harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bigoplus_{E' \in \mathcal{X}_l}(B_{d,m}\downarrow_{E'}) \uparrow^{E}_{E'}\simeq_{\mathcal{X}_{l-1}}\bigoplus_{E'<E: |E'|=p^l}(B_{d,m}(E') \oplus B_{d-p^l,m}(E') \oplus \ldots \oplus B_{d-ap^l,m}(E')) \uparrow^{E}_{E'}
\int_{\tilde{\Omega}}\nabla \tilde{u}(x)\nabla \phi(x) d x=\int_{\tilde{\Omega}}\nabla \tilde{u}(x)\nabla(\phi(x) \varphi_\delta(x_n)) d x+\int_{\tilde{\Omega}}\nabla \tilde{u}(x)\nabla (\phi(x) (1-\varphi_\delta(x_n)))d x=I_1+I_2
\frac{1}{\tau}\Big\|(v_n-v_{n-1})_{n=0}^N\Big\|_{L^p(X)}\le \sum_{m=0}^N |\chi_m|\, \|(\eta_{n-m}\dot v_{n-m})_{n=0}^N \|_{L^p(X)}\le \sum_{m=0}^N |\chi_m|\, \|(\dot v_{n})_{n=0}^N \|_{L^p(X)}\le C\big\|(\dot v_n )_{n=0}^N\big\|_{L^p(X)}.
\int\limits_0^T\int\limits_0^{\ell}|v^{\tau}-\tilde v|^2\,dx\,dt =\sum\limits_{k=1}^n\sum\limits_{i=0}^{m-1}\tau\int\limits_{x_i}^{x_{i+1}}\Big|v_i(k)+v_{ix}(k)(x-x_i)-v_i(k)\Big|^2\,dx = \sum\limits_{k=1}^n\sum\limits_{i=0}^{m-1}\frac13\tau h^3 v_{ix}^2(k) \longrightarrow 0 ,
\widetilde{P}_{N,M,k}^{(0^{2})}(z; \lambda_1) = -4k ( k+N+M+1 ) P_{k}^{(N,M)}(z) \big( \lambda_{1}+Q_{0}^{(N,M)}(z) \big) \hphantom{\widetilde{P}_{N,M,k}^{(0^{2})}(z; \lambda_1) =}{}+(1-z)^{N+1}(1+z)^{M+1}P_{0,k}^{( N,M)}(z) , k\ne 0, \widetilde{P}_{N,M,0}^{(0^{2})}(z; \lambda_1) =1,
\textup{Osc}^\lambda_{\mu}(x, r) = \frac{1}{(40)^{2(n-1)}(11t)^{n-1}}\,\textup{Osc}^\lambda_{\mu_1^l}(w_0+11tx, 11tr) &\leq \frac{2^{2n+1}\,}{(40)^{2(n-1)}(11t)^{n-1}}\,\bar C(\lambda)^2\,\tau\,(11tr)^{n-1}\leq\delta_{\ref{t:Reif discreto}}^2(\lambda)\,r^{n-1}.
|| u_h^{n}||^2_{L^2 \left( \Omega_{t^{n+1}}\right)}= || u_h^{n}||^2_{L^2( \Omega_{t^{n}})}+ \int_{t^{n}}^{t^{n+1}}\int_{\Omega_{t}}\nabla \cdot \mathbf{w}_{h}|u_h^{n}|^2~dx~ dt,
\sum_{i=1}^{N_n}E\left[\max_{1\leq k\leq d_n}\left|\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)W^{(i)}_j\right|^3\right]\leq d_n^{3/p}\eta^{-3}\sum_{i=1}^{N_n}\max_{1\leq k\leq d_n}\left(\sum_{j=1}^{N_n}\gamma_{n,k}(i,j)^2\right)^{3/2}.
S(z) \begin{pmatrix}1 0 0 0 z^{-\beta}0 0 0 z^{-\beta}\end{pmatrix}= \mathcal{O}\begin{pmatrix}1 h_{-\alpha-\frac{1}{2}}(z) h_{-\alpha-\frac{1}{2}}(z) 1 h_{-\alpha-\frac{1}{2}}(z) h_{-\alpha-\frac{1}{2}}(z) 1 h_{-\alpha-\frac{1}{2}}(z) h_{-\alpha-\frac{1}{2}}(z)\end{pmatrix},
|J+V|_g = |\chi J_1 + (1-\chi)J_2|_g & \le \chi |J_1|_{g_1}+ (1-\chi) |J_2|_{g_2}+ \chi(1-\chi) (|g_1-g_2|_{g_1}|J_1|_{g_1}+ | g_1-g_2|_{g_2}|J_2|_{g_2}),
\begin{array}{c}\sum_{k=1}^m \left(\beta_{i,k}-\beta_{j,k}\right) + \alpha_{ij}+ \gamma_i = 0 \implies \gamma_i = \sum_{k=1}^m \left(\beta_{j,k}-\beta_{i,k}\right) - \alpha_{ij}\leq 2mp_{\max}, \end{array}
{\Pr}_S\left[ |C_S(v)| \ge x \mid V_j \cap S = W \right] \le{\Pr}_S\left[ |S'(v)| \ge \frac{x}{\Delta+1}\mid V_j \cap S = W \right] \le{\Pr}_{S}\left[ |C_{S_2}(v)| \ge \frac{x}{\Delta+1}\right].
W[C]=\sum_{n}\int_{s_{1}<s_{2}\ldots <s_{n}}\Gamma^{(n)}_{\mu_{1}\ldots \mu_{n}}(x(s_{1}),\ldots x(s_{n}))\, \dot{x}_{\mu_{1}}(s_{1})\ldots \dot{x}_{\mu_{n}}(s_{n})\, ds_{1}\ldots ds_{n}
\lVert \mathfrak{I}_{n+1}\rVert_{s_0+\mu}\le \sum_{k=1}^{n+1}\lVert \hat{\mathfrak{I}}_k \rVert_{s_0+\mu}\le C_* \varepsilon^{b_*}\gamma^{-1}\sum_{k \geq 1}N_{k-1}^{-\alpha_1}\le C_* \varepsilon^{b_*}\gamma^{-1}
f^{(\kappa)}(z)=\frac{1}{\kappa}\Bigl(1-\exp(-\kappa\tan^{-1}z)\Bigr)=z-\frac{\kappa}{\sqrt{2}}\frac{z^2}{\sqrt{2}}+\frac{\kappa^2-2}{2\sqrt{3}}\frac{z^3}{\sqrt{3}}+\cdots.
\mathrm{X}^{l m 0}\partial_{i}\mathrm{D}^{l m}- \partial_{m}( \mathrm{X}^{m l 0}\mathrm{D}_{l i}) - \partial_{l}( \mathrm{X}^{m l 0}\mathrm{D}_{m i}) - q{~}\partial_{i}\partial_{l}\partial_{m}\mathrm{X}^{l m 0}= 0 .
\mathbf{G}=\left(\begin{array}{cccccc}1 0 0 0 a_1 \underline{a_1}0 1 0 0 a_2 \underline{a_2}0 0 1 0 b_1 \underline{b_1}0 0 0 1 b_2 \underline{b_2}\end{array}\right).
{\cal S}^{D0}= \frac{1}{2 g_{S}}\int dt~{\rm Tr}\left(G_{ij}\dot{X}^i \dot{X}^j+ \frac{1}{2}\frac{1}{(2\pi)^2}G_{ij}G_{kl}[X^i,X^k] [X^j,X^l]+ \right.
J_{n}( k_{1}, \cdots , k_{n}) = \int_{0}^{1}d \tau_{1}\cdots \int_{0}^{1}d \tau_{n}\; \exp \left[ \frac{i}{2}\sum_{a < b}^{n}( k_{a}\times k_{b}) ( 2 \tau_{a b}- \epsilon ( \tau_{a b}) ) \right] .
E_{modi}^{S, low2}(t) = \sum_{\tau=p,N_1}\sum_{\mu, \nu, \mu',\nu' \in\{+,-\}}\mathfrak{Re}\Big[ \int \overline{\widehat{(SU)_{\mu}}(\xi)}\widehat{(SU)_{\nu}}(\xi-\eta) \widehat{{U}_{\mu'}}(\eta-\sigma) \widehat{{U}_{\nu'}}(\sigma)
\widetilde{y_1}\widetilde{y_2}\cdots \widetilde{y_n}\widetilde{g}=\widetilde{y_1}\widetilde{y_2}\cdots\widetilde{y_{n-1}}[\widetilde{y_n},\widetilde{g}]\widetilde{y_n}=[\widetilde{y_n},\widetilde{g}]\widetilde{y_1}\widetilde{y_2}\cdots\widetilde{y_{n-1}}\widetilde{g}\widetilde{y_n}.
K(T)=\tfrac{2^\rho}{\varepsilon^2}Te^{T(\rho-1)/\rho}\|H_0\|_\rho\left(1+4\left(\tfrac{2^\rho}{\varepsilon^\rho}T\|H_0\|_\rho+e^{-T(\rho-1)/\rho}\right)\right)\xrightarrow{T\rightarrow0}0,
C_1(\nu) :=\frac{1}{\sqrt{3}}\max\Bigl(1,\frac{2}{\nu}\Bigr)\quad C_2(\nu) :=\frac{1}{\sqrt{6}}\max\Bigl(1,\frac{2}{\nu}\Bigr)+\max\Bigr(\sqrt{2\nu},\sqrt{\frac{2}{\nu}}+\sqrt{\frac{\nu}{2}}\Bigr)\,.
\phi_{\mathcal{M_{\mathbf{m}}}}\oplus \phi_{\mathcal{A}}\oplus \phi_{\mathcal{O}}: (S^d)^k \rightarrow \oplus_{i=1}^k U_{k,i}^{\oplus m_i}\oplus_{i=1}^{a_i}V_{\mathbf{e}_i}^{a_i}\oplus V_{A(\mathcal{O})}.
\|\psi'\|_{L^p_{p-1}}=\gamma_n,\ \|\psi\|_{L^p_{\alpha_0}}=1,\ \|\psi_n\|_{X^{1,p}_{\infty}}\to1,\ \lim_{R\to\infty}\lim_{n\to\infty}\int_R^\infty|\psi_n|^pr^{\alpha_0}\mathrm dr=1.
2 \mathrm{Re}\langle u_{\alpha}\otimes \Phi_y\,|\, H_y R_y \rangle = 2 \mathrm{Re}\langle u_{\alpha}\otimes \Phi_y^{(0)}\,|\, H_y R_y \rangle + 2 \mathrm{Re}\langle u_{\alpha}\otimes 2 \alpha^{3/2}\eta \Phi_*^1\,|\, H_y R_y \rangle +2 \mathrm{Re}\langle u_{\alpha}\otimes R_*\,|\, H_y R_y \rangle.
H_c = \int [ \frac{1}{2}(\pi_i^a)^2 + \frac{1}{2}m^2(A_i^a)^2 + \frac{1}{4}(F_{ij}^a)^2 + \frac{m^2}{2}(A_0^a)^2 - A_0^a T_2^a ]
(III)' \hspace{7mm}\Psi_{\bf D}=\left(\begin{array}{c}a_1 +{\rm j}b_1 a_2 +{\rm j}b_2 a_3 +{\rm j}b_3 a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm}\leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C}=\left(\begin{array}{c}-a_1+{\rm i}a_3 -a_2+{\rm i}a_4 -b_1+{\rm i}b_3 -b_2+{\rm i}b_4\end{array}\right)
X = 3^{1/2}2^{-1/3}\begin{pmatrix}-\tfrac{1}{\sqrt{3}}x + y z_{3}\bar{z}_{1}\bar{z}_{3}-\tfrac{1}{\sqrt{3}}x - y z_{2}z_{1}\bar{z}_{2}\tfrac{2}{\sqrt{3}}x\end{pmatrix}.
\frac{\Gamma(\tilde{\sigma}_i\tilde{\sigma}\lambda)}{\Gamma(\tilde{\sigma}\lambda)}=\frac{[i(\tilde{\sigma}\lambda)_{i+1}-(\tilde{\sigma}\lambda)_i-c]}{[i(\tilde{\sigma}\lambda)_{i+1}-(\tilde{\sigma}\lambda)_i+c]},\,\,\frac{\Gamma(\bar{\sigma}_N\tilde{\sigma}\lambda)}{\Gamma(\tilde{\sigma}\lambda)}=-1.
\int_0^{U(N; \alpha)}e^{-x}\left(1 - \frac{\ln x}{\ln N}\right)^r dx\leq \int_0^1 e^{-x}\left(1 - \frac{\ln x}{\ln N}\right)^r dx+ \int_1^{U(N; \alpha)}e^{-x}dx = O(1)
C_3 = pE\sup_{t_0 \leq t < u \wedge \pi_R}\Big|\int_{t_0}^{t}|x_{s-}|^{p-2}x_{s-}\sigma_s( x_{s-}) dw_s\Big| & \leq KE\sup_{t_0 \leq t \leq u\wedge \pi_R}|x_{t-}|^{p-1}\left(\int_{t_0}^{u \wedge \pi_R}|\sigma_s(x_{s-})|^2ds\right)^{1/2}
\begin{multlined}[t][12.5cm]\Psi(x,t,\xi,\eta)=t\psi\left(x+\left(\xi\Phi_{G}(m)+\eta\Phi_{T}(m)+ O(\|(\xi,\eta)\|^3),\right.\right. \left.\left.,-\xi_{M}(m)-\eta_{M}(m)+ O(\|(\xi,\eta)\|^2)\right),x\right)-\nu_{T}\cdot\eta-\nu_{T}\cdot\vartheta_{j}.\end{multlined}
f(n) =\frac{n\left( n-1\right)}{2}+\frac{n\left( n-1\right) \left(n-2\right) \left( n-3\right)}{8}+\frac{n\left( n-1\right) \left(n-2\right)}{3}& =\allowbreak\frac{1}{24}n\left( 3n^{3}-10n^{2}+21n-14\right) .
n(R_1+R_2+R_3 -\epsilon_n) \leq \max_{\alpha\in[0,1]}\left\{n\mathcal{C}\left(\tfrac{\bar{\alpha}P_1}{1+\alpha P_1}\right) + n\mathcal{C}(a\alpha P_1+bP_2)\right\}-n\mathcal{C}(bP_2) + n\mathcal{C}(P_2) + 0.5n, &=n\mathcal{C}(\tfrac{aP_1}{1+bP_2}) + n\mathcal{C}(P_2)+0.5n,
S(D) = \sum^\infty_{n=0}{1 \over (n + 1)^2}{1 \over n!}{\Gamma (n + D - 3) \over \Gamma (D - 3)}{\Gamma (n + 4 - D/2) \over\Gamma ( 4 - D/2)}{\Gamma (D/2) \over \Gamma (n + D/2)}.
\Delta T_{i k}+D_X T_{i k}=-2 \sum_{j, l=1}^{n-1}R_{i j k l}T^{j l}-2 R \operatorname{Ric}_{i k}+2|\operatorname{Ric}(g)|^2\left(g_{i k}-\partial_i f \partial_k f\right)+O\left(r^{-\frac{5}{2}-16\delta}\right).
\begin{array}{rcl}T(z) = -\frac{1}{4}\partial \phi(z) \partial \phi(z) + i \alpha_0 \partial^2 \phi(z)~, & \langle\phi(z) \phi(w)\rangle= -2\ln (z - w)~, & c = 1 - 24\alpha^2_0~.\end{array}
K(\rho ,{\widetilde{V}})\ =\P_{\mathop{\mathrm{Ran}}{\widetilde{V}}}\,{\widetilde{V}}^{1/2}\,(\rho +h(\mathfrak{e}))^{-1}\,{\widetilde{V}}^{1/2}\,P_{\mathop{\mathrm{Ran}}{\widetilde{V}}}\;-\;P_{\mathop{\mathrm{Ran}}{\widetilde{V}}}.
\left.\frac{d\boldsymbol{w}(t)}{dt}\right|_{t=k\delta}\approx\lim_{\delta\rightarrow 0}\frac{\boldsymbol{w}_{k+1}-\boldsymbol{w}_k}{\delta}=-\mu\frac{(\boldsymbol{g}_k-\lambda_k\boldsymbol{b}_k)}{\delta}
f'(z) =e^{-i\theta}(u'_r+iv'_r)=e^{-i\theta}\bigg( \frac{R'_r(u+iv)}{R}+i\Theta '_r(u+iv)\bigg) =e^{-i\theta}(u+iv)\bigg( \frac{R'_r}{R}+i\Theta '_r\bigg) =e^{-i\theta}Re^{i\Theta}\bigg( \frac{R'_r}{R}+i\Theta '_r\bigg) =e^{i(\Theta -\theta )}(R'_r+iR\Theta '_r),
\begin{cases}\gamma_{4}+ \gamma_{5}+ \gamma_{10}+ \gamma_{11}= \gamma_{4}+ \gamma_{6}+ \gamma_{12}+ \gamma_{14}= \gamma_{5}+ \gamma_{6}+ \gamma_{13}+ \gamma_{15}=0 \gamma_{10}= \gamma_{12},\ \gamma_{11}= \gamma_{13},\ \gamma_{14}= \gamma_{15},\ \gamma_{16}= \gamma_{19},\ \gamma_{17}= \gamma_{20},\ \gamma_{18}= \gamma_{21}. \end{cases}
\left\{\frac{\gamma}{4 N}\left( \sum_{1}^{N}- \frac{\partial^{2}}{\partial \alpha_{i}^{2}}- N ( N^{2}- 1 ) / 1 2 \right) + V ( \alpha ) \right\}\tilde{\psi}( \alpha ) ={\cal E}\tilde{\psi}( \alpha ) .
10=(6,4),2,4,2,4,6,2; &6,4,2,4,6=(2,4),6,2; &6,4,2,6=(4,2),4,6,8=(2,6); &\ 4,2,4,2,4,8=(6,2); &6,4,6=(2,4),2,4,6,2; &6,6=(4,2),4,2,4,6,2; &6,4,2,4,2,10=(4,6),2.
X_{i}\hat{X}_{j}=v^{-d_i\delta_{ij}}\hat{X}_{j}\hat{X}_{i}\ (i\in J, j\in J_e) \hat{X}_{i}\hat{X}_{j}=v^{d_ib_{ij}}\hat{X}_{j}\hat{X}_{i}\ (i, j\in J_e).
\begin{cases}V(\xi) \sim m^{-\frac{1}{m-1}}\left(\dfrac{k}{c}\right)^{-\frac{1}{m-1}}\xi^{-\frac{1}{m-1}}, V'(\xi) \sim -(m-1)^{-1}m^{-\frac{1}{m-1}}\left(\dfrac{k}{c}\right)^{-\frac{1}{m-1}}\xi^{-\frac{m}{m-1}},\end{cases}{\rm{as}}\xi\to +\infty.
\alpha_{3}^{\prime}= \left( \begin{array}{c c c c}{{0}}&{{- i}}&{{0}}&{{0}}\\{{i}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{i}}\\{{0}}&{{0}}&{{i a}}&{{0}}\\ \end{array}\right) , \hspace{0 . 3 i n}\alpha_{4}^{\prime}= \left( \begin{array}{c c c c}{{- i}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- i}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- i}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{i a}}\\ \end{array}\right) ,
Z_{L}^{( m , n )}( \tau ) = Z_{n / 2}^{m / 2}( \tau )^{1 6}+ Z_{1 + n / 2}^{m / 2}( \tau )^{1 6}+ Z_{n / 2}^{1 + m / 2}( \tau )^{1 6}+ Z_{1 + n / 2}^{1 + m / 2}( \tau )^{1 6}.
4 dz_j \wedge d\bar z_k = d(z_j + z_k) \wedge d(\overline{z_j + z_k}) - d(z_j-z_k) \wedge d(\overline{z_j-z_k}) &+id(z_j + i z_k) \wedge d(\overline{z_j + i z_k}) - i d(z_j - i z_k) \wedge d(\overline{z_j - i z_k}), &:= \eta_1\wedge \bar \eta_1 - \eta_2\wedge \bar \eta_2 + i \eta_3\wedge \bar \eta_3 - i \eta_4\wedge \bar \eta_4 .
a_1 c_2 = a_2 c_1,b_1 d_2 = b_2 d_1, a_1 d_2 -a_2 d_1 + b_1 c_2 - b_2 c_1 = 0 , a_1 d_1 + a_2 d_2 - b_1 c_1 - b_2 c_2 =1.
\lambda_{\rm min}(D^2d(z)) = |z|^{p-2}, \lambda_{\rm max}(D^2d(z)) =(p-1)\, |z|^{p-2}, \lambda_{\rm min}(D^2s(z)) = (q-1)\, |z|^{q-2}, \lambda_{\rm max}(D^2 s(z)) = |z|^{q-2}.
\tilde{\delta}(x^2)= t(\mathcal M)-{\lim_{n \to \infty}\delta(x^2z_n)}=t(\mathcal M)-{\lim_{n \to \infty}\delta(xz_nxz_n)}= t(\mathcal M)-{\lim_{n \to \infty}(\delta(xz_n)(xz_n)+(xz_n)\delta(xz_n))}= \tilde{\delta}(x)x+x\tilde{\delta}(x).
{\cal I}_{\cal K}^{2}={\cal I}_{\cal K}^{3}= \left( \begin{array}{c c}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) , \qquad{\cal I}_{\cal K}^{4}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) .
|Rm|(x,t) \le C_0\left\{\frac{\rho^{\frac{4}{5}}e^{C(\rho+tK)}}{K^{\frac{2}{5}}V_{x_0}\left(\rho/\sqrt{2K}\right)^{\frac{2}{5}}\min(t,\rho^2/K)}\left[(K+(4\rho/\sqrt{K})^{\frac{1}{p}}E_p(t)^{\frac{1}{p}})t+1\right]\right\}^{\frac{5}{2p}}\cdot &\qquad\cdot (1+(4\rho/\sqrt{K})\sqrt{tV_{x_0}\left(2\rho/\sqrt{K}\right)}E_p(t)^{\frac{1}{2}})^{\frac{7}{2p}}
f(3^{2m+1}) = f(3\cdot3^{2m}) = f\big((3^m)^2 + (3^m)^2 + (3^m)^2 \big)= 3\,f(3^{2m}) *f(3^{2m+2}) = f\big(9 (3^m)^2 \big) = f\big( (3^m)^2 + 2^2(3^m)^2 + 2^2(3^m)^2 \big) = 3^2\,f(3^{2m}).
{\tilde{\delta}}\biggl[{\dot\varphi}^{a}-\omega^{ab}{\partial H\over\partial\varphi^{b}}\biggr]=\int~{\cal D}\lambda_{a}~exp~i\int\lambda_{a}\biggl[{\dot \varphi}^{a}-\omega^{ab}{\partial H\over\partial\varphi^{b}}\biggr]dt
l_{\pi_{\lambda,\beta}}(f(z)) =E_{\beta}(\lambda z)=\sum_{m=0}^{\infty}\frac{(\lambda z)^{m}}{\Gamma(m\beta+1)}=\sum_{m=0}^{\infty}\frac{z^{m}}{m!}\underbrace{\frac{\lambda^{m}m!}{\Gamma(m\beta+1)}}_{=:\tilde{M}_{\lambda,\beta}(m)}=\sum_{m=0}^{\infty}\frac{z^{m}}{m!}\tilde{M}_{\lambda,\beta}(m).
\partial_1\dot H^\pm_n=-\partial_2(\dot H^\pm_2\partial_1\hat\Phi^\pm)=-\partial_2\dot H^\pm_2\partial_1\hat\Phi^\pm-\dot H^\pm_2\partial_2\partial_1\hat\Phi^\pm\quad\mbox{in}\,\,\,\Omega_T\,.
\delta_\alpha (A) = r_0^\alpha-\sum_{n=1}^\infty r_n^{\alpha}= \lim_{m\to \infty}\left( (r_0^{(m)})^{\alpha}- \sum_{n=1}^\infty(r_n^{(m)})^{\alpha}\right) = \lim_{m\to \infty}\delta_\alpha(A^{(m)}).
\textstyle L_z' = \textstyle \smash{\sum_v (-1)^v{A-i \choose v}\sum_{w\geqslant v}X^w{A-v \choose w-v}= \sum_v (-X)^v{A-i \choose v}\sum_{u}X^{u}{A-v \choose u}}& = \textstyle \smash{(1+X)^{i}\sum_v{A-i \choose v}(-X)^v(1+X)^{A-i-v}= (1+X)^i = \sum_{w}X^w{i \choose w}}.
T_5^{-1}T_0^{-1}J(k)^{-1}M_0^{-1}T_3^{-1}T_4^{-1}=\begin{bmatrix}\frac{1}{k}(I_\mu+o(1)) o(1) o(1) I_{n-\mu}+o(1)\end{bmatrix},\;\; k\to0,\;\;k\in \overline{\mathbb{C}}^+.
\left\Vert x-x_{k-1}\right\Vert_{A^{T}A}^{2}-\left\Vert x-x_{k}\right\Vert_{A^{T}A}^{2}=\phi_{k}^{2}\left( \left[-\frac{\theta_k \phi_{k-1}}{\rho_k\phi_k}\right]w_{k}^{T}v_{k}-\frac{\theta_{k+1}}{\rho_{k}}v_{k+1}^{T}w_{k}\right).
(F \otimes F_2) \cdot w_s = 2 (p_1 - p_2 - (s-1)) F v_{p_1, p_2}^{p_1 - p_2 - 2(s-1)}\otimes 1 & = 2 (p_1 - p_2 - (s-1)) v_{p_1, p_2}^{p_1 - p_2 - 2s}\otimes 1 & = \frac{p_1 - p_2 - (s-1)}{p_1 - p_2 - s}(1 \otimes F_2) \cdot w_{s+1},
f^{i}= f_{{\hat{\alpha}}{\hat{\beta}}}^{i}{\hat{e}}^{\hat{\alpha}}\wedge{\hat{e}}^{\hat{\beta}}= 2{\hat{e}}^{5}\wedge{\hat{e}}^{i}+ \varepsilon_{i j k}{\hat{e}}^{j}\wedge{\hat{e}}^{k}
u_k(y) = c_2^{-1}u(y) - m_k V(\psi(y)) \geq (m_j - m_k)V(\psi(y)) &\geq (m_j - M_j + M_k - m_k) V(d_D(y)) \geq - (V(r_{j+1}/ 2)^\gamma - V(r_{k+1}/ 2)^\gamma ) V(r_j).
{\hat{\mathcal{X}}_k}=\Big\{\mathbf{x}(k): \mathbf{x}(k)=\mathbf{a}_d(\mathbf{x}(k-1),k-1)+\mathbf{w}(k-1) ~with~ \mathbf{x}(k-1) \in \mathcal{X}_{k-1}, ~\mathbf{w}(k-1) \in \mathcal{W}_{k-1}\Big\},
\chi (Y, \mathcal O_Y(mK_Y)\otimes \mathcal J(Y, |\!| (m-1)K_Y|\!|)) &= \chi (Y, f^*(\mathcal O_X(mK_X)\otimes \mathcal J (X, |\!|(m-1)K_X|\!|))) =\deg f \cdot \chi (X, \mathcal O_X(mK_X)\otimes \mathcal J(X, |\!| (m-1)K_X|\!|))
\mu_{\alpha}=lim_{Q\rightarrow P}\frac{P^{\prime}Q^{\prime}}{PQ}=lim_{Q\rightarrow P}\frac{\sqrt{\delta x^{2}+\delta y^{2}}}{\sqrt{a^{2}\delta\varphi^{2}+a^{2}cos^{2}\varphi\delta\lambda^{2}}}.
\frac{1}{h}\sum_{p = 1}^{h}e^{i \pi \frac{p \nu}{h}}\mathrm{L i}_{m}\left( e^{- \frac{i \pi p}{h}}x \right) = \sum_{n = 1}^{\infty}\frac{x^{n h + \nu}}{\left( n h + \nu \right)^{m}}.
Q_{2 s + 2 t}^{\mu | 2 s , 2 t}= ( - 1 )^{s t}( \sigma_{2 s + 2 t})^{\frac{3}{2}- t}( \sigma_{2 s}^{+})^{s - 2}( \sigma_{2 t}^{-})^{- 1}\det \mathcal{A}^{\mu}\mathcal{\,},
\delta_{\kappa}\Pi_{i}^{r}= 2 i \bar{\kappa}_{+}\Gamma^{r}\partial_{i}\theta + i \Pi_{i}^{s}( 2 \bar{\kappa}_{+}\Gamma^{r}\tilde{\Omega}_{s}\theta + \bar{\kappa}_{+}\Gamma^{1 1}\tilde{\Omega}_{1 1}\theta \delta_{s}^{r}) + M_{~ s}^{r}\Pi_{i}^{s}+{\cal O}( \theta^{3}) ~ ,
\sum_{m=n}^{\infty}\left( \left( L2^{m+2}\right)^{-\left( \frac{\varepsilon}{2}+1\right)}+\frac{\left( L2^{m+1}\right)^{-\left( \frac{\varepsilon}{2}+1\right)}}{2^{\left( 1+\frac{2}{\varepsilon}\right)}\left( 2^{\left( 1+\frac{2}{\varepsilon}\right)}-1\right)}\right) .
f_1(x) = \frac{1}{z^2}\left(\frac{x(z+1)}{z^2}+\frac{z+1}{xz^2}+1\right), f_2(x) = \frac{z^2+1}{z^2}\left(\frac{x(z+1)}{z^2}+\frac{z+1}{xz^2}+1\right),
\frac{\det D ( A )}{\det D ( 0 )}\, = \, \prod_{k}\left\{\, \frac{v_{+}^{\dagger}( k \mid 0 ) v_{+}( k \mid A )}{\mid v_{+}^{\dagger}( k \mid 0 ) v_{+}( k \mid A ) \mid}\frac{v_{+}^{\dagger}( k \mid A ) v_{-}( k \mid A )}{v_{+}^{\dagger}( k \mid 0 ) v_{-}( k \mid 0 )}\frac{v_{-}^{\dagger}( k \mid A ) v_{-}( k \mid 0 )}{\mid v_{-}^{\dagger}( k \mid A ) v_{-}( k \mid 0 ) \mid}\, \right\}\; ,
\begin{array}{|l|l|l|}\hline\emph{row}H\leqslant \emph{remark}\hline1 R.(3_+^{1+2}{:}8{:}2) q=3 2 R.(5_+^{1+2}{:}24{:}2) q=5 3 R.(57{:}9.2) q=8 4 R.(2^{3+6}{:}63{:}3).2 q=8 \hline\end{array}
\left<\varphi_{2}+\frac{\pi}{6\sqrt{3}}\bar{\varphi}_{1},\varphi_{2}+\frac{\pi}{6\sqrt{3}}\bar{\varphi}_{1}\right>_{L^{2}}^{2}= \left<\varphi_{2},\varphi_{2}\right>+\frac{\pi}{3\sqrt{3}}\left<\bar{\varphi}_{1},\varphi_{2}\right>+\frac{\pi^2}{108}= \frac{\pi^{2}}{27}-\frac{\pi^{2}}{54}+\frac{\pi^2}{108}= \frac{3\pi^{2}}{108}= \frac{\pi^{2}}{36}
k ={u_1 \over 2}\,\Big[ \,{{\cal E}_1 +\sqrt{{\cal E}_1^2 + \mu^2}\over K + \sqrt{K^2 + \mu^2}}-{K + \sqrt{K^2 + \mu^2}\over{\cal E}_1 +\sqrt{{\cal E}_1^2 + \mu^2}}\,\Big]\,.
\int \frac{\textup{d}^dk}{(2\pi)^d}e^{-i{\bf k}\cdot{\bf x}}k^q=\frac{2^q\Gamma\left(\frac{d+q}{2}\right)}{\pi^{d/2}\Gamma\left(\frac{-q}{2}\right)}\frac{1}{|{\bf x}|^{d+q}}
\mathcal{P}(v) := \partial_\xi^2 v + \frac{d+1}{\xi}\partial_\xi v - 3(d-2) (2Q +\xi^2Q^2) v +\bar B(v) + \theta(\tau) \Lambda_\xi Q + \theta(\tau) \Lambda_\xi v - b(\tau) \partial_\tau v,
f=\displaystyle \sum_{k \in \mathfrak{D}}\sum_{j=0}^{q-1}t_{j,k}T_{Ak}e_{\alpha_j}= \displaystyle \sum_{m \in \mathfrak{D}}\sum_{p=0}^{q-1}s_{p,m}T_{Am}\varphi_p= \displaystyle \sum_{v \in{\mathbb{Z}}^2_N}w_{v}F_{v},
\begin{cases}D_{a^+}^{\alpha,\beta}x(t)=f(t,x(t)), 0<\alpha<1,\, 0\leq\beta\leq1,t>{a}, \lim_{t\to{a^{+}}}{(t-a)}^{1-\gamma}x(t)=x_0, \gamma=\alpha+\beta(1-\alpha),\end{cases}
\left( \begin{array}{c}{\phi}\\{\chi}\\ \end{array}\right) = \frac{\sqrt{1 2}( P ( z ) Q^{\prime}( z ) - Q ( z ) P^{\prime}( z ) )}{| q | ( | P ( z ) |^{2}+ | Q ( z ) |^{2})^{3 / 2}}\left( \begin{array}{c}{P ( z )}\\{Q ( z )}\\ \end{array}\right) ,
\begin{multlined}\left\|i^\ast (f_\varepsilon (u_0-W_{\delta_\varepsilon (t),\xi}))-u_0+W_{\delta_\varepsilon (t) , \xi}\right\|_{P_g}= O\left(\left\| (f_\varepsilon (u_0-W_{\delta_\varepsilon (t),\xi}))-P_g(u_0-W_{\delta_\varepsilon (t) , \xi})\right\|_{L^\frac{2n}{n+4}}\right)\end{multlined}
\begin{array}{rl}{\rm E}[z] \geq \frac1{\beta}\sum_B p_B{\rm E}[c^t\mid E_B]x -\frac1{\beta}\sum_B p_B{\rm E}[c^t]x+\frac1{\beta}\sum_B p_B{\rm E}[c_B^t]x_B & & = \frac1{\beta}\sum_B p_B{\rm E}[c_B^t]x_B.\end{array}
{\tilde{\nu}}_{\tilde{\alpha}}^{( 1 )}\frac{\partial V_{( n = 1 )}^{( 1 )}( a_{i}, p_{i}, \theta )}{\partial{\tilde{\xi}}_{\tilde{\alpha}}^{( 1 )}}- \frac{\partial{\cal G}^{( n = 2 )}( a_{i}, p_{i}, \theta )}{\partial \theta}= 0 .
f_{|\Delta_n|^2}(x) =\begin{cases}\frac{1}{b_{0,n}}K_0\left(\sqrt{{2\tilde{A}_n}/{b_{0,n}}}\right)I_0\left({\sqrt{2x}}/{b_{0,n}}\right),\,x < \tilde{A}_n^2 \frac{1}{b_{0,n}}I_0\left(\sqrt{{2\tilde{A}_n}/{b_{0,n}}}\right)K_0\left({\sqrt{2x}}/{b_{0,n}}\right),\,x > \tilde{A}_n^2.\end{cases}
\Vert u_n-\tilde u'\Vert \le\Vert u_n-v_n\Vert+\Vert v_n-v\Vert+\Vert v-\tilde u'\Vert &\le\frac{\delta}{3}+\Vert v_n-v_{\tilde g(n)}\Vert+\frac{\delta}{12}\le\delta=\frac{\varepsilon}{2d(2+\tau)}.
K_{n,i}(x) \ = \x\sum_{j=0}^{\left\lceil \frac{n-1}{n}i\right\rceil -1}K_{n-1,j}(x)+\sum_{j=\left\lceil \frac{n-1}{n}i\right\rceil}^{2n-3}K_{n-1,j}(x)\,.
\prod_{i=1}^{n+1}u_i = u_{n+1}\prod_{i=1}^n u_i = \min(v_n,u_n) \left( \frac{\prod_{i=1}^n v_i}{\prod_{i=1}^n u_i}\right) \prod_{i=1}^n u_i & = v_{n+1}\prod_{i=1}^n v_i = \prod_{i=1}^{n+1}v_i.
\frac{1}{n}\sum_{i=1}^n \phi_b(b_{1,i}^0,...,b_{P,i}^0,...,b_{1,i}^t,...,b_{P,i}^t,w_i) \doteq \mathbb{E}\left[\phi_b(\sigma_1^0\breve{Z}_1^0,...,\sigma_P^0\breve{Z}_P^0,...,\sigma_1^t\breve{Z}_1^t,...,\sigma_P^t\breve{Z}_P^t,W)\right],
w(x,\tau) = \frac{e^{\frac{\tau}{m-1}}}k (\delta \rho^2)^{\frac1{m-1}}u(x,\Lambda^{-1}(\tau)) \le \frac{b_1^\theta}{k}(\delta \rho^2 )^{\frac 1{m-1}}u(x,\Lambda^{-1}(\tau)),
\hat{\Gamma}_{\hat{\mu}}= \hat{\Gamma}_{\hat{\mu}\hat{\nu}\hat{\lambda}}S^{\hat{\nu}\hat{\lambda}}= \frac{i}{4}( C_{\hat{\mu}\hat{\nu}\hat{\lambda}}- C_{\hat{\mu}\hat{\lambda}\hat{\nu}}- C_{\hat{\nu}\hat{\lambda}\hat{\mu}}) S^{\hat{\nu}\hat{\lambda}}
\| w \|_{X_{h}^{\ast}}:= \sup_{v_{h}\in X_{h}}\frac{(w,v_{h})}{\|\nabla v_{h}\|}, \; \;\| w \|_{V_{h}^{\ast}}:= \sup_{v_{h}\in V_{h}}\frac{(w,v_{h})}{\|\nabla v_{h}\|}.
\frac{1}{(2n+\alpha+\beta+2)}Q_n(x) - \frac{1}{(2n+\alpha+\beta+2)}Q_{n+1}(x)= \frac{x}{N(\alpha+1)}\hat Q_{n}(x), - \frac{(n+1)(n+\beta+1) (n+\alpha+\beta+N+2)}{(2n+\alpha+\beta+3)}\hat Q_{n}(x) \qquad{}+ \frac{(n+\alpha+\beta+2)(N-n-1)(n+\alpha+2)}{(2n+\alpha+\beta+3)}\hat Q_{n+1}(x)= N(\alpha+1) Q_{n+1}(x) .
\alpha_{i}=\frac{2|\tau_{i}|}{\tau_{1}-\tau_{2}}\alpha(r,\tau_{i},\Phi,\Psi) (r - r_{a,b}(\tau_{i},\Phi,\Psi)) \in S^0_{hom},\beta_{i}= \beta(r,\tau_{i},\Phi,\Psi),
\left ( \int_{-Q(v)}^{Q(v)}\frac{n^{1/\eta}}{q^{1/\eta}}\left ( \sum_{k=1}^{[n/q]}\frac{1}{k^{1+2\delta/r}}D_{k,q}(t)^{\delta}\right )^{1/(2\delta)}dt \right )^\eta \ll n x^{\eta-\beta/2-1}\int_0^v R_n^{\beta/2}(u) Q(u) du \, .
\begin{array}{rcl}T_c = \displaystyle +[3D_m(k_1)D_m(k_2)+D(k_1)D(k_2)]D_m(k_1+k_2)(1+K_{12}) & \displaystyle +4(N-1)D(k_1)D(k_2)D_m(k_1+k_2). \end{array}
{\cal L}_{\rm mass}^{W}= -{1 \over 2}\left( \begin{array}{ccc}\psi_X^{\dot\alpha}\psi_Y^{\dot\alpha}\psi_S^{\dot\alpha}\end{array}\right) M_\psi^W \left( \begin{array}{c}\psi_{X\dot\alpha}\psi_{Y\dot\alpha}\psi_{S\dot\alpha}\end{array}\right) +{\rm h.c.},
U_{_W}( t_{i}, t_{i - 1}) \cong \frac{1}{2 \pi}\int d \triangle_{i}d \xi_{i}\exp \{- i ( \xi_{i}x_{i}+ \triangle_{i}p_{i}) \}\textrm{T r}[ \{1 - i \epsilon \hat{H}( \hat{x}, \hat{p}) \}\exp i ( \xi_{i}\hat{x}+ \triangle_{i}\hat{p}) ] .
\partial_t \Gamma +b\cdot \nabla \Gamma -\Delta \Gamma +\frac{2}{r}\partial_r \Gamma \leq 0\textrm{in}\ \Pi\times (0,T], \partial_n\Gamma +2\Gamma \leq 0\textrm{on}\ \partial \Pi\times (0,T], \Gamma \leq 0\textrm{on}\ \Pi\times \{t=0\}.
\mu_{n}(c) =\sum\limits_{k=0}^{n}\genfrac{\{}{\}}{0pt}{}{n}{k}c^{k}\frac{( \vec{\alpha})_{k}}{(\vec{\beta}+1)_{k}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}+k \vec{\beta}+1+k\end{matrix};c\right] .
\pi(u)\cdot\pi(v) =\sum_{j=1}^m u_j v_j + \Bigl( 1-\sum_{j=1}^m u_j^2 \Bigr)^{\frac12}\cdot\Bigl(1-\sum_{j=1}^m v_j^2\Bigr)^{\frac12}&=1-\frac12\sum_{j=1}^m(u_j-v_j)^2+O\bigl(|u|^4+|v|^4\bigr)u,v\to 0\,.
\begin{array}{ll}V_\phi (1, \cdots, N) = exp \left\{\frac{1}{2}\sum_{r \neq s}\phi_0^{(r)}\, \tilde{N}_{0,0}^{r,s}\phi_0^{(s)}+ \right. begin{align*}2ex] \left. + \sum_{r,s}\sum_n \phi_0^{(r)}\tilde{N}_{0,n}^{r,s}\phi_n^{(s)}+ \frac{1}{2}\sum_{r,s}\sum_{n,m}\phi_0^{(r)}\tilde{N}^{r,s}_{n,m}\phi_m^{(s)}\right\}.\end{array}
\begin{array}{r c l}{d \Pi_{( 2 )}^{2}}&{=}&{\mathrm{c o s h}^{2}\chi d \phi^{2}- d \chi^{2},}\\{}&{}&{}\\{d \Omega_{( 2 )}^{2}}&{=}&{d \theta^{2}+ \sin^{2}{\theta}d \varphi^{2},}\\ \end{array}
\mathrm{div}_{\eta}(T(\nabla x)):= \big(\mathrm{div}_{\eta}(T(\nabla x_1)),\ldots,\mathrm{div}_{\eta}(T(\nabla x_m))\big) = (\mathrm{div}(T(\nabla x_1))-\langle \nabla\eta, T(\nabla x_1)\rangle,\ldots,\mathrm{div}(T(\nabla x_m))-\langle \nabla\eta,T(\nabla x_m)\rangle) = \mathrm{div}(T(\nabla x))-\mathrm{d}\eta\circ T(\nabla x).
{}_2F_1\bigg[\begin{matrix}\alpha \beta &\alpha-\beta+1\end{matrix}\bigg|\,z\bigg]=(1+z)^{-\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12+\frac12\alpha &\alpha-\beta+1\end{matrix}\bigg|\,\frac{4z}{(1+z)^2}\bigg],
-d \hat{Y}_s^{t,x;u_{\cdot}}= \hat{G}\bigl(s, X_s^{t,x;u_{\cdot}}, \hat{Y}_s^{t,x;u_{\cdot}}, Z_s^{t,x;u_{\cdot}}\bigr) ds - Z_s^{t,x;u_{\cdot}}dB_s,
I_\lambda(\omega) =I_\lambda(\omega)-\frac{1}{2}I'_\lambda(\omega)\omega &=\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx+\left(\frac{1}{2}-\frac{1}{2_\alpha^*}\right)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx.
(\frac{1}{b^{2}}\frac{\partial^{2}}{\partial z_{1}^{2}}+\sum_{i=2}^{N}(\frac{1}{z_{1}-z_{i}}\frac{\partial}{\partial z_{i}}+\frac{\Delta_{i}}{(z_{1}-z_{i})^{2}}))\langle V_{2,1}(z_{1})\prod_{i=2}^{N}V_{\Delta_{i}}(z_{i})\rangle=0.