harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E\prod_{j=0}^{n-1}\rho(x_j,\omega)\rho(x_{j+1},\omega) =M^{2n}E\prod_{j=0}^{n-1}\widetilde{\rho}(x_j,\omega)\widetilde{\rho}(x_{j+1},\omega) &\leq M^{2n}E\prod_{j=0}^{|x_n|-1}\widetilde{\rho}(y_j,\omega)\widetilde{\rho}(y_{j+1},\omega) &\leq M^{2n}[E{\widetilde{\rho}}^2]^{|x_n|-1}.
\|\omega\|_X^2 =\lambda\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx+\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx &\leq\lambda C(\|\omega\|_X^2+\|\omega\|_X^4)+C\|\omega\|_X^{2_\alpha^*}.
g([X,Y], Z) =-g(\widetilde\nabla_{X}\varphi Y-(\widetilde\nabla_{X}\varphi) Y,\varphi Z)-g(\widetilde\nabla_{Y}X, Z) =-g(\widetilde\nabla_{X}{tY}, \varphi Z)-g(\widetilde\nabla_{X}{nY}, \varphi Z)-g((\widetilde\nabla_{Y}\varphi) X,\varphi Z)- g(\widetilde\nabla_{Y}X, Z).
T \colon H^1_{\mathrm{dR}}(X) \rightarrow{\textstyle \bigwedge^3}H^1_{\mathrm{dR}}(X)^* T(a)(b_1,b_2,b_3) =(a,b_1,b_2,b_3)- \frac{1}{\mathrm{vol}(X)}\Big(\sum_{\mathrm{cycl}}(a,b_1)(b_2, b_3)\Big)
( W_{0}( n ) \overline{{{P}}}_{\beta}^{k}W_{0}( n )^{- 1})_{-}{}~ = ~ \overline{{{B}}}_{k \beta}^{K P}( n ) |_{t = ( x , 0 , \cdots )}~ \mathrm{f o r}~^{\forall}k \in \mathrm{~ \bf{Z}~}_{\geq 1}.
4 i \dot{u}(b b_{xxu}\!-\! b_x b_{xu}) \!-\! 8 V_c b_u^2 \!+\! 8 V_q b_x^2 \!+\! b_{xx}^2 \!-\! 2b_x b_{xxx}\!+\! b (4 V_c' b_u \!+\! 8 V_c b_{uu}\!-\! 4 V_q' b_x \!-\! 8 V_q b_{xx}\!+\! b_{xxxx}) = 0,
\lim_{T\rightarrow\infty}\int_0^T\langle f(\overrightarrow{x}_t)\rangle_{x_0}^0\,dt =\lim_{T\rightarrow\infty}\int_0^T e^{tL}f(x_0)\,dt &=\lim_{T\rightarrow\infty}L^{-1}e^{tL}f(x_0)\Big|_{t=0}^T &=-L^{-1}f(x_0)+L^{-1}\lim_{T\rightarrow\infty}e^{TL}f(x_0) &=-L^{-1}f(x_0)+L^{-1}\lim_{T\rightarrow\infty}\langle f(\overrightarrow{x_T})\rangle_{x_0}^0.
\big\langle \Phi,(\mathcal{N}_\perp^2+1)\Phi\big\rangle \le\; C \langle \Phi, \mathfrak{D}^2 \Phi\rangle \le\; CN \big\langle U_N^* \Phi,\,\mathcal{N}_-\, U_N^*\Phi\big\rangle \le\; C \min\{N,\frac{1}{T}\}
(a) z^2_{min}>3(X_u-x_i)^2-(Y_u-y_i)^2,\forall i\in I~~~~~~~~~~~~~~~~ (b) z^2_{min}>3(Y_u-y_i)^2-(X_u-x_i)^2,\forall i\in I~~~~~~~~~~~~~~~~ (c) z^2_{min}>3(X_u-x_i)^2+3(Y_u-y_i)^2,\forall i\in I~~~~~~~~~~~~~~~
\displaystyle v_{t}+\frac{1-{p}^{\prime}/c^{2}}{1-{p}^{\prime}v^{2}/c^{4}}vv_{r}+\frac{(1-v^{2}/c^{2})^{2}{p}^{\prime}}{q(1-{p}^{\prime}v^{2}/c^{4})}\rho_{r}=\frac{4\pi n(\sqrt{1-v^{2}/c^{2}})^{3}}{q(1-{p}^{\prime}v^{2}/c^{4})}\phi_{r}+\frac{2(1-v^{2}/c^{2})v^{2}{p}^{\prime}/c^{2}}{r(1-{p}^{\prime}v^{2}/c^{4})}\geq0
W(x_i) =-\frac{1}{4}\sum_{i=1}^{N}x_{i}^{4}+\alpha\sum_{\stackrel{i,j=1}{i<j}}^{N}\ln (x_{i}^{2}-x_{j}^{2})+\beta \sum_{i=1}^{N}\ln x_{i}^{2}\, .
\Phi^{\downarrow}:= (A, \{E_i\}_{i=0}^d, A^*, \{E^*_{d-i}\}_{i=0}^d), \Phi^{\Downarrow}:= (A, \{E_{d-i}\}_{i=0}^d, A^*, \{E^*_{i}\}_{i=0}^d), \Phi^{\downarrow\Downarrow}:= (A, \{E_{d-i}\}_{i=0}^d, A^*, \{E^*_{d-i}\}_{i=0}^d).
{\delta}{\lambda}_4(1235) = \frac{8}{3}g^4_4\int_{{\Lambda}}\frac{d^4\vec{p}_4}{(2{\pi})^4}\frac{{\Phi}(4;1235)}{(\vec{p}_4^2+m^2)((\vec{p}_1+\vec{p_2}+\vec{p}_4)^2+m^2)},
\sum_{\ell=2}^{6}\sum_{j=1}^{d}W_{\ell}^{j}\leq \varepsilon(\mathcal{G}_{1}(K)+\mathcal{G}_{2}(K))(E_{w}+E_{\mu})+\frac{1}{4}\int_{\mathbb{T}^{d}}|D\mu^{1}-D\mu^{2}|^{2}\dx.
e^1 = \begin{pmatrix}1 1 0 0 \end{pmatrix}, \,e^2 = \begin{pmatrix}1 0 1 0 \end{pmatrix}, \,e^3 = \begin{pmatrix}0 1 0 1 \end{pmatrix}, \,e^4 = \begin{pmatrix}0 0 1 1 \end{pmatrix},
D_{X^\prime}\cdot e_0 = -5b_0 +b_1 +6 \geq 0, D_{X^\prime}\cdot e_0^\prime = 5b_0 -b_1 +6 \geq 0, D_{X^\prime}\cdot e_1 = b_0 - 5b_1 +6 \geq 0, D_{X^\prime}\cdot e_1^\prime = -b_0 + 5b_1 +6 \geq 0.
\psi (x) = \frac{24}{\pi^4}\sum_{n}a_n \bigl( \sin n \pi x - n \pi x \bigr) & = \frac{96}{\pi^4}S(x) + 3 x^2 - \frac{84\zeta(3)}{\pi^3}x - \frac{96}{\pi^4}\sum_{n}\frac{d_n}{n^4}\bigl( \sin n \pi x - n\pi x \bigr) .
d_j^n = (n-j)^2 \, h_j^{n-1}+ j^2 (n-j)^2\, b^2_{j-1}(\beta) \, h_{j-2}^{n-3}+ j^2 \, h_{j-1}^{n-1}+ j^2 (n-j)^2\, b^2_{n-j-1}(\alpha) \, h_{j-1}^{n-3},
H_r(T)= \sum_{S \in UP(T)}(-1)^{|S|}|K_r(S)|=\sum_{S \in UP(T)}(-1)^{|S|}|i_{S}(K_r(S))| = \sum_{S\in UP(T)}(-1)^{|S|}(\sum_{P\in K_r(\{\{t_1\},\{t_2\},\cdots,\{t_m\}\})}\chi_S(P)) = \sum_{P\in K_r(\{\{t_1\},\{t_2\},\cdots,\{t_m\}\})}(\sum_{S\in UP(T)}(-1)^{|S|}\chi_S(P)).
\int_{v_1}^v\left(\frac{\eta}{r}\right)(v')dv' \geq \frac{\underline{\eta}}{\frac{1}{2}\tilde{\phi}(v_1)-\tilde{C}}\int_{v_1}^v\frac{dv'}{v^\ast-v'}&=\frac{\underline{\eta}}{\frac{1}{2}\tilde{\phi}(v_1)-\tilde{C}}\log\left(\frac{v^\ast-v_1}{v^\ast-v}\right).
\bigg\{(\bar{z},w) \in \mathbb{R}\times \mathbb{R}^{n-1}: |w| > r_{\ell}^{\vee}(\bar{z}) \bigg\}\bigg\backslash \{( \bar{z},\tau ) \}\subset E_{1}^{(0)}\cap ( (0,\tau) +E_{2})^{(0)}= E^{(0)}.
W_{CDB}{}^A \equiv \varepsilon^{ij}f^X_{jC}f^Y_{iD}{\cal R}_{XYB}{}^A = \frac 12 \varepsilon^{ij}f^X_{jC}f^Y_{iD}f_{kB}^Z f_W^{Ak}R_{XYZ}{}^W\,.
\left|f(w)-\sum_{j=0}^{n-1}a_jw^j\right| =\left|g(w^{1/q})-\sum_{j=0}^{n-1}b_{qj}(w^{1/q})^{qj}\right|=\left|g(w^{1/q})-\sum_{k=0}^{qn-1}b_{k}(w^{1/q})^{k}\right| &\le C_1A_1^{qn}M_{qn}|w^{1/q}|^{qn}.
\mathbb{E}\sum_{x \in V_{\sigma}(\mathbf{X}_0)}f(\mathbf{X}_0, x) = \mathbb{E}\sum_{t\in \mathbf{T}_{\sigma_1}}F(\mathbf{T}_{\sigma_1}, t)=\int_0^\infty \mathbb{E}F(\mathbf{T}_{\sigma_1}\cup \{z_1\}, z_1) \, dz_1.
\frac{d}{dz}W_z(k_1, ... k_R) = \sum_{1 \leqslant r_1 < r_2 < ... < r_l \leqslant R}k_{r_2}(k_{r_2}+ k_{r_3}) ... (k_{r_2}+ ... + k_{r_l}) W_z (k_1, ... \widehat{k_{r_1}}, ... \widehat{k_{r_l}}, ... k_R)
\begin{aligned}\|\zeta_c^m\|_{\mathcal{T}_h}^2+\Delta t\sum_{n=1}^m(\|\nabla \zeta_c^n\|_{\mathcal{T}_h}^2+\frac{1}{h}|\zeta_\lambda^n-\zeta_c^n|_{\partial\mathcal{T}_h}^2+|\zeta_\lambda^n-\zeta_c^n|_{u,\partial\mathcal{T}_h}^2)\leq C\Delta t\sum_{n=1}^m\|\zeta_\sigma^n\|_{\mathcal{T}_h}^2.\end{aligned}
\frac{\|x^{\delta}\|_2}{\|x\|_2}=\frac{\langle x^{\delta}, u(x')\rangle}{\langle x, u(x')\rangle}\geq \frac{\langle x'+\Delta^{x'}(\delta)u(x'), u(x')\rangle}{\langle x', u(x')\rangle}=1+\frac{\Delta^{x'}(\delta)}{\langle x', u(x')\rangle}\quad.
\Bigl| \int u_{1}\overline{v}_{2}w_{3,\pm}dt dx \Bigr| \sim \Bigl| \int \bigl( \widetilde{u}_{1}* \widetilde{w}_{3,\pm}\bigr) \overline{\widetilde{v}}_{2}d\tau d k \Bigr| & = \Bigl| \int \int \widetilde{u}_{1}(\tau-\tau', k-k')\widetilde{w}_{3,\pm}(\tau',k') d\tau' dk' \overline{\widetilde{v}}_{2}(\tau,k) d\tau d k \Bigr|.
\sum_{k=1}^{n-1}\left(k^2-(k-2(1-\beta))^2 \alpha^2{\rho}^2 \right)|b_k|^2{\rho}^{2k}+n^2|b_n|^2{\rho}^{2n}\leq 4(1-\beta)^2\alpha^2 \rho^2.
\langle\omega, \omega'a\rangle_{\mathcal D}=E(\langle\omega,\omega'\rangle_{\mathcal A}a)=E(a\langle\omega,\omega'\rangle_{\mathcal A})=E(\langle\omega a^*,\omega'\rangle_{\mathcal A})=\langle\omega a^*,\omega'\rangle_{\mathcal D},
|1,1\rangle_{ab}=\hat{a}^{\dagger}\hat{b}^{\dagger}|0,0\rangle_{ab}\rightarrow\frac{1}{2}(\hat{c}^{\dagger}+\hat{d}^{\dagger})(\hat{c}^{\dagger}-\hat{d}^{\dagger})|0,0\rangle_{cd}
\|\Pi_{\rm HHJ}\tau\|_{0,\infty,K}\lesssim\sum^3_{j=1}|M_{f_jf_j}(\Pi_{\rm HHJ}\tau)|\lesssim\sum^3_{j=1}\|M_{f_jf_j}(\tau)\|_{0,\infty,\partial K_j}\lesssim\|\tau\|_{0,\infty,K}.
e^{- S_{\mathrm{d u a l}}( A )}= \int D B_{\mu}D B_{\mu}^{\dagger}D \eta_{\mu}e^{- \left( S_{\mathrm{t o p}}( B , ( \partial + i \eta ) B ) + \int d^{3}x A_{\mu}\epsilon^{\mu \nu \rho}\partial_{\nu}\eta_{\rho}\right)}.
\nabla^{A}\partial_{A}\Phi^{a}= -{\frac{( \partial^{A}\Phi^{b}) ( \partial_{A}\Phi^{b})}{\eta^{2}}}\Phi^{a}+{\frac{\sqrt{- h}}{\sqrt{- g}}}{\frac{\partial \sigma_{i}( \Phi^{a})}{\partial \Phi^{a}}}\delta ( y_{i}) ,
\sum_{j=1}^{m_i}n_{i,j}= (d_{i,1}) + (d_{i,1}+ d_{i,2}) + \cdots + (d_{i,1}+ d_{i,2}+ \cdots + d_{i,m_i}) &= \sum_{j=1}^{m_i}(m_i - j +1) d_{i,j}
\lim_{i \to \infty}\frac{i(n_i+3)}{\ell_i}\le \lim_{i \to \infty}\frac{i(n_i+3)}{n_i^2}&= \left( \lim_{i \to \infty}\frac{n_i+3}{n_i}\right) \cdot \left( \lim_{i \to \infty}\frac{i}{n_i}\right) = 1 \cdot 0 = 0.
g_j^v(f_{jk}(z_k,s),s) \equiv_v g_{jk}(g_k^v(z_k,s),h^v(s)) \Lambda_{M_j}^{r,s}(g_j^v(z_j,s),h^v(s)) \equiv_{v}\sum_{\alpha,\beta=1}^n \Lambda_{N_j}^{\alpha,\beta}(z_j,s)\frac{\partial{g_j^r}^v}{\partial z_j^{\alpha}}\frac{\partial{g_j^s}^v}{\partial z_j^{\beta}}
\begin{gathered}\sum_{i=1}^{m_1}{}\beta_i p_i \ + \sum_{v\in \{0,1\}^{m_2}}\big( \sum_t b_{tv}\big)^2\cdot\prod_{j=1}^{m_2}q_j^{v_j}+ \prod_{k=1}^{m_3}r_k^{d_k}+ \sum_{v\in \{0,1\}^{m_3}}\big( \sum_t c_{tv}\big)^2\cdot \prod_{k=1}^{m_3}r_k^{v_k}+ \sum_w s_w^2 = 0 \end{gathered}
{\cal G}_{\rho z}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\left(-z\right)\Gamma_{\rm{bc}}\left(a=-A_l\right)}{\rho\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}
\phi_{o p t}( t , z , \theta ) = \int_{- \infty}^{+ \infty}d t^{\prime}\int_{S^{d}}d \mu ( \theta^{\prime}) e^{\frac{d z^{\prime}}{\beta_{H}}}\left( G \partial_{z^{\prime}}\varphi ( z^{\prime}, t^{\prime}, \theta^{\prime}) - \partial_{z^{\prime}}G \varphi ( z^{\prime}, t^{\prime}, \theta^{\prime}) \right)_{z^{\prime}= Z}\ ,
{\cal H}^{'}_{phys}(x)={1\over 2}[{\vec \pi}^2(x)+{\vec B}^2(x)]+{1\over 2}M^2{\vec A}^2(x)+{1\over 2}\vec \partial \cdot \vec \pi (x)\,{1\over{\triangle +M^2}}\, \vec \partial \cdot \vec \pi (x).
{\cal S}_{1}= \int \sqrt{- G}[ - \partial_{\xi}\phi_{( 1 )}\partial_{\xi}\phi_{( 0 )}- \frac{\partial V}{\partial \phi}\mid_{( 0 )}\phi_{( 1 )}+ K_{\xi}( - \frac{1}{2}\partial_{\xi}\phi_{( 0 )}\partial_{\xi}\phi_{( 0 )}- V ( \phi_{( 0 )}) ) \xi ] d^{3}\sigma d \xi .
\frac{2q}{1+q}{}_2\phi_1 \left(\genfrac{}{}{0pt}{}{-q,-q}{-q^2};q,q\right) = \frac{2q}{1+q}\frac{(-q^2;q)_\infty}{(q;q)_\infty}{}_2\phi_2 \left(\genfrac{}{}{0pt}{}{-q,q}{-q^2,-q^2};q,-q^2\right).
K_{\beta_{H}}\equiv \beta_{H}\left( \frac{\partial Q}{\partial \varphi}\right)_{\beta_{H}}= \frac{\tilde{d}\Omega_{\tilde{d}+ 1}( r_{+}r_{-})^{\tilde{d}/ 2}}{\pi}\frac{[ 1 - ( \tilde{d}- 1 ) ( r_{-}/ r_{+})^{\tilde{d}}]}{[ 1 - ( r_{-}/ r_{+})^{\tilde{d}}]}.
A_{ij}w_{j}=\begin{pmatrix}a_{ij}b_{ij}b_{ij}a_{ij}\end{pmatrix}v_{j}\mathbf{e=}\begin{pmatrix}a_{ij}+b_{ij}b_{ij}+a_{ij}\end{pmatrix}v_{j}=s_{ij}v_{j}\mathbf{e}
\left\lbrace\begin{array}{ll}\psi''-\Delta\psi +b(\psi)=0 \hbox{in}(0,T) \times \Omega, \partial_\nu \psi = e^{\lambda h}(e, DH) \hbox{on}(0,T) \times\Gamma, t=0: \psi=0, \psi'=0 \hbox{in}~ \Omega.\end{array}\right.
c(i^{*}_{D}W^{-}_{i}(\nu^{j})\oplus L_{1}^{\oplus s}\oplus L_{2}^{\oplus s'})=(1-x)^{r+s}e^{\frac{-\theta}{1-x}}(1-2x)^{r'+s'}e^{\frac{4\theta}{1-2x}},
\Lambda_{n_{12}}^{\mu_{12}}F_j^{\mu_1,\mu_2}(\phi)(\csc\phi)^{n_{12}+j}= \displaystyle\sum\limits_{n=0}^{n_{12}+j}C^{n_1, n_2}_{n_{12}, j}N_{\mu_1}(n)N_{\mu_2}(n_{12}+j-n)\cot^{n}\phi
{d\omega^F_n\over dm}={1\over\sqrt{k^2+m^2}}\left(m + k_n{dk_n\over dm}\right) ={1\over\sqrt{k^2+m^2}}\left(m +{1\over L}{k_n^2\over m}\left(\delta^\prime(k)+{\theta^\prime(k)\over2}\right)\right)
\mathcal{I}_{\mathrm{D}}(\delta_1, 0) = \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + V_{1, YY}^{1/2}\delta_1^2 \mathcal{I}_{\mathrm{D}}(0, \delta_2) = \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + V_{2, YY}^{1/2}\delta_2^2.
\sigma(z) \sum_{y \in B_{R_L}(z) \setminus \{z\}}\frac{\phi_{z, m_L}(y)}{\phi_{z, m_L}(z)}<\frac{1}{c - \delta_\sigma^{-1}}\sum_{k \geq 1}\left( \frac{1}{1 + \delta_\sigma (c - 2 \delta_\sigma^{-1})}\right)^{k-1}= \frac{1}{c - 2\delta_\sigma^{-1}}.
Y_t = Y_t^{b}\textbf{1}_{t<\tau}+ \xi^a_{\tau}\textbf{1}_{t\geq\tau}, Z_t = Z^{b}_t \textbf{1}_{t\leq\tau}, U_t = (\xi^a_t-Y_t^{b}) \textbf{1}_{t\leq\tau}.
\div(\Delta)=\sum\limits_{\substack{d|N 1\leq d\leq N}}\frac{N}{d}\frac{1}{\gcd(d,N/d)}\mathfrak{a}_{c/d}&\div(\Delta(N\cdot))=\sum\limits_{\substack{d|N 1\leq d\leq N}}\frac{d}{\gcd(d,N/d)}\mathfrak{a}_{c/d}.
\sum \limits_{\kappa}\kappa((a,b))\overline{\kappa((m,n))}=\sum \limits_{\chi}\chi(ab)\overline{\chi(mn)}&=\sum \limits_{\chi}\chi(a)\chi(b)\overline{\chi(m)}\overline{\chi(n)}&=\sum \limits_{\chi}\chi(a)\overline{\chi(m)}\chi(b)\overline{\chi(n)}&=\sum \limits_{\chi}1
\int_{(\ell_1-1)2^{-k_1}}^{\ell_1 2^{-k_1}}\int_{(\ell_2-1)2^{-k_2}}^{\ell_2 2^{-k_2}}f_{\rho,L}(x,y) \,dy \,dx &= 2^{-k_1-k_2}\int_0^1 \int_0^1 \bigl\{1 + \rho \sin(2^{L-k_1+1}\pi u) \sin(2^{L-k_2+1}\pi v) \bigr\}\,dv \,du = 2^{-k_1-k_2},
\mathbf{E}(\mathbf{x}) =\frac{1}{4\pi}\int\rho_e(\mathbf{x}')\frac{\mathbf{r}}{r^3}\,d^3x' +\frac{1}{4\pi}\int\mathbf{J}_m(\mathbf{x}')\,{\bf \times}\,\frac{\mathbf{r}}{r^3}\,d^3x
\frac{1}{2 \pi i}\int_{\lambda_1 - i \infty}^{\lambda_1 - i \infty}\frac{\Gamma(s) \Gamma\left(\frac{k}{2}-s \right)}{\Gamma\left(\frac{1-k}{2}+s \right)}X_n^{-s}{\rm d}s = G_{1,2}^{1,1}\left(\begin{matrix}1 \frac{k}{2},\frac{k+1}{2}\end{matrix}\Big| \frac{1}{X_n}\right).
a(b^{2}c) =a(bb\cdot c)=a(b\cdot bc)=bc\cdot ab=(ab\cdot c)b=(cb\cdot a)b =ba\cdot cb=b(ba\cdot c)=b(ca\cdot b)=b(b\cdot ca)=ca\cdot bb =(bb\cdot a)c=(b\cdot ba)c=(a\cdot bb)c=(ab^{2})c.
\begin{aligned}\ \sum_{1\le i\le n}(-1)^{\deg x_1+\dots+\deg x_{i-1}+S_1}f_n(x_1,\dots,x_{i-1},dx_i,x_{i+1},\dots,x_n)+ &\sum_{1\le i\le n-1}(-1)^{\deg x_1+\dots+\deg x_{i-1}+i-1+S_2}f_{n-1}(x_1,\dots,x_{i-1},m_2(x_i,x_{i+1}),x_{i+2},\dots,x_n)= &(-1)^{S_3}df_n(x_1,\dots,x_n)+\sum_{a+b=n}(-1)^{S_4}m_2(f_a(x_1,\dots,x_a),f_b(x_{a+1},\dots,x_n))\end{aligned}
M(\Phi,\Phi',T)=\left(\begin{array}{cc}-i\Phi_I({\bf X}^{0})\tilde{D}{\bf X}^I \overline{T}({\bf X}^{0}) T({\bf X}^{0}) -i\Phi_I'({\bf X}^{0})\tilde{D}{\bf X}^{I}\end{array}\right) ,\I=1,\dots, 9 \ .
g = \int_{S_{\infty}^{2}}d \vec{\sigma}\cdot \frac{1}{v}\Phi^{a}\vec{B}_{a}= \frac{1}{v^{2}}\int_{S_{\infty}^{2}}d \vec{\sigma}\cdot \Phi_{a}\Phi^{a}\vec{B}= \int_{S_{\infty}^{2}}d \vec{\sigma}\cdot \vec{B}= \frac{4 \pi n}{e}.
R_{a}b=a\times b R_{ga}=gR_{a}g^{-1}R_{a}.R_{b}=<a\mid b> A_{gx}(a)=gA_{x}(g^{-1}a):= Ad_{g}A_{x}(a) (x\mid R_{\xi}y)=(x\wedge y\mid \xi) (R_{\xi}x\mid R_{\eta}y)=(R_{\xi}x\wedge y\mid \eta)
a_{1}=1, a_{2}=\left\lceil \tfrac{\gamma}{1-\gamma}\right\rceil +1, a_{2k+1}=\left\lceil \tfrac{\beta}{1-\beta}e_{k}- o_{k}\right\rceil+1, a_{2k+2}=\left\lceil \tfrac{\gamma}{1-\gamma}o_{k+1}- e_{k}\right\rceil+1
\begin{array}{c}{A_{\mu}= \left( \frac 1 2 \left( l_{\mu}^{1}+ l_{\mu}^{2}\right) , \frac 1 2 \left( l_{\mu}^{1}+ l_{\mu}^{2}\right) \right) ,}{B_{\mu}= \left( \frac 1 2 \left( l_{\mu}^{1}- l_{\mu}^{2}\right) , - \frac 1 2 \left( l_{\mu}^{1}- l_{\mu}^{2}\right) \right) ,}\end{array}
\emph{Vec}(\nabla \mathbf{u}^+(\widetilde{X})) = M^-(\widetilde{X})\emph{Vec}(\nabla \mathbf{u}^-(\widetilde{X})), ~~~~~~\emph{Vec}(\nabla \mathbf{u}^-(\widetilde{X})) = M^+(\widetilde{X})\emph{Vec}(\nabla \mathbf{u}^+(\widetilde{X})).
u^{F,G}_{X,Y}(a(b_{i}\boxtimes c_{j})) =\sum_{k}c_{k}\boxtimes b_{i}\langle c_{k}\ |\ac_{j}\rangle &=\sum_{k}c_{k}\boxtimes \langle c_{k}\ |\ac_{j}\rangle b_{i}&=\sum_{k}c_{k}\langle c_{k}\ |\ac_{j}\rangle\boxtimes b_{i}&=ac_{j}\boxtimes b_{i}=a(c_{j}\boxtimes b_i) &=au^{F,G}_{X,Y}(b_{i}\boxtimes c_{j}).
S_{\lambda_1\cdots \lambda_{k_{\rm o}}}^{(k_{\rm o})}(n_2)= \!\! \sum_{(\epsilon_1,\cdots,\epsilon_{k_{\rm o}})\in\{-1,+1\}^{k_{\rm o}}}\!\!\!\! (-1)^{n_{{\rm o}-}}{\frac{n_2!}{(n_2\!-\!n_{{\rm o}+})!}}{\frac{n_2!}{(n_2\!-\!n_{{\rm o}-})!}}\;.
F_{j'}(\xi) =(F^{-M}_{j'}(\xi), \hdots F^{M}_{j'}(\xi)) \|\tilde{C}_{j'}(\xi)\|_{\ell^{2}}^{2}= \|VF_{j'}(\xi)\|_{\ell^{2}}^{2}\leq \|V\|_{2}^{2}\|F_{j'}(\xi)\|_{\ell^{2}}^{2}\|F_{j'}(\xi)\|_{\ell^{2}}^{2}\leq \|V^{-1}\tilde{C}_{j'}(\xi)\|_{\ell^{2}}^{2}\leq \|V^{-1}\|_{2}^{2}\|\tilde{C}_{j'}(\xi)\|_{\ell^{2}}^{2}.
\vert S_k\vert \leq \sum_{2^k/c_k<p\leq 2^k}\frac{2^k}{p}=2^k(\log\log 2^k-\log\log (2^k/c_k)+O(1/\log(2^k/c_k))) =2^k\log \frac{k\log 2}{k\log 2-\log c_k}+ O(2^k/k) =2^k\log\left(1+\frac{\log c_k}{k\log 2-\log c_k}\right)+O(2^k/k) =\frac{2^k\log c_k}{k\log 2}+o((2^k\log c_k)/k)
\left[\begin{array}{c}\mu' \nu' \end{array}\right]^H\left[\begin{array}{cc}1 0 0 -1 \end{array}\right]\left[\begin{array}{c}\mu' \nu' \end{array}\right] = 0, \left[\begin{array}{c}\mu' \nu' \end{array}\right]^H\left[\begin{array}{cc}0 -\delta -\delta 2\epsilon \end{array}\right]\left[\begin{array}{c}\mu' \nu' \end{array}\right] \leq 0,
t_{k + 1}= t_{1}- \frac{x_{3}^{( 0 )}}{\beta_{2}^{( 1 )}}+ \bigl ( x_{2}^{( 0 )}+ x_{3}^{( 0 )}\bigr ) \sum_{j = 1}^{k}\frac{1}{\beta_{2}^{( j )}}
\left( \frac{-D}{\ell}\right) = \left( \frac{-1}{\ell}\right) \prod_{i=1}^{2a}\left( \frac{q_i}{\ell}\right) = - \prod_{i=1}^{2a}\left( \frac{-\ell}{q_i}\right) = - \prod_{i=1}^{2a}-1 = -1.
{\bf I}_{J}= \lim_{\varepsilon \rightarrow + 0}\int_{0}^{2 \pi}{\frac{d \lambda}{2 \pi}}\exp \left\{{i \lambda \left({{\bf a}^{\dagger}{\bf a}- 2 J}\right) - \varepsilon{\bf a}^{\dagger}{\bf a}}\right\}
d |Y^1_s-Y^2_s|^r &=\frac{r(r-1)}{2}\big([\tilde{\sigma}(s,Y^1_s)-\tilde{\sigma}(s,Y^2_s)][\tilde{\sigma}(s,Y^1_s)-\tilde{\sigma}(s,Y^2_s)]^T\big)|Y^1_s-Y^2_s|^{r-2}ds+dM_s &\leq \frac{r(r-1)}{2}|\tilde{\sigma}(s,Y^1_s)-\tilde{\sigma}(s,Y^2_s)|^2|Y^1_s-Y^2_s|^{r-2}ds + dM_s &=|Y^1_s-Y^2_s|^rdA_s+dM_s
s(|D|,X^{(d)}) = c(N_{X^{(d)}}X^{(d+s)}|_{|D|})\cap s(|D|,X^{(d+s)}) = \frac{(1+h)^s}{(1+h)^{d+s-g-r}}\cap [|D|] = (1+h)^{g-d+r}\cap [|D|]
| \vec{k}> = a^{\dagger}( \vec{k}\, ) | 0 > \ \ \ , \ \ \ < \vec{k}| \vec{k}^{\prime}> = ( 2 \pi )^{3}2 \omega ( \vec{k}\, ) \, \delta^{( 3 )}\left( \vec{k}- \vec{k}^{\prime}\right) \ .
= 2 \binom{n+2}{4}+ \sum_{i=2}^{n+1}\frac{(i-1) i}{2}+ \sum_{\alpha=0}^{n-1}\frac{(n-\alpha) (n-\alpha+1)}{2}= 2 \binom{n+2}{4}+ \sum_{i=2}^{n+1}\frac{(i-1) i}{2}+ \sum_{j=1}^{n}\frac{j (j+1)}{2}
\left( x-a\right)^{l}p_{n-l,k-l}\left( x\right) = \binom{n-l}{k-l}\frac{\left( x-a\right)^{k}\left(b -x\right)^{n-k}}{\left( b-a\right)^{n -l}}=\frac{k!\left( n-l\right) !}{\left( k-l\right) !n!}\ \left( b-a\right)^{l}p_{n,k}\left( x\right),
\int_{B_1}|u|^p\, dx \leq C\left(\int_{B_1\times (B_3\setminus B_2)}\frac{|u(x)-u(z)|^p}{|x-z|^{N+ps}}\, dx\, dz+\int_{B_3\setminus B_2}|u(z)|^p\, dz\right) &\leq C\int_{B_3\setminus B_2}|D^su|^p+|u|^p\, dx,
\begin{array}{l}Y_1^{(\nu)}(x,\lambda)\!=\!(i\lambda)^{\nu}\exp\left(i\left(\lambda x-\int_0^xp(t)dt\right)\right)(1\!+\!O(\lambda^{-1})), Y_2^{(\nu)}(x,\lambda)\!=\!(-i\lambda)^{\nu}\exp\left(-i\left(\lambda x-\int_0^xp(t)dt\right)\right)(1\!+\!O(\lambda^{-1})),\end{array}
\sum\limits_{i=0}^{\nu}\varepsilon_{n}^{\sigma_{i}}\alpha_{i}\frac{\partial^{i}u}{\partial x_{n}^{i}}\left( x^{\prime},0,y,t\right) =0\text{,}\nu \in \left\{0,1\right\},x^{\prime}\in R^{n-1}y\in\left( 0,1\right)
\dot{\bar{x}}(t) = \bar{A}(\varepsilon) \bar{x}(t) + \bar{B}(\varepsilon) u(t) + \displaystyle \sum_{j=1}^{n_{\rm in}}u_j(t) \bar{N}_j(\varepsilon) \bar{x}(t) + \bar{H}(\varepsilon) ( \bar{x}(t) \otimes \bar{x}(t) ) \bar{y}(t) = \bar{c}^\top \bar{x}(t)
\begin{cases}\Psi_{\ell,1}(j_1,j_2,j_3,j_1') = 0, \partial_X \Psi_{\ell,1}(j_1,j_2,j_3,j_1')\cdot j_k' - \Psi_{\ell,k}(j_1,j_2,j_3,j_1') = 0 . \end{cases}
0 = \int_0^{\rho}r\phi^n(r) \mathrm{d}r \int_{\mathbb{S}^n}x_l \mathrm{d}A + \rho^2\phi^n(\rho) \int_{\mathbb{S}^n}ux_l \mathrm{d}A + O(\varepsilon)\|u\|_{L^2(\mathbb{S}^n)}.
\sup_x\mathbb{E}\vert\det\sigma_{X_{t,r}\circ X_{t-1,t}^{\mathcal{P},{M_{\mathcal{P}}}}(x)}-\det\sigma_{X_{t,r}\circ X_{t-1,t}(x)}\vert^{\frac{2}{1+\varepsilon_0}}=\sup_x\mathbb{E}\vert \det\sigma_{F_{r-t+1}^{\mathcal{P},{M_\mathcal{P}}}(x)}-\det\sigma_{F_{r-t+1}(x)}\vert^{\frac{2}{1+\varepsilon_0}}\leq C\vert{\mathcal{P}^{t-1,t}}\vert^{\frac{2}{(2+\varepsilon_0)(1+\varepsilon_0)}}.
\textup{smoothness term}:= 4\biggl(\sum_{i=1}^n(\alpha \gamma_{ii}+ \beta_i^2)+ \frac{1}{4}\sum_{i,j=1}^n \bigl(\alpha \gamma_{ij}^2+\beta_i \beta_j \gamma_{ij}+ 4\beta_i \gamma_{ij}\bigr)\biggr)^{1/2}&+ \frac{1}{4}\Big(\sum_{i=1}^n \beta_i^2\Big)^{1/2}\Big(\sum_{i=1}^n \gamma_{ii}^2\Big)^{1/2}+ 3\sum_{i=1}^n \gamma_{ii}+ \log 2\, ,
S_{s,t}(z;z_1,\ldots,z_{s+t-2}) := z^{2s-t-1}\prod_{i=1}^{s+t-2}\frac{(1-z+ z_i z)(1-z_i^{-1})}{(z_i-z)}, T_{s,t}(z;z_1,\ldots,z_{s+t-2}) := (1-z^{-1})^s z^{t-2}\prod_{i=1}^{s+t-2}\frac{1-z_i + z_i z}{(z-z_i) z_i}.
u(x) \leq \lim_{j \to \infty}\left( \inf_{y\in B_{r_j}(x)}\min\big\{j,u(y)\big\}-\frac1{j}\right) &\le \lim_{j \to \infty}\left( \inf_{y\in B_{r_j}(x)}\Big\{\min\big\{j,u(y)\big\}+ j^2 |x-y| \Big\}- \frac1{j}\right)= \lim_{j \to \infty}\widetilde \psi_j(x).
\lim_{k\rightarrow\infty}\frac{f(\alpha n_k^2)-1}{\left(\frac{Tc}{n_k^{1/\delta}}+\Lambda_{1,k}\right)^{\delta}}=\lim_{k\rightarrow\infty}\frac{f\left(\epsilon_k\left(\frac{Tc}{n_k^{1/\delta}}+\Lambda_{1,k}\right)\right)-1}{\left(\frac{Tc}{n_k^{1/\delta}}+\Lambda_{1,k}\right)^{\delta}}=q.
\sum_{k=0}^{(n-1)/2}\frac{(aq^r;q^d)_k (q^{d-r}/a;q^d)_k (q^d;q^{2d})_k q^{dk}}{(aq^d;q^d)_k(q^d/a;q^d)_k (q^{2d};q^{2d})_k}\equiv 0\pmod{\Phi_n(q)}.
\max\left(\left|\tfrac{1-\gamma\sigma}{1+\gamma\sigma}\right|,\left|\tfrac{1-\gamma \beta}{1+\gamma \beta}\right|\right) =\max\left(\tfrac{1-\gamma\sigma}{1+\gamma\sigma},\tfrac{\gamma \beta-1}{1+\gamma \beta}\right) &=|1-\alpha|+\alpha\max\left(\tfrac{1-\gamma\sigma}{1+\gamma\sigma},\tfrac{\gamma \beta-1}{1+\gamma \beta}\right).
\partial_t \partial_x^m \tilde{w}_\epsilon - \partial_y^2 \partial_x^m \tilde{w}_\epsilon - \epsilon \partial_x^m\partial^2_x \tilde{w}_{\epsilon}= - \partial_x^m \big((u^s + \tilde{u}_\epsilon)\partial_x \tilde{w}_\epsilon \big) - \partial_x^m \big( \tilde{v}_\epsilon (\partial_y\tilde{w}_\epsilon + u^s_{yy}) \big),
{\cal A}= \{{\sf a}\in C([0,T]),\ |{\sf a}(t)-{\sf a}_0(t)|\leq(T-t)^{\frac{{\sf d}_1}{2}}{\sf a}_0(t)\},{\sf a}_0(t) = \frac{6-n}{2{\sf A}_1}\int_t^T \eta(t_1)^\frac{2}{1-q}dt_1 = \kappa_1 (T-t) \eta^\frac{2}{1-q}.
\begin{aligned}\frac{dx_i}{dt}=r_i x_i\left(1 - \frac{x_i}{K_i}\right) - \frac{a_i x_i y_i}{1 + x_i}\frac{dy_i}{dt}= \frac{a_i x_i y_i}{1 + x_i}-d_i y_i \end{aligned}
|\ddot{z}_1(t)-\ddot{z}_2(t)| \leq 10\epsilon(2x(t))(\frac{|\lambda|}{4\pi x(t)}+5\epsilon)+(6\epsilon) 20\epsilon x(t)+(10\epsilon)2x(t) = 220\epsilon^2x(t)+\epsilon(20x(t)+\frac{5|\lambda|}{\pi}).
(h_1+h_2)^{-1/2}-(2e-\Delta_{x_1}-\Delta_{x_2})^{-1/2}=\frac{1}{2\pi i}\int_{\sigma}z^{-1/2}(h_1+h_2-z)^{-1}\left(V(x_1)+ V(x_2)\right)(2e-\Delta_{x_1}-\Delta_{x_2}-z)^{-1}\,dz,
\begin{array}{ll}\alpha_1={1\over2}(e_1-e_2-e_3-e_4-e_5-e_6+\sqrt2 e_7), \alpha_2=e_1+e_2, \alpha_3=-e_1+e_2, \alpha_4=-e_2+e_3, \alpha_5=-e_3+e_4, \alpha_6=-e_4+e_5, \alpha_7=-e_5+e_6, \alpha_0=-\sqrt{2}e_7. \end{array}
p_0 = 5^3 \bigl( t (5 t^6 + 5 t^3 + 8) (5 t^6 - 40 t^3 - 1) (40 t^6 - 5 t^3 + 1) \bigr)^3 p_{\infty}= 2^6 (25 t^{12}+ 275 t^9 - 165 t^6 - 55 t^3 + 1)^5 p_1 = - \bigl( (5 t^6 + 1) (25 t^{12}- 1750 t^9 - 2190 t^6 + 350 t^3 + 1) \cdot \quad\quad(200 t^{12}- 500 t^9 + 2055 t^6 + 100 t^3 + 8) \bigr)^2 ,
C(\beta;\delta) =2\,\int_0^{1-\delta}\left(1-\varrho^{N-s\,p-\beta\,(p-1)}\right)\,\varrho^{s\,p-1}\,(1-\varrho^{\beta})^{p-1}\,\Phi(\varrho)\,d\varrho &+2\,\int_{1-\delta}^{1/(1+\delta)}\varrho^{s\,p-1}\,(1-\varrho^{\beta})^{p-1}\,\Phi(\varrho)\,d\varrho.
F_{n}(k)^2-kF_{n-1}(k)F_{n}(k)-F_{n-1}(k)^2+(-1)^n begin{align*}5pt] =(kF_{n-1}(k)+F_{n-2}(k))^2-kF_{n-1}(k)(kF_{n-1}(k)+F_{n-2}(k))-F_{n-1}(k)^2+(-1)^n begin{align*}5pt] =kF_{n-1}(k)F_{n-2}(k)+F_{n-2}(k)^2-F_{n-1}(k)^2+(-1)^n begin{align*}5pt] =-(F_{n-1}(k)^2-kF_{n-1}(k)F_{n-2}(k)-F_{n-2}(k)^2+(-1)^{n-1}) &=0.
A_n(\omega) := \frac{1}{n-1}\sum_{k=2}^n\left[\frac{2c_k(\alpha)}{k^{\frac{1}{\alpha}}}\right]^{\alpha(\varphi^{n-k}\omega) - \alpha}&- \frac{1+ \alpha}{2}\frac{1}{n-1}\sum_{k=2}^n \left[\frac{2c_k(\beta)}{k^{\frac{1}{\beta}}}\right]^{2\alpha(\varphi^{n-k}\omega) - \alpha}
\lim_{N \rightarrow \infty}\frac 1 N \, K_{N}( \frac{x}{N^{2}}, \frac{y}{N^{2}}) = \frac{u ( 1 )}{2}\frac{\sqrt{x}J_{\alpha + 1}( u ( 1 ) \sqrt{x}) J_{\alpha}( u ( 1 ) \sqrt{y}) - J_{\alpha}( u ( 1 ) \sqrt{x}) \sqrt{y}J_{\alpha + 1}( u ( 1 ) \sqrt{y})}{x - y}.
\mathcal{J}^{+}(n^2_1n_2,m,q) = \displaystyle\int_{\mathbb{R}}\,V_1\left(\frac{\nu}{K}\right)\,\displaystyle\int_{\mathbb{R}}W(u)\,g(u,q)\,\eth_1^{+}(n^2_1n_2,u,q,p_1)\,\mathcal{I}^{+}(m,u,q,p_1,p_2)\,du\,d\nu.
\mathbf{(A)}\{0,1\}\subset\big\{{\mathbb S}_k(u,\,\cdot\,)\leq c_n(k)-\lambda\big\}, \mathbf{(B)}\forall\,s\in\big\{{\mathbb S}_k(u,\,\cdot\,)>c_n(k)-\lambda\big\},\ \exists\, j\in J\ \mathbf{(B1)}\ \u(s)\in\mathcal W_j,\mathbf{(B2)}\ \c_n(k)-{\mathbb S}_k(u,s)={\mathbb S}_k^{\gamma_j}(\gamma_j)-{\mathbb S}^{\gamma_j}_k(u(s))\,.
\eqref{term-2-mora}= -\sum_{j,k}\int \partial_j [x_j \psi_R(x)] \partial^2_k (|u(t,x)|^2) dx &= -\sum_k \int [3\phi_R(x) + 2(\psi_R-\phi_R)(x)] \partial^2_k (|u(t,x)|^2) dx &= \sum_k \int \partial_k [3\phi_R(x) + 2(\psi_R-\phi_R)(x)] \partial_k (|u(t,x)|^2) dx.
\langle \Omega | \Phi^{( h )}( z ) \Phi^{( h )}( w ) | \Omega \rangle = \sum_{n = 0}^{\infty}w^{n}z^{- 2 h - n}\phi_{- h - n}^{( h , 0 , h )}\phi_{h - n}^{( h , h , 0 )}