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\section{#1}}
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\newcommand{\mbox{\raisebox{-1.2ex}{$\stackrel{\displaystyle\otimes}{\mbox{\raisebox{-1.2ex}{$\stackrel{\displaystyle\otimes}
{\mbox{\scriptsize{$\rho$}}}$}}}
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\newcommand{{\widetilde{Q}}}{{\widetilde{Q}}}
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\pagestyle{empty}
\rightline{May 1999}

\vfill

\begin{center}
{\LARGE\bf Yangian symmetry in the
Nonlinear Schr{\"o}dinger hierarchy}
\\[2.1em]
\end{center}

{\Large
M. Mintchev$^{a}$\footnote{mintchev@difi.unipi.it},
E. Ragoucy$^{b}$\footnote{ragoucy@lapp.in2p3.fr},
P. Sorba$^{b}$\footnote{sorba@lapp.in2p3.fr} and
Ph. Zaugg$^{c}$\footnote{zaugg@CRM.UMontreal.CA}}\\

\null

\noindent
{\it $^a$ INFN, Dipart. di Fisica dell'Univ. di
  Pisa, Piazza Torricelli 2, 56100 Pisa, Italy\\[2.1ex]
$^b$ LAPTH, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux
  cedex, France\\[2.1ex]
$^c$ CRM, Univ. de Montr{\'e}al, CP 6128
Succ. Centre-ville, Montr{\'e}al
(Qc) H3C 3J7, Canada}

\vfill

\begin{abstract}
We study the Yangian symmetry of the multicomponent Quantum Nonlinear Schr\"odinger
hierarchy in the framework of the Quantum Inverse Scattering Method. We
give an explicit realization of the Yangian generators in
terms of the deformed oscillators algebra which naturally occurs in
this framework.
\end{abstract}

\vfill
\rightline{LAPTH-733/99}
\rightline{IFUP-TH 24/99}
\rightline{CRM-2608}
\rightline{\tt hep-th/9905105}
\newpage
\pagestyle{plain}
\setcounter{page}{1}
\sect{Introduction}

\null

An increasing number of integrable systems with internal degrees of
freedom have been shown to exhibit a Yangian symmetry.  One of the
earliest example of this is perhaps the Haldane-Shastry spin
chain~\cite{BGHP} and the spin generalisation of the
Calogero-Sutherland model investigated in~\cite{ABB}.

In the realm of integrable systems the quantum nonlinear
Schr{\"o}dinger (NLS) model distinguishes itself by being one of the
most studied system and its simplest version played an important role
in the development of the Quantum Inverse Scattering Method (QISM).
Recently the authors of~\cite{Wad} have considered the quantum NLS
model with spin $\frac{1}{2}$ fermions and repulsive interaction on
the line and have unraveled the presence of a Yangian symmetry
$Y(sl(2))$.

In this paper we consider the most general case of the quantum NLS
model with $N$-components bosons or fermions and prove that the
Yangian symmetry is {$Y(sl(N))$}.  In addition we provide an explicit
realisation of the Yangian generators using the algebra of creation
and annihilation operators for scattering states that is an essential
part of the QISM, also known as the Zamolodchikov-Faddeev (ZF)
algebra~\cite{ZF}.  This approach makes it clear that the Yangian is
actually a symmetry of the whole quantum integrable hierarchy whose
lowest instance is the NLS model. As a by-product, we also obtain all
the solutions to the equations of motion of the quantum NLS hierarchy.

The structure of the paper is the following.  In section 2 we
summarize the QISM applied to the NLS model.  Section 3 in devoted to
the study of the Yangian symmetry of the NLS model.  In section 4 we
outline the construction of the Yangian generators, as well as the
higher Hamiltonians of the NLS hierarchy, in terms of creation and
annihilation operators of the ZF algebra, the technical details being
gathered in the appendices.  Next we present a connection between the
action of the Yangian algebra restricted to the subspaces of fixed
particle number
and a class of finite $W$-algebras.  We conclude with possible extensions of this work.

\sect{The multicomponent nonlinear Schr\"odinger model}

We shall consider the $N$-components quantum nonlinear Schr\"odinger model with bosons or fermions and repulsive coupling.  We first collect a few known results from the Quantum Inverse Scattering Method applied to the model, most of which can be found in \cite{PWZ,Gut} with more details.

The Hamiltonian of the NLS model is
\begin{equation}
H = \int dx \left(\frac{\partial \phi^{\dag i}}{\partial x} \frac{\partial \phi_i}{\partial x} - \rho g \phi^{\dag i} \phi^{\dag j} \phi_i \phi_j \right),
\label{hamiltonian}
\end{equation}
where the fields operators satisfy the equal time canonical commutation relations
$$
[ \phi_i(x), \phi^{\dag j}(y)]_\rho = \delta_i^j \delta(x-y),
\qquad
[ \phi_i(x), \phi_j(y)]_\rho = [ \phi^{\dag i}(x), \phi^{\dag j}(y)]_\rho = 0.
$$
The conventions are that $\rho=-1$ for bosons, $\rho=1$ for fermions
and $[\;,\;]_\rho$ stands respectively for the commutator or anti-commutator.  Latin indices run from 1 to $N$, whereas Greek indices run from 1 to $N+1$.  Repeated indices are summed over their appropriate range.

The linear operator of the QISM is
\begin{equation}
L(x|\lambda) = i \frac{\lambda}{2} \Sigma + \Omega(x), \quad\hbox{with}\quad \Omega(x)= i \sqrt{g} \left(\phi_j(x) E_{j,N+1} - \phi^{\dag j}(x) E_{N+1,j}\right).
\end{equation}
In this equation $E_{\alpha \beta}$ is the standard $(N+1) \times (N+1)$ matrix $(E_{\alpha \beta})_{\mu\nu} = \delta_{\alpha \mu} \delta_{\beta \nu}$ and $\Sigma$ is a diagonal matrix $\Sigma = I_{N+1}-2 E_{N+1,N+1}$, where $I_{N+1}$ is the identity matrix.

The quantum monodromy matrix $T(x,y|\lambda)$ is defined by the equations
\begin{equation}
\frac{\partial}{\partial x} T(x,y|\lambda) = \ :\! L(x|\lambda) T(x,y|\lambda)\! :,
\qquad
T(x,y|\lambda)\vert_{x=y}=I_{N+1},
\label{dTLT}
\end{equation}
where $:~:$ denotes the usual normal order for the field operators
$\phi(x)$ and $\phi^\dag(x)$.  The infinite volume limit is a delicate
issue in the QISM~\cite{Gut}. The infinite volume monodromy matrix
$T(\lambda)$ is formally defined by
$$
T(\lambda) = \lim_{x \to \infty,y \to -\infty} E(-x|\lambda) T(x,y|\lambda) E(y|\lambda),
$$
where $E(x|\lambda)=\exp(i \lambda x \Sigma/2)$.
Using the implicit representation for $T(x,y|\lambda)$,
$$
T(x,y|\lambda) = E(x-y|\lambda) + \int_y^x dx_1 \,  E(x-x_1|\lambda) : \! \Omega(x_1) T(x_1,y|\lambda)\! :
$$
the monodromy matrix $T(\lambda)$ can formally be computed through an iterative procedure and is expressed as
\begin{equation}
T(\lambda) = I_{N+1} + \sum_{n=1}^\infty \int_{-\infty}^\infty d^nx \ \theta(x_1 > \dots > x_n) E(2 \sum_{i=1}^n (-)^i x_i|\lambda) :\!\Omega(x_1) \dots \Omega(x_n)\! :
\label{iteT}
\end{equation}

The commutation relations for infinite volume are encoded in the exchange relation
\begin{equation}
R_\rho^+(\lambda-\mu)\ T(\lambda) \mbox{\raisebox{-1.2ex}{$\stackrel{\displaystyle\otimes T(\mu) =
T(\mu) \mbox{\raisebox{-1.2ex}{$\stackrel{\displaystyle\otimes T(\lambda)\
R_\rho^-(\lambda-\mu)
\label{RTT}
\end{equation}
with the $R$-matrices
\begin{eqnarray}
 R_\rho^\pm(\mu) &=& \frac{i\rho g}{\mu+i\rho g}{\mbox{pv}}\frac{1}{\mu} \ E_{jj}\otimes E_{kk}+ \frac{1}{\mu+i\rho g}\ E_{\alpha j}\otimes E_{j\alpha} \nonumber\\
 &+&\frac{\mu-i\rho g}{(\mu+i0)^2}\ E_{j,N+1}\otimes E_{N+1,j} -\frac{\rho(\mu-i g)}{\mu+i\rho g}{\mbox{pv}}\frac{1}{\mu}\ E_{N+1,N+1}\otimes E_{N+1,N+1}\nonumber\\
 &\pm& i\pi\delta(\mu)(E_{jj}\otimes E_{N+1,N+1}
-E_{N+1,N+1}\otimes E_{jj}).
\end{eqnarray}
The term $i0$ is a consequence of the principal value regularisation adopted when $\mu$ goes to zero, according to the relation (in the sense of distributions)
$$
{\mbox{pv}}\frac{1}{\mu} = \frac{1}{\mu \pm i0} \pm i \pi \delta(\mu).
$$

It will be convenient to rename some elements of the monodromy matrix such as $D(\lambda)=T_{N+1,N+1}(\lambda)$ and $b^j(\lambda)=T_{N+1,j}(\lambda)$.  Further examination of some components of (\ref{RTT}) yields the following relations
\begin{eqnarray}
D(\lambda) D(\mu) &=&  D(\mu) D(\lambda), \label{DD} \\
D(\lambda) b^j(\mu) &=& \frac{\lambda-\mu+ig}{\lambda-\mu+i0}\ b^j(\mu) D(\lambda), \label{Db}\\
b^j(\lambda) b^k(\mu)& =&
\frac{-\rho(\lambda-\mu)}{\lambda-\mu-ig}\ b^k(\mu)b^j(\lambda)-
\frac{ig}{\lambda-\mu-ig}\ b^j(\mu)b^k(\lambda).
\label{bb}
\end{eqnarray}
The matrix element $D(\lambda)$ serves as a generating operator-function for the commuting integrals of motion of the NLS model.  This is most easily seen by performing an asymptotic expansion for large $\lambda$ in the solution (\ref{iteT}), whereby one gets
\begin{eqnarray}
 D(\lambda) &=& 1 + \frac{i g}{\lambda} \hat{N} + \frac{i g}{\lambda^2} \left( P+ i \frac{g}{2} \hat{N}(\hat{N}-1)\right) \label{Dexp} \\
 &+& \frac{i g}{\lambda^3} \left(H + i g (\hat{N}-1)P - \frac{g^2}{6}\hat{N}(\hat{N}-1)(\hat{N}-2) \right) + O(\frac{1}{\lambda^4})
\nonumber
\end{eqnarray}
with
$$
\hat{N} = \int dx \ \phi^{\dag j} \phi_j, \qquad
P = - i \int dx \ \phi^{\dag j} \partial\phi_j ,
$$
and $H$ is given in (\ref{hamiltonian}).
Consequently, eq.~(\ref{DD}) implies that these integrals of motion are all in involution.

In the multicomponent NLS model the commutation relations amongst $b^j(\lambda)$ and their adjoint have to be deduced from another type of exchange relation (see \cite{PWZ} for details).  Moreover these operators hardly make sense as operators on the Hilbert space~\cite{Gut} and it is necessary to consider instead the scattering states operators
\begin{equation}
a^{\dag j}(\lambda) = \frac{i}{\sqrt{g}} b^j(\lambda) D^{-1}(\lambda),
\label{defa}
\end{equation}
and their adjoint $a_k(\lambda)$.
Then the commutation relations amongst $a(\lambda)$ and $a^\dag(\lambda)$ are nicely encoded in the form of a Zamolodchikov-Faddeev (ZF) algebra \cite{ZF} as (a sort of deformed oscillator algebra):
\begin{eqnarray}
a_j(\lambda) a_k(\mu) &=& R_{kj}^{nm}(\mu-\lambda)\ a_m(\mu) a_n(\lambda), \\
a^{\dag j}(\lambda) a^{\dag k}(\mu) &=&  a^{\dag m}(\mu) a^{\dag n}(\lambda)\ R_{mn}^{jk}(\mu-\lambda),\\
a_j(\lambda) a^{\dag k}(\mu) &=&  a^{\dag m}(\mu)\ R_{jm}^{kn}(\lambda-\mu)\ a_n(\lambda) + \delta_j^k \delta(\lambda-\mu),
\end{eqnarray}
where the $R$-matrix given by
\begin{equation}
R_{jm}^{kn}(\mu) = \frac{1}{\mu+ig} \left( -\rho \mu\ \delta_m^k \delta_j^n + ig\ \delta_j^k \delta_m^n \right)
\label{RmatZF}
\end{equation}
is the two-body scattering matrix of the $N$-components NLS model.

The operators $a^{\dag}(\lambda)$ and $a(\lambda)$ play the role of creation and annihilation operators and as such can be used to build a Fock space through their action on the vacuum.  Owing to (\ref{Db}) these states are simultaneous eigenstates of the conserved quantities.  Actually, the set of (asymptotic scattering) states $\prod^n_{i=1} a^{\dag i}(\lambda_i)|0 \rangle$ for all $n$ is dense in the Hilbert space of the NLS model~\cite{Gut,fock}.  This property will be useful later on.

The original field operator $\phi_k(x)$ can be recovered from the
knowledge of the scattering data.  This is achieved by solving a
system of quantum Gel'fand--Levitan equations, whose result is a
quantum version of the Rosales expression~\cite{Ros,solQ} (given here for $t=0$):
\pagebreak[3]
\begin{eqnarray}
\phi_k(x) &=& \sum_{n=0}^\infty (-g)^n \int \frac{d^n\nu\ d^{n+1}\mu}{(2\pi)^{2n+1}} a^{\dag i_1}(\nu_1) \dots a^{\dag i_n}(\nu_n) a_{i_n}(\mu_n) \dots a_{i_1}(\mu_1) a_k(\mu_0) \nonumber \\
&&\frac{e^{i \mu_0 x} \prod_{l=1}^n e^{i(\mu_l-\nu_l) x}}{\prod_{l=1}^n (\nu_l - \mu_{l-1}+i\epsilon)(\nu_l - \mu_{l}+i\epsilon)}.
\label{solutionphi}
\end{eqnarray}

\sect{Yangian algebra in NLS}

In this subsection we shall show that the NLS model contains an infinite set of conserved charges having the structure of a Yangian algebra.  There are several equivalent definitions of the Yangian {$Y(sl(N))$}\ and we present two of them in Appendix~A.

The Yangian symmetry of the NLS model manifests itself already in the exchange relation (\ref{RTT}).  Indeed, denoting by ${\widetilde{T}}(\lambda)$ the $N \times N$ submatrix of $T(\lambda)$, ${\widetilde{T}}(\lambda)=T_{ij}(\lambda) E_{ij}$, and examining the appropriate components of (\ref{RTT}), one deduces the following relations
\begin{equation}
\widetilde{R}(\lambda-\mu) \ {\widetilde{T}}(\lambda) \otimes {\widetilde{T}}(\mu) =
 {\widetilde{T}}(\mu) \otimes {\widetilde{T}}(\lambda) \ \widetilde{R}(\lambda-\mu)
\label{RTtTt}
\end{equation}
with yet another $R$-matrix
\begin{equation}
\widetilde{R}(\lambda-\mu) =
(\lambda-\mu) E_{jk} \otimes E_{kj} +i \rho g I_N \otimes I_N  .
\label{Rtilde}
\end{equation}
This coincides precisely with the defining relation of the Yangian {$Y(gl(N))$}.

The fact that the Yangian algebra commutes with the Hamiltonian of the NLS model is a consequence of the exchange relation as well, since one extracts from (\ref{RTT}) that
\begin{equation}
[{\widetilde{T}}_{ij}(\lambda),D(\mu)] = 0
\label{TDzero}
\end{equation}
and the Hamiltonian is just one of the conserved quantities in the asymptotic expansion (\ref{Dexp}).

It is of some interest to get an explicit representation of the Yangian generators in terms of the field operators $\phi_i(x)$.  This is achieved with the help of the iterative solution (\ref{iteT}) of the monodromy matrix $T(\lambda)$, and by looking at its asymptotic expansion for large $\lambda$ one finds that
\begin{eqnarray}
 {\widetilde{T}}_{jk}(\lambda) &=& \delta_{jk} + \frac{i \rho g}{\lambda} \int dx \ \phi^{\dag k}(x) \phi_j(x) + \frac{\rho g}{\lambda^2} \Big( \int dx\ \phi^{\dag k}(x) \partial\phi_j(x) \label{expTij} \\
 && \mbox{}+ g\int d^2x \theta(x_1>x_2) \phi^{\dag n}(x_1) \phi^{\dag k}(x_2) \phi_j(x_1) \phi_n(x_2) \Big) + O(\frac{1}{\lambda^3})\nonumber \\
 & \equiv & \delta_{jk} + i \rho g \sum_{n=0}^\infty \frac{{\widetilde{T}}_{jk}^{(n)}}{\lambda^{n+1}} .
\end{eqnarray}
In particular this shows that in this model the formal series expansion of the Yangian generators (see \ref{formalt}) is to be understood as an asymptotic expansion for large $\lambda$.

The most important relations for our purpose are the commutators of the Yangian generators ${\widetilde{T}}_{jl}^{(n)}$ with the creation operators $a^{\dag k}(\mu)$.  Using the exchange relation (\ref{RTT}), the definition (\ref{defa}) and the symmetry property (\ref{TDzero}) one finds that
\begin{equation}
[ {\widetilde{T}}_{jl}(\lambda), a^{\dag k}(\mu) ] = \frac{i \rho g}{\lambda-\mu-i0} a^{\dag l}(\mu) {\widetilde{T}}_{jk}(\lambda),
\label{Ta}
\end{equation}
which, upon expanding in $1/\lambda$, yields
\begin{eqnarray}
 {[{\widetilde{T}}_{jl}^{(0)},a^{\dag k}(\mu)]} &=& a^{\dag l}(\mu) \delta_{jk}, \label{Tna*}\\
 {[{\widetilde{T}}_{jl}^{(1)},a^{\dag k}(\mu)]} &=& \mu \,  a^{\dag l}(\mu) \delta_{jk} + i \rho g \, a^{\dag l}(\mu) {\widetilde{T}}_{jk}^{(0)} .\nonumber
\end{eqnarray}

In the next section we shall need a more convenient basis of {$Y(sl(N))$}\ (see Appendix~A).  It will also be convenient to deal with self-adjoint generators for the Yangian algebra.  It turns out that these two requirements can be fulfilled in a single operation.  Let us denote by ${\widetilde{T}}(\lambda)^\ddag$ the hermitian conjugate of the $N \times N$ matrix ${\widetilde{T}}(\lambda)$ obtained by transposing the matrix and taking the adjoint of its entries (which are operators in a Hilbert space), namely
\begin{equation}
{\widetilde{T}}(\lambda)^\ddag = I_N - i \rho g \sum_{n=0}^\infty \frac{{\widetilde{T}}^{(n) \dag}_{jk} E_{kj}}{\lambda^{n+1}}
.
\end{equation}
Its anti-hermitian part is simply
\begin{equation}
\frac{1}{2}(T(\lambda)-T(\lambda)^\ddag) = i \rho g \sum_{n=0}^\infty  \frac{1}{\lambda^{n+1}} \frac{1}{2} \left({\widetilde{T}}^{(n)}_{jk} + {\widetilde{T}}^{(n) \dag}_{kj} \right)E_{jk} \equiv i \rho g \, U(\lambda),
\end{equation}
where $U(\lambda)=U(\lambda)^\ddag$ is now hermitian.  As such it can be expanded as
\begin{equation}
U(\lambda) = \sum_{n=0}^\infty \frac{U^{(n)}_{jk} E_{jk}}{\lambda^{n+1}} = \sum_{n=0}^\infty \frac{\sum_{a=1}^{N^2-1}{\widetilde{Q}}_{n}^a t_a + {\widetilde{Q}}_{n}^0 I_N}{\lambda^{n+1}},
\end{equation}
where ${\widetilde{Q}}_{n}^a = {\widetilde{Q}}_{n}^{a\dag}$ are self-adjoint generators.  The matrices $t_a = (t_a)^{*T}$ are in the fundamental representation of $su(N)$ and normalized to
$$
[t_a,t_b] = i {f_{ab}}^c t_c,
\qquad
\eta_{ab} = tr( t_a t_b).
$$
Therefore we have that
\begin{equation}
{\widetilde{Q}}_{n}^a = tr( U^{(n)} t^a) = U^{(n)}_{jk} (t^a)_{kj}.
\label{defQtn}
\end{equation}
In the ${\widetilde{Q}}^a_n$ basis, only the $n=0,1$ grades are necessary and from (\ref{expTij}) we find that
\begin{eqnarray}
 U^{(0)}_{jk} &=&{\widetilde{T}}^{(0)}_{jk}, \label{U0U1} \\
 U^{(1)}_{jk} &=& {\widetilde{T}}^{(1)}_{jk} - \frac{i \rho g}{2}\, {\widetilde{T}}^{(0)}_{lk}{\widetilde{T}}^{(0)}_{jl}  + \frac{i \rho g N}{2}\, {\widetilde{T}}^{(0)}_{jk}. \nonumber
\end{eqnarray}
The commutation relations of ${\widetilde{Q}}_{n}^a$ and $a^{\dag k}(\mu)$ are then readily computed using (\ref{Tna*}), (\ref{defQtn}) and (\ref{U0U1})
\begin{eqnarray}
 {[{\widetilde{Q}}_0^a,a^{\dag k}(\mu)]} &=& (a^{\dag}(\mu) t^a)^k, \label{Q0a}\\
 {[{\widetilde{Q}}_1^a,a^{\dag k}(\mu)]} &=& \mu\, (a^{\dag}(\mu) t^a)^k-\frac{\rho g}{2}
{f^a}_{bc} (a^{\dag}(\mu) t^c)^k {\widetilde{Q}}^b_{0}. \label{Q1a}
\end{eqnarray}

The operators ${\widetilde{Q}}^a_{0,1}$ generates the Yangian {$Y(sl(N))$}\ and the operators ${\widetilde{Q}}^0_n$ are related to its center.  They are also connected in some intricate way to the integrals of motion.

It is instructive to study the type of Yangian representations that appear in the Hilbert space of the NLS model.  The vacuum is invariant under the action of ${\widetilde{Q}}_{0,1}^a$ and the one-particle state $a^{\dag k}(\mu) |0 \rangle$ transforms as an evaluation representation in the fundamental representation of $sl(N)$~\cite{CP}, denoted by $V(\mu)$, since
\begin{eqnarray*}
 {\widetilde{Q}}_0^a \ a^{\dag k}(\mu) |0 \rangle &=& {(t^a)_j}^k a^{\dag j}(\mu)|0 \rangle,\\
 {\widetilde{Q}}_1^a \ a^{\dag k}(\mu) |0 \rangle &=& \mu\, {(t^a)_j}^k a^{\dag j}(\mu)|0 \rangle .
\end{eqnarray*}
The two-particles state $a^{\dag k_1}(\mu_1) a^{\dag k_2}(\mu_2) |0 \rangle$ transforms as a tensor product of two such representations
\begin{eqnarray}
 {\widetilde{Q}}_0^a \ a^{\dag k_1}(\mu_1) a^{\dag k_2}(\mu_2) |0 \rangle &=& ({(t^a)_{j_1}}^{k_1} \delta_{j_2}^{k_2} + \delta_{j_1}^{k_1} {(t^a)_{j_2}}^{k_2})a^{\dag j_1}(\mu_1) a^{\dag j_2}(\mu_2) |0 \rangle , \nonumber\\
 {\widetilde{Q}}_1^a \ a^{\dag k_1}(\mu_1) a^{\dag k_2}(\mu_2) |0 \rangle &=& \big(\mu_1 \,{(t^a)_{j_1}}^{k_1} \delta_{j_2}^{k_2} + \mu_2 \,\delta_{j_1}^{k_1} {(t^a)_{j_2}}^{k_2} \label{repVV}\\
 &&\mbox{} -\frac{\rho g}{2} {f^a}_{bc} {(t^c)_{j_1}}^{k_1} {(t^b)_{j_2}}^{k_2}\big)  a^{\dag j_1}(\mu_1) a^{\dag j_2}(\mu_2) |0 \rangle . \nonumber
\end{eqnarray}
In particular, the second term on the rhs of (\ref{Q1a}) ensures that the action of ${\widetilde{Q}}_{1}^a$ on the tensor product is consistent with the comultiplication of the Yangian
\begin{eqnarray*}
 \Delta({\widetilde{Q}}_0^a) &=& {\widetilde{Q}}_0^a \otimes 1 + 1 \otimes {\widetilde{Q}}_0^a ,\\
 \Delta({\widetilde{Q}}_1^a) &=& {\widetilde{Q}}_1^a \otimes 1 + 1 \otimes {\widetilde{Q}}_1^a - \frac{\rho g}{2} {f^a}_{bc} {\widetilde{Q}}_0^c \otimes {\widetilde{Q}}_0^b .
\end{eqnarray*}
Therefore $n$-particles states will carry a $n$-fold tensor product of $V(\mu_i)$ representations.

\sect{Yangian generators and deformed oscillators}

In view of formul{\ae} (\ref{solutionphi}) and (\ref{expTij}), it is
clear that trying to reconstruct the Yangian generators (in term of
oscillators) by direct calculation is a difficult task. Instead we
shall define two operators $Q_0^a$ and $Q_1^a$ that have the same
commutation relations with $a^\dag(\mu)$ as in (\ref{Q0a}) and
(\ref{Q1a}).  Therefore their action on the Fock space spanned by the
$a^\dag(\mu)$ will coincide with that of ${\widetilde{Q}}_{0,1}^a$ and, as this
Fock space is dense in the Hilbert space of the NLS model, we shall
identify these operators.  All the other Yangian generators are built
from these two sets of elements.

In order to simplify the presentation, we adopt a more compact notation for the ZF algebra.  We drop the explicit mention of the indices $i,j$ and the momenta $\mu_i$, and introduce instead a new index refering to an $N$-dimensional auxiliary space.  More explicitly
$$
a_1 \equiv a_i(\mu_1) \, e_1^i,
$$
where $e_1^i$ is some basis of the $N$-dimensional auxiliary space labeled by 1.
For instance, the $R$-matrix (\ref{RmatZF}) reads
\begin{equation}
R_{12} \equiv R_{12}(\mu_1-\mu_2) = \frac{1}{\mu_1-\mu_2+ ig}(-\rho (\mu_1-\mu_2) 1 \otimes 1 +ig P_{12}),
\end{equation}
where $P_{12}$ is the permutation operator in the auxiliary spaces.  The inverse of $R_{12}$ is $R_{21}=R_{21}(\mu_2-\mu_1)$.  With this notation, the ZF algebra relations read
\begin{eqnarray}
a_1 a_2 &=& R_{21} \, a_2 a_1, \nonumber \\
a_1^\dag a_2^\dag &=&  a_2^\dag a_1^\dag \, R_{21},  \label{ZFa}\\
a_1 a_2^\dag &=& a_2^\dag \, R_{12} \, a_1 + \delta_{12} .\nonumber
\end{eqnarray}
We also rename the operators we are looking for as $J^a=Q_0^a$ and $S^a=Q_1^a$.  Thus (\ref{Q0a}) and (\ref{Q1a}), and their conjugate, translate to
\begin{eqnarray}
{[J^a, a^\dag_0]} &=& a^\dag_0 t^a_0, \label{Ja*}\\
{[J^a, a_0]} &=& - t^a_0 a_0, \label{Ja}\\
{[S^a, a^\dag_0]} &=&\mu_0\, a^\dag_0 t^a_0 + \frac{\rho g}{2} {f^{a}}_{bc} a^\dag_0 t^b_0 J^c, \label{Sa*} \\
{[S^a, a_0]} &=& -\mu_0\, t^a_0 a_0 + \frac{\rho g}{2} {f^{a}}_{bc} J^b t^c_0 a_0. \label{Sa}
\end{eqnarray}
Here $t_0^a$ means that the $su(N)$ matrix $t^a$ is acting in the auxiliary space labeled by $0$.

We first construct the operator $J^a$. The idea is to built it
recursively in such a way that the relations (\ref{Ja*}-\ref{Ja})
are fulfilled. The expansion parameter is not the coupling
constant $g$ but rather the number of oscillators. More precisely, we start
with the expression
\begin{equation}
J^a = \sum_{n=1}^\infty \frac{(-)^{n+1}}{n!} J^a_{(n)}
= \sum_{n=1}^\infty \frac{(-)^{n+1}}{n!} a^\dag_{1\dots n}
T^a_{1\dots n} a_{n\dots 1},
\label{defJ}
\end{equation}
where $a_{n\dots 1}=a_n(\mu_n) \cdots a_2(\mu_2) a_1(\mu_1)$ and the integration on $\mu_1, \mu_2,\dots,\mu_n$ is implied in $J^a$.
We then determine the tensors $T^a_{1\dots n}$ recursively.
The details of the calculation are relegated in Appendix~B, and we find as a result that
\begin{equation}
T^a_{1\dots n}= \sum_{j=1}^n \alpha_j^n\ t^a_j
\qquad \mbox{with}
\qquad \alpha_j^n=
(-)^{j-1}
\bin{n-1}{j-1}.
\label{defT}
\end{equation}
These generators verify as well
\begin{equation}
{[J^a, J^b]} = i{f^{ab}}_c J^c.
\end{equation}
They form the $sl(N)$ subalgebra of {$Y(sl(N))$}.
\null

We then look for operators $S^a$ of a similar form
\begin{equation}
S^a = \sum_{n=1}^\infty \frac{(-)^{n+1}}{n!} S^a_{(n)}
= \sum_{n=1}^\infty \frac{(-)^{n+1}}{n!} a^\dag_{1\dots n}
{\widetilde{T}}^a_{1\dots n} a_{n\dots 1}
\label{defS}
\end{equation}
satisfying (\ref{Sa*}-\ref{Sa}).  In this case, the procedure is simpler since $S^a$ lives in the adjoint representation of the subalgebra generated by $J^a$.  This implies that
\begin{equation}
S^a = -\frac{i}{c_2} {f^a}_{bc} \, [S^b, J^c],
\label{SfSJ}
\end{equation}
where $c_2$ is as usual the second Casimir in the adjoint
representation, $c_2 \delta_a^{b}= f_{acd} f^{bcd}$.  Using the explicit expression (\ref{defJ}) for $J^a$ and imposing that $S^a$ satisfies (\ref{Sa*}-\ref{Sa}) enables us to compute the right-hand side of (\ref{SfSJ}) and to determine the tensors ${\widetilde{T}}^a_{1\dots n}$.

As anticipated from (\ref{Sa*}) the tensors depend on the momenta and their expression is (see Appendix C):
\begin{equation}
{\widetilde{T}}^a_{1\dots n}=\sum_{j=1}^n \alpha^n_j\,\left(
\mu_j\, t^a_j\, +\frac{\rho g}{2}{f^{a}}_{bc}\,
\sum_{i=1}^{j-1}  t^{b}_i t^c_j\right) .
\end{equation}
Therefore the expressions (\ref{defJ}) and (\ref{defS}) provide a realisation of the Yangian generators in terms of the ZF algebra.

A similar procedure can be applied to the integrals of motion (or higher Hamiltonians) of the NLS model.  The lowest Hamiltonians $\hat{N}, P, H$ are explicitly known, and from (\ref{Db}), (\ref{Dexp}) and (\ref{defa}) we find that
\begin{eqnarray*}
{[\hat{N},a^\dag_0]} &=& a^\dag_0 ,\\
{[P,a^\dag_0]} &=& \mu_0 \ a^\dag_0 ,\\
{[H,a^\dag_0]} &=& \mu_0^2 \ a^\dag_0 .
\end{eqnarray*}
Let us now define the infinite set of commuting operators ${\hat{I}}_n, n \geq 0$ by
\begin{equation}
{\hat{I}}_n = \int\! d\mu_1 \ \mu_1^n a_1^{\dag} a_1
\end{equation}
which enjoy the following commutation relations
\begin{eqnarray}
{[{\hat{I}}_n,a^\dag_0]} &=& \mu_0^n \ a^\dag_0 ,\label{Ina*a} \\
{[{\hat{I}}_n,a_0]} &=& -\mu_0^n \ a_0 .\nonumber
\end{eqnarray}
This implies that acting on a $m$-particles state
\begin{equation}
{\hat{I}}_n \, a_1^\dag \dots a_m^\dag |0 \rangle = (\sum_{j=1}^m \mu_j^n) \, a_1^\dag \dots a_m^\dag |0 \rangle
\end{equation}
which is precisely the definition of the higher Hamiltonians in the quantum NLS model~\cite{Gut}.  According to (\ref{Ina*a}) the lowest ones are obviously identified with
$$
\hat{N} = {\hat{I}}_0,
\qquad
P = {\hat{I}}_1,
\qquad
H = {\hat{I}}_2.
$$

Moreover we can show that $D(\lambda)$ is a generating
operator-function for the integrals of motion, that is, it can be
expressed entirely in terms of the ${\hat{I}}_n$ operators.  Indeed we can prove that
\begin{equation}
D(\lambda) = \exp(d(\lambda)),
\qquad
\hbox{where}
\qquad
d(\lambda) = \sum_{n=0}^\infty \ \frac{d_n}{\lambda^{n+1}},
\label{Dexpd}
\end{equation}
and
\begin{equation}
d_n = ig \sum_{j=0}^n \ \frac{(-i g)^{n-j}}{n+1}
\bin{n+1}{j} \ {\hat{I}}_j.
\label{dn}
\end{equation}
As the ${\hat{I}}_n$ commute with each other and satisfy (\ref{Ina*a}), it is straightforward to show that
$$
\exp(d(\lambda)) \ a_0^\dag \ \exp(-d(\lambda)) = \left( 1 + \frac{i g}{\lambda-\mu_0)} \right) a_0^\dag
$$
which is precisely the relation between $D(\lambda)$ and $a_0^\dag$ as deduced from (\ref{Db}).  This proves the assertion (\ref{Dexpd}).

Owing to the explicit expressions (\ref{defJ}) and (\ref{defS}), it is
very easy to check that the operators ${\hat{I}}_n$ commute with $J^a$ and
$S^a$.  As $J^a, S^a$ generate the Yangian algebra, then obviously the
${\hat{I}}_n$ commute with the whole Yangian algebra.  This is just another
way of expressing the content of (\ref{TDzero}). It also means that
the Yangian is a symmetry of all the quantum systems defined with the
help of the higher Hamiltonians.

As usual, the time evolution of the quantum field $\phi_k(x)$
of eq. (\ref{solutionphi}) is given by the conjugation by the NLS
Hamiltonian (\ref{hamiltonian}) which is nothing but
${\hat{I}}_2$. According to the commutation relations (\ref{Ina*a}), this
amounts to multiply in (\ref{solutionphi}) the creation operators
$a^\dag(\nu)$ by $e^{i\nu^2 t}$ and the annihilation operators
$a(\mu)$ by $e^{-i\mu^2 t}$. Although the expression
(\ref{solutionphi}) was originally obtained as the solution to the NLS
equation of motion, it also provides the solution to the higher flows
of the hierarchy. Simply, the $t_n$-time evolution is now induced by a
conjugation by $\exp(i{\hat{I}}_n t_n)$. Consequently, the phases
multiplying the creation and annihilation operators are respectively $e^{i\nu^n t_n}$
and $e^{-i\mu^n t_n}$. It is remarkable to get in such an easy way the
solutions to all the equations of motion of the hierarchy.

\sect{Connection with finite $W$-algebras}

It has already been shown that there is a strong connection between
Yangians {$Y(sl(N))$}\ and finite $W$ algebras~\cite{13}. We show in this section
that the NLS hierarchy offers a natural framework to illustrate this
relation.

For such a purpose, we focus on the Fock space $\cal F$
spanned by the $a^{i \dag}(\mu)$ and detail its structure.  Let us
recall that the $p$-particles subspace ${\cal F}_p(\mu_1,\dots,\mu_p)$
with fixed momenta $(\mu_1,\dots,\mu_p)$ is a tensor product of $p$
evaluation representations $\otimes_{i=1}^p V(\mu_i)$, all in the
fundamental representation of $su(N)$.  The $p$-particles subspace
${\cal F}_p$ is just the span over all momenta $\mu_i$.

On ${\cal F}_p$ it is straightforward to see that the Yangian
generators $t_{ij}^{(n)}$ with $n\geq p$ act as zero operators.  We thus
have a representation of a truncated Yangian, which is known to be
isomorphic, at the algebra level, to a finite $W(gl(Np),N.sl(p))$
algebra~\cite{BR}.  Thus on each $p$-particles subspace the Yangian
acts as a $W(gl(Np),N.sl(p))$ algebra.  This is another nice application of
finite $W$ algebras (more examples can be found in~\cite{BRS} and
references therein).

Let us illustrate this point on the simplest case, namely $p=2$ and
$N=2$.  Consequently, the only independent Yangian generators are
$Q_0^a$, $Q_1^a$ and their action on ${\cal F}_2$ is given in
(\ref{repVV}).  Let us denote their representation on ${\cal F}_2$ by
$J^a$ and $S^a$ respectively.  Up to an innocuous shift $S^a \to S^a -
\frac{1}{2}P J^a$, the full set of commutation relations satisfied by
these operators is
\begin{eqnarray}
{[J^a,J^b]} & = & i {f^{ab}}_c \ J^c, \nonumber\\
{[J^a,S^b]} & = & i {f^{ab}}_c \ S^c, \\
{[S^a,S^b]} & = & i {f^{ab}}_c \ \left( \frac{1}{2}H - \frac{1}{4} P^2
  - g^2 \frac{c_2}{8N} \right) J^c. \nonumber
\end{eqnarray}
One recognizes here the relations of the $W$-algebra
$W(gl(4),2.sl(2))$.

$W(gl(4),2.sl(2))$ and $W(sl(4),2.sl(2))$ algebras
essentially differ by one central element. In the present context,
this element is nothing but $P$, the total momentum.
Thus, the transition between the two $W$-algebras amounts to describe
the system in its center of mass frame.

\sect{Conclusions and outlook}

We deliberately imposed the restriction that the coupling constant $g$
be positive.  When $g$ is negative, the quantum spectrum also contains
bound states (solitons) and the asymptotic scattering states
$\prod^n_{i=1} a^{\dag i}(\lambda_i)|0 \rangle$ are no longer complete
in the Hilbert space of the NLS model.  Our construction relies
crucially on the completeness of those states in order to identify the
generators of the Yangian symmetry (\ref{defQtn}) with the generators
(\ref{defJ}) and (\ref{defS}) expressed in term of the Zamolodchikov-Faddeev algebra elements.

In both regions of the coupling constant, the Yangian generators
commute with the scattering matrix and the Yangian corresponds to a
symmetry of the scattering matrix.  The nuance is that in the second
situation we have only defined the action of the Yangian generators on
the asymptotic scattering states and the best we can say about the
operators (\ref{defJ}) and (\ref{defS}) is that they generate an
asymptotic symmetry of the NLS model.  Extending their definition to
the full Hilbert space would require the complete knowledge of the
bound states spectrum of the NLS model, something not yet achieved.

From a more general point of view, when considering our construction
as solely based on the existence of a ZF algebra we may conclude that
it is possible to realise a Yangian algebra on the Fock space
generated by the ZF algebra whenever the $R$-matrix is of the rational type given in~(\ref{RmatZF}).  For the NLS model, the $S$-matrix is invariant under $SU(N)$ and the symmetry algebra turns out to be a Yangian {$Y(sl(N))$}.  Let us mention that the $SU(N)$-Thirring model is another quantum system with such rational $R$-matrix, thus providing a relativistic example of a system with Yangian symmetry~\cite{Aris}.

It would be interesting to exhibit the symmetry algebra of more
general cases where the $R$-matrix is not rational but is still
invariant under some Lie group.  As the $R$-matrix in the ZF algebra
can be interpreted as the two-body $S$-matrix of integrable systems in
1+1 dimensions, this question reduces to the problem of identifying
the largest $S$-matrix symmetry of such integrable systems.  The study
of this sort of relationship in a more general situation is certainly
interesting and we intend to return to it in a future paper.

The NLS model considered in the present work is defined on the line.
Usually the presence of boundaries strongly influences the symmetry
and we are currently investigating this issue in the NLS model on the
half line.  Its quantification as a quantum field theory, as carried
out in~\cite{GLM}, reveals the presence of a boundary exchange algebra
generalizing the ZF algebra, thus fitting well within the more general
scheme of integrable systems with boundaries developed
in~\cite{Bound}.  We still expect the system to possess a large
internal symmetry, possibly in the form of a twisted Yangian
algebra~\cite{Molev} and we also expect to be able to express these
symmetry generators in terms of the boundary exchange
algebra~\cite{MRSZ2}.

\null

\noindent{\bf Acknowledgements:}
P.S. would like to express his gratitude to the members of CRM in Montreal for
their kind invitation for an extended visit this spring. Ph.Z. would
like to thank L. Vinet and Y. Saint-Aubin for interesting discussions.

redpajama_set_name: RedPajamaArXiv


text:
Justia Lawyer Directory Intellectual Property Texas Levelland Attorneys
Levelland, Texas Intellectual Property Lawyers
Thomason Blake Bush
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University of Oklahoma College of Law
Darin Michael Klemchuk
Dallas, TX Intellectual Property Lawyer with 22 years experience
(214) 367-6000	8150 N Central Expressway
Vanderbilt University Law School and University of Washington School of Law
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Southlake, TX Intellectual Property Lawyer with 21 years experience
(214) 999-0080	550 Reserve Street
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1550 Norwood Drive, Suite 107
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Donald Rudolph Taylor
Eden, TX Intellectual Property Attorney with 16 years experience
Eden, TX 76837
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text:
Business feature: Isaac Construction
Isaac Construction has been a long-standing supplier of construction materials to the Christchurch region for many years with much of the company's quarry product consumed in concrete and asphaltic concrete production.
Isaac Construction is one of the companies that took on the challenge of meeting the demand to rebuild Christchurch. Initially, the company achieved the additional aggregate volume required through a lot of hard work, driving its production staff and existing equipment to their limits. However, it soon became apparent that the need to expand the company's quarry production facilities was inevitable to ensure its ability to reliably meet customer's expectations.
Established by Isaac's family in 1957, Isaac Construction's McLean's Island quarry is one of the oldest established quarries in Christchurch. The quarry converts its raw material feed of hard, ancient river gravel into a coarse natural concrete aggregate, blended sand (natural and Barmac dust), sealing chip for roads and aggregates for asphaltic concrete.
Located just 18km from the centre of Christchurch, the quarry is ideally placed to efficiently supply its products for the city's reconstruction work.
Caring for the interests of the community and environment has long been part of Isaac Construction's business approach. The company's slogan 'Combining Construction with Conservation' is proudly demonstrated through the work of The Isaac Conservation and Wildlife Trust that was established by Sir Neil and Lady Diana Isaac in 1977.
The trust focuses on three areas: flora, fauna, and historic building conservation. In addition to the trust's conservation of rare plant species, its captive breeding programme rears for release a wide variety of New Zealand's most endangered birds and reptiles. An important source of funding for these visionary conservation projects is derived from the profits made in the quarrying business.
Today, the trust's facilities consist of a Heritage Village, the Isaac Conservation Park, Peacock Springs Conservation area and farmland, which are all co-located with the quarry on 1100 hectares of land at McLean's Island.
Upgrading to meet market demands
After the 2011 earthquakes, Isaac Construction ramped up production and was eventually running double shifts to meet the increased demand for concrete production and road building materials. However, the plant's production capacity and reliability started to become a major issue.
This sometimes resulted in the need to buy in products to satisfy supply obligations. So towards the end of 2011, the company's management team agreed that an upgrade was required and boldly decided to treble the quarry's production capacity from 100 to 300 tonnes per hour.
A tender process was initiated in February 2012 and after months of detailed bid reviews, Isaac Construction awarded a $NZ11m contract to MIMICO—Metso's exclusive distributor in New Zealand.
The contract included the design, supply, installation, and commission of a new plant based on Metso's crushing and screening equipment as well as its wear protection system.
The plant was designed, fabricated, and installed by QMI Engineering, a fully-owned subsidiary of MIMICO. Commissioning was undertaken by MIMICO's in-house service team.
The upgrade was completed in three stages. Work on the primary and secondary sections started in early 2013 and was completed in September 2014. The tertiary section was started in November 2014 and was completed by May 2015.
Financial prudence and unique payment terms
This project represented a significant investment for Isaac Construction and so flexibility around the payment terms was a high priority for the company's management.
Rex Davies, MIMICO's managing director explains, "As part of the project conditions, we offered Isaac Construction a monthly payment arrangement.
"This was a great result for Isaac's, as quite uniquely, the arrangement allowed them to make all of their payments from cash flow. I am quite certain that our payment package was an important factor in the decision to award us the contract." Stu Cameron, Isaac Construction's maintenance supervisor, explains from his perspective why choosing MIMICO as the supplier of the new plant was a good move.
"A big factor was the confidence we had in the working relationship. All the key people on the project from QMI Engineering and MIMICO knew our plant and company," he says. "In particular, Colin and Scott Welsh have been involved in maintenance and plant improvements here for many years prior to the start of the project.
"Another important factor was the reliability of our existing Metso crushers. In our old plant, we had a Metso HP100 and an HP200. Both machines were close to 20 years old and were still running strong."
Colin Welsh founded QMI in 1989 and has been personally involved with Isaac Construction since 2000.
He says, "When we first started doing work for Isaac Construction, it was just maintenance over their Christmas shutdown period. Over the years, this grew into doing refurbishments and small upgrades to improve production rates or to add the capability to produce different products. There is no doubt that our long-term relationship played a big part in us being awarded the contract."
Project challenges
The feedstock from McLean's Island quarry is infamous for its hardness. This factor alone made designing an optimal plant complex. Contributing to this complexity was shifting market demands both in terms of volume and variety of product. This forced Isaac Construction to change the required project specifications. While changes such as this are understandably unpopular with contractors, David McCaffrey, MIMICO's process equipment engineer believes that his company's ability to quickly interpret changes, adapt process models, and select suitable equipment helped them to win the project.
David elaborates, "BRUNO is Metso's easy-to-use software tool for planning and simulating the crushing process. It helps us to quickly explore various machine combinations for different applications. We use it extensively. On this project, I created many alternative models in BRUNO before proposing our overall design.
"When Isaac advised us about expected increases in sales production volumes for various products, I quickly made changes to input and output parameters and within minutes was able to assess the performance of the various models I had previously built. This allowed me to quickly understand what changes we needed to make our offer. We also excelled in addressing the uncertainty of Isaac's feedstock hardness, as we have our own laboratory which conducted extensive crush tests for us. The combination of using accurate crushability data and presenting BRUNO process models gave the management at Isaac Construction a high degree of confidence in the reliability of our design," he says.
A strong focus on safety
Talking about the project and its outcomes, Stu says, "Overall, things went really well, so it is hard to try to single out anything that stands out. I would say though, that the advanced safety levels in the new plant have really impressed me.
"Based on what I have seen, I think it has possibly worked out cheaper to install a new plant than to try to get the existing plant up to the same safety standards required to comply with New Zealand's latest safety rules."
Mike Higgins, Isaac Constructions industries manager is responsible for the running of all of the company's production facilities. Mike believes that the high level of safety provided by the new plant has ignited a wave of safety awareness. He says, "Suddenly, we all started looking at things differently and have become proactive in increasing our safety performance."
Plant efficiency and flexibility delivers a competitive advantage
The project delivered numerous optimisations, cost savings, and process efficiencies for Isaac Construction. Rex says, "An example of how the new plant has reduced costs is the elimination of the load and carry an aspect of the process. Previously, the plant consisted of three sections, and so at any one time, there would be four or five wheel loaders moving product between sections for the next stage of processing, doing up to 120 km a day. Not only did this mean high fuel and maintenance costs but it was also a big capital outlay. The new plant has a network of conveyors, removing this requirement completely."
For Mike , the flexibility of the new plant is what impresses him most. "We have a fairly unique situation in this region. At times, it is a real challenge to produce what the market needs from the feed gravel we have available to us. The gravel is very hard, and this makes it expensive to produce large volumes of fine crushed material.
"Thanks to this project, we have been able to optimise our costs far better than other quarries.
"It is a credit to David McCaffrey and the MIMICO team that they were able to build in additional flexibility to accommodate a wide variety of product specifications; a variety that even increased during the course of the project," he says.
Colin talks fondly of his involvement in the project and QMI Engineering's work at McLean's Island quarry.
"It is 15 years of history for me. If I look back, there were three different plants each dedicated to making a specific product. Today, I see a single plant that takes its feedstock on the one end and produces multiple products in multiple stockpiles.
"Thanks to the high degree of automation and advanced SCADA user interface, deciding and changing what product gets made and where it is stockpiled is a matter of keystrokes in the control room rather than hours of work out in the field. What this project has delivered to our long-term customer is certainly a distinction in my career."
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AVELING BARFORD 4X4 99H
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1986 CHAMPION 720A 720A
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Owen gambles on saving image
By Sam Wallace 20 January 2003 • 12:01am
Michael Owen admitted last night that he had lost up to £40,000 betting on horse racing, but the Liverpool striker denied newspaper reports that his debts reached £130,000 in an unprecedented defence of his lucrative clean-cut image.
The allegations in a Sunday newspaper that Owen had spent £2.2 million betting over the last three years and emerged with losses of £130,000 were vigorously denied by the player and his management company, SFX Sports.
The reports came at the end of a bad week for the England striker who was also named as the big loser in the World Cup squad's card school. Although Owen denied losing a reported £30,000, he is believed to have endured some expensive defeats during England's six-week campaign .
At stake amid all the allegations was the pristine image cultivated by Owen's management company, and the selling point for many of the 23-year-old's extremely profitable endorsements. SFX are reported to take 20 per cent of the striker's annual income of £2.5 million from commercial deals.
Since he burst dramatically into English football in the 1997-1998 season - and topped his debut year with a stunning World Cup goal against Argentina - Owen has been paid handsomely to advertise Umbro sportswear, Tissot watches and Jaguar cars. His picture has adorned boxes of Sporties cereals and Lucozade drinks, while Walkers Crisps also paid Owen to advertise their product.
On top of his Liverpool salary of around £40,000 a week - excluding bonuses and England royalties - Owen could count on a further £30,000 a week from his endorsements. It was these deals that the player, and SFX, would appear keen to defend when they issued yesterday's statement.
"I fully accept that high profile sports people like me are a role model for youngsters and I would never encourage anyone to gamble," Owen said. "I have gambled occasionally but to nowhere near the levels I have seen reported. This publicity will not stop my love for horses. They are my main hobby outside of football. My partner, Louise, and I are expecting our first baby soon and this and our new home - complete with stables - will keep me more than occupied in the future."
Owen, who owns a racehorse, was alleged to have bet with the bookmaker Victor Chandler and also kept an account with the spread betting firm IG Index. SFX maintained that this account was opened by the player's father, Terry, "two or three years ago" and used to "place bets for friends and family".
"Michael's gambling losses probably total around £30-40,000 over the last couple of years," SFX said. "Although the sums represent a tiny percentage of his earnings Michael and his family know how important even £100 is to a normal household. It was not long ago that their family of seven lived together in a tiny house in a north Wales village."
That is no longer the case for Owen, who bought his family five separate houses in the north Wales village of Ewloe and who owns a Grade II listed house, Lower Soughton Hall, set in 12 acres of land.
The details of Owen's gambling habits are likely to prompt further investigations into how widespread the practice is among Premiership footballers. Ruud van Nistelrooy revealed last week that he had been surprised by the extent to which his Manchester United team-mates played cards for money and Eidur Gudjohnsen, the Chelsea striker, has admitted he lost £400,000 gambling in casinos.
Gordon Taylor, the chief executive of the Professional Footballers' Association, yesterday said that though modern stars needed to be able to relax away from the game it was important any potential problems were dealt with by clubs.
"Footballers are kept in such a bubble that horse racing is a release and as long as it's kept in proportion, it's just recreation for millionaire sportsmen," Taylor said. "It may look a lot more serious because players these days do have a lot more money than before."
By Telegraph Sport 17 Jan 2022, 10:46pm


Board index Forex Brokers What is Forex Broker?
Please read more about what is Forex Broker.
A forex broker(retail forex broker) or currency trading broker in present time financial and commercial trading means an intermediary who buys and sells a particular asset or assets for a commission in the Forex Market. Thus, a broker may be thought of as a salesman of financial assets.
Re: What is Forex Broker?
Also, a broker is playing a role of the "gateway" for the beginners. If you took the right one and you understood how the things are working, you will be able to gain something on it.
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Enjoy these pictures & wallpapers of the Freightliner Century Class ST. It's one of many models produced by the manufacturer. Depending on the make, there may or may not be new / preowned Freightliner Century Class ST's available.


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Best Inventions In The Last 25 Years
When someone asks me what I think the best invention in the last couple of decades I say smart phone. We call it a "smart phone" but it's really a mini computer that can make phone calls. Just think about everything you can do with it. It can get you where you need to go, check weather, get news and information, play games, do your banking and shopping. It's pretty amazing when you think about it.
A survey asked what people thought the ten most important inventions in the last 35 years. Here's what they said.
Wi-Fi. (Internet is over 25 years old)
Online search engines, l
Online shopping platforms, like Amazon.
Stem cell technology.
GPS.
Flat-screen TVs.
Contactless payment.
Tablet devices.
YouTube was #11 on the list, 3D printers was at #12, Netflix was at #13, Skype was at #15, electric cars was at #16, Facebook was at #19, and wireless headphones came in #20.
(Product Focus)
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Hope You're Not Still Using These Passwords Minnesota


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ION FLOW THROUGH THE MEMBRANE PowerPoint Presentation
ION FLOW THROUGH THE MEMBRANE
Factors Affecting Ion Transport Through the Membrane under normal resting conditions . The flow of ions through the cell membrane depends mainly on three factors: the ratio of ion concentrations on both sides of the membrane the voltage across the membrane,and the membrane permeability .
Elvis Clint
membrane current
membrane voltage
sodium conductance
membrane ion flow
biological membrane consisted solely
Movement Through the Membrane - . mr. luis a. velázquez academic biology. diffusion. diffusion motion of particle for
Diffusion through the Membrane - . questions – pg 2. #9 – based on your knowledge of diffusion, predict what will
Movement Through the Membrane - . which diagram most likely represents the fruits in fresh water? which diagram most
Transport through Membrane - . diffusion. osmosis. active transport. membrane protein. endocytosis. exocytosis.
Transport through the Cell Membrane - . passive transport. movement across a cell membrane that does not require energy 2
Membrane Potential and Ion Channels - Colin nichols. membrane potential and ion
Movement Through the Cell Membrane - . important vocabulary. concentration gradient : difference in concentration from
Transport through the Cell Membrane - . passive transport. does not expend cellular energy for the movement to take place
Movement Through the Cell Membrane
Global Ion Exchange Membrane Market - Ion exchange membrane technology plays an important role because of its various
Ion Exchange Membrane of All Vanadium Redox Flow Battery Sales Market - Download a free sample copy @
NOTES – Transport Through The Cell Membrane - . what is cell transport?. cell transport – when substances move into
Cellular Transport Through the Cell Membrane - . sb1.d. explain the impact of water on life processes (i.e. osmosis,
Ion Channels are responsible for the membrane potential. - Ion channels are responsible for the membrane potential. when
MOVEMENT THROUGH THE MEMBRANE Bubble Lab - . section 7-3. the cell membrane. function : regulates what enters and leaves
Diffusion Through a Membrane - . diffusion through a membrane. semi-permeable – some molecules are able to pass
BLOOD FLOW through the HEART - . biology version:. the right and left sides of the heart work together right side blood
ENERGY FLOW THROUGH THE ENVIRONMENT - . energy is lost. energy is added. energy must constantly be added to the biosphere
FLOW THROUGH BEDS - . rapid filtration. rapid filtration. rapid filtration. rapid filtration. rapid filtration. rapid
FLOW THROUGH TUBES - . by joshua bower peer support 2013/2014 j.bower@warwick.ac.uk. warning. i am not a physicist. where
Factors Affecting Ion Transport Through the Membrane • under normal resting conditions. • The flow of ions through the cell membrane depends mainly on three factors: • the ratio of ion concentrations on both sides of the membrane • the voltage across the membrane,and • the membrane permeability. • The effects of concentration differences and membrane voltages on the flow of ions may be made commensurable if, instead of the concentration ratio, the corresponding Nernst voltage is considered. The force affecting the ions is then proportional to the differencebetween the membrane voltage and the Nernst voltage. • Regarding membrane permeability, we note that if the biological membrane consisted solely of a lipid bilayer, as described earlier, all ionic flow would be greatly impeded. However, specialized proteins are also present which cross the membrane and contain aqueous channels. Such channels are specific for certain ions; they also include gates which are sensitive to membrane voltage. The net result is that membrane permeability is different for different ions, and it may be affected by changes in the transmembrane voltage, and/or by certain ligands. • As mentioned in Section 3.4.1, Hodgkin and Huxley (1952a) formulated a quantitative relation called the independence principle. According to this principle the flow of ions through the membrane does not depend on the presence of other ions. Thus, the flow of each type of ion through the membrane can be considered independent of other types of ions. The total membrane current is then, by superposition, the sum of the currents due to each type of ions. ION FLOW THROUGH THE MEMBRANE
Membrane Ion Flow in a Cat Motoneuron We discuss the behavior of membrane ion flow with an example. For the cat motoneuron the following ion concentrations have been measured
For each ion, the following equilibrium voltages may be calculated from the Nernst equation: VNa = -61 log10(15/150) = +61 mV VK = -61 log10(150/5.5) = -88 mV VCl = +61 log10(9/125) = -70 mV The resting voltage of the cell was measured to be -70 mV.
When Hodgkin and Huxley described the electric properties of an axon in the beginning of the 1950s, they believed that two to three different types of ionic channels (Na+, K+, and Cl-) were adequate for characterizing the excitable membrane behavior. The number of different channel types is, however, much larger. In 1984, Bertil Hille summarized what was known at that time about ion channels. He considered that about four to five different channel types were present in a cell and that the genome may code for a total number of 50 different channel types. Now it is believed that each cell has at least 50 different channel types and that the number of different channel proteins reaches one thousand.
Graphical Illustration of the Membrane Ion Flow The flow of potassium and sodium ions through the cell membrane (shaded) and the electrochemical gradient causing this flow are illustrated. For each ion the clear stripe represents the ion flux; the width of the stripe, the amount of the flux; the inclination (i.e., the slope), the strength of the electrochemical gradient.
Membrane conductance changes during a propagating nerve impulse K. S. Cole and H. J. Curtis (1939) showed that the impedance of the membrane decreased greatly during activation and that this was due almost entirely to an increase in the membrane conductance. That is, the capacitance does not vary during activation. This is a numerical solution of Equation 4.31 (after Hodgkin and Huxley, 1952d)
An electric circuit representation of a membrane patch The equivalent circuit model of an axon
This figure illustrates the components of the membrane conductance, namely GNa and GK, and their sum Gmduring a propagating nerve impulse and the corresponding membrane voltage Vm.
The components of the membrane current during the propagating nerve impulse The figure illustrates the membrane voltage Vm during activation, the sodium and potassium conductances GNa and GK, the transmembrane current Im as well as its capacitive and ionic components ImC and ImI, which are illustrated for a propagating nerve impulse (Noble, 1966).
From the figure can be made: • The potential inside the membrane begins to increase before the sodium conductance starts to rise, owing to the local circuit current originating from the proximal area of activation. In this phase, the membrane current is mainly capacitive, because the sodium and potassium conductances are still low. • The local circuit current depolarizes the membrane to the extent that it reaches threshold and activation begins. • The activation starts with an increasing sodium conductance. As a result, sodium ions flow inward, causing the membrane voltage to become less negative and finally positive.
The potassium conductance begins to increase later on; its time course is much slower than that for the sodium conductance. • When the decrease in the sodium conductance and the increase in the potassium conductance are sufficient, the membrane voltage reaches its maximum and begins to decrease. At this instant (the peak of Vm), the capacitive current is zero (dV/dt = 0) and the membrane current is totally an ionic current.
The terminal phase of activation is governed by the potassium conductance which, through the outflowing potassium current, causes the membrane voltage to become more negative. Because the potassium conductance is elevated above its normal value, there will be a period during which the membrane voltage is more negative than the resting voltage - that is, the membrane is hyperpolarized. • Finally, when the conductances reach their resting value, the membrane voltage reaches its resting voltage..


Gibson 1975 Les Paul catalogue. The front cover features Keith Richards of the Rolling Stones playing a Les Paul Custom. Keith was, of course, well-known for playing Gibson Les Paul guitars in his early days.


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16 June 1915 – Bellewaarde Farm
Battle is chaos. Military leadership mainly consists of managing people and weapons within a state of chaos. Oxymorons like "fragmentary order" and "military intelligence" contain the difficult, unstable, uncertain nature of what warriors do in a battle; murder is the clarifying act, but modern militaries are creatures of extreme complexity. Take the above photograph, which shows men of the Liverpool Scottish, a Territorial (Reserve) Army unit, during an attack on Bellewaarde Farm today. It was taken near Objective Y13, a position marked in the map below, by a wounded soldier with a forbidden camera.
The banner those men are holding is a signal to British artillery spotters that they have secured the objective behind another attacking unit that, in turn, is prepared to take its next objective. The guns will now shift their barrage to another objective, hopefully avoiding 'friendly fire' casualties during the assault of the position that they have just bombarded. Known as 'leapfrogging,' it is the emerging British tactical order of movement on the Western Front.
A map of today's operation. Note the complex German trench system and meticulous, step-by-step planning
When the Second Battle of Ypres ended, the British-held salient was greatly reduced, and there were a few dents left in it. One of these lies in the angle of two roads, where a bulge in the German line is centered on a now-destroyed farm and a small lake. Beyond the German lines, Bellewaard Ridge provides overwatch of British positions in the salient, leaving any men in the open vulnerable to artillery fire during daylight hours. Sir John French is restricting offensive operations to conserve lives and store up a reserve of shells, but deems the reduction of the Bellewaarde Salient worthwhile, and moreover he must support his French allies by pinning down German reserves if he can.
To this end, troops have buried redundant telephone lines, laid up forward stores of ammunition, and built support trenches for the last ten days. Shortly before 3 AM, an intense eighty-minute bombardment with three deceptive pauses precedes the 9th Brigade going over the top in the predawn darkness, followed by the 7th. British artillery fire is very accurate today, reflecting improved training and weapons, but there are still not enough heavy guns in the British Army, and too many of them are engaged in battles elsewhere down the line. The 4th Royal Fusiliers attack from Y16 to Y17; the 1st Royal Scots Fusiliers and 1st Northumberland Fusiliers attack between Y15 and Y16, pressing on towards Y17.
Bellewaarde Lake was really a pond at the bottom of the shallow valley below Bellewaarde Ridge. Note the stripped vegetation
But as the methodical battle-plan unfolds into the late morning, German counterfire is already incurring casualties in the advance, and a lack of experience in new units is showing. By noon, the 'Y Wood' between Y16 and Y17 is a jammed mess of men taking cover from the shellfire, with not enough room for all, so the troops dig more trenches rather than retreat. By mid-afternoon, German chlorine gas shells are falling on their positions along with shrapnel and high explosives. In his book Sanctuary Wood and Hooge, historian Nigel Cave records the testimony of a Private Clapham as he braved the dangers of the battle to take a look around and noted the effects of chlorine on the vegetation around him.
Since the morning most of the branches of the trees in the wood had gone, and many of the trunks had become mere splintered poles. Something else had changed also, and for a time I could not make out what it was. Then it suddenly flashed across my mind that the thick hedge at the back of the trench had entirely disappeared. It was right in the path of the storm of gas shells and they had carried it away.
As they are equipped with rudimentary defensive measures, the gas does not kill very many British soldiers, but does add to the confusion and claustrophobia and general horror. At 5 PM, the 42nd Brigade retires on Ypres rather than moving up in turn, for the roads and support trenches are too congested by the traffic of four thousand casualties being transported to the rear as anxious reservists line the ways, awaiting orders to advance. The German bombardment intensifies with the approaching evening, peaking at ninety shells per minute by 8 PM.
The day does end with some progress, for the first and second objectives are taken, while German counterattacks are all beaten back. The 7th Brigade war diary records a gain of 250 yards along an 800-yard front, an area of about 41 acres, at a cost of 25% casualties — the same as other units which attack today — while killing less than one thousand Germans, who defend their front lines only sparingly. An acute shortage of grenades is greatly compounding the shortage of shells by limiting 'bombing parties' as they clear their objectives.
An overhead view from above the Menin Road. Bellewaarde Ridge is in the background and Y Ridge is visible below the lake
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15 June 1915 - The Duck's Bill
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