harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\widetilde{e}\left( \frac{c_3}{\varpi^{l-h-1}}\right) \sum_{c_2 \bmod{\varpi^{l-h-1}}}\widetilde{e}\left( \frac{c_2}{\varpi^{l-h-1}}\right)=\sum_{c_2 \bmod{\varpi^{l-h-1}}}\widetilde{e}\left( \frac{c_2+c_3}{\varpi^{l-h-1}}\right)=\sum_{c_2 \bmod{\varpi^{l-h-1}}}\widetilde{e}\left( \frac{c_2}{\varpi^{l-h-1}}\right).
u^N - \mathrm{id}\in L^\infty(0,T_\epsilon;H^4(\mathbb{R}^3)),u^N_t \in L^\infty(0,T_\epsilon;H^3(\mathbb{R}^3)), \phi^N \in L^\infty(0,T_\epsilon;H^3(\mathbb{R}^3)),\phi_t^N \in L^2(0,T_\epsilon;H^2(\mathbb{R}^3)),
\delta_w G_{u_0}(\eta) (x) = \int_{\mathbb{R}^3}DK_3\big( \eta(x) - \eta(y) \big) (w(x)-w(y)) D\eta(y) \omega_0(y) \, dy + \int_{\mathbb{R}^3}K_3\big( \eta(x) - \eta(y) \big) Dw(y) \omega(y) \, dy.
\frac{1}{n\rho_{sc}(x)}K^{GUE(n)}(x,y)-\frac{\sin(n\pi \rho_{sc}(x)(x-y) )}{n\pi \rho_{sc}(x)(x-y)}=O\left(\frac{1}{n}\right)+O\left(a_j\right)+O\left(na_j^2\right), &\frac{2\pi}{n}K^{CUE(n)}({2\pi}\rho_{sc}(y_j)x,{2\pi}\rho_{sc}(y_j)y)-\frac{\sin(n\pi \rho_{sc}(y_j)(x-y) )}{n\pi \rho_{sc}(y_j)(x-y)}=O\left(\frac{a_j}{n}\right),
V(x, k) = \frac{\Gamma\left( \frac{k}{2}\right)}{\pi^k}\sum_{n=1}^\infty \frac{\mu(n) n^{k-1}}{X_n^{\frac{k}{2}}}\left\{{}_1F_{1}\left( \frac{k}{2}; \frac{1}{2}; - \frac{1}{X_n}\right) -1 \right\}.
=1-\sum_{i=1}^{3}\sigma_{i}(u)+\sum_{1\leq i<j\leq3}\sigma_{i}(u)\sigma_{j}(u)-N_{K/\mathbf{Q}}(u) & =\sum_{i<j}\sigma_{i}(u)\sigma_{j}(u)-\sum_{i=1}^{3}\sigma_{i}(u)
\left \|\sum\limits_{j=N}^{\infty}\mu_{j}b_{j}\right \|_{h_{\omega}^{p}}\leq C\left \|\sum\limits_{j=N}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}.
\mathbb P\begin{bmatrix}{\frac 1k}\sum_{s\in S}A_s\notin [{\frac 14}I,{\frac 34}I]\end{bmatrix}\leq{\frac{2^{\frac{n+4}2}}{n!\eta}}\exp\begin{pmatrix}{\frac{-k}{16\ln 2}}\end{pmatrix}
H^{\prime}= \frac 1 2 \int d^{3}x \left( B_{a}^{i j}F_{i j}^{a}- A_{i}^{a}A_{a}^{i}+ \left( \left( D^{i}\right)_{b}^{a}B_{0 i}^{b}\right) \left( D_{j}\right)_{a}^{c}B_{c}^{0 j}\right) \equiv \int d^{3}x h^{\prime},
p_k^{CG}=- g_k +\frac{g_k^T g_k}{g_{k-1}^Tg_{k-1}}p_{k-1}^{CG}= -\big(I+\frac{1}{g_{k-1}^T p_{k-1}^{CG}}p_{k-1}^{CG}g_k^T\big)g_k.
[ X^{\alpha}, X^{\beta}] = g f^{\alpha \beta \gamma}X^{\gamma}= g f^{\alpha \beta \gamma}\int d^{4}z \delta^{( 4 )}( x - y ) \delta^{( 4 )}( x - z ) X^{\gamma}= g f^{\alpha \beta \gamma}\delta^{( 4 )}( x - y ) X^{\gamma}
\Sigma_{\pm}( q_{3}, \left|{\bf q}\right| ) = - \frac{1}{4 \pi}\ln \left[ \frac{1}{2 m}\left( - \frac{1}{2}\delta \pm i q_{3}+ \frac{\left|{\bf q}\right|^{2}}{2 m}\right) \right] + O \left( \frac{1}{m^{2}}\right) .
\sup_{0\le j\le N}\ \|P_{n}T^{\,j}P_{l}\,x\|\le \dfrac{\beta_{l}}{4}\cdot \Bigl(\prod_{i=\Delta^{(k)}-N+1}^{\Delta^{(k)}-1}\!\!\!|w_{i}^{(k)}|\ \Bigr)\,\cdot\, \|P_{l}\,x\|.
\left( \begin{array}{cc}a_1 b_1 c_1 d_1 \end{array}\right) \dotplus\left( \begin{array}{cc}a_2 b_2 c_2 d_2 \end{array}\right) =\left( \begin{array}{cccc}a_1 b_1 0 0 c_1 d_1 0 0 0 0 a_2 b_2 0 0 c_2 d_2 \end{array}\right),
\leq C e^{48c_{d,\sigma}k^2T(1+|x|)^2}E\left[e^{48c_{d,\sigma}k^2T\sum_{i=1}^d \sup_{0\leq t\leq T}|\sigma_i\cdot B^i_t|^2}\right]^{\frac{1}{2}}\leq C_{T,d,\sigma,M_2}e^{48c_{d,\sigma}k^2T(1+|x|)^2}\prod_{i=1}^{d}E\left[e^{48c_{d,\sigma}k^2T\sup_{0\leq t\leq T}|\sigma_i\cdot B^i_t|^2}\right]^{\frac{1}{2}}.
d\nu(s)=\frac{18}{\pi^{4}}\sum\nolimits_{m\in\mathbb{Z}}\frac{1}{m^{2}}I_{[m,m+1]}(s)\left\{\sum\nolimits_{n\in\mathbb{N}}\frac{1}{2^{n}n^{2}}\sum\nolimits_{j=1}^{2^{n}}d\tau\left(s-m-j2^{-n}\right)\right\},
\tilde{V}(x) :=\frac{2\pi}{N(\pi + gN)}\;\left(\mbox{K}_0\left(\sqrt{\frac{e^2 N}{\pi\!+\!gN}}|x|\right)+ \ln\left(\frac{1}{2}\sqrt{\frac{e^2 N}{\pi\!+\!gN}}\right)+\gamma\right) \; .
R_{u,v}^{J,q}(q)= (q-1)R_{u,vs_j}^{J,q}(q) begin{align*}5pt]= (-1)^{\ell(v)-\ell(u)}(1-q)\left(1-q+\delta_{u^{-1}(i-1),v^{-1}(i-1)}q^{1+a_{v^{-1}(i-1)}(u,v)}\right)\prod_{k\in D(u,v)}\left(1-q^{a_k(u,v)}\right).
d x_{j}^{\prime}= \left( \frac{\partial x_{j}^{\prime}}{\partial x_{k}}\right) d x_{k}\equiv J_{j k}\d x_{k}, \hspace{1 c m}A_{i}= A_{j}^{\prime}\left( \frac{\partial x_{j}^{\prime}}{\partial x_{i}}\right) \equiv J_{i j}^{t}A_{j}^{\prime},
\Big|\Re \left( \vec{\varpi}^{(2)}(z) \right) \Big| \leq \sum\limits_{n=1}^{\infty}\left|\varpi_{n,0}^{(2)}\right| e^{-s_0n}\leq \mathcal{C}k^{\alpha}\sum\limits_{n=1}^{\infty}n^{\alpha-1}e^{-s_0n}\leq \mathcal{C}_{\alpha,s_0}k^{\alpha}.
0=\frac{\partial}{\partial t}\int \dot{\varphi}_v(t)\omega_{\varphi_v(t)}^n = \int (\ddot{\varphi}_v(t) - \langle \partial \dot{\varphi}_v(t), \bar{\partial}\dot{\varphi}_v(t) \rangle_{\varphi_v(t)})\omega_{\varphi_v(t)}^n.
\textbf{p}(t)=\left( \textbf{p}(t_0)-\frac{\partial \bar{c}}{\partial \textbf{x}}(t,t_0,\textbf{x}) \right) \left[ \frac{\partial \bar{\textbf{g}}}{\partial \textbf{x}}(t,t_0,\textbf{x}) \right]^{-1}.
\o_{\tilde \nu_D}(u,b^*J_{\nu_D}u) =\o_{\tilde \nu_D}(u,(b_*)^{-1}J_{\nu_D}b_* u) = \o_{\tilde \nu_D}(u,J_{\tilde F}u) = \O_{\tilde F}(u,J_{\tilde F}u) >0 \, .
\left( \begin{matrix}{\alpha}&{\beta}\\{\gamma}&{\delta}\\ \end{matrix}\right) \left( \begin{matrix}{E_{x}}\\{E_{y}}\\ \end{matrix}\right) = \left( \begin{matrix}{\alpha E_{x}+ \beta E_{y}}\\{\gamma E_{x}+ \delta E_{y}}\\ \end{matrix}\right) ,
\left\Vert u^{\varepsilon}-\sum_{k=0}^{K}\varepsilon^{k}u_{k}-m^{\varepsilon}\sum_{m=K+1}^{M}\varepsilon^{m}u_{m}\right\Vert_{\mathcal{V}^{\varepsilon}}\le C\left(\varepsilon^{M-1}+\varepsilon^{M}+\sum_{m=K+1}^{M}\left(\varepsilon^{m-\frac{1}{2}}+\varepsilon^{m+\frac{1}{2}}\right)\right),
\mathbb{E}^\dagger [\mathbb{W}_{t_n,t_{n+1}}^\dagger] = \frac{1}{2}\mathbb{E}^\dagger [W^\dagger_{t_n,t_{n+1}}\otimes W^\dagger_{t_n,t_{n+1}}- [W^\dagger_{t_n},W^\dagger_{t_n,t_{n+1}}] ] - \frac{\Delta t}{2}I = 0 ,
\displaystyle \lambda_n(x_i- x_j) \leq \;\;(d_i+d_j)(x_i-x_j)- C_S(i,j)(x_i-x_j)-\frac{1}{2}C_{\frac{1}{2}}(i,j)(x_i-x_j) -\frac{1}{4}C_{\frac{1}{4}}(i,j)(x_i-x_j)-\left(\sum_{l=2}^{r-1}lC_{l}(i,j)\right)(x_i-x_j).
\Lambda_{- 1 , 1}= \lambda^{- 1}( e_{1 , 2 p}- e_{2 , 2 p + 1}) + ( e_{2 , 1}- e_{2 p + 1 , 2 p}) + 2 \sum_{k = 1}^{p - 1}e_{2 + k , 1 + k}- 2 \sum_{k = 1}^{p - 1}e_{p + 1 + k , p + k},
\langle u_nxu_gy u_n^*,x'u_gy'\rangle= \tau(u_g^*(x'^*u_nx)u_g(yu_n^*y'^*))=\tau(E_N(u_g^*(x'^*u_nx)u_g)yu_n^*y'^*) = \tau(E_N(u_g^*E_{L(\Sigma)}(x'^*u_nx)u_g)yu_n^*y'^*).
\sum_{i,j=1}^n \|W_{i,j}\|_{L^q (Q_r)}\leq \sum_{k=1}^n \| u_k\|_{W^{2,1}_q(Q_r)}+ \sum_{k=1}^n \| \tilde{u}_k \|_{W^{2,1}_q(Q_r)}\leq C \gamma + \sum_{k =1}^n \|u_k\|_{W^{2,1}_q (Q_1)}
\int_X g_\alpha(x)\mathrm{Tr}\,e^{-t\widetilde\Box^+_{b, m}}(x, x)dv_X(x)\sim \sum_j\sum_{s=n, n-1,\cdots}t^{-s}\int_0^{2\pi}J_s^{(j)}(g_\alpha(x))du
\int_{V}f d\mu_{A}=\int_{U}fdT^{-1*}\mu_{A}=\int_{U}fdL_{\sigma}^{(m-n)*}\mu_{A}&=e^{(m-n)\delta_{A}}\int_{U}fd\mu_{A}=e^{(m-n)\delta_{A}}\int_{V}fdT^{*}\mu_{A},
Y = ( y_{1}+ i y_{2}) / 2 = \sum_{n}\left[{\frac{1}{\sqrt{4 p^{+}\omega_{n}}}}\, \alpha_{n}^{+}e^{- i \omega_{n}\tau + i n \sigma / \alpha^{\prime}p^{+}}+{\frac{1}{\sqrt{4 p^{+}\omega_{- n}}}}\left( \alpha_{n}^{-}\right)^{\dagger}e^{i \omega_{- n}\tau - i n \sigma / \alpha^{\prime}p^{+}}\right] \ .
\displaystyle \frac{\displaystyle\ \ \sum_{j=1}^L \lambda_j^2\ \}{\displaystyle\sum_{j=1}^{N_h}\lambda_j^2}\ge \epsilon_1,\frac{\ \ \displaystyle \sum_{j=1}^{\mu_j}d_{j, k}^2\ \}{\displaystyle \sum_{j=1}^{m}d_{j, k}^2}\ge \epsilon_2,
\hat S_{MCS}=\int d^{3}x[-\frac{1}{2}(\partial_{\alpha}(\hat A +\hat a )_{\beta}-\partial_{\beta}(\hat A +\hat a )_{\alpha})*\partial^{\alpha}(\hat A +\hat a )^{\beta}+\frac{2\pi}{g^2}\epsilon^{\alpha\beta\gamma}(\hat A +\hat a )_{\alpha}*\partial_{\beta}(\hat A +\hat a )_{\gamma}].
N[G*H] \models k(\mathbb{S}) = \mathbb{S}= \{z \in \big( H_{\kappa^{++}}\big)^{N[G*H]}\ | \ \Big( \big( H_{\kappa^{++}}\big)^{N[G*H]}, \in \Big) \models \phi(z,r) \}<\mu \delta \le \delta.
\limsup_{e \to \infty}\frac{p^{s_F^\ell(L^{m_e'};x)}- 1}{m_e'/(\ell+1)}= \frac{\ell+1}{\ell+n}\cdot \lim_{e \to \infty}\frac{p^{e-1}- 1}{p^e - \frac{n+1}{n+\ell}}= \frac{\ell+1}{(\ell+n)p}.
P_{X}(z)\leq 32\sum_{p\in \mathcal{P}_{X}}\sum_{\eta\in\Gamma_{X,p}\backslash\Gamma_{X}}\Im(\sigma_{p}^{-1}\eta z)^{2}= 32\sum_{p\in \mathcal{P}_{X}}\mathcal{E}_{X,\mathrm{par},p}(z,2),
\underline{{A}}^{\mu}= \frac 1 \ell \underline{{\Theta}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu \rho}\underline{{\Omega}}^{\nu \rho}\qquad , \qquad \underline{{\tilde{A}}}^{\mu}= - \frac 1 \ell \underline{{\Theta}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu \rho}\underline{{\Omega}}^{\nu \rho}\qquad .
\det\left[\phi_{i}(x_{j})\right] = \sum_{1 \leq i < j \leq m}(-1)^{i+j+k+1}\det\begin{bmatrix}\phi_{i}(x_{1}) \phi_{i}(x_{k}) \phi_{j}(x_{1}) \phi_{j}(x_{k}) \end{bmatrix}\det\begin{bmatrix}\phi_{r}(x_{s}) \end{bmatrix}_{r \notin \{i, j\}, s \notin\{1, k\}}.
\mathcal{F}^{S,\pi}_{\preceq_0}(F_1^\ast,\dots,F_d^\ast):= \bigg\{cF : \begin{array}{l}c\in[0,1] F \mathbb{R}^d cF_i\preceq_0 F_i^\ast i=1,\dots, d cF(s)\leq \pi_s s\in S \end{array}\bigg\},
Z [ J ] = \mathcal{N}\int \big [ \mathcal{D}\Phi^{\prime}\big ] \mathrm{e x p}\Big ( \frac{\mathrm{i}}{\hbar}\Gamma_{\mathit{c l}}[ \Phi^{\prime}] + \frac{\mathrm{i}}{\hbar}J^{i}\Phi_{i}^{\prime}\Big ) .
\frac{\partial\,_{1}\phi_{1}(0;w;q,z)}{\partial z}=\frac{1}{(w;q)_{\infty}}\frac{\partial(z;q)_{\infty}}{\partial z}\!\left(1+O(z^{-1})\right)+\frac{(z;q)_{\infty}}{(w;q)_{\infty}}\, O(z^{-2})+O\!\left(\!\exp\!\left(\frac{\ln^{2}(z)}{2\ln(q)}\right)\!\right)\!.
W=\left|\begin{array}{cccc}0 iC_{-}0 C_{-}0 -C_{+}0 iC_{+}iC_{-}0 C_{-}0 iC_{+}0 C_{+}0\end{array}\right|; \quad C_{\mp}\equiv \frac 12(C\mp 1), \quad C\Psi \equiv \Psi^{*}, \quad C\mathcal{E}\equiv \mathcal{E}^{*}.
Z(z)= e^{\frac{-N\ell}{2}\sigma_3}Y(z)\,e^{-Ng(z)\sigma_3}e^{\frac{N\ell}{2}\sigma_3}\begin{bmatrix}1 0 \displaystyle \star\,\Big(\frac{z}{z-a}\Big)^{c}e^{N\phi(z)}1\end{bmatrix},
S_{c l}( e_{\mu}^{i}, \omega_{\mu}) = \int{d^{2}}x \; e \bigg ( \frac{1}{1 6 \alpha}R_{\mu \nu}{}^{i j}R^{\mu \nu}{}_{i j}- \frac{1}{8 \beta}T_{\mu \nu}{}^{i}T^{\mu \nu}{}_{i}- \gamma \bigg ) ,
g_{n}({\sigma}_{0}) = e x p \left\{J_{3}\sum_{\triangle}{\sigma}_{0}{\sigma}_{1}{\sigma}_{1}+ h \sum{\sigma}_{1}\right\}{[ g_{n - 1}({\sigma}_{1}^{( 1 )}) ]}^{\gamma - 1}{[ g_{n - 1}({\sigma}_{1}^{( 2 )}) ]}^{\gamma - 1}.
E(\Sigma,X(T_0),T_0)^{(5)}=\frac{1}{90}\{(\frac{1}{2}\sum_{k, m, n}\bar W_{0kmn}^2+ \sum_{m, n}\bar W_{0m0n}^2 )a^0 + 2\sum_i \sum_{m, n}\bar W_{0m0n}\bar W_{0min}a^i + \sum_{m, n}\frac{\bar W_{0m0n}^2}{2a^0}\}.
C_{\gamma}\left(\overline{z}^{m-j}z^{n-j}\left(1- |{z}|^{2}\right)^{j}\right) = \dfrac{-z^{n}\overline{z}^{m+1}}{(m-j+1)}\left(1-|z|^2\right)^{\gamma+1}& \times \left(\dfrac{1-|z|^2}{|z|^{2}}\right)^{j}{_2F_1}\left( \begin{array}{c}1, \gamma+m +2 m-j+2\end{array}\bigg ||z|^2\right).
\hat\Phi =\left(\begin{array}{c}\hat{P}^1 \ \vdots \hat{P}^m\end{array}\right) &=\left(\begin{array}{cccc}1 P^1_1 \cdots P^1_n \vdots \vdots \vdots 1 P^m_1 \cdots P^m_n\end{array}\right)\in\mathbb{R}^{m\times(n+1)},
Q_1 = \ \sum_{i = 1}^k s y_1 \dots y_{i-1}[y_i, x_1, x_2] y_{i+1}\dots y_k , & Q_2 = \ \sum_{1 \le i <i' \le k}s y_1 \dots y_{i-1}\bigl( [y_i, x_1] y_{i+1}\dots y_{i' - 1}[y_{i'}, x_2] + [y_i, x_2] y_{i+1}\dots y_{i' - 1}[y_{i'}, x_1] \bigr) y_{i'+1}\dots y_k .
[\varphi_n]^p_{W^{s,p}(\mathbb{R}^N)}=[(u\circ\Psi_{1/n})\ast\varrho_n]^p_{W^{s,p}(\mathbb{R}^N)}\le [u\circ\Psi_{1/n}]^p_{W^{s,p}(\mathbb{R}^N)},
g_{x_0}^T := (\partial_2 - a \partial_1^2+1) u_T - \sigma_i E f_{iT}(\cdot, a_0 ) & + (a - a(x_0)) \partial_1^2 u_T + (\sigma_i - \sigma_i(x_0)) E f_{iT}(\cdot, a_0 ) & + \sigma_i(x_0) ( E f_{iT}(\cdot,a_0 ) - f_{iT}(\cdot, a(x_0) )).
2\ell^{\prime}\left( 0,\chi\right) =-\chi\left( -1\right) \overline{E}_{1,\overline{\chi}}\left( 1\right) +\chi\left( -1\right)\int\limits_{1}^{\infty}\frac{\overline{E}_{1,\overline{\chi}}\left(x\right)}{x^{2}}dx.
\eta \leq -(m+p)(\partial_t - \tilde X_j\cdot \nabla_Y)\phi(\tilde X_j, \tilde Y_j, \tilde t_j) + (p-2)\lambda(E^X) + \textrm{tr}(E^X) = -(m+p)(\partial_{\tilde t}- \tilde X_j\cdot \nabla_{\tilde Y})\Psi(P_j) + (p-2)\lambda(E^X) + \textrm{tr}(E^X).
\overline{\widetilde{S}^{R,R}_{X,Y}}= \frac{\overline{\widetilde{S}^{R,R}_{1,Y}}}{\widetilde{S}^{R,R}_{1,Y}}\widetilde{S}^{R,R}_{X^*,Y}= \frac{\overline{\widetilde{S}^{R,R}_{1,Y}}}{\widetilde{S}^{R,R}_{1,\bar{Y}}}\widetilde{S}^{R,R}_{X,\bar{Y}},
C(\mathsf{P}) := \mathrm{Tr}\left(\pi(K_{2\rho}^{-1})(2\mathsf{P}-\mathrm{Id})\otimes\mathsf{P}\otimes\mathsf{P}\right), C(\mathsf{Q}) := \mathrm{Tr}\left(\pi(K_{2\rho})(2\mathsf{Q}-\mathrm{Id})\otimes\mathsf{Q}\otimes\mathsf{Q}\right).
i \frac{\partial}{\partial \overline{{{\psi}}}^{\prime \prime}}\langle \zeta^{\prime}| \zeta^{\prime \prime}\rangle = \gamma^{\mu}\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}\langle \zeta^{\prime}| \psi | \zeta^{\prime \prime}\rangle \, .
\langle V ({\bf x}) V ({\bf y}) \rangle = g M^{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{e^{i{\bf p}\cdot ({\bf x}-{\bf y})}}{p^{2}+ M^{2}}= \frac{g M^{2}}{4 \pi}\int_{0}^{\infty}\frac{d s}{s}e^{- M^{2}s - ({\bf x}-{\bf y})^{2}/ 4 s}.
C : \quad \left( \gamma_{\mu \nu}\, D_{( + )}^{\mu}\, D_{( + )}^{\nu}\, + \, m^{2}\right) \, u ( x ) \, = \, 0 \, \, \, \longrightarrow \, \, \, \left( \gamma_{\mu \nu}\, D_{( - )}^{\mu}\, D_{( - )}^{\nu}\, - \, m^{2}\right) \, v ( x ) \, = \, 0 \quad ,
\cosh || \vec{a}+ \vec{b}|| \le \cosh || \vec{a}|| \, \cosh || \vec{b}|| - \cos \theta\sinh || \vec{a}|| \, \sinh || \vec{b}||, \cos \theta := -{\vec{a}\cdot \vec{b}\over || \vec{a}|| \, || \vec{b}||},
\sum_{x,y=0}^n(\delta_{x,y}-M_{x,y})f_yf_x =\sum_{x,y=0}^n \sum_{j,j'=0}^n \left(1- \Theta(\mu_j,\mu_{j'})\right) \psi_j(x)\psi_{j'}(x)\psi_j(y)\psi_{j'}(y) f_yf_x &= \sum_{j,j'=0}^n \left(1- \Theta(\mu_j,\mu_{j'})\right) \left(\sum_{x=0}^n \psi_j(x)f_x\psi_{j'}(x)\right)^2.
\gamma=\frac{1}{\sqrt{1-(v/c)^{2}}}=\sum_{n=0}^{\infty}(\frac{v}{c})^{2n}\prod_{k=1}^{n}(\frac{2k-1}{2k})=1+\frac{1}{2}(\frac{v}{c})^{2}+\frac{3}{8}(\frac{v}{c})^{4}+\frac{5}{16}(\frac{v}{c})^{6}+\cdot\cdot\cdot
n_{g,r,d,a}= g! \cdot n (n^2-1) \frac{\prod\limits_{i=2}^t \dfrac{i! \cdot (n-i)}{(s-1+i)!}\cdot \prod\limits_{j=2}^{r+1-t}\dfrac{j! (n+j)!}{(s+n-1+j)! (n-t-1+j)!(n-1+j)}}{(s-1)!(s+n-1)!(t-1)!(r-t)!}
K^{\ast}=(\cos\varphi)\theta^{\prime}(1+\varphi^{\prime2})+\sin\varphi(\varphi^{\prime}\theta^{\prime\prime}-\theta^{\prime}\varphi^{\prime\prime}) & =\frac{1}{v^{3}}\left\{(\cos\varphi)\dot{\theta}(v^{2}+\dot{\varphi}^{2})+\sin\varphi(\dot{\varphi}\ddot{\theta}-\dot{\theta}\ddot{\varphi})\right\}
\left\|\frac{w(s)^p}{\Psi_{\beta,L}(w(s))}\right\|_{L^\infty({\bf R}^N)}\le C\gamma^{\frac{\theta}{N}}s^{-1}\biggr[\log\biggr(e+\frac{1}{s}\biggr)\biggr]^{-1-\beta}
g_I (x) = \prod_{i = 1}^m (1 - \frac{w_i}{4}x )^2 \sim 1 - \left(\sum_{i}\frac{w_i}{2}\right) x , x \to 0+ ; g_{II}(x) = \prod_{i = 1}^m \frac{4}{4 + v_i x}\sim 1 - \left(\sum_i \frac{v_i}{4}\right)x , x \to 0+ ; g_{III}(x) = \left(\frac{4}{4 + v x}\right)^m \sim 1 - m \cdot \frac{v}{4}x , x \to 0+ .
\chi_2(x) ={\hat{T}_2}(s_2(x))-s_2(T(x))={\hat{T}_2}(\zeta(s_1(x)))-\zeta(s_1(T(x))) =\zeta({\hat{T}_1}(s_1(x))-s_1(T(x)))=\zeta(\chi_1(x)) =\chi_1(x).
\Lambda_{\alpha}={\rm diag}\big(\lambda_i^{(k, \alpha)}\big)_{i=1}^{n}\textrm{with}\lambda_i^{(k,\alpha)}=\lambda_k -2\alpha^{\frac{1}{n}}\theta_n^{i}+\lambda_k \alpha^{\frac{2}{n}}\theta_n^{2i}\textrm{and}\theta_n =\exp\Big(\frac{2\pi{\bf i}}{n}\Big).
V(y,s) = 2(\psi_\alpha(y,s) - 1), G(Q) = (\alpha - 1)\left[\frac{|\nabla Q + \nabla \psi_\alpha|^2}{Q + \psi_\alpha}- \frac{|\nabla \psi_\alpha|^2}{\psi_\alpha}\right], R(y,s) = -\partial_s \psi_\alpha + \Delta \psi_\alpha - \frac{y}{2}\cdot\nabla \psi_\alpha+\psi_\alpha^2 - \psi_\alpha + (\alpha - 1)\frac{|\nabla \psi_\alpha|^2}{\psi_\alpha}.
e^{i(\omega_{N-1}-\omega_{N-1}')}\cdot\left[\begin{pmatrix}-\sin(\psi_{N-1}) 0 \ldots 0\end{pmatrix}B(B')^{-1}\begin{pmatrix}-\sin(\psi_{N-1}') 0 \vdots 0\end{pmatrix}\right]+ \cos(\psi_{N-1})\cos(\psi_{N-1}')e^{-i(N-1)(\omega_{N-1}-\omega_{N-1}')}=1.
W_{12}^*W_{23}^*(1\otimes1\otimes x)W_{23}W_{12}=W_{23}^*W_{13}^*W_{12}^*(1\otimes1\otimes x)W_{12}W_{13}W_{23}&=W_{23}^*W_{13}^*(1\otimes1\otimes x)W_{12}^*W_{12}W_{13}W_{23}&=W_{23}^*W_{13}^*(1\otimes1\otimes x)W_{13}W_{23}W_{23}^*W_{23}&=W_{23}^*W_{13}^*(1\otimes1\otimes x)W_{13}W_{23},
\Big(\prod_{{j}=1}^{k}\big({j}+\alpha\cdot(\mathbf a +\mathbf 1)\big)\Big)\phi_{\mathbf a,\mathbf b}(z^\prime)\int_{\mathbb B^1}\frac{(1-\|\eta\|^2)^{\alpha\cdot\mathbf 1}\eta^{\mathbf c}h(z,w,\eta)^{\mathbf a}}{\pi^k(1-\langle w,\eta\rangle)^{1+k+\alpha\cdot \mathbf 1}}dV(\eta).
\Psi_{\rm WKB}(x)=\begin{pmatrix}\dfrac{1}{2}(1+q_2)x-\alpha -1 \dfrac{1}{4}(1-q_2^2) 0\end{pmatrix}\left[I+\frac{M_0}{x}+\cdots\right]e^{\frac{x^3}{6}-\frac{tx}{2}}e^{\left(\frac{x^3}{6}-\frac{tx}{2}\right)\sigma_3},
G(z)= \sum_{k=0}^\infty \binom{(r+1)k}{k}\frac{1}{z^{k+1}}= \frac{1}{z}\_r F_{r-1}\left(\frac{1}{r+1}, \frac{2}{r+1}, \ldots, \frac{r}{r+1}; \frac{1}{r}, \frac{2}{r}, \ldots, \frac{r-1}{r}\big\vert \frac{(r+1)^{r+1}}{r^r z}\right),
(n-\delta-N) R_n(x) - (n+\gamma+1) R_{n+1}(x) =- \frac{(N-x)(x+\gamma+\delta+N+1)}{N}\hat R_{n}(x), (n+1) \hat R_{n}(x) -(n-N+1) \hat R_{n+1}(x) = N R_{n+1}(x),
{\cal W}_{ML}({\cal A}_P)=\exp \left(- i \frac{N-1}{2N}g^2{\cal A}_P \right)~\frac{1}{N}\oint\frac{dz}{2 \pi i}~\exp \big( - i g^2{\cal A}_P z \big)\bigg( \frac{z+1}{z}\bigg)^N.
\tilde{B}_{d - 1 , i}= \frac{G_{d - 1 , i}}{G_{d - 1 , d - 1}}, \tilde{B}_{i j}= B_{i j}+ \frac{G_{i , d - 1}B_{d - 1 , j}+ B_{i , d - 1}G_{d - 1 , j}}{G_{d - 1 , d - 1}}.
V^{inst}_0 = \frac{R}{2\pi}\sum_{n \ne 0}e^{i\cdot n\theta_{e}}K_0(2\pi |nc|) A^{inst}_0 = -\frac{R}{4\pi}\left( \frac{d c}{c}- \frac{d \overline{c}}{\overline{c}}\right) \sum_{n \ne 0}sign(n) \cdot e^{i\cdot n\theta_{e}}|c| K_1(2\pi |nc|) \omega^{inst}_0 = -2\pi \cdot [ dc_2 \wedge A^{inst}_0 + V^{inst}_0 d(\frac{\theta_e}{2\pi R}) \wedge dc_1]
\int_{0}^{T}{\bf a}^{\top}(t)\widehat{\bf z}(t)dt =\sum_{j=1}^{q}\int_{0}^{T}a_{j}(t)\widehat{z}_{j}(t)dt & \geq\sum_{j=1}^{q}\sum_{u=1}^{N}(t_{u}-t_{u-1})a_{j}^{(u)}\bar{z}_{j}^{(u)}=\sum_{u=1}^{N}(t_{u}-t_{u-1})({\bf a}^{(u)})^{\top}\bar{\bf z}^{(u)}.
\partial(z_{K}(v))=\sum_{i\in K}U^{h_K(v)-h_{K-i}(v)}z_{K-i}(v)+U^{h_{K}\left(v^*-\Lambda_{\overline{K}}\right)-h_{K-i}\left(v^*-\Lambda_{\overline{K}}\right)}z_{K-i}\left(v-\Lambda_{i}\right).
(\lambda-w(v))(\frac{W(P_{u,v})\phi(A-P_{u,v},\lambda)}{\phi(A-u,\lambda)})W(P_{u,v})\phi(A-P_{u,v},\lambda)= \displaystyle\sum_{z\sim v}(\frac{W(P_{u,v})\phi(A-P_{u,v},\lambda)}{\phi(A-u,\lambda)})w(zv)(W(P_{u,z})\phi(A-P_{u,z},\lambda))
\mathcal{T}\widetilde H^\alpha\mathcal{T}^*= \underset{m\in \mathbb Z}\bigoplus \big( 1 -\alpha V_{|m|- 1/2}(\cdot + \mathrm i/2)\big)R^1 =:\underset{m\in \mathbb Z}\bigoplus\widetilde H^\alpha_m,
\begin{array}{lcr}\partial_{\sigma}^2X^9+ \sum\limits_{\nu=1}^8{1\over 2}(\{X^{\nu+(-)^{\nu+1}},\{X^{\nu+(-)^{\nu+1}},X^9\}\}+\{X^{\nu},\{X^{\nu},X^9\}\})=0,\end{array}
\Delta E^{(1)}_{n}[\chi\,]= \frac{\langle V \rangle (2-n)}{(4\pi)^\frac{n}{2}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty (\omega(k)-m) k^{n-3}\, dk = \frac{\langle V \rangle m^{n-1}}{(4\pi)^\frac{n+1}{2}}\Gamma\Bigl(\frac{1-n}{2}\Bigr)
\begin{alignedat}{4}\hat a_0 =16n+2, \hat a_1 =8n+1, \hat a_2 =\phantom09n+1, \hat a_3 =10n+1, \hat b_0 =11n+2, \hat b_1 =1, \hat b_2 =16n+2, \hat b_3 =16n+2,\end{alignedat}
\Big\|\left( \begin{array}{cc}F_{11}-\cos \alpha F_{12}-\sin \alpha F_{21}+\sin \alpha F_{22}-\cos \alpha \end{array}\right)\Big\|^2=(F_{11}-\cos\alpha)^2+(F_{12}-\sin \alpha)^2+(F_{21}+\sin \alpha)^2+(F_{22}-\cos \alpha)^2\,.
\sigma (\Sigma (S))= 2 \sigma (X) - \frac{1}{2}S \circ S - \sigma (L)+ \sigma (L'), b_2 (\Sigma (S )) = 2b_2(X) + b_1(S) , b^+ (\Sigma (S )) = 2b^+(X) + \frac{1}{2}b_1(S) -\frac{1}{4}S \circ S - \frac{1}{2}\sigma (L) + \frac{1}{2}\sigma (L'), b_1(\Sigma(S) ) = b_3(\Sigma(S) ) =0.
A=\frac{1}{\sqrt{2k(k-\gamma)}}\sqrt{\frac{C}{n-|k|+\gamma}\frac{(n-|k|-1)!}{\Gamma(n-|k|+2\gamma+1)}\frac{1}{2}((\frac{Ek}{\gamma\mu c^{2}})^{2}+\frac{Ek}{\gamma\mu c^{2}})}
\omega := p^m \left( x^k y^l dx - d \left( \sum^{3}_{i=0}\sum^{k+5}_{j \geq 7-i}b_{i,j}x^j y^i \right) \right) &\phantom{:}= p^m \left( \sum^{3}_{i=1}\sum^{2}_{j=0}a_{i,j}x^j y^i dx + d \left( \sum^{3}_{i=0}\sum^{6-i}_{j=0}b_{i,j}x^j y^i \right) \right)
\dot{x}_i\sim\mu_i,\quad\dot{\mu}_i+\sum^n_{\substack{j=1, j\not=i}}\frac{a_{ij}}{(x_i-x_j)^3}+\sum_{\substack{j,k=1, j\not=i}}^n\frac{b_{ijk}\mu_k}{(x_i-x_j)^3}\sim 0,
Z = \int \mathrm{D}X \; \mathrm{D}Y \; \exp \left( - N \left[ \frac{( 1 - c )}{2}\mathrm{T r}\; X^{2}+ \frac{( 1 + c )}{2}\mathrm{T r}\; Y^{2}- \frac{\widehat{g}}{3}( \mathrm{T r}\; X^{3}+ 3 \mathrm{T r}\; X Y^{2}) \right] \right) .
\sup_{\substack{\psi \in \mathcal V_{01}\|\psi\|_{\ell^2(\square)}= 1}}\sup_{\substack{\varphi \in \mathcal V_0^\perp \|\varphi\|_{\ell^2(\square)}= 1}}\left| \left\langle \psi, V^\square \varphi \right\rangle \right|^2= 0.
\big[ \frac{d^2}{d x^2}+ a_1 (|\phi_1|^2+|\phi_2|^2) + U_1(x) \big] \phi_1 = \nu_1 \phi_1\ , \big[ \frac{d^2}{d x^2}+ a_2 (|\phi_1|^2+|\phi_2|^2) + U_2(x) \big] \phi_2 = \nu_2 \phi_2\ ,
\mathcal{A}_{\varepsilon}= f^{\varepsilon}(\mathbf{x})^* b(\mathbf{D})^* g^{\varepsilon}(\mathbf{x}) b(\mathbf{D}) f^{\varepsilon}(\mathbf{x}), \widehat{\mathcal{A}}_{\varepsilon}= b(\mathbf{D})^* g^{\varepsilon}(\mathbf{x}) b(\mathbf{D}).
{\bf y}_t ={\bf M}_t{\bf x}_t = \sum_{k\in [K]}{\bf M}_{t,k}\underbrace{\begin{bmatrix}V_{t,s_1}^k V_{t,s_2}^k \vdots V_{t,s_n}^k \end{bmatrix}}_{{\bf V}_{t,k}}{\bf W}_k,
\begin{aligned}\varphi(s,r,u,v) \geq\sum_{i=1}^{k}\sum_{j=1}^{k}(h(\xi_i+)-h(\xi_i-))\, (h(\xi_j+)-h(\xi_j-)) &\qquad\qquad\qquad\times \frac{1}{2\pi}\,e^{-\frac{a_i^2}{2}}\, \int_0^{\rho}\frac{1}{\sqrt{1-x^2}}\, e^{-\frac{(b_j-a_i x)^2}{2 (1-x^2)}}dx.\end{aligned}
\left[ F , \gamma_{i_{1}\ldots i_{p - 2 k - 2}}^{( 1 )}\right] \approx 0 , \; \left[ F , \gamma_{i_{1}\ldots i_{p - 2 k - 2}}^{( 2 )}\right] \approx 0 , \; k = 0 , \cdots , c ,
d\breve{s}^{2}=2T^{h}dt^{2}=2(T-T^{\omega})dt^{2}=2(T-\frac{\omega^{2}}{2\rho^{2}})dt^{2}& =d\bar{s}^{2}=2\bar{T}dt^{2}=d\rho^{2}+\frac{\rho^{2}}{4}(d\varphi^{2}+\sin^{2}\varphi d\theta^{2})
0 \ge \sum\limits_{ij = 1}^n{F^{ij}G_{ij}}- G_t \geq \big[\frac{1}{{32(1 + 2M_0 )}}- C_4 d \big] F^{11}w_1^2 - (C_3+C_4)w_1 \sum\limits_{i = 1}^n{F^{ii}}-C_4.
\| f^{0}_{x}\|^2 \leq \sup_{h \in H}\big\| L_{h^{-1}}f_{hx}\big\|^2 \, \mu_{H}(Kx^{-1}\cap H) = \sup_{h \in H}\big\| f^{L, V^x}_{(h,e) \cdot (e,x )}\big\|^2 \, \mu_{H}(Kx^{-1}\cap H) < \infty,
C_1(q) =\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{n(n+1)/2}}{1-q^n}=\sum_{k=0}^{\infty}\frac{kq^{k^2}}{(q)_k^{2}}, R_1(q) =\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}q^{n(3n+1)/2}}{1-q^n}.
\left\langle \begin{array}{c c}{{r}}\\{{t}}\\ \end{array}\right\rangle_{n - 1}\left\langle \begin{array}{c c}{{r}}\\{{q}}\\ \end{array}\right\rangle_{n - 1}= \left\langle \begin{array}{c c}{{r}}\\{{q}}\\ \end{array}\right\rangle_{n - 1}\left\langle \begin{array}{c c}{{r}}\\{{t}}\\ \end{array}\right\rangle_{n - 1}
\partial_{\mu_i^\prime}w_{\mu_i',\zeta_i'}=\frac{\alpha_3 \,\bigl(|y-\zeta_i'|^2-\mu_i'^2\bigr)}{2\,\sqrt{\mu_i'}\,\bigl(|y-\zeta_i'|^2+\mu_i'^2\bigr)^{\frac{3}{2}}}, D_{\zeta_i^\prime}w_{\mu_i',\zeta_i'}=\frac{\alpha_3\, \sqrt{\mu_i'}\,(y-\zeta_i')}{\bigl(|y-\zeta_i'|^2+\mu_i'^2\bigr)^{\frac{3}{2}}}.
\begin{cases}\bar{\Phi}_{\rm{dn}}|^{\tau^{-}_{i}}_{\tau_{i-1}}> \bar{\Phi}_{\rm{dn}}|^{\tau_{i}}_{\tau_{i-1}}\bar{\Phi}_{\rm{dn}}|^{\tau^{+}_{i}}_{\tau_{i-1}}> \bar{\Phi}_{\rm{dn}}|^{\tau_{i}}_{\tau_{i-1}}.\end{cases}
\frac{\partial}{\partial r}h(r,\theta) = \sum_{k=a}^{\infty}r^{k-1}\sin( k \theta)= \frac{1}{2 i}\sum_{k=a}^{\infty}r^{k-1}e^{ik \theta}- \frac{1}{2 i}\sum_{k=a}^{\infty}r^{k-1}e^{- ik \theta}&= \frac{r^{a-1}}{2 i}\left( \frac{e^{i a \theta}}{1- r e^{i \theta}}- \frac{e^{-ia \theta}}{1-r e^{-i \theta}}\right) .
\widehat{X}_{\overline{\mathbf{c}}}=\{(g,x) \in G \times G, \,\, x^{-1}g x \in \overline{S}\cdot U_P \}, & X_{\overline{\mathbf{c}}}=\{(g,xP) \in G \times G/P, \,\, x^{-1}g x \in \overline{S}\cdot U_P \},
h(p) - I(X_0;X_t) \le 3 \sum_{i=1}^k p_i \sum_{j \neq i}\|p\|_\infty e^{-0.085 \frac{\alpha m^2}{e^{2\alpha t}-1}}&= 3(k-1) \|p\|_\infty e^{-0.085 \frac{\alpha m^2}{e^{2\alpha t}-1}}.
(L^n_{p,q}f\ast \Psi_j)(z)=z+\sum\limits_{k=2}^\infty u_k[k]_{p,q}^na_{k}{z^k}+(-1)^{n+j}\sum\limits_{k=1}^\infty v_k[k]_{p,q}^nb_{k}{\overline{z}^k}.