harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_2^{i}(x_1,x_2) :={\chi_{2^{i}R}(x_1,x_2)\over |2^i R|}\iint_{\widetilde R}f_2(y_1,y_2)dy_1dy_2, f_2^{i}(x_1,x_2) := g_2^{i-1}(x_1,x_2)-g_2^{i}(x_1,x_2), \alpha_2^i :=\|f_2^i\|_\infty |2^iR| \quad{\rm and}a_2^i := (\alpha_2^i)^{-1}f_2^i.
\sum_{i \ge 0}Ce^{-c(t+i)^\gamma \mu(Y)^\gamma}\le \frac{C e^{-ct^\gamma \mu(Y)^\gamma}}{c^{1/\gamma}\mu(Y) \gamma}\sum_{k=0}^n \frac{n!}{k!}(ct^\gamma \mu(Y)^\gamma)^{n-k}\le \frac{e C n!\, t \,e^{-ct^\gamma \mu(Y)^\gamma}}{\gamma c^{\frac{1}{\gamma}-n}\mu(Y)^{1-\gamma n}}.
\frac{7 ! 3 !}{2}( C \Gamma^{[ \nu_{1}})_{\alpha{'}\beta{'}}( C \Gamma^{\nu_{2}\nu_{3}]})_{\delta{'}\epsilon{'}}+ \frac{6 ! 4 ! 7}{4 !}( C \Gamma^{\mu_{7}\nu_{1}\nu_{2}\nu_{3}}\Gamma_{1 1})_{\alpha{'}\beta{'}}( C \Gamma_{\mu_{7}}\Gamma_{1 1})_{\delta{'}\epsilon{'}}= 0 \quad ,
[x^{k}]s_{n}(x)=(-1)^{k}\frac{n!}{k!2^{n}}\sum_{n=l_{1}+l_{2}+\cdot\cdot\cdot+l_{k}}C_{(l_{1}-1)/2}C_{(l_{2}-1)/2}\cdot\cdot\cdot C_{(l_{k}-1)/2}
\begin{array}{ll}\mathcal{P}(u^{n+1},v^{n+1}) - \mathcal{P}(u^n,v^n)= \mathcal{P}_{(u_{\theta_n}^n, v_{\theta_n}^n)}^{\prime}(\delta u^n, \delta v^n) + e_n,\end{array}
\delta \Delta =-\Delta \biggl[(\epsilon^{2n},{\cal Z})_{E}-str\Bigl(\bigl(\sum_{l=0}^{n-1}\kappa^{(l)}\tilde{W}D^{(2l+1)}(\epsilon^{st})+\sum_{l=0}^{n}\kappa^{(l) \prime}W D^{(2l)}(\epsilon^{st})\bigr)W^{-1}\Bigr)\biggr],
M_{3}= \kappa \sigma \frac{\exp\big\{\mu t + \frac{1}{2}\alpha^{2}p_{\alpha_{+}}^{2}\big\}}{(p_{\alpha_{+}}- \sigma t)}\;\Phi(\alpha p_{\alpha_{+}})- \kappa \sigma \frac{\exp\big\{\mu t + \frac{1}{2}\alpha^{2}\sigma^{2}t^{2}\big\}}{(p_{\alpha_{+}}- \sigma t)}\; \Phi(\alpha \sigma t).
u_z=(u-\lambda(1-2^{-z}))^+ \textrm{and}U_z=\sup_{t\in[0,T]}\|u_z(t)\|^2+\int_0^T\left( \|u_z(t)\|_{\eta}^2+\sum_{k=1}^{d_1}\|L_ku_z(t)\|^2 \right)dt.
\kappa_{nl}(\mathbb{R}^n) = \sup\Bigg\{\min_{R\in SO(n)}\frac{\|\nabla u - R\|_{L^2(\mathbb{R}^n)}}{\|\mbox{dist}(\nabla u, SO(n))\|_{L^2(\mathbb{R}^n)}};~~ u\in W^{1,2}_{loc}(\mathbb{R}^n, \mathbb{R}^n), \qquad\mathrm{dist}(\nabla u, SO(n))\in L^2(\mathbb{R}^n) \setminus \{0\}\Bigg\}.
I(Y;\tilde{X})\ge \frac{1}{2}\log\frac{1}{1-\tilde{\rho}^2}\ge \frac{1}{2}\log\frac{1}{1-\rho^2}-\frac{\log e}{2}\cdot\frac{\rho^2 C}{1-\rho^2}\sqrt{\frac{\ln k}{k}}= I(Y;X)-\frac{\log e}{2}\cdot\frac{\rho^2 C}{1-\rho^2}\sqrt{\frac{\ln k}{k}}
{\tilde{D}}= 1 +{\frac{k_{1 1}^{2}T_{6}}{4 \pi^{2}}}\left({\frac{1}{[ ( x - x_{0})^{2}+ ( x^{1 1}- x_{0}^{1 1})^{2}]^{3 / 2}}}+{\frac{1}{[ ( x - x_{0})^{2}+ ( x^{1 1}+ x_{0}^{1 1})^{2}]^{3 / 2}}}\right) \ .
T_1 = \frac{y_1'}{y_1}= d_0 \frac{u'}{u}+ d_1, \frac{f'}{f}= d_0 \frac{g'}{g}+ d_1 = d_0 \left( \frac{g'}{g}- \frac{u'}{u}\right) + \frac{y_1'}{y_1}.
W_{1 2}\psi = ( h_{1}+ h_{2}+ \Lambda_{1 2}^{+ +}V_{1 2}\Lambda_{1 2}^{+ +}+ \Lambda_{2}^{+}V_{2}^{+}\Lambda_{2}^{+}+ \Lambda_{1}^{+}V_{1}^{+}\Lambda_{1}^{+}) \psi
\| \sigma_I \|^2 = \langle \sigma_I,\sigma_I \rangle = \langle \sigma_I^2, 1 \rangle = a_I \langle \sigma_I, 1 \rangle = a_I \, \overline{\sigma_{s^r}(\zeta^I)}= a_I/\sigma_{s^r}(\zeta^I),
\int \prod_{i = 1}^{4}\frac{d^{2}k_{i}}{( 2 \pi )^{2}}\cos [ k_{1}\wedge k_{2}- k_{3}\wedge k_{4}] ( 2 \pi )^{2}\delta ( k_{1}+ k_{2}+ k_{3}+ k_{4}) \tilde{\varphi}_{i}( k_{1}) \tilde{\varphi}_{j}( k_{2}) \tilde{\varphi}_{i}( k_{3}) \tilde{\varphi}_{j}( k_{4}) ,
\int_{X}\alpha_{x}~d\nu(x) = \int_{Y}\int_{X}\delta_{x}\times \delta_{\pi(x)}~dD_{\pi}(y)(x)~d\eta(y) &= \int_{Y}\int_{X}\delta_{x}\times \delta_{y}~dD_{\pi}(y)(x)~d\eta(y) &= \int_{Y}D_{\pi}(y) \times \delta_{y}~d\eta(y) = \alpha
I = \int dx^+ dx^- ( 4 \Omega \partial_+ \partial_- \rho+ \frac{\mu}{2}e^{2 \rho}\Omega^{1-\lambda -\gamma/4}+ \Omega^{\delta}\partial_+ f \partial_- f - e^{-2 \rho}\Omega^{(\gamma -2\epsilon)/4 +1}F^2_{-+})
\begin{aligned}{}\big\|(v_n )_{n=k}^N \big\|_{L^p(D)}&{}\le C\Big(\big\|(f_n)_{n=k}^N \big\|_{L^p(X)}+ \frac{1}{\tau}\big\|(v_i)_{i=0}^{k-1}\big\|_{L^p(X)}+ \big\|(v_i )_{i=0}^{k-1}\big\|_{L^p(D)}\Big) .\end{aligned}
K_{\alpha(\tau)(z)}(\alpha(\tau)(w)) =(\rho(\tau^{-1}) K_{\alpha(\tau)(z)})(w)=\langle \rho(\tau^{-1}) K_{\alpha(\tau)(z)},K_w\rangle_{L^2(\Omega)}&=\langle K_{\alpha(\tau)(z)},\rho(\tau) K_w\rangle_{L^2(\Omega)}=\overline{(\rho(\tau) K_w)(\alpha(\tau)(z))}=\overline{K_w(z)}=K_z(w).
\Sigma^R_> ={}\{-1 + re^{2\pi i/3}\mid 0 \leq r \leq R \}\cup \{-1 + re^{\pi i/3}\mid -R \leq r \leq 0 \}, \mathcal{C}^R_< ={}\{1 + re^{4\pi i/3}\mid -R \leq r \leq 0 \}\cup \{1 + re^{5\pi i/3}\mid 0 \leq r \leq R \}.
m(f_{n+1}(C_{f_{n+1}, \delta}\cap \tilde I_i^{i'})) = \sum_{j=1}^{N_{i,i'}}\sum_{j' = 1}^{\tilde N_{i,i',j}}m(f_{n+1}(C_{f_{n+1}, \delta}\cap \tilde I_{i,j}^{i',j'}))\leq (1-K) \mu(\tilde I_i^{i'})) + \epsilon_2
sin^{4}\theta=\frac{1}{(2i)^{4}}(e^{i\theta}-e^{-i\theta})^{4}=\frac{1}{16}(e^{i4\theta}-4e^{i2\theta}+6-4e^{i2\theta}+e^{-i4\theta})=\frac{1}{16}(2cos4\theta-8cos2\theta+6)
e^{-2\pi i\nu T p}\,(Z_{1/\Omega}\circ H)g(t + T p,\nu) & = \Omega\,\sum_{q,\,m=0}^{L-1}(T^q\,M^m c)_p\, e^{-2\pi i\nu T q}\,\eta^{QP}_H(t + T q,\nu + \Omega m).
-4i\Omega L_\xi = \xi\eta(\eta^2 + \zeta^2 - 2\xi^2) J(\zeta)(J(\zeta) - J(\xi) - J(\eta)) &+ \xi^2\eta(\zeta - \eta) J(\xi)(J(\xi) - J(\eta) - J(\zeta)) &+ \xi\eta^2(\zeta - \eta) J(\xi) (J(\xi) - J(\eta) + J(\zeta)),
\delta(s) =d(x,y)\le d(x,\Phi^b_a(x',t))+d(\Phi^b_a(x',t),\varphi^b_a(y',t))+d(\Phi^b_a(y',t),y) \le \lambda|t-s|/\Delta^b_a(M^b_a(\varphi))+\delta(t)+|t-s|/\Delta^b_a(M^b_a(\varphi)) =(1+\lambda)|t-s|/\Delta^b_a(M^b_a(\varphi))+\delta(t).
d_{n}= \sum_{i = 1}^{n}{\left( - 1 \right)^{i}\left( \sum_{\begin{array}{c}{{0 < k_{1}, k_{2}, \ldots , k_{i}\leq n}}\\{{k_{1}+ k_{2}+ \ldots + k_{i}= n}}\\ \end{array}}\frac{1}{\left[ k_{1}\right] ! \left[ k_{2}\right] ! \ldots \left[ k_{i}\right] !}\right)}
\left| \left. q \left( T \right) \right\rangle \hspace{- 0 . 0 3 i n}\right\rangle = \prod_{k = - \infty}^{\infty}e^{i G_{k}^{1}\left( T \right)}\prod_{s = - \infty}^{\infty}e^{i G_{s}^{2}\left( T \right)}\left| \left. q \right\rangle \hspace{- 0 . 0 3 i n}\right\rangle ,
\begin{cases}v_{\tau \tau}-\mathrm{e}^{2\tau}\Delta v -\frac{\delta}{4}v=\mathrm{e}^{\frac{\mu+3}{2}\tau}f(\mathrm{e}^\tau-1,x), x\in \mathbb{R}^n, \ \tau>0, v(0,x)=u_0(x), x\in \mathbb{R}^n, v_\tau(0,x)=\frac{\mu-1}{2}u_0(x) +u_1(x), x\in \mathbb{R}^n.\end{cases}
C^-(b_1,h)=\frac{h+\delta^+ b_1+ \min\{1,a^-,a^-_0\}(-h+b_1\varepsilon^-)}{\varepsilon^-+\delta^+}C^+(b_2,h)=\frac{h+\delta^- b_2+ \min\{1,a^+,a^+_0\}(-h+b_2\varepsilon^+)}{\varepsilon^++\delta^-}
\frac 1{2 4}\times \frac 1{2 4}= \frac{2 5}{2 4}\times \frac{2 5}{2 4}= 0 + 1 + \frac 1 6 \quad \quad \frac 1{2 4}\times \frac{2 5}{2 4}= \frac{2 5}{2 4}\times \frac 1{2 4}= \frac 2 3 + 2 \left( \frac 3 2 \right)
G(s,y) = 1 +\frac{y^{-s}}{\log y}\bigg(- \sum_{|\rho| \le T}\frac{y^{\rho}}{\rho-s}+ \frac{\sqrt{y}}{2s-1}+ O_{\varepsilon}\left(\frac{\sqrt{y}}{\log y}+ \frac{y \log^2( yT)}{T}\right)\bigg) &=1 + \frac{y^{-s}}{\log y}\left( \psi(y)-y + O_{\varepsilon}(\sqrt{y})\right)=1 + O_{\varepsilon}\left(\frac{\log x\log (u+1)}{\sqrt{y}}\right)
\begin{array}{l l}{{K_{1}^{\theta}= - \sin \phi}}&{{K_{1}^{\phi}= - \cot \theta \cos \phi}}\\{{K_{2}^{\theta}= \cos \phi}}&{{K_{2}^{\phi}= - \cot \theta \sin \phi}}\\{{K_{3}^{\theta}= 0}}&{{K_{3}^{\phi}= 1 .}}\\ \end{array}
\psi_1(u,v,w)= v(\alpha u+(1-\alpha)u^2+\beta w), \psi_2(u,v,w)= u(\alpha v+(1-\alpha)v^2+\beta w), \psi_3(u,v,w)= u(\alpha v+(1-\alpha)u^2+\beta w) \shortintertext{and}\psi_4(v,w)= v(\alpha u+(1-\alpha)v^2+\beta w).
\Gamma_+^{-1}C_3^{-2}B = \frac{1}{2}\frac{\alpha}{1 + \mu \alpha k}\left[ (k_2 \mu - k_1) + (k - \mu^2 k_2) C_3^{-1}+ (\frac{2}{\alpha}+ k \mu + k_1 \mu^2)C_3^{-2}\right]Y.
\sup_{R \geq 1}\frac{1}{R^{2\alpha}}\displaystyle\inf_{l \in \textrm{Span}\{1, x_1 \}}\| u - \sigma_i(x_0) v_{i}(\cdot, a(x_0)) - l \|_{B_R(x_0)}\lesssim \epsilon^{\alpha -2}(K + \|\sigma_i \| N_i)+ \epsilon^{2\alpha}M + \tilde{N}.
\int_\varepsilon^R G(\rho)\frac{d}{d\rho}\left(\int_{B_\rho(0)}F(z)\, dz \right)d\rho =\int_\varepsilon^R G(\rho)\left(\int_{\partial B_\rho(0)}F(s) \, dS_{\rho}\right) d\rho &=\int_{R \geq |z| \geq \varepsilon}F(z) G(|z|)\,dz.
\widehat{T}_{f}\widehat{T}_{g}=\begin{bmatrix}T_{f}T_{g}+{\Gamma^{*}_{\bar{f}}}{\Gamma_{g}}T_{f}{\Gamma^{*}_{\bar{g}}}+{\Gamma^{*}_{\bar{f}}}{S_{g}}\Gamma_{f}T_{g}+S_{f}{\Gamma_{g}}\Gamma_{f}{\Gamma^{*}_{\bar{g}}}+S_{f}S_{g}\end{bmatrix}.
\textrm{maximize}\quad \sum_{i=1}^T c_i \textrm{s.t.}Tc_t+\alpha\sum_{i=t+1}^T c_i\le 1, 1\le t \le T-m & (2T-m+1-t)c_t+\sum_{i=t+1}^T c_i\le 1, T-m+1 \le t \le T &c_i\ge 0, i=1,\dots,T.
R_p(\varphi, s) = \sum_{k = 0}^\infty \frac{\varphi(p^k)}{p^{ks}}= 1 + \sum_{k = 1}^\infty \frac{p^k - p^{k - 1}}{p^{ks}}&= 1 + \frac{p - 1}{p^s}\sum_{k = 0}^\infty \frac{1}{p^{k(s - 1)}}= 1 + \frac{p - 1}{p^s - p}&= \frac{p^s - 1}{p^s - p}.
Z_{\delta}^{\pm}= \int [ d x ] \ \theta \left( \pm \int_{- \beta / 2}^{\beta / 2}x d t \right) \exp \left( - \int_{- \beta / 2}^{\beta / 2}d t \left\{{\frac{1}{2}}\left[ \dot{x}^{2}+ ( m^{2}+ 2 \lambda_{\pm}) x^{2}\right] + \delta ( i g x^{3}- \lambda_{\pm}x^{2}) \right\}\right) .
Q_f^{\Sigma}(\varphi,\varphi)= \int_{\Sigma}\left(|\nabla \varphi|^2-(|A|^2+Ric_f(\nu,\nu))\varphi^2\right)e^{-f}dv = -\int_{\Sigma}\varphi L_f\varphi e^{-f}dv,
L \left( \frac{1}{2}, \chi_c \right) = 2\sum_{0 \neq \mathcal{A}\subset \mathcal{O}_K}\frac{\chi_c(\mathcal{A})}{N(\mathcal{A})^{1/2}}V \left(\frac{2 \pi N(\mathcal{A})}{(3N(c))^{1/2}}\right).
\frac{d^{2}\psi ( \zeta )}{d \zeta^{2}}+ \frac{2 \zeta - 1}{\zeta ( \zeta - 1 )}\frac{d \psi ( \zeta )}{d \zeta}- \frac{q + r ( 1 - 2 \zeta )^{2}}{\zeta^{2}( \zeta - 1 )^{2}}\psi ( \zeta ) = 0
|Dw|^2\log|Dw|^2\frac{\partial\varphi}{\partial\gamma}(x_0)= (g'(0)+h'\psi)|Dw|^2\log|Dw|^2-2\sum_{1\leq i,k,l\leq n}c^{kl}u_iu_l(\gamma^i)_k.
Z_{2}= \Biggl ( \frac{a_{1}+ Z_{1}}{{\bar{a}}_{1}+ Z_{1}^{- 1}}\Biggr ) = Z_{0}^{- 2}\Biggl ( \frac{a_{0}+ Z_{0}}{{\bar{a}}_{0}+ Z_{0}^{- 1}}\Biggr ) \Biggl ( \frac{\mu_{+}+ Z_{0}}{{\bar{\mu}}_{+}+ Z_{0}^{- 1}}\Biggr ) \Biggl ( \frac{\mu_{-}+ Z_{0}}{{\bar{\mu}}_{-}+ Z_{0}^{- 1}}\Biggr )
\begin{aligned}\begin{bmatrix}N M I_n S\end{bmatrix}=\begin{bmatrix}N M I_n S\end{bmatrix}\begin{bmatrix}-S 0 I_n I_n\end{bmatrix}\begin{bmatrix}-S^{\mathsf{T}}0 S^{\mathsf{T}}I_n\end{bmatrix}, &=\begin{bmatrix}M-NS M 0 S\end{bmatrix}\begin{bmatrix}-S^{\mathsf{T}}0 S^{\mathsf{T}}I_n\end{bmatrix}, \end{aligned}
\int_{x_{0}}^{x}\int_{x_{0}}^{x_{1}}\cdot\cdot\cdot\int_{x_{0}}^{x_{n-1}}f(x_{n})dx_{n}\cdot\cdot\cdot dx_{2}dx_{1}=\int_{x_{0}}^{x}f(t)\frac{(x-t)^{n-1}}{(n-1)!}dt.
h_i d_j = \begin{dcases*}d_j h_{i+1}if $i \geq j$ d_{j+1}h_i if $i <j$ \end{dcases*}h_i s_j = \begin{dcases*}s_j h_{i-1}if $i > j$ s_{j+1}h_i if $i \leq j$ \end{dcases*}
\frac{1}{2}\Delta| \nabla^{k}F_A|^2 \leq -|\nabla^{k+1}F_A|^2+C_k\sum_{m=0}^k |\nabla^{m}{\cal R}||\nabla^{k-m}F_A||\nabla^{k}F_A| &+C_k\sum_{m=0}^k |\nabla^{m}F_A||(\nabla^{k-m}F_A)^B||\nabla^{k}F_A |.
\textrm{rk}(D) = \max \left\{r \in \mathbb{Z}_{\geq 0}\, \Big| \begin{array}{l}\ |D-E| \not = \emptyset, \ \forall \E \in \textrm{Div}(\Gamma), \ \textrm{supp}(E) \subseteq A, \E \geq 0, \ \textrm{deg}(E)=r \end{array}\right\}.
M_{NN}=Nm_0\,\frac{\sqrt{2}}{3}\left(\,\sqrt{1-\sqrt{1-\left(\frac{g^2N}{4\pi m_0^2\alpha_0}\right)^2}}+\,\frac{4\pi m_0^2\alpha_0}{g^2N}\,\sqrt{1+\sqrt{1-\left(\frac{g^2N}{4\pi m_0^2\alpha_0}\right)^2}}\;\right)
\mathcal{K}^{\rm{Bog}}= \frac{1}{2}\sum_{k \neq 0}\mathcal{A}_k \big( a_k^\dagger a_k + a_{-k}^\dagger a_{-k}\big) + \frac 1 2 \sum_{k \neq 0}\mathcal{B}_k \big( a_k^\dagger a_{-k}^\dagger + a_k a_{-k}\big),
\Pr[\Phi(\alpha^n) \le n t] \le \exp\left(- n \beta\right), \ \ \gamma'(s_1) < t < \gamma'(0)=\bar{\phi}\Pr[\Phi(\alpha^n) > n t] \le \exp\left(- n \beta\right), \ \ \gamma'(0)=\bar{\phi}\le t < \gamma'(s_2)
(p-1)\eta^2 u^{p-2}(a\nabla u,\nabla u)+2u^{p-1}\eta (a\nabla u,\nabla \eta) = \frac{4(p-1)}{p^2}(a\nabla (\eta u^{p/2}),\nabla (\eta u^{p/2})) \;\;\;\;-\frac{4(p-2)}{p^2}u^{p/2}(a\nabla (\eta u^{p/2}),\nabla \eta)-\frac{4}{p^2}u^p(a\nabla \eta,\nabla \eta).
SINR_{1,K}= \frac{|\mathbf{v}_{1,K}^H\mathbf{H}_{1,K}\mathbf{p}_1 |^2 \alpha_{1,K}^2}{\sum^{K-1}_{l=1}|\mathbf{v}_{1,K}^H\mathbf{H}_{1,K}\mathbf{p}_1 |^2 \alpha_{1,l}^2 +\sum_{m=2}^{M}|\mathbf{v}_{1,K}^H\mathbf{H}_{1,K}\mathbf{p}_{m}|^2 +|\mathbf{v}_{1,K}|^2\frac{1}{\rho}},
\begin{array}{ccc}(z_j\mapsto -z_j) \hat{=}(1,0,0) begin{align*}3pt](z_j\mapsto -z_j+\frac{\tau_j}{2}) \hat{=}(0,1,0) begin{align*}3pt](z_j\mapsto -z_j+\frac{1}{2}) \hat{=}(0,0,1),\end{array}
I_{\varepsilon}(u_{k})=\left(\frac{1}{2}-\frac{1}{p}\right)\|u_{k}\|_{\varepsilon}^{2}+\left(\frac{1}{2}-\frac{2}{p}\right)\frac{\omega^{2}q}{\varepsilon^{3}}\int_{M}u_{k}^{2}\psi_{k}d\mu_{g}+\frac{\omega^{2}q^{2}}{\varepsilon^{n}p}\int_{M}u_{k}^{2}\psi_{k}^{2}d\mu_{g}\rightarrow c,
\begin{cases}|| \delta_{x}||_{-2,\alpha}\leq || \delta_{x}||_{-1,\alpha}\leq C_{1}(1+ M^{\alpha}), || D_{x,y}||_{-2,\alpha}\leq || D_{x,y}||_{-1,\alpha}\leq C_{2}(1+ M^{\alpha}).\end{cases}
\lim_{N \to \infty}N R_v\left( \pmb{\rho}^{(N)}\right) = \frac{e^{-\gamma_u\left({\rho}^{(\infty)}\right)}\sum_{k=0}^N\sum_{n_1+\cdots+ n_V = k}\prod_{u=1}^V \frac{\gamma_u^{n_u}\left({\rho}^{(\infty)}\right)}{n_u!}\bar r_v(k)}{e^{- \gamma_u\left({\rho}^{(\infty)}\right)}+ \tau\left( 1 - e^{- \gamma_u\left({\rho}^{(\infty)}\right)}\right)}
\begin{aligned}\int \nabla^k u_t \cdot \nabla^{k+1}\varrho dx+\int |\nabla^{k+1}\varrho|^2 dx &=\int \nabla^k [\mu \Delta u+(\mu+\nu)\nabla{\rm div}u]\nabla^{k+1}\varrho dx+ \int \nabla^k S_2 \cdot \nabla^{k+1}\varrho dx.\end{aligned}
\frac{1}{\sqrt{N}\pi}=\sum_{k=0}^\infty\frac{\left(\frac{1}{2}\right)_k^3}{(k!)^3}\left(\widehat{b}_N k+\widehat{a}_N\right)\widehat{X}_N^k,\intertext{where}\widehat{b}_{58}= -6930\sqrt{2}+1287\sqrt{58}, \widehat{a}_{58}= -\frac{6351}{2}\sqrt{2}+\frac{68403}{116}\sqrt{58}, \widehat{X}_{58}= -192119201+35675640\sqrt{29}.
\|(A(x_j^1,\ldots,x_j^n))_{j=1}^k\|_{\mathcal{Y}(F)}\le C\cdot\sup\limits_{T\in B_{\mathcal{L}(E_1,\ldots,E_n)}}\|(T(x_j^1,\ldots,x_j^n))_{j=1}^k\|_{\mathcal{X}(\mathbb{K})},
\delta^sS\simeq \int_{t_1}^{t_2}dt\;\epsilon \;\left[ U_{_{(\nu )}}^s\Omega_{_{(\nu )}}-y^s\Omega_{_{(1)}}-z_{_{(\nu )}}^s\left({\frac{\partial\Omega_{_{(\nu )}}}{\partial \pi}}\pi -\Omega_{_{(\nu )}}\right) \right]\;.
(\lambda_1(U,a) - \lambda_1(\Omega,a)) = J_{\mu}(U,a) - J_{\mu}(\Omega,a) + \mu(|\Omega| - |U|) \leq \sigma |U|^{-\frac{2}{d}}+\mu\, \textup{dist}_\omega(\Omega,U) +\mu(|\Omega| - |U|) \leq 2\sigma |U|^{-\frac{2}{d}}+\mu(|\Omega| - |U|)
\{v \in \bar{v}: ~u_i < v < v_k \}\cup \{w \in \bar{w}: ~u_i < w < w_j \}< w_j < v^\prime_k < w^\prime_j < v_k < \{v \in \bar{v}: v_k < ~ v < u_{i+1}\}\cup \{w \in \bar{w}: ~ w_j < w < u_{i+1}\}.
H(r,\sigma)=\left(\frac{R_0}{R_A}\right)^6\Delta^{-1}(r,\sigma)e^{-6\rho(r,\sigma)}\,,\qquad \mathcal{Z}(r,\sigma)=e^{-2\rho(r,\sigma)}\left(e^{2\rho(r,\sigma)}-\frac{1}{4}\right)\,.
N(B_\varepsilon(q))=\{t\ge0\,|\, f(t,x)\in B_{\varepsilon}(q)\textrm{for}1\le i\le l\}\intertext{and}N(B_\varepsilon(\mathcal{C}_x))=\{t\ge0\,|\, f(t,x)\in B_{\varepsilon}(\mathcal{C}_x)\textrm{for}1\le i\le l\}.
{\bf M}_1(uy) = \left[\begin{array}{cccccc}ay_{00}-y_{20}+by_{02}ay_{10}-y_{30}+by_{12}ay_{01}-y_{21}+by_{03}ay_{10}-y_{30}+by_{12}ay_{20}-y_{40}+by_{22}ay_{11}-y_{31}+by_{13}ay_{01}-y_{21}+by_{03}ay_{11}-y_{31}+by_{13}ay_{02}-y_{22}+by_{04}\end{array}\right],
S = ( - 1 )^{s}\int \left( - \nabla^{\mu}h^{\mu_{1}\ldots \mu_{s}}\nabla_{\mu}h_{\mu_{1}\ldots \mu_{s}}+ [ s^{2}+ ( d - 6 ) s - 2 ( d - 3 ) ] h^{\mu_{1}\ldots \mu_{s}}h_{\mu_{1}\ldots \mu_{s}}\right)
q^{-h}C\Phi_{h}(q^{24})|_{-\frac{1}{2}}\rho=\alpha q^{-\frac{(c,24)^2}{576}h}\left(b_0+b_1q^{\frac{(c,24)^2}{24}}+\cdots\right)\left(\sum_{n=0}^{\infty}a_{g_h}\left(\rho : \frac{n}{48h}\right)q^{\frac{n}{48h}}\right)
q_n^{\epsilon,\Delta t}-q_n^{0,\Delta t}=Q_n^{\epsilon,\Delta t}-q_n^{0,\Delta t}-P_n^{\epsilon,\Delta t}&=q_0^\epsilon-q_0^0+\Delta t\sum_{k=0}^{n-1}\bigl(f(q_k^{\epsilon,\Delta t})-f(q_k^{0,\Delta t})+\sum_{k=0}^{n-1}\bigl(\sigma(q_k^{\epsilon,\Delta t})-\sigma(q_k^{0,\Delta t})\bigr)-\epsilon p_n^{\epsilon,\Delta t}.
\begin{aligned}L_0 = (1-Q) a\partial_a + a^2 \partial_a^2 + a( a + \frac{1}{4}|X|^2 ) \Delta_Z + a \Delta_X L_1 = a^2 \sum_{m+1}^{m+k}\sum_{i=1}^m <[X,e_i] , e_r > \partial_r \partial_i.\end{aligned}
H^0(A,\Omega_A^2)=\wedge^2 H^0(A,\Omega_A^1) = \wedge^2H^0(A_1,\Omega_{A_1}^1)\oplus\wedge^2 H^0(A_2,\Omega_{A_2}^1)\oplus (H^0(A_2,\Omega_{A_2}^1)\otimes H^0(A_1,\Omega_{A_1}^1)).
\partial_{\alpha}(\hat \Psi_{\alpha}\circ F_{\alpha, n_0}^{-1})(z)=\psi'\left(\frac{z_{n_0}+1}{2}\right)\partial_{\alpha}z_{n_0}+\sum_{j=1}^{n_0}\psi'(z_{j})\partial_{\alpha}z_{n_j}-(n_0+1) \partial_{\alpha}\hat \psi_{\alpha}
\langle{\cal O}^{n_{2}}_{W_{2}}{\cal O}^{n_{3}}_{W_{3}}\cdots{\cal O}^{n_{D}}_{W_{D}}\rangle =d^{n_{2}}\langle{\cal O}^{n_{3}}_{W_{3}}\cdots{\cal O}^{n_{D}}_{W_{D}}\rangle
F(x,y) := f(-x-y) \,D_{L}^{p}\phi(x), G(y,z) := e_{N(z)}(y) \mathop{\mathrm{sgn}}g(z) \sqrt{|g(z)|}\,D_{L}^{2p'}\phi(y+z), H(z,x) := e_{N(z)}(z+x) \sqrt{|g(z)|}\,D_{L}^{2p'}\phi(x)
\frac{2\delta}{1+2\gamma\sigma+(\gamma\beta)^2}=\frac{\sqrt{(1-2\gamma\sigma+(\gamma\beta)^2)(1+2\gamma\sigma+(\gamma\beta)^2)}}{1+2\gamma\sigma+(\gamma\beta)^2}&=\sqrt{\frac{1-2\gamma\sigma+(\gamma\beta)^2}{1+2\gamma\sigma+(\gamma\beta)^2}}&=\sqrt{1-\frac{4\gamma\sigma}{1+2\gamma\sigma+(\gamma\beta)^2}}
=t\cdot v-tl_{0}+\stackrel[i=2]{n-1}{\sum}t^{i}(\stackrel[j=i]{n-1}{\sum}T^{j-i+1}l_{j})-\stackrel[i=1]{n-2}{\sum}t^{i+1}(\stackrel[j=i+1]{n-1}{\sum}T^{j-i}l_{j})-\stackrel[i=1]{n-2}{\sum}t^{i+1}l_{i}=
\left( \gamma^{(p-1)m_1}\right)^{p^i}- \frac{1}{\gamma^{(p-1)m_1}}= \left( \gamma^{(p-1)m_2}\right)^{p^i}- \frac{1}{\gamma^{(p-1)m_2}}, \mbox{that is}, \ \ \gamma^{(p-1)(m_1 + m_2)}\left( \gamma^{(p-1)m_1}- \gamma^{(p-1)m_2}\right)^{p^i}= - \left( \gamma^{(p-1)m_1}- \gamma^{(p-1)m_2}\right).
\sum_{j=1}^{N-1}\left. \{c_{ih}^n \}^2 |\widehat{\partial_xp^n_{ih}}|^2\right|_{x_{j+\frac12}}\leq \frac{2}{h}\max\{\beta_0, \Gamma(\beta_1, c_{ih}^n ) \{c_{ih}^n \}\}\|c_{ih}^n(\cdot)\|_\infty \|p^n_{ih}\|_{c_{ih}^n, E}^2 =\frac{2\beta_0}{h}\|c_{ih}^n(\cdot)\|_\infty \|p^n_{ih}\|_{c_{ih}^n, E}^2.
\theta(t)=\frac{(\alpha+\beta t)^{(a-b)}e^{t(\frac{ka\beta \gamma - b\beta \gamma}{k})}\Gamma(\alpha+\beta t)^a}{k^{-\frac{b\beta t}{k}}\Gamma_k(\alpha+\beta t)^b}, t\in (0,\infty), k\geq1
\left.{\frac{\partial S}{\partial \Phi^{\prime A}}}\right|_{K^{\prime}}= \left.{\frac{\partial_{r}\Phi^{B}}{\partial \Phi^{\prime A}}}\right|_{K^{\prime}}{\frac{\partial S}{\partial \Phi^{B}}}={\frac{\partial S}{\partial \Phi^{B}}}{( M^{- 1})^{B}}_{A}.
\delta_K \beta (k_1,k_2,k_3)(x) = \epsilon(k_2,k_3,x) \epsilon(k_1k_2,k_3,x)^{-1}\epsilon(k_1,k_2k_3,x) \epsilon(k_1,k_2,k_3x)^{-1}= \delta_K \epsilon(k_1,k_2,k_3,x) \epsilon(k_1,k_2,k_3)^{-1}= \widehat{F}(k_1,k_2)((F(k_3,x)) \bar{\epsilon}(k_1,k_2,k_3)(x)^{-1}.
\lim_{j\to\infty}T[u_j](z) = T[\delta_{\pi/2}+\delta_{-\pi/2}](z) = \frac{i+z}{i-z}+\frac{-i+z}{-i-z}= 2\frac{1-|z|^4 - 2\imath \Im (z^2)}{|1+z^2|^2}.
\delta S_{1}= \int_{\cal M}\left( \frac{1}{\gamma}\partial_{\mu}F^{\mu \nu}- \frac{k}{8 \pi}\epsilon^{\mu \nu \lambda}F_{\mu \lambda}\right) \delta A_{\nu}+ \int_{\partial \cal M}n_{\perp}\Pi^{\perp i}\delta A_{i}
B = \frac{\Gamma \left( 1 + n - m \alpha_{-}^{2}\right)}{\Gamma \left( n + ( 1 - m ) \alpha_{-}^{2}\right) \Gamma \left( 1 - \alpha_{-}^{2}\right)}\, C_{n m , 1 2}^{n , m - 1}A_{n , m - 1}^{\alpha}
\widetilde \alpha_r^{(n)}(z):= I+z v_n^*R_n^*(\bar z)H_{r,n}^{-1}u_{r,n}, \widetilde\beta_r^{(n)}(z):= -z v_n^*R_n^*(\bar z)H_{r,n}^{-1}v_n, \widetilde\gamma_r^{(n)}(z):= z u_{r,n}^*R_n^*(\bar z)H_{r,n}^{-1}u_{r,n}, \widetilde\delta_r^{(n)}(z):= I-z u_{r,n}^*R_n^*(\bar z)H_{r,n}^{-1}v_n.
{\cal J}= - i \int{d{\bf x}\, \psi^{\dagger}\epsilon_{l n}x_{l}( \partial_{n}- i e A_{n}) \psi}- \int{d{\bf x}\, \psi^{\dagger}\frac{\gamma_{0}}{2}\psi}- \frac{1}{2}\int{d{\bf x}\, x_{n}F_{n 0}\epsilon_{i j}F_{i j}},
\mathcal{H}_{(+,0)}^{(0,0,\pm |M|)}(t) =:\left(x(t),0,\tilde{z}(t)\right), \mathcal{H}_{(-,0)}^{(0,0,\pm |M|)}(t) =:\left(-x(t),0,\tilde{z}(t)\right), \mathcal{H}_{(0,+)}^{(0,0,\pm |M|)}(t) =:\left(0,y(t),z(t)\right), \mathcal{H}_{(0,-)}^{(0,0,\pm |M|)}(t) =:\left(0,-y(t),z(t)\right), ~~ t\in\mathbb{R}.
\|T_{a}(z)\|\leq \|T_{a}(z)\|_2=\Bigl\{\sum_{|K|=a}\,\sum_{|J|=m-a}|z_{JK}|^2\Bigr\}^{1/2}=\Bigl\{\sum_{|M|=m}|z_{M}|^2\Bigr\}^{1/2}=\|z\|.
\xi = - \sqrt{\alpha^{\prime}}\frac{1}{\partial_{\tau}}( u_{2}\psi^{2}+ i u_{4}\psi^{4}) \ , \overline{{\xi}}= - \sqrt{\alpha^{\prime}}\frac{1}{\partial_{\tau}}( u_{2}\psi^{2}- i u_{4}\psi^{4}) \ ,
\beta_{n,q}\left(x+y\right) =\sum_{l=0}^{n}\dbinom{n}{l}q^{lx}\beta_{l,q}\left(y\right)\left[x\right]_{q}^{n-l}=\sum_{l=0}^{n}\dbinom{n}{l}q^{\left(n-l\right)x}\beta_{n-l,q}\left(y\right)\left[x\right]_{q}^{l}.
0 = (\frac{1}{2}(\delta_1 + \delta_2)) \oplus(\frac{1}{4}(\varepsilon_1 + \dots + \varepsilon_5)) = &\frac{1}{4}(\delta_1 \oplus(\varepsilon_1 + \varepsilon_2) + \delta_2 \oplus (\varepsilon_3 + \varepsilon_4) + \delta_1 \oplus (\varepsilon_5 + \varepsilon_6) + \delta_2 \oplus \varepsilon_7).
2\sin(x)\left(\frac{1}{2}+\sum_{k=1}^n\cos(2kx)\right) begin{align*}5pt] =\sin(x)+2\sin(x)\cos(2x)+2\sin(x)\cos(4x)+\cdots+2\sin(x)\cos(2nx) begin{align*}5pt] \overset{\eqref{eq-2.3}}{=}\sin(x)+\left(\sin(3x)-\sin(x)\right)+\left(\sin(5x)-\sin (3x)\right) begin{align*}5pt] \quad+\cdots+\left(\sin((2n+1)x)-\sin ((2n-1)x)\right) begin{align*}5pt] =\sin((2n+1)x).
J(\phi,\omega,r;t) = \left| \frac{\partial z}{\partial \omega}(\phi,\omega,r;t) \det(M_1) - \frac{\partial z}{\partial \phi}(\phi,\omega,r;t) \det(M_2) + \frac{\partial z}{\partial r}(\phi,\omega,r;t)\det(M_3) \right|
\frac{\partial \ln g_{k}}{\partial \ln k^{2}}= - \frac{2 . 6 \cdot 3 g_{k}^{2}/ ( 4 \pi )^{2}\cdot ( 2 . 7 - 3 g_{k}^{2}/ ( 4 \pi )^{2})}{( 4 . 8 - 3 g_{k}^{2}/ ( 4 \pi )^{2}) ( 0 . 8 1 - 3 g_{k}^{2}/ ( 4 \pi )^{2})}\: ,
(x_1^2 +y_1^2+z_1^2)(x_2^2 +y_2^2+z_2^2) = \frac{w_1^2w_2^2||\vec{a}||^4}{(a^2+e^2)^2}(U_1^2+1) \; \left(\frac{1}{U_1^2}+1\right) = \frac{w_1^2w_2^2||\vec{a}||^4}{(a^2+e^2)^2}\left( 2+ U_1^2+ \frac{1}{U_1^2}\right) \geq \frac{4w_1^2w_2^2||\vec{a}||^4}{(a^2+e^2)^2}.
u(x,t)=A+O\left(t^{-\frac{1}{2}}e^{-16t\xi^{3/2}}\right), x>0, \t>0, \ \frac{\kappa^2}{3}<\xi<\kappa^2. u(x,t)=O\left((-t)^{-\frac{1}{2}}e^{16t\xi^{3/2}}\right), x<0, \t<0, \ \frac{\kappa^2}{3}<\xi<\kappa^2,
{{\boldsymbol{\Omega}}_{k;t}}{}\left[{\begin{array}{*{20}{c}}{{{\boldsymbol{\Omega}}_{x;k;t}}}{{{\boldsymbol{\Omega}}_{yx;k;t}}}{{{\boldsymbol{\Omega}}_{xy;k;t}}}{{{\boldsymbol{\Omega}}_{y;k;t}}}\end{array}}\right]
\frac{1}{2T}\sum_{v\in [n]}^{}\mathbb{E}[k_v\mathbb{1}_{\{k_v>T \}}] =\frac{1}{2T}\sum_{v\in [n]}^{}[T\mathbb{P}(k_v>T)+\sum_{\ell >T}^{}\mathbb{P}(k_v\geq \ell)].
{}^{i}\hat{O}_{\lambda , \mu}^{r_{1}r_{2}}={}^{i}\hat{O}_{\lambda}^{r_{1}}\otimes{}^{i}\hat{O}_{\mu}^{r_{2}}\equiv{}^{i}\hat{O}_{\lambda , \mu}^{r}={}^{i}O_{\lambda , \mu}^{r}( \alpha_{\lambda}\otimes \alpha_{\mu}) ,
\int_{0}^{\infty}d x \frac{\cos a x}{\cosh^{\nu}\beta x}= \frac{2^{\nu - 2}}{\beta \Gamma ( \nu )}\Gamma \left( \frac{\nu}{2}+ \frac{a i}{2 \beta}\right) \Gamma \left( \frac{\nu}{2}- \frac{a i}{2 \beta}\right) .
B'(t)=\frac{1}{K}\int_{0}^1\frac{e^{-G(t)}}{(y^2+e^{-2G(t)})^2}dy\geq \frac{1}{K}\int_{0}^1\frac{e^{-G}}{(\frac{K^2}{4}e^{-2G}+e^{-2G})^2}dy\geq\frac{1}{K(K^2/4+1)^2}\frac{e^{3A}}{3B}(1-e^{-3B}).
Q(x)B(x_{1})\cdots B(x_{m}) = q_m(x)B(x_{1})\cdots B(x_{m})W_{2m,0}(x) +\sum_{k=1}^{m}q_{k-1}(x)B(x_{1})\cdots B(x_{k-1})X_{2(k-1),0}(x,x_{k})B(x_{k+1})\cdots B(x_{m}) \,,
\dot{\epsilon}_{\delta \sigma}= \epsilon_{\delta \sigma}\left[ 3{\frac{\dot{\bar{a}}_{R}}{\bar{a}_{R}}}( 1 + w ) + 2{\frac{\dot{\bar{c}}_{4}}{\bar{c}_{4}}}{\frac{1 + \epsilon_{\delta \sigma}}{1 + \epsilon}}+ \left({\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}- 4{\frac{\dot{\bar{c}}_{4}}{\bar{c}_{4}}}\right) ( 1 + \epsilon_{\delta \sigma}) \right] .
\mathcal{S}_n := \{x\in [0,1]: |f_n(x) - \alpha_1| < \varepsilon_0,~\, |\overline{f}_n(x) - \alpha_2| < \varepsilon_0\}, \mathcal{L}_n := \{x\in [0,1]: |f_n(x) - \beta_1| < \varepsilon_0,~\, |\overline{f}_n(x) - \beta_2| < \varepsilon_0\}, \mathcal{T}_n := [0,1]\setminus (\mathcal{S}_n \cup \mathcal{L}_n).
\begin{cases}X= \dfrac{c_6 u^6 + c_4u^4+c_2u^2+c_0}{3u^2\left( \left(3x_1+a_2z\right) u^2 -\left(3x_2+a'_2z\right) \right)}, Y= \dfrac{d_{10}u^{10}+d_6u^6+d_5u^4+d_0}{2u^3\left(\left(3x_1+a_2z\right) u^2 -\left(3x_2+a'_2z\right) \right)^2}, \end{cases}