
$l2h_cache_key = q/$A_{st}$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_72.png"
 ALT="$A_{st}$"></SPAN>|;

$l2h_cache_key = q/$g(t,\tau)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_71.png"
 ALT="$g(t,\tau)$"></SPAN>|;

$l2h_cache_key = q/$$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f +  f^g \log f\partial_x g\cr
 &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f +  f^g \log f\partial_x g\cr
 &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_39.png"
 ALT="\begin{displaymath}\eqalign{\partial_x f^g &amp;= g f^{g-1}\partial_x f + f^g \log f\partial_x g\cr
&amp;= f^{g-1}(g\partial_x f + f\partial_x g)\cr}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\delta\eta$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_7.png"
 ALT="$\delta \eta$"></SPAN>|;

$l2h_cache_key = q/$A_{st} + A_{st}^\dagger$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $A_{st} + A_{st}^\dagger$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_73.png"
 ALT="$A_{st} + A_{st}^\dagger$"></SPAN>|;

$l2h_cache_key = q/$\tau$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_66.png"
 ALT="$\tau$"></SPAN>|;

$l2h_cache_key = q/$10^{-3}$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_24.png"
 ALT="$10^{-3}$"></SPAN>|;

$l2h_cache_key = q/$$
\varphi(t) = f(\eta(t),\varphi(t))
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\varphi(t) = f(\eta(t),\varphi(t))
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_4.png"
 ALT="\begin{displaymath}
\varphi(t) = f(\eta(t),\varphi(t))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$Q$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_58.png"
 ALT="$Q$"></SPAN>|;

$l2h_cache_key = q/$\varphi=f(\eta(t-dt)+d\varphi$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $\varphi=f(\eta(t-dt)+d\varphi$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_21.png"
 ALT="$\varphi=f(\eta(t-dt)+d\varphi$"></SPAN>|;

$l2h_cache_key = q/$m_k$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_34.png"
 ALT="$m_k$"></SPAN>|;

$l2h_cache_key = q/$$\pmatrix{A & B\cr
-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\pmatrix{A & B\cr
-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="42" BORDER="0"
 SRC="mini_ker_9.png"
 ALT="\begin{displaymath}\pmatrix{A &amp; B\cr
-C^+ &amp; I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\eta_1(t)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_44.png"
 ALT="$\eta_1(t)$"></SPAN>|;

$l2h_cache_key = q/$\omega$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_41.png"
 ALT="$\omega$"></SPAN>|;

$l2h_cache_key = q/$\eta_{pred}$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_20.png"
 ALT="$\eta_{pred}$"></SPAN>|;

$l2h_cache_key = q/$h(t)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_49.png"
 ALT="$h(t)$"></SPAN>|;

$l2h_cache_key = q/$T$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_51.png"
 ALT="$T$"></SPAN>|;

$l2h_cache_key = q/$\varphi$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_3.png"
 ALT="$\varphi$"></SPAN>|;

$l2h_cache_key = q/$c$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_17.png"
 ALT="$c$"></SPAN>|;

$l2h_cache_key = q/$f$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_10.png"
 ALT="$f$"></SPAN>|;

$l2h_cache_key = q/$\Phi(t+\delta t,t)=\exp{A_{st} \delta t}$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $\Phi(t+\delta t,t)=\exp{A_{st} \delta t}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_78.png"
 ALT="$\Phi(t+\delta t,t)=\exp{A_{st} \delta t}$"></SPAN>|;

$l2h_cache_key = q/$$\partial_{\varphi} f(\eta(t),\varphi(t));
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\partial_{\varphi} f(\eta(t),\varphi(t));
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="27" BORDER="0"
 SRC="mini_ker_28.png"
 ALT="\begin{displaymath}\partial_{\varphi} f(\eta(t),\varphi(t));
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$U$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_75.png"
 ALT="$U$"></SPAN>|;

$l2h_cache_key = q/$W$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_61.png"
 ALT="$W$"></SPAN>|;

$l2h_cache_key = q/$\delta \eta$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_7.png"
 ALT="$\delta \eta$"></SPAN>|;

$l2h_cache_key = q/$$\partial_{\eta} g(\eta(t),\varphi(t));
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\partial_{\eta} g(\eta(t),\varphi(t));
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="27" BORDER="0"
 SRC="mini_ker_25.png"
 ALT="\begin{displaymath}\partial_{\eta} g(\eta(t),\varphi(t));
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$J$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_50.png"
 ALT="$J$"></SPAN>|;

$l2h_cache_key = q/$N$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_37.png"
 ALT="$N$"></SPAN>|;

$l2h_cache_key = q/$r_k$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_35.png"
 ALT="$r_k$"></SPAN>|;

$l2h_cache_key = q/$\mu$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_57.png"
 ALT="$\mu$"></SPAN>|;

$l2h_cache_key = q/$tau$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_66.png"
 ALT="$\tau$"></SPAN>|;

$l2h_cache_key = q/$t + \delta t$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_6.png"
 ALT="$t + \delta t$"></SPAN>|;

$l2h_cache_key = q/$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\partial_t \eta (t) = g(\eta(t),\varphi(t))
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_1.png"
 ALT="\begin{displaymath}\partial_t \eta (t) = g(\eta(t),\varphi(t))\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\eta_1(t=0)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_45.png"
 ALT="$\eta_1(t=0)$"></SPAN>|;

$l2h_cache_key = q/$t=0$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_46.png"
 ALT="$t=0$"></SPAN>|;

$l2h_cache_key = q/$R$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_60.png"
 ALT="$R$"></SPAN>|;

$l2h_cache_key = q/$N+1$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_38.png"
 ALT="$N+1$"></SPAN>|;

$l2h_cache_key = q/$\Phi(t,0)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_80.png"
 ALT="$\Phi(t,0)$"></SPAN>|;

$l2h_cache_key = q/$g(\tau;t)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_67.png"
 ALT="$g(\tau;t)$"></SPAN>|;

$l2h_cache_key = q/$$\partial_{\varphi} g(\eta(t),\varphi(t));
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\partial_{\varphi} g(\eta(t),\varphi(t));
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="27" BORDER="0"
 SRC="mini_ker_26.png"
 ALT="\begin{displaymath}\partial_{\varphi} g(\eta(t),\varphi(t));
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$$\varphi _{meet} = \eta _{prey}\eta _{pred}$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\varphi _{meet} = \eta _{prey}\eta _{pred}
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="27" BORDER="0"
 SRC="mini_ker_15.png"
 ALT="\begin{displaymath}\varphi _{meet} = \eta _{prey}\eta _{pred}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\Phi$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_79.png"
 ALT="$\Phi$"></SPAN>|;

$l2h_cache_key = q/$V$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_77.png"
 ALT="$V$"></SPAN>|;

$l2h_cache_key = q/$h$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_42.png"
 ALT="$h$"></SPAN>|;

$l2h_cache_key = q/$g$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_11.png"
 ALT="$g$"></SPAN>|;

$l2h_cache_key = q/$\varphi_{meet}$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $\varphi_{meet}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_18.png"
 ALT="$\varphi_{meet}$"></SPAN>|;

$l2h_cache_key = q/$$
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="43" BORDER="0"
 SRC="mini_ker_52.png"
 ALT="\begin{displaymath}
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$x\/\Delta$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_40.png"
 ALT="$x/\Delta$"></SPAN>|;

$l2h_cache_key = q/$\eta_{prey}$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_19.png"
 ALT="$\eta_{prey}$"></SPAN>|;

$l2h_cache_key = q/$$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
}$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
}
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_13.png"
 ALT="\begin{displaymath}\eqalign{\partial_t \eta (t) &amp;= g(\eta(t),\varphi(t))\cr
\varphi(t) &amp;= f(\eta(t),\varphi(t))\cr
}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$$
\omega = h ( \eta , \varphi) 
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\omega = h ( \eta , \varphi)
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_43.png"
 ALT="\begin{displaymath}
\omega = h ( \eta , \varphi)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\varphi=f(\eta),\varphi)$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $\varphi=f(\eta),\varphi)$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_23.png"
 ALT="$\varphi=f(\eta),\varphi)$"></SPAN>|;

$l2h_cache_key = q/$$\partial_{\eta} f(\eta(t),\varphi(t));
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\partial_{\eta} f(\eta(t),\varphi(t));
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="27" BORDER="0"
 SRC="mini_ker_27.png"
 ALT="\begin{displaymath}\partial_{\eta} f(\eta(t),\varphi(t));
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$(I-D)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_29.png"
 ALT="$(I-D)$"></SPAN>|;

$l2h_cache_key = q/$\sqrt{\sqrt{2}}$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $\sqrt{\sqrt{2}}$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_70.png"
 ALT="$\sqrt{\sqrt{2}}$"></SPAN>|;

$l2h_cache_key = q/$t=s_i$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_56.png"
 ALT="$t=s_i$"></SPAN>|;

$l2h_cache_key = q/$\delta \varphi$/;
$l2h_cache{$l2h_cache_key} = q|<!-- MATH
 $\delta \varphi$
 -->
<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_8.png"
 ALT="$\delta \varphi$"></SPAN>|;

$l2h_cache_key = q/$0.2$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_69.png"
 ALT="$0.2$"></SPAN>|;

$l2h_cache_key = q/$$\left\{\eqalign{\partial_t \eta _{k} ^{pos}  &=  \eta _{k} ^{vel} \cr
\partial_t \eta _{k} ^{vel}  &= ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,\/m_k  \cr}\right.$$
$$\left\{\eqalign{
\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\cr
\varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.$$
$$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
\eta ^{vel}_{0} &= 0\cr
\varphi  ^{spr}_{N+1} &= 0\cr
\varphi ^{dmp}_{N+1} &= 0\cr}\right.$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{\partial_t \eta _{k} ^{pos}  &=  \eta _{k} ^{vel} \cr
\partial_t \eta _{k} ^{vel}  &= ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k  \cr}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="32" BORDER="0"
 SRC="mini_ker_31.png"
 ALT="\begin{displaymath}\left\{\eqalign{\partial_t \eta _{k} ^{pos} &amp;= \eta _{k} ^{ve...
... + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k \cr}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{
\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\cr
\varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="28" BORDER="0"
 SRC="mini_ker_32.png"
 ALT="\begin{displaymath}\left\{\eqalign{
\varphi_k ^{spr} &amp;= -k_k (\eta _{k} ^{pos}- ...
...spr} &amp;= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
\eta ^{vel}_{0} &= 0\cr
\varphi  ^{spr}_{N+1} &= 0\cr
\varphi ^{dmp}_{N+1} &= 0\cr}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="32" BORDER="0"
 SRC="mini_ker_33.png"
 ALT="\begin{displaymath}\left\{\eqalign{\eta ^{pos}_{0} &amp;= 0\cr
\eta ^{vel}_{0} &amp;= 0\...
...arphi ^{spr}_{N+1} &amp;= 0\cr
\varphi ^{dmp}_{N+1} &amp;= 0\cr}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$a$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_16.png"
 ALT="$a$"></SPAN>|;

$l2h_cache_key = q/$l$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_54.png"
 ALT="$l$"></SPAN>|;

$l2h_cache_key = q/$\eta$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_2.png"
 ALT="$\eta$"></SPAN>|;

$l2h_cache_key = q/$d\varphi$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_22.png"
 ALT="$d\varphi$"></SPAN>|;

$l2h_cache_key = q/$$\eqalign{
\partial_t \eta (t) &=  g(\eta(t),\varphi(t)) + W(t) \mu\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
}$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\eqalign{
\partial_t \eta (t) &=  g(\eta(t),\varphi(t)) + W(t) \mu\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
}
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_55.png"
 ALT="\begin{displaymath}\eqalign{
\partial_t \eta (t) &amp;= g(\eta(t),\varphi(t)) + W(t)...
...varphi(t))\cr
\omega(t) &amp;= h ( \eta(t) , \varphi(t)) + \nu\cr
}\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$$
 U w V^\dagger
$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
U w V^\dagger
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="22" BORDER="0"
 SRC="mini_ker_74.png"
 ALT="\begin{displaymath}
U w V^\dagger
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\eta_i(t=0)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_48.png"
 ALT="$\eta_i(t=0)$"></SPAN>|;

$l2h_cache_key = q/$\nu$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_59.png"
 ALT="$\nu$"></SPAN>|;

$l2h_cache_key = q/$k$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_30.png"
 ALT="$k$"></SPAN>|;

$l2h_cache_key = q/$d_k$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_36.png"
 ALT="$d_k$"></SPAN>|;

$l2h_cache_key = q/$$\left\{\eqalign{\partial_t \eta _{prey} &=  a \eta _{prey} - a \varphi _{meet} \cr
\partial_t \eta _{pred} &=  -c \eta _{pred} + c \varphi _{meet}\cr}\right.$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{\partial_t \eta _{prey} &=  a \eta _{prey} - a \varphi _{meet} \cr
\partial_t \eta _{pred} &=  -c \eta _{pred} + c \varphi _{meet}\cr}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="27" BORDER="0"
 SRC="mini_ker_14.png"
 ALT="\begin{displaymath}\left\{\eqalign{\partial_t \eta _{prey} &amp;= a \eta _{prey} - a...
... \eta _{pred} &amp;= -c \eta _{pred} + c \varphi _{meet}\cr}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$\delta t$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_12.png"
 ALT="$\delta t$"></SPAN>|;

$l2h_cache_key = q/$g(\tau)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_65.png"
 ALT="$g(\tau)$"></SPAN>|;

$l2h_cache_key = q/$\psi$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_53.png"
 ALT="$\psi$"></SPAN>|;

$l2h_cache_key = q/$\phi_{j}(t)$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="mini_ker_47.png"
 ALT="$\phi_{j}(t)$"></SPAN>|;

$l2h_cache_key = q/$w$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_76.png"
 ALT="$w$"></SPAN>|;

$l2h_cache_key = q/$$\left\{\eqalign{
\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
\partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
\partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.$$
$$\left\{\eqalign{
\varphi _1 &= \eta _1\cr
\varphi _2 &= \eta _2\cr
\varphi _3 &= \eta _3
}\right.$$
$$\left\{\eqalign{
\omega _1 &= \varphi _1 + \nu_1\cr
\omega _2 &= \eta _2 + \nu_2 \cr
\omega _3 &= \eta _3 + \nu_3
}\right.$$/;
$l2h_cache{$l2h_cache_key} = q|<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{
\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
\partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
\partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_62.png"
 ALT="\begin{displaymath}\left\{\eqalign{
\partial_t \eta_1 &amp;= a_{11} \eta_1 + a_{12} ...
...varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{
\varphi _1 &= \eta _1\cr
\varphi _2 &= \eta _2\cr
\varphi _3 &= \eta _3
}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_63.png"
 ALT="\begin{displaymath}\left\{\eqalign{
\varphi _1 &amp;= \eta _1\cr
\varphi _2 &amp;= \eta _2\cr
\varphi _3 &amp;= \eta _3
}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER" CLASS="mathdisplay">
<!-- MATH
 \begin{displaymath}
\left\{\eqalign{
\omega _1 &= \varphi _1 + \nu_1\cr
\omega _2 &= \eta _2 + \nu_2 \cr
\omega _3 &= \eta _3 + \nu_3
}\right.
\end{displaymath}
 -->

<IMG
 WIDTH="100" HEIGHT="26" BORDER="0"
 SRC="mini_ker_64.png"
 ALT="\begin{displaymath}\left\{\eqalign{
\omega _1 &amp;= \varphi _1 + \nu_1\cr
\omega _2 &amp;= \eta _2 + \nu_2 \cr
\omega _3 &amp;= \eta _3 + \nu_3
}\right.\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>|;

$l2h_cache_key = q/$2$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_68.png"
 ALT="$2$"></SPAN>|;

$l2h_cache_key = q/$t$/;
$l2h_cache{$l2h_cache_key} = q|<SPAN CLASS="MATH"><IMG
 WIDTH="100" HEIGHT="12" ALIGN="BOTTOM" BORDER="0"
 SRC="mini_ker_5.png"
 ALT="$t$"></SPAN>|;
1;