harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ u'(r)^{p-1-\alpha}\right]' \geq \frac{\delta}{n-1}\,f_1(r) g_1(v(r)) - \frac{\delta^2}{(n-1)(\delta+1)}\, f_1(r)g_1(v(r)) &= \frac{p-1-\alpha}{n(p-1-\alpha) + \alpha}\,f_1(r) g_1(v(r))
\theta_n^{\mathrm{causal}}=\left(\sum_{i=1}^n E\left[W_{n,i}^{XX}\right]\right)^{-1}\sum_{i=1}^nE\left[W_{n,i}^{XX}\right]\theta_{n,i}, \shortintertext{and, with probability approaching one,}\theta_n^{\mathrm{causal}, \mathrm{sample}}=\left(\sum_{i=1}^n R_{n,i}E\left[W_{n,i}^{XX}\right]\right)^{-1}\sum_{i=1}^n R_{n,i}E\left[W_{n,i}^{XX}\right]\theta_{n,i},
\widetilde{\Phi}_d(A,\nu; \widetilde{h})=\ln \left(\sum_{i\in [q]}\nu(i)\left(\sum_{j \in [q]}a_{ij}\widetilde{h}(j)\right)^d\right)-\frac{d}{2}\ln \left(\sum_{i,j\in [q]}a_{ij}\widetilde{h}(i)\widetilde{h}(j)\right).
\Delta_\alpha ( \phi \circ \mathbf{F}) = \sum_{i=1}^n \frac{\partial \phi}{\partial x_i}(\mathbf{F}) \, \Delta_\alpha F_i + \sum_{i,j=1}^n \frac{\partial^2 \phi}{\partial x_i \partial x_j}(\mathbf{F}) \, \Gamma_{\alpha}(F_i, F_j),
\Psi (x_1 , \cdots , x_M | z_1 , \cdots , z_N )=\vartheta (x_1 , \cdots , x_M | z_1 , \cdots , z_N )\prod_{\mu =1}^{M}\prod_{j=1}^{N}\psi \Bigl(\frac{x_{\mu}}{z_j}\Bigr),
\phi_{n}^{\pm}(x)=\frac{e^{\pm i \left( 2n\pi x/a-\theta \right)}}{\sqrt{a}}; E_{n}^{\pm}=\frac{\hslash^{2}}{2m}\left(\frac{2n\pi}{a}\right)^{2}+\frac{\beta \hslash^{4}}{3m}\left(\frac{2n\pi}{a}\right)^{4},
\left. \tilde{{\cal R}}( g ) \right\vert \sb{\phi ( \tau ) = n x ( \tau )}={\cal R}( g ) , \quad \left. \tilde{K}\right\vert \sb{\phi ( \tau ) = n x ( \tau )}= K , \quad \left. \tilde{B}\right\vert \sb{\phi ( \tau ) = n x ( \tau )}= B ,
E_1(t):=\frac{1}{2}\int_0^k\left(u+\mu r^2(\log\rho)_x\right)^2dx+\frac{Ca}{2}\frac{1}{\gamma-1}\int_0^kH(\rho)dx+\frac{Ca}{2}\frac{4\mu}{\gamma}\int_0^t\int_0^k\left(r^2(\rho^{\frac{\gamma}{2}})_x\right)^2dxd\tau.
\left\lbrace \begin{tabular}{l l}$u_t + u_{xxx}+ av_{xxx}=0$ in $(0,L)\times (0,T)$, $v_t +\frac{r}{c}v_x+\frac{ab}{c}u_{xxx}+ \frac{1}{c}v_{xxx}=0$ in $(0,L)\times (0,T)$, $u(x,0)=u^0(x), v(x,0) = v^0(x)$, in $(0,L)$\end{tabular}\right.
(D-CA^{-1}B)[D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1}] = I+C(A-BD^{-1}C)^{-1}BD^{-1}-CA^{-1}BD^{-1}-CA^{-1}[(BD^{-1}C-A)+A](A-BD^{-1}C)^{-1}BD^{-1}= I
\|(G,G_1,G_2,y_0)\|_Z^2 := \|\rho G\|^2_{L^2(Q)}+\|\rho_3 G_{t}\|^2_{L^2(Q)}+ \|G(0)\|_{H_0^1(I)}^2 + \|\rho G_1\|^2_{L^2(Q)}&+ \|\rho G_2\|^2_{L^2(Q)}+ \|y_0\|^2_{H^3(I)}.
\frac{f(x)}{(x-z_1)\cdots(x-z_m)}= f(x)\cdot\left(\sum_{n\geq 1}z_1^{n-1}x^{-n}\right)\cdots\left(\sum_{n\geq 1}z_m^{n-1}x^{-n}\right) = \sum_{n\geq -N}a_nx^{-n}.
\rho^+(n) =\rho(n,z)-\sum_{\substack{ph=n z<p<x^{1/4}}}\rho(h,z)+\sum_{\substack{p_1p_2h=n z<p_2<p_1<x^{1/4}}}\rho(h,z) &=\sum_{m h=n}a_m\rho(h,z)
\hat{Q}_k({\mu}_k)-\hat{Q}_k(0)= \int^{{\mu}_k}_0 \hat{Q}'_k(\mu)\,d\mu \ge{\mu}_k\left(\frac{\delta_k}{N-1}- \frac{1}{2}{\mu}_k\big\|s_{j_{k+1}}^k - g^{k}_{j_{k+1}}\big\|^2_{D^{-1}}\right).
\langle m^{j}; d^p \rangle = \int_{[0,+\infty)}(|y| + d(x_k))^p dm^{j}(y) &\leq 2^{p-1}\int_{[0,+\infty)}|y|^p dm^{j}(y) +2^{p-1}d(x_k)^p m(e_j).
\mathcal{C}_0 = \left\{i \in [t] \, \Big| \, \{|N(x) \cap C_i| \,|\, x \in D \}= \{0\}\right\}, \mathcal{C}_{\frac{1}{2}}= \left\{i \in [t] \, \Bigg| \, \{|N(x) \cap C_i| \,|\, x \in D \}= \left\{\frac{1}{2}|C_i| \right\}\right\}, \mathcal{C}_1 = \left\{i \in [t] \, \Big| \, \{|N(x) \cap C_i| \,|\, x \in D \}= \{|C_i|\}\right\}.
q_T^{(<)}\le \sup_{\omega\in A^{(<)}_{T,r_1(T)}}\mathcal{E}_T(\omega), q_T^{(>)}\le E^{P_T}\Big[\mathbf{1}_{A^{(>)}_{T,r_2(T)}}\Big] = P_T\big(A^{(>)}_{T,r_2(T)}\big).
g_{\mu \nu}^{( 1 )}= -{\frac{l^{2}}{2}}\left({\frac{1}{a^{2}}}- 1 \right) \left( R_{\mu \nu}-{\frac{1}{6}}h_{\mu \nu}R \right) -{\frac{l}{2}}\left({\frac{1}{a^{4}}}- 1 \right) \chi_{\mu \nu}\ ,
\Theta ={\frac{1}{1 6 \pi^{2}}}\left[ 2 ( 2 a - c ) R_{\mu \nu}R^{\mu \nu}+{\frac{1}{3}}( c - 3 a ) R^{2}+ ( c - a ) R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}\right] +{\frac{c}{6 \pi^{2}}}( F_{\mu \nu})^{2}.
R(r)=(-1)^{l}\frac{(l!)^{2}r_{s}^{l}}{(2l)!}P_{l}(1-\frac{2r}{r_{s}})+(-1)^{l+1}\frac{2(2l+1)!}{(l)!^{2}r_{s}^{l+1}}Q_{l}(1-\frac{2r}{r_{s}}).
P_{t,t+s}(\mathbf{n},\mathbf{m}) = \sum_{\mathbf{n}_1\in\mathbb{N}_0}\sum_{\mathbf{n}_2\in\mathbb{N}_0}\dots \sum_{\mathbf{n}_{s-1}\in\mathbb{N}_0}P(\mathbf{n},\mathbf{n}_1,\dots,\mathbf{n}_{s-1},\mathbf{m}).
= \sum_{\substack{\mu\subset n^m 0\leq r\leq m}}(-1)^{m(n+1)-|\mu |-r}\Bigl( P_{m^n-\mu^\prime}(z_1,\ldots ,z_n;t,q,\mathbf{t}) \langle z;t_0\rangle_{t,m-r}& \times \sum_{\substack{\lambda\subset (n+1)^m \lambda\sim_{r}\mu}}C^{\lambda,m}_{\mu , r}(q,t,\mathbf{t}) P_{\lambda}(x_1,\ldots ,x_m;q,t,\mathbf{t}) \Bigr) ,
S = \frac{1}{1 6 \pi G}\int_{M}\sqrt{-^{( 3 )}g}(^{( 3 )}R + \frac{2}{l^{2}}) d^{3}x + \frac{1}{8 \pi G}\int_{\partial M}\sqrt{-^{( 3 )}g}( 2 D_{\mu}n^{\mu}+ \frac{1}{l}) d^{2}x .
\int \frac{x}{(-x^2;q^2)_\infty}\widetilde{h}_n(x;q) d_qx = \dfrac{(1-q)x}{[n-1]_q(-x^2;q^2)_\infty}\left( \frac{\widetilde{h}_n(x;q)}{qx}-q^{-n}[n]_q \widetilde{h}_{n-1}(x;q)\right),
\| \varphi(t) \|^2_H = \langle \varphi(t), \varphi(t) \rangle_{V^*, V}\leq \| \varphi(t) \|_{V^*}\| \varphi(t) \|_{V}\leq \frac{\delta}{2}\| \varphi(t) \|^2_{V}+ \frac{1}{2\delta}\| \varphi(t) \|^2_{V^*}.
L=nA+nH\int_{\Sigma}u\,dvol_{\Sigma}\geq nA+\frac{L}{n+1}\left( \frac{\int_{\Sigma}u\,dvol_{\Sigma}}{A}\right)^2= nA+\frac{L}{n+1}\left( \frac{L-nA}{nHA}\right)^2\,.
\sum_{m=0}^{\infty}\frac{\left[w_{1}w_{2}\right]_{q}^{m+1}\left(abt\right)^{m+1}}{m!}T_{2,m}\left(w_{3}-1|q^{w_{1}w_{2}}\right) + \left(q-1\right)\sum_{m=0}^{\infty}\frac{\left[w_{1}w_{2}\right]_{q}^{m+1}\left(abt\right)^{m}}{m!}T_{1,m}\left(w_{3}-1|q^{w_{1}w_{2}}\right).
9 . x^{2}= \frac{( 8 0 \times 1 1 8 \times 2 9 - 3 6 \times 1 1 7 )^{2}}{3 5 \times 4 5 \times 5 4 \times 4 6}\approx 2 1 7 4 < 3 . 8 4 1
|\textup{var}\{f^{(2)}(\tau_1)E(\widetilde{X}_{1,2}^2\mid X_1)-f^{(2)}(\tau_3)E(\widetilde{Z}_{1,1}^2\mid X_1)\}|=O\bigg(p^{-4}\sum_{k=1}^{q}\{s_{kk}(\mu_{k,3}^{(1)}-\mu_{k,3}^{(2)})\}^2\bigg),
(e^{\lambda \Lambda_1}-1)f_1(x)/\breve g_1(x)-\lambda \Lambda_1 g_1(x)/\breve g_1(x) = \sum_{n=2}^\infty ((\lambda \Lambda_1)^n/n!)\, g^{\ast n}_1(x)/\breve g_1(x) &= \sum_{n=2}^\infty ((\lambda \Lambda_1)^n/n!)\, \breve g^{\ast n}_1(x)/\breve g_1(x)
\lambda_{\rm min}(D^2F_\delta(z)) =\frac{\lambda_{\rm min}\big(D^2F(P_\delta(z))\big)}{1+\delta\, \lambda_{\rm min}\big(D^2F(P_\delta(z))\big)}, \lambda_{\rm max}(D^2F_\delta(z)) =\frac{\lambda_{\rm max}\big(D^2F(P_\delta(z))\big)}{1+\delta\, \lambda_{\rm max}\big(D^2F(P_\delta(z))\big)}.
\left|x_i-y\right|= \left|x_i-x_p\right|+\left|x_p-y\right| \ge\left|x_i-x_p\right|+\tilde{\lambda}\ge \frac{\tilde{\lambda}}{\lambda}\left|x_i-x_p\right|+\tilde{\lambda}\ge \frac{\left|x_i-x_p\right|+\lambda}{\lambda}\tilde{\lambda}\ge \frac{\left|x_i-x\right|}{\lambda}\tilde{\lambda}.
\Lambda ={\frac{1}{\sqrt{2}}}\left( \begin{array}{r r}{1}&{i}\\{1}&{- i}\\ \end{array}\right) \quad \mathrm{{a n d}}\quad \tilde{{\cal G}}_{a b}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) .
g_{{\rm SO}(M)}\big(\alpha_{\mathfrak{m}}^{\ast}+L^{-1}(\xi\cdot\alpha_{\mathfrak{m}})^{h'},\beta^{\ast}\big)={\bf B}(\alpha_{\mathfrak{m}},\beta)+{\bf B}\big(\xi_{L^{-1}(\xi\cdot\alpha_{\mathfrak{m}})},\beta\big)=0.
B_{m,\ell}=g_{1,\ell,0,m}+\sum_{s=1}^\ell g_{1,\ell,s,m}&=g_{1,\ell,1,m-1}+g_{1,\ell-1,-1,m-1}+\sum_{s=1}^\ell g_{1,\ell,s+1,m}+\sum_{s=1}^\ell g_{1,\ell-1,s-1,m}&=g_{1,\ell,1,m-1}+g_{1,\ell,0,m-1}+\sum_{s=2}^{\ell+1}g_{1,\ell,s,m}+\sum_{s=0}^{\ell-1}g_{1,\ell-1,s,m}&=B_{m-1,\ell}+B_{m-1,\ell-1}.
\int_{Q_1}\epsilon_1^{-1}\left(v+\xi\right) \cdot \left(v+\xi\right)=\int_{Q_1}\epsilon_1^{-1}\bigl({\rm curl}(P_{V^\perp}U_v)+\xi\bigr) \cdot \bigl({\rm curl}(P_{V^\perp}U_v)+\xi\bigr).
A_{[\alpha\beta\gamma]\delta\cdot\cdot\cdot}=\frac{1}{3!}(A_{\alpha\beta\gamma\delta\cdot\cdot\cdot}+A_{\gamma\alpha\beta\delta\cdot\cdot\cdot}+A_{\beta\gamma\alpha\delta\cdot\cdot\cdot}-A_{\alpha\gamma\beta\delta\cdot\cdot\cdot}-A_{\gamma\beta\alpha\delta\cdot\cdot\cdot}-A_{\beta\alpha\gamma\delta\cdot\cdot\cdot})
A\left( z,-2N;\chi_{1},G_{2}\right) =\sum\limits_{n=1}^{\infty}G\left(n,\chi_{2}\right) n^{-2N-1}\sum\limits_{v=0}^{k-1}\chi_{1}\left( v\right)e^{2\pi ivnz/k}\sum\limits_{h=0}^{\infty}e^{2\pi inzh}& =\sum\limits_{n=1}^{\infty}\frac{G\left( n,\chi_{2}\right) G\left(nz,\chi_{1}\right)}{1-e^{2\pi inz}}n^{-2N-1}.
H^{\prime}=-lnp_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdot\cdot\cdot p_{R}^{p_{R}}=ln(\frac{1}{p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdot\cdot\cdot p_{R}^{p_{R}}})=ln(\frac{1}{\prod_{i=1}^{R}p_{i}^{p_{i}}})
\int_{\Omega}\varphi^{\prime}\varphi \; \d x + \int_{\Omega}\lvert \nabla \varphi \rvert^2 \; \d x = - \int_{\Omega}u \cdot \nabla \varphi \varphi \; \d x + 2 \int_{\Omega}\lvert \nabla d \rvert^2 \varphi^2 \; \d x.
\left\langle A B_{1}B_{2}\right\rangle - \left\langle A B_{1}\right\rangle \left\langle \tilde{B}_{2}\right\rangle - \left\langle A B_{2}\right\rangle \left\langle \tilde{B}_{1}\right\rangle + \left\langle A \right\rangle \left\langle \tilde{B}_{1}\right\rangle \left\langle \tilde{B}_{2}\right\rangle
\sum_{x\in B_l}(2m-1)^{-|x|_*}\leqslant \sum_{1\leqslant k \leqslant l}l\binom{l-1}{k-1}2m(2m-2)^{k-1}(2m-1)^{-k}\leqslant l \frac{2m}{2m-1}2^{l-1}.
\mathbb{E}\left[\delta_{0}(X(h)){\rm e}^{-I\left(\frac{h}{\varepsilon}\right)}\right] =\mathbb{E}\left[\delta_{0}(X(h)){\rm e}^{-G_{1}^{\varepsilon}}\right]\cdot\mathbb{E}\left[{\rm e}^{-G_{2}^{\varepsilon}}\right].
| M_{\xi}^{2}( 0 ) | - |{\cal M}_{\xi}^{2}( X_{0}) | = \lambda e_{\xi}\vec{\varphi}^{\, 2}( X_{0}) < < | M_{\xi}^{2}( 0 ) | \; \; ( e_{\pi}= 1 / 6 , \, e_{\sigma}= 1 / 2 ) ,
F ( P , l_{1}, l_{2}) = ( 2 \pi )^{2}\delta^{( 4 )}( P - p_{i}- q_{i}) \int d^{4}y | \psi_{1}( y ) |^{2}| \psi_{2}( y ) |^{2}\exp \bigl [ - i ( l_{1}- l_{2}) y \bigr ]
u\left( x,x_{0}\right) \mid_{x\in \Gamma ,x_{0}\in \left[ 0,1\right]}=g_{0}\left( x,x_{0}\right) ,\partial_{n}u\left( x,x_{0}\right) \mid_{x\in \Gamma ,x_{0}\in \left[ 0,1\right]}=g_{1}\left( x,x_{0}\right) ,
\lambda_{1;p,f}\int_{\Omega}|u|^{\alpha+p}\,d\mu =-\int_{\Omega}|u|^{\alpha+1}\Delta_{p,f}u\,d\mu &=\int_{\Omega}|\nabla u|^{p-2}\langle\nabla u,\nabla |u|^{\alpha+1}\rangle\,d\mu &=(\alpha+1)\int_{\Omega}|u|^{\alpha}|\nabla u|^{p}\,d\mu,
\langle \phi \{\cal Q_{\mathrm{G R S Z}}}| \Xi \rangle ={\frac{1}{2 i}}\langle f \circ \phi ( 0 ) \Bigl ( ( f^{\prime}( i ) )^{- 1}c ( i \infty ) - ( f^{\prime}( - i ) )^{- 1}c ( - i \infty ) \Bigr ) \rangle .
\sqrt{q}(r')^{-2/q}\leq \max(2 \frac{\log(1/\gamma)}{2},\log(1/\gamma)\left(\frac{\log(1/\gamma)}{2}\right)^{2/\log(1/\gamma)})^{1/2}=\sqrt{\log(1/\gamma)}\sqrt{\max(1,2)}=\sqrt{2}\sqrt{\log(1/\gamma)},
\varphi(((1-t_1a_1)(1-\rho a_1(t_1)a_2(t_2))(1-t_2a_2))^{-1}&\hskip 2 in ((1-s_1b_1)(1-\sigma b_1(s_1)b_2(s_2))(1-s_2b_2))^{-1}) &= \frac{(\varphi((a_1(t_1)b_1(s_1)) + h_{a_1}(t_1)h_{b_1}(s_1))(\varphi((a_2(t_2)b_2(s_2)) + h_{a_2}(t_2)h_{b_2}(s_2))))}{1-\rho\sigma\varphi(a_1(t_1)b_1(s_1))\varphi(a_2(t_2)b_2(s_2))}\,.
Q_k(s,t)= \frac{\exp \Big( (M + 1)F(t; x_N(k)) \Big)}{\exp \Big( (M + 1)F(s; x_N(k)) \Big)}\frac{\Gamma(t)}{\Gamma(s)}, q_k(s, t; \xi, \eta) = \frac{\exp((t+k-1)\xi\sqrt{\frac{M+1}{N}})}{\exp((s+k-1)\eta \sqrt{\frac{M+1}{N}})}= \frac{e^{\xi \tau}}{e^{\eta \sigma}},
d e t^{\prime}\Bigl ( \ast d_{\scriptscriptstyle 1}+ ( \ast d_{\scriptscriptstyle 1})^{\scriptscriptstyle 2}\Bigr ) = e^{i \pi \psi}d e t^{\prime}( \ast d_{\scriptscriptstyle 1}) d e t^{\prime}( \mathrm{1 \m k e r n - 5 m u I}+ \ast d_{\scriptscriptstyle 1})
=b(z,y,z)+\alpha b\left(\boldsymbol z, \boldsymbol{curl}\, \boldsymbol y,\boldsymbol{curl}\, \boldsymbol z\right)+2\alpha\sum_{k=1}^2\left(\nabla \boldsymbol z_k\times \nabla \boldsymbol y_k,\boldsymbol{curl}\,\boldsymbol z\right)+\alpha I,
T(y, z) =\frac{1-\frac1{1+4y}}{\frac{2}{1+4y}}-\frac{\sqrt{1-\frac{1}{1+4y}}}{\frac{2}{1+4y}}\sqrt{1-(4y+1)z}+ O(1- (4y+1)z) = 2y - \sqrt{y(1+4y)}\sqrt{1-(4y+1)z}+ O(1- (4y+1)z)
\operatorname{Im}\left(T_{\left(g,p\right)}\left(\rho\circ \left(id \times f\right)\right)\right)= T_{gf\left(p\right)}\left(\O_{f\left(p\right)}\right)+T_{f\left(p\right)}\left(\rho_g\right)\left(\operatorname{Im}\left(T_p\left(f\right)\right)\right).
\begin{aligned}d\ss_2\circ \ss_1^{-1}(H) =d\ss_2(G^c_1\cap (c,d))=d\ss(G_2\cap G_1^c\cap (c,d)) &=d\ss(G_2\cap (c,d))-d\ss(G_1\cap (c,d))>0. \end{aligned}
G_{k}(\textbf{a}, \textbf{v})=\left( \begin{array}{cccc}v_{1}v_{2}\cdots v_{n}v_{1}a_{1}v_{2}a_{2}\cdots v_{n}a_{n}\vdots \vdots \ddots \vdots v_{1}a_{1}^{k-1}v_{2}a_{2}^{k-1}\cdots v_{n}a_{n}^{k-1}\end{array}\right).
T_{3n+2+j}^2=1+3\delta_{j,2}+\sum_{k=1}^n\left\{T_{3k+j+1}^2+3T_{2k+j}^2+4\!\!\!\sum_{i=0}^{3k+j-3}(T_{3k+j-i}+T_{3k+j-i-1})T_{i+2}^2\right\}.
\partial_{t}(\partial_{t}^{s}z)_{\delta}- \partial_{t}\big(\bar{a}_{ij}\big(\partial_{x_{i}}\partial_{x_{j}}\partial_{t}^{s - 2}z\big)_{\delta}\big)= \partial_{t}(h^{s - 2})_{\delta}+ \partial_{t}\eta^{s}(\cdot, \cdot; \delta),
\left((\mathbf t_1,\dots, \mathbf t_r),[y_1:\dots :y_r]\right) =\left( (t_{11},\dots, t_{1,\lambda_1-1}\, ;\, \dots \, ;\, t_{r1},\dots, t_{r,\lambda_r-1}),[y_1:\dots :y_r]\right)
E_i = e_{i,i+1}- e_{n+i+1,n+i}F_i = e_{i+1,i}- e_{n+i,n+i+1}\displaybreak[1] X_{j}=e_{j,n+j}Y_{j}=e_{n+j,j}\displaybreak[1] Z_{j}= e_{jj}- e_{n+j,n+j}H_i =Z_{i}- Z_{i+1}
S_{12}=R_{2}^{2}(\alpha_{2}-\alpha_{1})+b^{2}\int_{\lambda_{1}}^{\lambda_{2}}(\frac{1}{2(1-e^{2}sin^{2}\varphi)}+\frac{tanh^{-1}(esin\varphi)}{2esin\varphi}-\frac{R_{2}^{2}}{b^{2}})sin\varphi d\lambda,
\Xi_{[s]}(\rho_x)=\rho_{x^{(0)}}\otimes\rho_{\Lambda_x^{(-)}}\otimes\rho_{\Lambda_x^{(+)}}\quad\Xi_{[s']}(\rho_{x'})=\rho_{x'^{(0)}}\otimes\rho_{\Lambda_{x'}^{(-)}}\otimes\rho_{\Lambda_{x'}^{(+)}}.
\left[ \omega_{W P}^{( n )}\right]^{n - 4}\cap \left[ \overline{{V}}^{( k + 2 )}\times \overline{{V}}^{( n - k )}\right] = \left[ \omega_{W P}^{( k + 2 )}+ \omega_{W P}^{( n - k )}\right]^{n - 4}\cap \left[ \overline{{V}}^{( k + 2 )}\times \overline{{V}}^{( n - k )}\right] ,
M_{ij}=\left(\begin{array}{c c}{\partial\beta^\lambda_{(2i)}\over\partial\lambda_{(2j)}}{\partial\beta^\lambda_{(2i)}\over\partial\xi_{(2j)}}{\partial\beta^\xi_{(2i)}\over\partial\lambda_{(2j)}}{\partial\beta^\xi_{(2i)}\over\partial\xi_{(2j)}}\end{array}\right).
\psi_{\mathrm{W K B}}( A ) = \frac{\exp [ - 2 \, \pi \, n \, ( A / L_{p})^{2}]}{| - 1 - A^{2}/ \ell^{2}+ G^{2}) |^{1 / 4}}\, e^{\pm i \, \int^{A}p \, d x},
E = -{\frac{2}{\pi a}}\sum_{l = 0}^{\infty}( l + 1 ) \int_{0}^{\infty}d x \, x \cos x \epsilon \,{\frac{d}{d x}}\ln [ ( e_{l}^{2}+ e_{l + 1}^{2}) ( s_{l}^{2}+ s_{l + 1}^{2}) ] .
{\sum_{i,j,k,l=1}^h\omega_l^2\partial_z\left({\omega_{i}/\omega_l}\right)A_{ij}A_{lk}\overline\omega_k^2\partial_{\bar z}\left({\overline\omega_j/\overline\omega_k}\right)}=\left(\sum_{i,j=1}^h\omega_iA_{ij}\overline\omega_j\right)^3.
\int_{\Delta'_Y}p_i\nu_i u_t^*P_{DH}'d\sigma = \int_{\tilde{\Delta}'_Y}(u_t^*P_{DH,i}'p_i+ru_t^*P_{DH}+u^*_{t,i}p_iP_{DH}')dp & = \int_{\tilde{\Delta}'_Y}(nu_t^*-u_t^*\sum_{\alpha}\frac{2\chi_i\alpha^{\vee,i}}{(2\chi-p)(\alpha^{\vee})}+u^*_{t,i}p_i)P_{DH}'dp
\left\| \mu_Y - \mu_U \right\|_2 = \left\| \frac{H^T H}{2^k}\left(\mu_Y - \mu_U \right) \right\|_2 & = \frac{1}{2^{k/2}}\left\| \frac{H^T}{2^{k/2}}\left(\chi(Y) - \chi(U) \right) \right\|_2 & \leq \left\| \chi(Y) - \chi(U) \right\|_1 & = \sum_{\mathbf{b}\in (\mathbb{F}_2)^k \setminus \mathbf{0}}\varepsilon_{\mathbf{b}\cdot Y}
W_m=\left[\frac{w\left(\frac{x_i-x_j}{l_m}\right)}{l_m}\cdot \frac{1}{n}\right]_{i,\j=0}^{n-1}=\left[\frac{w\left(\frac{i-j}{(n-1)l_m}\right)}{l_m}\cdot \frac{1}{n}\right]_{i,\j=0}^{n-1}
\mathbb{E}\left\vert \mathcal{M}_{n}^{\varepsilon}-\mathcal{M}_{n-1}^{\varepsilon}\right\vert^{2}=\mathbb{E}\left[ V^{\varepsilon}\left( \eta_{0}^{\varepsilon}\right) \left( I+q^{\varepsilon}\right)\left( I-q^{\varepsilon}\right)^{-1}V^{\varepsilon}\left( \eta_{0}^{\varepsilon}\right) \right] .
\int \frac{d^3q}{(2\pi)^3}\; d^2\theta_1 \, d^2 \theta_2 \; \Phi(-p,\theta_1) \, \left[ -\frac{e^2}{4}G_3(p,P,q) D^\alpha (q) D^\beta (q) \Delta (q;12) \right] \Delta_{\alpha\beta}(P;12) \, \Phi^*(p,\theta_2).
\left\{\widetilde{e}\right\}^{\prime}(T_0) = \frac{1}{k_BT^2_0}\int_{\mathbb{R}^3}\int_0^\infty \left\{cp^0 \left(1+\frac{\mathcal{I}}{mc^2}\right)- \widetilde{e}(T_0) \right\}^2F_E^0\phi(\mathcal{I})\,d\mathcal{I}dp
S = \int \!\! d^{3}x \Biggl\{-\frac{1}{4g^2}{\rm tr}F_{\mu\nu}F^{\mu\nu}- \Phi_i^{\dag}\left( D_\mu D^\mu + m^2_i \right) \Phi_i + \overline{\Psi}_a\left( iD\!\!\!\! / - m_a \right)\Psi_a + G(A_\mu,\Phi,\Psi) \Biggr\}\ ,
\frac{1}{\beta}\int_{0}^{\beta}d \tau_{1}\cdots \int_{0}^{\beta}d \tau_{n}\, \langle \eta^{k_{1}}( \tau_{1}) \cdots \eta^{k_{n}}( \tau_{n}) \rangle_{0 , c}\propto \beta^{n - 1 + ( k_{1}+ \cdots + k_{n}) / 2}.
P_{\lambda}( \vec{q}, \vec{k}) : ={\frac{| \Sigma_{\lambda}|^{2}}{p_{\perp}^{4}- 2 p_{\perp}^{2}( q_{\perp}^{2}+ k_{\perp}^{2}) + ( q_{\perp}^{2}- k_{\perp}^{2})^{2}}}\, .
\sum_{i\in I_1}\sum_{j\in I_l}\lambda_{ij}= ~\lambda_{1,(l-1)(p+q)-p+2}+2\lambda_{1,(l-1)(p+q)-p+3}+\cdots+(p-1)\lambda_{1,(l-1)(p+q)}&+ p~ \lambda_{1,(l-1)(p+q)+1}+(p-1)\lambda_{1,(l-1)(p+q)+2}+\cdots+ \lambda_{1,(l-1)(p+q)+p}.
\int_{{\bf R}}f(t) e^{-2\pi i u t}dt = \lim_{m \rightarrow +\infty}\int_{{\bf R}}f_{m}(t) e^{-2\pi iu t}dt &=\lim_{m \rightarrow +\infty}\sum_{1 \leq n \leq m}a_n \int_{{\bf R}}\chi_{n}(t) e^{-\kappa t}e^{-2\pi i ut}dt \; ,
[s\cdot o,a\mapsto s\cdot \delta(a)] = i(s\cdot o) + \sum_{b\in \Sigma}\big(j(b)\cdot (s\cdot \delta(b))\big) = s\cdot i(o) + \sum_{b\in \Sigma}s\cdot \big(j(b)\cdot \delta(b)\big) = s\cdot \left(i(o) + \sum_{b\in \Sigma}\big(j(b)\cdot \delta(b)\big)\right) = s\cdot [o,\delta]
B ( \theta ) ={\frac{\sqrt{2}}{\sqrt{\cosh 2 \lambda \theta}}}\exp \left[ - \int_{- \infty}^{\infty}{\frac{d \omega}{\omega}}e^{i ( 2 P - 1 ) \omega \theta / \pi}~ ~{\frac{2 \sinh [ ( P - 3 / 2 ) \omega ] + \sinh [ ( P - 1 / 2 ) \omega ]}{4 \sinh [ \omega ] \cosh [ ( P - 1 / 2 ) \omega ) ]}}\right]
\left<\vec{t}, \vec{c}\right>=\left<\frac{\vec{v}-\lambda}{m}, c\right> +\left<\frac{\vec{x}-\vec{v}}{m}, c\right> =\left<\vec{t}_0, \vec{c}\right> + \sum_{i=1}^{d-1}c_i \tilde{u}_i+\beta\left<\vec{c},\vec{v}\right>,
\hat{J}_{a}^{( r )}( z ) \hat{J}_{b}^{( s )}( w ) = \frac{\hat{G}_{a b}\delta_{r + s , 0 m o d \lambda}}{( z - w )^{2}}+ \frac{i f_{a b}^{c}\hat{J}_{c}^{( r + s )}( w )}{( z - w )}+ O ( ( z - w )^{0})
\sum_j \partial^{\alpha}f_{>j-2}D^{s,\alpha}g_j = \sum_j \partial^{\alpha}f_j D^{s,\alpha}g_{<j+2}& =\sum_j \partial^{\alpha}f_j D^{s,\alpha}g_{<j-2}+ \sum_{j}\partial^{\alpha}f_jD^{s,\alpha}g_{j-2\le \cdot <j+2}.
\varepsilon(f_2f_1;g_1,\dots,g_N)=\sum_{0\le m\le N}(-1)^{|f_1|(|g_{m+1}|+\dots+|g_N|+N-m)}\varepsilon(f_2;g_{m+1},\dots,g_N)\star\varepsilon(f_1;g_1,\dots,g_m)
I_{T,t}^{(i_1 i_2)}=\frac{T-t}{2}\left(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{i=1}^{\infty}\frac{1}{\sqrt{4i^2-1}}\left(\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right)\right),
\lvert u-w \rvert+H_d\big(A+F^{\alpha}(w),B+F^{\alpha}(u)\big)=0 ~ \lvert u-w \rvert=0 H_d\big(A+F^{\alpha}(w),B+F^{\alpha}(u)\big)=0 ~ u=w ~H_d\big(A+F^{\alpha}(w),B+F^{\alpha}(u)\big)=H_d(A,B)=0 ~ u=w A=B ~ (u,A)=(w,B).
f(u(x+jh))-f(u(x)) ={}\sum_{\ell=1}^{k-1}\frac{f^{(\ell)}(u(x))}{\ell !}\left(\Delta_{jh}^1 u(x) \right)^\ell &+ \int_0^1 \frac{f^{(k)}\left(u(x)+t \Delta_{jh}^1 u(x)\right)}{(k-1)!}(1-t)^{k-1}dt \left(\Delta_{jh}^1 u(x))\right)^k.
\int_0^{+\infty}t^{d-1}e^{-ht}F(a_1,b_1,\alpha t)F(a_2,b_2,\beta t)dt=h^{-d}\Gamma(d)F_2(d;a_1,a_2;b_1,b_2;\frac{\alpha}h,\frac{\beta}h)
(N_{+}^{-1}\psi_2)_{x}-ik[N_{+}^{-1}\psi_2, \sigma_3]=N_{+}^{-1}(U-U_{+})\psi_2, (N_{+}^{-1}\psi_2)_{x}-4ik^3[N_{+}^{-1}\psi_2, \sigma_3]=N_{+}^{-1}(V-V_{+})\psi_2.
{\varpi^\sigma}_{\mu\mu}(\mu,\sigma) = \frac{\partial_\mu^2 D^\prime(p^{(\sigma)}||p_{\mu,\sigma}) - g_{\mu\mu}(\mu,\sigma)}{\partial_\sigma D^\prime(p^{(\sigma)}||p_{\mu,\sigma})}&= \frac{3\mu^2 + \sigma^2 - [(\sigma+\delta \sigma)^2+\mu^2] - 2\mu^2}{\sigma(\mu^2 + \sigma^2 - [(\sigma+\delta \sigma)^2 + \mu^2])}&= \frac{1}{\sigma}.
\mathcal{L}_{\mathrm{b u l k}}= i \, \bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi^{i}+ \bar{\psi}_{2}\partial_{5}\psi^{1}- \bar{\psi}_{1}\partial_{5}\psi^{2}- \omega ( \bar{\psi}_{1}\psi^{1}+ \bar{\psi}_{2}\psi^{2})
\sum_{N_r'}\sum_{p=1,2}\int\limits_{C_p^{(r)}}\chi_{N_r'}(t)dt\left[E_{p,N_r}^{(-2)}(Y_{p,N_r'}(t))-\partial_{q_{N_r}}Y_{p,N_r'}(t)+(-1)^{e(N_r')e(N_r)}\partial_{q_{N_r'}}Y_{p,N_r}(t)\right]=0
S_{1}^{B W}= \left( \alpha{\frac{\partial}{\partial \alpha}}- 1 \right) W_{1}[ Q_{\alpha , \epsilon_{z}}] ={\frac{1}{6 \alpha}}\epsilon_{z}+ \frac 1 2 \ln{\frac{\pi \alpha}{\epsilon_{z}}}+ o ( \epsilon_{z}^{- 1}) .
f_i(\tau+1,z_1,z_2,t) = f_i(\tau,z_1,z_2,t) \mbox{for}i = 1,2, f_3 (\tau+1,z_1,z_2,t) = -(- i)^{4k}f_4(\tau,z_1,z_2,t), f_4 (\tau+1,z_1,z_2,t) = -(- i)^{4k}f_3(\tau,z_1,z_2,t).
\hat{G}(\lambda,t)={T}_t\widetilde{G}(\lambda,t)^{-1}\widetilde{T}_t^{-1}{P}_{t}+(I-{P}_{t})={T}_t(I-\widetilde{F}(\lambda,t))\widetilde{T}_t^{-1}{P}_{t}+(I-{P}_{\tau}) &={P}_{t}-{T}_t\widetilde{F}(\lambda,t)\widetilde{T}_t^{-1}{P}_{t}+(I-{P}_{t}) &=I-{T}_t\widetilde{F}(\lambda,t)\widetilde{T}_t^{-1}{P}_{t},
k'= (p,p+1,k_2-1,p+1,p+1,p,p+1,p+1), k^{\mu_1}= (p,p+1,k_2+1,p+1,p+1,p,p+1,p+1), k^{\mu_2}= (p,p+1,k_2-1,p+1,p+3,p,p+1,p+1), k^{\mu_3}= (p,p+1,k_2-1,p+1,p+1,p,p+1,p+3).
\frac{d{\cal J}_p(u,v;B(x_0,\varepsilon))}{d |D^c u|(B(x_0,\varepsilon))}= \liminf_{\varepsilon \to 0^+}\liminf_{n\to +\infty}\frac{1}{|D^cu|(B(x_0,\varepsilon))}\int_{B(x_0,\varepsilon)}h(v, Du\ast \varrho_n)dx &+\limsup_{\varepsilon \to 0^+}\frac{1}{|D^cu|(B(x_0,\varepsilon))}\int_{B(x_0,\varepsilon)}f(x_0, u(x_0),0,0)dx + O(\delta).
\mathcal{E}_n = \Bigl\{\max_{1\leq i\leq p}\bigl|V_{n,i}^2/n-1\bigr| \leq \varepsilon_{n1}n^{(\beta-1)/2},\max_{1\leq i\leq p}|\Delta_{n,i}| \leq\varepsilon_{n2}\Bigr\}.
\left(p_2p_3- p_1 p_2^2 + p_1 q_1^{-1}q_2^{-2}\right) \partial_{p_2}+\left(p_3^2- q_2 p_2^2 q_1^{-1}- q_1^{-2}q_2^{-1}\right) \partial_{p_3}\;,
\begin{array}{l l c c l}{{J_{L}}}&{{:}}&{{T_{L}^{*}({\cal M}_{n})}}&{{\rightarrow}}&{{T_{L}({\cal M}_{n})}}\\{{}}&{{}}&{{U}}&{{\mapsto}}&{{\ ( L U )_{+}\, L \, - \, L \, ( U L )_{+}\, + \, L \, \Phi_{U}( L ) \, + \, \bar{\Phi}_{U}( L ) \, L}}\\ \end{array}
\hat{\mu}=\frac{\sum_{i=1}^{n}w_{i}X_{i}}{\sum_{i=1}^{n}w_{i}},\frac{1}{\hat{\lambda}}=\frac{1}{n}\sum_{i=1}^{n}w_{i}(\frac{1}{X_{i}}-\frac{1}{\hat{\mu}}).
\partial_t u = \Delta (u\gamma(v)) \;\;\;\; (0,\infty)\times \Omega\,, \partial_t v = \Delta v - uv \;\;\;\; (0,\infty)\times \Omega\,, \nabla (u\gamma(v))\cdot \mathbf{n}= \nabla v\cdot \mathbf{n}= 0 \;\;\;\; (0,\infty)\times \partial\Omega\,, (u,v)(0) = (u^{in},v^{in}) \;\;\;\; \Omega\,,
E_{1s}=\langle(\frac{\zeta^{3}}{\pi})^{0.50}e^{-\zeta r}|-(\frac{\zeta^{3}}{\pi})^{0.50}e^{-\zeta r}[\frac{-2r\zeta+r^{2}\zeta^{2}}{2r^{2}}]\rangle+\langle\psi_{1s}|-\frac{Z}{r}|\psi_{1s}\rangle
\gamma B_{a}^{0 i}= \partial_{j}\eta_{a}^{( 2 ) i j}, \gamma B_{a}^{i j}= \eta_{a}^{( 1 ) i j}, \gamma H_{i}^{a}= C_{i}^{( 1 ) a}, \gamma H_{0}^{a}= - \partial^{i}C_{i}^{( 2 ) a},
\Delta_{g_{C_\ell}}\left( u - \frac{f_\ell}{2m}r^2\right) = \bar{f}+ E_\ell\left(\frac{f_\ell}{2m}r^2\right) + E_\ell \left( u - \frac{f_\ell}{2m}r^2\right), E_\ell = \Delta_{g_{C_\ell}}- \Delta_g = (g_{C_\ell}- g) \ast \nabla^2_{g_{C_\ell}}+ \nabla_{g_{C_\ell}}g \ast \nabla_{g_{C_\ell}}.
\frac{d}{dt}[m(t,q(t,x))q_{x}(t,x)]=(m_t+m_xq_t)q_x+m(t,q)q_{tx}&=(m_t+(uv-u_xv_x)m_x+(uv-u_xv_x)_xm)(t,q(t,x))q_x(t,x) &=\frac{1}{2}m(uv_x-vu_x)(t,q(t,x)),
T = \left( \begin{matrix}{- \frac{\Delta_{\phi}}{d}}&{0}&{\ldots}&{0}\\{1}&{- \frac{\Delta_{\phi}}{d}}&{\ldots}&{\vdots}\\{0}&{1}&{\ldots}&{0}\\{\vdots}&{\ldots}&{\ddots}&{0}\\{0}&{\ldots}&{1}&{- \frac{\Delta_{\phi}}{d}}\\ \end{matrix}\right) .
B-\hat\nu(s) = \begin{cases}A_1 \xi'(s) +\frac1{A_1q +A_2(1-q)+\Delta}, s\le q, A_1 \xi'(q)+ A_2 (\xi'(s)-\xi'(q)) + \frac1{A_1q +A_2(1-q)+\Delta}, q\le s \le 1.\end{cases}
xW_f = \frac{x(H^2 \ast f)}{H^2}- \frac{2H^2 \ast f}{H}= \frac{xH^2 \ast f}{H^2}+ \frac{H^2 \ast xf}{H^2}- \frac{2H^2 \ast f}{H}&=\frac{2H^3 \ast f}{H^2}+ \frac{H^2 \ast xf}{H^2}- \frac{2H^2 \ast f}{H}&= \frac{H \ast xf}{H}= W_{xf}\:.
\Vert [(\mathbf{D}\Lambda )^\varepsilon] S_\varepsilon \Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant M_2, &\Vert [(\mathbf{D}\widetilde{\Lambda})^\varepsilon] S_\varepsilon \Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \vert \Omega \vert^{-1/2}C_a n^{1/2}\alpha_0^{-1}\Vert g^{-1}\Vert_{L_\infty}=:\widetilde{M}_2.