harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sqrt{\lambda_k}\langle \tilde u_j,u_k\rangle= \tilde x\frac{\lambda_k}{\tilde \lambda_j-\lambda_k}\big(\sqrt{\lambda_j}\langle\tilde u_j,u_j\rangle+\sum_{l\neq j}\sqrt{\lambda_l}\langle\tilde u_j,u_l\rangle\big),
\varepsilon_{\frac{1}{2}}\left(x\right) =\frac{1}{1+\sqrt{c_{z}}x}\frac{1}{\Gamma(1+z)}\Big\vert_{z=0}= \sum_{r=0}^{\infty}\frac{\left(-x\right)^{r}}{\Gamma\left(\frac{r}{2}+1\right)}.
I_{k}\left( \mu ,d,\lambda \right) :=-\frac{\left( 2\mu \right)^{\frac{d}{2}-1}}{\left( 2k\lambda \right)^{\frac{d}{2}}}\Gamma \left( \frac{d}{2}+k\right) W_{-k,\frac{d-1}{2}}\left( 4\mu k\lambda \right) .
S = \int d^{2}x \hspace{1 m m}\mathrm{t r}\left( - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}D_{\mu}\phi D^{\mu}\phi + i \bar{\Psi}\gamma^{\mu}D_{\mu}\Psi - 2 i g \phi \bar{\Psi}\gamma_{5}\Psi \right) ,
\frac{\tilde \ell_i}{i m_i}= \frac{{m_i}^{\rho(\varphi(m_i)/\log m_i)}\varphi(m_i) \log m_i}{i m_i}\ge \frac{\varphi(m_i) \log m_i}{i}&> \frac{\varphi(m_1)}{i}\sum_{j=i_0}^{i-1}\log \log m_j \ge \frac{\varphi(m_1)}{i}\sum_{j=i_0}^{i-1}\log \log j &\ge \frac{\varphi(m_1)}{i}\int_{i_0}^{i-1}\log \log x \, \mathrm{d}x \to \infty
C=\begin{pmatrix}c(0) c(q-1) \cdots c(2) c(1) c(1) c(0) c(q-1) c(2) \vdots c(1) c(0) \ddots \vdots c(q-2) \ddots \ddots c(q-1) c(q-1) c(q-2) \cdots c(1) c(0) \end{pmatrix}.
f_{1}( L , m , d ) = \frac{1}{2 ( 2 \pi )^{d - 1}L}\sum_{n = - \infty}^{\infty}\int d^{d - 1}p \frac{1}{( \vec{p}^{\, 2}+ ( \frac{n \pi}{L})^{2}+ m^{2})}
\left| \sigma_n^{(p+1)}\right| \le 2R^{n+p+3}\frac{(n+2+p)!}{(n+2)!}+ nR^{n+p+3}\frac{(n+p+2)!}{(n+2)!}& = R^{n+p+3}\frac{(n+p+2)!}{(n+1)!},
\begin{array}{rclcrcl}H_{17}(r) = 1+\frac{\vert g^{17}\vert}{r}; H_{24}(r) = 1+\frac{\vert g^{24}\vert}{r}H_{18}(r) = 1+\frac{\vert e_{18}\vert}{r}; H_{23}(r) = 1+\frac{\vert e_{23}\vert}{r}\end{array}
\Theta_{n}^{+}=\{\omega_{1}+\omega_{n-1}=\varepsilon_{1}-\varepsilon_{n},\ \omega_{1}=\varepsilon_{1},\ \omega_{n-1}=-\varepsilon_{n}, \ \ \ \ \ \ 2\omega_{1}=2\varepsilon_{1},\ 2\omega_{n-1}=-2\varepsilon_{n},\ \omega_{2}=\varepsilon_{1}+\varepsilon_{2},\ \omega_{n-2}=-\varepsilon_{n-1}-\varepsilon_{n},\ 0\}.
\psi_{\nu}(R_{\nu}^{-1}\d 'B^+_r(z))=\psi_{\nu}(\d 'B^+_{R_{\nu}^{-1}r}(R_{\nu}^{-1}z))\subset D_{2R_{\nu}^{-1}r}(\psi_{\nu}(R_{\nu}^{-1}z))\,.
(\widehat\rho^{(\vec m)}_i)_Z\circ\varpi(g_1,\dots,g_n)\circ\varpi(f_1,\dots,f_n) = g_i\circ(\widehat\rho^{(\vec l)}_i)_Y\circ\varpi(f_1,\dots,f_n) &= g_if_i\circ(\widehat\rho^{(\vec k)}_i)_X\ ,
\dfrac{1}{2}\dfrac{d}{dt}\| \vec U \|_{P \otimes P_{\Omega}}^2 + \sum_{j=1,N}\lbrack \vec W^T \Lambda \vec W + 2 (\vec W^-)^T \Sigma(\sqrt{|\Lambda^-|}\vec W^- - R \sqrt{\Lambda^+}\vec W^+ - S \vec G) \rbrack_j ds_j=0.
m_{p h y s .}^{2}= m^{2}+ g^{2}\delta m_{0}^{2}+ \frac{g^{2}}{3 2 \pi^{2}}\int_{0}^{1}d x \int_{{\cal M}_{0}^{2}}^{\infty}d{\cal M}^{2}\frac{f_{\lambda_{0}}^{2}}{{\cal M}^{2}- m^{2}}= m^{2}+ g^{2}\delta \tilde{m}_{0}^{2}\; .
J^{\mu}_{a}\stackrel{\rm d}{\equiv}\sum_{n=0}^{N-1}\left(\partial^{A, \mu\alpha_{1}\cdots\alpha_{n}}{\cal L}\right)\stackrel{\leftrightarrow}{\partial}_{\alpha_{1}\cdots\alpha_{n}}\left( N_{aA}- \partial_{\nu}\varphi_{A}\cdot X^{\nu}_{a}\right)+{\cal L}X^{\mu}_{a},
\left\{e^{\frac{1}{24}A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}(\widetilde{T_CM}-\widetilde{L_C}-8+W_i)-e^{\frac{1}{24}A_1}A_1\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}&=-264\left\{e^{\frac{1}{24}A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}.
\mathbf V_{h}^{b}:=\left\{\mathbf v^{b}\in (L^{2}(\Omega))^2:\hat{\mathbf v}^{b}(\xi,\eta)=\mathbf v^{b}|_{K}\circ F_{K}\in \{\xi^2-1,\eta^2-1\}^2,\ \K\in\mathcal{T}_{h}\right\}.
\left\lbrace\begin{tabular}{l l l l}$u(0,t) = 0$, $u(L,t) = h_1(t)$, $u_{x}(L,t) = h_2(t)$ in $(0,T)$, $v(0,t) = 0$, $v(L,t) = g_1(t)$, $v_{x}(L,t) = g_2(t)$ in $(0,T)$,\end{tabular}\right.
\langle 0 | \mathrm{T r}\, \lambda^{2}| 0 \rangle^{2}= \frac{2^{1 0}\pi^{4}}{5}\, e^{- 8 \pi^{2}/ g^{2}}\, \frac{M_{\mathrm{P V}}^{6}}{g^{4}}= \frac{1 4 4}{5}\, \Lambda_{G}^{6}\, ,
[P_{(H_{i},K_{i})}: O_{(H_{i},K_{i})}] = p_{i}^{n_{i}-1}{\prod_{\stackrel{1< k < \frac{p_{i}^{n_{i}}}{2}}{(k,p_{i})=1}}o_{p_{i}^{n_{i}}}(k)p_{i}^{n_{i}-1}n_{H_{i},K_{i}}}=p_{i}^{n_{i}-1}\mathfrak{o}_{i}.
\int_{\{2R<|x|\}}\left| \widetilde{B}_{Q/p}\left(\frac{x}{2}\right)\right|^{p^{\prime}}dx=\int_{\{2R<|x|<2\}}\left| \widetilde{B}_{Q/p}\left(\frac{x}{2}\right)\right|^{p^{\prime}}dx+\int_{\{|x|\geqslant 2\}}\left| \widetilde{B}_{Q/p}\left(\frac{x}{2}\right)\right|^{p^{\prime}}dx.
2\ell\left( m+1,\chi\right) =-\frac{\left( -\pi/2k\right)^{m+1}}{m!}\sum_{n=1}^{2k-1}\left( -1\right)^{n}\chi\left( n\right) \cot^{\left(m\right)}\left( \frac{\pi n}{2k}\right) ,m\geq0,
\int \prod_{\mu = 0}^{p}\left( \frac{d \bar{\lambda}_{\mu}d \lambda_{\mu}}{2 i}\right) \exp \left[ - \left( \bar{\lambda}_{\mu}+ \alpha_{\mu}\right) M^{\mu \nu}\left( \lambda_{\nu}+ \beta_{\nu}\right) \right] = \frac{\pi^{p + 1}}{\det M^{\mu \nu}}~ ,
\mathbf{v}_{i}(\rho, x, t) =\tilde{\mathbf{v}}_{i}(\rho, x, t)+\eta(\rho) d_{\Gamma}(x, t) \hat{\mathbf{v}}_{i}(x, t),i=0,1,2, p_{j}(\rho, x, t) =\tilde{p}_{j}(\rho, x, t)+\eta(\rho) d_{\Gamma}(x, t) \hat{p}_{j}(x, t), j=-1,0,1,
D^{\mu}F_{\mu \nu}^{a}= e \varepsilon_{a b c}\phi^{b}D_{\nu}\phi^{c}+{\it F}_{\nu}^{a}; \qquad D^{\mu}D_{\mu}\phi^{a}= \lambda ( \phi^{2}- a^{2}) \phi^{a}+{\it F}^{a}.
\langle \phi_{1}(z_{1},\bar{z}_{1})\phi_{2}(z_{2},\bar{z}_{2})\phi_{3}(z_{3},\bar{z}_{3})\rangle =C_{12}^{3}z_{12}^{\gamma_{12}}z_{13}^{\gamma_{13}}z_{23}^{\gamma_{23}}\bar{z}_{12}^{\bar{\gamma}_{12}}\bar{z}_{13}^{\bar{\gamma}_{13}}\bar{z}_{23}^{\bar{\gamma}_{23}}
\begin{cases}D^{\alpha_1}_0y_s(t) = w_s(t), y_s(0) = -b_1/a_1, D^{1-\alpha_1}_0w_s(t) = z_s(t), w_s(0) = 0, D^{\alpha_2-1}_0z_s(t) = f_s(t,y(t),w(t)), z_s(0) = 1.\end{cases}
H=(\begin{matrix}1&0&1&1&&0&1&0&0&&1&0&1&1\\ 0&1&0&0&&0&1&1&1&&0&1&0&0\\ 0&1&1&1&&0&1&0&0&&1&0&1&0\\ 1&1&0&0&&1&0&1&1&&0&0&0&1\end{matrix})
\nu^{-1}(\tilde \varepsilon )={4}\,{\tilde \varepsilon^{-1}}-8 \frac{n-4}{n-2}\,\tilde \varepsilon^{-2}+16 \frac{n-4}{n-2}\,\tilde \varepsilon^{-3}-32 \left\{\left[52+108\zeta(3)\right]-\left[16+36\zeta(3)\right]n+n^2 \right\}\,\frac{\tilde \varepsilon^{-4}}{(n-2)^3}+\dots~.
u=\left(\begin{array}{c}u_1 u_2 u_3 \end{array}\right)\, , u_i\in{\rm H}^{1}(\Omega),\P=\left(\begin{array}{ccc}P_{11}P_{12}P_{13}P_{21}P_{22}P_{23}P_{31}P_{32}P_{33}\end{array}\right)\, P^Te_i\in{\rm H}({\rm curl}\,; \Omega)\, \P^Te_i\in{\rm H}({\rm div}\,; \Omega)
\sum_i \tilde\alpha_i^{\prime2}- \tilde\alpha_i'\sum_j\tilde\alpha_j' +{1\over D-2}\varphi^{\prime2}+ \sum_a \alpha_a^{\prime2}={Q^2 \over 2}\exp\left(-\tilde\Phi+2\sum_k \tilde\alpha_k \right) \ .
k^{i}\cdot k^{j}= - \omega^{2}b^{i}b^{j}c o s ( \theta^{i}- \theta^{j}) , \qquad \qquad \varepsilon^{i}\cdot k^{j}= - \omega b^{j}s i n ( \theta^{i}- \theta^{j}) \qquad \varepsilon^{i}\cdot \varepsilon^{j}= - c o s ( \theta^{i}- \theta^{j})
\sum_{k=0}^{n-1-(n+r)/d}\frac{(q^{d+r};q^d)_k^{d-r-1}(q^{d+r};q^d)_{k-2}^{r+1}(1-q^{dk-d+r})^rq^{dk}}{(q^d;q^d)_k^d}\equiv0\pmod{\Phi_n(q)^2}.
\gamma^{0}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \, \quad \gamma^{i}= \left( \begin{array}{c c}{{0}}&{{\sigma^{i}}}\\{{- \sigma^{i}}}&{{0}}\\ \end{array}\right) ,
\mathrm{gr}(\mu)\circ \mathrm{can}\circ \big(\mathrm{gr}\,E(\iota_V)\otimes \mathrm{gr}\, E(\iota_W)\big) = \mathrm{gr}(\mu)\circ \mathrm{gr}(E(\iota_V)\otimes E(\iota_W))\circ \mathrm{can}&= \mathrm{gr}(\phi)\circ \mathrm{can}
- \frac{1}{2 \pi i (u-v)}\left[ \left( e^{-i[ \theta_p(u) - \theta_p(v) + \mbox{arg}\, \Gamma(1+iu) - \mbox{arg}\, \Gamma(1+iv)]}- 1 \right){\left( \frac{a}{2}\right)}^{i(u-v)}\right.
\eqref{eq:M_3_V12_1}=\langle \Phi,U_N\sum_{j=1}^{N_1}\sum_{k=1}^{N_2}Q^{(1)}_{x_j}\otimes Q^{(2)}_{y_k}V^{(12)}(x_j-y_k)Q^{(1)}_{x_j}\otimes P^{(2)}_{y_k}U_N^*\,\Phi\rangle+\operatorname{h.c.}.
\begin{array}{l}{\tau_{1}\equiv \pi_{0}+ \eta^{1}\approx 0}\\{\tau_{2}\equiv E^{\prime}+ \phi^{\prime}+ \pi + A_{1}+ \eta^{2}\approx 0}\\{\tau_{3}\equiv E + \eta^{3}\approx 0}\\{\tau_{4}\equiv - \pi - \phi^{\prime}- 2 A_{1}+ A_{0}+ \eta^{4}\approx 0}\\ \end{array}
u_{\lambda,\nu}^A = \frac{(-1)^k k! \pi^{\frac{p+q}{2}}}{\Gamma(\frac{\nu+\rho'}{2})}\times \begin{cases}\frac{\Gamma(\frac{2\nu+p'}{4})}{\Gamma(\frac{2\nu+p'}{2})}u^C_{\lambda,\nu}\mbox{for $m>0$,}\frac{\Gamma(\frac{\nu}{2}-\lfloor\frac{k}{2}\rfloor)}{2^k\Gamma(\nu-k)}u^C_{\lambda,\nu}\mbox{for $m=0$.}\end{cases}
D_sY_t^{\theta}(t_k,x)=\sigma(Y_s^{\theta}(t_k,x))+\int_{s}^t\nabla_xb(\theta,Y_u^{\theta}(t_k,x))D_sY_u^{\theta}(t_k,x)du &+\sum_{i=1}^{d}\int_{s}^t\nabla_x\sigma_i(Y_u^{\theta}(t_k,x))D_sY_u^{\theta}(t_k,x)dW_{u}^i+\int_{s}^t\int_{\mathbb{R}_0^d}\nabla_xc(Y_{u-}^{\theta}(t_k,x),z)D_sY_u^{\theta}(t_k,x)\widetilde{M}(du,dz),
\delta_{\kappa}A^{i j}= \frac 1 2 S t r \left( \left( \begin{array}{c c}{0}&{\left( \not{X}_{k}\kappa^{( i}\right) \varepsilon^{j ) k}}\\{- \left( \tilde{\kappa}^{( i}\not{X}_{k}\varepsilon^{j ) k}\right)}&{0}\\ \end{array}\right) \partial_{\tau}t \left( \Theta \right) t^{- 1}\left( \Theta \right) \right) .
0<\int_{t_0}^t (t-\sigma) \Big( \lambda u(\sigma) f\big(\sigma,u(\sigma),\dot{u}(\sigma)\big) + \dot{u}(\sigma)\varphi\big(\dot{u}(\sigma)\big) \Big)\,d\sigma=\int_{t_0}^t u(\sigma) \varphi(\dot u(\sigma)) \,d\sigma\leq 0,
\frac{d P}{d \Omega_{8}}= 2 \kappa^{2}T_{0 i}\hat{k}_{i}={\frac{2 \kappa^{2}T_{0}}{1 0 5^{2}\, \Omega_{8}^{2}}}\omega^{8}\,{\frac{1}{3 !}}\, \left[ J^{i j k}J_{i j k}^{*}- 3 \, \hat{k}_{l}\hat{k^{m}}J^{l i j}J_{m i j}^{*}\right]
G(t;\zeta,Q)=Q^3-\frac{\zeta^2}{2}-Q\left(\frac{3v_2^2}{4}+\frac{\ddot{v}_2}{2}\right)-\frac{v_2^3}{4}-\frac{v_2\ddot{v}_2}{2}+\frac{\dot{v}_2^2}{2}\ ,
(16 \, \pi)^{\frac{3}{2}}\, F' (A) \, q^2 \, A^{\frac{1}{2}}& = \int_\Sigma (R + 6) + \int_\Sigma |\mathring h|^2 + 2 \int_\Sigma |\nabla_\Sigma \log u|^2 + \frac{1}{2}\, (q^2 - 1) \, (16 \, \pi + 12 \, A - 3 \, A \, H^2) & = O (A^{-\frac{3}{2}}) + O (A^{-2}) + 2 \, \int_\Sigma |\nabla_\Sigma \log u|^2 + (q^2 - 1) \, O (A^{- \frac{1}{2}})
\frac{n^{-2m}|K_n(z+\frac{u}{\sqrt{n}},z+\frac{v}{\sqrt{n}})|^2e^{-2n\varphi_n(z+\frac{u}{\sqrt{n}})}e^{-2n\varphi_n(z+\frac{v}{\sqrt{n}})}}{\det(dd^c(\varphi_n(z))^2\exp(-\sum_{j=1}^m\lambda_j^n|u_j-v_j|^2)}\to1
\Gamma^{( 1 )}[ G , B , \varphi ] = \int d^{D}z \sqrt{G}\exp ( - 2 \varphi ) \{R ( G ) + 4 \partial_{M}\varphi \partial^{N}\varphi - \frac{1}{1 2}H_{M N P}H^{M N P}+ \Lambda \}\, \, .
\delta_{t}U^{n}_{ij}-\mu\Delta_{h}U^{n-\frac{1}{2}}_{ij}- \mathfrak{w}_{n,1}\Delta_{h}U^{1}_{ij}-\sum_{m=2}^{n}\mathfrak{w}_{n,m}\Delta_{h}U^{m-\frac{1}{2}}_{ij}=b^{n}_{ij}+\frac{g(U^n_{ij})+g(U^{n-1}_{ij})}{2}, \qquad(x_i,y_j)\in\Omega_{h},2\leq n\leq \mathcal{N},
\mathrm{Fitt}_{\Lambda}( M ) = \mathrm{char}_{\Lambda}( M ) = \mathrm{char}_{\Lambda}( N ) \cdot \mathrm{char}_{\Lambda}( M/N ) = \mathrm{Fitt}_{\Lambda}( N ) \cdot \mathrm{char}_{\Lambda}( M/N ) \subseteq \mathrm{Fitt}_{\Lambda}( N )
\int_{\mathbb{R}^{2n}}\widetilde{U}(t,x) \, \mathrm{d}\gamma(x) = \frac{(2\pi)^{\frac{n}{2}(1 - \alpha_1 - \alpha_2)}}{(1-\rho^2)^{\frac{n}{2}}}\int_{\mathbb{R}^{2n}}u_1(\tau, L_1z)^{\alpha_1}u_2(\tau,L_1^\perp z)^{\alpha_2}u_3(\tau,L_3z)^{\alpha_3}\, \mathrm{d}z,
de_{1}=\nabla e_{1}+\sum h_{1j}^{k^{*}}\theta_{j}e_{k^{*}}=h_{11}^{1^{*}}\theta_{1}e_{1^{*}}\in V_{1}, &de_{1^{*}}=Jde_{1}=h_{11}^{1^{*}}\theta_{1}Je_{1^{*}}=-h_{11}^{1^{*}}\theta_{1}e_{1}\in V_{1},
\mathcal{U}_3=\int_{\kappa_{\rm b}}^{\kappa_{\rm t}}e^{-\frac{1}{2}\left(x+\frac{2\psi_{1}\tau_{\rm d}}{N_{\rm a}}\right)^2}\,\mathrm{d}x=\sqrt{{2\pi}}\left[Q\left(\kappa_{\rm b}+\frac{2\psi_{1}\tau_{\rm d}}{N_{\rm a}}\right)-Q\left(\kappa_{\rm t}+\frac{2\psi_{1}\tau_{\rm d}}{N_{\rm a}}\right)\right].
\omega^{(1)}= D^{\ell_1}(u-u_0) \mbox{with}\, 0 \le \ell_1 \le \ell+4, \omega^{(2)}\in \left\{u_0, D^{\ell_2}(u-u_0)\right\}\mbox{with}\, 0 \le \ell_2 \le \frac{\ell+4}{2}, \omega^{(j)}\in \left\{1, u_0, D^{\ell_j}(u-u_0)\right\}\mbox{with}\, 0 \le \ell_j \le \frac{\ell+4}{3}\, \mbox{for}\, j \ge 3.
\Vert \mathcal{L}_{\tau,s,\rho}\Vert_{\mathrm{HS}}^2 = \sum_{b\in \mathcal{A}}\sum_{\substack{\textbf{a},\textbf{b}\in Y(\tau) \textbf{a},\textbf{b}\to b}}\mathrm{tr}\left( \rho(\gamma_{\textbf{a}}^{-1}\gamma_{\textbf{b}}) \right) \mathcal{I}_{\textbf{a},\textbf{b}}^{(b)},
\psi = - 2^{-\frac{4}{3}}\frac{1}{2}\omega^2 + 2^{-\frac{4}{3}}dr^2 \wedge dr^3 \wedge \omega + 2^{\frac{2}{3}}dr^3 \wedge dr^1 \wedge \Re (\Omega) + 2^{\frac{2}{3}}dr^1 \wedge dr^2 \wedge \Im (\Omega) T^3 \times X^4,
\delta x = \dot{x}\xi \, \, , \, \, \, \, \, \, \delta e = \frac{d}{d t}( e \xi ) \, \, , \, \, \, \, \, \, \delta \psi^{n}= \dot{\psi}^{n}\xi \, \, , \, \, \, \, \, \, \delta \chi = \frac{d}{d t}( \chi \xi ) \, ,
0 =\langle p(v_i)+L(\gamma_i)-p(v), p(v_i)+L(\gamma_i)-p(v)\rangle -\langle q(v_i)+L(\gamma_i)-q(v),q(v_i)+L(\gamma_i)-q(v) \rangle &=\langle x_i-x_v, x_i-x_v\rangle - \langle y_i-y_v, y_i-y_v\rangle &=(\|x_i\|^2-\|y_i\|^2)-2\langle x_i, x_v\rangle +2\langle y_i, y_v\rangle,
\left|\int_{\mathbb{R}^{m}}|z|^{2}\left[U^{2_{m}^{*}-\varepsilon}(z)-U^{2_{m}^{*}}(z)\right]dz\right|\le|\varepsilon|\int_{\mathbb{R}^{m}}|z|^{2}U^{2_{m}^{*}-\theta\varepsilon}(z)\ln U(z)dz=o(1).
[{L}_{u^{\alpha ( i , j )}}] \Delta_{\nu}u^{\alpha}( x )^{( i , j )}+ \Sigma_{\mu = 1 , 2}\{\Delta_{\mu}E_{\mu}^{- 1}{{T}_{D}}_{\nu}^{\mu ( i , j )}+ \frac{\partial{{L}_{D}}^{( i , j )}}{\partial x^{\nu ( i , j )}}\}= 0 .
T_1 \le \Gamma(2m-4)\frac{\Gamma(m-1/2)\Gamma(m-7/2)}{\Gamma(2m-4)}\int_{0}^1y^{2}(1-y)^{m-3/2}\,dy &=\frac{\Gamma(m-7/2)\Gamma(3)\Gamma(m-1/2)^2}{\Gamma(m+5/2)}.
\sum_{\ell=0}^{k}\alpha_{k-\ell}U_h^{j-\ell}= h \sum_{\ell=0}^k \beta_{k-\ell}f(U_h^{j-\ell}) + \sum_{\ell=1}^k\gamma_{k-\ell}g( U_h^{j-\ell}) \Delta_h W^{j-\ell+1},
\begin{array}{c}M_{ABC}=\frac 1{12}(M^{ab}\left( \sigma_{ABC}\right)_{ba}+M_{ab}(\widetilde{\sigma}_{ABC})^{ba}) M^{ab}=M^{ABC}(\widetilde{\sigma}_{ABC})^{ab}\ ,\quad M_{ab}=M^{ABC}\left(\sigma_{ABC}\right)_{ab}\end{array}
\delta L(q, \dot{q})=\frac{1}{2}\dot{q}^{\mu}(\delta q^{\alpha}\partial_{\alpha}g_{\mu\nu}+g_{\mu\alpha}\partial_{\nu}\delta q^{\alpha}+g\_{\alpha\nu}\partial_{\mu}\delta q^{\alpha}) q^{\nu}++\frac{1}{4}\left[ \left[ \partial_{\alpha}\delta q^{\mu}, \dot{q}^{\alpha}\right] g_{\mu\nu}, \dot{q}^{\nu}\right]-\delta q^{\alpha}\partial_{\alpha}U_{q}(q)\,.
\begin{cases}u_t+u\cdot\nabla u+\nabla P=-\nabla\cdot\big(\nabla d\odot\nabla d\big)+\nabla \cdot(\sigma^L(u,d)), \nabla \cdot u=0, \lambda_1(d_t+u\cdot\nabla d-\Lambda d)+\lambda_2Ad=\Delta d+|\nabla d|^2d+\lambda_2(d^TAd)d,\end{cases}
\frac{f_w'(z)}{f_w(z)}= - \frac{V[g_w](z)}{g_w(z)}= \frac{g_1(w) V[g_2 ](z) - g_2(w) V[g_1] (z)}{g_2(w) g_1 (z) - g_1(w) g_2(z)}.
\alpha = \operatorname{c o s}{a}={\frac{\operatorname{c o s}{\frac{\mu_{2}}{2}}\operatorname{c o s}{\frac{\mu_{3}}{2}}- \operatorname{c o s}{\frac{\mu_{1}}{2}}}{\operatorname{s i n}{\frac{\mu_{2}}{2}}\operatorname{s i n}{\frac{\mu_{3}}{2}}}}
\begin{array}{rcll}B_{0}(t) = 1, B_{n+1}(t_{1}, \ldots, t_{n+1}) = \displaystyle\sum_{i=0}^{n}\binom{n}{i}B_{n-i}(t_{1}, \ldots, t_{n-i})t_{i+1}\left(n=0,1,\dots\right). \end{array}
- \frac{1}{M^{2}}[ C^{2}( \frac{6 f_{1}( s )}{c o s h^{2}s}+ \frac{\phi_{2}^{( 2 )}( s )}{c o s h^{6}s}) + 4 K_{a}^{b}K_{b}^{a}( \frac{f_{2}( s )}{c o s h^{2}s}+ f_{3}( s ) \phi_{2}^{( 2 )}( s ) ) ] ,
\begin{cases}\partial_t\boldsymbol u + \boldsymbol u\cdot \nabla \boldsymbol u + \nabla\pi = T \boldsymbol e_2,\boldsymbol e_2=(0,1),{\mathrm{div}\,}\boldsymbol u=0, \partial_t T - \kappa \Delta T + \boldsymbol u\cdot \nabla T =0,\end{cases}
\bigg{[}\mathop{\otimes}_{i=1}^{d}\mathcal{N}\bigg{(}\bigg{(}1-\frac{1}{(d/i)^{2k+1}+1}\bigg{)}x_{i},\varepsilon^{2}\bigg{(}1-\frac{1}{(d/i)^{2k+1}+1}\bigg{)}\bigg{)}\bigg{]}\otimes\bigg{[}\mathop{\otimes}_{i=d+1}^{\infty}\mathcal{N}(0,0)\bigg{]},
\tilde{\Phi}^{A}= \frac{\delta F ( \Phi , \tilde{\Phi}^{*})}{\delta \tilde{\Phi}_{A}^{*}}\hspace{2 c m}\Phi_{A}^{*}= \frac{\delta F ( \Phi , \tilde{\Phi}^{*})}{\delta \Phi^{A}}\ .
{\mathcal R}(x)\frac{\delta}{\delta\varphi(y)}S =-\frac{\delta F}{\delta\varphi(x)}\int dz\, \Delta_{S}^{\mathrm ret}(z,x) \frac{\delta^2 S}{\delta\varphi(x)\delta\varphi(z)}= -\frac{\delta F}{\delta\varphi(x)}\delta(x-y)\ .
\frac{1}{\prod_{m=1}^{\kappa(\ell_1, \ldots, \ell_j)}g_m!}\prod_{k=1}^j{\ell-\sum_{i=1}^{k-1}\ell_i \choose \ell_k}^2 \frac{\left( \ell_k ! \right)^2}{2 \ell_k}.
\mathcal{B}(s_1,s_2) = \mathcal{B}(0,s_2)+ \mathcal{B}^{(1,0)}(0,s_2)s_1 + \frac{1}{2!}\mathcal{B}^{(2,0)}(0,s_2)s_1^2 +\cdots \zeta(s_1+1)^{k}= s_1^{-k}(\alpha_{0,k}+ \alpha_{1,k}s_1 + \alpha_{2,k}s_1^2 + \cdots ), \alpha_{0,k}=1, X^{s_1}= 1 + (\log X) s_1 + \tfrac{1}{2}(\log X)^2 s_1^2 + \cdots
Y = \begin{pmatrix}Y_1 Y_2\end{pmatrix}, X = \begin{pmatrix}X_1 X_2\end{pmatrix}, A = \begin{pmatrix}A_{11}A_{12}A_{21}A_{22}\end{pmatrix}, Z = \begin{pmatrix}Z_1 Z_2\end{pmatrix}, W = \begin{pmatrix}W_1 W_2\end{pmatrix},
\alpha_{k,j}=\begin{cases}\sum\limits_{i=1}^{n}a_i\cdot\mathtt{b}^+(\phi_i)\geqslant1-c t_{k,j}=f^\sharp_{k,j}\sum\limits_{i=1}^{n}a_i\cdot\mathtt{b}^+(\phi_i)<-c \end{cases}
u(x)=\mathbb E_x u(X_0)=\mathbb E_xY^x_0 =\mathbb E_xg(X_{\tau_D})+\mathbb E_x\int_0^{\tau_D}f(X_t,u(X_t))\,dt+\mathbb E_xA^\mu_{\tau_D}&=P_D (g)(x)+R^Df(\cdot,u)(x)+R^D\mu(x)
I_p(f) \cdot I_q(g) = \sum_{r=0}^{p \wedge q}r! \cdot \binom{p}{r}\cdot \binom{q}{r}\cdot \frac{\sqrt{(p+q-2r)!}}{\sqrt{p! \cdot q!}}\cdot I_{p+q-2r}(f \ \widetilde{{\otimes}_r}\g).
(G_{\lambda}^{\mathrm{low}}(b'_+), \cdot)_{\lambda}=(G^{\mathrm{low}}(b_+).v_{w\lambda}, \cdot)_{\lambda}&=(v_{w\lambda}, \varphi(G^{\mathrm{low}}(b_+))(\cdot))_{\lambda}=f_{w\lambda}.\varphi(G^{\mathrm{low}}(b_+)).
\inf_{x \in \mathcal{X}}\ \sum_{i \in [m]}\max\{0, x_i - \overline{b}_i, x_i - \overline{b}_i + \Delta b_i - x_k + \overline{b}_k -\Delta b_k \}+ \Gamma (x_k - \overline{b}_k + \Delta b_k) \mathrm{s.t.}\x_k \geq b_k.
\widehat{M}= M_0\otimes M_0 D_1; \widehat{A}= A\otimes D_1^{-1}M_0+ M_0\otimes A_1 + A_2 \otimes D_2^{2}D_1^{-1}M_0 - C D_3\otimes D_2 C^T - D_3 C^T \otimes C D_2.
h\left(0\right) =\int_{0}^{1}\varphi\left(y\right)F\left(-y\right)dy,h\left(1\right)=\int_{0}^{1}\varphi\left(y\right)F\left(1-y\right)dy h'\left(0\right) =\int_{0}^{1}\varphi\left(y\right)F'\left(-y\right)dy,\quad\,\,\, h'\left(1\right)=\int_{0}^{1}\varphi\left(y\right)F'\left(1-y\right)dy;
N^{2}M \int_{0}^{M}{\frac{d^{3}k}{( 2 \pi )^{3}}}{\frac{k^{2}}{k^{2}+ \sigma^{2}}}={\frac{N^{2}}{2 \pi^{2}}}M \left[{\frac{1}{3}}M^{3}- \sigma^{2}M + \sigma^{3}\operatorname{a r c t a n}{\frac{M}{\sigma}}\right]
N_\lambda =\{x\in G|x\cdot yz=xy\cdot z \ \forall \y,z\in G\}N_\mu =\{y\in G|x\cdot yz=xy\cdot z \ \forall \x,z\in G\}N_\rho =\{z\in G|x\cdot yz=xy\cdot z \ \forall \x,y\in G\}
g_1(U,V)\xi =\mathcal{T}_{U}\phi V+\hat{\nabla}_{U}\phi V+\mathcal{T}_{U}\omega V+\mathcal{H}\nabla^{^{M_1}}_{U}\omega V &-\mathcal{B}\mathcal{T}_{U}V-\mathcal{C}\mathcal{T}_{U}V-\phi \hat{\nabla}_{U}V-\omega \hat{\nabla}_{U}V.
H_{\mathbb G}([\mu_m;f;x,e_m])= \left[\mu_m;\gamma_{\mathcal N}\left(\Theta(H)(f);\lambda_{\vec a,x^{-1}(1)}^{-1},\dots,\lambda_{\vec a,x^{-1}(m)}^{-1}\right);x,e_m\right]\ .
\sum_{k=0}^a\frac{(-a-p)_{p+k}(\frac12)_{p+k}}{(1)_{p+k}^2}\cdot \bigg(\frac{4z}{(1+z)^2}\bigg)^k\equiv-\frac{\phi(0)}{2}\cdot(4^{1-p}+p\cdot H_{a})-\frac{p}{2}\cdot\phi'(0)\pmod{p^2}.
([p,v]+_1 [q,u])+_1[p',v'] =[q,g^{-1}v+_1u]+_1[p',v']=[q,g^{-1}v+_1u+_1hv'] &=[p,v]+_1[q,u+_1hv']=[p,v]+_1([q,u]+_1[p',v']),
\mathring{E}_p^2 =M^{k_0}(\mathring{E}_p^2)=M^{k_0}(\{M\frac{e_1}{p},M^2\frac{e_1}{p},\cdots,M^{p^2-1}\frac{e_1}{p}\}) \;({\rm mod}\;\mathbb{Z}^2) &=\{M(\lambda+v_0),M^{2}(\lambda+v_0),\cdots,M^{p^2-1}(\lambda+v_0)\}\;({\rm mod}\;\mathbb{Z}^2) &=\{M\lambda,M^{2}\lambda,\cdots,M^{p^2-1}\lambda\}\;({\rm mod}\;\mathbb{Z}^2).
\begin{aligned}\phi G\leq \inf_{C_{4}>0}\{nA+\frac{n^{2}}{4}(C_{4}+\frac{C_{1}^{2}}{C_{4}R^{2}})^{2}+2Kn+n(p-2)\lambda_{1}+np\lambda_{1}\}= nA+\frac{n^{2}C_{1}^{2}}{R^{2}}+2Kn+n(p-2)\lambda_{1}+np\lambda_{1},\end{aligned}
\|\partial^2_x u\|^{1/2}_{L^2}=(\|\partial^2_x u\|_{L^2}^2)^{1/4}=(\|\partial_x u\|_{L^4}^4+\|u\times\partial_x^2 u\|^2_{L^2})^{1/4}\leq (\|\partial_x u\|_{L^4}+\|u\times\partial_x^2 u\|^{1/2}_{L^2})\, .
L_{0}^m = q^{1/2}m [(\dot x_{\mu}+q^{1/2}i \chi \psi_{\mu})^2 ]^{1/2}+ \frac{i}{2}(\psi_{\mu}\;\dot \psi^{\mu}- \psi_{5}\dot \psi_{5})- i \chi \psi_{5}m.
[M_{\hat{a}\hat{b}},M_{\hat{c}\hat{d}}]=\eta_{\hat{b}\hat{c}}M_{\hat{a}\hat{d}}-\eta_{\hat{b}\hat{d}}M_{\hat{a}\hat{c}}+\eta_{\hat{a}\hat{d}}M_{\hat{b}\hat{c}}-\eta_{\hat{a}\hat{c}}M_{\hat{b}\hat{d}}.
\chi_{\alpha}\equiv \Pi_{\alpha}- \frac{i}{2}\left( \gamma^{0}\right)_{\alpha}^{\beta}\bar{\psi}_{\beta}\approx 0 , \bar{\chi}^{\alpha}\equiv \bar{\Pi}^{\alpha}- \frac{i}{2}\left( \gamma^{0}\right)_{\beta}^{\alpha}\psi^{\beta}\approx 0 ,
\sum_{\substack{n\geq 0 k\geq 1}}\frac{u^{n+k}x^{k}(1-x)^{n}(1-u)^{2k}}{(1+u)^{n+2k}(1-ux)^{n+2k}}n^{\underline{d}}C_{k-1}\binom{n+2k-2}{n}2^{n}= \frac{(1-x)^{d}u^{d}2^{d}d!\, \tilde N_{d-1}(ux)}{(1-u)^{d-1}(1+u) (1-ux)^{2d}}
|D^2v Dv |^2= v_{x_1x_1}(\frac{|Dv |^2}2)_{x_1}v_{x_1}+v_{x_2x_2}(\frac{|Dv |^2}2)_{x_2}v_{x_2}+v_{x_1x_2}(\frac{|Dv |^2}2)_{x_1}v_{x_2}+ v_{x_2x_1}(\frac{|Dv |^2}2)_{x_2}v_{x_1},
\dot{x}= c_1\kappa_1 x^{a_1}y^{b_1}+ c_2\kappa_2 x^{a_2}y^{b_2}+ c_3\kappa_3 x^{a_3}y^{b_3}, \dot{y}= d_1\kappa_1 x^{a_1}y^{b_1}+ d_2\kappa_2 x^{a_2}y^{b_2}+ d_3\kappa_3 x^{a_3}y^{b_3}
\begin{array}{l}{{u = u_{L}v_{R}+ a_{L}b_{R}\qquad a = a_{L}u_{R}- u_{L}a_{R}}}\\{{b = b_{L}v_{R}- v_{L}b_{R}\, \qquad v = v_{L}u_{R}+ b_{L}a_{R}\, \, ,}}\\ \end{array}
p_{(j+1)(j+1)}= 1, p_{i,(j+1)}= \theta_K(p_{ij}) - j\frac{(1+2K)}{6(1-K)}p_{ij}+ p_{(i-1)j}, p_{1(j+1)}= \theta_K(p_{1j}) - j\frac{(1+2K)}{6(1-K)}p_{1j},
(N-M+1)_M(N-M+\frac{3}{2})_{M}= = \sum_{l=0}^M(-1)^{l}{M\choose l}(j-l+1)_l(N-M-j+l+1)_{M-l}\times \times (N-M-j+l+\frac{3}{2})_{M-l}(j-l-2N+M-\frac{1}{2})_{l},
P_{+}(x) = P_{-}(x) \begin{pmatrix}0 x^\beta 0 -x^{-\beta}0 0 0 0 1\end{pmatrix}, x\in\Delta_{1}\cap A(0; r_n, R), P_{+}(x) = P_{-}(x) \begin{pmatrix}1 0 0 0 0 -1 0 1 0\end{pmatrix}, x\in\Delta_{2}\cap A(0; r_n, R), P_+(z) = P_-(z), z \in \Delta_1^{\pm}\cap A(0; r_n, R).
\mathcal B_{ij}= \begin{bmatrix}(A_1)_{ii}(B_1)_{jj}-(C_1)_{ii}(D_1)_{jj}& \ddots \ddots & \ddots -(C_{r-1})_{ii}(D_{r-1})_{jj}& (A_r)_{ii}(B_r)_{jj}\end{bmatrix}.
\alpha_1 = |k+\sqrt{k^2-1}|, \alpha_2 = |4k^2-2k-1+\sqrt{(16k^3-4k)(k-1)}| \alpha_3 = \left|\frac{\sqrt{16k^3-4k}(\sqrt{k-1}+\sqrt{k+1})}{\sqrt{k+1}(x_1\sqrt{16k^3-4k}+z_1\sqrt{k+1})}\right|.
S_{\mathrm{main}}\geq \sum_{L=1}^{\infty}\phi(L) |y_L|^2 \bigg({\frac{((k-1)!)^2}{2}}-\frac{1}{2}+\frac{\delta}{2}\bigg)\log p+\sum_{L=1}^{\infty}\phi(L) |y_L|^2(c_0-c_1).
\sum_{e\,\cup f:\,v\in e\,\cup f}B(v,e\cup f) =\sum_{e\,\cup f:\,v\in e\,\cup f}B_1(v,e)w_2(f) & =\sum_{e\in E(G_1):\,v\in e}\sum_{f\in E(G_2)}B_1(v,e)w_2(f) & =\Bigg(\sum_{e\in E(G_1):\,v\in e}B_1(v,e)\Bigg)\Bigg(\sum_{f\in E(G_2)}w_2(f)\Bigg) & =1.
\phi_{H(n)}(t,s) = \sum_{k=0}^\infty \frac{\Gamma(-\lambda+1)}{\Gamma(-\lambda-k+1)\Gamma(k+1)}(\cosh x)^{-\lambda-k}\left(-\frac{1}{2(n-1)}\sum_{i=1}^{n-1}t_i\right)^k = \sum_{k=0}^\infty (2(n-1))^{-k}\frac{\Gamma(\lambda+k)}{\Gamma(\lambda)\Gamma(k+1)}(\cosh x)^{-\lambda-k}\left(\sum_{i=1}^{n-1}t_i\right)^k,
b_{33}= b'_{33}+ b^0_{33}= b'_{55}+ b'_{56}+ a_3 + a_5 = b_{55}+ b_{56}, b_{33}+ b_{34}= b'_{33}+ b^0_{33}+ b'_{34}+ b^0_{34}= b'_{44}+ b'_{45}+ a_3 + a_4 + a_5 = b_{44}+ b_{45}, b_{44}= b'_{44}+ b^0_{44}= b'_{55}+ a_5 = b_{55}.
(Z_s = i,Z_t = j) \subseteq \cup_{n=1}^\infty (s<T_n\le t,Z_s = i,Z_t = j) &\subseteq \cup_{n=1}^\infty (s<T_n\le t,U_{T_n}= 0, Z_s = i,Z_t = j) &\subseteq \cup_{n=1}^\infty (s<T_n\le t,U_t \le t-T_n,U_{T_n}= 0, Z_s = i,Z_t = j) &\subseteq (U_t \le t-s).
F_{2 n}( T , \mu ) = \prod_{j = 1}^{2 n}\int_{0}^{\beta}d \tau_{j}\int_{- L / 2}^{L / 2}d x_{j}\prod_{1 \leq k < j \leq 2 n}\left[ Q^{2}( x_{j}- x_{k}, \tau_{j}- \tau_{k}) \right]^{\epsilon_{j}\epsilon_{k}\lambda^{2}/ 4 \pi}\exp \left[ i \mu \frac{\lambda^{2}}{4 \pi}\sum_{j = 1}^{2 n}\epsilon_{j}x_{j}\right] ,
Z_{\scriptscriptstyle C S}( J ) = e^{- \frac{i \pi}{4}\eta ( 0 , \hspace{3 p t}\ast d_{\scriptscriptstyle 1})}\tau_{\scriptscriptstyle R S}( M )^{\scriptscriptstyle 1 / 2}e^{i \langle J , \hspace{3 p t}\frac{1}{\ast d_{\scriptscriptstyle 1}}\hspace{4 p t}J \rangle}