harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{a b}= \frac{1}{6}F_{a c d e f}F_{b}^{~ c d e f}+ \frac{1}{8}\left( G_{a c d}G_{b}^{* c d}+ G_{a c d}^{*}G_{b}^{~ c d}- \frac{1}{6}g_{a b}G_{c d e}G^{* c d e}\right) ,
\frac{N}{\sqrt{c^*}}=\int_{\mathbb{R}}(g+\frac{b}{\sqrt{c^*}})_+\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}g^2}dg =\int_{-\lambda}^{+\infty}(g+\lambda)\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}g^2}dg &=\frac{1}{\sqrt{2\pi}}e^{-\lambda^2/2}+\lambda\int_{-\lambda}^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-g^2/2}dg.
\Delta^{(-2)}_{\mu}=\begin{cases*}\frac{1}{16}\mu (3 \mu + 8) if \(\mu \le -2\), \frac{1}{16}(\mu + 4) (3 \mu + 4) if \(\mu \ge -2\)\end{cases*}\Delta^{(1)}_{\mu}=\begin{cases*}\frac{1}{16}(\mu - 4) (3 \mu - 4) if \(\mu \le 2\), \frac{1}{16}\mu (3 \mu - 8) if \(\mu \ge 2\),\end{cases*}
\|B_{4,M_n,n}\|^2_{L^2(\Omega)}=\sum_{p=0}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}\widetilde{H}_{np}(s)ds\Bigg]^2 \le T^2 C_2^2(\gamma, \widetilde E_2) \frac{\sum_{p=0}^{M_n}e^{2(T-t)p^{2\beta}}}{n^{2\gamma}}.
U\bigl(A_i^j(b,c)\bigr) = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i-j)!_{qp}}U\bigl(U^j(b)\alpha^j\bigl(U^{i-j}(c)\bigr)\bigr) & = \frac{p^{(j-i)j}\chi(h)^j}{(j)!_{qp}(i-j)!_{qp}}\Bigl(q^j U^j(b)\alpha^j\bigl(U^{i+1-j}(c)\bigr) + p^{j-i}\chi(h) U^{j+1}(b)\alpha^{j+1}\bigl(U^{i-j}(c)\bigr)\Bigr)
\alpha_{- 1}^{\mu}\bar{\alpha}_{- 1}^{2 5}| \Omega ; p > , \alpha_{- 1}^{2 5}\bar{\alpha}_{- 1}^{\mu}| \Omega ; p > , \alpha_{- 1}^{2 5}\bar{\alpha}_{- 1}^{2 5}| \Omega ; p > \ ,
\Theta \, \cot \left( \Theta \left[ \ln \left( \frac{z}{2}\right) + \gamma \right] \right) \stackrel{( a \rightarrow 0 )}{=}\tilde{k}a \, \frac{J_{l}^{\prime}( \tilde{k}a )}{J_{l}( \tilde{k}a )}\, \left[ 1 + O \left( [ \kappa a ]^{2}\right) \right] \; .
\zeta(-1) = 2^{-1}\pi^{-2}(-1)\Gamma(2)\zeta(2) = \frac{1}{2}\cdot\frac{1}{\pi^2}\cdot(-1)\cdot 1\cdot\frac{\pi^2}{6}= -\frac{1}{12}\; , \intertext{i.e.,}\zeta_R(-1) = -\frac{1}{12},
g^{(1)}=\frac{1}{\mu \cosh(\rho)}\left[\begin{array}{cc}e^{u_0}(\mu^2e^{\rho}+\alpha^2e^{-\rho}) \alpha^2-\mu^2 \alpha^2-\mu^2 e^{-u_0}(\alpha^2e^{\rho}+\mu^2e^{-\rho})\end{array}\right],
\int_{1}^{\infty}x^{-1-1/\beta}w(x^{-1/\beta}, \beta, 1)dx = \int_{1}^{\infty}x^{-1-1/\beta}Cdx = C \left(\lim_{K \rightarrow \infty}K^{-1/\beta}- 1^{-1/\beta}\right) = C.
d s^{2}\approx \left[ 1 - \left( \frac{4 \pi G_{D}- 2 | \Lambda |}{d ( p + d - 1 )}- \frac{\hat{R}}{d ( p + 1 )}\right) r^{2}\right] \hat{g}_{\mu \nu}( x ) d x^{\mu}d x^{\nu}+ \frac{d r^{2}}{1 + \bar{B}_{2}r^{2}}+ r^{2}d \Omega_{d - 1}^{2},
\mathbb{P}(\mathcal{A}^c) = \P \left ( \sup_{\theta\in \Theta_T}|\left \langle \phi_T(\theta),w_T \right \rangle_{L^2(\lambda_T)}| \geq \kappa\right ) \leq 3 \cdot \left ( \frac{\sqrt{g_\infty}|\Theta_T|}{\sigma_T\tau \sqrt{\log (\tau)}}\vee \frac{1}{\tau}\right ).
f = -\Delta_\zeta a - \nabla_\zeta(\nabla\beta\cdot a) - \nabla\alpha\times \nabla_\zeta\times a - i\omega\sigma\mu a + i\omega \sigma_0 \mu_0 a & = -\Delta a - \nabla(\nabla\beta\cdot a) - \nabla\alpha\times \nabla\times a - i\omega (\sigma\mu - \sigma_0 \mu_0) a- 2i \zeta\cdot \nabla a - (\nabla\beta\cdot a) i\zeta - \nabla\alpha \times (i\zeta \times a).
\left\|T\left(\sum_{j\in \mathbb{S}_{\mathcal{X}}}c_jx_j \right)\right\|^p=\left\|\sum_{j\in \mathbb{S}_{\mathcal{X}}}c_jy_j \right\|^p\leq b\sum_{j\in \mathbb{S}_{\mathcal{X}}}|c_j|^p\leq \frac{b}{a}\left\|\sum_{j\in \mathbb{S}_\mathcal{X}}c_jx_j \right\|^p.
\mathcal Q(z) = \frac 1 2 \rho_z^2\vert \Lambda \vert \widehat g(0) + \frac{1}{2}\sum_{k \neq 0}\Big( \sqrt{k^4 + 2 k^2 \rho_z \widehat g(k)}- k^2 - \rho_z \widehat g(k) \Big) \quad+ \frac 1 2 \rho_z^2\vert \Lambda \vert \widehat{g\omega}(0) + \mathcal K^{\rm{diag}}.
\mathrm{LC}_{s=1}^r [\mathcal{G}_{D,0}(z,s)]^{\mathrm{hol}}= -\mathrm{LC}_{s=1}^r\bigg[2^{s-2}\pi^{-\frac{s}{2}-1}|D|^{\frac{s}{2}}L_D(s) \Gamma\biggl(\frac{s}{2}+1\biggr)\bigg] + \sum_{0 > d \equiv 0, 1 (4)}\mathrm{Tr}_{d,D}(F_{0,0,r}) q^{-d}.
\mathbf{\tilde{U}_n^{*}}= \mathbf{\tilde{M}_{0n}^*S_n^*}+ (\mathbf{{S^*_n}}' \mathbf{\tilde{M}_{1n}^*S_n^*},\ldots, \mathbf{{S^*_n}}'\mathbf{\tilde{M}_{pn}^*S_n^*})'
\tau^*_v=\frac1{\tau_v},\tau^*_u=\begin{cases}\tau_u(1+\tau_v)^{\#(u,v)}\quad, \tau_u\frac{\tau_v^{\#(v,u)}}{(1+\tau_v)^{\#(v,u)}}\qquad, \tau_u\qquad\qquad\qquad\quad,\end{cases}
\Big \langle \int_{y \in Q_j}\int_{x \in Q_k}|\nabla_x G(x,y)|^2 \,dx \,dy \Big \rangle = \Big \langle \int_{y \in Q_0}\int_{x \in Q_{k-j}}|\nabla_x G(x,y)|^2 \,dx \,dy \Big \rangle.
RHS =t^{n(\lambda)}\prod_{b=(r,c)\in \lambda}\frac{1-q^{\mathrm{coarm}_\lambda(b)}t^{n-\mathrm{coleg}_\lambda(b)}}{1-q^{\mathrm{arm}_\lambda(b)}t^{\mathrm{leg}_\lambda(b)+1}}=t^{n(\lambda)}\prod_{r=1}^n \prod_{c=1}^{\lambda_r}\frac{1-q^{c-1}t^{n-(r-1)}}{1-q^{\lambda_r-c}t^{\lambda_c'-r+1}}
(Lw)(z) = h x^4\int_{S_zM}w(\gamma_{x,y,\lambda,\omega})g_{\mathrm{sc},\mathrm{1c},h}(\dot{\gamma}_{x,y,\lambda,\omega}(0)) \otimes g_{\mathrm{sc},\mathrm{1c},h}(\dot{\gamma}_{x,y,\lambda,\omega}(0)) d\lambda d\omega,
\left(\begin{matrix}x=0 \, \left( \frac12 \right) x=t_1 x=\infty \, \left( \frac12 \right) \overbrace{\begin{matrix}\sqrt{t_2}0 -\sqrt{t_2}0\end{matrix}}\begin{matrix}0 \theta^{t_1}\end{matrix}\overbrace{\begin{matrix}1 \theta^\infty_1/2 -1 \theta^\infty_1/2\end{matrix}}\end{matrix}\right),
\lambda_{k}= \lambda_{1}(p,\Omega^{+}_{k})\ge\frac{C_{n,F}}{|\Omega^{+}_{k}|^{\frac pn}},\lambda_{k}= \lambda_{1}(p,\Omega^{-}_{k})\ge\frac{C_{n,F}}{|\Omega^{-}_{k}|^{\frac pn}}.
\exp \left\{\mbox{$\cal{F}$}_{A_{2N-1}}(t ; \hbar) \right \}=\langle 0 | \exp \left\{\sum_{k \not\equiv 0 ~mod2N ,~ k \geq 1}\frac{t_k}{\hbar}J_k \right\}g_0| 0 \rangle ,
\Psi_2'(0)= 2z^a\cdot\frac{d}{d x}\bigg({}_2F_1\bigg[\begin{matrix}-a -a-x &1\end{matrix}\bigg|\,\frac1z\bigg]\bigg)\bigg|_{x=0}= 2z^a\cdot\frac{d}{d x}\bigg(\sum_{k=0}^a\frac{(-a)_{a-k}(-a-x)_{a-k}}{(1)_{a-k}^2}\cdot\frac{1}{z^{a-k}}\bigg)\bigg|_{x=0}.
P\Big(\widehat{\zeta}_t^M+\widehat{g}_t^M>0t\geq 0\Big|\widehat{\zeta}_0^M=n, \widehat{g}_0^M=m\Big) &=1-\Bigg(\frac{1+\delta+\gamma}{\frac{(2d-M)\lambda}{2d}\gamma}\Bigg)^n\Bigg(1-\frac{\frac{(2d-M)\lambda}{2d}\gamma-(1+\delta+\gamma)}{\frac{(2d-M)\lambda}{2d}(\gamma+1)}\Bigg)^m.
\sum_{j\geq 0}\frac{f(j) e^{-sj}\left(\sum_{l=j+1}^{\infty}f(l) e^{-sl}\right)^{k-1}}{\left(\sqrt{1-e^{-s}}-\sum_{l=j+1}^\infty f(l) e^{-sl}\right)^k}\sim\ln\frac1{\sqrt s}
\int_0^\infty\, J^2_\nu(a\, t)\t^{-\beta}\, dt \ =\ \frac{a^{\beta-1}\, \Gamma(\beta)\ \Gamma(\nu +\frac{1-\beta}{2})}{2^\beta\, \Gamma^2(\frac{1+\beta}{2})\ \Gamma(\nu +\frac{1+\beta}{2})}\, , 2\nu+1 > \beta>0.
E_2(x)=\sum_{x\leq p\leq 2x}\frac{1}{p}\sum_{\substack{0< a< p gcd(n,p-1)=1}}e^{\frac{i2\pi a(\tau^{n}-u)}{p}}\cdot \frac{1}{p}\sum_{\substack{0< b< p \gcd(m,p-1)=1}}e^{\frac{i2\pi b(\tau^{em}-v)}{p}}\ll x^{1-2\varepsilon}.
\sum_{i=1}^{n-1}\frac{i}{(v+i)^3}= \int_1^n \frac{x}{(v+x)^3}\dx+ O\left(\frac{1}{(v+1)^2}\right)+ O\left(\frac{1}{n^2}\right) &= \frac{1}{2}\left[\frac{(v+2)}{(v+1)^2}- \frac{(v+2n)}{(v+n)^2}\right]{+}\, O\left(\frac{1}{(v+1)^2}+ \frac{1}{n^2}\right){=}\, O\left(\frac{1}{v+1}\right){+}\, O\left(\frac{1}{n}\right),
\phi_k\in \mathcal S:=\Big\{\psi\in H_{loc}^1(\mathbb R_+, \mathbb C^2)\cap L_r^2(\mathbb R_+, \mathbb C^2)\,:\, \int_0^\infty\Big[\,|\psi'|^2+\frac{1}{r^2}|\psi|^2\,\Big]r\,{\mathrm d}r<+\infty\Big\}.
U^a_b = e^a_\mu (x) \, \frac{\partial x^\mu}{\partial q^\alpha}\, e^\alpha_b (q) = \left(\begin{array}{ccc}\cos \theta \cos \phi - \sin \phi \sin \theta \cos \phi \cos \theta \sin \phi \cos \phi \sin \theta \sin \phi - \sin \theta 0 \cos \theta\end{array}\right).
\left \lbrace \begin{array}{ll}\mathrm{div}\, T(\omega, Du )= 0 \mathrm{in}\{\mathbb{S}}[\Omega_Q]^-\,, u(x+qe_j) =u(x) +Be_j \textrm{$\forall x \in \overline{\mathbb{S}[\Omega_Q]^{-}}, \forall j\in \{1,\dots,n\}$}, T(\omega,Du(x))\nu_{\Omega_Q}(x)+ b[k](x)u(x)=g(x) \textrm{$\forall x \in \partial \Omega_Q$}\,. \end{array}\right.
g(x,t)=i\kappa\breve{M}^{r(1)}(x,t,i\kappa)-c_1(x,t)\breve{M}^{r(2)}(x,t,i\kappa), h(x,t)=i\kappa\breve{M}^{r(2)}(x,t,0)+c_0(\xi)\breve{M}^{r(1)}(x,t,0),
{\frac{[ a^{\prime}]_{0}}{a_{0}b_{0}}}= -{\frac{8 \pi G_{5}}{3 c^{4}}}\left[ \sigma + \varrho_{m}c^{2}( 1 + \zeta e^{- 2 \psi}) + \varrho_{r}c^{2}e^{- 2 \psi}+ \varrho_{\psi}c^{2}\right] ,
P_{1}:= x(1-x)\partial_x^2+y(1-x)\partial_x \partial_y+\{\gamma-(\alpha+\beta+1) x\}\partial_x -\beta y \partial_y - \alpha \beta, P_{2}:= y(1-y) \partial_y^2 +x(1-y) \partial_x \partial_y +\{\gamma-(\alpha+\beta'+1) y\}\partial_y -\beta' x \partial_x -\alpha \beta'
T_2 = (\widetilde{\boldsymbol{u}}_h^n \cdot \nabla \boldsymbol{\eta}_u^n, \widetilde{\boldsymbol{e}}_u^n)= - (\widetilde{\boldsymbol{u}}_h^n \cdot \nabla \widetilde{\boldsymbol{e}}_u^n, \boldsymbol{\eta}_u^n) - ((\nabla \cdot \widetilde{\boldsymbol{u}}_h^n)\widetilde{\boldsymbol{e}}_u^n, \boldsymbol{\eta}_u^n) =: T_{21}+T_{22}.
|\psi (t)-\psi (u)| =1-\psi (t) =1-\dfrac{t-a_{2n-1}}{b_{2n-1}-a_{2n-1}}=\dfrac{b_{2n-1}-t}{b_{2n-1}-a_{2n-1}}\leq \dfrac{u-t}{b_{2n-1}-a_{2n-1}}\leq \dfrac{t^{1/k}}{b_{2n-1}-a_{2n-1}}<\dfrac{1}{2^{2n-1}}.
\frac{E_{0}\left( \ell ,\mu ,d\right)}{L^{d-1}}\approx -\frac{C\left( d\right) \, \left( 2^{d}-1\right) \Gamma \left( d\right) \zeta_{R}\left( d+1\right)}{2^{3d-1}\pi^{d/2}\Gamma \left( \frac{d}{2}\right) \, \ell^{d}}.
\mathbb{E}_Y\{(\theta_j - \theta)Z_j\}= \mathbb{E}_{Y}\{(\theta_j - \theta)s(\theta_j - Y_j)\}= \mathbb{E}_Y \{(\theta_j - \theta) \mathbb{E}[ s(\theta_j - Y_j)| \theta_j]\}= \mathbb{E}_Y[(\theta_j - \theta) \{\mathbb{P}_{Y_j|\theta_j}(Y_j < \theta_j) - \mathbb{P}_{Y_j|\theta_j}(Y_j > \theta_j)\}].
i\hbar\dot{\varrho}_a = \big[\widehat{H}_e(q_a),{\varrho}_a\big]+\frac{\hbar^2}{2M}\sum_{b}\left[{\varrho_b},{\varrho}_a\right]{\int \frac{K'(r-q_a)K'(r-q_b)}{\sum_{c}w_c K(r-q_c)}\,r}\,,
X_i :=\{x: c(g_i^{-1}, x)=h_0^{-1}\},X_j:=\{x: c(g_j^{-1}, x)=h_1^{-1}\}, X_k :=\{x: c(g_k^{-1}, x)=h_2^{-1}\},X_s:=\{x: c(g_s^{-1}, x)=h^{-1}\}, X_t :=\{x: c(g_t^{-1}, x)=\bar{h}^{-1}\}.
G_{\mu\nu}(q) = \frac{-i}{q^2+i\epsilon}\left[ g_{\mu\nu}-\frac{q\cdot N^{\ast}(q_{\mu}N_{\nu}+q_{\nu}N_{\mu})}{q\cdot N q\cdot N^{\ast}+i\epsilon}+ \frac{N^2(q\cdot N^{\ast})^2q_{\mu}q_{\nu}}{(q\cdot N q\cdot N^{\ast}+i\epsilon)^2}\right].
\hat{\hat{k}}_{( m )}{}^{\hat{\hat{\mu}}}\partial_{\hat{\hat{\mu}}}= \partial_{m}\rightarrow \hat{\hat{k}}_{( m )}{}^{\hat{\hat{\mu}}}\hat{\hat{k}}_{( n )}{}^{\hat{\hat{\nu}}}\hat{\hat{g}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}= G_{n m},
(\mathcal{I}_{N/K}/\mathcal{I}_{K})\bigcap\ker(N_{N/M})=\bigoplus_{i=s_2+1}^{s_2+s_4}\,\left(\mathbb{F}_5\,\mathfrak{K}_1^{(\ell_i)}\oplus\mathbb{F}_5\,\mathfrak{K}_2^{(\ell_i)}\right)\,,
\cfrac{1}{a_n}- \cfrac{1}{a_n^0}= \cfrac{1}{\frac{\pi}{2}+ s_n}- \cfrac{1}{\frac{\pi}{2}}= - \cfrac{2}{\pi}\cdot \cfrac{s_n}{1 + \frac{\pi}{2}s_n}= - \cfrac{2}{\pi}\cdot s_n + q_n,
\begin{aligned}0 =\bold{E}^{\vartriangle}(L(m,n,[u])) &=u_{-1,0}-u_{0,1}-\frac{a(m-1)-b(n)}{u_{0,0}-u_{-1,1}}-\left(u_{0,-1}-u_{1,0}-\frac{a(m)-b(n-1)}{u_{1,-1}-u_{0,0}}\right).\end{aligned}
\overline{Ric}_{\rho\rho,\rho}=0, &-\frac{1}{2}\overline{R}_{\rho\rho,kk}=\frac{R_{,ii}}{4(1-n)}, &-\frac{1}{12}\overline{Ric}_{\rho\rho,\rho\rho}-\frac{1}{6}(\overline{R}_{\rho i\rho j})^2=\frac{\hat{h}^{ij}Ric_{ij}[h_{\rho}]_{,\rho\rho}|_{\rho=0}}{8(2-n)}=\frac{R''[h_\rho]|_{\rho=0}}{8(2-n)}.
G(X_t) =G(x_0)+\int_0^t(G'(X_s)\cdot \mu(X_s)+\tfrac{1}{2}G''(X_s)\cdot\sigma^2(X_s))\,ds+\int_0^t G'(X_s)\cdot\sigma(X_s)\,dW_s &=G(x_0)+\int_0^t \widetilde\mu(G(X_s))\,ds +\int_0^t \widetilde\sigma(G(X_s))\,dW_s.
\rho_{1}( L_{i_{1}}\cdots L_{i_{n}}) ={\frac{1}{n}}\sum_{( i_{1}, \ldots , i_{n})}\Bigl ( L_{i_{1}}\rho_{1}( L_{i_{2}}\cdots L_{i_{n}}) + P ( L_{i_{1}}, L_{i_{2}}\cdots L_{i_{n}}) \Bigr ) .
\frac{m}{r_{H}^{2}}-\frac{M}{r^{2}}(1+2\frac{r_{H}}{r})+\frac{M}{r^{2}}(1-\frac{r_{H}}{r})=\frac{m}{r_{H}^{2}}-\frac{M}{r^{2}}(3\frac{r_{H}}{r})\approx0.
\tilde{\Theta}_\pm D_aB\varphi =-\tilde{\Theta}_\pm BD_a\varphi &=-\tilde{\Theta}_\pm\gamma^{(2n+1)}(1-P_{\rm F}-D_aP_{\rm F}D_a)D_a\varphi=B\tilde{\Theta}_\pm D_a\varphi.
P=\frac{2}{3}\frac{q^{2}}{4\pi\epsilon_{0}c}(\frac{\dot{v}}{c})^{2}=\frac{2}{3}\frac{q^{2}a^{2}}{4\pi\epsilon_{0}c^{3}}=\frac{q^{2}a^{2}}{6\pi\epsilon_{0}c^{3}}=\mu_{0}\frac{q^{2}a^{2}}{6\pi c}(SIunits)
{\cal O}_{\pm}^{a}( \{x \}) : = \frac{1}{( N ! )^{2}}\sum_{\pi , \pi^{\prime}}\prod_{a = 1}^{N}\overline{{\psi}}^{( a )}( x^{( \pi ( a ) )}) P_{\pm}\psi^{( a )}( x^{( \pi ( a ) )}) .
B_1(x) = \int_{0}^{x}\df_{t+h}^\star\left( \frac{1}{Df_{t+h}|Df_{t+h}|}\nu_1 \right) - \int_{0}^{x}df_{t}^\star\left( \frac{1}{Df_{t}|Df_{t}|}\nu_1 \right)
\frac{\partial}{\partial t}H = \mathcal{L}H+\ddot{F}^{kl,rs}\nabla^i h_{kl}\nabla_i h_{rs}+\dot{F}^{kl}h_{km}h^m_l H + (1-\alpha)F|h|^2 begin{align*}5 pt]\frac{\partial}{\partial t}G = \mathcal{L}G + [\dot{G}^{ij}\ddot{F}^{kl,rs}-\dot{F}^{ij}\ddot{G}^{kl,rs}]\nabla_i h_{kl}\nabla_j h_{rs}+\dot{F}^{kl}h_{km}h_l^m\dot{G}^{ij}h_{ij}&+ (1-\alpha)F\dot{G}^{ij}h_{im}h^m_j
\log Z \left( \beta \right) = - \frac{\beta}{2 r}\sum_{\ell = 0}^{\infty}D_{\ell}M_{\ell}\left( \mu , \chi \right) - \sum_{\ell = 0}^{\infty}D_{\ell}\log \left( 1 - e^{- \beta M_{\ell}\left( \mu , \chi \right) / r}\right) ,
R_{i,k}=\mathrm{log}\left(1+\frac{\mathrm{SNR}\left|\mathbf{r}_{i,k}^{H}\mathbf{U}_{i}^{H}\mathbf{H}_{i,k}\mathbf{v}_{i,k}\right|^{2}}{1+\mathrm{SNR}\sum_{n=1}^{K}\left|\mathbf{r}_{i,k}^{H}\mathbf{U}_{i}^{H}\mathbf{G}_{i',n}\mathbf{v}_{i',n}\right|^{2}}\right),
\sharp\{\underline{g}= \underline h_k \dots \underline h_1 \in G^*_{kp}:\underline{g}\mbox{not $\gamma$-acceptable}\}\leq\! \displaystyle\sum_{l\geq [\gamma k]}^k\sharp\{\underline{g}\in G^*_{kp}:\sharp\{0\leq j\leq k:\underline{h}_j\in G^*_p\backslash\tilde G_p\}=l\}.
\sum_{m_1=1}^\infty\sum_{m_3=0}^\infty\frac{z^{m_1+m_3}}{(m_1+m_3+a)^{s}}=\sum_{m_1,m_3=1}^\infty\frac{z^{m_1+m_3}}{(m_1+m_3+a)^{s}}+\sum_{m_1=1}^\infty\frac{z^{m_1}}{(m_1+a)^{s}}.
\sum_{k=0}^{v+3}\varphi(k)\bigl(1-D(v+3-k)\bigr) &=\varphi(v+1)+\varphi (v+2)+\varphi(v+3)-\varphi(0)D(v+3)-b_0c_0\varphi(2)A(v+1) &-b_0c_1\varphi (1)A(v+1) +c_0\varphi(1) \Biggl( \sum_{k=0}^{v+2}a_k \overline{B}(v+2-k)+A(v+2)\Biggr).
\Psi(\alpha^*)\leq \Psi(\bar \alpha) \leq (\tau\delta)(C_\tau^{\frac1{\nu+1}}\delta^{-\frac{1}{\nu+1}}\rho^{\frac{1}{\nu+1}})= C_\tau^{\frac1{\nu+1}}\tau\delta^{\frac{\nu}{\nu+1}}\rho^{\frac{1}{\nu+1}}.
t^{\prime}=f(t),r_{S}(t)={r^{\prime}}_{S}(t^{\prime}),\dot{r}_{S}(t)=\frac{d{r^{\prime}}_{S}(t^{\prime})}{dt}=\frac{d{r^{\prime}}_{S}(t^{\prime})}{dt^{\prime}}\cdot\frac{dt^{\prime}}{dt}=\dot{r}_{S}^{\prime}(t^{\prime})\dot{f}(t)
\begin{array}[c]{ll}-d{p}(t)= \{b_x(\bar{X}{(t)},\bar{u}(t))^{}p(t)-\beta^0f_x(\bar{X}{(t)},\bar{u}(t))\}dt,\t\in(t_{i-1},t_{i}), &p(t_{i})= -\beta^0\Psi_{x}(\bar{X}(t_n)1_{i=n}(i)-\beta^i+p(t_{i}^{+}),\i=1,2,\ldots,n,\end{array}
\mathcal{K}_{4}= \, -\langle \nabla^3 u^{h},\nabla^3 \mathrm{div}\tau\rangle \lesssim \, \|\nabla^3 u^{h}\|_{L^2}\|\nabla^4 \tau\|_{L^2}\leq \, \frac{\mu}{6}\|\nabla^4 \tau\|_{L^2}^2 + C\|\nabla^3 u^{h}\|_{L^2}\leq \, \frac{\mu}{6}\|\nabla^4 \tau\|_{L^2}^2 + C(1+t)^{-\frac{25}{4}}.
D_{6}:=[\begin{matrix}1&1&1&1&1&1\\ 1&-1&i&-i&-i&i\\ 1&i&-1&i&-i&-i\\ 1&-i&i&-1&i&-i\\ 1&-i&-i&i&-1&i\\ 1&i&-i&-i&i&-1\\ \end{matrix}]\in H(4,6)
\xi (x_\alpha \otimes t^k) = \frac 1 2 \varphi(h_\alpha \otimes t; x_\alpha \otimes t^{k-1}) \xi (y_\alpha \otimes t^k) = -\frac 1 2 \varphi(h_\alpha \otimes t; y_\alpha \otimes t^{k-1})
(\Lambda_{N}(t)-\mid\gamma+t\mid^{2})(\Psi_{N,t},e^{i\left\langle\gamma+t,x\right\rangle})=\sum_{\gamma_{1},\gamma_{2},...,\gamma_{m}}\frac{q_{\gamma_{1}}q_{\gamma_{2}}...q_{\gamma_{m}}(q\Psi_{N,t},e^{i\left\langle \gamma+t-\gamma(m),x\right\rangle})}{{\textstyle\prod\limits_{s=1,2,...,m}}[\Lambda_{N}(t)-\left\vert \gamma+t-\gamma(s)\right\vert^{2}]},
\; \; \; \mathrm{i n d}^{D = 1 0}( \mathrm{S p}( 2 N ) ) = \mathrm{n u m b e r ~ o f ~ p a r t i t i o n s ~ o f}~ 2 N ~ \mathrm{i n t o ~ d i s t i n c t ~ e v e n ~ p a r t s}
|M(z_{0})-1|^{2}= \left|\cosh \sqrt{w(z_{0})}-1 + \frac{z_{0}w'(z_{0})\sinh \sqrt{w(z_{0})}}{2\sqrt{w(z_{0})}(\eta \cosh \sqrt{w(z_{0})}+\gamma)}\right|^{2}& =\left|\cosh e^{it/2}- 1 + \frac{k e^{it/2}\sinh e^{it/2}}{2 (\eta \cosh e^{it/2}+ \gamma)}\right|^{2}& = \frac{A_{N}(\tau)}{A_{D}(\tau)}(\tau=t/2, -\pi \leq \tau \leq \pi),
r_{p_1^ip_2^j}=0, i<n_1-1 j<n_2-1 &r_{p_1^{n_1-1}p_2^{n_2-1}}=\frac{24}{(p_1-1)(p_2-1)}=x &r_{p_1^{n_1-1}p_2^{n_2}}=-p_2x &r_{p_1^{n_1}p_2^{n_2-1}}=-p_1x &r_{p_1^{n_1}p_2^{n_2}}=p_1p_2x
h^{p^{\ast}}_{ijkl}-h^{p^{\ast}}_{ijlk}=\sum_mh^{p^{\ast}}_{mj}R_{mikl}+\sum_m h^{p^{\ast}}_{im}R_{mjkl}+\sum_{q}h^{q^{\ast}}_{ij}R_{{q^{\ast}p^{\ast}}kl}.
* \wedge \Bigl ( \delta \Omega , \Omega \Bigr ) = - \frac{1}{4 \pi}\Bigl [{\bf F}_{\mu \nu}\nabla_{\sigma}{\bf F}^{\sigma \nu}+ ( *{\bf F})_{\mu \nu}\nabla_{\sigma}( *{\bf F})^{\sigma \nu}\Bigr ] d x^{\mu}\otimes \varepsilon^{1}\wedge \varepsilon^{2}.
\tilde{\mathcal{F}_l}\big(x_\mu\otimes 1 - \big(\tilde\phi^{-1}\big)^\tau_\mu\otimes\hat{x}_\tau\big) = (x_\mu\otimes 1 - 1\otimes x_\mu)\cdot\tilde{\mathcal{F}}_L\in I_{U(\gg)},
\begin{array}{l}{{{\cal D}_{+ k}{\cal X}^{+ k}-{\cal X}^{+ k}{\cal D}_{+ k}= \mu_{+ k}~ , \mu_{+ k}{\cal X}^{+ k}= q^{2}{\cal X}^{+ k}\mu_{+ k}~ ,}}\\{{\mu_{+ k}{\cal D}_{+ k}= q^{- 2}{\cal D}_{+ k}\mu_{+ k}~ , \mu_{+ k}\equiv 1 + ( q^{2}- 1 ){\cal X}^{+ k}{\cal D}_{+ k}~ ,}}\\ \end{array}
y^{k+1}=\underset{y\in C}{\textup{argmin}}\bigg\{\lambda \langle Ay^k,y-y^k\rangle+\dfrac{1}{2}\|y-x^{k}\|^2\bigg\}, &=\underset{y\in C}{\textup{argmin}}\bigg\{\dfrac{1}{2}\|y-(x^{k}-\lambda Ay^k)\|^2\bigg\}&=P_C(x^{k}-\lambda Ay^k).
\left\langle i\right|Y_0 \left| i \right\rangle =\frac{e^2}\pi\left\langle i \right| \beta m \left( \frac 32\ln(A_0) +\frac 1{4A_0^2}- \frac 14\right)\left| i \right\rangle = 1.115\frac{e^2}\pi\left\langle i \right|\beta m\left| i \right\rangle.
V(C)V(A,B,C[n-2]) =\frac{2}{n(n-1)}\sum_{i<j}V_2( \pi_{i,j}(A), \pi_{i,j}(B)) &\leq \frac{1}{n(n-1)}\sum_{i<j}\left( |\pi_i(A)| \pi_j(B)|+|\pi_j(A)| |\pi_i(B)|\right) &=\frac{1}{n(n-1)}\sum_{i \neq j}|\pi_i(A)| |\pi_j(B)|\leq \frac{1}{n(n-1)}\left( \sum_i |\pi_i(A)| \right) \left( \sum_j |\pi_j(B)| \right) &=\frac{n}{n-1}V(A,C[n-1])V(B,C[n-1] ).
n = \frac{3}{5 - w / ( 1 - w )}+ \frac{3}{1 2 5}\frac{( 1 - w ) ( 1 - 3 w / 2 )}{( 1 - 6 w / 5 )^{3}}( 1 - Q ) + \mathcal{O}[ ( 1 - Q )^{2}]
T^{s}_{(a', [g, x']), (a, [g, x])}=T^{s}_{(a', [sh^{-1}, x']), (a, [sh^{-1}, x])}&=T^{s}_{(a', [s, h^{-1}x']), (a, [s, h^{-1}x])}&=T^{e}_{(s^{-1}a', [e, h^{-1}x']),(s^{-1}a, [e, h^{-1}x])}&=T^{e}_{(h^{-1}g^{-1}a', [e, h^{-1}x']),(h^{-1}g^{-1}a, [e, h^{-1}x])}
F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^vF_s(X_s)dX(s) + \frac{1}{2}\int_0^t\nabla^{v,2}F_s(X_s)d[X,X](s); t\ge 0
\Bigg \lvert \int\limits_{\lvert x-e\rvert+\lvert x\rvert=2\tau}\frac{q(x)}{\lvert 2\tau x-\lvert x\rvert e\rvert}dS_{x}\Bigg\rvert \leq K \int\limits_{\lvert x-e\rvert+\lvert x\rvert\leq 2\tau}\frac{\lvert q(x)\rvert}{\lvert x\rvert \lvert x-e\rvert}dx,\ \forall\tau\in(1/2,T/2].
\bullet [\sigma_1^{-1}\sigma_j, \sigma_1^{-1}\sigma_k] = 0, & \bullet [\sigma_1^{-1}\gamma_{1j}, \sigma_1^{-1}\gamma_{1k}] = 0, & \bullet [\sigma_1^{-1}\sigma_k, \sigma_1^{-1}\gamma_{1j}] = [\sigma_1^{-1}\sigma_j, \sigma_1^{-1}\gamma_{1k}].
\left(\begin{array}{c}x^1 x^0\end{array}\right)\simeq \left(\begin{array}{c}x^1 x^0\end{array}\right) + \left( \begin{array}{c}2\pi R_s 0\end{array}\right), \hspace{3ex}x^i \simeq x^i, \hspace{3ex}p^1=\frac{N}{R_s}\,.
(d(S(T)y_1,S(T)y_2))^2\leq C[d(y_1,y_2)^2-(d(S(T)y_1,S(T)y_2))^2 +g_1(\varrho_T^1(y_1,y_2),\varrho_T^2(y_1,y_2),\cdots,\varrho_T^m(y_1,y_2))]^\beta +g_2(\varrho_T^1(y_1,y_2),\varrho_T^2(y_1,y_2),\cdots,\varrho_T^m(y_1,y_2)).
\displaystyle\frac{z'-x}{\vert z'-x\vert}\cdot\frac{p-x}{\vert p-x\vert}=1-\frac{1}{2}\left\vert\frac{p-x}{\vert p-x\vert}-\frac{z'-x}{\vert z'-x\vert}\right\vert^2\le 1-\frac{1}{2}A_{\delta}^2.
\frac{(k-1)! S_f(x)}{x(\log x)^{k-1}}= \sum_{d\le x}\frac{g(d)}{d}+ \sum_{j=1}^{k-1}(-1)^j \binom{k-1}{j}(\log x)^{-j}\sum_{d\le x}\frac{g(d)}{d}(\log d)^j + O\left((\log x)^{-1}\right).
\langle \Psi_{k}| k^{\prime}\rangle_{\epsilon}= c_{k}\frac{2^{\epsilon + i k^{\prime}}}{4 \sqrt{2 \pi}}\Gamma \left( \frac{\epsilon + i k^{\prime}- i k}{2}\right) \Gamma \left( \frac{\epsilon + i k^{\prime}+ i k}{2}\right) ,
\gamma_{1}=\frac{\mu_{3}}{\mu_{2}^{3/2}}=\frac{K_{0}^{2}(a)K_{3}(a)-3K_{0}(a)K_{1}(a)K_{2}(a)+2K_{1}^{3}(a)}{(K_{0}(a)K_{2}(a)-K_{1}^{2}(a))^{3/2}}
\frac{n}{B_n}\left((1+\hat b^{(n)})^2 - (1+ b^{(n)})^2 \right) = \frac{n}{B_n}\langle \Xi^{(n)},v\rangle + o_{\mathbb P}(1),\; v = \left(\frac{1}{\sqrt{2}},0,-\frac{1}{2},-\frac{1}{2},0\right)
\mathbb{E}\left[ (\sigma_p^l\breve{Z}_p^l)(\sum_{r=0}^{t-1}\hat{\gamma}^t_{p,r}\sigma_p^{r}\breve{Z}_p^{r}+ (\sigma_p^t)_{\perp}Z_p) \right]\overset{(a)}{=}\sum_{r=0}^{t-1}\hat{\gamma}^t_{p,r}\tilde{E}_p^{l,t}\overset{(b)}{=}[\tilde{C}^t_p\hat{\gamma}^t_p]_{l+1}\overset{(c)}{=}\tilde{E}_p^{l,t},
\epsilon(s) := \epsilon := M\rho (s) |\log M\rho (s)| \delta_1 (s) := \delta_1 := \frac{1}{e^{2Y}}, \delta_2 (s) := \delta_2 := (X^2+Y^2)^{\frac{3}{4}}\sqrt{\epsilon}\delta_3 (s) := \delta_3 := \frac{1}{\sqrt{X + Y}},
\mathbb{E}\left[\frac{p_X(x_1,X_2)}{p_{X_1}(x_1)p_{X_2}(X_2)}\bigg|X_{2k}=x_{2k}\right] =\frac{p_{X_1,X_{2k}}(x_1,x_{2k})}{p_{X_1}(x_1)p_{X_{2k}}(x_{2k})}.
\mathrm{Pr}(\hat x_n \neq \bar x_n|x_n = \bar x_n) = \mathrm{Pr}(\hat x_n \neq x_n) = \mathrm{Pr}(|y_n - \sqrt{P}x_n|^2 > |y_n + \sqrt{P}x_n|^2) = \mathrm{Pr}(|z_n+v_n|^2 > |2 \sqrt{P}x_n + z_n+v_n|^2),
\begin{aligned}\log \det \Delta_\beta= -\frac{\log(\beta+1)}{6(\beta+1)}+\left( \frac 1 6 \log (8\pi) -4\log A\right)\frac 1{\beta+1}&\qquad\quad\ -\log(\beta+1)-\log 2 +O(\beta+1)\quad \quad\beta\to-1^+,\end{aligned}
\renewcommand{\arraystretch}{1.4}\begin{array}{rcl}L(f(x)) = \delta^n(f(x)) + a_{n-1}(x) \delta^{n-1}(f(x)) + \ldots + a_0(x) f(x) = 0 \mbox{and}S(f(x)) = \sigma^m(f(x)) + b_{m-1}(x) \sigma^{m-1}(f(x)) + \ldots + b_0(x)f(x) = 0,\end{array}
\tilde{G}^a_{\mu_1\ldots \mu_p}\equiv p!\, \frac{\partial L}{\partial \bar{F}^{a\,\mu_1\ldots \mu_p}}~,~~\tilde{\bar{G}}^a_{\mu_1\ldots \mu_p}\equiv p!\, \frac{\partial L}{\partial F^{a\,\mu_1\ldots \mu_p}}~.
|F_i(x)-F_\infty(g_i(\Phi_i(x)))| =|g_i(F_1(x)-F_\infty(\Phi(x)))| &\leq C_\delta \lambda^{-(1+\delta)d_1i}|F_1(x)|^2 \leq C_\delta^2 \lambda^{-(1+\delta)d_1i + 2(1-\delta)d_1 i}|F_i(x)|^2.
S = \int \mbox{d}x \; \left\{- \frac{1}{12}F_{\mu \nu \rho}F^{\mu \nu \rho}+ B^{* \mu \nu}\left( \partial_{\mu}C_{\nu}- \partial_{\nu}C_{\mu}\right) + C^{* \mu}\partial_{\mu}C \right\}\;,
\sum_{C \leq p \leq w}\sum_{\substack{m = -p_1 + p_2 - p_3 p_j \in I_j, \forall j \leq 3 p^2 | A - p_1 + p_2}}1 & = \sum_{C \leq p \leq w}\sum_{\substack{p_3 \in I_3 p^2 | A + m + p_3}}|\{n \in I_1 \cap (I_2 - m - p_3): n, m+n+p_3 \hbox{prime}\}|.
S = \int d^{6}x \left[{\frac{1}{1 2}}{\epsilon}^{0 i j k l m}F_{0 i j}F_{k l m}-{\frac 1 6}F_{k l m}F_{k l m}+{\frac 1 6}G^{{\mu}{\nu}{\sigma}}( F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]}) \right] ,
\begin{array}{l l}{{\Gamma^{i}= \sigma^{i}\otimes \gamma^{( 7 )}}}&{{\mathrm{f o r ~}~ i = 1 , 2 , 3}}\\{{{}}}&{{{}}}\\{{\Gamma^{a}= 1 \otimes \gamma^{a}}}&{{\mathrm{f o r ~}~ a = 4 , 5 , 6 , 7 , 8 , 9 \, .}}\\ \end{array}
|\Lambda^2| =h_{aj}H_{jib}\Lambda_{aib}=\frac{1}{3}h_{aj}h_{ak}H_{jib}H_{kib}+\frac{2}{3}h_{aj}h_{bk}H_{bji}H_{kai}&=\frac{4}{3}\mu|h|^2+\frac{8}{3}\langle\mathring{R}(h),h \rangle,
\begin{aligned}\rho_t^{k+1}+ \nabla\cdot\left( \rho^{k+1}Q\ast \mathbf{u}^k \right) = 0, & \mathbf{u}_t^{k+1}+ (\mathbf{u}^k\cdot\nabla ) \mathbf{u}^{k+1}= \rho^{k}\left( Q\ast \mathbf{u}^{k+1}- \mathbf{u}^{k+1}\right),\end{aligned}
W(\underline{u}(E_k^-,\theta),\underline{u}(E_k^-+\delta,\theta))(n) = \lambda(E_k^-+\delta)^n b_{11}(E_k^-+\delta;\theta+n \alpha) b_{11}(E_k^-;\theta+(n+1) \alpha) -\lambda(E_k^-+\delta)^{n+1}b_{11}(E_k^-+\delta;\theta+(n+1) \alpha)b_{11}(E_k^-;\theta+n \alpha).
{\tilde{G}=\sum_{^{|\alpha|=m}_{\alpha\neq\hat{\alpha}}}[\tilde{f}_\alpha]^{\ast}_{a, Q_{4R}^+}+{[\tilde{f}_{\hat{\alpha}}]_{\frac{a}{2m},a,Q_{4R}^+}}+\sum_{|\alpha|\le m}\|\tilde{f}_\alpha\|_{L_\infty(Q_{4R}^+)}.}
f_i =P_D\lambda_i, f_i^{pR,qR}=P_{D_{pR}}[\mathbf{1}_{D_{pR,qR}}P_D^*\lambda_i],\enskip \widetilde{f}_i^{pR,qR}=P_{D_{pR}}[\mathbf{1}_{D_{qR}^R}P_D^*\lambda_i], f_i^* =P_D^*\lambda_i, f_i^{pR,qR*}=P_{D_{pR}}^*[\mathbf{1}_{D_{pR,qR}}P_D^*\lambda_i],\enskip \widetilde{f}_i^{pR,qR*}=P_{D_{pR}}^*[\mathbf{1}_{D_{qR}^R}P_D^*\lambda_i].