harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta_{B}^{r}L_{B}^{r}= \eta \frac{d}{d \tau}\Bigl [ q^{1 / 2}\lambda p_{\mu}\dot{x}^{\mu}- \frac{\lambda e}{1 + q^{2}}( p^{2}- m^{2}) +{\cal B}\frac{d}{d \tau}( \lambda e ) \Bigr ] .
\omega^{sf}= \left[ R (S_1-\ln|c| ) + \frac{t_1^2 (m^2 + n^2)}{R(S_1 - \ln|c|)^3}\right] dc_1 \wedge dc_2 - \frac{t_1}{R(S_1 - \ln|c|)^2}[n dc_1 -m dc_2] \wedge dt_1 + \frac{t_1}{R(S_1 - \ln|c|)^2}[m dc_1 +n dc_2] \wedge dt_2 - \frac{1}{R(S_1 - \ln|c|)}dt_1 \wedge dt_2
\phi(\gamma\eta,\eta)= \sqrt{\gamma^2\eta^2+2p+2}-\sqrt{\eta^2+2m+2}-\sqrt{(\gamma-1)^2\eta^2+2m+2}= |\gamma|\sqrt{\eta^2+\frac{2p+2}{\gamma^2}}-\sqrt{\eta^2+2m+2}-|\gamma-1|\sqrt{\eta^2+\frac{2m+2}{(\gamma-1)^2}}.
\Gamma_\nu =\varepsilon^{\mu ,[\mu \nu ]}+\varepsilon^{[\mu \nu],\mu}+\varepsilon^{\nu ,0}+\varepsilon^{o,\nu}+\varepsilon^{\widetilde{\nu}, \widetilde{0}}+\varepsilon^{\widetilde{0},\widetilde{\nu}}+\frac 12e_{\mu \nu \rho \omega}\left( \varepsilon^{\widetilde{\mu},[\rho \omega ]}+\varepsilon^{[\rho \omega ],\widetilde{\mu}}\right)
-[\alpha (x_n),g(x_m)]+[\alpha (x_m),g(x_n)]+g([x_n,x_m])+\bar{f}(1,\varphi (x_n,x_m))=0. -\varphi (\alpha (x_n),g(x_m))+\varphi (\alpha (x_m),g(x_n))+\hat{v}((1,[x_n,x_m])+\bar{v}(1,\varphi (x_n,x_m))=0.
\begin{array}{c c c c}{\left\{P_{\phi}^{\tau}, X_{\phi}^{\tau}\right\}= 1 ,}&{\left\{P_{\phi}^{r}, X_{\phi}^{s}\right\}= - \delta^{r s},}&{\left\{P_{\phi}^{r}, X_{\phi}^{\tau}\right\}= 0 ,}&{\left\{P_{\phi}^{\tau}, X_{\phi}^{r}\right\}= 0 ,}\\ \end{array}
h_{\ell}^{\prime \prime}( i t , r ) = - 2 t \xi_{\ell}( t r ) h_{\ell}^{\prime}( i t , r ) + \left[ \sigma ( r ) - 2 t^{2}\left. \frac{d \xi_{\ell}( \tau )}{d \tau}\right|_{\tau = t r}\right] h_{\ell}( i t , r ) .
\left\vert R^{d}\widehat{\chi_{\Omega}}(Rn)\right\vert =\frac{1}{\pi}R^{\frac{d-1}{2}}\left\vert n\right\vert^{-\frac{d+1}{2}}K^{-\frac{1}{2}}\left( \sigma\left( n\right) \right) \left\vert\sin\left( 2\pi Rn\cdot\sigma\left( n\right) \right) \right\vert +O\left(R^{\frac{d-3}{2}}\left\vert n\right\vert^{-\frac{d+3}{2}}\right) .
\sigma(H^{\epsilon,\kappa})\cap \left [a_0,b_N\right ]\subset \bigcup_{k=0}^N [a_k,b_k]\,,\quad{\rm dim}\big({\rm Ran}E_{[a_k,b_k]}(H^{\epsilon,\kappa}) \big )= + \infty\,, & b_k-a_k\leq C_0 \,\epsilon\big(\kappa + C_1\, \epsilon^{1/3}\big)\, , \; 0\leq k\leq N\,,{\rm and}a_{k+1}-b_k\geq\frac{1}{C_2}\epsilon \, ,\; 0\leq k\leq N-1\,.
\|B_{N}(f)\|^{2}_{L^{2}(m)}/\|f\|_{L^2(m)}^{2}\ll N^{-2\alpha}+\left(N^{-\frac{1-\alpha}{d}}+N^{-\frac{1-\alpha-\epsilon}{q}}\right) &+ N^{4\beta L}e^{-N^\beta}.
\hat{B}(C,t)=3D\mbox{exp}[ig(\int d^3x[\hat{\vec{E}}(\vec{x},t)\int_C= ds\,{\vec{A}}^A (\vec{y}(s))+{\hat{\Pi}}_1^* (\vec{x},t)\int_C ds\, \phi^A_1 (\vec{y}(s))])]
\frac{1}{n}\sum_{i=1}^{n}W\left( \frac{d_{i}}{S}\right) \left(\mathbf{x-\mu}\right) \left( \mathbf{x-\mu}\right)^{\prime}=\mathbf{\Sigma~\ \}\frac{1}{n}\sum_{i=1}^{n}W\left( \frac{d_{i}}{S}\right) \left(\mathbf{x-\mu}\right) =0 \frac{1}{n}\sum_{i=1}^{n}\rho\left( \frac{d_{i}}{S}\right) =\delta
p_{\hat{S}^m|S^m}(\hat{s}^m|s^m)=\sum\limits_{x^{\rho m},y^{\rho m}}p_{X^{\rho m}|S^m}(x^{\rho m}|s^m)p_{Y^{\rho m}|X^{\rho m}}(y^{\rho m}|x^{\rho m})p_{\hat{S}^m|Y^{\rho m}}(\hat{s}^m|y^{\rho m})
|I\cap I_w| = \frac{1}{4}\sum_{x\in\ \mathbb{Z}/(p)}(1+\big(\frac{w+x^2}{p}\big))+\frac{1+(\frac{w}{p})}{4}+\frac{1+(\frac{-w}{p})}{4}&=\frac{p-1}{4}+\frac{2+(\frac{w}{p})+(\frac{-w}{p})}{4}&=\frac{p+1+(\frac{w}{p})+(\frac{-w}{p})}{4}
J_\alpha= \int_{B(p_{\alpha},\delta)}\frac{\chi_{E_\alpha}(x)-\chi_{E_\alpha^c}(x)}{|x-p_{\alpha}|^{N+s}}\, d x=\frac{1}{\alpha^s}\int_{B(0,\delta/\alpha)}\frac{\chi_{F_\alpha}(\tilde x) -\chi_{F_\alpha^c}(\tilde x)}{|\tilde x|^{N+s}}d \tilde x
\begin{aligned}\|(u^n,\,Q^n,\,\nabla Q^n)\|_{L^\infty(0,T;L^2_x)}+ \|(\nabla u^n,\,\Delta Q^n) \|_{L^2(0,T;L^2_x)}\lesssim &\lesssim(\| u_0 \|_{L^2_x}+\|Q_0\|_{L^2_x}+ \|Q_0\|_{H^1}^6 ) \exp\{\tilde{C}t\},\end{aligned}
\begin{aligned}h_1''(r) =r^{-n}e^{I(r)}[(n-1)\tau(r)-rI'(r)\tau(r)-r\tau'(r)] &+r^{-n}E_+(r)(n-1+R(r))\int_0^r\rho^{n-1}\,R(\rho)\,B[v](\rho)\,d\rho -B[v](r).\end{aligned}
\langle (\varphi^{\dagger}\varphi)^{aI}_{bJ}\rangle^{{\rm 1-loop,WZ}}_{{\rm FT}}\sim g^{2}\delta^{I}_{J}\delta^{a}_{b}p^{2}\int \frac{d^{4}k}{(2\pi)^{4}}\frac{1}{\left(k+\frac{p}{2}\right)^{2}\left(k-\frac{p}{2}\right)^{2}}\, .
2 \mathrm{Re}\langle u_{\alpha}\otimes \Omega\,|\, H_y u_{\alpha}\otimes 2\alpha^{3/2}\Phi_*^1\rangle = - 4 \alpha^2 \mathrm{Re}\langle P u_{\alpha}\otimes A_y^+\Omega\,|\,u_{\alpha}\Phi_*^1 \rangle = 0,
(h\perp h')^{\otimes d}(\pi(u\otimes v),u'\otimes v') = (h\perp h')^{\otimes d}((x_{\pi^{-1}(1)}\otimes \cdots \otimes x_{\pi^{-1}(d)})\pi, (y_1\otimes \cdots \otimes y_d)) = \pi^{-1}(h\perp h')(x_{\pi^{-1}(1)}, y_1)\otimes \cdots \otimes (h\perp h')(x_{\pi^{-1}(d)},y_d)
\mathcal{E}^{0,\gamma}_{\tilde{\alpha}-\gamma;I}:=C(I;\mathcal{B}_{\tilde{\alpha}-\gamma})\cap C^{\gamma}(I;\mathcal{B}_{\tilde{\alpha}-2\gamma}), \ \ \ \mathcal{E}^{\gamma,2\gamma}_{\tilde{\alpha};I}:=C^{\gamma}_{2}(I;\mathcal{B}_{\tilde{\alpha}-\gamma})\cap C^{2\gamma}_{2}(I;\mathcal{B}_{\tilde{\alpha}-2\gamma}).
\int_0^1 \int_t^\infty\left|e^{-|O(at)x-y|^2/4s}-e^{-1/4s}\right|\frac{ds}{s^2}\,dt &\leq \int_0^1 \int_t^\infty\left(e^{-|x-y|^2/4s}\, e^{|a||x|t(|x|+|y|)/2s}+e^{-1/4s}\right)\,\frac{ds}{s^2}\,dt &\leq e^{|a||x|(|x|+|y|)/2}\int_0^1 \int_t^\infty e^{-|x-y|^2/4s}\,\frac{ds}{s^2}\, dt +4
D_q(T)= (1-\phi(q)T) \prod_{i=1}^g(1-\phi(q)\alpha_{q,i}^{- 1}T)(1-\phi(q)\alpha_{q,i}T) \mbox{if}q \nmid N, D_q(T)= \prod_{i=1}^g(1-\alpha_{q,i}T) \mbox{if}q \mid N.
\int_0^1{\frac{{g(x)}}{{\left({1 - \alpha x^\eta}\right)^\xi}}dx = \sum\limits_{n = 0}^\infty{\frac{{\left( \xi \right)_n}}{{n!(1 - \lambda )^{n + \xi}}}}}\sum\limits_{j = 0}^n{n\choose j}( - \lambda )^{n - j}b_j (\alpha,\eta).
\det(-B_j) =\mathbb{P}^{CUE(n)}(\theta_i\not\in [0,2\pi\rho_{sc}(y_j)a_j],1\leq i\leq n) =D_n(\pi\rho_{sc}(y_j)a_j)=D_n\left(a_j\sqrt{4-y_j^2}/2\right),
{\frac{1}{g^{2}( q^{2})}}={\frac{R}{g_{5}^{2}}}+ \lambda_{k}( 2 k ) + \lambda_{T}( T / 2 ) -{\frac{1}{4 8 \pi^{2}}}\left[ \ln \left({\frac{2 - \nu}{2 \nu}}\right) + \nu \ln{\frac{k}{T}}\right] .
\int_{-\pi/2}^{\pi/2}y_{1,2}y_{1,3}= \frac{(n-1)^2(n-3)^2}{24^2}\frac{1}{160}\left(-15570\pi+2220\pi^3-67\pi^5+\frac{13\pi^7}{168}+\frac{4\pi^9}{315}\right) & \approx \frac{(n-1)^2(n-3)^2}{24^2}0.1766
|E(u - u_I, \varphi_h)| \leq\sum_{e \in \Gamma_{DI}}\int_e \Big| \Big\{\varepsilon \nabla \varphi_h \cdot \nu\Big\}\Big| \Big| [u - u_I] \Big|\, ds \leq\varepsilon_M \sum_{e \in \Gamma_{DI}}\|\{\nabla \varphi_h \cdot \nu\}\|_{L_2(e)}\| [u - u_I]\|_{L_2(e)}.
\sigma_{k+1}=\sigma_{k+1;i}+\lambda_{i}\sigma_{k;i}, \sum_{i=1}^{n}\lambda_{i}\sigma_{k;i}=(k+1)\sigma_{k+1}, \sum_{i=1}^{n}\sigma_{k;i}=(n-k)\sigma_{k}, \sum_{i=1}^{n}\lambda_{i}^{2}\sigma_{k;i}=\sigma_{1}\sigma_{k+1}-(k+2)\sigma_{k+2}.
\langle\langle \widehat\chi \rangle\rangle ={\varepsilon^\alpha}_{\mu\beta}\left({\cal F}_{+\alpha}\partial^\mu{\cal F}^\beta_+ +{\cal F}_{-\alpha}\partial^\mu{\cal F}^\beta_- \right) + m \left({\cal F}_{+\gamma}{\cal F}^\gamma_+ -{\cal F}_{-\gamma}{\cal F}^\gamma_- \right).
\mathrm{supp}\,m_1\subseteq\{(\xi,\eta)\in\mathbb{R}^2:\,|\xi-\eta|\leq |\eta|/2^4\}, &\mathrm{supp}\,m_2\subseteq\{(\xi,\eta)\in\mathbb{R}^2:\,|\xi-\eta|/|\eta|\in[2^{-20},2^{20}]\}.
\int_\mathbb{R}\sum_{i,j=1}^n |\rho_{xxt}||w_iw_j|dx\leqslant O(1)\varepsilon_0^2 \int_\mathbb{R}|\rho_{xxt}|dx\leqslant O(1) \varepsilon_0 \delta (1+t)^{-{3}/{2}}.
g_{\alpha \beta \gamma}: = \int_{\sum_{2}}d^{2}\varphi Y_{\alpha}( \frac{\partial Y_{\beta}}{\partial \varphi^{1}}\frac{\partial Y_{\gamma}}{\partial \varphi^{2}}- \frac{\partial Y_{\gamma}}{\partial \varphi^{1}}\frac{\partial Y_{\beta}}{\partial \varphi^{2}})
E_{c}= \frac{2 n r_{\mathrm{B H}}^{n - 1}\Omega_{n}}{1 6 \pi G_{n + 2}}, \ \ \ 2 E - E_{c}= - \frac{2 n r_{\mathrm{B H}}^{n + 1}\Omega_{n}}{1 6 \pi G_{n + 2}l_{n + 2}^{2}}.
\ll q^\varepsilon \sum_{l<M^{1-\Upsilon}}\frac{1}{l}\sum_{M^{1-\Upsilon}<n<X(\log q)^2}\frac{1}{n}\sum_{c\equiv0(q)}\frac{(c,n,l^2)^{1/2}\tau(c)}{c^{1/2}}\cdot \frac{l\sqrt{n}}{c}\ll M^{1-\Upsilon}q^{-1+\varepsilon}\ll M^{-(1-\Upsilon)}.
\begin{cases}\left( -\Delta+\frac{N^{3\beta-1}}{2}V(N^\beta x) \right) f^{(\beta)}_{l}(x)=\lambda_l^{(\beta)}f_l^{(\beta)}(x) |x|\le l \partial_{|x|}f_l^{(\beta)}(x)=0\quad |x|=l f_l^{(\beta)}(x)=1 |x|\ge l\end{cases}
[\hspace{-0.7mm}[S^{(i)}]\hspace{-0.7mm}]=[S_{a_i}][S_4^{(i)}][S_{b_i}]^{-1}[S_3^{(i)}][S_{a_i}]^{-1}[S_2^{(i)}][S_{b_i}][S_1^{(i)}].
\sum_{\mu\nu\gamma}(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{matrix})(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}^{\prime}&\nu_{2}^{\prime}&\nu\end{matrix})=\delta_{\nu_{1}\nu_{1}^{\prime}}\delta_{\nu_{2},\nu_{2}^{\prime}}
P\left(\left|\sum_{t=1}^m\beta^2_t-\sum_{t=1}^mE(\beta^2_t)\right|>n^{1-r}-\sum_{t=1}^mE(\beta^2_t)\right) &\qquad\leq 2\exp\left(-L_1\min\left\{\frac{\left(n^{1-r}-\sum_{t=1}^mE(\beta^2_t)\right)^2}{L^4_2m},\frac{n^{1-r}-\sum_{t=1}^mE(\beta^2_t)}{L^2_2}\right\}\right),
\sum_{i=0}^{2c-2}a_iq^i \le \sum_{i=2}^{2c}c^iq^{2c-i}= q^{2c-2}c^2\sum_{i=0}^{2c-2}\frac{c^i}{q^i}\le \frac{q^{2c-2}c^2}{1-c/q}<q^{2c-1}
\sum p(n)X^n=\frac{1}{\alpha}=\frac{\alpha(X^{25})}{\alpha(X^5)^6}\alpha(\zeta X)\alpha(\zeta^2X)\alpha(\zeta^3X)\alpha(\zeta^4X) &=\frac{\alpha(X^{25})}{\alpha(X^5)^6}(\alpha_0+\zeta\alpha_1+\zeta^2\alpha_2)(\alpha_0+\zeta^2\alpha_1+\zeta^4\alpha_2)(\alpha_0+\zeta^3\alpha_1+\zeta\alpha_2)(\alpha_0+\zeta^4\alpha_1+\zeta^3\alpha_2).
t=\pm\int\frac{sin\theta d\theta}{\sqrt{C_{1}sin^{2}\theta-1}}=\pm\int\frac{dcos\theta}{\sqrt{C_{1}(1-cos^{2}\theta)-1}}=\pm\frac{1}{\sqrt{C_{1}}}\int\frac{dcos\theta}{\sqrt{(1-\frac{1}{C_{1}})-cos^{2}\theta}}.
\bigl|\log |\gamma|_w - z_w\bigr| = \Bigl|\log |\gamma \mu|_w - |W_v(l/k)|^{-1}\sum_{x | v}\log |\mu|_x\Bigr| = \Bigl|\log |\gamma \mu|_w - |W_v(l/k)|^{-1}\sum_{x | v}\log |\gamma \mu|_x\Bigr|.
\Delta(H)=H\otimes 1+1\otimes H, \Delta(E)=E\otimes 1+K\otimes E, \Delta(F)=F\otimes K^{-1}+1\otimes F, \epsilon(H)=\epsilon(E)=\epsilon(F)=0, S(H)=-H,S(E)=-K^{-1}E,S(F)=-FK.
(-\lambda-2)^{2c-2}(\lambda^5+(4c-4)\lambda^4+(12c^2+36c+2)\lambda^3+(24c^2+88c+28)\lambda^2+(8c^2+72c+40)\lambda+16c+16) &=(-\lambda-2)^{2c-2}(\lambda^2+(2c+4)\lambda+4)(-\lambda^3+6c\lambda^2+(12c+6)\lambda+4c+4).
\|Df^{-1}- Dg^{-1}\|_{L^1(\Omega^*)}\; =\; \|D(f^{-1}|_{\mathcal{O}^*}) - D\phi^{-1}\|_{L^1(\mathcal{O}^*)}\leq\; c_1\left[\mathcal{E}_1(f^{-1}|_{\mathcal{O}^*}) + \mathcal{E}_1(\phi^{-1})\right] \leq\; c_1\left[\mathcal{E}_1(f^{-1}|_{\mathcal{O}^*}) + 4\mathcal{E}_p(\phi)^{\frac{1}{p}}\right] \ .
{\frac{1}{2}}\gamma_{b}^{a}\left( \xi_{i j}^{b}+ \xi_{j i}^{b}- \partial_{i}\rho_{j}^{b}- \partial_{j}\rho_{i}^{b}- \nu_{i c}^{b}\rho_{j}^{c}- \nu_{j c}^{b}\rho_{i}^{c}\right) = 0 .
\bullet D^{(y)}\psi=0\mbox{on}M\setminus\Sigma. &\bullet |\psi| \mbox{extends across}\Sigma \mbox{as a}C^{0,\frac{1}{2}}\mbox{function on}M \mbox{}\mbox{}\mbox{with its zero locuscontaining}\Sigma. &\bullet \liminf_{p\rightarrow x}\frac{|\psi|(p)}{\mbox{dist}(p,\Sigma)^{\frac{1}{2}}}>0\mbox{for all}x\in\Sigma.
TT^{\dag}_{E,F}=T^3(T^{\sharp})^2T^{\dag}_{E,F}=T^3T^{\dag}_{E,F}T(T^{\sharp})^2T^{\dag}_{E,F}=T^3(T^{\dag}_{E,F}T^{\sharp}T^{\dag}_{E,F}) =T^3((T^{\dag}_{E,F})^2T^{\sharp})=T^3T^{\dag}_{E,F}(T^{\sharp})^2=T^3T^{\dag}_{E,F}T(T^{\sharp})^3=TT^{\sharp}.
G^{ij}=\frac{\partial G}{\partial\nabla_{ij}u}\quad\textrm{and}\tilde G^i=\frac{\partial G}{\partial\nabla_i u}+g^{ik}\sigma\left( \nabla \psi(N), \frac{1}{u^2w}e_k-\frac{\nabla_lu}{u^3w}x\right)
\delta_{( 2 )}\Phi^{A}( \theta ) = ( \Phi^{A}( \theta ) , Y )^{a}\mu_{a}, \, \, \, \delta_{( 2 )}\bar{\Phi}_{A}( \theta ) = ( \bar{\Phi}_{A}( \theta ) , Y )^{a}\mu_{a}, \, \, \, \varepsilon ( Y ) = 0
\int_0^\pi \Psi(|v'_*|^2) d \varphi =\pi \Psi (Y(\pi-\theta)) +Z^2\int^{\frac{\pi}{2}}_0 (\sin\varphi - \varphi \cos\varphi )\sin\varphi & \times \{({\Psi}''(Y(\pi-\theta)+ Z\cos\varphi ) +{\Psi}''(Y(\pi-\theta)-Z \cos\varphi ) \}\d\varphi .
f = 2 \zeta ( 3 ) \tau_{2}^{3 / 2}+{\frac{2 \pi^{2}}{3}}\tau_{2}^{- 1 / 2}+ 2 \tau_{2}^{3 / 2}\sum_{m , n \neq 0}\int_{0}^{\infty}d y \exp \left( -{\frac{\pi^{2}m^{2}}{y}}+ 2 \pi i m n \tau_{1}- y n^{2}\tau_{2}^{2}\right) ,
S_{[m(p+1)+j,j]}^{(q)}=\sum_{i_{1}=q}^{j+q}\,\,\sum_{i_{2}=q}^{i_{1}+p}\,\,\sum_{i_{3}=q}^{i_{2}+p}\,\,\cdots\,\,\sum_{i_{m}=q}^{i_{m-1}+p}\,\,\prod_{k=1}^{m}a_{i_{k}}, T_{[m(p+1)+j,j]}^{(q)}=\sum_{i_{1}=-j-p-q}^{-p-q}\,\,\sum_{i_{2}=i_{1}-p}^{-p-q}\,\,\sum_{i_{3}=i_{2}-p}^{-p-q}\,\,\cdots\,\,\sum_{i_{m}=i_{m-1}-p}^{-p-q}\,\,\prod_{k=1}^{m}a_{i_{k}}.
G_i= \left| \frac{\textbf{a}_{r}^H (\xi_{r_o},N)}{\sqrt{N}}{\bf{A}}\left(\xi_{r_{i,o}},\varphi_{t_{i,o}}\right) \frac{\textbf{a}_{t}(\varphi_{t_i},N)}{\sqrt{N}}\right|^2.
F_{1}=(\begin{matrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{matrix}),F_{2}=(\begin{matrix}0&0&1\\ 0&0&0\\ -1&0&0\end{matrix}),F_{3}=(\begin{matrix}0&-1&0\\ 1&0&0\\ 0&0&0\end{matrix}).
\chi_{2}^{i}( F , G ;{\bf p^{\prime}, q}) = - \frac{3 \mu + 1}{( 1 + \mu )^{3}}\frac{q^{i}}{1 2 \pi^{2}}\tilde{f}{\bf ( p^{\prime}}) \tilde{g}({\bf - p^{\prime}- q}) .
G^{tttt}(\omega,q) = - 4{\delta^2 S_{\epsilon}\over \delta (H_{tt}^0)^2}={3N^2\pi^2 T^4 q^2\over 2 (3\omega^2 - q^2)}-{3 \pi^2 \over 8}N^2 T^4 = 3P\frac{5q^2-3\omega^2}{3\omega^2-q^2}\,.
r_{23}(q_1)L_{12}(x_{i};t_{j})\left( \begin{smallmatrix}1 0 0 q_1\end{smallmatrix}\right)_{1}L_{13}(x_{i};t_{j^{\prime}})=L_{13}(x_{i};t_{j^{\prime}})\left( \begin{smallmatrix}1 0 0 q_1\end{smallmatrix}\right)_{1}L_{12}(x_{i};t_{j})r_{23}(q_1),
\left[ G_{h/2}\ast, \phi \right] w = \int G_{h/2}(z) \left(\phi(\,\cdot\,-z)-\phi(\,\cdot\,)\right) w(\,\cdot\,-z) \,dz = \Big(\Big(- \partial_i\phi z_i + \frac12 \partial_{ij}\phi z_i z_j - \frac16 \partial_{ijk}\phi z_iz_jz_k\Big) G_{h/2}\Big) \ast w + O\Big( |z|^4 G_{h/2}\ast |w|\Big),
[\nabla^{k,g_P}\eta]_{C^{\alpha}(B^{g_P}(p,\delta R))}\leq \;{\epsilon[\nabla^{k,g_P}\eta]_{C^{\alpha}(B^{g_P}(p,R))}}+ C [\nabla^{k-1,g_P}L^{g_P}\eta]_{C^{\alpha}(B^{g_P}(p,R))}{+}\sum_{j=0}^k CR^{-k+j-\alpha}\|\nabla^{j,g_P}\eta\|_{L^\infty(B^{g_P}(p,\delta R))}.
\limsup_{k\to\infty}\left|\int_0^t\langle (hu_s+\widetilde U)\otimes (w_k-w),\nabla w_k\rangle d\tau\right| &\leq (CY(T)+1)\,\varepsilon+\lim_{k\to\infty}\int_0^T\|hu_s+\widetilde U_\varepsilon\|_\infty\|\chi_{B_L}(w_k-w)\|_2\|\nabla w_k\|_2 d\tau &=(CY(T)+1)\,\varepsilon.
\delta{\cal B}_{0 1}= - \frac{m}{\Delta_{m , n}^{1 / 2}}\, \frac{Q_{1}}{r^{6}}~ ~ ~ , ~ ~ ~ \delta{\cal C}_{0 1}= - \frac{n}{\Delta_{m , n}^{1 / 2}}\, \frac{Q_{1}}{r^{6}}~ ~ .
e(G) = \begin{cases}e(H)+e(O_{k-2})+e(K_2)+k k\ge4 e(H)+k k=3 \end{cases}&= \begin{cases}ex_{_{\mathcal{P}}}(n-k,C_k)+(2(k-2)-3)+1+k k\ge4 ex_{_{\mathcal{P}}}(n-k,C_k)+k k=3 \end{cases}&=ex_{_{\mathcal{P}}}(n-k,C_k)+3k-6.
\int_0^t \gamma(s) \psi(s)ds =\phi(t) \exp \left\{\int_0^t \beta(\theta) \gamma(\theta)d\theta \right\}& \leq \int_0^t \alpha(s)\gamma(s) \exp \left\{\int_s^t \beta(\theta) \gamma(\theta)d\theta \right\}ds.
y \in\partial V(u-\sum_{t=0}^{T-1}x_t\Delta s_{t+1}-\bar c\cdot\bar x+S_0(\bar x)), E_t[y\Delta s_{t+1}] \in N_{D_t}(x_t)t=0,\dots,T, \frac{E[y\bar c]}{E[y]}\in\partial S_0(\bar x)
E(g)\cdot B(g) = \sum_{j=1}^{2n}E_j(g)\, B_j(g)= \sum_{j=1}^{2n}E_j(g)\, R_j(\psi)(g) &= \sum_{j=1}^{2n}\bigg( E_j(g)\, R_j(\psi)(g) - \psi(g) R_j^t(E_j)(g) \bigg).
\int_{\mathbb{R}}\!dx\:\lvert \hat{f}^{\,\sharp}_\beta(x)\rvert^2 S(x)=\frac{1}{\beta^2}\!\int_{\mathbb{R}}\!dx\:\lvert\hat{h}(x/\beta)\rvert^2\,S(x) =\frac{1}{\beta}\!\int_{\mathbb{R}}\!dx\:\lvert\hat{h}(x)\rvert^2\,S(\beta \mspace{1.5mu}x).
\mathcal{Z}^\alpha\mbox{div}^\varphi S^\varphi v=\mbox{div}^\alpha\mathcal{Z}^\alpha S^\varphi v-\partial{_z^\varphi}S^\varphi v\cdot\nabla^\varphi\mathcal{Z}^\alpha\eta+\mathcal{D}^\alpha(S^\varphi v),
\displaystyle - \left( \frac{9 1 9}{6 3 0}+ \frac{7 1}{7 0}\zeta ( 3 ) + \frac{1 1 1}{5 6 0}\zeta ( 4 ) + \frac{1}{1 2}\zeta ( 5 ) + \frac{1}{5 6 0}\zeta ( 3 )^{2}+ \frac{1}{1 1 2}\zeta ( 6 ) \right)
\rho c(T_{P}-{T_{P}}^{0})\Delta V=\int_{t}^{t+\Delta t}[(K_{e}A\frac{T_{E}-T_{P}}{\delta x_{PE}})-(K_{w}A\frac{T_{P}-T_{W}}{\delta x_{WP}})]dt+\int_{t}^{t+\Delta t}\overline{S}\Delta Vdt
\frac{0.432\tau^2}{T}\left(\frac{1}{2\pi}\log{(2T)}+\frac{1}{2\pi}\log{\frac{200}{\pi e}}\right. & \left.-\frac{0.634}{3T}\log(2^\frac{1}{3}T) -\frac{1}{3T}(0.317\log{2}+0.634\log{100}+12.802)\right ) & \coloneqq K_{F,3}(T,\tau) & >0.
\mathcal{A}(x^*)\coloneqq\ I_{g+}(x^*,\lambda^*)\cup I_{g0}^=(x^*,\lambda^*,d), \mathcal{B}(x^*,y^*)\coloneqq\ I_{0+}(x^*,y^*)\cup I_{01}(x^*,y^*), M\coloneqq\ |\mathcal{A}(x^*)|+p+|\mathcal{B}(x^*,y^*)|
\Psi_{k(T)}(t)=\begin{cases}1/t, \quad\mbox{for $t\in[t_0+2,T]$;}0, \quad\mbox{for $t\le t_0$ and $t\ge T+2^{-k(T)}$}\end{cases}\intertext{and, for $0\le k\le k(T)-1$,}\Psi_{k}(t)=\begin{cases}1/t, \quad\mbox{for $t\in[t_0+2,T-2^{-k}]$;}0, \quad\mbox{for $t\le t_0$ and $t\ge T$.}\end{cases}
K=-\frac{1}{E}(\frac{\partial}{\partial u}\Gamma_{12}^{2}-\frac{\partial}{\partial v}\Gamma_{11}^{2}+\Gamma_{12}^{1}\Gamma_{11}^{2}-\Gamma_{11}^{1}\Gamma_{12}^{2}+\Gamma_{12}^{2}\Gamma_{12}^{2}-\Gamma_{11}^{2}\Gamma_{22}^{2})
| f |_{C^k_b( E_1, E_2)}= \sup_{x \in E_1}\left\| f^{( k )}( x ) \right\|_{L^{k}( E_1, E_2 )}, \| f \|_{C^k_b(E_1,E_2)}= \| f(0) \|_{E_2}+ \sum^k_{l = 1}| f |_{C^l_b(E_1,E_2)}
\mathcal{J}_n(\gamma) =\left\{(\boldsymbol{j},\boldsymbol{k}):|\theta_{\boldsymbol{j},\boldsymbol{k}}^{0}|>\prod_{l=1}^d\min\left\{2^{-\alpha_lj_l\left(\frac{1}{d}+\frac{1}{2\alpha^{*}}\right)}, \gamma\left(\frac{\log{n}}{n}\right)^{\frac{1}{2d}}\right\}\right\},
\sum_{n\ge 0}\chi^{(k)}(X^n, G_n)\cdot t^n=\left(\prod\limits_{r_1, \ldots,r_k\geq 1}\left(1- t^{r_1r_2\cdots r_k}\right)^{r_2r_3^2\cdots r_k^{k-1}}\right)^{-\chi^{(k)}(X, G)}\,.
\sum_{k,m=1}^n \alpha_m v_k^m \gamma_k(r\eta) =\sum_{m=1}^{n-1}w_m\left(\sum_{k=1}^n v_k^m \gamma_k(r\eta)\right)+\sqrt{1-|w|^2}e^{i\psi}\left(\sum_{k=1}^n v_k \gamma_k (r\eta)\right) =w_1 \sum_{k=1}^n \eta_k \gamma_k(r\eta)+ \sum_{m=2}^{n-1}w_m\left(\sum_{k=1}^n v_k^m \gamma_k(r\eta)\right)+\sqrt{1-|w|^2}e^{i\psi}\left(\sum_{k=1}^n v_k \gamma_k (r\eta)\right)
\displaystyle\,(\,(L)\,(0))=\left(\frac{1}{\vert p-x^*\vert}+\frac{1}{\vert x-x^*\vert}\right)^2\vert \vartheta\cdot\nu_{x^*}\vert^2=\left(\frac{\vert p-x\vert}{\vert p-x^*\vert\,\vert x-x^*\vert}\right)^2\vert\vartheta\cdot\nu_{x^*}\vert^2.
C_{p}^{(1)}:=\lim_{n\rightarrow\infty}\left\{\sum_{j=1}^{n-1}\frac{p^{2}+\lambda_{j}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{j}^{2}}\cdot\frac{1}{\lambda_{j}}-\log\left(\lambda_{n}\right)\right\}.
\begin{aligned}y\left[ n \right] = \Big({\sum\nolimits_{l = 1}^L{{\bf{h}}_l^H{{\bf{f}}_l}}}\Big)s\left[{n -{n_{\max}}}\right]+ \sum\nolimits_{l = 1}^L{\sum\nolimits_{l' \ne l}^L{{\bf{h}}_l^H{{\bf{f}}_{l'}}s\left[{n -{n_{\max}}+{n_{l'}}-{n_l}}\right]}}+ z\left[ n \right]. \end{aligned}
\max_{\boldsymbol \beta}{\displaystyle \min_{i\in\mathcal{N}}}\; R_{i,i}^{\textrm{NOMA}}(\boldsymbol \beta), &\textrm{s.t.}\;\; \sum_{j=1}^N\beta_j \leq 1, &0 \leq \beta_j,\;\;\;\textrm{for}\;j\in\mathcal{N}.
\langle A_\mu^a(x) A_\nu^b(y) \rangle_{\mbox{\scriptsize Feyn}}= \delta^{ab}\, D_{\mu\nu}^{\mbox{\scriptsize Feyn}}(x,y)= -{\Gamma(D/2-1) \over 4\pi^{D/2}}\,g_{\mu\nu}\,\delta^{ab}\, [(x-y)^2]^{1-D/2}\,.
\hat{K}^\beta_{(1/2)}(s\mid\tau,\mbox{\boldmath $x$};\tau',\mbox{\boldmath $x'$}) =\sum_{n=-\infty}^\infty (-1)^n\hat{K}_{(1/2)}(s\mid\tau,\mbox{\boldmath $x$};\tau'+\beta n,\mbox{\boldmath $x'$})
D_{a d}^{2}( b ) = 2 \cot \left( \frac{\pi x}{2 h}\right) \tan \left( \frac{\pi \left( 1 - x \right)}{2 h}\right) \tan \left( \frac{\pi}{2 h}\right) \mathcal{T}_{0}\mathcal{T}_{q - 1}\mathcal{T}_{h / 2 - q}
I = \frac{m_P^2}{16 \pi}\int d^4x \sqrt{-g}\Bigl[ R - 2 \Lambda\Bigr] + \frac{m_P^2}{8 \pi}\int d^3x \sqrt{h}K + \int d^4x\sqrt{-g}\Bigl[\frac{1}{2}g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi - V(\phi) \Bigr],
\Big( \sin \frac{\phi}{2}\Big)^{\alpha+1}\Big(\cos \frac{\phi}{2}\Big)^{\beta}P_{\ell}^{(\alpha+1,\beta)}(\cos \phi)=\frac{\Gamma(\ell+\alpha+2)}{\ell !}\Big( \frac{\phi}{\sin \phi}\Big)^{1/2}\Big[\sum_{n=0}^{m-1}A_n(\phi) \frac{J_{\alpha+n+1}(N \phi)}{N^{\alpha+n+1}}+\sigma_m \Big]
w+y_1+\cdots+y_{p_1}=(c_2-1)z_2+\cdots+(c_{p_2}-1)z_{p_2}+d_1t_1+\cdots+d_{p_3}t_{p_3}, &w+z_2+\cdots+z_{p_2}+t_1+\cdots+t_{p_3}=v_1, &u_1+\cdots+u_{p_4}+w=(c_2-1)z_2+\cdots+(c_{p_2}-1)z_{p_2}+(d_1-1)t_1+\cdots+(d_{p_3}-1)t_{p_3}.
\begin{cases}\rho(x) u_{tt}- \operatorname{div}(K(x) \nabla u) + V_1(x,t)u=V_3(x,t)v ~\hbox{in}~ \Omega \times (0,T), \rho(x) v_{tt}- \operatorname{div}[K(x) \nabla v] + V_2(x,t)v=V_3(x,t)u ~\hbox{in}~ \Omega\times (0,T), u=v=0 ~\hbox{on}~ \omega, \end{cases}
R_1(t) := \{(x_1, x_2) \mid t/2 \le x_1 + x_2 \le t \wedge t/2 \le x_1-x_2 \le t/2 \}, R_2(t) := \{x_2 \mid 0 \le x_2 \le t\}, R_3(t) := \{x_2 \in R_2(t) \mid \{x_1 \in \mathbb{R}\mid (x_1, x_2) \in R_1(t)\}= \varnothing\}.
\rho(u, z, b) =\begin{cases}\int\limits_{0}^{T}2ph(u+t-T, lp)h(T-u, lq)h(T-t, x)dt, x\geq 0, \int\limits_{u}^{T}2qh(u, lp)h(T-u, lq)h(T-t, x)dt, x<0,\end{cases}&=\begin{cases}2ph(T-u, bq)h(u, lp+x), x\geq 0, 2qh(u, lp)h(T-u, lq-x), x< 0.\end{cases}
|Df^n(x)(v)| =|Df^{m_q}(x)(v)||Df^{n-m_q}(f^{m_q}(x))(v)| &\geq C \lambda_0^{2n}\frac{|\xi_{m_q}(x)-c(\lambda^{m_q-1}z_0)|}{|w_0-c(z_{0}/\lambda)|}&\geq C \lambda_0^{2n}\frac{\min_{i=0}^{n}|\xi_{i}(x)-c(\lambda^{i-1}z_0)|}{|w_0-c(z_{0}/\lambda)|}.
\begin{gathered}\| u \|_{L^p_{l+k,\nu+k}}\leq C\left( \| Pu\|_{L^p_{l,\nu}}+ \| u \|_{L^p_{0,\nu+k}}\right) ,\| u \|_{C^{l+k,\alpha}_{\nu+k}}\leq C\left( \| Pu\|_{C^{l,\alpha}_{\nu}}+ \| u \|_{C^{0,\alpha}_{\nu+k}}\right), \| u \|_{C^{l+k,\alpha}_{\nu+k}}\leq C \left( \| Pu\|_{C^{l,\alpha}_{\nu}}+ \| u \|_{L^2_{\nu+k}}\right)\end{gathered}
\sum_{t=1}^n (g(\bar{c}_{i,t}-\bar{c}_{j,t}) + \bar{Z}_t)^2 = \sum_{t=1}^n g^2(\bar{c}_{i,t}-\bar{c}_{j,t})^2 + \sum_{t=1}^n \bar{Z}_t^2 + 2\sum_{t=1}^n g(\bar{c}_{i,t}- \bar{c}_{j,t})Z_t \;.\,
\Big(\frac{\zeta^2-\alpha^2}{Q(\lambda,x)}\Big)^{-\frac{1}{4}}\phi(\lambda,x)=\Big\{[c_1+o(1)]D_{\nu}({e}^{\pi i/4}\sqrt{2x}\zeta)+[c_2+o(1)]D_{-\nu-1}({e}^{-\pi i/4}\sqrt{2x}\zeta)\Big\}
y_{1}=\frac{b_{0}}{(-2)\times(-1)}(\frac{(\alpha)_{3}(\beta)_{3}}{(3!0!}x^{3}+\frac{(\alpha)_{4}(\beta)_{4}}{4!1!}x^{4}+\frac{(\alpha)_{5}(\beta)_{5}}{5!2!}x^{5}+\cdot\cdot\cdot)
log\Gamma_{p}(a)=log(\pi^{\frac{p(p-1)}{4}}\prod_{j=1}^{p}\Gamma(a+\frac{1-j}{2}))=\frac{p(p-1)}{4}log\pi+\sum_{j=1}^{p}log\Gamma[a+\frac{1-j}{2}]
\frac{P_{\lambda}\big(q^rt^{N-1}, t^{N-2}, \dots, t, 1; q, t\big)}{P_{\lambda}\big(t^{N-1}, \dots, t, 1; q, t\big)}= \frac{(q; q)_r}{\big(t^N; q\big)_r}\frac{1}{2\pi\sqrt{-1}}\oint_C{\frac{1}{y^{r+1}}\prod_{i=1}^N{\frac{\big(q^{\lambda_i}t^{N-i+1}y; q\big)_{\infty}}{\big(q^{\lambda_i}t^{N-i}y; q\big)_{\infty}}{\rm d}y}},
\sum_{j=0}^{n-1}\frac{h}{|z_j - U_j| \, |\tilde z_j - \tilde U_j|}\leq \left[ \sum_{j=0}^{n-1}\frac{h}{|z_j - U_j|^2}\right]^{1/2}\, \left[ \sum_{j=0}^{n-1}\frac{h}{| \tilde z_j - \tilde U_j|^2}\right]^{1/2}\leq \log(y/\delta) + O(\delta).
Z_*(x,y) = \Lambda\left( \begin{pmatrix}(\eta_*^{r_0}\circ\xi_*)^{r_1}\circ\eta_*(x) (\eta_*^{r_0}\circ\xi_*)^{r_1-1}\circ\eta_*(x) \end{pmatrix}, \begin{pmatrix}\eta_*^{r_0}\circ\xi_*(x) x \end{pmatrix}\right).
\frac{\lambda_y(u)}{\rho(u)}= 1 - \int_{0}^{u-1}\frac{\rho'(u-v)}{\rho(u)}\frac{\{y^v\}}{y^v}dv+ O\left( y^{-u}\right)= \int_{0}^{u-1}\frac{\rho(u-v-1)}{(u-v)\rho(u)}\frac{\{y^v\}}{y^v}dv+ O\left(1\right).
\|E\|_{L^1\cap L^\infty}\leq c\big(\lambda\eta_t + \lambda^2\gamma_t + |\mu-m^2|\big)\Big(\frac{1}{m^2_t}{\bf 1}_{d=2}+ \frac{1}{m_t}{\bf 1}_{d=3}+ \frac{1}{m^4_t}\Big) &+ \frac{c\|E\|_{L^1\cap L^\infty}}{m^2_t}\big(\lambda\eta_t + \lambda^2\gamma_t + |\mu-m^2|\big)+ c\sum_{i=1}^3\lambda^i R^{(i)}_{t}\big(\|E\|_{L^1\cap L^\infty}\big),
(D^{F}(C^{F}))(X, Y) (C_{Z}) = \mathrm{pr}^{F}(C_{Y}D_{X}(C_{Z}))- C_{Y}\mathrm{pr}^{F}(D_{X}(C_{Z})) & - \mathrm{pr}^{F}(C_{X}D_{Y}(C_{Z}))+ C_{X}\mathrm{pr}^{F}(D_{Y}(C_{Z})).
\begin{aligned}0 = R_a^2 + L_a R_a+ L_a^2 - R_a^3 -L_a R_a^2 -L_a = R_a^2 + L_a R_a + R_a^2 - R_a^3 -L_a R_a^2 -R_a = 2R_a^2 - R_a^3 +L_a (R_a -R_a^2) -R_a ,\end{aligned}
\begin{array}{l}\dot{V}=-\left[ \frac{{{k}_{v1}}}{{{m}_{c}}}{{V}^{2}}+{{k}_{v2}}\cos \left( \alpha(t) \right) \right]sgn\left( V \right) -g\sin \left( \alpha (t) \right)+\frac{1000}{{{m}_{c}}\left( V+{{\varepsilon}_{V}}\right)}\left[ P_{CDD,wh}^{v}-{{P}_{FR}}\right] \end{array}
\|\frac{d}{dt}\bar{z}_\nu(t,\omega)\|^2\leq\frac{2}{N}\sum\limits_{j=1}^{N}\Big(e^{-2O_t^{(j)}(\omega)}\|f^{(j)}(e^{O_t^{(j)}(\omega)}\bar{x}_\nu^{(j)}(t,\omega))\|^2+|O_t^{(j)}(\omega)|^2\|\bar{x}_\nu^{(j)}(t,\omega)\|^2\Big),