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

The aim of this review is to study gluon scattering amplitudes of four dimensional planar maximally super-symmetric Yang-Mills (MSYM).  We hope that the study of such amplitudes would teach us something about scattering amplitudes of QCD, but at the same time they are much more tractable. The reason for such tractability is twofold. On one hand, perturbative computations are much simpler than in QCD, due to the high degree of symmetry. In fact enormous progress has been made in the last few years. On the other hand, the strong coupling regime of the theory can be studied by means of the AdS/CFT duality, by studying a weakly coupled string sigma-model.

In this review we focus on how to use the AdS/CFT duality in order to compute gluon scattering amplitudes of planar MSYM at strong coupling, referring the reader to \cite{chapAmp,chapDual} for details on the perturbative side of the computation . In section two we set up the problem of computing scattering amplitudes at strong coupling. The problem boils down to the computation of the area of certain minimal surfaces in $AdS$. For the particular  case of four gluons, such surface, and its area, can be explicitly computed. Furthermore, the strong coupling computation hints at some symmetries that actually appear to be symmetries at all values of the coupling. This is briefly reviewed at the end of section two. In section three we focus on the mathematical problem of computing the area of minimal surfaces in $AdS$. The integrability of the model allows the introduction of a spectral parameter. By studying the problem as a function of the spectral parameter we are able to give a solution in the form of a set of integral equations. These equations have the precise form of thermodynamic Bethe ansatz (TBA) equations. The area turns out to coincide with the free energy of such TBA system. Finally, In section four, we end up with some conclusions and a list of open problems.

\section{Gluon scattering amplitudes at strong coupling}

Four dimensional MSYM, the theory whose amplitudes we want to
consider, turns out to be dual to type IIB string theory on
$AdS_5 \times S^5$. This duality receives the name of
$AdS/CFT$ duality \cite{Maldacena:1997re} and is the main focus of this review.
A remarkable feature of this duality is that it allows
to compute certain observables of MSYM at strong coupling by
doing geometrical computations on $AdS$. A well known example
is the computation of the expectation value of super-symmetric
Wilson loops, which reduces to a minimal area problem
\cite{Rey:1998ik,Maldacena:1998im}. In this section we will
show that this is also the case for the computation of
scattering amplitudes at strong coupling! \footnote{In this
section, we follow closely \cite{Alday:2007hr}, to which we
refer the reader for the details.}

As in the gauge theory, we will need to introduce a regulator
in order to define properly scattering amplitudes. In order to
set-up our computation we introduce a D-brane as IR regulator,
as we explain in detail below. Another convenient regulator is
the strong coupling/super-gravity analog of dimensional
regularization. This regulator will be used in order to compare our results with expectations from the perturbative side.

\subsection{Set-up of the computation}

In order to set up the computation at strong coupling, it is convenient to introduce a regularization as follows. We start from a $U(N+k)$ theory, with $k \ll N$, and then consider a vacuum breaking the symmetry to $U(N) \times U(k)$ by giving to a scalar field a vacuum expectation value $m_{IR}$ which plays the role of an infrared cut-off.\footnote{See \cite{Alday:2009zm} for perturbative computations using this regulator.} When we take the 't Hooft limit we keep $k$ fixed, so that the low energy $U(k)$ theory becomes free. We then scatter gluons of this $U(k)$ theory. We are interested in the regime where all kinematic invariants are much larger than the IR cut-off, $s_{ij} \gg m_{IR}^2$. It turns out that the leading exponential behavior at strong coupling can be captured simply by considering $k=1$. At strong coupling this corresponds to consider a $D3$-brane localized in the
radial direction. More precisely, we start with the $AdS_5$
metric written in Poincare coordinates

\begin{equation}
\label{origads}
ds^2 = R^2 {dx_{3+1}^2 +dz^2 \over z^2}
\end{equation}
and place a $D3$-brane at some fixed large value of $z=z_{IR} $
and extending along the $x_{3+1}$ coordinates. The asymptotic
states are open strings that end on that D-brane. We then
consider the scattering of these open strings, that will have
the interpretation of the gluons that we are scattering.

The proper momentum of the strings is $k_{pr}=k z_{IR}/R$, where
$k$ is the momentum conjugate to $x_{3+1}$, plays the role of
gauge theory momentum and will be kept fixed as we take away the
IR cut-off, $z_{IR} \to \infty$. Therefore, due to the warping of the metric, the
proper momentum is very large, so we are considering the
scattering of strings at fixed angle and with very large momentum.

Amplitudes in such regime were studied in flat space by Gross and
Mende \cite{Gross:1987ar}. The key feature of their computation is that the amplitude
is dominated by a saddle point of the classical action. In our
case we need to consider classical strings on $AdS$. Hence, we need to consider a world-sheet with the topology of a disk
with vertex operator insertions on its boundary, which correspond
to the external states (see fig. 1).  A disk amplitude with a fixed ordering of the open string vertex operators corresponds to a given color ordered amplitude.

\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{branescattering.pdf}
\caption{World-sheet corresponding to the scattering of four open
strings. In the figure on the left we see four open strings ending on the IR D-brane, the world-sheet has then the topology of a disk, shown on the right, with four vertex operator insertions.}
\label{fig:1}
\end{figure}

What are the boundary conditions for such world-sheet? since the open strings are attached to the D-brane, $z=z_{IR}$
at the boundary. Furthermore, in the vicinity of a vertex operator, the momentum of the
external state should fix the form of the solution.

In order to state more simply the boundary conditions for the
world-sheet, it is convenient to describe the solution in terms
of T-dual coordinates $y^\mu$, defined as follows

\begin{equation}
\label{tdual}
ds^2=w^2(z)dx_\mu dx^\mu +... ~~~\rightarrow ~~~
\partial_\alpha y^\mu = i w^2(z) \epsilon_{\alpha \beta} \partial_\beta x^\mu
\end{equation}
The presence of the $i$ is due to the fact that we are considering a Euclidean world-sheet in Minkowski space-time. Note that we do not T-dualize along the radial direction. After
defining $r=R^2/z$ the dual metric takes the form

\begin{equation}
\label{dualads}
ds^2=R^2 {dy_\mu dy^\mu+dr^2 \over r^2}
\end{equation}
Note that this metric is equivalent to the same $AdS_5$ metric
we started with! A crucial difference is that now, in terms of
the dual coordinates, the boundary of the
world-sheet is located at $r=R^2/z_{IR}$, which is very small. Furthermore, the $T-$duality we performed interchanges Neumann by Dirichlet boundary conditions. This means that the boundary of the world-sheet sits at a fixed point in the space of the dual coordinates. When a vertex operator with momentum $k^\mu$ is inserted, the location of such point gets shifted by an amount proportional to $\Delta y^\mu=2 \pi k^\mu$.

Summarizing, the boundary of the world-sheet is located at
$r=R^2/z_{IR}$ and is a particular line constructed as follows

\begin{itemize}
\item For each particle of momentum $k^\mu$, draw a
segment joining two points separated by $\Delta y^\mu=2 \pi
k^\mu$.
\item Concatenate the segments according to the insertions
    on the disk.
\end{itemize}

Since gluons are massless, the segments are light-like.
Furthermore, due to momentum conservation, the segments form a
closed polygon. The world-sheet, when expressed in T-dual
coordinates, will then end in such sequence of light-like
segments (see fig. 2) located at $r=R^2/z_{IR}$.

\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{originaltdual.pdf}
\caption{Comparison of the world sheet in original and
T-dual coordinates. The hyperplane on the picture to the right should not be interpreted as a D-brane, but rather as a radial slice where the boundary of the world-sheet it located.}
\label{fig:2}
\end{figure}

As we take away the IR cut-off, $z_{IR} \rightarrow \infty$,
the boundary of the world-sheet moves towards the boundary of
the T-dual metric, at $r=0$. This computation would then be formally equivalent to the computation of the
expectation value of a Wilson loop given by a
sequence of light-like segments at strong coupling \cite{Rey:1998ik,{Maldacena:1998im}}. \footnote{As explained in detail in \cite{chapDual}, this remarkable duality between Wilson loops and scattering amplitudes was also observed in perturbative computations.}

Our prescription is that the leading exponential behavior of
the $N-$point scattering amplitude is given by the area $A$ of the
minimal surface that ends on a sequence of light-like segments on
the boundary

\begin{equation}
\label{finalpresc}
{\cal A}_N \sim e^{-{\sqrt{\lambda} \over 2
\pi}A(k_1,...,k_N)}
\end{equation}
An important comment is in order. Note that the strong coupling
computation is blind to the type or polarization of the
external particles. Such information will contribute to
prefactors in (\ref{finalpresc}) and will be subleading in a
$1/\sqrt{\lambda}$ expansion, relative to the leading exponential term\footnote{As discussed in more detail in \cite{McGreevy:2008zy}, when computing the disk amplitude in the saddle point approximation, one can neglect the polarization of the gluon vertex operators.}.
These differences should be
visible once we consider quantum corrections to the classical
area. This is still an open problem.

Some generalizations to the above picture were developed. In \cite{finitetemp,finitetemp2} finite temperature was introduced while in \cite{Dorn:2009hs} the authors considered solutions with non trivial motion on the $S^5$. Finally, in \cite{quarks,quarks2,quarks3}, the scattering of quarks at strong coupling was considered. Unfortunately, due to space limitations, we wont discuss these interesting developments here, but refer the reader to the original literature.

We have then reduced the problem of computing scattering
amplitudes at strong coupling to the problem of finding minimal
surfaces in $AdS$. In the following we will show that such
surface can be found for the particular case of the scattering
of four gluons. To find and understand this solution in detail
will be quite instructive. Then, in the next section, we will
use the integrability of the problem in order to give a general
solution, for any number of gluons, in the form of a set of
integral equations.

\subsection{Scattering of four gluons}

Consider the scattering of two particles into two particles,
$k_1+k_3 \rightarrow k_2+k_4$ and define the usual Mandelstam
variables

\begin{equation} s=-(k_1+k_2)^2,~~~~~t=-(k_2+k_3)^2\end{equation}

According to our prescription we need to find the minimal surface
ending in the following light-like polygon

\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{4lines.pdf}
\caption{Polygon corresponding to the scattering of four gluons}
\end{figure}

In order to write the Nambu-Goto action it is convenient to use
Poincare coordinates $(r,y_0,y_1,y_2)$, setting $y_3=0$ and
parametrize the surface by its projection to the $(y_1,y_2)$
plane. In this case we obtain an action for two fields, $r$ and
$t$, living in the space parametrized by $y_1$ and $y_2$
\begin{equation} S = {R^2 \over 2 \pi } \int dy_1 dy_2 { \sqrt{ 1 + (
\partial_i r)^2 - (
\partial_i y_0)^2 - (
\partial_1 r \partial_2y_0 - \partial_2 r \partial_1 y_0 )^2 } \over
r^{ 2 } } \end{equation}
where $r=r(y_1,y_2)$ and $\partial_i r=\partial_{y_i} r$, etc.
Our aim is to find a solution to the classical equations of
motion with the appropriate boundary conditions. Let us
consider first the case $s=t$, where the projection of the
polygon lines to the $(y_1,y_2)$ plane
 is a square. By scale invariance we can choose the edges of the
square to lie at $y_1,y_2=\pm 1$. The boundary conditions are
then given by

\begin{equation}
\label{bcsquare}
 r(\pm 1,y_2)=r(y_1,\pm 1)=0,~~~~y_0(\pm 1,y_2)=\pm
y_2,~~~y_0(y_1,\pm 1)=\pm y_1 \end{equation}
In \cite{Kruczenski:2002fb} the solution corresponding to a
single cusp was considered. One can make educated guesses using
such solution as a guidance and propose
\begin{equation} \label{squaresol} y_0(y_1,y_2)=y_1
y_2,~~~~~r(y_1,y_2)=\sqrt{(1-y_1^2)(1-y_2^2)}\end{equation}
Remarkably this turns out to be a solution of the equations of
motion! This is the solution for the case $s=t$, how can we obtain the most general solution?

The dual $AdS_5$ space has a $SO(2,4)$ group of isometries. This symmetry is sometimes referred to as "dual conformal
symmetry" and should not be confused with the
original $SO(2,4)$ symmetry associated to the original $AdS$ space. This dual symmetry can be used in order to map the particular solution we have just found to the most general solution with four edges, in particular with $s \neq t$.
The general solution can be conveniently written as
\begin{eqnarray}r={a \over \cosh u_1 \cosh u_2+b \sinh u_1 \sinh u_2},~~~~
y_0= {a \sqrt{1+b^2} \sinh u_1 \sinh u_2 \over \cosh u_1 \cosh
u_2+b \sinh u_1 \sinh u_2} \\ y_1={a \sinh u_1 \cosh u_2 \over
\cosh u_1 \cosh u_2+b \sinh u_1 \sinh u_2},~~~~ y_2={a \cosh u_1
\sinh u_2 \over \cosh u_1 \cosh u_2+b \sinh u_1 \sinh u_2}
\end{eqnarray}
where $u_{1,2}$ parametrize the world-sheet and we have written the surface as a solution to the equations
of motion in conformal gauge.

$a$ and $b$ encode the kinematical information of the scattering
as follows

\begin{equation}-s(2 \pi)^2 = {8 a^2 \over (1-b)^2},~~~~~-t (2 \pi)^2 ={8 a^2
\over (1+b)^2},
 ~~~~~{ s \over t } = { (1+b)^2 \over
(1-b)^2 } \end{equation}

According to the prescription, we should now plug the classical
solution into the classical action to compute the area and obtain the four
point scattering amplitude at strong coupling. However, in
doing so, we obtain a divergent answer. That is of course the
case, since we have taken the IR regulator away. In order to
obtain a finite answer we need to reintroduce a regulator.
Since we want to compare our results to field theory
expectations, it is convenient to introduce the strong
coupling analog of dimensional regularization.

Gauge theory amplitudes are regularized by considering the
theory in $D=4-2\epsilon$ dimensions. More precisely, one
starts with ${\cal N}=1$ in ten dimensions and then
dimensionally reduce to $4-2\epsilon$ dimensions. For integer
$2\epsilon$ this is precisely the low energy theory living on a
$Dp-$brane, where $p=3-2\epsilon$. We regularize the amplitudes
at strong coupling by considering the gravity dual of these
theories and then analytically continuing in $\epsilon$. The
string frame metric is
 \begin{eqnarray}ds^2=f^{-1/2}dx_{4-2\epsilon}^2+f^{1/2}\left[ dr^2+r^2 d\Omega^2_{5+2\epsilon} \right],~~~~~f=(4 \pi^2 e^\gamma)^\epsilon \Gamma(2+\epsilon) \mu^{2\epsilon}
 \frac{\lambda}{r^{4+2\epsilon}}\end{eqnarray}
Following the steps described above, we are led to the following
action \begin{equation}S=\frac{\sqrt{c_\epsilon \lambda}
\mu^\epsilon }{2\pi} \int \frac{{\cal
L}_{\epsilon=0}}{r^\epsilon}\end{equation}

Where ${\cal L}_{\epsilon=0}$ is the Lagrangian density for
$AdS_5$. The presence of the factor $r^\epsilon$ will have two
important effects. On one hand, previously divergent integrals
will now converge (if $\epsilon < 0$). On the other hand, the
equations of motion will now depend on $\epsilon$ and we were
not able to compute the full solution for arbitrary $\epsilon$.
However, we are interested in computing the amplitude only up
to finite terms as we take $\epsilon \rightarrow 0$. In that
case, it turns out to be sufficient to plug the original
solution into the $\epsilon$-deformed action \footnote{Up to a
contribution from the regions close to the cusps that adds an
unimportant additional constant term.}. After performing the
integrals and expanding in powers of $\epsilon$ we get the
final answer

\begin{eqnarray} \label{finalfour}{\cal A}=e^{-\frac{\sqrt{\lambda}}{2\pi}A},~~~~,-\frac{\sqrt{\lambda}}{2\pi}A=i
S_{div}+\frac{\sqrt{\lambda}}{8\pi}\left(\log{\frac{s}{t}}
\right)^2+\tilde{C} \\ \nonumber S_{div}=2S_{div,s}+2S_{div,t}\\ \nonumber i
S_{div,s}=-\frac{1}{\epsilon^2}\frac{1}{2\pi}\sqrt{\frac{\lambda
\mu^{2\epsilon}}{(-s)^\epsilon}}
-\frac{1}{\epsilon}\frac{1}{4\pi}(1-\log 2) \sqrt{\frac{\lambda
\mu^{2\epsilon}}{(-s)^\epsilon}} \end{eqnarray}

This answer has the correct general structure (see {\it e.g.} \cite{chapAmp,chapDual}) from field theory expectations. Furthermore, once we use the
strong coupling behavior for the cusp anomalous dimension
\cite{Gubser:2002tv},
$f(\lambda)=\frac{\sqrt{\lambda}}{\pi}+...$ we see that the
leading divergence, as well as the finite piece, have not only
the correct kinematical dependence but also the correct overall
coefficient in order to match the BDS ansatz  \cite{Bern:2005iz,chapDual}. As we will see shortly, this result is actually a consequence of the symmetries of the problem!

\subsection{\texorpdfstring{$T-$}{T-}duality and dual conformal symmetry at strong coupling}

An important ingredient of the previous computation was the
existence of a dual $SO(2,4)$ symmetry \footnote{Actually, this
symmetry was first noticed in perturbative computations
\cite{Drummond:2006rz} and then independently in the strong
coupling computation described here.}, associated to the
isometry group of the dual $AdS_5$ space. This symmetry allowed
the construction of new solutions and fixed somehow the finite
piece of the scattering amplitude. \footnote{Naively, this
conformal symmetry would imply that the amplitude is
independent of
 $s$ and $t$, since they can be sent to arbitrary values by a dual conformal transformation.
 The whole dependence on $s$ and $t$ arises due to the necessity of introducing an
 IR regulator. However after keeping track of the dependence on the
 IR regulator, the amplitude is still determined by the dual conformal symmetry. Hence, this regulator breaks the dual conformal symmetry, but in a controlled
 way!}

To a symmetry we associate a Ward identity and in particular dual conformal symmetry will impose some constraints on the amplitudes. Quite remarkably, this duality was also (actually before!) observed at week coupling and is by now believed to be a duality of scattering amplitudes at all values of the coupling. You can see \cite{chapDual} for a detailed account of this symmetry and the constraints it imposes on the amplitudes.\footnote{These constraints have also been derived at strong coupling \cite{wardstrong,wardstrong2}.} Here we will just mention that dual conformal symmetry fixes the answer for the four-point function to have the form (\ref{finalfour}), actually, to all values of the coupling! and hence its agreement with the BDS ansatz. Furthermore, dual conformal symmetry does not fix the answer for the scattering of more than six gluons, hence, in general, the answer deviates from the BDS ansatz. The need for such a deviation, usually called remainder function, was established in \cite{Alday:2007he,Bern:2008ap}. See \cite{chapDual} for more details.

In the last section we have seen that existence of a dual $AdS$ space, is related to the fact that $AdS_5$ goes to itself after a sequence of four T-dualities, followed by the inversion of the radial coordinate, see (\ref{origads}) vs (\ref{dualads}). This set of T-dualities, however, does not leave the full $AdS_5 \times S^5$ sigma model invariant.  For instance, Buscher rules for T-dualities \cite{Buscher:1987qj} imply a shift on the dilaton of the form
\begin{equation}
\label{bosshift}
\phi \rightarrow \phi +4 \times \log z
\end{equation}
where $z$ is the radial coordinate of the original metric (\ref{origads}). The factor of $4$ is due to the fact that we are making four $T-$dualities. In addition to the usual, "bosonic", $T-$dualities, one can introduce a fermionic $T-$duality \cite{Berkovits:2008ic,Beisert:2008iq}. This duality is a non local redefinition of the fermionic world-sheet fields, very much like the bosonic T-duality is a redefinition of the bosonic fields. These T-dualities change the fields of the sigma model according to precise rules. For instance, each fermionic $T-$duality shifts the dilaton by an amount
\begin{equation}
\phi \rightarrow \phi - \frac{1}{2}\times \log z
\end{equation}
We see that by doing eight fermionic $T-$dualities we can undo the shift (\ref{bosshift}) on the dilaton. Actually, one can check that  a combination of the four bosonic $T-$dualities plus eight fermionic $T-$dualities maps the full sigma model to itself! Note also that this argument does not depend on the value of the coupling. One of the implications is that the dual model has the same conformal symmetry group as the original, helping to understand the origin of dual conformal symmetry. Actually, as the construction suggests, dual conformal symmetry extends to a full dual super conformal symmetry. In addition, one has a map between the full set of conserved charges of the two models, in such a way that some of the local charge of one model are mapped to non local charges of the dual model, and viceversa, see for instance \cite{Berkovits:2008ic,Beisert:2008iq}.

The structure of dual super conformal symmetry was also seen at weak coupling and is explained in detail in  \cite{chapDual}, for which we refer the reader for more details.

\section{Minimal surfaces on \texorpdfstring{$AdS$}{AdS}}

In the previous section we have seem how the problem of
computing gluon scattering amplitudes at strong coupling
reduces to the computation of the area of certain minimal
surfaces in $AdS$. In this section we show how the
integrability of the system can be used in order to give a
solution to the problem, in the form of a set of integral
equations. We will follow closely
\cite{Alday:2009ga,Alday:2009yn,Alday:2009dv,Alday:2010vh}, see also \cite{Burrington:2009bh,hiss}, to
which we refer the readers for the details. \footnote{Some of the key ideas used below may be found in relation to the study of wall crossing \cite{gmn,gmn2}. Actually, the method of the first paper in \cite{gmn,gmn2} where instrumental in deriving the expression for the eight gluon amplitude at strong coupling in \cite{Alday:2009ga}.} For this review, we
will focus mostly on a particular kinematic configuration, in which
the minimal surfaces are actually embedded into an $AdS_3$
subspace of the full $AdS_5$. However, the full problem has
been solved and it will be briefly mentioned at the end of the section.

The mathematical problem is to find the area of the minimal
surface ending on the boundary of $AdS$ at a given polygon of
light-like edges. The polygon is parametrized by the location of its cusps $x_i$, which
 are null separated, namely $x_{i,i+1}^2=0$.

We will focus on certain regularized area that is invariant
under conformal transformations. As such, it will depend only
on cross-ratios, of the form $\chi_{ijkl}=\frac{x_{ij}^2
x_{kl}^2}{x_{ik}^2 x_{jl}^2}$. Given the cross-ratios, we want
to compute the area as a function of those. The full problem involves minimal surfaces on
$AdS_5$, in which case there are $3N-15$ independent cross-ratios, where
$N$ is the number of cusps/gluons. We
will restrict to special kinematical configurations in which
the minimal surfaces involved are embedded in $AdS_3$. In this
case, we have $N-6$ independent cross-ratios \footnote{For the general
scattering in four dimensions we have $4N$ coordinates, minus
$N$, since the distance between consecutive points has to be
light-like, minus 15, that is the dimension of the conformal
group $SO(2,4)$. In the case of $AdS_3$, we have $2N-N$ minus
$6$, which is the dimension of $SO(2,2)$.} and the polygon is a
zig-zaged polygon living in one plus one dimensions, which correspond to the boundary of $AdS_3$, see figure
\ref{octagon}.

\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{octagoncrossratios.pdf}
\caption{A zig-zaged null polygon in $1+1$ dimensions is parametrized by $n$ $x^+_i$ coordinates and $n$ $x^-_i$ coordinates. If you want a closed polygon, you can fold the figure in a cylinder.} \label{octagon}
\end{figure}

Since we want a closed contour, and we are in $1+1$ dimensions, we can consider only polygons
with an even number of sides, hence $N=2n$. As one can see in
figure \ref{octagon}, the contour is parametrized by $n$
coordinates $x_i^+$ and  $n$ coordinates $x_i^-$. With each set
of coordinates we can form $n-3$ invariant cross-ratios, of the
form $\chi_{ijkl}^\pm=\frac{x_{ij}^\pm x_{kl}^\pm}{x_{ik}^\pm
x_{jl}^\pm}$.

In order to consider minimal surfaces in $AdS_3$ we need to consider the world-sheet of classical strings on $AdS_3$. This is the subject of the following subsection.

\subsection{Strings on \texorpdfstring{$AdS_3$}{AdS3}}

Classical strings on $AdS_3$ can be described in terms of
embedding coordinates, where $AdS_3$ is the following surface
embedded in $R^{2,2}$
\begin{equation}
\label{Ynorm}
Y.Y \equiv -Y_{-1}^2-Y_0^2+Y_1^2+Y_2^2=-1
\end{equation}
we take the world-sheet to be the whole complex plane.  Since we are interested in classical
solutions, the fields have to satisfy the conformal gauge
equations of motion and the Virasoro constraints
\begin{equation}
\label{Yeom}
\partial \bar \partial Y-(\partial Y . \bar \partial Y) Y=0,~~~~\partial Y . \partial Y= \bar \partial Y . \bar \partial Y=0
\end{equation}
where $\partial Y=\partial_z Y$, etc. An efficient way to focus only in the physical degrees of
freedom, similar to fixing light-cone gauge, is by performing
the so-called Pohlmeyer kind of reduction, see for instance
\cite{po1,po2,po3}, and consider the "reduced" fields
\begin{equation}
\alpha = \log \partial Y . \bar \partial Y,~~~~~p^2 = \partial^2 Y .\partial^2 Y
\end{equation}
As a consequence of the equations of motion and Virasoro
constraints, $p$ can be seen to be a holomorphic function,
$p=p(z)$ while $\alpha(z,\bar z)$ can be seen to satisfy a
generalized version of the Sinh-Gordon equation
\begin{equation}
\label{gsg}
\partial \bar \partial \alpha-e^\alpha+p(z) \bar p(\bar z) e^{-\alpha}=0
\end{equation}
From the definition of the reduced fields, it is clear that they are
invariant under space-time conformal transformation. This means that they describe only the essential part of the problem, without redundancies.

Before proceeding, let us make the following remark. Since
$p(z)$ is a holomorphic function, it is possible to make a
change of coordinates from the $z-plane$ to the  $w-plane$,
where $dw=\sqrt{p(z)} dz$.  In the $w-$plane, after a simple
field redefinition, the generalized Sinh-Gordon equation takes
the usual form
\begin{equation}
\alpha=\hat \alpha+\frac{1}{2} \log p \bar p \rightarrow \partial_w \bar \partial_w \hat \alpha = 2\sinh \hat \alpha
\end{equation}
It would seem that we got rid of all the information on $p(z)$.
However, this is not the case, since the $w-$plane will have in
general a complicated structure (for instance, it will have a
branch cuts, etc, depending on $p(z)$). So, we can choose
between a complicated equation on the complex plane, or a
simple equation on a more complicated space. Depending which
questions we want to answer, one description may be more
convenient than another.  Finally, we are interested in the area of the classical World-sheet. Written in terms of the reduced fields it becomes
\begin{equation}
\label{areaw}
{\cal A}=\int e^\alpha d^2 z=\int e^{\hat \alpha} d^2 w
\end{equation}

\subsection{Classical solutions corresponding to minimal surfaces ending on null polygons}

What are the properties of the holomorphic function $p(z)$ and
$\alpha(z,\bar z)$ for solutions corresponding to minimal surfaces ending on null polygons?
 In order to answer this question we can start by
considering the four cusps solution found in the previous
section and perform the Pohlmeyer reduction. We find
\begin{equation}
p(z)=1,~~~~~\alpha =\hat \alpha=0
\end{equation}
Hence, the four cusps solution simply correspond to the vacuum
solution of the Sinh-Gordon equation!  What about solutions
with a higher number of cusps? First of all we propose that the
field $\alpha$ is regular everywhere, since we are looking for
smooth space-like solutions. Second, we expect a general
solution to be similar to the four cusps solution when
approaching the boundary, so we expect that $\hat \alpha
\rightarrow 0$ as $|z|$ becomes large.

Finally, if we are interested on a minimal surface  ending on a
polygon with $2n$ cusps, we propose $p(z)$ to be a polynomial
of degree $n-2$ \footnote{The motivation for this proposal, comes from the fact that an homogeneous polynomial of degree $n-2$ possesses all the symmetries to correspond to a symmetric polygon of $n$ edges.}
\begin{equation}
p(z)=z^{n-2}+c_{n-4}z^{n-4}+...+ c_1 z+ c_0
\end{equation}
we have used rescalings and translations in order to set the
coefficients of $z^{n-2}$ and $z^{n-3}$ to one and zero
respectively. Such polynomial contains $n-3$ complex
coefficients, or $2n-6$ real coefficients, which exactly agrees
with the amount of expected independent cross ratios for a
polygon with $2n$ cusps!

Summarizing: minimal surfaces ending on a light-like polygon
with $2n$ cusps correspond to a holomorphic polynomial of
degree $n-2$ and a field $\hat \alpha$ satisfying the
Sinh-Gordon equations and with boundary conditions such that it
decays at infinity and diverges logarithmically at the zeroes
of $p(z)$, which amounts to say that $\alpha$ is regular everywhere.

Since $\hat \alpha$ decays at infinity, the integral defining
the area (\ref{areaw}) diverges. We define a regularized area by
subtracting the behavior at infinity
\begin{equation}
A_{reg}=\int (e^{\hat \alpha}-1)d^2w
\end{equation}
 As the reduced fields
are invariant under space-time conformal transformations, the
regularized area will be a function of the cross-ratios only.
\footnote{The full answer would include also the integral of
the one we have subtracted. In order to compute it one would
need to introduce a physical regulator and this part of the
answer will not be conformal invariant. Anyway, its explicitly
form can be worked out and turns out to be quite universal. In
this review we will focus on the "interesting" part of the
answer $A_{reg}$.} The computation of this regularized area is
the main focus of the remaining of this review.

\subsubsection{Reconstructing the space-time solution and its behavior at infinity}

In the following we would to check that the world-sheet we are considering has the desired form. In particular, we would like to understand the shape, in space-time, of the boundary of our world-sheet. For that, we first review a general procedure to reconstruct the world-sheet from the reduced fields, and then study its boundary.

Given an holomorphic function $p(z)$ and a field
$\alpha$ satisfying (\ref{gsg}) it is possible to reconstruct a
space-time solution satisfying (\ref{Yeom}), and
(\ref{Ynorm}). The procedure amounts to solve two auxiliary
linear problems, which we denote as left and right
\begin{eqnarray}
\label{linear}
(d+B^L) \psi^L_{a}=0,~~~~(d+B^R) \psi^R_{\dot a}=0
\end{eqnarray}
where the flat connections $B^{L,R}$ are two by two matrices
constructed from $p(z)$ and $\alpha(z,\bar z)$. For instance
\begin{equation}
B_z^L=\left(\begin{matrix} \partial \alpha/4 & \frac{1}{\sqrt{2}}e^{\alpha/2} \\ \frac{1}{\sqrt{2}}p e^{-\alpha/2} & -\partial \alpha/4 \end{matrix}  \right)
\end{equation}
we denote different components of the connections by
$B^L_{\alpha \beta}$ and $B^R_{\dot \alpha \dot \beta}$. On the
other hand, the indices $a$ and $\dot a$ in (\ref{linear})
denote independent solutions of the auxiliary linear problems.
Each $\psi_a^L$ or $\psi_{\dot a}^R$ is then a doublet. We
denote the components of this doublet by $\psi^L_{\alpha,a}$,
etc.

Given the solutions of these two auxiliary linear problems, one
can show that the space-time coordinates are simply given by
\begin{equation}
Y_{a,\dot a}=\left(\begin{matrix} Y_{-1}+Y_2 & Y_1-Y_0 \\ Y_1+Y_0 & Y_{-1}+Y_2 \end{matrix}  \right)_{a,\dot a}=\psi^L_{\alpha,a} \delta^{\alpha \dot \beta} \psi^R_{\dot \beta, \dot a}
\end{equation}
 One can show that $Y$
constructed this way satisfies all the required properties. If we see $\psi^L$ and $\psi^R$ as two by two matrices, then
the space-time coordinates would be given by $Y = (\psi^L)^T
\psi^R$. On the other hand, note that given a solution to the
left problem, $\psi^L$, then $\psi^L U^T$ is an equally good
solution, and the same happens with the right problem. Hence,
given $Y$, we obtain a family of space-time solutions $U Y V$.
These are nothing but the space-time conformal transformations.

Now we would like to understand the behavior of the solutions of
the linear auxiliary problems for very large values of $|z|$,
or $|w|$. This will tell us the behavior of the world-sheet
near its boundary. Let us start, by simplicity, with the case
of a homogeneous polynomial, $p(z)=z^{n-2}$. Hence $w \approx
z^{n/2}$. As a result, as we go once around the $z-$plane, we
go around the $w-$plane $n/2$ times.

Due to the boundary conditions for the reduced fields, the flat connections $B^{L,R}$
drastically simplify at infinity and we can solve the auxiliary
linear problems. A general solution will be of the form
\begin{eqnarray}
\label{approxsol}
\psi^L_a \approx c_a^+ \left( \begin{matrix} 1 \\0  \end{matrix} \right) e^{w +\bar w}+c_a^- \left( \begin{matrix} 0 \\1  \end{matrix} \right)e^{-(w+\bar w)} \cr
\psi^R_{\dot a} \approx d_{\dot a}^+ \left( \begin{matrix} 1 \\0  \end{matrix} \right) e^{\frac{w -\bar w}{i}}+d_{\dot a}^- \left( \begin{matrix} 0 \\1  \end{matrix} \right)e^{-\frac{w -\bar w}{i}}
\end{eqnarray}
The $w$-plane is naturally divided into quadrants, see figure
\ref{wplane}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{wplanecusps.pdf}
\caption{When looking at the left problem, (each sheet of) the $w-$plane is naturally divided into two parts, according to the sign of $Re(w)$. In the same way, when looking at the right problem, the $w-$plane is naturally divided into two parts, according to the sign of $Im(w)$. Hence, the $w-$plane is naturally divided into four quadrants. Large values of $|w|$ in each of these angular sectors correspond to a cusp.} \label{wplane}
\end{figure}
In each quadrant one of the two solutions of each problem (left
and right) dominates. For instance, in the upper right
quadrant, the solution proportional to $c_a^+$ dominates in the
left problem, while the solution proportional to $d_{\dot a}^+$
dominates in the right problem. This means that for large
values of $|w|$, the whole quadrant corresponds to a single
point in the boundary, given by $Y_{a \dot a} \approx (Large)
\times c_a^+ d_{\dot a}^+$.

As we change quadrant, one and only one of the two dominant
solutions change and we jump a light-like distance to the next
cusp. In each quadrant/cusp we can write
\begin{equation}
Y_{a,\dot {a}} \approx \lambda_a \tilde \lambda_{\dot a}
\end{equation}
where $\lambda$ is given by the leading contribution to the
left problem and $\tilde \lambda$ by the leading contribution
to the right problem. As we change quadrant, one of the two
solutions, $\lambda$ or $\tilde \lambda$, changes. As we go
around the $w-$plane $n/2$ times, we get the expected $2n$
cusps!

In the general case in which the polynomial is not homogeneous,
the picture is very much the same. In general, the degree of
the polynomial determines the number of cusps, while the
coefficients on the polynomial determine the shape of the
polygon.

Let us finish this section with a very important observation. Since we know that the classical equations describing strings on $AdS_5 \times S^5$ are integrable, see for instance \cite{Bena:2003wd},  we expect the present problem to be integrable as well. Indeed, it is possible to
introduce a spectral parameter $\zeta$
\begin{eqnarray}
B_z \rightarrow B_z(\zeta)=\frac{1}{4}\left(\begin{matrix} \partial \alpha &0 \\0& -\partial \alpha \end{matrix} \right) + \frac{1}{\zeta} \frac{1}{\sqrt{2}}\left(\begin{matrix} 0&e^{\alpha/2} \\ p e^{-\alpha/2}& 0\end{matrix} \right) \\
B_{\bar z} \rightarrow B_{\bar z}(\zeta) =\frac{1}{4} \left(\begin{matrix}- \bar \partial \alpha &0 \\0& \bar \partial \alpha \end{matrix} \right) + \zeta \frac{1}{\sqrt{2}}\left(\begin{matrix} 0&\bar p e^{- \alpha/2} \\  e^{\alpha/2}& 0\end{matrix} \right)
\end{eqnarray}
such that the flat connections are still flat, namely the satisfy $\partial B_{\bar z}-\bar \partial B_z+[B_z,B_{\bar z}]=0$, for all values of $\zeta$. The introduction of the spectral parameter, also allows to
study both linear problems in a unified manner. The left and
right connections are just particular cases of the above flat
connection, more precisely
\begin{equation}
B(\zeta =1) = B^L,~~~~B(\zeta=i)=B^R
\end{equation}
The existence of the spectral parameter played a key role in the spectrum problem, see for instance \cite{SchaferNameki:2010jy}. In that context, the key object are the eigenvalues of the monodromy matrix constructed out of the flat connection. In the present case the key object are certain cross-ratios constructed from the holonomy of the connections.

\subsection{Y-system for minimal surfaces}

Let us focus on the left problem. We see that each sheet on the
$w-$plane is naturally divided into two sectors, one with
$Re(w)>0$ and the other with $Re(w)<0$. In each sector the
small solution is well defined (up to a normalization
constant). On the other hand, the large solution is not, as we
can add to it a part of the small solution. Let us then
introduce the following terminology:
\begin{itemize}
\item The $w-$plane is divided into $n$ sectors, since each sheet contains two sectors. We label this sectors by $i=0,...,n-1$.
\item We call $s_i^L$ the small solution at the $i-th$
    sector. This is the solution with the
    fastest decay along the line in the center of the
    $i-th$ sector, for increasing $|w|$.
\end{itemize}
In order to understand why these small solutions are important, we need to introduce a new element. Given that our connections are $SL(2)$ matrices, we can introduce a $SL(2)$ invariant product
\begin{equation}
\label{psinor}
\psi_a^L \wedge \psi_b^L \equiv \epsilon^{\alpha \beta} \psi^L_{\alpha,a}  \psi^L_{\beta,b} = \epsilon_{ab}
\end{equation}
The second equality corresponds to setting a normalization factor for our solutions. This can be done since one can check the above product is independent on the world-sheet coordinate $z$.

As already seen, the location of the cusps is determined by the
large solutions. The large component of a solution, on a given
sector, can be extracted by using the small solution on such
sector and the $SL(2)$ invariant product just introduced, more
precisely
\begin{equation}
\psi_a^L \wedge s_i^L \approx \lambda_a^i
\end{equation}
where the symbol $\approx$ means up to factors that will cancel out in the final expression for the cross-ratios. How do we construct space-time cross-ratios? we have see that
the location of the cusps is given by $Y^i_{a \dot
a}=\lambda_a^i \tilde \lambda_{\dot a}^i$. The space time
cross-ratios involve distances like
\begin{equation}
Y^i . Y^j = \epsilon^{ab} \epsilon^{\dot a \dot b} Y^i_{a \dot a} Y^j_{b \dot b}=<\lambda^i \lambda^j> <\tilde \lambda^i \tilde \lambda^j>,~~~~~<\lambda^i \lambda^j> =\epsilon^{ab} \lambda^i_a \lambda^j_b
\end{equation}
Given the normalization condition (\ref{psinor}), one can
easily show
\begin{equation}
<\lambda^i \lambda^j> \approx < (\psi_a^L \wedge s_i^L)(\psi_a^L \wedge s_j^L)> = s_i^L \wedge s_j^L
\end{equation}
Which means that space-time cross-ratios can be constructed from inner products of the small solutions in the corresponding sectors! more precisely
\begin{equation}
\frac{x_{ij}^+ x_{kl}^+ }{x_{ik}^+ x_{jl}^+}=\frac{(s_i^L \wedge s_j^L)(s_k^L \wedge s_l^L)}{(s_i^L \wedge s_k^L)(s_j^L \wedge s_l^L)}
\end{equation}
Small solutions are defined up to a normalization constant.
Note that such normalization constants cancel out when
computing cross-ratios.

The strategy we will follow is to introduce the spectral
parameter $\zeta$ as shown in the previous section and study
the small solutions of the corresponding connection
\begin{equation}
(d+B(\zeta))s_i(\zeta)=0
\end{equation}
then, we can consider the cross-ratios as a function of such spectral parameter
\begin{equation}
\chi_{ijkl}(\zeta)=\frac{(s_i \wedge s_j)(s_k \wedge s_l )}{(s_i \wedge s_k )(s_j \wedge s_l)}
\end{equation}
The physical cross-ratios are then obtained by setting the
spectral parameter to appropriate values
\begin{equation}
\chi_{ijkl}(\zeta=1)=\chi_{ijkl}^+,~~~~~\chi_{ijkl}(\zeta=i)=\chi_{ijkl}^-
\end{equation}
A very important property of the flat connection $B(\zeta)$ is
that it possesses a $Z_2$ symmetry: $B(e^{i\pi} \zeta)=\sigma_3 B(\zeta) \sigma_3$, where $\sigma_3$ is the usual Pauli matrix. This symmetry
allows to relate small solutions at different values of the
spectral parameter, for instance $s_{i+1}(\zeta)=\sigma_3
s_i(e^{i\pi} \zeta)$, and in particular, it implies
\begin{equation}
\label{z2rel}
s_i \wedge s_j (e^{i \pi} \zeta)=s_{i+1} \wedge s_{j+1}(\zeta)
\end{equation}
This identity is crucial in deriving the equations below.
Besides, in order to simplify subsequent expressions, we will
assume $s_i \wedge s_{i+1}=1$.

Now we have all the elements to derive the so called Hirota
equations and the Y-system equations. The trick is to choose
$s_0$ and $s_1$ as a complete basis of flat sections, and
express two arbitrary consecutive small solutions $s_k$ and
$s_{k+1}$ in terms of these
\begin{eqnarray}
s_k& =& (s_k \wedge s_1) s_0-(s_k \wedge s_0) s_1 \\
s_{k+1} &=& (s_{k+1} \wedge s_1) s_0-(s_{k+1} \wedge s_0) s_1
\end{eqnarray}
Next, use (\ref{z2rel}) in order to express every wedge as a wedge involving $s_0$ and consider $1=s_k \wedge s_{k+1}$, we obtain
\begin{equation}
-(s_{k-1} \wedge s_0)^{++} (s_{k+1} \wedge s_0)+(s_k \wedge s_0)^{++} (s_k \wedge s_0)=1
\end{equation}
where we have introduced the notation $f^\pm=f(e^{\pm i
\pi/2}\zeta)$, $f^{++}=f(e^{ i \pi}\zeta)$, etc. Let us
introduce $T_k=s_0 \wedge s_{k+1} (e^{-i (k+1) \pi/2} \zeta)$.
In terms of these we obtain
\begin{equation}
T_s^+ T_s^-=T_{s+1}T_{s-1} +1
\end{equation}
which has the form of the so called Hirota equations! from the
definition of $T_s$, we see that it is non trivial for
$s=0,...,n-2$. The $Y-$system equations can be obtained by
introducing $Y_s \equiv T_{s-1} T_{s+1}$
 %
\begin{equation}
\label{Ysystem}
Y_s^+ Y_s^-=(1+Y_{s+1})(1+Y_{s-1})
\end{equation}
$Y_s$ is non trivial for $s=1,...,n-3$. Note that this agrees
with the amount of (complex) cross-ratios of our scattering
problem. These are functional equations for $Y_s(\zeta)$ and
are valid for any value of $\zeta$. Note that they followed from
a chain of rather trivial facts!

One could reintroduce the normalized factors $s_i \wedge
s_{i+1}$ and check that the $Y-$functions are given by the
usual cross-ratios introduced above. The physical cross-ratios,
are then obtained by evaluating $Y_s(\zeta)$ at $\zeta=1$ and
$\zeta=i$.

Such equations are not the whole story. In particular, note
that that the holomorphic function $p(z)$ does not enter at all
in such equations! The point is the following. There are many
solutions to such equations. The correct solution is then
picked by specifying the analytic properties and boundary
conditions of $Y_s(\zeta)$ as we move on the $\zeta-$plane.
This is how the information about the holomorphic polynomial
enters and will be the subject of the following section.

\subsection{Integral equations}

In order to pick the appropriate solution to the $Y-$system
equations (\ref{Ysystem}) we need to specify the analytic
properties of $Y_s(\zeta)$. By analyzing the auxiliary linear
problems and the definition of $Y_s(\zeta)$ one can show that
$Y_s(\zeta)$ are analytic away from $\zeta=0,\infty$. On the
other hand, As $\zeta \rightarrow 0,\infty$, the flat
connection simplifies and the inverse problem can be solved by
using a $WKB$ approximation, where the role of $\hbar$ is
played by $\zeta$ or $1/\zeta$ . By calling $\zeta=e^\theta$,
one can show that for large $\theta$ the solution behaves as \footnote{Note that even though we used the WKB approximation, this is the behavior of the exact solution.}
\begin{equation}
\log Y_s \approx - m_s \cosh \theta+...
\end{equation}
where $m_a$ is given by the periods of  $p(z)^{1/2}$ along the
cycles $\gamma_a$, namely $m_a \approx - \oint_{\gamma_a}
\sqrt{p(z)} dz$. This is how the information of the polynomial
$p(z)$ enters into the problem. This periods are usually
complex, and there are $n-3$ of them, which exactly agrees with
the quantity of expected cross-ratios. These $m_a$ should be
seen as the boundary conditions for the above equations.

The strategy now is well known from the study of integrable
systems. We can combine the $Y-$system equations with the
analytic properties and boundary conditions for the
$Y-$functions, in order to write a system of integral equations
for them. The solutions to these integral equations will
automatically satisfy the $Y-$system equations and have the
required boundary conditions. The system of integral equations
is given by \footnote{Considering $l_s \equiv \log(Y_s/e^{-m_s \cosh \theta})$, which is analytic in the strip $Im(\theta)<\pi/2$ and vanishes as $\theta$ approaches infinite. The integral equations can be obtained by convoluting the equation $l_s^++l_s^-=\log(1+Y_{s+1})(1+Y_{s-1})$ with the kernel in (\ref{tba}).}
\begin{equation}
\label{tba}
\log Y_s= -m_s \cosh \theta +\frac{1}{2\pi}  \int_{-\infty}^\infty \frac{1}{\cosh(\theta-\theta')} \log (1+Y_{s+1}(\theta'))(1+Y_{s-1}(\theta'))
\end{equation}
The system of equations (\ref{tba}) has the form of TBA
equations, that arise when studying integrable models in finite
volume, see {\it e.g.} \cite{Zamolodchikov:1989cf}. Even
though, for the sake of clarity, some overall coefficients have
been suppressed in the derivation of these equations, the final
form of the equations is given with all the correct
coefficients. From the TBA point of view, the parameters $m_a$
enter as masses. Once the masses are given, the solution of the
above system is unique. The physical cross-ratios can the been
read off by looking at $Y_s(\theta)$ for appropriate values of
$\theta$.

These integral equations can also be written in terms of the physical cross-ratios $y_s^{+}=Y_s(\zeta=1)$ and $y_s^{-}=Y_s(\zeta=i)$ only, without resorting to the auxiliary parameters $m_s$. In order to achieve that, one simple evaluates (\ref{tba}) at the physical values of the spectral parameter in order to eliminate the masses, see \cite{Alday:2010ku}. We obtain
\begin{equation}
\label{tbanew}
\log Y_s= \cosh \theta \log y_s^+ -i \sinh \theta \log y_s^- +\int d\theta' \frac{\sinh 2\theta}{\cosh(\theta'-\theta)\sinh 2\theta'}  \log (1+Y_{s+1}(\theta'))(1+Y_{s-1}(\theta'))
\end{equation}
Note that having solved (\ref{tbanew}), we could read off the masses from the asymptotic behavior of the solutions.

How do we compute the regularized
area, once we have solved the above system of integral
equations? It turns out that the area can be written in terms
of the $Y-$functions in a very simple form
\begin{equation}
\label{regarea}
A_{reg}=\sum_s \int d\theta \frac{m_s}{2\pi} \cosh \theta \log(1+Y_s(\theta))
\end{equation}
In order to derive this expression, one expands the $Y$ functions a few orders for small and for large values of $\zeta$. The expansion coefficients are written in terms of period integrals that also appear in the expression for the area, see \cite{Alday:2010vh} for details. This expression has exactly the form of the free energy of the TBA system. \footnote{It can be shown \cite{Alday:2010ku} that actually this area coincides with the extremum of the Yang-Yang functional for the modified TBA equations (\ref{tbanew})}.

The strategy to solve the full problem is then clear. For a choice of the cross-ratios we solve the integral equations (\ref{tbanew}), and from their solution we compute the area (\ref{regarea}). Hence, we have
the area for these values of the cross-ratios.

In this review we have treated in detail the case of minimal surfaces in $AdS_3$. However, the general case of minimal surfaces in $AdS_5$ can also be solved \cite{Alday:2010vh}. Much of what we have said can be carried out for the general case. In this case we get a bigger system of $Y-$functions, denoted by $Y_{a,s}$, where $a=1,23$ and $s=1,...,N-5$. Note that their number equals the number of independent cross-ratios. Very much as before, one can obtain $Y-$system equations, which supplemented with the appropriate boundary conditions can be written as a system of integral equations. Again, this system of equations has the form of a TBA system, and the regularized area coincides with the free energy of such system.

These equations can be solved numerically, see for instance \cite{Alday:2010vh}. On the other hand, it is very hard to find analytical solutions. However, some limits, for instance the so called small masses/CFT limit and the large masses limit , are more tractable, see \cite{limits,limits2,limits3}.

\section{Conclusions}

We reviewed the computation of scattering amplitudes of planar
maximally super-symmetric Yang-Mills at strong coupling. By
using the $AdS/CFT$ duality the problem boils down to the
computation of the area of certain minimal surfaces on $AdS$.

Then we showed how the integrability of the model can be efficiently used in
order to give an answer for the problem in terms of a set of
integral equations. Integrability allows to introduce a one parameter deformation (the spectral parameter $\zeta$) and study such deformed problem. One can then write down a system of functional equations, or $Y-$system, valid for any value of $\zeta$. One can combine these functional equations with the knowledge of the analytic behavior of the $Y-$functions in the $\zeta-plane$, in order to write a set of integral equations which can be solved iteratively, and give the desired answer. There are many directions one could try to follow, some of the most interesting are the following

\begin{itemize}
\item It would be nice to find a physical connection between the integrable system that the TBA equations describe and the original integrable system.
\item It would be very interesting to extend the present construction to the full quantum problem. As a first step, one could try to compute one loop (from the strong coupling point of view) corrections to the above picture. This would allow, for instance, to distinguish between different amplitudes.
\item It would also be interesting to look for similar structures (for instance, the analogous of the spectral parameter, etc) in perturbative computations. Related to this, in \cite{Alday:2010ku} and subsequent papers, an operator product expansion for Wilson loops have been developed. This allows to use certain tools of integrability \cite{Basso:2010in} in order to make predictions at all values of the coupling.
\item One could hope that similar technology can be applied to related problems. One such problem is the computation of form factors, in which progress have been made recently, see \cite{Alday:2007he}\cite{Maldacena:2010kp}\cite{Brandhuber:2010ad}.
\end{itemize}

{\bf Acknowledgments}

We would like to thank Davide Gaiotto, Amit Sever, Pedro Vieira
and specially Juan Maldacena for collaboration on the material
exposed in this review.

\phantomsection
\addcontentsline{toc}{section}{\refname}

redpajama_set_name: RedPajamaArXiv


text:
Q: TypeScript types to infer via keyof within React Context Provider I have setup the TypeScript types to infer the value based on an object key using keyof.  They appear to work outside of the React Context Provider however I am having problems when using similar functions inside the context provider.  The TypeScript Playground link demonstrates the inference error best.
I figure I am missing Type annotations somewhere but cannot find any solutions that work. Any assistance appreciated!
TypeScript playground demo

type IndividualSport = {
  name: string;
}

type TeamSport = {
  name: string;
  players: number;
}

type Sports = {
  cricket: TeamSport;
  golf: IndividualSport;
}

type SportsKey = keyof Sports;

let sampleSports: Sports = {
  cricket: {
    name: "Cricket",
    players: 11
  },
  golf: {
    name: "Golf",
  },
};

function extractSport<K extends SportsKey>(sports: Sports, key: K) {
  return sports[key];
}

const cricket = extractSport(sampleSports, 'cricket');
const golf = extractSport(sampleSports, 'golf');

// THIS WORKS
console.log(cricket.players);
console.log(golf.name);

// REACT CONTEXT PROVIDER

type SportsContextProps<K extends SportsKey> = {
  getSportInfo: (key: K) => Sports[K];
}

const SportsContext = createContext<SportsContextProps<SportsKey>>({
  getSportInfo: (key) => extractSport(sampleSports, key),
});

const useSportsContext = () => useContext(SportsContext)!;

function SportsProvider({
  children,
}: PropsWithChildren<{}>): ReactElement {

  const [sports, setSports] = useState<Sports>(sampleSports);

  function getSportInfo(key: SportsKey): Sports[SportsKey] {
    return extractSport(sports, key);
  }

  return (
    <SportsContext.Provider
      value={{
        getSportInfo,
      }}
    >
      {children}
    </SportsContext.Provider>
  );
}

function App() {
  const { getSportInfo } = useSportsContext();

  // Why is cricket not inferred correctly from the context?
  const cricket = getSportInfo('cricket');

  console.log(cricket.name);
  console.log(cricket.players); // THIS FAILS as cricket is inferred as TeamSport | IndividualSport

  return (

App

  );
}

ReactDOM.render(
  <React.StrictMode>
    <SportsProvider>
      <App />
    </SportsProvider>
  </React.StrictMode>,
  document.getElementById('root')
);

A: First, you don't have to create IndividualSport and TeamSport. You can create an intersection types:
type Sport = {
  name: string;
  players?: number;   // don't forget the question mark
};

But if you intentionally create two types, just extend an interface:
interface IndividualSport {
  name: string;
}

interface TeamSport extends IndividualSport {
  players: number;
}

But in your case, you just need one type which is Sport. So, your Sports types become:
type Sports = {
  cricket: Sport;
  golf: Sport;
};

Next, I think we don't have to make the key generics. If you write SportsKey is keyof Sports, the SportsKey already string or numeric literal union of Sports keys
type SportsKey = keyof Sports;
// idem with:
type SportsKey = 'name' | 'players'

So, your context creation becomes:
function extractSport(sports: Sports, key: SportsKey) {
  return sports[key];
}

type SportsContextProps = {
  getSportInfo: (key: SportsKey) => Sports[SportsKey];
};

const SportsContext = createContext<SportsContextProps>({
  getSportInfo: (key: SportsKey) => extractSport(sampleSports, key),
});

const useSportsContext = () => useContext(SportsContext);

function SportsProvider({ children }: PropsWithChildren<{}>): ReactElement {
  const [sports] = useState<Sports>(sampleSports);

  function getSportInfo(key: SportsKey): Sports[SportsKey] {
    return extractSport(sports, key);
  }

  return (
    <SportsContext.Provider
      value={{
        getSportInfo,
      }}
    >
      {children}
    </SportsContext.Provider>
  );
}

Finally, you can access your sport context and log the result as follow:

function App() {
  const sports = useSportsContext();

  useEffect(() => {
    if (sports) {
      const cricket = sports.getSportInfo("cricket");
      console.log(cricket);
    }
  }, [sports]);

  return (
    <SportsProvider>

App

    </SportsProvider>
  );
}

UPDATED
But if you still want to use your approach, the short solution is you need to cast the cricket as TeamSport even though the type of cricket already:
const cricket: TeamSport | IndividualSport
// which is comes from:
const getSportInfo: (key: keyof Sports) => IndividualSport | TeamSport

The weakness of your approach is that you will always be asked to manually cast both the cricket and golf to tell typescript which type that will be used.
function App() {
  const { getSportInfo } = useSportsContext();

  useEffect(() => {
    const cricket = getSportInfo("cricket");
    const { name, players } = cricket as TeamSport;
    console.log(name, players);
  }, [])

  return (

App

  );
}

redpajama_set_name: RedPajamaStackExchange


text:
Q: How to execute Wix custom action after installation? I am using Wix3 to install WCF service to IIS.
How can I use my custom action (c#) function after installation completed? i.e. I need to open installed web.config file and replace hostname with real one.
Any ideas?

A: There is a sequence of Actions in Windows Installer. The WiX tutorial has a good section on events (and is a great resource anyway).
A typical example of getting something to run after InstallFinalize is to get the installed app to start.
<InstallExecuteSequence>
   <Custom Action='LaunchFile' After='InstallFinalize'>NOT Installed</Custom>
</InstallExecuteSequence>

A: Why would you:
1) Need a custom action?
2) Do it after the install instead of during the install?
WiX has a built-in extension for handling what you are trying to do:
XmlFile Element (Util Extension)
It will update your XML after the file has been installed and handle rollback scenarios as well.
What you will have to write a CA for though is reading the XML value back into a property ti handle repair and upgrade situations.  Read:
The WiX toolset's "Remember Property" pattern.

A: You can schedule it after InstallFinalize action in InstallExecuteSequence.


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I'm going to say it, profanity or not, Monken is EXACTLY what we need right now. Gundy is very buttoned down and respectable; I mean, the man puts our exhibition guys in so as not to run up the score. Bill Young is the sage mentor of the group, and Monken is bringing a proper amount of swagger to our group that hasn't been seen in Stillwater since Lester Miles.


text:
Горчи́чное зерно́ (; ) — зерно горчичного семени ().

 Горчичное зерно — в русскоязычной Библии неточный перевод для термина «горушечное зерно», упоминаемого в Евангелии (—32), — семени дерева Salvadora persica L., имеющего мелкие цветы и такие же 1—2-семянные ароматические плоды.
 Горчичное зерно — упоминается в Коране в теме «весов Мизан» (весов Судного Дня).
 Горчичным зерном в перстне — назван наш мир в книге каббалистов «Бахир».

Современная литература
 «Горчичное зерно» (; 1982) — название одного из сборников лекций болгаро-французского оккультиста Омраама Айванхова.
 «Горчичное зерно» (1989) — произведение украинского писателя Мирослава Дочинца.

Персонажи
 Горчичное Зерно () — имя эльфа в комедии Уильяма Шекспира «Сон в летнюю ночь» (1594—1596).
 Горчичное Зерно () — персонаж романов-фэнтези «Сёстры Гримм» (2005—2012) американского писателя Майкла Бакли.

См. также
 Хардал (хардаль, горчичное зерно) — мера веса, распространённая в мусульманских странах, равная весу одного горчичного зерна.
 Библейская «притча о зерне горчичном»
 «Слово о живописи из Сада с горчичное зерно» (XVII век) — сочинение о традициях китайской живописи.
 Синапиус | Горчица | Горчичка

Примечания

redpajama_set_name: RedPajamaWikipedia


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text:
Preventing employee burnout with meaningful recognition programs
Workplace burnout has been an afterthought for a long time.
But in 2021, an alarming 89% of employees reported experiencing burnout. This is an urgent problem companies must address. As we move into a new year, organizations need to get serious about what they can do to prevent employee burnout, rather than responding to cases reactively.
In this post, we explore exactly what burnout is and how it happens—and explain how recognition can keep your employees engaged, healthy, and fulfilled in their jobs.
What is employee burnout? Here's a helpful framework.
First, what is employee burnout? According to the Stanford Social Innovation Review:
"Burnout reflects an uneasy relationship between people and their work. Like relationship problems between two people, those between people and their work usually indicate a bad fit between the two, rather than just individual weaknesses, or just evil workplaces."
There's a common misconception that burnout manifests as a sudden mental breakdown. That it looks like an employee who is sleep deprived, tired, and in need of an extended vacation. But burnout is more insidious than that—it's a chronic problem that builds over a long period of time and has symptoms that extend beyond just exhaustion, including:
Cynicism
Loss of concentration and productivity
Sadness, anger, or irritability
Sudden physical ailments (headaches, bowel problems, etc.)
To understand what causes workplace burnout in the first place, the Stanford Social Innovation Review surveyed more than 10,000 people across a wide range of organizations. They discovered that the person-job mismatch that leads to burnout falls into six categories:
Workload: Too much work, not enough resources.
Control: Micromanagement, lack of influence, accountability without power.
Reward: Not enough pay, acknowledgment, or satisfaction.
Community: Isolation, conflict, disrespect.
Fairness: Discrimination, favoritism.
Values: Ethical conflicts, meaningless tasks.
Let's explore how you can use this framework to address the source(s) of burnout in your workplace.
How to prevent employee burnout with meaningful recognition.
To address burnout as an organization, your leadership team needs to identify which of the six categories your employees are struggling with the most. This requires intentional dialogue with your people. Then, it's about prioritizing your highest risk areas and introducing initiatives to prevent burnout.
Thankfully, there are ways to address multiple sources of workplace burnout at the same time. Meaningful employee recognition can address four of the six variables that lead to burnout. Here's how.
Recognition creates more opportunities for rewards.
The desire to be recognized—and occasionally rewarded—by others is universal. But when companies fail to recognize employees in a timely and impactful way, it can be demotivating.
Imagine an employee pours their heart and soul into a project. They go above and beyond to make sure it's a success. But they don't get a single 'thank you' from their manager. Maybe this has happened to you at some point in your career. It feels awful, right?
Research shows that a lack of recognition and rewards is strongly correlated to burnout. Reducing recognition makes it 48% more likely that employees will report burnout. Similarly, when there's no company-wide recognition strategy in place, employees are 29% more likely to report burnout.
But recognize and reward your employees in a timely and meaningful way? They're going to feel valued, seen, and heard.
A culture of recognition builds community.
People find strength in community, especially when things are hard. A connected workplace will help your employees feel supported and sustained as they continue to navigate uncertain times. But new Blueboard research shows just 31% of HR professionals feel they've addressed connection challenges at work.
Recognition can be a powerful way to build a sense of community within your organization. When employees are celebrated (and can celebrate each other) for their efforts and achievements, they're more likely to feel seen and valued. And they're more likely to see and value each other. You can foster this sense of community even more with different types of recognition. Experiential rewards, for example, can help employees create powerful bonds.
Unlike cash bonuses, experiential rewards aren't awkward for employees to talk about with their teammates. Meaningful experiences provide more opportunities to connect with coworkers—whether that's bonding over photos from recent travels or sharing activity recommendations with colleagues who have similar interests. It's these small, but impactful, moments of connection that build community within teams.
Meaningful recognition is built on fair practices.
According to the framework, an important way to prevent employee burnout is fairness. In other words, your organization must be committed to giving everyone equal opportunities.
It can be detrimental when deserving employees don't receive promotions, opportunities, and recognition, either because they're not seen or face discrimination. This leads to demotivation, resentment, and a desire to search for a better work environment. There's a reason why one-third of workers who feel unappreciated are looking for employment elsewhere.
One way to ensure fairness is to introduce an inclusive recognition program within your organization. That is, a structured method of recognizing high-performing employees in a way that ensures everyone—regardless of their role, team, seniority, or background—has the opportunity to be considered.
For more in-depth tips to build morale, equity, and connection in a hybrid workforce, grab a copy of our free ebook.
Bring company values to life through recognition.
Your employees want to feel connected to the values and mission of your organization. Without this alignment, their day-to-day work may feel meaningless—a sentiment that can eventually lead to burnout.
To keep your employees feeling connected to the organization's work, you need to find ways to bring your mission, vision, and core company values to life.
Company values awards are a great way to accomplish this. This type of award helps you highlight what your organization stands for, while allowing your team members to feel like they're an integral part of the company by living out the core values.
The Trade Desk, for example, leans heavily on their refreshed company values, which act as an anchor for their spot recognition program. This program encourages and recognizes employees who demonstrate the company's values—Vision, Grit, Agility, Generosity, Openness, and Full-Hearted. These values serve as a North Star, helping everyone stay focused, globally aligned, and motivated toward a shared goal.
Vision: Illustration by Ana Duje, Hong Kong
Grit: Illustration James Gulliver Hancock, New York
Agility: Illustration by Neil Stevens, London
Generosity: Illustration by Asuka Watanabe, Tokyo
Openness: Illustration by MaryKate McDevitt, Los Angeles
Full-Hearted: Illustration by Shauna Lynn Panczyszyn, Chicago
Company values awards also tend to facilitate fairness. These awards are a more inclusive form of recognition because anyone can live out the company values, regardless of their status, role, or seniority in the organization.
Get ahead of employee burnout in 2022 with the power of recognition.
It's time to shift how we handle burnout at work. To move away from reactive responses to this widespread phenomenon, and move toward a model where an organization shapes its culture, processes, and policies with the employee's wellbeing top of mind.
We know through research and first-hand experience that authentic recognition is a powerful tool in preventing burnout. It's an essential tool you can use proactively. So if you don't already have a recognition program in place for 2022, now's the time to start planning.
Not sure where to begin? That's where we come in. Schedule some time to connect with our team and learn more about our experiential recognition programs.
The Biggest Mistake Managers Make With Recognition
Friend of Blueboard
11 ways to show employee appreciation during the holidays in 2021
Derrich Mosley
The Power Of Employee Recognition During Times Of Separation [Webinar Recap]

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text:
Working Our Network
Participants at the ITC Executive Forum 2001 in September were asked: is your network working? They could have turned the question back to us and asked ITC: is your own network working? Our answer would have been: yes, most definitely.
The Executive Forum on National Export Strategies started in September 1999 as an innovative experiment in South-South technical cooperation. On the shores of Lake Geneva, we gathered together leading strategy-makers and business people from 30 developing and transition countries for three days of intensive brainstorming. We hoped to develop some guidelines for best practice in designing and managing a national export strategy. In a commercial world where big fish eat little fish, but where fast fish quickly leave the slow fish behind-big or small-we felt that by working together we could perhaps give some slower fish the strategic tools to swim a lot faster-and help them to go in the proper direction.
ITC has a long history of practical work with developing countries, giving managers of public and private sector trade support institutions (TSIs) the information and tools they need to assist local companies to be better exporters. We have a good track record of providing market data and studies, operating guidelines and training kits to our developing country partners. All of these are worked out in close conjunction with institutions in the countries themselves, so that we are not just passing on knowledge, but also using and developing local expertise in a true partnership. We do not carry out this effort alone. In many cases we work with others in the United Nations family.
The product-network approach
Our concern is not just to produce these competitiveness support products, but to ensure that national trade support institution networks receive the backing to adapt our competitiveness tool kit to local requirements. This is our product-network approach - and it is working. By the end of 2001, specialized product-networks were operating in more than 120 developing countries and economies in transition.
The Executive Forum represents the next step in the product-network approach. Its objective is to build on these operational support tools and to develop best-practice scenarios at the strategic level. The Executive Forum itself was innovative because it was aimed at the strategy-maker rather than the practitioner, and deliberately gathered together government, TSI and business community leaders. We recognized that it was not enough to simply assemble TSI executives, because few of them play the pivotal role in conceiving and developing strategy.
At the same time, we realized that they are key to strategy implementation and therefore must be brought into the picture at the earliest possible stage, as should the business community. We also recognized that our network partners in developing and transition countries not only need best-practice guidelines for putting together a strategy; they also have a great deal of grass-roots experience to share. For its part, ITC needed to tap into this knowledge to learn what it should be doing to help improve the national strategy-making process. In what became the pattern for subsequent Executive Forums, the emphasis was on analysis, experience-sharing and networking. We also deliberately avoided being prescriptive in approach.
Changing perceptions of trade promotion
The first Executive Forum, which focused on the theme Redefining Trade Promotion: the Need for a Strategic Response, was an experiment because, at that time, trade promotion strategy-making was not a priority concern for developing countries. Even three years later, very few developing countries have put a national export strategy in place. However, the reali-zation is growing among trade promotion organizations and specialized TSIs that a strategic approach is critical.
To provide a starting point for discussions we presented participants with a number of case studies setting out the strategies which successful exporting countries in the industrialized or developing world (whether Finland or the Philippines) had adopted. All of this was available on our web site (http://www.intracen.org/execforum) well before the consultation. The lessons are still valid and the case studies continue to be available on our web site.
Redefining roles and strategies
The three days of discussions made it clear that to keep up with the rapidly changing environment, national strat-egies must be redefined and TSIs must embrace new roles. In order to be effective, a national export strategy must be comprehensively integrated into the overall economic planning framework. It should not simply deal with developing commercial opportunities in international markets. It should also encompass the longer-term "border-in" challenge of establishing a national competitiveness framework which would create an export culture, sharpen business competitiveness and develop new export industries. The experiment was a success. The relevance of an export strategy which addressed "border-in" and "border-out" issues was acknowledged. The notion of a global network of global strategy-makers was born.
Expanding the Executive Forum reach
One clear concern among strategy-makers in developing and transition economies was the national challenge posed by the global digital revolution. For some, simply getting an overall picture of the new e-world of commerce was difficult, let alone predicting the future, or seeing what strategy-makers should be doing to make their exporting community e-aware and e-competent.
Export Development in the Digital Economy became the theme of Executive Forum 2000, and the ITC network was broadened to include national strategy-makers concerned with developing telecommunications strategies. Equally important was the fact that we applied cyber-technology ourselves to extend our outreach and to stimulate further development of our network. A series of e-discussions was organized to support the Executive Forum debate. Some 600 participants from 85 countries joined in. What started out as a single event was already turning into a network of interested professionals. It also gave us a pattern we could apply to other ITC activities, such as the Business Sector Round Table organized for the United Nations Conference on the Least Developed Countries in May 2001 in Brussels, Belgium.
Ensuring the network promotes trade
Executive Forum 2001 tackled strategy implementation: the logical follow-up to such concerns. Of course, an effective strategy formulation process is essential. It is also fundamentally important to ensure that the formulated strategy addresses major shifts in business practice, such as e-trade. But the proof of the pudding is in the eating. Is strategy effectively implemented? That was the focus of the Executive Forum 2001 debate, which posed the question: Is Your Trade Support Network Working?
Different national approaches to implementing strategies were analysed. Mr. David Syz, the Swiss Secretary of State for Economic Affairs, explained in detail why Switzerland has devolved responsibility for trade support services to the private sector.
From Canada, we heard how the federal Government has segmented the business community into "potential", "preparing" and "experienced" exporter categories and has established separate institutional networks for each group, all linked in cyberspace.
The key point made in all discussions was that one size doesn't fit all in developing strategic solutions in trade promotion. But the strategy-maker must be aware of all the options. Who should coordinate the national trade support network? How should it be financed? Is the one-stop shop approach (or, in cyberspace, the one-stop screen) an effective model? Should trade and investment promotion be merged into a single network?
The challenge for the national trade promotion organization
The challenge confronting the national trade promotion organization is to develop a new set of services, or to facilitate specialized trade support institutions providing such services. The orientation of such programmes should be towards:
developing the overall competitiveness of the enterprise, rather than increasing its immediate export sales;
supporting industry associations and clusters of firms, rather than concentrating on the individual enterprise;
providing long-term support, rather than one-off assistance;
developing capabilities, both production and managerial, rather than simply marketing competence; and
promoting technology acquisition, investment and subcontracting, rather than focusing on just "getting the goods out of the country".
From Redefining Trade Promotion-the Need for a Strategic Response (2000)
A full exchange of ideas
Executive Forum 2001 proved a very business-like gathering, with national groups participating in break-out sessions to outline their priorities, needs, objectives and measures to be taken. The 2001 meeting was more than a simple dialogue. As we had always planned, it was a full exchange of ideas among professionals on the issues of their profession. One participant suggested that through the Executive Forum on strategy-making, ITC was turning what had been an art into a science.
We are not there yet. But we have certainly expanded our scope once more. The 2001 Executive Forum was followed by the first Regional Executive Forum at the end of November in Nairobi, Kenya. It shared the best-practice ideas from the Montreux Executive Forums with representatives from eight eastern and southern African nations. It also obtained additional feedback about the support required to enable TSIs to implement strategies effectively.
Similarly, this issue of Forum takes our effort a step further. It is not so much a report on the Executive and Regional Forums, but rather a quick panoramic look at the lessons learned in trade promotion; from the craft firm to the strategic level. It also offers snapshots of national, industry and company experiences that may help our readers from all groups to understand the issues and how the ideas apply in practice.
We were heartened to hear that readers have already used Forum material to find their own niche in international markets and we would like to see this issue of the magazine make a similar contribution to better business for developing and transition economies.
For this reason, the Market Profile in this issue concentrates on artisanal products, one of the obvious sectors where rich or poor countries have their own individual marketable products. The message is that, if approached in the right way, it is fairly easy, or at least possible, for a network in this sector to be effective and efficient.
Elements of a viable national competitiveness framework
A stable, predictable, macroeconomic environment for enterprise development characterized by low budget deficits, tight inflation control and competitive real exchange rates.
An outward-oriented, market-friendly trade and industrial regime emphasizing the dismantling of import controls and tariffs.
A proactive foreign investment strategy which targets a few realistic sectors and host countries, views overseas promotional offices as public-private partnerships, and provides competitive investment incentives and streamlined investment-approval processes.
Sustained investment in human capital at all levels (particularly tertiary scientific, information technology and engineering education) and increased enterprise training (including assistance for industry associations to launch training schemes, information campaigns to educate firms about the benefits of training and tax breaks for training).
Comprehensive technology support for quality management, productivity improvement, metrology and technical services for small and medium-sized enterprises or SMEs (including grants to obtain ISO 9000 certification), the creation of productivity centres and commercialization of public technology institutions.
Access to ample industrial finance at competitive interest rates through prudent monetary policy management, competition in the banking sector, training for bank staff in assessing SME lending risks and specialist soft loans for SMEs.
An efficient and cost-competitive infrastructure with respect to air and sea cargo, telecommunications, Internet access and electricity.
Contributed to the 1999 Executive Forum by Ganeshan Wignaraja of the Commonwealth Secretariat, London.
New ITC book and CD-ROM
In a forthcoming publication on the more strategic complex questions, we will set out the broader considerations at greater length.
This is the third in our series of Executive Forum publications, and will also include a CD-ROM of useful material from the three Forums held so far. The CD-ROM will be arranged in an easy-to-use format for all strategy-makers. With all of these products, we hope to export the network idea well beyond the borders of trade support institutions and bring businesses, as well as government ministry planners, within the net of strategy development and implementation.
We believe that without a network linked by continuous dialogue, any trade promotion effort, particularly one focused on poverty alleviation-something more and more donor organizations are beginning to demand-is bound to fail. Of course, we would rather have them succeed; certainly our future development of the Executive Forum process will be designed to help move developing and transition economies along the path to success. To produce this result, we will keep working our network-and keep our network working.
The Swiss State Secretariat for Economic Affairs (seco) has been our solid partner for the last two Executive Forums. I wish to thank them publicly for their ongoing support. Certainly seco is at the centre of the ITC network.
J. Denis Bélisle is Executive Director of ITC.


text:
Nick Sibilla: Nonsensical legislation hurting small businesses
Nick Sibilla
Red tape is holding back Florida's economy. A new report by the Institute for Justice, "Florida's Dirty Dozen," spotlights 12 harmful and ridiculous laws that burden small businesses.
Many of these laws are occupational licenses, where someone needs to obtain permission from the government before they can start their own business or work in an occupation. In the past, licensure was reserved for careers like becoming a doctor or a lawyer. Back in the 1950s, only about one out of every 20 workers in the United States needed a license to work. Over the years, that number has soared to one out of every three workers. Now would-be auctioneers, barbers, interior designers and make-up artists all require a permission slip from the state.
These occupations have costly requirements. Before they can legally cut or trim hair, would-be barbers have to complete at least 1,200 hours of training at a barber college. Schooling can cost anywhere from $10,000 to $15,000. That is a huge barrier to entry for aspiring entrepreneurs, especially in this current economic climate.
Moreover, under Florida law, being paid to cut hair without a license is actually a crime.. In a series of raids that made national headlines in late 2010, armed officers for the Orange County Sheriff's Office arrested 35 people for "barbering without an active license." Conducted without warrants and in conjunction with the Florida Department of Business and Professional Regulation, police swept through barbershops owned by African Americans and Hispanics. Cops handcuffed barbers and ordered customers out of the shops. One officer even had his gun drawn during the raids.
Other occupations have regulations that are wildly unrelated to any potential health and safety risks. A cosmetologist must complete 1,200 hours of coursework. For a prospective interior designer, it is six years of education. Incredibly, the license requirements for these occupations are more onerous than becoming an emergency medical technician. In Florida, EMTs need only about a month of training before they can be licensed.
Unlike what EMTs face, applying make-up or tastefully decorating a living room is not a life-or-death situation. However, apparently, some interior designers believe so. In a story worthy of an Onion article, in 2011, one interior designer testified before the Florida House that deregulating her trade could contribute to 88,000 deaths per year..
If that were the case, unlicensed interior designers would kill more than twice as many Americans as car crashes every year. That would also be news to the 47 states that do not license the occupation. No wonder Florida has the fourth most burdensome licensing laws in the entire nation, according to "License to Work," a study also published by the Institute for Justice.
Then there are other regulations on the books that are just downright baffling. The Sunshine State is just one of a handful of states that bans 64-ounce growlers (the industry-standard size). Growlers are reusable, glass jugs that have seen a renaissance among craft beer aficionados. Bars or microbreweries fill them straight from the tap. With growlers, consumers save money on their favorite beers while breweries enjoy lower packaging costs, widening their distribution.
Yet Florida's growler prohibition is a buzzkill for the state's craft beer industry. Fortunately, a reform bill (HB 283)) is already advancing in the House..
As for other burdensome restrictions, the Florida legislature has a special incentive to curb government intrusion. Legislators introduce so many laws they are limited from filing more than six bills each session. But if they introduce a bill that would only repeal or delete an existing law, that "repealer" does not count towards their bill introduction limit. Rather than piling on even more regulations (Florida passes about 300 new laws every year), repealers encourage legislators and citizens to focus more on undoing nonsensical laws.
To create new jobs and restore common sense, the Florida legislature needs to clean house on the dirty dozen.
Nick Sibilla is a writer at the Institute for Justice. The full report, "Florida's Dirty Dozen," can be read online at http://www.ij.org/FlaDirtyDozen
Florida Legislature


text:
Q: R - ggplot passing color by variable file.csv
group   metric
"1"       1.1
"0"       2.2
"0"       3.3
"1"       4.4

I'm reading the above file.csv as below in to a data frame
df <- read.csv("file.csv", header=T)
# store col names for using later
group_name <- colnames(df)[1]
metric_name <- colnames(df)[2]

While creating a box plot, I need to pass the color through variable group_name
p <- ggplot(data=df, aes(x=df[,group_name], y=df[,metric_name], color=group_name ) ) +
  geom_boxplot(outlier.colour="darkred", outlier.shape = NA, fill = fill, color=line, alpha = 0.5) +
  theme(legend.position = c(1, 1), legend.justification = c(1, 1))
p

The issue is that the legend in the plot shows as group_name instead of the value of the variable group_name - what is needed to done so that color takes the value of the variable group_name ?
The reason for using x=df[,group_name] in aes is so as to make it generally applicable for any csv files whole column name may not be known.

A: This is just how the ggplot paradigm works. Normally the grouping variable is a factor or character vector describing an attribute rather than a number. Let's take some random data similar to your .csv:
group <- c(1,0,1,1,0,0,0,1,0,1)
metric <- rnorm(10, 1)
df <- data.frame(group, metric)

In your case you could use dplyr::mutate to add a new column as suggested in the comments. You can call the new column and variables whatever you like.
library(dplyr)
df.gp <- df %>% mutate(Group.new = paste("Group", group))

# df.gp
#    group     metric  Group.new
# 1      1  1.3100608    Group 1
# 2      0  2.7120827    Group 0
# 3      1 -0.2310713    Group 1
# etc

Now plot the data:
library(ggplot2)
ggplot(df.gp, aes(x=Group.new, y=metric, fill=Group.new)) + geom_boxplot() +
theme(legend.position = c(1, 1), legend.justification = c(1, 1))

A: As it seems from the comments, user3206440's intention is to pass the column names to a function. The ggplot way to do it, as I mentioned in the comments, is as follows:
customBoxPlot <- function(df, group_var, metric_var) {

  ggplot(df, aes_string(x=group_var, y=metric_var, color=group_var)) +
    geom_boxplot(outlier.colour="darkred", outlier.shape = NA, alpha = 0.5) +
    theme(legend.position = c(1, 1), legend.justification = c(1, 1))

}

customBoxPlot(df, names(df)[1], names(df)[2])
# or
customBoxPlot(df, "group", "metric")

This is how the output will look using your data:


text:
02.December, 2019
Change at the top in Havyard Group
Geir Johan Bakke and the board have agreed to terminate the employment relationship.
Geir Johan Bakke has been employed in Havyard Group since 2001, and he has been CEO since 2004. Geir Johan Bakke has been central to the development of the company. From being a standalone shipyard, under Geir Johan Bakke's leadership Havyard has developed into a fully integrated shipbuilding group with design, shipyards and equipment production, as well as with almost 600 permanent employees and an order book of over NOK 4300 million.
Vegard Sævik, Chairman of the Havyard Group, points out that Geir Johan Bakke has been a dynamic and committed CEO with whom he has worked well, both in his time as a Board Member and later as Chairman. "Geir Johan has been incredibly important to the development Havyard Group has been through. He has shown an impressive commitment to developing the Group, which is of great importance to the communities in which Havyard operates."
Geir Johan Bakke has been working for Havyard for over 18 years, and the last 15 years he served as top leader of the group.
"It has been an amazing journey that I am both grateful and proud to have been a part of. I wish the company good luck for the future." The retiring CEO has stated his willingness to be available to the company for some time to come if needed.
The parties have agreed not to comment further on the matter. The Board of Havyard Group has appointed Gunnar Larsen as interim CEO. He will immediately begin the work of recruiting a permanent CEO.
Gunnar Larsen comes from the position of General Manager of the Havyard Production & Service business area, he is Senior Vice President of Market & Business Development in Havyard Group and has been with Havyard for 12 years.
Photo: Interim CEO Gunnar Larsen


Ex-Army Job In Jharkhand Police SAP Recruitment 2018 – Latest vacancy notice has been broadcast for Jharkhand police recruitment for rent and eligible 66 persons. Constable (SAP) has been started for the application process. Candidates interested in Jharkhand Police Bharti can apply here.
Before applying to Jharkhand Police Constable Recruitment 2019. you have to go deeper through Jharkhand Police notification and ensure eligibility norms. Eligible candidates can apply further and participate in walk-in interviews.


The form Miscellanea represents a specific category or genre of resources found in Lewisville Public Library.
Cowboys essential : everything you need to know to be a real fan!
Dig and sow! how do plants grow?
Do whales have belly buttons?
Dog on a bike! : and more true stories of amazing animal talents!
Guinness world records : man-made marvels!


48. Some on ground floor, plus 3-bed cottage in grounds, suitable for disabled.
Great Hall, library, 2 restaurants, private dining rooms, free Wi-Fi, in-room TV (Freeview), civil wedding licence, spa, swimming pools, terrace, tennis, 376-acre National Trust gardens, public areas wheelchair accessible.
not allowed in restaurants, spa or parts of garden.
per room B&B £495–£2,175, D,B&B £615–£2,295. Tasting menu £98, a la carte £73. .
Three nights for the price of two. Complimentary night stay. Three nights from £890.00 to £3,070. Prices are per room per stay and inclusive of full English breakfast and VAT. The above rates are valid from Sunday to Thursday, weekend rates are available at a supplement.
The details you provide will be forwarded directly to Cliveden House.


text:
Q: Facebook Audience Network show banner ad When I want to show banner in my iOS app on my test device,following the rules,I add the code like this:
[FBAdSettings addTestDevice:@"HASHED ID"];

Then the ad works normally.I want to know if I need to remove the code above when I release my app.
Thanks.

A: Yes, you need to remove the code so that your app user will see real ads rather than test ads.

A: I think you can leave this code. It's only useful when you test on your device and just tell the framework that your device is a test device.
But in production, it should not impact other users at all.


text:
Hermann Weyl (* 7. November 1866 in Berlin; † 20. November 1925 ebenda) war ein deutscher Arzt und Politiker (SPD, USPD). Er war verheiratet mit der Sozialpolitikerin Klara Weyl. Gerda Weyl war ihre gemeinsame Tochter.

Leben

(Chajim) Hermann Weyl entstammte einer strenggläubigen jüdischen Familie, einer seiner Brüder war Dr. Max Weyl, Rabbiner in Berlin. Hermann Weyl trat nach 1900 aus der jüdischen Gemeinde aus. Er besuchte das Sophien-Gymnasium, wo er 1885 das Abitur ablegte. Nach einem Studium der Medizin in Berlin, promovierte er 1890 zum Dr. med und arbeitete danach als praktischer Arzt mit dem Schwerpunkt Wasserheilverfahren und übte diese Tätigkeit trotz seines starken politischen Engagements bis zu seinem Lebensende aus. Er war Mitgründer und Vorsitzender des Deutschen Ärztevereins für physikalisch-diätische Therapie.

1886 trat Hermann Weyl in die SPD ein. 1917 wechselte er in die USPD, ging jedoch bei deren Spaltung im Jahre 1922 zurück in die SPD.

Von 1902 bis 1925 war Weyl Mitglied der Berliner Stadtverordnetenversammlung. In dieser Zeit war er von 1917 bis 1922 für die USPD Abgeordneter, davor und danach für die SPD. 1919 wurde er von der Fraktion der USPD in der Stadtverordnetenversammlung zum Fraktionsvorsitzenden gewählt. In der »Liste des Radikalismus im Landespolizeibezirk Berlin wohnhaft« des Berliner Polizeipräsidenten von 1918 wurde Dr. Hermann Weyl mit weiteren führenden Sozialdemokraten aufgeführt. Von 1919 bis 1920 war er gemeinsam mit Hugo Heimann (SPD) Stadtverordnetenvorsteher.

In der Zeit von 1919 bis 1925 war Weyl parallel auch im Preußischen Landtag vertreten – bis 1922 für die USPD, anschließend für die SPD. Nach seinem Tod rückte Helene Schmitz in den Landtag nach.

Nach Umbettung im Jahr 1951 befindet sich das Grab von Hermann Weyl auf dem Zentralfriedhof Friedrichsfelde an der Ringmauer der Gedenkstätte der Sozialisten.

Literatur
 Weyl, Hermann, in: Joseph Walk (Hrsg.): Kurzbiographien zur Geschichte der Juden 1918–1945. München : Saur, 1988, ISBN 3-598-10477-4, S. 386

Belege

Mediziner (20. Jahrhundert)
Alternativmediziner
Politiker (19. Jahrhundert)
Mitglied der Stadtverordnetenversammlung (Berlin)
Mitglied des Preußischen Landtags (Freistaat Preußen)
SPD-Mitglied
USPD-Mitglied
Deutscher
Geboren 1866
Gestorben 1925
Mann


And, as Nastassia Mironchyk-Ivanova could tell you, long hair is particularly poorly suited to long jump. The Belarusian missed out on the gold medal at the World Championships in Daegu, South Korea, on Sunday when her wayward ponytail touched the sand behind the landing point of her body.
Mironchyk-Ivanova's ponytail left a small indentation in the sand, and it was from this point that the judges measured the length of her jump.
Mironchyk-Ivanova's jump was measured at 6.74m, putting her in fourth place. Had it not been for her hair, the jump would have measured at 6.90m, earning her first place and a $60 000 prize.
Instead, American Brittney Reese, who sports a compact bob style, took the gold with a jump of 6.82m.


94819 – Nearest neighbor scaling?
Bug 94819 - Nearest neighbor scaling?
2016-04-04 14:48 UTC by N. W.
Whoops, didn't notice the different module for this bug vs. 94818, sorry. Reopening.
> Whoops, didn't notice the different module for this bug vs. 94818, sorry.
Yeah, sorry for using the wrong one initially.
Anyway, as being discussed in both threads, it would be interesting to know if the dedicated hardware scaler inside the GPU could also be used for this instead of using xrandr.
And would there be any chance you could point the NVIDIA Windows driver team to this issue as well? It would be nice if nearest neighbor scaling would be available for NVIDIA cards on Windows as well.
Would be really nice if you could do something about that.
Older GPUs certainly can't do nearest neighbor scaling in the display hardware. I'll have to look into whether it's possible with newer GPUs.
> hardware. I'll have to look into whether it's possible with newer GPUs.
> it's roughly a box filter, so that's a lower quality scaler.
So maybe the same shader method that is used for DSR could also be used to scale in software on the hardware with a nearest neighbor filter instead of a gaussian filter?