text:
\section{Introduction}
\label{sec:1}

     Evaluating the effective degrees of freedom of a physical system is a fundamental  problem in phisics. It is helpful to determine the phases of quantum many-body systems or to study the holographic principle which states that the degrees of freedom of a gravitational system are equal to those of a  system that is one dimension lower  compared to the gravitational system. Entanglement entropy(EE) is a  quantitative measure of the effective degrees of freedom and  the correlation  beween the sub-systems of a physical system ; thus, it has been investigated  from viewpoints of thermodynamics, statistical mechanics, and information theory. Generally, the difficulty in estimating the value of EE depends on the complexity of the structure of  a theory or the form of the sub-systems.  Therefore, despite, the difficulty in ecaluating the EE of general quantum field theory, EEs of two-dimensional conformal field theories are well studied  owing to their  abundant symmetries.  Particularly, the global conformal symmetry determines the EE of a single interval regardless of the  intricacies of the theories. However, when we deal with a two disjoint intervals  sub-system, it is hard to evaluate the EE unless it is a simple  theory such as  the free field \cite{Calabrese:2009ez}.

    The entanglement Renyi entropy(ERE) is a $q$-analogue of the EE. The ERE $S_A(q)$ of the sub-system $A$ is defined as
\begin{align}\label{eq:entanglement Renyi entropy}
    S_A(q) = \frac{1}{1-q} \log \mathrm{tr}_A \left(\rho_A^{\ q} \right),
\end{align}
    where $\rho_A$ is the partial density matrix on $A$. The partial density matrix is normalized as $\mathrm{tr}_A \left(\rho_A \right) = 1$, and then the EE can be defined as $S_A = \lim_{q \to 1}S_A(q)$. The ERE is rewritten as
\begin{align}
    S_A(q) = \frac{1}{1-q} \left(  \log Z_A(q) - q  \log Z \right),
\end{align}
    where $Z_A(q)$ and $Z$  denote the partition function of the $q$-replicated theory and that of the original theory, respectively \cite{Calabrese:2004eu}. The Liouville  conformal field thoery (CFT) has preferable properties for this formulation, which is studied in the context of the non-critical string theory, higher dimensional theory, etc. \cite{Harlow:2011ny}. The Liouville CFT  exhibits the semi-classical limit as the large $c$ limit. In this limit,  the evaluation of EREs is easier because the saddle points of the path integral represent the respective partition functions.  Previous studies have reported that there exist two saddle points for $Z_A(q)$ for the two disjoint intervals system  in the case of the large $c$ Liouville CFT with $q \sim 1$, or in the adjacent interval limit \cite{Faulkner:2013yia,Hartman:2013mia,Asplund:2014coa}.  Then, it has been assumed that only the minimal valued saddle point contributes to $Z_A(q)$.

    In this paper, we  numerically calculate the ERE for $q \sim 1$  using the Heun function. The Liouville CFT has  postulates that the correlation functions with the null vector  satisfy the linear differential equation known as the BPZ equation prefixed with Belavin, Polyakov and Zamolodchikov \cite{Belavin:1984vu}. As the replica partition function $Z_A(q)$ is given by the correlation function of the twist operators,  this correlation function  can be obtianed by solving the BPZ equation.  Further, we show that the solution  is consistent with the structure of the sub-system. For the two disjoint intervals, the BPZ equation is equivalent to the Heun's differential equation. We determine the ERE by imposing an appropriate condition on the monodromy matrices  of the Heun's differential equation, and  find the two saddle points  that were obtained by the previous studies \cite{Faulkner:2013yia,Hartman:2013mia,Asplund:2014coa}. However, we will point out that  these two saddle points should be treated carefully when  applying the $q \to 1$ limit  for the large $c$, because they contribute equally to $Z_A(q)$,  which can be understood by considering the quantum state corresponding to multiple saddle points.

     This paper is  structured as follows. In section~\ref{sec:2}, we will review the replica trick and  establish the relationship between the geometry of the replica manifold and the correlation function  related to the ERE. In section~\ref{sec:3}, we discuss how to treat multiple saddle points. In section~\ref{sec:4}, we show the EREs for $q \sim 1$ based on the  conditions specified in section~\ref{sec:3}. Finally, section~\ref{sec:5} is the conclusion.

\section{Entanglement Renyi entropy  (ERE) and replica trick}
\label{sec:2}

    We will review the replica trick and the ERE of two disjoint intervals $A = [z_1,z_2] \cup [z_3,z_4]$ for a $2$-dimensional CFT on the extended complex plane $\Sigma = \mathbb{C} \cup \{ \infty \}$ \cite{Calabrese:2004eu, Calabrese:2009qy}. To evaluate $\mathrm{tr}_A \left(\rho_A^{\ q} \right)$, it is useful to consider the replica manifold and the replica field theory. Fig.~\ref{fig:ReplicaManifold} shows a schematic picture of the replica manifold $\Sigma_A(q)$ of the ERE for two disjoint intervals, the original manifold $\Sigma$ with the twist operators $\mathcal{T}_q, \mathcal{\tilde{T}}_q$, and the conformal map $w: \Sigma_A(q) \to \Sigma$.  The left panel depicts the replica manifold $\Sigma_A(q)$ which  comprises $q$ sheets  and  a single field. The right panel  depicts the replica field theory defined on $\Sigma$, which  comprises $q$ fields on the single sheet with the twist operators.  The replica field theory provides the equivalent partition function to that of the theory on the replica manifold.
\begin{figure}
\begin{center}
    \includegraphics[width=0.9\linewidth]{ReplicaManifold.pdf}
    \caption{
    The left panel depicts the replica manifold $\Sigma_A(q)$ with $q=3$ sheets. Each sheet is connected to the same -colored lines. The right panel depicts the original manifold $\Sigma$ with the twist operators $\mathcal{T}_q, \mathcal{\tilde{T}}_q$ at the boundary of the sub-system. The conformal map $w$ maps $\Sigma_A(q)$ to $\Sigma$.
    \label{fig:ReplicaManifold}
    }
\end{center}
\end{figure}
    Let $Z$ be the partition function of the CFT on $\Sigma$, and $Z_A(q)$ be partition function of the same CFT on the replica manifold $\Sigma_A(q)$.  Because $Z_A(q)$ is constituted to satisfy $\mathrm{tr}_A \left(\rho_A^{\ q} \right) =  Z_A(q)/Z^q$, the ERE $S_A(q)$ is  calculated using the partition functions as  follows:
\begin{align}\label{eq:replica trick}
    S_A(q) = \frac{1}{1-q} \left(  \log Z_A(q) - q \log Z \right).
\end{align}
    The replica field theory is constructed so that the above ERE is expressed as the following $4$-point correlation function on $\Sigma$ :
\begin{align}\label{eq:n replicated partition function}
    S_A(q) = \frac{2}{1-q} \log \langle \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{T}_q(z_3)\mathcal{\tilde{T}}_q(z_4) \rangle_{\Sigma},
\end{align}
    where $\mathcal{T}_q$ and $\mathcal{\tilde{T}}_q$  represent the primary twist operators with the same conformal weight $h_q = c(q^2-1)/(24q)$.  Because we deal only with the holomorphic part of the $4$-point correlation function, we multiplied it by the factor $2$ in advance.

    For  a $2$-dimensional CFT, there are  some preferable properties to determine the correlation function. On  the original manifold $\Sigma$, a correlation function  incorporating the energy momentum tensor $T(z)$, and the primary operators $O_i(z_i)$ with the conformal weight $h_i$ satisfies  the following relation:
\begin{align}\label{eq:correlation with emt}
    \langle T(z) \prod_{i=1}^N O_i(z_i) \rangle_{\Sigma} = \sum_{i=1}^N \left[ \frac{h_i}{(z-z_i)^2} + \frac{\partial_{z_i}}{z-z_i} \right] \langle \prod_{i=1}^N O_i(z_i) \rangle_{\Sigma}.
\end{align}
    For an arbitrary operator $\mathcal{O}(z)$, the following relation holds from the definition of the twist operators.
\begin{align}\label{eq:correlation with twist}
    \frac{\langle \mathcal{O}(z) \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{T}_q(z_3)\mathcal{\tilde{T}}_q(z_4) \rangle_{\Sigma}}{\langle \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{T}_q(z_3)\mathcal{\tilde{T}}_q(z_4) \rangle_{\Sigma}} = q \langle \mathcal{O}(z) \rangle_{\tilde{\Sigma}_A(q)},
\end{align}
    where $\tilde{\Sigma}_A(q)$ is one of the $q$ sheets of the replica manifold $\Sigma_A(q)$ and we use the same complex coordinate on both $\tilde{\Sigma}_A(q)$ and $\Sigma$. If we obtain the conformal transformation $w(z):\ \Sigma_A(q) \to \Sigma$, the energy momentum tensor on the replica manifold is given by the Schwarzian derivative of $w(z)$ as  follows:
\begin{align}\label{eq:Schwarzian derivative}
    \langle T(z) \rangle_{\tilde{\Sigma}_A(q)} = \frac{c}{12}\left[ \frac{w'''(z)}{w''(z)} - \frac{3}{2}\left(\frac{w''(z)}{w'(z)}\right)^2 \right].
\end{align}
    From Eq.~\eqref{eq:correlation with emt}, \eqref{eq:correlation with twist} and \eqref{eq:Schwarzian derivative}, we obtain the following differential equation which relates the conformal transformation, the conformal weight, and the $4$-point correlation function  as follows:
\begin{align}
    \frac{q c}{12}\left[ \frac{w'''(z)}{w''(z)} - \frac{3}{2}\left(\frac{w''(z)}{w'(z)}\right)^2 \right] = \sum_{i=1}^4 \left( \frac{h_q}{(z-z_i)^2} - \frac{c_i}{z-z_i} \right),
\end{align}
    where $c_i = -\partial_{z_i} \log \langle  \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{T}_q(z_3)\mathcal{\tilde{T}}_q(z_4) \rangle_{\Sigma}$. The global conformal symmetry restricts the correlation function as
\begin{gather}
    \langle \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{T}_q(z_3)\mathcal{\tilde{T}}_q(z_4) \rangle_{\Sigma} = (z_3-z_1)^{-h_q}(z_4-z_2)^{-h_q} \langle  \mathcal{T}_q(0)\mathcal{\tilde{T}}_q(x)\mathcal{T}_q(1)\mathcal{\tilde{T}}_q(\infty) \rangle_{\Sigma} \\
    \sum_{i} c_i = 0,\ \ \ \sum_{i} c_i z_i = 4 h_q,\ \ \ \sum_{i} c_i z_i^2 = 2 h_q \sum_{i} z_i,
\end{gather}
    where $x=(z_4-z_3)(z_2-z_1)(z_4-z_2)^{-1}(z_3-z_1)^{-1}$  denotes one of the cross ratios. Therefore, it is enough to deal with the following equation:
\begin{align}\label{eq:ODE}
    &\frac{w'''(z)}{w''(z)} - \frac{3}{2}\left(\frac{w''(z)}{w'(z)}\right)^2 = \frac{12 h_q}{q c}Q(q,z),\\
    &Q(q,z) = \frac{1}{z^2}+\frac{1}{(z-x)^2}+\frac{1}{(z-1)^2} +2
   \left(\frac{1}{z}-\frac{1}{z-1}\right) + \frac{2 a(q,x)}{z(z-x)(z-1)},
\end{align}
    where $a(q,x)$ is defined as:
\begin{align}\label{eq:derivative of action}
    \frac{ 2 h_q\, a(q,x)}{x(1-x)} = -\partial_x \log \langle  \mathcal{T}_q(0)\mathcal{\tilde{T}}_q(x)\mathcal{T}_q(1)\mathcal{\tilde{T}}_q(\infty) \rangle.
\end{align}
    The solution of the third order non-linear differential equation \eqref{eq:ODE} for $w(z)$ is described as:
\begin{align}\label{eq:conformal map}
    w(z) = \frac{\alpha \Psi_1(z) + \beta \Psi_2(z)}{\gamma \Psi_1(z) + \delta \Psi_2(z)},\ \ \  \alpha \delta - \beta \gamma = 1,
\end{align}
    where $\Psi_1(z),\Psi_2(z)$ are the linearly independent solutions of the following linear differential equation :
\begin{align}\label{eq:Linearized equation}
    \frac{d^2}{dz^2} \Psi(z) + \frac{6 h_q}{q c}Q(q,z) \Psi(z) = 0.
\end{align}
    We can confirm that Eq.~\eqref{eq:conformal map} is the solution of Eq.~\eqref{eq:ODE} by substituting it.  Therefore, the ERE is equivalent to the correlation function of the twist operators, the energy momentum tensor on the replica manifold, the conformal map $w(z):\ \Sigma_A(q) \to \Sigma$,  and the linearly independent solutions of Eq.~\eqref{eq:Linearized equation}.  Thus, the ERE can be obtained by evaluating  any one of these entities. Next, we will evaluate $a(q,x)$ for the semi-classical Liouville CFT. The function $a(q,x)$ is  comparable to the derivative of the ERE. In what follows, we call $a(q,x)$  as the derivative of the ERE. To evaluate the derivative of the ERE $a(q,x)$ for the semi-classical Liouville CFT, we solve Eq.~\eqref{eq:Linearized equation} with the condition that $\Psi(z)$ goes  to the next or previous sheet when crossing the sub-region $A$, as depicted in the left panel of Fig.~\ref{fig:ReplicaManifold}.  Considering the twist operators, this condition  implies that $\Psi(z)$ is a $q$ valued function on $\Sigma$ and the phase of $\Psi(z)$ varies  with $\pm 2\pi/n$ when $\Psi(z)$ goes around the twist operators $\mathcal{T}_q$ and $\mathcal{\tilde{T}}_q$.

    As a practice of the above procedure, let us consider the ERE of the single interval $A = [u,v]$. In this system, two twist operators are inserted at $z_1=u$ and $z_2=v$. From the global conformal symmetry, we can immediately obtain $\langle \mathcal{T}_q(u)\mathcal{\tilde{T}}_q(v) \rangle_{\Sigma} \propto (u-v)^{-2h_q}$ without any other conditions.  Subsequently, we obtain the derivative of the ERE $c_1=-c_2 = 2h_q \, (u-v)^{-1}$ from the definition $c_i = - \partial_{z_i} \log \langle \mathcal{T}_q(u)\mathcal{\tilde{T}}_q(v) \rangle_{\Sigma}$ and the linearized equation corresponding to Eq.~\eqref{eq:Linearized equation} as:
\begin{align}
    &\frac{d^2}{dz^2} \Psi(z) + \frac{6 h_q}{q c}Q(q,z) \Psi(z) = 0,\\
    &Q(q,z) = \frac{1}{(z-u)^2}+\frac{1}{(z-v)^2} -
   \frac{2(u-v)^{-1}}{z-u} + \frac{2(u-v)^{-1}}{z-v}.
\end{align}
     The solution for this equation and the corresponding conformal map  are determined as follows:
\begin{align}
    &\Psi(z) = (z-u)^{ \frac{1}{2}\left(1 \pm \sqrt{1-\frac{24 h_q}{q c}}\right) } (z-v)^{ \frac{1}{2}\left(1 \mp \sqrt{1-\frac{24 h_q}{q c}}\right) },\\
    &w(z) = \frac{\alpha (z-u)^{ \sqrt{1-\frac{24 h_q}{q c}} }+ \beta (z-v)^{ \sqrt{1-\frac{24 h_q}{q c}} }}{\gamma (z-u)^{ \sqrt{1-\frac{24 h_q}{q c}} } + \delta (z-v)^{ \sqrt{1-\frac{24 h_q}{q c}} }}.
\end{align}
     Because $\Psi(z)$ and $w(z)$ are $q$ valued functions on $\Sigma$, the local behavior of the conformal map should  be consistent with $w(z \sim u) \sim (z-u)^{\pm 1/q}$ ; we further find the conformal weight $h_q = c(q^2-1)/(24q)$ again from the  condition $\pm 1/q = \sqrt{1-24 h_q/(q c)}$. Note that if $\alpha = \delta = 1,\  \beta = \gamma = 0$, we retrieve the well known conformal map $w(z) = (z-u)^{1/q}(z-v)^{-1/q}$. This method works extraordinarily in this example because the behavior of the solutions $\Psi(z)$ is completely determined by the conformal weight of the twist operators  owing to the global conformal symmetry. However,  because general multi point correlation functions depend on  the characteristics of each CFT, we can at most determine the local behavior of $\Psi(z)$ without  applying any other conditions related to the global structure of $\Sigma$. Therefore,  an additional condition  is required to be imposed to determine the global behaviour of $\Psi(z)$. We evaluate the ERE for the large $c$ Liouville CFT on the condition that the $\Psi(z)$ in Eq.~\eqref{eq:Linearized equation}  serves as the $1$-point correlation function on the replica manifold.

\section{  ERE with multiple saddle points }
\label{sec:3}

     In this section, we discuss the treatment of the ERE in the semi-classical approximation , within, the saddle points of the partition function represent the path integral in Eq.~\eqref{eq:replica trick}. According to previous studies \cite{Faulkner:2013yia,Hartman:2013mia,Asplund:2014coa}, the derivative of the ERE is given by $a(q \sim 1,x) = 1-x,\ -x$ from Eq.~\eqref{eq:Linearized equation} for the large $c$ Liouville CFT. We often come across the statement \cite{Dong:2016hjy} that the leading term of the derivative of the ERE for the large $c$ limit is proportional to $(q-1)c$, then $(q-1)c$ must be large enough for the saddle point approximation, and only the minimal -valued action contributes to the path integral for the partition function. However, we  show that all the saddle points may comparably contribute to the ERE for $q \sim 1$. At least, we point out that only one of them does not represent the EE. First, we consider the case  of two saddle points for two disjoint intervals system. From the saddle point approximation, the partition function $Z_A(q)$ is described as  follows:
\begin{align}
    Z_A(q) = p_1 \exp\left[ -I_{A,1}(q,x) \right] + p_2 \exp\left[ -I_{A,2}(q,x) \right],
\end{align}
    where $p_1,p_2$ are some constants and $I_{A,1}(q,x),\ I_{A,2}(q,x)$ are classical actions. Even after taking the large $c$ limit, the normalization condition $\lim_{q \to 1} \mathrm{tr}_A \left( \rho_A^{\ q} \right) = 1$ must hold. This means $\lim_{q \to 1} (Z_A(q) / Z^q) = 1$, and then
\begin{align}\label{eq:normalization}
     \lim_{q \to 1} \left( \, p_1 \exp\left[ qI -I_{A,1}(q,x) \right] + p_2 \exp\left[ qI -I_{A,2}(q,x) \right] \,\right)= 1,
\end{align}
    where we assume that $I = - \log Z$ is the unique Euclidean classical action of the original theory ; it is  the $c$  order term.  Because the replica field theory  involves the $q$ replicated field of the original theory, the effective action $I_{A,i}(q,x)$ may  be comparable to $I_{A,i}(q,x) = q I + O(q-1)$ for $q \sim 1$. Thus, $p_1 + p_2 = 1$ from Eq.~\eqref{eq:normalization}. One may concern that the two saddle points merge into the  saddle point of the original theory for $q \sim 1$ and become indistinguishable. Although the values of the two classical actions are infinitely close in  the $q \to 1$ limit, the two saddle points are distinct  considering the path integral because the configuration of the classical fields $\Psi(z)$ are topologically different as  observed in previous studies. Thus, there exist the two distinct saddle points,  out of which   the leading term behaves like  the $c$  order term, as long as $q \neq 1$. Then, each classical action are described as $I_{A,i}(q,x) = q I + b_i(q,x)$, $\lim_{q \to 1} b_i(q,x) = 0$. Therefore, Eq.\eqref{eq:replica trick} for an arbitrary $q$ in the semi-classical limit becomes
\begin{align}\label{eq:ERE with multi saddles}
    S_A(q) = \frac{1}{1-q} \left(  \log \sum_{i=1}^2 p_i \exp \left[-b_i(q,x) \right] \right).
\end{align}
    As a result, the term in the parenthesis is proportional to $(q-1)c$. However, note that the leading terms of the saddle points for $Z_A(q)$ are proportional to $q I$, and then Eq.\eqref{eq:ERE with multi saddles} is derived from the cancellation between $q I$,  which originated from $\log Z_A(q)$ and one from $q \log Z$. Thus, the saddle point approximation is valid for  a large $c$ independent of  the magnitude of $(q-1)c$.

    We obtained Eq.\eqref{eq:ERE with multi saddles} as the ERE in the semi-classical limit with an arbitrary $q$  based on the assumption that the replica field theory has two saddle points for the large $c$. If we adopt  a large enough but finite $c$ for the saddle point approximation and keep $q-1$ finite, we can  consider that only the minimal saddle point contributes to the ERE for  a large $(q-1)c$.  Conversely, if $(q-1)c \sim 0$ with large finite $c$ and $q \sim 1$, the ERE becomes
\begin{align}
    S_A(q \sim 1) \sim \frac{1}{1-q} \log  \sum_{i=1}^2 p_i \left( 1-b_i (q,x) \right)
    \sim \frac{1}{1-q} \left[\log \sum_{i=1}^2 p_i - \sum_{i=1}^2 \frac{p_i b_i(q,x)}{ \sum_{i=1}^2 p_i }\right].
\end{align}
     Owing to the normalization condition  of $p_1+p_2=1$, the EE is  defined as:
\begin{align}\label{eq:entanglement entropy}
    S_A = \lim_{q \to 1} S_A(q) = \lim_{q \to 1} \sum_{i=1}^2 \frac{p_i\, b_i(q,x)}{q-1}.
\end{align}
     Thus, the EE  determined in the semi-classical limit is  a summation of all  the $(q-1)c$  order terms of the classical actions.  The following two nuances should be noted: First, in the semi-classical approximation, the leading terms of the classical action of the two partition functions cancel each other  owing to the structure of the replica theory and the normalization condition of the density matrix.  Second, the $q \to 1$ limit is  adopted so that $(q-1) c \sim 0$ is satisfied. Therefore, because of the exquisite relationship between the two limits, the multiple saddle points equally contribute to the EE. The above cautions are specific to the EE in the semi-classical approximation. Thus,  we do not need to concern about it in other  scenarios, such as the thermal phase transition of physical systems.

    We identify the relation between the derivative of the ERE $a(q,x)$ and the order $(q-1)c$ term of the classical action $b_i(q,x)$, and then determine $p_1$ and $p_2$.  We should pay attention for the quantum state of the replica field theory to relate them. As there are two classical saddle points, it is natural that the replica field theory also has the two quantum states corresponding to respective them. We assume that the quantum state of the replica field theory is expressed as $|\Omega\rangle = \sqrt{p_1} |\Omega_1\rangle  + \sqrt{p_2} |\Omega_2\rangle$, $\langle \Omega_1|\Omega_2\rangle \sim 0$, where $|\Omega_1\rangle$ and $|\Omega_2\rangle$  represent the states corresponding to the respective classical actions in the semi-classical limit.  Subsequently, the partition function is described as $Z_A(q) = p_1 Z_{A,1}(q) + p_2 Z_{A,2}(q)$, where we defined $Z_A(q) = \langle \Omega|\Omega\rangle,\ Z_{A,i}(q) = \langle \Omega_i|\Omega_i\rangle$. We can  associate the weights $p_1$ and $p_2$ to the probability amplitude of $|\Omega_1\rangle$ and $|\Omega_2\rangle$, respectively. According to the above argument, Eq.~\eqref{eq:derivative of action} is written as:
\begin{align}
    \frac{ 2 h_q }{x(1-x)} a(q,x) = -\frac{p_1 \partial_x Z_{A,1}(q) + p_2 \partial_x Z_{A,2}(q)}{Z_A(q)} = -\frac{p_1 \partial_x e^{-b_1(q,x)} + p_2 \partial_x e^{-b_2(q,x)}}{p_1 e^{-b_1(q,x)}+ p_2 e^{-b_2(q,x)}}.
\end{align}
     Because the ERE for $q \sim 1$ is  equivalent tothe summation of the derivative of the classical actions, as  described in Eq.~\eqref{eq:entanglement entropy}, it  is natural that the derivative of the ERE also  decomposes into $a(q,x) = p_1 a_1(q,x) + p_2 a_2(q,x)$ at least for $q \sim 1$. In particular, we relate $a_i(q,x)$ and $b_i(q,x)$ as  follows:
\begin{align}\label{eq:a and b}
    \frac{ 2 h_q }{x(1-x)} a_i(q,x) = - \frac{\partial_x e^{-b_i(q,x)}}{p_1 e^{-b_1(q,x)}+ p_2 e^{-b_2(q,x)}} \sim \partial_x b_i(q\sim 1,x)
\end{align}
     Note that the two candidates of the derivative of the ERE $a(q \sim 1,x) = 1-x,\ -x$ are obtained just by analyzing Eq.~\eqref{eq:Linearized equation} independent of the quantum state. Therefore, we assume
\begin{align}
    p_1 a_1(q \sim 1,x) = 1-x, \ \ \  p_2 a_2(q \sim 1,x) = -x.
\end{align}
    Next, we determine the probability amplitude $p_1$ and $p_2$ because we can only obtain $p_i a_i(q,x)$  and not $a_i(q,x)$ itself.  Furthermore, the ERE is described by the $4$-point function of the twist operators Eq.~\eqref{eq:n replicated partition function}. Consider the $4$-point correlation function $G_{1234}(x) = \langle \phi_1(0)\phi_2(x)\phi_3(1)\phi_4(\infty) \rangle_{\Sigma}$, where $\phi_i$ is a general operator.  Because $G_{1234}(x)$ is independent of the way of the operator product expansion,  and $G_{1234}(x)$  exhibits the crossing symmetry $G_{1234}(x) = G_{3214}(1-x)$.  The first and the third operators, in the ERE in Eq.~\eqref{eq:n replicated partition function},  are identical twist operators; therefore, we obtain $G_{1234}(x) = G_{3214}(x)$  in addition to $G_{1234}(x) = G_{1234}(1-x)$. Therefore, the ERE $S_A(q)$ in Eq.~\eqref{eq:ERE with multi saddles} is invariant with the replacement $x \to 1-x$; thereby, allowing $p_1 = p_2 = 1/2$  to true. Finally, the EE of the two disjoint intervals for the large $c$ Liouville CFT from Eq.~\eqref{eq:entanglement entropy} is
\begin{align}
    S_A = \lim_{q \to 1} \frac{4 h_q}{q-1} \int \frac{ 1-2x}{x(1-x)} dx  = \frac{c}{3} \log \frac{x(1-x)}{\epsilon^2},
\end{align}
    where $\epsilon$  denotes the UV cut off scale.  Consequently, the obtained EE is equivalent to that of the free compactified boson at the leading order of the large $c$. Thus, it  shall not be in contradiction to any  postulate of  the CFT. Note that we do not need the weights $p_i$ to calculate the EE, and we exploited the symmetry between the two saddle points to determine the weights $p_i$. However, if there are more than two saddle points, we need some extra information to evaluate the weights $p_i$ and the ERE.

\section{  Determination of ERE for the semi-classical Liouville CFT}
\label{sec:4}

     In this section, we see that $\Psi(z)$ in Eq.~\eqref{eq:Linearized equation} should behave as the $1$-point correlation function on the replica manifold for the large $c$ Liouville CFT,  and then determine the ERE of the two disjoint intervals. For the Liouville CFT, the BPZ equation helps us to analyze the structure of the correlation functions. Let $\psi_\chi(z)$ denote the operator corresponding  to the level $2$ light null vector with the conformal weight $h_{\chi}$ ; wherein, the following BPZ equation holds :
\begin{align}\label{eq:BPZ equation}
    \left[ \frac{3 \partial^2_z}{2(2h_{\chi}+1)} - \sum_{i=1}^4 \left( \frac{h_q}{(z-z_i)^2} + \frac{\partial_{z_i}}{z-z_i} \right) \right] \langle \psi_\chi(z) \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{T}_q(z_3)\mathcal{\tilde{T}}_q(z_4) \rangle = 0
\end{align}
    As we treat  the $q$-replicated Liouville CFT, the central charge is $q$ times  that of the original theory, that is, $h_{\chi} = (5-qc+\sqrt{(qc-1)(qc-25)})/16$.  We can choose $(z_1,z_2,z_3,z_4)=(0,x,1,\infty)$ without  the loss of generality. In the large $c$ semi-classical limit, we can rewrite this equation  in a simple form  through the following steps. The conformal weight $h_{\chi}$ is $h_{\chi} = -1/2 - 9/(2qc) + O(c^{-2})$ and $\psi_\chi(z)$ is a light operator  whose expectation value  can be considered as a $1$-point correlation function $\Psi_\chi(z)$ on the replica manifold. This means that the above $5$-point correlation function behaves as  follows:
\begin{align}
    &\Psi_\chi(z) = \frac{\langle \psi_\chi(z) \mathcal{T}_q(0)\mathcal{\tilde{T}}_q(x)\mathcal{T}_q(1)\mathcal{\tilde{T}}_q(\infty) \rangle_{\Sigma}}{\langle \mathcal{T}_q(0)\mathcal{\tilde{T}}_q(x)\mathcal{T}_q(1)\mathcal{\tilde{T}}_q(\infty) \rangle_{\Sigma}} \\
    &\Longrightarrow \langle \psi_\chi(z) \mathcal{T}_q(0)\mathcal{\tilde{T}}_q(x)\mathcal{T}_q(1)\mathcal{\tilde{T}}_q(\infty) \rangle_{\Sigma} = \Psi_\chi(z) \, \sum_{i} p_i \, e^{-I_{A,i}(q,x)}.
\end{align}
    Thus, we will deal with the following equation assuming that $\Psi_\chi(z)$ behaves as the $1$-point correlation function on the replica manifold:
\begin{align}\label{eq:large c BPZ}
    &\frac{d^2}{dz^2} \Psi_\chi(z)  + \frac{q^2-1}{4q^2}Q(q,z) \Psi_\chi(z)  = 0,\\
    \label{eq: Q}
    &Q(q,z) = \frac{1}{z^2}+\frac{1}{(z-x)^2}+\frac{1}{(z-1)^2} +2
   \left(\frac{1}{z}-\frac{1}{z-1}\right) + \frac{2 a(q,x)}{z(z-x)(z-1)},\\
   \label{eq: ai}
    &a(q,x) = -\frac{12 q}{c(q^2-1)}x(1-x) \partial_x \log Z(q,x).
\end{align}
    $\Psi_\chi(z)$ satisfies the same differential equation as Eq.~\eqref{eq:Linearized equation}, but now we have  an additional global condition that $\Psi_\chi(z)$ behaves as a $1$-point correlation function on the replica manifold.  Furthermore, we evaluate $a(q,x)$ for $q \sim 0$ first as we can find  an analytical expression of $\Psi_\chi(z)$  using the WKB  approximation, and then numerically evaluate $a(q,x)$ for $q \sim 1$.

    First, as just a practice, we calculate the ERE for $q \sim 0$  using the WKB method because it  enables in for understanding the relation between the structure of the replica manifold and the global behavior of $\Psi_\chi(z)$ on it. Consider the following WKB solution of Eq.~\eqref{eq:large c BPZ} in the leading order of the WKB approximation for $q \sim 0$:
\begin{align}
    \Psi_\chi(z) = \frac{1}{Q(q,z)^{1/4}} \exp \left[ \pm \frac{1}{2q} \int^z \sqrt{Q(q, \zeta)} d\zeta \right].
\end{align}
    As we have the integral expression for $\Psi(z)$, it is easy to analyze its global behavior which is determined by the residues of $\sqrt{Q(q,z)}$. Note that we can rewrite $\sqrt{Q(q,z)}$ as
\begin{align}
    \sqrt{Q(q,z)} = \frac{\sqrt{z^4 -2z^3 +2z^2 -2xz +x^2 + 2 a(q,x) z(z-x)(z-1) }}{z(z-x)(z-1)}.
\end{align}
    The residues of $\sqrt{Q(q,z)}$ at $z=0,x,1$ are $\pm 1$ independent of $a(q,x)$. From the requirement that $\Psi(z)$ behaves as a $1$-point correlation function on the replica manifold as depicted in Fig.\ref{fig:ReplicaManifold}, $a(q,x)$ is determined so that $\sqrt{Q(q,z)}$  transforms into a rational function and its Riemann surface is single sheeted, that is,  $\mathrm{Res}\sqrt{Q(q,z=0)}=-\mathrm{Res}\sqrt{Q(q,z=x)}=\mathrm{Res}\sqrt{Q(q,z=1)}$ should holds. Therefore, we find the unique derivative of the ERE $a(q,x) = 1-2x$, and then, $\sqrt{Q(q,z)}$ and the ERE for $q \sim 0$ is  determined as follows:
\begin{gather}
    \sqrt{Q(q \sim 0, z)} = \pm \left( \frac{1}{z} - \frac{1}{z-x} + \frac{1}{z-1} \right), \\
    S_A(q \sim 0) = \lim_{q \to 0}\frac{4 h_q}{q-1} \int \frac{1-2x}{x(1-x)} dx  = \frac{c}{6q} \log \frac{x(1-x)}{\epsilon^2}.
\end{gather}
    We can express the conformal map as $w(z) = z^{\frac{1}{q}}(z-x)^{-\frac{1}{q}} (z-1)^{\frac{1}{q}}$ ; we obtain the energy momentum tensor for $q \sim 0$ as  follows:
\begin{align}
    \frac{12}{q\,c} T(z)
    &=\frac{q^2-1}{2 q^2 z^2}
    +\frac{q^2-1}{2 q^2 (z-x)^2}
    +\frac{q^2-1}{2 q^2 (z-1)^2} - \frac{\left(q^2-1\right) (x+z-1)}{q^2z (z-x) (z-1) } -\frac{6 (x-1) x}{(z^2 - 2xz + x)^2}\nonumber\\
    &\sim -\frac{1}{2q^2} \left[ \frac{1}{z^2} + \frac{1}{(z-x)^2} + \frac{1}{(z-1)^2} + 2\left(\frac{1}{z} -\frac{1}{z-1}\right) + \frac{2(1-2x)}{z (z-x) (z-1)} \right]
\end{align}
     The form of this energy momentum tensor  is consistent  with Eq.\eqref{eq:correlation with emt} for $q \sim 0$,  that is, it has the same poles. For  a finite $q$, the sub-leading terms of the WKB solution may cancel the extra poles at $z^2 - 2xz + x=0$.  Additionally, for $a(q,x) = \pm 1$, $\sqrt{Q(q,z)}$ also becomes a rational function :
\begin{align}
    &a(q,x) = 1 \ \Longleftrightarrow\ \sqrt{Q(q \sim 0, z)} = \pm \left( \frac{1}{z} - \frac{1}{z-x} - \frac{1}{z-1}\right),\\
    &a(q,x) = -1 \ \Longleftrightarrow\ \sqrt{Q(q \sim 0, z)} = \pm \left( \frac{1}{z} + \frac{1}{z-x} - \frac{1}{z-1}\right).
\end{align}
    From the relative sign of the poles, the $4$-point correlation functions corresponding to them are  given as:
\begin{align}
    &a(q,x) = 1\ \Longleftrightarrow\ \langle \mathcal{T}_q(z_1)\mathcal{\tilde{T}}_q(z_2)\mathcal{\tilde{T}}_q(z_3)\mathcal{T}_q(z_4) \rangle_{\Sigma}, \\
    &a(q,x) = -1 \ \Longleftrightarrow\ \langle \mathcal{T}_q(z_1)\mathcal{T}_q(z_2)\tilde{\mathcal{T}}_q(z_3)\tilde{\mathcal{T}}_q(z_4) \rangle_{\Sigma}.
\end{align}
    This practice clearly demonstrates the relation between each $4$-point correlation function and the geometry of each replica manifold. We may be able to precisely analyze by considering the higher order term of the WKB solution.  Thus, we confirm the one-to-one correspondence between each saddle point $a_i(q,x)$ and each replica manifold. However, we obtain multiple saddle points for  the general $q$.

    Second, we  consider the $q \sim 1$ case. Let $\Phi(z) = g(x)\Psi_\chi(z) $ and $g(z) = z^{\frac{q-1}{2q}}(z-x)^{\frac{q+1}{2q}}(z-1)^{\frac{q-1}{2q}}$, then Eq.~\eqref{eq:large c BPZ} is transformed into the Heun's differential equation  as follows:
\begin{align}\label{eq:Heun standard}
    &\frac{d^2}{d z^2} \Phi(z) + \left(\frac{\gamma}{z}+\frac{\epsilon}{z-x}+\frac{\delta}{z-1}\right) \frac{d}{d z} \Phi(z) + \frac{\alpha \beta z - p }{z(z-x)(z-1)} \Phi(z) = 0,\\
    &\alpha = 1,\
    \beta = 1-\frac{1}{q},\
    \gamma = 1-\frac{1}{q},\
    \delta = 1-\frac{1}{q},\
    \epsilon = 1+\frac{1}{q},\\
    &p = \frac{q-1}{2 q^2} \left[ 1-2x+q-(q+1) a_i(q,x) \right]
\end{align}
    The solution of this equation is called as the Heun function. The Heun's differential equation has four regular singular points at $z=0,x,1,\infty$ and the Frobenius solutions of the Heun's differential equation  are known as the local Heun functions. For example, two independent local Heun functions around $z=0$ can be expressed as :
\begin{align}
    &\Phi(z \sim 0) \sim \text{HeunG}[x,p,\alpha ,\beta ,\gamma ,\delta ,z],\\
    &\Phi(z \sim 0) \sim z^{1-\gamma } \text{HeunG}[x,p+(1-\gamma ) (\delta  x+\epsilon ),\alpha -\gamma +1,\beta -\gamma +1,2-\gamma ,\delta ,z],
\end{align}
    where the local Heun function is normalized as $\text{HeunG}[x,p,\alpha ,\beta ,\gamma ,\delta ,z=0]=1$ \cite{HeunG}. We denote a local Heun function near $z=z_i$ with the characteristic exponent $s$ as $y_{z_i}^{s}(z)$. The connection matrix describes  the relationships between the local Heun functions. For example, $y_{x}^{s}(z)$ and $y_{0}^{s}(z)$ are connected by the connection matrix $C_{x0}$ as  follows:
\begin{align}\label{eq:connection matrix}
    \left(
    \begin{array}{c}
    y_{x}^{0}(z)\\
    y_{x}^{1-\epsilon}(z)
    \end{array}
    \right)
    =
    \frac{1}{W(y_{0}^{0},y_{0}^{1-\gamma})}
    \left(
    \begin{array}{cc}
    W(y_{x}^{0},y_{0}^{1-\gamma}) & W(y_{0}^{0},y_{x}^{0}) \\
    W(y_{x}^{1-\epsilon},y_{0}^{1-\gamma}) & W(y_{0}^{0},y_{x}^{1-\epsilon})
    \end{array}
    \right)
    \left(
    \begin{array}{c}
    y_{0}^{0}(z)\\
    y_{0}^{1-\gamma}(z)
    \end{array}
    \right),
\end{align}
    where $W(y_{x}^{0},y_{x}^{1-\epsilon}) = y_{x}^{0}(z)\partial_z y_{x}^{1-\epsilon}(z) - \partial_z y_{x}^{0}(z) y_{x}^{1-\epsilon}(z)$ is the Wronskian of $y_{x}^{0}(z)$ and $y_{x}^{1-\epsilon}(z)$ and the others are the same. The ratio of these Wronskians  attains a constant value with respect to $z$,  contrary to the Wronskians themselves. We utilized the Mathematica to calculate these Wronskians, see \cite{Hatsuda:2020sbn}.

    The derivative of the ERE $a(q,x)$ determines the connection matrices.  Additionally, we  need to find the condition that the connection matrices must satisfy. To formulate it, consider the paths $P_{0x}$ and $P_{1x}$ which encircle the interval $[0,x]$ or $[x,1]$  once in  the counterclockwise  direction. And let $R_{0} = R_{x}^{-1} = R_{1} = R_{\infty}^{-1} = \mathrm{diag}(1,\exp[2\pi i/q])$ and the connection matrix $C_{x0}$ be given by Eq.~\eqref{eq:connection matrix} and the others be  defined in the same manner. Then, the analytic continuation along $P_{0x}$ for the local Heun functions $y_{x}^{0}$ and $y_{x}^{1-\epsilon}$ are described as :
\begin{align}
    \left(
    \begin{array}{c}
    y_{x}^{0}(z)\\
    y_{x}^{1-\epsilon}(z)
    \end{array}
    \right)
    =
    M_{0x}
    \left(
    \begin{array}{c}
    y_{x}^{0}(z)\\
    y_{x}^{1-\epsilon}(z)
    \end{array}
    \right),
\end{align}
    where we define the monodromy matrix $M_{0x} = C_{x0}R_{0}C_{0x}R_{x}$ as depicted in Fig.~\ref{fig:path}.
\begin{figure}[ht]
\begin{center}
    \includegraphics[width=0.85\linewidth]{path.pdf}
    \caption{
        The dots represent $z=0,x,1,\infty$ on $\Sigma$ from left to right. The analytic continuation along each magenta line  is described as  a matrix,  such as $C_{x0},\cdots$ and $R_{0},\cdots$. The paths $P_{0x}$ and $P_{1x}$ correspond to the monodromy matrices $M_{0x} = C_{x0}R_{0}C_{0x}R_{x}$ and $M_{1x} = C_{x1}R_{1}C_{1x}R_{x}$, respectively.
        \label{fig:path}
    }
\end{center}
\end{figure}
     Similarly, the analytic continuation along $P_{1x}$ is expressed  comparable to the other monodromy matrix $M_{1x} = C_{x1}R_{1}C_{1x}R_{x}$.  One may hope that both the monodromy matrices  transforms into the identity matrix like the WKB analysis for $q \sim 0$. However, both cannot transforms into the identity matrix simultaneously for general $q$ whatever $a(q,x)$ is chosen. Instead, one of  two should be the identity matrix,  also known as the Schottky uniformization \cite{Faulkner:2013yia,Krasnov:2000zq,Zograf_1988}.  Moreover, it is trivial  for the analytic continuation along the path which encircles all the four regular singular points  once in  the counterclockwise  direction. Therefore, $M_{0x}=I$ is equivalent to $M_{\infty 1}=I$ because $C_{x1} M_{\infty 1} C_{1x} M_{0x}=I$ and $C_{1x} C_{x1} = I$.  Comparably, $M_{1x}=I$  implies $M_{0 \infty}=I$ ; thus, it is  sufficient to deal with the monodromy matrices $M_{0x}$ and $M_{1x}$. On this condition, the monodromy matrices $M_{0x}$ and $M_{1x}$ are commutative. Thus, if we perform the analytical continuation via $q$ times $P_{0x}$ and $q$ times $P_{1x}$ for integer $q$, $\Phi(z)$  retains its original value  because this is the first time that $\Phi(z)$ is back to the starting point from the viewpoint of the replica manifold. For $0 < q \in \mathbb{Q}$, let $q = t/u$ with $t,u \in \mathbb{Z}^+$,  while considering the analytical continuation via $u$ times $P_{0x}$ and $u$ times $P_{1x}$ in random order, the same discussion holds because $\Phi(z)$ is $t$ times back to the starting point. Therefore, we accept the Schottky uniformization for an arbitrary $q \in \mathbb{R}$.

    For  an arbitrary $q$ near $x=0$ or $x=1$, the derivative of the ERE behaves as $a(q,x \sim 0) \sim 1$ or $a(q,x \sim 1) \sim -1$, respectively \cite{Hartman:2013mia,Asplund:2014coa}. Then, we regard the former as $p_1 a_1(q,x)$ if there are only two saddle points. We numerically calculate $p_1 a_1(q,x)$  in case all the components of the commutation relation between the two monodromy matrices $[ M_{0x},M_{1x} ]$ vanish  simultaneously. Then, we obtain the ERE from Eq.\eqref{eq:ERE with multi saddles} and \eqref{eq:a and b} with $a_1(q,x) = -a_2(q,1-x)$.
\begin{figure}[ht]
\begin{center}
    \includegraphics[width=\linewidth]{ERE.pdf}
    \caption{
        The left panel shows $p_1 a_1(q,x)$ for  $q=0.6,0.8, 1.0, 1.2, 1.4$. The right panel shows the corresponding ERE normalized as $S_A(q)=1$ at $x=1/2$ with the central charge $c=1$ and the UV cutoff  $\epsilon=0.1$.
        \label{fig:ERE}
    }
\end{center}
\end{figure}
    Fig.~\ref{fig:ERE} shows $p_1 a_1(q,x)$ and the ERE  $S_A(q)$ for $q \sim 1$. For $q \to 1$, we can  consider $p_1 a_1(q \to 1,x) \to 1-x$. As mentioned before, we obtained the same EE as  that of a compactified boson. Note that the central charge $c$ should be large enough  because Eq.\eqref{eq:ERE with multi saddles} is based on the saddle point approximation. For $(q-1)c \sim 0$, the EREs depend on $c$ only linearly, and then the EREs with $c=1$ in Fig.~\ref{fig:ERE} is meaningful. It is difficult to compute the ERE not for $q \sim 1$. In particular for $q < 0.5$, the number of saddle points  increases with a decreasing $q$, and we cannot determine the weights of the contribution for each saddle point to the ERE. Moreover, we cannot calculate each saddle point for small $q$  owing to the lack of numerical accuracy. The WKB analysis  could be considered to calculate the ERE for this region.

\section{Conclusion}
\label{sec:5}

    In this study, we reviewed the relationship between the ERE and the geometrical structure of the replica manifold and saw that some additional conditions must be imposed to determine the ERE of two disjoint intervals system in general. Then, we  considered the treatment of the EE in the semi-classical approximation in general. Because of the exquisite relationship between the large $c$ and $q \to 1$, we pointed out that the multiple saddle points contribute equally to the EE. The leading terms of the classical action of the two partition functions $Z_A(q)$ and $Z$ for large $c$ cancel each other due to the structure of the replica theory and the normalization condition of the density matrix. For the  case of general ERE,  the method to evaluate the contribution weights of each saddle point  is not known; and for the EE we do not need to know the weights as shown in Eq.~\eqref{eq:entanglement entropy}.  Thus, we numerically evaluated the ERE of the two disjoint intervals for the large $c$ Liouville CFT for $q \sim 1$ by analyzing the BPZ equation  by satisfying the criterion that its solution behaves like a $1$-point correlation function on a replica manifold. This condition is expressed by the condition that one of the monodromy matrices  transforms into the identity matrix for any real number $q$.

    In future work, it  shall be of interest to reconsider ERE in other  scenarios and entanglement measures. For  instance, there is a growing interest in the reflected Renyi entropy,  which signifies that the corresponding replica manifold  exhibits a rather complex geometry \cite{Dutta:2019gen}.  Additionally, we can consider the ERE of a single intercal on the torus as it is also well known for the expression of the Heun's differential equation. Conversely, it would be interesting to evaluate the higher order terms of the WKB method and the large $c$. Considering the higher order terms of the WKB method for finite $q$, we may check the consistency between the WKB method and the numerical method for  the ERE of the two disjoint intervals. For higher order corrections of large $c$, it may be necessary to evaluate the contribution of the cross term between multiple quantum states corresponding with each saddle point  in case of multiple saddle points.

\begin{acknowledgments}
    The author J.T. was financially supported by JSPS Fellows (22J14390). The author Y.N. was supported in part by JSPS KAKENHI Grant No. 19K03866.
\end{acknowledgments}

\bibliographystyle{JHEP}

redpajama_set_name: RedPajamaArXiv


text:
G-Dragon's Fans Donates to Charity to Celebrate GD's Birthday
Big Bang G-Dragon's fans donated to charity to celebrate the rapper's birthday.
On August 18, Kwon Ji Yong Supporters donated a total of 24 million won(about 23,500 dollars) to charities. The fan club donated 6 million won to Seoul University Children's Hospital, 9 million won to Purme Foundation Children's Rehabilitation Hospital, 2.4 million won to Moderato Scholarship Foundation, and 3 million won to Love Aid foundation.
Always-GD and Number-G also donated 8.18 million won to Seungil Hope Foundation, while G-Dragon himself donated 81.8 million won to the foundation. Park Seungil, who once was a basketball player, is now suffering from Lou Gehrig's Disease, and Jinusean's Sean introduced him to G-Dragon years ago.
Meanwhile, G-Dragon performed at YG Family Concert held in Seoul on August 15, and stated, "I'll do my best to be able to release Big Bang's new album within this year," increasing fans' expectations.
boy group gd gdragon News YG

redpajama_set_name: RedPajamaCommonCrawl


text:
Der S.E.A. Write Award oder auch Southeast Asian Writers Award ist ein wichtiger Literaturpreis im südostasiatischen Raum. Er wird für ein herausragendes Werk oder als Anerkennung des Gesamtwerkes eines Autors verliehen.

Er wird seit 1979 jährlich im Oktober in Bangkok an Schriftsteller aus den ASEAN-Staaten vergeben, dazu gehören Brunei, Indonesien, Kambodscha, Laos, Malaysia, Myanmar, die Philippinen, Singapur, Thailand and Vietnam.

Liste der Preisträger:

Weblinks
 Offizielle Homepage (Englisch und Thai)

Literaturpreis
Südostasien
Erstverleihung 1979

redpajama_set_name: RedPajamaWikipedia


Most popular carpet for and inspirations ideas pictures with bedroom options are best with most popular carpet for also its 2017 picture best carpets home design ideas pineloon cool. These are some purposes and can nevertheless be an incredible selection.
4 directory of most popular carpet for bedrooms outlined in this article to help your comprehensive the plan without any problem.


text:
GameConfiguration::GameConfiguration(void)
{
}

GameConfiguration::~GameConfiguration(void)
{
}

void GameConfiguration::getConfiguration(void)
{
}

redpajama_set_name: RedPajamaGithub


The human resources ("HR") function is at the forefront of a company's efforts with respect to two of the key elements of organizational design—people and organizational structure. While HR practices differ around the world, it would seem to be universally true that in order for companies to successfully achieve their strategic goals and objectives they must strive to attract, motivate, and retain those employees who are best qualified to carry out the necessary activities of the company and make sure that they are placed into the right spots in the most effective organizational structure. The traditional role of HR was perceived as being largely administrative—recruiting and interviewing prospective employees, administering benefit plans and writing policies—and is often referred to as the "personnel administration" approach; however, forward thinking companies now realize that the HR function must be part of the company's strategic planning activities and that HR leaders must engage in what is common referred to as "human resources management", or "HRM", strategy, and proactively suggest new policies and initiatives to the senior executives of the company to ensure that the company has access to appropriate knowledge and talent in each of the markets where it is active. The personality profile for HR managers and specialists has also changed radically in recent years due to the fact that the HR function now provides a wide array of services to persons throughout the company's organizational structure including training and development, job analysis, oversight of workplace conditions and mediation of disputes between employees and the company.


text:
Inaugural Cape 1000 ready to host the cream of the classic crop
13 January 2022 - 11:43 By Staff Writer
Some 60 classic cars are expected to take part in a new Mille Miglia-esque race to be held in Cape Town.
The Cape 1000 is SA's own "Mille Miglia"-styled classic car event. It is set to take place from March 8 to 13 in the Western Cape.
"We're very pleased with the response to our call for entries," says organiser Ross Crichton. "For this inaugural 1,000 Mile Tribute event, we decided on a limit of 60 cars, and I'm pleased to announce that just about half of those available slots have already been snapped up"
Some of the exotic cars confirmed to compete in the four categories of the event, which pays homage to the iconic Mille Miglia race that took place in Italy from 1927 to 1957, include a 1961 Mercedes-Benz 300 SL Roadster, a 1955 Jaguar XK140 OTS, 1974 Porsche 911 2.7 RS and a 1990 Ferrari F40.
The event, which kicks off with registration at the V&A Waterfront on March 8, will be run over the course of four days. The vehicles will first head for Hermanus along some of SA's most scenic coastal routes. The route then goes through Franschhoek and St Helena Bay on the West Coast, before returning to Cape Town the after day.
It will include regularity stages, noncompetitive track events and grand touring.
A 1990 Ferrari F40 is just one of the jaw-dropping cars on the entry list.
Entries are limited to certain brands, models and by invitation or application (and subject to approval by the admissions committee to ensure a line-up of top-end entrants). The categories are:
1927-1957: Mille Miglia Specific
Pre-1976: Classic
1977-1996: Modern Classic
1997-2021: Sports
"We call on any automotive enthusiast interested in participating in this exclusive event to get in touch as soon as possible to avoid disappointment," says Crichton. "We have also negotiated special rates for owners who need to transport competing vehicles from Johannesburg to Cape Town and back," he adds.
The Cape 1000 is organised by Super Car Lifestyle which also hosts the Round Table Cannonball Run Africa, Concours SA together with Cars.co.za, SCL Travel and Silvercrest Super Car Insurance.
Beneficiaries of the Cape 1000 include the Motorsport Legends Benevolent Fund and Quadpara Association of SA (QASA).
Registrations are open on www.cape1000.com or by contacting info@cape1000.com. Entries close on January 24.
Star ratings and the importance of safety when buying a car
Vehicle occupant safety is a subject of enduring relevance.
CLASSICS | The oddball VW MPV you rarely hear of
In the early 1960s, the German postal authority asked Volkswagen to build a bespoke vehicle for transporting mail and packages.
Four used diesel hatchbacks that offer maximum range for your money
Thanks to the extortionate price of fuel, adding a couple of hundred bucks to your tank, it doesn't get you nearly as far as it once did.
Meet the CityBug – SA's smallest and most affordable electric car New Models
WATCH | Avoid costly repairs: Five tips to extend the life of your car's engine Features
WATCH | New 2022 Ford Ranger shows off its impressive versatility New Models
Enjoying the evolution of the Toyota Corolla Features
Yes, you can now renew your vehicle licence disc at Pick n Pay news


THESSALONIKI, Greece (AP) — Thousands of police officers have been deployed to Greece's second-largest city ahead of anti-austerity protests at a trade fair where Prime Minister Alexis Tsipras is set to outline his post-bailout economic policy platform.
The Thessaloniki police department said more than 4,000 officers will be on duty in the northern city on Saturday as many previous protests turned violent.
U.S. personnel are helping with security since Commerce Secretary Wilbur Ross will be in Thessaloniki with American corporate delegations participating in the annual fair. The U.S. is the honored country this year.
Tsipras is due to meet Ross later Friday. He will inaugurate the fair on Saturday, delivering a keynote speech.
Trade unions, left-wing and anarchist groups plan protests around the fair venue, where right-wing groups also are set to rally.


text:
RAYMOND v. RAYMOND
Family Court, Albany County, New York.
Marie E. RAYMOND, Petitioner, v. Renee M. RAYMOND, Respondent.
Decided: September 19, 1997
Arthur Kahn, Kingston, for respondent. Ian R. Arcus, Albany, for petitioner. Robert T. Entin, Albany, Law Guardian.
This matter is before the court on motion of the Respondent (Mother) for disqualification of Petitioner's (Grandmother's) Attorney due to a potential conflict of interest.
By the underlying petition, the Mother seeks to end the Grandmother's visitation with the five-year-old grandchild because of a criminal indictment pending against the Grandfather. The indictment charges the Grandfather with two felony counts of sexual abuse of the Child. The Mother is asking that there be no contact between her Child and either of the Grandparents.
The basis for this motion to disqualify is that the Attorney representing the Grandmother in this visitation proceeding is also representing the Grandfather in the criminal matter. This dual representation, the Mother argues, creates a conflict of interest which should disqualify the Attorney from representing the Grandmother in this proceeding. She contends that the Attorney's independent professional judgement will be adversely affected by representing two clients with interests that appear to be similar but do, in fact, conflict. This conflict, the Mother contends, could impede a considered finding of the Child's best interest. The Grandmother argues that the representation will not be adverse because the two proceedings are unrelated.1
The Grandmother claims that no conflict of interest could arise in this case because she is not seeking any visitation on behalf of the Grandfather nor to have visitation take place in his presence. Accordingly, there would be no reason why the Grandfather would be called as a witness to testify in this proceeding. She also adds that, even if the Mother wishes to call the Grandfather to testify, he could not be compelled to testify to any incriminating acts. The Mother contends that the Grandfather may be compelled to testify and the breadth of his testimony would be determined by the Court while the Grandfather is on the witness stand. The Grandfather, in the end, could be prejudiced by his testimony given in the Family Court proceeding, even if not criminally incriminating. This would require decisions by the Grandparents' Attorney of which strategies to use to best protect the Grandfather's interests while potentially foregoing the interests of the Grandmother, or vice versa.2
An attorney's loyalty may be divided between two clients as long as it will not "adversely affect * * * [his or her] independent professional judgment in behalf of [his or her present] client [or clients]" (Code of Professional Responsibility DR 5-105[A], [B], [C]; 22 NYCRR 1200.24[a]; [b]; [c]; Greene v. Greene, 47 N.Y.2d 447, 418 N.Y.S.2d 379, 391 N.E.2d 1355). Yet, the mere fact that an attorney may represent two clients simultaneously does not make it proper or advisable to do so (Allegretti-Freeman v. Baltis, 205 A.D.2d 859, 613 N.Y.S.2d 449). An unethical conflict will arise when an attorney finds himself aiding one client against the other or is compelled to choose between them (Field v. Moore, 189 A.D. 709, 178 N.Y.S. 842). An attorney may represent multiple clients when it is so obvious that he or she may represent the clients' interests adequately and when the clients' consents to the representation are after full disclosure of possible effects on the attorney's independent professional judgement on behalf of both clients. (Code of Professional Responsibility DR 5-105[A], [B], [C]; 22 NYCRR 1200.24[a]; [b]; [c] ).3 However, doubts as to the existence of an asserted conflict of interest are to be resolved in favor of disqualification (Flushing Sav. Bank v. FSB Properties, Inc., 105 A.D.2d 829, 482 N.Y.S.2d 29; Matter of Kelly, 23 N.Y.2d 368, 296 N.Y.S.2d 937, 244 N.E.2d 456).4
In this case, both clients may not be adequately represented without infringing upon the other's interests. If the Grandfather is called to testify in the visitation proceeding, the Attorney may find himself in a situation of divided loyalty; namely that the testimony of one client will likely prejudice the other client. For example, at what point should the attorney advise the Grandfather to assert his Fifth Amendment rights if, at that point, the assertion would be adverse to the Grandmother's case? What if the Grandfather's testimony was essential to rebut a claim of the Mother's but the Attorney was hesitant to call him to testify for fear of opening the door to an incriminating answer? These are the types of circumstances that can occur in the heat of trial and, being unpredictable, become unwaivable on the client's part.
The Grandmother has also requested psychological evaluations of the Child, the Mother, the Mother's paramour, and of herself to determine what is in the best interest of the child. The strategy that the Attorney has chosen in requesting these evaluations may hurt his other client, the Grandfather, in his criminal trial or may be calculated to aid the Grandfather while harming the Grandmother's visitation interest. Again, this creates the potential divide of the Attorney's loyalty between his clients.
It is clear that this Attorney, by representing both Grandparents, even though in different proceedings, will be put in a position of choosing or aiding one client against the other when making strategic decisions such as deciding what witnesses to call and what evaluations to order. A strategy chosen to aid one client may adversely affect the other. Furthermore, while this inherent conflict may be waived in accordance with DR 5-105(C), the Court finds that the Attorney cannot adequately represent the interests of one client without adversely affecting the other client. It is inevitable that choices will have to be made which are inherent in this conflict and, accordingly, unwaivable.
This case presents a quite different situation from that described in the recent Court of Appeals decision in Tekni-Plex Inc. v. Meyner and Landis, 89 N.Y.2d 123, 651 N.Y.S.2d 954, 674 N.E.2d 663. Tekni-Plex involved an analysis under DR 5-108(A)(1) (22 NYCRR 1200.27[a][1] ) regarding the representation of a new client against a former client in "a substantially related matter in which that person's interests are materially adverse to the interests of the former client." While the guidance of Tekni-Plex is helpful, the analysis in this case is somewhat different. In this case, although the matters are substantially related, the Grandparents do not consider their interests as materially adverse. The Court is holding, however, that the dual representation by the Attorney of both Grandparents is inherently conflictual and materially adverse to the best interest of the Child and the integrity of the judicial process. The potential conflict that exists in this case presents a distinct possibility that evidence relevant to the Child's best interest will not be available to the Court because of the need of the Grandmother's counsel to serve two masters.
Based on the above, the motion of the Mother is granted and it is
ORDERED that the attorney for the Grandmother is removed from the case due to the apparent and potential conflict of interest.
1. The Court is not deciding the issue of whether the Mother has standing to bring this motion as the Court, sua sponte, is deciding this conflict issue (Booth v. Continental Insurance Co., 167 Misc.2d 429, 634 N.Y.S.2d 650; Matter of Vera, 49 A.D.2d 883, 373 N.Y.S.2d 210, app. den. 38 N.Y.2d 709, 382 N.Y.S.2d 1029, 346 N.E.2d 828). The Court has the duty to protect the integrity of the judicial system and to enforce the ethical standards of the bar (United States v. Flanagan, 679 F.2d 1072, rev'd on other grounds 465 U.S. 259, 104 S.Ct. 1051, 79 L.Ed.2d 288). This duty is incurred whether or not the parties raise the ethical issue.
2. This case differs markedly from Matter of Abrams, 62 N.Y.2d 183, 476 N.Y.S.2d 494, 465 N.E.2d 1. In Abrams, the Attorney-General moved to disqualify an attorney who represented several persons who were the target of the Attorney-General's investigation. The claim was that the multiple representation was frustrating the investigation because the attorney was advising his clients not to testify without immunity. The Attorney-General argued "that the public has an overriding interest in ensuring the swift and effective investigation of criminal activity which must prevail over the right of suspects to be represented by an attorney of their choice whenever that attorney's advice to his multiple clients hampers an ongoing investigation" (Abrams, supra, at 196-197, 476 N.Y.S.2d 494, 465 N.E.2d 1). The present case presents a much different situation. Here, the best interests of a child are involved and those interests should not be prejudiced by an attorney who may be utilizing this case to gain otherwise privileged information helpful to the defense of the criminal case against his other client. The potential for damage to the child is too great for the Court to ignore the potential conflict of interest, even acknowledging the right of a party being able to have representation by her attorney of choice. The rights she potentially foregoes, if the Court decides to disqualify her attorney, are not of the same magnitude as those in a criminal trial. Therefore, the Court has an obligation to exercise its discretion to determine if the conflict of interest presented will cast a shadow over the integrity of the judicial process or have adverse consequences to the best interest of the child and legal profession. (See United States v. Dolan, 570 F.2d 1177, 1184; see also Code of Professional Responsibility Canons 5, 9, DR 5-101 [22 NYCRR 1200.20]; EC 5-14, 5-15).
3. There is no evidence in this record that full disclosure has been made to the Grandfather.
4. The Grandfather is not a party to this proceeding nor has he appeared in Family Court in any other capacity. Accordingly, the Court has not had an opportunity to explore the conflict issue with him.
W. DENNIS DUGGAN, Judge.


(MENAFN - Gulf Times) KidZania Doha and Al Aziziyah Boutique Hotel have partnered to create a kid-sized hotel inside the 5,500sqm educational and entertainment facility, it was announced. KidZania is an award-winning educational and learning concept while Al Aziziyah is a five-star property currently acquired by Aspire Zone Foundation and managed by The Torch Hospitality.
The co-operation between the two companies will help children pay close attention to details, boost their organisational skills, the ability to communicate well with guests and their colleagues, and the discipline of upholding five stars service excellence.
Inside the hotel, children can role-play as housekeepers, waiters/waitresses or receptionists and engage in tasks like arranging the bed with linens, pillowcases, duvet cover and bedspread, ensuring the room is cared for and inspected according to standards.
As waiters they will learn how to arrange tables to five-star standards and appropriately communicate with the guest to ensure satisfaction. Also, as receptionists they will learn how to greet and register guests. 'KidZania strives to co-operate with international and local brands and companies. Our partnership with Al Aziziyah Boutique Hotel will provide a strong platform for communicating with future generations and upholding service excellence, KidZania governor Ric Fearnett said.
The Torch Hospitality Group's area general manager Sherif Sabry said: 'We are confident Al Aziziyah Boutique Hotel activities within the city will offer children a unique experience in immersing them in a real-life setting and will help them to think and react like a professional of the hospitality industry.
KidZania Doha, which is expected to open its doors by the end of 2018, will provide a unique educational and entertainment experience with 42 establishments. Children aged between four and 14 years old will be able to role play 'adult jobs' in up to 60 pretend roles, learning a variety of values, including self-reliance, team work, honesty, integrity and giving back to society.
KidZania is the world's fastest growing family entertainment brand, with KidZania Doha expecting to welcome more than 300,000 visitors year-on-year. The establishment will be able to accommodate up to 1,000 children with approximately 250 employees at a time.


text:
$NA
CME Group Inc.
Acquiror:
Bursa Malaysia Berhad
Advised CME Group, Inc., the operator of a leading global derivatives marketplace, on the structured sale of its 25% equity stake in Bursa Malaysia Derivatives Berhad to Bursa Malaysia Berhad
TT International
Sumitomo Mitsui Financial Group, Inc.
Advised TT International, a leading independent asset management firm focused on managing high alpha, high conviction active management strategies for institutional investors, on its sale to Sumitomo Mitsui Financial Group, Inc.
Martabe gold and silver mine
PT United Tractors Tbk
Advised EMR Capital, a specialist resources private equity manager, on the sale of the Martabe gold and silver mine to United Tractors
Suncorp Group Limited's Australian Life Insurance Business
Advised TAL, Australia's largest and leading insurance business that is 100% owned by Dai-ichi Life of Japan, on the acquisition of Suncorp Group Limited's Australian Life Insurance business
Coal and Allied Industries Limited
Rio Tinto Limited / Mitsubishi Corporation
Advised Coal and Allied Industries Limited, an Australian based coal mining company, in relation to an acquisition proposal from Rio Tinto Limited and Mitsubishi Corporation to acquire all the outstanding shares in the company not already held by Rio Tinto and Mitsubishi
Tyndall Investments
Nikko Asset Management
Advised Suncorp, on the sale of its investment management business, Tyndall Investments, in Australia and New Zealand to Nikko Asset Management, a leading Asian mutual funds management company
TOWER Australia Group Limited
The Dai-ichi Life Insurance Company, Limited
Advised TOWER Australia Group Limited, the largest mono-line life insurance company in Australia, in relation to the offer from The Dai-ichi Life Insurance Company, Limited to acquire all of the shares in TOWER that it does not already own by way of a scheme of arrangement
Close Brothers Corporate Finance Holdings Ltd.
Daiwa Securities SMBC Europe Ltd.
Advised Close Brothers Group plc, an independent financial services group, on the sale of its corporate finance subsidiary to Daiwa Securities SMBC Europe Ltd.
Nikko Cordial Corporation
Citigroup, Inc.
Advised Nikko Cordial, one of Japan's largest securities firms, on its share exchange with Citigroup Inc., which resulted in Citigroup's acquisition of the remaining 32% of Nikko Cordial that it did not already own
Republic Companies Group, Inc.
Delek Capital Ltd.
Advised Republic Companies Group, Inc., a regional provider of personal and commercial property and casualty insurance products
Rothschilds Continuation Holdings
Jardine Strategic Holdings
held by Royal & Sun Alliance
Advised Royal & Sun Alliance on the sale of its 21.5% shareholding in Rothschilds Continuation Holdings AG to Jardine Strategic Holdings


- That by their content can be considered documentary.
- Produced after January 2017.
- Best short documentary (minimum duration of 10 minutes and maximum of 39 minutes).
- That have not been broadcast on VOD platforms, TV or screened in film theaters.
- Documentaries will be presented in original version and must have subtitles in Spanish or Catalan when this is not the language of the film.
- The Festival will not accept edits that are not finalised.
Independent producers La Produktiva Films (Sabadell) and Plans Films (Caldes de Montbui) organize inDOCumentari, an international festival dedicated to documentary film in any of its expressions.
The two winning documentaries will receive a commemorative statuette.
The 3rd edition of the festival will be held during a day at the end of May 2019.
Registration deadline: May 1, 2019.
- They can be considered documentaries given their content.
- They were produced later than January 2017.
- - Preferably not available in cinemas, television or VOD platforms in Spain.
- Documentaries will be presented in their original version and they must have subtitles in Spanish or Catalan when this is not the language of the film.
- The Festival will not accept unfinished films.
- The participant guarantees that the submitted film is original and unpublished and that it is not subject to any contracts, assignments or third party licenses relating to copyright, intellectual property or other rights other than those belonging to the creators of the film.
- Documentaries that do not meet all these requirements will not be eligible for selection.
Registration will be online through the platform www.festhome.com.
DVDs or other physical formats will not be accepted.
Registrations will only be accepted if they are received on time. Incomplete applications will not be taken into consideration.
The organizers of the Festival and several professionals in the field will select the six documentaries (3 short films and 3 feature films) that will participate in the Festival and will decide each film's screening dates and times, one short film and one feature film per day.
The organization of the Festival has no obligation to make known to the participants the reasons for the selection or rejection of the films presented.
The Festival must receive the screening copy of the documentary before May 1, 2019.
The Festival will not accept copies that do not meet the minimum requirements for exhibition.
All selected films will be screened in the original versions and they must have subtitles in Spanish or Catalan when this is not the language of the film.
Those selected must provide the festival with the press kit (photograms, digital poster, synopsis, etc.), trailer and other promotional material to promote the film. The Festival reserves the right not to return the submitted material.
The winning documentaries (best long and best short documentary) will receive a commemorative statuette. The organization reserves the right to incorporate new awards and to replace the prizes with others of equal or greater value when there is a justified cause. Any decisions made by the Festival and the jury are final.
Registration and participation in the Festival implies acceptance of all the aforementioned regulations. Any doubt that may arise will be solved by the management of the Festival organization.


PRAYER TIME 9:30 A. M.
WELCOME GUESTS: Welcome to Green River Baptist Church. We are excited that you have chosen to worship with us today. We hope you will be encouraged and challenged every time you come to Green River Baptist Church. So that we may know you better, please fill out the Guest Worship Registration Card located in the pew and place it in the offering plate when it is passed or give it to the pastor at the door.
PRAYER REQUESTS: Please let us know about any prayer requests that you have by filling out the Congregation's Prayer Request Card on the back of the Guest Registration Card. We will be praying for all requests.
PARENTS OF YOUNG CHILDREN: Nursery is available for small children; Children's Church is available for preschool children. Please feel free to ask for information.
tee may be asked for assistance.
On the evening of September 2, we will have Committee Breakouts to elect chairpersons of the various committees.
The PBA Fall Revival will be September 23-26 at Columbus Baptist Church. Pre-Service Music will begin at 6:45 pm with the Service beginning at 7:00 pm. Dr. Mack Testor will be the Speaker.
The Global Missions Offering has been moved to October 14. Begin now to pray that God will lead you to to support this important gift to missions.
We will observe Baptism and The Lord's Supper today.
Also, those involved should remember that Church Council will meet on Tuesday, August 7, at 6:30 pm to plan the church calendar for September through December. I M P O R T A N T!!
The Church Council is made up of the Pastor (Rev. Gary Curtis),Sunday School Director (Jill Smith), Deacon Chairman (Larry Honeycutt), Church Clerk (Jane Edwards), Church Treasurer (Larry Honeycutt), WMU Director (Louise Allison), Outreach Director (Tabitha Ashley), and the Church Staff [as listed in the bulletin].


You can remove an application-specific access policy at anytime. The default access policy is permanent. You cannot remove the default policy.
Configure the appropriate authentication methods for your deployment. Configuring User Authentication in VMware Identity Manager.
Click the policy to edit.
If this policy applies to Web or desktop applications, click Edit Apps to add or delete applications in this policy.
In the Policy Rules section, Authentication Method column, select the rule to edit.
The Edit a Policy Rule page appears with the existing configuration displayed.
To configure the authentication order, in the then the user must authenticate using the following method drop-down menu, select the authentication method to apply first.
(Optional) To configure a fallback authentication method if the first authentication fails, select another enabled authentication method from the next drop-down menu.
Click Save and click Save again on the Policy page.
The edited policy rule takes effect immediately.


text:
<?php

/**
 * The ezcQueryExpressionSqlite class is used to create SQL expression for SQLite.
 *
 * This class reimplements the methods that have a different syntax in
 * SQLite (substr) and contains PHP implementations of functions that are
 * registered with SQLite with it's PDO::sqliteRegisterFunction() method.
 *
 * @package Database
 * @version //autogentag//
 */
class ezcQueryExpressionSqlite extends ezcQueryExpression
{
    /**
     * Contains an interval map from generic intervals to SQLite native intervals.
     *
     * @var array(string=>string)
     */
    protected $intervalMap = array(
        'SECOND' => 'seconds',
        'MINUTE' => 'minutes',
        'HOUR' => 'hours',
        'DAY' => 'days',
        'MONTH' => 'months',
        'YEAR' => 'years',
    );

    /**
     * Returns part of a string.
     *
     * Note: Not SQL92, but common functionality. SQLite only supports the 3
     * parameter variant of this function, so we are using 2^30-1 as
     * artificial length in that case.
     *
     * @param string $value the target $value the string or the string column.
     * @param int $from extract from this characeter.
     * @param int $len extract this amount of characters.
     * @return string sql that extracts part of a string.
     */
    public function subString( $value, $from, $len = null )
    {
        $value = $this->getIdentifier( $value );
        if ( $len === null )
        {
            $len = $this->getIdentifier( $len );
            return "substr( {$value}, {$from}, 1073741823 )";
        }
        else
        {
            return "substr( {$value}, {$from}, {$len} )";
        }
    }

    /**
     * Returns the current system date and time in the database internal
     * format.
     *
     * @return string
     */
    public function now()
    {
        return '"' . date( 'Y-m-d H:i:s' ) . '"';
    }

    /**
     * Returns the SQL that performs the bitwise XOR on two values.
     *
     * @param string $value1
     * @param string $value2
     * @return string
     */
    public function bitXor( $value1, $value2 )
    {
        $value1 = $this->getIdentifier( $value1 );
        $value2 = $this->getIdentifier( $value2 );
        return "( ( {$value1} | {$value2} ) - ( {$value1} & {$value2} ) )";
    }

    /**
     * Returns the SQL that converts a timestamp value to a unix timestamp.
     *
     * @param string $column
     * @return string
     */
    public function unixTimestamp( $column )
    {
        if ( $column == 'NOW()' )
        {
            return " strftime( '%s', 'now' ) ";
        }
        else
        {
            $column = $this->getIdentifier( $column );
            return " toUnixTimestamp( {$column} ) ";
        }
    }

    /**
     * Returns the SQL that subtracts an interval from a timestamp value.
     *
     * @param string $column
     * @param numeric $expr
     * @param string $type one of SECOND, MINUTE, HOUR, DAY, MONTH, or YEAR
     * @return string
     */
    public function dateSub( $column, $expr, $type )
    {
        $type = $this->intervalMap[$type];

        $column = $this->getIdentifier( $column );
        return " datetime( {$column} , '-{$expr} {$type}' ) ";
    }

    /**
     * Returns the SQL that adds an interval to a timestamp value.
     *
     * @param string $column
     * @param numeric $expr
     * @param string $type one of SECOND, MINUTE, HOUR, DAY, MONTH, or YEAR
     * @return string
     */
    public function dateAdd( $column, $expr, $type )
    {
        $type = $this->intervalMap[$type];

        $column = $this->getIdentifier( $column );
        return " datetime( {$column} , '+{$expr} {$type}' ) ";
    }

    /**
     * Returns the SQL that extracts parts from a timestamp value.
     *
     * @param string $column
     * @param string $type one of SECOND, MINUTE, HOUR, DAY, MONTH, or YEAR
     * @return string
     */
    public function dateExtract( $column, $type )
    {
        switch ( $type )
        {
            case 'SECOND':
                $type = '%S';
                break;
            case 'MINUTE':
                $type = '%M';
                break;
            case 'HOUR':
                $type = '%H';
                break;
            case 'DAY':
                $type = '%d';
                break;
            case 'MONTH':
                $type = '%m';
                break;
            case 'YEAR':
                $type = '%Y';
                break;
        }

        if ( $column == 'NOW()' )
        {
            $column = "'now'";
        }

        $column = $this->getIdentifier( $column );
        return " strftime( '{$type}', {$column} ) ";
    }
}
?>


text:
Posts Tagged 'Law Society of Ontario'
Law Society elections send a message on diversity and it's not what you'd hope
June 27th, 2019 by Shelina Ali
This article was first published on rabble.ca
The Law Society of Ontario (LSO) is once again facing a court challenge claiming that it has violated Section 2 of the Charter of Rights and Freedoms, which protects the right to freedom of conscience, speech and religion. This court challenge follows the LSO's successful defence of its decision to refuse to accredit the proposed law school at Trinity Western University (TWU) because of a requirement that attendees sign a covenant agreeing not to engage in homosexual activities. The LSO took the position that this prevented equal access to the legal profession in Ontario by excluding individuals who identified as LGBTQ.
The new battle relates to a Statement of Principles that the LSO requires lawyers to provide as of last year. It's another example of the LSO attempting to enshrine principles of diversity, anti‑oppression and anti‑discrimination in a profession that is known for its lack of inclusiveness and diversity. In this case, the opposition to advancing these values is coming from other lawyers and is proving to be divisive for the governing council of the LSO — the democratically elected body that oversees its governance Read the rest of this entry
The Law Society is flirting with the idea of doing away with articling. Should it?
July 26th, 2018 by Celia Chandler
One hundred thirty-seven — yes, that's right: 137. And last year, 150!
These are the number of applicants we received for one articling position for the period July 2019 to May 2020. For readers not in law in Ontario, articling is a 10-month work placement under the supervision of a lawyer. Completing articles is a condition to practising law in Ontario.
The competition for securing articles is so intense that the Law Society of Ontario (LSO) has explored alternatives and is flirting with the idea of giving up on articling altogether.
Bad idea. We don't want to lose an important training ground for progressive lawyers. Our law firm, Iler Campbell LLP, is unusual as a place where young lawyers can gain experience working for non‑profits, charities, co‑ops, social enterprise and the like. We see hiring articling students as an obligation to the legal profession and to the progressive organizations that we serve. Our tagline is "A law firm for those who want to make the world a little bit better." Articling here and in other like‑minded firms helps build a cadre of advocates for that better world.
We aim to 'walk the talk': our Statement of Principle regarding Equality, Diversity and Inclusion
November 9th, 2017 by Iler Campbell
As many of you have read in the media, and last week on our blog, the Law Society of Upper Canada has decided to take action to improve the circumstances of racialized lawyers and paralegals.
As lawyers, we are already required to follow the Law Society's Rules of Professional Conduct that apply to all lawyers in Ontario. Among other things, those Rules say that, "A lawyer has a duty to carry on the practice of law and discharge all responsibilities to clients, tribunals, the public and other members of the profession honourably and with integrity." The Commentary on this particular Rule includes the following: "A lawyer has special responsibilities by virtue of the privileges afforded the legal profession and the important role it plays in a free and democratic society and in the administration of justice, including a special responsibility to recognize the diversity of the Ontario community, to protect the dignity of individuals, and to respect human rights laws in force in Ontario" (emphasis added).
The Responsible Housing Provider — Who can appear at the Landlord and Tenant Board?
March 28th, 2013 by Celia Chandler
A recent decision of the Superior Court of Justice may have an impact on housing providers who use the services of property managers.
In The Law Society of Upper Canada v. Enzo Vincent Chiarelli, Mr. Justice Goldstein ruled that Chiarelli can no longer appear on behalf of his clients before the Ontario Landlord and Tenant Board (LTB) because he is not licensed under the Law Society Act to provide legal services.


Such passion, such power, beautiful rolling R's. She just sends me.
And GORGEOUS too. With an absolute heart of gold, no -beyond measure. and a Soul to match. Her Beauty in voice, body and demeanor truly, surely come from within. Than you God for Mimi.


This is European factory of SILK PLASTER. We launched our second factory under strict compliance to our internal quality control procedures, management policy and environmental sustainability. We aim to reduce any production waste in our production process to 0%. The factory abilities are to supply large orders and ship immediately worldwide.
Besides production unit, the company maintains large warehouse in the premises. The production site is located near to port of Riga and several more ports, such as Ventspils, Liepaja, Klaipeda, Parnu, Talinn. Our company has well developed relationships with leading shipping and transportation companies. That's why we are offering variety of INCOTERMS solutions to satisfy customer needs.
At last, SILK PLASTER's European factory produces goods "Made in Europe". This means cheap logistics and no import taxes within the Europe Union. Furthermore, it gives a possibility to use double trade treaties between Europe and lots of countries, such as Chile, Mexico, Egypt, Turkey, Israel, Norway, Morocco, Algeria, South Korea, Serbia, Montenegro, Switzerland, Albania and South Africa.


text:
Level 3 Chairman Scott to retire in 2014
Walter Scott, Jr., the chairman of Level 3 Communications' (NYSE: LVLT) board of directors, announced he will retire when the service provider holds its annual stockholder meeting next May by not standing for reelection.
Since 1979, Scott served as the chairman of Level 3 and its predecessor Kiewit Diversified Group Inc. (KDG), a subsidiary of Peter Kiewit Sons', Inc. (PKS), 100-plus year old construction, mining, information services and communications company.
Taking Scott's place will be James O. Ellis, Jr, who has been a board member since March 2005 and is the chairman of the Board's Classified Business and Security Committee and a member of the Nominating and Governance Committee. A retired U.S. Navy admiral, Ellis is also currently a member of the Board of Directors of Lockheed Martin Corporation, Dominion Resources, Inc. and Inmarsat PLC.
Scott's impending departure from the company is part of an ongoing transition of its management team and its ongoing movement to become profitable.
The biggest change came last April when the company appointed Jeff Storey as its new president and CEO. Storey took the reins over from company co-founder James Crowe, who announced he would step down in March.
Special report: The 25 Most Powerful People in U.S. Wireless and Wireline 2013
Level 3 adds space to Bogota data center to accommodate cloud growth
Level 3 expands Latin America fiber, Ethernet network to accommodate surging business growth
Level 3 expands CDN footprint in EMEA, Latin America, Asia
Level 3 CNS revenues climb to $1.4B, helping narrow losses


text:
Update Actor's Biography
John David Washington (born July 28, 1984) is an American actor and former football running back. He played college football at Morehouse College and signed with the St. Louis Rams as an undrafted free agent in 2006. Professionally, Washington spent four years as the running back for the United Football League's Sacramento Mountain Lions. Washington shifted to an acting career like his father, Denzel Washington and mother, Pauletta Washington. He was part of the main cast of the HBO comedy series Ballers (2015–2019) and starred as Ron Stallworth in Spike Lee's crime film BlacKkKlansman (2018). For the latter, he received both Golden Globe and Screen Actors Guild Award nominations.


text:
Q: how i can add title to my featured image? I want to add a title to my featured image, how I can add this item:
$attr = array(
    'title' => 'howdy partner',
    'alt'   => 'a nice view of the ocean',
);

// pick one, note 2 or 3 arguments:
the_post_thumbnail( 'post-thumbnail', $attr );

echo("permalink'>". // null for current $post->ID (use within loop)
    get_the_post_thumbnail( null, 'post-thumbnail', $attr ).
    "");

my site is : https://abzarwp.com

A: I'm not sure about your question, but the below code is more correct.
$attr = array(
    'title' => get_the_title(),
    'alt'   => 'a nice view of the ocean',
);

echo sprintf( '%s', get_the_permalink(), get_the_post_thumbnail( null, 'post-thumbnail', $attr ) );

redpajama_set_name: RedPajamaStackExchange


Imagine being told you may not survive beyond your mid-20s.
This is too often the reality for young men like 24-year-old Ryan Ballou, who was diagnosed more than 20 years ago with Duchenne muscular dystrophy (DMD) – a devastating disease that causes muscle breakdown and heart failure in boys and young men.
But because of Ryan's indomitable spirit and researchers at The Ohio State University Wexner Medical Center, we are on the cusp of imagining a new future for DMD patients.
For more than 20 years, Ryan and his dad have traveled from Pittsburgh to Ohio State's Wexner Medical Center for treatment by neurologist Dr. John Kissel. Five years ago, Dr. Kissel referred Ryan to Ohio State's Heart and Vascular Center, where a team of physician and researchers is collaborating on a revolutionary drug treatment for DMD that shows potential for delaying heart failure.
Ryan's success holds promise for thousands of men and their families affected by DMD.
Learn more about new findings in Duchenne Muscular Dystrophy here.


When medicine cannot be given by mouth, it may need to be given to a child by an injection or shot. A child of any age should be allowed to say he doesn't like the shot, and may be told it is OK to cry or say "ouch."
1. Roll the bottle back and forth gently between your hands.
2. Clean off the top of the bottle with alcohol.
3. Take off the needle cover.
4. Pull back the plunger to put air into syringe. Put in the same amount of air as the amount of medicine you need.
5. Put the needle into the bottle and push all the air in.
6. Leave the needle in the bottle. Turn the syringe and bottle so the bottle is on top. Make sure the tip of the needle is in the medicine. Pull back slowly on the plunger a little past the line that marks the dose amount.
7. Check syringe for air bubbles. If there are air bubbles, tap at them with your finger until they go to the top.
8. Push the plunger so the air goes out of the needle and the top of the black rubber plunger is right at the line that marks the dose amount.
9. Take the needle out of the bottle and replace needle cover.
Images indicate the proper places to give the shot.
1. Wipe the skin with alcohol where you are going to give the shot.
2. Remove the needle cover. Hold the syringe like a dart. Quickly push the needle straight in all the way.
3. Hold the syringe firmly with your hand. With your free hand, pull back on the plunger a little. If you see blood, take the needle out, throw syringe away, and draw up a new dose.
4. If you do not see blood, slowly push the plunger all the way down. Put cotton on the skin next to the needle and quickly pull out the needle. Apply gentle pressure to the site; be careful not to stick yourself.
5. Throw the syringe away in a puncture proof container.
Allow your child to squeeze your hand and distract his or her attention with a toy.
Ask the nurse to give you tips on holding your child for the injection.
Keep all items together in one place.
Caution: Be sure to keep syringes and supplies out of the reach of children.


text:
CVB SECRET SOUND
WEATHER & PASS REPORTS
Subscribe to 92.9 on YouTube
#1 FOR NEW COUNTRY IN YAKIMA
Shows and DJs
Brian Stephenson
Got Our App?
Listen to The Bull on Alexa
Road and pass reports
Federated Auto Parts
Contact Our Yakima Sales Staff
Advertising With TSM
92.9 The Bull92.9 The Bull
Maren Morris Hopes Baby Hayes Will Get to See Her 2020 ACM Awards Performance
When Maren Morris takes the stage at the 2020 ACM Awards on Wednesday night (Sept. 16), it'll be her first big awards show performance since giving birth to her son, Hayes Andrew Hurd, in late March. She's hoping he'll be able to watch the special moment, too.
"I don't know what time in the show I am, but he can stay up a little late to watch Mom," Morris tells Taste of Country Nights, even though "I don't think it's quite registered [to him] that the people in the TV are real."
Morris will be performing "To Hell & Back," from her sophomore album, Girl, from the Ryman Auditorium. The album is one of five nominated for Album of the Year at the ACMs in 2020; if it wins, it'll be a posthumous honor for Busbee, the late producer who died in September of 2019 of brain cancer.
"This is, in a lot of ways, the last remaining piece of art that I will make with him, so if it received that honor tomorrow, that would be amazing," Morris reflects. She adds, "I'll always treasure those years and albums that I had with Busbee. He was such a champion of my sound forming in that beginning time that we had together."
Morris has continued her friendship with Busbee's family — his wife and three daughters — since his death. The two "text all the time," the singer shares.
"Me being a new mother, she's just been so supportive of me," Morris says of Busbee's wife, "and I just love that I'll forever have a friend in her."
In addition to Album of the Year, Morris is nominated for Female Artist of the Year at the 2020 ACM Awards. She's already a winner at the show this year, as "Fooled Around and Fell in Love," a collaboration between Miranda Lambert, Morris and the rest of the women who were on Lambert's 2019 Roadside Bars & Pink Guitars Tour, earned Music Event of the Year.
The 2020 ACM Awards will air live from Nashville, for the first time ever, on CBS. The show is scheduled to begin at 8PM ET and will take place at the Ryman, the Bluebird Cafe and the Grand Ole Opry House.
See the Best Red Carpet Looks From the 2019 ACM Awards:
Source: Maren Morris Hopes Baby Hayes Will Get to See Her 2020 ACM Awards Performance
Filed Under: maren morris
Heard on 92.9
What Do You Need To Do To Discover Your Trucker Name? Answers!
Local Grandma Advice in Honor of Betty White's 99th Birthday
3 Yakima Valley Restaurants That Should Be A National Franchise
7 Things That Washington State Is Missing During The Pandemic
Yakima Business Listings
2021 92.9 The Bull, Townsquare Media, Inc. All rights reserved.


Our organization offers 24PetWatch microchips, which include free registration into the 24PetWatch pet recovery service. For more information visit www.24petwatch.com or call 1-866-597-2424.
For information on any animals, please call LHS at 303.772.1232 ext 0 during our regular hours of operation.


text:
Stories from the Shoemaker, Reebok's Co-Founder
Reebok co-founder Joe Foster recounts pivotal stories from his 31 years running the business.
Free Book Preview: Unstoppable
Get a glimpse of how to overcome the mental and physical fatigue that is standing between you and your full potential.
David Meltzer
Entrepreneur Leadership Network VIP
Co-Founder of Sports 1 Marketing, Speaker, Author and Business Coach
Joe Foster is the co-founder of Reebok and the author of Shoemaker: The Untold Story of the British Family Firm that Became a Global Brand. In this episode of #ThePlaybook, Foster talks with host David Meltzer about starting Reebok, its early successes and his advice to entrepreneurs about embracing change.
Foster talks about his family's shoemaking legacy and how he and his brother left the family business to begin their own company. He retells how they landed on "Reebok" as the name of the company based on a perusal of an American dictionary and how they redid the logo after Adidas claimed their earlier logo was too similar to theirs.
Foster talks about running's explosive growth in the U.S. during the 70s and how Reebok capitalized on the popularity of Runner's World by developing a highly rated running shoe.
Foster advises entrepreneurs to embrace change and growth, noting that there is no plateau in business — the other side of the hill is downhill. He also illustrates the importance of innovation with a story about introducing the aerobics shoe to the market through instructors in Los Angeles, propelling the brand's popularity globally.
Related: Unlocking Human Performance


text:
Home/Geneal NEWS/2023: Nigeria is driven by greedy individuals in the opposite direction – Liberal Party leader Musa Ogedei
2023: Nigeria is driven by greedy individuals in the opposite direction – Liberal Party leader Musa Ogedei
Ahead of the 2023 general election, Labor leader Dr Moses Paul Ogedei insisted that Nigeria was being pushed in the opposite direction by greedy individuals.
Ogedei said this while speaking at an event titled "Good Governance YellOut 4 in Nigeria" on Friday in Abuja.
According to him, politics should be about people and how to make life easier and richer.
Ogedei, who is the organizer/coordinator of the group, said, "Politics should be about the people and how to make life easier and richer. It should recognize the interest of the people and work in ways that enable both individual and collective dreams to come true.
"Unfortunately, Nigeria has been pushed in the opposite direction by individuals and systems who, through greed, nepotism and incompetence, have overseen the impoverishment of the country with over 63% of Nigerians declaring themselves multidimensionally poor in 2022, according to the Nigerian Bureau of Statistics.
"They have sowed tribalism, religious intolerance, violence, and bigotry in the sands of our homeland, turning poor and vulnerable Nigerians against each other and against the state, and weaponizing poverty in a cycle of frustration."
He added, "The stories of the victims have been very unfortunate over the years with names pouring in from Chibok, Dapchi, Boni Yadi, Parkin Ladi, Zango Kataf, Agatu, Mobi, Nyanya and a host of other communities affected by the hands of terrorists. These are crimes committed by young people, many of whom have been deceived by the lies of a corrupt political system."
Premier League: Potential next Chelsea boss revealed, and he's already waiting
English Premier League: Arsenal and Chelsea agreed on the transfer of Jorginho, and the details of the agreement were revealed
Fuel Scarcity: Students, DSS Return Team Alert, Falana blows up
Rivers Timber Market Fire
APC will retain the presidency, Sokoto wins, all Northwest States – Lukman


The National Union of Independent Itinerant Transport and Allied Workers (SINTRALOC) was founded on November 23, 1987. Throughout its 24 years of existence, SINTRALOC has fought tirelessly for the recognition and dignity of street vendors. In recent years, thanks to the perseverance of the union's struggle, we have managed to get a bill passed to legally develop our work nationwide.
The Alliance for Zambian Informal Economy Associations (AZIEA) is the national umbrella organization for associations of informal workers in Zambia. It was launched in October 2002, as the result of a project on organizing in the informal economy undertaken by the Workers' Education Association of Zambia (WEAZ) and the Zambia Congress of Trade Unions (ZCTU).
UGSEIN is a national union that had 30,150 members as of November 2013. Members include street vendors and other informal workers. UGSEIN's objectives include: organize the economic actors of the informal economy; defend the ethical and material interests of members; negotiate agreements with the employers of the sector; struggle for decent work conditions and health and security for the workers in the sector.
The National Confederation of Trade Unions of Guinea (Confédération Nationale des Travailleurs de Guinée - CNTG) was founded in June 1960. It has 20,000 members from the informal sector, including 12,000 contributing members.
The Trades Union Congress (TUC) of Ghana is the main umbrella organization for trade union activities in Ghana, made up of 17 affiliated autonomous national unions. The TUC is the official mouthpiece of unionized labour in its dealing with government and with the employers' association. It currently has a total membership of about 500,000 members.


text:
Summary The MPEG-4 codec in the Windows Media codecs in Microsoft Windows XP SP2 and SP3, Server 2003 SP2, Vista SP1 and SP2, and Server 2008 Gold and SP2 does not properly handle crafted media content with MPEG-4 video encoding, which allows remote attackers to execute arbitrary code via a file in an unspecified "supported format," aka "MPEG-4 Codec Vulnerability."
cpe:2.3:o:microsoft:windows_server_2003:*:sp2:*:*:*:*:*:*
Code Injection
An adversary exploits a weakness in input validation on the target to inject new code into that which is currently executing. This differs from code inclusion in that code inclusion involves the addition or replacement of a reference to a code file, which is subsequently loaded by the target and used as part of the code of some application.
Leverage Executable Code in Non-Executable Files
An attack of this type exploits a system's trust in configuration and resource files. When the executable loads the resource (such as an image file or configuration file) the attacker has modified the file to either execute malicious code directly or manipulate the target process (e.g. application server) to execute based on the malicious configuration parameters. Since systems are increasingly interrelated mashing up resources from local and remote sources the possibility of this attack occurring is high.
Manipulating User-Controlled Variables
This attack targets user controlled variables (DEBUG=1, PHP Globals, and So Forth). An attacker can override environment variables leveraging user-supplied, untrusted query variables directly used on the application server without any data sanitization. In extreme cases, the attacker can change variables controlling the business logic of the application. For instance, in languages like PHP, a number of poorly set default configurations may allow the user to override variables.
impact Remote Code Execution
title Vulnerability in MPEG-4 Codec Could Allow Remote Code Execution
organization G2, Inc.
description The MPEG-4 codec in the Windows Media codecs in Microsoft Windows XP SP2 and SP3, Server 2003 SP2, Vista SP1 and SP2, and Server 2008 Gold and SP2 does not properly handle crafted media content with MPEG-4 video encoding, which allows remote attackers to execute arbitrary code via a file in an unspecified "supported format," aka "MPEG-4 Codec Vulnerability."
title MPEG-4 Codec Vulnerability


The sun and skin aging are the number one causes of dark spots on the skin.
PHYTOMER spot correction skin care effectively fights pigmentation spots on the face and décolleté to even your complexion. CC Creams, serums and dry oil provide a boost of radiance.
Applying a special spot correction skin care daily improves the evenness of your complexion. CC Creams are formulated to protect the skin from UV rays and improve its radiance.
Serum boosts the effectiveness and enhances the action of the spot correction cream. Apply it before the day cream for a bright, even complexion.
Complete your spot correction routine with a clarifying mask or replenishing dry oil to restore your skin to its original luminosity and vitality.
A light and comfortable dry oil that leaves a delicate rose fragrance on the skin.