harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(G^{\{v \}})^{\mu,t}_n= \sqrt{-1}\sqrt{2|k+h^\vee|}:a \Phi_{[e,v]}:^\mu_n+2t\sqrt{-1}\sqrt{2|k+h^\vee|}(\Phi_{[e,v]})^\mu_n -2(k+1) ( T \Phi_{[e,v]})^\mu_n +{G^{\{v\}}}^\mu_n.
d(x^{g^{n}}, y) = d(v_1^{w_1 g^{n}}, v_2^{w_2}) = d(v_1, v_2^{w_2 g^{-n}w_1^{-1}}) = d(v_1, v_2^{c}) & \le d(v_1, v_2) + d(v_2, v_2^c) \le D + D(\|c\|_* +1) & = D(\|c\|_* +2)\le D(2r+7).
\begin{aligned}K(N, \ell,t;r):=[\xi^{r-\ell+t}z^{N-\ell}]\,(1-\xi)^{t-1}(1-z)^{-t+1}(1-\xi z)^{-\ell-1}= \sum_j(-1)^{r-\Delta-j}\binom{\ell+j}{j}\binom{t-1}{r-\Delta-j}\binom{N-\Delta-j-2}{t-2},\end{aligned}
\mathrm{E}\left[\int_{\mathcal{D}(T)}\int_{\mathcal{D}(T)}e_{k}^{(\beta)}(\xi(\mathbf{x}))e_{m}^{(\beta)}(\xi(\mathbf{y}))~d\mathbf{x}d\mathbf{y}\right]=\delta_{k,m}~\newline \sigma_{k}^{2}(T),
S_3 = \int_{\mathcal{X}_n}f(x) \int_0^{\frac{a_n}{n-1}}\mathrm{B}_{k,n-k}(s) \biggl\{\log^2 u_{x,s}- \log^2\biggl( \frac{(n-1)s}{e^{\Psi(k)}f(x)}\biggr)\biggr\}\, ds \, dx &= O\biggl\{\max\biggl( \frac{k^{\beta/d}}{n^{\beta/d}}\log n \, , \, \frac{k^{\frac{\alpha}{\alpha+d}-\epsilon}}{n^{\frac{\alpha}{\alpha+d}-\epsilon}}\biggr)\biggr\}.
\bar{G}=\mathbb{E}\left[G_i\right]=\frac{1}{N^2}\mathbb{E}\left[ \frac{{1 - \cos (N{\mathcal K}_1(\xi_{r_{i,o}}))}}{{1 - \cos ({\mathcal K}_1(\xi_{r_{i,o}}))}}\right] \times \mathbb{E}\left[\frac{1 - \cos ({N}{{\mathcal K}_2}(\varphi_{t_{i,o}},{\varphi_{{t_i}}}))}{1 - \cos ({{\mathcal K}_2}(\varphi_{t_{i,o}},{\varphi_{{t_i}}}))}\right].
K_X\cdot \Delta +\Delta^2=\sum_{i=1}^k(K_X\cdot \Delta_i+\Delta_i^2)+2\sum_{1\leq i<j\leq k}\Delta_i \cdot \Delta_j \geq \sum_{i=1}^k(K_X\cdot \Delta_i+\Delta_i^2)=\sum_{i=1}^k(2\mathrm{p}_a(\Delta_i)-2) \geq -2k \geq -10,
\begin{array}{lll}e_{n-2}= \{\sigma(0,n-2,n-1),\sigma(0,n-2,n)\}; e_{n-1}= \{\sigma(1,n-1,n),\sigma(0,1,n-1)\}; e_{n}= \{\sigma(0,2,n),\sigma(1,2,n)\}; \end{array}
e_{\alpha_{1},\alpha_{2},\cdots ,\alpha_{2n-k},\alpha_{2n-k+2},\cdots,\alpha_{k-1},\alpha_0}= \frac{u_{\alpha_{1},\alpha_{2},\cdots ,\alpha_{2n-k},\alpha_{2n-k+2},\cdots,\alpha_{k-1},\alpha_0}}{\left\|u_{\alpha_{1},\alpha_{2},\cdots ,\alpha_{2n-k},\alpha_{2n-k+2},\cdots,\alpha_{k-1},\alpha_0}\right\|}
0 < m_i < \min\big\{\dfrac{1}{2},\dfrac{p-\alpha}{\alpha(p-1)}\},\C > \dfrac{h}{m_i}\ \ 0 < \varepsilon < \dfrac{d^{2(p-1)}a_1\gamma_{BL}}{h(1+C/2)}, \i=1,2,
\left|G_{\rho_{\epsilon}}(t,\,F_{\rho}(t,\,x))-x\right| =\left|G_{\rho_{\epsilon}}(t,\,F_{\rho}(t,\,x))-G_{\rho_{\epsilon}}(t,\,F_{\rho_{\epsilon}}(t,\,x))\right| \le\frac{\lambda+1}{\lambda}\left|F_{\rho}(t,\,x)-F_{\rho_{\epsilon}}(t,\,x)\right| \le\frac{1}{\lambda}||\rho_{\epsilon}(t,\,\cdot)-\rho(t,\,\cdot)||_{L_{1}}
\Theta_{[\mathbf n], t}^{L, \Lambda, A}= \sum_{\substack{[j] \in \mathcal J L \cdot \mathbf n - L_j \leq t}}\widehat\Xi_{[\mathbf n - e_j], t}^{L, \Lambda, A}\widehat A_{[j]}^\Lambda(t - L \cdot \mathbf n + L_j).
\frac{d}{dt}{F}=- \sum_{j=1}^N \int_{I_j}\sum_{i=1}^m c_{ih}|p_{ihx}|^2 dx - \sum_{j=1}^{N-1}\sum_{i=1}^m ( \{c_{ih}\}[p_{ih}] (\widehat{p_{ihx}}+\{p_{ihx}\}))_{j+1/2}& =- \sum_{i=1}^m A_{c_{ih}}(p_{ih}, p_{ih}) & \leq 0,
\lambda^{\frac{2}{r}}\| u \|_{\Phi}^{2}+\sum_{j=1}^{N}\| X_{j}^{\Omega}u\|_{\Phi}^{2}\leq C \left( \langle P^{\Omega}u , u\rangle_{\Phi}+ \lambda^{\alpha}\|u\|_{\Phi, \Omega\setminus\Omega_{1}}^{2}\right),
\epsilon_n = \epsilon_n^w(d,\theta) := \frac{\sup_{k \in \{1,\ldots,k^*\}}\sup_{f \in \mathcal{F}_{d,\theta}}\Bigl(2\mathbb{E}_f\bigl[\{\tilde{V}_n^w - V(f)\}^2\bigr]\Bigr)^{1/3}}{\inf_{f \in \mathcal{F}_{d,\theta}}V(f)^{2/3}},
\langle \Phi_{H . W .}^{( \lambda_{1}, \mu_{1})}( z_{1}, \bar{z_{1}}) \Phi_{H . W .}^{( \lambda_{2}, \mu_{2})}( z_{2}, \bar{z_{2}}) \Phi_{T_{z}^{3}, Y^{3}}^{( \lambda_{3}, \mu_{3})}( z_{3}, \bar{z_{3}}) \tilde{\Phi}_{L . W .}^{( \lambda_{4}, \mu_{4})}( z_{4}, \bar{z}_{4}) \rangle =
\begin{aligned}\int_{0}^T\left<(1-P_R)S\left(W_\epsilon(x_j,x_k)u_\epsilon\right),\sum_{1\leq l\leq s}\mathcal{K}_{\mathcal{I}_l}v\right>dt \lesssim_{\alpha,\widetilde{p},p,\widetilde{q},q}T^{1/\theta}/R\left(\sum_{1\leq l\leq s}\|\mathcal{K}_{\mathcal{I}_l}u_\epsilon\|_X\right)\left\|\sum_{l}\mathcal{K}_{\mathcal{I}_l}v\right\|_{L_t^{\theta_q'}(L^{q',2}_D)}.\end{aligned}
\xi_+(x) + \xi_-(x) = 0, x \in \mathcal{S}, 2\xi(x) < 0, x \in (4,\infty), \xi_+(x) - \xi_-(x) = 2\pi i \int^{4}_x \rho(s) ds, x \in \mathcal{S}, \xi_+(x) - \xi_-(x) = 2\pi i, x \in (-\infty,0).
\mbox{Re}(\Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})-\widehat{\tilde \Psi},\delta)_{L^2(\widetilde\Gamma)}+\mbox{Re}(\bar\Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})-\widehat{\Psi},\widetilde \delta)_{L^2(\widetilde\Gamma)}=0
\Big [{\bf b}_{i}( \vec{K}) \, , \,{\bf b}_{i}^{\dagger}({\vec{K}}^{\prime}) \Big ] \, = \, 2 ( 2 \pi )^{3}\, w_{i}( \vec{K}) \, \delta^{3}( \vec{K}-{\vec{K}}^{\prime}) \, ,
\epsilon_j(\xi, t) = \frac{(-1)^{p+1}}{2}[[U_j]] \left[ R^-_{p+1}(\xi) + \sum_{k=1}^\infty \left(\frac{\omega_{0}h}{2}\right)^k R^{-,(-k)}_{p+1}(\xi) \right] + \mathcal{O}(h^{2p+1}),
\begin{array}{rll}w_p := w_p / \widehat{a}_{pq}^2 \noalign{\smallskip}w_i := w_i - 2 (\widehat{a}_{iq}/ \widehat{a}_{pq}) \tau_i + (\widehat{a}_{iq}/ \widehat{a}_{pq})^2 w_p i\ne p\end{array}
\beta=\frac{2}{1-\alpha},A=\pm\left( \frac{1}{\beta\left(\beta-1\right)}\right)^{\frac{1}{1-\alpha}}=\pm\left( \frac{\left(1-\alpha\right)^{2}}{2\left( 1+\alpha\right)}\right)^{\frac{1}{1-\alpha}}.
\begin{array}[c]{rl}C(t)= (\beta_{1}(t)p(t)+\beta_{2}(t)q(t)+\beta_{3}(t))\varphi(t)+p(t)L_{1}(t)+q(t)L_{2}(t)+L_{3}(t) & +(\gamma_{1}(t)p(t)+\gamma_{2}(t)q(t)+\gamma_{3}(t))(1-p(t)\gamma_{2}(t))^{-1}(\beta_{2}(t)p(t)\varphi(t)+p(t)L_{2}(t)+\nu(t)).\end{array}
\left( \cdots \left( \varphi_{k_1l_1}^{r_2'\cdots r_s'}\right) \cdots \right)_{k_sl_s}(X_1,\cdots, X_j+2\pi r_j',\cdots, X_n,Y_1,\cdots, Y_n) &=\left( \cdots \left( \varphi_{k_1l_1}^{r_2'\cdots r_s'}\right) \cdots \right)_{k_sl_s}(X_1,\cdots, X_n,Y_1,\cdots, Y_n),
\frac{\tau ( (i,j) (j,k))}{(x_{i}-x_{j}) (x_{j}-x_{k})}+\frac{\tau((i,k) (j,i))}{(x_{i}-x_{k})(x_{j}-x_{i})}+\frac{\tau ( (i,k) (j,k))}{(x_{i}-x_{k}) (x_{j}-x_{k})}\qquad{}=\frac{\tau((i,j,k))}{( x_{i}-x_{k}) (x_{j}-x_{k})}+\frac{\tau((j,i,k))}{(x_{i}-x_{k})( x_{j}-x_{k})},
\begin{gathered}\tau_L = 0 \;, \sigma_L = -1 \;, \delta_L = \frac{\delta_R + 2}{\delta_R \left( \delta_R^2 + 2 \delta_R + 2 \right)}\;, \tau_R = -\frac{\delta_R^2 + \delta_R + 2}{\delta_R + 2}\;, \sigma_R = 0 \;, \end{gathered}
{\check{Q}}_{i}= e_{i}\check{\theta}_{i}^{\ast}( \tau ) \check{\theta}_{i}( \tau ) = Q_{i}= e_{i}\theta_{i}^{\ast}( \tau ) \theta_{i}( \tau ) ~ ~ \big [ \dot{Q}_{i}( \tau ) = 0 \Rightarrow Q_{i}( \tau ) \equiv Q_{i}\big ] .
\sum\limits_r \eta_r(x)\eta_r(R^a)\eta_r(z^{-1})= n\sum\limits_r \big{(}\eta(R^{a+a_1+a_2})+\eta_r(R^{a+a_1-a_2})+\eta(R^{a-a_1+a_2})+\eta(R^{a-a_1-a_2})\big{)},
{\sf M}\left\{\left(J^{*}[\psi^{(5)}]_{T,t}-\sum\limits_{j_1, \ldots, j_5=0}^{p}C_{j_5 \ldots j_1}\zeta_{j_1}^{(i_1)}\ldots\zeta_{j_5}^{(i_5)}\right)^2\right\}\le \frac{C}{p^{1-\varepsilon}}
I = \int_{\Omega}\frac{1}{2}\left( \epsilon_{\mu \nu \lambda}B_{\mu}^{*}( \partial_{\nu}- i g a_{\nu}) B_{\lambda}+ M B_{\mu}^{*}B^{\mu}+ \epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\right) ,
M_N(a, b, \lambda) = \sum\limits_{\substack{I_{kl}=-\lambda k<l=1\cdots N}}^\lambda(-1)^{\sum\limits_{k<l}I_{kl}}\prod\limits_{k<l}^N \binom{2\lambda}{\lambda+I_{kl}}\prod\limits_{k=1}^N\binom{a+b}{a+\sum\limits_{l\neq k}I_{kl}}.
u=u_{0}-\delta^{2}\frac{z^{2}}{2}u_{0,xx}+\mathcal O(\epsilon^{2},\delta^4, \epsilon \delta^{2}), w=-zu_{0,x}+\delta^{2}\frac{z^3}{6}u_{0,xxx}+\mathcal O(\epsilon^{2},\delta^4, \epsilon \delta^{2}).
{{\cal L}^2}^{\rm eff}(H,\mu,T)= \frac{eH}{2\pi^2}\frac{\alpha}{2\pi}\sum_{n=0}^\infty b_n \int_{-\infty}^\infty d\! p_{z}~p_{z}[\ln (\frac{p_{z}^2}{eH})+1] ~(f_+(T) + f_-(T)).
\Phi_{p}(y)=\frac{(2\pi)^2}{l^2\tau_2}\sum_{j_1j_2}<\psi|e^{\frac{2i\pi}{l}(j_1(\hat{y_1}-y_1)+\frac{j_2-\tau_1j_1}{\tau_2}(\hat{y_2}-y_2)}|\psi>
V ( r ) = E \left[ 1 - \frac{1 + E H_{0}/ 2 m_{p}}{( 1 + E H_{0}/ m_{p})^{2}}\right] + \frac{L^{2}}{2 m_{p}r^{2}}\frac{1}{( 1 + E H_{0}/ m_{p})^{2}}.
\ \mathrm{I m}{\cal A}_{\mathrm{a n n u l u s}}^{( o o \to o o )}\ \ \stackrel{v \to 1_{-}}{\sim}\ \ \ \Bigl \vert{\cal A}_{\mathrm{d i s k}}^{( o o \to o o )}( \pi / 2 ) \Bigr \vert^{2}\sim e^{- 2 s \alpha^{\prime}\log 2}\ .
\sum_{t=1}^{T}t\left( \frac{1}{\gamma_t}V_{x_t}(x) - \frac{1}{\gamma_t}V_{x_{t+1}}(x) - \alpha V_{x_t}(x) + \frac{\gamma_t G^2}{2}\right) \leq \frac{G^2 T}{\alpha}- \frac{\alpha\, T(T+1)}{2}V_{x_{T+1}}(x) \leq \frac{G^2 T}{\alpha}.
g(u) = \xi(1)-\xi(u)-\frac1{A_1^2}\log\Big(1+\frac{A_1(q-u)}{A_2(1-q)+\Delta}\Big) + \frac{q-u}{A_1[A_1q+A_2(1-q)+\Delta]}& - \frac1{A_2^2}\log\Big(1+\frac{A_2(1-q)}{\Delta}\Big) +\frac{1-q}{A_2(A_2(1-q)+\Delta)}-\frac{(1-q)q}{[A_2(1-q)+\Delta][A_1q+A_2(1-q) +\Delta]};
\begin{aligned}\int_0^1 \lim_{s\to0}\frac{\partial}{\partial s}\zeta_k(s,a)d_qa=\lim_{s\to0}\frac{\partial}{\partial s}\left(\frac{1}{[1-s]_q}\right) &+\sum_{r=0}^{k-1}\binom kr \sum_{l=0}^\infty \frac1{[l+1]_q}\lim_{s\to0}\frac{\partial}{\partial s}\left[\binom{-s}{l}\zeta_{k-r}(s+l,k-r)\right],\end{aligned}
\frac{1}{3}\frac{2 \omega - 4}{2 \omega - 3}\oint d \tau_{1}\, d \tau_{2}\, d \tau_{3}\, \epsilon ( \tau_{1}\, \tau_{2}\, \tau_{3}) \frac{\sin \tau_{1 2}+ \sin \tau_{2 3}+ \sin \tau_{3 1}}{\alpha \beta ( 1 - \cos \tau_{1 2}) + \beta \gamma ( 1 - \cos \tau_{2 3}) + \gamma \alpha ( 1 - \cos \tau_{1 3})}.
\limsup_{k\rightarrow \infty}n!k^{-n}\dim H_{(2)}^{0,j}({X_c},{L}^k)\leq\int_{X_c(j,h^L)}(-1)^j c_1(L,h^L)^n= \int_{M(j)}(-1)^j c_1(L,h^L)^n.
\begin{aligned}\lambda_{ij}v_{ij}= 0,\forall i,j\in\{1,\ldots,n\},\i\neq j. &\gamma_i\left(\sum_{j\neq i}v_{ij}- x_0{_i}\right) = 0,\forall i\in\{1,\ldots,n\}.\end{aligned}
y(t) =\int_{0}^{T}{s(\tau)p_{\rm obs}(t-\tau) d\tau}=s(t)*p_{\rm obs}(t) &=\iiint_{\Omega_{\rm rx}}{{s(t)*C(\bar{r},t|{{\bar{r}}_{\rm tx}},{t_0=0})}\rho d\rho \varphi dz},
\mathcal{I}(r_+)=\hat ue^{\mu r_+}+ \sigma\! \int_{-r_+}^{0}\!{e^{-\mu \theta}\eta_{(k)}(\theta)d\theta}, \tfrac{\partial}{\partial r_+}\mathcal{I}(r_+)=e^{\mu r_+}\bigl[\mu \hat u +\sigma \eta_{(k)}(-r_+)\bigr].
F(t)=(-1)^{\frac{k}{2}}(it)^{\frac{2-k}{2}+m}(k-1)\sum_{0\leq j\equiv \frac{k-2}{2}-m\;({\rm mod}\;2)}\binom{k}{j}\int_{S^1}s(\mu_m)(\kappa(\theta))\frac{(it)^j(-\sin\theta)^{k-j}\cos\theta^j}{(\sin^2\theta+t^2\cos^2\theta)}d\theta.
\omega_{r}^{( 2 )}= 4 8 \rho^{3}d t_{0}\int_{S_{r}^{2}}{\frac{\rho d x_{0}^{i}+ d \rho ( x - x_{0})^{i}}{( t_{0}^{2}+ \rho^{2}+ ({\bf x}-{\bf x}_{0})^{2})^{4}}}\varepsilon_{i j k}d x^{j}d x^{k}.
\begin{cases}D_t^2\zeta-iA\zeta_{\alpha}=-i \frac{d}{dt}z_j(t)=(v-\frac{\lambda_j i}{2\pi(\overline{z-z_j})})\Big |_{z=z_j}(I-\mathcal{H})(D_t\bar{\zeta}+\sum_{j=1}^N \frac{\lambda_j i}{2\pi(\zeta(\alpha,t)-z_j(t))})=0 (I-\mathcal{H})(\bar{\zeta}-\alpha)=0.\end{cases}
[c, x_1] [x_2, x_3, x_4, x_5] = [c, x_1] [x_2, x_3, x_4] x_5 - [c, x_1] x_5 [x_2, x_3, x_4] = \ [c, x_1] [x_2, x_3, x_4] x_5 - x_5 [c, x_1] [x_2, x_3, x_4] - [c, x_1, x_5] [x_2, x_3, x_4].
\begin{aligned}\dot{x}= \omega+\xi(y,z,\omega,\sigma,\mu)+f(x,y,z,\omega,\sigma,\mu), \dot{y}= \sigma+\eta(y,z,\omega,\sigma,\mu)+g(x,y,z,\omega,\sigma,\mu), \dot{\sigma}= \Lambda y, \dot{z}= Q(\omega,\mu)z+\zeta(y,z,\omega,\sigma,\mu)+h(x,y,z,\omega,\sigma,\mu)\end{aligned}
{\mathcal{H}}_{i,u}={\mathcal{F}}_{i,u-1}^{X}\vee\sigma\big ((L_{j}^{X})_{1\leq j\leq i-1},(L_{k}^{Y})_{i+1\leq k\leq n}\big )\vee\sigma\big ({D}_{i,u+1}^{\ast},\dots,{D}_{ik_{i}}^{\ast}\big )\,.
(X_k, C_k, D_k) = \left( \left[ \begin{smallmatrix}X_k^{(1)}0 0 X_k^{(2)}\end{smallmatrix}\right], \left[ \begin{smallmatrix}C_k^{(11)}C_k^{(12)}C_k^{(21)}C_k^{(22)}\end{smallmatrix}\right], \left[ \begin{smallmatrix}D_k^{(11)}D_k^{(12)}D_k^{(21)}D_k^{(22)}\end{smallmatrix}\right] \right).
\int_{\Omega\setminus B(\xi,\rho)}\Big|V^{p-1}- \sum\limits_{i=1}^k(-1)^i(PU_{\mu_i,\xi_i})^{p-1}\Big|^{\frac{n}{2}}dx \leq \sum\limits_{i=1}^k\int_{\Omega\setminus B(\xi,\rho)}U_{\mu_i,\xi_i}^{(p-1)\frac{n}{2}}dx \leq C \sum\limits_{i=1}^k \mu_i^n =O \bigg(\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big)^{\frac{n}{n-2}}\bigg).
\begin{aligned}\delta\psi_2 = 2(\psi_2\psi_3+\psi_2\psi_4-\psi_3\psi_4), \delta\psi_3 = 2(\psi_2\psi_3+\psi_3\psi_4-\psi_2\psi_4), \delta\psi_4 = 2(\psi_2\psi_4+\psi_3\psi_4-\psi_2\psi_3). \end{aligned}
J^{T}WJ=(\begin{matrix}m\sum w_{i}&\sum w_{i}z_{i}&\sum w_{i}z_{i}^{2}\\ \sum w_{i}z_{i}&\sum w_{i}z_{i}^{2}&\sum w_{i}z_{i}^{3}\\ \sum w_{i}z_{i}^{2}&\sum w_{i}z_{i}^{3}&\sum w_{i}z_{i}^{4}\end{matrix})
\Delta^{a}= \int d^{2}\theta \frac{\partial^{2}\delta_{L}}{\partial \theta^{2}\delta \Phi^{A}( \theta )}\, \theta^{a}\frac{\delta}{\delta \bar{\Phi}_{A}( \theta )}= ( - 1 )^{\epsilon_{A}}\frac{\delta_{L}}{\delta \phi^{A}}\, \frac{\delta}{\delta \phi_{A a}^{*}}.
\mathbb{E}\bigg(\int_t^\infty z_ndW_s-\int_t^\infty zdW_s\bigg)^2=\mathbb{E}\int_t^\infty |z_n-z|^2ds\leq \mathbb{E}\int_t^\infty e^{-\beta s}|z_n-z|^2ds\rightarrow0,\ \n\rightarrow\infty.
(A_{\alpha}^{(r)})_{ij}= t_{j, m}(x_m^{(r+m-1)}x_m^{(r+m-2)}\cdots x_m^{(r+m- \alpha_i)}) &= s_j s_{j+1}\cdots s_{m-2}s_{m-1}(x_m^{(r+m-1)}x_m^{(r+m-2)}\cdots x_m^{(r+m- \alpha_i)}),
{}~~~~~~~~~~~~~\times \Bigg\{\sum_{\sigma \in Z_N}\frac{w(x_3,x_{13},x_1|d,h+\sigma)w(x_4,x_{24},x_2|a,g+\sigma)}{w(x_4,x_{14},x_1|e,c+\sigma)w(x_3/\omega,x_{23},x_2|f,b+\sigma)}\Bigg\}_0.
P^1(f^*)=\frac{Q(f^*,g^*)}{||\mu||}+I=\begin{matrix}\begin{pmatrix}p_1 p_2 \cdots p_{|S|}p_1 p_2 \cdots p_{|S|}\vdots \vdots \vdots p_1 p_2 \cdots p_{|S|}\end{pmatrix}\end{matrix}.
\limsup_{K(N)\to\infty}\frac{\left|\sum_{k=1}^K\Xi_k\right|}{m\sqrt{2s_K^2\log(\log(s_K^2))}}\leq \frac{\sqrt{\Sigma_{f,M_n}+Cd^{1/2}\log(d)G^{-1/2}}}{m\sqrt{\Sigma_{f,M_n}-Cd^{1/2}\log(d)G^{-1/2}}}.
\prod_{j=1}^r\left(\frac{\ln(1+t)}{(1+t)^{a_j}-1}\right)(1+t)^x =\sum_{n=0}^\infty D_n(x|a_1,\dots,a_r)\frac{t^n}{n!}\,, \prod_{j=1}^r\left(\frac{(1+t)^{a_j}\ln(1+t)}{(1+t)^{a_j}-1}\right)(1+t)^x =\sum_{n=0}^\infty\widehat D_n(x|a_1,\dots,a_r)\frac{t^n}{n!}\,,
\int_{\mathbb{R}_{\geqslant0}}x^{n}m_{\alpha,\beta}(\mathrm{d}x) \leqslant c_{1}\frac{e^{-\alpha}}{\beta^{n}}\left(\alpha+\alpha^{2}+\cdots+\alpha^{n-1}+\alpha^{n}\sum_{m=0}^{\infty}\frac{\alpha^{m}}{m!}\right) \leqslant c_{2}\left(\frac{1}{\beta^{n}}+\frac{\alpha^{n}}{\beta^{n}}\right),\quad\alpha,\beta>0,
S \ = \ \lim_{\epsilon \to 0}\sum_{(b,M)=1}\frac{R(1,b)R(m^2,b)}{\varphi(b) b^{\epsilon}}\int_{0}^{\infty}J_{k-1}(y) \hat{\phi}\left( \frac{2\log(by\sqrt{Q}/4\pi m)}{\log R}\right) \frac{dy}{\log R}.
\Big(\frac{\frac14 + Y^2}{\frac94+Y^2}\Big)^{k/2}= \exp\Big(\frac{k}{2}\log\Big(\frac{1 + \frac{1}{4Y^2}}{1 + \frac{9}{4Y^2}}\Big) \Big) = \exp(\tfrac{k}{2}(-2Y^{-2}+ O(Y^{-4}))).
\Pr[X_1=1|\mathcal{D}]=\Pr[X_2=1|\mathcal{D}]=p^2+pq=p, &\Pr[X_1=X_2|\mathcal{D}]=\Pr[X_1=1|\mathcal{D}]\Pr[X_2=1|\mathcal{D}]+\Pr[X_1=0|\mathcal{D}]\Pr[X_2=0|\mathcal{D}]=p^2+(1-p)^2=2-2p.
E_{k,l}=\Bigg\{e^{ilt}\left( \begin{array}{c}ae^{i\frac{1}{2}\theta}\mathfrak{I}_{k-\frac{1}{2},l}(r)-be^{i\frac{1}{2}\theta}l\mathfrak{I}_{-k+\frac{1}{2},l}(r) -ae^{-i\frac{1}{2}\theta}l\mathfrak{I}_{k+\frac{1}{2},l}(r)+be^{-i\frac{1}{2}\theta}\mathfrak{I}_{-k-\frac{1}{2},l}(r)\end{array}\right)\mbox{}\Bigg| \mbox{}a,b\in\mathbb{R}\Bigg\},
\limsup_{k\to\infty}I_k^{(4)}\leq \limsup_{k\to\infty}\frac{c}{\varepsilon^2}\int_{B_{1-\varepsilon}\setminus B_{1-2\varepsilon}}|z_\infty-z_{j_k}|^2\,d\mu_a + O(\varepsilon)\leq O(\varepsilon).
S(\sigma)(N) = \int_1^N\sigma(x)dx + \frac{1}{2}\left(\sigma(N)+\sigma(1)\right) + \sum_{k=2}^K\frac{B_k}{k!}\,\left(\sigma^{(k-1)}(N)- \sigma^{(k-1)}(1)\right) + \frac{(-1)^{K+1}}{K!}\int_1^N \overline{B_{K}}(x)\,\sigma^{(K)}(x)\, dx
J_{x,y}(\omega_1,\omega_2) = 8 \pi^2 \mathrm{Re}(\omega_1\omega_2^2 + \omega_1^2\omega_2^{-1}- \omega_1^2\omega_2 - \omega_1\omega_2^{-2}) & = 4 \pi^2 (\omega_1\omega_2^2 + \omega_1^{-1}\omega_2^{-2}+ \omega_1^2\omega_2^{-1}+ \omega_1^{-2}\omega_2 - \omega_1^2\omega_2 - \omega_1^{-2}\omega_2^{-1}- \omega_1\omega_2^{-2}- \omega_1^{-1}\omega_2^2).
\lim_{M\rightarrow\infty}\lim_{n\rightarrow\infty}II_2 \leq C \bigg\{\lim_{n\rightarrow\infty}\mathbb{E}\int_0^{T}\Vert X_n(s) - X(s) \Vert_H ds \bigg\}^{\frac{\alpha-\theta}{\alpha}}= 0 .
{\cal S}_{E}= \frac{\pi}{4}\tau \nu \sum_{n = 0}n^{2}X_{n 0}^{2}+ \frac{\pi}{2}\tau \sum_{n = 0}\sum_{k = 1}\left( \rho \left( \frac{2 \pi k}{\tau}\right)^{2}+ n^{2}\nu \right) | X_{n k}|^{2},
P_{\chi}(f)(g) =\int_{B_{n,*}}f(\mathfrak{s}(b_*)g)\delta^{1/2}_{B_n}(b_*)\chi^{-1}(\mathfrak{s}(b_*))\, d_lb_* &=\int_{T_{n,*}}\int_{N_{n}}f(\mathfrak{s}(tn)g)\delta^{1/2}_{B_n}(t)\chi^{-1}(\mathfrak{s}(t))\, dn\, dt.
C_{[m,0]}=[w^{m(p+1)+1}]\,h(w) & =\frac{1}{m(p+1)+1}\,[t^{m(p+1)}]\,(t^{p+1}+1)^{m(p+1)+1}=\frac{1}{m(p+1)+1}\binom{m(p+1)+1}{m}=\frac{1}{mp+1}\binom{m(p+1)}{m}.
\mu_{\rm II}(\eta )=\frac{1}{n}\sqrt{\frac{\pi}{\epsilon \vert \eta_{\rm i}\vert}}\cos \biggl(n\eta -n\eta_1 +\frac{2\pi}{\epsilon}\ln \biggl \vert \frac{\eta_1}{\eta_{\rm i}}\biggr \vert \biggr).
\frac{d}{d\lambda}S_{\omega,\mathbf{c}}(\Phi_0^{\lambda})=-(\sqrt{\lambda}-1)(2L(\Phi_0)+\mathbf{c}\cdot \mathbf{P}(\Phi_0))\left(\lambda +\frac{\mathbf{c}\cdot \mathbf{P}(\Phi_0)}{2L(\Phi_0)+\mathbf{c}\cdot \mathbf{P}(\Phi_0)}\sqrt{\lambda}+\frac{\mathbf{c}\cdot \mathbf{P}(\Phi_0)}{2L(\Phi_0)+\mathbf{c}\cdot \mathbf{P}(\Phi_0)}\right)
R_{\theta,n}=n^{-\sigma}\left\{\cos\left(\frac{\theta}{2}t\right)\Omega_{n-1}^{-1}\cos\left(\varphi_n-\varphi_1\right) + \sin\left(\frac{\theta}{2}t\right)\Omega_{n+1}^{-1}\sin\left(\varphi_{n+1}+\varphi_1\right)\right\},
m\Theta_{m+1}(f)= -\frac{1}{2\pi i}\int_{0}^{2 \pi}(e^{im\theta}+e^{-im\theta})\frac{d}{d\theta}F^T_f(\theta)d\theta & +\frac{1}{2}\int_{-\infty}^{\infty}e^{-m|t|}(\mathrm{sgn}(t)) \frac{d}{dt}F^M_f(a_t)dt +\frac{1}{2}(-1)^m\int_{-\infty}^{\infty}e^{-m|t|}(\mathrm{sgn}(t))\frac{d}{dt}F_f^M(-a_t)dt
\prod_{n=0}^\infty\left[ (1-e^{-\alpha\epsilon n+(\pi -\alpha )\epsilon +i\epsilon m\beta_{12}})(1-e^{-\alpha\epsilon n+(\pi -\alpha )\epsilon -i\epsilon m\beta_{12}})(1-e^{-\alpha\epsilon n-\alpha\epsilon +i\epsilon m\beta_{12}})(1-e^{-\alpha\epsilon n-\alpha\epsilon -i\epsilon m\beta_{12}})\right].
\sum_{k\geq 0}\Im G_{\ell\ell}(E+\mathbf{i}2^k\eta) =\sum_{k=0}^{k_1}\Im G_{\ell\ell}(E+\mathbf{i}2^k\eta)+\sum_{k=k_1}^{k_2}\Im G_{\ell\ell}(E+\mathbf{i}2^k\eta)+\sum_{k=k_2+1}^{\infty}\Im G_{\ell\ell}(E+\mathbf{i}2^k\eta) &\prec \frac{\eta_0}{\eta}\sum_{k=0}^{k_1}\frac{1}{2^k}\Im G_{\ell\ell}(E+\mathbf{i}\eta_0)+(k_2-k_1)+1\prec \frac{\eta_0}{\eta}
U^{- 1}( z_{1}, \dots , z_{N_{p}}) = P \exp \left[ -{\frac{1}{2 \pi \kappa}}\int_{\Gamma}\sum_{\alpha}d z_{\alpha}^{\prime}\sum_{\beta \not = \alpha}\hat{Q}_{\alpha}^{a}\hat{Q}_{\beta}^{a}\frac{1}{z_{\alpha}^{\prime}- z_{\beta}^{\prime}}\right]
\sum_{k=1}^l \frac{(\tilde{C_2}t)^k}{k! (k-1)!}\frac{(l-1)!}{(l-k)!}\leq \sum_{k=1}^l (\tilde{C_2}t)^k \frac{(l-1)!}{(l-k)! k!}\leq (1+\tilde{C}_2 t)^l\,,
B(\widetilde{\varepsilon}\vert \zeta \vert^{1/2};\vert \zeta \vert^{-1/2})=B(\widetilde{\varepsilon};1), B^0(\widetilde{\varepsilon}\vert \zeta \vert^{1/2};\vert \zeta \vert^{-1/2})=B^0(\widetilde{\varepsilon};1).
I_{\mathrm{C S , e d g e}}^{i}= \int J_{\mathrm{C S , e d g e}}^{i}d x_{2}={\frac{\sigma_{x y}}{2}}\int \partial_{2}f \epsilon^{i j}A_{j}d x_{2}= -{\frac{\sigma_{x y}}{2}}\epsilon^{i j}A_{j},
\begin{aligned}\operatorname{Pf}(D^g) = \frac{1}{8\pi^2}(\|W^+ +\frac{s_g}{12}I\|^2 + \|W^- + \frac{s_g}{12}I\|^2 - \|R_0^*\|^2 - \|R_0\|^2)\operatorname{vol}_g &=\frac{1}{8\pi^2}(\|W^+\|^2 + \|W^-\|^2 + \frac{1}{24}s_g^2 - 2\|R_0\|^2)\operatorname{vol}_g &= \frac{1}{8\pi^2}(\|W^+\|^2 + \frac{1}{24}s_g^2 - 2\|R_0\|^2)\operatorname{vol}_g\end{aligned}
\left|\frac{a+b}2\right|^p + \left|\frac{a-b}2\right|^p \leq \left( \frac12(a^2 + b^2) \right)^{p/2}&\leq \frac12 ((a^2)^{p/2}+ (b^2)^{p/2}) &= \frac12(|a|^p + |b|^p),
\mu(f)(x_1,x_2) = (1+x_1)(1+x_2)\left[ \frac{x_1}{1+x_1}+ \left(\frac{x_1}{1+x_1}\right)\left(\frac{x_2}{1+x_2}\right) \right] = & = (1+x_2)x_1 + x_1x_2,
a(z^{-n-1})^* Y(s^{L_0}\xi,w)\eta = Y(s^{L_0}a(s^{1/2}z^{-1}(sz^{-1}+\overline{w})^n)^*\xi,w)\eta+(-1)^{p(\xi)}Y(s^{L_0}\xi,w)a(z^{-n-1})^*\eta
\hat{T}^{( r )}( z ) \hat{T}^{( s )}( w ) = \frac{( \hat{c}/ 2 ) \delta_{r + s , 0 \hspace{. 0 5 i n}m o d \hspace{. 0 3 i n}\lambda}}{( z - w )^{4}}+ \frac{2 \hat{T}^{( r + s )}( w )}{( z - w )^{2}}+ \frac{\partial_{w}\hat{T}^{( r + s )}( w )}{z - w}+ O ( ( z - w )^{0})
|F\chi_{R_i}(l)|\le \prod_{j=1}^d \frac{1}{\max(|\pi M^{-\frac{1}{d}}A\, \rho(l)_j|,1)}=\prod_{j=1}^d \frac{(A\pi)^{-1}M^{\frac{1}{d}}}{\max(| \rho(l)_j|,(A\pi)^{-1}M^{\frac{1}{d}})}.
E_1(t) = 1, \ \ \E_2(t) = t+1, \ \ \E_3(t) = t^2 + 4t + 1, &E_4(t) = t^3 + 11t^2+ 11t+ 1, \ \ \E_5(t) = t^4+26 t^3 + 66 t^2 + 26t+1.
\left\langle\sum_{k=0}^m P_k(\lambda)(f_+^{\lambda}(\log f_+)^k\varphi_k),\,\,\phi\right\rangle =\lim_{\tau\rightarrow +0}\sum_{k=0}^m \int_V f_+^{\lambda}(\log f_+)^k\varphi_k{}^tP_k(\lambda)\Bigl(\chi\bigl(\frac{f}{\tau}\bigr)\phi\Bigr)\,dx=0.
\frac{q^{2j\langle n,\tilde m \rangle_1}-1}{q^{2ja}-1}= \frac{q^{2\alpha j/sd(n)}-1}{q^{2j/sd(n)}-1}= \begin{cases}\displaystyle\sum_{p=0}^{\alpha-1}q^{2jp/sd(n)}\alpha > 0, 0 \alpha=0, \displaystyle-\sum_{p=1}^{\alpha}q^{-2jp/sd(n)}\alpha< 0.\end{cases}
-\left(\int_{\Sigma}\overline{q}^2d\sigma_\Sigma\right)L+\left(-\int_{\Sigma}\overline{q}^2d\overline{\sigma_\Sigma}+\int_{\Sigma}Ad\sigma_\Sigma+\sum_{i=1}^n\int_{\gamma_i}k^{\infty,s}_{\gamma_i,\Sigma}d\overline{s}\right)+O(L^{-\frac{1}{2}})=2\pi\frac{\chi(\Sigma)}{\sqrt{L}}.
\begin{aligned}\int_\Omega Y_{l,m}\,\overline{Y}_{l^{\prime},m^{\prime}}\,d\hat{\mathbf{r}}=\delta_{ll^{\prime}}\delta_{mm^{\prime}}, \int_\Omega\mathbf{Y}_{l,m}\cdot\overline{\mathbf{Y}}_{l^{\prime},m^{\prime}}\,d\hat{\mathbf{r}}=l\left(l+1\right)\delta_{ll^{\prime}}\delta_{mm^{\prime}}, \end{aligned}
A_{k}= f_{1}\, \gamma_{k}\otimes 1 \! \! 1 + f_{2}\, \tilde{\gamma}\,^{m}h_{m k}\otimes \sigma_{3}+ f_{3}\, \Pi^{k}\otimes \sigma_{1}+ f_{4}\, \tilde{\Pi}\,^{m}h_{m k}\otimes \sigma_{2}
\begin{aligned}\Phi_k(z,s,a) =\frac1{a^s}+\sum_{r=0}^{k-1}\binom{k}r z^{k-r}&\quad\times\sum_{m_{r+1},\ldots,m_k=1}^\infty\frac{z^{(m_{r+1}-1)+\dots+(m_k-1)}}{((m_{r+1}-1)+\dots+(m_k-1)+a+(k-r))^{s}}&=\frac1{a^s}+\sum_{r=0}^{k-1}\binom{k}r z^{k-r}\Phi_{k-r}(z,s,a+(k-r))\end{aligned}
\begin{cases}\mathbf{S}= \left\{(x_0, x_1, x_2) \in W\, | \, x_0^2 + x_1^2 + x_2^2 = 1 \right\}, \mathbf{n}= (0, 0, 1), \mathbf{P}= \{(x_0, x_1, 0) \in W \mid x_0 + x_1 = 1 \}.\end{cases}
\beta_1 =-\frac{1}{4}(3+(N+3)x+2\check{D})(H_1/J)a_{01}-\frac{1}{2}\frac{\partial(H_1/J)}{\partial t}+(H_1/J)(\check{D}a_{01}+a_{00}), \beta_2 =(H_1/J)\dot{D}a_{00}, \beta_3 =-(H_1/J)\dot{D}J+\dot{D}H_1+\sqrt{x(1-x)}(b_1+a_{21}), \beta_4 =b_0,\beta_5=a_{20}.
L_{C M}( u )_{j}^{i}= ( p_{i}- \frac{l}{n}\frac{\partial}{\partial q_{i}}\ln \Delta ( z ) ) \delta_{j}^{i}- \frac{l}{n}\sigma^{\prime}( 0 ) ( 1 - \delta_{j}^{i}) \frac{\sigma ( u + z_{j i})}{\sigma ( u ) \sigma ( z_{j i})}
\alpha \xi^{m}+\alpha \eta^{m}+q_{m}(\xi,\eta)=(\alpha \xi^{m}+\alpha \eta^{m})\left(1+\frac{q_{m}(\xi,\eta)}{\alpha \xi^{m}+\alpha \eta^{m}}\right )\leq \frac{\alpha}{2}( \xi^{m}+ \eta^{m})<0<|\eta|.
\Omega(x') =\bigg[\Omega(x')-\frac1{\omega_{n-1}}\int_{\mathbf S^{n-1}}\Omega(y')d\sigma(y')\bigg]+\frac1{\omega_{n-1}}\int_{\mathbf S^{n-1}}\Omega(y')d\sigma(y'):=\Omega_0(x')+C(\Omega,n),
G^{\mathrm{r e t}}( \omega ,{\bf x}) = \frac{i^{D}\pi}{2 ( 2 \pi )^{( D - 1 ) / 2}}\left( \frac{\omega}{r}\right)^{( D - 3 ) / 2}H_{( D - 3 ) / 2}^{1}( \omega r ) \, ,
\begin{aligned}\| \mu_{s,t}^f - \mu_{s,t}^g \|_{B^{\alpha - \gamma - 1}_{p,q}}\le \| f - g \|_{L^\infty}\| \mu_{s,t}\|_{B^{\alpha - \gamma - 1}_{p,q}}+ \|\mu \|_{V^r([s,t],B^\alpha_{p,q})}\Big( \|g\|_{V^{r_1}}^\gamma \| f - g\|_{L^\infty}+ \|f-g\|_{V^{r_1}}\Big), \end{aligned}
r_{0,1}(\theta,\theta) = \frac12[\mathbf D'_{m}(2\theta)-\mathbf D'_{d}(2\theta)]=-\sum_{k= d+1}^{m}k\sin(2k\theta)\,, r_{1,1}(\theta,\theta) =\sum_{k= d+1}^{m}k^{2}(1-\cos(2k\theta))\,.
\begin{aligned}f_{k,\ell}- f_{0,k+\ell}- \varrho ( f_{k,\ell-1}- f_{0,k+\ell-1}) begin{align*}2mm] = \beta \sum_{j=1}^k \eta_{j,k+\ell-j}+ \frac{\beta^2}{2}\sum_{j=1}^k \sum_{i=0}^{2(k+\ell-j)}\varrho^i - \beta \sum_{j=0}^J b_j q_{j,k,\ell}\end{aligned}
\pi \epsilon^{2}= \mathcal{H}^{2}( B(y,\epsilon) \cap \Omega) = \mathcal{H}^{2}(B(y, \epsilon) \cap \Omega \cap S_1), < \mathcal{H}^{2}(B(y, \epsilon) \cap \Omega \cap S_1) + \mathcal{H}^{2}(B(y, \epsilon) \cap \Omega^{c}\cap S_1), = \mathcal{H}^{2}(B(y, \epsilon) \cap S_1) = \pi \epsilon^{2},