harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|\mathcal{T}_t(f)\|_{A^p}\leq \frac1{t^{1-\frac2p}(1-t)^{\frac2p}}\left(\int_{\mathbb{D}}\left|f(y)\right|^{p(m-1)}d\mu(y)\right)^\frac1p = \frac1{t^{1-\frac2p}(1-t)^{\frac2p}}\left(\left(\int_{\mathbb{D}}\left|f(y)\right|^{p(m-1)}d\mu(y)\right)^\frac1{p(m-1)}\right)^{m-1}= t^{\frac2p-1}(1-t)^{-\frac2p}\|f\|_{A^{p(m-1)}}^{m-1}.
\big|\partial^{\alpha_1}y_i \partial^{\alpha_2}y_j\big| \le C\sum_{|\alpha'_1|\le|\alpha_1|}\|\partial^{\alpha'_1}f\|_{L^2_{p,\mathcal{I}}}\sum_{|\alpha'_2| \le|\alpha_2|}\|\partial'^{\alpha_2}f\|_{L^2_{p,\mathcal{I}}}.
P_{C}(K_{o}x,w_{o})=E[|y(i,n)|^{2}]=\frac{1}{\sum_{\alpha=0}^{N-1}\sum_{\beta=0}^{M-1}\sum_{l=0}^{N-1}\sum_{m=0}^{M-1}\psi_{e}(l,\alpha;m,\beta)}
\mathbb{E}\Big(\Big\|\sum_{j=1}^m \epsilon_j a_j u_j \Big\| \Big)= \mathbb{E}\Big(\Big\|\sum_{j=1}^m \epsilon_j |a_j| u_j \Big\| \Big)\geq C \Big\|\sum_{j=1}^m |a_j| u_j\Big\|\geq Cx^*\big(\sum_{j=1}^m |a_j| u_j\big)\geq \geq C\varepsilon\sum_{j=1}^m |a_j|.
\boldsymbol{\omega}^*_j,\nu^*_j= \mbox{arg}\min_{\substack{\boldsymbol{\omega}_j \in \mathbb{R}^n \nu_j \in \mathbb{R}}}\sum_{i=1}^N I_{\{\mathcal{H}_j\}}(p_i) \mbox{s.t.}\boldsymbol{\omega}_j \neq 0 \mathcal{M}\subseteq \mathcal{H}_j p_i \in \mathcal{L}, \ \ \i=1,\ldots,N
H_1=\sum_{n=L}^{M_-}E_M(t,n)|C_n| \leq c \sum_{n=L}^{M_-}\frac{n^{d-1}}{M^d}\big(1-P_{\check x_n}(Y_t\in B_M)\big)\leq c_2\,{\rm e}^{-c_3 (\gamma t)^{2\varepsilon}}.
\psi(\xi)=\eta(\xi) \sum_{|\gamma|=m+1}\xi^\gamma\left(\frac{1}{m!}\int_{0}^{1}(1-t)^m \partial_{\gamma}\psi(t\xi)\,dt\right) + \frac{1-\eta(\xi)}{|\xi|^{2(m+1)}}\psi(\xi)\sum_{|\gamma|=m+1}c_{\gamma}\xi^{2\gamma}.
\begin{aligned}\frac{1}{2}\frac{d}{dt}\int |\nabla^k n|^2 dx+\int |\nabla^{k+1}n|^2 dx &=\int \nabla^k \left[- u \cdot \nabla n+|\nabla n|^2 (n+w_0)\right]\cdot \nabla^k n \dx=I_1+I_2.\end{aligned}
\frac{d}{dt}\int_0^{+\infty}\!\!\!x|u|^2dx = -2\,\int_0^{+\infty}\!\!\!u\bar{u}_xdx &=\frac{\alpha}{\gamma}\int_0^{+\infty}\!\!\!v^2-\mathcal{Q}^+_0 + \int_0^t\Big(2|u_x(0,s)|^2 +\frac{\alpha}{\gamma}v_x^2(0,s)\Big)ds &>-\mathcal{Q}^+_0,
\gamma_{o}(G) =\gamma_{o}(G_{0})+\sum_{i=2}^{i=\frac{k}{2}}\gamma_{o}(T_{i}) & =\frac{n(G_{0})-l(G_{0})+s(G_{0})}{3}+\sum_{i=2}^{\frac{k}{2}}\frac{n(T_{i})-l(T_{i})+s(T_{i})+1}{3}& =\frac{n(G_{0})-l(G_{0})+s(G_{0})}{3}+\sum_{i=2}^{\frac{k}{2}}\frac{n(T_{i})-l(T_{i})+s(T_{i})}{3}+\frac{\frac{k}{2}-1}{3}
\left(\begin{array}{cc}B-\widehat{\lambda}_{k}I -\mathbf{x}_{k}-\mathbf{x}_{k}^{T}0\end{array}\right) \left(\begin{array}{c}\Delta \mathbf{y}_{k}\delta_{k}\end{array}\right) =\left[\begin{array}{c}(\widehat{\lambda}_{k}I-B)\,\mathbf{x}_{k}0\end{array}\right] ,
\int_X f(x_1,x_2) d\mu(x_1,x_2) = \int_{X_1}\left( \int_{X_2}f(x_1,x_2)d\mu_2(x_2)\right) d\mu_1(x_1) & = \int_{X_2}\left( \int_{X_1}f(x_1,x_2)d\mu_1(x_1)\right) d\mu_2(x_2).
\begin{aligned}\lim_{\epsilon\rightarrow0}\big(\mathbf{D}_{26}+\mathbf{D}_{27}\big)=\mathbb{E}\int_0^\infty e^{-\beta t}\bar{f}\mathcal{Z}_1dt+\mathbb{E}\int_0^\infty e^{-\beta t}\bar{F}_uv dt. \end{aligned}
H^{(2)}=\int d^3{x}\,\biggl[\,\, \frac{1}{2}\mbox{tr}\,\left(P - \frac{\theta}{8 \pi^2}\, B^{(+)}\right)^2+\left(a_s^{(0)}(S, P) - \frac{\theta}{8\, \pi^2}\, B^{(-)}_s\right)^2 + \,\frac{1}{2}V(S) \ \biggr]\,,
\Big|\sum_{\widetilde\alpha}(-1)^{\widetilde{q}(\widetilde\alpha)}Z_{\widetilde\alpha,\widetilde{D}_0}(\widetilde{G})\Big|=\Big(\sum_{w_1(\alpha)=0}(-1)^{\frac{q(\alpha)}{2}}Z_{\alpha,D_0}(G)\Big)^2+\Big(\sum_{w_1(\alpha)=1}(-1)^{\frac{q(\alpha)-1}{2}}Z_{\alpha,D_0}(G)\Big)^2\,,
x y^{2}= \left[ x \prod_{a = 1}^{n_{c}}( x - \phi_{a}^{2})^{2}+ 2 \Lambda^{2 n_{c}+ 2 - n_{f}}m_{1}\cdots m_{n_{f}}\right]^{2}- 4 \Lambda^{2 ( 2 n_{c}+ 2 - n_{f})}\prod_{i = 1}^{n_{f}}( x + m_{i}^{2}) , \quad U S p ( 2 n_{c}) .
u(z) = \log|z| - \log|z-c\kern .3pt| + C + \sum_{k=1}^N \left[a_k^{}\kern 1pt\hbox{Re\kern 1pt}((z-c)^{-k}) + b_k^{}\kern 1pt\hbox{Im\kern 1pt}((z-c)^{-k})\right],
Q/Q(2) \downarrow X = (3^{[s]}\otimes 1^{[s+1]}) + (1^{[r]}\otimes 4^{[s]}) + (1^{[r]}\otimes 2^{[s+1]}), Q(2)/Q(3) \downarrow X = (1^{[r]}\otimes 3^{[s]}\otimes 1^{[s + 1]}) + 0, Q(3)/Q(4) \downarrow X = 3^{[s]}\otimes 1^{[s+1]}, Q(4) \downarrow X = 0.
\mathrm{Tr\,}e^{-t\Box^+_{b,m}}(x,x)-\mathrm{Tr\,}e^{-t\Box^-_{b,m}}(x,x) &\sim\Bigr(\sum\limits^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi}\Bigr)\sum^\infty_{j=0}t^{-n+j}\Bigr(\mathrm{Tr\,}\alpha^+_{n-j}(x)-\mathrm{Tr\,}\alpha^-_{n-j}(x)\Bigr)\mod O\Bigr(e^{-\frac{\varepsilon_0\hat d(x,X_{\mathrm{sing\,}})^2}{t}}\Bigr).
\alpha\circ \mathcal{T}(a) =\alpha( T\varrho(x)(a)+[T(a),x])=T(\beta(\varrho(x)(a))+\alpha([T(a),x])) &=T(\varrho(\alpha(x))(\beta(a)))+[\alpha(T(a)),\alpha(x)]))=T(\varrho(x)(\beta(a)))+[T(\beta(a)),x])) &=\mathcal{T}(\beta(a)).
X\to q^{m{\rho}^{\vee}}X{\bar q}^{m{\rho}^{\vee}}\,,\,\,\,{\rho}^{\vee}=\sum_i{\Lambda}^{\vee}_i\,\,\,(\,\Rightarrow\,\,E_i\to q^mE_i\,,\,\,F_i\to{\bar q}^mF_i\,)\,.
m^{GP}=\Delta_L(k)=\frac{1}{2}\left( \begin{array}{cc}\chi_L(k)+\chi_L^{-1}(k) i\left(\chi_L(k)-\chi_L^{-1}(k)\right) -i\left(\chi_L(k)-\chi_L^{-1}(k)\right) \chi_L(k)+\chi_L^{-1}(k) \end{array}\right),
1\cdot\mathbb{P}(X<0.5)+0\cdot\mathbb{P}(X=0.5)+\frac{1}{3}\cdot\mathbb{P}(X>0.5)=1\cdot\frac{1}{6}+0\cdot\frac{1}{3}+\frac{1}{3}\cdot(\frac{1}{6}+\frac{1}{3})=\frac{1}{3},
\left( \begin{array}{c}{{\sigma^{i}[ \omega_{0}^{+ 0 i}\tilde{\epsilon}_{2}- i F^{+ 0 i}\tilde{\epsilon}^{1}]}}\\{{\sigma^{i}[ - \omega_{0}^{+ 0 i}\tilde{\epsilon}^{1}- i F^{+ 0 i}\tilde{\epsilon}_{2}]}}\\ \end{array}\right) = 0 \, ,
S^{( N )}\left[{\bf r}_{1}, \dots{\bf r}_{N - 1}\right] ({\bf r}^{\prime \prime},{\bf r}^{\prime}; t^{\prime \prime}, t^{\prime}) \equiv S^{( N )}\left[{\bf r}_{1}, \dots{\bf r}_{N - 1}\right] ({\bf r}^{\prime \prime},{\bf r}^{\prime}; t^{\prime \prime}- t^{\prime}) ,
b_1(2n-1)^2-b_1(2n-2)b_1(2n) =b_1(2n-2)^2-b_1(2n-2)b_1(2n) =b_1(2n-2)\left(b_1(2n-2)-b_1(2n)\right)=-b_1(2n-2)b_1(n).
\left\| \nabla (\lambda +{\bf A})^{-1}\mathbb P\nabla\cdot(u\otimes v)\right\|_{L^q_{uloc}}\leq C|\lambda|^{-\frac12}\big ( 1 + |\lambda|^{\frac{d}{2}(\frac1q-\frac1p)}\big )\big ( \|u\nabla v\|_{L^q_{uloc}}+ \| v\nabla u \|_{L^q_{uloc}}\big )
\begin{cases}\frac{(P+ (u_\epsilon (x ) )_x )^2}{2}+V (x,\frac{x}{\epsilon})=\ln m_\epsilon (x )+\overline{H}_\epsilon (P ), \ \ \mathbb{T}, - (m_\epsilon (x ) (P+ (u_\epsilon (x ) )_x ) )_x=0, \ \ \mathbb{T}, \int_0^1 u_\epsilon (x )dx=0, \int_0^1 m_\epsilon (x )dx=1.\end{cases}
\frac{M(P,\vec{\nu})}{P^{n-rd}}=P^{rd}\sum_{(\vec{a},q)\in \mathcal{M}(\theta_0)}\frac{S_{\vec{a},q}(\vec{\nu})}{q^n}J_{\widetilde{C}^{r-1}P^{\eta}}(P^{-d}\vec{\nu}) + \mathfrak{O}_1+\mathfrak{O}_2.
n - 2\lfloor n/4 \rfloor -1=\begin{cases}n/2-1 \textrm{if $n = 0 \pmod 4$,}(n-1)/2 \textrm{if $n = 1 \pmod 4$,}n/2 \textrm{if $n = 2 \pmod 4$,}(n+1)/2 \textrm{if $n = 3 \pmod 4$.}\end{cases}
\left| |\xi^1_i + \xi^2_i|^{1/2}- |\xi^1_i|^{1/2}\right| = \frac{| |\xi^1_i + \xi^2_i| - |\xi^1_i||}{|\xi^1_i + \xi^2_i|^{1/2}+ |\xi^1_i|^{1/2}}& \leq C \left( |\xi^1_i|^{-1/2}|\xi^2_i|{\bf 1}_{|\xi^1_i| > \epsilon_n}+ |\xi^2_i|^{1/2}{\bf 1}_{|\xi^1_i| \leq \epsilon_n}\right),
{1\over 2\pi}L^{ab}_\infty J_a J_b, \hskip 10pt{1\over 2\pi}L^{ab}_\infty \bar J_a \bar J_b, \hskip 10pt{1\over 2\pi}\tilde L^{ab}_\infty J_a J_b, \hskip 10pt{1\over 2\pi}\tilde L^{ab}_\infty \bar J_a \bar J_b
\chi = \frac{-1}{8\pi^2}\int_M |W_{00}^+|^2+(60v^2-72 kv + 18k^2) \frac{3}{2}\sigma = \frac{1}{8\pi^2}\int_M 2|W_F^+|^2 + |W_{00}^+|^2 + 6(2k-3v)^2 \enskip\geq\enskip 0
\Sigma_1(f,z_1, z_2) = \int_{0}^{\infty}\int_0^\infty(f(0,0,t,u)+ f(0,0,t,-u))t^{z_1-1}u^{z_2-1}dt du \Sigma_2(f,z) = \int_0^\infty f(0,0,0,u)u^{z-1}du \Sigma_3(f,z,u) = \int_0^\infty f(0,0,t,u)t^{z-1}dt.
E ( \eta , \vartheta ) = - \frac{M}{2 \cos \frac{\pi}{2 \lambda}}\left( \cos \left( \frac{\eta}{\lambda}\right) + \cosh \left( \frac{\vartheta}{\lambda}\right) - \frac{1}{2}\cos \left( \frac{\pi}{2 \lambda}\right) + \frac{1}{2}\sin \left( \frac{\pi}{2 \lambda}\right) - \frac{1}{2}\right) \, .
\frac{\langle x, x \rangle^\frac{-1}{2}}{\sqrt{n}}\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}+\left(\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\right)\frac{\langle x, x \rangle^\frac{-1}{2}}{\sqrt{n}}=2-c_x
B_{j,k}=\frac{\Gamma(j+k+a)}{\Gamma(j+k+b+c)}\sqrt{\frac{\Gamma(j+b)\Gamma(j+c)\Gamma(k+b)\Gamma(k+c)}{\Gamma(j+a)\, j!\,\Gamma(k+a)\, k!}},\j,k\in\mathbb{Z}_{+}.
\lim_{\epsilon \rightarrow 0}\left( -{\frac{1}{1 6 \pi}}\int_{( M \backslash M_{\epsilon})_{t}}\ \sqrt{| g |}R \right) + H_{m a t t e r}^{\prime}+ \beta_{H m}^{- 1}{\frac{A}{4}}-{\frac{M_{t o t}}{2}}= 0 .
\|S_{1/N}-T_{m_{N}}:VN(\Gamma)\to VN(\Gamma)\| \leq \sum_{r=N^2+1}^\infty \|e^{-\frac{r}{N}}p_r\|\lesssim \beta\sum_{r=N^2+1}^\infty \frac{N^6}{r^6}(r+1) & \lesssim \beta\frac{1}{(N+1)^2}.
{\sf M}^{ab}[A^\omega] ={\sf D}^{ac}[\omega] \cdot\left.\frac{\delta\chi^c[A^{\tilde{\Omega}}]}{\delta \tilde{\varphi}^d}\right|_{\varphi=0}\cdot\left.\frac{\delta\tilde{\varphi}^d}{\delta\varphi^b}\right|_{\varphi=0}.
\leq k \frac{e}{2 \pi}\frac{\left( \frac{k}{2}\right)^{k/2 + 1/2}}{(a_4k)^{a_4k + 1/2}\left(\frac{k}{2}-a_4 k \right)^{k/2-a_4k+1/2}}(s^2-2)^{a_4k}2^{k/2 - a_4 k}2^{-1}&= \frac{e}{2 \pi}\frac{1}{\sqrt{a_4}\sqrt{1-2a_4}}\frac{1}{2}\cdot\frac{1}{(2a_4)^{a_4k}(1-2a_4)^{(1/2-a_4)k}}.
\| \mathcal{J}(t,\zeta) \| \le C_6 t^4 (c_* t^2 + \zeta)^{-2}+ C_7 t^2 (c_* t^2 + \zeta)^{-1},|t|\le t^0,\ \zeta >0; &C_6 = \beta_6 \delta^{-1}\bigl(\|X_1\|^4 + c_*^{-1}\|X_1\|^6 \bigr),C_7 = \beta_7 \delta^{-1}\bigl(\|X_1\|^2 + c_*^{-1}\|X_1\|^4\bigr).
\int_{B(0,s)}\log (1-|\xi|^2) d\mu(\xi)=2n\int_0^{s}\log(1-r^2) \frac{r^{2n-1}}{(1-r^2)^{n+1}}\; dr=-n\int_0^{\frac{s^2}{1-s^2}}x^{n-1}\log(1+x)\; dx.
\sum_{n_1 , ... , n_m = 0}^\infty \| T_m(\zeta_m,\zeta_m') V_{n_m}(A_m;z_m ,z_m') \cdots T_1(\zeta_1,\zeta_1') V_{n_1}(A_1;z_1 , z_1') \Psi \| < \infty .
\mathbf{S}_{}^*\triangleq \mathbf{Q}^{-\frac{1}{2}}\Tilde{\mathbf{V}}\Tilde{\boldsymbol{\Lambda}}_{\rm o}\Tilde{\mathbf{V}}^{\rm H}\mathbf{Q}^{-\frac{1}{2}}=\Tilde{\mathbf{V}}\left(\frac{\ln2}{\mu_2}\left(\nu_2\mathbf{I}_{r}-\boldsymbol{\Lambda}^{\rm H}\boldsymbol{\Lambda}\right)\right)^{-1}\Tilde{\boldsymbol{\Lambda}}_{\rm o}\Tilde{\mathbf{V}}^{\rm H},
f(z) =\sum_{\alpha\in A}c_\alpha(f)z^\alpha &= \sum_{\alpha\in \mathbb{N}^n \cap A}c_\alpha(f)z^\alpha+\sum_{\alpha\in A \setminus\mathbb{N}^n}c_\alpha(f)z^\alpha &=f_1(z)+f_2(z).
\mathbb{E}[X_{t}]_{L^{2}}^{2}-\sum_{k=1}^{N}\mathbb{E}[|Z_{k}|^{2}]\le\mathbb{E}[X_{t}]_{L^{2}}^{2}-\sum_{k=1}^{N}\mathbb{E}[|W_{k}|^{2}]
I_3 \leq\sum_{j=2}^\infty\dfrac{(\mu C^{p'})^{p-1+j}}{\Gamma(p+j)}\int_R^\infty r^{\alpha_0-\frac{(\alpha_0+1)(p-1+j)}{p}}\mathrm dr \leq\dfrac{pR^{\alpha_0+1}}{\alpha_0+1}\exp_{p+2}(\mu C^{p'}R^{-\frac{\alpha_0+1}{p}})\overset{R\to\infty}\longrightarrow0.
\frac{1}{|b_i|}\int_{b_i}u^{s'}(P)ds(P)= u^{s}(X)+\nabla u^{s}(X)\cdot (M_i-X) + \mathcal{R}_i(X) &+\frac{1}{|b_i|}\int_{b_i}\left(\left( M^{s}(\widetilde{Y}_i)-I \right)\nabla u^{s}(X)\right) \cdot (P-\widetilde{Y}_i) ds(P), ~ i\in \mathcal{I}^{s'},
\widetilde{B_2}(g)\widetilde{B_2}(h) =\sum \alpha_iB_2(g_i)^{-1}B_2(h)^{-1}=\sum \alpha_iB_2(B_1(g_i)hB_2(g_i))^{-1}&=\sum \alpha_iB_2(B_1(g_i)hS(B_2(g_i)^{-1}))^{-1}=\widetilde{B_2}(\widetilde{B_1}(g_1)hS(\widetilde{B_2}(g_2))).
\left\langle R_{s, l}\, ,\, \partial_{\theta}e^{(t-s)L_{\delta}^{\ast}}h\right\rangle_{d}= \left\langle R_{s, l}\, ,\, \partial_{\theta}e^{(t-s)L_{\delta}^{\ast}}h\right\rangle_{2, d}= - \left\langle e^{(t-s) L_{\delta}}\partial_{\theta}R_{s, l}\, ,\, h\right\rangle_{2, d}.
\det \partial J_L^l/\partial \omega =\frac{\prod_{j=-L}^{L-1}\sum_{i=0}^{\alpha-1}f_iu_{-L}^2(\alpha j+i,\omega,E)}{\prod_{k=-L}^{L-2}R^2_{-L}(\alpha k+\alpha -1,\omega, E)}\left(\sum_{i=-\alpha L}^{\alpha L-1}u_{-L}^2(i,\omega,E)\right)^{-2L}\times \det A,
Q^{(5)}\left( A_{2},A_{1}\right) =3\,\varepsilon_{abcde}\left( -\frac{1}{3\ell^{3}}\alpha_{0}\,R^{ab}e^{c}e^{d}e^{e}-\frac{1}{10\ell^{5}}\alpha_{0}\,e^{a}e^{b}e^{c}e^{d}e^{e}\right) .
U(x,t) = \frac{e^{-k_\ast\,t}}{\sqrt{4\,\pi\,K_{\alpha}\,t^{\alpha}}}\,H^{2,0}_{1,2}\left[ \frac{x^2}{4\,K_{\alpha}\,t^{\alpha}}\left|\begin{array}{ll}\displaystyle \left( 1-\frac{\alpha}{2},\alpha\right) [10pt](0,1) \left( \frac{1}{2},1\right) \end{array}\right.\right].
\sum_{n=1}^{N+1}\left( \frac{N}{2}q_n^2+m_n\right)=\frac{1}{2N}\sum_{n=1}^{N+1}\left( N\left(q_n+\frac{1}{2}\right)+(1-n)\right)^2 - \frac{(N+1)(N+2)}{24}.
N_2(0)=\frac14\sum_{n\in N}\big\{\big|(1\otimes Q^2)\;\!v\;\!(f_n\otimes1)\big\rangle\big\langle v\;\!(f_n\otimes1)\big|+\big|v\;\!(f_n\otimes1)\big\rangle\big\langle(1\otimes Q^2)\;\!v\;\!(f_n\otimes1)\big| &-2\;\!\big|(1\otimes Q)\;\!v\;\!(f_n\otimes1)\big\rangle\big\langle(1\otimes Q)\;\!v\;\!(f_n\otimes1)\big|\big\}.
\begin{array}{lc}(P_jQ_{jk}P_{k}w)(x,t) = c_j(x_j,x,t) [2mm]\displaystyle \quad\times\displaystyle \int_{0}^{1}r_{jk}(\xi,\omega_j(x_j,\xi,t))c_k(x_k,\xi,\omega_j(x_j,\xi,t))\int_{0}^{1}w(\eta,\omega_k(x_k,\xi,t))\, d\eta d\xi. \end{array}
\begin{aligned}I_1^2= [D^\alpha,u\partial_x+v\partial_y]u-[D^\alpha,h\partial_x+g\partial_y]h &+[D^\alpha,(u_e\phi+u_b(1-\phi))\partial_x]u-[D^\alpha,h_e\phi\partial_x]h =: I^2_{1,1}+I^2_{1,2}.\end{aligned}
q(x) =A \times \mathrm{HeunT}(-10^{\frac{1}{3}}s,3,-3 \,10^{\frac{2}{3}},-\frac{10^{\frac{1}{3}}(x-3)}{3}) &+ B \times{\mathrm e}^{-\frac{10 x^{2}(x-9)}{27}}\times \mathrm{HeunT}(-10^{\frac{1}{3}}s,-3,-3 \,10^{\frac{2}{3}},\frac{10^{\frac{1}{3}}(x-3)}{3})+\frac{1}{s},
R_{\mu \nu}- \nabla_{\mu}\Phi \nabla_{\nu}\Phi -{\frac{1}{4}}e^{4 \Phi}H_{\mu \rho \sigma}H_{\nu}^{~ ~ \rho \sigma}+{\frac{1}{6}}g_{\mu \nu}e^{4 \Phi}H^{2}+{\frac{4}{k}}g_{\mu \nu}e^{- 2 \Phi}= 0 ,
\int_{-\infty}^{+\infty}dy \frac{e^{iky}}{\cosh^\mu u}=\frac{2^{\mu-1}}{\Gamma(\mu)}\, \,\Gamma\left(\frac{\mu+ik}{2}\right)\Gamma\left(\frac{\mu-ik}{2}\right)~~~~~\mbox{for}~~~~~\mbox{Re}\,\mu>0,
\|\nabla(y^h-y_h)\|_{L^2(\Omega)}\le \|\nabla(y^h-B_hu^h)\|_{L^2(\Omega)}= \|\nabla y_0^h\|_{L^2(\Omega)}\le c\|f\|_{H^{-1}(\Omega)}+ \|\nabla (B_hu^h)\|_{L^2(\Omega)}.
\tau^{j,*}_{0}=0,\xi^{j,*}_{0}=j,\tau_k^{j,*}=\inf\{t\ge \tau^{j,*}_{k-1}:Y^{\xi^{j,*}_{k-1}}_t=H^{\xi^{j,*}_{k-1}}_t\}\wedge T,k\ge1,
Y_{l_{1},m_{1}}Y_{l_{2},m_{2}}={\displaystyle\sum\limits_{\substack{l=|l_{1}-l_{2}| l\equiv l_{1}+l_{2}(\operatorname{mod}2)}}^{l_{1}+l_{2}}}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{2l+1}}C_{m_{1},m_{2},m}^{l_{1},l_{2},l}C_{0,0,0}^{l_{1},l_{2},l}Y_{l,m}
i) \hbox{}\bigcup_{p+q=k}\Sigma_{\delta , p}(\rho_1)\times\Sigma_{\pi , m-q}(\rho_2)\subseteq\Sigma_{\delta ,k}(\tilde{\rho})\subseteq Sp_{\delta , k}(\tilde{\rho})\subseteq &\bigcup_{p+q=k}Sp_{\delta , p}(\rho_1)\times Sp_{\pi ,m-q}(\rho_2),
-\mathrm{div}(F_*(x)^\gamma \,F(\nabla u)^{p-1}\,\nabla F(\nabla u)) &=\beta^{p-1}\,\Big[N-p-\beta\, (p-1)+\gamma\Big]\, u^{p-1}\, F_*(x)^{\gamma-p},\mbox{in}\mathbb{R}^{N}\setminus\{0\}.
E_{r}|_{\mathcal{V}_{n}}=\sum_{j=0}^{r}e_{j}(t_{1},\ldots ,t_{n-r+j})\tilde{E}_{r-j,n}\tilde{E}_{r,n}=\sum_{j=0}^{r}(-1)^{j}\det (e_{1-a+b}(t_{1},\ldots,t_{n-r+b}))_{1\leq a,b\leq j}~E_{r-j}|_{\mathcal{V}_{n}}\;.
\Phi(r)(l)\pi(l^{-1}. r\#1) =\sum_{h \in H}\Phi(r)(h) \pi(1\#\delta_{l^{-1}h})=\sum_{h \in H}\Phi(r)(lh) \pi(1\#\delta_h) &= \pi\left( 1\#\sum_{h \in H}\Phi(r)(lh)\delta_h\right)
\acute{p}(1) |0\rangle\langle 0|^{\mathfrak{B}}+ \acute{p}(2) |1\rangle\langle 1|^{\mathfrak{B}}&= \acute{p}(1) |1\rangle\langle 1|^{\mathfrak{B}}+ \acute{p}(2) |2\rangle\langle 2|^{\mathfrak{B}}
\begin{aligned}\bold{pr}X =\xi^iD_i+\sum_{\alpha,J_1,J_2}\left(D_{J_1}Q^{\alpha}_{J_2}\right)\partial_{u_{J_1;J_2}^{\alpha}}&=\xi^iD_i+\sum_{\alpha,J_1,J_2}\left(D_{J_1}S_{J_2}Q^{\alpha}\right)\partial_{u_{J_1;J_2}^{\alpha}},\end{aligned}
\begin{array}{| l | l | l | l | l | l | l | l |}\hline{{s}}&{{s = 1}}&{{s = 2}}&{{s = 3}}&{{s = 4}}&{{s = 6}}&{{s = 8}}&{{s = 1 2}}\\ \hline{{\mathrm{q u o t . s i n g .}}}&{{A_{1}}}&{{A_{1}A_{2}}}&{{A_{1}^{2}A_{3}}}&{{A_{1}^{4}A_{4}}}&{{A_{1}^{3}A_{6}}}&{{A_{1}^{4}A_{8}}}&{{A_{1}^{6}A_{1 2}}}\\ \hline{{}}\\ \end{array}
0 = C_s \int_0^{\frac{1}{2\sqrt{N}}}\frac{\left(1+\tau^2+\frac{2}{\sqrt{N}}\tau\right)^{-\tilde \gamma/2}+\left(1+\tau^2-\frac{2}{\sqrt{N}}\tau\right)^{-\tilde \gamma/2}-2}{\tau^{1+2s}}\,d\tau + C_s O(1),
\sum_{i=0}^j (-1)^it_1^i\binom{j}{i}B_{n+1-i}&=\sum_{k=0}^{n-j}\sum_{\ell=0}^j \sum_{r=0}^{j-\ell}(-1)^rt_1^rt_{k+\ell+1}\binom{n-j}{k}\binom{j}{\ell}\binom{j-\ell}{r}B_{n-k-\ell-r}.
S [ \phi , g ] = S_{c}[ \phi_{c}, g ] + \left. \frac{\delta^{2}S [ \phi , g ]}{\delta \phi^{2}}\right|_{\phi = \phi_{c}}\frac{\tilde{\phi}^{2}}{2}+ \mathrm{~ h i g h e r ~ o r d e r ~ t e r m s ~ i n ~}\tilde{\phi}, \nonumber
\frac{d H}{d t}\simeq \frac{2}{5}M ( \langle R \rangle ) ( 3 2 \pi \langle R \rangle^{3}\Gamma ) \frac{\langle R \rangle}{R}\sim \frac{2}{5}M ( \langle R \rangle ) \frac{\frac{4 \pi}{3}\langle R \rangle^{3}}{\Delta V}( \Delta V - \Delta V^{\prime}) \Gamma ,
\sum\limits_{j=1}^{M}\Big(\sum\limits_{i=1}^{N}\varphi_{ij}\Big)^{2}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}\varphi_{ij}\Big(\sum\limits_{k=1}^{N}\varphi_{kj}\Big) begin{align*}2 pt] =\sum\limits_{i=1}^{N}\|\varphi_{i}\|^{2}+2\sum\limits_{1\leq i<k\leq N}\langle\varphi_{i},\varphi_{k}\rangle.
i\nabla_{\Omega\setminus s_\alpha}(e^{-\frac{i}{2}\theta_a}\varphi_a)=e^{-\frac{i}{2}\theta_a}(i\nabla+A_a)\varphi_a,i\nabla_{\Omega\setminus s_\alpha}(e^{-\frac{i}{2}\theta_0^a}\varphi_0)=e^{-\frac{i}{2}\theta_0^a}(i\nabla+A_0)\varphi_0,
[\Psi_G]_q=q^{N_G-1}-[\Psi^1]_q+ q^2 [\Psi^{12},\Psi^1_2,\Psi^2_1,\Psi_{12}]_q+ q [\Psi^{1,2}]_q - q [\Psi^{12},\Psi^2_1]_q.
v_i(\tau, \xi) = \int_0^\tau \sum_{j=1}^n \Big( C_{ij}\big(x_i(s, \tau, \xi)\big) v_j\big(s, x_i(s, \tau, \xi)\big) + D_{i j}\big(x_i(s, \tau, \xi)\big) v_j(s, 0) begin{align*}6pt] + f_i\big(s, x_i(s, \tau, \xi) \big) \Big) \, ds + v_{0, i}\big(x_i(0, \tau, \xi) \big),
U_n(\theta)=\sum_{i,j}(X^1_{t^1_i}-X^1_{t^1_{i-1}})(X^2_{t^2_j}-X^2_{t^2_{j-1}})1_{\{(t^1_{i-1},t^1_i]\cap(t^2_{j-1}-\theta,t^2_j-\theta]\neq\emptyset\}}.
h(j,t)= \begin{cases}2J(t) +\sum^j_{i=1}(1-2\eta_i(t)) \textrm{for}j\geq 1, 2J(t) \textrm{for}j=0, 2J(t) -\sum^0_{i=j+1}(1-2\eta_i(t)) \textrm{for}j\leq -1, \end{cases}
f_{(i_1,\ldots,i_n,j)}(t) = ~ s!\binom{s+2}{i_1}\cdots \binom{s+2}{i_n}\cdot \frac{g^{(j)}(t)}{j!}\cdot n!^{2+j}\binom{t+n}{n}^{2+j}&\qquad\times \prod_{k=1}^{n}\left((k-1)!(n-k)! \binom{t+k-1}{k-1}\binom{t+n}{n-k}\right)^{i_k}.
\dim M_{z}=\dim (A+\ker ad_{z})\cr =\dim A+\dim \ker ad_{z}-\dim (A \cap \ker ad_{z})\cr =\dim A+\dim \ker ad_{z}-\dim \ker ad_{z}|_{A}\cr =\dim A+(\dim L-b(z))-\dim C_{A}(z)\cr =\dim A+(\dim L-b(z))-\dim Z(L) \cr =\dim L+(\dim A-\dim Z(L))-b(z)\cr = \dim L+1-b(z)
0 = f_c(y)\equiv \frac{1}{2}y^-\star y^++\frac{a}{2}y^0\star y^0-c, begin{align*}6pt] 0= \mathrm{d}f_c =\frac{1}{2}(y^-\star\eta^++\eta^-\star y^+) +ay^0\star\eta^0, begin{align*}6pt] 0= y^-\star E^+-y^+\star E^--\sqrt{a}\,y^0\star H+\mathrm{i}\nu y^+\star H-2\mathrm{i}\nu (1+\mathrm{i}\nu) y^+\star E^+.
[f_n-g_n]_{W^{1,p},\varphi_n(K_n)}\leq[f_n-g_n]_{W^{1,p},\varphi_n(\mathsf{U}_n)}&=\left(\sum_{B\in\mathcal{B}_n}\int_{\varphi_n(B)}|Df_n-Dg_n|^p\,d\mu\right)^{\frac{1}{p}}&\leq\sum_{B\in\mathcal{B}_n}\left(\int_{\varphi_n(B)}|Df_n-Dg_n|^p\,d\mu\right)^{\frac{1}{p}}\ .
q_{2}^{( i )}- q_{1}^{( i )}= \frac{1}{\pi}\int_{0}^{\infty}d \omega \, \sinh \pi \omega \, s^{( i )}- \int_{0}^{\infty}d \omega \frac{I_{\omega}( k \xi_{1}) I_{- \omega}( k \xi_{2})}{D_{\omega}( k \xi_{1}, k \xi_{2})}S^{( i )},
\rho(L_{-1}) \, =\, h(z)^{-1}\, , \rho(L_{0}) \, =\, -\frac{h(z)}{h'(z)}\partial + b(z) \, , \rho(L_1) \, =\, h(z) \left(\rho(L_0)^2 - \rho(L_0) -c(c+1)\right)\, .
\omega_{i}= \sigma_{1}^{-1}\sigma_{2}^{-2}\sigma_{3}\sigma_{4}\cdots\sigma_{i+1}\tau_{i,k}= \sigma_{1}^{2}\sigma_{2}\cdots\sigma_{k-1}\sigma_{k}^{-2}\sigma_{k+1}\sigma_{k+2}\cdots\sigma_{i+k-1}.
{(J_\varepsilon(u^{\varepsilon, \alpha}))^2-(J_\varepsilon(u^{\varepsilon}))^2\over J_\varepsilon(u^{\varepsilon, \alpha})+J_\varepsilon(u^{\varepsilon})}=J_\varepsilon(u^{\varepsilon, \alpha})-J_\varepsilon(u^{\varepsilon})\ge -\sqrt{\varepsilon}d(u^{\varepsilon, \alpha}, u^{\varepsilon})\ge -C\alpha \sqrt{\varepsilon},
l_{0}^{e}+ l_{1}^{e}+ l_{2}^{e}+ l_{3}^{e}+1 \geq0, l_{0}^{e}+ l_{1}^{e}- l_{2}^{e}- l_{3}^{e}+1 \leq0 , & l_{0}^{e}- l_{1}^{e}+ l_{2}^{e}- l_{3}^{e}+1 \leq0, l_{0}^{e}=(l_{0}-l_{1}-l_{2})/2-1 \geq-1,
F_{\mu \nu}\equiv \partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu}\; , \; D_{\mu}\varphi_{1}\equiv \partial_{\mu}\varphi_{1}+ e A_{\mu}\varphi_{2}\; , \; D_{\mu}\varphi_{2}\equiv \partial_{\mu}\varphi_{2}- e A_{\mu}( \varphi_{1}+ \frac m e )
w^{(1)}(t,x)~=~\begin{cases}\overline{w}(x) \qquad\mathrm{if}~~x\in (t,0)\cup (0,\infty), begin{align*}2mm]\overline{w}(x+\tau_0-t) \qquad\mathrm{if}~~x\in (-\infty,t),\end{cases}
0 = \frac{1}{4}[ \frac{1 6 \pi^{4}\theta^{2}}{g^{2}\eta^{2}}z \int \frac{d^{2}p}{( 2 \pi )^{2}}[ p^{- 2}G^{- 2}( p ) - \frac{4 \theta^{2}g^{2}}{\eta^{2}}( 1 - K^{2}( p ) ( p^{2}+ \lambda \eta^{2}) ) ] + ( k^{2}+ \lambda \eta^{2}) - K^{- 2}( k ) ]
\underline{S_2}(p, N) = N \sum\limits_{i = (1-\varepsilon)\mu}^{(1+\varepsilon)\mu}{\left[ \binom{N}{i}{{p}^i}{{\left({1 - p}\right)}^{N - i}}\underline{L}\left( \frac{i + 1 - \mu}{\mu}\right) \right]}
(k \:.\: k \:.)^{q_1}\;p\:.\:k\:.\; (k\:.\:k\:.)^{q_2}\;p\:.\:k\:.\; (k\:.\:k\:.)^{q_3}\:\cdots\:(k\:.\:k\:.)^{q_{2\alpha+1}}\;p\; (.\:k\:.\:k)^{q_{2\alpha+2}}
J_0(t,|D|)u_1(x)-J_0(t,x)P_{u_1}=\int_{|y|\leqslant t^{\alpha_0}}\big(J_0(t,x-y)-J_0(t,x)\big)u_1(y)\mathrm{d}y &\quad+\int_{|y|\geqslant t^{\alpha_0}}J_0(t,x-y)u_1(y)\mathrm{d}y-J_0(t,x)\int_{|y|\geqslant t^{\alpha_0}}u_1(y)\mathrm{d}y
Q=(\begin{matrix}*&\kappa_{1}\pi_{G}&\pi_{C}&\pi_{T}\\ \kappa_{1}\pi_{A}&*&\pi_{C}&\pi_{T}\\ \pi_{A}&\pi_{G}&*&\kappa_{2}\pi_{T}\\ \pi_{A}&\pi_{G}&\kappa_{2}\pi_{C}&*\end{matrix})
L_1=2f_G^{d-1}\sum_i\epsilon^{\mu_1\mu_2\cdots\mu_{d}}\epsilon_{a_1a_2\cdots a_{d+1}}{\cal O}_{i,i+1}^{a_1n}(x)D_{\mu_1}{\cal O}_{i,i+1;n}^{a_2}(x) e_{\mu_2}^{a_3}(x,i)e_{\mu_3}^{a_4}(x,i)\cdots e_{\mu_d}^{a_{d+1}}(x,i)\, ;
\lambda_{\widetilde{Q}}^{i}:=\widetilde C\frac{\omega(\widetilde Q)^{\frac{1}{p}}}{\vert\widetilde Q\vert^{\frac{1}{q}}}\left \|\left\{\sum_{Q\subset\widetilde{Q}}\vert\psi_{Q}\ast h(u_{Q})\vert^2\chi_{Q}\right\}^{1/2}\right \|_{L^{q}}.
J_{orb}(\vec{R})=2\pi\kappa\frac{\sigma_{1}}{q_{1}}R^{i}\varepsilon_{ij}v_{j}^{(2)}(\vec{R}) -2\pi\kappa\frac{\sigma_{2}}{q_{2}}R^{i}\varepsilon_{ij}v_{j}^{(1)}(-\vec{R})\;\;,
{{\bar G}_S}= \sum\nolimits_{\ell ,k \in \left\{{{},{}}\right\}}{{G_{\ell}^S G_k}_{\ell k}^S},{{\bar G}_A}= \sum\nolimits_{\nu ,k \in \left\{{{},{}}\right\}}{{G_{\nu}^A G_k}_{\nu k}^A}.
z ( \lambda | \lambda_{j}) = M \left( \sum_{h = 1}^{N_{h}}e^{\pm \lambda_{h}}- \sum_{c = 1}^{N_{c}}e^{\pm \lambda_{c}}+ \Theta ( 1 - p ) \sum_{w}e^{\pm \lambda_{w}}\left( 1 + e^{\mp i \pi ( p + 1 ) \mathrm{s i g n}( \Im m \lambda_{w})}\right) - 2 \sum_{s = 1}^{N_{s}}e^{\pm \lambda_{s}}\right)
I(t) \leq (t+1)^{2-3(\gamma-1)}e^{-\frac{a_1}{t+1}}\left( e^{a_1}I(0)+a_2e^{a_1}\ln(t+1) \right) \displaystyle \leq C\left((t+1)\ln(t+1)+t+1\right), \t\in [0,T_m).
Z_n \coloneqq \frac{1}{n!}\sum_{\sigma\in S_n}\prod_{i=1}^n t_i^{j_i(\sigma)}= \sum_{j\in J_n}\frac{1}{\prod_{i=1}^n i^{j_i}j_i!}\prod_{i=1}^n t_i^{j_i}\in \mathbb Q[t_1,\dots,t_n]
\begin{array}{r c l r c l}{{T_{1}}}&{{=}}&{{M_{1}\, , \hspace{1 c m}}}&{{\tilde{T}_{1}}}&{{=}}&{{R_{2}P_{3}\, ,}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{T_{2}}}&{{=}}&{{R_{2}P_{1}\, , \hspace{1 c m}}}&{{\tilde{T}_{2}}}&{{=}}&{{R_{2}P_{2}\, ,}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{T_{3}}}&{{=}}&{{R_{2}P_{0}\, , \hspace{1 c m}}}&{{\tilde{T}_{3}}}&{{=}}&{{M_{2}\, ,}}\\ \end{array}
{\cal A}_{L}^{( 1 ) \, \mathrm{s u b d o m}}\simeq \ - \frac m{8 \pi}\left( \frac{5 \lambda^{2}}{6 4 m^{4}}- \frac{e^{4}}{m^{2}\Theta^{2}}\right) \frac{{\bf p}^{2}}{m^{2}}\left[ \ln \left( \frac{\Lambda^{2}}{{\bf p}^{2}}\right) + i \pi \right] \; .
\tilde{Q}_{X^nY^n}W_{X^nY^n}[\pi] = \big(1_{X^n}\times Q_{Y^n|X^n}\big) \big( W_{X^n}[\pi] \times W_{Y^n}[\pi] \big) = ( 1_{X^n}W_{X^n}[\pi] ) \times \big( W_{X^n}[\pi^{-1}] Q_{Y^n|X^n}W_{Y^n}[\pi] \big) = 1_{X^n}\times \big( W_{X^n}[\pi^{-1}] Q_{Y^n|X^n}W_{Y^n}[\pi] \big) \,.
g_{i j , k l}= \frac{- q^{- 1}}{\left[ 2 \right]}D_{s i}\hat{R}_{j s , k l}^{\epsilon}= \frac{- q^{- 1}}{\left[ 2 \right]}\epsilon_{i m}\hat{R}_{j m , k t}\epsilon_{l t}^{- 1}\quad , \quad ( g = \frac{- q^{- 1}}{\left[ 2 \right]}D_{1}{\cal P}\hat{R}_{1 2}^{\epsilon}) \quad ,
d_\ell = 2a_0 + 2\sum_{k'=1}^{n_1 - 1}(\omega^{n_2})^{k' \ell}a_{n_2 k'}+ 2 \sum_{k'=1}^{n_2 - 1}(\omega^{n_1})^{k' \ell}a_{n_1 k'}+ 2 \sum_{\substack{1 \leq k < n (k,n) = 1}}\omega^{k\ell}a_k.