harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a_{k_0}\leq C'r_0^s2^{k_0}\leq C''b^{-1/p}\left(\sum_{j=-\infty}^{l-2}2^{s'jp}\left(\int_{2B_0}g_j^p\,d\mu\right)^{\frac{q}{p}}\right)^{\frac{1}{p}}.
\mathcal{F}(t_0,t_1,t_2,Q) = \frac{1}{2}\left(t_0 t_1^2 + t_0^2 t_2 \right) + \sum_{d > 0}\sum_{n \geq 0}\sum_{\alpha_1+3d-1=n}N_d \frac{(d \, t_1)^{\alpha_1}}{{\alpha_1}!}\frac{t_2^{3d-1}}{(3d-1)!}Q^d
\begin{array}{l}\dot{S}_1=\displaystyle{\frac{1}{3}(1+3S_1)(3S_2-3S_1+2)\sin(2\theta)}, \dot{S}_2=\displaystyle{\frac{1}{9}\left(27 S_2^2-27 S_1S_2-9S_2+3 S_1-2\right)\sin(2\theta)}, \dot{\theta}=\displaystyle{\frac{(4+3S_1+9S_2)\cos(2\theta)}{9(S_1-S_2)}},\end{array}
A_{\infty}(x) = \sum_{\gamma=1,2}\int_{\mathbb{R}^3}\mathrm{d}k \; \frac{\chi_{\Lambda}(k)}{2\pi |k|^{1/2}}\mathbf{e}_{\gamma}(k) (a_{\gamma}(k) e^{i kx}+ a^{\dagger}_{\gamma}(k) e^{-ikx}).
\begin{array}{c}{x^{\mu}( \sigma , \tau ) = x_{R}^{\mu}( \sigma , \tau ) + x_{L}^{\mu}( \sigma , \tau )}\\{x_{R , L}^{\mu}( \sigma , \tau ) = \frac 1 2 x_{0}^{\mu}+ \frac 1 2 l^{2}p^{\mu}( \tau \mp \sigma ) + \frac{i l}2{\sum}\frac 1 n \alpha_{n , \pm}^{\mu}e^{- 2 i n ( \tau \mp \sigma )}}\\ \end{array}
\sqrt{-g}L \equiv{1\over 4\pi{\cal A}_P}\left( \sqrt{-g}{\cal G}\right) ={1\over 4\pi{\cal A}_P}\left(\sqrt{-g}\, g^{ik}\left(\Gamma^m_{i\ell}\Gamma^\ell_{km}-\Gamma^\ell_{ik}\Gamma^m_{\ell m}\right)\right).
\begin{array}{lllllllllll}g \cdot r_1 = -r_1, g \cdot r_2 = -r_2, g \cdot r_3 = -r_3, g \cdot r_4 = r_4, g \cdot r_5 = r_5, g \cdot r_6 = -r_6, x \cdot r_1 = 0 , x \cdot r_2 = 0, x \cdot r_3 = r_5, x \cdot r_4 = r_3-r_6, x \cdot r_5 = 0, x \cdot r_6 = r_5. \end{array}
\langle[\begin{matrix}x_{1}\\ \vdots\\x_{n}\end{matrix}],[\begin{matrix}y_{1}\\ \vdots\\y_{n}\end{matrix}]\rangle=x^{T}y=\sum_{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+\cdot\cdot\cdot+x_{n}y_{n},
\psi^{( + )}\stackrel{r \sim \infty}{\rightarrow}{\frac{\Gamma ( c ) \Gamma ( b - a )}{\Gamma ( b ) \Gamma ( c - a )}}e^{( - Q r )}+{\frac{\Gamma ( c ) \Gamma ( a - b )}{\Gamma ( a ) \Gamma ( c - b )}}e^{( + Q r )}
\mathcal{H}_m(u)(q_0) = \lambda_{n}(q_0) \sum\limits_{1\leq j_2<\dots<j_m\leq n-1}\lambda_{j_1}(q_0) \cdots\lambda_{j_{m-1}}(q_0)+ \sum\limits_{1\leq j_1<\dots<j_m\leq n-1}\lambda_{j_1}(q_0)\cdots\lambda_{j_m}(q_0).
\begin{aligned}\Delta r_{O2}=\max_{P_i,\alpha_i}\sum_{i=1}^K\Delta r_i \mathrm{s.t.}\sum_{i=1}^KP_i+N_0\sum_{i=1}^K\frac{(e^{\frac{r^*}{\alpha_i}}-1)\alpha_i}{|h_i|^2}\leq P , &\Delta r_i=\alpha_i\ln \Big(1+\frac{P_i|{h_i}|^2}{\alpha_ie^\frac{r^*}{\alpha_i}N_0}\Big), \sum_{i=1}^K\alpha_i=1.\end{aligned}
\rho \to \frac{a \rho + b}{c \rho + d}, \rho \equiv \frac{{\tilde{V}}_{2}}{( 2 \pi )^{2}\alpha^{\prime}}\left( B_{p - 1 , p}+ i \sqrt{G_{( p - 1 ) ( p - 1 )}G_{p p}}\right) ,
\sum_{i=1}^n u'_i = \left( \sum_{\substack{i=1 i \neq \ell-1, \ell}}^n u'_i \right) + u'_{\ell-1}+ u'_{\ell}& < \left( \sum_{\substack{i=1 i \neq \ell-1, \ell}}^n u'_i \right) + u_{\ell-1}+ u_{\ell}= \sum_{i=1}^n u_i.
\bar{Z}_{+}( z_{2}) Z_{+}( z_{1}) = F_{+}( z_{1 2}) = \left( \begin{array}{c c}{i x_{\bar{2}1}{\cdot \tilde{\sigma}}\tau_{+}}&{- 2 \bar{\theta}_{2 1 b}p_{+}}\\{- 2 \theta_{2 1}^{a}\tau_{+}}&{\delta_{b}^{a}p_{+}}\\ \end{array}\right)
\mathbb{E}_{\eta^{(L)}}\left[D(\eta(t),\xi^{(\ell_1, \ldots, \ell_n)})\right] \prod_{m=1}^n q^{-2N_{\ell_m +1}(\eta^{(L)})}= \mathbb{E}_{\xi^{(\ell_1, \ldots, \ell_n)}}\left[D(\eta^{(L)},\xi(t))\right] \prod_{m=1}^n q^{-2N_{\ell_m +1}(\eta^{(L)})}
a_2=2a_1-a_0,\; a_3=\left( \frac{\nu a_0-1}{2}\right) (2a_1-a_0), \; a_4=\left( \frac{\nu a_0-1}{2}\right) (2a_1-a_0)+a_1 -a_0, d_1=\left( \frac{\nu a_0-1}{2}\right) (2a_1-a_0)+a_1,\; d_2=\left( \frac{\nu a_0+1}{2}\right) (2a_1-a_0),\; \mbox{where}\; \gcd\left(\frac{\nu a_0-1}{2},a_1-a_0\right) =1.
\frac{(a+p_{1}-q_{1})(c+p_{2}-q_{2})}{(a-p_{1}+q_{1})(c-p_{2}+q_{2})}=\frac{(b+p_{2}-q_{1})(d+p_{1}-q_{2})}{(b-p_{2}+q_{1})(d-p_{1}+q_{2})}.
\bigl\{\mathbf{p}\bigl(\tfrac{\partial_1a}{a},\dots,\tfrac{\partial_ma}{a}\bigr) \ \big| \a\in A\bigr\}\neq\{0\}, \intertext{and such that there exists an element $f\in K$ satisfying}\mathbf{p}(\partial_1u,\dots,\partial_mu)-\mathbf{q}(\partial_1r_1,\dots,\partial_mr_1)=\delta_xf.
\begin{aligned}\|\tilde u - u_h\|_{L^\infty(\Omega_h)}\le Ch^2|\log h| \, \|u\|_{W^{2, \infty}(\Omega)}, \|\tilde u - u_h\|_{W^{1,\infty}(\Omega_h)}\le Ch \, \|u\|_{W^{2, \infty}(\Omega)},\end{aligned}
\delta \Psi_{M}= 0 \qquad \Longrightarrow \qquad \hat{\nabla}_{M}\epsilon ( x , y ) = 0 \Longrightarrow \left( \begin{matrix}{{\hat{\nabla}_{\mu}\eta ( x ) = 0}}\\{{\hat{\nabla}_{\alpha}\varsigma ( y ) = 0}}\\ \end{matrix}\right) \ .
1_{A_{1}\cup\cdot\cdot\cdot\cup A_{n}}\ge\sum_{j=1}^{k}(-1)^{j-1}\sum_{1\le i_{1}<\cdot\cdot\cdot<i_{j}\le n}p_{i_{1}}...p_{i_{j}}1_{A_{i_{1}}\cap\cdot\cdot\cdot\cap A_{i_{j}}}.
H=-\sum_{i=1}^{4}\frac{\partial^{2}}{\partial x_{i}^{2}}-\frac{\eta}{\sqrt{\sum_{i=1}^4 x_{i}^{2}\}}+\lambda \sum_{i<j}^3 \frac{1}{(x_{i}-x_{j})^{2}}+\frac{g}{(x_1+x_2+x_3- 3 x_4)^2}+\frac{\mu}{\sum_{i=1}^{4}x_{i}^{2}},\eta >0
-2 \int_{t+2(r+1)}^{\infty}ds \lambda''(s)(s-t) \left(\frac{(r^{2}-1-(s-t)^{2})}{\sqrt{\beta}(1+r^{2}+(s-t)^{2}+\sqrt{\beta})}\right) &= \int_{t+2(r+1)}^{\infty}ds \frac{\lambda''(s)}{(s-t)}+ E_{1,\partial_{r}v_{1}}(t,r)
S_Q^{(1)}=-\mu^{-\epsilon}\int d^4\theta d^d x\partial_{\alpha\dot{\alpha}}\sigma\partial^{\alpha\dot{\alpha}}\bar{\sigma}(\frac{18\xi_1^2{(4\pi)}^4}{Q^4\epsilon}+fin)\equiv S^{(1)}_{Q_{div}}+S^{(1)}_{Q_{fin}}
Q^- \sim \sqrt{g^2 N}\times \frac{K}{\sqrt{N}}\times \sum_{n,l=1}^{\infty}\left(\frac{1}{n}+ \frac{1}{l}\right) \times \left(\Gamma^{\dagger}(l+n) \Gamma(l) \Gamma(n)+ \Gamma^{\dagger}(n)\Gamma^{\dagger}(l) \Gamma (n+l)\right),
\begin{cases}t_{i,ex}(x_0,t_0)=\max\left(0,t_0-\dfrac{x_0-R}{\lambda_i}\right) \forall\: i\in [1,p], t_{i,ex}(x_0,t_0)=\max\left(0,t_0-\dfrac{x_0+R}{\lambda_i}\right) \forall\: i\in [p+1,n].\end{cases}
I_{P^{(\mathcal{C}_n)}}(M;\mathbf{Z})\leq\frac{1}{|\mathcal{M}_n|}\sum_{m\in\mathcal{M}_n}\mathsf{D}\Big(P^{(\mathcal{C}_n)}_{\mathbf{Z}|M=m}\Big|\Big|p^n_Z\Big).
(i +x) \sum_{j=1}^{\infty}\lambda_{l,j}(l+x)^{-j}\ = \ \ \sum_{j=1}^{\infty}\lambda_{l,j}(l+x)^{-j+1}+ \sum_{j=1}^{\infty}\lambda_{l,j}(i - l) (l+x)^{-j}\ = \ \ \lambda_{l,1}+ \sum_{j=1}^{\infty}\left( \lambda_{l,j+1}+ \lambda_{l,j}(i - l) \right) (l+x)^{-j}.
\int d^3 x \, \frac{e^{\pm i \vec{k}\vec{x}}}{| \vec{x}|}= \frac{4 \pi}{| \vec{k}|^2}\, , \int d^3 x \, e^{\pm i \vec{k}\vec{x}}\Delta \frac{1}{| \vec{x}|}= - 4 \pi \, , \Delta \frac{1}{| \vec{x}|}= - 4 \pi \delta^{(3)}(\vec{x}) \, ,
\sum_{p=1}^{k}\frac{|\langle \tau_p,\psi_\ell\rangle|^2}{\mathrm{vol}(\Omega_p)}=\sum_{p=1}^{k}\left|\left\langle \frac{\tau_p}{\sqrt{\mathrm{vol}(\Omega_p)}},\psi_\ell\right\rangle\right|^2\leq \Vert \psi_\ell\Vert^2=1.
\mathbb{P}_\varphi (X^{(T)}=x^{(T)}| \mu^{(T)}=m) & = \mathbb{P}_\varphi(X_0=x_0 | \mu^{(T)}=m) \prod_{t=0}^{T-1}\mathbb{P}_\varphi(X_{t+1}=x_{t+1}| \mu^{(T)}=m ,X^{(t)}=x^{(t)}) = m_0(\{x_{0}\}) \prod_{t=0}^{T-1}\mathbb{P}_\varphi(X_{t+1}=x_{t+1}| \mu^{(T)}=m ,X^{(t)}=x^{(t)}).
\frac{z \mathtt{f}'_{a, \nu}(z)}{\mathtt{f}_{a, \nu}(z)}=\frac{a^{a/2}}{a\nu-a+1}\frac{z\;{}_{a}\mathtt{B}'_{2a-1, a \nu-a+1,1}(a^{a/2}z)}{{}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(a^{a/2}z)}&= \frac{a^{a/2}}{a\nu-a+1}\left(\frac{z\mathtt{J}_{\nu-1}\left(z\right)}{\mathtt{J}_{\nu}\left(z\right)}-\nu(2-a)+1-a\right).
KiG(fg)\alpha_m = KiG(f)KiG(g)\alpha_m = KiG(f)\alpha_k\alpha_k^\dagger KiG(g)\alpha_m = \alpha_n\alpha_n^\dagger KiG(f)\alpha_k\alpha_k^\dagger KiG (g)\alpha_m = \alpha_n\alpha_n^\dagger KiG(f) KiG(g)\alpha_m = \alpha_n\alpha_n^\dagger KiG(fg)\alpha_m.
\delta_{f}\,\left( \left\{g,h\right\}_{{\footnotesize PB}}\right)\equiv\left\{f,\left\{g,h\right\}_{{\footnotesize PB}}\right\}_{{\footnotesize PB}}=\left\{\left( \delta_{f}\,g\right) ,h\right\}_{{\footnotesize PB}}+\left\{g,\left( \delta_{f}\,h\right) \right\}_{{\footnotesize PB}}\;.
\left[\begin{array}{ccccc}1 2 2^2 \ldots 2^{m-2}1 3 3^2 \ldots 3^{m-2}\vdots \vdots \vdots \ddots \vdots 1 m m^2 \ldots m^{m-2}\end{array}\right]\left[\begin{array}{c}a_0 a_1 \vdots a_{m-2}\end{array}\right]= \left[\begin{array}{c}2\cdot2^1-m\cdot2^2 3\cdot2^2-m\cdot2^3 \vdots m\cdot2^{m-1}-m\cdot2^m\end{array}\right].
\partial^s f \cdot \mathcal R g = \sum_j \partial^s f_{<j-2}\mathcal R g_j + \sum_j \partial^s f_{\ge j-2}\mathcal R g_j & = \sum_j \partial^s f_{<j-2}\mathcal R g_j + \sum_j \partial^s f_{j}\mathcal R g_{\le j+2}.
g^\ddag=\left(\begin{array}{ccc}1+{1\over 4}\eta\eta^\diamond{1\over 2}(b\eta^\diamond+a^\diamond\eta) -{1\over 2}(a\eta^\diamond-b^\diamond\eta{1\over 2}\eta^\diamond a^\diamond(1-{1\over 8}\eta\eta^\diamond) b^\diamond(1-{1\over 8}\eta\eta^\diamond){1\over 2}\eta -b(1-{1\over 8}\eta\eta^\diamond) a(1-{1\over 8}\eta\eta^\diamond)\end{array}\right)
Z=\int e^{\overline{\psi}M\psi+\overline{\eta}\psi+\overline{\psi}\eta}D\overline{\psi}D\psi=\int e^{(\overline{\psi}+\overline{\eta}M^{-1})M(\psi+M^{-1}\eta)-\overline{\eta}M^{-1}\eta}D\overline{\psi}D\psi=Det(M)e^{-\overline{\eta}M^{-1}\eta},
\kappa(\gamma_1 \vee \cdots \vee \gamma_n \otimes m)=\sum_{k=0}^n \sum_{\sigma\in Sh(k,n-k)}\epsilon(\sigma) \gamma_{\sigma(1)}\vee \cdots \vee \gamma_{\sigma(k)}\otimes \kappa_{n-k}(\gamma_{\sigma(k+1)}\vee \cdots \vee \gamma_{\sigma(n)}\otimes m).
{T_3}=\binom{5n}{4n}\frac{1}{{T_2}{T_1}}&>\binom{5n}{4n}\frac{1}{2^{\frac{n}{2}}AB}\cdot\frac{1}{{(5n)}^{\frac{2.51012\sqrt{5n}}{\log(5n)}}}&>\frac{0.054886}{2^{\frac{n}{2}}n^{\frac{3}{2}}}{\left(\frac{3125}{256}\right)}^{\frac{n}{6}}\cdot\frac{1}{{(5n)}^{\frac{2.51012\sqrt{5n}}{\log(5n)}}}>1,
S = \int~ d^{4}x \sqrt{-g}~\left[~\frac{\tilde{R}(g,T)}{\kappa}- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{2}\tilde{H}_{\mu \nu \lambda}\tilde{H}^{\mu \nu \lambda}+ \frac{1}{\sqrt{\kappa}}T^{\mu \nu \lambda}\tilde{H}_{\mu \nu \lambda}~\right]
P=\langle\psi_{e}^{\prime}\psi_{v}^{\prime}\psi_{s}^{\prime}|\mu|\psi_{e}\psi_{v}\psi_{s}\rangle=\int\psi_{e}^{\prime*}\psi_{v}^{\prime*}\psi_{s}^{\prime*}(\mu_{e}+\mu_{N})\psi_{e}\psi_{v}\psi_{s}d\tau
\begin{aligned}\left(\Psi^--e^{-ic_-(x)\sigma_3}\right)-\left(\widetilde{\Psi}^--e^{-ic_-(x)\sigma_3}(\tilde{u})\right) &=(I-F)^{-1}(T-\widetilde{T})+(I-F)^{-1}(F-\tilde{F})(I-\tilde{F})^{-1}\widetilde{T},\end{aligned}
\times \prod_{b = 1}^{N}\prod_{j = 1}^{n_{b}}\exp \left( 2 e \sum_{I = 1}^{N}\Big ( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, K_{I I}\; U_{I b}\; \delta ( x_{j}^{( b )}) \Big ) \right)
\| u(t_n) - U^{n}\|_{L^{p}(\Omega;H)}\leq \|( \bar E(t_n) - B_nP_h) u_0 \|_{L^p(\Omega;H)}+ \|\sum_{j=0}^{n-1}\int_{t_j}^{t_{j+1}}(\bar{E}(t_n-t)-B_{n-j}P_h)\d W(t) \|_{L^p(\Omega;H)}:={\rm I}+{\rm II}.
\Phi(\lambda)+\lambda(1-\Phi(\lambda))\frac{\Phi(\lambda)}{{\phi(\lambda)}}\geq \Phi(\lambda)+\lambda(1-\Phi(\lambda))\frac{\lambda}{1+\lambda^2}&=\frac{\lambda^2}{1+\lambda^2}+\Phi(\lambda)\frac{1}{\lambda^2+1}\geq \frac{\lambda^2}{1+\lambda^2}\geq \frac{1}{2}.
\prod_{\lambda\in \mathrm{Ext}_{\Sigma\setminus \Sigma H_I}(\mu;F)}(s_{r(F)}^\Sigma-s_{\mu\lambda}^\Sigma{s_{\mu\lambda}^\Sigma}^*)=(s_{r(F)}^\Sigma-s_\mu^\Sigma{s_\mu^\Sigma}^*)+s_\mu^\Sigma\Delta(s^\Sigma)^{\mathrm{Ext}_{\Sigma\setminus \Sigma H_I}(\mu;F)}{s_\mu^\Sigma}^*.
\triangle_m\left( \binom{t+2^m-1}{2^m-1}^2 \right) \geqslant -(m-1)+1=-m+2, \triangle_m\left( \binom{t+2^{m-1}-1}{2^{m-1}-1}\right) \geqslant -(m-2)=-m+2, \triangle_m\left( \binom{t+2^m-1}{2^{m-1}-1}\right) \geqslant -(m-2)=-m+2.
\jmath^\natural(F\otimes^{\mathrm{RD}}G)(U_I) =(F\otimes^{\mathrm{RD}}G)(\jmath(U_I)) &\simeq \bigoplus_{I_1\sqcup I_2=I, I_j\neq\varnothing}F(\jmath(U_{I_1}))\otimes G(\jmath(U_{I_2})) &=\bigoplus_{I_1\sqcup I_2=I, I_j\neq\varnothing}\jmath^\natural F(U_{I_1})\otimes \jmath^\natural G(U_{I_2}) &\simeq (\jmath^\natural F\otimes^\mathrm{RD}\jmath^\natural G)(U_I),
\|P_{2N}(fg)\|\le \sum_{\ell=1}^{N}{2N-1\choose 2\ell-1}\|fg\|^{2N+2\ell - 1}= \|fg\|^{2N-1}\sum_{\ell=1}^{N}{2N-1\choose 2\ell-1}\|fg\|^{2\ell}
H = C \left\{\sum_{n = 1}^{N - 1}h_{n , n + 1}+ \frac{1}{2}\dot{K}_{1}^{-}( 0 ) + \frac{t r_{0}[ K_{0}^{+ t}( - \eta ) h_{N 0}]}{t r [ K^{+}( - \eta ) ]}\right\}
\int_{\Pi}\varrho \mathbf{u}\cdot\partial_t\boldsymbol{\xi}\, dxdt=-{\Gamma^{(1)}}-\Gamma^{(2)}-\int_{Q_T}\zeta\varrho\mathbf u\cdot\nabla \Delta^{-1}\big[\zeta\psi(\varrho)\mathbf u\big]_h\, dxdt.
F_l^\mathfrak d =\sum_{d=0}^\mathfrak d\sum_{j=0}^\mathfrak d\gamma_d^\mathfrak d\,a_l^j(f^{(d)})\,Y_l^j =\sum_{j=0}^\mathfrak d\sum_{d=0}^\mathfrak d\gamma_d^\mathfrak d\,a_l^j(f^{(d)})\,Y_l^j, G_l^\mathfrak d =\sum_{j=0}^\mathfrak d\sum_{d=0}^\mathfrak d\gamma_d^\mathfrak d\,a_l^j(g^{(d)})\,Y_l^j.
\mathbf{E}\left(F\left((\eta^N_n)_{n=1}^{N}\right)\right)=\frac{\mathbf{E}\left(\nu(\eta_0)^{-1}F\left((\eta_n)_{n=1}^{N}\right)\:\vline\:\eta_N=\eta_0,\:S_N>0\right)}{\mathbf{E}\left(\nu(\eta_0)^{-1}\:\vline\:\eta_N=\eta_0,\:S_N>0\right)},
\frac{1}{2\pi}\int_{\delta\le|t|\le\pi}|f(x-t)-f(x)|F_{n}(t)dt\le\frac{1}{2\pi}\int_{\delta\le|t|\le\pi}2MF_{n}(t)dt=\frac{M}{\pi}\int_{\delta\le|t|\le\pi}F_{n}(t)dt
\vartheta_X(y,\nu,\ell)-\sum_{i=1}^{s-r}\left( \kappa(p_i) b_y + \kappa(p_i) \right) \langle w_i , (\nu,\ell) \rangle = (\vartheta_X-\vartheta_{X'})(y,\nu,\ell)-\sum_{i=1}^r \langle \nu,u_i \rangle -\sum_{j=1}^r \langle (\nu,\ell),w_j \rangle.
\frac{1}{i \partial_{-}- e \bar{B}_{-}}- \frac{1}{i \partial_{-}}= \left( \frac{1}{i \partial_{-}- e \bar{B}_{-}}e \bar{B}_{-}\right) * \frac{1}{i \partial}= \frac{1}{i \partial}\ast \left( e \bar{B}_{-}\frac{1}{i \partial_{-}- e \bar{B}_{-}}\right) \,
\delta_t U^n_j - \frac{1}{3}\mathcal{Q}_{n-\frac{1}{2}}^{(1)}(\delta_x^2U_j) - \frac{2}{3}\mathcal{Q}_{n-\frac{1}{2}}^{(2)}(\delta_x^2U_j) = f^{n-1/2}_j, n\geq 1, 1\leq j \leq \mathcal{M}-1, U_j^0=u_0(x_j), 1\leq j \leq \mathcal{M}-1, U_{0}^n = U_{\mathcal{M}}^n = 0, n\geq 1.
|a_{m+k}| \lesssim \sum_{j=1}^{\infty}|\tilde{c}_{j+k}|\lambda_{-}^{|m-j|}\lesssim \left(\sum_{j=1}^{\infty}|\tilde{c}_{j+k}|^{2}\lambda_{-}^{|m-j|}\right)^{1/2}\underbrace{\left(\sum_{j=1}^{\infty}\lambda_{-}^{|m-j|}\right)^{1/2}}_{\leq C},
\Omega_{i}= - \textstyle{\frac{1}{4 \sqrt{3}}}\gamma^{+}\left[ \lambda_{j}\gamma^{j}\gamma_{i}+ \lambda_{i}\right] , \Omega^{-}= \textstyle{\frac{1}{4 \sqrt{3}}}\lambda_{i}\gamma^{i}\left( \gamma^{+}\gamma^{-}+ 1 \right) .
(\partial_{t}^+ \phi_h^n,\phi_h^n)_{\mathcal{T}_h}=\frac{1}{2\Delta t}\left(\|\phi_h^n\|^2_{\mathcal{T}_h}-\|\phi_h^{n-1}\|^2_{\mathcal{T}_h}+\|\phi_h^n-\phi_h^{n-1}\|^2_{\mathcal{T}_h}\right),
p^{16}=\left[\begin{array}{ccccc}1 1 3 3 2 2 2 2 2 1 3 3 1 1 3 \end{array}\right],p^{17}=\left[\begin{array}{ccccc}1 1 2 2 2 2 2 3 3 1 3 3 1 1 3 \end{array}\right],p^{18}=\left[\begin{array}{ccccc}1 1 2 2 3 2 2 3 3 2 3 3 1 1 1 \end{array}\right],
S_{\mathrm{T}}= \int d^{4}z \, d^{4}\bar{z}\, \mathrm{t r}\Bigr \{\mathrm{~ \frac{1}{1 2}~}\Bigr ( \epsilon^{a b}F_{\! ~ a b}^{A B}\epsilon^{c d}F_{A B c d}- \epsilon_{A B}F_{\! ~ a b}^{A B}\epsilon_{C D}F^{C D a b}- F_{\! ~ a b}^{A B}F_{A B}^{\! ~ b a}\Bigr ) \Bigr \},
\frac{|\nabla f|^2}{H^{2+\sigma}}\partial_0 S^2 =|\nabla f|^2\left(\partial_0 U_{\sigma}+\frac{2+\sigma}{H}(\partial_0H)U_{\sigma}\right) &=|\nabla f|^2(\partial_0U_{\sigma})+\frac{2+\sigma}{H}\langle \nabla H, \nabla f\rangle U_{\sigma}.
\Delta \, P = ( \delta_{1}\otimes 1 + 1 \otimes \delta_{1})^{a_{1}}\, ( \delta_{2}\otimes 1 + 1 \otimes \delta_{2}+ R_{1})^{a_{2}}\ldots ( \delta_{n}\otimes 1 + 1 \otimes \delta_{n}+ R_{n - 1})^{a_{n}}\, .
\sum_{k=1}^{\infty}\frac{(2^{-k}t )^n}{\psi (t 2^{-k})^{n-1}}\le C(n, \alpha) \frac{t^n}{\psi(t)^{n-1}}, C(n, \alpha) = \frac{2^{\alpha(n-1)}}{2^n - 2^{\alpha(n-1)}}.
(F, F_1, \dots,F_m)=\bigg(\Big(\sum_j (f^j)^{s}\Big)^\frac1{s},\Big(\sum_j (f_1^j)^{s_1}\Big)^\frac1{s_1},\dots,\Big(\sum_j (f_m^j)^{s_m}\Big)^\frac1{s_m}\bigg)
\|(\mathcal{M}(\tilde u_0,a_n,b)u)(t,\cdot)-(\mathcal{M}(\tilde u_0,a_0,b)u)(t,\cdot)\|_{C^0(\bar\Omega)}= \| \int_s^t e^{-A(t-\tau)}u(\tau,\cdot)\big[a_n(\tau,\cdot)-a_0(\tau,\cdot)\big] d\tau\|_{C^0(\bar\Omega)}&\le R \|a_n-a_0\|_{\mathcal{X}_T}T.
\mathbb E\left[\|\Delta_n^{-\frac 12}(3)_t\|_{\mathcal H}^2\right] = 2\Delta_n\sum_{i=1}^n (i-1)\int_{(i-1)\Delta_n}^{i\Delta_n}\mathbb E\left[\|\sigma_s^{\mathcal S_n}\|_{L_{}(U,H)}^2\right]ds \leq 2\sup_{r\in [0,T]}\|\mathcal S(r)\|_{}^2.
\begin{array}{c}\displaystyle{\sum_{\nu=3/2}^{\infty}\nu^{-(s-k-1)}}\ \Re \left\{\displaystyle{\frac{- e^{\displaystyle{-\imath \frac{\pi}{2}(s+1)}}}{\pi}}\, \displaystyle{\int_{- \imath z}^{\infty - \imath z}}t^{-s}\, \displaystyle{D_{TE}^{(k)}(t)}\dt \right\},\end{array}
\Phi_{k,x,\nu}(u)=\left((\rho^t_{a\to b,w,j}\cdot u)\circ (A^{t-t(k,x)+6\delta_0}_{a\to c}|_{R^t_{a\to b,w}})^{-1}\right)_{w\in Q_\nu(k,x)}\intertext{and}\Psi_{k,x,\nu}\left((u_w)_{w\in Q_\nu(k,x)}\right)=\sum_{w\in Q_\nu(k,x)}u_w.
\widetilde{a}_+( (\widetilde{\varphi}, \widetilde{w}), (\vartheta \tau_{1,-s}\widetilde{\psi}, \vartheta \tau_{1,-s}\widetilde{v}))=\widetilde{F}_+ ( \vartheta \tau_{1,-s}\widetilde{\psi}, \vartheta \tau_{1,-s}\widetilde{v})
*_gH = \frac{h_1\left(a^4+ c^2\right) - 2a^2c h_3}{\mu\left(a^2b^2 - c^2\right)}\, e^{23}+ \frac{h_1\left(b^4 + c^2\right) - 2b^2ch_3}{\mu\left(a^2b^2 - c^2\right)}\, e^{45}&-\frac{h_4}{\mu}\, \left(e^{24}+e^{35}\right) - \frac{h_3\left( a^2 b^2 + c^2\right) -c h_1\left( a^2 + b^2\right)}{\mu\left(a^2b^2 - c^2\right)}\left(e^{25}-e^{34}\right),
\frac54 \frac{5^6}{4^6}a^6 N^{6/4}+ a'(t)N^{5/4}\frac54 aN^{1/4}= aN^{6/4}\frac54\Big(\frac{5^6}{4^6}a^5+a'(t)\Big) &= aN^{6/4}\frac54\Big(\frac{5^6}{4^6}a^5-ka^5\Big)= a^6N^{6/4}\Big(\frac54 \frac{5^6}{4^6}-\frac{5^6}{4^6}\Big(\frac32+\frac{25}{4\cdot5}\Big)\Big)<0
\||x|^{\gamma_1}|x_1|^{\alpha}u\|_{L^s(\mathbb{R}^n)}\le \||x|^{\gamma_1}|x_1|^{\alpha}\zeta u\|_{L^s(\mathbb{R}^n)}+\||x|^{\gamma_1}|x_1|^{\alpha}(1-\zeta)u\|_{L^s(\mathbb{R}^n)}=: I_1+I_2.
p_{-2}(a)p_{-2}(1)-p_{-2}(a+1) =2p_{-2}(a)-p_{-2}(a+1) &=2\sum_{k=0}^a p(k)p(a-k)-\sum_{k=0}^{a+1}p(k)p(a+1-k) &=\left(\sum_{k=0}^a p(k)\big(2p(a-k)-p(a+1-k)\big)\right)-p(a+1) &\ge p(0)+p(1)+\cdots+p(a-2)+p(a)-p(a+1) &> p(a-3)+p(a-2)+p(a)-p(a+1) &\ge p(a-1)+p(a)-p(a+1) &\ge 0.
\left.\big(\int_{B_2}{\hat\rho}(\cdot,t)\, d\|V_t\|-\int_J{\hat\rho}(\cdot,t)\, d{\mathcal H}^1\big)\right|_{t=t_1}^{t_0}\leq \int_{t_1}^{t_0}\int_{B_2}-\frac{\hat\rho}{2}\big|h-\frac{(\nabla{\hat\rho})^{\perp}}{\hat\rho}\big|^2+{\hat\rho}\frac{|u|^2}{2}\, d\|V_t\|dt+c\beta_*^2.
R_{s,z,f}\varphi(v) = \mathbb{E}_{\mathbb{Q}_{s,f}^v}\left[ e^{z \left( \log \frac{\langle f, g v \rangle}{\langle f, v \rangle}- q \right)}\varphi(g \cdot v) \right], v \in \mathcal{S}_\epsilon.
\lim_{i \to \infty}\frac{\Vert H_{\Gamma_{i,\ell}}\Vert_{C^0(\Gamma_{i,\ell})}}{\varepsilon_i |\log \varepsilon_i|}= 0 \ell = 1, 2, \liminf_{i \to \infty}\frac{\inf_{\Sigma}f_i}{\varepsilon_i |\log \varepsilon_i|}> 0.
\sigma_1\leqslant a_{m+2}=\left(a_{m+1}^2-\left|\operatorname{det}\left(a_{m+1}^2-A^{H}A\right)\right|\left(\frac{n-1}{(n+1) a_{m+1}^2-\|A\|_{F}^{2}}\right)^{n -1}\right)^{1/2}\leqslant a_{m+1}
|g(p^{\nu})| < \frac{\max \{2STp+T-pS^2(2\nu-1), pS^2(2\nu+1)-2STp\}}{p^{\nu}T(Tp+2\nu+1)}& \leq \frac1{p^{\nu}}\max \left\{\frac{STp+T}{T(Tp+1)},\frac{pS^2(2\nu-1)}{T^2p}\right\}& \leq \frac1{p^{\nu}}\max \{1,2\nu-1 \}= \frac{2\nu-1}{p^{\nu}},
\partial_{\chi_{( A )}}\Gamma_{Y_{1}}( 0 ) \; \; \Gamma_{\varphi_{1}\varphi_{1}}( 0 ) + \partial_{\chi_{( A )}}\Gamma_{Y_{1}\varphi_{1}}( 0 ) \; \; \Gamma_{\varphi_{1}}( 0 ) = - \partial_{\xi_{( A )}}\Gamma_{\varphi_{1}}( 0 )
\bar\kappa^{\pm}_{a-i}=2\kappa^{\pm}_{a}-\kappa^{\pm}_{a+i}, i\geq 2, \mathcal{K}^{\pm}= \big\{\kappa^{\pm}_{i}\big\}_{i\geq a}\bigcup\big\{\bar\kappa^{\pm}_{a-i}\big\}_{i\geq 1}.
S(P_{\mu})=\big({-}P_{\mu}-a_{\mu}p^{\rm L}_{\alpha}P^{\alpha}\big)Z, S(M_{\mu\nu})=-M_{\mu\nu}-a_{\mu}\big(p^{\rm L}\big)^\alpha M_{\alpha\nu}+a_{\nu}\big(p^{\rm L}\big)^\alpha M_{\alpha\mu}, \epsilon(P_{\mu})=\epsilon(M_{\mu\nu})=0,
\cos(\theta_k) = \frac{f(x_k) - f^*}{s_k \|g_k\|}= \tanh(\eta_k \|g_k\|) / \tanh(s_k), \sin(\theta_k) = \sinh(s_{k+1}) / \sinh(s_k), \cosh(s_{k+1}) = \cosh(s_k) \sqrt{1 - \frac{1}{\zeta_{s_k}^2}\cdot \frac{(f(x_k) - f^*)^2}{\|g_k\|^2}},
E_P(X)=E_Q\bigl\{f\,X\bigr\}=\frac{E_Q(X^+)-t\,E_Q(X^-)}{Q(X\geq 0)+t\,Q(X< 0)}=\frac{E_Q(X)-k\,E_Q(X^-)}{Q(X\geq 0)+t\,Q(X< 0)}\leq 0.
\Delta^{{\rm cl}}=\int d^3x\left( [A_\mu^{*},A^\mu ]-\left[ c^{*},c\right]+\left( \overline{\Psi}^{*}T^a\,\Psi +\overline{\Psi}\,T^a\,\Psi^{*}\right) \tau_a\right) .
\dfrac{x^{2}}{y}-\dfrac{y}{x^{2}}=\dfrac{4q}{K}, xy^{2}-\dfrac{q^{2}}{xy^{2}}=K, \dfrac{y^{3}}{x}+\dfrac{q^{2}x}{y^{3}}=K-2q+\dfrac{4q^{2}}{K}.
x_{\pm}= \pm \frac{m \, ( 2 \, \sqrt{1 + \sqrt{9 - \frac{8 \, b \, q^{2}}{m^{2}}}}\mp ( 1 + \sqrt{9 - \frac{8 \, b \, q^{2}}{m^{2}}}) )}{4 \, q^{2}\, A}
\alpha(x) = (\alpha(u) d' u) x (u d \alpha(u))= (\alpha(u) d' u) v x v (u d \alpha(u))= \alpha(u) d' v x v d \alpha(u) &= \alpha(v) ( \alpha(u) d' v x v d \alpha(u) ) \alpha(v)= (\alpha(v) d' v) x (v d \alpha(v) ) = b' x a'.
F(p,s) \leq F(p, 2)^{\frac{8-3s}{2}}F(p, 8/3)^{\frac{3s-6}{2}}&\leq \Big(G(p,2)\Big)^{\frac{8-3s}{2}}\Big(e^{-p/6}G(p,2)\Big)^{\frac{3s-6}{2}}&= e^{-p\frac{s-2}{4}}2^{p-1}\Gamma(p/2).
\begin{aligned}\left|e^{-ic_-(x)}-e^{-i\tilde{c}_-(x)}\right| =\left|e^{\frac{1}{2i}\int_{-\infty}^x\left(|u_y(y)|^2-|\tilde{u}_y(y)|^2\right)dy}-1\right|\leq 2\delta c_1 \|u_y-\tilde{u}_y\|_{L^2}.\end{aligned}
\bar K^{(n,\ell)}({x,y}):=\frac{1}{4(n+1)^2}\sum_{j,j'=-n-1}^n \Phi\left(\frac{j}{n+1}, \frac{j'}{n+1}\right) \cos\left(\frac{\pi j(y-x)}{n+1}\right) \cos\left(\frac{\pi j'(y-x-\ell)}{n+1}\right).
\left( F^{\mathcal{D}_\mathbf{a}}\circ G^{\mathcal{D}_\mathbf{a}}\right) (\boldsymbol\beta) = F^{\mathcal{D}_\mathbf{a}}(\boldsymbol\beta) \star G^{\mathcal{D}_\mathbf{a}}(\boldsymbol\beta) .
B^{2}\begin{pmatrix}\rho 0 \end{pmatrix}=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\widehat{m}_{j}\rho\begin{pmatrix}1-\cos{(\gamma_{j}-\gamma_{j})}\sin{(\gamma_{j}-\gamma_{i})}\end{pmatrix}h\left(2\rho\sin\left(\frac{1}{2}|\gamma_{j}-\gamma_{i}|\right)\right).
\left|\times_{j=1}^{d}{\widetilde K}_j\right|\ll_d \prod_{j=1}^{d}(c_7\,r_j)^{21r_j^2/2}\,m_j\le\prod_{j=1}^{d}(c_7\,r_j)^{21r_j^2/2}\, w_j ,
Q(s)=\sum_n\frac{1}{2\Gamma(s)}\sqrt{\frac{b}{a}}\int_0^{\infty}dt\;e^{-t\left(c-\frac{b^2}{8a}\right)}\;t^{s-1}\;K_{\frac{1}{4}}\left(\frac{b^2t}{8a}\right)
S_{3}^{\psi^{2}}\propto T_{3}{\frac{( 2 \pi \alpha^{\prime})^{2}}{9 6}}\int d^{4}\xi \sqrt{- G}e^{- \Phi}\Big ( \partial^{c}\lambda \partial_{c}\lambda \Big ) \Big ( \psi^{a b}\psi_{a b}\Big ) + \cdots ,
\bar c(|p|) =\begin{cases}\int_0^1 g(\chi_p(z))dz=\frac{\int_0^1 g(s)ds}{\int_0^1 (\chi_p'(z))^2 dz}p\neq 0 \left(\int_0^1 \frac{1}{g(s)}ds\right)^{-1}p=0.\end{cases}
\sum_{j = 0}^k{( - 1)^{rj}\binom kj\left({\frac{{F_r}}{{F_n}}}\right)^j G_{m - rk + (n + r)j}}= ( - 1)^{rk}\left({\frac{{F_{n + r}}}{{F_n}}}\right)^k G_m ,n\ne 0\,,
\begin{aligned}c_0f\left(\xi(t_1)\right) \leq \left(T_{x, \xi}-t_1\right)p(1-c_0) f(\xi(t_1)) \frac{c_0}{p(1-c_0)}\leq T_{x, \xi}-t_1 \frac{c_0}{p(1-c_0)}\leq T_{x, \xi}.\end{aligned}
\begin{array}{l}\bullet~ (x-1)^{\epsilon_1}h_1(x)=(x-1)(1+(x-1)), \bullet~ (x-1)^{\epsilon_4}h_2(x)=-1+(x-1), \bullet~ (x-1)^{\min\{k_1,r-r_1+k_4\}+\hat e_3}h_3(x)=0, \bullet~ (\epsilon_2,\epsilon_3)=(r-k_1+k_3,k_2)=(2,0), \bullet~ (\epsilon_5,\epsilon_6)=(r_1-k_1+k_3,k_5)=(1,1), \end{array}
h^2_{[-u_1,u_1]\oplus_2\cdots\oplus_2[-u_m,u_m]}(x) =\sum_{i=1}^m \langle u_i, x \rangle^2=\sum_{i=1}^m \langle U e_i, x\rangle^2 = \sum_{i=1}^m \langle e_i, U^* x\rangle^2=|U^*x|^2 = h^2_{U B_2^m}(x) &=\langle U^*x, U^*x\rangle=\langle\sqrt{UU^*}x, \sqrt{UU^*}x\rangle=|\sqrt{UU^*}x|^2 =h^2_{\sqrt{UU^*}B_2^n}(x).
\{k \tau \}+ \{F_n \tau \}= \{(F_{a_1}+ \ldots + F_{a_s}) \tau \}+ \{F_n \tau \}&\leq \tau^{a_1}+ \tau^{a_2}+ \ldots + \tau^{a_s}+ \tau^n &< \tau + \tau^3 + \tau^5 + \ldots &= \frac{\tau}{1 - \tau^2}&= 1.
Z(\rho_V, \rho_R) = \sum\limits_{(k,l) \in \chi}\rho_V^k \frac{\rho_R^l}{l!}\frac{\partial Z}{\partial \rho_R}= (1-\pi_S)Z Z \left(\frac{1}{\rho_V}, \rho_R \right) \rho_V^K = Z(\rho_V,\rho_R\rho_V) \rho_V>0.
f_j\left(-\frac1\tau\right) = (-i\tau)^{\kappa_1}\sum_{1\le k\le N}\chi_{j,k}\: f_k(\tau),g_\ell\left(-\frac1\tau\right)= (-i\tau)^{\kappa_2}\sum_{1\le m \le M}\psi_{\ell,m}\: g_m(\tau).
(e_0 \otimes f_1) \cup (e_0 \otimes f_2) = 0, (e_0 \otimes f_1) \cup (e_1 \otimes f_1) = 2d, (e_1 \otimes f_0) \cup (e_0 \otimes f_2) = 1, (e_1 \otimes f_0) \cup (e_1 \otimes f_1) = 0.
\frac{1}{\sqrt{1-([\eta]_{m}-2^{-j})^2}}=\sqrt{\frac{\eta^2+2m+2}{2m+2}}\frac{1}{\sqrt{1+2^{-j}\frac{1}{m+1}\eta\sqrt{\eta^2+2m+2}-2^{-2j}\frac{\eta^2+2m+2}{2m+2}}}