# 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| $(2N_{tr}+1) (N_{tr}+1)$ |; $key = q/f^{(n)}=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $f^{(n)}=0$ |; $key = q/Ige3N_{tr}+1,かつJge3N_{tr}+1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $I \ge 3N_{tr}+1, かつ J \ge 3N_{tr}+1$ |; $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| $\displaystyle \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \tilde{A}_n^m Y_n^m (\lambda_i,\phi_j)$ |; $key = q/JleKle2J-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $J \le K \le 2J-1$ |; $key = q/I=128;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/dot{calA}_{dif};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\dot{\cal A}_{dif}$ |; $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| $\displaystyle \vspace{4cm} \times \sum_{i=1}^I \exp(i(m'+m''-m) \lambda_i) \sum... ...^J P_{n'}^{m'}(\sin \phi_j) P_{n''}^{m''}(\sin \phi_j) P_n^{m}(\sin \phi_j) w_j$ |; $key = q/displaystyle(mbox{{sl{UA})_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (\mbox{\sl UA})_k$ |; $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| $\displaystyle - ( \zeta_k + f ) U_k - \frac{1}{2 \Delta \sigma_k} [ \dot{\sigma}_{k-1/2} ( V_{k-1} - V_k ) + \dot{\sigma}_{k+1/2} ( V_k - V_{k+1} ) ]$ |; $key = q/textstyle+;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle +$ |; $key = q/Y_n^m(lambda',phi');MSF=1.6;AAT/; $cached_env_img{$key} = q| $Y_n^m(\lambda',\phi')$ |; $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| $\displaystyle + \left\{ \alpha_k \left[ \sigma_{k-1/2} \Dvect{v}_k \cdot \nabla... ...l=k}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi ) \Delta \sigma_l \right] \right.$ |; $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| $\displaystyle - \sum_{i=1}^I \sum_{j=1}^J f_{n'}^{m'} \left. \DD{P_{n}^{m}}{\mu}\right\vert _j \exp(-im \lambda_i) P_{n'}^{m'}(\mu_j) \exp(im' \lambda_i) w_j$ |; $key = q/Dvect{D}={D_{k}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{D}=\{ D_{k} \}$ |; $key = q/displaystyle+tilde{calD}_{H,n}^mtilde{T}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \tilde{\cal D}_{H,n}^m \tilde{T}_n^m$ |; $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| $\tilde{A}_n^m \; (m=-M,-M+1, \cdots,M, \; n=\vert m\vert,\vert m\vert+1,\cdots,N(m))$ |; $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| $\displaystyle A_{ij} \equiv \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \tilde{A}_n^m Y_n^m (\lambda_i,\phi_j)$ |; $key = q/%latex2htmlidmarker7180setcounter{footnote}{5}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7180 \setcounter{footnote}{5}\fnsymbol{footnote}$ |; $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| $\begin{displaymath} \tilde{\cal D}_{M,n}^m = - K_{HD} \left[ \left( \frac{-n(n... ...\right)^{N_D/2} - \left( \frac{2}{a} \right)^{N_D} \right] . \end{displaymath}$ |; $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| $\begin{eqnarray*} h_{ij} &=& \frac{1}{r \cos^2 \phi_i} g_{ij} \\\\ ∴ h_{ij} &... ...2 \phi_j} \sum_{m,n} im \tilde{f}_n^m Y_n^m (\lambda_i, \phi_j) \end{eqnarray*}$ |; $key = q/-1leqmuleq1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $-1 \leq \mu \leq 1$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} A_{ij} Y_n^{m*} (\lambda_i,\phi_j) w_j$ |; $key = q/mu=mu_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mu=\mu_j$ |; $key = q/r;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{displaymath} \dot{\sigma}^{NG}_{k-1/2} = - \sigma_{k-1/2} \left( \DP{\p... ...um_{l=k}^{K} \Dvect{v}_{l} \cdot \nabla \pi \Delta \sigma_{l} \end{displaymath}$ |; $key = q/phi_j（j=1,2,3,cdots,J）;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\phi_j（j=1,2,3,\cdots,J）$ |; $key = q/%latex2htmlidmarker7404setcounter{footnote}{9}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7404 \setcounter{footnote}{9}\fnsymbol{footnote}$ |; $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| $\displaystyle \frac{1}{2} \int_{-1}^1 \frac{\tilde{P}_J(\mu)} {(\mu-\mu_j) \tilde{P}^{'}_J(\mu_j)} d \mu$ |; $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| $\displaystyle L(\mu) \equiv \sum_{j=1}^{J} f(\mu_j) \prod_{k=1,k \neq j}^{J} \frac{ \mu-\mu_k}{\mu_j-\mu_k}$ |; $key = q/displaystylewidetilde{left(DP{f}{lambda}right)_n^m};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \widetilde{ \left( \DP{f}{\lambda} \right)_n^m }$ |; $key = q/displaystyle(mbox{{sl{VA})_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (\mbox{\sl VA})_k$ |; $key = q/displaystylembox{div}_HDvect{v};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \mbox{div}_H \Dvect{v}$ |; $key = q/displaystyleDP{tilde{q}_n^m}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\tilde{q}_n^m}{t}$ |; $key = q/displaystyle(u,v)=left(frac{U}{cosphi},frac{V}{cosphi}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (u,v) = \left( \frac{U}{\cos \phi}, \frac{V}{\cos \phi} \right)$ |; $key = q/displaystylew_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle w_j$ |; $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| ${\displaystyle \left( \int_a^b f_k (\mu) f_{k'} (\mu) w(\mu) d \mu = \lambda_k \delta_{kk'} \right) }$ |; $key = q/P_n^m(sinphi)exp(imlambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n^m(\sin \phi) \exp(im \lambda)$ |; $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| $\begin{displaymath} \DP{\pi}{t} =　 \left( \frac{\partial \pi}{\partial t} \right)_{NG} - \Dvect{C} \cdot \Dvect{D} , \end{displaymath}$ |; $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| $\displaystyle \zeta = \frac{1}{r(1-\mu^2)} \DP{V}{\lambda} - \frac{1}{r} \DP{U}{\mu}$ |; $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| ${\displaystyle (g_{ij}=) \sum_{m,n} im \tilde{f}_n^m Y_n^m = \sum_{m,n} \tilde{g}_n^m Y_n^m }$ |; $key = q/displaystyle(n+1)left|array{{ll}tilde{P}_{n+1}(x)tilde{P}_n(x)tilde{P}_{n+1}(y)tilde{P}_n(y)array{right|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (n+1) \left\vert \begin{array}{ll} \tilde{P}_{n+1}(x) & \tilde{P}_n(x) \\\\ \tilde{P}_{n+1}(y) & \tilde{P}_n(y) \end{array}\right\vert$ |; $key = q/[a,b];MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{displaymath} \left( \DP{D}{t} \right)^{NG} = \frac{1}{a(1-\mu^{2})} \D... ...a} \sum_{k=1}^{K} W_{kl} ( T_{v,l}-T_{l} ) + {\cal D}(D_{k}) \end{displaymath}$ |; $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| $\displaystyle (1-\mu^2) \DD{}{\mu} P_n^m = (n+\vert m\vert) \sqrt{ \frac{(n-\vert m\vert)(2n+1)}{(n+\vert m\vert)(2n-1)} } P_{n-1}^m - n \mu P_n^m$ |; $key = q/Jslash2;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/Y_n^m(phi,lambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $Y_n^m(\phi,\lambda)$ |; $key = q/displaystyleDP{Dvect{T}}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\Dvect{T}}{t}$ |; $key = q/|m'-m|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\vert m'-m\vert$ |; $key = q/0lelambda<2pi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le \lambda < 2 \pi$ |; $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| $\displaystyle \sum_{m=-M}^M \sum_{n=\vert m\vert}^N \tilde{A}_n^m P_n^{m}(\sin \phi_j) \exp(im \lambda_i)$ |; $key = q/{displaymath}nabla^{2}_{sigma}=-frac{n(n+1)}{a^{2}}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \nabla^{2}_{\sigma} = - \frac{n(n+1)}{a^{2}} \end{displaymath}$ |; $key = q/max[N];MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/I=2M+1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleU;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle U$ |; $key = q/tilde{B}_n^m,tilde{C}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{B}_n^m,\tilde{C}_n^m$ |; $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| ${\displaystyle \frac{1}{r} \DP{f}{\phi} \left( = \frac{\cos \phi}{r} \DP{f}{\mu} \right) }$ |; $key = q/m,m',n,n';MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/I=2M;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \sum_{n=0}^{J-1} P_n (\mu_j) P_n (\mu_{j'}) = \frac{1}{w_j} \delta_{jj'}$ |; $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| $\displaystyle \frac{1}{r (1-\mu^2)} \DP{V}{\lambda} - \frac{1}{r} \DP{U}{\mu}$ |; $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| $\nabla_H^2 Y^m_n(\lambda, \phi) =- \frac{n(n+1)}{r^2} Y^m_n(\lambda,\phi)$ |; $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| $\displaystyle I \sum_{j=1}^{J} P_n^m (\mu_j) P_{n'}^{m} (\mu_j) w_j$ |; $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| $\displaystyle (n-\vert m\vert+1) \sqrt{ \frac{1}{2n+3} \frac{(n+1+\vert m\vert)... ...) \sqrt{ \frac{1}{2n+1} \frac{(n+\vert m\vert)!}{(n-\vert m\vert)!} } \mu P_n^m$ |; $key = q/0lemleM,|m|lenleN;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le m \le M, \vert m\vert \le n \le N$ |; $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| $\begin{displaymath} \delta_{t} \Dvect{T} = \left( \DP{\Dvect{T}}{t} \right)_{N... ... ( \Dvect{T}^{t-\Delta t} + 2 \Delta t \delta_{t} \Dvect{T} ) \end{displaymath}$ |; $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| $\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda = \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$ |; $key = q/{displaymath}Zequiv-sum_{k=1}^{K}Dvect{v}_kcdotnablapi.{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} Z \equiv - \sum_{k=1}^{K} \Dvect{v}_k \cdot \nabla \pi . \end{displaymath}$ |; $key = q/tilde{P}_J;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_J$ |; $key = q/I,J;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \left[ \frac{1}{\cos \phi} \DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right) + \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda} + n(n+1) \right] Y_n^m =0$ |; $key = q/n=0,1,cdots,n-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $n=0,1,\cdots,n-1$ |; $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| $\begin{displaymath} R_k = q_k D_k - \frac{1}{2 \Delta \sigma_k} [ \dot{\sigm... ... ( q_{k-1} - q_k ) + \dot{\sigma}_{k+1/2} ( q_k - q_{k+1} ) ] \end{displaymath}$ |; $key = q/J=63,N=21;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{eqnarray*} q \equiv (1-\mu^2) \DP{f}{\mu} = (1-\mu^2) p \end{eqnarray*}$ |; $key = q/M,N;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle + \left. \beta_k \left[ \sigma_{k+1/2} \Dvect{v}_k \cdot \nabla \... ...\nabla \pi ) \Delta \sigma_l \right] \right\} \frac{1}{\Delta \sigma_k} T_{v,k}$ |; $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| $\displaystyle w_j = \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 }$ |; $key = q/0le|m'|len';MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le \vert m'\vert \le n'$ |; $key = q/zeta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\zeta$ |; $key = q/{displaystylesin^{-1}left(cosfrac{j-1slash2}{J}piright)};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\displaystyle \sin^{-1} \left( \cos \frac{j-1/2}{J}\pi \right) }$ |; $key = q/tilde{P}_n^m(mu)(n=|m|,|m+1,cdots);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_n^m(\mu) (n=\vert m\vert,\vert m+1, \cdots)$ |; $key = q/{displaystylesum_{i,j}Y_n^{m*}(lambda_i,phi_j)w_j};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum_{i,j} Y_n^{m*}(\lambda_i,\phi_j)w_j$ |; $key = q/Leftrightarrow;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Leftrightarrow$ |; $key = q/displaystyleleft(frac{sigma_{k-1slash2}}{sigma_k}right)^{kappa}-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( \frac{ \sigma_{k-1/2} } { \sigma_k } \right)^{\kappa} -1$ |; $key = q/frac{I}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{I}{2}$ |; $key = q/P_n^m(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n^m(\mu)$ |; $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| $\displaystyle (n+1) \tilde{P}_{n+1}(\mu) = (2n+1) \mu \tilde{P}_n(\mu) - n \tilde{P}_{n-1}(\mu) (n=0,1,2,\cdots)$ |; $key = q/m=-M,cdots,M,n=|m|,cdots,N(m);MSF=1.6;AAT/; $cached_env_img{$key} = q| $m=-M,\cdots,M, n=\vert m\vert,\cdots,N(m)$ |; $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| $\displaystyle C= \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \left( \tilde{C}_n^m \exp(im \lambda) \right) P_n^m(\sin \phi)$ |; $key = q/displaystyleint_{-1}^{1}f(mu)dmu;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \int_{-1}^{1} f(\mu) d \mu$ |; $key = q/displaystyle+frac{Q_k}{C_{p}}+{calD}(T_k)+{calD}'(Dvect{v});MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \frac{Q_k}{C_{p}} + {\cal D}(T_k) + {\cal D}'(\Dvect{v})$ |; $key = q/tilde{P}_0=1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_0 = 1$ |; $key = q/displaystylefrac{(2J-1)(1-sin^2phi_j)}{(JP_{J-1}(sinphi_j))^2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{(2J-1)(1-\sin^2 \phi_j)} {(J P_{J-1}(\sin \phi_j))^2 }$ |; $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| $\displaystyle \sum_{i=0}^{I-1} \exp(i m \lambda_i) \exp(-i m' \lambda_i) = I \delta_{mm'}$ |; $key = q/2J-1gemax[n+n'+n'']=3max[N];MSF=1.6;AAT/; $cached_env_img{$key} = q| $2J-1 \ge max[n+n'+n''] = 3 max[N]$ |; $key = q/displaystylefrac{1}{rcosphi}DP{v}{lambda}-frac{1}{rcosphi}DP{}{phi}(ucosphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{r \cos \phi} \DP{v}{\lambda} - \frac{1}{r \cos \phi} \DP{}{\phi}(u \cos \phi)$ |; $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| $\displaystyle \sigma_k = \left\{ \frac{1}{1+\kappa} \left( \frac{ \sigma^{\kapp... ...a +1}_{k+1/2} } { \sigma_{k-1/2} - \sigma_{k+1/2} } \right) \right\}^{1/\kappa}$ |; $key = q/hat{calA}^{t+Deltat};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\hat{\cal A}^{t+\Delta t}$ |; $key = q/Deltat;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Delta t$ |; $key = q/B;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/U_{ij},V_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $U_{ij},V_{ij}$ |; $key = q/displaystyleD;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle D$ |; $key = q/displaystylePhi_{s}+C_{p}(sigma_{1}^{-kappa}-1)T_{v,1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Phi_{s} + C_{p} ( \sigma_{1}^{-\kappa} - 1 ) T_{v,1}$ |; $key = q/tilde{P}_J(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_J(\mu)$ |; $key = q/dot{calA}_{phy};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\dot{\cal A}_{phy}$ |; $key = q/n+n'+n'';MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/chi_n^m,psi_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\chi_n^m,\psi_n^m$ |; $key = q/A;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleg(lambda)=sum_{m=-M}^{m=M}g_mexp(imlambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle g(\lambda) = \sum_{m=-M}^{m=M} g_m \exp( i m \lambda )$ |; $key = q/{calA};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal A}$ |; $key = q/displaystylehat{T}_{k-1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{T}_{k-1/2}$ |; $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_j1lemleM,mlenleNarray{right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{A}_n^m = \left\{ \begin{array}{ll} {\displaystyle \frac{1}... ...}(\lambda_i,\phi_j) w_j & 1 \le m \le M, m \le n \le N \end{array}\right.$ |; $key = q/f_J(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $f_J(\mu)$ |; $key = q/x=-1,alpha,1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $x=-1,\alpha,1$ |; $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| $\displaystyle \tilde{A}_n^m = \frac{1}{2} \int_{-1}^1 A(\mu) P_n^m(\mu) d \mu$ |; $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| $\displaystyle C_{p} \alpha_k T_{v,k} + C_{p} \beta_{k-1} T_{v,k-1}$ |; $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| $\begin{figure}\begin{center} \begin{picture}(300,100)(10,-10) \put(20 , 10){\v... ...00,-20) \{ shortstack\{三角形切断\}\} \par \end{picture}\end{center}\end{figure}$ |; $key = q/P_n^mP_{n'}^{m'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n^m P_{n'}^{m'}$ |; $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| $\displaystyle - \Delta t \nabla^{2}_{\sigma} \left\{ ( 1-2\Delta t {\cal D}_H )... ...}^{t-\Delta t} + \Delta t \left( \DP{\Dvect{T}}{t} \right)_{NG} \right] \right.$ |; $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| $\begin{picture}(300,150)(50,10) \put(50,20){\line(1,0){220}} \put(50,40){\line... ...){\shortstack{$\sigma=1$}} \put(280,136){\shortstack{$\sigma=0$}} \end{picture}$ |; $key = q/Y_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{eqnarray*} & & \sum_{i} \sum_{j} f_{n'}^{m'} P_{n}^{m}(\mu_j) \exp(im ... ...}^{m} P_{n}^{m}(\mu) \DD{P_{n'}^m}{\mu} d \mu \delta_{m m'} \end{eqnarray*}$ |; $key = q/mu;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mu$ |; $key = q/displaystyleint_{-1}^1P_n^m(mu)P_{n'}^m(mu)dmu=2delta_{nn'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \int_{-1}^1 P_n^m(\mu) P_{n'}^m(\mu) d \mu = 2 \delta_{nn'}$ |; $key = q/displaystylesigma_{k-1slash2}-sigma_{k+1slash2}(1lekleK);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sigma_{k-1/2} - \sigma_{k+1/2} ( 1 \le k \le K )$ |; $key = q/2Deltat;MSF=1.6;AAT/; $cached_env_img{$key} = q| $2 \Delta t$ |; $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| $\begin{displaymath} \delta_{t} \pi = \left( \DP{\pi}{t} \right)_{NG} - \Dvect{C} \cdot \overline{ \Dvect{D} }^{t} \end{displaymath}$ |; $key = q/{displaystyleint_{-1}^1fdmu};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\int_{-1}^1 f d \mu$ |; $key = q/0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{ {\displaystyle \sum_{k=0}^{J-1} (2k+1) \tilde{P}_k(\mu) \tilde{P}_k(\mu_j) } } {J \tilde{P}_{J-1}(\mu_j)}$ |; $key = q/P_{J};MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_{J}$ |; $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| $\displaystyle \hspace*{2.5cm} - \sqrt{ \frac{(2n+1)(2n+3)}{(n-\vert m\vert+1)(n... ...+1)} } \sqrt{ \frac{(n-\vert m\vert)(n+\vert m\vert)}{(2n+1)(2n-1)} } P_{n-1}^m$ |; $key = q/displaystylefrac{1}{rcos^2phi}DP{V}{lambda}-frac{1}{rcosphi}DP{U}{phi};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{r \cos^2 \phi} \DP{V}{\lambda} - \frac{1}{r \cos \phi} \DP{U}{\phi}$ |; $key = q/L;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/K=2N_{tr},J=M=N_{tr};MSF=1.6;AAT/; $cached_env_img{$key} = q| $K=2N_{tr}, J=M=N_{tr}$ |; $key = q/displaystyletilde{D}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{D}_n^m$ |; $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| $\displaystyle \alpha_k \left[ 1- \left( \frac{ \sigma_k }{ \sigma_{k-1} } \right)^{\kappa} \right]^{-1}$ |; $key = q/N=64;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{displaymath} \delta_{t} {\cal A} \equiv \frac{1}{2 \Delta t} \left( {\cal A}^{t+\Delta t} - {\cal A}^{t-\Delta t} \right) \end{displaymath}$ |; $key = q/Jge65;MSF=1.6;AAT/; $cached_env_img{$key} = q| $J \ge 65$ |; $key = q/{displaymath}C_{k}=Deltasigma_{k}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} C_{k} = \Delta \sigma_{k} \end{displaymath}$ |; $key = q/sigma=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma=0$ |; $key = q/f''=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/max[n+n'];MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleDeltasigma_{1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta \sigma_{1/2}$ |; $key = q/displaystyleDP{tilde{pi}_n^m}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\tilde{\pi}_n^m}{t}$ |; $key = q/{P_n^m|n=|m|,|m+1|,cdots};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ P_n^m \vert n=\vert m\vert,\vert m+1\vert,\cdots \}$ |; $key = q/D;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylefrac{1}{Jtilde{P}_{J-1}(mu_j)tilde{P}^{'}_{J}(mu_j)};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{J \tilde{P}_{J-1}(\mu_j) \tilde{P}^{'}_{J} (\mu_j)}$ |; $key = q/M=N_{tr},N(m)=N_{tr}+|m|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $M= N_{tr}, N(m) =N_{tr}+\vert m\vert$ |; $key = q/A(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $A(\mu)$ |; $key = q/f(lambda,phi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $f(\lambda,\phi)$ |; $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| $\displaystyle \sum_{m,n} \left( \frac{1}{r} im \tilde{\psi}_n^m P_n^m (\sin \ph... ...cos^2 \phi_j \left. \DD{P_n^m}{\mu} \right\vert _{j} \right) \exp(im \lambda_i)$ |; $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| $\displaystyle w_j = \frac{1}{2} \int_{-1}^{1} \frac{P_J(\mu)}{(\mu-\mu_j) P^{'}_J (\mu_i)}d \mu = \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 } .$ |; $key = q/R;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle P_n^m(\mu)\equiv \sqrt{\frac{(2n+1)(n-\vert m\vert)!}{(n+\vert m\... ...mu^2)^{\frac{\vert m\vert}{2}} }{2^n n!} \DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n$ |; $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| $\displaystyle \sum_{n=1}^{I} \left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1} =0 (m \neq 0)$ |; $key = q/displaystyletilde{g}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{g}_n^m$ |; $key = q/P_J(sinphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_J(\sin \phi)$ |; $key = q/displaystyle+hat{kappa}_kDvect{v}_kcdotnablapiT_{v,k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \hat{\kappa}_k \Dvect{v}_k \cdot \nabla \pi T_{v,k}$ |; $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| ${\displaystyle w_j = \frac{1}{2} \int_{-1}^1 \frac{\tilde{P}_J(\mu)} {(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu }$ |; $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| $\displaystyle \frac{I}{2} \int_{-1}^1 P_n^m (\mu) P_{n'}^{m} (\mu) d\mu$ |; $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| $\displaystyle \int_{-1}^1 Y_n^m(\lambda, \phi) Y_{n'}^{m'*} (\lambda, \phi) d (\sin \phi) d \lambda = 4 \pi \delta_{mm'} \delta_{nn'}$ |; $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| $\begin{eqnarray*} p_n^m &=& \frac{1}{I} \sum_{i,j} p_{ij} Y_n^{m*} w_j \\\\ &=&... ...eft. \DD{P_n^m}{\mu}\right\vert _j \exp(-im \lambda_i) w_j \\\\ \end{eqnarray*}$ |; $key = q/-1|; $key = q/displaystylea_kT_k+b_{k-1}T_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle a_k T_k + b_{k-1} T_{k-1}$ |; $key = q/displaystyleJtilde{P}_J(mu)tilde{P}_{J-1}(mu_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle J \tilde{P}_J (\mu) \tilde{P}_{J-1} (\mu_j)$ |; $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| $\displaystyle \sum_{m,n} \left( \frac{1}{r} im \tilde{\psi}_n^m P_n^m (\sin \ph... ...de{\chi}_n^m \left. \DD{P_n^m}{\mu} \right\vert _{j} \right) \exp(im \lambda_i)$ |; $key = q/x=-1,1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{n(n+1)}{a^{2}} ( \Phi_n^m + C_{p} \hat{\kappa}_k \bar{T}_k \pi_n^m ) + \tilde{\cal D}_{M,n}^m \tilde{D}_n^m ,$ |; $key = q/displaystyleleft(DP{f}{mu}right)_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left(\DP{f}{\mu}\right)_{ij}$ |; $key = q/{P_n^m}(n=|m|,|m|+1,|m|+2,cdots,N);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ P_n^m \} (n=\vert m\vert,\vert m\vert+1,\vert m\vert+2,\cdots,N)$ |; $key = q/A_{i,j}-A_{i,J-j};MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{i,j}-A_{i,J-j}$ |; $key = q/displaystylefrac{1}{rcos^2phi}DP{U}{lambda}+frac{1}{rcosphi}DP{V}{phi};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{r \cos^2 \phi} \DP{U}{\lambda} + \frac{1}{r \cos \phi} \DP{V}{\phi}$ |; $key = q/P_n^mP_{n'}^{m};MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n^m P_{n'}^{m}$ |; $key = q/chi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\chi$ |; $key = q/bar{calA};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\bar{\cal A}$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} im (\mbox{\sl UA})_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/{displaystylesum_{i=1}^{I}sum_{j=1}^{J}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum_{i=1}^{I} \sum_{j=1}^{J}$ |; $key = q/displaystylenabla_H^2chi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \nabla_H^2 \chi$ |; $key = q/P_n^0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylefrac{1}{rcosphi}DP{u}{lambda}+frac{1}{rcosphi}DP{}{phi}(vcosphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{r \cos \phi} \DP{u}{\lambda} + \frac{1}{r \cos \phi} \DP{}{\phi} (v \cos \phi)$ |; $key = q/1leileI,1lejleJ;MSF=1.6;AAT/; $cached_env_img{$key} = q| $1 \le i \le I,1 \le j \le J$ |; $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| $\begin{displaymath} \frac{\partial \pi}{\partial t} = - \sum_{k=1}^{K} ( D_k + \Dvect{v}_k \cdot \nabla \pi ) \Delta \sigma_k \end{displaymath}$ |; $key = q/I=64,J=65;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{1}{I} \sum_{ij} \left( im V_{ij} P_n^m + U_{ij} \cos^2 \phi... ...m}{\mu} \right\vert _j \right) \exp(- im \lambda_i) \frac{w_j}{r \cos^2 \phi_j}$ |; $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| $\displaystyle \left[ \DP{}{\lambda} \left( \sum_{m,n} \tilde{f}_n^m Y_n^m (\lambda, \phi) \right) \right]_{ij}$ |; $key = q/tilde{P}_J(mu)S(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_J(\mu) S(\mu)$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J A_{ij} P_n^{m}(\sin \phi_j) \exp(-im \lambda_i) w_j$ |; $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| $\overline{P_n^1} = P_n^1/\sqrt{2}, \overline{P_n^2} = P_n^2/\sqrt{2}$ |; $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| $\displaystyle \tilde{A}_n^m = \frac{1}{4 \pi} \int_{-1}^1 d (\sin \phi) \int_0^{2 \pi} d \lambda A(\lambda, \phi) Y_n^{m*}(\lambda, \phi)$ |; $key = q/displaystyle+hat{kappa}_{k}T_{v,k}Dvect{v}_{k}cdotnablapi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \hat{\kappa}_{k} T_{v,k} \Dvect{v}_{k} \cdot \nabla \pi$ |; $key = q/displaystyletilde{h}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{h}_n^m$ |; $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| $\displaystyle \frac{1}{I} \sum_{n=1}^{I} \sum_{m=-M}^{M} g_m \exp(i m \lambda_n)$ |; $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| $\displaystyle A(\lambda, \phi) = \sum_{m=0}^{\infty} \sum_{n=\vert m\vert}^{\infty} \tilde{A}_n^m Y_n^m(\lambda,\phi) ,$ |; $key = q/M=21,N=42;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{displaymath} \dot{\sigma}_{k-1/2} = - \sigma_{k-1/2} \frac{\partial \pi... ...}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi ) \Delta \sigma_l \end{displaymath}$ |; $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| $\begin{eqnarray*} & & \sum_{i=1}^I \sum_{j=1}^J f_{n'}^{m'} (1-\mu_j^2) P_{n}... ...xp(-im \lambda_i) P_{n'}^{m'}(\mu_j) \exp(im' \lambda_i) w_j \end{eqnarray*}$ |; $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| $\begin{displaymath} W_{kl} = C_{p} \alpha_{l} \delta_{k \geq l} + C_{p} \beta_{l} \delta_{k-1 \geq l} \end{displaymath}$ |; $key = q/|m|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\vert m\vert$ |; $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| $\displaystyle \frac{1}{r^2} \left[ \frac{1}{\cos \phi} \DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right) + \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda} \right] Y_n^m$ |; $key = q/f=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/sinphi=mu;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sin \phi = \mu$ |; $key = q/{eqnarraystar}pequivDP{f}{mu}{eqnarraystar};MSF=1.6;TAGS=R;AAT/; $cached_env_img{$key} = q| $\begin{eqnarray*} p \equiv \DP{f}{\mu} \end{eqnarray*}$ |; $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| $\begin{displaymath} \frac{\partial \zeta_k}{\partial t} = \frac{1}{a(1-\mu^{2... ...partial (\mbox{\sl UA})_k}{\partial \mu} + {\cal D}(\zeta_k) \end{displaymath}$ |; $key = q/displaystylefrac{1-mu_j^2}{(Jtilde{P}_{J-1}(mu_j))^2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1-\mu_j^2}{(J \tilde{P}_{J-1}(\mu_j))^2 }$ |; $key = q/displaystyleint_{-1}^1L(mu)dmu;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \int_{-1}^1 L(\mu) d \mu$ |; $key = q/I;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{figure}\begin{center} \begin{picture}(300,170)(10,-20) \put(20 , 10){\v... ...00,-20) \{ shortstack\{五角形切断\}\} \par \end{picture}\end{center}\end{figure}$ |; $key = q/displaystyle(mbox{rot}Dvect{v})_r;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (\mbox{rot} \Dvect{v})_r$ |; $key = q/{displaymath}dot{sigma}_{1slash2}=dot{sigma}_{K+1slash2}=0{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \dot{\sigma}_{1/2} = \dot{\sigma}_{K+1/2} = 0 \end{displaymath}$ |; $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| $\begin{eqnarray*} q_{ij} = (1-\mu_j^2) \sum_{m,n} \tilde{f}_n^m \left. \DD{P_n^m}{\mu} \right\vert _j \exp(im \lambda_i) \end{eqnarray*}$ |; $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| $\begin{eqnarray*} h \equiv \frac{g}{r \cos^2 \phi} = \frac{1}{r \cos^2 \phi} \... ...left[ = \DP{}{x} \left( \frac{f}{\cos \phi} \right) \right] \end{eqnarray*}$ |; $key = q/displaystyleDP{tilde{zeta}_n^m}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\tilde{\zeta}_n^m}{t}$ |; $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| $\nabla_H^2 \equiv \frac{1}{r^2} \left[ \DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right) + \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda} \right]$ |; $key = q/displaystyleint_{-1}^1f(mu)P_n(mu)dmu=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \int_{-1}^1 f(\mu) P_n(\mu) d \mu = 0$ |; $key = q/I=128,J=65;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/K;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/M,N(m);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylehat{H}_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{H}_k$ |; $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| $\displaystyle A(\mu) = \sum_{n=\vert m\vert}^{\infty} \tilde{A}_n^m \tilde{P}_n^m(\mu),$ |; $key = q/{f_k(mu)}(k=0,1,2,cdots);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ f_k(\mu) \}(k=0,1,2,\cdots)$ |; $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| $\displaystyle \sum_{j=0}^{J-1} f_k (\mu_j) f_{k'} (\mu_{j}) w_j = \lambda_k \delta_{kk'}$ |; $key = q/P_J(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_J (\mu)$ |; $key = q/Idelta_{mm'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $I \delta_{mm'}$ |; $key = q/{displaymath}S_{kl}=sigma_{k-1slash2}Deltasigma_{l}-Deltasigma_{l}delta_{kleql}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} S_{kl} = \sigma_{k-1/2} \Delta \sigma_{l} - \Delta \sigma_{l} \delta_{k \leq l } \end{displaymath}$ |; $key = q/n=0,1,2,cdots,N|; $key = q/S(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $S(\mu)$ |; $key = q/128times64=8192;MSF=1.6;AAT/; $cached_env_img{$key} = q| $128 \times 64 = 8192$ |; $key = q/-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| $\begin{displaymath} \frac{\partial \Dvect{D}}{\partial t} = \left( \DP{\Dvect... ...ine{W} \Dvect{T} + \Dvect{G} \pi ) + {\cal D}_M \Dvect{D} , \end{displaymath}$ |; $key = q/A_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{ij}$ |; $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| $\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} \left( \hat{H}_{ij} + \frac{Q_{ij}}{C_{p}} \right) Y_n^{m *} ( \lambda_i, \mu_j ) w_j$ |; $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| $\begin{eqnarray*} & & \sum_{i} \sum_{j} f_{n'}^{m'} P_{n}^{m}(\mu_j) \exp(im ... .... \DD{P_{n}^{m}}{\mu}\right\vert _j \exp(-im \lambda_i) w_j \end{eqnarray*}$ |; $key = q/f-L;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/x;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/B,C;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{eqnarray*} h_{ij} = \sum_{m,n} \left. \tilde{h}_n^m Y_n^m \right\vert _{ij} \\\\ \end{eqnarray*}$ |; $key = q/h=nabla_H^2f;MSF=1.6;AAT/; $cached_env_img{$key} = q| $h = \nabla_H^2 f$ |; $key = q/tilde{P}_J(mu_j)=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_J(\mu_j)=0$ |; $key = q/-1lemule1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $-1 \le \mu \le 1$ |; $key = q/displaystyleoverline{calA}^{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \overline{\cal A}^{t}$ |; $key = q/NG;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle D = \frac{1}{r(1-\mu^2)} \DP{U}{\lambda} + \frac{1}{r} \DP{V}{\mu}$ |; $key = q/U,V;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/J;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/j;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle2sum_{j=1}^{J}f(mu_j)w_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 2 \sum_{j=1}^{J} f(\mu_j) w_j$ |; $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| $\displaystyle V = \frac{1}{r} \DP{\psi}{\lambda} + \frac{1-\mu^2}{r} \DP{\chi}{\mu}$ |; $key = q/displaystylefrac{partialT_k}{partialt};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{\partial T_k}{\partial t}$ |; $key = q/m=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{Y_n^m(lambda',phi')|m=-n,-n+1,cdots,n};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ Y_n^m(\lambda', \phi') \vert m=-n,-n+1, \cdots, n \}$ |; $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| $\displaystyle g_0 + \sum_{m=-M, m\neq0}^{M} \frac{g_m}{I} \sum_{n=1}^{I} \left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1}$ |; $key = q/k;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylebeta_kleft[left(frac{sigma_k}{sigma_{k+1}}right)^{kappa}-1right]^{-1}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \beta_k \left[ \left( \frac{ \sigma_k }{ \sigma_{k+1} } \right)^{\kappa} - 1 \right]^{-1} .$ |; $key = q/[];MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylenabla_H^2psi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \nabla_H^2 \psi$ |; $key = q/displaystylefrac{1}{I}sum_{n=1}^{I}g(lambda_n);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$ |; $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| $\nabla^{'2}_H Y_n^m = -\frac{n(n+1)}{r^2} Y_n^m$ |; $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| $\displaystyle \frac{1}{I} \sum_{i,j} im \left( \frac{f}{r \cos^2 \phi} \right)_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$ |; $key = q/displaystyle-sum_{k=1}^{K}(tilde{D}_n^m)_kDeltasigma_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \sum_{k=1}^{K} (\tilde{D}_n^m)_k \Delta \sigma_k$ |; $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| $\displaystyle - \frac{1-\mu^2}{r} \DP{\psi}{\mu} + \frac{1}{r} \DP{\chi}{\lambda}$ |; $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| $\displaystyle \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} \tilde{P}_n^m \right\} + \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\} \tilde{P}_n^m = 0$ |; $key = q/Ige2M+1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $I \ge 2M+1$ |; $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| $\displaystyle I \sum_{j=1}^{J} P_n^m (\mu_j) P_{n'}^{m'} (\mu_j) w_j \delta_{mm'}$ |; $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| $\displaystyle p_{ij} = \sum_{m,n} \tilde{f}_n^m \left. \DD{P_n^m}{\mu} \right\vert _j \exp(im \lambda_i)$ |; $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| ${\displaystyle w_j \equiv \frac{(2J-1)(1-\sin^2 \phi_j)} {(J P_{J-1}(\sin \phi_j))^2 } }$ |; $key = q/-frac{n(n+1)}{r^2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $- \frac{n(n+1)}{r^2}$ |; $key = q/displaystyleint_{-1}^1P_n(mu)P_n'(mu)dmu=2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \int_{-1}^1 P_n(\mu) P_n'(\mu) d \mu = 2$ |; $key = q/displaystyleU_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle U_{ij}$ |; $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| $\displaystyle \frac{1}{I} \sum_{m'=-M}^{M} \sum_{n'=\vert m'\vert}^N \tilde{A}_... ...i(m'-m) \lambda) \sum_{j=1}^J P_n^{m}(\sin \phi_j) P_{n'}^{m'}(\sin \phi_j) w_j$ |; $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| $\begin{displaymath} \left( \DP{\pi}{t} \right)^{NG} = - \sum_{k=1}^{K} \Dvect{... ...k} \cdot \nabla \pi \Delta \sigma_{k} \nonumber \\\\ = Z_{k} \end{displaymath}$ |; $key = q/M=21;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleb_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle b_k$ |; $key = q/displaystylesum_{n=1}^{I}exp(imlambda_n)=left{array{{ll}I(m=0)0(0<|m|array{right.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sum_{n=1}^{I} \exp(i m \lambda_n) = \left\{ \begin{array}{ll} I & (m=0) \\\\ 0 & (0 < \vert m\vert < I) \\\\ \end{array}\right.$ |; $key = q/n-m;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{1}{r \cos \phi} \DP{v_1}{\lambda} + \frac{1}{r \cos \phi} \DP{}{\phi} (v_2 \cos \phi)$ |; $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| $\displaystyle \frac{1}{I} \sum_{i,j} im f_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$ |; $key = q/{Y_n^m|m=0,1,2,cdots,n=|m|,|m+1|,cdots};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ Y_n^m \vert m=0,1,2,\cdots, n=\vert m\vert,\vert m+1\vert,\cdots \}$ |; $key = q/N;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle{calA}^{t-Deltat}+delta_{t}{calA}Deltat,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle {\cal A}^{t-\Delta t} + \delta_{t} {\cal A} \Delta t ,$ |; $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| $\begin{displaymath} (\mbox{\sl KE})_k = \frac{U^{2}_k+V^{2}_k}{2(1-\mu^{2}) } \end{displaymath}$ |; $key = q/-1|; $key = q/Dvect{T}={T_{k}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{T}=\{ T_{k} \}$ |; $key = q/mu_j,mu_{j'}(1lej,j'leJ);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mu_j, \mu_{j'}(1 \le j,j' \le J)$ |; $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| $\displaystyle \sum_{m,n} \left( - \frac{1}{r} \tilde{\psi}_n^m \cos^2 \phi_j \l... ...+ \frac{1}{r} im \tilde{\chi}_n^m P_n^m(\sin \phi_j) \right) \exp(im \lambda_i)$ |; $key = q/A_{ij}^{odd};MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{ij}^{odd}$ |; $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| $\displaystyle \sum_{j=1}^{J} \sum_{i=1}^{I} P_n^m (\mu_j) P_{n'}^{m'} (\mu_j) \exp(i m \lambda_i) \exp(-i m' \lambda_i) w_j = I \delta_{nn'} \delta_{mm'}$ |; $key = q/sigma=1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma=1$ |; $key = q/J=K=M=N_{tr};MSF=1.6;AAT/; $cached_env_img{$key} = q| $J=K=M =N_{tr}$ |; $key = q/displaystylePhi_k-Phi_{k-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Phi_k - \Phi_{k-1}$ |; $key = q/m,n;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{P_n}(n=0,1,2,cdots,J-1);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ P_n \}(n=0,1,2,\cdots,J-1)$ |; $key = q/tilde{B}_n^m=tilde{B}_n^{m*},{itetc.};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{B}_n^m = \tilde{B}_n^{m*}, {\it etc.}$ |; $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| $\begin{eqnarray*} \tilde{f}_{n'}^{m'} = -\frac{n(n+1)}{r^2} \tilde{h}_{n'}^{m'} \end{eqnarray*}$ |; $key = q/{displaymath}underline{h}=underline{Q}underline{S}-underline{R}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} \underline{h} = \underline{Q}\underline{S} - \underline{R} \end{displaymath}$ |; $key = q/textstyleequiv;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle \equiv$ |; $key = q/(2times21+1)times(21+1)=946;MSF=1.6;AAT/; $cached_env_img{$key} = q| $(2 \times 21 +1) \times (21+1)= 946$ |; $key = q/P_J;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{n(n+1)}{a^{2}} \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} (\mbox{\sl KE})_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) w_j$ |; $key = q/f(mu)-L(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $f(\mu)-L(\mu)$ |; $key = q/displaystylewidetilde{(BC)_n^m};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \widetilde{(BC)_n^m}$ |; $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| $\begin{displaymath} Q_{kl} = \frac{1}{\Delta \sigma_{k}} ( \hat{\bar{T}}_{k-1... ...{k}} ( \bar{T}_{k} - \hat{\bar{T}}_{k+1/2} ) \delta_{k+1=l} \end{displaymath}$ |; $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| $\displaystyle \sum_{m=-M}^M \sum_{n=\vert m\vert}^N im \tilde{f}_n^m Y_n^m (\lambda_i,\phi_j)$ |; $key = q/{displaystylex_j=cosfrac{j-1slash2}{n}pi};MSF=1.6;AAT/; $cached_env_img{$key} = q| $x_j=\cos \frac{j-1/2}{n}\pi$ |; $key = q/displaystyleDP{tilde{T}_n^m}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\tilde{T}_n^m}{t}$ |; $key = q/f'(alpha)=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $f'(\alpha)=0$ |; $key = q/{displaymath}T_{v,k}=T_k-bar{T}_k{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} T_{v,k} = T_k - \bar{T}_k \end{displaymath}$ |; $key = q/C;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \vspace{4cm} \times \left( \sum_{m''=-M}^{M} \sum_{n''=\vert m''\... ...P_{n''}^{m''}(\sin \phi_j) \right) P_n^{m}(\sin \phi_j) \exp(-im \lambda_i) w_j$ |; $key = q/%latex2htmlidmarker7274setcounter{footnote}{8}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7274 \setcounter{footnote}{8}\fnsymbol{footnote}$ |; $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| $\displaystyle \frac{1}{r \cos \phi} \DP{v_2}{\lambda} - \frac{1}{r \cos \phi} \DP{}{\phi}(v_1 \cos \phi)$ |; $key = q/P_n(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n(\mu)$ |; $key = q/(n,m);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyletilde{calD}_{M,n}^mtilde{zeta}_n^m,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{\cal D}_{M,n}^m \tilde{\zeta}_n^m ,$ |; $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| $\displaystyle \widetilde{ \left[ \DP{}{\lambda} \left( \frac{f}{r \cos^2 \phi} ... ...ht) \right]_n^m } = im \widetilde{ \left( \frac{f}{r \cos^2 \phi} \right)_n^m }$ |; $key = q/n=|m|,|m|+1,cdots,|m|+J-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $n=\vert m\vert, \vert m\vert+1, \cdots, \vert m\vert+J-1$ |; $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| $\displaystyle \sum_{m=-M}^M \sum_{n=\vert m\vert}^N \tilde{f}_n^m \left. \DD{P_n^m}{\mu} \right\vert _j \exp(im \lambda_i)$ |; $key = q/f'=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/%latex2htmlidmarker7276setcounter{footnote}{8}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7276 \setcounter{footnote}{8}\fnsymbol{footnote}$ |; $key = q/Depsf[10cm]{zahyouslashlg1.ps};AAT/; $cached_env_img{$key} = q| $\Depsf[10cm]{zahyou/lg1.ps}$ |; $key = q/m=n=0;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleleft|array{{ll}(2n+1)xtilde{P}_n(x)-ntilde{P}_{n-1}(x)tildytilde{P}_n(y)-ntilde{P}_{n-1}(y)tilde{P}_n(y)array{right|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left\vert \begin{array}{ll} (2n+1)x \tilde{P}_n(x)-n\tilde{P}_{n... ...+1)y \tilde{P}_n(y)-n\tilde{P}_{n-1}(y) & \tilde{P}_n(y) \end{array}\right\vert$ |; $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| $\displaystyle \sum_{k=0}^{J-1} \frac{f_k (\mu_j) f_k (\mu_{j'}) }{\lambda_k} = \frac{1}{w_j} \delta_{jj'}$ |; $key = q/Jgefrac{2N_{max}+1}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $J \ge \frac{2N_{max}+1}{2}$ |; $key = q/displaystylePhi_{s}+C_{p}alpha_{1}T_{v,1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Phi_{s} + C_{p} \alpha_{1} T_{v,1}$ |; $key = q/1leileI,1lejleJ,;MSF=1.6;AAT/; $cached_env_img{$key} = q| $1 \le i \le I , 1 \le j \le J,$ |; $key = q/L-f;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle - \alpha_k \sum_{l=k}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi ) \Delta \sigma_l \frac{T_{v,k}}{\Delta \sigma_k}$ |; $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| $\displaystyle \frac{1}{r} \DP{\psi}{\lambda} + \frac{1-\mu^2}{r} \DP{\chi}{\mu}$ |; $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| $\displaystyle (1-\mu^2) \DP{\tilde{P}_n}{\mu} = n \tilde{P}_{n-1}(\mu) - n \mu \tilde{P}_n(\mu)$ |; $key = q/g(lambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $g(\lambda)$ |; $key = q/Dvect{v};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{v}$ |; $key = q/P_n;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/tilde{P}_n^0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_n^0$ |; $key = q/displaystyleDP{tilde{D}_n^m}{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \DP{\tilde{D}_n^m}{t}$ |; $key = q/M=42,N=42;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{tilde{P}_n^m|n=|m|,|m+1|,cdots};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{ \tilde{P}_n^m \vert n=\vert m\vert,\vert m+1\vert,\cdots \}$ |; $key = q/sigma_{k-1slash2}(k=1,2,cdots,K);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma_{k-1/2} (k=1,2,\cdots,K)$ |; $key = q/n=J,mu=mu_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $n=J, \mu=\mu_j$ |; $key = q/displaystyleA_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle A_{ij}$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J B_{ij} C_{ij} P_n^{m}(\sin \phi_j) \exp(-im \lambda_i) w_j$ |; $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| $\displaystyle U = - \frac{1-\mu^2}{r} \DP{\psi}{\mu} + \frac{1}{r} \DP{\chi}{\lambda}$ |; $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| $\displaystyle \frac{1}{r^2} \left[ \DP{}{\mu} \left\{ (1-\mu^2) \DP{}{\mu} \right\} + \frac{1}{1-\mu^2} \DP[2]{}{\lambda} \right] f$ |; $key = q/phi=0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\phi=0$ |; $key = q/exp(imlambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\exp(i m \lambda)$ |; $key = q/displaystylebeta_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \beta_k$ |; $key = q/f'=2nx(x^2-1)^{n-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $f'=2nx(x^2-1)^{n-1}$ |; $key = q/displaystyle(2n+1)(x-y)tilde{P}_n(x)tilde{P}_n(y)+nleft|array{{ll}tilde{P}_{n}(x}(x)tilde{P}_{n}(y)tilde{P}_{n-1}(y)array{right|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y) + n \left\vert \begin{arr... ...de{P}_{n-1}(x) \\\\ \tilde{P}_{n}(y) & \tilde{P}_{n-1}(y) \end{array}\right\vert$ |; $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| $\displaystyle ∴ P_{n+1}^m = \sqrt{ \frac{(2n+1)(2n+3)}{(n-\vert m\vert+1)(n+\vert m\vert+1)} } \mu P_n^m$ |; $key = q/displaystyleleft(DP{f}{lambda}right)_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \left( \DP{f}{\lambda} \right)_{ij}$ |; $key = q/{Y_n^m|m=-n,-n+1,cdots,n};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\{Y_n^m \vert m= -n, -n+1,\cdots,n \}$ |; $key = q/frac{I-1}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{I-1}{2}$ |; $key = q/phi_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\phi_j$ |; $key = q/tilde{P}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_n^m$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} (\mbox{\sl VA})_{ij} (1... ...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $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| $\displaystyle - \frac{\alpha_{k}}{\Delta \sigma_{k} } T'_{v,k} \sum_{l=k}^{K} D... ...{\beta_{k}}{\Delta \sigma_{k} } T'_{v,k} \sum_{l=k+1}^{K} D_l \Delta \sigma_{l}$ |; $key = q/J-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{displaymath} \frac{\partial q_k}{\partial t} = - \frac{1}{a(1-\mu^{2})}... ...al V_k q_k}{\partial \mu} + R_k + S_{q,k} + {\cal D}(q_k) \end{displaymath}$ |; $key = q/M|; $key = q/mu_j（j=1,2,cdots,J）;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mu_j（j=1,2,\cdots,J）$ |; $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| $\begin{displaymath} {\cal A}^{t+\Delta t} = \hat{\cal A}^{t+\Delta t} + 2 \D... ...t \dot{\cal A}_{cnd}\left( \hat{\cal A}^{t+\Delta t} \right) \end{displaymath}$ |; $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| $\begin{displaymath} \bar{\cal A}^{t} = ( 1-2 \epsilon_f ) {\cal A}^{t} + \eps... ...eft( \bar{\cal A}^{t-\Delta t} + {\cal A}^{t+\Delta t} \right) \end{displaymath}$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} (\mbox{\sl UA})_{ij} (1... ...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/max[n+n'+n''];MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/Y_n^m(lambda,phi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $Y_n^m(\lambda,\phi)$ |; $key = q/P_n,P_n^1,P_n^2;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/(lambda,phi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\lambda, \phi)$ |; $key = q/tilde{A}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{A}_n^m$ |; $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| $\displaystyle \frac{ \sigma_{k-1/2}( \sigma^{\kappa}_{k-1/2} - \sigma^{\kappa}_... ...^{\kappa}_{k+1/2} ) } { \sigma^{\kappa}_k ( \sigma_{k-1/2} - \sigma_{k+1/2} ) }$ |; $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| $\displaystyle \hspace*{2cm} +(n+\vert m\vert) \sqrt{ \frac{1}{2n-1} \frac{(n-1+\vert m\vert)!}{(n-1-\vert m\vert)!} } P_{n-1}^m = 0$ |; $key = q/A(lambda,phi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $A(\lambda,\phi)$ |; $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| $\displaystyle \sum_{m,n} \left( - \frac{\cos^2 \phi_j}{r} \tilde{\psi}_n^m \lef... ...+ \frac{1}{r} im \tilde{\chi}_n^m P_n^m(\sin \phi_j) \right) \exp(im \lambda_i)$ |; $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| $\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} V_{ij} T'_{ij} (1-\mu... ...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/im'tilde{f}_{n'}^{m'}=tilde{g}_{n'}^{m'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $im' \tilde{f}_{n'}^{m'} = \tilde{g}_{n'}^{m'}$ |; $key = q/sigma_k(k=1,2,ldotsK);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma_k (k=1,2,\ldots K)$ |; $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| $\displaystyle w_j = \frac{1-\mu_j^2} {(J \sqrt{ \frac{1}{2J-1} } P_{J-1}(\mu_j))^2 } = \frac{(2J-1)(1-\mu_j^2)} {(J P_{J-1}(\mu_j))^2 }$ |; $key = q/displaystyleA(mu)=sum_{n=|m|}^{infty}tilde{A}_n^mP_n^m(mu),;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle A(\mu) = \sum_{n=\vert m\vert}^{\infty} \tilde{A}_n^m P_n^m(\mu),$ |; $key = q/{displaystylefrac{I}{3}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{I}{3}$ |; $key = q/displaystyleimtilde{f}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle im \tilde{f}_n^m$ |; $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| $\displaystyle \frac{1}{r^2} \left[ \frac{1}{\cos \phi} \DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right) + \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda} \right] f$ |; $key = q/f';MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/%latex2htmlidmarker7479setcounter{footnote}{9}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7479 \setcounter{footnote}{9}\fnsymbol{footnote}$ |; $key = q/displaystylea_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle a_k$ |; $key = q/epsilon_f;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\epsilon_f$ |; $key = q/K=42,J=M=21;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylesigma_{1slash2}-sigma_{1}=1-sigma_{1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sigma_{1/2} - \sigma_{1} = 1 -\sigma_{1}$ |; $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| $\begin{displaymath} \delta_{t} \Dvect{D} = \left( \DP{\Dvect{D}}{t} \right)_{N... ...( \Dvect{D}^{t-\Delta t} + 2 \Delta t \delta_{t} \Dvect{D} ) \end{displaymath}$ |; $key = q/y;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle1-left(frac{sigma_{k+1slash2}}{sigma_k}right)^{kappa}.;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle 1- \left( \frac{ \sigma_{k+1/2} } { \sigma_k } \right)^{\kappa} .$ |; $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| $\displaystyle \frac{1}{r \sqrt{1-\mu^2} } \DP{v_2}{\lambda} - \frac{1}{r} \DP{}{\mu}(\sqrt{1-\mu^2}v_1)$ |; $key = q/I=64,J=64;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd};MSF=1.6;AAT/; $cached_env_img{$key} = q| $A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd}$ |; $key = q/alpha;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\alpha$ |; $key = q/{displaystylesum_{i,j}Y_{n'}^{m'*}|_{ij}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum_{i,j} Y_{n'}^{m'*}\vert _{ij}$ |; $key = q/displaystylefrac{1}{2pi}int_0^{2pi}g(lambda)dlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda$ |; $key = q/I=64;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/frac{1}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{1}{2}$ |; $key = q/P_0simP_{n-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_0 \sim P_{n-1}$ |; $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| $\displaystyle - \frac{1}{a(1-\mu^{2})} \frac{\partial U_k T_k'}{\partial \lambda} - \frac{1}{a} \frac{\partial V_k T_k'}{\partial \mu} + \hat{H}_k$ |; $key = q/{displaystyle-frac{I-1}{2}simfrac{I-1}{2}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $-\frac{I-1}{2} \sim \frac{I-1}{2}$ |; $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| $\begin{displaymath} \tilde{\cal D}_{H,n}^m = - K_{HD} \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2} . \end{displaymath}$ |; $key = q/n=J,x=mu,y=mu_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $n=J,x=\mu,y=\mu_j$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J \left( \sum_{m'=-M}^{M} \su... ...ert}^N \tilde{B}_{n'}^{m'} \exp(im' \lambda_i) P_{n'}^{m'}(\sin \phi_j) \right)$ |; $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| $\displaystyle ( \zeta_k + f ) V_k - \frac{1}{2 \Delta \sigma_k} [ \dot{\sigma}_{k-1/2} ( U_{k-1} - U_k ) + \dot{\sigma}_{k+1/2} ( U_k - U_{k+1} ) ]$ |; $key = q/0leKle2J-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le K \le 2J-1$ |; $key = q/I=128,J=64;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/N_{max};MSF=1.6;AAT/; $cached_env_img{$key} = q| $N_{max}$ |; $key = q/(U,V);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\begin{displaymath} \hat{\cal A}^{t+\Delta t} = \bar{\cal A}^{t-\Delta t} + ... ...t \dot{\cal A}_{phy}\left( \bar{\cal A}^{t-\Delta t} \right) \end{displaymath}$ |; $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| $\displaystyle = ( 1-2\Delta t {\cal D}_H )( 1-\Delta t {\cal D}_M ) \Dvect{D}^{... ... t} + ( 1-2\Delta t {\cal D}_H ) \Delta t \left( \DP{\Dvect{D}}{t} \right)_{NG}$ |; $key = q/0le|m|len,n';MSF=1.6;AAT/; $cached_env_img{$key} = q| $0\le \vert m\vert \le n,n'$ |; $key = q/delta_{kleql};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\delta_{k \leq l}$ |; $key = q/displaystylesum_{m,n}imtilde{f}_n^mY_n^m(lambda_i,phi_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sum_{m,n} im \tilde{f}_n^m Y_n^m (\lambda_i,\phi_j)$ |; $key = q/tilde{P}^m_n(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}^m_n(\mu)$ |; $key = q/displaystylefrac{tilde{P}_J(mu)}{mu-mu_j};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{\tilde{P}_J (\mu) }{\mu-\mu_j}$ |; $key = q/-MlemleM,|m|lenleN;MSF=1.6;AAT/; $cached_env_img{$key} = q| $-M \le m \le M, \vert m\vert \le n \le N$ |; $key = q/{eqnarraystar}g&equiv&DP{f}{lambda}{eqnarraystar};MSF=1.6;TAGS=R;AAT/; $cached_env_img{$key} = q| $\begin{eqnarray*} g&\equiv& \DP{f}{\lambda} \end{eqnarray*}$ |; $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| $\displaystyle - \beta_k \sum_{l=k+1}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi ) \Delta \sigma_l \frac{T_{v,k}}{\Delta \sigma_k}$ |; $key = q/tilde{P}_n^m(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_n^m(\mu)$ |; $key = q/{displaystylefrac{1}{rcosphi}DP{f}{lambda}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\frac{1}{r \cos \phi} \DP{f}{\lambda}$ |; $key = q/dot{calA}_{cnd};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\dot{\cal A}_{cnd}$ |; $key = q/M=42;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylealpha_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \alpha_k$ |; $key = q/0le|m|len;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le \vert m\vert \le n$ |; $key = q/{displaymath}D^{t+Deltat}=2overline{Dvect{D}}^{t}-D^{t-Deltat}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} D^{t+\Delta t} = 2\overline{ \Dvect{D} }^{t} - D^{t-\Delta t} \end{displaymath}$ |; $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| $\displaystyle \sum_{i=1}^I \sum_{j=1}^J f_{n'}^{m'} P_{n}^{m}(\mu_j) \exp(im \lambda_i) \left. \DD{P_{n'}^{m'}}{\mu}\right\vert _j \exp(-im' \lambda_i) w_j$ |; $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| $\begin{displaymath} \left( \DP{T_{k}}{t} \right)^{NG} \equiv - \frac{1}{a(1-\... ... + H_{k} + {\cal D}(T_{k}) + {\cal D}^{\prime}(\Dvect{v}) \end{displaymath}$ |; $key = q/Ige3M+1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $I \ge 3M+1$ |; $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| $\displaystyle \sum_{j=1}^{J} f(\mu_j) \int_{-1}^1 \frac{\tilde{P}_J(\mu)} {(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu$ |; $key = q/int_0^{2pi}exp(imlambda)dlambda;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\int_0^{2\pi} \exp(i m \lambda) d \lambda$ |; $key = q/M=64;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/tilde{A}_n^m（-MlemleM,|m|lenleN）;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{A}_n^m （ -M \le m \le M, \vert m\vert \le n \le N）$ |; $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| $\displaystyle - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} im U_{ij} T'_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/displaystyleDeltasigma_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta \sigma_k$ |; $key = q/Y_n^mexp(imlambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $Y_n^m \exp(im\lambda)$ |; $key = q/tilde{P_n};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P_n}$ |; $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| $\displaystyle \frac{ \left[ \left( \frac{ \sigma_{k-1/2} } { \sigma_k } \right)... ...ight] \sigma_k^{\kappa} T_{k-1} } { \sigma_{k-1}^{\kappa} - \sigma_k^{\kappa} }$ |; $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| $\displaystyle \sum_{k=0}^{J-1} (2k+1) (\mu-\mu_j) \tilde{P}_k(\mu) \tilde{P}_k(\mu_j).$ |; $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| $\begin{eqnarray*} \tilde{q}_n^m &=& \frac{1}{I} \sum_{i,j} q_{ij} Y_n^{m*} w_j ... ... (1-\mu^2)P_n^m \right) \right\vert _j \exp(-im \lambda_i) w_j \end{eqnarray*}$ |; $key = q/{displaystyleM=left[frac{I}{2}right]};MSF=1.6;AAT/; $cached_env_img{$key} = q| $M= \left[ \frac{I}{2} \right]$ |; $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| $\displaystyle \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} P_n^m \right\} + \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\} P_n^m = 0$ |; $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| $\displaystyle \left. \hspace*{20mm} + ( 1-2\Delta t {\cal D}_H ) \Dvect{G} \left[ \pi^{t-\Delta t} + \Delta t \left( \DP{\pi}{t} \right)_{NG} \right] \right\} .$ |; $key = q/-MlemleM,|m|lenleN(m);MSF=1.6;AAT/; $cached_env_img{$key} = q| $-M \le m \le M, \vert m\vert \le n \le N(m)$ |; $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| $\displaystyle +n (- \tilde{P}_{n-1}(x) \tilde{P}_n(y) + \tilde{P}_{n-1}(y) \tilde{P}_n(x) )$ |; $key = q/%latex2htmlidmarker7488setcounter{footnote}{9}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7488 \setcounter{footnote}{9}\fnsymbol{footnote}$ |; $key = q/displaystyleV_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle V_{ij}$ |; $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| $\displaystyle p_{ij} \equiv \left[ \DP{}{\mu} \left( \sum_{m,n} \tilde{f}_n^m Y_n^m \right) \right]_{ij}$ |; $key = q/m;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/A_{ij}=A(lambda_i,phi_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{ij}=A(\lambda_i,\phi_j)$ |; $key = q/(U_{ij},V_{ij});MSF=1.6;AAT/; $cached_env_img{$key} = q| $(U_{ij},V_{ij})$ |; $key = q/L(mu_j)=f(mu_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $L(\mu_j) = f(\mu_j)$ |; $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| $\displaystyle - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} im U_{ij} q_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/{displaystyleexp(imlambda_n)=expleft(frac{2piim(n-1)}{I}right)};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\displaystyle \exp(i m \lambda_n) = \exp \left( \frac{2\pi i m (n-1)}{I} \right) }$ |; $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| $\begin{eqnarray*} h_{ij} = \left. \left(\nabla_H^2 \sum_{m,n} \tilde{f}_n^m... ...{n(n+1)}{r^2} \left. \tilde{f}_n^m Y_n^m \right\vert _{ij} \\\\ \end{eqnarray*}$ |; $key = q/displaystyleH_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle H_k$ |; $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| $\{Y_n^m \vert n=0,1,2,\cdots, m= -n, -n+1,\cdots,n \}$ |; $key = q/sigma;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sigma$ |; $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| $\displaystyle \sum_{j=1}^{J} P_n^m (\mu_j) P_{n'}^m (\mu_j) w_j = \delta_{nn'}$ |; $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| $\displaystyle Y^m_n(\lambda, \phi) = \sum_{m'=-n}^{n} A_n^{m'} Y^{m'*}_n(\lambda',\phi')$ |; $key = q/displaystyle(u,v)equivleft(frac{U}{cosphi},frac{V}{cosphi}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (u,v) \equiv \left( \frac{U}{\cos \phi}, \frac{V}{\cos \phi} \right)$ |; $key = q/displaystyletilde{A}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{A}_n^m$ |; $key = q/(N_{tr}+1)^2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $(N_{tr}+1)^2$ |; $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| $\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} Z_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) w_j ,$ |; $key = q/displaystylenleft|array{{ll}tilde{P}_n(x)tilde{P}_{n-1}(x)tilde{P}_n(y)tilde{P}_{n-1}(y)array{right|;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle n \left\vert \begin{array}{ll} \tilde{P}_n(x) & \tilde{P}_{n-1}(x) \\\\ \tilde{P}_n(y) & \tilde{P}_{n-1}(y) \end{array}\right\vert$ |; $key = q/displaystylesum_{i=1}^Iexp(i(m'-m)lambda)=Idelta_{mm'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sum_{i=1}^I \exp(i(m'-m) \lambda) = I \delta_{mm'}$ |; $key = q/K-J;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaystylefrac{pi}{2}>phi_1>phi_2>cdots>phi_{J}>-frac{pi}{2}};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\displaystyle \frac{\pi}{2} > \phi_1 > \phi_2 > \cdots > \phi_{J} > -\frac{\pi}{2} }$ |; $key = q/displaystyleg_{ij};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle g_{ij}$ |; $key = q/i,j,m,m',n,n',I,J,M,N(m);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaystylelambda_i=frac{2pi(i-1)}{I}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\lambda_i=\frac{2\pi(i-1)}{I}$ |; $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| $\displaystyle \frac{1}{r} \DP{Y_n^m}{\phi} = \frac{1}{r} \DP{}{\phi} \left( P_n... ...) \right) = \frac{\sqrt{1-\mu^2} }{r} \DD{}{\mu} P_n^{m} (\mu) \exp(im \lambda)$ |; $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| $\displaystyle \frac{1}{r (1-\mu^2)} \DP{U}{\lambda} + \frac{1}{r} \DP{V}{\mu}$ |; $key = q/P_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/(lambda_i,phi_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\lambda_i,\phi_j)$ |; $key = q/tilde{P}_n;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{P}_n$ |; $key = q/displaystylenabla_H^2Y_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \nabla_H^2 Y_n^m$ |; $key = q/x=-1,beta_1,beta_2,1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $x=-1,\beta_1,\beta_2,1$ |; $key = q/{displaystyleP_n=frac{1}{2^nn!}DD[n]{}{mu}(mu^2-1)^n};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\displaystyle P_n= \frac{1}{2^n n!} \DD[n]{}{\mu} (\mu^2-1)^n }$ |; $key = q/n;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/I>M;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyletilde{zeta}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{\zeta}_n^m$ |; $key = q/w(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $w(\mu)$ |; $key = q/tilde{D}_n^m,;tilde{zeta}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\tilde{D}_n^m, \; \tilde{\zeta}_n^m$ |; $key = q/{calF}_lambda,{calF}_varphi,Q,S_q;MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal F}_\lambda, {\cal F}_\varphi, Q, S_q$ |; $key = q/mu_j;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mu_j$ |; $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| $\displaystyle \frac{1}{I} \sum_{i,j} im f_{ij} Y_n^{m*} (\lambda_i \phi_j) \frac{w_j}{r \cos^2 \phi_j}$ |; $key = q/w_j;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/nabla_H^2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\nabla_H^2$ |; $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| $\begin{displaymath} R_{kl} = - \left( \frac{ \alpha_{k} }{ \Delta \sigma_{k} } ... ... \Delta \sigma_{l} \delta_{k+1 \leq l} \right) \bar{T}_{k} . \end{displaymath}$ |; $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| $\displaystyle - \frac{\alpha_{k}}{\Delta \sigma_{k} } T_{v,k} \sum_{l=k}^{K} \D... ...{k} } T_{v,k} \sum_{l=k+1}^{K} \Dvect{v}_{l} \cdot \nabla \pi \Delta \sigma_{l}$ |; $key = q/Dvect{v}=(v_1,v_2);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\Dvect{v}=(v_1,v_2)$ |; $key = q/displaystylesum_{i=1}^{I}exp(imlambda_i)exp(-im'lambda_i)=Idelta_{mm'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sum_{i=1}^{I} \exp(im \lambda_i) \exp(-im' \lambda_i) = I \delta_{mm'}$ |; $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| $\displaystyle \frac{1}{I} \sum_{m'=-M}^{M} \sum_{n'=\vert m'\vert}^N \sum_{m''=-M}^{M} \sum_{n''=\vert m''\vert}^N \tilde{B}_{n'}^{m'} \tilde{C}_{n''}^{m''}$ |; $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| $\displaystyle - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a} ( 1 - \mu^{2} ) \frac{\partial \pi}{\partial \mu} + F_{\varphi k} \cos \varphi$ |; $key = q/textstyle=;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle =$ |; $key = q/int_0^{2pi}exp(imlambda)exp(-im'lambda)dlambda=2pidelta_{mm'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\int_0^{2 \pi} \exp(im\lambda) \exp(-im'\lambda) d \lambda = 2 \pi \delta_{mm'}$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J \left( \sum_{m'=-M}^{M} \su... ... \phi_j) \exp(im' \lambda) \right) P_n^{m}(\sin \phi_j) \exp(-im \lambda_i) w_j$ |; $key = q/J=65;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J im f_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$ |; $key = q/Y^m_n(lambda,phi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $Y^m_n(\lambda, \phi)$ |; $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| $\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} im (\mbox{\sl VA})_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/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| $\displaystyle \frac{1}{I} \sum_{ij} \left( \frac{im}{r \cos^2 \phi_j} U_{ij} P_... ...} V_{ij} \left. \DD{P_n^m}{\mu} \right\vert _j \right) \exp(- im \lambda_i) w_j$ |; $key = q/{displaystylesum_{m=-M}^{M}sum_{n=|m|}^{N}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N}$ |; $key = q/overline{Dvect{D}_n^m}^{t};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\overline{ \Dvect{D}_n^m }^{t}$ |; $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| $\displaystyle \tilde{A}_n^m = \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} A_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$ |; $key = q/4pi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $4 \pi$ |; $key = q/2J+2|m|-2;MSF=1.6;AAT/; $cached_env_img{$key} = q| $2J+2\vert m\vert-2$ |; $key = q/(lambda',phi');MSF=1.6;AAT/; $cached_env_img{$key} = q| $(\lambda', \phi')$ |; $key = q/displaystyleA=BC;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle A= BC$ |; $key = q/lambda_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\lambda_k$ |; $key = q/M=N_{tr},N=N_{tr};MSF=1.6;AAT/; $cached_env_img{$key} = q| $M= N_{tr}, N =N_{tr}$ |; $key = q/{displaystylesum_{m,n}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum_{m,n}$ |; $key = q/m,n,n';MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/N_{max}=2M;MSF=1.6;AAT/; $cached_env_img{$key} = q| $N_{max}=2M$ |; $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| $\displaystyle P_n^m \equiv \sqrt{ \frac{(2n+1)(n-\vert m\vert)!}{(n+\vert m\ver... ...^2)^{\frac{\vert m\vert}{2}} }{2^n n!} \DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n ,$ |; $key = q/displaystyleIdelta_{nn'}delta_{mm'};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle I \delta_{nn'} \delta_{mm'}$ |; $key = q/displaystylesigma_{K}-sigma_{K+1slash2}=sigma_{K};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \sigma_{K} - \sigma_{K+1/2} = \sigma_{K}$ |; $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| $\displaystyle T_k' D_k - \frac{1}{\Delta \sigma_k} [ \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k ) + \dot{\sigma}_{k+1/2} ( T_k - \hat{T}_{k+1/2} ) ]$ |; $key = q/displaystylefrac{1}{2}left({calA}^{t+Deltat}+{calA}^{t-Deltat}right);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{2} \left( {\cal A}^{t+\Delta t} + {\cal A}^{t-\Delta t} \right)$ |; $key = q/m,m';MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \sum_{j=1}^{J} f(\mu_j) \int^1_{-1} \frac{ {\displaystyle \prod_{... ... } { (\mu-\mu_j) {\displaystyle \prod_{k=1,k \neq j}^{J}(\mu_j-\mu_k) } } d \mu$ |; $key = q/f(x)=(x-1)^n(x+1)^n;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyleV;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle V$ |; $key = q/displaystylehat{kappa}_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \hat{\kappa}_k$ |; $key = q/{displaystyleMleleft[frac{I}{2}right],N(m)leJ-1};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\displaystyle M \le \left[ \frac{I}{2} \right] , N(m) \le J-1 }$ |; $key = q/f;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \int_{-1}^1 \tilde{P}_n^m(\mu) \tilde{P}_{n'}^m (\mu) d \mu = \frac{2}{2n+1} \frac{(n+\vert m\vert)!}{(n-\vert m\vert)!} \delta_{nn'}$ |; $key = q/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| $\displaystyle \left[ \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} \right\} + J(J+1) \right] P_J(\mu) = 0$ |; $key = q/displaystyle(2n+1)(x-y)tilde{P}_n(x)tilde{P}_n(y);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)$ |; $key = q/f(mu_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $f(\mu_j)$ |; $key = q/-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/n-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylelambda_n=frac{2pi(n-1)}{I}(n=1,2,cdots,I);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \lambda_n = \frac{2\pi (n-1)}{I} (n=1,2,\cdots,I)$ |; $key = q/P_n^m(mu)(n=|m|,|m+1,cdots);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n^m(\mu) (n=\vert m\vert,\vert m+1, \cdots)$ |; $key = q/displaystyle+tilde{calD}_{H,n}^mtilde{q}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle + \tilde{\cal D}_{H,n}^m \tilde{q}_n^m$ |; $key = q/2J-1gemax[n+n']=2max[N];MSF=1.6;AAT/; $cached_env_img{$key} = q| $2J-1 \ge max[n+n'] = 2 max[N]$ |; $key = q/m(n-1);MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \tilde{A}_n^m = \frac{2n+1}{2} \frac{(n-\vert m\vert)!}{(n+\vert m\vert)!} \int_{-1}^1 A(\mu) \tilde{P}_n^m(\mu) d \mu$ |; $key = q/chi,psi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\chi ,\psi$ |; $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| $\displaystyle A_{ij} = \sum_{m=0}^{M} \sum_{n=m}^{N} \Re \tilde{A}_n^m Y_n^m(\lambda_i, \phi_j)$ |; $key = q/K+frac{1}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $K+\frac{1}{2}$ |; $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| $\displaystyle \frac{1}{2 J \tilde{P}_{J-1}(\mu_j) \tilde{P}^{'}_{J} (\mu_j)} \sum_{k=0}^{J-1} (2k+1) \tilde{P}_k(\mu_j) \int_{-1}^1 \tilde{P}_k(\mu) d \mu$ |; $key = q/f(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $f(\mu)$ |; $key = q/g=DP{f}{lambda};MSF=1.6;AAT/; $cached_env_img{$key} = q| $g = \DP{f}{\lambda}$ |; $key = q/displaystylelambda_i=frac{2pi（i-1)}{I}（i=1,2,cdots,I）;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \lambda_i = \frac{2 \pi （i-1)}{I} \ （i=1,2,\cdots,I）$ |; $key = q/displaystyleT_{k}^{prime}D_{k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle T_{k}^{\prime} D_{k}$ |; $key = q/nabla^2_Hf=-frac{n(n+1)}{r^2}f;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\nabla^2_H f = -\frac{n(n+1)}{r^2} f$ |; $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| $\displaystyle \sum_{m=-M}^{M} \frac{1}{2\pi} g_m \int_0^{2\pi} \exp(i m \lambda) d \lambda =g_0$ |; $key = q/M;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/psi;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\psi$ |; $key = q/{displaystylekappa=frac{R}{C_p}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\kappa=\frac{R}{C_p}$ |; $key = q/Depsf[][7cm]{zahyouslashlg2.ps};AAT/; $cached_env_img{$key} = q| $\Depsf[][7cm]{zahyou/lg2.ps}$ |; $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| $\displaystyle \left[ \DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right) + \frac{1}{1-\mu^2} \DP[2]{}{\lambda} + n(n+1) \right] Y_n^m =0$ |; $key = q/f_k（k=0,1,2,cdots,K-1）;MSF=1.6;AAT/; $cached_env_img{$key} = q| $f_k（k=0,1,2,\cdots,K-1）$ |; $key = q/A_{ij}^{even};MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{ij}^{even}$ |; $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| $\displaystyle \int^{1}_{-1} f(\mu) d \mu = 2 \sum_{j=1}^{J} f(\mu_j) w_j$ |; $key = q/A_{i,j}+A_{i,J-j};MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{i,j}+A_{i,J-j}$ |; $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| $\displaystyle \tilde{h}_n^m = -\frac{n(n+1)}{r^2} \tilde{f}_n^m$ |; $key = q/dot{calA}_{adv};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\dot{\cal A}_{adv}$ |; $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| $\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} \left( \hat{R}_{ij} + S_{q,ij} \right) Y_n^{m *} ( \lambda_i, \mu_j ) w_j$ |; $key = q/{calA}^{t+Deltat};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\cal A}^{t+\Delta t}$ |; $key = q/T=bar{T}_k;MSF=1.6;AAT/; $cached_env_img{$key} = q| $T=\bar{T}_k$ |; $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| $\displaystyle \int_{-1}^1 \tilde{P}_n(\mu) \tilde{P}_{n'}(\mu) d \mu = \frac{2}{2n+1} \delta_{nn'}$ |; $key = q/C_p;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/{displaystylesum_{i,j}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\sum_{i,j}$ |; $key = q/displaystylePhi_{1};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Phi_{1}$ |; $key = q/h;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \sum_{j=1}^J P_n^{m}(\sin \phi_j) P_{n'}^{m}(\sin \phi_j) w_j = \delta_{nn'}$ |; $key = q/3slash4;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \frac{1}{r \sqrt{1-\mu^2} } \DP{v_1}{\lambda} + \frac{1}{r} \DP{}{\mu}( \sqrt{1-\mu^2} v_2 )$ |; $key = q/J=63,N=42;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/kleql;MSF=1.6;AAT/; $cached_env_img{$key} = q| $k \leq l$ |; $key = q/t+Deltat;MSF=1.6;AAT/; $cached_env_img{$key} = q| $t+\Delta t$ |; $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| $\displaystyle B= \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \left( \tilde{B}_n^m \exp(im \lambda) \right) P_n^m(\sin \phi)$ |; $key = q/N_{max}=M;MSF=1.6;AAT/; $cached_env_img{$key} = q| $N_{max}=M$ |; $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| $\displaystyle \sum_{m=-\frac{I}{2}+1}^{\frac{I}{2}} \exp(i m \lambda_i) \exp(-i m \lambda_{i'}) = I \delta_{ii'}$ |; $key = q/g;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystyle-frac{n(n+1)}{r^2}Y_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle - \frac{n(n+1)}{r^2} Y_n^m$ |; $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| $\begin{displaymath} \frac{\partial D}{\partial t} = \frac{1}{a(1-\mu^{2})} \... ...ppa}_k \bar{T}_k \pi + (\mbox{\sl KE})_k ) + {\cal D}(D_k) \end{displaymath}$ |; $key = q/lambda;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\lambda$ |; $key = q/Jgefrac{3N_{max}+1}{2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $J \ge \frac{3N_{max}+1}{2}$ |; $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| $\displaystyle \sum_{j=1}^{J} \sum_{i=1}^{I} P_n^m (\mu_j) P_{n'}^{m'} (\mu_j) \exp(i m \lambda_i) \exp(-i m' \lambda_{i}) w_j$ |; $key = q/K=J+M;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/w_j=w(mu_j);MSF=1.6;AAT/; $cached_env_img{$key} = q| $w_j=w(\mu_j)$ |; $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| $\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} {\cal D}'_{ij}(\Dvect{v}) Y_n^{m *} ( \lambda_i, \mu_j ) w_j ,$ |; $key = q/L=f;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle \sum^{I}_{n=1} \exp \left\{ im \frac{2 \pi (n-1)}{I} \right\} = \... ...im 2 \pi}{I} } } = \frac{1 - e^{im 2 \pi} } {1 - e^{ \frac{im 2 \pi}{I} } } = 0$ |; $key = q/{displaystyleleft(tilde{A}^m_nexp(imlambda)right)^*=tilde{A}^{-m}_nexp(-imlambda)};MSF=1.6;AAT/; $cached_env_img{$key} = q| ${\displaystyle \left(\tilde{A}^m_n \exp(im\lambda) \right)^* = \tilde{A}^{-m}_n \exp(-im\lambda) }$ |; $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| $\displaystyle A(\lambda,\phi) \equiv \sum_{m=-M}^{M} \sum_{n=\vert m\vert}^{N} \tilde{A}_n^m Y_n^m (\lambda, \phi)$ |; $key = q/2J-1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/displaystylewidetilde{left(DP{f}{mu}right)_n^m};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \widetilde{\left(\DP{f}{\mu}\right)_n^m}$ |; $key = q/1+(2times1+1)+cdots+(2times42+1)=43^2=1849;MSF=1.6;AAT/; $cached_env_img{$key} = q| $1+ (2 \times1 +1) + \cdots + (2 \times 42 +1) = 43^2 = 1849$ |; $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| $\displaystyle \sum_{k=0}^{n-1} (2k+1)(x-y)\tilde{P}_k(x)\tilde{P}_k(y)$ |; $key = q/1;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle (1-\mu^2) \DD{}{\mu}\tilde{P}_{n}^{m} = (n+\vert m\vert) \tilde{P}_{n-1}^m - n \mu \tilde{P}_n^m$ |; $key = q/%latex2htmlidmarker7487setcounter{footnote}{9}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7487 \setcounter{footnote}{9}\fnsymbol{footnote}$ |; $key = q/displaystylefrac{sigma_{k-1slash2}alpha_k+sigma_{k+1slash2}beta_k}{Deltasigma_k};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{ \sigma_{k-1/2} \alpha_k + \sigma_{k+1/2} \beta_k } { \Delta \sigma_k }$ |; $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| $\displaystyle \tilde{P}_n^m \equiv \frac{ (1-\mu^2)^{\frac{\vert m\vert}{2}} }{2^n n!} \DD[n+\vert m\vert]{}{\mu} (\mu^2-1)^n$ |; $key = q/{displaystyleM=left[frac{I-1}{3}right]};MSF=1.6;AAT/; $cached_env_img{$key} = q| $M= \left[ \frac{I-1}{3} \right]$ |; $key = q/0le|m|,|m'|leM,|m|lenleN,|m'|len'leN;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le \vert m\vert,\vert m'\vert \le M, \vert m\vert \le n \le N, \vert m'\vert \le n' \le N$ |; $key = q/P_n^m(sinphi);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P_n^m(\sin \phi)$ |; $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| $\displaystyle - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a} \frac{\partial \pi}{\partial \lambda} + F_{\lambda k} \cos \varphi$ |; $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| $\displaystyle C_{p} \left[ \left( \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{... ...eft( \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} } \right)^{\kappa} \right] T_{v,k-1}$ |; $key = q/displaystylef(mu)-L(mu)=tilde{P}_J(mu)S(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle f(\mu) - L(\mu) = \tilde{P}_J(\mu) S(\mu)$ |; $key = q/%latex2htmlidmarker7357setcounter{footnote}{9}fnsymbol{footnote};AAT/; $cached_env_img{$key} = q| $% latex2html id marker 7357 \setcounter{footnote}{9}\fnsymbol{footnote}$ |; $key = q/dot{calA}_{dis};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\dot{\cal A}_{dis}$ |; $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| $\begin{figure}\begin{center} \begin{picture}(300,170)(10,-20) \put(20 , 10){\v... ...20) \{ shortstack\{平方四辺形切断\}\} \par \end{picture}\end{center}\end{figure}$ |; $key = q/textstyle-;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\textstyle -$ |; $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| $\displaystyle \left( \DP{\Dvect{T}}{t} \right)_{NG} - \underline{h} \Dvect{D} + {\cal D}_H \Dvect{T} ,$ |; $key = q/A,B,C;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/mathaccent"707Erelax{P}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\mathaccent ''707E\relax {P}_n^m$ |; $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| $\displaystyle \frac{1}{I} \sum_{ij} \left( im U_{ij} P_n^m - V_{ij} \cos^2 \phi... ...m}{\mu} \right\vert _j \right) \exp(- im \lambda_i) \frac{w_j}{r \cos^2 \phi_j}$ |; $key = q/displaystyletilde{g}_n^m=imtilde{f}_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \tilde{g}_n^m = im \tilde{f}_n^m$ |; $key = q/N_{tr};MSF=1.6;AAT/; $cached_env_img{$key} = q| $N_{tr}$ |; $key = q/0le|m|len-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0\le \vert m\vert \le n-1$ |; $key = q/A_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $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| $\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J} V_{ij} q_{ij} (1-\mu_... ...rtial }{\partial \mu} Y_n^{m *} ( \lambda_i, \mu_j ) \frac{w_j}{a(1-\mu_j^{2})}$ |; $key = q/mneq0;MSF=1.6;AAT/; $cached_env_img{$key} = q| $m \neq 0$ |; $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| $\displaystyle - \frac{1}{\Delta \sigma_{k}} [ \dot{\sigma}^{NG}_{k-1/2} ( \hat{... ...{T}_{k} ) + \dot{\sigma}^{NG}_{k+1/2} ( \bar{T}_{k} - \hat{\bar{T}}_{k+1/2} ) ]$ |; $key = q/P^m_n(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $P^m_n(\mu)$ |; $key = q/displaystylenabla^2_Hf;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \nabla^2_H f$ |; $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| ${\displaystyle \tilde{P}_{J-1} =\sqrt{ \frac{1}{2(\mbox{J}-1)+1} } P_{J-1} }$ |; $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| $\displaystyle \left\{ ( 1-2\Delta t {\cal D}_H )( 1-2\Delta t {\cal D}_M ) \und... ...vect{G} \Dvect{C}^{T} ) \nabla^{2}_{\sigma} \right\} \overline{ \Dvect{D} }^{t}$ |; $key = q/M=32;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/A_{ij};(i=1,2,cdots,I,;j=1,2,cdots,J);MSF=1.6;AAT/; $cached_env_img{$key} = q| $A_{ij} \; (i=1,2,\cdots,I, \; j=1,2,\cdots,J)$ |; $key = q/displaystyleDeltasigma_{K+1slash2};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \Delta \sigma_{K+1/2}$ |; $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| $\displaystyle - \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J f_{ij} \left. \DD{P_n^m}{\mu}\right\vert _j \exp(-im \lambda_i) w_j$ |; $key = q/{displaymath}G_{k}=hat{kappa}_{k}C_{p}bar{T}_{k}{displaymath};MSF=1.6;AAT/; $cached_env_img{$key} = q| $\begin{displaymath} G_{k} = \hat{\kappa}_{k} C_{p} \bar{T}_{k} \end{displaymath}$ |; $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| $\displaystyle - \frac{1}{\Delta \sigma_{k}} [ \dot{\sigma}_{k-1/2} ( \hat{T^{\p... ...e}_{k} ) + \dot{\sigma}_{k+1/2} ( T^{\prime}_{k} - \hat{T^{\prime}}_{k+1/2} ) ]$ |; $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| $\displaystyle \frac{1}{I} \sum_{ij} \left( \frac{im}{\cos^2 \phi} V_{ij} P_n^m ... ...} U_{ij} \left. \DD{P_n^m}{\mu} \right\vert _j \right) \exp(- im \lambda_i) w_j$ |; $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| $\displaystyle (n-\vert m\vert+1) \tilde{P}_{n+1}^m - (2n+1) \mu \tilde{P}_n^m +(n+\vert m\vert) \tilde{P}_{n-1}^m = 0$ |; $key = q/0leKleJ-1;MSF=1.6;AAT/; $cached_env_img{$key} = q| $0 \le K \le J-1$ |; $key = q/L(mu);MSF=1.6;AAT/; $cached_env_img{$key} = q| $L(\mu)$ |; $key = q/displaystyleY_n^m(lambda,phi)equivP_n^m(sinphi)exp(imlambda);MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle Y_n^m(\lambda,\phi) \equiv P_n^m(\sin \phi) \exp(im \lambda)$ |; $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]Y_n^m;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \frac{1}{r^2} \left[ \DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right) + \frac{1}{1-\mu^2} \DP[2]{}{\lambda} \right] Y_n^m$ |; $key = q/displaystylezeta;MSF=1.6;AAT/; $cached_env_img{$key} = q| $\displaystyle \zeta$ |; $key = q/{displaystyleIge3N_{tr}+1,かつJgefrac{3N_{tr}+1}{2}};MSF=1.6;AAT/; $cached_env_img{$key} = q| $I \ge 3N_{tr}+1, かつ J \ge \frac{3N_{tr}+1}{2}$ |; $key = q/64times64=4096;MSF=1.6;AAT/; $cached_env_img{$key} = q| $64 \times 64 = 4096$ |; $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| $\displaystyle \frac{1}{r \cos \phi} \DP{Y_n^m}{\lambda} = \frac{1}{r \cos \phi}... ...im \lambda) \right) = \frac{im}{r \cos \phi} P_n^m (\sin \phi) \exp(im \lambda)$ |; $key = q/n+n';MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/2M;MSF=1.6;AAT/; $cached_env_img{$key} = q|

|; $key = q/i=1,2,cdots,I,j=1,2,cdots,J;MSF=1.6;AAT/; $cached_env_img{$key} = q| $i=1,2,\cdots,I, j=1,2,\cdots,J$ |; 1;