text:
\section{Introduction}

The ideas of gluon saturation\cite{GLR} have been vigorously pursued in the last ten years or so \cite{Mueller,mv,reggeon,balitsky,JIMWLK,cgc,Kovchegov}.
The result of these efforts is the set of evolution equations which describes the change of the scattering amplitude at high energy in the regime where the hadronic wave function resembles a dense gluonic medium - the so called JIMWLK equation\cite{balitsky,JIMWLK,cgc,Kovchegov} (for reviews see \cite{IV,Weigert,JK,zakopane}).

The latest developments in this dynamic field are related to what is sometimes called Pomeron loops,
or Fluctuations or alternatively Saturation effects in the projectile wave function
\cite{IM,kozlov,LL1,shoshi1,IMM,balitsky1,janik,
LL2,IT,MSW,LL3,kl,kl1,something,blaizot,kl4,genya,
Braun05,SMITH,MMSW,HIMS,Balitsky05,Kovchegov05}. The corrections due to these effects
clearly must be important for the preasymptotic regime, but should also affect the more exclusive observables at very high energy. The last year or so has seen a lot of activity in this direction.
Much has been understood during the last year, but many questions remain open.
In particular the structure of the "Hilbert space" on which the evolution kernel acts \cite{kl} as well as the precise physical meaning of the Dense-Dilute Duality (DDD) introduced in \cite{something} are being actively debated \cite{kl4,SMITH,MMSW,HIMS,Balitsky05}.

This paper presents several remarks on these issues. The remarks are not necessarily causally or logically connected, but they all pertain to the questions mentioned above.

As regards the first issue, one of the questions is the following. The evolution is represented usually as acting on the weight function $W[\rho(x,x^-)]$, where the necessity of introducing the longitudinal coordinate $x^-$ stems from
the desire to represent quantum noncommuting operators $\hat\rho^a(x)$ in terms of commuting "classical fields". It was shown in \cite{kl} that this is achievable by endowing the quantum operator with an additional "ordering" variable. One also needs the weight functional $W$ to ensure that the correlators of  the type $\langle\rho(x_1^-)...\rho(x_n)^-\rangle=\int D[\rho]W[\rho]\rho(x_1^-)...\rho(x_n^-)$ depend only on the ordering of the variables $x^-_i$ rather than on their values, and that an exchange of two such variables must lead to the appearance of the $SU(N)$ structure constant. Namely if in the correlation function the two values of the ordering variable $x^-_i,x^-_j$are adjacent to each other and $x^-_i>x^-_j$, the following property must hold
\begin{equation}
\int D[\rho]W[\rho]...\rho^a(x_i^-)\rho^b(x_j^-)...=\int D[\rho]W[\rho]...\rho^a(x_j^-)\rho^b(x_i^-)...
+if^{abc}\int D[\rho]W[\rho]...\rho^c(x_i^-)...
\label{property}
\end{equation}

It was also shown in \cite{kl} that these requirements can be satisfied by a "weight function" $W$ which contains
a  Wess-Zumino term. The explicit representation of such a Wess - Zumino term is awkward, since it has to be local in transverse space and has to allow for an arbitrary representation of the colour charge at every transverse position\cite{kl}.

On the other hand in \cite{kl4} we have introduced a different representation of the weight functional
\begin{equation}
W[\rho^a]\,=\,\Sigma[R]\,\delta[\rho(x,x^-)]
\label{hilbert}
\end{equation}
where
\begin{equation}
R^{ab}(x)\,=\,\left[P\exp\{\int dx^-T^c{\delta\over\delta\rho^c(x,x^-)}\}\right]^{ab}
\end{equation}
and $\Sigma$ - an arbitrary functional. A similar representation was introduced earlier in the framework of the dipole model\cite{IM}, with the crucial difference that the exponential in functional derivatives was truncated at a low order in the Taylor expansion.
Although the formula eq.(\ref{hilbert}) has a very natural interpretation from the point of view of partonic content of the wave function, the question whether it satisfies the property eq.(\ref{property}) has not been addressed before.
Our observation is that the Hilbert space for the action of the high energy evolution kernel
is indeed naturally given by the vector space eq.(\ref{hilbert}). In particular we show that
the necessary condition eq.(\ref{property}) is satisfied by any element of the space defined by eq.(\ref{hilbert}).
We thus note that this form constitutes a simple alternative to the Wess-Zumino term as far as the correct representation of the charge density commutation relations is concerned. This discussion along with some additional remarks which are mostly meant to clarify and explain in more detail our earlier results  \cite{kl,kl1,something,kl4}
is contained in Section 3.

In Section 4 we discuss a dipole toy model with no transverse coordinates. We ask the question what is the DDD transformation
and what is the analogue of the Seld-Duality condition in terms of the dipole degrees of freedom.
By applying the DDD transformation we construct the dual to the Mueller`s dipole evolution.
In the framework of this model we explicitly construct a self dual evolution which is valid both in the dilute and dense limits. In the intermediate regime the model is incomplete and misses correct treatment of multiple dipole emissions. This is reminiscent of the self dual variant of the high energy evolution suggested in \cite{SMITH} and discussed recently in \cite{Balitsky05}. The value of this discussion is mainly in the simplicity of the model and transparency of the mathematics involved which thus allows us to clearly see some features of the evolution which are much more difficult in real QCD.

We start in Section 2 by a short review of the elements we would like to discuss.

\section{JIMWLK and other letters }

We start by describing the general setting of the high energy evolution.
We consider the leftmoving projectile with the wave function $|P\rangle$ colliding with the rightmoving target with wave function $|T\rangle$. The total rapidity of the process is $Y$, of which the target carries rapidity $Y_0$.
The $S$-matrix for the process is given by
\begin{equation}
{\cal S}(Y)\,=\,\int\, D\alpha_T^{a}[\tau]\,\, W^T_{Y_0}[\alpha_T(x,\tau)]\,\,\Sigma^P_{Y-Y_0}[\alpha_T(x,\tau)]\,.
\label{ss}
\end{equation}
Here $\alpha$ denotes the $gA^-$ component of the vector potential due to the target charge density $\rho_T$.  It defines the scattering matrix of a single  gluon projectile at transverse position $x$ on the target, which in the eikonal approximation is given by\footnote{In our convention
the variable $\tau$ is rescaled to run from 0 to 1.
Thus $\tau=0$ corresponds to $x^-=-\infty$
while $\tau=1$ to $x^-=+\infty$.}
\begin{eqnarray}\label{S}
S(x;0,\tau)\,\,=\,\,{\cal P}\,\exp\{i\int_{0}^{\tau} d\tau' T^a\alpha^a_T(x,\tau')\}\,.
\end{eqnarray}
where $T^a_{bc}=if^{abc}$ is the generator of the $SU(N)$ group in the adjoint representation.
The field $\alpha_T$
obeys the classical equation of motion  \cite{JIMWLK,cgc}
\begin{equation}
\alpha^a_T(x,\tau)T^a\,\,=g^2\,\,{1\over \partial^2}(x-y)\,
\left\{S^\dagger(y;0,\tau)\,\,\rho^{-a}_T(y,\tau)\,T^a\,\,S(y;0,\tau)\right\}
\end{equation}
The function $\Sigma^P$ in eq.(\ref{ss})
 is the eikonal scattering matrix for a composite projectile averaged over the projectile wave function, see \cite{something}
\begin{equation}
\Sigma^P[\alpha_T]\,\,=\,\,\int d\rho_P\,\,W^P[\rho_P]\,\,
\,\exp\left\{i\int_{0}^{1} d\tau\int d^2x\,\rho_P^a(x,\tau)\,\alpha_T^a(x,\tau)\right\}\label{s}
\end{equation}
with $x_i$ - the transverse coordinates of the gluons in the projectile. $W^P$ is the "weight functional"
which determines the distribution of the charge density in the projectile.

The evolution of the $S$-matrix with energy is given by the general expression of the form:
\begin{equation}
\frac{d}{d\,Y}\,{\cal S}\,=\,\int\, D\alpha_T^{a}\,\, W^T_{Y_0}[\alpha_T(x,\tau)]\,\,\,
\chi\left[\alpha_T,\frac{\delta}{\delta\,\alpha_T}\right]\,\,\,
\Sigma^P_{Y-Y_0}[\alpha_T(x,\tau)]\,.
\label{hee}
\end{equation}
The kernel of the evolution $\chi$ can be viewed as acting
either to the right on $\Sigma$ or to the left on $W$:
\begin{equation}
{\partial\over\partial Y}\,\Sigma^P\,\,=\,\,\chi\left[\alpha_T,\,{\delta\over\delta\alpha_T}\right]\,\,\Sigma^P[\alpha_T]\,;
\ \ \ \ \ \ \ \ \
{\partial\over\partial Y}\,W^T\,\,=\,\,
\chi^\dagger\left[\alpha_T,\,{\delta\over\delta\alpha_T}\right]\,\,W^T[\alpha_T]\,.
\label{dsigma}
\end{equation}

Note that according to eq.(\ref{s}), the projectile-averaged $S$-matrix $\Sigma^P$ is the functional Fourier transform of the projectile weight functional $W^P$. Thus $\alpha_T$ plays the role of conjugate to $\rho_P$.
 The evolution of $\Sigma^P$ determines the evolution of its Fourier image $W^P$:
\begin{equation}
{\partial\over\partial Y}\,W^P\,\,=\,\,\chi\left[-i{\delta\over\delta\rho_P},i\rho_P\right]\,\,W^P[\rho_P]\,.
\label{dwp}
\end{equation}
Since the functional form of the evolution of the target and projectile weight functionals should be the same (projectile-target
democracy - PTD) comparing eq.(\ref{dwp}) with eq.(\ref{dsigma}) we conclude that the kernel $\chi$ must be selfdual
\begin{equation}
\chi\left[\alpha,\,{\delta\over\delta\alpha}\right]\,=\,\chi\left[-i{\delta\over\delta\rho},i\rho\right]\,.
\label{sd}
\end{equation}
where now $\alpha$ and $\rho$ both refer to the same object, either target or projectile.
Thus if the target and wave function are treated on equal footing, the kernel $\chi$ must be invariant under the Dense Dilute Duality (DDD) transformation
\cite{something}:
\begin{equation}
\rho(x,x^-)\rightarrow -i{\delta\over\delta\alpha(x,x^-)}; \ \ \ \ \ \ \ {\delta\over\delta\rho(x,x^-)}\rightarrow i\alpha(x,x^-)
\label{duality}
\end{equation}

So far the approximations used in the literature to derive the kernel do not treat the target and the projectile equally. The approach \cite{JIMWLK,cgc} assumes that the the color charge density in the target is parametrically large ($\rho^a=O(1/\alpha_s)$). This yields the
 the so called JIMWLK evolution operator \cite{JIMWLK,cgc}
\begin{eqnarray}\label{JIMWLK}
\chi^{JIMWLK}&&=\,
\frac{\alpha_s}{2\pi^2}\,\int_{x,y,z} {(z-x)_i\,(z-y)_i
\over (z-x)^2\,(z-y)^2} \,\,\times  \\
&&\left\{ {\delta\over \delta \alpha^a(x,0)}{\delta\over \delta \alpha^a(y,0)}\,+\,
{\delta\over \delta \alpha^a(x,1)}{\delta\over \delta \alpha^a(y,1)}
\,- \,2\,{\delta\over \delta \alpha^a(x,0)}\,S(z;0,1)^{ab}\,{\delta\over
\delta \alpha^b(y,1)}\right\}\,. \nonumber
\end{eqnarray}
In this expression the functional derivative in \eq{JIMWLK} do not act on $S$ in the kernel,
but on external
factors only.
The same result follows from the calculation \cite{balitsky} which assumes that the projectile carries only small color charge density.
The derivation of \cite{kl} on the other hand assumes that the target is dilute (small charge density). This calculation yields
\begin{eqnarray}\label{KL}
\chi^{KLWMIJ}&&=\,-\,
\frac{\alpha_s}{2\pi^2}\int_{x,y,z} {(z-x)_i(z-y)_i
\over (z-x)^2(z-y)^2}\,\,\times  \\
&&\left\{\rho^a(x,0)\rho^a(y,0)+  \rho^a(x,1)\,\rho^a(y,1)-2\,  \rho^a(x,0)R(z,0,1)^{ab}\,\rho^b(y,1)\right\}\,.\nonumber
\end{eqnarray}
where the "dual Wilson line" $R$ is defined as
\begin{equation}
R(z;0,x^-)^{ab}=
\left[{\cal P}e^{\int_{0}^{x^-} d z^-T^c{\delta\over\delta\rho^c(z,\,z^-)}}\right]^{ab}
\label{rr}
\end{equation}
Just like in eq.(\ref{JIMWLK}) the functional derivatives in $R$  act only on external factors
 and not on the explicit factors $\rho^a(0)$ and $\rho^a(1)$ in Eq. (\ref{KL}).
Neither one of these expressions is self dual, rather they are related by the DDD transformation eq.(\ref{duality}).

Note that $R$ acts as the finite shift operator. When acting on a function of $\rho$ it shifts the charge density by the charge density of a single gluon
\begin{equation}\label{rs}
R(z,0,1)\,\Sigma^P[\rho^a(x,x^-)]\,=\,\Sigma^P[\rho^a(x,x^-)+T^a\delta^2(x-z)]\,\, .
\end{equation}

More recently a step forward has been made by resumming part of the corrections due to
finite density effects in the wave function of a more dilute object (projectile in the case of JIMWLK, and target in the case of KLWMIJ)\cite{kl1,something,SMITH,Balitsky05}.
The reulting improved expressions are

\begin{eqnarray}
\chi^{JIMWLK+}&=&
\frac{1}{2\pi}\int_z\left\{ b^a_i(z,0,[{\delta\over \delta \alpha}])\,
b^a_i(z,0,[{\delta\over \delta \alpha}])\,+\,
 b^a_i(z,1,[{\delta\over \delta \alpha}])\,b^a_i(z,1,[{\delta\over \delta \alpha}])
\right.\nonumber\\
&-&2\, \left. b^a_i(z,0,[{\delta\over \delta \alpha}])\,S^{ab}(z,0,1)\,b^b_i(z,1,[{\delta\over
\delta \alpha}])\right\}
\label{notlarge}
\end{eqnarray}
\begin{eqnarray}\label{notsmall}
\chi^{KLWMIJ+}=&-&
\frac{1}{2\pi}\int_{z} \{ b^a_i(z,0,[\rho])\,b^a_i(z,0,[\rho])\,
+\,b^a(z,0,[\rho])\,b^a(z,0,[\rho])
\nonumber \\
&-& 2\,b^a_i(z,0,[\rho])\,R^{ab}(z,0,1)\,b^b_i(z,1,[\rho])\}\,.
\end{eqnarray}
The field $b^a_i$ satisfies the "classical" equation of motion
\begin{eqnarray}\label{b}
&&\left\{\partial_i+gf^{abc}b^c_i(z,[\rho])\right\}\partial^+ b_i^b(z,[\rho])=g\rho^a(z,x^-)\,;\nonumber\\
&&\epsilon_{ij}[\partial_ib^a_j(z,[\rho])-\partial_jb^a_i(z,[\rho])+gf^{abc}b^b_i(z,[\rho])b^c_j(z,[\rho])]=0\,.
\end{eqnarray}
Expressions eqs.(\ref{notlarge}) and (\ref{notsmall}) are again related by the
DDD transformation eq.(\ref{duality}).

The KLWMIJ evolution equation was derived in \cite{kl} by directly considering the evolution of the projectile wave function.
In this framework the charge density operator depends only on the transverse coordinates $\hat\rho^a(x)$. As quantum operators the charge densities at the same transverse coordinate do not commute
\begin{equation}
[\hat\rho^a(x),\hat\rho^b(y)]=if^{abc}\hat\rho^c(x)\delta^2(x-y)
\label{com}
\end{equation}
Thus in general the averages of powers of $\hat\rho$ can not be represented simply by
a functional integral with a weight functional which depends on the classical field $\rho^a(x)$ only. This is unlike JIMWLK, where such a representation is possible.
It is however possible to have a functional integral representation if the "classical field" is endowed with an additional "ordering" coordinate
\begin{equation}\label{wf}
\langle \hat\rho^{a_1}(x_1)... \hat\rho^{a_n}(x_n)\rangle \,\equiv\,
\int d\rho(x,x^-)\, \rho^{a_1}(x_1,x_1^-)... \rho^{a_n}(x_n,x_n^-)\,\,W^T[\rho(x,x^-)]
\end{equation}
On the right hand side of this expression the values of the additional ordering coordinate inherit the ordering of the operator factors on the left hand side: $x^-_1>x^-_2>...>x^-_n$.
Clearly the weight functional $W$ can not be arbitrary. It must ensure two basic properties of the "classical" correlation functions. First, the correlation
function can not depend on the values of the coordinates $x^-_i$, but only on their relative ordering. Thus the correlation function should be a piecewise constant function of $x^-_i$. Second, under the interchange of the two adjacent values of $x^-_i$ the correlation function must emulate the effect of the commutator eq.(\ref{com})
\begin{equation}
\int D[\rho]W[\rho]...\rho^a(x_i^-)\rho^b(x_j^-)...=\int D[\rho]W[\rho]...\rho^a(x_j^-)\rho^b(x_i^-)...
+if^{abc}\int D[\rho]W[\rho]...\rho^c(x_i^-)...
\end{equation}
In particular this last property means that $W$ can not be real, and thus cannot be given the meaning of probability density. It was pointed out in \cite{kl} that a similar problem has appeared in the literature before \cite{polyakov} where it was shown that these properties are satisfied by a weight functional which contains an exponential of a Wess-Zumino term.

\section{On the operator ordering, $x^-$ vs $x^+$ and the Wess-Zumino-less weight functional.}
In this section we make several remarks on the structure of the evolution described in Section 2.

\subsection{On intimate relation between target and projectile}

A somewhat curious feature of the JIMWLK equation which is not frequently stressed in the literature, is that the same equation arises as the result of two apparently unrelated derivations. The original
derivation \cite{JIMWLK,cgc}
was given directly for the target wave function (equation for $W^T$ in \eq{dsigma}),
assuming explicitly that the target is dense, namely that the average of the single gluon $S$ matrix is close to zero. Nothing has been explicitly assumed there about the properties of the projectile. On the other hand, one can deduce the kernel
$\chi^{JIMWLK}$ in a different way assuming  the projectile is small and deriving the evolution of the
projectile wave function \cite{MuJIM,kl1}. This then yields the evolution equation for the target $W_T$ via the functional Fourier transform, cf. eq.(\eq{dsigma}). In this derivation we apparently do not assume anything about the properties of the target, instead we have assumed that the projectile is dilute. Nevertheless the two ways lead to the same evolution equation for $W_T$.
The question is how come one obtains the same equations under seemingly unrelated conditions?

The answer is that these two conditions - diluteness
of the projectile and dense nature of the target are actually inextricably related through the form of the $S$-matrix
eq.(\ref{s}). The JIMWLK kernel gives a correct approximation for the full kernel of the evolution of $W_T$ in the limit when $\alpha_T$
is large, or equivalently when $S$ is close to zero. On the other hand the evolution  of ${\cal S}$ eq.(\ref{hee}) is given by the functional
integral  over all $\alpha_T$. For $W^T$ which is sharply peaked around $S=0$, the integral picks out the large $\alpha_T$ region of $\Sigma^P[\alpha_T]$.
Thus the approximation is safest when $\Sigma^P[\alpha_T]$ is also peaked around large $\alpha_T$.

This however is precisely the condition for the projectile to be dilute. This
is simply seen from \eq{s}. Mathematically
it is the general nature of Fourier transformation that  $\Sigma^P[\alpha_T]$ is  peaked
at large $\alpha_T$ if its Fourier image $W^P$ is dominated by small values of
$\rho_P$.
The diluteness of the projectile is simply the statement that the projectile wave function contains only Fock space components with small number of gluons.
Parametrically, if $\alpha_T$ is of order one as is assumed in the derivation of \cite{JIMWLK,cgc}, the projectile charge density that is picked out by the Fourier transform
is of order $O(1)$ also, which corresponds in our normalization to $O(1)$ number of particles in the projectile.
In this case the projectile averaged $S$-matrix $\Sigma^P$ has only
several first terms in the Taylor expansion in powers of  $S$, as each projectile gluon contributes a factor of $S$. Thus for a dilute projectile, the Taylor series expansion of $\Sigma^P$ has only small number of terms and is indeed dominated by $S\sim 0$.

The diluteness of projectile, and the dense nature
of the target therefore  go hand in hand. The corollary to this observation is that if one wishes to discuss scattering in the situation where the projectile is not dilute, the JIMWLK approximation to the target weight function is not necessarily applicable even if the target is dense. The projectile averaged $S$ matrix $\Sigma^P$ is then itself peaked around $S\sim 1$, and so even if the target is mostly dense, the tail of the distribution in $W^T$ at $\alpha_T\ll 1/g$ will contribute significantly to the
total cross section. Physically this is very reasonable. If the projectile is dense, it has large probability of scattering on a variety of configurations in the target wave function: the ones with large charge density as well as the ones with low charge density. Thus the tail of the distribution $W_T$ which determines the weight of dilute configurations in the target wave function  becomes important.
A discussion of the evolution of the total scattering amplitude in this situation is to some extent superfluous, as it vanishes at all energies. However there are other less inclusive quantities which we believe will be strongly affected by this tail of the distribution. An example of such a quantity is diffractive scattering which is sensitive to the dilute peripheral region of the target. The  single gluon emission spectrum also may get sizeable contributions from the target configurations in the tail of the distribution.

\subsection{$x^+$ or $x^-$?}

Our second remark is essentially semantic. However since there has been some lack of clarity on this point in the recent literature, we thought it worth a comment.

The issue in question is whether to call the ordering variable entering the KLWMIJ kernel $x^-$ or $x^+$. We prefer simply to think of it as of an ordering variable and clearly in practical terms its name does not matter. There is however a certain physics picture that suggests its analogy with $x^-$.
In particular consider the KLWMIJ+ equation eq.(\ref{notsmall}). It involves the factors depending on the "classical" field $b_i(x,\tau)$, which satisfies an equation of motion identical to the classical Yang-Mills equation in the presence of the source $\rho(x,\tau)$, where $\tau$ is the ordering variable. In our wave function approach \cite{kl} this equation arises in the following way. To begin with one has to solve operator equations for the {\it operator} $\hat b_i(x)$ which has an {\it operator } source $\hat\rho(x)$  without any $\tau$-dependence.  Solving these equations, for example in the perturbation theory, one has to keep track of the ordering of operators $\hat\rho^a$ as explained in the previous section. This procedure is streamlined by introducing the ordering variable $\tau$ and treating all fields as classical. The resulting "classical" equation is identical to the classical equation solved in \cite{first} if one identifies the ordering coordinate $\tau$ with the longitudinal spatial coordinate $x^-$. It is actually more consistent to interpret the ordering variable as the rapidity $\ln x^-/x^-_0$ rather than the longitudinal coordinate itself, since it is dimensionless, but we will keep referring to it $x^-$ for simplicity. This identification is natural in the sense that $x^-$ and not $x^+$ is the spatial coordinate of the target. As the whole action of KLWMIJ takes place in the projectile Schroedinger (light cone) wave function there is no room for the time coordinate in this framework.

Once we have identified the ordering coordinate of $b_i$ as $x^-$, we have no choice but to identify the ordering coordinate of $R$ in eq.(\ref{rr}) as $x^-$ as well. This is quite clear,  the  operator $R$ simply shifts the charge density of the target by the charge of an extra gluon (\ref{rs}). Again, since this is a gluon in the light cone wave function, its coordinate is naturally identifiable with the longitudinal coordinate rather than the light cone time.

In fact this interpretation goes hand in hand with the usual spatial picture of the low $x$ gluon field. One normally thinks of the valence gluons as concentrated in a "pancake"
of small extent in $x^-$, while the wee gluons are distributed in a wider longitudinal slab at smaller values of the spatial longitudinal coordinate $z$, and so at smaller values $x^-$.
Eq.(\ref{notsmall}) conforms nicely to this picture.
In eq.(\ref{notsmall}) the values of $x^-$ in the various factors are ordered from left to write. Thus the values of $x^-$ in the factors of the charge density in the leftmost classical field $b_i$ are the smallest, the values of $\delta/\delta\rho(x^-)$ in the factor $R$ are larger and the values of $x^-$ in $\rho$'s of the rightmost $b_i$ are the largest. Following the derivation of \cite{kl} we can think of the rightmost $b_i$ and $R$ as belonging to the wave function, while the leftmost $b_i$ belonging to the conjugate wave function.
The rightmost $b_i$
in eq.(\ref{notsmall}) corresponds to the valence gluons in the wave function and thus takes up larger values of $x^-$. The wee gluons are the ones created by the shift operators $R$, and consistently with our interpretation take up smaller values of the longitudinal coordinate. From the action of $R$ eq.(\ref{rs}) it is obvious that the gluon emission probability is independent on $x^-$, which is again consistent with the common intuition that the distribution of the wee gluons is flat in rapidity.

The weight function $W$ on which the action of the kernel is defined therefore depends on the field $\rho(x^-)$, where $x^-$ takes values in three intervals necessary for representation of both factors of $b_i$ and $R$. This contour is
visualised in Fig. \ref{fig1}. The same structure is true for the JIMWLK+ equation if it is understood as the equation for the evolution of the target wave function in the regime of dense projectile. This is the logic adopted in Refs. \cite{kl,something,kl4}.
Of course if one writes the equation for the evolution of the projectile weight function the natural identification of the ordering coordinate there would be $x^+$.
\begin{figure}[htbp]
\begin{center}
\epsfig{file=RSRS.eps,width=160mm}
\end{center}
\caption{\it Ordering structure of (a) JIMWLK+ kernel, (b) KLWMIJ+ kernel, and (c) $W$ }
\label{fig1}
\end{figure}

The expression that is at the first sight somewhat puzzling within this interpretation, is eq.(\ref{s}), as there both $\alpha_T$ and $\rho_P$ carry the same ordering variable. However one should remember how this equation arises\cite{something}. The operator expression for the eikonal scattering matrix of the projectile charge density of the target field is
\begin{equation}
\hat\Sigma^P\,\,={\cal P}\,\,\,\exp\left\{i\int_{0}^{1} dx^-\int d^2x\,\hat\rho_P^a(x)\,\alpha_T^a(x,x^-)\right\}\label{ops}
\end{equation}
where the path ordering is along $x^-$.
The natural variable here is $x^-$, since it is the spatial variable from the point of view of the target fields and  time coordinate as far as the projectile is concerned. At this point $\hat\rho_P$ is an operator and does not carry any
longitudinal label. To average this expression over the projectile wave function we should expand eq.(\ref{ops}) in powers of $\hat\rho_P$, endow $\rho_P$ with an ordering coordinate, which we would now call $x^+$ (as it is the spatial variable in the projectile wave function), and average with the weight function $W[\rho_P(x^+)]$. We observe however that since eq.(\ref{ops}) is path ordered with respect to $x^-$, the ordering of the operators $\hat\rho_P$ in any term in the expansion is the same as the ordering of $x^-$ coordinates in the factors $\alpha_T$ which multiply them. Thus the values of variable $x^+$ are always ordered in exactly the same way as the vaues of $x^-$. Since the correlators of $\rho_P(x_i^+)$ depend only on the ordering of $x^+_i$'s, we can simply set the value of the longitudinal coordinate of $\rho_P$ equal to that of $x^-$ in $\alpha_T$. If one wishes to find a physical picture behind this mathematical statement, the proper words are that the projectile charges and the target fields always interact at the point $t=0$, where $x^+=x^-$. Thus at the interaction point indeed the longitudinal labels of the projectile density and the target field are equal.

\subsection{Life without Wess-Zumino.}

Though the explicit form of the WZ term is known (see, e.g. \cite{kl}), it is not clear if this
knowledge can be of any use to perform the actual averages for physical observables. In this remark
we show that the following compact parameterization of the weight-functional $W$ \cite{kl4}
\begin{equation}
W[\rho(x,x^-)]\,=\,\Sigma[R]\,\delta[\rho(x,x^-)]
\label{ansa}
\end{equation}
with $\Sigma$ - an arbitrary functional  of $R$, does not need to be augmented by the WZ term.
This form is a generalization of the form used in \cite{IM} in the framework of the dipole model with the difference that it explicitly incorporates the ordering variable, and the expression for $R$ should not be truncated at low order in derivatives as in \cite{IM}.

The logic for this "ansatz" is the following.
Consider a hadronic state containing energetic gluons in its wave function. When scattering on any hadronic target, its $S$-matrix must be some function of the eikonal factor for the propagation.
As the $S$-matrix is given via eq.(\ref{s}) we must have
\begin{equation}
\int D\rho^a\,e^{i\int \rho^a(x,\tau)\,\alpha^a(x,\tau)}\,W[\rho]\,=\,\Sigma\left[S\right]
\label{ft}
\end{equation}
Eq.(\ref{ansa}) is specifically devised to satisfy this requirement and thus is a natural representation for a weight functional.
We will now show that it also automatically satisfies the requirement eq.(\ref{property}) and thus does not have to be augmented by a Wess-Zumino term.
The $x^-$ ordering is relevant for $\rho$`s which are at the same transverse coordinate, thus we omit the transvers coordinate dependence in the following.

W will use the fact that $\rho$ acts on $R$ as rotation \cite{kl4}:
\begin{equation}\label{ror}
\rho^a(x^-)\,R_{bc}(y_1^-,y_2^-)\,=\,\left[R(y_1,x^-)\,T^a\,R(x^-,y^-_2)\right]_{bc} \,\Theta(y^-_2\,-\,x^-)
\,\Theta(x^-\,-\,y^-_1)
\end{equation}

To illustrate the general pattern, we consider first the one and  two point functions

\begin{eqnarray}\label{1ro}
\langle \rho^a(x^-)\,\rangle&=&\int d\rho\,\rho^a(x^-)\,\,\Sigma[R]\,\delta[\rho]=\int d\rho\,\,\left[R(0,x^-)\,T^a\,R(x^-,1)\right]_{cd}
 \,\frac{\delta\Sigma[R]}{\delta R_{dc}}\,\delta[\rho]\,\nonumber \\
&=&T^a_{cd}\,\frac{\delta\Sigma[R]}{\delta R_{dc}}|_{R=1}
\end{eqnarray}

\begin{eqnarray}\label{2ro}
\langle \rho^a(x^-_1)\,\rho^b(x^-_2)\rangle&=&\int d\rho\,\rho^a(x^-_1)\,\rho^b(x^-_2)\,\,\Sigma[R]\,\delta[\rho] \nonumber \\
&=&\int d\rho\,\rho(x^-_1)\,\left[R(0,x^-_2)\,T^b\,R(x^-_2,1)\right]_{cd}
 \,\frac{\delta\Sigma[R]}{\delta R_{dc}}\,\delta[\rho]\,\nonumber \\
&=&\int d\rho\,\left\{\left[R(0,x^-_1)\,T^a\,R(x^-_1,x^-_2)\,T^b\,R(x^-_2,1)\right]_{cd}
 \,\Theta(x_2^-\,-\,x^-_1)\right.\nonumber \\
&+&\left.\left[R(0,x^-_2)\,T^b\,R(x^-_2,x^-_1)\,T^a\,R(x^-_1,1)\right]_{cd}\,\Theta(x_1^-\,-\,x^-_2)
\right\}
 \,\frac{\delta\Sigma[R]}{\delta R_{dc}}\,\delta[\rho]\nonumber \\
&+&\left[R(0,x^-_1)\,T^b\,R(x^-_1,1) \right]_{ef}
\left[R(0,x^-_2)\,T^b\,R(x^-_2,1)\right]_{cd}
 \,\frac{\delta^2\Sigma[R]}{\delta R_{dc}\,\delta R_{fe}}\,\delta[\rho]\nonumber \\
&=&\left\{[T^a\,T^b]_{cd}\,\Theta(x_2^-\,-\,x^-_1)\,+\,[T^b\,T^a]_{cd}\,\Theta(x_1^-\,-\,x^-_2)\right\}\,
\frac{\delta\Sigma[R]}{\delta R_{dc}}|_{R=1}\nonumber \\
&+&T^a_{cd}\,T^b_{ef}\,\frac{\delta^2\Sigma[R]}{\delta R_{dc}\,\delta R_{fe}}|_{R=1}
\end{eqnarray}
The correlator is clearly piecewise constant, and depends only on the ordering of the coordinates. Moreover, the last term is symmetric under exchange of $x_1$ and $x_2$, while the interechange in the first term yields precisely the one point function, as required by eq.(\ref{property})
\begin{equation}
\langle \rho^a(x^-_1)\,\rho^b(x^-_2)\rangle=\langle \rho^a(x^-_2)\,\rho^b(x^-_1)\rangle+if^{abc}\langle\rho^c(x^-_1)\rangle{\rm sign}(x^-_2-x^-_1)
\end{equation}

Quite obviously
the same holds also for all higher correlators. The result for the three point function for example is
\begin{eqnarray}\label{3ro}
&& \langle \rho^{a_1}(x^-_1)\,\rho^{a_2}(x^-_2)\,\rho^{a_3}(x^-_3)
\rangle\,=\,\int d\rho\,\rho(x^-_1)\,\rho(x^-_2)\,\rho^c(x^-_3)\,\,\Sigma[R]\,\delta[\rho] \nonumber \\
&&=\,\left\{[T^{a_1}\,T^{a_2}\,T^{a_3}]_{cd}\,\Theta(x_2^-\,-\,x^-_1)\,\Theta(x_3^-\,-\,x^-_2)\,
+\,[T^{a_2}\,T^{a_1}\,T^{a_3}]_{cd}\,\Theta(x_1^-\,-\,x^-_2)\,\Theta(x_3^-\,-\,x^-_1)\right.\nonumber \\
&&+\,
[T^{a_1}\,T^{a_3}\,T^{a_1}]_{cd}\,\Theta(x_3^-\,-\,x^-_1)\,\Theta(x_2^-\,-\,x^-_3)\,
+\,[T^{a_3}\,T^{a_1}\,T^{a_2}]_{cd}\,\Theta(x_1^-\,-\,x^-_3)\,\Theta(x_2^-\,-\,x^-_1)\nonumber \\
&&+\,\left.
[T^{a_3}\,T^{a_2}\,T^{a_1}]_{cd}\,\Theta(x_2^-\,-\,x^-_3)\,\Theta(x_1^-\,-\,x^-_2)\,
+\,[T^{a_2}\,T^{a_3}\,T^{a_1}]_{cd}\,\Theta(x_3^-\,-\,x^-_2)\,\Theta(x_1^-\,-\,x^-_3)
\right\}\, \nonumber \\
&& \times\, \frac{\delta\Sigma[R]}{\delta R_{dc}}|_{R=1}
\,+\, \left\{[T^{a_1}\,T^{a_2}]_{cd}\,T^{a_3}_{ef}\,\Theta(x_2^-\,-\,x^-_1)\,+\,
[T^{a_2}\,T^{a_1}]_{cd}\,T^{a_3}_{ef}\,\Theta(x_1^-\,-\,x^-_2)\,\right. \nonumber \\
&&+\,\left. T^{a_1}_{cd}\,[T^{a_2}\,T^{a_3}]_{ef}\,\Theta(x_3^-\,-\,x^-_2)\,+\,
T^{a_1}_{cd}\,[T^{a_3}\,T^{a_2}]_{ef}\,\Theta(x_2^-\,-\,x^-_3)
\right\}
\,\frac{\delta^2\Sigma[R]}{\delta R_{dc}\,\delta R_{fe}}|_{R=1}\nonumber \\
&&+\,T^{a_1}_{cd}\,T^{a_2}_{ef}\,T^{a_3}_{bg}\,\frac{\delta^3\Sigma[R]}{\delta R_{dc}\,\delta R_{fe}\,
\delta R_{gb}}|_{R=1}
\end{eqnarray}
Again we see explicitly that eq.(\ref{property}) is satisfied.
This illustrates the general pattern: the factors $\rho^{a_i}(x^-_i)$
in the correlator are replaced by the color matrices  $T^{a_i}$ ordered according to the ordering of $x^-$.
That is for the correlator  $\langle \rho^{a_1}(x^-_1)\,\ldots \rho^{a_i}(x^-_i)\,\rho^{a_n}(x^-_n)\rangle $
with  strictly order $x^-_1\,<\,x^-_2\,<\,\ldots\,<\,x^-_n$ we get
\begin{eqnarray}
\label{nro}
\langle \rho^{a_1}\,\ldots \,\rho^{a_n}\rangle \,&=&\,\sum_{k=1}^n\,\,\sum_{1\le i_1<i_2<\cdots<i_{k-1}\le n}
[T^{a_1}\,\ldots\,T^{a_{i_1}}]_{cd}[T^{a_{(i_1+1)}}\,\ldots\,T^{a_{i_2}}]_{ef}\cdots[T^{a_{(i_{k-1}+1)}}\,\ldots\,T^{a_{n}}]_{bg}\,\,\nonumber\\
&\times&
\frac{\delta^k\Sigma[R]}{\delta R_{dc}\,\ldots\,
\delta R_{gb}}|_{R=1}
\end{eqnarray}
For different orderings of $x^-_i$, the n-point function is obtained from eq.(\ref{nro}) by the rule eq.(\ref{property}).

In the dense regime the introduction of the longitudinal coordinate is not necessary \cite{JIMWLK, kl1} and the correlators must become symmetric under the interchange the factors of $\rho$. This should be seen directly from eq.(\ref{nro}). Indeed the wave function of a dense system
is characterized by a large number of gluons, say $K\gg1$. In this case $\Sigma[R]$ behaves roughly as $R^K$. The n-point function (as long as $n\ll K$)
is dominated by the  $n$-th derivative of $\Sigma$, which as we have seen above, does not depend on $x^-$ at all:
\begin{equation}\label{nrod}
\langle \rho^{a_1}\,\ldots \,\rho^{a_n}\rangle \,\simeq\,
T^{a_1}_{cd}\,\ldots\,T^{a_n}_{bg}\,\frac{\delta^n\Sigma[R]}{\delta R_{dc}\,\ldots\,
\delta R_{gb}}|_{R=1}
\end{equation}

In accordance with discussion in the previous sunsection, we note that the variable $R$ in eq.(\ref{ansa}) is defined as the path ordered product along all three segments of the contour in Fig.\ref{fig1}. That is
we must understand $R$ in \eq{ansa} as the product $R=R_I R_{II}R_{III}$ (Fig. \ref{fig1},c). This does not affect the derivation of this section, and also correctly reproduces
the correlation functions involving factors of $\rho$ that appear due to expansion of the fields $b_i[\rho]$ in KLWMIJ+, eq.(\ref{notsmall}).

\section{Self-duality and the dipole toy model.}
A simplified version of the high energy evolution - the dipole model was introduced by Al Mueller \cite{Mueller}. It describes the leading high energy behaviour in the large $N_c$ limit as long as the densities in the wave functions are not too large. As shown in \cite{kl1} the dipole evolution equation can be obtained as a well defined limit of the JIMWLK evolution.
We can define the projectile dipole creation operator $s$ and the target dipole creation
operator $r$:
\begin{equation}\label{pdipole}
s(x,y)\,\equiv\,Tr[S_F^\dagger(x)\,S_F(y)]\,;\ \ \ \ \ \ \ \ \ \ \ \ \ \
r(x,y)\,\equiv\,Tr[R_F^\dagger(x)\,R_F(y)]
\end{equation}
where $F$ indicates the fundamental representation.
The canonical annihilation operators are simply $\frac{\delta}{\delta s}$ and
$\frac{\delta}{\delta r}$ for projectile and target respectively.

If  the target weight function is a function of $s$ only, that is $W^T=W^T[s]$, the action of the JIMWLK kernel on it
in the large $N_c$ limit is equivalent to the action of the dipole kernel \cite{kl1} (see also \cite{janik}):
\begin{equation}\label{jimdipole}
\chi^{JIMWLK}\,\,W^T[s]\,=\,\chi^{dipole}_{dp}\left[s,\frac{\delta}{\delta s}\right]\,\,W^T[s]
\end{equation}
The dipole kernel in the dilute projectile regime can be obtained by reformulating the original Muller`s
model and it has the following form  \cite{LL1}
\begin{equation}\label{chidip}
\chi^{dipole}_{dp}\left[s,\frac{\delta}{\delta s}\right]\,=\,\bar\alpha_s\,\int_{x,y,z} K_{x,y,z}\,\,
\left[s(x,y)\,-\,s(x,z)\,s(y,z)\right]\frac{\delta}{\delta s(x,y)}
\end{equation}

On the other hand for dilute target which contains only dipoles in its wave function $W^T=W^T[r]$, and the action of the KLWMIJ kernel
in the large $N_c$ limit is equivalent to\cite{kl4}:
\begin{equation}\label{jimdipole1}
\chi^{KLWMIJ}\,\,W^T[r]\,=\,\chi^{dipole}_{dt}\left[r,\frac{\delta}{\delta r}\right]\,\,W^T[r]
\end{equation}

Just like the JIMWLK and KLWMIJ kernels, the dipole kernels in the two regimes is related by the duality transformation eq.(\ref{duality}). Since the dipole model is much simpler than the complete evolution equation, one may hope to gain some insight into some features of the "holy grail" self dual evolution by studying it. So far unfortunately there is no exact formulation of duality in the framework of the dipole model. We are unable to provide such a formulation here, instead we will consider an even further simplified toy model of the evolution, which does not involve transverse coordinates. Physically this corresponds to considering the evolution of  distribution of dipoles of fixed size at a given impact paramter.

The projectile averaged $S$ matrix $\Sigma^P$ of the toy model is given by
\begin{equation}\label{Z}
\Sigma^P\,=\,\sum_n P_n\, s^n
\end{equation}
Here $s$ is the scattering matrix for the scattering of a single dipole on the target and $P_n$ is the probability to find $n$ dipoles in the projectile.  This probability plays the role of the projectile weight function
\begin{equation}\label{W}
W^P(n)\,\equiv\,P_n\ \ \ .
\end{equation}
In the dilute projectile regime the evolution of $\Sigma^P$  is given by
\begin{equation}\label{Z1}
\frac{d\Sigma^P}{dY}\,=\,\chi^{dipole}_{dp}\,\,\,\Sigma^P
\end{equation}
with the dipole kernel $\chi^{dipole}_{dp}$
\begin{equation}\label{chidipole}
\chi^{dipole}_{dp}\left[s,\frac{d}{d s}\right]\,=\,\bar\alpha_s\,
(s\,-\,s^2)\frac{d}{d s}
\end{equation}
where we can think of $\frac{d}{d s}$ as of the dipole annihilation operator, while $s$ as of creation operator.
The evolution (\ref{chidipole}) is equivalent to the evolution of the probabilities $P_n$ \cite{LL1}
\begin{equation}\label{Pn}
\frac{\partial P_n}{\partial Y}\,=\,\,\bar\alpha_s\,
(n\,-\,1)\,P_{n-1}\,\,-\,\bar\alpha_s\,n\,P_n\,=\,
\,\bar\alpha_s\,(e^{-\frac{d}{d n}}\,-\,1)\,n\,P_n
\end{equation}
This is a simple hierarchy of equations which has an interpretation that in every step in the evolution one additional dipole is emitted with probability $\bar\alpha_s$.
Note that $n$ is a discrete variable and thus the action of $d/dn$ is not well defined. However the action of the shift operator $e^{-\frac{d}{d n}}$ on any function $F_n$ is perfectly well defined and gives
\begin{equation}
e^{-\frac{d}{d n}}\,F_n\,=\,F_{n-1}
\end{equation}
Thus the analog of the dipole version of the KLWMIJ equation is
\begin{equation}\label{chidilute}
\chi^{dipole}_{dt}\left(n,e^{-\,\frac{d}{d n}}\right)\,=\,\,\bar\alpha_s\,
(e^{-\frac{d}{d n}}\,-\,1)\,n
\end{equation}
The shift operator $e^{-\frac{d}{d n}}$
is the analog of the  ``dual Wilson line'' $R$ and creates an extra projectile dipole.

Following Ref. \cite{Mueller} we can define the dipole densities $\rho^p_k$:
\begin{equation}
\rho^p_k \,\equiv\,\frac{1}{k!}\,\frac{\delta^k}{\delta s^k}\,\Sigma^P[s]_{|_{s=1}}
\end{equation}
The evolution \eq{Pn} or its equivalent \eq{chidip} leads to the equation for $\rho^p_k$ \cite{LL2}
\begin{equation}\label{rho}
\frac{d\rho^p_k}{dY} \,=\bar \alpha_s[\,k\,\rho^p_k\,+\,(k\,-\,1)\,\rho^p_{k-1}]
\end{equation}

It is sometimes useful to define the amplitude of  simultaneous  interaction of $k$ dipoles on the target - $\gamma_k$.
The $S$-matrix can be written in terms of  $\rho_k$ and $\gamma_k$  as\cite{Kovchegov}
\begin{equation}\label{Sd2}
{\cal S}(Y)\,=\,1\,+\,\sum_k (-1)^k\,\rho^p_k\,\gamma_k
\end{equation}
As shown in  \cite{LL2},
the evolution \eq{rho} can be mapped onto the Balitsky`s hierarchy for the amplitudes $\gamma$:
\begin{equation}\label{gam}
\frac{d\gamma_k}{dY} \,=\bar\alpha_s\,k\,[\gamma_k\,-\,\gamma_{k+1}]
\end{equation}

It is instructive to see how the duality transformation works in this simple model.
It is useful to keep in mind
the following set of anologies (for the definition of $\sigma$ see below):
$$\rho\, \rightarrow \,n\,; \,\,\,\,\,\,\,\,\, \ \ \alpha\,\rightarrow\,n\,\ln\sigma\,;\ \ \ \ \ \ \  \ \
S\,\rightarrow\,e^{\,n\,\ln\sigma}; \,\,\,\,\,\,\,\,\,\ \ R\,\rightarrow\,e^{-\,\frac{d}{d\,n}}
$$

The $S$-matrix is computed by averaging $\Sigma^P$ over target distribution $W^T[s]$
\begin{equation}\label{Sd}
{\cal S}(Y)\,=\,\int ds\,\Sigma^P_{Y-Y_0}\,[s]\,W^T_{Y_0}[s]
\end{equation}
Written in this form $W^T$ gives a distribution of probabilities that a single projectile dipole scatters off the target
with the amplitude $s$. Though we have written integral over $s$, in fact $s$ is discrete, and the integral over $s$ simply should be understood as the sum over postive integers $n$ with $s=\sigma^n$.

Let us
denote by $\sigma$ a cross section of a single projectile dipole to scatter on a single dipole of the
target (an analog of two gluon exchange in real QCD), $\sigma=1-\gamma\alpha_s^2$ with $\gamma$ of order unity.
If we allow a projectile dipole to multiply rescatter on $m$ dipoles of the target,
the natural model for the scattering amplitude on a $m$-dipole target state is
$s_m=\sigma^m$.
The target by itself has a distribution of dipoles
\begin{equation}\label{Wt}
W^T(m)\,\equiv\,T_m
\end{equation}
Then \eq{Sd} can be written in a manifestly symmetric form
\begin{equation}\label{Sd1}
S(Y)\,=\,\sum_{n,m} P_n(Y\,-Y_0)\,T_m(Y_0)
\,\sigma^{m\,n}\,=\,\,\sum_{n,m}\, P_n\,T_m\,e^{m\,n\,\ln \sigma}
\end{equation}

To derive the duality transformation in our toy model we assume that $P_n$ evolves according to the action
 of some kernel
$\chi^{dipole}(n,\,e^{-\frac{d}{d n}})$
acting on $ P_n$:
\begin{equation}\label{Pn1}
\frac{\partial P_n}{\partial Y}\,=\,\chi^{dipole}\left(n,\,e^{-\frac{d}{d n}}\right)\,P_n
\end{equation}
By ``integrating by parts'' we take the kernel $\chi^{dipole\,\dagger}$ to act
on $\sigma^{n\,m}$.  The result of this integration by parts is that $d/dn\rightarrow-d/dn$, and the ordering of factors containing $n$ and $d/dn$ is reversed. We then notice that
\begin{equation}\label{dddd}
n\,e^{\frac{d}{d n}}\, \sigma^{n\,m}\,=\,e^{\,m\,\ln \sigma}\frac{1}{\ln\sigma}\,\frac{d}{d m} \sigma^{n\,m}
\end{equation}

Integrating by parts  again we take the kernel to act now on $T_m$. The result of this three step procedure is the evolution equation for $T_m$
\begin{equation}\label{Pm1}
\frac{\partial T_m}{\partial Y}\,=\,\tilde\chi^{dipole\,}\left(
-\frac{1}{\ln\sigma}\,\frac{d}{d m},\,e^{\,\,m\,\ln \sigma}\right)\,T_m
\end{equation}
where the function $\tilde\chi$ differs from the function $\chi$ by the opposite ordering of the factors containing $n$ and $d/dn$.
In order to obtain \eq{Pm1} we used the Lorenz invariance (LI), which
requires that ${\cal S}$ does not depend on $Y_0$. Now  the projectile-target democracy  says that $P$ and $T$ have to obey the very same
evolution law. Hence we deduce
\begin{equation}\label{dddd1}
\chi^{dipole}\left(n,\,e^{-\frac{d}{d n}}\right)\,=\,\tilde\chi^{dipole}\left(-
\frac{1}{\ln\sigma}\,\frac{d}{d n},\,e^{\,n\,\ln \sigma}\right)
\end{equation}
The DDD transformation can therefore be formulated as the transformation
\begin{equation}\label{dddt}
n\,\rightarrow-\,\frac{1}{ \ln\sigma}\,\frac{d}{d n} \,;\ \ \ \ \ \ \ \ \ \ \  \
e^{-\frac{d}{d n}}\,\rightarrow\,e^{\,\,n\,\ln \sigma}
\end{equation}
accomapnied by the interchange of the order of $n$- and $d/dn$-dependent factors.

We remark  that the DDD transformation (\ref{dddt}) as well as the selfduality condition (\ref{dddd1})
have been derived in a specific model for the projectile-target interaction, namely ($e^{m\,n\,\ln \sigma}$).
Although this is a reasonable ansatz, in our toy model we are not bound to choose this particular form.
Have we choosen another ansatz for $s$ the resulting transformation would be different. Thus although the existence of duality transformation follows from the Lorentz invariance and PTD, its mathematical action depends on the
form of the projectile-target interaction.

Let us now apply the DDD transformation (\ref{dddt}) to the ``dilute'' kernel eqs.(\ref{chidipole},\ref{Pn}).
The resulting kernel is analogue of the JIMWLK equation.
\begin{equation}\label{chiden}
\chi^{dipole}_{dense}\,=\,\,\bar\alpha_s\,
\frac{1}{\ln\sigma}\,\frac{d}{d n}\left(1\,-\,e^{\,\,n\,\ln \sigma}\right)
\end{equation}
To make sense of the operator $\frac{d}{d n}$ we should represent it in terms of shift
operators $\exp\{-k\,\,d/dn\}$. This reresentation is obtained using the trivial property $x=-\ln e^{-x}$:
\begin{equation}\label{dn}
\frac{d}{d n}\,=\,\sum_{k=1}^{\infty}\,\frac{1}{k}\,\left(1-e^{-\frac{d}{d n}}\,\right)^k
\end{equation}
The evolution equation then can be written entirely in terms of  the creation operators of $k$ dipoles $e^{-k\,\frac{d}{d n}}$  (this interpretation is in analogy with eq.(\ref{Pn})).
\begin{eqnarray}
\label{chiden11}
\chi^{dipole}_{dense}\,&=&\,\,\bar\alpha_s\,
\frac{1}{\ln\sigma}\,\sum_k\,\frac{1}{k}\,\left(1-e^{-\frac{d}{d n}}\,\right)^k\left(1\,-\,e^{\,\,n\,\ln \sigma}\right)
\end{eqnarray}
The expansion (\ref{dn}) has a remarkable
propery: each  term in the expansion separatly conserves probability. The subtraction of the unity
takes into account virtual contributions for $k$-dipole emission\footnote{ One should be able to obtain a similar
representation for the JIMWLK equation by representing $\frac{\delta}{\delta \alpha}$ as an infinite series in powers of $(R-1)$.}.

The \eq{chiden11} suggests the interpretation that in the dense regime, arbitrarily many
dipoles are created during one step of evolution.
This interpretation is indeed correct, and is consistent with the actual evolution in QCD \cite{inprep,zakopane}. However it is also clear that the kernel does not correctly account for
the propagation of these newly born gluons through the target, as only one eikonal factor is present in eq.(\ref{chiden}). This is  an analog of the situation discussed in the beginning of the previous section. Assumption of the dense wave function of one of the colliding objects (multiple powers of $\exp\{-\frac{d}{d n}\}$) goes together with the assumption of the dilute wave function of the other
(only first power of $\,\exp\{\,\,n\,\ln \sigma\}$). In the full kernel higher powers of
the dipole creation operator must be also accompanied by higher powers of the eikonal scattering amplitude. The following interpretation of \eq{chiden11} looks natural. Real emission of $k$ dipoles in the full evolution kernel is accompanied by virtual corrections to all wave function components with numbers of dipoles smaller than $k$. The component of the wave function with $k$ emitted dipoles contributes to the $S$-matrix with the factor $s^k=\sigma^{nk}$, while the virtual terms contribute with lower powers of $s$, corresponding to the number of dipoles in the state whose probability is being "repaired" by the virtual correction. In the dense limit $s\ll 1$ and so the leading contribution to the $S$-matrix comes from the propagation of the state with only one emitted dipole and the state with no emitted dipoles. Thus although only one emitted dipole shows up as the "real" emission term, all the higher dipole contributions show up in the virtual correction piece: the norms of the "no extra dipole state" and "one extra dipole state" are decreased by the total probability of emitting more than one dipole in one step of the evolution. This interpretation is further strengthened by the observation that the series in the one dipole creation operator in \eq{chiden11} is an alternating sign one, which is what one naturally expects from a virtual correction.

In the dense limit where $n$ is large and $\frac{d}{d n}$ is small we can actually simplify the kernel by keeping only the first, single dipole term in the expansion (\ref{dn}).
\begin{equation}\label{chiden1}
\chi^{dipole}=\,
\bar\alpha_s\,\frac{1}{ \ln\sigma}\,\left(1-e^{-\frac{d}{d n}}\right)\,
\left(1\,-\,e^{\,\,n\,\ln \sigma}\right)
\end{equation}
The evolution eq. (\ref{chiden1}) possesses a remarkable property: it is selfdual under the DDD
transformation (\ref{dddt}). The kernel eq.(\ref{chiden1}) is expressed entirely in terms of the
``Wilson lines'' and in the two known limits of either dilute target or dilute projectile
reduces to the ``JIMWLK'' and ``KLWMIJ'' equations.
The kernel \eq{chiden1} leads to the following evolution equation for the probabilities
\begin{equation}\label{Pnsd}
\frac{\partial P_n}{\partial Y}\,=\,\,{\bar\alpha_s\over|\ln\sigma|}\,\left[(1-\sigma^{n-1})\,P_{n-1}\,-
(1-\sigma^{n})\,P_{n}\right]
\end{equation}
This has an interesting interpretation. The coefficient in front of $P_n$ on the right hand side has the meaning of the probability per unit rapidity to emit an extra dipole from the $n$-dipole state. In this respect \eq{Pn} is clearly problematic, since for large $n$ the emission probability grows without bound. Therefore it can only be considered informative for relatively small $n$, such that $\bar\alpha_sn<1$. The behavior of \eq{Pnsd} is much better, since the emission probabilities are finite even for large $n$ and so the gluon emission probability "saturates". Since by definition $\sigma<1$, the limiting value of the probability in \eq{Pnsd} is $\alpha_s/|\ln\sigma|$. We note however that this limiting value generically is still larger than unity. Within our present model $\alpha_s$ is in principle unrelated to $\sigma$. However in QCD the analog of $\sigma$ is the dipole-dipole scattering amplitude, which behaves as $\sigma=1-\gamma\bar\alpha_s^2$ with $\gamma$ of order unity. With this parametric dependence the limiting value of the coefficient in \eq{Pnsd} is $O(1/\alpha_s)\gg1$. Thus although the emission probability given by the selfdual form of the evolution saturates, it still has the probabilistic interpretation only for $\bar\alpha_sn<1$ just like the initial expression eq.(\ref{Pn}).

The expression \eq{Pnsd} is analogous to the self dual variant of JIMWLK+ evolution recently
proposed in Ref. \cite{SMITH}. The lesson we learn from our toy model is that self duality is not sufficient to gurantee the accuracy beyond the asymptotic limits large and small density limits. In general we should expect higher powers of both $R$ and $S$ to enter the fray (see also Ref. \cite{Balitsky05} for discussion of the selfdual form of the QCD evolution).

A somewhat surprising aspect of the evolution \eq{Pnsd} is that just like eq.(\ref{Pn}) it describes only "splitting" of dipoles and does not contain "mergings" in the parlance of \cite{MSW,IT,LL3}. One could naively expect to find merging terms which lead to growth of the probability
$P_n$ proportinal to probabilities  $P_k$ with $k>n$. Such terms do not appear in eq.(\ref{dn}) nor in the self dual limit of the evolution eq.(\ref{chiden1}).
On the other hand the common wisdom is that the "splitting" evolution of the projectile dipoles is somehow equivalent to "merging" evolution in the target. The resolution of this paradox is that even though there are no merging terms in the evolution equation for the probabilities $P_n$, they do indeed apear in the equation for the $n$-dipole densities $\rho_n$ and the $n$-dipole scattering amplitudes $\gamma_n$.
In particular
if instead of canonical dipole model \eq{Pn} we start from the selfdual evolution \eq{Pnsd} we obtain the following evolution
in terms of $\rho_k$
\begin{equation}\label{rho1}
\frac{d\rho^p_k}{dY} \,=\,\,{\bar\alpha_s\over|\ln\sigma|}\,\left[
\rho^p_{k-1}\,-\,\sum_{n=k-1}^{\infty}\rho_n^p\,\sum_{m=n+1-k}^{\infty} (\ln\sigma)^m\,C(k,m,n)\right]
\end{equation}
where the coefficients $C(k,m,n)$ are computable combinatorial factors. Since the expansion in $\ln\sigma$ is essentially an expansion
in $\alpha_s^2$ truncating it at  second order we obtain{\footnote{Strictly speaking this truncation is not justified. Eq.(\ref{rho1}) was derived in the approximation of large density. In this limit the higher dipole number densities carry extra powers of $\alpha_s^{-2}$ which cancel explicit powers of $\ln\sigma$ in eq.(\ref{rho1}). Thus we can only truncate the sum over $k$ but not the sum over $n$ in \eq{rho1}. The following expression therefore has to be understood within these limitations.}
\begin{equation}\label{rho2}
\frac{d\rho^p_k}{dY} \,=\,\,\bar\alpha_s\,\left\{ k\,\rho_k^p \,[1\,+\,(k-{1\over 2})\,\ln\sigma]\,+\,(k-1)\,
\rho^p_{k-1}\,[1\,+\,(k\,-{3\over 2})\,\ln\sigma]\,+\,{k\,(k+1)\over 2}\,\ln\sigma\,\rho_{k+1}^p\right\}
\end{equation}
The first two terms are the same as in Balitsky`s hierarchy amended by $\alpha_s^2$ corrections.
The last term corresponds to the merging of Refs. \cite{IT,MSW,LL3}. When translated
to the equation for the amplitudes $\gamma_k$, this term leads to a contribution of the form
\begin{equation}
\frac{d\gamma_k}{dY} \,\sim\,\,\bar\alpha_s\,|\ln\sigma| \,\gamma_{k-1}
\end{equation}
which is generally understood as Pomeron merging. Thus curiously enough we see that "Pomeron merging" term in the evolution of the amplitudes $\gamma_k$, when translated into the language of probabilities $P_n$ is in fact no different from dipole splittings. Physically this is perfectly reasonable, as boosting the hadronic state can never lead to the decrease in number of gluons or dipoles in the wave function. These numbers always grow, but the rate of their growth decreases at high energy. The Pomeron mergings describe the decrease in the rate of growth of the cross section, and not the decrease in the number of partons.

\section*{Aknowledgments}

We thank Ian Balitsky, Yoshi Hatta, Genya Levin, Larry McLerran and Urs Wiedemann for discussions related to the subject of this paper.

redpajama_set_name: RedPajamaArXiv


text:
Meet the Marine colonel who's looking to turn Alabama Senate race on its head: 'This is now a three-way gunfight'
"This is now a three-way gunfight," he said.
ByJOHN VERHOVEK
Retired Marine Colonel Lee Busby is running for Senate in Alabama against Roy Moore.
Courtesy Lee Busby/ElectLeeBusby.com
— -- Retired Marine Col. Lee Busby wanted to make sure one thing was clear as he rolled out his U.S. Senate campaign Monday in Alabama.
"I want to get something in — how to spell my name. L-e-e B-u-s-b-y. I want the voters to know that," he said.
As the race for Attorney General Jeff Sessions' old Senate seat has become consumed with sexual misconduct allegations against Republican nominee Roy Moore, Busby launched his eleventh-hour campaign hoping to capitalize on voters' dissatisfaction with the choices on the ballot.
"It started off innocuously enough. I was just dissatisfied with the choices I had as a voter," Busby told ABC News of his decision to run as a write-in candidate. "I didn't know at the beginning if that was just me or if it was a more widespread feeling. Then I talked with some people, and it turned out it was as widespread as I thought it was."
Roy Moore: Sexual misconduct allegations against me are 'completely false,' 'malicious'
Roy Moore's defiant road in pursuit of a US Senate seat
Meet Roy Moore, the controversial Alabama Senate candidate
With the election just two weeks away, Busby recognizes the long odds against him. But he believes his qualifications — he served in the military for 31 years — make him a viable alternative to Moore and the Democratic nominee, Doug Jones.
"I'm glad to let the voters look at the other two candidates' resumes, and they can look at mine on my LinkedIn page, and they can decide for themselves," Busby said. "The two other candidates are well steeped in Alabama judiciary experience. I don't think that means squat in terms of a U.S. Senate campaign."
When it comes to the allegations of sexual misconduct against Moore, Busby said he isn't interested in that topic of discussion.
"I view it as I don't have an opinion and don't care. I mean, it's a mess. It's not my mess, and as a voter, I've decided not to try to get immersed in figuring out what happened and what didn't. I have no interest in it," he said.
He is targeting voters outside the traditional Republican and Democratic bases, which he believes will remain loyal to the two major candidates.
"I do think that there's this huge swath between those margins that is very open to a candidacy like this one," he said.
Busby is a former top aide to White House chief of staff John Kelly, serving as his vice chief of staff when Kelly was a three-star lieutenant general. Busby said he has not spoken to Kelly since deciding to run and did not coordinate his candidacy with Kelly.
"Gen. Kelly and I haven't talked in about five years. But I've very much enjoyed watching his ascendancy to the spot that he is," Busby said.
Busby lives in Tuscaloosa, Alabama, and has spent time as a small-business owner and an entrepreneur. Over the last few years, he has been sculpting bronze memorial busts of Alabamians who were killed fighting in Iraq and Afghanistan, places he served.
Sculptor, Tuscaloosa resident and retired Marine Col. Lee Busby checks the image of Mark Forester he sculpted outside the Univ. of Alabama Foundry in Tuscaloosa, Ala., Nov. 3, 2017.
Gary Cosby Jr./AP
The special election will be held on Dec. 12, and Busby is relishing the chance to make his case to Alabama voters.
"With my throwing my name in the hat, this is now a three-way gunfight," he said.

redpajama_set_name: RedPajamaCommonCrawl


Happy to announce that the latest issue of Making is in!
Desert landscapes are the theme of this issue, which features projects for making of all kinds – quilting, embroidery, sewing, weaving, macrame, cooking, needle felting, and, yes, knitting!
I spotted plenty of familiar names in the knitting department – designers like Bristol Ivy, Susan B. Anderson, Norah Gaughan, Helga Isager, and others contributed to this issue.
Beautifully photographed and lovingly produced, Making is a treat of a publication.
Look for it on the teacart here at the shop, surrounded by the best in new books and magazines for makers. See you there!
Pom Pom Quarterly: Spring 2019.
Back in stock: the Lilly Brush.


text:
Maša Jazbec, PhD
Website: http://masajazbec.si/
Maša Jazbec, PhD, completed her studies in fine arts at the Faculty of Education in Maribor under the mentorship of prof. Dušan Zidar. She then pursued a master's degree in new media art at the University of Art and Design in Linz (Kunstuniversität Linz) in Austria. As part of her master's degree, she completed a study exchange at IAMAS (Institute of Advanced Media Arts and Science) in Japan. After completing her master's degree, she continued her doctoral studies at the University of Tsukuba (of 大学 / Tsukuba daigaku) in Japan in the laboratory of Professor Hiroo Iwata. As part of her doctoral studies, she completed an engineering residency at the renowned Advanced Telecommunications Research Institute International (ATR) in Professor Hiroshi Ishiguro's laboratory in Kyoto, Japan. Since 2013, he has been co-creating the international festival of new media culture Speculum Artium in Trbovlje. She has presented her art projects on international intermedia platforms, such as Ars Elektronica, Athens Media Art Festival, Sonica, Lab30, Tsukuba Media Art Festival, ISEA, Speculum Artium, MFRU, Kiblix, and ArtStay. In 2019, she presented her work in two comprehensive solo exhibitions in Slovenia, one in the Equrna Gallery in Ljubljana and the other in the Velenje Gallery in Velenje. In June of the same year, she co-created a new spatial arrangement in the Trbovlje park, where, in cooperation with the Municipality of Trbovlje, she set up a larger intermedia installation Monolit, which is her second outdoor sculptural installation. In 2019, she was a guest of the FrekvencaX show on Val 202 several times, and an extensive article about her work was published in the Delo newspaper. She also gave several guest lectures in Slovenia and abroad, lecturing on connecting art and science on the platform of android science. She was the initiator and organizer of the first international Brain-Computer Hackathon in Slovenia, which took place in Trbovlje. Since October 2019, she has been the head of DDTlab (Trbovlje), one of the three laboratories within the RUK network (Network of Research Arts and Culture Centers). In the laboratory, she and her colleagues create in the field of connecting art, science, and new technologies. They are currently developing new applications in the field of interfaces between the brain and the computer/machine, thus providing the general public with access to knowledge in this field not only in theory but also in practice. She has lectured at UCLA – Design Media Arts Department, DEWESoft Summer School, School of Integrative and Global Majors Tokyo Forum, Rhizomatiks Research – Flying Tokyo Event, Large Space Tsukuba Opening Symposium, Rikkyo University Tokyo, Sonica Artist Talk, and TEDx Trg Svobode. In 2018, she was a member of the expert jury for the prestigious Zlata Nika International Award given by Ars Electronica in Linz. Since 2019, she has been a member of the National Council for Culture of Slovenia. Since 2018, she has been a visiting professor at the Master's degree in Interface Culture at the University of Linz (Kunstuniversität Linz). She has presented research papers at professional conferences such as CHI (2016), Consciousness Reframed (2016), ISEA (2017), Human Robot Interaction (2017), and Systems Man and Cybernetics (2017).


Racket Sport Services is pleased to announce that as from Saturday 13th May 2017 that it will be providing an 'on-site' stringing service and shop at Frinton Lawn Tennis Club.
*weekend opening will alternate i.e. open Saturday, closed Sunday and the following week closed Saturday and open Sunday, commencing with Saturday opening on Saturday 13th May.
In addition we will be open when the club have one off events and tournaments and provide a stringing service for the four LTA torunaments held at the club this year.


text:
Best Movie Musicals
5 Shows & Movies | by Caitlin Leonard
A Chorus Line
A group of dancers congregate on the stage of a Broadway theatre to audition for a new musical production directed by Zach. After the initial eliminations, seventeen hopefuls remain, among them Cassie, who once had a tempestuous romantic relationship with Zach. She is desperate enough for work to humble herself and audition for him; whether he's willing to let professionalism overcome his personal feelings about their past remains to be seen.
Murderesses Velma Kelly and Roxie Hart find themselves on death row together and fight for the fame that will keep them from the gallows in 1920s Chicago.
In this hit Russian musical, a group of friends flaunts Soviet authority in 1950s Moscow by embracing jazz. When Communist Mels falls for Polly, a free-spirited jazz fan, he risks losing his party membership by associating with her rebellious crew.
Pleasantly plump teenager Tracy Turnblad auditions to be on Baltimore's most popular dance show - The Corny Collins Show - and lands a prime spot. Through her newfound fame, she becomes determined to help her friends and end the racial segregation that has been a staple of the show.
This lavishly produced and critically acclaimed screen adaptation of the international stage sensation tells the life-affirming story of Tevye (Topol), a poor milkman whose love, pride and faith help him face the oppression of turn-of-the-century Czarist Russia. Nominated for eight Academy Awards.


text:
Koka – west coast satisfaction
Koka is a small intimate restaurant with experienced staff who will offer an interactive dialogue with their guests to make it a learning and savoury experience. The focus is on modern Swedish gastronomy with what the west coast has to offer. It is not an open kitchen, but the chefs are present and will often come out and serve and explain the dishes. The restaurant is situated in the centre of Gothenburg, which is a small city yet with an interesting selection of fine dining places.
Koka opened up in 2014, but it has a longer history than that. It started already in 2000 by Björn Persson under the name of Kock & Vin. After 14 years the team at Koka decided to refine the concept of the restaurant to better reflect the dedication to contemporary western Sweden dining.
The focus of the restaurant is on modern Swedish gastronomy, and the aim is to work with local ingredients. Being situated on the Swedish west coast gives Koka a great access to the sea, farming areas and forest. It is a regional pantry just waiting to be discovered. So you will find local ingredients such as lamb from Bitterna and regional fish like mackerel. Even the music played in the restaurant is only local. Being local often means being sustainable and here Koka has its energy supplied by wind power and is also working with environmental certification.
The restaurant has a three, five or seven course tasting menu which is changed on a continuous basis according to season. The five and seven course menus are longer taste menus, while the three courses are for those who also plan to do something else that evening. The dishes are light and the focus is on key ingredients harmonising together where vegetables play an important role. So the scallop might be matched with carrot, elderflower and roasted buckwheat seed and in a dessert the chervil ice cream and rhubarb with zucchini.
Koka works predominantly with wines from Europe and in the wine list there are some vin nature, but it is mostly a classical wine list. The staff at Koka have a long history and experience of pairing food and wine so expect to find some good matches. There are some non-alcoholic beverages, but from a contemporary restaurant maybe you could expect a complete matching non-alcoholic beverage menu?
The interior is a mix of light wood with contrasting purple panels and the relaxed atmosphere is underlined with cutlery already in a wooden box on the table. Innovative and flexible are the loose table tops which make it possible to play around with the size of the tables and you can adjust the lamps to your own liking in height and light intensity.
Often neglected at restaurants is the selection of tea and coffee after the meal. At Koka the restaurant has a good selection of both tea and coffee. An excellent finish after a journey through some of the best stuff you will find in this part of Sweden.
Koka scored 80 points of 100 points in our rating system. View our Top 60 of the world. Back to the reviews.
By Joakim & Suzanne
Viktoriagatan 12
411 25 Gothenburg
Phone: 46 31 701 79 79
www.restaurangkoka.se
© wbpstars.com - images and films


With the fast growth of cross-border retail e-commerce market and the increasing consumption strength in China, the Australian natural health brand Swisse announced its official entry into this market on 8 April 2017.
A girl from Chile bought her father a watch as a gift at US$41.01 from Alibaba's retail platform AliExpress on 10 April 2017. What she did not know is that she is recorded by Alibaba as the 100 millionth customer.
China cross-border e-commerce total transactions reached RMB 751.2 billion in 2015, an increase of 69% YoY according to Alibaba.
Shopping guide websites are the top referral of China cross-border online shoppers (59.8%), followed by family & friends (51.2%) and search (47.3%) according to iReseach.
On March 24 2016, China Ministry of Finance, the General Administration of Customs and the State Administration of Taxation jointly issued the "notice on cross-border e-commerce retail import tax policies".
China had more than 5000 different types of cross-border online platforms serving over 200,000 companies of e-commerce operations in 2015. Driven by the rapid growth of e-commerce, small and medium e-commerce companies has become a new growth focus in China.


text:
Answer EOC's Questionnaire
EOC Interview
Jeremy Fox – Fox's Food
February 11, 2019 | By cbusentrepren
What is your full name, title, and business name?
Jeremy Fox, Owner, Fox's Food LLC
Follow Short North Bagel Deli on Social: Facebook | Instagram
Follow Blocks Bagels on Social: Facebook | Instagram | Twitter
Give us a summary of your business in 200 words or less.
Foxs Food LLC is the parent company that owns Short North Bagel Deli and Blocks Bagels Bexley. Short North Bagel Deli is a mobile food business selling steamed deli sandwiches on bagels from food carts, food trucks and concession stands inside Ohio Stadium, the Schottenstein Center and inside office buildings managed by Aramark. Blocks Bagels Bexley is the newest Blocks Bagels in Columbus and formed by entering into a licensing agreement with the Block Family.
How did the idea for your business come about?
It all started with a food cart in 2011. I went to Butler University in Indianapolis and there is a bagel deli there called Ripple Bagel Deli that was extremely popular in the area. After college, I moved back to Columbus and saw a lot of similarities between Broad Ripple (the part of Indy that the bagel deli was located in) and the Short North–the only thing that was missing was a bagel deli. The first plan was to open a brick and mortar bagel deli in the Short North called Short North Bagel Deli, but after drafting a business plan and meeting with some financial institutions and ECDI it was clear that a brick and mortar would be too risky considering I had no experience (was doing internet marketing at the time) and no money. So I decided to start with a food cart instead to mitigate risk, learn the industry, and build a following. I kept the name Short North Bagel Deli because the plan was to use the cart to start and open a brick and mortar in the Short North down the road (which was still the plan until the Blocks Bagels opportunity presented itself). The cart led to operations inside Aramark managed office buildings (Nationwide Insurance, Grange, IGS, etc), that led to concession stands inside the Schottenstein and the 'Shoe, that led to a food truck and most recently the license agreement with Blocks Bagels to own and operate my own independent location.
What was the turning point for your business? Was there a moment you knew you had something special?
I think the real turning point was when I added the food truck to the mix; this meant I could double my bookings and events but also meant I had to hire employees–as many entrepreneurs will attest, as hard as we might try, it is impossible to be in two places at once. We added the truck to meet demand, I was turning down gigs because I was booked solid with the food cart, growing demand meant I had a product people wanted and would have to expand to meet the demand.
What does it mean to you to be an entrepreneur and business owner?
Sounds corny but to me it means everything. I don't delineate between me and my business; I am my business and my business is me, its my identity. I tried working for someone else before going out on my own and realized quickly it wasn't for me. Although I work harder than I've ever worked before, I truly feel like I haven't worked a day since opening the cart in 2011. It just feels like I have a lot of stuff to do all the time but doesn't feel like work.
What does the city of Columbus mean to your business?
Columbus was integral in the success of the business. Since I started out in the mobile food industry which was just taking off in 2011, I relied heavily on the early adapters who flocked to mobile food and embraced it.
What's the number one piece of advice you'd give to anyone wanting to start a business?
Picture the hardest you've ever worked before, then multiply that by 10. If you think you can handle that while making no money then you should be okay.
What do you wish you knew about entrepreneurship before starting your business?
I wish I knew that my plan and vision would be constantly changing. It's vital to have a business plan, but starting out I figured I'd just continue down the path I set out on in the beginning. It's fun looking back on my original business plan to see what came true and where I deviated completely.
What's the most challenging part of your business (i.e, what keeps you up at night)?
If you ask anyone that owns a restaurant, they will all tell you the same thing: the biggest challenge is staffing. While I've been lucky and have been able to retain majority of my staff, it's still a constant challenge and something that requires daily attention. In the beginning, it was only me and there was comfort in knowing that it was all up to me. However, with growth you have to delegate and learn to trust in your staff, which leads us to my greatest flaw.
Every business owner has a flaw. What's yours?
I have a hard time delegating. It might be because in the beginning it was only me and I got used to that, or it might just be the way I am wired. But the hardest and most unnatural thing I've had to deal with is relying on other people to get work done.
Why do you think most business owners fail? What has made you different?
I think most business owners that fail have an idealized view of owning a business; they see success and accolades but don't see the hard work involved. I think my sports background makes me different, I've played sports my whole life and soccer at the collegiate level. When things gets tough, I call upon my memories of conditioning and the long hours and work put in to succeeding in athletics–compared to that the physical work I used to do, business is easy!
What was your biggest mistake and what did it cost you?
Honestly, I feel like my biggest mistake was over-paying for my food cart when I first started. If I opened another food cart tomorrow, I would be able to do it for 75% less then I paid when I first opened. Now one of my mottos is you don't know what you don't know until you know you don't know it. At the time I didn't think I was over-paying, but in hindsight, it did cost me a considerable amount of money and made starting up that much more stressful.
What tool has helped you the most for your business (invoicing, accounting, shipping, plugin for website, etc.)?
The backbone and foundation of any business is accounting and record keeping; I knew this from the beginning and have kept excellent records for any information I determined useful. I start every day with entering in data in to the spreadsheets I've built and they have become a tool we use to make decisions on a daily basis.
When did you know it was time to expand your business, make your first hire, etc.?
Already touched on this above, but when I had just the cart and had to turn down gigs because I was already booked I knew I had to make my first hire.
What is something that you did that was a game changer for your business?
To me it was a no brainer, but the decision to use Blocks Bagels for Short North Bagel Deli set things in motion to get me to where I am at today. Aside from them just being the best bagels in Columbus, I also had a personal relationship with the Blocks and grew up with them.
What was an idea that you spent a lot of time on or thought would make a big difference in your business that didn't pan out.
Recently, I've been spending a lot of time working on Catering for Blocks. I still think it's a good idea, but finding it hard to increase our catering business. I created a LinkedIn campaign trying to connect with pharmaceutical sales reps who call on offices and bring them food in an attempt to have them use Blocks for their catering. Unfortunately, that campaign proved less then fruitful but I will continue to work on it.
What is something that your business spends a lot of money on that's worth it?
Labor, other than product labor, makes up the biggest portion of our expenses. The restaurant industry is a very personal face-to-face interaction business, even if you have the best product in the world people won't eat if you don't have friendly people serving it.
What is something you're working on now that you're very excited about?
Continued expansion of the Blocks brand. The plan is to open more Blocks Bagels locations around Columbus.
What form of marketing is the most valuable for you?
Word of mouth. I have been hesitant to spend money on traditional print or online advertising, I believe that word of mouth is the truest form of marketing there is. One, it's free, which is great, but also is consistent with our core values, providing the best possible food while having the best possible customer service. If we do our job, our customers will tell their friends and network about our business, which in turn will create more customers, and opportunity to gain repeat customers who will tell their friends and network about us.
Who do you vent to when you have a business problem?
Hands down, my wife. As stated before, I am my business, which means its constantly on the forefront of my mind. Unfortunately for my wife, that means she gets to hear about everything going on and luckily for me, she's a great shoulder to lean on.
Where do you see your business in the next 10 years?
I see multiple Blocks locations around Columbus, expansion of Blocks Bagels in the wholesale market and the merging of Short North Bagel Deli in to Blocks Bagels Mobile.
Who is an entrepreneur that you look up to?
Ryan Vessler of Homage, he is a regular at our Blocks location and we talk weekly about business and different ideas. I'm very impressed by what he's been able to build by staying true to himself and his business' core values.
What other entrepreneur do you look up to most?
Mikey from Mikey's Late Night Slice because he started in the mobile food like I did and made his way into the brick and mortar is something I can relate too and trying to do as well. I also follow what the guys at Fusian Sushi are doing and I am extremely impressed at how they've been able to grow in seemingly a short amount of time.
If you had to tell a visitor one thing to do/see/eat in Columbus, what would it be?
I would say the obvious thing to do is go to an Ohio State Football game. If you've never been to one it's a must see. While you're there, grab a steamed deli sandwich on a Blocks bagel from our concession stand.
cbusentrepren
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About Entrepreneurs of Columbus
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Weekend Update: Pawhuska was the winner of Collinsville's Wood Bat Tourney ... which was finished in Bixby due to a sloppy field here Fri-Sat.
Collinsville was already down 0-4 in the first inning when these two card base runners (Jonathan Beweley and David Chester) were left stranded as they tried to advance above on a long fly out for the 3rd out.
Jonathan Beweley puts the tag on this Oologah base runner in the second inning with the throw from Cards' catcher Ryan Coe.
The Collinsville dugout had the rally caps out early (above) but were still trailing 1-4 when I left. Heard later at the soccer game that the baseball final was 3-9.
Shortstop Branden George drifted back into left field to make this out for the Cards (left).


In my 3 years work experience customer service has always been my top skill.
I have worked in Walmart, 7 eleven and have also sold alarm systems for GE. Patience is my middle name, and i believe I have the patience to deal with all and every customer. I also speak proper English as well as creole.
As a cart attendent in Walmart my main concern was pushing carts to the cart area. However I always was ready to help a customer finding a item while I was inside or help a customer load there car.
As a sales associate in 7 eleven I was in charge of the cash register, and making sure the store always looked clean.
As a sales rep for GE security I would walk door to door explaining to people and convincing them that GE security was the right security for there family, as well as taking calls in the office and making appointments.


The MA in TESOL has been offered since 2000 in a field-based format for ELIC teachers. Two weeks of courses are taken in January each year in Thailand, and 2-4 weeks of courses are taken each year in July on Wheaton's campus. During the rest of the year, the students work as teachers in various Asian countries. Combined with the TEFL Certificate, this MA takes three years to earn.
Through a cooperative arrangement between English Language Institute/China (ELIC) and Wheaton College, students accepted by both ELIC and Wheaton College will receive a tuition discount. Advance work, including reading, research, projects, is required for all intensive courses. It is necessary to apply to BOTH Wheaton College and ELIC because the application processes are distinct from each other. Acceptance by one does not guarantee acceptance by the other.
If you are with an agency other than ELIC, you are welcome to join the field-based program. This is especially helpful for teachers in Asian contexts.
If you are interested in the M.A. in Intercultural Studies/TESOL with ELIC, read about Professional Development on the ELIC website or contact the Wheaton/ELIC Program Director, Pam Barger.


asian paints interior colors? Asian Paints Net Worth is 6. 3 Billion. Asian Paints (BSE: , NSE: ASIANPAINT) is an Indian chemicals company headquartered in Mumbai, India.
Color& Lighting. Lighting plays an important role for the entire look of the interior. Three types of lighting mainly come into consideration from day to day life.
Interior wall color combinations asian paints bedroom decorating is posted on the category Design My Bedroom in the awanshop. co website. This post of" interior wall color combinations asian paints bedroom decorating" was published on by dave and has been viewed 533, 887 times.
Que rica estás.me encantaría lamertela y que te vinieras en mi cara(; grabarnos y subirlo aquí hermosa.soy mujer.envíame mensajito.que rica de verdad!


text:
class Task::BaseForm < ::Form
  attribute :task, :model, primary: true, new_records: true
  attribute :user, :model
  attribute :working_group, :model

  attribute :name,             :string
  attribute :working_group_id, :string
  attribute :due_date,         :date,   default: proc{ Date.today + 1.week }
  attribute :description,      :string

  attribute :primary_location, :string, default: proc{ (task.primary_location || task.circle.address.location).try :address }
  attribute :organizer_id,     :integer, default: proc { task.organizer.try(:id) || user.id }

  attribute :duration,      :integer

  attribute :scheduled_time_type,  :string, required: false
  attribute :scheduled_time_start, :string, required: false
  attribute :scheduled_time_end,   :string, required: false

  attribute :volunteer_count_required, :integer, default: proc { 1 }

  attribute :ability, :model
  attribute :circle, :model

  def duration_unit_options
    [
      [ "Hours",   'hour' ],
      [ "Minutes", 'minute' ]
    ]
  end

  def scheduled_time_type_options
    ['on_date', 'at', 'between'].map do |val|
      [I18n.t("activerecord.attributes.task.scheduled-time-select.#{val}"), val]
    end
  end

  def scheduled_time_options
    ("0".."23").map do |hour|
      %w(00 15 30 45).map do |min|
        "#{hour}:#{min}"
      end
    end.flatten
  end

  def duration_options
    Task.durations.map do |key, val|
      [I18n.t("activerecord.attributes.task.duration-text.#{key}"), val]
    end
  end

  def working_group
    @working_group ||= begin
      new_working_group = circle.working_groups.find_by(id: working_group_id) || task.working_group
      if ability.can? :create_task, new_working_group
        new_working_group
      else
        task.working_group
      end
    end
  end

  def available_working_groups
    @available_working_groups ||= begin
      working_groups = circle.working_groups.asc_order.to_a
      working_groups.select! { |wg| ability.can?(:manage, wg) } unless ability.can?(:manage, circle)
      working_groups << task.working_group unless working_groups.present?
      working_groups
    end
  end

  def available_working_groups_disabled?
    available_working_groups.size == 1 && ability.cannot?(:manage, available_working_groups.first)
  end

  class Submit < ::Form::Submit
    def validate
      add_error(:name, :too_short)                   if name.length < 5
      add_error(:description, :too_short)            if description.length < 5
      add_error(:due_date, :empty)                   if due_date.blank?
      add_error(:primary_location, :empty)           if primary_location.blank?
      add_error(:volunteer_count_required, :too_low) if volunteer_count_required < 1
    end

    def execute
      task.tap do |t|
        t.name          = name
        t.due_date      = due_date
        t.description   = description
        t.working_group = working_group

        t.duration      = duration

        t.scheduled_time_type   = scheduled_time_type
        t.scheduled_time_start  = scheduled_time_start
        t.scheduled_time_end    = scheduled_time_end

        t.volunteer_count_required = volunteer_count_required

        t.save

        t.roles.send('task.organizer').destroy_all

        organizer = User.find_by(id: organizer_id)
        organizer_ability = Ability.new(organizer)

        if organizer_ability.can?(:read, t)
          t.roles.send('task.organizer').create user_id: organizer_id
        else
          t.roles.send('task.organizer').create user_id: user.id
        end

        volunteers_to_remove = t.volunteers.select do |volunteer|
          Ability.new(volunteer).cannot? :read, t
        end
        t.roles.where(user: volunteers_to_remove).delete_all if volunteers_to_remove.present?

        t.location_assignments.destroy_all
        t.location_assignments.create primary: true, location: Location.location_from(primary_location)

        t.save
      end
    end
  end
end

redpajama_set_name: RedPajamaGithub


When I am trying to extract a data from the winrar5 archive which is protected with a password and have encrypted filenames, I get the issue: "Error: The password is incorrect". I can only open this file in PowerArchiver to view.
Program could not open the *.wim files from Windows 8.1 april updated ISO image.
I could open it only when I replaced your 7z.dll with the new one from 7-Zip 9.32 alpha published on 2013-12-01.
In russian version of your program not all text is visible.
The word Переименовать on the button.
The word Профили near the profiles field.
The word Программы above the orange rectangle.
Добавить в "Просмотр последних сообщений"
Localization errors still on 14.05.04 version.
let us know if that works. Thanks.
The problem with winrar5 archive with encrypted filenames extraction is solved.
The problem with new *.wim files opening is solved. May be it could be a little bit faster, but it's working fine.
I hope the problems with russian text will be solved later.
Thank you for the quality work.


text:
Al-Mashareq Al-Mashareq
Al-Mashareq
IRGC, regime embroiled in Deir Ezzor dispute
The Syrian regime's demand that fighters from IRGC-affiliated militias report for mandatory military service has fueled tensions in Deir Ezzor.
By Waleed Abu al-Khair in Cairo
Nawaf al-Bashir, commander of the IRGC-affiliated al-Baqir Brigade, stands with fighters from the brigade. [Photo circulated online]
Tension between Iran's Islamic Revolutionary Guard Corps (IRGC) and militias affiliated with the Syrian regime have reached a boiling point over the issue of control of the eastern city of Deir Ezzor, a Syrian activist said.
There also has been widespread anger over the Syrian regime's demand that youth fighting in the ranks of IRGC-affiliated militias report for compulsory military and reserve duty to serve the Syrian state.
Tensions between the two sides have rippled out from the city of Deir Ezzor to the surrounding areas, especially those dominated and controlled by local tribes, activist Jamil al-Abed told Diyaruna.
Al-Bakara tribal leader Nawaf al-Bashir, who answers directly to the IRGC, is trying to expand his militia's area of control in Deir Ezzor and its hinterland and beyond to the city of Albu Kamal, at the expense of the Syrian regime.
"There was an initial confrontation between Sattouf al-Merhi, a prominent commander of the Syrian regime-affiliated militias, and al-Bashir," al-Abed said.
"The dispute subsequently expanded to involve Sobhi Zayed al-Hanan, who is known to have strong relations with the regime's intelligence apparatus," he said.
Al-Hanan is a member of al-Bakara tribe as well, but refuses to be subordinate to the IRGC, al-Abed said. He has maintained his relations with the regime over the past years, and is a supporter of the decree on military and reserve service.
Summoned by Syrian regime
Al-Bashir was summoned to Damascus a few days ago, and he remains there.
The summons came on account of the disputes and al-Bashir's refusal to follow the orders issued by the Syrian regime with regard to the enforcement of mandatory military and reserve service, al-Abed said.
Al-Bashir publicly announced his refusal to follow these orders at a rally in the city of Deir Ezzor, he noted.
"From al-Bashir's point of view, enforcing the decree would cost him much of his strength, since all elements of the militia he heads are wanted for mandatory military service," al-Abed said.
He is therefore trying to assemble his fighters and move them to the village of Muhaymidah, his birthplace, which is outside the control of both the regime and the IRGC, to help them evade military and reserve service, he added.
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Syrian army subsumes IRGC-affiliated militia
The IRGC-affiliated Liwa Imam al-Baqir tribal militia has been absorbed into the 137th Infantry Brigade of the Syrian army's 17th Division.
Regime summons Syrians with IRGC-affiliated militias for military service
The regime issued a circular in Deir Ezzor instructing all Syrian nationals with IRGC-affiliated militias to report for military and reserve service.
Former regime defector joins forces with IRGC in Syria desert
By recruiting a prominent tribal leader, the IRGC seeks to control the Badiya region by establishing proxy militias to fight alongside the Syrian regime.
Glorification of slain Iranian general Soleimani rankles Afghans
Slain IRGC Quds Force commander Qassem Soleimani is responsible for thousands of Afghan deaths and should not be honoured as a hero, Afghans say.
Former Syrian military officer gets life sentence in historic torture trial
Prosecutors say he oversaw rape and sexual abuse, 'electric shocks', beatings with 'fists, wires and whips' and 'sleep deprivation' at the prison.
Iran demands maritime security while engaging in piracy, smuggling
A confidential UN report has concluded that thousands of weapons and rocket launchers seized in the Arabian Sea by the US Navy in recent months likely originated from the Iranian port of Jask.
USAID helps al-Raqa women farmers revive their lands
USAID is providing more than 150 women with the technical know-how and the tools to rehabilitate their dormant, war-damaged fields and become self sufficient.
Do you support a boycott of the upcoming Beijing Olympics due to China's repression of Muslims in Xinjiang?
Yes. Countries should send neither athletes nor officials
Yes, but a diplomatic boycott only
Pro-Iran groups fingered in attacks on coalition bases in Iraq, Syria
The attacks by Iran-backed groups undermine the international effort to eradicate the threat of ISIS in the region, and prove to be more dangerous to civilians than to soldiers.
Chinese projects in Iraq provoke increasing resentment among locals Global Imams urge Muslims to boycott Beijing Olympics Iran demands maritime security while engaging in piracy, smuggling
US calls on Iraq to dismantle armed groups Virtual reality game explores Mosul relics, promotes message of peace Kawtharani 'unwelcome' in Iraq, activists say


text:
Emmett calls Dome demo proposal a "silly idea"
Politics // Houston Politics
Kiah Collier
July 17, 2014 Updated: July 17, 2014 9:05 p.m.
Hoping to jump-start another discussion about redeveloping the Astrodome, Harris County Judge Ed Emmett on Thursday called the Houston Texans and the Rodeo's proposal to demolish the iconic stadium and replace it with a park-like green space "a silly plan" and pleaded with members of the Hotel & Lodging Association of Greater Houston to give repurposing a second chance.
"The Astrodome is the only building in the world that's 350,000 square feet of column-free space," he told a luncheon crowd at the Four Seasons Hotel downtown. "There are a lot of creative people in the world who would love to figure out ways to use the space if we just keep it and make it an option for them."
The county's top elected official was not advocating for a specific redevelopment proposal and told association members there is not one currently under consideration by the Harris County Commissioners Court, which will have the final say on any plan to renovate or demolish the structure. Rather, he said that tearing it down would be a waste of a valuable taxpayer-funded asset and that demolition would come back to haunt him in retirement.
The 49-year-old structure "is going to become a critical piece of who we are as a community. It's not about nostalgia," he continued. "It's an asset that belongs to the taxpayers of Harris County and it would be a shame, because I know that I would wake up in retirement at my log cabin 10 or 12 years later and somebody would come forth and say 'If we just had the Astrodome.' So, I just wanted to bring that out. I'm starting that discussion again."
'Astrodome Hall of Fame'
In the last week, Emmett has expressed opposition to a $66 million proposal by the Houston Livestock Show and Rodeo and the NFL's Texans to demolish the stadium and turn it into a grassy, outdoor space like downtown's Discovery Green where festivals, tailgating and concerts could take place. He did so again on Thursday.
"It's a silly plan," he said after his remarks. "I told them: If you're going to tear it down, just tear it down. I mean, is anybody going to care this is where Harmon Killebrew hit a home run?"
Titled the "Astrodome Hall of Fame," the proposal is designed to pay tribute to the Dome's history, calling for dozens of plaques commemorating the major events, athletes and entertainers who played and performed there throughout the decades. The stadium, declared unfit for occupancy in 2009, has not housed a sports team since 2000 when the Houston Astros moved to Minute Maid Field downtown.
Emmett said he would oppose the plan even if the Texans and the Rodeo agreed to foot the entire bill. Top officials there have said they would consider paying for some of it, but would not say how much.
The Rodeo declined comment through a spokeswoman; The Texans did not respond to a request by press time.
The hotel association supported the event center proposal that appeared on the ballot last fall, but has not yet taken a position on the latest idea.
"We're waiting to hear the plan, as well as the details, before we can really make a decision," President Stephanie Haynes said. "We haven't even discussed it as an association."
Emmett is alone on the five-member court in his staunch opposition to the plan.
While some commissioners have said they would like to avoid demolition, all four said the proposal is worth considering, especially if it somehow could be paid for without tax dollars.
"I still believe that with the improving economy, the opportunity of finding someone who wants to invest in this historic, classic structure will continue to increase," said Precinct 4 Commissioner Jack Cagle. "However, I have not discarded by any means the proposal that was submitted by the Rodeo and Texans."
If they offered to pay for it, "it certainly would pique my interest," he said.
One factor is working in Emmett's favor: The state historical commission is slated to consider designating the stadium a "state antiquities landmark" at a meeting at the end of the month. While the title would not outright protect the former Eighth Wonder of the World from the wrecking ball, it would make it more difficult to tear it down because the county would have to get permission from the commission to do so. In fact, the county already would have to do that because the landmark designation application still is under consideration.
'A wonderful building'
In his speech Thursday, Emmett said the "rumor is that they are going to table" the application, meaning it could be pending indefinitely.
"So long as it is being considered by the historical commission, we're not allowed to do anything to it without getting their sign off," he said.
Asked how much of his opposition to demolition of the historic stadium has to do with political legacy - not being known as the county judge who tore it down - Emmett said "really not much."
"I just think it's a wonderful building that can be used and we need to find a use for it," he said.


The most major upgrade is the addition of a round blue OLED display for at-a-glance track and feature selection even though the size of the player hasn?t increased.
It seems like only days ago that Creative announced the launch of its iPod Shuffle killer the Zen Stone, but here's the follow up; the Zen Stone Plus.
The most major upgrade is the addition of a round blue OLED display for at-a-glance track and feature selection even though the size of the player hasn't increased. Admittedly, the screen is small, but it's enough to see what is going on with your tracks, choose the track that you want next and adjust the volume. It also means you can scroll through the players options, including a newly added FM radio, a recording option, timer and stopwatch.
The Stone Plus comes in six high-gloss colours: black, white, red, blue, pink and green. Creative has upped the ante with a 2GB memory capacity and tracks can be transferred by the included USB cable, without the need for any software.
Creative does offer some simple software via its website so you can manage the player, however you can drag and drop tracks to the Zen Stone Plus like the Zen Stone either via a PC or a Mac as it if was a standard USB drive.
The songs can be organised in favourite artist or playlists folder and can be played in normal or shuffle modes, but the player has lost the skip to next folder option that we really liked on the original Zen Stone.
The Zen Stone Plus takes what we think is an already really good mini MP3 player, adds a screen, adds a radio, and more memory without changing the size or the functionality of the player. Top marks.