harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta^\alpha\partial_\xi^\beta (\sigma_1\sharp_0\sigma_2)(\xi)=\sum_{\substack{\alpha_1+\alpha_2=\alpha \beta_1+\beta_2=\beta}}\binom{\alpha}{\alpha_1}\binom{\beta}{\beta_1}(\delta^{\alpha_1}\partial_\xi^{\beta_1}\sigma_1)\sharp_0(\delta^{\alpha_2}\partial_\xi^{\beta_2}\sigma_2)(\xi).
\pi_0\left(\mathcal{M}^{2-\textrm{conv}}(S^{n-1}\times S^1)\right) \cong\begin{cases}\mathcal{K}\left(\mathcal{M}^{2-\textrm{conv}}(S^{1}\times S^1)\right), n=2, 0, n\geq 3,\end{cases}
f(i)=\frac{1}{2\pi^{2}}\int_{0}^{\infty}\tilde{f}(\lambda)\lambda d\lambda\int_{-\infty}^{\infty}\frac{sin\lambda t/2}{sinht}cosh\frac{t}{2}dt=\frac{1}{2\pi^{2}}\int_{-\infty}^{\infty}\tilde{f}(\lambda)\frac{\lambda\pi}{2}tanh(\frac{\pi\lambda}{2})d\lambda.
Q(M,P_i,\tau)=\left\{e^{\frac{1}{24}E_2(\tau)A_1}\left(\prod_{j=1}^{7}\frac{x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\right)\frac{\sqrt{-1}\theta(u,\tau)}{\theta_1(0,\tau)\theta_2(0,\tau)\theta_3(0,\tau)}\right. &\left.\frac{1}{2}\left(\prod_{l=1}^8\theta_1(y_l^i,\tau)+\prod_{l=1}^8\theta_2(y_l^i,\tau)+\prod_{l=1}^8\theta_3(y_l^i,\tau)\right)\right\}^{(14)}.
\begin{matrix}a^t_1 = \frac{e^{-u}(\xi+c)+e^{u}h^t}{\sqrt{2}}, b^t_1 = \frac{e^{-u}(\xi+c) - e^{u}h^t}{\sqrt{2}}, a^t_2 = \frac{e^{-u}\xi+e^{u}h^t}{\sqrt{2}}, b^t_2 = \frac{-e^{-u}\xi - e^{u}h^t}{\sqrt{2}}, \end{matrix}
M^{n-1}(t,x) := \frac{\langle w \rangle_{f^{n-1}_t,\varphi}}{\langle 1 \rangle_{f^{n-1}_t,\varphi}}= \frac{1}{\rho^{n-1}(t,x)}\int_{\mathbb{T}}\!\mathrm{d}y \int_{\mathbb{R}}\!\mathrm{d}w\, f^{n-1}_t(y,w)\, \varphi(x-y) \, w\,,
\P\left(\mathcal{E}(\rho_0,(1+\varepsilon)n)\right)\geq\P\left(S_j>g(j)\textup{for all}j=\rho_0,\ldots,\lfloor(1+\varepsilon)n\rfloor\right)\geq \exp\left(-16(\rho^2n)^{1/3}\right).
\nabla_{\Gamma}u:=\left. \left( \nabla u^{\star}\right) \right\vert_{\Gamma}\overrightarrow{\operatorname*{curl}\nolimits_{\Gamma}}u:=\nabla_{\Gamma}u\times\mathbf{n}\operatorname*{div}\nolimits_{\Gamma}\mathbf{v}=\left. \left( \operatorname*{div}\mathbf{v}^{\ast}\right) \right\vert_{\Gamma}\qquad\Gamma
\frac{\phi(x+1,r)}{\phi(x,r)}=\frac{e^{-r}(rx+r)^{x+2}}{(x+1)(rx)^{x+1}}= r e^{-r}\left(1+\frac{1}{x}\right)^{x+1}\le re^{-r+1+1/x}\le re^{-r+2},
I^{( \pm )}= 2 \pi \int_{0}^{\infty}d q \, \frac{q \, J_{0}( q | \vec{x}- \vec{x}^{\prime}| )}{k^{2}- q^{2}\pm i \epsilon}= - i \pi^{2}\, H_{0}^{( 1 )}( \pm k | \vec{x}- \vec{x}^{\prime}| + i \tilde{\epsilon}) \; ,
\{\tilde{\Theta}_{\alpha}^{\mu}, \tilde{\Theta}_{\beta}^{\nu}\}= \{\tilde{\Theta}_{\alpha}^{\mu}, \tilde{\Psi}_{l}\}= \{\tilde{\Theta}_{\alpha}^{\mu}, \tilde{H}\}= 0 ; \{\tilde{\Psi}_{l}, \tilde{\Psi}_{n}\}\approx 0 ; \{\tilde{\Psi}_{l}, \tilde{H}\}\approx 0 .
\mathbf{d}\widetilde \Omega = \widetilde{d \Omega}+ A^{\alpha}\widetilde{L_{\alpha}\Omega}- F^{\alpha}\widetilde{i_{\alpha}\Omega}= \widetilde{d \Omega}+ \widetilde{A^{\alpha}L_{\alpha}\Omega}- \widetilde{F^{\alpha}i_{\alpha}\Omega}
\sigma_N= -\sum_{n=1}^N\frac{s'_n(x)}{(2i\lambda)^n}- \sum_{n=0}^{N-1}\frac{s_{n+1}(x)}{(2i\lambda)^n}& + \varphi_2\sum_{n=2}^N\Big(\sum_{k+ l=n}s_ks_l\Big)\frac{1}{(2i\lambda)^n}+ \varphi_2 \sum_{n=N+1}^{2N}\Big(\sum_{k+l= n}s_ks_l\Big) \frac{1}{(2i\lambda)^n}-\varphi_1.
{\delta_{1}^{( 0 )}L_{1}^{( 3 )}= - i \,{\det \, e}\, \left( \bar{\epsilon}^{*}\gamma^{\mu}\psi_{\mu}^{*}\right) \, \lambda^{1 6}\, \left( 8 f^{( 1 2 , - 1 2 )}+ 6 \cdot 1 4 4 \, D_{1 1}f^{( 1 1 , - 1 1 )}\right) ,}
(N-1)c_1^2+m_{2,j}(c_2^2-c_1^1) =(m_{1,j}+m_{2,j})c_1^2+ m_{2,j}(c_2^2-c_1^2) &= m_{1,j}c_1^2+m_{2,j}c_2^2 &= m_{1,l}c_1^2+m_{2,l}c_2^2 &=(m_{1,l}+m_{2,l})c_1^2+m_{2,l}(c_2^2-c_2^1) &= (N-1)c_1^2+m_{2,l}(c_2^2-c_1^2).
\quad\,\mathcal M_{\alpha+j, \beta+i, k}\mathcal L_x^j\mathcal L_x^i \frac{1}{|x-z|^{2\alpha}|y-w|^{2\beta}}&=c_{i, j, k}(\alpha, \beta) \left( \frac{\langle x-z, y-w\rangle}{|x-z|^{2}|y-w|^{2}}\right)^k \frac{1}{|x-z|^{2\alpha+2j}|y-w|^{2\beta+2i}},
^{\left( \alpha \right)}J=\frac{3\left( 4\alpha-1\right)^{2}|\delta|^{2\alpha-2}}{2w^{4\alpha+3}}\left[ \kappa v^{4}+\delta^{2}\left(\kappa-\lambda \right) v^{2}+\delta^{2}\delta \,\acute{}\,v\right] .
\|T^{\sharp_A}T+TT^{\sharp_A}\|_A \leq \|\widetilde{T}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}^2+\|(\widetilde{T})^2\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}=\|\widetilde{T}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}^2+\|\widetilde{T^2}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}(\,\eqref{prosum}) =\|T\|_A^2+\|T^2\|_A\quad( \ref{lr2}(b)).
\begin{cases}\mathcal{Y}^{n}\widetilde{\chi}^{m}+(\widetilde{\chi}^{m})^{t}\mathcal{Y}^{n}-(\widetilde{\chi}^{m})^{n}\,_{k}\mathcal{Y}^{k}=0,\cr (\widetilde{\chi}^{i})^{j}\,_{p}\widetilde{\chi}^{p}+(\widetilde{\chi}^{i})\widetilde{\chi}^{j}-\widetilde{\chi}^{j}\widetilde{\chi}^{i}=0.\end{cases}
\|u_{\Omega_n}-u_{\Omega_n}\circ p_n\|_{L^2(U_h(\Gamma_n))}^2+ \|\nabla u_{\Omega_n}-(\nabla u_{\Omega_n})\circ p_n\|_{L^2(U_h(\Gamma_n))}^2 ={\operatorname{o}}_{h\to 0}(h).
t^{\frac{N+2A_k}{2}}\partial_r^\ell\frac{[e^{-tL_V^k}\phi^{k,i}](|x|)}{U_k(|x|)}=\left[M_{k,i}+o(1)\right]\delta_{0\ell}-\left[\frac{N+2A_k}{2}M_{k,i}+o(1)\right]t^{-1}(\partial_r^\ell F_k)(|x|) +t^{-2}O(|x|^{4-\ell}) =\left[M_{k,i}+o(1)\right]\delta_{0\ell}+O(t^{-1}|x|^{2-\ell})\quad\mbox{as}t\to\infty,
\sqrt{\int_{\widetilde{X}}|\widetilde{u}|_{\pi^*h}^2\,dV_{\pi^*\omega}}=\sqrt{\int_{\widetilde{X}}|\pi^*u|_{\pi^*h}^2\,dV_{\pi^*\omega}}=\sqrt{\ell\cdot \int_{X}|u|_{h}^2\,dV_{\omega}}
C^k_{{\rm diff},\lambda}(G):=\{c\in C^\infty(G^{\times(k+1)},\mathbb{C}),~c(gg_0,\ldots,gg_k)=c(g_0,\ldots,g_k),~c(g_0,\ldots,g_k)=(-1)^kc(g_k,g_0,\ldots,g_{k-1})\},
F(n,k) =(-1)^{n}[3n-2k+1]{2n-2k\brack n}\frac{(q;q^2)_{n}(q;q^2)_{n-k}}{(q;q)_{n}(q^2;q^2)_{n-k}}, begin{align*}5pt]G(n,k) =(-1)^{n+1}[n]{2n-2k\brack n-1}\frac{(q;q^2)_{n}(q;q^2)_{n-k}q^{n+1-2k}}{(q;q)_{n}(q^2;q^2)_{n-k}}.
d_z(f^n) \le d^{n-1}d_z(f)+\sum_{k=1}^{n-1}d^{n-1-k}\frac{d}{2}&\le d^{n-1}(\frac{d}{2}-1)+\frac{d^n}{2}\frac{1}{d-1}(1-\frac{1}{d^{n-1}}) &<\frac{d^n}{2}.
[ F_{1}^{+}( z , \tau ) , F_{2}^{+}( z , \tau ) , \ldots , F_{p - 2}^{+}( z , \tau ) , F_{p - 1}^{-}( z , \tau ) , F_{p - 2}^{-}( z , \tau ) , \ldots F_{2}^{-}( z , \tau ) ]
b_{1}( u^{2}, w , \theta ) = ( \frac 1 6 - \xi ) R ( \theta ) +{\frac{1}{6}}u^{2}\sin^{2}( w / 2 ) R_{i i}( \theta ) +{\frac{1}{2 4}}u^{4}\sin^{2}w R_{i j i j}( \theta ) .
\delta_{f}L = \frac{d}{d t}\left[ \frac{\kappa}{2 \pi}\int_{D}d^{2}x \epsilon^{i j}A_{i}^{a}D_{j}( f^{k}A_{k})^{a}\right] + \frac{\kappa}{2 \pi}\oint_{\partial{D}}d \varphi f^{i}A_{i}^{a}{\dot{A}}_{\varphi}^{a}.
\int_0^T \int_{I_\varepsilon^+}\partial_t s(t,x) s(t,x) dx dt = \varepsilon^{-2\gamma-\frac{3}{2}}\int_0^T \int_{I_\varepsilon^+}\Delta s(t,x) s(t,x) dx dt &+ \varepsilon^{-2\gamma-\frac{3}{2}}\int_0^T \int_{I_\varepsilon^+}\{f(v(t,x))-f(m(x)) \}s(t,x) dx dt,
G\left( \frac{\varpi^{-\rho}}{2}\left(\begin{matrix}-b -b -b -b\end{matrix}\right), \varpi^{r +\frac{t}{2}}\left(\begin{matrix}x -\frac{b}{vx}\end{matrix}\right) \right) = \gamma_F(-\frac{b}{2},\rho)q^{-\frac{\rho}{2}}
S = \int d^{4}x \sqrt{| \hat{g}_{E}|}\left[ \hat{R}( \hat{g}_{E}) -{\textstyle \frac{1}{2}}( \partial \hat{\phi})^{2}-{\textstyle \frac{1}{2}}e^{2 \hat{\phi}}( \partial \hat{a})^{2}-{\textstyle \frac{1}{4}}e^{- \hat{\phi}}\hat{F}^{I}\hat{F}^{I}+{\textstyle \frac{1}{4}}\hat{a}\hat{F}^{I}{}^{\star}\hat{F}^{I}\right] \, .
vol(G) = \int\limits_G \Dg = \int\limits_{\raise 1pt \hbox{$\scriptstyle G$}/ \lower 1pt\hbox{$\scriptstyle H_x$}}D[g] \hskip 4ptvol(H_x) = vol(H_x) vol\bigl(\!\!\raise 4pt \hbox{$\G$}\! / \!\! \!\lower 4pt\hbox{$\H_x$}\bigr)
\tau \partial_{t}\big(\mathbf{H}(t, \cdot) - \mathbf{F}(\mathbf{0})\big) +\big(\mathbf{H}(t, \cdot) - \mathbf{F}(\mathbf{0})\big) =\mathbf{F}\big(\nabla \mathbf{u}(t, \cdot)\big) - \mathbf{F}(\mathbf{0}) t > 0.
J_n=\left\{{n\mathcal{T}}_{\delta,M}(X^{(n)}) <{\mathcal{T}}^{\mathcal{E}}_{(M+c)\sqrt{n}}\right\}\cap \left\{{\mathcal{T}}^{\mathcal{E}}_{(M+1-c)\sqrt{n}}< n{\mathcal{T}}_{\delta,M+1}(X^{(n)}) \right\}.
( \Phi_{1}^{2}, \cdots , \Phi_{k}^{2}, \cdots , \Phi_{n_{c}- \tilde{n}_{c}}^{2}) = \Lambda^{2}( \omega , \cdots , \omega^{2 k - 1}, \cdots , \omega^{2 ( n_{c}- \tilde{n}_{c}) - 1}) ,
E(x,\dot{x},u)=\lambda\left( \sqrt{(u^{1})^{2}+(u^{2})^{2}}-\frac{\dot{x}^{1}u^{1}+\dot{x}^{2}u^{2}}{\sqrt{(\dot{x}^{1})^{2}+(\dot{x}^{2})^{2}}}\right) =\frac{\lambda}{\| \dot{x}\|}\left( \| u\| \| \dot{x}\| -(\dot{x}^{1}u^{1}+\dot{x}^{2}u^{2})\right).
Q^{(j)}(x'):=\Big(Q^{(j)}_{\alpha\beta}(x')\Big)_{1\leq\alpha,\beta\leq M}:=\Big((\partial_j E_{\alpha\beta})(x',1)\Big)_{1\leq\alpha,\beta\leq M},\qquad\forall\,x'\in{\mathbb{R}}^{n-1}.
\sum_{a=1}^{n-2}\sum_{m=1}^{n-a-1}f(a,m)=\sum_{a=1}^{n-2}\sum_{m=1}^{n-a}f(a,m)=\sum_{a=1}^{n-2}\left(a\binom{n+1}{a+2}-(a+1)\binom{n}{a+1}+n-a\right) &=\sum_{a=0}^{n-1}\left((a+2)\binom{n+1}{a+2}-(a+1)\binom{n}{a+1}-2\binom{n+1}{a+2}+n-a\right) &=(n+1)(2^n-1)-n2^{n-1}-2(2^{n+1}-n-2)+\binom{n+1}{2}&=(n-6)2^{n-1}+n+3+\binom{n+1}{2}.
\mathcal{L}_Z J_X= Z(J_X)-J_{\mathcal{L}_Z X}=\Im(D_Z\phi \cdot \overline{D_X\phi}+\phi\cdot\overline{D_Z D_X\phi}-\phi\cdot \overline{D_{[Z, X]\phi}}) &=\Im(\phi_1 \cdot \overline{D_X\phi}+\phi\cdot\overline{D_X\phi_1}+\phi\cdot \overline{([D_Z, D_X]-D_{[Z, X]})\phi}) &=\Im(\phi_l\cdot\overline{D_X\phi_{1-l}})-F_{ZX}|\phi|^2.
{\rm RC}(I^\vee)=\left(\bigcap_{i=1}^{s+1}H^+_{e_i}\right)\bigcap\left(\bigcap_{i=1}^qH^+_{(\delta_i,-f_i)}\right)\bigcap\left(\bigcap_{i=q+1}^{p_1}H^+_{(\delta_i,-f_i)}\right),
\|Tf\|_2 = \left(\sum_{n=0}^{\infty}\left(1-\dfrac{1}{n+1}\right)^2|a_n|^2\right)^{\dfrac{1}{2}}& < \Big(\sum_{n=0}^{\infty}|a_n|^2\Big)^{\frac{1}{2}}& =\|f\|_2 & \leq \|f\|_{\infty}.
\left( \begin{array}{c}{L}\\{M - g x g^{- 1}{\bf 1}_{B}}\\ \end{array}\right) = \left( \begin{array}{c c}{\tilde{g}}&{0}\\{0}&{\rho \cdot g}\\ \end{array}\right) \left( \begin{array}{c}{L}\\{M - x{\bf 1}_{B}}\\ \end{array}\right) g^{- 1}\cdot \rho^{- 1}
\frac{\partial}{\partial x_j}F(u(x)) = g'(|u(x)|) \frac{\partial|u(x)|}{\partial x_j}&= \frac{g'(|u(x)|)}{2|u(x)|}\left(u(x)\frac{\partial \overline{u}(x)}{\partial x_j}+\frac{\partial u(x)}{\partial x_j}\overline{u}(x)\right) &={\rm Re}\left( f(u(x)) \frac{\partial \overline{u}(x)}{\partial x_j}\right).
\mathcal{R}^+_2 = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{\frac{C_2}{m}\leq c \leq \frac{C_1}{m}}\frac{S(n, p; c)}{c}H_{m, n}^+ \Bigl( \frac{4 \pi \sqrt{n p}}{c}\Bigr),
{\cal L}={\frac{1}{2}}\partial_{\mu}\phi_{x}\partial^{\mu}\phi_{x}-{\frac{1}{2}}m_{0}^{2}\phi_{x}^{2}-{\frac{m^{2}}{\beta^{2}}}( 1 - c o s ( \beta \phi_{x}) ) \equiv{\frac{1}{2}}\partial_{\mu}\phi_{x}\partial^{\mu}\phi_{x}- U ( \phi_{x}) ,
g_k((t+s)/2) = \| A_k^{(t+s)/2}Z_kB_k^{t+s}Z^*_kA_k^{(t+s)/2}\|_{\infty}^{1/2}&=\rho^{1/2}( A_k^{t}Z_kB_k^{t+s}Z^*_kA_k^{s}) &\le \| A_k^{t}Z_kB_k^{t+s}Z^*_kA_k^{s}\|_{\infty}^{1/2}&\le \| A_k^{t}Z_kB_k^{t}\|_{\infty}^{1/2}\| B_k^{s}Z^*_kA_k^{s}\|_{\infty}^{1/2}&=\{g_k(t)g_k(s)\}^{1/2}.
\begin{array}{ll}Z= \frac{1}{2}( \mbox{Tr\thinspace}e^{-\beta(H^{(+)}-\mu\hat N)}+\mbox{Tr\thinspace}e^{-\beta(H^{(-)}-\mu\hat N)}+ \mbox{Tr\thinspace}\tau e^{-\beta(H^{(+)}-\mu\hat N)}- \mbox{Tr\thinspace}\tau e^{-\beta(H^{(-)}-\mu\hat N)})\end{array}
{\frac{\partial{\cal O}[ \tilde{\phi}( \kappa ) , \kappa ]}{\partial \kappa}}+ \int \tilde{\delta}_{i \B R S}[ \tilde{\phi}( \kappa ) , \kappa ] \Theta^{\prime}[ \tilde{\phi}( \kappa ) ]{\frac{\delta{\cal O}[ \tilde{\phi}( \kappa ) , \kappa ]}{\delta \tilde{\phi}_{i}( \kappa )}}= 0 ,
\tilde{a}_{\tilde{n}}^{\alpha}= \int_{0}^{\pi}\frac{d \sigma_{-}}{\pi}\exp \left[ 4 i \tilde{n}\frac{e_{\mu}X_{R}^{\mu}( \sigma_{-})}{e_{\mu}P_{R}^{\mu}}\right] \xi_{i}^{\alpha}\partial_{-}X_{R}^{i}( \sigma_{-}) ,
E[\prod_{i=\tau+1}^{n-1}F(S_i,\widehat{{S}}_i;S_{i+1},\widehat{S}_{i+1})] = E\prod_{i=\tau+1}^{n-1}[\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})}]E\prod_{i=\tau+1}^{n-1}[\frac{\lambda\rho(\widehat{S}_i)\rho(\widehat{S}_{i+1})}{1+\lambda\rho(\widehat{S}_i)\rho(\widehat{S}_{i+1})}].
\sum_{\substack{D \leq \exp((\log \log T)^2) \gcd(D,Q)=1}}\frac{\bar f_i(D)}{D}\ll\prod_{w(N)< p \leq \exp((\log \log T)^2)}\bigg(1 + \frac{|f(p)|}{Hp}\bigg)^{H-1}&\ll \exp\left((H-1) \sum_{\substack{p \leq \exp((\log \log T)^2)}}\frac{1}{p}\right) \ll (\log \log T)^{2H},
C_3\left[\left(\frac{m^2}{j}+m\right)\left(\frac{m^3}{(\beta j)^{5}}+\frac{m}{\beta j}\right)\right]=C_3\left(\frac{m^5}{j^{6}}+\frac{m^4}{j^5}+\frac{m^3}{j^2}+\frac{m^2}{j}\right)
\alpha^{\prime \prime \prime}(s)=\left(\begin{array}{c}0,\kappa^{\prime}\cosh \left( m\int \kappa (s)\,ds\right) +m\sinh \left(m\int \kappa (s)\,ds\right) \kappa^{2}(s), \sinh \left( m\int \kappa (s)\,ds\right) \kappa^{\prime}+m\cosh \left(m\int \kappa (s)\,ds\right) \kappa^{2}(s)\end{array}\right).
r \sum_{\tau = t+h-q}^n Z_{N,J}^{\tau}+n Z^{t+h-q-1}_{I,J}+\sum_{\tau=t}^{t+h-q-2}n^{t+h-q-\tau}Z^{\tau}_{i_{\tau},J}\leq r(q+1)+\sum_{\tau=1}^{h-q-1}n^{h-q-\tau}.
\begin{aligned}\Phi_{j}[(v_{1},v_{2})](t) &= i\int_{t}^{\infty}U(t-\tau)\left\{\tilde{{\mathcal N}}_{j}(v_{1}+u_{\mathrm{ap},1},v_{2}+u_{\mathrm{ap},2})-{\tilde{\mathcal N}}_{j}(u_{\mathrm{ap},1},u_{\mathrm{ap},2})\right\}(\tau)d\tau &+i\int_{t}^{\infty}U(t-\tau)\mathcal{E}_{j}(\tau)d\tau + \mathcal{R}_j\end{aligned}
\sum_{k=1}^{K+1}\mathbb{E}^2[R_k|\mathcal{F}_{k-1}]\leq \sum_{k=1}^{K+1}||R_k||_{\infty}^2\leq C\eta^{-2}C_{d,G}^{2(1+\eta)}(\log(G))^2K^{1+2\eta}.
{\cal F}_0(\beta )=(\det\zeta )^{-\frac 12}\exp\left[\frac 12\beta\eta^{*}\zeta^{-1}\beta +\beta ({\bf d}^{*}-\eta^{*}\zeta^{-1}{\bf d})+\frac 12{\bf d}\eta^{*}\zeta^{-1}{\bf d}-|{\bf d}|^2\right].
\Delta E=\frac{1}{2}\rho(v_{3}-v_{2})^{2}=\frac{1}{2}\rho(\frac{1}{\mu}-1)^{2}v_{2}^{2}=\frac{1}{2}\rho(\frac{1}{\mu}-1)^{2}(\frac{A_{1}}{A_{2}})^{2}v_{1}^{2}.
\#E(Q,Y_1) \le (2l-3)^q \prod_{i=1}^{2l-3}\#E(Q_i, Y_1) \le(2l-3)^q\prod_{i=1}^{2l-3}(32l^2)^{bq_i}\cdot (qb)^{2lq_i}\cdot\sigma^{6lq_i+4lb}&\le (2l)^q\cdot (32l^2)^{bq}\cdot (qb)^{2lq}\cdot\sigma^{6lq+8l^2b},
\mathbb{P}_{q,p}\left( \Omega_{\#}\right) =\mathbb{P}_{q,p}\left(\Omega_{\#}|\left\{s_{\#}\right\}\right) \mathbb{P}_{q,p}\left\{s_{\#}\right\}=\mathbb{P}_{q,p;s_{\#}^{\ast}}^{}\left( \Omega_{\#}\right) \mathbb{P}_{q,p}\left\{s_{\#}\right\}\ ,\ \#=b,f\ .
(\overline{\nabla}_V W-\overline{\nabla}_W V-[V, W])^K=\overline T_{IJ}{}^K V^I W^J, (\overline{\nabla}_V\overline{\nabla}_W Y-\overline{\nabla}_W\overline{\nabla}_V Y-\overline{\nabla}_{[V, W]}Y)^J=\overline R_I{}^J{}_{KL}Y^I V^K W^L
\lim_{m_{\pi}^{2}\rightarrow 0}m_{\pi}^{2}\int d \vec{x}\, \omega_{\pi}^{( 1 )}= - \lambda \int d \vec{x}\, ( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) \ne 0 .
\begin{array}{rcl}R(X,Y)\xi = d\alpha(X)(Y-\eta(Y)\xi)-d\alpha(Y)(X-\eta(X)\xi)+ \alpha\eta (X)(\alpha Y+\phi hY) begin{align*}+4pt] - \,\alpha \eta (Y)(\alpha X+\phi hX)+(\nabla_{X}\phi h)Y-(\nabla_{Y}\phi h)X. \end{array}
\left[\mathbf{a}\mathbf{b}\right]_{i}^{\,\,j}=a_{i}b^{j}\,\,\,\,\,\,\,\,\,\,\underrightarrow{\mathrm{contraction\,\,}}\,\,\,\,\,\,\,\,\,\,\mathbf{a}\cdot\mathbf{b}=a_{i}b^{i}
Z(u)=\prod_{p=1}^{\infty}\frac{\sin\left[\widetilde{\gamma}((2p-1)i\pi -u)\right]\sin\left[\widetilde{\gamma}(2pi\pi +u)\right]}{\sin\left[\widetilde{\gamma}((2p-1)i\pi +u)\right]\sin\left[\widetilde{\gamma}(2pi\pi - u)\right]}.
b_e(x_0)=\frac{3}{2}\frac{a(x_0)(e\cdot \nu_{x_0})}{(\nu_{x_0}\cdot A(x_0)\nu_{x_0})^{1/2}},b_{n+1}(x_0)=\frac{3}{2}\frac{a(x_0)}{(a^{n+1,n+1}(x_0))^{1/2}}.
\tilde{j}^{\mu_{1}\cdots \mu_{D - d}}= ( \frac 1{\sqrt{g}}) J^{\mu_{1}\cdots \mu_{D - d}}( \frac \phi x ) \sum_{i = 1}^{l}\beta_{i}\eta_{i}\int_{N_{i}}\delta^{D}( x - z_{i}( u ) ) \sqrt{g_{u}}d^{( D - d )}u ,
t \cdot [p_{\gamma_1, \alpha_1}(z_1), \; \ldots, \; p_{\gamma_n, \alpha_n}(z_n)] & =[p_{\gamma_1, \alpha_1}^\prime(z_1), \; \alpha_1^\vee(\lambda_{\alpha_1})s_1(t)p_{\gamma_2, \alpha_2}(z_2), \;p_{\gamma_3, \alpha_3}(z_3), \;\ldots, \; p_{\gamma_n, \alpha_n}(z_n)].
2 \int_{\mathcal Q_t}D(u) : D(w) = - \int_{0}^t \int_{0}^{L}\partial_t h(x,s)q_s(x,s){\rm d}x{\rm d}s + \int_{\mathcal Q_t}\partial_{xx}\psi ( \partial_x u_2 - 2\partial_{y}u_1).
\sum_{n=1}^{\nu_i(r)}Y_{i,q_i(n)}=\sum_{n=1}^{\nu^{(k)}(r)}Y_{i,b^{(k)}_n}\,\,\,\,\mbox{and}\,\,\,\,\sum_{n=1}^{\nu_i(r)}Y_{i,q_i(n),2^u}=\sum_{n=1}^{\nu^{(k)}(r)}Y_{i,b^{(k)}_n,2^u}
h^i_j =\frac{1}{\rho^2\sqrt{\rho^2+|D\rho|^2}}(e^{ik}-\frac{\rho^i\rho^k}{\rho^2+|D\rho|^2})(-\rho D_kD_j\rho+2\rho_k\rho_j+\rho^2e_{kj}) &=-\rho^{-2}\rho^i_{j}+\rho^{-1}\delta^i_{j}.
\rho^{(\vec k)}_i(j) =\begin{cases}-\infty j\le \sum_{s<i}k_s\ , begin{align*}.5ex]j- \sum_{s<i}k_s \sum_{s<i}k_s<j\le\sum_{s\le i}k_s\ , begin{align*}.5ex]\infty j>\sum_{s\le i}k_s\ .\end{cases}
\varphi (g^{n}_1 g_3) = \varphi \big( g^{j}_1 \big) \varphi \big( g^{i}_1 g_3 \big) \overset{(\ref{eq:5})}{=}\varphi \big( g^{j}_1 \big) \varphi (g_3) \varphi \big( g^{i}_1 \big) \overset{(\ref{eq:9})}{=}\varphi \big( g_3 g^{j}_1 \big) \varphi \big( g^{i}_1 \big) \,.
-i\frac{\delta}{\delta J_+(t_1)}\langle t_2|t_2\rangle \Bigg|_{+=-}= i\frac{\delta}{\delta J_-(t_1)}\langle t_2|t_2\rangle\Bigg|_{+=-}=\langle t_2|\Phi(t_1)|t_2 \rangle.
\omega(p) =\begin{cases}\#\{1\le n\le p: \prod_{i=1}^k L_i(n)\equiv 0\pmod{p}\}, p\nmid B, 0, p|B,\end{cases}\phi_\omega(d) =\prod_{p|d}(p-\omega(p)), \mathfrak{S}_D(\mathcal{L}) =\prod_{p\nmid D}\Bigl(1-\frac{\omega(p)}{p}\Bigr)\Bigl(1-\frac{1}{p}\Bigr)^{-k}.
- 2 i \int \frac{d^{\, 4}P}{(2 \pi)^4}\, e^{- i P \cdot (x - y)}\sum_{\tau = \pm}\theta (\tau p_0) \left[ P \cdot \partial_X f^{(\tau)}(X;{\bf p}) \right] \hat{\Delta}^{(\tau)}(X; P) = 0 ,
\max\left(\frac{h(b-1)}{4b-2},\frac{h}{3b},\frac{h(b+1)}{4b}\right) \le a < \frac{h(bc+1)}{3b+bc}, r(N) =\frac{1}{N^{a+\frac{a}{3b-1}-\frac{h}{3b-1}}\log^{\frac{h}{3b-1}}(N)}\rightarrow 0.
1= \left(\frac{b}{p^\sharp}\right)^{\frac{p}{2\cdot p^\sharp}}(S^\sharp)^{-1}(B(\varepsilon_g))^{1-\frac{p}{2\cdot p^\sharp}}+\varepsilon_g^{\frac{p}{2\cdot p^\sharp}}(S^*)^{-\frac{p_s^*}{2\cdot p^\sharp}}(B(\varepsilon_g))^{\frac{p_s^*-p}{2\cdot p^\sharp}}.
| \varphi^{0}\rangle = \int d \varepsilon A_{0}( \varepsilon ) | Y_{1 , 0}^{\varepsilon}\rangle + \frac{1}{\sqrt{2}}D_{0}( 2 \sqrt{2}) \tilde{Q}| 0 , 2 \sqrt{2}\rangle^{x , \phi}| 0 \rangle^{b , c}.
\begin{array}{lll}|u_{n}(t,x) - u_{n}(t,x')| \leq \int_{\mathcal{C}^d}|K(x-X_t(\omega))-K(x'-X_t(\omega))| \exp(M_{\Lambda}T) dm^{n}(\omega), \leq \exp(M_{\Lambda}T) L_K|x-x'|, \end{array}
Y(W,[G]) \leq Y^1(W,G) \leq \inf_{\substack{V\in Gr_{u_i}^1(C^{\infty}_c(W)) v\in V-\{0\}}}\frac{\int_W a_k|\nabla v|^2_G+s_Gv^2dv_G}{\int_Wu^{p_{k}-2}_iv^2dv_G}&\leq \int_W a_k|\nabla u_i|^2_G+s_Gu_i^2dv_G=J(u_i)\underset{i\to +\infty}{\longrightarrow}Y(W,[G]),
\begin{array}{| c | c | c | c |}\hline{}&{[ e ]}&{[ r^{k}]}&{[ s ]}\\ \hline{J_{0}}&{1}&{1}&{1}\\{J_{1}}&{1}&{1}&{- 1}\\{\alpha_{j}}&{2}&{q^{j k}+ q^{- j k}}&{0}\\ \hline{}\\ \end{array}
\mathcal{L}(v) = \sum_{i=1}^n cp_i\max\{-x_i-y_i,-d_i\}+ \sum_{i\neq j}\alpha_{ij}v_{ij}- \langle\lambda,v\rangle +\sum_{i\in I}\gamma_i(\sum_{j\neq i}v_{ij}- x_{0i}).
a_{p}(t)=-i\int_{t_{0}}^{t}dt^{\prime}\langle p|e^{-i\int_{t^{\prime}}^{t}\hat{H}(t^{\prime\prime})dt^{\prime\prime}}\hat{H}_{I}(t^{\prime})e^{-i\int_{t_{0}}^{t^{\prime}}\hat{H}_{0}(t^{\prime\prime})dt^{\prime\prime}}|\psi(t_{0})\rangle(1.4)
\begin{aligned}\frac{\chi_{\alpha}}{\chi'_{\alpha}}(\iota_{F_{\alpha}}\alpha(\mathrm{Nm}_{S(E)/S(F)}t)) =\frac{\chi_{\alpha}\circ \mathrm{Nm}_{E_{\alpha}/F_{\alpha}}}{\chi'_{\alpha}\circ \mathrm{Nm}_{E_{\alpha}/F_{\alpha}}}(\iota_{E_{\alpha}}\alpha(t)) &=\frac{\chi_{\mathrm{BC},\alpha}}{\chi'_{\mathrm{BC},\alpha}}(\iota_{E_{\alpha}}\alpha(t)),\end{aligned}
qL_{X\wedge Y}i_Z\omega =(-1)^{l}i_Y(\nabla_X\omega)(Z)+(-1)^{kl}\omega(Y\wedge \nabla_X Z) &+(-1)^{k(l-1)}i_X(\nabla_Y\omega)(Z)+\omega(X\wedge \nabla_YZ) &+(-1)^l\omega(T(X,Y)\wedge Z),
\tilde \varrho_N^\varphi(x) = \frac 1N \sum_{j=1}^N \varphi (x-y_j)\,, \tilde \varrho_N^\varphi \tilde u_N^\varphi(x) = \frac 1N \sum_{j=1}^N \varphi (x-y_j) w_j\,, \tilde \varrho_N^\varphi \tilde T_N^\varphi(x) = \frac 1{Nd}\sum_{j=1}^N \varphi (x-y_j) |w_j - \tilde u_N^\varphi(x)|^2 \,,
\mathcal{F}_o: = \{(\hat{g},\hat{b}_a)\in SE(3) \times \mathbb{R}^6 : \mathcal{U}(\tilde{g}) - \min_{g_q\in \mathbb{Q}}\mathcal{U}(\tilde{g}g_q) \leq \delta \}, \mathcal{J}_o: = \{(\hat{g},\hat{b}_a)\in SE(3)\times \mathbb{R}^6: \mathcal{U}(\tilde{g}) - \min_{g_q\in \mathbb{Q}}\mathcal{U}(\tilde{g}g_q) \geq \delta \},
\sum q^{\frac{1}{2}p^2}=\sum_n q^{\frac{15}{2}\left(2n+\frac{5m_1+3m_3}{30}\right)^2}=\sum_n q^{30\left(n+\frac{5m_1+3m_3}{60}\right)^2}=\theta_{5m_1+3m_3,30}\:(\tau).
w\ast\rho(x) & \leq k \int_{-\infty}^x w(x-y) \rho(x) dy + k \int_x^{x_1}w(x-y) e^{\frac{\tilde{\delta}_1(c)+\epsilon}{\sigma}(y-x)}\rho(x) dy &{}+ \int_{x_1}^{x_2}w(x-y) dy \max_{x_1\leq y \leq x_2}\rho(y) + k \int_{x_2}^{\infty}w(x-y) e^{\frac{\tilde{\delta}_2(c)+\epsilon}{\sigma}(y-x_2)}\rho(x_2) dy & =: I_1+I_2+I_3+I_4.
\frac{1}{2 \pi}\: \int_{0}^{2 \pi}\! \Phi_{\omega}( \sigma ) \: \Phi_{\omega^{\prime}}( \sigma ) \star 1 = ( \delta_{\omega , \omega^{\prime}}+ \delta_{\omega , - \omega^{\prime}}) < \Phi_{\omega}, \Phi_{\omega^{\prime}}>
\sum_{m=k-1}^{n-1}{m \choose k-1}S_{1,\lambda}(n-1,m) E[(U_1+\cdots +U_{k-1}+1)^{m-k+1}] &={n-1 \choose k-1}S_{1,\lambda}(n-1,n-1) E[(U_1+\cdots+U_{k-1}+1)^{n-k}&+ \sum_{m=k-1}^{n-2}{m \choose k-1}S_{1,\lambda}(n-1,m) E[(U_1+\cdots +U_{k-1}+1)^{m-k+1}].
\varphi\left(\frac{\partial}{\partial x_{i}}\right) =\frac{\partial}{\partial y_{i}},\, \varphi\left(\frac{\partial}{\partial y_{i}}\right)=\frac{\partial}{\partial x_{i}},\, \varphi \left(\frac{\partial}{\partial t}\right)=0,\,\xi=\frac{\partial}{\partial t},\,\eta=dt, g =\sum(dx_{i})^{2}-\sum(dy_{i})^{2}+(dt)^{2},\ \forall i\in\{1,2\}.
d s^{2}= d \tilde{x}^{\tilde{m}}\tilde{g}_{\tilde{m}\tilde{n}}d \tilde{x}^{\tilde{n}}= e^{m}\eta_{m n}e^{n}+ e^{\rho}e^{\rho}= \rho^{2}d x^{2}+ R^{2}\left( \frac{d \rho}\rho \right)^{2},
\begin{array}{l c r}{{( | W_{1}|^{2}+ | Z_{2}|^{2}) - ( | Z_{1}|^{2}+ | W_{2}|^{2}) = R_{2}}}\\{{Z_{1}W_{1}+ Z_{2}W_{2}= \overline{{{Z}}}_{1}\overline{{{W}}}_{1}+ \overline{{{Z}}}_{2}\overline{{{W}}}_{2}= 0 ,}}\\ \end{array}
\lim_{m\to\infty}\frac{1}{m!}\sum_{\sigma\in[m]}|d(F^\sigma, G^\sigma)|^\alpha =\lim_{m\to\infty}\frac{1}{m!}\sum_{\sigma\in[m]}|d((F-G)^\sigma, 0)|^\alpha &= \sum_{n=1}^\infty\frac{1}{n^\alpha}(1-p)^{n\choose 2}(1-(1-p)^n) &\leq p\sum_{n=1}^\infty\frac{1}{n^{\alpha-1}}(1-p)^{n\choose 2}&\leq p\sum_{n=1}^\infty\frac{1}{n^{\alpha-1}}
D_{pert,s\chi_0\eta_0}(\psi+s\mathfrak{c}_0^g-s\mathfrak{h}^g_0+s\mathfrak{e}^g_0 +s\mathfrak{k}^b_{0,s}+s\mathfrak{k}^{\perp}_{0,s}) &=s^2\mathcal{A}+s^2\mathcal{B}+s\mathcal{C}\mbox{}mod(\ker(D|_{L^2_1}))
V=G\big(\frac{m_{1}m_{2}}{\mid\pmb{x}_{2}-\pmb{x}_{1}\mid}+\frac{m_{1}m_{3}}{\mid\pmb{x}_{3}-\pmb{x}_{1}\mid}+\frac{m_{2}m_{3}}{\mid\pmb{x}_{3}-\pmb{x}_{2}\mid}\big)
T_{1,\alpha}= \frac{1}{2 \pi i}\oint_{C}\frac{1 - z^{\alpha}}{(1-z)z^{4g+1}}\sum_{D \in \mathcal{H}_{2g+1}}\mathcal{L}\left( \frac{z}{\sqrt{q}}, \chi_D \right)^4 \, du,
\int_{-\infty}^{\infty}y^2 p_{_1}(y) dy = \frac{2}{\sqrt{2 \pi}\sigma_w}e^{-\rho_0 -1}\int_{0}^{\infty}y^2 e^{-\left(\frac{1}{2 \sigma_w^2}+\rho_1\right) y^2}dy &=\frac{e^{-\rho_0 -1}\sigma_w^2}{(1 + 2\rho_1 \sigma_w^2)^{3/2}},
2C_j\,[C_{ik},C_{k\ell}] +(C_{ij}-C_i-C_j)[C_{k\ell},C_{jk}] +(C_{j\ell}-C_j-C_\ell) [C_{jk},C_{ik}] -(C_{jk}-C_j-C_k)[C_{ij},C_{j\ell}] =0\,.
\textrm{Pr}\left\{\left(d^{[k,i]}\right)^2\delta_1\cdot\mathsf{SNR}\le \epsilon'\right\}=1-\left(1-{\epsilon'}^{S-1}\delta_1^{-S+1}\left(\mathsf{SNR}\right)^{-(S-1)}\right)^{N_f}.
- 4 m^{d + 1}\, \, \frac{q ( \vec{x})}{| \vec{v}( \vec{x}) |}\sum_{m = 1 , 3 , 5 , . .}^{d / 2}{\binom{\frac{d}{2}}{m}}( \partial_{i}\hat{n}^{a}\, \partial_{i}\hat{n}^{a})^{( \frac{d}{2}- m )}| \vec{v}( \vec{x}) |^{m}
J^{*}[\psi^{(3)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p\to \infty}}$\cr}}}\sum\limits_{j_1, j_2, j_3=0}^{p}C_{j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)},
c = \left(1\ 2\ 3\ 3\ 1\ 2\right) A = \left(\begin{array}{cccccc}1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 \end{array}\right).
\hat{F}(p) =\int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty F(p\omega+t_1\omega_1^\bot+\cdots+ t_{n-1}\omega_{n-1}^\bot)dt_1...dt_{n-1}=\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}