harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}{k_{1}^{\mu}T_{\rho \sigma \mu \nu}^{( 4 ) a b c d}( p_{1}, p_{2}; k_{1}, k_{2}) = C_{1}\gamma_{\rho \sigma \nu}^{( 1 )}( k_{1}) + C_{2}\gamma_{\rho \sigma \nu}^{( 2 )}( k_{1})}\\{+ C_{3}\gamma_{\rho \sigma \nu}^{( 3 )}( k_{1})}\\ \end{array}
\int_{B_{x_\alpha}(\mu_\alpha)}(qv_\alpha-1)^2 \Psi_\alpha dv_g = \mu_\alpha^2 \int_{B_0(1)}(q\hat{v}_\alpha-1)^2 \hat{\Psi}_\alpha dv_{\hat{g}_\alpha}, = o(\mu_\alpha^2).
S_n = \sum_{i=1}^n \left(p_{\pi(i)+1}-i\right) & = 2+3+(5+5)+\cdots+(p_m-p_{m-1})p_m+p_{m+1}-\frac{n(n+1)}{2}& = 2+ \sum_{k=2}^m p_k(p_k-p_{k-1})+p_{m+1}- \frac{n(n+1)}{2}& = \frac1{2}\left(p_1^2+ 2 \sum_{k=2}^m p_k^2 - 2\sum_{k=2}^m p_kp_{k-1}+2p_{m+1}- n^2-n\right) & = \frac1{2}\left(2p_{m+1}-n+ \sum_{k=1}^{m-1}(p_{k+1}-p_k)^2\right)
(S_\varepsilon(t)-p_{\varepsilon}(t))^2 = (S_\varepsilon(0)-p_{\varepsilon}(0))^2 + 2\int_0^t (\dot{S_\varepsilon}(\tau) - \dot{p}_{\varepsilon}(\tau)) (S_\varepsilon(\tau)-p_{\varepsilon}(\tau)) d\tau,
\mathsf{f}_n = \vec f \phi_n - (a D_v \phi_n) u, &\mathsf{g}_n = g \phi_n - \vec f \cdot D_v \phi_n+ u (\partial_t \phi_n - v \cdot D_x \phi_n) - (a D_v \phi_{n}) \cdot D_v u,
d s^{2}={\frac{x^{2}}{R_{1}^{2}}}( - d t^{2}+ d z^{2}) + R_{1}^{2}{\frac{d x^{2}}{x^{2}}}+ R_{1}^{2}d \Omega_{3}^{2}+ H_{5}^{\prime}( y ) ( d y^{2}+ y^{2}d{\Omega_{3}^{\prime}}^{2}) ,
\left.\frac{{\rm d}\mathbb{P}^\circ_x}{{\rm d}\mathbb{P}_x}\right|_{\mathcal{F}_t}: = \frac{\sin(\pi\alpha\hat\rho)\mathbf{1}_{(X_t>0)}+ \sin(\pi\alpha\rho)\mathbf{1}_{(X_t<0)}}{\sin(\pi\alpha\hat\rho)\mathbf{1}_{(x>0)}+ \sin(\pi\alpha\rho)\mathbf{1}_{x <0)}}\left|\frac{X_t}{x}\right|^{\alpha-1}\mathbf{1}_{(t<\tau^{\{0\}})},t\geq 0,
s_1 = \left\vert \chi^{(2)}(1+\rho e^{i \beta \pi})+\sqrt{\left(\chi^{(2)}(1+\rho e^{i \beta \pi})\right)^2-1}\right\vert, s_2 = \left\vert \chi^{(2)}(1+\rho e^{- i \beta \pi})+\sqrt{\left(\chi^{(2)}(1+\rho e^{-i \beta \pi})\right)^2-1}\right\vert.
\mathbb{\mu}_{n}=\mathbb{E}\left\Vert \frac{1}{\sqrt{n}}\sum_{i=1}^{n}f(X_{i})\right\Vert_{\mathcal{F}}\leqslant M\int_{0}^{1}\sqrt{1+\log N_{\left[ \ \right]}(\mathcal{F},M\varepsilon,d_{P})}d\varepsilon.
\left(\frac{1}{8\omega^{3/2}}\right)\left((\omega^2C^2+\omega k)^{1/2}(2\omega C^3+kC)-k^2\ln(\omega C+(\omega^2C^2+\omega k)^{1/2})\right) &\qquad=\frac{\lambda_1\lambda_2\lambda_3^2}{\lambda_5}t+k_2,
\boldsymbol{F}_u(a)= \int_{-\infty}^{+\infty}e^{i v a}\exp\!\Big(\!-\frac{v^2}{2}\,\Big)f(\sqrt{u}v) \frac{dv}{\sqrt{2\pi}}\,, &\boldsymbol{G}_u(a) =\int_{-\infty}^{+\infty}e^{i v a}\exp\!\Big(\!-\frac{v^2}{2}\,\Big)\big(1-f(\sqrt{u}v)\big) \frac{dv}{\sqrt{2\pi}}\,.
ds^2={1\over g_s}\left(\mu_6 V Z_2 - \mu_2 Z_6\right)dx_\perp^2 ={\alpha'^{-3/2}\over (2\pi)^2 g}\Biggl(\frac{V}{V_*}- 1 - \frac{g_s N \alpha'^{1/2}}{r}\Biggr)(dr^2+r^2d\Omega^2_2)\ .
E_q \Bigl[ \triangle_2(X_k, Y_k) , \, Q_K \Bigr] = \frac{Z_q^{\prime}}{Z_q}\int_{Q_K}\Delta_2 d \nu_q + \frac{Z_q^{\prime}}{Z_q}\int_{Q_K}\Delta_2 ( e^{n\beta H_q^{(3)}}- 1) d\nu_q,
0 = \nabla^2 = d_1^2 - d_1\{I^\nabla,\cdot\}+ \{I^\nabla, \{I^\nabla,\cdot\}\}= d_1^2 + \{d_1 I^\nabla,\cdot\}- \frac{1}{2}\{\{I^\nabla,I^\nabla\},\cdot\}.
\widehat{G_{r}}\left(\lambda\right) =\int_{-\infty}^{\infty}G_{r}\left(x\right)e^{-i\lambda x}dx =\frac{2e\cdot\left(\lambda^{2}+r^{2}\right)+2(r-1)\left(\cos(\lambda)\left(\lambda^{2}-r\right)+\lambda(r+1)\sin(\lambda)\right)}{e\cdot\left(\lambda^{2}+1\right)\left(\lambda^{2}+r^{2}\right)};
\begin{aligned}p_1 = \left(x_1,\frac{u_{11}}{v_{11}},t_2,\dots, t_{n+1}\right), p_k = \left(t_1,\dots, t_{k,k-1},\frac{v_{k,k-1}}{u_{k,k-1}},\frac{u_{k,k}}{v_{k,k}},t_{k+1},\dots, t_{n+1}\right).(1<k\leqslant n)\end{aligned}
\mathcal{S}=\int \frac{1}{2l_{11}^{9}}\left( d^{11}x \sqrt{-g}R - \frac{1}{2\cdot 4!}d\mathcal{C}\wedge *d\mathcal{C}\right) + \int \frac{\sqrt{2}}{2^7\cdot 3^3}\mathcal{C}\wedge d\mathcal{C}\wedge d\mathcal{C}\ .
\frac{dw}{du}=1-\frac{1}{2\pi}\left( \frac{t_1}{u}+\frac{t_3}{u+n}\right), \frac{d^2 w}{du^2}= \frac{1}{2\pi}\left( \frac{t_1}{u^2}+\frac{t_3}{(u+n)^2}\right) >0.
w(Y_i,Y_j)=w(R_{\alpha}Y_i,R_{\alpha}Y_j)=w(\tilde Y_i,\tilde Y_j)=\tilde X_i^T(\tilde Q_j-\tilde Q_i)\tilde X_j,\;\tilde Q_k=\tilde U_k \tilde X_k^{-1},\,k=i,j.
\left( \begin{array}{c c c c}{{( E - m )}}&{{0}}&{{- P_{z}}}&{{- \pi_{-}}}\\{{0}}&{{( E - m )}}&{{- \pi_{+}}}&{{P_{z}}}\\{{- P_{z}}}&{{- \pi_{-}}}&{{( E + m )}}&{{0}}\\{{- \pi_{+}}}&{{P_{z}}}&{{0}}&{{( E + m )}}\\ \end{array}\right) \left( \begin{array}{c}{{\varphi_{+}}}\\{{\varphi_{-}}}\\{{\chi_{+}}}\\{{\chi_{-}}}\\ \end{array}\right) = 0
\left(\partial_{0,\nu}\left(\begin{array}{cc}1 0 0 0\end{array}\right)+\left(\begin{array}{cc}0 0 0 1\end{array}\right)+\left(\begin{array}{cc}0 -\partial_{1}-\partial_{1}0\end{array}\right)\right)\left(\begin{array}{c}u v\end{array}\right)=\left(\begin{array}{c}\partial_{0,\nu}^{-1}f 0\end{array}\right),
u(P)=\frac{1}{4\pi}a^{3}(1-\frac{\rho^{2}}{a^{2}})\int_{0}^{2\pi}\int_{0}^{\pi}\frac{g(\theta^{\prime},\varphi^{\prime})sin\theta^{\prime}}{(a^{2}+\rho^{2}-2a\rho cos\Theta)^{\frac{3}{2}}}d\theta^{\prime}d\varphi^{\prime}
g(\tau, r) = \left[2r^2 (u_r^2 + u_t^2) + 4r^2 u_r u_t + 2\rho u u_r r + 2\rho u u_t r\right] + \frac{e^{2\tau}}{3}\frac{r}{t_+}|u|^6 + \left(\frac{r(r^2 + t_+^2)}{t_+}- 2r^2\right)(u_r^2 + u_t^2) + \left(\frac{r^2}{t_+}-r\right)(2\rho u u_r) = g_1 (\tau,r) + g_2 (\tau,r) + g_3 (\tau,r) + g_4 (\tau,r).
\omega(a_t k)\phi= ~\xi^{\mu_1+\frac{3}{2}}(\eta\zeta)^{\mu_2+\frac{3}{2}}\bar{\gamma}^{\nu+\frac{3}{2}}(\cosh t)^{-\mu_2-\nu-3}e^{\pi \tanh t (z_{11}z_{31}+z_{12}z_{32}+z_{13}z_{33})}&\times (z_{11}\cosh^{-1}t\cos\theta -z_{21}\zeta\sin\theta)^r(z_{11}z_{22}-z_{21}z_{12})^{\mu_2}z_{33}^\nu.
N_{105}'(f,H)=-\frac{1}{576}\times\begin{cases}437d+139 \hbox{if $d\equiv 1\pmod{12}$}, 535d-644 \hbox{if $d\equiv 5\pmod{12}$}, 97d-324 \hbox{if $d\equiv 7\pmod{12}$}, 195d+13 \hbox{if $d\equiv 11\pmod{12}$.}\end{cases}
\Delta P-\frac{n}{\alpha}\bar{g}(\nabla P,\nabla \log u) =u^{\frac{\alpha+1}{\alpha}}(|h|^{2}-nH^{2})-u^{\frac{1}{\alpha}}\overline{\mathrm{Ric}}(\nu,\lambda\partial_{r}^{T}) &+\frac{n-1}{\alpha}u^{\frac{\alpha+1}{\alpha}}|\nabla\log u|^{2}.
\left( \sum_{i = 1}^{2}\left( \frac{1}{r_{i}^{2}}\frac{\partial}{\partial r_{i}}r_{i}^{2}\frac{\partial}{\partial r_{i}}+ \frac{1}{r_{i}^{2}\, \sin \psi}\, \frac{\partial}{\partial \psi}\, \sin \psi \, \frac{\partial}{\partial \psi}- \frac{r_{i}^{2}}{g^{2}}\right) \! + \frac{2}{g^{2}}\, E \right) \Psi ( r_{1}, r_{2}, \psi ) = 0 \; .
\mathbf{p}\{B_i^+,B_j^+\}=\sum_{k=1}^{i}\sum_{s=1}^{i-k+1}\sum_{I\in \mathcal{I}_{ki}(s)}\sum_{l=1}^{j}\sum_{t=1}^{j-l+1}\sum_{J\in \mathcal{I}_{lj}(t)}E^{e_I+e_J}\{B_k^+,B_l^+\}H_{IJ}^+
\psi_{1}= \int d^{D}x \left( - \mu_{3 a}^{( \varphi )}\partial^{\alpha}\mu_{2 \alpha}^{a}+ \left( \partial^{\alpha}\mu_{3 \alpha}^{a}\right) \mu_{2 a}^{( \varphi )}+ \left( \partial^{\alpha}A_{\alpha}^{a}\right) \nu_{1 a}^{( \varphi )}- \left( \partial^{\alpha}\varphi_{a}\right) \nu_{1 \alpha}^{a}\right) \, ,
h(1) =\gamma_-(\theta(1)u^++u^-)=I'(u^-)[u^-]+\theta(1)^2\lambda\int_{\mathbb R^3}\phi_{u^+}(u^-)^2dx <I'(u^-)[u^-]+\lambda\int_{\mathbb R^3}\phi_{u^+}(u^-)^2dx=\gamma_-(u)\leq0.
r_{jj^{\prime}}(q)L_{ij}(x_{i};t_{j})\left( \begin{smallmatrix}1 0 0 q\end{smallmatrix}\right)_{i}L_{ij^{\prime}}(x_{i};t_{j^{\prime}})=L_{ij^{\prime}}(x_{i};t_{j^{\prime}})\left( \begin{smallmatrix}1 0 0 q\end{smallmatrix}\right)_{i}L_{ij}(x_{i};t_{j})r_{jj^{\prime}}(q),
{\frac{1}{\sqrt{- g}}}\partial_{\mu}[ \sqrt{- g}\partial^{\mu}\sigma ] = - e^{- 2 \sigma}{({\partial}_{\mu}b )}^{2}+{\frac{2 \alpha^{\prime}}{3}}\left({\frac{\delta \Delta}{\delta \sigma}}\right){\cal{R}}_{G B}^{2}+{\frac{2 \alpha^{\prime}}{3}}\left({\frac{\delta \Theta}{\delta \sigma}}\right){\cal{R}}{\tilde{\cal{R}}}\,
H(X) =\tfrac{1}{(g!)^{2}}\int_{X^{g}}\log\|\theta\|(2P_{1}+P_{2}+\dots+P_{g-1}-P_{g})\Phi^{*}\nu^{g}&=S_{g}(X)+\int_{X}g(\sigma(P),P)\mu(P)+\left(\tfrac{g(g+1)}{2}-1\right)\tfrac{1}{2g(g-1)}\varphi(X) &=S_{g-1}(X)+\tfrac{(g+2)}{4g}\varphi(X).
\Phi^{( 1 )}\rightarrow{\frac{T}{2}}i k_{\alpha}[ \dot{x}_{\beta}( \tau_{2}) - \dot{x}_{\beta}( \tau_{1}) ] \, - \,{\frac{T}{2}}i k_{\beta}[ \dot{x}_{\alpha}( \tau_{2}) - \dot{x}_{\alpha}( \tau_{1}) ] ,
d=2,~\Omega=(0,1)\times(0,1),~T=1,~\psi_1({x})=\psi_0({x})=x_2(1-x_2)x_1(1-x_1), &a({x})=(30+\sin(x_1)^2)(30+\sin(x_2)^2), &f({x},t)=\exp(t)[x_1(1-x_1)x_2(1-x_2)-\sin(2x_1)(30+\sin(x_2)^2)(1-2x_1)x_2(1-x_2) &\quad\quad\quad-\sin(2x_2)(30+\sin(x_1)^2)(1-2x_2)x_1(1-x_1)+2a({x})(x_1(1-x_1)+x_2(1-x_2))].
\langle \mathbf h_2^{(2)}, \mathbf h_2^{(2)}\rangle = \langle \mathbf h_2,\mathbf h_2 \rangle, \langle \breve{\mathbf g}_2, \breve{\mathbf g}_2 \rangle = \langle \mathbf g_2, \mathbf g_2 \rangle, \langle \pmb{\phi}_2^{(2)}, \pmb{\phi}_2^{(2)}\rangle = 2 \langle \pmb{\phi}_2, \pmb{\phi}_2\rangle.
Z_N(\zeta)=\zeta\left(\prod_{i=1}^mK_{N,L_i}\right)\mathrm{Pf}\begin{bmatrix}{[}\zeta^2\alpha_{j,l}+\beta_{j,l}] [\mu_j]{[}-\mu_l] 0\end{bmatrix}_{j,l=1,\ldots,N}.
T_1^{-1}T_\pi T_\pi^\vee = T_\pi^\vee T_\pi T_{n-1}\qquad\hbox{and}T^{-1}_{n-1}\cdots T_1^{-1}T_\pi (T_\pi^\vee)^{-1}= q(T_\pi^\vee)^{-1}T_\pi T_{n-1}\cdots T_1.
\int_{-\infty}^{F_2(y^G_1,y^G_2)}\!\!\! p_{Y_1,Y_2}(F_1(y^G_1),\,\cdot\,) \; \frac{\partial F_1}{\partial y^G_1}(y^G_1) = \int_{-\infty}^{y^G_2}\!\! p_{Y^G_1,Y^G_2}(y^G_1,\,\cdot\,)
Ps_{n}^{m}\left({x,\gamma^{2}}\right) =\frac{\left({-1}\right)^{n}\sin\left({\gamma\xi+\gamma\sigma E\left({\sigma;\sigma^{-1}}\right)-{\frac{1}{2}}n\pi}\right) +{O}\left({\gamma^{-1}}\right)}{\gamma\left({n-m}\right) !V_{n}^{m}\left( \gamma\right) \left[{\left({x^{2}-1}\right) \left({x^{2}-\sigma^{2}}\right)}\right]^{1/4}},
\begin{matrix}1&0&0&0&0&0\\ (1-p)&0&p&0&0&0\\ 0&(1-p)&0&p&0&0\\ 0&0&(1-p)&0&p&0\\ 0&0&0&(1-p)&0&p\\ 0&0&0&0&0&1\end{matrix}
e^{i\alpha P_1}=e^{\alpha(a_1^{\dagger}-a_1)}=e^{-\alpha^2/2}e^{\alpha a_{1}^{\dagger}}e^{-\alpha a_{1}},e^{i\beta P_2}=e^{\beta(a_2^{\dagger}-a_2)}=e^{-\beta^2/2}e^{\beta a_{2}^{\dagger}}e^{-\beta a_{2}},
2 \pi \alpha^{\prime}S_{b}= \int d \tau \int_{0}^{2 \pi \alpha^{\prime}p^{+}}d \sigma \left( \,{\frac{1}{2}}G_{M N}\, \partial_{a}X^{M}\partial^{a}X^{N}+ B_{M N}\partial_{\tau}X^{M}\partial_{\sigma}X^{N}\right) .
\widehat{s}\cdot \widehat{t}=(s_1x_1 + s_2 x_2 + s_3 1)(-t_1 1 + t_2 x_2 - t_3 x_1) &=(-s_1t_1 -s_3t_3) x_1 + (s_1t_2+s_2t_3)x_1x_2 + (-s_2t_1+s_3t_2)x_2 + (-s_3t_1) 1,
\hat{f}(\xi)=\int_{\mathbb{R}}f(x)e^{-ix\xi}dx=\int_{\mathbb{R}}f(x+\frac{\pi}{\xi})e^{-ix\xi}e^{-i\pi}dx=-\int_{\mathbb{R}}f(x+\frac{\pi}{\xi})e^{-ix\xi}dx
(\partial_{\tau^1}\partial_{\tau^{i-1}}\partial_{\tau^e}F)\eta^{ef}(\partial_{\tau^f}\partial_{\tau^{j}}\partial_{\tau^k}F) = (\partial_{\tau^1}\partial_{\tau^{j}}\partial_{\tau^{e}}F)\eta^{ef}(\partial_{\tau^f}\partial_{\tau^{i-1}}\partial_{\tau^k}F).
\overline{\tau_{\alpha_1-1}(\gamma_1)\cdots\tau_{\alpha_{\ell}-1}(\gamma_{\ell})}=\sum_{P \{1,\ldots,\ell\}}\ \prod_{S\in P}\ \sum_{\widehat{\alpha}}\tau_{\widehat{\alpha}}(\widetilde{\mathsf{K}}_{\alpha_S,\widehat{\alpha}}\cdot\gamma_S) \ .
\mathbb{P}\left(z_{i+1},\dots,z_{j}\in \mathcal{C}_\mu\mid z_i\in\mathcal{C}_\mu\right) \leq \left(\frac{2}{n}\right)^{\left\lceil (j-i)/2\right\rceil}\frac{2^{\left\lfloor (j-i)/2\right\rfloor}}{n(n-1)\cdots (n-\left\lfloor (j-i)/2\right\rfloor + 1)}& = \frac{2^{j-i}(n-\left\lfloor(j-i)/2\right\rfloor)!}{n^{\left\lceil (j-i)/2\right\rceil}n!}.
\delta\, \left( \sum_k \xi(k)\right) \leq \sum_{\{k\,:\, \xi(k)=1\}}\bigl(I_f(x^{\bar k},3\lambda^{k-1})-I_f(x^{\bar k},\lambda^{k+1})\bigr) \leq C(\lambda)\,I_{f}(x^{\bar{k}}, 1)
(M_s,M_p)=\alpha\beta(M_{s_1},M_{p_{1,1}})+\alpha(1-\beta)(M_{s_1},M_{p_{1,2}})+ &(1-\alpha)\gamma(M_{s_2},M_{p_{2,1}})+(1-\alpha)(1-\gamma)(M_{s_2},M_{p_{2,2}}).
E_{\alpha}= ( - 1 )^{\alpha_{1}}\sigma_{\alpha}( 0 ) \sum_{j}\prod_{k \neq j}\frac{\sigma_{\alpha}( z_{j k})}{\sigma_{0}( z_{j k})}\left[ \frac{l}{n}\sum_{i \neq j}\frac{\sigma_{\alpha}^{\prime}( z_{j i})}{\sigma_{\alpha}( z_{j i})}- \partial_{j}\right] , \alpha \equiv ( \alpha_{1}, \alpha_{2}) \neq ( 0 , 0 ) \equiv ( n , n ) ,
A = \frac{i}{2}\{\frac{1}{2}d \operatorname{l n}( \frac{d \overline{{g}}_{2}}{d \overline{{z}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{g}}_{2}- \overline{{g}}_{2}d g_{2})}{( 1 - g_{2}\overline{{g}}_{2})}\},
0 = \int_\Omega\{[\partial_\rho(\rho\eta) + \kappa_M + \kappa_E(\frac{1}{2}|u|_2^2 +\partial_\rho(\rho\epsilon))]\sigma + [\rho\partial_\theta\eta + \kappa_E \rho\partial_\theta\epsilon]\vartheta\}dx & + \int_\Omega \{\kappa_E\rho u\cdot v + [\rho\partial_\tau\eta + \kappa_E \rho \partial_\tau \epsilon]\nabla d:\nabla\delta + \kappa_G d\cdot\delta \}dx.
I_{z_\varepsilon}(\gamma_\varepsilon)=\gamma_\varepsilon^{-4}+\frac{A(\gamma_\varepsilon)}{2}+ \frac{4 B(\gamma_\varepsilon)}{\gamma_\varepsilon^3 \exp(1+\mathcal{H}_{z_\varepsilon}(z_\varepsilon))}\int_{\Omega}G_{z_\varepsilon}(y)F(4\pi G_{z_\varepsilon}(y))~dy\,,
I_{e f f}= \frac{1}{\pi}\int d^{2}z [ \frac{k}{2}\ \ \frac{\partial w \overline{{{\partial}}}w}{w^{2}}+ \frac{w^{2}}{w^{2}- 4 \lambda}( \partial f \overline{{{\partial}}}f - c^{2}\partial y \overline{{{\partial}}}y ) + \frac{c w^{2}}{w^{2}- 4 \lambda}( \partial f \overline{{{\partial}}}y - \partial y \overline{{{\partial}}}f ) ]
A=(\begin{matrix}2&3&6&2\\ 5&6&1&6\\ 8&3&1&1\end{matrix})H=(\begin{matrix}1&0&50&-11\\ 0&3&28&-2\\ 0&0&61&-13\end{matrix})U=(\begin{matrix}9&-5&1\\ 5&-2&0\\ 11&-6&1\end{matrix})
0 \geq{\dot R}= -\iint_{\mathbb T \times \mathbb R}\sin(\theta-\phi)\omega f(\theta,\omega,t) ~ d\theta d\omega+KR \iint_{\mathbb T \times \mathbb R}\sin^{2}(\theta-\phi)\big)\rho(\theta,t) ~ d\theta d\omega \geq-M+K\underline{R}\cos^{2}\gamma(1-{\mathcal M}(L^+_{\gamma}) -{\mathcal M}(L^-_{\gamma})).
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{\ell_1}0 c_2 \, a^{\ell_1}\, b^{p^{e_2}}\, d^{p^{e_2}}0 a^{\ell_1}0 0 0 d^{- \ell_3}\end{array}\right) .
\mathbb{E}_{\lambda}^d\xi_t(O) \leq e^{-t}\sum_{n=0}^{\infty}\frac{t^n\lambda^nM^{2n}(d+1)^n}{n!E{\widetilde{\rho}}^2}[\frac{dE({\widetilde{\rho}}^2)}{d+1}+\frac{1}{(d+1)E({\widetilde{\rho}}^2)}]^n &=\big(E({\widetilde{\rho}}^2)\big)^{-1}\exp\big\{t\big[\lambda M^2\big(dE({\widetilde{\rho}}^2)+\frac{1}{E({\widetilde{\rho}}^2)}\big)-1\big]\big\}.
\left. \frac{\partial^{2}V_{e f f}^{( 2 )}( T = 0 , \mu , m )}{\partial m^{2}}\right|_{m = m_{1}}= - \frac{1}{\pi}\frac{2 \mu - m ( 0 )}{m ( 0 ) - \mu}< 0 , \ \mathrm{w h e n}\ \frac{m ( 0 )}{2}< \mu < m ( 0 )
\bigg( \sum_{\vert \alpha \vert = m}\Vert x_{\alpha}\Vert^{q}\bigg)^{1/q}\leq C\, \bigg(\int_{\mathbb{T}^{\infty}}\Big\Vert \sum_{\vert \alpha \vert = m}x_{\alpha}z^{\alpha}\Big\Vert^2 dz\bigg)^{1/2}\,.
\nabla S=\frac{\partial}{\partial q^{k}}[S_{ij}b^{i}\otimes b^{j}]\otimes b^{k}=[\frac{\partial S_{ij}}{\partial q^{k}}-\Gamma_{ki}^{l}S_{lj}-\Gamma_{kj}^{l}S_{il}]b^{i}\otimes b^{j}\otimes b^{k}
\Big| \int\langle\zeta, (1-\chi_{i+1})\mathcal{C}\rangle\Big| =\Big| \int\langle(1-\chi_{i+1})\zeta, \mathcal{C}\rangle\Big| &\leq C\frac{\mathfrak{r}}{T^i}\|\mathcal{C}\|_{L^2}&\leq C(\frac{\mathfrak{r}}{T^i})^{\frac{5}{2}}P^i\kappa_1(\frac{\mathfrak{r}}{T^i})^{\frac{1}{8}}&\leq \frac{\varepsilon}{8}P^i\kappa_1(\frac{\mathfrak{r}}{T^{i+1}})^{\frac{5}{2}}
\phi(t\otimes f_{e,g})(a u_k)\phi(t\otimes f_{h,e})=x_g^*(au_k)x_h=r_eu_g^*au_ku_hr_e=u_g\alpha_g(r_e)au_ku_hr_e=u_g^*r_gau_{kh}r_e &=u_g^*r_ga\alpha_{kh}(r_e)u_{kh}=u_g^*r_gar_{kh}u_{kh}=u_g^*ar_gr_{kh}u_{kh}=\delta_{g,kh}u_g^*ar_g^2u_g=\delta_{g,kh}u_g^*au_g\alpha_{g^{-1}}(r^2_g) &=\delta_{g,kh}\alpha_{g^{-1}}(a)r_e^2=\delta_{g,kh}r_e\alpha_{g^{-1}}(a)r_e.
2\Phi(p)= 2(1-z)^{\frac a2}\cdot{}_2F_1\bigg[\begin{matrix}\frac12+\frac12(p-1-a) -\frac12(p-1-a) &1\end{matrix}\bigg|\,\frac{z^2}{4z-4}\bigg]_{p-1}= (1-z)^{a-\frac{p-1}{2}}\cdot\big(\Omega_1(0)+\Omega_2(0)\big),
f' =g'+\frac{\pi^2-\Pi\pi'}{\pi^2}g'+\frac{\Pi}{\pi}g''=2g'-\frac{\Pi\pi'}{\pi^2}g'+\frac{\Pi}{\pi}g'' &=2g'+\frac{\Pi}{\pi}\left(g''+\frac{b'}{b}g'-\frac{2a}{b}g'\right)=2g'+\frac{\Pi}{\pi}\left(-\frac{2\lambda g}{b}-\frac{2\pi}{\Pi}g' \right) &=-2\lambda g \frac{\Pi}{\pi}\frac{1}{b}=-2\lambda g M s
\begin{gathered}\begin{cases}\int_{\Omega}\mathcal{A}\big(x,\nabla u(x)\big)\cdot\nabla\phi(x)\,dx = \int_{\Omega}\frac{f(x)}{u^\delta}\phi(x)\,dx, u > 0\Omega,\,\,u = 0 \partial\Omega,\end{cases}\end{gathered}
f R_{i j k l}=\frac{1}{2(n-2)}\left(g_{i k}-\partial_i f \partial_k f\right)\left(g_{i k}-\partial_i f \partial_k f\right) -\frac{1}{2(n-2)}\left(g_{i l}-\partial_i f \partial_l f\right)\left(g_{j k}-\partial_j f \partial_k f\right) +O(r^{-8\delta}).
{\mathbf{y}}^{[\sf bs]}_{i}[t] =\sum_{j=1}^{K}\sum_{k=1}^{N}\mathbf{{f}}_{i,jk}[t]x^{[\sf u]}_{jk}[t]+\sum_{j=1}^K\mathbf{{B}}_{ij}[t]{\mathbf{x}}^{[\sf bs]}_{j}[t]+\mathbf{{z}}^{[\sf bs]}_i[t],
\begin{array}{l}{\delta \theta^{\alpha}= \epsilon^{\alpha}, \qquad \delta \bar{\theta}_{\alpha}= \bar{\epsilon}_{\alpha},}{\delta x^{\bar{\mu}}= - i \bar{\epsilon}_{\alpha}\tilde{\Gamma}^{{\bar{\mu}}\alpha \beta}\bar{\theta}_{\beta}+ i \epsilon^{\alpha}\Gamma_{\alpha \beta}^{\bar{\mu}}\theta^{\beta}, \qquad \delta x^{1 1}= 0 ,}\end{array}
\nabla(\tilde\omega) = \tilde u \otimes \xi, \nabla(\tilde \omega^\vee) = 0, \nabla(\tilde u) = 0, \nabla(\tilde u^\vee) = \tilde \omega^\vee \otimes \xi, \varphi(\tilde \omega) = p \tilde\omega, \varphi(\tilde \omega^\vee) = p^{-1}\tilde\omega^\vee, \varphi(\tilde u) = \tilde u, \varphi(\tilde u^\vee) = \tilde u^\vee.
\big(\mathcal T\mathbf x^{m}\big)_i=\langle\operatorname{Sym}(\mathcal T_i),\mathbf x^{\otimes (m)}\rangle:=\sum_{i_1,\dots,i_m=1}^n\big(\operatorname{Sym}(\mathcal T_i)\big)_{i_1\dots i_m}x_{i_1}\dots x_{i_m}\ \mathbf x\in\mathbb C^n,
\begin{cases}\begin{array}{ll}u_m(.,t)\in SPSH(\Omega) \dot{u}_m=\log\det (u_m)_{\alpha\bar{\beta}}-Au_m+f(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), u_m\searrow\varphi \mbox{on}\;\partial\Omega\times [0,T), u_m\searrow u_0 \mbox{on}\;\bar{\Omega}\times\{0\}, \end{array}\end{cases}
p_n(x)=\frac{1}{\sqrt{D_n D_{n-1}}}\begin{vmatrix}c_0 c_1 \dots c_n c_1 c_2 \dots c_{n+1}\vdots \vdots \vdots c_{n-1}c_n \dots c_{2n-1}1 x \dots x^n\end{vmatrix}
\sum_{n=0}^{\infty}\left(a;q\right)_{n}S_{n}\left(x;q\right)z^{n}=\frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}}\left(a;q\right)_{n}\left(-xz\right)^{n}}{\left(q,az;q\right)_{n}}.
S = - \frac{1}{2 \kappa_{P}^{2}}\int d^{4}x \sqrt{- g}\left[ \frac{1}{2}R + \frac{1}{4}( \nabla \varphi )^{2}+ \frac{3}{4}( \nabla \beta )^{2}+ \frac{q_{5}}{2}e^{( \beta - \varphi )}( \nabla z )^{2}\right] \; ,
{\rm Pr}(s_{t+1}|s_t,a_{\pi}(s_t)) = \Big( \prod_{k=1}^K{\rm Pr}(h_{k,t+1})\prod_{i=1}^n{\rm Pr}(\beta_{k,t+1,i})\Big) ~~~~~~~~~~~~~~~~~~ \times{\rm Pr}(q^{}_{t+1},\{q_{k,t+1}^{},\forall k\}| s_t,a_{\pi}(s_t)).
e^{-\mathcal{A}_{D,\varepsilon}t}-e^{-\mathcal{A}_D^0t}=-\frac{1}{2\pi i}\int_\gamma e^{-\zeta t}\left((\mathcal{A}_{D,\varepsilon}-\zeta I)^{-1}-(\mathcal{A}_D^0-\zeta I)^{-1}\right)\,d\zeta .
\mathcal{H}=\{H_{(1,1)}^1,H_{(1,1)}^2,H_{(1,2)}^1,H_{(1,2)}^2,H_{(2,1)}^1,H_{(2,1)}^2,H_{(2,2)}^1,H_{(2,2)}^2,\dots,H_{(4,4)}^1,H_{(4,4)}^2\}
\tilde X_k^r := (X_{k\delta}^{r(m+1)},\cdots,X_{k\delta}^{r(m+1)+m}), \tilde Y_k^r := (Y_{k\delta+t}^{r(m+1)}-Y_{k\delta}^{r(m+1)},\cdots,Y_{k\delta+t}^{r(m+1)+m}-Y_{k\delta}^{r(m+1)+m})_{t\in[0,\delta]}.
W_{h_{1}}=\inf_{t\in T}\bigl(X(t) - h_{1}(t)\bigr) \le \inf_{s\in T}\bigl(X(s) - h_{2}(s)\bigr)+\|h_{2}- h_{1}\|_{\infty}&=W_{h_{2}}+\|h_{2}-h_{1}\|_{\infty}.
| \varphi , \beta > = e x p \{i \int_{x , y}[ ( \phi_{x}- \varphi_{x}) g_{x y}( \beta ) \tilde{\Pi}_{y}- \Pi_{x}g_{x y}( \beta ) ( \tilde{\phi}_{y}- \varphi_{y}) ] \}| \varphi \tilde{\varphi}> ,
\sum_{i=1}^{N-1}\sup_{h\in\mathrm{Lip}(1)}\left|\tilde{Q}_{i-1}\big(P_{\eta}-\tilde{Q}_{1}\big) P_{(N-i)\eta}h(x)\right| \leq C(1+|x|)\eta^{2/\alpha}\sum_{i=1}^{N-1}[(N-i)\eta]^{-1/\alpha}\leq C\frac{\alpha}{\alpha-1}\,(1+|x|)\eta^{2/\alpha-1}.
{\frac{\alpha^{( I )}\alpha^{( J )}}{2}}+ s -{\frac{d_{I}d_{J}}{D - 2}}+ \sum_{i = 1}^{N}r_{i}\Delta_{a i}^{( I )}\Lambda_{b^{\prime}i}^{( J )}= 0 , \; I , \; J = 1 , \; 2
\left\|\psi^{(n)}_j(f)a-a\psi^{(n)}_j(f)\right\| \to 0 \left\|\psi^{(n)}_j(\mbox{\texttt{Lt}}_g(f))-\alpha_g(\psi^{(n)}_j(f))\right\| \to 0 \left\|\psi^{(n)}_j(f_1)\psi^{(n)}_j(f_2)\right\| \to 0 \left\|\sum\limits_{j=0}^d\psi^{(n)}_j(1_{C(G)})-1_A\right\| \to 0
\sum_{k = 1}^{2 m}\frac{1}{1 - \omega^{2 k + 1}}= \sum_{k = 1}^{m}\frac{1}{1 - \omega^{2 k + 1}}+ \sum_{k = m + 1}^{2 m}\frac{1}{1 - \omega^{2 k + 1}}.
\begin{pmatrix}0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0\end{pmatrix}\begin{pmatrix}x y x^q+y^{q^2}x^{q^2}+y^q+y^{q^2}\end{pmatrix}=\begin{pmatrix}z t z^{q^{3}}+z^{q^{2}}+t^{q^{2}}z^{q^{2}}+t^{q^{3}}\end{pmatrix},
R_1 \Phi_{n}^{(e_1,e_2)}(\phi)=(1-2e_1)\Phi_{n}^{(e_1,e_2)}(\phi),R_2 \Phi_{n}^{(e_1,e_2)}(\phi)=(1-2e_2)\Phi_{n}^{(e_1,e_2)}(\phi),
b_i\ge \alpha mb_{i+1}\ge \sum_{r=i+1}^{k-1}\frac{\alpha m}{k}\cdot b_{i+1}\ge\sum_{r=i+1}^{k-1}\frac{\alpha^{r-i}m^{r-i}}{k}\cdot b_r\ge \sum_{r=i+1}^{k-1}\left(\frac{\alpha m}{k}\right)^{r-i}b_r\ge \sum_{r=i+1}^{k-1}\binom{\left\lfloor\frac{\alpha m}{k}\right\rfloor}{r-i}b_r.
I(u) \ge \frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|u|^{p}dx & \ge \frac{1}{2}\left\|\nabla u\right\|_{2}^{2}-\frac{1}{p}C_{p}\left\|u\right\|_{2}^{\frac{6-p}{2}}\left\|\nabla u\right\|_{2}^{\frac{3p}{2}-3}.
\begin{aligned}F^{kk}\widetilde{\lambda}_{1,k\bar k}\geq F^{kk}g_{k\bar k1\bar 1}+ F^{kk}(W_{1\bar 1k\bar k}- W_{k\bar k1\bar 1}) &+ \frac{1}{8n\lambda_1}\sum_{p > 1}F^{kk}\Big( |g_{1\bar pk}|^2 + |g_{p\bar 1k}|^2\Big) &- C(F^{kk}|g_{1\bar 1k}| + \lambda_1\mathcal{F}),\end{aligned}
a_{h,3}(\ell,k)=\frac 1{(2\pi i)^4}\int_{\Gamma_{D_1}}dz\int_{\gamma_{r_1}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1,\mu_1,\xi_1}(z)G_{\Delta n,\Delta\mu,\Delta\xi}(w)\left(\frac 1{z-w}-h\right)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(z-\zeta)(w-\omega)}.
a_i(\tau)\, =\frac 1\tau\bigl|\{x\in K_i\setminus\{-1\}: |S^{-1}(x)|=\tau-1\}\bigr|,2\le \tau\le k, a_i(1)\, =k-\delta_i-\sum_{\tau=2}^k\tau a_i(\tau), a_i(0)\, =n-\sum_{\tau=1}^ka_i(\tau).
G_{L}\left( z \right) = \left( \begin{array}{c c}{1}{0}{\theta_{L}}{1}\end{array}\right) \left( \begin{array}{c c}{H_{L}}{0}{0}{g_{L}}\end{array}\right) \left( \begin{array}{c c}{1}{\tilde{\theta}_{L}}{0}{1}\end{array}\right)
\varphi(x) g(x,D_x)\psi(x) =\varphi(x)\eta(x_n) g(x,D_x)\psi(x) + \varphi(x)(1-\eta(x_n))g(x,D_x)\psi(x) \ =\varphi(x)\eta(x_n) g(x,D_x)(\eta(x_n)\psi(x)) + \varphi(x)\eta(x_n) g(x,D_x)(1-\eta(x_n))\psi(x) \ \ \ + \varphi(x)(1-\eta(x_n))g(x,D_x)\psi(x).
C_{\Omega}= \{w\}= \bigcup\limits_{n\geq 1}\{wT_k\}= \bigcup\limits_{k\geq 1}\{\tau(p)=k,T_k<\zeta\}\bigcup\bigcup\limits_{k\geq1}\{T_k=\zeta-1,\tau(p)>k\}.
\langle w^*,{\mathcal N}v^* \rangle_{V^*,V}= \langle w^*, v \rangle_{V^*,V}= \int_{\Omega}\nabla w \cdot \nabla v dx & = \langle v^*, w \rangle_{V^*,V}= \langle v^*,{\mathcal N}w^* \rangle_{V^*,V}.
{R}_{1 2}( u - v ) Z_{2 1}^{- 1}L_{1}( u ) \tilde{Z}_{2 1}L_{2}( v ) = Z_{1 2}^{- 1}L_{2}( v ) \tilde{Z}_{1 2}L_{1}( u ){R}_{1 2}( u - v ) \ll{r z l z l}
\tilde{\epsilon}(t) = \frac{1}{t}\sum_{k=1}^t \epsilon(R(j),R_u(j)) &= \frac{t-1}{t}\frac{1}{t-1}\sum_{k=1}^{t-1}\epsilon(R(j),R_u(j)) + \frac{\epsilon(R(t),R_u(t))}{t}&= \frac{t-1}{t}\tilde{\epsilon}(t-1) + \frac{\epsilon(R(t),R_u(t))}{t}
\varphi_{\Sigma_2}\Big(\vec{\alpha}, \vec{\beta}\; \big| \, w_4, w_3, w_2, w_1\Big): =\phi_{\Sigma_2}\left(\vec{\alpha}, \vec{\beta}\; \big| \,w_4, w_3, w_2, w_1, 1, 1, 1, 1\right) .
0 = \frac{\delta M}{\delta \Omega ( m )}= \frac{\delta}{\delta \Omega ( m )}\sum_{n , \mu}\left\{\Omega ( n ) U_{\mu}\Omega^{\dagger}( n + 1 ) \sigma_{3}\Omega ( n + 1 ) U_{\mu}^{\dagger}\Omega^{\dagger}( n ) \sigma_{3}\right\}
\sum_{(e,\tilde{e})}\frac{C^n_{e,\tilde{e}}M^{\tilde{e}}}{L^eM^e}\Big(\frac{1}{M^e}+\frac{1}{M^{\tilde{e}}}\Big)\leq C\,\sum_{(e,\tilde{e})}\frac{M^{\tilde{e}}}{M^e}\Big(\frac{1}{M^e}+\frac{1}{M^{\tilde{e}}}\Big)\to 0,
-(\partial_y+\kappa\partial_z)\Delta^{-1}(\partial_yV\rho_1\Delta u_{\neq}^3)+\rho_1\nabla V\cdot\nabla u_{\neq}^3 =-[(\partial_y+\kappa\partial_z)\Delta^{-1},\partial_yV\rho_1]\Delta u_{\neq}^3 =-[\partial_y\Delta^{-1},\partial_yV\rho_1]\Delta u_{\neq}^3-\kappa[\partial_z\Delta^{-1},\partial_yV\rho_1]\Delta u_{\neq}^3.