harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C(L,n) =\frac{L^2}{4\sqrt{\pi}\Gamma(n)}\left(\sum_{k=1}^\infty\frac 1{(Lk)^{n+1/2}}\right) (1+\textrm{o}(1)) = \frac{1}{4\sqrt{\pi}\Gamma(n)}\frac{\zeta(n+1/2)}{L^{n-3/2}}(1+\textrm{o}(1))\,
vol(L') \cdot \sum_{x'}f(x') x'.v = vol(L') \cdot \sum_{x'}\frac{1}{\left[L':L\cap L'\right]}\left( \sum_{l'_1}f(x'l'_1) \x'l'_1.v\right) &= vol(L\cap L') \cdot \sum_{x'}\sum_{l'_1}f(x'l'_1) \x'l'_1.v &= vol(L\cap L') \cdot \sum_{x}\sum_{l_1}f(xl_1) \xl_1.v &= vol(L) \cdot \sum_{x}f(x) \x.v.
Z_n(q,t,Q) = \sum_{|\nu|+|\mu|=n}\frac{1}{N_{\nu,\mu}(Q) N_{\mu,\nu}(Q^{-1}) N_{\nu,\nu}(1) N_{\mu,\mu}(1)}, N_{\nu,\mu}(Q) = \prod_{t \in \mu}(1-Q q^{a_{\nu}(t)}t^{l_\mu(t) +1}) \prod_{s \in \nu}(1-Q q^{-a_{\mu}(s)-1}t^{-l_\nu(s)})
\bar{a}_i^be_b^\mu D_{\mu L}|a\rangle=0\,,\quad(a_i^b\bar{a}_i^b-h_i)|a\rangle=0\,,\quad\bar{a}_i^b\bar{a}_i^b|a\rangle=0\,,\quad\varepsilon^{ij}a_i^b\bar{a}_j^b|a\rangle=0\,,
\pi^{i j}= \mathrm{d i a g}\left( \frac{1}{2}e^{- 2 \mu}\pi_{\mu}, ~ \frac{1}{4}e^{- 2 \lambda}\pi_{\lambda}, ~ \frac{1}{4}e^{- 2 \lambda}s i n^{- 2}\theta \pi_{\lambda}\right)
\begin{array}{ll}\widetilde{\gamma}^*\omega^1=ds, \ \ \ \widetilde{\gamma}^*\omega^\ell=0\ 2\leq \ell \leq 2n, & \widetilde{\gamma}^*\omega^{2n+1}=\langle\frac{d\gamma(s)}{ds},T\rangle ds=\tau(s)ds.\end{array}
K_{n-1}' = \frac{1}{\sqrt{n}}\exp\left( \frac{1}{2}n \left(\mu_0 - \frac{\mu_0 + X_2 + \dots + X_n}{n}\right)^2 \right) = \frac{1}{\sqrt{n}}\exp\left( \frac{((n-1)\mu_0 - (S_n - S_1))^2}{2n}\right).
\sum_{j=0}^{N}a^j =\Big[\Big(\frac{1- a^{n_1+1}}{1-a}, \frac{1- a^{n_2+1}}{1-a}, \frac{1- a^{n_3+1}}{1-a}, ...\Big)\Big] &= \frac{1- a^{N+1}}{1-a}.
\forall \vec v u w_1 \dotsm w_m \, \bigl[ \bigl( \widehat \beta_r(\vec v,w_1,\dotsc,w_m) \And u = w_j \bigr) \leftrightarrow \widehat \beta_s(\vec v,u,w_{\theta(1)},\dotsc,w_{\theta(m)}) \bigr]
\varepsilon_{R}^{\mathrm{K}}= E [{\psi}_{\mathrm{K}}] + \Delta M_{K}+ o ( \hbar^{2}) = E [{\psi}_{\mathrm{K}}] + \lim_{s \rightarrow - \frac{1}{2}}\left[ \Delta_{1}\varepsilon^{\mathrm{K}}( s ) + \Delta_{2}\varepsilon^{\mathrm{K}}( s ) \right] + o ( \hbar^{2})
\frac{1}{X}\sum_{\substack{0\le a<X a\in\mathcal{M}}}S_{\mathcal{A}}\Bigl(\frac{a}{X}\Bigr)S_{\mathcal{R}_X}\Bigl(\frac{-a}{X}\Bigr)=\kappa_\mathcal{A}\frac{\#\mathcal{A}}{X}\sum_{n< X}\Lambda_{\mathcal{R}_X}(n)+O_{C,\eta}\Bigl(\frac{\#\mathcal{A}}{(\log{X})^C}\Bigr).
g_i^kg_j^l(T_s^m)\,=\,g_i^k(T_{\sigma_j(s)}^{\alpha_l(m)})=\,T_{\sigma_i\sigma_j(s)}^{\alpha_k\alpha_l(m)}g_{\sigma_i(j)}^{\alpha_k(l)}\,g_{\gamma_j(i)}^{\beta_l(k)}(T_s^m)=g_{\sigma_i(j)}^{\alpha_k(l)}(T_{\sigma_{\gamma_j(i)}(s)}^{\alpha_{\beta_l(k)}(m)})\,=\,T_{\sigma_{\sigma_i(j)}\sigma_{\gamma_j(i)}(s)}^{\alpha_{\alpha_k(l)}\alpha_{\beta_l(k)}(m)}
{\cal N}(en)\mbox{Im}\tau\overline{\eta}^{2}\int_{C_{1}}[\theta_{1}(z_{1}-w)/\theta'_{1}(0)]^{-e_{0}/k}[\theta_{1}(z_{2}-w)/\theta'_{1}(0)]^{e_{0}/k}[\theta_{1}(z_{1}-z_{2})/\theta'_{1}(0)]^{-2e_{0}(\widetilde{n_{0}}-1/2)/k}
\mathbb{P}\biggl(\sup_{x \in \mathbb{R}}\log \hat{f}_n(x) > \frac{t}{2}\log n\biggr) \leq \mathbb{P}\biggl(\frac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2 < \frac{C}{n^{t/2}}\biggr) \leq \mathbb{P}\biggl(|X_1 - \bar{X}| < \frac{C^{1/2}}{n^{t/4 - 1/2}}\biggr),
\frac{\log\overline{\mu}(\eta^{k-1})}{k\log\eta}-\frac{\log\overline{\mu}(\eta^{k})}{k\log\eta},\frac{\log\overline{\mu}(\eta^{k})}{(k-1)\log\eta}-\frac{\log\overline{\mu}(\eta^{k})}{k\log\eta}=O\left(\frac{1}{k\log\eta}\right) .
\kappa_{\ell,j,m,n_1,n_2}=\sqrt{\mu +A_{\ell,j,m,n_1,n_2}\}=\sqrt{\mu + \left(\ell+2j+2m+3n_1 + 3n_2+ d+3a_1+3a_2+\frac{11}{2}\right)^{2}\}
\epsilon \leq \langle e_i, g \cdot e_j \rangle= \frac{\langle e_i, g e_j \rangle}{\|g e_j\|}= \frac{g^{i,j}}{\sum_{i=1}^d \langle e_i, g e_j \rangle}= \frac{g^{i,j}}{\sum_{i=1}^d g^{i,j}}.
+ \sum_{n}\pi_{n , 1}\frac{\partial \overline{{{H}}}_{0}}{\partial q_{n , 0}}+ \sum_{n}\sum_{j_{n}= 2}^{m_{n}- 1}\pi_{n , j_{n}}\left[ \frac{\partial \overline{{{H}}}_{0}}{\partial q_{n , j_{n}- 1}}- \mu_{n , j_{n}- 1}\right] \ .
[\begin{matrix}ct^{\prime}\\x^{\prime}\\y^{\prime}\\z^{\prime}\end{matrix}]=[\begin{matrix}cosh\phi&0&0&-sinh\phi\\ 0&1&0&0\\ 0&0&1&0\\ -sinh\phi&0&0&cosh\phi\\ \end{matrix}][\begin{matrix}ct\\x\\y\\z\end{matrix}].
\begin{array}{r l}{{F_{i j}^{\mu \nu}d x_{\mu}d x_{\nu}\; \equiv}}&{{- \frac{1}{2}\left[ \nabla^{\mu}, \nabla^{\nu}\right]_{i j}d x_{\mu}d x_{\nu}=}}\\{{= \frac{1}{2}\{\left( \partial^{\mu}A^{I \nu}- \partial^{\nu}A^{I \mu}\right) \otimes \sigma_{i j}^{I}\; +}}&{{( A^{I \mu}A^{J \nu}- A^{I \nu}A^{J \mu}) \otimes \sigma_{i k}^{I}\sigma_{k j}^{J}\}d x_{\mu}d x_{\nu}\; ,}}\\ \end{array}
\P([X_g]_b=z|X_0=x) = \sum_{d^g\in\{0,1\}^g}\P([X_g]_b=z,J^g=d^g|X_0=x) &\stackrel{(a)}{=}\sum_{d^g\in\{0,1\}^g}\P([X_g]_b=z|J^g=d^g,X_0=x)\P(J^g=d^g).
\Lambda(F,G) := \sum_{j=1}^m \int_{\mathbb{R}^4}F(x+u,y)G(x,y+u)F(x+v,y)G(x,y+v) begin{align*}-1ex]\phi_{2^{k_{j}}}(u)(\phi_{2^{k_{j}}}-\phi_{2^{k_{j-1}}})(v) \,dx dy du dv .
\left\langle (x_{1},y_{1}) | (x_{1},y_{1})\right\rangle = \Big[\left\langle x_{1}| x_{1}\right\rangle_{K}+ \left\langle y_{1}| y_{1}\right\rangle_{K}\Big] + \Big[\left\langle x_{1}| y_{1}\right\rangle_{K}- \left\langle x_{1}| y_{1}\right\rangle_{K}\Big] \cdot n &= \|x_{1}\|^{2}+ \|y_{1}\|^{2}& \geq 0.
\frac{P_h(\epsilon\eta+\xi)}{\eta^k}= \exp\left(-\xi\frac{d^2}{d\epsilon^2}- \eta\epsilon\frac{d^2}{d\epsilon^2}- \eta(\alpha + 1)\frac{d}{d\epsilon}\right)(\epsilon^kp(\epsilon\eta)) &= \exp\left(-\xi\frac{d^2}{d\epsilon^2}- \eta\Lambda_\epsilon\right)(\epsilon^kp(\epsilon\eta))
\begin{array}{cl}\langle x_t, p(x_t) \rangle = \displaystyle \frac{1}{\Xi}\int_{-h}^0 \int_{\theta}^0 x_t^{\top}(s) R \int_{-h}^0 \int_{\theta_1}^0 x_t(s_1) ds_1 d\theta_1 ds d\theta & = \displaystyle \frac{1}{\Xi}\int_{-h}^0 \int_{\theta}^0 x_t^{\top}(s) ds d\theta R \int_{-h}^0 \int_{\theta}^0 x_t(s) ds d\theta,\end{array}
(R,V)=s\left(h||F_{2}-F_{1}||^{2}+\frac{s^{2}}{2}||F_{2}-F_{1}||^{2}\right)=s||F_{2}-F_{1}||^{2}\left(h+\frac{s^{2}}{2}\right)=s||F_{2}-F_{1}||^{2}\left(h+\bar{\bar{o}}(h)\right)
\inf_{\omega \in O(\frac{n(n-1)}{2})}\sup_{B_{10n H(x_2,t_2)^{-1}}(x_2)}\max_\alpha \bigg | K_\alpha^{(x_1,t_1)}- \sum_{\beta=1}^{\frac{n(n-1)}{2}}\omega_{\alpha\beta}K_\beta^{(x_2,t_2)}\bigg | \leq C \, 2^{-j}\, H(x_2,t_2)^{-1}.
{\rm e}^{L_{X_{(z,a,b,c)}}}X_{(\bar{z},\bar{a},\bar{b},\bar{c})}= \left( \bar{z}+ z_1 + \frac{z_2}{2!}+ \dots + \frac{z_n}{n!}+ \dots \right) X_{1}+ \left( \bar{a}+ a_1 + \frac{a_{2}}{2!}+ \dots + \frac{a_{n}}{n!}+ \dots \right)X_{q}- \bar{b}X_{p}+ \bar{c}\, X_{S}\,,
\psi(x_{k,n+1}-x_{k,n})-\psi(x_{k,n}-{x}_{k,n-1})= \phi({x}_{k+1,n-1}-x_{k,n})-\phi(x_{k,n}-x_{k-1,n+1}) +\psi_0(x_{k+1,n}-x_{k,n})-\psi_0(x_{k,n}-x_{k-1,n})
\left. - \frac{8}{9}\mathrm{T r}( T_{f}^{a}T_{f}^{b}T_{f}^{c}T_{f}^{d}) \right] F_{\mu \nu}^{a}F_{\nu \mu}^{b}F_{\rho \sigma}^{c}F_{\sigma \rho}^{d}+ \left[ \frac{4 1}{6 0}f^{e f a}f^{f g b}f^{g h c}f^{h e d}+ \right.
\tilde{z}_i-z_i =-\delta_{ki}\log(1+\hat{y}_k^{\varepsilon})+\delta_{ki}\log(1\oplus{y}_k^{\varepsilon}), \tilde{z}'_i-z'_i =-\delta_{k'i}\log(1+\hat{y}'_{k'}{}^{\varepsilon'})+\delta_{k'i}\log(1\oplus{y}'_{k'}{}^{\varepsilon'}).
\pi_b \cdot \pi_{[a,b]}= (b-a+1) \pi_{[a,b]}y_{b+1}+ (b-a+2) \pi_{[a-1,b]}&= \pi_{[a-1,b]}+ (b-a+1)\Big(\pi_{[a-1,b]}+ \pi_{[a,b]}y_{b+1}\Big).
h^{-1}v =h^{-1}(\varphi_+(v)+\pi_+(v)+\varphi_-(v)+\pi_-(v)) & =h^{-1}(\widetilde{\varphi_+}\pi_+(v)+\pi_+(v)+\widetilde{\varphi_-}\pi_-(v)+\pi_-(v)) & =h^{-1}(h\pi_+(v)+h\pi_-(v))=\pi_+(v)+\pi_-(v) \in U_9\cap U.
P(Z_n(\tilde\delta_n)>\epsilon_n)\le \frac{2}{\epsilon_n}E_{P\times P^W}\left[\sup_{f\in\mathcal M_{P,\tilde\delta_n}}\Bigl\vert\frac{1}{\sqrt n}\sum_{i=1}^nW_i f(X_i)\Bigr\vert\right],
|Ff(l)|\le \sum_{i=1}^M |c_i| \frac{\sqrt{M}}{\sqrt{a_1^i\cdots a_d^i}}|F \chi_{R_i}(l)|\le \sqrt{M/A^d}\left(\sum_{i=1}^M |F \chi_{R_i}(l)|^2\right)^\frac12.
k^{\prime \prime}= ( ( 1 - x - y ) p^{+},{\frac{( ( 1 - x - y ) p_{\perp}- k_{\perp})^{2}+ m^{2}}{2 ( 1 - x - y ) p^{+}}}, ( 1 - x - y ) p_{\perp}- k_{\perp}) \; ,
\left[\begin{matrix}\mathbf{a}_1 \cdots \mathbf{a}_{2g}\end{matrix}\right]=\left[\begin{matrix}\mathbf{b}_1 \cdots \mathbf{b}_{2g}\end{matrix}\right]\alpha\quad\textrm{for some}~\alpha=\left[\begin{matrix}A B C D\end{matrix}\right]\in M_{2g}(\mathbb{Z})\cap\mathrm{GSp}_{2g}(\mathbb{Q}).
V_{z\ll b,\,x\ll\sqrt{Q_pb}}\to E-{{m_f}\over{2Q_pQ_F}}z(x^2+4Q_pz)^{n\over 2}+{{m_fJ^2_{\perp}}\over{2E^2Q^2_pQ^2_F}}(x^2+4Q_pz)^n,
a = \Psi_{0}(\mathrm{Int}({^\vee}\delta)_{|{^\vee}G}) = \Psi_{0}(\varsigma \circ \mathrm{Int}({^\vee}\delta)\circ \varsigma^{-1}_{|{^\vee}G}) = \Psi_{0}(\varsigma_{|{^\vee}G}) \circ a \circ\Psi_{0}(\varsigma_{|{^\vee}G})^{-1}
S_i =\left\{\frac{\theta^{n-u}q_{ij}(\theta)}{p^{\lfloor{y_u+\frac{Y_{ij}-jV(\Phi_i(x))}{e}}\rfloor}}~:~ n-e\mu_i < u\le n, b_{it}-e_{it}f_{it}< j\le b_{it}, 1\le t\le z_i\right \},
a_{d-1}' = a_{d-1}+ \binom{d}{1}t_i a_d a_{d-2}' = a_{d-2}+ \binom{d-1}{1}t_i a_{d-1}+ \binom{d}{2}t_i^2 a_d a_{d-3}' = a_{d-3}+ \binom{d-2}{1}t_ia_{d-2}+ \binom{d-1}{2}t_i^2 a_{d-1}+ \binom{d}{3}t_i^3 a_d.
\rho^2(x,y,z,\lambda)\begin{bmatrix}\frac{\partial F_1}{\partial \lambda}\frac{\partial F_2}{\partial \lambda}\frac{\partial F_3}{\partial \lambda}\end{bmatrix}=\begin{bmatrix}a_{11}a_{12}a_{13}a_{21}a_{22}a_{23}a_{31}a_{32}a_{33}\end{bmatrix}\begin{bmatrix}F_1 F_2 F_3\end{bmatrix},
S(A|X)_{\hat{\sigma}_{AX}}= \int_{\mathbb{R}^{2m}}S\left(\hat{\sigma}_{A|X=\mathbf{x}}\right)\mathrm{d}p_X(\mathbf{x}) = S(\hat{\rho}_A)\;, S(B|X)_{(\Phi\otimes\mathbb{I}_X)(\hat{\sigma}_{AX})}= \int_{\mathbb{R}^{2m}}S\left(\Phi(\hat{\sigma}_{A|X=\mathbf{x}})\right)\mathrm{d}p_X(\mathbf{x}) = S(\Phi(\hat{\rho}_A))\;,
\frac{\partial \Lambda_t}{\partial t}:=\sum_{\alpha,\beta=1}^n\left[\sum_{\lambda=1}^{m}c_{\lambda}\frac{\partial g_{\alpha \beta}^{j}(z_j,t)}{\partial t_{\lambda}}\right] \frac{\partial}{\partial z_{j}^{\alpha}}\wedge \frac{\partial}{\partial z_{j}^{\beta}}
I[f]=\int_{x_{0}}^{x_{1}}L(x,f,f^{\prime},f^{\prime\prime},...,f^{(k)})dx;f^{\prime}:=\frac{df}{dx},f^{\prime\prime}:=\frac{d^{2}f}{dx^{2}},f^{(k)}:=\frac{d^{k}f}{dx^{k}}
(\omega_{k+1,k+2})_{11}=I_q, (\omega_{k+1,k+2})_{12}=J_{k+1,k+1}A-h_{k}^{-1}(\hat{H}_{k+1})_{12}A^{\top}A, (\omega_{k+1,k+2})_{21}=0, (\omega_{k+1,k+2})_{22}=I_q-h_{k}^{-1}(\hat{H}_{k+1})_{22}A^{\top}A.
L_s^2 = 1 - q^{-k}(E_s - L_sE_s) \underbrace{L_sL_tL_sL_t\dots}_{m_{st}}= \underbrace{L_tL_sL_tL_s \dots}_{m_{st}}R_tL_s = L_sR_{t'}, t' = sts^{-1}(t \in T(\mathbb{F}_{q^k})) R_{t_1}R_{t_2}= R_{t_1t_2}(t_1, t_2 \in T(\mathbb{F}_{q^k})).
h(z) =\left(\frac{z-a}{z\zeta(z)}\right)^{c}\left(\frac{a(1-a^2)^{c-1}}{N^{1-c}\Gamma(c)}\frac{1}{z-a}\right)-\frac{1}{\zeta(z)\Gamma(c)}={\cal O}\left(\frac{c}{N}\right).
\mathbf{h}_l^{(g_k)}=\left(\rho_l^{(g)}\right)^{1/2}\mathbf{U}_l^{(g)}\left(\boldsymbol{\Lambda}_l^{(g)}\right)^{1/2}\mathbf{c}_l^{(g_k)},\;l=0,\ldots,L_g-1,
+ \frac 5 8 F^{\lambda \nu}F^{\mu \rho}F_{\lambda \mu}F_{\nu \rho}+ 2 F_{\mu \rho}~^{; \rho}F^{\mu \lambda}~_{; \lambda}- 2 ( F_{\rho}~^{\nu}F^{\rho \mu})_{; \nu \mu}+ ( F_{\nu \rho}F^{\nu \rho})^{; \mu}~_{\mu}~ ~ .
\mathbf{x}_{\mathrm{m}}(i) = \sum_{z=1}^{Z}\alpha_z(i)(\mathbf{a}_d(\gamma_z)\otimes \mathbf{a}_{\tau}(\eta_z)) &=\mathbf{A}(\boldsymbol{\gamma},\boldsymbol{\eta})\boldsymbol{\alpha}(i)
P_{-3K_{X^{(1)}}}:= \left<(0,-2,-3), (-3,0,0), (-3,1,3), (0,0,3), (3,-2,0),\right. &\left. (0,2,3), (0,0,-3), (-3,2,0), (-3,3,3), (3,0,0), (3,-1,-3), (3,-3,-3)\right>
\frac{\rho_2}{\rho_0}=1+{2}\epsilon+O(\epsilon^2), &\frac{\mbox{\scriptsize{U}}_2}{c_0}={2\kappa_0\cos\alpha\cos(\theta-\alpha)}{\epsilon}+O(\epsilon^2), &\frac{\mbox{\scriptsize{V}}_2}{c_0}={-2\kappa_0\cos\alpha\sin(\theta-\alpha)}{\epsilon}+O(\epsilon^2), &\tan\psi=\tan(\pi/2-\alpha)+O(\epsilon),~~~~~~ \alpha<\theta<2\alpha.
\Delta_0 \colon F_0 \to F_0\otimes F_0 \Delta_0(1) = 1\otimes 1, \Delta_1 \colon F_1 \to (F_1\otimes F_0)\oplus (F_0\otimes F_1) \Delta_1(1) = \underbrace{1\otimes s}_{F_1\otimes F_0}+ \underbrace{1\otimes 1}_{F_0\otimes F_1}
B_{[H_1]}:=O_1 \cup O_2, \B_{[H_2]}:=O_3, \B_{[H_3]}:=O_4, \B_{[H_4]}:=O_5, \B_{[H_5]}:=O_6 \cup O_7; & \alpha_{[H_i]}(G;A) = 2, i \in \{1,5 \}, \alpha_{[H_i]}(G;A) = 1, i \in \{2,3,4 \}.
u^{S^N(\mathbf{\xi})}_t(y)= \displaystyle \frac{1}{N}\sum_{j=1}^N K(y-\xi^{j,N}_t) \exp \left \{\int_0^t\Lambda \big (s,\xi^{j,N}_s,u^{S^N(\mathbf{\xi})}(s,\xi^{j,N}_s) \big ) ds\right \},
\epsilon^{\mu \nu \rho \sigma}\partial_\nu R_{\rho \sigma}= \epsilon^{\mu \nu \rho \sigma}\partial_\nu (\vec{\Gamma}_\rho \times \vec{\Gamma}_\sigma) \vec{T}= - \frac{1}{2}\epsilon^{\mu \nu \rho \sigma}\partial_\nu (\partial_\rho \Gamma_\sigma - \partial_\sigma \Gamma_\rho) = 0
\int d^3 \zeta - \frac{1}{6}\epsilon^{ijk}\Pi_i^A \Pi_j^B \Pi_k^C B_{CBA}= \int \Big[ h^{-1}- h^{-2/3}\bar{\theta}(\Gamma^{\tilde{0}}\partial_0 + \Gamma^{\tilde{1}}\partial_1 + \Gamma^{\tilde{2}}\partial_2 ) \theta
\widehat{\tilde{T}}\left( \xi ,s\right) =\hat{T}_{0}\left( \xi \right) \frac{\Phi \left( s\right)}{\xi^{2}+s\Phi \left( s\right)}\;\;\;\;\widehat{\tilde{q}}\left( \xi ,s\right) =-\hat{T}_{0}\left( \xi \right) \frac{\mathrm{i}\xi}{\xi^{2}+s\Phi \left( s\right)}.
\Delta \geq \sum_{j=1}^r\left[\left( N(l_j,1)-\left(1-\frac{1}{l_j}\right)N(1,1)\right)+ \left( N(1,l_j)-\left(1-\frac{1}{l_j}\right)N(1,1)\right)\right]-2 \varepsilon(q-1).
u^{*}\frac{\partial u^{*}}{\partial s^{*}}+v^{*}\frac{\partial u^{*}}{\partial y^{*}}=[\frac{g\beta(T_{s}-T_{0})L_{c}^{3}}{\nu^{2}}]\frac{T^{*}}{Re_{L}^{2}}+\frac{1}{Re_{L}}\frac{\partial^{2}u^{*}}{\partial{y^{*}}^{2}}
f_i(P_{k,\ell}^m) = \binom{k+\ell+m+2}{i+1}- \binom{\ell+m+1}{i-k}- \binom{k+m+1}{i-\ell}+ \binom{m+1}{i-k-\ell}\qquad\textrm{for}-1\le i\le k+\ell+m.
\left( Y^{\mu}\right) ( \tau , \sigma^{1}, \sigma^{2}) = \left( \begin{array}{c}{{\tau}}\\{{Y^{1}( \tau , \sigma^{1}, \sigma^{2})}}\\{{Y^{2}( \tau , \sigma^{1}, \sigma^{2})}}\\{{Y^{3}( \tau , \sigma^{1}, \sigma^{2})}}\\ \end{array}\right) ,