harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bar{\mathsf h}_{ij}=\cfrac{\frac{\rho_{ij}}{2\mu^2}}{\rho_{ij}\frac{\rho_{ij}}{2\mu^2}+ 2 t^2 \frac{\rho_{ij}}{2\mu^2}+ t^2 \frac{t^2-\sigma_i^2}{2\mu^2}+ \sigma_i^2 \frac{t^2 - \sigma_j^2}{2\mu^2}}(\bar{w}_{ij}(\rho_{ij}+ t^2) - \bar{w}_{ji}\sigma_i \sigma_j)\to \frac{t^*}{\zeta_i + \zeta_j}(\bar{w}^*_{ij}- \bar{w}^*_{ji}).
T r [ Q_{2}\lambda_{S U ( 2 )}^{2}] = - \frac{2}{3}\sum_{k = 1}^{3}T r [ \gamma_{k , 7_{4}, ( 0 )}\lambda_{U ( 1 )_{2 , ( 0 )}}] T r [ \gamma_{k , 7_{4}, ( 0 )}^{- 1}\lambda_{S U ( 2 )}^{2}]
\Psi_j(t) =\frac{t^{p_j}}{p_j};\; \Phi_j(t) = \frac{t^{q_j}}{q_j},\; q_j = \frac{p_j}{p_j - 1},\;j = 0, 1, \; \nu(\tau) = \min(\tau^\alpha,1)
\begin{cases*}\square \tilde{\Lambda}=Q_0(\ln \alpha, \tilde{\Lambda})-2\sinh(2\lambda +2\tilde{\Lambda})Q_0(\phi,\phi), \square \phi =Q_0(\ln \alpha, \phi)+\dfrac{\sinh(2\lambda +2\tilde{\Lambda})}{\sinh^2(\lambda +\tilde{\Lambda})}Q_0(\phi,\tilde{\Lambda}).\end{cases*}
=\sum_{i=0}^{\infty}\exp\left(-\sum_{j=i+1}^{\infty}\left(W\left(p^{j}\right)-W\left(p^{j+1}\right)\right)V^{j}\left(0\right)t\right)\dfrac{V_{0}\left(0\right)}{V_{i}\left(0\right)}f_{i+1,0,0}\left(x\right).
\nu\left(d'_jQ_{\ell_1}^j\right)=\nu_\ell\left(d'_jQ_{\ell_1}^j\right)+j\left(\beta_{\ell_1}-\beta_\ell\right)\geq\nu\left(d_{\delta, i}z^j\right)+j\left(\beta_{\ell_1}-\beta_\ell\right)=\nu\left(d_{\delta, i}\right)+2j\beta_{\ell_1}-j\beta_\ell.
\begin{aligned}E_{\nu^n_\rho}[f(\overrightarrow{W}_{j+1}^\ell)^2 (W_{j+i-1}- W_{j+i})] & = - E_{\nu^n_\rho}[\sigma_{j+i-1, j+i}(\overrightarrow{W}_{j+1}^\ell)^2(\nabla_{j+i-1, j+i}f) W_{j+i-1}] - E_{\nu^n_\rho}[f (\nabla_{j+i-1, j+i}(\overrightarrow{W}_{j+1}^\ell)^2) W_{j+i-1}] \end{aligned}
\tilde{\ell_{2}}(x) \leq \bar{\ell_{2}}(x)+\frac{2\gamma\varepsilon}{\lambda_{2}^2}(\lambda_{2}+ \|x_{2}\|), \ -\tilde{a_{2}}\|x\|^2=-\bar{a}_{2}\|x\|^2+\frac{\varepsilon}{\lambda_{2}^2}\|x\|^2\leq -\bar{a}_{2}\|x\|^2+\frac{\varepsilon}{\lambda_{1}^2}\gamma^2 .
\frac{D-2}{2}\mbox{Tr}\,\ln(-\partial_\alpha\alpha^{\alpha\beta}\partial_\beta)\,=\,\frac{D-2}{2}T\,\sum_{n}^{}\int\limits_{-\infty}^{+\infty}\frac{dq_0}{2\pi}\ln\left(\alpha^{00}q_0^2\,+\,\alpha^{11}{\omega_n^2}\right)\,=\,{T}\sqrt{\frac{\alpha^{11}}{\alpha^{00}}}\,E_{\rm C}{.}
\begin{cases}q_{1,j}^{t+1}q_{2,j}^{t+1}= q_{2,j}^t q_{1,j+1}^t, q_{1,j+1}^{t+1}+ q_{2,j}^{t+1}= q_{2,j+1}^t + q_{1,j+1}^t. \end{cases}
A_k(m_{k,1},\dots,m_{k,u_k}) \leq\mu_{E,a}(I_k)^{-r_k}M^{r_k-u_k}\sideset{}{''}\sum \prod_{\alpha=1}^{u_k}|a_{\Xi_{k,n_{k,\alpha}}}|^{2}\leq \mu_{E,a}(I_k)^{-r_k}M^{r_k-u_k}\sideset{}{'''}\sum \prod_{\alpha=1}^{u_k}|a_{\Xi_{k,n_{k,\alpha}}}|^{2}= \left(\frac{M}{\mu_{E,a}(I_k)}\right)^{r_k-u_k}.
\int_{\Omega}\left(\lambda\frac{sign(u_n)u_n^-}{|u_n|^{\gamma}}+\tilde{f}(x,u_n)u_n^-\right)dx=\int_{Q}\frac{(u_n(x)-u_n(y))(u_n^-(x)-u_n^-(y))}{|x-y|^{N+2s}}dxdy &\Rightarrow\lambda\int_{\Omega^-}|u_n^-|^{1-\gamma}dx\leq -\|u_n^-\|^2<0.
\langle \varphi^{\hat{0},\hat{i}}_{j,s}, \varphi_{F_i,t}\rangle_{\lambda_j}:= \frac{1}{|F_i|}\int_{F_i}\lambda_j\hat{\varphi}_{j,s}^{\hat{0},\hat{i}}\varphi_{F_i,t}= \delta_{st}t = 1, \cdots, C_{n-1+k}^k,
Q_{1}^{n}= (\alpha_{1},(n-1)\alpha_{1}) \hspace{5pt},\hspace{5pt}Q_{2}^{n}= (\alpha_{2},n\alpha_{2})\hspace{50pt}Q_{1}^{n}=(\alpha_{1},(n-1)\alpha_{1}) \hspace{5pt},\hspace{5pt}Q_{2}^{n}=(\alpha_{2},n\alpha_{2})
\beta ( g_{\mathrm{R}}) : = \mu{\frac{\partial g_{\mathrm{R}}}{\partial \mu}}= - g_{\mathrm{R}}\mu{\frac{\partial}{\partial \mu}}\ln Z_{g}\cong - g_{\mathrm{R}}\mu{\frac{\partial}{\partial \mu}}Z_{g}^{( 1 )}.
S_{S G}( \theta ) = \frac{S_{0}( \theta )}{x p - p^{- 1}x^{- 1}}\left( \begin{array}{c c c c}{{x p - p^{- 1}x^{- 1}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{p - p^{- 1}}}&{{x - x^{- 1}}}&{{0}}\\{{0}}&{{x - x^{- 1}}}&{{p - p^{- 1}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{x p - p^{- 1}x^{- 1}}}\\ \end{array}\right) \,
y_k^n e_k = y_k^{k-1}(\psi_k \cdots \psi_n) (\psi_n \cdots \psi_k) e_k+\sum_{j=1}^{n-k+1}(-1)^{j+1}E_j(x_k, \ldots, x_n) .( y_k^{n-j}e_k).
\exp\Bigl(\bigl(\mathcal{S}_{M-1}\log S_M\bigr)(0\,|a,\,b) - \bigl(\mathcal{S}_{M-1}\log S_M\bigr)(q\,|a,\,b)\Bigr) = \eta_{M,M-1}(q\,|\,a, b) \eta_{M,M-1}(-q\,|\,a, \bar{b}), = \eta_{M, M-1}(q|a, b, \bar{b}),
- \int d^{10}x\ \sqrt{-g}e^{-2\phi}R \, =\, \int d^{10}x\ \sqrt{-g}e^{-2\phi}\biggl (\omega_a{}^{ac}\omega_b{}^{bc}+ \omega_a{}^{bc}\omega_b{}^{ca}+ 4 (\partial_a\phi)\omega_b{}^{ba}\biggr ) \ ,
\deg m_{\mu_{i}, \nu_{i}}\left( \frac{\gamma}{\sqrt{i}}\left( z_{1}^{i}+ \ldots + z_{r}^{i}\right) , - \frac{\gamma}{\sqrt{i}}\left( z_{1}^{- i}+ \ldots + z_{r}^{- i}\right) \right) = i \left( \mu_{i}- \nu_{i}\right) .
I_3\colon x_{11}^2+x_{12}x_{21}=\frac{1}{4}(\epsilon^a\chi_1(\gamma)-\epsilon^b\chi_2(\gamma))^2, I_4\colon y_{11}^2+y_{12}y_{21}=\frac{1}{4}(\mu_1(\delta)-\mu_2(\delta))^2.
{\cal K}( z ) = \sum_{\mathrm{\footnotesize \boldmath \alpha}\in G_{m}}\omega^{2 \alpha_{1}\alpha_{2}}u_{2 \mathrm{\footnotesize \boldmath \alpha}}^{( n )}( z , v ) I_{2 \mathrm{\footnotesize \boldmath \alpha}}= \sum_{\mathrm{\footnotesize \boldmath \alpha}\in G_{m}}u_{2 \mathrm{\footnotesize \boldmath \alpha}}^{( n )}( z , v ) J_{\mathrm{\footnotesize \boldmath \alpha}},
\Pi_{\gamma}(m) \leq \frac{\prod_{j=1}^m\Gamma (w_j +1)}{\Gamma \left(\sum_{j=1}^{m}w_j + \left(H-\frac{1}{2}-\gamma \right)\sum_{j=1}^{m-1}\varepsilon_j + m \right)}&\leq \frac{\prod_{j=1}^m\Gamma (w_j +1)}{\Gamma \left(\sum_{j=1}^{m}w_j + \left(H-\frac{1}{2}-\gamma \right)\sum_{j=1}^{m}\varepsilon_j + m \right)},
\frac{\partial\overline{u_{i}}}{\partial t}+\frac{\partial}{\partial x_{j}}(\overline{u}_{i}\overline{u}_{j})=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}+2\nu\frac{\partial}{\partial x_{j}}\overline{S}_{ij}-\frac{\partial\tau_{ij}}{\partial x_{j}}
\widehat{A}_{ij}=\sum^{iN_1+(i-1)M_1}_{l=(i-1)(N_1+M_1)+1}\widehat{x}_{lj},\widehat{B}_{ij}=\sum^{i(N_1+M_1)}_{l=iN_1+(i-1)M_1+1}\widehat{x}_{lj},
\begin{aligned}Lu = \sum_{i=1}^n\left(x_i u_{x_ix_i}+b_i(z) u_{x_i}\right) +\sum_{i,j=1}^n x_ix_ja_{ij}(z)u_{x_ix_j}+\sum_{l,k=1}^md_{lk}(z)u_{y_ly_k}&+\sum_{i=1}^n\sum_{l=1}^mx_ic_{il}(z)u_{x_iy_l}+ \sum_{l=1}^me_l(z)u_{y_l},\end{aligned}
\varepsilon_1(x,S) = 2(S \cdot Q(x)) + \langle S \circ\{\tau(\xi(x)) + Q(x)\}, \nabla_{\partial \Omega}(\tau-\nu)(\xi(x)), \tilde{x}-a \rangle, &\varepsilon_2(x,S) = -2\left\{Q(x)\left(\dfrac{\tilde{x}-a}{\tilde{r}}\right)\right\}\cdot \left\{S \circ i_x\left(\dfrac{\tilde{x}-a}{\tilde{r}}\right)\right\}
\mathfrak{a}(y) = \left\{\nabla_{X_1}\cdots \nabla_{X_k}R^{\nabla}_\chi(Y_1, Y_2) |_y \, : \, \begin{array}{c}Y_1, Y_2 \in \Gamma(TM), X_i \in \Gamma(D) , k=0,1,2 \dots \end{array}\right\},
\Big ( \sum_{|t_j| \leq CM (\log k)^{\varepsilon}}|\langle |F_k|^2 , u_j \rangle|^2\Big )^{1/2}\cdot \Big ( \sum_{j}|\langle u_j, \phi \rangle|^2 \Big )^{1/2}\ll (CM)^{3/2+\varepsilon}(\log k)^{\varepsilon}\cdot \prod_{p \leq k}\Big (1 - \frac{n(p)}{p}\Big ) \cdot \| \phi \|_2.
\begin{array}{ll}L^{-1}(T,X,Y,Z)=(\cosh t\cos\frac{r_{+}-r_{-}}{2}, \sinh t\cos\frac{r_{+}+r_{-}}{2}, & \sinh t\sin\frac{r_{+}+r_{-}}{2},-\cosh t\sin\frac{r_{+}-r_{-}}{2}).\end{array}
f_{1n}(t)=R_{n}e^{r_{n}t}+C_{n}e^{i\omega_{n}t}+\overline{C_{n}}e^{-i\overline{\omega_{n}}t}+D_{n}e^{i\zeta_{n}t}+\overline{D_{n}}e^{-i\overline{\zeta_{n}}t}\,,
\begin{array}{r l}{{a_{l}= s u ( l \! + \! 1 ) \hspace{0 . 2 i n}}}&{{2 , 3 , \dots ( l + 1 )}}\\{{b_{l}= s o ( 2 l \! + \! 1 ) \hspace{0 . 2 i n}}}&{{2 , 4 , \dots 2 l}}\\{{c_{l}= s p ( 2 l ) \hspace{0 . 2 i n}}}&{{2 , 4 , \dots 2 l}}\\{{d_{l}= s o ( 2 l ) \hspace{0 . 2 i n}}}&{{2 , 4 , \dots ( 2 l - 2 )}}\\ \end{array}
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_{i < j}^{n}( k_{i}\times k_{j}) ( 2 \tau_{i j}- \epsilon ( \tau_{i j}) ) \right]
\partial_{\mu}J{^{5 , \mu}_{3}}= \partial_{+}J{_3^{R}}- \partial_{-}J{_3^{L}}={\frac{g}{\pi}}( \partial_{-}A_{3}- \partial_{+}v ) ={\frac{g}{2 \pi}}\epsilon^{\mu \nu}F_{\mu \nu}^{3}\; ,
D_{2 ; 0}= 4 \cdot \left( \begin{array}{c c c c c c}{{+ 1}}&{{- 1}}&{{}}&{{}}&{{}}&{{}}\\{{- 1}}&{{+ 2}}&{{- 1}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{- 1}}&{{+ 2}}&{{- 1}}\\{{}}&{{}}&{{}}&{{}}&{{- 1}}&{{+ 1}}\\ \end{array}\right) .
\tilde{S}(m_1, n_1, n_2, D_1, D_2) = \textbf{1}_{D_1 | D_2}\underset{\substack{C_1 (\bmod{D_1}),\C_2(\bmod{D_2}) (C_1, D_1) = 1 = (C_2, D_2/D_1)}}{\sum \sum}e\left(\frac{m_1 C_1 + n_1 \overline{C_1}C_2}{D_1}\right) e\left(\frac{n_2 \overline{C_2}}{D_2/D_1}\right),
\left( \begin{array}{c}h_+ h_- h_3 1\end{array}\right) =\left( \begin{array}{cclc}N^{-1/2}N^{-1/2}1 N/2 1\end{array}\right)\left( \begin{array}{c}N \Sigma^{(N)}_+ N \Sigma^{(N)}_- N \Sigma^{(N)}_3/2{\bf 1}^{(N)}\end{array}\right) ,
\mathbf{P}_{\varepsilon}^{\beta}u =\sum_{p\in\mathbb{N}}\left(\frac{1}{T}\log\left(1+\gamma^{-1}\left(T,\beta\right)e^{\overline{M}T\mu_{p}}\right)-\overline{M}\mu_{p}\right)\left\langle u,\phi_{p}\right\rangle \phi_{p}.
\int_{-\infty}^{\infty}dq' U^{-1}(q,q') \phi(q')= \frac{(1-i)}{2}\sqrt{\pi}\mbox{e}^{\frac{i\pi a}{2}}\mbox{sgn}(q) \left|kq\right|^{\frac{1}{2}}J_a(k\left|q\right|).
M(u(t)) = \int |u(t,x)|^2 dx = M(u_0), E(u(t)) = \frac{1}{2}\int |\nabla u(t,x)|^2 dx -\frac{c}{2}\int |x|^{-2}|u(t,x)|^2 dx -\frac{1}{\alpha+2}\int |u(t,x)|^{\alpha+2}dx,
\Vert J_{44}^{\left( 3\right)}\Vert_{p}=\mathcal{O}\left( \frac{1}{m}\right) (v(|z_{1}|)+(v(|z_{2}|))\left( \frac{\omega (h_{1})}{v(h_{1})}+\frac{\omega (h_{2})}{v(h_{2})}\right) .
\sum_{j_1\in\mathcal{Q}_1}\cdots\sum_{j_d\in\mathcal{Q}_d}2^{\sum_{l=1}^dj_l}\omega_{\boldsymbol{j},n}\leq\sum_{j_1\in\mathcal{Q}_1}\cdots\sum_{j_d\in\mathcal{Q}_d}2^{-\sum_{l=1}^dj_l\mu_l}.
{\tilde{g}}= - \frac{1}{1 6}\frac{1 + 3 y}{\sqrt{y^{3}( 1 + y )^{3}}}- \frac{3}{1 6}C_{1}\frac{1 + 2 y}{\sqrt{y^{3}( 1 + y )}}+ C_{3}\left( \frac{1 + y}{y}\right)^{3 / 2}.
I_2^{(1)}(a) =\frac{1}{2i T}\left[\int_{\frac{1}{2}-iT}^{c-iT}+\int_{c+iT}^{\frac{1}{2}+iT}\right]\chi(1-s)\Bigg(\zeta(s+ia)\zeta(s)-\sum_{1\leq n\leq T/2\pi}\frac{\sigma_{-ia}(n)}{n^s}\Bigg)ds +\frac{1}{2iT}\int_{c-iT}^{c+iT}\chi(1-s)\sum_{n>T/2\pi}\frac{\sigma_{-ia}(n)}{n^s}ds-\frac{\pi}{T}\zeta(1-ia)\chi(ia).
(\begin{matrix}v_{11}^{2}&v_{11}v_{12}&v_{12}^{2}\\v_{21}^{2}&v_{21}v_{22}&v_{22}^{2}\\v_{31}^{2}&v_{31}v_{32}&v_{32}^{2}\\ \end{matrix})(\begin{matrix}a_{1}\\a_{2}\\a_{3}\end{matrix})=(\begin{matrix}x_{1}\\x_{2}\\x_{3}\end{matrix})
\langle x_r',x_s'\rangle= \sum_{t \in T}\lambda^{r}_t \langle y_t', x_s'\rangle = \sum_{t \in T}\lambda^{r}_t C_{s,t}= \sum_{t \in T}\lambda^{r}_t\langle y_t'',x_s''\rangle = \langle x_r'',x_s''\rangle
d\epsilon_{ab}=\left(\begin{array}{cccc}0 -dv_1 -dv_2 -dv_3 dv_1 0 -d\theta_{12}d\theta_{31}d v_2 d\theta_{12}0 -d\theta_{23}dv_3 -d\theta_{31}d\theta_{23}0 \end{array}\right )
\begin{cases}{dy(t)}/{dt}=Ay(t) + \alpha A y(t-r) +\displaystyle\int^0_{-r}\beta(\theta)A y(t+\theta)d\theta,\,\,\,\,t\ge 0, y(0)=\phi_0,\,\,\,\,y_0=\phi_1,\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X},\end{cases}
E_{K}=\frac{1}{2}(\sum_{i=1}^{n}m_{i}(\omega\times\Delta r_{i})\cdot(\omega\times\Delta r_{i}))+(\sum_{i=1}^{n}m_{i}V_{C}\cdot(\omega\times\Delta r_{i}))+\frac{1}{2}(\sum_{i=1}^{n}m_{i}V_{C}\cdot V_{C}).
\begin{aligned}\left|\int_{\Omega}u\cdot\nabla u\mathbb{P}\Delta udx\right| \leq C\|u\|_{L^{4}}\|\nabla u\|_{L^{4}}\|\mathbb{P}\Delta u\|_{L^{2}}\leq C\|u\|_{L^{2}}^{\frac{1}{2}}\|\nabla u\|_{L^{2}}\|\mathbb{P}\Delta u\|_{L^{2}}^{\frac{3}{2}}\leq C\|u\|_{L^{2}}^{2}\|\nabla u\|_{L^{2}}^{4}+\frac{1}{4}\nu\|\mathbb{P}\Delta u\|_{L^{2}}^{2}.\end{aligned}
|{}^{\vee}\mathfrak{m}_{m,m',l,l'}(y_1) | =\Big|\frac{1}{(iy_1)^M}\int_{-2}^2 e^{i y_1 \xi_1}\big( \frac{d}{d\xi_1}\big)^M \mathfrak{m}_{m,m',l,l'}(\xi_1)\, d\xi_1\Big| & \le C_M ( 1 + |y_1| )^{-M}.
({\bf e}_{\underline{a}}, s k_{\underline{a}}, s k_{{\underline{a}}- 1}, s ) \to ( \mathrm{i}{\bf e}_{\underline{a}}, \mathrm{i}s k_{\underline{a}}, \mathrm{i}s k_{{\underline{a}}- 1}, ( - \mathrm{i})^{\delta_{1{\underline{a}}}}s ) .
\mathcal{P}_2\left({x}\right)=\exp \left\{- 2\pi{\lambda_e}\int_0^\infty (1-{{f_{\Pr}}({r_e})}){r_e}\sum\limits_{\ell,n \in \left\{{{},{}}\right\}}{\rm{\mathbf{1}}}\left({\max \{{r_e},d\}}< \big(\frac{{P_t}{G_\ell}{G_n^e}\beta}{x \sigma_e^2}\big)^{\frac{1}{{\alpha_{\mathrm{NLoS}}}}}\right){{{\Pr}_{\ell n}}}d{r_e}\right\}
\int_{\partial M}\delta\frac{\partial}{\partial\nu}V_{\delta,q}\Phi d\mu_{g}\le\delta\left\Vert \frac{\partial}{\partial\nu}V_{\delta,q}\right\Vert_{L^{\frac{2(n-1)}{n}}(\partial M)}\|\Phi\|_{L^{\frac{2(n-1)}{n-2}}(\partial M)}\le O(\delta)\|\Phi\|_{H}=o(\varepsilon^{2}).
w_1(t) =e^{-tA}\Big[-u_1\Big]-\int_0^te^{-(t-s)A}V_{0}'(s)\,ds &=-e^{-tA}u_1-\int_0^te^{-(t-s)A}(-Ae^{sA}v_0)\,ds &=-e^{-tA}u_1+tAe^{-tA}v_0 &=V_1^{(1)}(t)+V_1^{(2)}(t).
\sigma_{g^{-1}}(y^*)_{((i',j'),1_H)}= (y^*)_{g((i',j'),1_H)}& = (y^*)_{((i,j)+\varphi_h(i',j'),h)}& = (y^{h^{-1}})_{(i',j')+\varphi_{h^{-1}}(i,j)}& = (\sigma_{-\varphi_{h^{-1}}(i,j)}(y^{h^{-1}}))_{(i',j')}.
h(\alpha_3) \leq \frac{1}{4}\left(\log (b^2(c-a)^2)+\log \frac{c(\sqrt{a}+\sqrt{b})^2}{b(c-a)}\right) &=\frac{1}{4}\log(cb(c-a)(\sqrt{a}+\sqrt{b})^2) &<\frac{1}{4}\log c^4= \log c.
p_k(\mathbf{x}_1, \ldots, \mathbf{x}_m) s_{\lambda}^{(r)}(\mathbf{x}_1, \ldots, \mathbf{x}_m) = \sum (-1)^{{\rm ht}(\mu / \lambda)}s_{\mu}^{(r)}(\mathbf{x}_1, \ldots, \mathbf{x}_m),
\begin{array}{l c r}{{\{X^{\mu}, X^{\nu}\}= \sum_{\stackrel{m_{1}, n_{1}}{m_{2}, n_{2}}}X_{m_{1}n_{1}}^{\mu}( \sigma ) X_{m_{2}n_{2}}^{\nu}( \sigma ) \{L_{m_{1}n_{1}}, L_{m_{2}n_{2}}\},}}\\ \end{array}
\epsilon_{1}^{i}= e^{{\frac{1}{2}}A}\tilde{\eta}_{+}^{i}\ , \qquad \epsilon_{2}^{i}= \left( e^{-{\frac{1}{2}}A}- 2 \kappa^{D - 2}W ( \phi_{0}) e^{{\frac{1}{2}}A}x^{\mu}\Gamma_{\underline{{\mu}}}\right) \tilde{\eta}_{-}^{i}\ .
\displaystyle{\it F_n}\left( 0 \right) = \Gamma \left( -b+f \right) \Gamma \left( f-c \right) \Gamma \left( f-a \right)\left( -{\frac{\sin \left( \pi\,f \right){\it F_p}\left( 4 \right)}{\pi\,\Gamma \left( 1-s \right)}}+{\frac{{\it F_n}\left( 5 \right)}{\Gamma \left( 1-b \right) \Gamma \left( 1-c \right) \Gamma \left( 1-a \right)}}\mbox{}\right) \,.
[{f},{f}](X,Y) ={f}^2 [X,Y] + [{f}X,{f}Y] -{f}[{f}X,Y] -{f}[X,{f}Y],\X,Y\in\mathfrak{X}_M , d\eta^i(X,Y) = \frac12\,\{X(\eta^i(Y)) - Y(\eta^i(X)) - \eta^i([X,Y])\},X,Y\in\mathfrak{X}_M .
S=S_{cm}-{1\over 2}\, \int_{\Sigma_{p+1}}d^{p+1}\sigma\, \sqrt{\, \overline g\,}\, \left[\, \overline{g}^{\, m n}\, \partial_m \, Y^\mu\, \partial_n\, Y_\mu -m_{p+1}\, p\, \right]\ .
x_{i,1}+ x_{i,2}+ \cdots + x_{i,n}= 1 \textrm{\, for all $i$ \; \textrm(\emph{row sums equal one})}, x_{1,j}+ x_{2,j}+ \cdots + x_{n,j}= 1 \textrm{\, for all $j$ \; (\emph{column sums equal one})}.
\tilde{e}''_k:=\mathcal{B}'({\mathbf V}_k|_{x_1=0},\psi_k)(\delta{\mathbf V}_k|_{x_1=0},\delta\psi_k)-\mathcal{B}'(S_{\theta_k}{\mathbf V}_k|_{x_1=0},S_{\theta_k}\psi_k)(\delta{\mathbf V}_k|_{x_1=0},\delta\psi_k).
\left<F \right| V^{(r)}_{\mathrm{CD}}(m,\Lambda) \left| I \right> = \left[ \langle F\vert\delta V^{(r)}\hspace{-.07cm}(m,\Lambda,\Lambda^{\prime})\vert I\rangle - \sum_{s=2}^{r-1}B_{r-s,s}\left<F \vert V^{(r-s)}(m,\Lambda) \vert I \right>\right]_{\Lambda \hspace{.1cm}\mathrm{terms}}.
\xi_{\small o u t}^{\small T E}( \mu ) ={\frac{\pi}{4}}\left[{\frac{\mu - 1}{\mu + 1}}\right] ={\frac{\pi}{4}}\left[{\frac{\mu_{1}- \mu_{2}}{\mu_{1}+ \mu_{2}}}\right] = - \xi_{\small i n}^{\small T E}( \mu ) .
{{\boldsymbol{K}}_{k;t}}={\left({{\boldsymbol{W}}_k^T{{\boldsymbol{\Omega}}_{k;t}}{{\boldsymbol{W}}_k}}\right)^{- 1}}\left({{\boldsymbol{\bar P}}_{k|k - 1;t}^{yx}+{\boldsymbol{H}}_k^T{\boldsymbol{\bar P}}_{k|k - 1;t}^y}\right).
\rho_D(U) = 7n(D[U]) - 3m(D[U]) - 2\pi(D[U]) &= \rho(\Tilde{D}) + 7(n(D[U]) - n(\tilde{D'})) - 3(m(D[U]) - m(\Tilde{D})) - 2(\pi(D[U])-\pi(\Tilde{D'})) &\leq \rho(\Tilde{D'}) + 7 - 3 + 2 &\leq \rho(\Tilde{D'}) + 6
\sum\limits_{i = 1}^k{({\lambda_{k + 1}}-{\lambda_i}}{)^2}\le \frac{{2\sqrt{n + 1}}}{n}{\left[{\sum\limits_{i = 1}^k{({\lambda_{k + 1}}-{\lambda_i}})\lambda_i^{\frac{1}{2}}}\right]^{\frac{1}{2}}}{\left[{\sum\limits_{i = 1}^k{{{({\lambda_{k + 1}}-{\lambda_i})}^2}}\lambda_i^{\frac{1}{2}}}\right]^{\frac{1}{2}}}.
\Lambda-\Lambda^{-1}=(a^{-1}-a)(\eta\cdot\eta^{t}-I), \quad\mathrm{where}\eta^{t}=( \begin{array}{cccc}0 ... 0 (1-\frac{c-c^{-1}}{a-a^{-1}})^{\frac{1}{2}}\end{array}).
Q^c(g,R)(v,w,w,v;x,y)= Q(g,R)(v,w,w,v;x,y)+Q(g,R)(v,w,w,v;Jx,Jy) &-4 g(Jx,y)R(Jv,w,w,v)-4 g(Jx,y)R(v,Jw,w,v) = Q(g,R)(v,w,w,v;x,y)+Q(g,R)(v,w,w,v;Jx,Jy).
\frac{G(z+\omega_1\,|\,\omega_1,\widetilde\omega_1,\omega_2)}{G(z\,|\,\omega_1,\widetilde\omega_1,\omega_2)}= F(z+\overline\omega_1\,|\,\widetilde\omega_1,\omega_2)^{-1}, \frac{G(z+\widetilde\omega_1\,|\,\omega_1,\widetilde\omega_1,\omega_2)}{G(z\,|\,\omega_1,\widetilde\omega_1,\omega_2)}= F(z+\overline\omega_1\,|\,\omega_1,\omega_2)^{-1}.
TVaR_{\alpha}^{right}(L)=E[L|L\ge VaR_{\alpha}(L)]=\frac{1}{1-\alpha}\int_{\alpha}^{1}VaR_{\gamma}(L)d\gamma=\frac{1}{1-\alpha}\int_{F^{-1}(\alpha)}^{+\infty}yf(y)dy.
\begin{array}{r c c c l}{\dot{\varphi}_{1}}{=}{[ \varphi_{1}, H ]}{=}{\varphi_{2},}{\dot{\varphi}_{2}}{=}{[ \varphi_{2}, H ]}{=}{\partial_{i}^{\makebox [ 1 m m ]{}2}\varphi_{1}.}\end{array}
d \Sigma^{2}= - \left( 1 - \frac{2 M}{r}\right) d t^{2}+ \left( r^{2}( \theta^{'})^{2}+ \left( 1 - \frac{2 M}{r}\right)^{- 1}\right) d r^{2}+ r^{2}\sin^{2}\theta \; d \phi^{2}
z_t(x,t) \cdot n = u(z(x,t),t) \cdot n \Omega \langle z^{\perp}(x,t), z_{x}^{\perp}(x,t) \rangle = \Omega \langle z(x,t), z_{x}(x,t) \rangle = \langle u(z(x,t),t), z_{x}^{\perp}(x,t) \rangle.
\delta_0 = \max_{t_0\in\mathbb{C}}\left\{\max\left\{\min_{\lambda\in S(\Lambda_1)}|\lambda-t_0|-\max_{\mu\in S(\Lambda_2)}|\mu-t_0|,\min_{\lambda\in S(\Lambda_2)}|\lambda-t_0|-\max_{\mu\in S(\Lambda_1)}|\mu-t_0|\right\}\right\},
\tilde{P}(\tilde{\mathcal{C}}_n,w,i,\tilde{\mathbf{u}}_1,\mathbf{y}_2|\mathbf{u}_0,\mathbf{u}_2)=\tilde{\lambda}(\tilde{\mathcal{C}}_n)\tilde{P}^{(\tilde{\mathcal{C}}_n)}(w,i,\tilde{\mathbf{u}}_1,\mathbf{y}_2|\mathbf{u}_0,\mathbf{u}_2).
M_{ZM}^2 = \frac{k^2}{\frac{1}{4\tanh(kz_0)}\Bigl(\cosh^2(kz_0)+(\frac{\beta k}{\alpha})^2\frac{1}{\cosh^2(kz_0)}\Bigr)\log\Bigl[\frac{1+\tanh(kz_0)}{1-\tanh(kz_0)}\Bigr]-\frac{\cosh^2(kz_0)}{2}}.
\det\left( \partial_{p_j}f_i \right)\left( \prod_{k=1}^n \frac{m_k}{p_k}\right) = \det\left( \partial_{p_j}f_i\left( \frac{m_i}{p_j}\right) \right) = \det\left( v_{p_j}(m_i)f_i(m_i) \right) = \left(\prod_{i=1}^n f_i(m_i) \right)\det\left( v_{p_j}(m_i) \right).
\frac{d \psi_{n}^{\lambda}(u)}{du}=\left(\frac{n+\lambda}{1-u^2}u\right)\psi_{n}^{\lambda}(u) -\frac{2\lambda+n}{1-u^2}\frac{N_{n}^{\lambda}}{N_{n+1}^{\lambda}}\psi_{n+1}^{\lambda}(u)
\psi(\bar{\xi}_k) = \sum_{i = 0}^k \bar{\xi}_i \otimes \bar{\xi}_{k - i}^{p^i}\psi(\bar{\tau}_k) = 1 \otimes \bar{\tau}_k + \sum_{i = 0}^k \bar{\tau}_i \otimes \bar{\xi}_{k - i}^{p^i}.
M(p;q_1,q_2)=\int\frac{dx\;dy}{(1+x+y)^3}exp[\frac{xy}{1+x+y}\frac{p^2}{\Lambda^2}+\frac{x}{1+x+y}\frac{q_1^2}{\Lambda^2}+\frac{y}{1+x+y}\frac{q_2^2}{\Lambda^2}],
\sum_{k=1}^n{\bf E}\sup_{0\leqslant t\leqslant 1}\left|z_i(u_k,t)-z_{i+1}(u_k,t)\right|\leqslant \sum_{k=1}^n \sum_{l=1}^k \sum_{j=1}^l 4(l-j) \cdot \sqrt{\varepsilon}= =\sum_{k=1}^n \dfrac{2k(k^2-1)}{3}\cdot \sqrt{\varepsilon}\leqslant \sum_{k=1}^n \dfrac{2k^3}{3}\cdot \sqrt{\varepsilon}\leqslant \dfrac{2n^4}{3}\cdot \sqrt{\varepsilon}.
f_{s,k}:=-\Big({f_{s,k}^*}+\sum_{m=1}^{s}\frac{1}{m!}\sum_{\substack{n_0+\ldots+n_m=s n_0\ge0 n_1,\ldots,n_m\ge1}}\left.\{\{\ldots\{\{f_{n_0}+f_{n_0}^*,G^0_{n_1}\},G^0_{n_2}\}\ldots\},G^0_{n_m}\}\right|_{p=\tilde{p}_k}\Big)\,.
R_1(z)=r_1^{(1)}z+r_1^{(0)}, \ \R_0(z)=r_0^{(2)}z^2+r_0^{(1)}z+r_0^{(0)}, \ \R_{-1}(z)=r_{-1}^{(2)}z^2+r_{-1}^{(1)}z+r_{-1}^{(0)},
\hat{\Delta}{\hat{G}_{N}}( \xi , \xi^{\prime}) ={\frac{{\delta^{2}}( \xi - \xi^{\prime})}{\sqrt{\hat{g}( \xi )}}}-{\frac{1}{\int{d^{2}}\xi^{\prime \prime}\sqrt{\hat{g}( \xi^{\prime \prime})}}}, \quad{\partial_{\hat{n}}}{\hat{G}_{N}}( \xi , \xi^{\prime}) = 0 ,
\left( \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a b c d \end{pmatrix}p \right)^n = \begin{pmatrix}1 n 0 1 \end{pmatrix}+ \begin{pmatrix}an + \frac{c(n-1)n}{2}\frac{(a+d+c(n-1))(n-1)n}{2}- c \sum_{k=0}^{n-1}k^2 + bn cn dn + \frac{c(n-1)n}{2}\end{pmatrix}p
h_{\omega,\kappa}''(\xi) = \kappa(\kappa+1) m_\psi^{-(\kappa+2)}(\beta^{-1}(\omega)\xi+\omega) m_\psi'(\beta^{-1}(\omega)\xi+\omega)^2\beta^{-2}(\omega) & - \kappa m_\psi^{-(\kappa+1)}(\beta^{-1}(\omega)\xi+\omega)m_\psi''(\beta^{-1}(\omega)\xi+\omega)\beta^{-2}(\omega).
\pi_{be_{k}}\beta \psi(n) = \pi_{be_{k}}\beta \left(-n_{1}-c_{k}^{-1}i_{2}\sigma_{1}\overline{\pi}_{3}(n_{2})-c_{k}^{-1}\sigma_{3}\overline{\pi}_{4}(n_{3})-i_{3}\sigma_{2}\overline{\pi}_{5}(n_{4})\right)
\overline{g}((\nabla'_{\widehat{W}}\mathcal{A}_{\widehat{W}})Z_{i},Z_{j})= \overline{g}(\overline{\nabla}_{\widehat{W}}\widehat{W},\overline{Z_{i}}Z_{j})+\overline{g}(\widehat{W},\overline{\nabla}_{\widehat{W}}\overline{Z_{i}}Z_{j}),
|P|^2=I(P,\mathcal S) <{\frac{|P||\mathcal S|}{q}+|P|^{1/2}|\mathcal S|^{1/2}q^{d/2}}&=\frac{|P|\sum_{p\in{P}}|\triangle_p(P)|}{q}+|P|^{1/2}\left(\sum_{p\in{P}}|\triangle_p(P)|\right)^{1/2}q^{d/2}.
\gamma^{ba^{\prime}}(x,y,z^{\prime})=\int d^4x^{\prime}d^4u^{\prime}S_F(x-x^{\prime})\Gamma^{b^{\prime}a^{\prime}}(x^{\prime},y,u^{\prime},z^{\prime})\Delta^{b^{\prime}b}(u^{\prime}-x)
\|H^{(k)}\|_{S+C,\infty}\leq \|H^{(0)}\|_{S+C,\infty}+ \|H^{(k)}- H^{(0)}\|_{S+C,\infty}&< S+C + C \sum_{l=0}^{k-1}(C_1)^l &< S+C + \frac{C}{1-C_1}\leq T.
f^{(n)}(\omega,s,y,z,u):=\mathrm{sign}\left(\hat{f}^{(n)}(\omega,s,y,z,u)\right) &\times\left [ F(\omega, s)\wedge n +(K_1(\omega,s)\wedge n)|y| +(K_2(\omega,s)\wedge n)(|c_n(z)|+\left\|\tilde{c}_n(u)\right\|)\right]
4 \Lambda^{2 n_{c}- n_{f}}\prod_{i = 1}^{n_{f}}( \Lambda \omega^{2 \ell - 1}+ \delta_{\ell}+ m_{i}) = 4 \Lambda^{2 n_{c}}( \omega^{2 \ell - 1})^{n_{f}}\left[ 1 + \sum_{i = 1}^{n_{f}}\frac{\delta_{\ell}+ m_{i}}{\Lambda \omega^{2 \ell - 1}}\right] .
\left[ \begin{array}{c c}{0}&{- \left({\frac{2 \pi n}{m_{l}- m_{r}}}\right)^{2}b_{i}}\\{b_{i}}&{0}\\ \end{array}\right] , \left[ \begin{array}{c c}{0}&{- \left({\frac{2 \pi n}{m_{l}- m_{r}}}\right)^{2}b_{i}^{*}}\\{b_{i}^{*}}&{0}\\ \end{array}\right]
| p_{( L_{1}, \cdots , L_{k})}|^{2}- | p_{( L_{1}, \cdots , L_{k}, L_{k + 1})}|^{2}- | p_{( L_{1}, \cdots , L_{k}, L_{k + 2})}|^{2}+ | p_{( L_{1}, \cdots , L_{k}, L_{k + 1}, L_{k + 2})}|^{2}= 0
N_{X\underset{x,y}{\star}Y}(m)= \left(N_X(m)+\sum N_{X}(k_1)N_{Y}(k_2)+\sum N_{X}(k_1)N_{Y}(k_2)N_{X}(k_3)+\cdots\right) + \left(N_Y(m)+\sum N_{Y}(k_1)N_{X}(k_2)+\sum N_{Y}(k_1)N_{X}(k_2)N_{Y}(k_3)+\cdots\right)
V_{0}^{(\mathrm{A})}=\left[\begin{array}{cccc|cccc|ccc|cccc}1 2 \cdots n\, \,1 2 \cdots n\, \,\bullet \bullet \bullet\, \,1 2 \cdots n 1 1 \cdots 1\, \,2 2 \cdots 2\, \,\bullet \bullet \bullet\, \,n n \cdots n\end{array}\right],
\big\vert \big\vert \tilde{\mu}^{(n)}\big( v_{\Sigma}^{(n)}- \tilde{v}_\Sigma^{(n)}\big) \big\vert \big\vert_{L^2(\Sigma)}^2 = O\left(\frac{1}{n\log^4n}\right), \big\vert \big\vert \tilde{\mu}^{(n)}\big( v_{\Sigma}^{(n)}- \tilde{v}_\Sigma^{(n)}\big) \big\vert \big\vert_{L^2(\Sigma)}= O\left(\frac{1}{n^{1/2}\log^2n}\right),
\int_{G}t_{m n}^{( \rho )}( g ) d g = \sum_{j = 1}^{\ell}P^{( j ) m}P_{n}^{( j )}, \qquad P^{( j ) m}= \eta^{m}( w^{( j )}) , \quad P_{n}^{( j )}= \theta^{( j )}( v_{n}) .
-\pi\leq\sigma\leq \pi, ~~~~ \psi^{i}(\sigma )=\begin{cases}i\psi^{i+1}(-\sigma -\pi),\quad-\pi\leq\sigma\leq -\frac{\pi}{2}, -i\psi^{i-1}(-\sigma -\pi),\quad -\frac{\pi}{2}\leq\sigma\leq 0, i\psi^{i-1}(\pi -\sigma ),\quad 0\leq\sigma\leq\frac{\pi}{2}, -i\psi^{i+1}(\pi -\sigma ),\quad\frac{\pi}{2}\leq\sigma\leq\pi.\end{cases}
\Delta q-2s^{2}\nabla q\int\limits_{s}^{\infty}\nabla q\left( x,\tau\right) d\tau +2s\big(\int\limits_{s}^{\infty}\nabla q\left( x,\tau \right)d\tau \big)^{2}=0,x\in \Omega ,s\geq \underline{s}\left( d\right) .
\left(\begin{array}{c}\phi_A \varphi_A \end{array}\right) = \left(\begin{array}{cc}\frac{3}{4}- \frac{\sqrt{7}}{4}- \frac{\sqrt{7}}{4}- \frac{3}{4}\end{array}\right) \left(\begin{array}{c}\phi_B \varphi_B \end{array}\right).
g(a_{k+1}^2,\ldots,a_n^2; \tilde{x}_k)=\frac{G(a;b,W)}{G(\tilde{a}_k; B_k,W_k)}\frac{e^{-\d B^*_kM^{-1}_{\tilde{x}_k}B_k}}{\sqrt{\det(M_{\tilde{x}_k})}}\textbf{1}_{C_{W_k}}(\tilde{x}_k)
I_{n}(r)\geq \frac{1}{2}\int_{r_{s}}^{r^2}\rho^{n}(\rho-1)^{2}\, d\rho\geq \frac{r_{s}^{n}}{2}(r_s-1)^{2}\int_{r_{s}}^{r^2}d\rho=\frac{r_{s}^{n}}{2}(r_s-1)^{2}(r^2-r_{s})= \frac{r_{s}^{n}}{2}s(1-s)^{2}(r^2 -1)^{3}.