harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L\Big(\int_{S_{t,r}}\rho dS_{r}\Big)=L\Big(\int_{\mathbb{S}^{2}}r^{2}\rho(t,\rho,\cdot)d\mu_{\mathbb{S}^{2}}\Big)=\int_{\mathbb{S}^{2}}L\big(r^{2}\rho(t,\rho,\cdot)\big)d\mu_{\mathbb{S}^{2}}=0.
|g(r+s)-g(s)|= \left|\int_s^{r+s}\rho^{-\delta_1}\tilde{h}(\rho)\,d\rho\right|\le C\left|\int_s^{r+s}\rho^{-\delta_1}\,d\rho\right| \le C\left|\left(r+s\right)^{1-\delta_1}-s^{1-\delta_1}\right| \le Cr^{1-\delta_1}\left|\left(1+\frac{s}{r}\right)^{1-\delta_1}-\left(\frac{s}{r}\right)^{1-\delta_1}\right| \le Cr^{1-\delta_1}\le Cr^{\delta_0}
\tilde{g}_{\mu \nu}= \left( \begin{array}{l c}{e^{- U}}&{e^{- U}b_{j}}\\{e^{- U}b_{i}}&{e^{- U}b_{i}b_{j}+ e^{- U}\tilde{\gamma}_{i j}}\\ \end{array}\right) \ , B_{\mu \nu}= \left( \begin{array}{l c}{0}&{\omega_{j}}\\{- \omega_{i}}&{b_{i j}}\\ \end{array}\right) \ .
\begin{array}{c c c}{{g_{A B}}}&{{=}}&{{\tilde{g}_{A B}( y^{C}) ,}}\\{{}}&{{}}&{{}}\\{{g_{i j}}}&{{=}}&{{a_{k}( y^{C}) a_{l}( y^{C}) \eta_{i j}^{k l},}}\\{{}}&{{}}&{{}}\\{{g_{A i}}}&{{=}}&{{0 .}}\\ \end{array}
B(y, \pi(\phi(\overrightarrow{x\xi})), f(\xi)) = B(y, \pi(\phi(\overrightarrow{x_0\xi})), f(\xi)) + B(\pi(\phi(\overrightarrow{x_0\xi})), \pi(\phi(\overrightarrow{x\xi})), f(\xi)) = B(y, \pi(\phi(\overrightarrow{x_0\xi})), f(\xi)) + B(x_0, x, \xi)
(P_V - \mu^2)A(\mu) = \chi_1 + [P_0,\chi_2]R_V(\lambda)\chi_1 + \chi_2(\lambda^2 - \mu^2)R_V(\lambda)\chi_1 - (1+VR_0(\mu))[P_0,\chi_2]R_V(\lambda)\chi_1 = \chi_1 + \chi_2(\lambda^2 - \mu^2)R_V(\lambda)\chi_1 -VR_0(\mu)[P_0,\chi_2]R_V(\lambda)\chi_1.
E_{p,q}(u_{t})=\frac{1}{2}t^{2}\|\Delta u\|_{2}^{2}-\frac{\mu}{q}t^{\frac{N(q-2)}{4}}\|u\|_{q}^{q}-\frac{1}{p}t^{\frac{N(p-2)}{4}}\|u\|_{p}^{p}
\alpha^{(2n)}(z):= Q_{4,n}^*(\bar z)Q_{4,n}^{*-1}(a), \beta^{(2n)}(z):= \frac{1}{b-a}Q_{3,n}^*(\bar z)P_{3,n}^{*-1}(a), \gamma^{(2n)}(z):= (z-a)P_{4,n}^*(\bar z)Q_{4,n}^{*-1}(a), \delta^{(2n)}(z):= \frac{b-z}{b-a}P_{3,n}^*(\bar z)P_{3,n}^{*-1}(a).
\left( \stackrel{[ 0 , 0 ]}{z}^{A}, \stackrel{[ 0 , 0 ]}{z}^{A_{1}}\right) , \; \left( \stackrel{[ 1 , 0 ]}{\cal P}_{1 a_{0}}, \stackrel{[ 0 , 1 ]}{\cal P}_{2 a_{0}}\right) , \; \left( \stackrel{[ 1 , 1 ]}{\lambda}_{a_{0}}\right) ,
\left( D^2 L(\nabla u) \nabla u_{x_j}\right)^i = l(\nabla u)^{p-2}u_{x_i x_j}+ (p-2) l(\nabla u)^{p-4}\langle \nabla u_{x_j}, \nabla u \rangle u_{x_i}, \frac{\partial}{\partial x_i}\gamma^j = l(\nabla u)^{2-q}u_{x_i x_j}+ (2-q) l(\nabla u)^{-q}\langle \nabla u_{x_i}, \nabla u \rangle u_{x_j},
\rho \frac1Nx_i^* \check{C}_N(\rho)^{-1}x_i =\rho\frac1Nx_i^* \check{B}_{(i)}(\rho)^{-1}x_iF_N\left(\left[\frac1{n}\sum_{i=1}^n \frac{\Vert x_i\Vert^2}{\frac1Nx_i^*\check{B}_{(i)}(\rho)^{-1}x_i}\right]^{-1};\rho\right)
\bigl[\mathcal{H},[\mathcal{H},\eta]\bigr] =\eta R_0(\mathcal{H}) +[\mathcal{H},\eta]R_1(\mathcal{H})+R_{-1}(\mathcal{H}), \quad \bigl[\widetilde{\mathcal{H}},[\widetilde{\mathcal{H}},\eta]\bigr] =\eta R_0(\widetilde{\mathcal{H}}) +[\widetilde{\mathcal{H}},\eta]R_1(\widetilde{\mathcal{H}}) +R_{-1}(\widetilde{\mathcal{H}}),
\left[ - \frac{1}{N^{2}}\, \lambda_{i}\, \frac{\partial^{2}}{\partial \lambda_{i}^{2}}- \frac{1}{N^{2}}\, \sum_{j \neq i}\lambda_{j}\, \frac{1}{\lambda_{j}- \lambda_{i}}\, \left( \frac{\partial}{\partial \lambda_{j}}- \frac{\partial}{\partial \lambda_{i}}\right) + \frac{1}{N}\, ( 2 \eta - 1 ) \, \frac{\partial}{\partial \lambda_{i}}+ 1 \right] Z ( \lambda ) = 0 .
\begin{array}{l}2\int_{0}^{t}\int_{M}\left \Vert \nabla_{b}\eta^{\left( i\right)}\right\Vert^{2}\left( x,s\right) d\mu ds\leq \int_{M}\left \Vert \eta^{\left(i\right)}\left( x,0\right) \right \Vert^{2}d\mu =\int_{M}\left \Vert \rho^{\left( i\right)}Ric\right \Vert^{2}d\mu .\end{array}
\Phi([T_-,f,T_+][U_-,g,U_+]) =\Phi([T_-,fg,U_+])=((T_-,fg,U_+)) =((T_-,f,T_+))((U_-,g,U_+))=\Phi([T_-,f,T_+])\Phi([U_-,g,U_+])
\Gamma_{-1}(\frac{x+p'(\xi)}{\sqrt{h}})\left[\tilde{\theta}_h(x)( -\partial_{\xi}b(x,\xi)d^2\varphi(x) - \partial_xb(x,\xi))\right] + & + \Gamma'_{-1}(\frac{x+p'(\xi)}{\sqrt{h}})(\frac{x+p'(\xi)}{\sqrt{h}}) \chi(h^{\beta}\xi)\tilde{\theta}_h(x) \left[ \frac{-d^2\varphi(x)p''(\xi) - 1}{\xi -d\varphi(x)}\right] \, ,
\left\| \sum_{h=1}^{H_L}\mathbb{I}_L^{(\textup{even})}(h \cdot) \right\|_2\leq \left\| \sum_{h=1}^{H_L}p_L (h \cdot) \right\|_2 + \left\|\sum_{h=1}^{H_L}r_L (h \cdot) \right\|_2.
\frac1Nz_i^*\hat{C}_N^{-1}z_i = \frac{\frac1Nz_i^*\hat{C}_{(i)}^{-1}z_i}{1+(1-\rho)c\frac{\frac1Nz_i^*\hat{C}_{(i)}^{-1}z_i}{\frac1Nz_i^*\hat{C}_N^{-1}z_i}}
\vert S_i(\lambda) \vert \ = \ \left( \sum_{k_1 + \hdots + k_i = n}C_{k_1}C_{k_2}\hdots C_{k_i}\right) + \left( \sum_{l_1 + \hdots + l_{i+1}= n}C_{l_1}C_{l_2}\hdots C_{l_{i+1}}\right)
\tilde{g}(x)= g(x+q_1) + c\left(Z_1(x+q_1) + \sum\limits_{j=2}^k Z_j(x+q_1) \right) &+ p\left[\left(w(x) + \sum\limits_{j=2}^k w(x+q_1-q_j)\right)^{p-1}- w^{p-1}(x) \right]\tilde{\varphi}.
\sigma_{\mathrm{\footnotesize \boldmath ~ \a l p h a ~}}^{( n )}( z ) = \vartheta \left[ \begin{array}{c}{{\alpha_{2}/ n + 1 / 2}}\\{{\alpha_{1}/ n + 1 / 2}}\\ \end{array}\right] ( z , \tau ) , \theta_{n}^{( j )}( z ) = \vartheta \left[ \begin{array}{c}{{1 / 2 - j / n}}\\{{1 / 2}}\\ \end{array}\right] ( z , n \tau ) ,
\frac{c_0^2k n /2}{\left(k(n-k)+\binom{k}{2}\right)p+Cc_0k \sqrt{n \log n}/6}\geq \frac{c_0^2k n /2}{2C\max\{kn\rho,k \sqrt{\rho n \log n},k \log n\}}& \geq \frac{\max\{\rho kn ,k \log n\}/2}{2C\max\{kn\rho,k \sqrt{\rho n \log n},k \log n\}}, c_0, & \geq \frac{1}{4C}.
\frac{1}{m}\sum_{k=0}^{m-1}\sum_{s=1}^{n-k}e^{-\beta(n-k-s)}e^{-\beta d(v, \partial K_{\sigma(s)}(v))}~=~ \frac{1}{m}\sum_{k=0}^{m-1}\sum_{s=0}^{n-k-1}e^{-\beta s}e^{-\beta d(v, \partial K_{\sigma(n-k-s)}(v))}& ~\leq~ \frac{1}{m}\sum_{k=0}^{m-1}\sum_{s=0}^{n-1}e^{-\beta s}e^{-\beta d(v, \partial K_{\sigma(n-k-s)}(v))}
\tilde\omega(\theta^*y\mu) = \tilde\omega(y_{(2)}\theta^*(y_{(1)}\cdot S(y_{(3)}))\mu) = \omega(y_{(2)})\theta^*(y_{(1)}S(y_{(3)}))\mu (1) &= \omega(y_{(1)})\theta^*(y_{(2)}S(y_{(3)}))\mu (1) &= \omega(y_{(1)})\theta^*(\epsilon(y_{(2)})1)\mu(1) &=\omega(y)\overline{\theta(1)}\mu (1)
Z(G_{4,1}\vee C^{(1)}\vee G_{4,1}) = \begin{cases}Z(C(9))=9 (C^{(1)}=C(\ddot5)) Z(C(3,6))=19 (C^{(1)}=C(\dot1,\dot4)) Z(C(3,3,3))=33 (C^{(1)}=C(\dot1,3,\dot1)) Z(C(5,4))=21 (C^{(1)}=C(\ddot1,4)). \end{cases}
lim_{x\rightarrow\infty}\frac{x^{\frac{1}{2}}+x^{-\frac{1}{2}}}{x^{\frac{1}{2}}-x^{-\frac{1}{2}}}=lim_{x\rightarrow\infty}\frac{x+1}{x-1}=lim_{x\rightarrow\infty}\frac{1}{1}=1.
d^nf(s_1,\ldots,s_{n+1}):=\kappa_{s_1}f(s_2,\ldots,s_{n+1}) +\sum_{i=1}^n{(-1)^nf(s_1,\ldots,s_i+s_{i+1},\ldots,s_{n+1})}+(-1)^{n+1}f(s_1,\ldots,s_n)
T_0^{t_0}T_1^{s_{\ell_0}- t_0}X^{j_{\ell_0}}Y^{k_{\ell_0}}g = \sum \Phi'(Q^{t,\ell}) \hskip1pt T_0^{t}T_1^{s_{\ell}- t}X^{j_{\ell}}Y^{k_{\ell}}g
\begin{array}{lc}F_\alpha (X,Y,Z)=\frac{1}{4n}\left[g(X,Y)\theta_\alpha (Z)+g(X,Z)\theta_\alpha(Y)\right. \qquad\qquad\left.+g(X,J_\alpha Y)\theta_\alpha (J_\alpha Z)+g(X,J_\alpha Z)\theta_\alpha (J_\alpha Y)\right] ,\end{array}
R \le E\Bigg[E\bigg(\prod_{i=1}^k e^{-\alpha t_i}\phi_0(t_i)\biggm|\xi_0\bigg) &\quad+\varrho_k\sum_{\emptyset\ne H\subset\{1,\dots,k\}}E\bigg(\prod_{i\notin H}e^{-\alpha t_i}\phi_0(t_i)\biggm|\xi_0\bigg)\sum_{\substack{1\le j_i\le \xi_0(t_i), \forall i\in H}}\ \prod_{i\in H}e^{-\alpha \sigma_{j_i}}\Bigg]
\nabla_{j}= \left( \begin{array}{c c}{i \theta_{j k}^{- 1}\hat{x}^{k}}&{0}\\{0}&{\partial_{j}}\\ \end{array}\right) + i \left( \begin{array}{c c}{( D_{j})_{i}^{l}}&{0}\\{0}&{( \hat{D}_{j})_{\beta}^{\alpha}}\\ \end{array}\right)
\cos\!\left( r_k\omega_n -(2\nu + 1)\frac{\pi}{4}\right) = \cos\!\left(\frac{kn\pi}{N}\right)\cos\!\left((2\nu+1)\frac{\pi}{4}\right) + \sin\!\left(\frac{kn\pi}{N}\right)\sin\!\left((2\nu+1)\frac{\pi}{4}\right).
U_{( 1 2 )}\diamond \hat{X}\diamond U_{( 1 2 )}^{- 1}= \sum_{k = 0}^{\infty}\left( \frac{i}{\hbar}\right)^{k}\frac{1}{k !}\underbrace{[ A_{( 1 2 )}, [ A_{( 1 2 )}, \ldots [ A_{( 1 2 )},}_{\mathrm{k - t i m e s}}\hat{X}] \ldots ] ]
\widehat{\omega}((k_1, \rho_1), (k_2,\rho_2),(k_3 ,\rho_3)) := \tilde{\nu}(k_1,k_2)(1;(1,k_3)) \ \rho_1(F(k_2,k_3)) = \epsilon(k_1,k_2,k_3) \ \rho_1(F(k_2,k_3)).
|f(x)|^2 = \frac{1}{4\pi^2}\left| \int_{\mathbb{R}}\mathcal{F}^\mu(f)(\xi)e^{\mu \xi x}d\xi \right|^2 &\leq \frac{1}{4\pi^2}\int_{\mathbb{R}}(\rho + \xi^2) \left|\mathcal{F}^\mu(f)(\xi)\right|^2\frac{e^{2\mu \xi x}}{\rho + \xi^2}d\xi .
\prod_{1\leq j \leq \ell}^{\to}\left(\prod_{i\in Q_0^j}^{\to}y_{e_i}^{\gamma(i)}\right)= q^{\sum \gamma(ta)\gamma(ha)}y_{e_1}^{\gamma(1)}\cdots y_{e_n}^{\gamma(n)}= \Gamma^{-1}\cdot y_{e_1}^{\gamma(1)}\cdots y_{e_n}^{\gamma(n)}
R(e_1,e_2)e_1 =\dfrac{-2a(a-b-c)-(a-b+c)(a+b-c)}{4}\,e_2; R(e_1,e_2)e_2 =\dfrac{2a(a-b-c)+(a-b+c)(a+b-c)}{4}\,e_1; R(e_1,e_3)e_1 =\dfrac{2b(a-b+c)+(a-b-c)(a+b-c)}{4}\,e_3; R(e_1,e_3)e_3 =\dfrac{-2b(a-b+c)-(a-b-c)(a+b-c)}{4}\,e_1; R(e_2,e_3)e_2 =\dfrac{2c(a+b-c)+(a-b+c)(a-b-c)}{4}\,e_3; R(e_2,e_3)e_3 =\dfrac{-2c(a+b-c)-(a-b+c)(a-b-c)}{4}\,e_2.
g(x_{+})=T_{n}\left( \begin{array}{cc}1 0 \frac{qx}{2\pi}1\end{array}\right) \quad ;\qquad J(x_{+})=n\left( \begin{array}{cc}0 -1 1 0\end{array}\right) +\frac{q}{2\pi}\left( \begin{array}{cc}\frac{1}{2}\sin (2nx_{+}) -\sin^{2}(nx_{+}) \cos^{2}(nx_{+}) -\frac{1}{2}\sin (2nx_{+})\end{array}\right) \, \, .
\left [\frac{z^{42k + 24}}{t^{7k + 4}x^{7k + 4}y^{28k + 16}}\right]= \left [\frac{z^3(z^7)^{6k + 3}}{t^{7k + 4}x^{7k + 4}y^{28k + 16}}\right]= \left [\frac{z^3(x^7 + y^7)^{6k + 3}}{t^{k + 1}x^{7k + 4}y^{28k + 16}}\right]
h_{-1-L}^{\mathcal{L}}(t)= h_{-1-L}^{\mathcal{L}_{1,2,..,L}\otimes\mathcal{K}_{L+1}}(t) =h_{-1}^{\mathcal{K}_{L+1}}(t)h_{-L}^{\mathcal{L}_{1,2,..,L}}(t)=\cdots=\prod_{\alpha=1}^{L+1}h_{-1}^{\mathcal{K}_\alpha}(t).
\limsup_{N\to\infty}\frac{ND_N(M_1x,\ldots,M_Nx)}{\sqrt{2N\log(\log(N))}}=\limsup_{N\to\infty}\frac{ND^*_N(M_1x,\ldots,M_Nx)}{\sqrt{2N\log(\log(N))}}=\frac{1}{2}\textnormal{a.e.}
\frac{d}{dt}\left\Vert U_{t}\right\Vert_{H_{0}^{-1}(\Omega)}^{2}=2(U_{tt},U_{t})_{H_{0}^{-1}(\Omega)}& =2(\nabla\Delta_{N}^{-1}U_{tt},\nabla\Delta_{N}^{-1}U_{t})_{L^{2}(\Omega)}=-2\left( U_{tt},\Delta_{N}^{-1}U_{t}\right)
H_k^{(s,t)}(z)\left[\begin{array}{c}k+1 k+1\end{array}\right]H_k^{(s,t)}(z)\left[\begin{array}{c}1 k\end{array}\right] &\quad=H_k^{(s,t)}(z)\left[\begin{array}{c}k+1 k\end{array}\right]H_k^{(s,t)}(z)\left[\begin{array}{c}1 k+1\end{array}\right]+H_k^{(s,t)}(z)H_k^{(s,t)}(z)\left[\begin{array}{cc}k+1 1 k+1 k\end{array}\right].
W^{a c}\sum_{n = 0}^{\infty}( - 1 )^{n}\left( \Lambda^{n}\right)^{c b}= \sum_{p = 1}^{\infty}a_{p}\left[ ( g \bar{\theta})^{p}\right]^{a c}\sum_{n = 0}^{\infty}c_{n}\left[ ( g \bar{\theta})^{2 n}\right]^{c b},
\beta_{0, 1}= \frac{1}{2}\left(4 r^4+23 r^3+53 r^2+ 58 r+30\right), \beta_{1, 0}= -\frac{1}{2}(r+2)(r+1)^3 \beta_{0, 1}= \frac{1}{2}\left(4r^6+35 r^5+144 r^4+345 r^3+513 r^2+451 r+195\right), \beta_{0, 2}= \frac{1}{2}(r^4+6 r^3+17 r^2+24 r+17)^2.
{\frac{1}{\sigma_{R}\sqrt{\pi}}}e^{-{\frac{( E - E_{R})^{2}}{\sigma_{R}^{2}}}}= \int d E^{\prime}w ( E^{\prime}){\frac{1}{\sigma_{Q}\sqrt{\pi}}}e^{-{\frac{( E - E^{\prime}- E_{Q})^{2}}{\sigma_{Q}^{2}}}},
H_{x}(x^{*},u^{*},\lambda^{*},t)=[\begin{matrix}\frac{\partial H}{\partial x_{1}}|_{x=x^{*},u=u^{*},\lambda=\lambda^{*}}&\cdot\cdot\cdot&\frac{\partial H}{\partial x_{n}}|_{x=x^{*},u=u^{*},\lambda=\lambda^{*}}\end{matrix}]
\xi_{ijk}=(\Phi_{ij})_{z_{k}}+ (\Phi_{ij})_{w}\xi_{k}+ \sum_{l=1}^n (\Phi_{ij})_{\xi_{l}}\xi_{lk}& = (\Phi_{ij})_{z_{k}}+ (\Phi_{ij})_{w}\xi_{k}+ \sum_{l=1}^n (\Phi_{ij})_{\xi_{l}}\Phi_{lk}, 1 \leq i,j ,k \leq n.
H_j = \left[\begin{array}{l}H_{j,e}H_{j,o}\end{array}\right]=\left(\prod_{s=1}^{u}\left[\begin{array}{ll}I_{n_j}-U_j^{[s]}0 I_{n_j'}\end{array}\right]\left[\begin{array}{ll}I_{n_j}0 P_j^{[s]}I_{n_j'}\end{array}\right]\right)\left[\begin{array}{l}D_j 0\end{array}\right],
\tau \left( I + \begin{pmatrix}\alpha 0 0 \delta \end{pmatrix}p \right)^{- \frac{a-d}{\alpha - \delta}}= \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a - \alpha \frac{a-d}{\alpha - \delta}* c d - \delta \frac{a-d}{\alpha- \delta}\end{pmatrix}p
\left(\frac{\ln(1+t)}{t}\right)^r(1+t)^x =\sum_{n=0}^\infty D_n^{(r)}(x)\frac{t^n}{n!}\,, \left(\frac{(1+t)\ln(1+t)}{t}\right)^r(1+t)^x =\sum_{n=0}^\infty\widehat D_n^{(r)}(x)\frac{t^n}{n!}\,,
\begin{aligned}\|u_2 (t)\|_{H^1}+ \|Ju_2(t)\|_{L^2}\le (1+C\varepsilon^2_1 \langle T_0 \rangle^{1+2C_1 \varepsilon_1^2}e^{C\varepsilon^2_1 \langle T_0 \rangle^{1+2C_1 \varepsilon_1^2}}) \varepsilon_2 <2\varepsilon_2\end{aligned}
A_k^n:=\int_{t_{k-1}}^{t_k}\mu_n(\bar{t}_n){\biggl\langle}Y^n_{\bar{t}_n}-Z_{\bar{t}_n},\bigl(\sigma\bigl(Y^n_t\bigr)-\sigma\bigl(Y^n_{\bar{t}_n}\bigr)\bigr)\dot{w}^n_t-\frac{1}{2}(\nabla\sigma)\sigma\bigl(Y^n_{\bar{t}_n}\bigr){\biggr\rangle}\,\d t,
f^{( C )}( \theta ) = - \frac{i \eta}{\sqrt 2 k}\frac{g_{\omega}( \theta )}{( \cos \omega - \cos \theta )^{3 / 2}}\frac{( 1 - \cos \theta )^{q}\Theta_{\omega}( \theta )}{[ ( \cos \omega - \cos \theta )^{1 / 2}- i \sqrt 2 \sin ( \omega / 2 ) ]^{2 q}}\ ,
\Tilde{e}_{i}^{F}=\mathrm{KR}^{\prime -1}\circ (\mathrm{Id},\Tilde{e}_{i}^{\pm P}) \circ \mathrm{KR}^{\prime}, \Tilde{f}_{i}^{F}=\mathrm{KR}^{\prime -1}\circ(\mathrm{Id},\Tilde{f}_{i}^{\pm P}) \circ \mathrm{KR}^{\prime},
N(r,\min\{\nu_{(f^u,H_i),\leqslant k_i}(z) ;1\leqslant u\leqslant 3\})-N^{(1)}_{(f,H_i),\leqslant k_i}(r) &\geqslant\sum_{u=1}^3N^{(n)}_{(f^u,H_i),\leqslant k_i}(r)-(2n+1)N^{(1)}_{(f,H_i),\leqslant k_i}(r)=N_i(r).
Z^{4}=e^{-2mt}\left( \partial_{t}-H-cK^{1}+\left( 2m^{2}\left( \int \frac{dx}{\sigma \left( x\right)}\right)^{2}+4mc\left( \int \frac{dx}{\sigma\left( x\right)}\right) +2c^{2}-m\right) F\partial_{F}\right)
T =p_{sd}\left(1-\pi_s(0) \right)+p_{sr}\left(1-\pi_s(0) \right) (1-\pi_r(B_r)), & \mathbb{E}\{D\}= \frac{1+L_s}{\mu_s}+\frac{(n-2+L_r)(1-\pi_r(B_r))}{p_{sd}+p_{sr}(1-\pi_r(B_r))},
F(\mathcal{L}_V) g(r,y)=\sum_{\nu\in\chi_\infty}\sum_{\ell=1}^{d(\nu)}\varphi_{\nu,\ell}(y) \int_0^\infty F(\rho^2) (r\rho)^{-\frac{n-2}2}J_\nu(r\rho)b_{\nu,\ell}(\rho)\rho^{n-1}d\rho
|J_3| =\Big|n c_{r+1}\mathrm{tr}\left( (D^{\frac{1}{2n}}g_r^*g_rD^{\frac{1}{2n}})^{n-1}D^{\frac{1}{2n}}g_r^*x_{i_{r+1}}D^{\frac{1}{2n}}\right) & + n c_{r+1}\mathrm{tr}\left( (D^{\frac{1}{2n}}g_r^*g_rD^{\frac{1}{2n}})^{n-1}D^{\frac{1}{2n}}x_{i_{r+1}}^*g_rD^{\frac{1}{2n}}\right)\Big| \leq \frac{n|c_{r+1}|}{r+1}.
E_{\nu,T}(z) = \frac{(-1)^T 2^{-2T-\nu}z^{2T+\nu}}{(T+\nu)!T!}{}_1\!F_2\left(\begin{matrix}1 T+1\!\!\!\!\! \!\!\!\!\!T+\nu+1\end{matrix};-\frac{z^2}{4}\right)\approx \frac{(-1)^T 2^{-2T-\nu}z^{2T+\nu}}{(T+\nu)!T!},
\int_{M/(M\cap H)}\left|\int_{A/(A\cap H)}a^{\lambda-\rho_{Q}}f(kmaH_{\emptyset})\,da\right|^{2}\,dm &\qquad=\sum_{[\sigma]\in \widehat{M}}\dim(V_{\sigma})\sum_{\eta\in E_{\sigma}}\left\|\int_{M/(M\cap H)}\int_{A/(A\cap H)}a^{\lambda-\rho_{Q}}f(kmaH_{\emptyset})\sigma(m)\eta\,da\,dm\right\|^{2}_{\sigma}.
\begin{aligned}\partial_tv_i - d_i\Delta v_i = \nabla f_i(u_{\infty})\cdot v =: L_iv, x\in\Omega, t>0, \nabla v_i \cdot \nu = 0, x\in\partial\Omega, t>0, v_i(x,0) = u_{i,0}(x) - u_{i,\infty}, x\in\Omega,\end{aligned}
I_1(P^{\dagger}) =\frac{1}{24}\int_Y c_2 \cdot p_1 + \frac{1}{12}\int_Y -2c_2c_1^2 + 2c_3c_1 + c^2_2 - 2c_4 & I_2(P^{\dagger}) + I_3(P^{\dagger}) = \pm \frac{1}{2}\int_{\Sigma^{(4)}_Y}c_2.
\hat{M}= \tilde{V}^{T}M \tilde{V},\ \hat{D}= \tilde{V}^{T}D \tilde{V}, \ \hat{K}= \tilde{V}^{T}(K+Z)\tilde{V}= \tilde{\hat{K}}+ \tilde{V}^{T}Z \tilde{V}, \hat{F}= \tilde{V}^{T}F, \ \hat{C}_{p}= C_{p}\tilde{V},\ \C_{v}= C_{v}\tilde{V}.
\quad[1-A_{U^M}]^L &=\left[1-\frac{(L\exp(-\gamma)A_{U^M})\exp(\gamma)}{L}\right]^L &\le \left[1-\frac{\exp(\gamma)}{L}\mathbb{E}[\pi(U^M;\tilde{V})|U^M]\right]^L &\le 1-\mathbb{E}[\pi(U^M;\tilde{V})|U^M]+e^{-\exp(\gamma)},
i\gamma_{d}\cdot \partial_{i}\left( \begin{array}{c}\psi_{u}\psi_{lL}\end{array}\right) =\frac{g}{2}\gamma_{d}^{\alpha}\left( \begin{array}{cc}Z_{\alpha}/\cos \theta_{w}W_{\alpha}^{1}-iW_{\alpha}^{2}W_{\alpha}^{1}+iW_{\alpha}^{2}-Z_{\alpha}/\cos \theta_{w}\end{array}\right) \left( \begin{array}{c}\psi_{u}\psi_{lL}\end{array}\right) -
\tilde L^{(1)}\tilde V( x-y ) ={\rm sign}(x-y) \cdot( -x^3 -x + y^3 +y) & \leq - |x-y|= -\tilde V( x -y), \tilde L^{(2)}\tilde V( x-y ) ={\rm sign}\left(x-y\right)(-\frac{1}{2}x +\frac{1}{2}y) &=-\frac{1}{2}|x-y|=-\frac{1}{2}\tilde V(x-y),
\omega_{\varphi \cdot X}(\tau) = \sum_{\substack{\sigma \in X^{(\dim X)}\tau < \sigma}}\omega(\sigma) \varphi_\sigma(\tilde{v}_{\sigma / \tau}) - \varphi_\tau\Big( \sum_{\substack{\sigma \in X^{(\dim X)}\tau < \sigma}}\omega(\sigma) \tilde{v}_{\sigma / \tau}\Big).
\frac{E_0(L\Omega,\beta, \rho L^2)}{L^2|\Omega|}\leq \frac{1}{L^2|\Omega|}\sum_{j =1}^{N_{\ell}}E_0(\ell Q, \beta,\rho L^2|\Omega| N^{-1}_{\ell}) &= \frac{\ell^2}{L^2|\Omega|}\sum_{j =1}^{N_{\ell}}E_0(\ell Q, \beta, (1 + o(1)) \rho \ell^2)/\ell^2 = (1 + o(1)) e(\beta,\rho),
+ 2 z_{\alpha}( \xi ) \frac{\partial}{\partial y_{\alpha}( \xi )}D_{m}^{( 4 )}( y ( \xi ) - y ( \xi^{\prime}) ) + z_{\alpha}( \xi ) z_{\beta}( \xi^{\prime}) \frac{\partial^{2}}{\partial y_{\alpha}( \xi ) \partial y_{\beta}( \xi^{\prime})}D_{m}^{( 4 )}( y ( \xi ) - y ( \xi^{\prime}) ) \Biggr ) \Biggr ] .
\frac{1}{q}\frac{d}{dt}\int_\Omega|v|^qdxdydz+ \int_\Omega|v|^{q-2}\Big(|\nabla_Hv|^2+(q-2) \big|\nabla_H|v|\big|^2 + \varepsilon|\partial_zv|^2+(q-2)\varepsilon \big|\partial_z|v|\big|^2\Big)dxdydz=I_1+I_2,
A_{1}=\{kp|k= 1,2,3,...,p^{n-1}-1\, \, p\nmid k\}A_{2}=\{lp^2|l =1,2,3,...,p^{n-2}-1\}A_{i}=\{mp^i|m =1,2,3,...,p^{n-i}-1\}A_{n-1}=\{np^{n-1}|n =1,2,3,...,p-1\}
\lim_{\lambda\in\Lambda}\frac{P_{n+1,k}(z)}{P_{n,k}(z)}=\widetilde{F}_{k}^{(\rho)}(z)=\frac{S_{k}^{(\rho)}(z)}{S_{k}^{(\rho)}(\infty)},z\in\overline{\mathbb{C}}\setminus[a_{k},b_{k}],
J(W^1_{-n}\cup \cdots \cup W^1_{n}) \cong J(W^1_{-n}) \vee \cdots \vee J(W^1_{n}), J(W^2_{-n}\cup \cdots \cup W^2_{n}) \cong J(W^2_{-n}) \vee \cdots \vee J(W^2_{n}).
\sup_{f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}}\frac{Q\left(f;\phi\right)}{T_{A}(f)}= 1, \sup_{f\in H^{1}\left(\mathbb{S}^{2}\right)\setminus\{0\}}\frac{Q\left(f;\psi\right)}{T_{B}(f)}= 1.
\frac{1}{\|\mathbf{x}\|}= \frac{1}{\sqrt{\pi}}\int_{\mathbb{R}}e^{- t^2\|\mathbf{x}\|^2}\,\mathrm{d}t \approx \sum_{k=-M}^{M}g_k e^{- t_k^2\|\mathbf{x}\|^2}\mbox{for}\|\mathbf{x}\| > 0,
I(g_{u}(\theta))=\frac{\theta^{6}}{2}A(u)+\frac{\theta^{6}}{4}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{1-e^{-\theta^{-2}|x-y|}}{|x-y|}u(x)^{2}u(y)^{2}dxdy+\frac{\theta^{4p-6}}{p}C(u).
\rho_{0,k}:=\frac{|\phi_k|^{q-2}\,(\phi_k)_-}{\displaystyle\int_{\Omega_k}|\phi_k|^{q-2}\,(\phi_k)_-\,dx}\cdot\mathcal{L}^N=\frac{|\phi|^{q-2}\,\phi_-}{\displaystyle\int_{\Omega_k}|\phi|^{q-2}\,\phi_-\,dx}\cdot\mathcal{L}^N.
0\le J[v]-J[u]= \int_\Omega |\nabla v|^2 - \int_\Omega |\nabla u|^2+\Phi_0({\mathcal{M}}_2(v))-\Phi_0({\mathcal{M}}_2(u)) &\qquad\qquad=\int_{B_{d/2}\cap\{\psi>u\}}|\nabla \psi|^2 - \int_{B_{d/2}\cap\{\psi>u\}}|\nabla u|^2+\Phi_0({\mathcal{M}}_2(v))-\Phi_0({\mathcal{M}}_2(u)).
\digamma \left( \lambda \right) =\overset{\infty}{\underset{n=0}{\sum}}\digamma \left( \lambda_{n}\right) \frac{h\left( \lambda \right) \varpi\left( \lambda \right)}{\left( \lambda -\lambda_{n}\right) h\left( \lambda_{n}\right) \varpi^{\prime}\left( \lambda_{n}\right)}.
\begin{array}{l l l}{{L_{N S}}}&{{=}}&{{\frac{1}{2}( R_{D 5}- L_{D 5}) + ( L_{D 3}- R_{D 3})}}\\{{L_{D 5}}}&{{=}}&{{\frac{1}{2}( R_{N S}- L_{N S}) + ( L_{D 3}- R_{D 3})}}\\ \end{array}
T_jH_{3,j}= -v_j u_j^* - (I-bT_j)H_{1,j}, H_{3,j}T_j^* = - u_j v_j^* - H_{1,j}(I-bT_j^*), \stepcounter{parentequation}\gdef\theparentequation{\arabic{parentequation}}\setcounter{equation}{0}T_jH_{4,j}= v_j u_j^* + (I-aT_j)H_{1,j}, H_{4,j}T_j^* = u_j v_j^* + H_{1,j}(I-aT_j^*).
\langle V_{i_1}\dots V_{i_n}\rangle_g = \int{\cal D}\rho{\cal D}X e^{-S^* -g^i\int_\Sigma d^2\sigma V_i + Q^2\partial \rho{\overline \partial}\rho - Q^2\rho \int_\Sigma d^2\sigma \rho R^{(2)}}V_{i_1}\dots V_{i_n}
f_1 \, (e_1^{-1}\rhd f_2) = \underbrace{f_1 \, (e_1^{-1}\rhd f_2)\, f_1^{-1}}_{}\, f_1 = ((\partial f_1 e_1^{-1}) \rhd f_2) \, f_1 = (e_4 e_5 \rhd f_2) \, f_1.
F = \dfrac{\exp{z^1 + \bar z^1}|\eta|^2 + \exp(z^2 + \bar{z}^2)|\eta^2|^2 + \exp(z^1 + z^2+z^3+\bar{z}^3)|\eta^3|^2}{\exp{z^1 + \bar z^1}|\eta|^2 + \exp(z^2 + \bar{z}^2)|\eta^2|^2 + \exp(z^1 + z^2+z^3+\bar{z}^3)|\eta^3|^2 - \dfrac{1}{2}(\exp^{z^2}\eta^2 + \exp^{\bar{z}^2}\eta^2)}.
\zeta_{P{\cal K}}(s)=\left( \frac{\sqrt{2}}{m}\right)^{2s}\zeta_{P\bar{\cal K}}(s) \hspace{1cm}, \hspace{1cm}\zeta_{\cal V}(s)=\left( \frac{\sqrt{2}}{m}\right)^{2s}\zeta_{\bar{\cal V}}(s)\,.
\mathbb{E}\left[U\left(\sum_{t=2}^T{\psi}_t (X_t-X_{t-1})\right)\vert\mathcal{F}_{1}\right]\leq \mathbb{E}\left[U\left(\sum_{t=2}^T \bar{\phi}_t (X_t-X_{t-1})\right)\vert\mathcal{F}_{1}\right].
\psi_{\beta}^{(2)}\left( 0,x;u_{\beta}^{(2)}\right) =k_{\beta}\left( \frac{1}{2\beta}\left( \varphi \left( 0,x;z\right) -\varphi \left( 0,x;\frac{1}{z}\right) \right) -u_{\beta}^{(2)}\right) .
\mathcal{T}_{V}\big( z(k) \big) = \textup{\textbf{w}}p \otimes \mathcal{R}(k^{2}) \textup{\textbf{w}}^{\ast}+M \big( z(k) \big) \textup{\textbf{w}}p \otimes \mathcal{R}(k^{2}) \textup{\textbf{w}}^{\ast}& + \Big( 1 + M \big( z(k) \big) \Big) \textup{\textbf{w}}R_{1}\big( z(k) \big) Q \textup{\textbf{w}}^{\ast}.
M_2=\begin{pmatrix}1 0 b_0 b_1 c_0 c_1 0 0 b_1 b_2 c_1 c_2 \vdots \vdots \vdots \vdots \vdots \vdots 0 1 b_{d-2}b_{d-1}c_{d-2}c_{d-1}\end{pmatrix}.
H(t) =\int_{\eta}^{t}\int_{\mathbb{T}}\partial_{t}\left[\sum_{c=1}^{m}v_{c}(s,\,x)\log\frac{v_{c}(s,\,x)}{u_{c}(s,\,x)}\right]dxds =\int_{\eta}^{t}\int_{\mathbb{T}}\left[\log\frac{\tilde{v}}{\tilde{u}}\right]^{\dagger}\partial_{t}\tilde{v}-\left(\frac{\tilde{v}}{\tilde{u}}\right)^{\dagger}\partial_{t}\tilde{u}dxds
\Phi(\varrho)=\mathcal{H}^{N-2}(\mathbb{S}^{N-2})\,\int_{-1}^1 \frac{(1-t^2)^\frac{N-3}{2}}{(1-2\,t\,\varrho+\varrho^2)^\frac{N+s\,p}{2}}\, dt,
[\begin{matrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ \end{matrix}]
\operatorname{tr}(\hat{\Pi}_{W_1}(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^j\hat{\Pi}_{W_1}) =\operatorname{tr}((\hat{\Pi}_{V_2}\hat{\Pi}_{V_1})^{j-1}\hat{\Pi}_{V_2}\hat{\Pi}_{W_1}\hat{\Pi}_{V_2}) &=\operatorname{tr}((\hat{\Pi}_{V_2}\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^{j-1}\hat{\Pi}_{V_2}\hat{\Pi}_{W_1}\hat{\Pi}_{V_2}).
L ( \tilde{x}_{0}) - \frac{g}{1 + g}L \Bigl ( \tilde{x}_{0}^{1 + \frac 1 g}\Bigr ) > L \Bigl ( \frac{1}{1 + \tilde{y}_{0}}\Bigr ) \, , \qquad \mathrm{f o r}\quad 0 < g < 1 \, .
I^{\mu}({\rm^R D}^\mu v)(t) =v^0_++\sum_{i=1}^{n-1}\,[v]^i+\sum_{j=1}^{n-1}\,(v^j-v^{j-1}_+)+v(t)-v^{n-1}_+=v(t)~~{\rm for}~~t \in I_n\,.\Box
\left({\dot{a}\over a}\right)^2+{{kc^2}\over a^2}={{8\pi m_0G_5}\over{3(1-\Omega^2_R)}}\left[{{\tilde{\varrho}_1\Omega_R+\varrho_2\Omega^5_R}\over\sqrt{\Omega^2_R-\dot{R}^2/c^2}}+V_r/c^2\right].
\frac{\left(aq,q/a;q\right)_{\infty}}{\left(q,q;q\right)_{\infty}}=\frac{1}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}\right)dx}{\left(q,-q^{1/2}/(ae^{ix}),-q^{1/2}(ae^{ix});q\right)_{\infty}},
\mathrm{t r}\left[ Q \Gamma^{, \rho}\Gamma^{, \sigma}\right] = - 4 k^{\rho}k^{\sigma}t^{2}\xi^{- 2}- \frac{2}{k^{2}}P^{\rho \sigma}\left[ \cosh \left( k^{2}( - 1 + \xi^{- 1}) t \right) - 1 \right]
\frac{\delta}{\delta B^{\alpha\beta}}S=\int_{M}\frac{\delta}{\delta B^{\alpha\beta}}B\wedge\delta B+\int_{M}B\wedge\delta\frac{\delta}{\delta B^{\alpha\beta}}B=\int_{M}\frac{\delta}{\delta B^{\alpha\beta}}B\wedge\delta B-\int_{M}\delta B\wedge\frac{\delta}{\delta B^{\alpha\beta}}B=-2\int_{M}\delta B\wedge\frac{\delta}{\delta B^{\alpha\beta}}B
\frac{\vert I^{\complement,1}\vert}{N}\leq\bar{e}^{N}(t+h)-\bar{e}^{N}(t-)\leq\frac{\vert I^{\complement,1}\vert}{N}+\biggl(\frac{\vert I\vert}{N^2}\sum_{j=1}^{N}\bigl(M_{t+h}^{j}\bigr)^{2}\biggr)^{1/2}.
\|B^k\|_{L^2(0,T; H^{\frac52-\frac{\alpha}2})}\leq \ 2C(T) \|B(0)\|_{H^{\frac{5}{2}-\alpha}}\|B^k\|_{L^\infty(0,T; H^{\frac52-\alpha})}\leq \ (1+C(T)) \|B(0)\|_{H^{\frac{5}{2}-\alpha}}
S^{I}_{p,q}\!\left(N,z^{\beta}\right) \; \equiv (-1)^N p^{-1}\, z_{-2}^{\beta (N-1)}\int_C \frac{s^{N+q/p-1}\; e^{-s^{1/p}}}{s+z_{-2}^{-\beta}}\; ds - \frac{2\pi i}{p}\; z_{-2}^{-\beta q/p}& \times \sum_{j=1}^2 e^{-i(2j-1)q \pi/p}\exp \!\left(-z_{-2}^{-\beta/p}e^{-i(2j-1) q\pi/p}\right) \;\; ,
\tau_{k, i}= \tau_k(p_i) = \sum_{j=1}^\infty \frac{\ell_k(p_i^j)}{p_i^j}=\sum_{j=1}^\infty \frac{(j+1)^k - j^k}{p_i^k}=\left[\left(\frac{1}{x}-1\right)\left(x \frac{\partial}{\partial x}\right)^k\frac{1}{1-x}-1\right]_{x = \frac{1}{p_i}}.