harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathfrak{f}_1(x) = (x+5)^{41}(x-11)^{13}(x-13)^3 (x^2-23x+106), \mathfrak{f}_2(x) = (x+5)^{41}(x-11)^{13}(x-13)^2 (x^3-36x^2+405x-1382), \mathfrak{f}_3(x) = (x+5)^{41}(x-11)^{13}(x-13)^2 (x-17) (x^2-19x+82).
S_{\overline{e}}D_{s}C v = S_{\overline{e}}D_{s}\left( \sum_{n=0}^{\infty}c_{mn}v_{n}\right)_{m}&= \sum_{m = 0}^{\infty}s^{m}\left( \sum_{n=0}^{\infty}c_{mn}v_{n}\right) \overline{e_{m}}
\sum\limits_{k=0}^{m-1}k\binom{n}{k}\;=\; \frac{n}{2}\binom{n}{<m}\;-\; \frac{m}{2}\binom{n}{m}&\sum\limits_{k=0}^{m-1}k(k-1)\binom{n}{k}\;=\; \frac{n(n-1)}{4}\binom{n}{<m}\;-\; \frac{m(2m+n-3)}{4}\binom{n}{m},
[\tau \cdot (a\psi) ] (m) = a\psi (\tau^{-1}\cdot m) = \psi (a (\tau^{-1}\cdot m)) = \psi (\tau^{-1}\cdot ( (\tau \cdot a) m)) = = [\tau \cdot \psi] ( (\tau \cdot a) m) = (\tau \cdot a) [\tau \cdot \psi] (m) = [(\tau \cdot a) (\tau \cdot \psi)] (m)
\mu(C_i) = \mu(C_i) \int_M{\mathbb E_\mu\left[\eta\mid \mathcal I\left(\frac{1}{\lambda^m}\mathcal P^i,\mu\right)\right]}\mu(\d x) \leq \frac{1}{m}\int_M \mathbb E_\mu\left[\frac{1}{\lambda^i}\mathcal P^i(\cdot,M)\mid \mathcal I\left(\frac{1}{\lambda^m}\mathcal P^i,\mu\right)\right](x) \mu(\d x) \leq \frac{1}{m}\i\in\{0,1,\ldots,m-1\}.
f_{C\cap A'}(x) = -\sin d(x,\partial\alpha)= -\sum_{j=1}^k a_j\sin d(v_{i_j},\partial\alpha) =-2\sum_{j=1}^{r-1}a_j\lambda_{i_j}-\sum_{i=r}^ka_j\lambda_{i_j}& > f_{C\cap A}(x) = \max_{\beta\in\Lambda_{C\cap A}}\{-\sin d(x,\partial\beta)\}=\max_{\beta\in\Lambda_{C\cap A}}\{-\sum_{j=1}^k a_j\sin d(v_{i_j},\partial\beta) \}.
\left\{b_{\underline{n}\sigma},b_{\underline{n}\sigma}^\dagger\right\}= \delta_{\underline{n}'\underline{n}}\delta_{\sigma'\sigma}\,,\;\;\left[a_{\underline{n}},a_{\underline{n}}^\dagger\right]= \delta_{\underline{n}'\underline{n}}\,.
\mathcal{K}_{\nu}(x)=2^{\nu}x^{-\nu}\Gamma\left(\nu+\frac{1}{2}\right)\mathbf{K}_{\nu}(x)=\frac{2}{\sqrt{\pi}}\int_0^{\infty}(1+t^2)^{\nu-\frac{1}{2}}e^{-xt}dt.
\frac{y^{2}}{x_{1}^{6}}= \frac{x_{1}^{2}x_{2}^{3}}{x_{1}^{6}}+ \frac{x_{1}^{4}x_{2}^{2}}{x_{1}^{6}}= ( \frac{x_{2}}{x_{1}})^{3}\frac{1}{x_{1}}+ ( \frac{x_{2}}{x_{1}})^{2}.
\begin{array}{l}{\nu_{r}^{\left( l \right)}\mid \Phi \rangle = 0 \qquad r > 0}\\{J_{n}^{\left( l \right)}\mid \Phi \rangle = 0 \qquad n > 0}\\ \end{array}\begin{array}{c}{l = 1 , 2 , \ldots N}\\ \end{array}
(\tau^p(x)^{pq},\phi(\tau^p(x)^{pq})) =\prod_{i=0}^{pq-1}(\tau^i(\tau^p(x)),\phi(\tau^i(\tau^p(x))))=\prod_{i=p}^{p(q+1)-1}(\tau^i(x),\phi(\tau^i(x))) =\Big(\prod_{i=p}^{2p-1}(\tau^i(x),\phi(\tau^i(x)))\Big)\dotsm\Big(\prod_{i=pq}^{p(q+1)-1}(\tau^i(x),\phi(\tau^i(x)))\Big)=v^q\in S.
\frac{\left| z\left(\left(\frac{1}{2}-\sigma\right) + j\omega\right)\right|}{\left|z\left(\left(\frac{1}{2}+\sigma\right) - j\omega \right)\right|}= \frac{\left|\left(\frac{1}{2}-\sigma\right) + j\omega\right|}{\left|\left(\frac{1}{2}+\sigma\right) - j\omega\right|}
{\lambda_{\ell}^{2}\frac 1{4!}A_{7777,\ell}=\lambda_{\ell}^{2}\frac{1}{3\cdot 4}\left[ \frac{3 \cdot 2^{11}}{\pi^2}\frac{1}{\ell^2}\log \ell +O(\ell^{-2})\right] = \frac{2^{9}}{\pi^2}\ell^2 \log \ell +O(\ell^{2}).}
\begin{bmatrix}c_0 c_1 -c_0 c_3 2 -c_2 -1 c_0 c_1 -c_0 -c_3 2 -c_2 -1 d_0 d_1 -d_0 d_3 2 -d_2 -1 d_0 d_1 -d_0 -d_3 2 -d_2 -1 \end{bmatrix}
\chi_{\hat{n}}( x ) \chi_{\hat{n}^{\prime}}( y )_{| x - y | > > \eta}= \chi_{\hat{n}^{\prime}}( y ) \chi_{\hat{n}}( x ) \exp \{i ( \Theta ( x - y ) - \Theta ( y - x ) ) \}= - \chi_{\hat{n}^{\prime}}( y ) \chi_{\hat{n}}( x )
\int_{\mathbb{R}^{\mathbb{N}}}\pi^{\star}d\nu_{\mathbf{f}}=\int_{\mathbb{R}^{\mathbb{N}}}1_{E}\pi^{\star}d\nu_{\mathbf{f}}=\int_{\mathbb{R}^{\mathbb{N}}}\lim_{n}1_{E}\pi_{n}d\nu_{\mathbf{f}}=\lim_{n}\int_{\mathbb{R}^{\mathbb{N}}}1_{E}\pi_{n}d\nu_{\mathbf{f}}=\lim_{n}\int_{\mathbb{R}^{\mathbb{N}}}\pi_{n}d\nu_{\mathbf{f}}.
\sum_{\rho}\hat{g}(\rho)-\int_{0}^\infty \Big\{g(x)+\frac{1}{4}g^*(x)\Big\}dx +\sum_{n=1}^\infty \Lambda(n)g(n) &+\int_1^\infty \Big\{g(x)+g^*(x)-\frac{2}{x}g(1)\Big\}\frac{x\dx}{2(x^2-1)}+\frac12(\log4\pi-\gamma)g(1),
\left\langle N_{\nu_{\alpha_{1}}}^{( \alpha_{1})}, \cdots , N_{\nu_{\alpha_{n}}}^{( \alpha_{n})}\right\vert \left. N_{\nu_{\alpha_{1}}}^{( \alpha_{1})}, \dots , N_{\nu_{\alpha_{n}}}^{( \alpha_{n})}\right\rangle ={\cal A}^{\{{\cal N}\}}\, .
U(t) \geqslant \exp\left((pq)^{\frac{j-1}{2}}\log (E_1\varepsilon^p)\right)(R+t)^{-\alpha_j}(t-L)^{\beta_j}&=\exp\left((pq)^{\frac{j-1}{2}}\left(\log(E_1\varepsilon^p)-\left(\frac{(n-1)p}{2}+n\right)\log(R+t)+\left(\frac{2p+1}{pq-1}+n\right)\log(t-L)\right)\right) &\quad\times(R+t)^{n}(t-L)^{-(2p+1)/(pq-1)}
{\cal H}=\frac{1}{2\mathcal{C_R}}\sum_{\mu\in\mathcal{R}}\ell^+_{\mu}\ell^-_{\mu}+ \sum_{\rho\in\Delta_{+}}g_{|\rho|}\left(\omega +\frac{1}{2}{|\rho|^2\over{(\rho\cdot q)^2}}\right)(\check{s}_{\rho}-1),
{\mathrm m}_\omega(e^{i\xi}) M_\lambda^{(c)}(\xi ) =U_{\lambda,\omega}(t) M_\lambda^{(c)}(\xi )+ \sum_{\substack{\nu\in W_0\omega \lambda+\nu\in{P_c}}}V_{\lambda,\nu}(t) M_{\lambda + \nu}^{(c)}(\xi ) .
\lim_{\lambda\rightarrow\infty}\sup_{i\geq 1}\mathbf{P}^{G}_0\left(\max_{x,y\in B_d(0,\kappa i)}\max_{0\leq t\leq Ti^\beta}\frac{i^{-(\beta-\alpha)}\left|L^{G}_{t}(x)-L_t^{G}(y)\right|}{\sqrt{{R}^{(i)}_{G}(x,y)\left(1+\ln_+{R}^{(i)}_{G}(x,y)^{-1}\right)}}\geq \lambda\right)=0.
F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^wF_s(X_s)dX(s) - \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s)~a.s
\lim_{\beta \rightarrow 2 \pi}\mathrm{T r}_{i}f_{\beta}^{\pm}( x , x ) = \lim_{\beta \rightarrow 2 \pi}\mathrm{T r}_{i}\bar{f}_{\beta}^{\pm}( x , x ) = \frac 1 2 \left( \delta ( x , x_{1}) + \delta ( x , x_{2}) \right) ,
-\frac{1}{2}\bar{\Omega}_{\bar{i}\bar{j}\bar{k}\bar{l}}F^{(2,0)\bar{k}\bar{l}}=\lambda F^{(0,2)}_{\bar{i}\bar{j}},\quad-\frac{1}{2}\Omega_{ijkl}F^{(0,2)kl}=\lambda F^{(2,0)}_{ij}.
[e_r, e_r]=\theta_re_n, [e_i, e_j]=\sum^{n-1}_{m=k+1}\alpha^t_{ij}e_t+\beta_{ij}e_n, [e_j, e_i]=-\sum^{n-1}_{m=k+1}\alpha^t_{ij}e_t+\gamma_{ji}e_n, [e_p, e_s]=\sum^{n-1}_{m=n-t+1}\alpha^t_{ij}e_t+\theta_{ij}e_n, [e_s, e_p]=-\sum^{n-1}_{m=n-t+1}\alpha^t_{ij}e_t+\delta_{ji}e_n,
c_4 (X){}= \sum_{r,s,t,u=1}^m c_4^{rstu}J_r J_s J_t J_u{}= \sum_{r,s,t,u=1}^m\frac{1}{4}\left[ -(n_r+1)\delta^{rstu}+ \sum_{\alpha=1}^K q_\alpha^r q_\alpha^s q_\alpha^t q_\alpha^u + 2 c_2^{rs}c_2^{tu}\right] J_r J_s J_t J_u.
\mathcal{S}_2= \{U=P+\mathbf{i}Q\in \mathcal{S}:\;P<0\}, \mathcal{S}_3= \{U=P+\mathbf{i}Q\in \mathcal{S}:\;Q>0\}, \mathcal{S}_4= \{U=P+\mathbf{i}Q\in \mathcal{S}:\;Q<0\}.
I_{b-}^{\alpha ;\psi}f\left( x\right) :=\frac{1}{\Gamma \left( \alpha\right)}\int_{x}^{b}\psi^{\prime}\left( t\right) \left( \psi \left(t\right) -\psi \left( x\right) \right)^{\alpha -1}f\left( t\right) dt.
{\textstyle \frac{1}{4}}\nabla_{a}k_{( I )}^{b}\Gamma^{a}{}_{b}={\textstyle \frac{1}{4}}f_{i a}{}^{b}\Gamma_{\mathrm{A d j}}( u^{- 1})^{i}{}_{I}\gamma^{a}{}_{b}= - \Gamma_{\mathrm{A d j}}( u^{- 1})^{i}{}_{I}\Gamma_{s}( M_{i}) ,
\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma\biggl(\frac{s}{2}\biggr)\zeta(s)={e^{b_{0}+b_{1}s}}\,\prod_{\rho}\biggl(1-\frac{s}{\rho}\biggr)\,e^{\frac{s}{\rho}}.
v_1\ =\ \frac{x_1+x_2-x_3-w}{x_1x_2-x_3w}\ =\ \frac{c-x_1x_2x_3(x_1+x_2-x_3)}{x_3(c-x_1^2x_2^2)}, v_2\ =\ \frac{x_1-x_2-x_3+w}{x_1w-x_2x_3}\ =\ \frac{c-x_1x_2x_3(-x_1+x_2+x_3)}{x_1(c-x_2^2x_3^2)}, v_3\ =\ \frac{x_1-x_2+x_3-w}{x_1x_3-x_2w}\ =\ \frac{c-x_1x_2x_3(x_1-x_2+x_3)}{x_2(c-x_1^2x_3^2)}.
\Delta v_1 =4(k_{11}e^{u_0^{'}+v_1}+k_{12}e^{u_0^{''}+v_2}-1)+\dfrac{4\pi N_1}{|\Omega|}\Delta v_2 =4(k_{21}e^{u_0^{'}+v_1}+k_{22}e^{u_0^{''}+v_2}-1)+\dfrac{4\pi N_2}{|\Omega|}.
{\widehat \Psi}_{k}{'^{\prime}}+ \left[ k^{2}- ( S^{1 / 2}){'^{\prime}}S^{- 1 / 2}\right]{\widehat \Psi}_{k}= 0 \, , ~ ~{\widehat \Pi}_{k}{'^{\prime}}+ \left[ k^{2}- ( S^{- 1 / 2}){'^{\prime}}S^{1 / 2}\right]{\widehat \Pi}_{k}= 0 .
X_t =x+ \int_0^t AX_r\d r +\int_0^t F(r,X_r)\d r + \int_0^t BX_r\d B^H_r &\qquad+\int_0^t\alpha_H\int_0^T\int_r^t|v-w|^{2H-2}BU_Y(v,r)F'_x(r,X_r)D^H_w X_r\d v\d w\d r
k = \frac{4 \sqrt{\pi}}{\Gamma ( 5 / 6 ) \Gamma ( 2 / 3 )}\left[ \frac{\Gamma ( 4 / 5 ) \Gamma ( 3 / 5 )}{1 0 0 \Gamma ( 1 / 5 ) \Gamma ( 2 / 5 )}\right]^{5 / 2 4}
\int d\mu_x^{\rm new}: = \lim_{N\rightarrow \infty}\left[ \prod^{N+1}_{n=2}\int \frac{d^D ( \Delta x_n)}{(2 \pi i \hbar \Delta t/M)^{D/2}}\right] \exp \left[ \frac{i}{\hbar}\sum_{n=1}^{N+1}M \frac{( \Delta x^i_n)^2}{2 \Delta t}\right],
\partial_t g_t(\partial_i,\partial_j)= 2\bar{\nabla}^2_{ij}t, \bar{g}|_{t=0}= g, \partial_t (\bar{\nabla}^2_{ij}t)-(\bar{\nabla}^2t)^2_{ij}= 0, \bar{\nabla}^2_{ij}t|_{t=0}= k_{ij}
{\cal Q}_{\bar{i}\bar{j}}( \psi , \bar{\psi}) \equiv c \alpha \beta_{\bar{i}\bar{j}}+ \bar{c}\alpha^{\prime}\beta_{\bar{i}\bar{j}}^{\prime}= c ( \psi^{\dag}\zeta ) ( \zeta^{\dag}\gamma_{\bar{i}\bar{j}}\psi ) + \bar{c}( \psi^{T}\bar{\zeta}) ( \zeta^{\dag}\gamma_{\bar{i}\bar{j}}\bar{\psi})
{\vec v}'=\gamma^{-1}\vec{v}+\gamma^{-2}\vec{V}\left[\displaystyle\frac{\vec{V}\vec{v}}{c^2\left(\gamma+\sqrt{\gamma^2+\displaystyle\frac{\vec{V}^2}{c^2}}\right)}-\frac{\vec{\sigma}\vec{v}}{c^2}\gamma_{0}^{-2}-1\right],
{\cal L}_{\rm b}~=~- \frac{1}{4l^2}\partial_{\mu}l\partial^{\mu}l + \frac{1}{4l^2}h_{\mu}h^{\mu}- G_{i\bar{\jmath}}\partial_{\mu}{z^i}\partial^{\mu}{\bar{z}^{\bar{\jmath}}}-\frac{i}{2}(G_{lj}\partial_{\mu}{z^j}-G_{l\bar{\jmath}}\partial_{\mu}{\bar{z}^{\bar{\jmath}}})h^{\mu},
A(n) P_n(x) - b_n P_{n-1}(x) = \frac{-(n-N)}{N}R_n(\lambda(x);\gamma,\delta,N-1) + \frac{n}{N}R_{n-1}(\lambda(x);\gamma,\delta,N-1) \hphantom{A(n) P_n(x) - b_n P_{n-1}(x)}{}= R_n(x).
I ( k ) = \frac{1}{( 2 \pi )^{4}}\int d p_{\parallel}p_{\parallel}d \varphi_{1}\int d p_{\perp}p_{\perp}d \varphi_{2}\frac{e^{i \Theta p_{\parallel}k_{\parallel}\sin \varphi_{1}}e^{i \Theta p_{\perp}k_{\perp}\sin \varphi_{2}}}{p_{\parallel}^{2}+ p_{\perp}^{2}}.
W_{\pi}(g_{-2,l,v}) = q\zeta_F(1)^{-2}\int_{(\mathcal{O}^{\times})^3}\chi(y_1y_2)\psi(y_1\varpi^{-1}+y_2\varpi^{-1}+y_3 v \varpi^{-l}) \sum_{\mu\in\mathfrak{X}_l}\mu(y_1y_2y_3^{-1})d^{\times}y_3d^{\times}y_2d^{\times}y_1.
\left(e_{i(k-1)}e_{-i(k-1)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}x_-\right)\left(e_{i(k)}e_{-i(k)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}x_-\right)^{-1}=e_{-i(k)}e_{i(k)}^{-1}\in B_+s_{i(k)}B_+,
i \gamma_{d}\cdot \partial_{i}\left( \begin{array}{c}{{\psi_{u}}}\\{{\psi_{l L}}}\\ \end{array}\right) = \frac{g}{2}\gamma_{d}^{\alpha}\left( \begin{array}{c c}{{Z_{\alpha}/ \cos \theta_{w}}}&{{W_{\alpha}^{1}- i W_{\alpha}^{2}}}\\{{W_{\alpha}^{1}+ i W_{\alpha}^{2}}}&{{- Z_{\alpha}/ \cos \theta_{w}}}\\ \end{array}\right) \left( \begin{array}{c}{{\psi_{u}}}\\{{\psi_{l L}}}\\ \end{array}\right) -
d \widetilde{\omega}= \sum_{I, J}d(d\widetilde{x}^I \wedge d \widetilde{\xi}^J) \widetilde{\omega}_{IJ}+ (-1)^{|I| + |J|}\, d\widetilde{x}^I \wedge d \widetilde{\xi}^J \wedge d\widetilde{\omega}_{IJ}.
\dfrac{M_{2}(\mathcal{NC}(G))}{|e( \mathcal{NC}(G) )|}- \dfrac{M_{1}(\mathcal{NC}(G))}{|v( \mathcal{NC}(G) )|}= \dfrac{n^{2}(m^{3}(m-6)+m(13m-12)+4))}{3(m-1)(2m-1)}:= \dfrac{n^2h(m)}{g(m)}.
M_{1,\alpha}(V = 0) = \frac{q^{2g+1}}{10! \zeta_q(2)}\Bigg( (2g)^{10}Q_0(\alpha) + (2g)^9 Q_1(\alpha) +(2g)^8 Q_2(\alpha) \Bigg) +O\Big(q^{2g+1}g^{7+\epsilon}\Big),
\mathcal{E}(\theta_2\wedge \theta_3\wedge \theta_4) (\psi_0,\dots,\psi_{2}) = \frac{1}{6}\theta_1 \cdot ((y_0^2+x_0^2)(z_1-z_2)+(y_1^2+x_1^2)(z_2-z_0)+(y_2^2+x_2^2)(z_0-z_1))
\max_{j\in\{1,\dots,K\}}\frac{R_j(1-\delta_{\{-j\}})}{p_j(1-\delta_{\{1,\dots,K\}})}+\max_{\pi}\sum_{i=1}^K\frac{R_{\pi_i}}{1-\delta_{\{\pi_1, \dots, \pi_i \}}}\leq 1 ,
\left\Vert \exp\left[\begin{array}[c]{ccc}0 x_{1}x_{3}0 0 x_{2}0 0 0\end{array}\right] \right\Vert =\left( \left( x_{1}^{2}+x_{2}^{2}\right)^{2}+x_{3}^{4}\right)^{1/4}.
\begin{array}{ll}E'(t) =\frac{1}{2}(b'\circ\Delta u)(t)-\frac{1}{2}b(t)\left \| \Delta u \right \|_{2}^{2}-\int_{\Omega}u_{t}h(u_{t}) \medskip & \leq \frac{1}{2}(b'\circ\Delta u)(t)-\int_{\Omega}u_{t}h(u_{t}) \medskip & \leq 0.\end{array}
\delta S_{1}^{( F )}= \frac{b}{2}\int d^{4}x \sqrt{- g}\, h^{a b}\Biggl ( 2 R^{c d}C_{a c b d}\varphi + 4 \nabla^{c}\nabla^{d}( C_{a c b d}\varphi ) \Biggr ) \, ,
S_{0}={\frac{1}{2 \pi}}\int d^{2}x \sqrt{- g}\left[ e^{- 2 \phi}( R + 4 ( \nabla \phi )^{2}+ 4 \lambda^{2}) -{\frac{1}{2}}\sum_{i = 1}^{N}( \nabla f_{i})^{2}\right] \ \ ,
\limsup_{s\to +\infty}\ \dfrac{\max_{[-s,s]}\, \tilde{f}}{\max\big\{\tilde{f}(-s),\tilde{f}(s)\big\}}\, \le \, c_0.\footnote{Observe that in particular, if $\tilde{f}(s)$ is monotone, then (H1) is obviously satisfied with $c_0=1$.}
\chi^{2}(\mu_{\theta^{\prime}};\mu_{\theta})\ge\frac{(E_{\theta^{\prime}}[h]-E_{\theta}[h])^{2}}{Var_{\theta}[h]}=\frac{\langle v,E_{\theta^{\prime}}[\hat{g}]-E_{\theta}[\hat{g}]\rangle^{2}}{v^{T}Cov_{\theta}[\hat{g}]v}
B^k(A-A^2 S')^k =B^k(A^k- A^{k+1}S')=B^k A^k-B^k A^{k-1}A^2 S'=B^k A^k-B^k A^{k-1}A^k{T'}^{k-1}&=B^k A^k-B^k A^{2k-1}{T'}^{k-1}=B^k A^k-(B^k A^{k})^k{T'}^{k-1}=B^k A^k-(B^k A^{k})^2 S'.
\psi_{m,\ell=0}(r)=N_mr^{-\frac{(D-1)}{2}}\frac{(1-\sin{r})^{\frac{1}{2}(A-B)}(1+\sin{r})^{\frac{1}{2}(A+B)}}{(2A-1-2B\sin{r})}\hat{P}_{m+1}^{(A-B-\frac{1}{2}, A+B-\frac{1}{2})}(\sin{r})
A_n =S^1\!\setminus\!\bigcup_{j=1}^{j=N}\bigcup_{i=0}^{q_n-1}f^{-i}\big(J_{2n}(c_j)\big) &=\left\{x \in S^1:\forall\,\,\, 0 \leq i \leq q_n-1:f^i(x) \in S^1\!\setminus\!\bigcup_{j=1}^{j=N}J_{2n}(c_j)\right\},
m_{\mu}L_N^t (T_{N-1}T_{N-2}\dots T_{N-h}) &= m_{\mu}\big( q^h \Phi_t^+ (L_N, L_{N-1}, \dots, L_{N-h}) - q^{h-2}\Phi_t^+ (L_N,L_{N-1}, \dots, L_{N-h+1}) \big)
{\varepsilon_2'}_\#{\varepsilon_1}_\#(v) ={\varepsilon_1'}_\#{\varepsilon_2}_\#(v)=\begin{cases}v v\in V_1(\varepsilon_1)\cap V_1(\varepsilon_2); x_{e_1}^{-1}vx_{e_1}v\in V_1(\varepsilon_2)\setminus V_1(\varepsilon_1); x_{e_2}^{-1}vx_{e_2}v\in V_1(\varepsilon_1)\setminus V_1(\varepsilon_2).\end{cases}
h_{\beta}(x,y) = \begin{cases}r(x,y), x,y\not\in S_r(\beta) \hat{r}_\beta(x,y), x \in S_r(\beta) y\in S_r(\beta) (x,y) \not\in S_r(\beta)^2, \min\{\hat{r}_\beta(x,y),\hat{r}_\beta(y,x)\}, x,y\in S_r(\beta). \end{cases}
\ddot{p}_{i}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{\widehat{m}_{j}(p_{j}-\sigma(p_{i}\odot p_{j})p_{i})}{(\sigma -\sigma(p_{i}\odot p_{j})^{2})^{\frac{3}{2}}}-\sigma(\dot{p}_{i}\odot\dot{p}_{i})p_{i},\textrm{}i\in\{1,...,n\}.
\inf_{\substack{\varphi(t) \in C_{0T}([0,T], \mathbb{R}^d) \varphi(0)=x_0}}\biggl \{S_{0T}^{\mathcal{K}}(\varphi(t)) \, \Bigl \vert \,\varphi(t) \in D_{+\delta}\cup \partial D_{+\delta}, \,\, \forall t \in [0, T] \biggr \},
f(x;h,\mu,\sigma,\tau)=\frac{h\sigma}{\tau}\sqrt{\frac{\pi}{2}}exp(\frac{1}{2}(\frac{\sigma}{\tau})^{2}-\frac{x-\mu}{\tau})erfc(\frac{1}{\sqrt{2}}(\frac{\sigma}{\tau}-\frac{x-\mu}{\sigma})),
K=\begin{pmatrix}2 -1 0 \cdots 0 -1 -1 2 -1 0 \cdots 0 0 -1 2 -1 \ddots \vdots \vdots 0 \ddots \ddots \ddots 0 0 \vdots \ddots -1 2 -1 -1 0 \cdots 0 -1 2 \end{pmatrix}_{n \times n}.
C ( l )_{n , n^{\prime}}^{m , m^{\prime}}= \left( \begin{array}{c c}{{n}}\\{{l}}\\ \end{array}\right) \left[ \begin{array}{c}{{m^{\prime}+ n^{\prime}}}\\ \end{array}\right]_{l}- \left( \begin{array}{c c}{{n^{\prime}}}\\{{l}}\\ \end{array}\right) \left[ \begin{array}{c}{{m + n}}\\ \end{array}\right]_{l}
T\circ (\omega_1\big(\begin{bmatrix}a_2 b_2 c_2 d_2 \end{bmatrix},h) \otimes \omega_1(\begin{bmatrix}a_1 b_1 c_1 d_1 \end{bmatrix},h)\big) &=\omega(\begin{bmatrix}a_1 b_1 &a_2 b_2 &c_2 d_2 c_1 d_1\end{bmatrix},h)\circ T
J_{1}( 0 , 0 , k , l ; m ) = C^{( 3 )}\;_{3}F_{2}\left( \left. \begin{matrix}{{- k ,}}&{{- l ,}}&{{- k - l - \frac{D}{2},}}\\{{\quad \frac{- k - l}{2},}}&{{\quad \frac{1 - k - l}{2}}}\\ \end{matrix}\right| \frac{t}{4 m^{2}}\right)
\mathrm{P}^{out}_{m}\approx \tau_m\int^{\psi_m^*}_{0}\left(\eta x\right)^{m-1}\left(1-\eta x\right)^{M-m}\frac{1}{\mathcal{R}_D}\times \sum^{N}_{n=1}\beta_n \left(1-c_nx\right) dx\approx \frac{\tau_m}{m}\eta^{m}\left(\psi_m^*\right)^m,
\frac{\partial}{\partial \epsilon}F_n(t,\epsilon)|_{\epsilon = 0}= (-2)\mathbb{E}\left[\int_0^{t\wedge \tau_n}\left(h_s - \mu(s,\theta^H_s) \right)\frac{\partial}{\partial x}\mu(s,\theta^H_s)d[B]_s \right] = 0.
\left[\mbox{Tr}e^{-\bar{H}^2_st}\right]_{\mbox{div}}={\Gamma\left(1-\frac D2\right) \over (4\pi)^{(D-1)/2}}{m^{D-4}\over 2\kappa t^{3/2}}\int_{\Sigma}\left[\left(m^2-\left(\frac 16 -\xi\right)R\right)t-{P \over 4\kappa^2}\right]~~~,
G^{( 1 , 1 )}= g^{( 1 , 1 )}( x_{1 2}^{2}, u_{1}, u_{2}) + \theta_{1}^{+ \alpha}\bar{\theta}_{1}^{+ \dot{\alpha}}\gamma_{\alpha \dot{\alpha}}^{( - 1 , 1 )}( x_{1 2}^{2}, u_{1}, u_{2}) + \ldots
\delta_{i_1i_2\cdots i_{m+1}i_{m+2}\cdots i_k}^{j_1j_2\cdots j_{m+1}j_{m+2}\cdots j_k}\delta^{i_{m+2}}_{j_{m+2}}\cdots\delta^{i_k}_{j_k}= \delta_{i_1i_2\cdots i_{m+1}}^{i_1j_2\cdots j_{m+1}}{n-m-1\choose k-m-1}(k-m-1)!,
\Pr\{\mathcal{E}_{k,1}\cap \mathcal{E}_{k,2}\}\geq \frac{1}{{n \choose r}}{{|S_k|}\choose r}\left( 1 - |S_k| \frac{{rd \choose d}}{{k \choose d}}\right) &= \frac{1}{{n \choose r}}\frac{1}{r+1}\binom{\frac{r}{r+1}\frac{\binom{k}{d}}{\binom{rd}{d}}}{r}.
\begin{array}{l l}{\ell^{-}}&{= 0 , \frac{1}{2}, 1 , \dots , \frac{1}{2}( k^{-}- 1 )}\\{}&{}\\{\ell^{+}}&{= 0 , \frac{1}{2}, 1 , \ldots , \frac{1}{2}( k^{+}- 1 )}\\ \end{array}
r_l\big(\epsilon(X,{\rm ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(\Pi_{\widetilde{\chi_0}}\otimes \Pi_{\pi_E}), \psi_F)\big) =r_l\Big(\epsilon\big(X, (\Pi_{\chi_0}\otimes \Pi_{\pi_F}) \otimes{\rm ind}_{\mathcal{W}_E}^{\mathcal{W}_F}(1_E),\psi_F\big)\Big) =\Big(r_l\big(\epsilon(X, \Pi_{\chi_0}\otimes \Pi_{\pi_F}, \psi_F)\big)\Big)^l.
(1-c)\int_X -\chi (u_1-v) \theta_{\psi}^n =\int_X -\chi (u_1-v)(\theta_{u_1}^n-c\theta_{u_2}^n)+(1-c)I_1+cI_2 &\leq \int_X -\chi (u_1-v)(\theta_{u_1}^n-c\theta_{u_2}^n)+ C_1\varrho (A+B)^2(1-\tilde{\chi}(-1))^2 Q^{\circ n}(c_{\mu}) &\leq \varrho c_{\mu}+ C_1\varrho (A+B)^2 Q^{\circ n}(c_{\mu}) &\leq C\varrho (A+B)^2 Q^{\circ n}(c_{\mu}),
a_{1,2}\frac{d^2}{dz^2}\tilde u_1(z)+b_{1,2}\frac{d}{dz}\tilde u_1(z)+c_{1,2}\tilde u_1(z) =0 a_{1,3}\frac{d^2}{dz^2}\tilde u_1(z)+b_{1,3}\frac{d}{dz}\tilde u_1(z)+c_{1,3}\tilde u_1(z) =0 a_{2,3}\frac{d^2}{dz^2}\tilde u_1(z)+b_{2,3}\frac{d}{dz}\tilde u_1(z)+c_{2,3}\tilde u_1(z) =0,
-{\frac{3}{2}}\cos{4 \tilde{\theta}}+ 1 6 \tilde{k}^{4}\tilde{\gamma}^{4}- 1 2 \tilde{k}^{2}\tilde{\gamma}^{2}+ 1 9 2 \tilde{k}^{4}\tilde{\gamma}t - 6 9 1 2 \tilde{k}^{6}t^{2}+{\frac{3}{2}}~ ~ .
\left[ \frac{E ( x b , x^{6 0}) E ( x^{1 1}b , x^{6 0}) E ( x^{3 1}/ b , x^{6 0}) E ( x^{4 1}/ b , x^{6 0})}{E ( x / b , x^{6 0}) E ( x^{1 1}/ b , x^{6 0}) E ( x^{3 1}b , x^{6 0}) E ( x^{4 1}b , x^{6 0})}\right]^{N}= b^{2 N}.
S_{1 2}= \frac{3 ( \mathrm{\boldmath ~ \s i g m a ~}^{( 1 )}\cdot{\bf r}) ( \mathrm{\boldmath ~ \s i g m a ~}^{( 2 )}\cdot{\bf r})}{r^{2}}- \mathrm{\boldmath ~ \s i g m a ~}^{( 1 )}\cdot \mathrm{\boldmath ~ \s i g m a ~}^{( 2 )}
\int d^{4}x \left[ - \frac{1}{2}G_{\mu}G^{\mu}+ \frac{1}{8 \alpha}( \varepsilon^{\mu \nu \sigma \lambda}\partial_{\nu}B_{\sigma \lambda})^{2}\right] = \nonumber \frac{1}{4}\int d^{4}x B_{\mu \nu}L_{( \alpha )}^{\mu \nu \sigma \lambda}B_{\sigma \lambda},
\sum_{l=1}^{2k-2}\biggl|\tilde{a}_l-\sum_{j=0}^{l}\frac{\alpha_{l-j}}{j!}\tau^j\biggr|+\biggl|\tilde{a}_{2k-1}-\sum_{j=1}^{2k-1}\frac{\alpha_{2k-1-j}}{j!}\tau^j\biggr| &+\sum_{j=1}^{2k-2}\biggl|\tilde{b}_{j}^{+}-\sum_{l=0}^{2k-2}\frac{\alpha_{2k-2-l}}{(l+j+1)!}\tau^{l+j+1}\biggr|\leq\omega(\sum_{j=1}^{2k-2}|b_j^+-b_j^-|)
\begin{tabular}[c]{l}$x^{3}P_{n}=P_{n+3}+\left( \gamma_{n+2}+\gamma_{n+1}+\gamma_{n}\right)P_{n+1}$ $+\gamma_{n}\left( \gamma_{n+1}+\gamma_{n}+\gamma_{n-1}\right)P_{n-1}+\gamma_{n}\gamma_{n-1}\gamma_{n-2}P_{n-3},$\end{tabular}
(\sqrt n\log n)^{-p}\sum_{j=nu+1}^{nt}\Bigl[\sum_{k=1}^{nt-j+1}k^{-1}\Bigr]^p \le\frac1{n^{p/2}\log^pn}\sum_{j=nu+1}^{nt}[1+\log(nt-j+1)]^p &\le2\Bigl[\frac1{\log2}+\frac{\log(n|t-u|)}{\log n}\Bigr]^p\cdot|t-u|^{p/2}
{\cal F}= \frac{H^{2}}{1 + H^{2}r^{2}}\left( \begin{array}{c c c}{{0}}&{{2 E \dot{r}}}&{{0}}\\{{0}}&{{2 E^{2}- m^{2}\alpha^{2}+ H^{2}m^{2}r^{2}}}&{{0}}\\{{0}}&{{0}}&{{( 1 + H^{2}r^{2}) m^{2}\alpha^{2}}}\\ \end{array}\right) .
\limsup_{N \to \infty}\int \left| \frac{1}{N}\sum_{n=1}^N G_1(T^nx) G_2(T^{2n}x) \right|^2 d\mu & \leq \frac{C}{H}+ \frac{C}{(H+1)^2}\sum_{h=1}^{H}(H - h + 1) \Re \left(\lim_{N \to \infty}\int \frac{1}{N}\sum_{n=1}^{N}(G_1 \cdot G_1 \circ T^h)(T^nx)(G_2 \cdot G_2 \circ T^{2h})(T^{2n}x) d\mu\right).
ds^2=g_{tt}dt^2+g_{rr}dr^2+2g_{t\phi_i}dtd\phi_i+g_{\phi_i\phi_j}d\phi_id\phi_j+g_{\theta\theta}d\theta^2+2g_{\theta\psi_i}d\theta d\psi_i+g_{\psi_i\psi_j}d\psi_id\psi_j,
\mathcal H^3_g\left(\Omega_{\epsilon^{\gamma^k},\epsilon}\right)\leq \Lambda_0\sum_{i=1}^{k-1}\epsilon^{\gamma^i(2-\gamma)}<\Lambda_0\epsilon^{\gamma(2-\gamma)}\left(1-\epsilon^{(\gamma-1)(2-\gamma)}\right)^{-1},
\langle E_{v a c}\rangle = -{\frac{1}{3 2 \pi^{2}}}\sum_{n 1 , n 2 = 1}^{\infty}m_{\bf n}^{4}\Bigg ({\frac{1}{\epsilon}}+ 2 \log 2 -{\frac{1}{2}}- \log ({\frac{m_{\bf n}^{2}}{\mu^{2}}}) \Bigg )
\left( [ \delta_{\varepsilon_{1}}\, , \, [ \delta_{\varepsilon_{2}}\, , \, \delta_{\varepsilon_{3}}] ] + [ \delta_{\varepsilon_{3}}\, , \, [ \delta_{\varepsilon_{1}}\, , \, \delta_{\varepsilon_{2}}] ] + [ \delta_{\varepsilon_{2}}\, , \, [ \delta_{\varepsilon_{3}}\, , \, \delta_{\varepsilon_{1}}] ] \right) q^{r}= 0 ,
r \int_{0}^{1}d\beta \int_{t}^{\infty}ds \int_{B_{s-t}(0)\cap (B_{\frac{s}{2}}(-\beta x))^{c}}dA(y)\left(\frac{1}{\sqrt{(s-t)^{2}-|y|^{2}}}-\frac{1}{(s-t)}\right) |\partial_{2}v_{4,c}(s,|\beta x+y|)| &\leq \frac{C r}{t^{4}\log^{2b-2}(t)}
(2\mu c_\alpha t_\alpha - e^\alpha)(\theta^\beta_\pm e^\beta + e^{\alpha+\beta}) = (2\mu c_\alpha \theta^\beta_\pm a |\alpha \cap \beta| - b_{\alpha, \alpha+\beta})e^\beta & + (2 \mu c_\alpha a | \alpha \cap (\alpha + \beta)| - b_{\alpha, \beta}\theta^\beta_\pm) e^{\alpha+\beta}
\hat{P}^{(\mathcal{B}_n)}(i|w,\mathbf{s}_0,\mathbf{s})=\frac{Q_{S|U,S_0}^n\big(\mathbf{s}\big|\mathbf{u}(\mathbf{s}_0,w,i),\mathbf{s}_0\big)}{\sum\limits_{i'\in\mathcal{I}_n}Q_{S|U,S_0}^n\big(\mathbf{s}\big|\mathbf{u}(\mathbf{s}_0,w,i'),\mathbf{s}_0\big)}.
\|(I-P_N)A\|_{\mathcal H}=\|\sum_{k,l\leq N-1}\langle A, e_k\otimes e_l\rangle_{\mathcal H}e_k\otimes e_l\|_{\mathcal H}\leq \sum_{k,l\leq N-1}|\langle A, e_k\otimes e_l\rangle_{\mathcal H}| \|e_k\otimes e_l\|_{\mathcal H}= \sum_{k,l\leq N-1}|\langle A, e_k\otimes e_l\rangle_{\mathcal H}| = \sum_{k,l\leq N-1}|\langle A e_k,e_l\rangle|.
\Xi \equiv \left(\begin{array}{ll}\xi_a(x) \bar \xi^{\dot a}(x)\end{array}\right), \hspace{.3 true cm}X \equiv \left(\begin{array}{ll}\chi_a(x) \bar \chi^{\dot a}(x)\end{array}\right), \hspace{.3 true cm}\Lambda \equiv \left(\begin{array}{ll}\lambda_a(x) \bar \lambda^{\dot a}(x)\end{array}\right);
g_{\mu \nu}\, = \,{\frac{1}{( x^{0})^{2}}}\delta_{\mu \nu}\, \, , \, \, \sqrt{g}\, = \,{\frac{1}{( x^{0})^{n + 1}}}\, \, , \, \, g^{\mu \nu}\, = \, ( x^{0})^{2}\delta^{\mu \nu}\, \, .
||\mathbf{A}\mathbf{x}^k+\mathbf{B}\mathbf{z}^k-\mathbf{c}||_2 \leq \varepsilon_{}, \ ||\mathbf{x}^k-\mathbf{x}^{k-1}||_2+||\mathbf{z}^k-\mathbf{z}^{k-1}||_2 \leq \varepsilon_{},
\sum_{p_{n+1}+\cdots +p_N=r-2\nu \atop \rm even}^{\ \ \ \ \ *}\ \sum_{T_t\in{\cal C}_r(p_1,\ldots,p_N)}\omega_N\big( T_t \big)= \textstyle{N-n \choose r/2-\nu}\displaystyle \sum_{T_t\in{\cal D}_r(p_1,\ldots,p_n)}\omega_{n+r/2-\nu}\big( T_t \big),
Z(s,f,\chi,\Delta_5)= \sum_{n=1}^{\infty}\int\limits_{\pi^{3n}O_v^{\times}\times \pi^{2n}O_v^{\times}}\chi(ac \f(x,y))\ |f(x,y)|^s|dxdy| =\sum_{n=1}^{\infty}q^{-5n-12ns}\int\limits_{O_v^{\times 2}}\chi(ac ((y^3-x^2)^2+\pi^{8n}x^4y^4)) |(y^3-x^2)^2+\pi^{8n}x^4y^4|^s\ |dxdy|.
G_j=\left[{\begin{array}{*{20}{c}}{g_{j1}^1}{g_{j2}^1}{\cdots}{g_{jr_j}^1}{g_{j1}^2}{g_{j2}^2}{\cdots}{g_{jr_j}^2}{\vdots}{\vdots}{\vdots}{\vdots}{g_{j1}^p}{g_{j2}^p}{\cdots}{g_{jr_j}^p}\end{array}}\right]_{p\times r_j},
\kappa_m= -\frac{2}{729}\frac{\left(m+1+\lfloor\frac{m}{2}\rfloor\right)^3\left(3 m+1+\frac{1-(-1)^m}{2}\right)^3}{(2m+1)^2},(m\ge 0) \lambda_m= \frac{1}{108}\left(2(27 (m+1)^2+2)+\frac{2 (m+1)^2-1}{(2m+1)(2m+3)}\right),(m\ge -1).
\underline{\Omega}_{IJ}= \left( \begin{array}{cccc}0 0 0 1 0 0 \frac{1}{V}-\underline{\Omega}_{C++}0 -\frac{1}{V}0 -\underline{\Omega}_{C--}-1 \underline{\Omega}_{C++}\underline{\Omega}_{C--}0 \end{array}\right).
g(\beta,\theta_{c,1},0)=g(\beta,\theta_{c,2},0)=0, &g(\beta,\theta,0)>0,(\forall \theta\in (\theta_{c,1},\theta_{c,2})), &g(\beta,\theta,0)<0,(\forall \theta\in [0,\theta_{c,1})\cup (\theta_{c,2},4\pi/\beta]).