harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2\omega_{12}+\theta_{12}=-\frac{B^1_{11,3}}{\mu}\omega_3, &\omega_{13}=-\frac{B^1_{11,3}}{\mu}\omega_2,~~~~\omega_{1i}=0, i\geq 4, &\omega_{23}=\frac{B^1_{11,3}}{\mu}\omega_1, ~~~~\omega_{2i}=0, i\geq 4.
S_{v}p=[\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&\frac{1}{s}\end{matrix}][\begin{matrix}p_{x}\\p_{y}\\p_{z}\\ 1\end{matrix}]=[\begin{matrix}p_{x}\\p_{y}\\p_{z}\\ \frac{1}{s}\end{matrix}]
I_{\mathcal{M}}[u] =\frac{1}{p}\mathcal{M}(\|u\|_{D^{s,p}}^p)+ \frac{1}{p}\int_{\mathbb{R}^N}V(x)|u|^p dx &\phantom{=}-\frac{1}{2\cdot p_{r;s}^{\uparrow *}}\varepsilon_K\int_{\mathbb{R}^N}(K\ast g(u))g(u)dx-\varepsilon_W\int_{\mathbb{R}^N}W(x)f(u)dx.
\begin{array}{l c r}{x}&{=}&{\alpha_{1}\displaystyle \frac{\alpha_{3}\pm | | \alpha | |}{( \alpha_{1}^{2}+ \alpha_{2}{}^{2})}\ ,}\\{y}&{=}&{\displaystyle - \alpha_{2}\frac{\alpha_{3}\pm | | \alpha | |}{( \alpha_{1}^{2}+ \alpha_{2}^{2})}\ .}\\ \end{array}
\Pi(x)=\pi(x)+\frac{1}{2}\pi(x^{\frac{1}{2}})+\frac{1}{3}\pi(x^{\frac{1}{3}})+\frac{1}{4}\pi(x^{\frac{1}{4}})+\frac{1}{5}\pi(x^{\frac{1}{5}})+\frac{1}{6}\pi(x^{\frac{1}{6}})+\cdot\cdot\cdot
\sum_{\ell=1}^\Delta\sum_{m=1}^{\rho n}n^{m+1}n^{\Delta m}\exp\big(-\rho p^\ell n m/T\big) \le \Delta\sum_{m=1}^{\rho n}\exp\big(2\Delta m\log n-\rho p^\Delta n m/T\big) \le \Delta\sum_{m=1}^{\rho n}\exp\big(-(\Delta+3)m \log n \big)<\Delta n\exp\big(-(\Delta+2)\log n\big)<1/n\,,
f(x\textbf{*}) = e_1f_1(x_1\textbf{*})+e_2f_2(x_2\textbf{*}) &=e_1\overline{f_1(x_1)}+e_2\overline{f_2(x_2)}=(f(x))^{\dagger_3},\; \forall\; x \in \mathcal A.
D_{q}^{n}f(x)=\frac{1}{(1-q)^{n}x^{n}}\sum_{k=0}^{n}(-1)^{k}(\begin{matrix}n\\k\end{matrix})_{q}q^{(\begin{matrix}k\\ 2\end{matrix})-(n-1)k}f(q^{k}x).
\boldsymbol{\eta}(\boldsymbol{x},\,t,\,s) = \boldsymbol{\varphi}(\boldsymbol{x},\,t) - \boldsymbol{\varphi}(\boldsymbol{x},\,t - s),\forall\;\boldsymbol{x}\in \Omega,\ \forall\;t,\,s\in \mathbb{R}^{+}
R (z) = \frac{1}{2km}\sum_{p = 0}^{m-1}\sum_{s = -p}^{k+p}z^s = \frac{1}{2km}\sum_{p = 0}^{m-1}z^{-p}\frac{z^{k+2p+1}- 1}{z - 1}= \frac{1}{2km(z-1)}\sum_{p = 0}^{m-1}(z^{k+p+1}- z^{-p}) = \frac{1}{2km(z-1)}\left( \frac{(z^{k+1}- z^{-m+1})( z^m - 1)}{z-1}\right),
F_{t^c}(t_0) = \frac{1}{\pi-\theta_k}\bigg(\arccos\left(\frac{r_u^{}}{v_ut_0}\right) +\min\bigg\lbrace \arccos\left(\frac{r_u^{}}{r_{u,k}(\boldsymbol{x})}\right), \arccos\left(\frac{r_u^{}}{v_ut_0}\right)\bigg\rbrace\bigg).
g_\varepsilon=g\partial\Omega\cap B_R(x_0)\, &\|g_\varepsilon\|_{H^1(\partial B_R(x_0)\cap\Omega)}\leq C, &\int_{\partial B_R(x_0)\cap\Omega}W(g_\varepsilon)\leq C\varepsilon^2, & g_\varepsilon\to gH^s (\partial B_R(x_0)\cap\Omega),\ 0<s<1.
\frac{P_T\left(1-\rho^*\right)\sigma^{-2}+\sum_{k=1}^{r_s}[\boldsymbol{\Lambda}]_{k,k}^{-2}}{r_s\sum_{k=1}^{r_s}\left(\nu^*-\rho^*[\boldsymbol{\Lambda}]_{k,k}^2\right)^{-1}}=2^{\frac{{R}}{r_s}}\left(\textstyle\prod\limits_{k=1}^{r_s}\frac{[\boldsymbol{\Lambda}]_{k,k}^2}{\nu^*-\rho^*[\boldsymbol{\Lambda}]_{k,k}^2}\right)^{-\frac{1}{r_s}},
(y_1 s_{b_2}\cdots s_{b_2-t_2}y_2) ( \overbrace{u_1' s_{b_2}\cdots}^{u_1}s_{b_1}\cdots \underbrace{s_{b_2-t_2+1}}_{=s_{b_1-t_1}}\overbrace{u_2}^{=u_2'})(v_1 s_{b_1}\cdots s_{b_1-t_1+1}v_2)\;.
G_5(\lambda) = \lambda_5\sum_{\substack{a,b=0 (a,b)\neq(0,0)}}^\infty \frac{(2a+2b-2)!}{a!^2b!^2}\bigg(\frac{\lambda_1\lambda_3}{\lambda_5^2}\bigg)^a\bigg(\frac{\lambda_2\lambda_4}{\lambda_5^2}\bigg)^b.
L=\sum_{k=1}^{n}C_{k}(I_{k})+\pi[L(I_{1},I_{2},...,I_{n-1})-\sum_{k=1}^{n}I_{k}]+\sum_{l=1}^{m}\mu_{l}[F_{l}^{max}-F_{l}(I_{1},I_{2},...,I_{n-1})]
Var[X]=\sigma_{\kappa}^{2}=\frac{1}{\beta}\frac{2+\kappa}{2-\kappa}\frac{4\kappa}{4-9\kappa^{2}}[\frac{\Gamma(\frac{1}{2\kappa}+\frac{1}{4})}{\Gamma(\frac{1}{2\kappa}-\frac{1}{4})}]^{2}
K(s,t) = \frac{1}{\sqrt{\phi(s) \phi(t)}}\left( \frac{\theta(s)}{\theta(t)}\right)^{\frac{a}{2}+ \frac{1}{4}}\exp \left[ \frac{1}{\sqrt{\beta}}\int_{\theta(s)}^{\theta(t)}\frac{db_z}{\sqrt{z}}\right] \mathrm{1}_{s < t},
\int_{B_{1}}dy \int_{}dw\, \frac{g(|y|)}{|\bar X - Y^{N+1}(t)|^{2}}\chi\left(|w|\leq N+1\right)\leq &C \int_{|\bar{w}|\leq C N}d\bar{w}\int_{|\bar{X}-\bar{y}|\leq 4R^{N+1}(t)}d\bar{y}\, \frac{1}{|\bar X - \bar{y}|^{2}}\leq CN^3R^{N+1}(t)\leq CN^4 .
G_{+}^{\ell k}( q^{0}) = i \int_{- \infty}^{\infty}d x^{0}e^{i q^{0}x^{0}}\int d \vec{x}\left( \Phi_{p}, \left( j^{\ell}( \vec{x}, x^{0}) j^{k}( \vec{y}, 0 ) \right)_{+}\Phi_{p}\right)
I_{m-1}(t) = \int_0^\infty \left( a_{m-1}+ t \xi + \sum_{k=1}^{m-1}(-1)^{k-1}{m-1 \choose k}\frac{1}{k}e^{-kt \xi}\right) \xi^{-m}\pi(d\xi) = \int_0^\infty \int_0^{\xi t}\left( 1 + \sum_{k=1}^{m-1}(-1)^{k}{m-1 \choose k}e^{-kw}\right) dw \xi^{-m}\pi(d\xi),
\varphi(y)=A'\,e^{-iky}F\left(-s,s+1;1+k;\frac{i\,e^{-iy}}{2\,\sin y}\right)+B'\,e^{iky}F\left(-s,s+1;1-k;\frac{i\,e^{-iy}}{2\,\sin y}\right).
\P \{\ell^{(A)}_i \leq m-i \}= \sum\limits_{k=0}^\infty \mu(k) \left(\P\left\{\forall t > 0:t > \sum\limits^{i + S_{t}}_{z = i+1}\frac{1}{A(z)}S_{t}< m-i \right\}\right)^k,
\frac{\partial^{2}}{\partial t^{2}}\phi_{\alpha j}( t ) + 3 \frac{\dot{R}( t )}{R ( t )}\frac{\partial}{\partial t}\phi_{\alpha j}( t ) + \left( m^{2}+ \xi{\bf R}+ \frac{{\bf k}^{2}}{R^{2}( t )}\right) \phi_{\alpha j}( t ) = 0 ,
V_{\cal{C}}^{-1}({\bf{r}})=\exp\left(ig{\cal X}^\alpha({\bf{r}}){\textstyle\frac{\lambda^\alpha}{2}}\right)\,\exp\left(\,ig{\overline{{\cal{Y}}^\alpha}}({\bf{r}}){\textstyle\frac{\lambda^\alpha}{2}}\,\right)\;.
{V_{T}}^{G}( \psi , z ) = : \left( \prod_{a = 1}^{M}{\frac{i}{( n_{a}- 1 ) !}}{\frac{d^{n_{a}}}{d z^{n_{a}}}}C^{i_{a}}( z ) \right) e^{i \lambda \cdot C ( z )}: \gamma_{\lambda}\, ,
\frac{\partial^{3}f}{\partial\eta^{3}}+f\frac{\partial^{2}f}{\partial\eta^{2}}+\xi(\frac{\partial f}{\partial\xi}\frac{\partial^{2}f}{\partial\eta^{2}}-\frac{\partial^{2}f}{\partial\xi\partial\eta}\frac{\partial f}{\partial\eta})=0
x_1 = \frac{x_0}{2}- \frac{1}{2x_0}- r_0 &= \frac{k_0+1/\sqrt{3}}{2}- \frac{1}{2\big(k_0+1/\sqrt{3})}-\frac{1}{2}\bigg( \sqrt{3}+ 2k_{1}+ k_0 -\frac{1}{k_0+1/\sqrt{3}}\bigg) &= -\frac{1}{\sqrt{3}}- k_1.
\frac{1}{2}\Delta |\nabla \psi|^2 \ge |\mathrm{Hess}_{\psi}|^2 +\langle \nabla \Delta \psi, \nabla \psi\rangle +K|\nabla \psi|^2 &\ge \frac{(\mathrm{tr}(\mathrm{Hess}_{\psi}))^2}{n}+\langle \nabla \Delta \psi, \nabla \psi\rangle +K|\nabla \psi|^2 &=\frac{(\Delta \psi)^2}{n}+\langle \nabla \Delta \psi, \nabla \psi\rangle +K|\nabla \psi|^2.
\mathrm{Res}_{Q_{\tau}}f_{\theta}dz=\begin{cases}\frac{1}{2\pi i}\nu_{\tau}^{(N)}(f) \tau \in \mathbb{H}, \frac{\alpha_\tau}{2\pi i}\left(\nu_{\tau}^{(N)}(f)-\frac{k}{12}\right) \tau \in \mathcal{C}_N.\end{cases}
\int_0^\infty (e^{-ux}-1) x^{-1-\beta}\, dx = \int_0^u \left(\frac{d}{dv}\int_0^\infty (e^{-vx}-1) x^{-1-\beta}\, dx\right) \, dv = -\int_0^u v^{\beta-1}\Gamma(1-\beta) \, dv = -\frac{\Gamma(1-\beta)}{\beta}u^\beta
S_{P V}( \Phi , \Phi^{*}, \underline{\Phi}_{i}) = \frac{1}{2}\sum_{i = 1}^{N}{\underline{\Phi}}_{i}^{A}S_{A B}{\underline{\Phi}}_{i}^{B}- \frac{1}{2}M_{i}^{2}{\underline{\Phi}}_{i}^{A}T_{A B}{\underline{\Phi}}_{i}^{B},
\frac{d}{dr}\bigg\{J(ru_1'(r)+2, ru_2'(r)+2)+(1+a_1+a_2)r^2\bigg[a_2e^{u_1}+a_1e^{u_2}&\qquad-\frac{a_2(1+a_1)}{2}e^{2u_1}-\frac{a_1(1+a_2)}{2}e^{2u_2}+a_1a_2e^{u_1+u_2}\bigg]\bigg\}= (1+a_1+a_2)r\left[a_2(1+a_1)e^{2u_1}+a_1(1+a_2)e^{2u_2}-2a_1a_2 e^{u_1+u_2}\right].
\begin{aligned}0 \geq \dot{R}= -\iint_{\mathbb T \times \mathbb R}\sin(\theta-\phi)\omega f(\theta,\omega) ~ d\theta d\omega &+KR \iint_{\mathbb T \times \mathbb R}\sin^{2}(\theta-\phi)f(\theta,\omega) ~ d\theta d\omega &\geq-M+K\underline{R}\cos^{2}\gamma ( 1 -{\mathcal M}(L_{\gamma}^+) -{\mathcal M}(L_{\gamma}^-)),\end{aligned}
e^{-iH_{\hbar}t/\hbar}D\left( \left( H_{\hbar}+\widetilde{C}\right)^{\beta}\right) =D\left( \left( H_{\hbar}+\widetilde{C}\right)^{\beta}\right) \left\Vert e^{-iH_{\hbar}t/\hbar}\psi\right\Vert_{\left( H_{\hbar}+\widetilde{C}\right)^{\beta}}=\left\Vert \psi\right\Vert_{\left(H_{\hbar}+\widetilde{C}\right)^{\beta}}.
\mathfrak{a}_i\triangleright b_i \Rightarrow b_i=\mathfrak{a}_i(\tau_i(b_i))=\mathfrak{a}_i(\delta_i(\tau_0(b_0)))=\mathfrak{a}_i(\delta_i(\tau_0(a_0)))=\mathfrak{a}_i(\tau_i(a_i))=a_i.
\delta_v(\Theta(L)(x,y,z))(a)=\sum_{i+j=N+1;~ i,j>0}-f_0^{\lambda_i(x,\lambda_j(y,z))}+f_0^{\lambda_i(y,\lambda_j(x,z))}+f_0^{\lambda_i(\lambda_j(x,y),z)}(a).
||s||^2_{h^\mu\otimes\mathrm{det}\,h^\mu}\geq||s||^2_{h^\lambda\otimes\mathrm{det}\,h^\lambda}\mathrm{i.e.}f_{s,\mu,\lambda}:=||s||^2_{h^\mu\otimes\mathrm{det}\,h^\mu}-||s||^2_{h^\lambda\otimes\mathrm{det}\,h^\lambda}\geq0.
K_\Omega(\gamma_n^{-1}z_0, z_0) + K_\Omega(z_0, y_n) - K_\Omega(\gamma_n^{-1}z_0, y_n) & = K_\Omega(\gamma_n^{-1}z_0, z_0) + K_\Omega(z_0, y_n) - K_\Omega(z_0, \gamma_n y_n) & \geq K_\Omega(\gamma_n^{-1}z_0, z_0) - K_\Omega(\gamma_n y_n, y_n) \rightarrow \infty.
G ( z_{2}, \cdots , z_{n}, z_{1}\tau^{2 n})^{n \cdots \hat{i}\cdots 1 i}= \delta_{i}G ( z_{1}, \cdots , z_{n})^{i n \cdots \hat{i}\cdots 1}, ~ \delta_{i}= \delta_{1}\tau^{2 - 2 i}.
A_2 =\big\||x|^{-\tau}|u|^{p-1}(I_\alpha \ast|\cdot|^{-\tau-1}|u|^{p})\big\|_{L_x^{\frac{2n}{n+2}}}&\leq \||x|^{-\tau}|u|^{p-1}\|_{L_x^{a_2}}\||x|^{-\tau-1}|u|^p\|_{L_x^{b_2}}&\lesssim \|\nabla u\|^{2p-1}_{L_x^r}
\left(z_{3}s^{\prime}z_{1}\right)^{-1}s^{\prime-1}\left(z_{3}s^{\prime}z_{1}s^{\prime}\right) =\left[\left(z_{3}s^{\prime}z_{1}\right)^{-1},s^{\prime-1}\right]\mbox{and}s^{\prime-1}\left(z_{3}s^{\prime}z_{1}\right)^{-1}\left(z_{3}s^{\prime}z_{1}s^{\prime}\right) =e.
\partial T_*(\beta'_*(\lambda_x))(\gamma) = T_*(\beta'_*(\sum_y s^x_y\times \lambda_y)) + T_*(\beta'_*(\lambda_x))(\partial \gamma) &= \sum_y T_*(a^x_y\cdot \beta'_*(\lambda_y))(\gamma) + T_*(\beta'_*(\lambda_x))(\partial \gamma) &= \sum_y T_*(\beta'_*(\lambda_y))\circ T_*(a^x_y)(\gamma) + T_*(\beta'_*(\lambda_x))(\partial \gamma) &=\Psi(d(\gamma \otimes x)).
[ \tilde{C}^{2}] W ={\frac{1}{2}}\sin{3 \tilde{\theta}}+ \sin{\tilde{\theta}}({\frac{1}{2}}- 4 \tilde{k}^{2}\tilde{\gamma}^{2}) - 2 \tilde{k}\tilde{\gamma}\cos{\tilde{\theta}}- 4 8 \tilde{k}^{3}t \cos{\tilde{\theta}},
n_i=\inf\{n>n_{i-1}:\;2^{p_1 n}d_1(t,T_{n_{i-1}})+2^{p_2 n}d_2(t,T_{n_{i-1}})> & >2(2^{p_1 n}d_1(t,T_n)+2^{p_2 n}d_2(t,T_n))\}.
_{2}\phi_{2}\left(\sqrt{b},-\sqrt{b};\sqrt{c},-\sqrt{c};\sqrt{q},-\frac{c}{b}\right)=\frac{\left(c/b;q\right)_{\infty}}{\left(c;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}/2}}{\left(q;q\right)_{n}}\left(-\frac{c}{b}\right)^{n}A_{q}\left(-cq^{n-1/2}\right)
D(0, \frac{\varepsilon}{2}) =\sigma_{\frac{\varepsilon}{2}}([iP,Q]_{\ast})=\sigma_{\frac{\varepsilon}{2}}([\psi(iP),\psi(Q)]_{\ast}) &=\sigma_{\frac{\varepsilon}{2}}( i(\psi(P)\psi(Q)+\psi(Q)\psi(P))),
C_3=1+a_1+a_1 b+b^2+a_1^2 k, C_2(Y)=1+a_1+a_1 b+b^2+(1+b^2+a_1^2 k )Y, C_1(Y)=b+a_1 b+b^2+k+a_1 k+a_1 b k+b^2 k+a_1^2 k^2+(1+a_1+b^2+a_1^2 k)Y, C_0(Y)=b+a_1 b+b^2+a_1 k+a_1^2 k^2+(a_1+b+b^2+k +a_1^2 k +b^2 k +a_1^2 k^2 )Y.
\sum_{k=0}^{u+2}s_k\ \mathbb{P}\Biggl(\bigcap\limits_{n=1}^{\infty}\Biggl\{u+n+3-k-\sum_{i=1}^{n}Z_i>0\Biggr\}\Biggr)=\sum_{k=0}^{u+2}s_k \ \varphi(u+3-k),
\vert \beta_{\omega^{\prime}\omega}^{B}\vert^{2}= - \mathrm{R e}\int \! \! \! \int_{- \infty}^{\infty}d u d v \, \mathrm{e x p}\, [ i \omega u + i \omega^{\prime}f ( u ) - i \omega^{\prime}v - i \omega g ( v ) ] ,
\frac{dE}{d\Omega}=\frac{2{\cal G}}{(2\pi)^{D-2}}\frac{1}{D-2}\frac{\gamma_{1}^2m_{1}^2v_{1}^2(v_1+v_2)^2\sin{\theta_1}^4}{(1-v_1\cos\theta_1)^2(1+v_2\cos\theta_1)^2}\times \omega_{c}^{D-3}\,.
\begin{array}{c c c c c c c}{{{\theta}}}&{{=}}&{{\partial \,{\cal K}= \partial_{i}{\cal K}d z^{i}}}&{{;}}&{{{\bar{\theta}}}}&{{=}}&{{{\bar{\partial}}\,{\cal K}= \partial_{i^{\star}}{\cal K}d{\bar{z}}^{i^{\star}}}}\\ \end{array}
{_2}\phi_{1}\left[ \begin{array}{c}{\begin{array}{c c}{a ,}{b}\end{array}}{c}\end{array}; q , u \right] = \sum_{n = 0}^{\infty}\frac{( a ; q )_{n}( b ; q )_{n}}{( q ; q )_{n}( c ; q )_{n}}u^{n},
\varphi_k = \frac{1}{(l+k)!}\lim_{\lambda\rightarrow \lambda_0}\left(\frac{\partial}{\partial\lambda}\right)^{l+k}(c(\lambda)^{-1}P(\lambda)f_+^{\lambda+m}) =\sum_{j=0}^{l+k}Q_{kj}(f_+^{\lambda_0+m}(\log f)^j)
|F_t(z_1,t)-F_t(z_2,t)|= 2|Re F_t(z_2,t)-Re F_t(0,y,t)|x=2|Re F_{tx}(\tilde{x},y,t)|x(t) \leq 2\|F_{t\zeta}\|_{L^{\infty}(\Omega(t))}x(t)\leq 20\epsilon x(t).
E(q)\subseteq \bigcup_{k\in\mathbb N}\left(\bigcup_{j=1}^{2N+1}(a_{k,j}, b_{k,j})\right)\cup (a_{m_1}, \infty)=\bigcup_{j=1}^{2N+1}\left(\bigcup_{k\in\mathbb N}(a_{k,j}, b_{k,j})\right)\cup (a_{m_1}, \infty).
[\nabla^{(i)}]_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{i}}=\frac{\partial^{i}}{\partial r_{\alpha_{1}}\partial r_{\alpha_{2}}\cdot\cdot\cdot\partial r_{\alpha_{i}}}where\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{i}=1,2,\cdot\cdot\cdot,n.
_{{C}_i}^{\eta}(1_{B_n^{\eta}})= \P_\lambda \left(1_{B_n^{\eta}}(\boldsymbol{\omega},\boldsymbol{\varepsilon}) \neq 1_{B_n^{\eta}}(\boldsymbol{\omega_{[{C}_i]}},\boldsymbol{\varepsilon_{[{C}_i]}})\right),
\int_{\operatorname{supp}\varphi}|\Gamma_n |^{1+\vartheta}dx \leq \left\|\int_{\mathbb{R}^N}\kappa_\varphi (x,y) |v_n(y)|^{p_{r;s}^{\uparrow *}}dy\right\|_{L^\sigma_x(\operatorname{supp}\varphi)}^{1+\vartheta}&\phantom{=}\times \| |\varphi v_n|^{p_{r;s}^{\uparrow *}}\|_{\ell_r}^{1+\vartheta}\leq C_5.
\sum_{k=1}^2\left[ B_{ik}(t)\dot{B}_{jk}(t)-\dot{B}_{ik}(t)B_{jk}(t)\right]=\sum_{k=1}^2\left[B_{ik}^\ast(t)\dot{B}_{jk}^\ast(t)-\dot{B}_{ik}^\ast(t)B_{jk}^\ast(t)\right]=0\;.
P^{(2)}_+(z) \left(P^{(2)}_-(z)\right)^{-1}= \mathbb{I}+\frac{A_n^{(2)}(z)}{n^6 z}- \frac{A_n^{(1)}(0)A_n^{(1)}(z) A_n^{(1)}(0)}{n^9z^3}+ \mathcal{O}(n^{-1})
(-1)^{\ell_1+\ell_2+\cdots+\ell_r-r}\det_{1\leq k,\ell\leq n}\left( e_{\kappa_\ell-\ell+k}(\c) \right)= \det_{1\leq k,\ell\leq n}\left( e_{\gamma_\ell-\ell+k}(\c) \right) = s_{\gamma'}(\c)\,
{\cal{J}}_{1}=\frac{x^2}{\varrho}\,; \quad{\cal{J}}_{2}=n\varrho \,; \quad{\cal{J}}_{3}= 2 \alpha x^2 - v^2 \varrho + \frac{x v}{2}\varrho_z \,; \quad \varrho = \left( 1-2\alpha z^2 \right) \,.
M=(B_{n_1}|\cdots|B_{n_d})= \left(\begin{array}{cccc|c|cccc}x_{1,0}x_{1,1}\dots x_{1,n_1-1}\dots x_{d,0}x_{d,1}\dots x_{d,n_d-1}x_{1,1}x_{1,2}\dots x_{1,n_1}\dots x_{d,1}x_{d,2}\dots x_{d,n_d}\end{array}\right),
\Re(\widehat{w})-P\left(\widehat{z},\widehat{\zeta}\right)-f^*\left(\Re(\widehat{z_1})\right) =e^{-2a_1-na_2}\big(u-P(z,\zeta)\big)-f^*\big(e^{-a_1-a_2}x_1\big),
\|(w_n)_+\|_{H^1(\Omega)}^2 =\|(w_n)_+\|_{L^2(\Omega)}^2+(\nabla (w_n)_+,~\nabla w_n)_{L^2(\Omega)}&\to\|w_+\|_{L^2(\Omega)}^2+(\nabla w_+,~\nabla w)_{L^2(\Omega)}=\|w_+\|_{H^1(\Omega)}^2,
\|\partial_xw\|^2_{L^2\left(\Omega_1\right)}+\|\partial_zw\|^2_{L^2\left(\Omega_1\right)}+\|\partial_xw\left(\cdot,z_\iota\right)\|^2_{L^2\left(\Omega_x\right)}\leq c\|\partial_x\xi_2^\prime\|^2_{L^2\left(\Omega_2\right)}\leq c\varepsilon\left(z_\iota\right).
\lambda = K(u) \leq C \int |u|^3dx \leq C\|u\|_{H^{\frac{1}{2}}(\mathbb{R})}^{7/3}\|\partial_x^{-1}u\|_{L^2(\mathbb{R})}^{\frac{2}{3}}&\leq C \left( ||u||^2_{L^2(\mathbb{R})}+ ||\partial_x^{\frac{1}{2}}u||^2_{L^2(\mathbb{R})}+ ||\partial_x^{-1}u ||^2_{L^2(\mathbb{R})}\right)^{3/2}
f_{U U U}= 4 \pi^{2}\frac{( T -{\bar{T}})}{( U -{\bar{U}})^{2}}\int \frac{d^{2}\tau}{\tau_{2}^{2}}{\bar{F}}({\bar{\tau}}) \sum_{P_{L}, P_{R}}P_{L}P_{R}P_{R}P_{R}e^{\pi i \tau | P_{L}|^{2}}e^{- \pi i{\bar{\tau}}| P_{R}|^{2}}.
D F ( \xi , x , y )\left[\begin{matrix}0 1 X_3 ( \xi , x , y )\end{matrix}\right]= \frac{1}{\tau ' \left( \rho_{k (\xi)}(x) \right)}\left[\begin{matrix}0 1 X_3 \circ F ( \xi , x , y )\end{matrix}\right]
log(\frac{\{C\}^{c}\{D\}^{d}}{\{A\}^{a}\{B\}^{b}})=log(\frac{(\gamma_{C})^{c}(\gamma_{D})^{d}}{(\gamma_{A})^{a}(\gamma_{B})^{b}})+log(\frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}})
\sum_{k \in \mathbb N_0}\left|<f, T_k\varphi>\right|^2= \sum_{k \in \mathbb N_0}\left|\int_{\mathcal O}r(\xi)w(\xi)\chi_{u(k)}(\xi)d\xi\right|^2 = \int_{\mathcal O}|r(\xi)|^2|w(\xi)|^2 d\xi.
\frac{|-1 + z_{j ,n}|}{|z_{j ,n}|}~ \leq ~ \, \kappa(1,a_{\max}) =\frac{1 -\exp\bigl(\frac{- \pi}{a_{\max}}\bigr)}{2 \exp\bigl(\frac{\pi}{a_{\max}}\bigr) - 1}=0.171573\ldots
\Phi^{\left( chiral)\right)}=\exp \left( \frac 12\theta^\alpha \sigma_{\alpha \dot{\beta}}^\mu \bar{\theta}^{\dot{\beta}}\partial_\mu \tilde{\partial}_{+}\right) \left( \phi +\theta^\alpha \chi_\alpha +\theta^\alpha \theta_\alpha \tilde{\partial}_{+}f\right) (x,y).
\eta_{m \kappa}\left( 0 \right) + \eta_{- m , \kappa}\left( 0 \right) = \left( N_{\kappa}^{+}+ N_{\kappa}^{-}\right) \pi + \left( - 1 \right)^{l}\frac{\pi}{2}\left[ \sin^{2}\eta_{m \kappa}\left( 0 \right) + \sin^{2}\eta_{- m , \kappa}\left( 0 \right) \right] \, ,
u_s\cdot\nabla u_s=\Delta u_s-\nabla p_{u_s}-u_\infty\cdot\nabla u_s, &\mbox{div $u_s$}=0, &u_s|_{\partial\Omega}=-u_\infty, &u_s\to 0 \quad\mbox{as $|x|\to\infty$}.
\Psi_{\infty,u}'(t)=s_1 t^{2s_1-1}|\nabla_{s_1}u|_2^2+s_2t^{2s_2-1}|\nabla_{s_2}u|_2^2-\frac{d}{t^{d+1}}\displaystyle\int_{\mathbb{R}^{d}}\widetilde{G}(t^{\frac{d}{2}}u(x))dx=\frac{1}{t}P_\infty(t*u)
\zeta_{\infty,0}(s,\,c)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}}\left(2\gamma-\log\left(\frac{c}{4}\right)+8\,\sum_{n=1}^{\infty}\frac{1}{2n-1}\cdot\frac{1}{e^{\pi\sqrt{c}(2n-1)}-1}\right)+O\left(s-1\right).
\int_{\Omega}\eta\langle \Delta\big(|du|^{p-2}du\big), du\rangle d\mu =-\int_{\Omega}\eta\langle (dd^*+d^*d)\big(|du|^{p-2}du\big), du\rangle d\mu &=-\int_{\Omega}\langle d^*\big(\eta d\big(|du|^{p-2}du\big)\big), du\rangle d\mu &=-\int_{\Omega}\langle \eta d\big(|du|^{p-2}du\big), d(du)\rangle d\mu=0.
\frac{d q_{2}( t )}{d t}= \frac{p_{2}( t )}{m_{2}}\, , \qquad \frac{d p_{2}( t )}{d t}= - m_{2}\omega_{2}^{2}q_{2}( t ) - g ( t ) \langle q_{1}( t ) \rangle_{\psi_{1}}\, .
\|(1-P_R)u\|_{L^2}\leq 1/R\left\|\sum_{1\leq l\leq s}\prod_{i\in \mathcal{I}_l}(1+|\omega_i|^2)^{1/2}\widehat{u}\right\|_{L^2}=1/R\left\|\sum_{1\leq l\leq s}\mathcal{K}_{\mathcal{I}_l}u\right\|_{L^2}.
S_3(X)=\sum_{q\le P}\int_{|\beta|\le\frac{1}{qP}}\sum_{|v|\le P}\sigma(v;\beta,q)\sum_{\substack{a=1 (a,q)=1}}^qf(\frac{a}{q}+\beta)^3h(-\frac{a}{q}-\beta)e\Big(-\frac{\overline{a}\,v}{q}\Big)d\beta.
P_{12}(x,t)=\frac{g_1(x,t)h_1(x,t)}{g_1(x,t)h_2(x,t)-g_2(x,t)h_1(x,t)}, P_{21}(x,t)=-\frac{g_2(x,t)h_2(x,t)}{g_1(x,t)h_2(x,t)-g_2(x,t)h_1(x,t)}, P_{11}(x,t)=-\frac{P_{12}(x,t)g_2(x,t)}{g_1(x,t)}, P_{22}(x,t)=-\frac{P_{21}(x,t)g_1(x,t)}{g_2(x,t)}
- i \Sigma_{a b}( q , p ) = - ( i e )^{2}\epsilon_{a}\epsilon_{b}\int{\frac{d^{4}k}{( 2 \pi )^{4}}}{\frac{1}{i}}\left[ D^{( 0 )}\right]_{b a}^{\rho \sigma}( k ) \gamma_{\sigma}i S_{b a}^{( 0 )}({p}{\mkern - 9 . 0 m u /}- k ) \gamma_{\rho}.
\begin{aligned}\mu(A) = \frac{C^{u-n}}{\sum_{p=0}^n \lfloor C^p \rfloor C^{-p}}\mu(\hat{A}) & \leq \frac{C^{u-n}}{\sum_{p=0}^n (C^p-1) C^{-p}}\mu(\hat{A}) & \leq \frac{C^{u-n}}{n+1 - \frac{1}{1-1/C}}\mu(\hat{A}) & \leq C \frac{n+1}{n+1 - \frac{1}{1-1/C}}\sup_{k \in \mathbb{N}}\frac{1}{n+1}\sum_{j=k}^{k+n}\tau_A(C,j).\end{aligned}
\frac{1}{N^2}\mathbb{E}\left[ \left( \mathrm{tr\,}A^k\right)^2 \right] = \frac{1}{N^{2+k}}\sum_{\underline{i}, \underline{j}}\mathbb{E}\left[ X_N(\underline{i}) \, X_N( \underline{j}) \right]
\beta_\wedge : \mathsf{\Lambda}^2 (\mathbb{C}^3)\times \mathbb{C}^3\to \mathbb{C}^3, \left( \begin{bmatrix}0 a b -a 0 c -b -c 0\end{bmatrix},\begin{bmatrix}x y z \end{bmatrix}\right)\mapsto \begin{bmatrix}ay+bz -ax+cz -bx-cy\end{bmatrix}.
\Big(\sum_{w\in S_n}w\Big) c_{w_0}^{X^{-1}}E_\lambda =\mathbf{1}_0 E_\lambda \mathbf{1}_Y =p^\lambda_X c_{v_\lambda}^{X^{-1}}\mathbf{1}_\lambda E_\lambda \mathbf{1}_Y =p^\lambda_X c_{v_\lambda}^{X^{-1}}E_\lambda \mathbf{1}_\lambda \mathbf{1}_Y =t^{-\frac12\ell(w_\lambda)}W_\lambda(t)p^\lambda_X c_{v_\lambda}^{X^{-1}}E_\lambda\mathbf{1}_Y
\Lambda_{W}^{( 0 )}( \zeta ; r ) = 1 , \qquad \Lambda_{W}^{( 1 )}( \zeta ; r ) = \Lambda ( \zeta ; r ) = \frac{\Theta_{q^{4}}( r \zeta^{2}) \Theta_{q^{4}}( q^{2}r \zeta^{- 2})}{\Theta_{q^{4}}( r \zeta^{- 2}) \Theta_{q^{4}}( q^{2}r \zeta^{2})}\zeta^{- 2},
N_i\colon\varphi\longmapsto N_i\varphi=:v \;\Longleftrightarrow \;\begin{cases}\Delta v -v =0 \textrm{on $\Omega$}begin{align*}1mm]\frac{\partial v}{\partial \nu}=\varphi \textrm{on $\Gamma_i$}begin{align*}1mm]\frac{\partial v}{\partial \nu}=0 \textrm{on $ \partial \Omega \setminus \Gamma_i$.}\end{cases}
\int{\cal D}Q \;{\cal D}c \;{\cal D}{\bar{c}}~ \exp \! \left[ \frac{i}{2 g_{0}^{2}}\, S^{\mathrm{d i a m}}\right] = \left( \mathrm{d e t}^{- 1 / 2}\, D^{2}[ B ] \, \right)^{D - 2}.
[[\mathbf{u},\mathbf{v}],[\mathbf{u}',\mathbf{v}']]=0,\ \ [\mathbf{u},\mathbf{v}]\mathbf{w}+[\mathbf{v},\mathbf{w}]\mathbf{u}+[\mathbf{w},\mathbf{u}]\mathbf{v}=0,\ \ \mathbf{w}[\mathbf{u},\mathbf{v}]+\mathbf{u}[\mathbf{v},\mathbf{w}]+\mathbf{v}[\mathbf{w},\mathbf{u}]=0
d s^{2}= ( H^{\prime}H )^{1 / 4}\left[ H^{- 1}\left( - ( d x^{0})^{2}+ ( d x^{1})^{2}\right) + H^{- 1}\left( ( d x^{2})^{2}+ ( d x^{3})^{2}\right) + d y^{i}d y^{i}\right] ,
f_{\scriptscriptstyle{L , L}}( a ) = \frac{1}{2 \pi a^{L}}\int_{0}^{a}d r r^{L}\int_{0}^{2 \pi}d \theta e^{i ( 1 - L ) \theta}e^{2 \phi ( r , \theta )}\sum_{m = 0}^{\infty}b_{\scriptscriptstyle{m , L}}r^{m}e^{i m \theta}.
\sum_{n = 0}^{\infty}F ( n ) = \frac{1}{2}\Bigl ( F ( 0 ) + F ( \infty ) \Bigr ) + \int_{0}^{\infty}d x F ( x ) + \sum_{k = 1}^{\infty}\frac{B_{2 k}}{( 2 k ) !}\Bigl ( F^{( 2 k - 1 )}( \infty ) - F^{( 2 k - 1 )}( 0 ) \Bigr ) ,
f_*\widetilde{Q}(E) =f_* \sum_a (f^* Q_a) E^a = \sum_a Q_a f_* E^a = \sum_a Q_a \sum_{\ell=1}^d Z_\ell^{a}M_\ell = \sum_{a}\sum_{\ell=1}^d Q_a Z_\ell^{a}M_\ell = \sum_{\ell=1}^d{Q}(Z_\ell) M_\ell.
\widetilde{\Phi}_i=\sum_{k_{i-1}\in\mathcal{I}_{e_{i-1}}}\sum_{\ell_i\in\mathcal{I}_{e_{i}}}m_{e_i,\ell_i}^*\otimes m_{e_{i-1},k_{i-1}}\otimes \bigl((m_{e_{i-1},k_{i-1}}^*\otimes\textup{id}_{S_i})\Phi_i(m_{e_i,\ell_i})\bigr)
\left|\frac{A+A}{A+A}\right| \geq{\sum_{i=1}^{k-1}(n_{i+1}+n_i-1)}&=2\left(\sum_{i=1}^{k}n_i\right)-n_1-n_k-(k-1) &=2|A|^2-n_1-n_k-k+1 ,
L(e_{ij})=\sum_{x<i}C_{xi}^{ii}e_{xj}+C_{ij}^{ij}e_{ij}-\left(C_{ji}^{ii}+C_{ji}^{jj}\right)e_{ii}+C_{ji}^{ij}e_{ji}+\sum_{y>j}C_{jy}^{jj}e_{iy}.
G(q)\cdot H(q) = \prod_{t=0}^\infty \frac{1}{(1-q^{5n+1})(1-q^{5n+4})}\cdot \frac{1}{(1-q^{5n+2})(1-q^{5n+3})}= \prod_{t=1}^\infty \frac{(1-q^{5t})}{(1-q^t)}= \sum_{n=0}^\infty d_5(n)q^n.
U_n(X)_t = \sup_{s\in[0,t]}\|\Delta_n^{1-\frac m2}\sum_{i=1}^{\lfloor s/\Delta_n\rfloor-k+1}\bigotimes_{j=1}^k\tilde{\Delta}_{i+j-1}^n X^{\otimes m_j}-\int_0^t\rho_{\Sigma_s}^{\otimes k}(m_1,...,m_k)ds\|_{\mathcal H^m}.
E[(\psi^n(\alpha))^2 (\int_0^T e^{-\alpha(\overline{X(s)}}(e^{\alpha X(s)}- e^{\alpha X^n(s)})ds)^2]=(\psi^n(\alpha))^2 E[T^2 (e^{-\alpha(\overline{X(t_0)}}(e^{\alpha X(t_0)}- e^{\alpha X^n(t_0)})^2]
W_{\mathrm{T}}^{( \xi )}= \frac{1}{4 \xi}\int d^{4}x \, \mathrm{t r}\, F_{\mu \nu}\tilde{F}^{\mu \nu}- \frac{1}{2 4 \xi}\epsilon^{\mu \nu \rho \sigma}\bar{Q}_{\mu}^{( \xi )}\bar{Q}_{\nu}^{( \xi )}\bar{Q}_{\rho}^{( \xi )}\bar{Q}_{\sigma}^{( \xi )}\int d^{4}x \, \mathrm{t r}\, \phi^{2},
q_{X_1Y_1X_2Y_2}(x_1,y_1,x_2,y_2) =p_{XY}(x_1,y_1)p_{XY}(x_2,y_2) r_{X_1Y_1X_2Y_2}(x_1,y_1,x_2,y_2) =p_{XY}(x_1, y_2)p_{XY}(x_2, y_1)
\delta \, M_{\beta , \gamma}^{2}= - \frac{2 \, G_{N}}{\pi \, \ell_{s}^{2}}\int \frac{d^{d}\, k \d \, \omega}{( 2 \pi )^{d + 1}\, [ \mathrm{\boldmath ~ k ~}^{2}- \omega^{2}- i \varepsilon ]}\int d \tau_{1 2}\, d \sigma_{1}\, d \sigma_{2}\, e^{- i \, \ell_{s}^{2}M \omega \, \tau_{1 2}}\, \langle W ( k , 1 , 2 ) \rangle_{\beta , \gamma}\, ,
\min \left\{1 + \frac{h_\nu + \int \log \lambda \, d \nu}{\int \log \tau ' \, d \nu}, \frac{h_\nu}{- \int \log \lambda \, d \nu}\right\}= \frac{h_\nu}{\int \log \tau ' \, d \nu}+ \min \left\{1 , \frac{h_\nu}{- \int \log \lambda \, d \nu}\right\}\cdot \left( 1 + \frac{\int \log \lambda \, d \nu}{\int \log \tau ' \, d \nu}\right) .
U_0(\hat{x}):=\lim_{t \to 0^+}U_t(\hat{x})=\frac{\gamma}{1-\gamma}\left[\left(\textstyle\sum_i(\theta_0^i)^{\frac{1}{\gamma}}\right)^\gamma\left(\hat{\phi}+\frac{\eta}{\gamma}\,\hat{x}\right)^{1-\gamma}-1\right], \ \hat{x}\in \hat{X},
A_{1,1}^1 : = \sup_{t > 0}\bigg( \int_0^t [V_2]^{\frac{q}{p}}(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}V^{- 1}(t), A_{1,2}^1 : = \sup_{t > 0}W_*^{\frac{1}{q}}(t) \bigg( \int_0^t \bigg( \frac{V_2(\tau)}{V(\tau)}\bigg)^{\frac{1}{1-p}}v(\tau)\,d\tau \bigg)^{\frac{1 - p}{p}};
\int_{3R^N(t)\leq |\mu-x|}\frac{\rho^N(x,t)}{|\mu-x|^2}\,dy =\int_{3R^N(t)\leq |\mu-x|}\frac{f^N(x,v,t)}{|\mu-x|^2}\,dxdv\leq & \int_{2R^N(t)\leq |\mu-x|}\frac{f_0^N({x},{v})}{|\mu-X^N(t)|^2}\,dxdv