harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{Z}_{m,n}^{\gamma}(z,\bar{z}) = \frac{\left((\gamma+1)_{m+n}\right)^2}{(\gamma+1)_{m}(\gamma+1)_{n}}\overline{z}^m z^n{_2F_1}\left( \begin{array}{c}-m , -n -\gamma-m-n \end{array}\bigg | \frac 1{|z|^2}\right) = (\gamma+1)_{m+n}\overline{R_{m,n}^{(\gamma)}(z)}.
\zeta(t)=\frac{\gamma_3}{6\gamma_2^{3/2}}+ \frac{\gamma_4\gamma_2-3\gamma_3^2}{24\gamma_2^3}t+ \frac{\gamma_5\gamma_2^2-10\gamma_4\gamma_3\gamma_2 + 15\gamma_3^3}{120\gamma_2^{9/2}}t^2 + \cdots,
\int_{R^6}(\partial_tu)^2(x,t_k)dx=\sum_{j=1}^J\int_{R^6}\left((\partial_tU_j^L)^2(x,\frac{t_{jn}}{\lambda_{jn}})+(\partial_tw_{Jn})^2(x,0)\right)dx+o(1), \,\,\,{\rm as}\,\,n\to\infty.
f^{B,\alpha}(\xi) := \partial_x^\alpha\big|_{x=0}\alpha_{-x}\big( f(\xi+Bx) \big) &= \sum_{\beta+\gamma = \alpha}\binom \alpha \beta i^{|\beta|}\delta^\beta \big (\partial_x^\gamma\big|_{x=0}f(\xi+Bx) \big) &= \sum_{\beta+\gamma = \alpha}\binom \alpha \beta i^{|\beta|}\delta^\beta \partial_{B,\xi}^\gamma f(\xi) .
\gamma^0=\left( \begin{array}{cc}1 0 0 -1 \end{array}\right)\,,\qquad\gamma^1=\left( \begin{array}{cc}0 1 -1 0 \end{array}\right)\,,\qquad\gamma^2=\left( \begin{array}{cc}0 i i 0 \end{array}\right)
u_i^{DF}\left( \pmb \rho \right) =\frac{1}{2}\min \left({\log \left({1 + \gamma_{i,1}^{DF}}\right),\log \left({1 + \gamma_{i,2}^{DF}}\right)}\right) &=\frac{1}{2}\log \left({1 + \gamma_i^{DF}}\right) ,
\frac{h_{n}\left(\sinh\left(\xi+\eta\right)\vert q\right)}{\left(e^{2\xi};q\right)_{n}}=\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}q^{k^{2}/2}h_{k}\left(\sinh\left(\log q^{\left(n-1\right)/2}-\eta\right)\vert q\right)}{\left(-e^{\xi+\eta}\right)^{n}\left(q,e^{2\xi};q\right)_{k}\left(-q^{n/2}e^{2\xi+\eta}\right)^{-k}}.
{\bf e}_{1}= \left( \begin{matrix}{1}\\{0}\\{0}\\{\vdots}\\{0}\\ \end{matrix}\right) , \quad{\bf e}_{2}= \left( \begin{matrix}{0}\\{1}\\{0}\\{\vdots}\\{0}\\ \end{matrix}\right) , \quad \ldots \quad ,{\bf e}_{N}= \left( \begin{matrix}{0}\\{0}\\{0}\\{\vdots}\\{1}\\ \end{matrix}\right) .
\begin{matrix}{{\theta}}&{{}}&{{+ 1}}\\{{W_{\alpha}}}&{{}}&{{+ 1}}\\{{\tau_{0}}}&{{}}&{{0}}\\{{\Phi}}&{{}}&{{0}}\\{{m}}&{{}}&{{+ 2}}\\{{K}}&{{}}&{{\mathrm{a r b i t r a r y}}}\\ \end{matrix}
\tau_{Y M}\equiv \frac{\theta_{Y M}}{2 \pi}+ i \frac{4 \pi}{g_{Y M}^{2}}= i \frac{2 N - M}{2 \pi}\log \frac{z}{\rho_{e}}, \rho_{e}= \epsilon \mathrm{e}^{\pi / ( 2 N - M ) g_{s}}
[\begin{matrix}A_{i+1,a}&A_{i+1,b}\\A_{i+1,b}^{T}&A_{i+1,d}\end{matrix}][\begin{matrix}X_{i+1,1}\\X_{i+1,2}\end{matrix}]=[\begin{matrix}B_{i+1,1}\\B_{i+1,2}\end{matrix}]
\frac{d}{dt}F (\varphi_t(x), T)\Big|_{t=0}=\langle d_xF(x,T),g(x)\rangle, \frac{d}{dt}J\varphi_t(x,T)\Big|_{t=0}=T:Dg(x), \frac{d}{dt}F (x, d\varphi_t (T))\Big|_{t=0}=\big\langle d_T F(x,T) ,\,T^\perp\circ Dg(x)\circ T +(T^\perp\circ Dg(x)\circ T)^*\big\rangle.
\begin{bmatrix}0 0 c(i+r,3r) c(i+3r,r) 0 0 c(i+r,r) c(i+3r,3r) c(i,r) c(i+2r,3r) 0 0 c(i,3r) c(i+2r,r) 0 0 \end{bmatrix}.
\begin{aligned}\frac 12\bigg |\int_{\Omega}\frac{|u|^2}{r^{a+b+1}}\left(r\widetilde{\Delta}r-(a+b)\right)dV_{\widetilde{g}}\bigg | =\bigg |\int_{\Omega}\left\langle\frac{uT}{r^{a+b}},\widetilde{\nabla}u\right\rangle_TdV_{\widetilde{g}}\bigg | & =\bigg |\int_{\Omega}\left\langle\frac{uT}{r^{a}},\frac{\widetilde{\nabla}u}{r^{b}}\right\rangle_TdV_{\widetilde{g}}\bigg |,\end{aligned}
a_k(x) \Big(\frac{\textup{d}}{\textup{d}x}\Big)^2\big( (1-x)^{p_k(k-j)/k}x^{p_k j/k}\big)+ b_k(x) \frac{\textup{d}}{\textup{d}x}\big( (1-x)^{p_k(k-j)/k}x^{p_k j/k}\big) & + p_k c_k(x) (1-x)^{p_k(k-j)/k}x^{p_k j/k}& = \frac{p_k^2}{k^2}\big( (1-x)^{p_k/k}- x^{p_k/k}\big) (1-x)^{p_k(k-j)/k}x^{p_k j/k}\big( j^2 (1-x)^{p_k/k}- (k-j)^2 x^{p_k/k}\big).
\chi(s)=\frac{\prod_{i=1}^{n}\Gamma^{\alpha_i}\left(1-a_i-A_is\right)\prod_{j=1}^{m}\Gamma^{\beta_j}\left(b_j+B_js\right)}{\prod_{i=n+1}^{p}\Gamma^{\alpha_i}\left(a_i+A_is\right)\prod_{j=m+1}^{q}\Gamma^{\beta_j}\left(1-b_j-B_js\right)}.
\lambda^2 - \lambda^1 = \nu^* \left(\frac{1}{2}\right)^{1+\lfloor \log_2(\nu^* (c(N))^2)\rfloor}< \nu^* \left(\frac{1}{2}\right)^{\log_2(\nu^*(c(N))^2)}= \frac{1}{(c(N))^2}.
{\cal A T}_{c l a s s}~ = ~ \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{1}}&{{1}}&{{0}}&{{0}}\\{{-{\frac{5}{6}}}}&{{-{\frac{5}{2}}}}&{{1}}&{{- 1}}\\{{{\frac{5}{2}}}}&{{5}}&{{0}}&{{1}}\\ \end{matrix}\right)
\exp \left\{- \Gamma_{\mathrm{W}}[ \tilde{A}\, ] \right\}\propto \, \prod_{k}\, \frac{\lambda_{k}}{\lambda_{k}+ i \Lambda_{0}}\, \left( \, \prod_{l = 1}^{\infty}\frac{\lambda_{k}^{2}+ M_{l}^{2}}{\lambda_{k}^{2}+ ( M_{l}+ \Lambda )^{2}}\, \right) \; ,
\sum_i \sum_{(x), (f^i(x_0))}h^i \otimes (Sx_{-1})f^i(x_0)_{-1}\otimes f^i(x_0)_0=\sum_{(x), (f(x_0))}(Sx_{-1})f(x_0)_{-2}\otimes (Sx_{-2})f(x_0)_{-1}\otimes f(x_0)_0.
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_1}y_4+\theta_2y_5, [y_2, y_1]=\alpha_6y_5, [y_2, y_2]=\beta_2y_5, [y_2, y_3]=\frac{\gamma_1}{\gamma_3}y_4+\theta_3y_5, [y_3, y_2]=y_4.
\operatorname*{l i m}_{m_{1}\rightarrow 0}\Bigl | \langle ( \bar{\psi}_{1}\psi_{1}\rangle_{A}|_{h = 0}\Bigr | = \operatorname*{l i m}_{m_{2}\rightarrow 0}\Bigl | \langle ( \bar{\psi}_{2}\psi_{2}\rangle_{A}|_{h = 0}\Bigr |
M^{-(a_1+a_2+a_4)(l+1)+ra_3(l+1)}\sum_{m=4}^N c_0^{\frac{m}{2}}M^{\frac{a_1(l+1)}{2}m}\alpha^m\|J_m^{l+1}\|_{l+1,r}\le 1,\ (\forall r\in \{0,1\}).
X_k(p_*,q_*)=\sqrt{\frac{\eta_k}M}\begin{pmatrix}e_k(x_1) \vdots e_k(x_M) 0 \vdots 0\end{pmatrix},\;[X_0,X_k](p_*,q_*)=\sqrt{\frac{\eta_k}M}\begin{pmatrix}-\hat{E}\begin{pmatrix}e_k(x_1) \vdots e_k(x_M)\end{pmatrix}(\frac1{h^2}A+\frac1MI)\begin{pmatrix}e_k(x_1) \vdots e_k(x_M)\end{pmatrix}\end{pmatrix}
| R_{1}^{( - )}( \nu , k ) | \leq \frac{2 ( 1 . 1 )}{\sqrt{\pi}}\sum_{\ell = 1}^{5}( \frac{4}{3})^{\ell}| \sum_{j = 0}^{5 - \ell}| C^{( 2 )}( \ell , j ) | ( - 1 )^{j}\Gamma ( j + \frac{1}{2}; \pi ) | \equiv C_{2}^{\prime},
2^{J_i}=2^{\mathfrak{k}_J}\prod_{j:\mathfrak{d}_j\leq i}(\mathfrak{s}_j)^{i-\mathfrak{d}_j}=2^{k}\prod_{j=1}^{J}(\mathfrak{s}_j)^{\mathfrak{d}_j}\prod_{j:\mathfrak{d}_j\leq i}(\mathfrak{s}_j)^{i-\mathfrak{d}_j}.
\sum_{i=1}^{d}[a_i(t,x,u,z) - a_i(t,x,\tilde{u},\tilde{z})](z_i - \tilde{z}_i) + [a_0(t,x,u,z)-a_0(t,x,\tilde{u},\tilde{z})](u-\tilde{u}) \geq - c ( f_3(t) + |u|^{\gamma}+ |\tilde{u}|^{\gamma})|u-\tilde{u}|^{2}.
\phi^{s_1-t, s_2}_{2(r_1-t)+r_2}(x) - \underset{m=1}{\overset{2t}{\sum}}{_{2t}}C_m \{_{(r_1-t)}}C_m \ 2^m \m! \ \phi^{s_1+t, s_2}_{2[r_1-t-m]+r_2}(x).
(a_y^+)|x_1,\cdots ,x_n\rangle = |y,x_1,\cdots x_n)\rangle a_y|x_1,\cdots ,x_n\rangle = \delta_{y,x_1}|x_2,\cdots ,x_n\rangle +\delta_{y,x_2}|x_1,x_3,\cdots ,x_n\rangle+\cdots + \delta_{y,x_n}|x_1,\cdots ,x_{n-1}\rangle
\begin{aligned}(\i\zeta_1+\nu_1\frac{d}{dz})\tilde u_1(z) =0 (\i\zeta_2+\nu_2\frac{d}{dz})\tilde u_1(z) =0 (\i\zeta_3+\nu_3\frac{d}{dz})\tilde u_1(z) =0.\end{aligned}
\left[\frac{1}{4}(\overline{p}\frac{l}{l_L})^2-\frac{3}{4}(\overline{q}\frac{l}{l_L})^2-\frac{1}{4}\right]L=-\overline{q}^2L-(\frac{1}{4}\overline{p}^2-\frac{3}{4}\overline{q}^2)\frac{(X_3u)^2}{l^2}+O(L^{-\frac{1}{2}})~~{\rm as}~~L\rightarrow +\infty,
x(Q_1)= \frac{9 (-7 + t)^2 (2 + t)^2 (1 + 2 t)^2 (-7 + 4 t)^2 (7 + 2 t^2)^2}{(-7 - 4 t + 2 t^2)^2}, x(Q_2)= \frac{(-7 + t)^2 (2 + t)^2 (1 + 2 t)^2 (-7 + 4 t)^2 (7 + 2 t^2)^2}{(-7 + 14 t + 2 t^2)^2}, x(Q_3)= \frac{(-7 + t)^2 (2 + t)^2 (1 + 2 t)^2 (-7 + 4 t)^2 (-7 + 14 t + 2 t^2)^2}{(-7 - 4 t + 2 t^2)^2}.
\gamma(s_{pj},s_{qj}^\prime,t) = \exp\left(\frac{t}{2}\left(\frac{\partial^2}{\partial s_{pj}^2}+\frac{1}{s_{pj}}\frac{\partial}{\partial s_{pj}}\right) \right) \, \frac{\delta(s_{pj}-s_{qj}^\prime)}{\sqrt{s_{pj}s_{qj}^\prime}}
\tfrac{\binom{Md_j}{Md_1}}{\binom{d_j}{Md_1}}= \tfrac{(Md_j)(Md_j-1)\cdots (Md_j-Md_1+1)}{(d_j)(d_j-1)\cdots (d_j-Md_1+1)}= \tfrac{M^{Md_1}(d_j)(d_j-\tfrac{1}{M})\cdots (d_j-\tfrac{Md_1}{M}+\tfrac{1}{M})}{(d_j)(d_j-1)\cdots (d_j-Md_1+1)}.
\left( \begin{array}{c}{{\psi_{1}}}\\{{\psi_{2}}}\\ \end{array}\right) \equiv \left( \begin{array}{c c c c}{{1}}&{{\psi_{1}}}&{{}}&{{}}\\{{0}}&{{0}}&{{}}&{{}}\\{{}}&{{}}&{{1}}&{{0}}\\{{}}&{{}}&{{\psi_{2}}}&{{0}}\\ \end{array}\right) .
\left( \begin{array}{c c}{{2 A_{[{\tt i}}{}^{[{\tt I}}\delta_{{\tt j}]}{}^{{\tt J}]}}}&{{\Sigma_{\tt i j K L}}}\\{{\Sigma^{\tt k l I J}}}&{{2 A_{[{\tt k}}{}^{[{\tt K}}\delta_{{\tt l}]}{}^{{\tt L}]}}}\\ \end{array}\right)
P_{k+1}^{a+1,b}(x)-P_{k+1}^{a,b+1}(x) =P_{k}^{a+1,b+1}(x) (1-x)P_{k}^{a+1,b}(x)+(1+x)P_{k}^{a,b+1}(x) =2P_{k}^{a,b}(x) (k+a+b+2)P_{k+1}^{a,b+1}(x)+(k+a+1)P_{k}^{a,b+1}(x) =(2k+a+b+3)P_{k+1}^{a,b}(x) P_k^{a+1,b}(x)=\frac{\Gamma(k+b+1)}{\Gamma(k+a+b+2)}\sum_{j=0}^k(2j+a+b+1) \frac{\Gamma(j+a+b+1)}{\Gamma(j+b+1)}P_j^{a,b}(x).
\sum_{N_0 \leq |k|\leq m}C_k e^{ikx}=\sum_{s=1}^{s_0}\left[\sum_{N_{s-1}\leq |k|<N_s}C_k^{(s)}e^{ikx}\right]+\sum_{N_{s_0-1}\leq |k|\leq m}C_k^{(s_0+1)}e^{ikx}.
2^{|\lambda|-\ell(\lambda)}{f}_{\lambda/\mu}=\sum\limits_{\lambda^-:\substack{\lambda\supseteq\lambda^-\supseteq \mu \ell(\lambda^-)<\ell(\lambda)}}2^{|\lambda^-|-\ell(\lambda^-)}{f}_{\lambda^-/\mu}+2\sum\limits_{\lambda^-:\substack{\lambda\supseteq\lambda^-\supseteq \mu \ell(\lambda^-)=\ell(\lambda)}}2^{|\lambda^-|-\ell(\lambda^-)}{f}_{\lambda^-/\mu}.
(-1)^{\ell(w)-\ell(u)}d^w_u\ =\ (-1)^{\ell(w)-\ell(u)}c^{w\varsigma}_{u,\varsigma}\ =\ (-1)^{\ell(w\varsigma)-\ell(u)-\ell(\varsigma)}c^{w\varsigma}_{u,\varsigma}\ >\ 0\,,
a_{s}^{l}= - \frac{1}{s ( 2 l - 2 s + 1}\sum_{r = 0}^{s - 1}a_{r}^{l}[ ({}_{l - 2 s}^{l - 2 r}) \ -{\lambda}^{2}\ ({}_{l - 2 s + 1}^{l - 2 r + 1}) ] \ \lambda^{2 s - 2 r - 2}\ .
\frac h{2}\sum_{n=1}^{N_T}\int_{\Omega}\Bigl\vert \frac{v_n (x) \, - \, v_{n-1}(x-hv_{n-1}(x))}{h}\Bigr\vert^2 \, dx\, + \, \frac 12 \int_\Omega | D v_n (x) |^2 \, dx &\; \le \; C e^{CT}\int_\Omega | D a (x) |^2 \, dx
\Psi^{r_1, r_2}_{k_1, k_{1 z}; k_2, k_{2 z}; I, I_z}(\theta,\alpha, \beta, \gamma, \alpha_I, \beta_I, \gamma_I) = \sqrt{\frac{p !}{m_1 ! m_2 ! m_3 ! m_4 !}}\Psi_1^{m_1}\Psi_2^{m_2}\Psi_3^{m_3}\Psi_4^{m_4},
Q1(U,\overline{U},\widehat{\overline{U}},\widehat{U};\alpha,\beta;0)=0 ~\Leftrightarrow~ \cfrac{(U-\overline{U})(\widehat{U}-\widehat{\overline{U}})}{(U-\widehat{U})(\overline{U}-\widehat{\overline{U}})}=\cfrac{\alpha}{\beta}\,;
\alpha:=-2^{r-3}\left( \frac{8}{r}+r+3\right) ,\quad\beta:=2^{r-1}\left(r+\frac{6}{r}+2\right) , \gamma:=-2^{r-3}\left( 5r+\frac{24}{r}+3\right) ,\quad\delta:=2^{r-2}\left(\frac{8}{r}+r-1\right) .
\mathrm{deg}\big(\alpha(\widetilde{Z})\big)=-K_{\widetilde{Q}_u}\cdot\widetilde{Z}=\Big(3\pi^*\big(H_{Q_u}\big)-E_{Q_u}\Big)\cdot\widetilde{Z}=\Big(\textbf{s}+\frac{n+20}{2}\textbf{l}\Big)\cdot\widetilde{Z}.
\sigma_{\mathrm{i n}\to \mathrm{a l l}}( \theta ) ={\frac{1}{2 \pi p}}\left({\frac{1}{\sin^{2}{\theta}}}\right) 2 \left( 1 - \mathrm{R e}\langle \mathrm{i n}|{\cal R}^{2}| \mathrm{i n}\rangle \right) \; ,
\! \{[0\!:\! 0\!:\! z_3\!:\!\cdots\!:\! z_{n+1}]: z_3,\!\cdots\!,\! z_{n+1}\!\in\!\mathbb{C}\}\setminus \!\{[0\!:\! 0\!:\! z_3\!:\!\cdots\!:\! z_{n}\!:\!1]: z_3,\!\cdots\!,\! z_{n}\!\in\!\mathbb{C}\}= &\{[0\!:\! 0\!:\! z_3\!:\!\cdots\!:\! z_{n}\!:\!0]: z_3,\!\cdots\!,\! z_{n}\!\in\!\mathbb{C}\}=\Lambda_0.
(D df)(B) = \sum_{i,k=1}^n\left( \hat{\kappa}(A_i) A_k^*(B) \right) \hat{\nabla}_{A_i}df(A_k) + \sum_{i=1}^n \sum_{s=1}^\nu \left( \hat{\kappa}(A_i) Z_s^*(B) \right) \hat{\nabla}_{A_i}df(Z_s).
\begin{cases}c(\mu,d,s)\mathcal{H}^s_{\infty}(\Omega)\leq \mathcal{H}^{\tilde{\mu},s}_{\infty}(\Omega)\leq \mathcal{H}^{s}_{\infty}(\Omega) s<\alpha \mathcal{H}^{\tilde{\mu},s}_{\infty}(\Omega)=0 s>\alpha.\end{cases}
\Delta \Omega = -{\frac{1}{\beta}}\sum_{m = 1}^{\infty}\frac{( - 1 )^{m}}{m !}\int_{0}^{\beta}d \tau_{1}. . . d \tau_{m}\langle T_{\tau}( H_{I}( \tau_{1}) . . . H_{I}( \tau_{m}) ) \rangle_{c o n n}^{\beta}\; .
S_{k+1}:= \frac{(-1)^{k+1}k}{(k-1)!}\sum_{p_1\neq \dotsb \neq p_k}\Bigl[\oint_M w^{p_1}w^{p_2}_{n}\dotsm w^{p_{k}}_n u_n\sigma_{k-1}(L)d\mu +\frac{1}{k}\oint_M u w^{p_1}_n \dotsm w^{p_{k}}_n \sigma_{k-1}(L )d\mu \Bigr] .
E[\rho,u](t) \;:=\;\int_\Omega\left(\dfrac{1}{{\rm Fr}^2}\,h(\rho)\,+\,\dfrac{1}{{\rm Fr}^2}\,h_c(\rho)\,+\,\dfrac{1}{2}\,\rho\,|u|^2\,+\,\dfrac{1}{2\,{\rm We}}\,|\nabla\rho|^2\right)dx\,,
u^{{{f}}}_{{m}}={1\over 32}\left[v^{{\hat a}}_{{A}}\left(C\Gamma_{{m}}\right)v_{{\hat b}{A}}+v^{{\hat a}{\dot A}}\left(C\Gamma_{{m}}\right)v_{{\hat b}{\dot A}}\right] (\gamma^{f})^{{{\hat b}}}_{{\hat a}},
\begin{cases}\left< w^{\ell + 1}, \varphi v \right> - \frac{\Delta t}{2}\left< \left( \frac{w^{\ell}+ u^{n}}{2}\right)^{2}, (\varphi v)_{x}\right> + \Delta t \left< \left( \mathcal{H}\frac{w^{\ell + 1}+ u^{n}}{2}\right)_{x}, (\varphi v)_{x}\right> = \left< u^{n}, \varphi v \right>, w^{0}= u^{n},\end{cases}
etz^{-2}~\stackrel{(\ref{2.18})}{=}[(z^{\ast})^{-1}et^2]z^{-2}= (z^{\ast})^{-1}(et^2z^{-1})z^{-1}~\stackrel{(\ref{2.18})}{=}(z^{\ast})^{-1}[(z^{\ast})^{-1}et^2]z^{-1}=(z^{\ast})^{-2}et^2z^{-1}\stackrel{(\ref{2.18})}{=}(z^{\ast})^{-2}et,
\overline{p}(n)=\frac{1}{2\pi}\underset{2 \nmid k}{\sum_{k=1}^{N}}\sqrt{k}\underset{(h,k)=1}{\sum_{h=0}^{k-1}}\dfrac{\omega(h,k)^2}{\omega(2h,k)}e^{-\frac{2\pi i n h}{k}}\dfrac{d}{dn}\biggl(\dfrac{\sinh \frac{\pi \sqrt{n}}{k}}{\sqrt{n}}\biggr)+R_{2}(n,N),
\bar{\partial}_{A}\alpha = \bar{\partial}_{A}\beta = \bar{\partial}_{A}^{*}\beta = \alpha u = 0 , F_{A'}^{0,2}= \bar{\partial}^{*}u = \bar{\alpha}\beta = 0 , i \Lambda F_{A'}^{1,1}= \frac{1}{8}\left( |u |^2 + |\beta|^2 - |\alpha|^2 \right)
N_{X_{\tilde{H},V_Y}}(n,n) = \sum_{k=1}^{n}\sum_{i_\ell | i_1 + \dots + i_k = n}N_{X_{H,V_Y}}(i_\ell,n) \leq \sum_{k=1}^n \binom{n+k-1}{n}N_{X_{H,V_Y}}(n,n) = \binom{2n}{n+1}N_{X_{H,V_Y}}(n,n).
(A_{i}+\partial_i \varphi)(\vec x) \rightarrow iD_{i}(\vec x)\equiv i\delta_i (\vec x) + \frac{1}{2\pi k}\sum_{a}s_{a}\varepsilon_{ij}\frac{(x-x_{a})^{j}}{|\vec x - \vec x_{a}|^{2}},
w_{n,i}^* \in \left(Q_{a_{n,i}}(\alpha-1,A_i,u_i + a_i f_{q(i)})\right)^{(\alpha-1)}\subseteq \left( \bigcap_{j=1}^\infty K(\alpha-1,A_j,u_j + a_j f_{q(j)}) \right)^{(\alpha-1)}= K^{(\alpha-1)}.
\int \exp \{a^4 \sum_{i=1}^2 \sum_x \bar{\psi}_i(x)(\hat{D}+m) \psi_i(x) \}d \bar{\psi}d \psi = \lim_{L \rightarrow \infty, b \rightarrow 0}\int \exp \{-S \}d \phi^*_n d \phi_n d \chi^* d \chi
\tilde{r}_{0,n}:= \frac{1}{n}\log |\tilde{{\cal M}}_0| + \Delta_n + \frac{\log\log |{\cal X}| +2}{n}, \tilde{r}_{2,n}:= \frac{1}{n}\log |\tilde{{\cal M}}_2| + \frac{1 + \log |{\cal Y}|}{n}, \delta_n := \Delta_n + \frac{\log\log |{\cal X}| + 3 + \log |{\cal X}|}{n},
H(\omega ,0)=\frac{\phi_{1}f_{2}\partial_{r}(\phi_{2}^{(-)}/A_{-})-(\phi_{2}^{(-)}/A_{-})f_{1}\partial_{r}\phi_{1}}{\phi_{1}f_{2}\partial_{r}(\phi_{2}^{(+)}/A_{+})-(\phi_{2}^{(+)}/A_{+})f_{1}\partial_{r}\phi_{1}}\ .
W(\Delta,z)=\sum_{\Gamma,\Gamma^{\prime}\vdash d}\frac{d!}{|C_{\Gamma^{\prime}}|}z^{d+l(\Gamma^{\prime})-l(\Delta)-l(\Gamma)}\mu_{0}^{h_+,d}(\Gamma^{\prime},\Delta,\Gamma)p_{\Gamma}\frac{\partial}{\partial p_{\Gamma^{\prime}}},
Spec\left(\{\hat{x}^\mu_k ,\,\hat{y}^\mu_k\}^4_{\mu=1}\right)\to TM^k_4,\{|x^\mu_k,y^\nu_k\rangle\}^4_{\mu=1}\mapsto (x^\mu_k,y^\mu_k),\,\mu=1,2,3,4;\,\,k=1,...,N.
\int_{\mathbb{CP}^n}\left(r \omega_{FS}+ i \partial \bar{\partial}\phi_{P_1}\right)\wedge \cdots \wedge \left(r \omega_{FS}+ i \partial \bar{\partial}\phi_{P_n}\right) =\int_{(\mathbb{C}^*)^n}(i \partial \bar{\partial}H_{P_1}) \wedge \cdots \wedge (i \partial \bar{\partial}H_{P_n}).
|a_{k,n}^{(\alpha)}|\leq |\gamma_{k,n}^{(\alpha)}|+\frac{c_{9}}{\rho_2^{k-n}}\sum_{w=1}^{d}\sum_{y=1}^{m_w}|\gamma_{n-m_w+y,n}^{(w)}|,\alpha=1,2,\ldots,d, k\geq n+1.
\phi(x) = \frac{1}{2\pi\sqrt{R_2 \sin\theta}}\sum_{n_+=0}^\infty\sum_{n_2=-\infty}^\infty [e^{-i(\frac{n_+}{R_+}Z_+ +\frac{n_2}{R_2}Z_2)}A(n_+,n_2)+ e^{i(\frac{n_+}{R_+}Z_+ + \frac{n_2}{R_2}Z_2)}A^\dagger(n_+,n_2){]}\,,
\Delta^2 u_1 =\left( R_1^2+R_2^2 \right) \frac{2(u_1+u_2)}{1+|u|^2}- \frac{20 u_1}{1+|u|^2}|\nabla u|^4 , \Delta^2 u_2 =\left( R_1^2+R_2^2 \right) \frac{2(u_2-u_1)}{1+|u|^2}- \frac{20 u_2}{1+|u|^2}|\nabla u|^4,
{\mathcal R}_{k,l,2}\geq \left( C_{\frac{1}{3}, 0,2}^{rad}\right)^{\frac{3(l+1-k)}{l+2}}\cdot \left( \frac{3(k+1)}{4 d_2}\right)^{\frac{k+1}{l+2}}\cdot \left( \frac{4}{3}\right)^{\frac{3k-2l-1}{l+2}}= C_{k,l,2}^{rad}\ ,
Y^+_{J_2 ,n_2}(1)W^+_{J_3 ,n_3}(x)=\frac{1}{1-x}f^{J_2 +J_3 -1,n_2 +n_3}_{J_2n_2 ,J_3 n_3}Y^+_{J_2 +J_3 -1,n_2 +n_3}(1) + regular~terms
N^{(2l-1)}=\delta_{l,1}\,\int dx\,\frac{\theta^\prime}{2\pi}-\sqrt{\frac{2mT}{\pi}}\, e^{-m/T}\, \frac{1}{(2l-1)!}\, \int dx\, \left(\frac{\theta^\prime}{2T}\right)^{2l-1}+\dots
u^{0}= x^{0}- x^{3}, \; u^{1}= q \left( u^{0}\right) r^{2}, \; u^{2}= \varphi , \; u^{3}= x^{0}+ x^{3}- u^{0}u^{1}, \; q \left( u^{0}\right) = \left[ \left( u^{0}\right)^{2}+ a \right]^{- 1},
Q_{m_{1},\ldots,m_{i+1}}^{(j)}=Q_{m_{1},\ldots,m_{i}}\cap\left[x_{d-i+1}=\frac{i+1}{j-i}x_{d-i}-\frac{i+1}{j-i}\left(x_{d-i}^{0}+\cdots+\frac{1}{i+1}\binom{j+1}{i}x_{d}^{0}+m_{i+1}\right)\right],
\tilde{b}^{\rho}={\frac{k^{2}b^{\rho}- ( b \cdot k ) k^{\rho}}{b^{2}k^{2}- ( b \cdot k )^{2}}}\ , \quad \tilde{b}^{\rho}k_{\rho}= 0 \ , \quad \tilde{b}^{\rho}b_{\rho}= 1 \ .
d\boldsymbol{Y}_t=d\boldsymbol{\tilde{W}}_t\sqrt{\frac{I+\boldsymbol{Y}_t^2}{2}}+\sqrt{\frac{I+\boldsymbol{Y}_t^2}{2}}d\boldsymbol{\tilde{W}}_t^*+\left[(1-N-2\Re(s))\boldsymbol{Y}_t+2\Im(s)\boldsymbol{I}+ Tr\left(\boldsymbol{Y}_t\right)\boldsymbol{I}\right]dt,
((ax)^{q}-(ax))^{q-1}=(a^{q}x^{q}-ax)^{q-1}=a^{q-1}(x^{q}-x)^{q-1}, ((x+b)^{q}-(x+b))^{q-1}=(x^{q}+b^{q}-x+b)^{q-1}=(x^{q}-x)^{q-1}.
\mathbf{v}= \left( 0, 0, \frac{(1-\lambda_1)^2}{16 \, d_1}, \frac{1}{2}- \frac{(1+\lambda_1)(1+\lambda_2)}{8 \, d_1}, \frac{(1-\lambda_2)^2}{16 \, d_1}, 1, 1, 1 \right) \;.
{\cal L}_{C o u l}= \frac{i g}{2}[ ( \partial \partial^{*}A_{-}^{a}) ( \frac{1}{\partial_{+}}A_{+}) T^{a}A_{+}+ ( \partial \partial^{*}A_{+}^{a}) ( \frac{1}{\partial_{-}}A_{-}) T^{a}A_{-}] .
(1-\beta)\delta_3^{(s)}-\beta\delta_4^{(s)}+\sum_{i=1}^{s-2}\delta_3^{(s-i)}\leq -\gamma\beta^22^{-2s^2}\left(1-2^{1+2s-2s^2}-\sum_{i=1}^{s-2}2^{1+2(s-i)-2s^2}\right) &<-\gamma\beta^2 2^{-2s^2}\left(1-2^{-3}\right),
\delta \Psi_{\hat{a}}= \partial_{\hat{a}}\epsilon +{\frac{1}{8}}H^{- 1}( \partial_{\hat{a}}H ) \epsilon +{\frac{1}{8}}W H^{{\frac{m - n}{4}}}\gamma^{\underline{{u x y}}}\gamma_{\underline{{\hat{a}}}}( i \sigma_{2}) \epsilon = 0 .
L=\frac{ml^{2}}{2}(\frac{1}{4}\dot{\theta}_{1}^{2}+\dot{\theta}_{1}^{2}+\frac{1}{4}\dot{\theta}_{2}^{2}+\dot{\theta}_{1}\dot{\theta}_{2}cos(\theta_{1}-\theta_{2}))+\frac{ml^{2}}{24}(\dot{\theta}_{1}^{2}+\dot{\theta}_{2}^{2})-mg(y_{1}+y_{2})
\partial_t \big(T_k(\rho_\epsilon)\big)+\nabla\cdot(T_k(\rho_\epsilon)u_\epsilon)+\big(T'(\rho_\epsilon)\rho_\epsilon-T_k(\rho_\epsilon)\big)\nabla\cdot u_\epsilon=0\ \mbox{in}\ \mathcal D'(\mathbb R^3\times (0,T)).
\langle \varphi_{i_1}(X_1) ... \varphi_{i_n}(X_n) \rangle_\zeta = (- i)^n \prod_{j = 1}^n \frac{\delta}{\delta y_{i_j}(X_j)}{\cal G}[\vec{y}, \vec{\varphi}] \, \rule[-3mm]{.14mm}{8.5mm}\raisebox{-2.85mm}{\scriptsize{$\; \vec{y}= 0$}}.
h_{p}^{\left( 0\right)}=\frac{\left( -1\right)^{l}}{2\pi i}\frac{1}{\left( p+1\right) \left( p+2\right) \cdots\left( p+l\right)}\int_{\mathbb{T}}\left( \widetilde{\varphi}_{0}^{-}\left( u\right) \right)^{\left( l\right)}u^{p+l}du.
{\bf E}\bigl[\beta_{2,2}^q(1, b_0, b_1, b_2)\bigr] = \frac{G(b_0)}{G(q+b_0)}\frac{G(q+b_0+b_1)}{G(b_0+b_1)}\frac{G(q+b_0+b_2)}{G(b_0+b_2)}\times \times\frac{G(b_0+b_1+b_2)}{G(q+b_0+b_1+b_2)}.
T_{\alpha}\Phi_{\alpha}(n^3 f(z)) D_0^n(z) = \frac{2\pi}{\sqrt{3}}\left(n^3 f(z)\right)^{-\frac{2\beta}{3}}\left( \mathbb{I}+ \frac{A_{\alpha}}{n^3 f'(0) z}+ O\left(n^{-3}\right)\right)F_n(z),
j_{e}=\frac{q}{2m}[(\Psi^{*}\hat{p}\Psi-\Psi\hat{p}\Psi^{*})-\frac{2q}{c}A|\Psi|^{2}]+\frac{\mu_{S}c}{s\hbar}\nabla\times(\Psi^{*}S\Psi)
\mathfrak{G}_{\mu, \nu;\mu',\nu'}^{\tau}= \mathfrak{Re}\Big[ \int \overline{\widehat{(SU)_{\mu}}(\xi)}\widehat{(SU)_{\nu}}(\xi-\eta) \widehat{{U}_{\mu'}}(\eta-\sigma) \widehat{{U}_{\nu'}}(\sigma) e_{\mu, \nu;\mu',\nu'}^{\tau,1}(t,\xi,\eta, \sigma)
{\cal L}^{D}= \sqrt{- g^{D}}\left( R^{D}- \frac{1}{2}\partial_{\mu}\Phi \partial_{\nu}\Phi -{\frac{1}{2 . 3 !}}e^{- l \Phi}F_{\mu \nu \sigma}^{( 3 )}F^{( 3 ) \mu \nu \sigma}\right) \, ,
\begin{array}{r c l}{T_{c}}&{=}&{\displaystyle + [ 3 D_{m}( k_{1}) D_{m}( k_{2}) + D ( k_{1}) D ( k_{2}) ] D_{m}( k_{1}+ k_{2}) ( 1 + K_{1 2})}\\{}&{}&{\displaystyle + 4 ( N - 1 ) D ( k_{1}) D ( k_{2}) D_{m}( k_{1}+ k_{2}) .}\\ \end{array}
H = 1 + \frac{Q_{3}}{r^{4}}+ \frac{K_{3}^{2}}{2 \, r^{4}}\left[ \log \left( \frac{r^{4}}{\epsilon^{2}( r^{2}- | z |^{2})}\right) - 1 + \frac{| z |^{2}}{r^{2}- | z |^{2}}\right]
N^{\cal P}~=~\left ( 2 \over{k+2}\right) ~\sum_{l=1}^{k+1}sin^2\theta_l~\sum_{m_1 =-j_1}^{j_1}\cdots \sum_{m_p=-j_p}^{j_p}\exp \{2i(\sum_{n=1}^p m_n)~\theta_l \}~,
i\lambda\left(\frac{k'_{\perp}}{k^{'+}}e^{-i\lambda\varphi'}-\frac{k_{\perp}}{k^+}e^{+i\lambda\varphi}\right)\delta^{\lambda}_{\lambda'}+im\left(\frac{1}{k^{'+}}-\frac{1}{k^+}\right)\delta^{\lambda}_{-\lambda'}
\kappa_n(\delta) = \frac{1}{8}\sum\limits_{m=0}^\infty \frac{\bigl(-(\delta-1/2)\bigr)^m}{m!}\,32^{(m+n)/2}\,\Gamma\bigl(\frac{m+n}{2}\bigr) \,{\bf E}\bigl[C_2^{-(m+n)/2}\bigr].
a_{j}= \frac{1}{2}\left( \begin{array}{c}{{f_{j}}}\\{{0}}\\ \end{array}\right) + \frac{1}{2}\left( \begin{array}{c}{{0}}\\{{f_{j}^{\ast}}}\\ \end{array}\right) , \ \b_{j}= - \frac{i}{2}\left( \begin{array}{c}{{f_{j}}}\\{{0}}\\ \end{array}\right) + \frac{i}{2}\left( \begin{array}{c}{{0}}\\{{f_{j}^{\ast}}}\\ \end{array}\right)
\quad\theta''(g_0,\ldots, g_{n-1})(\eta) &=\sum_{s_0,\ldots, s_{n-1}}\theta'(h_{s_0},\ldots, h_{s_{n-1}})(L(\eta)|_{[s_0,\ldots, s_{n-1}]}) &=\sum_{s_0,\ldots, s_{n-1}}\sum_{t_0,\ldots, t_{n-1}}\theta(g_{t_0},\ldots, g_{t_{n-1}})(\pi(L(\eta)|_{[s_0,\ldots, s_{n-1}]})|_{[t_0,\ldots, t_{n-1}]}).
R ( \phi ) = \sum_{n = 0}^{\infty}r_{m_{1}\ldots m_{n}}^{( n )}\phi_{m_{1}}\ldots \phi_{m_{n}}, \tilde{R}( \tilde{\phi}) = \sum_{n = 0}^{\infty}r_{m_{1}\ldots m_{n}}^{( n )}\tilde{\phi}_{m_{n}}\ldots \tilde{\phi}_{m_{1}}
Lf(x,t) =\frac{1}{2\pi}\int_0^\infty \left[\underset{k=0}{\overset{\infty}{\sum}}\frac{\mu^{d_2}}{(2k+d_1)^{d_2}}\int_{S^{d_2-1}}P_k\left(\frac{\mu}{2k+d_1}\right)f^{\frac{\mu\epsilon}{2k+d_1}}(x)e^{-\frac{i\mu t\cdot\epsilon}{2k+d_1}}\,d\sigma(\epsilon)\right]\,d\mu &=\int_0^\infty \mu \mathcal{P}_\mu f(x,t)\,d\mu
\begin{array}{ccl}\xi_k^\alpha (x) = \left( \left( \frac 1{R\left( A(x)\right) R\left(A(x)\right)}\right)^{1/4}\right)_{kk^{^{\prime}}}^{\alpha \alpha^{^{\prime}}}B_{k^{^{\prime}}}^{\alpha^{^{\prime}}}(x) = \frac 1{\pi \sqrt{2}}\int_0^\infty d\lambda \lambda^{-1/4}\frac1{\lambda +R\bullet R}\bullet B(x) \end{array}
A_{p a r e n t}^{\mathrm{I I}}= - \frac{T_{p}}{2}\int d^{p + 1}{\xi}\left[ \sqrt{- G}\left( - G_{i j}h^{i j}+ ( p + 3 ) \right) +{\Lambda}^{i j}\left( G_{i j}- g_{i j}\right) \right] .
\left(q\frac{d}{dq}\right)^r F(q)\biggr\rvert_{q=\zeta}= \left(q\frac{d}{dq}\right)^r F(q, m)\biggr\rvert_{q=\zeta}= \left(q\frac{d}{dq}\right)^r F(q, (r+1)p-1)\biggr\rvert_{q=\zeta}
SINR_{ijk}= \frac{{\bf u}^{T}_{ijk}{\bf T}_{ijk}{\bf u}_{ijk}}{{\bf u}_{ijk}^{T}{\bf F}^{'}_{ijk}{\bf u}_{ijk}},\ \forall\{i,j\}\in\mathcal{L},\ \forall k\in\mathcal{K},
\hat{T}_{\mu \nu}= - \frac{1}{2}\left( \hat{F}_{\mu \lambda}\star \hat{F}_{\nu}^{\lambda}+ \hat{F}_{\nu \lambda}\star \hat{F}_{\mu}^{\lambda}- \frac{1}{2}\eta_{\mu \nu}\hat{F}_{\lambda \rho}\star \hat{F}^{\lambda \rho}\right)
f^{\left( 1 \right)}( \theta ) = \sum_{l}\left( 2 \pi i k \right)^{- \frac{1}{2}}e^{i l \theta}\left[ - i \pi \alpha \mathrm{s g n}\left( l \right) \left( 1 - \delta_{l , 0}\right) - i \pi \alpha \delta_{l , 0}\cos \beta \right] ,
\lim_{v \to \infty}v^{1 - \alpha}\!\!\int_{0}^{\infty}h(u) \Phi(v - u) u^{\alpha -1}du =\lim_{v \to \infty}C \v^{1 - \alpha}\!\!\int_{0}^{\infty}\! \Phi(v - u) u^{\alpha - 1}du = C \int_{\mathbb{R}}\Phi(u) du .
\begin{aligned}\|\varrho_t\|_{L^2}=\|{\rm div}u+\varrho{\rm div}u+u\cdot \nabla \varrho\|_{L^2}&\le \|{\rm div}u\|_{L^2}+\|\varrho\|_{L^\infty}\|{\rm div}u\|_{L^2}+\|\nabla \varrho\|_{L^3}\|u\|_{L^6}&\le C(1+t)^{-\frac{5}{4}}.\end{aligned}
\sum_{\xi,\eta}\langle X_\alpha ,[X_\xi,X_\eta]\rangle\langle X_\beta,[\tilde RX_\xi,\tilde TX\eta]\rangle= \sum_{\xi,\eta}r_\xi t_\eta \langle X_\alpha ,[X_\xi,X_\eta]\rangle\langle X_\beta,[X_\xi,X\eta]\rangle=0,
(w', L_A w) = (w', w' + g) = \| w'\|_2^2 + (w', g),\ \ \ \ (w', L_A w) = ( L_A w - g, L_A w) = \| L_A w\|_2^2 - (g, L_A w) = \| L_A w\|_2^2 - (g, w' +g) = \| L_A w\|_2^2 - \| g\|_2^2 - (g,w').