harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a^{ijkl}=\frac{\partial x^i}{\partial \overline{x}^a}(p)\frac{\partial x^j}{\partial \overline{x}^b}(p)\frac{\partial x^k}{\partial \overline{x}^c}(p)\frac{\partial x^l}{\partial \overline{x}^d}(p)a^{abcd}=A^i_aA^j_bA^k_cA^l_da^{abcd}.
[ P_i,U_j]=\frac{\hbar}{\sqrt{-1}}\frac{1}{2}\delta_{ij},[ U_i,U_j]=\frac{\hbar}{\sqrt{-1}}\frac{-1}{2}d_i \omega_{ij},[ P_i,P_j]=\frac{\hbar}{\sqrt{-1}}\frac{1}{2d_j}b_{ij}.
\alpha_0 := 1-\frac{\sqrt{\alpha}}{1-\alpha^2}, \alpha_1 := 1-\frac{4\alpha}{(1-\alpha)(1+6\alpha+\alpha^2)}, \tilde{\alpha}_0 := 1+\frac{\sqrt{\alpha}}{1-\alpha^2},\tilde{\alpha}_1 :=1+\frac{4\alpha}{(1-\alpha)(1+6\alpha+\alpha^2)}.
\partial_{S}\omega ( S ) = 2 i \pi \left( E_{2}( S ) - 2 E_{2}( 2 S ) \right) , \; E_{2}( T ) = 1 - 2 4 q - 7 2 q^{2}- 9 6 q^{3}- 1 6 8 q^{4}+ \dots .
\widehat{\Lambda}_k(\lambda) := \left[ \begin{array}{ccccc}-1 -\lambda -\lambda^{2}\cdots -\lambda^{k-1}& -1 -\lambda \ddots \vdots & -1 \ddots -\lambda^{2}& \ddots -\lambda & -1 0 \cdots \cdots \cdots 0\end{array}\right]
\omega_2 = \left[\begin{matrix}\cos(\varphi_1) \sin(\varphi_1) \cos(\varphi_2) \vdots \sin(\varphi_1) \sin(\varphi_2) \ldots \sin(\varphi_{n-2}) \cos(\varphi_{n-1}) \sin(\varphi_1) \sin(\varphi_2) \ldots\sin(\varphi_{n-2}) \sin(\varphi_{n-1})\end{matrix}\right] ,
\frac{d \alpha}{d \phi}= \ell \, \tanh ( \eta / \ell ) \, \sin \chi \, \sin \beta \, \sin \theta \, \, \, , \, \, \, \, \, \, \frac{d y}{d \eta}= \left( 1 + \frac{y^{2}}{\ell^{2}}\right)^{1 / 2},
F \left({\cal A}, G , p , t_{n , l}^{( A )}\right) = \sum_{n = - \infty}^{+ \infty}F^{( \mathrm{c o v )}}\left( G , p , r_{n , l}^{( A )}\right) \prod_{A = 1}^{p}\prod_{l = 1}^{\infty}\left[ t_{n , l}^{( A )}\right]^{r_{n , l}^{( A )}}\mathrm{e}^{- | r |_{n}{\cal A}v ( n )}.
\overline{f}=\left(\begin{array}{c}f_{0}f_{1}\vdots f_{n-1}\end{array}\right)\in \mathbb{R}^{n}\stackrel{t}{\longrightarrow}f(x)=f_{0}+f_{1}x+\cdots+f_{n-1}x^{n-1}\in \mathbb{R}[x]/ \langle\phi(x)\rangle
\frac{X_j(t)}{N}\leq \frac{X_j(t)}{X_0(t)}\leq \frac{\alpha x_j(t^*) e^{\int_{t^*}^t G_j(v) \: dv}}{x_0(t^*) e^{\int_{t^*}^t G_0(v) \: dv}}= \frac{\alpha \mu^j (e^{st^*}- 1)^j}{s^j j!}e^{sj(t - t^*)}\leq \frac{\alpha \mu^j e^{sjt}}{s^j j!}.
\sigma_{-q}(\xi)=\rho_q(\xi)^{-1}, \sigma_{-q-j}(\xi)=-\!\!\!\sum_{\substack{k+l+|\alpha|=j l<j}}\!\! \frac{1}{\alpha !}\rho_q(\xi)^{-1}\partial_\xi^\alpha\rho_{q-k}(\xi)\delta^\alpha\sigma_{-q-l}(\xi), j\geq 1 .
\int_{\Omega}\sum_{i}\Delta \overline{u}_{i}(x)\, dx= \int_{\partial \Omega}\sum_{i}\frac{\partial \overline{u}_{i}}{\partial n}\, ds \le \int_{\partial \Omega}\sum_{i}\frac{\partial \underline{u}_{i}}{\partial n}\, ds= \int_{\Omega}\sum_{i}\Delta \underline{u}_{i}(x)\, dx.
S = T r [ \frac{m_{1}^{2}}{2}( \phi^{1})^{2}+ \frac{m_{2}^{2}}{2}( \phi^{2})^{2}+ \frac{\lambda_{1}}{4 N}( \phi^{1})^{4}+ \frac{\lambda_{2}}{4 N}( \phi^{2})^{4}+ g \phi^{1}\phi^{2}]
\begin{aligned}|\mathcal{O}_{2n-2}| = |\{\pi \in \mathcal{RAI}_{2n-2}(1234) \mid \pi_3 = 2n-2 \}| = |\mathcal{P}_{2n-4}| + |\mathcal{Q}_{2n-4}| &= M_{n-1}- M_{n-2}- M_{n-3}+ M_{n-3}= M_{n-1}- M_{n-2}.\end{aligned}
( C \Gamma^{\mu_{6}})_{\alpha{'}\beta{'}}( C \Gamma_{\mu_{1}\dots \mu_{6}})_{\delta{'}\epsilon{'}}= ( C \Gamma^{\mu_{6}})_{\alpha{'}\beta{'}}( C \Gamma_{\mu_{1}\dots \mu_{5}}\Gamma_{1 1})_{\delta{'}\lambda}{( \Gamma_{1 1}\Gamma_{\mu_{6}})^{\lambda}}_{\epsilon{'}}
E(u_{0,n}) = \frac{1}{2}\|u_{0,n}\|^2_{\dot{H}^1_c}- \frac{d}{2d+4}\|u_{0,n}\|^{\frac{4}{d}+2}_{L^{\frac{4}{d}+2}}&= \frac{1}{2}\mu_n^2 \lambda^2_n \|Q\|^2_{\dot{H}^1_c}- \frac{d}{2d+4}\mu_n^{\frac{4}{d}+2}\lambda_n^2 \|Q\|^{\frac{4}{d}+2}_{L^{\frac{4}{d}+2}}&= \frac{1}{2}\left(1-\mu_n^{\frac{4}{d}}\right) \mu_n^2 \lambda_n^2 \|Q\|^2_{\dot{H}^1_c}.
{\frac{1}{4}}{\frac{\partial v ( x )}{\partial x}}\Bigg \vert_{x = x^{*}}={\frac{1}{G}}-{\mathrm{e}}^{- x^{*}}+ x^{*}E_{1}( x^{*}) -{\frac{3 \alpha ( \Lambda )}{8 ( 3 \pi - \alpha ( \Lambda ) \ln x^{*}) x^{*}}}= 0
A_1 =\frac{1}{r}\left\{(\alpha-\beta)(\tfrac{1}{4}-\tfrac{1}{2}\eta')-\eta\right\}, B_1 =\frac{\alpha-\beta}{2r}(\tfrac{1}{2}+\eta'), C_1 =\frac{\eta(\alpha-\beta)}{r^2}(\alpha-\beta-2\eta), A_2=B_2 =-\frac{1}{2\eta}(\tfrac{1}{2}+\eta'), C_2 =\frac{\alpha-\beta}{r}(\eta'-\tfrac{1}{2})
c_p(t_1, \ldots, t_p)=\sum_{m=1}^p \frac{(-1)^{m}}{m}\*\sum_{\substack{(p_1, \ldots, p_m): p_1+\ldots +p_m=p, \p_1, \ldots p_m\geq 1}}\frac{1}{p_1!\cdots p_m!}\* \sum_{\sigma \in S_p}\* j(p_1, \ldots, p_m; t_{\sigma(1)},\ldots, t_{\sigma(p)}),
\mathrm{\Sigma}_2= \left\lbrace v \in \mathbb{R}^2:~I_1+I_2 \geq R \right\rbrace &I_1=\log_2\left(1+\mathrm{SIR}\left(v,0\right)\right), &I_2=\log_2\left(1+\mathrm{SIR}\left(v,\eta_1\right)\right), &\eta_1=\left(\tilde{d}_1,0\right)
r_{s,p}=\frac{\overline{\beta'(b_1)}(\overline{a}_1-\overline{b}_s)r_{1,s}+\overline{\beta'(b_1)}(\overline{a}_p-\overline{b}_1)r_{1,p}}{\overline{\beta'(b_s)}(\overline{a}_p-\overline{b}_s)}
\partial_t\big[u(x',t)\big] =\partial_t\big[\big(P^L_t\ast f\big)(x')\big]=-\sum_{j=1}^{n-1}\partial_{x_j}\Big[\big(R^{(j)}_t\ast f\big)(x')\Big] begin{align*}0pt] =-\sum_{j=1}^{n-1}\Big[R^{(j)}_t\ast\big(\partial_jf\big)\Big](x')\,\,\mbox{for every}\,\,(x',t)\in{\mathbb{R}}^n_{+},
\begin{array}{l}M=a^4+b^4-2c^4, N=12a^3+\frac{76}{7}b^3-\frac{160}{7}c^3, R=54a^2+\frac{2166}{49}b^2-\frac{4800}{49}c^2, S=108a+\frac{27436}{343}b-\frac{64000}{343}c.\end{array}
\left(a_{h-2},a_{h-1},a_h\right)\in\left\{\begin{aligned}\left(n-5,n-4,n-3\right), \left(n-5,n-4,n-2\right), \left(n-5,n-3,n-1\right), \left(n-5,n-2,n-1\right), \left(n-4,n-3,n\right), \left(n-4,n-2,n\right), \left(n-3,n-1,n\right), \left(n-2,n-1,n\right) \end{aligned}\right\}.
\prod_{i<j}F_{min}(\theta_{ij}) \longrightarrow [F_{min}(i\pi)]^3\left(\frac{\sinh^2\theta_{12}}{\sinh^2\theta_{12}+ \sin^2\pi\alpha}\right)\left(\frac{\sinh^2\theta_{13}}{\sinh^2\theta_{13}+ \sin^2\pi\alpha}\right)\left(\frac{\sinh^2\theta_{23}}{\sinh^2\theta_{23}+ \sin^2\pi\alpha}\right)\,\,\,.
\varphi(b)\pi_C(c) =\Phi(b\otimes 1_C)\Phi(1_B\otimes c) =\Phi\big((b\otimes 1_C)(1_B\otimes c)\big) =\Phi\big((1_B\otimes c)(b\otimes 1_C)\big) =\Phi(1_B\otimes c)\Phi(b\otimes 1_C) &= \pi_C(c)\varphi(b),
\int_A f \left( T_{t_2}(\eta,x)(\omega)\right) \mathbb{P}(d\omega) = \int_A \int_{\Omega}f\left( \left( T_{t_2}^{t_1}(T_{t_1}(\eta,x)(\omega')) \right)(\omega) \right) \mathbb{P}(d\omega) \mathbb{P}(d\omega')
\{f,g\}~=~i_{X_g}^{}i_{X_f}^{}\omega_{\mathcal{H}}^{}\, + \, d \, \Bigl( i_{X_g}^{}f \, - \, i_{X_f}^{}g \, - \, i_{X_g}^{}i_{X_f}^{}\theta_{\mathcal{H}}^{}\Bigr)
\begin{aligned}f(x) =\sum\limits_{j\in\mathbb N}\psi_{j}\ast\psi_{j}\ast f(x) =\sum\limits_{j\in\mathbb N}\sum\limits_{Q\in\Pi_{j+N}}\int_{Q}\psi_{j}(x-u)(\psi_{j}\ast f)(u)du =:T_{N}(f)(x)+R_{n}(f)(x)\end{aligned}
\varphi^{\prime}\colon ( \Lambda \oplus P)^{\ast}\oplus \Lambda^{\infty}\longrightarrow \ker(d_{n-1}) = H_{n-1}(\widetilde{K'}^{[n-1]}) \xrightarrow[]{\h_{n-1}^{-1}\}\pi_{n-1}(\widetilde{K'}^{[n-1]}) x \longmapsto h_{n-1}^{-1}\big( [d_{n}(x)] \big)
\left( \begin{array}{c}{{\phi}}\\{{\psi}}\\ \end{array}\right) ( \theta ) | \frac{1}{2}\rangle = \left( \begin{array}{l l}{{\mathcal{A}_{+}}}&{{\mathcal{B}}}\\{{\tilde{\mathcal{B}}}}&{{\mathcal{A}_{-}}}\\ \end{array}\right) \left( \begin{array}{c}{{\phi}}\\{{\psi}}\\ \end{array}\right) ( - \theta ) | \frac{1}{2}\rangle
\mathbf{x}^{k+1}= \mathbf{W}_1 \mathbf{x}^{k}- \frac{2}{\rho}(2\mathbf{D}+\frac{2}{\rho}\mathbf{P})^{-1}(\nabla f(\mathbf{x}^{k+1})+ \mathbf{E}_{}^T\boldsymbol{\alpha}^{k}),
\langle \vartheta_{p_1},\ldots,\vartheta_{p_r}\rangle^{naive}:= \sum_{\textbf{A}\in NE(\tilde{X})}t^{\pi_*(\textbf{A})}N_\textbf{A}^{naive}(\textbf{p}) \in \mathbb{Z}[NE(X)].
Z_{2}: \qquad ( A_{\mu}, \phi , \bar{\phi}, \lambda_{A \alpha}, \bar{\lambda}^{A \alpha}, G^{\alpha \beta}) \rightarrow ( A_{\mu}, \bar{\phi}, \phi , - \bar{\lambda}_{A}^{\! ~ ~ \alpha}, - \lambda_{\! ~ ~ \alpha}^{A}, G_{\alpha \beta}) .
\overset{\sim}{\lambda}\overset{\sim}{P}(C') = \displaystyle \sum_{C}\overset{\sim}{\lambda}(C)\overset{\sim}{P}(C,C') =\displaystyle \sum_{C}\displaystyle \int_{C}P(x,C')\lambda(dx) =\displaystyle \int_{X}P(x,C')\lambda(dx) =\overset{\sim}{\lambda}(C').
d \pi^* \omega(X_1, \ldots, X_{\ell +1}) = \sum_j (-1)^{j-1}X_j \pi^* \omega (X_1, \ldots, \widehat{X_j}, \ldots, X_{\ell +1}) &+ \sum_{j<k}(-1)^{j+k-1}\pi^* \omega([X_j, X_k], X_1, \ldots, \widehat{X_j}, \ldots, \widehat{X_k}, \ldots, X_{\ell+1}).
Y_{n}(z) = -\frac{2^{n}(n-1)!}{\pi z^{n}}\left(1+\mathcal{O}(\frac{1}{n})\right),n \rightarrow +\infty, Y_{n}^{\prime}(z) = \frac{2^{n}n!}{\pi z^{n+1}}\left(1+\mathcal{O}(\frac{1}{n})\right),n \rightarrow +\infty.
t_{+}^{\ell}( \mu , \nu ; a , x ) = \frac{N ( \mu )}{N ( \nu )}\frac{\Gamma ( \nu - 2 \ell ) \Gamma ( \mu - \nu )}{\Gamma ( \mu - 2 \ell )}\, e^{a ( \mu - \ell )}\, x_{+}^{\nu - \mu}\, , \qquad 2 \ell < \Re ( \nu ) < \Re ( \mu ) \, .
\left|\frac{a_i}{r}-\frac{a_i'}{r'}\right|\geq \frac{|c_i^\bot|}{|rr'|}i=1,2 \quad\max_{i=1,2}\left\{|c_i|\left|\frac{a_i}{r}-\frac{a_i'}{r'}\right|\right\}\geq 1.
e^{\tau(\phi_{+}-\phi_{-})}=\frac{\lim_{x_{n+1}\rightarrow0}\|x_{n+1}^{1-2s}\partial_{n+1}\tilde{w}\|_{L^{2}(C_{s,1/2}')}^{1-\alpha}}{\|x_{n+1}^{\frac{1-2s}{2}}\tilde{w}\|_{L^{2}(C_{s,1/2}^{+})}^{1-\alpha}}\le\frac{1}{c_{0}}
2{\frac{\partial A^{i}}{\partial z}}= - \Omega_{-}F^{i}= \Omega_{-}K^{- 1 i j^{*}}\frac{\partial{\cal W}^{*}}{\partial A^{* j}}, \quad \Omega_{-}\equiv i \frac{\langle - i Z_{1}^{*}- Z_{2}^{*}\rangle}{| \langle - i Z_{1}^{*}- Z_{2}^{*}\rangle |}.
\begin{aligned}\delta_i^{(1)}\ge k_1, \,\, \Delta_i^{(1)}+\Delta_i^{(1,2)}\ge k_2,i\in [\nu], \delta_i^{(1,2)}+\delta_i^{(2)}\ge k_1,\,\, \Delta_i^{(2)}\ge k_2,i\in [N-\nu], \end{aligned}
(|(-\Delta_x)^{1/3}u_{}|^2)^{1/2}_{Q_r (z_0)}\le N \nu^{1+2d}(|(-\Delta_x)^{1/3}u_{}|^2)^{1/2}_{Q_{2 \nu r}(z_0)}&\le N \nu^{1+2d}\delta^{-\theta}\sum_{k=0}^\infty 2^{-2k}F_k (1).\, \,
\begin{array}{l}\delta\theta_\alpha=\epsilon^\alpha, \qquad \delta\bar\theta_\alpha=\bar\epsilon^\alpha, \delta x^{\bar\mu}=-i\bar\epsilon_\alpha \tilde\Gamma^{\bar\mu\alpha\beta}\bar\theta_\beta-i\epsilon^\alpha\Gamma^{\bar\mu}_{\alpha\beta}\theta^\beta, \qquad \delta x^{11}=0,\end{array}
\frac{r!}{N^2}r u^{-2}[(\lambda_{\bf 84}\phi + \mu_{\bf 84}) P_{Y_{(2r,2)}}+ (\lambda_{\bf 105}\phi + \mu_{\bf 105}) P_{Y_{(2r+2)}}+(\lambda_{\bf 175}\phi + \mu_{\bf 175}) P_{Y_{(2r+1,1)}}]
\nu_{\sigma, >n}=\delta_{b_{\sigma_{n+1}}^{-1}D_{\sigma_{n+1}}}\ast \delta_{\left(b_{\sigma_{n+1}}b_{\sigma_{n+2}}\right)^{-1}D_{\sigma_{n+2}}}\ast \delta_{\left(b_{\sigma_{n+1}}b_{\sigma_{n+2}}b_{\sigma_{n+3}}\right)^{-1}D_{\sigma_{n+3}}}\ast\cdots .
J^k(J^T)^l = \left( \sum_{j=1}^{n-k}e_{j+k}e_j^T\right)\left(\sum_{j=1}^{n-l}e_je_{j+l}^T\right), &= \sum_{j=1}^{n-\max \ \{k,l\}}(e_{j+k}e_j^T)(e_je_{j+l}^T), &= \sum_{j=1}^{n-\max \ \{k,l\}}e_{j+k}e_{j+l}^T.
\delta^{\frac{n-2}{2}}PW_{\delta}(\delta y-e_n) = -U_{1,0}(y)+\delta \varphi_0(y)+O(\delta^{2}), \delta^{\frac{n-2}{2}}PW_{\delta}(\delta y+e_n) = U_{1,0}(x)-\delta \varphi_0(-x)+O(\delta^{2}),
\, \, \, \, \partial^{\mu}A_{\mu}\, = \, 0 \, \, \, \, \, ; \, \, \, F_{\mu \nu}\, + \,{\tilde{F}}_{\mu \nu}\, = \, 0 \, \, \, \, \, ; \, \, \, \partial^{\mu}\Psi_{\mu}\, = \, 0
\left\lvert e^{-( s- s_0) A}\sum_{n=0}^{l-1}q_n\phi_n(y)\right\rvert =\left\lvert\sum_{n=0}^{l-1}e^{-( s- s_{0})\lambda_{n}}q_{n}\phi_{n}(y)\right\rvert \lesssim \sum_{n=0}^{l-1}|q_{n}|e^{-( s- s_{0})\lambda_{n}}y^{2\lambda_{n}}\lesssim e^{- s\lambda_{l}}y^{2\lambda_{l}}\left(e^{s_{0}\lambda_{l}}\sum_{n=0}^{l-1}|q_{n}|\right).
Q(x)=\frac{\left(-1-e^{\frac{1}{5}y}+e^{\frac{2}{5}y}\right)^5}{\left(e^{\frac{1}{5}y}-e^{\frac{2}{5}y}+2 e^{\frac{3}{5}y}-3 e^{\frac{4}{5}y}+5 e^{y}+3 e^{\frac{6}{5}y}+2 e^{\frac{7}{5}y}+e^{\frac{8}{5}y}+e^{\frac{9}{5}y}\right)}
\langle J^{\mu AB}\rangle_{bare}=\lim_{x \rightarrow y}Tr\left\{\gamma^\mu(x){\frac{i}{2}\sigma^{AB}}P_L\left[\frac{1}{i{{\cal D}\kern-0.15em\raise0.17ex\llap{/}\kern0.15em\relax}}\frac{1}{2}\left(1-\sigma^3\right)\right]\delta(x-y)\right\}
{\nu}_{\alpha}^{( 1 )}\frac{\partial V_{( n = 1 )}^{( 1 )}( a_{i}, p_{i}, \theta )}{\partial{\xi}_{\alpha}^{( 1 )}}- \frac{\partial{\cal G}^{( n = 2 )}( a_{i}, p_{i}, \theta )}{\partial \theta}= 0 .
\begin{array}{r c l}{{\delta_{\sigma}L_{m}{}^{i}}}&{{=}}&{{- \sigma L_{n}{}^{i}\mathsf{m}^{n}{}_{m}\, ,}}\\{{}}&{{}}&{{}}\\{{\delta_{\sigma}A^{m}{}_{\mu}}}&{{=}}&{{\sigma \mathsf{m}^{m}{}_{n}A^{n}{}_{\mu}\, ,}}\\{{}}&{{}}&{{}}\\{{\delta_{\sigma}B_{\mu \nu \, m}}}&{{=}}&{{- \sigma B_{m \, \mu \nu}\mathsf{m}^{n}{}_{m}+ 2 \eta_{m n}\partial_{[ \mu}\sigma A^{n}{}_{\nu ]}\, ,}}\\ \end{array}
\mathbb{P}_{p_n}(T \leq t) \leq \exp\left(\frac{2(\log n)^{2/3}}{\lambda^2}\log 2 - \frac{(r-2)}{\tau}\log n\right) &+ \exp\left((\log n)^{2/3}\log 2 -(\log n)^{2/3}\right) + o(1) &= o(1).
\theta_{t+1}= \theta_t - \beta_t \left( \phi(s_t, a_t)^T \omega_t \right) \frac{d}{d \theta}\log \pi_{\theta}(a_t|s_t) \omega_{t+1}= P_{\Omega}\left[ \omega_t + \alpha_t \left(c_t' + \gamma \phi(s_t'', a_t'')^T \omega_t - \phi(s_t',a_t')^T \omega_t \right) \phi(s_t', a_t') \right]
\langle z_+,j | n_{12},j \rangle = Q_{n_{12}}^{\mu_{12}}(z) \iint d\lambda_{x_\pm}d\lambda_{y_\pm}\langle z_+,j | x_\pm y_\pm \rangle \displaystyle\sum\limits_{n=0}^{j}\langle x_\pm y_\pm |n,j-n\rangle \langle n, j-n | 0,j \rangle,
{}_2 F_1\bigg(\frac{d-2}{2},\frac{d-1}{2};\frac{d}{2};\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg) = \sin^{-(d-3)}\bigg(\frac{\zeta}{2}\bigg) \,{}_2 F_1\bigg(1,\frac{1}{2};\frac{d}{2};\cos^2\bigg(\frac{\zeta}{2}\bigg)\bigg).
A_{\ell_{+},k}^p(u) = \big\{Z \in (x_{k+1}^{\ell,p,j_{\ell}-}(x_k^i,u), x_{k+1}^{\ell,p,j_{\ell}+}(x_k^i,u)) \big\}, Z \sim{\cal N}( 0; 1).
\mathbb{E}_{t_{n}}^{GH,Q,X^{n}}\left[ Y^{n+1}\Delta B_{n+1}^{\top}\right]=\sum_{i_{1},\ldots,i_{d}=1}^{L}\omega_{i_{1}}\cdots \omega_{i_{d}}\psi_{n}(\sqrt{\Delta t}P_{Q},\Delta tQ)\sqrt{\Delta t}P_{Q}^{\top}.
\frac{\frac{L_{1}}{2l}}{\frac{L_{2}}{2l}}\approx4{(\frac{L_{2}}{L_{1}})}^{2}\Rightarrow\frac{L_{1}}{L_{2}}\approx4{(\frac{L_{2}}{L_{1}})}^{2}\Rightarrow\frac{L_{1}}{L_{2}}\approx\sqrt[3]{4}
\delta T_{N}^{M}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{0}}&{{- \delta \rho}}&{{- ( \rho + p ) e^{\alpha_{0}}v_{, i}}}\\{{0}}&{{( \rho + p ) e^{- \alpha_{0}}v_{, i}}}&{{\delta p \: \delta_{i j}}}\\ \end{array}\right) \: \delta ( y ) .
\mathbf{v}_{i, j}:= \left[\begin{array}{cccc}\binom{f}{j}\lambda_i^{f-j}\binom{f+1}{j}\lambda_i^{(f+1)-j}\dots \binom{f+(m+1)n-1}{j}\lambda_i^{(f+(m+1)n-1) - j}\end{array}\right]^T
C(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm})Y=(Y,\nabla_yA_0(\hat{{\mathbf U}}^{\pm}))\partial_t\hat{{\mathbf U}}^{\pm}+(Y,\nabla_y\tilde{A}_1(\hat{{\mathbf U}}^{\pm},\hat{\Psi}^{\pm}))\partial_1\hat{{\mathbf U}}^{\pm}+(Y,\nabla_yA_2(\hat{{\mathbf U}}^{\pm}))\partial_2\hat{{\mathbf U}}^{\pm}\,,
H(G,\tau;A,\nu)=\sum_{u\in V(G)}\sum_{i\in [q]}\tau_u(i)\ln \tau_u(i)+\sum_{(u,v)\in E(G)}\sum_{(i,j)\in{q}\times [q]}\tau_{u,v}(i,j)\ln \frac{\tau_{u,v}(i,j)}{\tau_u(i)\tau_v(j)}.
\mathcal{M}^{\mathcal{P}}_{\Xi}:= \lbrace \mathcal{P}^{h}(\mu)\, |\, \mu \in \Xi \rbrace \subset \mathcal{M}^{\mathcal{P}}, \mathcal{M}^{\mathcal{A}}_{\Xi}:= \lbrace \mathcal{A}^{h}(\mu) \,|\, \mu \in \Xi \rbrace \subset \mathcal{M}^{\mathcal{A}}, \enspace \Xi \subset \mathcal{D},
\begin{array}{c c l}{{J_{n}^{\pm}}}&{{\rightarrow}}&{{J_{n \pm w}^{\pm}}}\\{{J_{n}^{3}}}&{{\rightarrow}}&{{J_{n}^{3}-{\frac{k}{2}}w \delta_{n}\qquad \qquad w \in{\bf Z}}}\\{{L_{n}}}&{{\rightarrow}}&{{L_{n}- w J_{n}^{3}-{\frac{k}{4}}w^{2}\delta_{n}.}}\\ \end{array}
I'=\langle E_{1,2}^{2},E_{1,3}^{2},E_{1,4}^{2},E_{2,3}^{2},E_{2,4}^{2},E_{3,4}^{2}, E_{1,2}E_{1,3}, E_{1,2}E_{1,4},E_{1,3}E_{1,4}, E_{2,3}E_{2,4}, E_{1,4}E_{2,3}, E_{1,3}E_{2,3},E_{1,4}E_{2,4},E_{1,4}E_{3,4},E_{2,4}E_{3,4}\rangle.
\int_{\mathbb{R}^{2N}}\frac{|w(x) - w(y)|^{p-2}\, (w(x) - w(y))\, (\varphi_h(x)-\varphi_h(y))}{|x - y|^{N+s\,p}}\, dx\, dy-\int_\Omega\,f(x)\cdot \Phi_{k}'(w)\varphi_h\geq 0.
\lambda_f=\inf_{u\in H^1_0(M)}\frac{\int_M|\nabla u|^2e^{-f}dM}{\int_M u^2e^{-f}dM}=\inf_{u\in C^{\infty}_c(M)}\frac{\int_M -u\Delta_fu\, e^{-f}dM}{\int_M u^2e^{-f}dM}.
|\varphi_{2}(\zeta_{1},\zeta_{2},\psi,\mu)|^{2}\leq ( l_{2}(\zeta_{1})+\kappa_{2}(\zeta_{2})+\omega_{2}(\psi))^{2}\leq 3|l_{2}(\zeta_{1})|^{2}+3|\kappa_{2}(\zeta_{2})|^{2}+3|\omega_{2}(\psi)|^{2}.
S_n(z_1, z_2) = (2n^2)^{2m -s -1}\sum_{j = 0}^{n-1}\frac{D_j^{(n)}(1- \frac{z_1}{2n^2}) D_j^{(n)}(1- \frac{z_2}{2n^2})}{\frac{h_j^{(s)}k_{j+2m}^{(s)}}{k_j^{(s)}}\delta_j^{(n)}\delta_{j+1}^{(n)}},
\times \; \prod_{i = 1}^{n_{b}^{\prime}}\overline{{{\psi}}}^{( b )}({x^{\prime}}_{i}^{( b )}) P_{+}\psi^{( b )}({x^{\prime}}_{i}^{( b )}) \prod_{i = 1}^{m_{b}^{\prime}}\overline{{{\psi}}}^{( b )}({y^{\prime}}_{i}^{( b )}) P_{-}\psi^{( b )}({y^{\prime}}_{i}^{( b )}) \Big \rangle_{0}
\lim \limits_{r\rightarrow0}\sup \limits_{x\in{\mathbb{R}^{n}}}\varphi_{1}(x,r)^{-1}\, \left \Vert f\right \Vert_{L_{p}\left( B\left( x,r\right)\right)}=0\lim \limits_{r\rightarrow0}\sup \limits_{x\in{\mathbb{R}^{n}}}\varphi_{2}(x,r)^{-1}\, \left \Vert T_{\Omega,b,\alpha}f\right \Vert_{L_{q}\left( B\left( x,r\right) \right)}=0.
= \int_{0}^{\frac 1 4}\d s \ \frac{1}{( 1 - 4 s )^{\frac 1 2}s^{\frac 1 2}}\ \exp \left( - t k^{2}s \right) = \frac{\pi}{2}\exp \left( - t \frac{k^{2}}{8}\right) I_{0}( t \frac{k^{2}}{8}) ,
\|f\|_{M^{p,q}}\coloneqq \begin{cases}\|f\|_{M^{p,q}_{}}(n ) \|f\|_{M^{p,q}_{}}(n ), \end{cases}\|f\|_{W^{p,q}}\coloneqq \begin{cases}\|f\|_{W^{p,q}_{}}(n ) \|f\|_{W^{p,q}_{}}(n ). \end{cases}
\hat{A}_{( d ) \, \mu_{1}\cdots \mu_{d}}= c a^{d}\frac{\epsilon_{\mu_{1}\cdots \mu_{d}}}{\sqrt{| g |}}\, , \Rightarrow \hat{F}_{( \hat{d}) \, \underline{{{y}}}\mu_{1}\cdots \mu_{d}}= c d a^{d}\log^{\prime}{a}\, \frac{\epsilon_{\mu_{1}\cdots \mu_{d}}}{\sqrt{| g |}}\, ,
P_{mn-1,1}(k)w_{(m-r)n}u_{(m-r)n}\left(\underline{b_r}\right)T_{m,n}(k) = P_{mn-1,1}(k)w_{(m-\ell)n}u_{(m-\ell)n}\left(\underline{b_\ell}\right)T_{m,n}(k).
\delta S_{a b}^{c d}\left( \vartheta \right) = - i g \frac{F_{\bar{c}\bar{d}a b}^{\Psi}\left( i \pi , \vartheta + i \pi , 0 , \vartheta \right)}{m_{a}m_{b}\sinh \vartheta}\quad , \quad \vartheta = \vartheta_{a}- \vartheta_{b}\: .
\|\mathcal{M}_\psi\left[\left({\Psi_n^{(s)}\ast\Psi_n^{(s)}}\right);\alpha+s/2\right]\|=\|\mathcal{M}_\psi\left[\left({\Psi_n^{(s)}\star\Psi_n^{(s)}}\right);\alpha+s/2\right]\|=\|\psi_n^{(s)}(\alpha)^2\|\;.
J_1 =x_{n}x_{n-1}[I(G_3)^{[k]}+x_{n-2}I(G_3)^{[k-1]}], J_2 =x_{n}x_{n-1}\sum_{j=1}^tx_{i_j}I(G_2)^{[k-1]}=x_{n}x_{n-1}(x_{i_1},\dots,x_{i_t})I(G_2)^{[k-1]}.
A:= (n-2)\begin{pmatrix}j^{-1}p_{n,x,u}^{(j)}(1-p_{n,x,u}^{(j)}) j^{-1/2}l^{-1/2}(p_\cap-p_{n,x,u}^{(j)}p_{n,y,v}^{(l)}) j^{-1/2}l^{-1/2}(p_\cap-p_{n,x,u}^{(j)}p_{n,y,v}^{(l)}) l^{-1}p_{n,y,v}^{(l)}(1-p_{n,y,v}^{(l)}) \end{pmatrix}.
\big\|\big(H-\xi_{N^{\kappa-1}}*H\big)(Nu)\big\|_{N^\kappa,q}\leq CN^{-\beta\kappa}\|H(N\cdot)\|_{W^{\beta,q}}=CN^{-\beta\kappa}\Big\|\Phi_{t,\delta}(Nu)(1+N^2u^2)^{\gamma/2}\Big\|_{W^{\beta,q}}&\leq CN^{-\beta\kappa+\gamma}\delta^{{1\over q}-\beta}.
Y_{a_{1},a_{2}}\left(-n_{1},-n_{2}\right)=\frac{1}{n_{2}+1}\sum_{k_{2}=0}^{n_{2}+1}\sum_{l_{1}=0}^{n_{1}+n_{2}+2-k_{2}}\sum_{l_{2}=0}^{k_{2}}\frac{\binom{n_{2}+1}{k_{2}}\binom{n_{1}+n_{2}+2-k_{2}}{l_{1}}\binom{k_{2}}{l_{2}}}{n_{1}+n_{2}+2-k_{2}}a_{1}^{l_{1}}a_{2}^{l_{2}}.
\Psi_{\Pi'}\Psi_{\Pi''}=\sum_{\Pi\in\Pi'\Cup\Pi''}\alpha_{\Pi',\Pi''}^{\Pi}\Psi_{\Pi}\mbox{and}\Delta(\Psi_{\Pi})=\sum_{\Pi'\uplus\Pi''=\Pi}\Psi_{\Pi'}\otimes\Psi_{\Pi''}.
Z ={V_{p + 1}}\Gamma ( 0 ){{( 2 \pi{\alpha^{2}}\mathrm{T})}^{- 1 / 2}}\int{\frac{d \tau}{\tau}}{{\left({\frac{\mathrm{T}}{4 \pi \tau}}\right)}^{( p + 2 ) / 2}}\exp \left( - \mathrm{T}{\mathcal{Y}^{2}}\tau \right){{| \eta ( i \tau ) |}^{1 - d}},
\frac{d}{dv}(K(x/v,v)^2A(v/x,v)) =K'(x/v,v)-\frac{x^2(2v-x)}{v^2(v-x)^2}(K(x/v,v)-1)-\frac{x^2}{v(v-x)}K'(x/v,v) &-\frac{x}{(1-v)^2}K(x/v,v)A(x,1)-\frac{x}{1-v}K'(x/v,v)A(x,1) &-\frac{x^3(x-2v+3v^2-2xv)}{v^2(1-v)^2(v-x)^2}C(x),
\psi_1(x,t,k)=N_{-}(k)+\int_{-\infty}^{x}G_{-}(x,y,t,k)\left(U(y,t)-U_{-}\right)\psi_{1}(y,t,k)e^{ik(x-y)\sigma_3}dy, \psi_2(x,t,k)=N_{+}(k)-\int_{x}^{+\infty}G_{+}(x,y,t,k)\left(U(y,t)-U_{+}\right)\psi_{2}(y,t,k)e^{ik(x-y)\sigma_3}dy
\mbox{I)}\quad{\cal L}=\partial_z\varphi\partial_{\bar{z}}\varphi-4\varphi^2+2\varphi^4\quad\mbox{and}\quad\mbox{II)}\quad{\cal L}=\partial_z\varphi\partial_{\bar{z}}\varphi+2\varphi^2-2\varphi^4\,.
\int_{Q_1}U_j((x,0),t)^2 dxdt \leq r_j^{a+1-N_1}\int_{Q_{r_j}}U((x,0),t)^2 dx dt &= r_j^{a+1-N_1}\int_{Q_{r_j}}u(x,t)^2 dx dt &\stackrel{\eqref{contra}}\leq C r_j^{a+1+N_1}.
R^{\, \prime}( u , v ) ~ = ~ ( g ( u ) \otimes g ( v ) ) \biggl ( R ( u , v ) \, - \, \sum_{j = 1}^{r}\, A_{j}( u ) \otimes H_{j}\biggr ) ( g^{- 1}( u ) \otimes g^{- 1}( v ) ) ~ .
( 3 d ) = - \frac{\lambda^{2}}{2^{2}. 3}i^{5}\int{\cal D}k_{1}{\cal D}k_{2}{\cal D}k_{3}{\cal D}k_{4}\frac{1}{k_{1}^{2}k_{2}^{2}k_{3}^{2}k_{4}^{2}( p + k_{1}+ k_{2}- k_{3}- k_{4})^{2}}.
\tilde{E}\big[F|\mathcal{F}_{r}^{H}\big]=\sum\limits_{l=0}^{+\infty}\left( -1\right)^{l}\sum\limits_{k=0}^{l}\frac{\left( -1\right)^{k}(T-r)^{\left( l-k\right) H}}{\left( l-k\right) !}h_{l-k}\left( \frac{B_{T}^{H}-B_{r}^{H}}{(T-r)^{H}}\right)\psi_{k}^{(r,T)}(D_{T}^{H,l}F).
\frac{i}{2}\oint_{\gamma_{\infty}}( \phi_{1}+ \alpha_{1}\phi_{B}) \partial_{\bar{z}}\phi_{1}d \bar{z}- \int ( \phi_{1}+ \alpha_{1}\phi_{B}) \alpha_{1}\partial_{z}\partial_{\bar{z}}\phi_{B}\frac{i d z \wedge d \bar{z}}{2}.
H = \frac{J}{2}\left[ \sum_{n = 1}^{N - 1}\left( \sigma_{n}^{x}\sigma_{n + 1}^{x}+ \sigma_{n}^{y}\sigma_{n + 1}^{y}+ c o s \gamma \sigma_{n}^{z}\sigma_{n + 1}^{z}\right) + \right.
\hat{C}_s(\{(\overline{W}_s,{V}_t) :s \in \overline{\theta}, t \in \theta\}) &= \lim_{n\rightarrow \infty}\frac{1}{n}\max_{\Lambda_n}\Bigl(\min_{s \in \overline{\theta}}\chi(p_U;B_s^{\otimes n})- \max_{t^n\in \theta^n}\chi(p_U;Z_{t^n})\Bigr)
\frac{1}{N}\sum_{n=1}^N F_1(U^{an}(x, y, z))F_2(U^{bn}(x, y, z)) \exp(-2\pi i (q_1 \Phi_{an-1}(y, \alpha) + q_2 \Phi_{bn-1}(y, \alpha)+ \psi(n))) &= \frac{C_{y, z}}{N}\sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)\exp(2\pi i c_{m+1}n^{m+1}\alpha^m).
H ( r_{0}) = ( 3 - p ) \frac{q_{1}^{2}}{r_{0}^{7 - 2 p}}\left( \frac{h_{2}^{\prime}}{h_{2}}- \frac{h_{1}^{\prime}}{h_{1}}\right)^{- 1}\Bigg \vert_{r = r_{0}}.
T^{2\varepsilon}\gg E_1(g_1(t+u) - g_1(t))^2 + E_2(g_2(t+u) - g_2(t))^2 = \left\| \left(E_1^{1/2}, E_2^{1/2}\right) g''(t) u\right\|^2 + O(T^{3\varepsilon})
\mathcal J_{\boldsymbol{k}}(\tau):=\sum_{\delta\in\{0,1\}}\mathcal I_{(k_1+p\delta,k_2+3p\delta)}(\tau),\quad\mathcal I_{\boldsymbol{k}}(\tau):=-\frac{\sqrt{3}}{8\pi}I_{\Theta_1(2p,k_1,2p;\cdot),\Theta_0(6p,k_2,6p;\cdot)}(\tau).
a_{(K,\Upsilon)}+ a_{(K,\Upsilon^c)}= \sum_{\stackrel{\eta,\eta'\in \Upsilon}{\eta\neq \eta'}}a_{\eta,\eta'}+ \sum_{\stackrel{\eta,\eta'\in \Upsilon^c}{\eta\neq \eta'}}a_{\eta,\eta'}+ \sum_{\eta:K\to P}a_{\eta,\eta}.
\overline{\mathbb{K}}_{\overline{\psi}_F^l}^{r_l(\pi_F)^{(l)}}(w)\xi\{\chi,0\}=\epsilon(\chi^{-1}r_l(\pi_F)^{(l)},\overline{\psi}_F^l) \xi \{\chi,-n(\chi^{-1}r_l(\pi_F)^{(l)},\overline{\psi}_F^l)\},
\mathcal G_{T}=L^2_{\mathcal F}(\Omega; C([0,T]; L^2(I)))\cap L^2_{\mathcal F}(0,T; H_{\gamma}^1(I)), &\mathcal H_T=L^2_{\mathcal F}(\Omega; C([0,T]; L^2(I)))\cap L^2_{\mathcal F}(0,T; H_0^1(I)).
\left[ A_{\{( n_{i}) \}}^{\{r , r_{i}\}}, V_{\{( n_{j}) \}}^{\{\lambda , \lambda_{j}\}}\right] = \frac{1}{( 2 \pi i )^{2}}\oint_{0}d z \oint_{z}\frac{d \xi}{\xi}U_{\{( n_{i}) \}}^{\{r , r_{i}\}}( z ) U_{\{( n_{j}) \}}^{\{\lambda , \lambda_{j}\}}( \xi )
X_{t-}^{i_{1}}+\alpha\frac{k}{N}+\biggl(\frac{\vert I\vert}{N^2}\sum_{j=1}^{N}\bigl(M_{t+h}^{j}\bigr)^{2}\biggr)^{1/2}+Ch\bigl(1+\sup_{s \in [0,t+h]}\vert Z_{s}^{i_{1}}\vert\bigr)+\sup_{s \in [t,t+h]}\vert W_{s}^{i_{1}}-W_{t}^{i_{1}}\vert\geq1,
\overline{\rho}(\sigma) = \left( \begin{array}{cc}\overline{\chi}(\sigma) 0 0 \overline{\chi}^w(\sigma) \end{array}\right) \sigma \in G_K; \overline{\rho}(w) = \left( \begin{array}{cc}0 \overline{\chi}(w^2) 1 0 \end{array}\right).
P'(V) :=S^d (U \oplus V)=\bigoplus_{i=0}^d S^{d-i}U \otimes S^i V, X'(V) :=X(U \oplus V) \subseteq P'(V), Z'(V) :=\left\{q \in X'(V) ~\middle|~ h((S^d(1_U \oplus 0_{V \to U})) q) \neq 0\right\};
\hat{D}_{\mu \nu \rho \underline{{{x}}}}={\textstyle \frac{3}{8}}\left( C_{\mu \nu \rho}- A_{[ \mu}^{( i )}B_{\nu \rho ]}^{( i )}- \epsilon^{i j}A_{[ \mu}^{( i )}A_{\nu}^{( j )}B_{\rho ]}\right) \, , \hspace{. 5 c m}\hat{\rho}_{\mu \nu \underline{{{x}}}}={\textstyle \frac{1}{2}}\chi_{\mu \nu}\, .