% Automatically generated
\csname tex4ht\endcsname
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 1 -->\Hnewline}
$$\partial_t \eta (t) = g(\eta(t),\varphi(t))$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 1 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 2 -->\Hnewline}
$$
\varphi(t) = f(\eta(t),\varphi(t))
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 2 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 3 -->\Hnewline}
$$\pmatrix{A & B\cr
-C^+ & I-D\cr} \pmatrix{\delta \eta\cr
\delta \varphi\cr} = \pmatrix{\Gamma\cr
\Omega\cr}$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 3 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 4 -->\Hnewline}
$$\eqalign{\partial_t \eta (t) &= g(\eta(t),\varphi(t))\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
}$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 4 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 5 -->\Hnewline}
$$\left\{\eqalign{\partial_t \eta _{prey} &=  a \eta _{prey} - a \varphi _{meet} \cr
\partial_t \eta _{pred} &=  -c \eta _{pred} + c \varphi _{meet}\cr}\right.$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 5 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 6 -->\Hnewline}
$$\varphi _{meet} = \eta _{prey}\eta _{pred}$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 6 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 7 -->\Hnewline}
$$\partial_{\eta} g(\eta(t),\varphi(t));
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 7 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 8 -->\Hnewline}
$$\partial_{\varphi} g(\eta(t),\varphi(t));
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 8 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 9 -->\Hnewline}
$$\partial_{\eta} f(\eta(t),\varphi(t));
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 9 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 10 -->\Hnewline}
$$\partial_{\varphi} f(\eta(t),\varphi(t));
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 10 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 11 -->\Hnewline}
$$\left\{\eqalign{\partial_t \eta _{k} ^{pos}  &=  \eta _{k} ^{vel} \cr
\partial_t \eta _{k} ^{vel}  &= ( \varphi_k ^{spr} -\varphi _{k+1} ^{spr} + \varphi _{k} ^{dmp}-\varphi _{k+1} ^{dmp})\,/m_k  \cr}\right.$$
$$\left\{\eqalign{
\varphi_k ^{spr} &= -k_k (\eta _{k} ^{pos}- \eta _{k-1} ^{pos})\cr
\varphi_k ^{spr} &= -d_k (\eta _{k} ^{vel}- \eta _{k-1} ^{vel})
\cr}\right.$$
$$\left\{\eqalign{\eta ^{pos}_{0} &= 0\cr
\eta ^{vel}_{0} &= 0\cr
\varphi  ^{spr}_{N+1} &= 0\cr
\varphi ^{dmp}_{N+1} &= 0\cr}\right.$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 11 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 12 -->\Hnewline}
$$\eqalign{\partial_x f^g &= g f^{g-1}\partial_x f +  f^g \log f\partial_x g\cr
 &= f^{g-1}(g\partial_x f + f\partial_x g)\cr}$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 12 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 13 -->\Hnewline}
$$
\omega = h ( \eta , \varphi) 
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 13 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 14 -->\Hnewline}
$$
J = \psi[\eta(T),\varphi(T) ,h(T)] + \int_0 ^T {l[\eta(\tau),\varphi(\tau),h(\tau)]}\, d\tau
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 14 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 15 -->\Hnewline}
$$\eqalign{
\partial_t \eta (t) &=  g(\eta(t),\varphi(t)) + W(t) \mu\cr
\varphi(t) &= f(\eta(t),\varphi(t))\cr
\omega(t) &= h ( \eta(t) , \varphi(t)) + \nu\cr
}$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 15 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 16 -->\Hnewline}
$$\left\{\eqalign{
\partial_t \eta_1 &= a_{11} \eta_1 + a_{12} \varphi_2 + a_{13} \varphi_3 + W_{11} \mu_1 + W_{12} \mu_2\cr
\partial_t \eta_2 &= a_{21} \varphi_1 + a_{22} \eta_2 + a_{23} \varphi_3 + W_{21} \mu_1 + W_{22} \mu_2\cr
\partial_t \eta_3 &= a_{31} \varphi_1 + a_{32} \varphi_2 + a_{33} \eta_3 + W_{31} \mu_1 + W_{32} \mu_2
}\right.$$
$$\left\{\eqalign{
\varphi _1 &= \eta _1\cr
\varphi _2 &= \eta _2\cr
\varphi _3 &= \eta _3
}\right.$$
$$\left\{\eqalign{
\omega _1 &= \varphi _1 + \nu_1\cr
\omega _2 &= \eta _2 + \nu_2 \cr
\omega _3 &= \eta _3 + \nu_3
}\right.$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 16 -->\Hnewline}
\HCode{\Hnewline <!-- tex4ht_begin mini_ker_tex4ht_tex tex 17 -->\Hnewline}
$$
 U w V^\dagger
$$
\HCode{\Hnewline <!-- tex4ht_end mini_ker_tex4ht_tex tex 17 -->\Hnewline}

\bye
