harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A' \ = \ 2^{n-1}\sum_{\omega = 0}^{a-1}2^{-\omega}(-1)^{\omega}I(\omega, 0) \sum_{\alpha = 0}^\omega \sum_{\ell = 0}^{a - 1 - \omega}\sum_{i= 0}^{a - \ell - \omega - 1}(-1)^i \binom{n}{\ell + \omega +i}\binom{\ell + \omega +i}{\ell + i}\binom{\omega}{\alpha}\binom{\ell + i}{i}.
G(2,d):=\left(\begin{array}{cc}s t^d -s^{d+1}t^{d+1}-s^d t \end{array}\right)\equiv\left(\begin{array}{cc}1 -1 1 -1 \end{array}\right)\odot\left(\begin{array}{cc}s t^d s^{d+1}t^{d+1}s^d t \end{array}\right).
\lambda = \frac{(p -\alpha + m_1)(p-1-k_2) + k_1(p -k_3 +m_2)}{(p-1-\alpha)(p-1-k_2) -k_1 k_3}> 1, \mu = \frac{(p-1-\alpha)(p +m_2) + k_3(1 + m_1)}{(p-1-\alpha)(p-1-k_2) - k_1 k_3}> 1.
(\begin{matrix}d[X^{i},X^{j}]&d[X^{i},Y^{j}]\\d[Y^{i},X^{j}]&d[Y^{i},Y^{j}]\end{matrix})=\frac{\hbar}{2m}\delta^{ij}(\begin{matrix}1&1\\ 1&1\end{matrix})dt
n_{1}=\left\lceil \left( m+2+\alpha/4\right) \frac{\log\left( \frac{1}{h}\right)}{\log\left( \frac{2}{\Lambda_{K}}\frac{\operatorname*{dist}\left( \tau,\sigma\right)}{\max\left\{h_{\tau},h_{\sigma}\right\}}\right)}\right\rceil .
\int_M B_\alpha^2 dv_g = \int_{B_{x_\alpha}(\delta)}B_\alpha^2 dv_g + O(\mu_\alpha^{n-2}), &= \mu_\alpha^2\int_{B_0(\delta/\mu_\alpha)}B^2 dv_{\hat{g}_\alpha}+o(\mu_\alpha^2),
\frac{d}{dt}\int u^2\phi_N\,dx -\int u^2\,\partial_t\phi_N\,dx+\epsilon\int (\partial_x^2u)^2\partial_x\phi_N\,dx &-\frac32 \int u^2\partial_x^5\phi_N\,dx-\frac{25}{4(5-\epsilon)}\,\int u^2 \frac{(\partial_x^3\phi_N)^2}{\partial_x\phi_N}\,dx &\le 2\int F u\phi_N\,dx,
\min \sum_{i=\eta_1,\eta_2}\lambda_i\frac{N}{\lfloor\frac{N}{2}\rfloor-1+i}\sum_{i=\eta_1,\eta_2}\lambda_i\frac{i}{\lfloor\frac{N}{2}\rfloor-1+i}\leq \tilde{D}\lambda_{\eta_1}+\lambda_{\eta_2}=1 \lambda_{\eta_1},\lambda_{\eta_2}\geq 0.
\frac{1}{\gamma_0}\int^\infty_{\gamma_0}f_{\gamma_z}(\gamma)d\gamma =\frac{A_xA_y}{\gamma_0}\sum^\infty_{k=0}\sum^k_{n=0}\tilde{B}_x(n)\tilde{B}_y(k-n) &\times\Gamma\left(n+1,a_x\gamma_0\right)\Gamma\left(k-n+1,a_y\gamma_0\right).
\begin{array}{c}{G_{T}( z^{\prime \prime}) = G_{T}( z_{1}^{\prime \prime}) G_{T}( z_{s}( z^{\prime}; u ) ) ,}\\{{}}\\{G_{T}( z^{\prime}) = G_{T}( z_{1})^{- 1}G_{T}( z ) ,}\\ \end{array}
{P_{e,{\rm{OFDM}}}}= \frac{1}{K'}\sum\limits_{k = 1}^{K'}{{P_e}\left({\frac{{{K'}{p_k^{\star}}{{\left\|{{\bf{h}}\left[ k \right]}\right\|}^2}}}{{\left({{K'}+{{\tilde n}_{{\rm{span}}}}}\right){\sigma^2}/{K'}}}}\right)},
S_{\rm bosonic}= \frac{1}{2 \kappa_{11}^2}\int d^{11}x (-g)^{1/2}\bigl( R - \frac{1}{2}|F^M_4|^2 \bigr) - \frac{1}{6}\int A^M_3 \wedge F^M_4 \wedge F^M_4,
\begin{array}{l l}{< L_{1}^{\pm}, M_{2}> = R_{1 2}^{\pm}, \quad}&{\qquad < \tilde{L}_{1}^{\pm}, \tilde{M}_{2}> = R_{1 2}^{\pm},}\\{< L_{1}^{\pm}, \tilde{M}_{2}> = A_{1 2}, \quad}&{\qquad < \tilde{L}_{1}^{\pm}, M_{2}> = \tilde{A}_{1 2}.}\\ \end{array}
u_{n}=\sum_{k_{1}=1}^{n}k_{1}+\sum_{k_{1}=1}^{n}\sum_{k_{2}=1}^{k_{1}}k_{2}+...+\sum_{k_{1}=1}^{n}\sum_{k_{2}=1}^{k_{1}}...\sum_{k_{n}=1}^{k_{n-1}}k_{n}
\begin{array}{l}a_{ij}^{{\bf 4}}= \frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma}\chi_{\gamma}^{R_M \oplus{\bf 1}^2}\chi_{\gamma}^{(i)}\chi_{\gamma}^{(j)*}a_{ij}^{{\bf 6}}=\frac{1}{g}\sum\limits_{\gamma=1}^{r}r_{\gamma}\chi_{\gamma}^{R_M \oplus R_M \oplus{\bf 1}^2}\chi_{\gamma}^{(i)}\chi_{\gamma}^{(j)*}\end{array}
B_1= \begin{pmatrix}\frac{2\Delta^{2}}{\Delta +1}\frac{t_{v_{i}}\Delta +\Delta-2\Delta^{2}}{\Delta +1}\frac{t_{v_{i}}\Delta +\Delta-2\Delta^{2}}{n-\Delta -1}\frac{S-2 t_{v_{i}}\Delta +2\Delta(\Delta-1)}{n-\Delta -1}\end{pmatrix}.
\begin{aligned}X = \frac{g_s}{gh_s + dg_s + \frac{f_sg}{f}}\qquad Y = \frac{1}{gh_s + dg_s + \frac{f_sg}{f}}Z = \frac{\frac{f_sg}{f}}{gh_s + dg_s + \frac{f_sg}{f}}\qquad W = \frac{\frac{f}{g}}{gh_s + dg_s + \frac{f_sg}{f}}\end{aligned}
\widetilde{\cal F}= \left( \begin{array}{cc}F - \Phi \Phi^{\dag}+ \psi + \bar{\psi}+ m + \lambda + \bar{m}i ( D \Phi + \xi + \bar{\xi}) i ( D \Phi^{\dag}+ \xi^{\dag}+ \bar{\xi}^{\dag}) F - \Phi^{\dag}\Phi + \psi + \bar{\psi}+ m + \lambda + \bar{m}\end{array}\right)
{\tilde{\cal{E}}}_{m > 0}={\cal{E}}_{m > 0}( B ) -{\cal{E}}( 0 ) = \frac{B^{2}}{2}+ \frac{( e B )^{\frac{3}{2}}}{2 \pi}g \left( \frac{e B}{m^{2}}\right) + \frac{e B}{2 \pi}| m | \; .
\Omega = \left( \begin{array}{c c c c}{{J_{1 1}}}&{{\cdots}}&{{J_{1 N}}}&{{Q_{1}}}\\{{\vdots}}&{{\ddots}}&{{\vdots}}&{{\vdots}}\\{{J_{N 1}}}&{{\cdots}}&{{J_{N N}}}&{{Q_{N}}}\\{{-{\overline{{{Q}}}}_{1}}}&{{\cdots}}&{{-{\overline{{{Q}}}}_{N}}}&{{\sum_{\mu = 1}^{N}J_{\mu \mu}}}\\ \end{array}\right)
[\begin{matrix}F(0)\\F(1)\\F(2)\\F(3)\end{matrix}]=[\begin{matrix}1&1&1&1\\ 1&2&4&3\\ 1&4&1&4\\ 1&3&4&2\end{matrix}][\begin{matrix}f(0)\\f(1)\\f(2)\\f(3)\end{matrix}]
\int_D\int_D g_x(h)\> d(\delta_l*\delta_w)(h) \> \alpha(w) \> d\omega_D(w) =\int_D g_x(l*w) \alpha(w) \> d\omega_D(w) &=\int_D g_x(w)\alpha(\bar l*w)\> d\omega_D(w) &=\alpha(\bar l)\int_D g_x(w)\alpha(w) \> d\omega_D(w) =\alpha(\bar l)T_\alpha g(x).
H_3(1)(f^{-1})=\frac{1}{9216}(17c_1^6-102c_1^4c_2+ 32c_1^3c_3+180c_1^2c_2^2-144c_1^2c_4+192c_1c_2c_3-216c_2^3\quad\quad\quad +288c_2c_4-256c_3^2)
i\frac{\partial h}{\partial z}(-1,\alpha_{0})=i\frac{\partial^{2}h}{\partial z^{2}}(-1,\alpha_{0})=0,i\frac{\partial^{3}h}{\partial z^{3}}(-1,\alpha_{0})=-\frac{2\lambda(1-\lambda)}{(1+\lambda)^{3}}\neq0.
\|x\|_{\Gamma_p}^p =\displaystyle\sum_{n=1}^{\infty}\gamma_{n}\Big|\displaystyle\sum_{k=1}^{n}q_ka_{k}\Big|^{p}=\displaystyle\sum_{n=1}^{\infty}\gamma_{n}\Big|\displaystyle\sum_{k=1}^{n}\gamma_k^{-\frac{1}{p}}b_k-\gamma_{k-1}^{-\frac{1}{p}}b_{k-1}\Big|^{p}= \sum_{n=1}^{\infty}|b_n|^p=\|y\|_p^p=\|Tx\|_p^p.
\Omega^{j}= \big\{\bar{\xi}\in 2 \pi \lambda^{j(1-\sigma)}\mathbb{Z}^{n-1}\,:\, \lambda^{j}\leq | \xi_{m}|< \lambda^{j+1}, \, m=2,\ldots, n \big\}+ Q\Big( 0,\frac{\varepsilon_{1}}{\sqrt{n\!-\!1}}\Big) \, ,
\Tilde{A}_m(z_m)=(z_m-2\alpha_{m})p_m^2(z_m)\mbox{with}p_m(z_m)=z_m^{\ell_m}+\beta_{m,1}z^{\ell_m-1}+\!\ldots\!+\beta_{m,\ell_m}\mbox{and}\beta_{m,1}=\alpha_{m}.
Z_{\; \; \alpha_{L - 1}}^{\alpha_{L - 2}}Z_{\; \; \alpha_{L}}^{\alpha_{L - 1}}= C_{\alpha_{L}}^{\alpha_{L - 2}\beta_{0}}\frac{\delta S_{0}}{\delta \Phi^{\beta_{0}}}, \; \alpha_{L}= 1 , \cdots , M_{L},
\int_{\Omega}(y-T_k(y))\cdot \nabla \varphi(t,y) \rho_{h}^{k+1}(y)\,dy+h\int_{\Omega}\nabla(V[\rho_h^{k}])(y)\cdot \nabla \varphi(t,y) \rho_{h}^{k+1}(y) \,dy +h\int_\Omega \nabla(P(\rho_{h}^{k+1}))(y)\cdot \nabla \varphi(t,y) \,dy =0.
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{IV}^+}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_{d,s,k}C_{d,\alpha}T R \eta^{d-1}\end{aligned}
\left\|\sum_{j=1}^na_j\omega_j \right\|=\left\|\sum_{j=1}^na_jV\tau_j \right\|=\left\|V\left(\sum_{j=1}^na_j\tau_j \right)\right\|=\left\|\sum_{j=1}^na_j\tau_j \right\|=\left(\sum_{j=1}^n|a_j|^p\right)^\frac{1}{p}.
f\otimes \phi;\bullet_2;\kappa = 1_{A_1}\otimes \phi;f\otimes 1_{M_2};\bullet_2;\kappa = \bullet_1;\phi;\kappa = \bullet_1;\gamma;\kappa = 1_{A_1}\otimes \gamma; g \otimes 1_{M_2}; \bullet_2;\kappa = g\otimes \gamma;\bullet_2;\kappa
\mathbb{E}_{\mathbf{H}}\{\Vert\hat{\mathbf{x}}_{lmmse}-\mathbf{x}\Vert^2\}=\mathbb{E}_{\mathbf{H}}\textsf{Tr}\left(\mathbf{\Sigma}_{e,lmmse}\right) &=\mathbb{E}_{\mathbf{H}}\textsf{Tr}\left(\left(\mathbf{\Sigma}_x^{-1}+\mathbf{H}^*\mathbf{\Sigma}_z^{-1}\mathbf{H}\right)^{-1}\right)
\ell(x^*) =f(x^*) \geq f(x^k) & \geq f(x^k) + \sum\limits_{i=1}^m\lambda_i^*g_i(x^k) + \sum\limits_{i=1}^p\mu_i^* h_i(x^k) + \sum\limits_{i=1}^n \gamma_i^*x_i^k = \ell(x^k).
\bar k(s) = 2\alpha - (1+2\beta)\bar v(s) = \frac{\alpha}{1+\beta}- e^{\mu(s-t)}\left(\frac{1+2\beta}{1+\beta}\pi(s)\bar X(s) + \frac{\alpha(1+2\beta)}{(1+\beta)^2}\int_t^s \pi(\tau)d\tau \right)
gradF_{\lambda_{i}}\cdot gradF_{\lambda_{k}}=4(\frac{x^{2}}{(a^{2}-\lambda_{i})(a^{2}-\lambda_{k})}+\frac{y^{2}}{(b^{2}-\lambda_{i})(b^{2}-\lambda_{k})}+\frac{z^{2}}{(c^{2}-\lambda_{i})(c^{2}-\lambda_{k})})=0,
0 =\lim_{j\rightarrow\infty}\{v_{l_0}(x_j,t_j)-u_{l_0}(x_j,t_j+\tau_*)\}&=\lim_{j\rightarrow\infty}\{v_{l_0}(x_j-k_j,t_j-n\cdot k_j/c)-u_{l_0}(x_j-k_j,t_j-n\cdot k_j/c+\tau_*)\}&=v_{l_0}(x_*,t_*)-u_{l_0}(x_*,t_*+\tau_*).
F_{H}(t)={\mu^k\over k!}\Bigg[\exp\left\{-\lambda\big(1-{\rm e}^{-\mu}\big)t\right\}B_k\big(\lambda{\rm e}^{-\mu}t \big) +\sum_{j=0}^k S_2(k,j)\,{\gamma\left(j+1,\lambda (1-{\rm e}^{-\mu})t\right) \over ({\rm e}^{\mu}-1)^j}\Bigg].
W_{L}= \sum_{s,r =1}^{N_f}(\lambda_{r,s}M +{m_{Q}}_{r,s})V^{2r-1,2s}+ ({\rm Pf}V)^{\frac{1}{N_f - 2}}(2-N_f){\Lambda_L}^{\frac{N_f-6}{N_f-2}}+ \sum_{k=2}^{N_c}g_r{u^{cl}}_r ,
\sum_{\substack{c_{\nu\mu}(\nu, \mu) \not= (i, j)}}\Big| \sum_{c_{ij}}(...) \Big| \ll\big(\max_{1 \leq j \leq n}|\psi_j|^{-1/2}_p \big) p^{r_1 + \ldots + r_n -m_{ij}/2 + O(\mu_{ij})}.
H_{2,n}^{(\{1,1\})}(x)=x\,\frac{2^{n+3}n(n-1)(n-2)}{\prod_{1\leq i<j\leq n-1}(U_{j,n}-U_{i,n})}\det\big(M_n^{\{1,1\}}\big)+16(n-1)(n-2)H_{n-2}(0).
\int_{R}^{\infty}\frac{1}{2}\rho4\pi r^{2}dr(v\frac{R^{2}}{r^{2}})^{2}=2\pi\rho v^{2}R^{4}\int_{R}^{\infty}\frac{dr}{r^{2}}=2\pi\rho v^{2}R^{3}
G_{\beta}( \vec{x}, \tau ) ={\frac{1}{V}}\sum_{\vec{p}}{\frac{1}{2 E_{p}}}\left[{\frac{e^{i \vec{p}\cdot \vec{x}- E_{p}\tau}}{1 - e^{- \beta E_{p}}}}+{\frac{e^{- i \vec{p}\cdot \vec{x}- E_{p}\tau}}{e^{\beta E_{p}}- 1}}\right] .
C_m^k(t) =\left. (-1)^{k+kt+1}\sum_{l=0}^k \binom{k}{l}x^{l+kt}(l+kt)^{m-1}\right|_{x=-1}= (-1)^{k+kt+1}\left.\left( x\partial_x \right)^{m-1}((1+x)^kx^{kt}) \right|_{x=-1}= \begin{cases}0 m=1,\ldots, k, -k! m=k+1, \end{cases}
D_\xi^\alpha \hat{f}(\xi) = \int D_\xi^\alpha \left[ e^{-ix\cdot\xi}f(x)\right] dx = (-1)^{|\alpha|}\int x^\alpha e^{-ix\cdot\xi}f(x)dx = (-1)^{|\alpha|}\left[ x^\alpha f\right]^\wedge\!\!(\xi).
\begin{aligned}\xi y\left[ n \right] = \sqrt P s\left[{n -{n_{\max}}}\right] + \sqrt P \sum\limits_{l = 1}^L{\sum\limits_{l' \ne l}^L{\frac{{{\bf{h}}_l^H{{\bf{h}}_{l'}}}}{{\sum\nolimits_{k = 1}^L{{{\left\|{{{\bf{h}}_k}}\right\|}^2}}}}s\left[{n -{n_{\max}}+{n_{l'}}-{n_l}}\right]}}+ \xi z\left[ n \right]. \end{aligned}
p_{|h_1|,|h_2|}(x_1,x_2) =\frac{4x_1x_2}{\sigma^{4}(1-\mu_2^{2})}e^{-\frac{x_1^2+x_2^2}{\sigma^{2}(1-\mu_2^2)}}begin{align*}-0.2em] \!\!\!\!\!\!\times I_{0}\left ({\frac{2\mu_2 x_1x_2}{\sigma^{2}(1-\mu_2^2)}}\right)\!, x_1,x_2 \geq 0.
[\nabla u]^p_{W^{\tau,p}(B_r)}\le [\nabla (u\,\eta)]^p_{W^{\tau,p}(\mathbb{R}^N)}&\le C\,\left(\frac{h_0^{(\Gamma-1-\tau)\,p}}{\Gamma-1-\tau}\, \sup_{0<|h|<h_0}\left\|\frac{\delta_h \nabla (u\,\eta)}{|h|^{\Gamma-1}}\right\|_{L^p(\mathbb{R}^N)}^p+\frac{h_0^{-\tau\,p}}{\tau}\,\|\nabla (u\,\eta)\|^p_{L^p(\mathbb{R}^N)}\right),
(\begin{matrix}n\\ 3\end{matrix})(d(\tilde{P_{3}},G)+2d(P_{3},G))=(\begin{matrix}n\\ 2\end{matrix})(n-2)d(K_{2},G)\Rightarrow d(\tilde{P_{3}},G)+2d(P_{3},G)=3d(K_{2},G).
\frac{[G_n:B_{n}^{-}]}{|U_n|}\Psi_{\underline{\chi}_{n}}^{GL_{n}}(g_{1,n-1}(a_1,1))=\sum_{t\in T_n}f_{T_n}(e_{\psi_n}g_{1,n-1}(a_1,1)e_{\psi_n})(t)\underline{\chi}_n(t).
{\cal O}_i^{(q)}=\left.\left(\left.{\partial \Gamma_\infty^{(\beta)}\over \partial \beta_i}\right|_{\Phi,K}\right)\right|_{\beta=0}=\left.\left(\left.{\partial W_\infty^{(\beta)}\over \partial \beta_i}\right|_{J,K}\right)\right|_{\beta=0}=\left. <{\partial\Sigma_\infty^{(\beta)}\over \partial \beta_i}>_J\right|_{\beta=0}.
c_{n,j}q^{2(n-j)}\left[\theta_{j}(q^{2})-\mu_{j}(q^{2})\right] =c_{n-1,j-1}q^{2n-j}\left[\theta_{j}(q^{2})-\mu_{j}(q^{2})\right](q^{2n-j+1}-(-1)^{2n-j+1}) \Rightarrow c_{n,j}=c_{n-1,j-1}q^{j}(q^{2n-j+1}-(-1)^{2n-j+1}).
L'_x :=\{y\in D^+ \,|\, y_1\in(x_1,1) \,\,\&\,\, y_2\in(0,y_1-x_1+x_2)\}\subseteq L_t, L''_x := \big[ ((0,x_1]\cup[1,\infty))\times(0,x_2) \big] \setminus L'''_x,
{\cal L}= \overline{D_\mu\phi^i}D^\mu\phi^i + \sigma(\phi^{i\dagger}\phi^i -N \chi) + N (m^2 \chi +{\lambda \over 2}\chi^2 +{\eta \over 3}\chi^3)+ iN\epsilon_{\mu\rho\nu}A_\mu \partial_\rho A_\nu.
(\partial_T + P\cdot \nabla_X ) F = 4 \pi \int \d P_2 \d P_3 \d P_4 \delta(P + P_2 - P_3 - P_4 ) \delta ( P^2 + P_2^2 - P_3^2 - P_4^2 ) \times | \hat V(P - P_3) - \hat V (P - P_4)|^2 ( F F_2 \widetilde F_3 \widetilde F_4 - F_4 F_3 \widetilde F_2 \widetilde F ) \ .
\partial_{\rho}\varphi_F (\rho) = E_{h' (\rho)}[(u^{\ell}- \ell \rho) F] h'' (\rho), \partial^2_{\rho \rho}\varphi_F (\rho) = E_{h' (\rho)}[(u^{\ell}- \ell \rho)^2 F] (h'' (\rho))^2 - \ell E_{h' (\rho)}[F] h'' (\rho) + E_{h' (\rho)}[(u^{\ell}- \ell \rho) F] h''' (\rho) .
\begin{array}{lcl}a_1\mu_1 = (\alpha_2-\alpha_1)\log_1(\pi_2)+\beta\log_1(u(K)), k\lambda_1+a_1\mu_1 = -s\alpha_1 , a_2\mu_2 = (\alpha_1-\alpha_2)\log_1(\pi_2)-\beta\log_1(u(K)), k\lambda_2+a_2\mu_2 = -s\alpha_2.\end{array}
\|X\|_{*}=\min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{n\times d}:X=UV^{T}}\|U\|_{F}\|V\|_{F}&=\min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{n\times d}:X=UV^{T}}\frac{\|U\|^{2}_{F}+\|V\|^{2}_{F}}{2}.
L^{2}({\cal A}_{\gamma}\times{\cal F}_{\gamma}) = \bigotimes_{e \in E}\bigl ( \bigoplus_{\rho \in \mathrm{R e p}( G )}\rho \otimes \rho^{*}\bigr ) \otimes \bigotimes_{v \in V}\bigl ( \bigoplus_{\rho \in S}\rho \bigr ) .
S^{(b)}_{q,i}(N)=S^{(b)}_{q,i}(b^{k_{1}})-S^{(b)}_{q,i-b^{k_{1}}}(b^{k_{2}})-\cdots+(-1)^{r-1}S^{(b)}_{q,i-b^{k_{1}}-\cdots-b^{k_{r-1}}}(-1,b^{k_{r}}).
P_{F_{n_j}}(\varphi, \varepsilon_j) = P_{F_{n_j}}(\varphi) \cdot \frac{P_{F_{n_j}}(\varphi, \varepsilon_j)}{P_{F_{n_j}}(\varphi)}= P_{F_{n_j}}(\varphi) \cdot \frac{\overline{A}_{n_j}(\varepsilon_j)\overline{C}_{n_j}(\varepsilon_j)}{A_{n_j}C_{n_j}},
\max_{P_i\ge 0, \forall i}~~ \sum_{i=1}^K \ln\left(1+\frac{h_{i}P_i}{\sigma^2+\sum_{j=1,j\neq i}^K h_{j}P_j}\right), \mbox{s.t.}~~ \eqref{Con-PIPC},~\eqref{Con-PTPC}.
\begin{aligned}\frac{1}{2}(K_{0 2}w, w)_{L^2(\mathbb{T})}=\varepsilon^{-2 b}( (\nabla \mathcal{H})(T_{\delta}) , T_2 [w, w] )_{L^2(\mathbb{T})}+\frac{\varepsilon^{-2 b}}{2}((\partial_u \nabla \mathcal{H})(T_{\delta})[T_1 w] , T_1 w )_{L^2(\mathbb{T})}.\end{aligned}
b_0(t,y)-\pi(t,y)\sigma_0^2(t,y)- \frac{\lambda \pi(t,y) \gamma_0^2(t,y)}{1+\pi(t,y)\gamma_0(t,y)}&+\sigma_0(t,y)\Phi(t,y)+\frac{\lambda \gamma_0(t,y)}{1+\pi(t,y)\gamma_0(t,y)}\Psi(t,y)= 0.
G(t,\Delta+1)= -\frac{(\Delta+1)!(\Delta+2t-2)}{(N+2)^{(\Delta+1)}}\binom{N-\Delta-2}{t-3}, G(t,\Delta+2)= -\frac{(\Delta+2)!(\Delta+2t-2)(t+\Delta+1)}{(N+2)^{(\Delta+2)}}\binom{N-\Delta-3}{t-4}, G(t,\Delta+3)= -\frac{(\Delta+3)!(\Delta +2t-2)(t+\Delta+2)_2}{2(N+2)^{(\Delta+3)}}\binom{N-\Delta-4}{t-5}.
\sum_{\{n \}}\Psi_{P , \{n \}}( z^{\prime}) \overline{{{\Psi_{P , \{n \}}( z )}}}= \sum_{\{n \}}\Psi_{P , \{n \}}( z^{\prime}) \Psi_{P , \{n \}}( \tilde{z}) = < z^{\prime}, \tilde{z}>^{P}\: ,
H_n^f \otimes_{H_{n-1}^f}H_n^f \oplus \bigoplus_{r=0}^{\ell-1}H_n^f \rightarrow H_{n+1}^f, (a\otimes b, c_0,c_1,\dots,c_{\ell-1}) \mapsto a s_n b + \sum_{r=0}^{\ell-1}c_r x_{n+1}^r
\widetilde{\Delta}(x,m) =\Delta(x)+\rho_{\beta}(m)- \tau \circ (\rho_{\beta}(m)), \widetilde{\varphi}(x,m) =(\varphi_L(x),\varphi_M(m)), \widetilde{\alpha}(x,m) =(\alpha(x),\beta(m)),
3 2 T ={\frac{r_{0}^{2}}{2}}[ \dot{\xi}^{2}+ \tan^{2}( \xi ) \dot{\phi}^{2}] ={\frac{r_{0}^{2}}{2}}[ \dot{\xi}^{2}+ \cot^{2}( \xi ) p_{\phi}^{2}] ,
S = \frac i{4 \pi \alpha^{\prime}}\int \left\{g_{i \bar{\jmath}}( \partial x^{i}\bar{\partial}x^{\bar{\jmath}}+ \bar{\partial}x^{i}\partial x^{\bar{\jmath}}) - i B_{i \bar{\jmath}}( \partial x^{i}\bar{\partial}x^{\bar{\jmath}}- \bar{\partial}x^{i}\partial x^{\bar{\jmath}}) \right\}d^{2}z .
\sum_{\substack{p\leq z, p\equiv l\pmod{24}}}\frac{1}{p}= \frac{1}{8}\log\log z+M(24, l)+\frac{E_\theta(z; 24, l)}{z\log z}-\int_z^\infty \frac{(1+\log t)E_\theta(t; 24, l)}{t^2\log^2 t}dt > \frac{1}{8}\log\log z+M(24, l)-1.1304\times 10^{-6}-\frac{3.73844\times 10^{-7}}{\log z},
\prod_{\ell\in\mathcal{S}}\prod_{i=1}^{r(\ell)}\mathbf{G}^1[\ell,\mathcal{T}(\ell)_i]\backslash\left(\mathbf{G}^1[\ell,\mathcal{T}(\ell)_i]\Delta^{\mathcal{T}(\ell)_i}(\mathbf{G}^1(\mathbb{Q}_p))\mathbf{K}^1[\ell]/\mathbf{K}^1[\ell]\right),
\times \left\langle \exp ig\left[\sum_{i=1}^{r_1}\left\{\phi_1(x_{1,i})-\phi_1(y_{1,i})\right\}+\sum_{i=1}^{r_2}\left\{\phi_2(x_{2,i})-\phi_2(y_{2,i})\right\}+\cdots \cdots \right]\right\rangle.
\mathcal{T}_{0\neq;LH}= \int_1^T \int A Q^2 A\left( g_{Lo}(\partial_{Y}-t\partial_X) (Q^2_{\neq})_{Hi}+ (U^3_0)_{Lo}(\partial_{Z}Q^2_{\neq})_{Hi}\right) dV dt + \int_1^T \int A Q^2 A\left( g_{Lo}t \partial_{X}(Q^2_{\neq})_{Hi}\right) dV dt.
\left(m+\frac{1}{2r_1}+\frac{1}{2r_2}\right)dr^2+\left(\frac{1}{r_2}-\frac{1}{r_1}\right)drdR+\left(2M-\frac{2}{R}+\frac{1}{2r_1}+\frac{1}{2r_2}\right)dR^2
O_{3,3}=\mathbb{E}(\int_0^t\int_{B_{M_{{\mathcal{P}}_n}(r)}}\vert \nabla_x{c}(z,{X}^{M_{{\mathcal{P}}_n}}_{{T}_{i}^{k}-})\vert\vert D^Z{X}^{M_{{\mathcal{P}}_n}}_{{T}_{i}^{k}-}-D^Z{X}^{M_{{\mathcal{P}}_m}}_{{T}_{i}^{k}-}\vert_{l_2}N(dz,dr))^2.
\begin{aligned}R^{\nabla^t}_{X,Y}=(\nabla_X +t A_X)(\nabla_Y + t A_Y)-(\nabla_Y + t A_Y)(\nabla_X + t A_X) &= R^\nabla_{X,Y}+ t [\nabla_X , A_Y ] - t [\nabla_Y , A_X ] + t^2 [A_X, A_Y] &= R^\nabla_{X,Y}+ t \nabla_X (A_Y) - t \nabla_Y (A_X) + t^2 [A, A]_{X,Y}&= R^\nabla_{X,Y}+ t (d^\nabla A)_{X,Y}+ t^2 [A, A]_{X,Y}\, . \end{aligned}
\mathcal{M}[j]_{m_{1},m_{2},-m_{3}}^{l_{1},l_{2},l_{3}}=(-1)^{2j-m_{3}}\left[\begin{array}[c]{ccc}l_{1}l_{2}l_{3}m_{1}m_{2}m_{3}\end{array}\right] \!\!{[j]}
\tilde{\pi}_1(a_+b_+c) =[\tilde{v}(a),\tilde{g}(b_+c)]=(-1)^{|c|}[\tilde{v}(a),[\tilde{v}(b),\tilde{f}(\bar{c})]] &=(-1)^{|c|+|a||b|}[\tilde{v}(b),[\tilde{v}(a),\tilde{f}(\bar{c})]=(-1)^{|a||b|}\tilde{\pi}_1(b_+a_+c).
\sum_{a=2}^{n-3}\sum_{d=a+1}^{n-2}(2^{n-3-a}-1)=\sum_{a=2}^{n-3}(2^{n-3-a}-1)(n-2-a)=\sum_{a=1}^{n-4}a2^{a-1}-\binom{n-3}{2}&=(n-5)2^{n-4}+1-\binom{n-3}{2}
E_{0}=-\frac{\mu_{eff}^{2}}{\left(2\pi \right)^{2}}\sum_{r\, n=1}^{\infty}\frac{1}{n^{2}}K_{2}\left(2\pi n\mu_{eff}\right)+\frac{\mu_{eff}^{3}}{2\pi}\sum_{n=1}^{\infty}\frac{1}{n}K_{3}\left(2\pi n\mu_{eff}\right).
\beta_\lambda (r):= \frac{1}{\lambda}\bigl( r-J_\lambda (r) \bigr):=\frac{1}{\lambda}\bigl( r-(I+\lambda \beta )^{-1}(r)\bigr), \beta_{\Gamma, \lambda}(r):= \frac{1}{\lambda \varrho}\bigl( r-J_{\Gamma,\lambda}(r) \bigr ):=\frac{1}{\lambda \varrho}\bigl( r- (I+\lambda \varrho \beta_\Gamma )^{-1}(r) \bigr)
\bar{\theta}_R^{a}\theta_{L}^{b}|0,p^+\rangle_U \quad\longleftrightarrow \quad{1\over \sqrt{2JN^{J+2}}}(\sum_{l=0}^JTr[\chi^a \tilde{Z}^l \psi^b Z^{J-l}]+\sum_{l=0}^J Tr[\psi^a Z^l\chi^b \tilde{Z}^{J-l}]),
P(\lambda ) = \left[{\begin{array}{*{20}{c}}1 0 0 0 1 0 0 0 1\end{array}}\right]{\lambda^2}+ \left[{\begin{array}{*{20}{c}}{- 3}0 0 0{- 1}0 0 0 3\end{array}}\right]\lambda + \left[{\begin{array}{*{20}{c}}2 0 0 0 0 0 0 0 2\end{array}}\right].
\exp\left(-it\left((k-\delta)x-\frac{(k-\delta)^2x^2}{2}\right)-t(k-\delta)^2x^2\right) &= -\frac{1}{2t(k-\delta)^2x+it(k-\delta)(1-(k-\delta)x)}\frac{d}{dx}\exp\left(-it\left((k-\delta)x-\frac{(k-\delta)^2x^2}{2}\right)-t(k-\delta)^2x^2\right)
\left( \partial_{\alpha}\partial^{\alpha}+ M^{2}\right) T^{( \beta \gamma ) \nu}= \frac{1}{4}J^{( \beta \gamma ) \nu}+ \frac{1}{4}J^{( \gamma \beta \nu )}+ \frac{1}{6 M^{2}}\partial_{\alpha}\partial^{\alpha}J^{\left( \gamma \beta \nu \right)},
D(s) = \left( e^{i \frac{\pi}{2}(-s)}+ e^{-i \frac{\pi}{2}(-s)}\right) \frac{\Gamma \left(\frac{s+1}{2}\right) \Gamma \left(1 - \frac{s}{2}\right) \Gamma \left(\frac{1}{2}-s \right)}{{{\pi}}\Gamma(3-s) 2^{s+1}}.
\begin{array}{l}\displaystyle{-Q(z)\frac{d^2 \Psi (z)}{dz^2}+\left ((j-\frac{1}{2})Q'(z)+2P(z)\right )\frac{d\Psi (z)}{dz}}\,- -\displaystyle{\left ( \frac{1}{3}j(j-\frac{1}{2})Q''(z)+2jP'(z)\right )\Psi (z)=E\Psi (z)}\,,\end{array}
\lambda_1 (\Omega ) = \min \left\{\frac{\displaystyle\int_{\Omega}| \nabla \varphi |^p dx}{\displaystyle\int_{\Omega}|x|^{-\beta p}| \varphi (x) |^p dx}: \ \varphi \in W^{1,p}_0 ( \Omega )\setminus \{0\}\right\}\,.
\alpha_n'(q^2,q,q^2)=\frac{(-1)^nq^{n^2}(1+q^{2n+1})}{(1-q^2)}\left(\sum_{0\le j\le n}q^{-j(j+1)/2}+\sum_{0\le j\le n-1}q^{-j(j+1)/2}\right),
[Q_{[n]}, \Gamma_{\{i,k\}}] =\smashoperator{\sum_{\substack{j\in [n] j\neq k\neq i}}}\,\big([\Gamma_{\{i,j\}}^2, \Gamma_{\{i,k\}}]+[\Gamma_{\{j,k\}}^{2},\Gamma_{\{i,k\}}]\big) &=\smashoperator{\sum_{\substack{j\in [n] j\neq k\neq i}}}\,\big([\Gamma_{\{i,j\}}, \Gamma_{\{j,k\}}]+[\Gamma_{\{j,k\}},\Gamma_{\{i,j\}}]\big)=0.
f^*(1\times\cdots\times 1\times x_{M_j}^{1}\times 1\times\cdots\times 1)=\sum_{l=1}^s (1\times\cdots\times 1\times y_{M_l}^{1}\times 1\times\cdots\times 1) +\xi\cdot(1\times\omega_{S^1}),
R_{n,1}=\sum_{i=1}^{N_n}E\left[\max_{1\leq k\leq d_n}\left|\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)W^{(i)}_j\right|^3\right](E[|Y_i|^{3}]+E[|G_i|^3]), R_{n,2}=\max_{1\leq k,l\leq d_n}\left|\mathfrak{C}_n(k,l)-E[Q_{n,k}(G)Q_{n,l}(G)]\right|, R_{n,3}=\max_{1\leq k\leq d_n}\sqrt{E[Q_{n,k}(G)^4]-3E[Q_{n,k}(G)^2]^2}.
\partial ( \psi \gamma_{2 1}\tilde{\psi}) = \gamma^{\mu}\partial_{\mu}( \psi \gamma_{2 1}\tilde{\psi}) = \gamma^{\mu}( \partial_{\mu}\psi \gamma_{2 1}\tilde{\psi}+ \psi \gamma_{2 1}\partial_{\mu}\tilde{\psi}) ={\cal J}.
S \bigl ( K^{\otimes ( p - 2 n )}\otimes K_{[ 0 , 1 ]}^{\otimes n}\bigr ) T = \sum_{m = n}^{[ p / 2 ]}M_{n , m}^{( p )}\, S \bigl ( K_{[ p - 2 m ]}\otimes K_{[ 0 , 1 ]}^{\otimes m}\bigr ) T
\mathbb E_{x}\left[e^{-p\int_{0}^{e(q)}\rm{\bf{1}}_{\{X_s \geq b\}}ds}\rm{\bf{1}}_{\{X_{e(q)}< y\}}\right]-\mathbb P_x\left(X_{e(q)}< y\right)=J_2(x-b;b-y),\ \y \leq b,
R_{z \bar{z}z \bar{z}}^{\prime}\equiv R_{z \bar{z}z}^{\prime}{}^{z}g_{z \bar{z}}^{\prime}= - 2 g_{z \bar{z}}g_{z \bar{z}}^{\prime}+ C_{z z z}g^{z \bar{z}}\bar{C}_{\bar{z}\bar{z}\bar{z}}.
{\cal S}_{D ( - 1 )}= \frac{1}{2 \pi \alpha^{\prime}g_{s}}\left( \int B_{( 2 )}+ \mathrm{i}\int C_{( 2 )}\right) = - \frac{\mathrm{i}}{2 \pi \alpha^{\prime}g_{s}}\gamma = - 2 \pi \mathrm{i}\tau
\begin{array}[c]{l}G(s,\bar{X}^{t,x;\bar{u}}(s),W(s,\bar{X}^{t,x;\bar{u}}(s)),W_{x}(s,\bar{X}^{t,x;\bar{u}}(s)),W_{xx}(s,\bar{X}^{t,x;\bar{u}}(s)),\bar{u}(s)) =\min\limits_{u\in U}G\left( s,\bar{X}^{t,x;\bar{u}}(s),W(s,\bar{X}^{t,x;\bar{u}}(s)),W_{x}(s,\bar{X}^{t,x;\bar{u}}(s)),W_{xx}(s,\bar{X}^{t,x;\bar{u}}(s)),u\right) .\end{array}
R_{k,l}=R_k(\lambda(l))\ \ \ \ \ \ \tilde{R}_{k,l}=\frac{B(0)\dots B(l-1)}{D(1)\dots D(l)}\frac{A_0\dots A_{k-1}}{C_1\dots C_k}R_{k,l}\ \ \ \ \ \forall k,l=0,\dots,N\ .
M=\begin{pmatrix}a_0 a_1 a_2 a_3 a_4 a_5 \cdots a_0 a_1 a_2 a_3 a_4 a_5 \cdots 0 a_0 a_1 a_2 a_3 a_4 \cdots 0 0 a_0 a_1 a_2 a_3 \cdots 0 0 0 a_0 a_1 a_2 \cdots \vdots \ddots \ddots \ddots \ddots \ddots \ddots\end{pmatrix},
\begin{array}{l}e^{+2}\nabla\psi^{2-}_{+2q}+e^{-2}\nabla\psi^{2-}_{-2q}+e^+\nabla h^{2-}_{+q}+e^-\nabla\chi^2_q+T^{+2}\psi^{2-}_{+2q}+T^{-2}\psi^{2-}_{-2q}+T^+h^{2-}_{+q}+T^-\chi^2_q begin{align*}0.3cm]=\frac12\gamma^i_{q\dot p}\Omega^{-2i}\pi^{2+}_{\dot p},\end{array}
S_{0}^{\prime}[ X ] = \int_{0}^{T}d t \int d \theta e ( \theta \chi ) \left( - \frac{\kappa_{0}}{2}\right) D X \cdot D^{2}X = \int_{0}^{T}d t \left[ - \frac{\kappa_{0}}{2}\dot{x}^{2}+ i \zeta \cdot \dot{\zeta}+ \frac{1}{T}\zeta \cdot \dot{x}\chi \right] .
\left. \begin{array}{cl}\displaystyle{}\min_{\alpha_1,\ldots,\alpha_m}\displaystyle{}\sum_{i=1}^m \log\left(1+ \frac{\gamma_i}{\alpha_i}\right), \mbox{subject to}\displaystyle{}\sum_{i=1}^m \alpha_i=1-\alpha, \alpha_i>0. \end{array}\right.
g_{\alpha\beta}\biggl|_{r\to\infty}=-\nabla_{\alpha}u\nabla_{\beta}u-(\nabla_{\alpha}u\nabla_{\beta}r+\nabla_{\alpha}r\nabla_{\beta}u)+\frac{1}{2}(m_{\alpha}m^*_{\beta}+m^*_{\alpha}m_{\beta})
\mathbb{E}\left[ y_i^* y_i|\hat{\mathbf{H}}\right]= a_c^2\left(\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)+\sum_{j=1}^{K}a_j^2\left(\lvert\hat{\phi}^{\left(i,j\right)}\rvert^2+\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right)+\sigma_n^2.
-(-1)^{\epsilon{(X)}}i_X\delta_Q\omega=-(-1)^{\epsilon{(X)}}\delta\delta_Q\alpha-d\delta_Q\alpha' , -(-1)^{\epsilon{(X)}}i_Xd\omega_1=-(-1)^{\epsilon{(X)}}\delta d\alpha_1-d\delta_Q\alpha' , di_X\omega_1=d\delta\alpha_1+d\delta_Q\alpha' .
\|u_{12}\| =\|v_{11}v_{12}+v_{12}v_{22}\| \leq \|v_{12}\|\big(\|v_{11}\|+\|v_{22}\|\big) \leq \|v_{11}^2+v_{12}v_{12}^*\|^{1/2}\big(\|v_{11}^2+v_{12}v_{12}^*\|^{1/2}+\|v_{12}^*v_{12}+v_{22}^2\|^{1/2}\big) = \|u_{11}\|^{1/2}\big(\|u_{11}\|^{1/2}+\|u_{22}\|^{1/2}\big),
=|u|_{\overline{H}}^{(0)}+\sum_{0<|\alpha|<m-n}|D^{\alpha}_{x}u|^{\gamma}_{\overline{H}}+{\displaystyle\sum\limits_{j=0}^{j\leq n}}{\displaystyle\sum\limits_{|\alpha|=m-j}}|x_{N}^{n-j}D^{\alpha}u|_{\omega\gamma,\overline{H}}^{(\gamma)}.
S_{c l}= \frac{1}{2 \pi}\int d^{2}x \sqrt{- g}[ e^{- 2 \phi}( R + 4 ( \nabla \phi )^{2}+ 4 \lambda^{2}) - \frac{1}{2}\sum_{i = 1}^{N}( \nabla f_{i})^{2}]