harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_{a_1}(\theta_1)A_{a_2}(\theta_2)=\sum_{n=2}^{\infty}\,\sum_{\theta'_1{<}\dots{<}\theta'_n\atop{}}\!S^{b_1\dots b_n}_{~a_1a_2}(\theta_1,\theta_2;\theta'_1\dots\theta'_n)A_{b_1}(\theta'_1)\dots A_{b_n}(\theta'_n)~,
S=\,_1\langle \,_2\langle I||\Phi \rangle_1 Q^{(2)}|\Phi\rangle_2 +\frac{2}{3}\,_1\langle \,_2\langle \,_3\langle V||\Phi \rangle_1 |\Phi\rangle_2 |\Phi \rangle_3,
\frac{\partial MPK}{\partial K}=\frac{\partial^{2}Y}{\partial K^{2}}=\frac{\partial}{\partial K}(AL^{\beta}\alpha K^{\alpha-1})=AL^{\beta}\alpha(\alpha-1)K^{\alpha-2}=\alpha(\alpha-1)AL^{\beta}\frac{K^{\alpha}}{K^{2}}=\alpha(\alpha-1)\frac{Y}{K^{2}}
\prod_{j=x_1}^k |\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_1-k|}, \prod_{j=k}^{x_2}|\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_2-k|}c_{n,\ell}c_{n,\ell+1}.
Q_{2\gamma}= 2^{\frac{2(n-2)}{n}\gamma}\frac{\Gamma(\gamma)\Gamma\left(-\frac{\gamma}{n}\right)\Gamma\left(\frac{n+2\gamma}{2n}\right)}{n\Gamma(-\gamma)\Gamma\left(\frac{\gamma}{n}\right)\Gamma\left(\frac{3n-2\gamma}{2n}\right)}.
- \tau^t ( \Theta ( \Sigma^t ) ) = \lim_{M \to \infty}- \tau^t ( \Theta ( \Sigma^t_M ) ) = \lim_{M \to \infty}- \tau^t ( \Theta_M ( \Sigma^t_M ) ) = \lim_{M \to \infty}\lim_{N \to \infty}-\tau_N^t (\Theta_M (\Sigma^t_M ) ).
{\cal L}_{b}= - \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}+ \frac{1}{2}( D^{a})^{2}- \frac{1}{2}( D_{\mu}C^{a})^{2}- \frac{1}{2}( v_{\mu}^{a})^{2}+ \frac{k}{2}C^{a}D^{a}- \frac{k}{2}A_{\mu}^{a}v^{a \mu},
\sigma_{k}(p^j,s) = \frac{\sum_{i=0}^{\infty}\frac{\tau_{k}(p^{j+i})}{p^{is}}}{\sum_{i=0}^{\infty}\frac{\tau_{k}(p^i)}{p^{is}}}= (1-p^{-s})^{k}\sum_{i=0}^{\infty}\frac{\tau_{k}(p^{j+i})}{p^{is}}
\| \widehat{U}^{*}(\Phi^{\prime}(f\otimes a) + \sum_{j=1}^m\sum_{k=1}^m f(z_k)\rho (p_{j,k}\otimes a) ) \widehat{U}& -(\Psi^{\prime}(f\otimes a)+ \sum_{j=1}^m\sum_{k=1}^mf(z_k)\rho (p_{j,k}\otimes a)) \| < \frac{\varepsilon}{7}
\bigsqcup_{t \in T}S_t := \left( \bigsqcup_{t \in T^{+}}(S_t^+) \sqcup \bigsqcup_{t \in T^{-}}(S_t^-), \bigsqcup_{t \in T^{+}}(S_t^-) \sqcup \bigsqcup_{t \in T^{-}}(S_t^+) \right).
\theta_{f(n)}-\theta_{f(n+1)}=n^{-\frac{dq}{1-p}}-\left(n^{-\frac{dq}{1-p}}-\frac{dq}{1-p}\xi^{-\frac{dq}{1-p}-1}\right)+q\cdot n^{-\frac{d(1+q)}{1-p}}&=\frac{dq}{1-p}\xi^{-\frac{dq}{1-p}-1}+q\cdot n^{-\frac{d(1+q)}{1-p}}&\le\frac{dq}{1-p}n^{-\frac{dq}{1-p}-1}+q\cdot n^{-\frac{d(1+q)}{1-p}}.
\leq\sum_{l\in\mathbb{Z},\,\,a_{l-1}\neq0}\sum_{i\in\mathbb{Z},\,\,i\leq l-1}2^{ip}a^{-sp/Q}_{l-1}d_{l}=\sum_{l\in\mathbb{Z},\,a_{l-1}\neq0}\sum^{+\infty}_{k=0}2^{p(l-1-k)}a_{l-1}^{-sp/Q}d_{l}\leq S.
\sum\limits_{\delta|N}\delta'r_\delta=p_1p_2r+p_1p_2r-p_1p_2r-p_1p_2r=0 &\sum\limits_{\delta|N}r_\delta=r(1+p_1p_2-p_1-p_2)=\frac{24}{(p_1-1)(p_2-1)}(p_1-1)(p_2-1)=24.
\begin{pmatrix}Z_k Q_k \end{pmatrix}(t) = S_k(t,0) \begin{pmatrix}Z_k Q_k \end{pmatrix}(0) + \int_{0}^{t}S_k(t, \tau) \begin{pmatrix}c_k^{+}Z_{k+1}+ c_{k}^{-}Z_{k-1}d_k^{+}Z_{k+1}+ d_{k}^{-}Z_{k-1}\end{pmatrix}(\tau) d\tau.
\left[ \begin{array}{c}{{}}\\{{\sum \sum}}\\{{}_{j < l}}\\ \end{array}\alpha_{j}\alpha_{l}k_{j l}^{2}- \sum \alpha_{i}m_{i}^{2}\right] \Rightarrow - \left[ \sum \alpha_{i}^{2}m_{i}^{2}\! + \! 2 \! \begin{array}{c}{{}}\\{{\sum \sum}}\\{{}_{j < l}}\\ \end{array}\alpha_{j}\alpha_{l}m_{j}m_{l}c_{j l}\right] ,
\overrightarrow{C}_{(4:n)}=\bigoplus_{j=0}^3 \Gamma(j)=\bigoplus_{j=0}^3 \left(\Gamma(j)\oplus F_n(\Gamma(j))\oplus F_n(\Gamma(\pi(j))\right)=\bigoplus_{j=0}^3 \Lambda(j,\pi(j))
|\zeta^{(n)}(\sigma_0+i \tau, \alpha)-z_n| = \left| \frac{n!}{2\pi i}\oint_{|s-\sigma_0|=r}\frac{\zeta(s+i \tau, \alpha)-f(s)}{(s-\sigma_0)^{n+1}}\,ds \right| &\leq \frac{n!}{r^n}\sup_{s \in K}|\zeta(s+i \tau, \alpha)-f(s)|.
S_{0,t}(T) \coloneqq \sum_{n=1}^\infty \sum_{e^{n\ell(\gamma)}\leq T}\ell(\gamma) e^{n\ell_\psi(\gamma) + itn \ell_\phi(\gamma)}, \hat{S}_{0,t}(T) = \sum_{e^{\ell(\gamma)}\leq T}\ell(\gamma) e^{\ell_\psi(\gamma) + it\ell_\phi(\gamma)},
\begin{multlined}x^{k+1}\cdot x^{k+1}\cdot 17 x^4+x^{k+1}\cdot (2^k+k)x^k \cdot x^5+(2^{k}+k)x^{k}\cdot x^{k+1}\cdot x^5 = (2 k + 2^{k + 1}+ 17) x^{2 k + 6},\end{multlined}
\bar{F}(x) \leq \rho_{2}\bar{V}^{r}(x)+\rho_{1}f^{*}(\gamma)\bar{F}_{\gamma}(x) + \rho_{1}f^{*}(\gamma)\bar{B}^{r}(x-\Delta)F_{\gamma}(\Delta) &+ \rho_{1}f^{*}(\gamma) \int_{\Delta}^{x-\Delta}\bar{B}^{r}(x-u)dF_{\gamma}(u) + \rho_{1}f^{*}\left[\bar{F}_{\gamma}(x-\Delta) - \bar{F}_{\gamma}(x) \right].
1 \leq |\sin(\psi_{N-1})\sin(\psi_{N-1}')|\cdot\left|\left(B(B')^{-1}\right)_{11}\right|+|\cos(\psi_{N-1})\cos(\psi_{N-1}')| &\leq \sin(\psi_{N-1})\sin(\psi_{N-1}')+\cos(\psi_{N-1})\cos(\psi_{N-1}') &=\cos(\psi_{N-1}-\psi_{N-1}')\leq 1.
\mathring E\{T(\psi)\}=\eta^{1/2}\{\xi^{rst}\mathring\lambda_{rs,t}+\xi^{rs}\mathring\lambda_{rs}+O_p(n^{-3/2})\}&=\eta^{1/2}\{\xi^{rst}\lambda_{rs,t}+\xi^{rs}\lambda_{rs}+O_p(n^{-3/2})\}.
L_{\nu}=-(1+\left\vert z\right\vert^{2})\left( (1+\left\vert z\right\vert^{2})\frac{\partial^{2}}{\partial z\partial \bar{z}}+\nu \left( z_{j}\frac{\partial}{\partial z_{j}}-\bar{z_{j}}\frac{\partial}{\partial \bar{z_{j}}}\right) -\nu^{2}\right) -\nu^{2}\equiv -\frac{1}{4}\Delta_{\nu}.
{\mathrm{\bf ~ M}}( v ) = \left( \begin{array}{c c}{{\frac{1 + v^{2}}{1 - v^{2}}}}&{{\frac{+ 2 v}{1 - v^{2}}}}\\{{\frac{+ 2 v}{1 - v^{2}}}}&{{\frac{1 + v^{2}}{1 - v^{2}}}}\\ \end{array}\right) = \frac{1}{\gamma^{2}}\left( \begin{array}{c c}{{1 + v^{2}}}&{{+ 2 v}}\\{{+ 2 v}}&{{1 + v^{2}}}\\ \end{array}\right) \equiv \frac{1}{\gamma^{2}}{\mathrm{\bf ~ L}}( v ) \, .
( 1 , 3 ) ( 2 , 4 ) ( e - ( 1 , 3 ) ) ( e - ( 2 , 4 ) ) = ( e - ( 1 , 3 ) ) ( e - ( 2 , 4 ) ( 1 , 3 ) ( 2 , 4 ) ) = ( e - ( 1 , 3 ) ) ( e - ( 2 , 4 ) )
f^L(\lambda \vec x) =\int_0^{\lambda\|x\|_\infty}f(V_t^+(\lambda\vec x),V_t^-(\lambda\vec x))d t &=\int_0^{\|x\|_\infty}\lambda f(V_{s}^+(\vec x),V_{s}^-(\vec x))d s=\lambda f^L(\vec x).
T_{a , \alpha q}^{\underline{{\alpha}}}E_{\underline{{\alpha}}, \beta r}= T_{a , \alpha q}^{b}E_{b}^{\underline{{\alpha}}}E_{\underline{{\alpha}}, \beta r}- ( \nabla_{\alpha q}E_{a}^{\underline{{\alpha}}}+ \omega_{\alpha q , a}^{~ b}E_{b}^{\underline{{\alpha}}}) E_{\underline{{\alpha}}, \beta r}- ( \nabla_{a}E_{\alpha q}^{\underline{{\alpha}}}) E_{\underline{{\alpha}}, \beta r}.
a_2(x)=\frac{\Gamma(n-2)\operatorname{vol}(\mathbb{S}^{n-2})}{(2\pi)^{n-1}}\bigg( a_{2,1}H^2+a_{2,2}\sum_{\alpha}\kappa_\alpha^2+ a_{2,3}R+a_{2,4}R_{nn}\bigg)
\mathcal{L}_{I_{2,n}}(s,d)=\exp\left(-\frac{2\pi}{\alpha_2}T_{n}\lambda_{2}s^{\frac{2}{\alpha_2}}B^{'}\left(\frac{2}{\alpha_2},1-\frac{2}{\alpha_2},\frac{1}{1+sd^{-\alpha_2}}\right)\right),
\ell^\Phi_{\Pi_x}= \sum_{y\in \partial \Gamma_x}\ell^\Phi_{\Pi_y}\cdot \ell^\Phi_{\Gamma_{(\geq y)_x}}+ \sum_{y\in \notin \partial \Gamma}\ell^\Phi_{\Pi_y}\cdot \Phi_{\Gamma_{(\geq y)_x}}.
\tau \||x|^{-1}v\|_{L^2(A_{2 r_1, r_2}^+)}\lesssim (\ln(r_2/r_1))\|Av\|_{L^2(A_{r_1, r_2}^+)}, \tau \| |x|^{-1}v\|_{L^2(A_{2 r_2, r_3}^+)}\lesssim (\ln(r_3/r_2))\|Av\|_{L^2(A_{r_2, r_3}^+)}, \mbox{where}v=e^{\tau\phi}\tilde{w},
\begin{pmatrix}V_0 V_1 -\gamma W_0 -\gamma W_1\end{pmatrix}\cdot \begin{pmatrix}\alpha \beta\end{pmatrix}+ \begin{pmatrix}V_1 V_0 \gamma W_1 \gamma W_0\end{pmatrix}\cdot \begin{pmatrix}E_{\gamma}0 0 E_{\gamma}\end{pmatrix}\cdot \begin{pmatrix}\alpha \beta\end{pmatrix}=z.
{\cal E}= \frac{\sqrt{1 - \lambda_{9}^{2}}}{\lambda_{9}}\left( u - i v \right) = \frac{( 1 - \lambda_{9}^{2}){\cal E}_{0}}{1 - \lambda_{9}^{2}R e{\cal E}_{0}}, z = \frac{i}{\left( 1 - \lambda_{9}^{2}{\cal E}_{0}\right)}.
f^{\prime\prime}(x)\approx\frac{\nabla_{h}^{2}[f](x)}{h^{2}}=\frac{\frac{f(x)-f(x-h)}{h}-\frac{f(x-h)-f(x-2h)}{h}}{h}=\frac{f(x)-2f(x-h)+f(x-2h)}{h^{2}}.
F(t-s,x) = \P^{(2)}\left( M_{t-s}\geq \sqrt{2}(t-s) - \tfrac{1}{2\sqrt{2}}\log(t)- A - (x-\sqrt{2}s) \right) &\leq C \frac{\sqrt{t+1}}{(t-s+1)^{\frac{3}{2}}}\left(1 + \frac{\log(t)}{\sqrt{2}}+(-x)_+ \right) e^{-\sqrt{2}(\sqrt{2}s - x - A)}.
\int_{\Sigma}K_\Sigma \, dS = 2\pi \chi(\Sigma) - \int_{\partial \Sigma}\kappa \, d s, \int_{F(\Sigma)}\! \widetilde{K}_{F(\Sigma)}\, d \widetilde{S}= 2\pi \chi(F(\Sigma)) - \int_{\partial F(\Sigma)}\widetilde{\kappa}\, d \tilde{s}.
{\cal R}_{0}= q^{- \frac 1 2}\left( \begin{array}{c c c c}{{q}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{q}}\\ \end{array}\right) ,
\int_0^x t^\nu \mathbf{L}_\nu(t)\,\mathrm{d}t=\frac{x^{2\nu+2}}{\sqrt{\pi}2^{\nu+1}(\nu+1)\Gamma(\nu+\frac{3}{2})}{}_2F_3\bigg(1,\nu+1;\frac{3}{2},\nu+\frac{3}{2},\nu+2;\frac{x^2}{4}\bigg).
\pi^\infty_Y (a) U_n - U_n ( \pi_X^\infty (a) \oplus P_n \pi_Y^\infty (a) P_n ) =\pi^\infty_Y (a) P_n + \pi^\infty_Y (a) V_n - P_n \pi^\infty_Y (a) P_n - V_n \pi_X^\infty (a) &=\pi^\infty_Y (a) V_nV_n^* - V_nV_n^* \pi^\infty_Y (a) + \pi^\infty_Y (a) V_n - V_n \pi_X^\infty (a),
\sup_{u \in (-\infty,A]}\frac{\overline{\mu}(x-u)}{\overline{\mu}(x)}\le \frac{\overline{\mu}(x-A)}{\overline{\mu}(x)}\mbox{and}\lim_{x \to \infty}\frac{\overline{\mu}(x-A)}{\overline{\mu}(x)}= e^{\gamma A},
\dot{\hat{x}}^{\hat{\alpha}}=(\mathcal{J}^{\hat{\alpha}\hat{\beta}}-\mathcal{R}^{\hat{\alpha}\hat{\beta}})\delta_{\hat{\beta}}\hat{\mathcal{H}}+\mathcal{G}_{\xi}^{\hat{\alpha}}u^{\xi}+\mathcal{K}_{\eta}^{\hat{\alpha}}u_{o}^{\eta}\,, \hat{y}_{\xi}=\mathcal{G}_{\xi}^{\hat{\alpha}}\delta_{\hat{\alpha}}\hat{\mathcal{H}}\,,
\xi_1^{l_1^{(j)}}\cdots\xi_t^{l_t^{(j)}}=\left(\xi_{1,1}^{(j)}\cdots\xi_{1,l_1^{(j)}}^{(j)}\right)\cdots\left(\xi_{t,1}^{(j)}\cdots\xi_{t,l_t^{(j)}}^{(j)}\right)
e(G)\le 6-|U|+ex_{_\mathcal{P}}(n-4+|U|,\Theta_4^+) \le \begin{cases}6 +ex_{_\mathcal{P}}(n-k,\Theta_k) \, |U|=0 5+ex_{_\mathcal{P}}(n-k+1,\Theta_k) \, |U|=1 4 +ex_{_\mathcal{P}}(n-k+2,\Theta_k) \,|U|=2. \end{cases}\medskip
-2\pi-y_{p}+ 2\pi\coth\bigg(\frac{2\pi}{y_{p}}\bigg)=-2\pi-y_{p}+2\pi\bigg(\frac{y_{p}}{2\pi}+\frac{2\pi}{3y_{p}}+O\bigg(\frac{1}{y_{p}^{3}}\bigg)\bigg)=-2\pi+\frac{4\pi^{2}}{3y_{p}}+O\left(\frac{1}{y_{p}^{3}}\right).
\bigl\{T_a,\,T_b^{(1)}\bigr\}_{(\phi,\pi)}+\bigl\{T_a^{(1)},\,T_b\bigr\}_{(\phi,\pi)}+\bigl\{T_a^{(1)},\,T_b^{(2)}\bigr\}_{(\eta)}+\bigl\{T_a^{(2)},\,T_b^{(1)}\bigr\}_{(\eta)}=0\,.
\begin{aligned}\|\varphi_N (u_t- Lu)\|_{C^{k,\alpha}_{WF}([0,T]\times\bar S_{n,m})}\leq c \rho^N\left(r^{-(k+3)}+T_0^{-(k+3)}\right) \|u_t- Lu\|_{C^{k,\alpha}_{WF}([T_0/2,T]\times\bar B_{2r}(z^0))}.\end{aligned}
{\cal F}_{i j}= F_{i j}-{\cal B}_{i j}\, , \hspace{1 c m}F_{i j}= 2 \partial_{[ i}V_{j ]}\, , \hspace{1 c m}{\cal B}_{i j}= \partial_{i}Z^{M}\partial_{j}Z^{N}{\cal B}_{N M}\, .
\Psi_{\theta}\left( r , \varphi , t \right) = e^{- i E t / \hbar}\sqrt{\frac{\tilde{\zeta}}{\pi \left| \alpha \right| !}}\exp \left( - \frac{\tilde{\zeta}r^{2}}{2}\right) \left( \sqrt{\tilde{\zeta}}r \right)^{\left| \alpha \right|}e^{i \alpha \varphi}.
H_g^{(M_{24})}(\tau) =-2R^{[-1/8]}_{\Gamma_0(n_g), \rho_g\epsilon^{-3}, 1/2}(\tau) =-2q^{-1/8}+2\sum_{n>0}c_{\Gamma_0(n_g), \rho_g\epsilon^{-3}, \frac{1}{2}}q^{n-1/8}.
D_4^2f_{-2}= D_4^2u_{-2}+ c_2(v_0-u_0), &D_4^2f_{-1}= D_4^2u_{-1}+ c_1(v_0-u_0) + c_2(v_1-u_1), &D_4^2f_{0}= D_4^2v_{0}+ c_{-2}(u_{-2}-v_{-2}) + c_{-1}(u_{-1}-v_{-1}), &D_4^2f_{1}= D_4^2v_{1}+ c_{-2}(u_{-1}-v_{-1}).
I^{q}_{2,2}=\sum_{k=k_0+1}^{+\infty}\int_{2^{k}\leq|x|\leq 2^{k+1}}\left(\int_{B(0,2|x|)\setminus B\left(0,\frac{|x|}{2}\right)}|G_{\xi, \sigma}(y^{-1}x)u(y)|dy\right)^{q}\frac{dx}{|x|^{\beta q}}.
\prod_{i=1}^n|p_i|^\lambda \leq c \prod_{i=1}^{2n}(|u_{i}|^{\frac{\lambda}2}+|u_{i-1}|^{\frac{\lambda}2}) &=c\,\sum_{S_1}\prod_{i=1}^{2n}(|u_{i}|^{\frac{\lambda}2\gamma_{i}}|u_{i-1}|^{\frac{\lambda}2\overline{\gamma_{i}}}) &\leq c\,\sum_{S_2}\prod_{i=1}^{2n-1}(|u_{i}|^{\frac{\lambda}2\alpha_{i}}),
\begin{cases}f a_\pm, a_0, f'( a_\pm) =-p < 0,\f'(a_0)=\mu > 0, f(u)\leq C(1+|u|^q)C,q>0, f'(u)\leq cc>0, f(u)\leq -p(u-a_+) \ (u\geq a_+), f(u)\geq -p(u-a_-) \ (u\leq a_-), \end{cases}
V_{\cal N}^{-}( q ) = \frac{C_{1}^{2}}{2}q^{6}+ C_{1}C_{2}q^{4}+ \frac 1 2 \{C_{2}^{2}- ( 4{\cal N}- 1 - 2 C_{3}) C_{1}\}q^{2}+ \frac{C_{3}( C_{3}+ 1 )}{2}\frac{1}{q^{2}}.
\Delta_{\xi}( a ) = F \Delta ( a ) F^{- 1}, \quad \; S_{\xi}( a ) = u \, S ( a ) u^{- 1}, \quad \; u = \sum_{i}f_{i}^{( 1 )}S ( f_{i}^{( 2 )}) \, \quad \; ( a \in A ) \, .
\mathbb{B}_L=\Delta^{-1}\left(\begin{matrix}[d,a]-[b,c] [d,b] [a,c] 0\end{matrix}\right),\;\;\;\;\mathbb{B}_R=\left(\begin{matrix}[c,b] [b,a] [c,d] [d,a]\end{matrix}\right)\Delta^{-1}.
dq^\prime_t= (\tilde\gamma^T)^{-1}(t^*,q^\prime_t)\left(-\partial_t\psi(t^*,q^\prime_t)-\nabla_qV(t^*,q_t^\prime)+ \tilde F(t^*,q^\prime_t,\psi(t^*,q^\prime_t))\right) dt & +\tilde S^\prime(t^*,q^\prime_t)dt+(\tilde\gamma^T)^{-1}(t^*,q^\prime_t)\sigma(t^*,q^\prime_t)\circ dW_t,
\mathbb{P}_{T}\Big(\sup_{g\in S(r)}\Big(\frac{1}{n_r}\sum_{i\in A_{r k}}\sum_{\ell=1}^{n/m}\big(f_0(T_{\ell}^{(i)})g(T^{(i)}_\ell)-\mathbb{E}_{T}f_0(T_{\ell}^{(i)})g(T^{(i)}_\ell)\big)\Big)^2 \geq \frac{2c_1 2^r}{n_r}\Big)\lesssim 2^{-r}e^{-(c_3 2^{r}\wedge \sqrt{n_r})}.
F[x,y]=\frac{x^{\prime\prime}y^{\prime\prime\prime}-x^{\prime\prime\prime}y^{\prime\prime}}{(x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime})^{5/3}}-\frac{1}{2}[\frac{1}{(x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime})^{2/3}}]^{\prime\prime}
\textstyle m\alpha(\phi_{m+1})=2m\arccos\bigl(\cos\bigl(\frac{\pi}{m+2}\bigr)\bigr)=\frac{2\pi m}{m+2}; &\textstyle\beta(\phi_{m+1})=2\pi-2\arccos\bigl(2\cos^2\bigl(\frac{\pi}{m+2}\bigr)-1\bigr)=2\pi-2\arccos\bigl(\cos\bigl(\frac{2\pi}{m+2}\bigr)\bigr)=2\pi-\frac{4\pi}{m+2}=\frac{2\pi m}{m+2}.
R_{nm}''(r) + \frac{1}{r}R_{nm}'(r) -\left(\frac{m^2}{r^2}-\frac{4m}{1+r^2}-\frac{4(1-r^2)}{(1+r^2)^2}\right) R_{nm}(r) =-\varepsilon^2_{nm}R_{nm}(r).
\max_{j=1}^r \max_{\ell=0}^{k_j-1}\left\{|\lambda_j|^{m-\ell}{m\choose \ell}\right\}\leq \max_{l=0}^{d_p - 1}{m\choose l}= \begin{cases}{m\choose d_p-1}\quad\text{for $d_p -1 < \left\lfloor\frac{m}{2}\right\rfloor$}, begin{align*}3mm]{m \choose \left\lfloor\frac{m}{2}\right\rfloor}\text{for $d_p -1 \geq \left\lfloor\frac{m}{2}\right\rfloor.$}\end{cases}
\Big|\mathrm{Tr}(a_0\left(\prod_{k=1}^{p-1}\partial(a_k)\right) [|D|^{2-p}e^{-s^2D^2},\partial(a_p)])\Big| \leq\|a_0\|_{\infty}\prod_{k=1}^{p-1}\|\partial(a_k)\|_{\infty}\cdot\Big\|[|D|^{2-p}e^{-s^2D^2},\partial a_p])\Big\|_1.
I(\mu,\nu,\beta)= \frac{r!}{[(r+1)!]^2}\sum^\mu_{j=0}\sum^\nu_{k=0}{\mu\choose j}{\nu\choose k}(j+k)!(r-j-k)! &\times\int^1_0 (1-z)^{r-1-j}z F(1+r-j-k, 1+r+s, r+2,z)dz.
\begin{aligned}\left|\int_0^kKdx\right| \leq C_\epsilon(T)\left(1+\int_0^k(\rho^{\frac{\gamma-1}{2}})_t^2dx+\|ur^{\frac{1}{2}}\|_{L^\infty}^2\right)\left(\int_0^ku_t^2dx+\int_0^k(\rho^{\frac{\gamma-1}{2}})_t^2dx+\int_0^k\rho^{\gamma-1}\frac{u^2}{r^2}dx\right) &+\epsilon\int_0^k\rho(r^2u_{xt})^2dx+\epsilon\int_0^k\rho^{-1}\frac{u_t^2}{r^2}dx.\end{aligned}
\begin{aligned}G\left(\left|m_u\left(B\left(z, r_1\right)\right)-m_u\left(B\left(z, r_2\right)\right)\right|\right) & \leq 4 b^2 2^{s+\alpha p_{0}}[u]_{W^{\alpha,G}_{s}(X,d,\mu)}^{p^{0}}\left(r_1^{\alpha p_{0}-s}+\frac{r_2^{\alpha p_{0}}}{r_1^s}\right)\end{aligned}
i\hbar=i\hbar1=i\hbar(\frac{dp}{dp}-p\frac{d}{dp})=xp-px=x(-i\hbar\frac{d}{dx})-(-i\hbar\frac{d}{dx})x=-i\hbar(x\frac{d}{dx}-\frac{dx}{dx})=-i\hbar(-1)=i\hbar
\left ( \begin{array}{c}\tilde{\psi}_{0}\vdots \tilde{\psi}_{p}\end{array}\right ) = \sum_{k \in \mathbb{Z}}\tilde{B}_{k}\left ( \begin{array}{c}\tilde{\varphi}_{0}(2 \cdot - k) \vdots \tilde{\varphi}_{p}(2 \cdot - k) \end{array}\right ) .
P_{n}^t\times P_{n}^t=\bigcup_{\xi\in G_{2t+r}}\tilde{V}_t((\xi\times 1_{2m-2t})\times 1_{2n})=\bigcup_{\xi\in G_{2t+r}}\tilde{V}_t(1_{2n}\times(\xi\times 1_{2m-2t})),
\left|\epsilon\int_0^{r/\epsilon}B(X_{rs\epsilon/r}^\epsilon,y)(1_{\{Y_{s}=y\}}-\pi(y))ds\right|\leq r\frac{\eta}{2}+2rC\sum_{i=1}^{n^{\epsilon,r}+1}\left|\epsilon\int_0^{\sigma_i^{\epsilon,r}r/\epsilon}(1_{\{Y_{s}=y\}}-\pi(y))dy\right|.
\mathbb{E}[\|\mathbf{V}-\mathbf{W}\|_1] =\mathbb{E}[|\mathcal{V}-\mathcal{W}|+|\mathcal{V}'-\mathcal{W}'|] &=\mathbb{E}[|\mathcal{V}-\mathcal{W}|]+\mathbb{E}[|\mathcal{V}'-\mathcal{W}'|],
\sum_{(j_t,n_t)\in J_{-k_t}}(q^{n_t+1}_{j_t}|f)q^{j_t}_{n_t}=-\sum_{(j_t,n_t)\in J_{k_t-1}}(q^{j_t}_{n_t}|f)q^{n_t+1}_{j_t}=\delta_{k_t,0}x
\lambda_1 = \varpi_2 + \varpi_3, \lambda_2 = 2\varpi_1 + 2\varpi_2 + 2\varpi_3, \lambda_3 = 2\varpi_1 + 2\varpi_2 + 3\varpi_3, \lambda_4 = 4\varpi_1 + 4\varpi_2 + 7\varpi_3.
\lbrace A , B \rbrace{}^{*}= \lbrace A , B \rbrace - i [ \lbrace A , \pi_{\theta \, i}+{\frac{i}{2}}\theta_{i}^{*}\rbrace \lbrace \pi_{\theta^{*}\, i}+{\frac{i}{2}}\theta_{i}, B \rbrace + \lbrace A , \pi_{\theta^{*}\, i}+{\frac{i}{2}}\theta_{i}\rbrace \lbrace \pi_{\theta \, i}+{\frac{i}{2}}\theta_{i}^{*}, B \rbrace ]
\sum\limits_{\mathbf{p}_k\in \mathcal{P}(d,k)}f(\mathbf{p}_k) =\sum\limits_{1\le p_1<\cdots<p_k\le d}f(\mathbf{p}_k)= \sum_{p_{k}=1}^{d}\sum_{p_{k-1}=1}^{p_k-1}\cdots \sum_{p_1=1}^{p_2-1}f(\mathbf{p}_k).
\sum_{n=1}^\infty \,\Bigl\Vert f(A_n+B_n)-f(A_n)-\frac{d}{dt}\Big(f(A_n+tB_n)\Big) \Big|_{t=0}\Bigr\Vert_2^2 & \leq \Vert f''\Vert_\infty^2\,\Bigl(\sum_{n=1}\Vert B_n\Vert_2^2\Bigr)^2\,<\infty.
\bigl(NDu\bigr)(x) =\sum_{i=1}^\infty\int_{\mathbb R^c}n(x,x-y)\,d_i(y)u(y-h_i)\,dy &=\sum_{i=1}^\infty\int_{\mathbb R^c}n(x,x-y-h_i)\,d_i(y+h_i)u(y)\,dy &=\int_{\mathbb R^c}\Bigl(\sum_{i=1}^\infty n(x,x-y-h_i)\,d_i(y+h_i)\Bigr)u(y)\,dy.
\qquad\qquad\frac{(\frac{1}{3})_k(\frac{2}{3})_k}{(1)_{k}^2}=\frac{\binom{2k}{k}\binom{3k}{k}}{27^k}, begin{align*}1mm] H_k\bigg(-\frac{1}{3}\bigg)+H_k\bigg(-\frac{2}{3}\bigg)=3H_{3k}-H_k,
F(x,y,\frac{dy}{dx})-\frac{d^{2}}{dx^{2}}(I(x,y)-h(x))+\frac{d^{2}y}{dx^{2}}\frac{\partial J}{\partial x}+\frac{dy}{dx}\frac{d}{dx}(\frac{\partial J}{\partial x})=12x^{2}-0+0+0=12x^{2}
\left( \prod_{i=1}^{n+1}(1+x_i) \right) \sum_{J\subseteq [2,n]}(-1)^{|J|}\left(\prod_{j\in J}\frac{x_j}{1+x_j}\right) \sum_{K \subseteq [2,n]\setminus J}k_{n - |J|-|K|}(\underline x_n^{J\cup K}).
|f(x)-f(y)|^p\leq C \Big(\int_{B_{xy}}|\nabla f(u)|^p \omega(u)^{\frac{-p}{n}}\Big(\omega(B_{xu})^{-\frac{n-1}{n}}+\omega(B_{yu})^{-\frac{n-1}{n}}\Big) \omega(u) du\Big) & \cdot \Big(\int_{B_{xy}}\Big(\omega(B_{xu})^{-\frac{n-1}{n}}+\omega(B_{yu})^{-\frac{n-1}{n}}\Big) \omega(u)du\Big)^{p-1}.
LT(Q^{(1)}_1(y_1)\otimes t_1^{(1)}(x)) = q^{(1)}_{1,e}\cdot(t_1^{(1)}(x))^{[e]}= q^{(1)}_{1,e}\cdot \left((-q^{(1)}_{0,d}/q^{(1)}_{1,e})^{[m-e]}x^{[d-e]}\right)^{[e]}= -q^{(1)}_{0,d}x^{[d]}.
t_{1}=\frac{1}{2}[\begin{matrix}0&-i\\ -i&0\end{matrix}],t_{2}=\frac{1}{2}[\begin{matrix}0&-1\\ 1&0\end{matrix}],t_{3}=\frac{1}{2}[\begin{matrix}-i&0\\ 0&i\end{matrix}].
\begin{array}{r c l c r c l}{{| e_{R}^{c}\rangle}}&{{=}}&{{\displaystyle \bigg | 1 , \frac{1}{2}, \frac{1}{2}\bigg \rangle ,}}&{{\quad}}&{{- | \nu_{R}^{c}\rangle}}&{{=}}&{{\displaystyle \bigg | 1 , \frac{1}{2}, - \frac{1}{2}\bigg \rangle , \nonumber}}\\{{| e_{L}^{c}\rangle}}&{{=}}&{{| 2 , 0 , 0 \rangle ,}}&{{\quad}}&{{| \nu_{L}^{c}\rangle}}&{{=}}&{{| 0 , 0 , 0 \rangle ,}}\\ \end{array}
cos(\pi z)=\frac{zsin(\pi z)}{\pi}\sum_{n=-\infty}^{\infty}\frac{1}{z^{2}-n^{2}}=\frac{zsin(\pi z)}{\pi}(\frac{1}{z^{2}}+2\sum_{n=1}^{\infty}\frac{1}{z^{2}-n^{2}}).
\mathcal{K}_s(a_0\otimes\cdots\otimes a_p) = \mathrm{Tr}(\Gamma a_0\left(\prod_{k=1}^{p-1}[F,a_k]\right)[Fe^{-s^2D^2},a_p]), \mathcal{H}_s(a_0\otimes\cdots\otimes a_p) = \mathrm{Tr}(\Gamma a_0\left(\prod_{k=1}^{p-1}[F,a_k]\right)F[e^{-s^2D^2},a_p]).
u_n^{-1}(\hat{\theta}_{1,n}- \hat{\theta}_{n}) = (u_n^T J_n(\hat{\theta}_{0,n})u_n)^{-1}\left[u_n^T J_n(\hat{\theta}_{0,n}) u_n \right. \left. - u_n^T \int_0^1 J_n(\hat{\theta}_{n}+ t(\hat{\theta}_{0,n}- \hat{\theta}_{n}))dt \u_n \right] u_n^{-1}(\hat{\theta}_{0,n}- \hat{\theta}_{n}).
F(u,n) =\eta_T(t) n u q(t)= \eta_T(t ) \Bigl[\int_0^t e^{i(t-t^\prime)\Delta}F(u,n)\d t' \Bigr]_{x=0}n_\pm = \mp i \int_0^t e^{\pm i (t-t^\prime) D}G(u )\d t' G(u ) =\eta_T(t) D^{-1}|u|^2, z(t)= \eta_T(t) [n_++n_-]_{x=0}
\sum_{k+l=n}\sum_{(k,l)-\textnormal{unshuffles}}\left ( \pm \nabla_{(a_{\sigma(1)}, \cdots , a_{\sigma(k)})}\left(\nabla_{(a_{\sigma(k+1)}, \cdots, a_{\sigma(n-1)})}s \right) \right. \left.\pm \nabla_{(\{a_{\sigma(1)}, \cdots, a_{\sigma(k)}\}, a_{\sigma(k+1)}, \cdots, a_{\sigma(n-1)})}s\right) = 0.
c_{1}= \frac{1}{f_{1}}\sqrt{\frac{X ( g_{1}+ g_{2})}{p_{3}}}, \qquad c_{2}= \frac{\sqrt{X g_{1}p_{1}}}{f_{2}g_{1}}, \qquad c_{3}= \frac{\sqrt{X g_{2}p_{2}}}{f_{3}g_{2}}.
\langle \tilde{q}^{2}| q^{2}| q \tilde{q}| q \tilde{q}\rangle = ( - 2 A_{1}- B_{3}) [ 1 2 ]^{2}[ 3 4 ]^{2}+ B_{1}[ 1 4 ]^{2}[ 2 3 ]^{2}+ ( B_{1}+ B_{2}- B_{3}) [ 1 2 ] [ 3 4 ] [ 1 4 ] [ 2 3 ] \, .
M= \max\Big\{\sqrt{\frac{2C(1+T)W_{0}}{a^2 t_{\star}}},\, \sqrt{\displaystyle\frac{(2C(1+T))^{\frac{q}{q+1}}W_{0}^{\frac{1}{q+1}}}{a}}\ \displaystyle\frac{2}{t_{\star}^{(1/2)^+}}\Big)\Big\}\,,
\mathcal{A}:=\left(\begin{array}{llll}\frac{\partial x^1}{\partial r}\frac{\partial x^1}{\partial f^2}\cdots \frac{\partial x^1}{\partial f^n}\vdots \vdots \vdots \vdots \frac{\partial x^n}{\partial r}\frac{\partial x^n}{\partial f^2}\cdots \frac{\partial x^n}{\partial f^n}\end{array}\right) \,.
\mathbf{u}^{\mathcal{L}}(x)\mathcal{J}_n \mathbf{u}^{\mathcal{R}}(y)=\dfrac{\sqrt{d_{n}}[(y-i\omega)u_{n}^{\mathcal{R}}(y)u_{n-1}^{\mathcal{L}}(x)-(x+i\omega)u_{n-1}^{\mathcal{R}}(y)u_{n}^{\mathcal{L}}(x)]}{x-y}.
\gamma_{m,d}= \frac{2^{-2}\omega_d^2 \Gamma(m+\tfrac12 d) \Gamma(\tfrac12 d)}{2^{-1}\omega_{2d}\Gamma(m+d)}= \frac{\Gamma(m+\tfrac12 d) \Gamma(\tfrac12 d)}{\Gamma(m+d)}\cdot \frac{\Gamma(d)}{\Gamma(\tfrac12 d)^2}= \frac{\Gamma(m+\tfrac12 d)}{\Gamma(m+d)}\cdot \frac{\Gamma(d)}{\Gamma(\tfrac12 d)}.
\begin{array}{l l l}{{D_{u}V}}&{{=}}&{{U}}\\{{D_{u}U}}&{{=}}&{{e^{K}C_{u u u}G_{u{\bar{u}}}^{- 1}{\bar{U}}}}\\{{d_{u}{\bar{U}}}}&{{=}}&{{G_{u{\bar{u}}}{\bar{V}}}}\\{{d_{u}{\bar{V}}}}&{{=}}&{{0}}\\ \end{array}
S_{Q 2 f}( M_{n , i}) = \sum_{i = 0}^{L - 1}\frac{1}{N}\mathrm{T r}( 2 M_{1 , i}^{2}+ 2 M_{2 , i}^{2}) + \frac{2 g}{N^{2}}\mathrm{T r}M_{1 , i}^{2}\mathrm{T r}M_{2 , i}^{2}
b_0(t,Y)-\Pi(t)\sigma_0^2(t,Y)- \frac{\lambda\Pi(t) \gamma_0^2(t,Y)}{1+\Pi(t)\gamma_0(t,Y)}&+\sigma_0(t,Y)\Phi(t,Y)+\frac{\lambda\gamma_0(t,Y)}{1+\Pi(t)\gamma_0(t,Y)}\Psi(t,Y)= 0,
-\sum_{0\leq p\leq r-1}x_p \bigl( \|\gamma_{p+1}\|_{HS}^2 - \|\gamma_{p}\|_{HS}^2\bigr) =- \|\gamma_{r}\|_{HS}^2 + \sum_{1\leq p\leq r}(x_p-x_{p-1}) \|\gamma_{p}\|_{HS}^2 &=-\sum_{k\leq \kappa}d_{k}^2 + \int_{0}^1\! \|\pi(x)\|_{HS}^2\, dx
(\hat{Tv})^{i^{\prime}}=\hat{T}_{j^{\prime}}^{i^{\prime}}\hat{v}^{j^{\prime}}=[(R^{-1})_{i}^{i^{\prime}}T_{j}^{i}R_{j^{\prime}}^{j}][(R^{-1})_{k}^{j^{\prime}}v^{k}]=(R^{-1})_{i}^{i^{\prime}}(Tv)^{i}.
\langle e^{i\theta}e^{i \nu H}e^{i \mu |x|^2}S_\lambda h_0 \,,\, h_n \rangle = e^{i\theta}e^{i\nu(4n+2)}\langle e^{i \mu |x|^2}S_\lambda h_0 \,,\, h_n \rangle.
A(\eta)=\left({\begin{array}{*{20}c}a^2+\left(b^2-a^2\right)\eta_1^2 \left(b^2-a^2\right)\eta_1\eta_2 \left(b^2-a^2\right)\eta_1\eta_2 a^2+\left(b^2-a^2\right)\eta_2^2 \end{array}}\right)
G^{- 1}= ( G^{A B}) = \left( \begin{array}{c c}{\widetilde{{g}^{i j}}}&{g^{i l}\partial_{l}\theta_{\gamma}g^{\gamma \beta}}\\{g^{j l}\partial_{l}\theta_{\gamma}g^{\gamma \alpha}}&{- g^{\alpha \beta}}\\ \end{array}\right)
[ \hspace{- 0 . 7 m m}[ S^{( i )}] \hspace{- 0 . 7 m m}] = [ 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 )}] .
S = \int d^{4}x \sqrt{- \tilde{g}}\left[ \tilde{R}- \frac{1}{2}\left( \tilde{\nabla}\Phi \right)^{2}- \frac{1}{2}\left( \tilde{\nabla}y \right)^{2}- \frac{1}{2}Q^{2}e^{2 \Phi - \sqrt{3}y}- \frac{1}{2}m^{2}e^{2 \Phi + \sqrt{3}y}\right] ,
f(\eta) = C_Q \bigg(\frac{1-\cos\alpha}{1-\cos\eta}\bigg)^{\frac{d-1}{2}}\bigg(\frac{1-\cos\alpha}{\cos\alpha-\cos\eta}\bigg)^{\frac{1}{2}}\,{}_2 F_1\bigg (1, \frac{d-1}{2};\frac{1}{2}; \frac{\cos\alpha-\cos\eta}{1-\cos\eta}\bigg) + F(\eta), \alpha \leq \eta \leq \pi.
\varphi_n:= \widetilde\varphi_n/\|\widetilde\varphi_n\|_{\mathsf L^2(\mathbb R)}, \quad\textrm{with}\widetilde\varphi_n(s):= \mathrm e^{-n(s- \mathrm i/2)^2}\big(1 -\mathrm e^{-n^2(s- \mathrm i)^2}\big)
\left(F^{+}(x)\right)^{-1}= \frac{1}{\Delta_{+}}\left( \begin{array}{lr}(p(x) - q(x))^{+}(p(x) + q(x))^{+}(p(x) - q(x))^{+}- (p(x) + q(x))^{+}\end{array}\right),
( t \partial_{z}+ z^{- 1}( D_{\jmath_{1}}^{+}D_{\jmath_{1}}^{-}- 2 \jmath_{0}D_{\jmath_{1}}^{0}) ) - ( \tilde{J}^{+}( z ) D_{\jmath_{1}}^{-}+ 2 \tilde{J}^{0}( z ) D_{\jmath_{1}}^{0}+ \tilde{J}^{-}( z ) D_{\jmath_{1}}^{+})
C_1 =\{[0,\tfrac{3}{4}),(\tfrac{1}{4},1]\}, C_2 =\{[0,\tfrac{2}{3}),(\tfrac{1}{3},1]\}, C_3 =\{[0,\tfrac{1}{2}),(\tfrac{1}{4},\tfrac{2}{3}),(\tfrac{1}{3},\tfrac{3}{4}),(\tfrac{1}{2},1]\}.