harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta E_{\vec{k}}= \frac{3}{8}\frac{g^{2}L}{\ell^{2}}+\frac{64}{9\pi^{2}}\frac{1}{g^{2}L}\left[\sum_{\vec{\delta}}\sin^2\left(\frac{\pi \vec{\delta}\vec{k}}{K}\right) - 1 \right]
\sum_{r=0}^{n}a_{r}(\overline{\zeta})^{r}=\sum_{r=0}^{n}a_{r}\overline{\zeta^{r}}=\sum_{r=0}^{n}\overline{a_{r}\zeta^{r}}=\overline{\sum_{r=0}^{n}a_{r}\zeta^{r}}.
\xi_q^l \frac{\lambda}{1-\xi_q^k}= \frac 1k \Big( 1 + \frac{-1-2l+k}{2}\lambda + \frac{-1+k^2-6kl+6l^2}{12}\lambda^2 + \frac{-1+k^2-6kl-2k^2l+6l^2+6kl^2-4l^3}{24}\lambda^3 + \ldots\,,
\int\chi\Bigg{[}\frac{1}{A}\bigg{(}y+\frac{x(t)-x(T)}{\lambda(t)}\bigg{)}\bigg{]}\varepsilon^2\lesssim A\|\varepsilon\|_{L^{\infty}}^2\leq Ab_c^{\frac{149}{135}}\leq b_c^{\frac{1}{20}}, \int\chi\Bigg{[}\frac{1}{A}\bigg{(}y+\frac{x(t)-x(T)}{\lambda(t)}\bigg{)}\bigg{]}|Q_b|^2=\big{(}1+\delta_0(p)\big{)}\int |\mathcal{Q}_p|^2.
\int_{\mathcal{Y}^d}\frac{1}{\widetilde{m}}\widetilde{m}^2_{x_j}dy+\int_{\mathcal{Y}^d}\widetilde{m}\widetilde{w}_{y_ix_j}^2dy=\int_{\mathcal{Y}^d}V_{x_j}\widetilde{m}_{x_j}dy\leq\int_{\mathcal{Y}^d}\frac{1}{2\widetilde{m}}\widetilde{m}_{x_j}^2dy+\int_{\mathcal{Y}^d}\frac{V_{x_j}^2}{2}\widetilde{m}dy.
F^{\mu}= ( - g )^{- \tau}\partial_{\nu}\left( g^{\mu \nu}( - g )^{\tau}\right) + a_{1}\bar{\Gamma}_{~ \alpha \beta}^{\mu}g^{\alpha \beta}+ a_{2}\bar{\Gamma}_{~ \alpha \nu}^{\nu}g^{\alpha \mu}+ a_{3}\bar{\Gamma}_{~ \nu \alpha}^{\nu}g^{\alpha \mu}+ E_{~ \sigma}^{\mu \rho \lambda ~ \alpha \beta}\nabla_{\rho}\nabla_{\lambda}\bar{\Gamma}_{~ \alpha \beta}^{\sigma}
\mathrm{d}\varepsilon = \frac{1}{80\pi^2}|J| \, \left( \mathrm{d}^{(1/20,2/20)}+ \mathrm{d}^{(1/20,2/5)}+ \mathrm{d}^{(3/20,3/10)}+ \mathrm{d}^{(3/20,1/5)}\right),
P \left\lbrace \textbf{X}\in \widehat{\mathcal{C}}_{N}\Delta \mathcal{C}\Big| D_{n}\right\rbrace \leq I \{N>0 \}\frac{\mu(\widehat{\mathcal{C}}_{N}\Delta \mathcal{C})}{\mu(\mathcal{C})}+ I \{N = 0 \}.
R^\mathrm{J}=\log|\mathbf{I}+\mathbf{H}_\mathrm{r}\mathbf{Q}_\mathrm{s}\mathbf{H}_\mathrm{r}^\mathrm{H}(\mathbf{H}_\mathrm{z}\mathbf{Q}_\mathrm{z}\mathbf{H}_\mathrm{z}^\mathrm{H}+\sigma^2\mathbf{I})^{-1}|
\theta^{t+1}_i = \theta^{t}_i + \eta x^t_i y^t- \eta \sum_{j=1}^{d}x^t_i x^t_j \theta^{t}_j = \theta^t_i + \eta x^t_i \Big( \sum_{j=1}^d x^t_j \theta^*_j +\varepsilon{^t}-\sum_{j=1}^d x^t_j\theta^t_j \Big) = \theta^t_i - \eta \sum_{j=1}^d x^t_ix^t_j (\theta^t_j-\theta^*_j)+\eta x^t_i \varepsilon^t.
\lim_{t\rightarrow \infty}(a_t)^{1/t}=\lim_{t\rightarrow\infty}(h_t(\beta_t))^{-1/t}=\lim_{t\rightarrow\infty}\{\beta_t\exp[(t-1)u(\beta_t)]\}^{-1/t}=\lim_{t\rightarrow\infty}\exp[-u(\beta_t)] =\left(\int_0^\infty x^{-1}d\mu(x)\right)^{-1};
\left\lbrace\begin{aligned}-\Delta u =K(x)u^{\frac{n+2}{n-2}}\mbox{in}\,\Omega u >0 \,\,\,\,\,\,\,\,\,\,\,\,\mbox{in}\,\Omega u =0 \,\,\,\,\,\,\,\,\,\,\,\, \mbox{on}\,\partial\Omega,\end{aligned}\right.
\Theta^{d,T}(\cdot,x)=(\lfloor{dx}\rfloor+1-dx)\Theta^{d,T}\Big(\cdot,\frac{\lfloor{dx}\rfloor}{d}\Big)+(dx-\lfloor{dx}\rfloor)\Theta\Big(\cdot,\frac{\lfloor{dx}\rfloor+1}{d}\Big).
\begin{aligned}f_n^*{(y)}=\rho_nh e^{\tilde{u}_n^{(1)}}(\zeta_n{(y)}+O(\|\tilde{u}_n^{(1)}-\tilde{u}_n^{(2)}\|_{L^{\infty}(M)}))=\rho_nh e^{\tilde{u}_n^{(1)}{(y)}}(-b_0+o(1)),\end{aligned}
\alpha_n(a,b,c,q)=\frac{q^{n^2}(bc)^n(1-aq^{2n})(a/b)_n(a/c)_n}{(1-a)(bq,cq)_n}\sum_{0\le j\le n}\frac{(-1)^j(1-aq^{2j-1})(a)_{j-1}(b,c)_j}{q^{j(j-1)/2}(bc)^j(q,a/b,a/c)_j},
-\int_0^T \int_\Omega \big(\tfrac{1}{2}\bar \rho |\bar u|^2+ h_1(\bar \rho) + h_2(\bar n) + \tfrac{1}{2}|\nabla h_2^\prime(\bar n)| \big) \dot{\theta}(t) \dxdt & = \int_\Omega \big( \tfrac{1}{2}\bar \rho |\bar u|^2+h_1(\bar \rho) + h_2(\bar n) + \tfrac{1}{2}|\nabla h_2^\prime(\bar n)| \big) \big|_{t=0}\theta(0) \dx \ ,
-\langle1,f'(\phi)v + \mathcal{K}*v \rangle = \left\langle1,\left(c - f'(\phi) - \mathcal{K}* + k\partial_{\theta}\mathcal{K}_{1}- \frac{1}{2}k\partial_{\theta}\mathcal{K}_{2}*\right)v \right\rangle - c\langle1,v\rangle &= \langle\Psi_{2}^{0},(\mathcal{A}_{1}+ \mathcal{A}_{2}- c)v\rangle,
\frac 1N \sum_{j=0}^{2N-1}{(1-x_j^2 )}^{h-1}\le \int_{-1}^1 (1-x^2)^{h-1}\,dx ~+~ \frac 1N & = \frac{2^{2h-1}\cdot ((h-1)!)^2}{(2h-1)!}~+~ \frac 1N\,.
-\frac{N}{\gamma_N}\left(\sum_{k=1}^3\mu^N_{\tau_c^N,k}(f_k)-\sum_{k=1}^3\mu_{\tau_c^N,k}(f_k)\right) &=\frac{N}{\gamma_N}\left(\sum_{k=1}^3\mu_{\tau_c^N, k}(f_k)-\sum_{k=1}^3\mu_{\tau_c,k}(f_k)\right)+O(\gamma_N^{-1})=\sum_{k=1}^3\eta^N_{\tau_c^N,k}(f_k)+o(1),
:\!|g|^6\!: \, = \tfrac 18 H_6(h_1) + \tfrac 38 H_4(h_1) H_2(h_2)+ \tfrac 38 H_2(h_1) H_4(h_2)+ \tfrac 18 H_6(h_2) & = |g|^6 - 9|g|^4 + 18 |g|^2 - 6.
\begin{array}{l c c c c}{{\mathrm{f o r ~}1}}&{{:}}&{{1 ,}}&{{1 ,}}&{{1}}\\{{\mathrm{f o r ~}\epsilon}}&{{:}}&{{1 ,}}&{{1 ,}}&{{- 1}}\\{{\mathrm{f o r ~}\rho}}&{{:}}&{{2 ,}}&{{- 1 ,}}&{{0 .}}\\ \end{array}
\mathcal{I}(z_1,\dots,z_n;\vec{u}) = \prod_{j=1}^n \prod_{\alpha=1}^r \frac{-u_\alpha z_j}{(z_j - u_\alpha) (q_1 q_2 z_j - u_\alpha)}\; \prod_{1 \leq j \neq k \leq n}\frac{(z_j- z_k)(z_j - q_1q_2z_k)}{(z_j - q_1z_k)(z_j - q_2z_k)}.
|\zeta_0|\ge\begin{cases}\left |a_1 \right|^{\frac{1}{q}-\frac{2}{qp}}\cdot \left|\frac{q+1}{2}a_1^2-a_2 \right|^{\frac{1}{qp}}\left|\frac{q+1}{2}a_1^2-a_2 \right|^{\frac{1}{2q}}\left|a_2\left(a_1^2-a_2\right)\right|^{\frac{1}{4q}}\end{cases}
\sum_{k=1}^d ET(k) = 2 \sum_{k=1}^d\sum_{i=1}^{k-1}\sum_{j=i+1}^{k-1}\frac{1}{((x_k-x_i)+\epsilon^2)((x_k-x_j)+\epsilon^2)}- 2 \sum_{k=1}^d \sum_{i=k+1}^{d}\sum_{j=i+1}^{d}\frac{(x_i-x_k)}{((x_i-x_k)+\epsilon^2)((x_j-x_k)+\epsilon^2)((x_j-x_i)+\epsilon^2)}.
{\mathbb M}(\varphi,\varphi)={\mathbb M}(\varphi_0,\varphi_0)+ \sum_{j=1}^{\infty}{\mathbb M}(\varphi_j^1,\varphi_j^1)+\sum_{j=1}^{\infty}{\mathbb M}(\varphi_j^2,\varphi_j^2).
\begin{array}{r c l}{{\delta_{\omega}V^{a}}}&{{=}}&{{\omega^{b}\partial_{b}V^{a}- V^{b}\partial_{b}\omega^{a}+{\textstyle{\frac{1}{2}}}\partial_{b}\omega^{b}V^{a},}}\\{{\delta_{\omega}X}}&{{=}}&{{\omega^{a}\partial_{a}X .}}\\ \end{array}
\sum_{i \in I}v_{i}^{2}\langle \pi_{Z_{i}}C^{\prime}f, \pi_{Z_{i}}C f\rangle =\Vert v_{i}(C^{*}\pi_{{Z}_{i}}C^{\prime})^{\frac{1}{2}}f \Vert^{2}_{2}&\geq(\dfrac{(1-\lambda_{1})\sqrt{A}-\Vert\beta\Vert_{2}}{1+\lambda_{2}}\Vert f \Vert)^{2}
S_3(X)=\sum_{q\le P}\int_{|\beta|\le\frac{1}{qP}}\sum_{|v|\le P}\sigma(v;\beta,q)\sum_{\substack{a=1 (a,q)=1}}^q\frac{S(q,a)^3}{q^4}\mathcal{I}(\beta,q)e\Big(-\frac{\overline{a}\,v}{q}\Big)d\beta + \sum_{q\le P}\int_{|\beta|\le\frac{1}{qP}}\sum_{|v|\le P}\sigma(v;\beta,q)D(q,\beta)d\beta,
Z(\omega,\theta)=\sum_{n=0}^N \frac{Z_n(N,\omega)e^{\theta n}}{n!}=\sum_{\sigma_m=0,1}\exp \left( \sum_{0 \le m<k \le N-1}A_{mk}\sigma_m\sigma_k+(\theta+\eta) \sum_{m=0}^{N-1}\sigma_m \right),
G_{0,2}(x)= \sum_{n\geq 0}r_{n}x^n = \sum_{n=0}^\infty \frac{x^{2^n-1}}{1-x^{2^{n+2}}}= \sum_{n=0}^{\infty}x^{2^n-1}\left(\sum_{k\ge 0}(x^{2^{n+2}})^k\right) = \sum_{n,k\ge 0}x^{4k\cdot 2^{n}+2^{n}-1}.
f_1^{\mathrm{PNC}}\bar{R}_1^{\mathrm{PNC}}\ge \max(\lambda_1,\lambda_2) &f_2^{\mathrm{PNC}}\bar{R}_2^{\mathrm{PNC}}\ge \max(\lambda_1,\lambda_2) &P_{11}^{\mathrm{PNC}}g_{1r}=P_{12}^{\mathrm{PNC}}g_{2r}&f_1 + f_2 \leq 1
\frac{\rho^{n+1}_i-\rho^n_i}{\tau}= \nabla \cdot \left(D_i(\nabla \rho^{n+1}_i +z_i\nabla\phi^{n+1})\right)+O(\tau), & \nabla \cdot (\epsilon(x)\nabla \phi^{n+1})=f+\sum_{i=1}^sz_i\rho_i^{n+1}.
\sigma_{j}^{a}= \mathbf{1}_{V}^{\otimes ( j - 1 )}\otimes \sigma_{}^{a}\otimes \mathbf{1}_{V}^{\otimes ( N - j )}, \qquad{\cal V}= V^{\otimes N}= \underbrace{V \otimes V \otimes \ldots \otimes V}_{\mathrm{N f a c t o r s}},
b_{\ell,\varphi}(x,\xi',\xi_d =\pi_{j,2},\tau) =b_\ell(x,\xi'+i\tau d_{x'}\varphi, \pi_{j,2}+i\tau\partial_d\varphi) = b_\ell(x,\xi'+i\tau d_{x'}\varphi, i\alpha_j) =b_\ell(x,\xi'+i\tau d_{x'}\varphi, i|\xi'|_x+i(\alpha_j-|\xi'|_x)).
-\hbar'(p - 1) + a_s = t r_s + \hbar'(p_1 + \cdots + p_s - p + 1 - \dfrac{3}{2}) &= t m_{p_1 + \cdots + p_s - p +1}+ \hbar'(p_1 + \cdots + p_s - p + 1 - \dfrac{3}{2})
\frac{\partial \chi_w}{\partial w}(\mathcal{Z}) = -\frac{(1-a)}{4\sqrt{w}}\log\left(\frac{2\sin(\pi\mathcal{Z})}{\pi\sqrt{w}}\right)+\frac{\sqrt{w}(1-a)}{2}\frac{\pi\sqrt{w}}{2 \sin(\pi \mathcal{Z})}\frac{2\sin(\pi\mathcal{Z})}{2\pi w^{3/2}}&= O\left(\frac{\log(w)}{\sqrt{w}}\right)+O\left(\frac{1}{\sqrt{w}}\right) = O\left(\frac{1}{w}\right).
\sum_{i=1}^n(1+2 \rho|\mathcal{N}_i|)\frac{\sum_{l=0}^{T-1}x_i^{k+l}}{T}-\sum_{i=1}^{n}2\rho|\mathcal{N}_i|\frac{\sum_{l=0}^{T-1}\mathcal{Q}_\delta(x_i^{k+l})}{T}-\sum_{i=1}^nr_i=0,
F_{hkl}=\sum_{j=1}^{2}f_{j}e^{[-2\pi i(hx_{j}+ky_{j}+lz_{j})]}=[f_{Cs}+f_{Cl}e^{[-i\pi(h+k+l)]}]=[f_{Cs}+f_{Cl}(-1)^{h+k+l}]
y_1=x_1, \, y_2=x_1-x_2-\frac{1}{2\mu}x_3, \, y_3=x_1-x_2+\frac{1}{2\mu}x_3; &[y_1,y_2]=[y_3,y_1]=\frac{1+w}{\mu}y_1, \,[y_2,y_3]=-\frac{2w}{\mu}y_1+\frac{w-1}{2\mu}(y_2+y_3).
\partial_x^m \partial_y^{3-m}G(p_1,p_2) - \partial_x^m \partial_y^{3-m}L(p_1,p_2) = O \left(\frac{1}{|p_2 - p_1|^2}\right) \partial_x^m \partial_y^{4-m}G(p_1,p_2) - \partial_x^m \partial_y^{4-m}L(p_1,p_2) = O \left(\frac{1}{|p_2 - p_1|^3}\right)
x^{+}\simeq x^{+}+ \frac{2}{1 + \sqrt{1 + 4 R_{s}^{2}/ R^{2}}}\frac{R_{s}^{2}}{R}, x^{-}\simeq x^{-}+ \frac{1 + \sqrt{1 + 4 R_{s}^{2}/ R^{2}}}{2}R
|\Sigma_R, \xi \! >_{\epsilon}\: \equiv \: |\Sigma_R, \xi \! >_0 + \:\overbrace{\frac{\epsilon}{2 \pi i}\int_{P \in \Sigma_R/D_{\xi}}< \!\phi|\Sigma_R, \xi, u \! >_0 \, du|_P \wedge d\bar{u}|_P}^{\Delta |p\!>},
D = \partial_{u_3}+ a_2^2 U_1 + (\gamma_4 + \tfrac{7}{3}a_2^2) a_2 u_3 d(a_2,\gamma)^2 = \gamma_6 + \tfrac{5}{3}a_2 \gamma_4 + (\tfrac{5}{3}a_2)^3.
\{( \left( \begin{array}{cccc}1 \sigma(t) 1 \sigma(t^{-1})\end{array}\right) )_{\sigma \in I_F}\ \vert t=\omega_1^{p_1}\dots \omega_{d-1}^{p_{d-1}}, p_1, \dots, p_{d-1}\in \mathbb{Z}\}\hookrightarrow T_{1,L}(L),
\Gamma^{(2)}= \frac{n_6}{g_s (2\pi)^6}V_6 \int dt \left(-\frac{3 \epsilon^2 M^2}{4r^2}v^2-\frac{\sqrt{3}\epsilon M^2}{r^2}v^4-\frac{M^2}{r^2}v^6\right).
I_{i}(k_{x},k_{y})\propto|\vec{A}\cdot\vec{k}|^{2}\cdot|\tilde{\Psi}_{i}(k_{x},k_{y})|^{2}with|\vec{k}|^{2}=k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=\frac{2m}{\hbar^{2}}E_{kin}
y_0 = \sum_{n_i = m_i}\psi^{\otimes n_i}(\xi_i) \psi^{\otimes m_i}(\eta_i)^* = \sum_{n_i = m_i}(\rho \circ \psi_X)^{\otimes n_i}(\xi_i) (\rho \circ \psi_X)^{\otimes m_i}(\eta_i)^* &= \rho \left ( \sum_{n_i = m_i}\psi_X^{\otimes n_i}(\xi_i) \psi_X^{\otimes m_i}(\eta_i)^* \right) = \rho ( E (x)).
a_1 =2\int_0^1 \sin_p(\pi_p x) \sin(\pi x) \mathrm{d}x=4\int_0^\frac12 \sin_p(\pi_p x)\sin(\pi x) \mathrm{d}x &>4\int_0^\frac12 (2x) \sin(\pi x) \mathrm{d}x=\frac{8}{\pi^2}.
\frac{1}{(q)_{\infty}}\sum_{n=0}^{\infty}A_n (q^{\kappa -\iota})^{kn}q^{kn^2}= \chi^{(2\kappa +1, (2\kappa +1)k + 2\kappa -1)}_{\iota , \kappa k +\kappa -1}(q),
\|w\|_A = \|A^\frac{1}{2}w\| \le \|w\| = \|A^{-\frac{1}{2}}A^\frac{1}{2}w\| \le \|A^{-\frac{1}{2}}\| \cdot \|w\|_A = \kappa^\frac{1}{2}\|w\|_A
\begin{array}{l}{{T_{0}| \Omega \rangle = \frac{1}{2 ( k + \check{g})}S_{a 0}^{a}| \Omega \rangle = \frac{p}{2 ( k + \check{g})}| \Omega \rangle}}\\{{\}}\\{{J_{0}| \Omega \rangle = \frac{1}{( k + \check{g})}S_{a 0}^{a}| \Omega \rangle = \frac{p}{( k + \check{g})}| \Omega \rangle}}\\ \end{array}
|(j_{31}j_{2})jm\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}\sum_{m_{3}=-j_{3}}^{j_{3}}C_{m_{3},m_{1},m_{31}}^{j_{3},j_{1},j_{31}}C_{m_{31},m_{2},m}^{j_{31},j_{2},j}|j_{1}m_{1}j_{2}m_{2}j_{3}m_{3}\rangle
\sum_{n\ge0}(X_1)_n(X_2)_n(aq/X_1X_2)^n\beta_n(a,q)=\frac{(aq/X_1)_{\infty}(aq/X_2)_{\infty}}{(aq)_{\infty}(aq/X_1X_2)_{\infty}}\sum_{n\ge0}\frac{(X_1)_n(X_2)_n(aq/X_1X_2)^n\alpha_n(a,q)}{(aq/X_1)_n(aq/X_2)_n}.
\left| \bigcup_{i=1}^r L_i \right| = \sum_{1 \leq i < j \leq r}a_{i,j}= \sum_{j \in [r] \setminus \{1 \}}a_{1,j}+ \sum_{l,t \in [r] \setminus \{1 \}, l<t}a_{l,t}= (k+1) + \frac{(r-2)(k+1)}{2}= \frac{r(k+1)}{2}.
\lim_{\delta\rightarrow 0}\limsup_{N\rightarrow+\infty}\sup_{\sigma\in\mathcal{T}, t\leq \delta}\hat{P}^{N,F,G,H}_{f_1, f_2, f_3}\left(\left|\mu^N_{t+\sigma, k}(f)-\mu^N_{\sigma,k}(f)\right|>\epsilon\right)=0
D_{J}^{\, I}\Phi^{[ a_{1}, \ldots , a_{N - 1}]}( x_{A}, \theta_{p + 1}, \ldots , \theta_{N}, \bar{\theta}^{1}, \ldots , \bar{\theta}^{N - q}, u ) = 0 \; , \quad 1 \leq I < J \leq N \; .
(A)\qquad (a;q)_n = \frac{(a;q)_\infty}{(aq^n;q)_{\infty}}, (B)\qquad (a^{-1}q^{1-n};q)_{n}= (a;q)_n (-a^{-1})^n q^{-\frac{n(n-1)}{2}}, (C)\qquad (a;q)_{n-k}= \frac{(a;q)_n}{(a^{-1}q^{1-n};q)_k}(-q a^{-1})^k q^{\frac{k(k-1)}{2}- nk}.
\left( n_{1}a_{1}+n_{k}a_{k}\right)^{2}-n_{1}a_{1}^{2}-n_{k}a_{k}^{2}\leq\left( n_{1}a_{1}+n_{k}a_{k}\right)^{2}-\frac{1}{2l+1}\left(n_{1}a_{1}+n_{k}a_{k}\right)^{2}& =2l\left( 2l+1\right) \left( \frac{n_{1}}{2l+1}a_{1}+\frac{n_{k}}{2l+1}a_{k}\right)^{2}& \leq2l\left( 2l+1\right) b^{2}.
\renewcommand{\arraystretch}{1}\sum_{\begin{array}{c}{\scriptstyle \sigma_{1}, \ldots , \sigma_{L - 1}= 0 , 1}\\{\scriptstyle \sigma_{j}\sigma_{j + 1}= 0}\\ \end{array}}q^{\displaystyle \sum_{k = 1}^{L - 1}k \sigma_{k}}\qquad \qquad \sigma_{0}= a , \sigma_{L}= 0 ,
\mathrm{II}=\sum_{i=j+1}^{\infty}2^{(j-i)\widetilde{F}}\sum_{k\in\mathbb Z^n}\left(1+2^j|x_Q-2^{-i}k|\right)^{-\widetilde{D}}\sim\sum_{i=j+1}^{\infty}2^{(j-i)\widetilde{F}}\int_{\mathbb R^n}\left(1+2^j|x_Q-2^{-i}y|\right)^{-\widetilde{D}}\,dy \sim\sum_{i=j+1}^{\infty}2^{(j-i)(\widetilde{F}-n)}\sim 1.
\sum_{k=1}^\infty \sum_{\ell=1}^\infty\frac{\big(n_k+\frac{x}{x-1}\big) \big(\tilde{n}_\ell+\frac{y}{y-1}\big) - k\,\ell}{x^{n_k+\ell}y^{k+\tilde{n}_\ell}}> 0.
S_{\mathrm{I I B}}= \frac{1}{2 \kappa^{2}}\int d^{1 0}x \sqrt{- g}\left[ R + \frac{1}{4}\, \mathrm{T r}\left( \partial{\cal{M}}\, \partial{\cal{M}}^{- 1}\right) - \frac{1}{1 2}\,{\bf F}^{T}{\cal{M}}{\bf F}\right]
\begin{array}{r c l}{{d A + A^{2}+ \hat{\psi}^{2}- \psi^{2}+ \hat{A}}}&{{=}}&{{0 ,}}\\{{d \hat{\psi}+ [ A , \hat{\psi}] - \{\psi , \hat{A}\}}}&{{=}}&{{0 ,}}\\{{d \psi + \{A , \psi \}+ [ \hat{A}, \hat{\psi}]}}&{{=}}&{{0 ,}}\\{{d \hat{A}+ [ A , \hat{A}] + \{\hat{\psi}, \psi \}}}&{{=}}&{{0 .}}\\ \end{array}
y(y^{-1}(x\vee y)\vee w)=(yy^{-1}(x\vee y))\vee yw=(x\vee y)\vee yw=x\vee yw \intertext{and}y^{-1}(x\vee y)\vee w=y^{-1}(x\vee y)\vee y^{-1}yw=y^{-1}((x\vee y)\vee yw)=y^{-1}(x\vee yw).
= 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{\Lambda^{2}_{{\bf{k}}}({\bf{q}})}{{\tilde{\omega}}+ \omega_{{\bf{k}}}(-{\bf{q}})}- \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}\neq{\bf{k}}_{i}}\frac{\Lambda^{2}_{{\bf{k}}}(-{\bf{q}})}{{\tilde{\omega}}- \omega_{{\bf{k}}}({\bf{q}})}
\Upsilon_{+}(z) = \Upsilon_{-}(z) \begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in \Sigma_{2}, &\Upsilon_{+}(z) = \Upsilon_{-}(z) \begin{pmatrix}1 0 e^{\pi i \alpha}1\end{pmatrix}, z\in \Sigma_{1}, &\Upsilon_{+}(z) = \Upsilon_{-}(z) \begin{pmatrix}1 0 e^{-\pi i \alpha}1\end{pmatrix}, z\in \Sigma_{3},
H= \frac{1}{2}(J_{1}^{2}+ J^{2}_{2}+ 2 J^{2}_{3}) + c_{1}x_{1}+ c_{2}x_{2}+ c_{3}(x^{2}_{1}- x_{2}^{2}) + c_{4}x_{1}x_{2}+ c_{5}/ x^{2}_{3},
\Delta_kf(x)=\Delta f(x)+\sum_{\alpha\in R}k(\alpha)\left(\frac{\langle\nabla f(x),\alpha\rangle}{\langle\alpha,x\rangle}-\frac{|\alpha|^2}{2}\frac{f(x)-f(\sigma_\alpha(x))}{\langle\alpha,x\rangle^2}\right),
c_{g_j,j}(\lambda,l_{g_j}^{(j)})s_{g_j,j}^{\prime}(\lambda, l_{g_j}^{(j)})-c_{g_j,j}^{\prime}(\lambda, l_{g_j}^{(j)})s_{g_j,j}(\lambda,l_{g_j}^{(j)})=1.
\frac{\ddot{a}}{a}(t)=-\frac{4 \pi G_N}{3}\left[ 3(P_\star(t) +P(t) e^{-4 k b_0}+\frac{\hat{P}_B}{k}(t)(1-e^{-2kb}))+\rho_\star(t) +\rho(t) e^{-4kb}+\frac{\hat{\rho}_B}{k}(t)(1-e^{-2kb_0}) \right].
\frac{(M_{1}-1+M_{2}+\cdots+M_{\ell})!}{(M_{1}-1)!\cdots M_{\ell}!}+\cdots+\frac{(M_{1}+\cdots+M_{\ell-1}+M_{\ell}-1)!}{M_{1}!\cdots (M_{\ell}-1)!}= \frac{(M_{1}+\cdots+M_{\ell}-1)!}{M_{1}!\cdots M_{\ell}!}\left(M_{1}+\cdots+M_{\ell}\right) = \frac{(M_{1}+M_{2}+\cdots+M_{\ell})!}{M_{1}!M_{2}!\cdots M_{\ell}!}
f_{\gamma_z}(z)= A_xA_y\sum^\infty_{k=0}\sum^k_{n=0}\tilde{B}_x(n)\tilde{B}_y(k-n)\Big(a^{n+1}_x(k-n)! &\times \sum^{k-n}_{m_1=0}\frac{a^{m_1}_{y}}{m_1!}\gamma^{n+m_1}e^{-(a_x+a_y)\gamma}+a^{k-n+1}_y n! &\times \sum^n_{m_2=0}\frac{a^{m_2}_x}{m_2!}\gamma^{k-n+m_2}e^{-(a_x+a_y)\gamma}\Big).
(\pi\times \pi)^*\big(\sum_{h\in H''}\iota^{-1}(h)[\Gamma^Z_h\times\Delta_K]\big) = \sum_{h\in H''}\iota^{-1}(h)\big(\sum_{l\in L}[\Gamma^Z_h\times_{\Delta_B}\Gamma^A_l]\big) &= |L|\sum_{h\in H''}\iota^{-1}(h)[\Gamma^Z_h\times_{\Delta_B}\Delta_A] &= |L|\sum_{h\in H''}\iota^{-1}(h)[\Gamma^{Y'}_h] \in \mathrm{CH}^{\dim Y'}(Y'\times Y')_{\mathbb{C}}.
{\sf IJ}^{[1]}= &\{(i,j): 1\leq i< j\leq n, 1\leq|FI(i)|<|FI(j)| \leq c\}{\sf IJ}^{[2]}= \{(i,j): 1\leq i<j \leq n, 1\leq FI(i)\leq c,FI(j)=0\}{\sf IJ}^{[3]}= \{(i,j): 1\leq j<i \leq n, FI(i)=0, FI(j)=-c\}.
= \frac{1}{2}\int_{0}^{L}( \vec{l}- i \partial_{x}\vec{n})^{2}d x = \frac{1}{2}\int_{0}^{L}( \vec{l}^{2}- ( \partial_{x}\vec{n})^{2}) d x - i \int_{0}^{L}( \vec{l}\cdot \partial_{x}\vec{n}) d x
\widetilde{\nabla}^{\alpha}\big(\widetilde{Q}_{\alpha\beta}[\phi]\big(K_{0}\big)^{\beta}\big)=\Omega^{4}\Big(\Re\big(f\cdot\overline{\frac{1}{\Omega}D_{K_{0}}(\Omega\phi)}\big)+K_{0}^{\beta}F_{\beta\gamma}\Im\big(\phi\overline{\frac{1}{\Omega}D^{\gamma}(\Omega\phi)}\big)\Big).
\begin{aligned}\max_{\mathbf{W}_k}~ \sum_{m=1}^{M_{MS}}||({\mathbf{w}_k^{(m)}})^H\mathbf{H}_k||_1^2 s.t.~ \mathbf{w}_k^{(m)}\in \mathbf{D},~m = 1,\cdots, M_{MS}.\end{aligned}
\underbrace{\begin{bmatrix}N_{n}\vdots N_{n}N_{0}\vdots \ddots N_{0}\vdots N_{n}\ddots \vdots N_{0}\end{bmatrix}}_{b+1 \mbox{block columns}}\begin{bmatrix}B_{b}B_{b-1}\vdots B_0\end{bmatrix}=\begin{bmatrix}Q_{n+b}Q_{n+b-1}\vdots Q_0\end{bmatrix},
ds^2_{\L}={\ell^2\over 4}\left({d\rho\over \rho}\right)^2 +{1\over \rho}\left[\left(1-{\ell^2\rho\over 4}\right)^2 d\vec n\cdot d\vec n\right] ={\ell^2\over 4}{d\rho^2\over \rho^2}+{1\over \rho}\,g_{ab}(x,\rho)\,dx^a dx^b\,,
(- \Delta_p)^s\, u(x):= 2\, \lim_{\varepsilon \searrow 0}\int_{\mathbb{R}^N \setminus B_\varepsilon(x)}\frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{N+s\,p}}\, dy, x \in \mathbb{R}^N.
T_f(r)+T_g(r)\ge \sum_{v=i,\sigma (i)}\left ( N^{[n]}_{(f,a_v)}(r)+N^{[n]}_{(g,a_v)}(r)-nN^{[1]}_{(f,a_v)}(r) \right ) + \sum_{\underset{v\ne i,\sigma (i)}{v=1}}^{q}N^{[1]}_{(f,a_v)}(r)+o(T_f(r)).
\begin{aligned}\mathfrak{H}^{k}:=\{\mu\in H\Lambda^{k}(\Omega)\cap\mathring{H}^{\ast}\Lambda^{k}({\Omega})\mid d\mu=0,\ \delta\mu=0\}, \mathring{\mathfrak{H}}^{k}:=\{\mu\in \mathring{H}\Lambda^{k}(\Omega)\cap H^{\ast}\Lambda^{k}({\Omega})\mid d\mu=0,\ \delta\mu=0\}.\end{aligned}
A =\theta_{1}-\dfrac{\theta_{2}}{2}\in\left(-\frac{3\pi}{2},\frac{3\pi}{2}\right) B =\dfrac{1}{2}\arccos\left\{\dfrac{1}{2}\left[ \dfrac{\rho_{1}^{2}}{2\rho_{2}}-\sqrt{\dfrac{\rho_{1}^{4}}{4\rho_{2}^{2}}+4-2\dfrac{\rho_{1}^{2}}{\rho_{2}}\cos (2A)}\right] \right\}\in\left[0,\frac{\pi}{2}\right].
\overline{L}^{\diamond,\mbox{\tiny \sc Time}}f(\overline{\eta}^{\diamond}) = \lim_{\epsilon \downarrow 0}\frac{f((\overline{\eta}^{\diamond})^{\mbox{\tiny \sc Time}}+ \epsilon, (\overline{\eta}^{\diamond})^{\mbox{\tiny $\mathcal{D}$}}) - f(\overline{\eta}^{\diamond})}{\epsilon}
\begin{matrix}P^* = ( \,p_1^*,p_2^*,p_3^*,p_4^*\, \mid \,p_{12}^*,p_{13}^*,p_{23}^*,p_{14}^*,p_{24}^*,p_{34}^*\,\mid\,p_{123}^*,p_{124}^*,p_{134}^*,p_{234}^*\, ) = ( p_{234}, -p_{134}, p_{124}, -p_{123}\, \mid \, p_{34}, -p_{24}, p_{14}, p_{23}, -p_{13}, p_{12}\,\mid \, p_4, -p_3, p_2, -p_1 ).\end{matrix}
(e^{2\mathcal Rt}\Delta^h_k\partial^2_yu_\psi, \Delta^h_k(\partial_yu)_\psi)_{L^2}=-\frac 12\frac{d}{dt}||e^{\mathcal Rt}\Delta^h_h\partial_yu_\psi||^2_{L^2}-\theta'(t)2^k||e^{\mathcal Rt}\Delta^h_k\partial_yu_\psi||^2_{L^2}
\int_{0}^{t}s^{2\beta - 2 \beta/\alpha}(t-s)^{\beta - \beta/\alpha - 1}ds =t^{\beta - \beta/\alpha - 1}\int_{0}^{1}t^{2\beta -2\beta/\alpha + 1}z^{2\beta - 2\beta/\alpha}(1-z)^{\beta - \beta/\alpha - 1}dz =t^{3(\beta - \beta/\alpha)}B(2\beta - 2 \beta/\alpha + 1, \beta - \beta/\alpha).
|\tilde v(t)|^2 = |h(t)|^2+ 2\sum_{i=1}^m \int_{(0,t]}\langle v_i^{\ast}(s), v(s) \rangle\, dA(s) + 2 \int_{(0,t]}(\tilde v(s-)-h(-), dh(s)) & - \int_{(0,t]}\left|\sum_{i=1}^m v_i^{\ast}(s)\right|^2 \Delta A(s)\,dA(s).
(1-(-t)^{n/d})^d \cdot \sum_{j=0}^{n-1}\frac{j\omega_n^{i(j-1)}t}{1+t\omega_n^{ij}}= \sum_{r=0}^{n}\omega_n^{i(\binom{r}{2}-1)}t^r \Big( 1_{n \mid dr}\cdot \binom{r}{2}\binom{d}{dr/n}+\omega_n^i{n \brack r}^{'}_{\omega_n^i}\Big).
\eta^{i}( \sigma , t ) \equiv - \frac{1}{4 M}\sum_{n > 0}\frac{\epsilon}{\tau}\, W^{'}( n + \frac{\epsilon}{\tau}) \Biggl [ \frac{c_{n}^{i}}{\sqrt{n}}\, x_{n}^{+}+ \frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\, x_{n}^{-}+ h . c . \Biggr ] \, ,
\int (W_k^{\rm eff})'(x)f(x)\,d\mu_k^{aV}(x)= \iint \frac{f(x)-f(y)}{x-y}\,d\mu_k^{aV}(x)\,d\mu_k^{aV}(y)\forall\,k=1,\ldots,d,\,\,\forall\,f.
\hat{\phi}_{L^{\otimes m},*}R\hat{S}(a_*\omega_X\otimes\widehat{L^{\otimes m}}^{\vee})=RS(\phi_{L^{\otimes m}}^*(a_*\omega_X\otimes\widehat{L^{\otimes m}}^{\vee})) = RS(\hat{a}_{m,*}\omega_{\hat{X}_m}\otimes\bigoplus^{h^0(L^{\otimes m})}(L^{\otimes m}))
\alpha(f(x+h)) =v(x+h)+Df(x+h)\alpha(x+h) &=v(x)+Dv(x)h +Df(x)\alpha(x+h) + D^2f(x)\alpha(x+h)h+ O(|h|^{1+\epsilon}) &=v(x) +[Dv(x) +D^2f(x)\alpha(x)]h +Df(x)\alpha(x+h) + O(|h|^{1+\epsilon}).
V(\rho)=\frac{1}{G}\rho^{2}-i \mbox{tr}\int^{\rho}_{0}ds\S(x,x;s) -2i \int^{\rho}_{0}sds\ [G_{1}(x,x;s) +G_{2}(x,x;s)] +{\cal O}\left(\frac{1}{N}\right),
Q_{w_1}^{(1)}T_{w_1}Q_{w_1}^{(2)}\ldots Q_{w_r}^{(1)}T_{w_r}Q_{w_r}^{(2)}(P_{w_{r+1}}T_{w_{r+1}}P_{w_{r+1}})\ldots (P_{w_s}T_{w_s}P_{w_s}) \times (P_{w_{s+1}}^\perp T_{w_{s+1}}P_{w_{s+1}}) \ldots ( P_{w_{d}}^\perp T_{w_{d}}P_{w_{d}}).
f_{\varphi_{\omega_{\pi_{\nu}}},s}\left(\tilde{\varepsilon}t\left(I_m,xg_\nu\right)\right)=\sum_{j=1}^{N}\left<\xi_{\tau_{j},\tilde{\tau}_{j}}\left(x^*\right)\check{\xi}_{\tilde{\tau}_{j}}\right>f^{(j)}_{\varphi_{\omega_{\pi_{\nu}}},s}\left(\tilde{\varepsilon}t\left(I_m,g_\nu\right)\right),
2 (\Delta_{i}- \Delta_{i+1})= (i+1)\rho^i - (2i+2)\rho^{i+1}+(i+1)\rho^{i+2}+(n-i-1)\rho^{n-i-2}- 2(n-i-1)\rho^{n-i-1}+ (n-i-1)\rho^{n-i}&= (1-\rho)^2((i+1)\rho^i + (n-i-1)\rho^{n-i-2}).
\sup_{s \in [0,t]}\bigl(Z_{s}^{i}\bigr)_{+}\geq1-\frac{\varepsilon_{0}}{2}\Rightarrow C\int_{0}^{T}\bigl(1+\sup_{r \in [0,s]}\vert Z_{r}^{i}\vert\bigr)ds+\alpha T^{1/4}+\sup_{s \in [0,T]}\vert W_{s}^{i}\vert\geq\frac{\varepsilon_{0}}{2}.
\left|A(z,\zeta)\right|^{1+\mu_{0}}\delta_{\Omega}^{-\gamma\mu_{0}-\varepsilon}(\zeta)\lesssim\delta_{\Omega}(z)^{-\mu_{0}(\gamma+n)-\varepsilon}\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,\delta_{\Omega}(z)\right).
\lim_{n \to \infty}S_n(z_1, z_2) = (z_1z_2)^{- \frac{s}{2}}\cdot \frac{A_I^{(s,w)}(1, z_1) B_I^{(s,w)}(1, z_2) - A_I^{(s,w)}(1, z_2) B_I^{(s,w)}(1, z_1)}{2 \big[C_I^{(s, w)}(1) \big]^2 (z_1-z_2)}.
\tilde{H}(\hat{v}_2)=\biggl\{\hat{v}^{*}_1\in{\mathcal{S}_1}:\tilde{F}_1(i,\hat{v}^{*}_1(i),\hat{v}_2(i))=\inf_{v_1\in V_1(i)}\tilde{F}_1(i,v_1,\hat{v}_2(i))~\forall~i\in S\biggr\}.
S_{e f f}= \int d^{3}x \, \left\{\frac{| m |}{1 6 \pi}\, \mathrm{t r}\, ( \partial_{\mu}\hat{n})^{2}+ i \frac{\mathrm{s g n}( m )}{4 \pi}\, \epsilon^{\mu \nu \lambda}\, a_{\mu}\, \partial_{\nu}a_{\lambda}\right\},
|I_0(g)| =N_{q,n,m}(g)-C^{-1}q^{(g-1)nm^2}-1+O(1) = C^{-1}q^{gnm^2}(1-O(q^{-g}))-C^{-1}q^{(g-1)nm^2}-1+O(1) = C^{-1}(q^{gnm^2}-q^{(g-1)nm^2})(1-O(q^{-g})) \ .
\sum_{3r_0+2r_1+r_2=d_1+1}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}{d_1+1-r_0-r_1 \choose r_2}\frac{(d_2+3)!}{2^{r_2}6^{2r_0+r_1}}{n_1 \choose d_1+1}{n_2 \choose d_2+3}.
\begin{array}{l l}{{T_{+ +}^{L}= ( T_{0 0}^{L}+ T_{0 1}^{L}) / 2}}\\{{T_{--}^{L}= ( T_{0 0}^{L}- T_{0 1}^{L}) / 2}}\\{{T_{+ -}^{L}= T_{- +}^{L}= 0}}\\ \end{array}
r(t) = \sum\limits_{n = 1}^N \Bigg\{\sum\limits_{i = 0}^{p - 1}{x_A[n]h_i^A\left(t - (n + \tau_i )\right)}+ \sum\limits_{j = 0}^{s - 1}{x_B[n]h_j^B\left(t - (n + l_j + \Delta )\right)}\Bigg\}+ w(t),
\sigma(z+c)_{j}=\frac{e^{z_{j}+c}}{\sum_{k=1}^{K}e^{z_{k}+c}}=\frac{e^{z_{j}}\cdot e^{c}}{\sum_{k=1}^{K}e^{z_{k}}\cdot e^{c}}=\sigma(z)_{j}.
\chi\left(p',\{W_Q^{0}(a): a\in{\mathsf{A}}\}\right) &=-\frac{1}{2}q\log \frac{1}{2}q + \frac{1}{2}(1-q)\log\frac{1}{2}(1-q)-\frac{1}{2}\log\frac{1}{2}&+q\log q + (1-q)\log(1-q)
\int_{\delta_0*_t\delta_{\alpha}^{-1}}res\left(\frac{\omega_\beta}{f}\right)= \frac{\lambda^{A_{\beta'}-1}p(\{g=-1 \},\beta',\delta_{\alpha}^{-1})}{\zeta_{q}^{\gamma+1}-1}B\left(A_{\beta'}+\beta_{n+1},A_{\beta'}\right)\cdot \;\;\: F\left(A_{\beta'}+\beta_{n+1},1-A_{\beta'},2A_{\beta'}+\beta_{n+1};\frac{1}{\lambda}\right),
F^{m}{}_{\mu \nu}= 2 \partial_{[ \mu}A^{m}{}_{\nu ]}- f_{n p}{}^{p}A^{n}{}_{\mu}A^{p}{}_{\nu}= 2 \partial_{[ \mu}A^{m}{}_{\nu ]}- Q^{m q}\epsilon_{n p q}A^{n}{}_{\mu}A^{p}{}_{\nu}\, .