harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{\substack{N,L\to\infty N/L\to \sqrt E/ \pi}}-\frac{\left(2N g_N + g_N^2\right)\left(2(N+1) g_{N+1}+ g_{N+1}^2\right)}{\big( \left(N+1 +g_{N+1}\right)^2- \left(N +g_N\right)^2\big) \big((N+1)^2-N^2\big)}= -\frac{\delta^2(\sqrt E)}{\pi^2}.
(\kappa_{\tau}\frac{\partial}{\partial \kappa_{\tau}}+ \gamma_{\phi^2 (\tau)}^{*}(2 + t\frac{\partial}{\partial t})) \Gamma_{R (\tau)}^{(0,2)}= (\kappa_{\tau}^{-2 \tau})^{\frac{\epsilon_{L}}{2}}B_{\tau}(u_{\tau}^{*}) ,
\delta_{l s}\tilde{E}^{a}= - 2 i \tilde{E}^{\alpha}\Gamma_{\alpha \beta}^{a}\epsilon^{\beta}( x , \tilde{\theta}( x ) ) \; , \quad \delta_{l s}\tilde{E}^{\alpha}={\cal D}\epsilon^{\alpha}( x , \tilde{\theta}( x ) ) + \ldots \; \quad \ldots
Ci(x)=\gamma+lnx+\sum_{n=1}^{\infty}\frac{(-1)^{n}x^{2n}}{2n(2n)!}=\gamma+lnx-\frac{x^{2}}{2!\cdot2}+\frac{x^{4}}{4!\cdot4}\mp\cdot\cdot\cdot
k_{\gamma,\Sigma_1}^{\infty,s}=\frac{\overline{p}{\dot{\gamma}_3}+\frac{\sqrt{2}}{2}\overline{q}\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)}{|\omega(\dot{\gamma}(t))|},~~if ~~\omega(\dot{\gamma}(t))\neq 0,
\langle{T_{\k}^{j}}\rangle - \langle T_{_{\oslash}k}^{\, j}\rangle \sim{\frac{\pi M^{2}}{2}}\Phi^{2}\Big ({\frac{1}{c^{2}}}u^{j}u_{k}-{\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}\gamma_{_k}^{j}\Big ) \, \ln \Big \{{\frac{w_{\!_{\odot}}}{w}}\Big \}.
\epsilon_{RPA}({\bf{q}},{\tilde{\omega}}) = 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{\Lambda^{2}_{{\bf{k}}}({\bf{q}}) - \Lambda^{2}_{{\bf{k}}}(-{\bf{q}})}{{\tilde{\omega}}- \frac{{\bf{k.q}}}{m}}
T(z) = g_{ab}(z) \frac{\delta S}{\delta g_{ab}(z)}=\frac{\varepsilon\mu^{-\varepsilon}}{8\pi\alpha'}g^{ab}(z)\partial_ax^\mu\partial_bx^\nu G_{\mu\nu}-\frac{\mu^{-1-\varepsilon}}{4\pi\alpha'}H_{\mu\nu}\dot{x}^\mu\dot{x}^\nu \delta_{\partial M}(z)
\langle x^*-x,y-x^*\rangle+\langle x-x^*,z-x\rangle+\frac{1}{2}\big(|y-x^*|^2+|z-x|^2\big)-\langle y-x^*,z-x\rangle &=\frac{1}{2}\big(|y-z|^2+|x-x^*|^2\big)
G '' := \left\{( \xi ,x ) \in G ' : \begin{array}{r}\nu_{\left( \xi, q_\xi (x) \right)}\left( B_r (x) \cap G '_\xi \cap ( W - \ell^{ss}_{(\xi, x, W(x))})^{-1}(0) \right) \geqslant r^{\gamma_1 + \delta}\mbox{for}\forall r \in ( 0 , \varepsilon_1^{\frac{1}{1 + \alpha}})\end{array}\right\}
\widetilde{F}\left(\beta \right)=-\frac{1}{\beta}\log \widetilde{Z}\left(\beta \right)=-\frac{1}{\beta}\sum_{\ell =1}^{\infty}\ell^{2}\log \left(1-e^{-\beta \sqrt{\ell^{2}+\mu_{eff}^{2}}/r}\right).
Op_h^w\Big(\tilde{\theta}_h(x) (\xi -\sum_{l=1}^3 i_l d\varphi(x))\Big) = \frac{1}{2}hD \tilde{\theta}_h(x) + \frac{1}{2}\tilde{\theta}_h(x) hD - \tilde{\theta}_h(x)\sum_{l=1}^3 i_l d\varphi(x) & = h\frac{\tilde{\theta}'_h(x)}{2i}+ \tilde{\theta}_h(x) (hD -\sum_{l=1}^3 i_l d\varphi(x))\,,
\|x_{n+1}- x\|= \ \|(1-\lambda_n)(\beta_n x_n- x)+\lambda_n\big(T(\beta_nx_n)-T x\big)\| \leq \ (1-\lambda_n)\|\beta_nx_n- x\|+\lambda_n\|T(\beta_nx_n)-T x\| \leq \ \|\beta_nx_n- x\| = \ \|\beta_n(x_n- x)+(\beta_n-1) x\| \leq \ \beta_n\|x_n- x\|+(1-\beta_n)\| x\|.
\|x^{\frac{3}{2}}y^{\frac{1}{2}}+x^{\frac{1}{2}}y^{\frac{3}{2}}\|=\|x^{\frac{1}{2}}(x+y)y^{\frac{1}{2}}\| =\|(x+y)^{\frac{1}{2}}y^{\frac{1}{2}}x^{\frac{1}{2}}(x+y)^{\frac{1}{2}}\| \leq \|(x+y)^{\frac{1}{2}}\|^2\|x^{\frac{1}{2}}y^{\frac{1}{2}}\|=\|x+y\|\|x^{\frac{1}{2}}y^{\frac{1}{2}}\|<\delta.
\Sigma^{+}=\{e_{1}\pm e_{2}, 2e_{1}, 2e_{2}\},\W^{+}=\{e_{1}-e_{2}, e_{1}+e_{2}\}, \Pi =\{\alpha_{1}=e_{1}-e_{2}, \alpha_{2}= 2e_{2}\},\ \tilde{\alpha}=\alpha_{1}+\alpha_{2}=e_{1}+e_{2},
\frac{1 6 \pi^{2}}{g_{i , \mathrm{b a r e}}^{2}}= \frac{1 6 \pi^{2}}{g_{i}^{2}( \mu )}- b_{i}( 4 \pi )^{\epsilon}\int_{0}^{\infty}\frac{d t}{t^{1 - \epsilon}}e^{- t \frac{\mu^{2}}{M^{2}}}.
I_0 := \int_{\mathbb{R}^n}\left( |u_1(x)|^2 + |\nabla u_0(x)|^2 + |u_0(x)|^{p+1}+ |u_0(x)|^2 \right) \langle x \rangle^{2m}\,dx, m > 2\left( \frac{1}{p-1}- \frac{n}{4}\right)
F^{(\pi)}(m,T)= - T \sum_a m_a \int_{-\infty}^{\infty}\frac{d\beta}{2\pi}\left[e^{\beta}\ln\left( 1 + e^{-\epsilon_{aR}(\beta)/T}\right)+ e^{-\beta}\ln\left( 1 + e^{-\epsilon_{aL}(\beta)/T}\right)\right]
\frac{\phi^{\prime}}{\left[1-(\phi^{\prime})^2+(y^{\prime})^2\right]^{\frac{1}{2}}}=-\frac{e}{r^{p-1}},\;\;\;\frac{-y^{\prime}}{\left[1-(\phi^{\prime})^2+(y^{\prime})^2\right]^{\frac{1}{2}}}=\frac{c}{r^{p-1}}
h_k (s) = e^{-s^2/2}\sum_{m=0}^{[k/2]}d_{km}\s^{k-2m}, d_{km}= \frac{1}{\sqrt{2^k k!\sqrt{\pi}}}\frac{(-1)^m k! 2^{k-2m}}{m!(k-2m)!}& = e^{-s^2/2}\sum_{m=0}^{k}c_{km}\s^{m},
\frac{v_d^{(1)}}{v_d^{(0)}}= \sum_{a=2}^{d}(-1)^{d}a\binom{d}{a}\binom{d+a}{d}\sum_{r=1}^{a}(-2)^{r-2}&\Bigg(\sum_{b=0}^{a-r}(-1)^{b}\frac{1}{a+b}\binom{a+b}{2b+r}\binom{2b+r}{b}-\sum_{b=0}^{a-r}(-1)^{b}\frac{1}{a+b}\binom{a+b}{2b+r}\binom{2b}{b}\Bigg).
\sum_{i= 1}^4 U_{h,i}(y) \le M\sum_{i= 1}^4e^{-|y- ht_i|}\min\{|y-ht_i|^{-\left(\frac{N-1}{2}\right)},1\}&\le 4Me^{-|y- ht_1|}\min\{|y-ht_1|^{-\left(\frac{N-1}{2}\right)},1\}.
\chi_{\alpha \beta}^{\phi}( t , t^{\prime}; k ) = \int \frac{d^{3}k^{\prime}}{( 2 \pi )^{3}}g_{\alpha \beta}^{\phi}( t , t^{\prime}; k^{\prime}) g_{\alpha \beta}^{\phi}( t , t^{\prime}; k^{\prime}+ k ) ,
U^{\alpha_n(\phi)}_{nm^*}=\pi_{\alpha_n(\phi)}\big((ndn^*)nm^*(ndn^*)\big) = \iota_{n^*}(\pi_\phi(dm^*nd))=\iota_{n^*}(U^\phi(m^*n)) \in \mathcal{U}_0(D'_{A^{\delta}}/J_{\alpha_n(\phi)}).
(y^2 \!+\! u^qy)((h_0(y)/h_1(y))^2 \!\!+\! u(h_0(y)/h_1(y))) \!+\! a(y^2 \!\!+\! u^qy) \!+\! b((h_0(y)/h_1(y))^2 \!\!+\! u(h_0(y)/h_1(y))) \!+\! c = \frac{1}{h_1(y)^2}\big( (y^2 \!\!+\! u^qy)(h_0(y)^2 \!\!+\! uh_0(y)h_1(y)) \!+\! a(y^2 \!\!+\! u^qy)h_1(y)^2 \!+\! b(h_0(y)^2 \!\!+\! uh_0(y)h_1(y)) \!+\! ch_1(y)^2\big).
\phi(X_{j_1})^2 - \phi(X_{j_2})^2 = \sum_i (a_{ij_1}^2 - a_{ij_2}^2)x_i^2 + 2\sum_{i_1 < i_2}(a_{i_1j_1}a_{i_2j_1}- a_{i_1j_2}a_{i_2j_2})x_{i_1}x_{i_2}.
(f_{a'}^{(j)}v_{a'-1,j}, f_{a'}^{(j)}v_{a'-1,j}) = \frac{[1]_q \cdots [j]_q}{[j]_q \cdots [1]_q}(v_{a'-1,j}, v_{a'-1,j}) = \begin{bmatrix}m j \end{bmatrix}_q.
q_{j}^{( i )}= \frac{1}{\pi}\int_{0}^{\infty}d \omega \, \sinh \pi \omega \, f^{( i )}[ \tilde{D}_{i \omega}( k \xi , k \xi_{j}) ] - \int_{0}^{\infty}d \omega \frac{I_{\omega}( k \xi_{1}) I_{- \omega}( k \xi_{2})}{D_{\omega}( k \xi_{1}, k \xi_{2})}\frac{F^{( i )}[ D_{\omega}( k \xi , k \xi_{j}) ]}{I_{\omega}( k \xi_{j}) I_{- \omega}( k \xi_{j})},
E_n^{-1}(z) A_n^{(2)}(s) E_n(z) = n^3 \left( E_n^{-1}(z) A_n^{(1)}(0) E_n(z) \right)\left(E_n^{-1}(z) B_n^{(1)}(s) E_n(z) \right) &= n^3 \cdot \mathcal{O}(1) \cdot \mathcal{O}\left(n^{-\frac{1}{2}}\right)= \mathcal{O}\left(n^{\frac{5}{2}}\right)
\begin{aligned}E_{m,n}:= \{x \in \omega : |u_m(x)-u_n(x)| < 1 \}, e_{m,n}:= \left(|\nabla u_m|^{p-2}\nabla u_m - |\nabla u_n|^{p-2}\nabla u_n\right) \cdot \nabla (u_m-u_n).\end{aligned}
E_1 = \det T(\phi) T(\phi_R)^{-1}\cdots T(\phi_0)^{-1}, E_2 = \det T(\tilde{\phi}) T(\tilde{\phi}_R)^{-1}\cdots T(\tilde{\phi}_0)^{-1}, E_3 = \det T(\phi_{0})T(\phi_{0}^{-1}),
p\left({r\left(n\right) \leq c}\right) = p\left({d \geq \left({\sqrt{1 -{c^2}}}\right)}\right) = 1 - p\left({d \leq \left({\sqrt{1 -{c^2}}}\right)}\right) = 1 -{\left({1 -{c^2}}\right)^{n - 1}}.
{\nabla_{( - ) \, 1 2}^{2}f^{( 1 2 , - 1 2 )}= 4 D_{1 1}\bar{D}_{- 1 2}f^{( 1 2 , - 1 2 )}= \left( - 1 3 2 +{\frac{3}{4}}\right) f^{( 1 2 , - 1 2 )}}
(s^{-1}ds-g^{-1}dg)\omega_1+(s-g)\omega_2 = (s^{-1}ds-g^{-1}dg)\big{(}(s-g)\tau_1+g^{-1}dg\tau_2\big{)}+(s-g)g^{-1}dg\tau_1 =s^{-1}ds(s-g)\tau_1+{(}s^{-1}ds-g^{-1}dg{)}g^{-1}dg\tau_2 =s^{-1}ds\big{(}(s-g)\tau_1+g^{-1}dg\tau_2\big{)}.
u_{n}=\frac{1}{\lambda-\nu}\left( \left( u_{1}-\nu u_{0}\right)\lambda^{n}+\left( \lambda u_{0}-u_{1}\right) \nu^{n}+\sum_{k=1}^{n-1}\tau_{k-1}\left( \lambda^{n-k}-\nu^{n-k}\right) \right) \quad\left(B\neq0\right) , u_{n}=\frac{1}{\lambda}\left( u_{1}\lambda^{n}+\sum_{k=1}^{n-1}\tau_{k-1}\lambda^{n-k}\right) \quad\left( B=0\right) .
\begin{bmatrix}\mathbf{H}_2^H \mathbf{B}\end{bmatrix}=\mathbf{Q}_2\bar{\mathbf{R}}_2 =\begin{bmatrix}\mathbf{V}_2 \bar{\mathbf{V}}_2\end{bmatrix}\underset{\bar{\mathbf{R}}_2}{\underbrace{\begin{bmatrix}\mathbf{R}_2 \mathbf{C}\mathbf{0}_{(M-N)\times N}\mathbf{D}\end{bmatrix}}},
\alpha\cdot v=\rho^{\widetilde{v}_{1}}\sigma^{b_{1}v_{2}+b_{2}v_{3}}\tau^{b_{3}v_{2}+b_{4}v_{3}}, \ \widetilde{v}_{1}\stackrel{\mathrm{def}}{=}\mathrm{det}(\beta)v_{1}+\frac{1}{2}\left(b_{3}b_{1}v_{2}^{2}+b_{4}b_{2}v_{3}^{2}\right)+b_{2}b_{3}v_{2}v_{3}+r_{1}\left(b_{1}v_{2}+b_{2}v_{3}\right)+r_{3}\left(b_{3}v_{2}+b_{4}v_{3}\right)
\sup_{t\in[a_n,T-a_n]}\left|\frac{1}{n}\sum_{i=1}^n\Psi_h(t_{i-1}-t)-\int_{-\infty}^\infty \Psi_h(s)ds\right| &\leq\sup_{t\in[a_n,T-a_n]}\left\{\sum_{i=1}^n\int_{t_{i-1}}^{t_i}|\Psi_h(t_{i-1}-t)-\Psi_h(s-t)|ds+\int_{-\infty}^0|\Psi_h(s-t)|ds+\int_T^\infty|\Psi_h(s-t)|ds\right\}&=:\mathbb{I}_{n}+\mathbb{II}_{n}+\mathbb{III}_{n}.
\begin{array}{c}{{D^{B}H^{A}= 0 , \qquad \qquad D^{B}D^{A}\Phi_{A}= [ H^{A}, \Phi_{B}] ,}}\\{{D^{A}K^{B}= 0 , \qquad \qquad D^{A}D^{B}\Phi_{B}= [ K^{B}, \Phi_{A}] .}}\\ \end{array}
\left( \frac{2}{(1+\lambda t)^{\frac{1}{\lambda}}+1}\right)^r (1+\lambda t)^{\frac{x}{\lambda}}= \sum_{n=0}^\infty \mathcal{E}_{n,\lambda}^{(r)}(x) \frac{t^n}{n!},\,\,(r \in \mathbb{N}).
\ \int_{{\mathbb{D}}}{(g\triangle \log \left\vert{f}\right\vert -\log \left\vert{f}\right\vert \triangle g)}=\int_{{\mathbb{T}}}{(g\partial_{n}\log \left\vert{f}\right\vert -\log \left\vert{f}\right\vert \partial_{n}g)}
S_p^{' (m)}(\mathbf{e},\mathbf{r})=\sum_{\substack{\mathbf{d}\in\mathcal{D}_k' d_j|(e_j/r_j,p), d_m=1}}\frac{\mu(d)d}{\phi_L(d)}=\begin{cases}-1/(p-1), p\nmid a_mW'_j, 0, p|a_m, p\nmid W_j', 1, p|W'_j/W_j.\end{cases}
F_{\mu \nu}^{( H )}= \sum_{\jmath = 1}^{[ d / 2 ] - 1}H_{\jmath}\left( \delta_{\mu}^{2 \jmath}\delta_{\nu}^{2 \jmath - 1}- \delta_{\nu}^{2 \jmath}\delta_{\mu}^{2 \jmath - 1}\right) .
\alpha_+(F) = \liminf_{x\to +\infty}\frac{-\log \overline F(x)}{\log x}= \liminf_{x\to +\infty}\frac{-\log P_F(X>x)}{\log x}, & \alpha_-(F) = \liminf_{x\to -\infty}\frac{-\log F(x)}{\log (-x)}=\liminf_{x\to -\infty}\frac{-\log P_F(X\leq x)}{\log (-x)},
A_1 = Lh^{s-2}\cdot \int_{[0,1]^d}\frac{4g(x)^2}{(f(x)+Lh^s)^2}dx &\le 8Lh^{s-2}\cdot \int_{[0,1]^d}\frac{(g(x)-f(x))^2+f(x)^2}{(f(x)+Lh^s)^2}dx &\le 8Lh^{s-2}\cdot \left(\int_{[0,1]^d}\frac{(g(x)-f(x))^2}{L^2h^{2s}}dx + \int_{[0,1]^d}\frac{(f(x))^2}{(f(x))^2}dx\right) &= 8Lh^{s-2}\cdot \left(\frac{\|g-f\|_2^2}{L^2h^{2s}}+1\right) \lesssim Lh^{s-2}.
S = \frac{1}{4}A_{\mathrm{E H}}= \frac{1}{4}\left[ ( r^{2}C )^{\frac{\tilde{d}+ 1}{2}}D^{\frac{d - 1}{2}}\right]_{r = r_{+}}= \frac{1}{4}\Omega_{\tilde{d}+ 1}r_{+}^{\tilde{d}+ 1}\left[ 1 - \left( \frac{r_{-}}{r_{+}}\right)^{\tilde{d}}\right]^{\frac{\tilde{d}+ 2}{2 \tilde{d}}}.
{\cal R}_{( 1^{3})}\left( 1_{1 2}\right) = \left( \begin{array}{c c c c c c c c}{{1}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\pm 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\pm 1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{\pm 1}}\\ \end{array}\right) .
\begin{array}{c}{Z_{N}( + \infty ) = N \pi + \chi_{\infty}\left( N_{H}- 2 N_{S}- M_{C}- 2 \theta ( p - 1 ) M_{W}\right) +}\\{+ \pi \textrm{s i g n}( p - 1 ) M_{W}+ \displaystyle \int d x G ( x ){\cal Q}_{N}( + \infty ) + \alpha .}\\ \end{array}
\varepsilon_1({\bf a}) = \max\{a_3, a_1 - a_2 + 2 a_3\},\ \tilde{e}_1^{\max}{\bf a}= (\min\{a_1, a_2 - a_3\}, a_2, 0), &\varepsilon_2(\tilde{e}_1^{\max}{\bf a}) = a_2,\ \tilde{e}_2^{\max}\tilde{e}_1^{\max}{\bf a}= (\min\{a_1, a_2 - a_3\}, 0, 0),\{\rm and}&\varepsilon_1(\tilde{e}_2^{\max}\tilde{e}_1^{\max}{\bf a}) = \min\{a_1, a_2 - a_3\}
[\bar x]_{\mu}=\prod_{i=1}^n \left\{\dfrac{1}{(1-qt^{n-i})^{\mu_i}}\right\}(q^x;q^{-1}, t^{-1})_{\mu}= \prod_{i=1}^n \left\{\dfrac{(q^x t^{i-1};q^{-1})_{\mu_i}}{(1-qt^{n-i})^{\mu_i}}\right\}
c'_{1}=c'_{2}=\cdots=c'_{k_{i}}=i, \c'_{k_{i}+1}=c'_{k_{i}+2}=\cdots=c'_{k_{i}+k_{j}}=c'_{k_{i}+k_{j}+1}=j, \ \ldots, &\ldots, \c'_{k_{i}+k_{j}+\cdots+k_{n-1}+2}=c'_{k_{i}+k_{j}+\cdots+k_{n-1}+3}=\cdots=c'_{k_{i}+k_{j}+\cdots+k_{n}+1}=n.
u_{tt}+ u_{xxxx}+ F(u_t) + G(u) = f(x,t), (c,d) \times (0,T] & u(c,t)=0=u(d,t), t (0,T] & u_{xx}(c,t)=0=u_{xx}(d,t), t (0,T] & u(x,0)=u_0(x), u_t(x,0) = u_1(x) x (c,d).
\int_\Omega \rho^{in}\varphi(t=0)dx + \int_0^\infty \int_\Omega \rho\,\partial_t \varphi \,dx\,dt = \int_0^\infty \int_\Omega \rho( h_\alpha \varphi - \mathcal{L}_\alpha ( \varphi))dx\,dt ,
A_{1,1}= \Big\{(0,1), (1,0) \Big\}, A_{1,2}= \Big\{(0,1,1), (0,1,2), (1,0,1), (1,0,2), (1,1,0), (1,2,0) \Big\}, A_{2,1}= \Big\{(0,0,1), (0,1,0), (1,0,0) \Big\}, A_{0,3}= \Big\{(1,1,1), (1,1,2), (1,2,1), (1,2,2), (1,2,3) \Big\}
\sum_I (-1)^{\ell(I)-1}S^I \sum_{w\in S(I)}w = \sum_I (-1)^{\ell(I)-1}S^I \sum_{J\ge I}\sum_{w\in W(J)}w &=\sum_J\sum_{w\in W(J)}w\sum_{I\le J}(-1)^{\ell(I)-1}S^I
\omega^\mathrm{h}_{0k_l}= \mathop{\arg\max}\limits_{\omega \in \left[0,2\pi \right)}\frac{1}{M_{\mathrm{h}}}\left[ \sum_{m=1}^{M_\mathrm{h}}r_{mmk_l}+ \sum_{p=1}^{M_\mathrm{h}-1}\sum_{m=1}^{M_\mathrm{h}-p}r_{m\left(m+p\right)k_l}e^{-jp\omega}+ \sum_{p=1}^{M_\mathrm{h}-1}\sum_{m=1}^{M_\mathrm{h}-p}r_{\left(m+p\right)mk_l}e^{jp\omega}\right].
\psi^{out}(u, \Omega)\psi^{in}(v, \Omega')=-\{~1~-~16\pi\imath f(\Omega,\Omega')e^{(u-v)/4GM}\partial_{u}\partial_{v}~\})\psi^{in}(v, \Omega')\psi^{out}(u,\Omega)~.
\begin{array}{c}a^- a^+ -qa^+ a^-~=~q^{-N}a^{-}a^{+}-q^{-1}a^{+}a^{-}~=~q^{N}q^Na^+q^{-N}~=~qa^+ q^Na^-q^{-N}~=~q^{-1}a^- q^Nq^{-N}~=~q^{-N}q^N~=1~ \end{array}
\lim_{N \to \infty}\int \frac{1}{N}\sum_{n=1}^{N}\left(F_{1, h}\cdot F_{1, h}\circ T^k \right)(T^nx) \left(F_{2, h}\cdot F_{2, h}\circ T^{2k}\right)(T^{2n}x)d\mu(x) &= \lim_{N \to \infty}\int \left(F_{1, h}\cdot F_{1, h}\circ T^k \right)(x) \frac{1}{N}\sum_{n=1}^{N}\left(F_{2, h}\cdot F_{2, h}\circ T^{2k}\right) (T^{n}x) d\mu(x)
H^nR^m = \left(\begin{array}{cc}\lambda^n \lambda^{n-1}\alpha 0 0\end{array}\right)\left(\begin{array}{cc}\cos m\theta -\sin m\theta \sin m\theta \cos m\theta\end{array}\right) &= \left(\begin{array}{cc}\lambda^n(\cos m\theta+\alpha \lambda^{-1}\sin m\theta) \lambda^n(\alpha\lambda^{-1}\cos m\theta - \sin m\theta) 0 0\end{array}\right).
\int_{\mathbb{R}^2_+}\partial_y g_{1}\partial_x^{n - 1}v \, \langle y \rangle^{2\ell'}g_{n}= - \int_{\mathbb{R}^2_+}g_{1}\partial_x^{n - 1}v \, \langle y \rangle^{2\ell'}\partial_y g_{n}& + \int_{\mathbb{R}^2_+}g_{1}\partial_x^{n}u\, \langle y \rangle^{2\ell'}g_{n}dx dy,
\mathrm{R}_{\mathrm{D}}(\lambda, \delta ) = \langle\lambda, \delta\rangle + \sum_{1\leq \ell \leq 2}\frac{V_\ell / V_{\ell, X X}}{4 \lambda_\ell}+ \frac{1}{4}V_o^{\top}{\underbrace{\left(\lambda_1 V_{1, X X}^{-1}+\lambda_2 V_{2, X X}^{-1}\right)}_{= \Lambda_\lambda}}^{-1}V_o
\frac{(\Psi^\prime_{\ell^n}(X,Y))^2}{(\Psi^\prime_{\ell^{n-1}}(X,Y))^2}= \ell^2 X^{\ell^{2n-2}(\ell^2-1)}+ c X^{\ell^{2n-2}(\ell^2-1)-2}+ \cdot \cdot \cdot,
\begin{cases}\underline u_t\leq \underline u(t)( a_0(t)-a_1(t)\underline u(t) -a_2(t)\overline v(t)) \overline v_t\geq \overline v(t)( b_0(t)-b_1(t)\underline u(t) -b_2(t)\overline v(t)) \end{cases}
\Big|\sum_{\tau \in \acute I: \bar\tau \subset \varLambda}Ef_{\tau,j}^{\sharp}(x) \Big| \le \Big|\sum_{\tau: \bar\tau \subset \varLambda}Ef_\tau(x) \Big| + 16\max_{\tau}|Ef_\tau(x)| &+ \sum_{\tau \in \acute I: \bar\tau \subset \varLambda}|Ef_{\tau,j}^{\flat}(x)| + O(R^{-990}\sum_{\tau}\|f_\tau\|_{L^2(S)}) .
\left( \mathcal{N}+1\right)^{\beta}\mathcal{\bar{A}}\Omega_{n}=\left(\left( n+l\right)_{+}+1\right)^{\beta}\mathcal{\bar{A}}\Omega_{n}=\mathcal{\bar{A}}\left( \left( n+l\right)_{+}+1\right)^{\beta}\Omega_{n}& =\mathcal{\bar{A}}\left( \left( \mathcal{N}+l\right)_{+}+1\right)^{\beta}\Omega_{n}.
f^*Q_l = \sum_jw_{lj}P_j \bar{d_j}=\frac{d_j+w_{lj}-1}{w_{lj}},\; \; f(P_j)=Q_l \delta_l = max\{\bar{d_j}; f(P_j)=Q_l \}. \Delta = \sum_l \delta_l Q_l . M = L-\Delta .
\mathbb{E}f_\varepsilon(Z_t) = \varepsilon^3 + c_H^R\int_0^t \mathbb{E}\left[\psi_s\left(f''_\varepsilon(Z_s)(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + f'''_\varepsilon(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right]\,\mathrm{d}{s}
2 F^{\rm WK}_1(z_0,z_1) = F^{\rm even}_1(z_1) -\zeta'(-1) = \frac1{12}\log z_1, 2^g F^{\rm WK}_g(z_1,\dots,z_{3g-2}) = F_g^{{\rm even},[1-g]}(z_1,\dots,z_{3g-2}),g\geq 2.
\limsup_{t\to\infty}\| \bar{P}(t)\|_{L_\phi^2}^2 \le C\limsup_{t\to\infty}( h_2(t)^\frac12h_1(t)^\frac{a}{2-a}D(t)) \le C\limsup_{t\to\infty}( V(t)^\frac12 V(t)^\frac{a}{2-a}D(t)) &= C\limsup_{t\to\infty}( V(t)^{\kappa_0}D(t)).
\lim_{{\mathrm{\boldmath ~ \v a r e p s i l o n ~}}\to 0}\left( \frac{\mathrm{{\boldmath ~ \v a r e p s i l o n^{i}~}}\mathrm{{\boldmath ~ \v a r e p s i l o n^{j}~}}}{{\mathrm{\boldmath ~ \v a r e p s i l o n ~}}^{2}}\right) = \frac{1}{3}\delta^{i j}.
g|_\Sigma= (h^{-1}-hu^2_r)dr^2+(r^2-hu^2_\theta)d\theta^2+(r^2\sin^2\theta-hu^2_\phi)d\phi^2 &-2hu_ru_\theta drd\theta -2hu_ru_\phi drd\phi-2h u_\theta u_\phi d\theta d\phi
\begin{array}{r c l}{{{\cal L}_{\mathrm{g . f .}}}}&{{=}}&{{\displaystyle - \frac{1}{\xi}\mathrm{T r}( f f ) ,}}\\{{f}}&{{=}}&{{\partial^{\mu}G_{\mu}+ i g \xi ( \phi^{\dagger}t^{A}\phi_{0}- \phi_{0}^{\dagger}t^{A}\phi ) t^{A}.}}\\ \end{array}
\Big \{g , \partial_{\alpha_{1}\cdots \alpha_{k}}g , \partial_{\alpha_{1}\cdots \alpha_{k - 1}}A_{\alpha_{k}}^{a}\Big \}\rightarrow \Big \{g , \partial_{( \alpha_{1}\cdots \alpha_{k - 1}}A_{\alpha_{k})}^{a}, D_{\alpha_{1}\cdots \alpha_{k - 1}}^{( \Theta )}I_{\alpha_{k}}^{a}\Big \}.
\psi ( t ) = \frac{1}{2 E}\, \xi^{+}( 0 ) \xi^{-}( t ) \psi_{0}, \qquad \overline{{{\psi}}}( t ) = \frac{1}{2 E}\, \xi^{-}( 0 ) \xi^{+}( t ) \overline{{{\psi}}}_{0}.
{\cal L}_{sfield}= \frac 1 8 \int d^2 \theta~ \hbox{Tr}\overline{D{\bf G}}D{\bf G}=\frac 1 2 \left( Z^2 + J_{\mu}\cdot J^{\mu}+ i\overline{\chi}\cdot \partial\!\!\!/ \chi + i \varepsilon^{jkl}\overline{\chi}^j J\!\!\!\!/^k\chi^l+\frac 1 4 (\overline{\chi}\cdot\chi)^2\right).
f = \sum_{i=1}^{k}h_i \cdot \big( -\i \mathcal{G}_m(\psi_i, \psi'_i) , 0 , \psi_i \otimes \psi'_i - \psi'_i \otimes \psi_i , 0 , 0 ,\dots \big) \cdot \tilde{h}_i
\Phi_1^c= -\frac{F_A(\omega_B^c)}{(m_H(z))^2}\frac{1}{N}\sum_{i=1}^N \frac{1}{a_i-\omega_B^c}Q_i\,,\qquad\Phi_2^c= -\frac{F_B(\omega_A^c)}{(m_H(z))^2}\frac{1}{N}\sum_{i=1}^N \frac{1}{b_i-\omega_A^c}\mathcal{Q}_i\,,
^D E^{v_i}[\partial U_s \times [0, \mu t]] = \frac{\mu t}2 \left({^d E}^{v_t}[\partial U_{s}]+{^d E}^{u_i}[\partial U_{s}] \right) + \frac 1{\mu t}\int_{\partial U_{s}}D^2([v_t,1],[u_i,1]) d\sigma &< 24C \mu t /2+ C'\varepsilon/(\mu t) & < \delta.
H_t (u_t)= 2 \,\mbox{Re}\,\int\limits_{X_t}u_t\wedge\rho_{\omega_t}^{2,\,0}\wedge\overline{\rho_{\omega_t}^{2,\,0}}\wedge\bar\partial_t \frac{\omega_t^{n-3}}{(n-3)!}.
d_G(\mathcal{A}_\lambda+is\, I_H,\mathcal{A}_{\lambda_0}+is_0\, I_H)\leq d_G(\mathcal{A}_\lambda+is\, I_H,\mathcal{A}_{\lambda_0}+is\, I_H) + d_G(\mathcal{A}_{\lambda_0}+is\, I_H,\mathcal{A}_{\lambda_0}+is_0\, I_H).
\frac{\partial^{\beta_1}u(x,y,t)}{\partial x^{\beta_1}}=\frac{1}{\Gamma(2-\beta_1)}\frac{\partial^2}{\partial x^2}\int^x_a\frac{u(\xi,y,t)d\xi}{(x-\xi)^{\beta_1-1}}, \frac{\partial^{\beta_2}u(x,y,t)}{\partial y^{\beta_2}}=\frac{1}{\Gamma(2-\beta_2)}\frac{\partial^2}{\partial y^2}\int^y_c\frac{u(x,\xi,t)d\xi}{(y-\xi)^{\beta_2-1}},
(\nabla a,u\nabla (\eta^2 u^{p-1})) = uu^{p-1}(\nabla a,\nabla (\eta^2))+ (p-1)uu^{p-2}\eta^2(\nabla a,\nabla u) = uu^{p-1}(\nabla a,\nabla (\eta^2))+(p-1)( u^{p-1})\eta^2 (\nabla a,\nabla u) = (u^p)(\nabla a,\nabla (\eta^2))+\eta^2 (\nabla a,\nabla ( \frac{p-1}{p}u^p) ),
\dfrac{d}{dy}\int \varphi(\rho_{n_0}*\varphi)'(\cdot-y)=\int \varphi'(\rho_{n_0}*\varphi')(\cdot-y)\geq \dfrac{1}{4}e^{-1/2}\hbox{on}\left[-\tfrac{1}{2},\tfrac{1}{2}\right].
S_2 = \begin{bmatrix}35 -21 7 -1 -21 21 - 15 6 -1 7 -15 20 -15 6 -1 -1 6 -15 20 -15 6 -1 \ddots \ddots \ddots \ddots \ddots \ddots \ddots \end{bmatrix}_{N\times N},
\log \prod_{j=k}^{k'-1}\frac{p_j}{q_j}=\sum_{j=k}^{k'-1}\left\{\frac{-2G_j}{\sqrt{\beta j}}+\frac{1}{j}+\frac{2G_{j}^{(2)}}{\beta j}+O( ( \log n)^3 n^{-3/2})\right\}.
[(*,k_t)]^{r_t}(k_t,k_{t-1})\dots [(*,k_2)]^{r_2}(k_2,k_1)[(*,k_1)]^{r_1}&=[(\underbrace{*,\dots,*}_{r_t},k_t,\underbrace{*,\dots,*}_{r_{t-1}},\dots,k_{2},\underbrace{*,\dots,*}_{r_1},k_1)]+.
\frac{a^{2}}{a - 1}- \frac{1 - 2 a}{1 - a}= \frac{a^{2}}{a - 1}+ \frac{1 - 2 a}{a - 1}= \frac{a^{2}+ 1 - 2 a}{a - 1}= \frac{( a - 1 )^{2}}{a - 1}= a - 1
Z_{D_1}=\int_{\Omega}\inf_{D' \in \mathcal{H}}\left[ \exp\left(\epsilon d(D_1,D')\right) f_{\hat{\mathcal{A}}(D')}\right] d \mu &\leq \exp\left(\epsilon d(D_1,D_2) \right)\int_{\Omega}\inf_{D' \in \mathcal{H}}\left[ \exp \left( \epsilon d(D_2,D') \right) f_{\hat{\mathcal{A}}(D')}(\omega) \right] d\mu &=\exp \left( \epsilon d(D_1,D_2) \right) Z_{D_2}.
{\cal T}_{\mu / \lambda}( x ) = \mathrm{d e t ~}_{1 \le j , k \le \mu_{1}^{\prime}}( T_{\mu_{j}- \lambda_{k}- j + k}( x + i ( \mu_{1}^{\prime}- \mu_{1}+ \mu_{j}+ \lambda_{k}- j - k + 1 ) ) )
\overline{\alpha}_n \circ w_{n-1}={\phi_q^{n-1}}_{| W_n(\mathbf{E}_L)_L}\qquad w_{n-1}\circ \overline{\alpha}_n ={\phi_q^{n-1}}_{| \mathbf{A}_L / \pi_L^n \mathbf{A}_L}\ .
\int_{I_1}D_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)dy= \int_{a_1}^{b_1}D_n(z/S(I) \cdot G_n(x)/2)\frac{z}{\sqrt{4-z^2}}dz = S(I)(\ln n)^{-1}\int_{0}^{(b_1/a_1-1)\ln n}D_n((1+z/\ln n) G_n(x)/2)\frac{(1+z/\ln n)a_1}{\sqrt{4-(1+z/\ln n)^2a_1^2}}dz,
\begin{array}{lcl}\zeta_{+1}= \beta_{+1}^{-1}, \qquad \zeta_{-1}= \beta_{-1}, \zeta_{+2}= -\beta_{+1}^{-1}\mu_{+12}, \qquad \zeta_{-2}=\mu_{-21}\beta_{-1}, \zeta_{+3}= - \beta_{+1}^{-1}(\mu_{+13}- \mu_{+12}\mu_{+23}), \qquad \zeta_{-3}= \mu_{-31}\beta_{-1}.\end{array}
\sum_{n=1}^{N}a_{n}\sum_{\lambda \in (W), \atop |\lambda|= 1/R}(\lambda R)^{n}= \sum_{n=1}^{N}a_{n}\sum_{a=1}^{\Delta}e^{-2 \pi i an/\Delta}= \Delta \sum_{n=1}^{[N/\Delta]}a_{n \Delta}.
\chi_{a,\ell}(z,\tau) =\frac{\displaystyle \sum_{w\in \overline W}\mbox{sgn}(w) \;\Theta_{w(\overline{a}+\overline{\rho}),g+\ell}(z,\tau)}{\displaystyle \sum_{w\in \overline W}\mbox{sgn}(w) \;\Theta_{w(\overline{\rho}),g}(z,\tau)}\: ,
( d s )^{2}= - \frac{1}{\sqrt{\Lambda}}\left( 1 + \frac{2}{3}\delta \mathrm{c o s}R \right) \mathrm{s i n}^{2}R d^{2}T + \frac{1}{\Lambda}\left( 1 - \frac{2}{3}\delta \mathrm{c o s}R \right) d^{2}R + \frac{1}{\Lambda}\left( 1 - 2 \delta \mathrm{c o s}\right) d^{2}\Omega_{( 2 )}
\frac{1}{n}\|w_{\delta}- u_n\| \geq J(u_n) - J(w_{\delta}) =(1-s_n(\delta)) \langle J'(w_{\delta}),u_n \rangle + \delta s_n(\delta) \left\langle J'(w_{\delta}), \frac{J'(u_n)}{\|J'(u_n)\|}\right\rangle +o(\delta)
\alpha_1 =\frac{1}{2}\left(4 n^2+4 n \alpha+3 \alpha^2+4n\beta+2 \alpha\beta +3 \beta^2\right), \beta_1 =-\frac{i}{2}(\alpha -\beta ) (\alpha+\beta ) (2n+\alpha +\beta), \gamma_1 =\frac{1}{16}(2n+\alpha -\beta) (2n-\alpha+\beta) (2n+3 \alpha +\beta)(2n+\alpha +3 \beta).
\Gamma_{n + 1}^{\varepsilon^{0}. . . \varepsilon^{n}}( k^{0}, . . . , k^{n}) = \sum_{P}A_{i_{0}i_{1}. . . i_{n}}^{0 ;}E^{i_{0}i_{1}}. . . E^{i_{n - 1}i_{n}}+ o ( \omega^{d - 1})
\begin{aligned}E(x,y) =x^3+y^3-t_3x^2y^2-\frac{1-t_2}{t_3}(x^2y+y^2x)-\frac{1-t_2}{t_3}(x^2+y^2) &\quad-\frac{2-2t_2+t_2^2-t_3^2}{t_3}xy-t_3c^{(1)}_{1,1}(x+y)-t_3c^{(1)}_{1,0}\ ,\end{aligned}
\eta_{x}=\frac{1}{2k_{1}\epsilon}(\theta^{2})_{x}- \epsilon\left(\frac{2k_{2}+k_{1}^{2}}{2k_{1}}\right)(\eta^{2})_{x}-\delta^{2}\frac{k_{3}}{k_{1}}u_{0,xxx}+ \mathcal{O}(\epsilon^{2},\epsilon\delta^{2}),
q_\alpha(t) = \left(e^{- \alpha t}\dfrac{(1-e^{- \alpha t})^{\alpha -1}+e^{-\alpha t}}{(1-e^{-\alpha t})^{\alpha - 1}}\right)^{\frac{1}{\alpha}}= \dfrac{e^{-t}}{(1-e^{-\alpha t})^{1- \frac{1}{\alpha}}}\left( (1-e^{- \alpha t})^{\alpha -1}+e^{-\alpha t}\right)^{\frac{1}{\alpha}}.
\frac{|\xi|^{\frac{n-2}{2}}}{w(|\xi|)^\frac{1}{2}}\widehat{g}(\xi)=cJ_{\frac{n-2}{2}}(|\xi|)+\sum_{k\in\mathcal{K}}Y_k\left(\xi^\prime\right)J_{\frac{n-2}{2}+k}(|\xi|),
[e_1, e_1]=\alpha_1e_4+ \alpha_2e_5, [e_1, e_2]=\alpha_3e_3+\alpha_4e_4+ \alpha_5e_5, [e_2, e_1]=-\alpha_3e_3+ \beta_1e_4+ \beta_2e_5, [e_2, e_2]=\beta_3e_4+ \beta_4e_5, [e_1, e_3]=\gamma_1e_5, [e_2, e_3]=\gamma_2e_5, [e_3, e_1]=\gamma_3e_4+\gamma_4e_5, [e_3, e_2]=\gamma_5e_4+\gamma_6e_5, [e_3, e_3]=\gamma_7e_4+\gamma_8e_5.
\left( \big[\omega_{1},\omega_{2},f\big],\big[\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{f}\big]\right)_{\mathbf{H}_{\rho}}=(\Delta \omega_{1},\Delta \tilde{\omega}_{1})_{\Omega}+(\omega_{2},\tilde{\omega}_{2})_{\Omega}+\rho (\nabla \omega_{2},\nabla \tilde{\omega}_{2})_{\Omega}+(f,\tilde{f})_{\mathcal{O}}.
|\Delta_{l,1}^a \cap \Delta_{l,1}^b \cap \Delta_{c}| = |\Delta_{l,1}^a \cap \Delta_{l,1}^c \cap \Delta_{c}| = |\Delta_{l,1}^b \cap \Delta_{l,1}^c \cap \Delta_{c}| = = (q^5-q^2)(q^2-q)(q^2-q-2)(q^2-q-4).