harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bigg[\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}+x_{n+1}^{1-2s}P\bigg]\tilde{u}=0\quad\mathbb{R}_{+}^{n+1}, \tilde{u}=u\quad\mathbb{R}^{n}\times\{0\}, \lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2s}\partial_{n+1}\tilde{u}(x) =Vu\quad\mathbb{R}^{n}\times\{0\},
i\Pi^{(d)}_{\mu\nu}(p)=\frac{ie^{2}N_{f}}{6\pi^{2}\epsilon}(p_{\mu}p_{\nu}-\eta_{\mu\nu}p^{2})=\frac{ie^{2}}{(4\pi)^{2}\epsilon}\frac{-8N_{f}}{3}(\eta_{\mu\nu}p^{2}-p_{\mu}p_{\nu})
\intop_{0}^{a\left(x,t\right)}r\left(\frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}\right)dr=\intop_{0}^{a\left(x,t\right)}\frac{\partial\left(r\frac{\partial u}{\partial r}\right)}{\partial r}dr,
\mathcal{F}(u,s) =\frac{1}{\Gamma(s)}\int_{\varepsilon}^\infty t^{s-1}\mathcal{G}(u,t)\,dt+\frac{1}{\Gamma(s)(e^{2\pi i s}-1)}\int_{C_\varepsilon}t^{s-1}\mathcal{G}(u,t)\,dt.
P(y_{i})=\frac{e^{\beta_{0}+\beta_{1}x_{i1}+...+\beta_{k}x_{ik}}}{1+e^{\beta_{0}+\beta_{1}x_{i1}+...+\beta_{k}x_{ik}}}=\frac{1}{1+e^{-(\beta_{0}+\beta_{1}x_{i1}+...+\beta_{k}x_{ik})}}
\int_0^{m_0}\int_{|x^\prime|<1}\left|\frac{u_\varepsilon(x^\prime,\psi(x^\prime)+t)}{t}\right|^p dx^\prime dt= \bigg\{\int_0^{\varepsilon}\int_{|x^\prime|<1}+\int_{\varepsilon}^{m_0}\int_{|x^\prime|<1}\bigg\}\left|\frac{u_\varepsilon(x^\prime,\psi(x^\prime)+t)}{t}\right|^p dx^\prime dt.
\mathcal{Z}_0^\mathrm{prop}=\bigsqcup_{\substack{m_1 > 0 m_2 \ge 0 m_1+m_2=m}}\bigsqcup_{\substack{\mu_1\in L_0^\vee/L_0 \mu_2\in \Lambda^\vee/\Lambda (\mu_1,\mu_2) \in (\mu + L)/(L_0 \oplus \Lambda)}}\mathcal{Z}_0(m_1 , \mu_1 ) \times \Lambda_{m_2,\mu_2}
\begin{cases}H_\alpha(\beta,\beta) = \Delta_\alpha(\beta) \mu(0) \ \ \beta\ge 1, begin{align*}1mm]H_\alpha(\beta,k) = \Delta_\alpha( k) \mu(\beta-k) +\sum_{l=1}^{\beta-k}\Delta_\alpha(l) H_l(\beta-l,k) \ \ \beta\ge 2,\,1\le k\le \beta-1,\end{cases}
\widehat{G}(m,2, b, \epsilon,s) \leq 2\sum_{k=1}^{e^{(\log (m(1+\varepsilon)/2)^{1/b}}}k^{-ds}e^{- ds(\log (m-e^{(\log k)^b}))^{1/b}}\cdot \widehat{N}_{m,b,\epsilon}(k),
RT_e(\Phi) =\sum_{\gamma_0}\frac{\tau(H^{\sigma}\times H^{\sigma})}{d(H_{\gamma_0},H^{\sigma}\times H^{\sigma})}\sum_{\kappa_H}RO^{\kappa_H}_{\gamma_0}(\Phi) TRT_e(f) =\sum_{\delta_0}\frac{\tau(G^{\sigma}\times G^{\theta})}{d(G_{\delta_0},G^{\sigma}\times G^{\theta})}\sum_{\kappa}TRO^{\kappa}_{\delta_0}(f).
[(\psi^{u^{-1}x})^\ast(vu)] = [(\psi^x)^\ast(v)(\psi^{u^{-1}x})^\ast(u)] &= [(\psi^x)^\ast(v')(\psi^{u'^{-1}x'})^\ast(u')] &= [(\psi^{u'^{-1}x'})^\ast(v'u')]\ .
\displaystyle J ( \sigma , \tau ) ={\cal D}\mathrm{e t}( \bar{P}^{\dag}\bar{P})^{\frac{1}{2}}\left[ \det \left( \bar{k}^{i}, ( 1 - \bar{P}\frac{1}{\bar{P}^{\dag}\bar{P}}\bar{P}^{\dag}) \bar{k}^{j}\right) \right]^{\frac{1}{2}}\: .
\widehat{\tilde{\tilde{\varphi}}}_k(\xi):=\sum_{n}c_{n,k}e^{\frac{2 \pi i n \xi}{2^k}}=\sum_{n}c_{n,k}e^{\frac{2 \pi i n \xi}{2^k}}\widehat{\tilde \varphi}_k(\xi):=\sum_{n}c_{n,k}\widehat{\tilde \varphi}_{k, n}(\xi),
\kappa(\gamma_A,\gamma_H)=\lim_{K\uparrow H}\,\kappa(\gamma_A,\gamma_K)=\lim_{K\uparrow H}\,\int\kappa\gamma_A\,d\gamma_K=\lim_{K\uparrow H}\,\gamma_K(X)=\lim_{K\uparrow H}\,\|\gamma_K\|^2=\|\gamma_H\|^2,
\mathcal{Q}_{2}(V,V)=\lim_{\tau\to \infty}\frac{1}{\tau}\int_{0}^{\tau}\mathcal{L}(-s)\left(\begin{array}{c}0 \frac{1}{\sqrt{\rho_{0}}}\mathbb{H}^{\perp}_{\rho_{0}}B_{2}(V,V)\end{array}\right)ds
\MoveEqLeft \lim_{a\to 0}a^{-1}\int_{G(a)}\Big( f (\Theta_a^{Q_x}(t;aS),x+atu)- f (\Theta_u(t;S),x)\Big)dt &=\int_{-t_0^S}^{t_1^S}\Big(\Big\langle \nabla^1 f(\Theta_u (t;S),x),\Theta^{Q_x}(t; S)\Big\rangle +t\Big\langle \nabla^2 f(\Theta_u (t;S),x),u \Big\rangle\Big)dt.
I_{PPP}=|f_{z}^{\prime}f_{z}f_{z}\chi_{zzz}^{(2)}+f_{z}^{\prime}f_{i}f_{i}\chi_{zii}^{(2)}+f_{i}^{\prime}f_{z}f_{i}\chi_{zii}^{(2)}+f_{i}^{\prime}f_{i}f_{z}\chi_{iiz}^{(2)}|^{2},
\|(\tilde{u}_k,\tilde{v}_k)\|^p =\frac{p(\alpha+\beta-q)}{q(\alpha+\beta-p)}\|(u_k,v_k)\|^{q-p}\int_\Omega(\lambda|\tilde{u}_k|^q+\mu|\tilde{v}_k|^q)dx + o_k(1) &= \frac{p(\alpha+\beta-q)}{q(\alpha+\beta-p)}\|(u_k,v_k)\|^{q-p}\int_\Omega(\lambda|\tilde{u}|^q+\mu|\tilde{v}|^q)dx + o_k(1).
F(x_{1},...,x_{i},...,x_{a})=\sum_{k=0}^{n}\sum_{m_{1}=0}^{k}...\sum_{m_{a}=0}^{k}(\alpha_{m_{1},...,m_{a},k}R^{k}\prod_{i=1}^{a}(y_{i}^{k-m_{i}}(1-y_{i})^{m_{i}}))
H_{1}= 1 + \sum_{i}{\frac{Q_{i}}{[ | \vec{x}- \vec{x}_{0 i}|^{2}+{\frac{4 Q}{( p - 3 )^{2}}}| \vec{z}- \vec{z}_{0}|^{p - 3}]^{\frac{2}{p - 3}}}}, \H_{2}={\frac{Q}{| \vec{z}- \vec{z}_{0}|^{5 - p}}}.
P_{c}(z)= e^{\frac{\lambda}{\xi}z}z^{-(\frac{\mu j}{\xi}- c)}\left\lbrace -\frac{\gamma}{\xi}\ell_{1}(z) + \frac{\mu}{\xi}\sum_{n=1}^{c}n p_{c,n}\ell_{2}(z) - \frac{c(\xi - \mu)}{\xi}\sum_{n=0}^{c}p_{c,n}\ell_{3}(z)+ \sum_{n=1}^{c}n p_{c,n}\ell_{3}(z)\right\rbrace.
\left( T_{k}\mbox{\boldmath$ \phi$},\mbox{\boldmath$ \psi$}\right)_{\Gamma}=\left(\mbox{\boldmath$ \phi$},T_{k}^{\ast}\mbox{\boldmath$ \psi$}\right)_{\Gamma}\qquad\forall\mbox{\boldmath$ \phi$},\mbox{\boldmath$ \psi$}\in\mathbf{H}_{\operatorname*{curl}}^{-1/2}(\Gamma).
\Bigl ( \frac{1}{1 6 \pi G}\Bigr )^{2}S_{1}[ h ] = \Bigl ( \frac{1}{G \mu^{2}}\Bigr )^{2}\int d^{3}x \sqrt{h ( x )}\mu^{3}\biggl \{c_{1}+ c_{2}\frac{R ( x )}{\mu^{2}}\biggr \},
\begin{bmatrix}S_i S_{i-a_1+a_2}\hdots S_{i-a_1+a_n}S_{i-a_2+a_1}S_i \hdots S_{i-a_2+a_n}\vdots \vdots \ddots \vdots S_{i-a_n+a_1}S_{i-a_n+a_2}\hdots S_i \end{bmatrix}
\chi^{\pm}=\left( \pm u\right)^{n-1}\left[ \psi \mp \frac{1}{2}{\phi}^{\prime}-\frac{1}{4}\phi^{2}-2T^{\pm}-\frac{\phi T^{\pm}}{u}\mp \frac{1}{u}\frac{d T^{\pm}}{d \xi}-\frac{\left\{T^{\pm}\right\}^{2}}{u^{2}}\right] .
L=m(\dot{x},n)-im(n_{\alpha\beta}\theta^\alpha\dot{\theta}^\beta+n_{\alpha\beta}\chi^\alpha\dot{\chi}^\beta)+mb(\theta_\alpha\dot{\chi}^\alpha-\chi_\alpha\dot{\theta}^\alpha)-mb\theta^\alpha n_{\alpha\gamma}\dot{n}^\gamma{}_\beta\chi^\beta+is\frac{\bar{z}\dot{z}-z\dot{\bar{z}}}{1-z\bar{z}}\;,
p_k(r) := 2 (k+1) \exp \left( \varphi(\varepsilon/2) 2^{k p}r^p \right) \cdot \exp \left( - \frac{n}{8 (\log n)^{\frac{2}{\gamma}}}\cdot \frac{3 (2^{k-1}r)^2}{96 V (2^{k-1}r)^{\vartheta}+ 8 (2^{k-1}r)}\right).
a_{\scriptscriptstyle T}(\omega) = a(\omega)(\alpha^*(\omega))^2 - a^*(\omega) (\beta(\omega))^2 e^{-2j\phi}, b_{\scriptscriptstyle T}(\omega) = -\left(a^*(\omega)\alpha(\omega)\beta(\omega)+a(\omega)\alpha^*(\omega)\beta^*(\omega)e^{2j\phi}\right)
\frac{\partial}{\partial R}\log Z(R,\Sigma) = \frac{f_a(R,\Sigma^2)-R}{\Sigma^2}, \frac{\partial}{\partial \Sigma^2}\log Z(R,\Sigma) = \frac{(f_a(R,\Sigma^2)-R)^2+f_c(R,\Sigma)}{2\Sigma^4}.
\ln \Delta_{l}\tilde{\Delta}_{l}= \ln \left[ ( \tilde{s}_{l}( x^{\prime}) \tilde{e}_{l}^{\prime}( x ) - \tilde{s}_{l}^{\prime}( x^{\prime}) \tilde{e}_{l}( x ) )^{2}- \xi^{2}( \tilde{s}_{l}( x^{\prime}) \tilde{e}_{l}^{\prime}( x ) + \tilde{s}_{l}^{\prime}( x^{\prime}) \tilde{e}_{l}( x ) )^{2}\right] + \mathrm{c o n s t a n t}.
\begin{aligned}\left\| u(t) \right\|_{\infty}\leq K \left\| u_0 \right\|_{\infty^{\phantom{a}}}^{\frac{2(q_0-1)}{2q_0+N(m_2-1)}}\left( t^{-\frac{N}{2q_0+N(m_2-1)}}\, \Vert u_0 \Vert_{1}^{\frac{2}{2q_0+N(m_2-1)}}+ \Vert u_0 \Vert_{\infty^{\phantom{a}}}^{\frac{N(m_2-1)(q_0-1)}{q_0[2q_0+N(m_2-1)]}}\, \Vert u_0 \Vert_{1}^{\frac{1}{q_0}}\right) .\end{aligned}
v_p\left(\frac{N}{\gcd(C(uC+D),N)}\right) = v_p(N) - \min(v_p(uC+D),v_p(N)), v_p\left(\frac{m}{\gcd(C(uA+B),m)}\right) = v_p(m) - \min(v_p(A(uC+D)-1),v_p(m)).
Z_{d}^{v^{2}}(\nu;w_{1}, \ldots , w_{d}; u_{1}, \ldots , u_{d}) =\sum_{k_{1}, \ldots , k_{d}=-\infty}^{\infty}\left[ w_{1}(k_{1}-u_{1})^{2}+ \ldots +w_{d}(k_{d}-u_{d})^{2}+ v^{2}\right]^{-\nu}.
\begin{aligned}\mathcal S_2(k,3)= \left\{F\in\binom{[n]}{k}:[2]\subseteq F,F\cap[3,k-1]\neq\emptyset\right\}\cup\left\{F\in\binom{[n]}{k}:[2]\cup[k,k+3]\subseteq F\right\}&\cup\left\{[2,k+1],\, [2,k-1]\cup\{k+2,k+3\},\, [2,k]\cup\{k+3\},\, [2,k+2]\setminus\{k\}\right\}&\cup\left\{[3,k]\cup\{1,k+2\},\, [3,k-1]\cup\{1,k+1,k+3\}\right\},\end{aligned}
W_{\pi}(g_{t,l,v}) = \zeta_F(1)^{-2}q^{-\frac{t}{2}}q^{s(l_1-l_2)}\sum_{\mu\in \mathfrak{X}_l}G(\varpi^{-a_1},\mu\chi_1)G(\varpi^{-l_2},\mu\chi_2)G(v\varpi^{-l}, \mu^{-1}).
\sum_{s=n,n-1\ldots}t^{-s}\times\Big(\sum_\alpha\eta_s(g_\alpha)+D_{1}S^+_{1, s}+\big(D_2S^+_{2, s}\sqrt{t}^{e_2}+O(\sqrt{t}^{e_2+1})\big) &\qquad\qquad\qquad\qquad\qquad+\big(D_3S^+_{3, s}\sqrt{t}^{e_3}+O(\sqrt{t}^{e_3+1})\big)+\ldots\big)+\ldots\Big).
\MoveEqLeft \iint_{Q_\ell\times Q_\ell}g(x-y)\left|\phi_{\xi'}(y)\otimes\psi_\xi(x)-\phi_{\xi'}(x)\otimes\psi_\xi(y)\right|^2dxdy =2\iint_{Q_\ell\times Q_\ell}g(x-y)|\phi_{\xi'}(y)|^2|\psi_\xi(x)|^2\,dxdy-2\iint_{Q_\ell\times Q_\ell}g(x-y) ( \phi_{\xi'}(y)\otimes\psi_\xi(x))^*\cdot(\phi_{\xi'}(x)\otimes\psi_\xi(y))\,dxdy.
\mathcal{G}_{+ , l}( m e^{i \pi / 2}, r ) - \mathcal{G}_{- , l}( m e^{i \pi / 2}, r ) = \frac{i r}{\pi}e^{- i \pi | l |}( e^{2 i \delta_{l}^{+}}- e^{2 i \delta_{l}^{-}}) K_{W}^{2}( m r ) ,
\left( 1- \frac{\lambda}{m}+ \exp \left\{2 \pi i \frac{j + \theta_1 /2}{m}\right\}+ \frac{Tw}{m}\right)= e^{- \lambda/m}\left( 1+ \exp \left\{2 \pi i \frac{j + \theta_1 /2}{m}\right\}+ \frac{Tw + \lambda}{m}+ O(1/m^2) \right).
\Delta (y_m-y) = U(t,0)\Delta (x_m-x)+\int_0^tU(t,s)\{2i(D_j\nabla \tilde{b}^j(s)+\nabla \tilde{b}^j(s)D_j+\nabla \tilde{c}(s)) & \times \nabla(y_m-y)+i(D_j\Delta \tilde{b}^j(s)+\Delta \tilde{b}^j(s)D_j+\Delta \tilde{c}(s))(y_m-y) & -\lambda i\Delta [e^{(\alpha-1)W}(g(y_m(s))-g(y(s)))]\}ds,
y_k = \sum_{i=1}^I\widehat\alpha{}^y_{k,i}p^Y_{k,i},z_k = \sum_{i=1}^I\widehat\alpha{}^z_{k,i}p^Z_{k,i},\gamma_k = \sum_{i=1}^I\widehat\alpha{}^{\gamma}_{k,i}p^{\Gamma}_{k,i},
\left( \begin{array}{c c}{a^{\prime \prime}}&{b^{\prime \prime}}\\{c^{\prime \prime}}&{d^{\prime \prime}}\\ \end{array}\right) = \left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right) \left( \begin{array}{c c}{a^{\prime}}&{b^{\prime}}\\{c^{\prime}}&{d^{\prime}}\\ \end{array}\right)
\Delta_{2}={\frac{\omega}{2}}\left( 1 + ( 2 N_{2}+ 1 ) \left({\frac{\omega}{m}}\right) + \left( \ell_{2}+{\frac{1}{2}}\right)^{2}\left({\frac{\omega}{m}}\right)^{2}\right)^{{\frac{1}{2}}}.
\frac{1}{t}\left(\frac{1}{e^t-1}-\frac{1}{t}+\frac{1}{2}\right) = \frac{1}{t^2}\left(\frac{1}{1+t/2+t^2/6+\mathcal{O}(t^3)}-1+\frac{t}{2}\right) = 1/12 + \mathcal{O}(t).
{\frac{1}{2}}\sqrt{- \eta}\eta^{\mu \nu}={\frac{1}{1 - \lambda^{2}}}\left( \begin{array}{c c}{{\lambda_{--}}}&{{{\frac{1 + \lambda^{2}}{2}}}}\\{{{\frac{1 + \lambda^{2}}{2}}}}&{{\lambda_{+ +}}}\\ \end{array}\right)
\sigma_{2}(t): T^{*}M|_{Q,P}\times \mathbb{R}\rightarrow T^{*}M\times \mathbb{R}&;({\sigma_{2}^{*}}(t)\Theta^{\alpha}:=\epsilon^{\alpha},{\sigma_{2}^{*}}(t)\Theta_{\alpha}:=\epsilon_{\alpha},{\sigma_{2}^{*}}(t)Q^{i},{\sigma_{2}^{*}}(t)P_{i},{\sigma_{2}^{*}}(t)u=t) &\mapsto (\Theta^{\alpha},\Theta_{\alpha},Q^{i},P_{i},u),
\eta_{1}( v - \frac{m - 1}{2}) \xi_{1}( v - \frac{m - 2}{2}) \cdots \xi_{m - 1}( v ) \eta_{m - 1}( v + \frac{1}{2}) = ( x^{2 r}; x^{2 r})_{\infty}^{3 ( m - 1 )}c_{m}\eta_{m}( v )
D_{1 2}^{( l )}\stackrel{1}{\widehat{F}^{( l )}}\stackrel{2}{\widehat{F}^{( 1 )}}D_{1 2}^{( l ) - 1}= \left( \begin{array}{c c}{\widehat{F}^{( l + 1 )}}&{0}\\{0}&{- \alpha \beta \widehat{F}^{( l - 1 )}}\\ \end{array}\right) ,
\frac{1}{\prod_{(i,j)\in \lambda}(\lambda'_j+\mu_i-i-j+1+x)^2\prod_{(i,j)\in \mu}(\mu'_j+\lambda_i-i-j+1-x)^2}&=\prod_{i=1}^{K}\frac{\Gamma(x)^2\Gamma(1-x)^2}{\Gamma(m_i+K+1+x)^2\Gamma(l_i+K+1-x)^2}\prod_{i, j=1}^{K}(m_i-l_j+x)^2\ .
\int_{\mathfrak{m}}\big|S(\alpha)\big|d\alpha\le \int_{\mathfrak{m}}\int_{[0,1]^3}\Big|\mathcal{F}(\alpha,\boldsymbol{\beta})\mathcal{H}(\alpha,\boldsymbol{\beta}) \prod_{j=1}^{n-4}f_j(\alpha,\boldsymbol{\beta})\Big|d\boldsymbol{\beta}d\alpha.
L_{a b}= L_{a c}L_{c b}~ + ~ \frac{2}{l}\left( L_{c d}L_{e f}f^{c e a}f^{d f b}~ + ~ L_{c d}f^{c d f}f^{d f a}L_{b e}~ + ~ L_{c d}f^{c e f}f^{d f b}L_{a e}\right) .
\Vert Af\Vert^2_{H^{s-m}(\mathbb T)}= \frac1{2\pi}\sum\limits_{k\in\mathbb Z}(1+\vert k\vert)^{2(s-m)}\vert\widehat{Af}(k)\vert^2=\frac1{2\pi}\sum\limits_{k\in\mathbb Z}(1+\vert k\vert)^{2(s-m)}\vert A(k)\widehat{f}(k)\vert^2 &\le \frac{C^2}{2\pi}\sum\limits_{k\in\mathbb Z}(1+\vert k\vert)^{2s}\vert\widehat{f}(k)\vert^2=C^2\Vert f\Vert^2_{H^s(\mathbb T)}\,.
T^{\boldsymbol{\delta}_{l}(k_{1}+1)}\partial_{T}^{\boldsymbol{\delta}_l}= (T^{k_{1}+1}\partial_{T})^{\boldsymbol{\delta}_l}+\sum_{1 \leq p \leq \boldsymbol{\delta}_{l}-1}A_{\boldsymbol{\delta}_{l},p}T^{k_{1}(\boldsymbol{\delta}_{l}-p)}(T^{k_{1}+1}\partial_{T})^{p}
\begin{pmatrix}a_1 b_1 c_1 d_1\end{pmatrix},\begin{pmatrix}a_2 b_2 c_2 d_2\end{pmatrix},\begin{pmatrix}a_3 b_3 c_3 d_3\end{pmatrix},\ldots,\begin{pmatrix}a_n b_n c_n d_n\end{pmatrix}
\| gv \| = \sum_{i = 1}^d \sum_{j = 1}^d g^{i,j}v_j \geq \frac{1}{c}\sum_{i = 1}^d \sum_{j = 1}^d g^{i,i}v_j= \frac{1}{c}\sum_{i = 1}^d g^{i,i}\geq \frac{1}{c^2 d}\sum_{i = 1}^d \sum_{j = 1}^d g^{i,j}= \frac{1}{c^2 d}\|g\|,
\left<m^{\varepsilon},\phi\right> = -\frac{\varepsilon}{2}\,\displaystyle\sum_{j=1}^{d}\Big\{\int_{\Omega_{MT}}\,\mbox{sg}(u^{\varepsilon}-c)\left(B(u^{\varepsilon})\frac{\partial u^{\varepsilon}}{\partial x_{j}}\right)\frac{\partial\phi}{\partial x_{j}}\,dc\,dx\,dt\Big\}
\begin{array}[c]{c}(\omega_{N}+\sqrt{-1}\partial_{B}\overline{\partial}_{B}\phi_{T_{0},\varepsilon})^{2}\wedge\eta_{N}=C_{\varepsilon}\Omega_{\varepsilon}\wedge\eta_{N};\sup(\phi_{T_{0},\varepsilon}-\phi_{T_{0}})=\sup(\phi_{T_{0}}-\phi_{T_{0},\varepsilon})\end{array}
{M^g}' (t) = \int_0^t \sigma(s, X(s)) g(s) ds + \int_0^t \Big( \int_0^r U(s, r) g(s) ds\Big) dr + \int_0^t \Big( \int_0^r V(s, r) g(s) ds \Big) dW(r) &+\int_0^t \nabla_x b(r, X(r)){M^g}'(r) dr + \int_0^t \nabla_x \sigma (r, X(r)){M^g}'(r) dW(r).
\Theta(\hat x, \hat y) = F^{ij}(M[u](\hat y)) [D_zA_{ij}(\hat x, \bar z, Du(\hat y)) (u(\hat y)-v(\hat x)) + R^A_{ij}(\hat x, \hat y)] + D_zB(\hat x, \tilde z, Du(\hat y)) (u(\hat y)-v(\hat x)) + R^B(\hat x, \hat y),
P(A)=\frac{D^2 (k^2-1)}{2(1+q)Q}\left[ 1- \frac{2}{\Omega^{1+q}}\frac{k}{\sqrt{k^2-1}}Q(-\mu/2)^{-1/2}+ \frac{Q^2 (-\mu/2)}{\Omega^{2(1+q)}}\right].
L_{T}:=\tilde{\sigma}_{1}^{*}(t)\left[\Psi_{i}\dot{\Xi}^{i}-H_{T}\right] &\therefore L_{T}=\Psi_{1}\Xi^{3}+\frac{1}{2}\Xi^{2}(\Psi_{3})^{2}+\frac{d}{dt}\left(\Psi_{a'}\Xi^{a'}\right),
\left(\prod_{i=1}^q \left(m-d_i-(i-1)\right)\right) \prod_{i=q+1}^m \left(m-(i-1)\right) =m!\cdot \prod_{i=1}^q \frac{m-d_i-i+1}{m-i+1}&=m! \cdot \prod_{i=1}^q \left(1-\frac{d_i}{m-i+1}\right) &\geq m! \left(1- \sum_{i=1}^q \frac{d_i}{m-i+1}\right) &\geq m! \left(1- \sum_{i=1}^q \frac{d_i}{m-q}\right) &= m! \left(1-\frac{|E(H)|}{m-q}\right),
\Delta_{\Omega^{({D})}}= \sum_{j = 1}^{{D}- 1}\left[ \left( \prod_{k = 1}^{j - 1}\sin^{2}\theta_{k}\right) \sin^{{D}- j - 1}\theta_{j}\right]^{- 1}\frac{\partial}{\partial \theta_{j}}\left( \sin^{{D}- j - 1}\theta_{j}\frac{\partial}{\partial \theta_{j}}\right) ,
B_{1}( u ) \biggl |_{u \to \infty}= - 2 u ( \log u - \log 2 + 2 ) Q ( 1 ) + 4 u \int_{0}^{1}d \gamma \, \frac{\gamma^{2}}{1 - \gamma^{2}}\Bigl ( Q ( \gamma ) - Q ( 1 ) \Bigr ) + O ( \log u )
\Psi^{\iota_1\iota_2\iota_3}(\xi,\eta,\sigma) = (\partial_\eta\Psi^{\iota_1\iota_2\iota_3})(\xi,\eta,\sigma) = (\partial_\sigma\Psi^{\iota_1\iota_2\iota_3})(\xi,\eta,\sigma) = 0 .
\left\|\varphi^s(a)\right\|_{{c(s)}}^2 = \left\|\varphi^s(a)^\ast \varphi^s(a)\right\|_{{c(s)}}\leq \|\varphi^s(\Re a)^2 + \varphi^s(\Im a)^2\|_{{c(s)}}\leq 2 D(s)^2 \|a\|_{{c(s)}}^2
\lvert A(D) \rvert = \sum_{i \mbox{even}}n_i(n_1+n_3+\ldots +n_{i+1}) + \sum_{i \mbox{odd}}n_i(n_2+n_4+\ldots +n_{i+1}) &= n_0n_1+ \sum_{i\ge 2 \mbox{even}}n_i (n_o + n_{i-1}+n_{i+1}) &=(n_e-1) n_o +\sum_{i\ge 0}n_i n_{i+1}.
I_{1}=I_{1,1}+I_{1,2}\leq C(B)(\|f''\|_{L^\infty}+\|f''\|^2_{L^2})\|f'''\|^2_{L^2}-\frac{7\|f'''\|_{\dot{^H}^{1/2}}^2}{32(1+B^2)}-\frac{\|f'''\|^3_{L^3}}{2^7(1+B^2)^3\|f''\|_{L^\infty}}.
\Pr{A_k}= \ \Pr{s_i < k^2 \ \forall i\in\{1,\ldots,k\}}= \ (1-\Pr{s_1 < k^2})^k \leq \ (1-4c(k^2)^{-1/4})^k \leq \e^{-4ck^{1/2}}.
\int_0^\infty \int_0^x e^{f(t)-f(x)}\, dt\, dx = \int_0^\infty \int_t^\infty e^{f(t)-f(x)}\, dx\, dt &= \int_0^{R}\int_t^\infty e^{f(t)-f(x)}\, dx\, dt + \int_R^\infty \int_t^\infty e^{f(t)-f(x)}\, dx\, dt.
\begin{array}{ll}\displaystyle\min_{u,v,w}\tilde{\theta}(u,v,w): = p(u) + q(v) + \phi(w), begin{align*}8pt] \widetilde{\mathcal{A}}^*u + \widetilde{\mathcal{B}}^*v + \widetilde{\mathcal{C}}^*w = \tilde{c}.\end{array}
\mu \log\bigg( \frac{(t_{1}- 1)\theta + 1}{(t_{2}- 1)\theta + 1}\bigg) ~~~= \mu (1-\theta) \bigg[ \frac{t_{1}- 1}{(t_{1}- 1)\theta + 1}- \frac{t_{2}- 1}{(t_{2}- 1)\theta + 1}\bigg] - t_{3}.
a - \frac{1}{2}a = ( \frac{1}{2}+ \frac{1}{2^{2}}+ \frac{1}{2^{3}}+ \cdots + \frac{1}{2^{9 9}}+ \frac{1}{2^{1 0 0}}) - ( \frac{1}{2^{2}}+ \frac{1}{2^{3}}+ \frac{1}{2^{4}}+ \cdots + \frac{1}{2^{1 0 0}}+ \frac{1}{2^{1 0 1}})
\sum_{s=0}^{\infty}\|T_{s}\|_{1}=\sum_{s=0}^{\infty}\,\sum_{j=0}^{\infty}(T_{s})_{j,j}=\frac{1}{2}\sum_{j=0}^{\infty}\,\sum_{s=0}^{\infty}\frac{(s+1)\, j^{2s}}{(j+1)^{2s+4}}=\frac{\pi^{2}}{16}\,.
f\star g = \exp \Bigg[ \hbar\Bigg(\frac{\partial~}{\partial q}\frac{\partial~}{\partial\tilde p}-\frac{\partial~}{\partial p}\frac{\partial~}{\partial\tilde q}\Bigg)\Bigg] f({\bf x}) g({\bf \tilde x}) \vert_{{\bf x}={\bf\tilde x}},
\begin{array}{ccl}\int_{D}gf_{00}d\mu = -\int_{D}g_{0}f_{0}d\mu +2C_{n}\int_{\Sigma}\alpha gf_{0}d\Sigma_{p}= \int_{D}fg_{00}d\mu -2C_{n}\int_{\Sigma}\alpha fg_{0}d\Sigma_{p}+2C_{n}\int_{\Sigma}\alpha gf_{0}d\Sigma_{p}= \int_{D}fg_{00}d\mu .\end{array}
Q(X,V,\tau)=\left(\prod_{j=1}^{2k}\frac{x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\prod_{r=1}^{m_0}\left(\frac{\theta_1(u_r,\tau)}{\theta_1(0,\tau)}\frac{\theta_2(u_r,\tau)}{\theta_2(0,\tau)}\frac{\theta_3(u_r,\tau)}{\theta_3(0,\tau)}\right)\right)^{(4k)}.
\|w^{\ell+1}-w^{\ell}\|_{2,\varphi}\le \sqrt{2 C_{2}}\lambda \max\{\|w^{\ell}\|_{2,\varphi}, \|w^{\ell-1}\|_{2,\varphi}, \|u^{n}\|_{2,\varphi}\}\|w^{\ell}-w^{\ell-1}\|_{2,\varphi}&\le \sqrt{2 C_{2}}\lambda K \|u^{n}\|_{2,\varphi}\|w^{\ell}-w^{\ell-1}\|_{2,\varphi}\le L \|w^{\ell}-w^{\ell-1}\|_{2,\varphi}.
G^{Z \overline{{{Z}}}}= G^{p + 1 \, p + 1}+ G^{p + 2 \, p + 2}- 2 i G^{p + 1 \, p + 2}= \frac{2}{\varepsilon \left( 1 + \left( b_{( p + 2 ) / 2}\right)^{2}\right)}~ .
U=\exp\left( i~\theta~\boldsymbol{\hat{n}\cdot J}\right) =\sum_{k=0}^{2j}\frac{1}{k!}\left. A_{k}^{\left[ j\right]}\left( \theta\right)\right. \left( 2i~\boldsymbol{\hat{n}\cdot J}\right)^{k}~,
\begin{array}{l}{[{H}_{\pi_{\alpha}}] : ={\dot{u}}^{\alpha}( x ) - \frac{\partial{H}}{\partial \pi_{\alpha}},}\\{[{H}_{u^{\alpha}}] : ={\dot{\pi}}_{\alpha}( x ) + \frac{\partial{H}}{\partial u^{\alpha}}- \nabla_{a}( \frac{\partial{H}}{\partial ( \nabla_{a}u^{\alpha})}) .}\\ \end{array}
\Delta(p^2) \,=\, \frac{\Gamma(3-\omega)}{(4\pi)^\omega}\!\int_0^1 \! \! dx \! \int_0^1 \! \! \! dy \,\frac{[xy(1-y)]^{1-\omega/2}}{[ y(1-y)( x(1-x)p^2 - (1-x)m^2) -xm^2]^{3-\omega}}
\dfrac{d^{k}}{dx^{k}}\phi_{+\varepsilon ,\lambda}\left( x\right) =-\frac{\beta_{1}\gamma^{-1}}{\sqrt{\lambda}}\dfrac{d^{k}}{dx^{k}}\left( \sin \sqrt{\lambda}\left( x-a\right) \right) +O\left( \left( \sqrt{\lambda}\right)^{k-2}e^{\left\vert t\right\vert \left( x-a\right)}\right) ,
\frac{sin(\frac{\alpha}{2}-\frac{\beta}{2})}{sin(\frac{\alpha}{2}+\frac{\beta}{2})}=\frac{cot(\frac{\beta}{2})-cot(\frac{\alpha}{2})}{cot(\frac{\beta}{2})+cot(\frac{\alpha}{2})}=\frac{a-b}{2s-a-b}.
K_{1}(\epsilon) =\epsilon^{N-4}R^{2}(4-N)^{2}\omega\int_{0}^{2}\big[\frac{4r^{N-1}(|\varphi(r)|^{2}-1)}{(\epsilon^2+|r|^{2})^{N-2}}+\frac{4r^{N-1}}{(\epsilon^2+|r|^{2})^{N-2}}\big]dr &=L_{1}(\epsilon)+L_{2}(\epsilon).
f ( x ) = f ( z = x^{1}- i x^{2}, \bar{z}= x^{1}+ i x^{2}) \longrightarrow \hat{f}( a , a^{\dagger}) = \int \frac{d^{2}x d^{2}y}{( 2 \pi )^{2}}f ( x ) e^{i [ \bar{p}( \sqrt{2 \theta}a - z ) + p ( \sqrt{2 \theta}a^{\dagger}- \bar{z}) ]},
\begin{bmatrix}\mathcal G_{\mu}(r_C \xi_M^0) \mathcal G_{\mu}(r_C \xi_M^1) \vdots \mathcal G_{\mu}(r_C \xi_M^{M-1}) \end{bmatrix}= \begin{bmatrix}\xi_M^0 & \xi_M^1 & \ddots & \xi_M^{M-1}\end{bmatrix}\cdot F \cdot \begin{bmatrix}g_1 r_C g_2 r_C^2 \vdots g_{M}r_C^{M}\end{bmatrix},
(P_k)_i=\begin{bmatrix}\overbrace{0 \ldots 0}^{(k-1)}\eta_r U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\ldots \overbrace{0 \ldots 0}^{(k-1)}\eta_r \mu^{-\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r)\end{bmatrix}^{T},
{\rm Hilb}(6T_{3},t)=\big(1-3t+t^2\big)^{-1},{\rm Hilb}(6T_{4},t)=\big(1-6t+7t^2-t^3\big)^{-1},{\rm Hilb}(6T_{5},t)=\big(1-10t+25t^2-15t^3+t^{4}\big)^{-1},{\rm Hilb}(6T_{6},t)=\big(1-15t+65t^2-90t^3+31t^4-t^5\big)^{-1},{\rm Hilb}\big(6T_{3}^{(0)},t\big)= [2][3](1-t)^{-1},{\rm Hilb}\big(6T_{4}^{(0)},t\big)= [4](1-t)^{-2}\big(1-3t+t^2\big)^{-1}.
T^{f\circ\pi}g(x) = \int_X g(y) \cdot f(\pi(x,y))\> d\omega_X(y) =\int_D\int_X g(y) \cdot f(\pi(x,y))\> K_h(x,dy)\> d\omega_D(h) &= \int_D \int_X g(y)\cdot f(h) \> K_h(x,dy)\> d\omega_D(h)=T_fg(x).
|\partial_{t}^{2}\left(\langle\left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}}(r))-1}{r^{2}}\right)v_{4}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r=R\lambda(t)}, \phi_{0}\rangle\right)| \leq \frac{C}{t^{3}\log^{2N-2-b}(t)}
\Sigma(p)={\frac{i}{\theta}}\int_0^1 dx \; \int{\frac{d^3q}{(2\pi)^3}}{\frac{\left(q_\lambda+xp_\lambda\right)\left[ \left( p_\rho(1-x)-q_\rho\right)\; 2 \; g^\rho_\lambda-2M T^3\gamma_\lambda\right]}{\left[q^2+p^2x(1-x)-M^2x\right]^2}}
\int_{\Omega}(1-\sigma)D^2 u:D^2\varphi +\sigma\Delta u\Delta\varphi dx-\lambda\int_{\partial\Omega}\frac{\partial u}{\partial\nu}\frac{\partial\varphi}{\partial\nu}d\sigma=\mu(\lambda)\int_{\partial\Omega}u\varphi d\sigma\,,\ \ \ \forall\varphi\in H^2(\Omega),
\begin{array}[c]{l}\hat{{\psi}}\left( \eta\right) =\dfrac{1-4m^{2}}{16\eta}+\dfrac{m^{2}-1}{4\left({x^{2}-1}\right) \left({x^{2}-\sigma^{2}}\right)}+\dfrac{\left({1-\sigma^{2}}\right) \left({6x^{4}-\left({3+\sigma^{2}}\right) x^{2}-2\sigma^{2}}\right)}{16\left({x^{2}-1}\right) \left({x^{2}-\sigma^{2}}\right)^{3}}.\end{array}
2\rho\cdot \nabla a + (\nabla\beta^\sharp\cdot a) \rho + \nabla\alpha^\sharp \times (\rho \times a) = s_0 \tau^{-\theta}2\rho\cdot(\nabla \chi)(\tau^{-\theta}\,\cdot\,) \Psi^\sharp e^{\alpha^\sharp / 2}e^{\chi_\tau \Psi^\sharp}\overline{\rho}.
\sum\limits_{i = 1}^{{N_{\rm{R}}}}{\frac{1}{{{N_{\rm{R}}}{{\left|{\xi_i^\star}\right|}^2}}}}\approx \sum\limits_{i = 1}^{{N_{\rm{R}}}}{\frac{1}{{{N_{\rm{R}}}{{\left|{\xi_k^\star}\right|}^2}}}}= \frac{1}{{{{\left|{\xi_k^\star}\right|}^2}}}.
M(t)=\sum_{j=1}^3\sum_k\int_{\mathbb{R}}e^{\mu(t,\xi)+\lambda(t)(|k-k_0|+|\xi-\eta_0|)}|\widehat{g_j'}(t,k,\xi)|^2\,\mathrm{d}\xi,\mu(t,\xi)=-\int_{t}^{T_0}H(t',\xi)\,\mathrm{d}t'.
\sqrt{1+x_1}+\dots+\sqrt{1+x_{k}}+\sqrt{1+x_{k+1}}> k-1+\sqrt{1+x_1+x_2+\dots+x_{k}}+\sqrt{1+x_{k+1}}> k-1+1+\sqrt{1+x_1+x_2+\dots+x_{k}+x_{k+1}}= k+\sqrt{1+x_1+x_2+\dots+x_{k}+x_{k+1}}
H(B^{e-1}) = H((B^{e-1})^{\perp}) = N(\mathfrak{b})^{-1}\prod_{i=1}^p ||W^{(i)}\wedge Z_1^{(i)}\wedge\dots\wedge Z_m^{(i)}|| = N(\mathfrak{b})^{-1}\prod_{i=1}^p ||V_i||\;||Z_1^{(i)}\wedge\dots\wedge Z_m^{(i)}||\leq H(B^e) \prod_{i=1}^p ||V_i|| \leq C_5H(B^e)H^{(2y-1)/e}\leq C_5H^{(e+2y-1)/e}=H',
\mathfrak{d}^k_{\sigma,j}(f)=\sum_{0\leq n\leq \frac{k-j}{m_\sigma}}\frac{\binom{k}{j+nm_\sigma}\binom{\frac{(m_\sigma-1)-j}{m_\sigma}-1}{n}}{\binom{\frac{(k-j)-(m_\sigma-1)}{m_\sigma}}{n}}\frac{\partial^k(f\cdot V)}{\partial z^{j+nm_\sigma}\partial \overline{z}^{k-j-nm_\sigma}}(0,0).
{\frac{\Sigma \chi^{2}}{\sqrt{\tilde{G}}}}\frac{d}{d r}\left({\frac{\Sigma \chi^{2}}{\sqrt{\tilde{G}}}}\frac{d \theta}{d r}\right) ={\frac{\operatorname{c o s}\theta}{\Delta \operatorname{s i n}^{3}\theta}}Z \, ,
tr_{M_1}f =n~tr_{M_n(M_1)}~(vv^*) & = n~tr_{M_n(M_1)}(QE) = n~(1/n)\sum_itr_{M_1}(q_{ii}e_1) = \sum_i\tau~ tr(q_{ii}) ~~~~~~~~~~~ (\textrm{Markov property}) = \tau ~n~ tr_{M_n(N)}(Q) =1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(\textrm{by}~(1)).
\begin{array}{rcl}\dot{x}_1 = l_3 y_3x_2-y_2x_3, \dot{x}_2 = y_1x_3-y_3x_1, \dot{x}_3 = y_2x_1-l_3 y_1x_2,\end{array}\begin{array}{rcl}\dot{y}_1 = m_3x_2+(l_3-1)y_2y_3, \dot{y}_2 = -m_3x_1-(l_3-1)y_1y_3, \dot{y}_3 = 0,\end{array}
\begin{array}{r l}{T}&{= X - 2 s ,}\\{2 S}&{= \displaystyle \sum_{i = 1}^{s}( x_{i}- 2 ) ( x_{i}- 3 ) + 2 \sum_{1 \le i < j \le s}( x_{i}- 1 ) ( x_{j}- 1 ) ,}\\{}&{= X^{2}- ( 2 s + 3 ) X + s ( s + 5 ) ,}\\{P}&{= X - s .}\\ \end{array}
p_i+q_i =a_i(\ell_i-K_i+1)+(\ell_{i+1}-K_{i+1}+1)-\ell_i-2, p_n+q_n =a_n(\ell_n-K_n+1)-\ell_n-2, a_iK_i+K_{i+1}\geq (a_i-1)(\ell_i-1)+\ell_{i+1}, a_nK_n \geq (a_n-1)(\ell_n-1)-1, K_i+K_{i+1}\geq (1-a_i)(1+K_i),
\widetilde{\mathfrak C}_1 \Vert \mathbf{u}\Vert^2_{H^1(\mathcal{O})}\leqslant \Vert b(\mathbf{D})\mathbf{u}\Vert^2_{L_2(\mathcal{O})},\mathbf{u}\in H^1_\perp (\mathcal{O};\mathbb{C}^n).