harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde{T}^{L,\Xi,\widetilde{\mathbf{r}}}_{F,G,N}(g_1,\dots,g_{d}) := \frac{1}{N^{h-m}}\int\limits_{\mathbf{x}\in \mathbb{R}^h}\Big(\prod\limits_{j=1}^{d}g_j(\xi_j(\mathbf{x})+ \widetilde{\mathbf{r}_j})\Big)F(\mathbf{x})G(L\mathbf{x})\, d\mathbf{x}.
s = \frac{K}{2}+ \frac{K^2}{2(K+1)^2}\frac{\lambda}{\mu}+ O((\lambda/\mu)^2 ) , P_b = \frac{2}{K+1}+ \frac{2K}{(K+1)^3}\frac{\lambda}{\mu}+ O((\lambda/\mu)^2 ).
(\varphi \otimes \psi_n) \big( J_n(p_\xi) \cdot J_n(a) \cdot J_n(p_\xi) \big)= (\varphi \otimes \psi_n) \big( J_n(p_\xi a p_\xi) \big)= \varphi (p_\xi a p_\xi) = \omega_\xi(a) \,\varphi(p_\xi),
\alpha_1 = \sum_{m=2}^{M}\frac{p_m}{m}\binom{2n-2}{m-2},\alpha_2 = \sum_{m=2}^{M}\frac{p_m}{m}\binom{n-2}{m-2}, \alpha_3 = \sum_{m=2}^{M}\frac{p_m}{m}\left(\binom{2n-2}{m-1}- \binom{n-2}{m-1}\right).
{\bar{{\cal G}}}_{{\rm (A)}}^a\left[ \Delta (x)\cdot \Gamma \right]=0\,\,,\,\,\,\,\,{\cal G}_{{\rm (A)}}^a\left[ \Delta (x)\cdot \Gamma \right]=0\,\,,\;\;\;{\rm A}={\rm I},\,{\rm II}\,\,,
f\big(\frac{\alpha}{Mr}+iy\big)=\sum_{m=1}^\infty a(m) e^{2\pi im \frac{u}{r}}e^{2\pi im \frac{a}{M}+iy}= \frac{1}{\varphi(r)}\sum_{\chi\mod{r}}\overline{\chi}(u)\sum_{m=1}^\infty a(m) c_{\chi}(m) e^{2\pi im\left(\frac{a}{M}+iy\right)}.
\sum_{k\in\mathcal{R}}\left( 1-\left\vert z_{j}\right\vert^{2}\right)^{2}\mu\left( k\right) \leq\left( 1-R^{2}\right)^{2}\sum_{i=1}^{N}\sum_{z_{k}\in B_{i}:k\in\mathcal{R}}\mu\left( k\right) & \leq C\left( 1-R^{2}\right)^{2}N & \leq C\left( 1-R^{2}\right) <C\varepsilon,
-\frac{g_t}{g-g_x}= \frac{(\sum_{i=1}^Nc_i)g^{N+1}+\sum_{n=1}^N\epsilon^{(n-1)n}g^{N-n+1}d_ne^{(n-1)x}(e^xD_{N-n,t}^2-2D_{N-n+1,t}^0)+ o(\epsilon^{N(N+1)})}{g^{N+1}+\sum_{n=1}^N\epsilon^{(n-1)n}g^{N-n+1}d_ne^{nx}D_{N-n}^2 +o(\epsilon^{N(N+1)})}.
\phi(t,x) - x = \int_0^t u(s,\phi(s,x)) ds = u_0(x) t + \int_0^t \int_0^s \partial_{\tau}(u(\tau,\phi(\tau,x))) d\tau ds = u_0(x) t + \int_0^t (t-\tau) \partial_{\tau}( u(\tau, \phi(\tau,x) ) ) d\tau.
\frac{H_m^{(\delta )}(w, \gamma_m)}{H_m^{(\delta )}(y, \gamma_m)}H_0^{(\delta )}(y; A) = \frac{\Im\, \phi_m(w)}{\pi |\phi_m(w)-U_m|^2}\int_{-1}^{1}\frac{\Im\, \phi (y)}{\Im\, \phi_m (y)}\cdot \frac{|\phi_m (y) -U_m|^2}{|\phi(y)-x|^2}dx & +O(\epsilon^3 ).
1 = \int d^{2}{\bf r}\, | \Psi ({\bf r}) |^{2}= A_{l}^{2}\, 2 \pi \kappa^{- 2}\, \left\{{\mathcal K}_{i \Theta}( \kappa a ) + \left( \frac{\kappa}{\tilde{k}}\right)^{2}\, \left[ \frac{K_{i \Theta}( \kappa a )}{J_{l}( \tilde{k}a )}\right]^{2}\,{\mathcal J_{l}}( \tilde{k}a ) \right\}\; ,
L_{m}( \lambda ) = 1 + \Delta{\cal L}\left( m \frac{R}{N}, \lambda \right) + O ( \Delta^{2}) \quad , \quad \bar{L}_{m}( \lambda^{- 1}) = 1 + \Delta \bar{\cal L}\left( m \frac{R}{N}, \lambda^{- 1}\right) + O ( \Delta^{2}) \, ,
F_{R}(q;k,\nu)=\frac{\sqrt{2\pi}k\nu^{\nu/2}}{\Gamma(\nu/2)2^{(\nu/2-1)}}\int_{0}^{\infty}s^{\nu-1}\varphi(\sqrt{\nu}s)[\int_{-\infty}^{\infty}\varphi(z)[\Phi(z+qs)-\Phi(z)]^{k-1}dz]ds
\begin{aligned}\sum_{2\le|a|\le N}(f,e^{itH}\left(\frac{x}{t}-D\right)^2\tilde P^a e^{-itH}f) &\sim_{\epsilon_M}\sum_{2\le|a|\le N}\left[-\frac{4}{t^2}\int_0^ts(f,e^{isH}T_a\tilde P^ae^{-isH}f)ds+2(f,e^{itH}T_a\tilde P^ae^{-itH}f)\right] &+2\sum_{2\le|a|\le N}(f,e^{itH}H_0^a\tilde P^ae^{-itH}f)\end{aligned}
u_i =\alpha_i(x_i,x^c_i,x_j,x^c_j,w)+\beta_i(x_i,x^c_i,x_j,x^c_j,w)v, \dot{x}^c_i =\zeta_i(x_i,x^c_i,x_j,x^c_j,w)+\gamma_i(x_i,x^c_i,w)v,\,j\in\mathcal{N}_i
\begin{cases}u_t(x,t) - d\Delta u(x,t) = f, (x,t)\in \Omega_{T}, \nabla u(x,t) \cdot \nu(x) = 0, (x,t)\in \partial\Omega\times(0,T), u(x,0) = u_0(x), x\in\Omega,\end{cases}
M=\begin{pmatrix}\Gamma_{1,1}\Gamma_{1,0}\cdots \Gamma_{1,2-N}\Gamma_{2,1}\Gamma_{2,0}\cdots \Gamma_{2,2-N}\vdots \vdots \cdots \vdots \Gamma_{N,1}\Gamma_{N,0}\cdots \Gamma_{N,2-N}\end{pmatrix}.
S_\omega(\phi) = E(\phi)+\frac{\omega}{2}M(\phi) = \frac{1}{2}\|\nabla \phi\|^2_{L^2}-\frac{1}{p+1}\|\phi\|^{p+1}_{L^{p+1}}+ \frac{1}{q+1}\|\phi\|^{q+1}_{L^{q+1}}+ \frac{\omega}{2}\|\phi\|^2_{L^2}
e=e_1e_2e_3 \le 2\log_{q}(5g + 1+n)+e_1e_2 \le 2\log_{q}(5g + 1+n)+2\log_{q}\max\{n+1,2g + 1\}+e_1 \le 2\log_{q}(5g + 1+n)+2\log_{q}\max\{n+1,2g + 1\}+\log_q(n)+2.
S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1},k\right) =\frac{1}{k!}\sum_{j=0}^{k}\left( -1\right)^{j}\binom{k}{j}\left( a_{1}\left( k-j\right)+b_{1}\right)^{p_{1}}\left( a_{2}\left( k-j\right) +b_{2}\right)^{p_{2}}.
\begin{array}{rcl}(1+\lambda-\lambda^{1/2+\alpha}/2)^{-1}= (1+\lambda)^{-1}(1 -(1+\lambda)^{-1}\lambda^{1/2+\alpha}/2)^{-1}& \leq (1+\lambda)^{-1}(1 + 2(1+\lambda)^{-1}\lambda^{1/2+\alpha}/2) &\leq (1+\lambda)^{-1}+ (1+\lambda)^{-3/2+\alpha}\end{array}
y_1=\frac{1}{\sqrt{\mu}}x_3, \, y_2=x_2,\, y_3=x_1-x_2; &[y_1,y_2]=\frac{1-w}{\sqrt{\mu}}y_2, \,[y_2,y_3]=0, \,[y_3,y_1]=-\frac{2w}{\sqrt{\mu}}y_2-\frac{1+w}{\sqrt{\mu}}y_3.
I(U_*)= I(u_0)+k\left[A-\frac{B_1k^{N-2}}{r^{N-2}\lambda^{N-2}}+\frac{B_2u_0(r)}{\lambda^{\frac{N-2}{2}}}+O(\frac{1}{\lambda^{\frac{N-2}{2}(1+\delta)}})\right]
\begin{array}{ll}\int\limits_0^t \int\limits_{\mathbb{R}^3}- v \cdot\vec{\varphi}_2 - \frac{1}{k_1\varrho}\nabla\xi\cdot\vec{\varphi}_2\,\mathrm{d}x\mathrm{d}s= \tau^2\int\limits_0^t \int\limits_{\mathbb{R}^3}v_t \cdot \vec{\varphi}_2 + k_1 v\cdot\nabla v \cdot\vec{\varphi}_2\,\mathrm{d}x\mathrm{d}s.\end{array}
\frac{\partial}{\partial \bar{T}_{k}}d \Omega_{0}^{*}= \frac{\partial}{\partial X}d \Omega_{k}^{*}, \; \; \; \frac{\partial}{\partial T_{k}}d \Omega_{0}^{*}= \frac{\partial}{\partial X}d \tilde{\Omega}_{k}^{*}.
\ker(T_3)=\begin{cases}H_3/H_3^\prime\simeq\langle xy,s_2\rangle\simeq C_3\times C_3 a=w=z=0 a=1,\w=-1, G^\prime/H_3^\prime\simeq\langle s_2\rangle\simeq C_3 .\end{cases}
\tilde{\Theta}_-UA\varphi =\tilde{\Theta}_-US(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=-\tilde{\Theta}_-SU(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=\tilde{\Theta}_-S(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)U\varphi &=\tilde{\Theta}_-AU\varphi=A\tilde{\Theta}_-U\varphi
g(\chi_1\chi_2) =\sum_{a_1\bmod{q_1}}\sum_{a_2\bmod{q_2}}\chi_1(a_1)\chi_2(a_2)e_{q_1}(a_1\bar{q}_2)e_{q_2}(a_2\bar{q}_1) &=\chi_1(q_2)\chi_2(q_1)g(\chi_1)g(\chi_2).
\left[ \; \; X^{m}, \; \theta^{\alpha}, \; \bar{\theta}^{\dot{\alpha}}, \; p_{\alpha}, \; \bar{p}_{\dot{\alpha}}, \; \rho , \; \Gamma^{+ j}, \Gamma^{- j}, \; X^{+ j}, \; X^{- j}\; \; \right] \nonumber
\sum_{j=2l,}^k{k \choose j}\frac{j!2^{k-j}}{(j/2-l)!(j/2+l+1)!}= \frac{2^{k-2l}\Gamma(k+1)\Gamma(k+3/2)}{\Gamma(k-2l+1)\Gamma(k/2+2+l)\Gamma(k/2+3/2+l)}.
P(t, S) =K - S, S = \mathcal{B}(t), 0\leq t <T, \frac{\partial P}{\partial S}(t, S) =-1, S = \mathcal{B}(t), 0\leq t <T, P(T, S) = \max\{0, K-S \}, \lim_{S \rightarrow \infty}P(t, S) = 0,
u_{i}^{0}=\sum_{n=1}^{\infty}\alpha_{in}w_{n}\,,\qquad\quad\alpha_{in}=\langle u_{i}^{0},w_n\rangle \,,\|u_i^0\|^2_{D(\sqrt{L})}:=\sum_{n=1}^{\infty}\alpha_{in}^2\lambda_n\,, & u_{i}^{1}=\sum_{n=1}^{\infty}\rho_{in}w_{n}\,,\qquad\quad\rho_{in}=\langle u_{i}^{1},w_n\rangle \,,\|u_i^1\|^2_{H}:=\sum_{n=1}^{\infty}\rho_{in}^2\,.
8\pi-m_{u^2}(r,t) =8\pi-m_{\tilde{Q}_{\mu_2}}-m_{\varepsilon_2}\lesssim\frac{\lambda_2^2e^{-C_2\frac{r^2}{2}}}{r^2}+m_{\varepsilon_2}\Big(\frac{r}{\lambda_2},t\Big) &\lesssim\frac{\lambda_2^2e^{-C_2\frac{r^2}{2}}}{r^2}+\frac{\lambda_2^2}{(\lambda_2^2+r^2)t|\log t|},
\theta_{m_1}(x_0) = \frac{1}{3}, \theta_{m_1}(x_1) = \frac{4}{13},\theta_{m_1}(x_2) = \frac{2}{7}, \theta_{m_1}(x_3) = \frac{4}{15}, \theta_{m_1}(x_4) = \frac{1}{4}, \theta_{m_2}(x_0) = \frac{1}{5}, \theta_{m_2}(x_1) = \frac{4}{21}, \theta_{m_2}(x_2) = \frac{2}{11},\theta_{m_2}(x_3) = \frac{4}{23},\theta_{m_2}(x_4) = \frac{1}{6}.
m =\overline{\gamma}_{0}\sum_{n=1}^{\infty}\prod_{i=1}^{n}\Vert\gamma_{i}\Vert_{L^{1}}&=\overline{\gamma}_{0}\left(\Vert h\Vert_{L^{1}}+\Vert h\Vert_{L^{1}}\Vert g\Vert_{L^{1}}+\Vert h\Vert_{L^{1}}\Vert g\Vert_{L^{1}}\Vert h\Vert_{L^{1}}+\cdots\right) &=\frac{1+\Vert h\Vert_{L^{1}}}{1-\Vert h\Vert_{L^{1}}\Vert g\Vert_{L^{1}}}.
p_{2}-p_{1}=\frac{c_{0}^{2}\rho_{1}\rho_{2}(\rho_{2}-\rho_{1})}{[\rho_{2}-s(\rho_{2}-\rho_{1})]^{2}}=\frac{c_{0}^{2}(v_{1}-v_{2})}{[v_{1}-s(v_{1}-v_{2})]^{2}}.
[\begin{matrix}-26&-3&-6&2&2&-1&0&0\\ 0&-2&-4&1&1&0&0&0\\ -3&1&5&-1&-1&0&0&0\\ -4&1&2&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{matrix}]
\Psi = \exp \left\{\frac{\theta}{2}\Gamma_{\underline{{{r}}}\underline{{{\theta}}}}- A -{\frac{1}{4}}B + \int^{r}\mathrm{d}\rho \mathrm{e}^{- B ( \rho ) / 2}\left({\frac{1}{2 a}}- \mathrm{i}\Gamma_{7}{\frac{e}{2}}\Gamma_{\underline{{{r}}}\underline{{{\theta}}}}A_{\theta}( \rho ) \right) \right\}\psi ~ ,
\lambda_{app}^{ND,(n,k)}= i R_1 + h^{\frac 23}|a'_n| e^{i\pi/3}+ h (2k-1) \frac{e^{-i\pi/4}}{\sqrt{2 R_1}}+ h^{\frac 43}\frac{e^{\pi i/6}}{2|a'_n| R_1}+ \mathcal O(h^{\frac 53})\, .
\sum_{x \in [-p^{\alpha-1}, p^{\alpha-1}]}\min\left(f(k), \sum_{i=1}^{r_k}\psi_{y_i}(x) \right) = r_k \left( \frac{3p^{2\alpha-2}-2p^{\alpha-1}-1}{4}\right) + \frac{p^\alpha+p^{\alpha-1}}{2}&+ r_k - \frac{p+1}{2}+ 2b \geq r_k \left( \frac{3p^{2\alpha-2}-2p^{\alpha-1}-1}{4}\right) + \frac{p^\alpha+p^{\alpha-1}}{2}&+ r_k - \frac{p+1}{2},
\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^{r_3}}{n_1 \choose d_1+1}{n_2 \choose d_2+3},
\gamma(c) =\mu(c)\: u &=q(c,c\wedge m_\mu)\iota(c\wedge m_\mu,m_\mu)\: u \ref{rmkinvcone}&=q(c,c\wedge m_\gamma)\iota(c\wedge m_\gamma,m_\gamma) \gamma(m_\mu) m_\gamma=m_\mu.
\alpha_i = \begin{cases}\delta^j_i, Q_i = u_j -\delta^j_i, Q_i = w_j \end{cases}\mu_{2N}(Z(\eta)) =\begin{cases}\prod_{i=1}^n \frac{1 + \alpha_{2i-1}}{2N}, r(\eta) = v \prod_{i=0}^n \frac{1 + \alpha_{2i}}{2N}, r(\eta) \not= v. \end{cases}
\sum_{w\in C_n/C_{n-1}}\!\!\!w\cdot \frac{\prod_{1\le i\le 2n+1}\vartheta(z_n-y_i) \prod_{1\le i\le n}\vartheta(Y+z_i) \prod_{1\le i<n}\vartheta(Y-z_i)}{\vartheta(2z_n)\prod_{1\le i<n}\vartheta(z_n+z_i)\vartheta(z_n-z_i)}\hphantom{\sum_{w\in C_n/C_{n-1}}\!\!\!}{}= \prod_{1\le i\le 2n+1}\vartheta(Y-y_i),
\texttt{LHS}\eqref{id:9}\simeq (b-a)^{\gamma}\int_{b-a}^{b-a + a/4}x^{-\gamma}\, dx\simeq(b-a)^{\gamma}\begin{cases}b^{-\gamma + 1}\gamma < 1, \log \big( 1 + a/(b-a) \big) \gamma = 1.\end{cases}
4\Bigl(\sum_{k=1}^{n}x_{k}^{\:\left(t-1\right)}\Bigr)\Bigl(\sum_{r=1}^{\nu}y_{r}^{\:\left(t-1\right)}\Bigr)\leq\Bigl(\sum_{k=1}^{n}x_{k}^{\:\left(t-1\right)}+\sum_{r=1}^{\nu}y_{r}^{\:\left(t-1\right)}\Bigr)^{2}
-\frac{1}{2\pi i}\int_{0}^{y_{m-1}}\Psi_-(\xi)\sigma_+\Psi_-(\xi)^{-1}d\xi =\partial_{s_m}\Psi_1, -\frac{1}{2\pi i}\int_{0}^{y_{m-1}}\Psi_-(\xi)\sigma_+\Psi_-(\xi)^{-1}\xi d\xi = \partial_{s_m}\Psi_2-\partial_{s_m}\Psi_1\ \Psi_1.
\Vert \mathcal{A}_{D,\varepsilon}^{1/2}e^{-\mathcal{A}_{D,\varepsilon}t}\Vert_{L_2(\mathcal{O})\rightarrow L_2(\mathcal{O})}\leqslant \sup_{\mu \geqslant c_*}\mu^{1/2}e^{-\mu t}&\leqslant e^{-c_{*}t/2}\sup_{\mu \geqslant 0}\mu^{1/2}e^{-\mu t/2}\leqslant t^{-1/2}e^{-c_{*}t/2}.
\mathcal{R}_1^- = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{c \geq \frac{C}{m}}\frac{S(-n, p; c)}{c}H_{m, n}^- \Bigl ( \frac{4 \pi \sqrt{n p}}{c}\Bigr )
\varphi(w,z)=[\begin{matrix}u^{*}&v^{*}\end{matrix}]K_{p,q}[\begin{matrix}x\\y\end{matrix}]+j[\begin{matrix}u&-v\end{matrix}]K_{p,q}[\begin{matrix}y\\x\end{matrix}]=\varphi_{1}(w,z)+j\varphi_{2}(w,z),K_{p,q}=diag(I_{p,q},I_{p,q})
0 \, \leq 1-\frac{|A_1\cap A_2|}{\, |A_1|^{1/2}|A_2|^{1/2}\,}= 1 - \frac{\, |A_1|+|A_2|-|A_1\bigtriangleup A_2|\,}{2|A_1|^{1/2}|A_2|^{1/2}}& \, \leq \frac{|A_1\bigtriangleup A_2|}{\, 2|A_1|^{1/2}|A_2|^{1/2}\,}\leq \frac{1}{\, 4\,}\left( \frac{\, |A_1\bigtriangleup A_2|\,}{|A_1|}+\frac{\, |A_1\bigtriangleup A_2|\,}{|A_2|}\right)
F_{\widehat{K}^\mathbb{R}_{\mathrm{u-cpt}}|K}(\hat{k})=1-e^{-\frac{1+N\bar{\gamma}\hat{k}}{1+N\bar{\gamma}K}},\ \f_{\widehat{K}^\mathbb{R}_{\mathrm{u-cpt}}|K}(\hat{k})=\frac{N\bar{\gamma}}{1+N\bar{\gamma}K}e^{-\frac{1+N\bar{\gamma}\hat{k}}{1+N\bar{\gamma}K}},\ \ \hat{k}\ge -1/(N\bar{\gamma}).
\psi_{{\bf k}s}^{\pm}= \left[ - \gamma^{0}\left({\frac{d}{d \tau}}+{\frac{1}{2 \tau}}\right) - i \gamma_{\bf{{\perp}}}\cdot{\bf{k_{\perp}}}- i \gamma^{3}\pi_{\eta}+ m \right] \chi_{s}{\frac{f_{{\bf k}s}^{\pm}}{\sqrt \tau}}\, .
\frac{1}{N}\sum_{\ell = 1}^{j - k^* - 1}\ell X_{j - \ell}(t) \leq \sum_{\ell = 1}^{\infty}\alpha_{\ell}(t) \ell 2^{\ell}\bigg( \frac{\mu}{s}\bigg)^{\ell (\ell - 1)/6 k_N}e^{-s \ell (t - \gamma_j)}.
Z = \int D s_{i}D \phi D A_{i}\exp \left\{- \frac{1}{2}[ A_{i}G^{- 1}A_{i}+ A_{i}^{\phi , s}G^{- 1}A_{i}^{\phi , s}] - i \kappa [ ( s_{i}- \partial_{i}\phi ) \epsilon_{i j}A_{j}+ \epsilon_{i j}\partial_{i}s_{j}] \right\}
\delta S=\frac{1}{3}\int d^2xd_{ijk}(\partial_+\varphi^i\partial_+\varphi^j\partial_+\varphi^k\partial_-\lambda^{++}+\partial_-\varphi^i\partial_-\varphi^j\partial_-\varphi^k\partial_+\lambda^{--}).
{\cal L}= \overline{\psi}_a({\ooalign{\hfil/\hfil\crcr$\partial$}}-m)\psi_a+\overline{\psi}_b({\ooalign{\hfil/\hfil\crcr$\partial$}}-m)\psi_b +\frac{1}{2}gA_{\mu}j_a^{\mu}+\frac{1}{2}gB_{\mu}j_b^{\mu}-\frac{1}{2}gA_{\mu}B^{\mu}.
\overline{\Pi}^{\sharp}=\Pi^{\sharp}\circ\big(\operatorname{Id}-B^{\sharp}\circ \Pi^{\sharp}\big)^{-1}, \overline{\Pi}_{0,2}^{\sharp}=\Pi_{0,2}^{\sharp}\circ\big(\operatorname{Id}-B_{0,2}^{\sharp}\circ\Pi_{0,2}^{\sharp}\big)^{-1}.
y_{i+1}= \frac{q^{\alpha^i}}{2}(x_i + 1) + \frac{q^{\alpha^i}}{2}\left( C q^{|\mathbf A_k|_{0}^{i-1}}(q-1) 2^{k+1-i}\right) &= x_{i+1}+ C q^{|\mathbf A_k|_{0}^{i}}(q-1) 2^{k-i},
\frac{1}{4}\gamma_{a b}\psi_{A}\overline{{{\psi}}}_{B}\gamma^{a b}\psi_{C}\epsilon^{A C}- \frac{1}{2}\gamma_{a}\psi_{A}\overline{{{\psi}}}_{B}\gamma^{a}\psi_{C}\epsilon^{A C}+ 3 \psi_{C}\overline{{{\psi}}}_{B}\psi_{A}\epsilon^{B C}= 0
\varphi_{2n+1}(x;q^{-1})\mathcal{E}_{q}(q^{-1/2}x)-q^{n+1/4}\varphi_{2n+2}(q^{-1/2}x;q^{-1})\mathcal{E}_{q}(x)=q^{-n^{2}}\mathcal{E}_{q}(q^{n+1/2}x).
\phi(\xi) = \lambda^{-1}x\sim A_{8}e^{\tau}\sim A_{9}(\xi-\xi_{-})^{p}{\rm{as}}\xi \searrow \xi_{-}+0, \psi(\xi) =\lambda^{-1}\sim A_{10}e^{\gamma \tau}\sim A_{11}(\xi-\xi_{-})^{p-1}{\rm{as}}\xi \searrow \xi_{-}+0
(\alpha_{2 \underline{d}}\circ \varphi_{\widetilde{\underline{d}},\underline{d}})(s)=(\alpha_{2 \widetilde{\underline{d}}}\circ \varphi_{\underline{d},\widetilde{\underline{d}}}\circ \varphi_{\widetilde{\underline{d}},\underline{d}})(s)=(\alpha_{2 \widetilde{\underline{d}}}\circ 0)(s)=0\in V_{X_2}(-iA-lB),
\kappa(\epsilon) = \mu(I-e^{-\alpha}\tilde{P})^{-1}r + \epsilon e^{-2\alpha}\mu \tilde{P}(I-e^{-\alpha}\tilde{P})^{-2}\tilde{P}^{(1)}(I-e^{-\alpha}\tilde{P})^{-1}r + O(\epsilon^2(\log(1/\epsilon))^w)
F_N(e^{itQ}f)= a^{1/2}N^{-1/4}(\sum_{n=0}^{N/2-1}e^{itnaN^{-1/2}}f(naN^{-1/2})u(n)+ & \sum_{n=N/2}^{N-1}e^{it(n-N)aN^{-1/2}}f((n-N)aN^{-1/2})u(n))
0= [[\beta_{-1}, \beta_{-m+1}],\beta_n] +(m+n)(m-2) \beta_{n-m}+\sum_{j=1}^{m-3}(m-2)(j+1) \alpha_{n-1-j}\alpha_{-m+1+j}&+(m-2) \alpha_{n-1}\alpha_{-m+1}+(m-2)(m-1) \alpha_{n-m+1}\alpha_{-1}.
\begin{aligned}\int_{B^c_{r_{kk_0}}}P_{r_{kk_0}}\tilde{g}dy=\int_{B_{r_{(k-1)k_0}}\backslash B_{r_{kk_0}}}P_{r_{kk_0}}\tilde{g}dy+\int_{B^c_{r_{(k-1)k_0}}}P_{r_{kk_0}}\tilde{g}dy.\end{aligned}
\frac{\theta_\Gamma (c_{\mathbf{O}_{1m}})^2\theta_\Gamma(c_{\mathbf{O}_{2h}})^2}{\theta_\Gamma (c_{\mathbf{O}_{2m}})^2\theta_\Gamma(c_{\mathbf{O}_{1h}})^2}=\frac{\lambda_h-\lambda_m}{\lambda_h-\lambda_l}\frac{\lambda_k-\lambda_m}{\lambda_k-\lambda_l}
P(n,z) = - \pi z\int_0^1{y^n dy}+{\ln (2\pi z)}\int_0^1{y^{n - 1}dy}+ \int_0^1{y^{n - 1}\ln y}\,dy &\,- \int_0^1 y^{n - 1}\ln\!\big({1 - e^{- 2\pi yz}}\big)dy &= - \frac{\pi z}{{n + 1}}+ \frac{{\ln (2\pi z)}}n - \frac1{n^2}- \int_0^1{y^{n - 1}\ln\!\big({1 - e^{- 2\pi yz}}\big)dy}.
\left\vert \Phi(P^+) \right\rangle_f=\int_0^{P^+}dk_1 dk_2 \delta(k_1+k_2-P^+)\phi_{bf}(k_1,k_2){1 \over N}{\rm tr}[a^\dagger(k_1), b^\dagger(k_2)]\left\vert 0 \right\rangle ,
F_{q}^{(k)}(s,q^{\prime}) = \displaystyle \sum_{n=0}^{\infty}\frac{\left( - \frac{1}{1-q^{\prime}}\right)_{n}\left( \mp (1-q^{\prime}) \sigma_{\epsilon}\right)^{n}}{Q_{n+1}(2-q) \;\; n!}\frac{(-1)^{k}\Gamma(n+k+1)}{s^{n+k+1}}.
\mathcal G_{\mathcal Y}g(y) = \big(\mathcal G_{\mathcal X}(g\circ(-\log))\big)(e^{-y}) begin{align*}1em] = g\big(-\log(pe^{-y})\big) - g(y) + \alpha(e^{-y})\cdot\big(-\tfrac1xg'(-\log x)\big)\big|_{x=e^{-y}}begin{align*}1em] = g(y+\log\tfrac1p) - g(y) - \beta(y)g'(y).
\lim_{m \to \infty}\left| \prod_{0 \leq j \leq m-1}\left( 1+ e^{2 \pi i \frac{j + \theta /2}{m}}+ \frac{x + iy}{m}\right) \right| = 2 e^{x/2}\left| \sin \left( \frac{\pi \theta_1}{2}+ \frac{y}{2}+ \frac{x i}{2}\right) \right|.
E ={\frac{R^{2}}{2 \pi \alpha^{\prime}}}e \int_{0}^{2 \pi}d \sigma = 4{\frac{R^{2}}{2 \pi \alpha^{\prime}}}\int_{0}^{\theta_{0}}d \theta \left( 1 - \omega^{2}\sin^{2}\theta \right)^{- 1 / 2}\ ,
\zeta_{\mathcal{A}}(s_{d},\ldots,s_{1}) \mapsto \sum_{d'=0}^{d}(-1)^{s_{d'+1}+\ldots+s_{d}}\zeta^{}(s_{d'+1},\ldots,s_{d})\zeta^{}(s_{d'},\ldots,s_{1}) \mod \zeta^{}(2)
\sum_{i=1}^{N_n}\left(\max_{1\leq k\leq d_n}\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)^2\right)^{3/2}\leq\left(\max_{1\leq i\leq N_n}\sqrt{\Lambda_i}\right)\sum_{i=1}^{N_n}\Lambda_i,
\left\{\frac{\left( U_{n}- 2 + U_{n}^{- 1}\right)}{T^{2}}+ E^{2}\frac{\left( U_{n}+ 4 + U_{n}^{- 1}\right)}6 \right\}\tilde{\varphi}_{n}\left( \mathbf{p}\right) ={}_{{}_{\! \! \! \! \! \! \! c}}\tilde{j}_{n}\left( \mathbf{p}\right) ,
\inf\{\int_{X}(M_{\mathcal{T}}\phi)^{p}d\mu :\phi\geq0\text{measurable,}\int_{X}\phi d\mu=f\int_{X}\phi^{p}d\mu=F\}= & =f^{p}+\frac{N^{p}-1}{N^{p}-N}(F-f^{p}).
{\widetilde{D}}_{i}^{X}=({\widetilde{D}}_{i1},{\mathbf{0}}_{q},{\widetilde{D}}_{i2},{\mathbf{0}}_{q},\cdots,{\widetilde{D}}_{ik_{i}},{\mathbf{0}}_{q},{\mathbf{0}}_{r})\,\,{D}_{i}^{Y}=({D}_{i1}^{\ast},{\mathbf{0}}_{q},{D}_{i2}^{\ast},{\mathbf{0}}_{q},\cdots,{D}_{ik_{i}}^{\ast},{\mathbf{0}}_{q},{\mathbf{0}}_{r})\,.
\sum_{i \in M_P - i^*}\sum_{j \in N_i}a_{ij}x^*_{ij}+ \sum_{ij \in \bar P, i \neq i^*}(b-s) x^*_{ij}+ \sum_{j \in N_{i^*}}a_{i^* j}x^*_{i^* j}> b + (|M_P - M_0| - 2) (b-s) = s + (|\bar P| - 1) (b-s).
\hat H_{free}=\int\left\{\pi^+(\stackrel{\rightharpoonup}{x})\pi(x)+(\bigtriangledown\varphi^+(\stackrel{\rightharpoonup}{x}))\cdot(\bigtriangledown\varphi(\stackrel{\rightharpoonup}{x}))\right\}d^3\stackrel{\rightharpoonup}{x}
S_{DBI}= - \frac{T_{p}}{\kappa}\int_{V_{p+1}}d^{p+1}\xi~{\rm e}^{-\frac{\kappa\,(3-p)}{2\sqrt{2}}\,\phi}\sqrt{- \det\left[ \, g + \sqrt{2}\,\kappa\,A+{\hat F}\,{\rm e}^{-\frac{\kappa}{\sqrt{2}}\,\phi}\,\right]}~~.
\widehat{S}_{p}\equiv\sum_{J \subset \{p+1, \ldots , r\}}\alpha^{r-p}c(J) \prod_{i=\nu_{p}}^{l-|J|-1}(q^{p-1}+\delta[p-1]\widehat{V}_{i}){}\times\left\{\beta q^{p-1}\widehat{X}_{l-|J|}+\alpha(q^{p-1}+\delta[p-1]\widehat{V}_{l-|J|})\right\}(\widehat{T}_{l-|J|-1}\cdots \widehat{T}_{\nu_{p}})\widehat{Q}_{J}.
\begin{array}{r c l}{d{\tilde{s}}^{2}}&{=}&{( H^{2}+ \theta^{2}/ ( 2 \pi )^{2})^{1 / 2}[ H^{- 1 / 2}d s_{1 , 7}^{2}- H^{1 / 2}d \omega d{\bar{\omega}}] ,}\\{}&{}&{}\\{{\tilde{\lambda}}}&{=}&{- 2 \pi i / \log{\omega}.}\\ \end{array}
\int_0^1 \left( \int_0^{\frac{w}{t}}z e^{-z}e^{z t}dz \right) dt = \int_0^1 \left(\int_0^w \frac{y}{t}e^{- \frac{y}{t}}e^{y}\frac{dy}{t}\right) dt , & = \int_0^w y e^{y}\left(\int_0^1 e^{-\frac{y}{t}}\frac{dt}{t^2}\right) dy = \int_0^w y e^{y}\frac{e^{-y}}{y}dy = w.
(\phi_*\circ d)\ (\textbf{r}_0) = \phi_*(\textbf{a}_1 + \ldots + \textbf{a}_m) = \sum_{i = 1}^m k_i \cdot \textbf{a}_{\sigma(i)}(d \circ \phi_*)\ (\textbf{r}_0) = d(\kappa\cdot \textbf{r}_0 + \sum_{i = 1}^m \mu_i \cdot \textbf{r}_{\sigma(i)}) = \sum_{i = 1}^m (\kappa + p_i\mu_i) \cdot \textbf{a}_{\sigma(i)}
\lim_{\lambda \uparrow \Lambda}\varphi (\lambda )=\left\{\lim_{\lambda\uparrow \Lambda}\psi (\lambda )\ |\ \psi :\mathfrak{L}\rightarrow \mathcal{\mathbb{U}}_{n-1}\ \forall \lambda \in \mathfrak{L},\ \psi(\lambda )\in \varphi (\lambda )\right\}.
\omega_{4}^{3}= i \{\frac{1}{2}d \ln ( \frac{d \overline{{g}}_{2}}{d \overline{{z}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{g}}_{2}- \overline{{g}}_{2}d g_{2})}{( 1 + g_{2}\overline{{g}}_{2})}\},
\left(H^{[n]}\left(c_1^{[n]}\right)\right)^2>\frac{(d_1\dots d_{n-1}d_n)^2}{n^2 d_n^2}\geq\frac{d_1\cdots d_n n^3 d_n^2}{n^2 d_n^2}=n\cdot D.
\widehat{\ell}_{\alpha,\beta}[y](x) =\frac{1}{\widehat{w}_{\alpha,\beta}(x)}\left( -\left( \frac{(1-x)^{\alpha+1}(1+x)^{\beta+1}}{(x-b)^{2}}y^{\prime}(x)\right)^{\prime}\right. & +\left. \frac{2a(x-c)(bx-1)(1-x)^{\alpha}(1+x)^{\beta}}{(x-b)^{3}}y(x)\right) ,
i_p m\!+\!n_1 j_p\!-\!m n_1=\ell_p=\sum_w u_w= m\!\!\left(\sum_{w,h}k_{h,w}i_h+r_V a\right)\!+\!n_1\!\!\left(\sum_{w,h}k_{h,w}j_h+r_V b\right)\!-\!mn_1\!\!\left(\sum_{w,h}k_{h,w}+r_V n_2\right)\!.
\frac{\partial L_{d}D1(p\|q)}{\partial q_{j}}=\left\{\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]\frac{\partial A}{\partial q_{j}}\right\}&-\left\{\left[\frac{a-1}{a-b}(X+Y)^{a-2}-\frac{b-1}{a-b}(X+Y)^{b-2}\right]\frac{\partial (X+Y)}{\partial q_{j}}\right\}
\langle a_{\omega}^{R \dag}a_{\omega^{\prime}}^{R}\rangle_{\mathrm{{i n}}}= \frac{1}{2 \pi}\int_{0}^{\infty}d \omega^{\prime \prime}\tilde{\beta}_{\omega^{\prime \prime}\omega}^{R *}\tilde{\beta}_{\omega^{\prime \prime}\omega^{\prime}}^{R}.
J_{4,2}=6T_{x^2y}^2+T_{xy}^2T_{x^2}-3T_{xy}^2T_{x}^2-6T_{x^3}T_{xy^2}+2T_{x^2}T_{xy^2}T_{x}&+4T_{x^3}T_{xy}T_{y}-2T_{x^2}T_{xy}T_{x}T_{y}+2T_{xy}T_{x}^3T_{y}-4T_{x^2y}T_{x^2}T_{y}-T_{x^2}^2T_{y^2}&+T_{x^2}^2T_{y}^2+4T_{x^2}T_{x}^2T_{y^2}-T_{x^2}T_{x}^2T_{y}^2-T_{x}^4T_{y^2}-2T_{x^3}T_{x}T_{y^2}
\begin{cases}\partial_{t}\textbf{u}+ \sum_{j=1}^{d}A_{j}(\textbf{u})\partial_{x_{j}}\textbf{u}+(0, \nabla P)^{T}=0, & \nabla \cdot v = 0, & \rho(0,x)=\rho_{0}(x), ~~~~~~~~~~ v(0,x)=v_{0}(x) ~~~~\nabla \cdot v_{0}(x)=0,\end{cases}
\alpha_1(x'), \alpha_2(y'), \alpha_3(z) \Leftrightarrow f_2(x^{-1}), f_2(y^{-1}\xi^{-1}), f_1(z^{-1}\xi^{3k-1}) &\Leftrightarrow x^{-1}y^{-1}\xi^{-1}=z^{-1}\xi^{3k-1}&\Leftrightarrow xy\xi^{3k}= z &\Leftrightarrow x'*y'=z.
S [ \psi ] = - \int d^{2}x \left\{\frac{1}{2}\sum_{j = 1}^{N}\bar{\psi}^{j}\gamma \cdot \partial \psi^{j}+ \frac{\lambda}{8 N}\sum_{i j = 1}^{N}\bar{\psi}^{i}* \psi^{i}* \bar{\psi}^{j}* \psi^{j}\right\}
S_x^*S_{yw}=S_{x^{-1}(x \vee y)}(S_{y^{-1}(x \vee y)}^*S_{w}) &=S_{x^{-1}(x \vee y)}\big(S_{(y^{-1}(x\vee y))^{-1}(y^{-1}(x\vee y)\vee w)}S^*_{w^{-1}(y^{-1}(x\vee y)\vee w)}\big) &=S_{x^{-1}y(y^{-1}(x\vee y)\vee w)}S^*_{w^{-1}(y^{-1}(x\vee y)\vee w)}.
D_{\zeta}( \zeta_{1}, \zeta_{2}) \partial_{\zeta}D_{z}( \zeta , \zeta^{\prime}) F ( z ) = \partial_{\zeta_{1}}D_{z}( \zeta^{\prime}, \zeta_{1}) D_{\zeta}( z , \zeta_{2}) F ( \zeta ) + \partial_{\zeta_{2}}D_{z}( \zeta^{\prime}, \zeta_{2}) D_{\zeta}( \zeta_{1}, z ) F ( \zeta ) ,
\sum_{k=2}^p(-1)^{k+1}\left(-\sum_{k_2=k+1}^p(-1)^{k_2+1}\lambda_{\widehat{k,k_2},x_{p-1}^q,\sigma^{-1}(x)}-\sum_{k_1=2}^{k-1}(-1)^{k_1}\lambda_{\widehat{k_1,k},x_{p-1}^q,\sigma^{-1}(x)}\right) &=-\sum_{k=2}^p\sum_{l=k+1}^p ((-1)^{k+1}(-1)^{l+1}+(-1)^k(-1)^{l+1}) \lambda_{\widehat{k,l},x_{p-1}^q,\sigma^{-1}(x)}=0\,
T_{DD}(L,m,d,z)=\frac{2g}{L}\frac{1}{(2\pi)^{d-1}}\sum_{n=1}^{\infty}\sin^{2}(\frac{n\pi z}{L})\int d^{d-1}p\frac{1}{(\vec{p}^{\,2}+(\frac{n\pi}{L})^{2}+m^{2})}.
\det H = \det_{1\leq j,k\leq s}\left[\sum_{m=0}^{r-1}m^{j+k-2}\right]=\prod_{j=0}^{s-1}h_j= \prod_{j=0}^{s-1}\frac{(j!)^4 (r+j)!}{(2j)!(2j+1)!(r-j-1)!}.
E_{N}^{*}E_{N}= K_{N}F_{N}E_{N}= K_{N}E_{N}F_{N}- K_{N}\frac{K_{N}- K_{N}^{-1}}{q - q^{-1}}& \sim q^{2}E_{N}K_{N}F_{N}= q^{2}E_{N}E_{N}^{*}.
\langle \sigma \partial_t^{1/2}\eta,\partial_t^{1/2}z^{\perp}\rangle = \langle \sigma \partial_t \eta,z \rangle \forall \, \eta \in H^{1,1}_{0,per}(Q_T) \forall \, z \in H^{1,\frac{1}{2}}_{0,per}(Q_T).
T_{j,m,n}(f,g)(x):=\iint \hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)x}m_j(\xi,\eta)\hat{\Phi}\left(\frac{\xi}{2^{aj+m}}\right)\hat{\Phi}\left(\frac{\eta}{2^{bj+n}}\right)\,d\xi d\eta,
\P\big( \cap_{j=1}^k \{T_{s_{j-1}+1}\le t_{j1},\ldots,T_{s_j}\le t_{j\ell_j}\}\mid \cap_{j=1}^k \{N(t_{j-1},t_j] =\ell_j \}\big) = \prod_{j=1}^k \P\big( T_{s_{j-1}+1}\le t_{j1},\ldots,T_{s_j}\le t_{j\ell_j}\mid N(t_{j-1},t_j]=\ell_j \big).
u^{\beta ( \alpha )}( \vec{p}\, ) = ( - 1 )^{\alpha}\gamma^{2}\gamma^{5}u^{\alpha *}( \vec{p}\, ) \, , \qquad \mathrm{a n d}\qquad e_{\mu}^{- \ell}( \vec{q}) = e_{\mu}^{\ell *}( \vec{q}\, ) \, ,
J^TG_w(A_1^r,\dots,A_m^r)J =G_\omega(J^TA_1^rJ,\dots,J^TA_m^rJ) &\le G_w(\alpha_1^{2r}A_1^{-r},\dots,\alpha_m^{2r}A_m^{-r}) &=(\alpha_1^{w_1}\cdots\alpha_m^{w_m})^{2r}G_w(A_1^r,\dots,A_m^r)^{-1},