# LaTeX2HTML 2019.2 (Released June 5, 2019)
# Associate images original text with physical files.


$key = q/Deltat^k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img8.svg"
 ALT="$\Delta t^k$">|; 

$key = q/Deltat_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.12ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img7.svg"
 ALT="$\Delta t_i$">|; 

$key = q/c_{ij}^{kl}(tau);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.98ex; vertical-align: -0.97ex; " SRC="|."$dir".q|img10.svg"
 ALT="$c_{ij}^{kl}(\tau)$">|; 

$key = q/d(svector{x}_i,svector{x}^k,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img1.svg"
 ALT="$d(\svector{x}_i,\svector{x}^k,t)$">|; 

$key = q/displaystyleC_i(tau)equivsum_{jkl}c_{ij}^{kl}(tau)~~mbox{and}~~C^k(tau)equivsum_{ijl}c_{ij}^{kl}(tau);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.16ex; vertical-align: -2.96ex; " SRC="|."$dir".q|img17.svg"
 ALT="$\displaystyle
C_i (\tau) \equiv \sum_{jkl} c_{ij}^{kl}(\tau)&nbsp;&nbsp;\mbox{and}&nbsp;&nbsp;
C^k (\tau) \equiv \sum_{ijl} c_{ij}^{kl}(\tau)
$">|; 

$key = q/displaystyleC_i^k(tau)equivsum_{jl}c_{ij}^{kl}(tau).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.16ex; vertical-align: -2.96ex; " SRC="|."$dir".q|img16.svg"
 ALT="$\displaystyle
C_i^k (\tau) \equiv \sum_{jl} c_{ij}^{kl}(\tau) .
$">|; 

$key = q/displaystyleC_{ij}(tau)equivsum_{kl}c_{ij}^{kl}(tau)~~mbox{and}~~C^{kl}(tau)equivsum_{ij}c_{ij}^{kl}(tau).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.11ex; vertical-align: -2.91ex; " SRC="|."$dir".q|img18.svg"
 ALT="$\displaystyle
C_{ij} (\tau) \equiv \sum_{kl} c_{ij}^{kl}(\tau)&nbsp;&nbsp;\mbox{and}&nbsp;&nbsp;
C^{kl} (\tau) \equiv \sum_{ij} c_{ij}^{kl}(\tau) .
$">|; 

$key = q/displaystylec_{ij}^{kl}(tau)Leftarrowc_{ij}^{kl}(tau-Deltat_i+Deltat_j-Deltat^k+Deltat^l).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 3.09ex; vertical-align: -0.97ex; " SRC="|."$dir".q|img25.svg"
 ALT="$\displaystyle c_{ij}^{kl}(\tau ) \Leftarrow
c_{ij}^{kl}(\tau - \Delta t_i + \Delta t_j
- \Delta t^k + \Delta t^l ) .
$">|; 

$key = q/displaystylec_{ij}^{kl}(tau)equivc_{ji}^{lk}(-tau)equivint_{t_1}^{t_2}d(svector{x}_i,svector{x}^k,t-tau)~d(svector{x}_j,svector{x}^l,t)~dt.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.13ex; vertical-align: -1.98ex; " SRC="|."$dir".q|img11.svg"
 ALT="$\displaystyle c_{ij}^{kl}(\tau) \equiv
c_{ji}^{lk}(-\tau) \equiv
\int_{t_1}^{t...
...\svector{x}_i,\svector{x}^k,t-\tau)&nbsp;
d(\svector{x}_j,\svector{x}^l,t)&nbsp; d t .
$">|; 

$key = q/displaystyled(svector{x}_i,svector{x}^k,t)Leftarrowd(svector{x}_i,svector{x}^k,t-Deltat_i-Deltat^k).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img24.svg"
 ALT="$\displaystyle d(\svector{x}_i,\svector{x}^k,t )
\Leftarrow d(\svector{x}_i,\svector{x}^k,t - \Delta t_i - \Delta t^k ) .
$">|; 

$key = q/displaystyled(svector{x}_i,svector{x}^k,t)approxd(t-Deltat_i-Deltat^k)~~mbox{for}~~t_1<t<t_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.80ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img9.svg"
 ALT="$\displaystyle
d(\svector{x}_i,\svector{x}^k,t) \approx d(t - \Delta t_i - \Delta t^k )
&nbsp;&nbsp;\mbox{for}&nbsp;&nbsp;t_1 &lt; t &lt; t_2.
$">|; 

$key = q/displaystylemax_{Deltat_iDeltat^k}left[C_i^k(Deltat_i+Deltat^k)+C_i(Deltat_i)+C^^k)+C_{ij}(Deltat_i+Deltat_j)+C^{kl}(Deltat^k+Deltat^l)right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 4.50ex; vertical-align: -2.10ex; " SRC="|."$dir".q|img23.svg"
 ALT="$\displaystyle \max_{\Delta t_i \Delta t^k }
\left [
C_i^k( \Delta t_i + \Delta ...
...C_{ij}( \Delta t_i + \Delta t_j ) +
C^{kl}( \Delta t^k + \Delta t^l )
\right ]
$">|; 

$key = q/displaystylembox{find}~Deltat_i~mbox{and}~Deltat^k~mbox{tominimize}~sum_{ijkl}(tau_{ij}^{kl}-Deltat_i+Deltat_j-Deltat^k+Deltat^l)^2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.16ex; vertical-align: -2.96ex; " SRC="|."$dir".q|img15.svg"
 ALT="$\displaystyle \mbox{find}&nbsp;
\Delta t_i&nbsp;\mbox{and}&nbsp; \Delta t^k
&nbsp;\mbox{to minim...
...l} ( \tau_{ij}^{kl} -
\Delta t_i + \Delta t_j - \Delta t^k + \Delta t^l )^2 .
$">|; 

$key = q/displaystylembox{thelags}~~tau_{ij}^{kl}approxDeltat_i-Deltat_j+Deltat^k-Deltat^l~~mbox{willmaximize}~~c_{ij}^{kl}(tau_{ij}^{kl}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 3.09ex; vertical-align: -0.97ex; " SRC="|."$dir".q|img13.svg"
 ALT="$\displaystyle \mbox{the lags}&nbsp;&nbsp;
\tau_{ij}^{kl} \approx \Delta t_i - \Delta t_...
...\Delta t^k - \Delta t^l
&nbsp;&nbsp;\mbox{will maximize}&nbsp;&nbsp;
c_{ij}^{kl}(\tau_{ij}^{kl}) .
$">|; 

$key = q/displaystylemin_{Deltat_iDeltat^k}~~sum_{i,k}(tau_i^k-Deltat_i-Deltat^k)^2+sum_i(tau_i-Deltat_i)^2+sum_k(tau^k-Deltat^k)^2+;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.16ex; vertical-align: -2.96ex; " SRC="|."$dir".q|img21.svg"
 ALT="$\displaystyle \min_{\Delta t_i \Delta t^k }&nbsp;&nbsp;
\sum_{i,k} ( \tau_i^k - \Delta t...
...t^k )^2 +
\sum_i ( \tau_i - \Delta t_i )^2 +
\sum_k ( \tau^k - \Delta t^k )^2 +$">|; 

$key = q/displaystylesum_{i,j}(tau_{ij}-Deltat_i+Deltat_j)^2+sum_{k,l}(tau^{kl}-Deltat^k+Deltat^l)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 5.16ex; vertical-align: -2.96ex; " SRC="|."$dir".q|img22.svg"
 ALT="$\displaystyle \sum_{i,j} ( \tau_{ij} - \Delta t_i + \Delta t_j )^2 +
\sum_{k,l} ( \tau^{kl} - \Delta t^k + \Delta t^l )^2$">|; 

$key = q/displaystyle{tau_i^k,tau_i,tau^k,tau_{ij},tau^{kl}}~mbox{maximize}~C_i^k(tau_i^ki),~C^k(tau^k),~C_{ij}(tau_{ij}),~mbox{and}~C^{kl}(tau^{kl}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.87ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img20.svg"
 ALT="$\displaystyle \{ \tau_i^k , \tau_i , \tau^k , \tau_{ij} , \tau^{kl} \}
&nbsp;\mbox{...
... ,&nbsp; C^k(\tau^k) ,&nbsp; C_{ij} (\tau_{ij}) ,
&nbsp;\mbox{and}&nbsp;
C^{kl} (\tau^{kl}) .
$">|; 

$key = q/displaystyle|svector{x}_i-svector{x}_j|<x_{max}~,~|svector{x}^k-svector{x}^l|<x^j+svector{x}^l)slash2|<x^{raisebox{-4pt}{{scriptstylemax{}}~.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.87ex; vertical-align: -0.75ex; " SRC="|."$dir".q|img12.svg"
 ALT="$\displaystyle
\Vert \svector{x}_i - \svector{x}_j \Vert &lt; x_{\max}&nbsp;,&nbsp;
\Vert \s...
...r{x}_j + \svector{x}^l)/2 \Vert
&lt; x^{\raisebox{-4pt}{$\scriptstyle \max$}}&nbsp; .
$">|; 

$key = q/svector{x}^k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.10ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img4.svg"
 ALT="$\svector{x}^k$">|; 

$key = q/svector{x}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.57ex; vertical-align: -0.45ex; " SRC="|."$dir".q|img3.svg"
 ALT="$\svector{x}_i$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 1.62ex; vertical-align: -0.10ex; " SRC="|."$dir".q|img2.svg"
 ALT="$t$">|; 

$key = q/tau_{ij}^{kl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.98ex; vertical-align: -0.97ex; " SRC="|."$dir".q|img14.svg"
 ALT="$\tau_{ij}^{kl}$">|; 

$key = q/|svector{x}_i+svector{x}^k|slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img6.svg"
 ALT="$\Vert \svector{x}_i + \svector{x}^k \Vert/2$">|; 

$key = q/|svector{x}_i-svector{x}^k|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img5.svg"
 ALT="$\Vert \svector{x}_i - \svector{x}^k \Vert$">|; 

$key = q/|tau|<tau_{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="|."$dir".q|img19.svg"
 ALT="$\vert \tau \vert &lt; \tau_{\max}$">|; 

1;

