harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
dX_t = (r_t (X_t- \varphi_t^1 - \varphi_t^2)^+- R_t (X_t- \varphi_t^1 - \varphi_t^2)^-+\varphi_t^1 \mu^1_t +\varphi_t^2 \mu^2_t ) dt + \varphi_t' \sigma_t dW_t + \varphi_t' \beta_t d\tilde N_t & = (r_t X_t+\varphi_t' (\mu_t - r_t{\bf 1}) - (R_t -r_t)(X_t- \varphi_t^1 - \varphi_t^2)^-) dt+ \varphi_t' \sigma_t dW_t + \varphi_t' \beta_t d\tilde N_t.
\tilde{\sigma}_0(x,t):=\langle \lambda_0(\tilde{\mathcal{F}}_0-\gamma \mathcal{E}_{n+1})+\sum_{l=1}^p \lambda_l\tilde{\mathcal{F}}_l, \tilde{x}^{\otimes m}\rangle=\lambda_0(\tilde{f}_0(x,t)-\gamma t^m) +\sum_{l=1}^p \lambda_l\tilde{f}_l(x,t)
\xi_{m,k,i}^{(\alpha_k^*)}={( - 1)^i}\sin \left({{{\cos}^{- 1}}\left({z_{m,k,i}^{(\alpha_k^*)}}\right)}\right)\sqrt{{\mkern 1mu}\varpi_{m,k,i}^{(\alpha_k^*)}},i = 0, \ldots ,m;\,k = 0, \ldots ,n.
(\partial_iU_0f)(x) =\int_0^1\frac{1}{2\sqrt{t}}E\left[\partial_if(\sqrt{t}x+\sqrt{1-t}Z^*)\right]dt, (\partial_{i,j}^2U_0f)(x) =\int_0^1\frac{1}{2}E\left[\partial_{i,j}f(\sqrt{t}x+\sqrt{1-t}Z^*)\right]dt
\begin{array}{l}{{\cal X}^{0}\vert n , z \rangle_{0}= q^{n}z \vert n , z \rangle_{0},}\\{{\cal D}_{0}\vert n , z \rangle_{0}= \displaystyle \frac{q^{1 - n}}{q - 1}z^{- 1}\vert n - 1 , z \rangle_{0},}\\{\mu_{0}\vert n , z \rangle_{0}= \vert n - 1 , z \rangle_{0},}\\ \end{array}
-\frac{1}{2\pi^2{\cal M}}\cdot\frac{\Lambda^2Y+\left[q^2+k^2(1-\gamma)\right]/(2-\gamma)}{\Lambda^2Y(b\pm iq)+(2/\pi)\Lambda (1-Y)(b^2+q^2)-\left(b^3\pm (iq)^3\right) +O(1/\Lambda)},
\mathfrak{G}_{t}:=\{g_{t}(t_{1},t_{2},t_{3})(s_{1},s_{2},s_{3}) \{(s_{1},s_{2},s_{3})^{\pm1},(s_{1},s_{1}/s_{3},s_{1}/s_{2})^{\pm1},(s_{2}/s_{3},s_{2},s_{2}/s_{1})^{\pm1},(s_{3}/s_{2},s_{3}/s_{1},s_{3})^{\pm1}\}\}\simeq C_{2}^{3}
= \int_{0}^{\infty}\ldots \int_{0}^{\infty}\mathrm{d}s_{1 2}\ldots \mathrm{d}s_{( n - 1 ) n}\tilde{F}( s_{1 2}, \ldots , s_{( n - 1 ) n}, g^{2}, d ) e^{- ( s_{1 2}r_{1 2}^{2}+ \ldots + s_{( n - 1 ) n}r_{( n - 1 ) n}^{2})}
\int \left( \frac{v^2}{T_g}- \frac{d_3 v}{\sqrt{T_g}}- 1 \right)^2 g\mathrm{d}v = \frac{\rho_g T_g^2 (3+d_4)}{T_g}- \frac{2\rho_g d_3^2T_g^{3/2}}{T_g^{3/2}}+ \frac{\rho_g d_3^2 T_g}{T_g}-2 \frac{\rho_g T_g}{T_g}+ \rho_g = \rho_g \left( 2+d_4 -d_3^2\right).
\tilde{\varSigma}^{(k,l)}_N := \sum_{m=1}^M \beta_{m;N}^{k,l}\tilde{g}_{k,l;N}^m - \sum_{r,s=1}^M\alpha_{r,s;N}^{k,l}\tilde{g}_{k;N}^r\bigl(\tilde{g}_{l;N}^s\bigr)^T \in\mathbb{R}^{d_k \times d_l}; \tilde{\varSigma}_N := \bigl(\tilde{\varSigma}_N^{(k,l)}\bigr)_{k,l=1,\dots,K}\in \mathbb{R}^{d \times d},
|{\mathcal{R}}_R|\, \le \big| \{|x-y|\le R_0 \}\cap \{|x|,|y|\le 2R\}\big|+ 2\, \big| \{|x-y|\le R_0 \}\cap \{|x|\le 2R\le|y|\}\big| \le 3\, \int_{B_{R_0}}\left[ \int_{B_{2R}}\, dx\right]\,d\zeta \le C R^2,
u_{2\epsilon\kappa}''-\left[\frac{\kappa\left(\kappa-1\right)}{r^2}+\frac{\kappa}{r}\frac{V'}{\epsilon-m-V}-V^2+2\epsilon V-k^2\right]u_{2\epsilon\kappa}+\frac{V'}{\epsilon-m-V}\,u_{2\epsilon\kappa}'=0\,.
\int_{S^n}|X_*(x)|_{g_b}dx = \int_{S^n}\frac{|X_*(x)|_{g^n}}{d(f_*(x))}dx &=\int_{S^n}\frac{\sqrt{|X(x_1)|^2_g+\ldots+|X(x_n)|^2_g}}{d(f_*(x))}dx & \leq \int_{S^n}\frac{|X(x_1)|_g+\ldots+|X(x_n)|_g}{d(f_*(x))}dx & = n \int_{S^n}\frac{|X(x_1)|_g}{d(f_*(x))}dx.
L - H{\frac{\delta L}{\delta H}}- I{\frac{\delta L}{\delta I}}+ [ H{\frac{d}{d t}}+ H ( 3 H + D I ) - \dot{H}]{\frac{\delta L}{\delta \dot{H}}}+ [ I{\frac{d}{d t}}+ I ( 3 H + D I ) - \dot{I}]{\frac{\delta L}{\delta \dot{I}}}\equiv{\cal D}L = 0 .
\sum_{k=0}^{m}c^{2k}T(m+1,k+1) =c^{2}\sum_{k=0}^{m-1}c^{2k}T(m,k+1)-m^{2}\sum_{k=0}^{m-1}c^{2k}T(m,k+1) =(c^{2}-m^{2})\sum_{k=0}^{m-1}c^{2k}T(m,k+1)=0,
c_-^{(h,g)}= \mathrm{pexp}\left(- \dfrac{(Q^8-Q^{-8})}{(1 + s^{-\varphi(g)_{2,2}}p^{\varphi(g)_{2,1}})(1 + s^{\varphi(hg)_{1,2}}p^{-\varphi(hg)_{1,1}})(1 + s^{\varphi(hg)_{2,2}}p^{-\varphi(hg)_{2,1}})}\right)\qquad\in\quad\mathbf R.
(t^2at^{-1}\phi(a)^{-1})\phi(a)\left[(t^2at^{-1}\phi(a)^{-1})a^{-1}\right](t^2at^{-1}\phi(a)^{-1})^{-1}&=(t^2at^{-1}\phi(a)^{-1})\phi(a)\left[ta^{-1}tat^{-1}\phi(a)^{-1}\right](t^2at^{-1}\phi(a)^{-1})^{-1}&=(t^3at^{-1}\phi(a)^{-1})(t^2at^{-1}\phi(a)^{-1})^{-1}&=t
|u|(r,t)=\int_r^{\infty}|\partial_{\tau}u(\tau,t)|\,\tau^{\frac{5}{2}}\,\tau^{-\frac{5}{2}}\,d\tau\leq Cr^{-2}\left(\int_{R^6}|\nabla u|^2\,dx\right)^{\frac{1}{2}}.
P\left(X_t^{(h)}(\epsilon) \le -\frac 1 4 \eta x b_t\right) = P\left(e^{-\lambda X_t^{(h)}(\epsilon)}\ge e^{\frac{\lambda}{4}\eta x b_t}\right) \le E\left(e^{-\lambda X_t^{(h)}(\epsilon)}\right) e^{-\frac \lambda 4 \eta x b_t}.
\overset{g}{P}(X,Y)Z = \overset{g}{R}(X,Y)Z + \frac{1}{n+2}( \overset{g}{R}ic (X,Z)Y - \overset{g}{R}ic (Y,Z)X ) - \frac{1}{n+2}(\overset{g}{R}ic (X,AZ)AY - \overset{g}{R}ic (Y,AZ)AX + 2\overset{g}{R}ic (X,AY)AZ)
Rm(Y_1, Y_2, Y_1, Y_2)= \frac{\mathrm{h}^2\big(4- (\mathrm{h}')^2\big)- 3\mathrm{f}^2}{\mathrm{h}^4}> \frac{3(\mathrm{h}^2- \mathrm{f}^2)}{\mathrm{h}^4}\geq 0
f^{\ast}(x^{\ast})=\min\bigg\{\sum_{n\geq1}f_{n}^{\ast}(x_{n}^{\ast})\mid(x_{n}^{\ast})_{n\geq1}\subset(\operatorname*{dom}f_{n}^{\ast})_{n\geq1},\x^{\ast}=w^{\ast}\sum_{n\geq1}x_{n}^{\ast}\bigg\}
f_i(x,m,v_i) = \chi(v_i) + \Phi_{i,\Lambda}(x,m) \, , &\chi(v) = \frac{|v|^2}{2}, \Phi_{i,\Lambda}(x,m) = \sum_{k=1}^2 \Lambda_{i,k}\, \phi * \mu_k(x) \, ,
J^{*}[\psi^{(3)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p\to \infty}}$\cr}}}\sum\limits_{j_1, j_2, j_3=0}^{p}C_{j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}
\begin{aligned}\int_{0}^{t}\left[\Theta_{,i}(t-\tau)A\dot{\Theta}_{,j}(t+\tau)-\Theta_{,i}(t+\tau)A\dot{\Theta}_{,j}(t-\tau)\right]d\tau= \negmedspace=-\Theta_{,i}(t)\Theta_{,j}(t)+\beta\left[\Theta_{,j}(2t)\theta_{,i}^{0}-2\Theta_{,j}(t)\theta_{,i}(t)\right],\end{aligned}
a_i(x) ={1 \over \sqrt{2\pi L_1}}\sum_{n_1}\int d p_2\,{1 \over{p^0}}[ ({p_i \over \mu}g(\vec p)+ i{\epsilon_{ij}p_j \over 2}h (\vec p))e^{-ip\cdot x}+ c.c]_{p^0 = |\vec p|}\,,
\left( x_L^{(\ell+1)}, x_R^{(\ell+1)}\right) = \begin{cases}\left( \max (0, x_L^{(\ell)}), x_R^{(\ell)}\right), \ell \left( x_L^{(\ell)}, x_R^{(\ell)}+1 \right), \ell . \end{cases}
\begin{array}{r}\displaystyle0\ne \hat f(x_2)-\hat f(x_1)=\lim_\alpha f(x_{2\alpha})-\lim_\alpha f(x_{1\alpha})=\lim_\alpha f(q_j(x_{2\alpha}))-\lim_\alpha f(q_j(x_{1\alpha}))\medskip \displaystyle=\tilde f(q_j(x_2))-\tilde f(q_j(x_1))=0,\end{array}
\rho \colon \U_q(\mathfrak{sl}_2)_{(\alpha,\beta,\psi,\omega )}\otimes \mathbb{A}_{q,\alpha , \beta}^{2|0}\rightarrow \mathbb{A}_{q,\alpha , \beta}^{2|0}, \rho (h\otimes a)=h\triangleright a=\alpha (h)\cdot \beta_\mathbb{A}(a).
2 \alpha\mathrm{Re}\langle u_{\alpha}\otimes R_*\,|\, V_y \Phi_{\#}^y \rangle \geq - C \alpha \varepsilon_8 \|R_*\|^2 - C \alpha \varepsilon_8^{-1}|\kappa|^2 \|V_y\Phi_{\#}^y\|^2 &= - C \alpha \varepsilon_8 \|R_*\|^2 + O(\alpha^4).
\begin{aligned}a)\ \beta^N_r(X;\xi)= \ 0 b)\ \beta_r(X) =\ \mathcal J_{r-1}(X; \xi)(1) + \mathcal J_r(X; \xi)(1) c)\ \beta_r(X;(\xi,u)) =\ \mathcal J_{r-1}(X; \xi)(1/u) + \mathcal J_r(X; \xi)(u). \end{aligned}
A_3 =\left(\int_{\varphi_n(E_M)}|Dh_n\circ\phi_n-Dh_n\circ\phi_n D\phi_n|^p\,d\mu\right)^{\frac{1}{p}}&\leq\|\mathrm{id}-D\phi_n\|_{L^\infty}\|D\phi_n^0\|_{L^\infty}\left(\int_{\varphi_n(E_M)}|Df_n|^p\,d\mu\right)^{\frac{1}{p}}\ .
\mathbb{P}(\mbox{SIR}_0>T) \ge\mathbb{P}\left(\frac{F_0^{-1}}{\sum_{\ell:X_\ell\in\Phi_\mathrm{L}}F_\ell^{-1}}>T\right) &\stackrel{(a)}=\frac{\alpha_\mathrm{L}T^{-2/\alpha_\mathrm{L}}}{2\pi\sin(2\pi/\alpha_\mathrm{L})},
\delta \mathcal{P}_{a}^{( 1 )}= - \pi_{a}^{0}, \delta \mathcal{P}_{a}= \partial_{i}B_{a}^{0 i}, \delta P_{a}^{( 1 ) i}= p_{a}^{i}, \delta P_{a}^{i}= - \partial^{i}\varphi_{a},
\begin{aligned}r_2=0 \Leftrightarrow{}_2 x_1 \circ{}_1 x_2 - i{}_2 x_1 \circ{}_1 x_2 i =0 \Leftrightarrow z_1w_1+ z_i w_i=0, \; z_i w_1- z_1 w_i=0.\end{aligned}
f_2f_{2'}^{(3)}f_2 + f_{2'}^{(3)}f_2 = f_{2'}^{(2)}f_2^{(2)}f_{2'}, f_2f_{2'}^{(4)}f_2^{(2)}+ f_2^{(2)}f_{2'}^{(4)}f_2 = f_{2'}^{(2)}f_2^{(3)}f_{2'}^{(2)}.
\begin{aligned}\qquad\sum_{\mu \in P}m(\pi_\lambda, \sigma_\mu) \delta_{\mu}= \sum_{t\geq 0, h \geq 0}[ \sum_{j}m(\pi_{\lambda +\rho_n}^K, \pi_{\mu_j +\rho_\mathfrak z}^{K_\mathfrak z})\binom{t+c-1}{c-1}\binom{h+d-1}{d-1}\delta_{[(\mu_j +\rho_\mathfrak z)(Z_0)+ t+ 2h-1] \varphi}. \end{aligned}
\mathbf{II}\leq \frac{C_n}{t}\int^{\sqrt{t}}_{r_0}\exp\left(-\frac{r^2}{C_1t}\right) \frac{k(x_0,r)}{r}\,dr \leq \frac{C_4}{t}\int^{\sqrt{t}}_{r_0}\frac{k(x_0,r)}{r}\,dr
\mu_f(x,t) = f(x) + \frac{t^2}{6}\,\sum_{i=1}^n f_{ii}(x) + \left(\frac{1}{120}\sum_{i=1}^n f_{iiii}+ \frac{1}{36}\,\sum_{\substack{i,j=1 j>i}}^n f_{iijj}\right) t^4\,,
\overset{..}{q}_{j}(t)= \frac{d}{dt}\int_{\Omega}\mathrm{Im}\left( \overline{\psi}\partial_{x_{j}}\psi dx\right) =-\int_{\Omega}\partial_{x_{k}}T_{jk}(t,x)dx-\int_{\Omega}|\psi (t,x)|^{2}\partial_{x_{j}}V(x)dx = \int_{\partial \Omega}T_{jk}(t,x)\cdot n_{k}d\sigma -\int_{\Omega}|\psi(t,x)|^{2}\partial_{x_{j}}V(x)dx:=I_{1}+I_{2}
{\frac{d S^{( 1 )}}{d \ln M^{2}}}={\frac{\lambda}{6 4 \pi^{2}}}\int d^{4}x \mu^{2}\phi_{c l}^{2}( x ) +{\frac{\lambda^{2}}{2 5 6 \pi^{2}}}\int d^{4}x \phi_{c l}^{4}( x ) .
\sum_{xx_1\cdots x_r \in \mathcal{P}_r(x)}\frac{k}{d_{H}(x)^2}\prod_{j=1}^{r-1}\frac{1}{(d_{H}(x_j) -1)^2}+ \sum_{yy_1\cdots y_r \in \mathcal{P}_r(y)}\frac{k}{d_{H}(y)^2}\prod_{j=1}^{r-1}\frac{1}{(d_{H}(y_j) -1)^2}.
(0,0)=\langle \mathbf{F}(u^n_h,{\boldsymbol\sigma}^n_h), (\bar{u}_h,\bar{\boldsymbol\sigma}_h)\rangle=\left( \langle F_1(u^n_h,{\boldsymbol\sigma}^n_h),\bar{u}_h\rangle, \langle F_2(u^n_h,{\boldsymbol\sigma}^n_h), \bar{\boldsymbol\sigma}_h\rangle\right),
L^{(1)}v_1=0, \x\in (D_0^{(1)}\setminus B)\times(t_1,t_2), v_1\big|_{x_0=t_1}=\frac{\partial v_1}{\partial x_0}\Big|_{x_0=t_1}=0, v_1\big|_{\partial (D_0^{(1)}\setminus B)\times(t_1,t_2)}=f_1,
\Delta_{\beta_{m+1},\beta_{m-1}}(u_m^2) = \, (m)_q(1-q^{m-1}r)(u_mx_1+q^{m(m-1)}r^{m-1}sq_{21}x_1u_m)\otimes u_{m-1}= \,(m)_q(1-q^{m-1}r)q^{m(m-1)}r^{m-1}sq_{21}u_{m+1}\otimes u_{m-1}.
\begin{array}{l c l}{w_{3}}&{=}&{u_{3}- 2 \theta u_{2}^{\prime}}\\{w_{4}}&{=}&{u_{4}- \frac{9}{1 0 0}u_{2}^{2}- \theta u_{3}^{\prime}+ \frac{{4 \theta^{2}}}{5}u_{2}^{\prime \prime}}\\ \end{array}
u_i =-g_i(\xi_i,x_i)+x_{i(n_i^x+1)}, \dot{x}_{i(n_i^x+1)}=x_{i(n_i^x+2)}, \vdots \dot{x}_{in_0^w}=\sum\nolimits_{j=1}^{n_0^w}s_j^0x_{ij}+d_{n_0}^wv+d_{n_0}^w\sum\nolimits_{j=1}^{n_0^w}k_j^0(x_{ij}-\eta_{ij}), \dot{\xi}_i =A_i^0\xi_i+f_i(x_i), \dot{\eta}_i =S\eta_i+d v+ l_0c^{\top}\eta_{vi}
{F_{\alpha_1\alpha_2}}{F_{\alpha_2\alpha_3}}{F_{\alpha_3\alpha_4}}{F_{\alpha_4\alpha_1}}\rightarrow{F_{1\alpha_1\alpha_2}}{F_{2\alpha_2\alpha_3}}{F_{3\alpha_3\alpha_4}}{F_{4\alpha_4\alpha_1}}
\Phi_1(S_1,S_2,P) =9(I-P^*P)+(S_1^*S_1-S_2^*S_2)-6\text{Re}(S_1-S_2^*P)\,, \Phi_2(S_1,S_2,P) =9(I-P^*P)+(S_2^*S_2-S_1^*S_1)-6\text{Re}(S_2-S_1^*P)\,.
\frac{r(q_k -2,{\mathbf x})}{q_k-2}\le \frac{|W_k| + q_k + q_{k-1}-2}{q_k -2}= 1 + \frac{t_k +\eta_k}{1- \varepsilon_k}< 1 + t_k +\eta_k + 2\varepsilon_k.
1+b' = \dfrac{(r_p(r-p))_q + r_p j' - i'}{(r_p(r-p))_q}\ge \dfrac{(r_p(r-p))_q -(r_p -2)}{(r_p(r-p))_q}= \dfrac{(r_p(r-p-q))_q + 2}{(r_p(r-p))_q}> 0, 1-b' = \dfrac{(r_p(r-p))_q - r_p j' + i'}{(r_p(r-p))_q}\ge \dfrac{(r_p(r-p))_q -(r_p(r-p-q))_q}{(r_p(r-p))_q}= \dfrac{q}{r-p}> 0.
\tilde R^{(k_s)}= \left( \prod_{1\leq i<j\leq k_s}\frac{1-\delta_i\delta_j R_{ij}}{1+\delta_i \delta_j (R_{ij}-\beta T_i)}\right) \left(\prod_{i=1}^{k_s}(1-\beta\tilde T_i)^{k_s-i}\right),
\times \delta ( p_{\theta}- p^{\prime}+ \frac{\hbar m}{2}) \delta ( p_{\theta}- p^{\prime \prime}- \frac{\hbar n}{2}) e^{\frac{2 i}{\hbar}[ - ( p_{\theta}- p^{\prime}) ( \theta - \theta^{\prime \prime}) + ( p_{\theta}- p^{\prime \prime}) ( \theta - \theta^{\prime}) ]}f ( \theta^{\prime}, p^{\prime}) g ( \theta^{\prime \prime}, p^{\prime \prime}) .
\Pi (t_{\emptyset})\prod_{i=1}^{n}(u-x_{i})=(u|t)^{n}+\sum_{r=1}^{n}(-1)^{r}G_{1^{r}}(x|t)(u|t)^{n-r}\prod_{i=1}^{r-1}(1+\beta u\oplus t_{n+1-i})
e_1(i)=1, e_2(i)=j_k(i), e_3(i)=\begin{cases}1, i \neq i_1 j_k(i_1), i=i_1 \end{cases}, e_4(i) = \begin{cases}1, i \neq i_2 j_k(i_2), i=i_2 \end{cases}
P_{\mathrm{gen},p}(z)=\lim_{w\rightarrow z}\bigg(-\log\big{|}1-e^{2\pi i(\sigma_{p}^{-1}z-\sigma_{p}^{-1}w)}\big{|}^{2}- \log\bigg{|}\frac{\sigma_{p}^{-1}{z}-\overline{\sigma_{p}^{-1}w}}{\sigma_{p}^{-1}{z}-\sigma_{p}^{-1}w}\bigg{|}^{2}\bigg)+ O_{z}(1)= 4\pi\Im(\sigma_{p}^{-1}z)-\log\big(4\Im(\sigma_{p}^{-1}z)^2\big) + O_{z}(1),
p\mathcal{E}_{p,q}(k^\frac1q)\mathcal{K}_{q,p}(k^\frac1p)-q\mathcal{K}_{p,q}(k^\frac1q)\mathcal{E}_{q,p}(k^\frac1p)=\frac{(p-q)\pi_{p,q}\pi_{q,p}}{4}.
\begin{aligned}J_2 \leq C\Vert \langle y\rangle^{l+k}I^2_1\Vert_{L^2_y}\Vert (u+u_e\phi(y)+u_b(1-\phi(y)))^{\frac{1}{2}}\langle y\rangle^{l+k}D^\alpha u \Vert_{L^2_y}&+C\Vert \langle y\rangle^{l+k}I^2_2\Vert_{L^2_y}\Vert (u+u_e\phi(y)+u_b(1-\phi(y)))^{\frac{1}{2}}\langle y\rangle^{l+k}D^\alpha h \Vert_{L^2_y},\end{aligned}
\mathcal{H}^{2}(B(z,R) \cap \Omega) = \sum_{k=1}^{i}\mathcal{H}^{2}(S_k) + \mathcal{H}^{2}(S_{i+1}\cap B(z,R)), &=\pi{D_{S_{i+1}}(z)}^{2}+ \pi (R^{2}-{D_{S_{i+1}}(z)}^{2}), \mbox{by Lemma}\ref{geometriclemma}\mbox{and}\eqref{spherelocally2unifeq}, = \pi R^{2}.
\underset{[0,T]}{\sup}\int\limits_{-\infty}^{\infty}e^{a(t) x_{+}^{5/4}}|u(x,t)|^2\,dx\le c^{*}=c^{*}(a_0; \|u_0\|_{4,2};\|e^{a_0 x_{+}^{5/4}}u_0\|_2; T)
\begin{aligned}\mathcal{L}\mathcal{C}[\chi](\zeta) =\mathcal{L}W^{(p'-1)}[\chi](\eta)=\psi(\zeta)\ , \mathcal{L}\mathcal{C}[\psi](\eta) =\mathcal{L}W^{(p-1)}[\psi](\eta)=\chi(\eta)\ , \end{aligned}
\begin{bmatrix}\Lambda_\eta(\lambda^\ell)^T\otimes I_m -(\Lambda_\eta(\lambda^\ell)^T\otimes I_m)M(\lambda)(\widehat{\Lambda}_\epsilon(\lambda^\ell)\otimes I_n)\end{bmatrix}\mathcal{L}(\lambda)=\begin{bmatrix}e_{\epsilon+1}0\end{bmatrix}^T\otimes P(\lambda),
\tilde{Z}^{\Lambda , \Lambda_{0}}( \varrho \, ; J ) = \int d \mu_{\Lambda , \Lambda_{0}}( \phi ) e^{- \frac{1}{\hbar}\bigl ( \tilde{L}^{\Lambda_{0}, \Lambda_{0}}( \varrho \, ; \phi ) + \tilde{I}^{\Lambda_{0}, \Lambda_{0}}( \varrho ) \bigr ) + \frac{1}{\hbar}\langle \phi , J \rangle}.
-\log M_{i^*}(t) = \int_0^t \zeta(s,X_{i^*,t}(s)) \mathrm{d}s & \le \int_0^{t^{1/4}}\zeta(s,X_{i^*,t}(s))\mathrm{d}s + \int_{t^{1/4}}^t \frac{1}{t}\mathrm{d}s\, & \le Zt^{1/4}\log t + 1\, ,
\widehat{\rho}_\lambda \left(\begin{array}{cc}e^{\overline{a}}\mu(\overline{a})\overline{b}0 1\end{array}\right)=\left(\begin{array}{cc}e^{\overline{a}}\mu(\overline{a})(\lambda \overline{a}+ \overline{b}) 0 1\end{array}\right).
F(1,\, 0) = \frac{1}{2m}\sum_{i=1}^r \left\{-(m-m_i+\lambda) + [(m-m_i+\lambda)^2+4\lambda m_i]^{1/2}\right\}>0, F(1,\, 1) = \frac{1}{2m}\sum_{i=1}^r \left\{m_i-\lambda + [(m_i-\lambda)^2+4\lambda m_i]^{1/2}\right\}= \frac{1}{2m}\sum_{i=1}^r \{m_i-\lambda+m_i+\lambda\}=1.
-\frac12\Omega^{+2i}e^{-2}-\frac12\Omega^{-2i}e^{+2}-2i\pi^{1+}_q\gamma^i_{q\dot q}\pi^{1-}_{\dot q}-2i\pi^{2+}_q\gamma^i_{q\dot q}\pi^{2-}_{\dot q}=0
\varphi(t)=\sum_{\lambda=1}^n\sum_{v=1}^n \varphi_v^{\lambda}(z,t)d\bar{z}_v\frac{\partial}{\partial z_{\lambda}},\,\,\,\,\, \Lambda(t)=\sum_{\alpha,\beta=1}^n \Lambda_{\alpha\beta}(z,t)\frac{\partial}{\partial z_{\alpha}}\wedge \frac{\partial}{\partial z_{\beta}}
(B_1(a_1)\otimes a_2\otimes B_2(a_3))\cdot_{H^{\otimes 3}}(B_1(b_1)\otimes b_2\otimes B_2(b_3)) = B_1(a_1)B_1(b_1)\otimes B_1(a_2)b_2S(B_2(a_3))\otimes B_2(a_4)B_2(b_3),
g_0(p|\bar{P}) =\frac{b_0}{(1+b_0)(a_0+p)}, g_j(p|\bar{P}) =\frac{\alpha_{e_{ij}}c_j}{(1+b_j)(a_j+p)},~\forall{j}\in\{k|e_{ik}\in\mathcal{U}_e\},
\partial_t^2 \left( \begin{array}{c}\varphi{\pi_\chi/\kappa}\end{array}\right)= \left(\begin{array}{cc}\nabla^2 -\mu^2-{\kappa^2\over2}\kappa^2 \mu^2+{\kappa^2\over4}\nabla^2 -\mu^2-{\kappa^2\over2}\end{array}\right)\left( \begin{array}{c}\varphi{\pi_\chi/\kappa}\end{array}\right)
\theta \lambda^\frac{N-2}{2}\lesssim |t|^{-\frac{N}{N-6}}\quad\Rightarrow\theta \lesssim |t|^{-\frac{N}{N-6}+ \frac{N-2}{N-6}}= |t|^{-\frac{2}{N-6}}\ll |t|^{-\frac{1}{N-6}}.
\exp ( \mathrm{e}^{t}- 1 ) = \exp ( \mathrm{e}^{x + t}- \mathrm{e}^{x}) |_{x = 0}= \sum_{n = 0}^{\infty}\frac{F_{n}( 1 , 1 , \cdots , 1 )}{n !}t^{n}\equiv \sum_{n = 0}^{\infty}\frac{B ( n )}{n !}t^{n}.
\sum_{k = 1}^n{( - 1)^{k - 1}L_{mk}{}^4}= \sum_{k = 1}^n{( - 1)^{k - 1}L_{4mk}}& - 4\sum_{k = 1}^n{( - 1)^{k(m - 1)}L_{2mk}}+ 3(( - 1)^{n - 1}+ 1)\,,
\begin{array}{rclcl}X_iX_j = X_jX_i \mbox{for all $i,j=1,\ldots,n$,}begin{align*}0.1em]X_it_j = t_jX_i \mbox{for all $i,j=1,\ldots,n$,}begin{align*}0.1em]g_iX_ig_i = X_{i+1}\mbox{for all $i=1,\ldots,n-1$,}begin{align*}0.1em]g_iX_j = X_{j}g_i \mbox{if $j\neq i,i+1$.}\end{array}
\mathfrak{M}\bigl(\frac{q}{\tau}\,|\,\frac{1}{\tau},\tau\lambda_1,\tau\lambda_2\bigr) (2\pi)^{-\frac{q}{\tau}}\,\Gamma^{\frac{q}{\tau}}(1-\tau) \Gamma(1-\frac{q}{\tau}) = \mathfrak{M}(q\,|\,\tau,\lambda_1,\lambda_2) (2\pi)^{-q}\times \times\Gamma^{q}(1-\frac{1}{\tau}) \Gamma(1-q).
W_{\nu}(z)=w_{\nu}^{\prime}(-4z)=\frac{\sqrt{\pi}}{2}\sum_{n\geq0}\frac{\left( n+1\right)z^{n}}{\Gamma \left( n+\frac{3}{2}\right) \left( \nu +\frac{3}{2}\right)_{n}}=\frac{\sqrt{\pi}}{2}\sum_{n\geq0}\frac{\left( n+1\right)!}{\Gamma \left( n+\frac{3}{2}\right) \left( \nu +\frac{3}{2}\right)_{n}}\cdot\frac{z^n}{n!}.
R_{}= \frac{Nm}{N(p_1-p_2)}&= \frac{N^{K-1}(N-T)}{(N^K - T^K)- T^{K-M}(N^M-T^M)}&= \frac{1-\frac{T}{N}}{1-(\frac{T}{N})^{K-M}}& = \left( 1+\frac{T}{N}+ \cdots + \left(\frac{T}{N}\right)^{K-M-1}\right)^{-1}.
\widehat{g}^{t}(\xi,\omega) = \int f(p,\xi) \left[\int_{-\infty}^{\infty}e^{-2\pi i \left(\xi \cdot v(p)\right)t}e^{-2 \pi i \omega t}dt\right]d^3p &=\int f(p,\xi) \delta\left(\omega + \xi \cdot \frac{p}{\sqrt{1+\lvert p \rvert^2}}\right)d^3p,
C_{\textrm{F}}^{\textrm{NOMA}}=\begin{cases}\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^2 \left( 2^R-1\right)}{P_{\textrm{N}}}, \textrm{if}k > 2^R, \frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^2 \left( 2^R-1\right)}{P_{\textrm{F}}-P_{\textrm{N}}\left( 2^R-1\right)}, \textrm{if}k \le 2^R. \end{cases}
\widetilde{f}_i^{pR/2,qR}(x) =\int_{D_{qR}^R}\int_{D_{pR/2}}G_{D_{pR/2}}(x,z)j(|z-y|)dzf_i^*(dy) &\le (1+\varepsilon) \Lambda_{0,qR}(f_i^*) P_{D_{pR/2}}(x,0),
f(t)\overset{{\rm def}}{=}\left\langle u(t),v\right\rangle -\cos\angle\left(v,V^{\bot}\right)\,\left\Vert u(t)\right\Vert \left\Vert v\right\Vert =\varepsilon+t\left\Vert v_{2}\right\Vert^{2}-\left\Vert v_{2}\right\Vert \sqrt{1+t^{2}\left\Vert v_{2}\right\Vert^{2}}\,.
\cdots \quad \rightarrow H_{1}( \Sigma_{0}){\frac{\buildrel \alpha}{\rightarrow}}H_{1}( V ){\frac{\buildrel \beta}{\rightarrow}}H_{1}( V , \Sigma_{0}){\frac{\buildrel \partial}{\rightarrow}}{\tilde{H}_{0}}( \Sigma_{0}) = 0 \quad
\frac{1}{h}\int_{-\infty}^{t}\sum_{k=1}^{\infty}\frac{2((-1)^k-1)}{\Pi_1 k\pi}e^{-\nu(\frac{k\pi}{h})^2(t-\tau)}\frac{c_1 k^2}{vol_1 h}r_1 (0) r_2 (0)\sin{\frac{k\pi x_3}{h}}p_1(\tau)d\tau,
{M^2 \pi \over g^2N}=\left(\begin{array}{cccc}9x 0 0 0 0{81 \over 4}+ 3x + 6y -3i\sqrt{{y \over 2}}-{3i \over 2}\sqrt{y}0 3i\sqrt{{y \over 2}}{9 \over 2}x 0 0{3i \over 2}\sqrt{y}0{81 \over 4}+{9 \over 2}y \end{array}\right)
{\rm Tr}(Q(t,x)D^2\theta_n(x)) = \zeta''\left(\frac{|x|}{n}\right)\frac{\langle Q(t,x)x,x \rangle}{n^2|x|^2}+\zeta'\left(\frac{|x|}{n}\right)\frac{\textrm{Tr}(Q(t,x))}{n |x|}&- \zeta'\left(\frac{|x|}{n}\right)\frac{\langle Q(t,x)x,x \rangle}{n |x|^3},
A_1 = \left[\begin{array}{ccc}\frac{3}{10}\frac{i \sin \phi}{\sqrt{2}}-\frac{3}{10}-\frac{i \sin \phi}{\sqrt{2}}0 -\frac{i \sin \phi}{\sqrt{2}}-\frac{3}{10}\frac{i \sin \phi}{\sqrt{2}}\frac{3}{10}\end{array}\right],
\frac{\sum_{i=0}^n \sigma_i [-\mu_{i+1}^-]}{\sqrt{\sum_{k=0}^{n}\sigma^2_k}}\le \frac{\sqrt{\sum_{k=0}^{n}\sigma^2_k}\sqrt{\sum_{i=0}^n [\mu^-_{i+1}]^2}}{\sqrt{\sum_{k=0}^{n}\sigma^2_k}}=\sqrt{\sum_{i=1}^n [\mu^-_{i+1}]^2}\le \|\mu^-\|_{{\bf \ell}_2}.
A\cong\frac{\mathbb{C}}{\langle1\rangle}\oplus\frac{\mathbb{C}}{\langle1\rangle}\oplus\frac{\mathbb{C}}{\langle x-1\rangle}\oplus\frac{\mathbb{C}}{\langle(x-1)(x+1)^{2}\rangle}\cong\frac{\mathbb{C}}{\langle x-1\rangle}\oplus\frac{\mathbb{C}}{\langle(x-1)(x+1)^{2}\rangle}
E_{1 n o n - d i s p .}= ( \varepsilon - 1 )^{2}\Bigl ( - \frac{2 3}{1 2 8}\frac{V}{\pi^{2}\lambda^{4}}+ \frac{2 3}{3 8 4}\frac{S}{\pi^{2}\lambda^{3}}+ \frac{2 3}{1 5 3 6}\frac{1}{\pi a}\Bigr ) + \mathrm{c o n s t}
d_j = \frac{\gcd(p_{i_1}p_{i_2}\ldots p_{i_j}\ \ 1\le i_1 < i_2 < \ldots < i_j \le m)}{\gcd(p_{i_1}p_{i_2}\ldots p_{i_{j-1}}\ \ 1\le i_1 < i_2 < \ldots < i_{j-1}\le m)}.
\frac{d}{dt}\int_{\Omega}\Big(\frac{1}{2}\rho |u|^2+\frac{1}{2}H^2\Big) x+\int_{\Omega}\Big(\frac{1}{\sigma}|\nabla H|^2+\mu |\nabla u|^2+(\lambda+\mu)(u)^2\Big)x\leq C.
\begin{array}{lcl}\Delta^{+}|S| \geq |(S,\overline{S})_{D}|=\sum_{v\in\overline{S}}deg_{S}^{-}(v)\geq \sum_{v\in\overline{S}}(deg_{\overline{S}}^{-}(v)+k) begin{align*}0.3cm] = k|\overline{S}|+\sum_{v\in\overline{S}}deg^{-}(v)-\sum_{v\in\overline{S}}deg_{S}^{-}(v)\geq(k+\delta^{-})|\overline{S}|-\Delta^{+}|S|.\end{array}
\lim_{v\rightarrow l}\left(\sum_{r=1}^{k}\sum_{s=1}^{k}f_{kr}(v)e_{ks}(v+\delta^{kt})-\sum_{r=1}^{k}\sum_{s=1}^{k}e_{ks}(v)f_{kr}(v+\delta^{ks})\right) &=\lim_{v\rightarrow R}h_{k}(v)=h_{k}(R).
\langle\bar{P}^{n}\cdots\rangle ={\partial^{n}\over\partial\bar{t}^{n}}\, \langle\cdots\rangle=\left({\partial t_s\over \partial\bar{t}}\,{\partial\over\partial t_s}\right)^{n}= \left(-{1\over\langle PP\rangle_{0}}\,{\partial\over\partial t_s}\right)^{n}\ \langle\cdots\rangle \ ,
t^* \in \begin{cases}\{-\lambda_{i_m,m+1}\},{\rm if ~}x^*_{i_m}(t^*) > x^*_{m+1}(t^*), [-\lambda_{i_m,m+1}, \mu_{i_m,m+1}] ,{\rm if ~}x^*_{i_m}(t^*) = x^*_{m+1}(t^*), \{\mu_{i_m,m+1}\},{\rm if ~}x^*_{i_m}(t^*) <x^*_{m+1}(t^*).\end{cases}
\big | \widehat f_{2n_*}\big(\tfrac in \big) - f_0\big(\tfrac in \big) \big | \leq C_4 L_{n_*}^{\beta/(\beta+1)}+ C_4 \big( \widehat f_{2n_*}\big(\tfrac in \big) L_{n_*}\big)^{\beta/(2\beta+1)}\ \ \i=1,\ldots,n.
S_{e f f}={\frac{1}{1 6 \pi}}\, \int d^{4}x \, \sqrt{- g}\, \, ( - R +{\frac{1}{2}}{\frac{\partial_{\mu}\lambda \partial^{\mu}\bar{\lambda}}{( \mathrm{I m}\lambda )^{2}}}- \sum_{n = 1}^{N}{F_{\mu \nu}^{( n )}{}^{\star}\tilde{F}^{( n )}{}^{\mu \nu}}) ,
\gamma_3 = \eta_2 \frac{|g|^2}{|w|^2},\gamma_4 = \eta_3 \frac{|h|^2}{|v|^2},\gamma_5 = \frac{P s^{-\epsilon}|h|^2+P_{\mathrm{su}_1}l^{-\epsilon}|g|^2}{P r^{-\epsilon}|v|^2},
\alpha=\alpha\mu_1^TH(1)=\mu_n^TP(n)H(1)\alpha=\frac{1}{k}\mu_n^T\left(v_n+w_n\right)=\frac{1}{k}\mu_n^Tw_n=\frac{1}{k}\mu_n^Tb_n+ \mu_n^Ta_n,
\left( \begin{array}{c c}{R^{\alpha \beta}}&{0}\\{0}&{R^{\alpha \beta}}\\ \end{array}\right) - \frac{1}{2}\left( \begin{array}{c c}{g^{\alpha \beta}}&{0}\\{0}&{g^{\alpha \beta}}\\ \end{array}\right) R = \left( \begin{array}{c c}{T^{\alpha \beta}}&{0}\\{0}&{T^{\alpha \beta}}\\ \end{array}\right)
h_{(\sigma_{i})_{i},\mu}(\mathcal{G}_{A}|\mathcal{F}:\mathcal{X}) \leq \liminf_{n\to\infty}h_{(\sigma_{i})_{i},\mu}\left(\bigvee_{j=1}^{n}\mathcal{G}_{A_{j}}|\mathcal{F}:\mathcal{X}\right) &\leq \liminf_{n\to\infty}\sum_{j=1}^{n}h_{(\sigma_{i})_{i},\mu}(\mathcal{G}_{A_{j}}|\mathcal{F}:\mathcal{X}) &\leq 0.
\textsf{Tr}\left(\mathbf{\Sigma}_{e,lmmse}\right) =\sum_{k=0}^{\infty}\frac{\left(-1\right)^k}{\sigma_x^{2k}}\textsf{Tr}\left(\mathbf{S}^{-k-1}\right), &=\sum_{k=0}^{l}\frac{\left(-1\right)^k}{\sigma_x^{2k}}\textsf{Tr}\left(\mathbf{S}^{-k-1}\right) +o\left(\frac{1}{\sigma_x^{2l}}\right)
f_{T_n, \underline{\chi}_n}(e_{\psi_n}g_{n_1,\cdots, n_s}(c_1,\cdots,c_s)e_{\psi_n})=\frac{[G_n:B_{n}^{-}]}{|U_n|}\Psi_{\underline{\chi}_n}^{GL_n}(g_{n_1,\cdots, n_s}(c_1,\cdots,c_s)).
\begin{aligned}\dot{x}= - y + a_{20}x^2 - 2b_{20}x y - a_{20}y^2 + x(r_{20}x^2 + r_{11}x y -r_{20}y^2) = P_1(x,y), \dot{y}= x + b_{20}x^2 + 2a_{20}x y -b_{20}y^2 + y(r_{20}x^2 + r_{11}x y -r_{20}y^2) = Q_1(x,y). \end{aligned}
H_{n}(k,\ell;b;q) := \sum_{n = n_k \geq n_{k-1}\geq \ldots \geq n_1 \geq 0}\prod_{j=1}^{k-1}q^{n_j^2+(1-b)n_j}\begin{bmatrix}n_{j+1}-n_j - bj + \sum_{r=1}^j (2n_r + \chi_{\ell > r}) n_{j+1}-n_j \end{bmatrix}_q.
L_{2,\ell}(\tau) : =U_{\ell}\left(\phi_{\ell}^d(\tau)L_1(\tau)\right), L_{2,\ell}(\tau) =\prod_{n=1}^{\infty}(1-q^{\ell\cdot n})^d(1-q^n)^c\sum_{m\geq \left\lceil\frac{\delta_{\ell}\cdot d+\lceil\frac{\delta_{\ell}\cdot c}{\ell}\rceil}{\ell}\right\rceil}^{\infty}p_{[1^c\ell^d]}(\ell^2m-\delta_{\ell}\cdot \ell\cdot d -\delta_{\ell}\cdot c)q^m.
H' = \frac{1}{2m}\sum_n \left[\vec{p}_n - \frac{e}{c}\vec{A}(x_n) - \frac{e}{c}\; \vec{a}(x_n) \right]^2 + \sum_n e A_0 (x_n) + \sum_{n < n'}V(x_n - x_{n'})
\epsilon^{-1}\left(f^n(u_{h,1}^n)-f^n(u_{h,2}^n),U_h^n\right)_{\mathcal T_h}=\epsilon^{-1}\left((u_{h,1}^n)^3-(u_{h,2}^n)^3,U_h^n\right)_{\mathcal T_h}-\epsilon^{-1}\|U_h^n\|_{\mathcal{T}_h}^2.