harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pi_{\mu \nu}^{( \mathrm{t a d p o l e})}= \frac{1}{2}( 2 e^{2}\delta_{\mu \nu}) \frac{1}{\beta}\sum_{n}\int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{1}{k_{0}^{2}+ \omega_{\sigma}^{2}}= e^{2}\delta_{\mu \nu}\frac{\beta}{( 2 \pi )^{2}}\sum_{n}\int \frac{d^{2}k}{( 2 \pi )^{2}}\, \frac{1}{n^{2}+ ( \frac{\beta \omega_{\sigma}}{2 \pi})^{2}},
\widetilde{q}(t, x) =\widetilde{q}(t,x; r, \alpha_0) &\equiv \lim_{\widehat{\beta}_0 \to \infty}\widehat{q}(t, x; t, \alpha_0, i \widehat{\beta}_0)= \frac{\sin \zeta_1}{\sin \eta_1},
\mu^{2}( t ) = \frac{\lambda_{\mathrm{R}}}{2}\tilde{I}_{1}^{\phi}( t ) - \frac{1}{2}a^{2}( t ) \hat{m}^{2}\left( 1 + \frac{\lambda_{\mathrm{R}}}{1 6 \pi^{2}}\bar{c}\right) + O ( \lambda_{\mathrm{R}}^{2}) ,
0 = \nabla_{(\alpha}\overline{\xi}_{\overline{\beta})a}=\partial_{(\alpha}\overline{\xi}_{\overline{\beta})a}-H_{\alpha \overline{\beta}}^{\, \, \,\overline{\gamma}}\overline{\xi}_{\overline{\gamma}a}+H_{\alpha \overline{\beta}}^{\, \, \, \gamma}\xi_\gamma .
\sup_{|x-\theta|<1}|g^\prime(x)| = \sup_{|x-\theta|<1}|2|(a\lambda_2^2-b\mu_2^2)x+(a\lambda_1\lambda_2-b\mu_1\mu_2)| \leqslant 2\sqrt{ab}|\lambda_2\mu_1-\lambda_1\mu_2|+2|a\lambda_2^2-b\mu_2^2|+4|a\lambda_1\lambda_2-b\mu_1\mu_2|.
( (a-bt)^2 - (c-dt)^2 J ) \cdot bd - (ad + bc - 2bdt) \cdot (ab - cd J) = ( a^2 - c^2 J - 2abt + 2cdtJ ) \cdot bd - (a^2bd + ab^2c - 2ab^2dt - acd^2J - bc^2dJ + 2bcd^2tJ) = -ab^2c + acd^2 J = 0,
\|x_{2n+2}-x_{0}\| \le\|x_{2n+2}-x_{2n+1}\|+\|x_{2n+1}-x_{0}\|\le\|x_{2n+2}-x_{2n+1}\| &+\sum_{k=1}^{n}\left(\|x_{2k+1}-x_{2k}\|+\|x_{2k}-x_{2k-1}\|\right)+\|x_{1}-x_{0}\| &\le2\sum_{k=1}^{n}\|x_{2k}-x_{2k-1}\|+2\|x_{1}-x_{0}\| &\le 2\sum_{k=0}^{n}c^k\|x_{1}-x_{0}\|\le\frac{2}{1-c}\|x_1-x_0\|.
\tau^{-1}\geq\min_{1\leq i\leq N}\sum_{j=1}^{N}a_{ij}\Pr\left[X_{j}=S|X_{i}=I\right] \geq\min_{1\leq i\leq N}\left( \min_{\left(k,l\right) \in\mathcal{L}}\Pr\left[ X_{k}=S|X_{l}=I\right] d_{i}\right) & =\min_{\left( k,l\right) \in\mathcal{L}}\Pr\left[ X_{k}=S|X_{l}=I\right] d_{\min}
Q(4\eta; -8\eta, \eta) = \frac{\theta(4\eta; -8\eta) \theta'(0, -8\eta)}{\theta(2\eta; -8\eta) \theta(6\eta; -8\eta)}= -2\pi i e^{4\pi i \eta}\frac{\theta_0(4\eta; -8\eta) (-8\eta; -8\eta)^2}{\theta_0(2\eta; -8\eta) \theta_0(6\eta; -8\eta)}.
(\Phi^t)_{xy}=[(QQ^*)^t]_{xy}=\sum_{k}\mu_k^{2t}v_k(x)v_k(y) \leq \rho^{2t-2}\sum_{k}\mu_k^{2}|v_k(x)||v_k(y)| \leq \rho^{2t-2}\sqrt{(\Phi)_{xx}}\sqrt{(\Phi)_{yy}}\leq \frac{K^2 \rho^{2t}}{n},
\frac{(\begin{matrix}m\\k\end{matrix})(\begin{matrix}N-m\\n-k\end{matrix})}{(\begin{matrix}N\\n\end{matrix})}=\frac{(\begin{matrix}6\\ 3\end{matrix})(\begin{matrix}43\\ 3\end{matrix})}{(\begin{matrix}49\\ 6\end{matrix})}=\frac{20\cdot12341}{13983816}=0.0176504
A_{c, E}(\theta)=\frac{1}{f(\theta)}\left(\begin{matrix}\frac{E-v(\theta)}{g(\theta)}\ \ -\frac{\tilde{c}(\theta-\alpha)}{g(\theta)}f(\theta)\ \ 0\end{matrix}\right)\triangleq\frac{1}{f(\theta)}D_{c, E}(\theta).
\mu_K(\gamma)\!\!= \!\! \int_{K_{\rm min}}^{K_{\rm max}}\frac{dF}{dK}K\, dK =\!\!\!\frac{K_{\rm min}\exp(\gamma K_{\rm min})- K_{\rm max}\exp(\gamma K_{\rm max})}{\exp(\gamma K_{\rm min})-\exp(\gamma K_{\rm max})}-\frac{1}{\gamma}.
{\cal M}= \int \frac{d^D\! q}{(q-p)^2 (q\!\cdot\! n)}\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\!\!\!\! ML}\int \frac{d^D\! q \, (q\!\cdot\!n^*)}{(q-p)^2 \left[(q\!\cdot\! n) (q\!\cdot\! n^*) +i\epsilon\right]}.
S_{B F}[ A , \phi ] = \int d^{2}x \, t r \left( \phi^{0}F^{0}+ \phi^{a}F^{b}\right) = \int d^{2}x \, \left( - \phi^{0}F_{\alpha \beta}^{0}\epsilon^{\alpha \beta}+ \Lambda \delta_{a b}\phi^{a}F_{\alpha \beta}^{b}\epsilon^{\alpha \beta}\right) .
\begin{array}{r c c c c c c}{{}}&{{{\cal G}^{( 1 )}}}&{{\oplus}}&{{{\cal G}^{( 1 )}}}&{{\supset}}&{{{\cal G}^{( 1 )}}}&{{}}\\{{\mathrm{l e v e l ~}}}&{{\ell - s}}&{{}}&{{s}}&{{}}&{{\ell}}&{{,}}\\ \end{array}
D_{\hat\alpha}\xi_{\hat\beta}=-{\frac{1}{4}}(\Gamma^{\mu\nu})_{\hat\beta\hat\alpha}\partial_\mu\xi_\nu+{\frac{1}{12}}C_{\hat\beta\hat\alpha}\partial_\sigma\xi^\sigma+{\frac{1}{12}}(\Gamma^{ij})_{\hat\beta\hat\alpha}D_{\sigma[ij]}\xi^{\sigma},
T_{\mu \nu}( k ) = i \int \frac{d^{2}p}{( 2 \pi )^{2}}\int_{0}^{1}d \xi_{1}\int_{0}^{1}d \xi_{2}\delta ( 1 - \xi_{1}- \xi_{2}) \left[ \frac{P ( m )}{\left[ z - m^{2}+ i \varepsilon \right]^{2}}\right]_{z = \sum_{j = 1}^{2}\xi_{j}p_{j}^{2}},
{\frak C}^2 \sum_{x=0}^n |\tilde V_x(m)|^2\le \left[{\frak M}|\tilde V_0(m)| + \left\{\sum_{x=0}^n |\tilde v_{x}(m)|^2\right\}^{1/2}\right]^2 \le 2{\frak M}^2 |\tilde V_0(m)|^2 + 2 \sum_{x=0}^n |\tilde v_{x}(m)|^2.
|T^n|^2=(G^n)^*(R_n^*R_n)G^n \leq\|R_n\|^2(G^n)^*G^n\leq(1+4\varepsilon)^{2n}(G^n)^*G^n, |T^n|^2=(G^n)^*(R_n^*R_n)G^n \geq\|R_n^{-1}\|^{-2}(G^n)^*G^n\geq(1+4\varepsilon)^{-2n}(G^n)^*G^n.
\gamma\langle \nabla E(\mu)- \nabla E(\zeta),\mu-\zeta\rangle \le -\frac{1}{n}\sum_{i=1}^n\int_{\Omega}\left[(\phi^{\mu,\nu_i})^{cc}(x)-(\phi^{\zeta,\eta_i})^{cc}(x)\right]d(\mu-\zeta)(x) &=-\frac{1}{n}\sum_{i=1}^n \iint_{\Omega\times\Omega}\left[(\phi^{\mu,\nu_i})^{cc}(x)-(\phi^{\zeta,\eta_i})^{cc}(x)\right]d(\pi^{\mu,\nu_i}-\pi^{\zeta,\eta_i})(x,y),
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{\mathbf{y}_{k-1}^2}{\mathbf{x}_{k}^2}= \frac{A_{k,k-1}(1+a/(k-1))^{-1/2}}{A_{k,k+1}(1+a/k)^{1/2}}= \frac{1}{1+a/k}\frac{q_k}{p_k}(1+O(k^{-2})) = \frac{1}{1+a/k}\frac{s_{k+1}-s_k}{s_{k}-s_{k-1}}(1+O(k^{-2})) .
\delta \left( p\right) \star \delta \left( x\right) =\frac{1}{\pi \left|\theta \right|}\exp \left( -\frac{2ixp}{\theta}\right) ,\quad \delta\left( x\right) \star \delta \left( p\right) =\frac{1}{\pi \left| \theta\right|}\exp \left( \frac{2ixp}{\theta}\right) .
\Omega^{\scriptscriptstyle 1}( M ) = k e r ( \mathrm{1 \m k e r n - 5 m u I \!}\hspace{2 p t}+ \ast \hspace{2 p t}d_{\scriptscriptstyle 1}) \oplus k e r ( \mathrm{1 \m k e r n - 5 m u I \!}\hspace{2 p t}+ \ast \hspace{2 p t}d_{\scriptscriptstyle 1})^{\perp}
\lambda_j=q, & v_j= (\underbrace{1,\omega_j,\omega_j^2, \cdots, \omega_j^{q-1}}_{q}),\intertext{and if $\gcd(j,q)>1$, then}\lambda_j=0, & v_j= (\underbrace{1,\omega_j,\omega_j^2, \cdots, \omega_j^{q'-1}}_{q'},\underbrace{1,\omega_j,\omega_j^2, \cdots, \omega_j^{q'-1}}_{q'}, \cdots,\underbrace{1,\omega_j,\omega_j^2, \cdots, \omega_j^{q'-1}}_{q'}),
\|e^{u^\varepsilon}\|_{L^{a_\varepsilon}(\gamma)}= \| P_{\varepsilon}[e^{-\frac{\phi_\varepsilon}{2\varepsilon}}] \|_{L^{-p_\varepsilon}(\gamma)}^{-2\varepsilon}= \| P_{\varepsilon}[e^{-\frac{a_\varepsilon \phi_\varepsilon}{p_\varepsilon}}] \|_{L^{-p_\varepsilon}(\gamma)}^{-2\varepsilon}\le \|e^{\phi_\varepsilon}\|_{L^{-a_\varepsilon}(\gamma)}.
F_2=-\frac32 \int u_1\phi_N \partial_x(\partial_xw)^2\,dx =-\frac32 \int \partial_x(u_1\phi_N) w\partial_x^2w\,dx +\frac34\int \partial_x^3(u_1\phi_N)w^2\,dx &\equiv F_2^1+F_2^2.
\deg x_{01}= (1,0,0) \deg x_{11}= (1,0,0) \deg y_{1}= (2,0,0) \deg x_{02}= (0,1,0) \deg x_{12}= (0,1,0) \deg y_{2}= (0,2,0) \deg x_{03}= (0,0,1) \deg x_{13}= (0,0,1) \deg y_{3}= (0,0,2)
\frac{\partial\psi}{\partial t}=(-i\frac{mc^{2}}{\hbar}\phi+\frac{\partial\phi}{\partial t})e^{-\frac{i}{\hbar}mc^{2}t}\approx-i\frac{mc^{2}}{\hbar}\phi e^{-\frac{i}{\hbar}mc^{2}t}
\left\langle \exp \left( \; 2 \sum_{b = 1}^{N}\beta^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) \; : \cos \left( 2 \sqrt{\pi}\varphi^{( b )}( x ) \; \right) :_{Q^{W}}\; \right) \right\rangle \; .
\|\widehat{V}\|_{S_{\widehat{p}_{2}}}=\min_{V\in\mathbb{R}^{d\times d},W\in\mathbb{R}^{n\times d}:\widehat{V}=(VW^{T})^{T}}\|V\|_{S_{p_{2}}}\|W\|_{S_{p_{3}}}.
\varphi\wedge\omega_L=\sum_{j=1}^n\varphi_{jj}L_j\bigwedge_{k=1}^n\frac i{2\pi}\frac{dz_k\wedge d\bar z_k}{(1-|z_k|^2)^2}=\bigl(\sum_{j=1}^n L_j\varphi_{jj}\bigr) d\nu(z)\ ,
\alpha . J = - b I + a J , \ \, a l p h a^{- 1}= \frac{1}{a^{2}+ b^{2}}( a I - b J ) , \ \, a l p h a^{- 1}. J = \frac{1}{a^{2}+ b^{2}}( b I + a J ) , \quad d e t ( \alpha ) = a^{2}+ b^{2}
\bar{\partial}\varphi=\sum_{\lambda,v=1}^n \left(\sum_{\beta=1}^n\frac{\partial \varphi_v^{\lambda}}{\partial \bar{z}_{\beta}}d\bar{z}_{\beta}+\sum_{\mu=1}^m \frac{\partial \varphi_v^{\lambda}}{\partial \bar{t}_{\mu}}d\bar{t}_{\mu}\right) \wedge d\bar{z}_v\frac{\partial}{\partial z_{\lambda}}
\frac{\partial}{\partial\overline{z}_{j}}(f_{j}^{-1}\circ f)(z)=\sum_{k}\frac{\partial f_{j}^{-1}}{\partial w_{k}}(w)\frac{\partial f_{k}}{\partial\overline{z}_{j}}(z)+\sum_{k}\frac{\partial f_{j}^{-1}}{\partial\overline{w}_{k}}(w)\frac{\partial\overline{f}_{k}}{\partial\overline{z}_{j}}(z)
\chi_{\bar{k}}= \left({\frac{1 - \Gamma}{2}}\right) = \left({\frac{1 - \gamma^{0}\gamma^{9}}{2}}\right) \ , \qquad \chi_{k}= \left({\frac{1 + \Gamma}{2}}\right) = \left({\frac{1 + \gamma^{0}\gamma^{9}}{2}}\right) \ .
\langle T^{\mu}_{\ \mu}\rangle = \frac{1}{48\pi}\Biggl[ \left(\partial_{\mu}\partial_{\nu}h^{\mu\nu}-\partial_{\mu}\partial^{\mu}h^{\nu}_{\ \nu}\right) \mp\frac{1}{2}\varepsilon_{\mu\lambda}\partial^{\lambda}\partial_{\nu}h^{\mu\nu}\Biggr] \, .
{\rm either}\quad{\cal S}^{ab}= \tilde{S}^{ab}= -\frac{i}{4}[\tilde{a}^a, \tilde{a}^b] = -\frac{i}{4}[\tilde{\gamma}^a, \tilde{\gamma}^b] , \quad{\rm or}\qquad{\cal S}^{ab}= \tilde{\tilde{S}}{}^{ab}= -\frac{i}{4}[\tilde{\tilde{a}}{}^a, \tilde{\tilde{a}}{}^b].
{\mathbb P}\{Y_i^{(m)}-X_{i-1}^{(m)}=z''-z-\langle{\bf n},z''-z\rangle{\bf n}\mid X_{i-1}^{(m)}=z\}=~{\mathbb P}\{Y_i^{(m)}-X_{i-1}^{(m)}=z'-z-\langle{\bf n},z'-z\rangle{\bf n}\mid X_{i-1}^{(m)}=z\}
\sum_{h\in H_{x}}\mu(h) =\sum_{h\in H_{x}}\phi_0(s(h))\mu_0(h)+\phi_1(s(h))\mu_1(g) = \phi_0(x)\sum_{h\in (H_0)_x}\mu_0(h)+\phi_1(x)\sum_{h\in (H_1)_x}\mu_1(h) & \leq \phi_0(x)+\phi_1(x)\leq 1.
R_{\mathcal{BS}(\alpha)}(\mathcal{S^{*}(\beta)}) =\begin{dcases}\dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{1+16\alpha (1-\beta)^2}}, \ 0\leq \beta<\max\left\{0; \dfrac{9\alpha-1}{8\alpha}\right\}, \dfrac{1}{1+2(1-\alpha)(1- \beta)}, \ \max\left\{0; \dfrac{9\alpha-1}{8\alpha}\right\}\leq \beta< 1.\end{dcases}
T(z) = R(z)E_{-x_{1}}(z)\Phi_{\mathrm{Be},0}(rf_{-x_{1}}(z);0)s_{2}^{-\frac{\sigma_{3}}{2}}\begin{pmatrix}1 \frac{s_{2}}{2\pi i}\log \frac{r(z+x_{1})}{4}0 1\end{pmatrix}e^{\frac{\pi i \alpha}{2}\theta(z)\sigma_{3}}e^{-\sqrt{r}g(z)\sigma_{3}}.
\log \prod_{k=n+1}^{n^2}\frac{c k^\frac12}{1+ c k^\frac12}=\sum_{k=n+1}^{n^2}\log \Big(1- \frac{1}{1+ ck^\frac12}\Big)\le - \sum_{k=n+1}^{n^2}\frac{1}{1+ ck^\frac12}\lesssim - n^{\frac12},
\mu := \mathbb{E}(Y_i) = \begin{pmatrix}p_{n,x,u}p_{n,y,v}\end{pmatrix}V := \mathrm{Cov}(Y_i) = \begin{pmatrix}p_{n,x,u}(1-p_{n,x,u}) p_\cap-p_{n,x,u}p_{n,y,v}p_\cap-p_{n,x,u}p_{n,y,v}p_{n,y,v}(1-p_{n,y,v}) \end{pmatrix},
L_s \cdot \varepsilon_\theta = q^{-k}(E_{s}+ R_{s}\Psi_{s})(\varepsilon_\theta) = q^{-k}((q^k - 1)\varepsilon_\theta - R_{s}\varepsilon_\theta) = (1 - q^{-k})\varepsilon_\theta - q^{-k}R_{s}\cdot \varepsilon_\theta,
\inf_{x\in \mathbb R^d, i\in \mathcal M}P(\eta_{\alpha}>\eta+ \delta: \alpha_{\eta}=i, X_{\eta}=x )\geq 1-\frac{1}{4}P(B), \inf_{y\in \mathbb R^d, i\in \mathcal M}P(\eta_{\tilde\alpha}>\eta+ \delta: \tilde\alpha_{\eta}=i, Y_{\eta}=y )\geq 1-\frac{1}{4}P(B).
S_n(p^dq) =\sum_{j=0}^{p^{\delta}q-1}\sum_{i=1}^{p^c}(jp^c+i)^n= \sum_{j=0}^{p^{\delta}q-1}\sum_{i=1}^{p^c}\left(i^n+\sum_{k=1}^{n}\binom{n}{k}(j p^c)^k i^{n-k}\right) &= \sum_{j=0}^{p^{\delta}q-1}\sum_{i=1}^{p^c}i^{n}+\sum_{k=1}^{n}\binom{n}{k}p^{ck}\sum_{j=0}^{p^{\delta}q-1}j^k \sum_{i=1}^{p^c}i^{n-k}.
\sqrt{\frac{1+z}{1-z}}(\frac{1}{6}(\frac{z^{2}}{2})^{3}+\frac{z^{2}}{2}\frac{z^{4}}{4}+\frac{z^{6}}{6})=\frac{5}{16}z^{6}\sqrt{\frac{1+z}{1-z}}.
\begin{dcases}\sum_{e\in E(G)}w_i(e)=1, \sum_{e:\,v\in e}B_i(v,e)=1,\ \v\in V(G), w_i(e)^{1-r/p_i}\prod_{v\in e}(B_i(v,e))^{1/p_i}=\alpha_i^{1/p_i},\ \e\in E(G).\end{dcases}
z_{k}= \left( \begin{array}{c c}{{E - 2 \cos \left( ( k - 1 ) \frac{2 \pi}{N}+ k_{x}\right)}}&{{- e^{{\mathrm i}k_{y}}}}\\{{e^{-{\mathrm i}k_{y}}}}&{{0}}\\ \end{array}\right) \cdot z_{k - 1}
{\cal L}_{2}={\frac{1}{2}}E^{+ +}\wedge E^{--}- i \Pi^{\underline{{{m}}}}\wedge d \Theta \Gamma_{\underline{{{m}}}}\Theta +{\frac{1}{2}}E^{--}\wedge \Psi_{-}^{I}d \Psi_{-}^{I}, \qquad
\Gamma_{+}^{2}= \frac{1}{( 2 \pi )^{2}}\int d t ~ d^{2}{\bf x}\int d^{2}{\bf p}~ \left[ \frac{- e^{2}}{2}\pi \delta ( p_{1}, p_{2}) \epsilon_{i j}a_{j}\dot{a}_{i}\right] ~ e^{i{\bf p}\cdot{\bf x}},
\beta_{0 0}( t ) = \frac{\pi g \mathrm{e}^{- \pi g t / 4}}{1 6 \tilde{\omega}\tilde{\tilde{\omega}}}\left[ \left( 2 \pi g - i \tilde{\tilde{\omega}}\right) \mathrm{e}^{i \tilde{\tilde{\omega}}t}- \left( 2 \pi g - i \tilde{\tilde{\omega}}\right) \mathrm{e}^{- i \tilde{\tilde{\omega}}t}\right] \; .
e_{i}(v_{j_{1}}\otimes\cdot\cdot\cdot\otimes v_{j_{i}}\otimes v_{j_{i+1}}\otimes\cdot\cdot\cdot\otimes v_{j_{k}})=\delta_{j_{i},j_{i+1}}\sum_{j=1}^{n}v_{j_{1}}\otimes\cdot\cdot\cdot\otimes v_{j}\otimes v_{j}\otimes\cdot\cdot\cdot\otimes v_{j_{k}}.
\mathbb{E}_{\widetilde{p}_n^{(\beta)}}[{X_i X_j}] =\mathbb{E}_{\widetilde{p}_n}[X_i{X}_j] =\frac{1}{n^{1/2}2^d}\sum_{l=1}^n\frac{(2n^{-1/2d})^{d-2}[4n^{-1/2d}X_{il}][4n^{-1/2d}X_{jl}]}{4}&=\frac{1}{n}\sum_{l=1}^n X_{il}X_{jl}~~ i,j\in\{1,\dots,d\}~ ~ i< j.
\left \|\uppercase\expandafter{\romannumeral1}_{1}\right \|_{L_{\omega}^{p}}=\left \|\sum\limits_{j=1}^{\infty}\mu_{j}h_{j}(x)\right \|_{L_{\omega}^{p}}\leq C\left \|\sum\limits_{j=1}^{\infty}\frac{\mu_{j}\chi_{4P_{j}}}{\omega(P_{j})^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}.
\int_{DE}e^{sx}\overline{G}(s,t)ds=\int_{r}^{-r+\lambda_2-\lambda_1}e^{-x\lambda_1}e^{-wx}e^{t(c_1\lambda_1^{\alpha_1}+c_2\lambda_2^{\alpha_2})}e^{-t[c_1(w^{\alpha_1}e^{i\pi \alpha_1})+c_2(\lambda_2-\lambda_1+we^{i\pi})^{\alpha_2}]}dw.
\frac{\varepsilon(\lambda, x)}{\Vert x \Vert^2}= \int_{\mathbb{R}^n}\frac{\alpha(y, x)}{\Vert x \Vert^2}\eta_{\delta}(\lambda - y) dy \rightarrow 0 \,\, \,\, \Vert x \Vert \rightarrow 0, \,\, \forall \lambda \in \mathbb{R}^n.
\begin{aligned}\mathcal{R}((xw^*)*R(ww^*)^*) =\mathcal{R}((ww^*)*R(ww^*)^*), \mathcal{R}(\mathcal{F}((xw^*)*R(ww^*)^*)) =\mathcal{R}(\mathcal{F}((ww^*)*R(ww^*)^*)).\end{aligned}
A_{k}\left( \overrightarrow{\eta}( \overrightarrow{X}, t ) , t \right) = \frac{1}{2}\hat{A}_{l}( \overrightarrow{X}, t ) \left( \delta_{k l}+ \frac{e_{k l}( \overrightarrow{X}, t )}{1 + e \theta \widehat{B}}\right)
\chi_T =\left(\frac{\partial J}{\partial \Omega}\right)_T =2^{5/3}\pi^{1/3}\kappa^{-4/3}Nmr_+^{-1}\frac{r_+^4 +4l^2r_+^2 -l^4}{3r_+^4 +4l^2r_+^2 +l^4}\frac{3r_+^2 +l^2}{3r_+^2-l^2}.
\mathcal{D}_L M_\omega T_x f(t) = |\det(L)|^{-1/2}e^{2 \pi i \omega \cdot (L^{-1}t)}f(L^{-1}t - x) & = |\det(L)|^{-1/2}e^{2 \pi i (L^{-T}\omega) \cdot t}f( L^{-1}(t - Lx)) & = M_{L^{-T}\omega}T_{L x}\mathcal{D}_L f(t).
\frac{\partial u^A}{\partial t}-\mathcal{M}_\Sigma(u)^A=\left( e^{|X(u)|^2/4-|X(0)|^2/4}\sqrt{\frac{\det g_0}{\det g_u}}(h_u)^{AB}(h_0)_{BC}-\delta_C^A \right)\cdot \mathcal{M}_\Sigma(u)^C.
{\mathcal S}_{\gamma}= \Big\{(\widetilde u,\widetilde \pi,\widetilde \ell,\widetilde \omega) \mid \widetilde \rho(t,y) =\left\| (\widetilde u, \widetilde \pi,\widetilde \ell,\widetilde \omega)\right\|_{\mathcal{S}}\leqslant \gamma \Big\},
[\hat{x}^\mu_k,\hat{p}_{x l\nu}]= \,\imath\,\hbar\,\delta^\mu_\nu\,\delta_{kl},[\hat{y}^\mu_k,\hat{p}_{y l\nu}]=\, \imath\,\hbar\,\delta^\mu_\nu\,\delta_{kl},\mu,\nu=1,2,3,4,\, k=1,...,N.
\left|\det \left(N_1^{1/3}M_0\right)\right|\le C^LL^{L/2}\prod\limits_{j=1}^re^{-c(-x_j)^{3/2}}\prod\limits_{j=1}^se^{-c(-x'_j)^{3/2}}\prod\limits_{j=1}^re^{-cy_j^{3/2}}\prod\limits_{j=1}^te^{-c{y'}_j^{3/2}}.
\mathbb{P}(Y=1\, |\, X) = \left(1 + \frac{1-p}{p}\exp\left\{\langle X,m_0-m_1\rangle_{K}- \mathbb{E}_{m_1}[\langle X,m_0-m_1\rangle_{K}] - \frac{1}{2}\|m_0-m_1\|^2_{K}\right\}\right)^{-1}.
\begin{aligned}\dot x_J(t) =A_Jx_J(t)+B_{J1}y(t)+B_{J2}\bigl(\eta(t)+\eta_Q(t)\bigr) u(t) =C_{J1}x_J(t)+D_0y(t)+\eta(t)+\eta_Q(t) \epsilon(t) =C_{J2}x_J(t)+y(t) \end{aligned}
\Bigl ( \frac{d}{d u}+ 2 k ~ \mathrm{s n}( u , k ) \Bigr ) \Bigl ( - \frac{d}{d u}+ 2 k ~ \mathrm{s n}( u , k ) \Bigr ) \phi ( u ) = ( \frac{m_{F}}{\omega})^{2}\phi ( u )
{\bf r}_{i+1}^{T}\mathcal{M}(\gamma){\bf r}_{i}={\bf r}_{i}^{T}\mathcal{M}(\gamma){\bf r}_{i}-\frac{{\bf r}_{i}^{T}\mathcal{M}(\gamma){\bf r}_{i}}{{\bf p}_{i}^{T}M^{T}\mathcal{M}(\gamma){\bf p}_{i}}{\bf p}_{i}^{T}M^{T}\mathcal{M}(\gamma){\bf r}_{i}.
{\tilde{\cal{E}}}(B) = \frac{B^2}{2}+ \frac{1}{8 \pi^{\frac{3}{2}}}\;\int_0^{\infty}\frac{ds}{s^{\frac{5}{2}}}\; e^{- m^2 s}\; \left[ (eBs) \coth(eBs) - 1 \right] \; .
3^K < 2^{K+L}\leq (3+\mu)^K, K \cdot \log (3) < (K+L) \cdot \log (2) \leq K \cdot \log (3+\mu) &=K \cdot \left(\log(3) + \log\left(1+\frac{\mu}{3}\right)\right) &< K \cdot \left(\log(3) +\frac{\mu}{3}\right), \delta=\frac{\log(3)}{\log(2)}<\frac{K+L}{K}<\frac{\log(3)}{\log(2)}+\frac{\mu}{3\cdot \log(2)}.
F_{w}(f_{o})=\frac{C_{d}}{2}\rho u^{2}\pi R_{0}^{2}(\frac{B(R_{0})}{B_{mp}})^{2/f_{o}}=\frac{C_{d}}{2}\rho u^{2}\pi R_{0}^{2}(\frac{\sqrt{\mu_{0}}I_{c}C_{0}}{4R_{0}u\sqrt{\rho C_{SO}}})^{2/f_{o}}
\begin{aligned}\tfrac{2L\sigma_N}{B}= 1 + h_{N-1}\tfrac{1}{2-(1+\kappa)h_{N-1}}\,\,, \tfrac{\alpha_{N-1}}{B}= \tfrac{1-\kappa h_{N-1}}{2-(1+\kappa) h_{N-1}}\end{aligned}
\sum_{t\in \mathbb F_q}\nu^2(t) \leq |E|q^{-1}\sum_{s\in \mathbb F_q}\sum_{x\in E, y, y^\prime\in F}\psi(sx\cdot (y-y^\prime)) &=q^{-1}|E|^2|F|^2 + |E|q^{-1}\sum_{s\in \mathbb F_q^*}\sum_{x\in E, y, y^\prime\in F}\psi(sx\cdot (y-y^\prime)).
E_1^{(a)}E_2^{(b)}E_1^{(c)}= \sum_{{p+r=b \atop p\leq c}\atop r\leq a}\left[\begin{array}{c}a+c-b c-p\end{array}\right]E_2^{(p)}E_1^{(a+c)}E_2^{(r)}.
[[y,x,\overset{i-1}\ldots, x],x]^2 =[[y,x,\overset{i-1}\ldots, x]^x, [y,x,\overset{i-1}\ldots, x]^{-1}] &=[[y,x,\overset{i-1}\ldots, x][y,x,\overset{i}\ldots, x], [y,x,\overset{i-1}\ldots, x]^{-1}] &=[[y,x,\overset{i}\ldots, x], [y,x,\overset{i-1}\ldots, x]^{-1}]\in [H_k,H_k]\cap \gamma_{2i+1}(G_k),
e^{-\int_{\tau_j}^{u \wedge \zeta_j}G_j(v) \: dv}X_j^*(u \wedge \zeta_j) = \int_{\tau_j}^{u \wedge \zeta_j}\mu X_{j-1}^{\tau_j, \tau_{j+1}}(w) e^{-\int_{\tau_j}^w G_j(v) \: dv}\: dw + Z_j^*(u \wedge \zeta_j),
\int\limits_0^{k-1}x^{-\alpha}\phi_{n+k}^H(x)\,dx =(n+k)\int\limits_0^{\frac{k-1}{n+k}}(1-v)^{\alpha}v^{-\alpha}dv-(n+k-1)\int\limits_0^{\frac{k-1}{n+k-1}}(1-v)^{\alpha}v^{-\alpha}dv \leq \int\limits_0^{\frac{k-1}{n+k-1}}(1-v)^{\alpha}v^{-\alpha}dv \leq \frac{1}{1-\alpha}\left(\frac{k-1}{n+k-1}\right)^{1-\alpha}.
{\hat x}^i_a(\varepsilon)=e^{i{\hat H}\varepsilon}{\hat x}^i_a (0)e^{-i{\hat H}\varepsilon}\simeq{\hat x}^i_a(0)+{{\hat \pi}^i_a(0)\over m}\varepsilon +{e\over 2m^2}\sum_{b=1}^N\epsilon^{ij}[{\hat B}({\hat{\bf x}}_b)\delta_{ab}+{e\over\theta}\delta ({\hat{\bf x}}_a-{\hat{\bf x}}_b)]{\hat \pi}^j_b(0)\varepsilon^2
\frac{\partial}{\partial \lambda_{1}}\cdots \frac{\partial}{\partial \lambda_{p + q}}\int_{- \infty}^{\infty}\: d s \: \eta \left. T_{\bullet}^{( a_{1})}\cdots T_{\bullet}^{( a_{p})}\: \overline{{T_{\bullet}^{( b_{1})}}}\cdots \overline{{T_{\bullet}^{( b_{q})}}}\right._{| \lambda_{1}= \cdots = \lambda_{p + q}= 0}.
\mp \left[ - e^{3 A}\dot{\Psi}^{( 0 )}F_{w w}+ 2 e^{A}\dot{\varphi}+ 2 e^{2 A}( e^{A}\dot{\Psi}^{( 0 )})^{\cdot}\phi_{\pm w}\right] ={V_{\pm}^{\prime \prime}}^{( 0 )}( \varphi + e^{2 A}\dot{\Psi}^{( 0 )}\phi_{\pm w}) + 2{\alpha_{\pm}^{\prime}}^{( 0 )}e^{2 A}\bar{\tau}_{\pm ( Y )},
\underline{h}_n\mathcal{E}_\mathrm{s}\sqrt{\frac{4}{\pi(|\underline{h}_n|^2\mathcal{E}_\mathrm{s}+1)}}=\left[\sqrt{4\mathcal{E}_\mathrm{s}/\pi}+O(1/\sqrt{\mathcal{E}_\mathrm{s}})\right]\cdot\frac{\underline{h}_n}{|\underline{h}_n|},
\lim_{N{\rightarrow}\infty}\sum^{N}_{n=1}\phi_{n}(x)^{\dag}\gamma_{5}\phi_{n}(x)=\lim_{\Lambda\rightarrow\infty}\sum^{\infty}_{n=1}\phi_{n}(x)^{\dag}\gamma_{5}f(\lambda_{n}^{2}/\Lambda^{2})\phi_{n}(x)
S_{1, 2}(\lambda) = \sum\limits_{i=0}^{2\lambda}\sum\limits_{j=0}^{2\lambda}\sum\limits_{k=0}^{2\lambda}(-1)^k \binom{2\lambda}{i}\binom{2\lambda}{j}\binom{2\lambda}{k}\frac{1}{1+i+j}\frac{1}{1+2\lambda-i+k}\frac{1}{1+4\lambda-j-k.}
S_{\mathrm{off}}:= \sum_{i=1}^D\sum_{j=1}^D \alpha_i \bar{\alpha}_j\sum_{p|c}\frac{\mathcal{S}(i, j, c )}{c}\,J_{2k-1}\left(\frac{4\pi \sqrt{ij}}{c}\right)\,\big|L(f, n+1) \big|^2.
\hat \lambda_{{\cal A},j}^{(K)}\buildrel \Delta \over = \frac{1}{{\sum\limits_{l = 1}^K{{\alpha^{K - l}}}}}\sum\limits_{l = 1}^K{{\alpha^{K - l}}u_j^{(l)}}= \frac{{1 - \alpha}}{{1 -{\alpha^K}}}\sum\limits_{l = 1}^K{{\alpha^{K - l}}u_j^{(l)}}\;\; \alpha<1
\begin{array}{c c c c}{{z :}}&{{\left( x^{2 , 3}, x^{4 , 5}, x^{6 , 7}, x^{8 , 9}\right)}}&{{\to}}&{{\left( x^{2 , 3}, x^{4 , 5}, e^{- 2 \pi i z}x^{6 , 7}, e^{2 \pi i z}x^{8 , 9}\right)}}\\{{y :}}&{{\left( x^{2 , 3}, x^{4 , 5}, x^{6 , 7}, x^{8 , 9}\right)}}&{{\to}}&{{\left( e^{- 2 \pi i y}x^{2 , 3}, e^{2 \pi i y}x^{4 , 5}, x^{6 , 7}, x^{8 , 9}\right)}}\\ \end{array}
T_{DD}(L,m,d,z)=\frac{g}{2L}\frac{1}{(2\pi)^{d-1}}\sum_{n=-\infty}^{\infty}\int d^{d-1}p\frac{1}{(\vec{p}^{\,2}+(\frac{n\pi}{L})^{2}+m^{2})}-gf_{2}(d,L,m,z)
\frac{G^{\prime}( \frac{1}{2}+ \frac{1}{\sqrt{2}}- i{\beta}, \frac{1}{2}- \frac{1}{\sqrt{2}}- i{\beta}; \frac{i z_{0}}{{\hbar}c})}{G ( \frac{1}{2}+ \frac{1}{\sqrt{2}}- i{\beta}, \frac{1}{2}- \frac{1}{\sqrt{2}}- i{\beta}; \frac{i z_{0}}{{\hbar}c})}= - \frac{i}{2{\hbar}c}- ( \frac{1}{2}+ i{\beta}) \frac{1}{z_{0}}.
\frac{-\operatorname*{E}\left[ \ln(q(T))\right]}{n}\rightarrow2{\displaystyle\sum\limits_{k=1}^{\infty}}\frac{\ln(k)}{(k+1)(k+2)}\ =-\gamma-2{\displaystyle\int\limits_{0}^{1}}\frac{\left( (t-2)\ln(1-t)-2t\right) \ln(\ln(1/t))}{t^{3}}dt=1.2035649167....
\sigma_i e_{k_1 k_2 \ldots k_n}=\sum\limits_{l_i,l_{i+1}\in{\cal I}}{\hat{\cal R}}\left(k_i, k_{i+1};l_i, l_{i+1}\right)e_{k_1 k_2 \ldots k_{i-1}l_i l_{i+1}k_{i+2}\ldots k_n}\;.
I^0_\chi(u_1, v)=\varrho^{-1}\int_{\{u_1<u_2\}}-\chi (u_1-v)(\theta_{u_1}^n-\theta_{u_2}^n)\leq\varrho^{-1}\int_{\{u_1<u_2\}}-\chi (u_1-v)(\theta_{u_1}^n-c\theta_{u_2}^n)\leq c_{\mu}.
\tilde{E}_{i}(h,\theta)= \int\limits_{y_{i}+2n_{i-1}s}^{y_{i}}\big(W(U_{h,\theta}(ih-,y))-W(U_{h,\theta_i}(ih-,y))\big)dy+\int\limits_{y_{i}+(2n_{i-1}+d_i)s}^{y_{i}+2n_{i-1}s}W(U_{h,\theta}(ih-,y))dy.
\begin{aligned}R_N|_{t=0}= R_0, & \left[\int_{\mathbb T^d_{\ell}}R_NU_N \cdot \phi\right]|_{t=0}= \int_{\mathbb T^d_{\ell}}R_0U_0 \cdot \phi ,\forall \phi \in (X_N)^d.\end{aligned}
\begin{aligned}\|\sigma - \sigma_h\|_0 + \|p - p_h\|_0 + \|u_h - P_{V_h}u_h\|_0 \leq C h^{r+2}\|u\|_{r + 3}, \|u - u_h\|_0 \leq C h^{r + 1}\|u\|_{r + 2}, |\sigma - \sigma_h|_{\div}\leq C h^{r + 1}\|\div \sigma\|_{r + 1}. \end{aligned}
\tilde{a}^{ij}(y)=a^{ij}(\xi(y)) =a^{ij}(\xi_0)+ (a^{ij}(\xi)-a^{ij}(\xi_0)) &=a^{ij}(\xi_0) + \int\limits_{0}^{1}\nabla_x a^{ij}((1-t)\xi_0 + t \xi(y))dt \cdot (\xi_0 - \xi(y)) .
g^{\prime}(\rho,x)=e^{-i\rho x}\left( -\frac{z-1}{2(z+1)}+\frac{1}{2}\int_{-\infty}^{x}q(t)dt+(z+1)\sum_{n=0}^{\infty}\left( -1\right)^{n}z^{n}c_{n}(x)\right)
L=\sum_{a=1}^N\biggl({m\over 2}{\dot{\bf x}}^2_a +e{\dot{\bf x}_a}\cdot{\bf A}({\bf x}_a,t) -eA_0({\bf x}_a,t)\biggr) +\theta\int d^2 x \biggl({1\over 2}\epsilon^{ij}{\dot A}_iA_j -A_0B\biggr)\,.
w_{k \overline{{l}}}^{-}{'^{\prime}}- \left[ \frac{\overline{{l}}\left( \overline{{l}}+ 1 \right)}{r^{2}}- \frac{\kappa}{r}\frac{V^{\prime}}{2 m + V}- \frac{1}{2}\frac{V^{\prime \prime}}{2 m + V}+ \frac{3}{4}\left( \frac{V^{\prime}}{2 m + V}\right)^{2}- V^{2}- 2 m V - k^{2}\right] w_{k \overline{{l}}}^{-}= 0 \, ,
\rho_{2 c}( \lambda , \mu ) = -{\frac{1}{2 N^{2}\pi^{2}}}{\frac{1}{( \lambda - \mu )^{2}}}{\frac{4 a^{2}- \lambda \mu}{[ ( 4 a^{2}- \lambda^{2}) ( 4 a^{2}- \mu^{2}) ]^{1 / 2}}}
s ( \Gamma ) \equiv \int ( \frac{\delta \Gamma}{\delta \varrho}\frac{\delta \Gamma}{\delta A}+ B \frac{\delta \Gamma}{\delta \bar{c}}+ \frac{\delta \Gamma}{\delta \sigma}\frac{\delta \Gamma}{\delta c}+ \frac{\delta \Gamma}{\delta \bar{Y}}\frac{\delta \Gamma}{\delta \psi}+ \frac{\delta \Gamma}{\delta \bar{\psi}}\frac{\delta \Gamma}{\delta Y}= 0
m_{I}\; f_{L_{1}\ldots L_{r}}^{K_{1}\ldots K_{q}}( u ) = ( \delta_{K_{1}I}- \delta_{K_{1}N}- \delta_{L_{1}I}+ \delta_{L_{1}N}+ \ldots ) f_{L_{1}\ldots L_{r}}^{K_{1}\ldots K_{q}}( u )
\begin{array}{cc}R_{D,B}^{0,2}= \lambda_1, \ \R_{C,B}^{0,2}= \lambda_2, \ \P_A^1 = P_C^1 = 1, P_B^{-1}=P_D^{-1}= r, \ \P_C^{-2}= P_D^{-2}= 1,\end{array}
ln\prod_{p\in\mathbb{P}}\frac{1}{1-1/p}=\sum_{p\in\mathbb{P}}ln\frac{1}{1-1/p}=\sum_{p\in\mathbb{P}}(\frac{1}{p}+\frac{1}{2p^{2}}+\frac{1}{3p^{3}}+\cdot\cdot\cdot)=\sum_{p\in\mathbb{P}}\frac{1}{p}+K.
\mathcal{L}U=U_{xx}+(c_{\mu}+\partial_{x}(\chi_2V_2-\chi_{1}V_{1})(\cdot;u))U_{x}+(a+(\chi_2\lambda_2V_{2}-\chi_{1}\lambda_1V_{1})(\cdot;u)-(b+\chi_2\mu_2-\chi_1\mu_1)U)U .