harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
B ( z , \bar{z}) = w_{+}e^{\omega z + \bar{\omega}\bar{z}}+ w_{-}e^{- \left( \omega z + \bar{\omega}\bar{z}\right)}+ w e^{\omega z - \bar{\omega}\bar{z}}+ \bar{w}e^{- \left( \omega z - \bar{\omega}\bar{z}\right)},
\phi b^{m_0}xb^{m_k}\sim \phi b^{m_0-1}xb^{m_k}-\sum_{s\in S_b}\phi sb^{m_0-1}xb^{m_k}&\sim ... &\sim \phi b^{m_0-(m_0-n_0)}xb^{m_k}-\sum_{i=1}^{m_0-n_0}\sum_{s\in S_b}\phi sb^{m_0-i}xb^{m_k}&=\phi b^{n_0}xb^{m_k}-\sum_{i=n_0}^{m_0-1}\sum_{s\in S_b}\phi sb^ixb^{m_k}.
F_{n}^{\mathcal{O}| \mu_{1}\ldots \mu_{n}}( \theta_{1}+ \lambda , \ldots , \theta_{n}+ \lambda ) = e^{s \lambda}F_{n}^{\mathcal{O}| \mu_{1}\ldots \mu_{n}}( \theta_{1}, \ldots , \theta_{n}) \, \, ,
\langle H_{k}\rangle ={\frac{1}{2}}\left( \langle |{\widehat \Pi}_{k}|^{2}\rangle + k^{2}\langle |{\widehat \Psi}_{k}|^{2}\rangle \right) = k \left({\frac{S_{e x}}{S_{r e}}}+{\frac{S_{r e}}{S_{e x}}}\right) ,
|\partial^\zeta_x(\nabla \phi^2 \cdot Q(v))-\nabla \phi^2 \cdot \partial^\zeta_x Q(v)|_2 \leq C(|\nabla \phi|^2_\infty |\nabla v |_2+|\phi|_\infty|\nabla^2 \phi|_2 |\nabla v|_\infty\big)\leq Cc^3_3;
\gamma^{'}=(\overbrace{\beta_{1},\cdots,\beta_{1}}^{M_{1}},\cdots,\overbrace{\beta_{s},\cdots,\beta_{s}}^{M_{s}},\overbrace{-\gamma_{s},\cdots,-\gamma_{s}}^{N_{s}},\cdots,\overbrace{-\gamma_{1},\cdots,-\gamma_{1}}^{N_{1}})
\Delta_{G}= \frac 1{\sin \theta}\frac{\partial}{\partial \theta}\biggl ( \sin \theta \frac{\partial}{\partial \theta}\biggr ) + \frac 1{\sin^{2}\theta}\biggl ( \frac{\partial}{\partial \phi}- \cos \theta \frac{\partial}{\partial \psi}\biggr )^{2}+ \frac{\partial^{2}}{\partial \psi^{2}}.
P \c{k}{l}{m}{a}{b}{c}{x}= (\alpha_{1})_{i_{1}}(\alpha_{2})_{i_{2}}\dotsm (\alpha_{n})_{i_{n}}(-1)^{j_{1}}x^{j_{2}}(1 - x)^{j_{3}}\left(\varphi P \right) \c{k}{l}{m}{a}{b}{c}{x},
S = \frac{1}{\pi}\int d^2x \left[e^{-2\phi}(-2\partial_+\partial_-\rho +4\partial_+\phi\partial_-\phi+\lambda^2 e^{2\rho})+\frac{1}{2}\partial_+f\partial_-f+\mu^2(\cos f-1)e^{2\rho-2\phi}\right],
\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}^{2}b_{j}^{2}+\sum_{j=1}^{n-1}\sum_{i=j+1}^{n}a_{i}^{2}b_{j}^{2}-2\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}a_{i}b_{i}a_{j}b_{j}
\widehat{u}(t,\xi) = \dfrac{1}{1 - e^{-i2\pi\xi c_{j0}}}\int_{0}^{2\pi}e^{-i\xi H(t_j, \tau)}\widehat{f}_j(t_1,\ldots,t_j-\tau,\ldots, t_n,\xi) d\tau,
\left[ U \vert M^{(_c)}=n \right]{=}\left[_{U\vert \tau_1,t_j,M^{(_c)}=n}\left[ U^{\tau_1}{+}\sum_{j = 1}^{n-1}U^{t_j}\right] \right] &=\lambda\left( \left[ \tau_1 \vert M^{(_c)}=n \right] + \sum_{j = 1}^{n-1}\left[ t_j \vert M^{(_c)}=n \right]\right),
(\ln n)^{-k}\phi_{k,n}(y_1,\cdots,y_k)\geq (2n)^k(2\ln n)^{-\frac{k}{2}}\mathbb{P}(\xi^{(n)}(I_{n,k})=0) -(2n)^k(2\ln n)^{-\frac{k}{2}}\sum_{j=0}^k\mathbb{P}(\xi^{(n)}(I_{n,k})=\xi^{(n)}(J_{n,k,j})=0).
\tilde{\alpha}_{\mathrm{e f f}}^{\prime}= \frac{\tilde{\alpha}^{\prime}}{\epsilon}, \qquad G_{o ( 1 )}^{2}= \frac{g_{s}}{\epsilon}, \qquad \tilde{u}= \frac{r}{\epsilon \tilde{\alpha}_{\mathrm{e f f}}^{\prime}}
\delta_{\mathrm{S}}( k ) = \delta_{\mathrm{A}}( k ) - \operatorname{t a n}^{- 1}( \frac{2 \: \mathrm{I m}\: \beta^{\prime}( k , 0 )}{k + 2 \: \mathrm{R e}\: \beta^{\prime}( k , 0 )}) .
{\cal U}_{\Delta}^{\dagger}B_{i}^{3}\,{\cal U}_{\Delta}=\frac{1}{2}\epsilon_{ijk}\left\{\left( e^{-ig \Delta}F_{jk}e^{ig\Delta}\right)^{3}+ ig \,{\cal U}_{\Delta}^{\dagger}\left( [A_{j}, A_{k}]\right)^{3}{\cal U}_{\Delta}\right\}\ .
\operatorname{lp}\int_{\rho>\epsilon}i\partial\Upsilon\wedge \overline{\partial}\Upsilon\wedge \omega^n=\operatorname{lp}\int_{\rho=\epsilon}i\Upsilon\overline{\partial}\Upsilon\wedge \omega^n+\operatorname{lp}\int_{\rho>\epsilon}({\rm cpt\supp})=0.
L ( A_{\mu}, \phi , \theta ) ={\frac{- d}{F ( \phi )}}G_{\mu \nu \rho \sigma}G^{\mu \nu \rho \sigma}-{\frac{\theta}{4}}\epsilon_{\mu \nu \rho \sigma}G^{\mu \nu \rho \sigma}-{\frac{1}{2}}\partial^{\mu}\phi \partial_{\mu}\phi -{\frac{1}{2}}m^{2}\phi^{2},
\begin{array}{rcll}-\Delta_{\xi}\ \Xi(\xi) = 0, \xi \in \Pi_1 , \partial_{\xi_1}\Xi(\xi) = 0, \xi \in \partial_{\parallel}\Pi_1, \partial_{\xi_2}\Xi(\xi_1,0) = 0, (\xi_1,0) \in \partial\Pi_1 \setminus \partial_{\parallel}\Pi_1,\end{array}
\phantom{=}\sup_n\int_{\Omega}|W(x)f'(u_n)u_n|dx &\leq \sum_{j=1,2}\|W\|_{L^{\frac{p_s^*}{p_s^*-q_j}}(\Omega)}\sup_n(\|u_n\|_{p_s^*}^{q_j-1}\|u_n\|_{p_s^*}) &\leq C\sum_{j=1,2}\|W\|_{L^{\frac{p_s^*}{p_s^*-q_j}}(\Omega)}
b e g i n{e q n a r r a y}- i ( \Sigma ( x , y ) )_{1 1}= \left[ - i ( \Sigma ( y , x ) )_{2 2}\right]^{*}, \; - i ( \Sigma ( x , y ) )_{1 2 ( 2 1 )}= \left[ - i ( \Sigma ( y , x ) )_{1 2 ( 2 1 )}\right]^{*}.
\begin{aligned}N_\lambda^2 = \L(P_{\lambda}g(u_{\ll \lambda}), \partial^2 u_{<\lambda})+ L(P_\lambda g(u_{\ll\lambda}), \partial^2 u_{\lambda}) +\sum_{\mu \gg \lambda}L( P_\mu g(u_{\ll\lambda}),\partial^2 u_\mu) := \N_\lambda^{2,lo}+ N_\lambda^{2,med}+ N_\lambda^{2,hi}.\end{aligned}
\sum_{m=\max\{\ell_1,\ell_2\}}^{\infty}\sum_{k=\max\{0,\ell_1+\ell_2-m\}}^{\min\{\ell_1,\ell_2\}}A_{m,k}=\sum_{k=0}^{\min\{\ell_1,\ell_2\}}\sum_{m=\ell_1+\ell_2-k}^{\infty}A_{m,k}.
u*\bar\vartheta_z (\tau\otimes a_X) = u*(\bar\partial_z\tau_1\otimes a_X,\, (\tau_1-\bar\partial_z\tau_{01})\otimes a_X)) &=( \bar\partial (u*(\tau_1\otimes a_X)),\, u*(\tau_1\otimes a_X) -\bar\partial (u*(\tau_{01}\otimes a_X)) &=\bar\vartheta(u*(\tau\otimes a_X)).
\begin{array}{l}f_{-}{\bf U}_{a+1}{\bf X}^{'a+1}_{a+2,a}+c_{1a}(u_{-})f_{+}{\bf X}^{a+1}_{a+2,a}{\bf U}_{a-1}^{'}+c_{2a}(u_{+})f_{-}{\bf X}^{a+1}_{a+2,a}{\bf D}^{'}_{a+1}\mbox{}=f_{+}{\bf X}^{'a+1}_{a+2,a}{\bf U}_{a-1}\end{array}
(b \cdot f)(p_1,\dots,p_r) \coloneqq f(b^{-1}p_1,p_2,\dots,p_r), (b \cdot \tilde{f})((p_{k,l})_{k,l}) \coloneqq \tilde{f}(b^{-1}p_{1,1}, p_{1,2},\dots,p_{1,N_1},\dots,p_{r,N_r}).
\left( \begin{array}{c c}{\{\Phi_{0}, \Phi_{0}\}}&{\{\Phi_{0}, \Phi_{1}\}}\\{\{\Phi_{1}, \Phi_{0}\}}&{\{\Phi_{1}, \Phi_{1}\}}\\ \end{array}\right) ={\cal D}\delta ( z - z^{\prime}) ={\cal D}\delta ( x - x^{\prime}) \delta ( \theta - \theta^{\prime})
y_\ell (r) = -\frac{(2\ell-1)!!}{r^{\ell+1}}\left[ \sum_{k=0}^N \frac{(2\ell-2k-1)!!}{k!}\frac{1}{(2\ell-1)!!}\left( \frac{r^2}{2}\right)^k+ \mathcal O\left(\frac{1}{\ell^{N+1}}\right)\right].
\|x^{k+1}_i - x^*_i\|^2 \leq \|x_i^k-\alpha_{k,i}F_i(x_i^k,N\hat v^k_i) - x^k_i - x^*_i + \alpha_{k,i}F_i(x^*_i,\bar x^*)\|^2 = \|x_i^k-x^*_i \|^2 + \alpha_{k,i}^2\| F_i(x_i^k,N\hat v^k_i) -F_i(x^*_i,\bar x^*) \|^2 \cr -2\alpha_{k,i}(F_i(x_i^k,N\hat v^k_i) -F_i(x^*_i,\bar x^*))^T(x^k_i-x^*_i).
S'_{\mathbf{t}}=\left\{\Psi_7\left(G_d+\frac{q}{2}x_{m-2}+\frac{q}{2}\mathbf{t}\cdot\mathbf{y})\right): d \in \{1,2\}, \mathbf{y}\in \mathbb{Z}_{2}^{2}\right\}.
(\frac{\mathrm{d}}{\mathrm{d}t}\exp(t Y)^* \sigma^* \exp(-tY)^* \omega_{\varphi})|_{t=0}= \sigma^* (\frac{\mathrm{d}}{\mathrm{d}t}\exp(-tY)^* \omega_{\varphi})|_{t=0}+ (\frac{\mathrm{d}}{\mathrm{d}t}\exp(t Y)^* \sigma^*\omega_{\varphi})|_{t=0}& = \sqrt{-1}\partial \bar{\partial}[( f+\bar{f}) - ( f+\bar{f})\circ \sigma]
F_{1}({\bf \Delta P}, \Delta E ) = \frac{({\bf \Delta P})^{2}}{4}+ \frac{1}{({\bf v}_{i}-{\bf v}_{f})^{2}}\left[ \Delta E - \frac{1}{2}({\bf v}_{i}-{\bf v}_{f}){\bf \Delta P}\right]^{2}
\begin{array}{lll}H =4p_q^Tp_r=(p_q^T,p_r^T)(2\mathcal{I})\left(\begin{array}{c}p_q p_r\end{array}\right)=(p_x^T,p_s^T)J^{-1}(2\mathcal{I})(J^{-1})^T\left(\begin{array}{c}p_x p_s\end{array}\right) &= p_s^Tp_s+p_x^T \mathcal{V}^{-1}p_x\,.\end{array}
\tilde g_{00}= \frac{g_{00}}{\left [ 1+ (1+g_{00})\sinh^{2}\alpha\right ]^{2}}= -\frac{\left ( 1-\frac{2\eta}{r}\right )^{\frac{m+\sigma}{\eta}}}{\left [ 1+ (1- (1-\frac{2\eta}{r})^{\frac{m+\sigma}{\eta}})\sinh^{2}\alpha \right ]^{2}}
\begin{aligned}\mathbb{C}\setminus[\pm\alpha,\pm\beta] \longrightarrow(0,1)\times[0,\tau)\subset\mathbb{C}\;, w \longmapsto\sigma(w)=\frac{1}{2K}\int_{1}^{w/\alpha}\mathrm{d}t\left((1-t^2)(1-(\alpha t/\beta)^2)\right)^{-1/2}\ .\end{aligned}
\sum_{i= -\infty}^k 2^{-i\frac{r}{p-r}}V_r(0, x_i)^{\frac{pr}{p-r}}\approx \sum_{i= -\infty}^k 2^{-i\frac{r}{p-r}}V_r(x_{i-1}, x_i)^{\frac{pr}{p-r}}
H (i, i_{1}, \dotsc, i_{r}, j) := \frac{(q^{1 - s})_{I_{r}+ j}(q^{1 - i}/ a)_{j}(q^{i - n_{1}})_{i_{1}}\dotsm (q^{i - n_{r}})_{i_{r}}}{(q^{i - m})_{1 - s + I_{r}+ j}(q)_{j}(q)_{i_{1}}\dotsm (q)_{i_{r}}}(b q^{i})^{j}\left(\frac{b q}{c_{1}}\right)^{i_{1}}\dotsm \left(\frac{b q}{c_{r}}\right)^{i_{r}},
\tilde{\varphi}( z , \bar{z}, u ) = \tilde{\varphi}_{0}( z , \bar{z}) + y ( z , \bar{z}, u ) = \tilde{\varphi}_{0}( z , \bar{z}) + u \sum_{n = 0}^{\infty}\frac{1}{n !}\Lambda_{n}( z , \bar{z}) u^{n}.
q(x,y^0|z) = \prod_{j=1}^N \left[\prod_{k=1}^{H_0}\left(b^0_{kj}\prod_{i=1}^M (a^0_{ik})^{x_{i}}\right)^{y^0_k}\right]^{z_j}, p(x,y|z,A,B) = \prod_{j=1}^N \left[\prod_{k=1}^{H}\left(b_{kj}\prod_{i=1}^M (a_{ik})^{x_{i}}\right)^{y_k}\right]^{z_j}.
\frac{|\Omega|}{|\Omega\setminus A_{t,M}|}=\frac{|\Omega|}{|\Omega|-|A_{t,M}|}\le \frac{1}{2},\mbox{if we choose}t=\left(\frac{2}{|\Omega|}\right)^{1/r}\,\|\phi\|_{L^r(\Omega)}.
K ( x_{1}, x_{2}, x_{3}) = \frac{1}{\pi^{2}\det \Theta}\, \exp \left( - 2 i \bigl [ x_{1}^{\alpha}( \Theta^{- 1})_{\alpha \beta}x_{2}^{\beta}+ x_{2}^{\alpha}( \Theta^{- 1})_{\alpha \beta}x_{3}^{\beta}+ x_{3}^{\alpha}( \Theta^{- 1})_{\alpha \beta}x_{1}^{\beta}\bigr ] \right) ,
\sum^{2N-1}_{n=1}\frac{1}{\sin^2\frac{n\pi}{2N}}=\frac{4N^2}{3}-\frac{1}{3}&=\frac{1}{\sin^2\frac{\pi}{2N}}+\frac{1}{\sin^2\frac{2\pi}{2N}}+\cdots+\frac{1}{\sin^2\frac{(N-1)\pi}{2N}}+\frac{1}{\sin^2\frac{N\pi}{2N}}&\ \ \ +\frac{1}{\sin^2\frac{(N+1)\pi}{2N}}+\cdots+\frac{1}{\sin^2\frac{(2N-2)\pi}{2N}}+\frac{1}{\sin^2\frac{(2N-1)\pi}{2N}}\,.
\chi(\tilde Z_{\mathbf e; r}(M)) = \sum_{j\in J}\chi(\pi^{-1}(Z_j)) = \sum_{j\in J}\binom{m_k-r}{e_k-r}\chi(Z_j) = \binom{m_k-r}{e_k-r}\chi(Z_{\mathbf e'; r}(M)).
\xi_k=\frac{1}{2}\dot{\tau}a_{kl}x_l+\rho_k(t), \phi(x,t)=-\frac{1}{8}\ddot{\tau}|x|^2-\frac{1}{2}\dot{\rho}_k(t)a_{kl}x_l+\sigma(t)-\frac{n}{4}\dot{\tau}.
q_{\rho_{N}}\left( x,t\right) =q\left( x,t\right) & +2\left[ \int\psi_{\rho_{N}}\left( x,t;\mathrm{i}s\right) \psi\left(x,t;\mathrm{i}s\right) \mathrm{d}\rho_{N}\left( s\right) \right]^{2}& +4\int\psi_{\rho_{N}}\left( x,t;\mathrm{i}s\right) \psi^{\prime}\left(x,t;\mathrm{i}s\right) \mathrm{d}\rho_{N}\left( s\right) ,
h_t(\Phi_i) =\lim_{k \to \infty}\frac{1}{k}\log P_t(\Phi_i^k G)=\lim_{k \to \infty}\frac{1}{k}\log e^{k(1-N)t}e^{-k(1-N)t}P_t(\Phi_i^k G) &=(1-N)t+ \lim_{k \to \infty}\frac{1}{k}\log e^{-k(1-N)t}P_t(\Phi_i^k G)=(1-N)t
\int_{\Omega}\int_{|u| \geq M}\frac{F(y,u)}{|x-y|^{\mu}}F(x,u) dy ~dx=o(M), \;\int_{\Omega}\int_{|u_k| \geq M}\frac{F(y,u_k)}{|x-y|^{\mu}}F(x,u_k) dy ~dx = o(M),
Z(p,\chi_{},0) =p^{-n}\sum_{I\subset T}c_{I,\chi_{}}^{0}\prod_{i\in I}\frac{(p-1)p^{-\nu_{i}}}{1-p^{-\nu_{i}}}; Z(p,\chi_{},s) =p^{-n}\sum_{I\subset T}c_{I,\chi_{}}^{0}\prod_{i\in I}\frac{(p-1)t^{N_{i}}p^{-\nu_{i}}}{1-t^{N_{i}}p^{-\nu_{i}}}.
A ( \omega ) = \left( \begin{array}{c c c}{\alpha ( \omega )}&{\beta ( \omega )}&{\lambda ( \omega )}\\{j^{2}\lambda ( \omega )}&{\alpha ( \omega )}&{j \beta ( \omega )}\\{j^{2}\beta ( \omega )}&{j \lambda ( \omega )}&{\alpha ( \omega )}\\ \end{array}\right)
\mathbb{E}^{t}_{x,y}\left[\left( \int_{0}^{t}V(X_s)ds\right)^k\right] =t^{k}\mathbb{E}^{t}_{x,y}\left[\left( \int_{0}^{1}V(X_{ts})ds\right)^k\right] =k!t^{k}\mathbb{E}^{t}_{x,y}\left[\int_{I_{k}}V(X_{t\lambda_1})...V(X_{t\lambda_{k}})d\lambda^{(k)}\right].
x_{1}=f_{1,\beta}\left( x_{0}\right) =\beta\tan\left( x_{0}\right) x_{2}=f_{2,\beta}\left( x_{1}\right) =-\beta\tanh\left( x_{1}\right) x_{3}=f_{1,\beta}\left( x_{2}\right) x_{4}=f_{2,\beta}\left( x_{3}\right) . & \vdots
\frac{1}{4}\left[k^2-G^{-2}(k)\right]+\frac{\pi^2}{g^2}\left[ 2zk^{-2}G^{-3}(k)-4\pi^2\frac{\delta z}{\delta G(k)}\int \frac{d^2p}{(2\pi)^2}p^{-2}G^{-2}(p) \right]=0
\mathcal{H}^{p,r}_{\alpha}\left(I\right)=\{u\in L^p(I);~~~ x^{(\alpha+k+1/2)}\left(\frac{1}{x}\frac{d}{dx}\right)^{(k)}\left(x^{-(\alpha+1/2)}u\right)\in L^p(I)~\,,\forall\, 0\leq k\leq r\}.
S_{br}=\frac{1}{2k^{2}}\int d^{9}x \sqrt{-g^{0}}V_{0}(\Phi)\Bigg|_{y=0}+\frac{1}{2k^{2}}\int d^{9}x \sqrt{-g^{\infty}}V_{\infty}(\Phi)\Bigg|_{y=\infty}\ ,
E_{\omega_1}(q,t,T)f^{B_2}(x_1,x_2;s_1,s_2,q,t,T)=t T^{1/2}(s_1+s_2+s_1^{-1}+s_2^{-1})f^{B_2}(x_1,x_2;s_1,s_2,q,t,T).
\gamma\big((n_E)_{E\in \mathbf{E}}, (s_i)_{1\le i\le r}, (\ell_{E,v})_{v\in E\in \mathbf{E}}\big)={}\big((s_i)_{1\le i\le r},(\ell_{E,v_E}+\ell_{E,v'_E})_{E\in \mathbf{E}}\big), \delta\big( (n_{E,v})_{v\in E\in \mathbf{E}}\big) ={}\big(n_{E,v_E}-n_{E,v'_E}\big)_{E\in \mathbf{E}}.
F_{z}(x,t)=-G\Sigma\int_{-\infty}^{\infty}dy^{\prime}\int_{-\infty}^{\infty}\frac{[h(x,t)-h(x^{\prime},t)]}{[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}]^{3/2}}dx^{\prime}=-2\pi G\Sigma kh(x,t)
W\circ T (f\oplus \gamma_A\oplus \gamma_B)(a_1\otimes b_1\oplus a_2\oplus b_2) = b_{T((f\oplus \gamma_A\oplus \gamma_B)}(a_1\otimes b_1)\oplus \gamma_A(a_2)\oplus \gamma_B(b_2) &= b_{T((f\oplus 0\oplus 0)}(a_1\otimes b_1)\oplus \gamma_A(a_2)\oplus \gamma_B(b_2) &= f(a_1\otimes b_1)\oplus \gamma_A(a_2)\oplus \gamma_B(b_2).
\mathrm{Res}_Z(\Phi_{n-1}(f)d\log(1+\pi_L h)) = \mathrm{Res}_Z(\Phi_{n-1}(g(Z'))d\log(Z'))-\mathrm{Res}_Z(\Phi_{n-1}(f(Z))d\log(Z)) = \mathrm{Res}_{Z'}(\Phi_{n-1}(g(Z'))d\log(Z'))-\mathrm{Res}_Z(\Phi_{n-1}(f(Z))d\log(Z)) = \mathrm{Res}_{Z}([\Phi_{n-1}(g(Z))-\Phi_{n-1}(f(Z))]d\log(Z)) \ .
\psi ( x_{1}, x_{2}) = f \left( \frac{x_{1}}{\sqrt{\gamma}}+ i \sqrt{\gamma}x_{2}\right) e^{- \frac{1}{\theta}\left( \left( \frac{x_{1}^{2}}{\gamma}+ \gamma x_{2}^{2}\right) - z \left( \frac{x_{1}}{\sqrt{\gamma}}- i \sqrt{\gamma}x_{2}\right) \right)}
\widetilde{\Pi}_{x}\left[\sum_{\tau_{i}\le t}f(\tau_{i},\widetilde{\xi}_{\tau_{i}-},\widetilde{\xi}_{\tau_{i}})\right]=\widetilde{\Pi}_{x}\left[\int_{0}^{t}{\rm d}s\int_{E}\pi^{h}(\widetilde{\xi}_{s},{\rm d}y)q(\widetilde{\xi}_{s})f(s,\widetilde{\xi}_{s},y)\right]
p_{n_c,n_d,n_e}=\left\langle{T_{1}^{k}T_{2}^{\ell}v_{0}, \prod_{i=1}^{n_c}\psi_{1}^{+}(x_{i})\psi_{0}^{-}(x_{i}) \prod_{i=1}^{n_d}\psi_{2}^{+}(y_{i})\psi_{0}^{-}(y_{i}) \prod_{i=1}^{n_e}\psi_{2}^{+}(z_{i})\psi_{1}^{-}(z_{i})v_{0}}\right\rangle,
h <\frac{3.4753}{4}\left(C\cdot G(a_1')\cdot G(a_2')(1.001+3\cdot 10^{-4}C^{-1})+\frac{\log B_3}{\log 10^5 (\log 10^{5/2})^3}\right)\cdot &\cdot \left(h'+\frac{\lambda'}{\sigma}\right)^2\log^2 \alpha_2 \log c &=:G(h2)\left(h'+\frac{\lambda'}{\sigma}\right)^2 \log^2 \alpha_2\log c.
\theta (t)=\begin{cases}0, \mbox{if $b_{2n-2}\leq t <a_{2n-1}$}, \dfrac{(2n-1)(t-a_{2n-1})}{b_{2n-1}-a_{2n-1}}, \mbox{if $a_{2n-1}\leq t <b_{2n-1}$}, 2n-1, \mbox{if $b_{2n-1}\leq t <a_{2n}$,}\dfrac{(2n-1)(b_{2n}-t)}{b_{2n}-a_{2n}}, \mbox{if $a_{2n}\leq t <b_{2n}$}.\end{cases}
\div \nabla_{J(\,\cdot\,)}\beta = - \nabla_{e_1}\nabla_{J(e_1)}\beta - \nabla_{e_2}\nabla_{J(e_2)}\beta = - \nabla_{e_1}\nabla_{e_2}\beta + \nabla_{e_2}\nabla_{e_1}\beta= -R(e_1,e_2) \beta.
L^{\pm}(u)^{-1}= \begin{pmatrix}1 0 -e_{1}^{\pm}(u) \ddots \vdots \ddots * \ldots -e_{n-1}^{\pm}(u) 1\end{pmatrix}\begin{pmatrix}k_{1}^{\pm}(u)^{-1}0 &\ddots & \ddots 0 k_{n}^{\pm}(u)^{-1}\end{pmatrix}&\begin{pmatrix}1 -f_{1}^{\pm}(u) \ldots * &\ddots & \ddots -f_{n-1}^{\pm}(u) 0 1\end{pmatrix}
\sum_{i=1}^n \sum_{\substack{0 \le h_1 \le K+1-r 0 \le h_2 \le r-1 h_1 + h_2 = h}}A_{ij,h_1}y_{i, h_2+1}= \mathbf{b}_{j, h}+ \alpha_{j, h+1}t - \alpha_{j, h},
\alpha\circ P_d(L_m)=\alpha(f(m+d)L_{m+d})=f(m+d)\alpha(L_{m+d})=f(m+d)(1+q^{m+d})L_{m+d}, & P_d\circ \alpha(L_m)=P_d( \alpha(L_m))=P_d((1+q^{m})L_m)=(1+q^{m})P_d(L_m)=(1+q^{m})f(m+d)L_{m+d}, & \alpha\circ P_d(L_m)=P_d\circ \alpha(L_m) Leftrightarrow\forall\m\in\mathbb{Z}:\f(m+d)(q^{d}-1)=0,
\Big|\frac{d}{dt}A_\Phi(\gamma(t),y,z)\Big| &\lesssim \frac{\big( |\gamma(t)-y|+|\gamma(t)-z|+ \Phi(\gamma(t))+\Phi(y) +\Phi(z)\big)^{6\beta-1}|x'-x|}{\big((|\gamma(t)-y|+|\gamma(t)-z|)^{3\beta}+ \Phi(\gamma(t))^{3\beta}+\Phi(y)^{3\beta}+\Phi(z)^{3\beta}\big)^2}&\sim \frac{|x'-x|}{|x-y|+|x-z|+ \Phi(x)+\Phi(y) +\Phi(z)}.
\phantom{=}\;\;\binom{-n+1}{k}^p (-1)^{-kp-p}(L_{k-n}^p-\delta_{p\mid(k-n)}L_{kp-np}) =\binom{-n+1}{k}(-1)^{-k-1}(L_{k-n}^p-\delta_{p\mid(k-n)}L_{kp-np}).
J(g) = \sum_{r \sim R}[r,g]^{-1+\epsilon}\sideset{}{^*}\sum_{\chi \!\! \!\! \mod r}\max_{|\beta| \le P^{-1.99}}\bigg| \sum_{m \in \mathcal I}\rho_j(m)\chi(m)e( \beta m^2 ) \bigg|,
\{\begin{matrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&0\end{matrix}\}=\frac{\delta_{j_{2},j_{4}}\delta_{j_{1},j_{5}}}{\sqrt{(2j_{1}+1)(2j_{2}+1)}}(-1)^{j_{1}+j_{2}+j_{3}}\{\begin{matrix}j_{1}&j_{2}&j_{3}\end{matrix}\}.
\begin{cases}\|\Gamma_{\chi_{t},b,t,N,\omega}^{B,j}(x,\tau)\|_{L^{2}(\Omega_{t}(\omega))}\leq c\tau^{|\alpha|-3/2}e^{-\tau(s-t)a_{B}}, \|r_{\chi_{t},b,t,N,\omega}^{B,j}(x,\tau)\|_{L^{2}(\Omega_{t}(\omega))}\leq c\tau^{j-N+1/2},\end{cases}
f(z_1,\dots,z_{i-1},z_i+1,z_{i+1},\dots,z_n)= f(z_1,\dots,z_n), f(z_1,\dots,z_{i-1},z_i+\tau,z_{i+1},\dots,z_n) = C_i e \bigg({-}\sum_j Q_{ij}z_j\bigg) f(z_1,\dots,z_n),
\mathcal{G}[\varepsilon,W(\tau)+h(\tau)]= [W'(\tau)+h'(\tau)-K(\varepsilon,\tau,W+h),W(0)+h(0),W(t_1)+h(t_1)] = \mathcal{G}[\varepsilon,W(\tau)+h(\tau)]+[A_{\varepsilon}h,h(0),h(t_1)]+o(h),
\sum_{k}\sum_{m=0}^\infty \sum_{|k_0-\ell_0|>N}\cdots \sum_{|k_m-\ell_m|>N}2^{-|k-k_0|(\varepsilon/2-|\alpha|)}2^{-|k_0-\ell_0|(\varepsilon/2-|\alpha|)}&\hskip 5cm \times 2^{-|\ell_0-k_1|(\varepsilon/2-|\alpha|)}\cdots2^{-|k_m-\ell_m|(\varepsilon/2-|\alpha|)}2^{-|\ell_m-k'-j|(\varepsilon/2-|\alpha|)}&\le C2^{-N(\varepsilon/2 -|\alpha|)}.
-\alpha ( S(1) ) X\overset{\eqref{form:3}}{=}\alpha ( S(X) ) \overset{\eqref{form:2}}{=}\alpha \big({-}X^{2}S(1) ) \overset{\, \alpha}{=}-\alpha \big( X^{2}\big) \alpha ( S(1) ) \overset{\,\alpha}{=}-X^{4}\alpha ( S (1) ),
z = V \int_{0}^{\infty}d p_{n}e^{- \frac 1 2 \beta c \left( p_{n}+ \frac{m^{2}c^{2}}{p_{n}}\right)}\prod_{k = 1}^{n - 1}\int_{- \infty}^{\infty}d p_{k}e^{- \frac{\beta c}{2 p_{n}}p_{k}^{2}}.
\lim_{a \to 0, \; b/\sqrt{a}\to k}Q_n\left(\frac{x}{a}\right) &=(-1)^n\Gamma(\mu+\nu+1+n)\sum_{j=0}^n\frac{(-n)_j}{j!}\frac{x^{\mu+j}}{\Gamma(\mu+\nu+1+j)\Gamma(\mu+1+j)}k^{2j}&=(-1)^n\frac{(\mu+\nu+1)_n}{\Gamma(\mu+1)}{\;}_1 F_2 \left({-n \atop \mu+\nu+1, \mu+1}\Big{|}k^2x \right)x^\mu,
\frac{u_h^{n+1}- 2 u_h^n + u_h^{n-1}}{\tau^2}= \frac{v_h^{n+1}+ v_h^n}{2 \tau}- \frac{v_h^{n}+ v_h^{n-1}}{2 \tau}= & = \frac{1}{2}\left( \Delta_h\, \left(\frac{u_h^{n+1}+ 2 u_h^n + u_h^{n-1}}{2}\right) - \left( f'\left(u_h^{n+\frac{1}{2}}\right) + f'\left(u_h^{n-\frac{1}{2}}\right)\right) \right)
\begin{aligned}u_{\xi\xi}= u_{\xi'\xi'}+ 2 g(\xi, \nu) u_{\xi'\nu}+ \big(g(\xi, \nu)\big)^2 u_{\nu\nu}&= u_{\xi'\xi'}+ 2 g(\xi, \nu) \left( \nabla_{\xi'}(u_\nu) - \nabla_{\nabla_{\xi'}\nu}u \right) + (1 - |\xi'|^2) u_{\nu\nu}, \end{aligned}
\prod_{i=1}^k \prod\limits_{e \in E(\tau_i)}f_e \prod\limits_{\varepsilon \in E(R(\tau_i)) \setminus E(\tau_i)}(1+f_{\varepsilon}) = \prod\limits_{e \in E(\tau)}f_e \prod\limits_{\varepsilon \in E(R(\tau)) \setminus E(\tau)}(1+f_{\varepsilon}).
u \circ v = \sum_{s = 0}^{\infty}{\frac{\kappa^{2 s}}{2^{2 s}( 2 s + 1 ) !}}\omega^{i_{1}j_{1}}\ldots \omega^{i_{2 s}j_{2 s}}{\frac{\partial^{2 s}u}{\partial x^{i_{1}}\ldots \partial x^{i_{2 s}}}}{\frac{\partial^{2 s}v}{\partial x^{j_{1}}\ldots \partial x^{j_{2 s}}}}.
x_i : = \frac{\langle u, \tilde{x}_i \rangle - \langle u, x^*\rangle}{\langle u, x_0 \rangle - \langle u, x^*\rangle}x_0 +\frac{\langle u, x_0 \rangle - \langle u, \tilde{x}_i \rangle}{\langle u,x_0 \rangle - \langle u, x^*\rangle}x^*
F(m;z)\ |\T(l)=\sum_{n=0}^\infty\left( b_3\left( \frac{mln-1}{12}\right)+\left(\frac{3}{l}\right)l^{3m-4}b_3\left( \frac{mn-l}{12l}\right) \right)q^n\equiv0\ (\mathrm{mod}\m).
\mathbb{E}\left[\sum_{j=1}^{d_1}\left( \int\left\langle\frac{p(x_1,\cdot)}{p_1(x_1)p_2(\cdot)},(1-\hat{\Pi}_{V_2})f_2\right\rangle_n\phi_j(x_1)p_1(x_1)d\nu_1(x_1) \right)^2\right] &\leq \int\mathbb{E}\left[\left\langle\frac{p(x_1,\cdot)}{p_1(x_1)p_2(\cdot)},(1-\hat{\Pi}_{V_2})f_2\right\rangle_n^2 \right]p_1(x_1)d\nu_1(x_1),
\quad\Pr\left(>\theta|\Phi,\Phi(\mathcal{A})>0 \right) \approx \sum_{m=1}^{M}C_{M}^m(-1)^{m+1}e^{-m\eta\theta r_1^{\alpha}/\rho}\!\!\!\!\!\!\!\!\!\!\prod\limits_{x_{i}\in\Phi\backslash\{x_1\}\cap{\mathcal{A}}}\frac{1}{\left(1+\frac{m\eta\theta r_1^{\alpha}}{Mr_i^{\alpha}}\right)^M}.
\left(1+2^{-j}\frac{1}{m+1}\eta\sqrt{\eta^2+2m+2}+2^{-2j}\frac{\eta^2+2m+2}{2m+2}\right)^{-\frac{1}{2}}=1-2^{-j}\frac{\eta\sqrt{\eta^2+2m+2}}{2m+2}-2^{-2j-1}\frac{\eta^2+2m+2}{2m+2}+r,
|u(x,t)|\leq C\left( 1+D+\left( \frac{|\sigma |(\Omega )+|\mu |(\Omega_{T})}{D^{N}}\right)^{m_3}+\mathbb{I}_{2}^{2D}\left[ |\sigma |\otimes \delta_{\{t=0\}}+|\mu |\right](x,t) \right)
\mathcal{H}': \begin{cases}\dot{x}'~~\in F(x') x'\in \mathcal{F}_c':=\{x'\in \mathcal{S}: (\hat{g},\hat{b}_a)\in \mathcal{F}_o\}x'^+\in G(x') x'\in \mathcal{J}_c':=\{x'\in \mathcal{S}: (\hat{g},\hat{b}_a)\in \mathcal{J}_o\}\end{cases}
\Gamma(z)^\ast=\Gamma(\bar{z}),\qquad\forall\,z\in\rho(\mathcal{H}_0), \Gamma(z)-\Gamma(w)=(z-w)\mspace{1.5mu}G(\bar{z})^\ast G(w),\qquad\forall\,w,z\in\rho(\mathcal{H}_0), \forall\,z \in\mathbb{C}: \;\Re(z)<-\lambda_0\:\lor\: \Im(z)>0,0\in\rho(\Gamma(z)).
\omega^\mathrm{h}_{0k_l}= \mathop{\arg\max}\limits_{\omega \in \left[0,2\pi \right)}\frac{1}{M_{\mathrm{h}}}\left[ g\left(\omega\right) + g^*\left(\omega\right)\right] = \mathop{\arg\max}\limits_{\omega \in \left[0,2\pi \right)}\left\lbrace \Re\left[g\left(\omega\right) \right] \right\rbrace.
\left(\begin{array}{c}\frac{1}{p^T Hp}p^TH \frac{1}{p^T Bp}p^TB \end{array}\right)= \frac{1}{p^TBp}\left(\begin{array}{cc}-1 \ 1 -1 \ 0\end{array}\right)\left(\begin{array}{c}g_{k-1}^T g_k^T\end{array}\right).
S^{\pm}[ x ] \simeq S_{\mathrm{q c}}[ x_{\mathrm{q c}}] \mp \hbar \varphi [ x_{\mathrm{q c}}] + \frac{1}{2}\int_{0}^{t}\mathrm{d}\tau \left[ \dot{\eta}^{2}- \left( \Phi^{2}\right)^{\prime \prime}\Bigl ( x_{\mathrm{q c}}( \tau ) \Bigr ) \eta^{2}\right] .
H_{+}(t, x) = H_{+}({\alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}}; t, x) = \frac{\textit{Re}}{\pi}\int_{0}^{\infty}\hskip -2mm e^{-i x \omega}\exp\left(\!- t \omega^{\alpha_{1}}e^{\frac{i\pi}{2}\beta_{1}}- t \omega^{\alpha_{2}}e^{\frac{i\pi}{2}\beta_{2}}\right) d\omega,
\beta=\int_{\partial\Omega}\nu\cdot u\,d\sigma=0, \int_{\partial\Omega}y^\perp\cdot\left\{(u\otimes y^\perp)\nu\right\}\,d\sigma_y=a\!\int_{\partial\Omega}(\nu\cdot y^\perp)|y|^2\,d\sigma_y=0,
\frac{(\nabla\rho_{\epsilon})^{2}}{\rho_{\epsilon}}\le\left(\frac{(\nabla\rho)^{2}}{\rho}\right)_{\epsilon}(b^{\epsilon})^{2}\rho_{\epsilon}=\frac{(b\rho)_{\epsilon}^{2}}{\rho_{\epsilon}}\le(b^{2}\rho)_{\epsilon}
\beta_{q,p}=\left(\frac{1}{C_{2}(\vec{\varepsilon})}-\frac{(3p-1)!!}{6}\varepsilon_{p}^{3}\sigma^{3p}-\frac{(2q+p-1)!!}{2}\varepsilon_{q}^{2}\varepsilon_{p}\sigma^{2q+p}\right)^{-1}.
\frac{\partial}{\partial t}h_{ij}=\lambda\left(\begin{array}{l}\Delta h_{ij}-2Hh_{il}h^l_{\j}+ h_{ij}(|A|^2+R_{\nu\nu}) begin{align*}2mm]-h_{jl}R^{l\ \ \m}_{\mi}-h_{il}R^{l\ \ \m}_{\mj}+2h_{lm}R^{l\m}_{\i\j}+\nabla_j R_{\nu li}^{\ \ \l}+\nabla_l R^{\ \ \l}_{\nu ij}\end{array}\right) &\quad+H\lambda_{,ij}+H_{,i}\lambda_{,j}+H_{,j}\lambda_{,i}.
\sum_{k = -\infty}^\infty k^{2(m+1)}|c_k(f)|^2 \leq \sum_{\substack{k = -\infty k \neq 0}}^\infty k^{2m+2}\frac{C}{|k|^{2+2m+2\mu}}= \sum_{\substack{k = -\infty k \neq 0}}^\infty \frac{C}{|k|^{2\mu}}< \infty.
\|\, D \mbox{\boldmath $u$}(\cdot,t) \,\|_{\mbox{}_{\scriptstyle L^{q}(\mathbb{R}^{3})}}\;\!=\;\Bigl\{\,\sum_{i, \,j \,=\,1}^{3}\int_{\mathbb{R}^{3}}\!|\, D_{\!\;\!j}\;\!u_{i}(x,t)\,|^{q}\,dx\,\Bigr\}^{\!\!\:\!1/q}
\det(\bold{C}) = \left( \beta^2-\eta^2 \right)^{n(n-1)}\left(\alpha\,\gamma-\alpha\,\tau-{\delta}^{2}-2\,\delta\,\nu+\gamma\,\mu-\mu\,\tau-{\nu}^{2}\right)^n \left( \alpha\,\gamma+\alpha\,\tau-{\delta}^{2}+2\,\delta\,\nu-\gamma\,\mu-\mu\,\tau-{\nu}^{2}\right)^n.
\partial_{\mu,x}\Psi_{SV}= \partial_{\mu,x}(\Psi_{S}(x) \times\Psi_{V}(x))=\partial_{\mu,x}\Psi_{S}(x)\times\Psi_{V}(x)+\Psi_{S}(x)\times\partial_{\mu,x}\Psi_{V}(x)
\Delta^{*}: H^{2}(BU_n;\mathbb{Z})\rightarrow H^{2}(BU_2;\mathbb{Z}), c_1\mapsto \frac{n}{2}c_1,c_2\mapsto \frac{n}{2}c_{2}+{{n/2}\choose{2}}c_1^2=\frac{n}{2}c_{2}+\frac{n(n-2)}{8}c_1^2.
\tilde{a}^{\alpha}_{\tilde{n}}=\int\limits_{0}^{\pi}~\frac{d\sigma_{-}}{\pi}\exp\left[4i\tilde{n}\frac{e_{\mu}X^{\mu}_{R}(\sigma_{-})}{e_{\mu}P^{\mu}_{R}}\right] \xi^{\alpha}_{i}\partial_{-}X^{i}_{R}(\sigma_{-}) \,,
{\omega}_{2}({\tau}^{(1)},{\tau}^{(2)}) \equiv- \frac{i}{4\pi}(1-{\rm N}^2) (\frac{e_{+}{\rm L}}{2{\pi}{\hbar}})^2 \sum_{p>0}p ({\tau}_{-}^{(1)}{\tau}_{+}^{(2)}-{\tau}_{+}^{(1)}{\tau}_{-}^{(2)})