harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{-B_\tau}\mathcal{G}_{N}^{(\beta)}e^{B_\tau}=\; 4\pi (N-1)\mathfrak{a}_N^{(\beta)}&-\frac{1}{2}\sum_{p\in \Lambda_+^*}\left[ p^2+\widehat V(0)-\sqrt{|p|^4+2p^2 \widehat{V}(0)}-\frac{\widehat{V}^2(0)}{2p^2}\right] &+\sum_{p \in \Lambda_+^*}\sqrt{F_p^2-G_p^2}a^*_p a_p+\theta_{N}^{(\beta)}
F(x) = \frac{1}{2}- \frac{1}{\pi}\int_{0}^{\infty}\left( \frac{e^{-jtx}\phi(t) - e^{jtx}\phi(-t)}{2jt}\right) dt &= \frac{1}{2}- \frac{1}{\pi}\int_{0}^{\infty}\mathcal{I}m\left( \frac{e^{-jtx}\phi(t)}{t}\right) dt.
G_{Dn}[x_2,x_1] = \frac{\gamma^0 \Delta x_0 + \sum_{\alpha =1}^{D-1}\gamma^{\alpha}( \Delta x_{\alpha}- 2n_{\alpha}\pi)}{{\vert{\Delta x_0 + \sum_{\alpha = 1}^{D-1}( \Delta x_{\alpha}- 2n_{\alpha}\pi)}\vert}^{D}},
V_{N S 5}( x , b , c ) = \frac{A^{2}m^{3}}{1 8}b^{3 / 2}\left[ \frac{3 x^{2}}{3 x^{4}+ b}\right] \Biggl [ \left( x^{4}- 1 \right)^{2}+ c + \frac{3 c}{b}x^{4}\Biggr ] ,
A_{\sigma(UV)}A_{\sigma(UW)}^* =f(UV)\overline{f(UW)}A_VA_W^* , A_{\sigma(UV)}\Psi_{\sigma(UW)}^* =f(UV)\overline{f(UW)}A_V\Psi_W^*, \Psi_{\sigma(UV)}\Psi_{\sigma(UW)}^* =f(UV)\overline{f(UW)}\Psi_V\Psi_W^*
\mathrm{Ad}_{V_1}\circ\pi =\lambda \mathrm{Ad}_W\circ \mathrm{Ad}_{T_1^{-1}}\circ \mathrm{Ad}_{T^{\frac{1}{2}}V}\circ\pi =\lambda \mathrm{Ad}_{W}\circ \mathrm{Ad}_{T_1^{-1}}\circ\Psi =\mathrm{Ad}_W\circ\Phi_1 =\Phi
R_K :=\Delta u_h - u_h +f \quad{\rm for\each}\K\in\mathcal{T}_h, R_e :=[\nabla_h u_h] \quad{\rm for\each}\e \in{\cal E}_h^i,R_e:=\nabla_h u_h\cdot n +g\lambda_h \quad{\rm for\each}\e\in{\cal E}_h^2.
\int_{-\pi/2}^{\pi/2}y_{2,2}y_{2,3}= \frac{(n-1)^2(n-3)^2}{2\times 48^2}\frac{1}{80}\left(-\frac{7785\pi}{128}+\frac{555\pi^3}{16}-\frac{67\pi^5}{16}+\frac{13\pi^7}{672}+\frac{4\pi^9}{315}\right) &\approx \frac{(n-1)^2(n-3)^2}{48^2}0.249
B=\bigg(\frac{|J_{\rho_0}|e^{-T_{0}}}{2|J_{\rho_0}|e^{-T_{0}}+2}\bigg({\rm \exp}\bigg[\bigg(2+\frac{2}{|J_{\rho_{0}}|e^{-T_{0}}}\bigg)T_{0}\bigg] -1\bigg) + C_*{\rm \exp}\bigg[\bigg(2+\frac{2}{|J_{\rho_{0}}|e^{-T_{0}}}\bigg)T_{0}\bigg] \bigg)^{-1},
\! \! \! \! \! \! \! \! B_{N}( \lambda ) + \frac{N}{2}\frac{\mathrm{d}V^{( \alpha )}( \lambda )}{\mathrm{d}\lambda}= \frac{\lambda}{2 c_{N}}A_{N}( \lambda ) - ( - 1 )^{N}\frac{\alpha}{\lambda}- \frac{\alpha}{2}( T_{B}( N - 1 , \lambda ) - T_{B}( N , \lambda ) ) \ .
\aligned \Psi(t) := 4(e^{t/2}+e^{-t/2}-2) - \sum_{n \leq e^t}\frac{\Lambda(n)}{\sqrt{n}}(t-\log n) &+ \frac{t}{2}\left[ \frac{\Gamma'}{\Gamma}\left(\frac{1}{4}\right) - \log \pi \right] + \frac{1}{4}\left( C - e^{-t/2}\Phi(e^{-2t},2,1/4) \right), \endaligned
\begin{pmatrix}M_1 M_2 M_3 \end{pmatrix}= \begin{pmatrix}Q_1 R_1 M_2 M_3 \end{pmatrix}= \begin{pmatrix}\begin{pmatrix}Q_1 I \end{pmatrix}\begin{pmatrix}M_2 R_1\end{pmatrix}M_3 \end{pmatrix}= \begin{pmatrix}Q_{12}R_{12}M_3 \end{pmatrix}= \cdots= Q_{123}R_{123}.
\langle (b+k-l)(\pi_{\lambda+\square_1}\boxtimes 1)v,(1\boxtimes\pi_{\hat{\lambda}})u\rangle+\langle le_1(\pi_{\lambda}\boxtimes 1)v,(1\boxtimes\pi_{\hat{\lambda}})u\rangle +\langle (\pi_{\lambda}\boxtimes 1)v,(-b-k)e_1(1\boxtimes\pi_{\hat{\lambda}})u \rangle = (b+k-l+l-b-k)e_1 = 0,
\sum_{i = 1}^k \gamma_i W_i = \sum_{\substack{i = 1 i \neq i_0}}^s \gamma_i W_i + \gamma_{i_0}W_{i_0}\longmapsto \sum_{\substack{i = 1 i \neq i_0}}^k \gamma_i W_i + \gamma_{i_0}\sum_{\substack{j = 1 j \neq h}}^s (-\alpha_j\alpha_h^{-1}La_jR) &\textit{(and the further cancellations if there are any)}.
\begin{matrix}T_{A}&=&0&:&sec^{2}\frac{B}{2}&:&sec^{2}\frac{C}{2}\\T_{B}&=&sec^{2}\frac{A}{2}&:&0&:&sec^{2}\frac{C}{2}\\T_{C}&=&sec^{2}\frac{A}{2}&:&sec^{2}\frac{B}{2}&:&0.\end{matrix}
(1+ R_0(\lambda+i0) \, T(B,V))^{-1}= (1+G_0\, T(B,V))^{-1}\, - \qquad - (1+G_0\, T(B,V))^{-1}\, G_1\, T(B,V)\, (1+G_0\, T(B,V))^{-1}\, \lambda^{\mu(\alpha)}+ o(\lambda^{\mu(\alpha)})
|p-(\alpha_1 q \sigma_1(u)+\cdots+\alpha_d q \sigma_d(u))| =|p-(\alpha_1 q u_1+\cdots+\alpha_d q u_d)| &=|(\lambda-\alpha_1)q \sigma_1(u)+(\sigma_2(\lambda)-\alpha_2)q \sigma_2(u)+\cdots+(\sigma_d(\lambda)-\alpha_d)q \sigma_d(u)|.
D_n({\rm e}^h{\rm d}\mu)= \frac{1}{n!}\int_{\mathbb T}\cdots \int_{\mathbb T}{\rm e}^{\sum_{j=1}^n h(z_j)}\prod_{1 \leq j <k \leq n}|z_j-z_k|^2{\rm d}\mu(z_1) \cdots{\rm d}\mu(z_n).
\lambda_{0}^{\prime}= \lambda_{0}+ \sum_{i = 1}^{N}( \alpha_{i}- 1 ) \log | \omega_{i}c + d | , \quad \displaystyle \omega_{i}^{\prime}= \frac{a \omega_{i}+ b}{c \omega_{i}+ d}, \qquad \alpha_{i}^{\prime}= \alpha_{i}.
\frac{[G_n:B_{n}^{-}]}{|U_n|}\Psi_{\underline{\chi}_{n}}^{GL_{n}}(g_{n-1,1}(1,a_n))=\sum_{t\in T_n}f_{T_n}(e_{\psi_n}g_{n-1,1}(1,a_n)e_{\psi_n})(t)\underline{\chi}_n(t).
\Lambda^{N+1}_N(\lambda, \mu) = t^{|\mu|}\psi_{\lambda/\mu}(q, t)\frac{P_{\mu}\big(1, t, \dots, t^{N-1}\big)}{P_{\lambda}\big(1, t, \dots, t^N\big)}= \psi_{\lambda/\mu}(q, t)\frac{P_{\mu}\big(t, t^2, \dots, t^N\big)}{P_{\lambda}(1, t, \dots, t^N\big)},
g_{n,k}(\tau)=\frac{1}{|\tau|}\,\widetilde{g}_{n,k}(\tau),\widetilde{g}_{n,k}(\tau):=|\tau|\frac{|h_{n+1,k}(\tau)|}{|h_{n,k}(\tau)|}\frac{|P_{n,k-1}(\tau)|}{|P_{n+1,k-1}(\tau)|}\,\frac{|P_{n,k+1}(\tau)|}{|P_{n+1,k+1}(\tau)|}.
f^{\prime}(x_{0})=-\frac{f(x_{0}-h)-f(x_{0})}{h}+\frac{f^{(2)}(x_{0})}{2!}h-\frac{f^{(3)}(x_{0})}{3!}h^{2}+\frac{f^{(4)}(x_{0})}{4!}h^{3}+\cdot\cdot\cdot
\sum_{I \subset [1,m] \atop |I|=k}\Theta_{I}\prod_{i \in [1,m]{\setminus}I, \, j \in J \atop i < j}\big(1+t \beta-t^2 q_{ij}\big)={m \brack k}_{1+t \beta}+\sum_{A \subset [1,n], \, B \subset[1,n] \atop |A|=|B|=r}u_{A,B},
\begin{matrix}\forall C[\neg MC\leftrightarrow\exists F(\forall x[Mx\rightarrow\exists s(s\in C\wedge\langle x,s\rangle\in F)]\wedge\\ \forall x\forall y\forall s[(\langle x,s\rangle\in F\wedge\langle y,s\rangle\in F)\rightarrow x=y])].\end{matrix}
(\nabla_{\mu}v)^{n}=(\nabla_{\mu}(v^{l}e_{l}))^{n}=((\partial_{\mu}v^{l})e_{l}+v^{l}(\nabla_{\mu}e_{l}))^{n}=\partial_{\mu}v^{n}+\Gamma_{\mu l}^{n}v^{l}
\sum_{s>i, t \geq j}( A_k )_{is}( X_k )_{st}( B_k )_{tj}=\sum_{s > i}( A_k )_{is}\underbrace{\sum_{t \geq j}( X_k )_{st}( B_k )_{tj}}_{=: (X^B_{k})_{sj}}=:\sum_{s > i}( A_k )_{is}(X^B_{k})_{sj},
[[B,B], \underbrace{C, \dots ,C}_{2d-1}, C, \dots ] \leq \prod_{i+j=2d-1}[[B, \underbrace{C, \dots ,C}_{i}], [B,\underbrace{C, \dots ,C}_{j}], C, \dots ] \leq [[[B,\underbrace{C, \dots ,C}_{d}], B], C, \dots ] \leq [[[B,B],B],C,\dots ].
\mathbb{P}_{\hat{\mathbf{D}}^{\perp}}(\mathbf{D}) = \mathbf{D}- \frac{(\mathbf{D}: \hat{\mathbf{D}}) \hat{\mathbf{D}}}{\hat{\mathbf{D}}: \hat{\mathbf{D}}}\mathbf{D}\in \mathbb{R}^{k \times d}.
\mathcal{H}^k(\bar{M}):=\{\gamma\in H^1_{\operatorname{loc}}\Omega^k(\bar{M})|d\omega=0=\delta\omega\}, \mathcal{H}^k_N(\bar{M}):=\{\gamma\in \mathcal{H}^k(\bar{M})|n(\gamma)=0\}, \mathcal{H}^k_D(\bar{M}):=\{\gamma\in \mathcal{H}^k(\bar{M})|t(\gamma)=0\},
(h_n,h_k)\!=\!0,\,\,\,\,\, n\not=k,\,\,\,\,\,\, (h_n,h_n)\!\sim\!2(1-c_{1n})c_{2n}^2\!-\!(1-c_{1n})^2c_{2n}^2\!+\!1\!-\!c_{2n}^2.
f(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\bigg(\sum_{l=0}^{k}S_{2,\lambda}(k,l)|z|^{2l}\bigg)=\sum_{l=0}^{\infty}\bigg(\sum_{k=l}^{\infty}S_{2,\lambda}(k,l)\frac{t^{k}}{k!}\bigg)|z|^{2l}.
\cos \widehat{b x c}= \frac{( x - b )^{2}+ ( x - c )^{2}- ( b - c )^{2}}{2 \left| x - b \right| \left| x - c \right|}= \frac{\mathcal{B}^{2}+ \mathcal{C}^{2}- 3 \mathcal{X}\alpha}{2 \mathcal{B C}}
\tau_{n}=A\gamma_{n}=A\,\left( \Psi_{n}^{3/2}\,\,\max_{i\in \Upsilon_{3}}\,p_{i,n}^{1/2}+\max_{1\leq i\leq s_{n}}\,q_{i,n}^{1/2}\right)\rightarrow 0{\it ,\as\}n\rightarrow \infty ,
\nabla_X Y\in\Gamma^{\infty}(H), \nabla J=0, \nabla g_{\theta}=0, T_{\nabla}(Z,W)=0, T_{\nabla}(Z,W^\prime) =2\sqrt{-1}L_{\theta}(Z,W^\prime)T, T_{\nabla}(T,J(X))+J(T_{\nabla}(T,X))=0
\begin{cases}-\Delta v_t + \frac{\lambda}{|\Omega|}\int_{\partial \Omega}b g'(t + v) (1 + v_t) = \lambda Q [f_u(x,t + v)(1+v_t)] \mbox{in $\Omega$}, \dfrac{\partial v_t}{\partial \mathbf{n}}= \lambda b g'(t + v)(1+v_t) \mbox{on $\partial \Omega$}. \end{cases}
\left(\exp(-\nu m)^{-1}(\partial +\nu)\exp(-\nu m)\phi\right)(x) & =\exp(\nu x)\left((\partial +\nu)\exp(-\nu m)\phi\right)(x) & =\exp(\nu x)(-\nu e^{-\nu x}\phi(x)+e^{-\nu x}\phi'(x) +\nu\exp(-\nu x)\phi(x)) = \phi'(x) = \partial_{0,\nu}\phi(x)
{\cal P}_{R}= (q^{I},\;n^{i},\;m_{j})\left( \begin{array}{c}l_{I}\cdot{\bf e}_{\mu}^{(R)}{\bar k}_{i}\cdot{\bf e}_{\mu}^{(R)}k^{j}\cdot{\bf e}_{\mu}^{(R)}\end{array}\right){\bf e}_{\mu}^{R}.
\begin{aligned}\frac{|\nabla u(x)-\nabla u(y)|}{|x-y|^{\gamma}}\leq 2 \left (\frac{128}{r}\right )^\gamma \sup_{B_\rho}|\nabla u| & \leq C_6 \left (||u||_{L^1(B_1)}+ \int_{\mathbb{R}^n \setminus B_{1}}\frac{|u(y)|}{|y|^{n+2s}}dy + ||\mu||_{L^p(B_1)}\right ),\end{aligned}
\left(cq;q\right)_{\infty}\sum_{n=0}^{\infty}\frac{\left(-\sqrt{q};\sqrt{q}\right)_{n}q^{\binom{n}{2}}c^{n}}{\left(\sqrt{q},\sqrt{cq},-\sqrt{cq};\sqrt{q}\right)_{n}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}-n}c^{n}A_{q}^{2}\left(-cq^{n-1/2}\right)}{\left(q;q\right)_{n}}.
|\sum_{j=0}^{n}Q^{j}f_{m}| =\frac{1}{m}|\sum_{j=0}^{n}{\displaystyle\sum\limits_{k=1}^{m}}Q^{j+k}f|\leq\frac{1}{m}{\displaystyle\sum\limits_{k=1}^{m}}|\sum_{j=k}^{n+k}Q^{j}f| & \leq2\sup_{n}|\sum_{j=0}^{n}Q^{j}f|\leq2g_{f},
v_f^{4/3}\left(\frac{\gamma_n{v^2_0}\log(d+1)}{n}\right)^{1/3}+ v_f\left(\frac{1}{n}\sum_{i=1}^n|Y_i|\right)\left(\frac{\gamma_n{v^2_0}\log(d+1)}{n}\right)^{1/3}+ \frac{T_n}{n}.
E\|E[S_n(\theta)|\mathcal{G}_0]\|^2 =\lim_{n\to\infty}E\|E[S_n(\theta)|\mathcal{G}_0]-E[E[S_n(\theta)|\mathcal{G}_0]|\mathcal{G}_{-n}]\|^2 & = \sum_{j\geq 0}E\|\mathcal{P}_{-j}(E[S_n(\theta)|\mathcal{G}_0])\|^2=\sum_{j\geq 0}E\|\mathcal{P}_{-j}(S_n(\theta))\|^2.
\sum_{v}\textup{inv}_v(((\pi^a)^*(\alpha_i))(R_v)) = \sum_v\textup{inv}_v(\alpha_i(\pi^a(R_v))) = \sum_{v\in S}\textup{inv}_v(\alpha_i(Q_v)) = \sum_{v\in S}\textup{inv}_v(\alpha_i(P_v)) & = \sum_{v}\textup{inv}_v(\alpha_i(P_v))=0,
R_{s}= \overline{\gamma}_{M}\times \bigg\{\mathcal{W}_{o}\bigg(\frac{e^{\frac{1}{\overline{\gamma}_{M}}+\frac{1}{\overline{\gamma}_{W}}}}{\overline{\gamma}_{W}(1-\epsilon)}\bigg) - \frac{1}{\overline{\gamma}_{W}}\bigg\}^{+},
\#E(Q,Y) \le b^{2l}b\cdot \#E(Q,R)\cdot \#E(Q,Y\cap Z_c)\cdot \#E(Q,Y_1) &\le b^{2l}b\cdot\sigma\cdot \sigma^{q}\cdot[(4l)^q\cdot (128l^2)^{qb}\cdot (qb)^{4lq}\cdot\sigma^{12lq+32l^2b}] &=b^{2l+1}(4l)^q\cdot(128l^2)^{qb}\cdot(qb)^{4lq}\cdot\sigma^{1+q+12lq+32l^2b}.
e^C\left(vol(\partial D_R\times [-C,C])+vol(D_R\times\{-C\})+vol(D_R\times [-C,C])\right) & \leq e^C \left( 2\omega_{n-1}R^{n-1}C+\omega_nR^n+4\omega_nR^n C\right) & = e^C (2 \omega_{n-1}C + \omega_n(1+4C) R) R^{n-1}.
J_3= \int_0^\infty (f_u^{(1)},f_h^{(1)})\cdot(u_{px}^1,h_{px}^1) x^{1-2\sigma_1}{\rm d}y \leq \Vert (f_u^{(1)},f_h^{(1)})x^{\frac{1}{2}-\sigma_1}\Vert_{L_y^2}\Vert (u_{px}^1,h_{px}^1) x^{\frac{1}{2}-\sigma_1}\Vert_{L_y^2}\leq Cx^{-\frac{3}{2}}++\delta_0\Vert (u_{px}^1,h_{px}^1) x^{\frac{1}{2}-\sigma_1}\Vert_{L_y^2}^2,
\|x -x_+\| \geq \alpha\sqrt{1 -\theta}\min\Big\{\lambda, \frac{\mu}{\sqrt{1 +\mu^2}}\Big\}\max\{d_A(x), d_B(x)\}&\geq \frac{\alpha\sqrt{1 -\theta}}{\kappa}\min\Big\{\lambda, \frac{\mu}{\sqrt{1 +\mu^2}}\Big\}d_{A\cap B}(x) =\nu d_{A\cap B}(x),
E_0(x)-E_0(x+1)= \ln\frac{f(x)}{f(x+1)}+\ln\frac{\Phi_0(\nu;x)}{\Phi_0(\nu;x+1)}, E_k(x)-E_k(x+1)= \ln\frac{f(x)}{f(x+1)}+\ln\frac{\Phi_0(\nu;x)+\mathrm{MC}_k(x)}{\Phi_0(\nu;x+1)+\mathrm{MC}_k(x+1)},k\ge 1.
L(f)=lim_{N\rightarrow\infty}\sum_{i=1}^{N}|f(t_{i})-f(t_{i-1})|=lim_{N\rightarrow\infty}\sum_{i=1}^{N}|\frac{f(t_{i})-f(t_{i-1})}{\Delta t}|\Delta t=\int_{a}^{b}|f^{\prime}(t)|dt.
\frac{G_2}{SO(4)}, \,\,\, \frac{F_4}{Sp(3)\times Sp(1)}, \,\,\, \frac{E_6}{SU(6)\times Sp(1)}, \,\,\, \frac{E_7}{Spin(12)\times Sp(1)}, \,\,\, \frac{E_8}{E_7\times Sp(1)}\ ,
\displaystyle \prod_{\mu = 1}^{m}\oint_{C^{( N )}}{\frac{d x_{\mu}}{2 \pi i}}= \displaystyle \prod_{\mu = 1}^{m - 1}\oint_{C^{\prime}{}^{( N )}}{\frac{d x_{\mu}}{2 \pi i}}\left( \oint_{C^{\prime}{}^{( N )}}{\frac{d x_{m}}{2 \pi i}}+ m \mathrm{R e s}_{x_{m}= z_{N}\tau^{-}1}\right) ,
1_{N}* ((\eta_{\hat{\nu}}\lambda_{\nu})\uparrow^P_{P_{\nu}}) = (e - 1)(\lambda_{\nu}\!\uparrow_{N_{\nu}}^{N}) - \sum_{i = 2}^{e}({\chi'_i \!\downarrow_{N_{\nu}}\!\lambda_{\nu}})\!\uparrow_{N_{\nu}}^{N}+ \ ({\chi'_{\nu}\lambda_{\nu}})\!\uparrow_{N_{\nu}}^{N}.
\varrho ={\frac{p ( p - 1 )}{2 5 6 \pi G}}\left[ ( p - 3 )^{2}\bar{\cal E}^{2}\bar{\cal P}_{\phi}^{- 2{\frac{p - 1}{p - 3}}}a^{2{\frac{p + 1}{p - 3}}}- ( p + 1 ) ( p - 7 ){\frac{1}{a^{2}}}\right] ,
E(\xi_t^2(O))=F_t(O)\leq \sum_{y\in\mathbb{Z}^d}e^{tG_\lambda}(O,y)\frac{K(y)}{\inf_{x\in\mathbb{Z}^d}K(x)}=\frac{K(O)}{\inf_{x\in \mathbb{Z}^d}K(x)}
\frac{\partial u}{\partial t}(t,\xi)+(\mathcal{A}u(t,\cdot))(\xi) =g(t,\xi)\,\frac{\partial w}{\partial t}(t,\xi), \ \ \mbox{on}\ [0,T]\times\mathcal{O}u(t,\xi) =0, \ \ \ \ \ \ \ \ \ \ \ \ \ \t\in (0,T], \ \xi\in\partial \mathcal{O}u(0,\cdot) =0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \xi\in\mathcal{O}.
\lambda g_{+}\pm\omega_{+}(I_{+})\partial_{\theta_{+}}g_{+}\mp\mu_{+,\pm}^{\prime}(e_{+})\omega_{+}(I_{+})\partial_{\theta_{+}}\phi =0,\text{for}\pm v>0, \lambda g_{-}\pm\omega_{-}(I_{-})\partial_{\theta_{-}}g_{-}\pm\mu_{-,\pm}^{\prime}(e_{-})\omega_{-}(I_{-})\partial_{\theta_{-}}\phi =0,\text{for}\pm v>0,
W(\xi_1 \otimes \cdots \otimes \xi_r) W(\eta_1 \otimes \cdots \otimes \eta_s) &= W(\xi_1 \otimes \cdots \otimes \xi_r \otimes \eta_1 \otimes \cdots \otimes \eta_s) + \overline{\langle \overline \xi_r, \eta_1\rangle}\, W(\xi_1 \otimes \cdots \otimes \xi_{r - 1}) W(\eta_2 \otimes \cdots \otimes \eta_s)
\nabla^{nc}_X(\alpha \cdot \beta) = \sum_I \nabla_X^{nc}\left(\alpha_I \cdot s_I \cdot \beta \right) = \sum_I X(\alpha_I) \cdot s_I \cdot \beta + \sum_I \alpha_I s_I \cdot \nabla^{nc}_X \beta &=\left(\nabla_X^{nc}\alpha \right) \cdot \beta + \alpha \cdot \left(\nabla_X^{nc}\beta \right),
\Xi^{{(n)}}_{{m}_{{1}}\ldots{m}_{{5}}}\equiv Sp(v^{{T}}\tilde{\sigma}_{{m}_{{1}}{\bf ...}{m}_{{5}}}v\sigma^{{(n)}})\equiv v^{{a}}_{\alpha}(\tilde{\sigma}_{{m}_{{1}}{\bf ...}{m}_{{5}}})^{\alpha \beta}v^{{b}}_{\beta}(\sigma^{{(n)}})_{{ab}}= 0
y_\textbf{i}(t) = s_{\textbf{i}}(t)+s_{\textbf{s}}(t)e^{j2\pi\delta_t}+w_\textbf{i}(t), y_\textbf{s}(t) = s_{\textbf{s}}(t)+s_{\textbf{i}}(t)e^{-j2\pi\delta_t}+w_\textbf{s}(t).
\begin{bmatrix}\dot{x}\dot{y}\dot{\omega}\end{bmatrix}= \begin{bmatrix}r\dot{\cos(\theta)}r\dot{\sin (\theta)}-\frac{g\cos (\theta)}{r}\end{bmatrix}= \begin{bmatrix}-r\sin (\theta) \dot{\theta}r\cos (\theta) \dot{\theta}-\frac{g(r\cos (\theta))}{r^2}\end{bmatrix}= \begin{bmatrix}-y\omega x\omega -\frac{gx}{r^2}\end{bmatrix}
\check{f}_S(x,a,b) =\tilde{f}_S(x,a,b) +\tilde{h}(x,a,b,0)\check{f}_S(a,a,b) &=\frac{I_a^{(q,r)}(x)}{(I_a^{(q,r)})'(b+)}+\Big( J_a^{(q,r)}(x)-(J_a^{(q,r)})'(b)\frac{I_a^{(q,r)}(x)}{(I_a^{(q,r)})'(b+)}\Big) \frac{\check{f}_S(0,a,b)}{Z^{(q)}(-a)}.
-4 \int_{t}^{\infty}\frac{\lambda_{0,0}''(x) dx}{(\log^{(\alpha-1)b}(t)+x-t)(1+x-t)^{3}}= E_{v_{3},ip,f}+\frac{4b^{2}(\alpha-1)\log(\log(t))}{t^{2}\log^{b+1}(t)}
\begin{cases}B_1 :=\mathrm{exp}\left (2^{10n+4}\delta^{-2n}D^{2n+2}\right).\log (4R)\log\log (4R), B_2 :=(4n+3)D\delta^{-1}, B_3 :=\exp\left(2^{5n+4}\delta^{-n-1}D^{n+2}\log (4R)\right).\end{cases}
\frac{1}{(1-z)^2}- \left(\frac{z^{d_r}}{(1-z)^2}+ \frac{z^{d_{r-1}}+ z^{d_{r-2}}+ \cdots + z^{d_1}}{1-z}\right)= \frac{1 + z + \cdots + z^{d_r-1}}{1-z}- \frac{z^{d_{r-1}}+ z^{d_{r-2}}+ \cdots + z^{d_1}}{1-z}.
{\bf M}=\frac{1}{AD-BC}\left(\begin{array}{crc}\frac{1}{2}({A^2}+{B^2}+{C^2}+{D^2}) AB+CD \frac{1}{2}(-{A^2}+{B^2}-{C^2}+{D^2}) AC+BD AD+BC -AC+BD \frac{1}{2}(-{A^2}-{B^2}+{C^2}+{D^2}) -AB+CD \frac{1}{2}({A^2}-{B^2}-{C^2}+{D^2})\end{array}\right) \,\,\,.
\frac{\partial}{\partial t}\int_M R\log Rd\mu= \int_M\big(\frac{\partial}{\partial t}\log R\big)Rd\mu+\int_M\log R\frac{\partial}{\partial t}(Rd\mu) = \int_MR\big(R+\frac{\alpha'}{4}R^2-r\big)d\mu+\int_M\log R\Delta (R+\frac{\alpha'}{4}R^2)d\mu
m_{i,j}=a([j]_q-[i]_q)\mu_{i+j-1}=(q^i-q^j)\left(q, a, a^{-1}q ; q\right)_{\infty}\sum_{k=0}^{i+j-1}\left[\begin{array}{c}i+j-1 k\end{array}\right]_{q}a^{k+1}.
\begin{array}{l c r}{d s^{2}={\frac{1}{2}}K_{\rho \sigma}[ d \phi_{\rho}d \phi_{\sigma}+ a_{\rho}a_{\sigma}\sum_{i = 1}^{3}d \Omega_{\rho , i}d \Omega_{\sigma , i}] ;}{H = - 2 a_{\rho}d \Omega_{\rho},}\end{array}
w^n_{1,-Q^n(0)+1}= t^n_{-1, Q^n_1(0) - R_{1,-Q^n(0)+1}(0)}, w^n_{1,1}= \left(t^n_{-1, 1+Q^n_1(0)-R^n_1(t^n_{1,1}-)+R^n_{-1}(t^n_{1,1}-)}- t^n_{1,1}\right)^+.
V_\varepsilon(x,t) = \frac{(-1)^{i+1}}{\sqrt{1+\varepsilon^2|\bar{\tau}_\varepsilon^i(x,t)|^2}}\Bigl(\overline{V_\Gamma}+\varepsilon\,\overline{\partial^\circ g_i}+\varepsilon^2\bar{g}_i\bar{\tau}_\varepsilon^i\cdot\overline{\nabla_\Gamma V_\Gamma}\Bigr)(x,t).
f_{\chi_n, \chi_d}(x, y) = \frac{e^{-\frac{1}{2(1-\rho)}\left(\frac{x}{\sigma_n^2}+\frac{y}{\sigma_d^2}\right)}}{4 \sigma_n^2 \sigma_d^2 (1-\rho)}I_0\left(\frac{\sqrt{\rho}}{1-\rho}\frac{\sqrt{xy}}{\sigma_n \sigma_d}\right),
\langle m_1+1,l_1,m_2,l_2| 0,l_{12}\rangle = -\sqrt{\frac{(m_2+1)(2l_2+m_2)}{(m_1+1)(2l_1+m_1)}}\langle m_1,l_1,m_2+1,l_2| 0,l_{12}\rangle.
\begin{cases}-c(x)\frac{\partial}{\partial y}z_a(y;x)-d_A\frac{\partial^2}{\partial y^2}z_a(y;x) = M_{11}z_a+M_{12}z_b+c^\prime(x)w_a+a\;\partial_xH_A, begin{align*}5pt] -c(x)\frac{\partial}{\partial y}z_b(y;x)-d_B\frac{\partial^2}{\partial y^2}z_b(y;x) = M_{21}z_a+M_{22}z_b+c^\prime(x)w_b+b\;\partial_xH_B. \end{cases}
F^{m}{}_{\mu\nu}= 2\partial_{[\mu}A^{m}{}_{\nu]}-f_{np}{}^{p}A^{n}{}_{\mu}A^{p}{}_{\nu}= 2\partial_{[\mu}A^{m}{}_{\nu]}-Q^{mq}\epsilon_{npq}A^{n}{}_{\mu}A^{p}{}_{\nu}\, .
\rho\int_{C_{\rho}}|\nabla u|^2ds \le 2\int_{R^2}^R\int_{C_{\rho}}|\nabla u|^2ds\,d\rho\Big/\int_{R^2}^R\frac{d\rho}{\rho}\le 2\int_B|\nabla u|^2dz\Big/|\log(R)|=4 E(u)/|\log(R)|.
e^{-\pi i (\bar{\Delta}_{l}- \bar{\Delta}_{\bar{v}}-\bar{\Delta}_{\bar{w}})}\; (\bar{\zeta}- \bar{z})^{\bar{\Delta}_{l}-\bar{\Delta}_{\bar{v}}- \bar{\Delta}_{\bar{w}}}\widetilde{F}_{\psi \chi}(z,\zeta)
u_{xx}=v_{xt}w_{xx}-v_{xx}w_{xt}+w_{xt}^{2}-w_{xx}w_{tt}, &u_{xt}=v_{tt}w_{xx}-v_{xt}w_{xt}, &u_{tt}=v_{xt}^{2}-v_{xx}v_{tt}+v_{tt}w_{xt}-v_{xt}w_{tt},
Jm = (1/2^n)JMa &= (1/2^n)[2^{n-1}, -2^{n-1}, 0, \ldots , 0 ]{}^t[\chi_{\pi}(h_0),\chi_{\pi}(h_1), \ldots , \chi_{\pi}(h_n)] &= (1/2)(\chi_{\pi}(h_0) - \chi_{\pi}(h_1)).
\|\bar{\phi}_{k+1}(x)-\bar{\phi}_{k+1}(y)\|_p^p =\frac{1}{2}(\|\bar{\phi}^{(l,+)}_{k}(x)-\bar{\phi}^{(l,+)}_{k}(y)\|_p^p+\|\bar{\phi}^{(r,+)}_{k}(x)-\bar{\phi}^{(r,+)}_{k}(y)\|_p^p).
e^{(k,i)}(\xi)=\frac{r^{(k,i)}(\xi)}{(1-\xi)^{\alpha}}\left(\sum_{m=0}^{k-1}e_{m}^{(k,i)}s_{m}^{(k,i)}(\xi)+(\Delta t)^{\alpha}\delta f^{(k,i)}(\xi)+(\Delta t)^{\alpha}\tau^{(k,i)}(\xi)\right),
\tilde{Q}^\alpha= \begin{pmatrix}\tilde{Q}^{\alpha,\{k_1\}}_0 \tilde{Q}^{\alpha,\{k_1\}}_1 \cr\tilde{Q}^{\alpha,\{k_1\}}_{-1}\tilde{Q}^{\alpha,\{k_1\}}_0 \tilde{Q}^{\alpha,\{k_1\}}_1 \cr \ddots \ddots \ddots \end{pmatrix},
F = \left( \begin{array}{c c c c}{0}&{- B_{z}}&{B_{y}}&{E_{x}}\\{B_{z}}&{0}&{- B_{x}}&{E_{y}}\\{- B_{y}}&{B_{x}}&{0}&{E_{z}}\\{- E_{x}}&{- E_{y}}&{- E_{z}}&{0}\\ \end{array}\right) .
( T^{\ell_j - \ell_{j-1}})' T^{\ell_{j-1}}x \ge C \frac{T^{p_j}( \hat{b}_{p_j - 1}) - T^{p_j}( \hat{b}_{p_j})}{\hat{b}_{p_j - 1}- \hat{b}_{p_j}}= \frac{C}{2}\frac{1}{\hat{b}_{p_j - 1}- \hat{b}_{p_j}}.
S = - \sum{\mathrm{T r}({{U}_{\mu}{U}_{\nu}{U}_{\mu}^{\dagger}{U}_{\nu}^{\dagger}})}+ \frac{i}{2}\mathrm{T r}{{\bar{\psi}}{\Gamma^{\mu}}( U_{\mu}\psi U_{\mu}^{\dagger}- U_{\mu}^{\dagger}\psi U_{\mu})},
i\frac{\partial}{\partial t}\Phi (1,\cdots,N;t ) =-\frac{1}{2m}\sum_{p=1}^{N}\left\{\partial^{(p)}_i +{\cal A}^a_i ({\bf r}_p;1, \cdots, p-1, p+1, \cdots,N)T^a_{(p)}\right\}^2\Phi(1,\cdots,N;t).
\begin{pmatrix}Dr Dk_u Dk_s \end{pmatrix}= \begin{pmatrix*}[l] A_c Dk_c + Dg_c(K{}{}) DK{}{}- Dk_c\left(R\right) DR -A_u^{-1}Dg_u(K{}{}) DK{}{}+ A_u^{-1}Dk_u(R) DR A_s Dk_s(T)DT + Dg_s(K{}{}\circ T) DK{}{}(T) DT \end{pmatrix*},
II_1= \| \langle x\rangle^{a}(\widetilde{(\nabla \chi_k )u}) \|_{L^2(\mathbb R; \dot H^\frac12(\mathbb{R}^n))}\le C \| \langle x\rangle^{a}(\nabla \chi_k )u\|_{L^2(\mathbb R; H^\frac12(-\left. \Delta \right|_{\mathit{D}}))}.
\widetilde{T}(z_{0})\widetilde{T}(z_{1}) =e^{\tfrac{i}{\hslash}\sigma(z_{0},z_{1})}\widetilde{T}(z_{1})\widetilde{T}(z_{0}) \widetilde{T}(z_{0}+z_{1}) =e^{-\tfrac{i}{2\hslash}\sigma(z_{0},z_{1})}\widetilde{T}(z_{0})\widetilde{T}(z_{1}).
\Sigma\le \inf\limits_{\Psi}({\bf h}^{D}(k_0)\Psi,\Psi) \le \inf\limits_{\phi}(h^{D_1}(k_0)\phi,\phi) +\inf\limits_{\psi}(h^{D_2}(K-k_0)\psi,\psi) = z_1(k_0) + z_2(K-k_0).
(a)=\begin{pmatrix}1,0\end{pmatrix}, \;(a^*)=\begin{pmatrix}a_0^* a_1^*\end{pmatrix}, \;(\beta)=\begin{pmatrix}0 1 b_{10}b_{11}\end{pmatrix}, \;(\gamma)=\begin{pmatrix}c_{00}c_{01}c_{10}c_{11}\end{pmatrix}, \;(\delta)=\begin{pmatrix}d_{00}d_{01}d_{10}d_{11}\end{pmatrix}
dd^{\mathrm c}u =\sum_{i=1}^3\sum_{j=1}^3\partial_i\partial_ju\,e^i\wedge Je^j-\partial_2u\,e^{12}&=\sum_{i=1}^3\sum_{j=1}^3\partial_i\partial_j u\,e^i\wedge Je^j+d(ue^1)+\partial_3u\,e^{13}.
{\mathcal W}_{\mu_{1}\cdots \mu_{n}}^{( n )}( x_{1}, \cdots , x_{n}) = \left. \frac{\delta^{n}{\mathcal W}[ b_{\mu}]}{\delta b_{\mu_{1}}( x_{1}) \cdots \delta b_{\mu_{n}}( x_{n})}\right|_{b_{\mu}= 0}.
I_{r+1,r}(x) = -\sinh(x)\biggl(\frac{1}{r-2}\tanh^{r-2}(x)+\frac{r-1}{(r-2)(r-4)}\tanh^{r-4}(x) & +\frac{(r-1)(r-3)}{(r-2)(r-4)(r-6)}\tanh^{r-6}(x)+\dots + \frac{(r-1)!!}{2(r-2)!!}\tanh(x)\biggr) + \frac{(r-1)!!}{(r-2)!!}I_{2,1}(x).
T_{\varepsilon,\delta_{\varepsilon}(t),\eta}(\Phi)=L_{\varepsilon,\delta_{\varepsilon}(t),\eta}^{-1}\left\{N_{\varepsilon,\delta_{\varepsilon}(t),\eta}(\Phi)+R_{\varepsilon,\delta_{\varepsilon}(t),\eta}\right\}.
Y(z) = \left(\mathbb{I}+\mathcal{O}\left(\frac{1}{z}\right)\right) \begin{pmatrix}z^{n}0 0 0 z^{-\lceil \frac{n}{2}\rceil}0 0 0 z^{-\lfloor\frac{n}{2}\rfloor}\end{pmatrix}.
\big(\frac{\partial \tilde{h}_1(\tilde{w})}{\partial \tilde{w}}-2\tilde{h}_2(\tilde{w}) \big)d\tilde{w}^2= \big(\frac{\partial h_1(w)}{\partial w}-2{h}_2(w) \big)d{w}^2 +\frac{1}{6}\{\tilde{w}, w\}_2 dw^2,
S_{F}( p , t , t^{\prime}) \simeq{\frac{1 + \gamma_{5}T_{c}}{2}}\{\frac{1}{i p \! \! \! / + \phi \cdot T}}g a m m a_{5}\Lambda \e^{- i \Lambda ( \vert t \vert + \vert t^{\prime}\vert )}+ \dots
u \mapsto \begin{bmatrix}0 1 0 0 0 1 1 0 0 \end{bmatrix}v \mapsto \begin{bmatrix}1 0 0 0 \rho 0 0 0 \rho^2 \end{bmatrix}w \mapsto \begin{bmatrix}\rho 0 0 0 \rho 0 0 0 \rho \end{bmatrix}
\frac{\sum_{j=1}^d v_j g^{i,j}}{\sum_{i=1}^d \sum_{j=1}^d v_j g^{i,j}}\geq \frac{\sum_{j=1}^d v_j \min_{1 \leq i \leq d}g^{i,j}}{d \sum_{j=1}^d v_j \max_{1 \leq i \leq d}g^{i,j}}\geq \frac{1}{cd},
\Delta w_k=S_k\sum\limits_{r\geq 0}\left( 2\sum\limits_{i_1+\cdots+i_s=r}\frac{1}{s!}\binom{r}{i_1,\cdots,i_s}\bar S^T_{i_1}*\cdots*\bar S^T_{i_s}w_{i_1}w_{i_2}\cdots w_{i_s}\right.
K^{\phi}(y,z)=\frac{\phi'(z)-\phi'(y)}{\sqrt{2\pi (z-y)^3}}\left (-\frac{1}{2}\int_{y}^{z}(\phi'(u))^2 du \right ) \mathbb{E}\left[ \left (-\int_{y}^{z}\phi''(u) \textbf{e}(u)du \right)\right]
d\Psi(u)[\varphi] =t_{u}^{2}\int_{\mathbb{R}^{N}}\Delta u\cdot\Delta\varphi dx-t_{u}^{\frac{N(q-2)}{4}}\mu\int_{\mathbb{R}^{N}}|u|^{q-2}u\cdot\varphi dx-t_{u}^{\frac{N(p-2)}{4}}\int_{\mathbb{R}^{N}}|u|^{p-2}u\cdot\varphi dx &=dE_{p,q}(u_{s_{u}})[\varphi_{s_{u}}]
S_{i j k l}( \vec{x}, \vec{y}) = \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \Bigl ( \eta_{i j , k l}\, \nabla^{2}+ \partial_{i j , k l}^{2}\Bigr ) \, \delta ( \vec{x}- \vec{y})
r_{\textrm{dual}}^{i}(z^{i,(l)})=\nabla \bar{f}_i(w^{i,(l)})+\sum_{j=1}^{m_i}\nabla \bar{G}_j^i(w^{i,(l)})\lambda_j^i+(\bar{A}^i)^Tv^{i,(l)}+v_c^{i,(l)}.
\Big|\mathbf P({\mathcal N}_n(R)=0|\tilde{\mathcal F})- P_n(0)\Big|\le\Big|\mathbf E\Big[\eta_n(R)f({\mathcal N}_n(R)+1)-{\mathcal N}_n(R)f({\mathcal N}_n(R))\Big|\tilde{\mathcal F}\Big]\Big|
\left\| J_{\varphi}\left( \mathbf{x}\right) \right\|_{\infty}=\max_{i=1,...,d}{\{\left| \frac{{\partial \varphi}_{i}(\mathbf{x})}{{\partial x}_{1}}\right|+\mathellipsis +\left| \frac{{\partial \varphi}_{i}(\mathbf{x})}{{\partial x}_{d}}\right|\}}.
{\cal O}_\pm^a( \{x\}) := (-1)^{\frac{N(N-1)}{2}}\left[\frac{1}{N!}\sum_\pi \mbox{sign}(\pi) \prod_{b=1}^{N}\overline{\psi}^{(\pi(b))}_{\stackrel{{\scriptstyle 1}}{{\scriptstyle 2}}}(x^{(b)})\right]
\int_0^T\Vert (\tilde{v}_3)_k(t)-\xi_k(t) \Vert_{L^2}dt=\int_0^T\Vert \int_{t-\delta_k}^t(\dot{\tilde{v}}_3)_k(s)ds \Vert_{L^2}dt\leq\delta_k^{1/2}C\int_0^T \Vert (\dot{\tilde{v}}_3)_k \Vert_{L^2(L^2)}dt = C \delta_k^{1/2},
F_{A_{/1}}=0\Longleftrightarrow q^{a_{i/1}\left[ i\right]}=\phi^{a_{i/1}}\left( q^{b_{i/1}\left[ i-1\right]},q^{b_{i/2}\left[ i-1\right]},q^{b_{i/2}\left[ i\right]}\right) \,.