harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e T_{\mu \nu}= e \Theta_{\mu \nu}+ \frac{1}{2}g_{\mu \nu}\left({\tilde{\overline{{\Psi}}}}_{L_{0}}\frac{\delta{\tilde{S}}_{F}}{\delta{\tilde{\overline{{\Psi}}}}_{L_{0}}}+{\tilde{\Psi}}_{L_{0}}\frac{\delta{\tilde{S}}_{F}}{\delta{\tilde{\Psi}}_{L_{0}}}\right) .
\tilde{H}= ( 2 \pi \alpha^{\prime}) d \tilde{\cal F}- ( 2 \pi \alpha^{\prime}) G^{( 6 )}{\cal G}^{( 1 )}+ 3 ( 2 \pi \alpha^{\prime})^{2}{\cal H}^{( 3 )}d{\cal H}^{( 3 )}+{\textstyle \frac{m}{2}}{\cal G}^{( 7 )}\, .
P_{\lambda}(z_1,\ldots ,z_n,z;q,t,\mathbf{t}) = & \sum_{\substack{\mu\subset m^n, \, 0\leq r\leq m n^m- \mu^\prime \sim_{r}(n+1)^m-\lambda^\prime}}(-1)^{m-r+|\lambda|-|\mu|}C^{(n+1)^m-\lambda^\prime,m}_{n^m-\mu^\prime ,r}(t,q,\mathbf{t}) P_{\mu}(z_1,\ldots ,z_n;q,t,\mathbf{t}) \langle z;t_0\rangle_{q,m-r}.
\beta =i\left(a_0+\sum_{j=1}^{2^{n-2}-1}a_j(\alpha^j+\alpha^{-j})\right)= &=a_0\cdot i+\sum_{j=1}^{2^{n-2}-1}a_j(\alpha^{j+2^{n-2}}+\alpha^{-j+2^{n-2}}).
g^{( 4 )}\simeq \Lambda^{2}\, ( \zeta^{2}\, d T^{2}+ d \zeta^{2}+ d \rho^{2}) + \Lambda^{- 2}\, \rho^{2}\, d \phi^{2}, \qquad \Lambda \simeq 1 + \frac{B^{2}\rho^{2}}{4},
v_\ell(q^n + 1) = v_\ell(q^{2n}- 1) - v_\ell(q^n - 1) = v_\ell(q^{2n}- 1) = v_\ell(q^e - 1) + v_\ell(2n/e) = v_\ell(q^e - 1) + v_\ell(n) = w_n.
\left< L ( \gamma ) \frac{1}{2 e^{2}}\int t r F^{2}d x \right> = - \frac{\epsilon}{e^{2}}\left< t r ( \nabla_{\mu}F_{\mu \nu}( \gamma ( 0 ) ) \dot{\gamma}_{\nu}( 0 ) U ( \gamma ) ) \right> = - \frac{\epsilon}{e^{2}}\frac{\partial^{2}}{\partial \gamma_{\mu}^{2}( 0 )}W ( \gamma )
0\le u(x) \le \left(\theta\,\omega_N^{-\frac{\theta}{q}}\,\int_0^\infty t^{\theta-1}\,\mu_u(t)^\frac{\theta}{q}\,dt\right)^\frac{1}{\theta}\, |x|^{-\frac{N}{q}},\mbox{if}\theta<\infty, 0\le u(x) \le \left(\omega_N^{-\frac{1}{q}}\,\sup_{t>0}t\,\mu_u(t)^\frac{1}{q}\right)\,|x|^{-\frac{N}{q}},\mbox{if}\theta=\infty.
(C_{\mu})^{\sigma}_{\rho}(C_{\nu})^{\rho}_{\lambda}+ \Big[(C_{\mu})^{\sigma}_{\rho}(\tilde{C}_{\nu})^{\rho}_{\lambda}-(\tilde{C}_{\nu})^{\sigma}_{\rho}(C_{\mu})^{\rho}_{\lambda}\Big]= C_{\mu \nu}^{\rho}(C_{\rho})^{\sigma}_{\lambda}.
B_{\Gamma,p}^{(\ell)}(z) = \sum_{j=1}^{p}(-1)^{j-1}\sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}\sum_{\beta=0}^{p-j-2t}2^{p-j-2t-\beta}\binom{n-\ell}{\beta}\binom{\ell}{p-j-2t-\beta}\sum_{g=0}^{p-1}\sum_{i=0}^{j-1}\binom{\beta}{p+i-j-t-g}z^{2g-p}\;\Upsilon_{p-1-2g}^{(\ell)}(z).
P = |A|^4 -2 |A^2|^2 + \sum_{i,j,k,\ell}\left\{2\, \langle A_{j \ell}, A_{ik}\rangle \, \langle A_{\ell k}, A_{ij}\rangle \, - \langle A_{ij}, A_{k\ell}\rangle^2 \right\}= \left( \sum_{i}\lambda_i^2 \right)^2 -2 \sum_i \lambda_i^4 + 2 \sum_{i}\lambda_i^4 - \sum_{i,k}\lambda_i^2 \lambda_k^2 = 0 \, .
\Theta(p,N;a) = \inf \left\{\liminf_{s\nearrow 1}(1-s)\,[v_s]_{W^{s,p}(Q)}^p: \begin{array}{c}v_s \to \Psi_a L^p(Q) v_s-\Psi_a\in W^{s,p}_0(Q) \end{array}\right\}.
\begin{aligned}[b] g(u(x)+\rho v(x)) - g(u(x)) = \frac{1}{2}( (u_i + u_{i+1})( u(x)+ \rho v(x)) - u_i u_{i+1}) \MoveEqLeft[-1]- \frac{1}{2}( (u_i + u_{i+1})u(x) - u_i u_{i+1}) = \frac{\rho}{2}(u_i + u_{i+1}) v(x). \end{aligned}
|D_t^{(0)}+D_s^{(\ell)}|^2 \ge \big(|D_t|-|D_s|\big)^2 + 2|D_t|\cdot|D_s|\theta^2 &\ge\beta^{1/2}N^{1/2}\left[\big(t^{3/4}-s^{3/4}\big)^2 + 2t^{3/4}s^{3/4}\theta^2\right] .
\delta (u\otimes v)_{st}= \int_s^t \big(\dot{\mu}_r \otimes v_r + u_r \otimes \dot{\nu}_r\big) dr + \Gamma_{st}^1 (u \otimes v)_s + \Gamma_{st}^2 (u \otimes v)_s + (u\otimes v)_{st}^{\natural}
\frac{\partial}{\partial \phi}\Re{f\Big(\frac{1}{\tau}+ \frac{R}{\eta_{-}}e^{i\phi}\Big)}=- \Big(x- \frac{1}{1+R^2+ 2R \cos\phi}\Big) R \sin\phi < -\Big(x- \frac{1}{(R-1)^2}\Big) R \sin\phi <0.
\mathbf{f}= (k_s + 1) \mathbf{e}_2 - (k_s + 1) \mathbf{e}_2(0) + \nu \mathbf{\dot{\nu}}= (k_s + 1) \alpha_2 \mathbf{e}_2 + \beta \mathrm{sign}(\mathbf{e}_2) \mathbf{e}_2 = \mathbf{\dot{e}}_1 + \alpha_1 (\mathbf{p}_d - \mathbf{p}) \mathbf{u}= k_p \mathbf{q}_d + k_i \int \mathbf{q}_d dt + k_d \mathbf{\dot{q}_d}
\sum_{K\in\mathcal{T}_h}\sum_{i=1}^3(L_2+L_5)n_i \leq Ch\sum_{K\in\mathcal{T}_h}\Big|\Big|\nabla\Big(\frac{\partial^2 u}{\partial x_2^2}\Big|_{F_2}-\frac{\partial^2 u}{\partial x_2^2}\Big|_{F_5}\Big)\Big|\Big|_{L^2(K)}|w_h|_h &\leq Ch^2|u|_{H^4(\Omega)}|w_h|_h.
|F(z,s)| \leq |\varphi(s)| y^{1-\sigma}+ \sum_{\substack{\gamma \in \Gamma_{\infty}\backslash \Gamma \gamma \neq 1}}((\gamma z))^\sigma= |\varphi(s)| y^{1-\sigma}+ (E(z,\sigma) - y^{\sigma}).
V_{\theta}= \frac{i}{4}x_{i}^{\ast}\dot{\psi}_{i}- \frac{i}{4}e A_{i}^{\ast}\psi_{i}+ \frac{i}{4}e x_{i}^{\ast}\partial_{j}A_{i}\psi_{j}+ \frac{i}{4}( R \chi^{\ast}- i \dot{\xi}\chi^{\ast})
\mathrm{Hom}_{G}(\mathrm{Ind}_{H}^G(1) \otimes \varphi,\mathrm{Ind}_{H}^G(1) \otimes \varphi) \cong\mathrm{Hom}_{G}(\mathrm{Ind}_{H}^G(1) \otimes \varphi,\mathrm{Ind}_{H}^G(\varphi|_H)) &\cong\mathrm{Hom}_{H}(\oplus_{i=1}^n \varphi|_H^{\mathrm{deg}(\rho_i)^2},\varphi|_H)
N(m)=\frac{1}{2^{m+1}}\sum_{i=0}^{m}(-1)^{i}\sum_{a_{1},\cdots,a_{i+1}=1}^{\infty}\delta_{a_{1}+\cdots+a_{i+1},m+1}\prod_{j=1}^{i+1}\frac{(2a_{j})!}{a_{j}!}.
\tilde{J}_{k|k+\Delta,\infty}= \lim_{k\rightarrow\infty,\Delta\rightarrow\infty}J_{k|k+\Delta,\infty}, \tilde{J}_{k|k+\Delta,q}= \lim_{k\rightarrow\infty,\Delta\rightarrow\infty}J_{k|k+\Delta,q}.
\hat{M}_{1}(\lambda)=\left[\begin{array}{c}{\rm\bf diag}\!\left\{\! \lambda I_{m_{{\rm\bf x}i}}-A_{\rm\bf TT}(i)|_{i=1}^{N}\right\}{\rm\bf diag}\!\left\{\! C_{\rm\bf T}(i)|_{i=1}^{N}\right\}{\rm\bf diag}\!\left\{\! \Theta(i) A_{\rm\bf ST}(i)|_{i=1}^{N}\right\}\end{array}\right]
\|x_0\|_1\geq\|\hat{x}\|_1 =\|x_0+\eta\|_1 =\|x_{0,T}+x_{0,T^c}+\eta_T+\eta_{T^c}\|_1 \geq\|x_{0,T}\|_1-\|x_{0,T^c}\|_1+\|\eta_{T^c}\|_1-\|\eta_T\|_1.
\sum_{\tau_m}\left[{\tau_{m-1}+\tau_m-1 \atop \tau_m}\right]_q\left[{k_{m-1}-\tau_{m-1}+k_m-\tau_m-1 \atop k_m-\tau_m}\right]_qq^{\tau_{m-1}(k_m-\tau_m)}=\left[{k_{m-1}+k_m-1 \atop k_m}\right]_q
\mathop{\inf}\limits_{x\in X}\left\{{F\left({x,{0_{Y}}}\right)-\left\langle{{x^{\ast}},x}\right\rangle}\right\}=\mathop{\sup}\limits_{{y^{\ast}}\in{Y^{\ast}}}-{F^{\ast}}\left({{x^{\ast}},{y^{\ast}}}\right) ,
\frac{d}{dt}\Big[\sum_{k=0}^{n-1}\frac{(2t)^{k+1}}{(k+1)!}\|A^{\frac{k+1}{2}}w(t)\|^2\Big] =2\sum_{k=0}^{n-1}\left(\frac{(2t)^{k}}{k!}\|A^{\frac{k+1}{2}}w(t)\|^2-\frac{(2t)^{k+1}}{(k+1)!}\|A^{\frac{k+2}{2}}w(t)\|^2\right) &=\|A^{1/2}w(t)\|^2-\frac{(2t)^{n}}{n!}\|A^{\frac{n+1}{2}}w(t)\|^2.
\sum_{S\in \binom{U}{k-1}}\deg(S, V') \le |C|\binom{|U|}{k-1}+ |V'\setminus C| (2k-b)^2\binom{|U|}{k-2}&< |C|\binom{|U|}{k-1}+ (2k-b)^2 n\binom{|U|}{k-2}|V'\setminus C|< n &=\binom{|U|}{k-1}\left(|C| +\frac{(2k - b)^2 n (k-1)}{|U|-k+2}\right).
\mathbb{E}(\exp(\theta \sigma^2(s))) = \exp(K_{L^{\sigma}}(\theta)), \,\,\,\, \, K_{L^{\sigma}}(\theta) =\int_{\mathcal{X}^{\sigma}\times \mathbb{R}^d}\int_0^{\infty}\left(e^{\theta h(x,z) w}-1 \right)\nu^{\sigma}(dw)p^{\sigma}(dx)dz,
(w(s_{2}),\varphi(s_{2}))+\int_{s_{1}}^{s_{2}}[(\nabla w,\nabla\varphi)+(w\cdot\nabla w+w\cdot\nabla u^{c,\gamma}+u^{c,\gamma}\cdot\nabla w,\varphi)] &=(w(s_{1}),\varphi(s_{1}))+\int^{s_{2}}_{s_{1}}(w,\partial_{t}\varphi)dt
\varphi(U^{n+1})= \varphi(U^n)+\tau\left[D\varphi(U^n)\Phi^n+\frac12\tau D^2\varphi(U^n)(\Phi^n)^2\right]+\frac16D^3\varphi(U^n)(\tau\Phi^n)^3+R^\Phi_n =: \varphi(U^n)+\tau\mathcal{L}^\Phi\varphi(U^n)+\frac16D^3\varphi(U^n)(\tau\Phi^n)^3+R^\Phi_n,
c(\mathbf{H},q)^{2}= \left( \frac{\beta \left( \frac{1}{2}- \frac{1-H_{1}}{q},\frac{2(1-H_{1})}{q}\right)^{q}}{H_{1}(2H_{1}-1)}\right)^{-1}\ldots \left( \frac{\beta \left( \frac{1}{2}- \frac{1-H_{d}}{q},\frac{2(1-H_{d})}{q}\right)^{q}}{H_{d}(2H_{d}-1)}\right)^{-1}
|F_2+F_5+F_6| = |V_2(z_{-N},x_{-N+1})-V_2(x_{-N},x_{-N+1}|\cdot |V_2(z_{-N},x_{-N+1})+V_1(x_{-N+1},x_{-N+2})| \leq \frac{\kappa_2}{4}|x_{-N}-z_{-N}|.
\zeta_2(z\pm 2\ell)=\zeta_2(z)\pm\frac{\pi}{\delta},\wp_2(z\pm 2\ell)=\wp(z), f_2(z\pm 2\ell)=f_2(z)\pm \frac{2\pi}{\delta}\zeta_2(z)+\bigg(\frac{\pi}{\delta}\bigg)^2
\begin{pmatrix}a b c d\end{pmatrix}\stackrel{.}{\otimes}\begin{pmatrix}e f g h\end{pmatrix}:=\begin{pmatrix}a \otimes e + b \otimes g a \otimes f + b \otimes h c \otimes e + d \otimes g c \otimes f + d \otimes h\end{pmatrix}.
\left\langle \varphi_v ( \operatorname{Ind}^G v (f)) \cdot \xi \;,\; \eta \right\rangle_{L^2(H_v, (\lambda_H)_v)}= \int_{x \in H_v}\int_{t \in H_v}f(xt^{-1}) \cdot \xi(t) \cdot \overline{\eta(x)}\, d\lambda_v(t) \, d\lambda_v(x) = 0 \;
X(t) = X(0) + \sum_{k=1}^K \zeta_k \int_0^t 1_{\left[ \frac{q_{k-1}(s-)}{\lambda_0(X(s-))}, \frac{q_{k}(s-)}{\lambda_0(X(s-))}\right)}( \xi_{R_0(s-)})dR_0(s) R_0(s) = Y\big( \int_0^t \lambda_0(X(s)) ds\big),
i\mathcal{G}_{\alpha \beta}(x,y)=\sum_{m=0}^{\infty}\left(-\frac{i}{\hbar}\right)^{m}\frac{1}{m!}\int_{-\infty}^{\infty}dt_{1}\cdots\int_{-\infty}^{\infty}dt_{m}\langle\phi_{0}|T[\hat{H}_{1}(t_{1})\cdots\hat{H}_{1}(t_{m})\hat{\psi}_{\mathrm{H}\alpha}(x)\hat{\psi}_{\mathrm{H}\beta}^{\dagger}(y)]|\phi_{0}\rangle_{\mathrm{connected}}.
{\mathcal A}(M_{j_1 i_1}^{(t)}v_{i_1}, \ldots, M_{j_k i_k}^{(t)}v_{i_k}) = t^{-k}{\mathcal A}(M_{j_1 i_1}v_{i_1},\ldots, M_{j_k i_k}v_{i_k})
\int\limits_0^\infty{f\left( x \right)}\left[{g\left({nx}\right) + g\left({- nx}\right)}\right]dx = 2\sum\limits_{k = 0}^\infty{\psi \left({2k}\right)\phi \left({- 2k - 1}\right){{\left({-{n^2}}\right)}^k}},
\begin{gathered}\phi_k := \begin{bmatrix}-y_{k-1}-y_{k-2}u_{k-1}u_{k-2}\end{bmatrix}^\top, \theta^*_1 := \begin{bmatrix}0.5 -0.1 1 -0.4 \end{bmatrix}^\top, \theta^*_2 := \begin{bmatrix}-1.4 0.3 1 -1.3 \end{bmatrix}^\top,\end{gathered}
m(\sigma,I)=\dim\mathrm{Hom}_{K}(\sigma,I|_{K}) =\dim\mathrm{Hom}_{K}(\sigma,\mathrm{Ind}_{K\cap MA}^{K}(\det\otimes\chi_{-1,\beta})) =\dim\mathrm{Hom}_{K\cap MA}(\sigma,\det\otimes\chi_{-1,\beta}),
\frac{11v_{m}^{n+1}-18v_{m}^{n}+9v_{m}^{n-1}-2v_{m}^{n-2}}{6k}+a(1+\frac{h^{2}}{6}\delta^{2})^{-1}\delta_{0}v_{m}^{n+1}=f_{m}^{n+1}.
{\tilde \beta}_{\mu\nu}^G = R_{\mu\nu}+ 2\gamma \nabla_\mu \partial_\nu \Phi,\qquad{\tilde \beta}^\Phi = -R + 4\gamma^2 (\nabla_\mu \Phi)(\nabla_\nu \Phi)G^{\mu\nu}- 4\gamma \nabla^2 \Phi + Q^2
\Phi_{\lambda_{n}}\left( x\right) :=\left( \begin{array}{c}\phi_{\lambda_{n}}\left( x\right) R_{b}^{\prime}\left( \phi_{\lambda_{n}}\right) R_{c_{1}}\left( \phi_{\lambda_{n}}\right) R_{c_{2}}\left( \phi_{\lambda_{n}}\right)\end{array}\right)
\mathbb{E}\left[ \cos(W_{(k-1)L+1}) \right]= \Re\left[ \mathbb{E}\left[ e^{j W_{(k-1)L+1}}\right] \right]= e^{- \sigma^2_{W}/2}&\mathbb{E}\left[ \sin(W_{(k-1)L+1}) \right]= \Im\left[ \mathbb{E}\left[ e^{j W_{(k-1)L+1}}\right] \right]= 0
h(R,\omega) = \begin{cases}cR^{\gamma_1^+}w_1(\omega) + O(R^{\gamma_1^+-\epsilon})\ \mu_1 > (\frac{n-2}{2})^2 cR^{\gamma_1^+}\log R\w_1(\omega) + O(R^{\gamma_1^+-\epsilon})\ \mu_1 = (\frac{n-2}{2})^2 \end{cases}
A_2 =\{1,3,5,7\}, q(2,1) =d_1^{-1}d_{16-1}=d_1^{-1}d_{15}, q(2,3) =d_3^{-1}d_{16-3}=d_3^{-1}d_{13}, q(2,5) =d_5^{-1}d_{16-5}=d_5^{-1}d_{11}, q(2,7) =d_7^{-1}d_{16-7}=d_7^{-1}d_9;
\frac{1}{a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}}= \frac{\Gamma [ \alpha_{1}+ \alpha_{2}]}{\Gamma [ \alpha_{1}] \Gamma [ \alpha_{2}]}\int_{0}^{1}d x \, \frac{x^{\alpha_{1}- 1}( 1 - x )^{\alpha_{2}- 1}}{[ x a_{1}+ ( 1 - x ) a_{2}]^{\alpha_{1}+ \alpha_{2}}}
\begin{aligned}\int_{v_0}^{v_1}e^{-v^2}v^k \, dv = \frac{1}{2}\left( \Gamma\left(\frac{1+k}{2}\right)-\Gamma\left(\frac{1+k}{2},v_1^2\right)\right)^{1+k}(v_1) &\phantom{=}-\frac{1}{2}\left( \Gamma\left(\frac{1+k}{2}\right)-\Gamma\left(\frac{1+k}{2},v_0^2\right)\right)^{1+k}(v_0)\end{aligned}
\Pi_{i}( p_{i}^{2}- m_{i}^{2}) F T \frac{\delta^{4}}{\delta J^{4}}\begin{array}{c}{l i m}\\{\varepsilon \rightarrow 0}\\ \end{array}< < [ i \delta \alpha \int d^{4}x \phi \partial_{0}^{2}\phi ] > > \mid_{_{J = 0 ; \alpha \rightarrow \alpha_{0}^{-}}}
\left. + \frac{3 g f^{\prime}\sin f}{2 \left( E - g \cos f \right)}\frac{( 1 - u )}{\sqrt{1 - u^{2}}}- \frac{g^{2}f^{\prime}\sin^{2}f}{E - g \cos f}+ g f^{\prime}\cos f \right){\tilde{G}}( u ) = 0 .
\sum_n \frac{e^{in(\phi-\phi')}}{2\pi}\frac{\Gamma(\frac{|n|}{\alpha}+1 -\frac{i c t}{l_P})}{\Gamma(\frac{|n|}{\alpha}+1)}\rho^{\frac{|n|}{\alpha}+1}{_2F_1}(\frac{|n|}{\alpha}+1 - \frac{ict}{l_P};\frac{|n|}{\alpha}+\frac{1}{2}; 2 \frac{|n|}{\alpha}+1; 4 \rho)
\sum_{n=0}^{\infty}p_{-2}\left(5^{2j}n+\dfrac{11\times5^{2j}+1}{12}\right)q^{n}=\dfrac{1}{q^{3}E_{25}^{2}}\sum_{l=1}^{\infty}a(2j,l)T^{l}\zeta^{-(6l+2)}.
\begin{aligned}\frac{1}{(1-q)^{r}}\prod_{z=1}^{r}\frac{(1-t_z^2)^2}{(1-q^{2m}t_z) (1-q^{-2m}t_z)}\times &\prod_{1 \leq i < j \leq r}\frac{(1-t_i t_j)^2(1-t_i t_j^{-1})^2}{(1-q^2 t_i t_j) (1-q^{-2}t_i t_j) (1-q^2 t_i t_j^{-1}) (1-q^{-2}t_i t_j^{-1})}.\end{aligned}
\big [ M_{\mu \nu}, Q_{\alpha}\big ] = -{\frac{1}{2}}( \sigma_{\mu \nu})_{\alpha}{}^{\beta}Q_{\beta}, \qquad \big [ M_{\mu \nu}, \bar{Q}_{\dot{\alpha}}\big ] ={\frac{1}{2}}( \tilde{\sigma}_{\mu \nu})^{\dot{\beta}}{}_{\dot{\alpha}}\bar{Q}_{\dot{\beta}},
\Re \xi(z) =\Re \left( \int_{e^{i\theta}}^{\tau e^{i \theta}}\frac{\rho_{\alpha, \varepsilon}( \frac{1}{2}(w+\frac{1}{w})) - \rho_{\alpha, \varepsilon}(\cos \theta)}{w}dw \right) + \rho_{\alpha, \varepsilon}(\cos \theta) \log \tau.
a^{4231}=a^{4321},a_{1}^{4231}=a_{1}^{4321},a_{2}^{4231}=a_{2}^{4321},a_{3}^{4231}=a_{3}^{4321},a_{4}^{4231}=a_{4}^{4321}, a_{12}^{4231}=a_{12}^{4321},a_{13}^{4231}=a_{13}^{4321},a_{23}^{4231}=a_{23}^{4321},a_{24}^{4231}=a_{24}^{4321},a_{34}^{4231}=a_{34}^{4321}, a_{123}^{4231}=a_{123}^{4321},a_{234}^{4231}=a_{234}^{4321}
{\cal H}_{1 . c_{0}\neq 0}= \frac{s}{2 c_{1}^{2}c_{0}}\left[ 2 \imath c_{1}^{2}c_{0}{\bf p}({\bf z}-{\bf{\bar{z}}}) + ( c_{0}^{2}-{\bf p}^{2}) ({\bf z}{\bar{\bf z}}- c_{1}^{2}) \right] .
{\frac{1}{2}}m^{b a}e_{b}^{m}e_{a}^{n}E_{m}^{\underline{{\beta}}}\left[ E_{n}^{\underline{{a}}}\Gamma_{\underline{{a}}}( 1 - \bar{\Gamma}) \right]_{\underline{{\beta}}\underline{{\alpha}}}= 0 ,
\int_0^\infty \frac{dt}{t}\, t^\beta \cdot 2^{\gamma-2\zeta}\Gamma \left(\frac{\gamma+1}{2}\right) t^{\frac{\gamma-2\zeta}{2}}\,_1\tilde{F}_1\left(\frac{2\zeta-\gamma}{2};\zeta+\frac12;-\frac{1}{4t}\right) = C(\beta,\gamma,\zeta),
\lim_{r\to 0}\{b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7}\}= \{3,3,\frac{15}{2},3,\frac{9}{2},9,9\}\lim_{r\to \infty}\{b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7}\}= \{0,9,9,\frac{3}{2},\frac{9}{2},6,9\}
\sum_{k=0}^{(n+1)/2}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k\equiv0\pmod{\Phi_n(q)}.
\theta_{4}[exp(-\sqrt{2}\pi)]=\theta_{4}[q(\sqrt{2}-1)]=\sqrt[4]{2\sqrt{2}-2}\sqrt{2\pi^{-1}K(\sqrt{2}-1)}=2^{-1/4}cos(\frac{1}{8}\pi)^{1/2}\pi^{-1/2}\beta(\frac{3}{8})^{1/2}
|\mathcal{B}_p\rangle = \exp \left( \sum_{n = 1}^{\infty}\, \mp\frac{1}{n}\alpha_{-n}\cdot \tilde{\alpha}_{-n}- b_{-m}\tilde{c}_{-m}- \tilde{b}_{-m}c_{-m}\right) \int d^{25-p}q_{\bot}e^{-i q_{\bot}\cdot y_{\bot}}|q_{\bot},q_{\parallel}=0\rangle,
{S}_1\hookrightarrow{S}_2\hookrightarrow \cdots \hookrightarrow{S}_{k-1}\hookrightarrow{S}_k={\pi}^*(E) \stackrel{q_{1}}\twoheadrightarrow{Q}_1 \stackrel{q_{2}}\twoheadrightarrow{Q}_2 \stackrel{q_{3}}\twoheadrightarrow \cdots \stackrel{q_{k}}\twoheadrightarrow{Q}_k=0\,,
\int_{|M| \le 3 |E|}|M+E|^K \le \int_{|M| \le 3 |E|}(|M|+|E|)^K dU \le \int_{|M| \le 3 |E|}(4|E|)^K dU \ll_K \int_{U(N)}|E|^K dU = \mathbb E |E|^{K}= o\left( \mathbb E |M|^{K}\right)
\|f_{i}-f\|_{\tau(n)}=\|f_{i}-f\|_{1}+\|f_{i}\tau-f\tau\|_{1}+\cdots+\|f_{i}\tau^{n}-f\tau^{n}\|_{1}&=\|f_{i}-f\|_{1}+\|f_{i}\tau-g_{1}\|_{1}+\cdots+\|f_{i}\tau^{n}-g_{n}\|_{1}\longrightarrow 0.
\Biggr\langle n^{A_1}(x_1) \cdots n^{A_n}(x_n) \exp \left[ i \oint_C \omega \right]\Biggr\rangle_{TQFT_4}= \Biggr\langle n^{A_1}(x_1) \cdots n^{A_n}(x_n) \exp \left[ i \oint_C \omega \right]\Biggr\rangle_{NLSM_2},
\frac{f(\phi_{R^{-1}(p_{i})t}(x_i))}{|R^{-1}(p_{i})t|}= \frac{f(\phi_{R^{-1}(p_{i})t}(x_i))-f(x_i)}{|R^{-1}(p_{i})t|}+\frac{f(x_i)}{|R^{-1}(p_{i})t|}\to R_{\max}(1+\frac{C_0}{|t|}),~{\rm as}~i\to \infty.
\begin{dcases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)=\rho(t,1) t\in (0,T) , u(t,0)=0, \ \u(t,1)=0 t\in (0,T) , \rho(0,x)= \rho_0(x), \u(0,x)= u_0(x) x\in(0,1), \end{dcases}
\sum_{k=D}^{N}\frac{k!}{(k-D)!}=D!\sum_{k=D}^{N}(\begin{matrix}k\\D\end{matrix})=D!\sum_{k=0}^{N-D}(\begin{matrix}k+D\\D\end{matrix})=D!(\begin{matrix}N\\D+1\end{matrix})=\frac{N!}{(D+1)(N-D-1)!}
\textbf{g}(\nabla_{\partial t_k}T_i,T_j)+\textbf{g}(T_i,\nabla_{\partial t_k}T_j) = \frac{1}{-z}\textbf{g}(T_k,T_i\bullet T_j ) + \frac{1}{z}\textbf{g}(T_i, T_j \bullet T_j) = 0 = \partial_{t_k}\textbf{g}(T_i, T_j)
[\nabla_{j},\nabla_{k}]=\frac{2\pi i}{(\det\mathcal{T})^{2}}(\mathcal{C}_{3j}\mathcal{C}_{1k}+\mathcal{C}_{4j}\mathcal{C}_{2k}-\mathcal{C}_{3k}\mathcal{C}_{1j}-\mathcal{C}_{4k}\mathcal{C}_{2j})\mathcal{I},
\frac{\ell_{a}^{\mu}( \sigma )}{\sqrt{2}}= \sum_{n = 0}^{\infty}\ell_{a , 2 n + 1}^{\mu}\cos ( 2 n + 1 ) \sigma , \quad \frac{r_{a}^{\mu}( \sigma )}{\sqrt{2}}= \sum_{n = 0}^{\infty}r_{a , 2 n + 1}^{\mu}\cos ( 2 n + 1 ) \sigma .
\Omega = G_{A_{0}}( \stackrel{( 1 , 0 )}{\eta^{A_{0}}}+ \stackrel{( 0 , 1 )}{\eta^{A_{0}}}) + ({\raise . 5 p t \mathrm{\stackrel{( 0 , - 1 )}{{\cal P}_{A_{0}}}}}-{\raise . 5 p t \mathrm{\stackrel{( - 1 , 0 )}{{\cal P}_{A_{0}}}}}) \stackrel{( 1 , 1 )}{\pi^{A_{0}}}+ \cdots
\|\mathcal{T}[F_1](t) - \mathcal{T}[F_2](t)\|_{L^\infty}= \|\mathcal{T}[F_1\!-\!F_2](t)\|_{L^\infty}begin{align*}5pt] \leq 0 + T_1\big|Q^\varepsilon[F_1\!-\!F_2]\big| begin{align*}1pt] \leq \frac{4T_1}{\varepsilon^2}\sup_{t\in [0,T_1]}\|F_1(t)-F_2(t)\|_{L^\infty},
\partial^{+ +}= u^{+ \alpha^{\prime}}{\frac{\partial}{\partial u^{- \alpha^{\prime}}}}\ : \ \ \, p a r t i a l^{+ +}u^{+ \alpha^{\prime}}= 0 \ , \ \ \partial^{+ +}u^{- \alpha^{\prime}}= u^{+ \alpha^{\prime}}\ .
\left| \left| w_{\dot{\alpha}}^{u}\right| \right| = \left| \left| \bar{w}_{u}^{\dot{\alpha}}\right| \right| = \rho T \left( \begin{matrix}{0_{[ N - 2 ] \times [ 2 ]}}\\{1_{[ 2 ] \times [ 2 ]}}\\ \end{matrix}\right)
p_a = \frac{1}{\sqrt{k}}\times\frac{1}{l_s}\left(\frac{n_a}{r_a}+w^ar_a\right) ~,~~~\bar{p}_a = \frac{1}{\sqrt{k}}\times\frac{1}{l_s}\left(\frac{n_a}{r_a}-w^ar_a\right) ~,
\setlength{\arraycolsep}{4pt}\def\arraystretch{1}\Omega_n = \left(\begin{matrix}0 \frac{\mathrm{d}z}{1-z}0 \frac{\mathrm{d}z}{z}0 0 \ddots \ddots 0 0 \frac{\mathrm{d}z}{z}0 \end{matrix}\right).
\begin{aligned}\|v^{k+1}-v^*\|^2_{H}= \|(I-M)(v^k-v^*)+M(\widetilde{v}^k-v^*)\|^2_H = \underbrace{\|(I-M)(v^k-v^*)\|^2_H}_{:=A^k}+\underbrace{\|M(\widetilde{v}^k-v^*)\|^2_H}_{:=B^k}&+2\underbrace{(v^k-v^*)^T(I-M)^THM(\widetilde{v}^k-v^*)}_{:=C^k}.\end{aligned}
\epsilon([m_t^{-1}]) = \epsilon([g_t^{-1}m^{-1}g_t]) &= \epsilon([g_t^{-1}m m^{-1}g_t]) + \epsilon([g_t^{-1}])\epsilon([m^{-1}g_t]) \qquad[\textrm{Skein relation}] &= \epsilon([e]) + \epsilon([g_t^{-1}])\epsilon([m^{-1}g_t]) = 0.
[ \langle \Phi^{2}( t ) \rangle - \langle \Phi^{2}( 0 ) \rangle ] = \int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{1}{2 \omega_{<}( k )}[{\cal{U}}_{k}^{+}( t ){\cal{U}}_{k}^{-}( t ) - 1 ] \coth [ \beta_{i}\omega_{<}( k ) / 2 ]
\int_{\Omega}\left(\sum_{i,j=1}^Na_{ij}\frac{\partial u_n}{\partial x_j}\frac{\partial v}{\partial x_i}+ \sum_{i=1}^N \left(b_i \frac{\partial u_n}{\partial x_i}v + c_i\frac{\partial v}{\partial x_i}u_n\right) +du_nv\right) + \int_{\Omega}(g_n\circ u_n)v = \int_{\Omega}fv,
\dot f={}\frac{d}{dt}\bigl(\tau L+N+\partial_{\dot q}L\cdot(\xi-\tau \dot q) \bigr)= ={}\dot \tau L+\tau \dot L+\dot N+ \Bigl(\frac{d}{dt}\partial_{\dot q}L\Bigr)\cdot(\xi-\tau \dot q)+ \partial_{\dot q}L\cdot (\dot\xi-\dot \tau \dot q-\tau \ddot q).
((\mathbb{I}-\mathbb{P})D^{\alpha}v^{\varepsilon}, D^{\alpha}v^{\varepsilon})_{0}= ((\mathbb{I}-\mathbb{P})D^{\alpha}v^{\varepsilon}, D^{\alpha}((\mathbb{I}-\mathbb{P})v^{\varepsilon}+\mathbb{P}v^{\varepsilon}))_{0}= ||(\mathbb{I}-\mathbb{P})v^{\varepsilon}||_{0}^{2}.
s_{2^{k-1}(2l+1)}\equiv s_{2^{n-2}-2^{k-1}(2l+1)}=r_{2^{k-1}(2l+1)}-s_{2^{k-1}(2l+1)}\equiv &\equiv r_{2^{k-1}(2l+1)}+s_{2^{k-1}(2l+1)}\pmod{2}.
R^e_n(s_{v_0}) (p_b^*(v_0)) =R^e_n(s_{v_0})\left( ew_0+\sum_{i=1}^{e-1}(e-i)(w_{-i}+w_i)\right) \left(\mbox{by Equation \ref{b*}}\right) &=-e\left(\sum_{i=-(e-1)}^{e-1}w_{i}\right)+\sum_{i=1}^{e-1}(e-i)(w_{e-i}+w_{-(e-i)})\left(\mbox{by Equations \ref{S.1}, \ref{S.3}}\right) &=-\left(ew_0+\sum_{i=1}^{e-1}(e-i)(w_{-i}+w_i)\right)=-p_b^*(v_0).
\int_{m_e}^\infty dk\ (k^2-m_e^2)^{1-s}\partial_k\frac{t^j}{m^n}=-\frac{m_e^{2-2s}}{2}\frac{\Gamma(2-s)\Gamma\left(\frac{1+j-n}{2}\right)\Gamma(s+\frac{n-3}{2})}{(Rm_e)^{n-1}\Gamma(j/2)}\ ,
\Delta_A = \langle c_{100}c_{111}c_{201}c_{210}- c_{101}c_{110}c_{200}c_{211}, c_{000}c_{011}c_{101}c_{110}- c_{001}c_{010}c_{100}c_{111}, c_{000}c_{011}c_{201}c_{210}- c_{001}c_{010}c_{200}c_{211}\rangle .
\int_{\mathbb{R}^{2N}}\frac{|w_k(x) - w_k(y)|^{p-2}\, (w_k(x) - w_k(y))\, (\varphi(x) - \varphi(y))}{|x - y|^{N+s\,p}}\, dx\, dy=\int_{\Omega}\frac{f_n(x)}{(u^+_k+1/n)^\gamma}\,\varphi\, dx.
\alpha = \cos ( \vec{J}_{1}, \vec{J}_{2}) = \cos ( \vec{k}\times \hat{n}_{1}, \vec{k}\times \hat{n}_{2}) = \frac{( \vec{k}\times \hat{n}_{1}) \cdot ( \vec{k}\times \hat{n}_{2})}{| \vec{k}\times \hat{n}_{1}| | \vec{k}\times n_{2}|}\nonumber
f(x) \, = \, \displaystyle{\min_{1 \leq i \leq I}}\, \left\{\, q_i(x) + \mbox{dist}_1(x;P^{\, i}) \, \displaystyle{\max_{1 \leq j \leq I}}\, \| \nabla q_j(x) - \nabla q_i(x) \|_1 + \displaystyle{\frac{3 \, L_i}{2}}\, \left[ \, \mbox{dist}_1(x;P^{\, i}) \, \right]^2 \, \right\}.
\ddot{q}_1+2D_1\omega_1\dot{q}_1+\omega_1^2q_1+\frac{\omega_1^2}{2}(3q_1^2+q_2^2)+\omega_2^2q_1q_2+\frac{\omega_1^2+\omega_2^2}{2}q_1(q_1^2+q_2^2) =F_1=f_1\cos(\Omega t), \ddot{q}_2+2D_2\omega_2\dot{q}_2+\omega_2^2q_2+\frac{\omega_2^2}{2}(3q_2^2+q_1^2)+\omega_1^2q_1q_2+\frac{\omega_1^2+\omega_2^2}{2}q_2(q_1^2+q_2^2) =0.
\sup_{\nu \in \mathcal{M}_\sigma}\Lambda(\mathsf{A},\nu) \leq \sup_{\nu \in \mathcal{M}_\sigma}(2\ell)^{-1}h(\nu)+ \Lambda(\mathsf{A},\nu) &=(2\ell)^{-1}P(\mathsf{A},2\ell) &=\Lambda(\mathsf{A},\mu)\leq \sup_{\nu \in \mathcal{M}_\sigma}\Lambda(\mathsf{A},\nu)
B:=\frac{\mathbb{C}[x_{\epsilon_1+\epsilon_2},x_{\epsilon_1-\epsilon_2}, x_{\epsilon_2},x_{\epsilon_1}]}{( x_{\epsilon_1+\epsilon_2}^2, x_{\epsilon_1-\epsilon_2}^2,x_{\epsilon_1}^2,x_{\epsilon_2}x_{\epsilon_1+\epsilon_2}, x_{\epsilon_1}x_{\epsilon_1-\epsilon_2}, x_{\epsilon_1}x_{\epsilon_1+\epsilon_2},x_{\epsilon_2}^3, x_{\epsilon_2}^2 x_{\epsilon_1})},
a_{2}=f\sum_{i=1}^{i_{3}}{\cal U}_{i}^{(2,2)}\; I_{i}^{(2)}+f_{;N}\sum_{i=1}^{i_{2}}{\cal U}_{i}^{(2,3/2)}\; I_{i}^{(3/2)}+f_{;NN}\sum_{i=1}^{i_{1}}{\cal U}_{i}^{(2,1)}\; I_{i}^{(1)}.
\Phi_x=\vert\varphi^{(0)}_x\vert, ~~\Phi_{x}(t)=\sqrt{\Phi_x^2+2\ell_x(t)}, ~\forall x\in V, J_e(\Phi(t))=W_e \Phi_{e_{-}}(t)\Phi_{e_{+}}(t), ~\forall e\in E.
\|M_{kn}M_{k_1 n_1}\|\leq \|M_{kn}N\| +\|N N_1\| +\|N_1 M_{k_1 n_1}\|\leq & \leq 4\Lambda_{n}+ 18\frac{1}{8}\Lambda_{n}+4\Lambda_{n_1}\leq \left(4+18\frac{1}{8}+4\cdot 8\right) < 55\Lambda_{n}
J_{2}( q ) \simeq \int_{1 / q}^{\infty}\! d x \, \ln \left( 1 \, + \, \frac{1}{x^{2}\, + \, 2 x}\right) \simeq 2 \, \ln 2 \, - \, \frac{1}{q}\, \ln \frac{q}{2}\,{.}
\pi(\omega'_f \otimes \omega'_g) = G(\chi_f)^{-1}G(\chi_g)^{-1}\alpha_f^- \alpha_g^{(-1)^n}e_f^- \otimes e_g^{(-1)^n}+ G(\chi_f)^{-1}G(\chi_g)^{-1}\alpha_f^+ \alpha_g^{(-1)^{n+1}}e_f^+ \otimes e_g^{(-1)^{n+1}}& = G(\chi_f)^{-1}G(\chi_g)^{-1}(2\pi i)^{1-n}(\alpha_f^+ \alpha_g^{(-1)^{n+1}}e_1 + \alpha_f^- \alpha_g^{(-1)^n}e_2).
\left(\int_{B}h^s w^{s-1}w\right)^\frac{1}{s}\le\left(\int_{B}h^{p'}w\right)^\frac{1}{p'}\left(\int_{B}w^{\frac{s(p'-1)}{p'-s}}\right)^{\frac1{s}-\frac{1}{p'}}\lesssim\ |B|^{\frac{1}{s}}\left(\int_{B}w^{1- p'}\right)^{-\frac{1}{p'}}\lesssim\left(\int_{B}w\right)^{\frac{1}{p}}|B|^{-\frac{1}{s'}},
\hat{\gamma}(\hat{y}) = e^{-i \omega z}\hat{f}(\hat{y}) = \left( \frac{\alpha+\hat{y}}{\alpha-\hat{y}}\right)^{\frac{-i \zeta \omega}{2 \alpha (\alpha^2-1)}}\left( \frac{1-\hat{y}}{1+\hat{y}}\right)^{\frac{-i \zeta \omega}{2 (\alpha^2-1)}}\hat{f}(\hat{y}),
\phi_3(c,r,p) = (c^3-2c-10)^2+2r(4-c^2)(c^3-2c-10)(2p+2cp-6crp^2+3rc) +r^2(4-c^2)^2(4c^2+4+9c^2r^2+8c-12c^2rp-12crp).
{\cal L}_{E}=-\frac{1}{2}\left(\frac{\partial \phi}{\partial \tau}\right)^{2}-\frac{1}{2}(\mbox{\boldmath $\nabla$}\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}+{\cal L}_{ct}+J\phi,
- \int_{\Omega_{FEM_T}}\frac{\partial \bar{\lambda}}{\partial t}\frac{\partial u_h}{\partial t}~ dxdt - \int_{\partial \Omega_{FEM}}\partial_n u_h \bar{\lambda}~ dxdt+ \int_{\Omega_{FEM_T}}( a_h \nabla u_h) (\nabla \bar{\lambda}) ~ dxdt = 0,