harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal L}_{\mathrm{S C S}}= - \frac{1}{2}\epsilon^{i j}\left( A_{i}^{( 0 )}+ \sum_{n = 1}^{\infty}A_{i}^{( n )}\right) \left( \dot{A}_{j}^{( 0 )}+ \sum_{n = 1}^{\infty}\dot{A}_{j}^{( n )}\right) + A_{0}^{( 0 )}\epsilon^{i j}\partial_{i}\left( A_{j}^{( 0 )}+ \sum_{n = 1}^{\infty}A_{j}^{( n )}\right)
F_{\gamma_{SCHCN}}(\gamma,\lambda_{eq}) =\sum_{i=1}^{N_c+1}\frac{\alpha_i^*}{\lambda_{eq}^{i}}\left[ 1-e^{-\lambda_{0}\gamma}\sum_{m=0}^{i-1}\frac{(\lambda_{0}\gamma)^m}{m!}\right] &+\sum_{j=1}^{N-N_c}\frac{\beta_j^*}{(1+\frac{j}{N_c})\lambda_{0}}(1-e^{-(1+\frac{j}{N_c})\lambda_{0}\gamma}),
\begin{cases}r\varphi_r=\lambda r^2+\frac{r^2-1}{r^2+1}\varphi, \forall r\geq0 \varphi(0,\theta)=\varphi_0(\theta), \ \ \varphi_r(0,\theta)=0, \forall \theta\in{\mathbb{S}}^{n-1}. \end{cases}
T_{1}\left(W',Z'\right)=\left(\frac{1}{I_{q'^{2}}+R\left(W'\right)}\otimes W', V\otimes Z' \right),\quad\mbox{for suitable $V=V\left(W',\overline{W'}\right)\in \mathcal{M}_{q'N'\times q'N'}\left(\mathbb{C}\right)$.}
\mathcal T(k) = - \frac{\widehat g(k)^2}{(1-\varepsilon_K) \mathcal D_k (1-\alpha_k^2) \vert \Lambda \vert^2}\sum_{p,s \in \mathcal P_L}\frac{1}{\sqrt{1-\alpha_{p-k}^2}\sqrt{1-\alpha_{s-k}^2}}\times ( \bar z a_p^\dagger + \alpha_k \alpha_{p-k}z a_{-p}) b_{p-k}b_{s-k}^\dagger ( z a_s + \alpha_k \alpha_{s-k}\bar z a_{-s}^\dagger ).
\begin{aligned}\partial_t P_a^{\varepsilon}(t)+i[H_0,P_a^{\varepsilon}(t)]=\frac{1}{t}Q_a^{\varepsilon}(t)^*Q_a^{\varepsilon}(t)+R_a^{\varepsilon}(t), &Q_a^{\varepsilon}(t)^*Q_a^{\varepsilon}(t)\ge0,\Vert R_a^{\varepsilon}(t)\Vert\le C_{\varepsilon}t^{-2}.\end{aligned}
Q = \sum_{\sigma}\int \frac{d^{3}{\bf p}}{( 2 \pi )^{3}}\, \left[ a_{\sigma}^{\dagger}({\bf p}) a_{\sigma}({\bf p}) - b_{\sigma}({\bf p}) b_{\sigma}^{\dagger}({\bf p}) + c_{\sigma}^{\dagger}({\bf p}) c_{\sigma}({\bf p}) - d_{\sigma}({\bf p}) d_{\sigma}^{\dagger}({\bf p}) \right] \quad ,
D^\flat_3\mathcal{I}_+^{\mathcal{J}}(\phi_-,\phi_+,T)(\mathfrak t)= \tfrac{d}{d\tau}|_{\tau=0}B(\psi_+ + \phi_+, T+\tau{\mathfrak t})(v(\psi_- + \phi_-, \psi_+ +\phi_+,T)) + B(\psi_+ +\phi_+,T) \tfrac{d}{d\tau}|_{\tau=0}v(\psi_- + \phi_-, \psi_+ +\phi_+,T+\tau{\mathfrak t}).
S_{c l}( e_{\mu}^{i}, \omega_{\mu}) = \int{d^{2}}x e \bigg ( \frac{1}{1 6 \alpha}R_{\mu \nu}{}^{i j}R^{\mu \nu}{}_{i j}- \frac{1}{8 \beta}T_{\mu \nu}{}^{i}T^{\mu \nu}{}_{i}- \gamma \bigg ) ,
|u(x,t)|\leq C\left( \left( \frac{|\sigma |(\Omega )+|\mu |(\Omega_{T})}{d^{N}}\right)^{m_{1}}+|\sigma |(\Omega )+|\mu |(\Omega_{T})+1+\mathbb{I}_{2}^{2d}[|\sigma |\otimes \delta_{\{t=0\}}+|\mu |](x,t)\right) ,
\begin{aligned}A^m= \frac{\partial}{\partial \left(|\nabla u|^2\right)}\left. \left[(-1)^m\rho^m{n-m \choose k-m}\frac{\phi'^{k-m}}{D^{k+1}}\phi^{n-m+1}\right]\right|_{u=|\nabla u|^2=0}= (-1)^{m+1}\rho^{m+2}\frac{k+1}{2}{n-m \choose k-m}\phi'^{k-m}(\rho)\phi^{n-m-k-2}(\rho),\end{aligned}
\begin{array}{lc}F_1(X,Y,Z)=\frac{1}{2(2n-1)}\left[g(X,Y)\theta_1(Z)-g(X,Z)\theta_1(Y)\right. \qquad\qquad\qquad\left.-g(X,J_1Y)\theta_1(J_1Z)+g(X,J_1Z)\theta_1(J_1Y)\right] ,\end{array}
\Vert B_\varepsilon (\vartheta )^{1/2}{\mathcal J}_1(\varepsilon; \vartheta;\widehat{\zeta}) \Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \mathfrak{C}_4\varepsilon c(\phi )^2, & 0 < \vartheta \leqslant 1,0< \varepsilon \leqslant \vartheta^{-1},\widehat{\zeta}=e^{i\phi},0<\phi <2\pi,
\begin{array}[c]{ll}\beta^{n,0,\theta}=\displaystyle\frac{\big{[}J(u^{n,\theta}(\cdot))+\theta\big{]}^+}{J^{n,\theta}(u^{n,\theta}(\cdot))}, \beta^{n,i,\theta}=\displaystyle\frac{-\big{[}-X^{n,\theta}(t_i)\big{]}^+}{J^{n,\theta}(u^{n,\theta}(\cdot))},\i=1,2,\cdots,n, \end{array}
\frac{\left[a,p;q^2\right]_{\infty}}{\left[a,p;q\right]^2_{\infty}}=\frac{\vartheta_3\left((p/2-a)\pi z;q^{p/2}\right)}{\vartheta_4\left((p/2-a)\pi z;q^{p/2}\right)}=\frac{\textrm{dn}\left((p-2a)zK[pz],q^{p}\right)}{\sqrt{m_1^{*}\left(pz\right)}}.
{\displaystyle V_{\alpha}^{i}= \pi_{\alpha}^{i}+ i p_{\alpha \dot{\alpha}}\bar{\theta}^{\dot{\alpha}i}+ m \theta_{\alpha}^{i}- i e A_{\alpha}^{i}\approx 0 , \bar{V}_{\dot{\alpha}i}= \bar{\pi}_{\dot{\alpha}i}+ i \theta_{i}^{\alpha}p_{\alpha \dot{\alpha}}+ m \bar{\theta}_{\dot{\alpha}i}- i e \bar{A}_{\dot{\alpha}i}\approx 0 ,}
u_{\underline{{m}}}^{i}= G_{{\cal R}}^{i k}H_{L}^{k j}( - l_{\underline{{m}}}^{j}+ e^{2 W}r^{( + + ) \underline{{n}}}l_{\underline{{n}}}^{j}l_{\underline{{m}}}^{( -- )}) ~ , \qquad
\Theta^{(q)}(z) \triangleq \frac{2}{1+z^{-1}}\left( \frac{k^{\alpha_q}}{2}\widetilde{\mu}^{(q)}(z) - \widetilde{\kappa}^{(q)}(z) \right) = \frac{k^{\alpha_q}\mu_0^{(q)}- \kappa_{0,0}^{(q)}}{1+z^{-1}}- \kappa^{(q)}(z),
0\leq e^{-\psi(r)}u(r) \leq\frac{\max_{\lbrack0,r_{0}]}e^{-\psi}}{\min_{[0,r_{0}]}e^{-\psi}}\cdot\frac{1}{|H_0|}=e^{\sup_\Omega \psi-\inf_\Omega\psi}\frac{1}{|H_0|}.
\begin{array}{c}\psi =\psi_0\{th[\frac \gamma 2(r_{+}-R)]-\epsilon /\lambda \},z<0, \psi =\psi_0\{th[\frac \gamma 2(r_{-}-R)]-\epsilon /\lambda \},z>0, r_{\pm}=\sqrt{x^2+y^2+\left( z\pm b\right)^2}, b>R.\end{array}
R\left( \left[ n_{i}\right] ,\left[ a_{i}\right] \right) =\frac{l^{2}\left( l-1\right)}{2}\sum_{1\leq i<j\leq k}\left( a_{i}^{2}a_{j}+a_{j}^{2}a_{i}\right) +l^{3}\sum_{1\leq i<j<m\leq k}a_{i}a_{j}a_{m}& =\frac{l^{2}}{2}\left( \left( l-1\right) \sum_{i=1}^{k}a_{i}^{2}\left(1-a_{i}\right) +2l\sum_{1\leq i<j<m\leq k}a_{i}a_{j}a_{m}\right) .
\begin{cases}\partial_t \tilde \omega^j + \Delta^{-1}\nabla^{\perp}\tilde \omega^j \cdot \nabla \tilde \omega^j =0, 0<t\le 1,\, x \in \mathbb R^2; \tilde \omega^j(t=0,x)= f^j(x), x \in \mathbb R^2. \end{cases}
\psi^{F^{(kl)}}(v):= \begin{cases}\frac{1}{|F^{(kl)}|}\int_{F^{(kl)}}(u^{(k)})^{(k)}\,ds \; v\in W^{(k)}, \frac{1}{|F^{(kl)}|}\int_{F^{(kl)}}(u^{(l)})^{(k)}\,ds \; v\in W^{(l)}, 0 , \end{cases}
\tilde{h}_0(\beta,p,q) = - p, \: \: \tilde{h}_1(\beta,p,q) ={iq (1 - \kappa) \over 2 \pi \kappa^2}, \: \:\tilde{h}_2(\beta,p,q) = -{(1 - \kappa) \over 4 \pi^2 \kappa^3}(2 q^2 + \kappa p + \kappa^2(-1+p)p) , \tilde{h}_3(\beta,p,q) = -{iq (1 - \kappa) \over 48 \pi^3 \kappa^4}(-6 + 16 q^2 + \kappa (11 + 12 p) + 6 \kappa^2 (-1 - 2 p + 2 p^2) ).
\sum \bigl((\omega;\omega_1,\ldots,\omega_j,\ldots,); \nu_1,\ldots,\nu_i,\ldots,\nu_n\bigr) &=\sum \bigl(\nu; (\omega_1;\omega_{1,1},\ldots),\ldots, (\omega_i;\omega_{i,1},\ldots,\omega_{i,j},\ldots),\ldots, (\omega_n;\omega_{n,1},\ldots)\bigr)
\mathcal{G}_q(\mathcal{H},r,\alpha):=\bigoplus_{n=0}^\infty\mathcal{G}^{(n)}_q(\mathcal{H},r,\alpha),\mathcal{G}^{(n)}_q(\mathcal{H},r,\alpha):=\mathcal{H}^{\otimes n}r^n([n]_q!)^\alpha.
\widehat{W}={\widehat M}_{\infty}= \left( \begin{array}{cccccc}-1 0 0 2 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 \end{array}\right) \;.
\sum_{i = 1}^k [K_i: \mathbb{Q}] n_i^2 \log n_i \leq \sum_{i = 1}^k [K_i: \mathbb{Q}]^2 n_i^2 \log g &= \sum_{i = 1}^k x_i^2 \log g &\leq (1 - 2 \cdot 0.99 \cdot 0.01) g^2 \log g &\approx 0.98 g^2 \log g.
\mathbb{X}^{N}_t=1+\int_0^t\mathbb{X}^{N}_{s^-}\otimes d\mathbf{X}_s+\sum_{0<s\leq t}\mathbb{X}^{N}_{s^-}\otimes \{\exp^{(N)}(\log^{([p])}(\Delta \mathbf{X}_s))-\Delta \mathbf{X}_s\}.
f(x,y)= \left(\begin{array}{cc}-2\partial_{x^-}\alpha\partial_{x^-}^2 - 2\partial_{x^-}^2\phi - 2\partial_{x^-}\phi\partial_{x^-}-\alpha\partial_{x^-}^2 - 2\partial_{x^-}\phi\partial_{x^-}0\end{array}\right) \delta(x-y)
|Y_{t\wedge\tau_D}|-|Y_0| \ge\int^{t\wedge\tau_D}_0\mbox{sgn}(Y_{s-})\,dY_s=-\int^{t\wedge\tau_D}_0\mbox{sgn}(Y_s)(f(X_s,Y_s)-f(X_s,0))\,ds &\quad+\int^{t\wedge\tau_D}_0\mbox{sgn}(Y_s)(-f(X_s,0)\,ds+dA^{\mu}_s)+\int^{t\wedge\tau_D}_0\mbox{sgn}(Y_{s-})\,dM^x_s,
{}\displaystyle\sum_{k=1}^{n}a_kb_k = \sum_{k=1}^{n-1}\Big(\gamma_k^{\frac{1}{p}}\sum_{i=1}^{k}q_ia_i\Big)\Big(\gamma_k^{-\frac{1}{p}}(b_k-b_{k+1})\Big) + \Big(\gamma_n^{\frac{1}{p}}\sum_{i=1}^{n}q_ia_i\Big)\Big(\gamma_n^{-\frac{1}{p}}b_n\Big) = \displaystyle\sum_{k=1}^{n}h_{nk}\sigma_k,
\underset{\pmb{F}_\ell}{}\mathrm{tr}\left\{\pmb{F}_\ell^H\left(\sum\limits_{k=1,k \neq \ell}^{K+1}\pmb{H}_{k,\ell}^H(\pmb{I}- \pmb{U}_k\pmb{U}_k^H)\pmb{H}_{k,\ell}\right)\pmb{F}_\ell\right\}\pmb{F}_\ell^H\pmb{F}_\ell = \frac{P_t}{d}\pmb{I}, \,\,\, \forall \ell \in \mathcal{K}
{\bf S}= \int d^{8}z E^{- 1}\left[ \frac{1}{2}{\bf D}_{\alpha \dot{\alpha}}L{\bf D}^{\alpha \dot{\alpha}}L - \frac{\imath}{1 2}\left( \nabla_{\alpha}T_{\dot{\alpha}}+ \nabla_{\dot{\alpha}}T_{\alpha}\right){\bf D}^{\alpha \dot{\alpha}}L \right]
\tilde{L}^{( 0 )}= \pi_{i}\dot{a}_{i}+ ( a_{i}a_{i}- 1 ){\eta}- M - \frac{1}{8 \lambda}\pi_{i}\pi_{i}+ \frac{1}{4 \lambda}( a_{i}\pi_{i}) \theta - \frac{1}{8 \lambda}a_{i}^{2}\theta^{2},
\frac{\partial \mathbb{P}(t,\rho)}{\partial t}=\Pi(\rho)^{\frac{1}{2}}\nabla_\rho\cdot(L(\rho)\mathbb{P}(t,\rho)\big(d_\rho\mathcal{F}(\rho)+ \beta d_\rho\log\mathbb{P}(t,\rho)\big) \Pi(\rho)^{-\frac{1}{2}}).
S_1 =\{a,\,a^5,\,a^6b,\,a^6b^2\},\ \S_2=\{ab,\,(ab)^{-1}\}, S_3 =\{a^3,\,b,\,ab^2,\,a^4b^2\},\ \S_4=\{a^2b,\,(a^2b)^{-1}\},
{\cal J}^{\mu\nu}_{i}[\xi|s] = 4\pi\int d^4x\, \epsilon^{\mu\nu\rho\sigma}\left[\Omega_{\xi}(s,0)T_{i}\Omega_{\xi}^{-1}(s,0)\right]^{j}\dot{\xi}_{\sigma}(s)\widetilde{j}_{\rho}^{j}(x)\delta^{4}(x-\xi(s)).
{\frac{g_{\tiny \mathrm{U V}}^{( a )}}{g_{\tiny \mathrm{I R}}^{( a )}}}={\frac{\operatorname{s i n}{\frac{\pi a}{k + l + 2}}}{\operatorname{s i n}{\frac{\pi}{k + l + 2}}}}{\frac{\operatorname{s i n}{\frac{\pi}{k + 2}}}{\operatorname{s i n}{\frac{\pi a}{k + 2}}}}={\frac{g_{( 1 , a )}^{[ l , k ]}}{g_{( a , 1 )}^{[ l , k ]}}},
\frac{2\beta M^2}{(d-2)\Lambda}\,\sqrt{\rho_0\lambda_1}\,\left(\lambda_0\,+\,\lambda_1\right)\,=\,1\,{,}\quad\sqrt{\lambda_0\lambda_1}\,\left(\rho_0^{-1}\,-\,1\right)\,=\,\frac{(d-2)\pi}{6\beta^{2}M^2}\,\rho_0^{-1}\,{.}
\|m\|_{L_t^\infty L_x^q}^q + \|m\|_{L_t^q L_x^{dq/(d-2)}}^q \leq C\|m_0\|_\infty^q + CNq^2\sum_{i=1}^{N}\|m^{\beta_i}b_i\|_{r/(r-1)}^2 \|m\|_{r^*(q+\gamma_i)}^{q+\gamma_i},
\prod_{i = 1}^{L}\int d^{d}k_{i}\frac{\partial}{\partial k_{r \mu}}\left\{\left( \sum_{l}x_{l}\overline{{{k}}}_{l \mu}\right) \prod_{j = 1}^{N}P_{\overline{{{k}}}_{j}, m_{j}}^{\nu_{j}}\right\}\equiv 0 ,
\frac{1}{3}(u\frac{\partial}{\partial u}+L_0)\cdot F_H(u,q)= \sum_{g,n}(2g-2+n)\sum \left< \tau_{d_1}\dots \tau_{d_n}\lambda_j\right>_{g,n}u^{2j}\prod_{i=1}^n \widetilde{\phi_{d_i}}(u,q).
s_i(\pi_i) = s_i(x_i^{(1)}) s_i(x_i^{(2)}) \cdots s_i(x_i^{(n)}) = x_{i+1}^{(2)}\frac{\kappa_i^{(2)}}{\kappa_i^{(1)}}x_{i+1}^{(3)}\frac{\kappa_i^{(3)}}{\kappa_i^{(2)}}\cdots x_{i+1}^{(1)}\frac{\kappa_i^{(1)}}{\kappa_i^{(n)}}= \pi_{i+1}.
n\le \sum_{x\in V}g(N^-[x]) = g(V) + \sum_{x\in V}g(N^-(x)) & = g(V) + \sum_{ux\in E}g(u) & = g(V)+ \sum_{u\in V}g(u)|N^+(u)| &\leq g(V) + d\cdot g(V) & = (d+1)g(V) < n,
X\langle \xi_1 = \xi_2 \rangle : = \{(z_1, z_2, w_{12}) \in X \mid \xi_1 = \xi_2 \}\subseteq X C\langle \xi_1 = \xi_2 \rangle : = \{(z_1, z_2, w_{12}) \in C \mid \xi_1 = \xi_2 \}\subseteq C .
\chi_{(\mu \circ s_k)}=\ \frac{1}{2}\sum_{\sigma \in \mathfrak{S}_N}(\sigma)L_{\mu(1),\sigma(1)}\dots L_{\mu(k-1),\sigma(k-1)}& \times \big\{L_{\mu(k+1),\sigma(k)}L_{\mu(k),\sigma(k+1)}-L_{\mu(k+1),\sigma(k+1)}L_{\mu(k),\sigma(k)}\big\}L_{\mu(k+2),\sigma(k+2)}\dots L_{\mu(N),\sigma(N)}\,.
\mathbb{E}[Z^{2}]=\sum_{i=1}^{k}\sum_{j=1}^{l}\frac{|U_{i}|}{|U|}\frac{|W_{j}|}{|W|}d(U_{i},W_{j})^{2}=\frac{n^{2}}{|U||W|}q(P_{U},P_{W})
(X^2)_{mn}= \sum_{k=1}^x X_{m k}X_{k n}=\sum_{k=1}^x (\sum_{q=1}^Q c_q (m-k)) (\sum_{r=1}^Q c_r (k-n)) = \sum_{q=1}^Q \sum_{r=1}^Q \sum_{k=1}^x c_q(m-k) c_r(k-n) = \sum_{q=1}^Q \sum_{r=1}^Q x \delta (q,r) c_q(m-n) = x \sum_{q=1}^Q c_q(m-n)=x X_{mn}.
R_{a b c}^{i_{1}i_{2}i_{3}}R_{i_{1}i_{2}d}^{i_{4}i_{5}i_{6}}R_{i_{4}i_{3}i_{6}}^{e i_{7}i_{8}}R_{i_{5}i_{7}i_{8}}^{f g h}= R_{b c d}^{i_{1}i_{2}i_{3}}R_{a i_{2}i_{3}}^{i_{4}i_{5}i_{6}}R_{i_{4}i_{1}i_{6}}^{i_{7}i_{8}h}R_{i_{7}i_{8}i_{5}}^{e f g}
\sum_{j = 0}^k{\left({\frac{{F_r}}{{F_{n + r}}}}\right)^j G_{m - r + nj}}= ( - 1)^r \frac{{F_{n + r}}}{{F_n}}G_m - ( - 1)^r \frac{{F_r}}{{F_n}}\left({\frac{{F_r}}{{F_{n + r}}}}\right)^k G_{m + (k + 1)n},n+r\ne 0\,.
(\alpha_z(\lambda_{\eta(\lambda)\eta(\mu)^{-1}}s_\lambda s_\mu^*))\cdot b =u_z((s_\lambda s_\mu^*)\cdot_\sigma u_z(s_\nu s_\tau^*)) &=z^{d(\lambda)-d(\mu)+d(\nu)-d(\tau)}((s_\lambda s_\mu^*)\cdot_\sigma z^{-(d(\nu)-d(\tau))}(s_\nu s_\tau^*)) &=(z^{d(\lambda)-d(\mu)}s_\lambda s_\mu^*)\cdot_\sigma (s_\nu s_\tau^*),
W_2^2(T_\# \mu, \delta_{T(a)}) = \int_{X}\|x-T(a)\|^2 \, d(T_\# \mu )(x) = \int_{X}\|T(x)-T(a)\|^2 \, d\mu (x) = \int_{X}\|x-a\|^2 \,d\mu(x) = W_2^2(\mu, \delta_a).
(\tilde F_{\parallel})^{\mu\nu}\equiv\left(\begin{array}{*{4}{c}}0 0 0 B 0 0 0 0 0 0 0 0 -B 0 0 0\end{array}\right),\qquad(\tilde F_{\perp})^{\mu\nu}\equiv\left(\begin{array}{*{4}{c}}0 0 0 0 0 0 -E 0 0 E 0 0 0 0 0 0\end{array}\right) \vspace{7 mm}
( f_1,f_2 )_{q,t}= \mathrm{ct}\Big(\frac{f_1 \overline{f_2}}{\Delta_\infty^X \Delta_0^X \Delta_\infty^{X^{-1}}}\Big),\qquad\hbox{where}\mathrm{ct}(f) = (\hbox{constant term in $f$}),
K_{n,q}(w_1, \cdots w_{n-1}|i,t) = \prod_{l=1}^{n-1}\sum_{k_l=0}^{w_l-1}q^{(t+1) \sum_{j=1}^{n-1}\big(\prod_{\substack{i=1 i \neq j}}^{n-1}w_i \big) k_j}\bigg[ \sum_{j=1}^{n-1}\big(\prod_{\substack{i=1 i \neq j}}^{n-1}w_i \big) k_j \bigg]_q^i .
F^\varepsilon (\mathbf{x},t)= 2 e^{\tau/2}\tilde{F}^\varepsilon (\mathbf{x},\tau)= 2tH^{\varepsilon}( X^\varepsilon(\mathbf{x},t)) + \left\langle X^\varepsilon(\mathbf{x},t) , \boldsymbol{\nu}^\varepsilon(\mathbf{x},t)\right\rangle\,,
k_{2}^{+}(v)^{-1}e_{1}^{-}(u)k_{2}^{+}(v) =\frac{u_{-}-v_{+}+h}{u_{-}-v_{+}}e_{1}^{-}(u)-\frac{h}{u_{-}-v_{+}}e_{1}^{+}(v), k_{2}^{+}(v)f_{1}^{-}(u)k_{2}^{+}(v)^{-1}=\frac{u_{+}-v_{-}+h}{u_{+}-v_{-}}f_{1}^{-}(u)-\frac{h}{u_{+}-v_{-}}f_{1}^{+}(v).
\partial_{\mu}\xi^{i}= - \frac{\sigma^{\prime}}{2}H_{j}^{i}\gamma_{\mu}\xi^{j}- \frac{\sigma^{\prime}}{2}\gamma_{5}\gamma_{\mu}\xi^{i}- \frac{i \sigma^{\prime}}{2}L_{j}^{i}\gamma_{\mu}\xi^{j},
\beta(y) \geq |x-y|+ \gamma(x) \cdot (y-x) + \alpha(x)\,\, \forall x \in [0,1], \forall y \in [0,1] ; \beta(y) = |x-y| + \gamma(x) \cdot (y-x) + \alpha(x)\,\, \forall (x,y) \in \Gamma .
M(\|w\|^2)\kappa_\alpha \int_\mathcal C y^{1-2\alpha}\nabla w.\nabla \phi dzdy =\lambda \int_{\Omega\times\{0\}}f(z)|w(z, 0)|^{q-2}w(z, 0)\phi(z, 0) dz &+\int_{\Omega\times \{0\}}|w(z,0)|^{2^*_\alpha-2}w(z, 0)\phi(z, 0) dz.
\mathcal{F}(\delta) =F(t_\delta, x_\delta, u(t_\delta, x_\delta),\nabla_x\varphi(x_\delta, y_\delta),\nabla_x^2\varphi(x_\delta, y_\delta)) &\quad-F(s_\delta, y_\delta, v(s_\delta, y_\delta),-\nabla_y\varphi(x_\delta, y_\delta),-\nabla_y^2\varphi(x_\delta, y_\delta)).
\psi ( p ) = - \frac{\lambda}{\pi^{2}i}\int d^{4}q \frac{\psi ( q )}{\left[ \left( p - q \right)^{2}+ i o \right] \left[ \left( \frac{P}{2}- q \right)^{2}- m^{2}\right] \left[ \left( \frac{P}{2}+ q \right)^{2}- m^{2}\right]}.
a_{2c-i}=\sum_{j=0}^{\lfloor i/2\rfloor}\frac{c!\cdot 2^j}{(c+j-i)!(i-2j)!j!}\le c^i\sum_{j=0}^{\lfloor i/2\rfloor}\underbrace{\frac{2^j}{c^j(i-2j)!j!}}_{=:b_{ij}}.
\varphi(x) = \frac{q \cdot x}{|x|^n}+ O\left(\frac{1}{|x|^{n-1+\varepsilon}}\right), \nabla \varphi(x) = \nabla \left( \frac{q\cdot x}{|x|^n}\right) + O\left(\frac{1}{|x|^{n+\varepsilon}}\right),
F(x) = 1 - \frac{i\lambda}{4}\log{\left( \frac{1+x}{2}\right)}+ \frac{\lambda^2}{32}\left[ \left( \log{\left( \frac{1+x}{2}\right)}+8\right) \log{\left( \frac{1+x}{2}\right)}- 4\, \mbox{Li}_2\left(\frac{1-x}{2}\right)\right] + O(\lambda^3)\,.
+ \sum_{i=0}^{\frac{m}{2}-1}(\alpha^{i-\frac{m}{2}}+\alpha^{\frac{m}{2}+1-i})(\alpha-1)(\frac{m}{2}t_{\frac{m}{2}}+ \sum_{k=0}^{\frac{m}{2}-1}t_k (k\alpha^{k-\frac{m}{2}}+ (m-k)\alpha^{\frac{m}{2}-k}))p_i < 0.
\frac{1}{t-s}(f(t)-f(s)) =\frac{1}{t-s}\psi\left( e^{sa*}e^{(t-s)a*}e^{(t-s)a}e^{sa}- e^{sa*}e^{sa}\right) &=\psi\left( e^{sa*}\frac{1}{t-s}\left(e^{(t-s)a*}e^{(t-s)a}-I\right)e^{sa}\right).
\sum_v (3 g_v-3+n_v-\sum_{e\in E_v}b_{(e,v)}-\sum_{i\in L_v}b_i ) +\sum_{e=(v_1,v_2)}( b_{(e,v_1)}+ b_{(e,v_2)}+1) = (\sum_v 3 g_v) - 3|V|+ 3 |E| +m -\sum_i b_i = 3h-3+m -\sum_i b_i ;
f(t_{1}w_{1}+\cdots+t_{k}w_{k})=\sum_{\substack{\alpha=(\alpha_{1},\ldots,\alpha_{k}) \alpha_{1}+\cdots+\alpha_{k}=k}}t^\alpha f_\alpha(w_1,\dots,w_k), t^\alpha := t_{1}^{\alpha_{1}}\cdots t_{k}^{\alpha_{k}}.
\left( \begin{array}{c}U^{+}_{k,l}U^{-}_{k,l}\end{array}\right)=\left( \begin{array}{c}u_{k,l}^{+}\mathfrak{I}_{k-\frac{1}{2},l}(r)-u_{k,l}^{-}l\mathfrak{I}_{-k+\frac{1}{2},l}(r) -u^{+}_{k,l}l\mathfrak{I}_{k+\frac{1}{2},l}(r)+u^{-}_{k,l}\mathfrak{I}_{-k-\frac{1}{2},l}(r)\end{array}\right)
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|^m\geq \sqrt{\frac{\frac{1}{(G_{f,\tau}^{\circ^m})}\left(\sum_{j=1}^nf_j(\tau_j)^m\right)^2-\sum_{j=1}^n|f_j(\tau_j)|^{2m}}{n^2-n}}.
\partial_{\dot q}f={}\partial_{\dot q}N+ \partial_{\dot q}\bigl( \tau L+\partial_{\dot q}L\cdot (\xi -\tau \dot q)\bigr)= ={}-g(\xi -\tau \dot q)+ \partial_{\dot q}\bigl( \tau L+\partial_{\dot q}L\cdot (\xi -\tau \dot q)\bigr),
\begin{aligned}-\int{\rm curl}\left[g(\varrho)(B\cdot \nabla B)\right]Bdx &=-\int g(\varrho)(B\cdot \nabla B){\rm curl}Bdx &\lesssim \|g(\varrho)\|_{L^\infty}\|B\|_{L^\infty}\|\nabla B\|_{L^2}\|{\rm curl}B\|_{L^2}&\lesssim \|B\|_{H^2}\|\nabla B\|_{L^2}^2.\end{aligned}
\nabla^{2}=\frac{1}{h_{1}h_{2}h_{3}}\left[\frac{\partial}{\partial u_{1}}\left(\frac{h_{2}h_{3}}{h_{1}}\frac{\partial}{\partial u_{1}}\right)+\frac{\partial}{\partial u_{2}}\left(\frac{h_{1}h_{3}}{h_{2}}\frac{\partial}{\partial u_{2}}\right)+\frac{\partial}{\partial u_{3}}\left(\frac{h_{1}h_{2}}{h_{3}}\frac{\partial}{\partial u_{3}}\right)\right]
{\alpha}_{ab}=\bar W_{aLbL}\ \underline{\alpha}_{ab}= \bar W_{a\underline{L}b\underline{L}}\beta_a =\bar W_{a L\underline{L}L}\ \underline{\beta}_a= \bar W_{a\underline{L}\underline{L}L}\rho =\bar W_{\underline{L}L\underline{L}L}\ \sigma = \epsilon^{ab}\bar W_{ab\underline{L}L}.
\nabla\varphi=\sum_{i}\frac{\partial\varphi}{\partial q^{i}}b^{i}=\sum_{i}\sum_{j}\frac{\partial\varphi}{\partial q^{i}}g^{ij}b_{j}=\sum_{i}\frac{1}{h_{i}^{2}}\frac{\partial f}{\partial q^{i}}b_{i};\nabla v=\sum_{i}\frac{1}{h_{i}^{2}}\frac{\partial v}{\partial q^{i}}\otimes b_{i}
\sum_{d \mid a d \neq a}\mathcal{C}(k, \mathcal{F}_d(x - 1)) = \left[\sum_{d \mid a}\mathcal{C}(k, \mathcal{F}_d(x - 1))\right] - \mathcal{C}(k, \mathcal{F}_a(x - 1)) = \mathcal{C}(k, \mathcal{F}_a) - \mathcal{C}(k, \mathcal{F}_a(x - 1)) = \mathcal{C}(k + 1, \mathcal{F}_a(x - 1)),
\det M(s^{(r)},{p}^{(r)})^2=(\det T_{n}^{(r)})^2 \times{{\prod_{\rho \in CP_{r,n}}z_{\rho}^{-2}}}\mathop{=}_{\rm Thm \ref{23}}(\det T_{n}^{(r)})^2 \times r^{-2c_{r,n}}{{\prod_{\rho \in CP_{r,n}}\prod_{i\geq1}\rho_{i}^{-4}}},
R_6:=R_5[Y_{32};\alpha_6,\delta_6]; \alpha_6: Y_{31}\mapsto t^{-1}Y_{31}, \Y_{22}\mapsto t^{-1}Y_{22}, \Y_{12}\mapsto t^{-1}Y_{12}; \delta_6: Y_{21}\mapsto (t^{-1}-t)Y_{22}Y_{31},\Y_{11}\mapsto (t^{-1}-t)Y_{12}Y_{31}.
L(E,\mathrm{i}y)=\lim_{n\rightarrow \infty}\frac{1}{2|n|}\log (|f_{n}(E,x+\mathrm{i}y)|^{2}+|f_{n-1}(E,x+\mathrm{i}y)|^{2}),\x \in\mathbb{T}^{d}.
w_k^\mu=\frac{\sum_{\lambda=1}^r(S_{jk})_\lambda^\mu\cdot w_j^\lambda}{1+\sum_{\lambda=1}^ra_{jk, \lambda}w_j^\lambda}=\sum_{\lambda=1}^r(S_{jk})_\lambda^\mu\cdot w_j^\lambda-\sum_{\lambda=1}^r\sum_{\nu=1}^r(S_{jk})_\lambda^\mu\cdot a_{jk, \nu}w_j^\lambda w_j^\nu+O(|w_j|^3),
Q/Q(2) \downarrow L' = \lambda_3 + \lambda_4 + 0, Q(2)/Q(3) \downarrow L' = \lambda_1 + \lambda_4, Q(3)/Q(4) \downarrow L' = \lambda_1 + 0, Q(4)/Q(5) \downarrow L' = \lambda_3, Q(5)/Q(6) \downarrow L' = 0.
\Vert \partial_x p_tu \Vert_{1+\rho}^2 = \int (\partial_x p_tu)^2 (1+\rho) \, dx = \int p_t( u_x)^2 (1+\rho)\, dx \leq e^{\frac{c^2}{\nu}t}\int ( u_x)^2 (1+\rho) \, dx &= e^{\frac{c^2}{\nu}t}\Vert u_x \Vert_{1+\rho}^2\, .
{3 \over r_1^{( 1 +{3 b \over 4})}(r_1^3 - R_1^3)^{( 1 -{3 b \over 4})}}\,\left[ \left( 1 +{(4 + 3 b) \over 3(4 - 3 b)}\right) \, r_1^3 - R_1^3 \right] = -{V \over 4}\, ({4 \over 3}- b) \, e^{{3b \over 2}\, \Phi_0}.
\lim_{t\to \infty}\Big(\Vert \sigma(t)\Vert_{\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}}}+\Vert u(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}}\Big) = 0.
v_1 =\frac{(t_1 - t_3+2) (t_1 + t_3+2) (t_1 - t_2 t_3)}{D}, v_2 =\frac{(t_1 - t_3+2) (t_1 + t_3-2) (t_1 - t_2 t_3)}{D}, v_3 =\frac{(t_2-1) (2 - t_1 - t_3) (2 + t_1 + t_3) (t_1 - t_2 t_3)}{(t_2+1) D}
\frac{1}{\beta}\sum_{r=1}^{m}\frac{(-\delta)^{r-1}[r-1]!}{\prod_{p=1}^{r}(1+\beta\gamma[p-1])}\,\,\alpha^{r}\!\!\!\sum_{1\le b_{1}<\cdots <b_{r}\le m}\!\!\!q^{\sum_{p=1}^{r}(b_{p}-m)}=\frac{\alpha}{\beta}\sum_{d=0}^{m-1}\frac{1}{1+\beta\gamma[d]}
\alpha_1+\alpha_k= \delta_n^k, \{\alpha_j= \alpha_1+\alpha_{j-1}\}_{j=2}^k \{\alpha_{j+(i-1)}= \alpha_j+\alpha_{i-1}\}_{j=1}^{k+1-i}, \ \ \delta_{n}^k=\alpha_{k+2-i}+\alpha_{i-1}, \ \{\alpha_{j+k+2-i}= \alpha_{j}+ \alpha_{k+2-i}\}_{j=1}^{i-2}k\geq i \geq 2.
C_{32,11,1}= \langle RM(1,5),x_{32,7,2}, z_{32,8,1}, z_{32,9,1}, z_{32,10,1}, z_{32,11,1}\rangle, C_{32,11,2}= \langle RM(1,5),x_{32,7,2}, z_{32,8,1}, z_{32,9,2}, z_{32,10,2}, z_{32,11,2}\rangle,
{\frac{1}{{\widetilde g}_{\mathrm{{\small Y M}}}^{2}( p^{2})}}\equiv{\frac{1}{g_{\mathrm{{\small Y M}}}^{2}}}+{\frac{\beta_{0}}{1 6 \pi^{2}}}~ \ln \left({\frac{\Lambda^{2}}{p^{2}}}\right) ~ ,
\begin{array}{c}C_{ij}^k= C_{ia}^j=-C_{ai}^j=C_{ab}^i=0,\;\;\;C_{ij}^a=-B_{ij}^a,\;\;\;C_{ia}^c=-C_{ai}^c=c_{ab}^cA_i^b,\;\;\;C_{ab}^c=c_{ab}^c, \rho_i^j=\delta_{ij},\;\;\; \rho_i^a=\rho_a^i=\rho_a^b=0.\end{array}
\frac{2}{\pi}\int_{-1}^{1}\frac{(1+\rho^{2})^{3}+16\rho^{3}xyz-4\rho^{2}(1+\rho^{2})(x^{2}+y^{2}+z^{2})}{w_{3}(x,y,z|\rho)}\sqrt{1-z^{2}}dz=1,
\frac{\partial^{2}z}{\partial t^{2}}=\frac{\partial^{2}z}{\partial x^{2}}(\frac{\partial x}{\partial t})^{2}+\frac{\partial^{2}z}{\partial x\partial y}\frac{\partial x\partial y}{\partial t^{2}}+\frac{\partial^{2}z}{\partial y\partial x}\frac{\partial y\partial x}{\partial t^{2}}+\frac{\partial^{2}z}{\partial y^{2}}(\frac{\partial y}{\partial t})^{2}
- \int_{H_{\sigma_2}}^T N(\sigma_2,t){d}\Big( \frac{x^{-\frac{1}{R\log t}}}{t}\Big) \le - N(\sigma_2,T) \int_{t_0}^T{d}\Big( \frac{x^{-\frac{1}{R\log t}}}{t}\Big) = N(\sigma_2,T) \Big( \frac{x^{-\frac{1}{R\log t_0}}}{t_0}- \frac{x^{-\frac{1}{R\log T}}}{T}\Big) .
M^{i j}= \frac{1}{L_{x}L_{y}( 1 + \phi^{2}/ T^{2})}\left( \begin{array}{c c}{{L_{y}/ L_{x}}}&{{- \phi / T - ( 1 + \phi^{2}/ T^{2}) T \theta}}\\{{\phi / T + ( 1 + \phi^{2}/ T^{2}) T \theta}}&{{L_{x}/ L_{y}}}\\ \end{array}\right) .
\eta_{}\left(\theta,\phi=\frac{\pi}{2}\right) = -\min{\left\{12\left(\frac{\theta}{\theta_{}}\right)^2,\eta_{\max}\right\}}\eta_{}\left(\theta=0,\phi\right) = -\min{\left\{12\left(\frac{\phi-\frac{\pi}{2}}{\phi_{}}\right)^2,\eta_{\max}\right\}},
\lim_{n\rightarrow \infty}n^{2\alpha+1}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'(X_s) \Theta^n_s )^2 ds = \kappa_2^2 \int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'\sigma)^2(X_s) ds,
N_{\pm}=\pm \left( I_{H}^{\pm}-2I^{\pm}_{S}-I^{\pm}_{C}-I^{\pm}_{W}\right) \mp S^{\pm}2l_{W}^{\pm}\pm{\cal L}^{\pm}_{W}-2\left( S^{\pm}\right)^{2}\mp 2S^{\pm}\left( \displaystyle\frac{\delta}{2}-k_{W}^{\pm}+k_{\pm}\right) .
{\rm II}^2 \leq c \sum_{j=0}^{n-1}\int_{t_j}^{t_{j+1}}\| \bar E(t_n-t) - B_{n-j}P_h \|_{\mathcal{L}_2^0}^{2}\d t & \leq c\| A^{-\frac{s}{2}}\|_{\mathcal{L}_2^0}^2\sum_{j=0}^{n-1}\int_{t_j}^{t_{j+1}}\|A^\frac{s}{2}(\bar E(t_{n}-t)-B_{n-j}P_h) \|^{2}\d t.
x^0_{z\bar z}-\mu_1(x^0_{\bar z}x^2_z+x^0_zx^2_{\bar z}) =0, x^1_{z\bar z}-\mu_2(x^1_{\bar z}x^2_z+x^1_zx^2_{\bar z}) =0, x^2_{z\bar z}-\mu_1x^0_zx^0_{\bar z}e^{-2\mu_1x^2}+\mu_2x^1_zx^1_{\bar z}e^{-2\mu_2x^2}=0.
\int_0^\infty x^{\rho-1}(1+x)^{\sigma-1}{_2F_1}(\alpha,\beta;\gamma;-x) dx = \frac{\Gamma(\rho)\Gamma(\alpha-\sigma-\rho+1)}{\Gamma(\alpha-\sigma+1)}{_3F_2}(\alpha,\gamma-\beta,\rho;\gamma,\alpha-\sigma+1;1).
\big(x-q^{\lambda+1}\beta^{-1}\big)\widehat{y}_1(x/q) + q^{-\mu}\beta^{-1}(\alpha x- 1 )\widehat{y}_1(qx) &{}-\big\{\big(q^{-\mu}\alpha\beta^{-1}+1\big)x- \beta^{-1}\big(q +q^{\lambda-\mu}\big)\big\}\widehat{y}_1(x)=0,
B(k) = \frac{e^{5.449}(6.274)^k}{(\frac{k}{4}-1)^{\frac{k-1}{2}}}+ \frac{e^{10.905}(4.793)^k}{\sqrt{(k-2)!}}+ \frac{e^{6.511}(10.096)^k}{\sqrt{(k-2)!}}.
\hat h_{\mu\nu}= a^2\chi_{\mu\nu}+ 2a_0^2 a^2 \left(\int \frac{dz}{a^4}\right)\partial_{\mu}\partial_{\nu}\xi_z - 2 a^2 \left(\int \frac{dz}{a^2}\right)\partial_{\mu}\partial_{\nu}\xi_z-\frac{U_B}{2}a^2 \xi_z \eta_{\mu\nu}
{}~{\displaystyle\sum_{\{k_i\},\atop i=1,\cdots,6}}R(u_3,u_5,u_6)^{k_3,k_5,k_6}_{i_3,i_5,i_6}R(u_2,u_4,u_6)^{k_2k_4j_6}_{i_2i_4k_6}R(u_1,u_4,u_5)^{k_1j_4j_5}_{i_1k_4k_5}R(u_1,u_2,u_3)^{j_1j_2j_3}_{k_1k_2k_3}