harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
p\varphi^{-1}(S^{(n)}_{\xi}) \varphi^{-1}(S^{(m)}_{\eta}) p = p\sum_{k=0}^{\infty}S^{(k)}_{\sum_{\ell =0}^kU^X_{\ell,k - \ell}(\xi_{\ell}\otimes \eta_{k-\ell})}p & = \sum_{k=0}^{\infty}S^{(k)}_{\sum_{\ell =0}^kU^X_{\ell,k - \ell}(p \xi_{\ell}p \otimes p \eta_{k-\ell}p)}= p\varphi^{-1}(S^{(n)}_{\xi}) p \varphi^{-1}(S^{(m)}_{\eta}) p.
I_P = \bigl( a_0C_1, a_0C_3, a_0C_5, a_0C_7,\: b_0C_1, b_0C_3, b_0C_5, b_0C_7, \: a_1 C_0,\ldots a_0a_1, a_0b_1, b_0a_1, b_0b_1, C_0C_1, C_0C_2, \ldots, C_6C_7, a_0b_0 -(C_0 + C_2 + C_4 + C_6), a_1b_1 - (C_1 + C_3 + C_5 + C_7) \bigr).
D_{j}(\lambda) = \left( \frac{(\sqrt{\lambda}-i \sqrt{|\lambda_{j-1}|})(\sqrt{\lambda}+i \sqrt{|\lambda_{j}|})}{(\sqrt{\lambda}-i \sqrt{|\lambda_{j}|})(\sqrt{\lambda}+i \sqrt{|\lambda_{j-1}|})}\right)^{\frac{\log s_{j}}{2\pi i}}.
I({\cal M}) ={\int}_{\cal M}{\varepsilon}^{{\mu}_1{\mu}_2{\mu}_3}{\varepsilon}^{{\nu}_1{\nu}_2{\nu}_3}{\omega}^{(1)}_{{\mu}_1}(x){\omega}^{(1)}_{{\nu}_1}(y){\omega}^{(2)}_{{\mu}_2}(x){\omega}^{(2)}_{{\nu}_2}(y) G^{(\chi)}_{{\mu}_3{\nu}_3}(x, y) d^3 x d^3 y,
\Delta ( p^{2}) \, = \, \frac{\Gamma ( 3 - \omega )}{( 4 \pi )^{\omega}}\! \int_{0}^{1}\! \! d x \! \int_{0}^{1}\! \! \! d y \, \frac{[ x y ( 1 - y ) ]^{1 - \omega / 2}}{[ y ( 1 - y ) ( x ( 1 - x ) p^{2}- ( 1 - x ) m^{2}) - x m^{2}]^{3 - \omega}}
|\delta\mathcal{Y}_0^{v}(T)-L| =\lim_{T'\rightarrow\infty}|\delta\mathcal{Y}_0^{v}(T)-\delta\mathcal{Y}_0^{v}(T')| &=\lim_{T'\rightarrow\infty}\left|\delta\mathcal{Y}_0^{v}(T)-\mathbb{E}^{\mathbb{Q}}\left[h^{\alpha^{m(T')}_{T'}}(V_{T'}^v)-\mathbf{y}^{\alpha^{m(T')}_{T'}}(V_{T'}^{v})\right]\right|,
A_{0}^{3}(\alpha^{\prime}\rightarrow 0)=2g_{d}\,\,\varepsilon^{(1)}_{\lambda}\varepsilon^{(2)}_{\mu}\varepsilon^{(3)}_{\nu}\left\{\eta^{\lambda \mu}\left( p_{1}^{\nu}-p_{2}^{\nu}\right) +\eta^{\lambda \nu}\left(p_{3}^{\mu}-p_{1}^{\mu}\right) +\eta^{\mu \nu}\left( p_{2}^{\lambda}-p_{3}^{\lambda}\right) \right\}.
x\mapsto x^{11},\y\mapsto x^3 y\Longrightarrow\mathcal{I}_{1,1,\beta_1,\beta_2}=\!44\!\!\int_{\mathcal{D}}\!\hat{f}_{t,s}(x,y)^s x^{4(5\beta_1+3\beta_2+24s)}y^{3\beta_1+4\beta_2+21s}\frac{dx}{x}\!\frac{dy}{y},
\begin{array}{l}\underset{\epsilon \rightarrow 0}{\lim}\,\underset{\eta \rightarrow 0}{\lim}\,\left\vert \sum_{k=1}^{n}\left[ \breve{F}_{\epsilon}(y_{k})-\breve{F}_{\epsilon}(y_{k-1})\right] -\sum_{k=1}^{n}y_{k}^{\prime}\left[F_{\epsilon}(y_{k})-F_{\epsilon}(y_{k-1})\right] \right\vert =0.\end{array}
\underset{n=1}{\overset{n=N}{\sum}}\frac{n^{-s}}{x_0^{1-s}}-\frac{1}{1-s}-\frac{1}{\Gamma (s)}.\underset{K\rightarrow \infty}{lim}\underset{k=0}{\overset{K}{\sum}}\beta_{2.k+1}.\Gamma (s+2.k+1)=0
h_k'=\sum\limits_{\substack{|I|=k}}\prod\limits_{\substack{i\in I j\notin I}}\sqrt{\phi(z_j-z_i)}\prod_{i\in I}e^{-v_i/c}\prod\limits_{\substack{i\in I j\notin I}}\sqrt{\phi(z_i-z_j)}k=1,\dots,N\,,
\frac{\partial^2 \varphi_R}{\partial x_i \partial x_j}(x)= (\Delta^2 \varphi_R)(x)= \frac{\partial^2 \varphi_R}{\partial x_i^2}(x) -2 =2N-\Delta\varphi_R(x) =0, 2x- \nabla \varphi_R (x)=0,
\mathrm{E}[e_{k}^{(r/2)}(\chi_{r}^{2}(\mathbf{x}))~e_{m}^{(r/2)}(\chi_{r}^{2}(\mathbf{y}))]=\delta_{m,k}~B^{2m}(\left\Vert \mathbf{x}-\mathbf{y}\right\Vert),
\|h\|_{C^{0,\alpha}}:= \| h \|_{L^\infty}+ \sup_{x\neq y \in \mathbb R^d}\frac{|h(x)-h(y)|}{|x-y|^\alpha}& \|h\|_{C^{1,\alpha}}:= \| h \|_{L^\infty}+ \| \nabla h \|_{L^\infty}+ \sup_{x\neq y \in \mathbb R^d}\frac{|\nabla h(x)-\nabla h(y)|}{|x-y|^\alpha},
\psi_{15}(m) = \sum_{p_1,p_2,p_3}\bigg\{\sum_{p_2p_3 \ge V}+ \sum_{p_2p_3 < V}\bigg\}\psi(m/(p_1p_2p_3), p_3) = \psi_{17}(m) + \sum_{\substack{p_1,p_2,p_3 p_2p_3 < V}}\bigg\{\psi(m/(p_1p_2p_3),z) - \sum_{z \le p_4 < p_3}\psi(m/(p_1 \cdots p_4), p_4) \bigg\}= \psi_{17}(m) + \psi_{18}(m) - \psi_{19}(m), .
u_{\varepsilon, 1}(z,t) =-\left(\begin{array}{c}K_{1}(\varepsilon)x+\varepsilon K_{2}(\varepsilon)y +H_{3}h_{10}(t)+\varepsilon H_{1}h_{2 0}(t) (1/\varepsilon)\Big[ \left( P_{20}^{*}\right)^{T}x+P_{30}^{*}y+h_{2 0}(t)\Big]\end{array}\right)
\limsup_{k\to \infty}\bigg\|(\Phi_{R}u_{k})^{\frac{\alpha(\cdot)}{q(\cdot)}}(\Phi_{R}v_{k})^{\frac{\beta(\cdot)}{q(\cdot)}}\bigg\|_{L^{q(\cdot)}(\mathbb{R}^{N})}\geqslant \bigg( \frac{\nu_\infty}{4}\bigg)^{\frac{1}{q_{\infty}-\overline{\epsilon}}}
v_1=\Big(\frac{1}{2}+ \frac{-m}{n(1-\rho)}, \frac{1}{2}+ \frac{m}{n(1-\rho)}\Big), v_2 =\Big(\frac{1}{2}+ \frac{-m}{n(1-\rho)}, 1/2\Big), &v_3=\Big(\frac{1}{2}, \frac{1}{2}+ \frac{-m}{n(1-\rho)}\Big), v_4=\Big(\frac{1}{2}+ \frac{m}{n(1-\rho)}, \frac{1}{2}+ \frac{-m}{n(1-\rho)}\Big),
i \langle \phi ( k , \omega ) \phi ( - k , - \omega ) \rangle = 1 / \left( - \frac{2 \pi}{\log ( | k | / \Lambda )}+ \frac{e^{2}}{\pi}\frac{v_{F}k^{2}}{v_{F}^{2}k^{2}- \omega_{k}^{2}}\right) .
X_0' = \lbrace \lbrack \pm u\rbrack\rbrace \cup \lbrace \lbrack \pm u,v_1,\dots ,v_i\rbrack\in X\rbrace \cup \lbrace \lbrack v_1,\dots ,v_i\rbrack\in X\mid B_q(v_j,u)=0\,\,\,\, j\rbrace.
\hspace{- 1 . 5 c m}\sum_{\Sigma \rightarrow{\cal M}}\, \int_{{\cal M}}\, [ d x d \psi ] \, g_{s}^{- \chi ( \Sigma )}\exp \Bigl [ -{\frac{1}{4 \pi \alpha^{\prime}}}\int_{\Sigma}d^{2}\xi \, L ( x , \partial_{\xi}x , \psi , \partial_{\xi}\psi , \ldots ) \Bigr ] \prod_{i}\int d^{2}\xi_{i}\, V_{i}( \xi_{i})
\log m_t^{(n)}(x) = e^{-\frac{\sigma^2}{2}t}\log m_0(x) &- \int_0^t \frac{\sigma^2}{2}e^{-\frac{\sigma^2}{2}(t-s)}\left( \frac{2}{\sigma^2}\frac{\delta F}{\delta m}(m_s^{(n-1)},x) - \log \pi(x) - \operatorname{KL}(m_s^{(n-1)}|\pi) \right) ds \,.
\begin{aligned}\|(u, \mathbf{b}, \theta)\|_{C^{1/2, 1/4}(\overline Q_T)}\leq C, &\|\mathbf{w}\|_{C^{1/2, 1/4}([\delta, 1-\delta]\times[0, T])}\leq C,~\forall\delta\in\big(0, (b-a)/2\big) . \end{aligned}
P(r_3,r_4) = q_{44}r_4^2 + 2q_{34}r_3r_4 + q_{33}r_3^2 + 2(q_{13}r_1+q_{23}r_2)r_3 + 2(q_{14}r_1 + q_{24}r_2)r_4 &+ q_{11}r_1^2 + 2q_{12}r_1r_2 + q_{22}r_2^2 - mq_{11},
W(c,\iota;D,{\mathfrak a}_c):=-\frac{1}{n}\log N{\mathfrak a}_c \mathfrak f \cdot \left[\sum_{z \in ({\mathfrak a}_c \mathfrak f)^{-1}\cap D,\, (z){\mathfrak a}_c \mathfrak f \in c}\iota(z)^{-s}\right]_{s=0}.
(d_{\mathcal{F}}\mu)(X_{0},\ldots,X_{p}):= \sum_{i=0}^{p}(-1)^{i}L_{X_{i}}(\mu(X_{0},\ldots,\hat{X}_{i},\dots,X_{p})) &+\sum_{i<j}(-1)^{i+j}\mu([X_{i},X_{j}],X_{0},\ldots,\hat{X}_{i},\ldots,\hat{X}_{j},\ldots,X_{p}).
x^{k+1}=x^k-\alpha P^{-1}\nabla F(x^k) &=x^k-\alpha P^{-1}P(S_2S_1x^k-x^k) &=x^k-\alpha(S_2S_1x^k-x^k) &=(1-\alpha)x^k+\alpha S_2S_1x^k,
|J_1(t_1,t_2)| = \Big|\int_{\mathbb{R}}G_{t_1}(x-y) \Big( \int_{\mathbb{R}}G_{t_2-t_1}(y-z) \big[\psi_0(z)-\psi_0(y)\big] dz \Big) dy \Big| &\leq C \int_{\mathbb{R}}G_{t_1}(x-y) \Big( \int_{\mathbb{R}}G_{t_2-t_1}(y-z) |z-y| dz \Big) dy &= C (t_2-t_1)^{1/2}.
D(P)=\{F_{\psi}\in L^2(0,\infty): F_{\psi}\;{\rm abs.}\;{\rm cont.}, \int_0^{\infty}{\rm d}q\,|{{\rm d}F_{\psi}\over{\rm d}q}|^2 <\infty,\; F_{\psi}(0)=0\}.
\|\nabla\sqrt{E_\varepsilon \varphi^2}\|_2 =\|\frac{E_\varepsilon(|\varphi||\nabla|\varphi|)}{\sqrt{E_\varepsilon \varphi^2}}\|_2 &\leq \|\sqrt{E_\varepsilon|\nabla |\varphi||^2}\|_2=\|E_\varepsilon|\nabla |\varphi||^2\|_1^\frac{1}{2}& \leq\|\nabla|\varphi|\|_2\leq \|\nabla \varphi\|_2 ,
D(\phi) = \int_{M^*}\left( \phi^* - \phi_0^*\right) d\nu - \frac{1}{\lambda}\log \int_M e^{-\lambda(\phi-\phi_0)}\mu_0 M(\mu) = \lambda \inf_{\phi\in C^0(M,L)}F_{\mu,\nu}(\phi) + \int_M \log \frac{\mu}{\mu_0}d\mu,
d \mu = \frac{V \kappa^{2}}{2 \pi^{2}\hbar^{3}( a + 1 )^{2}}\, \left( 1 - e^{- ( a + 1 ) P_{0}/ \kappa}\right)^{2}\, e^{- ( 4 a + 1 ) P_{0}/ \kappa}\, d P_{0}.
\hat{\phi}={i\mu m\over \sqrt{\lambda}}+(c_{1,0}\bar{\rho}_1+c_{0,1}\bar{\rho}_2)\xi-\Biggl({i\mu\sqrt{\lambda}(c_{1,0}^2\bar{\rho}_1^2+c_{0,1}^2\bar{\rho}_2^2)\over 2m}+{3i\mu m\lambda c_{1,0}c_{0,1}\bar{\rho}_1\bar{\rho}_2\over (k_{1,0}k_{2,0}-{\bf k_1}.{\bf k_2}+m^2)}\Biggr)\xi^2+...,
\begin{cases}\bar p_\tau = \bar p_{xx}, (x,\tau)\in (0,1)\times (0,\infty), \bar p_x(0,\tau) = c \bar p(0,\tau), \tau\in(0,\infty), \bar p_x(1,\tau) = 0, \tau\in(0,\infty), \bar p(x,0) = \bar p_0(x), x\in(0,1).\end{cases}
\begin{gathered}\frac{d^2}{dz^2}f^{(1,2)}_++p(z)\frac{d}{dz}f^{(1,2)}_++q(z)f^{(1,2)}_+=0, begin{align*}5pt]p(z)=\frac{1-\theta_0}{z}+\frac{1-\theta_t}{z-t}-\frac{1}{z-\lambda}, begin{align*}5pt]q(z)=-\frac{1}{4}+\frac{C_0}{z}+\frac{C_t}{z-t}+\frac{\mu}{z-\lambda},\end{gathered}
L^2_i(M\setminus\Sigma; \mathbf{\mathcal{S}}\otimes \mathcal{I})=\mbox{closure of}\{\mathfrak{u}\in C^{\infty}(M\setminus\Sigma, \mathbf{\mathcal{S}}\otimes \mathcal{I})\mbox{}|\mbox{}\|\mathfrak{u}\|_{L^2_i}< \infty\}
[\gamma^{2}-\frac{(1-\gamma^{2})^{2}c^{2}}{\gamma^{2}v^{2}}]x^{2}-2\gamma^{2}vtx+y^{2}+z^{2}=(c^{2}\gamma^{2}-v^{2}\gamma^{2})t^{2}+2\frac{[1-\gamma^{2}]txc^{2}}{v}
E_n^{-1}(z) \frac{A_n^{(1)}(s) - A_n^{(1)}(0)}{s}E_n(z)= \frac{\mathcal{O}\left(1\right)}{n^{-\frac{5}{2}}}= \mathcal{O}\left(n^{\frac{5}{2}}\right)
U_{n}(r)=\frac{iP_{n}^{\prime}}{\rho n\omega}[1-\frac{J_{0}(\alpha n^{1/2}i^{3/2}\frac{r}{R})}{J_{0}(\alpha n^{1/2}i^{3/2})}]=\frac{iP_{n}^{\prime}}{\rho n\omega}[1-\frac{J_{0}(\Lambda_{n}\frac{r}{R})}{J_{0}(\Lambda_{n})}],
\begin{matrix}\sum_{n=1}^{\infty}\frac{H_{n}-(\gamma+lnn)}{n}=-\gamma_{1}-\frac{1}{2}\gamma^{2}+\frac{1}{12}\pi^{2}\\ \sum_{n=1}^{\infty}\frac{H_{n}^{2}-(\gamma+lnn)^{2}}{n}=-\gamma_{2}-2\gamma\gamma_{1}-\frac{2}{3}\gamma^{3}+\frac{5}{3}\zeta(3)\end{matrix}
x\otimes T_nj_P^-(E^P_{o_{+,P}}(w)) =x\otimes j_{P'}^+(E_{o_{+,P'}}^{P'}(nwn^{-1}))T_n & =x(n\sigma)(E_{o_{+,P'}}^{P'}(nwn^{-1}))\otimes T_n & =x\sigma(E_{o_{+,P}}^{P}(w))\otimes T_n.
\mathcal{F}^\prime_{k-1}=\sigma\left( \left(\xi^{(0)}_0,{\xi^{(0+)}_{0}},\xi^{(1)}_0, \xi^{(2)}_0\right), \dots, \left(\xi^{(0)}_{k-1},{\xi^{(0+)}_{k-1}},\xi^{(1)}_{k-1}, \xi^{(2)}_{k-1}\right), \left(\xi^{(1)}_k, \xi^{(2)}_k\right) \right)
(\mathcal{J}_{\ast}V)(y) =\int_{\lbrace z\leqslant1\rbrace}\left(V(y+z)-V(y)\right)\nu(\mathrm{d}z), (\mathcal{J}^{\ast}V)(y) =\int_{\lbrace z>1\rbrace}\left(V(y+z)-V(y)\right)\nu(\mathrm{d}z).
{f_{{\rm{low}}}}\left({\xi P}\right) = \frac{1}{2}\ln \left({1 + \frac{{2{\varsigma^2}{\sigma^2}}}{{\xi P}}}\right) - \frac{{\xi P +{\varsigma^2}{\sigma^2}}}{{{\varsigma^2}{\sigma^2}}}+ \frac{{\sqrt{\xi P\left({\xi P + 2{\varsigma^2}{\sigma^2}}\right)}}}{{{\varsigma^2}{\sigma^2}}}.
{\mathcal{D}}{\mathcal{F}}\equiv\prod\limits_{k=1}^{n_f}d{\bf{\chi}}_k^{\dag}d{\bf{\chi}}_kd{\bf{\varphi}}^{\dag}_kd{\bf{\varphi}}_k,\;\;{\mathcal{D}}{\mathcal{B}}\equiv\prod\limits_{l=1}^{n_b}d{\bf{s}}_l^{\dag}d{\bf{s}}_ld{\bf{p}}^{\dag}_ld{\bf{p}}_l
P_k^S \approx \left. \frac{16}{9}\frac{G^2 V^2}{c_s \dot{\phi}^2 F_{,X}}\right|_{c_s k = a H}= \left. \frac{16}{9}\left( \frac{3}{8 \pi}\right)^2\frac{H^4}{(c_s F_{,X}) \dot{\phi}^2}\right|_{c_s k = a H}~,
\psi (R ) = - \left( \frac{\bar{\mu}\mu_P}{2 g}+ \frac{g}{2}\bar{q}\epsilon_P \right) \sum_j \ln \left| \frac{R - r_j}{a}\right| +\frac{1}{4}\left[ \frac{\mu_P \bar{m}}{2 g}+ \frac{g}{2}\epsilon_P \bar{q}\right] R^2
\left. \left( - \psi_{+ \mu}\delta \psi_{+}^{\mu}+ \psi_{- \mu}\delta \psi_{-}^{\mu}\right) \right|_{\sigma = 0}= \left. \left( - \psi_{+ \mu}\delta \psi_{+}^{\mu}+ \psi_{- \mu}\delta \psi_{-}^{\mu}\right) \right|_{\sigma = \pi}= 0 .
Z_{b}= \int \prod_{p}d C_{p}\int \prod_{p}d D_{p}\exp \left\{- \frac{1}{2}( \begin{array}{l l}{{C_{p}}}&{{D_{p}}}\\ \end{array}) G ( p ) \left( \begin{array}{l}{{C_{p}}}\\{{D_{p}}}\\ \end{array}\right) \right\},
\sum_{p\leq X}\pi_{k_0,A}\left(\frac{X}{p}\right)=\sum_{n>[\sqrt{\log \log X}]}^{X-1}\pi_{1,P}(n)\left(\pi_{k_0,A}\left(\frac{X}{n}\right)-\pi_{k_0,A}\left(\frac{X}{n+1}\right) \right) + o\left(\frac{X(\log \log X)^{k_0}}{\log X}\right).
\Biggl\{\begin{pmatrix}y_{1}z_{1}-z_{1}y_{1}& \ddots & y_{n-1}z_{n-1}& -z_{n-1}y_{n-1}& 1 & y_{n}z_{n}& z_{n}y_{n}\end{pmatrix}\qquad:y_{1}^{2}+z_{1}^{2}=\cdots=y_{n-1}^{2}+z_{n-1}^{2}=y_{n}^{2}-z_{n}^{2}=1,\, y_{n}>0\Biggr\}
\mathcal{C}=\left\{\textbf{c}(\alpha,\beta)=\left({\rm Tr}_1^m(\alpha\pi^{\frac{p^k+1}{2}i}+\beta(-\pi)^{i})\right)_{i=0}^{p^m-2}\mid \alpha,\beta \in \mathbb{F}_{p^m}\right\}.
\int_{-\infty}^{\infty}P_j(x) Q_k(x) dx ={}\int_{-\infty}^{\infty}\mathcal W^{-1}p_j(x)\, \mathcal W q_k(x)dx ={}\int_{-\infty}^{\infty}\left(\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal W^{-1}p_j(x) e^{-\frac{1}{2}(t-x)^2}dx \right) q_k(t) dt.
\frac{e^{3}}{| e |}\left[ \prod_{a = 1}^{2}\delta \left( \frac{F^{a}}{| F |}+ \frac{e^{a}}{| e |}\right) + \prod_{a = 1}^{2}\delta \left( \frac{F^{a}}{| F |}- \frac{e^{a}}{| e |}\right) \right] ,
\nabla^{d_{1}}_{u_{1}}\circ \nabla^{d_{2}}_{u_{2}}Y_{t_{1},t_{2}}=\left(I-2u_{1}B_{1}+B_{1}^{2}\right)^{d_{1}}\circ\left(I-2u_{2}B_{2}+B_{2}^{2}\right)^{d_{2}}Y_{t_{1},t_{2}}=\varepsilon_{t_{1},t_{2}},
\pi_{\mu\nu}= -\frac{1}{4}\tau_{\mu\nu}\tau_{\nu}^{\alpha}+\frac{1}{12}\tau \tau_{\mu\nu}+\frac{1}{8}q_{\mu\nu}\tau_{\alpha\beta}\tau^{\alpha\beta}- \frac{1}{24}q_{\mu\nu}\tau^{2}
\int \sqrt{g}~\widehat{R}^{\mu \nu \lambda \rho}\widehat{R}_{\mu \nu\lambda \rho},\qquad \int \sqrt{g}~\widehat{R}^{\mu \nu}\widehat{R}_{\mu\nu},\qquad \int \sqrt{g}~\widehat{R}^{2}.
\begin{array}{l l}{\nabla \cdot \mathbf{E}^{\prime}= \rho_{e}^{\prime},}&{\nabla \times \mathbf{B}^{\prime}= \mathbf{j}_{e}^{\prime}+ \frac{\partial \mathbf{E^{\prime}}}{\partial t},}\\{\nabla \cdot \mathbf{B}^{\prime}= \rho_{m}^{\prime}= 0 ,}&{\nabla \times \mathbf{E}^{\prime}= - \frac{\partial \mathbf{B^{\prime}}}{\partial t}.}\\ \end{array}
\mathfrak{h}_{0}\Big(\varphi-\sum_{1\leq i<i_a}d^+_{i,0}(D^2\varphi)h_{i,2}-\sum_{1\leq i\leq g}d^+_{i,1}(D\varphi)h_{i,1}-\sum_{1\leq s<\gamma}d^0_{s,1}(D\varphi)c_{s,1}\Big) & = \sum_{1\leq i\leq g}d^+_{i,2}(\varphi)h_i+\sum_{1\leq s<\gamma}d^0_{s,2}(\varphi)c_s+\sum_{1\leq j\leq g}d^-_{-j,2}(\varphi)h_{-j}.
T_{\eqref{e_difference_cg_hamiltonian}}= \frac{1}{N}\sum_{l=1}^{M}\log \left( \mathbb{E}_{\mu_K (dx^{B(l)}| y_l)}\left[ \frac{1}{2}\sum_{l=1}^{M}\sum_{n \neq l}\sum_{\substack{i \in B(l) j \in B(n)}}|M_{ij}| x_i^2 \right] \right) .
S ( g ) ={\bf 1}+ \sum_{n = 1}^{\infty}\frac{1}{n !}\int \! \! d^{4}x_{1}\ldots d^{4}x_{n}\, T_{n}( x_{1}, \ldots , x_{n}) \, g ( x_{1}) \cdot \ldots \cdot g ( x_{n}) \quad ,
\sum_{a=1}^n \{\langle\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi,\phi\rangle+\langle\phi,\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi\rangle\}&\geq C\sum_{a=1}^n\{g_Q(\nabla_{V_a}H^{1,0},\bar V_a)+\overline{g_Q(\nabla_{V_a}H^{1,0},\bar V_a)}\}&\geq C\{\rm div}_\nabla(\kappa_B^\sharp).
\Delta^{\mu\nu,\lambda\sigma}(x -y)=<0|\;T({\bf a}^{\mu\nu}(x)\;\;{\bf a}^{\lambda\sigma}(y)\;)|0>\;=\;\frac{1}{(2\pi)^4}\int d^4p\;e^{i\;p\cdot(x-y)}\;\Delta^{\mu\nu,\lambda\sigma}(p),
\nabla^{TU}_{e_j}e_{\ell}=\Gamma^{TU}(e_j)e_{\ell},\ \j,\ell=1,2,\ldots,2n, &\nabla^{\mathrm{det\,}}(w_1\wedge\cdots\wedge w_n)=\Gamma^{\mathrm{det\,}}w_1\wedge\cdots\wedge w_n.
\frac{d}{dt}\alpha (\mathbf{r}(t))=\mathbf{\nabla}\alpha (\mathbf{r}(t))\cdot \frac{d\mathbf{r}}{dt}=\mathbf{\nabla}\alpha (\mathbf{r}(t))\cdot \mathbf{\vec{\omega}(r(}t\mathbf{))=}0
R(M_s,M_p)=\alpha\beta R_{1,1}(M_{s_1},M_{p_{1,1}})+ \alpha(1-\beta)R_{1,2}(M_{s_1},M_{p_{1,2}})+(1-\alpha)\gamma R_{2,1}(M_{s_2},M_{p_{2,1}})+ (1-\alpha)(1-\gamma)R_{2,2}(M_{s_2},M_{p_{2,2}}).
( \delta_{1 2})^{2}( \delta_{3 4})^{2}= \delta_{\{A_{1}B_{1}\}}^{A_{2}B_{2}}\delta_{\{A_{3}B_{3}\}}^{A_{4}B_{4}}\; , \qquad \delta_{1 3}\delta_{1 4}\delta_{2 3}\delta_{2 4}= \delta_{\{A_{1}B_{1}\}}^{\{A_{3}\{B_{4}}\delta_{\{A_{2}B_{2}\}}^{A_{4}\}B_{3}\}}\; ,
\theta_1 = - \frac{1}{t^5}-\frac{2}{t^8}-\frac{\kappa_1}{t^{11}}, \; \; \theta_2 = t+\frac{1}{t^5}+ \frac{3}{t^8}+\frac{\kappa_2}{t^{11}}\; \mbox{and}\; \theta_3 = t^4-2t-\frac{1}{t^8}-\frac{\kappa_3}{t^{11}}.
J_-(\tilde x)=\frac{\pi}{2\vert \mbox{det}\,\psi''(\tilde x)\vert^\frac 12}(-(\mathcal A B_1\tilde U_0, \tilde V_0)-(2\partial_{\bar z}\mathcal B\tilde U_0, \tilde V_0)-(\mathcal B\tilde U_0, B_2^*\tilde V_0)+ (\mathcal Q\tilde U_0,\tilde V_0))(\tilde x),
B_{n}(\lambda,b) :=(1-b^{2})\Lambda_{1}(\lambda,b)+b\big[\Omega_{n}(\lambda)-\Omega_{n}(\lambda b)\big], C_{n}(\lambda,b) :=b\left[\left(\Lambda_{1}(\lambda,b)-\frac{1}{b}\Omega_{n}(\lambda)\right)\Big(b\Omega_{n}(\lambda b)-\Lambda_{1}(\lambda,b)\Big)+\Lambda_{n}^{2}(\lambda,b)\right].
i{\cal G}_4(p)=-[8ie^4B^4] \left [{2 p_{\bot}^2 p_L^2 -3 (p_{\bot}^2)^2 \over (p^2-m^2)^7}\right] D -8 e^5B^5 \left[{15 ( p_{\bot}^2 )^2 -16 p_\bot^2 p_L^2 +2 p_L^2 \over (p^2-m^2)^8}\right] \bar D.
\mathfrak{q}_1 =(X_1 + X_2,X_4+(\zeta+1)X_6,X_3^3,X_2^3 + X_5^3 - X_6^3), \mathfrak{q}_2 =(X_1 + X_2,X_4-\zeta X_6,X_3^3,X_2^3 + X_5^3 - X_6^3), \mathfrak{q}_3 =(X_1 + X_2,X_4-X_6,X_3^3,X_2^3 + X_5^3 - X_6^3).
\begin{aligned}\delta'>|\mu_{n,j}(a)-\xi_{n,j}(a)| =|f_{a}^{n}(\xi_{0,j}(a))-f_{a}^{n}(\mu_{0,j}(a))| &\geq C\lambda^{n}|\xi_{0,j}(a)-\mu_{0,j}(a)| &=C\lambda^n |x_j(a)|.\end{aligned}
\log \left( \sum_{n=0}^\infty e^{|f(n)|}\pi(n) \right) = \log \left( \sum_{n=0}^\infty e^{|f(n) - f(0) + f(0) - \pi(f)|}\pi(n) \right) & \leq \log \left( \sum_{n=0}^\infty e^{|f(n) - f(0)|}\pi(n) \right) + |f(0)-\pi(f)| .
\chi_{\nu}( \rho ) \sim \left( \begin{array}{c}{{\sqrt{E + m}\ [ \sin \mu J_{\nu}( k \rho ) + ( - 1 )^{n}\cos \mu J_{- \nu}( k \rho ) ]}}\\{{i \sqrt{E - m}\ [ \sin \mu J_{\nu + 1}( k \rho ) + ( - 1 )^{n + 1}\cos \mu J_{- ( \nu + 1 )}( k \rho ) ]}}\\ \end{array}\right)
{\cal A}_{\rm NS}= \frac{T}{2\pi}{1\over \sqrt{8\pi^2\alpha'}}\int_0^\infty{dt}\left(\pi\over t \right)^{1\over 2}\,{\rm e}^{-b^2 /(2\alpha' t)}\,\left[ \left({f_4\over f_2}\right)^8 - \left({f_3\over f_2}\right)^8\right] = -{T\over 4\pi\alpha'}|b|~.
\tilde L_a f(\xi)= (2\pi)^{-d}\int_{\mathbb{R}^d}\int_{\mathbb{T}^d}e^{i\int_0^1 \nabla_\xi \varphi_a(x,\eta+\theta(\xi-\eta))d\theta \cdot (\xi-\eta)}f(\eta) d\eta dx .
\int_0^u \int_X G((u - s)/p, x, y) s^{M}|G(s/p, y, e^{-i \theta + 2u}x) | dv (y)ds \sim u^{- \frac{\dim X}{2}}e^{- p \theta^2/ u}\int_0^u e^{p u^2/s}s^{M}ds \ll u^{- \frac{\dim X}{2}}e^{- p \theta^2/ u}e^{p u}u^{M+1}.
\begin{aligned}\int_\Omega |f(x,u_{n})|^{q'}d\mu_g(x) \leq 2^{\frac{q+1}{q-1}}\beta^{\frac{q+1}{q-1}}\Big(\|u_{n}\|^{q}\|_{L^{\frac{p_s^*}{q}}(\Omega)}\|1\|_{L^{\frac{p_s^*}{p_s^*-q}}(\Omega)}+\mu(\Omega)\Big) &\leq C(\mu(\Omega))^{\frac{p_s^*-q}{p_s^*}}+C\mu(\Omega)\leq C(q, \beta,\Omega),\end{aligned}
\Big(\sum_{j=0}^{K-1}d(\gamma_{N_{j+1}},\gamma_{N_j})^r\Big)^{\frac1r}=\Big(\sum_{j=0}^{K-1}\|\sigma_{N_{j+1}}-\sigma_{N_{j}}\|_{}^r\Big)^{\frac1r}+O(\mathcal{V}^2_r(\sigma;[-R,R])).
\|g_j\| =\displaystyle\sup_{x\in \mathcal{X}, \|x\|\leq 1}|g_j(x)|=\displaystyle\sup_{x\in \mathcal{X}, \|x\|\leq 1}|f_j(V^{-1}x)|=\displaystyle\sup_{Vy\in \mathcal{X}, \|Vy\|\leq 1}|f_j(y)| &=\displaystyle\sup_{Vy\in \mathcal{X}, \|y\|\leq 1}|f_j(y)|=\|f_j\|=1, \forall 1\leq j \leq n.
Y_{[a^{\ast}w(^{\ast}b)]}(\mu_{k}P) =\rho^{-1}\left(X_{[a^{\ast}w(^{\ast}b)]}(\rho(\mu_{k}P))\right) &=\rho^{-1}\left(X_{[a^{\ast}w(^{\ast}b)]}(\rho(\Delta_{k}'))\right) &=\rho^{-1}\left(X_{[a^{\ast}w(^{\ast}b)]}(\rho(\Delta_{k}'))\right)
f_k =[ u^k] F(u)=\frac{N_3^{t-k+1}}{D_t}(-1)^{k-1}, g_k =[ u^k] G(u)=\frac{N_2^{t-k}}{D_t}(-1)^{k}, \mbox{and}h_k =[ u^k] H(u)=\frac{N_2^{t-k+1}}{D_t}(-1)^{k-1}.
p_1(\nabla) = \frac{1}{4\pi^2}\left( \frac{s_C^2}{4}+ \|W_F^+\|^2 + \|R_{00}\|^2 - \frac{s_g^2}{48}\right)\operatorname{vol}_g \operatorname{Pf}(\nabla) = \frac{1}{8\pi^2}\left(\frac{s_C^2}{4}+\|W_F^+\|^2 + \frac{s_g^2}{48}- 2\|R_F\|^2 - \|R_{00}\|^2 \right)\operatorname{vol}_g
E(L_1,\ell_\nu,\ell_\mu)+E(L_1\ell_\nu,\ell_{\mu'})= D(L_1,\ell_\nu,2\ell_\mu)+D(L_1,\ell_\nu,2\ell_{\mu'}) & +L_1-R(L_1,\ell_\nu,2\ell_\mu)-R(L_1,\ell_\nu,2\ell_{\mu'}).
[ \gamma_{l}, \bar{\gamma}_{m}] = \frac{1}{4 \pi k^{\prime}}G_{l m}\mathrm{o r}\ \ \ \, b a r{\gamma}_{l}= - \frac{1}{4 \pi k^{\prime}}G_{l m}\frac{\partial}{\partial \gamma_{m}}= - \frac{1}{4 \pi k^{\prime}}\frac{\partial}{\partial \gamma^{l}}
\left|{\frac{{\cal N}_{\mathrm{s t r a d d l e}}}{{\cal N}_{\mathrm{e s c a p e}}}}\right|^{2}={\frac{1}{{\operatorname{e x p}\left\{+{\frac{2 \pi ; \omega \; c_{H}}{g_{H}}}\right\}}- 1}}.
\forall 1\leq i,j,k,l\leq 2, R_{i \bar{j}k \bar{l}}=-g_{i \bar{j}k \bar{l}}+\sum_{1\leq \alpha,\beta \leq 2}g_{i k \bar{\alpha}}g^{\bar{\alpha}\beta}g_{\beta \bar{j}\bar{l}},
-\Delta(\phi_1(q_0)-\phi_1(q))+q_0(\phi_1(q_0)-\phi_1(q))-\lambda_1(q_0)(\phi_1(q_0)-\phi_1(q))+ (q_0-q)\phi_1(q)-(\lambda_1(q_0)-\lambda_1(q))\phi_1(q)=0 .
L_{1}\, = \, L_{0}\, + \, \dot{\lambda}_{1}^{i}\lbrace{\frac{( X_{1}^{i}- X_{0}^{i})}{\epsilon}}\, - \, 2 \pi \alpha^{\prime}\, M_{i j}^{- 1}{\cal B}^{j k}P_{0}^{k}\, \rbrace \, + \, \dot{\lambda}_{3}^{i}\lbrace{\frac{( X_{N}^{i}- X_{N - 1}^{i})}{\epsilon}}\, - \, 2 \pi \alpha^{\prime}\, M_{i j}^{- 1}{\cal B}^{j k}P_{N - 1}^{k}\, \rbrace
\lefteqn{V_i=\frac{(1-T_iT_i^*)\otimes T_i+T_i\otimes(1-T_iT_i^*)}{\sqrt{2}}e_+}+\frac{(1-T_iT_i^*)\otimes T_i-T_i\otimes(1-T_iT_i^*)}{\sqrt{2}}e_-.
(2\pi)^n\|\mathcal{S}_wg\|_{L^2(\mathbb{S}^{n-1})}^2=\sum_{k,m}\left(\int_0^\infty g_0^{(k,m)}(r)r^{\frac{1}{2}}J_{k+\frac{n-2}{2}}(r)w(r)^{\frac{1}{2}}\,\mathrm{d}r\right)^2=:\sum_{k,m}A_{k,m}.
Q_0(y,y'|x) = Q_0(y|x) Q_0(y'|x) Q_0(y|x) = \frac{W(y|x)^{(1-w_1)/2}\f(y)^{(1+w_1)/2}}{\sum_{\mathcal{Y}}W(y|x)^{(1-w_1)/2}\f(y)^{(1+w_1)/2}}Q_0(\mathbf{v,v'|u_0}) = Q_0(\mathbf{v|u_0}) Q_0(\mathbf{v'|u_0})
\mathbf{P_*}\Big(\big{|}\sum_{i = 1}^{n}x_{ij}x_{ik}\psi'(\bar{\epsilon}_i)(G_i^* - EG_i^*)\big{|}>n{\epsilon}\Big)=o(n^{-1/2}),\;\;\;\; j,k\in\{1,\ldots,p\}
T_2^{(n)}(x) = (n+1) 2^s \int_{-\infty}^{\infty}\left( 1 + \sqrt{\frac{x + 1}{x - 1}}e^{\tau}\right)^{-s}\Big( x + (x^2 - 1)^{\frac{1}{2}}\cosh \tau \Big)^{- n - 2}\times \times ( 1 + \frac{x}{\sqrt{x^2 - 1}}\cosh \tau ) d\tau .
I_{17}= -\int_{\Omega}(H\times H)_{tt}\cdot u_{tt}x=\int_{\Omega}\big(H\otimes H-\frac{1}{2}|H|^2I_3\big)_{tt}: \nabla u_{tt}x \leq C|\nabla u_{tt}|_2|H_t|^2_4+C|\nabla u_{tt}|_2|H_{tt}|_2| H|_{\infty}.
O_{r}(1) \sum_{c_2 \ldots c_d \leq x}c_2^{d-2}c_3^{d-3}\ldots c_{d-2}^2 c_{d-1}(x/(c_2 c_3 \ldots c_d))^{d-1}=\ O_r(1) x^{d-1}\sum_{c_2 c_3 \ldots c_d \leq x}c_2^{-1}c_3^{-2}\ldots c_{d-1}^{-(d-2)}c_{d}^{-(d-1)}\leq\ O_{d,r}(1) x^{d-1}\log x.
\int \left\|BAB^{-1}\right\|^s d\mu(A) \leq d^{\frac{s}{2}}\int \max_{1 \leq i \leq d}\left\|BAB^{-1}e_i\right\|^s d\mu(A) &\leq d^{\frac{s}{2}}\int \sum_{i=1}^d\left\|BAB^{-1}e_i\right\|^s d\mu(A) &= d^{\frac{s}{2}}\sum_{i=1}^d\int \left\|BAB^{-1}e_i\right\|^s d\mu(A) &\leq d^{1+\frac{s}{2}}\max_{1 \leq i \leq d}\int \left\|BAB^{-1}e_i\right\|^s d\mu(A),
\overline{{\psi}}^{( 1 )}( x ) P_{+}\psi^{( 1 )}( x ) \longleftrightarrow \frac{1}{2 \pi}c^{( 1 )}: e^{- i a \varphi^{( 1 )}( x )}:_{M^{( 1 )}}: e^{- i b \varphi^{( 2 )}( x )}:_{M^{( 2 )}}e^{i \frac{\theta}{2}},
\alpha_W[n]L[n] =\left( \begin{array}{ccc|ccc|ccc|ccc}-1 \cdots -1 0 \cdots 0 0 \cdots 0 0 \cdots 0 -1 \cdots -1 0 \cdots 0 \ddots 0 \cdots 0 0 \cdots 0 -1 \cdots -1\end{array}\right),
-D^\perp_\ell(q)=\sqrt{\frac{Z(q)Z(q^\ell)}{A(q)A(q^\ell)}}\bigg( \frac{1}{1+\alpha(q)}\sqrt{\frac{Z(q)}{Z(q^\ell)}}\alpha(q)(1-\alpha(q)) &-\frac{\ell}{1+\alpha(q^\ell)}\sqrt{\frac{Z(q^\ell)}{Z(q)}}\alpha(q^\ell)(1-\alpha(q^\ell))-4D_\ell(q)\bigg).
|\alpha_t -\alpha|^2 \leq C^2\Bigl(2N\Bigl(\Bigl|\frac{\psi_n}{\psi_{nt}}-1\Bigr|^2 + \sum_{k\neq n,k\leq N}\xi_k^4\Bigl|\frac{\psi_k}{\psi_{kt}}-1\Bigr|^2\Bigr) + 8\varepsilon^2\Bigr) \leq 2N C^2\max\bigl(1,\sup_{k\in\mathbb N}\xi_k^4\bigr)\sum_{k=1}^N\Bigl|\frac{\psi_k}{\psi_{kt}}-1\Bigr|^2 + 8C^2\varepsilon^2\ \bigcup_{n=1}^N\tau_n.
H_a\left( \frac{dS_a}{d\lambda_a},\lambda_a \right) = -2A(\lambda_a)\left( \frac{dS_a}{d\lambda_a}\right)^2- \frac{1}{2}\left( \lambda_a^{N+1}-(\alpha-1)\lambda_a^N+(1-\alpha+\beta)\lambda_a^{N-1}\right),
-\varepsilon d_A \frac{d^2}{d x^2}\phi_A=\big[\min_{0\leq s\leq F_A(0)}H_A\big(1,s,F_B(1)\big)\big]\phi_A\leq H_A\big(x,\phi_A(x),B_\varepsilon(x)\big) \phi_A.
\Delta^{\mathrm{c l}}= \int d^{3}x \left( [ A_{\mu}^{*}, A^{\mu}] - \left[ c^{*}, c \right] + \left( \overline{{\Psi}}^{*}T^{a}\Psi + \overline{{\Psi}}T^{a}\Psi^{*}\right) \tau_{a}\right) .
W_M(S;\varphi)=\mathbb{E}\sup_{x\in S}\bigg\langle x,\frac{1}{\sqrt{M}}\sum_{i=1}^M\epsilon_i\varphi_i\bigg\rangle\leq C\sigma_\mathrm{max}\cdot\mathbb{E}\sup_{x\in S}\langle x,g\rangle=C\sigma_\mathrm{max}\cdot w(S).
\begin{aligned}J_i(v_1(\cdot),v_2(\cdot))=\mathbb{E}^{v_1,v_2}[\int_0^Tl_i(t,x(t),y(t),z(t),z_1(t),z_2(t),v_{1}(t),v_{2}(t))dt+\Phi_i(x(T))+\gamma_i(y(0))] =\mathbb{E}[\int_0^TZ^{v_1,v_2}(t)l_i(t,x(t),y(t),z(t),z_1(t),z_2(t),v_{1}(t),v_{2}(t))dt+Z^{v_1,v_2}(T)\Phi_i(x(T))+\gamma_i(y(0))],\end{aligned}