harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{rank}\, \begin{bmatrix}t_{N-1, N-2}Q_{N-1}T P^\perp t_{N, N-2}Q_N T P^\perp \vdots \vdots t_{N-3, N-2}Q_{N-3}T P^\perp \end{bmatrix}= \mathrm{rank}\, R T R^\perp = 1,
\big|\int_M\langle\nabla \sqrt{1+|\nabla\triangle e_\lambda|^2}, \nabla\triangle e_\lambda\rangle dV\big|\leq\|\nabla^2\triangle e_\lambda\|_{L^2(M)}\|\nabla\triangle e_\lambda\|_{L^2(M)}&\lesssim\sqrt{\lambda}\sqrt{\lambda^3}\|\triangle e_\lambda\|_{L^2(\partial M)}^2=\lambda^2\|\triangle e_\lambda\|_{L^2(\partial M)}^2.
[\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{12}&\sigma_{22}&\sigma_{23}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{matrix}]=2\mu[\begin{matrix}\epsilon_{11}&\epsilon_{12}&\epsilon_{13}\\ \epsilon_{12}&\epsilon_{22}&\epsilon_{23}\\ \epsilon_{13}&\epsilon_{23}&\epsilon_{33}\end{matrix}]+\lambda I(\epsilon_{11}+\epsilon_{22}+\epsilon_{33})
\Psi^{Ai}(\zeta) \sim \frac{e^{\frac{i\pi}{12}}}{2\sqrt{\pi}}\zeta^{-\frac{\sigma_3}{4}}\sum_{j=0}^{\infty}\begin{pmatrix}(-1)^{j}s_{j}s_{j}-(-1)^{j}\nu_j \nu_j \end{pmatrix}e^{-\frac{i\pi}{4}\sigma_3}\left(\frac{2}{3}\zeta^{\frac{3}{2}}\right)^{-j}, \zeta\rightarrow\infty.
\sum_{x=0}^n |\tilde V_x(m)|^2= \sum_{x=0}^n \sum_{x'=1}^nM_{x,x'}(m) \tilde V_{x'}(m) \tilde V^{\star}_{x}(m)+ \sum_{x=0}^n\tilde v_x(m) \tilde V^{\star}_{x}(m).
\begin{cases}\|f\|_{\mathcal{L}^{\rho}_{T}(\dot{B}^{s}_{p,r})}\leq\|f\|_{L^{\rho}_{T}(\dot{B}^{s}_{p,r})}\ \ \ \ \ \ \rho\leq r, \|f\|_{L^{\rho}_{T}(\dot{B}^{s}_{p,r})}\leq \|f\|_{\mathcal{L}^{\rho}_{T}(\dot{B}^{s}_{p,r})}\ \ \ \ \ \r\leq \rho.\end{cases}
(X)_{ll'}= - \frac{\alpha}{n + \frac{\delta_1+\delta_2}{2}}\sum_{n_1=0}^{n-|m|-1}\left( 2n_1 - n + |m| + \frac{\delta_1-\delta_2}{2}+ 1 \right)\widetilde W_{nlm}^{n_1}(\delta_1,\delta_2)\widetilde W_{nl'm}^{n_1}(\delta_1,\delta_2).
J_{\lambda,\mu}(u,v) =\left(\frac{1}{p}-\frac{1}{p_s^\ast}\right)\|(u,v)\|^p-\left(\frac{1}{q}-\frac{1}{p_s^\ast}\right)\int_\Omega(\lambda|u|^q+\mu|v|^q)dx &=\frac{s}{n}\|(u,v)\|^p-\left(\frac{1}{q}-\frac{1}{p_s^\ast}\right)\int_\Omega(\lambda|u|^q+\mu|v|^q)dx.
\frac{\Im F_{\nu}(iy)}{y}\int_{\mathbb{R}}\frac{(1+s^{2})\,d\sigma_{\mu}(s)}{|F_{\nu}(iy)-s|^{2}}=\frac{-\Im G_{\nu}(iy)}{y}\int_{\mathbb{R}}\frac{(1+s^{2})\,d\sigma_{\mu}(s)}{|1-sG_{\nu}(iy)|^{2}}<1
u_i =-g_i(\xi_i,x_i)+x_{i(n_i^x+1)}, \dot{x}_{i(n_i^x+1)}=x_{i(n_i^x+2)}, \vdots \dot{x}_{in_0^w}=\sum\nolimits_{j=1}^{n_0^w}s_j^0x_{ij}+d_{n_0}^wv+d_{n_0}^wK\sum\nolimits_{j=0}^{N}a_{ij}(\hat x_{i}-\hat x_j), \dot{\xi}_i =A_i^0\xi_i+f_i(x_i)
\angle({y_{k,i}})\!=\!\angle{\beta\left(\textbf{x}_k\right)}{\rm{-}}\pi\! \left[\!{\frac{M \!-\! 1}{M}(\omega_{k,i1}\!-\!x_{k,1}) \!+\! \frac{N \!-\! 1}{N}{(\omega_{k,i2}\!-\!x_{k,2})}}\!\right]\!.
G_{{\cal Q}^4}(x_p) =\frac{3^{3}\, \Gamma(11)}{2^{7}\left(\pi^3 \Gamma(4) \right)^{4}}\: g^{8}e^{-\frac{8\pi^2}{g^2}+ i\theta}\, x_{12}^4 x_{34}^4 \prod_{p<q}\frac{\partial}{\partial x_{pq}^2}B(x_{pq}) \, ,
k! (f\tilde{\otimes}g)(z_1,\dotsc,z_{k}) = p(k-1)!\bigl(f(z_k,\cdot)\tilde{\otimes}g\bigr)(z_1,\dotsc,z_{k-1}) &\;+q(k-1)!\bigl(f\tilde{\otimes}\bigl(g(z_k,\cdot)\bigr)(z_1,\dotsc,z_{k-1})\,.
d e t \hat{K}={\frac{4 \coth^{2}({\frac{\omega}{2}}) \sinh \Bigl ( ( \nu - q_{+}){\frac{\omega}{2}}\Bigr ) \sinh ( q_{-}{\frac{\omega}{2}}) \sinh ({\frac{\omega}{2}})}{\sinh ({\frac{\nu \omega}{2}})}}.
\begin{array}{r c l}{{m_{0}^{2}}}&{{=}}&{{\displaystyle \frac{1}{2 a^{2}}L^{\alpha \beta}L_{\alpha \beta}}}\\{{}}&{{=}}&{{E^{2}-{\bf P}^{2}- \displaystyle \frac{{\bf L}^{2}}{a^{2}}~ ,}}\\ \end{array}
\int_1^\infty\phi(x)\cos(2\pi\alpha nx)x^{\frac12-s}\,dx=\sum_{j=0}^{m-1}\frac{\cos^{(j+1)}(2\pi\alpha n)}{(2\pi\alpha n)^{j+1}}\phi_j(1,s)+\int_1^\infty\frac{\cos^{(m)}(2\pi\alpha nx)}{(2\pi\alpha n)^m}\phi_k(x,s)x^{\frac12-m-s}\,dx
E[U_n(\vartheta_n)] =\sum_{I,J}K(I,J_{-\vartheta_n})\rho_{m^*}\int_{I\cap J_{-\vartheta_n}}\sigma_1(t)\sigma_2(t+\theta_{m^*})dt+o(1)=\rho_{m^*}\Sigma(\theta_{m^*})+o(1).
\begin{dcases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)=\rho(t,1)+ p(t) t\in (0,T) , u(t,0)=0, \ \u(t,1)=0 t\in (0,T) , \rho(0,x)= u(0,x)= 0 x\in(0,1), \end{dcases}
u_t =\Delta u -\chi\nabla v\cdot\nabla u +u(a(x,t)-u(b(x,t)-\chi\mu)-\chi\lambda v) &\ge \Delta u -\chi\nabla v\cdot\nabla u +u(a_{\inf}-\|u(\cdot,t;t_0,u_0)\|_\infty(b_{\sup}-\chi\mu)-\chi\lambda \frac{\mu}{\lambda}\|u(\cdot,t;t_0,u_0)\|_\infty) &=\Delta u -\chi\nabla v\cdot\nabla u +u(a_{\inf}-\|u(\cdot,t;t_0,u_0)\|_\infty b_{\sup})
c_{ij}(t, x) =\sum_{l=0}^kc_{ij}^l(t)L_l(\xi)=L^\top(\xi)c_{ij}(t), \psi_j(t, x) =\sum_{l=0}^k\psi_j^l(t)L_l(\xi)=L^\top(\xi)\psi_{j}(t), p_{ij}(t, x) =\sum_{l=0}^kp_{ij}^l(t)L_l(\xi)=L^\top(\xi)p_{ij}(t),
\alpha r_n (x) = \displaystyle \sum_{i= 1}^{n-1}\alpha \langle s_i(x) , P_i H_1 \rangle = \displaystyle \sum_{i=1}^m \langle s_i(x) , \alpha P_i H_1 \rangle +\sum_{i = m+1}^{n-1}\langle s_i(x) , P_{i,m}\left(\frac 1k{v_m}+\frac 1k w_{m}\right)\rangle.
\phi(n^*m^*mn)\alpha_{mn}(\phi)(d) =\phi(n^*m^*dmn)=\phi(n^*n)\alpha_n(\phi)(m^*dm) =\phi(n^*n)\alpha_n(m^*m)\alpha_m(\alpha_n(\phi))(d)=\phi(n^*m^*mn)\alpha_m(\alpha_n(\phi))(d).
Q_k^{(t)}=\frac{H_{k-1}^{(0,t)}H_{k}^{(0,t+1)}}{H_{k}^{(0,t)}H_{k-1}^{(0,t+1)}},k=1,2,\dots,m, &E_k^{(t)}=\frac{H_{k+1}^{(0,t)}H_{k-1}^{(0,t+1)}}{H_{k}^{(0,t)}H_{k}^{(0,t+1)}},k=1,2,\dots,m-1,
\int_M^\infty |df(t)| = o\big( M^{-\nu-1}\big) M\to \infty, \nu<\alpha+1/2 \nu>-1, \int_M^\infty t^{\nu-\alpha-1/2}|df(t)| =o\big(M^{-\alpha-3/2}\big) M\to \infty, \nu\geq \alpha+1/2\nu\leq -1.
\begin{array}{l}\sum_{i=1}^{l}\theta_i J_i(x,u)=\sum_{i=1}^{l}\theta_ig_i^0(x(T))+\int_{0}^{T}\sum_{i=1}^{l}\theta_i f^0_i(t,x(t),u(t))dt \le \sum_{i=1}^{l}\theta_ig_i^0(\overline{x}(T))+\sum_{i=1}^{l}\theta_iD_Hg_i(\overline{x}(T))\cdot(x(T)-\overline{x}(T))+ \int_{0}^{T}\sum_{i=1}^{l}\theta_i f^0_i(t,x(t),u(t))dt.\end{array}
\omega_{\rho}^{2}( q ) = q^{2}\frac{1 + \frac{2}{\pi}V_{( 0 )}^{0 0}+ \frac{1}{2 \pi^{2}}\{( U_{( 0 )}^{0 0})^{2}- 2 U_{( 0 )}^{0 0}U_{( 1 )}^{0 0}\}}{1 + \frac{2}{\pi}V_{( 1 )}^{0 0}- \frac{1}{2 \pi^{2}}( U_{( 1 )}^{0 0})^{2}},
\int_c^x (f(u,u) -f(u,c)) du +\int_c^b (f(u,x)-f(u,c)) du &=\int_c^x f(u,u) du +\int_c^b f(u,x) du -\int_c^b f(u,c) du &= \int_c^x f(u,u) du +F_2 (x) -F_2 (c).
\| \left(R_{L^*}\right)^{*}\|_{L^p_\omega \to L^p_\omega}= \| R_{L^*}\|_{L^{p'}_{\sigma}\to L^{p'}_{\sigma}}\lesssim [\omega^{1-p'}]_{A_{p'}}^{\max\{1,1/(p'-1)\}}\lesssim [\omega]_{A_{p}}^{(p'-1)\max\{1,1/(p'-1)\}}\lesssim [\omega]_{A_{p}}^{\max\{1,1/(p-1)\}}.
\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}\beta^2f^0(\lambda)-\lambda^{2n}C_{\mu,N}^{\beta,0}\bigl(e^{i\lambda}\bigr)\big|=\alpha_2|\lambda|^{n}\big|1-e^{i\lambda\mu}\big|^{n},
U(x,t) = \begin{cases}U_\ell x/t<-1 U_*=(\rho_{*,\ell}, J_*,a_\ell) -1<x/t<0 U_{**}=(\rho_{*,r}, J_*,a_r) 0<x/t<1 U_r x/t> 1\end{cases}
\frac{\lambda^c_x\left(t(\ell+m)\right)}{\lambda^c_{x^u}\left(\tau(\ell+m)\right)}=\frac{\lambda^c_x\left(t(\ell)\right)\lambda^c_{f^{t(\ell)}(x)}\left(t(\ell+m)-t(\ell)\right)}{\lambda^c_{x^u}\left(\tau(\ell)\right)\lambda^c_{f^{\tau(\ell)}(x^u)}\left(\tau(\ell+m)-\tau(\ell)\right)}
d |\eta_{n}(t)|^2 ={2\langle \eta_{n}(t), d \eta_{n}(t)\rangle}= 2 \langle b'_{n}(X_{n}(t)) \eta_{n}(t), \eta_{n}(t) \rangle dt + 2 \sum_{i=1}^d \langle \eta_{n}(t), (\sigma_{n})_i'(X_{n}(t)) \eta_{n}(t) \rangle dW^i_t+ \sum_{i=1}^d |(\sigma_{n})_i'(X_{n}(t)) \eta_{n}(t)|^2 dt
\sum_{i=\lfloor N-\eta^{-1}\rfloor+1}^{N-1}\sup_{h\in\mathrm{Lip}(1)}\left|\tilde{Q}_{i-1}\big(P_{\eta}-\tilde{Q}_{1}\big) P_{(N-i)\eta}h(x)\right| \qquad\leq C (1+|x|)\,\eta^{2/\alpha}\sum_{i=\lfloor N-\eta^{-1}\rfloor+1}^{N-1}[(N-i)\eta]^{-1/\alpha}\leq C\frac{\alpha}{\alpha-1}\,(1+|x|)\,\eta^{2/\alpha-1}.
\|Tf\|_{L^q(d\mu)}\le C\|f\|_{L^p(d\mu)}= C\|w^{-\frac{1}{s}}w^{\frac{1}{s}}f\|_{L^p(d\mu)}&\le C\|w^{-\frac{1}{s}}\|_{L^{rs}(d\mu)}\|w^{\frac{1}{s}}f\|_{L^s(d\mu)}=C\|w^{-1}\|_{{L^r(d\mu)}}^{\frac{1}{s}}\|w^{\frac{1}{s}}f\|_{L^s(d\mu)},
1 \; = \;{\frac{\int d A \exp \left\{- N \beta \, \mathrm{T r}\, \left[ \left( A - \sum_{i = 1}^{d}U_{i}\right) \left( A^{\dagger}- \sum_{i = 1}^{d}U_{i}^{\dagger}\right) \right] \right\}}{\int d A \exp \left\{- N \beta \, \mathrm{T r}\, \left[ A A^{\dagger}\right] \right\}}}\; .
\left|\sum_{t=0}^{k}P_t(\lambda)\right| \geq |P_k(\lambda)| - \sum_{t=0}^{k-1}|P_t(\lambda)| = \Theta(d^{k\alpha}) - \sum_{t=0}^{k-1}\Theta(d^{t\alpha})= \Theta(d^{k\alpha})
\widehat{S_p}= \frac{\sqrt{m_{2p}^{-1}\hat{\gamma}_{2p}(1/n;X)}}{m_p^{-1}\hat{\gamma}_p(1/n;X)}, m_s := \frac{2^{s/2}}{\sqrt{\pi}}\Gamma\left(\frac{s+1}{2}\right), p, s > 0,
|u+v\sqrt{D}|\geqslant 2|v|\sqrt{D}-|u-v\sqrt{D}|=2|v|\sqrt{D}+O_{\varepsilon,\eta}\left(\frac{1}{B}\right)\geqslant 2\sqrt{\frac{A^\prime |m|D}{\varepsilon^\prime(2\sqrt{D}+\frac{\varepsilon^\prime}{B^2})}}B+O_{\varepsilon,\eta}\left(\frac{1}{B}\right).
\left( - 1 \right)^{F}= 2^{4}d_{0}^{2}d_{0}^{3}d_{0}^{4}d_{0}^{5}d_{0}^{6}d_{0}^{7}d_{0}^{8}d_{0}^{9}\left( - 1 \right)^{\sum_{n > 0}d_{- n}^{i}d_{n}^{i}},
\psi ~=~{1\over{\kappa}}V^{-7/2}\partial_{\underline{i}}V\gamma_i{\cal C}^{(\bar k)}~dt~+~{1\over{\kappa}}V^{-3/2}\partial_{\underline{j}}V \gamma_j\gamma_i{\cal C}^{(\bar k)}~dx^{\underline{i}}\ \ . \relax
\Lambda = \left\{(s, t, \sigma, \tau; u, v, \xi, \eta) : \phantom{\frac{1}{2}}\right. t = \frac{2}{c_0}|R(u, v, s)|, & \sigma = \frac{2\tau}{c_0}\hat{R}(u, v, s)\cdot\dot{\gamma}(s), & \left. (\xi, \eta) = -\frac{2\tau}{c_0}\pi_{T\Psi}\hat{R}(u, v, s) \right\}.
\lim_{\varepsilon\to 0}{\rm K}(\varepsilon) \,=\, \lim_{\varepsilon\to 0}\left[\int_0^1\!\!d\phi\,\,\frac{\varepsilon}{f(\phi)}\, \exp{\!\left(-\frac{\varepsilon}{f(\phi)}\right)}\right] \,=\, 0\,.
{\rm BL}(f_1,\ldots,f_m) \coloneqq \frac{\int_{\mathbb{R}^d}e^{-\pi \langle x, \mathcal{Q}x\rangle}\prod_{j=1}^m f_j \circ L_j^{c_j}\, dx}{\prod_{j=1}^m \big(\int_{\mathbb{R}^{d_j}}f_j\, dx_j \big)^{c_j}}
A_1 (B_1 - A_1) = \frac 12 \int_0^x \int_0^{t_1}(e^{-2kt_1}+ e^{-2kt_2}- 2 e^{-2k(t_1+t_2)}- e^{-2k(x-t_1+t_2)}) V(t_1) V(t_2) \,dt_2 \,dt_1 + O( e^{-2\Re k x})
(\phi_1^{(p)}\circ \pi)(x,y) = \phi_1^{(p)}(x^p,y^p) &=(f^{(p)}(x^p,y^p),g^{(p)}(x^p,y^p)) &= ((f(x,y))^p, (g(x,y))^p) &= (\pi\circ \phi_1)(x,y).
{\rm tr}(\widetilde{Z}_{\alpha\alpha}) +{\textstyle\sum_{i=1}^{k-k_0}}\lambda_i\big(\mathcal{G}(\widetilde{Z}_{\beta\beta}\big)\le \tau \{\rm and}\{\rm tr}(\widetilde{Z}_{\alpha\alpha}) +\langle \overline{u}_{\beta}, \lambda(\mathcal{G}(\widetilde{Z}_{\beta\beta}))\rangle =\tau.
\left( \bigcap_{\sigma\in\Gamma}\sigma(V)\right) \cap P^{n}(k) =\bigcap_{\sigma\in\Gamma}\left( \sigma(V)\cap P^{n}(k)\right)=\bigcap_{\sigma\in\Gamma}\left( \sigma(V)\cap\sigma\left( P^{n}(k)\right)\right) & =\bigcap_{\sigma\in\Gamma}\sigma\left( V\cap P^{n}(k)\right)=\bigcap_{\sigma\in\Gamma}\left( V\cap P^{n}(k)\right) =V\cap P^{n}(k)
|\Lambda_\iota|\leq C|\mathcal{S}|+ C\Big(N^{\varepsilon}\frac{|\mathcal{S}|+ \hat{\Lambda}}{(N\eta)^{\frac13}}+N^{-\frac{\gamma}{4}}\hat{\Lambda}^2\Big)^{\frac{1}{2}}\leq CN^{-\varepsilon}\hat{\Lambda}\,, \iota=A, B \,,
X' := \{(\vec{g},\vec{z})\in G^d\times\Delta^L\mid \forall i\not=j: d_{\vec{g}}^{(\alpha_1,\ldots,\alpha_l)}(z_i,z_j)>2l+2 &\forall i: \exists j_1,j_2\in\{0,\ldots, l\}: (j_1\not=j_2 \nu_{j_1}(z_i) \nu_{j_2}(z_i) &)\}.
\mathbf{P}\approx \mathbf{P}_R =\sum_{k=-M}^{M}p_k \bigotimes_{\ell=1}^{3}{\bf b}^{(\ell)}(t_k)= \sum\limits_{k=-M}^{M}{\bf p}^{(1)}_k \otimes{\bf p}^{(2)}_k \otimes{\bf p}^{(3)}_k\in \mathbb{R}^{n\times n \times n},
\hat{J}_{a}^{( r )}( z ) \hat{J}_{b}^{( s )}( w ) = \frac{\hat{G}_{a b}\delta_{r + s , 0 \hspace{. 0 5 i n}m o d \hspace{. 0 4 i n}\lambda}}{( z - w )^{2}}+ \frac{i f_{a b}^{\ \c}\hat{J}_{c}^{( r + s )}( w )}{( z - w )}+ O ( ( z - w )^{0})
A \sharp_{\nu}B \leqslant A \sharp_{\nu}B + \sum_{k=0}^{\infty}r_{k}(A\sharp_{\frac{m_k}{2^k}}B -2 A\sharp_{\frac{2m_k+1}{2^{k+1}}}B+ A\sharp_{\frac{m_k+1}{2^k}}B) \leqslant A \nabla_{\nu}B.
\!\!\!\!\!\!\! \mathcal{O}_{\rm u}(R_d)= 1-\exp \left( \frac{N_o (1-2^{\frac{R_{d}}{B_{\rm u}^{\alpha{\rm D}}}})}{\rho}\right) \mathcal{L}_{I_{\rm u}}\left(\frac{2^{\frac{R_{d}}{B_{\rm u}^{\alpha{\rm D}}}}-1}{\rho |\mathcal{I}_{\rm u}(\alpha)|^2}\right) ,
B_{kk,k}C_{i,j}+b_k[C_{i,jk}-C_{k,j}\frac{B_{ki,k}}{b_k-b_i}-C_{i,k}\frac{B_{kj,k}}{b_k-b_j}] &=-C_i[C_{j,k}-C_k\frac{B_{jk,k}}{b_k-b_j}]-C_j[C_{i,k}-C_k\frac{B_{ik,k}}{b_k-b_i}].
\begin{array}{c}{{\displaystyle K_{a b}^{-}= \frac{\epsilon}{4}( E_{a a}- E_{b b}+ E_{\theta ( a ) \theta ( a )}- E_{\theta ( b ) \theta ( b )}) ,}}\\{{\displaystyle M_{a b}^{-}= \frac{\epsilon}{4}( E_{a a}- E_{b b}- E_{\theta ( a ) \theta ( a )}+ E_{\theta ( b ) \theta ( b )}) ,}}\\ \end{array}
\int_0^T \int_\omega (\ddot{u}_3(t,x')- \ddot{v}_3(t,x'))(\dot{u}_3(t,x')-\dot{v}_3(t,x'))\,dx'\,dt =\frac{1}{2}\| \dot{u}_3 (T)- \dot{v}_3 (T) \|_{L^2}^2 .
Z_{\mu_M}= \bigcup_{\alpha \in \Gamma_M}K\cdot \left( M^{\alpha}\cap \mu_M^{-1}(\alpha)\right); Z_{\mu_{M\times N}}= \bigcup_{\beta \in \Gamma_{M\times N}}K\cdot \left( (M\times N)^{\beta}\cap \mu_{M\times N}^{-1}(\beta)\right) ,
\displaystyle B_1 \int_{-1}^{0}\dfrac{\Phi_0^2}{H_p}\,dp +\mathcal{R}_3 \Phi_0(\hat{p})-\mathcal{R}_2 \Phi_0(0)= \int_{-1}^{0}\mathcal{R}_1 \Phi_0 \,dp.
\frac{1}{a\log a}\ \le \ \frac{1}{1+v}\frac{1}{a\log P(a)}= \frac{e^\gamma}{\mu_{P(a)}}\frac{1}{(1+v)a}\prod_{p<P(a)}\Big(1-\frac{1}{p}\Big) = \frac{e^\gamma}{\mu_{P(a)}}\frac{{\rm d}({\rm L}_a)}{1+v}.
R^{m}((x,\theta),(x^{\prime},\theta^{\prime}))=exp\{-\sum_{k=1}^{d}\omega_{k}^{m}(x_{k}-x_{k}^{\prime})^{2}\}exp\{-\sum_{k=1}^{r}\omega_{d+k}^{m}(\theta_{k}-\theta_{k}^{\prime})^{2}\}.
\begin{cases}V(k)=\lefteqn{-H(P_{1}(k+1))^{-1}G_{u}(P_{1}(k+1))^{T},}\mathbb{V}(k)=V(k)+\tilde{V}(k)=\tilde{H}(P_{1}(k+1),Q_{1}(k+1))^{-1}\tilde{G}_{u}(P_{1}(k+1),Q_{1}(k+1))^{T}, \end{cases}
\|u'\|_J K_L e_2 = K_L c_1 = - c_2 + (c_1)_x = -\begin{pmatrix}a b c d\end{pmatrix}+ \begin{pmatrix}0 (\|u'\|_J)_x 0 d\end{pmatrix}= -\begin{pmatrix}a 0 0 d\end{pmatrix}
\frac{1}{n^2}\left|\sum\limits_{i=1}^{k^*}\sum\limits_{j=k^*+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right)\right|\overset{P}{\longrightarrow}\tau(1-\tau)\int_{\mathbb{R}}\left(F(x+\Delta)-F(x)\right)dF(x)
\mathbb{P}[K_A=1, K_B=1|\mathbf{F}=\mathbf{f}] \geq \frac 12 -2\delta \mathbb{P}[K_A=2, K_B=2|\mathbf{F}=\mathbf{f}] \geq \frac 12 -2\delta \mathbb{P}[K_A=1, K_B=2|\mathbf{F}=\mathbf{f}] \leq 2\delta \mathbb{P}[K_A=2, K_B=1|\mathbf{F}=\mathbf{f}] \leq 2\delta.
(I + K^{1/2}X^{-1}K^{1/2})^{-1}\succeq Z \ \Leftrightarrow \I - K^{1/2}( X + K)^{-1}K^{1/2}\succeq Z & \ \Leftrightarrow \ \left[ \begin{array}{@{}c@{\,\,}c@{}}I - Z K^{1/2}K^{1/2}X + K \end{array}\right] \succeq 0
\breve{w}_\varepsilon = \phi_\varepsilon - \phi_0- \varepsilon\chi_k^*(\cdot/\varepsilon)S_\varepsilon(\psi_{10\varepsilon}\nabla_k\phi_0)\breve{z}_\varepsilon = \theta_\varepsilon - \theta_0- \pi_k^*(\cdot/\varepsilon)S_\varepsilon(\psi_{10\varepsilon}\nabla_k\phi_0)
M_{\mu\nu}=\int_{z>z_{0}}({\psi_{\nu}^{T}}^{*}E_{\mu}\psi_{\mu}^{S}+\psi_{\mu}^{S}\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial z^{2}}{\psi_{\nu}^{T}}^{*})dxdydz.
\int_{c-i\infty}^{c+i\infty}z^s f(s) \Gamma(1+s-N) \Gamma(N-s) \; ds +\int_{C}z^s f(s) \Gamma(1+s-N) \Gamma(N-s) \; ds & = \; \; 2 \pi i (-1)^N \sum_{k=N}^{\infty}(-z)^k f(k) \;\;.
\bigcirc_{w=0}^i (\theta_t+\gamma_{r-i+w})^{-1}\circ (\theta_t+\xi_{i,1})\circ \cdots \circ (\theta_t+\xi_{i,\ell_i})=\bigcirc_{w'=0}^j(\theta_t+\gamma_{r-j+w'})^{-1}\circ (\theta_t+\xi'_{j,1})\circ \cdots \circ (\theta_t+\xi'_{j,\ell'_j})\enspace,
\sum^{s-1}_{t=s-r}\binom{T}{t}(\Delta^{(s+1)}{\bf b}(r+t-s+1,r+1, t+1, s+1))_{k}= \alpha_{r+T-s+1,r+1,T+1,s+1}\sum^{r-1}_{t=0}\binom{r+T-s}{t}(\Delta^{(s+1)}{\bf c}(t, s))_k,
\begin{aligned}R^{u,1}= \sqrt{\epsilon}\left[(v^0_p+v^1_e)\partial_Y u^1_e-(g^0_p+g^1_e)\partial_Y h^1_e\right]-\nu \epsilon \partial_Y^2u^1_e &+\int_Y^\infty E_b(x,\theta)\mathrm{d}\theta-\sqrt{\epsilon}h^0_e\partial_x\rho-\epsilon\int_0^y\partial_x \rho(x,s)\mathrm{d}s\cdot\partial_Yh^0_e.\end{aligned}
\frac{dT(x,B)}{dx}=\frac{-\frac{dT(x,A)}{dx}(F_{1}-F_{2})+\frac{T(x,A)\frac{dT(x,A)}{dx}F_{1}^{2}}{\sqrt{D(x)}}}{F_{2}}=-\frac{dT(x,A)}{dx}\frac{(F_{1}-F_{2})-\frac{T(x,A)F_{1}^{2}}{\sqrt{D(x)}}}{F_{2}}.
\sum_{n=1}^{\infty}|d_{n(j+k-1)}|^2 \le \left(\frac{|A-B|}{2(j+k-1)B}\right)^2\sum_{n=1}^{\infty}\frac{B^{2n}}{n^2}& = \frac{|A-B|^2}{4(j+k-1)^2}\frac{{\rm Li\,}_{2}(B^2)}{B^2},
0 = \left({\int}_{- L}^{L}d x^{-}\left[{\Theta}_{\mu}^{+}( x ) \right]_{x^{+}= - T}^{x^{+}= T}+{\int}_{- T}^{T}d x^{+}\left[{\Theta}_{\mu}^{-}( x ) \right]_{x^{-}= - L}^{x^{-}= L}\right) ,
\hat\varsigma^2s^2_L(\theta)= \hat\varsigma^2\left(1-\mathbf r_L(\theta)'\mathbf R_L^{-1}\mathbf r_L(\theta)+\frac{(1-\mathbf 1'\mathbf R_L^{-1}\mathbf r_L(\theta))^2}{\mathbf 1'\mathbf R_L^{-1}\mathbf 1}\right).
g^{(h)}(\hat{x})\cdot d\log(\{1+aS^iT^j, S\}) &=-ajg^{(h)}(c')\sum_{k=0}^\infty(-a)^kS^{(k+1)i+\ell_1p^h-1}T^{(k+1)j+\ell_2p^h-1}dS\wedge dT.
\left(\partial_{t_j}\partial_{t_k}\partial_{t_\alpha}\mathcal{F}(\tau,Q) \right) \left( \partial_{t_i}G^{\alpha \beta}\right) = \left( \partial_{t_j}\partial_{t_k}\partial_{t_\alpha}\mathcal{F}(\tau,Q) \right) \sum_{\alpha',\beta'}\left( \partial_{t_i}G_{\alpha' \beta'}\right) G^{\alpha \alpha'}G^{\beta \beta'}
\widetilde{\mathbf{S}}= \left(\widetilde{S}_{ij}\right) = \begin{cases}\sqrt{\dfrac{4k_{B}T\mu}{\Delta V \Delta t}}\widetilde{\boldsymbol{\Re}}~i = j \sqrt{\dfrac{2k_{B}T\mu}{\Delta V \Delta t}}\widetilde{\boldsymbol{\Re}}~i \neq j\end{cases},
\tilde \ell_{k+j+1}- \tilde \ell_{k+j}= \tilde \ell_{k+j}\left( e^{\frac\alpha2 ( \varphi(m_{k+j+1})- \varphi(m_{k+j}))}-1 \right) &> \tilde \ell_{k+j}\frac{\alpha( \varphi(m_{k+j+1})- \varphi(m_{k+j}))}{2}> \frac{\alpha}{2}\tilde \ell_{k+j}
\begin{array}{c c c}{{\partial_{\sigma}\varrho = 0 \hfill}}&{{\Longrightarrow}}&{{\varrho = \rho ( x , \tau ) , \hfill}}\\{{\partial_{\sigma}\theta = 1 \hfill}}&{{\Longrightarrow \hfill}}&{{\theta = \Theta ( x , \tau ) + \sigma , \hfill}}\\ \end{array}
I_{1,y_{L,{N_y},i}^{(\alpha )}}^{(y)}\,{P_{{N_y},{N_t}}}\phi (y,t_{{t_f},{N_t},j}^{(\alpha )}) = \sum\limits_{s = 0}^{{N_y}}{{}_Lp_{B,{N_y},i,s}^{(1)}\,{\phi_{s,j}}}, I_{1,t_{{t_f},{N_t},j}^{(\alpha )}}^{(t)}\,{P_{{N_y},{N_t}}}\phi (y_{L,{N_y},i}^{(\alpha )},t) = \sum\limits_{k = 0}^{{N_t}}{{}_{{t_f}}p_{B,{N_t},j,k}^{(1)}\,{\phi_{i,k}}},
j = \left( \begin{array}{c c c c c c}{{\theta_{1}}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{\ddots}}&{{}}&{{}}&{{}}&{{}}\\{{}}&{{}}&{{\theta_{q + 1}}}&{{}}&{{}}&{{b}}\\{{}}&{{}}&{{}}&{{\ddots}}&{{}}&{{}}\\{{}}&{{}}&{{}}&{{}}&{{\theta_{p}}}&{{}}\\{{}}&{{}}&{{c^{t}}}&{{}}&{{}}&{{0}}\\ \end{array}\right) .
\Delta [ n x_{c l}, e_{0}] = \left( 1 + \frac{g e_{0}}{4}B_{\alpha \beta}\gamma^{[ \alpha}\gamma^{\beta ]}+ \frac{g^{2}e_{0}^{2}}{1 6}\bar{B}_{\alpha \beta}\bar{B}^{* \alpha \beta}\gamma^{5}\right) \bar{\Lambda}( e_{0}) + Q_{\alpha \beta}\gamma^{[ \alpha}\gamma^{\beta ]}
\frac{\partial A}{\partial z}=-\beta_{1}\frac{\partial A}{\partial t}-\frac{i}{2}\beta_{2}\frac{\partial^{2}A}{\partial t^{2}}+\frac{1}{6}\beta_{3}\frac{\partial^{3}A}{\partial t^{3}}+\gamma_{x}\frac{\partial A}{\partial x}+\gamma_{y}\frac{\partial A}{\partial y}
\omega_{\; b \mu}^{a}( x ) \longrightarrow \omega_{\; b \mu}^{\prime a}( x ) = \Lambda_{c}^{a}( x ) \Lambda_{b}^{d}( x ) \omega_{\; d \mu}^{c}( x ) + \Lambda_{c}^{a}( x ) \partial_{\mu}\Lambda_{b}^{c}( x ) .
\,\,\frac{\partial\delta}{\partial \bar z}=0 \quad\mbox{in}\,\Omega,\frac{\partial\widetilde \delta}{\partial z}= 0\quad\mbox{in}\,\Omega,(\nu_1+i\nu_2)\delta\vert_{\Gamma_0}=-(\nu_1-i\nu_2)\widetilde \delta\vert_{\Gamma_0}
\Delta(\mathfrak{u}) = 1 \otimes \mathfrak{u}+ \sum \mathfrak{u}_{(1)}\otimes \mathfrak{u}_{(2)}, \Delta(\mathfrak{v}) = 1 \otimes \mathfrak{v}+ \sum \mathfrak{v}_{(1)}\otimes \mathfrak{v}_{(2)}.
S_{F}^{\alpha}(T)=-\frac{1}{\sqrt{2 g_{F}^{\alpha}}}\int_{S_{F}^{\alpha}(y)}^{S_{F}^{\alpha}(0)}(S_{F}^{\alpha}(y)-S_{F}^{\alpha}(\eta))^{-1/2}f(S_{F}^{\alpha}(y))d_{F}^{\alpha}\eta.
x^i=x^i(t),y^{(1)i}(t)=\displaystyle\frac1{1!}\displaystyle\frac{dx^i}{dt}(t),\ldots,y^{(k)i}(t)=\displaystyle\frac1{k!}\displaystyle\frac{d^kx^i}{dt^k}(t),t\in I
\left. \frac{d^j}{dx^j}\Bigl( 1+x+x^2+ \cdots +x^{r-1}\Bigr) \right|_{x=1}= j!+\frac{(j+1)!}{1!}+ \cdots + \frac{(r-1)!}{(r-1-j)!}= j!\left( \binom{j}{j}+\binom{j+1}{j}+ \cdots +\binom{r-1}{j}\right) = j! \binom{r}{j+1}.
\alpha_{2}=\frac{\frac{x\alpha_{1}sinh(x\alpha_{1})-cosh(x\alpha_{1})+1}{\alpha_{1}}-\frac{cosh(x\alpha_{1})-1-y\alpha_{1}}{\alpha_{1}}}{\frac{x\alpha_{1}sinh(x\alpha_{1})-cosh(x\alpha_{1})+1}{\alpha_{1}^{2}}}
\Phi_{k}^{\nu ,m}(z):=\sqrt{\frac{\left( 2\nu +2m+1\right) \left( 2\nu+m\right) !m!}{(m+k)!(2\nu +m-k)!}}\left( 1+z\overline{z}\right)^{-\nu}z^{k}P_{m}^{\left( k,2\nu -k\right)}\left( \frac{1-z\overline{z}}{1+z\overline{z}}\right) ,
\mathbf{B}_k = \begin{bmatrix}\mathbf{H}(1,k) \mathbf{H}(2,k) \cdots \mathbf{H}(R,k) \mathbf{H}(2,k) \mathbf{H}(3,k) \cdots \mathbf{H}(R+1,k) \vdots \vdots \ddots \vdots \mathbf{H}(S,k) \mathbf{H}(S+1,k) \cdots \mathbf{H}(Q,k) \end{bmatrix}
g^{11}=\frac{1}{2}-sf+\frac{s^2}{2}\big(3f^2-2\textstyle(\frac{\partial f}{\partial \theta_1})^2\big)+O(s^3), g^{12}=-s^2\textstyle\frac{\partial f}{\partial \theta_1}\frac{\partial f}{\partial \theta_2}+O(s^3) , g^{22}=\frac{1}{2}-sf+\frac{s^2}{2}\big(3f^2-2\textstyle(\frac{\partial f}{\partial \theta_2})^2\big)+O(s^3) .
\AA^X := \frac{1}{2}\sum_{i = 1}^m \sum_{j = 1}^m a_{ij}(x) \partial_{x_i}\partial_{x_j}+ \sum_{i = 1}^m b_i(x) \partial_{x_i}\AA^Y := \frac{1}{2}\sum_{i = 1}^n \sum_{j = 1}^n \rho_{ij}(y) \partial_{y_i}\partial_{y_j}+ \sum_{i = 1}^n \gamma_i(y) \partial_{y_i}
\overline{H}_n(s_1,\ldots,s_r)=\sum\limits_{k_1=0\atop p=2k_1+1}^{n-r}\frac{c_{k_1}}{(2k_1+1)^{s_1}}+\sum\limits_{k_1=0\atop p\neq 2k_1+1}^{n-r}\frac{c_{k_1}}{(2k_1+1)^{s_1}}=\frac{c_{\frac{p-1}{2}}}{p^{s_1}}+\sum\limits_{k_1=0\atop p\neq 2k_1+1}^{n-r}\frac{c_{k_1}}{(2k_1+1)^{s_1}}.
\log\eta_{M, N}(\kappa\,q\,|\,\kappa\,a,\kappa\,b) = \bigl(\mathcal{S}_N L_M\bigr)(\kappa\,q\,|\,\kappa\,a,\kappa\,b) - \bigl(\mathcal{S}_N L_M\bigr)(0\,|\,\kappa\,a,\kappa\,b), & = \log\eta_{M, N}(q\,|\,a,\,b).
-{\cal L}_{\mathrm{m a s s}}= \sum_{i = 1}^{N_{f}}{\big [}\mu_{L i}^{2}| \phi_{i}|^{2}+ \mu_{R i}^{2}|{\bar{\phi}}_{i}|^{2}+ \mu_{i}^{2}( \phi_{i}{\bar{\phi}}_{i}+ \phi_{i}^{\ast}{\bar{\phi}}_{i}^{\ast}){\big ]}+ m_{\lambda}( \lambda \lambda + \bar{\lambda}\bar{\lambda}) .
\frac{E_V}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{dk_ydk_z}{(2\pi)^2}\sum_{n=1}^{\infty}((\frac{n\pi}{a\mu})^2+(\frac{k_y}{\mu})^2+(\frac{k_z}{\mu})^2)^{\frac12}e^{-((\frac{n\pi}{a\mu})^2+(\frac{k_y}{\mu})^2+(\frac{k_z}{\mu})^2)^{\frac12}t}\rfloor_{t=0}\,.
\alpha_{3}- i \beta_{3}= - \frac{1}{2}\gamma_{2}^{2}; \; \alpha_{4}- i \beta_{4}= - \gamma_{2}\gamma_{3}; \; \alpha_{5}- i \beta_{5}= - \gamma_{2}\gamma_{4}- \frac{1}{2}( \alpha_{3}^{2}+ \beta_{3}^{2}+ \gamma_{3}^{2}) , \dots
\tilde{\lambda}(\hat{\tau}/\hat{\sigma},g)\, F \left( \frac{w + \rho (\hat{\tau}/\hat{\sigma})}{\sqrt{(m + (\hat{\tau}/\hat{\sigma})^2) (1 - \rho^2)/(m+g-1)}}\, \Bigg | \,t_{m+g-1}\right ) &+ \left(1-\tilde{\lambda}(\hat{\tau}/\hat{\sigma},g) \right) \,F \left(\sqrt{\frac{m+g-1}{m}}\, w \, \Bigg| \, t_{m+g-1}\right) = \eta,
\Lambda' f=\sum_{j,k=0}^n \hat g^{jk}(y)\Big(\frac{\partial u'}{\partial y_j}-iA_j'(y)u'\Big)\nu_k(y)\Big(\sum_{p,r=0}^n \hat g^{pr}(y)\nu_r(y)\nu_p(y)\Big)^{-\frac{1}{2}}\Big|_{\hat S},
\begin{aligned}d(g(\nabla^g_Z X, \cdot))(U,V) = U(g(\nabla_Z^gX,V))-V(g(\nabla^g_ZX,U))-g(\nabla^g_ZX,[U,V]) &=g(\nabla_U^g\nabla^g_ZX,V)-g(\nabla_V^g\nabla^g_ZX,U) \stackrel{(\ref{syga})}{=}0.\end{aligned}
\frac{\overline{C}_{n_j}(\varepsilon_j)}{C_{n_j}}\geq 1- \sum_{t=1}^{\infty}\frac{(1+2p_j)^2}{u_t^2}- \mathcal{O}(\varphi^{n_j/5}) &> 1- \frac{1}{7}(1+2p_j)^2 - \mathcal{O}(\varphi^{n_j/5}).
\mathcal{P}_1(q)\times\{\lambda\in \mathcal{P}_\ell\mid p_c(\lambda)=q_c(\lambda)=0\}\rightarrow \{\lambda\in\mathcal{P}_\ell\mid p_c(\lambda)=0, q_c(\lambda)=q\}, (\mu,\lambda)\mapsto \tilde{a}_\mu\lambda
\left(\kern-.1em\left(\kern-.1em\mathbf{w}_{h},\mathbf{r}\kern-.1em\right)\kern-.1em\right) =\left(\kern-.1em\left(\kern-.1em\mathbf{w}_{h},\left( \Pi^{\operatorname*{curl},\ast}-\Pi_{h}^{\operatorname*{curl},\ast}\right) H_{\Omega}\mathbf{v}_{h}\kern-.1em\right)\kern-.1em\right) =0\quad\forall\mathbf{w}_{h}\in\nabla S_{h}.
[T_{\bar{z}^m},T_{z^n}](z^k) = \begin{cases}\frac{mn}{(k+1)(k+n+1)}\,z^{k+n-m}k\geq m \frac{k+n-m+1}{k+n+1}\,z^{k+n-m}m-n\leq k \leq m-1. 0 \end{cases}