harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
c_2^T=(2\pi)^{-1/6}3^{1/2}\exp \Bigg[\frac{1}{3\varepsilon}(1-\delta)^2+\frac{1}{4}-\frac{1}{2}\, \gamma +\frac{\zeta^{'}(2)}{\pi^2}-\frac{1}{9}\delta+\frac{37}{81}{\delta}^2 +{\cal O}\left({\delta}^3 \right) \Bigg]\, .
s_{W_{\mathcal{I}},X_{T^c}^k, Y_{T^c}^{k-1}}\stackrel{\eqref{eqnProperty3}}{=}p_{W_{\mathcal{I}}}p_{X_{T^c,k}|W_{T^c}, X_{T^c}^{k-1}, Y_{T^c}^{k-1}}\prod_{m=1}^{k-1}\left(p_{X_{T^c,m}|W_{T^c}, X_{T^c}^{m-1}, Y_{T^c}^{m-1}}s_{Y_{T^c,m}|X_{T^c,m}}^{(\lambda, T)}\right).
-A'_{n+1}\big(\psi^*,\, \psi,\, \phi'_{* n+1},\, \phi'_{n+1}\big)\, +\, p_{n+1}' =-A''_{n+1}\big(\psi^*,\, \psi,\, \phi'_{* n+1},\, \phi'_{n+1}\big)\, +\, p_{n+1}''
\begin{aligned}\sum_{i\in [\nu]}\delta_i^{(1)}=\sum_{i\in [\nu]}\Delta_i^{(1)}=\mu_1, \sum_{i\in [N-\nu]}\!\delta_i^{(1,2)}=\sum_{i\in [\nu]}\Delta_i^{(1,2)}=\mu_{1,2}, \sum_{i\in [N-\nu]}\!\delta_i^{(2)}=\sum_{i\in [N-\nu]}\!\Delta_i^{(2)}=\mu_2.\end{aligned}
\Psi(a_1)(a_2) = (\Psi(1_A)\cdot a_1)(a_2) = \Psi(1_A)(a_1\cdot a_2)=\Psi(1_A)(\alpha^{-1}(a_1)a_2), \Phi(a_2)(a_1) = (a_2\cdot \Phi(1_A))(a_1) = (-1)^{\bar a_1 \bar a_2}\Phi(1_A)(a_1\cdot a_2) = (-1)^{\bar a_1 \bar a_2}\Phi(1_A)(a_1\alpha(a_2)).
\sum_{\substack{k_{1}, k_2,\ldots, k_{l}\geq 0 k_{1}+k_2+\cdots+k_{l}=n}}\binom{n}{k_{1}, k_2,\ldots, k_{m}}\varphi_{k_{l}}(x)\varphi_{k_{2}}(x) \cdot \prod_{t=3}^{l}\varphi_{k_{t}}(0) = \sum_{k=0}^{n}\binom{n}{k}\varphi_{k}(x)\varphi_{n-k}(y).
q_i(x) = 4n(\log(\sum\limits_{s = 1}^{n}|w^i_s|^2)\lambda(4\frac{\sum\limits_{s = 1}^{n}|w^i_s|^2}{\delta_0^2}) + \log(\sum\limits_{s = 1}^{n}|v^i_s|^2)\lambda(4\frac{\sum\limits_{s = 1}^{n}|v^i_s|^2}{\delta_0^2})).
\alpha_{A}= (-1)^{(r+s)(t+1)}\prod_{i=1}^{r}\frac{f_{D \cup \{z_{r}, \ldots, z_{i}, x_{i-1}, \ldots, x_{1}\},S}(x_{i})}{f_{D \cup \{z_{r}, \ldots, z_{i+1}, x_{i}, \ldots, x_{1}\},S}(z_{i})},
I_{i,j}^t=\frac{f_{i-1,j}^{t}f_{i,j}^{t+1}}{f_{i,j}^{t}f_{i-1,j}^{t+1}},V_{i,j}^t=\frac{f_{i-1,j}^{t}f_{i+1,j+1}^{t}}{f_{i,j}^{t}f_{i,j+1}^{t}},
\begin{array}{l}\omega=(w_y-v_z,u_z-w_x,v_x-u_y)=\frac{1}{1-m^2a^2e^{-2ms}}\times \begin{pmatrix}\frac{m^2af}{k}e^{-ms}\sin\theta -c(m^2-k^2)ae^{-ms}\cos\theta+cma^2(m^2+k^2)e^{-2ms}fma(\cos\theta-mae^{-ms})e^{-ms}\end{pmatrix}^{T}. \end{array}
\hat{\mathcal{R}}_s:=\frac{-1}{1+s(\chi_z\eta+\chi_{\bar{z}}\bar{\eta})}[e_2(s^2\chi\chi_{\bar{z}}(\eta_t\bar{\eta}-\bar{\eta}_t\eta)\partial_t)+e_3(s^2\chi\chi_z(\eta \bar{\eta}_t-\eta_t\bar{\eta})\partial_t)]
\widetilde{\psi}_{k}^{(n^{2})}\big(x;\omega ,N,- \Vert \psi_{n}\Vert^{2}\big)=\frac{2\omega z^{(N+1/2) /2}\exp ( -z/2 )}{Q_{n}^{N}(z)}\widetilde{L}_{k,N}^{(n^{2})}( z ) ,
\mathbb{E}_{n,m}'[\beta \gamma] = \mathbb{E}_{\mathcal{M}}[\mathbb{1}_{X_0(M)}(x_1) \mathbb{1}_{X_1(M)}(x_2) \mathbb{E}_{n,m}[\chi_{n,m}(w,x_1) \chi_{n,m}(w, x_2) | M ]
\langle\Psi|\Phi\rangle_{\nu}:=\sum_{n}\langle\Psi_{n}|\Phi_{n}\rangle_{\nu}=a^{*}b+\sum_{ij}a_{i}^{*}b_{j}\langle\psi_{i}|\phi_{j}\rangle+\sum_{ijkl}a_{ij}^{*}b_{kl}\langle\psi_{i}|\phi_{k}\rangle\langle\psi_{j}|\phi_{l}\rangle_{\nu}+\cdot\cdot\cdot
\langle z|(a^{\dagger}a)_{k,\lambda}|z\rangle =e^{-\frac{|z|^{2}}{2}}\cdot e^{-\frac{|z|^{2}}{2}}\sum_{m,n=0}^{\infty}\frac{\bar{z}^{m}z^{n}}{\sqrt{m!}\sqrt{n!}}(n)_{k,\lambda}\langle m|n\rangle &=e^{-|z|^{2}}\sum_{n=0}^{\infty}\frac{|z|^{2n}}{n!}(n)_{k,\lambda}.
\tilde{Z}_{1 2}T_{1}^{[ j , k ]}\tilde{Z}_{2 1}T_{2}^{[ n , k ]}{R}_{t 1 2}^{+}= T_{2}^{[ n , k ]}\tilde{Z}_{1 2}T_{1}^{[ j , k ]}\ll{r z t z t +}
\begin{aligned}- \nabla \cdot (a(x,\xi) \nabla y(x,\xi)) + r(x,\xi) (y(x,\xi))^3 = u(x), (x,\xi) \in D \times \Xi, y(x,\xi) = 0, \phantom{tex}(x,\xi) \in \partial D \times \Xi. \end{aligned}
d_x - \nabla_{xy}^2g(x, y) \nabla_{yy}^2g(x, y)^{-1}d_y = 0 & -\nabla_{yy}^2g(x, y)^{-1}\nabla_{yx}g(x, y) \left(d_x - \nabla_{xy}^2g(x, y) \nabla_{yy}^2g(x, y)^{-1}d_y\right) + \beta \nabla_y g(x, y) = 0.
\|u_{k}\|_{\varepsilon}^{2}\le\|u_{k}\|_{\varepsilon}^{2}+\frac{q\omega^{2}}{\varepsilon^{n}}\int_{M}[2-q\psi(u_{k})]u_{k}^{2}\psi(u_{k})d\mu_{g}=|u_{k}^{+}|_{p,\varepsilon}^{p}\le C\|u_{k}\|_{\varepsilon}^{p},
|\partial_1 f (\xi, x_2,\dots, x_d)| = \left| \frac{f(\frac{i+1}{K}, x_2,\dots, x_d) - f(\frac{i}{K}, x_2,\dots, x_d)}{\frac{i+1}{K}- \frac{i}{K}}\right| \leq 2K \cdot \| f \|_{\sup}\leq 4 \cdot N^{\frac{1}{\sigma}}\cdot \| f \|_{\sup},
\sum_{b\in B}x_b & x_b \geq \sum_{j\in C}z_{j,b}b\in B \sum_{b\in B}w_{i,b}\leq 1 i\in H & \sum_{b\in B}z_{j,b}= 1 j\in C & \sum_{i\in H}w_{i,b}\cdot h_i \geq \sum_{j\in C}z_{j,b}\cdot c_j b\in B & x_b, w_{i,b}, z_{j,b}\in\{0,1\}b\in B, i\in H, j\in C
S \quad=(s_{i,j})_{i=1-p}^{d-p}\,\mbox{}_{j=1-p}^{d-p}\mbox{}\quad=\mathcal B'M \quad=\begin{bmatrix}B'B B'N \tilde{B}'B \tilde{B}' N\end{bmatrix}, &T \quad=(t_{i,j})_{i=1-p}^{d-p}\,\mbox{}_{j=1-p}^{d-p}\mbox{}\quad=\mathcal C'M \quad=\begin{bmatrix}C'B C'N \tilde{C}'B \tilde{C}' N\end{bmatrix}.
\int \prod_{\mu = 0}^{p}\left( \frac{d \bar{\lambda}_{\mu}d \lambda_{\mu}}{2 i}\right) \exp \left[ - \left( \bar{\lambda}_{\mu}+ \alpha_{\mu}\right) M^{\mu \nu}\left( \lambda_{\nu}+ \beta_{\nu}\right) \right] = \frac{\pi^{p + 1}}{\det M^{\mu \nu}},
V^{2}( r ) \equiv{\frac{\Delta ( r )}{R^{2}( r )}}\left\{\frac{\lambda}{R^{2}( r )}+ \frac{1}{R ( r )}\frac{\mathrm{d}}{\mathrm{d}r}\left[ \frac{\Delta ( r )}{R^{2}( r )}\frac{\mathrm{d}R ( r )}{\mathrm{d}r}\right] -{\frac{m^{2}a^{2}}{R^{6}( r )}}( r^{2}+ a^{2}- Q^{2}+ 2 M r ) \right\},
\partial_{t}\tilde{\textbf{u}}^{\varepsilon}+\sum_{j=1}^{d}J_{\varepsilon}\textbf{P}A_{j}(J_{\varepsilon}(\tilde{\textbf{u}}^{\varepsilon}+\bar{\textbf{u}}))\partial_{x_{j}}\tilde{\textbf{u}}^{\varepsilon}=0.
(p_2 - p_1 + s)(E_{1}^{n-1}F_{2}^{m}E_2 \otimes F_2) \otimes w_s \in - (E_{1}^{n-1}F_{2}^{m}E_1 \otimes F_2) \otimes w_{s+1}+ Z & = - (E_{1}^{n-1}(E_1 F_{2}^{m}- m F_{2}^{m-1}E) \otimes F_2) \otimes w_{s+1}+ Z.
\phi(e_u (q)) = ([\mathcal{D}- \mathcal{A}_u], (c_{u+1},\ldots,c_M,c_1,\ldots,c_u),O,A) , \phi(\tilde e_u (q)) = ([\mathcal{D}+ \mathcal{O}_u], (c_{M-u+1},\ldots,c_M,c_1,\ldots,c_{M-u}),O,A) .
\sum_{n=0}^N k^{-a}(n)k^s(n+m) = k^s(m)+\sum_{n=1}^N (k^{-a+1}(n)-k^{-a+1}(n-1))k^s(n+m) &= k^{-a+1}(N)k^s(N+m)+\sum_{n=0}^{N-1}(k^{s}(n+m)-k^{s}(n+m+1))k^{-a+1}(n) &= k^{-a+1}(N)k^s(N+m)-\sum_{n=0}^{N-1}k^{s-1}(n+m+1)k^{-a+1}(n).
p_{sr}\!=\!p_{rd}\!=\!\frac{1}{2 \varepsilon^2}\left \{\frac{m^2\!-\!\Gamma}{n\!-\!1}\left(1\!-\!\left(1\!-\!\frac{1}{m^2}\right)^{n\!-\!1}\right) \!- \!\left(1\!-\!\frac{\Gamma}{m^2}\right)^{n\!-\!1}\right \},
\int_{-L+2}^{L-2}\int_{-L}^L \max(|t-s|,2)^p \, ds\, dt ={\frac{2^{p+3}\bigl( ( L-1)^{p}{L}^{2}+ L{p}^{2}- 2\,( L-1)^{p}L + 2\, L p -2\,{p}^{2}+(L-1)^{p}- 4\,p -1 \bigr)}{{p}^{2}+3\,p+2}}.
\tilde f_1 (x,t)=\big\{-\tilde f_{AV}+\frac13(\tilde f_1-\tilde f_2)+\frac13(\tilde f_1-\tilde f_3)\big\}\Big|_{(-x,t)}=\frac13(\tilde f_1-2\tilde f_2-2\tilde f_3)(-x,t).
\mathbf{F}_{\frac{n}{2}}:= \left[\frac{(\lambda_{2k+1}- \lambda_{1}) \sin\left[\frac{(2k+1)\pi}{n + 3}\right]}{\lambda_{2k+1}- \phi_{\ell}}\right]_{k,\ell}\mathbf{G}_{\frac{n}{2}}:= \left[\frac{(\lambda_{2k}- \lambda_{n+2}) \sin\left(\frac{2k\pi}{n + 3}\right)}{\lambda_{2k}- \psi_{\ell}}\right]_{k,\ell}
\begin{aligned}\pi\Big[\int_{\tau_1}^\infty 1_{\{|x+\omega_{X_0}(s)-y-\omega_{Y_0}(s)|\leq 1\}}ds >K |X_{\tau_1},Y_{\tau_1}\Big] &=\pi\Big[\int_0^\infty 1_{\{x+\omega_{X_0}(\tau_1)+\tilde{\omega}_{X_{\tau_1}}(s)-y-\omega_{Y_0}(\tau_1)-\tilde{\omega}_{Y_{\tau_1}}(s)\}}ds>K|X_{\tau_1},Y_{\tau_1}\Big]<\frac12\end{aligned}
L =\frac{1}{2}\sum_{k=0}^{2m}\left[ a_{k}\partial^{k}+\left( -1\right)^{k}\partial^{k}M_{a_{k}}\right] & =\frac{1}{2}\left[ a_{2m}\partial^{2m}+\partial^{2m}M_{a_{2m}}\right] +R
\begin{array}{l l}{{[ h_{i}, h_{j}] = 0 ,}}&{{[ e_{i}, f_{j}] = \delta_{i j}h_{i},}}\\{{\mathrm{~ [ ~}h_{i}, e_{j}\mathrm{~ ] ~}= A_{i j}e_{j},}}&{{\mathrm{~ [ ~}h_{i}, f_{j}\mathrm{~ ] ~}= - A_{i j}f_{j},}}\\ \end{array}
I_2= \left[\overline u_e^i(x)\varphi(x,0)+\overline h_e^i(x)\psi(x,0)\right]x^{-\frac{1}{2}-2\sigma_i}\leq |(\varphi,\psi)(x,0)x^{-\frac{3}{4}+\sigma_{i-1}-\frac{1}{2}-2\sigma_i}| \leq \delta_0|(\varphi,\psi)(x,0)x^{-\frac{3}{4}-\sigma_i}|^2+C|x^{-\frac{1}{2}-\sigma'}|^2,
G_j(z):= \frac{F(x_j,\xi_j+\lambda_jz)-F(x_j,\xi_j)-\lambda_jF_{z}(x_j,\xi_j)[z]}{\lambda_j^2}= \int_0^1(1-t)F_{zz}(x_j,\xi_j+t\lambda_jz)[z,z].
d(f(x_0), f(x_n)) \leq \sum_{i = 0}^{m-1}d(f(x_i), f(x_{i+1})) \leq \sum_{i = 0}^{m-1}\left( \beta \, d_l(x_i, x_{i+1}) + \gamma \right) = \beta \, d_l(x_0, x_m) + m \gamma \leq \beta \, d_l(x_0, x_m) + |M| \gamma,
D(T_s) =\frac{\left(\mathbb{E}\{T_s|\mathcal{H}_1\}-\mathbb{E}\{T_s|\mathcal{H}_0\}\right)^2}{\mathrm{Var}\{T_s|\mathcal{H}_0\}}&=\left(\frac{\sigma_{\theta}^2\mathbf{a}^H\mathbf{h}\mathbf{h}^H\mathbf{a}}{\mathbf{a}^H\mathbf{F}\mathbf{V}\mathbf{F}^H\mathbf{a}+\sigma^2_n}\right)^2\;,
8 \delta \tilde{C}= 0 , \delta ( \tilde{\Delta}\: \frac{1}{\sqrt{- g}}\: \tilde{C}) = 0 , \delta [ \tilde{\Delta}\frac{1}{\sqrt{- g}}\tilde{\Delta}. . . \frac{1}{\sqrt{- g}}\: \tilde{\Delta}C ) ] = 0 ,
S^m U \in \langle \, S^i U \, \rangle_{0 \leq i \leq m}= \langle \, \Lambda^i Q \, \rangle_{0 \leq i \leq m}= \langle \, \Lambda^i Q \, \rangle_{0 \leq i \leq n-k}= \langle \, S^i U \, \rangle_{0 \leq i \leq n-k}.
\renewcommand{\arraystretch}{1.5}\begin{array}{lcl}{[}L_m,L_n{]}= (m-n)L_{m+n}+\delta_{m+n,0}\frac{m^3-m}{12}c{[}L_m,G_r{]}= (\frac{1}{2}m-r)G_{m+r}{[}G_r,G_s{]}_{+}= 2L_{r+s}+ \frac{1}{3}(r^2-\frac{1}{4}) \delta_{r+s,0}c \end{array}
\frac{r !}{N^{2}}r u^{- 2}[ ( \lambda_{\bf 8 4}\phi + \mu_{\bf 8 4}) P_{Y_{( 2 r , 2 )}}+ ( \lambda_{\bf 1 0 5}\phi + \mu_{\bf 1 0 5}) P_{Y_{( 2 r + 2 )}}+ ( \lambda_{\bf 1 7 5}\phi + \mu_{\bf 1 7 5}) P_{Y_{( 2 r + 1 , 1 )}}]
\Big \langle \overline{{{\psi}}}_{\alpha_{0}}^{( a_{0})}( x_{0}) \psi_{\beta_{0}}^{( b_{0})}( x_{0}) \overline{{{\psi}}}_{\alpha_{1}}^{( a_{1})}( x_{1}) \psi_{\beta_{1}}^{( b_{1})}( x_{1}) . . . . . \overline{{{\psi}}}_{\alpha_{n}}^{( a_{n})}( x_{n}) \psi_{\beta_{n}}^{( b_{n})}( x_{n}) \Big \rangle_{0 \; c}^{\theta}\; ,
|G^{\delta}_{R_n}(f) | =\Big|\frac{1}{R_n}\int^{R_n}_0[S_{r}^{\delta+1}(f)-S_{r}^{\delta}(f)]dr\Big|\le \Big ( \int^{R_n}_0 \big| S_r^{\delta+1}(f)-S_r^{\delta}(f) \big|^2 \frac{d r}{R_n}\Big )^{1/2}&\leq \Big ( \int^{\infty}_0 \big | S_r^{\delta+1}(f) - S_r^{\delta}(f) \big |^2 \frac{d r}{r}\Big )^{1/2}=:G^{\delta}(f).
\int_{S^{2}}f(\mathbf{n})\star_{\vec{c}}^{n}g(\mathbf{n)}d\mathbf{n=}\int_{S^{2}}f(\mathbf{n})\tilde{g}(\mathbf{n})d\mathbf{n=}\int_{S^{2}}\tilde{f}(\mathbf{n})g(\mathbf{n})d\mathbf{n,}
\sum_{i\in I}\sum_{j\in J_i}\left\langle \Gamma_{ij}\Lambda_iTf,\Gamma_{ij}\Lambda_iUf\right\rangle =\sum_{i\in I}\sum_{j\in J_i}\left\langle \Gamma^*_{ij}\Gamma_{ij}\Lambda_iTf,\Lambda_iUf\right\rangle &\ge \sum_{i\in I}C_i\left\langle \Lambda_iTf,\Lambda_iUf\right\rangle\ge CA\|f\|^2.
W ( N , \mu N ) = \frac{\exp \{N ( \ln f ( x ) - \mu \ln x ) \}}{\sqrt{2 \pi N h_{\mu}( x )}}\, \Bigl ( \frac{1}{x}+ O ( 1 / N ) \Bigr ) \qquad \mathrm{a s}\quad N \rightarrow \infty \, ,
= \displaystyle\sum_{j=1}^{m}\frac{\partial a_{j}}{\partial x^{i}}(p)\frac{\partial}{\partial v^{j}}|_{\phi(p)}- \displaystyle\sum_{j=1}^{m}\frac{\partial b_{j}}{\partial x^{i}}(p)\frac{\partial}{\partial u^{j}}|_{\phi(p)}
\displaystyle\sum_{j=1}^{d}\,\left(\sqrt{\varepsilon}\Big\| \frac{\partial u^{\varepsilon}}{\partial x_{j}}\Big\|_{L^{2}(\Omega_{T})}\right)^{2}\leq\frac{1}{2r}\|u_{0\varepsilon}\|^{2}_{L^{2}(\Omega)}\leq \frac{1}{2r}A^{2}\mbox{Vol}(\Omega).
\left(2\sigma_k^2\xi^3\right)\frac{\partial\eta_k(\xi)}{\partial \xi}= \xi^2\left(2k+4\log(C_k)-1\right)+\xi\sigma_k &-\left(2\sigma_k^2\left(2k\log(\sigma_k+\xi) +\log\left(\frac{1}{2}\left(1+\frac{\xi}{\sigma_k}\right)\right)\right)+4k(k+2\log(C_k))\right).
\begin{aligned}F(f)(t):=\int_0^te^{(t-s)\Delta}\Gamma^{-1}(s)[K(\Gamma(s)f(s))\cdot \nabla](\Gamma(s)f(s))ds-\int_0^t e^{(t-s)\Delta}\Gamma^{-1}(s)(\Gamma y(s)\cdot\nabla)(K(\Gamma y(s))ds,\t\geq0.\end{aligned}
\mbox{heterogamous if$(1-\pi_{11})(1-\pi_{22}) > (1-\pi_{12})(1-\pi_{21})$,}&\mbox{panmictic if\qquad\ \ $(1-\pi_{11})(1-\pi_{22}) = (1-\pi_{12})(1-\pi_{21})$,and}&\mbox{homogamous if\quad\ $(1-\pi_{11})(1-\pi_{22}) < (1-\pi_{12})(1-\pi_{21})$.}
\Psi\propto(\begin{matrix}(1+\gamma)r^{\gamma-1}e^{-Cr}\\ (1+\gamma)r^{\gamma-1}e^{-Cr}\\iZ\alpha r^{\gamma-1}e^{-Cr}(x-iy+z)/r\\iZ\alpha r^{\gamma-1}e^{-Cr}(x+iy-z)/r\end{matrix})
[ i \gamma_{\mu}\partial^{\mu}, M^{k l}] = i \gamma_{\rho}g^{\rho l}\partial^{k}- i \gamma_{\rho}g^{\rho k}\partial^{l}+ i \gamma_{\rho}Z_{\nu}^{l \rho}\partial^{\nu}\partial^{k}- i \gamma_{\rho}Z_{\nu}^{k \rho}\partial^{\nu}\partial^{l}
\displaystyle\lim_{s\to -\infty}\frac{\langle\overline{w}(s), e_1\rangle}{\|\overline{w}(s)\|}=1,\quad\mbox{and}\quad\lim_{s\to -\infty}\frac{\langle\underline{w}(s), e_1\rangle}{\|\underline{w}(s)\|}=-1\,,
\begin{cases}Ric_{g_S}=\frac{1}{u}D^2_{g_S}u+2u^{-4}(\omega^2-|\omega|^2_{g_S}\cdot g_S) \Delta_{g_S}u=2u^{-3}|\omega|^2_{g_S}\delta_{g_S}\omega+3u^{-1}\langle du,\omega\rangle_{g_S}=0 d\omega=0\end{cases},
\hat{Q}_X^s(x) \sim P_X(x) g_s ( P_{Y|X=x}\| \hat{Q}_Y^s )^{\frac{1}{s}}\textrm{and}\hat{Q}_Y^s(y) \sim P_Y(y) g_s ( P_{X|Y=y}\| \hat{Q}_X^s )^{\frac{1}{s}}\,.
\mathbf{c^{i,j}}=( b^i_1,b^i_1,\dots,b^i_1,b^i_2,b^i_2,\dots,b^i_2,\dots,b^i_{i-1},b^i_{i-1},\dots,b^i_{i-1}, & a^j_1,a^j_2,\dots,a^j_k,b^i_{i+1},b^i_{i+1},\dots,b^i_{i+1},\dots,b^i_n,b^i_n,\dots,b^i_n),
{\ddot{\hat{a}}_0\over\hat{a}_0}-{\ddot{\varepsilon}\over\varepsilon}=-{{16\pi^2\hat{G}^2_5}\over{9c^6_0}}(2\hat{\varrho}^2c^4_0+\hat{\sigma}\hat{\varrho}c^2_0+3\hat{\sigma}\hat{\wp}+3\hat{\wp}\hat{\varrho}c^2_0)-{{\hat{\cal C}c^2_0}\over\hat{a}^4_0}+{{4\pi\hat{G}_5}\over{3c^2_0}}\left(\hat{\Lambda}+{{4\pi\hat{G}_5}\over{3c^4_0}}\hat{\sigma}^2\right),
\sum_{h_0, i}\theta(g_i, g_j)(g_i(\pi(\xi)|_{X_j\cap g_i^{-1}X_i})) =\sum_{g}\sum_{h_0\in c(g, X)}\theta(g, g_j)(g(\pi(\xi)|_{X_j\cap Z_{g, h_0}})) &=\sum_{g}\theta(g, g_j)(g(\pi(\xi)|_{X_j})).
{\cal{V}}= -\frac{TV^2}{8\pi^4\lambda}\frac{m}{N(N+m)}+ \frac{1}{2\pi^2}(\frac{1}{N}(B^0)^2 + \sum_k (B^k)^2)+\frac{1}{\pi^2}\sum_i \sum_{j>i}|X^{(ij)}|^2) +\frac{1}{\pi^2}\sum_i|X^i|^2
\Delta u_{(x, 0), r}|_{\mathcal{O}_{(x,0), r}}(B_2^2(0,0)) \ge \int_{\partial_{red}\mathcal{O}_{(x,0), r}\cap B_2^2(0,0)}|y|^{\gamma}d\sigma \gtrsim \left|\sup_{0< \rho< 1}\{|x_2|: (x_1, x_2) \in \partial \mathcal{O}_{(x, 0), r}\cap B_2^2(0, 0), |x_1|=\rho \}\right|^{2\gamma}\rightarrow 0.
\theta'(v,\tau+1)=e^{\frac{\pi\sqrt{-1}}{4}}\theta'(v,\tau),~~ \theta'(0,-\frac{1}{\tau})=\frac{1}{\sqrt{-1}}\left(\frac{\tau}{\sqrt{-1}}\right)^{\frac{1}{2}}\tau\theta'(0,\tau).
V_{j-1,m}^{n}=v-h\frac{\partial v}{\partial x}+\frac{h^{2}}{2}\frac{\partial^{2}v}{\partial x^{2}}-\frac{h^{3}}{6}\frac{\partial^{3}v}{\partial x^{3}}+\frac{h^{4}}{24}\frac{\partial^{4}v}{\partial x^{4}}+...
(\begin{matrix}1&a&c\\ 0&1&b\\ 0&0&1\\ \end{matrix})(\begin{matrix}1&a^{\prime}&c^{\prime}\\ 0&1&b^{\prime}\\ 0&0&1\\ \end{matrix})=(\begin{matrix}1&a+a^{\prime}&c+ab^{\prime}+c^{\prime}\\ 0&1&b+b^{\prime}\\ 0&0&1\\ \end{matrix}).
\int_{D_r}\frac{n-4}{2}\Phi^{\frac{n+4}{n-4}}Q_{g}+ \frac{(n-4)}{(n-2)^2}|\mathrm{Ric}_{\tilde{g}}|^2_{\tilde{g}}\Phi \, dv_{\tilde{g}}= \int_{S_r}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\omega_{\tilde{g}}+ \frac{4}{n-2}\int_{S_r}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu}) d\omega_{\tilde{g}}
y^{(2)}_{1,2}(\theta) =3( d^{[0]}_0(E \omega^7) d^{[1]}_0(E \omega^3) -i d^{[0]}_0(E \omega^3) d^{[1]}_0(E \omega^7))d^{[2]}_0(-E\omega^3), y^{(1)}_{1,2}(\theta) =-3( d^{[0]}_0(-E \omega) d^{[1]}_0(-E \omega^5)+ i d^{[0]}_0(-E \omega^5) d^{[1]}_0(-E \omega))d^{[2]}_0(E\omega^5).
\left\langle \hat{p}\cdot\nabla_x P(f),\left(1+\frac{\mathcal{I}}{mc^2}\right)p^j\sqrt{F_E^0}\right\rangle_{L^2_{p,\mathcal{I}}}= k_BT_0\partial_{x^j}\left\{a(t,x)-\widetilde{e}(T_0)c(t,x)\right\}+ b_0mc^2\partial_{x^j}c(t,x)\cr =k_BT_0\partial_{x^j}\left\{a(t,x)+k_BT_0 c(t,x) \right\}
u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}\,\,\,\,\,\,\mbox{and}\,\,\,\,\,v=\frac{x_{12}^{2}x_{34}^{2}}{x_{14}^{2}x_{23}^{2}}.
\frac{1}{(mt_1+1)e^{mt_1}+c}-{1\over mn}+{1\over m^2nt_0}={(n+1)(1+mt_1-mt_0)+m^2t_0t_1^2c-mt_1^2c-m^2t_0t_1\over m^2nt_0(mnt_1+n+1+mt_1-cmt_1^2)}.
\begin{array}{l}{I_{+}}^{T}={I_{+}}^{*}={(I_{+})}^{+}, \quad{(I_{+})}^{2}=1, \quad\mid I_{+}\mid=1, ({I_{+}})^{(2,1,\alpha)}_{(1,1,\alpha)}=({I_{+}})^{(1,2,\alpha)}_{(2,2,\alpha)}=0, \quad({I_{+}})^{(1,1,\alpha)}_{(1,1,\alpha)}=({I_{+}})^{(2,2,\alpha)}_{(2,2,\alpha)}=1.\end{array}
n (D(n-i+\ell+2, n-i+j+1)+D(n-i+\ell+1)) =n+k_{n-i+\ell+2}-k_{n-i+j+1}-k_{n-j+\ell+1}+r+k_{n-i+\ell+1}-k_{n-i+\ell+2}=n-k_{n-i+j}-r+n-k_{n-j+\ell+1}+r+r+k_{n-i+\ell}=n+r+k_{n-j+\ell}-k_{n-j+\ell+1}=n (1+D(n+\ell-j)),
I_{0}= \frac 1{4 \pi \alpha^{^{\prime}}}\int d z^{\alpha}\left( \delta^{A}u^{\alpha}\delta_{A}u^{\beta}G_{\alpha \beta}( u ) + i \epsilon^{A B}\delta_{A}u^{\alpha}\delta_{B}u^{\beta}H_{\alpha \beta}( u ) \right)
R=(d-2)\left[\frac{(d-3)(1-A)}{r^2}-\frac{2}{r}\left(\frac{dA}{dr}+A\frac{d\Omega}{dr}\right)\right]-\left[\frac{d^2A}{dr^2}+2A\frac{d^2\Omega}{dr^2}+2A\left(\frac{d\Omega}{dr}\right)^2+3\frac{dA}{dr}\frac{d\Omega}{dr}\right].
f_{(t,s)}'(z) =t^{-b_v}\sum_{\alpha\in A}\xi_{\alpha}t^{\nu_{PR}^{(s)}(\alpha )}(z_1t^{-a_{v_i,\, 1}})^{\alpha_1}\ldots (z_nt^{-a_{v_i,\, n}})^{\alpha_n}&\phantom{f_{(t,s)}'(z)}=t^{-b_v}f_{(t;\, s)}\circ \Phi^{-1}_{\Delta_{v_i},\, t}(z),
\begin{gathered}-\pi^{-1}e_1 \otimes 1, \dotsc, -\pi^{-1}e_m \otimes 1, e_{m+1}\otimes 1 - \pi e_{m+1}\otimes \pi^{-1}, e_{m+2}\otimes 1,\dotsc, e_n\otimes 1, e_1\otimes 1,\dotsc, e_m \otimes 1, \frac{e_{m+1}\otimes 1 + \pi e_{m+1}\otimes \pi^{-1}}2, \pi e_{m+2}\otimes 1, \dotsc, \pi e_{n}\otimes 1.\end{gathered}
\tilde{\psi}_{u\mathbf{j}_{|u|}}(\mathbf{x}_{u})=\frac{(-1)^{|\mathbf{j}_{|u|}|}}{\phi_{u}(\mathbf{x}_{u};\mathbf{\Sigma}_{u})}\left(\frac{\partial}{\partial\mathbf{x}_{u}}\right)^{\mathbf{j}_{|u|}}\phi_{u}(\mathbf{x}_{u};\mathbf{\Sigma}_{u}),\;\mathbf{j}_{|u|}\in\mathbb{N}_{0}^{|u|},
{\frac{\beta^{2}}{1 6 \pi}}( \tilde{\mu}^{2}( \varphi_{0}) - \tilde{m}_{0}^{2}) \int{\frac{d \tilde{p}}{f^{2}( \tilde{p}) ( 2 f ( \tilde{p}) - \tilde{m}_{2})}}= 1 .
\big\|C\star \psi \star S\big\|_{L^1}\leq c\Big(\frac{1}{m^4_t}{\bf 1}_{d=2}+ \big(\frac{1}{m_t^{1/2}}+\frac{1}{m_t^{5/2}}\big){\bf 1}_{d=3}\Big)\Big(\|E\|_{L^1}+\frac{1}{m^2_t}\Big).
[\lambda_{{z'}a}, \lambda_b + 2\rho_b] = \lambda_{{z'}[a,b] + [{z'},b]a}- [\lambda_{z'}, \frac{1}{2}(\lambda_{[a,b]}+ 2\rho_{[a,b]}) - \frac{1}{2}[\lambda_a + 2\rho_a, \lambda_b + 2 \rho_b]].
d_{\mu}( q ) = ( 1 - \delta_{q_{3}, 0}) \frac{1}{2 q^{2}q_{3}^{2}}q_{\mu}\left( 1 - \delta_{\mu , 3}\right) + \delta_{q_{3}, 0}\frac{q_{\mu}}{( q^{2}+ i \epsilon )^{2}}( 1 - \xi ) \ .
\mu_A (C_0(U_i)) \cdot A = \mu_A (C_0(U_i)) \cdot \left( \mu_B ( C_0 (X) ) \cdot B \right) & = \mu_B \left( C_0 ( X) \right) \cdot \left( \mu_{B}\left( C_0 ( V_{y_i}) \right) \cdot B \right) & = \mu_{B}\left( C_0(X) \right) \cdot C_0 ( V_{y_i}, C ) & = C_0 (U_i , C ),
\sum_{\sqrt{n}<p\leq \frac{n}{2}}\log p = \sum_{p\leq \frac{n}{2}}\log p - \sum_{p\leq \sqrt{n}}\log p = \frac{n}{2}+O\left(\frac{n}{\log n}\right)+O(\log n \sqrt{n}) &= \frac{n}{2}+O\left(\frac{n}{\log n}\right)\,.
d s^{2}= e^{2 \Upsilon ( x^{m})}\eta_{\mu \nu}d x^{\mu}d x^{\nu}+ e^{- 2 \Upsilon ( x^{m})}h_{m n}d x^{m}d x^{n}\quad m , n = 1 ,{\ldots}, 6 ; \ \ \mu , \nu = 7 ,{\ldots}, 1 0
\mathcal{A}_{G,\psi}\coloneqq \left\{\{z=0 \}\right\}\cup \left\{\{x_{i}-x_{j}= 0 \}\mid \{i,j\}\in E_{G}\right\}\cup \left\{\{x_{i}=az\}\mid 1 \leq i \leq \ell, a \in \psi(i) \right\},
\bigg\| 1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(t)|\bigg\|_{L^{\frac{p}{2}}}\le \bigg\| 1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|X_{n}(t)-B(T_{k})|\bigg\|_{L^{\frac{p}{2}}}+\bigg\|1_{\{|T_{n}-1|\le 1\}}\sup_{t\in[0,1]}|B(T_{k})-B(t)|\bigg\|_{L^{\frac{p}{2}}}=: I_1 + I_2.
I(\phi,\widetilde{K}) =\textrm{span}\{a\bullet k-\phi(a)k : a\in A, k\in K\}&=\textrm{span}\{(ae_{J})\cdot k-\phi(ae_{J})k : a\in A, k\in K\}&\subseteq \textrm{Im}T=\textrm{Im}\widetilde{T}.
\mathcal S: \underset{m\in \mathbb Z}\bigoplus\mathsf L^2(\mathbb R_+, \mathbb C^2)\to \underset{\varkappa\in \mathbb Z+1/2}\bigoplus\mathsf L^2(\mathbb R_+, \mathbb C^2), \underset{m\in \mathbb Z}\bigoplus\binom{\varphi_m}{\psi_m}\mapsto \underset{\varkappa\in \mathbb Z+1/2}\bigoplus\binom{\varphi_{\varkappa-1/2}}{\psi_{\varkappa+1/2}}.
\mathrm{p g h}\left( V^{* \alpha ( \lambda )}\right) = \mathrm{p g h}\left( A_{( \lambda )}^{* \alpha}\right) = \mathrm{p g h}\left( \eta^{* ( \lambda )}\right) = \mathrm{p g h}\left( C_{( \lambda )}^{*}\right) = 0 ,
\sum_{i_1,\cdots,i_{k}\}\prod_{j=1}^k(\tau_{i_j}-x_j)_+ = (n/4)^k(2\ln n)^{\frac{k}{2}}\sum_{i_1,\cdots,i_{k}\}\prod_{j=1}^k(m_{i_j}-F_n(x_j))_+ = (n/4)^k(2\ln n)^{\frac{k}{2}}|\Sigma_k(F_n(x_1),\cdots,F_n(x_k))|.
Q_{\alpha}^{i}= \phi_{Q \alpha}^{i}+ \sqrt{2}\theta \psi_{Q \alpha}^{i}+ \theta^{2}F_{Q \alpha}^{i}, \qquad{\tilde{Q}}^{\alpha j}= \phi_{\tilde{Q}}^{\alpha j}+ \sqrt{2}\theta \psi_{\tilde{Q}}^{\alpha j}+ \theta^{2}F_{\tilde{Q}}^{\alpha j},
\frac{3}{2}(\partial^\mu \partial^\nu h_{\mu\nu}- \partial_\mu\partial^\mu \tilde h ) - \frac{15}{2}\partial_4 \sigma \partial_4 \tilde h- \frac{3}{2}\frac{\partial^2 \tilde h}{\partial{x^4}^2}-\frac{3}{2}\partial_\mu \partial^\mu \phi + 30 k^2 \phi -12 k \phi \tilde \delta =0.
\|\partial_z (\alpha v)\|_{L^p(C(x_0'; |\lambda|^{-1/2}))}\le \|\partial_z (\theta \alpha v)\|_{L^p(\Omega')}, \|\nabla\partial_z (\alpha v)\|_{L^p(C(x_0'; |\lambda|^{-1/2}))}\le \|\nabla\partial_z (\alpha v)\|_{L^p(C(x_0'; |\lambda|^{-1/2}))}.
\frac 1{2\pi i}\int_{\rho-i\infty}^{\rho + i\infty}n^{-s}T(s)^{k-j}\Gamma(m+s)\, ds = \sum_{\ell \ge \max\{0, - m - \rho\}}\frac{n^{m + \ell}(-1)^\ell}{\ell!}T(-\ell- m)^{k-j},
E_M[h(W_n)|X^\infty=x^\infty] =\int h(W^*_{n,x^\infty}(\tilde \omega))d\tilde{\mathbf P}^*(\tilde\omega),~\forall h\in BL_1 E[h(W)] =\int h(W^*_{x^\infty}(\tilde\omega))d\tilde{\mathbf P}^*(\tilde\omega),~\forall h\in BL_1,
|\mathrm{E}[(x_i'-x_i)^2 - e_i]| = |\mathrm{E}[(I^d-I^{d-})(y_i-x_i)^2]| &\leq \mathrm{E}[(\alpha(x,y)-\alpha^-(x,y))^2]^{1/2}\mathrm{E}[(y_i-x_i)^4]^{1/2}.
\left\{Q_{\alpha a}, Q_{\beta b}\right\}= \gamma_{\alpha \beta}^{\mu}\gamma_{a b}^{i j}P_{\mu i j}+ \gamma_{\alpha \beta}^{\mu_{1}\mu_{2}\mu_{3}}\delta_{a b}T_{\mu_{1}\mu_{2}\mu_{3}}^{+}+ \gamma_{\alpha \beta}^{\mu_{1}\mu_{2}\mu_{3}}\gamma_{a b}^{i j k l}Z_{\mu_{1}\mu_{2}\mu_{3}i j k l}^{+}
{\mathcal L}^{(<)}(\Theta_{l}) =-p\,\frac{\Theta_{l}}{\left( \lambda_{l}^{(\ast)}\right)^{1/2}}\tan\left\{p \left[\frac{\Theta_{l}}{\left( \lambda_{l}^{(\ast)}\right)^{1/2}}-\tan^{-1}\left( \frac{\Theta_{l}}{\left( \lambda_{l}^{(\ast)}\right)^{1/2}}\right)\right]-\frac{\pi}{4}\right\}+ O(1)\; .
f_{2n}(t)=\Upsilon_n\big(R_{n}e^{r_{n}t}+C_{n}e^{i\omega_{n}t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}+D_{n}e^{i\zeta_{n}t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\big)\,.
[u^i\omega d^lu] = [u^i\omega d^{l-2}(\alpha dud + d)] = \alpha [u^i\omega d^{l-1}ud] + [u^i\omega d^{l-1}] &=\alpha\frac{\alpha^{l-1}-1}{\alpha-1}[u^i\omega d^{l-2}d] + [u^i\omega d^{l-1}] &=\frac{\alpha^l-1}{\alpha-1}[u^i\omega d^{l-1}].
\left(\frac{\sqrt{\frac{\pi}{2}}\rho(y)}{e^{-2y^{2}}}-1\right)\frac{1}{\sqrt{\tau}}=\frac{8}{3}y^{3}-2y+\sqrt{\tau}\left(\frac{32}{9}y^{6}-\frac{31}{3}y^{4}+\frac{15}{2}y^{2}-\frac{37}{48}\right)+O(\tau)
A d ( g )^{T}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{\theta}}&{{1}}&{{0}}\\{{- \theta}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) , \ \ \A d ( \tilde{g})^{T}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{- \rho}}&{{e^{\sigma}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{{\rho}{e^{- \sigma}}}}\\{{0}}&{{0}}&{{0}}&{{e^{- \sigma}}}\\ \end{matrix}\right)
S_{I I}+ S_{I I I}\sim M^{2}\tau_{f}\left\{1 - \frac{2}{\pi r^{2}}\left( \frac{1}{\sigma_{0}}+ \frac{1}{2 \pi - \sigma_{0}}\right){\sin}^{2}\left( \frac{r \sigma_{0}}{2}\right) \right\}
F'_{m,n}(t) = \frac{d}{dt}\left( \frac{1}{||J_0(\gamma_m x) \sin(n \pi y)||_2^2}\int\limits_0^1 \int\limits_0^1 x f(x,y,t) J_0(\gamma_m x) \sin(n \pi y) dx dy\right) &= \frac{1}{||J_0(\gamma_m x) \sin(n \pi y)||_2^2}\int\limits_0^1 \int\limits_0^1 x f_t(x,y,t) J_0(\gamma_m x) \sin(n \pi y) dx dy,
\lim_{t \to \infty}\int_{I_\delta^+(t)}\rho(\theta, t) ~ d\theta = 1{\rm{and}}\cr \int_{I_{\delta}^{-}(t)}|\rho(\theta,t)|^{2}~ d\theta\leq e^{-\frac{R(0)(\cos\delta)}{2}K(t-T_{1})}\int_{I_{\delta}^{-}(T_{1})}|\rho(\theta,T_{1})|^{2}~ d\theta \qquad\forall~t\geq T_{1}.