harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g(\mathsf{x},T,\mathsf{y})=\hat{g}(\mathsf{x},T,\mathsf{y})+\sum_{\mathsf{n}\in \mathbb{N}^{d},\mathsf{n\neq 0}}\exp \left[ -TE_{\mathsf{n}}\right] \mathsf{h}_{\mathsf{n}}\left( \mathsf{x}\right) \mathsf{h}_{\mathsf{n}}\left( \mathsf{y}\right)
u\Delta(b)\,L_{\tilde{\Delta}(m)}\left(\sum_{i=1}^N\Delta(a_i)z_i\right) =u\Delta(b)\sum_{i=1}^N\Delta(ma_i)z_i=u\sum_{i=1}^N\Delta(bma_i)z_i &=u\Delta(bm)\sum_{i=1}^N\Delta(a_i)z_i=0.
\sum_{n\ge1}\frac{x^{n}}{1-q^{n}}\sum_{w\in\tilde{S}_{n}}t^{des(w)}q^{l(w)}=[\frac{x\cdot\frac{\partial}{\partial x}log(exp(x;q))}{1-texp(x;q)}]_{x\mapsto x\frac{1-t}{1-q}}
\forall i\in\{1,\ldots,n\}:\;\partial_t u^i=\partial_x(\partial_xu^i+\alpha_i(u^i\partial_x \mathcal{V})),\quad(t,x)\in(0,T)\times(0,1), \partial_{xx}\mathcal{V}=\sum_{i=1}^n\beta^i u^i+\zeta,\quad(t,x)\in(0,T)\times(0,1),
w ( z ) ={\frac{\hbar}{2 \pi^{2}}}\int_{0}^{\infty}d k_{z}\int_{0}^{\infty}d k_{\|}{\frac{k_{\|}}{k}}\left[ \left( k^{2}+ k_{\|}^{2}\right) \sin^{2}( k_{z}z ) + k_{z}^{2}\cos^{2}( k_{z}z ) \right] \left[{\frac{1}{2}}+{\overline{{n}}}( k ) \right] .
\frac{m_{\ell_{s+1}}}{q_{\ell_{s+1}}}< m_{\ell_{s+1}}^{a_{\ell_{s+1}}(b_{\ell_{s+1}}+\epsilon_{s+1})}= 2^{\log(m_{\ell_{s+1}})a_{\ell_{s+1}}(b_{\ell_{s+1}}+\epsilon_{s+1})}.
\langle 0 | H_{\mathrm{t r i v}}+ \Delta H_{\mathrm{t r i v}}| 0 \rangle = - \frac{1}{2}\delta m^{2}\sum \frac{1}{2 \tilde{\omega}}- \frac{\lambda}{8}( \sum \frac{1}{2 \tilde{\omega}})^{2}\frac{1}{L}
h ( T_{i}, k_{i}) \sim e^{- \frac{2}{3}\pi k_{i}^{2}T_{i}}\Bigl [ \Theta_{3}( i k_{i}T_{i}, 2 i T_{i}) \Theta_{3}( i k_{i}T_{i}, 6 i T_{i}) + \Theta_{2}( i k_{i}T_{i}, 2 i T_{i}) \Theta_{2}( i k_{i}T_{i}, 6 i T_{i}) \Bigr ]
j^{\mu \nu}= \frac{L_{p}}{2 \pi}\frac 1{\sqrt{g_{x}}}\epsilon^{\mu \nu \lambda \rho}\epsilon_{a b}\frac \partial{\partial \phi^{c}}\frac \partial{\partial \phi^{a}}( \ln \Vert \phi \Vert ) \partial_{\lambda}\phi^{c}\partial_{\rho}\phi^{b}.
\left[\begin{array}{cccc}g_{2}+w_{2}^{\ 2}h_{4}+n_{2}^{\ 2}h_{5}w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5}w_{2}h_{4}n_{2}h_{5}w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5}g_{3}+w_{3}^{\ 2}h_{4}+n_{3}^{\ 2}h_{5}w_{3}h_{4}n_{3}h_{5}w_{2}h_{4}w_{3}h_{4}h_{4}0 n_{2}h_{5}n_{3}h_{5}0 h_{5}\end{array}\right] ,
\mathcal{E}_{j}= -U(t){{\mathcal F}}^{-1}[t^{-1}{\tilde{\mathcal N}}_{j}(F_{1},F_{2})]+{\tilde{\mathcal N}}_{j}(M(t)D(t) F_1,M(t)D(t) F_2) &= -t^{-1}M(t)D(t)\mathcal{F}M(t)\mathcal{F}^{-1}\tilde{\mathcal{N}}_j(F_1, F_2)+t^{-1}M(t)D(t)\tilde{\mathcal{N}}_j(F_1, F_2) &= -t^{-1}M(t)D(t)\mathcal{F}(M(t)-1)\mathcal{F}^{-1}\tilde{\mathcal{N}}_j(F_1, F_2).
A ( z , z^{\prime}) = \frac{W_{+}( z ) W_{-}( z^{\prime})}{e^{- i \hbar \gamma^{2}h^{+}( z - z^{\prime})}}\delta^{+}( z - z^{\prime}) + \frac{W_{-}( z^{\prime}) W_{+}( z )}{e^{i \hbar \gamma^{2}h^{-}( z - z^{\prime})}}\delta^{-}( z - z^{\prime}) .
\int \varphi^s(A)d\mu_2(A) = \int \|A\|^{2-s}d\hat\mu_2(A) & \leq K_{d,2-s}e^{\mathbf{M}(\mu,2-s)}\int \|A\|^{2-s}d\hat\mu(A) &=2^{7-2s}e^{\mathbf{P}(\mu,s)}\int \varphi^s(A)d\mu(A)
R_n=\overline{C}(p,f,\pi_n)_T, &R=\int_0^T m_{c_s}(f) d{G_p}(s), &R_n(r)=n^{p/2-1}\sum_{i:t_{i,n}\leq T}f\big(\Delta_{i,n}C(r)\big), &R(r)=\int_0^T m_{c_s(r)}(f) d{G_p}(s),
\begin{array}{l}\lim_{s \to \infty}\,\phi (s) = \bar u_1 (B) := \displaystyle \frac{1}{2}- \frac{1}{2B}\,\bigg(\frac{c_1 + c_2}{c_1 - c_2}+ \sqrt{(B - 1)^2 + \frac{4 \,c_1 c_2}{(c_1 - c_2)^2}}\bigg). \end{array}
\mathcal{W}_j = \left\{(\underbrace{0,\ldots,0}_{k_1+\ldots + k_{j-1}},\tau_{j1},\ldots,\tau_{jk_{j}},\underbrace{0,\ldots,0}_{k_{j+1}+\ldots+k_{\nu}})\ :\ \sum_{i=1}^{k_j}\tau_{j i}= 0 \right\}
{\cal L}= \frac{1}{2}\cdot \frac{\left( \partial_{\mu}\vec{\varphi}\right)^{2}}{\left( 1 + \frac{g^{2}}{4}\vec{\varphi}^{2}\right)^{2}}\qquad \left( \vec{\varphi}= ( \varphi^{1}, \cdots , \varphi^{N}) \right) ,
\frac{\mathbf{P}\Big( \frac{1}{\sigma}(\hat{\theta}_n -\theta )\sqrt{\Sigma_{k=1}^n X_{k-1}^2}> x \Big)}{1-\Phi \left( x\right)}=\exp\bigg\{\theta_1 C \frac{\ln n +x^{3}}{\sqrt{n}}\bigg \}
\sigma_{\ell - \ell_P}(T_w^{P*}) = (-1)^{\ell_P(w)}(\sigma_{\ell - \ell_P})^{\iota_P}(T_w^P) & = (-1)^{\ell_P(w)}(\sigma^{\iota_P})_{\ell - \ell_P}(T_w^P) & = (-1)^{\ell(w) - \ell_P(w)}(-1)^{\ell_P(w)}\sigma^{\iota_P}(T_w^P) & = (-1)^{\ell(w) - \ell_P(w)}\sigma(T_w^{P*}).
(1-\beta T_i)^{-q_i}\cdot \tilde{c}_{q_i}(i) = \left[(1-\beta)^{-q_i}\cdot \tilde{c}(i)\right]_{q_i}&= \left[ (1-\beta)^{-q_i}\cdot c(i)\cdot c(E_{p_i}/D_i) \right]_{q_i}.
S(u^*_{\mu, \xi})(x, t) = \mu_0^{\frac{n-2s}{2}-1}\sum_{j=1}^k\dot{\mu}_jf_{j}+ \mu_0^{\frac{n-2s}{2}}\sum_{j=1}^k\dot{\xi}_j\cdot \vec{f}_{j}+ \mu_0^{\frac{n+2s}{2}}f,
[\begin{matrix}c_{2}&-c_{1}s_{2}&s_{1}s_{2}\\c_{3}s_{2}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{2}c_{3}s_{1}-c_{1}s_{3}\\s_{2}s_{3}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}\end{matrix}]
P_{n}(\lambda)= \lambda^2P_{n-1}-\lambda(\lambda^2-4)^{n-1}-4P_{n-1}-2(\lambda^2-4)^{n-1}-2(\lambda^2-4)^{n-1}-\lambda(\lambda^2-4)^{n-1}= (\lambda^2-4)P_{n-1}-2(\lambda+2)(\lambda^2-4)^{n-1}
0<\gamma_1:=\min_{1\leq i\leq m}\min\left\{\frac{r_i^-}{p_i^+},\frac{s_i^-}{q_i^+}\right\}-1\leq\gamma_2:=\max_{1\leq i\leq m}\max\left\{\frac{r_i^+}{p_i^-},\frac{s_i^+}{p_i^-}\right\}-1.
\langle \langle \delta^{(1)}j_\mu (x,t) \rangle \rangle = -i\int_{t_0}^{\infty}dt' \int_{-\infty}^{\infty}dx' A_0^{cl}(x',t') \langle \langle [j_\mu (x,t),j_0 (x',t')]\rangle \rangle \theta (t-t').
S^{i i}( \theta ) = \left( \begin{array}{c c c}{{( 2 )_{\theta}}}&{{( 3 )_{\theta}( 1 )_{\theta}}}&{{- ( 2 )_{\theta}}}\\{{( 3 )_{\theta}( 1 )_{\theta}}}&{{( 2 )_{\theta}^{2}}}&{{( 3 )_{\theta}( 1 )_{\theta}}}\\{{- ( 2 )_{\theta}}}&{{( 3 )_{\theta}( 1 )_{\theta}}}&{{( 2 )_{\theta}}}\\ \end{array}\right)
a=\frac{1-a}{2^{q+1}-2}\Longleftrightarrow a=\frac{1}{2^{q+1}-1}, 2+b=2+\frac{1}{2^q}-\frac{b}{2^{q+1}-2}\Longleftrightarrow b=\frac{1}{2^q}\frac{2^{q+1}-2}{2^{q+1}-1}, 2-c=2+\frac{c}{2^{q+1}-2}\Longleftrightarrow c=0, d=-\frac{d}{2^{q+1}-2}-\frac{1}{2^q}\Longleftrightarrow d=-\frac{1}{2^q}\frac{2^{q+1}-2}{2^{q+1}-1}.
L_{n}^{q}(f;x)=\frac{1}{\ell_{n}(x)}\sum_{k=0}^{n}f\left( \frac{[k]_{q}}{[n-k+1]_{q}q^{k}}\right) q^{\frac{k(k-1)}{2}}\left[\begin{array}{c}n k\end{array}\right]_{q}x^{k}
\frac{\delta S_{m}}{\delta \phi^{A}}\frac{\delta S_{m}}{\delta \phi_{A a}^{*}}+ V_{m}^{a}S_{m}= 0 , \qquad \frac{\delta S_{m}}{\delta \phi^{A}}\frac{\delta S_{m}}{\delta \eta_{B}}( \sigma_{\alpha})_{B}^{~ ~ \! A}+ V_{\alpha}S_{m}= 0 , \qquad \frac{\delta S_{m}}{\delta \phi^{A}}\frac{\delta S_{m}}{\delta \eta_{B}}\gamma_{A}^{B}+ V_{m}S_{m}= 0 ,
\varphi^* \left(\begin{array}{c c}a b 0 d \end{array}\right) := \left(\begin{array}{c c c}a^{\ell_1}a^{\ell_1}\, b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{- \ell_2}0 0 0 d^{- \ell_2}\end{array}\right) .
\prod U \equiv U_{2}U_{4}U_{3}^{- 1}\cdot U_{3}U_{5}U_{1}^{- 1}\cdot U_{1}U_{6}U_{2}^{- 1}\cdot U_{2}U_{6}^{- 1}U_{5}^{- 1}U_{4}^{- 1}U_{2}^{- 1}= 1 .
\begin{cases}-\partial_{tt}w + \Delta w = |\nabla w|^2 + 2 \nabla w \cdot \nabla F^\omega + |\nabla F^\omega|^2 \qquad ~ (t,x)\in \mathbb{R}^{1+d}w|_{t=0}=0, ~ \partial_t w|_{t=0}=0\end{cases}~.
\int \frac{\mathrm{d}\Omega}{4 \, \pi}\; \;{\cal L}_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left( k_{1}, k_{2}, k_{3}, \hat{Q}\right) = - i \; f^{a_{1}\, a_{2}\, a_{3}}\;{\cal A}_{\mu_{1}\mu_{2}\mu_{3}}\left( k_{1}, k_{2}, k_{3}\right) .
\| u_{x_1, r}- Q_{x_1}\|_{L^1(\partial B_1)}+ \| u_{x_2, r}- Q_{x_2}\|_{L^1(\partial B_1)}\leq C (-\log(r ))^{-\frac{1-\gamma}{2\gamma}}&= C (-\log |x_1-x_2| - \frac{1-\gamma}{2\gamma}\log( \log |x_1-x_2|))^{-\frac{1-\gamma}{2\gamma}}
\mathcal{J}_1\leqslant\frac{1}{\overline{F}_{\xi_{(\eta)}}(x)}\sum\limits_{n=1}^{K}\sum\limits_{k=1\atop k\notin\mathcal{K}}^{n}\overline{F}_{\xi_{k}}(xy)\mathbb{P}(\eta=n)+ \sum\limits_{n=1}^{K}\sum\limits_{k=1\atop k\in\mathcal{K}}^{n}\frac{\overline{F}_{\xi_{k}}(x/2)}{\overline{F}_{\xi_{(\eta)}}(x)}\mathbb{P}(\eta=n).
\int_{0}^{T}{\bf c}^{\top}(t)\bar{\bf w}^{(\epsilon)}(t)dt =\int_{0}^{T}{\bf c}^{\top}(t){\bf w}^{(\epsilon)}(t)dt=\lim_{r\rightarrow\infty}\int_{0}^{T}{\bf c}^{\top}(t)\widehat{\bf w}^{(k_{r})}(t)dt & \leq\lim_{r\rightarrow\infty}\int_{0}^{T}{\bf c}^{\top}(t){\bf w}^{(k_{r})}(t)dt=M.
\frac{\partial w_\epsilon}{\partial t}\,-\,\triangle w_\epsilon\,+\,\frac 2{\epsilon^{2(1-\kappa)}}f(w_\epsilon)\,-\,\frac{\dot\kappa}{K(\epsilon)}g(w_\epsilon) & \;=\;\frac{\dot{q}}{\epsilon^{1-\kappa}}\left[\frac{\partial r}{\partial t}\,-\,\triangle r\,+\,\frac{2q}{\epsilon^{1-\kappa}}(|Dr|^2\,-\,1)\right]\;\le\;0(x\ne 0).
H^{\epsilon}f(\varphi)=\frac{i}{\pi}\int_{\epsilon\le|\theta|\le\pi}\frac{f(\varphi-\theta)}{1-e^{i\theta}}d\theta=\frac{1}{\pi}\int_{|\zeta-e^{i\varphi}|\ge\delta}\frac{f(\zeta)}{\zeta-e^{i\varphi}}d\zeta,
I_{n}( \mu ) = - \left( \frac{\mu}{2}\right)^{4 - D}\frac{M}{( 4 \pi )^{3 / 2}}\frac{\Gamma ( \frac{2 n + D - 1}{2})}{\Gamma ( \frac{3}{2})}( - M E )^{( 2 n + D - 3 ) / 2}\Gamma \left( \frac{3 - D - 2 n}{2}\right) ,
\iota_{a,i}\tau_j=\left(\begin{array}{ccccccccccccccc}x_{1}\cdots x_{i-1}x_i x_{i+1}\cdots x_{j-1}x_j x_{j+1}x_{1}\cdots x_{i-1}ax_i x_{i+1}\cdots x_{j-1}x_{j+1}x_j\end{array}\right. \left.\begin{array}{ccc}x_{j+2}\cdots x_r x_{j+2}\cdots x_r\end{array}\right).
\Phi^{( b )}( u_{1}, u_{2}, \ldots , u_{b}, g , h ) = \sum_{G = 0}^{\infty}\sum_{\{k_{j}\}, n = 0}^{\infty}h^{G}{\cal N}_{G}^{( b )}( k_{1}, k_{2}, \ldots , k_{b}; n ) \; g^{n}u_{1}^{k_{1}}u_{2}^{k_{2}}u_{3}^{k_{3}}\cdots u_{b}^{k_{b}}
= - i \sum_{k = 1}^{n}\{f ({\bf{x}}_{k}) a ( \sigma_{k}) \nabla_{s}^{k}+ \frac{1}{2}[ \nabla_{s}^{k}f ({\bf{x}}_{k}) ] a ( \sigma_{k}) \}\varphi_{n}({\bf{x}}_{1}\sigma_{1},{\bf{x}}_{2}\sigma_{2}, . . . ,{\bf{x}}_{n}\sigma_{n})
\langle s^+_{1,k}, s^+_{1,l}\rangle = \delta_{kl}, \,\, \langle s^+_{1, k}, s^+_{2,1}\rangle = - \frac 12, \,\, \langle s^+_0, s^+_0 \rangle = \frac 12, \,\, \langle s^+_0, s^+_{i, j}\rangle = 0 \,\, (i =1, 2, j = 1,2, 3, 4)
+\sum_{k=1}^{d}[\sum_{j=1}^{m}\gamma^{ij}(\theta_{t})\int[b_{k}(x,t)-\int b_{k}(z,t)p(z,\theta_{t})dz]p(x,\theta_{t})\frac{\partial p(x,\theta_{t})}{\partial\theta_{j}}dx]\circ dY_{t}^{k}.
G_1(x)-G_1(x+1) = \sum_{k=0}^{\infty}\frac{1}{( k + x+\frac a3)\left(( k + x+\frac a3)^2-a^2+3b\right)}-\sum_{k=0}^{\infty}\frac{1}{(k + x+1+\frac a3)\left(( k + x+1+\frac a3)^2-a^2+3b\right)}= \frac{1}{x^3 + a x^2 + b x + c}.
\begin{array}{rl}du = \mp\star e\sp{u}\left(-d\delta +\alpha\wedge \beta -\sin\theta d\theta\wedge d\phi \right), \alpha = \mp\star e\sp{u}\left(-d\beta+\delta\wedge\alpha+\alpha\wedge\cos\theta d\phi \right), \beta = \mp\star e\sp{u}\left(d\alpha-\beta\wedge\delta+\beta\wedge\cos\theta d\phi\right).\end{array}
\tilde{U}_{i,i+1}(t_i)={\rm id}_{F_2\otimes M_N(\mathbb C)},\,\,\, \tilde{U}_{i, i+1}(t_{i+1})=(z_iw_i^*w_{i+1}z^*_{i+1})\otimes 1_{M_N(\mathbb C)},
S_{N}[ \Phi ] \ = \N \, \mathrm{t r}\left({\frac{\Phi^{2}}{2}}( 1 - b ) - g{\frac{\Phi^{4}}{4}}\right) \, + \,{\frac{N^{2}}{2}}\,{\frac{b^{2}}{x}}\, + \, N^{2}\, \rho
\bar{Q}_{\mu}\psi_{\nu}^{*}= 0 , \qquad \bar{Q}_{\mu}A_{\nu}^{*}= \partial_{\mu}\psi_{\nu}^{*}, \qquad \bar{Q}_{\mu}\phi^{*}= 0 , \qquad \bar{Q}_{\mu}C^{*}= \partial_{\mu}\phi^{*},
h_R(x_i^{n})= \begin{cases}(k+k(-1)^{(2k-1)|x_i|^2})y_ix_i^{2k-1}, if~n=2k, (k+1+k(-1)^{(2k-1)|x_i|^2})y_ix_i^{2k}, if~n=2k+1. \end{cases}
W_\lambda=\left[\begin{array}{cc}({q}_1, G\ast{q_1}) ({q}_1, G\ast{q_2}) ({q}_2, G\ast{q_1}) ({q}_2, G\ast{q_2})\end{array}\right] + \left[\begin{array}{cc}r_{11}r_{12}r_{21}r_{22}\end{array}\right],
J_{n n^{\prime}}( r ) \equiv \sum_{m = 0}^{\mathrm{m i n}( n , n^{\prime})}\frac{n ! n^{\prime}!}{m ! ( n - m ) ! ( n^{\prime}- m ) !}[ i \mathrm{s g n}( e H ) r ]^{n + n^{\prime}- 2 m}.
\delta_{\lambda^\dagger}\widehat\Lambda^\dagger(\lambda^\dagger,A^\dagger)-\delta_{\eta^\dagger}\widehat\Lambda^\dagger(\lambda^\dagger,A^\dagger)-i[\widehat\Lambda^\dagger(\lambda^\dagger,A^\dagger)\stackrel{*}{,}\widehat\Lambda^\dagger(\eta^\dagger,A^\dagger)]=\widehat\Lambda^\dagger(-i[\lambda^\dagger,\eta^\dagger],A^\dagger).
XS_{t_{-\xi}}\eta_{e}= (\dots,\, (wz_{\xi}t_{\xi})(z_{-\xi}t_{-\xi+\phi_{J}(-\xi)}) \,;\, 0,\,\dots,\,1) & = (\dots,\, \underbrace{(wz_{\xi}t_{\xi})(t_{-\xi}y)}_{=\kappa(XS_{t_{-\xi}}\eta_{e})}\,;\, 0,\,\dots,\,1),
\begin{aligned}\int_{\Lambda_{h}}(w-v)\partial_{\nu}u_{0}-u_{0}\partial_{\nu}(w-v)\mathrm{d}\sigma-\int_{\Gamma_h^{\pm}}\eta u_0v\mathrm d\sigma =k^{2}\int_{S_{h}}(q-1)vu_{0}\mathrm{d}\mathbf x. \end{aligned}
f \star_{r}^{r-1}g (t_1 ,\dots, t_{d-r},\gamma, s_1,\dots, s_{d-r}) = &=\sum\limits_{i_1,\dots, i_{r-1}=1}^{N}f(t_1,\dots, t_{d-r},\gamma, i_1,\dots, i_{r-1})g(i_{r-1},\dots, i_{1}, \gamma,s_1,\dots,s_{d-r});
\frac{\partial \Phi_{j-1}}{\partial s_j}= \frac{\zeta'(s_j)[\xi(s_j)-\xi(s_{j-1})] - \xi'(s_j)[\zeta(s_j)-\zeta(s_{j-1})]}{[\xi(s_j)\zeta(s_{j-1})-\xi(s_{j-1})\zeta(s_j)]^2}=: \frac{\zeta'(s_j)N_1 - \xi'(s_j)N_2}{D_1^2},
\frac{-c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta \left((b-1) \int_{0}^{\frac{t_{+}}{2}}\frac{du \sin(u)}{u \log^{b}(\frac{t_{+}}{u}) t_{+}^{2}}\right)= -\frac{c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta (b-1) \left(\frac{\pi}{2 \log^{b}(t_{+}) t_{+}^{2}}+ (t,r,\theta) \right)
q^{n\choose 2}C^\prime_n\left(\frac{1}{q}\right)= \sum_{k_0+k_1+\cdots k_n=n}q^{{k_0+k_1+\cdots k_n\choose 2}-\sum_{i=0}^{n}ik_i}P_1(k_0,k_1,\cdots,k_n) = q^{n\choose 2}\sum_{k_0+k_1+\cdots k_m=n}q^{-\sum_{i=0}^{n}ik_i}P_1(k_0,k_1,\cdots,k_n).
\begin{aligned}-\omega(x,t)=-\frac{\omega(x,t)}{2}+\int_{0}^{t}\int_{\partial\Omega}\bigg( \frac{\partial \varepsilon_{n}\big(x-\xi,b(t,\tau)\big)}{\partial\nu(\xi)}a(\tau) \omega(\xi,\tau) &-\varepsilon_{n}\big(x-\xi,b(t,\tau)\big)a(\tau) \frac{\partial \omega(\xi,\tau)}{\partial\nu(\xi)}\bigg)dS_{\xi}d\tau,\end{aligned}
\forall \varphi,\psi \in C_c^{\infty}(\mathcal{D}), \ \ \int_{\mathcal{D}\times\mathcal{D}}k(x,y)\partial^{\alpha}\varphi(x)\partial^{\alpha}\psi(y)dxdy = b_{\alpha}(\varphi,\psi) = \langle \mathcal{E}_k^{\alpha}\varphi,\psi\rangle_{L^2}
s_{m}= \sum_{n=1}^{r}\mu_{n}\lambda_{n}^{m}\ , \ \m \geq 0 \ , \ \mu_{n}= \left(\sum\limits_{k=0}^{r-1}\dfrac{P_{k}(\lambda_{n})^{2}}{D_{k}D_{k-1}}\right)^{-1}> 0 \ , \P_{r}(\lambda_{n}) = 0 \ , \ 1 \leq n \leq r \ .
\int_{\Sigma_g}\omega^j_z\bar\omega^k_{\bar z}d^2z=(\bar\Omega-\Omega)^{jk},\quad\,\,\int_{\Sigma_g}\mu^{i z}_{\bar z}\omega^j_{z}\omega^k_{z}d^2z=i{\partial \Omega^{jk}\over\partial m_i},\quad\,\,\int_{\Sigma_g}\bar\mu^{i\bar z}_z\bar\omega^j_{\bar z}\bar\omega^k_{\bar z}d^2z=-i{\partial\bar\Omega^{jk}\over\partial\bar m_i},
\mathrm{DTE}_{(p,q)}(X)=\mu+\sigma\frac{c_{1}\left[\overline{G}_{1}\left(\frac{1}{2}\xi_{p}^{2}\right)-\overline{G}_{1}\left(\frac{1}{2}\xi_{q}^{2}\right)\right]}{F_{Y}(\xi_{p},\xi_{q})},
\epsilon \colon 1_{\mathcal{F_{\mathbb{R}}}}\to\nu_{*}\nu^{*},\,H(S)\stackrel{H(\mathrm{pr}_{1})}{\longrightarrow}(\nu_{*}\nu^{*}H)(S)=H(\nu_{*}S_{\mathbb{C}}) \eta \colon \nu^{*}\nu_{*}\to1_{\mathcal{F}_{\mathbb{C}}},\,(\nu^{*}\nu_{*}F)(T)=F(T\sqcup\overline{T})\stackrel{F(j_{T})}{\longrightarrow}F(T)
\begin{array}{c c c c c}{\hline \mathrm{c u r r e n t s}}{J^{+ i j}}{J^{+ - i}}{J^{i j k}}{J^{- i j}}{\hline \hline \mathrm{d i m e n s i o n s}}{- 1}{1}{1}{3}{\hline}\end{array}
M_{1}=\int_{-t/2}^{t/2}r\times(\sigma_{11}e_{1}+\sigma_{12}e_{2}+\sigma_{13}e_{3})dx_{3}andM_{2}=\int_{-t/2}^{t/2}r\times(\sigma_{12}e_{1}+\sigma_{22}e_{2}+\sigma_{23}e_{3})dx_{3}
\frac{1}{|I|}\int_0^{|I|}\int_I|tD_{\lambda ,x}P_t^\lambda (f_1)(x)|^2\frac{dxdt}{t}\leq \frac{1}{|I|}\int_0^\infty |g_\lambda (f_1)(x)|^2dx\leq \frac{C}{|I|}\int_0^\infty |f_1(y)|^2dy &=\frac{C}{|I|}\int_{2I}|f(y)-f_{2I}|^2dy\leq C\|f\|_{BMO_{\rm o}(\mathbb{R})}^2,
e_{0}^{a}= \left( \begin{array}{c}{{C ( \eta )}}\\{{0}}\\{{0}}\\{{0}}\\ \end{array}\right) \qquad e_{1}^{a}= \left( \begin{array}{c}{{0}}\\{{C ( \eta )}}\\{{0}}\\{{0}}\\ \end{array}\right) \qquad e_{2}^{a}= \left( \begin{array}{c}{{0}}\\{{0}}\\{{C ( \eta )}}\\{{0}}\\ \end{array}\right) \qquad e_{3}^{a}= \left( \begin{array}{c}{{0}}\\{{0}}\\{{0}}\\{{C ( \eta )}}\\ \end{array}\right)
\int_I \left[u(t)-u_0\right]\vartheta{\rm d}x{\rm d}y+\int_{Q_t}\left(u_{x}\vartheta_x+x^{2\gamma}u_{y}\vartheta_y-\frac{\sigma}{x^2}u\vartheta\right){\rm d}x{\rm d}y{\rm d}t = \int_{Q_t}f\vartheta{\rm d}x{\rm d}y{\rm d}t+\int_{Q_t}F\vartheta{\rm d}x{\rm d}y{\rm d}B(t),\mathbb P-a.s.
= - 2 \pi b_{0}\int_{\Sigma}+ \beta^{- 1}\int_{\Sigma}\left[ \gamma_{1}+ \gamma_{2}R_{\Sigma}+ ( \gamma_{3}+ \gamma_{4}\beta^{- 2}) R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}+ ( \gamma_{5}+ \gamma_{6}\beta^{- 2}) R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}\right] ~ ~ ~
\{E_x,\E_y\}(u) =dE_y(u)\cdot X_{E_x}(u)=dJ_y(S^*u)\cdot S^*S X_{J_x}(S^*u) &=d J_y(S^*u)\cdot X_{J_x}(S^*u)=\{J_x,J_y\}(S^*u)=0\ .
E^2(\lambda) = \left({\mathrm{sot}}\lim_{n\to\infty}f_{\lambda,n}(A) \right)^2 &={\mathrm{sot}}\lim_{n\to\infty}f^2_{\lambda,n}(A) &\ge{\mathrm{sot}}\lim_{m\to\infty}f_{\lambda,m}(A)=E(\lambda)
\rho_{j}(\sigma^{m})(w_{\alpha}) =w_{\alpha-1}\,, \rho_{j}(\tau^{m})(w_{\alpha}) = (\zeta^{m})^{j(\alpha m+k)}w_{\alpha}\,, \rho_{j}([\sigma^{m},\tau^{m}]^{r}(\sigma^{m})^{s}(\tau^{m})^{t})(w_{\alpha}) = \rho_{j}([\sigma,\tau]^{m^{2}r}\sigma^{ms}\tau^{mt})(w_{\alpha}) &= (\zeta^{m})^{j((\alpha m+k)t+mr)}w_{\alpha-s}\,.
\mathcal{G}_n(u,t) =\sum_{m=0}^\infty \sum_{j=0}^{n}{n \brack j}e^{nt}\frac{{\rm Li}_{-m-j}(1-e^t)}{1-e^{t}}\frac{u^m}{m!}&=\sum_{m=0}^\infty \sum_{k=0}^\infty \sum_{j=0}^{n}{n \brack j}B_k^{(-m-j)}(n)\frac{(-t)^k}{k!}\frac{u^m}{m!}.
\mathcal Q(x)(v) =\sum\limits_{e\in E_v}\frac{(avg(x))(e)}{|e|}\frac{\delta_E(e)}{\delta_V(v)}& =\sum\limits_{e\in E_v}\sum\limits_{u\in e}x(u) & =\sum\limits_{e\in E}B_{ve}\sum\limits_{u\in V}B_{ue}x(u)=((BB^T)x)(v),
||P_{\lambda^t}(a)||_{s-t}^2 = \sum_{m}(1+|m_1|^2+\cdots+|m_n|^2)^{s-t}\lambda^{2t}(m)|a_m|^2 &= \sum_{m}(1+|m_1|^2+\cdots+|m_n|^2)^{s-t}(1+|m_1|^2+\cdots+|m_n|^2)^t|a_m|^2 &= \sum_{m}(1+|m_1|^2+\cdots+|m_n|^2)^s|a_m|^2 &= ||a||_s
M = 1- \frac{q}{p+q+n}\cdot\frac{q+n-1}{p+q+n-1}- \frac{p+n-1}{p+q+n-2}\cdot\frac{p}{p+q+n-1}, R_j = - \frac{q}{p+q+n}\cdot \frac{1}{q+j}\cdot \frac{q+n-1}{p+q+n-1}+ \frac{p+n-1}{p+q+n-2}\cdot \frac{1}{p+n-1-j}\cdot \frac{p}{p+q+n-1}.
S_{\mathrm{f r e e}}\! = \! \frac{1}{2}\int \! \! d^{D}\! x [ ( \partial_{x}\Phi )^{2}\! - \! m^{2}\Phi^{2}] = \sum_{j , k}\! \frac{1}{2}\int \! \! d^{D - 2}\!{\bf x}[ ( \partial_{\bf x}\phi_{j k}^{*}) ( \partial_{\bf x}\phi_{j k}) \! - \! m_{j}^{2}\phi_{j k}^{*}\phi_{j k}] ,
\int_{X(y,k)}V \cdot \hat n_{X_=}d{\mathcal H}^{m-1}= \int_{X_\le(y,k)}\nabla_X \cdot V d{\mathcal H}^m - \int_{\partial^* X \cap \overline{X_\le(y,k)}}V \cdot \hat n_X d{\mathcal H}^{m-1}
q_{s}^{2}{\frac{\partial^{2}H}{\partial q_{i}\partial q_{i + 2}}}= - ( c q_{s}/ ( 2 \Delta x ) )^{2}\neq 0 \quad \mathrm{a n d}\quad{\frac{\partial^{2}H}{\partial p_{i}\partial p_{i + 2}}}= 0 ,
S \left[ \vec{h}_{\mu \nu}\right] = \int d^{4}x \Biggl ( \frac{1}{4}\vec{h}_{\mu \nu}^{2}- \frac{i \theta g^{2}}{3 2 \pi^{2}}\vec{h}_{\mu \nu}\tilde{\vec{h}}_{\mu \nu}\Biggr ) + V \left[ \vec{h}_{\mu \nu}\right] .
\int_{0}^{z}\frac{u^{\alpha}}{1-u^{2}}\ln u\,du=\frac{1}{2}\ln z\,\mathrm{B}_{z^{2}}\left( \frac{1+\alpha}{2},0\right) -\frac{z^{\alpha +1}}{4}\,\Phi\left( z^{2},2,\frac{1+\alpha}{2}\right) .
\bar{\alpha}_A(ab, x) =\bar{\alpha}_A(a, \bar{\alpha}_A(b, x))=\beta(a, \bar{\alpha}_A(b, x)) \cdot \alpha_{A}(a, \bar{\alpha}_A(b, x)) =\beta(a, \bar{\alpha}_A(b, x)) \cdot a \cdot \beta(b, x) \cdot \alpha_{A}(b, x)
\psi_{LF}(k_{\perp},x)=\frac{1}{\sqrt{4\pi}}\int_{0}^{\infty}\frac{x(1-x)g(\gamma^{\prime},1-2x)d\gamma^{\prime}}{[\gamma^{\prime}+k_{\perp}^{2}+m^{2}-x(1-x)M^{2}]^{2}}.
[A,K] = \sum_{i=1}^{L+1}\alpha_i\{A^i,C\}+ 2\epsilon C + \sum_{i=1}^{L+1}\frac12 m_i\{A^i,C\}-\sum_{i=1}^{L+1}{\beta\alpha_i}[A^i,B] +\ell_1 C.
X^-( x s_{k,k}\otimes_k y) = X^-(x \otimes_k (s_{k,k+1}+ \xi_{k+1}s_{k+1,k+1}) y) &= X^-(x \otimes_k y) s_{k,k+1}+ X^-(x \otimes_k y (\xi_{k+1}s_{k+1,k+1})) &= s_{k,k}X^-(x \otimes_k y) - X^-(x \otimes_k y) \xi_{k+1}s_{k+1,k+1}+ X^-(x \otimes_k y (\xi_{k+1}s_{k+1,k+1})),
\Phi_{1}^{G}= e + \zeta \pi_{0}^{- 1}\; , \; \; \Phi_{2}^{G}= \chi \; , \; \; \Phi_{3 \mu}^{G}= b_{\mu}\; , \; \; \Phi_{4}^{G}= x_{0}- \zeta \tau \; , \; \; \Phi_{5}^{G}= \psi^{0}\; , \; \;
\psi_{\lambda_{1}, \lambda_{2}}^{m l}( \xi_{1}, \xi_{2}) = N \exp \left\lbrace{\frac{\xi_{1}^{2}}{2 c \theta}}-{\frac{\xi_{2}^{2}}{2 ( 1 + c ) \theta}}-{\frac{\lambda_{1}\xi_{1}}{c \theta}}+{\frac{\lambda_{2}\xi_{2}}{( 1 + c ) \theta}}\right\rbrace \quad .
[h_i,h_j] =0, \ \ [h_i,d]=0, &[e_i,f_j] = \delta_{ij}h_i, &[h_i, e_j] = a_{ij}e_j, \ \ [h_i, f_j] = -a_{ij}f_j, &[d, e_j]= \delta_{0,j}e_j, \ \ [d, f_j]=-\delta_{0,j}f_j, &(\, e_i)^3e_j = (\, f_i)^3f_j = 0, i \neq j.
lnK=\sum_{k}ln{a_{k}}^{m_{k}}-\sum_{j}ln{a_{j}}^{n_{j}};K=\frac{\prod_{k}{a_{k}}^{m_{k}}}{\prod_{j}{a_{j}}^{n_{j}}}\equiv\frac{{\{R\}}^{\rho}{\{S\}}^{\sigma}...}{{\{A\}}^{\alpha}{\{B\}}^{\beta}...}
x^{i}_{k}=T_{ii}x^{i}_{k-1}+\sum_{j=M+1}^{M+N}T_{ij}Q^{\lambda^j}_k+\mu c_i, \lambda^{j}_{k}=\lambda^j_{k-1}+\sum_{i=1}^MT_{ji}Q^{x^i}_k-\mu b_j
H_{-}\chi ( r , \theta ) = \left( \begin{array}{c c}{{m}}&{{i e^{i \theta}\left( \partial_{r}+ \frac{i}{r}\partial_{\theta}+ \frac{e B}{2}r \right)}}\\{{i e^{- i \theta}\left( \partial_{r}- \frac{i}{r}\partial_{\theta}- \frac{e B}{2}r \right)}}&{{- m}}\\ \end{array}\right) \chi ( r , \theta ) = E \chi ( r , \theta ) ,
\tilde{\zeta}_N^{R\prime}(t) = \left( \zeta_N\left( 1 \right) \,,\, \delta\!\left( \sqrt{\tfrac{2\pi}{6}}\, \tilde{r}_0\tilde{\alpha}_0 e^{\pi i\left( \frac{t}{4}+ \frac{1}{4}-\frac{\eta_N}{6}+ \frac{k_N}{3}\right)}\right) \right) \ .
H_\nu(I+ H_\nu)^{-1}(I+\varepsilon H)^{-n}=\Gamma(n)^{-1}\int^\infty_0ds\int^\infty_0dt\,e^{-(s+t)}s^{-1+n}S_{\varepsilon s}(I-S_{\nu^{-1}t})\;.
\begin{cases}dY_t^{i,k}=-\frac{\partial H^i(t,X_t,\gamma_t,Y^i_t,Q^i_t,R^i_t)}{\partial x^k}\,dt+\sum_{j=1}^nQ^{i,k,j}_t \,dW^j_t+\sum_{j=1}^n R^{i,k,j}_{t}\,d\widetilde N^j_t Y^{i,k}_T=\frac{\partial g^i}{\partial x^k}(X_T)\,,\end{cases}
{3\over b^2}{a^{\prime}\over a}\left({a^{\prime}\over a}+{n^{\prime}\over n}\right)-{3\over n^2}\left[{\dot{a}\over a}\left({\dot{a}\over a}-{\dot{n}\over n}\right)+{\ddot{a}\over a}\right]-{{3k}\over a^2}=\kappa^2_5\left[{4\over 3}n^{-2}\dot{\phi}^2+{4\over 3}b^{-2}\phi^{\prime\,2}-e^{-2\alpha\phi}\Lambda\right],
\bigg|\frac{E[L^2]}{2E[L]}+\frac{1}{2}\log_2{\delta}-\frac{1}{2}h(X)\bigg| < \bigg|\frac{E[L^2]}{2E[L]}-\frac{1}{2}{H[Q_{{\mathrm{uni}}}(X)]}\bigg| + \bigg|\frac{1}{2}{H[Q_{{\mathrm{uni}}}(X)]}+\frac{1}{2}\log_2{\delta}-\frac{1}{2}h(X)\bigg| < \frac{\epsilon}{2}+\frac{\epsilon}{2}= \epsilon.
\gamma_{x,v}^\lambda(t) = x*\left(\cos\theta\frac{\sin(\lambda t)}{\lambda}+ \sin\theta \frac{1-\cos(\lambda t)}{\lambda}, -\cos\theta \frac{1-\cos(\lambda t)}{\lambda}+ \sin\theta \frac{\sin(\lambda t)}{\lambda}, - \frac{\lambda t - \sin(\lambda t)}{\lambda^2}\right).
\frac{1}{4\xi}( \partial_\mu A_\mu^-)^2 \ + \{\bf u}D_\mu^- \partial_\mu \bar{\bf v}\ + frac{1}{4\alpha}( D_\mu^- \Phi_\mu )^2\ + \{\bf c}D_\mu^+ D_\mu^- \bar{\bf b}\ + bar{\bf b}D_\mu^+ D_\mu^-{\bf u}
\frac{d n_{i}}{d \rho_{i}}= \frac{1}{1 6 \pi G_{1 1}T_{2}}\int F_{\rho_{i}\phi_{i}\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}^{( 7 )}d \phi_{i}d^{5}\alpha = 8 N^{2}\frac{q_{i}}{L_{7}^{6}}\rho_{i}\, .
t^{a}\partial_{a}= \partial_{+}+ \frac{\partial x_{A}^{-}}{\partial x^{+}}\partial_{-}= \partial_{+}+ \left( \frac{1}{( 2 \lambda x^{+})^{2}}-{\frac{1}{\lambda^{2}}}\widetilde{T}_{+ +}\right) \partial_{-}
\lim_{d\rightarrow \infty}\frac{\sum_{i=1}^d |t_i|^r}{\left( \sum_{i=1}^d t_i^2 \right)^{r/2}}= 0 \Rightarrow \lim_{d\rightarrow \infty}\frac{\sum_{i=1, i \neq k}^d |t_i|^r}{\left( \sum_{i=1, i \neq k}^d t_i^2 \right)^{r/2}}= 0.
\mathbb F^0_{p,q}:=\mathbb F_{p,q}\cap\left(\mathbb C\times\{0\}^{\sigma(1)-1}\times\mathbb C\times\{0\}^{m-\sigma(1)}\right), \mathbb F^0_{\tilde p,q_{\sigma}/r}:=\mathbb F_{\tilde p,q_{\sigma}/r}\cap\left(\mathbb C^2\times\{0\}^{m-1}\right).
\hat{\Omega}= i \hbar \Delta_{\psi}+{\cal J}_{i}\frac{\delta}{\delta \varphi_{i}^{*}}+ L_{m}\sigma_{, A}^{m}\bigg ( \frac{\delta{\cal W}}{\delta{\cal J}}+ \frac{\hbar}{i}\frac{\delta}{\delta{\cal J}}, \psi \bigg ) \frac{\delta}{\delta \phi_{A}^{*}}.
\Delta_N= \big(L_1^{1/2}\oplus L_1^{-1/2}\big)\otimes\cdots \otimes\big(L_{N/2}^{1/2}\oplus L_{N/2}^{-1/2}\big) \hphantom{\Delta_N}{}= L_1^{1/2}\otimes\cdots\otimes L_{N/2}^{1/2}\oplus\cdots\oplus L_1^{-1/2}\otimes\cdots\otimes L_{N/2}^{-1/2},
\left(f^{(k)}_{j}\right)^{\star n}=\left(U_{k}\over V^{2}_{k}-4q^{k}\right)^{n-1}\sum_{s=0}^{n-1}(-q^{k})^{n-s-1}\beta_{n,s}(j+n)V_{k}^{s}f_{s+j}^{(k)},