harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{\alpha_\epsilon^3}\cdot (\alpha - \beta) \cdot \alpha_\epsilon^{2}= \frac{2 \epsilon \cdot \alpha^{2}\omega + \epsilon^2 \cdot (\alpha-\beta)\omega^2}{3\epsilon \cdot \alpha^{2}\omega + 3\epsilon^2 \cdot \alpha \omega^2 + \epsilon^3 \cdot \omega^3}\sim_{\epsilon \to 0}\frac{2 \epsilon \cdot \alpha^{2}\omega}{3\epsilon \cdot \alpha^{2}\omega}= \frac{2}{3}.
g_{ij}=\begin{pmatrix}1+\lambda (u^2)^2 -\lambda u^1 u^2 0 -\lambda u^1 u^2 4+\lambda(u^1)^2 0 0 0 \lambda\end{pmatrix}, w_{12}=\frac{2\lambda}{\sqrt{\det g}},w_{31}= w_{23}=0,
=\frac{15!}{2}\cdot(\frac{4!}{2})^{14}\cdot4\cdot\frac{64!}{2}\cdot3^{63}\cdot\frac{96!\cdot2}{2\cdot(4!)^{24}}\cdot\frac{2^{95}\cdot64!\cdot2}{2\cdot(8!)^{8}}
\frac{x^{i}}{\rho}F_{\hat{i}\hat{0}}= \partial_{\rho}\psi = - \frac{Q_{e l e c}}{\rho^{2}}+{\cal O}( \rho^{- 3}) \, \qquad \frac{x^{i}}{\rho}\tilde{G}_{\hat{i}\hat{0}}= \partial_{\rho}\chi = - \frac{P_{e l e c}}{\rho^{2}}+{\cal O}( \rho^{- 3}) \{}.
\int_a^b \left|\alpha - f(x)\right| dx = \int_a^b \alpha - f(x) dx = \int_a^b \alpha - \max\left(f(a),f(b)\right) + \max(f(a),f(b)) - f(x) dx = \int_a^b \left|\alpha-\max(f(a),f(b))\right|+\left|\max(f(a),f(b)) - f(x)\right| dx > \int_a^b\left|\max(f(a),f(b)) - f(x)\right| dx.
{{\langle :\dot{p}_i \dot{p}^i: \rangle}\over{r_1^2}}\sim{{\kappa^2 N^2}\over{L^4 (Q_1 Q_5)^2 Q_5}}; \ \{{\langle :\dot{\gamma}^2: \rangle}\over{r_1^2 r_5^2}}\sim{{\kappa^2 N^2}\over{L^4 (Q_1Q_5)^2}}
F=[\begin{matrix}F_{1}\\F_{2}\\F_{3}\end{matrix}]=[\begin{matrix}\kappa_{11}&\kappa_{12}&\kappa_{13}\\ \kappa_{21}&\kappa_{22}&\kappa_{23}\\ \kappa_{31}&\kappa_{32}&\kappa_{33}\end{matrix}][\begin{matrix}X_{1}\\X_{2}\\X_{3}\end{matrix}]=\kappa X
i \tilde{\Delta}( \rho , \sigma ) \equiv \sum_{{\bf p}, \,{\bf p}^{\prime}}\frac{1}{\sqrt{2 E_{p}V}\sqrt{2 E_{p^{\prime}}V}}\langle a_{{\bf p}^{\prime}}^{\dagger}\, a_{\bf p}\rangle \, e^{- i ( P \cdot \rho - P^{\prime}\cdot \sigma )}\, ,
\left[ T_{(A,0)},T_{(B,0)}\right] =C_{AB}{}^{C}T_{(C,0)},\quad\left[T_{(A,0)},T_{(B,1)}\right] =C_{AB}{}^{C}T_{(C,1)}, \quad\left[ T_{(A,1)},T_{(B,1)}\right] =-C_{AB}{}^{C}T_{(C,0)}.
P_{\Gamma^{(\pm n)}}^{B}:=\frac{1}{2\pi i}\int_{\Gamma^{(\pm n)}}(\lambda-{\mathcal A}_B)^{-1}\, d\lambda, P_{\Gamma^{(\pm n)}}^{0}:=\frac{1}{2\pi i}\int_{\Gamma^{(\pm n)}}(\lambda-{\mathcal A}_0)^{-1}\, d\lambda,
A_r = \begin{pmatrix}3.0291 0.0796 1.9073 0.0796 1.9821 -0.5467 1.9073 -0.5467 3.2192\end{pmatrix}, B_r = \begin{pmatrix}0.9263 1.1414 -1.0458 0.3119 -0.3533 0.9406 \end{pmatrix}, C_r = \begin{pmatrix}1.0185 1.0416 -0.5334 0.7696 1.3947 0.7315\end{pmatrix}.
\Phi^{[m;s]}_1|_S (\tau, z_1, z_2) = \frac{1}{\tau}e^{-\frac{2 \pi i (m+1)z_1 z_2}{\tau}}\Phi^{[m;s]}_1 \left(-\frac{1}{\tau}, \frac{z_1}{\tau}, \frac{z_2}{\tau}\right).
{\cal L}^{++}_{++}(\Lambda) ={1\over 2}\Lambda^i_+[\delta^{ij}D^{++}+ (V^{++})^{ij}]\Lambda^j_+ + \Lambda^{+Y'}_+[{1\over 2}(V)_{Y'Z'}\Lambda^{+Z'}_+ +(V^{++})^{-i}_{Y'}\Lambda^{i}_+ + m\Phi^+_{+Y'}]\; ,
\mathrm{\int_{0}^{1}d x \int_{0}^{x}d G \int_{G}^{1 - x + G}d y \int_{0}^{x - G}d v = - \int_{0}^{1}d x \int_{0}^{x}d t \int_{1 + t - x}^{t}d Y \int_{0}^{t}d v ~ ,}
\sum_{j_{3}m_{3}}(2j_{3}+1)(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{matrix})(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}^{\prime}&m_{2}^{\prime}&m_{3}\end{matrix})=\delta_{m_{1},m_{1}^{\prime}}\delta_{m_{2},m_{2}^{\prime}}.
\begin{array}{ll}\ [k_\lambda,k_\mu]_{\frak g'}=0, [k_\lambda, x_\alpha]_{\frak g'}=(\alpha|\lambda)x_\alpha, \ [x_\alpha,x_\beta]_{\frak g'}=[x_\alpha,x_\beta]_{\frak g}\ \ (\alpha\neq-\beta), [x_\alpha,x_{-\alpha}]_{\frak g'}=\varphi([x_\alpha,x_{-\alpha}]_{\frak g})\end{array}
\left\langle\phi_{m_1}\psi^{a_1}\cdots\phi_{m_n}\psi^{a_n}\cdot\phi_1\psi\right\rangle_{0,n+1}^\mathrm{FJRW}=(n-2)\left\langle\phi_{m_1}\psi^{a_1}\cdots\phi_{m_n}\psi^{a_n}\right\rangle_{0,n}^\mathrm{FJRW}.
G^{- 1}= ( G^{A B}) = \left( \begin{array}{c c}{\widetilde{{g}^{i j}}}{g^{i l}\partial_{l}\theta_{\gamma}g^{\gamma \beta}}{g^{j l}\partial_{l}\theta_{\gamma}g^{\gamma \alpha}}{- g^{\alpha \beta}}\end{array}\right)
\|D^2 v\|_{L^q(Q_{\sigma \rho})}\le \frac{C}{(1-\sigma)^2R^2}\left(R^2 \| c_2|D v|^\gamma + g \|_{L^q(Q_{\rho})}+ \| v \|_{L^q(Q_{\rho})}\right) &\le \frac{C}{(1-\sigma)^2}\left(R^{\gamma-\frac{N+2}{q}(\gamma-1)}\|D v\|_{p;Q_\rho}^\gamma+1\right).
S_{0}= -{\frac{N}{\lambda}}\mathrm{T r}( U + U^{\dagger}) + \sum_{l}{\frac{1}{2 \sigma_{l}^{2}}}\mathrm{T r}( X_{l}^{i}X_{- l}^{i}) - \sum_{l \not = 0}{\frac{1}{s_{l}}}\mathrm{T r}( \bar{\alpha}_{l}\alpha_{l}) \, .
{||C||}_{W^{2,1}_p(Q_T)}\le \mu_n\left({||C ||}_{L^p(Q_T)}+{||\psi ||}_{W^{2,1}_p(Q_T)}+||e^{-\sum_{k=0}^mh_k(x)B_k}g ||_{L^p(Q_T)}\right),~~ \forall\omega\in A_n,
\phi^2(t, u -w_i) - 2(u -w_i)w_i'' \geq \sum_{j \in \mathcal{I}_{i-1}(t)}\left(\tilde{\phi}_j^2 (t, u-w_i)\right) - 2|u-w_i ||w_i''| & = \sum_{j \in \mathcal{I}_{i-1}(t)}\left(\psi_j^2 (t)|u-w_i|\right) - 2|u-w_i||w_i''| & \geq |u -w_i|.
\frac{\partial u_3}{\partial x_3}=-i 2\pi \sum_{k\in \mathbb{Z}^2 \times \mathbb{N}}\left(\hat{u}_1(k)\frac{k_1}{\Pi_1}+\hat{u}_2(k)\frac{k_2}{\Pi_2}\right)\sin(\frac{\pi k_3x_3}{h}) e^{2\pi i (\frac{k_1x_1}{\Pi_1}+\frac{k_2x_2}{\Pi_2})};
\begin{array}{lllllllllll}0=\omega(C)-\omega(C) \le |E_{f=(0,0)}\cap E(C)|+\Omega|E(C)-E_{f=(0,0)}|-\omega(C) &\le \frac{\omega(C)}{4}+\Omega(\frac{\omega(C)}{4}+2-\Omega)-\omega(C) &=-3<0,\end{array}
\mathrm{E}(\mathrm{x}) - \omega^2\mu_\mathrm{m}\Big(\varepsilon_\mathrm{p}(\omega,\gamma)-\varepsilon_\mathrm{m}\Big)\int_\Omega \mathbb{G}^{(\mathrm{k})}(\mathrm{x},\mathrm{y})\mathrm{E}(\mathrm{y})d\mathrm{y}= \mathrm{E}^{\textbf{in}}(\mathrm{x})
\frac{\beta_k}{1-P_{T_k}}V(x_k, u) + [\Phi_k(\tilde x_k) - \Phi_k(u) ] \le & \frac{P_{T_k}}{1-P_{T_k}}\left[\beta_k V(x_{k-1}, u) \frac{M^2}{2\nu \beta_k}\sum_{i=1}^t (p_i^2 P_{i-1})^{-1}\right].
\begin{aligned}y\left[ n \right] = \sum\nolimits_{l = 1}^L{{\bf{h}}_l^H{{\bf{f}}_l}s\left[{n -{\kappa_l}-{n_l}}\right]}+ \sum\nolimits_{l = 1}^L{\sum\nolimits_{l' \ne l}^L{{\bf{h}}_l^H{{\bf{f}}_{l'}}s\left[{n -{\kappa_{l'}}-{n_l}}\right]}}+ z\left[ n \right]. \end{aligned}
\hat{H}= \frac{1}{2}\hat{p}_{z}^{2}+ \hbar \omega \left[ a_{+}^{\dagger}a_{+}+ a_{-}^{\dagger}a_{-}\right] \ , \ \hat{\phi}= \hat{p}_{z}+ \hbar g \left[ a_{+}^{\dagger}a_{+}- a_{-}^{\dagger}a_{-}\right] \ .
r_{n,n}= \overline{r}_{n,0}= \prod_{k=1}^{n}(1 + i c_k), \n \geq 1 \mbox{and}q_{n,n-1}= \overline{q}_{n,0}= 2d_1 \, \prod_{k=2}^{n}(1 + i c_k), \n \geq 2.
\mathop{{\rm Res}}_{s=\frac{m+1}{2}}E_{\frac{n-m}{2}}^{(n)}(Z,s) =\mathop{{\rm Res}}_{s=\frac{m+1}{2}}F_{n,m}(s)\cdot E_{\frac{n-m}{2}}^{(n)}(Z,0) = \gamma_{n,m}\left(\tfrac{m+1}{2}\right) \mathop{{\rm Res}}_{s=\frac{m+1}{2}}\xi_{n,m}(s)\cdot E_{\frac{n-m}{2}}^{(n)}(Z,0).
\zeta(x) = \sum_{n=1}^\infty \frac{1}{n^x}= \prod_{p}\frac{1}{1-p^{-x}}x>1 \frac{1}{1^x}+\frac{1}{2^x}+\frac{1}{3^x}+\dots = \prod_{p}\frac{p^x}{p^x-1}= \left(\frac{2^x}{2^x-1}\right)\left(\frac{3^x}{3^x-1}\right)\left(\frac{5^x}{5^x-1}\right)\dots
i\partial_{\tau}v_{\lambda}+ \tilde P^y_\lambda(iP_{\lesssim \epsilon^2}^yV_2 \partial_y+\frac{i}2 P_{\lesssim \epsilon^2}^y \partial_y V_2) \tilde P^y_\lambda v_{\lambda}+\partial^2_y v_{\lambda}= \tilde{f}_\lambda,
G_{bb}^{-2}(x,y) = \left[- \frac{{N^{2}}}{g^{2}}\frac{{4}}{f^{4}({\hat{\phi}})}\left({1- \frac{{3}}{2}\frac{{\hbar G_{aa}(z,z)}}{f({\hat{\phi}})}}\right) \nabla_{x}^{2}+ \frac{{1}}{2}\frac{{\hbar}}{f({\hat{\phi}})}G_{aa}^{-1}(z,z) \right ] \delta (x-y),
{W}( \varphi_{1}, \varphi_{2}) = \sqrt{2}\left[ \frac{m^{2}}{\lambda}\varphi_{1}- \frac{\lambda}{6}\varphi_{1}^{3}+ \frac{\lambda}{2}\varphi_{1}\varphi_{2}^{2}+ \frac{\mu}{\sqrt{2}}\varphi_{2}^{2}\right] ,
{\cal L}= \mathrm{i}\varphi^{\dagger}\partial_{+}\varphi + \mathrm{i}\chi^{\dagger}\partial_{-}\chi + \frac{\mathrm{i}}{\sqrt{2}}\left( \varphi^{\dagger}\tilde{\partial}_{m}\chi + \chi^{\dagger}\partial_{m}\varphi \right) +{\cal L}_{\mathrm{i n t}}( \varphi , \chi )
\frac{\Gamma_{r}\left\{t_{1}\,S_{n}+t_{2}\,U_{n}\right\}}{r!}=\sum_{k=0}^{r}\frac{\Gamma_{k,r-k}\left\{\left( S_{n},\,U_{n}\right)\right\}\,t_{1}^{k}t_{2}^{r-k}}{k!\,(r-k)!},r=1,2,\ldots .
f'(x^*;y'-x^*) =\nabla f(x^*)^\top(y'-x^*) &\le\|\nabla f(x^*)\|_\infty\|y'-x^*\|_1 &\le\left(\|\nabla f(0)\|_\infty+\|\nabla f(x^*)-\nabla f(0)\|_2\right)\|y'-x^*\|_1 &\le\left(\|\nabla f(0)\|_\infty+M\|x^*-0\|_2\right)\|y'-x^*\|_1 &\le\left(\|\nabla f(0)\|_\infty+MC\right)\|y'-x^*\|_1,
\sum_{Q\in \Gamma\backslash \mathcal Q_D}\chi_d(Q) \int_{C_Q}i\partial_z\varphi(z) y^{s}\, dz = \pi^{-s-\frac 12}\sqrt{|d|}\, \Gamma(\tfrac s2+\tfrac{ir}2+\tfrac 34)\Gamma(\tfrac s2-\tfrac{ir}2+\tfrac 34) L(s+\tfrac 12,\varphi\times\chi_d).
\frac{1}{|df_{n,p_j}v_j|}=\frac{|(df_{n,p_j})^{-1}(df_{n,p_j}v_j)|}{|df_{n,p_j}v_j|}\leq ||df_{n,p_j}(0)^{-1}|| & =||\left(T_n \circ df_{n,p_j}(0) \circ S_j^{-1}\right)^{-1}|| \leq 2 \max\left\{\frac{1}{|\alpha_{n,p_j}|},\frac{1}{|\beta_{n,p_j}|},\frac{|\gamma_{n,p_j}|}{|\alpha_{n,p_j}\beta_{n,p_j}|}\right\}.
Af(x) =\sum_{[\xi]\in\widehat{G}_0}d_{\xi}[\xi(x)\sigma(x,\xi)\widehat{f}(\xi)] &=\int_{M}[\sum_{[\xi]\in\widehat{G}_0}d_{\xi}[ \xi(y^{-1}x)\sigma(x,\xi)]]f(y)dy &=\int_{M}[\sum_{[\xi]\in\widehat{G}_0}d_{\xi}[ \xi(y)\sigma(x,\xi)]]f(xy^{-1})dy.
- i T r L n ( i D_{\mu}\gamma^{\mu}- \sigma ) = - \frac{i}{2}T r L n ( D^{2}+ \sigma^{2}) = \int \frac{i}{2 s}t r \langle x | e^{- i s ( D^{2}+ \sigma^{2})}| x \rangle d s d^{4}x
{f_Y}\left( y \right) = \sum\limits_{i = 1}^L{\sum\limits_{j = 1}^{{m_c}}{\frac{{{A_{i,j}}{y^{j - 1}}\exp \left({- \frac{{{m_c}y}}{{{P_i}{\Omega_i}}}}\right)}}{{\left({j - 1}\right)!}}}},
W_{2r,2}(Z)= (q-1)((q-1)q^{2r-1}-q^{r-1})Z^{q^m-q^{m-1}-q^{m-r-1}-1}&+(q-1)(q^{2r-1}+q^{r-1})Z^{q^m-q^{m-1}-q^{m-r-1}}&+(q-1)(q^m-q^{2r})Z^{q^m-q^{m-1}-1}+(q^m-q^{2r})Z^{q^m-q^{m-1}}&+(q-1)(q^{2r-1}+q^{r-1})Z^{q^m-q^{m-1}+q^{m-r-1}(q-1)-1}&+(q^{2r-1}-q^{r-1}(q-1))Z^{q^m-q^{m-1}+q^{m-r-1}(q-1)}.
\frac{d^{l-m}}{dx^{l-m}}(x^{2}-1)^{l}=\sum_{j=0}^{l-m}\frac{(l-m)!}{j!(l-m-j)!}\frac{l!}{(m+j)!}(x+1)^{m+j}\frac{l!}{(l-j)!}(x-1)^{l-j}
\sum_{q=1}^{2j+1}\left( V\left[ j\right] \right)_{m,q}\frac{1}{2^{q}}~t\left( 2n,q-1\right) =\sum_{l=1}^{j}\left( V\left[ j\right] \right)_{m,2l+1}\frac{1}{2^{2l+1}}~t\left( 2n,2l\right) =\frac{1}{2}\prod\limits_{k=0}^{n-1}\left( \left( j+1-m\right)^{2}-k^{2}\right) \ .
X^m_{t+1}=\Psi\left(t,X^m_t,\Phi^m\left(t,X^{m,(t)},\mu^{m,(t)}\right),\xi^m_{t+1}\right), \mathbb{P}_m &=\Psi\left(t,X^m_t,\Phi^m\left(t,X^m_t \right),\xi^m_{t+1}\right).
h_{(\zeta_{\bf p},\zeta_{\bf q})}\zeta_{\bf m}=\zeta_{\bf m}\Biggl(1 +i\frac{\Delta(\zeta_{\bf p},\zeta_{\bf q})}{R^2}+ O \left(\biggl(\frac{|\mbox{\boldmath$\delta$}|}{n}\biggr)^2\right)\Biggr),
C^{-1}E_{f,p}(X)\le \liminf_{s\nearrow 1}(1-s)\int_{X}\int_{X}\frac{|f(x)-f(y)|^p}{d(x,y)^{ps}\mu(B(y,d(x,y)))}\,d\mu(y)\,d\mu(x) &\le \limsup_{s\nearrow 1}(1-s)\int_{X}\int_{X}\frac{|f(x)-f(y)|^p}{d(x,y)^{ps}\mu(B(y,d(x,y)))}\,d\mu(y)\,d\mu(x)\le CE_{f,p}(X).
(A_2)_j^{j'}(l):=\begin{cases}-\dfrac{T_j^{j'}(l)}{\mathrm{i}(\omega\cdot l+m_3 (j'^3-j^3))}\mbox{if}\,\,\overline{\omega}\cdot l+j'^3-j^3\neq 0, begin{align*}3mm]0 \qquad\qquad\qquad\qquad\qquad\quad\mbox{otherwise}.\end{cases}
|\mathrm{II}| \lesssim\int_{|x-y|<4|h|}|x-y|^{-n-|\alpha|}|x-y|^{|\alpha|+1}\,dy &=\int_{|x-y|<4|h|}|x-y|^{-n+1}\,dy\sim\int_0^{4|h|}t^{-n+1}t^{n-1}\,dt\sim|h|.
p_{0\alpha}= p_{\alpha}- \frac{g}{2}(\tau^{+}A^{+}{}_{\alpha}+ \tau^{-}A^{-}{}_{\alpha}) + \frac{gg'}{\sqrt{g^2 +g'^2}}Q A_{\alpha}+ \frac{g^2}{\sqrt{g^2 + g'^2}}Q' Z_{\alpha}.
\left( \begin{array}{c}{{t^{\prime}}}\\{{z^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{0}}&{{- \frac{2 \lambda}{\sqrt{\mu}}}}\\{{\frac{\sqrt{\mu}}{2 \lambda}}}&{{\frac{2 \lambda}{\sqrt{\mu}}}}\\ \end{array}\right) \left( \begin{array}{c}{{t}}\\{{z}}\\ \end{array}\right) .
\int_0^{+\infty}\frac{\log\left({r}^{2}+2\,a\,r+1\right)}{{\left({r}^{2}+2\,a\,r+1\right)}^{\frac{3}{2}}}\,r\,dr=\int_0^{+\infty}\partial_r g(a,r)\,dr=g(a,+\infty)-g(a,0) &\qquad\qquad=\frac{2a\big(1-\log 2+\log(a+1)\big)-2}{a^2-1}=:h(a).
-\frac{\dot{\lambda}}{\lambda}=\frac{r}{k}(\gamma_{1}\frac{H(a)}{\lambda^{3}}+\gamma_{2}\frac{\Delta K(a)}{K(a)\lambda^{2}})(1+o_{\frac{1}{\lambda}}(1)) & +o(\frac{1}{\lambda^{3}})+O(\frac{\vert\nabla K(a)\vert^{2}}{\lambda^{2}}+\vert \delta J(u)\vert^{2})
\big(\theta_x(f)(h_x\otimes \zeta_x) \,\big|\,(h_x\otimes \zeta_x)\big) =\big(f*h_x\otimes \zeta_x \,\big|\,h_x\otimes \zeta_x\big) &=\big(\pi_x(\langle h_x,f*h_x\rangle) \zeta_x \,\big|\,\zeta_x\big) &=\psi_x\big(\langle h_x,f*h_x\rangle\big).
\widetilde \Pi_{( \widetilde p )}\Pi_{( p )}^{- 1}= - ( \mathrm{c o s}\theta \Gamma^{6}+ \mathrm{s i n}\theta \Gamma^{8}) ( \mathrm{c o s}\theta \Gamma^{7}+ \mathrm{s i n}\theta \Gamma^{9}) \Gamma^{6}\Gamma^{7}\ .
\zeta_{MT,2}(s_1,s_2,s_3) = \sum_{m \leq bt_3}\sum_{n \leq bt_3}\frac{1}{m^{s_1}n^{s_2}(m+n)^{s_3}}+O ( t_3^{\frac{3}{2}-\sigma_1-\sigma_2-\sigma_3}).
\mathbb{E}\{\langle X^{x}_{\tau'}, X^{x}_{\tau}\rangle\}= \mathbb{E}\left\{|X_{\tau}^{x}|^2\right\}\intertext{and thus}\mathbb{E}\left\{| X^{x}_{\tau'}- X^{x}_{\tau}|^2 \right\}= \mathbb{E}\left\{|X^{x}_{\tau'}|^2\right\}- \mathbb{E}\left\{|X^{x}_{\tau}|^2\right\}.
\sum_{k=1}^{n-2}p^{n-3}(p-2)(p-3) \binom{n-k-2}{2}= p^{n-3}(p-2)(p-3) \sum_{i=0}^{n-3}\binom{i}{2}&=p^{n-3}(p-2)(p-3) \binom{n-2}{3}
\begin{pmatrix}a_{1}a_{2}a_{3}a_{4}\end{pmatrix}\cdot\alpha_{1}= \alpha_{1}^{a_{4}}\alpha_{3}^{-a_{2}}, \ \begin{pmatrix}a_{1}a_{2}a_{3}a_{4}\end{pmatrix}\cdot\alpha_{3}= \alpha_{1}^{-a_{3}}\alpha_{3}^{a_{1}}.
\Sigma_{12}=\sum_{n}\frac{x_n^2(1-y_n)^2}{x_n(1-y_n)^2+y_n},\Vert F\Vert^2\sim\sum_{n}\frac{x_n^2}{x_n+y_n},\quad \Vert G\Vert^2\sim\sum_{n}\frac{y_n^2}{x_n+y_n},\Vert G-tF\Vert^2\sim\sum_{n}\frac{(y_n-tx_n)^2}{x_n+y_n}.
2 \pi \int_{\Sigma}\langle \hat{\phi}( z )^{2}\rangle ~ d^{2}z = 2 \pi \int_{\Sigma}{\cal G}( z , z ) ~ d^{2}z = - \frac 1 2 \ln \det ( ( - \nabla_{\Sigma}^{2}+ m^{2}) \epsilon^{2}) .
\delta_{\perp}\kappa_{1}= \kappa_{3}\kappa_{2}\Psi_{3}+ 2 \kappa_{2}\Psi_{2}{}^{\prime}+ \kappa_{2}^{\prime}\Psi_{2}+ \Psi_{1}{}^{\prime \prime}- \left( \kappa_{1}^{2}+ \kappa_{2}{}^{2}\right) \Psi_{1}\, .
{\cal L}_{\mathrm{b}}~ = ~ - \frac{1}{4 l^{2}}\partial_{\mu}l \partial^{\mu}l + \frac{1}{4 l^{2}}h_{\mu}h^{\mu}- G_{i \bar{\jmath}}\partial_{\mu}{z^{i}}\partial^{\mu}{\bar{z}^{\bar{\jmath}}}- \frac{i}{2}( G_{l j}\partial_{\mu}{z^{j}}- G_{l \bar{\jmath}}\partial_{\mu}{\bar{z}^{\bar{\jmath}}}) h^{\mu},
\lim_{\rho\to 0^{+}}\lim_{k \to +\infty}\int_{\mathbb{R}^{N}}\big[\Psi_{\rho,i}\mathrm{H}_{u}(x,u_k,v_k)u_k+ \Psi_{\rho,i}\mathrm{H}_{v}(x,u_k,v_k)v_k\big]\,dx=0.
\int_G u\,d\nu =\int_{K_1/K}\int_{K_2}u(k_1k_2)\rho(k_1k_2)\,dk_2\, d(k_1K) &=\int_{K_1K_2/K_2}\int_{K_2}u(k_1k_2)\rho(k_1k_2)\,dk_2\, d(kK_2) &=\int_{K_1K_2}u(k)\rho(k)\, dk=\int_G u\,d(\rho m_{K_1K_2})
\xi_1 + \xi_2 = \xi_1' + \xi_2' + O(R^{-1/2}), \zeta_1 + \zeta_2 = \zeta_1' + \zeta_2' + O(R^{-1/2}), \xi_1 \zeta_1 + \xi_2 \zeta_2 = \xi_1' \zeta_1' + \xi_2' \zeta_2' + O(R^{-1/2}).
p_{2}= \int_{0}^{\infty}\frac{d \omega}{2 \pi}\int_{0}^{\omega}\frac{d \omega^{\prime \prime}}{2 \pi}\vert \langle o u t \omega \omega^{\prime \prime}\vert i n \rangle \vert^{2}= \frac 1 2 \int_{0}^{\infty}\frac{d \omega}{2 \pi}p_{2 \omega}.
\mathfrak{h}= \bigoplus_{\alpha\in \Phi_{P^u}}\mathbb{C}e_{\alpha}\oplus \mathfrak{t}^{\sigma}\oplus \bigoplus_{\alpha\in \Phi_L^{\sigma}}\mathbb{C}e_{\alpha}\oplus \bigoplus_{\alpha\in \Phi_s^+}\mathbb{C}(e_{\alpha}+\sigma(e_{\alpha}))
\begin{array}{r l}\hat{v}_\alpha(x)= \mu_\alpha^{\frac{n-2}{2}}\varphi_\alpha(\mu_\alpha x)u_\alpha\circ\Phi_\alpha^{-1}(\mu_\alpha x) \hat{Z}_\alpha(x)= \mu_\alpha^{n-1}\varphi_\alpha(\mu_\alpha x)^{2-q}\left(\Phi_\alpha\right)_* W_\alpha(\mu_\alpha x),\end{array}
\beta_k = \int_{B_{S_1}}f_k(z) dz + \int_{\partial D}\sum^3_{\ell=1}(-\partial_{\ell}u_k (z) +\delta_{k\ell}\cdot \pi(z) + u_k (z) (\tau \cdot e_1 - \omega \times z)_{\ell}) &+ u_k(z) u_{\ell}(z)) n_{\ell}d0_z
\hat{G}_{lm,l^{\prime}m^{\prime}}\left( k\mathbf{\hat{e}}_{z}\right)=\delta_{m,m^{\prime}}\left( -1\right)^{m}\sum_{j=\left\vert l-l^{\prime}\right\vert}^{l+l^{\prime}}\left(\begin{array}[c]{ccc}l l^{\prime}j m -m^{\prime}0\end{array}\right) \tilde{g}_{l,l^{\prime}}^{j}\left( k\right).
\frac{dx}{de_{1}}\frac{e_{1}}{x}=\frac{-\frac{\partial v_{1}}{\partial e_{1}}\frac{e_{1}}{v_{1}}}{\frac{\partial v_{2}}{\partial x}\frac{x}{v_{2}}-\frac{\partial v_{1}}{\partial x}\frac{x}{v_{1}}}
T r_{L}\Phi^{\left[ 1 \right]}= D + \frac{{3 2}}{{T^{2}}}\bar{\xi}_{4}\xi_{4}\bar{\xi}_{3}\xi_{3}\bar{\xi}_{2}\xi_{2}\bar{\xi}_{1}\xi_{1}( \varepsilon^{1}\cdot \varepsilon^{4}\varepsilon^{2}\cdot \varepsilon^{3}- \varepsilon^{1}\cdot \varepsilon^{3}\varepsilon^{2}\cdot \varepsilon^{4}) \delta ( u_{4}- u_{3}) \delta ( u_{2}- u_{1}) + \, f . t .
S(0)= S_0 \geq 0 \, , L_T(0) = L_{T0}\geq 0 \, ,I_T(0) = I_{T0}\geq 0 \, , R_T(0) = R_{T0}\geq 0 \, , &I_H(0) = I_{H0}\geq 0 \, , A(0) = A_0 \geq 0 \, ,L_{TH}(0) = L_{TH0}\geq 0 \, , &I_{TH}(0) = I_{TH0}\geq 0 \, , R_{TH}(0) = R_{TH0}\geq 0\, , A_T(0) = A_{T0}\geq 0 \, .
\mathbb{E}\sum_{x \in V_{\sigma}(\mathbf{X}_0)}f(\mathbf{X}_0, x) = \int_0^\infty \dots \int_0^\infty \mathbb{E}\,f(\mathbf{X}^{z_1, \dots, z_l}, x^{z_1, \dots, z_l})\,dz_1\dots\,dz_l
| \partial_{r_n}D' h_k (p) -\partial_{r_n}D' h_k (p')| = \left| \int_{\theta_n(p)}^{\theta_n(q)}\frac{\partial h_{k,ir_n}}{\partial \theta_n}d\theta_n \right| \le C r_n(p)^{\frac 1 \beta - 1}\tau^{-k(\frac 1\beta -1)}\omega(2 \tau^k).
A_{\mu}^{p q}( x ) = - i \int_{\tilde{T}^{2}}\psi^{p}{}^{\dagger}D_{x}( \hat{A}) \left( D_{x}^{\dagger}( \hat{A}) D_{x}( \hat{A}) \right)^{- 1}\sigma_{\mu}^{\dagger}\psi^{q}+ R_{\mu}^{p q}.
\|u(t,x)-u(s,x)\|_{L^p}\le\sum_{n=1}^\infty (p-1)^{n/2}\sqrt{n!}\|h_n(\cdot,t,x)-h_n(\cdot,s,y)\|_{\mathcal H^{\otimes n}}\le \sum_{n=1}^\infty (p-1)^{n/2}\sqrt{n!}\sqrt{D_n+D_n'}\le C (t-s)^{[\alpha_2\wedge (1-\beta_0)]/2}.
0 = \int_{\mathbb R^n}\left( -\frac{u_\tau - u_0}{\tau}\Delta \varphi + \frac{g(u_\tau) - g(u_0)}{\tau}\varphi \right) &= \int_{\mathbb R^n}\frac{u_\tau - u_0}{\tau}\left( - \Delta \varphi + \frac{g(u_\tau) - g(u_0)}{u_\tau - u_0}\varphi \right).
\frac{\kappa_{j}+ i k_{j}}{\kappa_{j}- i k_{j}}= - \, \frac{A_{a}( \bar{\sigma}_{N}\sigma^{\prime \prime}q_{j}^{\prime \prime}, i d )}{A_{a}( \sigma^{\prime \prime}q_{j}^{\prime \prime}, i d )}\, e^{- i k_{j}L}.
\mu_{E_{k}}\left( \mathsf{R}_{k}>\vartheta_{k,\left\lfloor \frac{m+1}{M}n_{k}\right\rfloor}\right) =1-\widetilde{G}_{k}\left( \vartheta_{k,\left\lfloor \frac{m+1}{M}n_{k}\right\rfloor}\right) & \longrightarrow1-\widetilde{G}\left( t\left( 1-\frac{m+1}{M}\right)^{\alpha}\right)
\tilde{C}= -\frac{3}{2}g^{2}\delta^{ab}\int \frac{d^{4}k}{(2\pi)^{4}}d^{2}\theta d^{2}{\overline\theta}\, \frac{1}{(k^{2}+m^{2})}\left\{V^{a}(p,\theta,{\overline\theta}) V^{b}(-p,\theta,{\overline\theta}) \right\}\, .
{\rm dist}(\Gamma^{(\pm n)},\sigma({\mathcal A}_0)) =\min\big(|b_n-i\sqrt{\mu_n}|,|d_n-i\sqrt{\mu_n}|,|ia_{n+1}-i\sqrt{\mu_{n+1}}|,|ia_n-i\sqrt{\mu_{n-1}}|\big) &=\frac{\delta_{n-1}}{2},
\frac{1}{\sqrt{L}}d\sigma_{\Sigma,L}=\frac{1}{\sqrt{L}}e_1^{\star}\wedge e_2^{\star}=\frac{l}{l_L}(\overline{p}\omega_2-\overline{q}\omega_1)\wedge \omega+\frac{1}{\sqrt{L}}\overline{r_L}\omega_1\wedge \omega_2.
\int_r^{\infty}\frac{V(s)}{s\varphi(s)}ds = \int_r^\infty V(s) d(-\mathcal{P}_1)(s) &= V(r) \mathcal{P}_1(r) - \lim_{s \rightarrow \infty}V(s) \mathcal{P}_1(s) + \int_r^\infty V'(s) \mathcal{P}_1(s)ds &\le c_5 \left(\frac{1}{V(r)}- \lim_{s \rightarrow \infty}\frac{1}{V(s)}+ \int_r^\infty \frac{V'(s)}{V(s)^2}ds \right) = \frac{2c_5}{V(r)},
\begin{array}{c c c c c c c c c c l}{{D 2 :}}&{{1}}&{{2}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\quad \mathrm{p r o b e}\nonumber}}\\{{F 1 :}}&{{\_}}&{{\_}}&{{\_}}&{{4}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\quad \mathrm{s o l i t o n}\nonumber}}\\{{D 2 :}}&{{\_}}&{{\_}}&{{3}}&{{4}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\_}}&{{\quad \mathrm{s o l i t o n .}}}\\ \end{array}
\left| \bigcup_{i=1}^{r}T_{i}\right| =\sum_{\emptyset \subsetneqq I \subseteq [r]}(-1)^{|I| - 1}\cdot|T_{I}|=\sum_{\emptyset \subsetneqq I \subseteq [r]}(-1)^{|I| - 1}\cdot|U_{I}|=\left| \bigcup_{i=1}^{r}U_{i}\right| .
\mathbb{E}\left[\eta^{\kappa}\right] =\frac{-1}{\Gamma(\theta)}\int_{0}^{\infty}\frac{\partial}{\partial u}\mathbb{E}\left[e^{-u\eta}\right]u^{\theta-1}\mathrm{d}u =\frac{\alpha\beta}{\Gamma(\theta)\left(1-e^{-\alpha}\right)}\int_{0}^{\infty}\exp\left\lbrace \frac{-\alpha u}{\beta+u}\right\rbrace \frac{u^{\theta-1}}{(\beta+u)^{2}}\mathrm{d}u.
P_j(i) =\frac{2(q-1)}{q^2}g_\frac{1}{2}(j)\Sigma(i)=\frac{2(q-1)}{q^2}g_\frac{1}{2}(j)\frac{-q^2q^{-i}}{(q-1)^2}\left(1-q^{-f(i)}\right) &=\frac{-2q^{-i}}{q-1}g_\frac{1}{2}(j)(1-q^{-f(i)}),
H:=\{(X,Y) : \|X(\hat{\beta}(Y) - \beta_{0})\| \leq c_{1}\sqrt{p\log n}, (\underline{b}/2)^{2}I_{p}\preceq X^{\top}X / n \preceq ( 2 \overline{b})^{2}I_{p}\}&\cap \{(X,Y) : \Pi_{\sigma^{2}}(|\sigma^{2}/\sigma_{0}^{2}-1| \geq \delta_{1}\mid Y)\leq \delta_{2}\}.
\Phi \left( p^{\mu}\right) = \Lambda \left( \dot{p}^{\mu}\to p^{\mu}\right) \Phi \left( \dot{p}^{\mu}\right) = e^{i ( \vec{J}\cdot \vec{\vartheta}+ \vec{K}\cdot \vec{\varphi})}~ \Phi \left( \dot{p}^{\mu}\right) \, ,
\underset{T \rightarrow T^*}{\lim}\left(\|\nabla \rho\|_{L^\infty(0,T;L^q)}+\|u\|_{L^{s_1}(0,T;L_w^{r_1})}+\|\nabla d\|_{L^{s_2}(0,T;L_w^{r_2})}\right)=\infty,
g(\xi) =|\xi|^{2\sigma}-1+\{\cdots\}(\xi) &\leq|\xi|^{2\sigma}-1+\{\cdots\}(1)=|\xi|^{2\sigma}-1+g(1) &=|\xi|^{2\sigma}-1+(2^{2\sigma-1}-1)=|\xi|^{2\sigma}-(2-2^{2\sigma-1}) &<|\xi|^{2\sigma}\quad\textup{(by \eqref{elementary estimate for sigma})}.
\varepsilon_{k+1}= - A_{k+1}\frac{\mu}{2}\| (z_k - z_{k+1}) - \tau_k(z_k - x_k)\|^2 + \frac{A_{k+1}}{2\epsilon}\|x_{k}- y_{k+1}\|^2 &\qquad+ A_{k+1}\left(\frac{\tau_k}{2\epsilon}- \frac{\mu}{2\tau_k}\right)\|x_k - y_k\|^2 + A_{k+1}\psi(y_{k+1}) - A_k \psi(y_k) - \alpha_k \psi(z_{k+1}).
\mathbb{E}_f(\hat{H}_n^{w})-H(f)= O \biggl( \max \biggl\{\frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\, , \, \frac{k^{\frac{2d'}{d}}}{n^{\frac{2d'}{d}}}\, , \, \frac{k^\frac{\beta}{d}}{n^\frac{\beta}{d}}\biggr\}\biggr) = o(n^{-1/2}),
H^0(X, \mathcal O_X(mK_X)\otimes \mathcal J(X, |\!| mK_X|\!|)) = H^0(X, \mathcal O_X(mK_X)\otimes \mathcal J(X, |\!| (m-1)K_X|\!|)) =H^0(X, \mathcal O_X(mK_X))
\mathcal{P}( \Omega, \Xi, \hat{X}, \hat{B}, V, T) =\exp\Big{\{}-\sum_{j,k}\tilde{\mathfrak{s}}_{jk}Tr \Omega_j\Xi_k\Big{\}}\cdot\prod_j\frac{1}{\det^M\big(1+M^{-1}V_j^*\hat{X}_j^{-1}V_j \Omega_jT_j^{-1}\hat{B}_j^{-1}T_j\Xi_j\big)}&\times\prod_{k=p,q}\frac{\det\big(V_k^*\hat{X}_kV_k+M^{-1}\Omega_k T_k^{-1}\hat{B}_k^{-1}T_k\Xi_k\big)}{\det \hat{B}_k}
\mathbb{E}\left[ U_{n+1}^{2}\big| \mathcal{F}_{n}\right] = \frac{n (\ln (n))^{1+\delta}}{(n+1) (\ln (n+1))^{1+\delta}}U_{n}^{2}+ \frac{1}{(n+1) (\ln (n+1))^{1+\delta}}\mathbb{E}\left[ \left\| \xi_{n+1}\right\|^{2}\big| \mathcal{F}_{n}\right] & \leq U_{n}^{2}+ \frac{1}{(n+1) (\ln (n+1))^{1+\delta}}.
\sum_{n=0}^{\infty}u_{n}=\sum_{n=0}^{\infty}\sum_{k=1}^{m}\frac{a_{k}}{(n+b_{k})^{r_{k}}}=\sum_{k=1}^{m}\frac{(-1)^{r_{k}}}{(r_{k}-1)!}a_{k}\psi^{(r_{k}-1)}(b_{k}),
\left.\left( \dfrac{\Upsilon_{\mu p}\Upsilon-\Upsilon_p \Upsilon_{\mu}}{H_p^3}\right)\right|_{p=0}-\left.\left(\dfrac{\Upsilon_{\mu p}\Upsilon-\Upsilon_p \Upsilon_{\mu}}{H_p^3}\right)\right|_{p=\hat{p}}=\int_{\hat{p}}^{0}\rho_p \Upsilon^2(p;\mu)\;dp.
\hspace{- . 5 c m}S_{d = 2 6}= - \frac{1}{2 \kappa^{2}}\int d^{2 6}x \sqrt{G}e^{- 2 \Phi}\left( R - 4 D_{\mu}\Phi D^{\mu}\Phi + \frac{1}{1 2}H_{\mu \nu \lambda}H^{\mu \nu \lambda}\right) .
\frac{1}{2}\int_\Omega |Du|^2dx =\int_{\partial\Omega}\left[\big(\widetilde T(u,p)\nu\big)\cdot(u-u_\infty)+\frac{a\,(\nu\cdot x^\perp)}{2}|u-u_\infty|^2\right]d\sigma &+\int_\Omega f\cdot(u-u_\infty)\,dx
( 2 , 3 ) \, \leftrightarrow \, ( 4 , 5 , 6 ) \ ( i = 2 ) , \qquad ( 3 , 4 ) \, \leftrightarrow \, ( 5 , 6 , 2 ) \ ( i = 3 ) , \qquad ( 4 , 2 ) \, \leftrightarrow \, ( 3 , 5 , 6 ) \ ( i = 4 ) .
A_i =\frac{i}{2}[\frac{\phi_1}{s}\: \sigma_i \: \: +\frac{1}{s^2}(A_1 -\frac{\phi_1}{s}) \: x_i \vec x .\vec \sigma \: \: +(\frac{\phi_2 +1}{s^2})\: \varepsilon_{ijk}\sigma_j x_k]
\alpha^\star=\tfrac{8 L_{xy}\left(L^2-\mu^2\right) \sqrt{L \mu+L_{xy}^2}+\left(L^2-\mu^2\right)^2+16 L_{xy}^2 \left(L \mu+L_{xy}^2\right)}{\left((L+\mu)^2+4 L_{xy}^2\right)^2}.