harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{\theta}= \frac{\Pi_{\theta , E}}{R \operatorname{s i n}\theta}\sqrt{\frac{B^{2}+ R^{2}\operatorname{s i n}^{2}\theta ( R^{2}+ R_{\theta}^{2})}{\Pi_{\theta , E}^{2}+ R^{2}\operatorname{s i n}^{2}\theta}}.
Q = - 2 \left( \begin{array}{c c c}{N_{1}- N_{2}}&{N - N_{1}+ \frac 1 2 N_{2}}&{N - N_{1}+ \frac 1 2 N_{2}}\\{N - N_{1}+ \frac 1 2 N_{2}}&{N_{1}}&{N_{1}- N}\\{N - N_{1}+ \frac 1 2 N_{2}}&{N_{1}- N}&{N_{1}- 2 N}\\ \end{array}\right)
\begin{array}{rcl}{\rm integral}= \displaystyle -2m^2(4\pi)^{-2}\int_0^1 dx\int\frac{d^4k_1}{(2\pi)^4}D_m^2(k_1)\overline{D(k_1+p)}K_0\left(\sqrt{xm^2\tilde{k_1}^2}\right).\end{array}
\begin{array}{l l l}{{\{\theta^{0}, \theta^{0}\}= - 2 \theta^{1}\theta^{2},}}&{{\{\theta^{0}, \theta^{1}\}= - 2 \theta^{0}\theta^{1},}}&{{\{\theta^{0}, \theta^{2}\}= 2 \theta^{0}\theta^{2},}}\\{{\{\theta^{1}, \theta^{1}\}= 0 ,}}&{{\{\theta^{1}, \theta^{2}\}= 0 ,}}&{{\{\theta^{2}, \theta^{2}\}= 0 .}}\\ \end{array}
(T_0-v_1)(T_0-v_2)\dots (T_0-v_r) =0 ; (T_i-q)(T_i+1) =0\textit{for}1\leq i\leq n-1; T_0T_1T_0T_1 =T_1T_0T_1T_0; T_iT_{i+1}T_i =T_{i+1}T_iT_{i+1}\textit{for}1\leq i\leq n-2; T_iT_j =T_jT_i \textit{for}|i-j|\geq 2.
\begin{cases}\lambda \sqrt{\det h}X_z=-\partial_y(h_{11}X_x+h_{12}X_y)+\partial_x(h_{12}X_x+h_{22}X_y)\,, -\lambda \sqrt{\det h}X_y=\partial_xX_z-z\partial_z(h_{11}X_x+h_{12}X_y)\,, \lambda \sqrt{\det h}X_x=\partial_yX_z-z\partial_z(h_{12}X_x+h_{22}X_y)\,.\end{cases}
-\frac{\widetilde h_{\alpha,\theta}^\prime(s|\mu)}{\widetilde h_{\alpha,\theta}(s|\mu)}= \mu\dfrac{(1+s)^{\alpha-1}}{s^{\alpha-\theta}}\equiv \mu\,\rho_{\alpha,\theta}(s) \implies \widetilde h_{\alpha,\theta}(s|\mu) = \exp\left\{-\mu\int_0^s \rho_{\alpha,\theta}(t) dt \right\}
(\Box^h\Phi)_{A_p\overline{B}_q}= -h^{\bar{\varepsilon}\gamma}\bigg(\delta_\gamma\circ\delta_{\bar{\varepsilon}}(\phi_{A_p\overline{B}_q}) - \sum_i(-1)^{i-1}[\delta_\gamma,\delta_{\bar{\beta}_i}]\phi_{A_p\bar{\varepsilon}\bar{\beta}_1\dots\hat{\bar{\beta}}_i\dots\bar{\beta}_q}\bigg).
\mu_{2}(\theta) =\frac{1}{1-\cos^{2}\theta}\Big((1-4\cos^{2}\theta)\mu_{2}(\frac{\pi}{2}) &+\frac{3}{4}(\cos\theta+2\cos^{2}\theta)\mu_{2}(\frac{\pi}{3}) &+\frac{3}{4}(2\cos^{2}\theta-\cos\theta)\mu_{2}(\dfrac{2\pi}{3})\Big),
\Delta^{\mu \nu}= \frac 1 4 \left( \Pi_{\rho}^{\rho , \mu \nu}+ \tilde{\Gamma}^{\mu \nu}\right) -{\frac 1{3 2}}\left( \Pi_{\rho \sigma}^{\rho , \sigma}+ \tilde{\Gamma}_{\rho}^{\rho}\right) \eta^{\mu \nu}.
\prod_{p\leq (\log x)^A}\left(1-\frac{\chi_d(p)}{p}\right)^{-1}= \prod_{p\leq y}\left(1-\frac{\chi_d(p)}{p}\right)^{-1}\exp\left(\sum_{y<p<(\log x)^A}\frac{\chi_d(p)}{p}+O\left(\frac{1}{\sqrt{y}\log y}\right)\right).
p\left[x_1,x_2;(m_1,m_2,\rho)\right] = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp \left[-\frac{\left(x_1-m_1\right)^2 + \left(x_2 - m_2\right)^2 - 2\rho \left(x_1-m_1\right)\left(x_2 - m_2\right)}{2\left(1- \rho^2\right)}\right]
\int_{|z| >R_2}f(y-z) v(z)~dz = \int_{R_2}^\infty \int_{\partial B_{1}}f(y- \rho w) v(\rho w)\rho^{d-1}~S_1(dw)d\rho &= \int_{\partial B_{1}}\left[\int_{R_2}^\infty v(\rho w) \left(\frac{d}{d\rho}\int_{R_{2}}^\rho f(y- \gamma w) \gamma^{d-1}~d\gamma\right) S_1(dw) \right] d\rho.
V_{\mathrm{g r a v i t y}}^{1 / r^{8}}= -{\frac{1 0 5 R^{2}r_{i}}{4 r^{9}}}\left( \hat{T}_{I J}\tilde{T}^{I J ( i )}+ \hat{T}^{I J ( i )}\tilde{T}_{I J}-{\frac{1}{9}}( \hat{T}^{I ( i )}{}_{I}\tilde{T}^{J}{}_{J}+ \hat{T}^{I}{}_{I}\tilde{T}^{J ( i )}{}_{J}) \right) \,
\lim_{r \to \infty}-r^{-5}\int_{\Sigma_r}\alpha_{H}(V^2 \nabla \tau)d \Sigma_r = \int_{S^2}\left[\frac{4A}{3}W_0^2 +\frac{2}{3}C_iW_iW_0- (C_i \tilde{X}^i) |\beta|^2 \right]dS^2.
|H_0| = \frac{2}{r}+ r h_0^{(1)}+ r^2 h_0^{(2)}+ r^3 h_0^{(3)}+O(r^4) \alpha_{H_0}= r^2 \alpha_{H_0}^{(2)}+ r^3 \alpha_{H_0}^{(3)}+ r^4 \alpha_{H_0}^{(4)}+ O(r^{5}).
\sum_{\alpha \in S_{3, 2}}g_\alpha(p) = \frac{1}{6(p-1)}\bigg(6(n-1)p^{2n-5}+ (n-1)(n-2)(n-3)p^{n}& - 3(n-1)(n-2)(n-4)p^{n-1}+(n-1)(n-2)(5n-24) p^{n-2}&- 3(n-1)(n-3)(n-4)p^{n-3}\bigg).
M_{1,0}^{1,0}= 1 - g, M_{1,0}^{1,2}= M_{1,2}^{1,0}= 1, M_{j,1}^{j+g,1}= -M_{j+g,1}^{j,1}= 1 (j = 1, \dotsc, g), M_{j,k}^{j',k'}= 0 .
\int_{\Omega\setminus\Sigma_r}|h|^p dx = \int_{Z(0;r)}|h|^p dx = \int_{0}^{r}\int_{\{x\in\Omega:\delta(x)=t\}}\frac{|h|^p}{|\nabla\delta|}d\mathcal{H}^{d-1}dt =\int_{0}^{r}\int_{S_t}|h|^p dS_tdt,
(\gamma_{1})^{T}=\alpha(\begin{matrix}0.8182\\ 0.1818\end{matrix})\circ(\begin{matrix}0.5923\\ 0.4077\end{matrix})=\alpha(\begin{matrix}0.4846\\ 0.0741\end{matrix})=(\begin{matrix}0.8673\\ 0.1327\end{matrix})
|\phi(y)|\leq \frac{|\tilde{P}_{x_2-y}(\theta+(y+1)\alpha)|}{|\tilde{P}_{2q_n-1}(\theta+x_1\alpha)|}\prod_{j=x_1}^{y}|\cos(\pi(\theta+j\alpha))|\cdot |\phi(x_1-1)| &+\frac{|\tilde{P}_{y-x_1}(\theta+x_1\alpha)|}{|\tilde{P}_{2q_n-1}(\theta+x_1\alpha)|}\prod_{j=y}^{x_2}|\cos(\pi(\theta+j\alpha))|\cdot |\phi(x_2+1)|,
\psi([a_{i_0-1},\dots,a_{i_0-1}+t-1 \mid b_{i_0},\dots, b_{i_0}+t-1]) = X_{S_{i_0}}|N|, \chi(|N|) = X^{-1}_{S_{i_0}}[a_{i_0-1},\dots,a_{i_0-1}+t-1 \mid b_{i_0},\dots, b_{i_0}+t-1].
H_{q q q q}={\frac{g^{2}}{8 \pi}}\int d^{2}x^{\perp}\int_{- \infty}^{\infty}{\frac{d p^{+}}{( p^{+})^{2}}}{\tilde{j}^{+ a}}_{q}( p^{+}, x_{\perp}){\tilde{j}^{+ a}}_{q}( - p^{+}, x_{\perp}) .
\|\mathbf{O}_{a, b}\|_1 \le \int \|\mathbf{R}_{w, v}\|_1 \mathbf{P}_{a, b}(w, v) dw dv = \int \|R_w\|_2 \|R_v\|_2 \mathbf{P}_{a, b}(w, v) dw dv.
T^{+q}_{+2-2}=\frac12\gamma^i_{q\dot p}\left(\Omega^{+2i}_{+2}\psi^{1-}_{-2\dot p}-\Omega^{+2i}_{-2}\psi^{1-}_{+2\dot p}\right),\T^{+q}_{+p\pm2}=\frac12\gamma^i_{q\dot p}\Omega^{+2i}_{+p}\psi^{1-}_{\pm2\dot p},
C^{+}_{p}= \tau^{-1}\{(j_{r},l_{r})\in p: j_{r}\not \in \{0,2n-1,4n-2\}, \l_{r-1}>l_{r},l_{r+1}>l_{r}\}& \bigsqcup \{(n,l)\in \mathcal{D}: (2n-1,l-\epsilon)\in p (2n-1,l+\epsilon)\not\in p\},
\mathbb{E}_{\mathbb{Q}_{s,f}^{v}}\left[ \varphi(G_n \cdot v) e^{z \left( \log \frac{\langle f, G_n v \rangle}{\langle f, v \rangle}- n\Lambda'(s) \right)}\right]= \lambda^{n}_{s, z}\Pi_{s,f,z}\varphi(v) + N^{n}_{s,f,z}\varphi(v), \v \in \mathcal{S}_{\epsilon}.
\left\langle\rho,\tau^{x_{3}}\alpha_{1},y\alpha_{2}\alpha_{3}^{a}\right\rangle \cong M_{1}&\a,y_{3}=0,...,p-1,\x_{3},y_{2}=1,...,p-1 \ \ax_{3}- y_{2}-x_{3}y_{2}\not\equiv 0 \ \mathrm{mod}\p.
S_Q(p)\pm R_Q(p)=4\pi^2d_Q(R)\int \frac{d^4l}{{(2\pi)}^4}\delta(l^2-m_Q^2)\delta[{(l+p)}^2-m_Q^2] \sin^2[\theta(l^0+p^0,\mu_Q)\pm \theta(l^0, \mu_Q)]\geq 0.
\frac{d\mathbb{P}^p}{d\mathbb{P}^q}(X) = \frac{\frac{d\xi\mathbb{P}_{(\cdot|X;C)}}{d\xi\mathbb{P}}(p)}{\frac{d\xi\mathbb{P}_{(\cdot|X;C)}}{d\xi\mathbb{P}}(q)}\cdot \frac{d\mathbb{P}_{(\cdot|X;C)}^p}{d\mathbb{P}_{(\cdot|X;C)}^q}(X|_{E\backslash C}).
F(s; \mathbf{a}, \mathbf{z}) =(L(s, \chi_0)^{\frac{z_1+\cdots+z_l}{\phi(q)}}\prod_{\chi\neq\chi_0}(L(s, \chi))^{\frac{\bar{\chi}(b_1)z_1+\cdots+\bar{\chi}(b_l)z_l}{\phi(q)}}\prod_{j=1}^l G_1(s; b_j, z_j) &=(\zeta(s))^{\frac{z_1+\cdots+z_l}{\phi(q)}}H(s; \mathbf{a}, \mathbf{z}),
\begin{cases}-u_1'' = \frac{\mu_1}{\alpha^2}\left(\alpha - (\alpha u_{1}- d u_2)^+\right)(\alpha u_{1}- d u_2)^+ - k \omega u_1 u_2 -d u_2'' = \frac{\mu_2}{d^2}\left(d - (\alpha u_{1}- d u_2)^-\right)(\alpha u_{1}- d u_2)^- - \alpha k \omega u_1 u_2 \end{cases}
\mathrm{P}_{n,1}=e^{-\lambda_c\pi \left(\frac{\rho}{\epsilon_1}\right)^{\frac{2}{\alpha}}}\sum^{n-1}_{k=0}\frac{(\lambda_c\pi)^{k}\left(\frac{\rho}{\epsilon_1}\right)^{\frac{2k}{\alpha}}}{k!}.
\sum_{Q}m_{Q}^{4}K_{Q}( p ) = \tilde{k}, \ \ \sum_{Q}m_{Q}^{4}H_{Q}( p ) = \tilde{h}, \ \, s u m_{Q}m_{Q}^{4}S_{Q}( p ) = \tilde{s}, \ \ \sum_{Q}m_{Q}^{4}R_{Q}( p ) = \tilde{r}.
C_{0}^{\lambda,1}(x) =1 C_{1}^{\lambda,1}(x) =x-\lambda C_{2}^{\lambda,1}(x) =x^{2}-(1+2\lambda)x+\lambda^{2}C_{3}^{\lambda,1}(x) =x^{3}-(3+3\lambda)x^{2}+(2+3\lambda+3\lambda^{2})x-\lambda^{3}.
\omega^{\Lambda}= ( \mathrm{d}\varphi_{I}+ \vec{W}_{I J}\cdot \mathrm{d}\vec{x}^{J}) \wedge \mathrm{d}x^{\Lambda I}+ U_{I J}\, \varepsilon^{\Lambda \Sigma \Pi}\mathrm{d}x^{\Sigma I}\wedge \mathrm{d}x^{\Pi J}\ .
w_{0}( x , \rho ) = \sum_{m_{1}, \cdots , m_{k}\geq 0}{\frac{\Gamma ( - \sum ( ( m_{k}+ \rho_{k}) l_{0}^{( k )}+ 1 )}{\prod_{1 \leq i \leq p}\Gamma ( \sum ( m_{k}+ \rho_{k}) l_{i}^{( k )}+ 1 )}}x^{m + \rho}\; \; .
\sigma^\otimes: A \otimes_B A \longrightarrow A \otimes_B A a \otimes b \longmapsto \sigma(a) \otimes \sigma(b) \delta^\otimes: A \otimes_B A \longrightarrow A \otimes_B A a \otimes b \longmapsto \delta(a) \otimes \sigma(b) + a \otimes \delta(b)
\lim_{X\to\infty}\frac{\#\{K \in \mathcal{F}_{S_d}(X): \omega(K)-\log\log X \leq z \sqrt{\log\log X}\}}{N_{S_d}(X)}= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^z e^{-t^2/2}\,dt.
\frac{\theta(z-w+x+\alpha\hbar)\theta(\hbar)}{\theta(z-w+\hbar)\theta(x+\alpha\hbar)}a^-(z)a^+(w)- \frac{\theta(z-w+x)\theta(\hbar)}{\theta(z-w+\hbar)\theta(x)}a^-(w)a^+(z) = \sum_{k=1}^n \frac{\theta(z-w)}{\theta(z-w+\hbar)}A_k(z;x+\hbar)B_k(w;x).
R(2,1,1,1,2,2) = \left(-q;q^2\right)_{\infty}, R(2,2,1,1,2,2) = \left(-q^2;q^2\right)_{\infty}, R(m,m,1,1,1,2) = \frac{\left(q^{2m};q^{2m}\right)_{\infty}}{\left(q^m;q^{2m}\right)_{\infty}}.
\varphi+\frac{1}{t^b}=\frac{f(t_1, t_2)}{t_1^{a_1}t_2^{a_2}}+\frac{1}{t_1^{b_1}}=\frac{t_1^{b_1-a_1}f(t_1, t_2)+t_2^{a_2}}{t_1^{b_1}t_2^{a_2}}
\frac{\|\prod_{k=i+1}^{q_{n_l}-1}D^k\|}{|\prod_{k=i}^{q_{n_l}-1}f_k|}=\frac{\|\prod_{k=i+1}^{q_{n_l}-1}D^k\|\cdot |\prod_{k=0}^{i-1}f_k|}{|\prod_{k=0}^{q_{n_l}-1}f_k|}\leq q_{n_l+1}e^{(L(E)-\delta_c+3\epsilon)q_{n_l}}.
\tau_l(f_{a,b}) = \sum_{(\alpha,\beta) \in G\cdot (a,b)}\tau_l( e_{\alpha,\beta}) = \delta_{a,b}\sum_{\alpha\in G\cdot a}\mu(\bar a) e_{t(\alpha)}= \delta_{a,b}\sum_{w\in V^\mp}\sum_{\alpha \in G\cdot a : t( \alpha ) =w}\mu(\bar a) e_w & = \delta_{a,b}\mu(\bar a) \vert\{\alpha\in G\cdot a:t(\alpha)=v^\mp\}\vert.
E_{\ell_1,\ell_2,\ell_3}= -x_3 \cdot e_{2,(\ell_1-1,\ell_2),\ell_3 +1}+ x_2 \cdot e_{2,(\ell_1-1,\ell_2+1),\ell_3}+x_2 \cdot e_{1,(\ell_2+2,\ell_3),\ell_1-2}-x_3 \cdot e_{1,(\ell_2+1,\ell_3),\ell_1-1}.
\frac{1}{\binom{N}{S}}\sum_{j\in B}\Pr_{v\sim\mu_{t}}\big[|v H|=j\big] K_S(j)^2 \leq \frac{1}{\binom{N}{S}}\max_{j\in B}\left\{\Pr_{v\sim \mu_t}[|v H|=j]\cdot \frac{1}{\binom{N}{j}}\right\}\sum_{j\in B}\binom{N}{j}\cdot K_S(j)^2 &\leq 2^{N}\max_{j\in B}\left\{\Pr_{v\sim \mu_t}[|v H|=j] \cdot \frac{1}{\binom{N}{j}}\right\}.
S_1(x) =\frac{1}{x}\sum_{x/\log x<m\le x}\varphi(d)\pi(x;m,1)^2+O\left(x\log\log x\right) &\ll x\sum_{x/\log x<m\le x}\frac{\varphi(m)}{m^2}+x\log x\log x &\ll x\log\log x.
\chi(\omega_{C_1}\otimes h_{1,*}\omega_X)=\chi(h_{1,*}\omega_X)+{\rm rk}(h_{1,*}\omega_X)(2g(C_1)-2) = \chi(h_{1,*}\omega_X)+p_a(F)(2g(C_1)-2)\ge \chi(h_{1,*}\omega_X)+2p_a(F).
\tilde{H}_{n , k}= n^{- k}\prod_{s = 0}^{k}( s n - 1 ) \sum_{m = 0}^{\lfloor \frac{k}{2}\rfloor}\frac{1}{m ! ( k - 2 m ) ! n^{m}}\prod_{l = 0}^{m}{\frac{1}{l n - 1}}v^{m n}u^{k - 2 m}
\mathbb{E}[G_t] \le e^{-2\kappa\gamma t}G_{0}+\frac{m\tau_N^2 \gamma}{4\kappa}(1-e^{-2\kappa\gamma t})=e^{-2\kappa\gamma t}G_{0}+\frac{m\sigma^2 \Gamma}{4\kappa N}(1-e^{-2\kappa\gamma t}).
\mathcal{U}_{i}\zeta_{\alpha,T}=\left( \alpha_{i}+1+\kappa c\left(r_{\alpha}(i) ,T\right) \right) \zeta_{\alpha,T}, 1\leq i\leq N, r_{\alpha}(i) :=\#\{j\colon \alpha_{j}>\alpha_{i}\}+\# \{j\colon 1\leq j\leq i,\alpha_{j}=\alpha_{i}\};
\prec_t = \sum_{i \geq 0}t^i \prec_i, \succ_t = \sum_{i \geq 0}t^i \succ_i \curlyvee_t = \sum_{i \geq 0}t^i \curlyvee_i (\prec_0 = \prec,~\succ_0 = \succ \curlyvee_0 = \curlyvee)
{\mathcal V}( z , \bar{z}, q ) \, = \, g^{2}\Bigl [ \left( g_{i j^{\star}}k_{\Lambda}^{i}k_{\Sigma}^{j^{\star}}+ 4 h_{u v}k_{\Lambda}^{u}k_{\Sigma}^{v}\right) \bar{L}^{\Lambda}L^{\Sigma}
\sum_{k=1}^n\tbinom{\lambda-1}{k-1}(k-1)!B_{n,k}(1!z_1,2!z_2,\dots) = \sum_{k=1}^n\tbinom{\lambda+an-1}{k-1}(k-1)!B_{n,k}(1!x_1,2!x_2,\dots) \intertext{for any $\lambda\in\mathbb{C}$, and}\sum_{k=1}^n\tbinom{-1}{k-1}(k-1)!B_{n,k}(1!z_1,2!z_2,\dots) = \sum_{k=1}^n\tbinom{an-1}{k-1}(k-1)!B_{n,k}(1!x_1,2!x_2,\dots).
\left\| u\right\|_{\mathbb{L}_{H}^{1,p}}^{p}:=\mathbb{E}\left[ \left\|u\right\|_{L^{1/H}\left( \left[ 0,T\right] \right)}^{p}\right]+\mathbb{E}\left[ \left\| Du\right\|_{L^{1/H}\left( \left[ 0,T\right]^{2}\right)}^{p}\right] <\infty
\left(\frac{1}{n}\sum_{i=1}^n U^2_{ni}\right) =\frac{(U^2_{n1})}{n}+\frac{n-1}{n}\left(U_{n1}^2,U_{n2}^2\right) &=\frac{EU^4_{n1}-1}{n}+\frac{n-1}{n}\left(EU_{n1}^2U_{n2}^2-1\right)
V ( \overline{{{\rho}}}) = \frac{m_{R}^{2}}{2}\overline{{{\rho}}}^{2}+ \frac{\lambda_{R}}{2 4}\overline{{{\rho}}}^{4}+ \frac{3 e^{4}}{6 4 \pi^{2}}\overline{{{\rho}}}^{4}\ln \frac{e^{2}\overline{{{\rho}}}^{2}}{\mu^{2}}- \frac{3 e^{4}}{1 2 8 \pi^{2}}\overline{{{\rho}}}^{4}.
\tau=(\begin{matrix}r_{ac}&t_{bc}\\t_{ad}&r_{bd}\end{matrix})=e^{i\phi_{0}}(\begin{matrix}sin\theta e^{i\phi_{R}}&cos\theta e^{-i\phi_{T}}\\cos\theta e^{i\phi_{T}}&-sin\theta e^{-i\phi_{R}}\end{matrix}).
|je_{J + 1}(B) - je_{J + 1}(A)| = |\sum_{i = 0}^{J}(-1)^{i}(e_{J - i}(B)p_{i + 1}(B) - e_{J - i}(A)p_{i + 1}(A))| & \leq \max_{0 \leq i \leq J}|e_{J - i}(B)p_{i + 1}(B) - e_{J - i}(A)p_{i + 1}(A)|.
= \frac{1}{q^{m-1}-1}\left(q^{m-1}\zeta_\nu(n-1)\cdots\zeta_\nu(n-m+1) -\zeta_\nu(n)\cdots\zeta_\nu(n-m+2)\right) &= \frac{1}{q^{m-1}-1}\zeta_\nu(n-1)\cdots\zeta_\nu(n-m+2)\left(q^{m-1}\zeta_\nu(n-m+1)-\zeta_\nu(n)\right),
u\mapsto \frac{1}{p}\,[u]_{W^{s,p}(\mathbb{R}^N)}^p:=\frac{1}{p}\,\iint_{\mathbb{R}^N\times\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy,
\tilde{\mu}_{V X^{(n)}}(A) \le \mu_{V X^{(n)}}(A_\varepsilon) + \mathbb{P}\Big(\Big\lVert \Big(\frac{1}{\beta}\Big)^{1/p}\sum_{j=1}^n Z_j V_{\bullet, j}- \sum_{j=1}^n n^{\kappa/p}\frac{Z_j}{(\lVert Z^{(n)}\rVert_p^p + W_n)^{1/p}}V_{\bullet,j}\Big\rVert_2 \ge \varepsilon\Big).
\delta_{\mathsf{P-FS}}= \left(\frac{\mu_2 - \mu}{\mu_2 - \mu_1}\right)^+ \frac{(1-\mu_1 M)K}{Mr}+ \left(1 - \left(\frac{\mu_2 - \mu}{\mu_2 - \mu_1}\right)^+\right) \frac{(1-\mu_2)K}{Mr}= \frac{K}{Mr}\left[1 - \mu_2 - \left[\mu_1 M - \mu_2\right]\left(\frac{\mu_2-\mu}{\mu_2 - \mu_1}\right)^+ \right].
J^{( 4 )}( 4 ; 1 , 1 , 1 , 1 ) ={\textstyle{\frac{1}{1 2}}}\mathrm{i}\pi^{2}\frac{\Omega^{( 4 )}}{V^{( 4 )}}= \frac{2 \mathrm{i}\pi^{2}}{m_{1}m_{2}m_{3}m_{4}}\frac{\Omega^{( 4 )}}{\sqrt{D^{( 4 )}}}.
\widetilde{H}_{\Lambda_lx_j}=\int_{\mathcal{Y}^d}\frac{\widetilde{m}_{x_j}\widetilde{m}_{\Lambda_l}}{\widetilde{m}}dy+\int_{\mathcal{Y}^d}(\widetilde{\delta}_{il}+\widetilde{w}_{y_i\Lambda_l})\widetilde{w}_{y_ix_j}\widetilde{m}dy+\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{y_ix_j\Lambda_l}\widetilde{m}dy.
(\mathbf{x}, \mathbf{k})^2_{s,t}= \Bigl( \mathbf{x}^2_{s,t}, \,\, \int_s^t \mathbf{x}^1_{s,u}dk_u, \,\, \int_s^t \mathbf{k}^1_{s,u}d \mathbf{x}^1_{0,u}, \,\, \mathbf{k}^2_{s,t}\Bigr).
\prod_{n=1 \atop n \equiv 1-j ({\rm mod}2)}^{\infty}(1+q^n )=\displaystyle\sum_{r=0}^{\infty}\frac{q^{r^2 +jr}}{(q)_r (-q)_r}=\displaystyle\sum_{r=0}^{\infty}\frac{q^{r^2 +jr}}{(q^2 ; q^2 )_r},{}~~~~ j=0, 1.
m_1(t)= h_1t, &m_2(t)= h_2t+h_1^2t^2, &m_3(t)= h_3t+5h_1h_2\frac{t^2}{2}+h_1^3t^3, &m_4(t)= h_4t+\left( 3h_1h_3+\frac{3}{2}h_2^2 \right)t^2+\frac{13}{3}h_1^2h_2t^3+h_1^4t^4.
\frac{\ell_k}{\ell_{\hat \mu}}= \frac{{n \choose k}2^{k(n-k)}B_{n-k}}{{n \choose \hat \mu}2^{\hat \mu(n - \hat \mu)}B_{n-\hat \mu}}= \frac{(n - \hat \mu)!}{(n - \hat \mu - x)!}\left( \frac{(\hat \mu + x)!}{\hat \mu!}\right)^{-1}2^{x(n - 2\hat \mu)}2^{-x^2}\frac{B_{n - \hat \mu - x}}{B_{n - \hat \mu}}.
\sum_{k=0}^{m}(-1)^k q^{k^2}[4k+1]\frac{(aq;q^2)_k(q/a;q^2)_k(q;q^2)_k}{(aq^2;q^2)_k(q^2/a;q^2)_k(q^2;q^2)_k}\equiv q^{(n-1)^2/4}[n] (-1)^{(n-1)/2}.
\int_{S_1}\left|\frac{1}{|x-z|^2}-\frac{1}{|y-z|^2}\right| \rho^N(z,t) \,dz\leq & CN^{3\alpha}\int_{S_1}\left(\frac{1}{|x-z|^2}+\frac{1}{|y-z|^2}\right)\,dz\leq C N^{3\alpha}|x-y|.
{\mathcal L}_\pm(\phi)= -\Delta \phi^\pm \,+\,\Big[\,A_\pm\big({U^\pm}^2-{t^\pm}^2\big)+B\big({U^\mp}^2-{t^\mp}^2\big)\,\Big]\phi^\pm &\,+\,2A_\pm{\mathbf{Re}}\Big(w^\pm\overline{\phi^\pm}\Big)w^\pm\,+\,2B{\mathbf{Re}}\Big(w^\mp \overline{\phi^\mp}\Big)w^\pm.
\delta^{L}\Phi ( X ) = \frac{i}{2}\omega^{\alpha \beta}( X ) \star \left( \mathcal{D}_{\alpha \beta}\Phi \right) ( X ) = \frac{i}{2}\left( \omega^{\alpha \beta}( X ) \star \mathbf{Q}_{\alpha \beta}\right) \star \Phi ( X ) \equiv i \omega_{L}( X ) \star \Phi ( X ) .
\mathcal{L}_{(Q_0, P_0)}= \lambda P_0 \partial_{Q_0}- \gamma P_0 \partial_{P_0}- U'(\lambda^{-1}Q_0)\partial_{P_0}+ \gamma T \partial_{P_0}^2 &\approx \beta B \lambda^{\beta+1}Q_0^{-\beta-1}\partial_{P_0}.
f_{n}(T, U) = \zeta_0T^{n+1}- \zeta_1T^{n}U + \zeta_2T^{n-1}U^2 + \cdot \cdot \cdot + (-1)^n\zeta_nTU^{n}+ (-1)^{n+1}\zeta_{n+1}U^{n+1}.
\limsup_{n\to\infty}\int_{\mathbb{R}^N}(K\ast |u_n|^{p_{r;s}^{\uparrow *}}) |u_n|^{p_{r;s}^{\uparrow *}}dx = \int_{\mathbb{R}^N}d\xi+\xi_\infty; \limsup_{n\to\infty}\|u_n\|_{D^{s,p}}^p = \int_{\mathbb{R}^N}d\mu+\mu_\infty; \limsup_{n\to\infty}\int_{\mathbb{R}^N}|u_n|^{p_s^*}dx = \int_{\mathbb{R}^N}d\nu+\nu_\infty;
\Big(\kappa m-{\frac{\sigma^2}{2}}\Big)^2>{\frac{2\epsilon_3\mu}{2\epsilon_3\mu(1+2\gamma)-(2+2\gamma)}}\left(\gamma-{\frac{\beta}{\mu\sigma^2}}\right)^2\sigma^4={\tilde f}(2\epsilon_3\mu)\sigma^4.
I_1 :=\int_{B}W_{\delta}(x)(U_{\delta,e_n}^p-U_{\delta,-e_n}^p), I_2 := \int_{B}\delta^{-\frac{n-4}{2}}\Big(\varphi_0\Big(\frac{e_n-x}{\delta}\Big)-\varphi_0\Big(\frac{e_n+x}{\delta}\Big)\Big)(U_{\delta,e_n}^p-U_{\delta,-e_n}^p), I_3 :=\int_{B}\zeta_\delta(x)(U_{\delta,e_n}^p-U_{\delta,-e_n}^p).
\overline C_{ij}=C_{T+1-j,T+1-i},\overline C^T=a_0\begin{pmatrix}r_0 r_1 r_2 \ldots r_{T-1}r_1 r_0+r_2 r_1+r_3 \ldots .. r_2 r_1+r_3 r_0+r_2+r_4 \ldots .. \cdot \cdot \cdot \cdot \cdot \end{pmatrix},
Q(\hbar) = Q_A + \delta + D^\perp + \mathcal{R}+ \sum_{j \geq 2}T^j + \hbar\delta_+ + \hbar\mathcal{R}_+ + \hbar\sum_{i \geq 2}T_+^i, \bar{Q}(\hbar) = Q_A + \delta + D^\perp + \mathcal{R}+ \sum_{j \geq 2}T^j + \hbar\bar{\delta}_+ + \hbar\bar{\mathcal{R}}_+ + \hbar\sum_{i \geq 2}\bar{T}_+^i,
h = \frac{1}{Q^{3}}\left( \begin{array}{c}{\begin{array}{c c}{- 1 -{y^{\prime}}^{2}}&{y^{\prime}{\phi}^{\prime}}\\{y^{\prime}{\phi}^{\prime}}&{1 -{{\phi}^{\prime}}^{2}}\\ \end{array}}\\ \end{array}\right) ,{\hat{H}}^{\dagger}={\hat{H}},
(\pi^*\omega)_{\exp(x)}(Z_1,\bar{Z}_2) = \left.\frac{\partial^2}{\partial z_1 \partial \bar{z}_2}\right|_0 \left( u(x+x_1l_{j_1}+x_2l_{j_2})-2\left<\chi, x\right> \right) &= \left.\frac{1}{4}\frac{\partial^2}{\partial x_1 \partial x_2}\right|_0 u(x+x_1l_{j_1}+x_2l_{j_2}) & = \frac{1}{4}\frac{\partial^2 u}{\partial x_{j_1}\partial x_{j_2}}(x).
E[Y^p] \le p! \int_{0 \le s_1 \le \ldots \le s_p \le 1}\int_{0 \le t_1 \le \ldots \le t_p \le 1}E\Big[ \sum_{\sigma}\prod_{i=1}^p |\beta_{s_i}-\tilde{\beta}_{t_{\sigma(i)}}|^{-2}\Big]ds_1 \cdots ds_p dt_1 \cdots dt_p \,.
\begin{cases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)=\rho(t,1) t\in (0,T) , u(t,0)=0, \ \u(t,1)=0 t\in (0,T) , \rho(0,x)=\rho_0(x),\ \u(0,x)=u_0(x) x\in(0,1),\end{cases}
r_{n}^{l}=\mathrm{Q}_{n}(y_n^l)=\begin{cases}h_{n}x^{l}+z_{n}^{l}, \mathrm{if}\ \delta_n=1, \mathrm{sgn}(h_{n}x^{l}+z_{n}^{l}), \mathrm{if}\ \delta_n=0,\end{cases}
|\phi'(t)| =\Big|\frac1n \sum_{j = 1}^n \alpha(y_j) q_t(x, y_j)\Big| \leqslant \|\alpha\|_{L^2(\hat \nu)}\Big(\frac1n \sum_{j = 1}^n q_t(x, y_j)^2 \Big)^{1/2}.
D(0, \varepsilon) = \sigma_{\varepsilon}([S\bullet T,\varphi^{-1}(iI)]_{\ast})=\sigma_{\varepsilon}([\frac{I}{2}\bullet \varphi(T),iI]_{\ast}) &=\sigma_{\varepsilon}(i(\varphi(T)-\varphi(T)^{\ast})).
(\Delta_{\{1,2\}}(V) , \Delta_{\{1,3\}}(V) , \Delta_{\{1,4\}}(V)) = (1 , 0 , 1), &(\Delta_{\{1,2\}}(V) , \Delta_{\{2,3\}}(V) , \Delta_{\{2,4\}}(V)) = (1 , -1 , 0), &(\Delta_{\{1,3\}}(V) , \Delta_{\{2,3\}}(V) , \Delta_{\{3,4\}}(V)) = (0 , -1 , 1), &(\Delta_{\{1,4\}}(V) , \Delta_{\{2,4\}}(V) , \Delta_{\{3,4\}}(V)) = (1 , 0 , 1)
A_{\rho}(\alpha) =\{f \in C_0(\alpha\times L):f(\eta,y)=0(\eta,y)\in \alpha\times (L\setminus L_{\rho})\}, B_{\rho}(\alpha) =\{f \in C_0(\alpha\times L):f(\eta,y)=0(\eta,y)\in \alpha\times L_{\rho}\}
L(t, U(t,s)x) + \int_s^t \|H(\tau) U(\tau,s) x\|^p d\tau &\leq 2^{p-1}\int_s^{+\infty}\|H(\tau) U(\tau,s) P(s) x\|^p d\tau - 2^{p-1}\int_0^s \|H(\tau) U_{Q}(\tau,s)Q(s)x\|^p d\tau &= L(s,x),
\lim_{p_{0}\rightarrow 0}\Pi_{0 0}^{( \mathrm{r i s i n g \s u n}) ( T )}\rightarrow \frac{e^{2}}{2 \pi \beta}\left[ 2 + \frac{4 m^{2}( m^{2}- m_{\sigma}^{2})}{( m_{\sigma}^{2}- m_{+}^{2}) ( m_{\sigma}^{2}- m_{-}^{2})}\, \ln \frac{m_{\sigma}}{m_{-}}- \frac{4 m^{2}( m^{2}- m_{+}^{2})}{( m_{\sigma}^{2}- m_{+}^{2}) ( m_{+}^{2}- m_{-}^{2})}\, \ln \frac{m_{+}}{m_{-}}\right] .
\left((-1)^{m-1}\xi_1,\dots, (-1)^{m-i}\xi_i, (-1)^{m-i-1}\alpha^{-1}\xi_{i+1}, \dots, \alpha^{-1}\xi_m,\xi_{m+1}\dots, \xi_n\right) = \left(\alpha^{-1}\xi_m,\dots, \alpha^{-1}\xi_{i+1},\xi_i, \dots, \alpha\xi_1\right)\chi_\Gamma\left(\left(X'_f\right)_f\right).
\Delta_{\widehat{C}}= H(Z|A,B,C) + H(W|B,C,D) + 2H(X|A,C,D) + H(Y|W,X,Z) + H(C|B,X,Y) \ \ \ + H(A) + H(B) + H(C) + H(D) - H(A,B,C,D)
\displaystyle\sum_{i = 1}^N a_i^k \frac{\varepsilon}{\lambda_k^{\beta_k}}+ \frac{d_k}{(\lambda_1\lambda_2)^{\frac{N - 2\gamma}{2}}}= O(\varepsilon\varepsilon_{12}) + O(\varepsilon_{12}^{1 + \tau_1}) + O\left(\varepsilon\displaystyle\sum_{j = 1}^2\left(\frac{1}{\lambda_j^{\beta_j + \sigma}}+ |y^j - z^j|^{\beta_j + \sigma}\right)\right),
[ S_{n , n}^{+}, S_{n , n - 1}^{-}]_{{}_{+}}= - i ( D_{n}( 2 K_{n})^{n - 1}+ ( 2 K_{n})^{n - 1}D_{n}) + ( 2 K_{n})^{n - 1}( \alpha_{n}+ n \Sigma ) .
b^{-\frac{\gamma}{2}-\frac{1}{2}}\partial_\xi \phi_{i,int, \beta}\left( \frac{y_0}{\sqrt{b}}\right) = b^{-\frac{\gamma}{2}}\frac{\phi_{i,int, \beta}\left( \frac{y_0}{\sqrt{b}}\right)}{\phi_{i, out, \beta}(y_0)}\partial_y \phi_{i,out, \beta}(y_0),
\lim_{\epsilon \to 0}\left\{\frac{d}{d r}G \left( r , r^{\prime}\right) \vert_{r = r^{\prime}+ \epsilon}- \frac{d}{d r}G \left( r , r^{\prime}\right) \vert_{r = r^{\prime}- \epsilon}\right\}= \frac{1}{r^{3}g ( r )^{2}}
[B\centerdot B]= -8B_2B_3\eta_1\otimes\sigma_1-8B_3B_1\eta_2\otimes\sigma_2-8B_1B_2\eta_3\otimes\sigma_3, B\centerdot\theta_1+[B\centerdot B]\centerdot\theta_2= \eta_1\otimes\sigma_1\centerdot(B_1\theta_1-8B_2B_3\theta_2)+\eta_2\otimes\sigma_2\centerdot(B_2\theta_1-8B_3B_1\theta_2) &+\eta_3\otimes\sigma_3\centerdot(B_3\theta_1-8B_1B_2\theta_2).
\left( \sqrt{1-{d^{\max}_i}^2}\mathbf{F}_i +d^{\max}_i\nu_i\mathbf{T}_i \right)^{-1}=\mathbf{F}_i^{-1}- \mathbf{F}_i^{-1}\mathbf{T}_i\mathbf{F}_i^{-1}\nu_id^{\max}_i +\sum_{k=2}^{\infty}\mathbf{A}_k \left(d^{\max}_i\right)^k,
P_q^{(\alpha ,\beta )}(\cos\theta )\approx\frac{\Gamma (q+\alpha+1)}{N^{\alpha}n!}\left(\sin\frac{\theta}2\right)^{-\alpha}\left(\cos\frac{\theta}2\right)^{-\beta}\left(\frac{\theta}{\sin\theta}\right)^{\frac 12}J_{\alpha}(N\theta ),
{\mathbb E}\left\{1 - \frac{{{N_{{\rm{S}},k}}}}{\pi}\frac{{\left|{\tilde \Theta_{{k,{\rm{R}}}}^{\rm{D}}- \Theta_{{k,{\rm{R}}}}^{\rm{D}}}\right|}}{2}\right\}={1 -{2^{- 1 -{x_k}}}}.
a_k= \prod_p\Big( (1-p^{-1})^{k^2}\sum_{j=0}^{\infty}\Big(\frac{\Gamma(k+j)}{\Gamma(k)j!}\Big)^2\frac{1}{p^{j}}\Big) \mbox{and}C_k=\frac{2^{2-\frac1k}-1}{2-\frac1k}\cdot \frac{k^{k^2-1}}{\Gamma(k^2)}\cdot a_k.
< \phi_{\omega}, \phi_{\sigma}> ={\frac{1}{( \omega - \sigma )}}\lim_{r \rightarrow \infty}\int_{C_{r}}d \sigma^{i}\left( \nabla_{i}\phi_{\omega}^{*}\phi_{\sigma}- \phi_{\omega}^{*}\nabla_{i}\phi_{\sigma}- i ( \omega + \sigma ) a_{i}\phi_{\omega}^{*}\phi_{\sigma}\right) ~ ~ .
U_{m_1}(S_{i_1})U_{m_{2}}(S_{i_2})\cdots U_{m_k}(S_{i_k}) = I_{m}^{S}\big(e_{i_1}^{\otimes m_1}\otimes e_{i_2}^{\otimes m_2}\otimes \cdots \otimes e_{i_k}^{\otimes m_k}\big),
\sum_{n = N_1}^{N_2-1}\big| m_{n+1}(x, y) - m_n(x, y) \big|=\bigg(\frac{1}{\vartheta_B(N_1)}- \frac{1}{\vartheta_B(N_2)}\bigg) \prod_{j = 1}^{k''}\log y_j.
\frac{1}{2n}(\nabla^*Q_H)\,g\ =\ \mathcal L_Q\,g\ =\ \frac{d}{dt}\vert_{t=0}\left(\frac{1}{2h_t}g\right)\ =\ -\ \frac{h'_0}{2h_0}\,g\ =\ -\ 2h'_0\, g
\frac{1}{2}{\cal S}_{RN}= \beta_{RN}\Lambda \left[ \frac{r_{s}^{3}-r_{h}^{3}}{12}\right] \,+\, \frac{q^{2}}{4}\beta_{RN}\left( \frac{1}{r_{h}}- \frac{1}{r_{s}}\right) + 4\pi\sigma r_{s}^{2}\beta_{RN}\sqrt{f(r_{s})}
a_{n}^{\alpha}= \int_{0}^{\pi}~ \frac{d \sigma_{+}}{\pi}\exp \left[ 4 i n \frac{e_{\mu}X_{L}^{\mu}( \sigma_{+})}{e_{\mu}P_{L}^{\mu}}\right] \xi_{i}^{\alpha}\partial_{+}X_{L}^{i}( \sigma_{+}) \, ,
\begin{array}{rl}a_l= su(l\!+\!1)\hspace{0.2in}2,3,\dots (l+1) b_l= so(2l\!+\!1)\hspace{0.2in}2,4, \dots 2l c_l= sp(2l)\hspace{0.2in}2,4, \dots 2l d_l= so(2l)\hspace{0.2in}2,4, \dots (2l-2) \end{array}