harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_\mathrm{ot}^\mathrm{J}=\sum\limits_i \beta_i \! \log\!|\mathbf{I}\!+\!\mathbf{H}_i\mathbf{Q}_i\mathbf{H}_i^\mathrm{H}(\mathbf{H}_{\mathrm{z}i}\mathbf{Q}_{\mathrm{z}i}\mathbf{H}_{\mathrm{z}i}^\mathrm{H}\!+\!\sigma_i^2\mathbf{I})^{\!-1}\!|
{{\mathbf{\Omega}}_{p2}}= \left\{{\left.{\left({{{\sin}^2}{\theta_1}, \cdots ,{{\sin}^2}{\theta_k}}\right)}\right|\sin{\theta_1}\leq \delta ,0 \leq{{\sin}^2}{\theta_k}\leq \cdots \leq{{\sin}^2}{\theta_1}}\right\}.
\langle \Xi | \left( G_{- 1 / 2}+ \frac{1}{\sqrt{3}}\G_{1 / 2}+ \frac{1 1}{6}\G_{3 / 2}+ \frac{3}{2 \sqrt{3}}\G_{5 / 2}+ \frac{7}{7 2}\G_{7 / 2}+ \cdots \right) = 0 .
\hat\psi(c_q)=\begin{cases}s_{l_q}, \textrm{if}o(E_q)=1\textrm{and}O_q=1 s_{r_q}^{-1}, \textrm{if}o(E_q)=1\textrm{and}O_q=-1 1, \textrm{if}o(E_q)=-1\textrm{and}O_q=1 s_{l_q}s_{r_q}^{-1}, \textrm{if}o(E_q)=-1\textrm{and}O_q=-1 \end{cases}.
\begin{aligned}\partial_t|A|^2~=~ \alpha H^{-\alpha-1}\Delta |A|^2-2\alpha H^{-\alpha-1}|\nabla A|^2-2\alpha(\alpha+1)H^{-\alpha-2}h_{ij}\nabla^iH\nabla^jH +2\alpha H^{-\alpha-1}|A|^4-2(1+\alpha)H^{-\alpha}h_i^kh_k^jh_j^i +2(1-\alpha)H^{-\alpha+1}+4\alpha H^{-\alpha-1}|A|^2. \end{aligned}
\left|\int_{|\theta|\ge n^{-\delta}}z^{-(n+1)}p(z)\,dz\right|\le \,\rho^{-n}p(\rho)\int_{|\theta|\ge n^{-\delta}}e^{-\alpha_1n^{3/2-2\delta}}\,d\theta = \exp\left(bn^{1/2}-\alpha_2n^{3/2-2\delta}\right),
q'(A)= -\frac{4-A}{4-A}-\ln(4-A)-\left[\frac{A}{A}+\ln(A)\right]-\left[-\frac{2(1-A)}{1-A}-2\ln(1-A)\right] &-\left[\frac{2+A}{2+A}+\ln(2+A)\right]-\left[-\frac{3-A}{3-A}-\ln(3-A)\right] = -\ln(4-A)-\ln(A)+2 \cdot \ln(1-A)-\ln(2+A)+\ln(3-A).
G^{i j}={\frac{1}{\det ( G )}}\left( \begin{array}{c c}{{G_{2 2}}}&{{- G_{1 2}}}\\{{- G_{1 2}}}&{{G_{1 1}}}\\ \end{array}\right) ={\frac{4}{3}}\left( \begin{array}{c c}{{G_{2 2}}}&{{- G_{1 2}}}\\{{- G_{1 2}}}&{{G_{1 1}}}\\ \end{array}\right) ~ .
\int_{A(k, R)}|\nabla u|^{p(x)}+ b(x)|u|^{p(x)}\leq C_0K\left( \int_{A(k, R)}(1-\eta)^{p(x)}\left( |\nabla u|^{p(x)}+ b(x)|u|^{p(x)}\right)\right. &+ \left.\int_{A(k, R)}\left(\frac{u-k}{R-r}\right)^{p(x)}+ \int_{A(k, R)}\left(b(x)(\eta w)^{p(x)}+ b(x)k^{p(x)}+ g(x)\right)\right)
\delta_a (J_1) = -a_{13}P_1 \wedge P_2 + a_{12}P_1 \wedge P_3, \delta_a (J_2) = -a_{23}P_1 \wedge P_2 + a_{12}P_2 \wedge P_3, \delta_a (P_i) = 0, \delta_a (J_3) = -a_{23}P_1 \wedge P_3 + a_{13}P_2 \wedge P_3.
q^{-7/60}\varphi_1^{-6}(\varphi_2^2G_2-q^{-7/60}\varphi_1^{-6}\varphi_\frac{1}{2}^2H_\frac{1}{2}= q^{-7/60}\varphi_1^{-6}(\varphi_2^2G_2-\varphi_\frac{1}{2}^2H_\frac{1}{2})
p(u) := \sum_{j=1}^n\limits w_j (u) \phi_{\Delta_j}'(u)= \sum_{j=1}^n\limits w_j (u) \Delta_j\Psi'(u) \phi_{{\Delta_j}}(u)\text{and}q(u) := \sum_{j=1}^n\limits w_j(u) \Delta_j \phi_{\Delta_j}(u).
I_{n , B}= \frac 1{2 ( 4 \pi^{2})^{2}}\int_{0}^{\Lambda}\, | \varphi_{k}\dot{\varphi}_{k^{\prime}}- \dot{\varphi}_{k}\varphi_{k^{\prime}}|^{2}Q_{n}( k^{2}, k^{2}) k^{2}k^{2}\coth \frac{\beta W_{k}}2 \coth \frac{\beta W_{k^{\prime}}}2 \, d k d k^{\prime}\ .
A_{j}\mathbf{\perp \!\!\!\perp}_{\mathcal{G}}\left[ \overline{\mathbf{G}}_{j}\mathbf{\backslash}\overline{\mathbf{B}}_{j}\right]\mathbf{|}\overline{\mathbf{B}}_{j},\overline{\mathbf{A}}_{j-1}j=0,\dots ,p
\ = \q^{2}\int_{q}d{\hat{z}}d{\hat{\bar{z}}}\ ({\bar{\partial}}_{q}{\hat{\xi}}_{k}) ({\hat{z}},{\hat{\bar{z}}}) ( \partial_{q}\Phi ) ({\hat{z}}q^{2 k + 1}) ( \partial_{q}\Phi ) ({\hat{z}}) \ .
n_{uv}=(-\kappa t-\tau b)_{v}=(-\kappa_{v}+\lambda_{u}\tau)t+(\kappa\lambda\tau-\tau\gamma)n+(-\lambda_{u}\kappa-\tau_{v})b, n_{vu}=(-\lambda\tau t+\gamma b)_{u}=(-\lambda_{u}\tau-\lambda\tau_{u})t+(\kappa\lambda\tau-\tau\gamma)n+(\gamma_{u})b.
I_2 \le C \left(\int_{A_2}\frac{|v(x)|^p\, |\varphi_\delta(x) - \varphi_\delta(y)|^p}{|x - y|^{N+sp}}\, dx dy + \int_{A_2}\frac{|v(x) - v(y)|^p\, \varphi_\delta(y)^p}{|x - y|^{N+sp}}\, dx dy\right) =: C\, (I_7 + I_8).
\sum_{i=1}^{N}\int_{G_{N}}\left\langle \nabla f(x),\,\mathbf{V}(x)\right\rangle dx =-\int_{G_{N}}f(x)\left(\nabla\cdot\mathbf{V}\right)(x)dx-(h(1)-h(0))\sum_{i\neq j}\int_{F_{ij}}f_{ij}(x)dS_{ij}(x)
(\bar{\partial}^{*h}\circ\bar{\partial}^h\Phi)_{A_p\overline{B}_q}= - h^{\bar{\varepsilon}\gamma}\bigg((\delta_\gamma\circ\delta_{\bar{\varepsilon}})(\phi_{A_p\overline{B}_q}) - \sum_i(-1)^i(\delta_\gamma\circ\delta_{\bar{\beta}_i})\big(\phi_{A_p\bar{\varepsilon}\bar{\beta}_1\dots\hat{\bar{\beta}}_i\dots\bar{\beta}_q}\big)\bigg)
\frac{1}{|C|}\sum_{c \in C}h(cg) - h(g) = \Big\langle h \Big| \tfrac{1}{|C|}\sum_{c \in C}P^n_{cg}- P^n_g \Big\rangle = \langle h \mid \nabla^* \tau_n \rangle = \langle \nabla h \mid \tau_n \rangle
\int_{F}\tilde{h}(\mu,z)p(d\mu,dz) = \lim_{\mathcal{T}\rightarrow\infty}\frac{1}{\mathcal{T}}\int_0^\mathcal{T}\tilde h(\mu(t),z(t))dt\ \ \ \ \forall \tilde{h}(\mu,z)\in C(F) .
i_{V}\omega = \left. \int d x^{+}T r \left( s_{0}^{- 1}\Phi s_{0}\delta A + [ A , s_{0}^{- 1}\Phi s_{0}] s_{0}^{- 1}\delta s_{0}\right) \right\vert_{x^{-}= b}^{x^{-}= a}.
\begin{aligned}\sigma(t)= \, \alpha(T-t)^{-\alpha-1}+ \frac{C^{m-1}m (N-1) k \coth(k)}{a(m-1)}\, (T-t)^{-\alpha m +\beta}-\frac{\beta}{m-1}\, (T-t)^{-\alpha-1}\leq \, \frac{2 C^{m-1}m (N-1) k \coth(k)}{a(m-1)}\, (T-t)^{-\alpha m +\beta}\, .\end{aligned}
\widetilde{\nabla}\mathbf{d}_{t}-\Delta\widetilde{\nabla}\mathbf{d}=-\widetilde{\nabla}\mathbf{u}\cdot\nabla\mathbf{d}-\mathbf{u}\cdot\nabla\widetilde{\nabla}\mathbf{d}+|\nabla \mathbf{d}|^{2}\widetilde{\nabla}\mathbf{d}+ 2(\nabla \mathbf{d}\cdot\mathbf{d})\nabla\widetilde{\nabla}\mathbf{d}.
\left(u\alpha_{1}\right)\left(w\alpha_{3}\right) =\rho^{w_{2}}uw\alpha_{1}\alpha_{3}\ \left(w\alpha_{3}\right)\left(u\alpha_{1}\right) =\rho^{u_{3}+w_{3}u_{2}-u_{3}w_{2}}uw\alpha_{1}\alpha_{3},
{\Pr}_{s, \rm{OSTBC}}\leq \sum_{t = 0, t \neq u}^{M - 1}\left(\frac{1 - \mu_{u, t}}{2}\right)^{\!K}\, \sum_{r = 0}^{K - 1}\binom{K - 1 + r}{r}\left(\frac{1 + \mu_{u, t}}{2}\right)^{\!r}
\psi(w) = \mathrm{e}^{\frac12 w^2}\int_w^\infty g^{(3)}(s) \mathrm{e}^{-\frac12 s^2}\, \mathrm{d}s = - \mathrm{e}^{\frac12 w^2}\int_{-\infty}^w g^{(3)}(s) \mathrm{e}^{-\frac12 s^2}\, \mathrm{d}s.
\mathcal{U}_{-}(q^{1/2})=\left( -1\right)^{\mathsf{N}}_{q}M(1)I_{0},\mathcal{U}_{-}(iq^{1/2})=i(-1)^{\mathsf{N}+1}\frac{\zeta_{-}+1/\zeta_{-}}{\zeta_{-}-1/\zeta_{-}}_{q}M(i)\sigma_{0}^{z},
A^{\pm}= - i \sqrt{2}\cdot \alpha \left( x \right) \cdot e^{\pm \xi \left( x \right)}\left[ \sinh \rho_{\frac{s}{2}}\left( x \right) \cosh \frac{i}{2}\frac{d}{d x}\mp \cosh \rho_{\frac{s}{2}}\left( x \right) \sinh \frac{i}{2}\frac{d}{d x}\right]
B = \int \partial_xu(\partial_x^2u)^2 \chi(x+\nu t) \; dx + 3 \int u(\partial_x^2u)^2 \chi'(x+\nu t) \; dx &+ \frac43 \int (\partial_xu)^3 \chi''(x+\nu t) \; dx- \int u(\partial_xu)^2\chi'''(x+\nu t) \; dx.
s_k[X(w)] = \begin{cases}\hphantom{-}[X(w)] \textrm{if}\ell(ws_k)< \ell(w); -[X(w)] - \sum \langle \alpha_k, \beta^\vee \rangle [X(ws_k s_\beta)] \textrm{if}\ell(w s_k) > \ell(w) . \end{cases}
h_g(z, z,\dots,z,n_1, \dots, n_j)=z^{j-g}\,\P(Y_1=n_1, \dots, Y_j=n_j)\prod_{i=j+1}^c Y_i(z), &h_g(1, z,\dots,z,n_1, \dots, n_j)=\P(Y_1=n_1, \dots, Y_j=n_j)\prod_{i=g+1}^c Y_i(z).
G_{1 , 1 ,{\frac{1}{3}}}^{( 1 0 )}\, \left| \,{p \; \, q \; \; i \atop p \; p \; ( p + 2 q ) / 3}\, \right\rangle = \sqrt{\frac{( p + 1 ) \, ( p + q + 2 )}{p + 2}}\; \sum_{i^{\prime}}\; \left| \,{p + 1 \; \, q \; \; i^{\prime}\atop p \; p \; ( p + 2 q - 2 ) / 3}\, \right\rangle \; A_{i^{\prime}\; , i}\; .
E_1(x,\zeta) = e^{i (x_1\zeta_1 + x_2 \zeta_2-x_0(\zeta_1 e_1 +\zeta_2 e_2))}= e^{i (x_1\zeta_1 + x_2 \zeta_2)}e^{-ix_0(\zeta_1 e_1 +\zeta_2 e_2)}.
_{Q}\left\{\mathbf{\Sigma}_{\mu},\mathbf{\Sigma}_{\nu}\right\}=\mathbf{\Sigma}_{\lambda}C^{\lambda}\!_{\mu\nu},{}_{Q}\left\{\widehat{\mathbf{\Sigma}}_{\mu},\widehat{\mathbf{\Sigma}}_{\nu}\right\}=-\mathbf{\Sigma}_{\lambda}C^{\lambda}\!_{\mu\nu},{}_{Q}\left\{\mathbf{\Sigma}_{\mu},\widehat{\mathbf{\Sigma}}_{\nu}\right\}=0.
b_{p+1}-b_p= \int_M d(u-v)\wedge d^c_B(u-v)\wedge dd^c_B (\psi-\phi)\omega^p_\psi\wedge \omega^{n-p-2}_\phi\wedge \eta = -\int_M d(u-v)\wedge dd^c_B(u-v)\wedge d^c_B (\psi-\phi)\omega^p_\psi\wedge \omega^{n-p-2}_\phi\wedge \eta = -\int_M d(u-v)\wedge (\omega_u-\omega_v)\wedge d^c_B (\psi-\phi)\omega^p_\psi\wedge \omega^{n-p-2}_\phi\wedge \eta
\int_{\mathbb{T}}\frac{1-|\eta_{\mu_{1}}(z)|^{2}}{|t-\eta_{\mu_{1}}(z)|^{2}}\,d\sigma(t)=\beta\frac{1-|\eta_{\nu_{1}}(z)|^{2}}{|1-\eta_{\nu_{1}}(z)|^{2}},
{\beta^*_{2}}=\left(id-(t_{2}^*)^{2}\right)\cdot \left( id-t_{3,2}^*\right)^{-1},{\beta^*_{3}}=\left(id-(t_{1}^*)^{2}\cdot t_{3,2}^*\right)\cdot \left(id-(t_{1}^*)^{2}\right)\cdot \left( id-t_{2,1}^*\right)^{-1}\cdot \left( id-t_{3,1}^*\right)^{-1}.
\langle (q_{1/\varepsilon}f)(\mathbf{x}_{0}+\varepsilon \mathbf{x}),\varphi(\mathbf{x})\rangle = \langle \langle f(\mathbf{y}), q_{1/\varepsilon}(\mathbf{x}_{0}+\varepsilon \mathbf{x},\mathbf{y}) \rangle ,\varphi(\mathbf{x})\rangle &= \langle f(\mathbf{x}_{0}+\varepsilon \mathbf{y}), (J_{\varepsilon}\varphi)(\mathbf{y})\rangle,
\binom{a_1,a_2,\ldots,a_{m_1}}{b_1,b_2,\ldots,b_{m_2}}\sim\binom{a_1+1,a_2+1,\ldots,a_{m_1}+1,0}{b_1+1,b_2+1,\ldots,b_{m_2}+1,0},
\Gamma_{\underline{\alpha}\underline{\beta}}^{0}= \left( \begin{array}{c c}{1}&{0}\\{0}&{1}\\ \end{array}\right) , \quad \Gamma_{\underline{\alpha}\underline{\beta}}^{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \quad \Gamma_{\underline{\alpha}\underline{\beta}}^{2}= \left( \begin{array}{c c}{- 1}&{0}\\{0}&{1}\\ \end{array}\right) .
\alpha_{1}=\|B^{\dagger}\|_{2}^{2}\big(\|EB^{\dagger}\|_{F}^{2}-\|AA^{\dagger}EB^{\dagger}\|_{F}^{2}\big)+(n-s)\|A^{\dagger}\|_{2}^{2}, &\alpha_{2}=\|A^{\dagger}\|_{2}^{2}\big(\|EA^{\dagger}\|_{F}^{2}-\|BB^{\dagger}EA^{\dagger}\|_{F}^{2}\big).
D^{++}_1 G^{(1,1)}(x_1,\theta^+_1,u_1\vert x_2,\theta^+_2,u_2) = \delta^4(x_1-x_2)(\theta^+_1-(u^+_1u^-_2)\theta^+_2)^4(u^-_1u^+_2)\delta (u_1,u_2)\;.
{\widehat \gamma}^{1}= \left( \begin{array}{c c}{{\widehat E}_{2}}&{0}\\{0}&{-{\widehat E}_{2}}\\ \end{array}\right) ,{\widehat \gamma}^{2}= \left( \begin{array}{c c}{0}&{\sigma_{x}}\\{- \sigma_{x}}&{0}\\ \end{array}\right) ,
\langle e_i, g\!\cdot\! x \rangle = \frac{1}{|gx|}\sum_{j=1}^d x_j \langle e_i, g e_j \rangle = \frac{\sum_{j=1}^d x_j g^{i,j}}{\sqrt{\sum_{i=1}^d ( \sum_{j=1}^d g^{i,j}x_j )^2}}.
\exp ( \beta S_{H K}) = \prod_{p}\sum_{j = 0 , 1 / 2 , . .}( 2 j + 1 ) \frac{\sin ( j + \frac{1}{2}) \theta_{p}}{\sin \frac{\theta_{p}}{2}}\exp [ - j ( j + 1 ) / 2 \beta \, ] \, ,
\partial_{x_j}^p\frac{x_j^{2r}}{(1+\epsilon x^{2m})^2}=\sum_{\ell=0}^p\binom{p}{\ell}(2r)\cdots(2r-\ell+1) x_j^{2r-\ell}\partial_{x_j}^{p-\ell}\frac{1}{(1+\epsilon x^{2m})^2}.
A = \left( \begin{array}{c c c c c c c c}0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 \end{array}\right).
-2[z_t,\mathfrak{H}]\frac{\partial_{\alpha}z_t}{z_{\alpha}}= -2[\bar{f}, \mathfrak{H}]\frac{\partial_{\alpha}\bar{f}}{z_{\alpha}}-2[\bar{p}, \mathfrak{H}]\frac{\partial_{\alpha}\bar{f}}{z_{\alpha}}-2[\bar{f},\mathfrak{H}]\frac{\partial_{\alpha}\bar{p}}{z_{\alpha}}-2[\bar{p},\mathfrak{H}]\frac{\partial_{\alpha}\bar{p}}{z_{\alpha}}
w_1 v_1 = (a_{-1}- a_1)(2a_0 - (a_{-1}+ a_1) - 4s_{1}) &= (a_{-1}- a_1) - (a_{-1}- a_1) -4\left( -\tfrac{3}{4}(a_{-1}- a_1) + \tfrac{3}{8}(a_{-2}+ a_0 - a_0 - a_2) \right) &= -\tfrac{3}{2}\left( a_{-2}-2a_{-1}+ 2a_1 -a_2 \right),
(\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{{x_{i}}^{2}}{{x_{j}}^{2}})+(\sum_{i=2}^{n}\sum_{j=1}^{i-1}\frac{{x_{i}}^{2}}{{x_{j}}^{2}})+\sum_{i=1}^{n}1\ge n^{2}
\bar{\Psi}_{\mathrm{C T}}= - \frac{i}{2 e^{2}}\int d^{4}x \, \sqrt{g}\, \Bigr \{\xi_{-}^{\nu}( \nabla^{\mu}T_{\mu \nu}^{+}- i V^{\mu}T_{\mu \nu}^{+}- Y_{\nu}^{+}) - ( \nabla^{\mu}T_{\mu \nu}^{-}- i T_{\mu \nu}^{-}V^{\mu}- Y_{\nu}^{-}) \xi_{+}^{\nu}\Bigr \}.
{\cal S}={\cal S}(A_{\mu}^p, B_{\mu\nu}^p) = \int{d^4}x \{-\frac{1}{4}{\varepsilon}^{\mu\nu\rho\sigma}G_{\mu\nu}^p B_{\rho\sigma}^p + \frac{1}{2}A_{\mu}^p A^{p\mu}\},
J =-2\int_{\mathbb{R}^2}(r\partial_rV^{(\alpha,a)}+t\nabla\cdot H^{(\alpha,a)})\cdot\mu t\Delta\sum\limits_{l=0}^{\alpha}C_\alpha^l (-1)^{\alpha-l}V^{(l, a)}dx &\quad+\|SV^{(\alpha,a)}-tf^1_{\alpha a}\|^2_{L^2}.
\mspace{-8mu}E_{}( \mspace{-1mu}D \mspace{-1mu}) \mspace{-3mu}= \mspace{-3mu}\sigma_z^2 \mspace{-2mu}\cdot \mspace{-2mu}\log_e 2 \mspace{-2mu}\cdot \mspace{-2mu}\max \mspace{-2mu}\big\{2R_{S_1}(D), (1 \mspace{-3mu}+ \mspace{-3mu}\rho_z) R_{S_1,S_2}(D) \big\}\mspace{-1mu}.
\sum_{i=1}^d |D(\sigma_n)_i(x)h|^2 +| Db_n(x)h| = |Db \big( \Phi_n(x) \big) D\Phi_n(x) h| + \sum_{i=1}^d |D\sigma_i \big (\Phi_n(x)\big) D\Phi_n(x)h|^2\le c_1 \, V(\Phi_n(x)),
\frac1{{m \choose l}}\sum_{\substack{J\subseteq I |J|=l}}w_S(\varepsilon(J)) =\frac{1}{{m \choose l}}\sum_{j=0}^{l}(-1)^j\binom{|S|}{j}\binom{m-|S|}{l-j}&=\Bbbk_{l}^{(m)}(|S|),
\int_T^{2T}\bigg(\sum_{|\rho_a-s_0| \leq r-\delta}|\log |\sigma+it-\rho_a|| \bigg)^{2k}dt\ll \sum_{n=\lfloor T \rfloor}^{\lfloor 2T \rfloor+1}\Gamma(2k+1)\bigg(\sum_{\rho_a \in \mathcal D_n}1\bigg)^{2k}\leq T \, \Gamma(2k+1) (C \log M_{r+\delta}(s_0))^{2k},
\sup_{\eta\in \mathbb{U}_X}\operatorname{E}_\mu^{(OW)}[\pi,\eta]\, = \, \sup_{\eta\in \mathbb{U}_X}\limsup_{i\in I}\frac{H_\mu[\eta_{A_i}]}{\theta(A_i)}& = \, \sup_{\eta\in \mathbb{U}_X}\limsup_{i\in I}\frac{H_\mu[\eta_{KF_i}]}{\theta(A_i)}\, = \, \sup_{\eta\in \mathbb{U}_X}\limsup_{i\in I}\frac{H_\mu[\eta_{F_i}]}{\theta(A_i)}.
\sum_{k=0}^n \binom{a}{c+k}\binom{b}{n+d-k}\binom{c+k}{c}\binom{n+d-k}{d}H_{a-c-k}H_{b-d-n+k}={}\binom{a+b-c-d}{n}\binom{a}{c}\binom{b}{d}[H^{(2)}_{a+b-n-c-d}-H^{(2)}_{a+b-c-d}+ (H_{a+b-n-c-d}-H_{a+b-c-d}+H_{a-c})(H_{a+b-n-c-d}-H_{a+b-c-d}+H_{b-d})]
{\rm e}^{-N{\rm tr}_N\sum_{a=1}^P\ln(1-gM_a)}=\int d\psi d\psi^{\dagger}\,{\rm e}^{-\sum_{a=1}^P\sum_{i,j=1}^Q{\rm tr}_N\psi^{\dagger\,a}_i(1-gM_a)_{ij}\psi_j^a}.
\left( \begin{array}{c}{\{M_{A q}^{f}\}^{Y^{1}q}}\\{\{M_{A q}^{f}\}^{Y^{2}q}}\\ \end{array}\right) = \left( \begin{array}{c c}{0}&{i}\\{- i}&{0}\\ \end{array}\right) \left( \begin{array}{c}{\{M_{A q}^{b}\}^{Y^{1}q}}\\{\{M_{A q}^{b}\}^{Y^{2}q}}\\ \end{array}\right) .
{\check{\delta}}^{( m )}\ast{\check{\delta}}^{( n )}= \frac{i}{4 \pi}\int_{{\Gamma}_{1}}d k_{1}\; i^{m}k_{1}^{\lambda + m}\frac{\epsilon ( k_{1})}{2}\int_{{\Gamma}_{2}}d k_{2}\; \frac{i^{n}k_{2}^{\lambda + n}\epsilon ( k_{2})}{z - k_{1}- k_{2}}=
J_\nu(r) = \frac{r^\nu}{2^\nu \Gamma(\nu+1)}\left[ \sum_{k=0}^N \frac{(-1)^k \Gamma(\nu+1)}{k!\, \Gamma(k+\nu+1)}\left( \frac{r}{2}\right)^{2k}+ \sum_{k \geq N+1}\frac{(-1)^k \Gamma(\nu)}{k!\, \Gamma(k+\nu+1)}\left( \frac{r}{2}\right)^{2k}\right].
g^{2}\int \frac{d^{6}p}{( 2 \pi )^{6}}\, \frac{d^{6}k}{( 2 \pi )^{6}}\, \frac{e^{- i ( \theta_{1 2}( p_{1}\, k_{2}- p_{2}\, k_{1}) + \theta_{3 4}( p_{3}\, k_{4}- p_{4}\, k_{3}) )}}{p^{2}\, k^{2}\, ( p + k )^{2}}= g^{2}\frac{1}{\theta_{1 2}+ \theta_{3 4}}\, \frac{1}{\theta_{1 2}\, \theta_{3 4}}.
\begin{array}{c}{e^{- B}A e^{B}= \sum_{n = 0}^{\infty}\displaystyle \frac 1{n !}[ A , B ]_{( n )},}\\{{}[ A , B ]_{( 0 )}\equiv A , \qquad [ A , B ]_{( n + 1 )}= [ [ A , B ]_{( n )}, B ] .}\\ \end{array}
\begin{aligned}\varphi_{a}(x,\xi) &=\{(\phi_{a}^+(x,\xi)-x\cdot\xi)\chi_a^+(x,\xi)+(\phi_{a}^-(x,\xi)-x\cdot\xi)\chi_a^-(x,\xi)\}&\times\prod_{(ij)\not\le a}\chi_0(2\xi_{ij}/d)\chi_0(2x_{ij}/R_0)+x\cdot\xi\end{aligned}
G^{(n)}:y^{2}=x^3 - \Bigl(\frac{I_{4}}{12}+\frac{1}{t^{n}}\Bigr)x+\Bigl(\frac{I_{10}}{4}t^{n}+\frac{I_{2}I_{4}-3I_{6}}{108}+\frac{I_{2}}{24t^{n}}\Bigr).
(NN^{\ast}X)^{k_{2}}=U\begin{pmatrix}(\Sigma^{2}X_{1})^{k_{2}}(\Sigma^{2}X_{1})^{k_{2}-1}\Sigma^{2}X_{3}0 0\end{pmatrix}U^{\ast}, c_{2}(NN^{\ast})^{\ell}=U\begin{pmatrix}c_{2}\Sigma^{2\ell}0 0 0\end{pmatrix}U^{\ast}.
X_{1}={\frac{x_{1 4}}{x_{1 4}^{2}}}-{\frac{x_{2 4}}{x_{2 4}^{2}}}\; , \quad X_{2}={\frac{x_{1 4}}{x_{1 4}^{2}}}-{\frac{x_{3 4}}{x_{3 4}^{2}}}\; , \quad X_{3}={\frac{x_{3 4}}{x_{3 4}^{2}}}-{\frac{x_{2 4}}{x_{2 4}^{2}}}
s:= s_{[h]}:=: \Lambda_h tr_h \Theta_{[h]}:=: h^{k\bar{l}}\Theta^{(1)}_{k\bar{l}}:=: h^{i\bar{j}}\Theta^{(2)}_{i\bar{j}}; \; \; \; \; \; \; \; \; \; \; \hat s : ={\hat s}_{[h]}:=: h^{i \bar{l}}\Theta^{(3)}_{i\bar{l}}\; .
f(T) = \prod_{j = 1}^g (T - \sqrt{q}e^{i\theta_j})(T - \sqrt{q}e^{-i\theta_j}) &= T^{2g}+ c_1 T^{2g - 1}+ \dots + c_gT^g + c_{g - 1}qT^{g - 1}+ \dots + c_1q^{g-1}T + q^g,
\left|\int_{|\mu|\le |\eta|^{3/2}/2}e^{3i\mu^2/4}e^{ia\ln(\eta^2-\mu^2/\eta)}e^{2ia\ln|(\eta+\mu/\sqrt{|\eta|})|}\frac{e^{i\beta (\eta+\mu/\sqrt{\eta})^3}}{(\eta+\mu/\sqrt{\eta})^3}d\mu\right| \le \int_{-|\eta|^{3/2}/2}^{|\eta|^{3/2}/2}\frac{1}{(\eta+\mu/\sqrt{\eta})^3}d\mu = O(|\eta|^{-3/2}).
\mathrm{R e s}_{s = 1 / 2 - k}\ \zeta_{G}( s ) = \frac 1 2 \, \mathrm{r e s}\G^{k - 1 / 2}= \frac{( 2 k - 1 ) ! ! \, q^{k}}{k ! \, 2^{k}}, \, m b o x{R e s}_{s = - k}\ \zeta_{G}( s ) = \frac 1 2 \, \mathrm{r e s}\G^{k}= 0 , \k = 0 , 1 , 2 , \ldots ,
\Delta_{\alpha}= \sum_{( g_{o}, h_{o}) \in{\cal O}}b_{a}^{( h_{o}, g_{o})}\int_{\tilde{\cal F}}\frac{d^{2}\tau}{\tau_{2}}Z_{( h_{o}, g_{o})}( \tau ,{\bar{\tau}}) - b_{a}^{n = 2}\int_{\cal F}\frac{d^{2}\tau}{\tau_{2}}.
F_{\alpha,x,t}(u) = \frac{1}{\left(1- e^{-\alpha t}\right)^{\frac{d}{\alpha}}}\frac{p_\alpha(u)}{p_\alpha(x)}p_\alpha\left( \dfrac{x-ue^{-t}}{(1- e^{-\alpha t})^{\frac{1}{\alpha}}}\right).
\d f= \big(f^x(x)+x(f^x)'(x)+ yf^{xy|x}(x) +xy(f^{xy|x})'(x) +yf^{xy|y}(y)\big)\,\d x + \big(f^y(y)+y(f^y)'(y)+ xf^{xy|y}(y) +xy(f^{xy|y})'(y) +xf^{xy|x}(x)\big)\,\d y.
\lim_{N \rightarrow \infty}\langle\frac{1}{N}{\rm Tr}\xi^{2k_{1}}\cdots\frac{1}{N}{\rm Tr}\xi^{2k_{p}}\rangle=\frac{1}{N}{\rm Tr}{\hat \xi}^{2k_{1}}\cdots \frac{1}{N}{\rm Tr}{\hat\xi}^{2k_{p}}
iS_{F}(x-y) = \langle 0|T^{+}(\psi(x)\overline{\psi}(y))|0\rangle = \sqrt{2}i \int \frac{d^{n}p}{(2\pi)^{n}}e^{-ip\cdot (x-y)}\left(\frac{p\!\!\!\slash + m}{p^{2}-m^{2}+i\epsilon}- \frac{\gamma^{+}}{2p^{+}}\right)
z_{2}= x_2+\iota x_2 - \langle w_1, x_2+\iota y_2 \rangle w_1 = y_2+\iota y_2 - \langle x_1+\iota y_1, x_2+\iota y_2 \rangle w_1 + \mathcal{O}(\|H\|) = x_2+\iota y_2+\mathcal{O}(\|H\|).
\begin{cases}u_{tt}(x,t)-\Delta_{x}u(x,t)+bu_{t}(x,t)+mu(x,t)=f(u(x,t)),\,\,\,\,(x,t)\in X\times (0,T),\,\,T>0, u(x,0)=u_{0}(x),\,\,\,\,x\in X, u_{t}(x,0)=u_{1}(x),\,\,\,\,x\in X, \end{cases}
f_{{\bf{k}},{\bf{q}}}({\bf{x}})= e^{i{\bf{q.x}}}(\frac{1}{2 \mbox{}N\mbox{}\epsilon_{{\bf{q}}}})\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}}) + i\mbox{}U_{-{\bf{q}}}({\bf{x}})\Lambda_{{\bf{k}}}(-{\bf{q}})
\begin{array}{ll}\int_{[0, \Lambda]}D\phi \,\,exp\left\{- \langle{\phi}, L(\Lambda, \phi, g){\phi}\rangle \right\}= \int_{[0,{\Lambda'}]}D\psi \,\,exp\left\{- \langle{\psi}, L(\Lambda', \psi, g){\psi}\rangle \right\}\end{array}
\widetilde{\mathbf{y}}_\ell =\left[ \boldsymbol{\lambda}_\ell - \mathbf{u}_\ell \right] \bmod \Lambda_{C}\stackrel{(a)}{=}\left[ \mathcal{Q}_{\Lambda_{F}}\left({\mathbf{y}}_\ell +\mathbf{u}_\ell \right) -\mathbf{u}_\ell \right] \bmod \Lambda_{C}\stackrel{(b)}{=}\left[{\mathbf{y}}_\ell +{\mathbf{q}}_\ell \right] \bmod \Lambda_{C}
K=\left(\begin{array}{lcccc}u_1 u_2 u_3 u_4 m_1 \, 1 0 0 0 m_2 \, 0 1 0 0 m_3 \, 0 0 1 0 m_4 \, 0 0 0 1 m_5 \, 1 1 -1 0 m_6 \, 1 1 0 -1 \end{array}\right).
\ell_{s}= g^{- 1 / 4}\ell_{p}^{( 1 0 )}= g^{- 1 / 3}\ell_{p}\ \ \ \, e l l_{p}^{( 1 0 )}= ( \ell_{p}^{( 1 0 )})^{9 / 8}R_{1 1}^{- 1 / 8}, \ \ \R_{1 1}= g \ell_{s}= g^{2 / 3}\ell_{p}^{( 1 1 )}.
\frac{1}{\varepsilon_{k}^{n}}\int_{M}b(x)f^{\prime}(W_{\varepsilon_{k},\xi_{k}})u_{k}\varphi_{k}d\mu_{g}=b(\xi)\int_{B(0,\rho)}f^{\prime}(V^{\xi})\tilde{u}_{k}(z)\varphi(z)dz+o(1),
\int_{k_\epsilon}^k|s|^{p-2}(s-k)_-\,ds=\tfrac{1}{p-1}\, \mathfrak g_-(k_\epsilon,k)\geq \tfrac{1}{\boldsymbol\gamma (p)}\big(|k_\epsilon|+|k|\big)^{p-2}(k-k_\epsilon)^2\geq\tfrac{1}{\boldsymbol\gamma (p)}M^p.
d(\theta)=\frac{\lambda_1}{\lambda_2}e^{-2\pi i (\theta+\frac{\alpha}{2})}\left(e^{2\pi i (\theta+\frac{\alpha}{2})}-\frac{-1+\sqrt{1-4\lambda_1\lambda_3}}{2\lambda_1}\right)\left(e^{2\pi i (\theta+\frac{\alpha}{2})}-\frac{-1-\sqrt{1-4\lambda_1\lambda_3}}{2\lambda_1}\right).
\begin{array}{l}(1)\quad{\displaystyle{\frac{\delta}{\delta b_a}}}\Delta =0\,,\qquad(2)\quad \displaystyle{\int}d^3x\,{\displaystyle{\frac{\delta}{\delta \,c^a}}}\Delta =0\,,\qquad (3)\quad{\cal G}^a\Delta =0\,, begin{align*}3mm] (4)\quad{\cal W}_X\Delta =0\,,\qquad X={\rm diff.,\,Lorentz\,,\,rigid}\,.\end{array}
|\zeta-\zeta_0|^2+\tau^2 \leqslant \frac49\tau^2+\tau^2 = \frac{13}{9}\tau^2, |(\zeta-\zeta_0)^2+\tau^2| \geqslant \tau^2-|\zeta-\zeta_0|^2 \geqslant \frac59\tau^2.
N_0(t) =\frac{1}{V_F}\int_{\mathbb{R}}g_+p_0(t,g)dg=\frac{1}{V_F}\int_{\mathbb{R}}g_+\int_{\mathbb{R}}p_{t,0}(y)\bar{G}_{t,0}(g-y)dy &=\int_{\mathbb{R}}p_{t,0}(y)dy\int_{\mathbb{R}}\frac{1}{V_F}g_+\bar{G}_{t,0}(g-y)dg &=\int_{\mathbb{R}}p_{t,0}(y)N(B(t)+y,C(t))dy.
F = \left(\begin{array}{cc}f_1(x) 0 0 f_2(x)\end{array}\right) ~~,~~{\tilde F}= \left( \begin{array}{cc}f_2(x) 0 0 f_1(x)\end{array}\right), f_1(x), f_2(x) \in C^\infty( C,{\cal M}).
(n_1-n_2)_x= \frac{n_1^3E_1-\alpha n_1^2}{n_1^2-1}-\frac{n_2^3E_2-\alpha n_2^2}{n_2^2-1}= E_1\left(f(n_1)-f(n_2)\right)+f(n_2)(E_1-E_2)-\alpha\left(g(n_1)-g(n_2)\right) = \left(E_1 f'(\bar{\eta})-\alpha g'(\tilde{\eta})\right)(n_1-n_2)+f(n_2)(E_1-E_2),\quad\exists\bar{\eta},\tilde{\eta}\in(n_2,n_1),
\Phi_{1}\rightarrow \frac{\eta^{1 / 2}}{i \pi \nu}\left( \frac{( k_{x}\eta )^{\nu}}{2^{\nu}}e^{i \pi \nu}\Gamma ( 1 - \nu ) - 2^{\nu}\Gamma ( \nu + 1 ) ( k_{x}\eta )^{- \nu}\right)
\lim_{R \searrow 0}F(R) = \int_{R_0}^0 \frac{1}{f(z)}\, \mathrm{d}z = - \int_0^{R_0}\frac{1}{f(z)}\, \mathrm{d}z > 0, \lim_{R \nearrow \infty}F(R) = \int_{R_0}^{\infty}\frac{1}{f(z)}\, \mathrm{d}z < 0
\|\psi_s-\underline{\psi_s}\|_{L^{\frac{4}{3}}(M)}\geq \left(\int_{\{\psi_s=0\}}|\psi_s-\underline{\psi_s}|^\frac{4}{3}\omega^2\right)^{\frac{3}{4}}=\gamma(s)^{\frac{3}{4}}\underline{\psi_s}.
X_k' =X_k\setminus S_k(1)(k\geq 2) Y_k' =Y_k\setminus S_k(1)(k\geq 2) Z_k' =Z_k\setminus S_k(1)(k\geq 3). =\{(0,x)\mid x \in X_{k-1}\}, =\{(0,x)\mid x \in Y_{k-1}\}, =\{(0,x)\mid x \in Z_{k-1}\}.
\begin{aligned}(U_{ij})_t \equiv \, (U (e_i, e_j))_t = (\nabla_{ij}u)_t + A^{ij}_t (x, t, \nabla u) + A^{ij}_{p_l}(x, t, \nabla u) (\nabla_{l}u)_t \equiv \, \nabla_{ij}u_t + A_{t}^{ij}(x, t, \nabla u) + A^{ij}_{p_l}(x, t, \nabla u) \nabla_{l}u_t, \end{aligned}
\begin{aligned}P_{O2}^* =\min_{\sum_{i=1}^K\alpha_i=1}\sum_{i=1}^KP_i^* &=\min_{\sum_{i=1}^K\alpha_i=1}N_0\sum_{i=1}^K \frac{(e^{\frac{r^*}{\alpha_i}}-1)\alpha_i}{|h_i|^2}.\end{aligned}
\| \nabla^j (u\rho^{\frac{1}{2}}) \|^2_{L^2}= \int_\Omega |\nabla^j (u\rho^{\frac{1}{2}})|^2\, d\mu_g &\le 2 \int_\Omega | \nabla^j u|^2 \rho\, d\mu_g + C^2N^{2j}\sum_{i=1}^j \int_\Omega |\nabla^{j-i}u |^2 d^{-4i}\rho\, d\mu_g.
\langle S_{\xi\chi}f,g\rangle =\langle \int_{\Theta}v(w)s(w)\pi_{G(w)}\xi_{w}^{\ast}\chi_{w}\pi_{F(w)}fd\mu(w),g\rangle &=\int_{\Theta}v(w)s(w)\langle \chi_{w}\pi_{F(w)}f,\xi_{w}\pi_{G(w)}g\rangle d\mu(w).
\sum_{n_j=1}^{\infty}e^{-\left[\frac{\pi}{a_j}(n_j-c)\right]^{2s}t}\stackrel{\textrm{as}\, t\rightarrow 0^+}{\sim}\frac{a_j}{\pi t^{\frac{1}{2s}}}\Gamma\left(1+\frac{1}{2s}\right)-\frac{s}{2}, \,\, s\in (0,1].
H^{*}( \overline{\alpha}\otimes \overline{\beta})= H^{*}( \overline{\alpha})H^{*}( \overline{\beta})=\left(\begin{matrix}pq 0 0 pq\end{matrix}\right),\ \L_{\alpha,\beta}=L\left(H^{*}( \overline{\alpha}\otimes \overline{\beta}) \right)
\varepsilon_{N+1}-\delta = -\delta +\frac{1}{N+1}\left(\sum_{n=1}^{N+1}\varepsilon_n +\sum_{n=1}^N n(\varepsilon_{n+1}-\varepsilon_n)\right) = \frac{2}{N+1}\rho_1 +\varepsilon_{N+1}-\frac{1}{N+1}\sum_{n=1}^{N+1}\varepsilon_n.
d e t A^{( i_{1}, \dots , i_{n})}= \prod_{k = 1}^{n - 1}\{\prod_{( j_{1}, \dots , j_{k + 1})}[ 1 - \prod_{\alpha}\prod_{\beta}| q_{j_{\alpha}j_{\beta}}|^{2}] \}^{( k - 1 ) ! ( n - k ) !}
A^3_n:=\sum_{k: n\leq |\widehat{a}_k|\leq 2n+1n\leq |\widehat{b}_k|\leq 2n+1}\frac{\left(\left(\widehat{f}\cdot(\varphi_n-1)\right)(\widehat{a}_k)-\left(\widehat{f}\cdot(\varphi_n-1)\right)(\widehat{b}_k)\right)^2}{|\widehat{b}_k-\widehat{a}_k|}.
\|u-M_{1}\widetilde{f}_{0}\|_{L^{p}(D)}=\|\sum_{j=2}^{m}M_{j}\widetilde{f}_{j-1}\|_{L^{p}(D)}&\leq \sum_{j=2}^{m}\| M_{j}\widetilde{f}_{j-1}\|_{L^{p}(D)}&\leq C\sum_{j=1}^{m-1}\| f_{j}\|_{L^{p}(\partial D,\,wd\sigma)}
\begin{aligned}\nabla^2_U \eta (U,x,t) - (\nabla_V \tilde{\eta}) (A(U,x,t),x,t) \cdot \nabla^2_U A(U,x,t)= &=(\nabla_V^2 \tilde{\eta})(A(U,x,t),x,t) : ( \nabla_U A(U,x,t) , \nabla_U A(U,x,t) ) \;.\end{aligned}
W=A(\mathrm{det}\,T_{(0,2)}-M_{0}M_{1}M_{2}+\Lambda_{1}^{4}M_{2}+\Lambda_{2}^{4}M_{0})+m_{0}M_{0}+m_{1}M_{1}+m_{2}M_{2}+\mathrm{tr}\,(c\,T_{(0,2)})