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\begin{document}
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$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
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$\eta$
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$\varphi$
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$$
\varphi(t) = f(\eta(t),\varphi(t))
$$
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$t$
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$t + \delta t$
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$\delta \eta$
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$\delta \varphi$
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$$\pmatrix{A & B\cr
-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}$$
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$f$
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$g$
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$\delta t$
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$$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
}$$
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$$\left\{\eqalign{\partial_t \eta _{prey} &=  a \eta _{prey} - a \varphi _{meet} \cr
\partial_t \eta _{pred} &=  -c \eta _{pred} + c \varphi _{meet}\cr}\right.$$
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$$\varphi _{meet} = \eta _{prey}\eta _{pred}$$
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$a$
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$c$
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$\varphi_{meet}$
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$\eta_{prey}$
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$\eta_{pred}$
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$\delta\eta$
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$\varphi=f(\eta(t-dt)+d\varphi$
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$d\varphi$
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$\varphi=f(\eta),\varphi)$
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$10^{-3}$
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$$\partial_{\eta} g(\eta(t),\varphi(t));
$$
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$$\partial_{\varphi} g(\eta(t),\varphi(t));
$$
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$$\partial_{\eta} f(\eta(t),\varphi(t));
$$
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<!-- l2h_begin mini_ker_l2h 29 -->
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$$\partial_{\varphi} f(\eta(t),\varphi(t));
$$
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$(I-D)$
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$k$
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<!-- l2h_begin mini_ker_l2h 32 -->
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$$\left\{\eqalign{\partial_t \eta _{k} ^{pos}  &=  \eta _{k} ^{vel} \cr
\partial_t \eta _{k} ^{vel}  &= ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k  \cr}\right.$$
$$\left\{\eqalign{
\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\cr
\varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.$$
$$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
\eta ^{vel}_{0} &= 0\cr
\varphi  ^{spr}_{N+1} &= 0\cr
\varphi ^{dmp}_{N+1} &= 0\cr}\right.$$
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<!-- l2h_begin mini_ker_l2h 33 -->
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$m_k$
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$r_k$
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$d_k$
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$N$
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$N+1$
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<!-- l2h_begin mini_ker_l2h 38 -->
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$$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f +  f^g \log f\partial_x g\cr
 &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$
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<!-- l2h_end mini_ker_l2h 38 -->
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$x/\Delta$
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$\omega$
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$h$
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$$
\omega = h ( \eta , \varphi) 
$$
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<!-- l2h_end mini_ker_l2h 42 -->
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$\eta_1(t)$
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$\eta_1(t=0)$
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$t=0$
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$\phi_{j}(t)$
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$\eta_i(t=0)$
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$h(t)$
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$J$
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$T$
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<!-- l2h_begin mini_ker_l2h 51 -->
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$$
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
$$
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<!-- l2h_end mini_ker_l2h 51 -->
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$\psi$
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$l$
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<!-- l2h_begin mini_ker_l2h 54 -->
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$$\eqalign{
\partial_t \eta (t) &=  g(\eta(t),\varphi(t)) + W(t) \mu\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
}$$
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<!-- l2h_end mini_ker_l2h 54 -->
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$t=s_i$
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$\mu$
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$Q$
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$\nu$
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$R$
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$W$
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<!-- l2h_begin mini_ker_l2h 61 -->
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$$\left\{\eqalign{
\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
\partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
\partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.$$
$$\left\{\eqalign{
\varphi _1 &= \eta _1\cr
\varphi _2 &= \eta _2\cr
\varphi _3 &= \eta _3
}\right.$$
$$\left\{\eqalign{
\omega _1 &= \varphi _1 + \nu_1\cr
\omega _2 &= \eta _2 + \nu_2 \cr
\omega _3 &= \eta _3 + \nu_3
}\right.$$
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<!-- l2h_end mini_ker_l2h 61 -->
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\begin{rawhtml}
<!-- l2h_begin mini_ker_l2h 62 -->
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$g(\tau)$
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$\tau$
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$g(\tau;t)$
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$2$
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$tau$
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$0.2$
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$\sqrt{\sqrt{2}}$
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$g(t,\tau)$
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$A_{st}$
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$A_{st} + A_{st}^\dagger$
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$$
 U w V^\dagger
$$
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$U$
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$w$
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$V$
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<!-- l2h_end mini_ker_l2h 75 -->
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<!-- l2h_begin mini_ker_l2h 76 -->
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$\Phi(t+\delta t,t)=\exp{A_{st} \delta t}$
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$\Phi$
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<!-- l2h_begin mini_ker_l2h 78 -->
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$\Phi(t,0)$
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\end{document}
