text:
\section{Introduction}
        Quantum duality principle \cite{Drin,STS} asserts
        that quantization of a Lie bialgebra $ (A,A^*) $
        gives rise to a dual pair of Hopf algebras
        $ (U_p (A),U_p (A^*)) $ or in dual terms --
        ($ \mbox{Fun}_p (G)$, $\mbox{Fun}_p (G^*) $).
        In the standard form
        quantum algebras and groups do not exhibit this
        duality explicitly. This is clearly seen when the
        classical limit is concerned. The quantum algebra
        $ U_p(A) $ is meant as a member of 1-dimensional
        family of Hopf algebras -- the deformation curve.
        Its classical limit $ U(A) $ is a fixed point of the
        orbit $ \mbox{Orb}(U(A)) $ where the deformation curve
        starts. Due to quantum duality formulated in terms
        of quantum formal series Hopf algebras \cite{Drin}
        quantum algebra can be interpreted as a quantum group
        \[
        U_p(A) \approx (\mbox{Fun}(G^*))_{p}
        \]
        for the universal covering group $G^* $ with Lie
        algebra $ A^* $. So there must be another classical
        limit, i.e. another deformation curve that starts
        at a fixed point of $ \mbox{Orb}(\mbox{Fun}(G^*)) $ and
        contains the Hopf algebra $ U_p(A) $.

        Thus the natural form of deformation quantization
        of Lie bialgebra $ (A,A^*) $ must be a 2-parametric
        family of Hopf algebras with two dual classical limits.
        Within certain assumptions this family forms an
        analytic variety $Q$ and the classical limits -- its boundary.
        The existence of a variety $Q$ with such properties is equivalent
        to attributing its member the quantum duality.
        The Lie bialgebra appears here in the form of two vector
        fields tangent to $Q$. From the point of view of Lie-Poisson
        structures and their symmetries it can be shown
        natural to consider the varieties of this type and their
        boundaries entirely placed in the domain of noncommutative
        and noncocommutative Hopf algebras. Preserving the main
        property of quantum duality we find that in this case some
        other characteristics are not conserved. In particular the
        lifted tangent fields may not form a Lie-bialgebra any more.
        But dual parameters and dual limits are still present there and
        can play important role in applications.

        The paper is organised as follows. In the subsection \ref{gen21}
        we describe how to construct the dual classical limits and under
        what conditions it can be done. The explicit example is considered
        in subsection \ref{ex22} where it is also shown how inverting
        the dual variety $Q^*$ one can change the role of dual
        parameters. In the subsection \ref{sub31}
        lifted varieties $Q_{\varepsilon}$ (that are called the quantized
        Hopf pairs) are studied. Their Lie-Poisson properties are
        considered in \ref{LPstr} and in \ref{ntexamp} the nontrivial
        example of such $Q_{\varepsilon}$ variety is presented. In
        Appendix the two-parametric form for the standard quantization
        of $sl(n,{\bf C})$ is given explicitly.

        \section{Dual classical limits}
        \subsection{General scheme} \label{gen21}
        Let us construct the second classical limit for a quantum Lie
        algebra $ U_p(A) $. Consider the variety $ {\cal H} $ of Hopf
        algebras with fixed number of generators. Its points
        $ H \in {\cal H} $ are parametrized by the corresponding
        structure constants. We must find  a (smooth)
        curve in $ {\cal H} $ containing $ U_p(A) $ and
        intersecting with the orbit $ \mbox{Orb}(\mbox{Fun}(G^*)) $.
        In the limit to be obtained the multiplication
        $ m $ in $ U_p(A) $ must become Abelian. For the
        universal enveloping algebra $ U(A) $ such a procedure
        is trivially described by a linear contraction.
        The corresponding transformation of basis $ \{ a_i \} $,
        \begin{equation}
        B(t) : a_i \rightarrow a_i/t ,
                                        \label{e1}
        \end{equation}
        leads to new structure constants $ C_{ij}^{'k} $
        \begin{equation}
        C_{ij}^{'k} (t) = t C_{ij}^{k}.
                                        \label{e2}
        \end{equation}
        The costructure  of $ U(A) $ being primitive is
        insensitive to this transformation. Algebras
        $ U(A_t) $ form a line in $ \mbox{Orb} U(A) $ with the
        limit point  $ U(A_0 \equiv \mbox{Abelian}) $. Let us apply
        operators $ B(t) $ to $ U_p(A) $. Such a sequence
        of equivalence transformations (for each value
        of $ p $ fixed) generates the smooth one-parametric
        curve
        \begin{equation}
        B(t) U_p(A) \equiv U_p (A_t)
                                        \label{e3}
        \end{equation}
        belonging to the orbit $ \mbox{Orb}(U_p(A)) $.
        Note that in general case Hopf algebras $ U_p(A) $
        are not equivalent for different $ p $. We thus
        obtain in $ {\cal H} $ the 2-parametric subset
        \begin{equation}
        \{ U_p (A_t) \}_{p>0,t>0} \equiv Q(A,A^*)
                                        \label{e4}
        \end{equation}
        formed by the dense set of smooth curves.
        Being the deformation quantization
        $ \{ U_p(A) \}_{p>0} $ is an analytic family.
        So $ Q(A,A^*) $ is a smooth variety with the global
        co-ordinates $p$ and $t$. Nevertheless these
        co-ordinates are not appropriate for our task
        because the limit
        \begin{equation}
        lim_{t\rightarrow 0} \, \, U_{p>0} (A_t)
                                        \label{e4a}
        \end{equation}
        does not exist. This is clearly seen from the
        properties of the coproduct  in $ U_p(A) $,
        \begin{equation}
        \Delta (a_i) = a_i \otimes {\bf 1} + {\bf 1} \otimes a_i +
        p (D_{i}^{kl} a_k \otimes a_l + S_{i}^{kl} a_k
        \otimes a_l + D_{i}^{ \{k,m\} \{l\} } a_k  a_m
        \otimes a_l + \cdots ).
                                        \label{e5}
        \end{equation}
        Here the first deforming function is divided
        into symmetric and antisymmetric parts and
        $ D_{i}^{kl} $ are the structure constants of $ A^* $.
        The transformation (3) leads to the expression
        divergent for $ t \rightarrow 0 $:
        \begin{equation}
        \Delta (a_{i}) = a_{i} \otimes {\bf 1} + {\bf 1} \otimes
        a_{i} + p/t (D_{i}^{kl} a_{k} \otimes a_{l}
         + \cdots ).                      \label{e6}
         \end{equation}
        Here the unwritten terms in (\ref{e6}) may contain the
        higher negative powers of $t$.

        It is easy to define a class of deformation quantizations
        for which one can overcome this difficulty. For each $ U_p(A) $
        consider the family of groups parametrized by $ p $ and defined
        on the space $ \mbox{Mor} (U_p(A),{\bf K}) $
        by the convolution multiplication
        \begin{equation}
        \phi_1 * \phi_2 = ( \cdot_{\bf K} )( \phi_1 \otimes
                          \phi_2 ) \Delta, \rule{1cm}{0cm}
        \phi_1, \phi_2 \in  \mbox{Mor} (U_p(A),{\bf K})
                                        \label{e6a}
        \end{equation}
        (${\bf K}$ here is the main field).
        Let ${\cal P}$ be the subspace
        of functionals dual to the space of algebra $A$.
        On this space (when certain conditions are fulfilled \cite{Lya})
        the convolution (\ref{e6a}) also generates a family of groups
        parametrised by $p$.
        This analytic family describes the
        contraction of the obtained group ${\cal P}$ to the additive
        Abelian vector group on the space dual to that of Lie algebra $A$.

        Suppose now that
        \begin{description}
        \item[(a)] the system of equations
\[  ( \cdot_{U_p} )(\mbox{id} \otimes S) \Delta =
( \cdot_{U_p} )(\mbox{id} \otimes S) \Delta = \eta \varepsilon
\]
on the basic elements $a_i$ fixes the antipode $S$ of $ U_p(A)$
(see \cite{Lya}) .
        \item[(b)] for $U_p(A)$
the contraction ${\cal P_{\rm p}} \longrightarrow {\cal AB}$
is equivalent to the trivial, that is induced by such a contraction of
its algebra $A^*$ where the structure constants in the lowest order
are proportional to $p$.
        \end{description}
        It is easy to verify that for such deformation quantizations
        in every monome of the coproduct (\ref{e5}) the power of $p$ is
        less than the total degree
        of the basic elements $a_i$ by one.

        If the Hopf algebra $U_p(A)$ belongs to the class described above
        the second classical limit can be obtained as follows.
        Let us change the co-ordinates:
        \begin{equation}
        (p,t)  \Rightarrow  (h,t),   \rule{1cm}{0cm}
        h  =  p/t.                        \label{e7}
        \end{equation}
        Then the coproduct (\ref{e6}) becomes well defined in the limit
        $t \rightarrow 0 $ (with $h$ fixed)
        because now its structure constants can bare only positive
        powers of $t$ (and in the two lowest orders can depend only on $h$).

        The multiplication
        structure constants in these new co-ordinates also have
        the finite limit values. To see this let us go to the
        dual picture.
        Consider the space of linear functionals on the
        space of Hopf algebra  $ U_p (A_t) $ and the canonical
        dual basis $ \{f^{I}\} $ such that the elements
        $ f^i $ dual to the basic elements of $A$,
        \[ <a_i, f^j >  = \delta_{i}^{j} , \]
        form the basis of $ A^* $.
        Construct the dual Hopf
        algebras for the elements of $ Q(A,A^*) $
        \begin{equation}
        \{ (U_p (A_t))^* \}_{p>0,t>0} =
        \{ \mbox{Fun}_p (G_t) \}_{p>0,t>0} \equiv
        Q^* (A,A^*)
                                        \label{e8}
        \end{equation}
        and also for the border line
        \begin{equation}
        \{ (U(A_t))^* \}_{t\geq0} = \{ \mbox{Fun}(G_t) \}_{t\geq0}.
                                        \label{e9}
        \end{equation}
        We shall consider these Hopf algebras as quantum
        formal series groups \cite{Wor}. The basis transformation
        $ B^*(t) $ dual to (\ref{e1})
        \begin{equation}
        B^* (t) : f^i \rightarrow t f^i
                                        \label{e10}
        \end{equation}
        leads to the following compositions:
        \begin{equation}
        [ f^i,f^k ] = p/t ( D_{l}^{ik} f^l + tE_{su}^{ik} f^s f^u
        + \cdots ),
                                        \label{e11}
        \end{equation}
        and
        \begin{equation}
        \Delta f^i = f^i \otimes {\bf 1} + {\bf 1} \otimes f^i +
        t (C_{kl}^{i} f^k \otimes f^l + T_{kl}^{i} f^k
        \otimes f^l + t C_{\{k,l\}\{u\}}^{i} f^k f^l
        \otimes f^u + \dots )
                                        \label{e12}
        \end{equation}
        For simplicity the renormalization factors for high
        power monomials are incorporated in the corresponding
        structure constants. These coproduct structure constants
        (in general) depend on $p$
        but they all have well defined limits when
        $ p \rightarrow 0 $. These limits describe
        the power series expansion of the
        multiplication law in $ G_t $ in terms of exponential
        co-ordinate functions. So the corresponding Taylor
        series can be written for them in the neighbourhood
        of $ ((\cdot))p = 0 $. Substituting $ p=ht $ in (\ref{e12})
        and going to the limit $ t \rightarrow 0 $ we see
        that all coefficients are finite. (Note that in the
        framework of formal series Hopf algebras this
        conclusion is true for all $ p \geq 0 $.)

        We have shown that for algebras $ U_h (A_t) \in
        Q (A,A^*) $ of the described class the limit $ U_h (A_0) \equiv
        lim_{t\rightarrow 0} \, \, U_h (A_t) $  exists. According
        to the quantum duality
        $ U_h (A_t) \approx \mbox{Fun}_t (G_{h}^{*}) $,
        so
        \begin{equation}
        lim_{t \rightarrow 0} \, \,  U_h (A_t) \equiv U_h (A_0)
        \approx \mbox{Fun} (G_{h}^{*})
                                        \label{e13}
        \end{equation}
        Thus every such deformation quantization can be written
        in the form $ U_h (A_t) $ (respectively
        $ \mbox{Fun}_t (G_{h}^{*}) ) $ that reveals two canonical
        dual classical limits:
\begin{equation}
\begin{array}{cc}
\intt{5}  U_h (A_t) \intt{5} & \\
 \raisebox{2mm}{\mbox{\scriptsize $h \rightarrow 0$}}  \mbox{\LARGE $\swarrow$} \intt{7} \Updownarrow \intt{7}  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $t \rightarrow 0$}} & \\
\intt{3}  U(A_t) \intt{3} \intt{10}  \mbox{Fun}_h(G_t) \intt{10}  \mbox{Fun}(G^*_h) & \approx  U_h(\mbox{Ab}) \\
\Updownarrow \intt{8}  \mbox{\LARGE $\swarrow$} \intt{20}  \mbox{\LARGE $\searrow$} \intt{8} \Updownarrow &  \\
\intt{2}  \mbox{Fun}(G_t) \intt{38}  \mbox{Fun}_h ({\cal AB}) & \approx  U(A^*_h)
\end{array}                 \label{e14}
\end{equation}

        All the reasoning is invariant with respect to
        interchange $ A \rightleftharpoons A^* $.
        So the set $ \{ \mbox{Fun} (G_{h}^{*}) \} $ can be also
        considered as a straight line in the orbit
        $ \mbox{Orb} (\mbox{Fun}(G^*)) $ -- the trivial contraction of
        the group $ G^* $ into the abelian additive vector
        group $ {\cal AB} $. The lines $ \{ U (A_t) \} $ and
        $ \{ \mbox{Fun} (G_{h}^{*}) \} $ intersect in the point
                $ U(\mbox{Ab}) \approx \mbox{Fun} ({\cal AB}) $.

The conclusion can be formulated as follows.
\newtheorem{propo}{Proposition}
\begin{propo}
If $ U_p(A)$ is a deformation quantization of a Lie bialgebra $(A,A^*)$
with the properties (a)-(b) then in ${\cal H}$ there exists an analytic
submanifold $Q(A,A^*)$ (respectively $Q^*(A,A^*)$).It can be globally
parametrised by co-ordinates $(h,t)$. The trivial contraction lines
$ U(A_t)$ and $ \mbox{Fun}(G^*_h)$ (respectively $ U(A^*_h)$ and $ \mbox{Fun}(G_t)$) together with
their intersection point $U(\mbox{Ab}) =  \mbox{Fun} ({\cal AB}) $ form a boundary of
$Q(A,A^*)$ (respectively $Q^*(A,A^*)$) and supply its elements with
the dual classical limits:
         \begin{equation}
         \left.
         \begin{array}{l}
         lim_{h \rightarrow 0} H_{h,t} = U (A_t)  \\
         lim_{t \rightarrow 0} H_{h,t} = \mbox{Fun} (G_{h}^*)
         \end{array}
         \right\} \mbox{for} \, \, Q            \label{e15}
         \end{equation}
         and
         \begin{equation}
         \left.
         \begin{array}{l}
         lim_{h \rightarrow 0} H_{h,t}^{*} = \mbox{Fun} (G_t)  \\
         lim_{t \rightarrow 0} H_{h,t}^{*} = U (A_{h}^{*})
         \end{array}
         \right\} \mbox{for} \, \, Q^* .        \label{e16}
         \end{equation}
\end{propo}
        Suppose now that
        \begin{enumerate}
        \item $ t $ and $ h $ parametrize
        the intersecting trivial contraction lines
        $ U (A_t) $ and $ \mbox{Fun} (G_{h}') $ in the orbits
        $ \mbox{Orb} (U(A)) $ and $ \mbox{Orb} (\mbox{Fun}(G')) $
        (respectively
        $ \mbox{Orb} (\mbox{Fun}(G)), \mbox{Orb} (U(A')) $ )
        for a certain pair of inequivalent algebras $ (A,A^{'}) $
        of equal dimention and corresponding universal covering
        Lie groups $G$ and $G'$.
        The intersection
        point coincides with the contraction limit.
        \item In the variety
        $ {\cal H} $ of Hopf algebras with the generators
        $ \{ a_i, {\bf 1} \} $ ($ \{ a_i \} $ is the basic of
        $ A $) there exists the analytical 2-dimensional
        subvariety $ Q (A,A') $ (respectively
        $ Q^* (A,A') $) of Hopf algebras $ H $ and the
        disunqued union $ U(A_t) \cup \mbox{Fun} (G_{h}') $
        is the boundary of $ Q(A,A') $.
        \end{enumerate}
It is easy to check the validity of the following statement
\begin{propo}
If for algebras $ A $ and $ A' $ the conditions (1)-(2) are fulfilled
-- they are dual,
Hopf algebras $ H \in Q(A,A') $ are the deformation
         quantizations of the  Lie bialgebra
         $ (A,A' \approx A^* ) $ and the contraction curves
$ U(A_t)$ and $ \mbox{Fun}(G^*_h)$ supply the dual classical limits to the points of
$Q(A,A^*)$.
\end{propo}

         It is possible to invert the dual list $ Q^* $
         with respect to $ Q $ so that the dual limits
         will refer to dual algebras (respectively groups).
         This is due to the fact  that in the
         $ (h,t) $ - co-ordinates the Hopf algebras
         $ H_{h,t}^{*} $ and $ H_{t,h}^{*} $ are equivalent,
         they are characterised by the same parameter
         $ p = ht $ (see (\ref{e7})) and are connected
         by the transformation $ B^* (h/t) $.
        Thus one can always introduce the transformed nondegenerate
        bilinear form $ < , >_{h/t} $ such that the parameters
        on the list $ Q^* $ will be interchanged with respect to
        $ Q $,
        \begin{equation}
        H_{h,t} \stackrel{ <>h/t }{ \Longleftrightarrow } H_{t,h}^* .
                                        \label{e17}
        \end{equation}
        Now the correlation between the canonical classical limits
        for $ Q $ and $ Q^* $ will differ from
        that described by the diagram (\ref{e14}):
\begin{equation}
\begin{array}{c}
\intt{5}  U_h (A_t) \intt{5}  \\
 \raisebox{2mm}{\mbox{\scriptsize $h \rightarrow 0$}}  \mbox{\LARGE $\swarrow$} \intt{2}
\Updownarrow <> \mbox{\scriptsize $h/t$}  \intt{2}  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $t \rightarrow 0$}}  \\
\intt{4}  U(A_t) \intt{2}
\intt{3} \Updownarrow \intt{3}  \mbox{Fun}_t(G_h) \intt{3} \Updownarrow \intt{3}
 \mbox{Fun}(G^*_h)  \\
  \raisebox{2mm}{\mbox{\scriptsize $h \rightarrow 0$}}  \mbox{\LARGE $\swarrow$} \intt{20}  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $t \rightarrow 0$}} \\
\intt{3}  U(A^*_t) \intt{37}  \mbox{Fun}(G_h)
\end{array}                      \label{e18}
\end{equation}
        Note that here in contrast to (\ref{e14}) the duality holds only
        for points of $ Q $ and $ Q^* $.

\subsection{Example: $ U_h(sl_t(2)) \approx
        \mbox{Fun}_t (\widetilde{E}_h(2))$ }    \label{ex22}
        Consider the standard quantum algebra $ U_p(sl(2,{\bf C})) $:
        \begin{equation}
        \begin{array}{lll}
        \Delta L = L \otimes {\bf 1} + {\bf 1} \otimes L, & \rule{5mm}{0mm} &
        [L,M] = M,                                        \\
        \Delta N = N \otimes {\bf 1} + e^{-2pL} \otimes N, &  &
        [L,N] = -N,                                       \\
        \Delta M = M \otimes e^{2pL} + {\bf 1} \otimes M, &  &
        [M,N] = 2 \frac{\sinh 2pL}{1 - e^{-2p}}.
        \end{array}                     \label{e19}
        \end{equation}
        Applying the transformation $ B(t) $ and introducing the new
        co-ordinates $ (h=p/t, t) $ (see (\ref{e7})) we get the canonical
        form for Hopf algebras of the variety $ Q(sl(2), \widetilde{e(2)})$:
        \begin{equation}
        \begin{array}{lll}
        \Delta L = L \otimes {\bf 1} + {\bf 1} \otimes L, & \rule{5mm}{0mm} &
        [L,M] = tM,                                        \\
        \Delta N = N \otimes {\bf 1} + e^{-2hL} \otimes N, & &
        [L,N] = -tN,                                       \\
        \Delta M = M \otimes e^{2hL} + {\bf 1} \otimes M, & &
        [M,N] = 2t^2 \frac{\sinh 2hL}{1 - e^{-2ht}}.
        \end{array}                     \label{e20}
        \end{equation}
        The dual classical limits are
        \begin{equation}
        \begin{array}{c}
        \lim_{h \rightarrow 0} U_h(sl_t(2)) = U(sl_t(2)), \\
        \lim_{t \rightarrow 0} U_h(sl_t(2)) = \mbox{Fun} (\widetilde{E_h(2)}).
        \end{array}                     \label{e21}
        \end{equation}
        Here $ \widetilde{E(2)} $ is the group of flat motions with
        the quasiorthogonal rotation.

        The canonical dualization produces the variety $ Q^*(sl(2),
        \widetilde{e(2)}) $ whose points can be interpreted as
        $ \mbox{Fun}_h(SL_t(2)) $ and the parametrization is as follows:
        \begin{equation}
        \begin{array}{rcl}
       \rule{1mm}{0mm} [\lambda,\mu] & = & -2h \mu, \\
       \rule{1mm}{0mm} [\lambda,\nu] & = & -2h \nu, \\
       \rule{1mm}{0mm} [\mu, \nu]    & = & 0,       \\
        \Delta \mu    & = & {\bf 1} \otimes \mu + ( \mu \otimes e^{-t\lambda})
                ({\bf 1} \otimes {\bf 1} + t^2 \mu \otimes \nu )^{-1} \\
        \Delta \nu    & = & \nu \otimes {\bf 1} + ( e^{-t\lambda} \otimes \nu)
                ({\bf 1} \otimes {\bf 1} + t^2 \mu \otimes \nu )^{-1} \\
        \Delta \lambda &= & \lambda \otimes {\bf 1} + {\bf 1} \otimes \lambda
        -4ht\sum_{n=1}^{\infty} t^{2n-1}\frac{(- \mu \otimes \nu )^n }{ 1-
        e^{-2nht}}.
        \end{array}                     \label{e22}
        \end{equation}
        We have used here the realization of $ \mbox{Fun}_h (SL(2)) $
        obtained in \cite{CELL}. On the dual list $ Q^* $ the classical
        limits are
        \begin{equation}
        \begin{array}{c}
        \lim_{h \rightarrow 0} \mbox{Fun}_h(SL_t(2)) = \mbox{Fun}(SL_t(2)), \\
        \lim_{t \rightarrow 0} \mbox{Fun}_h(SL_t(2)) = U (\widetilde{e_h(2)}).
        \end{array}                     \label{e23}
        \end{equation}
        The first of these two limits is treated here as
        the formal series group whose generic elements are
        the basic exponential co-ordinate functions on
        $ SL_t(2) $ in the neighbourhood of unit.

        Using the equivalence transformation $ B^*(h/t) $ on $ Q^* $
        the parametric dualization can be defined:
        \begin{equation}
        <L,\lambda>_{h/t} =<M,\mu>_{h/t} =<N,\nu>_{h/t} = h/t.
                                        \label{e24}
        \end{equation}
        It inverts the dual list $ Q^* $ with respect to $ Q $
        and shows explicitly that quantum algebras of $A$ and $A^*$
        as well as quantum groups of $G$ and $G^*$ form the dual pairs of
        Hopf algebras
        (see (\ref{e18})):
\begin{equation}
\begin{array}{c}
\intt{5}  U_h (sl_t(2)) \intt{5}    \\
 \raisebox{2mm}{\mbox{\scriptsize $h \rightarrow 0$}}  \mbox{\LARGE $\swarrow$} \intt{8}
\Updownarrow <> \mbox{\scriptsize $h/t$}
\intt{8}  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $t \rightarrow 0$}} \\
\intt{4} U(sl_t(2)) \intt{1} \Updownarrow
\intt{4} \mbox{Fun}_t(SL_h(2)) \intt{4}  \Updownarrow
\mbox{Fun} (\widetilde{E_h(2)})   \\
 \raisebox{2mm}{\mbox{\scriptsize $h \rightarrow 0$}}  \mbox{\LARGE $\swarrow$} \intt{30}  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $t \rightarrow 0$}}   \\
\intt{6} U(\widetilde{e_t(2)}) \intt{44} \mbox{Fun} (SL_h(2))
\end{array}                      \label{e25}
\end{equation}

        In the Appendix the 2-parametric variety $ Q $ for quantum algebras
        $ U_h(sl(n,{\bf C})) $ is explicitly described.

        \section{Quantization of Hopf pairs}
        \subsection{ } \label{sub31}
        The geometric description of the quantum duality properties of the
        deformation quantization given by the criteria (1)-(2) leads to the
        natural generalisation of this notion.

        Let us call the deformation quantization of the Hopf pair
        $ (H_{(\theta, 0)},H_{(0,\tau)}) $ the following construction in
        the variety $ {\cal H} $ of Hopf algebras with the generators
        $ \{ a_i, {\bf 1} \} $ : \\
        (1') $ \theta $ and $ \tau $ parametrize the intersecting
        contraction curves $ H_{(\theta,0)} $ and $ H_{(0,\tau)} $ such that
        in the first order the coproducts of generators remain undeformed
        in $ H_{(\theta,0)}$ and the products of generators
        -- in $ H_{(0, \tau)}$;
        the intersection point $ H_{(0,0)} $
        coincides with the contraction limit for both
        $ H_{(\theta, 0)} $ and $ H_{(0,\tau)} $. \\
        (2') In the variety $ {\cal H} $ there exists the analitic
2-dimensional
        manifold $ Q(H_{(\theta, 0)},H_{(0,\tau)}) $ (respectively
        $ Q^* (H_{(\theta, 0)},H_{(0,\tau)}) $ of (noncommutative and
        noncocommutative) Hopf algebras $ H_{(\theta,\tau)} $ . The
        disjoint union
        $ H_{(\theta, 0)} \bigcup H_{(0,\tau)} \bigcup H_{(0,0)} $
        is the boundary of $ Q(H_{(\theta, 0)},H_{(0,\tau)}) $.
        (Respectively
        $ H^*_{(\theta, 0)} \bigcup H^*_{(0,\tau)} \bigcup H_{(0,0)} $ --
        the boundary of $ Q^* $ .) \\
        Then it is easy to check that on the curves
        $ H_{(\theta,0)} $ and $ H_{(0,\tau)} $
        the smooth vector fields of the first deforming functions
        $ V(\theta) $ and $ W(\tau) $ exist such that the limit vectors
        \begin{equation}
        \lim_{\theta \rightarrow 0} V(\theta) \equiv V(0), \label{e26}
        \end{equation}
        \begin{equation}
        \lim_{\tau \rightarrow 0} W(\tau) \equiv W(0), \label{e27}
        \end{equation}
        are the first deformation functions for $H_{(0,0)}$
        in the direction of $ H_{(0,\tau)} $ and $ H_{(\theta,0)} $
        respectively.

        To connect this purely geometric construction with the usual
        deformation quantization picture it is natural to formulate
        the additional condition. \\
        (3') In $ {\cal H} $ there exists a smooth 1-parametric family
        $ Q_{\varepsilon} $ of varieties
        $ Q(H_{(\theta, 0)},H_{(0,\tau)}) $ such that
        its limit
        \[
        \lim_{\varepsilon  \rightarrow 0} Q_{\varepsilon} \equiv
                 Q_0(H_{0(\theta, 0)},H_{0(0,\tau)})
        \]
        satisfies the conditions (1)-(2).

        The main difference with the canonical case is that the
        intersection point algebra $ H_{\varepsilon} (0,0) $ may be
        noncommutative and noncocommutative for $ \varepsilon \neq 0 $.
        From this point of view
        the canonical dualization is the dualization with respect to
        the Abelian and coAbelian Hopf algebra. While in the
        quantization of a Hopf pair no such restrictions on
        $ H_{\varepsilon} (0,0) $ are imposed.
        The parameters
        $\theta$ and $\tau$ are dual with respect to the common limit
        $ H_{0,0} $ of the boundary curves $ H_{(\theta,0)} $
        and $ H_{(0,\tau)} $.
        The pair of vector fields
        $ (V_{\varepsilon}(\theta),W_{\varepsilon} (\tau)) $
        plays here the role of a Lie-bialgebra.
        In the limit $ \varepsilon \rightarrow 0 $ the pair
         $ (V_0(0), W_0(0)) $
        becomes a Lie-bialgebra.

        The existence of the deformation quantization of a Hopf pair
        is tightly connected with the contraction properties of Hopf
        algebras. In \cite{BAL} it was demonstrated how the deformation
        parameters  of the quantum group can be dualized with the
        quantization parameters of the corresponding quantum algebra.
        The algebra $ U_{\phi} {\cal G}_{0,k_2} $ was constructed to
        illustrate this effect:
        \begin{equation}
        \begin{array}{l}
       \rule{1mm}{0mm} [P_3,P_1] =
        \frac{e^{i \phi P_2} - e^{-i \phi P_2}}{2i \phi}, \\
       \rule{1mm}{0mm} [P_3,P_2] = -k_2 P_1, \\
       \rule{1mm}{0mm} [P_1,P_2] = 0.
        \end{array}                     \label{e28}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \Delta(P_{1,3})& = &e^{-i \phi P_2 /2} \otimes P_{1,3} +
                        P_{1,3} \otimes  e^{i \phi P_2 /2}, \\
        \Delta(P_2)& = & P_2 \otimes {\bf 1} + {\bf 1} \otimes P_2.
        \end{array}                             \label{e29}
        \end{equation}
        The authors emphasise that in the dual Hopf algebra
        $ (U_{\phi} {\cal G}_{0,k_2})^* $ the contraction parameter
        $ k_2 $ measures the deformation of coproduct and thus obtain the
        features of the quantization parameter.

        From our point of view it must be also stressed that in the limit
        \[
        \lim_{k_2 \rightarrow 0} U_{\phi} {\cal G}_{(0,k_2)} =
        U_{\phi} {\cal G}_{(0,0)}
        \]
        we do not get the classical algebra of functions on the dual
        group . The co-ordinate functions remain noncommutative. So the
        parameters $ k_2 $ and $ \phi $ are not canonically dual.

        The situation becomes clear in terms of
        quantization of a Hopf pair. The contraction curves
        $ \mbox{Fun}_{\phi} (G^*_{\phi} ) \equiv H_{(0,\phi)} $ and
        $ U {\cal G}_{(0,k_2)} \equiv H_{(k_2,0)} $
        have the common limit
        $ U {\cal G}_{(0,0)} \equiv H_{(0,0)} $. Here $H_{(0,0)}$ is
        the Hiesenberg algebra. The union
        $ \mbox{Fun}_{\phi} (G^*_{\phi} ) \bigcup U {\cal G}_{(0,k_2)} $
        form the boundary of the 2-dimensional variety
        $ Q(H_{(k_2,0)},H_{(0,\phi)}) $ . The vector fields $ V $ and $W$
        are trivial in the sense that their projections
$ V_{\downarrow {\cal G} \wedge {\cal G} \rightarrow {\cal G}} $
        and
$ W_{\downarrow {\cal G} \rightarrow {\cal G} \wedge {\cal G}} $
do not depend on
        $ k_2 $ and $ \phi $ respectively. The first deforming function
        \[
        [P_3,P_2]_{(1)} = -P_1
        \]
        with the only nonzero co-ordinate $ V^1_{[3,2]} = -1 $ generates
        \[
        V(\phi) = V(0)
        \]
        and the first deforming function
        \[
        \Delta_{(1)} P_{1,3} = -i/2(P_2 \otimes P_{1,3} - P_{1,3} \otimes
                                P_2 )
        \]
        with nontrivial co-ordinates $ W_1^{[2,1]} = -i/2, \, \,
        W_3^{[2,3]} = -i/2 $ generates the field
        \[
        W(k_2) = W(0).
        \]
        Notice that here the pair
$ (V_{\downarrow {\cal G} \wedge {\cal G} \rightarrow {\cal G}},
 W_{\downarrow {\cal G} \rightarrow {\cal G} \wedge {\cal G}}) $
is just a Lie-bialgebra.
        This fact is tightly connected with the possibility to fulfil
        the condition (3'). The family $ Q_{\varepsilon} $ is
        obtained by a simple transformation of generators
        \[
        P_{1,3} = p_{1,3}/ \varepsilon, \, \, \, P_2 = p_2
        \]
        that doesn't touch the tangent vectors $(V,W)$.
        In terms of new generators $ p_i$ the left-hand side of the first
        commutator in (\ref{e28}) acquires the multiplier $ \varepsilon^2 $
        and in the limit $ \varepsilon \rightarrow 0 $ the 2-dimensional
        subvariety $ Q_0 $ will have all the properties characteristic
        to the canonical deformation quantization scheme (see
        the conditions (1)-(3)):
        \begin{equation}
        \begin{array}{llcl}
       \rule{1mm}{0mm} [p_3,p_1] = 0, &
        \Delta(p_{1,3})& = &e^{-i \phi p_2 /2} \otimes p_{1,3} +
                        p_{1,3} \otimes  e^{i \phi p_2 /2}, \\
       \rule{1mm}{0mm} [p_3,p_2] = -k_2p_1, &
        \Delta(p_2)& = & p_2 \otimes {\bf 1} + {\bf 1} \otimes p_2, \\
       \rule{1mm}{0mm} [p_1,p_2] = 0. & & &
        \end{array}                             \label{e30}
        \end{equation}
        We come to the conclusion that the variety described by
        (\ref{e28}) and (\ref{e29}) is the deformation
        of a Hopf pair $ (H_{(k_2,0)},H_{(0,\phi)}) $. The parameters $ \phi $
        and $k_2 $ are dual with respect to $ H_{\varepsilon (0,0)} $
        -- the universal enveloping algebra of the Hiesenberg algebra.
        In the limit $ \varepsilon \rightarrow 0 $ they become
        canonically dual.

        It is obviously clear that considering $ U_{\phi} {\cal G}_{(0,k_2)} $
        with fixed $ k_2 $ just as the canonical deformation quantization
        of $ U {\cal G}_{(0,k_2)} $, one can easily find the parameter $ \psi $
        canonically dual to $ \phi $. The standard procedure described in
        section 1 leads to the following variety $ Q( {\cal G}_{(0,k_2)},
        \widetilde{e(2)}) $,
        \begin{eqnarray*}
        \rule{1mm}{0mm} [P_3,P_1] & = & \frac{\psi}{2i \phi}
                                     (e^{i \phi P_2} - e^{-i \phi P_2}), \\
        \rule{1mm}{0mm} [P_3,P_2] & = & - \psi k_2 P_1, \\
        \rule{1mm}{0mm} [P_1,P_2] & = & 0,  \\
        \Delta P_{1,3} & = & e^{-i/2 \phi P_2} \otimes P_{1,3} +
                           P_{1,3} \otimes e^{i/2 \phi P_2}, \\
        \Delta P_2 & = & P_2 \otimes {\bf 1} + {\bf 1} \otimes P_2.
        \end{eqnarray*}
        In the considered example of the quantization of a Hopf pair
        one of the contraction curves
        -- $ H_{\varepsilon (k_2,0)} $ -- remains canonical
        (with a primitive coproduct) even for $ \varepsilon \neq 0 $.
        The reason is that the contraction described by the parameter
        $ k_2 $ does not touch the coproducts of basic coordinate functions.
        In terms of the quantization of a Hopf pair this means
        that only one of the contractions, namely the $ \mbox{Fun}
        ( G^*_{\phi} ) $ is "lifted" by the $\varepsilon$-deformation
        nontrivially. In the subsection \ref{ntexamp} we demonstrate
        an example
        where both contraction lines are nontrivially lifted.

        \subsection{Lie-Poisson structures in a case of quantized pair}
        \label{LPstr}
        In the deformation quantization of a Lie bialgebra the
        quantum algebra $ U_h(A_t) $ refers to the initial
        Lie-Poisson structure as to the Poisson-Hopf algebra
        $ (U(A_t), A^*) $, where the Poisson comultiplication is
        described by a Lie structure of $ A^* $. In construction
        of a Hopf algebra $ U_h(A_t) $ the comultiplication is
        given a preference -- it is quantized first. The deformations
        of commutation relations play an auxiliary role.
        It starts when the first order deformation with respect to
        parameter $h$ is already performed in $ \Delta_{\downarrow A} $'s
        by $ A^* $.

        Considering $ U_h(A_t) $ as a quantum group
        $ \mbox{Fun}_t(G^*_h) $ one obtains the dual scenario.
        Now one has a Lie-Poisson group $ (\mbox{Fun}(G^*_h),A) $.
        The multiplication is quantized first and the deformation
        of the costructure is auxiliary. For example, when the group
        $ G^*_h $ is Abelian the noncommutative co-ordinates can
        always be introduced without any deformation of $ G^*_h $.

        Having the applications in mind one can consider such a
        Lie-Poisson structure $ A^* $ on $ \mbox{Fun}(G_t) $
        that do not form a Poisson-Hopf algebra with $ U(A) $,
        but nevertheless the simultaneous Hopf deformations of both
        multiplication and comultiplication exist. In this case
        the first order deforming function of $ U(A_t) $ will
        have the nontrivial multiplication constituents, that
        naturally may not form a Lie algebra themselves. They
        depend on $t$ and tend to zero when $ t \rightarrow 0 $
        if $t$ is still dual to $h$ with respect to $ U(\mbox{Ab}) $.
        This means that the desired Poisson properties can be
        imposed on the group $G$ only together with the symmetry
        deformation.
        In other words the Poisson structure
        on $ \mbox{Fun} (G_t) $ defined by $ A'$ does not perform
        the group $ G_t $ into a Lie-Poisson group but does it for a
        certain infinitesimal deformation $ (G_t)_h $.
        From the Hopf pair quantization point of view this case
        is based on such a deformation quantization where the field
        $ V(t) $ ( or both $ V(t) $ and $ W(h) $ ) has not only
        comultiplicative but also multiplicative nontrivial parts. The Lie
        bialgebra is reobtained in the limit $ t,h \rightarrow 0 $
        (see (\ref{e26}), (\ref{e27})).

        It must be noticed that the canonical scheme is stable
        (in the sense of generalisation described above)
        for semisimple algebras $A$ (or $A^*$).
        For them a first deforming function can be always set to
        zero by a similarity transformation of $A$. But for
        physical applications the nonsemisimple algebras play an
        important role and for them a deformation with a nontrivial
        first order may exist.

        The other possibility is to consider deformations of
        quantum algebras induced by deformations of classical ones
        \cite{BAL}. The example considered in the \ref{sub31}
        illustrates this case. There the Hopf algebra $ H_{(0,0)} $
        is nontrivial but classical Lie universal enveloping.
        In general the pair of deforming functions $ (V(0),W(0)) $
        must not form a Lie bialgebra in such a case.
        The Lie bialgebra structure must be obtained as a limit
        for $ \varepsilon \rightarrow 0 $ if the condition (3')
        is fulfilled.
        This is the case when the Poisson structure that form a
        Poisson-Hopf algebra with $A$ and gives rise to a global
        deformation is applicable also for a classical deformation
        $A_t$ with the analogous results.

        The general case described by the conditions (1')-(3')
        can be considered as a combination of these two possibilities.
        Here instead of Poisson structures on Lie group $G$ a first
        deforming function of some initial Poisson structure on $G$ appears.
        And instead of constructing a Lie structure for the initial
        multiplication described by $A$ and supposed to become Poisson
        we see a deformation problem for some already existing
        Lie structure, compatible with $A$. The details of this
        situation are explicitly demonstrated in the following subsection.

        \subsection{Example of nontrivial lifting}  \label{ntexamp}

        To obtain a nontrivial lifting for both multiplications and
        comultiplications one must chose quantum algebras that have
        nontrivial deformations. A favourable situation may
        be found in case of Hopf algebras obtained as Drinfeld doubles
        \cite{Drin}.

        Let us construct the double $D$ for the Hopf algebras
        $ U_h (sl_t(2,{\bf C})) $ and $ \mbox{Fun}_h(\widetilde{E_t(2)}) $
        (see (\ref{e20},\ref{e22})). To make the compositions more
        transparent The modified notations for the basic elements,
        \[
        \begin{array}{c}
        \{ L,M\intt{2},N\intt{2},\intt{1}\lambda,
        \intt{1}\mu\intt{2},\nu\intt{3} \} \\
        \Downarrow \\
        \{ L,X_+,X_-,H,Y_+,Y_- \}
        \end{array}
        \]
        make the compositions of the double more transparent:
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [L,X_{\pm}] & = & \pm h X_{\pm}, \\
        \rule{1mm}{0mm} [X_+,X_-] & = & h^2 \frac{e^{tL} - e^{-tL} }
        {1- e^{-th}} ,
        \end{array}                     \label{e31}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [H,Y_{\pm}] & = & -t Y_{\pm}, \\
        \rule{1mm}{0mm} [Y_+,Y_-] & = & 0 ,
        \end{array}                     \label{e32}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [H,L] & = & 0, \\
        \rule{1mm}{0mm} [H,X_{\pm}] & = & tX_{\pm} \pm
        \frac{2th^2 }
        {1- e^{-th}} Y_{\mp} , \\
        \rule{1mm}{0mm} [Y_{\pm},L] & = & \pm h Y_{\pm}, \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\pm}] & = & ( \mp e^{\pm tL}
        \pm e^{-hH} ) \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\mp}] & = & \pm h^2 Y^2_{\pm}
        +( e^{-th} - 1 ) X_{\mp} Y_{\pm},
        \end{array}                     \label{e33}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \Delta L & = & L \otimes {\bf 1} + {\bf 1} \otimes L, \\
        \Delta X_+ & = & X_+ \otimes e^{tL}
        + {\bf 1} \otimes X_+, \\
        \Delta X_- & = & X_- \otimes {\bf 1}
        + e^{-tL} \otimes X_-,
        \end{array}                     \label{e34}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \Delta H & = & H \otimes {\bf 1} + {\bf 1} \otimes H
        -2t \sum_{n=1}^{ } \frac{(-h^2 (Y_- \otimes Y_{+}))^n}{1 - e^{-nth}},
\\
        \Delta Y_+ & = & Y_+ \otimes {\bf 1}
        + (e^{-hH} \otimes Y_+)({\bf 1} \otimes {\bf 1}
        + h^2 Y_{-} \otimes Y_{+})^{-1}, \\
        \Delta Y_- & = & {\bf 1} \otimes Y_-
        + (Y_{-} \otimes e^{-hH})({\bf 1} \otimes {\bf 1}
        + h^2 Y_{-} \otimes Y_{+})^{-1}.
        \end{array}                     \label{e35}
        \end{equation}
        The costructure of this quantum algebra describes the direct
        product $ \widetilde{E_t(2)} \times SL_h(2,{\bf C}) $.

        Consider a pair of new parameters $ ( \tau, \theta ) $ that describe
        the contractions
        \begin{equation}
        \begin{array}{rcl}
        SL_h(2,{\bf C}) &
        \stackrel{\mbox{\scriptsize
        contract}}{\tau \longrightarrow 0} & E_h(2,{\bf C}) \\
        sl_h(2,{\bf C}) &
        \stackrel{\mbox{\scriptsize
        contract}}{\theta \longrightarrow 0} & e_h(2,{\bf C})
        \end{array}                     \label{e36}
        \end{equation}
        lifted to the variety ${\cal H}$ of quantum algebras with six
        generators. After the necessary re\-para\-met\-ri\-za\-tion of the type
        described in section 1 we get the two-dimensional family
        $ D_{\tau ,\theta} $ (on this stage of construction the
        parameters $t$ and $h$ may be fixed).
        The elements of this variety differ from the initial Hopf algebra
        $D$ in the following compositions:
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [X_+,X_-] & = & \tau \theta^2 h^2
        \frac{e^{tL} - e^{-tL} }{1- e^{-th}} ,    \\
        \rule{1mm}{0mm} [H,X_{\pm}] & = & tX_{\pm} \pm  \tau \theta
        \frac{2th^2 }{1- e^{-th}} Y_{\mp} , \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\pm}] & = & \theta (
        \mp e^{\pm tL} \pm e^{-hH} ) \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\mp}] & = &
        \pm \tau \theta  h^2 Y^2_{\pm}
        +( e^{-th} - 1 ) X_{\mp} Y_{\pm},
        \end{array}                     \label{e37}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \Delta H & = & H \otimes {\bf 1} + {\bf 1} \otimes H
        -2t \sum_{n=1}^{ }
        \frac{(-\tau h^2 (Y_- \otimes Y_{+}))^n}{1 - e^{-nth}}, \\
        \Delta Y_+ & = & Y_+ \otimes {\bf 1}
        + (e^{-hH} \otimes Y_+)({\bf 1} \otimes {\bf 1}
        + \tau h^2 Y_{-} \otimes Y_{+})^{-1}, \\
        \Delta Y_- & = & {\bf 1} \otimes Y_-
        + (Y_{-} \otimes e^{-hH})({\bf 1} \otimes {\bf 1}
        + \tau h^2 Y_{-} \otimes Y_{+})^{-1}, \\
        \end{array}                     \label{e38}
        \end{equation}
        The smooth subvariety $ D_{\theta , \tau} $ contains two nontrivial
        contraction curves $ D_{\theta , 0} $ and $ D_{0, \tau} $
        having the noncommutative and noncocommutative Hopf algebra
        $ D_{0,0} $ as the common limit:
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [L,X_{\pm}] & = & \pm h X_{\pm}, \\
        \rule{1mm}{0mm} [H,Y_{\pm}] & = & -t Y_{\pm}, \\
        \rule{1mm}{0mm} [H,X_{\pm}] & = & tX_{\pm}, \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\mp}] & = &
        ( e^{-th} - 1 ) X_{\mp} Y_{\pm},
        \end{array}                     \label{e39}
        \end{equation}
        (All other pairs of generators commute.)
        \begin{equation}
        \begin{array}{rcl}
        \Delta L & = & L \otimes {\bf 1} + {\bf 1} \otimes L, \\
        \Delta X_+ & = & X_+ \otimes e^{tL}
        + {\bf 1} \otimes X_+, \\
        \Delta X_- & = & X_- \otimes {\bf 1}
        + e^{-tL} \otimes X_-,
        \end{array}                     \label{e40}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        \Delta H & = & H \otimes {\bf 1} + {\bf 1} \otimes H, \\
        \Delta Y_+ & = & Y_+ \otimes {\bf 1}
        + e^{-hH} \otimes Y_+, \\
        \Delta Y_- & = & {\bf 1} \otimes Y_-
        + Y_{-} \otimes e^{-hH}.
        \end{array}                     \label{e41}
        \end{equation}
        The picture of the smooth subvariety $ D_{\theta , \tau} $
        and its boundary looks like follows:
\begin{equation}
\begin{array}{c}
 \mbox{\LARGE $\searrow$} \intt{23} D_{\theta' ,\tau'} \intt{23}  \mbox{\LARGE $\swarrow$}  \\
 \raisebox{2mm}{\mbox{\scriptsize $\theta \rightarrow 0$}}  \mbox{\LARGE $\swarrow$} \intt{15}  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $\tau \rightarrow 0$}} \\
         D_{0,\tau} \intt{35}  D_{\theta ,0} \\
\raisebox{-2mm}{ \raisebox{2mm}{\mbox{\scriptsize $\tau \rightarrow 0$}}}
 \mbox{\LARGE $\searrow$} \intt{15}  \mbox{\LARGE $\swarrow$}
\raisebox{-2mm}{ \raisebox{2mm}{\mbox{\scriptsize $\theta \rightarrow 0$}}} \\
        D_{0,0}
\end{array}                      \label{e42a}
\end{equation}
        The points of the curve $ D_{(\theta ,0)} $ differ from that of
        $ D_{(0,0)} $ by a single commutator
        \begin{equation}
        [Y_{\pm}, X_{\pm}] = \theta (\mp e^{\pm tL} \pm e^{-hH}).
                                        \label{e42}
        \end{equation}
        For the contraction curve $ D_{(0,\tau)} $ the coproducts
        $ \Delta H, \Delta Y_{\pm} $ have the initial form
        (\ref{e34},\ref{e38})
        while the multiplication coincides with that of $D_{(0,0)}$
        (see (\ref{e39})).
        The vector field $ V(\theta) $ is described by the deformation
        function:
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [X_+,X_-]_{(1)} & = & \theta^2 h^2
        \frac{e^{tL} - e^{-tL} }{1- e^{-th}} ,    \\
        \rule{1mm}{0mm} [H,X_{\pm}]_{(1)} & = & \pm \theta
        \frac{2th^2 }{1- e^{-th}} Y_{\mp} , \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\mp}]_{(1)} & = &
        \pm \theta  h^2 Y^2_{\pm}
        +( e^{-th} - 1 ) X_{\mp} Y_{\pm},
        \end{array}                     \label{e43}
        \end{equation}
        \begin{equation}
        \begin{array}{rcl}
        (\Delta - \Delta^{\mbox{\scriptsize opp}})_{(1)} H  &
        = & \frac{2th^2}{1-e^{-th}}
        (Y_- \otimes Y_+ - Y_+ \otimes Y_-), \\
        (\Delta - \Delta^{\mbox{\scriptsize opp}})_{(1)} Y_{\pm} & = & \mp h^2
[
        (e^{-hH} \otimes Y_{\pm})(Y_{\mp} \otimes Y_{\pm}) -
        (Y_{\pm} \otimes e^{-hH})(Y_{\pm} \otimes Y_{\mp})].
        \end{array}                     \label{e44}
        \end{equation}
        In the vector field $ W(\tau) $ only some antisymmetric
        multiplication structure constants on $A \bigwedge A$
        are different from zero,
        \begin{equation}
        \begin{array}{rcl}
        \rule{1mm}{0mm} [H,X_{\pm}] & = &  \pm  \tau
        \frac{2th^2 }{1- e^{-th}} Y_{\mp} , \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\mp}] & = &
        \pm \tau h^2 Y^2_{\pm},
        \end{array}                     \label{e45}
        \end{equation}
        \begin{equation}
        \rule{1mm}{0mm} [Y_{\pm},X_{\pm}]  =
        (\mp e^{\pm tL} \pm e^{-hH} ) .  \label{e46}
        \end{equation}
        Both $ V(\theta) $ and $ W(\tau) $ nontrivially depend on the
        co-ordinates of contraction curves. The limit values $ V(0) $
        and $ W(0) $  are defined by the relations (\ref{e44}) and
        (\ref{e46}) respectively.

        To complete the construction we must find such an
        $\varepsilon$-dependent family $ D_{\varepsilon} $
        of 2-dimen\-sion\-al varieties that in the limit $ \varepsilon
        \rightarrow 0 $ satisfies the canonical conditions (1)-(2).
        This may be done using the parameters $h$ and $t$.
        In the structure relations for $ D_{\theta ,\tau} $
        we perform the substitution:
        \begin{equation}
        h \Rightarrow \varepsilon^2, \, \, \,
        t \Rightarrow \varepsilon^2, \, \, \,
        X_{\pm} \Rightarrow \varepsilon X_{\pm}, \, \, \,
        Y_{\pm} \Rightarrow \varepsilon Y_{\pm}
                                        \label{e47}
        \end{equation}
        and obtain in the limit $ \varepsilon \rightarrow 0 $ the following
        compositions. For the internal points of $ D_{\varepsilon
         (\theta ,\tau)} $:
        \begin{equation}
        D_{ 0( \theta ,\tau)} \left\{ \begin{array}{rcl}
        \rule{1mm}{0mm} [X_+,X_-] & = & 2 \tau \theta^2 L, \\
        \rule{1mm}{0mm} [Y_{\pm},X_{\pm}] & = & - \theta (L \mp H), \\
        \Delta & -- & \mbox{primitive}.
                                        \end{array}
                                \right.
                                        \label{e48}
        \end{equation}
        and for the points of the boundary:
        \begin{equation}
        D_{0(\theta ,0)} \left\{ \begin{array}{rcl}
        \rule{1mm}{0mm} [Y_{\pm},X_{\pm}] & = & - \theta (L \mp H), \\
        \Delta & -- & \mbox{primitive},
                                        \end{array}
                                \right.
                                        \label{e49}
        \end{equation}
        \begin{equation}
        D_{0(0,\tau)} \left\{ \begin{array}{c}
        \mbox{Abelian}, \\
        \Delta  --  \mbox{primitive}.
                                        \end{array}
                                \right.
                                        \label{e50}
        \end{equation}
        \begin{equation}
        D_{0(0,0)} \left\{ \begin{array}{c}
        \mbox{Abelian}, \\
        \Delta  --  \mbox{primitive}.
                                        \end{array}
                                \right.
                                        \label{e51}
        \end{equation}

        All the Hopf algebras here are the classical universal
        enveloping  and the points of the curve $ D_{\varepsilon (0,\tau)} $
        are found to be trivialised. The deformation diagram obtains
        the form
\begin{equation}
\begin{array}{ccc}
& D_{\theta' ,\tau'}  &     \\
& \mid &  \mbox{\LARGE $\searrow$}  \raisebox{2mm}{\mbox{\scriptsize $\tau \rightarrow 0$}} \intt{3}  \mbox{\LARGE $\swarrow$} \\
\mbox{\scriptsize $\theta \rightarrow 0$}
        & \mid & D_{\theta' ,0}    \\
&\downarrow &  \mbox{\LARGE $\swarrow$}  \raisebox{2mm}{\mbox{\scriptsize $\theta \rightarrow 0$}} \intt{9} \\
& D_{0,0} & \approx  U(\mbox{Ab}) \intt{15}
        \end{array}                      \label{e52}
        \end{equation}
        It reflects the fact that the algebra $ D_{0(\theta ,\tau)} $
        -- a two-step classical first order deformation (of Abelian
        algebra) -- can also be treated as the second order deformation
        for each $ \tau \neq 0 $.

        So the limit $ \varepsilon \rightarrow 0 $ leads us to the
        degenerated case. The quantization removes this degeneracy.

        \section{Acknowledgments}
        I am heartily grateful to Prof. J.Lukierski for fruitful
        discussions and to all the scientists of the Institute of
        Theoretical Physics of Wroclaw University for their warm
        hospitality.

        The work is supported in part by the International Science
        Foundation, Grant N U9J000, and by the Russian Foundation
        for Fundamental Research, Grant N 95-01-00569a.

        \section{Appendix}
        The transformation $ B(t) $ and the reparametrization (\ref{e7})
        lead to the following 2-parametric variety
        $ Q(sl(n,{\bf C}),sl^*(n,{\bf C}))$ for quantum $sl(n,{\bf C})$
        algebras. (We use here the basis introduced in \cite{Ross}.)
        \begin{eqnarray*}
        \rule{1mm}{0mm}
        [H_i,H_j] &  = & 0, \rule{2cm}{0cm} i,j=1,\dots,n-1; \\
        \rule{1mm}{0mm}
        [H_i,X_{\pm(j,j+1)}] & = & \pm t \alpha_{ij} X_{\pm(j,j+1)}, \\
        \rule{1mm}{0mm}
        [X_{+(i,i+1)},X_{-(j,j+1)}]_{\exp(ht/2(\delta_{j+1,n} +
        \delta_{i+1,n}))} & = & t^2 \delta_{ij} e^{(ht/2)\delta_{i+1,n}}
        \frac{e^{-hH_{i;n-1}} - e^{-hH_{i+1;n-1}}}{e^{-ht/2} - e^{ht/2}}, \\
        \rule{1mm}{0mm}
        [X_{\pm (i,i+1)},X_{\pm (j,j+1)}]_{\exp(\pm ht/2(\delta_{i+1,n}
        - \delta_{j+1,n}))} & = & 0, \rule{2cm}{0cm} \mbox{for}
        \mid i-j \mid >1;
        \end{eqnarray*}
        \[
        \begin{array}{l}
        e^{\pm ht(\delta_{j,i+1} + \delta_{j+1,n})}(X_{\pm (i,i+1)})^2
        X_{\pm (j,j+1)} \\
        - e^{\pm ht/2(\delta_{j,i+1} + \delta_{j+1,i} +
        \delta_{j+1,n} + \delta_{i+1,n})}(e^{ht/2} - e^{-ht/2})
        X_{\pm (i,i+1)}X_{\pm (j,j+1)}X_{\pm (i,i+1)}  \\
        + e^{\pm ht(\delta_{j+1,i} + \delta_{i+1,n})}X_{\pm (j,j+1)}
        (X_{\pm (i,i+1)})^2 = 0, \rule{2cm}{0cm}
        \mbox{for} \mid i-j \mid =1;
        \end{array}
        \]
        \begin{eqnarray*}
        \Delta H_i & = & H_i \otimes {\bf 1} + {\bf 1} \otimes H_i, \\
        \Delta X_{\pm(i,i+1)} & = & X_{\pm(i,i+1)} \otimes
                e^{-(h/2)H_{i+1;n-1}} +
                e^{-(h/2)H_{i;n-1}} \otimes X_{\pm(i,i+1)}, \\
        S(H_i) & = & - H_i, \\
        S(X_{\pm (i,i+1)}) & = & -e^{hH_{i;n-1}/2}
                               X_{\pm (i,i+1)} e^{hH_{i+1;n-1}/2}, \\
        \varepsilon (X_{\pm (i,i+1)}) & = & \varepsilon (H_i) = 0
        \end{eqnarray*}
        Here $ \alpha_{i,j} $ is the Cartan matrix and
        \[
        H_{i,n-1} = H_{i,i+1} + H_{i+1,i+2} + \cdots + H_{n-2,n-1}.
        \]
        The modified $(\pm)$-co-ordinate functions \cite{Lya2}
        were applied so that the relation
        \[
        te^{\mp ht/2(1+ \delta_{i,j})}X_{\pm(i,j)} e^{-hH_{i+1,n-1}/2} =
        [X_{\pm (i,i+1)},X_{\pm (i+1,j)}]_{\exp(\pm (ht/2)\delta_{j,n})}.
        \]
        holds.
        In these terms the dual group $ SL^*(n,{\bf C}) $ has the simplest
        form. The coproduct for an arbitrary basic element $ X_{\pm(i,j)}$
        looks like
        \begin{eqnarray*}
        \Delta X_{\pm(i,j)} & = & X_{\pm (i,j)} \otimes e^{-hH_{j;n-1}/2}
                                + e^{-hH_{i;n-1}/2} \otimes X_{\pm (i,j)} \\
                        & + & \frac{(1-e^{\pm ht})}{t} \sum_{k=i+1}^{j-1}
                              X_{\pm(i,k)} \otimes X_{\pm(k,j)}.
        \end{eqnarray*}
        In the limit $ t \rightarrow 0 $ it describes the compositions
        of the solvable group $ SL^*(n,{\bf C}) $:
        \begin{eqnarray*}
        \Delta H_i & = & H_i \otimes {\bf 1} + {\bf 1} \otimes H_i, \\
        \Delta X_{\pm(i,j)} & = & X_{\pm (i,j)} \otimes e^{-hH_{j;n-1}/2}
                                + e^{-hH_{i;n-1}/2} \otimes X_{\pm (i,j)} \\
                        & \mp & h \sum_{k=i+1}^{j-1}
                              X_{\pm(i,k)} \otimes X_{\pm(k,j)}.
        \end{eqnarray*}

redpajama_set_name: RedPajamaArXiv


The Gold Award is for young people aged 16 years and older and as such fits in very well with the Senior Scout and Rover Scout Programmes.
Expedition (training for, planning and completing a journey on foot, horseback, boat or cycle).
It would normally take a participant between 18 - 24 months to complete a Gold Award.
Below is a outline of the Gold Award requirements and the Senior Scout badges that fulfill them, you can find more ideas on The Award website.
Direct Entrants must undertake a further 6 months in either the service or the longer of the Skill or Physical Recreation sections.
Gain either the Senior Explorer badge or the Discovery Award.


The most mountainous region of Fauquier County, today, Linden is the location of several wineries. Linden played a prominent part in the rich history of northwestern Fauquier since it was the location of a mountain gap providing access from the Virginia Piedmont to the Shenandoah Valley.
According to historian Eugene Scheel, the gap was first called Calmes' Gap for Marquis Calmes who moved to the valley from the gap in the 1730s. Later it became known as Manassas Gap. A road marker notes that the area was first explored by John Lederer in 1670.
The Appalachian Trail crosses through Linden at Route 55 and Rt. 638.
The building of the Manassas Gap Railroad around 1850 spurred growth and facilitated the movement of apples to port for shipment to Europe. The area around Markham and Linden hosted 25 apple orchards which thrived around the mid-1700s until the early 1950s.
This area was a hotbed of activity in the Civil War as a strategic location in the mountains with a railroad. The Battle of Manassas Gap took place in the Linden, Belle Meade, and Markham vicinity.
The Linden Methodist church was built in 1849, and remodeled in 1892 and 1956. Additional architectural and historic research conducted in consultation with local residents can determine the age and significance of structures in Linden, including what historic resources remain in the vicinity of Route 55 at Fiery Run Road.
The area is speckled with buildings dating from the late 18th century through the 1940s. Linden spans the county line into Warren County but contains a number of significant buildings on the Fauquier County side.


– The master suite including double vanity, spa bath and large walk in wardrobe that is conveniently located on the ground floor.
– 3 more upstairs bedrooms, all of generous size with plenty of built ins.
– Delightful gourmet kitchen, fitted out with top of the range Miele appliances including steam oven.
– Adjacent family/meals area and Alfresco dining.
Entertain your guests in style in the formal dining room, the secluded courtyard with water feature, or on your sweeping balcony while watching the sun set over the beautiful parklands.
Garden maintenance is a breeze, all taken care of by your gardener (included in rent).
Enjoy living in this wonderful street by the park amongst historic properties, all within easy distance to North Adelaide Village, the City and the Golf Club.


DULZEVIA 60 SOBRES ALNATUR S.A.
DULZEVIA BOLSA 150grs. ALNATUR S.A.
K-7 EXTRACTO 120ml. ALNATUR S.A.
K-7 FACIL 21filtros. ALNATUR S.A.
K-7 INFUSION 200grs. Bote ALNATUR S.A.
LIMPIADOR UNIVERSAL ARCILLA BLANCA 300G(BPNATUR S.
OLIGEN SUGAR CONTROL 30 CAP ALNATUR S.A.
PAPEL HIGIENICO 8 UDS ALNATUR S.A.
SNACK SESAMO ECO 25UNDS(BP) ALNATUR S.A.


Yes! The Magic Fingers and self styled "Beat Boy" hits up the airwaves yet again with this cool track "pepper dem".
This song features indigenous rapper Bablow who did the first verse, while Ogshee2 the " La la Le Friday" crooner, jumps on the second verse to add more pep to the on going flow.
The track was mixed and mastered by Cahsemhimself and promises to get you reeled up and dancing as you sing along and "Pepper Dem" haters!


text:
Legends are all too many these days. Paddon, however, was always the real deal
20th November 2007 By Guest Blogger Leave a Comment
Not many players who pull on the yellow and green can really claim legendary status.
Of course, many great footballers have played for our club in their time, making their name at City before looking to pastures new.
By the same token, several lesser mortals have passed through Norwich City on their way to nowhere in particular. But only a relative few achieve an affection in the supporters' hearts and memories that attains to genuine greatness.
Without doubt, Graham Paddon was one of those.
For City supporters around and beyond Stan's age, Paddon is, and forever will be, a hero.
He was a player who could always be relied upon to give his all for the cause, who played through pain to dig his comrades out of trouble, and who kept the green-and-yellow flag flying through thick and thin.
He also had a cultured left foot, meaning he fitted seamlessly into the two very different teams built by Ron Saunders and John Bond over the 1970s.
Throughout his time at City, he was – with Kevin Keelan – the first name on the team sheet, week in week out. It was with Graham Paddon ensconced in the side that Norwich first won promotion to Division One; that City first visited Wembley; and that the Canaries became a cultured top flight team who garnered the respect of the football elite.
For those reasons, Paddon is an integral part of our club's and our own histories.
Of course, Paddon did briefly pop over to West Ham and win an FA Cup winners' medal. In so doing, he proved that the Canary-Hammer link worked two-ways – we helped them out just as much as they helped us.
Nevertheless, he soon returned to the fold, teaming up with the likes of Martin Peters and Colin Suggett to forge one of the best midfields ever to work its magic at Carrow Road.
He also looked fantastic – with a golden mane that could take an opponent's eye out whilst simultaneously putting to shame the array of dodgy perms bobbing along beside him. Even his moustache had style.
Three things come immediately to mind when Stan thinks of Graham Paddon.
First, he always played. In 1970-71, he played 45 out of 47 games; in 1971-72, he played 46 out of 48 games; in 1972-73, he played 54 our of 60 games. Then, on returning, he played 49 out of 51 games in 1979-80, and 48 out of 48 in 1980-81.
Imagine a Safri – or most of our lot today – dealing with that? Two hairs out of place and half the current team downs tools.
Second, take yourself back to Highbury, 21 November 1972. City are in the League Cup fifth round (such giddy heights), playing one of the great Arsenal teams – one that will finish second come the end of the season.
Norwich are new to Division One but giving it a go; showing some fight and some spirit, and even having a cup run. Across from the red ranks of the North Bank, the Norwich contingent pull their woollen scarves tight across their Ben Shermans or tank tops, light their cigarettes, and stamp their feet against the cold.
We could get a hiding, but we know this team can do almost anything. And they do. 3-0 City, Paddon scores one of the great hat-tricks of all time, and 'On the Ball, City' resounds across north London. Beautiful.
Third, Nottingham Forest come to Carrow Road on 25 February 1978. They are on their way to winning the league. We have been doing all season what John Bond teams tended to do – play well, but not always get the result that such culture deserved.
A few weeks back, Man City demolished us 3-1 at home; young Stan cried.
Now, it's 3.35 pm and we are 3-0 down. In such situations, there is always one player to whom the crowd and his team-mates looked: Graham Paddon.
A John Ryan penalty before half-time gets out hopes up, before a second-half performance, in which Paddon begins to cut into and work his way around the Forest defence, ensures we get a 3-3 draw.
For whatever reason, Stan will always remember Paddon leaving the pitch after that game – a gladiator leaving the coliseum, a true hero being applauded for his endeavour, a City legend showing the fight and the heart that will always make this club great.
Rest in peace Graham, we love you?
Filed Under: Column, Stanley

redpajama_set_name: RedPajamaCommonCrawl


text:
Query over use of former Kildare Hewlett Packard building as data centre
Kildare County Council has asked An Bord Pleanála to assess whether the use of one of the former Hewlett Packard buildings as a data centre requires planning permission.
The council wants to know if the proposed use of building seven at the campus "is or is not development, or is or is not exempted development".
The case is due to be decided by May 10.
In January 2020 EFIV Irish Property ICAV was granted permission for two changes at the 80.56 hectares site at Liffey Business Campus (formerly known as the Hewlett Packard Campus), Barnhall Road, Leixlip.
It was given the go ahead to change the use of building three from a warehouse to a manufacturing facility. Building four was also changed from manufacturing to office use.
Tweets by @leinsleadernews


text:
enum {
	IMAGE_FORMAT_RGBA = 1,
	IMAGE_FORMAT_RGB
};

enum {
	IMAGE_CODEC_PNG = 1,
	IMAGE_CODEC_JPEG,
};

struct Image {
	unsigned width, height;
	int format;
	void *data;
};

struct Image*
image_from_buffer(const void *data, size_t size, int codec);

struct Image*
image_from_file(const char *filename);

void
image_free(struct Image *image);

redpajama_set_name: RedPajamaGithub


text:
Canada adds 3,235 new coronavirus cases, 142 deaths as total infections reaches near 790K
Across the country health officials reported 3,235 new cases of the novel coronavirus on Wednesday, as the total number of infections reached 789,654.
Another 142 people have died in Canada after testing positive for COVID-19, pushing the country's death toll to 20,355.
However, more than 720,000 people have recovered after contracting the disease, and 997,265 vaccinations to protect against the virus have been administered.
Thousands of new cases in central Canada
In Ontario, 1,172 new cases and 67 new fatalities were reported.
To date, the province has seen 272,097 infections and 6,305 deaths related to COVID-19.
The new cases and fatalities come as provincial authorities announced some schools would begin to re-open.
Meanwhile, in Quebec, 1,053 new cases were detected, pushing the total number of infections in the province to 265,579.
Thirty-seven more deaths mean to date, 9,899 people have died in Quebec after falling ill.
In Saskatchewan, 194 new infections were reported, bringing the total number of cases in the province to 24,430.
Eight new deaths push the provincial death toll to 322.
Meanwhile, Manitoba saw three new deaths and 126 new cases.
To date, 29,858 people in the province have contracted the virus, while 835 have died.
New cases in Atlantic Canada
In Atlantic Canada, 17 new cases of the virus were detected.
New Brunswick health officials said another 14 people have tested positive for the virus, while Nova Scotia added one new infection.
Health authorities in Newfoundland and Labrador said two new cases have been detected in the province.
Prince Edward Island did not add any new cases on Wednesday. So far, 110 of the province's 112 confirmed cases are considered to be resolved.
No new fatalities were reported in the maritime provinces or Newfoundland and Labrador on Wednesday.
New infections in Western Canada
A total of 673 new cases of the virus were detected in Western Canada.
Health officials in Alberta said 259 more people have fallen ill, while 414 new cases were reported in British Columbia.
Eleven new deaths were reported in Alberta, while B.C. recorded 16 new fatalities.
None of Canada's territories reported a new case or death related to COVID-19.
Global cases top 104 million
The total number of coronavirus cases worldwide topped 104 million on Wednesday.
According to the latest tally from Johns Hopkins University, as of 7 p.m. ET, 104,269,426 confirmed cases of COVID-19 had been confirmed globally.
To date, the virus has claimed 2,264,118 lives around the world.
The United States remained the epicentre of the virus on Wednesday, with more than 26.5 million confirmed infections.
A total of 450,200 people have died in the U.S. after testing positive for the virus.
Previous articleJack Ma's Ant Group to restructure into financial holding company, face tough banking regulations
Next articleCanada will list the Proud Boys movement as a terrorist group


text:
Q: SecurityError: DOM Exception 18 after upgrade to Cordova iOS 4.0.1? I have upgraded my Cordova app to Cordova iOS 4.0.1, as mentioned here.
http://devgirl.org/2016/01/11/a-faster-hybrid-app-for-the-new-year/
But my app keep giving following error in iOS 9.1 after the splash-screen displayed.

I upgraded Cordova and its plugins (added wkwebview-engine). And also added meta tag into index.html file and update config.xml as well.
Error is throwing when initialising the database.
//// config.xml ////
<access origin="*" />
<allow-navigation href="http://acs.sdss.net/" />

//// Index.html ////
<head>
<meta http-equiv="Content-Security-Policy" content="default-src 'self' data: gap: https://ssl.gstatic.com 'unsafe-eval'; style-src 'self' 'unsafe-inline'; media-src *; connect-src http://api.fixer.io">
<meta charset="utf-8" />
<meta name="format-detection" content="telephone=no" />
<meta name="msapplication-tap-highlight" content="no" />
<meta name="viewport" content="user-scalable=no, initial-scale=1, maximum-scale=1, minimum-scale=1" />
<link rel="stylesheet" href="css/mobile-angular-ui-base.min.css" />

///// This is where error is throwing /////
// Initialize the db, create if it does not exist
self.init = function () {
    try {
        self.db = window.openDatabase(DB_CONFIG.name, '0.1', 'database', 10 * 1024 * 1024);
    } catch (e) {
        alert(e);
        return;
    }

    angular.forEach(DB_CONFIG.tables, function (table) {
        var columns = [];

        angular.forEach(table.columns, function (column) {
            columns.push(column.name + ' ' + column.type);
        });

        var query = 'CREATE TABLE IF NOT EXISTS ' + table.name + ' (' + columns.join(',') + ')';
        self.query(query);
    });
};

Xcode console display that it is using "WKWebView" too. Please tell me if you need more information on this.

redpajama_set_name: RedPajamaStackExchange


Caple, PH500H, Hydronic Plinth Heater. 500mm. Connects to your central heating system. Up to 24% more efficient than standard radiators. Featuring stainless steel grille, 2 speed fan, whilst the heater is easy install and generates little noise.


text:
New wave Democrats are mirroring Corbyn's moves, but is America really shifting left?
David Millward
She was born in the Bronx to a Puerto Rican mother and American father, but the campaign pitch adopted by Alexandria Ocasio-Cortez - the 28-year-old political novice who has shaken up the Democratic Party by ousting a powerful New York congressman - will have a familiar ring to British voters.
"A New York for the many is possible," Cortez said in a TV ad, in which she rounded on Joe Crowley, a ten term incumbent who had been considered a candidate to replace Nancy Pelosi as party leader, as elitist.
As divisions in the party widen, those on the left has been mirroring the Corbyn playbook, running grassroots campaigns on issues outside the Democratic mainstream and promising to work for "the many...


text: <input type="text" ng-model="searchText" placeholder="{{placeholder}}" typeahead="match.name for match in getMatchResults($viewValue)" typeahead-on-select="getSelectedItem($item)" typeahead-wait-ms="400" class="form-control">


Touareg is the first Volkswagen in this market segment, combining a high quality off-roader with the comfort of a top-class saloon and the dynamic of a sports car. It is well equipped with many advanced and sophisticated off-road features and vertically adjustable pneumatic suspension. Four engines are available, with the V10 diesel being the most powerful passenger car diesel engine in the world.


Search the newest Henie Hills Oceanside homes for sale in San Diego CA. To change how the search is displayed, click the gray button next to "Sort By." Using the button below, you can sort homes by prices or days on the market. For information about properties for sale by price in Henie Hills Oceanside California, or to request a showing, click "Schedule a Showing" when viewing any listing.
If you would like more information on any of these Henie Hills listings, just click the "Request More Information" button when viewing the details of that property. We can provide you with disclosures, past sales history, dates and prices of homes recently sold nearby, and more.
And, if you haven't already, be sure to register for a free account so that you can receive email alerts whenever new Henie Hills houses come on the market.
When searching Henie Hills Oceanside homes for sale you will find Henie Hills is a community located in the northeast corner of El Camino Real in Oceanside and by the 78 freeway and is surrounded by the El Camino Country Golf Course. This community was originally owned by Olympic ice skater and movie star, Sonja Henie and her brother, Leif. Sonja Henie and her brother, Leif purchased 1600 acres of the undeveloped land to host private retreats and entertain Hollywood friends. Many people gained interest in the land after the Henies built a custom home on the property. The Henies began selling parcels to developers and also developed some land into estate view homes. There are about 200 homes in Henie Hills in 5 different neighborhoods. It is a small, affordable community of single family homes. Most homes are located on about half acre lots, which provide a lot of privacy. The community also has its own pool, clubhouse, and tennis courts. Mira Costa College and Tri-City Hospital are next to the community and used to be part of the original land purchase by the Henies. Although there are no amenities within walking distance, shopping, restaurants, and services are within close proximity, including the Carlsbad mall, Carlsbad Village, and the freeway. Nearby parks include Palisades Park. Students living in Henie Hills attend schools in the Oceanside Unified School District. Schools include Palmquist Elementary, San Luis Rey Elementary, Martin Luther King, Jr. Middle School, and El Camino High School. Other options include Pacific View Charter School, Oceanside Adventist, and Montessori Arts and Sciences. When searching homes for sale in Jeffries Ranch Oceanside CA you will find Jeffries Ranch is an equestrian themed community in northeast Oceanside and was first developed by the Jeffries family, specially CW Jeffries. The community is filled with tree-lined streets with homes on large lots. It is located off the 76 freeway, so it provides quick access to Vista, downtown Oceanside, shopping, restaurants, and more. Although it is located off the freeway, there is only one major entrance in.


"Narrative Analysis" is EVERYWHERE in America today. It wasn't this way a decade ago. It IS the zeitgeist. Get used to it. AND learn about it, or be lost.
TOURNEY NARRATIVE: The New Yorker has a nice article today about how the coaches (and one wife) make for better storytelling these days than the players.
I'm tellin' ya, it wasn't like this just a decade ago. Nobody talked to the President about, "the Democratic party narrative," as Jon Stewart did a couple years ago on his show. News pundits didn't sit around and discuss, "the NRA narrative," as they have for the past couple months. And sports writers didn't break down a simple basketball tournament the way that Ian Crouch does today in The New Yorker.
But his analysis is actually very good. He explains why there's so much more interest in the basketball coaches rather than players in today's NCAA Basketball Tournament. It's because the coaches are interesting characters — they have backstories, personalities and distinctive styles. Players are young, faceless, temporary, and largely without any depth of character.
It's very true. The biggest matchup this past weekend was between coaches Bill Self (of my beloved Kansas Jayhawks) and Roy Williams, the former Kansas coach. THAT was a good story, but it had absolutely zero to do with the players. Same for the biggest story of the weekend — Cinderella team Florida Gulf Coast — it's all about the coach, Andy Enfield and his Victoria's Secret wife (who doesn't even get a name in the New Yorker article!).
The point to notice is this whole idea of "narrative analysis." It's everywhere. And at the center of it is storytelling. Which is why Dorie, Brian and I have developed our storytelling workshop over the past two years and are now writing a book about it. Welcome to "the narrative" of today's communication.


text:
Q: JQuery textbox value computation I am new to JQuery and i am looking for a script which fulfills the below need.
User has to type a number input in the textbox say:456 (Can be input of any length) the value has to change to 000000000000000456 (length has to be 18 with zeros appended in front of the user input).
Eg: 12 has to change to 000000000000000012
145 has to change to 000000000000000145
Function has to be invoked on click of button.
Below is the snippet:
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"     "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<title></title>
<script type="text/javascript" src="js/jquery-1.6.2.min.js"></script>
<script>
    $(document).ready(function () {
        $("#acb1").click(function () {
        });
    });
</script>
</head>
<body>
<form id="form1">
<div>
<input type="text" id="ac1" value=""/>
<input type="button" id="acb1" value="Search" />
</div>
</form>
 </body>
</html>

Any leads would be appreciated.
Thanks

A: function pad(number, length) {
    var str = '' + number;
    while (str.length < length) {
        str = '0' + str;
    }

    return str;
}

alert(pad(145, 18));

A: You could use a JavaScript for loop to increment zeroes in front of the string until the variable.length === 18. Tell me if you need an example. :)
Here is an example (using Manuels function): http://jsfiddle.net/BX5Xs/1/

A: This has been asked and answered before. You are wondering how to create a zero-filled String of the number in your input field. This is the proposed function by Peter Bailey:
function zeroFill( number, width )
{
  width -= number.toString().length;
  if ( width > 0 )
  {
    return new Array( width + (/\./.test( number ) ? 2 : 1) ).join( '0' ) + number;
  }
  return number;
}

In your case you can use it this way:
$(document).ready(function () {
    $("#acb1").click(function () {
         var num = this.value;
         var padded_num = zeroFill(num, 18);
         alert('Zero padded number: ' + padded_num);
    });
});

A: $("#acb1").click(function () {
  var inp = $('#ac1').val();
  var limit = 18;
  var inpLen = inp.length;
  var res = '';
  for(var i=0;i<(limit-inpLen);i=i+1){
res = res + '0';
  }
  res = res + inp;
});

Good luck ;)

A: Try this
$(document).ready(function () {
    $("#acb1").click(function () {
        var len = $("#ac1").val().length;

        if (len < 18 && len > 0)
        {
            var temp = '';
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John Pettigrew Scott, 73 in ill health for some time, died at 12:30 p.m. Sunday (24 Jul 1938) of peritonitis, in his home in Idaho, near Shelocta. A son of Zacchariah T. and Margaret (Olinger) Scott, he was born January 21, 1865. He was a well-known thresherman.


text:
\section{Introduction}

Visual understanding and language generation are two tasks that are intuitive for humans, but pose a challenge to computers. Recently, convolutional neural networks (CNNs) \citep{cnn-imagenet} and long short-term memory networks (LSTMs) \citep{lstm} have achieved state-of-the-art results on image understanding and natural language generation, respectively. For image captioning, one of the most successful approaches has been the encoder-decoder architecture, where the image is first ``encoded'' by a CNN into a latent semantic hidden vector, and then ``decoded'' into a natural language sentence using a recurrent, language generating LSTM.

For most people, describing a picture is an intuitive task that requires little effort. However, patients with neurodegenerative disorders have impaired brain function that inhibits some cognitive subtasks; this manifests as difficulties with picture description \citep{picture-description-alzheimer}. In fact, because linguistic impairment is one of the earliest signs of AD, picture description is useful as a cognitive test for AD \citep{detecting-alzheimer-picture}.

In this paper, we implement a variant of the ``Show and Tell'' neural network for image captioning \citep{show-and-tell}, and simulate the effects of neurodegeneration by adding dropout during inference, which randomly sets a subset of the neuron outputs in a layer to zero. We evaluate the effects of dropout on language generation, and compare the results to picture description by patients with diseases like AD and aphasia.

\section{Related Work}

\subsection{Image Captioning}

\citet{mao2014} were the first to apply deep recurrent neural networks (RNNs) to image captioning. Their architecture used a multimodal layer that combined image representations preprocessed by a CNN, the input word embedding, and the output of the RNN at each time step to generate output word embeddings.

\citet{show-and-tell} used an encoder-decoder architecture that only looked at the image once. The encoder was a CNN that encoded images into vector representations and the decoder was an LSTM that decoded the image features into natural language descriptions. By using an LSTM, their model was able to retain long-term dependencies and avoid having to show the image to the RNN multiple times. \citet{show-attend-tell} added a visual attention mechanism, which learned which part of the image to focus on at each time step.

\subsection{Picture Description}

Alzheimer's disease (AD) is a neurodegenerative disorder affecting 47 million people worldwide \citep{alzheimer-is-bad}. One of the earliest symptoms of AD is cognitive impairment, especially difficulty with language production. One widely-used cognitive test for AD is the {\em Cookie Theft} picture description task from the Boston Diagnostic Aphasia Examination \citep{boston-exam}. In this task, the patient is shown a drawing of a chaotic kitchen scene, and is asked to describe it in as much detail as possible. Language in patients with AD is characterized by semantic impairment, particularly difficulty finding words for concepts and ideas \citep{alzheimer-language-review}.

Wernicke's aphasia (WA), also known as fluent aphasia, is caused by damage to Wernicke's area, which is partially responsible for language production. Patients with WA tend to produce long stretches of words in a seemingly random order\footnote{Video of a patient with WA performing a picture description task: \url{https://www.youtube.com/watch?v=xzp-XUBknQI}} \citep{buckingham-fluent-aphasia}.

Recently, methods have been developed for computational methods to diagnose AD automatically from speech. For example, \citet{fraser2016} extracted a wide range of linguistic features to classify AD with about 82 percent accuracy, given a few minutes of speech from picture description.

\subsection{Dropout in neural networks}

Dropout is the technique of randomly selecting, with probability $p$, neurons in a layer and setting their outputs to zero. When used during training, dropout has been shown to be an effective regularization method and has an effect similar to averaging an ensemble of models \citep{dropout}. Dropout has been applied to RNN language models as well, with a similar regularizing effect \citep{zaremba-rnn-dropout}. While previous work considered dropout during training as a regularization mechanism, we consider dropout during {\em inference} and its effects on language generation. To our knowledge, dropout during inference in RNNs has not been studied.

AD may be caused by a misfolding in the beta-amyloid protein, causing beta-amyloid plaques to form in neurons \citep{century-of-alzheimer}. This causes inhibited electrochemical signal transmission in the synapses. Thus, the effect of AD in brain cells is similar to that of dropout in neural networks. The goal of our work is to simulate the effects of AD to produce pathological linguistic effects similar to picture description by patients with neurodegenerative disorders.

\section{Model}

\subsection{Image caption network}

We implement an encoder-decoder neural architecture. For the CNN, we use the VGG16 convolutional network \citep{vggnet} and initialize it with weights from Torchvision \cite{pytorch}, which were trained on ImageNet. During the training of our image captioning network, all of the weights for the CNN are frozen.

During inference, the image is processed by the VGG16 network up until the last hidden layer. This hidden layer is fed into a linear layer to produce a representation for the initial hidden layer of the LSTM. Then, a sentence is generated as follows: On each iteration, the LSTM hidden layer is used to produce a linear softmax classification to generate a probability distribution over the entire vocabulary to pick the next word. The word with the highest probability is picked as the next word in the sentence. We then use a word embedding lookup to convert this word into a vector, and feed it into the next iteration of the LSTM. This process continues until the LSTM generates a special end marker, or exceeds a fixed word limit of 20 words (fewer than 3\% of sentences in the training data exceed this length).

To train the network, we compute the {\em perplexity}, which represents the likelihood function of this image-caption pair according to our model, and is computed by feeding the image and caption sentence into the network and taking the sum of the cross entropy errors at each step. Once computed, the perplexity is minimized using backpropagation and stochastic gradient descent.

\subsection{Dropout in GRU}

The LSTM was the first recurrent neural network model that addressed the vanishing and exploding gradient problems, in which simple RNNs had difficulty learning long-term dependencies due to numerical instability \cite{bengio-rnn-difficult}. More recently, the gated recurrent unit (GRU) was found to achieve similar performance to the LSTM, while using only a hidden state and no cell state \citep{cho-gru}. We use a modified version of the GRU, in which a constant level of dropout is added to the hidden state between iterations. This is represented by the following equations:

$$r = \sigma(x_t U_r + h_{t-1} W_r + b_r)$$
$$z = \sigma(x_t U_z + h_{t-1} W_z + b_z)$$
$$\bar{h} = \tanh(x_t U_{\bar{h}} + (r \times h_{t-1}) W_{\bar{h}} + b_{\bar{h}})$$
$$h_t = \mathrm{dropout}(z \times h_{t-1} + (1-z) \times \bar{h})$$
where $\times$ denotes element-wise multiplication between vectors and $\sigma$ is the sigmoid. The variable $h_{t}$ is the hidden state at time $t$, and $x_t$ is the input vector. First, we compute vectors $r$ representing the {\em reset} gate, and $z$ representing the {\em update} gate, both in the range $[0,1]$. A temporary hidden state $\bar{h}$ is computed from the input $x_t$ and previous hidden state $h_{t-1}$ masked by the reset gate $r$. The next hidden state $h_t$ is set to a combination of the previous hidden state $h_{t-1}$ and the temporary hidden state $\bar{h}$ as determined by the update gate $z$. Finally, a dropout is applied to $h_t$. The variables $U$, $W$ and $b$ (with various subscripts) are unknown weights to be learned by backpropagation.

We define $d_t$ to be the dropout probability during training, and $d_e$ the dropout probability during evaluation. Typically, dropout is used during training for regularization and turned off during evaluation, so $d_t > 0$ and $d_e = 0$. However, in this work we explore the possibility of $d_e > 0$ and its effects on language generation.

\subsection{Evaluation metrics}

To test the performance of our model, we generate captions for images in a validation set, and consider the BLEU and METEOR scores; for BLEU, we consider $n=4$, which correlates most highly with human ratings of performance \citep{bleu-metric,meteor-metric}.

Next, we evaluate the effects of dropout on caption accuracy and vocabulary diversity. We use Kullback-Leibler (KL) divergence to measure the distance between the word frequency distribution of the ground-truth validation captions and the sentences generated by our network on the validation set. That is, $$D_{KL}(P || Q) = \sum_w P(w) \log \frac{P(w)}{Q(w)},$$ where $P$ is the word distribution of the generated captions and $Q$ is the word distribution of the most common 10,000 words among the ground-truth captions.

For each run of the experiment, we also calculate $|V|$, the number of unique words among all generated captions, and $p(len > 20)$, the proportion of generated captions that exceed the word limit of 20.

\section{Results and Discussion}

We train our model using the COCO2014 dataset \citep{mscoco}, which contains 82,783 training images and 40,504 validation images.
\begin{table*}
\centering
\caption{Effects of $d_t$ and $d_e$ on caption accuracy and vocabulary diversity.}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$d_t$ & $d_e$ & BLEU-4 & METEOR & $D_{KL}$ & $|V|$ & $p(len > 20)$ \\
\hline
0.0 & 0.0 & 20.1 & {\bf 19.8} & 0.453 & 733 & 0.00 \\
\hline
0.0 & 0.2 & 16.5 & 18.4 & 0.298 & 2158 & 0.02 \\
\hline
0.0 & 0.4 & 7.4 & 14.0 & 0.270 & 7500 & 0.35 \\
\hline
0.0 & 0.6 & 2.2 & 9.3 & 0.823 & 9303 & 0.78 \\
\hline
0.0 & 0.8 & 0.2 & 5.4 & 1.496 & 9585 & 0.95 \\
\hhline{|=|=|=|=|=|=|=|}
0.2 & 0.0 & {\bf 20.3} & 19.5 & 0.497 & 630 & 0.00 \\
\hline
0.2 & 0.2 & 19.0 & 19.0 & 0.409 & 1312 & 0.00 \\
\hline
0.2 & 0.4 & 15.4 & 17.3 & {\bf 0.260} & 7007 & 0.03 \\
\hline
0.2 & 0.6 & 3.3 & 9.6 & 1.837 & 9841 & 0.83 \\
\hline
0.2 & 0.8 & 0.1 & 3.0 & 3.106 & 9840 & 0.99 \\
\hline
\end{tabular}
\label{table:dropout-scores}
\end{table*}

The neural network model is implemented using PyTorch \citep{pytorch}. We use GLoVe embeddings from SpaCy \citep{glove, spacy} trained on Common Crawl. The network was trained using Adam \citep{adam}.

The LSTM version of our model achieves a BLEU-4 score of 20.6 and METEOR score of 20.0. The GRU version performed similarly, with a BLEU-4 score of 20.1 and METEOR score of 19.8.

Our model slightly underperforms the scores reported by \citet{show-and-tell}. We did not implement beam search to minimize perplexity across a sequence but instead used the greedy approach of picking the highest probability word at each step. Additional hyperparameter optimization may have improved our model accuracy, though that is not our purpose here.

Next, we evaluate the accuracy and vocabulary diversity of the model as dropout is added in inference. We train two versions of the GRU model, once with no training dropout ($d_t = 0$) and once with $d_t = 0.2$. For each model, we generate captions for the validation set, using evaluation dropouts $d_e = [0.0, 0.2, 0.4, 0.6, 0.8]$. The results are shown in Table~\ref{table:dropout-scores}.

For both versions of the model, the BLEU-4 and METEOR scores are maximized when $d_e = 0$; this is expected, since dropout is usually disabled during evaluation for best performance. When evaluation dropout is moderate ($d_e = 0.4$), the model trained with dropout performs better than the model trained without. When evaluation dropout is high ($d_e \geq 0.6$), both models perform poorly.

The standard model without dropout only generates a vocabulary of 733, out of a total possible vocabulary of 10,000; when dropout is added during inference, the generated vocabulary is more diverse. In both versions of the model, the KL divergence of word frequency distributions is minimized using a moderate dropout ($d_e = 0.4$). Thus, a moderate amount of dropout produces a word frequency distribution closer to the target distribution when dropout is added during inference. However, when evaluation dropout is too high, the generated captions have low BLEU-4 and METEOR scores, high $D_{KL}$, and high increased probability of exceeding the sentence length limit.

Next, we comment on the qualitative effects of dropout on language generation. A sample of captions generated with various levels of dropout is given in the appendix. Generally, errors follow into two common patterns:

\begin{enumerate}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item A caption starts out normally, then repeats the same word several times: {\em ``a small white kitten with red collar and yellow chihuahua chihuahua chihuahua''}
\item A caption starts out normally, then becomes nonsense: {\em ``a man in a baseball bat and wearing a uniform helmet and glove preparing their handles won while too frown''}
\end{enumerate}

Both of these patterns are sometimes observed in patients with paraphasic disorders, like fluent aphasia \citep{neurobiology-brain}. In particular, one symptom of Wernicke's Aphasia is the use of {\em jargon speech}: long streams of words with seemingly no meaning but retaining some phonological and grammatical structure. \citet{buckingham-fluent-aphasia} give some examples of jargon speech:

\begin{itemize}
\setlength{\itemsep}{0pt}
\setlength{\parsep}{0pt}
\item {\em ``I know a deprecol over american person churches such as no dish or penthenis''}
\item {\em ``I think my foremust acoushner looks ellington''}
\item {\em ``I would say that the mick daysas nosis or chpickters''}
\end{itemize}

\section{Conclusion and Future Work}

In this paper, we have implemented an encoder-decoder image captioning neural network, and applied dropout during inference to simulate the effects of neurodegeneration on language production. The resulting sentences qualitatively resemble speech produced by patients with language production disorders like fluent aphasia. Future research will include quantitative comparisons of our language model with speech by aphasic patients.

Our model applies dropout in the hidden layer of the GRU, but there are other ways to simulate neurodegeneration as well. For example, one may instead add Gaussian noise to the hidden layer, or deterministically dropout specific neurons instead of at random. Furthermore, it is unknown whether the effects we observed can be reproduced using different corpora. The next step is to train a recurrent neural language model on a picture description corpus, such as the DementiaBank Corpus in the TalkBank project \citep{dementiabank}.

This work may be suitable as a data augmentation preprocessing step for systems that automatically detect dementia and aphasia using speech. A neural language model is trained using normative data, then dropout is applied during inference to generate degenerate data. This synthetic data is then combined with real data from patients with neurodegenerative disorders using semi-supervised methods to train a classifier. However, the implementation of this concept is a topic for future research.