harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{F}_q(\mathcal{H},r,\alpha):=\bigoplus_{n=0}^\infty\mathbb{F}^{(n)}_q(\mathcal{H},r,\alpha),~~ \mathbb{F}^{(n)}_q(\mathcal{H},r,\alpha):=\mathbb{F}_q^{(n)}(\mathcal{H}) r^n([n]_{|q|}!)^\alpha.
\chi_{k}^{n}( \tau ) ={\frac{1}{( \eta ( \tau ) )^{3}}}\cdot 2 ( k + 2 ) \sum_{m = - \infty}^{\infty}\left( m +{\frac{n + 1}{2 ( k + 2 )}}\right) e^{2 \pi i ( k + 2 ) \tau \left( m +{\frac{n + 1}{2 ( k + 2 )}}\right)^{2}}.
\frac{\bar F_{b}(a,b)}{B}=\frac{\bar F(b)}{B}+\bar P_{b}(a,b,b)=\frac{\eta^{2}}{24\pi b^{4}}(c_{1}\Lambda'+c_{2}\Lambda)+\frac{\eta^{2}\Lambda'}{3}P_{b}(a,b,b).
e^{- W ( J , \xi )}= \int{\cal D}\phi_{i}{\cal D}\eta{\cal D}\overline{{\eta}}e^{- S - \frac 1{2 \xi}\int d^{d}x F^{2}+ \int d^{d}x \overline{{\eta}}( x ) \frac{\delta F}{\delta \phi_{i}( x )}\Delta_{i}\eta ( x ) + \int d^{d}x J_{i}( x ) \phi_{i}( x )},
\lambda_{k}^{\epsilon, v_k}= -\limsup_{t \rightarrow \infty}\frac{1}{t}\log \mathbb{P}_{x, k}^{\epsilon, v_k}\left \{\tau_{k}^{\epsilon, v_k}> t \right\}, \,\,\, x \in D, \,\,\, k=1,2, \ldots, n,
d_1(G, G') \leq \frac{1}{N}\sum_{i=1}^{N-j}|\lambda_i-\lambda'_i|+\frac{j}{N}\times 2 &\leq \frac{1}{N}\sum_{i=1}^{N+j}|\lambda_{i+k_2}-\lambda_{i-k_1}|+\frac{2j}{N}&\leq \frac{k_1+k_2+1}{N}\sum_{i=1}^{N}|\lambda_{i+1}-\lambda_{i}|+\frac{2j}{N}&\leq 2(k_1+k_2+j+1)\frac{1}{N}.
h_{M N | A}{}^{A}+ 2 R_{M A N B}^{( B )}h^{A B}- 2 R_{A ( M}^{( B )}h_{N )}^{A}+{\frac{2 ( 6 - N )}{N - 2}}\kappa^{2}\sigma_{i}\delta ( y_{i}) h_{M N}= 0 \, .
|\Re B_n| = \bigg|\frac{\alpha_n^{2\gamma}\delta^2 c \sqrt{\rho_1}}{a[c^2 \rho_1\alpha_n+ a\rho_3^2]}+\Re r_n\bigg| \noalign{\vskip1mm}\geq \frac{\alpha_n^{2\gamma}\delta^2 c \sqrt{\rho_1}}{a[c^2 \rho_1\alpha_n+ a\rho_3^2]}-|\Re r_n|\sim \frac{\alpha_n^{2\gamma-1}\delta^2}{ac\sqrt{\rho_1}}\to \infty.
\mathcal{I}(\delta) = \inf_{\lambda \in \mathbb{R}^2_+}\left\{\langle \lambda, \delta \rangle + \sup_{\pi \in \Pi(\mu_{13}, \mu_{23})}\int_{\mathcal{S}_1 \times \mathcal{S}_2}(f_\mathcal{S})_\lambda \, d\pi \right\},
\Delta_+(x''-x') = \int_{-\infty}^{\infty}dN(\tau''-\tau')\int{\cal D}p{\cal D}x\left( \prod_{\tau'}^{\tau''}\Theta(-p_0) \right) \exp\left( i \int_{\tau'}^{\tau''}d \tau (p \dot{x}-N{\cal H}_0)\right).
\begin{array}{l}{{\displaystyle x_{j}^{2}= \frac{\prod_{i = 1}^{D}( \rho_{i}- e_{j})}{\prod_{i \not = j}^{D}( e_{i}- e_{j})}\enspace , \qquad ( j = 1 , \dots , D )}}\\{{\displaystyle e_{1}< \rho_{1}< e_{2}< \dots < e_{D}< \rho_{D}\enspace .}}\\ \end{array}
\int_{1 / \Lambda^{2}}^{\infty}{\frac{d s}{s^{5 / 2}}}e^{- s \sigma^{2}}\left[ ( e B s ) \coth ( e B s ) - 1 \right] + \int_{1 / \Lambda^{2}}^{\infty}{\frac{d s}{s^{5 / 2}}}e^{- s \sigma^{2}}
\begin{array}{c}{{\sum_{0 \le p \le 1 - a_{i j}}( - 1 )^{p}\left[ \begin{array}{c l c r}{{}}&{{1 - a_{i j}}}\\{{}}&{{p}}\\ \end{array}\right]_{Q_{i}}e_{i}^{1 - a_{i j}- p}e_{j}e_{i}^{p}= 0}}\\{{\sum_{0 \le p \le 1 - a_{i j}}( - 1 )^{p}\left[ \begin{array}{c l c r}{{}}&{{1 - a_{i j}}}\\{{}}&{{p}}\\ \end{array}\right]_{Q_{i}}f_{i}^{1 - a_{i j}- p}f_{j}f_{i}^{p}= 0}}\\ \end{array}
\sup_{x\in\Omega}\int_0^t\int_{\Omega\,\cap\,B_{H_0}(x,1)}e^{-h(y,s)}H(\nabla u(y,s))\,dy\,ds \qquad \le C_1 t^{\sigma'}\sup_{x\in\Omega}\int_{\Omega\,\cap\,B_{H_0}(x,1)}e^{-H_0(y)^2}|\phi(y)|\,dy
b_A(\{4\}) = b_2^{(0,01,01,1)}= a_2^1 + a_2^2 + a_2^{2'}+a_2^3= 9 + 12 + 11 + 14 =46, b_B(\{4\}) = b_1^{(0,1,0,0)}= a_1^1 = 3, b_C(\{4\}) = b_2^{(0,01,0,0)}= a_2^0 + a_2^1 = 6+9=15, b_D(\{4\}) = b_4^{(0,0,0,0)}= a_4^0 = 66,
\left( \begin{array}{l l}{{\quad \tilde{d}}}&{{- \tilde{b}/ q}}\\{{- q \tilde{c}}}&{{\quad \tilde{a}}}\\ \end{array}\right) = \left( \begin{array}{l l}{{a^{*}}}&{{c^{*}}}\\{{b^{*}}}&{{d^{*}}}\\ \end{array}\right) \; .
\lim_{m \to \infty}Y_{m,n}\left( 1- \frac{\lambda}{m}, 1 , \frac{T}{m}\right) = \frac{e^{-\lambda}}{2}\sum_{\theta \in \{0,1\}^2}\prod_{w^n = (-1)^{\theta_2}}\lim_{m \to \infty}\prod_{j=0}^{m-1}\left| 1+ \exp \left\{2 \pi i \frac{j + \theta_1 /2}{m}\right\}+ \frac{Tw + \lambda}{m}\right|.
|w_s(s_i')|=\left|\frac{w(s_i)-w(s_{i+1})}{s_i-s_{i+1}}\right|\le\frac{|w(s_i)|+|w(s_{i+1})|}{2i+1}\Rightarrow\quad \lim_{i\to\infty}|w_s(s_i')|=0.
\frac{(2\pi)^{s + \frac{k-1}{2}}}{\Gamma(s + \frac{k-1}{2})}\langle P_h(\cdot, s), \widetilde{V}\rangle = \sum_{m \in \mathbb{Z}}\frac{r_{2k+1}(m^2+h)}{(2m^2+h)^{s+\frac{k-1}{2}}}.
\frac{1}{f(v)g(v)}\int_{0}^{v}g(s)\mkern+1.5mu df(s) = 1 - \frac{1}{f(v)g(v)}\int_{0}^{v}f(s) dg(s) = 1 - \frac{1}{f(v)g(v)}\int_{0}^{g(v)}f\bigl(g^{-1}(t)\bigr) dt,
\mathbf{F}\cdot\mathbf{G}=\left( \mathbf{E}+i\mathbf{H}\right)\cdot\left( \mathbf{D}+i\mathbf{B}\right) & =\left( \mathbf{E}\cdot\mathbf{D}-\mathbf{H}\cdot\mathbf{B}\right)+i\left( \mathbf{E}\cdot\mathbf{B}+\mathbf{H}\cdot\mathbf{D}\right).
\langle \varphi^{2}\rangle^{\mathrm{th}}_{M}=\frac{1}{2\pi^2}\int^{\infty}_{0}\frac{\left( \omega^{2}+2m\omega \right)^{1/2}d\omega}{e^{(m+\omega )/T}-1}, \quad T=\frac{1}{2\pi \xi}
H'(t) \leq -\epsilon E(t) - \frac{1}{4}\epsilon ||u_{xx}||^2_2 + a_2 \epsilon \gamma \delta ||u||^2_2 - [a_1 - (\frac{3}{2}+ a_2^2 B)\epsilon]||u_t||_2^2 - a_1[1- \frac{a_2}{a_1}\epsilon c(\delta)] ||u_t||_m^m,
-\frac{3cab_i}{4}{}=\langle\bar{R}(U_1,U_2)U_i,\xi\rangle=(\lambda_2-\lambda_i)\langle\nabla_{U_1}U_2,U_i\rangle-(\lambda_1-\lambda_i)\langle\nabla_{U_2}U_1,U_i\rangle.
\sum_{E\in\Psi'(G,H)}(-1)^{|E|}=\sum_{\mathcal{X}\in\Psi(G,H)}(-1)^{|\mathcal{X}|}=-\sum_{\mathcal{Y}\in\Psi(G,H)^\complement}(-1)^{|\mathcal{Y}|}& =-\mu_{\widehat{\mathcal{I}}(G,H)}(H,G)\,.
p_{OB,k,i}^{(1)}= \frac{{{x_k}+ 1}}{2}\sum\limits_{s = 0}^M{{\varpi_{M,s}^{(0.5)}}\,\mathcal{L}_{OB,m,i}^{(\alpha_k^*)}\left({x_{M,s}^{(0.5)}; - 1,{x_k}}\right)};
\psi_{a_1 \cdots a_{2k-2}}\;=\; F_\beta^{-1}\:(N^\dagger\:N)^{-1}\:N_a^{(\beta) \:\dagger}\:D_b^{(\beta)}\:F_\beta\:\psi_{a_1\cdots a_{2k-2}a b}\:+\: \psi^{[h]}_{a_1 \cdots a_{2k-2}}
G_{i_1i_2i_3}^{j_1j_2j_3}=\delta^{j_1k_1}\delta^{j_2k_2}\delta^{j_3k_3}\int_0^\infty (e^{-y\tilde \gamma})_{i_1k_1}(e^{-y\tilde \gamma})_{i_2k_2}(e^{-y\tilde \gamma})_{i_3k_3}dy,
-\dfrac{\partial y}{\partial t}+\triangle y\, =\,u\,\,\, in\,\, Q:=\Omega\times (0,T), y\, =\,h\,\,\, on\,\, \Sigma :=\Gamma\times (0,T), y\, =\,y_0\,\,\, at\,\, t=0.
\{I''_1\}\cup \emptyset \cup (B|_{I''_1 \cap I''_2})_{\rm max}\cup \emptyset\cup(B|_{(I''_1 \triangle I''_2) \setminus \{i''_1, i''_2\}})_{\rm max}&=\{\{1, 2, 3, 4\}, \{3\}, \{4\}, \{2\}\}
C_{r,n}=(1+r)^{n}+r^{n}+[n+1]_{q^{-1}}\sum_{j=1}^{n-1}\left[\begin{array}[c]{c}n j\end{array}\right]_{q}(1+r)^{n-j}r^{j}q^{j-1}\int_{0}^{1}p_{n-2,j-1}(q^{-1};q^{-1}t)d_{q^{-1}}t & =(1+r)^{n}+r^{n}+\sum_{j=1}^{n-1}\left[\begin{array}[c]{c}n j\end{array}\right]_{q}(1+r)^{n-j}r^{j}q^{j-1}.
\begin{array}{r l}{\delta e}&{= - \partial_{\tau}( a e ) -{i}\alpha ( \tau ) \psi ,}\\{\delta \psi}&{= - a ( \tau ) \dot{\psi}-{\frac{3}{2}}\dot{a}\psi -{\frac{1}{4}}\alpha ( \tau ) \dot{e}-{\frac{1}{2}}\dot{\alpha}( \tau ) e .}\\ \end{array}
\mathfrak{f}=\{\lambda \in E_r \mid \langle \lambda + \rho, \alpha^\vee\rangle = m_\alpha e \alpha \in R^0_+(\mathfrak{f}), & (m_\alpha-1)e < \langle \lambda + \rho, \alpha^\vee\rangle < m_\alpha e \alpha \in R^1_+(\mathfrak{f}) \},
V [ \rho ] = \sum_{n = - \infty}^{+ \infty}\int d^{3}x \left\{\rho \left[ \ln \left[ \frac{\rho}{2 \zeta}+ \sqrt{1 + \left( \frac{\rho}{2 \zeta}\right)^{2}}\right] + 2 \pi i n \right] - 2 \zeta \sqrt{1 + \left( \frac{\rho}{2 \zeta}\right)^{2}}\right\},
(\{g_j=f_j\widehat{S}_{f,\tau}^{-1}+h_jU-f_j\widehat{S}_{f,\tau}^{-1}\widehat{\theta}_\tau U\}_{j\in \mathbb{J}},\{\omega_j=\widehat{S}_{f,\tau}^{-1}\tau_j+Ve_j-V\theta_f\widehat{S}_{f,\tau}^{-1}\tau_j\}_{j\in \mathbb{J}})
N(F)|^3_0 (x_{0123}) = [N(F)|^3_0(x_{123}), \, N(F)|^3_0(x_{023}), \N(F)|^3_0(x_{013}),\, N(F)|^3_0(x_{012})] = [\phi_{x_{23},x_{12}}\bullet F(x_{123}),\ \phi_{x_{23},x_{02}}\bullet F(x_{023}),\ \phi_{x_{13},x_{01}}\bullet F(x_{013}),\ \phi_{x_{12},x_{01}}\bullet F(x_{012})]
E_{\lambda}(\Omega_{\infty};u,v,w) =\sum_{J\neq \emptyset}E_{\lambda}(\Omega_{J,\infty};u,v,w) &=\sum_{J\neq \emptyset}(E_{\lambda}((B^{*}\times T_{0}\times T_{J})_{\infty};u,v,w)-E_{\lambda}(Y_{J,\infty};u,v,w)) &=-\sum_{J\neq \emptyset}E_{\lambda}(Y_{J,\infty};u,v,w).
c^{++-}(\xi,0,-\xi) = -\frac{q_{+-}(\xi,-\xi) q_{++}(2\xi, \xi)}{|\xi|^{3/2}-|2\xi|^{3/2}+|\xi|^{3/2}}+\frac{2q_{--}(\xi,-\xi) q_{--}(2\xi, \xi)}{|\xi|^{3/2}+|2\xi|^{3/2}+|\xi|^{3/2}}+ d_1 |\xi|^{3/2}= d_2 |\xi|^{3/2},
T r K_{B_{3}^{\alpha}}= \left({\frac{V_{B_{3}^{\alpha}}}{l^{3}}}+{\frac{A_{+}}{l}}( 2 \pi \alpha ) c_{2}( \alpha ) \bar{s}\right){\frac{e^{- \mu \bar{s}}}{( 4 \pi \bar{s})^{3 / 2}}}+ E S ,
\int f\cdot \bar{g}\,dm = \sum_{\chi \in \widehat Z}\hat{f}(\chi) \cdot \overline{\hat g(\chi)}\int |f|^{2}\,dm = \sum_{\chi \in \widehat{Z}}|\hat{f}(\chi)|^{2}\widehat{f*g}=\hat{f}\cdot \hat{g}.
K ( \Phi ,{\bar{\Phi}}) = - \sum_{i = 1}^{3}\log \left( 1 - \Phi_{T}^{( i )}{\bar{\Phi}}_{T}^{( i )}\right) - \sum_{i = 1}^{3}\log \left( 1 - \Phi_{U}^{( i )}{\bar{\Phi}}_{U}^{( i )}\right) .
H ( t , M , \kappa_{\tau}) = \rho^{\tau [ \frac{d - \frac{m}{2}}{2}+ 1 ]}\nonumber \, \; \; H ( \frac{t}{\rho^{2 \tau}}, \frac{M}{\rho^{2 \tau [ \frac{d - \frac{m}{2}}{2}- 1 ]}}, \frac{\kappa_{\tau}}{\rho}) .
H^f_{11}= f_{11}- \Gamma^1_{11}f_1-\Gamma_{11}^2f_2, H^f_{12}= f_{12}-\Gamma_{12}^1f_1-\Gamma_{12}^2f_2, H^f_{22}= f_{22}-\Gamma_{22}^1f_1-\Gamma_{22}^2f_2,
\widetilde{\Delta}^b_{A}=\partial_b\partial_b^*+\overline{\partial}_b\overline{\partial}_b^*+\overline{\partial}_b^*\partial_b^*\partial_b\overline{\partial}_b+\partial_b\overline{\partial}_b\overline{\partial}_b^*\partial_b^*+\partial_b\overline{\partial}_b^*\overline{\partial}_b\partial_b^*+\overline{\partial}_b\partial_b^*\partial_b\overline{\partial}_b^*
{\cal D}( g ) = \left( \begin{array}{c c c}{z_{1}^{\prime}}&{- \bar{z}_{2}^{\prime}}&{- \chi}\\{z_{2}^{\prime}}&{\bar{z}_{1}^{\prime}}&{- \bar{\chi}}\\{\theta^{\prime}}&{- \bar{\theta}^{\prime}}&{\lambda}\\ \end{array}\right) , \quad \quad \sum_{i}| z_{i}^{\prime}|^{2}+ \bar{\theta}^{\prime}\theta^{\prime}= 1
\partial_{a_1}\ldots \partial_{a_n}\delta_n \phi =\epsilon^{\bar a_1}_{a_1}\ldots \epsilon^{\bar a_n}_{a_n}\bar{\partial}_{\bar a_1}\ldots \bar{\partial}_{\bar a_n}(\rho_0^{\bar c}\bar{\partial}_{\bar c}\phi )\; + \;{\cal O}( \phi^2 ).
F_{d, \ell}(z; it) =\sum_{a=0}^N\mathcal{D}_z^{2a}\left(\frac{\zeta^d}{1-\zeta^\ell}\right)\frac{(-2\pi t)^a}{a!}+O\left(t^{N+1}\right), G_{d, \ell}(z; it) =2i\sum_{a=0}^N\mathcal{D}_z^{2a}\left(\frac{\sin(2 \pi d z)}{1-\zeta^\ell}\right)\frac{(-2\pi t)^a}{a!}+O\left(t^{N+1}\right).
T=\begin{pmatrix}1 0 0 0 .... \ 0 0 1 1 0 0 .... \ 0 0 0 1 1 0 ....\ 0 0 \ddots \ddots \ddots \ddots 0 0 0 \ldots 1 0 0 0 0 \ldots 1 1 \end{pmatrix}
Z_k=\left(\sum_{i=1}^{Z_{k-1}}\xi_{i}^{(k)}-M_{k+1}\right)_+\leq\sum_{i=1}^{Z_{k-1}}\xi_{i}^{(k)}\leq \sum_{i=1}^{(W_{k-1}-M_{k}+1)_+}\xi_{i}^{(k)}=W_k.
\Delta J_{i}^{}({\bf{n}})=\frac{1}{x_i \Delta x_i}\left[J^{}_{i+1/2}({\bf{n}})- J^{}_{i-1/2}({\bf{n}})\right],\ \Delta{J}_{i}^{}({\bf{n}})=\frac{1}{x_i \Delta x_i}\left[J^{}_{i+1/2}({\bf{n}})- J^{}_{i-1/2}({\bf{n}})\right].
u(t) =e^{-(t-s)A(t)}u(s)+ \int_{s}^{t}{e^{-(t-r)A(t)}( \mathcal{A}(t)-B(r)\mathcal{A}(r))u(r)dr}+\int_{s}^{t}{e^{-(t-r)A(t)}[-P(r)u(r)+f(r)]}dr.
{m_{{v_{n + 1}}\nwarrow}}={m_{\swarrow{v_{n + 1}}}}{m_{\uparrow{v_{n + 1}}}}{m_{\swarrow{v_n}}}=\int d{{\tilde \theta}_{n + 1}}\sum\limits_{{s_{n + 1}}}{{m_{{v_{n + 1}}\nwarrow}}}\psi \left({{v_n},{v_{n{\rm{+ 1}}}}}\right)
p_i\left({\Delta t,\alpha ,\lambda}\right) ={\log_2}\frac{{f_i^{\left({\Delta t,\alpha ,\lambda}\right)}- f_i^{\left({2\Delta t,\alpha ,\lambda}\right)}}}{{f_i^{\left({\Delta t/2,\alpha ,\lambda}\right)}- f_i^{\left({\Delta t,\alpha ,\lambda}\right)}}}.
E_{\mu \mathrm{m}}={1\over 2}\biggl({\partial u^\mu\over \partial X^{\mathrm{m}}}+{\partial u^{\mathrm{m}}\over \partial X^\mu}+ \sum_{\mathrm{n}=1}^3{\partial u^{\mathrm{n}}\over \partial X^\mu}{\partial u^{\mathrm{n}}\over \partial X^{\mathrm{m}}}\biggr)
J=\sum_{m=1}^\infty\frac 1{m^2}\int\limits_{|z|^2<1-(\frac{\varepsilon_L}2)^{1/L}}(1-|z|^2)^{mL}d\mu(z)- \int\limits_{\stackrel{|z|^2<1-(\frac{\varepsilon_L}2)^{1/L}}{\rho_L(z,0)\leq \varepsilon_L}}(1-|z|^2)^{mL}d\mu(z)\ .
\Omega_{0}\, = \, \left( \begin{matrix}{{{\frac{1}{\sqrt{2}}}}}\\{{{\bf 0}}}\\{{-{\frac{\mathrm{i}}{\sqrt{2}}}}}\\{{{\bf 0}}}\\ \end{matrix}\right) \qquad{\hat{\Omega}}\, = \, \left( \begin{matrix}{{0}}\\{{Y^{I}}}\\{{0}}\\{{F_{J}}}\\ \end{matrix}\right) \qquad \Omega_{3}\, = \, \left( \begin{matrix}{{{\hat{Y}}_{0}}}\\{{{\bf 0}}}\\{{{\hat{F}}_{0}}}\\{{{\bf 0}}}\\ \end{matrix}\right)
\sum\limits_{|j|>K(\eta)}\langle j\rangle^{2}\|u_{j}\|_{L^2}^{2}<\eta, \sum\limits_{|j|=0}^{K(\eta)}\langle j\rangle^{2}\int_{|x-x(t)|\geq \frac{R(\eta)}{N(t)}}|u_{j}(x)|^{2}\mathrm{d}x <\eta, \sum\limits_{|j|=0}^{K(\eta)}\langle j\rangle^{2}\int_{|\xi-\xi(t)|\geq R(\eta)N(t)}|\hat{u}_{j}(\xi)|^{2}\mathrm{d}\xi <\eta.
\mathbb{P}^{(1)}_{\ell}:= \mathbb{P}\bigg[ \sup_{\substack{X_{\ell-1}< n\leqslant X_{\ell}1 \leqslant j\leqslant J}}V(n,y_j)> \frac{T_1(\ell)}{\ell^{K/2}}\bigg]
\alpha^{(N)}_n=(1-aq^{2n-N})(aq^{1-N})_N\sum_{j\in \mathbb{Z}}(-1)^j \frac{a^j q^{(2n-N)j-j(j+1)/2}}{(aq^{2n-N-j})_{N+1}}\left[{N\atop j}\right]\alpha_{n-j},
A^{n}_{t}(\varphi)=\frac{1}{n}\sum_{i=1}^{n}R\varphi(S^{n,i}_{t-}) \frac{\partial \Psi}{\partial y}(S^{n,i}_{t-},\overline{\gamma}(t)) \left( \sqrt{n}( \gamma^{n}_{t}-\overline{\gamma}(t)) + \sqrt{n}r^{n,i}_{t}\right),
\int_{0}^{T}\big\|\sigma(Y_s^{h_{\varepsilon}})-\sigma(Y_s^{h})\big\|^2ds=\int_{0}^{T}\int_{D}\big|\sigma(Y_s^{h_{\varepsilon}}(x))-\sigma(Y_s^{h}(x))\big|^2dxds\rightarrow 0.
d s^{2}={\frac{l_{1}l_{2}}{\sqrt{\alpha}}}e^{Q ( \alpha , \beta ) / 2}( - d \alpha^{2}+ d \beta^{2}) + \alpha{\left( e^{V ( \alpha , \beta )}d x^{2}+ e^{- V ( \alpha , \beta )}d y^{2}\right)},
AR_{24}= -\sum\limits_{j=1}^{m}\int_0^t AE(t-s) (B_jA^{-1}) G_j(s)ds - \sum\limits_{j=1}^{m}\int_0^t AE(t-s) \int_0^s \omega_{\alpha_j}(s-\sigma) H_j(\sigma) d\sigma ds,
\widetilde{O}_{p}\left[Z(1+\sqrt{\frac{\kappa d}{n}})+\sqrt{\kappa}d^{2}+d^{\omega}\right] =\widetilde{O}_{p}\left[Z+d^{\frac{p}{2}+1}+d^{\frac{4}{1+\frac{2}{p}}}+d^{\omega}\right] =\widetilde{O}_{p}\left[Z+d^{\frac{p}{2}+1}+d^{\omega}\right].
<f,K_W>_{H^2_\beta(\mathbb{U}_{\mathbb{B}\mathbb{C}})}=\sum_{n=0}^\infty{a_n\beta(n)\left(\frac{\left(W^{\dagger_3}\right)^n}{\beta(n)^2}\beta(n)\right)}^{\dagger_3}&= \sum_{n=0}^\infty{\frac{a_n W^n}{\beta(n)^2}\beta(n)^2}&= \sum_{n=0}^\infty{a_n W^n}=f(W).
J_{d,e}(x) =J'_{d,e}(x)+\sum_{j=1}^{d+e-1}\frac{x^{d+e+3}}{(1-x)^{j+e}}(C(x)-1),\textrm{\where}J'_{d,e}(x) =J''_{d,e}(x)+\frac{x^{d+e+2}}{(1-x)^e}(C(x)-1),\textrm{\and}J''_{d,e}(x) =\sum_{j=1}^e\frac{x^{d+e+4}}{(1-x)^{j}(1-2x)}C(x)\,.
O(\epsilon^k) = \left\| \sum_{1\le j\le m/2}j\tilde{P}^{m-j}\tilde{P}^{(1)}\tilde{P}^j - \sum_{1\le j\le m/2}j \tilde{\Pi}\tilde{P}^{(1)}\tilde{P}^j \right\| &= \left\| \sum_{1\le j\le m/2}j\tilde{\Pi}\tilde{P}^{(1)}\tilde{P}^j - \sum_{1\le j\le m/2}j \tilde{\Pi}\tilde{P}^{(1)}(\tilde{P}-\tilde{\Pi})^j \right\|
_3F_2\left(\left.\begin{array}{c}\frac{1}{3},\frac{1}{2},\frac{2}{3}1,1 \end{array}\right|\frac{27u^{2}}{4(1-u)^{3}}\right)=(1-u)\sum_{n=0}^\infty \frac{D_n}{4^n}u^n.
\pi_*[\tilde\Omega] = \pi_*R^{(\rho,s)}\tilde{c}(1)_{\tilde\lambda_1}\cdots \tilde{c}(s)_{\tilde\lambda_s}= R^{(\rho,s)}\pi_*\tilde{c}(1)_{\tilde\lambda_1}\cdots \tilde{c}(s)_{\tilde\lambda_r}= R^{(\rho,s)}{c}(1)_{\lambda_1}\cdots{c}(s)_{\lambda_s}= \Theta^{(\rho)}_\lambda(c(1),\ldots,c(s)).
\frac{x'(t)y''(t) - x''(t)y'(t)}{x'(t)^2 + y'(t)^2}= \frac{1}{2}(x(t)y'(t) - x'(t) y(t)) + \frac{(n-1)x'(t)}{y(t)}- \frac{(m-1)y'(t)}{x(t)}+ \lambda (x'(t)^2 + y'(t)^2)^{1/2}.
f_n(q)=\sum_{n=s_0,s_1,\cdots, s_n}\prod_{i= 0}^{n-1}{s_i-s_{i+2}-1 \choose s_{i+1}-s_{i+2}}q^{{s_0\choose 2}-(s_1+s_2+\cdots+s_{n-1}+s_n)},
\begin{cases}\theta_{A_4(C)}(q) =1 + 96 q^4 + (35634176 + 16777216 \alpha)q^6 + \cdots, \theta_{S}(q) =\alpha + (96 - 96 \alpha )q^2 +(- 4416 + 4512 \alpha) q^4 + \cdots,\end{cases}
\hat{\hat{K}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}={\textstyle \frac{1}{2}}\left( \hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{c}}\hat{\hat{b}}}+ \hat{\hat{T}}_{\hat{\hat{b}}\hat{\hat{c}}\hat{\hat{a}}}- \hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\right) ,
\mathcal{D}(T_{-2,\beta}) =\left\{f:(-1,1)\rightarrow\mathbb{C\mid}f,f^{\prime}\in AC_{\mathrm{loc}}(-1,1);\right. & \quad\left. f^{(j)}\in L^{2}((-1,1);(1-x)^{j-2}(1+x)^{\beta+j})\text{}(j=0,1,2)\right\}.
0 = [x,[y,z,w]] - [y,[x,z,w]]+[z,[x,y,w]]- [w,[x,y,z]] - [[x,y],z,w] + [[x,z],y,w] - [[x,w],y,z]- [[y,z],x,w] + [[y,w],x,z] - [[z,w],x,y].
\delta~\psi_1 = \xi^z \partial_z \psi_1 - (\partial_z\xi^z)\psi_1 + \xi^{zz}(\partial_z^2 \psi_1) -{1\over2}(\partial_z\xi^{zz}) (\partial_z\psi_1) + (({1\over6}\partial_z^2 -{2\over3}\tilde{u}_2) \xi^{zz})\psi_1 \quad,
\frac54\big(\partial_x^3P_N)^2=\frac54\Big\{\Big(\frac{5^3}{4^3}a^3N^{3/4}+\frac{75}{4^3}a^2N^{-2/4}-\frac{15}{4^3}aN^{-7/4}\Big) &+\Big(\frac{5^4}{4^4}a^4N^{4/4}+\frac{750}{4^4}a^3N^{-1/4}-\frac{225}{4^4}a^2N^{-6/4}+\frac{105}{4^4}aN^{-11/4}\Big)(x-N)\Big\}^2\,e^{2aN^{5/4}},
\lim_{t \to \infty}\Big(\|u(\cdot,t;t_0,u_0,v_0)-u^{**}(\cdot,t)\|_{C^0(\bar\Omega)}+\|v(\cdot,t;t_0,u_0,v_0)-v^{**}(\cdot,t)\|_{C^0(\bar\Omega)}\Big)=0,
\mathbf{E}\!\left[ X_{t}(x_{n})\,\big|\,\mathcal{F}_{\tau_{n}}\right]=S_{t-\tau_{n}}X_{\tau_{n}}(x_{n})=X_{t}(0)\mathsf{1}_{A^{\mathrm{c}}}+S_{t-\tau_{n}}X_{\tau_{n}}(x_{n})\mathsf{1}_{A\,}.
{\cal P}_{R}^{2}={\tilde{v}}^{T}{\tilde{\phi}}{\tilde{\phi}}^{T}{\tilde{v}}={\tilde{v}}^{T}{\phi}_{c}{\tilde{\phi}}_{c}^{\dagger}{\tilde{v}}= |{\tilde{v}}^{T}{\phi}_{c}|^{2}.
\hskip -2mm \sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(i+j+1)(2j+3)}+\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(i+j+1)(2j-1)}&\equiv 2-8p^2-2p^2\sum_{k=1}^{\frac{p-3}{2}}\frac{(-1)^k}{k^2}\pmod{p^3}.
\Big \langle : e^{i \varphi ( f )}:_{M}e^{i \varphi ( g )}:_{M}\Big \rangle_{C_{m = 0}}\; : = \; \lim_{m \rightarrow 0}\Big \langle : e^{i \varphi ( f )}:_{M}e^{i \varphi ( g )}:_{M}\Big \rangle_{C_{m}}\; .
\sum_{k=0}^nT(n,k)H_{2k}=\sum_{k=0}^nT(n,k)\sum_{j=1}^k\frac1{2j-1}+\frac{1}{2}\sum_{k=0}^nT(n,k)H_{k}&=\sum_{k=0}^nT(n,k)\sum_{j=1}^n\frac1{2j-1}-\sum_{j=0}^{n-1}\frac1{2j+1}\sum_{k=0}^jT(n,k)+H_n &=H_{2n}-\frac12H_n-H_{2n}+\frac52H_n-H_{\lfloor n/2\rfloor}+H_n &=3H_n-H_{\lfloor n/2\rfloor}.
{\frac{1}{1 6}}\omega^{2}( 3 + 4 \omega )^{2}+ \omega^{2}( \omega + 1 )^{2}\left({\frac{z_{0}^{2}}{l^{2}}}\right)^{2}+{\frac{\omega^{2}}{2}}( 3 + 4 \omega ) ( \omega + 1 ){\frac{z_{0}^{2}}{l^{2}}}> 0 .
\pi_\nu(\exp(a))f(y) = f(y-a), \pi_\nu(h)f(y) = \chi_{\nu+\tfrac{p}{2}}(h)f(h^{-1}y), \pi_\nu(\exp(b))f(y) = |\Delta(-b,y)|^{-2\nu-p}f(y^{-b}).
\mathbb{P}_{e,M|p,h}= \frac{1}{2}Q\left(\frac{R P_t \sqrt{T_s}\left(g_{11}h_{11}+g_{21}h_{21}\right)}{\sqrt{N_0}}\right) + \frac{1}{2}Q\left(\frac{R P_t \sqrt{T_s}\left(g_{11}h_{11}-g_{21}h_{21}\right)}{\sqrt{N_0}}\right).
\left\vert \theta_{1}^{(1)}-\frac{V}{U}\right\vert =\left\vert \frac{c+2}{c}-\frac{V^{2}}{U^{2}}\right\vert \left\vert \theta_{1}^{(2)}-\frac{V}{U}\right\vert^{-1}\leq\frac{2}{|c||U|^{2}}\sqrt{\frac{|c|}{|c|-2}}=\frac{2}{\sqrt{|c|(|c|-2)}}|U|^{-2}.
\langle s_1 \cdot \cdot \cdot s_N|\Psi \rangle =e^{-{1 \over 2B}\sum_i \left({\partial \over \partial s_i}\right)^2}\prod_{i<j}(e^{i2\pi s_i/L_x}-e^{i2\pi s_j/L_x})^m.
(\frac{\partial^{\alpha_1}\widetilde{E}\cdot\partial^{\alpha-\alpha_{1}}\nabla_{v}M}{\sqrt{\mu}},\frac{\partial^{\alpha}M}{\sqrt{\mu}})=(\partial^{\alpha}\widetilde{E}\cdot\nabla_{v}M,[\frac{1}{\mu}-\frac{1}{M}]\partial^{\alpha}M)+(\partial^{\alpha}\widetilde{E}\cdot\nabla_{v}M,\frac{1}{M}\partial^{\alpha}M).
g(x,y_1,v_\rho(y_1))-g(x,y_\lambda,v_\rho(y_\lambda)) = g_y(x,y_\tau,v_\tau)\cdot(y_1-y_\lambda) \quad\quad+ g_z(x,y_\tau,v_\tau)(v_\rho(y_1)-v_\rho(y_\lambda)),
\frac{1}{\sqrt{3}- \sqrt{2}}= \frac{\sqrt{3}+ \sqrt{2}}{( \sqrt{3}- \sqrt{2}) ( \sqrt{3}+ \sqrt{2})}= \frac{\sqrt{3}+ \sqrt{2}}{( \sqrt{3})^{2}- ( \sqrt{2})^{2}}= \frac{\sqrt{3}+ \sqrt{2}}{1}= \sqrt{3}+ \sqrt{2}
S^{i j}( \theta - \sigma_{i j}) = \left( \begin{array}{c c c}{{\eta_{i j}( - 1 )_{\theta}}}&{{\eta_{i j}^{2}( - 2 )_{\theta}}}&{{\eta_{i j}^{3}( - 3 )_{\theta}}}\\{{\eta_{i j}^{2}( - 2 )_{\theta}}}&{{- ( - 3 )_{\theta}( - 1 )_{\theta}}}&{{\eta_{i j}^{2}( - 2 )_{\theta}}}\\{{\eta_{i j}^{3}( - 3 )_{\theta}}}&{{\eta_{i j}^{2}( - 2 )_{\theta}}}&{{\eta_{i j}( - 1 )_{\theta}}}\\ \end{array}\right) ,
\frac{\langle s^\prime|e^{-\hat H_s \tau_f}\hat O^s_1(\tau_1)\hat O^s_2(\tau_2)\ldots \hat O^s_n(\tau_n)e^{\hat H_s\tau_i}|s\rangle}{\langle s^\prime|e^{-\hat H_s (\tau_f-\tau_i)}|s\rangle}\,.
n T_1^* =n \sum_{j=1}^b \int_0^1 \Big(\phi_j(t)-\widehat{\phi}_j(t) \Big)^2dt+n \sum_{j=1}^b \int_0^1 \phi_j^2(t)dt &+2n \sum_{j=1}^b \int_0^1 \Big( \widehat{\phi}_j(t)-\phi_j(t) \Big) \phi_j(t) dt.
\begin{aligned}(iii)\h(x,y) \geq h_0-Cx\vartheta_0^{-4}\left(P(C(u_e,u_b,h_e)+\Vert (u_0,h_0)\Vert_{H^m_l})+Cx\right)^{1/2}&\cdot\left\{1-C\vartheta_0^{-24}\left(P(C(u_e,u_b,h_e)+\Vert (u_0,h_0)\Vert_{H^m_l})+Cx\right)^2x\right\}^{-1/4},\end{aligned}
\begin{cases}\mu^{(0)}= \delta_{t,t}, \mu^{(m)}= \tfrac{1}{3}\delta_{2^{-m}t,-2^{-m}t}+ \tfrac{1}{6}\delta_{-2^{1-m}t,2^{1-m}t}+\tfrac{1}{2}\mu^{(m-1)}\end{cases}
\int_0^1 \|f_t\|^{q'}_{L^{q'}(\Omega)}\,dt \le \int_0^1 \left[\|f_0\|^{q'}_{L^{q'}(\Omega)}+t\,\left(\|f_1\|^{q'}_{L^{q'}(\Omega)}-\|f_0\|^{q'}_{L^{q'}(\Omega)}\right)\right]\,dt &=\frac{\|f_0\|^{q'}_{L^{q'}(\Omega)}+\|f_1\|^{q'}_{L^{q'}(\Omega)}}{2}.
\begin{aligned}\|u(x,t_2 )-u(x,t_1 )\|_{H^3\cap H^{2,1}}\leq c\|r_{1,2}( z)(e^{2i\eta^2 t_2}-e^{2i\eta^2 t_1})\|_{H^1\cap H^{2,1}}&\leq c|t_2-t_1| \|r_{1,2}(z)\|_{H^1\cap L^{2,1}},\end{aligned}
U(\beta,t) = E_{\frac{3}{4}, \frac{1}{3}, -\frac{2}{3}}\left(-\frac{1}{4}\beta^2 t^{1/4}\right) = 1+\sum_{k=1}^{\infty}\left(-\frac{\beta^2t^{1/4}}{2^2}\right)^k\prod_{j=0}^{k-1}\frac{\Gamma(\frac{j}{4}+\frac{1}{2})}{\Gamma(\frac{j}{4}+\frac{5}{4})}.
F_{p}(a):=\left\{\frac{e^{i\frac{\pi}{4}}}{\sigma\left(\sqrt{a}\,e^{i\frac{\pi}{4}}\right)\,e^{2\pi\sqrt{a}\,e^{i\frac{\pi}{4}}}-1}\right\}=\frac{G_{p}(a)}{2\sqrt{2}},
\frac{\delta C_{Q}(\kappa)}{\delta \sigma(x)}|_{\sigma=0}= -2 \lambda^2 \sqrt{g(x)}\Psi_{\kappa}(x) -2 \int d \mu C_{Q}(\mu) \Psi_{\kappa}(x) \Psi_{\mu}(x) \frac{\kappa^2}{\kappa^2 -\mu^2}\sqrt{g(x)}
m\geq\frac{1+\varepsilon}{(\sqrt{\delta+1}-1)^2}\left(\sum_{\stackrel{i,j=1}{j\neq i}}^{d_2+d_1}\frac{\lambda_i\lambda_j}{(\lambda_i-\lambda_j)^2}+(n-d_1-d_2)\sum_{i=1}^{d_2+d_1}\frac{\lambda_i\sigma^2}{(\sigma^2-\lambda_i)^2}\right),
Q=\left(\begin{array}{ccc}3 0 0 0 3 0 0 0 3\end{array}\right),A=\left(\begin{array}{ccc}1 2 2 1 0 1 1 1 0\end{array}\right),L=\left(\begin{array}{ccc}2 -2 -2 -1 3 -1 -1 -1 3\end{array}\right).
\lim_{M \to \infty}G_M^+(x) = \frac{1}{2\pi}\sum_{n \ge 1}\frac{a(n)\sin(2 \pi nx)}{n^2}; \lim_{M \to \infty}G_M^-(x) = \frac{1}{2\pi i}\sum_{n \ge 1}\frac{a(n)(\cos(2 \pi nx)-1)}{n^2}.
S ={\frac{1}{4}}\int_{0}^{T}d \tau \dot{x}^{2}( \tau ) +{\frac{1}{4 L}}\int_{0}^{T}d \sigma_{1}\int_{0}^{T}d \sigma_{2}B ( \sigma_{1}, \sigma_{2}| \tau_{1}, \tau_{2}) x ( \sigma_{1}) \cdot x ( \sigma_{2}) ,
(\nabla f)(x,y)= \frac 1{\big(y^2-x^2\big)^3}&\times \begin{pmatrix}(y^2-x^2)(y\delta(x\pi)\delta(y\pi)+xy\delta'(x\pi)\delta(y\pi)\pi)+ 4x^2y\delta(x\pi)\delta(y\pi) (y^2-x^2)(x\delta(x\pi)\delta(y\pi)+xy\delta(x\pi)\delta'(y\pi)\pi)-4 x y^2\delta(x\pi)\delta(y\pi) \end{pmatrix}&=: \frac 1{\big(y^2-x^2\big)^3}\, g(x,y).
\langle T_{i}^{k}\rangle_{N}^{( b )}\left( \xi_{1}, \xi_{2}, \xi \right) = \langle T_{i}^{k}\rangle_{N}^{( 1 b )}( \xi_{1}, \xi ) - A_{d}\delta_{i}^{k}\int_{0}^{\infty}d k \, k^{d}\int_{0}^{\infty}d \omega \frac{K_{\omega}^{\prime}( k \xi_{2})}{K_{\omega}^{\prime}( k \xi_{1})}\frac{F^{( i )}[ N_{\omega}( k \xi , k \xi_{1}) ]}{N_{\omega}^{\prime}( k \xi_{1}, k \xi_{2})},
\sum \mathcal{D}^{\mathrm{ret}}_1(x-x_2)\mathcal{D}^{\mathrm{ret}}_2(x-x_3):B_0(x)B_1(x_2)B_2(x_2)B_3(x_3)B_4(x_3):+(x_2\leftrightarrow x_3)
W_{F_{4}}^{\underline{{2 7}}}={\frac{a_{1}( x )}{2 4}}-{\frac{1}{2}}\left\{\left( - q + \sqrt{q^{2}+ 4 p^{3}}\right)^{1 / 3}+ \left( - q - \sqrt{q^{2}+ 4 p^{3}}\right)^{1 / 3}\right\},
\begin{aligned}x_i{}={}\frac{U_*}{L}+ \frac{U_*}{L}\sum\limits_{j=\textbf{\textit{i}}}^{N-1}h_j g_i{}={}U_* f_i{}={}\Delta - \frac{U_*^2}{2L}\sum\limits_{j=0}^{i-1}h_j \, \Big(2 - h_j \frac{-\kappa}{1-\kappa}\Big) \end{aligned}
c_1(d)\|g\|_{1}- C_d|\sigma + \|g\|_{2}| \sqrt{\frac{\log(2/\delta)}{m}}\leq \sum_{i=1}^{3}\bar{f}_{(i)}\leq C_1(d)\|g\|_{1}+ C_d|\sigma + \|g\|_{2}| \sqrt{\frac{\log(2/\delta)}{m}},