harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\partial_{t}G(z) = -6z_t^2Az-6zz_{tt}Az-12z z_t A z_t -3z^2 Az_{tt}&+ 6z_{tt}|\nabla z|^2+24z_t(\nabla z \cdot \nabla z_t) +12z|\nabla z_t|^2+12z(\nabla z_t \cdot \nabla z_{tt}).\end{aligned}
g \cdot g^{\prime}= \left( \begin{array}{r r}{a \cdot a^{\prime}+ b \cdot c^{\prime}}{a \cdot b^{\prime}+ b \cdot d^{\prime}}{c \cdot a^{\prime}+ d \cdot c^{\prime}}{c \cdot b^{\prime}+ d \cdot d^{\prime}}\end{array}\right)
\begin{aligned}x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0 & x_0^2 - \beta \gamma x_1^2 - \gamma x_2^2 + \beta x_3^2 = 0 x_0^2 + \gamma x_1^2 - \alpha \gamma x_2^2 - \alpha x_3^2 = 0 x_0^2 - \beta x_1^2 + \alpha x_2^2 - \alpha x_3^2 = 0.\end{aligned}
\Delta L = -{\frac{1}{1 2 \kappa^{2}}}\int d^{1 1}x \left( \partial_{1 1}C_{A B C}-{\frac{\sqrt{2}}{1 6 \pi}}\left({\frac{\kappa}{4 \pi}}\right)^{2 / 3}\overline{{\chi}}\Gamma_{A B C}\chi \ \delta ( x^{1 1}) \right)^{2}\ ,
\langle S_{I}[ \phi_{<}, \phi_{>}]_{2}\rangle_{0}= \int a ( t ) d t \int d^{3}\vec{k}( - \frac{1}{2}) \phi_{<}( \vec{k}, t ) \ \delta \tilde{m}^{2}( t , s )_{1}\, p h i_{<}( \vec{k}, t )
\forall x^X,{x'}^X,\gamma^1,{\gamma'}^1\big(\gamma>_\mathbb{R}0\land\tilde{\rho}>_\mathbb{R}-\gamma\land x\in\mathrm{dom}(J^A_\gamma) &\qquad\qquad\qquad\qquad\land\gamma=_\mathbb{R}\gamma'\land x=_Xx'\rightarrow J^A_\gamma x=_XJ^A_{\gamma'}x'\big).
{\cal L}= - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}( \partial_{\mu}\phi )^{2}+ e ( g^{\mu \nu}- \epsilon^{\mu \nu}) ( \partial_{\mu}\phi ) A_{\nu}+ \frac{1}{2}e^{2}\alpha A_{\mu}^{2},
\sum_{i=1}^{k r}\xi_i = \sum_{i=1}^{k r}\eta_i = \sum_{i=1}^r \sum_{j=0}^{k-1}v_{i,j}= \sum_{i=1}^r \sum_{j=0}^{k-1}\frac{v_i + j}{k}= \sum_{i=1}^r v_i + \frac{(k-1) r}{2},
\hat u_t = D\Delta \hat u+\lambda H( u+u_0) x\in \Omega, 0<t<T \hat v_t =\tilde D\Delta_{M}\hat v+\lambda F(u+u_0, v+v_0) x\in M, 0<t<T D\frac{\partial \hat u}{\partial \eta}= \lambda G(u+u_0, v+v_0) x\in M, 0<t<T \hat u = \lambda u_0 x\in\Omega , t=0 \hat v =\lambda v_0 x\in M ,\quad t=0
\lim_{n\rightarrow \infty}\left(\frac{\lambda_{j,n}}{\lambda_{j',n}}+ \frac{\lambda_{j',n}}{\lambda_{j,n}}+ \frac{|x_{j,n}-x_{j',n}|}{\lambda_{j,n}}+ \frac{|t_{j,n}-t_{j',n}|}{\lambda_{j,n}}\right) = +\infty.
{\rm mult}_\lambda(\mathbf{m}) =\sum_{\substack{\sigma\in S_n \sigma(1)=1}}\varepsilon(\sigma)Q\Big(\sum_{j=2}^n(\ell_1+1-\ell_j)e_j-\sigma\Big(\sum_{j=2}^nje_j\Big)\Big) =\sum_{\substack{\sigma\in S_n \sigma(1)=1}}\varepsilon(\sigma)Q\Big(\sum_{j=1}^{n-1}(\ell_1-\ell_{j+1})e_{j+1}-\sigma\Big(\sum_{j=1}^{n-1}je_{j+1}\Big)\Big).
2 ( 2 a )^{- s}\frac{1}{\Gamma ( s )}\sum_{k = 1}^{\infty}( - 1 )^{k}\int_{0}^{\infty}d t \, t^{s}\frac{e^{- ( \frac 3 2 + \frac{c}{2 a}) t}}{1 - e^{- t}}\frac{1}{( k \pi )^{2}+ ( \frac t 2 )^{2}}
\tilde g_{ij}:=[\frac{1}{2}(\bar p^*F)^2]_{\tilde y^i\tilde y^j}=\frac{1}{2}\frac{\partial^2((\bar p^*F)^2)}{\partial \tilde y^i\partial \tilde y^j}=\frac{1}{2}\frac{\partial^2(\bar p^*F^2)}{\partial \tilde y^i\partial \tilde y^j}.
I_3 = \int \sum_{k}\bar{R}_k w_{k,\alpha}\sum_{j \leq l}b_{l}r_{j,\alpha}\, d\alpha -\int \sum_{j,k \leq l}\bar{R}_k w_{k,\alpha}b_{l}r_{j,\alpha}\, d\alpha:= I'_3 - I''_3.
Q = \int d^{3}x \left[ \sum_{l = 1}^{1 2}\left({\mathbf{\sigma}}\cdot{\bf F}^{\dagger l}\psi^{l}( x ) \right) + \sigma^{\mu}\partial_{\mu}\phi^{\dagger}( x ) \cdot \psi_{L}( x ) + \sigma^{\mu}\partial_{\mu}\phi_{s}^{\dagger}( x ) \cdot \psi_{L}( x ) + \mathcal{W}~ \psi^{1}\right] .
0 =\sum_{i=1}^{2n}e^{-f}(\nabla_{\nabla_{\pi^*e_i}\pi^*e_i}\pi^*\Omega - \nabla_{\pi^*e_i}\nabla_{\pi^*e_i}\pi^*\Omega-\pi^*\theta(\nabla_{\pi^*e_i}\pi^*e_i)\pi^*\Omega &+ 2\pi^*\theta(\pi^*e_i)\nabla_{\pi^*e_i}\pi^*\Omega - (\pi^*\theta(\pi^*e_i))^2\pi^*\Omega +\pi^*e_i(\pi^*\theta(\pi^*e_i))\pi^*\Omega).
\begin{aligned}\widehat{F}(\widehat{x}) :=|(0,\widehat x)\cap \widehat{I}|,0\leq \widehat{x}\leq \widehat{r}, \widehat{F}(\widehat{x}) :=-|(\widehat x,0)\cap \widehat{I}|,\widehat{l}\leq \widehat{x}< 0,\end{aligned}
\sum_{l=n_{k}/2+1}^{n_{k}}\int_{E_{k}\cap\{\varphi_{E_{k}}>n_{k}-l\}}\widehat{T}^{l}u\,d\mu \leq\int_{E_{k}}\sum_{l=n_{k}/2+1}^{n_{k}}\widehat{T}^{l}u\,d\mu & =\mu(E_{k})o\left( a(n_{k})\right) & \longrightarrow0k\rightarrow\infty
\mathcal{S}:=\exp(\Pi_S^{\perp}(w \partial_x^{-1}))\Pi_S^{\perp}=\Pi_S^{\perp}(\mathrm{I}+w \partial_x^{-1})\Pi_S^{\perp}+\hat{\mathcal{S}}, \hat{\mathcal{S}}:=\sum_{k\geq 2}\frac{1}{k!}[\Pi_S^{\perp}(w\partial_x^{-1})]^k \Pi_S^{\perp},
T^{\mu}\_{\nu}( \eta ) = \left( \begin{array}{c c}{H ( \eta ) c^{- 1}( \eta )}&{0}\\{0}&{T_{\beta}^{\beta}- c^{- 1}( \eta ) H ( \eta ) \}\\ \end{array}\right) + c^{- 1}\left( \begin{array}{c c}{\zeta}&{\alpha^{\prime}}\\{- \alpha^{\prime}}&{- \zeta \}\\ \end{array}\right)
\hat\psi(c_{i_1})=\hat\psi(c_{i_3})=\cdots =\hat\psi(c_{i_{2m-1}})=1, \hat\psi(c_{i_{2j}})=s_{l_{i_{2j}}}s_{r_{i_{2j}}}^{-1},\textrm{for}j=1,\ldots, m, \hat\psi(c_{q})=s_{l_q},\textrm{for}i_{2j+1}< q< i_{2(j+1)},\textrm{and}\hat\psi(c_{q})=s_{r_q}^{-1},\textrm{for}i_{2j}< q< i_{2j+1}.
\frac{1}{2}(S^- - (S^-)^\dagger) = \int_M \left(d^4x \frac{i}{2}\partial_\mu({\tilde{\overline\Psi}}_L\gamma^\mu{\tilde\Psi}_L)-\frac{i}{4}{\tilde{\overline\Psi}}_L\gamma^A{\tilde\Psi}_Le^{-1}\epsilon_{ABCD}T^B\wedge e^C \wedge e^D \right).
\lambda_t\lambda_{j_1}= \lambda_t\lambda_{j_1^*}= \lambda_t\lambda_{j_2}= \lambda_t\lambda_{j_2^*}=\lambda_{j_1}\lambda_{j_2}=\lambda_{j_1}\lambda_{j_2^*}= \lambda_{j_1^*}\lambda_{j_2}=\lambda_{j_1^*}\lambda_{j_2^*}
c_n^c(u_0,f) = \frac{1}{2 \pi}\Big( 3 \left\| U \right\|_{L^{\infty}(n,n+1)}+ \left\| U' \right\|_{L^1(n,n+1)}\Big) \, m_n^{-1}& \leqslant \frac{\beta_- - 1}{2 \pi c_-}\Big( 3 \times 2^{-\mu+\alpha+\beta_-}\, M \, + 2^{-\mu + \beta_-}\left(2^{\alpha}M + M' \right) \Big) \, |n|^{\mu-2-\alpha+\beta_-}\; .
\begin{array}{c | c | c | c}{{H_{D}}}&{{H_{D}^{c}}}&{{H_{T}}}&{{H_{T}^{c}}}\\ \hline{{n / R}}&{{( n + 1 ) / R}}&{{( n + \frac{1}{2}) / R}}&{{( n + \frac{1}{2}) / R}}\\ \end{array}
\Phi_{\widetilde{F}}(z_0,x_0) = \Phi_{\widetilde{F}}\circ U\circ\boldsymbol{F}_{n^2+\kappa_0, n^2}(z,x) = \Phi_F \circ \boldsymbol{F}_{n^2+\kappa_0, n^2}(z,x) = \Phi_F \circ F^{\kappa_0}(z,x) +o(1) = \Phi_F(z,x) + \kappa_0 +o(1)
6P_{n}(\tfrac{x}{|x|}) = 6|x|^{-3}P_{n}(x) = \tfrac{(n+1)^{3}}{n^{3/2}(n+1)^{3/2}}\tfrac{(n-1)n^{3/2}(n+2)^{3/2}}{(n+1)^{3}}= n-1.
\phi - \phi_{0}= \pm \sqrt{\frac{2 \Omega_{\phi}}{3 \Omega_{B}}}F \left[ \operatorname{a r c c o s}\left( \frac{1 - \sinh ( a_{0}t )}{1 + \sinh ( a_{0}t )}\right) , \frac{1}{\sqrt{2}}\right] ,
p\left({{d_{pF}}\leq \delta}\right) = \frac{{M{\delta^{2kn - 2{k^2}}}}}{{k!\Gamma \left({nk -{k^2}+ 1}\right)}}\,\,\, \times \prod\limits_{i = 0}^{k - 1}{\Gamma \left({n - 2k + 1 + i}\right)\Gamma \left({2 + i}\right)}
q^{2 l}\prod_{{\stackrel{r = 1}{r \not = s}}}^{l}\frac{q \lambda_{r}^{2}- q^{- 1}\lambda_{s}^{2}}{q^{- 1}\lambda_{r}^{2}- q \lambda_{s}^{2}}= \left[ \frac{( 1 - \Delta^{2}\mu \lambda_{s}^{2}q ) \left( 1 - \Delta^{2}\frac{\mu}{\lambda_{s}^{2}}q^{- 1}\right)}{( 1 - \Delta^{2}\mu \lambda_{s}^{2}q^{- 1}) \left( 1 - \Delta^{2}\frac{\mu}{\lambda_{s}^{2}}q \right)}\right]^{N / 4}\, .
X_{a a}={\frac{e^{4}}{1 6 \, m^{2}( p k )^{2}}}\mathrm{T r}\left[ \gamma^{\beta}( \gamma^{\mu}( p + k )_{\mu}+ m ) \gamma^{\alpha}( \gamma^{\nu}p_{\nu}+ m ) \gamma_{\alpha}( \gamma^{\rho}( p + k )_{\rho}+ m ) \gamma_{\beta}( \gamma^{\lambda}{p^{\prime}}_{\lambda}+ m ) \right]
|G_{\lambda + m c^2}(x) - K_\lambda(x) P_+| \leq |t_1(x)| + |t_2(x)| \leq \frac{\kappa_5(m, \lambda)}{c}\begin{cases}|x|^{-2}, ~ |x| < R, e^{-\kappa_6(m, \lambda) |x|}, ~|x| \geq R. \end{cases}
\epsilon^{\alpha \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\beta \sigma \right\}}+ \epsilon^{\beta \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\sigma \alpha \right\}}+ \epsilon^{\sigma \mu \nu \xi}\Omega_{\left( \mu \nu \xi \right)}^{\left\{\alpha \beta \right\}}= 0 ,
\begin{aligned}\log |E_I(re^{I\theta})| =\frac1{2\pi}\int_{-\pi}^{\pi}P_r(\theta-t)\log |\tilde f_I(e^{It})|dt &=\frac1{2\pi}\int_{-\pi}^{\pi}P_r(\theta-t)\log |\tilde g_I(e^{It})|dt \ge \log |g_I(re^{I\theta})|\end{aligned}
f(x,\sigma^{2}|\mu,\gamma,\alpha,\beta)=\frac{1}{\sigma\sqrt{2\pi\gamma}}\frac{\beta^{\alpha}}{\Gamma(\alpha)}(\frac{1}{\sigma^{2}})^{\alpha+1}exp(-\frac{2\gamma\beta+(x-\mu)^{2}}{2\gamma\sigma^{2}})
\lim_{j \to \infty}\frac{\int_0^b \frac{| w' |^p F_j^p}{L_h(\gamma_j)^{p-1}}\,dt}{\int_0^b | w |^p L_h(\gamma_j) \,dt}= \frac{\int_0^b \frac{| w' |^p F_\gamma^p}{\ell^{p-1}}\,dt}{\int_0^b | w |^p \ell \,dt}\le \frac{\int_0^b \frac{| w' |^p F_\gamma^p}{|\gamma'|_h^{p-1}}\,dt}{\int_0^b | w |^p |\gamma'|_h \,dt}
F_k^{ij}=\dfrac{(f^k,H_i)}{(f^k,H_j)}\ (0\leqslant k\leqslant 2,\ 1\leqslant i,j\leqslant 2n+2), V_i =( (f^1,H_i),(f^2,H_i),(f^3,H_i))\in\mathcal M_m^3,
\big(\bar T_N^\varphi - \tilde T_N^\varphi\big)^2 \le \bigg(\sum_{j=1}^N (p_{1,j}+ q_{1,j})|w_j-\tilde u_N^\varphi|^2\bigg)^2 \le C \sum_{j=1}^N (p_{1,j}+q_{1,j}) |w_j|^4 + C |\tilde u_N^\varphi\big|^4\,,
\sum_{R=0}^{N-K}T(R) =\sum_{R=0}^{N-K}\sum_{m \ge 1}\frac{r(m+R)}{2^m}&\le \sum_{R=1}^\infty r(R) \cdot \sum_{m=\max\{R-N+K,1\}}^\infty \frac{1}{2^m}&\le \sum_{R=1}^N r(R) + \sum_{R=N+1}^\infty \frac{r(R)}{2^{R-N+K}}.
\wp(p_{C}^{(0)}(\tau)|\tau)=\frac{(\frac{g_{2}(\tau)}{2}-3e_{k}(\tau)^{2})(C_{1}\eta_{1}(\tau)-\eta_{2}(\tau))+\frac{g_{2}(\tau)}{4}e_{k}(\tau)(C_{1}-\tau)}{3e_{k}(\tau)(C_{1}\eta_{1}(\tau)-\eta_{2}(\tau))+(\frac{g_{2}(\tau)}{2}-3e_{k}(\tau)^{2})(C_{1}-\tau)}
\mathcal{F}(\mathcal{L};q,t)=\sum_{g\geq0}\left(\sum_{l=1}^{L}\frac{(-1)^{l-1}}{l}\sum_{\sum_{s=1}^lg_s=g}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^lh_{2g_s-I_s}^{\mathcal{L}_{\mathbf{I}_s}}(t)\right)z^{2g-L}.
A\subset \bigcap_{i=1}^{d}\left\{\left|B_{s_{1}}^{i}-B_{t_{0}}^{i}\right|<2\eta\right\}\cup\bigcup_{i=1}^{d}\left\{\sup_{s\in I}\left|B_{s_{1}}^{i}-B_{s}^{i}\right|>\eta\right\}\cup\bigcup_{i=1}^{d}\left\{\sup_{t\in J}\left|B_{t}^{i}-B_{t_{0}}^{i}\right|>\eta\right\},
\hat p_-(x_N) = -\mu_{-}\frac{(A+B_-)}{A}(i\xi'\cdot\beta'_- + B_-\beta_{-N})e^{Ax_N}&= -\mu_-(A+B_-)\Bigl\{\sum_{m=1}^{N-1}P^-_{m, 0}\hat h_\ell(0)+ AP^-_{N,0}\hat H(0)\Bigr\}e^{Ax_N},
\inf_{\substack{\eta,{\zeta}\in V, \boldsymbol{\tau},\boldsymbol{\rho}\in H(\emph{div},\Omega), v_{\zeta}= \mathbb{P}_{[u_a,u_b]}\{u_d - \frac{1}{\lambda}{\zeta}\}, \alpha,\beta > 0}}\mathcal{M}^\oplus(\alpha,\beta;\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho},v_{\zeta}) = 0.
\psi(x,t)=\psi_f(x,t)+\frac{\theta(t-|x-y|)}{4\pi|x-y|}\xi(t-|x-y|)-\frac{m}{4\pi}\int_0^t\frac{\theta(s-|x-y|)J_1(m\sqrt{s^2-|x-y|^2})}{\sqrt{s^2-|x-y|^2}}\xi(t-s)ds.
\widehat{\gamma}_{1}^{\left( BMN\right)}=\gamma_{1}+k^{-1/2}\Lambda\left(\mathbf{W}\right) +\frac{\mathbf{A}_{0}\left( n/k\right)}{1-\rho_{1}}\left( 1+o_{\mathbf{P}}\left( 1\right) \right) ,
M(\theta)= \sup_{x\in E}M(r,x)\leq c_f \gamma_0^{\frac{1}{\gamma_0 - 1}}\int_0^\theta \big\| M(r u^{\frac{1}{\gamma_0 - 1}}) \big\|_{\mathbf 1_{0\leq u\leq 1}\frac{du}{u};\gamma_0 - 1}dr.
s_j = \prod_{k=1}^n e^{2 \pi i a_{j,k}z_k'}= \prod_{k=1}^r e^{2 \pi i a_{j,k}\varphi_k(v^{-1,-1})}\cdot\prod_{k=r+1}^n \bigl( e^{2 \pi i z_k'}\bigr)^{a_{j,k}}.
f(r)=r^k-\frac{4}{3}\beta r^{k-1}-\frac{4}{3}\gamma r^{k-2}-\frac{4}{3}\alpha^2= r^{k-2}\left (r^2-\frac{4}{3}\beta r-\frac{4}{3}\gamma\right )-\frac{4}{3}\alpha^2 := r^{k-2}g(r)-\frac{4}{3}\alpha^2
\delta_{2}^{\prime}(n)=\frac{\pi}{4}(\frac{c_{1}(4n+1)}{1}-\frac{c_{3}(4n+1)}{3}+\frac{c_{5}(4n+1)}{5}-\frac{c_{7}(4n+1)}{7}+\cdot\cdot\cdot).
\int_\Omega \mathbb{P}[\mu] \nabla p \cdot \nabla \phi \,\delta_\Gamma(x)\, \d x = \int_0^1 \bar{C}\partial_s p \, \partial_s \phi \,\d s = G \int_0^1 \partial_s \phi(x(s))\, \d s = G(\phi(x^-)-\phi(x^+)).
p_{1,k_1,k_2}^{(m)}(x,t)= \sum_{k=1}^{\frac{m-1}{2}}\frac{c_{m,k}(-1)^k}{(2\pi)^{\frac{m-1}{2}}}\sum_{r=0}^{k_2}\binom{k_2}{r}\frac{(-1)^r (m-1-k)_r}{|t|^{m-1-k+r}}p_{1,k_1,k_2+k-r}^{(1)}(x,|t|),
\left(\begin{array}{c}\theta^{\mathrm{causal}}_n \gamma^{\mathrm{causal}}_n\end{array}\right)=\left(\begin{array}{cc}\Omega^{XX}_n \Omega^{XZ}_n \Omega^{ZX}_n \Omega^{ZZ}_n\end{array}\right)^{-1}\left(\begin{array}{c}\Omega^{XY}_n \Omega^{ZY}_n\end{array}\right).
\eta_{a}(C(\mathsf{Q}))=\sum_{i,j,k}q^{(2\rho,\lambda_{i})}(2c_{j}^{i}-\delta_{j}^{i})\varepsilon(F_{a}\triangleright\mathsf{Q}_{k}^{j})\varepsilon(E_{a}\triangleright\mathsf{Q}_{i}^{k}).
\partial_k S_{k}^I [\phi]={1\over2}{\delta S_{k}^I [\phi]\over \delta\phi}\, \partial_k G_k \,{\delta S_{k}^I [\phi]\over \delta\phi}-{1\over2}{\mathrm{Tr}}\left[\partial_k G_k{\delta^2 S_{k}^I [\phi]\over \delta\phi\delta\phi}\right].
\mathbb{G}^{(2m)}(f) =\mathbb{G}(f)- S_{1,}^{(m-1)}(f)^{t}\cdot\mathbb{G}[\mathcal{A}]- S_{2,}^{(m-2)}(f)^{t}\cdot\mathbb{G}[\mathcal{B}], \mathbb{G}^{(2m+1)}(f) =\mathbb{G}(f)- S_{1,}^{(m-1)}(f)^{t}\cdot\mathbb{G}[\mathcal{A}]- S_{2,}^{(m-1)}(f)^{t}\cdot\mathbb{G}[\mathcal{B}].
C_1 \ = \B_1 \ltimes A_1 \ = \ \{2 + 2, 6 + 2, 10 + 2 \}\ = \ \{4, 8, 12 \}, C_2 \ = \B_2 \ltimes A_2 \ = \ \{2 + 1, 4 + 1, 14 + 1 \}\ = \ \{3, 5, 15 \}, C_3 \ = \B_3 \ltimes [1] \ = \ \{0 + 1, 8 + 1, 12 + 1 \}\ = \ \{1, 9, 13 \}.
k-\Lambda=-\frac{k\sigma}{1-\sigma}+\left( \frac{k^2(k+1)\sigma +\theta^{t-3}((k^3-3k^2+k+2)-\sigma(k^3-2k^2+k+2))}{(k^2+k-k\theta^{t-3})(1-\sigma)}\right) \phi^u.
\sum_{n=1}^{\infty}\frac{q^n}{(q^{n};q)_{n+1}(q^{2n+2};q^2)_{\infty}}=\sum_{n=0}^{\infty}\frac{q^{2n^2+2n+1}}{(q;q^2)^2_{n+1}}, begin{align*}6pt]\sum_{n=0}^{\infty}q^n (-q^{n+1};q)_n (-q^{2n+2};q^2)_{\infty}=\sum_{n=0}^{\infty}\frac{q^{n^2+n}}{(q;q^2)_{n+1}}.
\widetilde{J_i}\left( q, \varphi^{-1}(Q) \right) = \sum_{\substack{a + b = i 0 \leq a, b \leq N}}(-1)^a \binom{\ell_q \left(\left( \frac{1-q}{z}\right)^{N+1}Q \right)}{a}J_b \left( q,\left( \frac{1-q}{z}\right)^{N+1}Q \right)
W_c := \sum_{i,j \ge 2}U_R(x_i-x_j) \prod_{k\ne 1,i,j}\theta_{2R}(x_j-x_k) W_d := \sum_{i,j \ge 2}U_R(x_i-x_j) \Big(1-\theta_{2R}(x_j-x_1)\Big) \prod_{k\ne 1,i,j}\theta_{2R}(x_j-x_k) .
{\frac{T_{Q}}{2 . 7 3 \mathrm{K}}}\simeq \left({\frac{\kappa_{5}^{4}\lambda_{0}^{2}- 3 6 \xi^{2}}{2 \kappa_{5}^{4}| \lambda_{0}| \rho_{0}}}\right)^{\frac{3}{9 - k^{2}}}.
\mathcal{Q}_H: = \left\{s \left| \begin{aligned}s_{in}= s_{ij}j \geq \tau(i,t_i(H)) & \frac{s_{ij+1}- s_{ij}}{m_{i(j+1) t_i(H)}- m_{ijt_i(H)}}\leq \frac{s_{ij}- s_{ij-1}}{m_{ij t_i(H)}- m_{i(j-1)t_i(H)}}\tau(i,t_i(H)-1) \leq j \leq \tau(i,t_i(H)) &s_{ij}= m_{ij t_i(H)}j \leq \tau(i,t_i(H)-1)\end{aligned}\right.\right\}
P_2(t) = (1 - 3 t)^2 (1 - t + 9 t^2 - 45 t^3 + 108 t^4 - 324 t^5 + 972 t^6 - 3645 t^7 + 6561 t^8 - 6561 t^9 + 59049 t^{10})
\exp [ \pm \sum_{n = 1}^{\infty}\sum_{j = 1}^{M}{\frac{\alpha_{- n}}{\sqrt{\beta}}}{\frac{( y_{j})^{n}}{n}}] = \sum_{\{\lambda \}}( - 1 )^{| \lambda |}J_{\{\lambda^{\prime}\}}^{( 1 / \beta )}( y_{j}) J_{\{\lambda \}}^{( \beta )}( \{\mp{\frac{\alpha_{- n}}{\sqrt{\beta}}}\}) .
{\cal E}_{\bf z}(\pi(f_{\bf z}^\epsilon))=\frac{1}{T}\sum_{i=1}^T\psi_q(\pi(f_{\bf z}^\epsilon)(x_i)-y_i) &\leq \frac{1}{T}\sum_{i=1}^T\psi_q^\epsilon(\pi(f_{\bf z}^\epsilon)(x_i)-y_i)+q(\|\pi(f_{\bf z}^\epsilon)\|_\infty+|y|)^{q-1}\epsilon &\leq{\cal E}_{\bf z}^\epsilon(\pi(f_{\bf z}^\epsilon))+q2^{q-1}\epsilon
\sum_{\gamma\in\mathcal{S}_{1}(n+1,0)}w(\gamma)=\sum_{k=0}^{p}a_{q}^{(k)}\Big(\sum_{\gamma\in\mathcal{S}_{1}(n,k)}w(\gamma)\Big)=\sum_{k=0}^{p}a_{q}^{(k)}\langle\mathcal{H}_{q}^{n}e_{k},e_{0}\rangle=\langle\mathcal{H}_{q}^{n+1}e_{0},e_{0}\rangle
\begin{array}{l}H_{2}=\dfrac{1}{(3h+2\alpha_{1}-2\alpha_{2})}\{q_{2}^2p_{1}^2+2q_{1}^2p_{1}+2q_{1}q_{2}p_{2}-q_{1}p_{1}^2+2q_{2}p_{1}p_{2}+2t_{1}q_{2}p_{1}+2t_{2}q_{2}p_{2}begin{align*}10pt]\phantom{H_{1}=}-t_{2}p_{1}^2+p_{2}^2+2\alpha_{1}q_{1}-2t_{2}^2p_{1}+2t_{1}p_{2}\}.\end{array}
\left( 1 - \frac{\mu+\tau}{\mu_{k_0+1}(\lambda) + \tau}\right) \sum_{k=1}^\infty \tilde{c}_k^2 \; \leq \; \left( 1 - \frac{\mu+\tau}{\mu_{k_0+1}(\lambda) + \tau}\right) \sum_{k=1}^{k_0}\tilde{c}_k^2 \; + \; \sum_{k=k_0+1}^\infty \left( 1 - \frac{\mu+\tau}{\mu_k(\lambda) + \tau}\right)\tilde{c}_k^2.
\mu \beta_0 =\phi(\beta_{m-1}) \mu\beta_{1}=\phi(\beta_{0})\gamma+\lambda_{1}\phi(\beta_{m-1}) \mu\beta_{i}= \phi(\beta_{i-1})+\lambda_{i}\phi(\beta_{m-1}), ~\mathrm{for}~ 2\leq i\leq m-1.
\mathbb{P}_{G_1 \sim G(n,m)}\left[ \mathcal{H}^c_{\rho,C}\right] \Omega(\frac{1}{n}) \leq \mathbb{P}_{G_1 \sim G(n,m)}\left[ \mathcal{H}^c_{\rho,C}\right] \mathbb{P}_{G \sim G_{n,p}}\left[E(G)=m\right]\leq O\left( \frac{1}{n^{\frac{C}{16}-2}}\right).
{I_{D}}^{\mathrm r}\equiv - V T{\frac{{\mathrm{t r}}{\bf 1}}{2 ( 4 \pi )^{D / 2}}}\int_{1 / \Lambda^{2}}^{\infty}\! d \tau \, \tau^{-{\frac{D}{2}}- 1}\left\{{\mathrm e}^{- \tau ( m^{2}+ \epsilon )}G_{D}( \tau F ) - 1 \right\}\ .
\begin{cases}r^{\frac{2}{\alpha}}h_m (r) \to (-1)^m \beta^{\frac{1}{\alpha}}\frac{2}{N-2}< \alpha < \frac{4}{N-2}|\log r|^{\frac{1}{\alpha}}r^{\frac{2}{\alpha}}h_m (r) \to (-1)^m ({\frac{2}{\alpha}})^{\frac{2}{\alpha}}\alpha = \frac{2}{N-2}\end{cases}
g(\tau)=\sum_{\ell=0}^{k-1}\alpha_k(\ell)\cdot F(\tau)^\ell\theta(2\tau)^{4(k-\ell)+2}+\sum_{\ell=k}^{2k}\beta_k(\ell)\cdot F(\tau)^{2k-\ell}F(2\tau)^{\ell-k}\theta(2\tau)^2+\gamma_k\cdot \dfrac{F(\tau)F(2\tau)^k}{\theta(2\tau)^2},
0= \langle \tau_{H}, \alpha_{2}\rangle \{m_{1}(1-(\cot \langle \alpha_{2}, H\rangle )^{2}) +(2m_{2}+4m_{3})(1-(\cot \langle 2\alpha_{2}, H\rangle )^{2}) &+n_{1}(1-(\tan \langle \alpha_{2}, H\rangle )^{2}) \}(\alpha_{1}+2\alpha_{2}).
\mu_{\omega,c}\le S_{\omega,c}(\lambda_0 \Phi_{\omega}) &=\lambda_0^2L(\Phi_{\omega})+\lambda_0^3N(\Phi_{\omega})+\lambda_0^2\omega Q(\Phi_{\omega})+\lambda_0^2cP(\Phi_{\omega}) &=\lambda_0^2S_{\omega,0}(\Phi_{\omega})+\lambda_0^2\{(\lambda_0-1)N(\Phi_{\omega})+cP(\Phi_{\omega})\}&=\lambda_0^3\mu_{\omega,0}<8\mu_{\omega,0}.
S_{( a )}[ J^{\mu}, \varphi , \theta ,{\tilde{\alpha}}^{r}, \beta_{r};{}^{4}g_{\mu \nu}] = \int d^{4}x \Big [ | J | a ({\frac{| J |}{\sqrt{{}^{4}g}}}, \partial_{\mu}\theta J^{\mu}) - J^{\mu}( \partial_{\mu}\varphi + \beta_{r}\partial_{\mu}{\tilde{\alpha}}^{r}) \Big ]
\omega\left[2\pi\iota n\left(r_n,\frac{1}{f}\right) +\frac{k-1}{2k}2\pi\iota n\left(r_n,\frac{1}{A}\right)\right]+o(1)=2\pi\iota n\left(r_n,\frac{1}{y}\right)+\frac{k-1}{2k}2\pi\iota n\left(r_n,\frac{1}{A}\right).
\mathfrak{h}_0 = \mathfrak{h}_1, \textrm{on}\{|x|=a_1,|y|\le b_1\}; \mathfrak{h}_0 =\mathfrak{h}_2,\textrm{on}\{|x|\le a_2,|y|=b_1\}; \mathfrak{h}_0 =h_s, \textrm{on}T_s,\, s\in [0,1].
\begin{aligned}\hat{A}_r^{-1}= (I - \hat{A}^{-1}A) (\hat{A}_s - A)^{-1}(I - A \hat{A}^{-\top}) = A^{-1}- (I - \hat{A}^{-1}A) A^{-1}(I - A \hat{A}^{-\top}).\end{aligned}
u^{(1)}= u^n + \Delta t L(u^n), u^{(2)}= \frac{3}{4}u^n + \frac{1}{4}u^{(1)}+ \frac{1}{4}\Delta t L \left( u^{(1)}\right), u^{n+1}= \frac{1}{3}u^n + \frac{2}{3}u^{(2)}+ \frac{2}{3}\Delta t L \left( u^{(2)}\right),
M_{a b , c}^{\hspace{. 1 8 i n}d}\equiv{\delta_{( a}}^{d}\hat{G}_{b ) c}+ \frac{1}{2}f_{e ( a}^{\ \ \d}f_{b ) c}^{\ \ \e}\hspace{. 3 i n}N_{a b , c}^{\hspace{. 1 8 i n}d e}\equiv \frac{i}{2}\delta_{( a}^{( d}f_{b ) c}^{\ \ \e )}.
\int_{\Omega}|(\mathbf{D}+ \mathbf{k}) \mathbf{u}|^2\, d\mathbf{x}= \sum_{\mathbf{b}\in \widetilde{\Gamma}}|\mathbf{b}+ \mathbf{k}|^2 |\hat{\mathbf{u}}_{\mathbf{b}}|^2, \mathbf{u}\in \widetilde{H}^1(\Omega; \mathbb{C}^n), \; \mathbf{k}\in \mathbb{R}^d,
\frac{1}{z^{3}}\left( - q^{2}- \partial_{z}^{2}+ \frac{3}{z}\partial_{z}+ \frac{m^{2}}{z^{2}}+ \frac{\lambda_{v 2}}{z_{v}}\delta ( z - z_{v}) + \frac{\lambda_{h 2}}{z_{h}}\delta ( z - z_{h}) \right) \Delta_{q}( z , z^{\prime}) = \delta ( z - z^{\prime}) .
S\left(\frac{1}{x^{i_1}},\frac{1}{x^{i_2}};\infty,1\right)=A\left(\frac{1}{x^{i_1}},\frac{1}{x^{i_2}};\infty,1\right)+A\left(\frac{1}{x^{i_1+i_2}};\infty,1\right),
\dot{\varrho}+3{\dot{a}_0\over a_0}\left(\varrho+{\wp\over c^2}\right)=-{\dot{G}_4\over G_4}\varrho+{{3kc\dot{c}}\over{4\pi G_4a^2_0}}-{{3Cc\dot{c}}\over{4\pi G_4a^4_0}}.
\mathbf{R}^{(g)}_{\boldsymbol{\xi}}\triangleq \mathbb{E}\left\{\boldsymbol{\xi}^{(g)}\left(\boldsymbol{\xi}^{(g)}\right)^H\right\}=\mathbf{I}_T \otimes \mathbf{R}^{(g)}_{\boldsymbol{\eta}}
[\,X_{0},\,Y_{0}\,] = X_{1},\,\,\, [\,X_{0},\,Y_{1}\,] = -X_{1},\,\,\, [\,X_{1},\,Y_{0}\,] = -Y_{0},\,\,\, [\,X_{1},\,Y_{1}\,] = 2Y_{1}.
M_{f,a,\beta}(x) := \max \biggl\{\max_{t=1,\ldots, m}\frac{\|f^{(t)}(x)\|}{f(x)}\, , \, \sup_{y \in B_x^\circ(r_a(x))}\frac{\|f^{(m)}(y)-f^{(m)}(x)\|}{f(x) \|y-x\|^{\beta- m}}\biggr\}.
\left( \begin{matrix}{{{\bf 1}}}&{{0}}\\{{{\tilde{\bf c}}}}&{{{\bf 1}}}\\ \end{matrix}\right) \qquad \qquad{\tilde{\bf c}}= \sum_{i}N_{i}g_{i}^{t}{\tilde{\bf c}}^{v}g_{i}+ \sum_{j}M_{j}g_{j}^{t}{\tilde{\bf c}}^{h}g_{j}
R_{\mu \nu \lambda}^{\kappa}= \Gamma_{{\nu \lambda}, \mu}^{\kappa}- \Gamma_{{\mu \lambda}, \nu}^{\kappa}+ \Gamma_{\mu \sigma}^{\kappa}\Gamma_{\nu \lambda}^{\sigma}- \Gamma_{\nu \sigma}^{\kappa}\Gamma_{\mu \lambda}^{\sigma}
\int_{\Sigma}(\dfrac{n}{2}+|{\bf H}|^2)u^2e^{\frac{|x|^2}{4}}d\sigma \leq \int_{\Sigma}|\nabla u|^2e^{\frac{|x|^2}{4}}d\sigma &=\frac n2\int_{\Sigma}u^2e^{\frac{|x|^2}{4}}d\sigma.
\lambda_j\int \left( \Lambda W\right)_{[\lambda_j]}\partial_t^2U= -\int\left( \Lambda W \right)_{(\lambda_j)}L_{W_{(\lambda_j)}}h &+2\int \sum_{j\neq k}(\Lambda W)_{(\lambda_j)}W_{(\lambda_k)}h+\int (\Lambda W)_{(\lambda_j)}h^2 &+\int (\Lambda W)_{(\lambda_j)}(2Mv_L+v_L^2+2v_Lh) &+ \int(\Lambda W)_{(\lambda_j)}(M^2+\Delta M).
(\mathcal{A}^{-1}h)(\varphi, y):=(1+\tilde{\beta}_y (\varphi, y))\,h(\varphi, y+\tilde{\beta}(\varphi, y)), (\mathcal{A}^T h)(\varphi, y)=h(\varphi, y+\tilde{\beta}(\varphi, y))
\frac{1 0 \times ( 0 . 4 x + 0 . 9 )}{1 0 \times 0 . 5}- \frac{1 0 \times ( 0 . 1 x - 0 . 5 )}{1 0 \times 0 . 2}= \frac{1 0 0 \times ( 0 . 0 3 + 0 . 0 2 x )}{1 0 0 \times 0 . 0 3}
\chi_{t o p}= \int \langle q ( x ) q ( 0 ) \rangle d^{2}x = \frac{e^{2}}{( 2 \pi )^{2}}\int \langle F_{1 2}( x ) F_{1 2}( 0 ) \rangle d^{2}x = \frac{e^{2}}{2 \pi}\hat{G}_{F F}( 0 ) ,
S = \int d^{d + 1}x \sqrt{\tilde{g}}\left[ - \tilde{R}+ \frac 1 2 \tilde{g}^{a b}\partial_{a}\phi^{I}G_{I J}( \phi ) \partial_{b}\phi^{J}- V ( \phi ) \right] + 2 \int d^{d}x \sqrt{g}\, H ~ .
P^{- 1}{}_{\! \! \mu_{1}\cdots \mu_{k}, \nu_{1}\cdots \nu_{k}}( p ) \, P_{\nu_{1}\cdots \nu_{k}, \lambda_{1}\cdots \lambda_{k}}( p ) ={\cal E}^{A}{}_{\! \! \mu_{1}\cdots \mu_{k}, \lambda_{1}\cdots \lambda_{k}}\,
\mathcal{C}(f,g,h) = \frac 16 \left( \bar{\mathcal{C}}(f,g,h) + \bar{\mathcal{C}}(f,h,g) + \bar{\mathcal{C}}(g,f,h) + \bar{\mathcal{C}}(g,h,f) + \bar{\mathcal{C}}(h,f,g) + \bar{\mathcal{C}}(h,g,f) \right),
I_n(\bar{\epsilon})\cap \left[y_n(\bar{\epsilon}),y_n(\bar{\epsilon})+f\left(\frac{\Psi(n)}{\beta^n}\right)\right) =\left\{x\in [0,1]: 0\leqslant x-y_n(\bar{\epsilon})< t_n(\bar{\epsilon})\right\}=[y_n(\bar{\epsilon}), y_n(\bar{\epsilon})+t_n(\bar{\epsilon}))
\Big \langle \prod_{a = 1}^{N}\prod_{i = 1}^{n_{a}}\overline{{{\psi}}}^{( a )}( x_{i}^{( a )}) P_{+}\psi^{( a )}( x_{i}^{( a )}) \prod_{i = 1}^{m_{a}}\overline{{{\psi}}}^{( a )}( y_{i}^{( a )}) P_{-}\psi^{( a )}( y_{i}^{( a )}) \Big \rangle
\lim_{\theta \to 1}\Bigl ( \Omega , S_{\theta ({\cal L}_{0}+ \epsilon{\cal L}_{1})}( \lambda f ) \Omega \Bigr ) = \lim_{\theta \to 1}\frac{\Bigl ( \Omega , S_{\theta{\cal L}_{0}}( \theta \epsilon{\cal L}_{1}+ \lambda f ) \Omega \Bigr )}{\Bigl ( \Omega , S_{\theta{\cal L}_{0}}( \theta \epsilon{\cal L}_{1}) \Omega \Bigr )}
\begin{array}{rcl}Z(t,\tau) = E(t,\zeta_{j})E(t_{j},\zeta_{j})^{-1}\prod\limits_{k=i+2}^{\rightarrow j}E(t_{k},\gamma(t_{k}))E(t_{k},\gamma(t_{k-1}))^{-1}& E(t_{i},\gamma(\tau))E(\tau,\gamma(\tau))^{-1},\end{array}
\{\lambda_1, \lambda_2\}= \frac{a^2+b^2+c^2+d^2 \pm \sqrt{(a^2+b^2+c^2+d^2)^2 - 4(ad-bc)^2}}{2}= \frac{a^2+b^2+c^2+d^2 \pm \sqrt{(a^2+c^2 - b^2-d^2)^2 + 4(ab-cd)^2}}{2}
d s_{1 1}^{2}={\frac{1}{({\frac{1}{6}}C_{a b c}H^{a}H^{b}H^{c})^{\frac{1}{3}}}}\left[ d u d v + H_{0}d u^{2}+{\frac{1}{6}}C_{a b c}H^{a}H^{b}H^{c}d \vec{x}^{2}+ H^{a}\omega_{a}\right] ,
\begin{array}{r l}{{< 0 | T \left( \hat{\phi}( x_{f}) \hat{\phi}( x_{i}) \right) | 0 > =}}&{{\theta ( x_{f}^{0}- x_{i}^{0}) \, < 0 | \hat{\phi}( x_{f}) \hat{\phi}( x_{i}) | 0 >}}\\{{}}&{{}}\\{{+}}&{{\theta ( x_{i}^{0}- x_{f}^{0}) \, < 0 | \hat{\phi}( x_{i}) \hat{\phi}( x_{f}) | 0 > \ ,}}\\ \end{array}
\frac{1}{\mathbb{V}(D_n)}\Sigma_n =\frac{1}{\mathbb{V}(D_n)}\begin{pmatrix}\mathbb{V}(D_n) (D_n,D'_n) (D_n,D'_n) \mathbb{V}(D'_n) \end{pmatrix}&\stackrel{a.s.}{\rightarrow}I,
\tilde{S}(r)=\frac{2}{(1+r^2)^2}+\frac 18 H_1\left(1,\frac{1+r^2}{4},\frac{1+r^2}{4};1\right)-\frac 18 H_1\left(0,\frac{r^2}{4},\frac{r^2}{4};1\right).