harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_th-|\partial_x|\phi\in O_{2,0},\partial_t\phi-h_{xx}\in O_{2,-1},\qquad\partial_t(G(h)\phi)+|\partial_x|^3h\in O_{2,-2}, &\partial_tB+|\partial_x|^3h\in O_{2,-2},\partial_tV-\partial_x^3h\in O_{2,-2}.
\big<\Gamma_1,\Gamma_2\big>_{\mathcal{L}_2(H)}=\sum_{j=1}^\infty \big<\Gamma_1 \phi_j,\Gamma_2\phi_j\big>,\|\Gamma\|^2_{\mathcal{L}_2(H)}=\sum_{j=1}^\infty \big\|\Gamma \phi_j\|^2,
\widetilde{\nu}(S)= \lim_{m\to \infty}\int_S 2^{-m\frac{\gamma^2}{2}}e^{\gamma \Gamma(\rho_{x,2^{-m}})}\, \nu(dx) = \lim_{m\to \infty}\int_S e^{\gamma \Gamma(\tau_{U,x})}2^{-m\frac{\gamma^2}{2}}e^{\gamma \Gamma^U(\rho_{x,2^{-m}})}\, \nu(dx).
\sum_{1\le n\le t}\widetilde{G}_{n,n}(t) =\sum_{1\le n\le t}\frac{1}{\sqrt{n+1}}\int_n^{n+1}v^{-1/2}\exp\left(-t\left(i\log{\frac{n+1}{v}}+\frac{1}{2}\log^2{\frac{n+1}{v}}\right)\right) dv &=\sum_{1\le n\le t}\int_1^{1+1/n}v^{-3/2}\exp\left(-t\left(i\log v+\frac{1}{2}\log^2v\right)\right) dv.
\begin{array}{rcl}i\Sigma_{ij}(p) = \displaystyle i\lambda^2\delta_{ij}\int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}[f_0(k_i,p)+f_{12}(k_i,p)K_{12}], i\Sigma_{00}(p) = \displaystyle i\lambda^2\int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}[g_0(k_i,p)+g_1(k_i,p)K_1+g_{12}(k_i,p)K_{12}], \end{array}
G_{B}( \tau , \tau^{\prime}) = \frac{| \tau - \tau^{\prime}|}{T}( T - | \tau - \tau^{\prime}| ) , \quad \frac{\partial}{\partial \tau}G_{B}( \tau , \tau^{\prime}) = s i g n ( \tau - \tau^{\prime}) -{\frac{2 ( \tau - \tau^{\prime})}{T}}.
\frac{Y}{2}=g_{string}^{-2}=S_R+\frac{1}{8\pi^2}\delta^{GS}\log(2T_R)\simeq \frac{2}{3( k_1\tilde{\beta}_{1}^{-1}- k_2\tilde{\beta}_{2}^{-1})}\log\frac{\tilde{d}_1k_1\tilde{\beta}_1^{-1}}{\tilde{d}_2k_2\tilde{\beta}_2^{-1}}\ -\ \frac{0.195}{4\pi^2}\ \delta^{GS}.
(U \oplus V)^\mathsf{T}\! A (U \oplus V) = \begin{bmatrix}\hat{C}\oplus \mathrm{O}\hat{B}\oplus \mathrm{O}- \hat{B}^\mathsf{T}\oplus \mathrm{O}\hat{G}\oplus \mathrm{O}\end{bmatrix}, \boldsymbol{b}= A [1, 2, \dots, n]^\mathsf{T},
\dfrac{\dfrac{s(t_1,t_2,\dotsc,t_{2n-1},t_{2n},t_{2n+1},t_{2n+2},\dotsc, t_{2l})}{s(t_1,t_2,\dotsc,t_{2n-1},t_{2n}+1,t_{2n+1}-1,t_{2n+2},\dotsc, t_{2l})}}{\dfrac{s(t_1-1,t_2,\dotsc,t_{2n-1},t_{2n},t_{2n+1},t_{2n+2},\dotsc, t_{2l})}{s(t_1-1,t_2,\dotsc,t_{2n-1}, t_{2n}+1,t_{2n+1}-1,t_{2n+2},\dotsc, t_{2l})}}=\dfrac{t_1+t_2+\dotsc+t_{2n}}{o_t-1}.
Pr \biggl( \lVert L^{-1}\sum_{i=1}^{L}\Pi\cdot\Pi_{X_i}\cdot X_i \cdot\Pi_{X_i}\cdot\Pi- \omega \rVert_1\allowdisplaybreaks &\leq 1-p(\mathsf{M}')+ 4\sqrt{1-p(\mathsf{M}')}+ 42 \sqrt[8]{\epsilon}\biggr)\allowdisplaybreaks \geq 1- 2D \exp\left( -p(\mathsf{M}')\frac{\epsilon^3Ld}{2\ln 2D}\right)
|\partial_\theta x_N| = \left|(\partial_\theta c(\theta_{N-1})) \cdot p(x_{N-1}) + \sum \limits_{j = 1}^{N-1}(\partial_\theta c(\theta_{j-1})) \cdot p(x_{j-1}) \prod \limits_{i=j}^{N-1}c(\theta_i) \cdot p'(x_i)\right|,
\pi\cdot Kf(x) = \lim_{\epsilon\to 0}\left(\int_0^{x-\epsilon}\frac{f(y)}{x-y}dy + \int_{x+\epsilon}^{x_0}\frac{f(y)}{x-y}dy \right) +\int_{x_0}^{\frac12}\frac{f(y)}{x-y}dy + \log\left(\frac12 - x \right).
\begin{cases}\partial y(x, t)/\partial t = \Delta y(x, t) + \alpha (-\Delta)^{\delta}y(x, t-1) + f(x)dB(t), \,\,\,x\in{\cal O},\,\,\,\, t\ge 0, y(x, t) =0,\,\,\,\,x\in \partial{\cal O},\,\,\,t\ge 0, y(x, t) =\varphi(x, t),\,\,\,\,(x, t)\in{\cal O}\times [-1, 0],\end{cases}
f\left( z\right) =\left( z-z_{0}\right)^{-m}\sum\limits_{s=0}^{\infty}{f}_{s}\left( z-z_{0}\right)^{s},\g\left( z\right) =\left( z-z_{0}\right)^{-p}\sum\limits_{s=0}^{\infty}{g}_{s}\left( z-z_{0}\right)^{s},
( \Phi , H_{\mathrm{c o u l}}\Psi ) = \int_{\Lambda}d A^{t r}\sigma \int d^{3}x 2^{- 1}[ ( E_{i}^{a}\Phi )^{*}E_{i}^{a}\Psi + \Phi^{*}B_{i}^{a}B_{i}^{a}\Psi ] \mathrm{,}
\rho_\ell( y_\ell , y_\ell^\prime ):= \inf_{x_\ell, x_\ell^\prime \in \mathcal{X}}\boldsymbol{d}_{\mathcal{S}_\ell}\left( (y_\ell, x_\ell) , (y_\ell^{\prime}, x_\ell^{\prime}) \right)^{p_\ell}.
\Sigma_{\mathrm{M}}(\delta_0):= \left\{\gamma \in \mathcal{P}(\mathcal{S}): \boldsymbol{K}(\mu_{(Y_1, \dotsc, Y_L)|X=x}, \gamma_{(Y_1, \dotsc, Y_L)|X=x}) \le \delta_0 x, \; \mu_{X}= \gamma_X\right\},
x_{0}(z)_{-1}x(z)=\frac{1}{4z^{2}}\left(\frac{(z_1^{-1/2}+z^{-1/2})^{2}}{(z_{1}z)^{-1/2}}X_{0}(z_1^{-1/2})X(z^{-1/2})\right)\Bigg|_{z_1=z}.
\Delta_i^{i+m}h =\sum_{\alpha=m}^{k+\gamma}c_{\alpha}\sigma_{\alpha-m}(\xi_i,\xi_{i+1},\dots,\xi_{i+m}) &=c_{m}+\sum_{\alpha=m+1}^{k+\gamma}c_{\alpha}\sigma_{\alpha-m}(\xi_i,\xi_{i+1},\dots,\xi_{i+m}),
\left( \partial^{2}+ M^{2}\right){\hat{T}}^{( \nu \rho ) \sigma}= 0 , \quad{\hat{T}}^{( \nu \rho )}{}_{\rho}= 0 , \quad \epsilon_{\kappa \nu \rho \sigma}\,{\hat{T}}^{( \nu \rho ) \sigma}= 0 , \quad \partial_{\alpha}\,{\hat{T}}^{({\alpha}\nu ) \sigma}= 0 , \quad \partial_{\alpha}\,{\hat{T}}^{( \nu \rho ) \alpha}= 0 .
\mbox{Res}_{s=1/2-k}\ \zeta_G (s) =\frac 1 2 \, \mbox{res}\G^{k-1/2}= \frac{(2k-1)!! \, q^k}{k! \, 2^k}, mbox{Res}_{s=-k}\ \zeta_G (s) =\frac 1 2 \, \mbox{res}\G^{k}= 0, \k=0,1,2, \ldots,
f_{\{a_1\},a_1}(x)=\begin{cases}a_1 a_1=x, x \end{cases}f_{\{a_1,a_2\},a_1}(x_1,x_2)=\begin{cases}a_1 \{a_1,a_2\}=\{x_1,x_2\}, x_1 \end{cases}
U=V=\sum_{k=0}^{\infty}\beta_{2}^{\prime}(k)\frac{1}{(j_{10A}(\tau)-4)^{k+\frac{1}{2}}}=\sum_{k=0}^{\infty}\beta_{3}^{\prime}(k)\frac{1}{(j_{10A}(\tau)+16)^{k+\frac{1}{2}}}
\delta\left({\sigma_{\tau}^{*}}(t)I\right)=\int^{t_{2}}_{t_{1}}\left[\dot{Q}^{a}-\frac{\partial H_{T}}{\partial P_{a}}\right]\delta P_{a}dt+\left[-\dot{P}_{a}-\frac{\partial H_{T}}{\partial Q^{a}}\right]\delta Q^{a}dt+\left[P_{a}\delta Q^{a}\right]^{t_{2}}_{t_{1}}.
{\cal D}^{*}= \Big\{(d_{1,2},d_{1,3},d_{2,1},d_{2,3},d_{3,1},d_{3,2}) \in \mathbb{R}_{+}^6: &~~~d_{p_1,p_2}+d_{p_1,p_3}\leq M_{p_1},~\forall{\bf p}&~~~d_{p_2,p_1}+d_{p_3,p_1}\leq M_{p_1},~\forall{\bf p}&~~~d_{p_1,p_2}+d_{p_1,p_3}+d_{p_2,p_3}\leq N,~\forall{\bf p}\Big\},
\begin{array}{ccccc}x_i > 0 \Longrightarrow w^{-1}(\alpha_i) \in \varPhi^+ \Longrightarrow l(s_i w) = l(w) + 1, x_i < 0 \Longrightarrow w^{-1}(\alpha_i) \in \varPhi^- \Longrightarrow l(s_i w) = l(w) - 1. \end{array}
\begin{array}{l l}{{g =}}&{{\cos \sqrt{q}\chi - i \cos \omega \sin \theta \sin \sqrt{q}\chi \sigma_{1}- i \sin \omega \sin \theta \sin \sqrt{q}\chi \sigma_{2}}}\\{{}}&{{- i \cos \theta \sin \sqrt{q}\chi \sigma_{3}\, ,}}\\ \end{array}
\varphi_{\alpha}( \lambda , \omega ) \sim{\frac{1}{( 4 \pi )^{d / 2}}}\sum_{n = 0}^{\infty}\left[ a_{n}( \omega ) ~{\frac{\lambda^{d / 2 - n + \alpha - 1}}{\Gamma \left( \frac d 2 - n + \alpha \right)}}+ b_{n}( \omega ) ~{\frac{\lambda^{( d - 1 ) / 2 - n + \alpha - 1}}{\Gamma \left({\frac{d - 1}{2}}- n + \alpha \right)}}\right] .
\Lambda_{j}= \Big( \prod_{\substack{k=1 k \neq j}}^{m}\widetilde{T}_{k,j}^{\beta_{k}}\Big) (4|\lambda_{j}|)^{\beta_{j}}e^{-\frac{2}{3}(r\lambda_{j})_{+}^{3/2}}r^{\frac{3}{2}\beta_{j}}c_{\lambda_{j}}^{\beta_{j}}.
\begin{aligned}\mathcal{F}:\mathcal{C}^{1,p}\longrightarrow \mathcal{E}^{p}\mathcal{F}(\mathsf{a}, u, \Phi) = \left(*F_{\mathsf{a}}- \mu_{1}\circ u, ~ \partial_{\mathsf{a}}u - \left(X_{\Phi}(u)\right)^{1,0},~ \ast\overline{\partial}\Phi^{1,0}+ \mu_{c}\circ u \right) \end{aligned}
\stackrel{x}{\partial}_{\mu}H ( x , x^{\prime}) = 2 \pi \int \frac{d^{n - 1}k}{( 2 \pi )^{n - 1}}e^{i k ( x - x^{\prime})}\frac{( 1 + f )}{2 | \omega |}\lbrack i k_{\mu}+ F_{\mu}- \Omega_{\mu}\rbrack
G_{m,n}=\sum\limits_{k=1}^n\frac{(-1)^{k-1}}{(m-1)!}\binom{n}{k}\sum\limits_{l=0}^{k-1}\binom{k-1}{l}\frac{l!(m+k-2)!}{(m+k+l-1)!}&=\sum\limits_{k=1}^n\frac{(-1)^{k-1}}{(m-1)!(m+k-1)}\binom{n}{k}\sum\limits_{l=0}^{k-1}\frac{(1)_l(1-k)_l(-1)^l}{l!(m+k)_l}&=\sum\limits_{k=1}^n\frac{(-1)^{k-1}}{(m-1)!(m+k-1)}\binom{n}{k}{_2}F_1(1,1-k;m+k;-1).
F(x,x')=F(x)-\frac{1}{2}F_{;\alpha}(x) \sigma^{;\alpha}+\frac{1}{2}F_{\alpha \beta}(x) \sigma^{;\alpha}\sigma^{;\beta}+\frac{1}{4}\{\frac{1}{6}F_{;\alpha\beta\gamma}(x)-F_{\alpha\beta;\gamma}(x)\}\sigma^{;\alpha}\sigma^{;\beta}\sigma^{;\gamma}+O(\sigma^{2})~,
\int_{Y_0(\lambda)\cup Y_k}\Phi\, d\mu = \lambda p w_0+\sum_{l=1}^k p_l(1) v_l = \sum_{l=1}^k p_l\left(\lambda w_0+\sum_{i=1}^l v_i\right) &= \sum_{l=1}^k p_l(l+\lambda) w_l.
H_{ph-e}=-\frac{e_{c}}{m_{e}}(a+a^{\dagger})a_{e}\cdot p_{e}=-(\frac{\hbar\omega_{ph,\alpha}}{2\epsilon_{o}V})^{1/2}(s_{ph,\alpha}\cdot e_{c}x_{e})(a+a^{\dagger})(ce^{i\kappa\cdot x}+c^{\dagger}e^{-i\kappa\cdot x}),
\|u\|_{X_{b,s}^{\beta,\gamma}}=\left(\sum_{k\in\mathbb{Z}}\int_{\mathbb{R}}\langle k\rangle^{2s}\langle \tau-p^{\beta,\gamma}(k)\rangle^{2b}|\widehat{u}(\tau,k)|^2d\tau\right)^{\frac{1}{2}}
T(0) = \prod_{j=l+1}^{P}\; \prod_{i=l}^{j-1}R_{ij}^{\iota^{P-j+i-l+1}}((\epsilon^{-j}-\epsilon^{-i})\theta') \;\; \prod_{j=l}^{P-1}\; \prod_{i=j+1}^{P}R_{ij}^{\iota^{P-j+i}}((\epsilon^{-j}-\epsilon^{-i})\theta')
\sum_{m=0}^{\infty}\frac{p^{m}\omega_{m}(a,p)}{(p;p)_{m}}\, z^{m}\,\sum_{n=0}^{\infty}m_{n}(a,p)z^{n}=\sum_{m=0}^{\infty}\frac{\omega_{m}(a,p)}{(p;p)_{m}}\, z^{m}
\Big\|\sum^{2^n-1}_{j=0}\lambda^n_j (m_r\circ R_{j/2^n}) - \sum^{2^n-1}_{j=0}\lambda^n_j(m_{n,r}\circ R_{j/2^n})\Big\|_1 \le \sum^{2^n-1}_{j=0}\lambda^n_j\big\|m_r\circ R_{j/2^n}- m_{n,r}\circ R_{j/2^n}\Big\|_1 = \big\|m_r - m_{n,r}\big\|_1,
\tilde{\Pi}^{(2)}= (\frac{4\pi}{\kappa})^2[ \frac{1}{96\pi^2}(1-\gamma) (\frac{4\pi}{\kappa})^2 R - \frac{1}{96\pi^2}(\frac{C_2 (G)}{2})^2 (5 -4\gamma )]
I_{\varepsilon}(t_{\varepsilon}w)=\frac{t_{\varepsilon}^{2}}{4}\|w\|_{\varepsilon}^{2}+\left(\frac{1}{4}-\frac{1}{p}\right)t_{\varepsilon}^{p}\ge\left(\frac{1}{4}-\frac{1}{p}\right)t_{\varepsilon}^{p}.
\frac{1}{2}\Delta \widetilde{\psi}=\left[-\frac{1}{2}\|\dot{x}_c\|^2 - \langle \dot{x}_c , Q [x-x_c]\rangle - \frac{1}{2}\langle Q^2 [x-x_c],[x-x_c]\rangle + \frac{i}{2}(Q)\right] \widetilde{\psi}\,.
\lambda_{N,k}:=2^{s+1}\,\mathcal{R}_{s}(2^{n_{k}+1})\,\sum_{j=k+1}^{\tau(N)}2^{n_{j}-n_{k}}+\mathcal{R}_{s}(2^{n_{k}})\left(1-\sum_{j=k+1}^{\tau(N)}2^{n_{j}-n_{k}+1}\right).
\mu_{t}(x) = \mu_t(z,y) = d^{g_{z,t}}(x,\partial(B \times Y))^\alpha \sup_{x' \in B^{g_{z,t}}(x,\frac{1}{4}d^{g_{z,t}}(x,\partial(B\times Y)))}\frac{|\eta_t(x) - \mathbf{P}^{g_{z,t}}_{x'x}(\eta_t(x'))|_{g_{z,t}(x)}}{d^{g_{z,t}}(x,x')^\alpha},
(\frac{2^{\nu}B_{\nu}}{\alpha_{0}})^{\frac{1}{1-2\beta'}}=(\frac{2^{\nu}B_{\nu}}{\alpha_{0}})^{\frac{1}{3(\kappa-1)}}= (\prod_{\mu=\nu}^{\infty}(\frac{2^{\nu}B_{\nu}}{\alpha_{0}})^{\frac{1}{3\kappa^{\mu+1}}})^{\kappa^{\nu}}\leq(\prod_{\mu=\nu}^{\infty}(\frac{2^{\mu}B_{\mu}}{\alpha_{0}})^{\frac{1}{3\kappa^{\mu+1}}})^{\kappa^{\nu}}.
\!\!\!\! T_{s}\!=\! (1\!-\!\theta) R \!=\! (1\!-\!\theta)\bigg[\!\log \bigg(1\!+\!\frac{p_{s}|h_{s}|^{2}}{\sigma_{s}^{2}}\bigg) \!-\! \log (1\!+\!\rho_{e})\!\bigg].
\int_{Q}|f|^p \leq C (\log \delta^{-1})^{3p}\times \left[ D_p(\delta^{1-\varepsilon})^p + q^{\frac{p}{2}+ \frac{k^2 + 7k - 4}{2}}\delta^{-(k^2+4k-2) \varepsilon}\delta^{-\frac{1}{k}(\frac{p}{2}+\frac{k(k-3)}{2})}D_{p-2k}(\delta^{1-\frac{1}{k}})^{p-2k}\right]\Big( \sum_{K \in P_{\delta_0}}\|f_K\|_{L^p(Q)}^2 \Big)^{p/2}.
{{\bar{\psi}}_{( 1 )}^{a}}\, \beta_{a b}\psi_{( 2 )}^{b}={{\bar{\phi}}_{( 1 )}^{\dot{K}}}\, \varphi_{( 2 ){\dot{K}}}+{{\bar{\varphi}}_{( 1 ) K}}\, \phi_{( 2 )}^{K}\, .
\int_{\epsilon\leq|\eta|\leq R}F(\eta)G(|\eta|)d\eta =-\int_{\epsilon}^{R}G'(\rho)\left[\int_{|\eta|\leq\rho}F(\eta)d\eta\right]d\rho \qquad+G(R)\int_{|\eta|\leq R}F(\eta)d\eta-G(\epsilon)\int_{|\eta|\leq\epsilon}F(\eta)d\eta.
\frac{\omega_{1}}{4}(e^{i(\omega_{0}+\omega)t}+e^{i(\omega_{0}-\omega)t})d(t)=i\dot{c}(t),\frac{\omega_{1}}{4}(e^{-i(\omega_{0}+\omega)t}+e^{-i(\omega_{0}-\omega)t})c(t)=i\dot{d}(t)
H_{\epsilon}f(\varphi)=\frac{i}{\pi}\int_{\epsilon\le|\theta|\le\pi}\frac{f(\varphi-\theta)}{1-e^{i\theta}}d\theta=\frac{1}{\pi}\int_{|\zeta-e^{i\varphi}|\ge\delta}\frac{f(\zeta)}{\zeta-e^{i\varphi}}d\zeta,
t^{0 0}= \sum \int_{\lambda}E_{\lambda}\psi_{\lambda}^{*}( \textbf{x}) \psi_{\lambda}( \textbf{x}) - ( \xi - 1 / 4 ) \triangle \sum \int_{\lambda}E_{\lambda}^{- 1}\psi_{\lambda}^{*}( \textbf{x}) \psi_{\lambda}( \textbf{x}) ,
\Theta ( D_{e}; D_{\bar{e}}) = - \int_{0}^{\infty}d l^{\prime}\frac{d f ( D_{e}^{\kappa}l^{\prime})}{d l^{\prime}}f ( D_{\bar{e}}^{\kappa}l^{\prime}) = - \int_{0}^{\infty}d z \frac{d f ( z )}{d z}f \left( \left( \frac{D_{\bar{e}}}{D_{e}}\right)^{\kappa}z \right) .
K^{*}_{20}>|K_{b0}|,K^{*}_{2i}>|K_{bi}|,i=2,3,5,K^{*}_{24}>C^{*}_{2},K^{*}_{14}>C_{2}^{*}, &K^{*}_{11}<\frac{1-K^{*}_{25}|K_{25}|}{|K_{21}|},K^{*}_{15}>K^{*}_{25}|K_{15}|+K^{*}_{11}|K_{11}|,
\begin{aligned}|f(x)-f(y)| \le \max\{|f(0) - f(|x-y|)|,|f(1-|x-y|)-f(1)|\}& = \max\{f(|x-y|),f(1-|x-y|)\}& \le f(|x-y|)+f(1-|x-y|) & \le f(|x-y|) + |x-y| & = \tilde f(|x-y|),\end{aligned}
V ( y_{i}) = \frac{- k \pi^{\frac{D - 2}{2}}}{\Gamma ( \frac{D - 2}{2}) V_{T}}\int_{0}^{\infty}d x \prod_{i = 1}^{D}\theta_{3}\Big ( \frac{y_{i}}{L_{i}}, \frac{i x}{L_{i}^{2}}\Big ) ,
( A_\mu, \psi_\mu, \bar{\psi}_\mu, \phi, V_\mu,T_{\mu\nu}^\pm, \lambda_{\mu\nu}^\pm, Y_\mu^\pm, \xi_\mu^\pm ) \rightarrow( - A_\mu, i \bar{\psi}_\mu, - i \psi_\mu, - \phi, - V_\mu,T_{\mu\nu}^\mp, \mp \lambda_{\mu\nu}^\mp, Y_\mu^\mp, \mp \xi_\mu^\mp ),
{\bar s}_{l_j}[i] = \Re\left[{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol H}_D[i]{\boldsymbol{\bar{s}}}_l +{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol n}_D[i]\right] = \Re\left[{\boldsymbol w}^\emph{H}_j[i]\boldsymbol{\bar{r}}_l[i]]+ e_{l_j}[i]\right]=\bar{s}'_{l_j}[i] + e_{l_j}[i],
( i g^{2})^{m}\Big ({\frac{1}{2 \pi i}}\Big )^{m}{\frac{1}{2^{m}}}\sum_{P_{m}}\int_{t_{P_{m}}^{-}< t_{1}< \dots < t_{r}< \dots < t_{m}< t_{P_{m}}^{+}}{\hspace{- 1 c m}}s ( P_{m}) \prod_{r = 1}^{m}{\frac{d z_{i_{r}}( t_{r}) - d z_{j_{r}}^{\prime}( t_{r})}{z_{i_{r}}( t_{r}) - z_{j_{r}}^{\prime}( t_{r})}}R ( P_{m}) ,
\dot{\Lambda}^{\frac{d}{p}-1}e^{(t - \tau)\Delta}\mathbb{P}\nabla .\big(u(\tau) \otimes v(\tau)\big) =\frac{1}{(t - \tau)^{\frac{\{\frac{d}{p}\}+d+1}{2}}}K\Big(\frac{.}{\sqrt{t - \tau}}\Big)*\Big(\dot{\Lambda}^{[\frac{d}{p}]-1}\big(u(\tau) \otimes v(\tau)\big)\Big).
\partial_{t}u-\Delta_{\mathbb{H}^{n}}u=\left(\frac{\beta_{0}}{\beta}\right)^{\frac{1}{p}}\big(\partial_{t}v-\Delta_{\mathbb{H}^{n}}v\big)=e^{\mu t}u\big( e^{\beta u^{p}}-1\big)~\hbox{in}\mathbb{H}^{n}\times (0, +\infty).
[P,Q] = P_x Q_Y-P_Y Q_x & = \sum_{i=0}^n i p_i x^{i-1}Y^{n-i}\sum_{j=0}^m (m-j) q_j x^j Y^{m-j-1}& - \sum_{i=0}^n (n-i) p_i x^i Y^{n-i-1}\sum_{j=0}^m j q_j x^{j-1}Y^{m-j}& = \sum_{i,j}p_i q_j (i(m-j)-(n-i)j) x^{i+j-1}Y^{m+n-i-j-1}&= \sum_{i,j}p_i q_j (mi-nj) x^{i+j-1}Y^{m+n-i-j-1}
L^C_+=\left(\begin{array}{ll}(qt^{-1})^{S_3}\Omega S^- 0 (qt)^{-S_3}\end{array}\right)\ \ \ ,L^C_-=\left(\begin{array}{ll}(qt)^{-S_3}0 -\Omega S^+ (qt^{-1})^{S_3}\end{array}\right)\ ,
\frac{\partial^{2}u_{\omega}}{\partial t^{2}}=\frac{\partial^{2}}{\partial t^{2}}(e^{-i\omega t}f(x))=-\omega^{2}e^{-i\omega t}f(x)=c^{2}\frac{\partial^{2}}{\partial x^{2}}(e^{-i\omega t}f(x)).
\int_D\frac{\partial U}{\partial y}(x,Y(T,x,z),z)Y(T,x,z)dx\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]=\tilde{p}(0,z) \exp(\int_0^T (b_0(t,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})dB(s) -\frac{1}{2}\int_0^T (b_0(s,z)\pi(s,z)-\frac{a_0(s,z)}{b_0(s,z)})^{2}ds).
{\cal L^{\prime}}_{\mathrm{s p i n}}^{( 2 )}[ F ] + \Delta{\cal L^{\prime}}_{\mathrm{s p i n}}^{( 2 )}[ F ] \sim \mathrm{l n}( m^{2}T_{0}){( 4 \pi )}^{- 4}e^{4}F_{\mu \nu}F^{\mu \nu}\quad
\displaystyle \int_{0}^{T}B_{t}\circ dB_{t}=\displaystyle \frac{1}{2}B_{T}^2=\displaystyle \lim_{\substack{n\longrightarrow +\infty}}\sum_{\substack{j\in J\setminus J_{K(n)}}}\frac{B_{t_{j+1}}+ B_{t_{j}}}{2}(B_{t_{j+1}}-B_{t_{j}}).
1+S^{\langle -1 \rangle}_{\mu_{T^*T}}\left(\frac{1}{r^2}\right) =1+\psi_{\mu_{T^*T}}\left(-\frac{1}{s(r,0)^2}\right) =\int_0^\infty\frac{s(r,0)^2}{s(r,0)^2+u^2}d\mu_{|T|}(u) =s(r,0) h(s(r,0)) = \frac{s(r,0)^2}{s(r,0)^2+r^2}.
( \ln G )^{\prime \prime}+{\frac{d - 1}{r}}( \ln G )^{\prime}+ ( \ln G )^{\prime}( \ln f )^{\prime}+{\frac{1}{r}}( \ln f )^{\prime}+{\frac{d - 2}{r^{2}}}\left( 1 -{\frac{F^{2}}{G^{2}}}\right)
u(rt,\rho) =\sum_{\mu=0}^{n}(-1)^{n-\mu}\binom{2n}{2\mu}(rt)^{2\mu}\rho^{2n-2\mu}, v(rt,\rho) =\sum_{\mu=0}^{n-1}(-1)^{n-\mu-1}\binom{2n}{2\mu+1}(rt)^{2\mu+1}\rho^{2n-2\mu-1},
(\widehat{S}_{f,\tau}^{-1}\widehat{\theta}_\tau+V(I_{\ell^p(\mathbb{J})}-\theta_f\widehat{S}_{f,\tau}^{-1}\widehat{\theta}_\tau))(\theta_f\widehat{S}_{f,\tau}^{-1}+(I_{\ell^p(\mathbb{J})}-\theta_f\widehat{S}_{f,\tau}^{-1}\widehat{\theta}_\tau)U) &=\widehat{S}_{f,\tau}^{-1}+VU-V\theta_f\widehat{S}_{f,\tau}^{-1}\widehat{\theta}_\tau U.
\int\limits_{a}^{b}P(x,t)f^{\prime}(t)dt=f\left( x\right) -\frac{1}{\alpha +\beta}\left[ \alpha \frac{N\left( a,x\right)}{m\left( a,x\right)}+\beta \frac{N\left( x,b\right)}{m\left( x,b\right)}\right]
\Gamma_{j \bar{\jmath}}\sim c_{j_{0}, j}c_{j_{0}, \bar{\jmath}}+ d_{j_{0}, j}d_{j_{0}, \bar{\jmath}}+ \frac{1}{2}c_{j_{0}, j}d_{j_{0}, \bar{\jmath}}+ \frac{1}{2}d_{j_{0}, j}c_{j_{0}, \bar{\jmath}},
d_2(v) =(c_1,e,g_1) d_1(v) =(c_2,e,g_2g_1) d_0(v) =(c_2-R^{g_2}_1(c_1)-R_2^{g}(e),R^{g_1}_1(e)+R^{x_1}_0(c_1),g_2)
L_{1 1}^{( \pm )}L_{1 2}^{( + )}= q^{\mp 1}L_{1 2}^{( + )}L_{1 1}^{( \pm )}, \; \; L_{2 2}^{( \pm )}L_{1 2}^{( + )}= q^{\pm 1}L_{1 2}^{( + )}L_{2 2}^{( \pm )}
\begin{array}{ccl}\tilde{T}^{\, (+)}= i^{\, M}\, T\,\left(P_{\,(+)}g_{\, 2M-2}^{\ast}g_{\, 2M-1}^{\ast}\,-\, i P_{\, (-)}g_{\, 2M-2}g_{\, 2M-1}\right) \tilde{T}^{\, (-)}= i^{\, M}\, T\,\left(P_{\, (+)}g_{\, 2M-2}g_{\, 2M-1}\,+\, i P_{\, (-)}g_{\, 2M-2}^{\ast}g_{\, 2M-1}^{\ast}\right)\end{array}
ds^{2}= (1 -{qcos\theta\over r})^{2\over 1 + \alpha^{2}}\lbrace dr^2+r^2d\theta^2+r^2sin^2\theta d\varphi^2\rbrace -{dt^2\over (1 -{qcos\theta\over r})^{2\over 1 + \alpha^{2}}}\, .
Y_k^{(m)}(t) = Y^{(m)}_k(0) + g_k^{(m)}t + B^{(m)}_k(t) + \tfrac12L_{(k-1, k)}^{(m)}(t) - \tfrac12L_{(k, k+1)}^{(m)}(t),\ \k = 1, \ldots, m^2,
\begin{pmatrix}r+s-p r \end{pmatrix}= \frac{(r+s-p)(r+s-1-p)\cdots (s+1-p)}{r(r-1)\cdots 2\cdot 1}= \frac{(r+s)(r+s-1) \cdots (s+1)}{r(r-1)\cdots 2 \cdot 1}\equiv \begin{pmatrix}r+s r \end{pmatrix}\mod p.
A_n =|2F_n\sin(\pi \varphi^n)| , B_n = \prod_{t=1}^{F_n-1}\left|\frac{s_{nt}}{2\sin(\pi t/F_n)}\right| , C_n = \prod_{t=1}^{(F_n-1)/2}\left(1-\frac{s_{n0}^2}{s_{nt}^2}\right) ,
\frac{1}{n}\sum_{p=0}^{\lfloor (n-1)/2 \rfloor}n \binom{n-1}{2}^p 2^{Q(n,n-2p)-Q(n,n)}\le \sum_{p=0}^{\lfloor (n-1)/2 \rfloor}\bigg(\frac{n^2}{2^n}\bigg)^p &= \frac{1 - (n^2/2^n)^{\lfloor (n-1)/2 \rfloor +1}}{1 - n^2/2^n}.
\left( - \frac{h^{2}}{8 \pi^{2}M_{1}}\bigtriangledown_{1}^{2}- \frac{h^{2}}{8 \pi^{2}M_{2}}\bigtriangledown_{2}^{2}+ V_{n}( r ) \right) \psi_{\mathrm{v i b - r o t}}= E_{t}\psi_{\mathrm{v i b - r o t}}~ ,
{\rm det}\left[{{\cal Y}^{\sigma}}_{\alpha}\right] = \frac{(j^2)^2}{({\tilde j}^2)^2}\,{\rm det}\left[{{\cal Y}^{\sigma}}_{\alpha}\right] = \frac{({\cal J}^2)^2}{({\tilde j}^2)^2}\,{\rm det}\left[{{\cal Y}^{\sigma}}_{\alpha}\right] = a \frac{({\cal X}^2)^4}{({\tilde j}^2)^2}.
V_{c}^{\prime}( x ) = \frac{1}{T_{c}}\, ( x^{3}- 4 c_{1}x^{2}+ 2 c_{2}x + 8 c_{1}) , \quad T_{c}= 1 + 4 c_{1}^{2}\, ; \qquad V_{c}( 0 ) = 0 \, ,
\dot{\Lambda}^{(2)}_{m^{\varepsilon},(lh,\mathbb{I})}(f_{1},f_{2},f_{3}):=\int_{0}^{1}\int_{0}^{1}\sum_{\vec{P}\in\vec{\mathbb{P}}}\frac{C^{\varepsilon}_{Q_{\vec{P}}}}{|I_{\vec{P}}|^{\frac{1}{2}}}\langle f_{1},\Phi^{1,\nu}_{\vec{P}_{1}}\rangle\langle f_{2},\Phi^{2,\nu}_{\vec{P}_{2}}\rangle \langle f_{3},\Phi^{3,\nu}_{\vec{P}_{3}}\rangle d\nu,
\omega_{j}=tr(\gamma^{-1}d\gamma )e^{2\pi \sqrt{-1}\frac{\sqrt{3}}{3}aq(j)\mathbf{\sigma}_{3}}(\gamma^{-1}d\gamma )e^{-2\pi \sqrt{-1}\frac{\sqrt{3}}{3}aq(j)\mathbf{\sigma}_{3}},
\begin{cases}f_1(A,\lambda,\mathbf x):=a_{11}x_1^2+a_{12}x_1x_2+a_{13}x_2^2+a_{14}x_1x_3+a_{15}x_2x_3+a_{16}x_3^2-\lambda x_1^2=0, f_2(A,\lambda,\mathbf x):=a_{21}x_1^2+a_{22}x_1x_2+a_{23}x_2^2+a_{24}x_1x_3+a_{25}x_2x_3+a_{26}x_3^2-\lambda x_1^2=0, f_3(A,\lambda,\mathbf x):=a_{31}x_1^2+a_{32}x_1x_2+a_{33}x_2^2+a_{34}x_1x_3+a_{35}x_2x_3+a_{36}x_3^2-\lambda x_1^2=0.\end{cases}
j^{\mu}= -(D_{\nu}c^{a})(F_{a}^{\mu \nu}+ B_{a}\eta^{\mu \nu}) - \frac{1}{2}{\varphi_{m n}}^{a}c^{m}c^{n}\partial^{\mu}{\bar{c}}_{a}- B_{a}D^{\mu}c^{a}~,
T^{(d)}_z(u)=C_zu^{h+1}\sum_{n=(n_1,\ldots,n_d)\in(\mathbb{Z}_{\ge 0})^d}\frac{\bigl(\frac{h}{2}+\frac{1}{2}\bigr)_{|n|}\bigl(\frac{h}{2}+1\bigr)_{|n|}}{(z_1+1)_{n_1}\cdots (z_d+1)_{n_d}}\frac{2^{2|n|}u^{2|n|}}{n_1!\cdots n_d!}\bigl(1+(2d-1)u^2\bigr)^{-(h+1+2|n|)}.
\tilde{R}( x , t ) = \sum_{n}m \omega_{n}^{2}\partial_{x}Q_{n}( x , t ) \Big \{( q_{n 0}- Q_{n 0}) \cos [ \omega_{n}( t - t_{0}) ] + \frac{\dot{q}_{n 0}}{\omega_{n}}\sin [ \omega_{n}( t - t_{0}) ] \Big \},
\Phi_+(z)=\Phi_-(z)\begin{pmatrix}1 0 -e^{\pi i\alpha}1\end{pmatrix}, z\in \Sigma_1, &\Phi_+(z)=\Phi_-(z)\begin{pmatrix}0 -1 1 0\end{pmatrix}, z\in \Sigma_2, &\Phi_+(z)=\Phi_-(z)\begin{pmatrix}1 0 -e^{-\pi i\alpha}1\end{pmatrix}, z\in \Sigma_3.
s ( y_{3}, \upsilon , \bar{\upsilon}) = - 2 \ln \upsilon + \frac{4}{( e^{y_{3}}+ e^{- y_{3}})^{2}}\cdot \frac{\upsilon{\bar{\upsilon}}-{\bar{\upsilon}^{2}}}{\upsilon^{2}+{\bar{\upsilon}}^{2}- \upsilon{\bar{\upsilon}}}+ . . . .
\lim_{N\to\infty}\frac{\#\left( \mathcal{A}_N\cap \Gamma_\alpha^{m,\varepsilon}\right)}{\mathcal{S}_{\mathcal{A}_N,\alpha}}= \mathcal{R}(\Gamma_\alpha^{m,\varepsilon}) , 1\le m\le M_\alpha^{(\varepsilon)}
\limsup\limits_{n\rightarrow+\infty}\frac1{\int_\Omega \frac{\eta_n^2}{|x|^2}}= \frac1{\liminf\limits_{n\rightarrow+\infty}\int_\Omega \frac{\eta_n^2}{|x|^2}}\le\frac1{\int_\Omega\frac{\eta^2}{|x|^2}}
\begin{cases}U_0(x,z):= \varphi_0(x)+ A(\log (|z|^2+3)-\log(|z|^2+1) -\log 2); U_1(x,z):=\varphi_1(x)+ A(\log |z|^2-\log(|z|^2+1) +\log (e^2+1) -2).\end{cases}
\left( F_{n}^{N}(z)-\frac{\psi_{n}(x;\omega,N) \psi_{n-1}\left( x;\omega ,N+1\right)}{\sqrt{2\omega}n}\right) \Big/\psi_{n-1}( x;\omega ,N+1 ){}=C\Psi_{n-1}^{(n-1)}\big(x;\omega ,N+1,- \Vert \psi_{n-1}\Vert^{2}\big)
\begin{array}{ll}f_1:= abc + ~ bc +~a^2b^2c^2, begin{align*}4pt] f_2:= ab +~ abc^2+~t^7~bc^2+~t^3~b^2c^4+~t^2~ab^2c^4+~t^5~a^2b^2c^4, begin{align*}4pt] f_3:= t~a +~ ab+~abc+~a^2bc. \end{array}
\begin{aligned}x_{\frac{n}{4}}= \frac{\sin\left(\beta - \frac{\pi}{n}\right)}{\sin \frac{2\pi}{n}}= -\frac{1}{2}, y_{\frac{n}{4}}= \frac{\sin\left(\frac{\pi}{n}- \beta\right)}{\sin \frac{2\pi}{n}}= \frac{1}{2}.\end{aligned}
\biggl ( \varepsilon + \sigma^{2}\biggr ) c_{\sigma}+ \frac{B}{2}\biggl [ \sqrt{( S - \sigma ) ( S + \sigma + 1 )}~ c_{\sigma + 1}+ \sqrt{( S + \sigma ) ( S - \sigma + 1 )}~ c_{\sigma - 1}\biggr ] = 0 ,
\int_{B_R(0) \setminus S}\rho |\nabla \varphi|^2 \, dx = \int_{B_R(0) \cap S}\left( A_{+}\cdot N_+ + A_{-}\cdot N_- \right) \, dS + \int_{\partial B_R(0) \setminus S}A \cdot N \, dS =: \mathbf{I}+ \mathbf{II}.
L_{1}= \frac{1}{(\rho-r)^2}\int_{Q^{+}(\rho)}|u|^{2}dxds, L_{2}= \frac{1}{(\rho-r)}\int_{Q^{+}(\rho)}|u|^{3}\phi dxds, L_{3}= \frac{1}{(\rho-r)}\int_{Q^{+}(\rho)}u(\Pi-\overline{\Pi}_{B^{+}(\rho)}) dxds.
\begin{aligned}|\mu(X_{t_{i-1}}+u)-\mu(\widetilde X_{t_{i-1}}+u)| \qquad\qquad\leq c\, \bigl(|X_{t_{i-1}}-\widetilde X_{t_{i-1}}|+\sum_{j=1}^k 1_{\{(X_{t_{i-1}}+u-\xi_j)\, (\widetilde X_{t_{i-1}}+u-\xi_j)\leq 0\}}\bigr).\end{aligned}
X_i=\left\{-\frac{n}{2}+{\left(\frac{n^2}{4}+\frac{m^3}{27}\right)}^{1/2}\right\}^{1/3}+ \left\{-\frac{n}{2}-{\left(\frac{n^2}{4}+\frac{m^3}{27}\right)}^{1/2}\right\}^{1/3}\equiv{x^*},
p_1(\lambda,\mu) = 1 + 2\lambda + 3\lambda + 4\lambda^2 +5\lambda \mu + 6\mu^2 + 7\lambda^3 + 8\lambda^2\mu +9\lambda\mu^2 + 10\mu^3 = 0, p_2(\lambda,\mu) = 10 + 9\lambda + 8\mu + 7\lambda^2 + 6\lambda\mu + 5\mu^2 + 4\lambda^3 + 3\lambda^2\mu + 2\lambda\mu^2 + \mu^3 = 0.
\Sigma_{2m}:=(D^{-1}_m(\lambda,c)\mu,\mu)=\frac{\Gamma(f_m^{(2)})+\Gamma(f_m^{(2)},g_m^{(2)})+\Gamma(f_m^{(2)},h_m^{(2)})}{1+\Gamma(g_m^{(2)})+\Gamma(h_m^{(2)})+\Gamma(g_m^{(2)},h_m^{(2)})},
r\dfrac{d}{dr}\left(\int_{M\cap B_r}fS_1 dM \right) - \dfrac{m-1}{2}\int_{M\cap B_r}fS_1 dM = r^{\frac{m+1}{2}}\dfrac{d}{dr}\left(\frac{1}{r^{\frac{m-1}{2}}}\int_{M\cap B_r}fS_1 dM\right),
\begin{array}{l}{{\epsilon ( k_{i}) = 1 ~ , ~ \epsilon ( X_{i}^{-}) = 0 = \epsilon ( X_{i}^{+}) ~ ,}}\\{{S ( k_{i}) = k_{i}^{- 1}~ , S ( X_{i}^{-}) = - X_{i}^{-}k_{i}^{- 1}~ , S ( X_{i}^{+}) = - k_{i}X_{i}^{+}~ .}}\\ \end{array}
\omega_2(x,y,z)-\omega_1(x,y,z)=(\varphi_2-\varphi_1)([x,y],z)-\rho(z)(\varphi_2-\varphi_1)(x,y) \quad\;\;\; =\partial_{\rho}(\alpha)([x,y],z)-\rho(z)\partial_{\rho}(x,y) \quad\;\;\;=\delta^{1}(\alpha)(x,y,z),\;\alpha\in C^{1}_{Lie}(L,V),
\sum_{i,j=1}^3 \sigma^{ij}\big(u_0\hat{p}_i-u_i\big)\big(u_0\hat{p}_j-u_j\big)\geq \sum_{i,j=1}^3\frac12u_0^2\sigma^{ij}\hat{p}_i\hat{p}_j-\sum_{i,j=1}^3\sigma^{ij}u_iu_j.