text:
\section*{Introduction}
In this paper we study the hypergeometric solutions of the quantized \KZv/
(\qKZ/) equation with values in a tensor product of vector representations
of $U_q(sl_n)$, see Section 1 for the precise formulation of the problem.
It is known that the \qKZ/ equation respects the weight decomposition of
the tensor product. For each weight subspace we construct a fundamental matrix
solution of the \qKZ/ equation and explicitly calculate the corresponding
determinant, see Theorem \ref{thm6.1}.

Formal integral representations for solutions of the \qKZ/ equation in
the $sl_n$ case, both in the rational and trigonometric situation, were
constructed in \cite{TV1}. Though to write down the phase function explicitly
in the trigonometric situation it had been assumed in \cite{TV1} that
the multiplicative step $p$ of the \qKZ/ equation is inside the unit circle:
$0<|p|<1$, all the construction in \cite{TV1} used only difference equations
for the phase function and after obvious modifications remained valid for
an arbitrary step $p\ne 0,1$. However, the problem of integrating the formal
integral representations suitably and getting in this way actual solutions of
the \qKZ/ equation is much more analytically involved; one can see this looking
at the $sl_2$ case.

In the last four years the hypergeometric solutions of the \qKZ/ equation
in the $sl_2$ case were studied quite well. The generic situation was
considered in \cite{TV2} (the rational case) and in \cite{TV3}
(the trigonometric case for ${0<|p|<1}\,$). The construction was generalized
to the trigonometric case for $|p|=1$ in \cite{MT1} and to the elliptic case
of the quantized \KZv/-Bernard (\qKZB/) equation in \cite{FTV1}, \cite{FTV2}.

If some of the representations are finite-dimensional, the situation is no
more generic. Rather detailed study of this case has been done in \cite{MV1};
see also \cite{S}, \cite{JKMQ}, \cite{NPT}, \cite{T1} for some important
particular cases.

Less is known for the case of ${n>2}$. Some integral formulae for solutions of
the \qKZ/ equation were obtained in \cite{S}, \cite{KQ}, \cite{N}, \cite{MT2},
but in all the considered cases solutions takes values in a tensor product of
vector representations. Recently Varchenko and the third author have managed to
extend the costruction of \cite{TV2}, \cite{TV3} to the higher rank case and
get solutions taking values in a tensor product of arbitrary highest weight
representations \cite{TV4}. Let us also mention a paper \cite{M}, where
integral formulae for solutions of another type of the \qKZ/ equation were
suggested.

In this paper we evaluate a determinant of a certain matrix whose entries are
given by multidimensional integrals of $q$-hypergeometric type. In the case
of ordinary multidimensional hypergeometric integrals a problem of evaluating
similar determinants appears, say, in studying arrangements of hyperplanes,
and several results have been obtained in this direction, see for instance
\cite{V1}, \cite{L}, \cite{LS}, \cite{DT}, \cite{MTV}, \cite{MV2}. In some
particular cases these determinant formulae have another meaning; namely they
imply that under certain assumptions the hypergeometric solutions of the
differential \KZv/ equation form a basis of solutions \cite{SV}, \cite
{V2}. There are similar determinant formulae for solutions of the \qKZ/
equation in the $sl_2$ case.  They have been obtained for the rational case in
\cite{TV2}, \cite {T1}, and for the trigonometric case in \cite{TV3} for
${0<|p|<1}$ and in \cite{MT1} for $|p|=1$. It turns out that there is a nice
connection of constructions given in \cite{TV3} and \cite {MT1}, which in
particularly allows to derive the determinant formula for $|p|=1$ from the
determinant formula for ${0<|p|<1}$. This subject will be addressed elsewhere
\cite{T2}.

The paper is organized as follows. The first section contains preliminaries
and precise definitions on the \qKZ/ equation. In Section 2 we construct the
hypergeometric pairing and give integral formulae for solutions of the \qKZ/
equation. The main result of the paper is formulated in Section 3, see Theorem
\ref{thm6.1}. We show that both the left hand side and the right hand side of
formula (\ref{107}) satisfy the same system of difference equations and have to
be proportional. To compute the proportionality coefficient we study suitable
asymptotics of the hypergeometric solutions. We see that the proportionality
coefficient splits into a product of contributions of each tensor factor, which
are calculated in Section 5. In the last Section we complete the proof of
Theorem \ref{thm6.1}. A short Appendix contains the necessary information of
the double sine function for the convenience of the reader.

\section*{Acknoledgement}
V\&Tarasov thanks the RIMS, Kyoto University for warm hospitality during his
stay there, when most of the results of this paper had been obtained.
He also thanks A\&Varchenko for valuable remarks.

\section{The quantized \KZv/ equation}
Consider the vector representation $V$ of $sl_n$:
$$
\tsize V\,=\,\bigoplus\limits_{j=0}^{n-1}\,{\Bbb C}\,v_j\,.
$$
Let $\eps_1,\ldots,\eps_n$ be the fundamental weights of $sl_n$. We consider
the basis vectors $v_0,\ldots v_{n-1}$ as weight vectors with respect to
the Cartan subalgebra of $sl_n$ with weights $\eps_1,\ldots,\eps_n$,
respectively.

Fix a complex number $\rho$ and a weight $\bar\mu=\Sum\limits_{j=1}^n\mu_j\eps_j$.
Consider a diagonal matrix
$$
D(\bar\mu)\,=\,\mathop{\rm diag\;\!}(e^{2\pi i\mu_1},\ldots,e^{2\pi i\mu_n})
$$
and a matrix $R^{(\rho)}(\beta)\in\mathop{\rm End\;\!}(V\otimes V)$ with the following entries:
$R^{(\rho)}(\beta)_{jj}^{jj}=1$,
\begin{eqnarray}
R^{(\rho)}(\beta)_{jk}^{jk}\,=\,
{\mathop{\rm sh}{\pi\over\rho}\beta\over\mathop{\rm sh}{\pi\over\rho}(\beta-{2\pi i\over n})} \label{eq01}
\end{eqnarray}
for $j\ne k$,
\begin{equation}
R^{(\rho)}(\beta)_{kj}^{jk}\,=\,
-\;{e^{{\pi\over\rho}\beta}\,\mathop{\rm sh}{2\pi^2i\over\rho n}\over
\mathop{\rm sh}{\pi\over\rho}(\beta-{2\pi i\over n})}\;,\qquad\quad
R^{(\rho)}(\beta)_{jk}^{kj}\,=\,
-\;{e^{-{\pi\over\rho}\beta}\,\mathop{\rm sh}{2\pi^2i\over\rho n}\over
\mathop{\rm sh}{\pi\over\rho}(\beta-{2\pi i\over n})}
\label{eq02}
\end{equation}
for $j<k$, and $R^{(\rho)}(\beta)_{lm}^{jk}=0$, otherwise. We have
$$
R^{(\rho)}(\beta)\,v_l\otimes v_m\,=\,\Sum\limits_{j,k=0}^{n-1}
R^{(\rho)}(\beta)^{lm}_{jk}\,v_j\otimes v_k\,.
$$

Fix a complex number $\la$ and define the \qKZ/ operators
$K_1^{(\rho)},\ldots K_N^{(\rho)}$ acting in the tensor product
$V^{\otimes N}$:
\begin{eqnarray}
K^{(\rho)}_m(\beta_1,\ldots,\beta_N)\,=\,
R^{(\rho)}_{m,m-1}(\beta_m-\beta_{m-1}-\la i)
\ldots R^{(\rho)}_{m1}(\beta_m-\beta_1-\la i)
\nonumber
\\[4pt]
{}\times\,D_m(-\bar\mu/\rho)\,R^{(\rho)}_{mN}(\beta_m-\beta_N)\ldots
R^{(\rho)}_{m,m+1}(\beta_m-\beta_{m+1})\,.\kern-17pt
\end{eqnarray}
In this paper we consider the \qKZ/ equation for a function
$f(\beta_1,\ldots,\beta_N)$ taking values in $V^{\otimes N}$, which is
the following system of difference equations:
\begin{equation}
f(\beta_1,\ldots \beta_m-\la i,\ldots \beta_N)\,=\,
K^{(\rho)}_m(\beta_1,\ldots,\beta_N)\,f(\beta_1,\ldots,\beta_N)\,,
\quad m=1,\ldots N.\kern-2em
\label{qKZ}
\end{equation}
We also consider the mirror \qKZ/ equation for a similar function
$g(\beta_1,\ldots,\beta_N)$:
\begin{equation}
g(\beta_1,\ldots \beta_m-\rho i,\ldots \beta_N)\,=\,
K^{(\la)}_m(\beta_1,\ldots,\beta_N)\,g(\beta_1,\ldots,\beta_N)\,,
\quad m=1,\ldots N.\kern-2em
\label{qKZm}
\end{equation}
The \qKZ/ operators respects the $sl_n$ weight decomposition of the tensor
product. Therefore, one can consider solutions of the \qKZ/ and mirror \qKZ/
equations taking values in a weight subspace $(V^{\otimes N})_\xi$ for any
given weight $\xi$.

In this paper for any given weight $\xi$ such that $(V^{\otimes N})_\xi$
is nontrivial we will construct a function
$\Psi_\xi(\beta_1,\ldots,\beta_N)$ taking values in
$(V^{\otimes N})_\xi\otimes(V^{\otimes N})_\xi$, which solves the \qKZ/
equation (\ref{qKZ}) in the first tensor factor and solves the mirror \qKZ/
equation (\ref{qKZm}) in the second tensor factor. Also we will compute the
determinant $\det\Psi_\xi(\beta_1,\ldots,\beta_N)$.

The matrix $R^{(\rho)}(\beta)$ is the $R$-matrix associated with the tensor
product of the evaluation vector representations of $U_q(\widehat{sl_n})$ for
$$
q\,=\,e^{-{2\pi^2 i\over \rho n}}\,.
$$
Similarly, $R^{(\la)}(\beta)$ is associated with $U_{q'}(\widehat{sl_n})$ for
$$
q'\,=\,e^{-{2\pi^2 i\over\la n}}\,.
$$
All over this paper we assume that $\rho$ and $\la$ are real positive.
Thus, under our assumptions we have that
$$
|q|\,=\,|q'|\,=\,1\,.
$$
However, it is clear from the consideration that all our construction remain
valid if $\rho$ and $\la$ have small enough imaginary parts of arbitrary sign.
Therefore, $q$ and $q'$ can deviate from the unit circle and vary in a narrow
annulus.

In addition to the reality and positivity of $\rho$ and $\la$ we assume that
both of them and their ratio are not rational. Sometimes we take $\rho$ and
$\la$ to be sufficiently large.
\section{Integral formulae for solutions}

For non-negative integers $\nu_{1}, \cdots , \nu_{n-1}$ satisfying
\begin{eqnarray}
N=\nu_{0} \ge \nu_{1} \ge \cdots \ge \nu_{n-1} \ge \nu_{n}=0,
\end{eqnarray}
we denote by ${\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}$ the set of all $N$-tuples
$J=(J_{1}, \cdots , J_{N}) \in \left( {\Bbb Z}_{\ge 0} \right)^{N}$
such that
\begin{eqnarray} \# \{ r\,;\,J_{r} \ge j \}=\nu_{j}.
\end{eqnarray}
For $J=(J_{1}, \cdots , J_{N}) \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}$, we set
\begin{eqnarray}
{\cal N}_{j}^{J}=\{ r\,;\,J_{r} \ge j \}
\end{eqnarray}
and define integers
$r_{j, m}^{J}\,, \ (\,0 \le j \le n-1\,,\ \ 1 \le m \le \nu_{j}\,)$ as follows:
\begin{eqnarray}
{\cal N}_{j}^{J}=\{ r_{j, 1}^{J}, \cdots , r_{j, \nu_{j}}^{J} \},
\quad r_{j, 1}^{J}< \cdots < r_{j, \nu_{j}}^{J}.
\end{eqnarray}
We have, in particular, $r_{0, m}^{J}=m$.

Now we set
\begin{eqnarray}
& & w_{J}^{(\rho)}(\{ \gamma_{j, m} \} ; \beta_{1}, \cdots , \beta_{N})=
{\rm Skew}_{n-1} \circ \cdots \circ {\rm Skew}_{1}
g_{J}^{(\rho)}(\{\gamma_{j, m}\}; \beta_{1}, \cdots , \beta_{N}),
\\[6pt]
& & g_{J}^{(\rho)}(\{\gamma_{j, m}\}; \beta_{1}, \cdots , \beta_{N})\nonumber
\\[2pt]
& & =\Prod\limits_{j=1}^{n-1} \Bigl\{
\Prod\limits_{1 \le m<m' \le \nu_{j}}
\mathop{\rm sh}\frac{\pi}{\rho}(\gamma_{j, m'}-\gamma_{j, m}-\frac{2\pi i}{n})
\Prod\limits_{m=1}^{\nu_{j}} \Bigl\{
e^{-\frac{\pi}{\rho}(\gamma_{j, m}-\gamma_{j-1, m^*(J, j, m)})} \nonumber \\
& & {}\times
\Prod\limits_{m' \atop r_{j-1, m'}^{J}<r_{j, m}^{J}}
\mathop{\rm sh}\frac{\pi}{\rho}(\gamma_{j, m}-\gamma_{j-1, m'}+\frac{\pi i}{n})\!\!\!\!\!
\Prod\limits_{m' \atop r_{j, m}^{J}<r_{j-1, m'}^{J}}
\mathop{\rm sh}\frac{\pi}{\rho}(\gamma_{j, m}-\gamma_{j-1, m'}-\frac{\pi i}{n})
\Bigr\} \Bigr\}.
\end{eqnarray}
The notation in the above formulae is as follows.
The operator ${\rm Skew}_{j}$ is the skew-symmetrization
with respect to the variables $\{ \gamma_{j, m}\}_{m=1, \cdots , \nu_{j}}$:
\begin{eqnarray}
{\rm Skew}_{j}X(\gamma_{j, 1}, \cdots , \gamma_{j, \nu_{j}})=
\Sum\limits_{\sigma \in S_{\nu_{j}}}(\mathop{\rm sgn\;\!}\sigma)
X(\gamma_{j, \sigma(1)}, \cdots , \gamma_{j, \sigma(\nu_{j})}).
\end{eqnarray}
The integer $m^*(J,j,m)$ is uniquely determined by the condition
\begin{eqnarray}
r_{j,m}^{J}=r_{j-1,m^*(J, j, m)}^{J},
\end{eqnarray}
and $\gamma_{0, m}=\beta_{m}$. We abbreviate
$w_{J}^{(\rho)}(\{\gamma_{j, m}\}; \beta_{1}, \cdots , \beta_{N})$ to
$w_{J}^{(\rho)}(\beta_{1}, \cdots , \beta_{N})$ when the dependence on
the abbreviated variables is irrelavant.

Set
\begin{eqnarray}
{\cal F}_{\nu_{1}, \cdots , \nu_{n-1}}^{(\rho)}(\beta_{1}, \cdots , \beta_{N})=
\Sum\limits_{J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}{\Bbb C} \,\,w_{J}^{(\rho)}(\beta_{1}, \cdots,
\beta_{N}).
\end{eqnarray}
In the following, we define a pairing between
\begin{eqnarray}
w_{J}^{(\rho)} \in {\cal F}_{\nu_{1}, \cdots , \nu_{n-1}}^{(\rho)}(\beta_{1}, \cdots ,
\beta_{N}) \quad
{\rm and} \quad
w_{J}^{(\lambda)} \in {\cal F}_{\nu_{1}, \cdots , \nu_{n-1}}^{(\lambda)}(\beta_{1}, \cdots ,
\beta_{N}).
\end{eqnarray}
We use
\begin{eqnarray}
\varphi(x)=\frac{1}{S_{2}(ix-\frac{\pi}{n})S_{2}(-ix-\frac{\pi}{n})}, \quad
\psi(x)=\frac{1}{S_{2}(ix+\frac{2\pi}{n})S_{2}(-ix+\frac{2\pi}{n})},
\end{eqnarray}
where $S_{2}(x)=S_{2}(x | \rho , \lambda )$ is the double sine function with
periods $\rho$ and $\lambda$.

For $J, J' \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}$, we set
\begin{eqnarray}
& &
I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})=\left( \Prod\limits_{j=1}^{n-1}\Prod\limits_{m=1}^{\nu_{j}}
\int_{C_{j}} d\gamma_{j, m} \right)
K(\{\gamma_{j, m}\}; \beta_{1}, \cdots , \beta_{N}; \mu_{1}, \cdots , \mu_{n}) \nonumber \\[4pt]
& & \quad{}\times
w_{J}^{(\rho)}(\{\gamma_{j, m}\}; \beta_{1}, \cdots , \beta_{N})
w_{J'}^{(\lambda)}(\{ \gamma_{j, m}\} ; \beta_{1}, \cdots , \beta_{N}). \label{001}
\end{eqnarray}
Here the function $K$ is defined by
\begin{eqnarray}
&&K(\{\gamma_{j,m}\};\beta_{1},\cdots,\beta_{N};\mu_{1},\cdots,\mu_{n})=
e^{\frac{2\pi}{\rho \lambda}\Sum\limits_{j=0}^{n-1}\Sum\limits_{m=1}^{\nu_{j}}
\gamma_{j,m}(\mu_{j+1}-\mu_{j})}\nonumber\\[6pt]
&&{}\times\Prod\limits_{j=1}^{n-1}\left\{\Prod\limits_{m=1}^{\nu_{j}}
\Prod\limits_{m'=1}^{\nu_{j-1}}\varphi(\gamma_{j,m}-\gamma_{j-1,m'})
\Prod\limits_{1\le m<m'\le\nu_{j}}\psi(\gamma_{j,m}-\gamma_{j,m'})\right\},
\label{Kdef}
\end{eqnarray}
where $\mu_{0}=0$. We say the variable $\gamma_{j,m}$ belongs to the point
$r_{j,m}$. Then, $J_r$ is the number of integral variables which belong
to the point $r$.

The contour $C_{j}$ for $\gamma_{j,m}, \,(m=1, \cdots , \nu_{j})$
is a deformation of the real line $(-\infty, \infty)$
such that the poles at
\begin{eqnarray}
\gamma_{j-1, m'}-\frac{\pi i}{n}+\rho i {\Bbb Z}_{\ge 0}+\lambda i {\Bbb Z}_{\ge 0}, \quad
\gamma_{j, m'}+\frac{2\pi i}{n}+\rho i {\Bbb Z}_{\ge 0}+\lambda i {\Bbb Z}_{\ge 0}
\label{CONT1}
\end{eqnarray}
are above $C_{j}$ and the poles at
\begin{eqnarray}
\gamma_{j-1, m'}+\frac{\pi i}{n}-\rho i {\Bbb Z}_{\ge 0}-\lambda i {\Bbb Z}_{\ge 0}, \quad
\gamma_{j, m'}-\frac{2\pi i}{n}-\rho i {\Bbb Z}_{\ge 0}-\lambda i {\Bbb Z}_{\ge 0}
\label{CONT2}
\end{eqnarray}
are below $C_{j}$, where $\gamma_{0, m}=\beta_{m}$.

These conditions are not compatible if all the poles really exist.
Pinching of the integration contours by poles occurs for each triple
of variables $\gamma_{j,m_1},\gamma_{j,m_2},\gamma_{j-1,m}$. However,
we can improve the definition (\ref{001}) as follows. We have that
$$
I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})=
\Sum\limits_{{\poorbold{$\scriptstyle\sigma$}}, {\poorbold{$\scriptstyle\sigma$}'}\in\,
{\bf S}_{\nu_{1}} \times \cdots \times {\bf S}_{\nu_{n-1}}}\!\!
(\mathop{\rm sgn\;\!}{\poorbold{$\sigma$}})\,(\mathop{\rm sgn\;\!}{\poorbold{$\sigma$}'})\, \left( \Prod\limits_{j=1}^{n-1}\Prod\limits_{m=1}^{\nu_{j}}
\int_{C_{j}} d\gamma_{j, m} \right)
F_{J,J'\!,\,\poorbold{$\scriptstyle\sigma$},\,\poorbold{$\scriptstyle\sigma$}'}(\{\gamma_{j,m}\})
$$
where ${\poorbold{$\sigma$}}=(\sigma_{1},\cdots,\sigma_{n-1})\in{\bf S}_{\nu_{1}}
\times\cdots\times{\bf S}_{\nu_{n-1}}\,$,
\ $\mathop{\rm sgn\;\!}{\poorbold{$\sigma$}}=\Prod\limits_{j=1}^{n-1}(\mathop{\rm sgn\;\!}\sigma_{j})\,$ and
\begin{eqnarray}
F_{J,J'\!,\,\poorbold{$\scriptstyle\sigma$},\,\poorbold{$\scriptstyle\sigma$}'}(\{\gamma_{j,m}\}) & \!\!\!=\!\!\! &
K(\{\gamma_{j, m}\};\beta_{1},\cdots,\beta_{N}; \mu_{1},\cdots,\mu_{n})
\nonumber
\\[4pt]
& \!\!\!\times\!\!\! &
g_{J}^{(\rho)}(\{\gamma_{j,\sigma_{j}(m)}\};\beta_{1},\cdots,\beta_{N})\,
g_{J'}^{(\lambda)}(\{\gamma_{j,\sigma_{j}'(m)}\};\beta_{1},\cdots,\beta_{N})\,.
\nonumber
\\[-.9\bls]\nonumber
\end{eqnarray}
A partial integrand $F_{J,J'\!,\,\poorbold{$\scriptstyle\sigma$},\,\poorbold{$\scriptstyle\sigma$}'}(\{\gamma_{j,m}\})$
does not have poles at some points (\ref{CONT1}) and (\ref{CONT2})
because of the zeros of
$g_{J}^{(\rho)}(\{\gamma_{j, \sigma(m)}\};\beta_{1},\cdots,\beta_{N})$.
So, given $J$ and $\poorbold{$\sigma$}$ there is a choice of integration contours
$C_j^{(J,\poorbold{$\scriptstyle\sigma$})}$ satisfying the required conditions for the actual poles of
$F_{J,J'\!,\,\poorbold{$\scriptstyle\sigma$},\,\poorbold{$\scriptstyle\sigma$}'}(\{\gamma_{j,m}\})$ for arbitrary
$J'\!,\,\poorbold{$\sigma$}'$.
Similarly, given $J'$ and $\poorbold{$\sigma$}'$ there is a choice of integration contours
$C_{j,(J'\!,\poorbold{$\scriptstyle\sigma$}')}$ satisfying the required conditions for the actual poles
of $F_{J,J'\!,\,\poorbold{$\scriptstyle\sigma$},\,\poorbold{$\scriptstyle\sigma$}'}(\{\gamma_{j,m}\})$ for arbitrary $J,\,\poorbold{$\sigma$}$.
Finally, one can easily check that the integrals of the term
$F_{J,J'\!,\,\poorbold{$\scriptstyle\sigma$},\,\poorbold{$\scriptstyle\sigma$}'}(\{\gamma_{j,m}\})$ over the contours
$C_j^{(J,\poorbold{$\scriptstyle\sigma$})}$ and over the contours $C_{j,(J'\!,\poorbold{$\scriptstyle\sigma$}')}$ are equal.

\vsk.5>
In this paper we assume that $\rho$ and $\lambda$ are large positive.
Then, as we will see in the next section, there is a region of the parameters
$\mu_{1}, \cdots , \mu_{n}$ where the integral (\ref{001}) is absolutely
convergent (see (\ref{B})).
\vsk.5>

Consider the vector representation $V$ of $\,sl_n\,$:
\begin{equation}
\tsize V\,=\,\bigoplus\limits_{j=0}^{n-1}\,{\Bbb C}\,v_j\,.
\end{equation}

\begin{thm}\label{thm2.2}
For $J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}$, we set
\begin{eqnarray}
f_{J}=\Sum\limits_{J'\in{\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}I(w_{J'}^{(\rho)}, w_{J}^{(\lambda)})v_{J'},
\quad {\it where} \quad v_{J'}=v_{J_{1}'}\otimes\cdots\otimes v_{J_{N}'}.
\end{eqnarray}
Then $f_{J}$ is a solution to (\ref{qKZ}).
\end{thm}

\noindent{\it Proof.}\quad
For $J=(J_{1}, \cdots , J_{N}) \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}$,
we set
$w_{J_{1}, \cdots , J_{N}}^{(\rho)}=w_{J}^{(\rho)}$.
In the same way as the proof of Lemma1 and Lemma 3 in \cite{MT2},
we can show the following formulae:
\begin{eqnarray}
& & w_{J_{1}, \cdots , J_{k+1}, J_{k}, \cdots , J_{N}}^{(\rho)}
(\beta_{1}, \cdots , \beta_{k+1}, \beta_{k}, \cdots , \beta_{N})\nonumber \\
& & {}=\Sum\limits_{J_{k}', J_{k+1}'}R^{(\rho)}(\beta_{k}-\beta_{k+1})_{J_{k}, J_{k+1}}^{J_{k}', J_{k+1}'}
w_{J_{1}, \cdots , J_{k}', J_{k+1}', \cdots , J_{N}}^{(\rho)}
(\beta_{1}, \cdots , \beta_{k}, \beta_{k+1}, \cdots , \beta_{N}), \label{pf00} \\
& & I(g_{J_{N}, J_{1}, \cdots , J_{N-1}}^{(\rho)}(\beta_{N}, \beta_{1}, \cdots , \beta_{N-1}),
w^{(\lambda)}(\beta_{1}, \cdots , \beta_{N}))|_{\beta_{N} \to \beta_{N}-\lambda i} \nonumber \\
& & {}=e^{-\frac{2\pi i}{\rho}\mu_{J_{N}+1}}
I(g_{J_{1}, \cdots , J_{N}}^{(\rho)}(\beta_{1}, \cdots , \beta_{N-1}, \beta_{N}),
w^{(\lambda)}(\beta_{1}, \cdots , \beta_{N})), \label{pf01}
\end{eqnarray}
where $w^{(\lambda)} \in {\cal F}_{\nu_{1}, \cdots , \nu_{n-1}}^{(\lambda)}(\beta_{1}, \cdots , \beta_{N})$ and
the left hand side of (\ref{pf01}) is understood
as the analytic continuation of the integral.
It is easy to prove Theorem \ref{thm2.2} from (\ref{pf00}) and (\ref{pf01}).
\quad$\square$
\vsk>

We note that the weight of the solution $\psi_{J}$ is given by
\begin{eqnarray}
\Sum\limits_{j=1}^{n}\lambda_{j}\eps_{j},\quad{\rm where}\quad\lambda_{j}=\nu_{j-1}-\nu_{j}.
\end{eqnarray}
Now we set
\begin{eqnarray}
\Psi_\xi(\beta_{1}, \cdots , \beta_{N})=
\Sum\limits_{J, J' \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})v_{J}^{(\rho)} \otimes v_{J'}^{(\lambda)},
\end{eqnarray}
where
\begin{eqnarray}
\xi=\Sum\limits_{j=1}^{n}\lambda_{j}\eps_{j}.
\end{eqnarray}
Then $\Psi_\xi$ is the fundamental matrix solution mentioned in Section 1.

\section{Determinant formula for the solutions}

In the following sections we calculate the determinant
\begin{eqnarray}
D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N})=
\det{\left(I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})
\right)_{J, J' \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}}.
\end{eqnarray}
The result is as follows.
\begin{thm}\label{thm6.1}
\begin{eqnarray}
& &
\det{\left(I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})
\right)_{J, J' \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}} \nonumber \\
& & {}=
2^{-d_{\nu_{1}, \cdots , \nu_{n-1}}\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}}
\exp{\Bigl( (N^{2}-N-2) \left(\frac{1}{\rho}+\frac{1}{\lambda}
\right) \frac{\pi^{2}i}{n}\;\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)} \Bigr)}
\nonumber \\
& & {}\times
\left(\frac{\sqrt{\rho\lambda}}{S_{2}(-\frac{2\pi}{n})} \right)^{\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}\Sum\limits_{j=1}^{n}(j-1)\lambda_{j}} \nonumber \\
& & {}\times
\Prod\limits_{1 \le r' < r \le n}
\left\{ \Prod\limits_{a=0}^{\min\{\lambda_{r}-1, \lambda_{r'}\}}
\left( \frac{S_{2}(\mu_{r}-\mu_{r'}-\frac{\pi}{n}(\lambda_{r}+\lambda_{r'}-2a))}
{S_{2}(\mu_{r}-\mu_{r'}+\frac{\pi}{n}(\lambda_{r}+\lambda_{r'}-2a))} \right)
^{\lambda_{r}+\lambda_{r'} \atopwithdelims() a} \right\}
^{\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}\big/
{\lambda_{r}+\lambda_{r'} \atopwithdelims() \lambda_{r}}} \nonumber \\
& & {}\times
\exp{\Bigl( \frac{2\pi}{\rho \lambda}
\left( \Sum\limits_{j=1}^{n}
{N-1 \atopwithdelims() \lambda_{1}, \cdots , \lambda_{j}-1, \cdots , \lambda_{n}}\mu_{j} \right)
\Sum\limits_{m=1}^{N}\beta_{m} \Bigr)} \nonumber \\
& & \times{}
\left( \Prod\limits_{1 \le r < s \le N}\frac{S_{2}(i(\beta_{r}-\beta_{s})+
\frac{2\pi}{n})}{S_{2}(i(\beta_{r}-\beta_{s})-\frac{2\pi}{n})}
\right)^{\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}},
\label{107}
\end{eqnarray}
where
\begin{eqnarray}
& & \lambda_{j}=\nu_{j-1}-\nu_{j}, \quad d_{\nu_{1}, \cdots , \nu_{n-1}}=\Sum\limits_{j=1}^{n-1}(2\nu_{j}\nu_{j-1}+\nu_{j}^{2}-3\nu_{j}), \nonumber \\
& & \Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}={N \atopwithdelims() \lambda_{1}, \cdots , \lambda_{n}}, \quad
\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}=
\Sum\limits_{1 \le j<k \le n}
{N-2 \atopwithdelims() \lambda_{1}, \cdots , \lambda_{j}-1, \cdots , \lambda_{k}-1, \cdots ,\lambda_{n}}. \nonumber
\end{eqnarray}
\end{thm}

Note that
\begin{eqnarray}
\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}\bigg/
{\lambda_{r}+\lambda_{r'} \atopwithdelims() \lambda_{r}}=
{N \atopwithdelims() \lambda_{1}, \cdots , \lambda_{r'-1}, \lambda_{r'+1},
\cdots , \lambda_{r-1}, \lambda_{r+1}, \cdots , \lambda_{n}, \lambda_{r'}+\lambda_{r}}
\end{eqnarray}
is a positive integer.

First, we determine the dependence on $\beta_{1}, \cdots , \beta_{N}$
of $D_{\lambda_{1}, \cdots , \lambda_{n}}$.
{}From Theorem \ref{thm2.2}, we find that
\begin{eqnarray}
\frac{D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{m}-\lambda i, \cdots , \beta_{N})}
{D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{m}, \cdots , \beta_{N})} &=&
\det_{\lambda_{1}, \cdots , \lambda_{n}}K_{m}^{(\rho)}(\beta_{1}, \cdots , \beta_{N}), \label{201} \\
\frac{D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{m}-\rho i, \cdots , \beta_{N})}
{D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{m}, \cdots , \beta_{N})} &=&
\det_{\lambda_{1}, \cdots , \lambda_{n}}K_{m}^{(\lambda)}(\beta_{1}, \cdots , \beta_{N}). \label{202}
\end{eqnarray}
Here $\det_{\lambda_{1}, \cdots , \lambda_{n}}K_{m}$ stands for the determinant of the operator which
is a restriction of the operator $K_{m}$ to the weight subspace of the weight
$\Sum\limits_{j=1}^{n}\lambda_{j}\eps_{j}$.

Using formulae (\ref{eq01}) and (\ref{eq02}), we have
\begin{eqnarray}
& & \det_{\lambda_{1}, \cdots , \lambda_{n}}K_{m}^{(\rho)}(\beta_{1}, \cdots , \beta_{N})=\exp{\Bigl( -\frac{2\pi i}{\rho}\Sum\limits_{j=1}^{n}
{N-1 \atopwithdelims() \lambda_{1}, \cdots , \lambda_{j}-1, \cdots , \lambda_{n}}\mu_{j} \Bigr)} \nonumber \\
& & {}\times
\left( \Prod\limits_{m'=1}^{m-1}\frac{\mathop{\rm sh}\frac{\pi}{\rho}(\beta_{m}-\beta_{m'}-\lambda i+\frac{2\pi i}{n})}
{\mathop{\rm sh}\frac{\pi}{\rho}(\beta_{m}-\beta_{m'}-\lambda i-\frac{2\pi i}{n})}
\Prod\limits_{m'=m+1}^{N}\frac{\mathop{\rm sh}\frac{\pi}{\rho}(\beta_{m}-\beta_{m'}+\frac{2\pi i}{n})}
{\mathop{\rm sh}\frac{\pi}{\rho}(\beta_{m}-\beta_{m'}-\frac{2\pi i}{n})} \right)
^{\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}}.
\end{eqnarray}

Now we set
\begin{eqnarray}
E_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N})&=&
\exp{\Bigl( \frac{2\pi}{\rho \lambda}
\left( \Sum\limits_{j=1}^{n}
{N-1 \atopwithdelims() \lambda_{1}, \cdots , \lambda_{j}-1, \cdots , \lambda_{n}}\mu_{j} \right)
\Sum\limits_{m=1}^{N}\beta_{m} \Bigr)} \nonumber \\
& & \times{}
\left( \Prod\limits_{1 \le r < s \le N}\frac{S_{2}(i(\beta_{r}-\beta_{s})+\frac{2\pi}{n})}
{S_{2}(i(\beta_{r}-\beta_{s})-\frac{2\pi}{n})} \right)
^{\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}}.
\label{202.1}
\end{eqnarray}
Then by using (\ref{app3}) we can check that
$E_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N})$ satisfies (\ref{201})
and (\ref{202})\,.  Therefore, we have

\begin{prop}\label{prop4.1}
\begin{eqnarray}
D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N})=c_{\lambda_{1}, \cdots , \lambda_{n}}(\mu_{1}, \cdots , \mu_{n} ; \rho, \lambda)
E_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N}),
\label{202.2}
\end{eqnarray}
where $c_{\lambda_{1}, \cdots , \lambda_{n}}(\mu_{1}, \cdots , \mu_{n} ; \rho, \lambda)$ is a constant
independent of $\beta_{1}, \cdots , \beta_{N}$.
\end{prop}

In order to determine $c_{\lambda_{1}, \cdots , \lambda_{n}}(\mu_{1},\cdots ,\mu_{n};\rho,\lambda)$,
we consider the asymptotics of $D/E$ as
\begin{eqnarray}
\beta_{1}\ll \cdots\ll \beta_{N}\,. \label{202.9}
\end{eqnarray}
This is in the next section.

\section{Asymptotics of the solutions}

First, we consider the asymptotics of $D_{\lambda_{1}, \cdots , \lambda_{n}}$.

We denote the set of variables
\begin{eqnarray}
\gamma_{j,m}\quad(0\leq j\leq n-1\,;\ \ 1\leq m\leq\nu_j)
\end{eqnarray}
by $\gamma$.
Fix a set of permutations $\sigma=(\sigma_1,\ldots,\sigma_{n-1})$;
$\sigma_j\in S_{\nu_j}$ $(1\leq j\leq n-1)$. We use $\sigma_0={\rm id}$.
We denote
\begin{eqnarray}
\gamma_{j,\sigma_j(m)}\quad(0\leq j\leq n-1\,;\ \ 1\leq m\leq\nu_j)
\end{eqnarray}
by $\gamma_\sigma$.

Consider
\begin{eqnarray}
F_{J,J',\sigma}(\gamma)
=K(\gamma)g_J(\gamma)g_{J'}(\gamma_\sigma),
\end{eqnarray}
where $K(\gamma)$ is given by (\ref{Kdef}).

In the following, we use the abbreviation $\beta_{ij}=\beta_{i}-\beta_{j}$.

\begin{prop}\label{prop1}
Suppose that $\beta_1<\cdots<\beta_n$ and $\gamma_{j,m}$'s are all real.
If $\lambda$ is sufficiently large, then
there exist positive constants $\varepsilon,C,\kappa$ independent
of the variables $\beta$ and $\gamma$ such that the following estimate holds.
\begin{eqnarray}
|F_{J,J',\sigma}|&<&C
\exp{\Bigl( -\kappa\Sum\limits_{1\leq j\leq n-1\atop1\leq m\leq\nu_j}
|\gamma_{j,m}-\gamma_{j-1,m^*(J,j,m)}|\Bigr)}\nonumber\\
&\times&\exp{\Bigl( -{2\pi^2\over\rho\lambda n}
\Sum\limits_{1\leq r<s\leq N}(1-\delta_{J_r,J_s})\beta_{sr}
+{2\pi\over\rho\lambda}\Sum\limits_{r=1}^{N}\mu_{J_r+1}\beta_r \Bigr)}
\end{eqnarray}
if
\begin{eqnarray}
{2\pi\over\rho\lambda}(\mu_{j+1}-\mu_j)>\varepsilon,\quad
{2\pi\over\rho\lambda}(\mu_n-\mu_1)<n\varepsilon.
\label{B}
\end{eqnarray}
\end{prop}

\noindent{\it Proof.}\quad
Throughout the proof, we set $r_{j, m}=r_{j, m}^{J}$.
We define new variables $\tilde\gamma$ by
\begin{eqnarray}
\gamma_{j,m}=\tilde\gamma_{j,m}+\beta_{r_{j,m}}.
\end{eqnarray}
Note that $\tilde\gamma_{0,m}=0$. From (\ref{app2}), we have
\begin{eqnarray}
e^{{2\pi\over\rho\lambda}\Sum\limits_{0\leq j\leq n-1\atop1\leq m\leq\nu_j}
(\mu_{j+1}-\mu_j)\gamma_{j,m}}
&=&
e^{{2\pi\over\rho\lambda}\Sum\limits_{1\leq j\leq n-1\atop1\leq m\leq\nu_j}
(\mu_{j+1}-\mu_j)\tilde\gamma_{j,m}
+{2\pi\over\rho\lambda}\Sum\limits_{1\leq r\leq N}\mu_{J_r+1}\beta_r},\nonumber\\
|\varphi(\gamma_{j,m}-\gamma_{j-1,m'})|&\leq&{\rm const.}\,
e^{-\pi({1\over\rho}+{1\over\lambda}+{2\pi\over\rho\lambda n})
|\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m'}+\beta_{r_{j,m}r_{j-1,m'}}|},
\nonumber\\
|\psi(\gamma_{j,m}-\gamma_{j,m'})|&\leq&{\rm const.}\,
e^{-\pi({1\over\rho}+{1\over\lambda}-{4\pi\over\rho\lambda n})
|\tilde\gamma_{j,m}-\tilde\gamma_{j,m'}+\beta_{r_{j,m}r_{j,m'}}|},
\\
|g_J(\gamma)|
&\leq&{\rm const.}\,
\Prod\limits_{1\leq j\leq n-1}
\Bigl\{
\Prod\limits_{1\leq m\leq\nu_j}
e^{-{\pi\over\rho}(\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m^*(J,j,m)})}\nonumber
\\
&\times&\!\!\!\!\!\!\!\!
\Prod\limits_{1\leq m\leq\nu_j \atop {1 \le m' \le \nu_{j-1} \atop m'\not=m^*(J,j,m)}}
\!\!\!\!\!\!
e^{{\pi\over\rho}|\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m'}
+\beta_{r_{j,m}r_{j-1,m'}}|}
\!\!\!\!\Prod\limits_{1\leq m<m'\leq\nu_j}\!\!\!\!
e^{{\pi\over\rho}
|\tilde\gamma_{j,m}-\tilde\gamma_{j,m'}+\beta_{r_{j,m}r_{j,m'}}|}
\Bigr\},\nonumber
\end{eqnarray}
\begin{eqnarray}
|g_{J'}(\gamma_\sigma)| &\leq&{\rm const.}\!\!\Prod\limits_{1\leq j\leq n-1}
\Biggl\{\Prod\limits_{1\leq m\leq\nu_j}e^{-{\pi\over\lambda}
(\tilde\gamma_{j,\sigma_j(m)}-\tilde\gamma_{j-1,\sigma_{j-1}(m^*(J',j,m))}
+\beta_{r_{j,\sigma_j(m)}r_{j-1,\sigma_{j-1}(m^*(J',j,m))}})}
\nonumber
\end{eqnarray}
\begin{eqnarray}
&&\times\Prod\limits_{1\leq m\leq\nu_j\atop m'\not=m^*(J',j,m)}\!\!\!\!
e^{{\pi\over\lambda}|\tilde\gamma_{j,\sigma_j(m)}
-\tilde\gamma_{j-1,\sigma_{j-1}(m')}
+\beta_{r_{j,\sigma_j(m)}r_{j-1,\sigma_{j-1}(m')}}|} \\
&&\times\Prod\limits_{1\leq m<m'\leq \nu_j}\!\!\!\!
e^{{\pi\over\lambda}|\tilde\gamma_{j,m}-\tilde\gamma_{j,m'}
+\beta_{r_{j,m}r_{j,m'}}|}
\Biggl\}. \nonumber
\end{eqnarray}
Therefore, we have
\begin{eqnarray}
|F_{J,J',\sigma}(\gamma)|&\leq&{\rm const.}\,
e^{{2\pi\over\rho\lambda}\Sum\limits_{1\leq j\leq n-1\atop1\leq m\leq\nu_j}
(\mu_{j+1}-\mu_j)\tilde\gamma_{j,m}
+{2\pi\over\rho\lambda}\Sum\limits_{1\leq r\leq N}\mu_{J_r+1}\beta_r}
\nonumber\\&\times&
\Prod\limits_{{1\leq j\leq n-1\atop1\leq m\leq\nu_j}\atop1\leq m'\leq\nu_{j-1}}
e^{-{2\pi^2\over\rho\lambda n}
|\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m'}+\beta_{r_{j,m}r_{j-1,m'}}|}
\nonumber\\&\times&
\Prod\limits_{1\leq j\leq n-1\atop1\leq m<m'\leq\nu_j}
e^{{4\pi^2\over\rho\lambda n}
|\tilde\gamma_{j,m}-\tilde\gamma_{j,m'}+\beta_{r_{j,m}r_{j,m'}}|}\label{D}
\\&\times&
\Prod\limits_{1\leq j\leq n-1\atop1\leq m\leq \nu_j}
e^{-{\pi\over\rho}\xi(\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m^*(J,j,m)})}
\nonumber\\&\times&
\Prod\limits_{1\leq j\leq n-1\atop1\leq m\leq \nu_j}
e^{-{\pi\over\lambda}\xi(\tilde\gamma_{j,\sigma(m)}
-\tilde\gamma_{j-1,\sigma(m^*(J',j,m))}
+\beta_{r_{j,\sigma(m)}r_{j-1,\sigma(m^*(J',j,m))}})}\nonumber
\end{eqnarray}
Here
\begin{eqnarray}
\xi(x)=x+|x|.\label{A}
\end{eqnarray}

We apply $-|A+B|\leq|A|-|B|$ to the second line of (\ref{D}),
and $|A+B|\leq|A|+|B|$ to the third line. Then, we use
\begin{eqnarray}
\Sum\limits_{{1\leq j\leq n-1\atop1\leq m\leq\nu_j}\atop1\leq m'\leq\nu_{j-1}}
|\beta_{r_{j,m}r_{j-1,m'}}|&=&
\Sum\limits_{1\leq r<s\leq N}(2\min(J_r,J_s)+1-\delta_{J_r,J_s})
\beta_{sr},\\[12pt]
\Sum\limits_{1\leq j\leq n-1\atop1\leq m<m'\leq\nu_j}
|\beta_{r_{j,m}r_{j,m'}}|&=&
\Sum\limits_{1\leq r<s\leq N}\min(J_r,J_s)\beta_{sr}.
\end{eqnarray}
We ignore the last line of (\ref{D}). After all these steps, it is enough to
show
\begin{eqnarray}
&&{2\pi\over\rho\lambda}\Sum\limits_{1\leq j\leq n-1\atop1\leq m\nu_j}
(\mu_{j+1}-\mu_j)\tilde\gamma_{j,m}
+{2\pi^2\over\rho\lambda n}\!\!\!\!
\Sum\limits_{{1\leq j\leq n-1\atop1\leq m\leq\nu_j}\atop1\leq m'\leq\nu_{j-1}}
|\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m'}|
\nonumber\\&&
+{4\pi^2\over\rho\lambda n}\!\!\!\!
\Sum\limits_{1\leq j\leq n-1\atop1\leq m<m'\leq\nu_j}
|\tilde\gamma_{j,m}-\tilde\gamma_{j,m'}|
-{\pi\over\rho}\Sum\limits_{1\leq j\leq n-1\atop1\leq m\leq\nu_j}
\xi(\tilde\gamma_{j,m}-\tilde\gamma_{j-1,m^*(J,j,m)})
\nonumber\\&&\quad
<-\kappa\Sum\limits_{1\leq j\leq n-1\atop1\leq m\leq\nu_j}
|\gamma_{j,m}-\gamma_{j-1,m^*(J,j,m)}|.\label{C}
\end{eqnarray}
The left hand side is not larger than
\begin{eqnarray}
\Sum\limits_{1\leq j\leq n-1\atop1\leq m\leq\nu_j}
\Bigl\{
{2\pi\over\rho\lambda}
(\mu_{j_{\rm max}(J,j,m)+1}-\mu_j)(\gamma_{j,m}-\gamma_{j-1,m^*(J,j,m)})
\nonumber\\
+{K\over\rho\lambda}|\gamma_{j,m}-\gamma_{j-1,m^*(J,j,m)}|
-{\pi\over\rho}\xi(\gamma_{j,m}-\gamma_{j-1,m^*(J,j,m)})\Bigr\},
\end{eqnarray}
where
\begin{eqnarray}
K&=&{2\pi^2\over n}\!\!\!\!
\Sum\limits_{{1\leq j\leq n-1\atop1\leq m\leq\nu_j}\atop1\leq m'\leq\nu_{j-1}}1
+{4\pi^2\over n}\!\!\!\!
\Sum\limits_{1\leq j\leq n-1\atop1\leq m<m'\leq\nu_j}1,
\end{eqnarray}
and
\begin{eqnarray}
j_{\rm max}(J,j,m)&=&\max\{j';r_{j,m}\in{\cal N}_{j'}^{J}\}.
\end{eqnarray}

Choose $\varepsilon,\kappa$ so that
\begin{eqnarray}
n\varepsilon+{K\over\rho\lambda}-{2\pi\over\rho}&<&-\kappa,\\
\varepsilon-{K\over\rho\lambda}&>&\kappa.
\end{eqnarray}
This is possible if
\begin{eqnarray}
{2K\over\rho\lambda}<{2\pi\over\rho}.
\end{eqnarray}
Then, the estimate (\ref{C}) follows from (\ref{A}) and (\ref{B}).
\quad$\square$
\vsk>

For $J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}$, we set
\begin{eqnarray}
P_{J}=\exp{ \Bigl(
\frac{2\pi^2 }{\rho \lambda n}\Sum\limits_{1 \le r < s \le N}(1-\delta_{J_{r}, J_{s}})\beta_{sr}
-\frac{2\pi}{\rho \lambda}\Sum\limits_{r=1}^{N}\mu_{J_{r}+1}\beta_{r} \Bigr)}.
\label{PJ}
\end{eqnarray}

The following is an obvious consequence of Proposition \ref{prop1}.
\begin{coro}
The integral (\ref{Kdef}) is absolutely convergent. The convergence
is uniform in the variables $\beta$ if we multiply $P_J$ to the integrand.
\end{coro}

Define a partial order in ${\cal Z}_{\nu_1,\ldots,\nu_n}$:
\begin{eqnarray}
J\leq J'\quad\hbox{if and only if}\quad J_r+\cdots+J_N\leq J'_r+\cdots+J'_N
\quad\hbox{for all }\ r\,.
\end{eqnarray}

\begin{prop}\label{prop2}
If $J\not\leq J'$, then we have
\begin{eqnarray}
\mathop{\rm lim\;\!}\nolimits_{\beta_1\ll \cdots\,\ll \beta_N} P_{J}
\left( \Prod\limits_{j=1}^{n-1}\Prod\limits_{m=1}^{\nu_{j}}
\int_{C_{j}} d\gamma_{j, m} \right)
F_{J,J',\sigma}(\gamma)=0.
\label{E}
\end{eqnarray}
\end{prop}

\noindent{\it Proof.}\quad
We follow the estimate in the proof of Proposition \ref{prop1}.
When we go from (\ref{D}) to (\ref{C}), we dropped the last line
in (\ref{D}). This time we use that term. Namely, we can claim that (\ref{E})
holds unless for some $\sigma=(\sigma_1,\ldots,\sigma_{n-1})$
\begin{eqnarray}
r_{j,\sigma_j(m)}^{J}\leq
r_{j-1,\sigma_{j-1}(m^*(J',j,m))}^{J}\label{F}
\end{eqnarray}
holds for all $j$ and $m$. This is clear because
\begin{eqnarray}
\xi(x+y)\,=\,2(x+y)\quad{\rm if}\quad y>-x\,.
\end{eqnarray}
We show that (\ref{F}) implies $J\leq J'$. This will complete the proof.

First we prove
\begin{eqnarray}
r_{j,\sigma_j(m)}^{J}\leq r_{j,m}^{J'}\quad\hbox{for all }\ m \label{G}
\end{eqnarray}
by induction on $j$. The case $j=0$ is obvious. Suppose that (\ref{G}) is
true for $j-1$. Then we have
\begin{eqnarray}
r_{j,\sigma_{j}(m)}^{J}\,\leq\,r_{j-1,\sigma_{j-1}(m^*(J',j,m))}^{J}
\,\leq\,r_{j-1,m^*(J',j,m)}^{J'}\,=\,r_{j,m}^{J'}\,.
\end{eqnarray}
Therefore, (\ref{G}) is true for all $j$. It follows from (\ref{G}) that
\begin{eqnarray}
r_{j,m}^{J}\leq r_{j,m}^{J'}\quad\hbox{for all \ $j,m$}\,.
\end{eqnarray}
Finally, we prove $J\leq J'$. This is clear because
\begin{eqnarray}
J_r+\cdots+J_N\,=\,\#\{(j,m);r_{j,m}^{J}\geq r.\}
\end{eqnarray}
The proof of Proposition \ref{prop2} is over.\quad$\square$
\vsk>

\noindent
This proposition shows that in the asymptotic limit the matrix
$\left(I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})\right)_{J,J'}$
is triangular.

We have also
\begin{prop}\label{prop3}
\begin{eqnarray}
\mathop{\rm lim\;\!}\nolimits_{\beta_1\ll \cdots\,\ll \beta_N} P_{J}
\left( \Prod\limits_{j=1}^{n-1}\Prod\limits_{m=1}^{\nu_{j}}
\int_{C_{j}} d\gamma_{j, m} \right)
F_{J,J,\sigma}(\gamma)=0
\end{eqnarray}
unless $\sigma_j={\rm id}$ for all $j$.
\end{prop}

\noindent{\it Proof.}\quad
Suppose that
\begin{eqnarray}
r_{j,\sigma_j(m)}^{J}\leq r_{j-1,\sigma_{j-1}(m^*(J,j,m))}^{J}
\end{eqnarray}
for all $j,m$. From the proof of Proposition \ref{prop2}
we have
\begin{eqnarray}
r_{j,\sigma_j(m)}^{J}\leq r_{j,m}^{J}\quad\hbox{for all \ $j,m$.}
\end{eqnarray}
This implies that $\,\sigma_j={\rm id}\;$ for all $j$.
\quad$\square$
\vsk>

Define
\begin{eqnarray}
\nu^{J, +}_{j,r}=\#\{s\in{\cal N}_{j}^{J};r<s\}\,,
\qquad\nu^{J, -}_{j,r}=\#\{s\in{\cal N}_{j}^{J};r>s\}.
\end{eqnarray}
{}From (\ref{app2}), we have
\begin{prop}\label{prop4}
\begin{eqnarray}
&&\mathop{\rm lim\;\!}\nolimits_{\beta_1\ll \cdots\,\ll \beta_N}
P_{J}
\left( \Prod\limits_{j=1}^{n-1}\Prod\limits_{m=1}^{\nu_{j}}
\int_{C_{j}} d\gamma_{j, m} \right)
F_{J,J,{\rm id}}(\gamma)\nonumber\\
&&=
{e^{({1\over\rho}+{1\over\lambda}){\pi^2 i\over n}\Sum\limits_{j=1}^{n-1}
\{\nu_j(\nu_{j-1}-1)-\nu_j(\nu_j-1)\}}\over 2^{d_{\nu_{1}, \cdots , \nu_{n-1}}}}\Prod\limits_{r=1}^N
G_{J_r}(\tilde\mu_{1, r}^{J},\ldots,\tilde\mu_{n, r}^{J})
\label{301}
\end{eqnarray}
where
\begin{eqnarray}
G_k(\mu_1,\ldots,\mu_n)&=&
\Prod\limits_{j=1}^k\int_{C_{j}}{d\gamma_j\over2\pi i}\Prod\limits_{j=1}^k
\varphi(\gamma_j-\gamma_{j-1})
e^{{2\pi\over\rho\lambda}\Sum\limits_{j=1}^k(\mu_{j+1}-\mu_j)\gamma_j},\\
\tilde\mu_{j, r}^{J}&=&
\mu_j+{\pi\over n}\Sum\limits_{\varepsilon=\pm}\varepsilon
(\nu^{J, \varepsilon}_{j,r}-\nu^{J, \varepsilon}_{j-1,r})
-\delta_{j, J_{r}+1}\frac{\rho+\lambda}{2}.
\end{eqnarray}
In the above formula of $G_{k}$,
$\gamma_{0}=0$ and
the contour $C_{j}$ for $\gamma_{j}$ is a deformation
of the real line $(-\infty, \infty)$ such that the poles at
\begin{eqnarray}
\gamma_{j-1}-\frac{\pi i}{n}+\rho i {\Bbb Z}_{\ge 0}+\lambda i {\Bbb Z}_{\ge 0}
\end{eqnarray}
are above $C_{j}$ and the poles at
\begin{eqnarray}
\gamma_{j-1}+\frac{\pi i}{n}-\rho i {\Bbb Z}_{\ge 0}-\lambda i {\Bbb Z}_{\ge 0}
\end{eqnarray}
are below $C_{j}$.
\end{prop}
This proposition shows that in the asymptotic limit the diagonal element
$I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})$ reduces to the one point functions
$G_{J_r}$ $(1\leq r\leq N)$.

Now we consider the asymptotics of $D_{\lambda_{1}, \cdots , \lambda_{n}}/E_{\lambda_{1}, \cdots , \lambda_{n}}$.
Hereafter we use the notation $\sim$ as follows:
\begin{eqnarray}
f(\beta_{1}, \cdots , \beta_{N}) \sim g(\beta_{1}, \cdots , \beta_{N})
\quad \stackrel{\rm def}{\Longleftrightarrow} \quad
\mathop{\rm lim\;\!}\nolimits_{\beta_1\ll \cdots\,\ll \beta_N}\left\{\,
\frac{f(\beta_{1},\cdots, \beta_{N})}{g(\beta_{1},\cdots,\beta_{N})}\,\right\}
\,=\,1\,.
\end{eqnarray}
{}From (\ref{app1}), we have
\begin{eqnarray}
E_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N}) &\sim&
\exp{ \Bigl( \frac{2\pi}{\rho \lambda}
\left( \Sum\limits_{j=1}^{n}
{N-1 \atopwithdelims() \lambda_{1}, \cdots , \lambda_{j}-1, \cdots , \lambda_{n}}\mu_{j} \right)
\Sum\limits_{m=1}^{N}\beta_{m} \Bigr)} \\
& & {}\times
\exp{ \Bigl( \left\{
\left( \frac{1}{\rho}+\frac{1}{\lambda} \right)\frac{2\pi^2 i}{n}
-\frac{4\pi^{2}}{n\rho \lambda}\Sum\limits_{1 \le r<s \le N}\beta_{sr} \right\}
\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)} \Bigr)}. \nonumber
\end{eqnarray}
We note that
\begin{eqnarray}
& &\Prod\limits_{J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}} \!\!\!\! P_{J} \\
& &{}=
\exp{\Bigl( \frac{4\pi^{2}}{n\rho \lambda}\Sum\limits_{1 \le r<s \le N}\beta_{sr}\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}
-\frac{2\pi}{\rho\lambda}
\left( \Sum\limits_{j=1}^{n}
{N-1 \atopwithdelims() \lambda_{1}, \cdots , \lambda_{j}-1, \cdots , \lambda_{n}}\mu_{j} \right)
\Sum\limits_{m=1}^{N}\beta_{m} \Bigr)}. \nonumber
\end{eqnarray}
Hence we find
\begin{eqnarray}
& & \frac{D_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N})}{E_{\lambda_{1}, \cdots , \lambda_{n}}(\beta_{1}, \cdots , \beta_{N})} \nonumber \\
& & {}\sim
\exp{\Bigl(-\left(\frac{1}{\rho}+\frac{1}{\lambda}\right)\frac{2\pi^{2}i}{n}
\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)} \Bigr)}
\det{\left(P_{J} I(w_{J}^{(\rho)}, w_{J'}^{(\lambda)})
\right)_{J, J' \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}}.
\end{eqnarray}
{}From Propositions \ref{prop2}, \ref{prop3}, and \ref{prop4}, we see that
\begin{eqnarray}
\det{\left(P_{J} I(w_{J}^{(\rho)},w_{J'}^{(\lambda)})
\right)_{J,J'\in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}}\sim
\Prod\limits_{J\in{\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}(\hbox{the right hand side of \,(\ref{301})}).
\end{eqnarray}
Therefore, we get
\begin{prop}\label{prop4.2}
\begin{eqnarray}
\kern-4em &&
c_{\lambda_{1}, \cdots , \lambda_{n}}(\mu_{1}, \cdots , \mu_{n} ; \rho , \lambda )\,=\,
2^{-d_{\nu_{1}, \cdots , \nu_{n-1}}\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}}
\label{203}
\\[6pt]
\kern-4em &&
{}\times\,\exp{\Bigl((N^{2}-N-2)\left(\frac{1}{\rho}+\frac{1}{\lambda}
\right) \frac{\pi^{2}i}{n}\;\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)} \Bigr)}\,
\Prod\limits_{J \in{\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}\Prod\limits_{r=1}^{N}G_{J_{r}}
(\widetilde{\mu}_{1, r}^{J}, \cdots , \widetilde{\mu}_{n, r}^{J})\,.\nonumber
\end{eqnarray}
\end{prop}

\noindent{\it Proof.}\quad
Note that
\vv-.6>
\begin{eqnarray}
\# {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}=\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}.
\end{eqnarray}
We get the term
\vv-.5>
\begin{eqnarray}
\exp{\Bigl( (N^{2}-N-2)\left( \frac{1}{\rho}+\frac{1}{\lambda}\right)
\frac{\pi^{2}i}{n}\;\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}\Bigr)}
\end{eqnarray}
by using the following formulae:
\begin{eqnarray}
& & \Sum\limits_{j=1}^{n-1}\{ \nu_{j}(\nu_{j-1}-1)-\nu_{j}(\nu_{j}-1) \}=
\frac{1}{2}\left( N^{2}-\Sum\limits_{j=1}^{n}\lambda_{j}^{2} \right),
\nonumber \\
& & \frac{1}{2}\left(N^{2}-\Sum\limits_{j=1}^{n}\lambda_{j}^{2} \right)\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}=
N(N-1)\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(2)}.
\end{eqnarray}
$\square$

\section{Proof of Theorem \protect\ref{thm6.1}}

First, we find an explicit formula for
$G_{k}(\mu_{1}, \cdots , \mu_{n})$. We set
\begin{eqnarray}
H_{k}(x_{1}, \cdots , x_{k})=\left( \Prod\limits_{j=1}^{k}\int_{C_{j}}d\gamma_{j} \right)
\Prod\limits_{j=1}^{k}\varphi(\gamma_{j}-\gamma_{j-1})
e^{\frac{2\pi}{\rho\lambda}\Sum\limits_{j=1}^{k}x_{j}(\gamma_{j}-\gamma_{j-1})},
\label{500}
\end{eqnarray}
where $\gamma_{0}=0$. Then we have
\begin{eqnarray}
G_{k}(\mu_{1}, \cdots , \mu_{n})=H_{k}(\mu_{k+1}-\mu_{1}, \cdots,
\mu_{k+1}-\mu_{k}).
\label{501}
\\[-.9\bls]\nonumber
\end{eqnarray}
The integral (\ref{500}) is absolutely convergent if
\begin{eqnarray}
|\mathop{\rm Re\;\!} x_{j}|\,<\,\frac{\rho+\lambda}{2}+\frac{\pi}{n},\qquad(j=1,\cdots,n).
\end{eqnarray}
By changing the integration variables $\gamma_{j}$ to
\vv.3>
\begin{eqnarray}
u_{j}=\gamma_{j}-\gamma_{j-1}, \qquad (j=1, \cdots , n)\,,
\\[-.7\bls]\nonumber
\end{eqnarray}
we can see that
\vv->
\begin{eqnarray}
H_{k}(x_{1}, \cdots , x_{k})=\Prod\limits_{j=1}^{k}H(x_{j}),
\label{relH}
\end{eqnarray}
where
\vv->
\begin{eqnarray}
H(x)=\int_{C}du\, \varphi(u)\, e^{\frac{2\pi}{\rho\lambda}xu}.
\label{defH}
\end{eqnarray}
In the above formula, the contour $C$ is a deformation of the real line $(-\infty, \infty)$
such that the poles at
\begin{eqnarray}
-\frac{\pi i}{n}+\rho i{\Bbb Z}_{\ge 0}+\lambda i {\Bbb Z}_{\ge 0}
\end{eqnarray}
are above $C$ and the poles at
\begin{eqnarray}
\frac{\pi i}{n}-\rho i {\Bbb Z}_{\ge 0}-\lambda i {\Bbb Z}_{\ge 0}
\end{eqnarray}
are below $C$.

The explicit formula for the function $H$ is obtained in \cite{MT1}.

\begin{prop}\label{5.1}
\begin{eqnarray}
H(x)=\frac{\sqrt{\rho\lambda}}{S_{2}(-\frac{2\pi}{n})}
     \frac{S_{2}(x+\frac{\rho+\lambda}{2}-\frac{\pi}{n})}{S_{2}(x+\frac{\rho+\lambda}{2}+\frac{\pi}{n})}
\label{form1}
\end{eqnarray}
\end{prop}

\nt
{}From (\ref{501}), (\ref{relH}) and (\ref{form1}), we get

\begin{prop}\label{prop5.4}
\begin{eqnarray}
G_{k}(\mu_{1}, \cdots , \mu_{n})=
\left( \frac{\sqrt{\rho\lambda}}{S_{2}(-\frac{2\pi}{n})} \right)^{k}
\Prod\limits_{j=1}^{k}\frac{S_{2}(\mu_{k+1}-\mu_{j}+\frac{\rho+\lambda}{2}-
\frac{\pi}{n})}
{S_{2}(\mu_{k+1}-\mu_{j}+\frac{\rho+\lambda}{2}+\frac{\pi}{n})}\;.
\label{512}
\end{eqnarray}
\end{prop}

\nt
Now it remains to calculate
\begin{eqnarray}
\nonumber\\[-24pt]
\Prod\limits_{J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}} \Prod\limits_{r=1}^{N}
G_{J_{r}}(\widetilde{\mu}_{1, r}^{J}, \cdots , \widetilde{\mu}_{n, r}^{J}).
\end{eqnarray}
We set
\begin{eqnarray}
M_{j, k}^{J, +}= \# \{ r ; J_{r}=j, k<r \}, \quad
M_{j, k}^{J, -}= \# \{ r ; J_{r}=j, k>r \}.
\end{eqnarray}
Note that
\begin{eqnarray}
\Sum\limits_{r=1}^{n}J_{r}=\Sum\limits_{j=1}^{n}(j-1)\lambda_{j}, \quad {\rm for \,all} \quad
J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}.
\end{eqnarray}
{}From (\ref{301}) and (\ref{512}), we have
\begin{eqnarray}
& &
\Prod\limits_{J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}} \Prod\limits_{r=1}^{N}
G_{J_{r}}(\widetilde{\mu}_{1, r}^{J}, \cdots , \widetilde{\mu}_{n, r}^{J}) \\
& & {}=
\left( \frac{\sqrt{\rho\lambda}}{S_{2}(-\frac{2\pi}{n})} \right)
^{\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}\Sum\limits_{j=1}^{n}(j-1)\lambda_{j}}\!\!\!\!\!\!
\Prod\limits_{1 \le r'<r \le n}\Prod\limits_{J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}\Prod\limits_{k \atop J_{k}+1=r}
\frac{S_{2}(\mu_{r}-\mu_{r'}+\frac{\pi}{n}(D_{r', r, k}^{J}-1))}
{S_{2}(\mu_{r}-\mu_{r'}+\frac{\pi}{n}(D_{r', r, k}^{J}+1))}, \nonumber
\end{eqnarray}
where $D_{r', r, k}^{J}$ is given by
\begin{eqnarray}
D_{r', r, k}^{J}=\Sum\limits_{\eps=\pm}
\eps (M_{r'-1, k}^{J, \eps}-M_{r-1, k}^{J, \eps})
=\lambda_{r'}-\lambda_{r}+1-2(M_{r'-1, k}^{J, -}-M_{r-1, k}^{J, -})
\end{eqnarray}
for $k$ satisfying $J_{k}+1=r$.

Now we rewrite
\begin{eqnarray}
\Prod\limits_{J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}}}\Prod\limits_{k \atop J_{k}+1=r}
\frac{S_{2}(\mu_{r}-\mu_{r'}+\frac{\pi}{n}(D_{r', r, k}^{J}-1))}
{S_{2}(\mu_{r}-\mu_{r'}+\frac{\pi}{n}(D_{r', r, k}^{J}+1))}\;.
\label{601}
\end{eqnarray}
Let us consider the following set:
\begin{eqnarray}
{\cal F}^{(r', r)}=
\bigsqcup_{(J, k) \atop {J \in {\cal Z}_{\nu_{1}, \cdots , \nu_{n-1}} \atop J_{k}+1=r}}\{M_{r'-1, k}^{J, -}-M_{r-1, k}^{J, -}\},
\quad (1 \le r' < r \le n),
\end{eqnarray}
where $\bigsqcup$ means a disjoint union.
For $a \in {\bf Z}$, we set
\begin{eqnarray}
{\rm mult}^{(r', r)}(a)=\# \{ t \in {\cal F}^{(r', r)}; t=a \}.
\end{eqnarray}
Then we have
\begin{eqnarray}
(\ref{601})=\Prod\limits_{a \in {\bf Z}}S_{2}\left(
\mu_{r}-\mu_{r'}+\frac{\pi}{n}(\lambda_{r'}-\lambda_{r}-2a)\right)
^{{\rm mult}^{(r', r)}(a)-{\rm mult}^{(r', r)}(a+1)}.
\end{eqnarray}
We can show
\begin{eqnarray}
& & {\rm mult}^{(r', r)}(a)-{\rm mult}^{(r', r)}(a+1) \nonumber \\[6pt]
& & {}=\left\{
\begin{array}{ll}
\displaystyle{
-\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}
{\lambda_{r}+\lambda_{r'} \atopwithdelims() \lambda_{r}+a}\bigg/
{\lambda_{r}+\lambda_{r'} \atopwithdelims() \lambda_{r}} }\,,&
-\lambda_{r} \le a \le \min\{-1, \lambda_{r}-\lambda_{r'}\}, \\[12pt]
\displaystyle{
\Lambda_{\lambda_{1}, \cdots , \lambda_{n}}^{(0)}
{\lambda_{r}+\lambda_{r'} \atopwithdelims() \lambda_{r'}-a}\bigg/
{\lambda_{r}+\lambda_{r'} \atopwithdelims() \lambda_{r}} }\,,&
\max\{0, \lambda_{r'}-\lambda_{r}+1\} \le a \le \lambda_{r'}, \\[12pt]
0\,, &{\rm otherwise}.
\end{array}
\right. \nonumber
\end{eqnarray}
This completes the proof. \quad$\square$

\addtocounter{section}{1}
\setcounter{equation}{0}

\section*{Appendix}

Here we summarize the property of the double sine function
$S_2(x)=S_{2}(x|\om_{1},\om_{2})$ following \cite{JM}.

We assume that $\mathop{\rm Re\;\!}\om_{1} >0, \mathop{\rm Re\;\!}\om_{2}>0$.
$S_{2}(x|\om_{1},\om_{2})$ is a meromorpic function of $x$ and
symmetric with respect to $\om_{1},\om_{2}$. Its zeros and poles are
given by
$$
\hbox{zeros at } \ x=\om_{1}{\Bbb Z}_{\le 0}+\om_{2}{\Bbb Z}_{\le 0}\,,\qquad
\hbox{poles at } \ x=\om_{1}{\Bbb Z}_{\ge 1}+\om_{2}{\Bbb Z}_{\ge 1}\,.
$$
Its asymptotic behavior is as follows:
\begin{eqnarray}
\label{app1}
\log S_{2}(x)\,=\,\pm \pi i \! \left( \frac{x^2}{2 \om_{1} \om_{2}}-
\frac{ \om_{1}+
\om_{2}}{2 \om_{1} \om_{2}}\,x-\frac{1}{12}\left(\frac{\om_{1}}{\om_{2}}+
\frac{\om_{2}}{\om_{1}}+3 \right) \right)\! \! + \! o(1)\,, &&
\\[8pt]
(x \to \infty, \,\pm\mathop{\rm Im\;\!} x > 0)\,. && \nonumber
\\[-2\bls] \nonumber
\end{eqnarray}
This implies that
\begin{eqnarray}
\log S_{2}(a+x)S_{2}(a-x)\,=\,
\pm \pi i \frac{2a-\om_{1}-\om_{2}}{\om_{1} \om_{2}}\,x+o(1),
\quad (x \to \infty\,,\;\ \pm \mathop{\rm Im\;\!} x >0)\,. \label{app2}
\end{eqnarray}
The following formulae hold:
\begin{eqnarray}
\frac{S_{2}(x+\om_{1})}{S_{2}(x)} &=& \frac{1}{2 \sin \frac{\pi x}{\om_{2}}},
\label{app3} \\
S_{2}(x) &=& \frac{2 \pi}{\sqrt{\om_{1}\om_{2}}}x+O(x^2) \quad (x \to 0)\,.
\label{app5}
\end{eqnarray}

\ifUS\else\newpage\fi

redpajama_set_name: RedPajamaArXiv


Take on the role of a Pokemon Trainer uncover new tales, Unravel the mystery behind the two forms reminiscent of the legendary kiro Shadows Die Twice Official Gameplay Overview Trailer Revealed March 18, encounter Pokemon FromSoftware shared the official gameplay overview trailer of the game. Oct 24, · Rick Astley - Never Gonna Give You Up ( Official Video) - Listen On Spotify: Learn more about the brand new album ' Beautiful.
Pokémon entered its third generation with the release of Pokémon Ruby continued with the Game Boy Advance remakes of Pokémon Red , LeafGreen ( Red , Green, Pokémon FireRed , Green representing the original Japanese first generation games; territories outside Japan instead saw releases of Red , Sapphire for Game Boy Advance Blue). & Let' s Go, Eevee! → Hauptartikel: Pokémon Bank und PokéMover Pokémon Omega Rubin und Alpha Saphir sind wie ihre Vorgänger Pokémon X und Y kompatibel mit Pokémon Bank.
A trade ( Japanese: 交換 exchange) also known as a Link Trade ( Japanese: 通信交換 Link Exchange), is a process in which a Pokémon Trainer sends one of his her Pokémon to another Trainer in exchange for one of the other Trainer' s Pokémon. Jogos de consoles: Stadium • Stadium 2 • Colosseum • XD: Gale of Darkness • Battle. Pokémon Pinball ( Japanese: ポケモンピンボール 1999, Hepburn: Pokemon Pinbōru) is a pinball- based Pokémon spin- off video game for the Game Boy was released in Japan on April 14, in North America on June 28 1999.
BATTLENET EU CD KEY: European Key to download and play Overwatch on EUROPEAN SERVERS. ( in America Red Blue).
An enhanced remake of Pokémon Ruby. In every Pokémon game, trading is necessary to collect all Pokémon. Explore a brand new world where friendly tamed, hostile monsters can be captured trained! Released in 1999 it offered updated graphics , as well as new rosters for trainers throughout the game including most significantly the Gym Leaders.
What' s up everybody. Mit dieser Software können Pokémon über eine Cloud in Online- Boxen gelagert werden. I' m DarkChild316, the bad boy of. Free ROMS of Pokemon X and Y for PC are out Now.
They are Nintendo' s best remake, for a pokemon game. Wait is Over Guys. I' m a really fun loving person and easy to get along with so don' t be afraid to ask me anything.
While between Pokémon Omega Ruby & Alpha Sapphire games you can do when you connect with players of Pokémon X & Y, use anything there are several restrictions. Play pokemon omega ruby online no download.
Catch all Pokemon and try to be the best trainer in the world! We don' t have words to describe how awesome this game is. Qualifying players who logged into their Pokémon ar Twitpic Community - thank you for all the wonderful photos you have taken over the years.
House is no place for rats! It gives us a quick overview of the game' s story as well as giving a few tips about the game and introducing the protagonist' s prosthetic arm.
Rats have invaded your house! A new light shines on the Alola region! This Miltank was distributed to players who inserted a serial code that was given away to players who completed at least three battles during the Johto Classic competition on the Pokémon Global Link.
- Rats Invasion Free Online. Play pokemon omega ruby online no download.
Série principal: Red & Blue • Gold & Silver • Ruby & Sapphire • Diamond & Pearl • Black & White • X & Y • Sun & Moon ( Ultra Sun & Ultra Moon) • Sword & Shield: Remakes: FireRed & LeafGreen • HeartGold & SoulSilver • Omega Ruby & Alpha Sapphire • Let' s Go, Pikachu! Like any pinball game the main objective is to get points using.
This game is much better then Pokemon Black 2 and White 2. Pokemon Yellow was the only fourth version title in Japan and first in a line of third version games in other territories. The competition was held from May 27 to May 29,.
We have now placed Twitpic in an archived state. In it the ball is a Poké Ball most of the objects on the table are Pokémon- related. Welcome to Pokemon Emulators Website Introduction We have heard known so much information about the Pokemon Emulators but maybe we don' t know clearly about them. There isn' t much changed with the story, but plenty of.
Pokemon Fire Red Leaf Green are remakes of the classic Red Green games. The official source for Pokémon news animation, video games, apps, information on the Pokémon Trading Card Game the Pokédex. Play pokemon omega ruby online no download.
Basically, Pokemon Emulators are a type of. So, exactly what are they? Play Pekemon Fire Red Gameboy Advance GBA Online Emulator in Browser. You must Activate the CD Key on your Account FREE NCLEX exams for the NCLEX, FREE resources for the NCLEX, FREE NCLEX Quizzes for the NCLEX, Prioritization for the NCLEX, create legation strategies for the NCLEX, Infection Control for the NCLEX Failed the NCLEX - Help is here.
Starting from Generation II items can be held by Pokémon allowing. Battles: Online and wireless battles in the games remains exactly the same as before.
A list of eligible players was posted on the Pokémon Global Link on June 7,. Try to banish the rats from the rooms using different means: smoke- bomb clinging- bomb, screw- nut, cheese bait poison.


We will Manage your Traveling arrangements …While you get on with business OR leisure vacation.
(FIVE HUNDRED AND FIFTY THOUSAND NAIRA ONLY P/P).
We can also organize BUSINESS or CAREER workshop for the group which will attract $850 (Eight hundred and Fifty Dollar) per head. The fee will cover the seminar hall venue, facilitation, certification, Manual, Lunch/refreshments, etc.
Prices are subject to change due to fluctuations in dollar exchange rates and airlines fee.
Thank you for choosing TOUR CONCEPTS as your preferred Tour/Travel consultant.


text:
Pierpaolo dalle Masegne (Venecia,) fue un escultor italiano del período gótico, hermano de Jacobello dalle Masegne.

Biografía
Pierpaolo trabajó frecuentemente junto a su hermano Jacobello en el taller que ambos tenían en Venecia. En 1368 se documentó la presencia de Pierpaolo en Udine para la construcción de la catedral de dicha localidad.

Los dos hermanos juntos crearon entre 1388 y 1392 el altar de la Basílica de San Francisco en Bolonia y el iconostasio en la Basílica de San Marcos en Venecia. En esta última obra las esculturas de las partes laterales se atribuyen a Pierpaolo.

En 1400, Pierpaolo ayudó a Jacobello en la construcción de la fachada de la Catedral de San Pedro Apóstol en Mantua, obra posteriormente destruida que, no obstante, se puede observar en una pintura de Domenico Morone conservada en el Palazzo Ducale de dicha ciudad. También en Mantua creó la tumba de Margherita Malatesta, esposa de Francesco I Gonzaga. En Vicenza, en la Iglesia de San Lorenzo, creó el Arca de Bartolomeo da Porto en la capilla absidial izquierda.

También en 1400 se le encomendó la decoración de una ventana de la Sala del Maggior Consiglio en el Palazzo Ducale en Venecia.

Referencias

Escultores de Italia del siglo XIV
Arquitectos de Italia del siglo XIV
Nacidos en Venecia
Fallecidos en Venecia

redpajama_set_name: RedPajamaWikipedia


text:
World's most liveable city: Melbourne takes top spot for seventh year running
By Stephanie Chalkley-Rhoden
Posted Wed 16 Aug 2017 at 1:58am Wednesday 16 Aug 2017 at 1:58am Wed 16 Aug 2017 at 1:58am , updated Wed 16 Aug 2017 at 6:07am Wednesday 16 Aug 2017 at 6:07am Wed 16 Aug 2017 at 6:07am
Melbourne once again beat Vienna and Vancouver for top spot. (ABC Open contributor Shayne T Wright)
abc.net.au/news/melbourne-named-worlds-most-liveable-city-for-seventh-year/8812196
Melbourne has once again been named the world's most liveable city by The Economist, receiving a perfect score for healthcare, education and infrastructure.
The Economist Intelligence Unit's (EIU) Liveability Index ranks 140 cities each year on those topics, as well as stability, culture and environment.
Top five most liveable cities:
1.Melbourne
2.Vienna
3.Vancouver
4.Toronto
5.Adelaide, Calgary
Vienna once again came second and Vancouver third.
Adelaide was the next best Australian city finishing in fifth spot, followed by Perth at number seven. Sydney was ranked 11th.
It was a record seventh time in a row in the top spot for Melbourne, which scored 97.5 out of a possible 100.
The top five remained unchanged from last year.
City of Melbourne Lord Mayor Robert Doyle said the EIU's index was the most widely accepted city rankings system.
Mayor Robert Doyle said the city did not take the accolade for granted. (ABC News: Margaret Burin )
"This world record is an amazing feat that all Melburnians should be extremely proud of today," Cr Doyle said.
"This accolade is an important selling point for Melbourne internationally: for businesses to invest or move here, for the best and brightest people to make Melbourne their home and for tourists to visit us.
"There will always be naysayers and whingers, and of course we are not perfect. No great world city is, but we should be very proud of the work we all do together to make Melbourne the best city in the world."
Low crime rate gets Melbourne across the line
The report said the ongoing threat of terrorism around the world affected a number of scores.
It noted Melbourne's relatively low crime rate as one of the main reasons it outranked other high-performing cities, despite an increase in robberies and thefts in the last 12 months.
However, Melbourne is not without its problems.
The city has been struggling to deal with a rising and increasingly visible homelessness crisis, with more people sleeping rough in the CBD than ever before.
There's been an increase in the number of rough sleepers in Melbourne. (ABC News: Margaret Burin )
The issue of housing affordability is also causing stress as property prices continue to rise, locking many out of being able to afford their own home.
The price of renting is also high, with almost all suburbs considered unaffordable for those on low incomes.
Public transport is also bursting at the seams. Last month, the entire train network ground to a halt during peak hour after a computer glitch.
Rating 'glosses over realities of life in Melbourne'
Victorian Council of Social Service chief executive Emma King said the Economist's index painted a distorted picture of life in Melbourne.
"These blunt measures gloss over the realities of life in Melbourne for many people," she said.
"Did The Economist survey anybody who's living under a bridge or skipping meals to pay their power bill?
"Melbourne is a great city. But, for many, it provides anything but an easy life."
But the Victorian Government spruiked the city's "world-class health care system" and strong economy, saying 100,000 new jobs were created in the state last year.
"There's a buzz about the city that keeps bringing the world's best to enjoy Melbourne," the Government said in a statement.
"The biggest exhibitions, the best events, world-renowned restaurants and all-night public transport to get you home safe."
Tourism groups welcome the ranking but say Melbourne cannot afford to rest on its laurels. (ABC News: Seraphine Charpentier-Andre)
Posted 16 Aug 2017 16 Aug 2017 Wed 16 Aug 2017 at 1:58am , updated 16 Aug 2017 16 Aug 2017 Wed 16 Aug 2017 at 6:07am
Housing affordability: Is it time to move to Adelaide?
Melbourne 'exceeds expectations': World's most liveable city — again
Travel and Tourism (Lifestyle and Leisure)

redpajama_set_name: RedPajamaCommonCrawl


text:
You may know Pete Holmes as the creator, writer, and star of the Judd Apatow-produced HBO series Crashing—the semi-autobiographical series about an aspiring standup comedian whose life turns upside-down when he discovers his wife cheating on him.
As someone who wears his vulnerabilities on his sleeve, Holmes isn't phased by exposing his personal life to scrutiny: "It feels really good to make jokes, find the lighter side, and share some of the wisdom that comes through painful experiences by bringing them into a story."
Holmes also hosts You Made It Weird, a podcast dedicated to having real conversations with fellow comedians. The fact that he can start with a question as basic as, "How was finding parking?" and reach something as complex and deep as "What is the meaning of life?" by the end, keeps him, and us, coming back for more.
Holmes joins Off Camera to discuss what it's really like to get divorced, why a little delusion goes a long way in pursuing your dreams, why being vulnerable is the best way to connect to others, what we can learn from comedians, and why he'll never tire of having amazing conversations.
As Holmes says, "I feel qualified to talk about Pete Holmes, because I'm Pete Holmes-ing the hell out of this Pete Holmes." We're glad we get to talk to an expert.


Westview is dotted with white sand bunkers, ponds, and hundreds of mature trees. The design and layout of this 27-hole Illinois golf course will challenge the skilled player or provide an enjoyable experience for the average player or beginner.
Thank you for choosing Westview Golf Course for your round of Golf. We hope that you enjoy your round. Below you will find a map of the golf course as well as a few of the rules and procedures that we strive for during a round of golf.
Please keep pace with the group in front of you. The recommended pace of play for each nine is 2 hours or less. Players who cannot keep pace may be required to allow groups to play through, or moved to the appropriate location on the golf course.
Please do your best to take care of the golf course. Divots should be replaced, or filled with the seed mix located on the carts and par three tees. Ball marks should be repaired.
Steel penetrating spikes are not allowed at Westview Golf Course.
Please proceed immediately from the #9 green to the #10 tee. If you go the clubhouse and slow the pace of play you may lose your turn on the 10th tee.
Keep your receipt. Golf Course personnel periodically will check them on the golf course.
Personal coolers are not allowed on the premises. Our concession will provide you with a cooler after a FIVE DOLLAR deposit.
All alcoholic beverages must be purchased from our concession. Please do not attempt to bring alcohol from your car onto the course.
All players are required to wear a shirt at Westview Golf Course.
Washrooms are located behind the fifth green, in the maintenance facility and behind the 22nd green.
All players are required to have their own set of golf clubs. If you do not have a set of clubs, rentals are available in the Pro Shop.


This gallery is a resource for those lovely souls who have purchased my patterns. Clicking on the photos below will link you to further images of the pattern you have purchased.
at the 'Made with Bloom' Flickr group.


The opening words of the Gospel of Luke gives us contrasting figures: King Herod of Judea, considered to be one of the most evil and cruel figures of history along with two figures of faith—Zechariah and Elizabeth.
It was two years ago today that I had my sermon written on Mary. And then came the horrendous news of the tragic mass shooting at Sandy Hook Elementary School, in which 20 children and 6 staff were killed. Our nation was shaken. I remember that Saturday night tossing out my sermon and starting over…longing to bring some kind of comfort to a community of faith shaken to the core. On that Sunday morning, we had twenty of our youngest children up-front singing the Christmas song that they had been working on for weeks in preparation for that service. Tears were shed throughout the congregation…for we all got it! These children were about the same ages as the one's lost in CT. The children reminded us of just how unbelievable the loss was not only for their parents, families, community and for our nation.
Zechariah and Elizabeth were both descendants of priestly families. When they married Zechariah's portrait was hung in the long line-up of temple priests, while Elizabeth stayed at home and was expected to produce a new generation of priests. Not only was becoming a priest not an option for her, we are told she was barren, and in those days the blame was placed directly on her. This was reason enough for Zechariah to have divorced her. But scripture tells us that they had not only kept their marriage intact, but they were considered righteous before God.
Elizabeth was used to Zechariah's treks from their home in the hills to the Temple. As a priest, he had certain duties to carry out, but on this particular day, he had shown up at the Temple and upon casting the lots to see who got the honor of burning the incense on the holy alter, it was Zechariah who had won. This was most probably a once in a lifetime opportunity.
As all the folks are gathered outside waiting for the sweet fragrance and smoke to drift up from the altar representing their prayers being lifted to the heavens, Zechariah is confronted with an angelic presence. As he holds the sacred elements in his hands, the divine angel appears. He is both shocked and terrified. The angel offers words to comfort him. "Do not be afraid, your prayer has been answered." Now, surely, Zechariah had given up praying for a son a long time ago. Not only is he scrambling to make sense out of the mystery of this visit, but he is stunned to hear that a son will be born to him and to Elizabeth. He did what anyone of us would do, he tried to bring some reason into the encounter by reminding this angel that he and his wife are well beyond that dream, and, after all, they are near retirement. I imagine it was difficult to even hear Angel Gabriel going on and on about this future son…especially when he got to the litany about how he was to be raised…even down to dietary issues. I can picture a glassy-eyed Zechariah stumbling to just keep his balance and stay upright.
Zechariah's arguing with Gabriel did not go well. He probably re-played his reply a thousand times during the next nine months. In his humanness, Zechariah asked for a sign. He got his request. He was silenced. Gabriel made it clear that because he did not embrace the message, that he would become mute, unable to speak, until the birth of his son, John, who all of us know as John the Baptist.
Meanwhile, we can all imagine what is happening to the crowd outside who are growing weary of waiting to see the white smoke of the incense rising up. There is no fragrance greeting them. And, I suspect that some of the women are nervous because they timed the lamb shanks to be done based on when worship is always over.
Some scholars view Zechariah's going mute as punishment for his lack of faith. My sense is that it was a powerful sign of God's presence in Zechariah's life. Not a day could go by without him being fully aware of the gift of new life that was coming and the expectations that were his for raising this future leader and prophet…the one who would point to the coming of the Messiah.
We're offered a lesson that says that silence can be more powerful than words. And, it is in that silence that watching and waiting becomes far more intentional. In the midst of the outer world of darkness around the leadership of Herod was the power of God bringing forth new leadership, new hope and a new vision.
Divine calls do not immediately inspire folks to throw a party. The response is usually one that elicits mouth and stomach grabbing and a heavy intake of breath that often tends to push our shoulders up around our ears. Divine calls are usually not the type of information that one races around excitedly wanting to make public. They often demand a gestation period—a period of entering the silence and listening. There is generally a season of wrestling with God…a period of doubts and fears before one is able to claim, embrace and own the call. It is only after the internal process is well in place that it is time for the public announcement. It makes perfect sense to me why Elizabeth went into seclusion.
Silence comes into our lives in a variety of ways: the silence of an empty home after a death. An illness or a heart attack can bring the silence that comes with sitting in the hospital between blood draws and visitors while trying to make sense out of our body shutting us down. And, sometimes, the illness or injury is a wake-up call to the unfinished business that we may be carrying from a childhood trauma, or some painful encounter or event in our lives.
On the eighth day following the long hoped for child, the baby was gratefully carried into the gathered crowd of relatives and friends for the ritual of circumcision. The time came for the public to hear this newborn's name. As his mother, Elizabeth, broke the rules and announced that his name would be John. The crowd was shocked. They called out with indignation, "You have no relatives by that name." They sought out Zechariah, and scrambled to find a tablet. With his trembling hand, he wrote, "His name is John." Any fears that he had had of living the rest of his life in silence ended in that moment as his tongue was loosened, and the party took on an additional piece of gratitude and celebration. From the nervous fearful responses of those living around them, there is the feeling that something has shifted, not only in Zechariah and Elizabeth's life, but in their relationship with God, that has indeed already impacted neighbors, relatives and friends.
Aren't we familiar with these actions? Let's face it, truth speaking isn't popular. It is important to remember that every divine call is wrapped in the words, "Do not be afraid." Calls have an element of risk. They demand some kind of leap of faith. And, they demand support from our faith community.
Friday's paper featured members of our church from Mountain Meadows on the front row protesting against the proposed pipeline. If the picture had been bigger, I would have spotted many more of our community who took a stand.
As always, we're invited into the story. When and where have the lessons from seasons of waiting and watching come into your life? Did it follow a period of grieving? When have you been invited into a silence or at least a season of quieting down? What support do you need from this community to live out God's call on your life?
Do not be afraid, First Congregational United church of Christ.
Do not be afraid Search Committee.
How silently, how silently, the wondrous gift is given. Amen.


text:
« Marketers: TV Advertising Less Effective Than Two Years Ago
K-C, Unilever Turn Down TV to Ramp Up ROI »
P&G's New Austerity Plan: A Small-Scale Restructuring
Posted by Mort Greenberg on February 23, 2008
source: AdvertisingAge
Cutting Underperforming Brands, 15% of General Managers in Bid to Maintain Spending
By Jack Neff
BOCA RATON, Fla. (AdAge.com) — Procter & Gamble Co. will eliminate 15% of its general-manager positions and an unspecified number of underperforming brands while keeping overall employment flat to negative, in a bid to accelerate productivity amid rising commodity costs and tougher competition.
P&G Chairman-CEO A.G. Lafley said that while most of P&G's brands will be limited to zero overhead growth, layoffs are not in the plan.
Chairman-CEO A.G. Lafley laid out some details of what sounded very much like a standard-issue restructuring in a presentation to the Consumer Analyst Group of New York today, though both he and Chief Financial Officer Clayton Daley said the restructuring would fall well short of the bigger revamps rivals such as Unilever, Kimberly-Clark Corp. and Colgate-Palmolive Co. have undergone in recent years.
ZOG, NOG, HOG
Under the plan, most of P&G's brands will be limited to "zero overhead growth" (or ZOG), where employment won't rise regardless of sales growth. The highest-priority businesses, such as China and Central and Eastern Europe, along with beauty care, will be limited to "half overhead growth" (HOG), where overhead costs can rise no more than half as fast as sales. The lowest-priority businesses, including an unspecified number of brands P&G will look to divest, will aim for "negative overhead growth" (NOG), in which employment and other overhead costs decline as a share of sales.
The goal is to boost P&G's productivity growth, which already has averaged 6% annually under Mr. Lafley (well ahead of the rate for the U.S. economy), to 7% to 8%, while continuing to provide strong marketing support for the 41 brands that account for 80% of P&G's sales and 90% of its net earnings.
The new austerity doesn't mean P&G will stop hiring, but it does mean it will "create additional attrition" of more senior employees, as Mr. Daley put it. Exactly how P&G will get people to leave faster, however, isn't clear.
Accelerated attrition doesn't mean layoffs, Mr. Lafley said in an interview after the presentation. "The only time we'll announce a layoff is when we're shutting down a manufacturing site," Mr. Lafley said. "But most of our brands we're handling from attrition. We're handling it through growth [without commensurate hiring]. I don't rule anything out, because we need to do what's right for the shareholders. But [layoffs are] not the plan."
While P&G will continue to hire entry-level marketers into its associate brand manager ranks, it may hire fewer than before. "We're [already] not hiring as many [associate brand managers] as we used to," Mr. Lafley said, because the company's bigger brands are driving more of its sales. "And if we weed the brand portfolio," he said, that will result in fewer marketers who were formerly assigned to divested brands. He declined to say which brands are expected to be divested.
45 jobs globally
The move to eliminate 15% of general-manager positions should result in elimination of around 45 of 300 such positions globally, said people familiar with the company. P&G also will cut costs by using 40% to 50% fewer expatriate managers, reducing the considerable relocation and sometimes salary expense required to move managers outside their home countries.
P&G will also cut costs by eliminating duplication between its global business units, where most marketing executives now work, and its market development organizations, which handle sales, shopper marketing and media buying. For example, Mr. Lafley said some market research done on shoppers, which was duplicated in both units, has been consolidated within the global business units in North America.
"We feel we can eliminate 10% to 20% of initiative development [new product and marketing efforts] among the global and regional teams," Mr. Lafley said. One example is in point-of-sale marketing and store signage, once handled on a brand-by-brand basis, but now being consolidated into a single global organization, he said.
P&G executives didn't commit to spending any of the savings from overhead controls on marketing — but they could provide room for the company to maintain spending while improving operating margins in the face of rising commodity costs.
Overall, Mr. Lafley said P&G is looking to apply the same analytics and spending discipline it applies to the $10 billion it spends annually on advertising and consumer promotion to the $10 billion it spends on trade promotion through retailers, most of which appears as a reduction of net sales on P&G's books.
Smaller in scope
The moves sound in many ways similar to the waves of global restructuring P&G rival Unilever has undergone the past three years, but Messrs. Lafley and Daley said it will be considerably smaller in scope. P&G plans to keep its annual restructuring costs, which include writedowns of divested facilities and separation costs for employees, to $300 million to $400 million — about double the rate P&G has had in years past, but well below the multi-billion-dollar writeoffs it has had in prior decades and competitors have undergone more recently. "We've said many times that we're not eager to do major restructurings like some of our competitors," Mr. Lafley said.
But the upcoming split off of Folgers could provide some room for P&G to do restructuring next year above its $400 million limit. Mr. Daley noted that the transaction, in which P&G shareholders likely will be able to opt for shares of Folgers Coffee Co., will generate a one-time gain to be offset at least in part by a one-time charge for writedown of Folgers assets.
P&G's belt tightening is another testament to the legacy of former Gillette Co. Chairman-CEO Jim Kilts, who similarly deployed ZOG and NOG at Gillette prior to its acquisition by P&G in 2005, though HOG is a kinder, gentler P&G formulation. Just prior to P&G's presentation, former Gillette and current Avon Products Chief Financial Officer Charles Cramb outlined how ZOG and NOG had freed funds for a tripling of ad spending by that company in the past three years.
This entry was posted on February 23, 2008 at 4:57 am	and is filed under Ad Spending, Brand Advertising. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


A medicine or alcoholism impacts every location of a person's life. In addition to career as well as relationship issues, a drug abuse problem can increase some very uncomfortable questions regarding identity, and the significance as well as function of life. Faith-based recuperation choices effort to offer a spiritual– typically Christian– perspective on dependency and therapy, using characteristics and also direction that stand out from nonreligious programs.
For individuals who rely on the assistance of a spiritual tradition, or that are looking for a spiritual structure for recovery, faith-based recuperation programs offer an atmosphere that nourishes this element of therapy. Unlike nonreligious recuperation programs, which could understate the function of spirituality in order to interest a larger range of customers, faith-based programs are freely structured around the values, customs, and mentors of a certain religious beliefs. Customers can go over issues of belief and dependency with therapists and also peers in a safe, helpful environment that verifies their beliefs.
The defining attribute of Christian-centered treatment choices is that the methods and techniques are based on, or affected by, Christian teachings. This indicates that they encourage spiritual representation as component of recovery as well as therapy, as well as make use of Christian scripture (as found in the Scriptures) to assist and guide people with detoxing, in addition to specific, team, or family counseling. Such programs often make use of testimonies and testimonies of Christian tenets of God's love as well as acceptance as resources of strength as well as inspiration for recovering people.
The crossway between the Christian faith and substance misuse therapy goes back a lengthy means. In enhancement to his work in psychiatry, Jung additionally studied globe religions, creating the idea that an individual's life was reliant upon spiritual wellbeing.
Jung thrilled this perspective into an alcoholic patient of his, informing the man that the only guaranteed method of defeating the condition (since all other methods had failed) was to find a spiritual experience. That conversation eventually made its way to the ears of Costs Wilson, the man who would incorporate his determined wish to stop alcohol consumption with his newly found Christian belief to establish Twelve step programs. Jung wrote to Wilson directly to discuss that spiritualism was just what was required to overcome alcoholism.
Twelve-Step programs like Twelve step programs have come under objection and analysis for their lack of openness and eschewment of standardized scientific practices, yet the core ideas of peer support, mutual assistance, and liability are still supported.
The principle of having communal assistance as well as a network is one of the structures of faith-based therapy. While definitely not unique to Christian-oriented programs, the concept of coming to be a mate of people finding out as well as improving together is mirrored in the impacts seen at church events. The New York Times reports on a North Carolina research that discovered regular church guests had extra social assistance, and resulting much better health and wellness, compared to people that did not participate in church.
Faith-based healing has been gaining appeal recently, but exactly how effective is this method to therapy? The National Survey important Misuse Therapy Solutions reported in 2008 that there were 527 faith-based rehab centers in the USA. Given that the staff of such treatment facilities tend to be "unlicensed as well as nonprofessional," as the Journal important Misuse Treatment kept in mind, and therapy does not always comply with standard scientific techniques, there exists a lack of methodical empirical study right into the performance of faith-based programs.
However, the supervisors of those programs concentrate less on the numbers and data of their centers' success rates and even more on their efficiency. Talking to The Fix, one supervisor claims his program is "off the graphes" effective; an additional says that both secular and religious treatment programs think very extremely of his operation. A third director states 70-80 percent of the grads of his program did not regression within the past year; as well as another directed at a two-year, in-house research that located 48 percent of its grads preserved their sobriety after completing their treatment.


text:
|-
| 96601 - || || 15 de desembre, 1998 || Socorro || LINEAR
|-
| 96602 - || || 23 de desembre, 1998 || Xinglong || BAO Schmidt CCD Asteroid Program
|-
| 96603 - || || 20 de desembre, 1998 || Ondřejov || A. Kolář, L. Šarounová
|-
| 96604 - || || 25 de desembre, 1998 || Kitt Peak || Spacewatch
|-
| 96605 - || || 7 de gener, 1999 || Kitt Peak || Spacewatch
|-
| 96606 - || || 7 de gener, 1999 || Kitt Peak || Spacewatch
|-
| 96607 - || || 14 de gener, 1999 || Kitt Peak || Spacewatch
|-
| 96608 - || || 15 de gener, 1999 || Anderson Mesa || LONEOS
|-
| 96609 - || || 9 de gener, 1999 || Mérida || O. A. Naranjo
|-
| 96610 - || || 16 de gener, 1999 || Socorro || LINEAR
|-
| 96611 - || || 18 de gener, 1999 || Kitt Peak || Spacewatch
|-
| 96612 - || || 5 de febrer, 1999 || Xinglong || BAO Schmidt CCD Asteroid Program
|-
| 96613 - || || 12 de febrer, 1999 || Gekko || T. Kagawa
|-
| 96614 - || || 6 de febrer, 1999 || Višnjan Observatory || K. Korlević
|-
| 96615 - || || 10 de febrer, 1999 || Socorro || LINEAR
|-
| 96616 - || || 10 de febrer, 1999 || Socorro || LINEAR
|-
| 96617 - || || 10 de febrer, 1999 || Socorro || LINEAR
|-
| 96618 - || || 10 de febrer, 1999 || Socorro || LINEAR
|-
| 96619 - || || 12 de febrer, 1999 || Socorro || LINEAR
|-
| 96620 - || || 12 de febrer, 1999 || Socorro || LINEAR
|-
| 96621 - || || 11 de febrer, 1999 || Socorro || LINEAR
|-
| 96622 - || 1999 DY || 18 de febrer, 1999 || Višnjan Observatory || K. Korlević
|-
| 96623 Leani || || 14 de març, 1999 || Monte Agliale || M. M. M. Santangelo
|-
| 96624 - || || 12 de març, 1999 || Kitt Peak || Spacewatch
|-
| 96625 - || || 16 de març, 1999 || Kitt Peak || Spacewatch
|-
| 96626 - || || 22 de març, 1999 || Anderson Mesa || LONEOS
|-
| 96627 - || || 19 de març, 1999 || Socorro || LINEAR
|-
| 96628 - || || 19 de març, 1999 || Socorro || LINEAR
|-
| 96629 - || || 19 de març, 1999 || Socorro || LINEAR
|-
| 96630 - || || 19 de març, 1999 || Socorro || LINEAR
|-
| 96631 - || || 23 de març, 1999 || Kitt Peak || Spacewatch
|-
| 96632 - || || 6 d'abril, 1999 || Fountain Hills || C. W. Juels
|-
| 96633 - || || 7 d'abril, 1999 || Socorro || LINEAR
|-
| 96634 - || || 9 d'abril, 1999 || Prescott || P. G. Comba
|-
| 96635 - || || 10 d'abril, 1999 || Višnjan Observatory || K. Korlević
|-
| 96636 - || || 6 d'abril, 1999 || Anderson Mesa || LONEOS
|-
| 96637 - || || 7 d'abril, 1999 || Anderson Mesa || LONEOS
|-
| 96638 - || || 10 d'abril, 1999 || Anderson Mesa || LONEOS
|-
| 96639 - || || 11 d'abril, 1999 || Anderson Mesa || LONEOS
|-
| 96640 - || || 15 d'abril, 1999 || Socorro || LINEAR
|-
| 96641 - || || 6 d'abril, 1999 || Socorro || LINEAR
|-
| 96642 - || || 6 d'abril, 1999 || Socorro || LINEAR
|-
| 96643 - || || 7 d'abril, 1999 || Socorro || LINEAR
|-
| 96644 - || || 7 d'abril, 1999 || Socorro || LINEAR
|-
| 96645 - || || 6 d'abril, 1999 || Socorro || LINEAR
|-
| 96646 - || || 7 d'abril, 1999 || Socorro || LINEAR
|-
| 96647 - || || 7 d'abril, 1999 || Socorro || LINEAR
|-
| 96648 - || || 12 d'abril, 1999 || Socorro || LINEAR
|-
| 96649 - || || 12 d'abril, 1999 || Socorro || LINEAR
|-
| 96650 - || || 15 d'abril, 1999 || Socorro || LINEAR
|-
| 96651 - || || 12 d'abril, 1999 || Socorro || LINEAR
|-
| 96652 - || 1999 HA || 16 d'abril, 1999 || Prescott || P. G. Comba
|-
| 96653 - || || 17 d'abril, 1999 || Socorro || LINEAR
|-
| 96654 - || || 8 de maig, 1999 || Catalina || CSS
|-
| 96655 - || || 8 de maig, 1999 || Catalina || CSS
|-
| 96656 - || || 8 de maig, 1999 || Catalina || CSS
|-
| 96657 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96658 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96659 - || || 8 de maig, 1999 || Catalina || CSS
|-
| 96660 - || || 8 de maig, 1999 || Catalina || CSS
|-
| 96661 - || || 14 de maig, 1999 || Catalina || CSS
|-
| 96662 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96663 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96664 - || || 15 de maig, 1999 || Catalina || CSS
|-
| 96665 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96666 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96667 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96668 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96669 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96670 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96671 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96672 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96673 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96674 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96675 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96676 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96677 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96678 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96679 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96680 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96681 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96682 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96683 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96684 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96685 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96686 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96687 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96688 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96689 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96690 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96691 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96692 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96693 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96694 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96695 - || || 10 de maig, 1999 || Socorro || LINEAR
|-
| 96696 - || || 13 de maig, 1999 || Socorro || LINEAR
|-
| 96697 - || || 13 de maig, 1999 || Socorro || LINEAR
|-
| 96698 - || || 13 de maig, 1999 || Socorro || LINEAR
|-
| 96699 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
| 96700 - || || 12 de maig, 1999 || Socorro || LINEAR
|-
6


As of this post, all art sent! Thanks for a great thursday Drawing Request Show. This montage includes other drawings/ commissions I worked on today including this mornings personal daily warmup practice drawing and some DRS commission upgrade inks.
See folks out 2pmET at twitch.tv/eryckwebb for more drawing request action live on Twitch!


The United States is in the middle of a dilemma that involves real estate markets. It's making its way to New Jersey as well. It's striking fear in the hearts of people who are based in a particularly crowded section of the country. New Jersey's real estate markets are behind those of most other parts of the nation. The difference is substantial, too. People are trying to make positive changes in the state, however. Quite a few contracting and development firms exist these days. These businesses are striving to enhance the state's real estate scene. This information is available via a publication that's known as Philly Purge.
According to Patch, Boraie Development LLC is a company that specializes in marketing, sales, property management and real estate development. It gives people access to numerous services that zero in on the real estate market in urban areas. The team at Boraie Development concentrates on the construction of impressive properties. It collaborates with imaginative architects and financial institutions alike. Boraie Development is headquartered in New Brunswick, New Jersey. The company has an excellent reputation in New Jersey's development community. It's been working on development projects of all kinds for roughly three decades now. The team members who work for Sam Boraie have extensive development knowledge. They focus on projects that draw in financial partners, residents and tenants alike. For more details visit Crunchbase.
The Aspire is the name of a Boraie Development project that's located in New Brunswick, New Jersey. It's a residential structure that consists of a total of 238 spacious and bright units. It's extremely close to the local train station and therefore is ideal for people who commute on a daily basis to work. Some of the amenities that are accessible to residents of the Aspire are a rooftop garden, a rooftop sundeck, a round-the-clock doorman, a yoga center, a modern exercise room and connected parking.
Boraie Development spends a lot of time working on residential projects of all sizes. The company also spends a lot of time concentrating on commercial matters. It worked on Albany Street Plaza. This is also located in New Brunswick. One of the project's towers was finished back in 1988. The second one, on the other hand, was finished more recently in 2004. Albany Street Plaza is right in front of the city's train station. It gives businesses access to a lot of retail space, too. It has 20,000 square feet for retail applications.
Since co-founding the Israeli-American Council in California (IAC), Adam Milstein has dedicated his life to supporting and strengthening both the State of Israel and the Jews. It has also been steadfast in improving US-Israel relations. This is considered the fastest organization that is concerned with the Jews of the United States.
As a native of Haifa, Adam Milstein was born in 1952 but relocated to California in 1981. In California, he served as a prolific real estate agent. He today holds the position of managing partner at Hager Pacific Properties, an organization that has specialized in retail, office, industrial and multi-family properties. In 2000, together with his wife Gila, they created the Adam and Gila Milstein Family Foundation to encourage the young Jews to be true to their Jewish roots.
Through Adam Milstein, Sheldon Adelson and Haim Saban came together in June 2015 for the first ever Campus Maccabbees Summit hosted in Las Vegas. This summit convened well over 50 organizations with the agenda of fully harnessing the power of the pro-Israel community so as bring anti-Semitism to an end and also quell the lies that are said about Israel especially in the various colleges in America.
Being a renowned philanthropist is a member of the board for various important organizations. Some of these organizations include Israel on Jewish Funders Network, StandWithUs, and Birthright Israel among others. He has been part of more than 100 not-for profit organizations for decades. According to Adam Milstein, philanthropy is not just about money donations.
One must embrace various practices so that his/her philanthropic investment makes an impact. There has to be sacrifice and love in the giving according to Adam Milstein. One should also stay focused on his philanthropic duties to ensure that he achieves his goals. The philanthropist also has to add his expertise and talents to the contributions to achieve that which is desirable.
Hussain Sajwani is a billionaire and owner and Chairman of DAMAC Properties, the largest luxury private real estate developer in the Middle East. He has built an incredible career that began at humble roots. He continues to develop and invest in real estate around the world. He has even worked with current US President Donal Trump. His work and impact have been felt worldwide.
The Damac owner, Sajwani's start began at humble roots. His family was a conservative, middle-class family. Sajwani watched his father work at a watch and pen store in creek-side Deira district. He began working with father at a very young age. Watching his father was very influential. Sajwani began to learn lots about business and was inspired to enter this world himself. After getting his degree in economics from the University of Washington, Sajwani would slowly began making his way into the business world.
After a short stint there he would leave to begin working with the Abu Dhabi National Oil Company. Sajwani was determined to be his own business man. He would eventually go on to begin DAMAC in 1982. The Hussain Sajwani family was about to experience a whole new level of success.
Sajwani's work expands worldwide. He has even worked with current US President Donald Trump. The two have a common interest in real estate and even worked together on Trump's Trump International Golf Club. The two recently spent New Years Eve together celebrating this massive real estate venture that consists of luxury villas that have racked up almost $2 billion in sales. With Trump currently President, the two have postponed work for the meantime. However Sajwani has collaborated with Ivanka Trump along with Trumps sons.
Hussein Sajwani has made a worldwide impact. With his tremendous track record and experience in real estate, he is sure to continue to make an impact on the business world for many years to come.
A philanthropist and real estate developer, Sam Boraie is one of the Vice President's for Boraie Development. As a leader in the firm's business development, Sam Boraie plays a crucial role of reinventing the real estate market and communities of New Brunswick, Newark and Atlantic City. Within the past five years Sam Boraie and Boraie Development revitalized the community and attracted new businesses, apartments, retail stores and office spaces. Along with real estate development Sam Boraie actively participates in decreasing poverty in the New Brunswick area with charities like Elijah's Promise.
In an article on PR NewsWire, the Aspire, a completed project from Boraie Development comprises 238 units near the New Brunswick Train Station includes luxury amenities and 10,000 square feet of retail space. Also near the New Brunswick Train Station is the Albany Street Plaza providing over 270,000 square feet of office and retail space. Comprising two towers the Albany Street Plaza generated two billion dollars in development for New Brunswick. Completed in 2007, One Spring Street includes 121 home residences in 25-stories with amenities that include a health club and swimming pool. Currently under construction in the New Brunswick area, The Rector will house residents within a 26-story building. Although Sam Boraie and Boraie Development primarily build in the New Brunswick area, other communities in the Atlantic City and Newark area are being developed under Boraie Development.
Partnered with basketball superstar Shaquille O'Neal, Boraie Development will add residential space to the Newark community. With an estimated investment of over $70 million dollar, Boraie Development focuses on adding a rental complex, movie theater and a new retail section. According to NY Times, after the economic downfall of Atlantic City, Sam Boraie works to renew landmarks such as The Revel Casino Hotel Tower. By rebuilding Atlantic City and Newark Sam Boraie gives communities affordable housing and enhances socialization and enjoyment among community members and families. A development project known as The Beach gives the community of Atlantic City 250 luxury residences comprising a swimming pool, workout facility and lounge. Equipped with luxury amenities, The Beach also offers a 400-space parking garage, retail shops, a grocery store and restaurants for building residents and members in the community.
Apart from operating development projects through Boraie Development Sam Boraie devotes himself to charity organizations in the New Brunswick area. As an Advisory Board Member of Elijah's Promise Same Boraie works to decrease poverty in New Brunswick. The organization not only provides non-profit meals for the New Brunswick community, but works to lower poverty by providing access to fresh food. To further decrease poverty Elijah's Promise also offers employment training in the culinary arts and baking. Sam Boraie is also involved in The State Theatre New Jersey as a Board of Trustees, https://www.statetheatrenj.org/board-of-trustees. As the second largest non-profit charity in New Jersey, the theater hosts a variety of music and family events for the community to attend. In the summer of 2017, Boraie Development is sponsoring free film screenings for children at the theater during the weekdays.