text:
\section{Introduction}

The concept of deformation quantization as defined by
F.~Bayen, M.~Flato, C.~Fronsdal, A.~Lichnerowicz, and D.~Sternheimer
in 1978 (cf. \cite{bayen}) consists in a formal local deformation of the
commutative algebra (a so-called star product)
of all smooth complex-valued functions on a symplectic
manifold such that the first order commutator equals the Poisson
bracket and pointwise complex conjugation remains an antilinear involution.
The existence of star products on every symplectic manifold was proved by
M.~DeWilde and P.~B.~A.~Lecomte in 1983 (cf. \cite{DL 83}) and independently
by B.~Fedosov in 1985 \cite{Fed 85}, \cite{Fed 94}.
A third existence proof was given
by H.~Omori, Y.~Maeda and A.~Yoshioka \cite {OMY 91}.

One of the problems with these star products is the fact that the formal
series involved is shown to never converge on the space of {\em all}
complex-valued smooth functions (see e.g.\cite{Rub 84}),
i.e. for every complex
number there are two functions whose star product diverges when the
formal parameter is substituted for that number.

This paper is a continuation of our work \cite{uns} in which we gave
a closed formula for a star product of Wick type on complex projective
space by a version of quantum phase space reduction.
Star products on complex projective space have already been
constructed by
H.~Omori, Y.~Maeda and A.~Yoshioka \cite {OMY 93} and by
C.~Moreno \cite {Mor 86} in a less explicit way.
$\CP^n$ can also be regarded as a coadjoint orbit of the unitary
group $U(n+1)$ (see the work of D.~Arnal, J.~Ludwig and M.~Masmoudi
for the existence of covariant star products on more general coadjoint
orbits in \cite {ALM 94} and references therein).

The aim of the present paper is twofold: firstly, we use our formula to
explicitly compute the star product
for a uniformly dense subalgebra of representative functions for
the $U(n+1)$-action on complex projective space. Secondly, we would
like to use the example $\CP^n$ in order to illustrate
the following {\em algebraic} procedure to deal with the above convergence
problem:
\begin{enumerate}
\item Find a complex subalgebra ${\cal U}$ of the $*$-algebra of
      all formal power series in $\nu$ with smooth coefficients
      such that  1.) all elements of ${\cal U}$ are power series with
      infinite radius of convergence in $\nu$ and 2.) their coefficients
      may be choosen in a ``sufficiently large'' (e.g. uniformly
      dense) subspace of the space of all smooth complex-valued functions.
\item Verify whether the subspace ${\cal I}_{\alpha}$ of ${\cal U}$
      defined by
      \begin{equation}
          {\cal I}_{\alpha} := \{\Phi(\nu) \in {\cal U} | \Phi(\alpha)=0 \}
      \end{equation}
      is a star-ideal of ${\cal U}$.
\item Identify the quotient ${\cal U}/{\cal I}_{\alpha}$ with the
      associative algebra of quantum observables related to
      the ``$\hbar$-value'' $\alpha$ and try to find a
      representation of this quotient in some Hilbert space.
\end{enumerate}
{}From the physical point of view it is often required that the subalgebra
$\cal U$ contains certain ``important observables'' which are in some cases
related to the presence of additional symmetries of the classical phase
space. For a general symplectic manifold the viability of the above
procedure (in particular the existence of a sufficiently large subalgebra
$\cal U$) is --to our best knowledge-- an open problem in the theory of
deformation quantization.

Nevertheless, in several examples having a large symmetry group
(e.g. the Moyal product on the 2-torus or the Wick product on
$\Bbb{C}^{n+1}$) the above programme may be carried out.
For such manifolds the space of representative functions
of the symmetry group plays a prominent r\^ole for the construction
of $\cal U$: in a remarkable article of M.~Cahen, S.~Gutt, and J.~Rawnsley
\cite{CGR II} the convergence of a star product for these functions has
been proved for all compact Hermitean spaces (in particular for $\CP^n$)
by analytic methods of complex differential geometry.
They start from the
finite-dimensional operator algebras of geometric quantization
in tensor powers of a very ample regular prequantum line bundle over
a compact K\"ahler manifold and use coherent states (see \cite{Ber 74},
\cite{Raw 77}) to first construct star products for the Berezin-Rawnsley
symbols (\cite{Ber 74}, \cite{Raw 77}) for each tensor power separately.
In a second step
an asymptotic expansion of these star products in the inverse tensor
power is shown to define a local star product on the manifold where
the formal parameter appears as a sort of interpolation of the inverse
tensor powers.

The approach of this paper is in some sense reverse
to the programme of M.~Cahen, S.~Gutt, and J.~Rawnsley
(cf. \cite{CGR I}, \cite{CGR II},\cite{CGR III}, \cite{CGR IV})
and only makes use of elementary algebraic methods:
We start from the explicit star product on $\CP^n$ (see \cite{uns})
and define $\cal U$ as a certain proper subspace of the space of all
polynomials in the formal parameter with coefficients in the
uniformly dense subspace of representative functions
for the unitary group $U(n+1)$. Since all occurring star products
can explicitly be computed the analysis of the ideals ${\cal I}_\alpha$
and the quotient algebras ${\cal U}/{\cal I}_\alpha$
becomes relatively simple. The main result is that for inverse integer
values of
the formal parameter $\nu$ the quotient algebras turn out to be
finite-dimensional full complex matrix-algebras whereas all
noninteger quotients
are of infinite dimension and define converging star products on the
space of representative functions for these values as has been stated in
\cite{CGR II}.

{}From the physical point of view one can
regard the finite-dimensional quantum algebras as the set of all
quantum observables restricted to the eigenspace of integral energy of a
harmonic oscillator of $n+1$ degrees of freedom (where the ground state
energy is zero for the Wick quantization rule).

The advantage of the above algebraic programme of prescribing a real value
to the formal parameter of deformation quantization is that one may hope
to transfer it to physical situations with an infinite number of degrees
of freedom, i.e. field theories, where the powerful analytical methods
in the theory of finite-dimensional manifolds are no longer well-defined.

The paper is organised as follows: after briefly reviewing the concepts
and formulas of our last paper \cite{uns} in section 2 we compute the
star product of two Berezin-Rawnsley symbols in section 3 and discuss
the unitary symmetry action on the star product.
Section 4 is then devoted to define the algebra ${\cal U}$, to compute the
ideals ${\cal I}_{\alpha}$ and the quotient algebras
${\cal U}/{\cal I}_{\alpha}$.
In section 5 we briefly consider other phase spaces already dealt with
in the literature for which the above
programme works: the complex vector space $\Bbb{C}^{n+1}$ with the
Wick product, the $2n$-torus $T^{2n}$ with the Moyal product, and the
Poincar\'e disk. For this last example we can recover the formula of the
star product for the corresponding Berezin-Rawnsley symbols given by
Cahen, Gutt, and Rawnsley in \cite{CGR III}.

\vspace {0.3cm}
\noindent

{\it Notation:}
Throughout this paper we use the Einstein summation convention,
i.e. summation over repeated indices is automatic. Moreover, the symbol
$F(z)$ for a complex-valued function $F$ of a complex vector $z$ does
{\em not} necessarily imply that $F$ is holomorphic.

\section{Review of star products on complex projective space}

In this section we shall give a short review of earlier work \cite{uns}
in which we derived an explicit formula for a star product
of Wick-type on the complex projective space $\CP^n$.

Let $\pi: \CNull \to \CP^n$ be the canonical projection of a complex
vector $z$ onto the complex ray through it. Let $x:=\bar{z}^iz^i$.
The usual Wick product on $\CNull$ of two complex-valued functions
$F, G \in C^\infty(\CNull)$ is given
as the following formal power series in the parameter $\lambda$:
\begin{equation} \label{Wickusual}
    F\ast G = \sum_{r=0}^\infty
    \frac{\lambda^r}{r!}\frac{\partial^r F}{\partial z^{i_1}\cdots\partial z^{i_r}}
    \frac{\partial^rG}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}.
\end{equation}
We have called a function $F \in C^\infty (\CNull)$ {\em homogeneous}
iff it is invariant under the natural action of the group
$\Bbb{C}\setminus \{0\}$. These functions are precisely given by
pull backs of functions $f \in C^\infty(\CP^n)$,
$F = f \circ \pi = \pi^*f$. We have called
a function $R \in C^\infty(\CNull)$ {\em radial} iff it is a function
of $x$, i.e. iff there is a smooth function $\rho: \Bbb{R}^+ \to \Bbb{C}$
such that $R = \rho \circ x$.
We have defined a formal differential operator
$S: C^\infty(\CNull)[[\lambda]] \to C^\infty(\CNull)[[\lambda]]$
depending only on $x$ and
$\partial_x := \frac{1}{2x}(z^i\partial_{z^i}+\bar{z}^i\partial_{\bar{z}^i})$
whose standard symbol
$\hat{S}(x,\alpha) := (Se_\alpha)(x) e^{-\alpha x}$ ($e_\alpha$ denoting
the exponential function $x \mapsto e^{\alpha x}$ for $\alpha\in\Bbb{R}$)
is given by (setting the series $D$ in \cite [eqn. 9] {uns} equal to $1$):
\begin{equation} \label{Ssymb}
    \hat{S}(x,\alpha) = \exp\left(\frac{x}{\lambda}
    \log(1+\lambda\alpha) - \lambda\alpha)\right).
\end{equation}
$S$ and its inverse $S^{-1}$ act trivially on homogeneous functions, i.e.
$SF = F = S^{-1}F$, but do in general transform radial functions
into radial ones, in particular (for a nonnegative integer $r$)
\cite [eqn. 14] {uns}
\begin{equation} \label{xrel}
    Sx=x,
    ~~~~
    Sx^r = x^r\prod_{k=0}^r \left(1 - k\frac{\lambda}{x}\right),
    ~~~~
    Sx^{-r} = x^{-r}\prod_{k=0}^r \left(1 + k\frac{\lambda}{x}\right)^{-1}.
\end{equation}
We have used $S$ as an equivalence transformation for a modified
Wick product of two functions $F,G\in C^\infty(\CNull)$:
\begin{equation} \label{Wickmodif}
   F \newstar G := S (S^{-1}F \ast S^{-1}G).
\end{equation}
For two radial functions $R_1,R_2$
and a homogeneous function $F$ on $\CNull$ this new star product is
just pointwise multiplication \cite [eqn. 12] {uns}:
\begin{equation} \label{radradhom}
    R_1 \newstar R_2 = R_1R_2 = R_2 \newstar R_1,
    ~~~~
    R_1 \newstar F = R_1F = F \newstar R_1,
\end{equation}
whereas for two smooth homogeneous functions $F, G$ we get
\begin{equation} \label{homhom}
  (F \newstar G) (z)
  = \sum_{r=0}^\infty\frac{1}{r!}
  \left(\frac{\lambda}{x}\right)^r
  \prod_{k=1}^r \left( 1+k\frac{\lambda}{x} \right)^{-1}
  x^r\frac{\partial^r F}{\partial z^{i_1}\cdots\partial z^{i_r}}(z)\frac{\partial^r G}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}(z).
\end{equation}
The main result of \cite{uns} was the fact that this formula can
directly be projected to $\CP^n$ by phase space reduction via
the $U(1)$-momentum map $J: \CNull \to \Bbb{R}: z \mapsto -\frac{x}{2}$
of the canonical $U(1)$-action on $\CNull$: for a negative real
number $\mu$ and a $U(1)$ invariant function $F$ in
$C^\infty(\CNull)$ we write $F_\mu$ for the unique function in
$C^\infty(\CP^n)$ obtained by first restricting $F$ to the odd
sphere $J^{-1}(\mu)$ and then projecting it to $\CP^n$
\cite [4.3] {AM 85}. Then the formula
\begin{equation} \label{reduct}
    F_\mu \mustar G_\mu:=(F \newstar G)_\mu
\end{equation}
was shown to define a star product on $\CP^n$. The explicit form
of $\mustar$ is obtained by replacing $\lambda/x$ by $\lambda/(-2\mu)$
in (\ref{homhom}) and noting that the bidifferential operator
$\tilde{M}_r(f,g)(\pi(z)) := x^r\frac{\partial^r \pi^*f(z)} {\partial z^{i_1}\cdots\partial z^{i_r}}
\frac{\partial^r \pi^*g(z)} {\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}$
is well-defined on $f, g \in C^\infty(\CP^n)$. For simplicity we shall
work with the redefined formal parameter $\nu:=\lambda/(-2\mu)$ in what
follows.

\begin{lemm} :
   The standard symbol of $S^{-1}$ is described by the formula
    \begin{equation} \label {zeitent}
      \widehat{S^{-1}}(x,\alpha)
      = e^{\frac{x}{\lambda}(e^{\alpha\lambda}-1-\alpha\lambda)}
      = e^{\ast\alpha x}e^{-\alpha x}
    \end{equation}
   where the last term involves the star-exponential \cite{bayen} of
   the function $x$ with respect to the usual Wick product
   (\ref{Wickusual}).
\end{lemm}
{\sc Proof}: Since $Se_\beta=e_{\frac{1}{\lambda}\log(1+\beta\lambda)}$
we obviously get $e_\beta=S^{-1}e_{\frac{1}{\lambda}\log(1+\beta\lambda)}$ which
proves the first equation after the substitution
$\alpha:=\frac{1}{\lambda}\log(1+\beta\lambda)$. Secondly, note that
\[
  e^{\ast\alpha x} = S^{-1}Se^{\ast\alpha S^{-1}x} =
  S^{-1}e^{\newstar\alpha x}
  \stackrel{(\ref{radradhom})} {=} S^{-1}e^{\alpha x} =
  \widehat{S^{-1}}(x,\alpha)e^{\alpha x}
\]
which proves the second equality. \hfill $\Box $

Remark: Note that the function $H := \frac{1}{2} x$ equals the usual
Hamiltonian function of an isotropic harmonic oscillator in $n+1$
degrees of freedom. The above star-exponential of $x$ for
$\alpha = - it / 2\hbar$ and $\lambda = 2\hbar$
then corresponds to the quantum mechanical time development
operator for this system.

\section{A star product for representative functions on
         complex projective space}

Let $p: \Bbb{C}^{n+1} \to \Bbb{C}: z \mapsto P(z)$ be a polynomial
function (in the $2n+2$ variables
$z^0, \ldots, z^n$, $\bar{z}^0, \ldots, \bar{z}^n$).
We shall call $p$ {\em homogeneous of degree $(k,k)$} for a nonnegative
integer $k$ iff $p(\lambda z) = (\lambda\bar{\lambda})^k p(z)$
for all $\lambda\in\Bbb{C}\setminus\{0\}$.
We denote by ${\cal E}_k$ the following subspace of $C^\infty (\Bbb{C} P^n)$:
\begin{equation} \label{dieEks}
   {\cal E}_k := \left\{\phi \left|
   \mbox{ there is a homogeneous polynomial $p_k$ of degree $(k,k)$ s.t.~}
           (\pi^\ast\phi)(z)=\frac{1}{x^k}p_k(z) \right\}\right..
\end{equation}

\begin{lemm} :
\begin{enumerate}
\item For each integer $k \geq 0$: ${\cal E}_k\subset{\cal E}_{k+1}$.
\item ${\cal E}:=\bigcup_{k=0}^\infty{\cal E}_k$ is a filtered
      subalgebra of $C^\infty(\Bbb{C} P^n)$
      with respect to the pointwise multiplication. It is closed under
      complex conjugation.
\item $\cal E$ separates points and is therefore a dense subalgebra
      of $C^\infty(\Bbb{C}P^n)$ with respect to the uniform topology.
\end{enumerate}
\end{lemm}
{\sc Proof}:
i)   Clearly $\frac{p_k}{x^k}=\frac{xp_k}{x^{k+1}} \in \pi^* {\cal E}_{k+1}$.
ii)  This is obvious.
iii) Consider the complex rays $\pi(z_{(1)})\not=\pi(z_{(2)})$ for
$z_{(1)},z_{(2)}\in\Bbb{C}^{n+1}\setminus 0$. This is true iff
$z_{(1)},z_{(2)}$ are linearly independent iff there is a
$y\in\Bbb{C}^{n+1}$ such that $\langle y,z_{(1)}\rangle = 1$
and $\langle y,z_{(2)}\rangle = 0$ where $\langle y,z \rangle$ denotes
the standard sesquilinear form $\langle y,z\rangle=\bar{y}^kz^k$. Then
$\phi\in {\cal E}_1$ defined by $\phi(\pi(z)):=\frac{|<y,z>|^2}{x}$
separates $\pi(z_{(1)})$ and $\pi(z_{(2)})$. The density of $\cal E$
follows from the Stone-Weierstrass Theorem. \hfill $\Box $ \\

Consider the standard action of the unitary group $U(n+1)$ on $\CNull$:
$(g,z)\mapsto gz=:\Phi_g(z)$ and its induced action on $\CP^n$:
$(g,\pi(z))\mapsto \pi(gz)=:\Psi_g(\pi(z))$. A smooth
complex-valued function $f$ on $\CP^n$ is called {\em representative} with
respect to the $U(n+1)$ action iff
\begin{equation} \label{rep}
    \Bbb{C}{\rm -span} \{f \circ \Psi_g | g\in U(n+1)\}
    \quad \mbox{ is finite-dimensional.}
\end{equation}
We now get a characterization of $\cal E$ which should be fairly standard
(see \cite[sec. 3, Lemma 1]{CGR II}):
\begin{lemm} \label{Eistrep}:
  The algebra $\cal E$ is equal to the set of all representative
  functions on $\CP^n$.
\end{lemm}
{\sc Proof}:
Since $x$ and the finite-dimensional space of all homogeneous
polynomials of degree $(k,k)$ are invariant under $U(n+1)$
it follows that $\cal E$ consists of representative functions.
In order to prove the reversed inclusion we can use a more general argument:
$\CP^n$ is a homogeneous space $G/H$ for the compact Lie group
$G=U(n+1)$ with compact isotropy subgroup $H=U(1)\times U(n)$. Now the
space of all representative functions $f$ on $G/H$ (defined as in (\ref{rep})
with $U(n+1)$ replaced by any compact Lie group $G$) is clearly in
one-to-one correspondence with the space of its pull-back
to $G$ under the natural projection $G\rightarrow G/H$: this in turn is given by
the space of all $H$ right invariant representative functions on $G$ with
respect to left multiplication. Both this space and the pull-back of $\cal E$
to $G$ are $G$-modules and are therefore closed in the space of all
representative functions on $G$ with respect to the uniform topology on $G$
(see e.g. \cite{BtD 85}, p.126, Prop.(1.4) (iii)). Since the pull-back
obviously is a continuous closed linear map with respect to the uniform
topologies
and $\cal E$ is dense in the space of all representative functions on
$G/H$ thanks to the previous Lemma it follows that $\cal E$ is equal to that
space.
\hfill $\Box $ \\

Yet another equivalent description of $\cal E$ is obtained in terms
of the {\em Berezin-Rawnsley symbols} known from geometric quantization
(\cite{Ber 74}, \cite{Raw 77}, \cite{CGR I}): for a fixed nonnegative
integer $k$ take the vector space ${\cal H}^{(k)}$ of complex-valued
holomorphic polynomials $\psi$ on $\Bbb{C}^{n+1}$, which are
homogeneous of degree $k$, i.e. $\psi(\lambda z)=\lambda^k\psi(z)$ for
all complex numbers $\lambda$.
The dimension of this space is clearly $N:={n+k \choose k}$ and we shall
henceforth identify ${\cal H}^{(k)}$ with $\Bbb{C}^N$ by means of the base
$(z^{i_1}\cdots z^{i_k})$. Consider
the space $B({\cal H}^{(k)})$ of complex linear endomorphisms of
${\cal H}^{(k)}$. Any $A\in B({\cal H}^{(k)})$ can be viewed as
a complex $N\times N$ matrix
$A_{i_1\cdots i_k,j_1\cdots j_k}$ where each of the indices
$i_1,\ldots,i_k,j_1,\ldots,j_k$ ranges over $0,1,\ldots,n$ and the
matrix elements are symmetric with respect to all permutations among
the $i_1,\ldots, i_k$ and among the $j_1,\ldots, j_k$. To each
$A\in B({\cal H}^{(k)})$ one can associate
the polynomial function
\begin{equation} \label{sigmaoben}
   \tilde{\sigma}(A):\Bbb{C}^{n+1}\rightarrow\Bbb{C}:z\mapsto
      \bar{z}^{i_1}\cdots\bar{z}^{i_k}z^{j_1}\cdots z^{j_k}
                 A_{i_1\cdots i_k,j_1\cdots j_k}.
\end{equation}
Clearly, $\tilde{\sigma}(A)$ is homogeneous of degree $(k,k)$, and
by counting dimensions it can be seen that every homogeneous polynomial
function of degree $(k,k)$ is of that form.
The {\em Berezin-Rawnsley symbol $\sigma (A)$ associated to $A$}
is then defined by
\begin{equation} \label{BerRaw}
    \sigma(A) : \CP^n \to \Bbb{C}:
    \pi(z) \mapsto \frac{\tilde{\sigma}(A)(z)}{x^k}.
\end{equation}
Then the following corollary is clear:
\begin{cor} :
    For each nonnegative integer $k$ the space ${\cal E}_k$ is spanned
    by all the Berezin-Rawnsley symbols $\sigma(A)$ associated to
    $A\in B({\cal H}^{(k)})$.
\end{cor}

Both on $\CNull$ and $\CP^n$ there are momentum maps (see \cite{AM 85} for
general definitions) for the $U(n+1)$-action which can be expressed in
terms of Berezin-Rawnsley symbols contained in ${\cal E}_1$:
\begin{lemm} : Let $\goth{u} (n+1)$ denote the space of
    complex antihermitean $(n+1)\times (n+1)$-matrices,
    i.e the Lie algebra of the unitary group $U(n+1)$. Then:
    \begin{enumerate}
    \item[i)] The following map
              \begin{equation}
                   \tilde{P}:\CNull\to \goth{u}(n+1)^* :
                   z \mapsto (A \mapsto \tilde{\sigma}(A))
              \end{equation}
              is a momentum map for the $U(n+1)$-action on $\CNull$.
    \item[ii)] The following map
               \begin{equation}
                   P:=\tilde{P}_\mu:\CP^n \to \goth{u}(n+1)^*:
                   \pi(z) \mapsto (A\mapsto -2\mu \sigma(A)(\pi(z))
                   = \tilde{\sigma}(A)_\mu (\pi(z))~)
               \end{equation}
               is a momentum map for the $U(n+1)$-action on $\CP^n$.
   \end{enumerate}
   Recall that the index $\mu$ refers to the $U(1)$-phase space reduction
   of $\CNull$ by the $U(1)$ momentum map $J(z)=-\frac{x}{2}$.
\end{lemm}
{\sc Proof}:
i) The Hamiltonian vector field of $P(A)$ is given by
$X_{\tilde{\sigma}(A)}(z)$ $=$
$\frac{2}{i} (A_{ij}z^j\partial_{z^i}
 + \bar{A}_{ij}\bar{z}^j\partial_{\bar{z}^i})$,
since $A_{ij}=-\overline{A}_{ji}$ which obviously equals the infinitesimal
generator $A_{\CNull}$ of the $U(n+1)$-action.
The $Ad^\ast$-equivariance of this map is obvious.\\
ii) Recall the projection $\pi_\mu:J^{-1}(\mu)\rightarrow \CP^n$ to the reduced
phase space (compare \cite[4.3]{AM 85}). Since
$\Phi_g\circ\pi_\mu=\pi_\mu\circ\Psi_g$ it is clear that
$T\pi_\mu A_{\CNull}=A_{\Bbb{C}P^n}\pi_\mu$ where $A_{\Bbb{C}P^n}$ is the
infinitesimal generator of the $U(n+1)$-action on $\CP^n$. Using the
identity
$A_{\Bbb{C}P^n}\pi_\mu(z)=X_{\tilde{\sigma}(A)_\mu}\pi(z)$ we get by
phase-space reduction
\[
  T_z\pi_\mu X_{\tilde{\sigma}(A)_\mu}(z)=
  T_z\pi_\mu A_{\CNull} (z)=
  X_{\tilde{\sigma}(A)_\mu}(\pi(z)).
\]
The $Ad^\ast$-equivariance of this map follows at once from
the $Ad^\ast$-equivariance of $\tilde{P}$. \hfill $\Box $\\

We shall now compute star products for elements of $\cal E$:
Set $\nu := \frac{\lambda}{-2\mu}$, $\nu^{(0)} := 1$, $\nu^{(1)} := 1$
and
\begin{equation} \label {nudef}
    \nu^{(k)} := (1-\nu) \dots (1-(k-1)\nu)
\end{equation}
\begin{theo}\label{one} :
For $f\in{\cal E}_k$, $g\in{\cal E}_l$
we get
\begin{equation} \label{oneeins}
    (\pi^\ast f)\newstar (\pi^\ast g)(z)=\sum_{r=0}^{min(k,l)}
    \frac{\lambda^r}{r!}\frac{S(x^{k+l-r})}{(Sx^k)(Sx^l)}\frac{1}{x^{k+l-r}}
    \frac{\partial^r(x^k\pi^\ast f)}{\partial z^{i_1}\cdots\partial z^{i_r}}(z)
    \frac{\partial^r(x^l\pi^\ast g)}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}(z)
\end{equation}
and
\begin{equation} \label {StarEqn}
    f \mustar g = \sum_{r=0}^{min(k,l)}
    \frac{\nu^r}{r!}\frac{\nu^{(k+l-r)}}{\nu^{(k)}\nu^{(l)}}
    M_r^{(k,l)}(f,g)
\end{equation}
with the following bidifferential operator on $C^\infty(\CP^n)$
\begin{equation} \label{dieHs}
    \pi^\ast M^{(k,l)}_r(f,g)
    := \frac{1}{x^{k+l-r}}
    \frac{\partial^r(x^k\pi^\ast f)}{\partial z^{i_1}\cdots\partial z^{i_r}}
    \frac{\partial^r(x^l\pi^\ast g)}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}
\end{equation}
\end{theo}
{\sc Proof} : The usual Wick product (\ref{Wickusual}) of the two
{\em polynomials}
$\tilde{f_k}:=x^k\pi^\ast f,\tilde{g_l}:=x^l\pi^\ast g$ is given by
\begin{equation}
     \tilde{f}_k\ast\tilde{g}_l =
     \sum_{r=0}^{min(k,l)} \frac{\lambda^r}{r!}
     \frac{\partial^r\tilde{f}_k} {\partial z^{i_1}\cdots\partial z^{i_r}}
     \frac{\partial^r\tilde{g}_l}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}.
\end{equation}
Using definition (\ref{Wickmodif}) and the formulas
(\ref{radradhom}) and (\ref{xrel}) we find
\begin{eqnarray*} {(Sx^k)(Sx^l)(\pi^\ast f)\tilde{\ast} (\pi^\ast g) }
 & = & \left( S(x^k\pi^\ast f)\right)\tilde{\ast}
       \left( S(x^l\pi^\ast g) \right) = S((x^k\pi^\ast f)\ast (x^l\pi^\ast g)) \\
 & = & S\left( \sum_{r=0}^{min(k,l)}\frac{\lambda^r}{r!}
       \frac{\partial^r\tilde{f}_k}{\partial z^{i_1}\cdots\partial z^{i_r}}
       \frac{\partial^r\tilde{g}_l}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}
       \right) \\
 & = & \sum_{r=0}^{min(k,l)}\frac{\lambda^r}{r!}
       S\left( x^{k+l-r}\frac{1}{x^{k+l-r}}
       \frac{\partial^r\tilde{f}_k}{\partial z^{i_1}\cdots\partial z^{i_r}}
       \frac{\partial^r\tilde{g}_l}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}
       \right) \\
 & = & \sum_{r=0}^{min(k,l)}\frac{\lambda^r}{r!}
       S\left(x^{k+l-r} \right) \frac{1}{x^{k+l-r}}
       \frac{\partial^r\tilde{f}_k}{\partial z^{i_1}\cdots\partial z^{i_r}}
       \frac{\partial^r\tilde{g}_l}{\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_r}}
\end{eqnarray*}
which proves the first equation. The second equation immediately follows
by the reduction formula (\ref{reduct}).
\hfill $\Box $ \\

Remark: Thanks to the straight forward recursion formulas
$M^{(k+1,l)}_r = M^{(k,l)}_r+r(l-(r-1)) M^{(k,l)}_{r-1}$ and
$M^{(k,l+1)}_r = M^{(k,l)}_r+r(k-(r-1)) M^{(k,l)}_{r-1}$ it can
easily be checked that the right hand side
of (\ref{oneeins}) is well-defined, i.e. if e.g. $f$ is regarded as an
element of ${\cal E}_{k+a}$ for a positive integer $a$.

\begin{cor}\label{COR}: For $A, B \in B({\cal H}^{(1)})$ we have
\begin{equation}
    \sigma(A)^k \mustar \sigma(B)^l =
    \sum_{r=0}^{min(k,l)}
    \frac{\nu^r}{r!}\frac{k! \: l!} {(k-r)!(l-r)!}
    \frac{\nu^{(k+l-r)}}{\nu^{(k)}\nu^{(l)}}
    \sigma(AB)^r\sigma(A)^{k-r}\sigma(B)^{l-r}
\end{equation}
\end{cor}
{\sc Proof}: This is easily seen by setting $f = \sigma(A)^k$ and
$g = \sigma(B)^l$ and using Theorem \ref{one}.
\hfill $\Box $ \\

We shall now show that the momentum map $P$ for the $U(n+1)$-action
on $\CP^n$ is even a quantum momentum map. More precisely:
\begin{lemm} \label{quantmom}:
   $P$ is a quantum momentum mapping on $\CP^n$ for the
   $U(n+1)$-action, i.e.
   for every smooth function $\phi: \CP^n \to \Bbb C$ the
   following equation holds:
   \begin{equation} \label {invarianz}
        P(A) \mustar \phi - \phi \mustar P(A)
        = \frac{i\lambda}{2}\{P(A), \phi\}_\mu ,
   \end{equation}
   i.e. the star product $\mustar$ is strongly $U(n+1)$-invariant.
\end{lemm}
{\sc Proof}:
First we prove the equation
$\pi^*\sigma(A) \newstar f - f \newstar \pi^*\sigma(A) = \frac{i\lambda}{2}
\{\pi^*\sigma(A), f\}$ for $f = \pi^*\phi$. Then eqn. (\ref{invarianz})
follows by eqn.(\ref{reduct}).
We have the strong invariance of the Wick product
\[
   \tilde{\sigma}(A) * f - f * \tilde{\sigma} (A) =
   \frac{i\lambda}{2} \{\tilde\sigma (A), f \}.
\]
With the equivalence transformation $S$ we find
\[
   S^{-1} \Big( S\tilde\sigma (A) \newstar Sf -
                 Sf \newstar S\tilde\sigma (A) \Big)
   = \frac{i\lambda}{2} \{\tilde\sigma (A), f \}
\]
and with $\pi^*\sigma (A) = x \tilde \sigma (A)$ and $Sf = f$
this leads to
\[
   Sx \Big(
   \pi^*\sigma (A) \newstar f - f \newstar \pi^*\sigma (A) \Big)
   =
   \frac{i\lambda}{2} S \left(\{x \pi^*\sigma (A), f \}\right).
\]
The Poisson bracket $\{x\pi^*\sigma (A), f\} = x \{\pi^*\sigma (A), f\}$ is
again homogeneous. Hence the right hand side of the last equation is simply
$x \{\pi^*\sigma(A), f\}$. With $Sx = x$ the proof is complete.
\hfill $\Box $

Remark: The case $k=l=1$ in Cor.2 (\ref{COR}) shows that the functions
$\sigma(A)$ and $\sigma(B)$ commute with respect to $\mustar$ iff
the operators $A$ and $B$ commute. Let $A_1,\ldots, A_n$ be a linearly
independent set of commuting traceless Hermitean matrices. It follows in
particular that the functions $\sigma(A_1), \ldots, \sigma(A_n)$
are functionally
independent and in involution, i.e. the Poisson bracket of $\sigma(A_i)$
with $\sigma(A_j)$ vanishes for all $i,j$. In other words they form a
{\em completely integrable system} in the sense of Liouville on $\CP^n$.
Note that this system is nontrivial in the sense that there is no global
chart of action-angle-variables, i.e. $\CP^n$ is not symplectomorphic to
some $T^r \times \Bbb{R}^{2n-r}$ for a nonnegative integer $r\leq n$. The
above Corollary now
implies that these functions also commute with respect to the star
product and can be viewed as a {\em quantum integrable system} on
$\CP^n$.

\section{Construction of the subalgebra $\cal U$, the ideals
        $\cal I_\alpha$, and the quotients ${\cal U}/{\cal I}_\alpha$}

If we take two functions $\phi_k \in {\cal E}_k$ and
$\psi_l \in {\cal E}_l$ and multiply them with the $\nu$-polynomial
$\nu^{(k)}$ and $\nu^{(l)}$, respectively, then formula
(\ref{StarEqn}) shows that their $\mustar$-product
contains only {\em polynomials} in the parameter $\nu$.
Moreover, that
particular combination is restored, i.e. the functions
$M_t^{(k,l)} (\phi_k, \psi_l) \in {\cal E}_{k+l-t}$ appear only
in combination with $\nu^{(k+l-t)}$.
This motivates the definition of the following subspaces of
$C^\infty(\CP^n)[[\nu]]$:
\begin{equation} \label{dieUks}
    {\cal U}_0 := {\cal E}_0 \mbox{ and }
    {\cal U}_k := \nu^{(k)}{\cal E}_k + \nu\nu^{(k-1)} {\cal E}_{k-1}
                  + \cdots + \nu^{k-1}\nu^{(1)}{\cal E}_1 +
                  \nu^k{\cal E}_0 \qquad \forall~k \in \Bbb{N}
\end{equation}
where each ${\cal U}_k$ is a (not necessarily direct) sum of subspaces of
$C^\infty(\CP^n)[[\nu]]$.
We denote by ${\cal E}[\nu]$ the {\em polynomials} in $\nu$ with
coefficients in ${\cal E}$.
The following theorem describes the structure of ${\cal U}$.
\begin{theo} :
\begin{enumerate}
\item For each integer $k$: ${\cal U}_k\subset {\cal U}_{k+1}$.
      Define ${\cal U}:=\bigcup_k^\infty {\cal U}_k$. Then
      $\cal U$ is a proper $\Bbb{C}[\nu]$-submodule of ${\cal E}[\nu]$.
\item $\cal U$ is a filtered subalgebra of
      $C^\infty(\Bbb{C}P^n)\left[ [\nu ] \right]$
      with respect to $\mustar$, i.e.
      \[ {\cal U}_k \mustar {\cal U}_l\subset {\cal U}_{k+l} \]
\end{enumerate}
\end{theo}
{\sc Proof :}
{\it i.)} Let $\Phi\in {\cal U}_k$. $\Phi$ is of the form
$\Phi=\sum_{r=0}^k\nu^r\nu^{(k-r)}\phi_{k-r}$ with
$\phi_{k-r} \in {\cal E}_{k-r}$.
The filtration of $\cal E$ implies $\phi_{k-r} \in {\cal E}_{k-r+1}$ so
$\nu^r\nu^{(k-r+1)}\phi_{k-r} \in {\cal U}_{k+1}$. The element
$\nu^{r+1} \nu^{(k-r)}\phi_{k-r}$ is also contained in ${\cal U}_{k+1}$.
By eqn. (\ref{nudef}) we get
$\nu^{(k-r+1)} = (1-(k-r)\nu)\nu^{(k-r)}$. Hence
\[
    \nu^r\nu^{(k-r)} \phi_{k-r} =
    \nu^r\nu^{(k-r+1)} \phi_{k-r} + (k-r) \nu^{r+1} \nu^{(k-r)} \phi_{k-r}
    \in {\cal U}_{k+1}
\]
which proves the inclusion ${\cal U}_k \subset {\cal U}_{k+1}$. It is
clear from the definition (\ref{dieUks}) that
$\nu {\cal U}_k \subset {\cal U}_{k+1} \subset {\cal E}[\nu]$
whence $\cal U$ is $\Bbb{C}[\nu]$-submodule of ${\cal E}[\nu]$.
Note that for fixed $y\in\CNull$ the function
$\pi(z) \mapsto \frac{|\langle y,z \rangle|^4}{x^2}\in{\cal E}_2$ is
not contained in $\cal U$.
{\it ii.)} By Theorem (\ref{one}) the star product for arbitrary
$\Phi_k=\sum_{r=0}^k\nu^{k-r}\nu^{(r)}\phi_r \in {\cal U}_k$ and
$\Psi_l=\sum_{s=0}^l\nu^{l-s}\nu^{(s)}\psi_s \in {\cal U}_l$
with $\phi_r \in {\cal E}_r$ and $\psi_s \in {\cal E}_s$ is given by
\[
    \Phi_k \mustar \Psi_l =
    \sum_{r=0}^k\sum_{s=0}^l\sum_{t=0}^{min(r,s)}
    \frac{1}{t!} \nu^{k-r+l-s+t} \nu^{(r+s-t)}
    M_t^{(r,s)} (\phi_r, \psi_s),
\]
hence each summand is an element of ${\cal U}_{k+l}$.
This proves the second part.
\hfill $\Box $\\

We should now like to substitute the formal parameter $\nu$ for a fixed
nonzero real number $\alpha$. In the subalgebra $\cal U$ this
is well-defined because $\cal U$ only contains polynomials in $\nu$.
The kernel of the substitution homomorphism
\begin{equation}
    {\cal I}_\alpha:=\{\Phi(\nu)\in {\cal U}| \Phi(\alpha)=0\}
\end{equation}
will turn out to be a $\mustar$-ideal.
\begin{lemm}
    ${\cal I}_\alpha$ is a $\mustar$-ideal and the general form of an
    element $\Phi \in {\cal I}_\alpha \cap {\cal U}_k$ is
    \begin {enumerate}
    \item for $\alpha \not\in \{1, \frac{1}{2}, \frac{1}{3}, \ldots \}$
          \begin{equation} \label {alphaR}
            \Phi (\nu) = (\nu - \alpha) u_{k-1} (\nu)
            \quad \mbox { with } u_{k-1} (\nu) \in {\cal U}_{k-1}
          \end{equation}
    \item for $\alpha = \frac{1}{K}$ with $K \in \Bbb N\setminus \{0\}$
          for $k \le K$
          \begin{equation}
               \Phi (\nu) = (\nu - \frac{1}{K}) u_{k-1}(\nu)
               \label {alphakK}
          \end{equation}
          and for $k > K$
          \begin{equation}
               \Phi (\nu) = \nu^{(k)} \phi_k + \cdots +
               \nu^{k-K-1} \nu^{(k-K+1)} \phi_{k-K+1}
               + (\nu - \frac{1}{K}) u_{k-1} (\nu)
               \label {alphaK}
          \end{equation}
          with some $u_{k-1}(\nu) \in {\cal U}_{k-1}$ and
          $\phi_r \in {\cal E}_r$.
    \end {enumerate}
\end {lemm}

{\sc Proof :}
    It is clear that ${\cal I}_\alpha$ is a $\mustar$-ideal if the
    eqn. (\ref {alphaR}) resp. (\ref {alphakK}, \ref {alphaK})
    are valid because of the
    form of $\nu^{(k)}$ in eqn. (\ref {nudef}) and eqn. (\ref {StarEqn})
    for the $\mustar$-product. So we only have to prove eqn. (\ref {alphaR})
    and eqs. (\ref {alphakK}, \ref{alphaK}).

    Case {\it i.)}
    $\alpha \not\in \{ 1, \frac{1}{2}, \frac{1}{3},\ldots \}$
    We prove (\ref {alphaR}) by induction on k.
    With ${\cal U}_1 = \nu{\cal E}_0 + {\cal E}_1$ we get
    $\Phi (\nu) = c + \nu\sigma(A)$ for some $c \in \Bbb{C}$ and
    $\sigma (A) \in {\cal E}_1$. $\Phi (\alpha) = 0$ implies
    $A = - \frac{c}{\alpha} \mbox{\bf 1}$ where {\bf 1} is the unit matrix.
    So we have $\Phi(\nu) = c (1-\frac{\nu}{\alpha})$. This proves the
    case $k = 1$.

    Consider now the induction step $k\!-\!1 \to k$. Since
    ${\cal U}_k = \nu^{(k)} {\cal E}_k + \nu {\cal U}_{k-1}$ we write for
    $\Phi \in {\cal U}_k$
    \[
        \Phi (\nu) = \nu^{(k)} \phi_k + \nu u_{k-1} (\nu)
        \qquad \mbox {~ with~} \phi_k \in {\cal E}_k \mbox {~and~}
        u_{k-1} (\nu) \in {\cal U}_{k-1}.
    \]
    Then $\Phi (\alpha) = 0$ implies $\phi_k = -
    \frac{\alpha}{\alpha^{(k)}} u_{k-1} (\alpha) \in {\cal E}_{k-1}$
    because $\alpha^{(k)} \ne 0$.
    Writing $\nu^{(k)} = (1-(k-1)\nu)\nu^{(k-1)}$ we get
    \[
        \Phi (\nu) = \nu^{(k-1)} \phi_k + \nu \left(
        -(k-1)\nu^{(k-1)} \phi_k + u_{k-1} (\nu) \right)
        =: \nu^{(k-1)} \phi_k + \nu u_{k-1}'(\nu)
    \]
    with $u_{k-1}'(\nu) \in {\cal U}_{k-1}$.
    In $\Phi (\nu) = (\nu-\alpha) u_{k-1}'(\nu) + \nu^{(k-1)} \phi_k +
    \alpha u_{k-1}'(\nu)$ the first term vanishes at $\nu = \alpha$ so
    we have
    \[
        \left. \nu^{(k-1)} \phi_k + \alpha u_{k-1}'(\nu)
        \right|_{\nu=\alpha} = 0 .
    \]
    But $\nu^{(k-1)} \phi_k + \alpha u_{k-1}'(\nu)$ is an element in
    ${\cal U}_{k-1}$ so we can apply the induction and conclude
    \[
        \Phi (\nu) = (\nu - \alpha) u_{k-1}'(\nu) +
                     (\nu - \alpha) u_{k_2}'' (\nu)
    \]
    with some $u_{k-2}''(\nu) \in {\cal U}_{k-2}$. This proves the first
    part.

    Case {\it ii.)}
    Let $\alpha = \frac{1}{K}$ with $K \in \Bbb{N} \setminus \{0\}$
    and $\Phi (\nu) \in {\cal U}_k$. We have to consider
    the two cases $k \le K$ and $k > K$ separately:

    a.) For $k \le K$ we have $\left(\frac{1}{K}\right)^{(k)} \ne 0$ and
    we can apply the same arguments as in the first part, hence
    $\Phi (\nu) = (\nu - \frac{1}{K}) u_{k-1} (\nu)$ with
    $u_{k-1} (\nu) \in {\cal U}_{k-1}$.

    b.) For $k > K$ we have for $r \ge 1$
    $\left. \nu^{(K+r)}\right|_{\nu = \frac{1}{K}} = 0$ according to the
    definition (\ref{nudef}) of $\nu^{(k)}$.
    Hence in every $\Phi(\nu) \in {\cal U}_k$
    \[
        \Phi(\nu) = \nu^{(k)}\phi_k + \cdots + \nu^{k-K-1} \nu^{(K+1)}
                    \phi_{K+1} + \nu^{k-K} u_K (\nu)
    \]
    the first terms are automatically elements of
    ${\cal I}_{\frac{1}{K}}$ and $\Phi(\frac{1}{K}) = 0$ implies
    $u_K(\frac{1}{K}) = 0$. But this is an element of ${\cal U}_K$ and we can
    apply the case a.) Hence $u_K(\nu) = (\nu - \frac{1}{K}) u_{K-1}'(\nu)$
    which proves the second part. \hfill $\Box $ \\

The quotient algebras ${\cal U}/{\cal I}_\alpha$ can now easily
be described:
\begin{theo}
   The quotient algebra ${\cal A}_\alpha:={\cal U}/{\cal I}_\alpha$
   is isomorphic to one of the following algebras:
   \begin{enumerate}
   \item For $\alpha$ not equal to one of the rational numbers
         $1,\frac{1}{2},\frac{1}{3},\ldots$ the algebra ${\cal A}_\alpha$
         is isomorphic to the vector space of representative
         functions $\cal E$ equipped with the multiplication $\ast_\alpha$
         defined by ($f\in{\cal E}_k,g\in{\cal E}_l;k,l\in\Bbb{N}$):
         \begin{equation}
             f\ast_\alpha g:=\sum_{r=0}^{min(k,l)}\frac{\alpha^r}{r!}
             \frac{\alpha^{(k+l-r)}}{\alpha^{(k)}\alpha^{(l)}}
             M^{(k,l)}_r(f,g)
         \end{equation}
         where the real number $\alpha^{(k)}$ is defined by the formula
         (\ref{nudef}) and $M_r^{(k,l)}(f,g)$ is given in (\ref{dieHs}).
   \item Let $\alpha$ be equal to $\frac{1}{K}$ with $K$ a positive integer.
         Then ${\cal A}_\alpha$ is isomorphic to the finite-dimensional
         complex algebra of linear endomorphisms of $\Bbb{C}^N$ with
         $N:={n+K \choose K}$. The isomorphism
         is given by the map
         \begin{equation} \label{finrep}
             A \mapsto  \frac{\nu^{(K)}}{(1/K)^{(K)}}\sigma(A)
                                      \bmod {\cal I}_{1/K}
         \end{equation}
         where $\sigma(A)$ is the Berezin-Rawnsley symbol of the complex
         $N\times N$-matrix $A$ (see (\ref{BerRaw})). The matrix product
         $(AB)_{i_1\cdots i_K,j_1\cdots j_K}$ is given by
         $A_{i_1\cdots i_K,a_1\cdots a_K}$ $B_{a_1\cdots a_K,j_1\cdots j_K}$.
   \end{enumerate}
\end{theo}

{\sc Proof:} i) According to the preceding Lemma the ideal ${\cal I_\alpha}$
is equal to $(\nu-\alpha){\cal U}$ which amounts to simply substituting
$\nu=\alpha$ in (\ref{StarEqn}) which is obviously well-defined.

ii) According to the second part of the preceding Lemma we get
${\cal U}_k/{\cal I}_{1/K}={\cal U}_K/{\cal I}_{1/K}$ for $k\geq K$.
For ${\cal U}_K$ we may substitute $\nu$ for $1/K$ since $(1/K)^{(k)}\neq 0$
for $k\leq K$. Hence $\dim {\cal A}_{1/K}=\dim {\cal E}_K={n+K \choose K}^2$.
Moreover,
\begin{eqnarray*}
  \lefteqn {\frac{\nu^{(K)}}{(1/K)^{(K)}}\sigma(A)\mustar
          \frac{\nu^{(K)}}{(1/K)^{(K)}}\sigma(B) \bmod {\cal I}_{1/K}} \\
                 & = &
  \sum_{r=0}^K\frac{\nu^r}{r!}\frac{\nu^{(2K-r)}}{(1/K)^{(K)}(1/K)^{(K)}}
               M^{(K,K)}_r(\sigma(A),\sigma(B)) \bmod {\cal I}_{1/K} \\
                 & = &
          \frac{(1/K)^K}{K!}\frac{\nu^{(K)}}{(1/K)^{(K)}(1/K)^{(K)}}
               M^{(K,K)}_K(\sigma(A),\sigma(B)) \bmod {\cal I}_{1/K} \\
                 & = &
        \frac{\nu^{(K)}}{(1/K)^{(K)}}\sigma(AB) \bmod {\cal I}_{1/K} ,
\end{eqnarray*}
since a simple calculation shows that
\[
  \pi^\ast(M^{(K,K)}_K(\sigma(A),\sigma(B)))(z)
    =
     K!K!\frac{\tilde{\sigma}(AB)(z)}{x^K}.
\]
This proves the Theorem.
\hfill $\Box $\\

Remarks: Note that for each $A\in\goth{u}(n+1)$ the Berezin-Rawnsley
  symbol $\sigma(A)$ is contained in ${\cal E}_K$ and thus uniquely
  corresponds to a linear operator in $B({\cal H}^{(K)})$ which is mapped
  via the linear map (\ref{finrep}) to ${\cal A}_{1/K}$. By Lemma
  \ref{quantmom} it follows that this defines a
  {\em representation} of $\goth{u}(n+1)$ in $\Bbb{C}^N$, which is
  {\em irreducible}:
  in fact, by Lemma \ref{quantmom} we know that the momentum map
  $P(A)$ star-commutes with some function iff it Poisson commutes
  with that function which is only possible iff that function is constant
  since the unitary group acts transitively on $\CP^n$. Since the
  Poisson bracket with $P(A)$ obviously preserves each ${\cal E}_k$,
  it follows that this reasoning carries over to the quotient by
  ${\cal I}_{1/K}$.

  From the physical point of view the second part of the preceding Theorem
  can be viewed as follows:
         The classical phase space reduction of $\CNull$ to $\CP^n$
         was motivated by the $U(1)$-action on $\CNull$ which is
         just the flow of the classical isotropic $(n+1)$-dimensional
         harmonic
         oscillator with Hamiltonian $H = \frac{1}{2} x$ (The frequency
         and mass are normalised to 1).
         Passing from $\CNull$ to $\CP^n$ for a fixed value
         $\mu \in \Bbb{R}^-$ of the momentum mapping
         $J = -\frac{1}{2} x$ means classically that the
         harmonic oscillator is considered at a fixed energy $E = -\mu$.\\
         This is also true in the quantum mechanical case but now the
         energy is quantised. Only for the discrete values
         $\frac{1}{K}$, $K$ a positive integer, of the formal parameter
         $\nu = \frac{\lambda}{-2\mu}$ one obtains
         {\em finite dimensional} algebras of observables
         ${\cal U}/{\cal I}_{1/K}$ as one would physically expect for the
         {\em compact phase space} $\CP^n$ because the phase volume is
         finite and each state `occupies a phase cell of volume not smaller
         than $\hbar^n$' which results in a finite-dimensional
         Hilbert space for the quantum mechanical states.
         With $\lambda = 2\hbar$ the quantised energy is given by
         \[
            E_K = \hbar K
         \]
         (where the usual ground state energy $\frac{1}{2} \hbar (n+1)$
         is absent because of the Wick ordering). Note that in this
         interpretation the
         formal parameter $\lambda = 2\hbar$ is {\em not} quantised
         but the energy $E = -\mu$
         is. The dimensions of the operator algebras for a
         fixed $K\in \Bbb{N}$ correspond to the well-known degeneracy of
         the $K$th energy eigenvalue of the isotropic harmonic oszillator
         (see e.g. \cite[eqn XII.64]{Mes 61}).

\section{Other Examples}

In this section we briefly sketch how the programme mentioned in the
introduction applies to the deformation quantization of other phase spaces
which have been dealt with in the literature:

{\bf 1.} Consider complex $n+1$-space $\Bbb{C}^{n+1}$ as a symplectic
manifold in the usual manner, i.e. with symplectic form
$\omega= \frac{{\bf i}}{2}\sum_{i=0}^n dz^i\wedge d\bar{z}^i$. The Wick product
(\ref{Wickusual}) then defines a star product on this space. It is
natural to consider the action of $\Bbb{C}^{n+1}$ on itself by translations.
Suppose that the smooth complex-valued function $F$ on $\Bbb{C}^{n+1}$ is
representative with respect to this group action, i.e. there is a finite
number $L$ of linearly independent smooth complex-valued
functions $F_1,\ldots,F_L$ on $\Bbb{C}^{n+1}$ such
that $F(z+v)=\sum_{a=1}^L\beta_a(v)F_a(z)$ with smooth complex valued
coefficients $\beta_a$. The same equation holds for each $F_a$ thus giving
rise to a coefficient matrix $\beta_{ab}(v)$. Since $\Bbb{C}^{n+1}$ is
abelian, $\beta_{ab}(v)$ commutes with each $\beta_{ab}(w)$ whence
all these matrices can simultaneously be transformed to Jordan normal form.
It can easily be seen that the generalized eigenvectors of $\beta_{ab}(v)$
are of the form $p(z)e^{b_iz^i+c_i\bar{z}^i}$ where $p$ is a complex-valued
polynomial function of $(z,\bar{z})$ and $b_i,c_j$ are complex numbers.
Conversely, it is easy to see that each linear combination of the functions
of this form
is indeed representative. Since this space of functions
is a subalgebra of the algebra of all smooth complex-valued functions
on $\Bbb{C}^{n+1}$ (with respect to pointwise multiplication) which
clearly is closed under complex conjugation, contains a unit element,
and separates points it is dense in $C^\infty(\Bbb{C}^{n+1})$ with
respect to the uniform topology on compacta thanks to the
Stone-Weierstrass Theorem. Note that the space of polynomials is
a dense proper $\Bbb{C}^{n+1}$ submodule of representative functions
which would be impossible for compact Lie groups (compare the proof of
Lemma \ref{Eistrep}). \\
Writing $e_{(a,b)}$ for the exponential function
$z\mapsto e^{a_iz^i+b_i\bar{z}^i}$ parametrized by two complex vectors
$a,b\in\Bbb{C}^{n+1}$ we easily obtain the following formula:
\begin{equation}
    e_{(a+\rho,b+\sigma)}\ast e_{(a'+\rho',b'+\sigma')}
           =
          e^{\lambda(a_i+\rho_i)(b'_i+\sigma'_i)}
           e_{(a+\rho,b+\sigma)}e_{(a'+\rho',b'+\sigma')}
\end{equation}
where $a,a',b,b',\rho,\rho',\sigma,\sigma'\in\Bbb{C}^{n+1}$. After
differentiating this formula a finite number of times with respect
to the components of $\rho,\rho',\sigma,\sigma'$ at
$\rho=\rho'=\sigma=\sigma'=0$ (which generates polynomial prefactors)
we see that the Wick-product of two representative functions
is again representative and entire analytic in the formal parmater
$\lambda$. Therefore the algebra $\cal U$ can be chosen to be the
subalgebra of $C^\infty(\Bbb{C}^{n+1})[[\lambda]]$ which consists of
polynomials in $\lambda$ with coefficients in the space of representative
functions. Substituting $\lambda$ for a real number $\alpha$ then is
straight-forward.

{\bf 2.} Consider now the $2n$-torus $T^{2n}:=S^1\times\cdots\times S^1$
($2n$ factors).
Let $(\vp^1,\ldots,\vp^{2n})=:\vp$, $0\leq\vp^i <1$, denote the standard
angle co-ordinates on $T^{2n}$ and take any nondegenerate
complex $2n\times 2n$-matrix $(\Lambda^{ij})$. The Moyal product may then
be defined by for two smooth, complex-valued functions $f,g$ on $T^{2n}$
as follows:
\begin{equation} \label{Moyal}
    f\ast g:=\sum_{r=0}^\infty\frac{(\lambda/2\pi{\bf i})^r}{r!}
                     \Lambda^{i_1j_1}\cdots\Lambda^{i_rj_r}
                     \frac{\partial^r f}{\partial\vp^{i_1}\cdots\vp^{i_r}}
                     \frac{\partial^r g}{\partial\vp^{j_1}\cdots\vp^{j_r}}
\end{equation}
It is known that this formula defines an associative deformation
for the pointwise multiplication in $C^\infty(T^{2n})$ (see e.g.
\cite{bayen}). The space of
representative functions for the torus action on itself is spanned by
the Fourier modes $T_k(\vp):=e^{2\pi{\bf i}k_i\vp^i}$, $k\in\Bbb{Z}^{2n}$:
indeed, it is clear
that the complex span of the Fourier modes consists of representative
functions. Since it is a subalgebra of $C^\infty(T^{2n})$ which is
closed under complex conjugation, contains a unit element, and separates
points it is a dense $T^{2n}$-submodule of the space of all representative
functions (by the Stone-Weierstrass Theorem) which has to be
equal to that space since $T^{2n}$ is compact (compare again the proof of
Lemma (\ref{Eistrep})). The Moyal product of two Fourier modes is then
simply be computed by
\begin{equation} \label{MoyalFourier}
    T_k\ast T_{k'}=e^{2\pi{\bf i}\lambda\Lambda^{ij}k_ik'_j}T_kT_{k'}
\end{equation}
which again is an entire function in the formal parameter $\lambda$,
whence the choice of $\cal U$ and the substitution of the formal parameter
is completely analogous to the previous example. \\
Suppose now that the matrix $(\Lambda^{ij})$ is integral and the
greatest common divisor of the matrix elements is equal to $1$. Choosing
$\lambda=\frac{1}{K}$, $K$ a positive integer, it is easily seen from the above
formula (\ref{MoyalFourier}) that the subspace ${\cal J}_{1/K}$ spanned
by all elements of the form $T_k-T_{k+Kk'}$ with $k,k'\in\Bbb{Z}^{2n}$
is a $\ast$-ideal in the $\ast$-algebra of representative functions.
It can be shown (see \cite{BHSS 91}) that the quotient algebra is a
simple complex algebra of dimension $K^{2n}$, which is related to the
geometric quantization on the $2n$-torus in the theta-bundle and its
tensor powers.

{\bf 3.} Let $\Bbb{D}:=\{v\in\Bbb{C}||v|^2<1\}$ be the Poincar\'e disk
in the complex plane. As we have indicated in the last section of
\cite{uns} we can use the projective representation of $\Bbb{D}$:
in $\Bbb{C}^2\setminus \{0\}$ consider the open subset defined by
the inequality $0<y:=|z^0|^2-|z^1|^2$ (the function $y$ was defined
with an erroneous sign in \cite{uns}).
The image of this open set under
the projection $\pi$ is an open set of $\CP^1$ which is holomorphically
diffeomorphic to $\Bbb{D}$ via $\pi(z)\mapsto v:=\frac{z^1}{z^0}$. In their
article \cite{CGR III} Cahen, Gutt, and Rawnsley have considered
the following functions $f_{p,q}(v)$ ($p,q\in\Bbb{N}$) on $\Bbb{D}$
\begin{equation}
  f_{p,q}(v):=v^p(\frac{\bar{v}}{1-|v|^2})^q.
\end{equation}
Their pull-back to $\Bbb{C}^2\setminus \{0\}$ is simply given by
$(\pi^*f_{p,q})(z)=\frac{(z^1/z^0)^p(z^0\bar{z}^1)^q}{y^q}$.
In \cite{uns} we have defined a star product
on $\Bbb{D}$ by essentially replacing $x$ by $y$ and the operators
$M_r$ by
\begin{equation}
    \check{M}_r(G,H):=y^rg^{i_1j_1}\cdots g^{i_rj_r}
                      \frac{\partial^rG}{\partial z^{i_1}\cdots\partial z^{i_r}}
                      \frac{\partial^rG}{\partial \bar{z}^{j_1}\cdots\partial \bar{z}^{j_r}}
\end{equation}
with $g^{ij}:={\rm diag}(1,-1)$.
Observing that eqs (\ref{oneeins}) and (\ref{StarEqn}) remain valid
for arbitrary smooth complex-valued functions when the upper bound
of the sum is $\infty$ and that these formulas pass to the noncompact
case with the above replacements and adapting the sign and ordering
conventions of \cite{CGR III} we obtain the following star product:
\begin{eqnarray*}
    { (f_{p,q}\ast f_{r,s})(\pi(z)) }
        & = &
                 \sum_{m=0}^\infty\frac{(-\nu)^m}{m!}
                         \frac{(-\nu)^{(q+s-m)}}{(-\nu)^{(q)}(-\nu)^{(s)}}
                            y^{m-q-s}
                         g^{i_1j_1}\cdots g^{i_mj_m} \\
        &   & \;\; \times \;\;
              \frac{\partial^r(y^q\pi^*f_{p,q})}
              {\partial \bar{z}^{i_1}\cdots\partial \bar{z}^{i_m}}(z)
              \frac{\partial^r(y^s\pi^*f_{r,s})}
              {\partial z^{j_1}\cdots\partial z^{j_m}}(z) \\
        & = & \sum_{m=0}^\infty\frac{(-\nu)^m}{m!}
              \frac{(-\nu)^{(q+s-m)}}{(-\nu)^{(q)}(-\nu)^{(s)}}
              y^{m-q-s}(-1)^m \\
        &   & \;\; \times \;\;
              \frac{\partial^m(((z^1/z^0)^p(z^0\bar{z^1})^q)}{\partial (\bar{z^1})^m}(z)
              \frac{\partial^m(((z^1/z^0)^r(z^0\bar{z^1})^s)}{\partial (z^1)^m}(z)   \\
        & = & \sum_{m=0}^{\min (q,r)}\frac{\nu^m}{m!}
              \frac{(-\nu)^{(q+s-m)}}{(-\nu)^{(q)}(-\nu)^{(s)}}
              \frac{q!}{(q-m)!}\frac{r!}{(r-m)!}
              f_{p+r-m,q+s-m}(\pi(z))
\end{eqnarray*}
which reproduces the result of \cite{CGR III}. In \cite{CGR IV} star products
were computed on more general bounded symmetric domains.

\vspace{0.5cm}

\noindent
{\large\bf Acknowledgment}

\vspace{2mm}

\noindent The authors would like to thank D.~Arnal, J.~Hoppe, M.~Masmoudi
and W.~Soergel for valuable discussions.

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text:
HOME / Blog / shared value
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Mandela Day: Making a Difference
"What counts in life is not the mere fact that we have lived. It is what difference we have made to the lives of others that will determine the significance of the life we lead." – Nelson Mandela In 2009, the United Nations General Assembly declared July 18 as Nelson Mandela International Day in commemoration...
Gugu McLaren, Sustainability Strategy, Discovery | Jul 18, 2016
67 minutes, Discovery, education, environment, food security, freedom, infrastructure, literacy, Nelson Mandela, Nelson Mandela Foundation, Nelson Mandela International Day, peace, shared value, shared value insurance, shelter, social justice
Health: The New Driver of Business Value
Revenues, profits, and cash flows: These are common concerns for chief financial officers (CFO) conducting business across sectors and geographies. Historically, workforce health has not featured among the top concerns for CFOs and other C-Suite executives, including the chief executive officer (CEO) or chief operating officer (COO). For many companies, this is changing. Workforce health...
Brett Tromp, Chief Financial Officer, Discovery Health | Jun 30, 2016
chief executive officer, chief financial officer, chief operating officer, Discovery, Health, king report on corporate governance, National Academy of Medicine, S&P 500 index, shared value, South Africa, united nations global compact, well-being
Placing Health at the Heart of Sustainability: Lessons from the Environment
In 2000, the United Nations Secretary-General Kofi Annan announced the establishment of the United Nations Global Compact (UNGC), a unique organization dedicated to collaborating with businesses across sectors to promote responsible practices. At the launch, Kofi Annan outlined nine guiding principles for responsible businesses that focused on human rights, labor standards, and the environment. Anti-corruption...
Gillian Christie & Derek Yach | Jun 23, 2016
Dow Jones Sustainability Index, ESG, FTSE4Good Index Series, global health, Health, King III, national academies of sciences, shared value, Sustainable Development Goals, UN Global Compact, workplace health

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text:
Spotlight Open House: Audition Technique with Director
by Spotlight
Thank you for booking on to this event. Please remember that members are limited to two events across Open House and to one ticket per person per event. If you have registered for more than this, please remove enough tickets so that you only have the allowed allocation.
Mon, 22 October 2018
7 Leicester Pl
WC2H 7RJ
Please note: Members are limited to two events per person, and one ticket to each event of these events during Open House. This is to ensure as many members as possible get the chance to experience the events on offer.
Age Range: 18+
Clothing Req: Comfortable
Prep Req: Please come prepared with one short monologue of no longer than 1 minute. Please bring a copy with you too.
During this workshop, you will have the chance to perform in front of theatre director Caroline Steinbeis and receive feedback and small directorial changes. Along with this, you will be able to feedback on your peer's performances and observe their pieces being directed.
About Caroline Steinbeis:
Caroline is a freelance theatre director and dramaturge. She is currently Associate Director at Sheffield Theatres.
Caroline was born in Munich and now lives in London with her family. She studied at the London University of Royal Holloway and completed her MA at the Central School of Speech and Drama.
In 2009 Caroline won the prestigious JMK Award for Outstanding Directors. She directed Caryl Churchill's Mad Forest. Her production Fatherland was invited to the Radikal Jung Festival in Munich in 2011.
Her production of Brilliant Adventures won the prize for Best Studio Production at the Manchester Theatre Awards in 2013.
Recent Directing credits include: Love and Information (Sheffield Theatres), The Tempest (Royal & Derngate), The Crucible (Manchester Royal Exchange), We Want You To Watch (National Theatre)The Broken Heart (Globe Theatre, Sam Wannamaker), Brilliant Adventures (Manchester Royal Exchange) A Time To Reap (Royal Court, Upstairs) Earthquakes in London (UK Tour for Headlong Theatre and NT) and many more.
If you have any accessibility requirements please do let us know. (Please note - unfortunately Spotlight Rooms and Studios are not wheelchair accessible. We will be holding workshops on other dates at a fully accessible venue. Please email marketing@spotlight.com if you wish to attend one of these.)
Spotlight Open House: Audition Technique with Director at Spotlight
7 Leicester Pl, London, WC2H 7RJ, United Kingdom


You are currently browsing the tag archive for the 'beat box' tag.
If you're in Melbourne, you MUST go to see the Tom Tom Crew. One week left. If you live in a major city worldwide, they travel and when they come to your city you MUST go. And if you just can't wait to see what I'm talking about, well there's always youtube.


The Archive One technical team offers custom data migration services for existing datasets.
We have migrated more than 50 million records from older systems and made them accessible. This has enabled our customers to use all the previous investments made on data.
Some of the records we have migrated are medical records, personnel records, historical mining maps and financial records.
Planning needs to be done properly. There will be a technical evaluation to understand the current storage and meta data associated with existing files. A discussion will the stakeholders on what can be extracted and what needs to be extracted will take place to determine the final specification of the migration activity.


Katsucon, in coordination with the Gaylord National, have made the decision for Katsucon to be what is called a "closed profile" event.
This means that for the duration of the convention weekend the Gaylord National Resort and Convention Center will only be accessible to people wearing – or in the process of obtaining – a Katsucon membership badge, registered Gaylord National guests, or guests with a confirmed restaurant or spa appointments. This is not in response to any particular threat or incident but is instead a proactive measure. Katsucon's closed profile will allow a greater degree of safety for all attendees, guests and staff and brings our convention in line with other large events held at the Gaylord National throughout the year.
We realize this will be a change for all of us who have become used to a more casual atmosphere in areas outside of the Katsucon programming rooms, and we're taking steps to make the transition easier for everyone.
Additional signage and staff will be available during Katsucon weekend to help all convention attendees be quickly and easily directed to our main registration hall on the bottom level of the convention center upon arrival. We're also making it faster and easier to replace lost badges in light of the reinforced importance of the badge. Just make sure to keep your registration confirmation email or at-door receipt and Katsucon staff will be able to quickly replace any lost badge from an easily accessible kiosk.
Both Katsucon and the Gaylord National are committed to making Katsucon a safe space for everyone. On this end, we are reiterating our zero-tolerance policy towards all forms of harassment. Any attendee found violating this policy will be ejected from the convention with no refunds. Any incidents of assault will automatically trigger a call to local law enforcement.
If any attendee feels they have been the target of any form of harassment or assault, we ask that you find the nearest Katsucon staff member and ask them to "Page a Chair". Our Page-a-Chair program allows staff to call the nearest Katsucon representative with the full authority of the Convention Chairman to your location immediately. This representative can render assistance with deploying additional security, ejecting offending persons, as well as providing access to counseling and law enforcement services.
We come together every year to Katsucon first and foremost to have fun; providing a space where our attendees, staff, guests, and friends feel safe and protected is the best way to ensure that having fun (and repairing the occasional cosplay malfunction) is all you need to worry about. We, at Katsucon, are very much looking forward to sharing that fun with all of you in 2020.


Saturday was projected to be gloomy well before the day came, and it was no different in the morning.
I didn't mind the gloomy weather because I landed club level seats. Upon arrival at Nationals Park, the racing presidents were out on half street as usual. Since I'm always decked out in Marlins garb, I get taunted by them in some way.
To my disappointment there was no batting practice Saturday due to the lousy weather. The last two games I have been to there have lacked BP (I am still yet to see Mike Stanton take batting practice; I'm hoping to get that chance in July when the Fish return to D.C.). So arriving an hour and a half early was now useless, so I went down by the dugout where there were some more Marlins fans I could hang with.
It was pups in the park promotion day in D.C., so at about noon, a parade of dogs and dog owners sauntered around the perimeter of the field. Throughout the hour leading up to the game, Craig Minervini gave his pre-game analysis right outside the dugout. Although I couldn't understand what he was saying, it was enjoyable anyway. Also, right before the first pitch I saw him right outside the elevators, but he didn't get on mine.
At about 12:30 players started filing one-by-one out of the dugout, greeted by kids' autograph requests. I've already gotten most of my favorite players to sign various items for me, so I'm not too interested in signatures. However I did get Scott Cousins to sign a baseball because I had one on me.
With about five minutes before first pitch, I raced to the elevators, where I saw Craig, and proceeded to the seats. Despite being low mezzanine on the first base side, no foul balls came our way during the game. Anibal Sanchez and Livan Hernandez both pitched phenomenally . Sanchez had a three-hit shutout going through the eighth, but had to exit. Anibal also had his first hit of the year, in the third inning.
With the club level comes a lot better concourse full of cuisines and amenities. The prices are insanely high, but the food is good regardless.


text:
Q: Hide/show and make bigger while clicking one div I have a page where there are four content and each content has a image. Now, my task is to make the content bigger
with the image when the user clicks on a particular content. Also, I need to hide all the other content.
Like this is one sample

Here every image is small and the content is also cut. When the user clicks on the image I need to hide all the other content
as well as I need to make the clicked content appear full and the making the image bigger.
Like this
For this what I am doing is I am loading all the contents with a class and on clicking I am adding hiding the div of both small content and
the small image and making the other bigger content and image as visible.
Also, if clicked again I need to show all the small content with image again.
Now my problem is I have to write the specific div id here so it only makes that particular id bigger. How can I make it more generic?
EDIT:  Added the code
$($(this).find(".view-content")).click(function(){

        if ( $( this ).hasClass( 'active' ) ) {
            $(this).removeClass('active');
                  $($(this).find(".Boardofadviser1")).show();
            $($(this).find(".Boardofadviser")).hide();

            $($(this).find(".views-row-3").show());

        }
        else
        {
           $(this).addClass('active');
           $($(this).find(".Boardofadviser1")).hide();
           $($(this).find(".Boardofadviser")).show();
            $($(this).find(".views-row-3").hide());
            $(".Boardofadviser1").nextAll().hide();
            $(".views-row-1.Boardofadviser1").hide();
           $(".views-row-1.Boardofadviser").show();

        }
         });

This is my HTML Page where all the small content is shown and large content are hidden

A: To avoid having additional if/else condition to check if views-row holds the active class, You have to use $(this) context to tell which element you want those classes and show the image under that specific class without affecting other elements that wasn't been called by the click event handler yet.
Example:
var oViewsRow = $('.view-content .views-row');
oViewsRow.click(function(event) {
    event.preventDefault();

    // Call all DOM elements that have '.views-row' class
    oViewsRow.removeClass('active');
    oViewsRow.find(".Boardofadviser1").show();
    oViewsRow.find(".Boardofadviser").hide();
    oViewsRow.find(".views-row-3").show();

    $(".Boardofadviser1").nextAll().hide();
    $(".views-row-1.Boardofadviser1").hide();
    $(".views-row-1.Boardofadviser").show();

    // Then call that specific '.views-row' DOM element using $(this)
    $(this).addClass('active');
    $(this).find(".Boardofadviser1").hide();
    $(this).find(".Boardofadviser").show();
    $(this).find(".views-row-3").hide();
});

Hope this helps for your case

A: You can use the jQuery .toggle() to show/hide content

$(document).ready(function(){
  $('.small').on('click', function(event) {
    $('.small').parent().toggle('show');
    $(this).parent().next('div').toggle('show');
  })
  $('.big').on('click', function(event) {
    $(this).parent().toggle('show');
    $('.small').parent().toggle('show');
  })
});

      Adviser1

Blah Blah Blah

      Adviser1

Blah Blah Blah Blah Blah Blah

      Adviser2

Blah Blah Blah

      Adviser2

Blah Blah Blah Blah Blah Blah

Codepen example: https://codepen.io/anon/pen/VbbBwg
Hope this helps!

redpajama_set_name: RedPajamaStackExchange


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Beautiful place located in downtown Ottawa with great view, food and service.
To book the space please call (613) 594-4545 Or email at info@thaliottawa.ca.
We will make a custom menu with buffet option, plated or Family style.
Make your next event flavourful at Thali. We need 48 hours notice.


text:
Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia
J. Willem L. Tideman, Olavi Pärssinen, UK Biobank Eye and Vision Consortium, The Consortium for Refractive Error and Myopia, Annechien E.G. Haarman, Anthony P. Khawaja, Juho Wedenoja, Katie M. Williams, Ginevra Biino, Xiaohu Ding, Mika Kähönen, Terho Lehtimäki, Olli T. Raitakari, Ching Yu Cheng, Jost B. Jonas, Terri L. Young, Joan E. Bailey-Wilson, Jugnoo Rahi, Cathy Williams, Mingguang HeDavid A. Mackey, Jeremy A. Guggenheim
Importance: Uncertainty currently exists about whether the same genetic variants are associated with susceptibility to low myopia (LM) and high myopia (HM) and to myopia and hyperopia. Addressing this question is fundamental to understanding the genetics of refractive error and has clinical relevance for genotype-based prediction of children at risk for HM and for identification of new therapeutic targets. Objective: To assess whether a common set of genetic variants are associated with susceptibility to HM, LM, and hyperopia. Design, Setting, and Participants: This genetic association study assessed unrelated UK Biobank participants 40 to 69 years of age of European and Asian ancestry. Participants 40 to 69 years of age living in the United Kingdom were recruited from January 1, 2006, to October 31, 2010. Of the total sample of 502682 participants, 117279 (23.3%) underwent an ophthalmic assessment. Data analysis was performed from December 12, 2019, to June 23, 2020. Exposures: Four refractive error groups were defined: HM, -6.00 diopters (D) or less; LM, -3.00 to -1.00 D; hyperopia, +2.00 D or greater; and emmetropia, 0.00 to +1.00 D. Four genome-wide association study (GWAS) analyses were performed in participants of European ancestry: (1) HM vs emmetropia, (2) LM vs emmetropia, (3) hyperopia vs emmetropia, and (4) LM vs hyperopia. Polygenic risk scores were generated from GWAS summary statistics, yielding 4 sets of polygenic risk scores. Performance was assessed in independent replication samples of European and Asian ancestry. Main Outcomes and Measures: Odds ratios (ORs) of polygenic risk scores in replication samples. Results: A total of 51841 unrelated individuals of European ancestry and 2165 unrelated individuals of Asian ancestry were assigned to a specific refractive error group and included in our analyses. Polygenic risk scores derived from all 4 GWAS analyses were predictive of all categories of refractive error in both European and Asian replication samples. For example, the polygenic risk score derived from the HM vs emmetropia GWAS was predictive in the European sample of HM vs emmetropia (OR, 1.58; 95% CI, 1.41-1.77; P = 1.54 × 10-15) as well as LM vs emmetropia (OR, 1.15; 95% CI, 1.07-1.23; P = 8.14 × 10-5), hyperopia vs emmetropia (OR, 0.83; 95% CI, 0.77-0.89; P = 4.18 × 10-7), and LM vs hyperopia (OR, 1.45; 95% CI, 1.33-1.59; P = 1.43 × 10-16). Conclusions and Relevance: Genetic risk variants were shared across HM, LM, and hyperopia and across European and Asian samples. Individuals with HM inherited a higher number of variants from among the same set of myopia-predisposing alleles and not different risk alleles compared with individuals with LM. These findings suggest that treatment interventions targeting common genetic risk variants associated with refractive error could be effective against both LM and HM..
JAMA Ophthalmology
https://doi.org/10.1001/jamaophthalmol.2021.0497
conducted using the UK Biobank Resource (applications 17351). UK Biobank was established by the Wellcome Trust, the UK Medical Research Council, the Department for Health (London, United Kingdom), Scottish Government (Edinburgh, United Kingdom), and the Northwest Regional Development Agency (Warrington, United Kingdom). It also received funding from the Welsh Assembly Government (Cardiff, United Kingdom), the British Heart Foundation, and Diabetes UK. Collection of eye and vision data was supported by grant BRC2_009 from the Department for Health through an award made by the National Institute for Health Research to the Biomedical Research Centre at Moorfields Eye Hospital National Health Service Foundation Trust and UCL Institute of Ophthalmology, London, United Kingdom. Additional support was provided by grant ST 12 09 from the Special Trustees of Moorfields Eye Hospital, London, United Kingdom. Data analysis was performed using the HAWK computing cluster, maintained by Supercomputing Wales and Cardiff University Advanced Research Computing at Cardiff. This work was supported by grant 24WG201 from the Welsh Government and Fight for Sight (Dr Guggenheim), the Silmäsäätiö Foundation and Evald and Hilda Nissi Foundation (Dr Pärssinen), grant R01EY014685 from the National Institutes of Health, National Eye Institute; a University of Wisconsin Centennial Scholars Award; and an unrestricted grant from Research to Prevent Blindness Inc to the Department of Ophthalmology and Visual Sciences, University of Wisconsin, Madison (Dr Young); and the Intramural Research Program of the National Institutes of Health (Dr Bailey-Wilson).
reported receiving personal fees from Allergan, Novartis, Santen, Google Health, and Aerie outside the submitted work. Dr Jonas reported having a pending patent for use in the therapeutic or prophylactic treatment of myopia or hyperopia. Dr Young reported receiving grants from Research to Prevent Blindness Inc, University of Wisconsin Centennial Scholars Fund, and Research to Prevent Blindness Inc and consulting fees from Aerpio Pharmaceuticals Consultant outside the submitted work. Dr Guggenheim reported being an unpaid consultant for CooperVision Inc (consultancy fee paid directly by company to a charity selected by the company). No other disclosures were reported.
© 2021 American Medical Association. All rights reserved.
10.1001/jamaophthalmol.2021.0497
https://jamanetwork.com/journals/jamaophthalmology/articlepdf/2778382/jamaophthalmology_tideman_2021_oi_210013_1623191631.44966.pdf
Dive into the research topics of 'Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia'. Together they form a unique fingerprint.
Myopia Neuroscience 100%
High Myopia Medicine and Dentistry 43%
Hyperopia Medicine and Dentistry 39%
Hypermetropia Pharmacology, Toxicology and Pharmaceutical Science 39%
Sample Biochemistry, Genetics and Molecular Biology 26%
Odds Ratio Medicine and Dentistry 21%
Refractive Error Medicine and Dentistry 21%
Refraction Error Pharmacology, Toxicology and Pharmaceutical Science 21%
Tideman, J. W. L., Pärssinen, O., UK Biobank Eye and Vision Consortium, The Consortium for Refractive Error and Myopia, Haarman, A. E. G., Khawaja, A. P., Wedenoja, J., Williams, K. M., Biino, G., Ding, X., Kähönen, M., Lehtimäki, T., Raitakari, O. T., Cheng, C. Y., Jonas, J. B., Young, T. L., Bailey-Wilson, J. E., Rahi, J., Williams, C., ... Guggenheim, J. A. (2021). Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia. JAMA Ophthalmology, 139(6), 601-609. https://doi.org/10.1001/jamaophthalmol.2021.0497
Tideman, J. Willem L. ; Pärssinen, Olavi ; UK Biobank Eye and Vision Consortium et al. / Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia. In: JAMA Ophthalmology. 2021 ; Vol. 139, No. 6. pp. 601-609.
@article{ece67628be1d4973820f5c1d7497286a,
title = "Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia",
abstract = "Importance: Uncertainty currently exists about whether the same genetic variants are associated with susceptibility to low myopia (LM) and high myopia (HM) and to myopia and hyperopia. Addressing this question is fundamental to understanding the genetics of refractive error and has clinical relevance for genotype-based prediction of children at risk for HM and for identification of new therapeutic targets. Objective: To assess whether a common set of genetic variants are associated with susceptibility to HM, LM, and hyperopia. Design, Setting, and Participants: This genetic association study assessed unrelated UK Biobank participants 40 to 69 years of age of European and Asian ancestry. Participants 40 to 69 years of age living in the United Kingdom were recruited from January 1, 2006, to October 31, 2010. Of the total sample of 502682 participants, 117279 (23.3%) underwent an ophthalmic assessment. Data analysis was performed from December 12, 2019, to June 23, 2020. Exposures: Four refractive error groups were defined: HM, -6.00 diopters (D) or less; LM, -3.00 to -1.00 D; hyperopia, +2.00 D or greater; and emmetropia, 0.00 to +1.00 D. Four genome-wide association study (GWAS) analyses were performed in participants of European ancestry: (1) HM vs emmetropia, (2) LM vs emmetropia, (3) hyperopia vs emmetropia, and (4) LM vs hyperopia. Polygenic risk scores were generated from GWAS summary statistics, yielding 4 sets of polygenic risk scores. Performance was assessed in independent replication samples of European and Asian ancestry. Main Outcomes and Measures: Odds ratios (ORs) of polygenic risk scores in replication samples. Results: A total of 51841 unrelated individuals of European ancestry and 2165 unrelated individuals of Asian ancestry were assigned to a specific refractive error group and included in our analyses. Polygenic risk scores derived from all 4 GWAS analyses were predictive of all categories of refractive error in both European and Asian replication samples. For example, the polygenic risk score derived from the HM vs emmetropia GWAS was predictive in the European sample of HM vs emmetropia (OR, 1.58; 95% CI, 1.41-1.77; P = 1.54 × 10-15) as well as LM vs emmetropia (OR, 1.15; 95% CI, 1.07-1.23; P = 8.14 × 10-5), hyperopia vs emmetropia (OR, 0.83; 95% CI, 0.77-0.89; P = 4.18 × 10-7), and LM vs hyperopia (OR, 1.45; 95% CI, 1.33-1.59; P = 1.43 × 10-16). Conclusions and Relevance: Genetic risk variants were shared across HM, LM, and hyperopia and across European and Asian samples. Individuals with HM inherited a higher number of variants from among the same set of myopia-predisposing alleles and not different risk alleles compared with individuals with LM. These findings suggest that treatment interventions targeting common genetic risk variants associated with refractive error could be effective against both LM and HM..",
author = "Tideman, {J. Willem L.} and Olavi P{\"a}rssinen and {UK Biobank Eye and Vision Consortium} and {The Consortium for Refractive Error and Myopia} and Haarman, {Annechien E.G.} and Khawaja, {Anthony P.} and Juho Wedenoja and Williams, {Katie M.} and Ginevra Biino and Xiaohu Ding and Mika K{\"a}h{\"o}nen and Terho Lehtim{\"a}ki and Raitakari, {Olli T.} and Cheng, {Ching Yu} and Jonas, {Jost B.} and Young, {Terri L.} and Bailey-Wilson, {Joan E.} and Jugnoo Rahi and Cathy Williams and Mingguang He and Mackey, {David A.} and Guggenheim, {Jeremy A.}",
note = "Funding Information: conducted using the UK Biobank Resource (applications 17351). UK Biobank was established by the Wellcome Trust, the UK Medical Research Council, the Department for Health (London, United Kingdom), Scottish Government (Edinburgh, United Kingdom), and the Northwest Regional Development Agency (Warrington, United Kingdom). It also received funding from the Welsh Assembly Government (Cardiff, United Kingdom), the British Heart Foundation, and Diabetes UK. Collection of eye and vision data was supported by grant BRC2_009 from the Department for Health through an award made by the National Institute for Health Research to the Biomedical Research Centre at Moorfields Eye Hospital National Health Service Foundation Trust and UCL Institute of Ophthalmology, London, United Kingdom. Additional support was provided by grant ST 12 09 from the Special Trustees of Moorfields Eye Hospital, London, United Kingdom. Data analysis was performed using the HAWK computing cluster, maintained by Supercomputing Wales and Cardiff University Advanced Research Computing at Cardiff. This work was supported by grant 24WG201 from the Welsh Government and Fight for Sight (Dr Guggenheim), the Silm{\"a}s{\"a}{\"a}ti{\"o} Foundation and Evald and Hilda Nissi Foundation (Dr P{\"a}rssinen), grant R01EY014685 from the National Institutes of Health, National Eye Institute; a University of Wisconsin Centennial Scholars Award; and an unrestricted grant from Research to Prevent Blindness Inc to the Department of Ophthalmology and Visual Sciences, University of Wisconsin, Madison (Dr Young); and the Intramural Research Program of the National Institutes of Health (Dr Bailey-Wilson). Funding Information: reported receiving personal fees from Allergan, Novartis, Santen, Google Health, and Aerie outside the submitted work. Dr Jonas reported having a pending patent for use in the therapeutic or prophylactic treatment of myopia or hyperopia. Dr Young reported receiving grants from Research to Prevent Blindness Inc, University of Wisconsin Centennial Scholars Fund, and Research to Prevent Blindness Inc and consulting fees from Aerpio Pharmaceuticals Consultant outside the submitted work. Dr Guggenheim reported being an unpaid consultant for CooperVision Inc (consultancy fee paid directly by company to a charity selected by the company). No other disclosures were reported. Publisher Copyright: {\textcopyright} 2021 American Medical Association. All rights reserved.",
doi = "10.1001/jamaophthalmol.2021.0497",
journal = "JAMA Ophthalmology",
Tideman, JWL, Pärssinen, O, UK Biobank Eye and Vision Consortium, The Consortium for Refractive Error and Myopia, Haarman, AEG, Khawaja, AP, Wedenoja, J, Williams, KM, Biino, G, Ding, X, Kähönen, M, Lehtimäki, T, Raitakari, OT, Cheng, CY, Jonas, JB, Young, TL, Bailey-Wilson, JE, Rahi, J, Williams, C, He, M, Mackey, DA & Guggenheim, JA 2021, 'Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia', JAMA Ophthalmology, vol. 139, no. 6, pp. 601-609. https://doi.org/10.1001/jamaophthalmol.2021.0497
Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia. / Tideman, J. Willem L.; Pärssinen, Olavi; UK Biobank Eye and Vision Consortium et al.
In: JAMA Ophthalmology, Vol. 139, No. 6, 06.2021, p. 601-609.
T1 - Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia
AU - Tideman, J. Willem L.
AU - Pärssinen, Olavi
AU - UK Biobank Eye and Vision Consortium
AU - The Consortium for Refractive Error and Myopia
AU - Haarman, Annechien E.G.
AU - Khawaja, Anthony P.
AU - Wedenoja, Juho
AU - Williams, Katie M.
AU - Biino, Ginevra
AU - Ding, Xiaohu
AU - Kähönen, Mika
AU - Lehtimäki, Terho
AU - Raitakari, Olli T.
AU - Cheng, Ching Yu
AU - Jonas, Jost B.
AU - Young, Terri L.
AU - Bailey-Wilson, Joan E.
AU - Rahi, Jugnoo
AU - Williams, Cathy
AU - He, Mingguang
AU - Mackey, David A.
AU - Guggenheim, Jeremy A.
N1 - Funding Information: conducted using the UK Biobank Resource (applications 17351). UK Biobank was established by the Wellcome Trust, the UK Medical Research Council, the Department for Health (London, United Kingdom), Scottish Government (Edinburgh, United Kingdom), and the Northwest Regional Development Agency (Warrington, United Kingdom). It also received funding from the Welsh Assembly Government (Cardiff, United Kingdom), the British Heart Foundation, and Diabetes UK. Collection of eye and vision data was supported by grant BRC2_009 from the Department for Health through an award made by the National Institute for Health Research to the Biomedical Research Centre at Moorfields Eye Hospital National Health Service Foundation Trust and UCL Institute of Ophthalmology, London, United Kingdom. Additional support was provided by grant ST 12 09 from the Special Trustees of Moorfields Eye Hospital, London, United Kingdom. Data analysis was performed using the HAWK computing cluster, maintained by Supercomputing Wales and Cardiff University Advanced Research Computing at Cardiff. This work was supported by grant 24WG201 from the Welsh Government and Fight for Sight (Dr Guggenheim), the Silmäsäätiö Foundation and Evald and Hilda Nissi Foundation (Dr Pärssinen), grant R01EY014685 from the National Institutes of Health, National Eye Institute; a University of Wisconsin Centennial Scholars Award; and an unrestricted grant from Research to Prevent Blindness Inc to the Department of Ophthalmology and Visual Sciences, University of Wisconsin, Madison (Dr Young); and the Intramural Research Program of the National Institutes of Health (Dr Bailey-Wilson). Funding Information: reported receiving personal fees from Allergan, Novartis, Santen, Google Health, and Aerie outside the submitted work. Dr Jonas reported having a pending patent for use in the therapeutic or prophylactic treatment of myopia or hyperopia. Dr Young reported receiving grants from Research to Prevent Blindness Inc, University of Wisconsin Centennial Scholars Fund, and Research to Prevent Blindness Inc and consulting fees from Aerpio Pharmaceuticals Consultant outside the submitted work. Dr Guggenheim reported being an unpaid consultant for CooperVision Inc (consultancy fee paid directly by company to a charity selected by the company). No other disclosures were reported. Publisher Copyright: © 2021 American Medical Association. All rights reserved.
N2 - Importance: Uncertainty currently exists about whether the same genetic variants are associated with susceptibility to low myopia (LM) and high myopia (HM) and to myopia and hyperopia. Addressing this question is fundamental to understanding the genetics of refractive error and has clinical relevance for genotype-based prediction of children at risk for HM and for identification of new therapeutic targets. Objective: To assess whether a common set of genetic variants are associated with susceptibility to HM, LM, and hyperopia. Design, Setting, and Participants: This genetic association study assessed unrelated UK Biobank participants 40 to 69 years of age of European and Asian ancestry. Participants 40 to 69 years of age living in the United Kingdom were recruited from January 1, 2006, to October 31, 2010. Of the total sample of 502682 participants, 117279 (23.3%) underwent an ophthalmic assessment. Data analysis was performed from December 12, 2019, to June 23, 2020. Exposures: Four refractive error groups were defined: HM, -6.00 diopters (D) or less; LM, -3.00 to -1.00 D; hyperopia, +2.00 D or greater; and emmetropia, 0.00 to +1.00 D. Four genome-wide association study (GWAS) analyses were performed in participants of European ancestry: (1) HM vs emmetropia, (2) LM vs emmetropia, (3) hyperopia vs emmetropia, and (4) LM vs hyperopia. Polygenic risk scores were generated from GWAS summary statistics, yielding 4 sets of polygenic risk scores. Performance was assessed in independent replication samples of European and Asian ancestry. Main Outcomes and Measures: Odds ratios (ORs) of polygenic risk scores in replication samples. Results: A total of 51841 unrelated individuals of European ancestry and 2165 unrelated individuals of Asian ancestry were assigned to a specific refractive error group and included in our analyses. Polygenic risk scores derived from all 4 GWAS analyses were predictive of all categories of refractive error in both European and Asian replication samples. For example, the polygenic risk score derived from the HM vs emmetropia GWAS was predictive in the European sample of HM vs emmetropia (OR, 1.58; 95% CI, 1.41-1.77; P = 1.54 × 10-15) as well as LM vs emmetropia (OR, 1.15; 95% CI, 1.07-1.23; P = 8.14 × 10-5), hyperopia vs emmetropia (OR, 0.83; 95% CI, 0.77-0.89; P = 4.18 × 10-7), and LM vs hyperopia (OR, 1.45; 95% CI, 1.33-1.59; P = 1.43 × 10-16). Conclusions and Relevance: Genetic risk variants were shared across HM, LM, and hyperopia and across European and Asian samples. Individuals with HM inherited a higher number of variants from among the same set of myopia-predisposing alleles and not different risk alleles compared with individuals with LM. These findings suggest that treatment interventions targeting common genetic risk variants associated with refractive error could be effective against both LM and HM..
AB - Importance: Uncertainty currently exists about whether the same genetic variants are associated with susceptibility to low myopia (LM) and high myopia (HM) and to myopia and hyperopia. Addressing this question is fundamental to understanding the genetics of refractive error and has clinical relevance for genotype-based prediction of children at risk for HM and for identification of new therapeutic targets. Objective: To assess whether a common set of genetic variants are associated with susceptibility to HM, LM, and hyperopia. Design, Setting, and Participants: This genetic association study assessed unrelated UK Biobank participants 40 to 69 years of age of European and Asian ancestry. Participants 40 to 69 years of age living in the United Kingdom were recruited from January 1, 2006, to October 31, 2010. Of the total sample of 502682 participants, 117279 (23.3%) underwent an ophthalmic assessment. Data analysis was performed from December 12, 2019, to June 23, 2020. Exposures: Four refractive error groups were defined: HM, -6.00 diopters (D) or less; LM, -3.00 to -1.00 D; hyperopia, +2.00 D or greater; and emmetropia, 0.00 to +1.00 D. Four genome-wide association study (GWAS) analyses were performed in participants of European ancestry: (1) HM vs emmetropia, (2) LM vs emmetropia, (3) hyperopia vs emmetropia, and (4) LM vs hyperopia. Polygenic risk scores were generated from GWAS summary statistics, yielding 4 sets of polygenic risk scores. Performance was assessed in independent replication samples of European and Asian ancestry. Main Outcomes and Measures: Odds ratios (ORs) of polygenic risk scores in replication samples. Results: A total of 51841 unrelated individuals of European ancestry and 2165 unrelated individuals of Asian ancestry were assigned to a specific refractive error group and included in our analyses. Polygenic risk scores derived from all 4 GWAS analyses were predictive of all categories of refractive error in both European and Asian replication samples. For example, the polygenic risk score derived from the HM vs emmetropia GWAS was predictive in the European sample of HM vs emmetropia (OR, 1.58; 95% CI, 1.41-1.77; P = 1.54 × 10-15) as well as LM vs emmetropia (OR, 1.15; 95% CI, 1.07-1.23; P = 8.14 × 10-5), hyperopia vs emmetropia (OR, 0.83; 95% CI, 0.77-0.89; P = 4.18 × 10-7), and LM vs hyperopia (OR, 1.45; 95% CI, 1.33-1.59; P = 1.43 × 10-16). Conclusions and Relevance: Genetic risk variants were shared across HM, LM, and hyperopia and across European and Asian samples. Individuals with HM inherited a higher number of variants from among the same set of myopia-predisposing alleles and not different risk alleles compared with individuals with LM. These findings suggest that treatment interventions targeting common genetic risk variants associated with refractive error could be effective against both LM and HM..
U2 - 10.1001/jamaophthalmol.2021.0497
DO - 10.1001/jamaophthalmol.2021.0497
JO - JAMA Ophthalmology
JF - JAMA Ophthalmology
Tideman JWL, Pärssinen O, UK Biobank Eye and Vision Consortium, The Consortium for Refractive Error and Myopia, Haarman AEG, Khawaja AP et al. Evaluation of Shared Genetic Susceptibility to High and Low Myopia and Hyperopia. JAMA Ophthalmology. 2021 Jun;139(6):601-609. Epub 2021 Apr 8. doi: 10.1001/jamaophthalmol.2021.0497


The MixBox is very sturdy as a product but it isn't bulletproof. It's definitely not waterproof either, so no exposure to water, please!!
But with every electronic piece of equipment, please look after the MixBox and make sure it returns in the sturdy case we supply to you.
If you seem to have any difficulties with the MixBox you can go to the Troubleshooting page which will have more information for you.


text:
DTE Energy breaks ground on $1B Blue Water Energy Center
By The Building Tradesman
EAST CHINA TWP. - DTE Energy on Aug. 21 broke ground on a new $1 billion natural gas-fueled plant, the Blue Water Energy Center. The utility said it will be the most efficient power plant in the state when it begins producing electricity for 850,000 homes starting in 2022.
"Just over a year ago, DTE first announced a broad sustainability initiative to reduce carbon emissions by more than 80 percent and do it in a way that maintains affordability and reliability for our customers," said Gerry Anderson, chairman, and CEO, DTE Energy. "In fact, we're already doing it. Since 2009 we have driven investments of more than $2.5 billion in renewable energy with more investments coming."
The plant will be built by Kiewit Engineering (MI) Co. The company will spend a minimum of $200 million on Michigan-based labor and materials and has begun the pre-qualification process for local sub-contractors. Additionally, GE Power, a world leader in power generation and water technologies for utilities, independent power producers, and industrial applications, has been chosen to provide the major generation equipment. DTE Energy says more than 500 Michigan skilled labor jobs will be created during construction beginning in spring 2019 and an additional 35 full-time positions will be filled when the plant becomes operational in 2022.
"As we continue to retire coal-fired power plants – all of them by 2040 – we need to complement wind farms and solar arrays with high-reliability assets," Anderson said. "As we Michiganders know well, the wind doesn't always blow, and the sun doesn't always shine in our beautiful state. And that's why we need natural gas-fueled plants like the Blue Water Energy Center. When it begins operations in 2022, it will represent our single largest step in reducing carbon emissions to date."
Already a significant producer of renewable energy, DTE has announced plans to double its renewable energy capacity by 2022 to 2,000 megawatts – providing enough additional electricity to power 800,000 homes while driving investments of more than $1.7 billion in Michigan. Within the next year, DTE will begin operating the Pine River and Polaris wind parks, which will have a combined capacity of 330 megawatts and be DTE's largest and most efficient wind parks to date.
DTE Energy serves 2.2 million customers in Southeastern Michigan and is a natural gas utility serving 1.3 million customers in Michigan.


Notebook: " The Most Beautiful Love Story" Upcoming Film Production By Salman Khan Films.
Salman Khan, actor- producer is known for his face value in movies and also for launching fresh faces in the industry. His production house, Salman Khan films has launched Sooraj Pancholi and Athiya Shetty in 'Hero', Daisy Shah in 'Jai Ho', Sonakshi Sinha in 'Dabangg' and his Brother-In-Law Aayush Sharma and Warina Hussain in 'Loveyatri'.
As per our sources in the year 2019 Salman Khan is all set to launch two new faces, Pranutan Behl and Zaheer Iqbal .Pranutan is the granddaughter of late actress Nutan and daughter of Mohnish Behl. Salman and Mohnish have worked together in Hum Saath Saath Hai and Hum Aapke Hai Koun.
Zaheer does not belong to a family background but is the son of a family friend of Salman Khan. The title of the film is "Notebook" and the story is set in Kashmir.
The film is being directed by Nitin Kakkar. Salman Khan films is producing the film with Murad Khetani and Ashvin Varde of Cine1 Studios. The film is set to release on 29th March 2019.
The other details about the film have not been disclosed yet.
Next articleTeaser Of The Most Sizzling And Hot Track Of The Year Is Out Now.


text:
HomeTop HeadlinesPolitics
Houenipwela becomes 17th Prime Minister of Solomon Islands
adminNovember 15, 2017 12:02 am
Prime Minister elect Hon Rick Hounipwela presenting his speech outside Parliament. Photo: Lowen Sei.
Members of Parliament elected Rick Houenipwela, MP for Small Malaita, as Prime Minister of Solomon Islands in a 33-16 vote in Parliament.
John Moffat Fugui, MP for Central Honiara, was the opposing candidate in the election today.
Speaking outside Parliament this morning, Mr Houenipwela said his priority as Prime Minister would be to maintain political stability, rectify the country's fiscal situation and stamp out corruption.
"I stand here today as your newly elected Prime Minister," he said with his new Government flanking him. "This is an awesome responsibility which I take with sincere humility.
"I would like to thank you for your prayers and your support . . . and for the citizens for maintaining peace and order.
"The road ahead is challenging. I am fully aware of the enormity of the task before us.
"My good people, I can ensure you that our new administration will quickly put in place a strategy that entails a set of priorities deemed achievable in this short period.
"It is my Government's priority to stabilise our ailing fiscal situation. It is a priority to immediately address the cash-flow situation, to make sure there is fiscal discipline."
Acting Governor General, Hon Ajilon Nasiu making the announcement . Photo: Lowen Sei.
He said his officials had already identified key areas of prioritisation in the short-term.
"Given the limited time frame we will concentrate on those infrastructure programs that are considered achievable in the period, projects that will resuscitate the Government's finances and the economy in general."
Prime Minister Hou also highlighted the reopening of closed health clinics and the submarine cable as key priorities.
He also spent much of his speech addressing the public backlash over the anti-corruption bill.
"Corruption has caused inefficiencies in Government operations (and) it has undermined public trust in Government," he said.
"It has robbed vital resources from our schools and hospitals. I can guarantee our administration will deliver this important legislation and we are committed its implementation before the end of the 10th Parliament."
"In conclusion I call on all Solomon Islanders to embrace peace and national unity. I call on all members of Parliamnet (to) work together in governing our beloved country. We can move the country forward if we put our differences aside and work towards the national interests."
Mr Houenipwela is a seven-year member of Parliament who has been governor of the Central Bank, minister of Finance and Treasury and chairman of the Public Accounts Committee.
Prior to entering Parliament, Mr Houenipwela worked for the World Bank in the United States where he was a senior advisor to the executive director.
He resigned from the World Bank to contest the election for the Small Malaita Constituency in August 2010.
Since then, Mr Houenipwela has been a member of the Bills and Legislation Committee and a member of the Public Accounts Committee. He was minister of Public Service from April 2011 to November 2011 and minister of Finance and Treasury from November 2011 to September 2014. He was re-elected to Parliament in November 2014.
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