harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H'((\mathcal{S},\mathcal{T}),\gamma)=\limsup_{n\rightarrow\infty}\frac{1}{n}\log r'(n,\gamma),H'((\mathcal{S},\mathcal{T}))=\lim_{\gamma\rightarrow 0}H'((\mathcal{S},\mathcal{T}),\gamma).
| \Phi_{\beta,\varepsilon}(t,x;t_0) | \le K_{\beta,\varepsilon}(t_0+t)^{-\beta}\left( 1+ \frac{\widetilde{\gamma}_{\varepsilon}A_{\varepsilon}(x)}{t_0+t}\right)^{-\beta}\le C \left( t_0 + t + A_{\varepsilon}(x) \right)^{-\beta}= C \Psi(t,x;t_0)^{-\beta}
\lim_{t \to \infty}t^{- \kappa_1}\P_x \left( \tau_C > t \right) = \frac{\Gamma \left( \kappa_1 + \frac{d}{2}\right)}{\Gamma \left( 2 \kappa_1 + \frac{d}{2}\right)}\left( \frac{|x|^2}{2}\right)^{\kappa_1}M(x),
\int_{Q\times \mathbb{R}^{N}}f_{\varepsilon}\left( \frac{\partial \phi}{\partial t}+\varepsilon v\cdot \nabla \phi +(\boldsymbol{u}_{\varepsilon}-v)\cdot \nabla_{v}\phi \right) dxdvdt+\int_{\Omega \times \mathbb{R}^{N}}f^{0}\phi (0,x,v)dxdv=0;
F_{n}^{N}(0)=-\frac{1}{\sqrt{2\omega}}\int_{0}^{+\infty}d\zeta \zeta^{N}\big( L_{n}^{N}(\zeta) \big)^{2}\exp ( -\zeta ) =-\frac{1}{\sqrt{2\omega}}\frac{(n+N) !}{n!}<0,
g^{k}_{\beta,\epsilon}=\Sigma_{j=1}^{k}(ds_{j}^2+a_{j,\epsilon}s_{j}^2d\theta_{j}^2)+\Sigma_{j=k+1}^{n}dz_{j}\otimes d\bar{z}_{j},\ \beta_{j}^{2}<a_{j,\epsilon}\leq 1.
c_1(q,2q,p^0(0)) = \frac 14 \left( \frac{1}{-q}f(-qr) + \frac{1}{3q}f(3qr) - \frac{2}{q}f(qr) - H(q,r)\frac 1q f(qr)\right) &=\frac 1q g(qr),
\frac{q^{2}}{2q-1}\mu (F_{h}^{-1}{\bf D}) = \frac{q^{2}}{2q-1}\sum_{b_{0}\in \mathbb F_{q}}\frac{1}{q^{2}}\mu ({\bf B})= \frac{q}{2q-1}\mu (\mathbf B) &= \frac{q}{2q-1}\mu(a +{\bf B}) = \frac{q}{2q-1}\mu({\bf D}) ,
\rho_i \equiv \ \underset{j=0}{\stackrel{d}{\textstyle{\prod}}}[x_i^{-1},x_j^{-1}]^{c_{ij}}\cdot [x_i,x_0,x_0]^{c_{i0}(c_{i0}-1)/2}\cdot \underset{a < b}{\textstyle{\prod}}[x_i,x_a,x_b]^{c_{ia}c_{ib}}[x_a,x_b,x_i]^{c_{iab}}& \bmod{F_3^2 F_4(F_2)_2N}
\nabla \left[k^j (\mathbf y^j T(\mathbf y^{-1}))(\nabla k) \right] = (\nabla k^j) \cdot (\mathbf y^j T(\mathbf y^{-1}))(\nabla k) + k^j \nabla \left[ (\mathbf y^j T(\mathbf y^{-1}))(\nabla k) \right].
\frac{1}{n!}\sum_{\pi\in S_{2n}}\prod_{j=1}^n |m_{\pi(2j-1)\, \pi(2j)}|\leq C_{d,\mu}^n \cdot |\{B: B\subset [2n], |B|=n\}| \cdot e^{-\frac{\mu}{2\sqrt{d}}D_s(X)}\leq C_{d,\mu}^n e^{-\frac{\mu}{2\sqrt{d}}D_s(X)},
\underline\lim_{\,n\to\infty}{\mathbb P}(\varphi_n(f_1,\ldots,f_s)\in \bigcup_{\mathbf b\in M}\bigcup_{\mathbf c\in O_K(\mathbf b)}J_K(\mathbf c)\,|\, \xi_d=\beta)\geqslant\sum_{\mathbf b\in M}V(J_K(\mathbf b)).
u_0^{(h)}(x) = \sum_{M\le j \le M+\sqrt M}2^{-mj}(\nabla^{\perp}a_0)(2^j x), &\omega_0^{(h)}(x) = \sum_{M\le j \le M+\sqrt M}2^{-(m-1)j}(\Delta a_0)(2^j x).
\sum_{n\geq 2}^{m}(-1)^n\left(\sum_{\cup_{i=1}^{n-1}S_{i}=\{1,..,m-1\}}1+\sum_{\cup_{i=1}^{n}S_{i}=\{1,..,m-1\}}1\right) &=1+\sum_{k=2}^{m-1}(-1)^k\left(\sum_{\cup_{i=1}^{k}S_{i}=\{1,..,m-1\}}1-\sum_{\cup_{i=1}^{k}S_{i}=\{1,..,m-1\}}1\right)+0 &=1
\left( \begin{array}{c}\vec{F}^{(1)}-\vec{F}^{(2)}\end{array}\right) =\frac{1}{\sqrt{2}}\left( \begin{array}{cc}1 1 -1 1 \end{array}\right)\left(\begin{array}{c}\vec{F^R}\vec{F}^L \end{array}\right)\, .
\bigoplus_{\substack{u\prec w \ell(u)=\ell(w)-1}}i_{P_{I_u}\cap L_I}^{L_I}\to\cdots \to \bigoplus_{\substack{u\prec w \ell(u)=1}}i_{P_{I_{u}}\cap L_I}^{L_I}\overset{d_{\ell(w)-1}}{\longrightarrow}i_{B^*\cap L_I}^{L_I}
\frac{\rho(\omega \lambda(t)^{2})}{\rho(\omega \lambda(x)^{2})}\leq C \omega \lambda(t)^{2}\omega \lambda(x)^{2}\log^{2}(\omega \lambda(x)^{2}) &\leq C \frac{\lambda(t)^{2}}{\lambda(x)^{2}}\left(\omega \lambda(x)^{2}\right)^{2}\log^{2}(\omega \lambda(x)^{2}) &\leq \frac{C \lambda(t)^{2}}{\lambda(x)^{2}}
l_{\dot{J}}^{\dot{I}}l_{\dot{L}}^{\dot{K}}= \delta_{\dot{J}}^{\dot{K}}l_{\dot{L}}^{\dot{I}}+ \sum_{\dot{J}_{1}J_{2}= \dot{J}}\delta_{\dot{J}_{1}}^{\dot{K}}l_{\dot{L}J_{2}}^{\dot{I}}+ \sum_{\dot{K}_{1}K_{2}= \dot{K}}\delta_{\dot{J}}^{\dot{K}_{1}}l_{\dot{L}}^{\dot{I}K_{2}}.
\iota_g(\eta) = \gamma^\ast g \circ \eta \circ g^{-1}&= \gamma^\ast (\alpha_z \circ f) \circ \eta \circ f^{-1}\circ \alpha_z^{-1}&= \gamma^\ast \alpha_z \circ \iota_f(\eta) \circ \alpha_z^{-1}&= \alpha_{\gamma^\ast z}\circ \iota_f(\eta) \circ \alpha_z^{-1}
\int_{t_o}^x \left[ G^*(e^{t^2/2}) e^{-t^2/2}- t \right]dt \leq - \int_{t_o}^x \frac{1.228}{W^{-1}(t^2/2)}dt \leq - \int_{t_o}^x \frac{1.228}{t}dt \, = \, - 1.228 \log \left( \frac{x}{t_o}\right) .
X_j(t) \approx \int_{\tau_j}^t \mu \cdot \frac{s}{\mu}e^{s Q(\tau_j) (u - \tau_j)}\cdot e^{s(Q(\tau_j) + 1)(t - u)}\: du &= s e^{s (Q(\tau_j) + 1) (t - \tau_j)}\int_{\tau_j}^t e^{-s (u - \tau_j)}\: du &\approx e^{s (Q(\tau_j) + 1) (t - \tau_j)},
\eqref{eq-too}\times ( v/mj ) < (mjv^m - m^2jv^2w + mjvw^2 + mv - 2w)w^2b^2 & + (mjv^mw^2 - m^2jv^2w^m + mjvw^4 + 2v^2w - mvw^2)b^2 & - 2(mjv^mw - m^2jv^2w^2 + mjvw^m + v^2 - w^2)wb^2 =0.
{\cal A}_{2}( x ) = \frac{1}{4 \pi}\mathrm{t r}\; \gamma_{5}{\not \! \! D}*{\not \! \! D}= \frac{1}{4 \pi}\mathrm{t r}\left( \gamma_{5}\gamma^{\mu}\gamma^{\nu}\right) D_{\mu}* D_{\nu}\; .
\aligned 0 <s_2-s_1<\gamma_1/2, s_1 \geq r >s_2-\gamma_1, \gamma_1 \geq (s_2-s_1)+(s_2-r), r =-1+\gamma_2+M-n/p, n/p >\gamma_2/2+M, s_2 \geq \gamma_2/2+M.\endaligned
c^{2}=\frac{1}{1-\frac{r_{s}}{r}}\frac{E^{2}}{m^{2}c^{2}}-\frac{1}{1-\frac{r_{s}}{r}}(\frac{dr}{d\tau})^{2}-\frac{1}{r^{2}}\frac{L^{2}}{m^{2}},
. . . Q^{( 1 )}{\bar{Q}}^{( 5 )}Q^{( 3 )}{\bar{Q}}^{( 4 )}Q^{( 2 )}{\bar{Q}}^{( 1 )}Q^{( 5 )}{\bar{Q}}^{( 3 )}Q^{( 4 )}{\bar{Q}}^{( 2 )}. . .
x(t) = x_0 + v_0 t_1 + v_1 (t_2-t_1) +v_2 (t_3-t_2) \cdots\,, y(t) = x_0 + v_0 t_1 + \xi_1+ w_1 (t_2-t_1) + \xi_2 + w_2 (t_3-t_2) \cdots
E(A_\alpha(G))\geq\begin{cases}(1-\alpha)n,\,\,\,\, s_n=0\,\, \,\, G\cong K_{\frac{n}{2},\frac{n}{2}}2(1-\alpha)nr\dfrac{\sqrt{s_n}}{r+s_n}, \,\, \,\, s_n>0. \end{cases}
\left[M_0 \prod_{j=1}^{d-1}L_{0,j}\,\,:\,\, M_1 \prod_{j=1}^{d-1}L_{1,j}\,\,:\,\, M_2 \prod_{j=1}^{d-1}L_{2,j}\,\,:\,\, X_0^{d-2}X_1 X_2 \,\,:\,\, M_3 \prod_{j=1}^{d-1}L_{3,j}\right]
\int_{0}^{\infty}\! d t I m T_{1}( t ) = - \frac{m^{2}}{4}\int_{4 m^{2}}^{\infty}\! \! \frac{d t}{t^{2}}\left( 1 - \frac{4 m^{2}}{t}\right)^{\frac{1}{2}}= - \frac{1}{2 4}.
RQ_3(a,b;p;z):=q^{(a^2-b^2)/(2p)-(a-b)/2}\frac{(-q^a;q^p)_{\infty}(-q^{p-a};q^p)_{\infty}}{(-q^p;q^p)_{\infty}(-q^{p-b};q^p)_{\infty}}.
\begin{aligned}\partial_t \psi - \Delta \psi + \psi \cdot \nabla \psi + \nabla \pi + \phi( \partial_t \eta - \Delta \eta) - 2 \nabla \phi \cdot \nabla \eta + (\psi \cdot \nabla \eta) \phi =0 \div \psi = - \nabla \eta \cdot \phi \end{aligned}
\nabla_{j}= \nabla_{j}^{0}+ i \left( \begin{array}{c c}{( A_{j})_{i}^{k}}{\langle ( b_{j})_{i}^{\beta}|}{| ( b_{j})_{\alpha}^{k}\rangle}{( \hat{D}_{j})_{\alpha}^{\beta}}\end{array}\right) \, .
x_N(t)-x_{N-1}(t) = ~ \bar{x}_N-\bar{x}_{N-1}+ \int_0^t \left[w_{N-1}-v_{N-1}\left(\dfrac{1}{x_N(s)-x_{N-1}(s)}\right)\right]{\rm{d}}s = ~ \bar{x}_N-\bar{x}_{N-1}+ \int_0^t p\left(\dfrac{1}{x_N(s)-x_{N-1}(s)}\right){\rm{d}}s \geq \bar{x}_N-\bar{x}_{N-1}\geq \dfrac{1}{R_{N-1}},
U\left[ v\right] =\mathcal{T}\left( \dot{Q},Q\right) +\mathcal{V}\left(\dot{Q},Q\right) ,W_{\mathrm{dis}}\left[ v\right] =2\mathcal{R}\left( \dot{Q}\right) ,W\left[ v\right] =\operatorname{Re}\left( \dot{Q},F\right)
| -\partial_s w_n - \Delta w_n| = | - \theta_n h(\bar x_n + r_n y, \bar t_n + r_n^2 s) |Dw_n|^\gamma + g_n| \stackrel{\eqref{thetabound}}{\le}h_1 K^{\gamma-1}|Dw_n|^\gamma + 1,
0 =\ \Big(\sum_{k=0}^{n-1}(C_n^k)^t (C_n^t-C_n) C_n^k \Big)\, x = \Big(C_n^t \sum_{k=0}^{n-1}(C_n^k)^t C_n^k- (\,\sum_{k=0}^{n-1}(C_n^k)^t C_n^k\,) C_n \Big)\, x =\, \Big(C_n^t \sum_{k=0}^{n-1}(C_n^k)^t C_n^k- C_n^t(\,\sum_{k=0}^{n-1}(C_n^k)^t C_n^k\,) C_n^2 \Big)\, x =\C_n^t\sum_{k=0}^{n-1}(C_n^k)^t C_n^k \,(I_d-C_n^2)\, x,
H=H(\gamma,K;\epsilon,\alpha) \overset{\bigtriangleup}{=}(\gamma K)E\left\{F_\alpha\left(\left(\frac{\epsilon^\alpha}{\eta_{ij}}\right)^{1/(1-\alpha)}\right)\right\}= (\gamma K)\sum_{k=0}^{K}F_\alpha\left(\left(\frac{\epsilon^\alpha}{k}\right)^{1/(1-\alpha)}\right) \binom{K}{k}\gamma^k(1-\gamma)^{K-k}
H= \pi^t_x x_t+\pi^m_x x_m+\pi^t_S S_t-\tilde{L}, = u x_t+px_m+rS_t-\left(-\frac{1}{2}u^2+u x_t+px_m+r S_t-w\right) = \frac{1}{2}u^2+w,
\vec{S}= \left[ \begin{array}{c}{\mathrm{R e}\left({\mathcal{W}}\right)}{\mathrm{I m}\left({\mathcal{W}}\right)}\end{array}\right] , \qquad \vec{a}= - \left[ \begin{array}{c}{\mathrm{I m}( \phi \partial_{x}\bar{\phi})}{\mathrm{I m}( \phi \partial_{t}\bar{\phi})}\end{array}\right] ,
M_{CM}=\left(\begin{array}{lll}\mathcal{A}_{CM}\mathcal{B}_{CM}\mathcal{E}_{CM}\mathcal{B}_{CM}\mathcal{A}_{CM}\mathcal{E}_{CM}(\mathcal{E}_{CM})^{t}(\mathcal{E}_{CM})^{t}\mathcal{G}_{CM}\end{array}\right) ,
\sum\limits_{k=2}^\infty\frac{\lambda_k[k]^m_{p,q}-\alpha u_k[k]^n_{p,q}}{1-\alpha}|a_{k}|+\sum\limits_{k=1}^\infty\frac{\mu_k[k]^m_{p,q}-(-1)^{n+j-(m+i)}\alpha v_k[k]^n_{p,q}}{1-\alpha}|b_{k}|\leq 1,
\begin{array}{rcl}T_1 = \displaystyle\mu_h(x_k,t)= h(x_k) + \frac{t^2}{6}\,h''(x_k) + \frac{t^4}{120}h^{(4)}(x_k) & &= \displaystyle f(x)+ \frac{t^2}{6}\,f_{kk}(x) + \frac{t^4}{120}f_{kkkk}. \end{array}
\tilde H''_1(s) =h'''g_H(T_* N,\dot\eta)^2g_H(\nu,T_*N)+h''\frac{d}{ds}g_H(T_*N,\dot\eta)g_H(\nu,T_*N) &\quad+h''g_H(T_*N,\dot\eta)\frac{d}{ds}g_H(\nu,T_*N)+h''g_H(T_*N,\dot\eta)\frac{d}{ds}g_H(\nu,T_*N) &\quad+h'\frac{d^2}{ds^2}g_H(\nu,T_*N).
\hat{\nu}(j) = \frac{2^{j-1}(-1)^{m+j}(m-j+1){{N-1}\choose{m}}{{m}\choose{j-1}}}{(N-j)\sum_{k=0}^{m-1}{{N-1}\choose{k}}}and pgf_{\hat{T}_{j,N}}(s)=\prod_{k=j}^{N-1}\left[{(1-\frac{k-1}{N-1})s\over 1-\frac{k-1}{N-1}s}\right].
{}S^{R}\left[ h_{i j}, P_{\mu}, X^{\nu}, N , N^{i}, [{\bf \bar{n}}]_{L}\right] = \int_{\cal M}d t \, d^{3}x \left( \dot{X}^{\mu}P_{\mu}- N H^{R}- N^{i}H_{i}^{R}\right) ,
\omega^{(g)}_{m,0}=[\hbar^{2g}]\sum_{r_1,\dots,r_m\ge0}\prod_{i=1}^m D_{x_i}^{r_i}[w_i^{r_i}]\sum_{\Gamma}\hbar^{2g(\Gamma)}\;\prod_{(i,j)\in E(\Gamma)}W_{i,j}\prod_{i\in V(\Gamma)}W_{i}.
\langle u(t),\varphi\rangle +\int_{0}^{t}\langle u(\tau), A\varphi\rangle d\tau-\int_{0}^{t}\langle B(u(\tau), \varphi), u(\tau)\rangle d\tau=\langle u_{0},\varphi\rangle &+\int_{0}^{t}\langle f(\tau),\varphi\rangle d\tau -\mu \int_{0}^{t}\langle R_{h}(u(\tau)-U(\tau)), \varphi\rangle d\tau+\langle W(t),\varphi\rangle
\begin{cases}du_n(t) = - \left[ P_n \mathrm{A}u_n(t) + P_n B(u_n(t)) +|\nabla u_n(t)|_{L^2}^2u_n(t) \right]dt + \sum_{j=1}^m P_n C_j u_n(t) \circ dW_j(t), \;\;\;\;\;\;\; t \in [0,T], u_n(0) = \frac{P_n u_0}{|P_n u_0|}.\end{cases}
A_{+}= \frac{1}{2}\left( \begin{array}{c c}{\sigma_{-}}&{0}\\{0}&{- \sigma_{+}}\\ \end{array}\right) , A_{-}= \frac{1}{2}\left( \begin{array}{c c}{\sigma_{+}}&{0}\\{0}&{- \sigma_{-}}\\ \end{array}\right) .
\frac{d}{dt}|H|^r_r+\frac{r(r-1)}{\sigma}\int_{\Omega}|H|^{r-2}|\nabla H|^2x = r \int_{\Omega}\big(H\cdot \nabla u-u\cdot \nabla H-H u\big) \cdot H|H|^{r-2}x = r \int_{\Omega}\big(H\cdot D( u)-u\cdot \nabla H-H u\big) \cdot H|H|^{r-2}x.
n_{t}+ \nabla\cdot (n Q^\epsilon\ast \mathbf{v}) + O(\epsilon) = 0, & \mathbf{v}_{t}+ (\mathbf{v}\cdot\nabla ) \mathbf{v}= \frac{n(Q^\epsilon\ast \mathbf{v}- \mathbf{v})}{\epsilon}+ n(Q^\epsilon\ast \mathbf{u_1}- \mathbf{u_1}) + \rho_1(Q^\epsilon\ast \mathbf{v}- \mathbf{v}) + O(\epsilon).
\{D_\alpha^+,D_\beta^+\}=(C\Gamma^\mu \mathcal{P}_+)_{\alpha\beta}X_\mu+ (C\Gamma^{\mu_1\dots \mu_5}\mathcal{P}_+)_{\alpha\beta}Z_{\mu_1\dots \mu_5}+ (C\Gamma^{\mu_1\dots \mu_9}\mathcal{P}_+)_{\alpha\beta}Z_{\mu_1\dots \mu_9}\quad.
\mu'_{p}=\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\left(\lambda_{p}\hat{A}_{p}v''_{p}-\hat{A}_{12}\hat{Q}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]\right).
\ln \left[ \underbrace{\frac{e^{q_1}\, b_1 \, q_1}{e^{q_2}\, b_2 \, q_2}}_{A}\cdot \frac{e^{b_1 x}}{e^{b_2 x}}\cdot \frac{e^{-q_1 e^{b_1 x}}}{e^{-q_2 e^{b_2 x}}}\right]= \ln A + x \, (b_1 - b_2) + q_2 \, e^{b_2 x}- q_1 \, e^{b_1 x}
\begin{aligned}\frac{1}{\tau}\big\|(v_n-v_{n-1})_{n=k}^N\big\|_{L^p(X)}+ \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}
\begin{array}{lll}f_L(L\circ s) = f^M_L(\mathcal{M}_{\mathbb{C},\mathsf{S}}(L\circ s)) &= f^M_L((L,R)\star\mathcal{M}_{\mathbb{C},\mathsf{S}}(s)) &= f^M_L(L,R)\circ f^M_L(\mathcal{M}_{\mathbb{C},\mathsf{S}}(s)) &= f^M_{L}(L,R)\circ f_L(s).\end{array}
{U^{\prime}}^{\dagger}RU^{\prime}=(\begin{matrix}cos\phi&-sin\phi&0\\sin\phi&cos\phi&0\\ 0&0&1\\ \end{matrix})withU^{\prime}=U(\begin{matrix}\frac{1}{\sqrt{2}}&\frac{i}{\sqrt{2}}&0\\ \frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\ 0&0&1\\ \end{matrix}).
\Psi(x)\Psi(y) =(1_{B_\omega}-e)\widehat{\Psi}(x)(1_{B_\omega}-e)^2\widehat{\Psi}(y)(1_{B_\omega}-e) &=(1_{B_\omega}-e)^{2}\widehat{\Psi}(x)\widehat{\Psi}(y)(1_{B_\omega}-e)^2=0,
\frac12\int_{\Sigma}\int_{\Sigma}\frac{(\phi(x) - \phi(y))^2}{|x-y|^{N+s}}d x d y =\int_{\Sigma}A(x)\phi(x)^2 dx & +\frac12\int_{\Sigma}\int_{\Sigma}\frac{(\psi(x) - \psi(y) )^2 w(x) w(y)}{|x-y|^{N+s}}d x d y .
\alpha_{10}\overline{\alpha_{30}}+ \alpha_{11}\overline{\alpha_{31}}+ \alpha_{12}\overline{\alpha_{32}}= \alpha_{10}\overline{\alpha_{30}}+ \overline{\lambda}|\alpha_{11}|^2 + \overline{\lambda}|\alpha_{12}|^2 &= \alpha_{10}\overline{\alpha_{30}}+ \overline{\lambda}(1 - | \alpha_{10}|^2) &= 0,
\partial_t \delta u + \delta u \partial_x u_1 + u_2 \partial_x \delta u =- \partial_x ((\rho_1)- (\rho_2) ) + \rho_1^{-1}\partial_x(\mu(\rho_1) \partial_xu_1)- \rho_2^{-1}\partial_x(\mu(\rho_2) \partial_xu_2), &\partial_t \delta \rho + \partial_x (u_1 \delta \rho + \rho_2 \delta u ) = 0, &(\delta \rho,\delta u)|_{t = 0}= (0,0)
Z ( x_{1}, . . . , x_{M}; \Gamma ) = \sum_{N = 0}^{\infty}\, \sum_{n_{1}+ . . . n_{M}= N}\; d_{n_{1}, . . . , n_{M}}\, x_{1}^{n_{1}}\cdot \cdot \cdot x_{M}^{n_{M}}\, ,
T := \left( \mathbb{T}^q_{1}\times \mathbb{T}^q_{1}\right) \cup \left( \mathbb{T}^q_{1}\times \mathbb{T}^q_{2}\right) \cup \left( \mathbb{T}^q_{2}\times \mathbb{T}^q_{1}\right) \cup \left( \mathbb{T}^q_{2}\times \mathbb{T}^q_{2}\right).
\int_{0}^{T}d \tau_{1}\int_{0}^{T}d \tau_{2}\, \Bigl ({\dot{G}_{B}}^{2}( \tau_{1}, \tau_{2}) - G_{F}^{2}( \tau_{1}, \tau_{2}) \Bigr ) = -{\frac{2}{3}}\, T^{2}\quad
a d \left( \hat{G}_{\mu \nu}\right) \rightarrow 2 \left( x_{\mu}\frac{\partial}{\partial x_{\nu}}- x_{\nu}\frac{\partial}{\partial x_{\mu}}- w_{\mu \lambda}\frac{\partial}{\partial w_{\lambda \nu}}+ w_{\nu \lambda}\frac{\partial}{\partial w_{\lambda \mu}}\right) .
J_1= - \left(\begin{array}{ccc}D^{-1}0 0 0 0 \frac{\textstyle{1}}{\textstyle{q_{xx}}}[2mm]0 - \frac{\textstyle{1}}{\textstyle{q_{xx}}}\frac{\textstyle{1}}{\textstyle{q_{xx}}}D\frac{\textstyle{r_x}}{\textstyle{q_{xx}}}+ \frac{\textstyle{r_x}}{\textstyle{q_{xx}}}D\frac{\textstyle{1}}{\textstyle{q_{xx}}}\end{array}\right)
f(x) >|x|-\sum_{k=n}^\infty(b_k-a_{k+1})\ge|x|-\left(\max_{k\ge n}\frac{b_k-a_{k+1}}{a_k-b_k}\right) \sum_{k=n}^\infty(a_k-b_k) &>|x|-\left(\max_{k\ge n}\frac{b_k-a_{k+1}}{a_k-b_k}\right) \sum_{k=n}^\infty(a_k-a_{k+1})\ge|x|\left(1-\max_{k\ge n}\frac{b_k-a_{k+1}}{a_k-b_k}\right).
-\frac{d^{n-1}}{dv^{n-1}}\left(\frac{1}{3}v^2A\right)=-\frac{1}{3}v^2\frac{d^{n-1}A}{dv^{n-1}}-\frac{2}{3}(n-1)v\frac{d^{n-2}A}{dv^{n-2}}-\frac{1}{3}(n-1)(n-2)\frac{d^{n-3}A}{dv^{n-3}}.
\overline{k}\otimes_k F \simeq \overline{k}\otimes_k k[T]/k[T]P \simeq \overline{k}/ \overline{k}[T]P =\overline{k}[T]/ \overline{k}[T](T-\alpha_1)(T-\alpha_2)\cdots (T-\alpha_u) \simeq \oplus_{i=1}^u \ \overline{k}[T]/ \overline{k}[T](T-\alpha_i) \simeq \oplus_{i=1}^u \ \overline{k}.
\bigg[\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}+x_{n+1}^{1-2s}\sum_{j,k=1}^{n}\partial_{j}a_{jk}\partial_{k}\bigg]\tilde{w}=0\quad\mathbb{R}_{+}^{n+1}, \tilde{w}=w\quad\mathbb{R}^{n}\times\{0\},
t_1 = \lfloor \beta \rfloor,t_2 = \lfloor \beta \{\beta\}\rfloor = \lfloor \beta T_{\beta}(1)\rfloor,t_3 = \lfloor \beta \{\beta \{\beta\}\}\rfloor = \lfloor \beta T_{\beta}^{2}(1)\rfloor,\ldots
\frac{dB}{dt}=\frac{\lambda_1\lambda_2^2\lambda_3}{\lambda_4B^2C}=\frac{\lambda_1\lambda_2^2\lambda_3}{\lambda_4B^2}\left(\frac{1}{\omega}(B^2)\right)^{-1/2}B^3\frac{dB}{dt}=\frac{\lambda_1\lambda_2^2\lambda_3\omega^{1/2}}{\lambda_4}B(t) =\lambda_2\left(\frac{4\lambda_1\lambda_3\omega^{1/2}}{\lambda_2^2\lambda_4}t+1\right)^{1/4}.
e(4,5) = e_2^{(4)}= q_1^{(1)}q_2^{(1)}+ q_2^{(3)}q_3^{(3)}+ q_1^{(1)}q_3^{(3)}, e(3,3) = e_1^{(3)}= q_1^{(3)}+ q_2^{(2)}+q_3^{(1)}, &e(a,a+2) = e_3^{(a)}= q_1^{(i)}q_2^{(i)}q_3^{(i)}, i \equiv a \mod 3,
b_{jk}= R_j\tilde c_{jk}R_k^{-1}= \begin{pmatrix}z_j^{p_j}z_k^{-p_k}dz_j/dz_k \lambda_{jk}0 z_j^{q_j}z_k^{-q_k}(dz_j/dz_k)^2 \end{pmatrix},
\int\int_{T}f(r)dr=6V\int_{0}^{1}\int_{0}^{1-\lambda_{3}}\int_{0}^{1-\lambda_{2}-\lambda_{3}}f(\lambda_{1}r_{1}+\lambda_{2}r_{2}+\lambda_{3}r_{3}+(1-\lambda_{1}-\lambda_{2}-\lambda_{3})r_{4})d\lambda_{1}d\lambda_{2}d\lambda_{3}
\begin{array}{rl}\lambda^* = \max \lambda \textnormal{subject to:}\displaystyle p_{_G}- \lambda = v^T Q_0 v + \sum_{g \in G}v^T Q_g^+ v(1+X_g) + \sum_{g \in G}v^T Q_g^- v (1-X_g) & \lambda \in \mathbb{R}& \displaystyle Q_0,Q_g^+,Q_g^- \succeq 0 \textnormal{for all}g\in G.\end{array}
a(\tau)\, =\frac 1\tau\bigl|\{x\in \Bbb Z_r^*\setminus\{-1\}: |S^{-1}(x)|=\tau-1\}\bigr|,2\le \tau\le k, a(1)\, =nk-1-\sum_{\tau=2}^k\tau a(\tau), a(0)\, =n^2-\sum_{\tau=1}^ka(\tau).
S = \frac{1}{2}\int [ ( e^{- \Phi}G_{0}^{+}e^{\Phi}) ( e^{- \Phi}\tilde{G}_{0}^{+}e^{\Phi}) - \int_{0}^{1}d t ( e^{- t \Phi}\partial_{t}e^{t \Phi}) \left\{e^{- t \Phi}G_{0}^{+}e^{t \Phi}, e^{- t \Phi}\tilde{G}_{0}^{+}e^{t \Phi}\right\}] .
S_{m,n}\cong \prod_{\alpha \in \mathbb N^{m},\, \beta \in \mathbb N^{n},\, |\alpha| = |\beta|}Kx^{\alpha}y^{\beta}&=\left\{\sum_{|\alpha| = |\beta|}a_{\alpha,\, \beta}x^{\alpha}y^{\beta}\; \Big | \; a_{\alpha,\, \beta}\in K, \, \alpha \in \mathbb N^{m},\, \beta \in \mathbb N^{n}\right\}\subset K[[x_1, \dotsc,x_{m}, y_1, \dotsc, y_{n}]].
\sum\limits_{k=0}^{N_n-1}\left(\tau(t^n_{k+1})-\tau(t^n_k)\right) L(t^n_{k+1})(u) &=\sum\limits_{k=0}^{N_n-1}\tau'(\xi_k^n )(t^n_{k+1}-t^n_k) L(t^n_{k+1})(u) &=\sum\limits_{k=0}^{N_n-1}\tau'(\xi_k^n )(t^n_{k+1}-t^n_k) L(\xi_{k}^n)(u) -\sum\limits_{k=0}^{N_n-1}\tau'(\xi_k^n)(t^n_{k+1}-t^n_k) \big(L(\xi_k^n)(u)-L(t^n_{k+1})(u)\big).
f_{kj, n+1}=\left( \begin{array}{c}f_{kj, n+1}^1 f_{kj, n+1}^2 \vdots f_{kj, n+1}^r \end{array}\right):=\sum_{|\alpha|=n+1}\left( \begin{array}{c}f_{kj, \alpha}^1 f_{kj, \alpha}^2 \vdots f_{kj, \alpha}^r \end{array}\right)\cdot e_j^\alpha,
\sum_{m=4}^N2^{3m}c_0^{\frac{m}{2}}M^{\frac{a_1(l+1)}{2}m}\|J_m^{l+1}\|_{l,r}\le c M^{(a_1+a_2+a_4)(l+1)-ra_3(l+1)}\alpha^{-4}, &\sum_{m=4}^N2^{2m}c_0^{\frac{m}{2}}M^{\frac{a_1 l}{2}m}\alpha^m\|J_m^{l+1}\|_{l,r}\le c M^{(a_1+a_2+a_4)(l+1)-ra_3(l+1)-2a_1}, &\ (\forall r\in \{0,1\}).
\tilde{\Phi}^{1}= c \partial c , \tilde{\Phi}^{2}= \frac{1}{3}c \partial c T_{B}, \tilde{\Phi}^{3}= \frac 1 2 \partial c \partial^{2}c , \tilde{\Phi}^{4}= \frac 1 6 \partial^{3}c c .
b_{a}= \lim_{\tau_{2}\rightarrow \infty}{\cal B}_{a}( \tau , \bar{\tau}) = \frac{- 1 1}{3}t r_{V}Q_{a}^{2}+{\frac{2}{3}}t r_{F}Q_{a}^{2}+{\frac{1}{3}}t r_{S}Q_{a}^{2}.
\left\{\begin{array}{ccc}j_1 j_2 j_{12}j_3 j j_{23}\end{array}\right\}^S=\sum_{j_{13}}(-1)^{\Theta^{R}}\left\{\begin{array}{ccc}j_1 j_3 j_{13}j_2 j j_{23}\end{array}\right\}^S~\left\{\begin{array}{ccc}j_2 j_1 j_{12}j_3 j j_{13}\end{array}\right\}^S\,.
\begin{array}{|c|c||cccc|}\hline 1 i j k \hline\hline1 e_{4}U_{4}U_{4}U_{4}U_{4}e_{1}e_{5}U_{4}U_{4}U_{4}U_{4}e_{2}e_{6}U_{4}U_{4}U_{4}U_{4}e_{3}e_{7}U_{4}U_{4}U_{4}U_{4}\hline\end{array}
Q^{\mathcal{O}|2s+\tau ,2t+\tau^{\prime}}=Q^{\mathcal{O}_{\tau ,\tau^{\prime}}^{\mu ,\nu}|2s+\tau ,2t+\tau^{\prime}}=Q_{2s+\tau ,2t+\tau^{\prime}}^{\mu ,\nu}\sim g_{2s+\tau ,2t+\tau^{\prime}}^{\tau^{\prime}-1-\nu ,\tau +1-\mu}\,\,\det \mathcal{A}_{2s+\tau ,2t+\tau^{\prime}}^{\mu ,\nu}\,\,
\min_{\tau_n=\pm1}\Big\|\sum_{n=1}^m \tau_n a_n x_n \Big\| \leq\mathbb{E}\Big(\Big\|\sum_{n=1}^m \epsilon_n a_n x_n \Big\| \Big)\leq \max_{\tau_n=\pm1}\Big\|\sum_{n=1}^m \tau_n a_n x_n \Big\|
L_{1}=\frac{c_{1}\left[\xi_{p}\overline{G}_{(1)}\left(\frac{1}{2}\xi_{p}^{2}\right)-\xi_{q}\overline{G}_{(1)}\left(\frac{1}{2}\xi_{q}^{2}\right)\right]}{F_{Y}(\xi_{p},\xi_{q})},~ L_{2}=\frac{F_{Y_{(1)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},
b_n =\min\left\{j=2,3,\dots,n:\;\sum_{i=j+1}^n\frac{1}{i-2}\le\frac{1}{2}\right\}, u_n =(b_n-2)(2n-4)\sum_{i=b_n}^n\frac{1}{i-2}, f_n(x) =3x^2-(1+4n)x+(n-2)b_n+2(n+1)+u_n, a_n =\min\left\{j=2,3,\dots,n:\;f_n(j)\le0\right\}.
\Omega_s=-{\rm d}x^a\wedge{\rm d}p_a+\Omega_{\rm m}\qquad\Omega_{\rm m}=\frac{s}2\frac{\epsilon^{abc}p_a{\rm d}p_b\wedge{\rm d}p_c}{(-p^2)^{3/2}}\qquad(p^2<0)\,,
{\cal L}_{L}={\frac{\alpha}{3 2 \pi^{2}}}\{R^{\alpha \beta \gamma \delta}R_{\alpha \beta \gamma \delta}- 2 R^{\alpha \beta}R_{\alpha \beta}+ R^{2}\}+{\frac{\beta}{8 \pi^{2}}}\{F^{\mu \nu}\tilde{F}_{\mu \nu}\}.
\mathcal{Q}^g_n(z_1;z_2,\ldots, z_n):=\omega_{n+1}^{g-1}(z_1,\iota(z_1),z_2,\ldots, z_n) +\sum_{\substack{J\subseteq I h+h'=g}}\omega_{|J|+1}^h(z_1,J)\otimes \omega_{|I-J|+1}^{h'}(\iota(z_1),I-J),
T=\frac{1}{2}\left[(A+\lambda^{2}B)\dot{\lambda}^{2}+\frac{p_{\theta}^{2}C}{\lambda^{2}A(C-\lambda^{2}A/4)}+\frac{p_{\alpha}(p_{\alpha}+p_{\theta})}{C-\lambda^{2}A/4}\right]
\left( \sum_{j_{1},\ldots,j_{=m}}^{n}|T(e_{j_{1}},\ldots,e_{j_{m}})|^{\frac{1}{1-\left|\frac{1}{\mathbf{p}}\right|}}\right)^{1-\left|\frac{1}{\mathbf{p}}\right|}\leq \Vert T\Vert,
\alpha^{E[\hbar]}(e) - \overline{\alpha^{E[\hbar]}}(e) = \hbar((Q_A + D^\perp + D^E)((\Psi_1 \circ D^E)(e)) + (W \circ D^E)(e)) = \hbar(\partial_A \circ W + W \circ \partial_A)(e) = \hbar \partial_A W(e),
(\mu-\Delta + b \cdot \nabla)^{-1}f :=(\mu-\Delta)^{-1}f &- (\mu - \Delta)^{-\frac{1}{2}-\frac{1}{q}}Q_{p}(1 + T_p)^{-1}G_{p}(\mu - \Delta)^{-\frac{1}{2}+\frac{1}{r}}f,
\begin{array}{l l}{{l_{k}^{\prime}= k ,}}&{{\qquad \qquad k^{\prime}= 0 ,}}\\{{p_{s}^{\prime}= s \, p_{s - 1}}}&{{\qquad \qquad \operatorname{a r c t a n}^{\prime}= \frac{1}{1 + p_{2}},}}\\{{\exp^{\prime}= \exp ,}}&{{\qquad \qquad \ln^{\prime}= p_{- 1},}}\\ \end{array}
\delta E \! \simeq \!{\frac{M |{\frac{| \delta \omega |}{2 \pi}}}{r \! + \!{\frac{| \delta x |}{2}}}}\! - \!{\frac{M{\frac{| \delta \omega |}{2 \pi}}}{r \! - \!{\frac{| \delta x |}{2}}}}\! \simeq \! - \!{\frac{M | \delta \omega |}{2 \pi r^{2}}}| \delta x | \! \simeq \! - \!{\frac{M}{2 \pi r^{2}}},
\exp \left[ i \sum_{k}\frac{p_{k}^{\xi^{2}}}{2 m}+ \frac{p_{k}^{\eta^{2}}}{2 m}- E ( \xi_{k}^{2}+ \eta_{k}^{2}) - \frac{\hbar^{2}}{8 m}\left[ \frac{1}{\xi_{k}^{2}}+ \frac{1}{\eta_{k}^{2}}\right] \right] K_{\phi}
\Big\{\frac{\langle w,(\hat\Sigma-\tilde\Sigma) w\rangle}{\langle v,v\rangle}\geq 0\Big\}\cap\Big\{\tilde x\geq \frac{2z}{\sqrt{n}}\Big\}=\Big\{\frac{\langle w,(\hat\Sigma-\tilde\Sigma) w\rangle}{\langle v,v\rangle^2}\geq 0\Big\}\cap\tilde{\mathcal{E}}_{2z},
A_0 =\dfrac{1}{T}\displaystyle\int_{0}^{T}X(t) dt, A_k =\dfrac{2}{T}\displaystyle\int_{0}^{T}X(t) \cos \left(k \dfrac{2\pi}{T}t\right)dt,~~~~~~~k=1,...,\infty. B_k =\dfrac{2}{T}\displaystyle\int_{0}^{T}X(t) \sin \left(k \dfrac{2\pi}{T}t\right)dt,
\frac{\partial F}{\partial a_i}=\oint_{\beta_i}\lambda, \ \ frac{\partial F}{\partial T_n}=-2 \pi i mbox{res}\Big(z^{-n}\lambda\Big), \ \ frac{\partial F}{\partial T_0}=-2 \pi i\, \int_{z_*=0}^{z=0}\lambda ,
\|F\|_{L^{p(\cdot)}(\Omega)}=\|\tilde F\|_{L^{p(\cdot)}(\Omega)}\le C\|f\|_{L^{p(\cdot)}(\Omega)}, &\| G \|_{L^{q(\cdot)}(\Omega)}\le \| \tilde{G}\|_{L^{q(\cdot)}(\Omega)}+\|I+S-P\|_{L^{q(\cdot)}(\Omega)}\le C\|g\|_{L^{q(\cdot)}(\Omega)}.
\tilde{g}_2(s,y) =s^{2H-1}y^{-\tilde{\theta}\rho+\tilde{\theta}+1}\Big(\alpha_2\frac{1}{\tilde{\theta}^{\rho}}-(H+\epsilon)\tilde{\sigma}(\tilde{\theta}+1)y^{\tilde{\theta}\rho-\tilde{\theta}-2}\Big) &\ge h_1s^{2H-1}y^{-\alpha}
{\bf T}^{(1)}_1(E)=\xi_0 \frac{{\mathbf Q}(E q^{\frac{5}{4}})}{{\mathbf Q}(E q^{\frac{1}{4}})}+ \xi_1 \frac{{\mathbf Q}(E q^{-\frac{3}{4}})}{{\mathbf Q}(E q^{\frac{1}{4}})}\frac{\bar{{\mathbf Q}}(E q^{\frac{3}{4}})}{\bar{{\mathbf Q}}(E q^{-\frac{1}{4}})}+\xi_2 \frac{\bar{{\mathbf Q}}(Eq^{-\frac{5}{4}})}{\bar{{\mathbf Q}}(E q^{-\frac{1}{4}})}.
\exp \left[ - \frac{i}{2}\sum_{a = 1}^{3}( P_{a}^{-}\wedge P_{a}^{+}) - \frac{i}{2}P_{1}\wedge P_{2}\right] = \exp \left[ - \frac{i}{2}\sum_{a = 1}^{3}\left( P_{a}^{-}\wedge P_{a}^{+}+ \frac{1}{3}P_{a}\wedge P_{a + 1}\right) \right] ~ ,
\mathcal{K}_{t, \mathcal{Q}^{\flat}}(=K_t(\mathcal{C}_{\mathcal{Q}^{\flat}})):=\bigoplus_{m\in \mathbb{B}^{\xi, \flat}}\mathbb{Z}[t^{\pm 1/2}]F_t(m)\subset \mathcal{K}_{t}.