harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{F}^+L^{\epsilon,E}_d(q,\bar q) = \mathbb{F}L^\epsilon\big(q^\epsilon_E(h),\dot q^\epsilon_E(h) \big)=(\bar q, \epsilon \dot{\bar q}^\epsilon+\alpha(\bar q)), \mathbb{F}^-L^{\epsilon,E}_d(q,\bar q) = \mathbb{F}L^\epsilon\big(q^\epsilon_E(0),\dot q^\epsilon_E(0) \big)=(q, \epsilon \dot{q}^\epsilon+\alpha(q)),
\widetilde W''_i = (\alpha \widetilde V_i \pm \beta \widetilde V_{i+k-1}) \wedge \widetilde V_{i+1}\wedge \ldots \wedge \widetilde V_{i+k-2}=(\lambda \widetilde V_{i+k-1}\pm \mu \widetilde V_{i+2k-2}) \wedge \widetilde V_{i+k}\wedge \ldots \wedge \widetilde V_{i+2k-3}.
\biggl[\bar{z},u_1+\sum_{j=1}^{n-1}\bar{z}^{m_j}u_{j+1}\biggr]=0\qquad\Rightarrow\qquad u_1=-\sum_{j=1}^{n-1}\bar{z}^{m_j}u_{j+1}+f(t,\bar{z}),
\Big[ \frac{1}{\sqrt{k}l}\underset{i=1}{\overset{k}{\sum}}\Big( \underset{j=1}{\overset{l}{\sum}}B^j_i - lp_i \Big)^2 \Big] = \frac{1}{kl^2}\sum_{i=1}^k \Big[ \Big(\sum_{j=1}^l B_i^j - lp_i \Big)^2\Big] \leq 1/8.
\begin{aligned}\dfrac{d^{B^{(1)}_{n+1}}_{k,l}(z)}{(z-q^{\mathtt{h}^\vee})^{\delta_{kl}}}= D_{k,l}(z) \times D_{k,l^*}(z) = D_{k,l}(z) \times D_{k^*,l}(z) & = D_{k^*,l^*}(z) \times D_{k,l^*}(z) = D_{k^*,l^*}(z) \times D_{k^*,l}(z)\end{aligned}
\int_{\mathbb{R}^N}V|u_n|^p\eta_R dx = \int_{\mathbb{R}^N}((V-\tau_0)_{+}+\tau_0)|u_n|^p\eta_R dx-\int_{\mathbb{R}^N}(V-\tau_0)_{-}|u_n|^p\eta_R dx &\geq \int_{\mathbb{R}^N}\tau_0|u_n|^p\eta_R dx-\|(V-\tau_0)_{-}\|_{L^{\frac{N}{ps}}(\mathbb{R}^N\setminus B_R(0))}\|u_n\|_{p_s^*}^{p}
K_0 = \tfrac{1}{2}+2\kappa\big(\log(2\kappa) -\tfrac{1}{2}(\log p_1 +\log p_2-\log d)\big), K_1 = 2\kappa\big(1-\log(2\kappa) +\tfrac{1}{2}(\log p_1 +\log p_2-\log d)\big)-1.
S_{ij}: =\{(a_ic_1,b_ic_1)+(a_jc_2+b_jc_2):c_1,c_2\in{\mathcal{C}'}\}&=\{(a_ic_1+a_jc_2,b_ic_1+b_jc_2):c_1,c_2\in{\mathcal{C}'}\},
\chi_{2}(\omega)=-\frac{2}{\pi}P\int_{0}^{\infty}\frac{\omega\chi_{1}(\omega^{\prime})}{\omega^{\prime2}-\omega^{2}}d\omega^{\prime}=-\frac{2\omega}{\pi}P\int_{0}^{\infty}\frac{\chi_{1}(\omega^{\prime})}{\omega^{\prime2}-\omega^{2}}d\omega^{\prime}.
A=\begin{pmatrix}0 1 0 \cdots 0 0 0 1 \cdots 0 \vdots \vdots \vdots \cdots \vdots 0 0 0 \cdots 1 -\frac{p_n}{p_0}-\frac{p_{n-1}}{p_0}-\frac{p_{n-2}}{p_0}\cdots -\frac{p_1}{p_0}\end{pmatrix}
G_2(n, f) \ = \ 2(-1)^{n+1}\binom{n}{f}\sum_{\ell = 0}^n (-1)^\ell \sum_{c = 0}^\ell \binom{f}{c}\binom{n - c}{n - \ell}&= \ 2(-1)^{n+1}\binom{n}{f}\sum_{c = 0}^n \binom{f}{c}\sum_{\ell = c}^n (-1)^\ell \binom{n - c}{n - \ell}.
\int_{\mathbb{R}^{m}}a_{k}\left\langle \nabla\left( \zeta_{k}v_{k}\right),\nabla\varphi\right\rangle \,dx =\int_{\mathbb{R}^{m}}a(p)\left\langle\nabla v,\nabla\varphi\right\rangle \,dx+o(1), \int_{\mathbb{R}^{m}}c_{k}|\zeta_{k}v_{k}|^{2^{\ast}-2}\left( \zeta_{k}v_{k}\right) \varphi\,dx =\int_{\mathbb{R}^{m}}c(p)|v|^{2^{\ast}-2}v\varphi\,dx+o(1).
G_m(x_1, x_2, x_3, x_4) = [\prod_{i=1}^m\Gamma(\alpha_i)]^{-1}r_{12}^{\alpha_4-\frac{1}{2}D}r_{14}^{\frac{1}{2}D-\alpha_1-\alpha_4}r_{24}^{\frac{1}{2}D-\alpha_2-\alpha_4}r_{34}^{-\alpha_3}F(u,v)
b_k = p \cdot (\hat{\alpha}_k, \hat{\mu}_k + (0, \rho, h^{\vee})) = p \cdot ((m_k, \alpha_k, 0),(n_k, \mu_k+\rho, \kappa' + h^{\vee})) = p \cdot (\kappa' + h^{\vee}) m_k + p \cdot (\mu_k + \rho, \alpha_k),
\sup_{n\geq 0}\|X_t^{n,\cdot}\|_{L^2(\Omega, W^{k,p}(U))}^2 = \sup_{n\geq 0}\sum_{i=0}^{k}\textrm{E}\left[\|\frac{\partial^i}{\partial x^i}X_t^{n,\cdot}\|_{L^{p}(U)}^2\right] \leq \sum_{i=0}^{k}\left( \int_{U}\sup_{n\geq 0}\textrm{E}\left[ \|\frac{\partial^i}{\partial x^i}X_t^{n,x}\|^{p}\right]dx\right)^{\frac{2}{p}}< \infty
\|\widehat{g}\|^2_{L^2(S_j, d\sigma)}= \frac{1}{|S_j|(q-1)}\sum_{\substack{t\in \mathbb F_q^*, x\in \mathbb F_q^d : x_1^2+\cdots+x_d^2=jt^2}}\left| \sum_{m\in \mathbb F_q^d}\chi(-m\cdot x) g(m) \right |^2 \leq \frac{1}{|S_j|(q-1)}\sum_{(x, t)\in H_j}\left| \sum_{m\in \mathbb F_q^d}\chi(-m\cdot x) g(m) \right |^2.
\|Rx-Ry\| =\|(1-\alpha)x+\alpha Tx-(1-\alpha)y-\alpha Ty\| &\leq |1-\alpha|\|x-y\|+|\alpha|\|Tx-Ty\| &\leq |1-\alpha|\|x-y\|+|\alpha|\delta\|x-y\| &= (|1-\alpha|+|\alpha|\delta)\|x-y\|.
d s^{2}= \left( \prod_{i = 1}^{i = n}H_{i}^{- 2 r_{i}}\right) d t^{2}- \left( \prod_{i = 1}^{i = n}H_{i}^{- 2 r_{i}}\right)^{- \frac{1}{d - 3}}d \vec{x}^{\ 2}\, ,
\frac{1}{2 \kappa^{2}}\left(n_{\mu}\Gamma^{\mu}_{\rho \sigma}-\frac{1}{2}g_{\rho \sigma}n_{\mu}\Gamma^{\mu}_{\alpha \beta}g^{\alpha \beta}-\Gamma_{\alpha (\rho}^{\alpha}n_{\sigma)}+\frac{1}{2}\Gamma^{\alpha}_{\beta \alpha}n^{\beta}g_{\rho\sigma}\right) =T^{bra}_{\rho \sigma}.
D \left( \begin{array}{l}f^q_\pm f^{-q}_\pm \end{array}\right)= d \left( \begin{array}{l}f^q_\pm f^{-q}_\pm \end{array}\right)- i q A_\pm \sigma_3\left( \begin{array}{l}f^q_\pm f^{-q}_\pm \end{array}\right).
m^{\underline{\ell_i}}m^{\underline{\ell_j}}=\sum_{k=\max\{0,\ell_i+\ell_j-m\}}^{\min\{\ell_i,\ell_j\}}G_{\ell_i,\ell_j,k}m^{\underline{\ell_i+\ell_j-k}}.
\begin{aligned}{\omega}_k=\frac{\sup_{Q_{k-1}}u-\inf_{Q_{k-1}}u}{2}\leq \frac{\sup_{\Gamma_{1,a}}u_{k-2}-\inf_{\Gamma_{1,a}}u_{k-2}}{2}\omega_{k-2}\leq \theta\omega_{k-2}. \end{aligned}
\bigl( W_{n+1}(z) \bigr)_{\rho<\varepsilon_n}= \bigl( \overline{(O_{n+1}z)_{\rho<\delta_{n+1}}}^\circ \bigr)_{\rho<\varepsilon_n}\subseteq \overline{(O_{n+1}z)_{\rho<\delta_{n+1}+ \varepsilon_n}}= F_{n+1}(z).
\int_0^{\infty}v^3 g^T(x,v) \mathrm{d}v = \int_0^{\infty}\int_0^1 v^3 \frac{e^{-(1-y)/\kappa |v|}}{\kappa |v| (1-e^{-1/\kappa|v|})}\rho_g(x+y) \left( \alpha\mathcal{M}_{T(y+x)}+ (1-\alpha) \mathcal{M}_{\tau(y+x)}\right) \mathrm{d}y \mathrm{d}v.
v_{1}\wedge v_{-1}+q^{2}v_{-1}\wedge v_{1}-q^{2}v_{0}\wedge v_{0}=0,v_{1}\wedge v_{-1}+q^{-4}v_{-1}\wedge v_{1}+q^{-1}(q+q^{-1})v_{0}\wedge v_{0}=0.
\mathcal{O}(r) = \left\{(\alpha,\beta,\gamma): \begin{array}{cc}\frac{\alpha\beta-\gamma^2}{\alpha+\beta-2\gamma}\leq r, \gamma<\min(\alpha,\beta),\end{array}\begin{array}{cc}0\leq\alpha\leq a, 0\leq\beta\leq b, 0\leq\gamma\leq c\end{array}\right\}.
\log \frac{|x_j|}{r_j}\ge \frac\lambda{n}\log \frac1{\rho_j}\ge \frac\lambda{n}\log \left(\frac{\lambda A}n |x_j|^{-a/2}\right) &= \frac\lambda{n}\left(\frac{a}{2}\log \frac1{|x_j|}+ \log \frac{\lambda A}n\right).
\mathbf{x}^{k+1}= (\mathbf{W}\otimes \mathbf{I}_p) \mathbf{x}^k - \xi \nabla f(\mathbf{x}^{k+1}) + \sum_{t=0}^{k}(\mathbf{W}- \tilde{\mathbf{W}})\otimes \mathbf{I}_p \mathbf{x}^{t},
(e_{\alpha}+ \alpha \|\lambda_y\|^2)\, 2 \mathrm{Re}\langle \kappa \Phi_{\#}^y\,|\, R^{\#}_y\rangle \geq - C \varepsilon_6 \alpha \|R^{\#}_y\|^2 -C \varepsilon_6^{-1}\alpha |\kappa|^2 \|\Phi_{\#}^y\|^2 &= - C \varepsilon_6 \alpha \|R^{\#}_y\|^2 + O(\alpha^4 \log(\alpha^{-1})).
\lambda_{2N}(\alpha)={1+\sum_{m=0}^{N-1}\left({i\alpha\over 2}\right)^{2m+1}{(2N+2m)!\over (2m+1)!(2N-2m-2)!}\over \sum_{m=0}^{N-1}\left({i\alpha\over 2}\right)^{2m}{(2N+2m-1)!\over (2m)!(2N-2m-1)!}-i\alpha(4N+1)}\times\lambda_0(\alpha)
\begin{alignedat}{4}\hat a_0 =\hat\alpha_0n+2, \hat a_1 =\hat\alpha_1n+1, \hat a_2 =\hat\alpha_2n+1, \hat a_3 =\hat\alpha_3n+1, \hat b_0 =\hat\beta_0n+2, \hat b_1 =\hat\beta_1n+1, \hat b_2 =\hat\beta_2n+2, \hat b_3 =\hat\beta_3n+2,\end{alignedat}
\delta_{ls}\tilde{E}^a = - 2i \tilde{E}^{\alpha}\Gamma^{a}_{\alpha\beta}\epsilon^{\beta}(x, \tilde{\theta}(x))\; , \quad \delta_{ls}\tilde{E}^{\alpha}={\cal D}\epsilon^{\alpha}(x, \tilde{\theta}(x))+\ldots \; \quad \ldots
A_0=\begin{pmatrix}0 1\end{pmatrix}\begin{pmatrix}1-p_1-p_2 \theta^0\end{pmatrix},A_{t_i}=\begin{pmatrix}q_i 1\end{pmatrix}\begin{pmatrix}p_i \theta^{t_i}-p_i q_i\end{pmatrix}, N= \begin{pmatrix}0 1 0 0 \end{pmatrix},N_i=\frac{q_i(p_i q_i-\theta^{t_i})}{t_i}N, i=1,2.
\begin{alignedat}{2}\Gamma_4 = -(\Gamma_1 + \Gamma_2 + \Gamma_3), \Gamma_6 = -(\Gamma_5 + \Gamma_7 + \Gamma_8), &\Gamma_{11}= -(\Gamma_9 + \Gamma_{10}+ \Gamma_{12}), \Gamma_{13}= -(\Gamma_{14}+ \Gamma_{15}+ \Gamma_{16}).\end{alignedat}
A_{D}= \frac{{\cal N}_D}{2\pi^2}\int_0^1 d\nu \int_0^1 \frac{dq}{q}\left(\frac{\sin^2(\pi\nu)}{\pi^2}\right) \left(\frac{-\ln(q)}{2\pi^2}\right)^{(k+1-D)/2}\exp\left[ (Y^2 + 2 l \sin^2(\pi\nu))/\ln(q) \right]
\Bigl\{b^r(k,\sigma),b^{s\dagger}(p,\sigma')\Bigr\}=\Bigl\{d^r(k,\sigma),d^{s\dagger}(p,\sigma')\Bigr\}= 16\pi^3 k^+ \delta_{rs}\delta_{\sigma \sigma'}\delta^3(k-p) \;.
\prod_{v:v\in e_{m}}B(v,e_{m}) =\frac{2\alpha}{1-\alpha^{\frac13}}\cdot\left(1-2\alpha^{\frac23}\right)^2 & =\frac{2\alpha\left(4\alpha^{\frac43}-4\alpha^{\frac23}+1\right)}{1-\alpha^{\frac13}}& =\alpha+\frac{\alpha(2\alpha^{\frac13}-1)(\alpha^{\frac13}+1)(4\alpha^{\frac23}-2\alpha^{\frac13}-1)}{1-\alpha^{\frac13}}.
[G(k,0) +iG'(k,0)]E^{-1}[G(k,0)^\dag-iG'(k,0)^\dag] = G(k,0)E^{-1}G(k,0)^\dag+G'(k,0)E^{-1}G'(k,0)^\dag -i[G(k,0)E^{-1}G'(k,0)^\dag-G'(k,0)E^{-1}G(k,0)^\dag] =I_n
\mu \int_{B_s}|\nabla u|^2\,dx \le \frac{1}{\mu}\L{2}{\nabla u}{B_t\setminus B_s}\L{2}{\nabla \eta}{B_1}+ \Big | \int_{B_t}f(x) (\eta u)\,dx \Big | &\le \frac{C(\mu)}{2\tau^{2}}\L{2}{\nabla u}{B_t\setminus B_s}^{2}+ 2\tau^{2}\L{2}{\nabla \eta}{2}^2 +C(N).
G_{D}[ x_{2}, x_{1}] = \sum_{n_{1}}\sum_{n_{2}}. . . \sum_{n_{D - 1}}e^{i ( n_{1}\theta_{1}+ n_{2}\theta_{2}. . . + n_{D - 1}\theta_{D - 1})}G_{n_{1}n_{2}. . . n_{D - 1}}[ x_{2}, x_{1}] ,
\Psi(x_1, x_2) =\frac{(x_1^2 + x_2^2 +2x_1 - 2x_2 -2, x_1^2 + x_2^2 - 2x_1 + 2x_2 -2)}{x_1^2 + x_2^2 - 2x_1 - 2x_2 + 2}.
\mathrm{sf}(\hat{H}^\sharp|_{\partial \overline{D}_i}) = \sum_{j = 1}^\ell \epsilon(f_j) I(f_j, \partial \overline{D}_i)+ \sum_{j = 1}^{\ell^c}\epsilon(f^c_j) I(f^c_j, \partial \overline{D}_i).
G_{1}(s) = \int_{s}^{1}\frac{1}{G_{0}(\tau)}\d\tau\quad\mbox{and}G_{0}(s) =\begin{cases}s \quad\hbox{if \quad\eqref{203+}\mbox{holds}}, s\,G'(s) \quad\hbox{if \quad\eqref{203}\mbox{holds}}.\end{cases}
{\cal L}=-\frac{1}{12}(\partial_\mu A_{\nu \lambda}+ \partial_\lambda A_{\mu \nu}+ \partial_\nu A_{\lambda\mu})^2+\frac{m^2}{4}A_{\mu\nu}^2-\overline{\psi}(\gamma_\mu\partial^\mu+M)\psi-\frac{1}{2}g\varepsilon_{\mu\nu\lambda\varrho}\overline{\psi}\gamma^\nu\psi\partial^\mu A^{\lambda\varrho}.
Q_{2,\vec{v}_{s,t,f,0}^{(1)}}= \left(\left(\begin{array}{rrr}s^{2}- t 1 -s -s 0 1 s t + 1 0 -t\end{array}\right), \left(\begin{array}{rrr}1 0 0 0 1 0 0 0 1\end{array}\right), \left(\begin{array}{rrr}-s 0 1 1 0 0 -t 1 0\end{array}\right)\right),
\begin{array}{l}{{e^{+ 2}\nabla \psi_{+ 2 q}^{2 -}+ e^{- 2}\nabla \psi_{- 2 q}^{2 -}+ e^{+}\nabla h_{+ q}^{2 -}+ e^{-}\nabla \chi_{q}^{2}+ T^{+ 2}\psi_{+ 2 q}^{2 -}+ T^{- 2}\psi_{- 2 q}^{2 -}+ T^{+}h_{+ q}^{2 -}+ T^{-}\chi_{q}^{2}}}\\{{= \frac 1 2 \gamma_{q \dot{p}}^{i}\Omega^{- 2 i}\pi_{\dot{p}}^{2 +},}}\\ \end{array}
\left( \partial_\mu 1\otimes 1 +\frac{1}{4}\epsilon_{\alpha \beta \delta}\omega^{\alpha \beta}_{\mu}1\otimes \gamma^{\delta}+\frac{i}{2}A^{c}_{\mu}\sigma_c \otimes 1 - \frac{1}{2\ell}e_{\mu\alpha}1\otimes \gamma^{\alpha}\right) \varepsilon_r = 0 \, ,
Y^{(n)}(t_{n, i+1})=Y^{(n)}(t_{n, i})+ \int_{t_{n, i}}^{t_{n, i+1}}g(s, W_0^{(n), t_{n, i}}, Y_0^{(n), t_{n, i}}) ds + B(t_{n, i+1})-B(t_{n, i}),
\partial_{j}^{2}+\bar{\partial}_{j}^{2}=\frac{1}{4}(\tilde{X}_{j}-i\tilde{Y}_{j})^{2}+\frac{1}{4}(\tilde{X}_{j}+i\widetilde{Y}_{j})^{2}=\frac{1}{2}(\tilde{X}_{j}^{2}-\tilde{Y}_{j}^{2}), \partial_{j}^{2}-\bar{\partial}_{j}^{2}=\frac{1}{4}(\tilde{X}_{j}-i\tilde{Y}_{j})^{2}-\frac{1}{4}(\tilde{X}_{j}+i\tilde{Y}_{j})^{2}=-i\tilde{X}_{j}\tilde{Y}_{j}
{\bf E}\bigl[\beta_{2, 2}(\tau, b)^{-k}\bigr] =\prod\limits_{l=0}^{k-1}\Bigl[\frac{\Gamma\bigl((-(l+1)+b_0)/\tau\bigr)\,\Gamma\bigl((-(l+1)+b_0+b_1+b_2)/\tau\bigr)}{\Gamma\bigl((-(l+1)+b_0+b_1)/\tau\bigr)\,\Gamma\bigl((-(l+1)+b_0+b_2)/\tau\bigr)}\Bigr], \; k<b_0.
\sum_{m=0}^{+\infty}\sum_{\ell=-\infty}^{+\infty}\,\psi_{m,\ell,p}(r_f,\theta_f,z_f)\,e^{-\frac{i}{\hbar}\Delta t\left[\frac{1}{2}\hbar^2 g^2\ell^2+\hbar\omega(2m+|\ell|+1)\right]}\,\psi^*_{m,\ell,p}(r_i,\theta_i,z_i)\ \ \ ,
S^{T}= M_{0}^{2}\int_{0}^{1 / T}d t \int_{0}^{R}d r \left[ 1 + \frac{1}{2}{\bf u}\left( 1 - \frac{\alpha}{M_{0}^{2}}\triangle \right) ( - \triangle ){\bf u}\right] - \beta \int_{0}^{1 / T}d t \ddot{\bf u}\acute{\bf u}
\frac{d}{dt}G(v) + \frac{b_0}{8}Z_0(v) \leq C \lambda Z_0 (v) & + \frac{1}{2}\sum_{\mu=0}^{L-1}\int_{\partial \Omega_\lambda}h \cdot \sigma | \sigma \cdot \nabla \partial_t^\mu v|^2 dS + \bar{C}_\lambda \sum_{\mu=0}^{L-1}\langle \partial_t^\mu F_\lambda, \partial_t^{\mu+1}v + \partial_t^\mu v + [h ; \nabla \partial_t^\mu v] \rangle,
\overline{u}_\beta(x,t) =(\limsup_{\alpha\to\beta,\alpha\neq\beta}{}^{*}u_\alpha)(x,t) &=\lim_{\delta\to0^+}\sup\{u_\alpha(y,s)\mid(y,s)\in Q_{T,0}\cap\overline{B_\delta(x,t)},\ 0<|\alpha-\beta|<\delta\}
(\frac{y^{\prime}}{x^{\prime3}})^{2}=\frac{x^{\prime}+1}{x^{\prime}}(\frac{x^{\prime}+1}{x^{\prime}}+1)(\frac{x^{\prime}+1}{x^{\prime}}+2)(\frac{x^{\prime}+1}{x^{\prime}}+3)(\frac{x^{\prime}+1}{x^{\prime}}-1)(\frac{x^{\prime}+1}{x^{\prime}}-2).
D ( P ) = \{F_{\psi}\in L^{2}( 0 , \infty ) : F_{\psi}\; \mathrm{a b s .}\; \mathrm{c o n t .}, \int_{0}^{\infty}\mathrm{d}q \, |{\frac{\mathrm{d}F_{\psi}}{\mathrm{d}q}}|^{2}< \infty , \; F_{\psi}( 0 ) = 0 \}.
\frac{d^{n}}{dz^{n}}\left\lbrace{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z)\right\rbrace =\frac{(b)_{n}}{(c)_{n}}\left[{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b+n;c+n;z)\right],
c_1(p,s) \left(\frac{\log(n)}{\log(1+d/\log(n))}\right)^{s/p}n^{-\frac{c_1(p) r}{p\log(1+d/\log(n))}}~~~~~~~~~\leq a_n \leq c_2(p,s)\left(\frac{\log(n)}{\log(1+d/\log(n))}\right)^{s/p}n^{-\frac{c_2(p) r}{p\log(1+d/\log(n))}}.
S_{{\mathrm{\scriptsize ~{g a u g e d}}}}= \frac 1 2 \int_{M}d^{2}x \left( G_{\mu \nu}D^{a}X^{\mu}D_{a}X^{\nu}+ i Y_{\alpha}\epsilon^{a b}( \partial_{a}V_{b}^{\alpha}- \partial_{b}V_{a}^{\alpha}) \right) \, ,
\gamma_{k,n}^{(\alpha)}-a_{k,n}^{(\alpha)}=\sum_{j=1}^{q}\sum_{t=0}^{\tau_j-1}\beta_n(j,t) \left( (z-\lambda_j)^{\tau_j}F_\alpha(z) \Phi^{n-k}(z) \right)^{(t)}_{z=\lambda_j}, \alpha=1,2,\ldots,d, k\geq 0.
0 \rightarrow{\cal O}_{\pi^*{\cal S}}(\sigma|_{\pi^*{\cal S}})\rightarrow{\cal O}_{\pi^*{\cal S}}(2\sigma|_{\pi^*{\cal S}})\rightarrow{\cal O}_{\pi^*{\cal S}}(2\sigma|_{\pi^*{\cal S}})|_{\sigma|_{\pi^*{\cal S}}}\rightarrow 0
\begin{cases}\partial_tv -\mu\Delta v-\nabla\cdot G= -\nabla p-v\cdot\nabla v + \nabla\cdot(GG^\top), begin{align*}-4mm] \partial_tG-\nabla v =-v\cdot\nabla G+ \nabla v G, begin{align*}-4mm] \nabla\cdot v = 0,\nabla\cdot G^\top=0.\end{cases}
C_n(0) =\sum_{y_n\in\{0,1\}}\log\Big(\frac{q_n(y_n|1,0)}{\nu^{\pi^*}_n(y_n|0)}\Big)q_n(y_n|1,0)=\log\Big(\frac{q_n(0|1,0)}{\nu^{\pi^*}_n(0|0)}\Big)q_n(0|1,0)+\log\Big(\frac{q_n(1|1,0)}{\nu^{\pi^*}_n(1|0)}\Big)q_n(1|1,0) &=\gamma_n\log\big(\frac{1-c_0(n)}{c_0(n)}\big)+\log\big(\frac{1}{1-c_0(n)}\big)-H(\gamma_n).
L(\zeta_{i_2+1}-\zeta_{i_3+1})\frac{\zeta_j -\zeta_{i_1+1}}{\zeta_j -\zeta_{j+1}}+ L(\zeta_{i_3+1}-\zeta_{i_1+1})\frac{\zeta_j -\zeta_{i_2+1}}{\zeta_j -\zeta_{j+1}}+ L(\zeta_{i_1+1}-\zeta_{i_2+1})\frac{\zeta_j -\zeta_{i_3+1}}{\zeta_j -\zeta_{j+1}}= 0.
\dot{x}_i =-\prod_{j\in N_i^-}x_i^{w_{ji}}+\prod_{j\in N_i^+}x_i^{w_{ij}}, \dot{x}_i = -\sum_{j\in N_i^+}w_{ij}x_i\ln\frac{x_i}{x_j}, \ \ \dot{x}_i =\sum_{j\in N_i^+}w_{ij}(\ln x_j-\ln x_i).
\begin{vmatrix}xI_{2^{n-1}}-B_{2^{n-1}}-C_{2^{n-1}}C^T_{2^{n-1}}xI_{2^{n-1}}\end{vmatrix}= x^{2^{n-1}}\bigg[(x - \frac{2^{n-1}}{x}+1)(x -(2^{n-1}-2))((x +1)^{(2^{n-1}-2)}) - (1+x)(1+x)^{(2^{n-1}-2)}\bigg] & = x^{2^{n-1}-1}(1+x)^{(2^{n-1}-2)}\bigg[x^3 + x^2(2-2^{n-1})x^2- (1-2^n)x + 2^{2n-2}-2^n\bigg].
\varphi(E_{ij})= y_{ij}E_{ij}+y_{i,\sigma^{-1}(j)}E_{i,\sigma^{-1}(j)}+y_{\sigma^{-1}(i),j}E_{\sigma^{-1}(i),j}&+y_{\sigma^{-1}(i),\sigma^{-1}(j)}E_{\sigma^{-1}(i),\sigma^{-1}(j)}
\varphi(y,a^*)^{(p-1)/p}- \varphi(y,a_*)^{(p-1)/p}= \frac{p-1}{p}\ \int_{\varphi(y,a_*)}^{\varphi(y,a^*)}z^{-1/p}\dz & \le \frac{p-1}{p}\varphi(y,a_*)^{-1/p}G(y) & \le \frac{(p-1)(2p-1)}{2p}G(y)\ .
P_{\tilde{p}}={\frac{\Omega_{p + 2}}{\sqrt{2}\kappa_{D}}}( p + 1 ) ( r_{+}r_{-})^{\frac{p + 1}{2}},{\cal M}_{\tilde{p}}={\frac{\Omega_{p + 2}}{2 \kappa_{D}^{2}}}[ ( p + 2 ) r_{+}^{p + 1}- r_{-}^{p + 1}] .
\alpha_{i}^{\pm}= \frac{1}{\sqrt{2}}\left[ \alpha_{i}^{1}\mp i \alpha_{i}^{2}\right] ,{\alpha_{i}^{\pm}}^{\dagger}= \frac{1}{\sqrt{2}}\left[{\alpha_{i}^{1}}^{\dagger}\pm i{\alpha_{i}^{2}}^{\dagger}\right] \ ,
\int_a^b h(s)^p ds = bh(b)^p -a h(a)^p + p \int_a^b h(s)^{p-1}\frac{s^2 f^{**}(s)}{[n \omega_n (\sinh F(s))^{n-1}]^2}ds \leqslant p \Big(\int_a^b h(s)^p ds\Big)^{(p-1)/p}\Big(\int_a^b \Big[\frac{s^2 f^{**}(s)}{[n \omega_n (\sinh F(s))^{n-1}]^2}\Big]^p ds\Big)^{1/p}& +b h(b)^p -a h(a)^p.
(1-x^2) \Phi(x;s|-a,a,-q^{1/2}a,q^{1/2}a|q)=(1-x^2)\sum_{j\geq 0}{(a^2;q)_j (s;q)_j\over (q;q)_j (qs/a^2;q)_j}(qx^2/a^2)^j &=\sum_{j\geq 0}{(a^2/q;q)_j (s/q;q)_j\over (q;q)_j (qs/a^2;q)_j}{1-q^{2j-1}s\over 1-q^{-1}s}(qx^2/a^2)^j.
\nabla^n \varphi^n_j = \frac{n}{2}(\varphi^n_{j + 1}- \varphi^n_{j-1}) , \Delta^n \varphi^n_j = n^2 (\varphi^n_{j + 1}+ \varphi^n_{j -1}- 2 \varphi^n_j ) .
M := [k+1], \n_1 = \ldots = n_k := 1, \n_{k+1}:= \beta+1, & a_{i1}:= \alpha_i \ \forall i \in [k], \a_{k+1,1}:= 3, \a_{k+1,2}= \ldots = a_{k+1,\beta+1}:= 1, & b := \beta + 2, \c := a, & x^*_{11}= \ldots = x^*_{k1}:= \frac{2\beta-3}{6 \beta}, \x^*_{k+1,1}:= 1, \x^*_{k+1,2}= \ldots = x^*_{k+1,\beta+1}:= \frac{1}{3}.
\mathbb{P}\Big( \log \langle f, G_n v \rangle \geq n q \Big)= \frac{r_s(v) r_s^*(f)}{\nu_s(r_s)}\frac{\exp(- n\Lambda^{*}(q))}{s\sigma_{s}\sqrt{2\pi n}}\big[ 1 + o(1) \big].
K=\sum_{anc}(b_1^+)^{n_1}\cdots (b_r^+)^{n_r}\; C_{n_1,\cdots,n_r}^{m_1,\cdots,m_r}(b^+_1b_1,b_2^+b_2,\cdots,b_r^+b_r)\;(b_1)^{m_1}\cdots (b_r)^{m_r},
C(t)=\sum_{k=1}^m \int_0^t(J_{s,0}^{-1}b_{\cdot,k}(X_s))(J_{s,0}^{-1}b_{\cdot,k}(X_s))'\d s + \sum_{q=1}^l \int_0^t\int_0^t \varphi_H(s,u)(J_{s,0}^{-1}c_{\cdot,q}(X_s))(J_{u,0}^{-1}c_{\cdot,q}(X_u))'\d s\d u.
\sum_{l=1}^{k-1}|g_{l}|^{p-2-\delta}\mathbf{P}(\tau_x>l) \le C\sum_{l=1}^{k-1}l^{p/2-1-\gamma}\mathbf{P}(\tau_x>l) &\le C\mathbf{E}[\tau_x^{p/2-\gamma/2}]\le C(1+|x|^{p-\gamma}).
\chi_{\lambda p_0}^{0,\alpha,\beta}({\bf 0}, z)=\begin{cases}\exp [ - \frac{1}{2}C_{\alpha, \beta}p_0 \lambda(h_\alpha(z)^2 + h_\beta(z)^2 ) ], \overline{h}_0(z)h_\beta (z)>0, \exp [ - \frac{1}{2}C_{\alpha, \beta}p_0 \lambda h_\alpha(z)^2 ], \overline{h}_0(z)h_\beta (z)>0,\end{cases}
F(t,X_{t}) \ = \F(0,X_{0}) + \int_{0}^{t}\partial_t F(s,X_s)ds + \int_{0}^{t}\partial_{x}F(s,X_s)d^-X_s &\ + \frac{1}{2}\int_{0}^{t}\partial^2_{x\, x}F(s,X_s)d[X]_s, \qquad\qquad0\leq t\leq T.
f(x_{i},\beta)=f^{k}(x_{i},\beta)+\sum_{j}J_{ij}\Delta\beta_{j}+\frac{1}{2}\sum_{j}\sum_{k}\Delta\beta_{j}\Delta\beta_{k}H_{jk_{(i)}},H_{jk_{(i)}}=\frac{\partial^{2}f(x_{i},\beta)}{\partial\beta_{j}\partial\beta_{k}}.
\left\Vert a+b \right\Vert_{\mathcal{P}_n}\le\sum_{j=1}^l \left\Vert a_j \right\Vert^n + \sum_{j=1}^m \left\Vert b_j\right \Vert^n\le\left\Vert a \right\Vert_{\mathcal{P}_n}+ \left\Vert b \right\Vert_{\mathcal{P}_n}+\varepsilon,
H_{D}= \int{d^{3}\sigma \Big [ - A_{\tau}( \tau , \vec{\sigma}) \Gamma^{\ast}( \tau , \vec{\sigma}) + \lambda^{\mu}( \tau , \vec{\sigma}){\cal H}_{\mu}^{\ast}( \tau , \vec{\sigma}) + \mu_{\tau}( \tau , \vec{\sigma}) \pi^{\tau}( \tau , \vec{\sigma}) \Big ]}
\aligned \frac d{dt}\mathcal{YM}_e^0(\nabla^t)_{\big{|}_{s =0}}= \frac d{dt}\mathcal{YM}_e (\nabla^t)_{\big{|}_{s =0}}& =\int_M\exp\, (\frac12||R^\nabla ||^2)\langle R^\nabla ,d^\nabla A \rangle \, dv &=\int_M\langle\delta^\nabla \big(\exp\, (\frac 12||R^\nabla ||^2)R^\nabla \big), A \rangle \, dv\, .\endaligned
\lim_{m\to\infty}\dfrac{\beta(m)}{\|z\|_{\alpha}}= \Big(\lim_{m\to\infty}\dfrac{\beta(m)}{m^{t}}\Big)\Big(\lim_{m\to\infty}\dfrac{m^{t}}{\|z^m\|_{\alpha}}\Big) & = \ell\lim_{m\to\infty}\dfrac{m^{-(\alpha+1)/2}}{\|z^m\|_{\alpha}}.
d s^{2}= ( H_{1}H_{2}. . . H_{n})^{- 4 t / r}\eta_{\mu \nu}d y^{\mu}d y^{\nu}+ ( H_{1}H_{2}. . . H_{n})^{4 u / r )}[ \sum_{i = 1}^{n}H_{i}^{- 2 / r}d z_{i}^{m_{i}}d z_{i}^{m_{i}}+ d x^{\gamma}d x^{\gamma}] ,
\partial_{x}R(x,x^{\prime})+\partial_{x^{\prime}}R_{aj}(x,x^{\prime})=M^{-1}\left[-\frac{(x+x^{\prime})}{2\pi}+A(x,x^{\prime}) +{\Large\sum_{n\geq 1}}\frac{m^{2}M^{-1}T^{2}\sin n(x+x^{\prime})}{n\pi[n^{2}-m^{2}M^{-1}T^{2}]}\right].
S^{(3)}_{source}= \int d^4x \left( \sqrt{\det (G_{E}+e^{-\phi/2}{\cal F})}+{i\over 4}\chi{\cal F}*{\cal F}+{i \over 2}B^R *{\cal F}+{i\over24}\epsilon^{\alpha\beta\gamma\delta}A^{(4)}_{\alpha\beta\gamma\delta}\right),
{\cal A}= \frac 1{\sinh | v_{1}- v_{2}|}\int_{0}^{\infty}d \tau \int_{0}^{\infty}d l^{\prime}\int \frac{d^{2}\vec{k}_{T}}{( 2 \pi )^{2}}e^{i \vec{k}\cdot \vec{b}}e^{- \frac{q^{2}}2 \tau}e^{- \frac{k^{2}}2 l^{\prime}}< e^{i p \cdot X}>_{o s c}{\cal N}
\mathfrak{C}_{00}=\mathfrak{C}_{01},\;\mathfrak{C}_{11}=\mathfrak{C}_{01},\; \mathfrak{C}_{1a}=\mathfrak{C}_{0a},\;\mathfrak{C}_{ab}=- \frac{\mathfrak{C}_{01}}{g_{11}}g_{ab},\; \mathfrak{\widetilde{C}}=-\frac{n-1}{g_{11}}\mathfrak{C}_{01};\;a,b=2,...,n;
\hphantom{{}={}}D_{(o,o)}(\{f,g\}.\{f',g'\}) * =\begin{pmatrix}Df(f'(o))Df'(o) Dg(f'(o))Df'(o)+Hf(f'(o))Df'(o)g'(o)+Df(f'(o))Dg'(o) Df(f'(o))Df'(o)\end{pmatrix}
\widehat{w'}(\xi,y_d)=\int^{\infty}_0 \frac{\xi}{|\xi |}\frac{e^{-|\xi|y_d}- e^{-\omega_\lambda(\xi)y_d}}{\omega_\lambda(\xi)(\omega_\lambda(\xi)-|\xi|)}\,e^{-\omega_\lambda(\xi)z_d}\xi \cdot \widehat{f}'(\xi,z_d)dz_d
\mathbb{E}\bigl [ \mathbb{E}\bigl(e^{i y_1 (M(T_3-T_2))}|T_2,T_3\bigr) \mathbb{E}\bigr(e^{i y_2 (M(T_2-T_1))}|T_1,T_2\bigr)\bigr]= \mathbb{E}e^{iy_1M(T_3-T_2)}\mathbb{E}e^{i y_2M(T_2-T_1)},
S\begin{pmatrix}z_1^k-x_1^k (z_2^k-x_2^k)-(z_{1}^k-x_{1}^k) +\lambda b_2^k \vdots (z_{n-1}^k-x_{n-1}^k)-(z_{n-2}^k-x_{n-2}^k)+\lambda b_{n-1}^k x_n^k \end{pmatrix}\ni\begin{pmatrix}x_1^k-x_n^k x_2^k-x_n^k \vdots x_{n-1}^k-x_n^k x_1^k-x_n^k + \lambda\displaystyle\sum_{i=1}^{n-1}b_{i+1}^k\end{pmatrix},\small
G ( \epsilon ) = \exp \left[ \frac{V}{( 2 \pi )^{( n - 1 )}}\sum_{\sigma}\int d^{n - 1}p \epsilon ( p ) [ d_{p}^{( \sigma )}\bar{d}_{\tilde{p}}^{( - \sigma )}- d_{p}^{( \sigma ) \dagger}\bar{d}_{\tilde{p}}^{( - \sigma ) \dagger}] \right]{,}
\mathcal{R}^+_2 = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{\frac{C_2}{m}\leq c \leq \frac{C_1}{m}}\frac{S(n, p; c)}{c}H_{m, n}^+ \Bigl( \frac{4 \pi \sqrt{n p}}{c}\Bigr)
r(u, 2, 2, 2; n) = \frac1{(q)_n}\sum_{j\geq 0}\frac{\left(q^{-n}, q^{1-n}; q^2\right)_j}{\left(q^2, -q^2; q^2\right)_j}\left(-q^{1+2n-2u}\right)^j & = \frac{1}{(q)_n}{}_2 \phi_1 \left(\begin{array}{cc}q^{-n}, q^{1-n}-q^2 \end{array}; q^2; -q^{1+2n-2u}\right).
\sum_{i=1}^d \left( \lambda_i-\lambda_k \right)^2 \langle q_k, p_i \rangle^2 \leq 2B\left(\lambda_1 \right)^2 \sum_{i=1}^d \left( \lambda_i-\widehat \lambda_k \right)^2 \langle q_k, p_i \rangle^2 \leq B\left(\lambda_1 \right)^2,
R_{x}^{\alpha}\psi_{n}(x) =F_{1}(x)=-\frac{1}{2\Gamma(-\alpha)\cos\alpha\pi/2}\int_{-a}^{a}\frac{\psi_{n}(x^{\prime})}{(x-x^{\prime})^{\alpha+1}}& =-\frac{A}{2\Gamma(-\alpha)\cos\alpha\pi/2}\int_{-a}^{a}\frac{\sin\frac{n\pi}{2a}(x^{\prime}+a)}{(x-x^{\prime})^{\alpha+1}},x\geq a
C^{\diamond}\leq \max_{p(x_1,x_2)}\min_{p(u|x_1,x_2,y)=p(u|y)}\min\left\{\begin{array}{l}C_1+C_2, C_1+I(X_2;Y|X_1), C_2+I(X_1;Y|X_2), I(X_1X_2;Y), \frac{1}{2}\left( C_1+C_2+I(X_{1}X_{2};Y|U)+I(X_{1};U|X_{2})+I(X_{2};U|X_{1})\right)\end{array}\right\}.
2 \pi x \int_{1}^{\infty}{\frac{\pi^{2}B_{2}( t - [ t ] )}{( x^{2}+ ( 2 \pi t )^{2})^{{\frac{3}{2}}}}}d t = 2 \pi x \sum_{m = 1}^{\infty}\int_{0}^{1}{\frac{\pi^{2}B_{2}( t )}{( x^{2}+ ( 2 \pi ( t + m ) )^{2})^{{\frac{3}{2}}}}}d t \ ,
T_{i k}= ( 1 - 2 \xi ) \partial_{i}\varphi \partial_{k}\varphi + ( 2 \xi - 1 / 2 ) g_{i k}\partial^{l}\varphi \partial_{l}\varphi - 2 \xi \varphi \nabla_{i}\nabla_{k}\varphi + ( 1 / 2 - 2 \xi ) m^{2}g_{i k}\varphi^{2}.
y_n =\frac{1}{n+1}\sum_{k=1}^{n}\; \sum_{\ell\le k}(-1)^\ell \binom{n+k}{k}\binom{k}{\ell}\frac{(k-\ell)!}{(n-q\ell)!}B_{n-q\ell,k-\ell}(\bar x),
\alpha_1+\tau=\tilde{a}_1,\ \alpha_2+\alpha_1\tau+\frac{1}{2}\tau^2=\tilde{a}_2,\ \alpha_3+\alpha_2\tau+\frac{1}{2}\alpha_1\tau^2+\frac{1}{6}\tau^3=\tilde{a}_3, \alpha_4+\alpha_3\tau+\frac{1}{2}\alpha_2\tau^2+\frac{1}{6}\alpha_1\tau^3+\frac{1}{24}\tau^4=\tilde{a}_4.
\||x|^{\frac12}f_{1,+}\|_{L^2}^2+t\|\partial_xf_{1,+}\|_{L^2}^2=\|L_-f_{1,+}\|_{L^2}^2+4t\Im\int(|x|^{\frac14}f_{1,+})\partial_x\overline{(|x|^{\frac14}f_{1,+})}\,dx.
\|(r_{\textrm{primal,1}}^{i,(l)},r_{\textrm{primal,2}}^{i,(l)})\|^2 \leq\frac{\epsilon_{\textrm{feas}}^2}{N}, \|r_{\textrm{dual}}^{i,(l)}\|^2 \leq\frac{\epsilon_{\textrm{feas}}^2}{N}, \hat{\eta}^{i,(l)}=(s^{i,(l)})^T\lambda^{i,(l)}\leq\frac{\epsilon}{N},
M_t = Y_t -Y_0 +A^{W,W}_t(\nabla u^*\, b ) + A^{W,W}_t(\tilde f) = Y_t -Y_0 +A^{W,W}_t(\nabla u^*\, b + \tilde f) = Y_t -Y_0 -A^{W,W}_t(-\nabla u^*\, b - \tilde f).