harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
z_1 =z_0+\O{}\left(\frac{\sup_{|z-z_0|\leq 1}\left|a(z,mq)\left[1+\exp\left(-\frac{4imq}{3}(-\zeta(c^{-1}z))^{3/2}\right)\right]\right|}{\inf_{|z-z_0|\leq 1}|\partial_zH(z,mq)|}\right) &=z_0+\O{}((mq)^{-5/3})
U^\lambda \omega^\lambda (U^\lambda)^* = \omega^{(0)}+ \lambda U^{(0)}\omega^{(1)}(U^{(0)})^* &\qquad- \lambda \Big[\omega^{(0)}, \sum_{i}\pi_i v \pi_i (-\i\tau) + \sum_{i \neq j}\pi_i v \pi_j \Big( \frac{\exp(-\i\tau(E_i-E_j))-1}{E_i-E_j}\Big)\Big] + O(\lambda^2).
\Psi^*g_0 = V(dx_1)^2 + \frac{1}{4x_1^2}[\rho^2 -2\rho^2(m) + g_{x_1,0,0}(\xi_3(m),\xi_3(m))]((dx_2)^2+(dx_3)^2)+ + \beta_3dx_2 - \beta_2dx_3 + g_{x_1,0,0}+ \frac{1}{V}\eta^2.
\mathbf{E}_{\omega,T}|Z_{T}-f_{\omega}^{*}| \geq \frac{1}{2}\mathbf{E}_{\omega,T}|f_{\omega}^{*}-f_{\hat{\omega}}^{*}| =\frac{1}{2}\big[\frac{h_{T}^{2\beta-2}}{\tilde{\alpha}}\big]\mathbf{E}_{\omega,T}\rho(\hat{\omega},\omega),
d\widetilde{\beta}(\tilde{x},\tilde{y},\tilde{z}) =-\widetilde{\beta}([\tilde{x},\tilde{y}],\tilde{z})-\widetilde{\beta}([\tilde{z},\tilde{x}],\tilde{y})-\widetilde{\beta}([\tilde{y},\tilde{z}],\tilde{x}) &=-\beta([{x},{y}],{z})-\beta([{z},{x}],{y})-\beta([{y},{z}],{x}).
b (x,y,z) - b\left(\frac{x+y}{2},\frac{x+y}{2},z\right) = \left(\rho^{x/2}-\rho^{y/2}\right)^2 - \left(\rho^{(n-x)/2}-\rho^{(n-y)/2}\right)^2 = (1 - r^{n-x-y})\left(\rho^{x/2}-\rho^{y/2}\right)^2 \geq 0,
P_n(x)=\left|\begin{array}{cccc}m_0 \cdots m_{n-1}\mathbb{I}_p \vdots \vdots \vdots m_{n-1}\cdots m_{2n-1}x^{n-1}\mathbb{I}_p m_n \cdots m_{2n-1}\boxed{x^n\mathbb{I}_p}\end{array}\right|.
\left|U(m_1,m_2, a_r)\right| = \left| S_{r,j}(m_1,m_2,a_r) - \sum_{x=0}^{\tau_{r,j}-1}e\Big(2\pi i \Big( m_1 \alpha \lambda_j^{n_{r,j}+x}+ \frac{m_2 x}{\tau_{r,j}}\Big)\Big)\right|.
z ( 1 - z ) \frac{\partial^{2}}{\partial z^{2}}U_{\epsilon}( z , \bar{z}) - \left[ c_{\epsilon}- ( a_{\epsilon}+ b_{\epsilon}+ 1 ) z \right] \frac{\partial}{\partial z}U_{\epsilon}( z , \bar{z}) - a_{\epsilon}b_{\epsilon}U_{\epsilon}( z , \bar{z}) = 0 ,
T(\Omega) =\frac{1}{n+2}\int_{\partial\Omega}X\cdot\nu_{\Omega}(X)|\overline{\nabla}U(X)|^{2}d\mathcal{H}^{n-1}(X) &=\frac{1}{n+2}\int_{\S^{n-1}}h_{\Omega}(x)|\overline{\nabla}U(\nu^{-1}_{\Omega}(x))|^{2}dS_{\Omega}(x).
{\alpha}^2 - \frac{3}{4}(\sqrt{2}-1)^2 N^2+ \frac{1}{2}\sqrt{C(M_0) \left( \frac{9}{4}- \frac{\alpha^2}{N^2}\right)}M_0^{-\frac{1}{2}}N^2 =0, &{\beta}^2 - \frac{3}{4}(\sqrt{2}-1)^2 N^2- \frac{1}{2}\sqrt{C(M_0) \left( \frac{9}{4}- \frac{\beta^2}{N^2}\right)}M_0^{-\frac{1}{2}}N^2 =0.
\begin{aligned}m_{{i_0}+1}=\sum_{c \in C-F}m_{i_0+1}(c) &=\sum_{c \in C}m_{i_0+1}(c) - \sum_{c \in F_1 \cup F_2}m_{i_0+1}(c) & \ge k|R_{{i_0}+1}|-(|R_{{i_0}+1}|-1)(k-{i_0})-|R_{{i_0}+1}|{i_0}=k-{i_0},\end{aligned}
\begin{array}{ll}{\cal A}_c(\overline{Y},\overline{\mu},\overline{\lambda},W_1,W_2):= \nabla^2_{xx}L(\overline{x},\overline{Y},\overline{\mu},\overline{\lambda})+{\rm D}F(\overline x)^*W_1{\rm D}F(\overline x) begin{align*}6pt] + c{\cal J}h(\overline{x})^T{\cal J}h(\overline{x})+c{\rm D}g(\overline{x})^*W_2{\rm D}g(\overline{x}).\end{array}
\int_{0}^{\infty}\frac{\partial}{\partial u}\mathbb{E}\left[e^{-u\eta}\right]u^{\theta-1}\mathrm{d}u =-\int_{0}^{\infty}\mathbb{E}\left[Ye^{-u\eta}\right]u^{\theta-1}\mathrm{d}u =-\mathbb{E}\left[\int_{0}^{\infty}\eta e^{-u\eta}u^{\theta-1}\mathrm{d}u\right]=-\mathbb{E}\left[\Gamma(\theta)\eta^{1-\theta}\right],
\ell_j =\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle+\left\langle\frac{1}{2}+\frac{j}{p-1}\right\rangle+\left(\frac{j}{p-1}\right) &=1-\left\lfloor\frac{1}{2}-\frac{2j}{p-1}\right\rfloor-\left\lfloor\frac{1}{2}+\frac{j}{p-1}\right\rfloor.
\begin{array}{l}\textbf{Km}= \textbf{BR}[4,\{2,1,3,2,2,1,3,2,2,1,3,2, -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1\}] \textbf{ColouredJones}[\textbf{Km},1][q^4](q^4-q^{-4})/(q^2-q^{-2}) -q^{-30}+q^{-6}+q^{-2}+q^2+q^6+q^{10}-q^{22}-q^{26}-q^{30}+q^{38}\end{array}
y_i^{(l)}=\textrm{Pr}\left\{Z\leq\dfrac{X}{\mu}-\dfrac{1}{\rho}\right\}=\displaystyle\int_0^\infty\displaystyle\int_0^{\frac{x}{\mu}{-}\frac{1}{\rho}}p_{X,Z}(x,z) dzdx,
T r \left[ \left( U \overline{{{U}}}({\cal D}_{\alpha}^{+}\overline{{{V}}}V ) + \overline{{{V}}}V ({\cal D}_{\alpha}^{+}U \overline{{{U}}}) \right)_{\j \B}^{i \A}( U^{3}V^{2})^{k}\right]
\frac{d}{dt}\int_\Omega\left(1+\lambda \gamma(v)\right)u^2+ 2\lambda\int_\Omega|\nabla(u\gamma(v))|^2+2\int_\Omega\gamma(v)|\nabla u|^2+\lambda\int_\Omega \gamma''(v) u^2|\nabla v|^2 - \lambda\int_\Omega u^3\gamma'(v)=-\int_\Omega\left(2+2\lambda\right)u\gamma'(v)\nabla u\cdot\nabla v-\lambda\int_\Omega u^2\gamma'(v)v.
\begin{aligned}T_{\lambda, k}((F*G)_{\lambda}^{(g_1,g_2;h_1,h_2)})(y) &=T_{\lambda,\mathbf{s}(g_1k,h_1)}(F)\big(\mathcal Z_{g_1}(y,\cdot)\big)T_{\lambda,\mathbf{s}(g_2k,h_2)}(G)\big(\mathcal Z_{g_2}(y,\cdot)\big). \end{aligned}
\alpha^{(m)}={\frac{\prod_{i=1}^{s+1}b_i^{(m)}}{\prod_{i=1}^ra_i\cdot\bigl(\alpha^{(m)}\bigr)^{s-r}}}\ge{\frac{\prod_{i=1}^rb_i}{\prod_{i=1}^ra_i}}\biggl({\frac{b_s}{\alpha^{(m)}}}\biggr)^{s-r}b_{s+1}^{(m)}\ge b_{s+1}^{(m)}\,,
\int_{\{\Psi<-t\}}|\frac{f_1+f_2}{2}|^2_hc(-\Psi)+\int_{\{\Psi<-t\}}|\frac{f_1-f_2}{2}|^2_hc(-\Psi) =\frac{1}{2}(\int_{\{\Psi<-t\}}|f_1|^2_hc(-\Psi)+\int_{\{\Psi<-t\}}|f_1|^2_hc(-\Psi))=G(t),
x \mapsto \psi(x_1,x_2)=\left(\begin{array}{cc}1/\sqrt{v_{11}}0 0 1\end{array}\right) \left[V^{1/2}\left(\begin{array}{c}x_1 x_2\end{array}\right) + \left(\begin{array}{c}-v_{12}{-v_{22}/2}\end{array}\right)\right].
C_m = \frac{12^{3m}}{m!}\cdot\frac{(a)_m(a+1/3)_m(a+2/3)_m}{(a-b+1)_m(a-c+1)_m}=\ \frac{(e-1)n+3r}{m!}\cdot\left(\frac{2^8}{n}\right)^m\cdot \prod_{i=2m+e}^{3m+e-2}(ni+3r).
L_{1}=\langle 1\rangle^{\perp n-2}\perp \langle 1,-\Delta \rangle,L_{2}=\langle 1\rangle^{\perp n-2}\perp \langle\pi,-\Delta\pi \rangle, &L_{3}= \langle 1\rangle^{\perp n-2}\perp \langle -1, -\Delta\pi \rangle, L_{4}=\langle 1 \rangle^{\perp n-2}\perp \langle -\Delta,-\pi \rangle.
\varphi (f\circ g) =((f\circ g)_{11},\ldots,(f\circ g)_{nn}) =(f_{11}\circ g_{11},\ldots,f_{nn}\circ g_{nn}) =(f_{11},\ldots,f_{nn})\circ (g_{11},\ldots,g_{nn}) =\varphi (f)\circ \varphi (g).
{\cal P}^{a \mu}= \alpha_{2}\sqrt{- \gamma}\left[ \left( K_{i}K^{i}\gamma^{a b}- 2 K_{i}K^{a b \, i}- 2{\cal G}^{a b}\right) e^{\mu}{}_{b}+ 2 ( \widetilde \nabla^{a}K^{i}) n^{\mu}{}_{i}\right] \, .
\int_{\mathbb{R}^{2N}}\frac{|w(x) - w(y)|^{p-2}\, (w(x) - w(y))\, (\varphi(x) - \varphi(y))}{|x - y|^{N+s\,p}}\, dx\, dy=\int_{\Omega}\frac{f_n(x)}{(u^++1/n)^\gamma}\,\varphi\, dx.
I ( \mu^{2}) = - \mathrm{T r}_{-}\left( \Gamma_{7}\frac{\mu^{2}}{- ( \Gamma \cdot \Delta )^{2}+ \mu^{2}}\right) = - \mathrm{T r}\left( P_{-}\Gamma_{7}\frac{\mu^{2}}{- ( \Gamma \cdot \Delta )^{2}+ \mu^{2}}\right)
\sum_{\substack{\lambda=1 \{\lambda:\, \varepsilon_j(\lambda,N,\Sigma)=h+1, \, \varepsilon_j(\lambda,N,\Sigma^{\ast})=h\}}}^{b^n}1=b^{j-1}(\nu_j(N,\Sigma^{\ast})-\nu_j(N,\Sigma))
\lim_{r\to 0^+}\frac{A(r)^2}{A'(r)}\cdot \left[\mathcal W(r)^{p-1-\alpha}\right]' = \lim_{r\to 0^+}A(r) \cdot \left[ Q_1(\mathcal V(r), v_0(r)) - \mathcal W(r)^{p-1-\alpha}\right] &= \lim_{r\to 0^+}\left[ r^\delta u_0'(r)^{p-1-\alpha}Q_1(\mathcal V(r), v_0(r)) -r^\delta u'(r)^{p-1-\alpha}\right]
\sum_{q > 0}\{\hat{\mathrm{G}}_{+}( q ) [ \hat{v}_{\mathrm{H , +}}( q ) , \hat{\tilde{\mathrm{G}}}_{\pm}( p ) ]_{-}+ \hat{\mathrm{G}}_{-}( q ) [ \hat{v}_{\mathrm{H , -}}( q ) , \hat{\tilde{\mathrm{G}}}_{\pm}( p ) ]_{-}\}\mp{\hbar}^{2}\frac{e^{2}\mathrm{L}^{2}}{8{\pi}^{2}}p \hat{v}_{\mathrm{H , \mp}}( p )
\left\Vert(\Delta+z^2)^{-d}(p)-(4\pi)^{-\frac{m}{2}}\sum_{l=0}^{k+m/2-d}z^{-2d+m-2l}u_l(p)\frac{\Gamma(-\frac{m}{2}+d+l)}{(d-1)!}\right\Vert &\leq \tilde{C}_k(K)z^{-2d+m-2l-2},
\begin{aligned}c_1 \left(t^{-d/2}\wedge (at)^{-d/\beta}\right) \wedge \left( p_0(t,c_2x, c_2y)+ \frac{at}{|x-y|^{d+\beta}}\right) \leq p_a(t, x, y) \leq c_3\left(t^{-d/2}\wedge (at)^{-d/\beta}\right) \wedge \left( p_0(t,c_4x, c_4y)+ \frac{at}{|x-y|^{d+\beta}}\right). \end{aligned}
\arraycolsep=2.5pt\def\arraystretch{1.2}\newcommand*{\temp}{\multicolumn{1}{|}{}}\left[\begin{array}{ccccccccccccccc}x_0 x_1 \cdots x_{s-1}x_s \cdots x_{m-1}\temp y_0 y_1 \cdots y_{n-1}\cline{1-12}0 0 \cdots 0 z_0 \cdots z_{n-1}\temp w_0 w_1 \cdots w_{n-1}\end{array}\right].
\varepsilon_{0}=\frac{E_{0}}{AL_{z}}=\frac{eB}{(2\pi)^{2}}\int_{-\infty}^{\infty}dp_{z}\left(\sqrt{m^{2}+p_{z}^{2}}- 2\sum_{n=0}^{\infty}\sqrt{m^{2}+p_{z}^{2}+2eBn}\right).
\int_{Q_{e,V}}\bar{\sigma}_{\underline{\vec{a}}_1}\cup \bar{\sigma}_{\underline{\vec{a}}_2}= \int_{Q_{e+r\ell,C}}\big(\bar{\sigma}_{\underline{\vec{a}}_1}\cup \bar{\sigma}_{1^r}^{(r+s)\ell_1-d_1}\big) \cup \big( \bar{\sigma}_{\underline{\vec{a}}_2}\cup \bar{\sigma}_{1^r}^{(r+s)\ell_2-d_2}\big).
\sum_{\substack{1 \leq b \leq \theta, (b,t)=1}}1 \ll \theta \prod_{\substack{p \mid t, p \leq \theta}}\left(1 - \frac{1}{p}\right) = \theta \frac{\varphi(t)}{t}\prod_{\substack{p \mid t, p > \theta}}\left(1 - \frac{1}{p}\right)^{-1},
p_{sd}\!= \!\frac{1}{\varepsilon^2}\left \{\frac{\Gamma\!-\!\frac{m^2}{n}}{n\!-\!1}\!+\! \frac{m^2\!-\!1\!-\!(\Gamma\!-\!1)n}{n(n\!-\!1)}\left(1\!-\!\frac{1}{m^2}\right)^{n\!-\!1}\right \},
\limsup_{y\uparrow1}\limsup_{x\to\infty}\mathcal{I}_2 &\leqslant \frac{2c_3}{\mathbb{P}(\eta=a)}\biggl(\limsup_{y\uparrow1}\limsup_{x\to\infty}\frac{\overline{F}_{\xi_1}(\mathit{xy})}{\overline{F}_{\xi_1}(x)}\biggr)\sum\limits_{n=K+1}^{\infty}n^{p+1}\mathbb{P}(\eta=n).
\Delta_g =\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij}\partial_j ) &=g^{ij}\partial_{ij}+\frac{1}{\sqrt{g}}\partial_i(\sqrt{g}g^{ij})\partial_j &=g^{ij}\partial_{ij}-g^{jp}\Theta_q u_{pq}\partial_j.
x_{i}+\sum_{j}\lfloor\overline{a}_{i,j}\rfloor x_{j}-\lfloor\overline{b}_{i}\rfloor=\overline{b}_{i}-\lfloor\overline{b}_{i}\rfloor-\sum_{j}(\overline{a}_{i,j}-\lfloor\overline{a}_{i,j}\rfloor)x_{j}.
\mathbf{t}^{(+)}(x_1)=\frac{G}{\pi}\sqrt{\frac{a}{x_1}}\frac{1}{(x_1+a)(H_{33}H_{44}-H_{34}^2)}\begin{pmatrix}(\gamma-\delta_4)H_{34}H_{44}\delta_4 H_{33}H_{44}-\gamma H_{34}^2\end{pmatrix}.
\sum_{r=0}^{t-1}\sum_{l=0}^{t-1}\hat{\gamma}^t_{u,r}\hat{\gamma}^t_{u,l}\tilde{E}_u^{r,l}= (\hat{\gamma}^t_u)^*\tilde{C}_u^t\hat{\gamma}^t_u = [(\tilde{E}_u^t)^*(\tilde{C}_u^t)^{-1}]\tilde{C}_u^t [(\tilde{C}_u^t)^{-1}\tilde{E}_u^t] = (\tilde{E}_u^t)^*(\tilde{C}_u^t)^{-1}\tilde{E}_u^t = (\sigma_u^t)^2 - (\sigma_u^t)^2_\perp.
k_{n + 1}= \frac{C^{(\alpha)}_{2 n}(1) N (D, 2 n + 2)}{C^{(\alpha)}_{2 n + 2}(1) N (D, 2 n)}\frac{(2 n - 1) (n + \alpha)}{(n + 1) (2 n + 2 \alpha + 3)}k_n .
f^{\lambda}= \left( f_{1}e_{a}^{\lambda}e^{b \nu}+ f_{2}e^{b \lambda}e_{a}^{\nu}+ f_{3}g^{\nu \lambda}\delta_{b}^{a}\right) \bar{\Omega}_{~ b \nu}^{a}+ N_{~ b}^{\mu \alpha \beta ~ b \nu}\nabla_{\alpha}\nabla_{\beta}\bar{\Omega}_{~ b \nu}^{a}
\beta=(\begin{matrix}\frac{\partial u_{1}}{x_{1}}&\frac{\partial u_{1}}{x_{2}}&\frac{\partial u_{1}}{x_{3}}\\ \frac{\partial u_{2}}{x_{1}}&\frac{\partial u_{2}}{x_{2}}&\frac{\partial u_{2}}{x_{3}}\\ \frac{\partial u_{3}}{x_{1}}&\frac{\partial u_{3}}{x_{2}}&\frac{\partial u_{3}}{x_{3}}\end{matrix}),r=(\begin{matrix}r_{1}\\r_{2}\\r_{3}\\ \end{matrix})
\bigg|\sum_{i=1}^d D_iq_{ij}^{(n)}\bigg| \leq \bigg|\sum_{i=1}^d D_iq_{ij}\bigg|+ \frac{2}{W_1}(|Q\nabla_y W_1| + \eta |\nabla_y W_1|) \leq (c_5+4c_9) w^{-\frac{1}{k}}W_2^\frac{1}{k}.
\overline\Delta_b(a_1 a_2 a_3) = a_2 a_3 \otimes a_1 + a_1 a_2 \otimes a_3 + a_1 | a_3 \otimes a_2 &+ a_1 \otimes a_2a_3 + a_2 \otimes a_1 | a_3 + a_3 \otimes a_1 a_2.
\frac{(1-z)^{-t+1}I(z)}{\tfrac{(-1)^{\ell-t}}{(t+1)^{(\ell-t)}}}=\,(-1)^{\ell-t}\sum_{j=1}^{\ell-1}\frac{(t+1)^{(j-1)}(j-1)_{\ell-t}}{(N+2)^{(j)}}(1-z)^{j-\ell}&+\frac{(t+1)^{(\ell-1)}}{(N+2)^{(\ell-1)}}\sum_{\mu=0}^{\ell-t}(-1)^{\mu}\binom{\ell-t}{\mu}(\ell-1)_{\mu}(1-z)^{\ell-t-\mu}\,\frac{d^{\ell-t-\mu}K_{\ell}(z)}{dz^{\ell-t-\mu}}.
(1 + \rho_{n}\alpha_{n-1}) (1 - \rho_{n-1}\alpha_{n-1}) = (1 + \overline{\rho}_{n}\overline{\alpha}_{n-1}) (1 - \overline{\rho}_{n-1}\,\overline{\alpha}_{n-1}) = 1 - |\alpha_{n-1}|^2, n \geq 0,
\widetilde{\boldsymbol{\mathrm{K}}}_{t}=\mathbb{E}\{\widetilde{\boldsymbol{\mathrm{x}}}_{t}\widetilde{\boldsymbol{\mathrm{x}}_{t}}^{\dagger}\}=\mathbb{E}\{\boldsymbol{\mathrm{V}}_t^{\dagger}\boldsymbol{\mathrm{x}}_t\boldsymbol{\mathrm{x}}_{t}^{\dagger}\boldsymbol{\mathrm{V}}_t\}=\boldsymbol{\mathrm{V}}_t^{\dagger}\boldsymbol{\mathrm{K}}_{t}\boldsymbol{\mathrm{V}}_t.
u^h(t)-v^h(t)= \int_{0}^{t}\big(\Delta u^h(s)-\Delta v^h(s)\big)ds+\int_{0}^{t}\big(u^h(s)\log|u^h(s)|-v^h(s) &\log|v^h(s)|\big)ds+\int_{0}^{t}\big(\sigma(u^h(s))-\sigma(v^h(s))\big)h(s)ds.
\int_0^\infty(AL_1^u(y)+BL_1^v(y))\mathrm{e}^{-\lambda y^2}\mathrm{d}y=\frac{1}{(1+\frac{4t^\alpha}{\alpha}\lambda)^{\frac{c+1+\sqrt{mn}}{2}}}\mathrm{e}^{-\frac{\lambda x^2}{(1+\frac{4t^\alpha}{\alpha}\lambda)}},
\left|P\cap \mathbb{Z}^n \right| = \frac{e^{g(z)}}{(2\pi)^{k_1+\ldots+k_{\nu}-\nu+1}}\int_{\Pi}e^{-i\left<t,b\right>}\prod_{j=1}^{n}\frac{1}{1+\zeta_j-\zeta_j e^{i\left<a_j,t\right>}}dt.
e_G(A',B') = e_{\overline{G}}(A') + e_{\overline{G}}(B') - \frac12 \left( |A'|^2 +|B'|^2 -n \left( D + 1\right) \right) & = e_{\overline{G}}(A') + e_{\overline{G}}(B') + \frac{(2D +2 - n )n}{4}- \frac{(|A'|-|B'|)^2}{4},
y_{n-1}^{(l)}|_{\mu=\tfrac{1}{n}}\stackrel{\rho}{\approx}\tfrac{n!}{(n{-}l)!(l{-}1)!}\displaystyle\sum_{k=0}^{n{-}l}\binom{n{-}l}{k}\tfrac{(-1)^k}{k{+}l}\left(\tfrac{n-l-k}{n}\right)^{n-l}= 1.
\begin{aligned}\dot{x}_1 =\tfrac{\bar{u}_2}{\bar{u}_1}\dot{x}_2 =\bar{u}_2 \dot{x}_3 =x_1+\tfrac{1}{2}\bar{u}_1\bar{u}_2\,,\end{aligned}\begin{aligned}\dot{\bar{u}}_1 =\bar{u}_{1,1}\dot{\bar{u}}_{1,1}=\bar{u}_{1,2}\dot{\bar{u}}_{1,2}=\bar{u}_{1,3}\dot{\bar{u}}_{1,3}=\bar{u}_{1,4}\end{aligned}
S_{22}=\left\langle \frac{2\pi}{H}\right\rangle\left\langle \frac{2\pi}{3}\right\rangle\left\langle \frac{2\pi}{H}+\frac{\pi}{3}\right\rangle\left\langle \frac{\pi}{3}-\frac{4\pi}{H}\right\rangle\left\langle \frac{\pi}{3}-\frac{6\pi}{H}\right\rangle\left\langle \frac{4\pi}{H}-\pi\right\rangle\left\langle \frac{6\pi}{H}-\pi\right\rangle \ ,
\mathcal{L}= \left(-t^2+52 t^4-624 t^6+1728 t^8\right)D_t^3+\left(-5 t+4 t^2+348 t^3-4080 t^5-576 t^6+12096 t^7\right)D_t^2 +\left(-4+16 t+516 t^2+96t^3-5520 t^4-2304t^5+17280 t^6\right)D_t +\left(8+132 t+96 t^2-1104 t^3-1152 t^4+3456 t^5\right),
I_{1}''+I_{1}'''=\delta^{2}2h_{ij}(q)\int_{\mathbb{R}_{+}^{n}}t\frac{\partial U}{\partial z_{i}}\frac{\partial v_{q}}{\partial z_{j}}dzdt+\frac{\delta^{2}}{2}\int_{\mathbb{R}_{+}^{n}}|\nabla v_{q}|^{2}dzdt+o(\delta^{2})
U_N a_0^* a_0 U_N^*=\; N-\mathcal{N}_+ U_N a^*_p a_0 U_N^*=\; a^*_p \sqrt{N-\mathcal{N}_+}=:\sqrt{N}b^*_p U_N a^*_0 a_p U_N^*=\; \sqrt{N-\mathcal{N}_+}a_p =:\sqrt{N}b_p U_N a^*_p a_q U_N^*=\; a^*_p a_q
\big(\overline{\psi}\gamma^\mu \psi\big) \gamma_\mu \psi = \begin{cases}\big(\overline{\psi}\psi \big) \psi n=2 \big(\overline{\psi}\psi \big) \psi - (\overline{\psi}\gamma^5 \psi)\gamma^5 \psi n=3. \end{cases}
\|(L_a\psi)\|^2=\int\left(\sum_{x\in [y]_E}|(L_a\psi)(x,y)|^2\right)d\mu(y)=\int\left(\sum_{x\in [y]_E}|\sum_{z\in [y]_E}a(x,z)\psi(z,y)|^2\right)d\mu(y).
\frac{\sum_{k=0}^n V_k - \frac{\mu_Q(n/\mu_{\sigma})}{\mu_{\sigma}}n}{n}~~ = \left( \frac{m_n}{n}\cdot \frac{\sum_{i=1}^{m_n}Q_i - m_n \mu_Q(m_n)}{m_n}\right)+ \frac{m_n \mu_Q(m_n) - \frac{n}{\mu_{\sigma}}\mu_Q(n/\mu_{\sigma})}{n}+ \frac{\sum_{k=0}^n V_k - \sum_{i=1}^{m_n}Q_i}{n}~~ \equiv (I) + (II) + (III).
r_i(x) = \Re\left[a_{t_ii}(x)^{\left(\frac{2\ell_in_i}{m_i}-n_i + \frac{2n_i(m_i+t_i)-2t_i}{m_i}\right)}h_{k_ii}(x)^{-2}\exp\int \frac{2n_i}{m_i}\frac{a_{(t_i-1)i}(x)}{a_{t_ii}(x)}dx\right]
\mu^*=2^{\frac{{R}}{r_s}}{\sigma^2}\left(\left(1-\rho^*\right)\left(\textstyle\prod_{k=1}^{r_s}\frac{[\boldsymbol{\Lambda}]_{k,k}^2}{\nu^*-\rho^*[\boldsymbol{\Lambda}]_{k,k}^2}\right)^{\frac{1}{r_s}}\right)^{-1}\ln2.
\alpha_{2}=\frac{1}{2}\alpha_{ab}\eta^{a}\eta^{b}+\alpha_{\;\;ab}^{\mu\nu}\eta^{a}\eta_{\mu \nu}^{b}+\frac{1}{2}\alpha_{\;\;ab}^{\mu \nu\;\lambda \rho}\eta_{\mu \nu}^{a}\eta_{\lambda \rho}^{b},
W = ( e^{i \theta}, e^{- i \theta}, \cdots , e^{i ( \pi - \theta )}, e^{- i ( \pi - \theta )}) , \quad X^{9}( 0 ) = \theta R_{9 I^{\prime}}, \quad X^{9}( 1 ) = ( \pi - \theta ) R_{9 I^{\prime}}, \quad \theta_{I}= i 2 \pi R_{9 I^{\prime}}A_{9}^{I}\quad ,
\beta_{5}=-(AB+2AC-DE)\qquad\qquad\qquad\beta_{6}=AB-2AC+DE \beta_{7}=-(AB+2AD-CE)\qquad\qquad\qquad\beta_{8}=AB-2AD+CE \beta_{9}=-(AB+2AE-CD)\qquad\qquad\qquad\beta_{10}=AB-2AE+CD
L_p(\mathrm{Sym}^2(f),s) = C^{-1}_{\kappa_0,[1]}L_p(\kappa_0,[k_0-s]), L_p(\mathrm{Sym}^2(f),s) = C \frac{\kappa(2)\tilde{L}_p(\kappa,[k-k_0+1])}{C_{\kappa_0,[1]}(1-\phi^{-2}\chi^2 \kappa(4)2^{-2k_0})}
A_{\mu}^{\mu^{\prime}}A_{\nu}^{\nu^{\prime}}( \delta_{\mu^{\prime}}^{\lambda^{\prime}}\delta_{\nu^{\prime}}^{\rho^{\prime}}-{\bar{B^{\prime}}}_{\mu^{\prime}\nu^{\prime}}^{\lambda^{\prime}\rho^{\prime}}) \frac{\partial{\bar{g}^{\prime}}_{\lambda^{\prime}\rho^{\prime}}}{\partial l_{i}}
\epsilon \, \,{}_3F_2\left(1-n, 1, \frac{3}{2}, \frac{3}{2}, 2; \epsilon\right) = \epsilon \, \,{}_2F_1(1-n, 1, 2; \epsilon) = \sum_{k=0}^{n-1}\frac{(1-n)_k}{k+1}\frac{\epsilon^{k+1}}{k!}& = \int_0^\epsilon \sum_{k=0}^{n-1}(1-n)_k \frac{u^{k}}{k!}du = \int_0^\epsilon (1-u)^{n-1}du = \frac{1-(1-\epsilon)^n}{n}.
\sum^{\infty}_{n,m=-\infty}q^{n^2+nm+2m^2}=\frac{\eta(q^2)^5\eta(q^{14})^5}{\eta(q)^2\eta(q^4)^2\eta(q^{7})^2\eta(q^{28})^2}+4\frac{\eta(q^4)^2\eta(q^{28})^2}{\eta(q^2)\eta(q^{14})}.
\langle \mu_{\hat X^1_t}(\omega), \mu_{\hat Y^1_s}(\omega^\prime) \rangle_{\mathcal{H}_{\mathcal{S}}}= \langle \mathbb{E}_{\mathbb{P}}[S(X)|\mathcal{F}_t](\omega), \mathbb{E}_{\mathbb{Q}}[S(Y)|\mathcal{G}_s](\omega^\prime) \rangle_{\mathcal{H}^1},
\log \bigg( \frac{N \mu^{k^* + 1}}{s^{k^* + 1}k^*!}k_N^{-d}e^{2s k^* a_N/k_N}\bigg) = (k_N - k^* - 1) \log \bigg( \frac{s}{\mu}\bigg) - k^* \log k^* + k^* + \frac{2 s k^* a_N}{k_N}+ o(k_N) &= \bigg( k_N - k^* - 1 + \frac{2k^*}{k_N}\bigg) \log \bigg( \frac{s}{\mu}\bigg) - k^* \log k^* + k^* + o(k_N).
\|f_2-\Pi_{V_2}f_2\|^2 \leq \bigg\|\sum_{j=1}^{q-1}(f_{2j}-\Pi_{V_{2j}}f_{2j})\bigg\|^2 &\leq (1+\epsilon_2')\sum_{j=1}^{q-1}\|f_{2j}-\Pi_{V_{2j}}f_{2j}\|^2 &\leq (1+\epsilon_2')C_2K_2^2(q-1)\left(\frac{c(1-\rho_0)(1-\epsilon_2)n}{8(q-1)\log^4 n}\right)^{-2\alpha_2}.
\left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right) \mapsto \frac{a i + b}{c i + d}, \quad \mathrm{w h e r e \}\left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right) \in S l ( 2 ) .
\widetilde{P}_{\pm}=\int_0^{x\pm}dy^{\pm}\, \widetilde{T}_{\pm\pm}(y^{\pm})\qquad\qquad\widetilde{\Delta}_{\pm}=\int_0^{x\pm}dy^{\pm}\,y^{\pm}\,\widetilde{T}_{\pm\pm}(y^{\pm})\ \ .
S ( B ) \approx S ( A ) +{\frac{1}{2}}\int \! d^{4}x \, d^{4}y \, b_{\mu}^{a}( x ) \,{\frac{\delta^{2}S ( A )}{\delta A_{\mu}^{a}( x ) \, \delta A_{\nu}^{b}( y )}}\, b_{\nu}^{b}( y ) .
\mu=\frac{\langle \hat{v},g_s^{-1}\hat{v}\rangle}{\langle \hat{v},h_s^{-1}\hat{v}\rangle}=\frac{\langle b\hat{v},A_g b\hat{v}\rangle}{\langle b\hat{v},b\hat{v}\rangle}=\frac{\langle b\hat{v},A_g A_h^{-1}b\hat{v}\rangle}{\langle b\hat{v},b\hat{v}\rangle}.
\Phi^V\wedge\Phi^V=0,\big[R_0^V,\Phi^V\big]=0,{\rm d}^{\nabla^V}\Phi^V=0,\nabla^V R_0^V+\Phi^V =\big[\Phi^V,R_{\infty}^V\big], \nabla^V R_{\infty}^V=0,
d s^{2}= d X \cdot d X = - \left( \frac{r^{2}+ 1}{2 r}\right)^{2}d t^{2}+ \frac 1{r^{2}}d r^{2}+ \left( \frac{r^{2}- 1}{2 r}\right)^{2}\left( d \Omega \right)^{2}{.}
\begin{multlined}[t][12.5cm]T_{\nu_{G},k\nu_{T}}[f](x,x')=\int_{X}\widetilde{\Pi}_{\nu_{G},k\nu_{T}}(x,y)f(y)\widetilde{\Pi}_{\nu_{G},k\nu_{T}}(y,x')dV_{X}(y) =\sum_{j}T_{\nu_{G},k\nu_{T}}[f](s_{j}^{k}(x))\overline{(s_{j}^{k}(x'))}\end{multlined}
E^I_{R, m}= \begin{cases}B_{R^{s_1}}^{1, 2}C^{i_{m - 1}, i_m}_{R^{s_m}}\cap B^{i_{m - 1}, i_m + 1}_{R^{s_m}}\cap B_{R^{s_{m + 1}}}^{i_m, i_m + 1}C^{i_{m - 1}, i_m}_{R^{s_m}}. \end{cases}
\frac{L^n}{(n+1)^n}\frac{(n+1)^{n-2}m^{n-2}}{n!}\cdot \frac{1}{2}\left(\left(\sum_{i=0}^ni\right)^2-\sum_{i=0}^n i^2\right) =\frac{L^n}{(n-1)!}\cdot\frac{(n-1)(3n+2)}{24(n+1)}\cdot m^{n-2}
R_n^{(\alpha,\beta)}(\cos 2\theta) R_n^{(\alpha,\beta)}(\cos 2\theta^\prime)= = \frac{\alpha}{\pi}\int_0^1 \int_{-\pi}^{\pi}R_n^{(\alpha, \beta)}(2|b+are^{i\varphi}|^2 -1) \cdot \frac{(b+are^{i\varphi})^\beta}{b^\beta}\cdot r(1-r^2)^{\alpha-1}dr d\varphi
\left\langle\rho,\tau^{x_{3}}\alpha_{1},\sigma^{y_{2}}\tau^{y_{3}}\alpha_{2}\alpha_{3}^{\left(1+x_{3}\right)y_{2}x_{3}^{-1}}\right\rangle \cong C_{p}^{3}& \y_{3}=0,...,p-1, \y_{2},x_{3}=1,...,p-1,
1 + \sum_{i \ge 1}(-1)^i \beta_i \, t^{i+e-1}= \sum_{i=0}^{m}f_{i-1}\, t^i \, (1-t)^{d-i}= \sum_{i=0}^{m}\left [ f_{i-1}\, t^i \, \sum_{j=0}^{d-i}(-1)^j \binom{d-i}{j}t^j \right ] begin{align*}1ex] = \sum_{i=0}^{m}t^i \left [\sum_{j=0}^{i}(-1)^{i-j}f_{j-1}\binom{d-j}{i-j}\right ]
\overline{{\delta}}( k ) = \sum_{\sigma \in \{+ , - \}}\sum_{G = 0}^{\infty}( 2 G + 1 ) \left( \delta_{G , \sigma}( k ) - \delta_{G , \sigma}^{( 1 )}( k ) - \delta_{G , \sigma}^{( 2 )}( k ) \right) + \delta_{\mathrm{l i m}}( k ) .
\lim_{n \rightarrow + \infty}\chi_{K_n}(x) = \chi_{(\bigcup_{n \geq 1}K_n)}(x) \leq \chi_A (x) \leq \chi_{(\bigcap_{n \geq 1}V_n)}(x) = \lim_{n \rightarrow + \infty}\chi_{V_n}(x) \ \ \ \ \ \forall \x \in X.
P^{\prime}_{L}= ({\rho}\frac{\tilde m}{2}+(G_{\bot}-B_{\bot}-\frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\tilde n}-\frac{1}{2}A_{\bot}^{t}C_{\bot}{\tilde l},\;{\tilde l}+A_{\bot}{\tilde n})
f_1= x_1^{c_1}- x_3x_4^{c_4-1}, f_2=x_2^{c_2}- x_1^{\alpha_{21}}x_4, f_3=x_3^{c_3}-x_1^{c_1-\alpha_{21}-1}x_2, f_4=x_4^{c_4}- x_1 x_2^{c_2-1}x_3^{c_3-1}, f_5=x_3^{c_3-1}x_1^{\alpha_{21}+1}-x_2 x_4^{c_4-1}.
(\tilde{\Delta}_E\tilde{h}_{4,i}^{(1)},\tilde{h}_{4,i}^{(1)})_{L^2(\tilde{g})}=\int_{J}\varphi^2f^{n-2}dr\cdot({\Delta_E}{h},{h})_{L^2({g})}+\int_{J}(\varphi')^2f^n dr\cdot \left\|h\right\|^2_{L^2(g)},
\begin{aligned}\|\nabla^k(\rho-1)(t)\|_{H^{3-k}}+\|\nabla^k u(t)\|_{H^{3-k}}\le C(1+t)^{-\frac{3+2k}{4}}, &\|\nabla^m B(t)\|_{H^{3-m}}\le C(1+t)^{-\frac{3+2m}{4}}, \end{aligned}
L_1=\langle S({\bf{v}})\partial_t\partial_y{\bf{v}}, \partial_y{\bf{v}}\rangle_{H^{2, 0}}+\langle \partial_yS({\bf{v}})\partial_t{\bf{v}}, \partial_y{\bf{v}}\rangle_{H^{2, 0}}=: L_1^1+L_1^2.
\ \prod_{j=j^*}^{\infty}\left(1- \lambda_{p_j}(\{0\})\right) \ge \prod_{j=j^*}^{\infty}\left(1- \frac{c}{p_j^2}\right) \ge 1-\sum_{j=j^*}^{\infty}\frac{c}{p_j^2}\ge 1-\sum_{i=p_{j^*}}^{\infty}\frac{c}{i^2}\ge \ 1-\sum_{i=p_{j^*}}^{\infty}c\left( \frac{1}{i-1}-\frac{1}{i}\right) =1-\frac{c}{p_{j^*}-1}>0\,,
E[Y_k] =\sum_{i=1}^k \sum_{j=i}^{k}\frac{k}{k-j+1}= \sum_{j=1}^k \sum_{i=1}^j \frac{k}{k-j+1}=\sum_{j=1}^k j \frac{k}{k-j+1}=k\sum_{j=1}^k \frac{j}{k-j+1}.
\check{\alpha}=\begin{cases}\underset{a\in R:a|_{\check{\underline{X}}}=\alpha}{\sum\check{a},}\mbox{if}2\alpha\notin\underline{R}\underset{a\in R:a|_{\check{\underline{X}}}=\alpha}{2\sum\check{a},}\mbox{if}2\alpha\in\underline{R}\end{cases}
\mathbb{E}_{\underline{\varepsilon}|\underline{X}}\exp\left\{\frac{1}{\gamma}\sum_{i=1}^n\varepsilon_ig(X_i)\right\}\leq \exp\left\{\frac{\sigma^2\sum_{i=1}^ng^2(X_i)}{2\gamma^2(1-\eta K/\gamma)}\right\}= \exp\left\{\frac{1}{\gamma A}\sum_{i=1}^ng^2(X_i)\right\},
P_i(q) - P_j(q) - (i-j) = \frac{(q^{2i-2}-1)q^{j-2}- (q^{2j-2}-1)q^{i-2}}{q^{j+i-4}(q^2-1)}= \frac{(q^{j+i-2}+1)( q^{i-2}- q^{j-2})}{q^{j+i-4}(q^2-1)}.
\left(\xi_{\bar C}\right)'_A(x)\left(\xi_{\bar D}\right)'_B(y)\hat I_{AB}(x)\delta(x,y)=\left(\xi_{\bar C}\right)'_A(x)\hat I_{AB}(x)\left[\left(\xi_{\bar D}\right)'_B\right]^{\rm ``adjoint''}(x)\delta(x,y),
L_{2i-1}=(q^{10};q^{10})_\infty \sum_{n=0}^\infty c\left(5^{2i-1}n+\frac{7\cdot 5^{2i-1}+1}{12}\right)q^{n+1}, L_{2i}=(q^2;q^2)_\infty \sum_{n=0}^\infty c\left(5^{2i}n+\frac{11\cdot 5^{2i}+1}{12}\right)q^{n+1}.
\begin{aligned}F_D(x^k) = \min_x\Big\{F(x) + \frac{1}{2}\big\|x - x^k\big\|^2_D \Big\}\geq\min_x \Big\{\tilde{F}^k(x) + \frac{1}{2}\big\|x - x^k\big\|^2_D \Big\}= \tilde{F}^k (z^k_{j_k}) + \frac{1}{2}\big\|z^k_{j_k}- x^k\big\|^2_D \geq F(x^k) - \varepsilon.\end{aligned}
\frac{\mathcal{H}^{m}(B_{r}^{g}(x))}{\mathcal{H}^{m}(B_{r}(x)\cap M)}=\frac{\omega_{m}r^{m}(1+O(r^{2}))}{\omega_{m}r^{m}(1+O(r^{2}))}\to1,r\to0^{+},
p_k \sim \begin{cases}{\rm Beta}\left(\frac{2n - k}{4}\beta, \frac{2n - k -2}{4}\beta + a + b + 2 \right), ,{\rm Beta}\left(\frac{2n - k - 1}{4}\beta + a + 1, \frac{2n - k -1}{4}\beta + b + 1 \right), .\end{cases}
\circ(1)= \int_{\Omega}v_j^-= (a_j)^\lambda \Big( 1-\big(1- \frac{\eta_j^-}{a_j}\big)^\lambda \Big)\ge (a_j)^{\lambda}\int_{\Omega}\frac{\eta_j^-}{a_j}= (a_j)^{\lambda-1}\int_{\Omega}\eta_j^-.
t_1^{\kappa-p/2+1}\int_{t_1}^{1/2}\frac{\displaystyle|Z_N(t)-\sigma B_N(t)|| B_N(t)|^{p-1}}{t^{\kappa}}dt=O_P\left( (Nt_1)^{-1/2+\zeta}\right)=o_P(1)