harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{\sigma}_u =\frac{1}{q!}\sum_{\tau\in\mathfrak{S}_q}\hat{\sigma}_{u\circ\tau}\textrm{for $G=O(q)$}, \hat{\sigma}_u =\frac{1}{(q-1)!}\sum_{\tau\in\mathfrak{S}^{\rm even}_q}\hat{\sigma}_{u\circ\tau}\textrm{for $G=SO(q)$}.
y^2= (x^2-1)\left(x^2-\left(\frac{1-\mu\sqrt{\lambda}}{1+\mu\sqrt{\lambda}}\right)^2\right)y^2= (1-x^2)\left(1-\left(\frac{1-\mu\sqrt{\lambda}}{1+\mu\sqrt{\lambda}}\right)^2 x^2\right)
\theta_{P_H}= \frac{\frac{(1+e^{ab})^{k_1-2}}{\left(1+e^{ab}\left(1-\zeta\right)\right)^{k_1}}-\frac{(1+e^{ab})^{2k_1-2}}{\left(1+e^{ab}\left(1-\zeta\right)\right)^{2k_1}}}{\frac{(1+e^{ab})^{k_1-1}}{\left(1+e^{ab}\left(1-\zeta\right)\right)^{k_1}}-\frac{1}{1+e^{ab}}}.
b =\frac{c_{\iota_{1}(Q),n}}{u_{\iota_{1}(Q)}^n}+ \ldots + \frac{c_{\iota_{1}(Q),2}}{u_{\iota_{1}(Q)}^2}+ \frac{c_{\iota_{1}(Q),1}}{u_{\iota_{1}(Q)}}+ \tilde{g}
\sum\limits_{\substack{b=1}}^{n^s}(b,n^s)_se^{\frac{2\pi ibj}{n^s}}= \sum\limits_{\substack{d^s\mid n^s \frac{n^s}{d^s}\mid j}}\Phi_s({d^s}){\frac{n^s}{d^s}}& =\sum\limits_{\substack{d^s\mid n^s{d^s}\mid j}}\Phi_s({\frac{n^s}{d^s}}){{d^s}}& =\sum\limits_{\substack{{d^s}\mid (j,n^s)_s}}\Phi_s({\frac{n^s}{d^s}}){{d^s}}.
G_0 =G_{0,0}-(G_{0,0}-P_0G_{0,0})vM_0^\dagger v^*(G_{0,0}- G_{0,0}P_0) &-P_0G_{0,0}-G_{0,0}P_0+P_0 G_{0,0}P_0 &=(I-P_0)\bigl[G_{0,0}&-G_{0,0}v(U+v^*G_{0,0}v)^\dagger v^*G_{0,0}\bigr](1-P_0).
- 2 \, \ddot g_\rho + \ddot{\bar g}+ 2\,g_\rho -\bar g + \xi_\rho =0 H^{-1}(I;L^2(\Omega)), g_\rho(0)=0,2 \, \dot g_\rho(T) - \dot{\bar g}(T) = 0.
X_{N}^{( K )}( a ) \ = \ \left( \frac{\alpha}{\pi}\right)^{K^{2}}\int d [ \sigma ] \exp \left( - \alpha{\mathrm{t r}}( \sigma - a ) ( \sigma^{\dagger}- a ) \right) \mathrm{d e t}^{N}\sigma \sigma^{\dagger}\: ,
\mathcal U(r)=\frac{u(r)}{u_0(r)}, \mathcal V(r)=\frac{v(r)}{v_0(r)}, \mathcal W(r)=\frac{u'(r)}{u'_0(r)}, \mathcal Y(r)=\frac{v'(r)}{v'_0(r)}
\begin{array}{l}{[{H}_{\pi_{\alpha}}] : ={\dot{u}}^{\alpha}( x ) - \frac{\partial{H}}{\partial \pi_{\alpha}},}{[{H}_{u^{\alpha}}] : ={\dot{\pi}}_{\alpha}( x ) + \frac{\partial{H}}{\partial u^{\alpha}}- \nabla_{a}( \frac{\partial{H}}{\partial ( \nabla_{a}u^{\alpha})}) .}\end{array}
E ={\rm tr}\big[D^2 H(\nabla\varphi)\cdot D^2\varphi\cdot\left(I_d-(I_d-D^2\varphi)^{-1}\right)\big] & =-{\rm tr}\big[D^2 H(\nabla\varphi)\cdot \big[D^2\varphi\big]^2\cdot (I_d-D^2\varphi)^{-1}\big].
{\rm Tr}\;SU[\vartheta_{a}^{(1)},P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-] &={\rm Tr}\;SP_+[\vartheta_{a}^{(1)},P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-] &-{\rm Tr}\;SP_-[\vartheta_{a}^{(1)},P_-][\vartheta_a^{(2)},P_-]\cdots[\vartheta_a^{(2n+1)},P_-].
\pi_*\mathrm{PD}(c_1(E_{\hat{H}})) = c_1(\hat{H}|_{\{k^0_x \}\times S^1 \times S^1}) [\overline{\alpha}_x]- c_1(\hat{H}|_{S^1 \times \{k^0_y \}\times S^1}) [\overline{\alpha_y}] &-\sum_{i = 1}^{n-1}\bigg(\sum_{j = 1}^{m_i}\int_{\partial \overline{D(w_i^j)}}c_1(E_{\hat{H}})\bigg)[\overline{\alpha}_i].
D_1=1,\;\;D_2 = 0,\;\;D_3=-\frac{3}{20},\;\;D_4=\frac{1}{6},\;\;D_5=-\frac{9}{70},\;\;D_6=\frac{3}{40},\;\; D_7=-\frac{199}{7200},\;\; D_8=-\frac{3}{700}.
\begin{array}{lcr}u^+_1=\cos{\theta\over2}\exp{i\over2}(\psi+\phi) u^+_2=\sin{\theta\over2}\exp{i\over2}(\psi-\phi) u^-_1=\sin{\theta\over2}\exp{-i\over2}(\psi+\phi) u^-_2=\cos{\theta\over2}\exp{-i\over2}(\psi-\phi).\end{array}
\underline{\underline{S}}=[\begin{matrix}\frac{1}{E}&-\frac{\nu}{E}&-\frac{\nu}{E}&0&0&0\\ -\frac{\nu}{E}&\frac{1}{E}&-\frac{\nu}{E}&0&0&0\\ -\frac{\nu}{E}&-\frac{\nu}{E}&\frac{1}{E}&0&0&0\\ 0&0&0&\frac{1}{G}&0&0\\ 0&0&0&0&\frac{1}{G}&0\\ 0&0&0&0&0&\frac{1}{G}\\ \end{matrix}].
\Phi^{\textbf{e}}\Big(\xi,\mathrm{t}; \mathrm{z},\tau - \dfrac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4\mathrm{m}^2}\Big) - \Phi^{\textbf{e}}\Big(\xi,\mathrm{t};\mathrm{z}, \tau \Big) = \displaystyle\int_{\tau}^{\tau - \frac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4\mathrm{m}^2}}\partial_{s}\Phi^{\textbf{e}}\Big(\xi,\mathrm{t};\mathrm{z}, \mathrm{s}\Big)d\mathrm{s}.
\theta (v;\gamma ,\delta ) = \sum_{n=-\infty}^\infty \exp \left\{\sum_{a,b=1}^g \pi i(n_a+\delta_a)\tau_{ab}(n_b+\delta_b)+\sum_{a=1}^g\left[ v_a(n_a+\delta_a)+2\pi i\gamma_a(n_a+\delta_a)\right] \right\}\ ,
a_{j}=\frac{2}{q_j q_{j-1}}+\frac{2}{q_{j}q_{j+1}}+\frac{2}{q_{j+1}q_{j+2}},b_j=\frac{2}{q_{j-1}(q_{j-1}+q_j)}, c_j=\frac{2}{q_{j+2}(q_{j+1}+q_{j+2})}.
\rho= \rho_{n}:=\sup_{A\in\mathcal{A}^{\mathrm{re}}}\left|\Pr \left(\sum_{i=1}^{n}Z_{i}/\sqrt{n}\in A\right)-\Pr\left(\sum_{i=1}^{n}\widetilde{Z_{i}}/\sqrt{n}\in A \right) \right|.
G = \left(1-\frac{x}{(e-1) e (x-1)}\right) \,_2F_1(1,1;e+1;x) + \frac{x (e (2 x-1)-3 x+1)}{(e-1) e (e+1) (x-1)}\,_2F_1(2,2;e+2;x)
0= [[\beta_{-1}, \beta_{-m+1}],\beta_n] +(n+m)(m-2) \beta_{n-m}+ \sum_{j=1}^{m-2}j (\alpha_{n-j}\alpha_{-m+1+j}\beta_{-1}-\beta_{-1}\alpha_{n-j}\alpha_{-m+1+j}) &+(n+1) \sum_{j=1}^{m-2}j \alpha_{n-1-j}\alpha_{-m+1+j}
q_p=c_p\frac{(w_{p-1}v_{\omega_p},wgX_{\beta_1}(-q_1)\ldots X_{\beta_{p-1}}(-q_{p-1})w_p v_{\omega_p})}{(w_{p-1}v_{\omega_p},wgX_{\beta_1}(-q_1)\ldots X_{\beta_{p-1}}(-q_{p-1})w_{p-1}v_{\omega_p})}.
S_{G F - g h o s t}= \int d^{d}x \left\{- \frac{1}{2 \xi}\partial_{\mu}A^{\mu}( x ) \star \partial_{\nu}A^{\nu}( x ) + \partial^{\mu}\overline{{{c}}}( x ) \star \left( \partial_{\mu}c ( x ) - i [ A_{\mu}( x ) , c ( x ) ]_{M}\right) \right\}
I_n^\zeta(a,b) :=\int_{\{a< r_1<r_2<\cdots<r_n<b \}}\Big[(r_2-r_1)(r_3-r_2)\cdots(b-r_n)\Big]^\zeta dr_1dr_2\cdots dr_n &= \frac{\Gamma(1+\zeta)^{n+1}(b-a)^{n(1+\zeta)}}{\Gamma\big(n(1+\zeta)+1\big)},
\pi i z^2 \tau + \tau \frac{z \, \partial_z \Theta(\tau z, \tau) + \partial_\tau \Theta(\tau z, \tau)}{\Theta(\tau z, \tau)}- \frac{1}{\tau}\frac{\partial_\tau \Theta\left(z, -\tfrac{1}{\tau}\right)}{\Theta\left(z, -\tfrac{1}{\tau}\right)}= -\frac{1}{2}.
\xi_\alpha(\lambda)=\tfrac1\pi \tan^{-1}\left(\tfrac{2}{\pi}\cosh^{-1}\left(\tfrac{\pi}{\sqrt{\lambda}}\right)\right)=\tfrac1\pi \tan^{-1}\left(\tfrac{2}{\pi}\log\left(\tfrac{\pi}{\sqrt{\lambda}}+\sqrt{\tfrac{\pi^2}{\lambda}-1}\right)\right).
\begin{aligned}|a_{l+1}| \le \left(\frac{2c_1\ln n}{nh^d}\right)^{-l}\ln\left(\frac{nh^d}{2c_1\ln n}\right)\cdot 2^{3c_2\ln n}&\le \left(\frac{2c_1\ln n}{nh^d}\right)^{-l}\cdot n^{3c_2\ln 2}\ln n. \end{aligned}
u =x_1\big(\textstyle\prod_{s=\ell+1}^{d-2}x_{n-s-1}\big)\big(\prod_{s=0}^{\ell}x_{n-s}\big),\ \ \ \ v=\big(\textstyle\prod_{s=\ell}^{d-1}x_{n-s-1}\big)\big(\prod_{s=0}^{\ell-1}x_{n-s}\big);
\ll{\mathop{\sum\nolimits^*}\limits_{u\,(\, mcn_1^{-1})}}(1+un_1, c)~=~ \sum_{d|c}d{\mathop{\sum\nolimits}\limits_{\stackrel{u\,(\, mcn_1^{-1})}{(1+un_1, c) = d}}}1 ~\ll~\sum_{d|c}d{\mathop{\sum\nolimits}\limits_{\stackrel{u\,(\, mcn_1^{-1})}{un_1 \equiv -1 (\!\!\!\mod d)}}}1.
{\cal R}_{0}= m \sqrt{u^{2}}+ e u^{\mu}A_{\mu}- \frac{u^{\mu}}{2}{\omega}_{\mu}^{a b}S_{a b}+ \frac{e g}{4 m}F^{a b}S_{a b}\sqrt{u^{2}}- \frac{\kappa}{8 m}R^{a b c d}S_{a b}S_{c d}\sqrt{u^{2}}\, .
U \ = \ \frac{1}{2\pi i}\int_{\Re(s) = 1}\left[ \sum_{v = 1}^\infty \hat{\phi}\left( \frac{\log v}{\log R}\right) \frac{\chi_0 (v) \Lambda(v)}{v^{(1+s)/2}\log R}\right]^{n-\eta}\left(\frac{4\pi m\sqrt{c}}{b \sqrt{N}}\right)^{-s}G_{k-1}(s) ds.
\begin{aligned}\quad\quad\quad\left.\frac{d^b\cal S(n,a,x)}{dx^b}\right|_{x=1}\!=(-1)^{a+b}(n+1) &\times \int_0^1\!\!\frac{\partial^b}{\partial x^b}\,\frac{x^b+(-1)^{n-a-b}x^{n-a+1}\left(\tfrac{t}{1-t}\right)^{n-a-b+1}}{1+\tfrac{xt}{1-t}}\biggr|_{x=1}\!\! t^{a+b}(1-t)^{n-a-b}\,dt.\end{aligned}
g_{n}=(-1)^{n}\alpha^{n}q^{-\frac{1}{2}n(n-1)}\sum_{k=0}^{\infty}\left(q^{-k}z^{-1}\alpha^{-1};q\right)_{n}\left(q^{k+1}z\alpha;q\right)_{\!\infty}\frac{q^{\frac{1}{2}k(k+1)+kn}}{(q;q)_{k}}(-1)^{k}z^{2k}
\|P_{N}f(x)\|_{L^{2}(X)}^{2}=\frac{1}{N^{2}}\sum_{n,m=1}^{N}\left<u_{p_{n}}.f,u_{p_{m}}.f\right> \leq \frac{\|f\|_{L^{2}(X)}^{2}}{N}+\frac{2}{N^2}\sum_{k=1}^{2N\log(N)}d_{2N\log(N)}(k)\left<u_{k}f,f\right>,
\theta(g) = \left[ \delta_j^i +\frac{1}{2!}c_{j k}^i g^k +\frac{1}{3!}c_{j k_1}^{h_1}c_{h_1 k_2}^i g^{k_1}g^{k_2}+ \frac{1}{4!}c_{j k_1}^{h_1}c_{h_1 k_2}^{h_2}c_{h_2 k_3}^{i}g^{k_1}g^{k_2}g^{k_3}+ \ldots \right] dg^j X_i \quad ,
\tfrac{\alpha\,\mathrm{Num}}{L_X(s,t)L_U(t,s)}= L_U(t+1,s+1)\left[\alpha L_X(s+2,t+1)-L_X(s+1,t)\right] - L_X(s+1,t+1)\left[\alpha L_U(t+2,s+1)-L_U(t+1,s)\right].
0= \lim_{n\to\infty}\ \sum_{i=1}^{n}\mathbb E\left[\|(\mathcal S(\Delta_n)-I)Y_{(i-1)\Delta_n})\|^4\right] \geq \lim_{n\to\infty}\ \sum_{i=1}^{n}\mathbb E\left[\|(\mathcal S(\Delta_n)-I)Y_{(i-1)\Delta_n})\|^2\right]^2 = \lim_{n\to\infty}\Delta_n^{2+4\mathfrak H}\sum_{i=1}^n(i-1)^2>0,
b_n J'_{n-1+{D\over 2}}(|\omega| r')+c_n J'_{-n+1-{D\over2}}(|\omega| r')-a_nJ'_{n-1+{D\over 2}}(|\omega| r')={(2n+D-2)\Gamma\left ({D-2\over 2}\right )\over 4 (\pi r')^{D\over 2}|\omega|}
\sum\limits_{i,j=1}^{n}a_{ij}\, u_{x_jx_i}= | \nabla u |^{2-p}\sum\limits_{i,j=1}^{n}\frac{\partial \mathcal{A}_i}{\partial \eta_j}(\nabla u ) u_{x_jx_i}\, = \, 0 \mbox{at}\,\, x_0, y_0, z_0.
T_\eta\xi^\alpha\!=\!\eta^\alpha(\xi),\,T_\eta^{-1}\xi^\alpha\!=\!T_\zeta\xi^\alpha\!=\!\zeta^\alpha(\xi),\quad T_\eta f (\xi) =f(\eta(\xi)),\,T_\eta^{-1}f(\xi)=T_\zeta f(\xi)=f(\zeta(\xi)),
\frac{1}{2}\frac{F''(v)}{F'(v)}\|\nabla v\|^2 =-\frac{1}{2}v\|\nabla v\|^2 +\frac{1}{2v}\|\nabla v\|^2 &=\sup_{a\in\mathbb{R}^n}\inf_{b\in\mathbb{R}^n}\bigg\{\langle a+cb,\nabla v+b\rangle-\frac{1}{2}v\|b\|^2\bigg\}+\sup_{\tilde a\in\mathbb{R}^n}\bigg\{\langle\tilde a,\nabla v\rangle -\frac{1}{2}v\|\tilde a\|^2\bigg\}.
\sigma=(1+h_x^2)^{-3/2}-1,\gamma=(1+h_x^2)^{-3/4}-1,p_1=\gamma,p_0=-(3/4)\gamma_x, & B= \frac{G(h)\phi + h_x\phi_x}{(1+h_x^2)}, V = \phi_x-Bh_x,\omega=\phi-T_BP_{\geq 1}h.
P\left(\frac{n-A_n^*}{f_2(n)}\leq z+\delta\right) -P\left(\frac{n-A_n^*}{f_2(n)}\leq z-\delta\right)\geq\mathrm{e}^{-(H(\ell_2(z-\delta))+\eta)b_c^{(n)}}-\mathrm{e}^{-(H(\ell_2(z+\delta))-\eta)b_c^{(n)}}&=\mathrm{e}^{-(H(\ell_2(z-\delta))+\eta)b_c^{(n)}}\left(1-\mathrm{e}^{-(H(\ell_2(z+\delta))-H(\ell_2(z-\delta))-2\eta)b_c^{(n)}}\right)
(\rho^{k}\otimes (^{k}\rho))\Delta(\alpha) \Diamond \rho(z) =(\rho^{k}\otimes (^{k}\rho))(1 \otimes \alpha - \alpha \otimes 1) \Diamond \rho(z) &=(\bar{e_{k}}\otimes \rho(\alpha)-\rho(\alpha) \otimes \bar{e_{k}}) \Diamond \rho(z) &=cyc(\rho(\alpha)\rho(z)-\rho(z)\rho(\alpha)) &=\rho(cyc(\alpha z - z\alpha)) &=\rho^{k}(\Delta(\alpha) \Diamond z)
0 = \langle A_1(u_1)+\lambda_1 B_1(u_1),(u_1-\overline{u}_1)_+\rangle_{\mathcal{H}_1}&- \langle F_1(\overline{u}_1,T_2u_2,\nabla \overline{u}_1,\nabla (T_2u_2))+G_1(\overline{u}_1,T_2u_2),(u_1-\overline{u}_1)_+\rangle_{\mathcal{H}_1}
\int_{-\infty}^{-X}e^{-u^2/2}\, \mathrm du = \int_X^\infty e^{-u^2/2}\, \mathrm du \le \frac1X \int_X^\infty u e^{-u^2/2}\, \mathrm du \ll \frac1Xe^{-X^2/2}.
\begin{cases}[j]=0,[H_N]=0, & j[u_N]+\vert N\vert^2[q]=0,j[u_{\tau}]=H_N[H_{\tau}], &H_N[u_{\tau}]=j[\frac{H_{\tau}}{\rho}], j[e+\frac{1}{2}\frac{|\mathbf{H}|^2}{\rho}]+[qu_N-H_N(\mathbf{H}\cdot\mathbf{u})]=0.\end{cases}
W^1=\phi_1 \left(1-\frac{\phi_1^4}{5}+2 \phi_1^2 \phi_2^2- \phi_2^4\right) \hspace{1cm}W^2= \phi_2\left(1-\phi_1^4+2\phi_1^2 \phi_2^2-\frac{\phi_2^4}{5}\right)
\sum_{n=1}^\infty \frac{q^{2n-1}}{(1+q^{2n-1})^2}-2\sum_{n=1}^\infty \frac{q^{2n}}{(1-q^{2n})^2}=\sum_{n=1}^\infty \left(\frac{(2n-1)q^{2n-1}}{1-q^{4n-2}}-\frac{4nq^{2n}}{1-q^{2n}}+\frac{2nq^{4n}}{1-q^{4n}}\right) &= \sum_{n=1}^\infty 6\tilde{R}_n q^n,
\begin{aligned}X_2=\frac{\nu_{R,I}\nu_{S,I}\nu_{T,1}}{\delta_{R}^2\Delta_{S}}[-\delta_{R}^2\nu_{T_1,T_2}\eta_{S,S,V}/\delta_{U}^2\nu_{T_1,T_2}+\delta_{R}^2\eta_{T_1,T_1,T_2}\nu_{S,V}\nu_{T,1,2}\nu_{S}/\delta_{R}^2 ]\end{aligned}
\int_{-1}^{1}\frac{4}{R(1-z+R)(1+z+R)}f(x)\mathrm{d}x &= \frac{4}{3z^3}\left[4z^2 +(z^3-3z)\log\left(\frac{1+z}{1-z}\right) -2\log\left((1+z)(1-z)\right)\right]\ ,
M_{\pm \gamma}=\left( M_{{ij}_{\pm \gamma}}\right) M_{{ij}_{\pm \gamma}}=\begin{pmatrix}\frac{s_{ij}\pm \gamma c_{ij}}{2}\frac{s_{ij}\mp \gamma c_{ij}}{2}\frac{s_{ij}\mp \gamma c_{ij}}{2}\frac{s_{ij}\pm \gamma c_{ij}}{2}\end{pmatrix},\ \ 1\leq i,j\leq n
f_{1}\star_{\vec{b}}^{n}f_{2}\ (\mathbf{n})=\int_{S^{2}}\left\{\int_{S^{2}}\mathbb{T}_{\vec{b}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})\star_{\vec{b}}^{n}f_{1}(\mathbf{n}_{1})d\mathbf{n}_{1}\right\}\star_{\vec{b}}^{n}f_{2}(\mathbf{n}_{2})d\mathbf{n}_{2}
\begin{aligned}0 = \langle \psi^{(0)}_{n}, H^{(1)}\psi^{(0)}_{n}\rangle 0 = \sum_{i \neq n}\frac{\left| \langle \psi^{(0)}_{i}, H^{(1)}\psi^{(0)}_{n}\rangle \right|^{2}}{\lambda_{n}^{(0)}-\lambda_{i}^{(0)}}\end{aligned}
{\left[{\overline{\bf{y}}}\right]_i}={a_i}{\lambda_i}\sum\limits_{j = 1}^2{{V_{ij}}{d_j}}i = 1,2{\left[{\overline{\bf{y}}}\right]_i}={a_i}{\lambda_i}\sum\limits_{j = 3}^4{{V_{ij}}{d_j}}i = 3,4 .
\Phi_{\bigstar_\Gamma h_v}(\lambda_{s_1}\cdots \lambda_{s_n}) = \Phi_{\bigstar_\Gamma h_v}(\lambda_{s_1\cdots s_n}) &=\lambda_{s_1\cdots s_n}(\bigstar_\Gamma h_v)_{s_1\cdots s_n}&=\Phi_{h_{v_1}}(\lambda_{s_1}) \cdots \Phi_{h_{v_n}}(\lambda_{s_n}).
V ( R ) = \lim_{T \rightarrow \infty}\, \frac{1}{T}\, \left[ \int_{0}^{T}\! \! d t \int_{0}^{R}\! \! d r \, M_{0}^{2}\, + \, \frac{D - 2}{2}\, \mathrm{T r}\ln G^{- 1}\right] \,{.}
\delta(C_4\wedge H_3\wedge G_3) =C_4\wedge d\delta B_2\wedge G_3 &=d\left(C_4\wedge \delta B_2\wedge G_3\right)-G_5\wedge \delta B_2\wedge G_3 &=d(\ldots)-\delta B_2\wedge G_5\wedge G_3.
R_2(t) = -i\lambda_6 t^{-1}\mathcal{F}M(t)^{-1}\mathcal{F}^{-1}\mathcal{N}_2 (\mathcal{F}M{{\mathcal F}}^{-1}v_1, \mathcal{F}M{{\mathcal F}}^{-1}v_2) (t) & +i \lambda_6 t^{-1}\mathcal{N}_2 (v_1,v_2)(t) ,
\gamma_{i j}d x^{i}d x^{j}= \left( 1 +{\frac{k}{4}}\delta_{m n}x^{m}x^{n}\right)^{- 2}\delta_{i j}d x^{i}d x^{j}={\frac{d r^{2}}{1 - k r^{2}}}+ r^{2}( d \theta^{2}+ \sin^{2}\theta d \psi^{2}) ,
\phi_y(t+h,x)-\phi_y(t,x) =\int_0^{t-\varepsilon}\left( \phi_y(t+h,x,s)- \phi_y(t,x,s)\right)ds &+ \int_{t-\varepsilon}^{t+h}\phi_y(t+h,x,s)\,ds- \int_{t-\varepsilon}^t\phi_y(t,x,s)\,ds\,.
\sup_{\Vert u\Vert=1}\Vert B(t)u\Vert_{H^1(1+\rho)}^2 \leq b^2 (\Vert f'(\hat{v})\Vert^2_\infty + \Vert f''(\hat{v})\Vert_\infty^2\Vert \hat{v}_x\Vert_\infty^2)
1+4C((i+\ell-1)(j+1))^{-2}+\left( 1+4C((i+\ell-1)(j+1))^{-2}\right) j^{-2}8C\sum_{k=1}^{\ell-1}(i+k-1)^{-2}&\leq 1+j^{-2}8C\sum_{k=1}^{\ell}(i+k-1)^{-2}.
= \int D j_{\mu}D \lambda_{\mu}\exp \left[ - \frac{1}{8 \pi^{2}}\int d^{4}x d^{4}x^{\prime}j_{\mu}( x ) \frac{1}{( x - x^{\prime})^{2}}j_{\mu}( x^{\prime}) - i \int d^{4}x \lambda_{\mu}j_{\mu}+ 2 \zeta \int d^{4}x \cos \left( \frac{\left| \lambda_{\mu}\right|}{\Lambda}\right) \right] ,
\Big((h_1,K_1,v_{h_1,K_1}),(h_2,K_2,v_{h_2,K_2})\Big)_{f}=\int_M \Big(\langle h_1,h_2 \rangle+\langle K_1,K_2 \rangle+ v_{h_1,K_1}v_{h_2,K_2}\Big)e^{-f}dV_g
\lambda_{1;p,f}\int_{\Omega}|u|^{\alpha}|\nabla u|^{p}\,d\mu= (\alpha-p+1)I_{p,\alpha}\int_{\Omega}|u|^{\alpha}|\nabla u|^{p}\,d\mu &+p\int_{\Omega}|u|^{\alpha-p+1}|\nabla u|^{2p-3}\langle\nabla u,\nabla|\nabla u|\rangle\,d\mu,
H ( x , y ) = 1 +{\frac{A}{( x + y )^{2}}}+{\frac{B}{( x + y )^{2}}}\left( \mathrm{l n}\left({\frac{( x + y )^{2}}{\rho_{0}^{2}y}}\right) - 1 +{\frac{x}{y}}\right) ~ ,
\Psi(0,z)={}_2F_1\bigg[\begin{matrix}-a 1+a &1\end{matrix}\bigg|\,z\bigg]={}_2F_1\bigg[\begin{matrix}-\frac12a \frac12+\frac12a &1\end{matrix}\bigg|\,4z(1-z)\bigg]=\Phi(0,z).
\begin{aligned}\frac{D}{Dt}\,H (H^2/2-2K) &=\big(\Delta_gw+w(H^2-2K)\big)(H^2/2-2K) &\quad+H\big(H(\Delta_gw+w(H^2-2K))-2w\,K H-2H\Delta_g w+ 2k^{\alpha\beta}w_{;\alpha\beta}\big) &=w_{;\alpha\beta}\big((-H^2/2-2K)g^{\alpha\beta}+2Hk^{\alpha\beta}\big)+w(3H^4/2-7KH^2+4K^2)\end{aligned}
J_{i;m}=\frac{\partial J_i}{\partial x^m}-J_l\Gamma^l_{im}-\frac{\partial J_i}{\partial y^l}N^l_m, J_{i|m}=\frac{\partial J_i}{\partial x^m}-J_l\bar{\Gamma}^l_{im}-\frac{\partial J_i}{\partial y^l}\bar{N}^l_m.
F(\mathbf{B}_B) = 4(\kappa-3\nu/4)(\kappa-3\nu/4-1)(\kappa-\nu/2-2)+4(\kappa-3\nu/4)(\nu/4)(\kappa-\nu/2-1) + 4(\nu/4)(\kappa-3\nu/4)(\kappa-\nu/2-1)+4(\nu/4)(\nu/4)(\kappa-\nu/2).
\zeta(3)=14\sum_{k=1}^{\infty}\frac{1}{k^{3}sinh(\pi k)}-\frac{11}{2}\sum_{k=1}^{\infty}\frac{1}{k^{3}(e^{2\pi k}-1)}-\frac{7}{2}\sum_{k=1}^{\infty}\frac{1}{k^{3}(e^{2\pi k}+1)}.
\begin{aligned}{\dot r}= -\frac{1}{N}\sum_{j=1}^{N}\sin (\theta_j- \phi) \Big( \omega_j - K r\sin (\theta_j - \phi) \Big),{\dot \phi}= \frac{1}{rN}\sum_{j=1}^{N}\cos (\theta_j - \phi)\Big( \omega_j - K r\sin (\theta_j - \phi) \Big).\end{aligned}
\lim_{\varepsilon\rightarrow0}\varepsilon^{2}\left\vert \left\vert \nabla Z_{\varepsilon,\xi}\right\vert \right\vert_{2,\varepsilon}^{2}=\int_{\mathbb{R}_{+}^{n}}\left\vert \nabla V\right\vert^{2}(y)dy
\# \left\{\{x_h\in\mathbb{Z}_p\mid s_{h}x_h \ne 0 \}\bigcap \{x_h\in\mathbb{Z}_q\mid x_h \ne x_l ,\ \l < h \}\right\}=\, q-\#\left\{\{x_h\in\mathbb{Z}_p\ |\s_{h}x_h=0\}\bigcup\{x_1,\ldots ,x_{h-1}\}\right\}=\, q-d_h-h+1.
\begin{aligned}U(\mathcal{O}f ) = U\bigg[\Big(-\frac{d}{dx}+2x\Big)\frac{d}{dx}f\bigg] &=\Big(-\frac{d}{dx}+x\Big)\Big(\frac{d}{dx}+x\Big)Uf &=\frac12\Big(\Big(\frac{d}{dx}+x\Big)\Big(-\frac{d}{dx}+x\Big)+\Big(-\frac{d}{dx}+x\Big)\Big(\frac{d}{dx}+x\Big)\Big)Uf- Uf &=\mathcal{H}(Uf)-Uf.\end{aligned}
\sum\limits_{i = 1}^k{({\lambda_{k + 1}}-{\lambda_i}}{)^\alpha}\le \sqrt{\frac{2}{n}}{\left({\sum\limits_{i = 1}^k{({\lambda_{k + 1}}-{\lambda_i}}{)^\beta}\sum\limits_{i = 1}^k{{{\left({{\lambda_{k + 1}}-{\lambda_i}}\right)}^{2\alpha - \beta - 1}}}{\lambda_i}}\right)^{\frac{1}{2}}}.
|sA\Delta A| = |sA\cup A\backslash sA\cap A| & = |sA\cup A|- |sA\cap A| & = |sA|+|A| - 2|sA\cap A| & = 2|A| - 2|sA\cap A| & \geqslant 2|A| - 2|SA\cap A| & \geqslant \left(2-\frac{2\beta}{\epsilon}\right)|A|.
\left\langle\mu_{\frac{k-2}{2}},(y^{k-2})^\vee\right\rangle'=(y^{k-2})^\vee\mu_{\frac{k-2}{2}}\left(\left|\begin{array}{cc}X Y x y\end{array}\right|^{k-2}\right)=(y^{k-2})^\vee\left(y^{k-2}\right)=1.
L_{n}( \lambda , t ) = \left( \begin{array}{c c}{{t^{Z_{n}- S_{n}^{3}}\; s h [ \gamma ( \lambda + i S_{n}^{3}) ]}}&{{i S_{n}^{-}\; s i n \gamma}}\\{{i S_{n}^{+}s i n \gamma}}&{{t^{- Z_{n}- S_{n}^{3}}\; s h [ \gamma ( \lambda - i S_{n}^{3}) ]}}\\ \end{array}\right) \; ,
J^{(1)}_{jl}(t, \tau) = t |t| \sum_{r=1}^{i_0}\sum_{q=1}^{k_r}\left( \int_0^{\tau}e^{i \tilde{\tau}\bigl( \sqrt{\mathstrut \lambda^{(r)}_q (t)}-\sqrt{\mathstrut \gamma^{\circ}_l}|t|\bigr)}d \tilde{\tau}\right) ( P_j G_* P_l \cdot, \varphi^{(r)}_{q}(t))\varphi^{(r)}_{q}(t).
F_{\mu \nu}= \mathrm{t r}\, \Psi^{\dagger}\left( \partial_{\left[ \mu \right.}\mathcal{D}_{z}^{\dagger}\frac{1}{\mathcal{D}_{z}\mathcal{D}_{z}^{\dagger}}\partial_{\left. \nu \right]}\mathcal{D}_{z}\right) \Psi \, .
S_{ii}\: = \: - \frac{1}{z + \frac{1}{n}\alpha_i^* S_i \alpha_i}= \frac{1}{-z - s^1(z)}+ \frac{- s^1(z) + \frac{1}{n}\alpha_i^* S_i \alpha_i}{(-z - \frac{1}{n}\alpha_i^* S_i \alpha_i)(-z-s^1(z))}.
F_y(u)\, =\, - |\pi_{-1}u|^2 +|\pi_{1}u|^2 + 5|\pi_{5}u|^2 + |\pi_0 u|^\frac{3}{2}+ 2|\pi_{-1}u|^2 &=\, |\pi_{-1}u|^2 +|\pi_{1}u|^2 + 5|\pi_{5}u|^2 + |\pi_0 u|^\frac{3}{2}.
\sigma^{2}=\sigma^{2}(K)\stackrel{\rm def}{=}||K^{2}||\int_{-1}^{1}\mbox{{\rm cov}}\left( \left| \sqrt{1-\rho^{2}(t)}\,Z_{1}+\rho(t)\,Z_{2}\right| ,\left| Z_{2}\right| \right) dt.
[L^{\pm}(u),c]=0, R(u-v)L_1^{\pm}(u)L_2^{\pm}(v)=L_2^{\pm}(v)L_1^{\pm}(u)R(u-v), R(u-v-\frac{1}{2}hc)L_1^{+}(u)L_2^{-}(v)=L_2^{-}(v)L_1^{+}(u)R(u-v+\frac{1}{2}hc),
R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}= -{\frac{2 r_{-}}{r_{+}^{3}}}, R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}={\frac{2 r_{+}- 4 r_{-}}{r_{+}^{3}}}~
\begin{aligned}C_{\phi,n}^2 =(\phi/\psi_{\gamma_n}^2, R^n_1(\phi/\psi_{\gamma_n}^2))_{H_n}=\int_0^\infty \mathrm{e}^{-t}\cdot (\phi/\psi_{\gamma_n}^2, Q^n_t(\phi/\psi_{\gamma_n}^2))_{H_n}dt \leq (\phi/\psi_{\gamma_n}^2, \phi/\psi_{\gamma_n}^2)_{H_n}=\int \left(\phi/\psi_{\gamma_n}\right)^2(x)dx\leq \int \left(\phi/\psi_{\gamma_1}\right)^2(x)dx=C_\phi^2.\end{aligned}
\varphi_{\mathrm{rc}}(0;x_{0},\omega) =x_{0}, \varphi_{\mathrm{rc}}(t;x_{0},\omega) = \Phi^{\pmb\alpha(\omega)(s_{n}(\omega))}_{t - s_{n}(\omega)}\varphi_{\mathrm{rc}}(s_{n}(\omega);x_{0},\omega),n\in\mathbb{N}t\in\left(s_{n}(\omega),s_{n+1}(\omega)\right] ,
S_{0}= \int d^{D}x \left[ \pi^{i}\dot{A}_{i}-{\frac{1}{2}}{\pi^{i}}^{2}-{\frac{1}{4}}F_{i j}^{2}+ i \bar{\psi}\gamma^{\mu}D_{\mu}\psi + \lambda^{1}( J^{0}+ \partial_{i}\pi^{i}) + \lambda^{2}( \partial_{i}A^{i}) \right] \, ,
= \frac{1}{2}\times \frac{3}{2}\times \frac{2}{3}\times \frac{4}{3}\times \cdots \times \frac{2 0 1 4}{2 0 1 5}\times \frac{2 0 1 6}{2 0 1 5}\times \frac{2 0 1 5}{2 0 1 6}
{\cal D}_{\alpha \beta}= \left( \begin{matrix}{{\partial_{t}^{2}+ \beta_{k}- i \alpha_{k}}}&{{\gamma_{k}\partial_{t}+ \textstyle{\frac{1}{2}}\dot{\gamma}_{k}+ i \alpha_{k}}}\\{{- \gamma_{k}\partial_{t}- \textstyle{\frac{1}{2}}\dot{\gamma}_{k}+ i \alpha_{k}}}&{{- \partial_{t}^{2}- \beta_{k}- i \alpha_{k}}}\\ \end{matrix}\right) \, ,
G_\omega^\theta(t-t')={e^{-i\theta/2}\over 4\omega}\biggl[{e^{i\omega(t-t'-\tau/2)}\over\sin({\omega\tau+\theta\over2})}+{e^{-i\omega(t-t'-\tau/2)}\over\sin({\omega\tau-\theta\over2})}\biggr]\;\; ;\;\; t-t' epsilon\;[0,\tau)
\begin{array}{l l l}{{\langle \alpha | \phi ( x ) \phi ( y ) | \alpha \rangle}}&{{=}}&{{N_{\alpha}^{2}\left( G_{E}( x , y ) + | \gamma |^{2}G_{E}( \bar{x}, \bar{y}) + \gamma G_{E}( \bar{x}, y ) + \gamma^{*}G_{E}( x , \bar{y}) \right)}}\\{{}}&{{\equiv}}&{{G_{\alpha}( x , y )}}\\ \end{array}
Q_{(q_1,\ldots, q_k)}=\sum_{p_1=0}^\infty\cdots \sum_{p_k=0}^\infty Q_{(q_1,\ldots, q_k)}\Phi_{T_1}^{p_1}\circ \cdots \circ \Phi_{T_k}^{p_k}({\bf \Delta_{T}}(I))
A_{ij}=\Gamma_{ij}{}^1,{B^i}_{jk}=2\Gamma_{1(j}{}^1\delta_{k)}{}^i-\Gamma_{jk}{}^i,{C^i}_j=\Gamma_{11}{}^1\delta_j{}^i-2\Gamma_{1j}{}^i, D^i=-\Gamma_{11}{}^i.
|j *_{\{0, s \}}A| \ge |A + ((j - 1) *_{\{0, s \}}A)| + |J| \ge |A| + |(j - 1) *_{\{0, s \}}A| + |J| - |H| \ge |A| + |(j - 1) *_{\{0, s \}}A| \ge |A| + (j-1) |A| = j |A|,
exp\bigg(+i\tilde{L}_{D}\bigg)=\int{\cal D}G{\cal D}B\exp\bigg(+i\int_{X}d^{3}x\varepsilon^{ijk}(aB_{i}^{AB}H_{jkAB}+bA_{i}^{AB}G_{jkAB}+cB_{i}^{AB}G_{jkAB})\bigg).
k_{y}^2 = 3(-u_{(1,3,2)}+ u_{(1,2,3)}- u_{(1,4,3)}+u_{(1,3,4)}-u_{(1,4,2)}+2u_\mathrm{Id}-u_{(2,4,2)}+ u_{(1,2,3)}-u_{(2,3,4)})
\frac{{n-t-k-1\choose k-t-1}}{{n-t-1\choose k-t-1}}=\frac{n-2k+1}{n-t-k}\cdot\frac{{n-t-k\choose k-t-1}}{{n-t-1\choose k-t-1}}=\ldots = \prod_{i=1}^{k}\frac{n-k+1-i}{n-t-i}.
E_1(Y_1)=-T,~~E_2(Y_1)=0,~~E_3(Y_1)=0, &E_1(T)=QY_1,~~E_2(T)=\frac{B_{12,1}}{b_1-b_2}T,~~E_3(T)=\frac{B_{13,1}}{b_1-b_3}T.
\mathcal{F}_h(u) = \frac{h}{2\tau}\sum_{j=1}^N \sum_{i=1}^{s}\frac{\hat{m}^2_{i,j}}{\rho_{i,j}}D^{-1}_{i,j}+h\sum_{j=1}^{N}\left( \sum_{i=1}^s \rho_{i,j}\log \rho_{i,j}+\frac{\epsilon_j}{8h^2}(\phi_{j+1}-\phi_{j-1})^2\right) & +\frac{1}{8\beta_a}(\phi_0+\phi_1)^2+\frac{1}{8\beta_b}(\phi_N+\phi_{N+1})^2,
\sum_{n=1}^{N+1}\frac{(q^2;q^2)_{n-1}(q^2;q^2)_N(zq;q^2)_{N-n+1}(zq)^{n}}{(zq^2;q^2)_{n}(q^2;q^2)_{N-n+1}(zq;q^2)_{N+1}}&=\frac{1}{(1-q^{2N+2})}\sum_{n=1}^{N+1}\left(\frac{(q;q)_{2n-2}z^{2n-1}q^{n(2n-1)}}{(zq;q)_{2n-1}}+\frac{(q;q)_{2n-1}z^{2n}q^{n(2n+1)}}{(zq;q)_{2n}}\right)\frac{\left(q^{2N-2n+4};q^2\right)_n}{\left(zq^{2N+3};q^2\right)_n}.
\min\{\iota(x),1\}\ge \sup \bigl\{t\in (0,1] \mid c_1 \mathrm{e}^{-c_2 b(x)}\geq t \mathrm{e}^{c_2}\bigr\}=\frac{c_1}{\mathrm{e}^{c_2}}\mathrm{e}^{-c_2 b(x)}=:r_0(x).
c_{n,k}(p(d)) = \sum_{j= 1}^{n-k+1}p_j(d) c_{n-j,k-1}(p(d)) = \sum_{j= 1}^{n-k+1}\binom{j+d-1}{j-1}\binom{n-j+d(k-1)-1}{n-j-k+1}= \sum_{j= 0}^{n-k}\binom{j+d}{j}\binom{n-j+d(k-1)-2}{n-k-j}= (-1)^{n-k}\sum_{j= 0}^{n-k}\binom{-d-1}{j}\binom{-d(k-1)-k+1}{n-k-j}= (-1)^{n-k}\binom{-dk-k}{n-k}= \binom{n + dk - 1}{n-k}.
\delta_{k}^{(\ell)}=\begin{cases}2^{1-2s\ell}\gamma\beta k=1 2^{2\ell-4s^2}\gamma\beta^2 k=2 2^{1+2\ell-4s^2}\gamma\beta^2 k=3 2^{-2s\ell}\gamma\beta k=4 \end{cases}.
(a) ~ \mbox{for all}T>0, & \lim\limits_{t\rightarrow\infty}\sup\limits_{k}\left(\|\sum\limits_{i=t}^{k-1}\alpha_{i+1}M({i+1}) \|:\sum\limits_{i=t}^{k-1}\alpha_{i+1}\leq T \right) = 0, (b) ~ \sup_{t\geq1}\|x(t)\|<\infty,
G^*(z\,|\,\omega_1,\widetilde\omega_1,\omega_2)\cdot G^*(z\,|\,\omega_1,\widetilde\omega_1,-\omega_2)= \prod_{k_1\geq 0, k_2\geq 0}\big(1-x_2 q_2^{k_1+\frac{1}{2}}\widetilde{q}_2^{k_2+\frac{1}{2}}\big)\cdot\prod_{k_1\geq 0, k_2\geq 0}\big(1-x_2^{-1}q_2^{k_1+\frac{1}{2}}\widetilde{q}_2^{k_2+\frac{1}{2}}\big).
u \tau \log(1/\tau) \le (24ku^3/m) \log( m / (24ku^2) ) \le 20^{3/4}48k \sqrt{n}\left(\frac{n}{m}\right)^{1/4}\log \frac{(m/n)^{1/4}}{(24k)^{1/2}(20)^{1/4}}=: f_1(m),
\eta^{[i,j]}_{\textrm{\textrm{SUM}}}= \sum_{k=1, k\neq i}^{K}\left\|{\mathbf{U}_k}^{H}\mathbf{H}_{k}^{[i,j]}\mathbf{w}^{[i,j]}\right\|^2\triangleq \left\| \mathbf{G}^{[i,j]}\mathbf{w}^{[i,j]}\right\|^2,