# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/(2N_{tr}+1)(N_{tr}+1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$ (2N_{tr}+1) (N_{tr}+1) $">|; 

$key = q/f^{(n)}=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$f^{(n)}=0$">|; 

$key = q/displaystylesum_{m=-M}^{M}sum_{n=|m|}^{N}tilde{A}_n^mY_n^m(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="$\displaystyle \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N}
\tilde{A}_n^m
Y_n^m (\lambda_i,\phi_j)$">|; 

$key = q/JleKle2J-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$J \le K \le 2J-1$">|; 

$key = q/I=128;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="$I=128$">|; 

$key = q/dot{calA}_{dif};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$ \dot{\cal A}_{dif} $">|; 

$key = q/displaystylevspace{4cm}timessum_{i=1}^Iexp(i(m'+m''-m)lambda_i)sum_{j=1}^JP_{n'}^{m'}(sinphi_j)P_{n''}^{m''}(sinphi_j)P_n^{m}(sinphi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$\displaystyle \vspace{4cm}
\times
\sum_{i=1}^I
\exp(i(m'+m''-m) \lambda_i)
\sum...
...^J
P_{n'}^{m'}(\sin \phi_j)
P_{n''}^{m''}(\sin \phi_j)
P_n^{m}(\sin \phi_j) w_j$">|; 

$key = q/displaystyle(mbox{{sl{UA})_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$\displaystyle (\mbox{\sl UA})_k$">|; 

$key = q/displaystyle-(zeta_k+f)U_k-frac{1}{2Deltasigma_k}[dot{sigma}_{k-1slash2}(V_{k-1}-V_k)+dot{sigma}_{k+1slash2}(V_k-V_{k+1})];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="436" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$\displaystyle - ( \zeta_k + f ) U_k
- \frac{1}{2 \Delta \sigma_k}
[ \dot{\sigma}_{k-1/2} ( V_{k-1} - V_k )
+ \dot{\sigma}_{k+1/2} ( V_k - V_{k+1} ) ]$">|; 

$key = q/textstyle+;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\textstyle +$">|; 

$key = q/Y_n^m(lambda',phi');MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$Y_n^m(\lambda',\phi')$">|; 

$key = q/displaystyle+left{alpha_kleft[sigma_{k-1slash2}Dvect{v}_kcdotnablapi-sum_{l=k}^{K}(D_l+Dvect{v}_lcdotnablapi)Deltasigma_lright]right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$\displaystyle + \left\{ \alpha_k
\left[ \sigma_{k-1/2} \Dvect{v}_k \cdot \nabla...
...l=k}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta \sigma_l
\right]
\right.$">|; 

$key = q/displaystyle-sum_{i=1}^Isum_{j=1}^Jf_{n'}^{m'}left.DD{P_{n}^{m}}{mu}right|_jexp(-imlambda_i)P_{n'}^{m'}(mu_j)exp(im'lambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$\displaystyle - \sum_{i=1}^I \sum_{j=1}^J
f_{n'}^{m'}
\left. \DD{P_{n}^{m}}{\mu}\right\vert _j
\exp(-im \lambda_i)
P_{n'}^{m'}(\mu_j) \exp(im' \lambda_i)
w_j$">|; 

$key = q/Dvect{D}={D_{k}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$\Dvect{D}=\{ D_{k} \}$">|; 

$key = q/displaystyle+tilde{calD}_{H,n}^mtilde{T}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$\displaystyle + \tilde{\cal D}_{H,n}^m \tilde{T}_n^m$">|; 

$key = q/tilde{A}_n^m;(m=-M,-M+1,cdots,M,;n=|m|,|m|+1,cdots,N(m));MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$\tilde{A}_n^m \;
(m=-M,-M+1, \cdots,M, \; n=\vert m\vert,\vert m\vert+1,\cdots,N(m))$">|; 

$key = q/displaystyleA_{ij}equivsum_{m=-M}^{M}sum_{n=|m|}^{N}tilde{A}_n^mY_n^m(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle A_{ij} \equiv \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N}
\tilde{A}_n^m
Y_n^m (\lambda_i,\phi_j)$">|; 

$key = q/{displaymath}tilde{calD}_{M,n}^m=-K_{HD}left[left(frac{-n(n+1)}{a^{2}}right)^{N_Dslash2}-left(frac{2}{a}right)^{N_D}right].{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="342" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="\begin{displaymath}
\tilde{\cal D}_{M,n}^m = - K_{HD} \left[
\left( \frac{-n(n...
...\right)^{N_D/2}
- \left( \frac{2}{a} \right)^{N_D}
\right] .
\end{displaymath}">|; 

$key = q/{eqnarraystar}h_{ij}&=&frac{1}{rcos^2phi_i}g_{ij}∴h_{ij}&=&frac{1}{rcos^2phi_j}sum_{m,n}imtilde{f}_n^mY_n^m(lambda_i,phi_j){eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="92" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="\begin{eqnarray*}
h_{ij} &amp;=&amp; \frac{1}{r \cos^2 \phi_i} g_{ij} \\\\
∴ h_{ij} &amp;...
...2 \phi_j}
\sum_{m,n} im \tilde{f}_n^m Y_n^m (\lambda_i, \phi_j)
\end{eqnarray*}">|; 

$key = q/-1leqmuleq1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$-1 \leq \mu \leq 1$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}A_{ij}Y_n^{m*}(lambda_i,phi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{i=1}^{I} \sum_{j=1}^{J}
A_{ij} Y_n^{m*} (\lambda_i,\phi_j) w_j$">|; 

$key = q/mu=mu_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$\mu=\mu_j$">|; 

$key = q/r;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$r$">|; 

$key = q/{displaymath}dot{sigma}^{NG}_{k-1slash2}=-sigma_{k-1slash2}left(DP{pi}{t}right)^{NG}-sum_{l=k}^{K}Dvect{v}_{l}cdotnablapiDeltasigma_{l}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="317" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="\begin{displaymath}
\dot{\sigma}^{NG}_{k-1/2}
= - \sigma_{k-1/2} \left( \DP{\p...
...um_{l=k}^{K} \Dvect{v}_{l} \cdot \nabla \pi
\Delta \sigma_{l}
\end{displaymath}">|; 

$key = q/phi_j（j=1,2,3,cdots,J）;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\phi_j（j=1,2,3,\cdots,J）$">|; 

$key = q/displaystylefrac{1}{2}int_{-1}^1frac{tilde{P}_J(mu)}{(mu-mu_j)tilde{P}^{'}_J(mu_j)}dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$\displaystyle \frac{1}{2} \int_{-1}^1
\frac{\tilde{P}_J(\mu)}
{(\mu-\mu_j) \tilde{P}^{'}_J(\mu_j)}
d \mu$">|; 

$key = q/displaystyleL(mu)equivsum_{j=1}^{J}f(mu_j)prod_{k=1,kneqj}^{J}frac{mu-mu_k}{mu_j-mu_k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$\displaystyle L(\mu) \equiv
\sum_{j=1}^{J} f(\mu_j)
\prod_{k=1,k \neq j}^{J}
\frac{ \mu-\mu_k}{\mu_j-\mu_k}$">|; 

$key = q/displaystylewidetilde{left(DP{f}{lambda}right)_n^m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$\displaystyle \widetilde{ \left( \DP{f}{\lambda} \right)_n^m }$">|; 

$key = q/displaystylembox{div}_HDvect{v};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="$\displaystyle \mbox{div}_H \Dvect{v}$">|; 

$key = q/displaystyle(mbox{{sl{VA})_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$\displaystyle (\mbox{\sl VA})_k$">|; 

$key = q/displaystyleDP{tilde{q}_n^m}{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$\displaystyle \DP{\tilde{q}_n^m}{t}$">|; 

$key = q/displaystyle(u,v)=left(frac{U}{cosphi},frac{V}{cosphi}right);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$\displaystyle (u,v) = \left( \frac{U}{\cos \phi}, \frac{V}{\cos \phi} \right)$">|; 

$key = q/displaystylew_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$\displaystyle w_j$">|; 

$key = q/{displaystyleleft(int_a^bf_k(mu)f_{k'}(mu)w(mu)dmu=lambda_kdelta_{kk'}right)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="${\displaystyle
\left( \int_a^b f_k (\mu) f_{k'} (\mu) w(\mu) d \mu
= \lambda_k \delta_{kk'} \right) }$">|; 

$key = q/P_n^m(sinphi)exp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$P_n^m(\sin \phi) \exp(im \lambda)$">|; 

$key = q/displaystylezeta=frac{1}{r(1-mu^2)}DP{V}{lambda}-frac{1}{r}DP{U}{mu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="$\displaystyle \zeta = \frac{1}{r(1-\mu^2)} \DP{V}{\lambda}
- \frac{1}{r} \DP{U}{\mu}$">|; 

$key = q/{displaystyle(g_{ij}=)sum_{m,n}imtilde{f}_n^mY_n^m=sum_{m,n}tilde{g}_n^mY_n^m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="${\displaystyle (g_{ij}=)
\sum_{m,n} im \tilde{f}_n^m Y_n^m
= \sum_{m,n} \tilde{g}_n^m Y_n^m }$">|; 

$key = q/displaystyle(n+1)left|array{{ll}tilde{P}_{n+1}(x)<col_mark>tilde{P}_n(x)<row_mark>tilde{P}_{n+1}(y)<col_mark>tilde{P}_n(y)array{right|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$\displaystyle (n+1) \left\vert
\begin{array}{ll}
\tilde{P}_{n+1}(x) &amp; \tilde{P}_n(x) \\\\
\tilde{P}_{n+1}(y) &amp; \tilde{P}_n(y)
\end{array}\right\vert$">|; 

$key = q/[a,b];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="$ [ a ,b ]  $">|; 

$key = q/{displaymath}left(DP{D}{t}right)^{NG}=frac{1}{a(1-mu^{2})}DP{(mbox{{sl{UA})_{k}}{sigma}sum_{k=1}^{K}W_{kl}(T_{v,l}-T_{l})+{calD}(D_{k}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="587" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="\begin{displaymath}
\left( \DP{D}{t} \right)^{NG}
= \frac{1}{a(1-\mu^{2})}
\D...
...a} \sum_{k=1}^{K} W_{kl} ( T_{v,l}-T_{l} )
+ {\cal D}(D_{k})
\end{displaymath}">|; 

$key = q/displaystyle(1-mu^2)DD{}{mu}P_n^m=(n+|m|)sqrt{frac{(n-|m|)(2n+1)}{(n+|m|)(2n-1)}}P_{n-1}^m-nmuP_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$\displaystyle (1-\mu^2) \DD{}{\mu} P_n^m
= (n+\vert m\vert)
\sqrt{ \frac{(n-\vert m\vert)(2n+1)}{(n+\vert m\vert)(2n-1)} } P_{n-1}^m
- n \mu P_n^m$">|; 

$key = q/Jslash2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$J/2$">|; 

$key = q/Y_n^m(phi,lambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$Y_n^m(\phi,\lambda)$">|; 

$key = q/displaystyleDP{Dvect{T}}{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$\displaystyle \DP{\Dvect{T}}{t}$">|; 

$key = q/|m'-m|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="$\vert m'-m\vert$">|; 

$key = q/0lelambda<2pi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$0 \le \lambda &lt; 2 \pi$">|; 

$key = q/displaystylesum_{m=-M}^Msum_{n=|m|}^Ntilde{A}_n^mP_n^{m}(sinphi_j)exp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="$\displaystyle \sum_{m=-M}^M \sum_{n=\vert m\vert}^N
\tilde{A}_n^m P_n^{m}(\sin \phi_j)
\exp(im \lambda_i)$">|; 

$key = q/{displaymath}nabla^{2}_{sigma}=-frac{n(n+1)}{a^{2}}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="\begin{displaymath}
\nabla^{2}_{\sigma} = - \frac{n(n+1)}{a^{2}}
\end{displaymath}">|; 

$key = q/max[N];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="$max[N]$">|; 

$key = q/I=2M+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="$I=2M+1$">|; 

$key = q/displaystyleU;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="$\displaystyle U$">|; 

$key = q/tilde{B}_n^m,tilde{C}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="$\tilde{B}_n^m,\tilde{C}_n^m$">|; 

$key = q/{displaystylefrac{1}{r}DP{f}{phi}left(=frac{cosphi}{r}DP{f}{mu}right)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="${\displaystyle \frac{1}{r} \DP{f}{\phi}
\left( = \frac{\cos \phi}{r} \DP{f}{\mu} \right) }$">|; 

$key = q/m,m',n,n';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$m,m',n,n'$">|; 

$key = q/I=2M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="$I=2M$">|; 

$key = q/displaystylesum_{n=0}^{J-1}P_n(mu_j)P_n(mu_{j'})=frac{1}{w_j}delta_{jj'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="$\displaystyle \sum_{n=0}^{J-1} P_n (\mu_j) P_n (\mu_{j'})
= \frac{1}{w_j} \delta_{jj'}$">|; 

$key = q/displaystylefrac{1}{r(1-mu^2)}DP{V}{lambda}-frac{1}{r}DP{U}{mu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$\displaystyle \frac{1}{r (1-\mu^2)} \DP{V}{\lambda}
- \frac{1}{r} \DP{U}{\mu}$">|; 

$key = q/nabla_H^2Y^m_n(lambda,phi)=-frac{n(n+1)}{r^2}Y^m_n(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$\nabla_H^2 Y^m_n(\lambda, \phi)
=- \frac{n(n+1)}{r^2} Y^m_n(\lambda,\phi)$">|; 

$key = q/displaystyleIsum_{j=1}^{J}P_n^m(mu_j)P_{n'}^{m}(mu_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$\displaystyle I \sum_{j=1}^{J}
P_n^m (\mu_j) P_{n'}^{m} (\mu_j) w_j$">|; 

$key = q/displaystyle(n-|m|+1)sqrt{frac{1}{2n+3}frac{(n+1+|m|)!}{(n+1-|m|)!}}P_{n+1}^m-(2n+1)sqrt{frac{1}{2n+1}frac{(n+|m|)!}{(n-|m|)!}}muP_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$\displaystyle (n-\vert m\vert+1)
\sqrt{ \frac{1}{2n+3} \frac{(n+1+\vert m\vert)...
...)
\sqrt{ \frac{1}{2n+1} \frac{(n+\vert m\vert)!}{(n-\vert m\vert)!} }
\mu P_n^m$">|; 

$key = q/0lemleM,|m|lenleN;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="$0 \le m \le M, \vert m\vert \le n \le N$">|; 

$key = q/{displaymath}delta_{t}Dvect{T}=left(DP{Dvect{T}}{t}right)_{NG}-underline{h}overl+{calD}_H(Dvect{T}^{t-Deltat}+2Deltatdelta_{t}Dvect{T}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="\begin{displaymath}
\delta_{t} \Dvect{T} =
\left( \DP{\Dvect{T}}{t} \right)_{N...
... ( \Dvect{T}^{t-\Delta t}
+ 2 \Delta t \delta_{t} \Dvect{T} )
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{2pi}int_0^{2pi}g(lambda)dlambda=frac{1}{I}sum_{n=1}^{I}g(lambda_n);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda
= \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$">|; 

$key = q/{displaymath}Zequiv-sum_{k=1}^{K}Dvect{v}_kcdotnablapi.{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="\begin{displaymath}
Z \equiv - \sum_{k=1}^{K} \Dvect{v}_k \cdot \nabla \pi .
\end{displaymath}">|; 

$key = q/tilde{P}_J;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="$\tilde{P}_J$">|; 

$key = q/I,J;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="$I,J$">|; 

$key = q/displaystyleleft[frac{1}{cosphi}DP{}{phi}left(cosphiDP{}{phi}right)+frac{1}{cos^2phi}DP[2]{}{lambda}+n(n+1)right]Y_n^m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="$\displaystyle \left[ \frac{1}{\cos \phi}
\DP{}{\phi}
\left( \cos \phi \DP{}{\phi} \right)
+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
+ n(n+1) \right] Y_n^m =0$">|; 

$key = q/n=0,1,cdots,n-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$n=0,1,\cdots,n-1$">|; 

$key = q/{displaymath}R_k=q_kD_k-frac{1}{2Deltasigma_k}[dot{sigma}_{k-1slash2}(q_{k-1}-q_k)+dot{sigma}_{k+1slash2}(q_k-q_{k+1})]{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="\begin{displaymath}
R_k = q_k D_k
- \frac{1}{2 \Delta \sigma_k}
[ \dot{\sigm...
... ( q_{k-1} - q_k )
+ \dot{\sigma}_{k+1/2} ( q_k - q_{k+1} ) ]
\end{displaymath}">|; 

$key = q/J=63,N=21;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="$J=63, N=21$">|; 

$key = q/{eqnarraystar}qequiv(1-mu^2)DP{f}{mu}=(1-mu^2)p{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="\begin{eqnarray*}
q \equiv (1-\mu^2) \DP{f}{\mu} = (1-\mu^2) p
\end{eqnarray*}">|; 

$key = q/M,N;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="$M,N$">|; 

$key = q/displaystyle+left.beta_kleft[sigma_{k+1slash2}Dvect{v}_kcdotnablapi-sum_{l=k+1}^v}_lcdotnablapi)Deltasigma_lright]right}frac{1}{Deltasigma_k}T_{v,k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\displaystyle + \left. \beta_k
\left[ \sigma_{k+1/2} \Dvect{v}_k \cdot \nabla \...
...\nabla \pi )
\Delta \sigma_l
\right]
\right\}
\frac{1}{\Delta \sigma_k} T_{v,k}$">|; 

$key = q/displaystylew_j=frac{(2J-1)(1-mu_j^2)}{(JP_{J-1}(mu_j))^2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="$\displaystyle w_j = \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 }$">|; 

$key = q/0le|m'|len';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="$0 \le \vert m'\vert \le n'$">|; 

$key = q/zeta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$\zeta$">|; 

$key = q/{displaystylesin^{-1}left(cosfrac{j-1slash2}{J}piright)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="${\displaystyle
\sin^{-1} \left( \cos \frac{j-1/2}{J}\pi \right)
}$">|; 

$key = q/tilde{P}_n^m(mu)(n=|m|,|m+1,cdots);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$\tilde{P}_n^m(\mu)  (n=\vert m\vert,\vert m+1, \cdots)$">|; 

$key = q/{displaystylesum_{i,j}Y_n^{m*}(lambda_i,phi_j)w_j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="${\displaystyle \sum_{i,j} Y_n^{m*}(\lambda_i,\phi_j)w_j}$">|; 

$key = q/Leftrightarrow;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="$\Leftrightarrow$">|; 

$key = q/displaystyleleft(frac{sigma_{k-1slash2}}{sigma_k}right)^{kappa}-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$\displaystyle \left( \frac{ \sigma_{k-1/2} }
{ \sigma_k } \right)^{\kappa} -1$">|; 

$key = q/frac{I}{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="$\frac{I}{2}$">|; 

$key = q/P_n^m(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$P_n^m(\mu)$">|; 

$key = q/displaystyle(n+1)tilde{P}_{n+1}(mu)=(2n+1)mutilde{P}_n(mu)-ntilde{P}_{n-1}(mu)(n=0,1,2,cdots);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$\displaystyle (n+1) \tilde{P}_{n+1}(\mu)
= (2n+1) \mu \tilde{P}_n(\mu) - n \tilde{P}_{n-1}(\mu)
    (n=0,1,2,\cdots)$">|; 

$key = q/m=-M,cdots,M,n=|m|,cdots,N(m);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$m=-M,\cdots,M,  n=\vert m\vert,\cdots,N(m)$">|; 

$key = q/displaystyleC=sum_{m=-M}^{M}sum_{n=|m|}^{N}left(tilde{C}_n^mexp(imlambda)right)P_n^m(sinphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="$\displaystyle C= \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N}
\left( \tilde{C}_n^m \exp(im \lambda) \right)
P_n^m(\sin \phi)$">|; 

$key = q/displaystyleint_{-1}^{1}f(mu)dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="$\displaystyle \int_{-1}^{1} f(\mu) d \mu$">|; 

$key = q/displaystyle+frac{Q_k}{C_{p}}+{calD}(T_k)+{calD}'(Dvect{v});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$\displaystyle + \frac{Q_k}{C_{p}}
+ {\cal D}(T_k)
+ {\cal D}'(\Dvect{v})$">|; 

$key = q/tilde{P}_0=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$\tilde{P}_0 = 1$">|; 

$key = q/displaystylefrac{(2J-1)(1-sin^2phi_j)}{(JP_{J-1}(sinphi_j))^2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="$\displaystyle \frac{(2J-1)(1-\sin^2 \phi_j)}
{(J P_{J-1}(\sin \phi_j))^2 }$">|; 

$key = q/displaystylesum_{i=0}^{I-1}exp(imlambda_i)exp(-im'lambda_i)=Idelta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="$\displaystyle \sum_{i=0}^{I-1}
\exp(i m \lambda_i) \exp(-i m' \lambda_i)
= I \delta_{mm'}$">|; 

$key = q/2J-1gemax[n+n'+n'']=3max[N];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="$2J-1 \ge max[n+n'+n''] = 3 max[N]$">|; 

$key = q/displaystylefrac{1}{rcosphi}DP{v}{lambda}-frac{1}{rcosphi}DP{}{phi}(ucosphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$\displaystyle \frac{1}{r \cos \phi} \DP{v}{\lambda}
- \frac{1}{r \cos \phi} \DP{}{\phi}(u \cos \phi)$">|; 

$key = q/displaystylesigma_k=left{frac{1}{1+kappa}left(frac{sigma^{kappa+1}_{k-1slash2}-ssh2}}{sigma_{k-1slash2}-sigma_{k+1slash2}}right)right}^{1slashkappa};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle \sigma_k = \left\{ \frac{1}{1+\kappa}
\left( \frac{ \sigma^{\kapp...
...a +1}_{k+1/2} }
{ \sigma_{k-1/2} - \sigma_{k+1/2} }
\right)
\right\}^{1/\kappa}$">|; 

$key = q/hat{calA}^{t+Deltat};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$ \hat{\cal A}^{t+\Delta t} $">|; 

$key = q/Deltat;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$\Delta t$">|; 

$key = q/B;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="$B$">|; 

$key = q/U_{ij},V_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="$U_{ij},V_{ij}$">|; 

$key = q/displaystyleD;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$\displaystyle D$">|; 

$key = q/displaystylePhi_{s}+C_{p}(sigma_{1}^{-kappa}-1)T_{v,1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\displaystyle \Phi_{s} + C_{p} ( \sigma_{1}^{-\kappa} - 1 ) T_{v,1}$">|; 

$key = q/tilde{P}_J(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$\tilde{P}_J(\mu)$">|; 

$key = q/dot{calA}_{phy};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$ \dot{\cal A}_{phy} $">|; 

$key = q/n+n'+n'';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="$n+n'+n''$">|; 

$key = q/chi_n^m,psi_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="$\chi_n^m,\psi_n^m$">|; 

$key = q/A;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$A$">|; 

$key = q/displaystyleg(lambda)=sum_{m=-M}^{m=M}g_mexp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$\displaystyle g(\lambda) = \sum_{m=-M}^{m=M} g_m \exp( i m \lambda )$">|; 

$key = q/{calA};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="${\cal A}$">|; 

$key = q/displaystylehat{T}_{k-1slash2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\displaystyle \hat{T}_{k-1/2}$">|; 

$key = q/displaystyletilde{A}_n^m=left{array{{ll}{displaystylefrac{1}{I}sum_{i=1}^{I}sum_{ij}Y_n^{m*}(lambda_i,phi_j)w_j<col_mark>1lemleM,mlenleNarray{right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="467" HEIGHT="119" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle \tilde{A}_n^m =
\left\{
\begin{array}{ll}
{\displaystyle \frac{1}...
...}(\lambda_i,\phi_j) w_j
&amp;    1 \le m \le M, m \le n \le N
\end{array}\right.$">|; 

$key = q/f_J(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$f_J(\mu)$">|; 

$key = q/x=-1,alpha,1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$x=-1,\alpha,1$">|; 

$key = q/displaystyletilde{A}_n^m=frac{1}{2}int_{-1}^1A(mu)P_n^m(mu)dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$\displaystyle \tilde{A}_n^m = \frac{1}{2}
\int_{-1}^1 A(\mu) P_n^m(\mu) d \mu$">|; 

$key = q/displaystyleC_{p}alpha_kT_{v,k}+C_{p}beta_{k-1}T_{v,k-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$\displaystyle C_{p} \alpha_k T_{v,k} + C_{p} \beta_{k-1} T_{v,k-1}$">|; 

$key = q/{figure}center{picture{(300,100)(10,-10)put(20,10){vector(1,0){85}}put(30,0){vec0,-20){shortstack{三角形切断}}{par{picture{center{{{{figure};MSF=1.6;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="139" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="\begin{figure}\begin{center}
\begin{picture}(300,100)(10,-10)
\put(20 , 10){\v...
...00,-20) \{ shortstack\{三角形切断\}\}
\par
\end{picture}\end{center}\end{figure}">|; 

$key = q/P_n^mP_{n'}^{m'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="$ P_n^m P_{n'}^{m'} $">|; 

$key = q/displaystyle-Deltatnabla^{2}_{sigma}left{(1-2Deltat{calD}_H)Dvect{Phi}_{S}+undervect{T}^{t-Deltat}+Deltatleft(DP{Dvect{T}}{t}right)_{NG}right]right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$\displaystyle - \Delta t \nabla^{2}_{\sigma}
\left\{ ( 1-2\Delta t {\cal D}_H )...
...}^{t-\Delta t}
+ \Delta t
\left( \DP{\Dvect{T}}{t}
\right)_{NG} \right]
\right.$">|; 

$key = q/{picture}(300,150)(50,10)put(50,20){line(1,0){220}}put(50,40){line(1,0){220}}put){shortstack{{sigma=1{}}put(280,136){shortstack{{sigma=0{}}{picture};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="192" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="\begin{picture}(300,150)(50,10)
\put(50,20){\line(1,0){220}}
\put(50,40){\line...
...){\shortstack{$\sigma=1$}}
\put(280,136){\shortstack{$\sigma=0$}}
\end{picture}">|; 

$key = q/Y_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$Y_n^m$">|; 

$key = q/{eqnarraystar}&&sum_{i}sum_{j}f_{n'}^{m'}P_{n}^{m}(mu_j)exp(imlambda_i)left.DD{P^{m}P_{n}^{m}(mu)DD{P_{n'}^m}{mu}dmudelta_{mm'}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="497" HEIGHT="161" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="\begin{eqnarray*}
&amp; &amp; \sum_{i} \sum_{j}
f_{n'}^{m'}
P_{n}^{m}(\mu_j) \exp(im ...
...}^{m}
P_{n}^{m}(\mu)
\DD{P_{n'}^m}{\mu} d \mu \delta_{m m'}
\end{eqnarray*}">|; 

$key = q/mu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\mu$">|; 

$key = q/displaystyleint_{-1}^1P_n^m(mu)P_{n'}^m(mu)dmu=2delta_{nn'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\displaystyle \int_{-1}^1 P_n^m(\mu) P_{n'}^m(\mu) d \mu = 2 \delta_{nn'}$">|; 

$key = q/displaystylesigma_{k-1slash2}-sigma_{k+1slash2}(1lekleK);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle \sigma_{k-1/2} - \sigma_{k+1/2} ( 1 \le k \le K )$">|; 

$key = q/displaystylefrac{{displaystylesum_{k=0}^{J-1}(2k+1)tilde{P}_k(mu)tilde{P}_k(mu_j)}}{Jtilde{P}_{J-1}(mu_j)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="120" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$\displaystyle \frac{
{\displaystyle
\sum_{k=0}^{J-1} (2k+1)
\tilde{P}_k(\mu) \tilde{P}_k(\mu_j) }
}
{J \tilde{P}_{J-1}(\mu_j)}$">|; 

$key = q/0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="$0$">|; 

$key = q/{displaystyleint_{-1}^1fdmu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="${\displaystyle \int_{-1}^1 f d \mu }$">|; 

$key = q/{displaymath}delta_{t}pi=left(DP{pi}{t}right)_{NG}-Dvect{C}cdotoverline{Dvect{D}}^{t}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="\begin{displaymath}
\delta_{t} \pi =
\left( \DP{\pi}{t} \right)_{NG}
- \Dvect{C} \cdot \overline{ \Dvect{D} }^{t}
\end{displaymath}">|; 

$key = q/2Deltat;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$2 \Delta t$">|; 

$key = q/P_{J};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$P_{J}$">|; 

$key = q/displaystylehspace*{2.5cm}-sqrt{frac{(2n+1)(2n+3)}{(n-|m|+1)(n+|m|+1)}}sqrt{frac{(n-|m|)(n+|m|)}{(2n+1)(2n-1)}}P_{n-1}^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="515" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$\displaystyle \hspace*{2.5cm}
- \sqrt{ \frac{(2n+1)(2n+3)}{(n-\vert m\vert+1)(n...
...+1)} }
\sqrt{ \frac{(n-\vert m\vert)(n+\vert m\vert)}{(2n+1)(2n-1)} } P_{n-1}^m$">|; 

$key = q/L;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$L$">|; 

$key = q/displaystylefrac{1}{rcos^2phi}DP{V}{lambda}-frac{1}{rcosphi}DP{U}{phi};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="$\displaystyle \frac{1}{r \cos^2 \phi} \DP{V}{\lambda}
- \frac{1}{r \cos \phi} \DP{U}{\phi}$">|; 

$key = q/K=2N_{tr},J=M=N_{tr};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="$ K=2N_{tr}, J=M=N_{tr} $">|; 

$key = q/displaystyletilde{D}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="$\displaystyle \tilde{D}_n^m$">|; 

$key = q/displaystylealpha_kleft[1-left(frac{sigma_k}{sigma_{k-1}}right)^{kappa}right]^{-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$\displaystyle \alpha_k
\left[ 1- \left( \frac{ \sigma_k }{ \sigma_{k-1} }
\right)^{\kappa} \right]^{-1}$">|; 

$key = q/N=64;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="$N=64$">|; 

$key = q/{displaymath}delta_{t}{calA}equivfrac{1}{2Deltat}left({calA}^{t+Deltat}-{calA}^{t-Deltat}right){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="\begin{displaymath}
\delta_{t} {\cal A} \equiv \frac{1}{2 \Delta t}
\left( {\cal A}^{t+\Delta t} - {\cal A}^{t-\Delta t} \right)
\end{displaymath}">|; 

$key = q/Jge65;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="$J \ge 65$">|; 

$key = q/{displaymath}C_{k}=Deltasigma_{k}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="\begin{displaymath}
C_{k} = \Delta \sigma_{k}
\end{displaymath}">|; 

$key = q/sigma=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\sigma=0$">|; 

$key = q/f''=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$f''=0$">|; 

$key = q/max[n+n'];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="$max[n+n']$">|; 

$key = q/displaystyleDeltasigma_{1slash2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$\displaystyle \Delta \sigma_{1/2}$">|; 

$key = q/displaystyleDP{tilde{pi}_n^m}{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$\displaystyle \DP{\tilde{\pi}_n^m}{t}$">|; 

$key = q/{P_n^m|n=|m|,|m+1|,cdots};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$ \{ P_n^m \vert n=\vert m\vert,\vert m+1\vert,\cdots \}$">|; 

$key = q/D;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="$D$">|; 

$key = q/displaystylefrac{1}{Jtilde{P}_{J-1}(mu_j)tilde{P}^{'}_{J}(mu_j)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$\displaystyle \frac{1}{J \tilde{P}_{J-1}(\mu_j)
\tilde{P}^{'}_{J} (\mu_j)}$">|; 

$key = q/M=N_{tr},N(m)=N_{tr}+|m|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$M= N_{tr},  N(m) =N_{tr}+\vert m\vert$">|; 

$key = q/A(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$A(\mu)$">|; 

$key = q/f(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$f(\lambda,\phi)$">|; 

$key = q/displaystylesum_{m,n}left(frac{1}{r}imtilde{psi}_n^mP_n^m(sinphi_j)+frac{1}{r}ti{chi}_n^mcos^2phi_jleft.DD{P_n^m}{mu}right|_{j}right)exp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="$\displaystyle \sum_{m,n}
\left(
\frac{1}{r} im \tilde{\psi}_n^m P_n^m (\sin \ph...
...cos^2 \phi_j \left. \DD{P_n^m}{\mu} \right\vert _{j}
\right)
\exp(im \lambda_i)$">|; 

$key = q/displaystylew_j=frac{1}{2}int_{-1}^{1}frac{P_J(mu)}{(mu-mu_j)P^{'}_J(mu_i)}dmu=frac{(2J-1)(1-mu_j^2)}{(JP_{J-1}(mu_j))^2}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="$\displaystyle w_j = \frac{1}{2} \int_{-1}^{1}
\frac{P_J(\mu)}{(\mu-\mu_j) P^{'}_J (\mu_i)}d \mu
= \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 } .$">|; 

$key = q/R;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$R$">|; 

$key = q/displaystyleP_n^m(mu)equivsqrt{frac{(2n+1)(n-|m|)!}{(n+|m|)!}}frac{(1-mu^2)^{frac{|m|}{2}}}{2^nn!}DD[n+|m|]{}{mu}(mu^2-1)^n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$\displaystyle P_n^m(\mu)\equiv
\sqrt{\frac{(2n+1)(n-\vert m\vert)!}{(n+\vert m\...
...mu^2)^{\frac{\vert m\vert}{2}} }{2^n n!}
\DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n$">|; 

$key = q/displaystylesum_{n=1}^{I}left(exp(frac{2piim}{I})right)^{n-1}=0(mneq0);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$\displaystyle \sum_{n=1}^{I}
\left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1} =0
    (m \neq 0)$">|; 

$key = q/displaystyletilde{g}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="$\displaystyle \tilde{g}_n^m$">|; 

$key = q/P_J(sinphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$P_J(\sin \phi)$">|; 

$key = q/displaystyle+hat{kappa}_kDvect{v}_kcdotnablapiT_{v,k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$\displaystyle + \hat{\kappa}_k \Dvect{v}_k \cdot \nabla \pi T_{v,k}$">|; 

$key = q/{displaystylew_j=frac{1}{2}int_{-1}^1frac{tilde{P}_J(mu)}{(mu-mu_j)tilde{P}^{'}_J(mu_j)}dmu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="${\displaystyle w_j
= \frac{1}{2} \int_{-1}^1
\frac{\tilde{P}_J(\mu)}
{(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu
}$">|; 

$key = q/displaystylefrac{I}{2}int_{-1}^1P_n^m(mu)P_{n'}^{m}(mu)dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$\displaystyle \frac{I}{2} \int_{-1}^1
P_n^m (\mu) P_{n'}^{m} (\mu) d\mu$">|; 

$key = q/displaystyleint_{-1}^1Y_n^m(lambda,phi)Y_{n'}^{m'*}(lambda,phi)d(sinphi)dlambda=4pidelta_{mm'}delta_{nn'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$\displaystyle \int_{-1}^1 Y_n^m(\lambda, \phi)
Y_{n'}^{m'*} (\lambda, \phi)
d (\sin \phi) d \lambda
= 4 \pi \delta_{mm'} \delta_{nn'}$">|; 

$key = q/{eqnarraystar}p_n^m&=&frac{1}{I}sum_{i,j}p_{ij}Y_n^{m*}w_j&=&frac{1}{I}sum_{i,j}j}left.DD{P_n^m}{mu}right|_jexp(-imlambda_i)w_j{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="497" HEIGHT="237" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="\begin{eqnarray*}
p_n^m &amp;=&amp; \frac{1}{I} \sum_{i,j} p_{ij} Y_n^{m*} w_j \\\\
&amp;=&amp;...
...eft. \DD{P_n^m}{\mu}\right\vert _j
\exp(-im \lambda_i) w_j \\\\
\end{eqnarray*}">|; 

$key = q/-1<beta_1<beta_2<1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="$-1 &lt; \beta_1 &lt; \beta_2 &lt; 1$">|; 

$key = q/displaystylea_kT_k+b_{k-1}T_{k-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$\displaystyle a_k T_k + b_{k-1} T_{k-1}$">|; 

$key = q/displaystyleJtilde{P}_J(mu)tilde{P}_{J-1}(mu_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="$\displaystyle J \tilde{P}_J (\mu) \tilde{P}_{J-1} (\mu_j)$">|; 

$key = q/displaystylesum_{m,n}left(frac{1}{r}imtilde{psi}_n^mP_n^m(sinphi_j)+frac{cos^2phj}{r}tilde{chi}_n^mleft.DD{P_n^m}{mu}right|_{j}right)exp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="$\displaystyle \sum_{m,n}
\left(
\frac{1}{r} im \tilde{\psi}_n^m P_n^m (\sin \ph...
...de{\chi}_n^m
\left. \DD{P_n^m}{\mu} \right\vert _{j}
\right)
\exp(im \lambda_i)$">|; 

$key = q/x=-1,1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$x=-1,1$">|; 

$key = q/displaystylefrac{n(n+1)}{a^{2}}(Phi_n^m+C_{p}hat{kappa}_kbar{T}_kpi_n^m)+tilde{calD}_{M,n}^mtilde{D}_n^m,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\displaystyle \frac{n(n+1)}{a^{2}}
( \Phi_n^m + C_{p} \hat{\kappa}_k \bar{T}_k \pi_n^m )
+ \tilde{\cal D}_{M,n}^m \tilde{D}_n^m ,$">|; 

$key = q/displaystyleleft(DP{f}{mu}right)_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\displaystyle \left(\DP{f}{\mu}\right)_{ij}$">|; 

$key = q/{P_n^m}(n=|m|,|m|+1,|m|+2,cdots,N);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="$\{ P_n^m \} (n=\vert m\vert,\vert m\vert+1,\vert m\vert+2,\cdots,N)$">|; 

$key = q/A_{i,j}-A_{i,J-j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$A_{i,j}-A_{i,J-j}$">|; 

$key = q/displaystylefrac{1}{rcos^2phi}DP{U}{lambda}+frac{1}{rcosphi}DP{V}{phi};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$\displaystyle \frac{1}{r \cos^2 \phi} \DP{U}{\lambda}
+ \frac{1}{r \cos \phi} \DP{V}{\phi}$">|; 

$key = q/P_n^mP_{n'}^{m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="$ P_n^m P_{n'}^{m} $">|; 

$key = q/chi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\chi$">|; 

$key = q/bar{calA};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$\bar{\cal A}$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}im(mbox{{sl{UA})_{ij}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im (\mbox{\sl UA})_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/{displaystylesum_{i=1}^{I}sum_{j=1}^{J}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="${\displaystyle \sum_{i=1}^{I} \sum_{j=1}^{J} }$">|; 

$key = q/displaystylenabla_H^2chi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$\displaystyle \nabla_H^2 \chi$">|; 

$key = q/P_n^0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$P_n^0$">|; 

$key = q/displaystylefrac{1}{rcosphi}DP{u}{lambda}+frac{1}{rcosphi}DP{}{phi}(vcosphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="$\displaystyle \frac{1}{r \cos \phi} \DP{u}{\lambda}
+ \frac{1}{r \cos \phi} \DP{}{\phi} (v \cos \phi)$">|; 

$key = q/1leileI,1lejleJ;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="$1 \le i \le I,1 \le j \le J$">|; 

$key = q/{displaymath}frac{partialpi}{partialt}=-sum_{k=1}^{K}(D_k+Dvect{v}_kcdotnablapi)Deltasigma_k{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="\begin{displaymath}
\frac{\partial \pi}{\partial t}
= - \sum_{k=1}^{K} ( D_k + \Dvect{v}_k \cdot \nabla \pi )
\Delta \sigma_k
\end{displaymath}">|; 

$key = q/I=64,J=65;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="$I=64, J=65$">|; 

$key = q/displaystylefrac{1}{I}sum_{ij}left(imV_{ij}P_n^m+U_{ij}cos^2phi_jleft.DD{P_n^m}{mu}right|_jright)exp(-imlambda_i)frac{w_j}{rcos^2phi_j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
im V_{ij} P_n^m
+ U_{ij}
\cos^2 \phi...
...m}{\mu} \right\vert _j
\right)
\exp(- im \lambda_i)
\frac{w_j}{r \cos^2 \phi_j}$">|; 

$key = q/displaystyleleft[DP{}{lambda}left(sum_{m,n}tilde{f}_n^mY_n^m(lambda,phi)right)right]_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$\displaystyle \left[
\DP{}{\lambda} \left(
\sum_{m,n} \tilde{f}_n^m Y_n^m (\lambda, \phi)
\right)
\right]_{ij}$">|; 

$key = q/tilde{P}_J(mu)S(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$\tilde{P}_J(\mu) S(\mu)$">|; 

$key = q/%latex2htmlidmarker7426setcounter{footnote}{9}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="% latex2html id marker 7426
\setcounter{footnote}{9}\fnsymbol{footnote}">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^Isum_{j=1}^JA_{ij}P_n^{m}(sinphi_j)exp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{i=1}^I \sum_{j=1}^J
A_{ij} P_n^{m}(\sin \phi_j)
\exp(-im \lambda_i) w_j$">|; 

$key = q/overline{P_n^1}=P_n^1slashsqrt{2},overline{P_n^2}=P_n^2slashsqrt{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$ \overline{P_n^1} = P_n^1/\sqrt{2},
\overline{P_n^2} = P_n^2/\sqrt{2}$">|; 

$key = q/displaystyletilde{A}_n^m=frac{1}{4pi}int_{-1}^1d(sinphi)int_0^{2pi}dlambdaA(lambda,phi)Y_n^{m*}(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$\displaystyle \tilde{A}_n^m = \frac{1}{4 \pi}
\int_{-1}^1 d (\sin \phi)
\int_0^{2 \pi} d \lambda A(\lambda, \phi)
Y_n^{m*}(\lambda, \phi)$">|; 

$key = q/displaystyle+hat{kappa}_{k}T_{v,k}Dvect{v}_{k}cdotnablapi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$\displaystyle + \hat{\kappa}_{k} T_{v,k} \Dvect{v}_{k} \cdot \nabla \pi$">|; 

$key = q/displaystyletilde{h}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="$\displaystyle \tilde{h}_n^m$">|; 

$key = q/displaystylefrac{1}{I}sum_{n=1}^{I}sum_{m=-M}^{M}g_mexp(imlambda_n);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{n=1}^{I} \sum_{m=-M}^{M}
g_m \exp(i m \lambda_n)$">|; 

$key = q/displaystyleA(lambda,phi)=sum_{m=0}^{infty}sum_{n=|m|}^{infty}tilde{A}_n^mY_n^m(lambda,phi),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$\displaystyle A(\lambda, \phi)
= \sum_{m=0}^{\infty} \sum_{n=\vert m\vert}^{\infty}
\tilde{A}_n^m Y_n^m(\lambda,\phi) ,$">|; 

$key = q/M=21,N=42;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$M=21, N=42$">|; 

$key = q/{displaymath}dot{sigma}_{k-1slash2}=-sigma_{k-1slash2}frac{partialpi}{partialt}-sum_{l=k}^{K}(D_l+Dvect{v}_lcdotnablapi)Deltasigma_l{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="\begin{displaymath}
\dot{\sigma}_{k-1/2}
= - \sigma_{k-1/2} \frac{\partial \pi...
...}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta \sigma_l
\end{displaymath}">|; 

$key = q/{eqnarraystar}&&sum_{i=1}^Isum_{j=1}^Jf_{n'}^{m'}(1-mu_j^2)P_{n}^{m}(mu_j)exp(imimlambda_i)P_{n'}^{m'}(mu_j)exp(im'lambda_i)w_j{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="116" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="\begin{eqnarray*}
&amp; &amp; \sum_{i=1}^I \sum_{j=1}^J
f_{n'}^{m'} (1-\mu_j^2)
P_{n}...
...xp(-im \lambda_i)
P_{n'}^{m'}(\mu_j) \exp(im' \lambda_i)
w_j
\end{eqnarray*}">|; 

$key = q/{displaymath}W_{kl}=C_{p}alpha_{l}delta_{kgeql}+C_{p}beta_{l}delta_{k-1geql}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="\begin{displaymath}
W_{kl} = C_{p} \alpha_{l} \delta_{k \geq l}
+ C_{p} \beta_{l} \delta_{k-1 \geq l}
\end{displaymath}">|; 

$key = q/|m|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="$\vert m\vert$">|; 

$key = q/displaystylefrac{1}{r^2}left[frac{1}{cosphi}DP{}{phi}left(cosphiDP{}{phi}right)+frac{1}{cos^2phi}DP[2]{}{lambda}right]Y_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="307" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$\displaystyle \frac{1}{r^2}
\left[ \frac{1}{\cos \phi}
\DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right)
+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
\right] Y_n^m$">|; 

$key = q/f=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$f=0$">|; 

$key = q/sinphi=mu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$\sin \phi = \mu$">|; 

$key = q/{eqnarraystar}pequivDP{f}{mu}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="\begin{eqnarray*}
p \equiv \DP{f}{\mu}
\end{eqnarray*}">|; 

$key = q/{displaymath}frac{partialzeta_k}{partialt}=frac{1}{a(1-mu^{2})}frac{partial(mboxfrac{partial(mbox{{sl{UA})_k}{partialmu}+{calD}(zeta_k){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="\begin{displaymath}
\frac{\partial \zeta_k}{\partial t}
= \frac{1}{a(1-\mu^{2...
...partial (\mbox{\sl UA})_k}{\partial \mu}
+ {\cal D}(\zeta_k)
\end{displaymath}">|; 

$key = q/displaystylefrac{1-mu_j^2}{(Jtilde{P}_{J-1}(mu_j))^2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$\displaystyle \frac{1-\mu_j^2}{(J \tilde{P}_{J-1}(\mu_j))^2 }$">|; 

$key = q/displaystyleint_{-1}^1L(mu)dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="$\displaystyle \int_{-1}^1 L(\mu) d \mu$">|; 

$key = q/I;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$I$">|; 

$key = q/{figure}center{picture{(300,170)(10,-20)put(20,10){vector(1,0){95}}put(30,0){vec0,-20){shortstack{五角形切断}}{par{picture{center{{{{figure};MSF=1.6;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="199" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="\begin{figure}\begin{center}
\begin{picture}(300,170)(10,-20)
\put(20 , 10){\v...
...00,-20) \{ shortstack\{五角形切断\}\}
\par
\end{picture}\end{center}\end{figure}">|; 

$key = q/displaystyle(mbox{rot}Dvect{v})_r;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="$\displaystyle (\mbox{rot} \Dvect{v})_r$">|; 

$key = q/{displaymath}dot{sigma}_{1slash2}=dot{sigma}_{K+1slash2}=0{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="\begin{displaymath}
\dot{\sigma}_{1/2} = \dot{\sigma}_{K+1/2} = 0
\end{displaymath}">|; 

$key = q/{eqnarraystar}q_{ij}=(1-mu_j^2)sum_{m,n}tilde{f}_n^mleft.DD{P_n^m}{mu}right|_jexp(imlambda_i){eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="\begin{eqnarray*}
q_{ij} = (1-\mu_j^2) \sum_{m,n} \tilde{f}_n^m
\left. \DD{P_n^m}{\mu} \right\vert _j
\exp(im \lambda_i)
\end{eqnarray*}">|; 

$key = q/{eqnarraystar}hequivfrac{g}{rcos^2phi}=frac{1}{rcos^2phi}DP{f}{lambda}left[=DP{}{x}left(frac{f}{cosphi}right)right]{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="\begin{eqnarray*}
h \equiv \frac{g}{r \cos^2 \phi}
= \frac{1}{r \cos^2 \phi} \...
...left[ =
\DP{}{x} \left( \frac{f}{\cos \phi} \right)
\right]
\end{eqnarray*}">|; 

$key = q/displaystyleDP{tilde{zeta}_n^m}{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$\displaystyle \DP{\tilde{\zeta}_n^m}{t}$">|; 

$key = q/nabla_H^2equivfrac{1}{r^2}left[DP{}{phi}left(cosphiDP{}{phi}right)+frac{1}{cos^2phi}DP[2]{}{lambda}right];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$\nabla_H^2 \equiv
\frac{1}{r^2}
\left[ \DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right)
+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
\right]
$">|; 

$key = q/displaystyleint_{-1}^1f(mu)P_n(mu)dmu=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$\displaystyle \int_{-1}^1 f(\mu) P_n(\mu) d \mu = 0$">|; 

$key = q/I=128,J=65;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="$I=128, J=65$">|; 

$key = q/K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$K$">|; 

$key = q/M,N(m);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="$M,N(m)$">|; 

$key = q/displaystyleA(mu)=sum_{n=|m|}^{infty}tilde{A}_n^mtilde{P}_n^m(mu),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$\displaystyle A(\mu) = \sum_{n=\vert m\vert}^{\infty}
\tilde{A}_n^m \tilde{P}_n^m(\mu),$">|; 

$key = q/displaystylehat{H}_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$\displaystyle \hat{H}_k$">|; 

$key = q/{f_k(mu)}(k=0,1,2,cdots);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$ \{ f_k(\mu) \}(k=0,1,2,\cdots) $">|; 

$key = q/displaystylesum_{j=0}^{J-1}f_k(mu_j)f_{k'}(mu_{j})w_j=lambda_kdelta_{kk'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$\displaystyle \sum_{j=0}^{J-1}
f_k (\mu_j) f_{k'} (\mu_{j}) w_j
= \lambda_k \delta_{kk'}$">|; 

$key = q/P_J(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$P_J (\mu)$">|; 

$key = q/Idelta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$I \delta_{mm'}$">|; 

$key = q/{displaymath}S_{kl}=sigma_{k-1slash2}Deltasigma_{l}-Deltasigma_{l}delta_{kleql}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="\begin{displaymath}
S_{kl} = \sigma_{k-1/2} \Delta \sigma_{l}
- \Delta \sigma_{l} \delta_{k \leq l }
\end{displaymath}">|; 

$key = q/n=0,1,2,cdots,N<J-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$n=0,1,2,\cdots,N &lt; J-1$">|; 

$key = q/S(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="$S(\mu)$">|; 

$key = q/128times64=8192;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$ 128 \times 64 = 8192 $">|; 

$key = q/-1<mu<1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$-1 &lt; \mu &lt; 1$">|; 

$key = q/{displaymath}frac{partialDvect{D}}{partialt}=left(DP{Dvect{D}}{t}right)_{NG}-nab_{S}+underline{W}Dvect{T}+Dvect{G}pi)+{calD}_MDvect{D},{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="\begin{displaymath}
\frac{\partial \Dvect{D}}{\partial t} =
\left( \DP{\Dvect...
...ine{W} \Dvect{T}
+ \Dvect{G} \pi )
+ {\cal D}_M \Dvect{D} ,
\end{displaymath}">|; 

$key = q/A_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$A_{ij}$">|; 

$key = q/displaystyle+frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}left(hat{H}_{ij}+frac{Q_{ij}}{C_{p}}right)Y_n^{m*}(lambda_i,mu_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
\left( \hat{H}_{ij} + \frac{Q_{ij}}{C_{p}} \right)
Y_n^{m *} ( \lambda_i, \mu_j )
w_j$">|; 

$key = q/{eqnarraystar}&&sum_{i}sum_{j}f_{n'}^{m'}P_{n}^{m}(mu_j)exp(imlambda_i)left.DD{Pft.DD{P_{n}^{m}}{mu}right|_jexp(-imlambda_i)w_j{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="208" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="\begin{eqnarray*}
&amp; &amp; \sum_{i} \sum_{j}
f_{n'}^{m'}
P_{n}^{m}(\mu_j) \exp(im ...
.... \DD{P_{n}^{m}}{\mu}\right\vert _j
\exp(-im \lambda_i)
w_j
\end{eqnarray*}">|; 

$key = q/f-L;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="$f-L$">|; 

$key = q/x;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$x$">|; 

$key = q/B,C;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="$B,C$">|; 

$key = q/{eqnarraystar}h_{ij}=sum_{m,n}left.tilde{h}_n^mY_n^mright|_{ij}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="\begin{eqnarray*}
h_{ij} = \sum_{m,n}
\left. \tilde{h}_n^m Y_n^m \right\vert _{ij} \\\\
\end{eqnarray*}">|; 

$key = q/h=nabla_H^2f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$h = \nabla_H^2 f$">|; 

$key = q/tilde{P}_J(mu_j)=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$\tilde{P}_J(\mu_j)=0$">|; 

$key = q/-1lemule1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$-1 \le \mu \le 1$">|; 

$key = q/displaystyleoverline{calA}^{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$\displaystyle \overline{\cal A}^{t}$">|; 

$key = q/NG;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$NG$">|; 

$key = q/displaystyleD=frac{1}{r(1-mu^2)}DP{U}{lambda}+frac{1}{r}DP{V}{mu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="$\displaystyle D = \frac{1}{r(1-\mu^2)} \DP{U}{\lambda}
+ \frac{1}{r} \DP{V}{\mu}$">|; 

$key = q/U,V;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$U,V$">|; 

$key = q/J;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$J$">|; 

$key = q/j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$j$">|; 

$key = q/displaystyle2sum_{j=1}^{J}f(mu_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$\displaystyle 2 \sum_{j=1}^{J} f(\mu_j) w_j$">|; 

$key = q/displaystyleV=frac{1}{r}DP{psi}{lambda}+frac{1-mu^2}{r}DP{chi}{mu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="$\displaystyle V = \frac{1}{r} \DP{\psi}{\lambda}
+ \frac{1-\mu^2}{r} \DP{\chi}{\mu}$">|; 

$key = q/displaystylefrac{partialT_k}{partialt};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\displaystyle \frac{\partial T_k}{\partial t}$">|; 

$key = q/m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="$m=0$">|; 

$key = q/{Y_n^m(lambda',phi')|m=-n,-n+1,cdots,n};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$\{ Y_n^m(\lambda', \phi') \vert m=-n,-n+1, \cdots, n \} $">|; 

$key = q/displaystyleg_0+sum_{m=-M,mneq0}^{M}frac{g_m}{I}sum_{n=1}^{I}left(exp(frac{2piim}{I})right)^{n-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\displaystyle g_0
+ \sum_{m=-M, m\neq0}^{M} \frac{g_m}{I}
\sum_{n=1}^{I}
\left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1}$">|; 

$key = q/k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$k$">|; 

$key = q/displaystylebeta_kleft[left(frac{sigma_k}{sigma_{k+1}}right)^{kappa}-1right]^{-1}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$\displaystyle \beta_k
\left[ \left( \frac{ \sigma_k }{ \sigma_{k+1} }
\right)^{\kappa} - 1 \right]^{-1} .$">|; 

$key = q/[];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$[  ]$">|; 

$key = q/displaystylefrac{1}{I}sum_{n=1}^{I}g(lambda_n);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$">|; 

$key = q/displaystylenabla_H^2psi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$\displaystyle \nabla_H^2 \psi$">|; 

$key = q/nabla^{'2}_HY_n^m=-frac{n(n+1)}{r^2}Y_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$\nabla^{'2}_H Y_n^m = -\frac{n(n+1)}{r^2} Y_n^m $">|; 

$key = q/displaystylefrac{1}{I}sum_{i,j}imleft(frac{f}{rcos^2phi}right)_{ij}Y_n^{m*}(lambda_i,phi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{i,j}
im
\left( \frac{f}{r \cos^2 \phi} \right)_{ij}
Y_n^{m*} (\lambda_i, \phi_j) w_j$">|; 

$key = q/displaystyle-sum_{k=1}^{K}(tilde{D}_n^m)_kDeltasigma_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\displaystyle - \sum_{k=1}^{K} (\tilde{D}_n^m)_k \Delta \sigma_k$">|; 

$key = q/displaystyle-frac{1-mu^2}{r}DP{psi}{mu}+frac{1}{r}DP{chi}{lambda};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$\displaystyle - \frac{1-\mu^2}{r} \DP{\psi}{\mu}
+ \frac{1}{r} \DP{\chi}{\lambda}$">|; 

$key = q/displaystyleDD{}{mu}left{(1-mu^2)DD{}{mu}tilde{P}_n^mright}+left{n(n+1)-frac{m^2}{1-mu^2}right}tilde{P}_n^m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$\displaystyle \DD{}{\mu}
\left\{ (1-\mu^2) \DD{}{\mu} \tilde{P}_n^m \right\}
+ \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\}
\tilde{P}_n^m = 0$">|; 

$key = q/Ige2M+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="$I \ge 2M+1$">|; 

$key = q/displaystyleIsum_{j=1}^{J}P_n^m(mu_j)P_{n'}^{m'}(mu_j)w_jdelta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$\displaystyle I \sum_{j=1}^{J}
P_n^m (\mu_j) P_{n'}^{m'} (\mu_j) w_j \delta_{mm'}$">|; 

$key = q/displaystylep_{ij}=sum_{m,n}tilde{f}_n^mleft.DD{P_n^m}{mu}right|_jexp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$\displaystyle p_{ij} = \sum_{m,n} \tilde{f}_n^m
\left. \DD{P_n^m}{\mu} \right\vert _j
\exp(im \lambda_i)$">|; 

$key = q/{displaystylew_jequivfrac{(2J-1)(1-sin^2phi_j)}{(JP_{J-1}(sinphi_j))^2}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="${\displaystyle w_j \equiv \frac{(2J-1)(1-\sin^2 \phi_j)}
{(J P_{J-1}(\sin \phi_j))^2 } }$">|; 

$key = q/-frac{n(n+1)}{r^2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$- \frac{n(n+1)}{r^2}$">|; 

$key = q/displaystyleint_{-1}^1P_n(mu)P_n'(mu)dmu=2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$\displaystyle \int_{-1}^1 P_n(\mu) P_n'(\mu) d \mu = 2$">|; 

$key = q/displaystyleU_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$\displaystyle U_{ij}$">|; 

$key = q/displaystylefrac{1}{I}sum_{m'=-M}^{M}sum_{n'=|m'|}^Ntilde{A}_{n'}^{m'}sum_{i=1}^p(i(m'-m)lambda)sum_{j=1}^JP_n^{m}(sinphi_j)P_{n'}^{m'}(sinphi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="491" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{m'=-M}^{M} \sum_{n'=\vert m'\vert}^N
\tilde{A}_...
...i(m'-m) \lambda)
\sum_{j=1}^J
P_n^{m}(\sin \phi_j)
P_{n'}^{m'}(\sin \phi_j) w_j$">|; 

$key = q/{displaymath}left(DP{pi}{t}right)^{NG}=-sum_{k=1}^{K}Dvect{v}_{k}cdotnablapiDeltasigma_{k}nonumber=Z_{k}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="\begin{displaymath}
\left( \DP{\pi}{t} \right)^{NG}
= - \sum_{k=1}^{K} \Dvect{...
...k} \cdot \nabla \pi
\Delta \sigma_{k} \nonumber \\\\
= Z_{k}
\end{displaymath}">|; 

$key = q/M=21;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="$M=21$">|; 

$key = q/displaystyleb_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$\displaystyle b_k$">|; 

$key = q/displaystylesum_{n=1}^{I}exp(imlambda_n)=left{array{{ll}I<col_mark>(m=0)<row_mark>0<col_mark>(0<|m|<I)<row_mark>array{right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="$\displaystyle \sum_{n=1}^{I} \exp(i m \lambda_n)
= \left\{
\begin{array}{ll}
I &amp;   (m=0) \\\\
0 &amp;   (0 &lt; \vert m\vert &lt; I) \\\\
\end{array}\right.$">|; 

$key = q/{displaymath}DP{pi}{t}=left(frac{partialpi}{partialt}right)_{NG}-Dvect{C}cdotDvect{D},{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="\begin{displaymath}
\DP{\pi}{t} =
\left( \frac{\partial \pi}{\partial t} \right)_{NG}
- \Dvect{C} \cdot \Dvect{D} ,
\end{displaymath}">|; 

$key = q/n-m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$n-m$">|; 

$key = q/displaystylefrac{1}{rcosphi}DP{v_1}{lambda}+frac{1}{rcosphi}DP{}{phi}(v_2cosphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$\displaystyle \frac{1}{r \cos \phi} \DP{v_1}{\lambda}
+ \frac{1}{r \cos \phi} \DP{}{\phi} (v_2 \cos \phi)$">|; 

$key = q/displaystylefrac{1}{I}sum_{i,j}imf_{ij}Y_n^{m*}(lambda_i,phi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{i,j} im f_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$">|; 

$key = q/{Y_n^m|m=0,1,2,cdots,n=|m|,|m+1|,cdots};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$ \{ Y_n^m \vert m=0,1,2,\cdots,  n=\vert m\vert,\vert m+1\vert,\cdots \}$">|; 

$key = q/N;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="$N$">|; 

$key = q/displaystyle{calA}^{t-Deltat}+delta_{t}{calA}Deltat,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$\displaystyle {\cal A}^{t-\Delta t} + \delta_{t} {\cal A} \Delta t ,$">|; 

$key = q/{displaymath}(mbox{{sl{KE})_k=frac{U^{2}_k+V^{2}_k}{2(1-mu^{2})}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="\begin{displaymath}
(\mbox{\sl KE})_k = \frac{U^{2}_k+V^{2}_k}{2(1-\mu^{2}) }
\end{displaymath}">|; 

$key = q/-1<alpha<1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$-1&lt;\alpha&lt;1$">|; 

$key = q/Dvect{T}={T_{k}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$\Dvect{T}=\{ T_{k} \}$">|; 

$key = q/mu_j,mu_{j'}(1lej,j'leJ);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$\mu_j, \mu_{j'}(1 \le j,j' \le J)$">|; 

$key = q/displaystylesum_{m,n}left(-frac{1}{r}tilde{psi}_n^mcos^2phi_jleft.DD{P_n^m}{mu}rht|_j+frac{1}{r}imtilde{chi}_n^mP_n^m(sinphi_j)right)exp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="$\displaystyle \sum_{m,n}
\left(
- \frac{1}{r} \tilde{\psi}_n^m
\cos^2 \phi_j \l...
...+ \frac{1}{r} im \tilde{\chi}_n^m P_n^m(\sin \phi_j)
\right)
\exp(im \lambda_i)$">|; 

$key = q/A_{ij}^{odd};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$A_{ij}^{odd}$">|; 

$key = q/displaystylesum_{j=1}^{J}sum_{i=1}^{I}P_n^m(mu_j)P_{n'}^{m'}(mu_j)exp(imlambda_i)exp(-im'lambda_i)w_j=Idelta_{nn'}delta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="$\displaystyle \sum_{j=1}^{J}
\sum_{i=1}^{I}
P_n^m (\mu_j) P_{n'}^{m'} (\mu_j)
\exp(i m \lambda_i) \exp(-i m' \lambda_i) w_j
= I \delta_{nn'} \delta_{mm'}$">|; 

$key = q/sigma=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\sigma=1$">|; 

$key = q/J=K=M=N_{tr};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="$ J=K=M =N_{tr} $">|; 

$key = q/displaystylePhi_k-Phi_{k-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$\displaystyle \Phi_k - \Phi_{k-1}$">|; 

$key = q/m,n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$m,n$">|; 

$key = q/{P_n}(n=0,1,2,cdots,J-1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$\{ P_n \}(n=0,1,2,\cdots,J-1) $">|; 

$key = q/tilde{B}_n^m=tilde{B}_n^{m*},{itetc.};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="$\tilde{B}_n^m = \tilde{B}_n^{m*}, {\it etc.}$">|; 

$key = q/{eqnarraystar}tilde{f}_{n'}^{m'}=-frac{n(n+1)}{r^2}tilde{h}_{n'}^{m'}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="\begin{eqnarray*}
\tilde{f}_{n'}^{m'}
= -\frac{n(n+1)}{r^2} \tilde{h}_{n'}^{m'}
\end{eqnarray*}">|; 

$key = q/{displaymath}underline{h}=underline{Q}underline{S}-underline{R}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="\begin{displaymath}
\underline{h} = \underline{Q}\underline{S} - \underline{R}
\end{displaymath}">|; 

$key = q/textstyleequiv;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\textstyle \equiv$">|; 

$key = q/(2times21+1)times(21+1)=946;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="$ (2 \times 21 +1) \times (21+1)= 946$">|; 

$key = q/P_J;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="$P_J$">|; 

$key = q/displaystylefrac{n(n+1)}{a^{2}}frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}(mbox{{sl{KE})_{ij}Y_n^{m*}(lambda_i,mu_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\displaystyle \frac{n(n+1)}{a^{2}}
\frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
(\mbox{\sl KE})_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
w_j$">|; 

$key = q/f(mu)-L(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="$f(\mu)-L(\mu)$">|; 

$key = q/displaystylewidetilde{(BC)_n^m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="$\displaystyle \widetilde{(BC)_n^m}$">|; 

$key = q/{displaymath}Q_{kl}=frac{1}{Deltasigma_{k}}(hat{bar{T}}_{k-1slash2}-bar{T}_{k})da_{k}}(bar{T}_{k}-hat{bar{T}}_{k+1slash2})delta_{k+1=l}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="\begin{displaymath}
Q_{kl} = \frac{1}{\Delta \sigma_{k}}
( \hat{\bar{T}}_{k-1...
...{k}}
( \bar{T}_{k} - \hat{\bar{T}}_{k+1/2} ) \delta_{k+1=l}
\end{displaymath}">|; 

$key = q/displaystylesum_{m=-M}^Msum_{n=|m|}^Nimtilde{f}_n^mY_n^m(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle \sum_{m=-M}^M
\sum_{n=\vert m\vert}^N im \tilde{f}_n^m
Y_n^m (\lambda_i,\phi_j)$">|; 

$key = q/{displaystylex_j=cosfrac{j-1slash2}{n}pi};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="${\displaystyle x_j=\cos \frac{j-1/2}{n}\pi }$">|; 

$key = q/%latex2htmlidmarker7548setcounter{footnote}{9}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="% latex2html id marker 7548
\setcounter{footnote}{9}\fnsymbol{footnote}">|; 

$key = q/displaystyleDP{tilde{T}_n^m}{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$\displaystyle \DP{\tilde{T}_n^m}{t}$">|; 

$key = q/f'(alpha)=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="$f'(\alpha)=0$">|; 

$key = q/{displaymath}T_{v,k}=T_k-bar{T}_k{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="\begin{displaymath}
T_{v,k} = T_k - \bar{T}_k
\end{displaymath}">|; 

$key = q/C;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$C$">|; 

$key = q/displaystylevspace{4cm}timesleft(sum_{m''=-M}^{M}sum_{n''=|m''|}^Ntilde{C}_{n''}_i)P_{n''}^{m''}(sinphi_j)right)P_n^{m}(sinphi_j)exp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="535" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="$\displaystyle \vspace{4cm}
\times
\left( \sum_{m''=-M}^{M} \sum_{n''=\vert m''\...
...P_{n''}^{m''}(\sin \phi_j)
\right)
P_n^{m}(\sin \phi_j) \exp(-im \lambda_i) w_j$">|; 

$key = q/displaystylefrac{1}{rcosphi}DP{v_2}{lambda}-frac{1}{rcosphi}DP{}{phi}(v_1cosphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$\displaystyle \frac{1}{r \cos \phi} \DP{v_2}{\lambda}
- \frac{1}{r \cos \phi} \DP{}{\phi}(v_1 \cos \phi)$">|; 

$key = q/P_n(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$ P_n(\mu)$">|; 

$key = q/(n,m);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="$(n,m)$">|; 

$key = q/displaystyletilde{calD}_{M,n}^mtilde{zeta}_n^m,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\displaystyle \tilde{\cal D}_{M,n}^m \tilde{\zeta}_n^m ,$">|; 

$key = q/displaystylewidetilde{left[DP{}{lambda}left(frac{f}{rcos^2phi}right)right]_n^m}=imwidetilde{left(frac{f}{rcos^2phi}right)_n^m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="$\displaystyle \widetilde{
\left[
\DP{}{\lambda} \left( \frac{f}{r \cos^2 \phi} ...
...ht)
\right]_n^m
}
= im \widetilde{ \left( \frac{f}{r \cos^2 \phi} \right)_n^m }$">|; 

$key = q/n=|m|,|m|+1,cdots,|m|+J-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="$n=\vert m\vert, \vert m\vert+1, \cdots, \vert m\vert+J-1$">|; 

$key = q/%latex2htmlidmarker7726setcounter{footnote}{5}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="% latex2html id marker 7726
\setcounter{footnote}{5}\fnsymbol{footnote}">|; 

$key = q/displaystylesum_{m=-M}^Msum_{n=|m|}^Ntilde{f}_n^mleft.DD{P_n^m}{mu}right|_jexp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$\displaystyle \sum_{m=-M}^M \sum_{n=\vert m\vert}^N \tilde{f}_n^m
\left. \DD{P_n^m}{\mu} \right\vert _j
\exp(im \lambda_i)$">|; 

$key = q/f'=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$f'=0$">|; 

$key = q/Depsf[10cm]{zahyouslashlg1.ps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="275" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="\Depsf[10cm]{zahyou/lg1.ps}">|; 

$key = q/m=n=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$m=n=0$">|; 

$key = q/displaystyleleft|array{{ll}(2n+1)xtilde{P}_n(x)-ntilde{P}_{n-1}(x)<col_mark>tildytilde{P}_n(y)-ntilde{P}_{n-1}(y)<col_mark>tilde{P}_n(y)array{right|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$\displaystyle \left\vert
\begin{array}{ll}
(2n+1)x \tilde{P}_n(x)-n\tilde{P}_{n...
...+1)y \tilde{P}_n(y)-n\tilde{P}_{n-1}(y)
&amp; \tilde{P}_n(y)
\end{array}\right\vert$">|; 

$key = q/displaystylesum_{k=0}^{J-1}frac{f_k(mu_j)f_k(mu_{j'})}{lambda_k}=frac{1}{w_j}delta_{jj'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="$\displaystyle \sum_{k=0}^{J-1}
\frac{f_k (\mu_j) f_k (\mu_{j'}) }{\lambda_k}
= \frac{1}{w_j} \delta_{jj'}$">|; 

$key = q/Jgefrac{2N_{max}+1}{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="$J \ge \frac{2N_{max}+1}{2}$">|; 

$key = q/displaystylePhi_{s}+C_{p}alpha_{1}T_{v,1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle \Phi_{s} + C_{p} \alpha_{1} T_{v,1}$">|; 

$key = q/1leileI,1lejleJ,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$1 \le i \le I , 1 \le j \le J, $">|; 

$key = q/L-f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$L-f$">|; 

$key = q/%latex2htmlidmarker7247setcounter{footnote}{5}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="% latex2html id marker 7247
\setcounter{footnote}{5}\fnsymbol{footnote}">|; 

$key = q/displaystyle-alpha_ksum_{l=k}^{K}(D_l+Dvect{v}_lcdotnablapi)Deltasigma_lfrac{T_{v,k}}{Deltasigma_k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$\displaystyle - \alpha_k \sum_{l=k}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta \sigma_l
\frac{T_{v,k}}{\Delta \sigma_k}$">|; 

$key = q/displaystylefrac{1}{r}DP{psi}{lambda}+frac{1-mu^2}{r}DP{chi}{mu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$\displaystyle \frac{1}{r} \DP{\psi}{\lambda}
+ \frac{1-\mu^2}{r} \DP{\chi}{\mu}$">|; 

$key = q/displaystyle(1-mu^2)DP{tilde{P}_n}{mu}=ntilde{P}_{n-1}(mu)-nmutilde{P}_n(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$\displaystyle (1-\mu^2) \DP{\tilde{P}_n}{\mu}
= n \tilde{P}_{n-1}(\mu) - n \mu \tilde{P}_n(\mu)$">|; 

$key = q/g(lambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$g(\lambda)$">|; 

$key = q/Dvect{v};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$\Dvect{v}$">|; 

$key = q/P_n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$P_n$">|; 

$key = q/tilde{P}_n^0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$\tilde{P}_n^0$">|; 

$key = q/displaystyleDP{tilde{D}_n^m}{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle \DP{\tilde{D}_n^m}{t}$">|; 

$key = q/M=42,N=42;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="$M=42, N=42$">|; 

$key = q/{tilde{P}_n^m|n=|m|,|m+1|,cdots};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$ \{ \tilde{P}_n^m \vert n=\vert m\vert,\vert m+1\vert,\cdots \}$">|; 

$key = q/sigma_{k-1slash2}(k=1,2,cdots,K);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\sigma_{k-1/2} (k=1,2,\cdots,K)$">|; 

$key = q/n=J,mu=mu_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="$n=J, \mu=\mu_j$">|; 

$key = q/displaystyleA_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$\displaystyle A_{ij}$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^Isum_{j=1}^JB_{ij}C_{ij}P_n^{m}(sinphi_j)exp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J
B_{ij} C_{ij}
P_n^{m}(\sin \phi_j) \exp(-im \lambda_i) w_j$">|; 

$key = q/displaystyleU=-frac{1-mu^2}{r}DP{psi}{mu}+frac{1}{r}DP{chi}{lambda};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="$\displaystyle U = - \frac{1-\mu^2}{r} \DP{\psi}{\mu}
+ \frac{1}{r} \DP{\chi}{\lambda}$">|; 

$key = q/displaystylefrac{1}{r^2}left[DP{}{mu}left{(1-mu^2)DP{}{mu}right}+frac{1}{1-mu^2}DP[2]{}{lambda}right]f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="$\displaystyle \frac{1}{r^2}
\left[
\DP{}{\mu}
\left\{ (1-\mu^2) \DP{}{\mu} \right\}
+ \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
\right] f$">|; 

$key = q/phi=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$\phi=0$">|; 

$key = q/exp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="$\exp(i m \lambda)$">|; 

$key = q/displaystylebeta_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$\displaystyle \beta_k$">|; 

$key = q/f'=2nx(x^2-1)^{n-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$f'=2nx(x^2-1)^{n-1}$">|; 

$key = q/displaystyle(2n+1)(x-y)tilde{P}_n(x)tilde{P}_n(y)+nleft|array{{ll}tilde{P}_{n}(x}(x)<row_mark>tilde{P}_{n}(y)<col_mark>tilde{P}_{n-1}(y)array{right|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="362" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$\displaystyle (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)
+
n \left\vert
\begin{arr...
...de{P}_{n-1}(x) \\\\
\tilde{P}_{n}(y) &amp; \tilde{P}_{n-1}(y)
\end{array}\right\vert$">|; 

$key = q/displaystyle∴P_{n+1}^m=sqrt{frac{(2n+1)(2n+3)}{(n-|m|+1)(n+|m|+1)}}muP_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="2" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$\displaystyle ∴ P_{n+1}^m
= \sqrt{ \frac{(2n+1)(2n+3)}{(n-\vert m\vert+1)(n+\vert m\vert+1)} }
\mu P_n^m$">|; 

$key = q/displaystyleleft(DP{f}{lambda}right)_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$\displaystyle \left( \DP{f}{\lambda} \right)_{ij}$">|; 

$key = q/{Y_n^m|m=-n,-n+1,cdots,n};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$\{Y_n^m \vert m= -n, -n+1,\cdots,n \}$">|; 

$key = q/frac{I-1}{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="$\frac{I-1}{2}$">|; 

$key = q/phi_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="$\phi_j$">|; 

$key = q/tilde{P}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="$\tilde{P}_n^m$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}(mbox{{sl{VA})_{ij}(1-mu_j^2)frac{partial}{partialmu}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
(\mbox{\sl VA})_{ij}
(1...
...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/displaystyle-frac{alpha_{k}}{Deltasigma_{k}}T'_{v,k}sum_{l=k}^{K}D_lDeltasigma_{ac{beta_{k}}{Deltasigma_{k}}T'_{v,k}sum_{l=k+1}^{K}D_lDeltasigma_{l};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$\displaystyle - \frac{\alpha_{k}}{\Delta \sigma_{k} } T'_{v,k}
\sum_{l=k}^{K} D...
...{\beta_{k}}{\Delta \sigma_{k} } T'_{v,k}
\sum_{l=k+1}^{K} D_l \Delta \sigma_{l}$">|; 

$key = q/J-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$J-1$">|; 

$key = q/{displaymath}frac{partialq_k}{partialt}=-frac{1}{a(1-mu^{2})}frac{partialU_kq_k}}frac{partialV_kq_k}{partialmu}+R_k+S_{q,k}+{calD}(q_k){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="\begin{displaymath}
\frac{\partial q_k}{\partial t}
= - \frac{1}{a(1-\mu^{2})}...
...al V_k q_k}{\partial \mu}
+ R_k
+ S_{q,k}
+ {\cal D}(q_k)
\end{displaymath}">|; 

$key = q/M<I;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$M &lt; I$">|; 

$key = q/mu_j（j=1,2,cdots,J）;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="$\mu_j（j=1,2,\cdots,J）$">|; 

$key = q/{displaymath}{calA}^{t+Deltat}=hat{calA}^{t+Deltat}+2Deltatdot{calA}_{dis}left(h)+2Deltatdot{calA}_{cnd}left(hat{calA}^{t+Deltat}right){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="\begin{displaymath}
{\cal A}^{t+\Delta t}
= \hat{\cal A}^{t+\Delta t}
+ 2 \D...
...t
\dot{\cal A}_{cnd}\left( \hat{\cal A}^{t+\Delta t} \right)
\end{displaymath}">|; 

$key = q/{displaymath}bar{calA}^{t}=(1-2epsilon_f){calA}^{t}+epsilon_fleft(bar{calA}^{t-Deltat}+{calA}^{t+Deltat}right){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="\begin{displaymath}
\bar{\cal A}^{t}
= ( 1-2 \epsilon_f ) {\cal A}^{t}
+ \eps...
...eft( \bar{\cal A}^{t-\Delta t} + {\cal A}^{t+\Delta t} \right)
\end{displaymath}">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}(mbox{{sl{UA})_{ij}(1-mu_j^2)frac{partial}{partialmu}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
(\mbox{\sl UA})_{ij}
(1...
...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/max[n+n'+n''];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="$max[n+n'+n'']$">|; 

$key = q/Y_n^m(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$Y_n^m(\lambda,\phi)$">|; 

$key = q/P_n,P_n^1,P_n^2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="$P_n,P_n^1,P_n^2$">|; 

$key = q/(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$(\lambda, \phi)$">|; 

$key = q/tilde{A}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\tilde{A}_n^m$">|; 

$key = q/displaystylefrac{sigma_{k-1slash2}(sigma^{kappa}_{k-1slash2}-sigma^{kappa}_k)+si_{k+1slash2})}{sigma^{kappa}_k(sigma_{k-1slash2}-sigma_{k+1slash2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle \frac{ \sigma_{k-1/2}( \sigma^{\kappa}_{k-1/2}
- \sigma^{\kappa}_...
...^{\kappa}_{k+1/2} ) }
{ \sigma^{\kappa}_k
( \sigma_{k-1/2} - \sigma_{k+1/2} ) }$">|; 

$key = q/displaystylehspace*{2cm}+(n+|m|)sqrt{frac{1}{2n-1}frac{(n-1+|m|)!}{(n-1-|m|)!}}P_{n-1}^m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$\displaystyle \hspace*{2cm}
+(n+\vert m\vert)
\sqrt{ \frac{1}{2n-1} \frac{(n-1+\vert m\vert)!}{(n-1-\vert m\vert)!} }
P_{n-1}^m = 0$">|; 

$key = q/A(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$A(\lambda,\phi)$">|; 

$key = q/displaystylesum_{m,n}left(-frac{cos^2phi_j}{r}tilde{psi}_n^mleft.DD{P_n^m}{mu}riht|_j+frac{1}{r}imtilde{chi}_n^mP_n^m(sinphi_j)right)exp(imlambda_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="417" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$\displaystyle \sum_{m,n}
\left(
- \frac{\cos^2 \phi_j}{r} \tilde{\psi}_n^m
\lef...
...+ \frac{1}{r} im \tilde{\chi}_n^m P_n^m(\sin \phi_j)
\right)
\exp(im \lambda_i)$">|; 

$key = q/displaystyle+frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}V_{ij}T'_{ij}(1-mu_j^2)frac{partial}{partialmu}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
V_{ij} T'_{ij}
(1-\mu...
...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/im'tilde{f}_{n'}^{m'}=tilde{g}_{n'}^{m'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="$im' \tilde{f}_{n'}^{m'}
= \tilde{g}_{n'}^{m'} $">|; 

$key = q/sigma_k(k=1,2,ldotsK);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\sigma_k (k=1,2,\ldots K)$">|; 

$key = q/displaystylew_j=frac{1-mu_j^2}{(Jsqrt{frac{1}{2J-1}}P_{J-1}(mu_j))^2}=frac{(2J-1)(1-mu_j^2)}{(JP_{J-1}(mu_j))^2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="$\displaystyle w_j = \frac{1-\mu_j^2}
{(J \sqrt{ \frac{1}{2J-1} } P_{J-1}(\mu_j))^2 }
= \frac{(2J-1)(1-\mu_j^2)}
{(J P_{J-1}(\mu_j))^2 }$">|; 

$key = q/displaystyleA(mu)=sum_{n=|m|}^{infty}tilde{A}_n^mP_n^m(mu),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$\displaystyle A(\mu) = \sum_{n=\vert m\vert}^{\infty}
\tilde{A}_n^m P_n^m(\mu),$">|; 

$key = q/{displaystylefrac{I}{3}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="${\displaystyle \frac{I}{3} }$">|; 

$key = q/displaystyleimtilde{f}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="$\displaystyle im \tilde{f}_n^m$">|; 

$key = q/displaystylefrac{1}{r^2}left[frac{1}{cosphi}DP{}{phi}left(cosphiDP{}{phi}right)+frac{1}{cos^2phi}DP[2]{}{lambda}right]f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$\displaystyle \frac{1}{r^2}
\left[ \frac{1}{\cos \phi}
\DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right)
+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
\right] f$">|; 

$key = q/%latex2htmlidmarker8039setcounter{footnote}{9}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="% latex2html id marker 8039
\setcounter{footnote}{9}\fnsymbol{footnote}">|; 

$key = q/{displaystyleJgefrac{3N_{tr}+1}{2}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="${\displaystyle J \ge \frac{3N_{tr}+1}{2} }$">|; 

$key = q/f';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$f'$">|; 

$key = q/displaystylea_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$\displaystyle a_k$">|; 

$key = q/epsilon_f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$\epsilon_f$">|; 

$key = q/K=42,J=M=21;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="$K=42, J=M=21$">|; 

$key = q/displaystylesigma_{1slash2}-sigma_{1}=1-sigma_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$\displaystyle \sigma_{1/2} - \sigma_{1}
= 1 -\sigma_{1}$">|; 

$key = q/{displaymath}delta_{t}Dvect{D}=left(DP{Dvect{D}}{t}right)_{NG}-nabla^{2}_{sigma}+{calD}_M(Dvect{D}^{t-Deltat}+2Deltatdelta_{t}Dvect{D}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="\begin{displaymath}
\delta_{t} \Dvect{D} =
\left( \DP{\Dvect{D}}{t} \right)_{N...
...( \Dvect{D}^{t-\Delta t}
+ 2 \Delta t \delta_{t} \Dvect{D} )
\end{displaymath}">|; 

$key = q/y;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$y$">|; 

$key = q/displaystyle1-left(frac{sigma_{k+1slash2}}{sigma_k}right)^{kappa}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\displaystyle 1- \left( \frac{ \sigma_{k+1/2} }
{ \sigma_k } \right)^{\kappa} .$">|; 

$key = q/displaystylefrac{1}{rsqrt{1-mu^2}}DP{v_2}{lambda}-frac{1}{r}DP{}{mu}(sqrt{1-mu^2}v_1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$\displaystyle \frac{1}{r \sqrt{1-\mu^2} } \DP{v_2}{\lambda}
- \frac{1}{r} \DP{}{\mu}(\sqrt{1-\mu^2}v_1)$">|; 

$key = q/I=64,J=64;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$I=64, J=64$">|; 

$key = q/A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd}$">|; 

$key = q/alpha;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$\alpha$">|; 

$key = q/{displaystylesum_{i,j}Y_{n'}^{m'*}|_{ij}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="${\displaystyle \sum_{i,j} Y_{n'}^{m'*}\vert _{ij} }$">|; 

$key = q/displaystylefrac{1}{2pi}int_0^{2pi}g(lambda)dlambda;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda$">|; 

$key = q/I=64;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="$I=64$">|; 

$key = q/frac{1}{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\frac{1}{2}$">|; 

$key = q/P_0simP_{n-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="$P_0 \sim P_{n-1}$">|; 

$key = q/displaystyle-frac{1}{a(1-mu^{2})}frac{partialU_kT_k'}{partiallambda}-frac{1}{a}frac{partialV_kT_k'}{partialmu}+hat{H}_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\displaystyle - \frac{1}{a(1-\mu^{2})}
\frac{\partial U_k T_k'}{\partial \lambda}
- \frac{1}{a}
\frac{\partial V_k T_k'}{\partial \mu}
+ \hat{H}_k$">|; 

$key = q/{displaystyle-frac{I-1}{2}simfrac{I-1}{2}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="${\displaystyle -\frac{I-1}{2} \sim \frac{I-1}{2} }$">|; 

$key = q/{displaymath}tilde{calD}_{H,n}^m=-K_{HD}left(frac{-n(n+1)}{a^{2}}right)^{N_Dslash2}.{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="\begin{displaymath}
\tilde{\cal D}_{H,n}^m
= - K_{HD} \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2} .
\end{displaymath}">|; 

$key = q/n=J,x=mu,y=mu_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="$n=J,x=\mu,y=\mu_j$">|; 

$key = q/%latex2htmlidmarker8037setcounter{footnote}{9}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="% latex2html id marker 8037
\setcounter{footnote}{9}\fnsymbol{footnote}">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^Isum_{j=1}^Jleft(sum_{m'=-M}^{M}sum_{n'=|m'|}^Ntilde{B}_{n'}^{m'}exp(im'lambda_i)P_{n'}^{m'}(sinphi_j)right);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J
\left( \sum_{m'=-M}^{M} \su...
...ert}^N
\tilde{B}_{n'}^{m'}
\exp(im' \lambda_i)
P_{n'}^{m'}(\sin \phi_j)
\right)$">|; 

$key = q/displaystyle(zeta_k+f)V_k-frac{1}{2Deltasigma_k}[dot{sigma}_{k-1slash2}(U_{k-1}-U_k)+dot{sigma}_{k+1slash2}(U_k-U_{k+1})];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="430" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$\displaystyle ( \zeta_k + f ) V_k
- \frac{1}{2 \Delta \sigma_k}
[ \dot{\sigma}_{k-1/2} ( U_{k-1} - U_k )
+ \dot{\sigma}_{k+1/2} ( U_k - U_{k+1} ) ]$">|; 

$key = q/0leKle2J-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$0 \le K \le 2J-1$">|; 

$key = q/I=128,J=64;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$I=128, J=64$">|; 

$key = q/N_{max};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$N_{max}$">|; 

$key = q/(U,V);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$(U,V)$">|; 

$key = q/{displaymath}hat{calA}^{t+Deltat}=bar{calA}^{t-Deltat}+2Deltatdot{calA}_{adv}lef)+2Deltatdot{calA}_{phy}left(bar{calA}^{t-Deltat}right){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="507" HEIGHT="39" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="\begin{displaymath}
\hat{\cal A}^{t+\Delta t}
= \bar{\cal A}^{t-\Delta t}
+ ...
...t
\dot{\cal A}_{phy}\left( \bar{\cal A}^{t-\Delta t} \right)
\end{displaymath}">|; 

$key = q/displaystyle=(1-2Deltat{calD}_H)(1-Deltat{calD}_M)Dvect{D}^{t-Deltat}+(1-2Deltat{calD}_H)Deltatleft(DP{Dvect{D}}{t}right)_{NG};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$\displaystyle = ( 1-2\Delta t {\cal D}_H )( 1-\Delta t {\cal D}_M )
\Dvect{D}^{...
... t}
+ ( 1-2\Delta t {\cal D}_H ) \Delta t
\left( \DP{\Dvect{D}}{t} \right)_{NG}$">|; 

$key = q/0le|m|len,n';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$0\le \vert m\vert \le n,n'$">|; 

$key = q/delta_{kleql};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$\delta_{k \leq l}$">|; 

$key = q/displaystylesum_{m,n}imtilde{f}_n^mY_n^m(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$\displaystyle \sum_{m,n} im \tilde{f}_n^m
Y_n^m (\lambda_i,\phi_j)$">|; 

$key = q/%latex2htmlidmarker7345setcounter{footnote}{8}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="% latex2html id marker 7345
\setcounter{footnote}{8}\fnsymbol{footnote}">|; 

$key = q/tilde{P}^m_n(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="$\tilde{P}^m_n(\mu)$">|; 

$key = q/displaystylefrac{tilde{P}_J(mu)}{mu-mu_j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="$\displaystyle \frac{\tilde{P}_J (\mu) }{\mu-\mu_j}$">|; 

$key = q/-MlemleM,|m|lenleN;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="$ -M \le m \le M, \vert m\vert \le n \le N$">|; 

$key = q/{eqnarraystar}g&equiv&DP{f}{lambda}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="\begin{eqnarray*}
g&amp;\equiv&amp; \DP{f}{\lambda}
\end{eqnarray*}">|; 

$key = q/displaystyle-beta_ksum_{l=k+1}^{K}(D_l+Dvect{v}_lcdotnablapi)Deltasigma_lfrac{T_{v,k}}{Deltasigma_k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$\displaystyle - \beta_k \sum_{l=k+1}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta \sigma_l
\frac{T_{v,k}}{\Delta \sigma_k}$">|; 

$key = q/tilde{P}_n^m(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$\tilde{P}_n^m(\mu)$">|; 

$key = q/{displaystylefrac{1}{rcosphi}DP{f}{lambda}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="${\displaystyle \frac{1}{r \cos \phi} \DP{f}{\lambda} }$">|; 

$key = q/dot{calA}_{cnd};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$ \dot{\cal A}_{cnd} $">|; 

$key = q/M=42;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="$M=42$">|; 

$key = q/displaystylealpha_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle \alpha_k$">|; 

$key = q/0le|m|len;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$ 0 \le \vert m\vert \le n$">|; 

$key = q/{displaymath}D^{t+Deltat}=2overline{Dvect{D}}^{t}-D^{t-Deltat}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="\begin{displaymath}
D^{t+\Delta t} = 2\overline{ \Dvect{D} }^{t} - D^{t-\Delta t}
\end{displaymath}">|; 

$key = q/displaystylesum_{i=1}^Isum_{j=1}^Jf_{n'}^{m'}P_{n}^{m}(mu_j)exp(imlambda_i)left.DD{P_{n'}^{m'}}{mu}right|_jexp(-im'lambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="$\displaystyle \sum_{i=1}^I \sum_{j=1}^J
f_{n'}^{m'}
P_{n}^{m}(\mu_j) \exp(im \lambda_i)
\left. \DD{P_{n'}^{m'}}{\mu}\right\vert _j
\exp(-im' \lambda_i)
w_j$">|; 

$key = q/{displaymath}left(DP{T_{k}}{t}right)^{NG}equiv-frac{1}{a(1-mu^{2})}DP{U_kT'_k}{lkT'_k}{mu}+H_{k}+{calD}(T_{k})+{calD}^{prime}(Dvect{v}){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="\begin{displaymath}
\left( \DP{T_{k}}{t} \right)^{NG}
\equiv - \frac{1}{a(1-\...
... + H_{k}
+ {\cal D}(T_{k})
+ {\cal D}^{\prime}(\Dvect{v})
\end{displaymath}">|; 

$key = q/Ige3M+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="$I \ge 3M+1$">|; 

$key = q/displaystylesum_{j=1}^{J}f(mu_j)int_{-1}^1frac{tilde{P}_J(mu)}{(mu-mu_j)tilde{P}^{'}_J(mu_j)}dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$\displaystyle \sum_{j=1}^{J} f(\mu_j)
\int_{-1}^1
\frac{\tilde{P}_J(\mu)}
{(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu$">|; 

$key = q/int_0^{2pi}exp(imlambda)dlambda;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="$\int_0^{2\pi} \exp(i m \lambda) d \lambda $">|; 

$key = q/M=64;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="$M=64$">|; 

$key = q/tilde{A}_n^m（-MlemleM,|m|lenleN）;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="$\tilde{A}_n^m （ -M \le m \le M,  \vert m\vert \le n \le N）$">|; 

$key = q/displaystyle-frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}imU_{ij}T'_{ij}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$\displaystyle - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im U_{ij} T'_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/displaystyleDeltasigma_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle \Delta \sigma_k$">|; 

$key = q/Y_n^mexp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="$Y_n^m \exp(im\lambda)$">|; 

$key = q/tilde{P_n};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="$\tilde{P_n}$">|; 

$key = q/displaystylefrac{left[left(frac{sigma_{k-1slash2}}{sigma_k}right)^{kappa}-1righta}right]sigma_k^{kappa}T_{k-1}}{sigma_{k-1}^{kappa}-sigma_k^{kappa}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\displaystyle \frac{ \left[ \left( \frac{ \sigma_{k-1/2} }
{ \sigma_k } \right)...
...ight] \sigma_k^{\kappa} T_{k-1} }
{ \sigma_{k-1}^{\kappa} - \sigma_k^{\kappa} }$">|; 

$key = q/displaystylesum_{k=0}^{J-1}(2k+1)(mu-mu_j)tilde{P}_k(mu)tilde{P}_k(mu_j).;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$\displaystyle \sum_{k=0}^{J-1} (2k+1) (\mu-\mu_j)
\tilde{P}_k(\mu) \tilde{P}_k(\mu_j).$">|; 

$key = q/{eqnarraystar}tilde{q}_n^m&=&frac{1}{I}sum_{i,j}q_{ij}Y_n^{m*}w_j&=&frac{1}{I}su((1-mu^2)P_n^mright)right|_jexp(-imlambda_i)w_j{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="544" HEIGHT="278" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="\begin{eqnarray*}
\tilde{q}_n^m &amp;=&amp; \frac{1}{I} \sum_{i,j} q_{ij} Y_n^{m*} w_j ...
... (1-\mu^2)P_n^m \right) \right\vert _j
\exp(-im \lambda_i) w_j
\end{eqnarray*}">|; 

$key = q/{displaystyleM=left[frac{I}{2}right]};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="${\displaystyle M= \left[ \frac{I}{2} \right] }$">|; 

$key = q/displaystyleDD{}{mu}left{(1-mu^2)DD{}{mu}P_n^mright}+left{n(n+1)-frac{m^2}{1-mu^2}right}P_n^m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$\displaystyle \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} P_n^m \right\}
+ \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\} P_n^m = 0$">|; 

$key = q/displaystyleleft.hspace*{20mm}+(1-2Deltat{calD}_H)Dvect{G}left[pi^{t-Deltat}+Deltatleft(DP{pi}{t}right)_{NG}right]right}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$\displaystyle \left. \hspace*{20mm}
+ ( 1-2\Delta t {\cal D}_H ) \Dvect{G}
\left[ \pi^{t-\Delta t}
+ \Delta t
\left( \DP{\pi}{t} \right)_{NG}
\right]
\right\} .$">|; 

$key = q/-MlemleM,|m|lenleN(m);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="$ -M \le m \le M, \vert m\vert \le n \le N(m)$">|; 

$key = q/displaystyle+n(-tilde{P}_{n-1}(x)tilde{P}_n(y)+tilde{P}_{n-1}(y)tilde{P}_n(x));MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="$\displaystyle +n (- \tilde{P}_{n-1}(x) \tilde{P}_n(y)
+ \tilde{P}_{n-1}(y) \tilde{P}_n(x) )$">|; 

$key = q/displaystyleV_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="$\displaystyle V_{ij}$">|; 

$key = q/displaystylep_{ij}equivleft[DP{}{mu}left(sum_{m,n}tilde{f}_n^mY_n^mright)right]_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="$\displaystyle p_{ij} \equiv \left[ \DP{}{\mu}
\left( \sum_{m,n} \tilde{f}_n^m Y_n^m \right)
\right]_{ij}$">|; 

$key = q/m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$m$">|; 

$key = q/A_{ij}=A(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$A_{ij}=A(\lambda_i,\phi_j)$">|; 

$key = q/(U_{ij},V_{ij});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$(U_{ij},V_{ij})$">|; 

$key = q/L(mu_j)=f(mu_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$ L(\mu_j) = f(\mu_j) $">|; 

$key = q/displaystyle-frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}imU_{ij}q_{ij}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$\displaystyle - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im U_{ij} q_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/{displaystyleexp(imlambda_n)=expleft(frac{2piim(n-1)}{I}right)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="${\displaystyle \exp(i m \lambda_n)
= \exp \left( \frac{2\pi i m (n-1)}{I} \right) }$">|; 

$key = q/{eqnarraystar}h_{ij}=left.left(nabla_H^2sum_{m,n}tilde{f}_n^mY_n^mright)right|_{c{n(n+1)}{r^2}left.tilde{f}_n^mY_n^mright|_{ij}{eqnarraystar};MSF=1.6;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="66" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="\begin{eqnarray*}
h_{ij}
= \left. \left(\nabla_H^2 \sum_{m,n}
\tilde{f}_n^m...
...{n(n+1)}{r^2}
\left. \tilde{f}_n^m Y_n^m \right\vert _{ij} \\\\
\end{eqnarray*}">|; 

$key = q/displaystyleH_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$\displaystyle H_k$">|; 

$key = q/{Y_n^m|n=0,1,2,cdots,m=-n,-n+1,cdots,n};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$\{Y_n^m \vert n=0,1,2,\cdots, m= -n, -n+1,\cdots,n \}$">|; 

$key = q/sigma;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\sigma$">|; 

$key = q/displaystylesum_{j=1}^{J}P_n^m(mu_j)P_{n'}^m(mu_j)w_j=delta_{nn'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="203" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle \sum_{j=1}^{J} P_n^m (\mu_j) P_{n'}^m (\mu_j) w_j
= \delta_{nn'}$">|; 

$key = q/displaystyleY^m_n(lambda,phi)=sum_{m'=-n}^{n}A_n^{m'}Y^{m'*}_n(lambda',phi');MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$\displaystyle Y^m_n(\lambda, \phi)
= \sum_{m'=-n}^{n} A_n^{m'} Y^{m'*}_n(\lambda',\phi')$">|; 

$key = q/displaystyle(u,v)equivleft(frac{U}{cosphi},frac{V}{cosphi}right);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$\displaystyle (u,v) \equiv
\left( \frac{U}{\cos \phi}, \frac{V}{\cos \phi} \right)$">|; 

$key = q/displaystyletilde{A}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="$\displaystyle \tilde{A}_n^m$">|; 

$key = q/(N_{tr}+1)^2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$(N_{tr}+1)^2$">|; 

$key = q/displaystyle+frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}Z_{ij}Y_n^{m*}(lambda_i,mu_j)w_j,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
Z_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) w_j ,$">|; 

$key = q/displaystylenleft|array{{ll}tilde{P}_n(x)<col_mark>tilde{P}_{n-1}(x)<row_mark>tilde{P}_n(y)<col_mark>tilde{P}_{n-1}(y)array{right|;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$\displaystyle n \left\vert
\begin{array}{ll}
\tilde{P}_n(x) &amp; \tilde{P}_{n-1}(x) \\\\
\tilde{P}_n(y) &amp; \tilde{P}_{n-1}(y)
\end{array}\right\vert$">|; 

$key = q/displaystylesum_{i=1}^Iexp(i(m'-m)lambda)=Idelta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="$\displaystyle \sum_{i=1}^I \exp(i(m'-m) \lambda) = I \delta_{mm'}$">|; 

$key = q/K-J;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="$K-J$">|; 

$key = q/{displaystylefrac{pi}{2}>phi_1>phi_2>cdots>phi_{J}>-frac{pi}{2}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="${\displaystyle \frac{\pi}{2} &gt; \phi_1 &gt; \phi_2 &gt;
\cdots &gt; \phi_{J} &gt; -\frac{\pi}{2} }$">|; 

$key = q/displaystyleg_{ij};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="$\displaystyle g_{ij}$">|; 

$key = q/i,j,m,m',n,n',I,J,M,N(m);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="$ i,j,m,m',n,n',I,J,M,N(m)$">|; 

$key = q/{displaystylelambda_i=frac{2pi(i-1)}{I}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="${\displaystyle \lambda_i=\frac{2\pi(i-1)}{I} }$">|; 

$key = q/displaystylefrac{1}{r}DP{Y_n^m}{phi}=frac{1}{r}DP{}{phi}left(P_n^m(sinphi)exp(imlambda)right)=frac{sqrt{1-mu^2}}{r}DD{}{mu}P_n^{m}(mu)exp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="484" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$\displaystyle \frac{1}{r} \DP{Y_n^m}{\phi}
= \frac{1}{r}
\DP{}{\phi}
\left( P_n...
...) \right)
= \frac{\sqrt{1-\mu^2} }{r}
\DD{}{\mu} P_n^{m} (\mu) \exp(im \lambda)$">|; 

$key = q/displaystylefrac{1}{r(1-mu^2)}DP{U}{lambda}+frac{1}{r}DP{V}{mu};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$\displaystyle \frac{1}{r (1-\mu^2)} \DP{U}{\lambda}
+ \frac{1}{r} \DP{V}{\mu}$">|; 

$key = q/P_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$P_n^m$">|; 

$key = q/(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$(\lambda_i,\phi_j)$">|; 

$key = q/tilde{P}_n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$\tilde{P}_n$">|; 

$key = q/displaystylenabla_H^2Y_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$\displaystyle \nabla_H^2
Y_n^m$">|; 

$key = q/x=-1,beta_1,beta_2,1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="$x=-1,\beta_1,\beta_2,1$">|; 

$key = q/{displaystyleP_n=frac{1}{2^nn!}DD[n]{}{mu}(mu^2-1)^n};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="${\displaystyle
P_n= \frac{1}{2^n n!}
\DD[n]{}{\mu} (\mu^2-1)^n }$">|; 

$key = q/n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$n$">|; 

$key = q/I>M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$I &gt; M$">|; 

$key = q/displaystyletilde{zeta}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="$\displaystyle \tilde{\zeta}_n^m$">|; 

$key = q/w(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="$w(\mu)$">|; 

$key = q/tilde{D}_n^m,;tilde{zeta}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$\tilde{D}_n^m, \; \tilde{\zeta}_n^m$">|; 

$key = q/{calF}_lambda,{calF}_varphi,Q,S_q;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="${\cal F}_\lambda, {\cal F}_\varphi, Q, S_q$">|; 

$key = q/mu_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$\mu_j$">|; 

$key = q/displaystylefrac{1}{I}sum_{i,j}imf_{ij}Y_n^{m*}(lambda_iphi_j)frac{w_j}{rcos^2phi_j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{i,j} im f_{ij}
Y_n^{m*} (\lambda_i \phi_j)
\frac{w_j}{r \cos^2 \phi_j}$">|; 

$key = q/w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$w_j$">|; 

$key = q/nabla_H^2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$\nabla_H^2$">|; 

$key = q/{displaymath}R_{kl}=-left(frac{alpha_{k}}{Deltasigma_{k}}Deltasigma_{l}delta_{kligma_{k}}Deltasigma_{l}delta_{k+1leql}right)bar{T}_{k}.{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="\begin{displaymath}
R_{kl} = - \left( \frac{ \alpha_{k} }{ \Delta \sigma_{k} } ...
... \Delta \sigma_{l} \delta_{k+1 \leq l}
\right) \bar{T}_{k} .
\end{displaymath}">|; 

$key = q/displaystyle-frac{alpha_{k}}{Deltasigma_{k}}T_{v,k}sum_{l=k}^{K}Dvect{v}_{l}cdotigma_{k}}T_{v,k}sum_{l=k+1}^{K}Dvect{v}_{l}cdotnablapiDeltasigma_{l};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$\displaystyle - \frac{\alpha_{k}}{\Delta \sigma_{k} } T_{v,k}
\sum_{l=k}^{K} \D...
...{k} } T_{v,k}
\sum_{l=k+1}^{K} \Dvect{v}_{l} \cdot \nabla \pi
\Delta \sigma_{l}$">|; 

$key = q/Dvect{v}=(v_1,v_2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="$\Dvect{v}=(v_1,v_2)$">|; 

$key = q/displaystylesum_{i=1}^{I}exp(imlambda_i)exp(-im'lambda_i)=Idelta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle \sum_{i=1}^{I} \exp(im \lambda_i) \exp(-im' \lambda_i)
= I \delta_{mm'}$">|; 

$key = q/displaystylefrac{1}{I}sum_{m'=-M}^{M}sum_{n'=|m'|}^Nsum_{m''=-M}^{M}sum_{n''=|m''|}^Ntilde{B}_{n'}^{m'}tilde{C}_{n''}^{m''};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{m'=-M}^{M} \sum_{n'=\vert m'\vert}^N
\sum_{m''=-M}^{M} \sum_{n''=\vert m''\vert}^N
\tilde{B}_{n'}^{m'} \tilde{C}_{n''}^{m''}$">|; 

$key = q/displaystyle-frac{C_{p}hat{kappa}_kT_{v,k}'}{a}(1-mu^{2})frac{partialpi}{partialmu}+F_{varphik}cosvarphi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a}
( 1 - \mu^{2} ) \frac{\partial \pi}{\partial \mu}
+ F_{\varphi k} \cos \varphi$">|; 

$key = q/textstyle=;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$\textstyle =$">|; 

$key = q/%latex2htmlidmarker7811setcounter{footnote}{6}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="4" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="% latex2html id marker 7811
\setcounter{footnote}{6}\fnsymbol{footnote}">|; 

$key = q/int_0^{2pi}exp(imlambda)exp(-im'lambda)dlambda=2pidelta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$\int_0^{2 \pi} \exp(im\lambda) \exp(-im'\lambda)
d \lambda = 2 \pi \delta_{mm'}$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^Isum_{j=1}^Jleft(sum_{m'=-M}^{M}sum_{n'=|m'|}^Nt'}(sinphi_j)exp(im'lambda)right)P_n^{m}(sinphi_j)exp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="555" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J
\left(
\sum_{m'=-M}^{M} \su...
... \phi_j)
\exp(im' \lambda)
\right)
P_n^{m}(\sin \phi_j)
\exp(-im \lambda_i) w_j$">|; 

$key = q/J=65;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="$J=65$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^Isum_{j=1}^Jimf_{ij}Y_n^{m*}(lambda_i,phi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J
im f_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$">|; 

$key = q/Y^m_n(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$Y^m_n(\lambda, \phi)$">|; 

$key = q/displaystylefrac{1}{I}sum_{ij}left(frac{im}{rcos^2phi_j}U_{ij}P_n^m-frac{1}{r}V_{ij}left.DD{P_n^m}{mu}right|_jright)exp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
\frac{im}{r \cos^2 \phi_j} U_{ij} P_...
...} V_{ij} \left. \DD{P_n^m}{\mu} \right\vert _j
\right)
\exp(- im \lambda_i) w_j$">|; 

$key = q/displaystylefrac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}im(mbox{{sl{VA})_{ij}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im (\mbox{\sl VA})_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/{displaystylesum_{m=-M}^{M}sum_{n=|m|}^{N}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="${\displaystyle \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} }$">|; 

$key = q/overline{Dvect{D}_n^m}^{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$\overline{ \Dvect{D}_n^m }^{t}$">|; 

$key = q/displaystyletilde{A}_n^m=frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}A_{ij}Y_n^{m*}(lambda_i,phi_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle \tilde{A}_n^m
= \frac{1}{I}
\sum_{i=1}^{I} \sum_{j=1}^{J}
A_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$">|; 

$key = q/4pi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$4 \pi$">|; 

$key = q/%latex2htmlidmarker8038setcounter{footnote}{9}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="% latex2html id marker 8038
\setcounter{footnote}{9}\fnsymbol{footnote}">|; 

$key = q/2J+2|m|-2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$2J+2\vert m\vert-2$">|; 

$key = q/(lambda',phi');MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$(\lambda', \phi')$">|; 

$key = q/displaystyleA=BC;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="$\displaystyle A= BC$">|; 

$key = q/lambda_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$\lambda_k$">|; 

$key = q/M=N_{tr},N=N_{tr};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$M= N_{tr},  N =N_{tr}$">|; 

$key = q/{displaystylesum_{m,n}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="${\displaystyle \sum_{m,n} }$">|; 

$key = q/m,n,n';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$m,n,n'$">|; 

$key = q/N_{max}=2M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$N_{max}=2M$">|; 

$key = q/displaystyleP_n^mequivsqrt{frac{(2n+1)(n-|m|)!}{(n+|m|)!}}tilde{P}_n^m=sqrt{fracm|)!}}frac{(1-mu^2)^{frac{|m|}{2}}}{2^nn!}DD[n+|m|]{}{mu}(mu^2-1)^n,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="595" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$\displaystyle P_n^m
\equiv \sqrt{ \frac{(2n+1)(n-\vert m\vert)!}{(n+\vert m\ver...
...^2)^{\frac{\vert m\vert}{2}} }{2^n n!}
\DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n ,$">|; 

$key = q/displaystyleIdelta_{nn'}delta_{mm'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$\displaystyle I \delta_{nn'} \delta_{mm'}$">|; 

$key = q/displaystylesigma_{K}-sigma_{K+1slash2}=sigma_{K};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle \sigma_{K} - \sigma_{K+1/2}
= \sigma_{K}$">|; 

$key = q/displaystyleT_k'D_k-frac{1}{Deltasigma_k}[dot{sigma}_{k-1slash2}(hat{T}_{k-1slash2}-T_k)+dot{sigma}_{k+1slash2}(T_k-hat{T}_{k+1slash2})];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$\displaystyle T_k' D_k
- \frac{1}{\Delta \sigma_k}
[ \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k )
+ \dot{\sigma}_{k+1/2} ( T_k - \hat{T}_{k+1/2} ) ]$">|; 

$key = q/displaystylefrac{1}{2}left({calA}^{t+Deltat}+{calA}^{t-Deltat}right);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$\displaystyle \frac{1}{2} \left( {\cal A}^{t+\Delta t}
+ {\cal A}^{t-\Delta t} \right)$">|; 

$key = q/%latex2htmlidmarker7473setcounter{footnote}{9}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="% latex2html id marker 7473
\setcounter{footnote}{9}\fnsymbol{footnote}">|; 

$key = q/m,m';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$m,m'$">|; 

$key = q/displaystylesum_{j=1}^{J}f(mu_j)int^1_{-1}frac{{displaystyleprod_{k=1}^{J}(mu-mu_k)}}{(mu-mu_j){displaystyleprod_{k=1,kneqj}^{J}(mu_j-mu_k)}}dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="120" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="$\displaystyle \sum_{j=1}^{J} f(\mu_j)
\int^1_{-1}
\frac{ {\displaystyle
\prod_{...
...
}
{ (\mu-\mu_j)
{\displaystyle
\prod_{k=1,k \neq j}^{J}(\mu_j-\mu_k) }
}
d \mu$">|; 

$key = q/f(x)=(x-1)^n(x+1)^n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$f(x)=(x-1)^n (x+1)^n$">|; 

$key = q/displaystyleV;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$\displaystyle V$">|; 

$key = q/displaystylehat{kappa}_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\displaystyle \hat{\kappa}_k$">|; 

$key = q/{displaystyleMleleft[frac{I}{2}right],N(m)leJ-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="${\displaystyle M \le \left[ \frac{I}{2} \right] ,
N(m) \le J-1 }$">|; 

$key = q/f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$f$">|; 

$key = q/displaystyleint_{-1}^1tilde{P}_n^m(mu)tilde{P}_{n'}^m(mu)dmu=frac{2}{2n+1}frac{(n+|m|)!}{(n-|m|)!}delta_{nn'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$\displaystyle \int_{-1}^1 \tilde{P}_n^m(\mu) \tilde{P}_{n'}^m (\mu) d \mu
= \frac{2}{2n+1} \frac{(n+\vert m\vert)!}{(n-\vert m\vert)!} \delta_{nn'}$">|; 

$key = q/Jge3N_{tr}+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$J \ge 3N_{tr}+1$">|; 

$key = q/displaystyleleft[DD{}{mu}left{(1-mu^2)DD{}{mu}right}+J(J+1)right]P_J(mu)=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle \left[
\DD{}{\mu}
\left\{ (1-\mu^2) \DD{}{\mu} \right\}
+ J(J+1) \right] P_J(\mu) = 0$">|; 

$key = q/displaystyle(2n+1)(x-y)tilde{P}_n(x)tilde{P}_n(y);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$\displaystyle (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)$">|; 

$key = q/f(mu_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$f(\mu_j)$">|; 

$key = q/-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$-1$">|; 

$key = q/n-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$ n-1$">|; 

$key = q/displaystylelambda_n=frac{2pi(n-1)}{I}(n=1,2,cdots,I);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$\displaystyle \lambda_n = \frac{2\pi (n-1)}{I}  (n=1,2,\cdots,I)$">|; 

$key = q/P_n^m(mu)(n=|m|,|m+1,cdots);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$P_n^m(\mu)  (n=\vert m\vert,\vert m+1, \cdots)$">|; 

$key = q/displaystyle+tilde{calD}_{H,n}^mtilde{q}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle + \tilde{\cal D}_{H,n}^m \tilde{q}_n^m$">|; 

$key = q/2J-1gemax[n+n']=2max[N];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$2J-1 \ge max[n+n'] = 2 max[N]$">|; 

$key = q/m(n-1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$m(n-1)$">|; 

$key = q/displaystyletilde{A}_n^m=frac{2n+1}{2}frac{(n-|m|)!}{(n+|m|)!}int_{-1}^1A(mu)tilde{P}_n^m(mu)dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$\displaystyle \tilde{A}_n^m = \frac{2n+1}{2} \frac{(n-\vert m\vert)!}{(n+\vert m\vert)!}
\int_{-1}^1 A(\mu) \tilde{P}_n^m(\mu) d \mu$">|; 

$key = q/chi,psi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\chi ,\psi $">|; 

$key = q/displaystyleA_{ij}=sum_{m=0}^{M}sum_{n=m}^{N}Retilde{A}_n^mY_n^m(lambda_i,phi_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle A_{ij} = \sum_{m=0}^{M} \sum_{n=m}^{N}
\Re \tilde{A}_n^m Y_n^m(\lambda_i, \phi_j)$">|; 

$key = q/K+frac{1}{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$K+\frac{1}{2}$">|; 

$key = q/displaystylefrac{1}{2Jtilde{P}_{J-1}(mu_j)tilde{P}^{'}_{J}(mu_j)}sum_{k=0}^{J-1}(2k+1)tilde{P}_k(mu_j)int_{-1}^1tilde{P}_k(mu)dmu;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$\displaystyle \frac{1}{2 J \tilde{P}_{J-1}(\mu_j)
\tilde{P}^{'}_{J} (\mu_j)}
\sum_{k=0}^{J-1}
(2k+1) \tilde{P}_k(\mu_j)
\int_{-1}^1 \tilde{P}_k(\mu) d \mu$">|; 

$key = q/f(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="$f(\mu)$">|; 

$key = q/g=DP{f}{lambda};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$g = \DP{f}{\lambda}$">|; 

$key = q/displaystylelambda_i=frac{2pi（i-1)}{I}（i=1,2,cdots,I）;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$\displaystyle \lambda_i = \frac{2 \pi （i-1)}{I}  \
（i=1,2,\cdots,I）$">|; 

$key = q/displaystyleT_{k}^{prime}D_{k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$\displaystyle T_{k}^{\prime} D_{k}$">|; 

$key = q/nabla^2_Hf=-frac{n(n+1)}{r^2}f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\nabla^2_H f = -\frac{n(n+1)}{r^2} f$">|; 

$key = q/displaystylesum_{m=-M}^{M}frac{1}{2pi}g_mint_0^{2pi}exp(imlambda)dlambda=g_0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$\displaystyle \sum_{m=-M}^{M} \frac{1}{2\pi} g_m
\int_0^{2\pi} \exp(i m \lambda) d \lambda
=g_0$">|; 

$key = q/M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$M$">|; 

$key = q/psi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$\psi$">|; 

$key = q/{displaystylekappa=frac{R}{C_p}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="${\displaystyle \kappa=\frac{R}{C_p} }$">|; 

$key = q/Depsf[][7cm]{zahyouslashlg2.ps};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="317" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="\Depsf[][7cm]{zahyou/lg2.ps}">|; 

$key = q/displaystyleleft[DP{}{mu}left((1-mu^2)DP{}{mu}right)+frac{1}{1-mu^2}DP[2]{}{lambda}+n(n+1)right]Y_n^m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$\displaystyle \left[
\DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right)
+ \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
+ n(n+1)
\right] Y_n^m =0$">|; 

$key = q/f_k（k=0,1,2,cdots,K-1）;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$f_k（k=0,1,2,\cdots,K-1）$">|; 

$key = q/A_{ij}^{even};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$A_{ij}^{even}$">|; 

$key = q/displaystyleint^{1}_{-1}f(mu)dmu=2sum_{j=1}^{J}f(mu_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$\displaystyle \int^{1}_{-1} f(\mu) d \mu = 2 \sum_{j=1}^{J} f(\mu_j) w_j$">|; 

$key = q/A_{i,j}+A_{i,J-j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$A_{i,j}+A_{i,J-j}$">|; 

$key = q/displaystyletilde{h}_n^m=-frac{n(n+1)}{r^2}tilde{f}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$\displaystyle \tilde{h}_n^m = -\frac{n(n+1)}{r^2} \tilde{f}_n^m$">|; 

$key = q/dot{calA}_{adv};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$ \dot{\cal A}_{adv} $">|; 

$key = q/displaystyle+frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}left(hat{R}_{ij}+S_{q,ij}right)Y_n^{m*}(lambda_i,mu_j)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
\left( \hat{R}_{ij} + S_{q,ij} \right)
Y_n^{m *} ( \lambda_i, \mu_j )
w_j$">|; 

$key = q/{calA}^{t+Deltat};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$ {\cal A}^{t+\Delta t} $">|; 

$key = q/T=bar{T}_k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$T=\bar{T}_k$">|; 

$key = q/displaystyleint_{-1}^1tilde{P}_n(mu)tilde{P}_{n'}(mu)dmu=frac{2}{2n+1}delta_{nn'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="$\displaystyle \int_{-1}^1 \tilde{P}_n(\mu) \tilde{P}_{n'}(\mu) d \mu
= \frac{2}{2n+1} \delta_{nn'}$">|; 

$key = q/C_p;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$C_p$">|; 

$key = q/{displaystylesum_{i,j}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="${\displaystyle \sum_{i,j} }$">|; 

$key = q/displaystylePhi_{1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$\displaystyle \Phi_{1}$">|; 

$key = q/h;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="$h$">|; 

$key = q/displaystylesum_{j=1}^JP_n^{m}(sinphi_j)P_{n'}^{m}(sinphi_j)w_j=delta_{nn'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="$\displaystyle \sum_{j=1}^J P_n^{m}(\sin \phi_j)
P_{n'}^{m}(\sin \phi_j) w_j
= \delta_{nn'}$">|; 

$key = q/3slash4;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$3/4$">|; 

$key = q/displaystylefrac{1}{rsqrt{1-mu^2}}DP{v_1}{lambda}+frac{1}{r}DP{}{mu}(sqrt{1-mu^2}v_2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$\displaystyle \frac{1}{r \sqrt{1-\mu^2} } \DP{v_1}{\lambda}
+ \frac{1}{r} \DP{}{\mu}( \sqrt{1-\mu^2} v_2 )$">|; 

$key = q/J=63,N=42;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="$J=63, N=42$">|; 

$key = q/kleql;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$ k \leq l$">|; 

$key = q/t+Deltat;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$t+\Delta t$">|; 

$key = q/displaystyleB=sum_{m=-M}^{M}sum_{n=|m|}^{N}left(tilde{B}_n^mexp(imlambda)right)P_n^m(sinphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$\displaystyle B= \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N}
\left( \tilde{B}_n^m \exp(im \lambda) \right)
P_n^m(\sin \phi)$">|; 

$key = q/N_{max}=M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="$N_{max}=M$">|; 

$key = q/displaystylesum_{m=-frac{I}{2}+1}^{frac{I}{2}}exp(imlambda_i)exp(-imlambda_{i'})=Idelta_{ii'};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$\displaystyle \sum_{m=-\frac{I}{2}+1}^{\frac{I}{2}}
\exp(i m \lambda_i) \exp(-i m \lambda_{i'})
= I \delta_{ii'}$">|; 

$key = q/g;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="$g$">|; 

$key = q/displaystyle-frac{n(n+1)}{r^2}Y_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$\displaystyle - \frac{n(n+1)}{r^2} Y_n^m$">|; 

$key = q/{displaymath}frac{partialD}{partialt}=frac{1}{a(1-mu^{2})}frac{partial(mbox{{sl{{p}hat{kappa}_kbar{T}_kpi+(mbox{{sl{KE})_k)+{calD}(D_k){displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="541" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="\begin{displaymath}
\frac{\partial D}{\partial t}
= \frac{1}{a(1-\mu^{2})}
\...
...ppa}_k \bar{T}_k \pi
+ (\mbox{\sl KE})_k )
+ {\cal D}(D_k)
\end{displaymath}">|; 

$key = q/lambda;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\lambda$">|; 

$key = q/Jgefrac{3N_{max}+1}{2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="$J \ge \frac{3N_{max}+1}{2}$">|; 

$key = q/displaystylesum_{j=1}^{J}sum_{i=1}^{I}P_n^m(mu_j)P_{n'}^{m'}(mu_j)exp(imlambda_i)exp(-im'lambda_{i})w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$\displaystyle \sum_{j=1}^{J}
\sum_{i=1}^{I}
P_n^m (\mu_j) P_{n'}^{m'} (\mu_j)
\exp(i m \lambda_i) \exp(-i m' \lambda_{i})
w_j$">|; 

$key = q/K=J+M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="$K=J+M$">|; 

$key = q/w_j=w(mu_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="$w_j=w(\mu_j)$">|; 

$key = q/displaystyle+frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}{calD}'_{ij}(Dvect{v})Y_n^{m*}(lambda_i,mu_j)w_j,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
{\cal D}'_{ij}(\Dvect{v})
Y_n^{m *} ( \lambda_i, \mu_j )
w_j ,$">|; 

$key = q/L=f;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="$L=f$">|; 

$key = q/displaystylesum^{I}_{n=1}expleft{imfrac{2pi(n-1)}{I}right}=frac{1-left(e^{frac{i)^I}{1-e^{frac{im2pi}{I}}}=frac{1-e^{im2pi}}{1-e^{frac{im2pi}{I}}}=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="85" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$\displaystyle \sum^{I}_{n=1} \exp \left\{ im \frac{2 \pi (n-1)}{I} \right\}
= \...
...im 2 \pi}{I} } }
= \frac{1 - e^{im 2 \pi} }
{1 - e^{ \frac{im 2 \pi}{I} } }
= 0$">|; 

$key = q/{displaystyleleft(tilde{A}^m_nexp(imlambda)right)^*=tilde{A}^{-m}_nexp(-imlambda)};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="${\displaystyle
\left(\tilde{A}^m_n \exp(im\lambda) \right)^*
= \tilde{A}^{-m}_n \exp(-im\lambda) }$">|; 

$key = q/displaystyleA(lambda,phi)equivsum_{m=-M}^{M}sum_{n=|m|}^{N}tilde{A}_n^mY_n^m(lambda,phi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$\displaystyle A(\lambda,\phi)
\equiv \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N}
\tilde{A}_n^m Y_n^m (\lambda, \phi)$">|; 

$key = q/2J-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$2J-1$">|; 

$key = q/displaystylewidetilde{left(DP{f}{mu}right)_n^m};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$\displaystyle \widetilde{\left(\DP{f}{\mu}\right)_n^m}$">|; 

$key = q/1+(2times1+1)+cdots+(2times42+1)=43^2=1849;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="$ 1+ (2 \times1 +1) + \cdots
+ (2 \times 42 +1) = 43^2 = 1849$">|; 

$key = q/displaystylesum_{k=0}^{n-1}(2k+1)(x-y)tilde{P}_k(x)tilde{P}_k(y);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$\displaystyle \sum_{k=0}^{n-1} (2k+1)(x-y)\tilde{P}_k(x)\tilde{P}_k(y)$">|; 

$key = q/1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$1$">|; 

$key = q/displaystyle(1-mu^2)DD{}{mu}tilde{P}_{n}^{m}=(n+|m|)tilde{P}_{n-1}^m-nmutilde{P}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$\displaystyle (1-\mu^2) \DD{}{\mu}\tilde{P}_{n}^{m}
= (n+\vert m\vert) \tilde{P}_{n-1}^m
- n \mu \tilde{P}_n^m$">|; 

$key = q/displaystylefrac{sigma_{k-1slash2}alpha_k+sigma_{k+1slash2}beta_k}{Deltasigma_k};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle \frac{ \sigma_{k-1/2} \alpha_k + \sigma_{k+1/2} \beta_k }
{ \Delta \sigma_k }$">|; 

$key = q/displaystyletilde{P}_n^mequivfrac{(1-mu^2)^{frac{|m|}{2}}}{2^nn!}DD[n+|m|]{}{mu}(mu^2-1)^n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$\displaystyle \tilde{P}_n^m
\equiv \frac{ (1-\mu^2)^{\frac{\vert m\vert}{2}} }{2^n n!}
\DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n$">|; 

$key = q/{displaystyleM=left[frac{I-1}{3}right]};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="${\displaystyle M= \left[ \frac{I-1}{3} \right] }$">|; 

$key = q/%latex2htmlidmarker7860setcounter{footnote}{6}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="4" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="% latex2html id marker 7860
\setcounter{footnote}{6}\fnsymbol{footnote}">|; 

$key = q/0le|m|,|m'|leM,|m|lenleN,|m'|len'leN;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="$0 \le \vert m\vert,\vert m'\vert \le M, \vert m\vert \le n \le N, \vert m'\vert \le n' \le N$">|; 

$key = q/%latex2htmlidmarker7812setcounter{footnote}{6}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="4" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="% latex2html id marker 7812
\setcounter{footnote}{6}\fnsymbol{footnote}">|; 

$key = q/P_n^m(sinphi);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$P_n^m(\sin \phi)$">|; 

$key = q/displaystyle-frac{C_{p}hat{kappa}_kT_{v,k}'}{a}frac{partialpi}{partiallambda}+F_{lambdak}cosvarphi;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$\displaystyle - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a}
\frac{\partial \pi}{\partial \lambda}
+ F_{\lambda k} \cos \varphi$">|; 

$key = q/displaystyleC_{p}left[left(frac{sigma_{k-1slash2}}{sigma_k}right)^{kappa}-1rightft(frac{sigma_{k-1slash2}}{sigma_{k-1}}right)^{kappa}right]T_{v,k-1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$\displaystyle C_{p}
\left[ \left( \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{...
...eft( \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} } \right)^{\kappa}
\right] T_{v,k-1}$">|; 

$key = q/displaystylef(mu)-L(mu)=tilde{P}_J(mu)S(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$\displaystyle f(\mu) - L(\mu) = \tilde{P}_J(\mu) S(\mu)$">|; 

$key = q/dot{calA}_{dis};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$ \dot{\cal A}_{dis} $">|; 

$key = q/{figure}center{picture{(300,170)(10,-20)put(20,10){vector(1,0){85}}put(30,0){vec0){shortstack{平方四辺形切断}}{par{picture{center{{{{figure};MSF=1.6;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="217" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="\begin{figure}\begin{center}
\begin{picture}(300,170)(10,-20)
\put(20 , 10){\v...
...20) \{ shortstack\{平方四辺形切断\}\}
\par
\end{picture}\end{center}\end{figure}">|; 

$key = q/textstyle-;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\textstyle -$">|; 

$key = q/displaystyleleft(DP{Dvect{T}}{t}right)_{NG}-underline{h}Dvect{D}+{calD}_HDvect{T},;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$\displaystyle \left( \DP{\Dvect{T}}{t} \right)_{NG}
- \underline{h} \Dvect{D}
+ {\cal D}_H \Dvect{T} ,$">|; 

$key = q/A,B,C;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$A,B,C$">|; 

$key = q/mathaccent"707Erelax{P}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$\mathaccent ''707E\relax {P}_n^m$">|; 

$key = q/displaystylefrac{1}{I}sum_{ij}left(imU_{ij}P_n^m-V_{ij}cos^2phi_jleft.DD{P_n^m}{mu}right|_jright)exp(-imlambda_i)frac{w_j}{rcos^2phi_j};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
im U_{ij} P_n^m
- V_{ij}
\cos^2 \phi...
...m}{\mu} \right\vert _j
\right)
\exp(- im \lambda_i)
\frac{w_j}{r \cos^2 \phi_j}$">|; 

$key = q/displaystyletilde{g}_n^m=imtilde{f}_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="$\displaystyle \tilde{g}_n^m = im \tilde{f}_n^m$">|; 

$key = q/%latex2htmlidmarker7343setcounter{footnote}{8}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="% latex2html id marker 7343
\setcounter{footnote}{8}\fnsymbol{footnote}">|; 

$key = q/N_{tr};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$N_{tr}$">|; 

$key = q/0le|m|len-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$0\le \vert m\vert \le n-1$">|; 

$key = q/A_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$A_n^m$">|; 

$key = q/displaystyle+frac{1}{I}sum_{i=1}^{I}sum_{j=1}^{J}V_{ij}q_{ij}(1-mu_j^2)frac{partial}{partialmu}Y_n^{m*}(lambda_i,mu_j)frac{w_j}{a(1-mu_j^{2})};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
V_{ij} q_{ij}
(1-\mu_...
...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$">|; 

$key = q/mneq0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$m \neq 0$">|; 

$key = q/displaystyle-frac{1}{Deltasigma_{k}}[dot{sigma}^{NG}_{k-1slash2}(hat{bar{T}}_{k-k})+dot{sigma}^{NG}_{k+1slash2}(bar{T}_{k}-hat{bar{T}}_{k+1slash2})];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$\displaystyle - \frac{1}{\Delta \sigma_{k}}
[ \dot{\sigma}^{NG}_{k-1/2} ( \hat{...
...{T}_{k} )
+ \dot{\sigma}^{NG}_{k+1/2} ( \bar{T}_{k}
- \hat{\bar{T}}_{k+1/2} ) ]$">|; 

$key = q/P^m_n(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$P^m_n(\mu)$">|; 

$key = q/displaystylenabla^2_Hf;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$\displaystyle \nabla^2_H f$">|; 

$key = q/{displaystyletilde{P}_{J-1}=sqrt{frac{1}{2(mbox{J}-1)+1}}P_{J-1}};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="${\displaystyle
\tilde{P}_{J-1}
=\sqrt{ \frac{1}{2(\mbox{J}-1)+1} } P_{J-1}
}$">|; 

$key = q/displaystyleleft{(1-2Deltat{calD}_H)(1-2Deltat{calD}_M)underline{I}-(Deltat)^{2}}+Dvect{G}Dvect{C}^{T})nabla^{2}_{sigma}right}overline{Dvect{D}}^{t};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$\displaystyle \left\{ ( 1-2\Delta t {\cal D}_H )( 1-2\Delta t {\cal D}_M )
\und...
...vect{G} \Dvect{C}^{T} ) \nabla^{2}_{\sigma}
\right\}
\overline{ \Dvect{D} }^{t}$">|; 

$key = q/M=32;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="$M=32$">|; 

$key = q/A_{ij};(i=1,2,cdots,I,;j=1,2,cdots,J);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="$A_{ij} \; (i=1,2,\cdots,I, \; j=1,2,\cdots,J)$">|; 

$key = q/displaystyleDeltasigma_{K+1slash2};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$\displaystyle \Delta \sigma_{K+1/2}$">|; 

$key = q/displaystyle-frac{1}{I}sum_{i=1}^Isum_{j=1}^Jf_{ij}left.DD{P_n^m}{mu}right|_jexp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$\displaystyle - \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J f_{ij}
\left. \DD{P_n^m}{\mu}\right\vert _j
\exp(-im \lambda_i) w_j$">|; 

$key = q/{displaymath}G_{k}=hat{kappa}_{k}C_{p}bar{T}_{k}{displaymath};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="\begin{displaymath}
G_{k} = \hat{\kappa}_{k} C_{p} \bar{T}_{k}
\end{displaymath}">|; 

$key = q/displaystyle-frac{1}{Deltasigma_{k}}[dot{sigma}_{k-1slash2}(hat{T^{prime}}_{k-1s})+dot{sigma}_{k+1slash2}(T^{prime}_{k}-hat{T^{prime}}_{k+1slash2})];MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$\displaystyle - \frac{1}{\Delta \sigma_{k}}
[ \dot{\sigma}_{k-1/2} ( \hat{T^{\p...
...e}_{k} )
+ \dot{\sigma}_{k+1/2} ( T^{\prime}_{k}
- \hat{T^{\prime}}_{k+1/2} ) ]$">|; 

$key = q/displaystylefrac{1}{I}sum_{ij}left(frac{im}{cos^2phi}V_{ij}P_n^m+frac{1}{r}U_{ij}left.DD{P_n^m}{mu}right|_jright)exp(-imlambda_i)w_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
\frac{im}{\cos^2 \phi} V_{ij} P_n^m
...
...} U_{ij} \left. \DD{P_n^m}{\mu} \right\vert _j
\right)
\exp(- im \lambda_i) w_j$">|; 

$key = q/displaystyle(n-|m|+1)tilde{P}_{n+1}^m-(2n+1)mutilde{P}_n^m+(n+|m|)tilde{P}_{n-1}^m=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$\displaystyle (n-\vert m\vert+1) \tilde{P}_{n+1}^m
- (2n+1) \mu \tilde{P}_n^m
+(n+\vert m\vert) \tilde{P}_{n-1}^m = 0$">|; 

$key = q/0leKleJ-1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="$0 \le K \le J-1$">|; 

$key = q/L(mu);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$L(\mu)$">|; 

$key = q/displaystyleY_n^m(lambda,phi)equivP_n^m(sinphi)exp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle Y_n^m(\lambda,\phi)
\equiv P_n^m(\sin \phi) \exp(im \lambda)$">|; 

$key = q/Ige3N_{tr}+1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$I \ge 3N_{tr}+1$">|; 

$key = q/%latex2htmlidmarker7859setcounter{footnote}{6}fnsymbol{footnote};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="4" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="% latex2html id marker 7859
\setcounter{footnote}{6}\fnsymbol{footnote}">|; 

$key = q/displaystylefrac{1}{r^2}left[DP{}{mu}left((1-mu^2)DP{}{mu}right)+frac{1}{1-mu^2}DP[2]{}{lambda}right]Y_n^m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$\displaystyle \frac{1}{r^2}
\left[ \DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right)
+ \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
\right] Y_n^m$">|; 

$key = q/displaystylezeta;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$\displaystyle \zeta$">|; 

$key = q/64times64=4096;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="$ 64 \times 64 = 4096 $">|; 

$key = q/displaystylefrac{1}{rcosphi}DP{Y_n^m}{lambda}=frac{1}{rcosphi}DP{}{lambda}left(Pnphi)exp(imlambda)right)=frac{im}{rcosphi}P_n^m(sinphi)exp(imlambda);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$\displaystyle \frac{1}{r \cos \phi} \DP{Y_n^m}{\lambda}
= \frac{1}{r \cos \phi}...
...im \lambda) \right)
= \frac{im}{r \cos \phi}
P_n^m (\sin \phi) \exp(im \lambda)$">|; 

$key = q/n+n';MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$n+n'$">|; 

$key = q/2M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="$2M$">|; 

$key = q/i=1,2,cdots,I,j=1,2,cdots,J;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$i=1,2,\cdots,I,  j=1,2,\cdots,J$">|; 

1;

