harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D_{n}=\begin{vmatrix}0 0 \ldots 0 s_{n}0 0 \ldots s_{n}s_{n+1}\ldots \ldots \ldots \ldots \ldots 0 s_{n}\ldots s_{2n-2}s_{2n-1}s_{n}s_{n+1}\ldots s_{2n-1}s_{2n}\end{vmatrix}= (-1)^{\tfrac{n (n+1)}{2}}s_{n}^{n+1}\ .
D_{2k}:=\langle x,y\, |\, x^{2k}=1, y^{2}=1, xyx=y\rangle, Dic_k :=\langle x,y\,|\, x^{2k}=1, y^2=x^k, xyx=y\rangle, \mathbb U_k :=\langle x,y\,|\,x^{2k}=1,y^{2}=1,xyx=yx^k \rangle.
\begin{array}{l}{{R_{1 2}e_{1}e_{2}= c e_{2}e_{1}\; , \; \; R_{1 2}A_{1}R_{2 1}A_{2}+ A_{2}R_{1 2}A_{1}R_{1 2}^{- 1}= 0 \; ,}}\\{{e_{1}A_{2}= ( \pm ) R_{2 1}A_{2}R_{1 2}e_{1}\; , \; \; R_{1 2}F_{1}R_{2 1}F_{2}= F_{2}R_{1 2}F_{1}R_{2 1}\; ,}}\\{{e_{1}F_{2}= R_{2 1}F_{2}R_{1 2}e_{1}\; , \; \; R_{2 1}A_{2}R_{1 2}F_{1}= F_{1}R_{2 1}A_{2}R_{1 2}\; .}}\\ \end{array}
A^{\mu\nu}_{(a)}=iP^{\alpha,\lambda\kappa}iP^{\gamma\delta,\rho\sigma}i\int{d^4\ell\over(2\pi)^4}{\tau_{\alpha\beta}(p,p'-\ell) \tau_{\gamma\delta}(p'-\ell,p')\tau^{\mu\nu}_{\rho\sigma,\lambda\kappa}(\ell,q)\over \ell^2(\ell-q)^2((\ell-p')^2-m^2)}
\pi_i^{(1)a}=-\,\frac{1}{m^2}\,\eta^{1b}\,G_{1i}^{(0)ba}+\frac{g}{2m^2}\,f^{bcd}\,\eta^{2b}\,(D_i\pi^i)^c\,G_{1i}^{(0)da}-\eta^{2b}\,G_{2i}^{(0)ba}\,,
S^{{\rm cl}}_{\rm M}(\hat g,g)=\int_M\Big(2\pi(1-\textbf{h})\phi \Delta_g\phi+(\frac{8\pi(1-\textbf{h})}{V_{g}}-K_{g})\phi+\frac{2}{V_{\hat g}}\omega e^{\omega}\Big) d{\rm v}_{g}
H = 1 + \frac{r_{0}^{7 - p}\sinh^{2}\alpha}{r^{7 - p}}, \ \ \f = 1 - \left( \frac{r_{0}}{r}\right)^{7 - p}, \ \ \h^{- 1}= \cos^{2}\theta + H^{- 1}\sin^{2}\theta .
\begin{array}{r l}\left(X-\pi_R(X)\right)_i(x)= \int_{B_0(R)}\mathcal{G}_{i,R}(x,y)_j\overrightarrow{\Delta}_\xi X(y)^j\,dx +\int_{\partial B_0(R)}\mathcal{L}_\xi X(y)^{kl}\nu_k(y)\mathcal{G}_{i,R}(x,y)_k\,d\sigma.\end{array}
\frac{1}{\sqrt{N}\pi}=\sum_{k=0}^\infty\frac{\left(\frac{1}{2}\right)_k^3}{(k!)^3}\left(\frac{1+\alpha_N}{1-\alpha_N}k+\widehat{a}_N\right)\left(-4\frac{\alpha_N}{(1-\alpha_N)^2}\right)^k,
\begin{cases}u_t + (u-\frac{\gamma}{\alpha^2})u_{x}= -\partial_xp*(\frac{\alpha^2}{2}u_x^2+u^2 + (c_0 + \frac{\gamma}{\alpha^2})u) , t > 0, x\in \mathbb{R}u(x,0) = u_0(x), x\in \mathbb{R}.\end{cases}
\tilde{e}'_k:=\mathcal{B}({\mathbf V}_{k+1}|_{x_1=0},\psi_{k+1})-\mathcal{B}({\mathbf V}_{k}|_{x_1=0},\psi_{k})-\mathcal{B}'({\mathbf V}_{k}|_{x_1=0},\psi_{k})(\delta{\mathbf V}_k|_{x_1=0},\delta\psi_k).
d\Gamma^{\prime}=\prod_{i=1}^{N}d^{3}q_{i}^{\prime}d^{3}p_{i}^{\prime}=[1-\gamma\delta t]^{3N}\prod_{i=1}^{N}d^{3}q_{i}d^{3}p_{i}=d\Gamma[1-3N\gamma\delta t].
\begin{pmatrix}A \cdots T^{3}A\end{pmatrix}= &\begin{pmatrix}0 1 0.75 0.25 0.5625 0.4375 0.609375 0.390625 0 1 0.50 0.50 0.6250 0.3750 0.593750 0.406250 1 0 0.75 0.25 0.5625 0.4375 0.609375 0.390625 1 0 0.50 0.50 0.6250 0.3750 0.593750 0.406250\end{pmatrix}
\left\lfloor\frac{b}{rm+1}\right\rfloor = \left\lfloor\frac{p(rm+1)+qr+j-p}{rm+1}\right\rfloor, &= p + \left\lfloor\frac{qr+j-p}{rm+1}\right\rfloor, &= \begin{cases}p-1 q=0 j<p , p .\end{cases}
\left\| \nabla' w \right\|_{L^2(B_{1})}+ \left\| y_{n+1}^{\frac{1-2s}{2}}\nabla' \nabla w \right\|_{L^2(B_{1}^+)}\lesssim \left\| y_{n+1}^{\frac{1-2s}{2}}w \right\|_{L^2(B_{2}^+)}+ \left\| w \right\|_{L^2(B_{2})}.
S_{\rm eff}\left[A_{\mu}\right] = S_{\rm YM}\left[A_{\mu},A_{3}=0\right] + S_{\rm gf}\Bigl[ \int_{0}^{L}dz\, A_{\mu}^{3}\Bigr] + M^{2}\int d^{4}x\,\, \Phi_{\mu}^{\dagger}(x) \Phi^{\mu}(x) .
\begin{array}{rcll}\partial^2_{\xi_1\xi_1}V^{(i,m,j)}_1(\xi_1,x_2,t) = 0, \xi_1\in I_{h_{i,m}}(b_{i,m}), begin{align*}1mm]\partial_{\xi_1}V^{(i,m,j)}_1(b_{i,m}\pm \tfrac{h_{i,m}}{2},x_2,t) = 0, \end{array}
\|z^n\|_{L^2(D_{1}\backslash{D_{r}})}\|z^{-2-n}\|_{L^2(D_{1}\backslash{D_{r}})}\geq\|z^0\|_{L^2(D_{1}\backslash{D_{r}})}\|z^{-2}\|_{L^2(D_{1}\backslash{D_{r}})}&=\pi\dfrac{1-r^2}{r}.
a_i^{({\rm eff})}(x)={\theta_i \over L_i}- (Q +1) \tilde x_i+\epsilon_{ij}\partial_{x^j}\int \! d \vec y G(\vec x, \vec y)b^{(1)}(y) -{e \over \mu c}\epsilon_{ij}\partial_{x^j}\int \! d \vec y G_p(\vec x, \vec y)J_0(y)\,.
\mathbb{P}\left( \max_{0\leqslant N\leqslant N_{0}}\left\Vert \alpha_{n}^{(N)}\right\Vert_{\mathcal{F}}\geqslant\lambda\right) \leqslant\sum_{N=1}^{N_{0}}\mathbb{P}\left( \left\Vert \alpha_{n}^{(N)}\right\Vert_{\mathcal{F}}\geqslant\lambda\right) \leqslant N_{0}\mathbb{P}\left(\left\Vert \alpha_{n}^{(N_{0})}\right\Vert_{\mathcal{F}}\geqslant\lambda\right) .
(\Gamma^{(\kappa)}_1)^* =G_0^{(\kappa)}(G_1^{(\kappa)})^{-1}Q_1^{(\kappa)}, (\Gamma^{(\kappa)}_2)^* =G_1^{(\kappa)}(G_2^{(\kappa)})^{-1}Q_2^{(\kappa)}-G_0^{(\kappa+1)}(G_1^{(\kappa+1)})^{-1}(\Gamma_1^{(\kappa+1)})^*(\Gamma_1^{(\kappa)})^*.
Q_{56}= b_{35}^{2}\left(k \cos \! \left(\theta \right)+1\right) \theta'=c_{56}& b_{11}b_{22}b_{35}\big(\ell_1^2+\ell_2^2\big)+1=\frac{\theta'+\gamma b_{35}\big(k^{2}+2\,k\, \cos \! \left(\theta \right) +1\big)}{\theta'}=0\ .
s = (s_1, s_2, \ldots, s_n) = (s_1^p q,\ldots, s_{k-1}^p q, q, s_2^q, \ldots, s_{n_q}^q), t = (t_1, t_2, \ldots, t_n) = (t_1^p, \ldots, t_{k-1}^p, p, p t_2^q, \ldots, p t_{n_q}^q)
-{\frac{\lambda^{2}}{4}}( \overline{{\psi}}_{+}\Gamma_{\mu}\partial_{\nu}\psi_{+}\overline{{\psi}}_{+}\Gamma^{\nu}\partial^{\mu}\psi_{+}+ \overline{{\psi}}_{-}\Gamma_{\mu}\partial_{\nu}\psi_{-}\overline{{\psi}}_{-}\Gamma^{\nu}\partial^{\mu}\psi_{-})
\frac{1}{2\delta}\int_{-\delta}^{\delta}|1+ e^{\xi}r|^{-\omega}+ |1- e^{\xi}r|^{-\omega}\,d\xi = \frac{1}{2\pi i}\int_{(c)}\mathcal{H}(z,\omega)r^{-z}\frac{e^{\delta z}-e^{-\delta z}}{2\delta z}\, dz,
P_{k}(x^2) = \sum_{n=1}^{\ell-1}\frac{\mu(n)}{n^k}\exp\left({-\frac{x^2}{n^2}}\right) + \sum_{n=\ell}^{\infty}\frac{\mu(n)}{n^k}\exp\left({-\frac{x^2}{n^2}}\right) := S_1(x^2) + S_2(x^2),
\Phi = e^{-i(m_2x_2^0+p_{10}x_1^0)}\psi(\bf{x}) = e^{-i(m_1+m_2+E)X^0}e^{-i(m_1-m_2)x^0/2}e^{-iEx^0/2}\psi(\bf{x}),\ \ \ \x^0<0,
Mc_{\sigma(\chi)}(\sum_g g(\alpha)g^{-1}) =\sigma(m)|G|^{-1}\sum_{h,g}\sigma(\overline{\chi(h)}) g(\alpha)hg^{-1}&=|G|^{-1}\sigma(m)\sum_{h,g}\sigma(\overline{\chi(hg)})g(\alpha)h &=|G|^{-1}\sigma(m)\sum_h\sigma(\overline{\chi(h)})\{\sum_{g}\sigma({\chi(g)})g^{-1}(\alpha)\}h.
\mathcal{E}\left( x \right) = \int \mathrm{d}^{3}k \sqrt{\frac{2 \omega}{\left( 2 \pi \right)^{3}}}\left\{\begin{array}{c}{{\left[ c^{1}e_{1}+ c^{3}\left( e_{3}+ e_{4}\right) \right] \mathrm{e}^{- i k x}+}}\\{{\left[ c^{* 2}e_{1}+ c^{* 4}\left( e_{3}+ e_{4}\right) \right] \mathrm{e}^{i k x}}}\\ \end{array}\right\}, \quad \omega \equiv \sqrt{\overrightarrow{k}^{2}},
G_{X_{m},\chi_{\beta}}(x,y;s)= e^{-2\pi i \beta \frac{\ell - (x-y)}{m}}\sum_{j=-\infty}^{\infty}e^{-2\pi i \beta j}\int\limits_{0}^{\infty}e^{-(s+1)t}I_{|\ell+jm|}(t)dt,
{\mathcal M}_{k_1+1}(\beta_1) \,\,{}_{{\rm ev}_i}\times_{{\rm ev}_0}{\mathcal M}_{k_2+1}(\beta_2)=(-1)^{\delta}{\mathcal M}_{k_1+1}(\beta_1) \,\,{}_{{\rm ev}_1}\times_{{\rm ev}_0}{\mathcal M}_{k_2+1}(\beta_2)
\limsup \limits_{n \to \infty}\sum \limits_{k = 1}^m E\big [( S_n (\tfrac{k}{m}) - S_n (\tfrac{k - 1}{m}))^4\big] \le \limsup \limits_{n \to \infty}\sum \limits_{k = 1}^m (4C)^2\big (\tfrac{1}{m}+ \tfrac{1}{n}\big)^2 = \frac{(4C)^2}{m}.
\tilde{C}_{\alpha}^{\alpha_{0}}( p ) : = \int_{\alpha_{0}}^{\alpha}d \alpha^{\prime}e^{- \alpha^{\prime}( p \eta p + ( \varepsilon + i ) m^{2})}\quad , \qquad \varepsilon > 0 \ , \ 0 < \alpha_{0}\leq \alpha < \infty \quad ,
\begin{aligned}a_2(h)= ~4h_*(\alpha_1)\beta_1^2+2h_*(\alpha_1^2+\alpha_2)\beta_1-2h_*(\alpha_1)\beta_2+2h_*(\alpha_1\alpha_2) 2(a_2(h)+a_{1,1}(h))= ~\delta^2-\beta_1\delta+(-2h_*(\alpha_1\alpha_2)-h_*(\alpha_3) -2h_*(\alpha_1^2+\alpha_2)\beta_1 & +h_*(\alpha_1)(-4\beta_1^2+3\beta_2)) \end{aligned}
h_{K_{N,\ell,q}}(\theta) \geq N^{-1/q}\bigg(\sum_{j=1}^k\sum_{i\in\sigma_j}|\langle X_i,\theta\rangle|^q\bigg)^{1/q}\cr \geq N^{-1/q}\bigg(\sum_{j=1}^k\Big(\max_{i\in\sigma_j}|\langle X_i,\theta\rangle|\Big)^q\,\bigg)^{1/q}.
\lim_{v \to \infty}v^{1 - \alpha}\int_{0}^{\infty}\frac{C_{i}}{u^{\alpha - \alpha_{i}}}\Phi(v - u) u^{\alpha -1}du &= \lim_{v \to \infty}\frac{1}{v^{\alpha - \alpha_{i}}}v^{1 - \alpha_{i}}\int_{0}^{\infty}C_{i}\Phi(v - u) u^{\alpha_{i}-1}du &= \ 0
\sum\limits_{J=1}^{3}\Big(he^{{W}^{J}}\bar{\varphi}_{1}\big({W}^{J}\big)-\frac{1}{2}h{W}^{J}\bar{\varphi}_{1}\big({W}^{J}\big)\varphi_{1}\big({W}^{J}\big)\Big) d\bar{\tilde{v}}_{n}^{J}\wedge d\tilde{v}_{n}^{J}=0.
(1-a)^2 \left( \frac{s}{1-a(N+1)}- 1 \right)^2 < (1-a)(1-a(N+1)) \left( \frac{s}{1-a(N+1)}- 1 \right)^2 +1 -(1-a)(1-a(N+1)) &\Leftrightarrow (1-a)N \left( \frac{s}{1-a(N+1)}- 2 \right) \frac{s}{1-a(N+1)}2-a
Q_{f}= f_{\mu_{1}\dots \mu_{r}}\Pi^{\mu_{1}}\psi^{\mu_{2}}\dots \psi^{\mu_{r}}+ \frac{i}{r + 1}( - 1 )^{r + 1}f_{[ \mu_{1}\dots \mu_{r}; \mu_{r + 1}]}\psi^{\mu_{1}}\dots \psi^{\mu_{r + 1}}. \nonumber
\|\sigma_N(a)-a\| =\Big\|\int_{[-\pi,\pi]^k}F_N(t)\alpha_{e^{it}}(a)\,dt-\int_{[-\pi,\pi]^k}F_N(t)a\,dt\Big\| &\leq \int_{(-\delta,\delta)^k}F_N(t)\|\alpha_{e^{it}}(a)-a\|\,dt +2\|a\|\int_{[-\pi,\pi]^k\backslash (-\delta,\delta)^k}F_N(t)\,dt;
|R-R(r)| =\Big| \int_0^T \int_{\mathbb{R}^2}(f(\sigma_{s-}x)-f(\sigma_s(r)x))\Phi_{0,I_2}(dx) d{G_p}(s)\Big| \leq \int_0^T \int_{\mathbb{R}^2}|(f \psi_A)(\sigma_{s-}x)-(f \psi_A)(\sigma_s(r)x)|\Phi_{0,I_2}(dx)d{G_p}(s) &~~~+\int_0^T \int_{\mathbb{R}^2}|(f \psi_A')(\sigma_{s-}x)-(f \psi_A')(\sigma_s(r)x)|\Phi_{0,I_2}(dx)d{G_p}(s).
\bar{g}_1 (x) = \lim_{K \to -\infty \atop{L \to +\infty}}\bar{g}_1^{\,[K,L]}(x) & = (1-q)(\alpha_3 -\alpha_1 )\sum_{n=-\infty}^{+\infty}q^n\xi \frac{\big(q^{\lambda+n+1}\xi /x, q^{n+1}\xi\alpha_1, q^n \xi\alpha_2, q^{n+1}\xi\alpha_3; q\big)_{\infty}}{\big(q^{n+1}\xi /x, q^n \xi\beta_1, q^n \xi\beta_2, q^n \xi\beta_3 ;q\big)_{\infty}}.
\ 0: \mathcal{H}_0=-\frac{d^2}{dx^2}+V_0(x),a<x<b, \mathcal{H}_0\varphi_0(x)=\tilde{\mathcal E}_0\varphi_0(x), \ 1: \mathcal{H}_1=-\frac{d^2}{dx^2}+V_1(x),c<x<d, \mathcal{H}_1\phi_m(x)={\mathcal E}_m\phi_m(x),m=1,\ldots,
\sum_{|\mathfrak{d}| < z}\mu(\mathfrak{d})^2 h(\mathfrak{d}) G \left(\frac{\log |\mathfrak{d}|}{\log z}\right)=\mathfrak{S}\frac{c_K^\kappa (\log z)^\kappa}{\Gamma(\kappa)}\int_0^1 G(x)x^{\kappa - 1}dx+ O_{K,A_1, A_2, \kappa}\left(L G_{max}(\log z)^{\kappa - 1}\right),
\sum_{n\neq0}\frac{1}{|n|}W_{s}(n\sigma_{p}^{-1}z)\overline{V_{\overline{s}}(n\sigma_{p}^{-1}w)}= -\log\big{|}1-e^{2\pi i(\sigma_{p}^{-1}z-\sigma_{p}^{-1}w)}\big{|}^{2}+ O\big(e^{-2\pi(\Im(\sigma_{p}^{-1}z)-\Im(\sigma_{p}^{-1}w))}\big).
\dot{y}_R(t) = 2R \Im \int_{\mathbb{R}^N}\nabla \varphi \left( \frac{x+X(t)}{R}\right) \cdot \nabla Q \bar{v}dx + 2R \Im \int_{\mathbb{R}^N}\nabla \varphi \left( \frac{x+X(t)}{R}\right) \cdot \nabla v Q dx &+ 2R \Im \int_{\mathbb{R}^N}\nabla \varphi \left( \frac{x+X(t)}{R}\right) \cdot \nabla v \bar{v}dx, v =\alpha Q +h,
I_{0}(\frac{2\sqrt{2}i m\omega}{\hbar \sin \omega \tau}(r_{a}r_{b}+ \vec{x}_{a}\vec{x}_{b})^{1/2}) = \int_{0}^{2\pi}\frac{d\theta}{2\pi}exp\{\frac{2\sqrt{2}im\omega (r_{a}r_{b}+ \vec{x}_{a}\vec{x}_{b})^{1/2}}{\hbar \sin \omega\tau}\cos \theta \}
\frac{d p^i}{dt}=\,{p'}^i =\,-\frac{\partial H(u,p)}{\partial u^i}=\,-2(1-\kappa(t,\tau))\,\sum^{8N}_{k=1}\,\frac{\partial \beta^k(u)}{\partial u^i}p_k +\frac{\partial\kappa(t,\tau)}{\partial u^i}\,\left(\sum^{8N}_{k=1}\,\beta^k(u)p_k\right) = \,-2(1-\kappa(t,\tau))\,\sum^{8N}_{k=1}\,\frac{\partial \beta^k(u)}{\partial u^i}p_k.
G(y,2t-s) =(2t-s)^{-\frac{N}{\theta}}G\left(\frac{y}{(2t-s)^{\frac{1}{\theta}}},1\right) \ge\left(\frac{s}{2t}\right)^{\frac{N}{\theta}}s^{-\frac{N}{\theta}}G\left(\frac{y}{s^{\frac{1}{\theta}}},1\right)=\left(\frac{s}{2t}\right)^{\frac{N}{\theta}}G(y,s)
\widetilde{W}_i^T=-{\beta\beta_H \over (4\pi)^{D/2+1}}\Gamma\left(1-\frac D2\right)n_im_i^{D-2}\left[\int_{0}^{\infty}{dt \over t^2}\sum_{k=1}^{\infty}\eta_i^{k+1}e^{-{\beta^2 k^2 \over 4t}}\right]{\cal A}^H+P_i(\epsilon,D)~~~.
\eta_i=\frac{\eta}{1+\eta(\rho_i)}, \Phi_i=\Phi-\frac{1}{1+\eta(\rho_i)}\Phi\rho_i\otimes \eta, \omega_i^T=\frac{1}{2}d\eta_i, g_i=\eta_i\otimes \eta_i+\omega_i^T(\mathbb{I}\otimes \Phi_i),
\nabla \sigma_t=\sigma_t^{1-q}\,\left[(1-t)\, \nabla u\, u^{q-1}+t\, \nabla v\, v^{q-1}\right]\mbox{and}\frac{d}{dt}\sigma_t=\frac{1}{q}\,\sigma_{t}^{1-q}\, (v^q-u^q),
\sum_{k=0}^{n}\left( -1\right)^{k}\left( \begin{array}{c}n k\end{array}\right) \frac{\Gamma \left( z+n-k\right) \Gamma \left( z+k-1\right)}{\Gamma\left( z-k\right) \Gamma \left( z+k\right)}=\frac{\Gamma \left( n+1\right)\Gamma \left( z-1\right) \Gamma \left( 2z+n-1\right)}{\Gamma \left(z+n\right) \Gamma \left( 2z-1\right)}.
\Phi^+_{m_j,\pm(j+1/2)}:=\left(\begin{array}{c}i\,\psi^{m_j}_{j\pm1/2}0\end{array}\right),\Phi^-_{m_j,\pm(j+1/2)}:=\left(\begin{array}{c}0 \psi^{m_j}_{j\mp1/2}\end{array}\right).
\mathfrak{D}^df=\mathfrak{DD}^{d-1}f,\,\,\overline{\mathfrak{D}}^df=\overline{\mathfrak{D}}\overline{\mathfrak{D}}^{d-1}f,\,\,\mathfrak{C}^df=\mathfrak{CC}^{d-1}f,\,\,\mathfrak{D}^0f=\overline{\mathfrak{D}}^0f=\mathfrak{C}f=f.
\frac{\partial e_j}{\partial[\tilde{w}_i]_{\alpha,j}}=\frac{[\tilde{v}_i]_{j,\alpha}}{m|g|\lambda_j}\bigg[c_{i-1}+\frac{i}{m}|g|-[\tilde{v}_i\tilde{w}_i]_{j,j}-\sum_{s=0}^{m-1}\frac{m-s}{m}[\tilde{v}_s\tilde{w}_s]_{j,j}\bigg].
(\mathcal{N}\phi)(e^{it})=-\int_{t-\pi}^{t+\pi}\frac{\psi'(t)-\psi'(s)}{\tan((t-s)/2)}ds=\int_{-\pi}^{\pi}\frac{\psi'(t)-\psi'(t+u)}{\tan(u/2)}du=\int_{-\pi}^{\pi}\int_{0}^{1}\frac{u\psi''(t+\tau u)}{\tan(u/2)}d\tau du
[D(m,m j+k):D(m+1,n)]_q =q^{(2j+1)(2k-m)}[D(m,mj+m-k):D(m+1,n)]_q &=q^{(2j+1)(2k-m)}q^{4(m-k)j}[D(m,m(j-1)+k):D(m+1,n)]_q &=\begin{cases}q^{j(m j + 2k)}n=k, q^{(j + 1)(m j+ 2k - m)}n=m-k, 0 \end{cases}
F_+=\int_{\mathbb{R}^2}\Phi_+(f)v_2dv_2dv_3,~ G=\int_{\mathbb{R}^2}\mathcal{M}_{\nu}(f)v_2dv_2dv_3,~ F_{L,+}(v_1)=\int_{\mathbb{R}^2}f_L(v)v_2dv_2dv_3.
{\rm dist}(z,\mathcal{N}_\Omega(\overline{y})) ={\rm dist}(\nabla\Xi(y)\xi,\mathcal{N}_\Omega(\overline{y})) \le\|\nabla\Xi(y)\xi-\nabla\Xi(\overline{y})\xi\| \le\|\xi\|L\|y-\overline{y}\| \le L(\widetilde{\delta}+\|\overline{\xi}\|)\|y-\overline{y}\|.
{\cal A}_{n + 1}( x ) = \frac{1}{n}{\cal D}{\cal A}_{n}( x ) = \frac{1}{n !}{\cal D}^{n + 1}\ln{\cal A}_{0}( x ) m b o x{w i t h}\{\cal A}_{0}( x ) = \frac{x - 1}{x \ln x}\,
B(n,a,b,\delta) := \{t_\nu d_\mu d'_\lambda u_\sigma b_\tau^{(b,a)}\;|\;0\leq i,j\leq \min(a,b), \delta=i-j, \lambda \in P(a-j), \mu \in P(b-j), \nu \in P(i), \sigma \in P(j), \tau \in P \}.
[a,b](t)-[a-1,b+1](t) = [a,b] - ([a]-t^{a-1})([b]+t^b) = t^{a-1}[b] - [a]t^b + t^{a+b-1}= &= ([a+b-1]-[a-1]) - ([a+b]-[b]) + t^{a+b-1}= &= -t^{a+b-1}- [a-1] + [b] + t^{a+b-1}= [b] - [a-1] \ge 0.
|\mathcal{M}_{\nu}(f)-\mathcal{M}_{\nu}(g)|\cr \qquad\leq C_{\ell,u}\Big\{|\rho_{f}-\rho_{g}|+|U_{f}-U_{g}|+|\mathcal{T}_{f}-\mathcal{T}_{g}|\Big\}e^{-C_{\ell,u}|v|^2}.
\max_{\pi \in \Omega'}\hat{B}\bigl(\psi(T)\bigr)\Bigl(Z\bigl(s(T)\bigr)\Bigr) \leq \phi \leq \min_{\pi \in \Omega'}\hat{B}(\psi)\bigl(Z(s)\bigr),\V_1^{-1}(s_1)\in(\ref{eq:SOS1}),\s_{21}\in [a_{20}, a_{21}],\s_{i0}=
I_0 = \int_{{\cal M}_0}{\cal L}^0_{p+1}\equiv\int_{{\cal M}_0}({1 \over (p+1)!}E^{a_0}\wedge E^{a_1}\wedge ... \wedge E^{a_p}\epsilon_{a_0 a_1... a_p}e^{-{p-3 \over 2}\phi}\sqrt{-det (\eta_{ab}+F_{ab})}
S ={\frac{k A_{H}}{4 \ell_{P}^{2}}}+ 4 \pi{\frac{k}{\hbar}}\int_{H}\; J^{\mu \nu \lambda \rho}\; g_{\mu \lambda}^{\perp}g_{\nu \rho}^{\perp}\; \sqrt{{}_{2}g}d^{2}x .
\tilde\lambda=\lim_{k_0<k\to\infty}\frac{F^{\bf a}_{k+1}}{F^{\bf a}_k}=\lim_{k_0<k\to\infty}\frac{D^{\bf a}_{k+1}+E^{\bf a}_{k+1}}{D^{\bf a}_k+E^{\bf a}_k}=\lim_{k_0<k\to\infty}\frac{D^{\bf a}_{k+1}}{D^{\bf a}_k+E^{\bf a}_k}.
\begin{aligned}\begin{pmatrix}f_{b}e_{b}\end{pmatrix}= \begin{pmatrix}(-1)^{n}\mathrm{tr}_{\Gamma}(\cdot) 0 0 E(\cdot)|_{\Gamma}\end{pmatrix}\begin{pmatrix}\frac{\delta H}{\delta v}\frac{\delta H}{\delta\Sigma}\end{pmatrix}. \end{aligned}
\sum_{\substack{r=1 (r,n)=1}}^{\lfloor n/4\rfloor}\frac{1}{r}\equiv -3q_2(n)+\frac{3}{2}nq_2^2(n)+(-1)^{\frac{n+1}{2}}n^{\phi(n)-1}\phi_{J_{4}}^{(2-\phi(n))}(n)E_{\phi(n)-2}\pmod{n^2};
(-\Delta )^s u = \lambda |u|^{q-2}u + \frac{2\alpha}{\alpha+\beta}|u|^{\alpha -2}u |v|^{\beta},\; \; (-\Delta )^s = \lambda |v|^{q-2}v + \frac{2\alpha}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v \; in \; \Omega, u =v=0 \; on \; \partial \Omega.
\begin{array}{c}{\epsilon_{c}= k^{a b}M_{a b}+ b^{a}P_{a}+ f^{a}K_{a}+ f D ;}\\{M_{a b}= x_{a}p_{b}- x_{b}p_{a}; P_{a}= p_{a}; K_{a}= x^{2}p_{a}- 2 ( x p ) x_{a}; D = ( x p ) .}\\ \end{array}
Z_g(A_1\otimes \cdots \otimes A_K) (i,j) := \sum_{\substack{\kappa : V \to [N] \kappa(v_{\mathrm{in}})=j, \kappa(v_{\mathrm{out}})=i}}\prod_{r=1}^K A_r(\kappa(w_r), \kappa(v_r)),
u_{1}^{\prime}=\frac{u_{1}-v}{1-u_{1}v/c^{2}},u_{2}^{\prime}=\frac{u_{2}}{\gamma(1-u_{1}v/c^{2})},u_{3}^{\prime}=\frac{u_{3}}{\gamma(1-u_{1}v/c^{2})}.
\Delta_{\widetilde{K}}(t) = \widetilde{a}_0 + \sum_{\widetilde{s}> 0}\widetilde{a}_{\widetilde{s}}(t^{\widetilde{s}}+ t^{-\widetilde{s}}), \mbox{and}\Delta_{K(t)}= a_0 + \sum_{s > 0}a_s (t^s + t^{-s}),
H_k^-(x^{(k)})\star1 (-E_k(-u), E_k(v))+ F_k(u)\star E_k(v) (1, H_k(x^{(k)})) = E_k(v)\star F_k(u) (1, H_k(x^{(k)}))+H_k^+(x^{(k)})\star1 (E_k(v), -E_k(-u)).
\begin{cases}-\Delta \psi = \lambda m \psi + \lambda (p-1)U_{j,\lambda}^{p-2}\psi + \gamma \psi \mbox{in $\Omega$}, \frac{\partial \psi}{\partial \mathbf{n}}= \lambda (q-1)U_{j,\lambda}^{q-2}\psi + \gamma \psi \mbox{on $\partial \Omega$}. \end{cases}
wx^2wx^4 = w\left(2c-2-\frac{1}{12}x^4\right)wx^4 & = 2 cwx^4 - 2 x^4 - 2 wx^4 & = 2 wx^4 + x^2wx^4 + \frac{1}{12}x^4wx^4 - 2 x^4 - 2 wx^4 & = x^2wx^4
\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|s_{l,m}^{(k,i)}| \le \tilde{c}_{m}^{(k,i)}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|\sum_{l=0}^{j}g_{j-l}^{(-\alpha)}|g_{l}^{(\alpha)}| &\le 2\tilde{c}_{m}^{(k,i)}\sum_{j=0}^{n-k}|r_{n-k-j}^{(k,i)}|g_{j}^{(-\alpha)},
\hat{p}(d(x_{0},\cdot\cdot\cdot,x_{k})|y_{0},\cdot\cdot\cdot,y_{k}):=\frac{1}{N}\sum_{i=1}^{N}\delta_{(\hat{\xi}_{0,k}^{i},\hat{\xi}_{1,k}^{i},\cdot\cdot\cdot,\hat{\xi}_{k,k}^{i})}(d(x_{0},\cdot\cdot\cdot,x_{k}))
\begin{array}{c}{{s \left( \lambda , \mu \right) = \frac t{\lambda - \mu}\left( \phi \left( \lambda \right)^{- 1}- \phi \left( \mu \right)^{- 1}\right) ,}}\\{{a \left( \lambda , \mu \right) = \frac t{\lambda - \mu}\left( \phi \left( \lambda \right)^{- 1}- \phi \left( \mu \right)^{- 1}\right) .}}\\ \end{array}
\overline{{{G}}}_{\sigma}^{( n )}( \zeta^{\prime}, x^{2}\zeta_{2 n})^{\varepsilon^{\prime}\, \varepsilon_{2 n}}= \overline{{{G}}}_{\sigma}^{( n )}( \zeta_{2 n}, \zeta^{\prime})^{\varepsilon_{2 n}\, \varepsilon^{\prime}}\prod_{j = 1}^{2 n - 1}\frac{\zeta_{2 n}}{\zeta_{j}}.
\bigg|\int\frac{G_1(z) F(z)\overline{F_1(z)}}{z-w}d\nu(z) \bigg|+\bigg|\int\frac{G_1(z) H(z)\overline{F_1(z)}}{z-w}d\nu(z) \bigg| =O\Big(\frac{1}{|w|}\Big),
\{\eta^{1/2}(\breve T_2-T_2)-{\textstyle{1\over 6}}(\breve \kappa_3-\kappa_3)\eta T_1^2\}-\{(\breve \kappa_1-\kappa_1)-{\textstyle{1\over 6}}(\breve \kappa_3-\kappa_3)\}=O_p(n^{-1}).
\frac{m_{i}}{m_{j}}= \frac{\sin \left[ ( \eta_{k}- \eta_{j}) \pi \vartheta_{h}+ ( \hat{\eta}_{k}- \hat{\eta}_{j}) \pi \vartheta_{H}\right]}{\sin \left[ ( \eta_{i}- \eta_{k}) \pi \vartheta_{h}+ ( \hat{\eta}_{i}- \hat{\eta}_{k}) \pi \vartheta_{H}\right]}\; .
\mu_{2r-1,\ell}=\left\lceil\frac{\ell\cdot c+d}{24}-\dfrac{c+\ell\cdot d}{24\cdot\ell^{2r-1}}\right\rceil, \mu_{2r,\ell}=\left\lceil\frac{c+\ell\cdot d}{24}-\dfrac{c+\ell\cdot d}{24\cdot{\ell}^{2r}}\right\rceil.
\mathbb{E}\big(e^{it\xi_{L_n,E}^\omega(I)}\big) =\bigg[1+\frac{N_{L_n}^d\big[\mathbb{E}(\eta_{p,E}^{\omega}(I))[e^{it}-1]+R_{L_n}\big]}{N_{L_n}^d}\bigg]^{N_{L_n}^d}.
f(u) \! \! = \! \! \prod_{j=1}^{N}\frac{\sin (u-\lambda \theta_{j})}{\sin (u-\lambda \theta_j-\mu)},\qquad Q(u) \! \! = \! \! \prod_{k=1}^{N/2}\sin \left( u- \frac{\mu}{2}-\lambda\alpha_k \right)
\left\{\stackrel{\wedge}{\psi}^{\dagger},\stackrel{\wedge}{\psi}\right\}=1,\quad\left\{\stackrel{\wedge}{\psi},\stackrel{\wedge}{\psi}\right\}=\left\{\stackrel{\wedge}{\psi}^{\dagger},\stackrel{\wedge}{\psi}^{\dagger}\right\}=0
|k_{x_i}(s,p,(\widehat x_i,y))| \,d\mu_{H_i}(\widehat x_i,y) \leq C \prod_{\stackrel{j=1}{j\neq i}}^{n_0}x_jx_j^{-1}\,dx_j\prod_{k=n_0+1}^n x_k^{b_k-1}\,dx_k\prod_{l=1}^mdy_l &\leq C \prod_{k=n_0+1}^n x_k^{b_k-1}\,dx_k \prod_{\stackrel{j=1}{j\neq i}}^{n_0}\,dx_j\prod_{l=1}^mdy_l,
g(x,v) = \int_0^1 \frac{e^{-(1-y)/ \kappa |v|}}{\kappa |v| (1-e^{-1/\kappa |v|})}\rho_g (x-y) \left( \alpha\mathcal{M}_{T(x-y)}(v) + (1-\alpha) \mathcal{M}_{\tau(x-y)}(v) \right) \mathrm{d}y.
v_{t}(x,t)= w_{t}(x,t)-\int_0^L\eta(y,t)[k_{y}(x,y)+k_{yyy}(x,y)]dy-\int_0^Lw(x,y)[s_{y}(x,y)+s_{yyy}(x,y)]dy &- \int_0^Ls_{y}(x,y)\eta(y,t)w(y,t)dy-\frac12\int_0^Lk_{y}(x,y)w^2(y,t)dy, v_{x}(x,t)= w_{x}(x,t)-\int_0^Lk_x(x,y)w(y,t)dy-\int_0^Ls_x(x,y)\eta(y,t)dy,
\overline{(\Delta n)^2}=\frac{n \, d \, (n+d)}{\left (d-n \,\displaystyle{\frac{\partial d}{\partial n}}\right )^2 -n \, d\,(n+d) \, \displaystyle{\frac{\partial^2 d}{\partial n^2}}\,\log\frac{n+d}{d}}\ \ .
E(k)=\frac{\pi}{2}(1-(\frac{1}{2})^{2}\frac{k^{2}}{1}-(\frac{1\cdot3}{2\cdot4})^{2}\frac{k^{4}}{3}-\cdot\cdot\cdot-(\frac{(2n-1)!!}{(2n)!!})^{2}\frac{k^{2n}}{2n-1}-\cdot\cdot\cdot).
m_1 =-\frac{1}{3}\bigg[\mu+\left(\mu+\sqrt{3}\mu^{1/2}-\frac{3}{2}+\frac{3\sqrt{3}}{8}\mu^{-1/2}+O(\mu^{-3/2})\right) &+\left(\mu-\sqrt{3}\mu^{1/2}-\frac{3}{2}-\frac{3\sqrt{3}}{8}\mu^{-1/2}+O(\mu^{-1})\right)\bigg] &=-\frac{1}{3}\left(3\mu-3+O(\mu^{-1})\right) &=-\mu+1+O(\mu^{-1}),
\int_{\mathbb R^2}F_0(x-\phi^{(l)}(t,0)) W(t,x) dx = \int_{\mathbb R^2}F_0(\phi^{(l)}(t,x)-\phi^{(l)}(t,0) ) \omega_0^{(h)}(x) dx = \sum_{M\le j\le M+\sqrt M}\int_{\mathbb R^2}F_0(2^{j}\tilde \phi^{(l)}(t,2^{-j}x)) (\Delta a_0)(x) dx,
b(u_h^i, q_h) - \delta_0\mathcal{C}_h(p_h^i, q_h) = 0, \forall q_h \in Q_h, b(\hat{u}_h^i, q_h) - \delta_0\mathcal{C}_h(\hat{p}_h^i, q_h) = b(u^i, q_h) =0, \forall q_h\in Q_h,
\left[ - \frac{d^{2}}{d z^{2}}+ \frac{\lambda}{2}\phi_{\mathrm{w a l l}}^{2}( z ) + \frac{\sigma}{4 !}\phi_{\mathrm{w a l l}}^{4}( z ) \right] \chi_{n}( z ) = \eta_{n}^{2}\chi_{n}( z ) ,
\dot \rho_1+m_{12}=(\Sigma_1-\Sigma_2)\theta_{12}(\rho)\dot W^{\delta}, \dot \rho_2+m_{21}+m_{23}=(\Sigma_2-\Sigma_1)\theta_{21}(\rho)\dot W^{\delta}+(\Sigma_2-\Sigma_3)\theta_{23}(\rho)\dot W^{\delta}, \dot \rho_3+m_{32}=(\Sigma_3-\Sigma_2)\theta_{32}(\rho)\dot W^{\delta}.
u_{ij}(t+1;x_1,\ldots,x_k;y_1,\ldots,y_k) &= \sum_{M\in\mathcal{E}}\rho(0,M)\left[u_{ij}(t;x_1-m_{1,\cdot},\ldots,x_k-m_{k,\cdot};y_1-m_{\cdot,1},\ldots,y_k-m_{\cdot,k}) + m_{ij}\right].
\begin{matrix}f_{1}=f_{2}=f_{3}&&e_{2}=e_{4}=e_{7}\\e_{1}=e_{2}+e_{3}&&f_{2}=f_{4}+f_{7}\\ &&\\e_{3}=e_{5}=e_{6}&&f_{7}=f_{6}=f_{8}\\f_{3}=f_{5}+f_{6}&&e_{7}+e_{6}=e_{8}\end{matrix}
\vec{\Gamma}_\nu \partial_\mu \vec{J}^\mu = - \vec{\Gamma}_\nu \vec{q}\, \frac{d \Lambda}{d q_0}= - \partial_\nu \alpha \sin \alpha \, \frac{d \Lambda}{d q_0}= \partial_\nu q_0 \, \frac{d \Lambda}{d q_0}= \partial_\nu \Lambda .
\begin{array}{l}{{\lbrack q^{-}, \alpha_{0}^{+}] = i \, ,}}\\{{\left[ \alpha_{n}^{-}\, , \alpha_{m}^{+}\right] = n \, \delta_{n + m , 0}\, \, \, ,}}\\{{\left[ s_{n}\, , s_{m}\right] = \left( \frac k 2 - 1 \right) \, n \, \delta_{n + m , 0}\, \, \, ,}}\\{{\left[ L_{n}^{\prime}\, , L_{m}^{\prime}\right] = ( n - m ) L_{n + m}^{\prime}}}\\ \end{array}
\stackrel{\circ}{T}{^a}=\stackrel{\circ}{D}e^a\equiv de^a+\stackrel{\circ}{\omega}{_{\,\cdot b}^{\,a}}\wedge e^b,\;\;\;\;\;\stackrel{\circ}{R}{^{ab}}=d\stackrel{\circ}{\omega}{^{ab}}+\stackrel{\circ}{\omega}{^a_{\cdot c}}\wedge\stackrel{\circ}{\omega}{^{cb}},
S_d(\delta_k \otimes c \otimes \delta_n \otimes b) = \delta_{k+1}\otimes c \otimes \delta_n \otimes b \mapsto \alpha_d^{k+1}(c)\cdot \delta_{(n,k+1)}\otimes b = S_d \alpha_d^k(c) S_d^* \cdot \delta_{(n,k+1)}\otimes b = S_d\left( \alpha_d^k(c) \cdot \delta_{(n,k)}\otimes b \right)
d_{k+2}= d_{k+1}(d_kd_{k+1}^{-1})^{r}&= d_{k+1}(d_k ((d_{k-1}d_k^{-1})^r)^{-1}d_k^{-1})^r &= d_{k+1}((d_{k-1}d_k^{-1})^{-1})^{r^{2}}&= d_{k+1}((d_0 d_1^{-1})^{(-1)^k})^{r^{k+1}}.
\begin{aligned}\left |\zeta'_B(0;a,1,1)-\left(\frac 1{12}-\zeta'_R(-1)\right)\frac 1 a +\frac 1 4 \log (2\pi)-\frac{\gamma a}{12}-\sum_{k=2}^{N-1}\frac{B_{2k}\zeta_R(2k-1)}{2k(2k-1)}a^{2k-1}\right| \leq \frac{|B_{2N}\zeta_R(2N-1)|}{2N(2N-1)}a^{2N-1},\end{aligned}
\sup_{f_1,f_2,f_3,F,G,H}\inf_{W\in C}\left(\mathcal{I}_2(W_0, f_1, f_2, f_3)+\mathcal{I}_1(W,F,G,H)\right) &=\inf_{W\in C}\sup_{f_1,f_2,f_3,F,G,H}\left(\mathcal{I}_2(W_0, f_1, f_2, f_3)+\mathcal{I}_1(W,F,G,H)\right)
\frac{F}{V}= - \sum_{\stackrel{n = 1}{n \mathrm{o d d}}}^{\infty}\sum_{k = 1}^{\infty}\frac{2}{k}\left( \frac{1}{4 \pi^{2}\alpha^{\prime}\tau_{2}}\right)^{5}e^{- n^{2}\beta^{2}/ 4 \pi \alpha^{\prime}\tau_{2}}\sum_{M}e^{- \pi \alpha^{\prime}\tau_{2}M^{2}}