harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{q}\left( 0,0\right) =1,\ \S_{q}\left( m,0\right) =0,\ \m\in N, S_{q}\left( m+1,s\right) =\left[ m\right]_{q}S_{q}\left( m,s\right)+q^{m}S_{q}\left( m,s-1\right) ,\ \ \m\in N_{0},\s\in N, S_{q}\left( m+1,m+1\right) =q^{m}S_{q}\left( m,m\right) ,\ \ \S_{q}\left( m,s\right) =0\ \ \s>m.
\sum_{\{\hat{\alpha},\hat{\beta}\}\in\mathcal{S}_{(\hat{d\varphi},\iota)}}\left(e^{\frac{1}{2}(\ell_{\hat{\alpha}}(\hat{\phi})+\ell_{\hat{\beta}}(\hat{\phi}))}+(-1)^{\hat{\alpha}\cdot\hat{\beta}}\right)^{-1}=0.
\frac{1}{2}\log \left(\frac{|u_s^1|^2-1}{|u_s^2|^2-1}\right) =\frac{1}{2}\log \left(\frac{|u_{\tau_0}^1|^2-1}{|u_{\tau_0}^2|^2-1}\right) + Q_s + R_s+M_s,
H ( z_{2}^{\prime}, . . , z_{\cal N}^{\prime}, P_{2}^{\prime}, . . , P_{\cal N}) \! = \! \! - \! \! \int_{\{z_{0}\}}^{\{z_{B}^{\prime}\}}\! \! \sum_{B}\frac{\partial \beta_{B}}{\partial \mu}( z_{2}^{\prime}, . . , z_{\cal N}^{\prime}, z_{A}^{\prime \prime}, . . , \mu ) d z_{B}^{\prime \prime}+ f ( z_{2}^{\prime}, . . , z_{\cal N}^{\prime}) .
D_{2}\pi^{\alpha_{1}}= \frac 1 2 C_{\beta_{1}\gamma_{1}}^{\alpha_{1}}\pi^{\beta_{1}}\eta_{2}^{\gamma_{1}}+ \frac 1 2 C_{\beta_{1}\gamma_{3}}^{\alpha_{1}}\left( \pi^{\beta_{1}}\eta_{2}^{\gamma_{3}}- \eta_{2}^{\beta_{1}}\pi^{\gamma_{3}}\right) ,
c^{k,1-k}_{i_0\ldots i_k}=\begin{pmatrix}(-1)^{k-1}a^{k,1-k}_{i_0\ldots i_k}0 (-1)^k\phi^{k,-k}_{i_0\ldots i_k}b^{k,1-k}_{i_0\ldots i_k}\end{pmatrix}.
\frac{\Phi_{x,y-1,z}(\textbf{a})}{\Phi_{x,y-1,z-1}(\textbf{a})}\frac{\Phi_{x,y,z-1}(\textbf{a})}{\Phi_{x,y,z}(\textbf{a})}+\frac{\Phi_{x+1,y-1,z-1}(\textbf{a})}{\Phi_{x,y-1,z-1}(\textbf{a})}\frac{\Phi_{x-1,y,z}(\textbf{a})}{\phi_{x,y,z}(\textbf{a})}=1.
\tilde\beta_0(a,b,b)=(\sigma_2\sigma_{1,3}\sigma_2) \sigma_2^{a-1}\sigma_{1,3}^{b-1}\quad\tilde\beta(a,b,b)=(\sigma_2\sigma_{1,3}\sigma_2)^3 \sigma_2^{a-1}\sigma_{1,3}^{b-1},
\sum_{f\in \mathcal{H}^*_k(N)}\omega_f \omega^z_{{\rm sym}^m f}(x) = \sum_{n=1}^{\infty}\frac{{\rm e}^{-n/x}}{n}\sum_{f\in \mathcal{H}^*_k(N)}\omega_f\lambda^z_{{\rm sym}^m f}(n) & = \sum_{n=1}^{\infty}\frac{\lambda^z_{{\rm sym}^m}(n)}{n}{\rm e}^{-n/x}+ O\Big(k^{-5/6}N^{-1+\varepsilon}\sum_{n=1}^{\infty}n^{m/4-1}\log(2n){\rm e}^{-n/x}r_m^z(n)\Big).
\left( \frac{1+ix}{1-ix}\right)^{1/3}=\sum_{e\geq0}A_{e}x^{e}+i\sum_{o>0}A_{o}x^{o}\,,~~~~\left( \frac{1+ix}{1-ix}\right)^{2/3}=\sum_{e\geq 0}B_{e}x^{e}+i\sum_{o>0}B_{o}x^{o}\,.
F_i=g_{i+1}(g_{i}-g_{i+2}-g_{i+4}-\cdots-g_{i+2k})+g_{i+3}(g_{i}+g_{i+2}-g_{i+4}-\cdots-g_{i+2k})+ g_{i+2k-1}(g_{i}+g_{i+2}+g_{i+4}+\cdots+g_{i+2k-2}-g_{i+2k}),
\left\{\begin{array}{c c c}{{l_{1}}}&{{l_{2}}}&{{l}}\\{{{}}}&{{{}}}&{{}}\\{{{\frac{L}{2}}}}&{{{\frac{L}{2}}}}&{{{\frac{L}{2}}}}\\ \end{array}\right\}\sim{\frac{( - 1 )^{l_{1}+ l_{2}+ l}}{\sqrt{L + 1}}}\left( \begin{array}{c c c}{{l_{1}}}&{{l_{2}}}&{{l}}\\{{{}}}&{{{}}}&{{}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right) + O ( L^{- 3 / 2})
\sum_{n=0}^{\infty}\frac{\left(a,b;q\right)_{n}z^{n}}{\left(q,c;q\right)_{n}}=\frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}\left(a,c/b;q\right)_{n}}{\left(q,c,az;q\right)_{n}}\left(-bz\right)^{n}.
_{B_{y_n/2}(y)}v =_{B_{y_n/2}(y)}\big(v-v(y^\prime )\big) &\leq 2 \sup_{z\in B_{y_n/2}(y)}\vert v(z)-v(y^\prime )\vert &\leq 2 \sup_{z\in B_{y_n/2}(y)}\vert v(z)-v(z^\prime )\vert + 2 \sup_{z\in B_{y_n/2}(y)}\vert v(z^\prime )-v(y^\prime )\vert ,
\tilde Qf(k,l) := \sum_{m,n\in \mathcal S}q_{(k,l)(m,n)}(f(m,n)-f(k,l)) =\mathbf 1_{\{k=l\}}\!\sum_{m\in \mathcal S}q_{km}(f(m,m)-f(k,l))+\mathbf 1_{\{k\neq l\}}\!\!\sum_{m\in \mathcal S,m\neq k}\!\!q_{km}(f(m,l)-f(k,l)) \quad+\mathbf 1_{\{k\neq l\}}\!\sum_{m\in \mathcal S,m\neq l}\!\!q_{lm}(f(k,m)-f(k,l)),
V_{\kappa,\alpha}(x) = -\frac{2^\alpha}{\Gamma(\kappa)}\Gamma\left( \frac{\delta+\alpha}{2}\right)\Gamma\left( \frac{\alpha}{2}+ \kappa \right)(1+|x|^2)^\kappa \,_2\textbf{F}_1\left( \left. \begin{array}{c}\frac{\delta+\alpha}{2}\frac{\alpha}{2}+ \kappa \frac{\delta}{2}\end{array}\right| -|x|^2 \right),
\bar\eta_{kl}=\frac{\langle u_k, Eu_l\rangle}{\sqrt{\lambda_k\lambda_l}}=\frac{1}{n}\sum_{i=1}^n\frac{\langle u_k, X_i\rangle}{\sqrt{\lambda_k}}\frac{\langle u_l, X_i\rangle}{\sqrt{\lambda_l}}-\delta_{k,l}, k,l\geq 1,
a_{d^{\prime}}^{\sigma, \theta_0^{\prime}}(v^2)= v^{\mathcal{E}(d^{\prime})}\frac{v^{- 2\mathcal{E}(d)}}{v^2-1}P_{\mathfrak{M}^{\sigma, \theta^{\prime}_0}_{d^{\prime}}(Q^{\prime})}(v).
F_n ={}_{1}F_{1}\!\left(\left.\begin{matrix}\mu+n \lambda+n\end{matrix}\right|x\right): F_n' =\frac{n+\mu}{n+\lambda}F_n; F_n ={}_{2}F_{1}\!\left(\left.\begin{matrix}\mu+n,a \lambda+n\end{matrix}\right|x\right): (1-x)F_n' =\frac{(n+\mu)(n+\lambda-a)}{n+\lambda}F_{n+1}+(n+\mu)F_n.
\int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty F\left(\sqrt{p^2+t_1^2+\cdots t_{n-1}^2}\right)dt_1...dt_{n-1}=\int\limits_{\mathbb{R}^{n-1}}F(\sqrt{p^2+|x|^2})dx &=|S^{n-2}|\int\limits_0^\infty F(\sqrt{p^2+t^2})t^{n-2}dt.
\sigma_{I I}={\frac{2}{\kappa^{2}[ B_{0}( y_{0}+ y_{*}) ]^{1 / 3}( y_{0}+ y_{*})}}={\frac{2}{\kappa^{2}( B_{0}y_{0})^{1 / 3}y_{0}e^{4 n}}}> 0 ,
\left( \Sigma^{\mu \nu}\right)^{\rho \tau \alpha \beta}=\frac 12\left[\eta^{\mu \rho}\eta^{\nu \alpha}\eta^{\tau \beta}+\eta^{\mu \tau}\eta^{\nu \beta}\eta^{\rho \alpha}+\left( \alpha \leftrightarrow \beta\right) -\left( \mu \leftrightarrow \nu \right) \right] \;\,,
-k\frac{d}{iy(1 + \frac{x}{iy})}+ \frac{kd^2}{2 (iy)^2(1 + \frac{x}{iy})^2}= \frac{ikd}{y}- \frac{xdk}{y^2}- \frac{k d^2}{2y^2}+ O\Big(\frac{k d^2}{y^3}\Big).
(\cot\langle \alpha_{1}, H\rangle )^{2}= \frac{m_{1}+2m_{2}+6m_{3}\pm \sqrt{(m_{1}+2m_{2}+6m_{3})-4(m_{1}+2m_{2}+m_{3})m_{3}}}{m_{1}+2m_{2}+m_{3}}.
\begin{aligned}\| v \|_{L^q (S_T)}\le C \left( \|u\|_{L^{q}(S_T)}^{\kappa-1}+ \|\tilde u \|_{L^{q}(S_T)}^{\kappa-1}\right) \| v \|_{L^q(S_T)}\textup{(by \eqref{4.34}and \eqref{1.2})}\le C 2^\kappa H_0(T)^{\kappa-1}\|v\|_{L^q(S_T)}\textup{(by \eqref{H2})}\le \frac 1 2\, \| v \|_{L^s_t L^q_x(S_T)}\textup{(by \eqref{5.16}).}\end{aligned}
\sum_{p=1}^{r^{(1)}_{i}}\frac{1}{2}\sum_{q=1}^{r^{(1)}_{i-1}}\mathrm{min}\{n_{p,i},n_{q,{i-1}}\}=\frac{1}{2}\sum_{s=1}^{k}r^{(s)}_{i-1}r_{i}^{(s)}, \sum_{p=1}^{r_{i}^{(1)}}(\sum_{p>p'>0}\mathrm{min}\{n_{p,i},n_{p',i}\}+\frac{1}{2}n_{p,i}) =\frac{1}{2}\sum_{s=1}^{k}r^{(s)^{2}}_{i},
\begin{aligned}\mathcal L^{x,y}_{i \leftrightarrow j}f(\xi,\omega) = \eta_i(x) \eta_j(y)\Big( f(\xi^{x,y},\omega^{x,y}) - f(\xi,\omega) \Big) +\eta_j(x)\eta_i(y) \Big( f(\xi^{x,y},\omega^{x,y}) - f(\xi,\omega) \Big)\, . \end{aligned}
\int_{\Omega}(1-\sigma)D^2v:D^2\varphi+\sigma\Delta v\Delta\varphi dx=\lambda\int_{\partial\Omega}\frac{\partial v}{\partial\nu}\frac{\partial\varphi}{\partial\nu}d\sigma\,,\ \ \ \forall\varphi\in H_c^2(\Omega).
\lim_{s\to0^+}s\,\int_E\,dx\, \int_{B_R\cap\Omega^c}\frac{dy}{|x-y|^{n+s}}\le \lim_{s\to0^+}s\,\int_\Omega\,dx\,\int_{B_R\cap\Omega^c}\frac{dy}{|x-y|^{n+s}}=0.
\partial_{z}\partial_{\bar{z}}\int_{\Sigma}d^{2}w ( \xi^{\prime}) G ( w ( \xi^{\prime}) , z ( \xi ) ) J ( w ( \xi^{\prime}) , \bar{w}( \bar{\xi}^{\prime}) ) g_{w \bar{w}}= -{\frac{J ( z , \bar{z})}{2}}g_{z \bar{z}}+
\langle x^{\prime \prime}| ( E - \hat{H}_{\pm})^{- 1}| x^{\prime}\rangle \simeq \frac{1}{\mathrm{i}\hbar \, \sqrt{| \dot{x}^{\prime}\dot{x}^{\prime \prime}|}}\sum_{x_{\mathrm{q c}}}^{\mathrm{f i x e d}\, E}\exp \left\{\frac{\mathrm{i}}{\hbar}\, W [ x_{\mathrm{q c}}] \mp \mathrm{i}\varphi [ x_{\mathrm{q c}}] - \mathrm{i}( n_{R}+ n_{L}) \frac{\pi}{2}\right\}
\left|A_Q\left(S_{\varphi}\vec f\right)_Q\right|\leq\left|A_Q\left\langle\vec f,\varphi_Q\right\rangle\right|=|Q|^{\frac12}\left|A_Q\left(\widetilde{\varphi}_{j_Q}*\vec f\right)(x_Q)\right|\leq\sup_{\mathbb{A},\widetilde{\varphi},Q}\left(\vec f\right),
{C_{\rho \varphi}}(\rho ,\varphi ,t|{{\bar r}_{\rm tx}},{t_0}) = \sum\limits_{n=0}^\infty \sum\limits_{m = 1}^\infty{H_{nm}J_n(\lambda_{nm}\rho)\cos (n(\varphi-\varphi_{\rm tx}))e^{-D\lambda_{nm}^2 (t-t_0)}u(t-t_0)},
K_T(\mathbb{T}(d)_v) \otimes_{\mathbb{K}}\mathbb{F}=\bigoplus_{\begin{subarray}{c}(d, v)=(d_1, v_1)+\cdots+(d_k, v_k), v/d=v_i/d_i\end{subarray}}\mathbb{F}\cdot [\mathcal{E}_{d_1, v_1}] \ast \cdots \ast [\mathcal{E}_{d_k, v_k}].
-\frac{x_{1}x_{2}+ x_{1}x_{3}+ x_{2}x_{3}}{x_{1}x_{2}x_{3}}= -\frac{2}{q^2}-15 +O(q), \frac{x_{1}+ x_{2}+ x_{3}}{x_{1}x_{2}x_{3}}= \frac{20}{q^2}+\frac{108}{q}+419+O(q), - \frac{1}{x_{1}x_{2}x_{3}}= \frac{8}{q^3}+\frac{62}{q^2}+\frac{316}{q}+1307 + O(q).
(q_n +1) (q_n +2) \tilde a_{q+ \overline{n}, \kappa}+ \sum_{i \neq n}(q_i + 1) (q_i + 2) \tilde a_{q+ 2 \overline{i}- \overline{n}, \kappa}- (\kappa+1) \tilde a_{q- \overline{n}, \kappa+1}+ c^{m,\ell}_{q,\kappa}\tilde a_{m,\ell}= 0.
\frac{\gamma(t)}{\lambda(t)}\frac{d}{dt}\big(\|\dot x(t)\|^2\big) = \ \frac{d}{dt}\left(\frac{\gamma(t)}{\lambda(t)}\|\dot x(t)\|^2\right)-\frac{\dot \gamma(t)\lambda(t)-\gamma(t)\dot\lambda(t)}{\lambda^2(t)}\|\dot x(t)\|^2
P_{t_1}g(z)=\int_{\Gamma_1}p_{t_1}(z_1-\xi_1)g(\xi_1 , z_2)d\xi_1 P_{t_2}g(z)=\int_{\Gamma_2}p_{t_2}(z_2-\xi_2)g(z_1, \xi_2)d\xi_2
2w \gamma_{[0,0]}= \gamma_{[1,1]}\;, 2(w-2) \gamma_{[1,1]}= 2 \gamma_{[2,2]}\;, \vdots 2(w-2 \Omega+2) \gamma_{[\Omega-1,\Omega-1]}= \Omega \gamma_{[\Omega,\Omega]}\;, 2(w-2\Omega) \gamma_{[\Omega,\Omega]}= (\Omega+1) \gamma_{[\Omega+1,\Omega+1]}= 0 \;.
|D^\alpha P_{Z_\circ}(rX) - D^{\alpha}P_{X_\circ}(rX)| \leq C_hr^{\ell + \beta - |\alpha|}= C_h|X_\circ - Z_\circ|^{\ell + \beta - |\alpha|}X \in B_{1/2}(r^{-1}X_\circ).
l^{*}={\frac{r}{\gamma + 1}}+ \left({\frac{1}{\gamma ( \gamma + 1 )}}{\frac{A_{1}}{A_{0}}}-{\frac{1}{\gamma^{2}}}{\frac{B}{A_{0}}}\right) +{\frac{1}{\gamma ( \gamma + 1 )}}{\frac{B}{A_{0}}}\ln r + . . . ~ ~ .
\partial_{x_jx_k}( I_{h,T}\mathbf{u}(X) - \mathbf{u}(X) )= -\partial_{x_jx_k}\mathbf{u}(X)+ \sum_{i\in\mathcal{I}}\partial_{x_jx_k}\Phi_{i,T}(X)\widetilde{\mathbf{R}}_i(X),
B_2 = \frac{1}{2}\sum_{y=1}^{\alpha-2}y \left( \frac{\sin(2 \pi \theta(4g+1-y))}{4g+1-y}- \frac{\sin(2 \pi \theta(4g+1+y))}{4g+1+y}\right) &= \sum_{y=1}^{\alpha-2}y \frac{y \sin(2 \pi \theta(4g+1)) \cos(2 \pi y \theta) - (4g+1) \sin(2 \pi y \theta) \cos(2 \pi (4g+1) \theta)}{(4g+1)^2-y^2}=O (g^{-1}\theta \alpha^3).
{\cal H}_{\rho}= \frac{1 6 \pi G}{\sqrt{- g}}\left( \frac{\pi^{2}}{d - 1}- \pi_{a b}\pi^{a b}\right) - \frac{\sqrt{- g}}{1 6 \pi G}\left( R + \frac{d ( d - 1 )}{\ell^{2}}\right) ,
p_{j}^{{*}}= - (1 -{\beta^{j}})^{- 1}\sum\limits_{l = 1}^N \sum\limits_{s = 1}^J{\frac{{\partial{U^l}}}{{\partial{\beta^{s}}}}\cdot\frac{{\partial{\beta^{s}}}}{{\partial \lambda_j}}= 0}\ .
{\widehat C}_{( p - 1 )}= \sqrt{2}\kappa \, C_{( p - 1 )}\simeq \frac{m}{n}\, \frac{n^{2}}{\Delta_{m , n}}\, \frac{Q_{p}}{r^{7 - p}}~ d x^{2}\wedge \ldots \wedge d x^{p}~ ~ .
M_\lambda(\zeta):=\begin{pmatrix}g_{\lambda}(\zeta_1) -G_{\lambda}(\zeta_1,\zeta_2) \ldots -G_{\lambda}(\zeta_1,\zeta_k) -G_{\lambda}(\zeta_1,\zeta_2) g_{\lambda}(\zeta_2) \ldots -G_\lambda(\zeta_2,\zeta_k) \vdots \vdots -G_\lambda(\zeta_1,\zeta_k) -G_\lambda(\zeta_2,\zeta_k) \ldots g_\lambda(\zeta_k)\end{pmatrix}.
\sum_{x \in V_x}u^\rho_{xy}= \sum_{r = 1}^k \sum_{x \in V_X}\delta_{x \sigma^r(y)}p_r + \sum_{s = 1}^k \sum_{x \in V_X}\delta_{x \tau^s(y)}q_s - 1= \sum_{r = 1}^k p_r + \sum_{s = 1}^k q_s - 1 = 1,
\begin{array}{lll}\mu\circ ((R\circ\eta)\otimes\eta) = \mu\circ (R\otimes id_M)\circ (\eta\otimes\eta) &= \mu\circ (id_M\otimes L)\circ(\eta\otimes \eta) &= \mu\circ (\eta\otimes id_M)\circ (id_I\otimes (L\circ\eta)) &= \lambda_M\circ (id_I\otimes (L\circ \eta)) &= L\circ\eta\circ \lambda_I\end{array}
[X_{i}^{+}(u),X_{j}^{-}(v)]=h\delta_{ij}\{\delta(u_{-}-v_{+})k_{i+1}^{+}(u_{-})k_{i}^{+}(u_{-})^{-1}-\delta(u_{+}-v_{-})k_{i+1}^{-}(v_{-}) &k_{i}^{-}(v_{-})^{-1}\}where ~\delta(u-v)=\Sigma_{k\in\mathbb{Z}}u^{-k-1}v^{k}.
S^{T}= M_{0}^{2}\int_{0}^{1 / T}d t \int_{0}^{R}d r \left[ 1 + \frac{1}{2}{\bf u}\left( 1 - \frac{\alpha}{M_{0}^{2}}\triangle \right) ( - \triangle ){\bf u}\right] - \beta \int_{0}^{1 / T}d t \, \ddot{\bf u}\acute{\bf u}
S^t \, \delta_1 = -\delta_1 \ , \S^t \, \delta_2 = -\delta_2 + \delta_1^2 \ , \S^t \, \delta_3 = -\delta_3 + 4 \, \delta_1 \, \delta_2 - 2 \, \delta_1^3 \ , \ \ldots
u=(\mu - \Delta)^{-1}f - (\mu - \Delta)^{-\frac{1}{2}-\frac{1}{2q}}Q_{p}(q) (1 + T_p)^{-1}G_{p}(r) (\mu - \Delta)^{-\frac{1}{2}+\frac{1}{2r}}f, f \in L^p
G^{c}( x - y ) = < 0 | T \{\psi^{( 0 )}( x ) \exp [ i \int_{y_{0}}^{x_{0}}d^{4}x^{\prime}\, j_{\mu}^{a s}( x^{\prime}) A^{\mu ( 0 )}( x^{\prime}) ] \overline{{{\psi}}}^{( 0 )}( y ) \}| 0 > \, \, ,
n^{2h(s)-3}n|\gamma_{0}(s)|\leq \sigma^2(s)\left( 1 + \frac{1}{2h(s)-1}\right) \intertext{and}n^{2h(s)-3}\sum_{k=1}^{n}(n-k)\left| \gamma_{k}(s)\right|\leq n^{2h(s)-3}\sigma^2(s)\sum_{k=1}^{n}(n-k) k^{1-2h(s)}c_{h}(s) \leq \sigma^2(s)\frac{1}{2}\left( \frac{1}{(1-h(s))^2}+ \frac{1}{(1-h(s))(2h(s)-1)}\right),
\left[ \frac{\xi_p}{H_p^3}\right]_p + \left[ \frac{\xi_q}{H_p}\right]_q =[{\cal J}_1^*]_p + [{\cal J}_2^*]_q, \frac{\xi_p}{H_p^3}- \xi ={\cal J}_1^* \mbox{when}\p = 1, \ \ \ \xi = 0 \ \ \mbox{when}\p = 0.
\big| |\xi| \widehat u(t,\xi) \big| \lesssim \begin{cases}(1+t)^{-\frac{b_0}2}\big( |\xi| \widehat u_0(t,\xi) + \widehat u_1(t,\xi) \big) |\xi|>N \left( \frac{1+t}{1+t_\xi}\right)^{-\frac{b_0}2}(1+t_\xi)^{\Re \mu_+}\big( \widehat u_0(t,\xi) + \widehat u_1(t,\xi) \big) |\xi|\le N \end{cases}
\eta^{1/2}\{(\breve\xi^{rst}-\xi^{rst})\lambda_{rs,t}+(\breve\xi^{rs}-\xi^{rs})\lambda_{rs}\}+\eta^{3/2}\{(\breve\xi^{rs1}-\xi^{rs1})\lambda^{1t}\lambda_{rs,t}+\breve\xi^{11}-\xi^{11}\}=O(n^{-1}),
\frac{\pi^2}{8}= \sum_{k \geq 0}\frac{1}{(2k+1)^2}= 1 + \frac{1}{3^2}+ \frac{1}{5^2}+ \frac{1}{7^2}+ \cdots \zeta(s) = \frac{1}{1-2^{-s}}\times \sum_{n \geq 0}\frac{1}{(2n+1)^s}.
\sum_{e}\sum_{m , n = 0}^{1}C_{e}( m , n ) \theta^{4}[ e ] F_{R , t w s}^{m , n}= E ( z , 0 )^{- 2}\sum_{i = 2}^{3}C_{i}\theta_{i}^{1 2}( 0 | \tau ) \frac{\theta_{i}^{2}( z | \tau )}{\theta_{i}^{2}( 0 | \tau )},
\mathbb{C}^{N}=\operatorname{Ker}\eta\oplus\operatorname{Ran}\eta,\operatorname{Ker}\eta=\operatorname{Ker}\sqrt{\eta},\operatorname{Ran}\eta=\operatorname{Ran}\sqrt{\eta}, \left( \operatorname{Ran}R\right)^{\bot}=\operatorname{Ker}R\text{and}\left( \operatorname{Ran}\eta\right)^{\bot}=\operatorname{Ker}\eta
\mu_{k}^i\left(B\right) = \frac{1}{Z_{k}^i}\int\limits_{\theta \in B}\prod\limits_{j=1}^{n}\left( \frac{d\mu_{k-1}^j\left(\theta\right)}{d\lambda(\theta)}\right)^{a_{ij}}\ell^i(s_{k}^i|\theta) d\lambda\left(\theta\right)
\bigl(~_{s+1}\mathcal{F}_s(\alpha)\bigr)^{q^d}=\sum_{n=0}^{\infty}\biggl(\prod_{m=1}^{n+d-1}(\theta^{q^m}-t)^{c(m-n)q^{n+d-m}}\alpha^{q^{n+d}}\biggr) =\sum_{n=0}^{\infty}\biggl(\prod_{l=1}^{d-1}(\theta^{q^{l+n}}-t)^{c(l)q^{d-l}}\alpha^{q^{n+d}}\biggr).
\int \partial_j \varphi \ \partial_j f \d \mu = - \int \sum_{k=1}^N \partial_k \partial_j \partial_j \varphi \ \partial_k \varphi \d \mu + \int \sum_{k=1}^N \partial_k \partial_j \varphi \ \partial_j H \ \partial_k \varphi \d \mu + \int \sum_{k=1}^N \partial_j \varphi \ \partial_k \partial_j H \ \partial_k \varphi \d \mu.
1+\frac{z\mathbb{F}_{\alpha_{j},\beta_{j},\lambda_{j}}^{\prime \prime}(z)}{\mathbb{F}_{\alpha_{j},\beta_{j},\lambda_{j}}^{\prime}(z)}=\sum_{j=1}^{n}\frac{1}{\lambda_{j}}\left( \frac{z\mathbb{E}_{\alpha_{j},\beta_{j}}^{\prime}(z)}{\mathbb{E}_{\alpha_{j},\beta_{j}}(z)}\right) +1-\sum_{j=1}^{n}\frac{1}{\lambda_{j}}.
\int{\mathcal{D}_{\hat{g}}}( \bar{\Psi},{\Psi_{0}}) \exp \{-{S_{c}^{0 0}}[ \bar{\Psi},{\Psi_{0}}, \hat{g}] \}={e^{\mathcal{F}_{B}^{0 0}}}{{\left({\frac{D e t^{\prime}{\tilde{K}_{D}}}{\oint{d}\tilde{s}}}\right)}^{- d / 2}}\int{d}{\Psi_{0}}{e^{-{Q_{0}}{\chi_{o}}{\Psi_{0}}/ 2}}
{\rm Res}_{s=1}\eta(s)=\lim_{s\to 1}(s-1)\eta(s)=\lim_{s\to 1}\frac{2\pi^{\frac{s}{2}}h(s)}{s\Gamma\left(\frac{s}{2}\right)}=\frac{\sqrt{\pi}}{\Gamma\left(\frac{1}{2}\right)}=1,
\mathbf{v}^{n}\left( x\right) \equiv \left( -1\right)^{n-1}\left[ \left(-1\right)^{k}\left( \begin{array}{c}n k\end{array}\right) \Gamma_{k}^{n}\left( \frac{x-1}{2}\right) \right]_{k=0}^{n-1},
\sum_{m = 0}^{2 r}D [ f_{2 l + m l}] - \sum_{m = 0}^{2 r}D [ f_{m l}] = 4 A \sum_{m = 0}^{2 r}\frac{d}{d \tau}\left\{K_{1}\sqrt{\dot{x}_{+ m l}^{2}( 0 )}+ K_{2}\sqrt{x_{l + m l}^{2}( l )}\right\}.
C_{2n}^{\lambda}(t)=\frac{(-1)^n}{(\lambda+n)B(\lambda,n+1)}\,{}_{2}F_{1}\Big(\!-n,n+\lambda;\frac{1}{2};t^2\Big); begin{align*}4pt] C_{2n+1}^{\lambda}(t)=\frac{(-1)^n 2t}{B(\lambda,n+1)}\,{}_{2}F_{1}\Big(\!-n,n+\lambda+1;\frac{3}{2};t^2\Big),
\mathfrak{g}= \mathfrak{g}_{-m}\oplus \cdots \oplus \mathfrak{g}_{-2}\oplus \mathfrak{g}_{-1}\oplus \mathfrak{g}_{0}\oplus \mathfrak{g}_{1}\oplus \mathfrak{g}_{2}\oplus \cdots \oplus \mathfrak{g}_{m},
Y_t=Y_{T\wedge a}+\int_t^{T\wedge a}f(r,Y_r)\,dr+\int_t^{T\wedge a}\,dV_r+\int_t^{T\wedge a}\,dK_r-\int_t^{T\wedge a}\,dM_r,t\in [0,T\wedge a]
\epsilon (1+t^2)^{1/4}\frac{\xi}{k^2}\left((\frac{\xi}{k^2})^2\right)^{-1/4}\approx \epsilon \left(\frac{\xi}{k}\right)^{1/2}\left(\frac{\xi}{k^2}\right)^{1/2}= \epsilon \frac{\xi}{k^{3/2}},
\mathrm{D}_{z}^{( s )}( E_{i}) = z^{s_{i}}E_{i}, \mathrm{D}_{z}^{( s )}( F_{i}) = z^{- s_{i}}F_{i}, \mathrm{D}_{z}^{( s )}( h_{i}) = h_{i}, i = 0 , \cdots , r
\widetilde{F}\left( \begin{array}{c}z x \end{array}\right) := U\circ F\circ U^{-1}\left( \begin{array}{c}z x \end{array}\right)= \left( \begin{array}{c}z+z^2+a_3 z^3 + O(z^4) +\dots b_0(x) + b_1(x) z + b_2(x) z^2 + \dots \end{array}\right),
\sqrt{\sigma_{1}^{2}/n+\sigma_{2}^{2}/n}/S=\sqrt{\frac{\sigma_{1}^{2}+\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\sigma_{1}\sigma_{2}\rho}}>1where\rho:=corr(Y_{i1},Y_{i2}).
T_k(N) := \sum^{\lfloor \frac{N}{2}\rfloor}_{j=1}l^*(h_{j,k}) \otimes E_{2j-1, 2j}(N) + \sum^{\lfloor \frac{N-1}{2}\rfloor}_{j=1}l(h_{j,k}) \otimes E_{2j, 2j+1}(N).
\mathcal G_{b_{i_0+1}-1,i_0+1}\mathcal F_{b_{i_0}-1,i_0}\cdots \mathcal F_{b_{1}-1,1}(U_{n,k}) = 0, \mathcal G_{a_{i_0+1},i_0+1}\mathcal F_{a_{i_0},i_0}\cdots \mathcal F_{a_{1},1}(U_{n,k-1}) = 0,
\mathbf{H}_{3}\left( r,\theta \right) =\left( \begin{array}{c}\frac{r^{2}}{2}\cos \left( 2\theta \right) -\frac{r^{4}}{4}\cos \left(4\theta \right) -\frac{r^{2}}{2}\sin \left( 2\theta \right) -\frac{r^{4}}{4}\sin \left(4\theta \right) \varphi \left( r\right) +a\theta\end{array}\right) ,
H_{\mathcal{L}}:=\left(\begin{array}{ccc}\frac{\partial^2 \Phi}{\partial x_1^2}\cdots \frac{\partial^2 \Phi}{\partial x_1\partial x_n}\vdots \ddots \vdots \frac{\partial^2 \Phi}{\partial x_n\partial x_1}\cdots \frac{\partial^2 \Phi}{\partial x_n^2}\end{array}\right).
\mathrm{` ` a "}= \sum ( - 1 )^{r + z + w + p - m}\left( \begin{array}{c c c}{j_{1}}&{j_{4}}&{j_{1 2}}\\{o}&{r}&{z}\\ \end{array}\right) \left( \begin{array}{c c c}{j_{1}^{\prime}}&{j_{1 3}}&{j_{1 4}}\\{- u}&{- w}&{- p}\\ \end{array}\right)
\partial_{U}\left( U^{8}\Big ( 1 -{\frac{U_{0}^{7}}{U^{7}}}\Big ) \partial_{U}\phi \right) + \left({\frac{c g_{Y M}^{2}N \omega^{2}U}{1 - U_{0}^{7}/ U^{7}}}- \ell ( \ell + 7 ) U^{6}\right) \phi = 0 .
\sup_{s \in [0,t]}\vert Z_{s}^i \vert \leq C_{T}\biggl[ 1+ \vert X_{0}^i \vert + \sup_{s \in [0,t]}\vert W_{s}^i \vert + \frac{1}{N}\sum_{j=1}^N \bigl( \vert X_{0}^j \vert + \sup_{s \in [0,t]}\vert W_{s}^j \vert \bigr) \biggr].
H = \begin{pmatrix}\psi_{vv}(0, p_0)[v,v] \psi_{vp}(0,p_0)[v, p'(0)] \psi_{vp}(0,p_0)[v, p'(0)] \psi_{pp}(0,p_0)[p'(0), p'(0)] + \psi_p(0,p_0)p''(0)\end{pmatrix}
(\partial_1^{s'_1}\partial_i^{\alpha_1}u)\, (\partial_1^{s'_2}\partial_j^{\alpha_2}u)\, (\partial_1^{s'_3}\partial^2_{ij}u )\ , & (\partial_1^{s'_1}\partial_i^{\alpha_1}u)\, (\partial_1^{s'_2}\partial_j^{\alpha_2}u)\, (\partial_1^{s'_3}\partial_h u )\,,
\prod_{i,j}\frac{1}{1-x_i y_j}= \sum_{\lambda \in \mathcal{P}}s_\lambda(\mathbf{x})s_\lambda(\mathbf{y})= \sum_{\mu \in \mathcal{P}}\frac{p_\mu(\mathbf{x}) p_\mu(\mathbf{y})}{z_\mu}
D_{1}\ =\ - \frac{2}{9}\left( 8 J_4J_2 + 3 J_3J_1\right) A_9 - \frac{2}{9}\left(8 J_5J_1 - 8 J_3J_2 - 11 J_2J_0\right) A_8\ - & \frac{4}{3}J_2J_1 A_7 - \frac{22}{9}J_2J_1 A_6 + \frac{4}{9}J_2J_1 A_5 + \frac{22}{9}J_2^2 A_4 - \frac{4}{9}J_1^2A_3 \ ,
\begin{aligned}\vert S_{\{b_{i,i+1}\},\{c_{i,i+1}\}}(\psi_p, \psi_p'; c,w_{G_n}) \rvert \leq C \times p^{a_1+a_2+\cdots+a_{n-2}+a_{n-1}-\frac{a_{k}}{2}+\frac{n(n-1)}{2}m}&= C \times p^{a_1+a_2+\cdots+a_{k-1}+\frac{a_k}{2}+a_{k+1}+\cdots+ a_{n-2}+a_{n-1}+\frac{n(n-1)}{2}m}.\end{aligned}
\frac{d}{dt}\sum_{|\alpha|=0}^{m}\langle A_{1}^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle\leq\frac{C\epsilon}{2}\|\nabla v\|_{m}^{2}+C\left\lbrace\|f_{1}\|_{m}^{2}+(\mu_{0}(t)+\mu_{1}(t))\left(\|u\|_{m}^{2}+\|v\|_{m}^{2}\right)\right\rbrace,
(2w_1 + n - 2s - 2)a_{s+1,t}= -(2w_1 + 2w_2 + n - 2k - 2s - 2t)a_{s,t}, (2w_2 + n - 2t - 2)a_{s,t+1}= -(2w_1 + 2w_2 + n - 2k - 2s - 2t)a_{s,t}
\sum_{n=0}^{\infty}\frac{\left(b;q\right)_{n}q^{n^{2}/2}S_{n}\left(aq^{-n-1/2};q\right)}{\left(c;q\right)_{n}}\left(\frac{c}{ab}\right)^{n}=\frac{\left(c/b;q\right)_{\infty}}{\left(c;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}/2}}{\left(q;q\right)_{n}}\left(\frac{c}{ab}\right)^{n}A_{q}\left(\frac{cq^{n-1/2}}{a}\right)
\omega\colon \ \mathfrak{gl}(H_+\oplus H_-)_{(2)}\times \mathfrak{gl}(H_+\oplus H_-)_{(2)}\to\mathbb C, \omega(X,Y)=\operatorname{tr}([A(X),A(Y)]-A([X,Y]))=\operatorname{tr}(B(Y)C(X)-B(X)C(Y)).
\widetilde{\eta}= p^{2}\, \frac{\lambda}{\lambda_{_\mathrm{( g s )}}^{( \ast )}}\, \left\{1 + \left( \lambda_{_\mathrm{( g s )}}^{( \ast )}\right)^{1 / 2}\left[ \ln \pi + \gamma + 2 \left( \lambda_{_\mathrm{( g s )}}^{( \ast )}\right)^{- 1 / 2}- \ln \left| \frac{E}{\mu^{2}}\right| \right] \, p^{- 1}+ O \left( p^{- 2}\right) \right\}\; .
h_{-1-L}^{\mathcal{L}_{1,2,..,L}\otimes\mathcal{K}_{L+1}}(t)z^{-L-1}+h_{1-L}^{\mathcal{L}_{1,2,..,L}\otimes\mathcal{K}_{L+1}}(t)z^{1-L}+\cdots =(h_{-L}^{\mathcal{L}_{1,2,..,L}}(t)z^{-L}+h_{2-L}^{\mathcal{L}_{1,2,..,L}}(t)z^{2-L}+\cdots)(h_{-1}^{\mathcal{K}_{L+1}}(t)z^{-1}+h_{1}^{\mathcal{K}_{L+1}}(t)z^{1}+\cdots)
S^{(3)}_{source}= \int d^4x \left(e^{-\phi}\sqrt{\det (G+{\cal F})}+{i\over4}\chi{\cal F}*{\cal F}+{i\over 2}B^R *{\cal F}+{i\over 24}\epsilon^{\alpha\beta\gamma\delta}A^{(4)}_{\alpha\beta\gamma\delta}\right),
0=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\|Q_{j}\|\|Q_{i}\|\sin{(\alpha_{j}-\alpha_{i})}}{(\sigma-\sigma(\|Q_{j}\|\|Q_{i}\|\cos{(\alpha_{j}-\alpha_{i})}+Z\odot_{k-2}Z)^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\}.
p_{j}(\left\|x-y\right\|_2,t) =\sum_{n,m \in \mathbb{Z}}\left(p_{\mathbb{R}}(\left\|x-y_{++}\right\|_2,t)+p_{\mathbb{R}}(\left\|x-y_{--}\right\|_2,t)\right) - \sum_{n,m \in \mathbb{Z}}\left(p_{\mathbb{R}}(\left\|x-y_{+-}\right\|_2,t)+p_{\mathbb{R}}(\left\|x-y_{-+}\right\|_2,t)\right)
\mathcal{A}(z) = \mathcal{\tilde A}(z) + \mathcal{R}_1 (z) + \mathcal{R}_1^*(z) + \mathcal{R}_2 (z) + \mathcal{R}_3 (z) + \mathcal{R}_3^* (z) +\mathcal{R}_4 (z) ,
F_{t}(q,t,w)=-Q\left[ a(x)\left\{q(t\phi_{1}+w)^{q-1}\phi_{1}-\phi_{1}\right\}\right] , & F_{q}(q,t,w)=-Q\left[ a(x)(t\phi_{1}+w)^{q}\log(t\phi_{1}+w)\right] ,
\langle a_{{\bf{k}}}^{t \dagger}( -{\bf{q}}) \rangle = U_{e x t}^{*}( -{\bf{q}}, t ) C_{{\bf{k}}}^{*}( -{\bf{q}}) + U_{e x t}({\bf{q}}, t ) D_{{\bf{k}}}^{*}( -{\bf{q}})
\Re \varrho_{ZZ}(L,L) =-a (b+1)^2 u^2 + a (b-1)^2 v^2 - b(1 + a)^2 x^2 + b(a-1)^2 y^2, \Im \varrho_{ZZ}(L,L) =-2 a \left(b^2-1\right) u v -2b \left(a^2-1\right) x y.
\alpha_{n,q}(l)= \sum_{k=q}^{l}c_{n,q,k}\binom{l-k+2n-2}{2n-2}= \sum_{k=q}^{l}c_{n,q,k}\sum_{i=0}^{2n-2}b_{2n-1,i}(l-k)^i \forall l\geq 2n-1.
c_1= \frac{|z_1|^4}{||z_1|^2-\sin \theta_\xi \cos \theta_\xi +\mathrm i \cos \theta_\xi (\cos\theta_\xi+\sin \theta_\xi )|^4}, z_4= \left(|z_1|^2-1/2\sin2\theta_\xi -\mathbf i \cos \theta_\xi (\cos\theta_\xi +\sin\theta_\xi ) \right)^2.
\int_{-\tau}^{T - \tau}\exp_{\tau}(D, s) \mathrm{d}s = \frac{1}{D}\left(1 + D \tfrac{T}{1!}+ D^{2}\tfrac{(T - \tau)^{2}}{2!}+ D^{3}\tfrac{(T - 3\tau)^{3}}{3!}+ D^{4}\tfrac{(T - 3\tau)^{4}}{4!}+ \dots + D^{k+1}\tfrac{(T - k\tau)^{k+1}}{(k + 1)!}- 1\right).
S_0^{(\Phi)}= - \sum_{t=1}^T \ln \left\{\sum_{r=0}^{\Omega}\frac{1}{r!}\left[\Phi^*(t)\right]^r\sum_{s=0}^r{r\choose s}\frac{(\Omega-s)! \left[\Omega-(r-s)\right]!}{(\Omega!)^2}x^{2(r-s)}\Phi^s(t) \Phi^{r-s}(t-1) \right\}\ .
\bigl\{\Phi(A^{\alpha r})^{1/2}\Psi(B^{(1-\alpha)r})\Phi(A^{\alpha r})^{1/2}\bigr\}^{1/r}&\qquad=\bigl\{\bigl(\Phi(A^{\alpha r})^{1/r}\bigr)^{r/2}\bigl(\Psi(B^{(1-\alpha)r})^{1/r}\bigr)^r\bigl(\Phi(A^{\alpha r})^{1/r}\bigr)^{r/2}\bigr\}^{1/r}&\qquad\longrightarrow\exp\{\alpha\Phi(\log A)+(1-\alpha)\Psi(\log B)\}
(gh^{-1})^{(n-1)}h^{n-1}r \pi_h = s_{n-1}r\pi_{h}+(-1)^{n-1}(n-1)! g(h')^{n-1}h^{-1}r\pi_{h}, (gh^{-1})^{(k-1)}r h^k =s_{k-1}rh+(-1)^{k-1}(k-1)!g(h')^{k-1}r .