# LaTeX2HTML 2002-2-1 (1.71)
# Associate images original text with physical files.


$key = q/18.44;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$18.44$">|; 

$key = q/u(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="$u(x,y)$">|; 

$key = q/{displaymath}omega_p^2=frac{n_0,e^2}{epsilon_0,m_e}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="\begin{displaymath}
\omega_p^2 = \frac{n_0 e^2}{\epsilon_0 m_e}
\end{displaymath}">|; 

$key = q/deltaz=LslashI;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="$\delta z = L/I$">|; 

$key = q/{displaymath}m,l,frac{d^2theta}{dt^2}+nu,frac{dtheta}{dt}+m,g,sintheta=A,cosomegat,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="\begin{displaymath}
m l \frac{d^2\theta}{dt^2}+ \nu \frac{d\theta}{dt} + m g \sin\theta = A \cos\omega t,
\end{displaymath}">|; 

$key = q/O(h^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$O(h^2)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{reject.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="324" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{reject.eps}}
\end{figure}">|; 

$key = q/{displaymath}b_N=-2+frac{beta_h}{alpha_h,deltax+beta_h},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="\begin{displaymath}
b_N = -2 + \frac{\beta_h}{\alpha_h \delta x+\beta_h},
\end{displaymath}">|; 

$key = q/{displaymath}a_i,u_{i-1}+b_i,u_i+c_i,(x_i,u_i+y_i)=w_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="\begin{displaymath}
a_i u_{i-1} + b_i u_i+c_i (x_i u_i+y_i) = w_i,
\end{displaymath}">|; 

$key = q/a_1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="$a_1=0$">|; 

$key = q/{displaymath}n_jrightarrown_j+left.left(frac{x_{j+1}-r_i}{x_{j+1}-x_j}right)rightslashdeltax,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="\begin{displaymath}
n_j \rightarrow n_j+\left.\left(\frac{x_{j+1}-r_i}{x_{j+1}-x_j}\right)\right/\delta x,
\end{displaymath}">|; 

$key = q/displaystyleT_0^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="$\displaystyle T_0^n$">|; 

$key = q/y(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$y(x)$">|; 

$key = q/5times5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$5\times 5$">|; 

$key = q/int_0^Ln(x),dx=n_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="$\int_0^L n(x) dx = n_0$">|; 

$key = q/|A|<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$\vert A\vert&lt;1$">|; 

$key = q/{figure}{epsfysize{=5in{centerline{epsffile{temp.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="575" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="\begin{figure}
\epsfysize =5in
\centerline{\epsffile{temp.eps}}\end{figure}">|; 

$key = q/a=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$a=0$">|; 

$key = q/T_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$T_c$">|; 

$key = q/v';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="$v'$">|; 

$key = q/displaystylefrac{Gamma_{h,j}^n,deltax+beta_h^n,hat{T}_{N,j}^n}{alpha_h^n,deltax+beta_h^n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="$\displaystyle \frac{\Gamma_{h j}^n \delta x + \beta_h^n \hat{T}_{N,j}^n}{\alpha_h^n \delta x +\beta_h^n},$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{farray.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{farray.eps}}
\end{figure}">|; 

$key = q/{displaymath}F_n=frac{r_{n-1}-r_{n-2}}{r_{n}-r_{n-1}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="\begin{displaymath}
F_n =\frac{r_{n-1}-r_{n-2}}{r_{n}-r_{n-1}}
\end{displaymath}">|; 

$key = q/displaystyle-frac{2,i,D}{1+i^2,D^2},hat{E}_i^n+frac{1-i^2,D^2}{1+i^2,D^2},hat{H}_i^n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="$\displaystyle -\frac{2 i D}{1+i^2 D^2} \hat{E}_i^n + \frac{1-i^2 D^2}{1+i^2 D^2} \hat{H}_i^n,$">|; 

$key = q/10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="$10^{-4}$">|; 

$key = q/10times10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$10\times 10$">|; 

$key = q/x_lleqxleqx_h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$x_l\leq x\leq x_h$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pd_phased.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pd_phased.eps}}
\end{figure}">|; 

$key = q/{displaymath}I={3,1,7,9,3,1,cdots}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="\begin{displaymath}
I=\{3, 1, 7, 9, 3, 1, \cdots\}.
\end{displaymath}">|; 

$key = q/z_i=i,deltaz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="$z_i = i \delta z$">|; 

$key = q/{displaymath}sigma=sqrt{N',p,(1-p)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="\begin{displaymath}
\sigma = \sqrt{N' p (1-p)}.
\end{displaymath}">|; 

$key = q/D_x=pi,c,deltatslash(2,L_x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="$D_x=\pi c \delta t/(2 L_x)$">|; 

$key = q/k=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$k=10$">|; 

$key = q/M=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="$M=10$">|; 

$key = q/displaystylefrac{Aleft[(1-omega^2),cosomegat+(omegaslashQ),sinomegatright]}{left[(1-omega^2)^2+omega^2slashQ^2right]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$\displaystyle \frac{A\left[(1-\omega^2) \cos\omega t+ (\omega/Q) \sin\omega t\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]},$">|; 

$key = q/{displaymath}nabla^2u({{bf{r})=v({{bf{r}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="\begin{displaymath}
\nabla^2 u({\bf r}) = v({\bf r}).
\end{displaymath}">|; 

$key = q/{displaymath}|P_x(x),dx|=|P_y(y),dy|.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="\begin{displaymath}
\vert P_x(x) dx\vert = \vert P_y(y) dy\vert.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{waveb.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="317" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{waveb.eps}}
\end{figure}">|; 

$key = q/x_0<x_1<x_2cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$x_0&lt;x_1&lt;x_2 \cdots$">|; 

$key = q/f_i=f(x_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="$f_i = f(x_i)$">|; 

$key = q/O(eta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$O(\eta)$">|; 

$key = q/h^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$h^{-4}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p2dd.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p2dd.eps}}
\end{figure}">|; 

$key = q/Q=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$Q=10$">|; 

$key = q/J_z=-mu_0,c^2,j_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="$J_z =-\mu_0 c^2 j_z$">|; 

$key = q/{displaymath}{{bf{M}=left(array{{rrrrrr}-2&1&0&0&0&01&-2&1&0&0&00&1&-2&1&0&00&0&1&-2&1&00&0&0&1&-2&10&0&0&0&1&-2array{right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="145" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="\begin{displaymath}
{\bf M} = \left(\begin{array}{rrrrrr}
-2 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0...
...1 &amp; -2 &amp; 1 \\\\
0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; -2 \\\\
\end{array} \right)
\end{displaymath}">|; 

$key = q/displaystyley(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$\displaystyle y(t=0)$">|; 

$key = q/2times10^4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="$2\times 10^4$">|; 

$key = q/hproptoN^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="$h\propto N^{-1}$">|; 

$key = q/0leqxleqL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="$0\leq x\leq L$">|; 

$key = q/gamma_h=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$\gamma_h=0$">|; 

$key = q/sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$\sin\theta$">|; 

$key = q/v(0)=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$v(0)=1$">|; 

$key = q/displaystyleT_{N+1,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="$\displaystyle T_{N+1,j}^n$">|; 

$key = q/{displaymath}F_n=frac{Q_{n-1}-Q_{n-2}}{Q_{n}-Q_{n-1}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="\begin{displaymath}
F_n = \frac{Q_{n-1}-Q_{n-2}}{Q_{n}-Q_{n-1}}
\end{displaymath}">|; 

$key = q/t=t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$t=t$">|; 

$key = q/{displaymath}(n+1)!=(n+1),n!{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="\begin{displaymath}
(n+1)! = (n+1) n!
\end{displaymath}">|; 

$key = q/{displaymath}F_j^S=2,{{rm{i},F_j.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="\begin{displaymath}
F_j^S = 2 {\rm i} F_j.
\end{displaymath}">|; 

$key = q/displaystylefrac{dr_i}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="$\displaystyle \frac{dr_i}{dt}$">|; 

$key = q/{displaymath}frac{d^2theta}{dhat{t}^2}+frac{1}{Q},frac{dtheta}{dhat{t}}+sintheta=hat{A},coshat{omega}hat{t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="\begin{displaymath}
\frac{d^2\theta}{d\hat{t}^2} + \frac{1}{Q} \frac{d\theta}{d\hat{t}}
+ \sin\theta = \hat{A} \cos\hat{\omega}\hat{t}.
\end{displaymath}">|; 

$key = q/rho,v,A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$\rho v A$">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$B$">|; 

$key = q/displaystylefrac{Gamma_{l,j}^n,deltax-beta_l^n,hat{T}_{1,j}^n}{alpha_l^n,deltax-beta_l^n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="$\displaystyle \frac{\Gamma_{l j}^n \delta x-\beta_l^n \hat{T}_{1,j}^n}{\alpha_l^n \delta x -\beta_l^n},$">|; 

$key = q/u(x_l)=u(x_h)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$u(x_l) = u(x_h)=0$">|; 

$key = q/displaystyleleft{theta(0)-frac{A,(1-omega^2)}{[(1-omega^2)^2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q}cosomega_astt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="$\displaystyle \left\{\theta(0) - \frac{A (1-\omega^2)}{[(1-\omega^2)^2+\omega^2/Q^2]}\right\}
{\rm e}^{-t/2Q}\cos\omega_\ast t$">|; 

$key = q/deltat=2.49times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\delta t = 2.49\times 10^{-3}$">|; 

$key = q/displaystyleg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="$\displaystyle g$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$t$">|; 

$key = q/displaystylev_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="$\displaystyle v_i,$">|; 

$key = q/{displaymath}frac{2}{L}int_0^Lcos(j,pi,yslashL),cos(k,pi,yslashL),dy=delta_{jk}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="\begin{displaymath}
\frac{2}{L}\int_0^L \cos(j \pi y/L) \cos(k \pi y/L) dy = \delta_{jk}.
\end{displaymath}">|; 

$key = q/displaystyle-frac{e,E(r_i)}{m_e},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="$\displaystyle - \frac{e E(r_i)}{m_e},$">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$C$">|; 

$key = q/J_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="$J_z$">|; 

$key = q/int_{x_l}^{x_h}f(x),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$\int_{x_l}^{x_h} f(x) dx$">|; 

$key = q/phi=0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$\phi = 0^\circ $">|; 

$key = q/O(h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$O(h)$">|; 

$key = q/{displaymath}v(x,y)=6,x,y,(1-y)-2,x^3{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="\begin{displaymath}
v(x,y) = 6 x y (1-y) - 2 x^3
\end{displaymath}">|; 

$key = q/displaystylesum_{j'=0,J-1}hat{rho}_{j'},{{rm{e}^{,{{rm{i},j,j',2,pislashJ},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="$\displaystyle \sum_{j'=0,J-1}\hat{\rho}_{j'}  {\rm e}^{ {\rm i} j j' 2 \pi/J},$">|; 

$key = q/T=3,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$T=3 T_0$">|; 

$key = q/Qsimeq1.3755;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="$Q\simeq 1.3755$">|; 

$key = q/{displaymath}y(0)=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="\begin{displaymath}
y(0) = 1.
\end{displaymath}">|; 

$key = q/intf,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="$\int f dV$">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$t=0$">|; 

$key = q/m=1slashn=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="$m=1/n=1$">|; 

$key = q/D=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$D=1$">|; 

$key = q/Q=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$Q=2$">|; 

$key = q/H_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$H_x$">|; 

$key = q/u(x,t)=hat{u}_k(t),exp(,{{rm{i},k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="$u(x,t)=\hat{u}_k(t) \exp( {\rm i} k x)$">|; 

$key = q/O(deltax)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="$O(\delta x)^2$">|; 

$key = q/E(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="$E(x)$">|; 

$key = q/Q=1.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$Q=1.3$">|; 

$key = q/{displaymath}{{bf{q}=-kappa,nablaT,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="\begin{displaymath}
{\bf q} = - \kappa \nabla T,
\end{displaymath}">|; 

$key = q/displaystyle-F(v),v,v_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$\displaystyle - F(v) v v_x,$">|; 

$key = q/beta=1slashkT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$\beta = 1/kT$">|; 

$key = q/{displaymath}V=V',frac{N}{N'}pmfrac{sqrt{V,(V'-V)}}{sqrt{N'}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="\begin{displaymath}
V = V' \frac{N}{N'} \pm \frac{\sqrt{V (V'-V)}}{\sqrt{N'}}.
\end{displaymath}">|; 

$key = q/{displaymath}w_N=v_N,(deltax)^2-frac{gamma_h,deltax}{alpha_h,deltax+beta_h}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="\begin{displaymath}
w_N = v_N (\delta x)^2 - \frac{\gamma_h \delta x}{\alpha_h \delta x+\beta_h}.
\end{displaymath}">|; 

$key = q/-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$-1$">|; 

$key = q/{displaymath}b_1=-2-frac{beta_l}{alpha_l,deltax-beta_l},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="\begin{displaymath}
b_1 = -2 - \frac{\beta_l}{\alpha_l \delta x-\beta_l},
\end{displaymath}">|; 

$key = q/v({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="$v({\bf r})$">|; 

$key = q/2times10^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$2\times 10^5$">|; 

$key = q/Q=10^{16};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$Q=10^{16}$">|; 

$key = q/theta=4^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\theta = 4^\circ $">|; 

$key = q/displaystyley_{{rm{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="$\displaystyle y_{\rm min}$">|; 

$key = q/displaystyleU_{0,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="$\displaystyle U_{0,j}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{q05p.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{q05p.eps}}
\end{figure}">|; 

$key = q/displaystyle(J_z)_{i,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="$\displaystyle (J_z)_{i,j}^n$">|; 

$key = q/deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="$\delta t$">|; 

$key = q/displaystylegamma_l,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$\displaystyle \gamma_l,$">|; 

$key = q/displaystylefrac{dy}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="$\displaystyle \frac{dy}{dt}$">|; 

$key = q/n(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="$n(x)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ising2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ising2.eps}}
\end{figure}">|; 

$key = q/{tth}=1times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="${\tt h}=1\times 10^{-4}$">|; 

$key = q/v(0)=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$v(0)=-1$">|; 

$key = q/omega=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$\omega =1$">|; 

$key = q/{displaymath}sinthetasimeqtheta{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="\begin{displaymath}
\sin\theta \simeq \theta
\end{displaymath}">|; 

$key = q/L=40;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="$L=40$">|; 

$key = q/n>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$n&gt;1$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$\epsilon $">|; 

$key = q/displaystylevarphi_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="$\displaystyle \varphi_0,$">|; 

$key = q/f(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="$f(y)$">|; 

$key = q/A=1.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$A=1.5$">|; 

$key = q/{displaymath}u(x,y)=(1-x^2),(2,y^3-3,y^2+1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="\begin{displaymath}
u(x,y) = (1-x^2) (2 y^3-3 y^2+1).
\end{displaymath}">|; 

$key = q/{displaymath}bar{N}=sum_{i=1,N'}bar{x}=N',bar{x},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="\begin{displaymath}
\bar{N} = \sum_{i=1,N'} \bar{x}= N' \bar{x},
\end{displaymath}">|; 

$key = q/displaystyle(H_y)_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="$\displaystyle (H_y)_i^n$">|; 

$key = q/{displaymath}{mitDelta}xsimsqrt{D,t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="\begin{displaymath}
{\mit\Delta}x \sim \sqrt{D t}.
\end{displaymath}">|; 

$key = q/0.46;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$0.46$">|; 

$key = q/A,I_n>10^{32}-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="$A I_n &gt; 10^{32}-1$">|; 

$key = q/2times2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="$2\times 2$">|; 

$key = q/j=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$j=1$">|; 

$key = q/displaystylefrac{gamma_l^n,deltax-beta_l^n,T_1^n}{alpha_l^n,deltax-beta_l^n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="$\displaystyle \frac{\gamma_l^n \delta x-\beta_l^n T_1^n}{\alpha_l^n \delta x -\beta_l^n},$">|; 

$key = q/{displaymath}alpha_l,U_j(x)+beta_l,frac{dU_j(x)}{dx}=Gamma_{l,j},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="\begin{displaymath}
\alpha_l U_j(x) + \beta_l \frac{dU_j(x)}{dx} = \Gamma_{l j},
\end{displaymath}">|; 

$key = q/{displaymath}epsilonsimfrac{eta}{h}+h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="\begin{displaymath}
\epsilon \sim \frac{\eta}{h} + h.
\end{displaymath}">|; 

$key = q/displaystylev,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="$\displaystyle v,$">|; 

$key = q/Q=Q_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$Q=Q_\infty$">|; 

$key = q/t=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$t=1$">|; 

$key = q/H_y=-c,B_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="$H_y=-c B_y$">|; 

$key = q/C=1283;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$C=1283$">|; 

$key = q/l=18.44,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$l=18.44 {\rm m}$">|; 

$key = q/theta_0(t_0)+deltatheta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="$\theta_0(t_0)+\delta\theta_0$">|; 

$key = q/|y_i|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$\vert y_i\vert$">|; 

$key = q/{displaymath}frac{partialT(x,0,t)}{partialy}=frac{partialT(x,L,t)}{partialy}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="\begin{displaymath}
\frac{\partial T(x,0,t)}{\partial y} = \frac{\partial T(x,L,t)}{\partial y} = 0.
\end{displaymath}">|; 

$key = q/bar{s}_{i+1}rightarrowbar{s}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="$\bar{s}_{i+1}\rightarrow \bar{s}_i$">|; 

$key = q/{displaymath}bar{s}=tanh(beta,mu,H_m).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="\begin{displaymath}
\bar{s} = \tanh(\beta \mu H_m).
\end{displaymath}">|; 

$key = q/displaystyleh,f(x_n+hslash2,y_n+k_1slash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="$\displaystyle h f(x_n+h/2, y_n+k_1/2),$">|; 

$key = q/10^{-5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$10^{-5}$">|; 

$key = q/y=f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$y=f(x)$">|; 

$key = q/{displaymath}f_j=sum_{k=0}^{J}F_k^C,cos(j,k,pislashJ).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="\begin{displaymath}
f_j= \sum_{k=0}^{J} F_k^C \cos(j k \pi/J).
\end{displaymath}">|; 

$key = q/2,J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="$2 J$">|; 

$key = q/{displaymath}overline{(x-bar{x})^2}=p-p^2=p,(1-p),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="\begin{displaymath}
\overline{(x-\bar{x})^2}= p -p^2 =p (1-p),
\end{displaymath}">|; 

$key = q/{displaymath}e_i=-frac{J}{2}sum_{k=1,z}s_k,s_i-mu,H,s_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="\begin{displaymath}
e_i = -\frac{J}{2}\sum_{k=1,z}s_k s_i -\mu H s_i,
\end{displaymath}">|; 

$key = q/displaystylek_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$\displaystyle k_2$">|; 

$key = q/{displaymath}A=frac{1-{{rm{i},(Cslash2),sin(k,deltax)}{1+{{rm{i},(Cslash2),sin(k,deltax)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="\begin{displaymath}
A = \frac{1 - {\rm i}  (C/2) \sin (k  \delta x)}{1 + {\rm i}  (C/2) \sin (k  \delta x)}.
\end{displaymath}">|; 

$key = q/v(0)=-3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$v(0)=-3$">|; 

$key = q/n_jequivn(x_j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="$n_j\equiv n(x_j)$">|; 

$key = q/NtimesN;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="$N\times N$">|; 

$key = q/displaystylev_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$\displaystyle v_y,$">|; 

$key = q/A=16807;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="$A=16807$">|; 

$key = q/deltax=LslashJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="$\delta x = L / J$">|; 

$key = q/P_{y,{{rm{max}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="$P_{y {\rm max}}$">|; 

$key = q/displaystyle-F(v),v,v_x+B,omega,(v_z,sinphi-v_y,cosphi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$\displaystyle - F(v) v v_x + B \omega (v_z \sin\phi-v_y \cos\phi),$">|; 

$key = q/1.37555;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="$1.37555$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{wave1da.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="317" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{wave1da.eps}}
\end{figure}">|; 

$key = q/displaystyleh,f(x_n+h,y_n+k_3),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$\displaystyle h f(x_n+h, y_n+k_3),$">|; 

$key = q/P_y(y)<x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="$P_y(y) &lt; x$">|; 

$key = q/deltay;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="$\delta y$">|; 

$key = q/{displaymath}hat{rho}_j=frac{1}{J}sum_{j'=0,J-1}rho_{j'},{{rm{e}^{-{{rm{i},j,j',2,pislashJ},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="\begin{displaymath}
\hat{\rho}_j = \frac{1}{J}\sum_{j'=0,J-1}\rho_{j'}  {\rm e}^{-{\rm i} j j' 2 \pi/J},
\end{displaymath}">|; 

$key = q/150,{{rm{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$150 {\rm g}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ising3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ising3.eps}}
\end{figure}">|; 

$key = q/w_i=v_i,(deltax)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="$w_i = v_i (\delta x)^2$">|; 

$key = q/displaystyleomega_0,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$\displaystyle \omega_0 t,$">|; 

$key = q/{displaymath}H_c=frac{k,T_c}{mu}=frac{z,J}{2,mu}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="\begin{displaymath}
H_c = \frac{k T_c}{\mu} = \frac{z J}{2 \mu}.
\end{displaymath}">|; 

$key = q/displaystyleT_{0,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="$\displaystyle T_{0,j}^n$">|; 

$key = q/{displaymath}T_i^{n+1}=T_i^n+C,left(T_{i-1}^n-2,T_i^n+T_{i+1}^nright){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="\begin{displaymath}
T_i^{n+1} = T_i^n + C \left(T_{i-1}^n-2 T_i^n+T_{i+1}^n\right)
\end{displaymath}">|; 

$key = q/displaystylehat{T}_{N+1,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="$\displaystyle \hat{T}_{N+1,j}^n$">|; 

$key = q/4.9pm0.20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$4.9\pm 0.20$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\phi$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{d1372.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{d1372.eps}}
\end{figure}">|; 

$key = q/{displaymath}f_D=6,pi,eta,r,v.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="\begin{displaymath}
f_D = 6 \pi \eta r v.
\end{displaymath}">|; 

$key = q/T_i^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="$T_i^{n+1}$">|; 

$key = q/{displaymath}u_i=-frac{a_i,u_{i-1}}{b_i+c_i,x_i}+frac{w_i-c_i,y_i}{b_i+c_i,x_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="\begin{displaymath}
u_i = - \frac{a_i u_{i-1}}{b_i+c_i x_i} + \frac{w_i-c_i y_i}{b_i+c_i x_i}.
\end{displaymath}">|; 

$key = q/{displaymath}x_i=-frac{a_{i+1}}{b_{i+1}+c_{i+1},x_{i+1}},mbox{hspace{1cm}}y_i=fr_{i+1}-c_{i+1},y_{i+1}}{b_{i+1}+c_{i+1},x_{i+1}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="\begin{displaymath}
x_i = - \frac{a_{i+1}}{b_{i+1}+c_{i+1} x_{i+1}},\mbox{\hspa...
...i =
\frac{w_{i+1}-c_{i+1} y_{i+1}}{b_{i+1}+c_{i+1} x_{i+1}}
\end{displaymath}">|; 

$key = q/E_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$E_z$">|; 

$key = q/0leqyleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="$0\leq y \leq 1$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{scanqva.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="359" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{scanqva.eps}}
\end{figure}">|; 

$key = q/+y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$+y$">|; 

$key = q/C=12345;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$C=12345$">|; 

$key = q/E_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="$E_b$">|; 

$key = q/{displaymath}deltat<frac{deltax}{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="\begin{displaymath}
\delta t &lt; \frac{\delta x}{v}.
\end{displaymath}">|; 

$key = q/i=N-2,1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="$i=N-2, 1$">|; 

$key = q/x=sqrt{k},nu^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="$x = \sqrt{k} \nu^2$">|; 

$key = q/omega=20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\omega =20$">|; 

$key = q/hat{rho}_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="$\hat{\rho}_j$">|; 

$key = q/displaystyle-frac{v}{Q}-theta+A,cosomegat.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="$\displaystyle -\frac{v}{Q} -\theta + A \cos\omega t.$">|; 

$key = q/{displaymath}frac{T(x,t_{n+1})-T(x,t_n)}{deltat}=D,frac{partial^2T(x,t_n)}{partialx^2}+O(deltat).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="\begin{displaymath}
\frac{T(x,t_{n+1}) - T(x,t_n)}{\delta t}
= D  \frac{\partial^2 T(x,t_n)}{\partial x^2}+ O(\delta t).
\end{displaymath}">|; 

$key = q/v_0=85;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$v_0=85$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{q130u.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{q130u.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{scanvd.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="322" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{scanvd.eps}}
\end{figure}">|; 

$key = q/R_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$R_e$">|; 

$key = q/gamma_l=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$\gamma _l=1$">|; 

$key = q/displaystylefrac{D}{2},frac{T_{i-1,j}^{n}-2,T_{i,j}^{n}+T_{i+1,j}^{n}}{(deltax)^+frac{D}{2}left(frac{partial^2T}{partialy^2}right)_{i,j}^{n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="$\displaystyle \frac{D}{2} 
\frac{T_{i-1,j}^{n}-2 T_{i,j}^{n}+T_{i+1,j}^{n}}{(...
...ta x)^2} +
\frac{D}{2}\left(\frac{\partial^2 T}{\partial y^2}\right)_{i,j}^{n},$">|; 

$key = q/{displaymath}{{bf{f}=-nabla,phi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="\begin{displaymath}
{\bf f} = - \nabla \phi.
\end{displaymath}">|; 

$key = q/{displaymath}omega_0=sqrt{frac{g}{l}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="\begin{displaymath}
\omega_0 = \sqrt{\frac{g}{l}}.
\end{displaymath}">|; 

$key = q/H_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="$H_m$">|; 

$key = q/displaystyle(E_z)_{i,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="$\displaystyle (E_z)_{i,j}^n$">|; 

$key = q/2,Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$2 Q$">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$E$">|; 

$key = q/{displaymath}F_j^S=frac{2}{J}sum_{k=1}^{J-1}f_k,sin(j,k,pislashJ){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="\begin{displaymath}
F_j^S = \frac{2}{J}\sum_{k=1}^{J-1} f_k \sin(j k \pi/J)
\end{displaymath}">|; 

$key = q/t_n=t_0+n,deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="$t_n=t_0+n \delta t$">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{basin2a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="542" HEIGHT="439" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{basin2a.eps}}
\end{figure}">|; 

$key = q/displaystyledeltav_0,cos(omega_ast{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="$\displaystyle \delta v_0 \cos(\omega_\ast {\mit\Delta} t)  {\rm e}^{-{\mit\Delta} t/2Q}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{resonance.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="321" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{resonance.eps}}
\end{figure}">|; 

$key = q/(E_x)_i^nequivE_x(z_i,t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="$(E_x)_i^n\equiv E_x(z_i,t_n)$">|; 

$key = q/{displaymath}T_c=frac{z,J}{2,k},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="\begin{displaymath}
T_c = \frac{z J}{2 k},
\end{displaymath}">|; 

$key = q/t=0.002;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$t=0.002$">|; 

$key = q/n=0,1,2,cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$n=0,1,2,\cdots$">|; 

$key = q/phi(0)=phi(L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="$\phi(0)=\phi(L)$">|; 

$key = q/{displaymath}f_D=frac{1}{2},C_D(v),rho,A,v^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="\begin{displaymath}
f_D = \frac{1}{2} C_D(v) \rho A v^2,
\end{displaymath}">|; 

$key = q/1slashL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="$1/L$">|; 

$key = q/3cdot2^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="$3\cdot 2^n$">|; 

$key = q/{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="${\bf c}$">|; 

$key = q/{displaymath}u_i^{n+1}=u_i^n-frac{C}{2},(u_{i+1}^n-u_{i-1}^n),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="\begin{displaymath}
u_i^{n+1} = u_i^n -\frac{C}{2} (u_{i+1}^n-u_{i-1}^n),
\end{displaymath}">|; 

$key = q/f_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$f_0$">|; 

$key = q/displaystylesum_{j=0,I}hat{H}_j^n,cos(i,j,pislashI),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="$\displaystyle \sum_{j=0,I} \hat{H}_j^n \cos(i j \pi/I),$">|; 

$key = q/{displaymath}theta(t+n,tau)=theta(t)+2pi,l{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="\begin{displaymath}
\theta(t+n \tau) = \theta(t) + 2\pi l
\end{displaymath}">|; 

$key = q/{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="${\rm m/s}$">|; 

$key = q/-x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="$-x$">|; 

$key = q/y=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$y=0.5$">|; 

$key = q/x_i=x_l+i,deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="$x_i = x_l + i \delta x$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p2db.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p2db.eps}}
\end{figure}">|; 

$key = q/C=0.408;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="$C=0.408$">|; 

$key = q/4.669ldots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$4.669\ldots$">|; 

$key = q/d^2slashdx^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="$d^2/dx^2$">|; 

$key = q/{displaymath}|A|^2=1-(1-C^2),sin^2(k,deltax).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="\begin{displaymath}
\vert A\vert^2 = 1 - (1-C^2) \sin^2(k \delta x).
\end{displaymath}">|; 

$key = q/theta<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="$\theta&lt;0$">|; 

$key = q/displaystylephi_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$\displaystyle \phi_j$">|; 

$key = q/{{tt{Nacc}geq100;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="${\tt Nacc}\geq 100$">|; 

$key = q/lambda_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="$\lambda_1$">|; 

$key = q/displaystylek_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$\displaystyle k_3$">|; 

$key = q/displaystylefrac{v}{t}-4,k,t^2,x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="$\displaystyle \frac{v}{t} - 4 k t^2 x,$">|; 

$key = q/{{bf{B}=[0,B_y(z,t),0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="${\bf B} = [0,B_y(z,t),0]$">|; 

$key = q/x_l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="$x_l=0$">|; 

$key = q/displaystyle-f_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="$\displaystyle - f_j,$">|; 

$key = q/5times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="$5\times 10^{-3}$">|; 

$key = q/Q=1.375;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$Q=1.375$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cavity4.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="324" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cavity4.eps}}
\end{figure}">|; 

$key = q/{{bf{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="${\bf f}$">|; 

$key = q/displaystylefrac{partialE_z}{partialt}+c,frac{partialH_y}{partialx}+c,frac{partialH_x}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="$\displaystyle \frac{\partial E_z}{\partial t} + c \frac{\partial H_y}{\partial x}
+ c \frac{\partial H_x}{\partial y}$">|; 

$key = q/T({{bf{r},t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="$T({\bf r}, t_0)$">|; 

$key = q/N=L^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="$N=L^2$">|; 

$key = q/displaystyle1mbox{hspace{1cm}for~~{|x-0.5|<0.1{~~and~~{|y-0.5|<0.1{,};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="$\displaystyle 1\mbox{\hspace{1cm}for&nbsp;&nbsp;$\vert x-0.5\vert&lt;0.1$&nbsp;&nbsp;and&nbsp;&nbsp;$\vert y-0.5\vert&lt;0.1$,}$">|; 

$key = q/{displaymath}x_{N-1}=-frac{a_N}{b_N},mbox{hspace{1cm}}y_{N-1}=frac{w_N}{b_N},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="\begin{displaymath}
x_{N-1} = -\frac{a_N}{b_N},\mbox{\hspace{1cm}}y_{N-1} = \frac{w_N}{b_N},
\end{displaymath}">|; 

$key = q/{displaymath}H_x=frac{partialH_y}{partialx}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="\begin{displaymath}
H_x = \frac{\partial H_y}{\partial x} = 0
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{diff2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="549" HEIGHT="444" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{diff2.eps}}
\end{figure}">|; 

$key = q/A=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$A=1$">|; 

$key = q/v=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$v=1$">|; 

$key = q/0leqzleqL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="$0\leq z \leq L$">|; 

$key = q/{displaymath}{{bf{E}=-nabla,phi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="\begin{displaymath}
{\bf E} = - \nabla \phi.
\end{displaymath}">|; 

$key = q/{displaymath}u_{i+1}=x_i,u_i+y_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="\begin{displaymath}
u_{i+1} = x_i u_i + y_i.
\end{displaymath}">|; 

$key = q/{displaymath}alpha_h,u(x)+beta_h,frac{du(x)}{dx}=gamma_h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="\begin{displaymath}
\alpha_h  u(x)+ \beta_h \frac{du(x)}{dx} = \gamma_h,
\end{displaymath}">|; 

$key = q/phi=135^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\phi = 135^\circ $">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pd_scanvud.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="330" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pd_scanvud.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{partialH_y}{partialt}+c,frac{partialE_z}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$\displaystyle \frac{\partial H_y}{\partial t} + c \frac{\partial E_z}{\partial x}$">|; 

$key = q/Q_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$Q_n$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{image3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="217" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{image3.eps}}\end{figure}">|; 

$key = q/f=0.7071;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$f=0.7071$">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="$V$">|; 

$key = q/O(h^2)timesO(h^{-1})=O(h)=O(N^{-1slash2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="$O(h^2)\times O(h^{-1}) =
O(h)= O(N^{-1/2})$">|; 

$key = q/{figure}{epsfysize{=5in{centerline{epsffile{wave1d.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="560" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="\begin{figure}
\epsfysize =5in
\centerline{\epsffile{wave1d.eps}}
\end{figure}">|; 

$key = q/eta=2.22times10^{-16};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$\eta = 2.22\times 10^{-16}$">|; 

$key = q/J+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="$J+1$">|; 

$key = q/displaystylefrac{dx_j}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$\displaystyle \frac{dx_j}{dt}$">|; 

$key = q/h_0sim3times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$h_0\sim 3\times
10^{-4}$">|; 

$key = q/theta_0(t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="$\theta_0(t_0)$">|; 

$key = q/gamma_h=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$\gamma _h=1$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{d1375.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{d1375.eps}}
\end{figure}">|; 

$key = q/sin(j,pi,yslashL);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="$\sin(j \pi y/L)$">|; 

$key = q/u({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$u({\bf r})$">|; 

$key = q/b_i=-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="$b_i = -2$">|; 

$key = q/l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="$l=0$">|; 

$key = q/eta=1.19times10^{-7};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$\eta=1.19\times 10^{-7}$">|; 

$key = q/beta_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="$\beta_l$">|; 

$key = q/+z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="$+z$">|; 

$key = q/displaystylefrac{gamma_h,deltax+beta_h,u_N}{alpha_h,deltax+beta_h}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="$\displaystyle \frac{\gamma_h \delta x + \beta_h u_N}{\alpha_h \delta x +\beta_h}.$">|; 

$key = q/{displaymath}P_y(y)=left{array{{lcl}{{rm{e}^{-y}&mbox{hspace{1cm}}&mbox{if{0leqyleqinfty{}0&&mbox{otherwise}array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="\begin{displaymath}
P_y(y) = \left\{\begin{array}{lcl}
{\rm e}^{-y}&amp;\mbox{\hspac...
...eq y \leq \infty$}\\\\
0&amp;&amp; \mbox{otherwise}
\end{array}\right..
\end{displaymath}">|; 

$key = q/x_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$x_j$">|; 

$key = q/v=partialxislashpartialt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="$v=\partial\xi/\partial t$">|; 

$key = q/gamma_h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="$\gamma_h$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{pic2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="395" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{pic2.eps}}
\end{figure}">|; 

$key = q/displaystyley(x_n)+h,y'(x_n)+frac{h^2}{2},y''(x_n)+cdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$\displaystyle y(x_n) + h y'(x_n)+ \frac{h^2}{2} y''(x_n)+\cdots$">|; 

$key = q/v_d=35,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$v_d = 35 {\rm m/s}$">|; 

$key = q/{displaymath}P_{y,{{rm{max}}=frac{1}{sqrt{2pi},sigma},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="\begin{displaymath}
P_{y {\rm max}} = \frac{1}{\sqrt{2\pi} \sigma},
\end{displaymath}">|; 

$key = q/displaystyle(J_z)_{i,j}^n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="$\displaystyle (J_z)_{i,j}^n,$">|; 

$key = q/s_i=+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="$s_i=+1$">|; 

$key = q/{displaymath}A=1-{{rm{i},C,sin(k,deltax).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="\begin{displaymath}
A = 1 - {\rm i}  C \sin (k  \delta x).
\end{displaymath}">|; 

$key = q/{mitDelta}E<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="${\mit\Delta}E&lt;0$">|; 

$key = q/{{tt{Nacc}=12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="${\tt Nacc}=12$">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="$p$">|; 

$key = q/{displaymath}hat{u}_k(t)=hat{u}_k(0),exp(-{{rm{i},k,v,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="\begin{displaymath}
\hat{u}_k(t) = \hat{u}_k(0) \exp(-{\rm i} k v t).
\end{displaymath}">|; 

$key = q/v(0)=-0.75;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$v(0)=-0.75$">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$y$">|; 

$key = q/d^2uslashdx^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$d^2u/dx^2$">|; 

$key = q/{displaymath}A=cos(k,deltax)-{{rm{i},C,sin(k,deltax).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="\begin{displaymath}
A = \cos (k \delta x) - {\rm i}  C \sin (k  \delta x).
\end{displaymath}">|; 

$key = q/n=m=L_x=L_y=c=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="$n=m=L_x=L_y=c=1$">|; 

$key = q/rho_jequivrho(x_j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="$\rho_j\equiv
\rho(x_j)$">|; 

$key = q/{displaymath}V_j(x)=frac{2}{L}int_0^Lv(x,y),cos(j,pi,yslashL),dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="\begin{displaymath}
V_j(x) = \frac{2}{L} \int_0^L v(x,y) \cos(j \pi y/L) dy,
\end{displaymath}">|; 

$key = q/T=20,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$T=20 T_0$">|; 

$key = q/Q=1.372;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$Q=1.372$">|; 

$key = q/H_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$H_y$">|; 

$key = q/omega=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$\omega =2$">|; 

$key = q/displaystylefrac{d^2phi(x)}{dx^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="$\displaystyle \frac{d^2\phi(x)}{dx^2}$">|; 

$key = q/deltay=LslashJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="$\delta y = L/J$">|; 

$key = q/lambda=deltaxslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="$\lambda = \delta x/2$">|; 

$key = q/{displaymath}y=left{array{{lcl}A,(z,{{rm{mod},q)-r,(zslashq)&mbox{hspace{1cm}}&m},q)-r,(zslashq)+M&&mbox{otherwise}array{right.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="\begin{displaymath}
y = \left\{\begin{array}{lcl}
A  (z {\rm mod} q) - r  (z...
...d} q) - r  (z/q) + M &amp;&amp; \mbox{otherwise}
\end{array}\right.,
\end{displaymath}">|; 

$key = q/{displaymath}v(x,y)=sum_{j=0}^{infty}V_j(x),cos(j,pi,yslashL),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="\begin{displaymath}
v(x,y) = \sum_{j=0}^{\infty} V_j(x) \cos(j \pi y/L),
\end{displaymath}">|; 

$key = q/F_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="$F_n$">|; 

$key = q/displaystylefrac{F_j(x_1,...,x_n,t)}{m_j}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$\displaystyle \frac{F_j(x_1,...,x_n,t)}{m_j}.$">|; 

$key = q/D=kappaslashc;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="$D=\kappa/c$">|; 

$key = q/{displaymath}intf,dV=frac{1}{N}sum_{i=1,N}f(x_i)+Oleft(frac{1}{sqrt{N}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="\begin{displaymath}
\int f dV = \frac{1}{N}\sum_{i=1,N} f(x_i) + O\left(\frac{1}{\sqrt{N}}\right).
\end{displaymath}">|; 

$key = q/{{bf{M};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="${\bf M}$">|; 

$key = q/{displaymath}P_a,P_{arightarrowb}=P_b,P_{brightarrowa},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="\begin{displaymath}
P_a P_{a\rightarrow b} = P_b P_{b\rightarrow a},
\end{displaymath}">|; 

$key = q/Q=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$Q=4$">|; 

$key = q/{{bf{M}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="${\bf M}^{-1}$">|; 

$key = q/1.2times10^{-5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$1.2\times 10^{-5}$">|; 

$key = q/textstylesim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$\textstyle \sim$">|; 

$key = q/{displaymath}V_j(x)=frac{2}{L}int_0^Lv(x,y),sin(j,pi,yslashL),dy.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="\begin{displaymath}
V_j(x) = \frac{2}{L} \int_0^L v(x,y) \sin(j \pi y/L) dy.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2phi(x)}{dx^2}=-frac{e}{epsilon_0},{n_0-n(x)},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="\begin{displaymath}
\frac{d^2\phi(x)}{dx^2} = -\frac{e}{\epsilon_0} \{n_0- n(x)\},
\end{displaymath}">|; 

$key = q/{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="${\bf b}$">|; 

$key = q/{displaymath}E_z=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="\begin{displaymath}
E_z=0
\end{displaymath}">|; 

$key = q/{displaymath}frac{P_b}{P_a}=exp[-beta,(E_b-E_a)],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="\begin{displaymath}
\frac{P_b}{P_a} = \exp[-\beta (E_b-E_a)],
\end{displaymath}">|; 

$key = q/trightarrowt+tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="$t\rightarrow t+\tau$">|; 

$key = q/{mitDelta}thetaslash2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="${\mit \Delta }\theta /2\pi $">|; 

$key = q/v_b=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$v_b = 3$">|; 

$key = q/x_0=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$x_0=5$">|; 

$key = q/displaystylehat{Y}_{i,j}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="$\displaystyle \hat{Y}_{i,j}^{n+1}$">|; 

$key = q/displaystyley_n+frac{k_1}{6}+frac{k_2}{3}+frac{k_3}{3}+frac{k_4}{6}+O(h^5).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$\displaystyle y_n + \frac{k_1}{6}+\frac{k_2}{3} +\frac{k_3}{3}+\frac{k_4}{6}
+O(h^5).$">|; 

$key = q/{displaymath}left(frac{theta}{tilde{A}}right)^2+left(frac{v}{omega,tilde{A}}right)^2=1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="\begin{displaymath}
\left(\frac{\theta}{\tilde{A}}\right)^2 + \left(\frac{v}{\omega \tilde{A}}\right)^2 =1,
\end{displaymath}">|; 

$key = q/N=6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$N=6$">|; 

$key = q/i=0,N+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="$i=0,N+1$">|; 

$key = q/displaystylebar{y}-4,sigma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="$\displaystyle \bar{y}-4 \sigma,$">|; 

$key = q/166^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$166^\circ $">|; 

$key = q/T_c=2,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="$T_c = 2 T_0$">|; 

$key = q/{displaymath}bar{s}=frac{{{rm{e}^{+beta,mu,H_m}-{{rm{e}^{-beta,mu,H_m}}{{{rm{e}^{+beta,mu,H_m}+{{rm{e}^{-beta,mu,H_m}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="\begin{displaymath}
\bar{s } = \frac{{\rm e}^{+\beta \mu H_m} - {\rm e}^{-\bet...
...m}}
{{\rm e}^{+\beta \mu H_m} + {\rm e}^{-\beta \mu H_m}},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{diff1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="319" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{diff1.eps}}
\end{figure}">|; 

$key = q/gamma_l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="$\gamma_l=0$">|; 

$key = q/{displaymath}u_1=y_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="\begin{displaymath}
u_1 = y_0,
\end{displaymath}">|; 

$key = q/{displaymath}E=sum_{i=1,N}e_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="\begin{displaymath}
E =\sum_{i=1,N} e_i,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{valid1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{valid1.eps}}
\end{figure}">|; 

$key = q/4.69pm0.04;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$4.69\pm 0.04$">|; 

$key = q/s=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$s=-1$">|; 

$key = q/displaystylehat{X}_{i,j}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="$\displaystyle \hat{X}_{i,j}^{n+1}$">|; 

$key = q/displaystylehat{H}_i^{n+1}-hat{H}_i^n+i,D,(hat{E}_i^{n+1}+hat{E}_i^n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="$\displaystyle \hat{H}_i^{n+1}-\hat{H}_i^n + i D (\hat{E}_i^{n+1}+\hat{E}_i^n)$">|; 

$key = q/Q=1.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$Q=1.5$">|; 

$key = q/T(x,y,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$T(x,y,t)$">|; 

$key = q/L=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="$L=1$">|; 

$key = q/displaystylefrac{dx}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="$\displaystyle \frac{dx}{dt}$">|; 

$key = q/N=bar{N}pmsigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="$N=\bar{N}\pm\sigma$">|; 

$key = q/1.2times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$1.2\times 10^{-4}$">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="$g$">|; 

$key = q/P_x(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="$P_x(x)$">|; 

$key = q/{displaymath}int_{x_l}^{x_h}f(x),dxsimeqsum_{i=1}^{N}f_i,h{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="\begin{displaymath}
\int_{x_l}^{x_h} f(x) dx\simeq \sum_{i=1}^{N} f_i h
\end{displaymath}">|; 

$key = q/displaystylehat{E}_{i,j}^{n+1}-hat{E}_{i,j}^n-i,D_x,(hat{Y}_{i,j}^{n+1}+hat{Y}_{i,j}^{n})-j,D_j,(hat{X}_{i,j}^{n+1}+hat{X}_{i,j}^{n});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="$\displaystyle \hat{E}_{i,j}^{n+1}-\hat{E}_{i,j}^n - i D_x (\hat{Y}_{i,j}^{n+1}+\hat{Y}_{i,j}^{n})
- j D_j (\hat{X}_{i,j}^{n+1}+\hat{X}_{i,j}^{n})$">|; 

$key = q/{displaymath}{{bf{w}=[v_1,(deltax)^2-u_l,,v_2,(deltax)^2,,v_3,(deltax)^2,,cdots,,v_{N-1},(deltax)^2,,v_N,(deltax)^2-u_h]{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="550" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="\begin{displaymath}
{\bf w} = [v_1 (\delta x)^2 - u_l,  v_2 (\delta x)^2,  v...
... \cdots,  v_{N-1}  (\delta x)^2, 
v_N (\delta x)^2 - u_h]
\end{displaymath}">|; 

$key = q/{displaymath}A=frac{1-2,C,sin^2(k,deltaxslash2)}{1+2,C,sin^2(k,deltaxslash2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="\begin{displaymath}
A = \frac{1 - 2 C \sin^2(k \delta x/2)}{1+2 C \sin^2(k \delta x/2)}.
\end{displaymath}">|; 

$key = q/theta(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\theta (0)$">|; 

$key = q/tilde{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$\tilde{A}$">|; 

$key = q/{displaymath}P_y(y)=frac{1}{|f'|}=x={{rm{e}^{-y},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="\begin{displaymath}
P_y(y) = \frac{1}{\vert f'\vert}= x = {\rm e}^{-y},
\end{displaymath}">|; 

$key = q/{displaymath}A=1-2,C,(1-cosk,deltax)=1-4,C,sin^2(k,deltaxslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="\begin{displaymath}
A = 1 - 2 C (1-\cos k \delta x) = 1 - 4 C \sin^2(k \delta x/2).
\end{displaymath}">|; 

$key = q/deltat=10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\delta t = 10^{-4}$">|; 

$key = q/t=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$t=0.$">|; 

$key = q/x-v,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="$x-v t$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{double.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="453" HEIGHT="324" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{double.eps}}
\end{figure}">|; 

$key = q/displaystylehat{X}_{i,j}^{n+1}-hat{X}_{i,j}^n+j,D_y,(hat{E}_{i,j}^{n+1}+hat{E}_{i,j}^{n});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="$\displaystyle \hat{X}_{i,j}^{n+1}-\hat{X}_{i,j}^n + j D_y (\hat{E}_{i,j}^{n+1}+\hat{E}_{i,j}^{n})$">|; 

$key = q/displaystyleh,f(x_n+hslash2,y_n+k_2slash2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$\displaystyle h f(x_n+h/2, y_n+k_2/2),$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{rejection.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="483" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{rejection.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{magnus.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="312" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{magnus.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ising1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ising1.eps}}
\end{figure}">|; 

$key = q/(1,1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="$(1,1)$">|; 

$key = q/x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="$x_1$">|; 

$key = q/r_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="$r_i$">|; 

$key = q/{displaymath}T(pmx_0,t)=sqrt{frac{t_0}{t}},exp!left(frac{-x_0^2}{4,D,t}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="\begin{displaymath}
T(\pm x_0,t) = \sqrt{\frac{t_0}{t}}  \exp\!\left(\frac{-x_0^2}{4 D t}\right).
\end{displaymath}">|; 

$key = q/{displaymath}nabla^2phi=4pi^2,G,rho,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="\begin{displaymath}
\nabla^2 \phi = 4\pi^2 G \rho,
\end{displaymath}">|; 

$key = q/2.4times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$2.4\times 10^{-3}$">|; 

$key = q/sim10^{24};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="$\sim 10^{24}$">|; 

$key = q/displaystylefrac{(E_x)_i^{n+1}-(E_x)_i^n}{deltat}+frac{c}{2}left(frac{partialH_y)_i^{n+1}+frac{c}{2}left(frac{partialH_y}{partialz}right)_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="$\displaystyle \frac{(E_x)_i^{n+1}-(E_x)_i^n}{\delta t} +\frac{c}{2}\left(\frac{...
...z}\right)_i^{n+1}
+ \frac{c}{2}\left(\frac{\partial H_y}{\partial z}\right)_i^n$">|; 

$key = q/f=0.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$f=0.6$">|; 

$key = q/(d-1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="$(d-1)$">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{rand3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="434" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{rand3.eps}}
\end{figure}">|; 

$key = q/O(N^3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="$O(N^3)$">|; 

$key = q/displaystyledeltatheta({mitDelta}t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$\displaystyle \delta\theta({\mit\Delta} t)$">|; 

$key = q/{displaymath}T_c=frac{z,J}{2,k}=2,T_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="\begin{displaymath}
T_c = \frac{z J}{2 k}= 2 T_0.
\end{displaymath}">|; 

$key = q/alpha_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="$\alpha_l$">|; 

$key = q/F=4.69;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="$F=4.69$">|; 

$key = q/{displaymath}E_x(z,0)=H_y(z,0)=exp[-100,(z-0.5)^2],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="\begin{displaymath}
E_x(z,0) = H_y(z,0) = \exp[-100 (z-0.5)^2],
\end{displaymath}">|; 

$key = q/{displaymath}u(x,y)=sum_{j=0}^{infty}U_j(x),cos(j,pi,yslashL),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="\begin{displaymath}
u(x,y) = \sum_{j=0}^{\infty} U_j(x) \cos(j \pi y/L),
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{f}_M=S(v),{mbox{{boldmath{{omega{}}times{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="\begin{displaymath}
{\bf f}_M = S(v) {\mbox{\boldmath$\omega$}}\times{\bf v},
\end{displaymath}">|; 

$key = q/displaystylefrac{dvarphi}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$\displaystyle \frac{d\varphi}{dt}$">|; 

$key = q/h=0.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$h=0.01$">|; 

$key = q/j=0,J-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="$j=0,J-1$">|; 

$key = q/{displaymath}xi(x,t)=F(x-c,t)+G(x+c,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="\begin{displaymath}
\xi(x,t) = F(x-c t) + G(x+c t),
\end{displaymath}">|; 

$key = q/(n+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="$(n+1)$">|; 

$key = q/v_{th}=sqrt{k_B,Tslashm_e};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="$v_{th} = \sqrt{k_B T/m_e}$">|; 

$key = q/delta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$\delta$">|; 

$key = q/{displaymath}R=sqrt{theta^2+(vslashomega)^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="\begin{displaymath}
R = \sqrt{\theta^2 + (v/\omega)^2}
\end{displaymath}">|; 

$key = q/rho({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$\rho({\bf r})$">|; 

$key = q/{displaymath}x_n=x_0+n,h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="\begin{displaymath}
x_n = x_0 + n h.
\end{displaymath}">|; 

$key = q/{displaymath}T_0=frac{J}{k}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="\begin{displaymath}
T_0 = \frac{J}{k}.
\end{displaymath}">|; 

$key = q/f(x,v),dx,dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="$f(x,v) dx dv$">|; 

$key = q/{displaymath}T_{i,0}^n=T_{i,J}^n=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="\begin{displaymath}
T_{i,0}^n = T_{i,J}^n = 0.
\end{displaymath}">|; 

$key = q/displaystylev_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$\displaystyle v_j,$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{sliderr.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{sliderr.eps}}
\end{figure}">|; 

$key = q/b^2<4,a,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$b^2&lt;4 a c$">|; 

$key = q/omega=40;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\omega = 40$">|; 

$key = q/{displaymath}u(x)=g+h,x+frac{x^2}{2}-frac{x^4}{6},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="\begin{displaymath}
u(x) = g + h x + \frac{x^2}{2} - \frac{x^4}{6},
\end{displaymath}">|; 

$key = q/t=0.45;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$t=0.45$">|; 

$key = q/-mu,H_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="$-\mu H_m$">|; 

$key = q/u(x_l)=u_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$u(x_l) = u_l$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$m$">|; 

$key = q/a_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="$a_1$">|; 

$key = q/Qrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$Q\rightarrow
\infty$">|; 

$key = q/displaystyleh,f(x_n,y_n),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$\displaystyle h f(x_n,y_n),$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{att2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="326" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{att2.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{bs3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{bs3.eps}}
\end{figure}">|; 

$key = q/displaystyleJ_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="$\displaystyle J_z,$">|; 

$key = q/V_{i,j}equivV_j(x_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="$V_{i,j} \equiv V_j(x_i)$">|; 

$key = q/displaystyle(E_x)_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="$\displaystyle (E_x)_i^n$">|; 

$key = q/displaystylefrac{dv_x}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$\displaystyle \frac{dv_x}{dt}$">|; 

$key = q/N-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="$N-1$">|; 

$key = q/v_0=65;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$v_0=65$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$T$">|; 

$key = q/j=0,2J-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="$j=0,2J-1$">|; 

$key = q/displaystyle-g-F(v),v,v_z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$\displaystyle -g - F(v) v v_z.$">|; 

$key = q/k_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="$k_B$">|; 

$key = q/epsilon_0sim10^{-8};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$\epsilon_0\sim 10^{-8}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cavity3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cavity3.eps}}
\end{figure}">|; 

$key = q/f_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="$f_j$">|; 

$key = q/sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="$\sigma$">|; 

$key = q/{{tt{Nacc}=100;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="${\tt Nacc} = 100$">|; 

$key = q/<ij>;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="$&lt;i j&gt;$">|; 

$key = q/u_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$u_i^n$">|; 

$key = q/a_i=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="$a_i=1$">|; 

$key = q/t=0.003;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$t=0.003$">|; 

$key = q/displaystyle+frac{Aleft[(1-omega^2),cosomegat+(omegaslashQ),sinomegatright]}{left[(1-omega^2)^2+omega^2slashQ^2right]},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="$\displaystyle +\frac{A\left[(1-\omega^2) \cos\omega t+ (\omega/Q) \sin\omega t\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]},$">|; 

$key = q/-pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="$-\pi$">|; 

$key = q/x=x_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="$x=x_l$">|; 

$key = q/eta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$\eta$">|; 

$key = q/h^1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$h^1$">|; 

$key = q/{mitDelta}t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="${\mit \Delta } t = 0$">|; 

$key = q/sigma_E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$\sigma_E$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{coulomb4.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="346" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{coulomb4.eps}}
\end{figure}">|; 

$key = q/omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="$\omega_0$">|; 

$key = q/{displaymath}AsimeqN,h^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="\begin{displaymath}
A \simeq N h^2.
\end{displaymath}">|; 

$key = q/deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="$\delta x$">|; 

$key = q/{{bf{q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="${\bf q}$">|; 

$key = q/displaystyle+frac{2,i,D}{1+i^2,D^2},hat{H}_i^n+frac{1-i^2,D^2}{1+i^2,D^2},hat{E}_i^n,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="$\displaystyle +\frac{2 i D}{1+i^2 D^2} \hat{H}_i^n + \frac{1-i^2 D^2}{1+i^2 D^2} \hat{E}_i^n,$">|; 

$key = q/omega=1800;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\omega =1800$">|; 

$key = q/-infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="$-\infty$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p1d.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="321" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p1d.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{rand2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="434" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{rand2.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{gamma_{l,j}^n,deltax-beta_l^n,T_{1,j}^n}{alpha_l^n,deltax-beta_l^n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="$\displaystyle \frac{\gamma_{l j}^n \delta x-\beta_l^n T_{1,j}^n}{\alpha_l^n \delta x -\beta_l^n},$">|; 

$key = q/displaystyley_{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$\displaystyle y_{n+1}$">|; 

$key = q/M=6075;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$M=6075$">|; 

$key = q/leq10^6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$\leq 10^6$">|; 

$key = q/Q=2.13;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$Q=2.13$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cavity5.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cavity5.eps}}
\end{figure}">|; 

$key = q/10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$10^{-6}$">|; 

$key = q/{displaymath}bar{s}_{i+1}=tanhleft{frac{T_c}{T}left(frac{H}{H_c}+bar{s}_iright)right}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="\begin{displaymath}
\bar{s}_{i+1} = \tanh\left\{\frac{T_c}{T}\left(\frac{H}{H_c} + \bar{s}_i\right)\right\}.
\end{displaymath}">|; 

$key = q/y_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$y_n$">|; 

$key = q/L=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="$L=10$">|; 

$key = q/displaystyleeta^{nslash(n+1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$\displaystyle \eta^{n/(n+1)},$">|; 

$key = q/{{bf{B}=[B_x(x,y,t),B_y(x,y,t),0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="${\bf B} = [B_x(x,y,t), B_y(x,y,t), 0]$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ts10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ts10.eps}}\end{figure}">|; 

$key = q/{displaymath}bar{s}=tanhleft{beta,mu,H+beta,z,J,bar{s}slash2right}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="\begin{displaymath}
\bar{s} = \tanh\left\{\beta \mu H + \beta z J \bar{s}/2\right\}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{scan0.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="384" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{scan0.eps}}
\end{figure}">|; 

$key = q/x_l=v,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="$x_l = v t$">|; 

$key = q/C_D,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$C_D v^2$">|; 

$key = q/{displaymath}N=N',frac{A}{A'}pmsqrt{N',frac{A}{A'}left(1-frac{A}{A'}right)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="\begin{displaymath}
N = N' \frac{A}{A'} \pm \sqrt{N' \frac{A}{A'}\left(1-\frac{A}{A'}\right)}.
\end{displaymath}">|; 

$key = q/{displaymath}h_{{rm{new}=h_{{rm{old}left|frac{epsilon_0}{epsilon}right|^{1slash5}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="\begin{displaymath}
h_{\rm new} = h_{\rm old}\left\vert\frac{\epsilon_0}{\epsilon}\right\vert^{1/5}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{knuckle1r.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="377" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{knuckle1r.eps}}
\end{figure}">|; 

$key = q/{displaymath}m,l,frac{d^2theta}{dt^2}+nu,frac{dtheta}{dt}+m,g,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="\begin{displaymath}
m l \frac{d^2\theta}{dt^2}+ \nu \frac{d\theta}{dt} + m g \sin\theta = 0,
\end{displaymath}">|; 

$key = q/3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="$3$">|; 

$key = q/{displaymath}frac{d^2u(x)}{dx^2}=v(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="\begin{displaymath}
\frac{d^2 u(x)}{dx^2} = v(x),
\end{displaymath}">|; 

$key = q/N^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="$N^{-1/2}$">|; 

$key = q/v(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="$v(x,y)$">|; 

$key = q/L_x=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$L_x=1$">|; 

$key = q/lnN_Agg1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="$\ln N_A \gg 1$">|; 

$key = q/T(x,t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="$T(x,t_0)$">|; 

$key = q/theta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="$\theta&gt;0$">|; 

$key = q/Jsim1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="$J\sim 1$">|; 

$key = q/Q=1.36;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$Q=1.36$">|; 

$key = q/displaystylehat{X}_{i,j}^n-j,D_y,(hat{E}_{i,j}^{n+1}+hat{E}_{i,j}^{n}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="$\displaystyle \hat{X}_{i,j}^n - j D_y (\hat{E}_{i,j}^{n+1}+\hat{E}_{i,j}^{n}),$">|; 

$key = q/displaystyle-frac{C}{2},hat{T}_{i-1,j}^{n+1}+left{1+C,(1+j^2,kappa^2slash2)right},hat{T}_{i,j}^{n+1}-frac{C}{2},hat{T}_{i+1,j}^{n+1}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="$\displaystyle -\frac{C}{2} \hat{T}_{i-1,j}^{n+1} + \left\{1+C (1+j^2 \kappa^2/2)\right\} 
\hat{T}_{i,j}^{n+1}-\frac{C}{2} \hat{T}_{i+1,j}^{n+1}
=$">|; 

$key = q/1.348;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="$1.348$">|; 

$key = q/T_c=2.27,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="$T_c=2.27 T_0$">|; 

$key = q/t_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="$t_n$">|; 

$key = q/{{tt{Nacc}=48;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="${\tt Nacc} = 48$">|; 

$key = q/Lsimsqrt{N_A}sim10^{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$L\sim \sqrt{N_A} \sim 10^{12}$">|; 

$key = q/displaystylefrac{gamma_l,(alpha_h+beta_h)-beta_l,[gamma_h-(alpha_h+beta_h)slash3]}{alpha_l,alpha_h+alpha_l,beta_h-beta_l,alpha_h},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="$\displaystyle \frac{\gamma_l (\alpha_h+\beta_h) - \beta_l [\gamma_h-(\alpha_h+\beta_h)/3]}
{\alpha_l \alpha_h + \alpha_l \beta_h-\beta_l \alpha_h},$">|; 

$key = q/deltat=10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$\delta t = 10^{-2}$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="$\mu$">|; 

$key = q/q=MslashA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="$q = M/A$">|; 

$key = q/Q=3.9;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$Q=3.9$">|; 

$key = q/displaystylevarphi(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$\displaystyle \varphi(t=0)$">|; 

$key = q/{displaymath}V=N,h^d+O(N^{-1slashd}),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="\begin{displaymath}
V = N h^d + O(N^{-1/d}),
\end{displaymath}">|; 

$key = q/{displaymath}E_x(0,t)=E_x(L,t)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="\begin{displaymath}
E_x(0,t) = E_x(L,t) = 0.
\end{displaymath}">|; 

$key = q/{displaymath}y(x)=exp(-alpha,x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="\begin{displaymath}
y(x) = \exp(-\alpha x).
\end{displaymath}">|; 

$key = q/alpha_l^nequivalpha_l(t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="$\alpha_l^n\equiv \alpha_l(t_n)$">|; 

$key = q/x_h=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$x_h = 1$">|; 

$key = q/displaystylefrac{omega}{omega_0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="$\displaystyle \frac{\omega}{\omega_0},$">|; 

$key = q/t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="$t_0$">|; 

$key = q/Arightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$A\rightarrow 0$">|; 

$key = q/Q_{n}-Q_{n-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="$Q_{n} - Q_{n-1}$">|; 

$key = q/{displaymath}Q_n=Q_1+(Q_2-Q_1)sum_{j=0}^{n-2}frac{1}{F^j},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="\begin{displaymath}
Q_n = Q_1 + (Q_2-Q_1)\sum_{j=0}^{n-2}\frac{1}{F^j},
\end{displaymath}">|; 

$key = q/x_j<r_i<x_{j+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="$x_j &lt; r_i &lt; x_{j+1}$">|; 

$key = q/H_x(x=0.5,y=0.0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$H_x(x=0.5,y=0.0)$">|; 

$key = q/s_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="$s_i$">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$r$">|; 

$key = q/varphi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$\varphi_0$">|; 

$key = q/rho(x)=n(x)slashn_0-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$\rho(x) = n(x)/n_0-1$">|; 

$key = q/+mu,H_m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="$+\mu H_m$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{montecarlo.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{montecarlo.eps}}
\end{figure}">|; 

$key = q/{displaymath}v(x,y)=sum_{j=0}^{infty}V_j(x),sin(j,pi,yslashL),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="\begin{displaymath}
v(x,y) = \sum_{j=0}^{\infty} V_j(x) \sin(j \pi y/L),
\end{displaymath}">|; 

$key = q/(x_0,y_0)=(0.5,0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="$(x_0,y_0) = (0.5,0.5)$">|; 

$key = q/N=20000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$N=20000$">|; 

$key = q/i=j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$i=j$">|; 

$key = q/E_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="$E_a$">|; 

$key = q/hat{u}_i^{n+1}=A,hat{u}_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="$\hat{u}_i^{n+1}=A \hat{u}_i^n$">|; 

$key = q/1.245;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="$1.245$">|; 

$key = q/theta(0)slashA=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$\theta (0)/A=0$">|; 

$key = q/E(0)=E(L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="$E(0) = E(L)$">|; 

$key = q/beta_l=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="$\beta_l=0$">|; 

$key = q/{displaymath}H_{{rm{eff}=H+frac{z,J,bar{s}}{2,mu}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="\begin{displaymath}
H_{\rm eff} = H + \frac{z J \bar{s}}{2 \mu}.
\end{displaymath}">|; 

$key = q/{displaymath}u_{i-1}-2,u_i+u_{i+1}=v_i,(deltax)^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="\begin{displaymath}
u_{i-1} -2 u_i + u_{i+1} = v_i (\delta x)^2,
\end{displaymath}">|; 

$key = q/{displaymath}Q_infty=Q_1+(Q_2-Q_1)sum_{j=0}^{infty}frac{1}{F^j}equivQ_1+(Q_2-Q_1),frac{F}{F-1}=1.3758.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="526" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="\begin{displaymath}
Q_\infty = Q_1 + (Q_2-Q_1)\sum_{j=0}^{\infty}\frac{1}{F^j} \equiv Q_1 + (Q_2-Q_1) \frac{F}{F-1}=
1.3758.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{fastballr.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{fastballr.eps}}
\end{figure}">|; 

$key = q/{ttNacc}=12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="${\tt Nacc}=12$">|; 

$key = q/h^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$h^5$">|; 

$key = q/displaystylev(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$\displaystyle v(t)$">|; 

$key = q/s=+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="$s=+1$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="$a$">|; 

$key = q/{displaymath}u(x,y)=y,(1-y),x^3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="\begin{displaymath}
u(x,y) = y (1-y) x^3.
\end{displaymath}">|; 

$key = q/delta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="$\delta_0$">|; 

$key = q/displaystyleomega.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="$\displaystyle \omega.$">|; 

$key = q/|omega|=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="$\vert\omega\vert=1$">|; 

$key = q/displaystylez(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$\displaystyle z(t=0)$">|; 

$key = q/C_D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$C_D$">|; 

$key = q/omega=2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\omega =2/3$">|; 

$key = q/{displaymath}x_{i-1}=-frac{a_i}{b_i+c_i,x_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="\begin{displaymath}
x_{i-1} = - \frac{a_i}{b_i+c_i x_i},
\end{displaymath}">|; 

$key = q/displaystylefrac{gamma_{h,j}^n,deltax+beta_h^n,T_{N,j}^n}{alpha_h^n,deltax+beta_h^n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="$\displaystyle \frac{\gamma_{h j}^n \delta x + \beta_h^n T_{N,j}^n}{\alpha_h^n \delta x +\beta_h^n},$">|; 

$key = q/{displaymath}T(x,t_0)=exp!left(frac{-x^2}{4,D,t_0}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="\begin{displaymath}
T(x,t_0) = \exp\!\left(\frac{-x^2}{4 D t_0}\right),
\end{displaymath}">|; 

$key = q/{displaymath}alpha_l,u(x,y)+beta_l,frac{partialu(x,y)}{partialx}=gamma_l(y),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="\begin{displaymath}
\alpha_l u(x,y) + \beta_l \frac{\partial u(x,y)}{\partial x} = \gamma_l(y),
\end{displaymath}">|; 

$key = q/{displaymath}h=frac{x_h-x_l}{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="\begin{displaymath}
h = \frac{x_h-x_l}{N}.
\end{displaymath}">|; 

$key = q/v_iequivv(x_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$v_i\equiv v(x_i)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{valid.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{valid.eps}}
\end{figure}">|; 

$key = q/E_z(x,y,t)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="$E_z(x,y,t)]$">|; 

$key = q/displaystyley_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$\displaystyle y_{\rm max}$">|; 

$key = q/{displaymath}H_{{rm{eff}=H+frac{J}{2,mu}sum_{k=1,z}s_k.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="\begin{displaymath}
H_{\rm eff} = H + \frac{J}{2 \mu}\sum_{k=1,z}s_k.
\end{displaymath}">|; 

$key = q/R=0.122859;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$R= 0.122859$">|; 

$key = q/x_{j+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$x_{j+1}$">|; 

$key = q/{displaymath}{{bf{u}={{bf{M}^{-1},{{bf{w},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="\begin{displaymath}
{\bf u} = {\bf M}^{-1} {\bf w},
\end{displaymath}">|; 

$key = q/10^7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$10^7$">|; 

$key = q/{displaymath}T_i^{n+1}-frac{C}{2},left(T_{i-1}^{n+1}-2,T_i^{n+1}+T_{i+1}^{n+1}riac{C}{2},left(T_{i-1}^n-2,T_i^n+T_{i+1}^nright).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="\begin{displaymath}
T_i^{n+1} - \frac{C}{2} \left(T_{i-1}^{n+1}-2 T_i^{n+1}+T_...
..._i^n + \frac{C}{2} \left(T_{i-1}^n-2 T_i^n+T_{i+1}^n\right).
\end{displaymath}">|; 

$key = q/(x,y)=(0.5,0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$(x,y)=(0.5,0.5)$">|; 

$key = q/N,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$N k$">|; 

$key = q/x=0,L_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="$x=0,L_x$">|; 

$key = q/{displaymath}deltat<frac{deltaz}{c},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="\begin{displaymath}
\delta t &lt; \frac{\delta z}{c},
\end{displaymath}">|; 

$key = q/{displaymath}1,,2,,2times2,,6,,5,,3,,2times3,,5,,6,,4,,6,,5,,6.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="\begin{displaymath}
1,  2,  2\times 2,  6,  5,  3,  2\times 3,  5,  6,  4,  6,  5,  6.
\end{displaymath}">|; 

$key = q/t=0.15;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$t=0.15$">|; 

$key = q/U_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="$U_{i,j}$">|; 

$key = q/C=v,deltatslashdeltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$C= v \delta t/\delta x$">|; 

$key = q/T_{i,j}^nequivT(x_i,y_j,t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="$T_{i,j}^n\equiv T(x_i,y_j,t_n)$">|; 

$key = q/displaystylev_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$\displaystyle v_x,$">|; 

$key = q/epsilon_0=10^{-8};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$\epsilon _0=10^{-8}$">|; 

$key = q/N=200;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$N=200$">|; 

$key = q/(x_0,y_0)=(0.25,0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$(x_0,y_0) = (0.25,0.5)$">|; 

$key = q/d=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="$d=2$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cavity2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cavity2.eps}}
\end{figure}">|; 

$key = q/{displaymath}VsimeqN,h^3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="\begin{displaymath}
V \simeq N h^3.
\end{displaymath}">|; 

$key = q/{displaymath}hat{phi}_j=-frac{hat{rho}_j}{j^2,kappa^2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="\begin{displaymath}
\hat{\phi}_j = - \frac{\hat{\rho}_j}{j^2 \kappa^2}
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{bs4.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="334" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{bs4.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{bs2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{bs2.eps}}
\end{figure}">|; 

$key = q/u_i^nequivu(x_i,t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="$u_i^n\equiv u(x_i,t_n)$">|; 

$key = q/{displaymath}|A|^2=1+C^2,sin^2(k,deltax)>1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="\begin{displaymath}
\vert A\vert^2 = 1 + C^2 \sin^2(k \delta x) &gt; 1.
\end{displaymath}">|; 

$key = q/t=0.48;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$t=0.48$">|; 

$key = q/n=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="$n=3$">|; 

$key = q/trightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$t\rightarrow\infty$">|; 

$key = q/x=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$x=1$">|; 

$key = q/{displaymath}alpha_h,U_j(x)+beta_h,frac{dU_j(x)}{dx}=Gamma_{h,j},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="\begin{displaymath}
\alpha_h U_j(x)+ \beta_h \frac{dU_j(x)}{dx} = \Gamma_{h j},
\end{displaymath}">|; 

$key = q/D_y=pi,c,deltatslash(2,L_y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="$D_y=\pi c \delta t/(2 L_y)$">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$S$">|; 

$key = q/y=0,L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="$y=0,L_y$">|; 

$key = q/P=exp(-beta,{mitDelta}E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="$P=\exp(-\beta {\mit\Delta}E)$">|; 

$key = q/t=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$t=3$">|; 

$key = q/{displaymath}J_z(x,y,t)=J_0,sin(m,pi,xslashL_x),sin(n,pi,yslashL_y),sin(2,pi,f,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="\begin{displaymath}
J_z(x,y,t) = J_0 \sin(m \pi x/L_x) \sin(n \pi y/L_y) \sin(2 \pi f t).
\end{displaymath}">|; 

$key = q/{displaymath}a_i,u_{i-1}+b_i,u_i+c_i,u_{i+1}=w_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="\begin{displaymath}
a_i u_{i-1}+ b_i u_i + c_i u_{i+1} = w_i,
\end{displaymath}">|; 

$key = q/displaystylek_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$\displaystyle k_1$">|; 

$key = q/u(x,0)=u(x,1)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="$u(x,0)=u(x,1)=0$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{image2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="217" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{image2.eps}}\end{figure}">|; 

$key = q/{displaymath}T_{i,j}^n=sum_{k=0}^{J}hat{T}_{i,k}^n,sin(j,k,pislashJ),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="\begin{displaymath}
T_{i,j}^n = \sum_{k=0}^{J} \hat{T}_{i,k}^n \sin(j k \pi/J),
\end{displaymath}">|; 

$key = q/T(0,y)=T(1,y)=T(x,0)=T(x,1)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="$T(0,y)=T(1,y)=T(x,0)=T(x,1)=0$">|; 

$key = q/1slash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="$1/4$">|; 

$key = q/{displaymath}omega,t=phi+k,2pi{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="\begin{displaymath}
\omega t = \phi + k 2\pi
\end{displaymath}">|; 

$key = q/t=0.001;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$t=0.001$">|; 

$key = q/v_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="$v_1(t)$">|; 

$key = q/delta(t)simdelta_0,exp(lambda,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="$\delta(t)\sim \delta_0 \exp(\lambda t)$">|; 

$key = q/displaystylebar{y}+4,sigma,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="$\displaystyle \bar{y}+4 \sigma,$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="$\lambda$">|; 

$key = q/nablaT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="$\nabla T$">|; 

$key = q/{displaymath}P_y(y)=frac{P_x(x)}{|f'(x)|},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="\begin{displaymath}
P_y(y) = \frac{P_x(x)}{\vert f'(x)\vert},
\end{displaymath}">|; 

$key = q/{displaymath}overline{x^2}=1^2timesp+0^2times(1-p)=p.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="\begin{displaymath}
\overline{x^2} = 1^2\times p + 0^2\times (1-p) = p.
\end{displaymath}">|; 

$key = q/5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$5$">|; 

$key = q/0leqyleqL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="$0\leq y \leq L$">|; 

$key = q/{displaymath}A=1-C,[1-cos(k,deltax)]-{{rm{i},C,sin(k,deltax).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="\begin{displaymath}
A = 1 - C [1-\cos (k  \delta x)]-{\rm i}  C \sin (k  \delta x).
\end{displaymath}">|; 

$key = q/T(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="$T(x,t)$">|; 

$key = q/H_y=c,B_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="$H_y = c B_y$">|; 

$key = q/x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="$x_2$">|; 

$key = q/v(0)=sqrt{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="$v(0) = \sqrt{k}$">|; 

$key = q/C=0.635;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="$C=0.635$">|; 

$key = q/C<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="$C&lt;1$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="$u$">|; 

$key = q/{displaymath}F_j=frac{1}{2J}sum_{k=0}^{2J-1}f_k,exp(-{{rm{i},j,k,pislashJ){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="\begin{displaymath}
F_j = \frac{1}{2J}\sum_{k=0}^{2J-1} f_k \exp(-{\rm i} j k \pi/J)
\end{displaymath}">|; 

$key = q/v(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$v(0)=0$">|; 

$key = q/displaystylerho_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="$\displaystyle \rho_j$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cv40.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cv40.eps}}
\end{figure}">|; 

$key = q/23,{{rm{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="$23 {\rm cm}$">|; 

$key = q/{{bf{E};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="${\bf E}$">|; 

$key = q/x(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="$x(0)=0$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$k$">|; 

$key = q/y=0,L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="$y=0,L$">|; 

$key = q/P=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="$P=1$">|; 

$key = q/T<T_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$T&lt;T_c$">|; 

$key = q/T=2.32,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$T=2.32 T_0$">|; 

$key = q/{displaymath}x_0=0,mbox{hspace{1cm}}y_0=frac{w_1-c_1,y_1}{b_1+c_1,x_1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="\begin{displaymath}
x_0 = 0,\mbox{\hspace{1cm}}y_0 = \frac{w_1-c_1 y_1}{b_1+c_1 x_1},
\end{displaymath}">|; 

$key = q/displaystyleE(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="$\displaystyle E(x)$">|; 

$key = q/v=2,sqrt{k},nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$v = 2 \sqrt{k} \nu$">|; 

$key = q/Q=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$Q=5$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{image5.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="217" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{image5.eps}}
\end{figure}">|; 

$key = q/x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$x_0$">|; 

$key = q/L_y=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$L_y=1$">|; 

$key = q/{displaymath}t_n=t_0+n,deltat,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="\begin{displaymath}
t_n = t_0 + n \delta t,
\end{displaymath}">|; 

$key = q/sigma_Eproptosqrt{C};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="$\sigma_E\propto \sqrt{C}$">|; 

$key = q/varphi_0=0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\varphi _0 = 0^\circ $">|; 

$key = q/t=0.00;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$t=0.00$">|; 

$key = q/c_N=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="$c_N=0$">|; 

$key = q/{displaymath}E=-Jsum_{<ij>}s_i,s_j-mu,Hsum_{i=1,N}s_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="\begin{displaymath}
E = - J\sum_{&lt;i j&gt;}s_i s_j -\mu H\sum_{i=1,N}s_i.
\end{displaymath}">|; 

$key = q/n_0=NslashL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="$n_0 = N/L$">|; 

$key = q/u_i=x_{i-1},u_{i-1}+y_{i-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="$u_i = x_{i-1} u_{i-1} + y_{i-1}$">|; 

$key = q/displaystylehat{H}_i^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="$\displaystyle \hat{H}_i^{n+1}$">|; 

$key = q/F_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="$F_j$">|; 

$key = q/{{bf{E}=-nablaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="${\bf E} = -\nabla \phi$">|; 

$key = q/{displaymath}u_i^{n+1}=u_i^n-C,(u_{i}^n-u_{i-1}^n),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="\begin{displaymath}
u_i^{n+1} = u_i^n -C (u_{i}^n-u_{i-1}^n),
\end{displaymath}">|; 

$key = q/i=0,I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="$i=0,I$">|; 

$key = q/gamma_l^nequivgamma_l(t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="$\gamma_l^n\equiv \gamma_l(t_n)$">|; 

$key = q/r=M<verb_mark>135<verb_mark>A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="$r=M\verb!%!A$">|; 

$key = q/t=0.005;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$t=0.005$">|; 

$key = q/H=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="$H=0$">|; 

$key = q/L=20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="$L=20$">|; 

$key = q/delta_k({mitDelta}t)simeqdelta_k(0),exp(lambda_k,{mitDelta}t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="$\delta_k({\mit\Delta}t)\simeq \delta_k(0) \exp(\lambda_k {\mit\Delta}t)$">|; 

$key = q/omega_p^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="$\omega_p^{-1}$">|; 

$key = q/{displaymath}sigma^2=N',overline{(x-bar{x})^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="\begin{displaymath}
\sigma^2 = N' \overline{(x-\bar{x})^2}.
\end{displaymath}">|; 

$key = q/A=107;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$A=107$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{coulomb2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="346" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{coulomb2.eps}}
\end{figure}">|; 

$key = q/T_i^nequivT(x_i,t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="$T_i^n \equiv T(x_i,t_n)$">|; 

$key = q/{displaymath}E(x)=-frac{dphi(x)}{dx}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="\begin{displaymath}
E(x) = - \frac{d\phi(x)}{dx}.
\end{displaymath}">|; 

$key = q/+infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="$+\infty$">|; 

$key = q/y=infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="$y=\infty$">|; 

$key = q/t=20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$t=20$">|; 

$key = q/displaystylealpha_h,u_{N+1}+beta_h,frac{u_{N+1}-u_N}{deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="$\displaystyle \alpha_h u_{N+1} + \beta_h \frac{u_{N+1}-u_N}{\delta x}$">|; 

$key = q/{displaymath}frac{partial^2phi(x,y)}{partialx^2}+frac{partial^2phi(x,y)}{partialy^2}=v(x,y)=-delta(x-x_0),delta(y-y_0),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="\begin{displaymath}
\frac{\partial^2 \phi(x,y)}{\partial x^2}+\frac{\partial^2 \...
...x,y)}{\partial y^2} =
v(x,y) = -\delta(x-x_0) 
\delta(y-y_0),
\end{displaymath}">|; 

$key = q/{displaymath}f(x,v)=frac{n_0}{2}left{frac{1}{sqrt{2,pi}},{{rm{e}^{-(v-v_b)^2slas1}{sqrt{2,pi}},{{rm{e}^{-(v+v_b)^2slash2}right}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="\begin{displaymath}
f(x,v) = \frac{n_0}{2}\left\{\frac{1}{\sqrt{2 \pi}}
 {\rm ...
.../2}+ \frac{1}{\sqrt{2 \pi}} {\rm e}^{-(v+v_b)^2/2}
\right\}.
\end{displaymath}">|; 

$key = q/I_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$I_2$">|; 

$key = q/56,mph;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$56 mph$">|; 

$key = q/u_{N+1}=u_h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="$u_{N+1} = u_h$">|; 

$key = q/{calE};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="${\cal E}$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{scanqv.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{scanqv.eps}}
\end{figure}">|; 

$key = q/theta+2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$\theta +2\pi$">|; 

$key = q/y_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="$y_{\rm max}$">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="$b$">|; 

$key = q/displaystyle0mbox{hspace{1cm}otherwise.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="$\displaystyle 0\mbox{\hspace{1cm}otherwise.}$">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$x$">|; 

$key = q/n_j,deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="$n_j \delta x$">|; 

$key = q/deltax=(x_h-x_l)slash(N+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$\delta x = (x_h-x_l)/(N+1)$">|; 

$key = q/displaystyle(H_y)_{i,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="$\displaystyle (H_y)_{i,j}^n$">|; 

$key = q/t_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="$t_{\rm max}$">|; 

$key = q/{displaymath}V=frac{N}{N'},V'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="\begin{displaymath}
V = \frac{N}{N'} V'.
\end{displaymath}">|; 

$key = q/O(h^{n+1});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$O(h^{n+1})$">|; 

$key = q/y_{{rm{min};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="$y_{\rm min}$">|; 

$key = q/L_xtimesL_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="$L_x\times L_y$">|; 

$key = q/v_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="$v_b$">|; 

$key = q/A=7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="$A=7$">|; 

$key = q/displaystylefrac{dz}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$\displaystyle \frac{dz}{dt}$">|; 

$key = q/1.4215;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="$1.4215$">|; 

$key = q/1.0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$1.0$">|; 

$key = q/67.5^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$67.5^\circ $">|; 

$key = q/{displaymath}u(x,0)=u(x,L)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="\begin{displaymath}
u(x,0) = u(x,L) = 0.
\end{displaymath}">|; 

$key = q/displaystyleleft{v(0)-frac{A,omega^2slashQ}{[(1-omega^2)^2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q}cosomega_astt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$\displaystyle \left\{v(0) - \frac{A \omega^2/Q}{[(1-\omega^2)^2+\omega^2/Q^2]}\right\}
{\rm e}^{-t/2Q}\cos\omega_\ast t$">|; 

$key = q/i=N+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="$i=N+1$">|; 

$key = q/trightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="$t\rightarrow 0$">|; 

$key = q/h>2slashalpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$h&gt;2/\alpha$">|; 

$key = q/{displaymath}alpha_h(t),T(x,t)+beta_h(t),frac{partialT(x,t)}{partialx}=gamma_h(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="\begin{displaymath}
\alpha_h(t)  T(x,t)+ \beta_h(t) \frac{\partial T(x,t)}{\partial x} = \gamma_h(t),
\end{displaymath}">|; 

$key = q/displaystylev_x(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$\displaystyle v_x(t=0)$">|; 

$key = q/log(|deltav|);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="$\log(\vert\delta v\vert)$">|; 

$key = q/{displaymath}n_{j+1}rightarrown_{j+1}+left.left(frac{r_i-x_j}{x_{j+1}-x_j}right)rightslashdeltax.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="\begin{displaymath}
n_{j+1} \rightarrow n_{j+1}+ \left.\left(\frac{r_i-x_j}{x_{j+1}-x_j}\right)\right/\delta x.
\end{displaymath}">|; 

$key = q/t=0.47;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$t=0.47$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cv20.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cv20.eps}}
\end{figure}">|; 

$key = q/varphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$\varphi$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$j$">|; 

$key = q/2times3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$2\times 3$">|; 

$key = q/deltax=L_xslashI;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="$\delta x = L_x/I$">|; 

$key = q/hat{T}_{i,j}^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="$\hat{T}_{i,j}^{n+1}$">|; 

$key = q/displaystylefrac{dtheta}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$\displaystyle \frac{d\theta}{dt}$">|; 

$key = q/A';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="$A'$">|; 

$key = q/r_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$r_n$">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{basin1a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="438" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{basin1a.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p2da.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="325" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p2da.eps}}
\end{figure}">|; 

$key = q/0.69;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$0.69$">|; 

$key = q/displaystylealpha_l,u_0+beta_l,frac{u_1-u_0}{deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$\displaystyle \alpha_l u_0 + \beta_l \frac{u_1-u_0}{\delta x}$">|; 

$key = q/-x_0leqxleqx_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="$-x_0 \leq x \leq x_0$">|; 

$key = q/{displaymath}frac{T(x,t_{n+1})-T(x,t_n)}{deltat}=frac{D}{2},frac{partial^2T(x,t_{partial^2T(x,t_{n+1})}{partialx^2}+O(deltat)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="530" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="\begin{displaymath}
\frac{T(x,t_{n+1}) - T(x,t_n)}{\delta t}
= \frac{D}{2}  \f...
... \frac{\partial^2 T(x,t_{n+1})}{\partial x^2}+ O(\delta t)^2.
\end{displaymath}">|; 

$key = q/{displaymath}overline{(x-bar{x})^2}=overline{(x^2-2,x,bar{x}+bar{x}^2)}=overline{x^2}-bar{x}^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="\begin{displaymath}
\overline{(x-\bar{x})^2}=\overline{(x^2-2 x \bar{x}+\bar{x}^2)} = \overline{x^2} - \bar{x}^2,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{screwballr.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{screwballr.eps}}
\end{figure}">|; 

$key = q/deltagga;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="$\delta \gg a$">|; 

$key = q/nablaucdotd{{bf{S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$\nabla u\cdot d{\bf S}$">|; 

$key = q/omega=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\omega =3$">|; 

$key = q/T_{i,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="$T_{i,j}^n$">|; 

$key = q/displaystylehat{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$\displaystyle \hat{t}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{euler.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{euler.eps}}
\end{figure}">|; 

$key = q/displaystyleepsilon_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$\displaystyle \epsilon_0$">|; 

$key = q/95,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="$95 $">|; 

$key = q/{displaymath}frac{d^2phi(x)}{dx^2}=rho(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="\begin{displaymath}
\frac{d^2\phi(x)}{dx^2} = \rho(x),
\end{displaymath}">|; 

$key = q/theta_0(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$\theta_0(t)$">|; 

$key = q/y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$y=0$">|; 

$key = q/{displaymath}C=lim_{{mitDelta}Trightarrow0}frac{{mitDelta}E}{{mitDelta}T}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="\begin{displaymath}
C = \lim_{{\mit\Delta}T\rightarrow 0}\frac{{\mit\Delta}E}{{\mit\Delta} T}
\end{displaymath}">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="$G$">|; 

$key = q/{displaymath}x_j=j,deltax,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="\begin{displaymath}
x_j = j \delta x,
\end{displaymath}">|; 

$key = q/h^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$h^{-1}$">|; 

$key = q/n=1,l=+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="$n=1, l=+1$">|; 

$key = q/|f'|=1slashx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="$\vert f'\vert = 1/x$">|; 

$key = q/T=1.8,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$T=1.8 T_0$">|; 

$key = q/1.37003;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="$1.37003$">|; 

$key = q/n(0)=n(L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="$n(0)=n(L)$">|; 

$key = q/x_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="$x_i$">|; 

$key = q/y=f(x)=-lnx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="$y=f(x)=-\ln x$">|; 

$key = q/83^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$83^\circ $">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{sketch.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{sketch.eps}}
\end{figure}">|; 

$key = q/1.34870;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="$1.34870$">|; 

$key = q/{displaymath}frac{partialT}{partialt}=D,nabla^2T,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="\begin{displaymath}
\frac{\partial T}{\partial t} = D \nabla^2 T,
\end{displaymath}">|; 

$key = q/{displaymath}T_c=frac{2,T_0}{ln(1+sqrt{2})}=2.27,T_0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="\begin{displaymath}
T_c = \frac{2 T_0}{\ln(1+\sqrt{2})} = 2.27 T_0,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ts5.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ts5.eps}}
\end{figure}">|; 

$key = q/s_i=bar{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="$s_i=\bar{s}$">|; 

$key = q/displaystyle(H_x)_{i,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="$\displaystyle (H_x)_{i,j}^n$">|; 

$key = q/{displaymath}F(v)=0.0039+frac{0.0058}{1+{{rm{exp}[(v-v_d)slash{mitDelta}]}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="\begin{displaymath}
F(v) = 0.0039 + \frac{0.0058}{1+{\rm exp}[(v-v_d)/{\mit\Delta}]}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{x1376.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{x1376.eps}}
\end{figure}">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$z$">|; 

$key = q/{displaymath}u_{i+1}=x_i,u_i+y_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="\begin{displaymath}
u_{i+1} = x_i u_i + y_i
\end{displaymath}">|; 

$key = q/v_0=95;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$v_0=95$">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$N$">|; 

$key = q/4.69pm0.01;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="$4.69\pm 0.01$">|; 

$key = q/{displaymath}x=sin(sqrt{k},t^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="\begin{displaymath}
x = \sin(\sqrt{k} t^2).
\end{displaymath}">|; 

$key = q/v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$v^2$">|; 

$key = q/y=bar{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="$y=\bar{y}$">|; 

$key = q/displaystyleF_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="$\displaystyle F_j,$">|; 

$key = q/v(0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$v(0)$">|; 

$key = q/B_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$B_y$">|; 

$key = q/{displaymath}f_j=sum_{k=0}^{2J-1}F_k,exp(,{{rm{i},j,k,pislashJ).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="\begin{displaymath}
f_j = \sum_{k=0}^{2J-1} F_k \exp( {\rm i} j k \pi/J).
\end{displaymath}">|; 

$key = q/{displaymath}x=sin(sqrt{k},t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="\begin{displaymath}
x = \sin(\sqrt{k} t).
\end{displaymath}">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="$L$">|; 

$key = q/fsimeq0.7;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="$f\simeq 0.7$">|; 

$key = q/t=10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$t=10^{-3}$">|; 

$key = q/displaystylefrac{n(x)}{n_0}-1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="$\displaystyle \frac{n(x)}{n_0}-1.$">|; 

$key = q/{displaymath}nabla^2phi=-frac{rho}{epsilon_0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="\begin{displaymath}
\nabla^2 \phi = - \frac{\rho}{\epsilon_0},
\end{displaymath}">|; 

$key = q/displaystylefrac{gamma_h^n,deltax+beta_h^n,T_N^n}{alpha_h^n,deltax+beta_h^n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="$\displaystyle \frac{\gamma_h^n \delta x + \beta_h^n T_N^n}{\alpha_h^n \delta x +\beta_h^n},$">|; 

$key = q/displaystylev_0,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$\displaystyle v_0 \cos\theta$">|; 

$key = q/x_h=1+v,t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="$x_h = 1 + v t$">|; 

$key = q/n(x)=int_{-infty}^{infty}f(x,v),dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="$n(x) = \int_{-\infty}^{\infty} f(x,v) dv$">|; 

$key = q/t=0.004;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$t=0.004$">|; 

$key = q/t=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$t=10$">|; 

$key = q/0.6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$0.6$">|; 

$key = q/j=0,infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="$j=0,\infty$">|; 

$key = q/displaystylehat{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$\displaystyle \hat{A}$">|; 

$key = q/e>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="$e&gt;0$">|; 

$key = q/{{tt{Naccgeq100};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="${\tt Nacc\geq 100}$">|; 

$key = q/Q=1.3757;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$Q=1.3757$">|; 

$key = q/y=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$y=1$">|; 

$key = q/6.1times10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$6.1\times 10^{-6}$">|; 

$key = q/{displaymath}frac{partial^2u(x,y)}{partialx^2}+frac{partial^2u(x,y)}{partialy^2}=v(x,y),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="\begin{displaymath}
\frac{\partial^2 u(x,y)}{\partial x^2}+\frac{\partial^2 u(x,y)}{\partial y^2} = v(x,y),
\end{displaymath}">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\theta$">|; 

$key = q/displaystylefrac{dx_i}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="$\displaystyle \frac{dx_i}{dt}$">|; 

$key = q/{displaymath}frac{2}{L}int_0^Lsin(j,pi,yslashL),sin(k,pi,yslashL),dy=delta_{jk}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="\begin{displaymath}
\frac{2}{L}\int_0^L \sin(j \pi y/L) \sin(k \pi y/L) dy = \delta_{jk}.
\end{displaymath}">|; 

$key = q/{displaymath}y'=f(x,y),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="\begin{displaymath}
y' = f(x,y),
\end{displaymath}">|; 

$key = q/i=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$i=0$">|; 

$key = q/phi_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="$\phi_j$">|; 

$key = q/{displaymath}U_{i-1,j}-(2+j^2,kappa^2),U_{i,j}+U_{i+1,j}=V_{i,j},(deltax)^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="\begin{displaymath}
U_{i-1,j} - (2+j^2 \kappa^2) U_{i,j} + U_{i+1,j} = V_{i,j} (\delta x)^2,
\end{displaymath}">|; 

$key = q/displaystylefrac{partialH_y}{partialt}+c,frac{partialE_x}{partialz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="$\displaystyle \frac{\partial H_y}{\partial t} + c \frac{\partial E_x}{\partial z}$">|; 

$key = q/(hat{E}_i^{n+1},hat{H}_i^{n+1})=lambda,(hat{E}_i^n,hat{H}_i^n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="$(\hat{E}_i^{n+1}, \hat{H}_i^{n+1}) = \lambda (\hat{E}_i^n, \hat{H}_i^n)$">|; 

$key = q/x_{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$x_{n+1}$">|; 

$key = q/v_0(t_0)+deltav_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$v_0(t_0)
+\delta v_0$">|; 

$key = q/v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$v_0$">|; 

$key = q/displaystylev_y(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="$\displaystyle v_y(t=0)$">|; 

$key = q/displaystyle0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="$\displaystyle 0.$">|; 

$key = q/0.52;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$0.52$">|; 

$key = q/A=10^{-16};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$A=10^{-16}$">|; 

$key = q/2N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="$2N$">|; 

$key = q/v(0)slashA=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$v(0)/A=0$">|; 

$key = q/j=1,10000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$j=1,10000$">|; 

$key = q/N=100;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$N=100$">|; 

$key = q/i=2,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="$i=2,N$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{mc.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{mc.eps}}
\end{figure}">|; 

$key = q/x_{n-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$x_{n-2}$">|; 

$key = q/P_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$P_a$">|; 

$key = q/displaystyle-frac{dphi(x)}{dx},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="$\displaystyle - \frac{d\phi(x)}{dx},$">|; 

$key = q/{displaymath}omega_ast=sqrt{1-frac{1}{4Q^2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="\begin{displaymath}
\omega_\ast = \sqrt{1-\frac{1}{4Q^2}},
\end{displaymath}">|; 

$key = q/T>T_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="$T&gt;T_c$">|; 

$key = q/y_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$y_i$">|; 

$key = q/1slashlambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="$1/\lambda$">|; 

$key = q/{displaymath}u(x,y)=sum_{j=0}^{infty}U_j(x),sin(j,pi,yslashL).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="\begin{displaymath}
u(x,y) = \sum_{j=0}^{\infty} U_j(x) \sin(j \pi y/L).
\end{displaymath}">|; 

$key = q/{displaymath}y_{n+1}=y_n+h,f(x_n,y_n)+O(h^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="\begin{displaymath}
y_{n+1} = y_n + h f(x_n,y_n) + O(h^2).
\end{displaymath}">|; 

$key = q/displaystylefrac{alpha_l,[gamma_h-(alpha_h+beta_h)slash3]-gamma_l,alpha_h}{alpha_l,alpha_h+alpha_l,beta_h-beta_l,alpha_h}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="$\displaystyle \frac{ \alpha_l [\gamma_h-(\alpha_h+\beta_h)/3]-\gamma_l \alpha_h}
{\alpha_l \alpha_h + \alpha_l \beta_h-\beta_l \alpha_h}.$">|; 

$key = q/C=D,deltatslash(deltax)^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="$C= D \delta t/(\delta x)^2$">|; 

$key = q/n=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$n=1$">|; 

$key = q/{displaymath}E=sum_{i=1,N}e_i=-N,k,T_cleft(frac{H}{H_c}+bar{s}right)bar{s},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="\begin{displaymath}
E=\sum_{i=1,N} e_i= -N k T_c\left(\frac{H}{H_c}+\bar{s}\right)\bar{s},
\end{displaymath}">|; 

$key = q/sin(omega_ast,t_0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="$\sin(\omega_\ast t_0)=0$">|; 

$key = q/t_0=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="$t_0=0.1$">|; 

$key = q/hat{T}_{i,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="$\hat{T}_{i,j}^n$">|; 

$key = q/0<null1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$0&lt;\nu\ll 1$">|; 

$key = q/beta_h=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$\beta _h=1$">|; 

$key = q/displaystylesum_{i'=0,I}^{j'=0,J}hat{E}_{i',j'}^n,sin(i,i',pislashI),sin(j,j',pislashJ),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="$\displaystyle \sum_{i'=0,I}^{j'=0,J}\hat{E}_{i',j'}^n \sin(i i' \pi/I) \sin(j j' \pi/J),$">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$f(x)$">|; 

$key = q/C_{{rm{max}slashN,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="$C_{\rm max}/N k$">|; 

$key = q/|lambda|>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="$\vert\lambda\vert&gt;1$">|; 

$key = q/{displaymath}A=N,h^2+O(N^{-1slash2}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="\begin{displaymath}
A = N h^2 + O(N^{-1/2}).
\end{displaymath}">|; 

$key = q/T=T_c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$T=T_c$">|; 

$key = q/Qsimeq1.370;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$Q\simeq 1.370$">|; 

$key = q/c_N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="$c_N$">|; 

$key = q/hat{rho}_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="$\hat{\rho}_0=0$">|; 

$key = q/86,mph;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$86 mph$">|; 

$key = q/u(x,t)=hat{u}(t),exp({{rm{i},k,x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="$u(x,t)= \hat{u}(t) \exp ({\rm i} k x)$">|; 

$key = q/J=1000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$J=1000$">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{rand4.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="434" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{rand4.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{basin0a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="487" HEIGHT="439" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{basin0a.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{(H_y)_{i,j}^{n+1}-(H_y)_{i,j}^{n}}{deltat}+frac{c}{2}left(frac{,j}+frac{c}{2}left(frac{partialE_z}{partialx}right)^{n}_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="$\displaystyle \frac{(H_y)_{i,j}^{n+1} -(H_y)_{i,j}^{n}}{\delta t}
+\frac{c}{2}\...
...^{n+1}_{i,j}
+\frac{c}{2}\left(\frac{\partial E_z}{\partial x}\right)^{n}_{i,j}$">|; 

$key = q/displaystyle+frac{omegaAleft[-(1-omega^2),sinomegat+(omegaslashQ),cosomegatright]}{left[(1-omega^2)^2+omega^2slashQ^2right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$\displaystyle +\frac{\omega A\left[-(1-\omega^2) \sin\omega t+ (\omega/Q) \cos\omega t\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]}.$">|; 

$key = q/deltay=L_yslashJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="$\delta y = L_y/J$">|; 

$key = q/N=J=128;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$N=J=128$">|; 

$key = q/v(0)=-2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$v(0)=-2$">|; 

$key = q/{{tt{Nacc}=96;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="${\tt Nacc} = 96$">|; 

$key = q/{displaymath}frac{u_i^{n+1}-u_i^n}{deltat}=-v,frac{u_{i+1}^n-u_{i}^n}{deltax},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="\begin{displaymath}
\frac{u_i^{n+1} - u_i^n}{\delta t} = - v \frac{u_{i+1}^n-u_{i}^n}{\delta x},
\end{displaymath}">|; 

$key = q/{displaymath}int_{-infty}^{+infty}P(x),dx=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="\begin{displaymath}
\int_{-\infty}^{+\infty} P(x) dx = 1.
\end{displaymath}">|; 

$key = q/M-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="$M-1$">|; 

$key = q/{ttNacc}=96;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="${\tt Nacc} = 96$">|; 

$key = q/h_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$h_0$">|; 

$key = q/t_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$t_0&gt;0$">|; 

$key = q/E_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$E_x$">|; 

$key = q/gamma_l=2,y^3-3,y^2+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="$\gamma_l=2 y^3-3 y^2+1$">|; 

$key = q/1-p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="$1-p$">|; 

$key = q/9.0times10^{-14};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$9.0\times 10^{-14}$">|; 

$key = q/displaystylefrac{Gamma_{h,j},deltax+beta_h,U_{N,j}}{alpha_h,deltax+beta_h},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="$\displaystyle \frac{\Gamma_{h j} \delta x + \beta_h U_{N,j}}{\alpha_h \delta x +\beta_h},$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ellipse.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ellipse.eps}}
\end{figure}">|; 

$key = q/k=1,2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="$k=1,2$">|; 

$key = q/y=sin^2x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$y=\sin^2 x$">|; 

$key = q/4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="$4$">|; 

$key = q/D=pi,c,deltatslash(2,L);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="$D=\pi c \delta t/(2 L)$">|; 

$key = q/|M|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="$\vert M\vert$">|; 

$key = q/displaystyleeta^{1slash(n+1)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$\displaystyle \eta^{1/(n+1)},$">|; 

$key = q/n(x),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="$n(x) dx$">|; 

$key = q/3.7times10^{-11};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$3.7\times 10^{-11}$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$R$">|; 

$key = q/10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="$10^{-3}$">|; 

$key = q/{mitDelta}E>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="${\mit\Delta}E&gt;0$">|; 

$key = q/theta=-c,partialxislashpartialx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="$\theta=-c \partial\xi/\partial x$">|; 

$key = q/{displaymath}w_1=v_1,(deltax)^2-frac{gamma_l,deltax}{alpha_l,deltax-beta_l},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="\begin{displaymath}
w_1 = v_1 (\delta x)^2 - \frac{\gamma_l \delta x}{\alpha_l \delta x-\beta_l},
\end{displaymath}">|; 

$key = q/s_i=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="$s_i=-1$">|; 

$key = q/{displaymath}frac{d^2u(x_i)}{dx^2}=frac{u_{i-1}-2,u_i+u_{i+1}}{(deltax)^2}+O(deltax)^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="\begin{displaymath}
\frac{d^2u(x_i)}{dx^2} = \frac{u_{i-1} - 2 u_i+u_{i+1}}{(\delta x)^2} + O(\delta x)^2.
\end{displaymath}">|; 

$key = q/1.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="$1.2$">|; 

$key = q/{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="${\bf a}$">|; 

$key = q/{displaymath}hat{T}^{n+1}=A,hat{T}^n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="\begin{displaymath}
\hat{T}^{n+1} = A \hat{T}^n,
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialT(x,y,t)}{partialt}=D,frac{partial^2T(x,y,t)}{partialx^2}+D,frac{partial^2T(x,y,t)}{partialy^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="\begin{displaymath}
\frac{\partial T(x,y,t)}{\partial t} = D \frac{\partial^2 T...
...)}{\partial x^2}+D \frac{\partial^2 T(x,y,t)}
{\partial y^2},
\end{displaymath}">|; 

$key = q/displaystylefrac{dv_i}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="$\displaystyle \frac{dv_i}{dt}$">|; 

$key = q/Q=1.24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$Q=1.24$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{curver.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{curver.eps}}
\end{figure}">|; 

$key = q/displaystylehat{omega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="$\displaystyle \hat{\omega}$">|; 

$key = q/{displaymath}frac{d^2U_j(x)}{dx^2}-frac{j^2,pi^2}{L^2},U_j(x)=V_j(x),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="\begin{displaymath}
\frac{d^2 U_j(x)}{dx^2} - \frac{j^2 \pi^2}{L^2} U_j(x) = V_j(x),
\end{displaymath}">|; 

$key = q/g=9.8,{{rm{mslashs}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$g=9.8 {\rm m/s}^{-2}$">|; 

$key = q/{{bf{M},{{bf{u}={{bf{w};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="${\bf M} {\bf u} = {\bf w}$">|; 

$key = q/theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="$\theta=0$">|; 

$key = q/displaystylefrac{A}{m,g},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$\displaystyle \frac{A}{m g},$">|; 

$key = q/{displaymath}bar{N}=N',p=N',frac{A}{A'},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="\begin{displaymath}
\bar{N} = N' p = N' \frac{A}{A'},
\end{displaymath}">|; 

$key = q/E_x=-H_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="$E_x=-H_y$">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$A$">|; 

$key = q/{mitDelta}=5,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="${\mit\Delta} = 5 {\rm m/s}$">|; 

$key = q/h=h_0simeta^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$h=h_0\sim \eta^{1/2}$">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$i$">|; 

$key = q/(j+1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="$(j+1)$">|; 

$key = q/k=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$k=1$">|; 

$key = q/{displaymath}lambda^2-2,frac{1-i^2,D^2}{1+i^2,D^2},lambda+1=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="\begin{displaymath}
\lambda^2 - 2 \frac{1-i^2 D^2}{1+i^2 D^2} \lambda + 1 = 0.
\end{displaymath}">|; 

$key = q/{displaymath}A=frac{N}{N'},A'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="\begin{displaymath}
A = \frac{N}{N'} A'.
\end{displaymath}">|; 

$key = q/{displaymath}y_{n+1}=(1-alpha,h),y_n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="\begin{displaymath}
y_{n+1} = (1-\alpha h) y_n.
\end{displaymath}">|; 

$key = q/xi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="$\xi$">|; 

$key = q/C=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="$C=0.5$">|; 

$key = q/theta(0)slashA=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\theta (0)/A=0.5$">|; 

$key = q/Q=intc,T,dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="$Q=\int c T dV$">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$n$">|; 

$key = q/v(x)=1-2,x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$v(x) = 1 - 2 x^2$">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$\tau $">|; 

$key = q/{{tt{Nacc}=24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="${\tt Nacc} = 24$">|; 

$key = q/d=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="$d=3$">|; 

$key = q/j=Jslash2+1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="$j=J/2+1$">|; 

$key = q/(0,1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="$(0,1)$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{x1372.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{x1372.eps}}
\end{figure}">|; 

$key = q/1slash12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$1/12$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cd.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="527" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cd.eps}}
\end{figure}">|; 

$key = q/j=J-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="$j=J-1$">|; 

$key = q/h_0sim10^{-8};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$h_0\sim 10^{-8}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{d13757.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{d13757.eps}}
\end{figure}">|; 

$key = q/{displaymath}alpha_h,u(x,y)+beta_h,frac{partialu(x,y)}{partialx}=gamma_h(y),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="\begin{displaymath}
\alpha_h  u(x,y)+ \beta_h \frac{\partial u(x,y)}{\partial x} = \gamma_h(y),
\end{displaymath}">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$2$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{q130d.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{q130d.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{u_i^{n+1}-u_i^n}{deltat}=-v,frac{u_{i}^n-u_{i-1}^n}{deltax},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="\begin{displaymath}
\frac{u_i^{n+1} - u_i^n}{\delta t} = - v \frac{u_{i}^n-u_{i-1}^n}{\delta x},
\end{displaymath}">|; 

$key = q/{displaymath}f_j=sum_{k=1}^{J-1}F_k^S,sin(j,k,pislashJ).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="\begin{displaymath}
f_j= \sum_{k=1}^{J-1} F_k^S \sin(j k \pi/J).
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{omega_ast}left{theta(0)+frac{v(0)}{2Q}-frac{A,[(1-omega^2)-^2)^2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q}sinomega_astt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$\displaystyle -\frac{1}{\omega_\ast}\left\{ \theta(0) + \frac{v(0)}{2Q}
-\frac{...
...2Q^2]}
{[(1-\omega^2)^2+ \omega^2/Q^2]}\right\}{\rm e}^{-t/2Q}\sin\omega_\ast t$">|; 

$key = q/P(x),dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="$P(x) dx$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{graph1.ps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="289" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{graph1.ps}}
\end{figure}">|; 

$key = q/displaystyleh_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$\displaystyle h_0$">|; 

$key = q/delta_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$\delta_k$">|; 

$key = q/{displaymath}y_{i-1}=frac{w_i-c_i,y_i}{b_i+c_i,x_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="\begin{displaymath}
y_{i-1} = \frac{w_i-c_i y_i}{b_i+c_i x_i}.
\end{displaymath}">|; 

$key = q/x=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$x=0$">|; 

$key = q/L=5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$L=5$">|; 

$key = q/displaystylefrac{(E_z)_{i,j}^{n+1}-(E_z)_{i,j}^{n}}{deltat}+frac{c}{2}left(frac{,j}+frac{c}{2}left(frac{partialH_y}{partialx}right)^{n}_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="$\displaystyle \frac{(E_z)_{i,j}^{n+1} -(E_z)_{i,j}^{n}}{\delta t}
+\frac{c}{2}\...
...^{n+1}_{i,j}
+\frac{c}{2}\left(\frac{\partial H_y}{\partial x}\right)^{n}_{i,j}$">|; 

$key = q/v(x_i,y_j)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="$v(x_i, y_j) = 0$">|; 

$key = q/t=0.0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$t=0.0$">|; 

$key = q/0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="$0$">|; 

$key = q/C=2.0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$C=2.0$">|; 

$key = q/{displaymath}y_{n+1}=y_n+y_n',h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="\begin{displaymath}
y_{n+1} = y_n + y_n' h,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=4in{centerline{epsffile{rand1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="434" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="\begin{figure}
\epsfysize =4in
\centerline{\epsffile{rand1.eps}}
\end{figure}">|; 

$key = q/hslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$h/2$">|; 

$key = q/{displaymath}F_j^C=frac{f_0}{J}+frac{2}{J}sum_{k=1}^{J-1}f_k,cos(j,k,pislashJ)+frac{(-1)^j,f_J}{J}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="\begin{displaymath}
F_j^C = \frac{f_0}{J} + \frac{2}{J}\sum_{k=1}^{J-1} f_k \cos(j k \pi/J)
+ \frac{(-1)^j f_J}{J}
\end{displaymath}">|; 

$key = q/{displaymath}frac{partialT(x,t)}{partialt}=D,frac{partial^2T(x,t)}{partialx^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="\begin{displaymath}
\frac{\partial T(x,t)}{\partial t} = D \frac{\partial^2 T(x,t)}{\partial x^2},
\end{displaymath}">|; 

$key = q/G(varphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$G(\varphi )$">|; 

$key = q/displaystyletheta_1(t_0+{mitDelta}t)-theta_0(t_0+{mitDelta}t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$\displaystyle \theta_1(t_0+{\mit\Delta} t)-\theta_0(t_0+{\mit\Delta} t),$">|; 

$key = q/displaystyle+frac{1}{omega_ast}left{deltav_0+frac{deltatheta_0}{2Q}right}sin(omega_ast{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="$\displaystyle + \frac{1}{\omega_\ast}\left\{\delta v_0 + \frac{\delta\theta_0}{2Q}\right\}
\sin(\omega_\ast {\mit\Delta} t)  {\rm e}^{-{\mit\Delta} t/2Q},$">|; 

$key = q/displaystyle-frac{v}{Q}-sintheta+A,cosomegat.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$\displaystyle -\frac{v}{Q} -\sin\theta + A \cos\omega t.$">|; 

$key = q/M=32768;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$M=32768$">|; 

$key = q/Q=1.30;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$Q=1.30$">|; 

$key = q/(x,y)=(0.25,0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$(x,y)=(0.25,0.5)$">|; 

$key = q/{displaymath}R_e=frac{rho,v,d}{eta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="\begin{displaymath}
R_e = \frac{\rho v d}{\eta}.
\end{displaymath}">|; 

$key = q/{displaymath}hat{phi}_j=hat{phi}^ast_{J-j}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="\begin{displaymath}
\hat{\phi}_j = \hat{\phi}^\ast_{J-j}
\end{displaymath}">|; 

$key = q/t=0.46;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$t=0.46$">|; 

$key = q/N^{-1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$N^{-1/3}$">|; 

$key = q/V';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="$V'$">|; 

$key = q/v_0(t_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="$v_0(t_0)$">|; 

$key = q/O(deltax)^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="$O(\delta x)^n$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{glatr.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="425" HEIGHT="378" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{glatr.eps}}
\end{figure}">|; 

$key = q/beta_l=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\beta _l=-1$">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="$F$">|; 

$key = q/displaystyleh;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="$\displaystyle h$">|; 

$key = q/U_{i,j}equivU_j(x_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="$U_{i,j} \equiv U_j(x_i)$">|; 

$key = q/O(J^2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="$O(J^2)$">|; 

$key = q/n=1,l=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="$n=1, l=-1$">|; 

$key = q/{displaymath}A=A',frac{N}{N'}pmfrac{sqrt{A,(A'-A)}}{sqrt{N'}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="\begin{displaymath}
A = A' \frac{N}{N'} \pm \frac{\sqrt{A (A'-A)}}{\sqrt{N'}}.
\end{displaymath}">|; 

$key = q/0.8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$0.8$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{fixed.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="324" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{fixed.eps}}
\end{figure}">|; 

$key = q/Qsimeq1.375;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="$Q\simeq 1.375$">|; 

$key = q/phi(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="$\phi(x,y)$">|; 

$key = q/displaystylesum_{j'=0,J-1}hat{phi}_{j'},{{rm{e}^{,{{rm{i},j,j',2,pislashJ},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="$\displaystyle \sum_{j'=0,J-1}\hat{\phi}_{j'}  {\rm e}^{ {\rm i} j j' 2 \pi/J},$">|; 

$key = q/displaystyle-k,x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$\displaystyle -k x,$">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$d$">|; 

$key = q/F=4.669ldots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="$F=4.669\ldots$">|; 

$key = q/lambda_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$\lambda_2$">|; 

$key = q/1.5times10^{-8};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$1.5\times 10^{-8}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{p76.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{p76.eps}}
\end{figure}">|; 

$key = q/partialu(x,0)slashpartialy=partialu(x,1)slashpartialy=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="$\partial u(x,0)/\partial y=\partial u(x,1)/\partial y=0$">|; 

$key = q/displaystylehat{T}_{0,j}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$\displaystyle \hat{T}_{0,j}^n$">|; 

$key = q/Q_infty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$Q_\infty$">|; 

$key = q/(x,y)=(0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="$(x,y) = (0,0)$">|; 

$key = q/H_y(x=0.0,y=0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$H_y(x=0.0,y=0.5)$">|; 

$key = q/0.4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$0.4$">|; 

$key = q/AAAAAAcdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$AAAAAA\cdots$">|; 

$key = q/v_0(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="$v_0(t)$">|; 

$key = q/+x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="$+x$">|; 

$key = q/1.37575;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="$1.37575$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{coulomb3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="346" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{coulomb3.eps}}
\end{figure}">|; 

$key = q/displaystyleF_{2J-j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="$\displaystyle F_{2J-j}$">|; 

$key = q/hsim10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="$h\sim 10^{-1}$">|; 

$key = q/displaystylesum_{i'=0,I}^{j'=0,J}hat{X}_{i',j'}^n,sin(i,i',pislashI),cos(j,j',pislashJ),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="$\displaystyle \sum_{i'=0,I}^{j'=0,J} \hat{X}_{i',j'}^n \sin(i i' \pi/I) \cos(j j' \pi/J),$">|; 

$key = q/j=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="$j=0$">|; 

$key = q/displaystylehat{Y}_{i,j}^n-i,D_x,(hat{E}_{i,j}^{n+1}+hat{E}_{i,j}^{n}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="$\displaystyle \hat{Y}_{i,j}^n - i D_x (\hat{E}_{i,j}^{n+1}+\hat{E}_{i,j}^{n}).$">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$f$">|; 

$key = q/1.37458;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="$1.37458$">|; 

$key = q/{ttNacc}=100;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="${\tt Nacc} = 100$">|; 

$key = q/rightarrow;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$\rightarrow$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{h.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="319" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{h.eps}}
\end{figure}">|; 

$key = q/x_{n-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$x_{n-1}$">|; 

$key = q/{displaymath}y_n'=f(x_n,y_n).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="\begin{displaymath}
y_n' = f(x_n,y_n).
\end{displaymath}">|; 

$key = q/P(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="$P(y)$">|; 

$key = q/n=4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="$n=4$">|; 

$key = q/10^{-12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$10^{-12}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{scanvud.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="322" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{scanvud.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{C}{2},hat{T}_{i-1,j}^{n}+left{1-C,(1+j^2,kappa^2slash2)right},hat{T}_{i,j}^{n}+frac{C}{2},hat{T}_{i+1,j}^{n},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="395" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="$\displaystyle \frac{C}{2} \hat{T}_{i-1,j}^{n} + \left\{1-C (1+j^2 \kappa^2/2)\right\} 
\hat{T}_{i,j}^{n}+\frac{C}{2} \hat{T}_{i+1,j}^{n},$">|; 

$key = q/bar{y};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="$\bar{y}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{upwind2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="323" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{upwind2.eps}}
\end{figure}">|; 

$key = q/10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="$10^{-2}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{x15.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="577" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{x15.eps}}
\end{figure}">|; 

$key = q/displaystyledeltat,hat{J}^{n}_{i,j},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="$\displaystyle \delta t \hat{J}^{n}_{i,j},$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="$c$">|; 

$key = q/displaystylehat{E}_i^{n+1}-hat{E}_i^n-i,D,(hat{H}_i^{n+1}+hat{H}_i^n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="252" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="$\displaystyle \hat{E}_i^{n+1}-\hat{E}_i^n - i D (\hat{H}_i^{n+1}+\hat{H}_i^n)$">|; 

$key = q/x=x_h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$x=x_h$">|; 

$key = q/{displaymath}E_j=frac{phi_{j-1}-phi_{j+1}}{2,deltax}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="\begin{displaymath}
E_j = \frac{\phi_{j-1}-\phi_{j+1}}{2 \delta x}.
\end{displaymath}">|; 

$key = q/H_{{rm{eff};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="$H_{\rm eff}$">|; 

$key = q/deltat=9.95times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\delta t = 9.95\times 10^{-3}$">|; 

$key = q/n_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="$n_j$">|; 

$key = q/y=(A,z),{{rm{mod}M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="$y = (A z) {\rm mod} M$">|; 

$key = q/c=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$c=1$">|; 

$key = q/{displaymath}T(x,t)=hat{T}(t),{{rm{e}^{,{{rm{i},k,x}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="\begin{displaymath}
T(x,t) = \hat{T}(t) {\rm e}^{ {\rm i} k x}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{nodisperse.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="323" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{nodisperse.eps}}
\end{figure}">|; 

$key = q/v+dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="$v+dv$">|; 

$key = q/dslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$d/dx$">|; 

$key = q/theta=1^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\theta = 1^\circ $">|; 

$key = q/phi_J=phi_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="$\phi_J=\phi_0$">|; 

$key = q/t=10.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$t=10.$">|; 

$key = q/Qsimeq1.245;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$Q\simeq 1.245$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cv10.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cv10.eps}}
\end{figure}">|; 

$key = q/M=2147483647;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="$M=2147483647$">|; 

$key = q/{displaymath}Gamma_{l,j}=frac{2}{L}int_0^Lgamma_l(y),cos(j,pi,yslashL),dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="\begin{displaymath}
\Gamma_{l j} = \frac{2}{L} \int_0^L \gamma_l(y) \cos(j \pi y/L) dy,
\end{displaymath}">|; 

$key = q/epsilon_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$\epsilon _0$">|; 

$key = q/{mbox{{boldmath{{omega{}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="${\mbox{\boldmath$\omega$}}$">|; 

$key = q/{displaymath}{{bf{M},{{bf{u}={{bf{w}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="\begin{displaymath}
{\bf M} {\bf u} = {\bf w}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{T_i^{n+1}-T_i^n}{deltat}=D,frac{T_{i-1}^n-2,T_i^n+T_{i+1}^n}{(deltax)^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="289" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="\begin{displaymath}
\frac{T_i^{n+1}-T_i^n}{\delta t} = D \frac{T_{i-1}^n-2 T_i^n+T_{i+1}^n}{(\delta x)^2},
\end{displaymath}">|; 

$key = q/x=x_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$x=x_n$">|; 

$key = q/C_{{rm{max}slashN,k=1.34;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="$C_{\rm max}/N k = 1.34$">|; 

$key = q/kappa;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="$\kappa$">|; 

$key = q/{displaymath}P_x(x)=left{array{{lcl}1&mbox{hspace{1cm}}&mbox{if{0leqxleq1{}0&&mbox{otherwise}array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="\begin{displaymath}
P_x(x) = \left\{\begin{array}{lcl}
1&amp;\mbox{\hspace{1cm}}&amp;\mb...
...$0 \leq x \leq 1$}\\\\
0&amp;&amp; \mbox{otherwise}
\end{array}\right..
\end{displaymath}">|; 

$key = q/{displaymath}f_0=frac{c}{2}sqrt{frac{1}{(n,L_x)^2}+frac{1}{(m,L_y)^2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="\begin{displaymath}
f_0 = \frac{c}{2}\sqrt{\frac{1}{(n L_x)^2}+\frac{1}{(m L_y)^2}}.
\end{displaymath}">|; 

$key = q/gamma_{l,j}^nequivgamma_l(y_j,t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="$\gamma_{l j}^n\equiv
\gamma_l(y_j,t_n)$">|; 

$key = q/v_{th}=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="$v_{th}=1$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{scan1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="384" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{scan1.eps}}
\end{figure}">|; 

$key = q/O(etaslashh);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$O(\eta/h)$">|; 

$key = q/O(J,lnJ);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="$O(J \ln J)$">|; 

$key = q/Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$Q$">|; 

$key = q/{ttNacc}=48;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="${\tt Nacc} = 48$">|; 

$key = q/displaystyle-g-F(v),v,v_z-B,omega,v_x,sinphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$\displaystyle -g - F(v) v v_z - B \omega v_x \sin\phi.$">|; 

$key = q/{displaymath}R=frac{A}{sqrt{(1-omega^2)^2+omega^2slashQ^2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="\begin{displaymath}
R = \frac{A}{\sqrt{(1-\omega^2)^2+\omega^2/Q^2}}.
\end{displaymath}">|; 

$key = q/3.0times10^{-13};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$3.0\times 10^{-13}$">|; 

$key = q/{displaymath}sigma^2=overline{(N-bar{N})^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="\begin{displaymath}
\sigma^2 = \overline{(N-\bar{N})^2},
\end{displaymath}">|; 

$key = q/{displaymath}frac{partial^2xi}{partialt^2}=c^2,frac{partial^2xi}{partialx^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="\begin{displaymath}
\frac{\partial^2\xi}{\partial t^2} = c^2 \frac{\partial^2 \xi}{\partial x^2},
\end{displaymath}">|; 

$key = q/{displaymath}e_i=-mu,H_{{rm{eff},s_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="\begin{displaymath}
e_i = -\mu H_{\rm eff} s_i,
\end{displaymath}">|; 

$key = q/I_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="$I_1$">|; 

$key = q/{displaymath}y(x_0)=y_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="\begin{displaymath}
y(x_0) = y_0.
\end{displaymath}">|; 

$key = q/{displaymath}hat{T}_{i,j}^n=frac{2}{J}sum_{k=0}^{J}T_{i,k}^n,sin(j,k,pislashJ).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="\begin{displaymath}
\hat{T}_{i,j}^n = \frac{2}{J}\sum_{k=0}^{J} T_{i,k}^n \sin(j k \pi/J).
\end{displaymath}">|; 

$key = q/22.5^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$22.5^\circ $">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$h$">|; 

$key = q/{displaymath}alpha_l(t),T(x,t)+beta_l(t),frac{partialT(x,t)}{partialx}=gamma_l(t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="\begin{displaymath}
\alpha_l(t) T(x,t) + \beta_l(t) \frac{\partial T(x,t)}{\partial x} = \gamma_l(t),
\end{displaymath}">|; 

$key = q/displaystylex(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$\displaystyle x(t=0)$">|; 

$key = q/t=0.37;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$t=0.37$">|; 

$key = q/deltav;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="$\delta v$">|; 

$key = q/i=1,N-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="$i=1, N-1$">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\omega $">|; 

$key = q/displaystylesum_{j=0,I}hat{E}_j^n,sin(i,j,pislashI),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="$\displaystyle \sum_{j=0,I} \hat{E}_j^n \sin(i j \pi/I),$">|; 

$key = q/{displaymath}Gamma_{l,j}=frac{2}{L}int_0^Lgamma_l(y),sin(j,pi,yslashL),dy,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="\begin{displaymath}
\Gamma_{l j} = \frac{2}{L} \int_0^L \gamma_l(y) \sin(j \pi y/L) dy,
\end{displaymath}">|; 

$key = q/x_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="$x_0=0$">|; 

$key = q/C=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="$C=0$">|; 

$key = q/v(0)slashA=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$v(0)/A=0.5$">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="$\textstyle =$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{graph2a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{graph2a.eps}}
\end{figure}">|; 

$key = q/displaystyley_n+h,f(x_n,y_n)+frac{h^2}{2},y''(x_n)+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$\displaystyle y_n + h f(x_n,y_n) + \frac{h^2}{2} y''(x_n)+\cdots.$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{diff3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="319" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{diff3.eps}}
\end{figure}">|; 

$key = q/{displaymath}hat{T}^{n+1},{{rm{e}^{,{{rm{i},k,x_n}=hat{T}^n,{{rm{e}^{,{{rm{i},k,,k,deltax}-2+{{rm{e}^{+{{rm{i},k,deltax})right],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="38" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="\begin{displaymath}
\hat{T}^{n+1} {\rm e}^{ {\rm i} k x_n} = \hat{T}^n {\rm...
...m i} k \delta x}-2+ {\rm e}^{+{\rm i} k \delta x})\right],
\end{displaymath}">|; 

$key = q/A=1103515245;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$A=1103515245$">|; 

$key = q/{displaymath}v(x,y)=-2,(2,y^3-3,y^2+1)+6,(1-x^2),(2,y-1){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="\begin{displaymath}
v(x,y) =-2 (2 y^3-3 y^2+1) + 6 (1-x^2) (2 y-1)
\end{displaymath}">|; 

$key = q/t=0.30;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$t=0.30$">|; 

$key = q/exp(-beta,E);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="$\exp(-\beta E)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cvmax.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="412" HEIGHT="332" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cvmax.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p213.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p213.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{x13757.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{x13757.eps}}
\end{figure}">|; 

$key = q/Q=1.348;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$Q=1.348$">|; 

$key = q/lnL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="$\ln L$">|; 

$key = q/displaystylefrac{(H_x)_{i,j}^{n+1}-(H_x)_{i,j}^{n}}{deltat}+frac{c}{2}left(frac{,j}+frac{c}{2}left(frac{partialE_z}{partialy}right)^{n}_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="$\displaystyle \frac{(H_x)_{i,j}^{n+1} -(H_x)_{i,j}^{n}}{\delta t}
+\frac{c}{2}\...
...^{n+1}_{i,j}
+\frac{c}{2}\left(\frac{\partial E_z}{\partial y}\right)^{n}_{i,j}$">|; 

$key = q/l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$l$">|; 

$key = q/1.8times10^{-12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$1.8\times 10^{-12}$">|; 

$key = q/O(h^3)timesO(h^{-2})=O(h)=O(N^{-1slash3});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="$O(h^3)\times O(h^{-2}) =
O(h)= O(N^{-1/3})$">|; 

$key = q/displaystyletheta(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$\displaystyle \theta(t)$">|; 

$key = q/O(h^2)timesO(h^{-1})=O(h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="$O(h^2)\times O(h^{-1}) =
O(h)$">|; 

$key = q/displaystyleu_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="$\displaystyle u_0$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{image4.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="217" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{image4.eps}}\end{figure}">|; 

$key = q/gamma_h=y,(1-y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$\gamma_h=y (1-y)$">|; 

$key = q/0leqxleq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="$0\leq x\leq 1$">|; 

$key = q/sin^2(k,deltaxslash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="$\sin^2(k \delta x/2)$">|; 

$key = q/displaystyleU_{N+1,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="$\displaystyle U_{N+1,j}$">|; 

$key = q/u(x_h)=u_h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="$u(x_h) = u_h$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{coulomb1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="346" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{coulomb1.eps}}
\end{figure}">|; 

$key = q/{displaymath}E(r_i)=left(frac{x_{j+1}-r_i}{x_{j+1}-x_j}right)E_j+left(frac{r_i-x_j}{x_{j+1}-x_j}right)E_{j+1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="\begin{displaymath}
E(r_i) = \left(\frac{x_{j+1}-r_i}{x_{j+1}-x_j}\right)E_j+\left(\frac{r_i - x_j}{x_{j+1}-x_j}\right)
E_{j+1}.
\end{displaymath}">|; 

$key = q/displaystylefrac{T_{i,j}^{n+1}-T_{i,j}^n}{deltat}-frac{D}{2},frac{T_{i-1,j}^{n+1rac{D}{2}left(frac{partial^2T}{partialy^2}right)_{i,j}^{n+1}=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="488" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="$\displaystyle \frac{T_{i,j}^{n+1}-T_{i,j}^n}{\delta t} - \frac{D}{2} 
\frac{T_...
... x)^2}
- \frac{D}{2}\left(\frac{\partial^2 T}{\partial y^2}\right)_{i,j}^{n+1}=$">|; 

$key = q/|lambda|=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="$\vert\lambda\vert=1$">|; 

$key = q/1.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="$1.3$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{bs1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{bs1.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{image1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="217" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{image1.eps}}
\end{figure}">|; 

$key = q/T=lslashv_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$T = l/v_0$">|; 

$key = q/f(x,y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$f(x,y)$">|; 

$key = q/N=10^6;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="$N=10^6$">|; 

$key = q/displaystylef_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="$\displaystyle f_j,$">|; 

$key = q/phi=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\phi =0$">|; 

$key = q/{{bf{u}=(u_1,,u_2,,cdots,,u_N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="${\bf u} = (u_1,  u_2,  \cdots,  u_N)$">|; 

$key = q/u_iequivu(x_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="$u_i \equiv u(x_i)$">|; 

$key = q/{displaymath}u(x,0)=exp[-100,(x-0.5)^2],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="\begin{displaymath}
u(x,0) = \exp[-100 (x-0.5)^2],
\end{displaymath}">|; 

$key = q/{displaymath}alpha_h(t),T(x,y,t)+beta_h(t),frac{partialT(x,y,t)}{partialx}=gamma_h(y,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="\begin{displaymath}
\alpha_h(t)  T(x,y,t)+ \beta_h(t) \frac{\partial T(x,y,t)}{\partial x} = \gamma_h(y,t),
\end{displaymath}">|; 

$key = q/{displaymath}tilde{A}=frac{A}{sqrt{(1-omega^2)^2+omega^2slashQ^2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="\begin{displaymath}
\tilde{A} = \frac{A}{\sqrt{(1-\omega^2)^2+\omega^2/Q^2}}.
\end{displaymath}">|; 

$key = q/t=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$t=2$">|; 

$key = q/C={mitDeltaE}slash{mitDeltaT};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$C={\mit\Delta E}/{\mit\Delta T}$">|; 

$key = q/t_{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="$t_{n+1}$">|; 

$key = q/deltat=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\delta t = 0.1$">|; 

$key = q/u_0=u_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="$u_0=u_l$">|; 

$key = q/displaystylev_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$\displaystyle v_z,$">|; 

$key = q/|hat{u}_k|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="$\vert\hat{u}_k\vert$">|; 

$key = q/{displaymath}deltat<frac{(deltax)^2}{2,D}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="\begin{displaymath}
\delta t &lt; \frac{(\delta x)^2}{2 D}.
\end{displaymath}">|; 

$key = q/{displaymath}bar{s}=tanhleft{frac{T_c}{T}left(frac{H}{H_c}+bar{s}right)right}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="\begin{displaymath}
\bar{s} = \tanh\left\{\frac{T_c}{T}\left(\frac{H}{H_c} + \bar{s}\right)\right\}.
\end{displaymath}">|; 

$key = q/{displaymath}T(x,t)=sqrt{frac{t_0}{t}},exp!left(frac{-x^2}{4,D,t}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="\begin{displaymath}
T(x,t) =\sqrt{\frac{t_0}{t}}  \exp\!\left(\frac{-x^2}{4 D t}\right).
\end{displaymath}">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="$J$">|; 

$key = q/displaystyleu_{N+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$\displaystyle u_{N+1}$">|; 

$key = q/kappa=2pislashL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="$\kappa = 2\pi / L$">|; 

$key = q/{displaymath}P_y(y)=frac{{{rm{exp}[(y-bar{y})^2slash2,sigma^2]}{sqrt{2pi},sigma},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="\begin{displaymath}
P_y(y) =\frac{{\rm exp}[(y-\bar{y})^2/2 \sigma^2]}{\sqrt{2\pi} \sigma},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{inst.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{inst.eps}}
\end{figure}">|; 

$key = q/O(N);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="$O(N)$">|; 

$key = q/theta_1(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$\theta_1(t)$">|; 

$key = q/P_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="$P_b$">|; 

$key = q/alpha_h=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$\alpha _h=1$">|; 

$key = q/Q>1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$Q&gt;1/2$">|; 

$key = q/C_{{rm{max}slashN,k=2.16;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="$C_{\rm max}/N k = 2.16$">|; 

$key = q/displaystylef_{2J-j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="$\displaystyle f_{2J-j}$">|; 

$key = q/{displaymath}bar{x}=1timesp+0times(1-p)=p.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="\begin{displaymath}
\bar{x} = 1\times p + 0\times (1-p) = p.
\end{displaymath}">|; 

$key = q/overline{(x-bar{x})^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1255.png"
 ALT="$\overline{(x-\bar{x})^2}$">|; 

$key = q/v=1-2,x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$v=1-2 x^2$">|; 

$key = q/alpha_l=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\alpha _l = 1$">|; 

$key = q/C_{{rm{max}slashN,k=0.95;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="$C_{\rm max}/N k = 0.95$">|; 

$key = q/z=L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="$z=L$">|; 

$key = q/{displaymath}u_i^{n+1}+frac{C}{4},(u_{i+1}^{n+1}-u_{i-1}^{n+1})=u_i^n-frac{C}{4},(u_{i+1}^{n}-u_{i-1}^{n}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="\begin{displaymath}
u_i^{n+1} + \frac{C}{4} (u_{i+1}^{n+1}-u_{i-1}^{n+1}) = u_i^n - \frac{C}{4} (u_{i+1}^{n}-u_{i-1}^{n}).
\end{displaymath}">|; 

$key = q/y_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$y_0$">|; 

$key = q/2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$2\pi$">|; 

$key = q/i=2,N-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="$i=2,N-1$">|; 

$key = q/S>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$S&gt;1$">|; 

$key = q/{displaymath}v(x_i,y_j)=-frac{1}{deltax,deltay}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="\begin{displaymath}
v(x_i, y_j) = -\frac{1}{\delta x \delta y}
\end{displaymath}">|; 

$key = q/(N')^{-1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="$(N')^{-1/2}$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{p78.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{p78.eps}}
\end{figure}">|; 

$key = q/x=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$x=0.5$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{d1376.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{d1376.eps}}
\end{figure}">|; 

$key = q/H;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$H$">|; 

$key = q/displaystylefrac{partialv}{partialt}+c,frac{partialtheta}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="$\displaystyle \frac{\partial v}{\partial t} + c \frac{\partial \theta}{\partial x}$">|; 

$key = q/{displaymath}B=frac{S}{m}=4.1times10^{-4}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="\begin{displaymath}
B= \frac{S}{m} = 4.1\times 10^{-4}
\end{displaymath}">|; 

$key = q/E_z(x=0.5,y=0.5);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$E_z(x=0.5,y=0.5)$">|; 

$key = q/{displaymath}P(x)=left{array{{lcl}1slash(x_2-x_1)&mbox{hspace{1cm}}&mbox{if{x_1leqxleqx_2{}0&&mbox{otherwise}array{right..{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="\begin{displaymath}
P(x) = \left\{\begin{array}{lcl}
1/(x_2-x_1) &amp;\mbox{\hspace{...
... \leq x \leq x_2$}\\\\
0&amp;&amp; \mbox{otherwise}
\end{array}\right..
\end{displaymath}">|; 

$key = q/P_y(y);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="$P_y(y)$">|; 

$key = q/deltatheta_0=deltav_0=10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\delta \theta _0=\delta v_0 = 10^{-6}$">|; 

$key = q/|A|=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="$\vert A\vert = 1$">|; 

$key = q/2cdot2^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$2\cdot 2^n$">|; 

$key = q/|A|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="$\vert A\vert$">|; 

$key = q/n=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="$n=2$">|; 

$key = q/displaystylefrac{partialE_x}{partialt}+c,frac{partialH_y}{partialz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="$\displaystyle \frac{\partial E_x}{\partial t} + c \frac{\partial H_y}{\partial z}$">|; 

$key = q/0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$0.2$">|; 

$key = q/C_D(v);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$C_D(v)$">|; 

$key = q/-2pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$-2\pi$">|; 

$key = q/displaystyley(x_n+h);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$\displaystyle y(x_n + h)$">|; 

$key = q/T_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="$T_i^n$">|; 

$key = q/displaystyley_n+k_2+O(h^3).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$\displaystyle y_n + k_2 + O(h^3).$">|; 

$key = q/{displaymath}sigma^2=sum_{i,j,=1,N'}overline{(x_i-bar{x})(x_j-bar{x})}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="\begin{displaymath}
\sigma^2 = \sum_{i,j,=1,N'} \overline{(x_i-\bar{x})(x_j-\bar{x})}.
\end{displaymath}">|; 

$key = q/j=1,Jslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="$j=1,J/2$">|; 

$key = q/40times40;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$40\times 40$">|; 

$key = q/m_j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$m_j$">|; 

$key = q/phi_jequivphi(x_j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$\phi_j\equiv\phi(x_j)$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{exam.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{exam.eps}}
\end{figure}">|; 

$key = q/{{tt{h}=1times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="${\tt h}=1\times 10^{-4}$">|; 

$key = q/displaystylev_z(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="$\displaystyle v_z(t=0)$">|; 

$key = q/1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$1/2$">|; 

$key = q/displaystyle+frac{c}{2}left(frac{partialH_x}{partialy}right)^{n+1}_{i,j}+frac{c}{2}left(frac{partialH_x}{partialy}right)^{n}_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="$\displaystyle +\frac{c}{2}\left(\frac{\partial H_x}{\partial y}\right)^{n+1}_{i,j}
+\frac{c}{2}\left(\frac{\partial H_x}{\partial y}\right)^{n}_{i,j}$">|; 

$key = q/Q=1.387976;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$Q=1.387976$">|; 

$key = q/A=106;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$A=106$">|; 

$key = q/{displaymath}x_i=x_l+frac{i,(x_h-x_l)}{N+1},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="\begin{displaymath}
x_i = x_l + \frac{i (x_h-x_l)}{N+1},
\end{displaymath}">|; 

$key = q/m=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$m=1$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{pitch.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="259" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{pitch.eps}}
\end{figure}">|; 

$key = q/T({{bf{r},t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="$T({\bf r},t)$">|; 

$key = q/epsilon_0sim3times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$\epsilon_0\sim 3\times 10^{-4}$">|; 

$key = q/{figure}{epsfysize{=3.75in{centerline{epsffile{basin3a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="532" HEIGHT="412" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="\begin{figure}
\epsfysize =3.75in
\centerline{\epsffile{basin3a.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ts20.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ts20.eps}}\end{figure}">|; 

$key = q/{displaymath}frac{f_y}{m,g}=G(varphi)=0.5left[sin(4varphi)-0.25,sin(8varphi)+0.08,sin(12varphi)-0.025,sin(16varphi)right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="621" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="\begin{displaymath}
\frac{f_y}{m g} = G(\varphi) = 0.5\left[\sin(4\varphi) - 0....
...arphi) + 0.08 \sin(12\varphi)-
0.025 \sin(16\varphi)\right].
\end{displaymath}">|; 

$key = q/{displaymath}epsilonsimfrac{eta}{h}+h^n.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="\begin{displaymath}
\epsilon \sim \frac{\eta}{h} + h^n.
\end{displaymath}">|; 

$key = q/displaystylefrac{m,g}{omega_0,nu},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$\displaystyle \frac{m g}{\omega_0 \nu},$">|; 

$key = q/displaystylefrac{(H_y)_i^{n+1}-(H_y)_i^n}{deltat}+frac{c}{2}left(frac{partialE_x)_i^{n+1}+frac{c}{2}left(frac{partialE_x}{partialz}right)_i^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="$\displaystyle \frac{(H_y)_i^{n+1}-(H_y)_i^n}{\delta t} +\frac{c}{2}\left(\frac{...
...z}\right)_i^{n+1}
+ \frac{c}{2}\left(\frac{\partial E_x}{\partial z}\right)_i^n$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$\rho$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pd_scanvd.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="330" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pd_scanvd.eps}}
\end{figure}">|; 

$key = q/{displaymath}int_{x_l}^{x_h}f(x),dxsimeqsum_{i=1}^{N}f_i,h+O(N^{-1}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="\begin{displaymath}
\int_{x_l}^{x_h} f(x) dx\simeq \sum_{i=1}^{N} f_i h + O(N^{-1}).
\end{displaymath}">|; 

$key = q/j=1,J-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="$j=1,J-1$">|; 

$key = q/Q=1.387977;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$Q=1.387977$">|; 

$key = q/Q=0.5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$Q=0.5$">|; 

$key = q/{displaymath}tau=frac{2pi}{omega}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="\begin{displaymath}
\tau = \frac{2\pi}{\omega}
\end{displaymath}">|; 

$key = q/displaystylefrac{dv}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="$\displaystyle \frac{dv}{dt}$">|; 

$key = q/{mitDelta}E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="${\mit\Delta}E$">|; 

$key = q/(H_x)_{i,j}^nequivH_x(x_i,y_j,t_n);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="$(H_x)_{i,j}^n\equiv H_x(x_i, y_j, t_n)$">|; 

$key = q/theta(0)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$\theta (0)=0$">|; 

$key = q/phi=45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\phi = 45^\circ $">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{knuckler.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{knuckler.eps}}
\end{figure}">|; 

$key = q/v_{th};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="$v_{th}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{att1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="326" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{att1.eps}}
\end{figure}">|; 

$key = q/y_j=j,deltay;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="$y_j = j \delta y$">|; 

$key = q/{displaymath}frac{d^2x_j}{dt^2}=frac{F_j(x_1,...,x_n,t)}{m_j}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="\begin{displaymath}
\frac{d^2 x_j}{dt^2} = \frac{F_j(x_1,...,x_n,t)}{m_j}
\end{displaymath}">|; 

$key = q/displaystylesum_{i'=0,I}^{j'=0,J}hat{J}_{i',j'}^n,sin(i,i',pislashI),sin(j,j',pislashJ);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="$\displaystyle \sum_{i'=0,I}^{j'=0,J} \hat{J}_{i',j'}^n \sin(i i' \pi/I) \sin(j j' \pi/J)$">|; 

$key = q/1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="$1$">|; 

$key = q/7.4times10^{-4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$7.4\times 10^{-4}$">|; 

$key = q/{displaymath}V=N,h^3+O(N^{-1slash3}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="\begin{displaymath}
V = N h^3 + O(N^{-1/3}).
\end{displaymath}">|; 

$key = q/N_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="$N_A$">|; 

$key = q/displaystyledeltatheta_0,cos(omega_ast{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="$\displaystyle \delta\theta_0 \cos(\omega_\ast {\mit\Delta} t)  {\rm e}^{-{\mit\Delta} t/2Q}$">|; 

$key = q/displaystylehat{E}^{n+1}_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="$\displaystyle \hat{E}^{n+1}_{i,j}$">|; 

$key = q/displaystylefrac{partialH_x}{partialt}+c,frac{partialE_z}{partialy};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="$\displaystyle \frac{\partial H_x}{\partial t} + c \frac{\partial E_z}{\partial y}$">|; 

$key = q/hsim10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$h\sim 10^{-3}$">|; 

$key = q/displaystyledeltav({mitDelta}t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="$\displaystyle \delta v({\mit\Delta} t)$">|; 

$key = q/6,5,3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="$6,5,3$">|; 

$key = q/i=N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="$i=N$">|; 

$key = q/Q=2,A=1,omega=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="$Q=2, A=1, \omega=3$">|; 

$key = q/{displaymath}|deltav({mitDelta}t)|simeqdeltav_0,{{rm{e}^{,lambda,{mitDeltat}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="\begin{displaymath}
\vert\delta v({\mit\Delta} t)\vert \simeq \delta v_0 {\rm e}^{ \lambda {\mit\Delta t}},
\end{displaymath}">|; 

$key = q/{displaymath}C=frac{sigma_{E}^{,2}}{k,T^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="\begin{displaymath}
C = \frac{\sigma_{E}^{ 2}}{k T^2},
\end{displaymath}">|; 

$key = q/n_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="$n_0$">|; 

$key = q/C=sigma_E^{,2}slashk,T^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$C=\sigma_E^{ 2}/k T^2$">|; 

$key = q/{displaymath}frac{u_i^{n+1}-u_i^n}{deltat}=-v,frac{u_{i+1}^n-u_{i-1}^n}{2,deltax},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="\begin{displaymath}
\frac{u_i^{n+1} - u_i^n}{\delta t} = - v \frac{u_{i+1}^n-u_{i-1}^n}{2 \delta x},
\end{displaymath}">|; 

$key = q/deltat=5times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$\delta t = 5\times 10^{-3}$">|; 

$key = q/{displaymath}{{bf{f}_D=-frac{1}{2},C_D(v),rho,A,v,{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="\begin{displaymath}
{\bf f}_D = -\frac{1}{2} C_D(v) \rho A v {\bf v}.
\end{displaymath}">|; 

$key = q/{mbox{{boldmath{{omega{}}=omega,(0,sinphi,cosphi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="${\mbox{\boldmath$\omega$}} = \omega (0,\sin\phi,\cos\phi)$">|; 

$key = q/{displaymath}N=sum_{i=1,N'}x_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="\begin{displaymath}
N = \sum_{i=1,N'} x_i,
\end{displaymath}">|; 

$key = q/N=125;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$N=125$">|; 

$key = q/rho_J=rho_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="$\rho_J=\rho_0$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{scan15.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="384" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{scan15.eps}}
\end{figure}">|; 

$key = q/1slashQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="$1/Q$">|; 

$key = q/{displaymath}frac{partialu}{partialt}=-v,frac{partialu}{partialx}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="\begin{displaymath}
\frac{\partial u}{\partial t} = -v \frac{\partial u}{\partial x}.
\end{displaymath}">|; 

$key = q/T=5,T_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$T=5 T_0$">|; 

$key = q/{displaymath}alpha_l(t),T(x,y,t)+beta_l(t),frac{partialT(x,y,t)}{partialx}=gamma_l(y,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="\begin{displaymath}
\alpha_l(t) T(x,y,t) + \beta_l(t) \frac{\partial T(x,y,t)}{\partial x} = \gamma_l(y,t),
\end{displaymath}">|; 

$key = q/O(h^3);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="$O(h^3)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pendulum.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pendulum.eps}}
\end{figure}">|; 

$key = q/displaystylegamma_h,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="$\displaystyle \gamma_h,$">|; 

$key = q/{displaymath}x_n=n,h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="\begin{displaymath}
x_n = n h,
\end{displaymath}">|; 

$key = q/x+dx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="$x+dx$">|; 

$key = q/I=J=32;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$I=J=32$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p39.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p39.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{dv_j}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$\displaystyle \frac{dv_j}{dt}$">|; 

$key = q/E_a<E_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="$E_a &lt; E_b$">|; 

$key = q/{displaymath}C<frac{1}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="\begin{displaymath}
C &lt; \frac{1}{2}.
\end{displaymath}">|; 

$key = q/beta_h=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="$\beta_h=0$">|; 

$key = q/z=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="$z=1$">|; 

$key = q/displaystylefrac{Gamma_{l,j},deltax-beta_l,U_{1,j}}{alpha_l,deltax-beta_l},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="$\displaystyle \frac{\Gamma_{l j} \delta x-\beta_l U_{1,j}}{\alpha_l \delta x -\beta_l},$">|; 

$key = q/displaystylehat{E}_i^{n+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="$\displaystyle \hat{E}_i^{n+1}$">|; 

$key = q/{displaymath}y_{n+1}=y_n+f(x_n,y_n),h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="\begin{displaymath}
y_{n+1} = y_n +f(x_n,y_n) h.
\end{displaymath}">|; 

$key = q/alpha>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$\alpha&gt;0$">|; 

$key = q/{displaymath}f(x,v)=frac{n_0}{2}left{frac{1}{sqrt{2,pi},v_{th}},{{rm{e}^{-(v-v_b}},{{rm{e}^{-(v+v_b)^2slash2,v_{th}^{,2}}right}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="\begin{displaymath}
f(x,v) = \frac{n_0}{2}\left\{\frac{1}{\sqrt{2 \pi} v_{th}}...
...,\pi} v_{th}} {\rm e}^{-(v+v_b)^2/2 v_{th}^{ 2}}
\right\}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{midpoint.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{midpoint.eps}}
\end{figure}">|; 

$key = q/';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$'$">|; 

$key = q/Q=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$Q=1/2$">|; 

$key = q/displaystyleQ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$\displaystyle Q$">|; 

$key = q/Q=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$Q=1$">|; 

$key = q/{{tt{Nacc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="${\tt Nacc}$">|; 

$key = q/x_h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="$x_h$">|; 

$key = q/2^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="$2^n$">|; 

$key = q/|y_{n+1}|>|y_n|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="$\vert y_{n+1}\vert&gt; \vert y_n\vert$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{q124.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="323" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{q124.eps}}
\end{figure}">|; 

$key = q/phi=225^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\phi = 225^\circ $">|; 

$key = q/Q=1.376;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$Q=1.376$">|; 

$key = q/H_x=c,B_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="$H_x=c B_x$">|; 

$key = q/{{bf{E}=[0,0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="${\bf E} = [0,0,$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{adaptive.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="324" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{adaptive.eps}}
\end{figure}">|; 

$key = q/D>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="$D&gt;0$">|; 

$key = q/displaystyle-F(v),v,v_y+g,G(varphi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$\displaystyle - F(v) v v_y + g G(\varphi),$">|; 

$key = q/j=0,J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="$j=0,J$">|; 

$key = q/{{tt{Nacc}geq48;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="${\tt Nacc}\geq 48$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{rescav.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{rescav.eps}}
\end{figure}">|; 

$key = q/N^{-1slashd};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="$N^{-1/d}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{graph2b.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{graph2b.eps}}
\end{figure}">|; 

$key = q/t=0.000;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$t=0.000$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{ts40.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{ts40.eps}}\end{figure}">|; 

$key = q/{ttNacc}=24;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="${\tt Nacc} = 24$">|; 

$key = q/{displaymath}lambda_D=frac{v_{th}}{omega_p},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="\begin{displaymath}
\lambda_D = \frac{v_{th}}{\omega_p},
\end{displaymath}">|; 

$key = q/d=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="$d=1$">|; 

$key = q/displaystyle+frac{1}{omega_ast}left{v(0)+frac{theta(0)}{2Q}-frac{A,(1-3,omega^2)^2)^2+omega^2slashQ^2]}right}{{rm{e}^{-tslash2Q}sinomega_astt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$\displaystyle +\frac{1}{\omega_\ast}\left\{ v(0) + \frac{\theta(0)}{2Q}
-\frac{...
...2)/2Q}
{[(1-\omega^2)^2+ \omega^2/Q^2]}\right\}{\rm e}^{-t/2Q}\sin\omega_\ast t$">|; 

$key = q/{displaymath}P_{arightarrowb}=exp[-beta,(E_b-E_a)],{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="\begin{displaymath}
P_{a\rightarrow b} = \exp[-\beta (E_b-E_a)],
\end{displaymath}">|; 

$key = q/{displaymath}frac{C_{{rm{max}}{k,N}proptolnL.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="\begin{displaymath}
\frac{C_{\rm max}}{k N} \propto \ln L.
\end{displaymath}">|; 

$key = q/(x_0,y_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="$(x_0, y_0)$">|; 

$key = q/displaystylefrac{omegaAleft[-(1-omega^2),sinomegat+(omegaslashQ),cosomegatright]}{left[(1-omega^2)^2+omega^2slashQ^2right]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$\displaystyle \frac{\omega A\left[-(1-\omega^2) \sin\omega t+ (\omega/Q) \cos\omega t\right]}
{\left[(1-\omega^2)^2+\omega^2/Q^2\right]}.$">|; 

$key = q/displaystyleT_{N+1}^n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="$\displaystyle T_{N+1}^n$">|; 

$key = q/f_0=1slashsqrt{2}=0.7071;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="$f_0=1/\sqrt{2} = 0.7071$">|; 

$key = q/displaystylefrac{gamma_l,deltax-beta_l,u_1}{alpha_l,deltax-beta_l},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="$\displaystyle \frac{\gamma_l \delta x-\beta_l u_1}{\alpha_l \delta x -\beta_l},$">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="$z=0$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$v$">|; 

$key = q/N';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="$N'$">|; 

$key = q/x_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$x_n$">|; 

$key = q/76,mph;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$76 mph$">|; 

$key = q/displaystylefrac{dv_z}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="$\displaystyle \frac{dv_z}{dt}$">|; 

$key = q/20times20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$20\times 20$">|; 

$key = q/{displaymath}frac{partialH_y(0,t)}{partialz}=frac{partialH_y(L,t)}{partialz}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="\begin{displaymath}
\frac{\partial H_y(0,t)}{\partial z} = \frac{\partial H_y(L,t)}{\partial z} = 0.
\end{displaymath}">|; 

$key = q/n_{j+1},deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="$n_{j+1} \delta x$">|; 

$key = q/displaystyle-F(v),v,v_y+B,omega,v_x,cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$\displaystyle - F(v) v v_y + B \omega v_x \cos\phi,$">|; 

$key = q/f'=dfslashdx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="$f'=df/dx$">|; 

$key = q/{displaymath}-frac{partialQ}{partialt}=int{{bf{q}cdotd{{bf{S},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="\begin{displaymath}
-\frac{\partial Q}{\partial t} = \int {\bf q}\cdot d{\bf S},
\end{displaymath}">|; 

$key = q/u(x,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="$u(x,t)$">|; 

$key = q/displaystyle0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$\displaystyle 0,$">|; 

$key = q/u(x_i,y_j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="$u(x_i, y_j)$">|; 

$key = q/{displaymath}frac{{{bf{f}_D}{m}=-F(v),v,{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="\begin{displaymath}
\frac{{\bf f}_D}{m} = -F(v) v {\bf v},
\end{displaymath}">|; 

$key = q/C_{{rm{max}slashN,k=1.77;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="$C_{\rm max}/N k = 1.77$">|; 

$key = q/N=J=64;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$N=J=64$">|; 

$key = q/x=nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$x=\nu$">|; 

$key = q/{displaymath}u(x,t)=F(x-v,t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="\begin{displaymath}
u(x,t) = F(x-v t),
\end{displaymath}">|; 

$key = q/0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="$0.1$">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$M$">|; 

$key = q/pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="$\pi$">|; 

$key = q/{displaymath}T(x,0,t)=T(x,L,t)=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="\begin{displaymath}
T(x,0,t) = T(x,L,t) = 0.
\end{displaymath}">|; 

$key = q/displaystylehat{Y}_{i,j}^{n+1}-hat{Y}_{i,j}^n+i,D_x,(hat{E}_{i,j}^{n+1}+hat{E}_{i,j}^{n});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="$\displaystyle \hat{Y}_{i,j}^{n+1}-\hat{Y}_{i,j}^n + i D_x (\hat{E}_{i,j}^{n+1}+\hat{E}_{i,j}^{n})$">|; 

$key = q/N^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="$N^{-1}$">|; 

$key = q/E_x=+H_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="$E_x=+H_y$">|; 

$key = q/|lambda|<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="$\vert\lambda\vert&lt;1$">|; 

$key = q/overline{(x_i-bar{x})(x_j-bar{x})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="$\overline{(x_i-\bar{x})(x_j-\bar{x})}$">|; 

$key = q/displaystylefrac{dv_y}{dt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$\displaystyle \frac{dv_y}{dt}$">|; 

$key = q/displaystylefrac{(1-i^2,D_x^2-j^2,D_y^2),hat{E}_{i,j}^n+2,j,D_y,hat{X}_{i,j}^n+2,hat{Y}_{i,j}^n+deltat,hat{J}^n_{i,j}}{1+i^2,D_x^2+j^2,D_y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="467" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="$\displaystyle \frac{(1-i^2 D_x^2-j^2 D_y^2) \hat{E}_{i,j}^n + 2 j D_y 
\h...
... D_x \hat{Y}_{i,j}^n
+\delta t  \hat{J}^n_{i,j}}
{1 + i^2 D_x^2+j^2 D_y^2}$">|; 

$key = q/O(h^{-2});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="$O(h^{-2})$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{p2dc.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="325" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{p2dc.eps}}
\end{figure}">|; 

$key = q/{displaymath}C=frac{dE}{dT}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="\begin{displaymath}
C=\frac{dE}{dT}.
\end{displaymath}">|; 

$key = q/t=t_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$t=t_0$">|; 

$key = q/(1,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="$(1,0)$">|; 

$key = q/displaystyleT(x,y,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="$\displaystyle T(x,y,0)$">|; 

$key = q/kappa=pi,deltaxslashL;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="$\kappa = \pi \delta x/L$">|; 

$key = q/i=1,N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="$i=1,N$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{float.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="453" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{float.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{x1375.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{x1375.eps}}
\end{figure}">|; 

$key = q/deltat=4times10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\delta t = 4\times 10^{-3}$">|; 

$key = q/45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$45^\circ $">|; 

$key = q/{displaymath}frac{partialu(x,y=0)}{partialy}=frac{partialu(x,y=L)}{partialy}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="\begin{displaymath}
\frac{\partial u(x,y=0)}{\partial y} = \frac{\partial u(x,y=L)}{\partial y} = 0.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cv5.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="331" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cv5.eps}}
\end{figure}">|; 

$key = q/|A|>1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="$\vert A\vert&gt;1$">|; 

$key = q/p=AslashA';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="$p=A/A'$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{upwind1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{upwind1.eps}}
\end{figure}">|; 

$key = q/2n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$2n$">|; 

$key = q/epsilon_0simeta^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$\epsilon_0\sim \eta^{1/2}$">|; 

$key = q/Hneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="$H\neq 0$">|; 

$key = q/c_i=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="$c_i = 1$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{decay.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="323" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{decay.eps}}
\end{figure}">|; 

$key = q/gamma_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="$\gamma_l$">|; 

$key = q/L=100;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$L=100$">|; 

$key = q/m_e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="$m_e$">|; 

$key = q/J-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="$J-1$">|; 

$key = q/P=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="$P=0$">|; 

$key = q/displaystylesum_{i'=0,I}^{j'=0,J}hat{Y}_{i',j'}^n,cos(i,i',pislashI),sin(j,j',pislashJ),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="$\displaystyle \sum_{i'=0,I}^{j'=0,J} \hat{Y}_{i',j'}^n \cos(i i' \pi/I) \sin(j j' \pi/J),$">|; 

$key = q/{displaymath}{calE}=1+frac{v^2}{2}-costheta{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="\begin{displaymath}
{\cal E} = 1 + \frac{v^2}{2} - \cos\theta
\end{displaymath}">|; 

$key = q/{displaymath}m,l,frac{d^2theta}{dt^2}+m,g,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="\begin{displaymath}
m l \frac{d^2\theta}{dt^2} + m g \sin\theta = 0,
\end{displaymath}">|; 

$key = q/{displaymath}hat{Gamma}_{l,j}^n=frac{2}{J}sum_{k=0}^{J}gamma_{l,k}^n,sin(j,k,pislashJ),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="59" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="\begin{displaymath}
\hat{\Gamma}_{l j}^n = \frac{2}{J}\sum_{k=0}^{J} \gamma_{l,k}^n \sin(j k \pi/J),
\end{displaymath}">|; 

$key = q/n,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$n,l$">|; 

$key = q/AAAAAAcdotsrightarrowABABABcdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="$AAAAAA\cdots\rightarrow ABABAB\cdots$">|; 

$key = q/displaystylev_1(t_0+{mitDelta}t)-v_0(t_0+{mitDelta}t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="$\displaystyle v_1(t_0+{\mit\Delta} t)-v_0(t_0+{\mit\Delta} t),$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cavity1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="328" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cavity1.eps}}
\end{figure}">|; 

$key = q/nu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$\nu$">|; 

$key = q/displaystyle-E(x_i),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="$\displaystyle - E(x_i),$">|; 

$key = q/{figure}{epsfysize{=2.25in{centerline{epsffile{p77.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="581" HEIGHT="247" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="\begin{figure}
\epsfysize =2.25in
\centerline{\epsffile{p77.eps}}
\end{figure}">|; 

$key = q/{displaymath}t>t_hsimfrac{1}{lambda}ln!left(frac{a}{delta_0}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="\begin{displaymath}
t&gt; t_h \sim \frac{1}{\lambda}\ln\!\left(\frac{a}{\delta_0}\right).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.75in{centerline{epsffile{baseball.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="313" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{baseball.eps}}
\end{figure}">|; 

$key = q/{displaymath}hat{phi}_0=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="\begin{displaymath}
\hat{\phi}_0 = 0
\end{displaymath}">|; 

$key = q/P(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="$P(x)$">|; 

$key = q/{{bf{E}=[E_x(z,t),0,0];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="${\bf E} = [E_x(z,t),0,0]$">|; 

$key = q/y+dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="$y+dy$">|; 

$key = q/{displaymath}y'=-alpha,y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="\begin{displaymath}
y' = - \alpha y,
\end{displaymath}">|; 

$key = q/t=0.1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$t=0.1$">|; 

$key = q/{displaymath}u_i^{n+1}=frac{1}{2},(u_{i+1}^n+u_{i-1}^n)-frac{C}{2},(u_{i+1}^n-u_{i-1}^n),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="\begin{displaymath}
u_i^{n+1} = \frac{1}{2} (u_{i+1}^n+u_{i-1}^n) -\frac{C}{2} (u_{i+1}^n-u_{i-1}^n),
\end{displaymath}">|; 

$key = q/{displaymath}M=musum_{i=1,N}s_i=mu,N,bar{s},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="\begin{displaymath}
M=\mu\sum_{i=1,N} s_i= \mu N \bar{s},
\end{displaymath}">|; 

$key = q/O(h^{-1});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$O(h^{-1})$">|; 

$key = q/displaystylefrac{partialtheta}{partialt}+c,frac{partialv}{partialx};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="$\displaystyle \frac{\partial \theta}{\partial t} + c \frac{\partial v}{\partial x}$">|; 

$key = q/displaystylev_0,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$\displaystyle v_0 \sin\theta,$">|; 

$key = q/{displaymath}|A|^2=1-2,C,(1-C),[1-cos(k,deltax)].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="\begin{displaymath}
\vert A\vert^2 =1-2 C (1-C) [1-\cos (k  \delta x)].
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{pic1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="395" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{pic1.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pd_poincared.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pd_poincared.eps}}
\end{figure}">|; 

$key = q/Q=1.387978;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$Q=1.387978$">|; 

$key = q/ABABABcdots;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="$ABABAB\cdots$">|; 

$key = q/x_i=i,deltax;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="$x_i = i \delta x$">|; 

$key = q/{displaymath}P_{brightarrowa}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="\begin{displaymath}
P_{b\rightarrow a} = 1.
\end{displaymath}">|; 

$key = q/displaystyle-F_j,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="$\displaystyle -F_j,$">|; 

$key = q/displaystyle-frac{1}{omega_ast}left{deltatheta_0+frac{deltav_0}{2Q}right}sin(omega_ast{mitDelta}t),,{{rm{e}^{-{mitDelta}tslash2Q},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$\displaystyle - \frac{1}{\omega_\ast}\left\{\delta \theta_0 + \frac{\delta v_0}{2Q}\right\}
\sin(\omega_\ast {\mit\Delta} t)  {\rm e}^{- {\mit\Delta} t/2Q},$">|; 

$key = q/displaystylek_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\displaystyle k_4$">|; 

$key = q/{displaymath}H_y=frac{partialH_x}{partialy}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="\begin{displaymath}
H_y = \frac{\partial H_x}{\partial y} = 0
\end{displaymath}">|; 

$key = q/{displaymath}I_{n+1}=(A,I_n+C),{{rm{mod},M,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="\begin{displaymath}
I_{n+1} = (A I_n+C) {\rm mod} M,
\end{displaymath}">|; 

$key = q/{displaymath}alpha_l,u(x)+beta_l,frac{du(x)}{dx}=gamma_l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="\begin{displaymath}
\alpha_l u(x) + \beta_l \frac{du(x)}{dx} = \gamma_l,
\end{displaymath}">|; 

$key = q/y_j=j,LslashJ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="$y_j=j L/J$">|; 

$key = q/{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="${\mit\Delta} t$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{q05.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="329" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{q05.eps}}
\end{figure}">|; 

$key = q/{displaymath}F_j^C=2,F_j.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="\begin{displaymath}
F_j^C = 2 F_j.
\end{displaymath}">|; 

$key = q/x_l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="$x_l$">|; 

$key = q/{displaymath}frac{partialT({{bf{r},t)}{partialt}=D,nabla^2T({{bf{r},t),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="\begin{displaymath}
\frac{\partial T({\bf r},t)}{\partial t} = D \nabla^2 T({\bf r},t),
\end{displaymath}">|; 

1;

