saving his people. Yet it manages to remain fresh and zip along with a zany lightning pace. What an inspired choice for a director. The risk taken will pay huge dividends. This has superb rewatch value and I already feel like buying another ticket just to catch all the one-liners one more time. This Thor is Thor-some!
Director Taika Waititi revealed that his character was named Korg, "like the piano".
Thor: I know what you're thinking. "Oh, no. Thor's in a cage. How did this happen?" Well, sometimes you have to get captured just to get a straight answer out of something. It's a long story. Basically, I'm a bit of a hero. See, I've spent some time on ...
In the final scene with Skurge, the dust covers on the M16s he was dual wielding can be seen closed. These open upon the first round being fired.
Part of the closing credits are a sequence of illustrated scenes from the film.
Thor: The Dark World Theme
from Thor: The Dark World (2013) (uncredited)
Synopsis (WARNING: Spoilers)
Action | Adventure | Comedy | Fantasy | Sci-Fi
Village Roadshow Studios, Oxenford, Queensland, Australia
$180,000,000 (estimated)
Opening Weekend:
AUD 10,135,906 (Australia) (29 October 2017)
Gross:
$315,058,289 (USA)
See detailed box office info on IMDbPro
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text:
\section{Introduction}
One of the definite goals of the research of the holographic principle,
or the AdS/CFT correspondence \cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj},
is to find what class of quantum field theories (QFTs) or quantum materials possesses
their gravity dual.
Is there any direct test for the existence of a gravity dual for a given material?

Among various gravitational physics, one of the most peculiar astrophysical
objects is the black hole.
Gravitational lensing is one of fundamental phenomena by strong
gravity.
Let us consider that there is a light source behind a gravitational
body.
When the light source, the gravitational body, and observers are in
alignment, the observers will see a ring-like image of the light source,
i.e., the so-called Einstein ring.
If the gravitational body is a black hole, some light rays are so strongly
bended that they can go around the black hole many times, and especially
infinite times on the photon sphere.
As a result, multiple Einstein rings corresponding to winding numbers of
the light ray orbits emerge and infinitely concentrate
on the photon sphere.
Recently, the Event Horizon Telescope (EHT)~\cite{EHT}, which is an observational
project for imaging black holes, has captured the first image of the
supermassive black hole in M87. (See the left panel of Fig.~\ref{EHTfig}.)
The dark area inside the photon sphere, in which light rays have been
absorbed by the black hole, is called black hole shadow~\cite{Falcke:1999pj}.
In  this paper, we propose a direct method to check the existence of a gravity dual from measurements in a given thermal QFT
--- {\it imaging the dual black hole as an ``Einstein ring''}.

\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{compare_to_EHT_ver2.eps}
\end{center}
 \caption{
(Left) Image of the black hole in M87 (This figure is taken from Ref.~\cite{EHT}.)
(Right) Image of the AdS black hole constructed from the response function.
 }
 \label{EHTfig}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{AdSBHshadow_setup_ver2.eps}
\end{center}
\caption{
Our setup for imaging a dual black hole, the Schwarzschild-AdS$_4$ spacetime.
An oscillating Gaussian source
$J_\mathcal{O}$
is applied at a point on the AdS boundary.
Its response $\langle {\cal O}(x)\rangle$ is observed at another point on the boundary.
}
\label{setupfig}
\end{figure}

We demonstrate explicitly construction of holographic ``images'' of the dual black hole
from the response function of the boundary QFT with external sources, as follows.
As the simplest example, we consider a $(2+1)$-dimensional
boundary conformal field theory on a 2-sphere $S^2$ at a finite temperature,
and study
a one-point function of a scalar operator ${\cal O}$ with its conformal dimension $\Delta_{\cal O} = 3$,
under a time-dependent localized source $J_{\cal O}$.
The gravity dual is a black hole in the global AdS$_4$ and a probe
massless bulk scalar field in the spacetime.
The schematic picture of our setup is shown in Fig.~\ref{setupfig}.
The source $J_{\cal O}$, for which we employ a time-periodic localized Gaussian source with the frequency $\omega$,
amounts to an AdS boundary condition for the scalar field.
Due to the time periodic boundary condition,
a bulk scalar wave is injected into the bulk from the AdS boundary.
The scalar wave propagates inside the black hole spacetime
and reaches other points on the $S^2$ of the AdS boundary.
We measure the local response function $e^{-i\omega t}\langle {\cal O}(\vec{x})\rangle$ which
has the information about the bulk geometry of the black hole spacetime.

Using a wave-optical method,
we find a formula which converts the response function $\langle {\cal O}(\vec{x})\rangle$ to the image
of the dual black hole $|\Psi_\mathrm{S}(\vec{x}_\mathrm{S})|^2$ on a virtual screen:
\begin{equation}
 \Psi_\mathrm{S}(\vec{x}_\mathrm{S})= \int_{|\vec{x}|<d}d^2x\, \langle {\cal O}(\vec{x})\rangle e^{-\frac{i\omega}{f}\vec{x}\cdot\vec{x}_\mathrm{S}}\ ,
\label{lenstrans0}
\end{equation}
where $\vec{x}=(x,y)$ and $\vec{x}_\textrm{S}=(x_\textrm{S},y_\textrm{S})$ are Cartesian-like coordinates on boundary $S^2$ and
the virtual screen, respectively,
and we have set the origin of the coordinates to an observation point.
This operation is mathematically
a Fourier transformation of the
response function on a small patch with the radius $d$
around the observation point.
Note that $f$ describes magnification of the image on the screen.
In wave optics, we have virtually used a lens with the focal
length $f$ and the radius $d$ to form the image.
The right panel of Fig.~\ref{EHTfig} shows a typical image of the AdS black hole computed
from the response function through our method.
The AdS/CFT calculation clearly gives a ring similar to the
observed Einstein ring.
Equation~(\ref{lenstrans0}) can be regarded as the dual quantity of Einstein ring caused by the
black hole in a thermal QFT.

Several criteria for QFTs to have a gravity dual have been proposed in some previous works.
For example, in a conformal field theory,
the existence of a planer expansion and
a large gap in the spectrum of anomalous dimensions
has been conjectured to be the criterion for the existence of a gravity dual~\cite{Heemskerk:2009pn}.
There is also recent progress on emergent gravity based on quantum entanglement~\cite{Faulkner:2013ica},
although the entanglement entropy itself is not a physical observable.
The strong redshift of the black hole
has been also used as the condition for the existence of gravity
dual~\cite{Shenker:2013pqa,Maldacena:2015waa,Kitaev-talk}.
Our approach, checking the dual black hole by its image, gives an alternative.
The method is simple and can be applied to any QFT on a sphere, thus probing efficiently a black hole
of its possible gravity dual.
Once we have a strongly correlated material on $S^2$, we can apply
a localized external source such as electromagnetic waves and measure its response in principle.
Then, from Eq.~(\ref{lenstrans0}),
we would be able to construct the image of the dual black hole if it exists.
The holographic image of black holes in a material, if observed by a tabletop experiment, may serve as a
novel entrance to the world of quantum gravity.

The organization of this paper is as follows. In Section~\ref{Sec:scalar}, we review
how to obtain the response function under a source in AdS/CFT, from a scalar field dynamics
in Schwarzschild-AdS geometry, and describe our time-oscillatory source in the boundary QFT.
In Section~\ref{imgwave},
we review image formation in wave optics.
This leads us to our main formula for the dual quantity of the Einstein ring~\eqref{lenstrans0}.
In Section~\ref{ngeo}, we discuss null geodesics in the
Schwarzschild-AdS on the basis of geometrical optics so that we will
gain intuitive insight into our holographic images.
Then in Section~\ref{ImAdSBH}, we provide our concrete images for the AdS black holes
seen from the QFT with the formula~\eqref{lenstrans0}.
They show clear
holographic images of black holes
as
well as the Einstein rings. In Section~\ref{RingAngle}, we evaluate the Einstein radius (the size of the rings) in the images,
and provide its consistent understanding by geodesics. Then in Section~\ref{Analy},
we provide analytic examples of the pure AdS$_4$ and the BTZ black hole, for a comparison.
In Section~\ref{EringGreen}, we describe the image of the Einstein ring from holography in terms of the retarded Green function. We see that
quasi-normal mode frequencies in the gravity side give major contribution for the formation of the Einstein ring.
Finally, Section~\ref{Concl} is for our conclusion and discussions.
Appendix~\ref{sec:detail} describes detailed numerical evaluation of solutions of the scalar field in the bulk.
Appendix~\ref{Val} is for the validity of the approximation of the geometric optics used in Section~\ref{ngeo}.

\section{Scalar field in Schwarzschild-AdS$_4$ spacetime}
\label{Sec:scalar}

We consider Schwarzschild-AdS$_4$ (Sch-AdS$_4$) with the spherical horizon:
\begin{align}
ds^2=-F(r)dt^2+\frac{dr^2}{F(r)}+r^2(d\theta^2+\sin^2\theta d\varphi^2)\ ,\quad
F(r)=\frac{r^2}{R^2}+1-\frac{2 G M}{r}\ ,
\label{SchAdS4}
\end{align}
where $R$ is the AdS curvature radius
(i.e., a negative cosmological constant $\Lambda = -3/R^2$)
and $G$ is the Newton constant.
Since this is a black hole solution in the global AdS$_4$,
we consider a CFT on $\mathbf{R}_t \times S^2$ as the dual field theory.
The event horizon is located at $r=r_\mathrm{h}$ defined by $F(r_\mathrm{h})=0$
\footnote{
Although the Sch-AdS$_4$ with $r_\mathrm{h} \lesssim R$ is unstable in the canonical ensemble,
it can be stable in the microcanonical ensemble~\cite{Hawking:1982dh}.
We will consider black holes with $r_\mathrm{h} \lesssim R$ as well as those with $r_\mathrm{h} \gtrsim R$ in this paper.
}.
Using the horizon radius,
the
mass $M$ is written as
\begin{equation}
M=\frac{r_\mathrm{h}(r_\mathrm{h}^2+R^2)}{GR^2}\ .
\label{Mass}
\end{equation}
This corresponds to the total energy of the dual CFT.
In what follows,
for simplicity and without loss of generality
we take the unit of $R=1$.

We focus on dynamics of a massless scalar field in the Sch-AdS$_4$.
The Klein-Gordon equation for the massless scalar field,
$\Box \Phi(t,r,\theta,\varphi)=0$, is written as
\begin{equation}
 -\frac{1}{F}\partial_t^2 \Phi+F\partial_r^2
  \Phi+\frac{(r^2F)'}{r^2}\partial_r \Phi+\frac{1}{r^2} D^2 \Phi=0\ ,
\label{scalareq0}
\end{equation}
where
the prime denotes $r$-derivative and
$D^2$ is the scalar Laplacian on unit $S^2$.

Near the AdS boundary ($r=\infty$),
the asymptotic solution of the scalar field becomes
\begin{align}
 \Phi(t,r,\theta,\varphi)
=J_\mathcal{O}(t,\theta,\varphi)
-\frac{1}{2r^2}(\partial_t^2 - D^2)J_\mathcal{O}(t,\theta,\varphi)
+\frac{\langle \mathcal{O}(t,\theta,\varphi)\rangle }{r^3} +\cdots \ .
\label{phiinf0}
\end{align}
In the asymptotic expansion, we have two independent functions, $J_\mathcal{O}$ and $\langle \mathcal{O}\rangle$,
which depend on boundary coordinates $(t,\theta,\varphi)$.
According to the AdS/CFT dictionary, the leading term
$J_\mathcal{O}$ and the subleading term $\langle \mathcal{O}\rangle$
correspond to the external scalar source and its response function in
the dual CFT, respectively~\cite{Klebanov:1999tb}.

We consider that an axisymmetric and monochromatically oscillating Gaussian source is
localized at the south pole ($\theta=\pi$) of the boundary $S^2$:
\begin{equation}
 J_\mathcal{O}(t,\theta,\varphi)=e^{-i\omega t} g(\theta)\ ,\quad g(\theta)=\frac{1}{2\pi \sigma^2}\exp\left[-\frac{(\pi-\theta)^2}{2\sigma^2}\right]\ .
\label{BC}
\end{equation}
Note that
we will ignore a tiny value of
the Gaussian tail at the north pole because we suppose $\sigma\ll \pi$.
In the bulk point of view, this source $J_\mathcal{O}$
determines the boundary condition of the scalar field at the AdS boundary.
We also impose the ingoing boundary condition on the horizon of the Sch-AdS$_4$.
A schematic picture of our setup is shown in Fig.~\ref{setupfig}.
The scalar wave generated at the south pole of the AdS boundary
propagates inside the bulk black hole spacetime
and reaches the other points at the AdS boundary.
Imposing these boundary conditions, we solve Eq.~(\ref{scalareq0}) numerically and
determine the solution of the scalar field in the bulk.
We summarize the detailed method to solve the Klein-Gordon equation in
Appendix \ref{sec:detail}.
We read off the response from the coefficient of $1/r^3$ in Eq.~(\ref{phiinf0}).
Because of the properties
of the source~(\ref{BC}), the response function does not depend on
$\varphi$ and its time dependence is just given by $e^{-i\omega t}$.
Hence, we can write the response function as
\begin{equation}
 \langle \mathcal{O}(t,\theta,\varphi)\rangle=e^{-i\omega t}\langle \mathcal{O}(\theta)\rangle\ .
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{O_psn.eps}
\end{center}
\caption{
Absolute square of the response for $r_\mathrm{h}=0.3$, $\omega=20$ and $\sigma=0.2$.
}
\label{vev_abs}
\end{figure}
As an example, Fig.~\ref{vev_abs} shows the absolute square of the response function for $r_\mathrm{h}=0.3$, $\omega=20$ and $\sigma=0.2$.
From this response function itself, we cannot directly find
``black-hole-like images'' as we desired.
We have observed only the interference pattern resulting from the
diffraction of the scalar wave by the black hole.
In order to obtain images, we need to look at the response
function through a certain kind of optical system with a convex lens as
we will see in the next section.

\section{Image formation in wave optics}
\label{imgwave}

We introduce an optical system consisting of a convex
lens and a spherical screen so that we will construct images of the
black hole from the response function on the basis of the wave optics~\cite{Optics}.
(See also Refs.~\cite{Nambu:2012wa,Kanai:2013rga,Nambu:2015aea}.)
What role the convex lens plays in the wave optics is as follows.
The lens is regarded as a ``converter'' between plane and spherical waves as in Fig.~\ref{lens_transmit}.
The lens with focal length $f$ is located at $z=0$
and its focus is at $z=\pm f$.
We assume that
the size of the lens is much smaller than the focal length $f$ and
the lens is infinitely thin.
Imagine that a plane wave is irradiated to the lens from the left hand
side as shown in the figure.
Such a plane wave is converted into (a part of) spherical wave and it
converges at the focus located at $z=f$.
Inversely, if we consider a spherical wave emitted from the focus, it
should be converted into the plane wave in $z<0$.
Let $\Psi$ and $\Psi_\mathrm{T}$ denote the incident wave and the
transmitted wave, respectively.
Then, mathematically, the role of the convex lens for the wave functions
with frequency $\omega$
on the lens can be simply expressed as
\begin{equation}
\Psi_\mathrm{T}(\vec{x})=e^{-i\omega\frac{|\vec{x}|^2}{2f}}\Psi(\vec{x})\ ,
\label{tranwave}
\end{equation}
where $\vec{x}=(x,y)$ are coordinates on the lens located at $z=0$.
For example, an incident plane wave propagating along $z$-axis is written as
$\Psi(\vec{x}) = 1$ on the lens, whose phase does not depend on $\vec{x}$.
Then, from Eq.~(\ref{tranwave}), the transmitted wave becomes
$\Psi_\mathrm{T}(\vec{x}) =
\exp(- i\omega |\vec{x}|^2/2f)
\simeq
\exp[-i\omega\sqrt{f^2+|\vec{x}|^2} + i\omega f]$.
This phase dependence describes that of the spherical wave converging on the
focus at $z=f$ in the Fresnel approximation ($|\vec{x}|\ll f$).

Let us consider a spherical screen located at $(x,y,z)=(x_\mathrm{S},y_\mathrm{S},z_\mathrm{S})$ with $x_\mathrm{S}^2+y_\mathrm{S}^2+z_\mathrm{S}^2=f^2$.
The transmitted wave converted by the lens is focusing and imaging on this screen.
When the transmitted wave on the lens is
$\Psi_\mathrm{T}(\vec{x})$,
the wave function on the screen is given by
\begin{equation}
 \Psi_\mathrm{S}(\vec{x}_\mathrm{S})=\int_{|\vec{x}|\leq d} d^2x \Psi_\mathrm{T}(\vec{x}) e^{i\omega L}\ .
\label{PsiI}
\end{equation}
where $d$ is the radius of the lens and
$L$ is the distance between $(x,y,0)$ on the lens and
$(x_\mathrm{S},y_\mathrm{S},z_\mathrm{S})$ on the screen:
\begin{equation}
\begin{split}
 L&=\sqrt{(x_\mathrm{S}-x)^2+(y_\mathrm{S}-y)^2+z_\mathrm{S}^2}\\
&=\sqrt{f^2-2\vec{x}_\mathrm{S}\cdot\vec{x}+|\vec{x}|^2}
\simeq f-\frac{\vec{x}_\mathrm{S}\cdot\vec{x}}{f}+\frac{|\vec{x}|^2}{2f}\ ,
\end{split}
\label{dist}
\end{equation}
where $\vec{x}_\mathrm{S}=(x_\mathrm{S},y_\mathrm{S})$.
Substituting Eqs.~(\ref{tranwave}) and (\ref{dist}) into Eq.~(\ref{PsiI}), we obtain
\begin{equation}
 \Psi_\mathrm{S}(\vec{x}_\mathrm{S})\propto \int_{|\vec{x}|<d}d^2x\, \Psi(\vec{x})e^{-\frac{i\omega}{f}\vec{x}\cdot\vec{x}_\mathrm{S}}\ .
\label{lenstrans}
\end{equation}
This implies that the image on the screen can be obtained by the Fourier
transformation of the incident wave within a finite domain of the lens.
Equation~\eqref{lenstrans} motivates us to regard the observable defined in \eqref{lenstrans0}
as the dual quantity of the Einstein ring.

\begin{figure}
  \centering
  \subfigure[Transmitted wave]
 {\includegraphics[scale=1.3]{lens_transmitted_ver2.eps}\label{lens_transmit}
  }
\subfigure[Image formation on the screen]
 {\includegraphics[scale=1.1]{image_formation_ver3.eps}\label{imgf}
  }
 \caption{
(a) The lens converts plane waves into spherical waves and vice versa.
(b) The screen is located at $\{(x_\mathrm{S},y_\mathrm{S},z_\mathrm{S})|x_\mathrm{S}^2+y_\mathrm{S}^2+z_\mathrm{S}^2=f^2\}$.
 }
 \label{lensfig}
\end{figure}

\section{Null geodesics: geometrical optics}
\label{ngeo}
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{typical_pot_geodesic_ver2.eps}
\end{center}
\caption{
Typical profile of the effective potential.
For $1 < \omega^2/\ell^2 < v_\textrm{max}$, the orbit of null
 geodesics can go between two points on the AdS boundary ($r=\infty$).
}
\label{pot_geo}
\end{figure}
To help intuitive understanding
of the image of the AdS black hole which
will be shown in the following sections,
we consider null geodesics in the Sch-AdS$_4$
(that is, geometrical optics)
in this section.
In Appendix \ref{Val}, we derive the geodesic equation from the Klein-Gordon equation
and examine the validity of the Eikonal approximation in asymptotically AdS spacetimes.

It is well known that in the spherically symmetric spacetime an orbit of
geodesics lies in a plane passing through the center of the black hole.
Therefore, we can always rotate an orbital plane of the null geodesic to
coincide with the equatorial plane, without loss of generality.
In this section, for simplicity, we will study null geodesics on the
equatorial plane ($\theta=\pi/2$).
Then, the conserved energy and angular momentum are written as
\begin{equation}
 \omega=F(r)\dot{t}\ ,\quad
 \ell=r^2\dot{\varphi}\ ,
\label{omegal}
\end{equation}
where ${}^\cdot=d/d\lambda$ and $\lambda$ is an affine parameter. From the null condition, we have
\begin{equation}
 \dot{r}^2=\omega^2-\ell^2 v(r)\ ,\quad v(r)\equiv \frac{F(r)}{r^2}\ .
\label{dotrsq}
\end{equation}
We note that, since affine parameters of null geodesics can be
rescaled by a constant factor, the null orbits depend only on the ratio of
the conserved quantities, $\ell/\omega$.
The effective potential $v(r)$ has a maximum value:
\begin{equation}
 v_\textrm{max}= \frac{(3r_\mathrm{h}^2+4)(3r_\mathrm{h}^2+1)^2}{27r_\mathrm{h}^2(r_\mathrm{h}^2+1)^2}\ ,
\label{vmax}
\end{equation}
at $r=r_\textrm{max}\equiv 3r_\mathrm{h}(r_\mathrm{h}^2+1)/2$.
The maximum of the effective potential corresponds to the photon sphere,
i.e., the unstable circular orbit of null geodesics.
The schematic functional profile of the effective potential $v(r)$ is shown in Fig.~\ref{pot_geo}.
We are now interested in null orbits between two points on the AdS
boundary.
An observer is located at one point and a light source is at
the other point.
Then, we have to tune the parameters so that $1 < \omega^2/\ell^2 < v_\textrm{max}$ is satisfied.
When $\omega^2/\ell^2$ is close to (but less than) $v_\textrm{max}$,
the geodesic goes through the vicinity of the photon sphere and can wind around the black hole.
Consequently, there exist infinite geodesics labeled by the winding
number, $N_\textrm{w}$, which connect fixed two points on the AdS boundary,
and in the vicinity of the photon sphere infinitely many orbits accumulate.
Figure~\ref{geodesic_fig} shows
some geodesics stretched between antipodal points on the AdS boundary for $r_\mathrm{h}=0.3$.
In particular, for the geodesic with $N_\textrm{w}=\infty$ (i.e. geodesic from the photon sphere), the angular momentum per unit energy becomes
\begin{equation}
 \left(\frac{\ell}{\omega}\right)_\textrm{photon sphere}=\frac{1}{\sqrt{v_\textrm{max}}}=\frac{r_\mathrm{h}(r_\mathrm{h}^2+1)}{(r_\mathrm{h}^2+1/3)\sqrt{r_\mathrm{h}^2+4/3}}\ .
\label{ellps}
\end{equation}

\begin{figure}
  \centering
  \subfigure[$N_\textrm{w}=0$]
 {\includegraphics[scale=0.6]{orbit0.eps}\label{orbit0}
  }
\subfigure[$N_\textrm{w}=1$]
 {\includegraphics[scale=0.6]{orbit1.eps}\label{orbit1}
  }
\subfigure[$N_\textrm{w}=2$]
 {\includegraphics[scale=0.6]{orbit2_ver2.eps}\label{orbit2}
  }
 \caption{
Null geodesics between $\varphi=0$ and $\pi$ for winding number $N_\textrm{w}=0,1,2$.
The horizon radius of the Sch-AdS$_4$ is fixed as $r_\mathrm{h}=0.3$.
$\vartheta_\mathrm{i}$ denotes the angle of incidence to the AdS boundary.
 }
 \label{geodesic_fig}
\end{figure}

We can naturally define the angle of incidence
of the null geodesic to the AdS boundary by
$\cos\vartheta_\mathrm{i} \equiv g_{ij} u^i n^j/(|u||n|)|_{r=\infty}$,
where $u^i$ is
the spatial component of the 4-velocity of the geodesic, $n^i$ is the
normal vector to the AdS boundary and $g_{ij}$ is the induced metric on
the $t=\text{const.}$ surface.
($|u|$ and $|n|$ are the norms of $u^i$ and $n^i$ with
respect to $g_{ij}$.)
Using Eqs.~(\ref{omegal}) and (\ref{dotrsq}), we can explicitly
calculate the angle of incidence as
\begin{equation}
 \sin\vartheta_\textrm{i} = \frac{\ell}{\omega}\ .
\label{incangle}
\end{equation}
Combining Eqs.~(\ref{ellps}) and (\ref{incangle}),
we can determine the angle of incidence of the null geodesic from the photon sphere as a function of the horizon radius $r_\mathrm{h}$.
In the geometrical optics, this angle $\vartheta_\mathrm{i}$ gives the angular
distance of the image of the incident ray from the zenith if an observer
on the AdS boundary looks up into the AdS bulk.
If two end points of the geodesic and the center of the black hole are
in alignment, the observer see a ring image with a radius corresponding
to the incident angle $\vartheta_\mathrm{i}$ because of axisymmetry.

\section{Imaging AdS black holes}
\label{ImAdSBH}
\begin{figure}
\begin{center}
\includegraphics[scale=1.2]{response_image_ver4.eps}
\end{center}
\caption{
How to construct the image of the AdS black hole.
}
\label{restoim}
\end{figure}
Figure~\ref{restoim} shows our procedure to obtain the image of AdS black hole.
The sphere of the AdS boundary is depicted at the left side of the figure.
We show the absolute square of the response function $\langle \mathcal{O}(\theta)\rangle$ on the sphere
as the color map.
The brightest point is the north pole, i.e. the antipodal point of the Gaussian source.
The response function has the interference pattern caused by the diffraction of the wave by the black hole.
We now define an ``observation point'' at
$(\theta,\varphi)=(\theta_\textrm{obs},0)$ on the AdS boundary, where an
observer looks up into the AdS bulk.

To make an optical system,
we virtually consider the flat 3-dimensional space $(x,y,z)$
as shown in the right side of Fig.~\ref{restoim}.
We set the convex lens on the $(x,y)$-plane.
The focal length and radius of the lens will be denoted by $f$ and $d$.
We also prepare the spherical screen at $(x,y,z)=(x_\mathrm{S},y_\mathrm{S},z_\mathrm{S})$
with $x_\mathrm{S}^2+y_\mathrm{S}^2+z_\mathrm{S}^2=f^2$.
We copy the response function around the observation point
as the wave function on the lens and observe its image.

We map the response function defined on $S^2$ onto the lens as follows.
We introduce new polar coordinates $(\theta',\varphi')$ as
\begin{equation}
 \sin\theta'\cos\varphi'+i\cos\theta'=e^{i\theta_\textrm{obs}}(\sin\theta\cos\varphi+i\cos\theta)\ ,
\end{equation}
such that the direction of the north pole is rotated to align with the
observation point:
$\theta'=0 \Longleftrightarrow (\theta,\varphi)=(\theta_\textrm{obs},0)$.
Then, we define Cartesian coordinates as
$(x,y)=(\theta'\cos\varphi',\theta'\sin \varphi')$
on the AdS boundary $S^2$.
In this coordinate system, we regard the response function around the observation point
as the wave function on the lens: $\Psi(\vec{x})=\langle \mathcal{O}(\theta)\rangle$.
The image on the screen can be obtained by Eq.~(\ref{lenstrans0}):
We perform the Fourier transformation within a finite domain on the
lens,
that is, applying an appropriate window function.

We now summarize our results on image formations of Sch-AdS black holes.
Figure~\ref{rh010306_view} shows images of the black hole observed at
various observation points:
$\theta_\textrm{obs}=0^\circ, 30^\circ, 60^\circ, 90^\circ$.
The horizon radius is varied as $r_\mathrm{h}=0.6, 0.3, 0.1$.
We fix other parameters as $\omega=80$, $\sigma=0.01$ and $d=0.5$.
For $\theta_\textrm{obs}=0^\circ$, a clear ring is observed.
As we will see for details in the next section, this ring corresponds to
the light rays from the vicinity of the photon sphere of the Sch-AdS$_4$.
Not only the brightest ring, we can also see some concentric striped patterns.
They are caused by a diffraction effect with imaging, which is not directly
related to properties of the black hole, because we find these patterns change
depending on the lens radius $d$ and frequency $\omega$.
As we change the angle of the observer, the ring is deformed.
We observed similar images as those for asymptotically flat black hole~\cite{Nambu:2012wa,Kanai:2013rga,Nambu:2015aea}.
In particular, at $\theta_\textrm{obs}\sim 90^\circ$, we can observe double image of the point source.
They correspond to light rays which are clockwisely and anticlockwisely
winding around the black hole on the plane of $\varphi=0,\pi$.
The size of the ring becomes bigger as the horizon radius grows.

\begin{figure}
  \centering
  \subfigure[$r_\mathrm{h}=0.6$, $\theta_\textrm{obs}=0^\circ$]
 {\includegraphics[scale=0.46]{rh=0p6_angle=0_ver3.eps}\label{rh06_angle0}
  }
\subfigure[$r_\mathrm{h}=0.6$, $\theta_\textrm{obs}\!=\!30^\circ$]
 {\includegraphics[scale=0.46]{rh=0p6_angle=30_ver3.eps}\label{rh06_angle30}
  }
\subfigure[$r_\mathrm{h}=0.6$, $\theta_\textrm{obs}=60^\circ$]
 {\includegraphics[scale=0.46]{rh=0p6_angle=60_ver3.eps}\label{rh06_angle60}
  }
\subfigure[$r_\mathrm{h}=0.6$, $\theta_\textrm{obs}=90^\circ$]
 {\includegraphics[scale=0.46]{rh=0p6_angle=90_ver3.eps}\label{rh06_angle90}
  }
\subfigure[$r_\mathrm{h}=0.3$, $\theta_\textrm{obs}=0^\circ$]
 {\includegraphics[scale=0.46]{rh=0p3_angle=0_ver3.eps}\label{rh03_angle0}
  }
\subfigure[$r_\mathrm{h}=0.3$, $\theta_\textrm{obs}=30^\circ$]
 {\includegraphics[scale=0.46]{rh=0p3_angle=30_ver3.eps}\label{rh03_angle30}
  }
\subfigure[$r_\mathrm{h}=0.3$, $\theta_\textrm{obs}=60^\circ$]
 {\includegraphics[scale=0.46]{rh=0p3_angle=60_ver3.eps}\label{rh03_angle60}
  }
\subfigure[$r_\mathrm{h}=0.3$, $\theta_\textrm{obs}=90^\circ$]
 {\includegraphics[scale=0.46]{rh=0p3_angle=90_ver3.eps}\label{rh03_angle90}
  }
\subfigure[$r_\mathrm{h}=0.1$, $\theta_\textrm{obs}=0^\circ$]
 {\includegraphics[scale=0.46]{rh=0p1_angle=0_ver3.eps}\label{rh01_angle0}
  }
\subfigure[$r_\mathrm{h}=0.1$, $\theta_\textrm{obs}=30^\circ$]
 {\includegraphics[scale=0.46]{rh=0p1_angle=30_ver3.eps}\label{rh01_angle30}
  }
\subfigure[$r_\mathrm{h}=0.1$, $\theta_\textrm{obs}=60^\circ$]
 {\includegraphics[scale=0.46]{rh=0p1_angle=60_ver3.eps}\label{rh01_angle60}
  }
\subfigure[$r_\mathrm{h}=0.1$, $\theta_\textrm{obs}=90^\circ$]
 {\includegraphics[scale=0.46]{rh=0p1_angle=90_ver3.eps}\label{rh01_angle90}
  }
 \caption{
Image of the Sch-AdS black hole for $\omega=80$, $\sigma=0.01$ and $d=0.5$.
The horizon radius and the observation point are varied as $r_\mathrm{h}=0.6,0.3,0.1$ and
$\theta_\textrm{obs}=0^\circ,30^\circ,60^\circ,90^\circ$.
 }
 \label{rh010306_view}
\end{figure}

Figure~\ref{rh03_view_om} shows the
the image of the Sch-AdS black hole for relatively small frequency, $\omega=10,20,30$.
Other parameters are fixed as $r_\mathrm{h}=0.3$, $\theta_\textrm{obs}=0$, $\sigma=0.01$ and $d=0.5$.
For $\omega=20,30$,
we can only see the blurred ring because the wave effect is not negligible.
For $\omega=10$, the ring disappears.
In the geometric optics, we can only see the outside of the photon sphere.
In the wave optics, however, we have a chance to probe the region between photon sphere and event horizon due to some wave effects.
Do these images change if we modify the metric inside the photon sphere?
This is one of the interesting direction as a future work.

\begin{figure}
  \centering
  \subfigure[$\omega=10$]
 {\includegraphics[scale=0.6]{rh=0p3_angle=0_omega=10_ver2.eps}\label{rh03_om10}
  }
\subfigure[$\omega=20$]
 {\includegraphics[scale=0.6]{rh=0p3_angle=0_omega=20_ver2.eps}\label{rh03_om20}
  }
\subfigure[$\omega=30$]
 {\includegraphics[scale=0.6]{rh=0p3_angle=0_omega=30_ver2.eps}\label{rh03_om30}
  }
 \caption{
Frequency dependence of the image of the Sch-AdS black hole for $r_\mathrm{h}=0.3$, $\theta_\textrm{obs}=0$, $\sigma=0.01$ and $d=0.5$.
The frequency is varied as $\omega=10,20,30$.
 }
 \label{rh03_view_om}
\end{figure}

\section{Einstein radius}
\label{RingAngle}

To study the property of the brightest ring,
we set the observation point at $\theta_\textrm{obs}=0$ and search
$x_\mathrm{S}=x_\textrm{ring}$ at which $|\Psi_\mathrm{S}(\vec{x}_\mathrm{S})|^2$ has maximum value.
(Since the image has the rotational symmetry in $(x_\mathrm{S},y_\mathrm{S})$-plane for $\theta_\textrm{obs}=0$,
we only focus on
the $x_\mathrm{S}$-axis
in this section.)
We will refer to
the angle of the Einstein ring  $\theta_\textrm{ring}=\sin^{-1}(x_\textrm{ring}/f)$ as the Einstein radius.
Figure~\ref{thetaE} shows the Einstein radius $\theta_\textrm{ring}$ by the purple points
varying the horizon radius $r_\mathrm{h}$.
Note that the horizon radius relates to the total energy of the system by Eq.~(\ref{Mass}).
Although the Einstein radius fluctuates as the function of $r_\mathrm{h}$,
it has an increasing trend as $r_\mathrm{h}$ is enlarged.

As we have seen in the geodesic analysis,
there is an infinite number of
geodesics connecting antipodal points on the AdS boundary, which are
labeled by the winding number $N_\textrm{w}$.
Which geodesic in the geometrical optics corresponds to the ring found
in the image in the wave optics?
From the analysis of null geodesics,
the (specific) angular momentum of the null
geodesic provides the angle of incidence $\vartheta_\mathrm{i}$ to the AdS boundary as shown in Eq.~(\ref{incangle}).
In the wave optics, the null geodesic with angular momentum $\ell$ should be
a wave packet realized by the superposition of the spherical harmonics $Y_{\ell' 0}(\theta)$ with
$\ell-\Delta \ell \leq \ell' \leq \ell+\Delta \ell$ ($\Delta \ell \ll \ell$).
For a sufficiently large $\ell$,
the spherical harmonics behaves as $Y_{\ell 0}\sim e^{i\ell \theta}$.
Applying the Fourier transformation~(\ref{lenstrans}) onto the spherical harmonics,
we have a peak in the image at $x_\mathrm{S}/f\simeq \ell/\omega$.
Thus, the angular distance of the image on the screen,
$\sin^{-1}(x_\mathrm{S}/f)$, coincides with the angle of incidence of
the null geodesic to the AdS boundary,
$\vartheta_\mathrm{i} = \sin^{-1}(\ell/\omega)$.

Since we found that the angular momentum of the null geodesics from the
photon sphere is given by Eq.~(\ref{ellps}), they are expected to form the ring at
$\theta_\textrm{ring}=\sin^{-1}(\ell/\omega)=\sin^{-1}(1/\sqrt{v_\textrm{max}})$.
The Einstein radius calculated from geodesic analysis
is shown by the green curve in Fig.~\ref{thetaE}.
The curve seems to be consistent with the Einstein radius
of the image constructed from the response function in the
wave optics.
This indicates that the major contribution to the brightest ring in the
image is originated by the ``light rays`` from the vicinity of the photon sphere, which
are infinitely accumulated.
Although there
are expected to
be multiple Einstein rings corresponding to light lays with winding
numbers $N_\textrm{w}=0,1,2,\ldots$, the contribution for the image from small $N_\textrm{w}$
may be
so small that
we cannot
resolve
them within our numerical accuracy.

The deviation of the Einstein radius $\theta_\textrm{ring}(r_\textrm{h})$
from the geodesic prediction can be considered as some wave effects.
In the AdS cases, whether the geometrical optics can adapt to imaging of
black holes is not so trivial even for a large value of $\omega$.
As studied in Appendix~\ref{Val},
the Eikonal approximation, which supports the geometrical optics, will
inevitably break down near the AdS boundary, while we have given
the source $J_\mathcal{O}$ and read the response
$\langle\mathcal{O}\rangle$ on the AdS boundary.
Our results based on the wave optics imply that the geometrical
optics is qualitatively valid but gives a non-negligible deviation even
for a large $\omega$.

\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{thetaE.eps}
\end{center}
\caption{
The Einstein radius $\theta_\textrm{ring}$ as a function of the horizon radius $r_\mathrm{h}$.
The green curve expresses the Einstein radius of the photon sphere, which is obtained by the geodesic approximation.
}
\label{thetaE}
 \end{figure}

\section{Analytic examples of the images}
\label{Analy}

To illustrate our methods, in this section we provide examples in which
images
are obtained analytically. The first example is a $(2+1)$-dimensional CFT
on a sphere at zero temperature, which is dual to the pure AdS$_4$ geometry.
The second example is a $(1+1)$-dimensional CFT on a circle at a finite temperature,
which is dual to the BTZ black hole.

\subsection{Imaging AdS$_4$}
\label{ImAdS4}

In the absence of the black hole horizon,
there should not be holographic images of the black hole.
To demonstrate it, here we study the case of pure AdS geometry. It is a gravity dual of
a CFT at zero temperature, and we have an analytic expression for the response function.

For the pure AdS$_4$ ($r_\mathrm{h}=0$), we can solve the scalar field equation analytically as
\begin{equation}
 \phi_\ell=\frac{2\Gamma(\frac{\ell+\omega+3}{2}) \Gamma(\frac{\ell-\omega+3}{2})}{\sqrt{\pi}\Gamma(\ell+\frac{3}{2})}
\left(\frac{r^2}{1+r^2}\right)^{\ell/2}F\left(\frac{\ell+\omega}{2},\frac{\ell-\omega}{2},\ell+\frac{3}{2};\frac{r^2}{1+r^2}\right)\ .
\end{equation}
Its asymptotic behavior is
\begin{equation}
 \phi_\ell=1+\frac{\omega^2-\ell(\ell+1)}{2r^2}+\frac{8\Gamma(\frac{\ell+\omega+3}{2}) \Gamma(\frac{\ell-\omega+3}{2})}
{3\Gamma(\frac{\ell+\omega}{2}) \Gamma(\frac{\ell-\omega}{2})}\frac{1}{r^3}+\cdots\ .
\end{equation}
Therefore, the response function is given by
\begin{equation}
 \langle \mathcal{O}(\theta)\rangle = \sum_\ell
\frac{8\Gamma(\frac{\ell+\omega+3}{2}) \Gamma(\frac{\ell-\omega+3}{2})}
{3\Gamma(\frac{\ell+\omega}{2}) \Gamma(\frac{\ell-\omega}{2})}
c_\ell Y_{\ell 0}(\theta)\ ,
\label{pureAdSO}
\end{equation}
where
$c_\ell=\int d\theta d\varphi \sin\theta\, g(\theta)Y_{l\,0}(\theta)$ whose explicit expression is in Eq.~(\ref{gexpand}).
The response diverges for $\omega=\ell+2n+3$ ($n=0,1,2,\cdots$).
This corresponds to the normal modes of the pure AdS$_4$.
Applying the Fourier transformation~(\ref{lenstrans0}) onto Eq.~(\ref{pureAdSO}),
we obtain the image of the pure AdS$_4$ as in Fig.~\ref{rh00_view}.
For $\theta_\textrm{obs}=0^\circ$,
we can observe the bright spot at the center.
This corresponds to the ``straight'' null geodesic from the south pole $\theta=\pi$ to the north pole $\theta=0$. This indicates that there is no black hole shadow in pure AdS$_4$ as expected.

Note that there exists
a ring in addition to the bright center.
The angle seems $\theta_\textrm{ring}=\pi/2$.
What is the origin of the ring in the pure AdS?
For $r_\mathrm{h}=0$, the effective potential for null geodesics is simply given by
$v(r)=1+1/r^2$. This effective potential has the ``maximum'' at $r=\infty$.
Therefore, if we tune the angular momentum per unit energy as $\ell/\omega=1$,
we can realize the null geodesic propagating along the AdS boundary ($r=\infty$).
As shown in \eqref{incangle} in Section \ref{RingAngle},
the incident angle $\vartheta_\mathrm{i} = \pi/2$ given by
$\ell/\omega=1$
corresponds to $\theta_\textrm{ring}=\pi/2$.
The ring found in Fig.~\ref{rh00_view} would be originate from the null lay along the AdS boundary.
Even for the Sch-AdS$_4$ spacetime,
there should be null geodesics along the AdS boundary.
However, the ring formed by such geodesics is much weaker than the ring by the photon sphere.

\begin{figure}
  \centering
  \subfigure[$\theta_\textrm{obs}=0^\circ$]
 {\includegraphics[scale=0.46]{rh=0p0_angle=0_ver3.eps}\label{rh00_angle0}
  }
\subfigure[$\theta_\textrm{obs}=30^\circ$]
 {\includegraphics[scale=0.46]{rh=0p0_angle=30_ver3.eps}\label{rh00_angle30}
  }
\subfigure[$\theta_\textrm{obs}=60^\circ$]
 {\includegraphics[scale=0.46]{rh=0p0_angle=60_ver3.eps}\label{rh00_angle60}
  }
\subfigure[$\theta_\textrm{obs}=90^\circ$]
 {\includegraphics[scale=0.46]{rh=0p0_angle=90_ver3.eps}\label{rh00_angle90}
  }
 \caption{
Image of the pure AdS$_4$ ($r_\mathrm{h}=0$) for $\omega=80.5$, $\sigma=0.01$ and $d=0.5$.
The observation point are varied as
$\theta_\textrm{obs}=0^\circ,30^\circ,60^\circ,90^\circ$.
 }
 \label{rh00_view}
\end{figure}

\subsection{Imaging BTZ black holes}
\label{ImBTZ}

Another analytic example of the image is the $(1+1)$-dimensional CFT at a finite temperature.
As its gravity dual, we consider the BTZ black hole:
\begin{equation}
 ds^2=-(r^2-r_\mathrm{h}^2)dt^2+\frac{dr^2}{r^2-r_\mathrm{h}^2}+r^2d\varphi^2\ .
\label{BTZmetric}
\end{equation}
This spacetime is locally AdS and
the Klein-Gordon equation $\Box\Phi=0$ can be analytically solved as~\cite{Cardoso:2001hn}
\begin{equation}
 \phi_m=\frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+1)}\, x^{-\frac{i\omega}{2r_\mathrm{h}}}\, F(a,b,1+a+b; x)\ ,
\label{BTZsol}
\end{equation}
where we decompose the scalar field as $\Phi=e^{-i\omega t + m\varphi}\phi_m(r)$ and define
\begin{equation}
 x=1-\frac{r^2}{r_\mathrm{h}^2}\ ,\quad
a=-\frac{i(\omega+m)}{2r_\mathrm{h}}\ ,\quad
b=-\frac{i(\omega-m)}{2r_\mathrm{h}}\ .
\end{equation}
This solution satisfies ingoing boundary condition at the horizon.
The asymptotic form of the solution near the AdS boundary is given by
\begin{equation}
\phi_m=1+\bigg[\frac{i\omega}{2r_\mathrm{h}}+ab\big\{\psi(a+1)+\psi(b+1)\big\}\bigg]\epsilon
+ ab\, \epsilon \ln (e^{2\gamma-1}\epsilon) + \mathcal{O}(\epsilon^2\ln\epsilon)\ ,
\label{phiinfBTZ}
\end{equation}
where $\epsilon=1-x=r_\mathrm{h}^2/r^2$, $\gamma$ is the Euler's constant and $\psi(x)=\Gamma'(x)/\Gamma(x)$ is the polygamma function.
We consider the time-periodic Gaussian source around $\varphi=\pi$ as $\Phi|_{r=\infty}=e^{-i\omega t}g(\varphi)$ where $g$ is the Gaussian function introduced in Eq.~(\ref{BC}).
Then, the response function becomes
\begin{equation}
 \langle \mathcal{O}(\varphi)\rangle = \sum_m \langle \mathcal{O}_m\rangle
c_m e^{im\varphi}\ ,\quad
\langle \mathcal{O}_m\rangle\equiv \left[\frac{i\omega}{2r_\mathrm{h}} + ab\big\{\psi(a+1)+\psi(b+1)\big\}\right]r_\mathrm{h}^2 \ ,
\end{equation}
where $c_m=(2\pi)^{-1/2}\int d\varphi g(\varphi) e^{im\varphi}\simeq (-1)^me^{-\sigma^2 m^2 /2}$.
Setting observation point at $\varphi=0$, we can obtain the image of the BTZ black hole as
\begin{equation}
 \Psi_\mathrm{S}(x_\mathrm{S})=\int_{-d}^d d\varphi \langle \mathcal{O}(\varphi)\rangle e^{-i\omega \varphi x_\mathrm{S}/f}
=2d \sum_m \langle \mathcal{O}_m\rangle
c_m \frac{\sin[d(m-\omega x_\mathrm{S}/f)]}{d(m-\omega x_\mathrm{S}/f)}\ .
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{BTZ_image3.eps}
\end{center}
\caption{
Image of BTZ black holes with $r_\mathrm{h}=0.3,0.6,0.9,1.2$.
}
\label{BTZimage}
\end{figure}
Figure~\ref{BTZimage} shows $|\Psi_\mathrm{S}|^2$ for $r_\mathrm{h}=0.3,0.6,0.9,1.2$.
We can always find peaks at $x_\mathrm{S}/f=1$ regardless of the horizon radius.
Corresponding angle is $\theta_\textrm{ring}=\pi/2$.
The effective potential for null geodesics in the BTZ spacetime is given by
$v(r)=1-r_\mathrm{h}^2/r^2$.
This implies that all null geodesics with $\omega^2/\ell^2>1$ fall into the black hole.
However, like as the pure AdS case,
if we tune the angular momentum per unit energy as $\ell/\omega=1$,
we can realize the null geodesic propagating along the AdS boundary.
The ring found in the image of the BTZ would originate from the null geodesic on the boundary.
Therefore, the ``ring'' found in the image of BTZ does not have much information about the black hole spacetime.
However, the dual gravitational theory does not need to be pure Einstein in general.
For example,
it has been conjectured that the dual gravitational theory for a high $T_c$ superconductor is given by Einstein-Maxwell-charged scalar system~\cite{Gubser:2008px,Hartnoll:2008vx,Hartnoll:2008kx}.
Also, there can be a higher derivative corrections
if we take into account the quantum gravity effect.
Then, the dual black hole space time will deviate from BTZ and
we would be able to observe images of black holes even for $(1+1)$-dimensional materials.

\section{Einstein ring from retarded Green function}
\label{EringGreen}

We have been constructing the images of the AdS black hole
from the response functions in the previous sections.
Since the response function is closely related to the retarded Green
function, in this section we reinterpret the image of the Einstein ring
from holography in terms of the retarded Green function.
We demonstrate that poles of the Green function, which correspond to
  quasi-normal mode
(QNM)
frequencies
on the AdS black hole
in the gravity side, give major contribution for the formation of the Einstein ring.
  We also estimate the ``Einstein radius'' for weakly coupled quantum field theories which unlikely have their gravity duals.
  We find that the temperature dependence of the Einstein radius of weakly coupled theories is suppressed and gives qualitative difference with theories which have their gravitational duals.

The linear response $\langle \mathcal{O}\rangle$ with respect to the external source $J_\mathcal{O}$ on unit $S^2$
is written as
\begin{equation}
 \langle \mathcal{O}(t,\theta)\rangle
= -2\pi\int dt' d\theta' \sin\theta'
G(t,t',\theta,\theta')J_\mathcal{O}(t',\theta')\ ,
\label{eq:response_func}
\end{equation}
where we have assumed that the source $J_\mathcal{O}$ is axisymmetric.
We introduce the retarded Green function $G(t,t',\theta,\theta')$.
It is well known that the Green function is given by the real time correlation function as
\begin{equation}
 G(t,t',\theta,\theta')=-i \Theta(t-t') \langle [\mathcal{O}(t,\theta),\mathcal{O}(t',\theta')]\rangle\ ,
\end{equation}
where $\Theta(t-t')$ is the step function and $\langle \cdots\rangle$ is the ensemble average with the equilibrium density matrix.
(For example, see Ref.~\cite{Natsuume:2014sfa} for the derivation.)
Let us suppose that the source $J_\mathcal{O}$ is monochromatic with
a frequency $\omega$.
The Green function and the source can be expanded in terms of
Fourier modes
and spherical harmonics
$Y_\ell(\theta)\equiv Y_{\ell\,m=0}(\theta)$ as
\begin{align}
 &G(t,t',\theta,\theta')=\sum_\ell \int \frac{d\omega'}{2\pi} e^{-i\omega' (t-t')}
 G_\ell(\omega) Y_\ell(\theta)Y_\ell(\theta')\ ,\\
&J_\mathcal{O}(t',\theta')=e^{-i\omega t'}\sum_\ell J_\ell Y_\ell(\theta')\ .
\end{align}
Thus, we can rewrite the response function (\ref{eq:response_func}) as
\begin{equation}
 \langle \mathcal{O}(t,\theta)\rangle
=-e^{-i\omega t} \sum_\ell  G_\ell(\omega) J_\ell Y_\ell(\theta)\ .
\label{OofG}
\end{equation}

For simplicity, we set the observation point at the north pole:
$\theta_\textrm{obs}=0$, which is the antipodal point of the
external source localized at $\theta = \pi$.
From the formula~(\ref{lenstrans0}) that we have proposed,
the image of the Einstein ring on the virtual screen is given by
\begin{equation}
 \Psi_S(t,\theta_\textrm{S})=\int_0^{2\pi} d\varphi \int_0^d d\theta\sin\theta
\langle \mathcal{O}(t,\theta)\rangle
\exp\left(-\frac{i\omega}{f}\vec{x}\cdot \vec{x}_S\right)\ ,
\label{Psi_S}
\end{equation}
where $d$ is the radius of the lens and we assume $d\ll 1$ in the unit of the radius of $S^2$.
We have introduced polar coordinates on the boundary $S^2$ and the screen as
\begin{equation}
\begin{split}
&\vec{x}=\sin\theta ( \cos\varphi, \sin\varphi)\ ,\\
&\vec{x}_S=f\sin\theta_S (\cos\varphi_S,\sin\varphi_S)\ .
\end{split}
\end{equation}
Note that the formula (\ref{Psi_S}) means the Fourier transform of
the response function multiplied by a window function which is nonzero
only within a small finite region on $S^2$.

Now, we will rephrase the formula in terms of the retarded Green
function.
We perform the integration with respect to $\varphi$ by using
$\vec{x}\cdot \vec{x}_S = f\sin \theta \sin\theta_S \cos(\varphi-\varphi_S)$,
and plug Eq.~(\ref{OofG}) into Eq.~(\ref{Psi_S}).
As a result, we obtain
\begin{align}
 \Psi_S(t,\vec{x}_S)&=
2\pi\int_0^d  d\theta \sin \theta \, \langle \mathcal{O}(t,\theta)\rangle \mathcal{J}_0(\omega \sin \theta_\textrm{S} \sin\theta)\nonumber\\
&= -2\pi e^{-i\omega t}  \sum_\ell  G_\ell(\omega) J_\ell \int_0^d  d\theta \sin \theta
 Y_\ell(\theta) \mathcal{J}_0(\omega \sin \theta_\textrm{S} \sin\theta)\ ,
\label{PsiS2}
\end{align}
where $\mathcal{J}_n(x)$ is the Bessel function of the first kind, which comes from the $\varphi$-integration.
For $\theta \le d \ll 1$, the spherical harmonics can be approximated by the Bessel function as
\begin{equation}
 Y_\ell(\theta)\simeq \sqrt{\frac{\ell+1/2}{2\pi}}\mathcal{J}_0((\ell+1/2) \theta)\ .
\end{equation}
Using the above expression and replacing $\sin\theta\simeq \theta$ in Eq.~(\ref{PsiS2}),
we can explicitly perform the $\theta$-integration and obtain
\begin{equation}
 \Psi_S(t,\vec{x}_S)
\simeq -\sqrt{2\pi}d^2
e^{-i\omega t}  \sum_\ell  (\ell+1/2)^{1/2}G_\ell(\omega) \, J_\ell \,
\Delta\left((\ell+1/2)d,\omega\sin\theta_\textrm{S} d\right)\ ,
\label{PsiS3}
\end{equation}
where we have defined
\begin{equation}
 \Delta(x,y)\equiv \frac{x\mathcal{J}_1(x)\mathcal{J}_0(y)-y\mathcal{J}_1(y)\mathcal{J}_0(x)}{x^2-y^2}\ .
\end{equation}
The function $\Delta(x,y)$ has the highest peak with width $\sim \pi$ at
$x=y$ and is damping as $|x-y|$ becomes large.
If $\omega d \gg 1$,
$\Delta\left((\ell+1/2)d,\omega\sin\theta_\textrm{S} d\right)$ in
Eq.~(\ref{PsiS3}) has
greater values around
$|\ell/\omega - \sin\theta_\textrm{S}| \lesssim \pi/\omega d \ll 1$.
Thus, we can eventually evaluate
\begin{equation}
 \Psi_S(t,\vec{x}_S)
\propto
e^{-i\omega t} G_\ell(\omega) J_\ell\bigg|_{\ell=\omega\sin\theta_\textrm{S}}  \ .
\end{equation}
Formally, Eq.~(\ref{PsiS3}) means that the image on the virtual screen
corresponds to convolution of the Green function and the window function
in wave-number space $\ell$.
Therefore, it turns out that the image resolution is characterized by
the window function as
$\Delta \ell/\omega \simeq \Delta \theta_\textrm{S} \simeq \pi/\omega d$.

In the view of the gravity side, poles of the retarded Green function in
the frequency domain correspond to
QNM
frequencies, $\omega=\Omega_\ell^n\in \bf{C}$, where $n=0,1,2,\ldots$ represent overtone numbers.
We can expect that $|\Psi_S|^2$ has a large value when the frequency
of the given monochromatic source, $\omega$, is close to the position of a
QNM frequency in the complex $\omega$-plane, so that the Einstein ring
is formed on the screen.
In other words, the condition
for the Einstein radius $\theta_\textrm{S}=\theta_\textrm{ring}$ is written as
\begin{equation}
 \omega\simeq \textrm{Re}\, \Omega^n_{\ell=\omega \sin\theta_\textrm{ring}} \ ,
  \label{thetaring_cond}
\end{equation}
for an overtone number $n$.

The field equation of the massless scalar field,
$\Phi(t,r,\theta,\varphi) \equiv e^{-i\omega t}\sum_\ell Y_\ell(\theta) \psi_\ell(r)/r$,
is given by
\begin{equation}
 \bigg[
 - \frac{d^2}{dr_*^2} + \ell(\ell+1)v(r)
+ \frac{F(r)F'(r)}{r}
\bigg]\psi_\ell(r)
= \omega^2\psi_\ell(r) ,
\end{equation}
where $v(r)$ is the effective potential for null geodesics defined in Eq.~(\ref{dotrsq})
and
we have introduced the tortoise coordinate $dr_* = dr/F(r)$.
According to the WKB analysis~\cite{Ferrari:1984zz},
QNMs are characterized by the behavior of the effective potential around
an extremum.
In the Eikonal limit $\omega \simeq \ell \gg 1$, the local maximum of
a part of the potential $v(r)$ plays a significant role
and is given by Eq.~(\ref{vmax}).
As a result, the QNMs that originate from this local maximum
are described by
\begin{equation}
\begin{split}
&\textrm{Re}\, \Omega_\ell^n = \sqrt{\ell(\ell+1)v_\textrm{max}-\alpha^2}\ ,\\
&\textrm{Im}\, \Omega_\ell^n = \alpha\left(k(n)+\frac{1}{2}\right)\ ,
\end{split}
\end{equation}
where $\alpha\equiv \sqrt{-(d^2v/dr_\ast^2)/(2v)}|_{r=r_\textrm{max}}$
and $k(n)$ is a real number of $O(1)$.
Even though this expression of the QNM frequencies is derived for asymptotically flat spacetime, this is still valid for asymptotically AdS case
since the QNM is highly oscillating as a function of $r_\ast$ for
$\omega ,\ell\gg 1$ and can be easily connect to the desired solution near the AdS boundary.
(For detailed WKB analysis in asymptotically AdS spacetimes, see Refs.~\cite{Festuccia:2008zx,Berti:2009wx,Berti:2009kk,Dias:2012tq}.)
For $\ell \gg 1$, we have $\textrm{Re}\, \Omega_\ell^n \simeq \ell\sqrt{v_\textrm{max}}$ and Eq.~(\ref{thetaring_cond}) gives
\begin{equation}
 \sin\theta_\textrm{ring}\simeq \frac{1}{\sqrt{v_\textrm{max}}}\ .
\end{equation}
This is consistent with our direct numerical calculations.

The retarded Green function $G_\ell(\omega)$ is a well-studied quantity in quantum field theories.
For example,  the Green function of a weakly coupled $\phi^4$ theory with mass $m$ and coupling $\lambda$
is given by
\begin{equation}
 G_\ell(\omega)=\frac{1}{-\omega^2+\ell(\ell+1)+m_T^2}\ ,
\end{equation}
where $m_T=m^2+\mathcal{O}(\lambda T^2)$ is the effective mass with the thermal effect.
Then, the Einstein radius for weakly coupled theory is given by
$\sin^2\theta_\textrm{ring} = 1-m_T^2/\omega^2$. It does not depend on the temperature for a sufficiently large $\omega$ and gives $\theta_\textrm{ring}\simeq \pi/2$.
This suggests that, from the temperature dependence of the Einstein ring,
we can diagnose if a given quantum field theory has its gravity dual.

\section{Conclusion and Discussion}
\label{Concl}

We have studied how we can construct
the holographic image of the AdS black hole using observables in its dual field theory.
We have considered a thermal CFT with a scalar operator, which corresponds in the AdS/CFT to
the massless scalar field in Sch-AdS$_4$ spacetime with a spherical horizon.
We  have put
a time periodic localized source for the operator
and have
computed its response function by the AdS/CFT dictionary.
Applying the Fourier transformation~(\ref{lenstrans0})
onto the response function,
we have observed the Einstein ring as the image of the AdS black hole (Fig.~\ref{rh010306_view}).
The Einstein radius has an increasing trend as a function of the horizon radius $r_\mathrm{h}$.
It is also consistent with the angle of the photon sphere calculated from
the geodesic analysis on the basis of the geometrical optics.

We have shown
that, if the dual black hole exists,
we can construct
the image of the AdS black hole from the observable in the thermal QFT.
In other words, being able to observe the image the AdS black hole in the thermal QFT
can be regarded as a necessary condition for the existence of the dual black hole.
Finding conditions for the existence of the dual gravity picture for a given quantum field theory is
one of the most important problems
in the AdS/CFT.
We would be able to use the imaging of the AdS black hole as a test
for it.
One of the possible  applications
is  superconductors.
It is known that some properties of high $T_c$ superconductors can be captured by the black hole physics
in AdS~\cite{Gubser:2008px,Hartnoll:2008vx,Hartnoll:2008kx}.
One of the
other interesting applications
is the Bose-Hubbard model.
It has been conjectured that the Bose-Hubbard model at the quantum critical regime has
a gravity dual~\cite{Sachdev:2011wg,Fujita:2014mqa,BHMOTOC}.
If we can realize these materials on $S^2$,
they will be appropriate targets for observing AdS black holes by experiments.
Applying localized sources on such materials and measuring  their
responses,
we would be able to observe Einstein rings by tabletop experiments.

Can we distinguish the AdS black hole from thermal AdS by the observation of the Einstein ring?
Results in Section~\ref{ImAdS4} indicate that a ``ring'' will be also observed even for the thermal AdS.
The angle of the ring is, however, always fixed at $\theta_\textrm{ring}=\pi/2$
irrespective of
the temperature of the thermal radiation in the global AdS.
This implies that we would be able to distinguish the AdS black hole from thermal AdS
by the temperature dependence of the Einstein radius.

Finally we make some comments on a relation to black hole chaos and AdS/CFT correspondence.
It has been suggested \cite{Maldacena:2015waa} that
in quantum field theories with their gravity dual, the quantum Lyapunov
exponent $\lambda$ defined by out-of-time-ordered correlators saturates the bound $\lambda \leq 2\pi T$
where $T$ is the temperature of the system. The saturation is due to the strong red shift near the horizon
of the black hole in the gravity dual \cite{Shenker:2013pqa}.
In our study, the photon sphere exists due to the strong curvature of the spacetime near the horizon,
and typical time delay is discretized because the winding number of the null geodesics characterizes
the observed image of the black hole. Since the time delay is nothing but the chaotic behavior of
the black hole, we expect some relation between the frequency $\omega$, temperature $T$
and the integration of the observed amplitude $\Psi_S$. It would be interesting to find a concrete
relation for these quantities, as another check for the existence of the gravity dual based on the choas,
as well as the
direct imaging we proposed in this paper.

\vspace{5mm}
\begin{acknowledgments}
We would like to thank
Paul Chesler,
Vitor Cardoso,
Sousuke Noda
and
Chulmoon Yoo,
for useful discussions and comments.
We would also like to thank organizers and participants of the YITP workshop YITP-T-18-05 ``Dynamics in Strong Gravity Universe''
for the opportunity to present this work and useful comments.
The work of K.~H.~was supported
in part by JSPS KAKENHI Grants No.~JP15H03658,
No.~JP15K13483, and No.~JP17H06462.
The work of K. M. was supported by JSPS KAKENHI Grant No. 15K17658 and
in part by JSPS KAKENHI Grant No. JP17H06462.
The work of S.~K.~was supported in part by JSPS KAKENHI Grants No.~JP16K17704.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Tip 13: Part 5: Psychic Sweatshops Requiring Psychic Readers Give Free Minutes.
Many psychic networks offer clients free minutes. Yet some, such as Sweatshopa and Sweatshop B, require all psychic readers give out 3 free minutes to all clients regardless if they are new or not. This is the company's way of letting clients try different psychics to see who they connect and/or like prior to paying.
When I walk into a supermarket that gives out food samples, the purpose is to give the consumer a chance to taste the product so that an idea is formed about the product. It allows the client to decide if they like it or not. Yet I see gluttons all the time at the supermarket. Once, I saw a woman devour almost 6 slices of pizza. She was chomping down on those samples with such passion and urgency. It was as if she were pregnant with triplets. Yet she was NOT pregnant.
Gluttons also occur on Sweatshopa. They will try to get freebies and manipulate the system. However, the company knows this and condones this because its' their marketing tactic.
Our network never requires a reader to give out any free minutes or settle for a lesser rate for a client intro. Instead, we bite the bullet on those marketing introductions. The reader gets paid for all minutes regardless of promotion or not.
There are many new clients on Sweatshopa that will sign up under different user names to try to get more freebies. In fact, many clients will sign up just to get freebies and close out the moment paid reading time begins.
We cannot give advice how readers should handle that as that is beyond our scope. Sure I have an opinion towards that such as I have an opinion about the pizza glutton I saw at the supermarket. However, one thing is for sure, not every network works the same.
We will continue this series sharing input on Sweatshopa and Sweatshop B . Stay tuned and be sure to apply to our network.


text:
Q: How to set margin right value for a Text in Flutter? I have a card view as follow

There, the red coloured text row has 2 Text fields. I want to keep a space between the Text field in the right and the right side of the card(As exact same as text lines in black color). Below is my implementation. How can I achieve this? Thanks in advance!
Container(
        margin: const EdgeInsets.only(top: 20.0),
         child: Center(
           child: Align(
             alignment: Alignment.topCenter,
             child: Card(
                 child: InkWell(
                   splashColor: Colors.blue.withAlpha(30),
                   onTap: (){
                     print('Card Tapped');
                   },
                   child: Container(
                     width: 380,
                     height: 140,
                     child: Align(
                       alignment: Alignment.centerLeft,
                       child: new Column(
                         children : <Widget>[
                           Container(
                             margin: const EdgeInsets.only(left: 25, top: 25),
                             alignment: Alignment.centerLeft,
                             child: Row(
                               mainAxisAlignment: MainAxisAlignment.spaceBetween,
                               children: [
                                 Text('Account Number',
                                     textAlign: TextAlign.left,
                                     style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15, color: Colors.redAccent),
                                   ),
//                                 Spacer(flex: 1),
                                 Text('12345678',
                                      textAlign: TextAlign.right,
                                      textDirection: TextDirection.rtl,
                                      style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15, color: Colors.redAccent),) //<------- This text field should keep a space from right
                               ],
                             ),
                           ),
                           Container(
                             margin: const EdgeInsets.only(left: 25, top: 15),
                             alignment: Alignment.centerLeft,
                             child: Text('Available Amount                                   LKR 1200.00', //<------- I will re-implement these too in a Row once I figure out this
                                 style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15),
                               ),
                           ),
                           Container(
                             margin: const EdgeInsets.only(right: 21, top: 15),
                             alignment: Alignment.centerRight,
                             child: Text('Active',
                               style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15),
                             ),
                           ),
                         ],
                       ),
                     ),
                   ),
                 ),
             ),
           ),
         )
     ),

A: You're doing it right. You just need a padding field:
                           Container(
     THIS ------------->     padding: EdgeInsets.only(left: 20, right: 20),
                             margin: const EdgeInsets.only(left: 25, top: 25),
                             alignment: Alignment.centerLeft,
                             child: Row(
                               mainAxisAlignment: MainAxisAlignment.spaceBetween,
                               children: [
                                 Text('Account Number',
                                     textAlign: TextAlign.left,
                                     style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15, color: Colors.redAccent),
                                   ),
                                 Text('12345678',
                                      textAlign: TextAlign.right,
                                      textDirection: TextDirection.rtl,
                                      style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15, color: Colors.redAccent),) //<------- This text field should keep a space from right
                               ],
                             ),
                           ),

A: Add right margin like below
margin: const EdgeInsets.only(left: 25, top: 25, right: 25),

Code:
                 Container(
                            margin: const EdgeInsets.only(left: 25, top: 25, right: 25),
                            alignment: Alignment.centerLeft,
                            child: Row(
                              mainAxisAlignment: MainAxisAlignment.spaceBetween,
                              children: [
                                Text('Account Number',
                                  textAlign: TextAlign.left,
                                  style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15, color: Colors.redAccent),
                                ),
//                                 Spacer(flex: 1),
                                Text('12345678',
                                  textAlign: TextAlign.right,
                                  textDirection: TextDirection.rtl,
                                  style: TextStyle(fontWeight: FontWeight.bold,fontSize: 15, color: Colors.redAccent),) //<------- This text field should keep a space from right
                              ],
                            ),

            ),

redpajama_set_name: RedPajamaStackExchange


Gloria Kui Melwani is Of Counsel to Wolf Haldenstein. Ms. Melwani has expertise in complex class actions representing clients including institutional investors such as public and labor pension funds, labor health and welfare benefit funds, and private institutional investors who suffered losses due to corporate fraud. She is a member of the Business Practice and Investor Protection practice groups with an emphasis on securities litigation, derivative litigation, transactional litigation, and copyright, IP & public domain protection.
Represented shareholders in their efforts to make significant corporate governance changes in the companies they invest in. Shareholder actions were brought in Delaware Chancery Court and the New York Commercial Division for matters including breach of fiduciary duty and breach of state fiduciary duties in cases such as Seinfeld v. Barrett, 1:05-cv-00298 (D. Del.); Resnik v. Woertz, 1:10-cv-00527 (D. Del.); In re Invacare Derivative Litigation, 1:11-cv-01893 (N.D. Ohio).
Represented shareholders in state and federal courts in seeking corrective disclosures in proxy statements regarding executive and director compensation, and material information regarding mergers and acquisitions.
Represented historic class action that challenges the copyright to "We Shall Overcome," a song that the Library of Congress has called the most powerful song in the United States history. We Shall Overcome Foundation v. The Richmond Organization, Inc.
Ms. Melwani is a co-founder of the Women's Entrepreneurial Plaintiffs Lawyers Network ("wePLn"), a group for women attorneys to foster collaboration, business development, and career growth. WePLn events and meetings are sponsored by Wolf Haldenstein.
Ms. Melwani is a member of the Chamber Music Committee of the New York City Bar Association.
Ms. Melwani is a member of the New York State Bar Association, the New York City Bar Association, the Asian American Bar Association of New York.


The global economy is one arena where nations have the opportunity to showcase their prowess in strategic development. Historically, the global West has been at the center of all conversations regarding economic growth. Notably, this is down to their strong economic stature and their share of the global economy.
However, this will change soon as more emerging markets continue to make significant progress regarding productivity and stimulating aggregate demand. Notably, China already surpassed the United States and Europe as the global manufacturing destination of choice. Almost all major global companies have their manufacturing plants in the country. Further, countries like Indonesia are climbing up the ladder and could soon beat many European countries by GDP.
Interestingly, economies tend slowing once they hit the point of saturation. This is to say that no more amount of capital injection can result in higher productivity growth. Notably, most, if not all of the critical sectors in the economies are operating at full capacity. To add salt to the wound, there are numerous risks to further economic growth for the developed economies.
For starters, once an economy begins to enjoy the fruits of development, the demographic trends begin to dip. Further, as most of the productive population ages, workforce shortages begin to be a big issue. Although developed economies are already countering this trend by automation, it is clear that they cannot automate demand. Notably, human workforce does not only contribute, but also they consume the products of the economy.
In the emerging economies, demographic metrics are mind-boggling. Notably, countries like India are on track to clocking a third of the world population. Interestingly, this implies that India will not only have plenty of productive people to join the workforce but also a robust aggregate demand.
Further, as the other huge economies like China continue to develop, production costs ratchet up. As such, manufacturers will begin looking for destinations where there are cheap labor and one that is readily available. Eventually, the emerging economies will grow at break-neck speeds and while at that, they will gain much more share of the global economy.


I shared some industry-insider tips with you last week for fryer clean up. I hope the advice helps make that grueling job easier!
This week, I want to share a note I received recently from a special client in New Orleans. But before I get to the note, I want to let you know that the work I did with this client was the most rewarding work I've done since I've been CEO of Safety Fry.
I wanted to take a moment to thank you for helping us open Bon Andouille again.
You see, I believe there are business people, and then there are great humans who happen to be in business. You, of course, fall into the last category.
As we neared the 10-year mark after Katrina, it was time to get Bon Andouille going again. A friend told me about Safety Fry, but I didn't even have fryers. I certainly didn't need protective fryer covers! I called anyway.
When I told you our story (and how I wasn't sure why I was calling), you listened to us and talked to us about our plans for the restaurant. You offered suggestions and made some phone calls. With your help, we got Bon Andouille up and running.
So, from the bottom of my heart, I say "thank you." Just know, there will always be a table ready for you at Bon Andouille.
P.S. The Safety Fry fryer covers are great. Our cooks love them.
Here's the thing, it takes little effort to help someone, especially when you have connections who are willing to jump in and lend a hand. I will be eternally grateful to my business friends who stepped up to help the Richards.
That's why I love what I do. I created Safety Fry to help restaurants keep their staff safe around hot fryers. In fact, I'll be in touch next week with a shocking "hot-oil" story -- a preventable tale. Stay tuned!


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text:
The Fliegerstaffel 16 (Fightersquadron 16) of the Swiss Air Force  was last equipped with Northrop F-5 F aircraft. Their home base was Sion Airport. The Fliegerstaffel 16 had a shield-shaped coat of arms showing the side view of a black dragon with a red eye, red claws, and a red firebeam in front of the white digit 16 and a dark blue background. The old badge was the same, but in round design.

History

The squadron was founded on January 1, 1927 as a Pilot Squadron 16 with Fokker D-Vll aircraft. From 1929 on, the Dewoitine D-9 C-1, the Dewoitine D-19 C-1, the Dewoitine D-27 and the Häfeli DH-5 were used. These aircraft were replaced by the Fokker CV in 1938, and from the year 1940 on there were only C-35 aircraft in the fliegerstaffel 16. In 1942 the conversion to the C-3603 was carried out and 1945 changed the designation into Fliegerstaffel 16. 1951 was the retraining to the P-51 Mustang. The P-51 was replaced 1952 by the first jet aircraft De Havilland DH-100 Vampire. But as early as 1953 the change to the De Havilland DH-112 Venom took place. Reconnaissance tasks was done from the Fliegerstaffel 16 from 1961 on with the De Havilland DH-112 MK 1 R Venom.

From 1968 to 1999, the Fliegerstaffel 16 used the Dassault Mirage III S from  Military Airfield Buochs. The J-2311 received in 1999, do the passing out of this aircraft type from service in the Swiss air force, a golden painting . This machine was scrapped before 2003 in Buochs.

From the year 2000 until the year 2002, the Fliegerstaffel 16 used the F-5E Tiger and was stationed on the Payerne Air Base.  After the BAE Hawk  were taken out of service and the Pilatus PC-21 not yet in service the Fliegerstaffel 16 filled the gap for the training of future F/A-18 pilots with F-5F at Sion Airport. With the introduction of the PC-21 at the Pilotenschule in Emmen, the Fliegerstaffel 16 was disbanded in 2007.

Airplanes
 Fokker D-Vll
 Dewoitine D-9 C-1
 Dewoitine D-19 C-1
 Dewoitine D-27
 Häfeli DH-5
 Fokker CV
 EKW C-35
 C-3603
 P-51 Mustang
 de Havilland Vampire
 de Havilland Venom
 de Havilland VenomR
 Mirage IIIS
 Northrop F-5E
 Northrop F-5F

References

Hermann Keist FlSt16

 Christophe Donnet: Hunter fascination. Schück, Adliswil 1995,

Squadrons of the Swiss Air Force
Military units and formations established in 1927
Military units and formations disestablished in 2007

redpajama_set_name: RedPajamaWikipedia


text:
Q: Strong law of large numbers for the conditional expectation of functions of random vectors Consider a collection of 0-1 random variables $Y_{n,N}$, for all $n$ and $N$.
The random variable $Y_{n,N}$ is a deterministic function of the collection of random variables in $\mathcal{F}_n = \{(U_k)_{k=1,\ldots,n}, (V_k)_{k=1,\ldots,n} \}$ where the $U_k$'s and the $V_k$'s are all independent among them and uniformly distributed over $[0,1]$.
Therefore, $Y_{n,N}=\mathbb{E}[Y_{n,N} | \mathcal{F}_n ]$.
Let me denote by $\mathcal{F}_n\setminus V_n$ the set $\mathcal{F}_n$ where $V_n$ has been removed.
My question is the following. Assuming that
$$
\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N Y_{n,N}
$$
exists almost surely and that
$$
\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \mathbb{E}[Y_{n,N} | \mathcal{F}_n\setminus V_n] = Z
$$
almost surely (where $Z$ is some other random variable), is it true that necessarily
$$
\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N  Y_{n,N}  = Z
$$
almost surely? Why?

Though different (because I am assuming the existence of the first limit), this question is related to the following questions:
Strong law of large numbers for function of random vector: can we apply it for a component only?
and Law of large numbers with one dependency

A: With the notations of the opening post,let $\mathcal G_n:=\mathcal F_{n+1}\setminus V_{n+1}$ and $D_{n,N}:=Y_{n,N}-\mathbb{E}\left[Y_{n,N} | \mathcal{F}_n\setminus V_n\right]$. Then $D_{n,N}$ is $\mathcal G_n$-measurable, $\mathbb{E}\left[D_{n,N} | \mathcal{G}_{n-1} \right]=0$ and $\left\lvert D_{n,N}\right\vert\leqslant 2$ hence by Hoeffding's inequality, we derive that
$$
\mathbb P\left\{\frac 1N\left\lvert \sum_{n=1}^ND_{n,N}\right\rvert \gt \varepsilon\right\}\leqslant\exp\left(-\frac{N^2\varepsilon^2}{4N}\right)
$$
hence an application of the Borel-Cantelli lemma shows that $\frac 1N  \sum_{n=1}^ND_{n,N}\to 0$ almost surely.


text:
Category: Volume 1 Issue 6 (June 2014)
FORECASTING THE PRIMARY EDUCATION: CASE STUDY OF GUJARAT By Kalpana Satija, Mohan Patel, and Keya Patel
June 1, 2014 May 28, 2019 naspublishers
Abstract Education is a vigorous motivation of the 2015 Millennium Development Goals; it help to visualize enlargement of human development. How education play a significant role in human development process! The answer provides the statistical improvement in educational data and performance. In this paper we emphasis on educational performance with forecasting of all indicators in time series analysis with the help of linear regression for future prediction of primary education in Gujarat. We selected total five indicators (Total Literacy Rate, Female Literacy Rate, Enrolment Rate, Dropout Rate and Gender Parity…
VOLUME 1, Volume 1 Issue 6 (June 2014)C53, Forecasting primary education with Linear Regression Model. JEL Classification: I21
INCULCATING VALUES FOR CAREER DEVELOPMENT By Dr. R. Vijaya Anuradha
Abstract Values are the guiding principles of life that contribute to the all round development of an individual. Values give meaning and strength to an individual's character by occupying a central place in his/her life. In ancient India, the Vedas, the Upanishads, the Epics manifested and upheld the values and were the backbone of every individual. But in the present day world, there is a drastic depletion of values among the individuals, due to population explosion, industrialization and globalization, influence of western culture and so on irrespective of caste, creed,…
VOLUME 1, Volume 1 Issue 6 (June 2014)
RURAL HOUSEHOLDS HEALTH STATUS AND DETERMINANTS OF HEALTH CARE EXPENDITURE IN KERALA: A CASE STUDY OF MANNARKKAD PANCHAYATH By Jins Varkey[1]
June 1, 2014 September 26, 2019 naspublishers
India's population continues to grow larger and older. The number of sixty plus in India is expected to increase 198 million in 2030 and to an enormous 330 million by 2050 (United Nations, 2006). These population changes will have a significant impact on India's economic growth and especially on private healthcare spending as public healthcare options are negligible and most of the household expenditure is out-of-pocket (Bhat and Jain, 2006). Even the poorest patients who are in actual need of government health services have no option but to avail of…
COGNITIVE MEDIATED INTERVENTIONS TO DEVELOP SELF CONCEPT AND ASSERTIVE SKILLS AMONG CHILDREN WITH CONDUCT DISORDERS By Dr.Sony Mary Varghese[1]
INTRODUCTION Conduct disorders one of the disruptive behaviour disorders involve a heterogeneous group of under controlled behaviors that range in type and severity (Frick, 1998. It encompasses a wide variety of under controlled behaviours like aggression, lying, destructiveness, vandalism, theft and truancy. The connecting thread in this array of behaviours is the violation of societal norms and the basic rights of others (Jimerson, 2002). Herbert and Webster Stratton (1994) pointed out that these anti-social behaviours have an enduring pattern and hence is deemed unmanageable by parents or teachers. The severity…
ATTITUDE OF M.ED STUDENTS TOWARDS THE USE OF CYBER RESOURCES: A NORMATIVE PERSPECTIVE By Dr. Shazli Hasan Khan[1]
Abstract Tremendous changes are taking place throughout the world in every sphere of life and education system is not left behind unaffected from that changing pace of technology. Similarly the teacher educaiton system too cannot remain immune to these global changes. Information and Communication Technology has severely impacted upon the way the teaching and learning is being done in the classrooms of twenty first century. Digital learning has opened the doors of a new classroom and has made knowledge and information accessible even for those who are sitting at home.…
TOURISM EDUCATION AND HERITAGE RESOURCES By Naveen Kumar T.G[1]
Abstract This paper attempts to probe the importance of Tourism education & dimensions of its development from the past few decades in India. Heritage resources are the valuable assets for any nation which tells about the glorious past & rich culture of a particular country. Tourism can help in shaping our society. It can have advantages on rural as well as urban communities. There is a scope of tourism in India. The government is also encouraging Private enterprises to promote tourism. For developing the tourism education we need to…
ON THE OCCURRENCE OF FAMILY RALLIDAE IN SOLAPUR, (MS) INDIA By Vanjari Rahul[1] and Vanjari Raghvendra[2]
Abstract Rallidae consisting of Rails, Crakes, Coot and Waterhens; these are the shy and surviving very secretive life in a specialized marshy habitat. The present study deals with their diversity, occurrence, population and their behavior. Gradually, threats to species and conservation measures were taken in account to give them a healthy and sustainable surrounding. Keywords: Birds, Family, Rallidae, Solapur, Wetlands, Rails, Crakes. [1] Sangmeshwar College, Solapur, Maharashtra. [2] D.B.F. Dayanand College of Arts and Science, Solapur.
HOLMESIAN REVERBERATIONS: A CRITIQUE OF THE PARADIGM SHIFT IN INVESTIGATIVE-SUSPENSE THRILLER MOVIES IN MALAYALAM By Arshad Ahammad A[1]
Abstract Investigative/detective fiction has been always the favourite art form in the history of writing and reading. Curiosity to know the truth along with a passion towards mystery and mysterious affairs allure human beings to this artefact. Realising this huge popularity of the genre, this paper is an attempt to trace the remarkable and positive changes in the nature and treatment of crime investigation in the Malayalam mainstream cinema after the advent of Jeethu Joseph's investigative-thriller movies. This change is more or less caused by the influence of the…
THEMATIC STUDIES OF CHETAN BHAGAT'S NOVELS By Dr. Dinesh H. Patel[1]
Abstract Chetan Bhagat has been acclaimed as one of the best-selling novelists of the popular literature. He takes upon the sensitive issues which concern to the society in his novels, ranging from the romantic love story to a deplorable condition of the present educational system. He has started a crusade against eradicating the evils of the society by his 'sugar coated novels'. Chetan Bhagat, in almost all his novels, highlights the problems faced by the youth. He presents before us a world which is full of optimism and comic…
VOLUME 1, Volume 1 Issue 6 (June 2014)Alienation, Ambition, Corruption, Exploitation, Modernity, Political and natural Calamities, Religious
FOOTPATH PROBLEM IN KOLKATA – A CASE STUDY IN BURRABAZAR (KOLKATA, WEST BENGAL, INDIA) By Dipanjan Mukherjee[1]
Abstract Footpath next to the road, or a wide flat road shoulder, can prevent pedestrian crashes. The safety benefits will be greatest if the footpath is separated from the road (for example, by a drain, a grass verge or a barrier). A rural footpath can be made relatively cheaply by using a road grader to flatten and clear one side, or preferably, both sides of the road. Pedestrian crossings are needed where rural footpaths pass through communities or trading centres (see pedestrian grade separation, pedestrian refuge island, pedestrian crossing –…

redpajama_set_name: RedPajamaCommonCrawl


text:
Oral vitamin C reduces arterial stiffness and platelet aggregation in humans
I B Wilkinson, I L Megson, H MacCallum, N Sogo, J R Cockcroft, D J Webb
Atherosclerosis is associated with stiffening of conduit arteries and increased platelet activation, partly as a result of reduced bioavailability of nitric oxide (NO), a mediator that normally has a variety of protective effects on blood vessels and platelets. Increased levels of oxygen free radicals are a feature of atherosclerosis that contributes to reduced NO bioavailability and might lead to increased arterial stiffness and platelet activation. Vitamin C is a dietary antioxidant that inactivates oxygen free radicals. This placebo-controlled, double-blind, randomized study was designed to establish whether acute oral administration of vitamin C (2 g), would reduce arterial stiffness and in vitro platelet aggregation in healthy male volunteers. Plasma vitamin C concentrations increased from 42+/-8 to 104+/-8 microM at 6 h after oral administration, and were associated with a significant reduction in augmentation index, a measure of arterial stiffness (by 9.6+/-3.0%; p = 0.016), and ADP-induced platelet aggregation (by 35+/-13%; p = 0.046). There was no change in these parameters after placebo. Vitamin C, therefore, appears to have beneficial effects, even in healthy subjects. The mechanism responsible is likely to involve protection of NO from inactivation by oxygen free radicals, but this requires confirmation. If similar effects are observed in patients with atherosclerosis or risk factors, vitamin C supplementation might prove an effective therapy in cardiovascular disease.
Journal of Cardiovascular Pharmacology
Dive into the research topics of 'Oral vitamin C reduces arterial stiffness and platelet aggregation in humans'. Together they form a unique fingerprint.
Vascular Stiffness Medicine & Life Sciences 100%
Platelet Aggregation Medicine & Life Sciences 86%
Ascorbic Acid Medicine & Life Sciences 85%
Free Radicals Medicine & Life Sciences 48%
Atherosclerosis Medicine & Life Sciences 41%
Reactive Oxygen Species Medicine & Life Sciences 41%
Platelet Activation Medicine & Life Sciences 37%
Wilkinson, I. B., Megson, I. L., MacCallum, H., Sogo, N., Cockcroft, J. R., & Webb, D. J. (1999). Oral vitamin C reduces arterial stiffness and platelet aggregation in humans. Journal of Cardiovascular Pharmacology, 34(5), 690-3.
Wilkinson, I B ; Megson, I L ; MacCallum, H et al. / Oral vitamin C reduces arterial stiffness and platelet aggregation in humans. In: Journal of Cardiovascular Pharmacology. 1999 ; Vol. 34, No. 5. pp. 690-3.
@article{0104ef6b4fce4530ae2d0b7ecb39a0c1,
title = "Oral vitamin C reduces arterial stiffness and platelet aggregation in humans",
abstract = "Atherosclerosis is associated with stiffening of conduit arteries and increased platelet activation, partly as a result of reduced bioavailability of nitric oxide (NO), a mediator that normally has a variety of protective effects on blood vessels and platelets. Increased levels of oxygen free radicals are a feature of atherosclerosis that contributes to reduced NO bioavailability and might lead to increased arterial stiffness and platelet activation. Vitamin C is a dietary antioxidant that inactivates oxygen free radicals. This placebo-controlled, double-blind, randomized study was designed to establish whether acute oral administration of vitamin C (2 g), would reduce arterial stiffness and in vitro platelet aggregation in healthy male volunteers. Plasma vitamin C concentrations increased from 42+/-8 to 104+/-8 microM at 6 h after oral administration, and were associated with a significant reduction in augmentation index, a measure of arterial stiffness (by 9.6+/-3.0%; p = 0.016), and ADP-induced platelet aggregation (by 35+/-13%; p = 0.046). There was no change in these parameters after placebo. Vitamin C, therefore, appears to have beneficial effects, even in healthy subjects. The mechanism responsible is likely to involve protection of NO from inactivation by oxygen free radicals, but this requires confirmation. If similar effects are observed in patients with atherosclerosis or risk factors, vitamin C supplementation might prove an effective therapy in cardiovascular disease.",
author = "Wilkinson, {I B} and Megson, {I L} and H MacCallum and N Sogo and Cockcroft, {J R} and Webb, {D J}",
journal = "Journal of Cardiovascular Pharmacology",
Wilkinson, IB, Megson, IL, MacCallum, H, Sogo, N, Cockcroft, JR & Webb, DJ 1999, 'Oral vitamin C reduces arterial stiffness and platelet aggregation in humans', Journal of Cardiovascular Pharmacology, vol. 34, no. 5, pp. 690-3.
Oral vitamin C reduces arterial stiffness and platelet aggregation in humans. / Wilkinson, I B; Megson, I L; MacCallum, H et al.
In: Journal of Cardiovascular Pharmacology, Vol. 34, No. 5, 1999, p. 690-3.
T1 - Oral vitamin C reduces arterial stiffness and platelet aggregation in humans
AU - Wilkinson, I B
AU - Megson, I L
AU - MacCallum, H
AU - Sogo, N
AU - Cockcroft, J R
AU - Webb, D J
N2 - Atherosclerosis is associated with stiffening of conduit arteries and increased platelet activation, partly as a result of reduced bioavailability of nitric oxide (NO), a mediator that normally has a variety of protective effects on blood vessels and platelets. Increased levels of oxygen free radicals are a feature of atherosclerosis that contributes to reduced NO bioavailability and might lead to increased arterial stiffness and platelet activation. Vitamin C is a dietary antioxidant that inactivates oxygen free radicals. This placebo-controlled, double-blind, randomized study was designed to establish whether acute oral administration of vitamin C (2 g), would reduce arterial stiffness and in vitro platelet aggregation in healthy male volunteers. Plasma vitamin C concentrations increased from 42+/-8 to 104+/-8 microM at 6 h after oral administration, and were associated with a significant reduction in augmentation index, a measure of arterial stiffness (by 9.6+/-3.0%; p = 0.016), and ADP-induced platelet aggregation (by 35+/-13%; p = 0.046). There was no change in these parameters after placebo. Vitamin C, therefore, appears to have beneficial effects, even in healthy subjects. The mechanism responsible is likely to involve protection of NO from inactivation by oxygen free radicals, but this requires confirmation. If similar effects are observed in patients with atherosclerosis or risk factors, vitamin C supplementation might prove an effective therapy in cardiovascular disease.
AB - Atherosclerosis is associated with stiffening of conduit arteries and increased platelet activation, partly as a result of reduced bioavailability of nitric oxide (NO), a mediator that normally has a variety of protective effects on blood vessels and platelets. Increased levels of oxygen free radicals are a feature of atherosclerosis that contributes to reduced NO bioavailability and might lead to increased arterial stiffness and platelet activation. Vitamin C is a dietary antioxidant that inactivates oxygen free radicals. This placebo-controlled, double-blind, randomized study was designed to establish whether acute oral administration of vitamin C (2 g), would reduce arterial stiffness and in vitro platelet aggregation in healthy male volunteers. Plasma vitamin C concentrations increased from 42+/-8 to 104+/-8 microM at 6 h after oral administration, and were associated with a significant reduction in augmentation index, a measure of arterial stiffness (by 9.6+/-3.0%; p = 0.016), and ADP-induced platelet aggregation (by 35+/-13%; p = 0.046). There was no change in these parameters after placebo. Vitamin C, therefore, appears to have beneficial effects, even in healthy subjects. The mechanism responsible is likely to involve protection of NO from inactivation by oxygen free radicals, but this requires confirmation. If similar effects are observed in patients with atherosclerosis or risk factors, vitamin C supplementation might prove an effective therapy in cardiovascular disease.
JO - Journal of Cardiovascular Pharmacology
JF - Journal of Cardiovascular Pharmacology
Wilkinson IB, Megson IL, MacCallum H, Sogo N, Cockcroft JR, Webb DJ. Oral vitamin C reduces arterial stiffness and platelet aggregation in humans. Journal of Cardiovascular Pharmacology. 1999;34(5):690-3.


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