harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_s^{t,x}= \v + (\lambda+1) \int_t^s \xi(r,\psi(r, V_r^{t,x})) \mathrm dr &+\int_t^s (\nabla \xi(r,\psi(r, V_r^{t,x})) + \mathrm{I}_d) \mathrm dW_r,
\vec{u}_n(x) = \sum_{j=1}^J g_n^j e^{it_n^j \Delta_{\mathbb{R}^2}}\vec{\phi}^j + \vec{w}_n^J(x) := \sum_{j=1}^J \frac1{\lambda_n^j}e^{ix\xi_n^j}(e^{it_n^j \Delta_{\mathbb{R}^2}}\vec{\phi}^j)\left(\frac{x-x_n^j}{\lambda_n^j}\right) + \vec{w}_n^J(x ),
(\sum_{j=0}^NX_j^*X_j u,u)=\sum_{j=0}^N|\!|X_ju|\!|_0^2\geq c_K|\!|u|\!|_{1/2}^2-C_K|\!|u|\!|_0^2,\,\,\,\,\forall u\in C_0^\infty(K),
B=\left(\begin{array}{cccccc}0 0 -1 1 -1 1 0 0 1 -1 1 -1 1 -1 0 1 0 -1 -1 1 -1 0 1 0 1 -1 0 -1 0 1 -1 1 1 0 -1 0 \end{array}\right) \;.
{\mathcal{L}}= - \frac{1}{8}\int{d^{2}\Theta 2{\mathcal{E}}\left( \bar{\mathcal{D}}^{2}- 8 R \right) f}- \frac{1}{8}\int{d^{2}\bar{\Theta}2 \bar{\mathcal{E}}\left({\mathcal{D}}^{2}- 8 R^{\dagger}\right) f}.
\aligned|X_t|^2 =|x_0|^2+\int_0^t\Big(2\langle X_s,b(s,X_s)\rangle+||\sigma(s,X_s)||^2\Big)ds &\quad+\int_0^t\Big(\int_U(|f_1|^2(s,X_s,u)+F_2(s,u)\Big)\nu(du)ds &\quad+2\int_0^t\langle X_s,\sigma(s,X_s)dB_s\rangle+\sum_{i=1}^2\int_0^{t+}\int_U F_i(s-,u)\tilde{N}_k(ds,du),\endaligned
\big|\pi_{i=1 \sim N_c}\big| = \Lambda_T^2, \ \ \big|\pi_{i=N_c+1 \sim N_f}\big| = \xi\big|\pi_{i=1 \sim N_c}\big| , \ \|\pi_B| = |\pi_{\bar B}| \sim \Lambda_T^{N_c}, \ big|\pi_\lambda\big| \sim \Lambda^3\xi^{\frac{\rho+N_f-N_c}{N_f-N_c}},
z_1(\xi) = r_{\xi}\left(\omega_{\xi}^{\ast}- \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}\right) & = r_{\xi}\left(\frac{1 - \alpha \xi - \frac{\alpha}{2}\xi^{\alpha}}{1 - \alpha}\right) & = \ldots = \frac{1}{(1-\alpha)^{1 - \alpha}\cdot\alpha^{\alpha}}\cdot \frac{\xi - 1 + \frac{\alpha}{2}\xi^{\alpha}}{(\xi - 1 + \frac{1}{2}\xi^{\alpha})^{\alpha}},
\begin{array}{rcl}\Delta(k_i)=k_i \otimes k_i~, begin{align*}2mm] \Delta(X^\pm_i)=X^-_i \otimes 1 +k_i^{-1}\otimes X^-_i~, begin{align*}2mm] \Delta(X^+_i)=X^+_i \otimes 1 + k_i^{-1}\otimes X^+_i~.\end{array}
\sum_{\vec{V}_{r}( L )}\cdot \prod_{i = 1}^{\nu - 2}\left[ \begin{array}{c}{\frac{1}{2}( K_{\nu - 2}\cdot \vec{m}+ \vec{u}_{r}( L ) )_{i}+ \frac{1}{2}L \delta_{i , \nu - 2}}\\{m_{i}}\\ \end{array}\right] =
\frac{\hbar^2D}{4M}\bigg\|\nabla\!\bigg(\frac{\tilde\rho}{D}\bigg)\bigg\|^2=\frac{\hbar^2D}{4M}\bigg\|\frac{\nabla\tilde\rho}{D}-\frac{\nabla D}{D^2}\tilde\rho\bigg\|^2=\frac{\hbar^2}{4M}\left(\frac{\|\nabla\tilde\rho\|^2}{D}-\frac{|\nabla D|^2}{D}\right).
|S| \lesssim \frac{1}{|w|\left|\left\{\vec{v}\,:\, \widetilde{b}(\vec{v}) \le |w| \right\}\right|^2}\, + \frac{1}{2^{N+1}}\left\vert \int_{\frac{\pi}{|u|}}^{\infty}e^{-iu\tau}\int_0^{\frac{\pi}{|u|}}\cdots\int_0^{\frac{\pi}{|u|}}F^{(N+1)}\left(\tau + s_1+\ldots+ s_{N+1}\right)\,ds_1\,\cdots\,ds_{N+1}\,d\tau \right\vert.
- \Delta \left( \phi_{a,R_{n}}* \varphi^{2}\right) + a^{2}\left( \phi_{a,R_{n}}* \varphi^{2}\right) = 4 \pi \left( m_{R_{n}}* \varphi^{2}- u^{2}_{a,R_{n}}* \varphi^{2}\right).
[\bigoplus_{I}\sum_{p \in \mathbb{Z}}(p + \iota_{E}(\omega))\eta_{I,p}] = L_{\omega}([\bigoplus_{I}\eta_{I}]) = \nabla_{\omega}(\iota_{E}([\bigoplus_{I}\eta_{I}])) = \nabla_{\omega}([\bigoplus_{I}\iota_{E}(\eta_{I})]).
\frac{\partial}{\partial y^1_k}= \frac{\partial}{\partial x^1}+ \sum_l \bigg(\frac{\partial h^k_l}{\partial y^1_k}\bigg) \frac{\partial}{\partial x^l}+ \sum_\tau \bigg(\frac{\partial \omega^k_\tau}{\partial y^1_k}\bigg) \frac{\partial}{\partial \xi^\tau},
M = \left( \begin{array}{c c}{{\cosh t \, \sin \theta \, e^{i \phi}}}&{{i ( \sinh t + \cosh t \, \cos \theta )}}\\{{i ( - \sinh t + \cosh t \, \cos \theta )}}&{{\cosh t \, \sin \theta \, e^{- i \phi}}}\\ \end{array}\right) \, ,
2 V^{- 1}< H > = \frac{1}{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\left[ D^{- 1}( p ) + ( p^{2}+ 4 \kappa^{2}) D ( p ) \frac{\xi}{\kappa p^{2}D ( p ) + \xi}\right]
\lim\limits_{n\to \infty}n\P\left(X_{t_{2r+1}}^{\left(n\right)}\right. \left.<0,\,\ldots, X_{t_{[Tn]-1}}^{\left(n\right)}<0,\, X_{T}^{\left(n\right)}=j|X_{t_{2r}}^{\left(n\right)}=0\right) &=\sqrt{\frac{2}{\pi}}\frac{2q|z|}{(T-t)^{3/2}}e^{-\frac{z^2}{2(T-t)}}=4qh(T-t, z).
\pi(x) = \frac{x}{\log x}+ \frac{x}{\log^2 x}+ \frac{2x}{\log^3 x}+ \frac{6x}{\log^4 x}+ \ldots + \frac{(n-1)! x}{\log^nx}+ O \left( \frac{x}{\log^{n+1}x}\right)
\sum_{x \in \mathbb N}\overline{\eta_t^n}(x) f\big(\tfrac{x}{n}\big)\; = \; \sum_{x \in \mathbb N}\overline{\eta_0^n}(x) f\big(\tfrac{x}{n}\big)\;+\;\frac{1}{n}\sum_{x \in \mathbb N}\overline{J_t^n}(x) \nabla^n_x f\;+\;f\big(\tfrac{1}{n}\big)\overline{J_t^n}(0).
\Gamma \sim \frac{\kappa_{5}^{2}L_{e f f}\cos^{2}\psi}{T_{e f f}^{2}\omega}\int_{- \infty}^{\infty}d p_{1}d p_{2}d p_{3}\delta ( p_{1}+ p_{2}+ p_{3}- \frac{\omega}{2}) \prod_{i = 1 , 3}\frac{p_{i}}{1 - e^{- \frac{p_{i}}{T_{L}}}}\times \frac{\omega}{2 ( 1 - e^{- \frac{\omega}{2 T_{R}}})},
X_{i j}( x , y ) = e^{2}\left( \begin{array}{c c c c}{- 1}&{0}&{0}&{0}\\{0}&{- 1}&{0}&{0}\\{e^{2}}&{0}&{1}&{0}\\{0}&{e^{2}}&{0}&{- 1}\\ \end{array}\right) \delta ( x - y ) \equiv X_{i j}^{0}\delta ( x - y ) .
\int_{0}^{R}r|\log r|^{p}dr =\int_{0}^{1}\left(-r^{\frac{1}{p}}\log r\right)^{p}dr+\int_{1}^{R}r(\log r)^{p}dr &\leq \int_{0}^{R}\left[\left(\frac{p}{e}\right)^{p}+R(\log R)^{p}\right]dr; &= \left(\frac{p}{e}\right)^{p}R+R^{2}(\log R)^{p}&\leq C_{p}R^{2}\left[1+|\log R|^{p}\right]
\begin{array}{l c l l}{{\mathrm{f l o o r}}}&{{\mathrm{S U ( 2 / 1 )}}}&{{\mathrm{f i e l d}}}&{{\mathrm{h e l i c i t y}}}\\{{\mid \mathrm{g n d}>}}&{{( 0 ~ + \frac{1}{2})}}&{{\phi}}&{{~ 0}}\\{{\mid \mathrm{1 s t}>}}&{{( 0 ~ + \frac{1}{2})}}&{{2 \lambda}}&{{- \frac{1}{2}}}\\{{\mid \mathrm{2 n d}>}}&{{( 0 ~ + \frac{3}{2})}}&{{A_{\mu}}}&{{- 1 .}}\\ \end{array}
L'_{ij}(\delta) = \frac{\prod_{k=1}^{j-i}c_{(\delta_k+1)(i-1+k)}\prod_{k=1}^{j-i}\prod_{t=\delta_{k-1}+1}^{\delta_k-1}p^{m_{(t+1)(i-1+k)}}}{\prod_{k=1}^{j-i}c_{(\delta_k+1)(i+k)}\prod_{k=1}^{\delta_{j-i}}p^{m_{(k+1)j}}}, R'_{ij}(\partial) = R_{ij}(\partial).
\frac{1_{F\times F^c}(x,y)}{|x-y|^{n+s_h}}\le 1_{(F\times F^c)\cap\{|x-y|>1\}}(x,y)+\frac{1_{F\times F^c\cap\{|x-y|\le 1\}}(x,y)}{|x-y|^{n+s_{**}}}=:g(x,y)\,,
[\psi(x, t)]_{\eta, B_{\mu_jR}(\xi_j)}\lesssim \frac{t_0^{-\varepsilon}}{R^{a-2s}}\sum_{j=1}^k\frac{\mu_j^{-\eta}\mu_0^{\frac{n-2s}{2}+\sigma}(t)}{1+|y_j|^{a-2s+\eta}}|y_j|\leq 2 R,
\left| \frac{f(x(1+z)) - f(x)}{x f'(x)}\right|= \left| \int_{|1+z|}^1 \frac{g(xy)}{g(x)}y^{-1}\d y \right|\leq 2 \int_{|1+z|}^1 y^{-1-\varepsilon}\d y\leq \frac{2}{\varepsilon}|1+z|^{-\varepsilon},
P_{Y^{i-1},{\bf S}}(y^{i-1},{\bf s}) = P_{Y_{i-1}| Y^{i-2},{\bf S}}(y_{i-1}| y^{i-2},{\bf s}) P_{Y^{i-2},{\bf S}}(y^{i-2},{\bf s}) & = \prod_{j=1}^{i-1}\beta(y_{j}, y^{j-1}).
\big(\tau_{k}^{(\alpha)}\big)^2\big(\tau_{k+2}^{(\alpha-1)}\tau_{k}^{(\alpha+1)}-\tau_{k+1}^{(\alpha-1)}\tau_{k+1}^{(\alpha+1)}\big)=\big(\tau_{k+1}^{(\alpha)}\big)^2 \big(\tau_{k+1}^{(\alpha-1)}\tau_{k-1}^{(\alpha+1)}-\tau_{k}^{(\alpha-1)}\tau_{k}^{(\alpha+1)}\big).
F_{2 k}( r ) ={\frac{r}{( r^{2}- n^{2})^{k}}}\int^{r}\left[{\frac{( s^{2}- n^{2})^{k}}{s^{2}}}+{\frac{2 k + 1}{l^{2}}}{\frac{( s^{2}- n^{2})^{k + 1}}{s^{2}}}\right] \, d s .
\begin{cases}B\partial_{t}u+Bu\partial_{x}u-ABh\partial_{x}\tilde{h}+(1-A)Bh\partial_{y}\tilde{h}\partial_{y}u-ABh^2\partial_{y}^2u=0, A\partial_{t}\tilde{h}+Au\partial_{x}\tilde{h}-ABh\partial_{x}u+(1-B)Ah(\partial_{y}\tilde{h})^2-ABh^2\partial_{y}^2\tilde{h}=0, (u, \tilde{h})|_{t=0}=(u_0, \tilde{h}_0)(x, y),\end{cases}
C_1:= \mathrm{Id}_2,\C_{2}:= \left( \begin{tabular}{cc}$0$ $0$ $1$ $0$\end{tabular}\right) ,\D:= \left( \begin{tabular}{cc}$1$ $0$ $0$ $-1$\end{tabular}\right) , \E := \left( \begin{tabular}{cc}$0$ $1$ $0$ $0$\end{tabular}\right) .
\sum_{k=0}^{p-1}\bigg(\frac{(\frac12)_k}{k!}\bigg)^2\cdot(-1)^k\equiv p^2\sum_{k=\frac{p-1}2}^{p-1}\bigg(\frac{k!}{(\frac32)_k}\bigg)^2\cdot(-1)^k\pmod{p^2}
\log \varrho\left(L_0^{\phi_1},L_1^{\phi_1}\right) = \Lambda(\phi_1,\mu) \leq \Lambda(\phi_2,\mu)+\Lambda(\phi_1-\phi_2,\mu) &\leq \Lambda(\phi_2,\mu) + |\phi_1-\phi_2|_\infty &\leq \log\varrho\left(L_0^{\phi_2},L_1^{\phi_2}\right)+|\phi_1-\phi_2|_\infty
d_{01}=\begin{cases}\min \left\{\frac{MN_T}{N_R},\frac{qMN_T}{N_T+qN_R-q}\right\},{\rm if~}\lfloor\frac{N}{M}\rfloor= q \min \left\{N,\frac{qNN_T}{qN_T+N_R-q}\right\},{\rm if~}\lfloor \frac{M}{N}\rfloor=q \end{cases},
\begin{array}{l}\bar{\Phi}_{i}\bar{\Phi}_{j}=q_{ij}\bar{\Phi}_{j}\bar{\Phi}_{i}, \Phi_{i}\Phi_{j}=p_{ij}^{-1}\Phi_{j}\Phi_{i}, \Phi_{i}\bar{\Phi}_{i}=1+p_{ij}q_{ij}\bar{\Phi}_{i}\Phi_{i}+(p_{ij}q_{ij}-1)\sum_{j=i+1}^{N}\bar{\Phi}_{j}\Phi_{j}, \Phi_{i}\bar{\Phi}_{j}=p_{ij}\bar{\Phi}_{j}\Phi_{i}, \Phi_{j}\bar{\Phi}_{i}=q_{ij}\bar{\Phi}_{i}\Phi_{j},\end{array}
\Psi_{t,\Theta_+}(\xi)=(1+\|\xi\|^2)^{t/2}\varphi_+\left (\frac{\xi}{\|\xi\|}\right )\quad\mbox{and}\Psi_{v,\Theta_-}(\xi)=(1+\|\xi\|^2)^{v/2}\varphi_-\biggl (\frac{\xi}{\|\xi\|}\biggr )\, .
\int_{V^{(4)}}\langle \beta_{\tilde g^{(4)}}\delta^*_{g'}\tilde Y,\tilde Y\rangle dvol_{\tilde g^{(4)}}=-\int_{V^{(4)}}\langle \beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}Y',\tilde Y\rangle dvol_{\tilde g^{(4)}}&=-\int_{V^{(4)}}\langle Y',\beta_{\tilde g^{(4)}}\delta^*_{\tilde g^{(4)}}\tilde Y\rangle dvol_{\tilde g^{(4)}}&=0.
\mathcal{C}_{-1}(\mathcal{a}^{(x)}, \hat{\mathcal{n}}^{(x)}, i', \lbrace 1, \ldots, i'-1 \rbrace, \lbrace i'+1, \ldots, \ell^{\circ}_x \rbrace) = \mathcal{C}_{-1}(\alpha, \nu, p_{(x,i')}, \lbrace 1, \ldots, p_{(x,i')}-1 \rbrace, \lbrace p_{(x,i')}+ 1, \ldots, \ell \rbrace).
\Gamma_{( 0 )}={\frac{1}{( p + 1 ) ! ~{\sqrt - d e t G}}}\epsilon^{i_{1}. . . . . . i_{p}+ 1}\partial_{i_{1}}X^{M_{1}}. . . . . . . \partial_{i_{p + 1}}X^{M_{p + 1}}\Gamma_{M_{1}}^{\prime}. . . . ._{M_{p + 1}}.
\begin{array}{rcl}\chi_1 = A\Lambda_{1,2}+ B(\partial\Lambda)_{1,2}\,, \chi_2 = \Lambda_{0,2}\,, \chi_3 = A'\Lambda_{1,2}+ B'(\partial\Lambda)_{1,2}\,, \chi_4 = \Lambda_{2,2}\,, \chi_5 = \frac{1}{2}\Lambda_{1,2}- \frac{1}{2}(\partial\Lambda)_{1,2}\,.\end{array}
\|A_{b}^{m,k}f\|_{L^{p}(\lambda)}\leq\sup_{\|g\|_{L^{p'}(\lambda)}=1}\sum_{Q\in\mathcal{S}}\left(\int_{Q}|g\lambda||b-b_{Q}|^{m-k}\right)\frac{1}{|Q|}\int_{Q}|b-b_{Q}|^{k}|f|.
\Z = H^{- 1}\left( \begin{array}{c c c}{( 1 - z ) ( i \cal E \mathrm{+ 1 ) - \mathrm{\Phi}^{2}}}&{}&{i \cal E \it + z}\\{i \cal E \it + z}&{}&{( 1 + z ) ( i \cal E \mathrm{- 1 ) + \mathrm{\Phi}^{2}}}\\ \end{array}\right) ,
c_{i,j}:= -2(c_i+c_j) +c_{|i-j|}+ c_{i+j}t_{i,j}:= -2(s_{i}+ s_{j}) + s_{|i-j|}+ s_{i+j}u_{i,j}:=-2( u_i + u_j)+ u_{|i-j|}+ u_{i+j}v_{i,j}:=-2( v_i+ v_j)+ v_{|i-j|}+ v_{i+j}z_{i,j}:=-2(z_{i}+z_{j})+z_{|i-j|}+z_{i+j}.
\phi_{\widehat{\pi}_v}^\varepsilon(t_v)=\int_{U_n(k_v)\backslash G_n(k_v)}t_v((j\times 1)(g))\varepsilon_v(\det(g))|\det(g)|_v^{s-\frac{1}{2}}dg,\;\;\;v\in S_\infty,
\bf{A}+ \bf{B}+ \bf{C}+ \bf{A}\times \bf{B}+ \bf{A}\times \bf{C}+ \bf{B}\times \bf{C}- ( \bf{A}\cdot \bf{B}) \, \bf{C}+ ( \bf{A}\cdot \bf{C}) \, \bf{B}- ( \bf{B}\cdot \bf{C}) \, \bf{A}= 0 .
\hat{\psi}{''}(0) = - \frac{g(1)}{2^{2\alpha+1}}\left(\frac{\alpha}{3}+ \beta \right) - \frac{g'(1)}{2^{2\alpha}},\hat{\phi}{''}(\pi) = - \frac{g(-1)}{2^{2\beta+1}}\left(\alpha + \frac{\beta}{3}\right) + \frac{g'(-1)}{2^{2\beta}}.
|q_{t,\gamma}(x,y)|\leq |q_t(x,y)|\leq \frac{C_N}{t^{n/2+1}}e^{-\frac{c|x-y|^2}{t}}\Big(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\Big)^{-N}
(r+2)W^{r+2}\dot{f}_{l_1}\ddot{f}_{l_1}S_r=\dddot{f}_{l_1}\ddot{f}_{l_2}\cdots\ddot{f}_{l_r}W^2\alpha\Rightarrow (r+2)\dot{f}_{l_1}(\ddot{f}_{l_1})^2=\dddot{f}_{l_1}W^2.
F_{QR,P}(\Omega)(\alpha_j \otimes \alpha_k)= c(\sigma(k)) G_{QR,P}(dx_{\sigma(k)})(\alpha_j) = c(\sigma(k))\left((2g_C+1) \int_{\alpha_j}\log(f) dx_{\sigma(k)}- N \int_{\alpha_j}W(dx_{\sigma(k)})\right).
[v_i,[v_j,v_k]]+[v_j,[v_k,v_i]]+[v_k,[v_i,v_j]] =0 \implies [v_i,z_b]+[v_j,0]+[v_k,0] =0 \mbox{(by Equation \ref{2bracket})}\implies [v_i,z_b] = 0 \mbox{for all $v_i$ in the closed path}.
~~~~(\beta M_1^H-\alpha M_2^H)^H(\beta M_1^H-\alpha M_2^H)-(\beta M_1^H+\alpha M_2^H)^H(\beta M_1^H+\alpha M_2^H) =-\overline{\beta}\alpha M_1M_2^H-\overline{\alpha}\beta M_2M_1^H-\overline{\beta}\alpha M_1M_2^H-\overline{\alpha}\beta M_2M_1^H =-2\alpha\overline{\beta}(M_1M_2^H+M_2M_1^H) \leq0.
Y^i_t=\xi_i+\int_t^Tf_i(s, X_s)ds-\int_t^TZ_s^idW_s+\Lambda_T^i-\Lambda_t^i, &Y^i_t\geq \max_{j\in A_i}\left (Y^j_t - c_{ij}(t,X_t)\right), \int_0^T \left ( Y^i_t-\max_{j\in A_i}\left (Y^j_t -c_{ij}(t,X_t) \right ) \right )d\Lambda_t^i=0,
s_t(z) = \sum_{k=-1}^m z^{-k}\sum_{\begin{subarray}{c}S \subseteq \{0,\dots, m\}|S| = k+1 \end{subarray}}\prod_{l \in S}\gamma_l \prod_{i \notin S}\big(1 - (1-t)\gamma_i \big) = z \prod_{k=0}^m\big( \gamma_k/z+ 1 - (1-t)\gamma_k\big) .
\tau(x) - \mathbf{P}_{x'x}^{g_P}(\tau(x')) = \int_0^\ell \frac{d}{dt}\mathbf{P}^{g_P}_{\gamma(\ell-t)x}(\tau(\gamma(\ell-t))) \, dt= -\int_0^\ell \mathbf{P}^{g_P}_{\gamma(\ell-t)x}[(\nabla_{\dot{\gamma}(\ell-t)}\tau)(\gamma(\ell-t))]\,dt.
\chi_{\gamma}\left( 2\right) =(n-2)!\ \sum_{i\in{\cal I}}\frac{D(p_1,\ldots,p_{i-1},p_i - 2,p_{i+1},\ldots,p_m)}{p_1!\cdots p_{i-1}!\ (p_{i}-2)!\p_{i+1}!\ \cdots p_m!},
b_{k,i}=-\frac{t_2(i)}{2(i-k+1)}+\frac{(i+1)(m-i)}{2(i-k+1)}d_{k,i+1}-\frac{k(m+1-k)}{2(i-k+1)}d_{k-1,i}, c_{k,i}=\frac{t_3(i)}{2(i-k-1)}-\frac{1}{2(i-k-1)}d_{k,i-1}+\frac{1}{2(i-k-1)}d_{k+1,i}.
\frac{w^{n+1}_j}{a_j}=\biggl(1-\frac{a_j\Delta t}{\Delta x}\biggr) \frac{w^{n}_j}{a_j}+\frac{a_j \Delta t}{\Delta x}\frac{w_{j-1}^n}{a_j}:=\mathcal{H}(w^n_{j-1},w^n_j),
\frac{1}{2}p^{i}p_{i}+ \sum_{n = 1}^{\infty}n ( N_{b n}+ N_{c n}+ \sum_{i = d + 1}^{2 5}N_{i n}) + \sum_{r = 1 / 2}^{\infty}\sum_{a = 0}^{d}r N_{a r}={\frac{1 5 - d}{1 6}}.
\int_M f \: d \delta{c}_i(u) = \int_M \langle \nabla f, d\exp_{u \nabla_x \phi_i(0)}(u \nabla_x \sigma_i) \rangle_{F_{i,u}(x)}\: d\mu_{i,0}(x) & = u \int_M \langle \nabla f, W_{\sigma_i}(u) \rangle \: d\mu_{i,u}.
\begin{cases}P_\theta(L) \X_t=Q_\theta(L) \ \varepsilon_t, &\varepsilon_t=\sigma_t \zeta_t,\mbox{with}\sigma_t^2=c_0+\sum_{i=1}^{p'}c_i \varepsilon_{t-i}^2+\sum_{i=1}^{q'}d_i \sigma_{t-i}^2\end{cases}
C_\Theta= \sup_{h\in P\cap\Omega_{2,0}}\sup_{A\in \mathcal{A}}\frac{\|Ah\|_{q/2,u}}{\|h\|_{p/2,v}}\quad C_\Theta\le \sup_{h\in P\cap\Omega_{2,0}}\bigg(\int_0^\infty h(t)^{q/2}u^o(t)\,dt\bigg)^{2/q}\bigg(\int_0^\infty h(t)^{p/2}v(t)\,dt\bigg)^{-2/p}.
p_n(x^2;q^{-1};q^2) =\frac{(-1)^n q^{-n(n-1)}}{(q;q^2)_n}\,h_{2n}(x;q), xp_n(x^2;q;q^2) =\frac{(-1)^n q^{-n(n-1)}}{(q^3;q^2)_n}\,h_{2n+1}(x;q).
\le \int_{T}^{2T}\prod_{1\le i \le j}(\exp (k \Re G_{i, j}(t))^2 (\beta_{j+1}^{3/4}|\Re G_{j+1,\ell}(t) |)^{M}dt &\ll (\beta_{j+1}^{3/2})^{[1/(10\beta_{j+1})]}\int_{T}^{2T}\prod_{1\le i \le j}\left( \sum_{0\le n \le 100k \beta_{i}^{-3/4}}\frac{(k\Re G_{i,j}(t))^n}{n!}\right)^2 (\Re G_{j+1,\ell}(t) )^{M}dt,
m_\lambda = P_\lambda(x;0,1) \hbox{are the \emph{monomial symmetric polynomials}}, &s_\lambda = P_\lambda(x;0,0) \hbox{are the \emph{Schur polynomials}}, &P_\lambda(x;0,t) \hbox{are the \emph{Hall-Littlewood polynomials}}.
\frac{\dot{\sigma}}{R^2-1}= \frac{\dot{\sigma}G^2}{R^2 W^2}=-\frac{\dot{\sigma}\eta_G^2 \eta_U}{2 \sigma_0 R^2 W^2}\frac{1}{U}\approx -\frac{\dot{\sigma}\eta_G^2 \eta_U}{2 \sigma_0}\frac{1}{\tilde{U}}.
\widehat{P_{\leq M}f}(\xi) := \varphi(M^{-1}\xi) \hat{f}(\xi), \widehat{P_{>M}f}(\xi) := (1-\varphi(M^{-1}\xi)) \hat{f}(\xi), \widehat{P_M f}(\xi) := (\varphi(M^{-1}\xi) - \varphi(2M^{-1}\xi)) \hat{f}(\xi),
|P_n^1| = n + 1 |P_n^2| = n^2 + 2n + 1 |P_n^3| = \frac{2}{3}n^3 + 2n^2 + \frac{7}{3}n + 1 |P_n^4| = \frac{1}{3}n^4 + \frac{4}{3}n^3 + \frac{8}{3}n^2 + \frac{8}{3}n + 1
G_n(x;h)= \frac{\left|\begin{array}{cccc}F(x) F(x+h) \cdots F(x+nh) f(x) f(x+h) \cdots f(x+nh) \vdots \vdots \vdots f(x+(n-1)h) f(x+nh) \cdots f(x+(2n-1)h)\end{array}\right|}{\left|\begin{array}{cccc}1 1 \cdots 1 f(x) f(x+h) \cdots f(x+nh) \vdots \vdots \vdots f(x+(n-1)h) f(x+nh) \cdots f(x+(2n-1)h)\end{array}\right|},
trace(T) =\sum<Te_{i},e_{i}>=\sum_{i}<T(\int_{\mathcal{M}_{j}}<e_{i},m>mdm,e_{i}> & =\sum_{i}<\int_{\mathcal{M}_{j}}<e_{i},m>Tmdm,e_{i}>=\sum_{i}\int_{\mathcal{M}_{j}}<Tm,\overline{<e_{i},m>}e_{i}>dm & =\int_{\mathcal{M}_{j}}<Tm,\sum_{i}<m,e_{i}>e_{i}>dm=\int_{\mathcal{M}_{j}}<Tm,m>dm
\Big|\frac{D(-r)(X;z^{\prime},\bar{z}^{\prime})}{-r(X;z^{\prime},\bar{z}^{\prime})}\Big| =\frac{-\sum_{j=1}^{n}\alpha_j\frac{|z_j|^2}{(1-|w|^2)^{\alpha}}r_j(X;z^{\prime},\bar{z}^{\prime})}{-r(X;z^{\prime},\bar{z}^{\prime})}&<\frac{c\sum_{j=1}^{n}|z_j|^{2}r_j(X;z^{\prime},\bar{z}^{\prime})}{-r(X;z^{\prime},\bar{z}^{\prime})}
2iH_A^{\ \C}\sigma_{C \dot B}^{\ \ \m}+2i H_{\dot B}^{\ \ \dot C}\sigma_{A \dot C}^{\ \ \m}-2i H_{A \dot B}^{\ \ \m}-\nabla_AH_{\dot B}^{\m}-\nabla_{\dot B}H_A^{\m}=0
|\pi_{\xi}^{\perp}(\eta_\circ)|^2 = |\eta_\circ|^2 - |\pi_{\xi}(\eta_\circ)|^2 = 1 - |\langle \xi_\circ, \eta_\circ \rangle \cdot \xi_\circ|^2 = 1 - |\langle \xi_\circ, \eta_\circ \rangle|^2 \, .
\Delta_{1}= \int d^{3}x \overline{{\psi}}D^{2}\psi \, , \qquad \Delta_{2}= \int d^{3}x ( \overline{{\psi}}\psi )^{2}\quad \mathrm{a n d}\quad \Delta_{3}= \int d^{3}x ( \overline{{\psi}}\gamma^{\mu}\psi ) ( \overline{{\psi}}\gamma_{\mu}\psi ) \, ,
S_{eff}=\int d^4x \, \sqrt{-g}\, \left(-\frac{1}{2\lambda_1 h^2}\sigma^2 +\frac{1}{2\xi_1 h^2}R \sigma^2 -\frac{1}{4! \lambda_2 h^4}\sigma^4 \right) -i \ln \det\left(i\gamma^\mu (x) \nabla_\mu - \sigma \right),
S_{\mathrm{\footnotesize g f}}= - \frac{1}{6 4 e^{2}}\int d^{4}x d^{4}\theta \Bigg ( V D^{2}\bar{D}^{2}\Big ( 1 + \frac{\partial^{2 n}}{\Lambda^{2 n}}\Big ) V + V \bar{D}^{2}D^{2}\Big ( 1 + \frac{\partial^{2 n}}{\Lambda^{2 n}}\Big ) V \Bigg ) ,
\frac{1}{n \log \alpha}\log \frac{\sigma(x)(B \cap B_{\alpha^n}^T)}{\sigma(x)(B)}\geq \frac{1}{n \log \alpha}\sum_{i=0}^{p-1}\tau(\alpha_X^{-i}x) - \frac{1}{n\log \alpha}\sum_{i=0}^{p-1}\phi_{\alpha^{n-i}}(\alpha_X^{-i}x)\text.
\begin{array}{c c}{{\cal B}{\cal M}{\cal B}^{- 1}= -{\cal M}^{\dagger}}&{{\cal B}= \left( \begin{array}{c c c}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{- 1}\\ \end{array}\right)}\\ \end{array}
\nabla T=(\nabla T_{x},\nabla T_{y},\nabla T_{z})=[\begin{matrix}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{matrix}],whereT_{uv}\equiv\frac{\partial T_{u}}{\partial v}.
R_{\mathrm{m a s s i v e}}^{\mathrm{b o u n d a r y}}( r ) = A_{\mathrm{o u t}}\frac{1}{r^{\beta}}\left( 1 - \frac{i}{\pi}\left( \frac{2 r_{H}}{r}\right)^{1 - 2 \beta}\right) + A_{\mathrm{i n}}\frac{1}{r^{\beta}}\left( 1 + \frac{i}{\pi}\left( \frac{2 r_{H}}{r}\right)^{1 - 2 \beta}\right) ,
Q(X,L,\tau)=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^m}(\widetilde{V_C})\right]\right\}^{(4k+2)}.
F_n(z) = L_{\alpha}(n^3 f(z)) D_2^n(z), E_n(z) = N(z) \begin{pmatrix}z^{\frac{2\beta}{3}}0 0 0 z^{-\frac{\beta}{3}}0 0 0 z^{-\frac{\beta}{3}}\end{pmatrix}F_n^{-1}(z) T_{\alpha}, E_{in}^{(1)}(z) = \frac{\sqrt{3}}{2\pi}n^{2\beta}\left( \frac{f(z)}{z}\right)^{\frac{2\beta}{3}}E_n(z), E_{out}^{(1)}(z) = \mathbb{I}.
A:=-\langle e_1,\nabla_H(\frac{X_3u}{|\nabla_Hu|})\rangle-\overline{p}[X_1(\overline{p})+X_2(\overline{q})]-\overline{p}^2\frac{(X_3u)^2}{l^2}+2\overline{q}\frac{X_3u}{l}.
f_t \circ \left( \int_{0}^{\infty}\int_{0}^{\infty}\alpha (t_1,t_2) dt_1 dt_2 \right) = \int_{0}^{\infty}\int_{0}^{\infty}f_{T(t_1,t_2)}\circ ( \alpha ( t_1 , t_2 ) ) (x) dt_1 dt_2
\rho_{2}u_{tt}- D \tfrac{1 - \mu}{2}P \triangle u + K u = 0 \;\, (0, \infty) \times \Omega, u = 0 \;\, (0, \infty) \times \Gamma, u(t, \cdot) = u_{0}:= P v_{0}, \; u_{t}(t, \cdot) = u_{1}:= P v_{1}\Omega.
n(x)-f(x,y,v)v =0, f(x,y,v)v-\frac{a}{G_{1}}\varphi_{1}(y)-\frac{p}{G_{1}}\varphi_{1}(y)\varphi_{2}(z) =0, k\varphi_{1}(y)-\frac{u}{G_{2}}v =0, c\varphi_{1}(y)\varphi_{2}(z)-\frac{b}{G_{3}}\varphi_{2}(z) =0.
\frac{\sqrt{5b_n'}(2j)! \omega^{2n+1}}{2(2n)!}\,{n\brace j}_{1/2}= \frac{1}{\sqrt{2\pi}}~ e^{-x^2/2}~\left( 1 + \frac{c_n' (x^3 - 3x)}{6\sqrt{n}}\right) + o\, \left( \frac{1}{\sqrt{n}}\right),
\lambda(\kappa_1 + \kappa_2 + 2) (\kappa_1 + \kappa_2 + 1)\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz \leq (p - 1) \| (y_1, y_2)\|_{1,p}+ (q - 1)\| (\nabla y_1, \nabla y_2)\|_{q,\eta}+ \nu\int_{\Omega}\left[a_1\vert y_1 \vert^{1 - \nu}+ a_2\vert y_2 \vert^{1 - \nu}\right]dz.
\Phi^{\alpha}\rightarrow V \Phi^{\alpha}= \left( \begin{array}{c}{{- f^{\alpha}( r ) e^{i \theta}+ f^{\beta}( r )}}\\{{f^{\alpha}( r ) e^{i \theta}+ f^{\beta}( r )}}\\ \end{array}\right) = \left( \begin{array}{c}{{\phi^{1}( r , \theta )}}\\{{\phi^{2}( r , \theta )}}\\ \end{array}\right) .
\frac{1}{\epsilon}\int_Cd^n\eta D_c(\eta_1 - \eta)\big{[}\frac{L^2 - \frac{1}{2}n(n-2)}{2a^2}\big{]}_{n=4}(cos{\theta_4}- cos{\theta_4^*})^lD_c(\eta - \eta_2);~~~l \in N
H_{\pm}= \left( \begin{array}{c c}{{m}}&{{i e^{\mp i \theta}\left( \partial_{r}\mp \frac{i}{r}\partial_{\theta}\mp \frac{e B}{2}r \right)}}\\{{i e^{\pm i \theta}\left( \partial_{r}\pm \frac{i}{r}\partial_{\theta}\pm \frac{e B}{2}r \right)}}&{{- m}}\\ \end{array}\right) .
\mathbf{P}_n^{\alpha,\theta,T}( \Gamma(x_i : i \in \mathcal{B}\sqcup \mathcal{O}\sqcup \mathcal{U})) = \lim_{m \to \infty}m^{\# \mathcal{B}}P^{1,1-\frac{\lambda}{m},\frac{T}{m}}_{m,n}\left( \sigma(\tilde{x}^1) = \tilde{y}^1,\ldots,\sigma(\tilde{x}^p) = \tilde{y}^p \right) ,
x_{j}^{\tau}\; = \; \max_{\lambda \in \Lambda}x_{j}^{\lambda}\;{\mathrm{~ f o r ~}}j = 2 , \ldots , p \;{\mathrm{a n d}}\; y_{k}^{\tau}\; = \; \max_{\lambda \in \Lambda}y_{k}^{\lambda}\;{\mathrm{~ f o r ~}}k = 2 , \ldots , q .
\alpha(t):= x_{a}(t^{\theta})x_{b}(t)x_{a+b}(t^{\theta+1})x_{2a+b}(t^{2\theta+1}), \beta(t):= x_{a+b}(t^{\theta})x_{3a+b}(t), \gamma(t):= x_{2a+b}(t^{\theta})x_{3a+2b}(t), \tau:=\tau(1) = \pi_{a+b}(1)\pi_{3a+b}(1),
\varepsilon^{( 1 )}( x ) = \frac{g_{\varepsilon}\lambda^{2}}{4 \pi c \tau_{S}}\frac{\cos ( \sqrt{\omega_{S}^{2}/ c^{2}+ 1 / \lambda^{2}}| \vec{x}- \vec{x}_{S}| - \omega_{S}t )}{| \vec{x}- \vec{x}_{S}|}.
\Big\| \sum_{j = 1}^\infty \alpha_j u_j\chi_{B_{R_{j+1}}}\Big\|_{p^*,q,\mu}\leq \sum_{j = 1}^\infty |\alpha_j| \|u_j\chi_{B_{R_{j+1}}}\|_{p^*,q,\mu}\leq \sum_{j = 1}^\infty |\alpha_j| \gamma_j &\leq \|\{\alpha_j\}_{j = 1}^\infty\|_{\ell_q}\|\{\gamma_j\}_{j = 1}^\infty\|_{\ell_{q'}}.
a(\mathfrak f_{\xi,N}) = \prod_{p\mid \mathfrak f_{\xi,N}}a(p^{e_p}) = \mathfrak f_{\xi,N}^{k-1/2}\prod_{p\mid \mathfrak f_{\xi,N}}\alpha_p^{e_p}= 2^{-\nu(\mathfrak f_{\xi,N})}\mathfrak f_{\xi,N}^{k-1/2}\prod_{p\mid \mathfrak f_{\xi,N}}\Psi_p(\xi;\alpha_p).
Z_{1{\rm inst}}=\frac{(-)^N}{\sqrt{2\pi N \frac{16\pi^4m^4}{A^2}\sqrt{\frac{A}{\pi^2 m^2}-1}}}({\rm e}^{i\frac{3}{4}\pi -\frac{2\pi^2 m^2N}{A}\gamma(\frac{A}{\pi^2m^2})}+{\rm e}^{-i\frac{3}{4}\pi +\frac{2\pi^2 m^2N}{A}\gamma(\frac{A}{\pi^2m^2})}) .
D^{\alpha-1}\left((P_t^{\nu_\alpha})^*(f)\right)(x) + \frac{1}{p_\alpha(x)}R^\alpha \left(p_\alpha, (P_t^{\nu_\alpha})^*(f) \right)(x) = \dfrac{- x e^{-\alpha t}}{(1-e^{-\alpha t})}(P^{\nu_\alpha}_t)^{*}(f)(x) &\quad+ \dfrac{e^{-t}}{\left(1-e^{-\alpha t}\right)}(P^{\nu_\alpha}_t)^{*}(xf)(x),
\frak m_k^{(b_0,\cdots,b_k)}(x_1,\cdots,x_k)= \sum_{\ell_0,\cdots,\ell_k}\frak m_{k+\ell_0+\cdots+\ell_k}(b_0^{\ell_0},x_1,b_1^{\ell_1},\cdots,b_{k-1}^{\ell_{k-1}},x_k,b_k^{\ell_k}).
\lambda_{( 0 ) \alpha}^{A}= \frac{1}{2}{\overline{F}}_{\mu \nu}\sigma_{\alpha}^{\mu \nu \beta}\frac{1}{\sqrt{\rho_{0}}}\left( \rho_{0}\eta_{\beta}^{A}+ ( x - x_{0})_{\mu}\sigma_{\beta{\dot{\beta}}}^{\mu}{\overline{\xi}}^{\dot{\beta}A}\right) .
T_{\psi}^{A,A,A}(X,X) =\sum_{i,j,k=1}^n \psi(\lambda_i,\lambda_k,\lambda_j)P_{\xi_i}XP_{\xi_k}XP_{\xi_j}&=\sum_{i=1}^n \psi(\lambda_i, \lambda_i,\lambda_i)P_{\xi_i}\times X^2 &=\sum_{i=1}^n \widehat{\psi}(\lambda_i)P_{\xi_i}\times X^2\,=\,\widehat{\psi}(A)\times X^2,
\left|\frac{3a\alpha^{k}2^{-n}}{1+2^{m-n}}-1\right|=\left|\frac{3\cdot2^{-n}}{1+2^{m-n}}\left(-b\beta^{k}-c\gamma^{k}-\frac{(-1)^{n}-(-1)^{m}}{3}\right)\right|.
\log D_\pm(s;1)=-\frac{1}{16}\pi^2 s^2\mp \frac{1}{4}\pi s-\frac{1}{8}\log\pi s \pm \frac{1}{4}\log 2 +\frac{1}{12}\log 2 +\frac{3}{2}\zeta^\prime(-1) + \mbox{o}(1)
\frac{n}{2}(B_{n-1}(x)+\sum_{k=1}^{n-1}\frac{B_{k}(x)}{k}\frac{B_{n-k}(x)}{n-k})-\sum_{k=0}^{n-1}(\begin{matrix}n\\k\end{matrix})\frac{B_{n-k}}{n-k}B_{k}(x)=H_{n-1}B_{n}(x).
|t_2(x)| = \bigg|\frac{2 m}{4 \pi |x|}\left( e^{i\sqrt{\lambda^2/c^2 + 2 m \lambda}|x|}- e^{i\sqrt{2 m \lambda}|x|}\right) P_+ \bigg| \leq \kappa_3(m, \lambda) \frac{2 m}{4 \pi c}e^{-\kappa_4(m, \lambda) |x|}.