harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Theta^K_{\mu}(\theta) = N^K_{\mu}\sin^K\theta\, C^{(K+\frac{1}{2})}_{\mu}(-\cos \theta), N^K_{\mu}= (-1)^{\mu}2^K \Gamma\left(K+\frac{1}{2}\right) \left(\frac{\mu! \left(\mu+K+\frac{1}{2}\right)}{\pi \Gamma(\mu+2K+1)}\right)^{1/2},
\mathcal{I}_{\mathrm{D}}(\delta) = \sup_{\gamma \in \Sigma_{\mathrm{D}}(\delta)}\int_{\mathcal{Y}_1 \times \mathcal{Y}_2}f(y_1, y_2) \, d \gamma(y_1, y_2) \mathcal{I}(\delta) = \sup_{\gamma \in \Sigma(\delta)}\int_{\mathcal{S}}f(y_1, y_2) \, d \gamma(y_1, y_2, x).
G = - \int_0^{\frac{\pi}{4}}\log\left( \tan x\right) \d x = \sum_{n=1}^{\infty}(4n-1)L\left( \tilde{P}_{2n-1}\right) \left( -\frac{2}{\pi}\int_0^{\frac{\pi}{4}}\tilde{P}_{2n-1}(x) \d x \right) = \sum_{n = 1}^{\infty}\frac{(-1)^{n-1}(4n-1)}{2^{2n}n}\binom{2n-2}{n-1}L\left( \tilde{P}_{2n-1}\right).
\begin{array}{rrcll}\min \multicolumn{4}{l}{\max_{c\in U}c^\top x}begin{align*}1.5ex] \textnormal{s.t.}\sum_{e\in \delta(i)}x_e = 2 \forall i\in N begin{align*}1ex] \sum_{e\in \delta(X)}x_e \ge 2 \forall \,\emptyset\neq X\subsetneq N begin{align*}1ex] x \in \{0,1\}^{|E|}\end{array}
\left\lbrace\begin{array}{lll}\psi''-\Delta\psi +b(\psi)=-e^{\lambda h}(e, AU) \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.
\frac{2}{2^{.5}}\approx1.414,\frac{3}{4^{.5}}=1.5,\frac{4}{6^{.5}}\approx1.633,\frac{6}{12^{.5}}\approx1.732,\frac{8}{24^{.5}}\approx1.633,\frac{12}{60^{.5}}\approx1.549
d^D\mathbf{x}'\,d^D\mathbf{p}'= d^D\mathbf{x}\,d^D\mathbf{p}\left[ 1 - \left\{2(D-1)\beta\left( \frac{1 + (\beta+\beta') p^2}{1 + \beta p^2}\right) + (2\beta + \beta') \right\}p_j\frac{\partial H}{\partial x_j}\delta t \right] \;.
\begin{cases}\partial_t w -\partial_x^2w = 0 , t>0, \x \in (0, \, \pi) w(t, 0)=u_0(t), w(t, \pi)=u_\pi(t), t>0 w(0, x)=f, x \in (0, \, \pi).\end{cases}&\begin{cases}\partial_t w -\partial_x^2w +x^2 w = 0 , t>0,\x \in (0, \, \pi) w(t, 0)=u_0(t), w(t, \pi)=u_\pi(t), t>0 w(0, x)=f, x \in (0, \, \pi).\end{cases}
\sum_{h=1}^H c_h e^{\beta c_h}\prod_{\substack{j = 1 j\not=h}}^{H}\bigl(e^{\beta c_j}-1\bigr) = \sum_{h=1}^{H}(-1)^{H-h}\sum_{i\in \mathcal{C}_h^{\{1,\dots, H\}}}(c_{i_1}+\dots +c_{i_h})\,e^{\beta(c_{i_1}+\dots+c_{i_h})}.
{{\partial W}\over{\partial\theta}}= \int{\cal D}\bar{K}^{4i}{\cal D}\bar{K}\raisebox{0.97ex}{\mbox{\scriptsize{$4i$}}}\raisebox{-0.36ex}{\mbox{\scriptsize{$,\!\theta$}}}{\cal D}\bar{t}_{4}{\cal D}\bar{t}_{4,\theta}{{\partial\bar{W}}\over{\partial\theta}}
\left[{\begin{array}{*{20}{c}}{{{\left|{{g_{11}}}\right|}^2}}{{{\left|{{g_{12}}}\right|}^2}}{{{\left|{{g_{21}}}\right|}^2}}{{{\left|{{g_{22}}}\right|}^2}}\end{array}}\right] = \left[{\begin{array}{*{20}{c}}{2.1713}{1.4836}{3.0937}{0.9773}\end{array}}\right],
b(\mathbf{D})^*g(\mathbf{x})b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x})+\sum \limits_{j=1}^dD_ja_j(\mathbf{x})^*=0,\int_{\Omega}\widetilde{\Lambda}(\mathbf{x})\,d\mathbf{x}=0.
-\frac12\Delta_B|\phi|^2= \sum_a\{|\nabla_{\bar V_a}\phi|^2 +|\nabla_{V_a}\phi|^2\}+\sum_{i=1}^r Ric^Q(E_{a_i},E_{a_i})|\phi|^2 +\frac12\sum_a\{\langle\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi,\phi\rangle +\langle\phi,\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi\rangle\}.
\P(\xi_n \geq3k) \leq\lim_{m \to\infty}\bigl(\P^{\prime}\bigl( F^m_n \bigr)\bigr)^k = \Bigl(\P\Bigl(\lim_{m \to\infty}F^m_n \Bigr)\Bigr)^k = \bigl(\P(F_n)\bigr)^k,
\{( r - s ) z^{1 2}z_{1}^{r - s}- 2 ( z_{1}^{r - s}z_{2}+ \cdots + z_{1}z_{2}^{r - s}) \}= z_{1 2}\{( r - s ) z_{1}^{r - s}+ \sum_{\ell = 1}^{r - s - 1}2 ( r - s - \ell ) z_{1}^{r - s - \ell}z_{2}^{\ell}\}\, .
P_{x}(\omega)=\frac{1}{2}(\frac{[\sum_{j}X_{j}cos\omega(t_{j}-\tau)]^{2}}{\sum_{j}cos^{2}\omega(t_{j}-\tau)}+\frac{[\sum_{j}X_{j}sin\omega(t_{j}-\tau)]^{2}}{\sum_{j}sin^{2}\omega(t_{j}-\tau)})
\pi(a,a)\cdot (p^3-1)(p^2-1) = \begin{cases}7 \a=0, 12(a+1)^3 \ 1\le a \le \frac{p-3}{2}, 6\left((a+1)^3- (p-a-1)^3\right) \ \frac{p-1}{2}\le a <p-1, p^3 \a=p-1. \end{cases}
\frac{2n\dbinom{4n}{2n}+\sum\limits_{i=0}^{n}\dbinom{4n}{2i+1}}{\dbinom{4n}{2n}}=2n+\frac{2^{4n-2}}{\dbinom{4n}{2n}}=k+\frac{2^{2k-2}}{\dbinom{2k}{k}}.
a_\varnothing(\xi)[u,v] := \int_{-\frac\pi2}^{\frac\pi2}\xi^2( 2 u_1 \overline{v_1}+ u_2 \overline{v_2}) + \partial_2 u_1 \cdot \overline{\partial_2 v_1}+ 2 \partial_2 u_2 \cdot \overline{\partial_2 v_2}& - \mathrm{i}\xi (\partial_2 u_1 \cdot\overline{v_2}- u_2 \cdot \overline{\partial_2 v_1}) \; \mathrm{d}x_2
\mathcal{G}'(\beta) =(p-1)\,\log\varrho\, \varrho^{N-s\,p-\beta\,(p-1)}\,\left[1-\varrho^\beta\right]^{p-1}&-(p-1)\,\varrho^\beta\, \log\varrho\,\left[1-\varrho^{N-s\,p-\beta\,(p-1)}\right]\,\left[1-\varrho^\beta\right]^{p-2},
\sum_{m \in \mathbb{Z}}r_{d-1}(m^2 + h) e^{-(2m^2 + h)/X}= \delta_{[d=3]}\delta_{[h = a^2]}C' X^{\frac{1}{2}}\log X + \sum_{0 \leq m < \lceil \frac{d}{2}- 1 \rceil}C_{m}X^{\frac{d}{2}- 1 -\frac{m}{2}}+ O(X^{\frac{d}{4}- \frac{1}{2}+ \epsilon}).
\mathbb{L}^\prime[\varphi_0]\varphi^\prime=\varphi^\prime_{tt}-{\left(\mu-2\phi_{0,x}\right)}\varphi^\prime_{xx}-2\left[\mathbb H\,;\phi_{0,x}\right]\phi^\prime_{xx}-2\mathbb H[\phi_{0,xx}\phi^\prime_x] +\big(\left[\phi^\prime;\mathbb H\right]\phi_{0,xx}+{\left[\phi_0;\mathbb H\right]\phi^\prime_{xx}\big)_x}\,,
\begin{array}{l l l}{{L_{I J}^{( + )}L_{J K}^{( + )}}}&{{-}}&{{( - 1 )^{P_{J}( P_{I}+ P_{K})}L_{J K}^{( + )}L_{I J}^{( + )}=}}\\{{}}&{{}}&{{( q^{- 1}- q ) ( - 1 )^{P_{I}P_{J}}L_{J J}^{( + )}L_{I K}^{( + )}, ( I < J < K )}}\\ \end{array}
\begin{multlined}[t][12.5cm]\widetilde{\Pi}_{\nu_{G},k\nu_{T}}\left(x_{1k},x_{2k}\right)=\frac{kd_{\nu_{G}}}{(2\pi)^{d_{T}}}\int_{W}\chi_{\nu_{G}}(g^{-1})e^{ik\Psi^{(1)}(t,\vartheta,x)}\cdot \cdot s\left(\tilde{\mu}^{G}_{g^{-1}}\circ\tilde{\mu}^{T}_{-\vartheta}\left(x_{1k}\right),x_{2k},kt\right)dV_{W}(w),\end{multlined}
: = \left\langle \prod_{b = 1}^{N}2 \left( \sum_{n_{b}= 0}^{\infty}\frac{1}{( 2 n_{b}) !}\Big ( \beta^{( b )}\Big )^{2 n_{b}}\left[ \chi_{+}^{( b )}[ Q^{W}] + \chi_{-}^{( b )}[ Q^{W}] \right]^{2 n_{b}}\right) \right\rangle_{Q^{\mu}}
\begin{aligned}g(x)= - \sum_{i=1}^N \min_{l_i\leq y_i\leq u_i}\Big \{(\tilde{B_1}x- \alpha)_i (y_i-x_i) + \beta y^2_i + h_i(y_i)\Big\}&+ \beta \big{(}\sum_{i=1}^Nx_i\big{)}^2 +\sum_{i=1}^N h_i(x_i). \end{aligned}
\int_{\Omega}^{}\nabla \bar{y}\cdot \nabla z dx + \int_{\Gamma}^{}\nabla_\Gamma \bar{y}_\Gamma \cdot \nabla_\Gamma z_\Gamma d\Gamma =\langle \bar{u}_\Gamma , z_\Gamma \rangle_{V^*_\Gamma , V_\Gamma}
{\frac{1}{z \, K}}\sqrt{\frac{A_{T^{2}}^{\mathrm{( O M )}}}{A_{T^{2}}^{\mathrm{( b u l k )}}}}\sim \left({\frac{l_{p}}{l_{g}}}\right)^{3}\propto{\frac{\alpha^{\prime}}{\alpha_{( e f f )}^{\prime}}}\, .
\begin{gathered}E_U(r^{n-1}\rho J_{1,1}) = 2 r^{n-1}\rho U,E_\rho(r^{n-1}\rho J_{1,1}) = 2 r^{n-1}U^2, E_S(r^{n-1}\rho J_{1,1}) = -2 r^{n-1}p',E^{(1)}_S(r^{n-1}\rho J_{1,1}) = \tfrac{2}{n}r^n p'\end{gathered}
|(T(e_{1})-T(e_{2}))''(t)| \leq \frac{|RHS(e_{1},t)-RHS(e_{2},t)|}{\alpha}+ \frac{2}{\alpha}\sup_{z \geq t}\left(|RHS(e_{1},z)-RHS(e_{2},z)|\right) &\leq \frac{3 C_{lip}}{\alpha}\frac{||e_{1}-e_{2}||_{X}}{t^{2}\log^{b+1}(t) (\log(\log(t)))^{3/2}}
S = \int_{- \infty}^{\infty}\Bigl [{\frac{1}{2}}( 1 +{\frac{x^{2}}{\tau^{2}}}( 1 - \varepsilon ) ) \left({\partial_{x}\phi}\right)^{2}+{\frac{\lambda}{4}}{\frac{1}{1 +{\frac{x^{2}}{\tau^{2}}}( 1 - \varepsilon )}}\phi^{4}\Bigr ] d x ,
\Lambda^{-}_{t}=\{\lambda_{1}+2t,a-t+ib,a-t-ib,\lambda_{4},\ldots,\lambda_{n}\}\in \mathcal{CR}_{n}\Lambda^{+}_{t}=\{\lambda_{1}+\delta t,a+t+ib,a+t-ib,\lambda_{4},\ldots,\lambda_{n}\}\in \mathcal{CR}_{n},
|\Lambda(M,0)|> |\{\,\,(t_{1},t_{2},n)\,\,|\,\,A(\epsilon_{(t_{1},t_{2},n)})<M\,\,,\,\,1\leq n\leq N\,\,\}| = \sum_{k=1}^{N}|\{\,\,(t_{1},t_{2})\,\,|\,\,t_{1}R_{1}+t_{2}R_{2}<M-T_{k}\,\,\}| = \sum_{k=1}^{N}\frac{(M-T_{k})^{2}}{2R_{1}R_{2}}+O(M-T_{k}) = \frac{(M)^{2}N}{2R_{1}R_{2}}+O(M)
G_\alpha^{(n)}\equiv \sum_{m=0}^n\{\tau_\alpha^{(n-m)}, \tilde H^{(m)}\}+ \sum_{m=0}^{n-2}\{\tau_\alpha^{(n-m)}, \tilde H^{(m+2)}\}_{(\Phi)}+ \{\tau_\alpha^{(n+1)}, \tilde H^{(1)}\}_{(\Phi)},~~n\geq 2
\int K d_{k,i}\varphi_{i}^{\frac{n+2}{n-2}}u_{\alpha, \beta}=d_{k}\frac{\alpha K_{i}\omega_{i}}{\lambda_{i}^{\frac{n-2}{2}}}+o_{\varepsilon}( \frac{1}{\lambda_{i}^{\frac{n-2}{2}}}),
\sum_i I(\tilde{P}_i,\tilde{L}_i) = \sum_i O(m_i^{2/3}n_i^{1/3}s^{1/3}+ m_i + n_i) & = O\left( \left( \sum_i m_i \right)^{2/3}\left( \sum_i n_i \right)^{1/3}s^{1/3}+ m + n\right)= O\left( m^{2/3}n^{1/3}s^{1/3}+ m + n\right) .
H_{2}^{- 1}\Big ( H_{6}\delta^{i j}\partial_{i}\partial_{j}b + 2 \delta^{i j}\partial_{i}H_{6}\partial_{j}b \Big ) - \frac{\kappa T_{2}}{4 \pi^{2}\alpha^{\prime}}( 4 N - M ) \delta ( x^{3}) \cdots \delta ( x^{5}) = 0 \, ,
\hat{T}( F ) = \frac{i}{\hbar}\frac{\delta F}{\delta \phi^{A}}\frac{\delta}{\delta \bar{\phi}_{A}}- \frac{1}{2}\varepsilon_{a b}\frac{\delta}{\delta \phi_{A a}^{*}}\frac{\delta^{2}F}{\delta \phi^{A}\delta \phi^{B}}\frac{\delta}{\delta \phi_{B b}^{*}},
\begin{cases}0=\nabla \cdot ue^{-v}\nabla \left(\log u - v \right) \mathrm{in}\ \Omega, begin{align*}1mm]0=\Delta v -v+u \mathrm{in}\ \Omega, u > 0, \v > 0 \mathrm{in}\ \Omega, \displaystyle \frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 \mathrm{on}\ \partial \Omega.\end{cases}
\begin{array}{l}\textbf{Kp}= \textbf{BR}[4,\{2,1,3,2,2,1,3,2,2,1,3,2,-1\}] \textbf{ColouredJones}[\textbf{Kp},1][q^4](q^4-q^{-4})/(q^2-q^{-2}) q^{14}+q^{18}+q^{22}+q^{26}-q^{42}-q^{46}-q^{50}+q^{58}\end{array}
\tilde{P}\; = \; \sum_{n = 0}^{\infty}\sum_{\ldots}{\mathrm{c o n s t}}( n , \ldots ) \; C_{n , \ldots}\:{\cal{B}}\: C_{n - 1 , \ldots}\:{\cal{B}}\: \cdots \:{\cal{B}}\: C_{0 , \ldots},
R_{v v}^{\ast}-{\sigma}^{i j}\frac{\partial \beta}{\partial y^{i}}\frac{\partial \beta}{\partial y^{j}}- \frac{2 \Lambda}{( n - 1 )}\beta^{2}= \frac{1}{\alpha^{3}}\frac{d^{2}\alpha}{d u^{2}}- \frac{1}{\alpha^{4}}\left( \frac{d \alpha}{d u}\right)^{2}= \frac{1}{\alpha ( v )}\frac{d^{2}\alpha}{d v^{2}}\quad ,
e_{0}\q^{\mu}\Gamma_{\mu}^{( 3 )}( p , P ; q , Q ) - \sigma Q \ 2 M \Gamma^{( 4 )}( p , P ; q , Q , k , k^{\prime}) = e_{0}^{2}\left[ \frac{1}{G^{( 2 )}( p^{\prime}, P^{\prime})}- \frac{1}{G^{( 2 )}( p , P )}\right] \ .
|u(x)|^s =|u(x)|^{(1-a)s}|u(x)|^{as}\le |u(x)|^{(1-a)s}(|u(x)-\bar{u}|+|\bar{u}-u(\xi)|+|u(\xi)|)^{as}\le (1+\epsilon) |u(x)|^{(1-a)s}|u(\xi)|^{as}+C|u(x)|^{(1-a)s}(|u(x)-\bar{u}|+|\bar{u}-u(\xi)|)^{as}.
|h_{\lambda,K'}- h_{\lambda, F}| = \bigl| \bigl[1-\mbox{$\frac{1}{\lambda}$}d(x,K')\bigr]^+ - \bigl[1-\mbox{$\frac{1}{\lambda}$}d(x,F)\bigr]^+| & \leq \bigl| \bigl[1-\mbox{$\frac{1}{\lambda}$}d(x,K')\bigr] - \bigl[1-\mbox{$\frac{1}{\lambda}$}d(x,F)\bigr] \bigr| & = \mbox{$\frac{1}{\lambda}$}|d(x,K')- d(x,F)| \leq \mbox{$\frac{1}{\lambda}$}\cdot d_H(K',F).
\langle{\cal W}( 1 ){\cal W}( 2 ) \rangle |_{\theta = 0}= \langle \phi ( 1 ) \phi ( 2 ) \rangle = \frac{z_{1 i}\delta^{i j}z_{2 j}}{x_{1 2}^{2}}\equiv \frac{( 1 2 )}{x_{1 2}^{2}}= \frac{( 2 1 )}{x_{2 1}^{2}}.
\partial_t \partial_s \Sigma(t,s) = L_0'(s) + k\ \frac{L_1'(s)}{L_1(s)^2}\ \left[ \frac{L_1(s)}{1 + t L_1(s)}- L_1(s) \right] = L_0'(s) - kt\ \frac{L_1'(s)}{1 + t L_1(s))}< 0
\mathcal H^{n-1}(\{\psi_\lambda =0 \}\cap B_{r}(p)) \leq \mathcal H^{n-1}(\{\psi_\lambda =0 \}\cap Q_r(p)) = \frac{1}{2r}\mathcal H^{n}(\{H =0 \}\cap \tilde Q_r (\tilde p)) & \leq \frac{\kappa}{2r}(2r)^n N(H, \tilde Q_r(\tilde p))^{2\alpha}& = \kappa ' r^{n-1}N(H, \tilde Q_r(\tilde p))^{2\alpha}
\, \, \equiv \, \, \, K_{1 2}^{T + +}( s_{1 2}, s_{1 2}, P_{0}\! - \! h_{3}) ( 1 - G_{1}^{0}G_{2}^{0}K_{1 2}^{T + +}( s_{1 2}, s_{1 2}, P_{0}\! - \! h_{3}) )^{- 1}.
\Gamma_{c}= \left( \begin{array}{c c c}{{\displaystyle{\gamma_{c b}^{a}}}}&{{\displaystyle{{\frac{1}{2}T_{g}^{a}{_c}}}}}\\{{\displaystyle{- \frac{{\dot{g}}_{a c}}{\sin^{2}\theta}}}}&{{\displaystyle{\mathrm{c o t}\theta \partial_{c}\theta}}}\\ \end{array}\right)_{a , b , c , = 1 \cdots M}
\delta\int_{S_{\mathcal{I}_{t-2}},y_{t-1}\leq -M}{y^2_{t-1}p(\mathcal{I}_{t-1}|z_t)d\mathcal{I}_{t-1}}\leq \delta \int_{S_{\mathcal{I}_{t-2}},-\infty<y_{t-1}<0}{y^2_{t-1}p(\mathcal{I}_{t-1}|z_t)d\mathcal{I}_{t-1}}&=\delta\frac{E_{t-1}[y^2_{t-1}|z_t]}{2}
L(\hat{A}^{tot}_\mu (\vec{X},t), \vec{X}, \dot{\vec{X}}, \vec{P},\dot{\vec{P}}) = L_{DH}(A_\mu^{CS}(\vec{\eta},t) + A_\mu^{ext}(\vec{\eta},t)\ , \vec{\eta}, \dot{\vec{\eta}}, \vec{\cal{P}},\dot{\vec{\cal{P}}})
\frac{\partial}{\partial t}\rho(x,p;t)=\langle\Psi(t)|x,p\rangle\frac{\partial}{\partial t}\langle x,p|\Psi(t)\rangle+\langle x,p|\Psi(t)\rangle(\frac{\partial}{\partial t}\langle x,p|\Psi(t)\rangle)^{*}
g_{\mu \nu}^{( 1 )}= -{\frac{l^{2}}{d - 2}}\left({\frac{1}{a^{2}}}- 1 \right) \left( R_{\mu \nu}-{\frac{1}{2 ( d - 1 )}}h_{\mu \nu}R \right) -{\frac{2 l}{d}}\left({\frac{1}{a^{d}}}- 1 \right) \chi_{\mu \nu}\ ,
\epsilon^{(j)}_n \geq 1 - \frac{H_2(\epsilon_n^{(j)})}{\log M_j}- \frac{nI(U_j;Y_j|U_{j+1})}{\log M_j}&\geq 1 - \frac{H_2(\epsilon_n^{(j)})}{(1+\delta)nI(U_j;Y_j|U_{j+1})}- \frac{1}{1+\delta}&=1- \frac{H_2(\epsilon_n^{(j)})+nI(U_j;Y_j|U_{j+1})}{(1+\delta)nI(U_j;Y_j|U_{j+1})}\,.
\langle y_{n}, z_{0}^{\ast}\rangle = \langle x_{n}, z_{0}^{\ast}\rangle + \langle y_{0}, z_{0}^{\ast}\rangle \frac{\delta}{3(1 + |\gamma| + |\langle y_{0}, z_{0}^{\ast}\rangle|)}\geq \alpha_{n}+ \frac{2 \delta}{3}
\beta_{o}^{\mu}= \sum_{e \neq 0}\beta_{e}^{\mu}U_{- e , o}\, \, , \quad \beta_{o}^{b}= \sum_{e \neq 0}\beta_{e}^{b}U_{- e , o}\, \, , \quad \beta_{o}^{c}= \sum_{e \neq 0}U_{o , - e}^{- 1}\beta_{e}^{c}\, \, .
\tilde{\Phi}_v^{\epsilon}(x)= \chi_l^{-1}(\alpha+\frac{1}{2}l^{2v}x+\epsilon l^v(\alpha x+\frac{1}{4}l^{2v}x^2)_{\frac{1}{2}})\psi_l(-\frac{1}{l^c}(\frac{1}{2}l^vx+\epsilon(\alpha x+\frac{1}{4}l^{2v}x^2)_{\frac{1}{2}})).
\nabla\phi(x,y)=\begin{pmatrix}1-4\tau\delta M+2\tau \sum_{j=1}^M\chi_{\{|g_j(y)-x|< \delta\}}-2\tau \sum_{j=1}^Mg_j'(y)\chi_{\{|g_j(y)-x|<\delta\}}-2\tau \sum_{i=1}^Nf_i'(x)\chi_{\{|f_i(x)-y|<\delta\}}1-4\tau\delta N+2\tau \sum_{i=1}^N\chi_{\{|f_i(x)-y|< \delta\}}\end{pmatrix}.
\mathcal L_\alpha b_{\kappa, p}(x,t) =(\partial_t u)(B_{\kappa,p}(u(x,t)))(\partial_t u)^T &+\sum_{j=1}^d(\partial_{x_j}u)(B_{\kappa,p}(u(x,t)))(\partial_{x_j}u)^T &+\sum_{j=1}^d\partial_{\eta_j}B_{\kappa,p}(u(x,t))\frac{2\alpha_j}{x_j^2}\mathbb{P}_t^{\alpha,j}({g_j})(x),
\omega_H(\mathbf{c}) = \omega_H((\mathbf{u}G'\ |\ \mathbf{u}A\ |\ \mathbf{u}\mathbf{a_1}\ |\ \mathbf{u}\mathbf{a_2}\ |\ \cdots\ |\ \mathbf{u}\mathbf{a_{i}})) \geq \omega_H((\mathbf{u}G'\ |\ \mathbf{u}A)) \geq d,
N_{P_i}(r)= \int_{S(r)}\log |P_i|\eta + O(1) \le \int_{S(r)}\log (|(f,a_i)|^2+|(f,a_{\sigma (i)}|^2)^{\frac{1}{2}}\eta + \int_{S(r)}\log (|(g,a_i)|^2+|(g,a_{\sigma (i)}|^2)^{\frac{1}{2}}\eta +O(1) \le T_f(r)+T_g(r) +o(T_f(r)).
e'''_k:=\mathcal{L}'(S_{\theta_k}{\mathbf V}_k,S_{\theta_k}\Psi_k)(\delta{\mathbf V}_k,\delta\Psi_k)-\mathcal{L}'({\mathbf V}_{k+\frac{1}{2}},\Psi_{k+\frac{1}{2}})(\delta{\mathbf V}_k,\delta\Psi_k),
0 =\det(B(|\xi|;\rho,\theta)-\lambda I_{4\times 4}) &=\lambda^4-2\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\lambda^3+\left(\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)^2+\left(a^2+b^2\right)|\xi|^2\right)\lambda^2 &\quad-\left(a^2+b^2\right)|\xi|^2\left(|\xi|^{2\rho}+|\xi|^{2\theta}\right)\lambda+a^2b^2|\xi|^4.
\displaystyle{F_{\mu \nu}^{+ I}\longrightarrow \tilde{F}_{\mu \nu}^{+ I}= U_{J}^{I}\, F_{\mu \nu}^{+ J}+ Z^{I J}\, G_{\mu \nu J}^{+}\; \; G_{\mu \nu I}^{+}\longrightarrow \tilde{G}_{\mu \nu I}^{+}= V_{I}^{J}\; G_{\mu \nu J}^{+}+ W_{I J}\, F_{\mu \nu}^{+ J}},
\frac{1}{r_1^2}\int_{\{\Psi_1\le2\log r_1\}\cap D_j}|f|^2 = \lim_{p\rightarrow1+0}\frac{1}{r_1^2}\int_{\{p\Psi_1<2\log r_1\}\cap D_j}|f|^2 \ge \limsup_{p\rightarrow1+0}G_{j,p}(0) \ge C_j
\Vert u_j\Vert^2=\frac{\prod\limits_{\ell}(\rho_j^2-\sigma_\ell^2)}{\prod\limits_{\ell\neq j}(\rho_j^2-\rho_\ell^2)}\ ,\ \Vert u'_k \Vert^2=\frac{\prod\limits_\ell (\rho_\ell^2-\sigma_k^2)}{\prod\limits_{\ell\neq k}(\sigma_\ell^2-\sigma_k^2)}\ .
\begin{array}{c}\nabla g=0, c_{ijk}+c_{ikj}=0, g_{ij, k}+g_{jk, i}+g_{ki, j}=0, w_{ij}+w_{ji}=0, w_{ij, l}-c^s_{ij}w_{sl}=0, c_{nml,k}+c^s_{ml}c_{snk}+w_{ml}w_{nk}=0.\end{array}
\frac{A^m B^{n+m}}{C^{n+2m}}= \frac{1}{r^{2m(n+m)}}(-1)^{n}\frac{\sin^m(\frac{m\theta}{2}) \sin^{n+m}(\frac{(n+m)\theta}{2})}{\sin^{n+2m}(\frac{(n+2m)\theta}{2})},
\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\exp\left({-\frac{x}{n^2}}\right) = \sqrt{\frac{\pi}{x}}\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\exp\left(- \frac{\pi^2}{n^2 x}\right)- \frac{1}{2 \sqrt{\pi}}\sum_{\rho}\left( \frac{\pi}{\sqrt{x}}\right)^{\rho}\frac{\Gamma\left(\frac{1-\rho}{2}\right)}{\zeta'(\rho)},
\begin{aligned}D^{\tau}_tD^{\zeta}_z(x_ia_{ii}u_{x_ix_i}),D^{\tau}_tD^{\zeta}_z(x_ix_j\tilde a_{ij}u_{x_ix_j}),D^{\tau}_tD^{\zeta}_z(b_iu_{x_i}), &D^{\tau}_tD^{\zeta}_z(x_ic_{il}u_{x_iy_l}),D^{\tau}_tD^{\zeta}_z(d_{lp}u_{y_ly_p}),D^{\tau}_tD^{\zeta}_z(e_lu_{y_l})\end{aligned}
\pi_M({\vec e}_k)=\sum_{a=1}^p \lambda_{k,a}{\vec v}_a=\sum_{a=w(k)}^p \lambda_{k,a}{\vec v}_{a}=\frac{1}{2}{\vec v}_{w(k)}+\sum_{a=w(k)+1}^p \lambda_{k,a}{\vec v}_{a}.
\ \left\langle \hat f(t),B^* \mathbb{T}^*_{\tau-t}\eta \right\rangle_U = \displaystyle\max_{v\in U,\; \|v\|_U \leqslant N^*(\tau)}\ \left\langle v,B^* \mathbb{T}^*_{\tau-t}\eta \right\rangle_U.
\frac{f_{n}(x_{1},...,x_{n}|K_{n})}{\prod_{m=1}^{n}f_{1}(x_{m})}=\sum_{S}(\prod_{1\leq i<j\leq n}\frac{\left( \rho_{ij}\right)^{s_{ij}}}{s_{ij}!}\prod_{m=1}^{n}H_{\sigma_{m}}(x_{m})),
d_U^{[0]}f=f\ \ \ \ \ \d_U^{[p+1]}f(\underline{z},\dots,\underline{z}^{(k)})=\sum_{m=0}^p\sum_{i=1}^n\frac{\partial d_U^{[p]}f}{\partial z_i^{(m)}}z_i^{(m+1)}\ \ \ \ \ 0\leq p<k,
\tau_1 =\frac{1}{2\rho_1}=\frac{\sqrt{2\sqrt3+3}}{\sqrt3}=\sqrt{\frac{2}{\sqrt3}+1}\doteq 1.467890, \tau_{2}= \frac{1}{2 \rho_{2}}= \frac{1}{7}\sqrt{57+40 \sqrt{2}+2 \sqrt{1635+1168 \sqrt{2}}}\doteq 2.158182 ,
\frac{1}{2|\Omega|}\sum_{r=1}^{R}k_ru_{\infty}^{y_r}\left(\sum_{i=1}^{N}(y_{r,i}' - y_{r,i})\frac{\overline v_i}{u_{i,\infty}}\right)^2 \geq \beta \sum_{i=1}^{N}\frac{\overline v_i^2}{u_{i,\infty}}.
[ T_{i}, T_{j}] = - \epsilon_{i j k}\mathsf{Q}^{k l}T_{l}\, , \, \, \, \, i , j , \cdots = 1 , 2 , 3 , \, , \hspace{1 c m}\mathsf{Q}= \mathrm{d i a g}\, ( + + - ) \, ,
\int_{0}^{\infty}d r ~ r^{\frac{n}{2}}~ ( c^{2}+ r^{2})^{m}~ J_{\frac{n - 2}{2}}( r | p | ) ~ = ~ ( 2 / | p | )^{m + 1}~ \frac{c^{\frac{n}{2}+ m}}{\Gamma ( - m )}~ K_{\frac{n}{2}+ m}( c | p | ) ,
{\cal K}^{T}M{\cal K}= \left( \begin{array}{c c}{{\cal K}^{( 1 )}}&{{\tilde{\cal K}}^{( 1 )}}\\ \end{array}\right) e^{\varphi}\left( \begin{array}{c c}{e^{- 2 \varphi}+ C^{( 0 ) 2}}&{C^{( 0 )}}\\{C^{( 0 )}}&{1}\\ \end{array}\right) \left( \begin{array}{c}{{\cal K}^{( 1 )}}\\{{\tilde{\cal K}}^{( 1 )}}\\ \end{array}\right) ,
\big\langle \mathfrak{Im}\langle x,\cdot\rangle,\mathfrak{Re}\langle y,\cdot \rangle\big\rangle_{L^2(m)}=-\big\langle \mathfrak{Re}\langle x,\cdot\rangle,\mathfrak{Im}\langle y,\cdot \rangle\big\rangle_{L^2(m)}=\frac{1}{2}\mathfrak{Im}\langle Rx,y\rangle.
\left< V{x},\sum\limits_{i=1}^{n}a_i\otimes{y_i}+K\right> =\left< 1\otimes U_1{x}+K,\sum\limits_{i=1}^{n}a_i\otimes{y_i}+K\right> =\left< U_1{x},\sum\limits_{i=1}^{n}\tau(\alpha(1)a_i){y_i}\right> &=\left<{x},\sum\limits_{i=1}^{n}U^*_1\tau(a_i){y_i}\right>=\left<{x},\sum\limits_{i=1}^{n}\tau(a_i)U^*_1{y_i}\right>.
SRO_{\gamma}(\Phi) = \frac{\tau(H^{\sigma}\times H^{\sigma})}{\tau(H_{\gamma}) d(H_{\gamma},H^{\sigma}\times H^{\sigma})}\sum_{\kappa_H}RO^{\kappa_H}_{\gamma}(\Phi) &=\frac{\tau(H^{\sigma}\times H^{\sigma})}{\tau(H_{\gamma}) d(H_{\gamma},H^{\sigma}\times H^{\sigma})}\sum_{\kappa_H}\prod_{v}RO^{\kappa_{Hv}}_{\gamma_v}(\Phi_v)
\sup_{n \ge 1}\frac{\,1\,}{\,\sqrt{n}\,}\sum_{i=1}^{n}\mathbb E [ \sup_{0 \le t \le T}\lvert X_{t, i}^{(u)}- \overline{X}_{t,i}\rvert] \le2C_{T}e^{2C_{T}T}\sup_{n \ge 1}\big( \frac{\, 2 T\,}{\sqrt{n}}( \mathbb E [ \lvert \overline{X}_{0,1}\rvert] + c) e^{c T}+ c \big) < \infty \, .
\begin{array}{r c l}\frac{df}{dt}= \frac{\partial f}{\partial t}+\{f,H_T\}&= \frac{\partial f}{\partial t}+\{f,H_T\}_D+\{f,\chi_s\}\left(\Delta^{-1}\right)^{ss'}\{\chi_{s'},H_T\}\approx\frac{\partial f}{\partial t}+\{f,H_T\}_D\ .\end{array}
L =\left(\begin{tabular}{rrr}1 0 0 $\frac{2}{3}$ 1 0 $\frac{1}{3}$ $\frac{1}{2}$ 1\end{tabular}\right)\textrm{and}D = \frac{1}{12}\left(\begin{tabular}{rrr}9 0 0 0 8 0 0 0 6\end{tabular}\right) .
\lbrace A,B\rbrace{}^{*}=\lbrace A,B\rbrace -i[\lbrace A,\pi_{\theta \, i}+{i\over 2}\theta^{*}_i\rbrace \lbrace \pi_{\theta^{*}\, i}+{i\over 2}\theta_i,B\rbrace +\lbrace A,\pi_{\theta^{*}\, i}+{i\over 2}\theta_i \rbrace \lbrace \pi_{\theta \, i}+{i\over 2}\theta^{*}_i,B\rbrace ]
t_{\epsilon,\alpha}\ln\frac{e}{t_{\epsilon,\alpha}}=\frac{x}{4\ln (1/x)}\ln\frac{4e\ln(1/x)}{x}< \frac{9-4\sqrt{2}}{4}x=\frac{9-4\sqrt{2}}{16}(1-\sqrt{\alpha})\epsilon\le f_{\epsilon,\alpha}^2.
\|a-\hat a\|_{l_1^d}\leq2(1+C_0)\cdot\frac{g'(0)\cdot\sigma_s^d(a)_1+\|u\|_{l_1^d}}{\|\tilde a\|_{l_1^d}}&\leq 2\sqrt{2}(1+C_0)\cdot\frac{g'(0)\cdot\sigma_s^d(a)_1+c_1h}{\|\tilde a\|_{l_1^d}}.
\widehat{R}(e_{i}\otimes e_{j}+ q^{-1}e_{j}\otimes e_{i}) = e_{j}\otimes e_{i}+ (q - q^{-1}) e_{i}\otimes e_{j}+ q^{-1}e_{i}\otimes e_{j}& = q(e_{i}\otimes e_{j}+ q^{-1}e_{j}\otimes e_{i}).
\|\mathbb{T}\|_\mathbb{A}=\frac{1}{\sqrt{2}}\sqrt{2+\|T\|_A^2+\sqrt{\|T\|_A^4+4\|T\|_A^2}}=\frac{1}{2}\sqrt{\|T\|_A^2+4}+\frac{1}{2}\|T\|_A.
\prod_{\substack{\gcd(j_1,j_2,j_3,k)=1 j_1,j_2,j_3<k j_1,j_2,j_3 \geq 1; k \geq 2}}\left(\frac{1}{1-y^{j_1+j_2+j_3}z^k}\right)^{\frac{1}{k}},
\lim_{s\rightarrow0}z^{s}\Gamma(s)\left( \frac{1}{\left( dz-c\right)^{s}}G\left( V\left( -1/z\right) ,s;A,B\right) -z^{-s}G\left(-1/z,s;B_{-b},A_{-c}\right) \right) & =2\pi is\left( d,c;B_{b},A_{c}\right) -\pi iz\frac{B_{0}(B)B_{2}(A)}{c\left( dz-c\right)}-2\pi i\frac{dz-c}{cz}B_{0}(A)P_{2}\left(0,B_{b}\right)
C_T \frac{\Gamma(c) \Gamma(a-b)}{\Gamma(c-b) \Gamma(a)}(\alpha-1)^{b-a}2^{a-b}= \frac{\Gamma(c) \Gamma(b-a)}{\Gamma(c-a)\Gamma(b)}\left( C_R + \frac{\Gamma(2-c)\Gamma(c-a)}{\Gamma(c)\Gamma(2-c-a)}\right).
\pi^2/6 =[[0,2n-1],[2,n^4]]=[[1,6n-3],[2,n^4]]=[[5/4,10n-5],[2,n^4]] =[[49/36,14n-7],[2,n^4]]=[[205/144,18n-9],[2,n^4]] =[[5269/3600,22n-11],[2,n^4]]\;,
f(s)=2+e^{-3s}(s+1)\left(1-\frac{s^2}{2}-\frac{s^3}{3}-\frac{s^4}{8}\right)+e^{-2s}\left(2s+\frac{5s^2}{2}+\frac{s^3}{2}\right)-e^{-s}\left(3+2s\right).
\begin{array}{c}\lbrack e_{i}, f_{j}\rbrack = \delta_{ij}\frac{k_{i}- k_{i}^{-1}}{Q_{i}-Q_{i}^{-1}}k_{i}e_{j}k_{i}^{-1}= Q_{i}^{a_{ij}}e_{j},~~k_{i}f_{j}k_{i}^{-1}= Q_{i}^{-a_{ij}}f_{j}k_{i}k_{i}^{-1}=k_{i}^{-1}k_{i}=1,~~k_{i}k_{j}= k_{j}k_{i}\lbrack e_{i}, e_{i}\rbrack = \lbrack f_{i}, f_{i}\rbrack = 0, \mbox{if}~~~a_{ii}= 0 \end{array}
E_{k-l,N}^{\mathbf{1}, \overline{\alpha_N}}(z,s) = \frac{(k-l-1)!N^{k-l}}{(-2\pi i)^{k-l}G(\overline{\alpha})}\sum_{(c,d)\neq(0,0)}\frac{\alpha_N(d)}{(cNz+d)^{k-l}|cNz+d|^{2s}}.
\begin{aligned}p_{[1^07^1]}\bigg(7^{2r}m+\frac{7-7^{2r+1}}{24}\bigg) \equiv 0\pmod{7^{r}}, p_{[1^47^{-3}]}\bigg(7^{2r}m+\frac{17\cdot7^{2r}-17}{24}\bigg) \equiv 0\pmod{7^{r}}, p_{[1^87^{-7}]}\bigg(7^{2r}m+\frac{41\cdot7^{2r}-41}{24}\bigg) \equiv 0\pmod{7^{r}}.\end{aligned}
i \sum_{n = 1}^{\infty}{\frac{1}{n}}\Bigl [ e^{i{\frac{n}{L}}( r + \ell )}- e^{i{\frac{n}{L}}( r - \ell )}\Bigr ] ={\frac{1}{L}}- i \ln{\frac{\sin{\frac{r + \ell}{2 L}}}{\sin{\frac{r - \ell}{2 L}}}}, \quad 0 < r - \ell \le r + \ell < 2 \pi L .
\beta(1+Dr^m)(1+Er^m)^{\frac{D-2E}{E}}+(1-\beta)r^m(1+Er^m)^{\frac{D-E}{E}}&+\sum_{n=N}^{\infty}\prod_{k=0}^{N-2}\frac{|E-D+Ek|}{k+1}r^n =(1-E)^{\frac{D-E}{E}},
\frac{d}{dt}|D \varphi_t (\omega,x)v|^2 = 2 \left\langle Db ( \varphi_t(\omega,x)) D\varphi_t (\omega,x)v, D\varphi_t (\omega,x)v \right\rangle &\leq 2 \left( 1 - \left| \varphi_t(\omega,x)\right|^2 \right) | D\varphi_t (\omega,x)v|^2.
I(\alpha^{(0)},\alpha^{(1)})=\dfrac{\sum\limits_{l_0=2}^N\sum\limits_{l_1=2}^NA(l_0,l_1)\alpha_{l_0}^{(0)}\alpha_{l_1}^{(1)}}{\sum\limits_{l_0=2}^N\sum\limits_{l_1=2}^NB(l_0,l_1)\alpha_{l_0}^{(0)}\alpha_{l_1}^{(1)}},
\left(2^{u_{i_1}}+\cdots+2^{u_{i_M}}\right)^n\left(2^{u_{j_1}}+\cdots+2^{u_{j_N}}\right)^n=\left(\sum_{(p, q)\in\{1, \ldots, M\}\times\{1, \ldots, N\}}2^{u_{i_p}+u_{j_q}}\right)^n
f(x,y,z,t) = \dfrac{1}{\sqrt{(4 \pi D_\theta t)^3}}\exp \bigg( -\dfrac{(x + G_x - W_x t)^2 + (y + G_y - W_y t)^2 + (z + G_z - W_z t)^2}{4 D_\theta t}\bigg),
\mathbb{E}\big(\widetilde{\nu}^{\widetilde{S}}(S)^{p}\big)^{\frac{1}{p}}\ \le\ \mathbb{E}\big(\widetilde{\nu}_{n}^{\widetilde{S}}(S)^p\big)^{\frac{1}{p}}+\sum_{m=n}^\infty \mathbb{E}\big(|\widetilde{\nu}_{m+1}^{\widetilde{S}}(S)-\widetilde{\nu}_{m}^{\widetilde{S}}(S)|^p\big)^{\frac{1}{p}}.