harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{Q}_A := \{A_1, \dots, A_K\}, \mathcal{Q}'_A := \{A_{1,1}, \dots, A_{K,L}\}, \mathcal{Q}_B := \{B_1, \dots, B_K\}, \mathcal{Q}'_B := \{B_{1,1}, \dots, B_{K,L}\}.
\sum_{\substack{n>x (n,a_2)=1}}\frac{1}{\ell_u(n)}=\sum_{\substack{n>x P(n)>y (n,a_2)=1}}\frac{1}{\ell_u(n)}+\sum_{\substack{n>x P(n)\leq y (n,a_2)=1}}\frac{1}{\ell_u(n)}.
\log | \mathcal{L}(w,\chi_D) \cdot \ldots \cdot \mathcal{L}(wu^{l-1},\chi_D) | \leq \frac{2g}{N+1}(\log 2+1) + \sum_{d(f) \leq N}\sum_{j=0}^{l-1}\frac{a_j(d(f)) \chi_D(f) \Lambda(f)}{\sqrt{|f|}}\cos((\theta+j \beta)d(f)) & + O(\gamma^{-1}).
P^z\left(\left(Z\left(\left(V_k^n+\cdot\right)\wedge T_k^n\right), Z\left(\left(V_k^n+\cdot\right)\wedge T_k^n\right)-X\left(\left(V_k^n+\cdot\right)\wedge T_k^n\right)\right)\in{\mathcal A}\right)=1.
\|B^k\|_{L^{\frac{\alpha}{(s+\alpha-\frac52)}}(0,T; H^s)}\leq \C(T) \|B(0)\|_{H^{\frac{5}{2}-\alpha}}+C \|B^{k-1}\|_{L^{\frac{\alpha}{(s+\alpha-\frac52)}}(0,T; H^s)}\|B^{k}\|_{L^{\frac{\alpha}{(s+\alpha-\frac52)}}(0,T; H^s)}
S_\infty^2 (w,0)=\sum_{j = 0}^\infty{w^j G_j^2}= \frac{{w^2 G_2^2 - (2w^2 + 2w - 1)G_0^2 - (2w^2 - w)G_1^2}}{{w^3 - 2w^2 - 2w + 1}}
\det P = \frac{(x-12)(x-2)(x+2)^5(x+4)^{10}(x+6)^5}{(x-22)(x+8)}; P^{-1}= \frac{I_{20}}{x+4}- \frac{10J_{20}}{(x-12)(x-2)(x+6)}+ \frac{X^2-(x+4)X}{(x+2)(x+4)(x+6)}.
P(\chi_{P}^{2}(\{p_{i}\})>T)\sim\sum_{\{k_{i}\}|\chi_{P}^{2}(\{k_{i}\},\{p_{i}\})>T}\prod_{i=1}^{m}(\frac{np_{i}}{k_{i}})^{k_{i}}\sqrt{\frac{2\pi n}{\prod_{i=1}^{m}2\pi k_{i}}}
\sum_{n=0}^{\infty}q^{n^{2}/2}S_{n}\left(xq^{n};q\right)t^{n}=\left(-tq^{1/2};q\right)_{\infty}\sum_{k=0}^{\infty}\frac{q^{5k^{2}/2}\left(-xt\right)^{k}}{\left(q;q\right)_{k}\left(-tq^{1/2};q\right)_{2k}}
\tilde{\mathbf{n}}_{(1)}=-\frac{1}{2}\Big(\frac{\alpha-\lambda_2}{x_P(\alpha-\beta)}\ , \frac{\beta-\lambda_2}{y_P(\beta-\alpha)}\Big), \tilde{\mathbf{n}}_{(2)}=-\frac{1}{2}\Big(\frac{\alpha-\lambda_1}{x_P(\alpha-\beta)}\ , \frac{\beta-\lambda_1}{y_P(\beta-\alpha)}\Big),
\frac{1}{( q )_{\infty}}\sum_{r = - \infty}^{\infty}( - 1 )^{r}q^{r ( ( 2 k + 1 ) r + 2 k - 2 i + 1 ) / 2}= \displaystyle \sum_{n_{1}\geq \cdots \geq n_{k - 1}\geq 0}\frac{q^{n_{1}^{2}+ \cdots + n_{k - 1}^{2}+ n_{i}+ \cdots + n_{k - 1}}}{( q )_{n_{1}- n_{2}}\cdots ( q )_{n_{k - 2}- n_{k - 1}}( q )_{n_{k - 1}}}.
\mathrm{P}\left(U_1\in \mathcal{S}_1, E\right) = \mathrm{P}\left(U_1\in \mathcal{S}_1,|\mathcal{S}_1|=1, E\right) +\sum^2_{i=1}\mathrm{P}\left(U_1\in \mathcal{S}_1,|\mathcal{S}_1|=2 , E_i\right),
\left|A+B+(\{0,1\}^d\times \{0\}^k)\right|^{1/(k+d)}=\left|(A+([0,1]^d\times \{0\}^k))+(B+([0,1]^d\times \{0\}^k))\right|^{1/(k+d)}&\geq \left|A+([0,1]^d\times \{0\}^k)\right|^{1/(k+d)}+\left|B+([0,1]^d\times \{0\}^k)\right|^{1/(k+d)}&=\left|A\right|^{1/(k+d)}+\left|B\right|^{1/(k+d)},
L_{N+1}^s L_{N+2}^{t+1}= (L_{N+1}L_{N+2})^s (T_{N+1}L_{N+1}T_{N+1}) L_{N+2}^{t-s}&= T_{N+1}(L_{N+1}L_{N+2})^s L_{N+1}\big\{L_{N+1}^{t-s}T_{N+1}+ (q-q^{-1}) \sum_{p=1}^{t-s}L_{N+1}^{t-s-p}L_{N+2}^p \big\}&= T_{N+1}L_{N+1}^{t+1}L_{N+2}^s T_{N+1}+ (q-q^{-1}) T_{N+1}\sum_{p=1}^{t-s}L_{N+1}^{t-p+1}L_{N+2}^{s +p}.
\langle\hat{\Phi}^*(x)\hat{\Phi}(x)\rangle_{Ren.}=\frac1{48\pi^2r^2}\frac{1-\alpha^2}{\alpha^2}-\frac1{8\pi^2\alpha r^2\ln(\frac{2r}R e^{-{\mathcal{C}}+\alpha/\delta})}\ .
\nu_0(\varepsilon, B)\sim\begin{cases}\varepsilon^{-\frac{n}{m}}\sigma>\frac{m}{n}\varepsilon^{-\frac{n}{m}}|\log\varepsilon| \sigma=\frac{m}{n}\varepsilon^{-\frac{1}{\sigma}}0<\sigma<\frac{m}{n}\end{cases}
A_{n,1/3}= \sum_{k=\frac{n}{2}}^{\infty}\left( \frac{1}{(2k)^{1/3}}- \frac{1}{(2k+1)^{1/3}}\right) \ \frac{1}{2(n-\frac{3}{8})^{1/3}}= \sum_{k=\frac{n}{2}}^{\infty}\left( \frac{1}{2(2k-\frac{3}{8})^{1/3}}- \frac{1}{2(2k+\frac{13}{8})^{1/3}}\right).
\frac{\partial f}{\partial d_j}=2\sum_{i=j}^{n-1}(\theta_1+d_1+\dots+d_i)-2\sum_{i=0}^{n-1}(\theta_1+d_1+\dots+d_i)\frac{n-j}{n}, \frac{\partial^2 f}{\partial d_j\partial d_k}=2\sum_{i=\max\{j,k\}}^{n-1}1-2\sum_{i=k}^{n-1}\frac{n-j}{n}-2\sum_{i=j}^{n-1}\frac{n-k}{n}+2\sum_{i=0}^{n-1}\frac{n-k}{n}\frac{n-j}{n}&=2(\min\{j,k\}-\frac{jk}{n})\ge 0.
\begin{matrix}\frac{\partial z}{\partial t}&=&\frac{\dot{x}y-x\dot{y}}{y^{2}}\\ &&\\ &=&\frac{a(1-Q)xy-x(aQx+by)}{y^{2}}\\ &&\\ &=&a(1-Q)z-(aQz^{2}+bz)\\ &&\\ &=&z(a(1-Q)-aQz-b)\\ \end{matrix}
\Delta(s(\delta(xp_m-xp_n))) =\Delta(s(\delta(x(p_m-xp_n))))\leq6\Delta(s(x(p_m-p_n))) &\leq6\Delta(l(x(p_m-p_n))\vee(p_m-p_n))\leq12\Delta(p_m-p_n) &\leq12\Delta(p_n^\bot).
\begin{array}{r l}\Delta_\xi v_\alpha =\Delta_\xi\left(\varphi_\alpha u_\alpha\circ \Phi_\alpha^{-1}\right) \quad=\varphi_\alpha^{q-1}(x)\left(\Delta_g u_\alpha+\frac{n-2}{4(n-1)}R(g)u_\alpha\right)\left(\Phi_\alpha^{-1}\right)\end{array}
\hat{\sigma}= \sqrt{\Big(\frac{\Gamma(\frac{1}{2})}{2^{\frac{1}{k}}\Gamma(\frac{1}{2}+\frac{1}{k})}\Big)^{k}\sum_{i=1}^{n-k}|\Delta_i^n X \Delta_{i+1}^n X \ldots \Delta_{i+k}^n X|^{\frac{2}{k}}}
\left(\begin{array}{cc}1 0 -w_\Delta^-/\rho 1 \end{array}\right) \left(\begin{array}{cc}0 \rho/w_\Delta^+ -w_\Delta^+/\rho 0 \end{array}\right) \left(\begin{array}{cc}1 0 w_\Delta^+/\rho 1 \end{array}\right) = \left(\begin{array}{cc}1 \rho/w_\Delta^+ 0 1 \end{array}\right).
S_1(a_1,\,n_1,\,d_1)+S_1(a_2,\,n_2,\,d_1) =S_1(b_1,\,n_1,\,d_2)+S_1(b_2,\,n_2,\,d_2), S_2(a_1,\,n_1,\,d_1)+S_2(a_2,\,n_2,\,d_1) =S_2(b_1,\,n_1,\,d_2)+S_2(b_2,\,n_2,\,d_2), S_3(a_1,\,n_1,\,d_1)+S_3(a_2,\,n_2,\,d_1) =S_3(b_1,\,n_1,\,d_2)+S_3(b_2,\,n_2,\,d_2),
U(t) := ~ \frac{\widetilde{A_n}^{\prime}(t)}{\widetilde{A_n}(t)}= ~ \frac{\delta}{t+n/2}+(s+2)\sum_{\ell=0}^{n-1}\left( \frac{1}{t+\ell+1/4}+ \frac{1}{t +\ell +3/4}\right) -(2s+4)\sum_{\ell=0 \atop \ell \neq k}^{n}\frac{1}{t+\ell}.
\|\Psi_{p,i}(g^1,\cdots,g^p)(t)\|_{H^N}+\|\Phi_{p,i}(g^1,\cdots,g^p)(t)\|_{H^N}\leq C^pt^{4p}\sum_{j=1}^p\|g^j(t)\|_{H^N}\prod_{i\neq j}\|g^i(t)\|_{H^{3}}.
(i\gamma^{\mu}\partial_{\mu}-m_{1})\Psi_{1}=2\pi \lambda \varepsilon_{\mu \nu}J_{2}^{\nu}\gamma^{\mu}\Psi_{1},\quad (i\gamma^{\mu}\partial_{\mu}-m_{2})\Psi_{2}=2\pi \lambda \varepsilon_{\nu \mu}J_{1}^{\nu}\gamma^{\mu}\Psi_{2},
\omega\left(F(\rho_1 S_1,\ldots \rho_k S_k)\right)\leq \omega\left(F(\rho_1 S_1,\ldots \rho_{k-1}S_{k-1}, 0)\right)\left[ 1+2\sum_{p_k=1}^{m_k}\rho_k^{p_k}\cos \frac{\pi}{\left[\frac{m_k}{p_k}\right]+2}\right].
B(d_s,\infty, a+\epsilon) =\bigcup_{t\geq s}B(d_t,d_{t+1},a+\epsilon) \subseteq\bigcup_{t\geq s}B\big(d_t,d_{t+1},a+\frac{1}{2^s}\big) \subseteq \bigcup_{t\geq s}B\big(d_t,d_{t+1},a+\frac{1}{2^t}\big),
L=n\frac{ds}{d\sigma}=n(x_{1},x_{2},x_{3})\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}+\dot{x}_{3}^{2}}=L(x_{1},x_{2},x_{3},\dot{x}_{1},\dot{x}_{2},\dot{x}_{3})
\psi_{T_M}(v^*(n)) \geq |(T_M, v^*(n))_{u_1 \downarrow}| &\geq \ell + 1 + |(T_M, v^*(n))_{w\downarrow}| &= \ell + 1 + |(T_M, v_{n+1})_{w\downarrow}| &= \ell + 1 + \psi_{T_M}(v_{n+1}),
\begin{aligned}{\omega}_k=\frac{\sup_{Q_{k-1}^a}u-\inf_{Q_{k-1}^a}u}{2}\leq \frac{\sup_{\Gamma_{1,a}}u_{k-1}-\inf_{\Gamma_{1,a}}u_{k-1}}{2}\nu_{k-1}\leq \theta\nu_{k-1}, ~~~~1\leq k \leq k_0 + 1. \end{aligned}
\frac{1}{\rho}\frac{\partial p}{\partial x}=\sum_{n=1}^{N}\tilde{U}_{n}|i\rho n\omega[\frac{J_{0}(\Lambda_{n})}{1-J_{0}(\Lambda_{n})}]|cos\{n\omega t+phase(\frac{i}{\rho n\omega}[\frac{1-J_{0}(\Lambda_{n})}{J_{0}(\Lambda_{n})}])\}.
\beta C_{n_{l_{q}},1}+ \sum_{k=2}^{m_{n_{l_{q}}}}\frac{2\mu_{k}\left(C_{n_{l_{q}},k}- (n_{l_{q}}-1)\mathbb{I}_{k=2}\right)- \mu_{k}^2}{4k}\left| \mathbb{P}_{n_{l_q}}\right. \stackrel{d}{\to}M\{n_{l_{q}}\}.
\Psi_{+ \bf k}^{n}( P ) = ( - 1 )^{n}C_{n}^{1 / 2}\left( \frac{P_{1}- i P_{2}}{P^{0}+ m}\right)^{n}\Psi_{+ \bf k}^{0}( P ) , \quad \Psi_{+ \bf k}^{0}( P ) = \delta ( P^{0}- \sqrt{{\bf P}^{2}+ m^{2}}) \delta ({\bf P}-{\bf k}) f ({\bf k}) ,
c_n^{(k)}=\underbrace{\int_0^1\cdots\int_0^1}_k(x_1\cdots x_k)_n dx_1\dots dx_k &=n!\underbrace{\int_0^1\cdots\int_0^1}_k\binom{x_1\cdots x_k}{n}dx_1\dots dx_k\,.
R_L(F_n)(x,y) =R_L(\sum_{i=1}^{m}f_i\otimes g_i)(x,y)=R(\sum_{i=1}^{m}(f_i\otimes g_i)\restriction_{[0,\alpha]\times{y}})(x) =\sum_{i=1}^{m}R(f_i)(x)\cdot g_i(y)=\sum_{i=1}^{m}(R(f_i)\otimes g_i)(x,y)
\int_{B(0, 1)\backslash B\left( 0,\rho_{m-2}\right)}\dfrac{d\mu \left( y\right)}{\left| x-y\right|^{n-2}}=\sum^{m-3}_{k=0}\int_{D_k}\dfrac{d\mu \left( y\right)}{\left| x-y\right|^{n-2}}.
|\underbrace{\Pi_{\beta}(I(\hat{\beta}(Y),\hat{R}_{\alpha}) \mid Y)}_{=1-\alpha}-\mathcal{N}(I(\hat{\beta}(Y),\hat{R}_{\alpha}) \mid \hat{\beta}(Y),\sigma_{0}^{2}(X^{\top}X)^{-1})|\leq \overline{\zeta}\,
u = q \prod_{\substack{n \geq 1 n \equiv \pm 3,\pm 4 (25)}}(1-q^n) / \prod_{\substack{n \geq 1 n \equiv \pm 2,\pm 11 (25)}}(1-q^n) v = q^{-1}\prod_{\substack{n \geq 1 n \equiv \pm 9,\pm 12 (25)}}(1-q^n) / \prod_{\substack{n \geq 1 n \equiv \pm 6,\pm 8 (25)}}(1-q^n).
ds^2=\frac{X^{1/2}}{\rho}e^{2A(r)}dx_{//}^2 - \frac{X^{1/2}}{\rho}\left(dr^2+\frac{L^2}{\rho^2}\left[d\theta^2+\frac{\sin^2\theta}{X}d\phi^2+\frac{\rho^6\cos^2\theta}{X}d\Omega^2_3\right] \right),
A_k := \frac{g_k^2-f_kh_k}{g_k^2}\ , \B_k := \frac{|g_k| |g_{k+1}|-f_{k+1}h_k}{|g_k| |g_{k+1}|}\ , \C_k := \frac{g_{k+1}^2-f_{k+2}h_k}{g_{k+1}^2},
S = \frac{T}{2 \pi}\int d x \sqrt{H^{- 1}( U ) + ( \partial_{x}U )^{2}+{U^{2}}/{9}( \partial_{x}\psi )^{2}+{U^{2}}/{9}( \partial_{x}\psi ) ( \partial_{x}\phi_{1}) +{U^{2}}/{6}( \partial_{x}\phi_{1})^{2}},
G_{\Omega}(z,z') =\int_{\mathbb{C}}G_{\Omega}(z-w,z')\mu_{\epsilon}(w)dV(w) &=\int_{\mathbb{C}}\int_{\mathbb{C}}G_{\Omega}(z-w,z'-w')\mu_{\epsilon}(w)\mu_{\epsilon}(w')dV(w)dV(w') &=\int_{\mathbb{C}}\int_{\mathbb{C}}G_{\Omega}(w,w')\mu_{\epsilon}(z-w)\mu_{\epsilon}(z'-w')dV(w)dV(w')
d_B(j,k) =\Bigg(\frac{j}{6}+\frac{j^2}{2}-\frac{j^2}{4N}+\frac{j^3}{3}-\frac{j^3}{2N}-\frac{j^4}{4N}&-\frac{k}{6}-jk+\frac{jk}{2N}+\frac{j^2k}{2N}+\frac{k^2}{2}-\frac{k^2}{4N}&-jk^2+\frac{jk^2}{2N}+\frac{j^2 k^2}{2N}+\frac{2k^3}{3}-\frac{k^3}{2N}-\frac{k^4}{4N}\Bigg)^{\frac{1}{2}}\,.
\phi_{T}(t, v, x, l)=\psi_{p, T}(t, v, x, l)e^{-\frac{m_1^2v+m^2_2(T-v)}{2}-l\left(m_1p-qm_2\right)+xm(x)},\, 0\leq v\leq t\leq T, l\geq 0,
A_\beta x_\beta^{m_\beta}+ K_1^\beta x_\beta^p \ = \K_2^\beta x_\beta^p - c(1,2), A_\beta m_\beta x_\beta^{m_\beta-1}+ K_1^\beta p x_\beta^{p-1}\ = \K_2^\beta p x_\beta^{p-1}.
\int_{- \infty}^{+ \infty}d \kappa \rho_{\textsf{f i n}}( \kappa ) \beta ( \kappa ) = \lim_{L \to \infty}\left[ \sum_{n = 1}^{L}\frac{1}{4 n}- \sum_{n = 1}^{2 L}\frac{n}{4 n^{2}- 1}\right] = \frac{1}{4}- \frac{3 \log 2}{4}.
S \equiv S_{1}= ( S_{1 , i j})_{i , j = - 1 , 0 , 1}= \left( \begin{array}{l l l}{1 ,}&{0 ,}&{0}\\{0 ,}&{- q ( q^{2}+ 1 )^{- 1},}&{0}\\{0 ,}&{0 ,}&{1}\\ \end{array}\right) .
\sum_{\rho + \Lambda_i\in{\cal F}_{l}}S_{\beta \rho + \Lambda_i}{\cal U}^{(p_i, q_i)}_{\rho + \Lambda_i \, \rho}= \exp \biggl({\pi i k d \over 4 l}{\rm sign}\Bigl({q_i \over p_i}\Bigr) \biggr){\cal U}^{(-q_i, p_i)}_{\beta \rho},
G^{E}( 2 \pi n T , k ) = \pi \, C_{\cal{O}}{\frac{( 2 \pi T n )^{2}+ k^{2}}{4}}\left[ \psi \left( 1 +{\frac{n}{2}}-{\frac{i k}{4 \pi T}}\right) + \psi \left( 1 +{\frac{n}{2}}+{\frac{i k}{4 \pi T}}\right) \right] \, .
2\varphi_{r,s}([G_{m,i},G_{n,j}]) = [\varphi_{r,s}(G_{m,i}),G_{n,j}]-[G_{m,i}, \varphi_{r,s}(G_{n,j})] &=d^1_{r,s}(m,i)[L_{m+r,i+s},G_{n,j}]-d^1_{r,s}(n,j)[G_{m,i}, L_{n+r,j+s}].
\left\Vert x-x_{k-1}\right\Vert_{A^{T}A}^{2}-\left\Vert x-x_{k}\right\Vert_{A^{T}A}^{2}=\phi_{k}^{2}\left(w_{k}^{T}v_{k}-\frac{\theta_{k+1}}{\rho_{k}}v_{k+1}^{T}w_{k}\right).
P\left(\sup_{i=1,\ldots,X_0}\bar{R}_{0}^i\leq k\right)=\sum_{l\ge1}P\left(\sup_{i=1,\ldots,l}\bar{R}_{0}^i\leq k\,,\,\,X_0=l\right)=\sum_{l\ge1}P(X_0=l)\bar{\alpha}_k^l.
\Phi_x(x_1\wedge\cdots\wedge x_r) = x\wedge x_1\wedge\cdots\wedge x_r, x\in X, \Phi_{x'}(x_1\wedge\cdots\wedge x_r) = 2\sum_{i=1}^r(-1)^{i-1}(x'|x_i)x_1\wedge\cdots\wedge\widehat{x}_i\wedge\cdots\wedge x_r, x'\in X',
\int_0^T\left\langle 2\frac{d}{ds}k(t+s,\cdot,\cdot),\varphi(s)\right\rangle\, ds + \int_0^T \int_P Q(k(t+s,\cdot,q),\varphi(s,\cdot,q))\, d\mu(q)\, ds &+ \int_0^T \int_P\widehat Q(k(t+s,p,\cdot),\varphi(s,p,\cdot))\,d\mu(p)\, ds = 0,
\frac{d}{dt}\int_\Omega \big( u(x,t)+v(x,t)\big) \;dx =- \int_\Omega \big( a u(x,t)+b v(x,t)\big) \;dx +\int_\Omega \kappa\;dx &\leq - \min\{a,b\}\int_\Omega \big( u(x,t)+v(x,t)\big)\;dx + \kappa |\Omega|.
\int_0^1 e^{-|x-y|^2/4t}\,\frac{dt}{t}=\left(\int_0^1+\int_1^{1/|x-y|^2}\right) e^{-1/4t}\,\frac{dt}{t}&\leq 4+\int_1^{1/|x-y|^2}\frac{dt}{t}=4+2\log\frac{1}{|x-y|}(0<|x-y|<1),
\sum_{x=1}^{\infty}P(x)^{2}+2\sum_{x=1}^{\infty}\sum_{n=2}^{\infty}P(x)P(nx)=\sum_{x=1}^{\infty}P(x)^{2}+2\sum_{x=1}^{\infty}P(x)(\sum_{n=2}^{\infty}P(nx))
\mathop{\textstyle \prod}\limits_{j=1}^{N}\mathcal{Z}_{\ell_{j},k_{j}}=\Big(\mathop{\textstyle \prod}\limits_{j=1}^{N}\lambda_{k_{j}}\Big)^{-\frac{1}{2}(1-\delta_{0})}\Big( \mathop{\textstyle \prod}\limits_{j=1}^{N}\emph{q}_{k_{j},k_{j}}^{\ell_{j},\ell_{j}}\Big)^{\frac{1}{2}}\mathcal{O}(\varepsilon^{-N\delta_{0}}).
\sin(z;q) :=\frac{(q;q)_\infty}{(q^{1/2};q)_\infty}z^{1/2}J^{(3)}_{1/2}(z(1-q);q^2) = \sum_{n=0}^{\infty}(-1)^n \frac{q^{n^2+n}z^{2n+1}}{\Gamma_q (2n+2)},
\mathcal{E}_n(k, \ell) = \begin{cases}f_n^\ast(k) \mbox{on the event}I_n \cap J_n 1 - F^\ast(k - 1) - F^\ast(\ell - 1) \mbox{on the event}I_n \cap J_n^c F^\ast(k) + F^\ast(\ell) - 1 \mbox{on the event}I_n^c \cap J_n f_n^\ast(\ell) \mbox{on the event}I_n^c \cap J_n^c. \end{cases}
K_2(\xi,\eta) = \ \frac12(\tanh \xi + \tanh \eta)(\xi+\eta) - \frac12(1 + \tanh \xi \tanh \eta)(\xi-\eta)\tanh(\xi-\eta) = \ \tanh \xi \tanh \eta \left (\frac{\xi}{\tanh \xi}+ \frac{\eta}{\tanh \eta}- (\xi-\eta) \tanh(\xi-\eta)\right ).
{\cal L}= - \frac{1}{4}F_{\mu \nu}^2 + \frac{1}{2}(\partial_\mu \varphi )(\partial^\mu \varphi ) - \frac{e}{{2\sqrt \pi}}\varepsilon^{\mu \nu}F_{\mu \nu}\varphi + + m\sum \left({\cos \left({2\pi \varphi + \theta}\right) - 1}\right) - A_0J^0 ,
{\bf U}_{2,1}^p{\bf H}_{r,1}={\bf U}_{1,2}^p{\bf H}_{r,2}, &{\bf U}_{3,1}^p{\bf H}_{r,1}={\bf U}_{1,3}^p{\bf H}_{r,3}, &{\bf U}_{3,2}^p{\bf H}_{r,2}={\bf U}_{2,3}^p{\bf H}_{r,3}, &{\bf U}_{3,1}^{c}{\bf H}_{r,1}+{\bf U}_{1,2}^c{\bf H}_{r,2}={\bf U}_{2,3}^{c}{\bf H}_{r,3}.
\partial_d( I_{h,T}u(X)-u(X))=\sum_{i\in\mathcal{I}^{s'}\cup\mathcal{I}^{int}}(\mathcal{E}_i + \mathcal{F}_i )\partial_d\phi_{i,T}(X)+\sum_{i\in\mathcal{I}}\mathcal{R}_i \partial_d\phi_{i,T}(X), ~\forall X \in T_{non}^s\cap T^s_p, ~s = \pm,
D_a =\partial_a +{1 \over 4}\partial_a x^m \big[\ (\omega_{\mu \nu m}-{1\over 2}H_{\mu \nu m}\rho_3) \Gamma^{\mu \nu}+({1 \over 3!}F_{\mu\nu\lambda}\Gamma^{\mu\nu\lambda}\rho_1 +{1 \over 2\cdot 5!}F_{\mu\nu\lambda\rho\kappa}\Gamma^{\mu\nu\lambda\rho\kappa}\rho_0 ) \Gamma_m\ \big]
\begin{matrix}A=[\begin{matrix}1&-2&2\\ 2&-1&2\\ 2&-2&3\end{matrix}]&B=[\begin{matrix}1&2&2\\ 2&1&2\\ 2&2&3\end{matrix}]&C=[\begin{matrix}-1&2&2\\ -2&1&2\\ -2&2&3\end{matrix}]\end{matrix}
E\sup_{t\in[0,T]}\|u(t)\|_m^2+\sum_{k=1}^{d_1}E\int_{0}^T\|L_ku(t)\|_m^2\,dt &\leq C\left\{E\|\underline{u}_0\|_m^2 + E\int_{0}^T \left(\|f(s)\|_m^2+\|g(s)\|_m^2+\|h(s)\|_m^2\right)\,ds\right\},
y_j(x_j)=\begin{cases}\displaystyle\frac{j(n-j-1)(n-j)}{n(n-1)(n-2)}, \mbox{if}\;\;x_j=1; begin{align*}4mm]\displaystyle\frac{2j(j-1)(n-j)}{n(n-1)(n-2)}, \mbox{if}\;\;x_j=2; begin{align*}4mm]\displaystyle\frac{j(j-1)(j-2)}{n(n-1)(n-2)}, \mbox{if}\;\;x_j=3; begin{align*}4mm]0, \mbox{otherwise}.\end{cases}
\phi_{\ell}(z_\ell', x'; x_\ell) = a_\ell z_\ell' - \lambda_\ell \ \begin{bmatrix}z_\ell' x' - x_\ell \end{bmatrix}^{\top}Q_{\ell}\begin{bmatrix}z_\ell' x' - x_\ell \end{bmatrix}.
\Pr \{d(X, \hat{g}(\hat{f}(X))) \leq D \}&= \sum^{k^{*}-1}_{i=1}\Pr \{X \in A_{D}(y_i) \}+ \Pr \{\hat{f}(X) = w_{k^*}, X \in A_{D}(y_{k^*}) \}&=\alpha + \beta &= 1 - \epsilon.
(A \log|t_1| + B)^{-1}= \begin{pmatrix}[c|c] Q^{-1}-Q^{-1}B_{12}B_{22}^{-1}\hline -B_{22}^{-1}B_{21}Q^{-1}B_{22}^{-1}+ B_{22}^{-1}B_{21}Q^{-1}B_{12}B_{22}^{-1}\end{pmatrix}\, .
f_{\mathbf{M^{\ast}}_{-\xi_{\boldsymbol{s}k}}}(\boldsymbol{w})=c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast}\exp\left\{-\frac{1}{2}\boldsymbol{w}^{T}\boldsymbol{w}-\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}\right\}=\phi_{n-1}(\boldsymbol{w}),~\boldsymbol{w}\in \mathbb{R}^{n-1},
\Phi_{ij}(x)=G\int_{V}\rho(x^{\prime})\frac{(x_{i}-x_{i}^{\prime})(x_{j}-x_{j}^{\prime})}{|x-x^{\prime}|^{3}}dx^{\prime},\Rightarrow\Phi_{ii}=\Phi=G\int_{V}\frac{\rho(x^{\prime})}{|x-x^{\prime}|}dx^{\prime}
\begin{cases}u_t+uu_x+vu_y+wu_z+\hat{f}w-fv=-\frac{1}{\rho_0}P_x, v_t+uv_x+vv_y+wv_z+fu=-\frac{1}{\rho_0}P_y, w_t+uw_x+vw_y+ww_z-\hat{f}u=-\frac{1}{\rho_0}P_z-g. \end{cases}
V_{1}(q)=\sum_{n\ge0}\frac{q^{(n+1)^{2}}(-q;q^{2})_{n}}{(q;q^{2})_{n+1}}=\sum_{n\ge0}\frac{q^{2n^{2}+2n+1}(-q^{4};q^{4})_{n}}{(q;q^{2})_{2n+2}}
0\leq u(t,x) \leq \int_{\mathbb{R}^d}t^{-\frac{d}{2}}e^{c_3(t+1)}e^{-\frac{|x-y|^2}{c_4t}}u_0(y)\,\mathrm dy &=(c_4\pi)^{\frac{d}{2}}e^{c_3(t+1)}(e^{\frac{c_4}{4}t\Delta}u_0)(x).
\begin{aligned}H(x,\xi)=H_0(\xi)+V(x),H_0(\xi)=T_a(\xi_a)+H_0^a(\xi^a)=\frac{1}{2}|\xi|^2, &T_a(\xi_a)=\frac{1}{2}\sum_{\ell=1}^k|\xi_\ell|^2,H_0^a(\xi^a)=\frac{1}{2}\sum_{\ell=1}^k\sum_{i=1}^{|C_\ell|-1}|\xi_i^{C_\ell}|^2.\end{aligned}
\Lambda_{0}^{5}= \frac{1}{\bar{\Delta}_{( m^{\prime}, n^{\prime})}^{1 / 2}}\left( \begin{array}{c c}{{n^{\prime}\qquad}}&{{m^{\prime}\, e^{- \phi_{0}}+ \chi_{0}\, ( n^{\prime}+ \chi_{0}m^{\prime}) \, e^{\phi_{0}}}}\\{{- m^{\prime}\qquad}}&{{( n^{\prime}+ \chi_{0}m^{\prime}) \, e^{\phi_{0}}}}\\ \end{array}\right) ,
\lim_{M\rightarrow \infty}\lim_{N\rightarrow \infty}G_{k_1,k_2,M,N}(\tau) = E_{k_1}(\tau) E_{k_2}(\tau)-\frac{1}{2}\delta_{k_2,1}\frac{E_{k_1-1}'(\tau)}{k_1-1}- \frac{1}{2}\delta_{k_1,1}\frac{E_{k_2-1}'(\tau)}{k_2-1}-(-1)^{k_2}E_{k_1+k_2}(\tau),
q(t) := \int_0^\infty f_{E_t}(\tau) h(\tau) d\tau= -\int_0^\infty \frac{\partial}{\partial\tau}\left\{\frac{1}{\tau^{1/\beta}}(Jf_1)\bigg(\frac{t}{\tau^{1/\beta}}\bigg)\right\}h(\tau) d\tau.
\Omega =- 2 V \int \frac{d^3p}{(2\pi)^3}\int_{C_0}\frac{dz}{2\pi i}\left [\mbox{tanh}(\frac 12 \beta z) f(z,\mu)-\lim_{\beta\rightarrow \infty}\mbox{tanh}(\frac 12 \beta z) f(z,0) \right ],
z_3 =- \frac{1}{2}\rho^{2}+ O(\rho^3) , z_4 = \frac{1}{2}(\lambda_{1}x_1^2 + \lambda_2 x_2^2) + O(\rho^3), , z_j = \frac{1}{2}( \mu_{1,j}x_1^2 + \mu_{2,j}x_2^2 + 2m_j x_1 x_2) + O(\rho^3).
\# \{(i,j) \in I \times ( [m] \setminus I ) : i > j \}=\Sigma(I) - \binom{m-k+1}{2}, \# \{(i',j') \in ([n]' \setminus J') \times J' : i' > j' \}=k(n-k) + \binom{n-k+1}{2}- \Sigma(J).
\Psi_{\epsilon}( \alpha ) \equiv \frac{\mu^{\epsilon}}{2 R}\sum_{n = - \infty}^{\infty}\int \frac{d^{d - \epsilon}p}{( 2 \pi )^{d - \epsilon}}\left[ p^{2}+ \left( \frac{2 \pi n}{R}\right)^{2}+ m^{2}\right]^{\alpha}.
M_{R}= \left( \begin{array}{c c c}{{1}}&{{r_{1}}}&{{r_{3}}}\\{{0}}&{{1}}&{{r_{2}}}\\{{0}}&{{0}}&{{1}}\\ \end{array}\right) , \; \; M_{D}= \left( \begin{array}{c c c}{{d_{1}}}&{{0}}&{{0}}\\{{0}}&{{d_{2}}}&{{0}}\\{{0}}&{{0}}&{{( d_{1}d_{2})^{- 1}}}\\ \end{array}\right) , \; \; M_{L}= \left( \begin{array}{c c c}{{1}}&{{0}}&{{0}}\\{{l_{1}}}&{{1}}&{{0}}\\{{l_{3}}}&{{l_{2}}}&{{1}}\\ \end{array}\right) .
\chi(A)(f_0, f_1, f_2)= \int_{SL_2(\mathbb{R})}\int_{SL_2(\mathbb{R})}f_0\big((g_1g_2)^{-1}\big) f_1(g_1) f_2(g_2) \operatorname{Area}_{\mathbb{H}}\big( \Delta([e], g_1[e], g_1g_2[e])\big)dg_1 dg_2.
\mathcal{E}_{n+m}^{\left( \alpha\right)}\left( \lambda\right) =\sum_{k=0}^{m}\sum_{j=0}^{n}\binom{n}{j}\genfrac{\{}{\}}{0pt}{}{m}{k}\binom{\alpha+k-1}{k}\frac{\left( -\lambda\right)^{k}k!k^{n-j}}{2^{k}}\mathcal{E}_{j}^{\left( \alpha+k\right)}\left( \lambda\right) .
\begin{bmatrix}e_{(m+1)p}(x) e_{(m+1)p+1}(x) \cdots e_{(m+1)p+m}(x)\end{bmatrix}^{T}= K^{p}\begin{bmatrix}d_0(x) d_1(x) \cdots d_m(x) \end{bmatrix}^T.
u_{i\mp\frac{1}{2},\beta}^{\pm}=\theta_{ij}(u_{i\mp\frac{1}{2},\beta}^{\pm,old}-\bar{u}_{ij}^n)+\bar{u}_{ij}^n, u_{\beta,j\mp\frac{1}{2}}^{\pm}=\theta_{ij}(u_{\beta,j\mp\frac{1}{2}}^{\pm,old}-\bar{u}_{ij}^n)+\bar{u}_{ij}^n.
K(\theta)=\frac{s_{\frac{1}{2}}(\theta)s_{\frac{1+a}{2}}(\theta) s_{\frac{2-a}{2}}(\theta)}{s_{\frac{1-E}{2}}(\theta)s_{\frac{1+E}{2}}(\theta) s_{\frac{1-F}{2}}(\theta)s_{\frac{1+F}{2}}(\theta)}\,\,\,,
\langle T_{\mu \nu}\rangle = \operatorname*{l i m}_{x^{\prime}\rightarrow x}\left[{\frac{2}{3}}{D}_{; \nu^{\prime}\mu}-{\frac{1}{3}}{D}_{; \nu \mu}-{\frac{1}{6}}g_{\mu \nu}{D}^{; \sigma^{\prime}}{}_{\sigma}\right]
\det \mathbf{T}_{n,m,a,b}(x;q)=q^{\varphi(n,a,b)}\prod_{i=1}^n\prod_{j=1}^{m+b-a}\frac{(1-q^{x+m+i-j})(1-q^{x+2i+j+a-b-2})}{(1-q^{x+m+2i-j-1})(1-q^{i+j-1})}.
0 < i \left[ e_{\alpha}^{A}\bar{h}_{A \bar{\beta}}- h_{A \alpha}\bar{e}_{\bar{\beta}}^{A}\right] = e_{\alpha}^{A}\left[ i \bar{h}_{A \bar{\gamma}}( \bar{e}^{- 1})_{B}^{\bar{\gamma}}- i ( e^{- 1})_{A}^{\gamma}h_{B \gamma}\right] \bar{e}_{\bar{\beta}}^{B}= e_{\alpha}^{A}\left[ i \bar{\Pi}_{A B}- i \Pi_{B A}\right] \bar{e}_{\bar{\beta}}^{B}\ ,
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^+}\; d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_d \left( s+k \right) T R^d \left[\alpha +\frac{y}{\eta T}+ C_{d,\alpha}\left(\frac{\eta}{R}\right)^{d-1}+C_{d,\alpha}\theta^{(d-1)/2}\right]\end{aligned}
s_2s_3s_1 s^{h}s_1s_3s_2 =(s_2s_3s_1)(s_1s_2s_3s_1)(s_1s_3s_2) = s_2s_3, s_2s_3s_1 s^{hv}s_1s_3s_2 =(s_2s_3s_1)(s_1s_2s_1s_2s_3s_1)(s_1s_3s_2) = (s_2s_3)(s_2s_1).
0 = < \nabla u(F_p(x)), w > = \int_{\partial X}p < \nabla B^{f(\xi)}(F_p(x)), w > \exp(pB(z, \pi(\phi(\overrightarrow{x\xi})), f(\xi))) d\mu(\xi) = - p \int_{\partial X}< \overrightarrow{F_p(x)f(\xi)}, w > \exp(pB(z, \pi(\phi(\overrightarrow{x\xi})), f(\xi))) d\mu(\xi).
P_1(k_0,k_1,k_2,\cdots, k_n)=\sum_{\tau_1,\tau_2,\cdots,\tau_n}P_1(\tau_0,\tau_1,\tau_2,\cdots,\tau_n)P_1(k_0-\tau_0,k_1-\tau_1,k_2-\tau_2,\cdots,k_n-\tau_n)
\displaystyle \frac{\pi |z_{j,n}|}{n \, s_{j,n}}< |z - z_{j,n}| ,\mbox{with}~~s_{j,n}= a_{\max}\Bigl[1 +\frac{a_{\max}^{2}(j - J_n)^2}{\pi^2 \,J_{n}^{2}}\Bigr]^{-1/2}.
\Gamma_0=\inf_{z\in\mathbb{R}^n}\sup_{u\in\mathcal{A}(F(\overline{x}),dF(\overline{x})(w))}\Big\{\langle u,d^2F(\overline{x})(w|z)\rangle-\Xi(z)\Big\}+\langle\overline{\xi},d^2F(\overline{x})(w)\rangle.
b =(\neg k_{2l+1}\wedge b)\vee(k_{2l+1}\wedge b)= v\vee(k_{2l+1}\wedge b) \leq v\vee(k_{2l+1}\wedge (\neg c_{2l+2,j}\vee b'))=v\vee(k_{2l+1}\wedge\neg c_{2l+2,j})\vee (k_{2l+1}\wedge b') \leq v\vee(k_{2l+1}- c_{2l+2,j})\vee b' \leq v\vee(k_{2l+1}- k_{2l+2})\vee b'=b.
P\times \mathbb{C}^* = \mathrm{Stab}_{G\times \mathbb{C}^*}((B\times \mathbb{C}^*)\cdot \tilde{x}_0) = \mathrm{Stab}_{G}((B\times \mathbb{C}^*)\cdot \tilde{x}_0) \times \mathbb{C}^* = \mathrm{Stab}_{G}(B\cdot \tilde{x}_0) \times \mathbb{C}^*
\Pr(S, K) = \frac{|Z(S,K)|}{|S|}+ \frac{1}{|S||K|}\underset{s\in S\setminus Z(S, K)}{\sum}|C_K(s)| &\geq \frac{1}{p}+ \frac{|S|-|Z(S,K)|}{n|S|}= \frac{n + p - 1}{np}.
\begin{cases}\Delta v^t = -\Delta \delta_t h^0 \mbox{in}\Omega^t v^t = -\frac 1 t \tilde u^0 \mbox{on}\Gamma^t \lim_{x\to \infty}v^t = \delta_t c^0, \lim_{x\to \infty}\frac{v^t(x)}{-\log |x|}= \delta_t c^0.\end{cases}
\begin{aligned}x\cdot x': =(\xi+\xi',\eta+\eta',s+s'+2\langle\eta,\xi'\rangle-2\langle\xi,\eta'\rangle) =(\xi_1+\xi'_1,\cdots,\xi_n+\xi'_n,\eta_1+\eta'_1,\cdots,\eta_n+\eta'_n,s+s'+2\sum_{i=1}^n(\eta_i\xi'_i-\xi_i\eta'_i)). \end{aligned}
\mathcal{Z}_t=\exp\Big(\int_0^1\int_0^tb(s,\zeta,X_{s,\zeta}) \mathrm{d}X_{s,\zeta}-\dfrac{1}{2}\int_0^1\int_0^t|b(s,\zeta,X_{s,\zeta})|^2\mathrm{d}s \mathrm{d}\zeta\Big)
\sum_{i\leq 3,\;\mu\geq 5}{\rm Tr}[Y^{i},Y^{\mu}]^{2}=-2{\rm Tr}\left(J^{(1)}_{i}A_{\mu}A_{\mu}^{\dag}J^{(1)}_{i}+A_{\mu}J^{(2)}_{i}J^{(2)}_{i}A_{\mu}^{\dag}-2J^{(1)}_{i}A_{\mu}J^{(2)}_{i}A_{\mu}^{\dag}\right),
G_{0}(y) := \frac{1}{2\pi i}\int_{(\sigma)}(\pi^3 y)^{-s}\, \prod_{i=1}^{3}\frac{\Gamma\left(\frac{1+s+{\bf \alpha}_{i}}{2}\right)}{\Gamma\left(\frac{-s-{\bf \alpha}_{i}}{2}\right)}\tilde{v_1}(-s)ds.
\psi(t)=\begin{cases}O\left(t^{\theta}\right) \hbox{if}\theta<-\frac{M}2+5 O\left(t^{-\frac{M}2+5}\right) \hbox{if}\theta\ne-\frac{M}2+5 O\left(t^{-\frac{M}2+5}|\log t|\right) \hbox{if}\theta=-\frac{M}2+5\end{cases}
\xi_{2}( \mathrm{z}) = \vartheta [ \begin{array}{c}{{\frac{1}{3}}}\\{{0}}\\ \end{array}] ( \mathrm{z}, \tau ) \ \ \ \mathrm{o r \}\ \ \vartheta [ \begin{array}{c}{{0}}\\{{\frac{1}{3}}}\\ \end{array}] ( \mathrm{z}, \tau ) \ .
\displaystyle\int_{\mathbb{R}^N}b(x)|\Psi_n(x,0)|^{2_\alpha^*}dx=\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx-\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_\lambda^+|^{2_\alpha^*}dx+o_n(1).
A= \bigcup_{i=0}^{n}\left(\frac{i}{n}+A_i\right),\textrm{and}\; \bigcup_{i=0}^{n}A_i \bmod 1 \,\subset J, \textrm{so in particular}\lambda\Big(\bigcup_{i=0}^{n}A_i\Big)\leq (1+\varepsilon)\frac{\lambda(A)}{n}.