harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{P}_{Q^{(\mathsf{B}_n)}}\Big(\mathbf{U}(I,\mathsf{B}_n)\notin\mathcal{T}_\epsilon^n(Q_U)\Big)=2^{-nR_1}\sum_{i\in\mathcal{I}_n}\mathbb{P}_{Q^{(\mathsf{B}_n)}}\Big(\mathbf{U}(i,\mathsf{B}_n)\notin\mathcal{T}_\epsilon^n(Q_U)\Big| I=i\Big)
\ddot{\phi}({\bf k}_n,t) +\omega_n^2 \phi({\bf k}_n,t)+\sum_{n'}\Gamma^D_{nn'}{\dot \phi}({\bf k}_{n'},t)+\omega_n \sum_{n'}\Gamma^C_{nn'}\phi({\bf k}_{n'},t)+\frac{\delta V}{\delta \phi (-{\bf k}_n,t)}= \eta({\bf k}_n,t),
x_{j_{k+1}}^{-}(q^{t_{l_1}+1}z)_{0}x_{j_k}^{-}(z)_\bullet \ldots x_{j_2}^{-}(z)_\bullet x_{j_1}^{-}(z)_\bullet \mathcal{Y}_i (z)\neq 0, &x_{j_{k+1}}^{-}(q^{t}z)_{0}x_{j_k}^{-}(z)_\bullet \ldots x_{j_2}^{-}(z)_\bullet x_{j_1}^{-}(z)_\bullet \mathcal{Y}_i (z)= 0 t\neq t_{l_1}+1.
\prod_{j=1}^{n}(\lambda + X_{j})= \lambda^{n}e_{0}^{n-1}+\lambda^{n-1}e_{1}^{n-1}+.....+\lambda e_{n-1}^{n-1}=\sum_{s=0}^{n-1}e_{s}^{n-1}(1,2,...,(n-1))(\lambda)^{n-s}.
{\rm Re}\langle u_{tt}, u \rangle ={\rm Re}\int_{\mathbb{H}^n}f(u)\overline{u}dx- \int_{\mathbb{H}^n}|\nabla_{H}u|^2 dx - m\int_{\mathbb{H}^n}|u|^2 dx - b{\rm Re}\int_{\mathbb{H}^n}\overline{u}u_t dx &= - I(u) - b{\rm Re}\int_{\mathbb{H}^n}\overline{u}u_t dx.
\langle V(T_{\alpha,N}) , \xi \rangle = \langle T_{\alpha,N}, V^*(\xi) \rangle = \lim_{\varepsilon \to 0}\int_{\vert f\vert \geq \varepsilon}\vert f\vert^{2\alpha}f^{-N}V^*(\xi) = \lim_{\varepsilon \to 0}\int_{\vert f\vert \geq \varepsilon}V(\vert f\vert^{2\alpha}f^{-N})\xi
\int_\Omega u^qdx =\int_\Omega (u^pv^{-r})^{\frac{q}{p}}v^{\frac{rq}{p}}dx \leq\big(\int_\Omega u^pv^{-r}dx\big)^{\frac{q}{p}}\big(\int_\Omega v^{\frac{rq}{p-q}}dx\big)^{\frac{p-q}{p}}, ~~0<t<T.
f(z){}=\int_{\mathbb{R}^d}p_{s-r}(z,y)|y-z|^{\gamma p}\,dy &{}\leq C_T\int_{\mathbb{R}^d}(s-r)^{-d/\alpha}Q\bigl((s-r)^{-1/\alpha}(z-y) \bigr)|z-y|^{\gamma p}\,dy &{}= C_T(s-r)^{\gamma p/\alpha}\int_{\mathbb{R}^d}Q(y)|y|^{\gamma p}\,dy.
e^{q_1}\, q_1 \, (b_1 - b_2) \, \int\limits_{0}^{\infty}b_1 \, e^{b x}\, e^{-q_1 \, e^{b_1 x}}x \; dx= \frac{e^{q_1}\, q_1 \, (b_1 - b_2)}{b_1}\, \int\limits_{1}^{\infty}\ln y \, e^{-q_1 y}\, dy
\lim_{\xi\rightarrow\overline{c_{0}}^{-}}\frac{p_{\mu_{1}\boxtimes\mu_{2}}(\xi)}{\left|\xi-\overline{c_{0}}\right|^{1/3}}=\frac{-2\left|a_{1}\right|}{\sqrt[3]{\left|c_{3}\right|}}\cos\left(\frac{\theta}{3}-\frac{7\pi}{6}\right),
\begin{aligned}\mathbb{E}^p\left[\int_{S_k}^{T_k}\exp(-rt)\left(P(t)-C)\right)\mathrm{d}t-\exp(-r S_k)(k_1P(S_k)+k_0)\right. \left.-\exp(-r T_k)(l_1P(T_k)+l_0)\right] \leq H(p).\end{aligned}
d R_{g}( A |_{h}) = a^{M}( h ) \frac{\partial x^{N}( h g )}{\partial x^{M}( h )}\frac{\partial}{\partial x^{N}( h g )}= a^{M}( h ) R_{M}^{N}( h g , h ) \frac{\partial}{\partial x^{N}( h g )},
T_{B}^{\mu\nu}=T^{\mu\nu}-\frac{i}{2}\partial_{\kappa}[\frac{\partial L}{\partial(\partial_{\kappa}\Psi^{l})}(J^{\mu\nu})_{m}^{l}\Psi^{m}-\frac{\partial L}{\partial(\partial_{\mu}\Psi^{l})}(J^{\kappa\nu})_{m}^{l}\Psi^{m}-\frac{\partial L}{\partial(\partial_{\nu}\Psi^{l})}(J^{\kappa\mu})_{m}^{l}\Psi^{m}]
\nabla_{i}\delta A^{i}+ 2 i{\frac{e^{3}}{\kappa}}( 2 | \phi |^{2}- v^{2}) ( \phi^{*}\delta \phi - \phi \delta \phi^{*}) = -{\frac{e}{\kappa}}( \phi^{*}\partial^{0}\delta \phi - \partial^{0}\delta \phi^{*}\phi ) .
x_{n+2}^2=-\sum_{i=1}^{n+1}\frac{(\lambda_{n+3}-\lambda_i)}{(\lambda_{n+3}-\lambda_{n+2})}x_i^2 \,\,\textrm{and}\,\, x_{n+3}^2=-\sum_{i=1}^{n+1}\frac{(\lambda_{n+2}-\lambda_i)}{(\lambda_{n+2}-\lambda_{n+3})}x_i^2.
\sum_{1 \leq h_2,\dots,h_k \leq H}\left|\sum_{1 \leq n \leq X}g_1(a_1 n + b_1) \prod_{j=2}^k g_j(a_j n + b_j + h_j)\right| \ll A^2k \left( \exp( - M / 80) + \frac{\log\log H}{\log H}+ \frac{1}{\log^{1/3000}X}\right) H^{k-1}X.
\mathcal{A}(y) =\int_B\Delta_x \mathcal{K}(x-y)(1-x\cdot y)\,dx+\int_B\nabla_x \mathcal{K}(x-y)\nabla_x(1-x\cdot y)\,dx =-s\int_B\frac{1-x\cdot y}{|x-y|^{n+s}}\,dx+\int_B\frac{|y|^2-x\cdot y}{|x-y|^{n+s}}\,dx =(1-s)\int_B\frac{1-x\cdot y}{|x-y|^{n+s}}\,dx+\int_B\frac{|y|^2-1}{|x-y|^{n+s}}\,dx\,.
S_{fi}=e^{-i[(u+w)/2]^2 T/2}\!\int\! d^9b \,e^{-i q_\mu b_\mu}\, \langle{\cal H}^3| \langle{\cal H}^4|e^{i\Gamma_{\rm (1)}(u_\mu+w_\mu/2,b_\mu,\theta^3)}|{\cal H}^1\rangle |{\cal H}^2\rangle
\|\tilde a-g'(0)a\|_{l_1^d}=\sum\limits_{i=1}^d\lvert\tilde a_i-g'(0)a_i\rvert=\sum\limits_{i=1}^d\lvert g'(\xi_{h,i})-g'(0)\rvert \lvert a_i\rvert &\leq \sum\limits_{i=1}^d c_1\lvert \xi_{h,i}\rvert \lvert a_i\rvert\leq \sum\limits_{i=1}^d c_1\lvert ha_i\rvert \lvert a_i\rvert=c_1 h\sum\limits_{i=1}^d\lvert a_i\rvert^2 &\leq c_1 h.
\lim_{t \to\infty}g_t(\mathbf{x}) = \|\mathbf{x}\|^{-\beta}a\bigl(\mathbf{x}/\|\mathbf{x}\|\bigr) \qquad\mbox{for all}\mathbf{x}\in\mathbb{R}^d, \mathbf{x}\neq\mathbf{0},
\begin{cases}u_t + uu_x+u_{xxx}+ a v_{xxx}+ a_1vv_x+a_2 (uv)_x =0, \,\, (0,L)\times (0,T), c v_t +rv_x +vv_x+abu_{xxx}+v_{xxx}+a_2buu_x+a_1b(uv)_x =0, \,\, (0,L)\times (0,T), u(x,0)= u^0(x), v(x,0)= v^0(x), \,\, (0,L),\end{cases}
\left|\Pi_{x}\left[\int_{0}^{s}F_{r}(\xi_{t-r})-F_{r}(\xi_{s-r}){\rm d}r\right]\right| &\le\int_{0}^{s}\Pi_{x}\left[\left|\Pi_{\xi_{s-r}}(F_{r}(\xi_{t-s}))-\Pi_{\xi_{s-r}}(F_{r}(\xi_{0}))\right|\right]{\rm d}r &=\int_{0}^{s}\mathfrak{S}_{s-r}\left(\left|\mathfrak{S}_{t-s}F_{r}-F_{r}\right|\right)(x){\rm d}r.
H_F=1+\sum_i{{Q_i}\over{[|\vec{x}-\vec{x}_{0\,i}|^2+{{4Q_p}\over{(p-4)^2}}|\vec{z}-\vec{z}_0|^{p-4}]^{{(p-3)^2+3}\over{2(p-4)}}}},\ \ \ \H_p={{Q_p}\over{|\vec{z}-\vec{z}_0|^{6-p}}}.
T_{2,3}\lambda((t,\ell)) =p(T_p\lambda)(\iota(pt,p\ell)) + \lambda(\iota(t/p^2,\ell/p)) + p^6\lambda(\iota(p^2t,p\ell)) +p^4(T_p\lambda)(\iota(pt,\ell)) + (p^3-1 + \delta_{|t|,|\ell|}(-p^3 + \epsilon_L(p)p^2))\lambda(\iota(t,\ell)).
\overline{L}^{n-1}\cdot\pi_i^{*}\overline{L}_i = (\overline{L}^{n-1}|\mathrm{div}(\pi_i^*1_a)) -(n-1)!\sum_{k=1}^n\sum_{v \in M(K)}\int_{H^{\mathrm{an}}_{\overline{K}_v}}\log \|\pi_i^*1_a\|_{\pi_i^{*}\overline{L}_i ,v}\prod_{j \neq k}c_1(\pi_j^*\overline{L}_{j,v}).
\begin{cases}dx_i =( Ax_i +BR^{-1}B^Ty_i +G m_i+\gamma p_i) dt +D dW_i, \dot{m_i}= (A + G) m_i +BR^{-1}B^T{\mathbb E}y_i +\gamma p_i, \dot{p_i}= -(A+G)^T p_i -(I-\Gamma)^T Q [m_i- (\Gamma m_i+\eta)] , dy_i=\left\{-A^T y_i +Q[x_i-(\Gamma m_i+\eta)]\right\}dt +\zeta_i dW_i,\end{cases}
\sum_{j\geq 0}q_j=|\{i: \p_i\geq 0\}|=R-k+1\hbox{and}&\sum_{j\geq 0}q'_j=R-k+1-\chi(p'_1=-1)-\chi(p'_{R-k+1}=-1)=R-k-1+\delta_\alpha.
\psi_{\frac{1}{2}\frac{1}{2}}= \psi^*_{-\frac{1}{2}-\frac{1}{2}}= \frac{i}{\sqrt{2}}f_2 e^{-i\phi},\quad\psi_{\frac{1}{2}-\frac{1}{2}}=-\psi_{\frac{1}{2}-\frac{1}{2}}= -\frac{i}{\sqrt{2}}f_1.
2\hat{b}_+\cdot\tilde\Sigma^{-1}b_-=\frac{B_0\beta(t,q)}{\gamma^2(t)+B_0^2}\left(\partial_{q^1}\beta^{-1}(t,q)(-\partial_{q^2}V(t,q)+\tilde F_2(t,q))-\partial_{q^2}\beta^{-1}(t,q)(-\partial_{q^1}V(t,q)+\tilde F_1(t,q))\right)
P_{n m}\left( \left| \alpha_{n}\right\rangle \otimes \left| \alpha_{m}\right\rangle \right) = \left| \alpha_{m}\right\rangle \otimes \left| \alpha_{n}\right\rangle , \qquad \left| \alpha_{n}\right\rangle \otimes \left| \alpha_{m}\right\rangle \in{\cal H}_{n}\otimes{\cal H}_{m}.
h_+((a:b), x) = \left(x,\, \frac{a}{b}\left(\frac{a}{b}x + 2\sqrt\pi\right),\, \frac{a}{b}x+\sqrt\pi\right), h_-((a:b), y) = \left(\frac{b}{a}\left(\frac{b}{a}y - 2\sqrt\pi\right),\, y,\, \frac{b}{a}y-\sqrt\pi\right), h_0((a:b), z) = \left(\frac{b}{a}\left(z-\sqrt\pi\right),\, \frac{a}{b}\left(z + \sqrt\pi\right),\, z\right).
\mathcal{I}(\delta_1, 0) = \widehat{\mathcal{I}}(\delta_1, \delta_2) = \inf_{\lambda_1 \in \mathbb{R}_{+}}\left[\langle\lambda, \delta\rangle +\sup_{\varpi \in \Pi\left(\mu_{13}, \mu_{23}\right)}\int_{\mathcal{V}}f_{\lambda}(s_1, s_2) \, d \varpi(s_1,s_2) \right] ,
\left( \sum_{I = 0}^{K}\alpha_{I}^{m}( \varphi^{I})^{2}\right) \left( \sum_{I = 0}^{K}\alpha_{J}^{n}( \varphi^{J})^{2}\right) \left( \sum_{I = 0}^{K}\alpha_{L}^{p}( \varphi^{L})^{2}\right)
\begin{aligned}r^{2}\left\|p\right\|_{L^{\infty}(B_{\frac74 r}^{+}\backslash B_{\frac54 r}^{+})}\leq C(1+r\left\|p\right\|_{L^{2}(B_{2r}^{+}\backslash B_{r}^{+})}+r^{2}\left\|\nabla p\right\|_{L^{2}(B_{2r}^{+}\backslash B_{r}^{+})}) &\cdot\sqrt{\log (e+r^{3}\left\|\Delta p\right\|_{L^{2}(B_{2r}^{+}\backslash B_{r}^{+})})}.\end{aligned}
\psi_{-\sigma}\left( x,t,k\right) =\psi\left( x,t,k\right) & +{\displaystyle\sum\limits_{n=1}^{N}}c_{n}^{2}e^{8\kappa_{n}^{3}t}y_{n}\left( x,t\right) \int_{x}^{\infty}\psi\left( s,t,k\right) \psi\left( s,t,\mathrm{i}\kappa_{n}\right)\mathrm{d}s
\vec{b}^{n}=\frac{1}{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}[\begin{matrix}0\\ \vec{b}^{n-1}\end{matrix}]-\frac{\epsilon_{b}^{n}}{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}[\begin{matrix}\vec{f}^{n-1}\\ 0\end{matrix}].
\rho_{0,1,n}=\left(n-\frac{1}{2}\right)\pi+\frac{q_1(0)}{n\pi}\sin \left(n-\frac{1}{2}\right)\pi a+\frac{\kappa_{0,1,n}}{n},\, \, \{\kappa_{0,1,n}\}\in l_2, n\in\mathbb{Z};
\frac{\rho^{2}}{\Delta_{\chi}}d \chi^{2}+ [ ( r^{2}+ a^{2})^{2}\Delta_{\chi}- a^{2}\sinh^{2}\chi \Delta_{r}] \frac{\sinh^{2}\chi}{\rho^{2}}d \phi^{2}+ r^{2}d \Sigma_{n - 3}.
z_i x^\alpha w_\alpha^{-1}v_T =\left(\alpha_i+1-(d_{\beta(T^{-1}w_\alpha(i))}-d_{\beta(T^{-1}w_\alpha(i))-\alpha_i-1})-r \mathrm{ct}(T^{-1}w_\alpha(i))c_0 \right) x^\alpha w_\alpha^{-1}v_T +.
{\hat{\lambda}_{N}}={\frac{1}{8 \pi}}\oint{d}\hat{s}( \xi ){\hat{V}_{N}}( \xi ){\partial_{\hat{n}}}{\hat{V}_{N}}( \xi ) ={\frac{1}{8 \pi}}\int{d^{2}}\sqrt{\hat{g}}{\hat{g}^{a b}}{\partial_{a}}{\hat{V}_{N}}{\partial_{b}}{\hat{V}_{N}}.
\partial_\nu\left\{\begin{array}{c}L\left(\begin{array}{c}G_\mu H_\mu\end{array}\right)\end{array}\right\}=\{\partial_\nu(LA^a_\mu)\}\left(\begin{array}{c}\frac{\partial n^a}{\partial\theta}\frac{1}{\sin\theta}\frac{\partial n^a}{\partial\varphi}\end{array}\right)+D_{\nu\mu}
\lim_{\lambda\to \infty}\int_{\lambda^{-1}\partial D_{\lambda}}\psi(t)d^cV_{\lambda}\wedge(dd^cV_{\lambda})^{n-1}\ = \ \lim_{\lambda\to \infty}\int_{\lambda^{-1}\partial D_{\lambda}}\phi(t\lambda)d^cV_{\lambda}\wedge(dd^cV_{\lambda})^{n-1}.
\rho_{N}( \lambda_{1}, \cdots , \lambda_{p}) \equiv \left\langle \prod_{a = 1}^{p}\frac{1}{N}\mathrm{t r}\, \delta ( \lambda_{a}- M ) \right\rangle \stackrel{\mathrm{\small ~ W i c k^{\prime}s ~ t h e o r e m}\atop{\downarrow}}{=}\det_{1 \leq a , b \leq p}K_{N}( \lambda_{a}, \lambda_{b})
E(e^{\lambda X_n}) =E(E(e^{\lambda X_n}|X_{n-1}))=E\left(e^{\lambda (X_{n-1}+1)}\frac{m-X_{n-1}}{m}+e^{\lambda X_{n-1}}\frac{X_{n-1}}{m}\right)= &=E\left(e^{\lambda}e^{\lambda X_{n-1}}\left(1-\frac{X_{n-1}}{m}\right)+e^{\lambda X_{n-1}}\frac{X_{n-1}}{m}\right)
\eta_q^{\epsilon,{\bf h}}(t) =h_q+\frac{1}{\epsilon}\int_0^{t}\eta_p^{\epsilon,{\bf h}}(s)ds &=h_q+\int_0^t Df(q^\epsilon(s)).\eta_q^{\epsilon,{\bf h}}(s)ds+\int_0^t D\sigma(q^{\epsilon,{\bf h}}(s)).\eta_q^{\epsilon,{\bf h}}(s)d\beta(s)+\epsilon\bigl(\eta_p^{\epsilon,{\bf h}}(0)-\eta_p^{\epsilon,{\bf h}}(t)\bigr).
( F_{\mu \nu})_{+ 1}^{( + )}= + ( F_{\mu \nu})_{- 1}^{( - )}, ( F_{\mu \nu})_{- 1}^{( + )}= + ( F_{\mu \nu})_{+ 1}^{( - )}, ( F_{\mu \nu})_{0}^{( + )}= - ( F_{\mu \nu})_{0}^{( - )}\quad .
\mathbf{q}_d(\mathbf{x}) =[1,x_1,x_2,x_1^2,x_1x_2,x_2^2]^\top \mathbf{A}_{k-1}=\left[\begin{matrix}0 2 0 -1 0 0 0 0 0.5 0 1 0 \end{matrix}\right] \mathbf{C}_{k-1}=\left[\begin{matrix}0 1 1 0 0 0 \end{matrix}\right]
\tilde P_r = -\frac{1}{2\pi i}\oint_{\gamma_r}R_{\Sigma}(\eta)d\eta - \frac{1}{2\pi i}\oint_{\gamma_r}\sum_{k\geq 1}(-1)^{k}[R_{\Sigma}(\eta)E]^k R_{\Sigma}(\eta)d\eta &=P_r - \frac{1}{2\pi i}\oint_{\gamma_r}\sum_{k\geq 1}(-1)^{k}[R_{\Sigma}(\eta)E]^k R_{\Sigma}(\eta)d\eta,
\dfrac{1}t\int^t_0I(\bar t)d\bar t= \dfrac{1}t\int^t_0I(\bar t){\bf1}_{\{I(\bar t)<\frac{\rho}2\}}d\bar t+\dfrac{1}t\int^t_0I(\bar t){\bf1}_{\{I(\bar t)\geq\frac{\rho}2\}}d\bar t \leq \frac{\rho}2+\frac N t \int^t_0{\bf1}_{\{I(\bar t)\geq\frac{\rho}2\}}d\bar t,
\frac{1}{m}\sum_{j=0}^{m-1}\frac{e^{2\pi i \frac{jr}{m}}}{s+2\sin^2\left(\pi\frac{(j+\beta)}{m}\right)}&= \frac{e^{-2\pi i \beta \frac{\ell}{m}}}{\sqrt{s^2+2s}}\frac{\sinh\left((m-\ell)\cosh^{-1}(s+1)\right)+e^{2\pi i \beta}\sinh\left(\ell\cosh^{-1}(s+1)\right)}{\cosh\left(m\cosh^{-1}(s+1)\right)-\cos2\pi \beta}.
\begin{aligned}\Im\langle v ,D_jDQ(\dot z+iEz)\rangle + \Im \langle D_j Q, DQ(\dot z+iEz)\rangle & = -\Im i\biggl[\langle F(Q+v)-F(Q),D_j Q\rangle - \langle v, D_j(|Q|^p Q)\rangle\biggr]. \end{aligned}
y^{(m)}_{r_1,\ldots,r_k}=\Bigl(\prod_{i=1}^k\mu(r_i)g(r_i)\Bigr)\sum_{\substack{d_1,\ldots,d_k r_i\mid d_i\forall i d_m=1}}\frac{\lambda_{d_1,\ldots,d_k}}{\prod_i\varphi(d_i)}.
\sum_{i=1}^K\frac{R_i}{r_i}\leq \frac{\max_{\mathbf{\Sigma}_{X}:\mbox{tr}(\mathbf{\Sigma}_{X})\leq P}E\left[\log \det(\textbf{I}_{r_K}+\textbf{H}_K^H\mathbf{\Sigma}_X\textbf{H}_K)\right]}{r_K}
\prod_{j=1}^{m-p}\frac{\big[A(p-\beta)-B(p-\beta+j-1)\big]^2}{j^2}=\frac{1}{(m-p)^2}\Bigg((A-B)^2(p-\beta)^2 +\sum_{k=2}^{m-p}\Big(\big[A(p-\beta)-B(k+p-\beta-1)\big]^2-(k-1)^2\Big) \prod_{j=1}^{k-1}\frac{\big[A(p-\beta)-B(p-\beta+j-1)\big]^2}{j^2}\Bigg)
\mathcal{L}_{I_{cd}}(\frac{\beta n \eta_L \Gamma r_0^{\alpha_{L,d}}}{P_{d}G_0})=\exp\bigg(-2\pi\lambda_B \bigg (\sum_{j \in \{L,N\}}\sum_{i=1}^3 p_{G_i}\times &\bigg(\int_0^{\infty}\bigg(1-1/\bigg(1+\frac{\beta n \eta_s \Gamma r_0^{\alpha_{s,d}}P_d G_i}{P_d G_0N_jt^{\alpha_{j,d}}}\bigg)^{N_j}\bigg)p_{j,d}(t)tdt\bigg)\bigg)\bigg)
\Bigg|\int_0^{T-T/n}\int_{S_{k-1,0,s_k}}\int_{(\mathbb{R}^d)^{k+1}}\Biggl(\prod_{i=1}^kh(y_i)\Biggr)f(z) J_{2k}dz \prod_{j=1}^k dy_j \prod_{i=1}^{k-1}ds_{i}ds_k \Bigg| &\leq C_T \frac{1}{(k-1)!}\|h\|^k\|f\| T^{k}\frac{\log n}{n}
\hat{\phi}_{n}^{( i ) \mu}= \sqrt{\frac{\alpha^{\prime}}{2 n}}i \left( \hat{a}_{n}^{( i ) \mu}- \hat{a}_{n}^{\dagger ( i ) \mu}\right) , \quad \hat{\pi}_{n \mu}^{( i )}= \sqrt{\frac{n}{2 \alpha^{\prime}}}g_{\mu \nu}\left( \hat{a}_{n}^{( i ) \nu}+ \hat{a}_{n}^{\dagger ( i ) \nu}\right) .
\frac{1}{2}\bigg ( \bigg | \frac{\alpha_{{k}}}{\beta_{{k}}}\bigg | - \bigg | \frac{\beta_{{k}}}{\alpha_{{k}}}\bigg | \bigg ) = \frac{\Omega_{{k}}}{| \Lambda_{{k}}|}, \quad | \alpha_{{k}}|^{2}+ | \beta_{{k}}|^{2}= 1 ,
m\log_2\Bigl(\frac{n}{2m}\Bigr) =f(m)\ge f\Bigl(\frac{k}{\log_2(n/k+1)}\Bigr) &=\frac{k}{\log_2(n/k+1)}\log_2\Bigl(\frac{n\log_2(n/k+1)}{2k}\Bigr)\ge k,
g_y'(x) = \frac{d}{dx}\log f_y(x) & = \frac{d}{dx}\frac{1}{x}\left( \log \sum_{n=1}^N a_n^x\right) & = \frac{1}{x^2}\left( x\cdot\frac{\left(\sum_{n=1}^N a_n^x\log a_n\right)}{\left(\sum_{n=1}^N a_n^x\right)}- \log \sum_{n=1}^N a_n^x \right ).
2e(A_1) - 2e(B_1) + e(A_1,W_2) - e(B_1,W_2) = D(|A_1|-|B_1|); 2e(A_2) - 2e(B_2) + e(A_2,W_1) - e(B_2,W_1) = D(|A_2|-|B_2|).
\omega^{\beta}(\alpha_T^V(A))=\omega^{\beta}(U(-T)\alpha_{-T}(U(i\beta)^{-1})U(-T)^{-1}U(i\beta) \alpha_{i\beta}A) =\omega^{\beta}(\alpha^V_{-T}(U(i\beta)^{-1}) U(i\beta) \alpha_{i\beta}A).
P_4\,L_-\,P_4=P_4\,(-L_+\,P_2)=-(L_+\,P_4)\,P_2=0, P_4\,L_-\,P_2=P_4\,(-L_+\,P_0)=0, &P_{-4}\,L_+\,P_2=P_{-4}\,(-L_-\,P_4)=0, P_{-4}\,L_+\,P_0=P_{-4}\,(-L_-\,P_2)=0.
VIF=\frac{\sum_{j\in subbands}I(\overline{C}^{N,j};\overline{F}^{N,j}|S^{N,j}=s^{N,j})}{\sum_{j\in subbands}I(\overline{C}^{N,j};\overline{E}^{N,j}|S^{N,j}=s^{N,j})}.
\phi_t|_{\{\beta=0\}}= \Psi_t - \frac1J(X_\alpha X_t+Y_\alpha Y_t)\Psi_\alpha - \frac1J(Y_\alpha X_t-X_\alpha Y_t)\Theta_\alpha, \frac12|\nabla\phi|^2|_{\{\beta=0\}}= \frac{1}{2J}(\Psi_\alpha^2+\Theta_\alpha^2).
\alpha \beta ( v_{i}) = \prod_{j = 1}^{n}q_{i e_{j}}\ \prod_{k = 1}^{m}q_{i o_{k}}^{- 1}\, p r o d_{j = 1}^{n}q_{e_{j}i}\ \prod_{k = 1}^{m}q_{o_{k}i}^{- 1}\v_{i}= q^{2}v_{i}
\widehat{A}_1(x) =1, \widehat{A}_2(x) =1+x, \widehat{A}_3(x) =2+2x+2x^2, \widehat{A}_4(x) =5+7x+7x^2+5x^3, \widehat{A}_5(x) =16+26x+36x^2+26x^3+16x^4.
\| T_j(t)[x]_j \|_j = \inf_{p_j(z)=0}p_j\Big( T(t)x - T(t)z - \big(z-T(t)z\big) \Big) & \leqslant \inf_{p_j(z)=0}\left \{p_j^{X}\big(T(t)\big) p_j(x-z) + p_j(z) + p_j\big(T(t)z\big) \right \}&= p_j^{X}\big(T(t)\big) \| [x]_j \|_j,
B_\infty = \left(\frac{1}{2}-p\right) \sum_{k=0}^{\infty}\left(\frac{1-p}{2}\right)^k \sum_{j=0}^k{k \choose j}^2 \left(\frac{p}{1-p}\right)^{j}= \left(\frac{1}{2}-p\right)\sum_{k=0}^\infty \left(\frac{1}{2}-p\right)^kP_k\left(\frac{1}{1-2p}\right) &= \left(\frac{1}{2}-p\right)\frac{1}{\sqrt{1+(1/2-p)^2-2(1/2-p)/(1-2p)}}&=1.
4K_1^2+4M+8C_4M^{-\frac{p-d}{d}}K_2^{\frac{2p}{d}}\alpha^2\varrho^{-2}=4K_1^2+4K_2^2(1+2C_4)\alpha^{\frac{2d}{p}}\varrho^{\frac{-2d}{p}}& \le (4K^2(1+2C_4))\alpha^{\frac{2d}{p}}\varrho^{\frac{-2d}{p}}.
\displaystyle\lim_{n \to +\infty}\int_{\mathbb{R}^N}\left(|\nabla v_n(x)|^2+V_{0,\infty}v^2_n(x)\right)dx = \lim_{n \to +\infty}\int_{\mathbb{R}^N}\left(|\nabla v_n(x)|^2+V_0(x)v^2_n(x)\right)dx.
u(c_{2})=\frac{\delta}{\gamma}\phi_{2\lambda}\left( c_{2}^{-}\right) ,u^{\prime}(c_{2})=\frac{1}{\gamma}\left\{\delta \phi_{2\lambda}^{\prime}\left( c_{2}^{-}\right) +\lambda \phi_{2\lambda}\left( c_{2}^{-}\right) \right\}.
\omega (h,g) = \lambda_{h_{(1)}}\lambda_g \omega (h_{(2)},g) =0 , \forall h\in H_4 , & \omega (xg,x) = \lambda_{xg}\lambda_x \omega (g,1_H)+\lambda_{xg}\lambda_g \omega (g,x)+\lambda_1 \lambda_x \omega (xg ,1_H) +\lambda_1 \lambda_g \omega (xg,x)=-\lambda_x^2.
M_{\underline{sss}}= \{(111,sss) \}, & M_{\underline{ss}}= \{(110,ss),(101,ss),(011,ss),(111,s^2s),(111,ss^2)\}, & M_{\underline{s}}= \{(100,s),(010,s),(001,s),(110,s^2),(101,s^2),(011,s^2),(111,s^3)\}, & M_{\emptyset}= \{(000,\emptyset) \}.
s_{n}(\theta^{m}(\omega)) =\sum_{k=1}^{n}t_{k+m}=\sum_{k=m+1}^{m+n}t_{k}=s_{n+m}(\omega)-s_{m}(\omega),\displaybreak[0] \pmb\alpha(\theta^{m}(\omega))(s_{n}(\theta^{m}(\omega))) =i_{n+m}=\pmb\alpha(\omega)(s_{n+m}(\omega)).
V_{2}^{[0,\ell]}(x) \coloneqq \sup\limits_{\tau_{2}\in \mathcal{T}}M^{x}_{2}(D_{[0,\ell]},\tau_{2}) = M^{x}_{2}(D_{[0,\ell]},D_{[r_{\ell},1]}),\forall x \in [0,1].
b'a' = (\alpha(v) d' v) (v d \alpha(v)) = \alpha(v) d' v d \alpha(v) = \alpha(v) \alpha(u) d' u v u d \alpha(u) \alpha(v) &=\alpha(v) (\alpha(u) d' u ) v (u d \alpha(u)) \alpha(v) =\alpha(v) \alpha(v) \alpha(v) = \alpha(v).
p_{\hat{S}(t)|S(t)}(\hat{s}|s)=\sum\limits_{\substack{s^m:s(t)=s \hat{s}^m:\hat{s}(t)=\hat{s}}}p_{\hat{S}^m|S^m}(\hat{s}^m|s^m)\prod\limits_{t':t'\neq t}p_S(s(t')).
\frac{2 \pi^{(d-2)/2}}{\Gamma ((d-2)/2)}\, \int_0^\alpha f(\eta_1)\, \sin^{d-2}\eta_1 \, d\eta_1 \, \int_0^\pi \frac{\sin^{d-3}\xi\,d\xi}{(2 - 2\gamma)^{(d-2)/2}}= F_Q - Q(\theta_1), 0 \leq \theta_1 \leq \alpha.
d \left(x_1 \wedge \ldots \wedge x_n\right)= \sum_{i < j}(-1)^{i+j}[x_i, x_j] \wedge x_1 \wedge \ldots \wedge \widehat{x}_i \wedge\ldots \wedge \widehat{x}_j \wedge \ldots \wedge x_n.
\alpha(D^{\mathcal{B}})(e_\lambda) = (D^{\mathcal{B}}\partial^N- \partial^{N\otimes_B J}D^{\mathcal{B}})(e_\lambda) = D^{\mathcal{B}}\left(\partial^{N}(e_\lambda)\right) = \sum_{\mu<\lambda}e_\mu \otimes \delta(b_{\mu\lambda}) = \Delta_N(e_\lambda)
\tilde{F}=\frac{1}{2}\big((-1)^{l_1}(\textrm{Im}F)^{l_1}\textrm{Re}F(\textrm{Im}F)^{l_2}+(-1)^{l_2}(\textrm{Im}F)^{l_2}\textrm{Re}F(\textrm{Im}F)^{l_1}\big),
a_1(k)=\frac{k-i\kappa}{k}\exp\left\{\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{{\rm log}\left(1-b^2(s)\right)}{s-k}ds\right\}, a_2(k)=\frac{k}{k-i\kappa}\exp\left\{-\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{{\rm log}\left(1-b^2(s)\right)}{s-k}ds\right\}.
2^{d_{k,j}}=\frac{4}{\varepsilon_4 a^k}2^{-j}\pi^{j-2k}D^{(k)}_jF_{3k-2j}\le\frac{4 A(\sigma)}{\varepsilon_4}\frac{\Gamma(k+1/2)^{1/2}}{(B_1 a \sqrt{\pi})^k},0\le j\le 3k/2.
\kappa_0(t,0)=\int_1^{+\infty}\frac{\left( 1+\frac{2\,\cos t}{r}+\frac{1}{r^2}\right)^{-1}-1}{r}\,dr=-\int_1^{+\infty}\frac{2\cos t+\frac1r}{r^2+2\,r\cos t+1}\,dr.
\hat{\mathbf{v}}_{\mathcal{I}}=\left(\mathbf{B}_{\mathcal{I}}^t\mathbf{\Sigma}_{\mathbf{\hat{z}}}^{-1}\mathbf{B}_{\mathcal{I}}\right)^{-1}\mathbf{B}_{\mathcal{I}}^t\mathbf{\Sigma}_{\mathbf{\hat{z}}}^{-1}\mathbf{y}_s &=\mathbf{v}_{\mathcal{I}}+\mathbf{e}_{BLUE},
\Phi_t (0,c_j) = \int_\Omega f_u (x, c_j) + \int_{\partial \Omega}b g'(c_j) & = \int_\Omega m + (p-1)c_j^{p-2}\int_\Omega a + (q-1)c_j^{q-2}\int_{\partial \Omega}b.
X \cdot Y \cdot Aa - Y \cdot X \cdot Aa= &=A (X \cdot Y \cdot a) + [XY,A]a - A (Y \cdot X \cdot a) - [YX,A]a &=A (X \cdot Y \cdot a - Y \cdot X \cdot a) + X \cdot [Y,A]a - [X,A]\cdot Ya - Y \cdot [X,A]a - [Y,A] \cdot Xa &=A \cdot [X,Y] a + [X , [Y,A]]a + [Y,[A,X]]a &= A \cdot [X,Y] a - [A , [X,Y]]a &=[X,Y] \cdot Aa.
\ (\| \theta_t(a)\|_{\psi_0}^\sharp )^2 &=\frac{1}{2}\psi_0 \circ \theta_t(a^*a+aa^*) &=\frac{1}{2}(\psi_0 \circ \theta_{t_j}(a^*a+aa^*)) -\frac{1}{2}((\psi_0\circ \theta_{t_j}-\psi_0\circ \theta_t)(a^*a+aa^*)) &\leq (\| a\|_{\psi_0\circ \theta_{t_j}}^\sharp )^2 +\| a\|^2 \| \psi_0\circ \theta_{t_j}-\psi_0\circ \theta_t\|
{\rm Coeff}_{m_{1},\cdots m_{n-1}}(\sum_{(\lambda_{1}\cdots \lambda_{n-1})\in\mathbb{N}^{n-1}}M_{2}x_1^{\lambda_1}\cdots x_{n-1}^{\lambda_{n-1}}) ={\rm Coeff}_{m_{1},\cdots m_{n-1}}(\sum_{(k_{1}\cdots k_{n-1})\in\mathbb{N}^{n-1}}N_{2}x_1^{k_1}\cdots x_{n-1}^{k_{n-1}}).
\frac{1}{\sqrt{m}}E\left[J_{s,t}^m\right] = -\int_s^t E\left[(-\nabla_q V(r,q_r)-\partial_r\psi(r,q_r)+\tilde F(r,q_r))\cdot \left(\int (\nabla_z\chi)(r,q_r,z) h(r,q_r,z)dz\right)\right]dr &-\int_s^tE\left[ \int \left((\nabla_q\chi)(r,q_r,z )\cdot z\right) h(r,q_r,z)dz \right]dr+O(m^{1/2}),
\sum_{j}{\frac{t_{j}^{k}d t_{j}}{s_{j}}}= 4 d \tau u ( \lambda ) \sum_{j}( - 1 )^{N - 1 - j}{\frac{t_{j}^{k}}{t_{j}- \lambda}}{\frac{D_{j}}{D}}= - 4{\frac{\lambda^{k}}{\prod_{l}( \lambda - a_{l})}}d \tau
\Pr(X \le \ell - 1) = \sum_{s=0}^{\ell - 1}\Pr(X = s) \le \sum_{s=0}^{\ell - 1}\left(\frac{C \log m}{s}\right)^s \Pr(X = 0) \le \ell \left(\frac{C \log m}{\ell}\right)^{\ell}\Pr(X = 0) \le \left(\frac{C \log m}{\ell}\right)^{C\ell}\Pr(X = 0),
\tilde{\Theta}_-A\varphi =\tilde{\Theta}_-S(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=S\tilde{\Theta}_-(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\varphi &=-S(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)\tilde{\Theta}_-\varphi=-A\tilde{\Theta}_-\varphi.
G_{l + \nu}( r^{\prime \prime}, r^{\prime}; E ) = \sum_{n = 0}^{\infty}\, \prod_{\alpha = 1}^{n}\left[ \int_{0}^{\infty}d r_{\alpha}V ( r_{\alpha}) \right] \, \prod_{\beta = 0}^{n}\left[ G_{l + \nu}^{( 0 )}( r_{\beta + 1}, r_{\beta}; E ) \right] \; .
e_{2,ik}\!+\!\gamma_{ik}\frac{\sum_{j\neq i}|e^{(n)}_{1,jik}|^{2}}{(1+\nu_{ik}\gamma_{ik})^{2}}\!-\!2\Re{\{e_{3,ik}e^{(n)}_{1,iik}\}}\!-\!2\nu_{ik}|e_{3,ik}|^{2}\! =\! 0.
\limsup_{N\to\infty}\Bigl\lVert \frac{1}{|I_N|}\sum_{n\in I_N}v_{n}\Bigr\lVert_{L^2(\mu)}^2\leq 4 \limsup_{H\to\infty}\frac{1}{H}\sum_{h=1}^{H}\limsup_{N\to\infty}\Big|\frac{1}{|I_N|}\sum_{n\in I_N}\langle{v_{n+h}, v_n \rangle}\Big|.
\begin{cases}-\mathcal{L}v+z^{2-1/s}v_{zz}=0, \hbox{in}~\mathcal{C}_{\Omega}^{\displaystyle\star}, v=0, \hbox{on}~\partial_{L}\mathcal{C}_{\Omega}^{\displaystyle\star}, -\underset{z\rightarrow0^{+}}{\lim}v_z=d_sf^{\displaystyle\star}, \hbox{on}~\Omega^{\displaystyle\star},\end{cases}
\left. \begin{array}{l c l}{{T_{\; \; z}^{z}}}&{{=}}&{{V ( \phi ) -{{{{{\phi^{\prime}}}^{2}}}{{e^{- 2 \, a}}}}}}\\{{T_{\; \, i}^{i}}}&{{=}}&{{V ( \phi ) +{{{{{\phi^{\prime}}}^{2}}}{{e^{- 2 \, a}}}},}}\\ \end{array}\right.
\|A(t)\|_1 = \int_0^t \int_\Omega (u\gamma(v))(s,x)\ \mathrm{d}x\mathrm{d}s \le \|\gamma'\|_{L^\infty(0,V)}\int_0^t \int_\Omega (uv)(s,x)\ \mathrm{d}x\mathrm{d}s & \le \|v^{in}\|_1 \|\gamma'\|_{L^\infty(0,V)}\,,
\limsup_{\delta\rightarrow 0}\limsup_{N\rightarrow+\infty}\frac{N}{\gamma_N^2}\log\sup_{\sigma\in \mathcal{T}}P\left(\sup_{0\leq t\leq \delta}\left(\eta^N_{t+\sigma}(\vec{f})-\eta^N_\sigma(\vec{f})\right)>\epsilon, (D_2^N)^c\right)=-\infty.
p_k^+ = \prod_{\substack{1 \leq i \leq m b_{(n+i),k}>0}}x_{l(n+i)}^{b_{(n+i),k}}= \prod_{\substack{1 \leq i \leq m b_{(n+i),k}>0}}y_{i}^{b_{(n+i),k}}\intertext{and}p_k^- = \prod_{\substack{1 \leq i \leq m b_{(n+i),k}<0}}x_{l(n+i)}^{-b_{(n+i),k}}= \prod_{\substack{1 \leq i \leq m b_{n+i,k}<0}}y_{i}^{-b_{(n+i),k}}
\left\{z_m<\frac{\epsilon_i}{\rho\xi_i}\right\}= \left\{z_m<\frac{\epsilon_i}{\rho\left(\alpha_i^2-\epsilon_i \sum^{{M_s}}_{j=i+1}\alpha_j^2\right)}\right\}= \left\{z_m<\frac{\epsilon_i}{\rho\bar{\xi}_i\max \left\{0, \frac{\rho z_t -\epsilon_1}{\rho (1+\epsilon_1)z_t}\right\}}\right\},
\sum_{i=1}^n \sum_{\substack{0 \le h_1 \le K+1-r 0 \le h_2 \le r-1 h_1 + h_2 = h}}A_{ij,h_1}y_{i, h_2+1}t^h = \mathbf{b}_{j, h}t^h + \alpha_{j, h+1}t^{h+1}- \alpha_{j, h}t^h.
H_{D}^{F}= \tilde{\lambda}^{\mu}( \tau ) \tilde{H}_{\mu}^{*}( \tau ) - \frac 1 2 \tilde{\lambda}^{\mu \nu}( \tau ) \tilde{H}_{\mu \nu}^{*}( \tau ) + \int{d^{3}\sigma \Big [ - A_{\tau}( \tau , \vec{\sigma}) \Gamma^{*}( \tau , \vec{\sigma}) + \mu_{\tau}( \tau , \vec{\sigma}) \pi^{\tau}( \tau , \vec{\sigma}) \Big ]},