harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
X_{\mathcal{S}\times \mathcal{P}}= \underset{\lambda \in \mathcal{S}}{\bigoplus}\Big(\underset{j \in \mathcal{P}: j \leq \mathcal{L}_{\lambda}}{\oplus}W_{d_{\mathcal{U}_{\lambda}(j)}, \mathcal{S}\backslash \lambda, \mathcal{P}\backslash j}\Big).
\psi((fx)(f'x'))=\psi(f(x\bullet f')xx')=\phi(f)\phi(x\bullet f')\alpha(x)\alpha(x') &=\phi(f)(\alpha(x)\circ\phi(f'))\alpha(x)\alpha(x')=\phi(f)\alpha(x)\phi(f')\alpha^{-1}(x)\alpha(x)\alpha(x') &=(\phi(f)\alpha(x))(\phi(f')\alpha(x')=\psi(fx)\psi(f'x').
\|b\|_{L^1_{\gamma}(H^1)}=\int_{\mathbb{R}^d}\|b(x_1,\cdot)\|_{H^{1}(Y,\mu)}\,d\gamma(x_1)\lesssim s \sum_{j}(\gamma\otimes \mu)(S_j)=C s (\gamma\otimes \mu)(\Theta^{s}).
\frac{df(m|n)}{dt}=f(m|n)\sum_{m^{\prime}=1}^{M}[\delta_{mm^{\prime}}-f(m^{\prime}|n)]\frac{\partial lnl(n|m^{\prime})}{\partial\vec{M}_{m^{\prime}}}\frac{\partial\vec{M}_{m^{\prime}}}{\partial\vec{S}_{m^{\prime}}}\frac{d\vec{S}_{m^{\prime}}}{dt}
m_p(\alpha)=\frac{2^{p/ \alpha}2^p \Gamma( \frac{p+1}{2}) \Gamma(1- \frac{p}{\alpha})}{\sqrt{\pi}\Gamma(1- \frac{p}{2})}\; \Gamma(a)= \int_0^{\infty}x^{a-1}e^{-x}dx.
\begin{array}{rclcrclcrcl}\{e_2^+,e_2^-\}= h+z, \{e_3^+,e_3^-\}= h-z, \{e_2^\mp,e_3^\pm\}= \pm2e_1^\pm , \{e_2^\pm,e_2^\pm\}= 0, \{e_3^\pm,e_3^\pm\}= 0, \{e_2^\pm,e_3^\pm\}= 0. \end{array}
\sum_{n=1}^{\infty}(\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{(\begin{matrix}n-1\\k\end{matrix})})^{2}=\frac{4}{9}\pi^{2}=\int_{0}^{\infty}4(Ei(1,-x)+i\pi)^{2}e^{-3x}dx.
\sum_{n=0}^{\infty}\frac{z^{2n}q^{n^2+n}}{(-zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}(-1)^mb^4_\nu(m,n)z^mq^n=\sum_{n=0}^{\infty}(zq;q^2)_n(-zq)^n,
P_{-1,l}(\mathfrak{v})=e^{ilt}\left( \begin{array}{c}-v_{-1,l}^{-}l\mathfrak{I}_{\frac{3}{2},l}(r) v^{-}_{-1,l}\mathfrak{I}_{\frac{1}{2},l}(r)\end{array}\right),\mbox{}P_{1,l}(\mathfrak{v})=e^{ilt}\left( \begin{array}{c}v_{1,l}^{+}\mathfrak{I}_{\frac{1}{2},l}(r) -v^{+}_{1,l}l\mathfrak{I}_{\frac{3}{2},l}(r)\end{array}\right).
{\widehat \Gamma}^{(n)}_{\Lambda_1\dots\Lambda_n}(k_1\dots k_n) =2^{1-n}P_{\Lambda_1 \alpha_1}(k^0_1)\dots P_{\Lambda_n \alpha_n}(k^0_n) \Gamma^{(n)}_{\alpha_1\dots \alpha_n}(k_1\dots k_n)\, ,
{\cal V}(x,y,z)=\int d^{2\omega}\zeta{\partial\over \partial z^\alpha}\left[{\partial\over \partial x^\alpha}{\partial\over \partial y^+}-{\partial\over \partial y^\alpha}{\partial\over \partial x^+}\right]F(x-\zeta)F(y-\zeta) G(z-\zeta)
\varphi_k(\gamma_1,\omega_{k,k+1})=(u,v)\cdot\textbf{n}_k=\Big(\Phi_{1}^{(1)}(\gamma_1;U_a),\Phi_{1}^{(2)}(\gamma_1;U_a)\Big)\cdot\Big(-\sin (\omega_{k,k+1}),\cos(\omega_{k,k+1})\Big),
\tilde{R}_n(T) \leq T\log_2\left(1+\tfrac{\eta\rho g_n\left(1-T\right)}{T}\right),\,\forall n\in\mathcal{N}\sum_{n\in\mathcal{M}_k}\tilde{R}_n(T) \leq T\log_2\left(1+\tfrac{\eta\rho \left(1-T\right)\sum g_n}{T}\right),\,\forall k: \mathcal{M}_k\subseteq\mathcal{N},
\mathbf j (z_1+ z_2\mathbf j)= \mathbf j(x_0+x_1\mathbf i) +\mathbf j(x_2 + x_3\mathbf i)\mathbf j = -(x_2-x_3\mathbf i)+(x_0-x_1\mathbf i)\mathbf j = -\overline z_2 +\overline z_1 \mathbf j
D(s,\xi) = \frac{L(s-ir_1+ir_2,\xi)L(s+ir_1-ir_2,\xi)L(s+ir_1+ir_2,\xi\chi_1^{-1}\chi_2)L(s-ir_1-ir_2,\xi\chi_1\chi_2^{-1})}{L(2s,\xi^2)E^{-1}(s,\xi)}
Q(u)=\frac{1}{p}\int_{\Omega}\left|\nabla u\right|^{p}dx +\int_{\partial\Omega}\frac{\lambda}{p}\left|u\right|^{p}\d\mathcal H^{N-1}- \int_{\partial\Omega}gu \d\mathcal H^{N-1}- \int_{\Omega}fu \dx.
\left\langle\mbox{\boldmath$ \phi$},\mbox{\boldmath$ \psi$}\right\rangle_{\mathbb{X}}:=\sum_{1\leq j\leq n_{\Omega}}\left\langle\mbox{\boldmath$ \phi$}_{j},\mbox{\boldmath$ \psi$}_{j}\right\rangle_{\mathbf{X}_{j}}.
\Omega_{- 2}^{- 2 i}- \Omega_{+ 2}^{+ 2 i}= 4 i \psi_{+ 2 q}^{1 +}\gamma_{q \dot{q}}^{i}\psi_{- 2 \dot{q}}^{1 -}- 4 i \psi_{- 2 q}^{1 +}\gamma_{q \dot{q}}^{i}\psi_{+ 2 \dot{q}}^{1 -}- 4 i \psi_{+ 2 q}^{2 -}\gamma_{q \dot{q}}^{i}\psi_{- 2 \dot{q}}^{2 +}+ 4 i \psi_{- 2 q}^{2 -}\gamma_{q \dot{q}}^{i}\psi_{+ 2 \dot{q}}^{2 +}.
\mathcal{F}(\mathcal{L}_+;q,t)-\mathcal{F}(\mathcal{L}_-;q,t) &=\sum_{l=1}^{L}\frac{(-1)^{l-1}}{l}\sum_{\cup_{s=1}^l \mathbf{I}_s=\mathbf{L}}\left(\prod_{s=1}^l\mathcal{H}((\mathcal{L}_+)_{\mathbf{I}_s};q,t)-\prod_{s=1}^l\mathcal{H}((\mathcal{L}_-)_{\mathbf{I}_s};q,t)\right)
F_j(g) =x_j\int_{\mathbb R}\Big[{\sinh(\lambda+i\phi)\over \lambda+i\phi}\Big]^{1\over 2}\cosh(\lambda+i\phi)(\cosh \lambda)^{-n-{3\over 2}}d\lambda &\quad-iy_j\int_{\mathbb R}(\lambda+i\phi)\Big[{\sinh(\lambda+i\phi)\over \lambda+i\phi}\Big]^{3\over 2}(\cosh \lambda)^{-n-{3\over 2}}d\lambda,
\frac{1}{2}\, \mathrm{dist}\left(B_\frac{r_i+r_{i+1}}{2},\mathbb{R}^N\setminus B_{r_i}\right)=\frac{r_i-r_{i+1}}{4}=\frac{1}{16\,i_0}\le\frac{1}{16}=\frac{1}{2}\,\mathrm{dist}\left(B_{\frac{3}{4}},\mathbb{R}^N\setminus B_\frac{7}{8}\right),
\left\vert \frac{\left( 1-\delta \right)^{\left\vert \eta\right\vert}-1}{\delta}+\left\vert \eta \right\vert \right\vert =\left\vert\sum_{k=2}^{|\eta|}\binom{|\eta|}{k}(-1)^{k}\delta^{k-1}\right\vert &=\delta\left\vert\sum_{k=2}^{|\eta|}\binom{|\eta|}{k}(-1)^{k}\delta^{k-2}\right\vert\leq \delta \sum_{k=2}^{|\eta|}\binom{|\eta|}{k}<\delta\cdot 2^{|\eta|}.
\tfrac{1}{2}g_t'={}g_0((\cosh(t)+A\sinh(t))(\cosh(t)+A\sinh(t))'\cdot,\cdot) ={}g_t((\cosh(t)+A\sinh(t))^{-2}(\cosh(t)+A\sinh(t))(\cosh(t)+A\sinh(t))'\cdot,\cdot) ={}g_t((\cosh(t)+A\sinh(t))^{-1}(\sinh(t)+A\cosh(t))\cdot,\cdot), \intertext{so}A_t={}(\cosh(t)+A\sinh(t))^{-1}(\sinh(t)+A\cosh(t)).
V_i = \frac{\pi^{d/2}}{\Gamma(1+\frac{d}{2})}\prod_{l=1}^d r^l_i = \frac{\pi^{d/2}}{\Gamma(1+\frac{d}{2})}\epsilon(x_i,k)^d\prod_{l=1}^d \frac{\sigma^l_i}{\sigma_i^1}.
H^{2}= \frac{\rho}{3}\left( 1 + \frac{\rho}{2\lambda}\right)+ \frac{1}{6}\left( \sqrt{\frac{6}{\lambda}}\Lambda_{(5)}+\frac{1}{2}\lambda \right) + \frac{\sigma_0^2}{6}a^{\-6}- k a^{-2}+ \frac{2{\cal U}_{0}}{\lambda}a^{-4}.
\mathrm{E}\{\overline{\mathrm{SNR}}\}\geq &\frac{1}{N_0P}\sum_{p=0}^{P-1}\frac{1}{\lambda_p^{-1}+\{\mathrm{Tr}[(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}]-2\lambda_p^{-1}\}\mathrm{E}\{\sin^2\theta_{p,p}\}},
\begin{cases}e^{-2Q}\delta_{st}\leq \bar{h}_{st}\leq e^{2Q}\delta_{st}; (r/2)^{1+\alpha}\left\|\frac{\partial}{\partial x^{j}}\bar{h}_{st}\right\|_{C^{0,\alpha}(B_{r/2}(\bar{p}))}\leq e^{2Q},\end{cases}
z^{2}\left[ \gamma_{n}^{\prime\prime}+2\left( 6\gamma_{n}-n\right) \left(2\gamma_{n}-n\right) \right]^{2}=4\left( z^{2}+2\gamma_{n}-n\right)^{2}\left[ \left( \gamma_{n}^{\prime}\right)^{2}+4\gamma_{n}\left(2\gamma_{n}-n\right)^{2}\right] .
{\tilde \mu}= \frac{-4e^{G/2}\partial_T \{\log \eta(T))\log \eta(\frac{T}{3})\}\partial_U \{\log \eta(\frac{U}{3}) \log\eta(U) \}}{9\eta^{2}(T) \eta^{2}(\frac{U}{3})\eta^{-2}(U) \eta^{-2}(\frac{T}{3})}.
\left\|u\right\|_{\mathcal{W}^{k,p}_{\boldsymbol{\beta}}}:=\left(\sum_{j=0}^{k}\int_{M}|\rho^{(-\boldsymbol{\beta}+j)}\nabla^{j}u|^{p}\rho^{-m}\,d\mu_{\gamma}\right)^{1/p}.
\widetilde{V}^{(-)}_1(r)= \frac{1}{4}\omega^2 r^2 +\frac{\ell(\ell+1)}{r^2}+\frac{2\omega^2 r^2+4\omega \ell -2\omega}{\omega r^2 +2\ell+1}+\frac{8\omega^2r^2}{(\omega r^2 +2\ell+1)^2}+ (\ell-\frac{1}{2})\omega
\nabla_a{\cal M}^{a\mu\nu}={1 \over 2}\left[ 2{\cal Q}^{ab}e^\mu{}_a e^\nu{}_b - \sqrt{-\gamma}\; L^{ab}{}_i K_{ab\,j}n^{\mu\,i}n^{\nu\,j}- (\mu \leftrightarrow \nu)) \right]\,.
f = x_0^d + x_0^{d-2 a_1}x_1^2 + x_0^{d-a_2-a_3}x_2 x_3 + \cdots + x_0^{d-a_{n-1}- a_n}x_{n-1}x_n + x_0^{d-a_3}x_1^3 + \cdots,
\Phi_{i}(P,f;C)=\displaystyle{\sum\limits_{C_{k}\in C^{i}}\sum\limits_{R \subseteq C\backslash C^{i}}\sum\limits_{T \subseteq C_{k}\backslash \{i\}}q(c_k,r,t)}\Bigl(f\bigl(T \cup S_R \cup \{i\}\bigr)- f(T \cup S_R)\Bigr),
\begin{multlined}[t][12.5cm]\left.\partial_{\sigma_{l}\sigma_{k}}^{2}\Psi(x,t,\sigma)\right|_{(t_{0},\sigma_{0})}=i\lambda_{\nu_{T}}\left[\Phi_{l}(m)\Phi_{k}(m)+\langle\gamma_{l},\gamma_{k}\rangle_{m}\right],\end{multlined}
\Psi(t,\theta,\phi)=e^{-iEt}\exp{i(n+\frac{1}{2}-\frac{1}{2}\sigma_{_{3}})\phi}\left ( \begin{array}{c}AJ_{_{|\frac{1}{\beta}(n+\frac{1-\beta}{2})|}}(\alpha cr) BJ_{_{|\frac{1}{\beta}(n+\frac{1-\beta}{2})+1|}}(\alpha cr) \end{array}\right ),
\frac{1}{|{\bf x - x^{\prime}}|}= 4 \pi \sum_{l = 0}^{\infty}\sum_{m = - l}^{l}\frac{1}{( 2 l + 1 )}\, \frac{r^{\prime l}}{r^{l + 1}}Y_{l m}^{*}( \theta^{\prime}, \phi^{\prime}) Y_{l m}( \theta , \phi ) ,
c(\theta)=\sum_{j=1}^\infty c_j \phi_j(\theta)=\; c_1\phi_1(\theta)\quad+\sum_{\{j\,:\,\lambda_j=d-1\}}c_j\,\phi_j(\theta)\quad+\sum_{\{j\,:\,\lambda_j=2d\}}c_j\,\phi_j(\theta)\quad+\sum_{\{j\,:\, \lambda_j>2d\}}c_j \phi_j(\theta)\,.
\nabla_z g_{r, x}=\nabla_{\chi_r(d(z))\Phi_x(g(z))}\Phi_x^{-1}\left(\chi_r'(d(z))\left(\nabla_z d\right) \Phi_x(g(z))+\chi_r(d(z))\nabla_{g(z)}\Phi_x \nabla_z g\right),
K(z,z'')= \Big(\int_{r'<\frac{r''}{2}}+\int_{\frac{r''}{2}\leq r'\leq2r''}+\int_{r'>2r''}\Big)G(z,z')Q(z',z'')\;d\mu(z') = K_{2,1}(z,z'')+K_{2,2}(z,z'')+K_{2,3}(z,z'').
V^{[n]}_{z_1z_1z_1}- V^{[n]}_{z_1}+U_{z_2}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, & \left(\Omega^{[1]}\,U\right)_{z_1}+U_{z_2}=0, & \Omega^{[j]}_{z_1}=UV^{[j]}_{z_1}, j=1\dots n.
\Gamma_{PV}(A)=(-1\pm{\rm sign}(m))\frac{i\pi}{2}\eta(0)+i\sum_{n=0}^\infty \frac{P_{2n+1}}{(m^2)^{n-(d-1)/2}}\frac{\Gamma\left(n-\displaystyle{\frac{d}{2}}+1\right)}{2n+1-d}.
\begin{array}{l}\partial^2_t u=\partial^2_x u+\frac{\kappa^2}{4}|u|^r-\frac{c_1^2+c_2^2}{2}u,t>0,x\in D, u(t,x)=0,\mbox{for}~x\in \partial D,~t\geq 0, u(0,x)=u_0(x),\partial_t u(0,x)=v_0(x),\end{array}
Y_t = e^{-\delta \theta T}c_T^{1-\gamma}+ \int_t^T F(s, c_s, Y_s) \, ds - \int_t^T Z_s dB_s, F(t, c_t, y):= \delta \theta e^{-\delta t}c_t^{1-\frac{1}{\psi}}y^{1-\frac{1}{\theta}}.
\int_{\mathbb{R}^{d}}\left(h(U^{1})-h(U^{0})\right)\partial_{i}\phi dx =\int_{0}^{1}\left(\int_{\mathbb{R}^{d}}Dh(U_{r})(U^{1}-U^{0})\partial_{i}\phi dx\right)dr, &=-\int_{\mathbb{R}^{d}}\int_{0}^{1}\partial_{i}\left(Dh(U_{r})(U^{1}-U^{0})\right)dr\phi dx
|v_{1,2}| \geq |q_2 | - |v_{1,2}+ q_2 | \geq \frac{1}{4}|q_2| + \frac{3}{4}|q_2| - |v_{1,2}+ q_2 | & \geq \frac{1}{4}|q_2| + \frac{3}{4 \sqrt{d}}|q|^\alpha - \frac{3}{4 \sqrt{d}}|q|^\alpha = \frac{1}{4}|q_2| .
\delta_{z \bar{z}}^{}( z_{1}^{}- z_{2}^{}) = - 2 z_{1}^{}\bar{z}_{2}^{}\delta ( z_{1}^{}\bar{z}_{1}^{}- z_{2}^{}\bar{z}_{2}^{}) \delta ( z_{1}^{}\bar{z}_{2}^{}- z_{2}^{}\bar{z}_{1}^{})
\tilde{\Theta}_\pm A\varphi =\tilde{\Theta}_\pm \gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\gamma^{(2n+1)}\tilde{\Theta}_\pm (P_{\rm F}-D_aP_{\rm F}D_a)\varphi=A\tilde{\Theta}_\pm \varphi.
NLP(1,3,0,\neq) = \int_1^T \int A Q^2 A \partial_Y^t\left( \left(\partial^t_Z U_0^1\right)_{Hi}\left(\partial_X U_{\neq}^3\right)_{Lo}\right) dV dt + \int_1^T \int A Q^2 A \partial_Y^t\left( \left(\partial^t_Z U_0^1\right)_{Lo}\left(\partial_X U_{\neq}^3\right)_{Hi}\right) dV dt = P_{HL}+ P_{LH}.
{}_N\!\langle \psi'_{n},\bar{\psi}'_{n}|\psi_{n},\bar{\psi}_{n}\rangle_N=\prod_{\mu=0}^{p}\delta\left(\mathrm{Re}\psi'_{n\mu}-\mathrm{Re}\psi_{n\mu}\right) \delta\left(\mathrm{Im}\psi'_{n\mu}-\mathrm{Im}\psi_{n\mu}\right)~.
\mathcal{M}_{3 / 2}^{\mathcal{N}= 3}= \frac{\mathrm{S U}( 1 , 1 )}{\mathrm{U}( 1 )}\times \frac{\mathrm{U S p}( 3 , 3 )}{\mathrm{S U}( 3 ) \times \mathrm{U}( 1 )}~ .
\frac{\mathrm{d}r^\pm}{\mathrm{d}\theta}\!=\!\frac{\sum_{k=2}^{n}R^\pm_k(\theta)r^k}{1+\sum_{k=2}^{n}\Theta^\pm_k(\theta)r^{k-1}}=\sum_{k=2}^{\infty}S^\pm_k(\theta)r^k,
\{R \in S^2(\wedge^2 V^*)\: | \: \mathfrak{S}_{Y,Z,W}R(X,Y,Z,W)=0 \}& \{S \in S^2(\wedge^2 V^*) \otimes V^* \: | \: \mathfrak{S}_{Z,W,U}S(X,Y,Z,W,U)=0 \},
{\rm E}|I_0^{d,p}(\lfloor2dq\rfloor)| ={\rm E}\sum_x1_{\{\sum_{y:y\sim x}\eta_0^{d,p}(y)\geq \lfloor2dq\rfloor\}}&=\sum_xP\big(\sum_{y:y\sim x}\eta_0^{d,p}(y)\geq\lfloor2dq\rfloor\big) &=r^dP\big(\sum_{y:y\sim O}\eta_0^{d,p}(y)\geq\lfloor2dq\rfloor\big),
\frac{5}{4^4}\Big( \frac{150}{24}a^3 N^{-1/4}-\frac{45}{24}a^2 N^{-6/4}+\frac{21}{24}a N^{-11/4}\Big)(x-N)^4 &-\frac5{4^3}\frac36 a N^{-7/4}(x-N)^3+\frac5{4^2}\frac12 a N^{-3/4}(x-N)^2\equiv R_N(x,t) &\ge c\Big\{(a^3N^{-1/4}+a^2N^{-6/4}+aN^{-11/4}) (x-N)^4 &\;\;\;\;\;\;\;\;+ aN^{-7/4}(x-N)^3 + a N^{-3/4}(x-N)^2\Big\},
L_1\int_{B^+_4(0)}|\nabla w_k|^2dz+o(1)=L_1\int_{B^+_4(0)}|\nabla v_k(s_k)|^2dz+o(1) \ge L_1\int_{B}|\nabla u(t_k+r_ks_k)|^2\varphi_k^2dz+o(1) \ge L_1\int_{B}|\nabla u(t_k)|^2\varphi_k^2dz \ge L_1\int_{B^+_2(0)}|\nabla v_k(0)|^2dz\ge L_1\delta.
\hat{\gamma}_{i j}d \hat{x}^{i}d \hat{x}^{j}= \left( 1 + \textstyle{\frac{\hat{k}}{4}}\delta_{m n}\hat{x}^{m}\hat{x}^{n}\right) \delta_{i j}d \hat{x}^{i}d \hat{x}^{j}={\frac{d \hat{r}^{2}}{1 - \hat{k}\hat{r}^{2}}}+ \hat{r}^{2}\left( d \hat{\theta}^{2}+ \sin^{2}\hat{\theta}d \hat{\phi}^{2}\right) .
\frac{1}{2}\int_{E_m}( \langle D^2 g(x)y_0, y_0\rangle +2o_x(|y_0|^2)) dx =\frac{1}{2}|y_0|^2\int_{E_m}\langle D^2 g(w_{x,y_0})\frac{y_0}{|y_0|},\frac{y_0}{|y_0|}\rangle dx &\le \frac{1}{2}||D^2g||_{L^{\infty}(a_qE_m)}m |y_0|^2.
\sum_{f\in E(G_1\times G_2)}w(f) =\frac{1}{r!}\sum_{f\in E(G_1\times G_2)}w_1(\pi_1(f))w_2(\pi_2(f)) & =\sum_{e_1\in E(G_1)}\sum_{e_2\in E(G_2)}w_1(e_1)w_2(e_2) & =1.
(u_{-}-v_{+}-\frac{1}{2}h)(u_{+}-v_{-}+ \frac{1}{2}h)H_{i}^{+}(u)H_{i+1}^{-}(v) &=(u_{-}-v_{+}+\frac{1}{2}h)(u_{+}-v_{-}-\frac{1}{2}h)H_{i+1}^{-}(v)H_{i}^{+}(u).
(A_1-A_2)^2 =I_f\otimes I_{\ell+1}\otimes S^2 &=I_f\otimes I_{\ell+1}\otimes (\frac{n-2\ell}{a}S+\frac{\ell(n-\ell)}{a^2}I_n) &=\frac{\ell(n-\ell)}{a^2}A_0+\frac{n-2\ell}{a}(A_1-A_2), (A_1+A_2)^2 =I_f\otimes I_{\ell+1}\otimes (nJ_n-I_n) &=(n-1)A_0+(n-2)(A_1+A_2).
\int \hat \Psi_{\alpha}\cdot \hat \Psi_{\alpha}\circ F_{\alpha}^k \cdot h_{\alpha}\:dm = \int P_{\alpha}^k(\hat \Psi_{\alpha}h_{\alpha}) \cdot \hat \Psi_{\alpha}\:dm &= \int P_{\alpha}\left(P_{\alpha}^k(\hat \Psi_{\alpha}h_{\alpha}) \cdot \hat \Psi_{\alpha}\right) \:dm
S_{\rm ct}=-\frac{1}{8\pi G}\int_{\cal \partial M}d^nx \sqrt{-h}\left[ \frac{n-1}{l}+\frac{l}{2(n-2)}{\cal R}+ \frac{l^3}{2(n-4)(n-2)^2}\left({\cal R}_{ab}{\cal R}^{ab}- \frac{n}{4(n-1)}{\cal R}^2\right) \right],
f_{e + f + g + h}^{( 2 )}= \frac{g_{1}^{2}}{8 \pi^{2}}\frac{g_{2}^{2}}{8 \pi^{2}}\int_{\lambda_{1}^{2}}^{1}\frac{d \alpha_{1}}{\alpha_{1}}\int_{\lambda_{1}^{2}/ \alpha_{1}}^{1}\frac{d \beta_{1}}{\beta_{1}}\int_{\lambda_{2}^{2}}^{1}\frac{d \alpha_{2}}{\alpha_{2}}\int_{\lambda_{2}^{2}/ \alpha_{2}}^{1}\frac{d \beta_{2}}{\beta_{2}}.
\sum_{0\le t<n}\left(\frac{\exp\left\{i\,\frac{2\pi}{n}\,u\,t\right\}}{\sqrt{n}}\right)\,\left(\frac{\exp\left\{-i\,\frac{2\pi}{n}\,t\,v\right\}}{\sqrt{n}}\right)=\begin{cases}\begin{array}{cc}1 \mbox{if}\:0\le u=v<n 0 \mbox{otherwise}\end{array}.\end{cases}
\mathcal{C}_{v}(\alpha)=\int\limits_{-B_{v}^{\alpha{\rm D}}/2}^{B_{v}^{\alpha{\rm D}}/2}S^{*}(f,B_{v}^{\alpha{\rm D}},b_{v})S(f-\Delta f,B_{\bar{v}}^{\alpha{\rm D}},b_{\bar{v}})df ,
\begin{aligned}\frac{(-zq,-q^2/z,q^3;q^3)_\infty}{(q)_\infty}= \sum_{s=0}^\infty \frac{(-z,-q/z;q)_s}{(q)_{2s}}q^{s^2}= \sum_{s=0}^\infty \frac{(-zq)_{s+1}(-1/z)_s}{(q)_{2s+1}}q^{s^2+s}. \end{aligned}
\frac{\langle v,Sv\rangle}{\langle v,v\rangle}=\tilde x \langle v,v\rangle+\frac{\langle w,(\hat\Sigma-\tilde\Sigma) w\rangle}{\langle v,v\rangle}>\frac{\tilde x}{2}\frac{\sqrt{n}}{z}+\frac{\langle w,(\hat\Sigma-\tilde\Sigma) w\rangle}{\langle v,v\rangle}.
L_{0}^{m}= q^{1 / 2}m [ ( \dot{x}_{\mu}+ q^{1 / 2}i \chi \psi_{\mu})^{2}]^{1 / 2}+ \frac{i}{2}( \psi_{\mu}\; \dot{\psi}^{\mu}- \psi_{5}\dot{\psi}_{5}) - i \chi \psi_{5}m .
\begin{aligned}u^\varepsilon (x) = \inf \Big\{\int_0^{\tau^\varepsilon}L(X^\varepsilon (t, x, \alpha), \alpha (t))e^{-\lambda t}\, dt + g^\varepsilon (X^\varepsilon (\tau^\varepsilon, x, \alpha))e^{-\lambda \tau^\varepsilon}\ | \ \alpha \in L^\infty (\mathbb R; \mathbb R^2) \Big\},\end{aligned}
\delta(\mathbf{L},\mathbf{A}) =\delta(\mathbf{L},\mathbf{A})+\delta(\mathbf{A},\mathbf{B})\ge \delta(\mathbf{L},\mathbf{B}), \delta(\mathbf{L},\mathbf{B}) =\delta(\mathbf{L},\mathbf{B})+\delta(\mathbf{A},\mathbf{B})\ge \delta(\mathbf{L},\mathbf{A}),
P^{(\alpha,\beta)}_{d,k}(x)=\frac{(|\alpha|+1)_k}{k!}F^{(d)}_C\left(\begin{array}{c}-k,k+|\alpha|+\beta+1 begin{align*}3pt] \alpha_1+1,\ldots,\alpha_d+1\end{array};\,\frac{1-x}{2},\ldots,\frac{1-x}{2}\right),
I_{N}^{1}= - \int_{z_{-}}^{z_{+}}\frac{dw}{2\pi i}\frac{\eta_{-}}{\varphi(x)}e^{\frac{\eta_{-}}{\varphi(x)}(v-u)(w - \Re{z_{-}})}= - \frac{\sin \pi (u-v)}{\pi (u-v)}.
\left(\int_{\mathbb{R}^3}\Phi(f)|v_1|dv\right)^2-\Big|\int_{\mathbb{R}^3}\Phi(f)vdv\Big|^2\cr \geq\left(\int_{\mathbb{R}^3}\Phi(f)|v_1|dv\right)^2-\Big\{\sum_{1\leq i\leq 3}\Big|\int_{\mathbb{R}^3}\Phi(f)v_idv\Big|\Big\}^2\cr \qquad=\left(\int_{\mathbb{R}^3}\Phi(f)|v_1|dv\right)^2-\Big(\int_{\mathbb{R}^3}\Phi(f)v_1dv\Big)^2-R\cr \qquad\equiv I-R.
{\mathcal{F}}_{\tilde{\mathcal{D}}}(J ,J'_{\sigma,k}) = \frac{{\mathcal{A}}_{j'_{\sigma(k)}}^{J\setminus J'_{\sigma,k}}}{x_{1}-x_{j'_{\sigma(k)}}+2}-\frac{\tilde{\mathcal{D}}_{j'_{\sigma(k)}}^{J\setminus J'_{\sigma,k}}}{x_{1}-x_{j'_{\sigma(k)}}+1}\,.
\int_M|\nabla h(x)|d\mu(x)=\int_M\left|\nabla\left(\int_0^{f(x)}\eta(t)dt\right)\right|d\mu(x) = \int_M|\nabla f(x)||\eta(f(x))|d\mu(x)\leq\Vert |\nabla f| \Vert_{L^2(\mu)}\Vert f-g_k \Vert_{L^2(\mu)}.
\sum_{j=1}^dE\left[\frac{\partial\varphi}{\partial x_j}(\sqrt{t}F+\sqrt{1-t}Z)\frac{Z_j}{\sqrt{1-t}}\right]=\sum_{i,j=1}^dE\left[\frac{\partial^2\varphi}{\partial x_i\partial x_j}(\sqrt{t}F+\sqrt{1-t}Z)\mathfrak{C}(i,j)\right].
\bigg\vert 2\,\zeta_i^{(k)}\circ \Delta_t \circ{\rm gw}_t(A)- \ell_i^{(k)}(A)\bigg\vert = \bigg\vert \frac{2}{t}\ln\vert\Delta_i^{(k)}(b_t)\vert - \ell_i^{(k)}(A)\bigg\vert & = \bigg\vert \frac{1}{t}\ln\vert\Delta_i^{(k)}(b_t)\vert^2 - \frac{1}{t}\ln\left(\prod_{j=1}^i\lambda_{j}^{(k)}(b_tb_t^*)\right)\bigg\vert
\mathrm{d}\psi_t+\big(\tfrac{1}{2}\mathrm{L}^\ast\mathrm{L}+\mathrm{i}H\big)\psi_t\mathrm{d}t= L^k\mathrm{d}w_k^t\psi_t,\psi_0=\psi\otimes\delta_\emptyset
| \Delta F |_{1 1}\leq \frac{2 | g | | b_{1}| e^{\frac{- x}{4}}}{\pi}\int_{- \infty}^{+ \infty}d \lambda \left\{\frac{2}{| 1 + ( \lambda + i / 4 )^{2}|}+ \frac{4 \delta}{| 1 - i k |^{2}}\right\},
\frac{\alpha_{3}( m_{\mathrm{s t r i n g}})}{\alpha_{Y}( m_{\mathrm{s t r i n g}})}= \frac{1}{3}+ \frac{1}{2}| x y | + | 1 + ( \hat{n}- n_{2}) y | + | ( x + \hat{n}- n_{2}) y | + 1
\begin{aligned}F_{N.M}\Xi({\eta_{1}, \cdots, \eta_{M}}) \lesssim e^{CM^{2}}(\prod_{j=1}^{M}e^{-\frac{\beta}{2}(N-M)V(\eta_{j})})e^{(N-M)\mu_{{\rm eq}}(h)}&\le Ce^{CM^2}e^{-\frac{\beta NM}{2}\int V(\eta)d\mu_{{\rm eq}}(\eta)}e^{-N\frac{\beta}{2}\sum_{j=1}^{M}{\mathcal{J}}^{V;\boldsymbol{B}}(\eta_{j})}\,.\end{aligned}
z^{k+1}= (1-\alpha)z^k+\alpha R_{\gamma A}R_{\gamma B}z^k &=(1-\alpha)z^k+\alpha \delta \begin{bmatrix}\cos{\xi}\sin{\xi}-\sin{\xi}\cos{\xi}\end{bmatrix}\begin{bmatrix}\cos{\xi}-\sin{\xi}\sin{\xi}\cos{\xi}\end{bmatrix}z^k &=(1-\alpha)z^k+\alpha \delta z^k &=|1-\alpha|z^k+\alpha \delta z^k
\frac{d x_{\mu}}{d s \,}\frac{d^{2}x_{\mu}}{d s^{2}\,}\, = \, 0 ; \qquad \frac{d x_{\mu}}{d s \,}\frac{d^{3}x_{\mu}}{d s^{3}\,}\, = \, k_{1}^{2}; \qquad \frac{d^{2}x_{\mu}}{d s^{2}\,}\frac{d^{3}x_{\mu}}{d s^{3}\,}\, = \, - \, k_{1}\, \frac{d k_{1}}{d s \,}\,{.}
\lambda^{-\frac{n-2}{2}}{\sf Q}(y) + \lambda^{-\frac{n-2}{2}}\sigma{\sf A}_1 = \pm \eta^\frac{2}{1-q}{\sf U}(\xi) |y|\to\infty,\ |\xi|\to0, \pm \eta^\frac{2}{1-q}{\sf U}(\xi) = \pm{\sf U}_\infty(x) + \Theta_J(x,t) |\xi|\to\infty,\ |z|\to0.
\begin{aligned}\partial^\circ\eta_0(r)-V_\Gamma\partial_r\eta_1(r)+k_dr|W|^2\partial_r\eta_0(r) -k_d\Delta_\Gamma\eta_0(r)+k_dH\partial_r\eta_1(r)-k_d\partial_r^2\eta_2(r) = f \end{aligned}
\rho \theta_{tt}-\kappa \theta_t= -\rho_t\theta_t -\rho_t u\cdot\nabla \theta-\rho u_t\cdot\nabla \theta-\rho u\cdot\nabla \theta_t &+ P_tu+Pu_t +Q(u)_t+\frac{1}{\sigma}|H|^2_t.
J_{\alpha\beta}^{i(2)}= \frac{1}{2}\big{(}x^\gamma X_\gamma(J_{\alpha\beta}^{i})+z^j Z_j(J_{\alpha\beta}^{i})\big{)}^{(2)}=\frac{1}{2}\big{(}x^\gamma \big(X_\gamma(J_{\alpha\beta}^{i})\big)^{(1)}+z^j Z_j(J_{\alpha\beta}^{i})(q)\big{)}=\frac{1}{2}\big{(}x^\gamma x^\delta X_\gamma X_\delta(J_{\alpha\beta}^{i})(q)+z^j Z_j(J_{\alpha\beta}^{i})(q)\big{)}.
(f-g)(y) = (f-g)\bigl(\sum_{j\in J}\frac{1}{m}\, m_{u_j,v_j}\bigr) \leq (\|f\|+\|g\|) \sum_{j\in J}\frac{1}{m}\|m_{u_j, v_j}\| &\leq (2+2\theta)\frac{2m}{mn}= \frac{4(1+\theta)}{n}.
\mathcal{J}^{v_1, v_2}_{\alpha, i}(\theta, x) \ := \ \dfrac{1}{\theta}\log E^{v_1, v_2}_x\Big[ e^{\theta \int^{\infty}_0 e^{-\alpha t}r_i(X(t), v_1(t,X(t)), v_2(t,X(t)) dt}\Big] , x \in \mathbb{R}^d,
\hat{C}_{( 2 n )}^{+}\; = \; C_{( 2 n )}^{+}\, - \, C_{( 2 n - 1 )}^{+}\left( d \underline{{{y}}}+ A^{( 2 )}\right) \, + \, \underline{{{y}}}\G_{( 0 )}^{+}\textstyle{\frac{1}{n !}}\hat{B}^{n}\; ,
G=J=\int_{\Gamma}(Wn_{1}-t_{i}\frac{\partial u_{i}}{\partial x_{1}})d\Gamma=\int_{C_{1}+C_{+}+C_{-}-C}(Wn_{1}-\sigma_{ij}n_{j}\frac{\partial u_{i}}{\partial x_{1}})\tilde{q}d\Gamma
Q_{XYU}^{(\alpha,\theta)}(x,y,u) := \frac{Q_{XYU}(x,y,u)\exp\Big(-\theta\omega_{Q_{XYU}}^{(\alpha)}(x,y|u)\Big)}{\sum_{x,y,u}Q_{XYU}(x,y,u)\exp\Big(-\theta\omega_{Q_{XYU}}^{(\alpha)}(x,y|u)\Big)}.
A = QR\Pi^T = Q \left(R^T\right)^T \Pi^T \approx Q \left(\begin{array}{c}\widehat{R}_{11}^T \widehat{R}_{12}^T\end{array}\right) \widehat{Q}_1^T \Pi^T = A \left(\Pi \widehat{Q}_1\right) \left(\Pi \widehat{Q}_1\right)^T.
\delta S = \int_{M}d^{2}X \mathrm{~}\left( \frac{\partial{\cal L}}{\partial \Lambda_{A}}\mathrm{~}\delta \Lambda_{A}+ \frac{\partial{\cal L}}{\partial \partial_{\alpha}\Lambda_{A}}\mathrm{~}\delta \left( \partial_{\alpha}\Lambda_{A}\right) \right) \mathrm{.}
\frac{1}{2}\sum_{ij}\frac{\partial^{2}\epsilon_{n}}{\partial k_{i}\partial k_{j}}q_{i}q_{j}=\frac{\hbar^{2}}{2m}q^{2}+\sum_{n^{\prime}\ne n}\frac{|\langle nk|\frac{\hbar^{2}}{m}q\cdot(-i\nabla)|n^{\prime}k\rangle|^{2}}{\epsilon_{nk}-\epsilon_{n^{\prime}k}}
\left(i\j\right)=V_{\mathbb{Z}\beta+\frac{i+j}{4k}\beta}\otimes V_{\mathbb{Z}\beta+\frac{i-j}{4k}\beta}+V_{\mathbb{Z}\beta+\frac{2k+i+j}{4k}\beta}\otimes V_{\mathbb{Z}\beta+\frac{2k-i+j}{4k}\beta}.
\mathcal P^{n_0}(x, I) = \mathbb P [Y_{n_0}^{a,b,\varepsilon_i}\in I\mid X_0 = x]= \frac{1}{(b-a)^{n_0}}\mathrm{Leb}^{\otimes n_0}\left(\{p\in [a,b]^{n_0};\F^{x,n_0}(p) \in I\}\right) >0.
\begin{array}{cc}A Q^3 \leftrightarrow a q^3 Q^6 \leftrightarrow q^6 A \bar{Q}^3 \leftrightarrow a \bar{q}^3 \bar{Q}^6 \leftrightarrow \bar{q}^6 A^3 Q^3 \leftrightarrow a^3 q^3 Q \bar{Q}\leftrightarrow M_0 A^3 \bar{Q}^3 \leftrightarrow a^3 \bar{q}^3 Q A^2 \bar{Q}\leftrightarrow M_2 A^4 \leftrightarrow a^4 \end{array}
ds_{\rm BTZ}^2 = -{{(\rho^2 - \rho_+^2) ( \rho^2 - \rho_-^2)}\over{\rho^2 \ell^2}}dt^2 + \rho^2 \left ( d \phi -{J \over{2 \rho^2}}dt \right )^2 +{{\rho^2 \ell^2}\over{(\rho^2 -\rho_+^2) ( \rho^2 - \rho_-^2)}}d\rho^2 .
A = \left[{\frac{- i \mu^{\epsilon}}{( 2 \pi )^{d}}}\right]^{\ell}\int \prod_{a = 1}^{\ell}( d^{d}k_{a}){\frac{S_{0}( q , p )}{\prod_{r = 1}^{N}( - q_{r}^{2}+ m_{r}^{2}- i \epsilon )}}\ ,
\int_0^1e^{2\pi im(x+u)}\prod_{k=1}^s\cos( 2\pi n_{k_j}u)du &=2^{-s}\int_0^1e^{2\pi im(x+u)}(\cos2\pi mu+\cos(-2\pi mu))du &=2^{1-s}\int_0^1e^{2\pi im(x+u)}\cos(2\pi mu)du &=2^{1-s}e^{2\pi imx}\int_0^1\cos^2(2\pi mu)du &+i\cdot 2^{1-s}e^{2\pi imx}\int_0^1\sin(2\pi mu)\cos(2\pi mu)du &=2^{-s}e^{2\pi imx},
\frac{\#\{1\leq i\leq n:\lambda_i=0\}}{n}=\frac{\#\{1\leq i\leq 2^{k}:\lambda_i=0\}}{n}&+\sum_{l=0}^{k-1}\frac{\#\{\sum_{j=k-l}^{k}\epsilon_j 2^j+1 \leq i \leq \sum_{j=k-l-1}^{k}\epsilon_j 2^j:\lambda_i=0\}}{n}.
L\varphi(v,v_*)= \frac \alpha 2 |v-v_*|^\gamma (1+|v|^2)^{\alpha/2-2}\Big[-2(1+|v|^2)|v|^2+ 2(1+|v|^2)|v_*|^2 & \hskip3cm+ \big((\alpha-2)+\alpha (1+|v|^2)^{\alpha/2}\big)\big(|v|^2|v_*|^2-(v\cdot v_*)^2\big)\Big]\varphi(v).
\sum_{n_{j}=1}^{p_{j}^{a_{j}}}\left(\frac{D(n_{j})}{p_{j}}\right)^{a_{j}}=\sum_{n_{j}=1}^{p_{j}^{a_{j}-1}}\sum_{d=1}^{p_{j}}\left(\frac{D(n_{j}p_{j}+d)}{p_{j}}\right)^{2b_{j}+1}=p_{j}^{a_{j}-1}J(p_j).
- (1-\kappa )\mathfrak{a}[\mathbf{u},\mathbf{u}] -c_0\Vert \mathbf{u}\Vert^2_{L_2(\mathbb{R}^d)}&\leqslant q[\mathbf{u},\mathbf{u}]\leqslant c_2 \mathfrak{a}[\mathbf{u},\mathbf{u}]+c_3\Vert \mathbf{u}\Vert^2_{L_2(\mathbb{R}^d)},\mathbf{u}\in H^1(\mathbb{R}^d;\mathbb{C}^n),
\left\langle W_{P},W_{Q}\right\rangle ={\textstyle\sum\limits_{l=0}^{n}}\frac{\gamma_{l}^{2}}{2l+1}\left\langle P_{l},Q_{l}\right\rangle ={\textstyle\sum\limits_{l=0}^{n}}\frac{(c_{l})^{2}}{n+1}\left\langle P_{l},Q_{l}\right\rangle
\tilde{f}_0^n := \frac{1}{\sqrt{\hat{H}(\rho_n)^2 + \hat{P}(\rho_n)^2}}[\hat{H}(\rho_n) e_0 - \hat{P}(\rho_n) e_1], \tilde{f}_1^n := \frac{1}{\sqrt{\hat{H}(\rho_n)^2 + \hat{P}(\rho_n)^2}}[\hat{P}(\rho_n) e_0 + \hat{H}(\rho_n) e_1].