harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
<{\cal{H}}> ={\frac{1}{8}}G^{- 1}( x x ; \chi ) +{\frac{1}{2}}\lim_{x \rightarrow y}\nabla_{x}\nabla_{y}G ( x , y ; \chi ) -{\frac{\alpha_{0}}{\beta^{2}}}~ ~ e^{-{\frac{\beta^{2}}{2}}G ( x , x ; \chi )}
\sum_{n=1}^{\infty}n^2 e^{-\alpha n^2t}={\pi^{1 \over 2}\over 4(\alpha t)^{3 \over 2}}+{\pi^{1 \over 2}\over 2(\alpha t)^{3 \over 2}}\sum_{n=1}^{\infty}\exp \Bigl({n^2\pi^2 \over \alpha t}\Bigr) - \left({\pi \over \alpha t}\right)^{5 \over 2}\sum_{n=1}^{\infty}n^2 \exp \Bigl({n^2\pi^2 \over \alpha t}\Bigr)
\int_E f = \int_E f(t)\,\d t - \frac{1}{2}\int_{E\cup E^c}f(t)\,\d t & = \frac{1}{2}\int_E f(t)\,\d t - \frac{1}{2}\int_{E^c}f(t)\,\d t & \le \frac{1}{2}\int_{0}^{1}|f(t)|\,\d t\le \frac{1}{2}
C^{\mathrm{ul},k}_{3}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{ul}}}{T}\right) \min\left\lbrace c^k_{\mathrm{u2s},1}\left( N_{\mathrm{scr}}\right), \right. \,\,\,\,\,\, \left. c^k_{\mathrm{s2b},3}\left(N_{\mathrm{sct}}\right) \right\rbrace.
Q_{A}( \alpha , \bar{\alpha}) = \sum_{K = 0}^{M}\frac{P !}{( P - K ) !}\left( 1 + \sum_{i}^{N - 1}| \alpha_{i}|^{2}\right)^{- K}A_{\nu_{1}\ldots \nu_{K}}^{\mu_{1}\ldots \mu_{K}}\alpha_{\mu_{1}}\ldots \alpha_{\mu_{K}}\bar{\alpha}_{\nu_{1}}\ldots \bar{\alpha}_{\nu_{K}}\: ,
\sum_{B'\in V}\mathcal{K}(B,B')g(B')^{q/(q-1)}=\frac{1}{|B|}\sum_{k=n}^\infty\sum_{B'\in V_k, \, 8B\cap 8B'\ne \emptyset}|B'|^{1+\alpha q/(q-1)}&\lesssim \sum_{k=n}^\infty 2^{-kQ\alpha q/(q-1)}\simeq 2^{-nQ\alpha q/(q-1)}\simeq g(B)^{q/(q-1)}
\left| E^{(u)}- E_j \right| \leq \left| E^{(u)}- \xi^{(u)}\right| +\left| \xi^{(u)}- \xi^{(x_j)}\right| + \left| \xi^{(x_j)}- E_j \right| \leq 6 \sqrt K \tau_n = \epsilon_n \, .
\frac{g^2}2\int d^4xd^4x^{\prime}\left\{-\left[ \phi (x)\right] D\left(x,x^{\prime}\right) \left\{\phi (x^{\prime})\right\}+i\left[ \phi(x)\right] N\left( x,x^{\prime}\right) \left[ \phi (x^{\prime})\right]\right\}
\widehat{Y}_{i}(qx) = \widehat{Y}_{i}(x) - \sum^{N}_{j = 0}B_{j}\widehat{Y}_{j}(x) + \frac{1}{1 - x/b_{i}}\biggl\{-\big(1 - q^{\lambda}\big)\widehat{Y}_{i}(x) + \sum^{N}_{j = 0}B_{j}\widehat{Y}_{j}(x) \biggr\}
D_{\hat{\alpha}}\xi_{\hat{\beta}}= -{\frac{1}{4}}( \Gamma^{\mu \nu})_{\hat{\beta}\hat{\alpha}}\partial_{\mu}\xi_{\nu}+{\frac{1}{1 2}}C_{\hat{\beta}\hat{\alpha}}\partial_{\sigma}\xi^{\sigma}+{\frac{1}{1 2}}( \Gamma^{i j})_{\hat{\beta}\hat{\alpha}}D_{\sigma [ i j ]}\xi^{\sigma},
\phi^{(0,0)}_{\lambda}= \frac{\sin(\hat{\lambda}_1\xi) \sin(2\hat{\lambda}_2\xi) \sin((\hat{\lambda}_1+2\hat{\lambda}_2)\xi) \sin((2\hat{\lambda}_1+2\hat{\lambda}_2)\xi)}{\sin(\xi) \sin(2\xi) \sin(3\xi) \sin(4\xi)}.
L_+= \frac{\ell^2}2 \left [\frac 3 2 \left ( \frac{f''_+}{f'_+}\right )^2- \frac{f'''_+}{f'_+}\right ] \qquad , \qquad L_-= \frac{\ell^2}2 \left [\frac 3 2 \left ( \frac{f''_-}{f'_-}\right )^2- \frac{f'''_-}{f'_-}\right ] \qquad .
H(U|X_1,Q=i) =p_{X_1|Q}(0|i)h_2\left(\frac{p_{X_1X_2|Q}(0,0|i)(1-\alpha)+\frac{p_{X_1X_2}(0,1)}{2}}{p_{X_1|Q}(0|i)}\right)+p_{X_1|Q}(1|i)h_2\left(\frac{p_{X_1X_2|Q}(1,1|i)(1-\alpha)+\frac{p_{X_1X_2}(1,0)}{2}}{p_{X_1|Q}(1|i)}\right) \leq h_2\left((1-q_i)(1-\alpha)+\frac{q_i}{2}\right) &=h_2(\alpha\star\frac{q_i}{2})
\left[ \frac{\delta(\hat{A}(\omega) \hat{p}, \hat{A}(\omega) \hat{q})}{\delta(\hat{p}, \hat{q})}\right]^\alpha = \left[ \frac{|s_1(A(\omega))s_2(A(\omega))|}{\|A(\omega)p\| \|A(\omega)q\|}\right]^\alpha &\leq \frac{|s_1(A(\omega))s_2(A(\omega))|^\alpha}{2}\left[ \frac{1}{\|A(\omega)p\|^{2\alpha}}+\frac{1}{\|A(\omega)q\|^{2\alpha}}\right].
\psi^{( + )}\stackrel{r \sim 0}{\rightarrow}\left[ c_{1}\; r^{\left({\frac{7}{4}}\right)}\left( 1 +{\cal O}( r ) \right) \, \, + \, c_{2}\; r^{\left( -{\frac{3}{4}}\right)}\left( 1 +{\cal O}( r ) \right) \, \right] .
N_1(a, R) :=a\left(\log c_2(\overline{\alpha}; R)+k\left(\log k+a_2(m)\varphi(m)+1\right)\right)+\frac{a_6(m)k\varphi(m)a}{\log a}\quad+2.5\log a+\log(k+1)+2.5\log k+\log H+ (k-1)\log\left(\max_{1\leq j \leq k}\{|\alpha_j|\}\right) \quad+\log\sqrt{2\pi}+1+k\left(a_2(m)\varphi(m)+2\right)+\frac{a_6(m)k\varphi(m)}{\log a}+\frac{19}{12a}.
a_N - 1 = \frac{1}{\sin (2 \pi / s)}\begin{cases}\frac{\sin (4 \pi / s)}{2 - 2 \cos (2 \pi / s)}, \text{if $s$ is even,}\frac{2 \sin (\pi / s) - \sin (2 \pi / s) + \sin (4 \pi / s)}{2 - 2 \cos (2 \pi / s)}, \end{cases}
\left(\mathrm{curl}\, \sigma(\bar y_\alpha),\phi\right)-\tfrac{\alpha}{\nu}\, b(\bar y_\alpha,\phi,\mathrm{curl}\, \sigma(\bar y_\alpha))=\left(\tfrac{\alpha}{\nu}\,\mathrm{curl}\, \bar u_\alpha+\mathrm{curl}\, \bar y_\alpha,\phi\right)\mbox{for all}\ \phi\in{\cal D}(\Omega).
f_{ext}= \chi_{> 2R}f + [(\partial_k T^w_{g^{kl}_{ext}}\partial_l + T^w_{b_{ext}}\cdot\nabla), \chi_{> 2R}] w + (\partial_k T^w_{g^{kl}_{ext}}\partial_l - \partial_k T^w_{g^{kl}}\partial_l)w_{ext}+(T^w_{b_{ext}}\cdot\nabla - T^w_{b}\cdot \nabla)w_{ext}.
w =\left(\begin{array}[c]{cc}0 c 0 0\end{array}\right) m =\left(\begin{array}[c]{cc}m_{1}0 0 m_{2}\end{array}\right) ,\forall m\in M_{+}x =\left(\begin{array}[c]{cc}x_{1}0 0 x_{2}\end{array}\right) ,\forall x\in algLat(M_{+})
\frac{1}{( q \! \cdot \! n )^{j}}= P V \left( \frac{1}{( q \! \cdot \! n )^{j}}\right) - i \pi \frac{( - 1 )^{j - 1}}{( j - 1 ) !}\delta^{( j - 1 )}( q \! \cdot \! n ) s g n ( q^{0}) ,
X_P=((\sigma_{j_b}P_{y_b})_{1\leq b\leq n}, -(\sigma_{j_b}P_{x_b})_{1\leq b\leq n},-\mathbf{i}(\sigma_jP_{\bar{z}_j})_{j\in\mathbb{Z}_*},\mathbf{i}(\sigma_jP_{z_j})_{j\in\mathbb{Z}_*})^T
\left\Vert \nabla u_{\lambda}\right\Vert_{p}=\lambda^{\frac{Nd-sd}{r}+\frac{p-N}{p}}\left\Vert \nabla u\right\Vert_{p}; \left\Vert u_{\lambda}\right\Vert_{k}=\lambda^{\frac{Nd-sd}{r}-\frac{N}{k}}\left\Vert u\right\Vert_{k},
\mu_f(x,t) = \frac{1}{2t}\int_{x_k-t}^{x_k+t}\left(\frac{1}{(2t)^{k-1}}\,\int_{x_{k-1}-t}^{x_{k-1}+t}\cdots\int_{x_1-t}^{x_1+t}f(\tau_1,\ldots,\tau_{k-1},\tau_k)\,d\tau_1\cdots d\tau_{k-1}\right) d\tau_k\,.
\mspace{-3mu}h \mspace{-3mu}\left(\!{\left.{{{\bf{Y}}^{m\! -\! 1}}}\right|{{\bf{X}}^{n\! -\! 1}},{\bf{Y}}_m^{n\! -\! 1},{{\bf{S}}_0}\! =\!{{\bf{s}}_0}}\right)\! =\!\frac{1}{2}\log\left(\!\left(2 \pi e\right)^{n_r m}\big|\mathsf{\tilde Q}\big|\right) \mspace{-2mu},
\begin{aligned}\delta(t) = \, -\frac{\beta}{m-1}(\tau+t)^{\alpha-1}+ \frac{C^{m-1}m}{a^2 (m-1)^2}(\tau+t)^{\alpha m -2 \beta}\geq \, \frac{C^{m-1}m}{2 a^2 (m-1)^2}(\tau+t)^{\alpha m - 2\beta}\,.\end{aligned}
S_{G , A}= \frac 1 2 \int_{0}^{T}d \tau \left[ \begin{array}{c}{\left( D_{\tau}X_{i}^{M}\right) \varepsilon^{i j}X_{j}^{N}G_{M N}\left( X_{1}, X_{2}\right)}\\{+ \left( D_{\tau}X_{i}^{M}\right) \varepsilon^{i j}A_{j N}\left( X_{1}, X_{2}\right) .}\\ \end{array}\right]
\begin{aligned}\phi_7^7(z) = g_7^2(7z)\{343\phi_7^6(z)+343\phi_7^5(z) + 147\phi_7^4(z)+ 49\phi_7^3(z)+21\phi_7^2(z) +7\phi_7(z)+1\}+g_7(7z)\{7 \phi_7^4(z)+35\phi_7^5(z)+49\phi_7^6(z)\}. \end{aligned}
G_{-1}=-G_{0,0}v\Bigl(m_0^\dagger +T+q_0^\dagger M_2(m_0^\dagger +T)M_2q_0^\dagger &-q_0^\dagger M_2(m_0^\dagger +T)-(m_0^\dagger +T) M_2q_0^\dagger \Bigr)v^*G_{0,0}&=-G_{0,0}v(I-q_0^\dagger M_2)m_0^\dagger (I-M_2q_0^\dagger)v^*G_{0,0}&-G_{0,0}v(I-q_0^\dagger M_2)T(I-M_2q_0^\dagger)v^*G_{0,0}.
N_\xi=\frac{1}{\pi i}\int_{\frac{3}{2}+iT_o}^{\frac{3}{2}+iT}d\nu\frac{\xi'}{\xi}+\frac{1}{\pi i}\int_{\frac{3}{2}+iT}^{iT}d\nu \frac{\xi'}{\xi}+\frac{1}{\pi i}\int_{iT_o}^{\frac{3}{2}+iT_o}d\nu\frac{\xi'}{\xi},=\frac{1}{\pi}\{\bigtriangleup arg\xi(\nu+1/2)\};
M_{\sigma}M_{\tau}=(\begin{matrix}0&1&0\\ 1&0&0\\ 0&0&1\end{matrix})(\begin{matrix}0&0&1\\ 1&0&0\\ 0&1&0\end{matrix})=(\begin{matrix}1&0&0\\ 0&0&1\\ 0&1&0\end{matrix})=M_{\sigma\tau}.
u_\alpha(x) +d_g(x_\alpha,x)|\nabla u_\alpha(x)|\leq C\mu_\alpha^{\frac{n-2}{2}}d_g(x_\alpha,x)^{2-n}, &|u_\alpha(x)-B_\alpha(x)|\leq C\mu_\alpha^{\frac{n-2}{2}}\left(r_\alpha^{2-n}+d_g(x_\alpha,x)^{3-n}\right)+\varepsilon_\alpha B_\alpha(x),
I_{22}\lesssim c_{\alpha,r}\nu \ell\sum_{p'> Q}\lambda_{p'}^\alpha\sum_{p\geq p'+2}\|R^\perp w_p\|_{\ell}\sum_{\substack{|p-q|\leq2 q>Q}}\|w_q\|_{\ell}^{\ell-1}&\lesssim c_{\alpha,r}\nu \ell\sum_{q> Q}\lambda_{q}^\alpha \|w_q\|_{\ell}^{\ell}.
g_1(x) =g(c^*)+ \kappa_1(s(x)-s(\ell)) +\int_{\ell}^r (s(x)-s(y))^+ \mu_1(dy), g_2(x) =\kappa_2(s(r)-s(x)) +\int_{\ell}^r (s(y)-s(x))^+ \mu_2(dy),
g_{k,M}(x) =\sum_{j=1}^{N_J}\int_{Q_j}D_k(x,y)g(y)d\mu(y) &=\sum_{j=1}^{N_J}\int_{Q_j}[D_k(x,y)-D_k(x,y_j)]g(y)d\mu(y) &\quad+\sum_{j=1}^{N_J}D_k(x,y_j)\int_{Q_j}g(y)d\mu(y) &:=g^1_{k,M,J}(x)+g^2_{k,M,J}(x),
\left [\mathbf{F}\right ]_{i,j}= \frac{2}{\sigma_{m}^{2}}\mathrm{Re}\Big \{\sum_{l=1}^{L}\frac{\partial \mbox{\boldmath$\mu$}\left [ l \right ]^{H}}{\partial \xi_{i}}\frac{\partial \mbox{\boldmath$\mu$}\left [ l \right ]}{\partial \xi_{j}}\Big \}, \i,j = 1, \cdots, 4,
\sqrt{\left(d\mathrm{vol}_{g},(\pi\circ\Phi_{i})^{\ast}d\mathrm{vol}_{g}\right)}=(2\pi\hbar)^{n/2}\left(1+\frac{1}{12}p^{j}p^{k}R_{jk}(q)\right)+O\left(\left\vert p\right\vert^{3}\right).
u_j(x) = \int_M \left(-\Delta_g G_M^\omega(x,y) +_y(F(y)G_M^\omega(x,y)) - \omega^2G_M^\omega(x,y)\right)u_j(y) d\mu_g(y), &= (\lambda_j-\omega^2) \sum_{k=1}^\infty \int_M v_k(x) u_k(y) u_j(y) e^{\phi(y)}d\mu_g(y), = (\lambda_j - \omega^2) v_j(x) \int_M |u_j(y)|^2 e^{\phi(y)}d\mu_g(y).
\|W\|_F^2 = \sum_{t=1}^{p}\|W_{:,t}\|^2_{2}= \sum_{t=1}^{p}\left\|\frac{S_{:,j_{t}}}{\sqrt{p P'_{j_{t}}}}\right\|^2_{2}= \sum_{t=1}^{p}\frac{\|S_{:,j_{t}}\|^2_{2}}{p\frac{\|S_{:,j_{t}}\|^2_{2}}{\|S\|_F^2}}= \sum_{t=1}^{p}\frac{\|S\|_F^2}{p}= \|S\|_F^2.
\renewcommand{\arraystretch}{1.4}T^{(2)}_i(s)= \left( \begin{array}{cc}\rho_i/s+O(1) O\left(s^{(k-n-1)/2}\right) O\left(s^{(k-n-1)/2}\right) T^{(1)}_i(1)+O(s) \end{array}\right), \quad i=1,2,3.
\delta_{\xi}g_{\mu \nu}= 2 \nabla_{( \mu}\xi_{\nu )}, \ \ \delta_{\xi}\Gamma_{\rho \sigma}^{\tau}= \frac{1}{2}g^{\tau \nu}\left( \nabla_{\rho}\delta_{\xi}g_{\sigma \nu}+ \nabla_{\sigma}\delta_{\xi}g_{\rho \nu}- \nabla_{\nu}\delta_{\xi}g_{\rho \sigma}\right) .
\left| \dfrac{d^{n+m}}{dx^{n+m}}\phi'(x) \right| = M(\phi, m+1, j)^n \left| M(\phi, m+1, j)^{-n}\phi^{n+m+1}(x) \right|\leqslant M^n \, \sup_{n}p_{(m+1,j)}\big( M^{-n}\phi^{(n)}\big),
{\cal{L}}_{\mathrm{g h .}}= -{\partial}^{\mu}{\bar{c}}_{a}D_{\mu}{c^{a}}= -{\partial}^{\mu}{\bar{c}}_{a}({\partial}_{\mu}c^{a}+{\varphi_{r s}}^{a}{A_{\mu}}^{r}{c^{s}}) ~ .
H_{D}^{f e r}= \frac{1}{\sqrt{2}}\int d^{3}x \xi^{\dagger}( \vec{x}) \left({\cal G}_{( - )}[ V_{-}] \ast \xi \right) ( \vec{x}) + \frac 1 2 \frac{e^{2}}{2 L}\int d^{2}x_{\perp}\Gamma^{i}( x_{\perp}) \left({\cal G}_{( \perp )}[{\cal M}^{2}] \ast \Gamma^{i}\right) ( x_{\perp})
\|\Psi^{\lambda_u,\lambda_w}\|_{\mathcal{S}_p}= (\sum_{v\in \Gamma}\langle |\Psi^{\lambda_u,\lambda_w}|^p(\lambda_{v}),\lambda_{v}\rangle )^{\frac{1}{p}}= \|\gamma_{u,w}^{\psi}\|_{\ell_p(\Gamma)}.
Z_{N}^{inf}(x_1,\cdots, x_M) = (x_1+\cdots+x_M)^N = \sum_{{n_i}\atop{\Sigma n_i=N}}~{N!\over n_1!n_2!\cdots n_M!}~x_1^{n_1}x_2^{n_2}\cdots x_M^{n_M}\,\,\,\,.
\int_{M}|\tilde{F}-(1-b_{t_0,B}(\Psi_1))fF^{1+\delta}|^2_{\tilde{h}}e^{v_{t_0,B}(\Psi_1)-\delta \tilde{M}}c(-v_{t_0,B}(\Psi_1)) \le \left(\frac{1}{\delta}c(T_1)e^{-T_1}+\int_{T_1}^{t_0+B}c(t)e^{-t}dt\right)\int_M \frac{1}{B}\mathbb{I}_{\{-t_0-B< \Psi_1 < -t_0\}}{|fF|}^2_{\tilde{h}},
\frac{h_{ii}}{n}\tilde{\theta}_i^3(\tilde{\theta}_i-\hat{\theta}_i) +\tilde{\theta}_i^3 \left( \sum_{j \neq i}^{}\frac{h_{ij}}{n}(\tilde{\theta}_i-\hat{\theta}_i) + \frac{h_{i,|k|+1}}{n}(\tilde{\phi}-\hat{\phi}) \right)- \frac{2}{n}\tilde{\theta}_i^2 + \frac{2\phi}{n}=0.
\langle T_{\mu}^{\mu}\rangle_{r e n}= \frac{1}{( 4 \pi )^{3}}\left[ \frac{1 3 5}{1 6}\Lambda^{3}- \frac{6 3}{4}\left( \frac{\Lambda^{2}}{r^{2}}\right) + \frac{2 5}{2}\left( \frac{\Lambda}{r^{4}}\right) - \frac{2 2 4 9}{8 4 0}\left( \frac{1}{r^{6}}\right) \right] ,
p(n)\sim\begin{cases}\begin{tabular}{ll}$c\: n^{-1}m^{-\frac{1}{h}}$\ \for $0<\alpha< \frac{2h}{h-1}$; $c\: n^{-\frac{h+1}{h-1}}$ \ \for $\alpha=\frac{2h}{h-1}$; $c\: n^{-\frac{1}{h-1}}m^{-\frac{1}{2}}$\ \for $\alpha> \frac{2h}{h-1}$.\end{tabular}\end{cases}
\frac{(2\kappa\beta)^{-m/\alpha}}{1+\kappa\frac{m}{2\alpha}}\frac{\Gamma(\frac{1}{\kappa}+\frac{m}{\alpha})\Gamma(1-\frac{m}{2\alpha})}{\Gamma(\frac{1}{\kappa}+\frac{m}{2\alpha})}
\mathcal{A}\left(\begin{array}{c}u w \theta q\end{array}\right) =\left(\begin{array}{c}w \frac{1}{m(x)}\left(p(x)\,u_{x}+ 2\,\delta(x)\,w_{x}- \eta\,\theta\right)_{x}-\kappa\,q_{x}- \eta\,w_{x}\frac{-1}{\tau}\left(\kappa\,\theta_{x}+ \beta\,q\right)\end{array}\right).
\mathcal S_1(k,3)= \left\{F\in\binom{[n]}{k}:[2]\subseteq F,\,F\cap[3,k-1]\neq\emptyset\right\}\cup\left\{F\in\binom{[n]}{k}:[2]\cup[k,k+2]\subseteq F\right\}&\cup\left\{[2,k]\cup\{i\}:i\in[k+1,n]\right\}\, \cup\, \left\{[2,k+2]\setminus\{k\},\, [k+1]\setminus\{2\},\, [k+2]\setminus\{2,k+1\}\right\}.
{\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) .
x^*(x)=\sum_{i=1}^n \lambda_i f_{S_i}(x)\leq \sum_{i=1}^n \vert \lambda_i\vert \vert f_{S_i}(x)\vert \leq \left(\sum_{i=1}^n \vert \lambda_i\vert^q \right)^\frac{1}{q}\left(\sum_{i=1}^n \vert f_{S_i}(x)\vert^p \right)^\frac{1}{p}& \leq \left(\sum_{i=1}^n \left\vert \sum_{t\in S_i}x(t) \right\vert^p \right)^\frac{1}{p}\leq 1
x^{\mu}- y^{\mu}+ \frac{2 i}{( 1^{1}2_{4})}\xi_{2 / 4}\sigma^{\mu}\bar{\theta}^{1}\, , \qquad \xi_{2 / 4}\equiv [ ( 1^{2}2_{4}) \theta_{2}+ ( 1^{3}2_{4}) \theta_{3}+ ( 1^{4}2_{4}) \theta_{4}]
\left({\Phi_{\alpha_1}\cdots\Phi_{\alpha_{N-2}}{\cal B}_{\beta_1\dots\beta_{N-4}}\over \Lambda^3}\right)^{3\over{2N-9}}= \left({\left[\Phi_{\alpha_1}\cdots\Phi_{\alpha_{N-2}}{\cal B}_{\beta_1\dots\beta_{N-4}}\right]^3\over \Lambda^9}\right)^{1\over{2N-9}}~.
\gamma_{a \dot{b}}^{i}\tilde{\gamma}_{\dot{b}c}^{j}+ \gamma_{a \dot{b}}^{j}\tilde{\gamma}_{\dot{b}c}^{i}= 2 \delta^{i j}\delta_{a c}\; , \qquad \tilde{\gamma}_{\dot{a}b}^{i}\gamma_{b \dot{c}}^{j}+ \tilde{\gamma}_{\dot{a}b}^{j}\gamma_{b \dot{c}}^{i}= 2 \delta^{i j}\delta_{\dot{a}\dot{c}}\; .
\begin{array}{l}{{\psi_{a}( z ) \psi^{b}( w ) = \frac{\delta_{a}^{b}}{( z - w )}+ \cdots}}\\{{\}}\\{{\psi_{a}( z ) \psi_{b}( w ) = \psi^{a}( z ) \psi^{b}( w ) = 0 + \cdots}}\\ \end{array}
\left.\begin{array}{ll}c_1|\xi|\leq\left|g^*_1(\xi,\eta)\right|=\left|g_1(\xi,-\eta)\right|\leq d_1|\xi|, c_2|\eta|\leq\left|g^*_2(\xi,\eta)\right|=\left|-g_2(\xi,-\eta)\right|\leq d_2|\eta|.\end{array}\right.
w_n(t,\cdot) =e^{t(\Delta-I)}w_n(0)+\underbrace{\int_{0}^{t}e^{(t-s)(\Delta-I)}\nabla\cdot (w_{n}(s,\cdot) b_{n}(s,\cdot))ds}_{I_1}\cr +\underbrace{\int_{0}^{t}e^{(t-s)(\Delta-I)}((1+f_n(s,\cdot)-\nabla \cdot b_n(s,\cdot))w_n(s,\cdot)+g_{n}(s,\cdot)v_{n}(s,\cdot)+h_{n}\cdot \nabla v_{n})ds}_{I_2},
{\cal G}_{a}^{( \pm )}\equiv \frac{L}{2}\eta_{a b}\epsilon^{i j}( \partial_{i}A_{j}^{( \pm ) b}- \partial_{j}A_{i}^{( \pm ) b}+ \epsilon_{\mathrm{}c d}^{b}A_{i}^{( \pm ) c}A_{j}^{( \pm ) d}) \approx 0 ,
\mathrm{I m}\int_{r_{o}}^{r_{f}}P_{-}d r = - \pi \int_{0}^{- \omega}{\frac{d{\cal H}- \Phi d Q}{\kappa ( M +{\cal H}, Q - e )}}= - \frac{1}{2}\int_{M , Q}^{M - \omega , Q - e}d S_{B H},
\int_{0}^{T}q^{T,k}_{n_{m_l}}(s,X)W_s \underset{l \to \infty}{\to}\int_{0}^{\tau^T_k(X)}\sigma(s,X)W_s = X_{\tau^T_k(X)}-X_0-\int_{0}^{\tau^T_k(X)}b(s,X)s \,\,\, \mathbb{P}
K_3(R,\rho):=\big[ R^2+ 2\rho^2 \big] \times \Big[ 1+ \frac{54}{\lambda_1^2}\,\big( R^2+\rho^2\big)^2e^{\frac{54}{\lambda_1^2}\,\big( R^2+\rho^2\big)^2}\Big].
u(x) = u(0,x_{n+1}) + |x_{n+1}|^{2s}\int_0^1 \sum_{i=1}^{n}\sum_{k=0}^{\lfloor \frac{(m-1)}2\rfloor}x_{n+1}^{2k}q_{i,m-1-2k}(tx') \, x_i\,\d t & = u(0,x_{n+1}) + |x_{n+1}|^{2s}\sum_{k=0}^{\lfloor \frac{(m-1)}2\rfloor}x_{n+1}^{2k}q_{m-2k}(x),
\nu^x_n(B) = \mu_n(B\vert x)=\frac{\mu_n(B\cap \lbrace x\rbrace)}{\mu_n(\lbrace x\rbrace)}\longrightarrow \frac{\mu(B\cap \lbrace x\rbrace)}{\mu(\lbrace x\rbrace)}=\mu(B\vert x) = \nu^x(B) n\rightarrow\infty.
{\cal L}={\cal L}_{0}+{\cal L}_{I}= \left\{\frac{1}{2}( \partial \Phi )^{2}- \frac{1}{2}m^{2}\Phi^{2}\right\}+ \left\{- \frac{\lambda}{4 !}\Phi^{4}+{\cal L}_{\mathrm{c t}}\right\},
\dot{A}= 2 - \frac{B}{A}- \frac{A^{2}}{C^{2}}, \quad \dot{B}= \frac{B^{2}}{A^{2}}- \frac{B^{2}}{C^{2}}, \quad \dot{C}= \frac{A}{C}+ \frac{B}{2 C},
\exp\left( i\theta H\right) =\left( H^{2}-iH\frac{d}{d\theta}-I\left(1+\frac{d^{2}}{d\theta^{2}}\right) \right) \sum_{k=0,1,2}\frac{\exp\left(\frac{2}{\sqrt{3}}~i\theta\sin\left( \phi+2\pi k/3\right) \right)}{1-2\cos2\left( \phi+2\pi k/3\right)}\ .
d s^{2}= \left( \frac{r}{R_{0}}\right)^{2 ( d_{\perp}- 2 ) / ( p + 1 )}d x^{\mu}\otimes d x^{\nu}\eta_{\mu \nu}+ \left( \frac{R_{0}}{r}\right)^{2}d y^{I}\otimes d y^{J}\delta_{I J}\ .
\sum_{n}\frac{e^{i n ( \phi - \phi^{\prime})}}{2 \pi}\frac{\Gamma ( \frac{| n |}{\alpha}+ 1 - \frac{i c t}{l_{P}})}{\Gamma ( \frac{| n |}{\alpha}+ 1 )}\rho^{\frac{| n |}{\alpha}+ 1}{_2 F_{1}}( \frac{| n |}{\alpha}+ 1 - \frac{i c t}{l_{P}}; \frac{| n |}{\alpha}+ \frac{1}{2}; 2 \frac{| n |}{\alpha}+ 1 ; 4 \rho )
JD_{A_\rho}J^{-1}= J D J^{-1}+ J A_\rho J^{-1}+ \epsilon' J^2 A_\rho J^{-2}, &= \epsilon' D + J A_\rho J^{-1}+ \epsilon' A_\rho, &= \epsilon' (D + \epsilon' J A_\rho J^{-1}+ A_\rho) = \epsilon' D_{A_\rho}
\int_{\mathbb{R}^{2N}}\frac{J_p(U(x)-U(y))\, \big(\varphi(x)-\varphi(y)\big)}{|x-y|^{N+s\,p}}\,dx\,dy=\mathcal{S}_{p,s}\, \int_{\mathbb{R}^N}|U|^{p^*-2}\,U\,\varphi\,dx,
I + \omega (A_{\varnothing+}^{(2)}(\xi) - \omega )^{-1}= \sum_{k=1}^\infty \left( 1 + \frac{\omega}{\zeta_k(\xi) - \omega}\right) P_k (\xi) & = \frac{\zeta_1(\xi)}{\zeta_1(\xi) - \omega}P_1(\xi) + \sum_{k=2}^\infty \left( 1 + \frac{\omega}{\zeta_k(\xi) - \omega}\right)P_k(\xi) ,
B((x,y,z),\alpha(d))= B(\alpha(x)(yz),\alpha(d))-B((xy)\alpha(z),\alpha(d)) = B(\alpha^{2}(x),(yz)d)-B(\alpha(x)\alpha(y),\alpha(z\alpha^{-1}(d))) = B(\alpha^{2}(x),(yz)d)-B(\alpha^{2}(x),\alpha(y)(z\alpha^{-1}(d))) = -B(\alpha^{2}(x),(y,z,\alpha^{-1}(d))).
Z_{p_{U, X, Y}}(U|Y) \triangleq 2\sum_{y\in \mathcal{Y}}p_{Y}(y)\sqrt{p_{U|Y}(0|y)p_{U|Y}(1|y)}& = 2\sum_{y\in \mathcal{Y}}\sqrt{p_{U,Y}(0, y)p_{U,Y}(1,y)},
g_2(z)^{-1}=\dot{l}(u_2(z)^{-1})\left (\dot{d}(u_2(z)^{-1})a_2^{-1}\right ) \left (a_2\dot{u}(u_2(z)^{-1}) a_2^{-1}l_2(z)^{-1}\right).
H^F_D = \tilde{\lambda}^\mu(\tau) \tilde{H}_\mu^{*}(\tau) -\frac 12 \tilde{\lambda}^{\mu \nu}(\tau) \tilde{H}^{*}_{\mu \nu}(\tau) +\int{d^3\sigma \Big[ - A_\tau (\tau, \vec{\sigma}) \Gamma^{*}(\tau, \vec{\sigma}) +\mu_\tau (\tau, \vec{\sigma}) \pi^\tau (\tau, \vec{\sigma}) \Big]},
W_x(1) =\prod_{D\in\mathcal{U}_{x}}(1- Q_{D,x,S}^t)\prod_{D\in\mathcal{S}_{x}}(1- Q_{D,x,U}^t)-\prod_{D\in\mathcal{U}_{x}}(1- Q_{D,x,*}^t)\prod_{D\in\mathcal{S}_{x}}(1- Q_{D,x,*}^t).
[[D_{\theta},D_{\mu}](x),\alpha^{k+s}(y)] =[D_{\theta}D_{\mu}(x),\alpha^{k+s}(y)]-(-1)^{\theta \mu}[D_{\mu}D_{\theta}(x),\alpha^{k+s}(y)] &=(-1)^{(\theta+\mu)x}([\alpha^{k+s}(x),D_{\theta}D_{\mu}(y)]-(-1)^{\theta \mu}[\alpha^{k+s}(x),D_{\mu}D_{\theta}(y)]) &=(-1)^{(\theta+\mu)x}[\alpha^{k+s}(x),[D_{\theta},D_{\mu}](y)].
\bar{\Xi}^{\lambda_1}_{\lambda_2}\otimes f^{\dot{I}}_{\dot{J}}\otimes \Xi^{\lambda_3}_{\lambda_4}\left( \bar{\phi}^{\rho_1}\otimes s^{\dot{K}}\otimes \phi^{\rho_2}\right) = \delta^{\rho_1}_{\lambda_2}\delta^{\dot{K}}_{\dot{J}}\delta_{\lambda_4}^{\rho_2}\bar{\phi}^{\lambda_1}\otimes s^{\dot{I}}\otimes \phi^{\lambda_3}.
( \mu \frac{\partial}{\partial \mu}+ \beta_{g}\frac{\partial}{\partial g}+ \beta_{\lambda}\frac{\partial}{\partial \lambda}- \gamma_{M}\frac{\partial}{\partial \ln M^{2}}+ \gamma ) F_{n}( x_{1}, . . . , x_{n}; m , M , g , \lambda , \mu ) = 0 ,
\Gamma_{\mu a b}( p , q ) = \gamma_{\mu a b}+ \int \frac{d^{3}\ell}{( 2 \pi )^{3}}\left[ S ( p + \ell ) \Gamma_{\mu}( p + \ell , q + \ell ) S ( q + \ell ) \right]_{d c}K_{c d , b a}( q + \ell , p + \ell , \ell ) ,
z\varphi_j^L - \rho_j^L \varphi_{j+1}^L = \alpha_j^\dag \varphi_j^{R,*}, z\varphi_j^R - \varphi_{j+1}^R \rho_j^R = \varphi_j^{L,*}\alpha_j^\dag, p^*(z) = z^{\deg( p )}p^\dag(z^{-1}).
\begin{aligned}f\left( M+m-\overline{a}\right) \le \sum\limits_{i=1}^{n}{{{w}_{i}}f\left( M+m-\left( \left( 1-t \right)\overline{a}+t{{a}_{i}}\right) \right)}\le f\left( M \right)+f\left( m \right)-\sum\limits_{i=1}^{n}{{{w}_{i}}f\left({{a}_{i}}\right)}\end{aligned}
p\left({d_{pF}\leq \delta}\right) = p\left({d_{pF}^2 \leq \delta^2}\right) = M\iint\limits_\Omega{\sin^{n - 4}\theta_1 \sin^{n - 4}\theta_2 \left({\sin^2 \theta_i - \sin^2 \theta_j}\right)^2 d\sin^2 \theta_1 d\sin^2 \theta_2}
\tilde{b}_{k}:=\frac{1}{z+s(z) +\Lambda_{k}}= \sum_{i=0}^{p}\frac{\Lambda_{k}^{i}}{(z+s(z))^{i+1}}+ \frac{\Lambda_{k}^{p+1}}{(z+s(z))^{p}(z+N^{-1}\mathrm{Tr}(G_{k}))}
B_0 =I\otimes I\otimes I, \ \ \ \ \ \ \ \B_4=I\otimes P\otimes P, B_1 =I\otimes I\otimes P , \ \ \ \ \ \ \B_5=P\otimes I\otimes P, B_2 =I\otimes P\otimes I, \ \ \ \ \ \B_6=P\otimes P\otimes I, B_3 =P\otimes I\otimes I, \ \ \ \ \ \B_7=P\otimes P\otimes P.
\lim_{i\to \infty}\int_\Omega\hat{\varphi}_i(t)dt =\int_\Omega\varphi(t)dt & \ge \int_\Omega L(t,x(t),x'(t))F\left(t,\int_0^t\psi(s)ds \right) & \ge \int_\Omega L(t,x(t),x'(t))F\left(t,\int_0^tf(s,x(s),x'(s))ds \right).
exp([\begin{matrix}3&-4\\ 4&-7\end{matrix}]t)=[\begin{matrix}4e^{t}/3-e^{-5t}/3&2e^{-5t}/3-2e^{t}/3\\ 2e^{t}/3-2e^{-5t}/3&4e^{-5t}/3-e^{t}/3\end{matrix}]
\Lambda\sum_{m\geq 0}(-\Lambda)^m =\Lambda(1-\Lambda+\Lambda^2 \ldots )=\frac{\Lambda}{1+\Lambda}, \Lambda^2\sum_{m\geq 0}(m+1)(-\Lambda)^m =\Lambda^2(1-2\Lambda+3\Lambda^2\ldots )=\Lambda^2 \frac{d}{d\Lambda}\frac{\Lambda}{1+\Lambda}=\frac{\Lambda^2}{(1+\Lambda)^2}.
q_2 = \frac{t_d}{t_{(\ell-1)d}}\frac{1 - \frac{\tan \left( \theta^+_{\chi}\right) + 1}{\tan \left( \theta^+_{\chi}\right) \ln \left( -\tan \left( \theta^+_{\chi}\right) \right)}}{1 + \frac{\tan \left( \theta^+_{\chi}\right) + 1}{\ln \left( -\tan \left( \theta^+_{\chi}\right) \right)}}q_1 \;.
x_{\sigma(\iota)}=\frac{w_{j_{k'},i}}{w_{j_k,i}}\Bigl(\frac{y_{\sigma(\iota_i)}}{w_{j_{k'},i}\bigl(1+\frac{\lambda_{\ell_{\iota_i}}}{\mu_{i}}\bigr)}-\frac{y_{\sigma(\iota_{i^*_{k'}})}}{w_{j_{k'},i^*_{k'}}\bigl(1+\frac{\lambda_{\ell(i^*_{k'})}}{\mu_{i^*_{k'}}}\bigr)}\Bigr);
{\textstyle\sum\limits_{k\geq0}}P_{x_{N}=ke}e ={\textstyle\sum\limits_{k\geq0}}{\textstyle\sum\limits_{n\geq0}}P_{N=ne}P_{x_{N}=ke}e={\textstyle\sum\limits_{k\geq0}}{\textstyle\sum\limits_{n\geq0}}P_{N=ne}P_{x_{n}=ke}e & ={\textstyle\sum\limits_{n\geq0}}P_{N=ne}{\textstyle\sum_{k\geq0}}P_{x_{n}=ke}e={\textstyle\sum\limits_{n\geq0}}P_{N=ne}e=e.
x_{k+2}=Px_{k+1}+Qx_k &=P\sum_{l=0}^k(a+(k-l)d)r^{k-l}x_l+Q\sum_{l=0}^{k-1}(a+(k-1-l)d)r^{k-1-l}x_l &=P\sum_{l=0}^k(a+(k-l)d)r^{k-l}x_l+Q\sum_{l=1}^k(a+(k-l)d)r^{k-l}x_{l-1}.
\Lambda(s,{\rm sym}^m f \times{\rm sym}^m g) & :={\rm Cond}({\rm sym}^m f\times{\rm sym}^m g)^{s/2}L_\infty(s,{\rm sym}^m f \times{\rm sym}^m g)L(s,{\rm sym}^m f \times{\rm sym}^m g)
S_{\infty}^{\chi_{r}^{2}}=\lim_{T\longrightarrow\infty}\frac{C_{1}^{L}}{a_{d,1}^{\chi_{r}^{2}}(\mathcal{D})\mathcal{L}(T)T^{d-\alpha}}\int_{\mathcal{D}(T)}e_{1}^{(r/2)}(\chi_{r}^{2}(\mathbf{x}))~d\mathbf{x}.
L_{i}(x)=L_{0i}+L_{1i}x+\cdot\cdot\cdot+L_{(n-1)i}x^{n-1}=\prod_{j\ne i}\frac{x-\lambda_{j}}{\lambda_{j}-\lambda_{i}}=\frac{p(x)}{(x-\lambda_{i})p^{\prime}(\lambda_{i})}.
\sum_{j=1}^m\frac{(-m-i+1)_{j-1}(-m+i)_{j-1}}{(\tfrac{\beta}{2}-m+1)_{j-1}(-\tfrac{\beta}{2}-m)_{j-1}}=-\frac{(\tfrac{\beta}{2}-m+1)(-\tfrac{\beta}{2}-m)}{(\tfrac{\beta}{2}+i)(\tfrac{\beta}{2}-i+1)}
(\overline{a}_{j}^{0})^{\top}x_{0}-\overline{b}_{j}-\Vert \big((\overline{a}_{j}^{1})^{\top}x_{0},\ldots ,(\overline{a}_{j}^{q_{j}})^{\top}x_{0}\big)\Vert >0,\,j=1,\ldots ,p.
\int_{D_{j_{0}}}\overset{3}{\mathbb{P}}\left(E^{T}_{j_{0}}\right)(x) \, dx - \eta \, \sum_{j=1 \atop j \neq j_{0}}^{\aleph}{\bf{P}}_{1}\cdot \Upsilon_{k}(z_{j_{0}},z_{j}) \cdot \int_{D_{j}}\overset{3}{\mathbb{P}}\left(E^{T}_j\right)(y)\, dy = \mathcal{S}_{ource, j_0}+\mathcal{O}(a^{\min(6,8-h-3t)}).
\begin{aligned}\sum_{s\in A-A_{}}L(s)Q^N(s)\le &\Big(\sum_{s\in A_{}:r_s=2}L(s)Q^N(s)\Big)\sum_{r=2}^\infty \big((1+2^{l-1}e^{-c_0N})^r-1\big)e^{-c_1(r-2)N}K_l^{r-2}+ &\sum_{r=2}^\infty \big((1+2^{l-1}e^{-c_0N})^r-1\big)e^{-c_1(r-2)N}K_l^{r-2}(2(l-1)(r-2)).\end{aligned}
\left|\mathcal{F}\left[f\cdot S_{l}g\right]\left(\omega\right)\right|^{2}=\left|\int_{-\infty}^{\infty}\!\!\!f\left(t\right)g\left(t-l\right)e^{-2\pi i\omega t}dt\right|^{2}=\left|\left\langle f,h\right\rangle \right|^{2}