harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C_{t}(0) =\mu_0(t)(\alpha_{t}-1)+C_{t+1}(0)+\log(1+2^{\mu_0(t)+\Delta{C}_{t+1}})-H(\alpha_{t}),~C_{n+1}(0)=0, ~C_{t}(1) =\mu_1(t)(\beta_{t}-1)+{C}_{t+1}(0)+\log(1+2^{\mu_1(t)+\Delta{C}_{t+1}})-H(\beta_{t}),~C_{n+1}(1)=0,~t\in\{n,\ldots,0\}.
\lambda_x(\lambda_y(z)) =\lambda_{\lambda_x(y)}(\lambda_{\rho_y(x)}(z)), \lambda_{\rho_{\lambda_y(z)}(x)}(\rho_z(y)) =\rho_{\lambda_{\rho_y(x)}(z)}(\lambda_x(y)), \rho_z(\rho_y(x)) =\rho_{\rho_z(y)}(\rho_{\lambda_y(z)}(x)).
\mathbf{x}^\ast A \mathbf{x}= \sum_{i=1}^n \sum_{j=1}^n A(Y_i,Y_j) \overline{x_i}x_j &= \sum_{i=1}^n \sum_{j=1}^n \sum_{y \in Y}I_y(Y_i) I_y(Y_j) \overline{x_i}x_j &= \sum_{y \in Y}\left | \sum_i I_y(Y_i) x_i \right |^2 \geq 0.
x^{-}={\frac{4 e}{a m ( 1 - c )}}{\frac{( 1 +{\frac{\lambda ( 1 - c )}{2}}( x^{+}- x_{0}^{+}) )}{( x^{+}+{\frac{\lambda ( 1 - c ) x_{0}^{+}}{2}}( x^{+}- x_{0}^{+}) )}}
\begin{array}{l l}{}&{R_{2}^{\mathrm{i n}}= e^{- i \omega v_{*}},}\\{}&{R_{2}^{\mathrm{o u t}}= e^{- i \omega v_{*}}e^{2 [ i ( \omega - \omega_{0}) - C ] r_{*}}, ( r > r_{H}) .}\\ \end{array}
\mathrm{D}^{f}_H(K) &=\{h_0\in G; \forall g_1,\ldots,\forall g_k\in G, \exists h\in Ghh_0h^{-1}K(hh_0h^{-1})^{-1}g_1Hg_1^{-1}\cup\cdots\cup g_kHg_k^{-1}\}.
|v_{x_i}\cdot w_{x_j}|^2= |\langle (\hat{x}_i-x_i)(\hat{x}_j-x_j)\rangle|^2= |\Delta(x_ix_j)+{\textstyle\frac{1}{2}}i\hbar \epsilon_{ij}{}^kx_k|^2= \Delta(x_ix_j)^2+ \frac{\hbar^2}{4}(\epsilon_{ij}{}^kx_k)^2\,.
\Phi_{4 , 0}= - \frac{5}{4}V_{\Phi}- \frac{4 e^{\Phi_{\infty}}V_{s}}{3 P}+ \frac{2 1}{2}M_{1}V_{1}- \frac{1 6 M_{2}V_{1}}{3 P e^{\Phi_{\infty}}}+ \frac{8 M_{2}V_{2}}{3 P^{2}e^{2 \Phi_{\infty}}}- \frac{1 1 M_{1}V_{2}}{2 P e^{\Phi_{\infty}}}+ O ( M_{i}^{4}) , \, [ 2 m m ]
Q_{S}(x)\cdot\left[ \begin{array}{ccc}x^{2}x 1 1 0 0 0 1 0\end{array}\right]^{n}=\left[ \begin{array}{ccc}Q_{T,n+4}(x) P_{T,n+3}(x) Q_{T,n+3}(x) Q_{T,n+3}(x) P_{T,n+2}(x) Q_{T,n+2}(x) Q_{T,n+2}(x) P_{T,n+1}(x) Q_{T,n+1}(x)\end{array}\right],
\Psi(u)=\, \sum^{N_A}_{k=1}\,e^{\imath\,\vartheta_{k}(\varphi^{-1}_{k}(x),z_k)}\frac{1}{\sqrt{N}}\,n_{k}(\varphi^{-1}_{k}(x),z_k)\,|\varphi^{-1}_{k}(x),z_k\rangle & +\sum^{N_A+N_B}_{l=1+N_A}\,e^{\imath\,\vartheta_{l}(\varphi^{-1}_{l}(x),z_l)}\frac{1}{\sqrt{N}}\,n_{l}(\varphi^{-1}_{l}(x),z_l)\,|\varphi^{-1}_{l}(x),z_l\rangle,
\phi(K) = \begin{bmatrix}\phi_1(K_1) A_1(K_1)^{\top}K_{12}+ K_{12}A_2 + K_1A_{12}+ Q_{12}* A_2^{\top}K_2 + K_2A_2 + K_{12}^{\top}A_{12}+ A_{12}^{\top}K_{12}+ Q_2 - K_{12}^{\top}B_1B_1^{\top}K_{12}\end{bmatrix}\,.
h_\mu^{\mathrm{KS}}(T)=\lim\limits_{d,n\to\infty}h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T})=\sup_{d,n\in{\mathbb N}}h_\mu(T,\mathcal{P}_d^{(X_i)_{i=1}^n,T}).
V ( \overline{{\rho}}) = \frac{m_{R}^{2}}{2}\overline{{\rho}}^{2}+ \frac{\lambda_{R}}{2 4}\overline{{\rho}}^{4}+ \frac{3 e^{4}}{6 4 \pi^{2}}\overline{{\rho}}^{4}\ln \frac{e^{2}\overline{{\rho}}^{2}}{\mu^{2}}- \frac{3 e^{4}}{1 2 8 \pi^{2}}\overline{{\rho}}^{4}.
\iota_i = \mathcal{C}_{-1}(\alpha, \nu, i, \lbrace 1, \ldots, i-1 \rbrace, \lbrace i+1, \ldots, \ell \rbrace) - \ell + 2i - 1 = \nu_i - (i-1) + (\ell -i) - \ell + 2i -1 = \nu_i
D_{{\cal F}}^{(N, \alpha)}(\tau; \Delta)=\frac{2\pi}{\alpha}\int_{0}^{\infty}\frac{ds}{(4\pi i s)^{N/2}}e^{-i(\tau^{2}-2 r^{2})/4s}\sum_{n=-\infty}^{\infty}I_{2\pi |n|/\alpha}\left(r^{2}/2is \right)e^{i2\pi n\Delta/\alpha}.
\begin{array}{c}\{p_i(x),p_j(y)\}=-\delta_{ij}\delta^{\prime}(x-y)\ 1\leq i,j\leq l \{p_i(x),p_j(y)\}=\delta_{ij}\delta^{\prime}(x-y)\l+1\leq i,j\leq k+1+2l \{p_i(x),p_j(y)\}=0\ 1\leq i\leq l,\l+1\leq j\leq k+1+2l\end{array}
1 +{\frac{1}{2}}{\frac{U}{k^{2}+ M^{2}}}-{\frac{1}{8}}{\frac{U^{2}- U^{\prime \prime}}{( k^{2}+ M^{2})^{2}}}-{\frac{1}{4}}{\frac{k^{2}U^{\prime \prime}}{( k^{2}+ M^{2})^{3}}}+ . . .
\begin{array}[pos]{rcl}x_1(k) \!\!=\!\! x_1\!(k\!-\!1)x_2(k\!-\!1)\!(x_1\!(k\!-\!1)+x_2\!(k\!-\!1)) \!+\!w_1(k\!-\!1), x_2(k) \!\!=\!\! x_1\!(k\!-\!1)x_2(k\!-\!1)\!(2x_1\!(k\!-\!1)+x_2\!(k\!-\!1))\! +\! w_2(k\!-\!1). \end{array}
\nu (H'|_g(h)) = - \tfrac{1}{2}R'(h) + \tfrac{1}{2}R'^\Sigma|_{g^\intercal}(h^\intercal) + A_g \cdot (A_g \circ h) &- A_g \cdot A'(h) -H_g H'(h) &+ \tfrac{1}{4}\big( - R_g +R^\Sigma_g - |A_g|^2 - H_{g}^2\big)h(\nu, \nu) &- \tfrac{1}{2}\Delta_\Sigma h(\nu, \nu) + g^{ab}\omega (e_a) e_b(H_g).
\overline E^{I^{(1)}, \dots, I^{(p)}}_R(t_{|E_{\gamma}|}) = \overline{E}^{I^{(1)}}_{R}\cap \overline{E}^{I^{(2)}}_{R}\dots \cap \overline{E}^{I^{(p)}}_{R}\cap (\epsilon, L)^{|E(\gamma)|}
\dot{x}= \frac{\i}{\hbar}[ H , x ] = K^{2}[ \frac{\i}{\hbar}{\textstyle ( 1 - ( \frac{q}{\xi})^{4}) p^{2}x + q \xi^{- 4}( ( \frac{q}{\xi})^{2}+ 1 ){\mit \Lambda}^{2}p}] ,
\begin{array}{c}\frac{2Li_{(k_1,...,k_r)}(1-(ab)^{-t})}{(a^{-t}+b^t)^r}e^{rxt}=\sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_1,...,k_r)}}(x,a,b){t^{n}\over n!}\end{array}.
\tfrac{1}{2}({\mathbb D}^{A}{}_{B}{\mathbb D}^{C}{}_{D}+ \,{\mathbb D}^{C}{}_{D}{\mathbb D}{}^{A}{}_{B})={\mathbb D}^{(A}{}_{(B}{\mathbb D}^{C)}{}_{D)}+{\mathbb D}^{[A}{}_{[B}{\mathbb D}^{C]}{}_{D]},
\sum_{m=m_0}^{\infty}\frac{1}{m^s}((k_1u(m))^{-s+1}-(k_2v(m))^{-s+1})= &k_1^{-s+1}\sum_{m=m_0}^{\infty}\frac{1}{m^s}u(m)^{-s+1}-k_2^{-s+1}\sum_{m=m_0}^{\infty}\frac{1}{m^s}v(m)^{-s+1}.
\rho( \int_{T}x^*(t) d\mu(t) - x^*) \leq \rho( \int_{T}\tilde{x}_n^*(t) d\mu(t) - x^*) + \int_{A_n}\| \tilde{x}_n^*(t)\| d\mu(t) + \int_{A_n}\|y^*(t)\| d\mu(t) \leq \varepsilon/3 +\varepsilon/3 +\varepsilon/3 = \varepsilon,
\begin{array}{lll}\zeta^2(x,t) \leq \int_{\mathbb R}|\zeta\zeta_x|dx=\frac{1}{\sqrt{\mu|c|}}\int_{\mathbb R}\sqrt{\mu|c|}|\zeta \zeta_x| dx\leq \frac{1}{2\sqrt{\mu |c|}}\int_{\mathbb R}(\zeta^2+\mu|c|\zeta_x^2)dx &\leq \frac{1}{(1-\gamma)\sqrt{\mu|c|}}H((\zeta(\cdot,t),v(\cdot,t))\leq \frac{1}{(1-\gamma)\sqrt{\mu|c|}}H(\zeta_0,v_0)\equiv \alpha^2\end{array}
\langle b(x, \hat v_1(\theta, x), \hat v_2(\theta, x)) , \nabla_x \psi_{\kappa}\rangle + \frac{1}{2}{\rm trace}(a(x) \nabla^2_x \psi_{\kappa}) = \lambda \psi_{\kappa}- r_k(x, \hat v_2(\theta, x)).
\|G_j-G\|_{L^1((0,1))}=\int_0^1 \Big|\int_{B_r}(\varphi_j-\varphi)\,dx\Big|\,dr\leq \|\varphi_j-\varphi\|_{L^1(B_1)}\xrightarrow{j\to \infty}0, \|G_j'-G'\|_{L^1((0,1))}=\int_0^1 \Big|\int_{\partial B_r}(\varphi_j-\varphi)\,d\mathcal{H}^{n-1}\Big|\,dr\leq \|\varphi_j-\varphi\|_{L^1(B_1)}\xrightarrow{j\to \infty}0,
=\frac{1}{(q^3;q^3)_{\infty}}\left(\frac{27}{4}q^2\frac{(q^9;q^9)_{\infty}^6}{(q^3;q^3)_{\infty}^2}+\frac{3q}{2}\frac{(q;q)_{\infty}^3(q^9;q^9)_{\infty}^3}{(q^3;q^3)_{\infty}^2}+\frac{1}{12}\frac{(q;q)_{\infty}^6}{(q^3;q^3)_{\infty}^2}-\frac{1}{12}\right)+\sum_{n\ge1}N_2(n)q^{3n}
\d(x_0,x_1)-\d(x_0,y_1) =n(x_1^{n+1}-y_1^{n+1})-(n+1)x_0(x_1^n-y_1^n) =(x_1-y_1)\sum_{j,k}(x_1^jy_1^{n-j}-x_0x_1^ky_1^{n-k-1}).
\begin{array}{lccc}\mbox{floor}\mbox{SU(2/2)}\mbox{SU(2)$\otimes$SU(2)}\mbox{field}\noalign{\vskip3pt}\noalign{\hrule}\noalign{\vskip3pt}\mid \mbox{ground}> (0~0~1) (0)(1) \lambda \mid \mbox{1st}> (1~-1~0) (1)(0) 2\phi \end{array}
\int_0^vv'^{k+l}g_0(v')dv' =\frac{1}{k+l+1}\left(v^{k+l+1}g_0-g_{k+l+1}\right), \int_0^vv'^{l-m}g_{k+m}(v')dv' =\frac{1}{l-m+1}\left(v^{l-m+1}g_{k+m}-g_{k+l+1}\right).
J= 0.5\Big[ \boldsymbol{e}^T(t_f) F(t_f) \boldsymbol{e}(t_f)\Big]+ 0.5 \Big[ \int_{t_0}^{t_f}\boldsymbol{e}^T(t)Q(t)\boldsymbol{e}(t)+\boldsymbol{u}^T(t)R(t)\boldsymbol{u}(t) dt \Big]
1-\frac 1\alpha= \frac{t^{2H}-s^{2H}-(t-s)^{2H}}{t^{2H}+s^{2H}-(t-s)^{2H}}\leqslant \frac{t^{2H}-s^{2H}}{t^{2H}+s^{2H}}\leqslant &\leqslant \frac{(t^{H}-s^{H})(t^{H}+s^{H})}{(t^{H}+s^{H})^2/2}\leqslant \frac{2(t^{H}-s^{H})}{t^H}\leqslant \frac{2(t-s)^H}{t^H}.
\sum_{\substack{nm>N (n,m)=1}}\frac{r(n)^2r(m)\omega(n)\omega'(m)}{\sqrt{m}}&\qquad\leqslant\prod_p\left(1+r(p)^2\omega(p)+\frac{r(p)}{\sqrt{p}}\omega'(p)\right)\exp\left(-\frac{C\log N}{(\log\log N)^3}\right)
m(L)>\pi^{-4}\cdot\frac{q^\frac{3m+l-k}{2}}{2^3\cdot 3\cdot 5}\cdot (1-2^{-4})\cdot (1-q^{-4})\cdot \zeta(4)=\frac{q^\frac{3m+l-k}{2}}{2^8\cdot 3^2\cdot 5}\cdot (1-q^{-4})
\leq \sum_{n=s+1}^\infty2^{-2n}\left[\int\int_T \left|\sum_{k,\nu=N_{s-1}}^{\overline{n}, \overline{m}}c_{k,\nu}^{(s)}\psi_k(x) \cdot \psi_\nu(y)\right| h_s^{-1}dxdy \right]<{1\over 3}2^{-2s}.
(\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\frac{{x_{i}}^{2}}{{x_{j}}^{2}})+(\sum_{i=2}^{n}\sum_{j=1}^{i-1}\frac{{x_{i}}^{2}}{{x_{j}}^{2}})+\sum_{i=1}^{n}\frac{{x_{i}}^{2}}{{x_{i}}^{2}}\ge n^{2}
\begin{array}{c}\mathbf{E}^{(k_1,k_2,\ldots, k_r)}_n(x)=\sum\limits_{i=0}^n\sum\limits_{0\le m_1\le m_2\le\ldots\le m_r \atop c_1+c_2+\ldots=r}\sum\limits_{j=0}^{m_r}\frac{2(rx-j)^{n-i}r!(-1)^{j+c_1+2c_2+\ldots}(c_1+2c_2+\ldots)^i\binom{m_r}{j}\binom{n}{i}}{(c_1!c_2!\ldots)(m_1^{k_1}m_2^{k_2}\ldots m_r^{k_r})}.\end{array}.
R(e_i,e_j,e_k,e_l)= \begin{cases}R_1(e_i,e_j,e_k,e_l), i,j,k,l \in \{1, \cdots, n_1\}, R_2(e_i,e_j,e_k,e_l), i,j,k,l \in \{n_1+1, \cdots, n_1+n_2\}, 0, \end{cases}
(\partial_{y_iy_j}v)_{(n-1)\times (n-1)}= (\partial_{x_ix_j}w)_{(n-1)\times (n-1)}- A(w)^t H(w)^{-1}A(w), (\partial_{x_ix_j}w)_{(n-1)\times (n-1)}= (\partial_{y_iy_j}v)_{(n-1)\times (n-1)}- A(v)^t H(v)^{-1}A(v).
\sum_{i\in I}\left\langle U^*\Gamma^*_i\Gamma_iTf,f\right\rangle =\sum_{i\in I}\left\langle U^*(\Gamma^*_i\Gamma_i-\Lambda^*_i\Lambda_i)Tf,f\right\rangle+\sum_{i\in I}\left\langle U^*\Lambda^*_i\Lambda_iTf,f\right\rangle &\le R\|f\|^2+B\|f\|^2=(R+B)\|f\|^2.
\widetilde{T}_{n,2n-i-1}(x) = \sum_{j=0}^{2n-3}x^{\chi\left(j<(2n-i-1)-1\right)+\chi\left(j<2n-1-(2n-i-1)\right)}T_{n-1,j}(x) begin{align*}5pt] = \sum_{j=0}^{2n-3}x^{\chi\left(j<2n-i-2\right)+\chi\left(j<i\right)}T_{n-1,j}(x) begin{align*}5pt] = \widetilde{T}_{n,i}(x),
U=e^{i\varphi/2}[\begin{matrix}e^{i\psi}&0\\ 0&e^{-i\psi}\end{matrix}][\begin{matrix}cos\theta&sin\theta\\ -sin\theta&cos\theta\\ \end{matrix}][\begin{matrix}e^{i\Delta}&0\\ 0&e^{-i\Delta}\end{matrix}].
\omega t-kx=\omega t-\frac{\omega}{v_{\phi}}vt=\omega t(1-\frac{v^{2}}{c^{2}})=\frac{\omega t}{\gamma^{2}}=\frac{1}{\gamma^{2}}\frac{\gamma m_{o}c^{2}}{\hbar}(\gamma t_{o})=\omega_{o}t_{o}=\omega_{c}t.
\begin{cases}\partial_t ( \frac{\omega^m}r ) + (u^m \cdot \nabla) ( \frac{\omega^m}r ) =0, 0<t\le 1, u^m = -\Delta^{-1}\nabla \times \omega^m, \omega^m \Bigr|_{t=0}= \sum_{i=1}^m g_i.\end{cases}
\mathcal{J}(u,\bar{u},r) = \left[\begin{array}{c}\bar{\pi}^{\bar{i}}-F(r) \bar{\pi}^{\bar{i}}- \bar{\Gamma}^{\bar{i}}_{\bar{j}\bar{k}}\bar{\pi}^{\bar{j}}\bar{\pi}^{\bar{k}}+ g^{k\bar{i}}\left(G(r)\partial_{k}V_{BH}+ H(r)\partial_{k}V\right) \end{array}\right] \: .
\begin{aligned}p(N,\vec\ell;2)= \frac{(-1)^{N+\ell}N\prod_j\ell_j!}{(N)_{\ell}}&\times [z^{N-\ell}] (1-z)^{-t+1}\int_0^1\frac{(1-u)^{N+1}u^{\ell-t}}{(1-u+zu)^{\ell+1}}\,du.\end{aligned}
P_{Y}^{( 6 ) . . . ( i_{1}i_{2}) . . .}= \frac{1}{2}( \delta_{i_{1}i_{1}^{\prime}}\delta_{i_{2}i_{2}^{\prime}}+ \delta_{i_{1}i_{2}^{\prime}}\delta_{i_{2}i_{1}^{\prime}}) P_{Y}^{( 6 ) . . . i_{1}^{\prime}i_{2}^{\prime}. . .}
{-i\mu^2 \over 8 H_5 \pi V}(2 \pi)^3 \delta^3(\vec{k}+\vec{k}') \int dt |t| \int\int du du' e^{i(u-u'){k_y\over H_5}}e^{-i({\rm cosh}u+{\rm cosh}u') mt}e^{-\epsilon |t|},
\{x_1,\,x_4,\,x_3\}=\left\{\partial_z,\,\partial_\zeta,\,z\partial_z\right\}\qquad\mbox{or by}\{x_1,\,x_4,\,x_6\}=\left\{\partial_z,\,\partial_\zeta,\,\zeta\partial_\zeta\right\}.
G(p_1,p_2) = -\frac{1}{4 \pi}\log \left( d^2(p_1,p_2) \right ) + \frac{1}{4 \pi}\log \left( d^2_{R_i}(p_1,p_2) \right) + \frac{1}{4 \pi}\log \left( d^2_{R_j}(p_1,p_2) \right) - \frac{1}{4 \pi}\log \left( d^2_{R_{ij}}(p_1,p_2) \right) + \omega(p_1,p_2)
S_{II}= k^2~ \frac{\sigma_0}{\pi}\int_{\tau_0}^{\tau_f}d\tau ~\sum_{n \neq 0}^{}~| F_{\nu}\left( \frac{2\pi}{\sigma_0}|n| \tau, \frac{2\pi}{\sigma_0}|n| \tau_0 \right) |^2 ~|\phi_n(\sigma_0)|^2
\int d^{4}x \left\{\sqrt{g_{1}^{( 4 )}}\lambda_{1}\left( \Phi^{2}( - l ) - v_{1}^{2}\right)^{2}+ \sqrt{g_{2}^{( 4 )}}\lambda_{2}\left( \Phi^{2}( l ) - v_{2}^{2}\right)^{2}\right\}\; .
i \left( k - 2 \right) : \partial_{z}g g^{- 1}: = \left( \begin{array}{c c}{- J^{2}( z )}&{J^{0}( z ) + J^{1}( z )}\\{- J^{0}( z ) + J^{1}( z )}&{J^{2}( z )}\\ \end{array}\right)
\langle \nabla f_{\delta}(x^g_k) - \nabla f(x^*), x_f^k - x_g^k \rangle \leq f(x_f^k) - f_{\delta}(x_g^k)-\langle \nabla f(x^*), x_f^k - x_g^k \rangle &\leq f(x_f^k) - f(x_g^k)-\langle \nabla f(x^*), x_f^k - x_g^k \rangle+\delta_x &= B_f(x_f^k,x^*)-B_f(x_g^k)+\delta_x.
\Big\| e_{\lambda}h e_{\lambda}- \sum_{i=n+1}^{n+m}[x_i, y_i] \Big\| < \frac\epsilon{3}, \sum_{i=n+1}^{n+m}\| x_i \|\cdot \| y_i \| \leq C \| e_{\lambda}h e_{\lambda}\| \leq C \| h \|.
q^{2(\lambda,\lambda+2\rho_0)}=\sum_{(\mu,2\kappa,-s)\in D^+}\sum_{s=0}^\infty (m^+_{\mu,s}-m^-_{\mu,s})q^{(\mu,\mu+2\rho_0)/2-s(2\kappa+g)}\frac{D_q[(\mu,2\kappa,-s)]}{D_q[(\lambda,\kappa,0)]}
\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\}\cup \{-\beta-s_2\}}(p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\}\cup \{-\alpha-s_1\}}(p^m) & = \tau_{A \smallsetminus \{\alpha \}\cup \{-\beta-s_1-s_2\}}(p^m) \tau_{B_{s_1+s_2}\smallsetminus \{\beta+s_1+s_2\}\cup \{-\alpha\}}(p^m).
\psi_{j}(z):=\langle(zI-\mathcal{W})^{-1}\hat{e}_{j},\hat{e}_{0}\rangle=\sum_{n=0}^{\infty}\frac{\langle\mathcal{W}^{n}\,\hat{e}_{j},\hat{e}_{0}\rangle}{z^{n+1}},0\leq j\leq p,
\sup\{W_r(t) - W_{n,r}(t) :{}j/2^n \le t < (j+1)/2^n\}= \sup\{W_r(t) - W_{n,r}(c^n_j) : j/2^n \le t < (j+1)/2^n\}= \sup\{W_r(t) - W_r(c^n_j) : j/2^n \le t < (j+1)/2^n\}\le \epsilon.
T_i(h_{i,1}) = T_i\big(\big[x_{i}^+,x_{i,1}^-\big]\big) = \big[{-}x_{i}^-, -x_{i,1}^+ + \tfrac{\hbar}{2}\big\{h_i, x_i^+ \big\}\big] = -h_{i,1}- \hbar \big\{x_i^+,x_i^-\big\}+ \hbar h_i^2.
\varphi^*\left(\begin{array}{c c}a b 0 d \end{array}\right) := \left(\begin{array}{c c c}a^{\ell_1}a^{\ell_1}\, b^{p^{e_1}}\, d^{p^{e_1}}a^{\ell_1}\, b^{p^{e_2}}\, d^{p^{e_2}}0 a^{\ell_2}0 0 0 d^{- \ell_3}\end{array}\right) .
|a_0^{-1}-a_1^{-1}|\geq\bigl|a_0^{-1}-(a_0+n^{-1}\nu_1/2)^{-1}\bigr|=|a_0|^{-1}|a_0+n^{-1}\nu_1/2|^{-1}\frac{1}{2}n^{-1}\nu_1\geq \frac{1}{2}\cdot \frac{1}{3}\cdot \frac{1}{2}n^{-1}\nu_1=\frac{1}{12}n^{-1}\nu_1.
T(\tilde{w})_1(t) = b_1 + y_1(t) + \int_0^t\left(-\phi_0(\tilde{w}_1(s)) + \phi_B\big(\tilde{w}_2(s) + \psi_0(\tilde{w}_1(s))\big)\right)ds, T(\tilde{w})_2(t) = b_2 + y_2(t) + \int_0^t\left(-\phi_B\big(\tilde{w}_2(s)+\psi_0(\tilde{w}_1(s))\big)\right)ds.
\widehat{\eta^1(t)p_1(t)}-\widehat{\eta^1}(t)\widehat{p}_1(t)=\Sigma^1(t)^{-1}\big(\widehat{\mu^1(t)p_1(t)}-\widehat{\mu^1}(t)\widehat{p}_1(t)\big)=\widehat{b^1(t)p_1(t)}-\widehat{b^1}(t)\widehat{p}_1(t).
\sum_{k=0}^{(n-2)/2}\frac{2k\pi}{n-1}\sin\left(\frac{2k\pi}{n-1}\cdot b\right)=\frac{\pi\csc\left(\frac{b\pi}{2a+1}\right)\left(2(a+1)+\sin\left(\frac{2b\pi(a+1)}{2a+1}\right)\csc\left(\frac{b\cdot\pi}{2a+1}\right)\right)}{4a+2}
-2\,G_{\mu\nu}\,{\delta\Gamma\over\delta G_{\mu\nu}}+\beta^i\,{\delta\Gamma\over\delta\phi^i}=-2\,\sqrt{G}\,({\cal W}_d+\nabla_{\mu}{\cal J}^{\mu}_d) -\beta^i\,{\delta S_{{\rm loc};\,0}\over \delta\phi^i}.
\widetilde{\Delta}=\,\int^\tau_0 dt \,\int_{M_3(1)}\,d^3 x\,\sqrt{\det g_3}\,g_3(\vec{f}_1(t,x)-\vec{f}_2(t,x),\vec{f}_1(t,x)-\vec{f}_2(t,x)) =\,\tau\,\Delta & =\,\tau\,\Delta E_G
\mathbb{P}\Bigl( \sup_{(s,t)\in\mathbf{E}}X(s,t)> u \Bigr) =\sum_{i=1}^n\mathbb{P}\Bigl( \sup_{(s,t)\in\mathbf{E}_i}X(s,t)> u \Bigr) \bigl(1+o(1)\bigr)
1) \alpha_{0},\alpha_{1}>0 &2) \alpha_{0}=8b_{1}^{2}\alpha_{1}(13\log3-12)\ \ (for\k=1) &3)\frac{k+1}{2(2\pi)^{k+1}(3^{k+1}-1)}\alpha_{0}=\alpha_{1}\frac{2b_{k}^{2}(k+1)k}{(2\pi)^{2}}\left(\frac{(k+1)(3^{k-1}-1)}{(k-1)(3^{k+1}-1)}-R_{k}\right) (for\k=2,3,4)
\sum_{n=0}^\infty |f(n)-f(0)|\pi(n) =\sum_{n=0}^\infty |\sum_{k=0}^{n-1}\nabla f(k)|\pi(n) & \leq \sum_{k=0}^\infty |\nabla f(k)| \pi(k) \sum_{n =k+1}^\infty \frac{\pi(n)}{\pi(k)}& \leq A \sum_{k=0}^\infty |\nabla f(k)| \pi(k)
I = i R \int_{0}^{\rho}s d s \int \frac{d^{4}p}{( 2 \pi )^{4}}\frac{1}{( p^{2}- s^{2})^{2}}\rightarrow - R \int_{0}^{\rho}s d s \int \frac{d^{4}p}{( 2 \pi )^{4}}\frac{1}{( p^{2}+ s^{2})^{2}}.
\left|\int_0^\infty e^{-\varepsilon t+iat}e^{-r^2/t}\,\frac{dt}{t^m}\right|+\left|\int_0^\infty e^{-\varepsilon t+iat}\int_t^\infty e^{-r^2/s}\,\frac{ds}{s^{m+1}}\,dt\right| &\leq\frac{C}{\sqrt{\varepsilon^2+a^2}\; r^{2m}}\leq\frac{C}{|a|r^{2m}}
\sum_{\substack{a \mod d \gcd(a, d)=1}}\left< \frac{a}{d}\right>^{\pm}e^{2 \pi i n \frac{a}{d}}=\pi \sum_{\substack{a \mod d \gcd(a, d)=1}}\left ( \Lambda(1, f, \frac{a}{d}) \pm \Lambda(1,f, -\frac{a}{d})\right ) e^{2 \pi i n \frac{a}{d}}.
u_1 = \ \frac{1}{3}( d_{112},d_{212}), u_2 = \ \frac{1}{2}( d_{112}+1,d_{212}), u_3 = \ \left( -\frac{1}{2},0 \right), u_4 = \ \left( \frac{1}{3},0 \right) u_5 = \ \frac{1}{3}( -d_{112}-1,-d_{212}), u_6 = \ \frac{1}{2}( -d_{112},-d_{212}).
\epsilon_{xx}^{topface}=-z\frac{d^{2}w_{b}}{dx^{2}}-(z-h-\frac{f}{2})\frac{d^{2}w_{s}}{dx^{2}};\epsilon_{xx}^{botface}=-z\frac{d^{2}w_{b}}{dx^{2}}-(z+h+\frac{f}{2})\frac{d^{2}w_{s}}{dx^{2}}
v:= S(x_{n_{1}}\otimes \cdots \otimes x_{n_{r'+1}}\otimes x_{n_{r'}}\otimes \cdots \otimes x_{n_{i+1}}) + S(\mu^{i-1}\alpha_{T}(x_{n_{1}}\otimes \cdots\otimes x_{n_{r'-1}})\otimes [x_{n_{r'}},x_{n_{r'+1}}]_{\mathfrak{g}}\otimes \alpha_{T}(x_{n_{r'+2}}\otimes \cdots \otimes x_{n_{i+1}})).
c({\tilde \lambda,\tilde \mu,\tilde B}) =I_{\tilde \lambda,\tilde \mu,\tilde B}(\mathbf{u}^*)=\max_{t>0}I_{\tilde \lambda,\tilde \mu,\tilde B}(t \mathbf{u}^*) &\geq \max_{t>0}I_{\lambda,\mu,B}(t \mathbf{u}^*)=I_{\lambda,\mu,B}(t^* u^*) \geq \inf_{\mathcal{N}_{\lambda,\mu,B}}I_{\lambda,\mu,B}=c({\lambda,\mu,B}).
\|u_i\|_\infty\le u(2^{i-1})\le \frac{C}{2^{b(i-1)}}\|f\|_r^{\frac{1}{p-1}},\quad\mbox{with}b:=\frac{N}{p-1}\left(\frac 1 r-\frac{ps}{N}\right)>0,
\Gamma_{\rm{bc}}\left(a\right)=\left({\cal C}_{66}a+{\cal C}_{44}\right)\left[\left(e_{31}+e_{15}\right)\left(e_{15}a+e_{33}\right)+\left(\eta_{11}a+\eta_{33}\right)\left({\cal C}_{13}+{\cal C}_{44}\right)\right] ,
\|\partial_{t}^{\alpha}(\tilde{u}(\cdot,t) - \tilde{u}(\cdot,0))\|_{L^{2}(\Omega)}= \|\partial_{t}^{\alpha}\tilde{u}(\cdot,t)\|_{L^{2}(\Omega)}= \|\partial_{t}(g_{1-\alpha}*\mu*v)(\cdot,t)\|_{L^{2}(\Omega)}\leq \|\mu\|_{L^{1}(0,T)}(\|\partial_{t}^{\alpha}(v-g)\|_{L^{\infty}((0,T];L^{2}(\Omega))}+ g_{1-\alpha}(t)\|g\|_{L^{2}(\Omega)}).
\langle I_{0}^{\prime}(U_{0},V_{0}),(\varphi,\psi)\rangle = \langle I_{0}^{\prime}(u_{\lambda_{n}},v_{\lambda_{n}}),(\varphi,\psi)\rangle-(u_{\lambda_{n}}-U_{0},\varphi)_{E_{1}}-(v_{\lambda_{n}}-V_{0},\psi)_{E_{2}}-\int_{\mathbb{R}^{N}}(f_{1}(u_{n})-f_{1}(U_{0}))\varphi\,\mathrm{d}x-\int_{\mathbb{R}^{N}}(f_{2}(v_{n})-f_{2}(V_{0}))\psi\,\mathrm{d}x=o_{n}(1),
|\bigcup_{p}A_{p}|=\sum_{p}|A_{p}|-\sum_{p<q}|A_{p}\cap A_{q}|+\sum_{p<q<r}|A_{p}\cap A_{q}\cap A_{r}|-\cdot\cdot\cdot\pm|A_{p}\cap\cdot\cdot\cdot\cap A_{s}|.
_3F_2\left(\left.\begin{array}{c}\frac{1}{3},\frac{1}{2},\frac{2}{3}1,1 \end{array}\right|\frac{27u^{2}}{4(1-u)^{3}}\right)=1+3\sum_{n=1}^\infty[(2n-1)!!]^2{3n-1\choose2n}\frac{1}{(n!2^n)^2}\frac{u^{2n}}{(1-u)^{3n}},
\begin{array}{rl}(\prod\limits_{i=1}^m a(i)\ast g_{w(l)}(t(i))) \ast a(m+1)= (\prod\limits_{i=1}^m a(i) \ast (\prod\limits_{p=1}^r c_{t(i)}(p) \ast h_l(z_{t(i)}(p))) \ast c_{t(i)}(r+1)))\ast a(m+1) = (\prod\limits_{i=1}^u d(i) \ast h_l(q(i))) \ast d(u+1).\end{array}
\int_{H}|\langle f,\mu(h(q,a))\psi\rangle|^2\,dh(q,a)=\int_D\int_{Y_j}[| f(x) \overline{\psi(a^{-1}x)}|^2+| f(-x) \overline{\psi(-a^{-1}x)}|^2 &\,\quad\quad\quad\quad+ 2{\mathcal Re}\,(f(x)\overline{\psi(a^{-1}x)}\overline{f(-x)}\psi(-a^{-1}x)]\frac{dx}{|J_{\Phi(x)}|}|\det \theta(a) a |^{-1}\!da.
tH_{\mathrm{FS}}^{A_3}\left({\alpha, \beta \atop \gamma, \delta};t;{q_1,p_1 \atop q_2,p_2}\right) =tH_{\mathrm{III}(D_6)}\big({\theta^\infty_3, \theta^\infty_3-\theta^\infty_1+1}; t; q_1, p_1 \big) \qquad{}+tH_{\mathrm{III}(D_6)}\big({-}\theta^0_2-\theta^\infty_2, \theta^\infty_1-\theta^\infty_2; t; q_2, p_2 \big) -p_1p_2(q_1q_2+t).
\lefteqn{\frac{1}{V(u)}\Pr[ u^{-1}X \in \{x : \rho(x) > \lambda, \, \theta(x) \in B \}]}= \frac{V(\lambda u)}{V(u)}\Pr[ \rho(X)^{-1}X \in B \mid \rho(X) > \lambda u ] \to \lambda^{-\alpha}\, H(B) = \mu ( \{x : \rho(x) > \lambda, \, \theta(x) \in B \}), u \to \infty.
\sigma_{e_\beta}(z_k)=\begin{cases}\phi_\beta^\gamma(z_1,\ldots, z_{k-1})+\psi_\beta^\gamma(z_1,\ldots,z_{k-1})z_k, \textrm{if}\;\gamma_k=s_k, -\phi_\beta^\gamma(z_1,\ldots, z_{k-1})z_k^2+\psi_\beta^\gamma(z_1,\ldots, z_{k-1})z_k, \textrm{if}\;\gamma_k=e.\end{cases}
\psi^{s}= \psi_{l}= \left( \begin{array}{c}{{E + m}}\\{{- i k}}\\ \end{array}\right) e^{i k x}+ l ( k ) \left( \begin{array}{c}{{E + m}}\\{{i k}}\\ \end{array}\right) e^{- i k x}, \qquad x \leq - \xi
\Vert g^{\mathbf{r}, \boldsymbol{\sigma}}_{N}\Vert_F^2 = \sum_{\boldsymbol{\rho}}\Vert \prod_{a \in \mathcal{I}_2}\langle h_a , h_{\nu(a)}\rangle \cdot \mu (\otimes_{j \in \mathcal{I}_3}h_{j, \boldsymbol{\rho}}) \otimes_{q_N}h_N \Vert_F^2.
\psi_{2}= \frac{e^{2}Q^{2}}{8 \pi^{2}}\ln ( q^{2}/ p_{1}^{2}) \ln ( q^{2}/ p_{2}^{2}) + \frac{g^{2}}{1 6 \pi^{2}}C_{W Z}\ln^{2}( s / M^{2}) ~
\Omega ={\frac{2 C_{F}C_{A}}{( 2 \pi )^{D}}}( 2{\cal A}_{C})^{4 - D}e^{\frac{- i \pi D}{2}}\cos \left({\frac{\pi D}{2}}\right){\frac{\Gamma ( D - 4 )}{( D - 4 )^{2}}},
R_{i}\left(\sum_{m=0}^{M}\varepsilon^{m}u_{1,m},...,\sum_{m=0}^{M}\varepsilon^{m}u_{N,m}\right)=\sum_{m=0}^{M}\varepsilon^{m}\bar{R}_{i}\left(u_{1,m},...,u_{N,m}\right)+\mathcal{O}\left(\varepsilon^{M+1}\right),
S_{\underset{k}{\underbrace{a,\ldots,a}}}\left(N\right)=\sum_{l=1}^{N}\left(\prod_{\genfrac{}{}{0pt}{}{n=1}{n\neq l}}^{N}\frac{n^{a}}{n^{a}-l^{a}}\right)\frac{1}{l^{ak}},
K^{\mu}\; : = \; \mathrm{d i a g}\left( \frac{\pi}{\pi + g N}\frac{1}{- \triangle + \frac{e^{2}N}{\pi + g N}}\; , \; \frac{1}{- \triangle + \mu}\; , \; . \; . \; . \; . \; , \; \frac{1}{- \triangle + \mu}\right) \; ,
\tilde{y}_{T_{(k,m,i,n)}}^{\mathcal{K}'}\circ \big(\tilde{y}_{t}^{\mathcal{K}'}\big)^{-1}= \tilde{y}_{T_{(k,m,i,n)}}^{\mathcal{K}}\circ \big(\tilde{y}_t^{\mathcal{K}}\big)^{-1}.
\mathrm{A_r}\triangleq \{\mathbf{Y}\in GF(q)^{k/\ell}|~\mathbf{G_r^TY}= \mathcal{P}_r\}, \mathrm{A_1^c}\triangleq \mathrm{A_1}\cap \mathrm{A_2}\cap \ldots \cap \mathrm{A_c}.
\begin{pmatrix}-\gamma^-\partial_t u 0 \end{pmatrix}+\begin{pmatrix}0 \frac12 I -\frac12 I 0\end{pmatrix}\begin{pmatrix}\varphi \psi\end{pmatrix}+ B(\partial_t) \begin{pmatrix}\varphi \psi\end{pmatrix}= \begin{pmatrix}-\partial_t \beta_0 0 \end{pmatrix}
\|\sin[\frac{\pi}{2}A]-\sin[\frac{\pi}{2}B]\|_{1}=\max_{j=1,\dots,p}\sum_{i=1}^p|\sin(\frac{\pi}{2}A_{ij})-\sin(\frac{\pi}{2}B_{ij})| \le \max_{j=1,\dots,p}\frac{\pi}{2}\sum_{i=1}^p|A_{ij}-B_{ij}| =\frac{\pi}{2}\|A-B\|_1.
\textrm{osc}(u,D_{\rho}\backslash D_{r/\rho})\leq \sum_{j=1}^\ell \textrm{osc}(u,D_{\rho_{j-1}}\backslash D_{\rho_j}) \leq C\sum_{j=1}^\ell \Big(|a_{-1}|+\max\big({2e\rho_j},2r/\rho_j\big)+\rho_j^{\frac{2p-2}{p}}\Big) \leq C \Big(|a_{-1}|\ell+\rho+\rho^{\frac{2p-2}{p}}\Big) \leq C\Big((A_0 +r)^{\frac{1}{2}}\ln\dfrac{1}{r}+\rho^{\frac{2p-2}{p}}\Big).
\prod_{\tilde{T}\subseteq T}\gamma^{-h_{\tilde{T}}}e^{- \eta C \tilde{\gamma}^{-h_{\tilde{T}}}M_{\tilde{T}}}\leq e^{- \frac{\eta}{2}C \tilde{\gamma}^{-h_T}}\prod_{\tilde{T}\subseteq T}\left(\gamma^{-h_{\tilde{T}}}e^{- \frac{\eta}{2}C \tilde{\gamma}^{-h_{\tilde{T}}/\tau}}\right)^{M_{\tilde{T}}}\, .
L_{F}= \exp \left( \begin{array}{c c}{}&{( P \Xi )_{\alpha N}}\\{( \overline{{P \Xi}})^{\beta M}}&{}\\ \end{array}\right) \exp \left( \begin{array}{c c}{}&{( Q \Xi )_{\alpha N}}\\{( \overline{{Q \Xi}})^{\beta M}}&{}\\ \end{array}\right)
X_\xi := 2k\log(\sigma_k+\xi)+\log\left(\frac{1}{2}\left(1+\frac{\xi}{\sigma_k}\right)\right)-k-2\log(C_k); Y_\xi := \frac{1}{2}-\frac{\sigma_k}{2\xi}; Z_\xi := \frac{\xi^2}{2k},
\partial_z u_1(t,x,z) = - \left( G_1 \partial_x f(t,x) + G_2\mu \partial_x g(t,x)- \sigma_2^c\mu\partial_x^3(f+g)(t,x)-\sigma_1^c\partial_x^3f(t,x)\right)(f(t,x)-z) begin{align*}5pt] +\displaystyle{\mu}\partial_z u_2(t,x,f).
\lim_{m \to \infty}I_m = \mathbf{1}_{\mathrm{Re}(z) > 0 , \delta = 0}- \mathbf{1}_{\mathrm{Re}(z) < 0 , \delta=1}+ F(z+\pi i \theta_1,0) - \int_{-1/2}^{1/2}F(z+\pi i \theta_1 + 2 \pi i u, 0 ) \mathrm{d}u.
\lim_{n\to\infty}n \widetilde f\,^\prime(s|\tfrac{\mu}{n}) = -\lim_{n\to\infty}\mu\widetilde\rho(s)\widetilde f(s|\tfrac{\mu}{n}) = -\mu \widetilde\rho(s) \widetilde f(s|0) \implies\mu\,\widetilde\rho(s) = -\lim_{n\to\infty}n\,\widetilde f\,^\prime(s|\tfrac{\mu}{n})/\widetilde f(s|0) = -\lim_{n\to\infty}n\,\widetilde f\,^\prime(s|\tfrac{\mu}{n})
|\sigma_n^2(y)\cdot (G''(x)-G''(y))|\leq \begin{cases}c\cdot (1+|y|^{2\ell_\sigma+2})\cdot|x-y|, (x,y)\in B^c, c\cdot (1+|y|^{2\ell_\sigma+2}), (x,y)\in B.\end{cases}