harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f^*(\vartheta_i) = f^*(\varphi_{n_1})^{c_1}\cdots f^*(\varphi_{n_r})^{c_r}= (\varphi_{n_1 + 1}^{dc_1}/\varphi^{c_1}) \cdots (\varphi_{n_r + 1}^{dc_r}/\varphi^{c_r}),
\delta_q \hat{P}(p) = \frac{i}{(2 \pi)^4}\: \big(\hat{\Lambda}_q * \hat{P})\Big(p - \frac{q}{2}\Big) \delta_{-q}^* \hat{P}(p) = -\frac{i}{(2 \pi)^4}\: \big(\hat{\Lambda}_{-q}^* * \hat{P})\Big(p + \frac{q}{2}\Big) \:.
|U_1|+|U_2| = O\biggl(\frac{k^{1/2}}{n}\max\biggl\{\frac{k^{\beta/d}}{n^{\beta/d}}\, , \, \frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\biggr\}\biggr).
\begin{aligned}r_2-r_1 \leq d_1 \leq r_2+r_1, r_4-r_3 \leq d_2 \leq r_3 +r_4, r_6-r_5 \leq d_3 \leq r_5+r_6, |d_1-d_2| \leq d_3 \leq d_1 +d_2\end{aligned}
g(z,u)=exp(uz-z+u^{2}\frac{z^{2}}{2}-\frac{z^{2}}{2}+u^{3}\frac{z^{3}}{3}-\frac{z^{3}}{3}+u^{6}\frac{z^{6}}{6}-\frac{z^{6}}{6}+log\frac{1}{1-z})
\displaystyle{\int_0^\infty f^T(x)\overline{g(x)}{\rm d}x=\lim_{n\rightarrow\infty}\sum_{j,m=1}^{2}\int_{-\infty}^{\infty}D_{n,jm}^\sigma(\rho)\omega_f^{m}(\rho)\eta_{\overline{g}}^j(\rho){\rm d}\rho}.
A[h](y,t) = \frac{1}{2t}\left[2u\cdot \int_{\mathbb{T}}h(z)dz + \int_{y-v}^{y+v}h(z)dz\right] &= \frac{u}{t}\cdot A[h](y,\frac{1}{2}) + \frac{v}{t}\cdot A[h](y,v) &\le \frac{u+v}{t}\cdot \sup_{0<s\le \frac{1}{2}}A[h](y,s) &= \sup_{0<s\le \frac{1}{2}}A[h](y,s)
S_{f i}= e^{- i [ ( u + w ) / 2 ]^{2}T / 2}\! \int \! d^{9}b \, e^{- i q_{\mu}b_{\mu}}\, \langle{\cal H}^{3}| \langle{\cal H}^{4}| e^{i \Gamma_{\mathrm{( 1 )}}( u_{\mu}+ w_{\mu}/ 2 , b_{\mu}, \theta^{3})}|{\cal H}^{1}\rangle |{\cal H}^{2}\rangle
\epsilon_{\mu}\Pi^{a}{}_{i}[ n^{\mu \, i}] = \epsilon_{\mu}\left[ - L^{a b}{}_{i}K_{b}{}^{c \, i}e^{\mu}{}_{c}+ \left( \widetilde{\nabla}_{b}L^{a b}{}_{i}\right) n^{\mu \, i}\right] \, .
\left| B_{X}\right\rangle \! = N_{p}\delta^{\left( d_{\bot}\right)}\left( \widehat{q}- y \right) \exp \left[ - \sum_{n = 1}^{\infty}A_{n}^{\dagger}\cdot S \cdot B_{n}^{\dagger}\right] \left| 0 \right\rangle \! ,
\gamma_{k,n}^{(\alpha)}-a_{k,n}^{(\alpha)}=\frac{1}{\Delta}\sum_{j=1}^{q}\sum_{t=0}^{\tau_j-1}\sum_{w=1}^{d}\sum_{y=1}^{m_w}\gamma_{n-m_w+y,n}^{(w)}C[g_{w,y}, h_{j,t}] \left( (z-\lambda_j)^{\tau_j}F_\alpha(z) \Phi^{n-k}(z) \right)^{(t)}_{z=\lambda_j}.
\frac{\gamma}{\ell}\int_{Q_t}f_{\varepsilon}\chi_{\varepsilon}\leq \frac{\gamma}{\ell}\int_{Q_t}|f_{\varepsilon}|^2 + \frac{\gamma}{4\ell}\int_{Q_t}|\chi_{\varepsilon}|^2= \frac{\gamma}{\ell}\int_{Q_t}|f_{\varepsilon}|^2 + \frac{\gamma}{4\ell}\int_0^t \|\chi_{\varepsilon}(s)\|_H^2 \ds.
d_jd_k =(1+s_j)(1+s_k)=1+s_j+s_k+s_{k+j}+s_{k-j}= &=(1+s_j)+(1+s_k)+(1+s_{k+j})+(1+s_{k-j})-3= &=d_j+d_k+d_{k+j}+d_{k-j}-3.
\begin{aligned}D_tu+D_x\left(\frac{1}{2}u^2+u_{xx}\right) =F, D_t\left(\frac{1}{2}u^2\right)+D_x\left(\frac{1}{3}u^3+uu_{xx}-\frac{1}{2}u_x^2\right) =uF, D_t\left(\frac{1}{3}u^3-u_x^2\right)+D_x\left(\frac{1}{4}u^4+u^2u_{xx}+2u_xu_t+u_{xx}^2\right) =\left(u^2+2u_{xx}\right)F,\end{aligned}
\left(\sum_{r = 1}^n \frac{p_{nr}(K)}{(1-K)^{\rho(n,r)}}\theta_K^r\right) + \sum_{i = 1}^{n-1}\sum_{j = 0}^i a_{ij}\left(\sum_{r = 1}^{n-i}\frac{p_{(n-i)r}(K)}{(1-K)^{j+\rho(n-i,r)}}\theta_K^r\right) + \sum_{j = 0}^n\frac{a_{nj}}{(1-K)^j}.
\varlimsup_{r\to+\infty}\inf\left\{\left. \int_0^re^{-2\frac{\phi(\gamma_t)}{n-m}}\d t\;\right| \left.\begin{array}{ll}\!\!\!\!\gamma: &\!\!\!\!{\gamma}(0)=p, L(\gamma)=r\end{array}\right.\right\}=+\infty.
\sum_j 2^{j (\theta s_1+(1-\theta)s_2 )}a_j = \sum_{j\le J_0}2^{js_1}a_j \cdot 2^{j(1-\theta)(s_2-s_1)}+ \sum_{j>J_0}2^{js_2}a_j \cdot 2^{-j(1-\theta)(s_2-s_1)}& \lesssim (2^{js_1}a_j)_{l_j^{\infty}}\cdot 2^{J_0 (1-\theta)(s_2-s_1)}+ (2^{js_2}a_j)_{l_j^\infty}\cdot 2^{-J_0(1-\theta)(s_2-s_1)}.
\begin{aligned}\mathbf{P}_x(T_{x_n}<\varepsilon_0/2, T_x>\varepsilon_0) \leq \mathbf{P}_x(T_{x_n}<\varepsilon_0/2, T_x\circ \theta_{T_{x_n}}>\varepsilon_0/2) &=\mathbf{P}_x(T_{x_n}<\varepsilon_0/2)\mathbf{P}_{x_n}(T_x>\varepsilon_0/2). \end{aligned}
\mathbb{P}\left(\sum_{i=1}^{M_{2}\left(n\right)}a_{n}W_{i}\leq\frac{\epsilon}{2}\right) <\frac{\epsilon}{8}, \mathbb{P}\left(\sum_{i=1}^{M_{2}\left(n\right)}a_{n}U_{i}\leq\frac{\epsilon}{2}\right) <\frac{\epsilon}{8}.
q_S = \begin{cases}\left( w^{+1}_{p,1}\right)^{\oplus k}s_{(+)}-s_{(-)}\equiv k(1-p) 8 \left( w^{+1}_{p,1}\right)^{\oplus k-1}\oplus w^{-1}_{p,1}s_{(+)}-s_{(-)}\equiv k(1-p) +4 8 \end{cases}
u(z,\gamma w)=\sinh^{2}\big(d_{\mathbb{H}}(z,\gamma w)\slash 2\big)= \frac{|z-\gamma w|^{2}}{4\Im(z)\Im(\gamma w)}\geq \frac{|\Im(z)-\Im(\gamma w)|^{2}}{4\Im(z)\Im(\gamma w)}.
f_{0}(\mathbf{x}) =\frac{q}{p}\mathbf{x}^{T}\mathbf{A}\mathbf{x}+\mathcal{L}_{c}(\mathbf{x}), f_{1}(\mathbf{x}) =f_{0}(\mathbf{x})+\frac{q}{p}x_{\pi(1)}, &\vdots f_{p-1}(\mathbf{x}) =f_{0}(\mathbf{x})+\frac{q}{p}(p-1) x_{\pi(1)}.
A_1 =\{1,3\}. q(1,1)^2 =d_1^{-2}d^2_7 \;\; \;\; q(1,3)^2=d_3^{-2}d^2_5, F_1 =\langle q(1,1)^{2}\rangle\times\langle q(1,3)^{2}\rangle.
F(\mu_N)\,=\,\frac{1}{N}\sum_{i=1}^N\varphi_1(X^{i,N}_{s_1},M^N_{s_1})\ldots\varphi_k(X^{i,N}_{s_k},M^N_{s_k})\big[(R_t^{i,N}-R_s^{i,N})+(\Delta_{t}^{i,N}-\Delta_{s}^{i,N})\big],
\rho_{\pm}= \frac{(1+\alpha^2) (M \pm \sqrt{M^{2}- \left( 1-\alpha^{2}\right) Q^{2}})}{\left(1 \pm \alpha^{2}\right)}, \; \; \; a = \frac{2 (1 + \alpha^2) J}{(1 + \alpha^2) \rho_{+}+(1-\alpha^{2}/3) \rho_{-}}.
z \ =\ \log(1 + (e^z-1))\ =\ \sum_{n=1}^\infty z^n \sum_{m=1}^n \sum_{\substack{\lambda_1 + \cdots + \lambda_m = n \lambda_j \ge 1}}\frac{(-1)^{m + 1}}{m}\frac{1}{\lambda_1 ! \cdots \lambda_m !}.
{\cal G}_{0}( \omega ,{\bf k}; z , z^{\prime}) = \int_{- \infty}^{\infty}\frac{d q}{2 \pi}\frac{e^{i q ( z - z^{\prime})}}{\omega^{2}+{\bf k}^{2}+ q^{2}+ m^{2}}= \frac{e^{- \sigma | z - z^{\prime}|}}{2 \sigma},
q_i\|f_i\|\;\|\tfrac{1}{A}f_i + \epsilon_{1}^{i}h_i \| =q_i\sqrt{\frac{1}{A^2}\|f_i\|^4 + (\epsilon_{1}^{i})^2\|f_i\|^2\|h_i\|^2 + \frac{2}{A}\epsilon_{1}^{i}\|f_i\|^2 Re( \langle f_i, h_i\rangle)}\;\;<\; q_i\frac{1}{A}\|f_i\|^2 = max_{i=1}^N \frac{q_i}{A}\|f_i\|^2 .
E_{\alpha}^{c}(\vec{x},\vec{u}) = \mathbb{E}_{\alpha}^{c}\Bigg[ \prod_{j=2}^{m}e^{u_{j}N_{(x_{1},x_{j})}}\Bigg] := \mathbb{E}_{\alpha}\Bigg[ \prod_{j=2}^{m}e^{u_{j}N_{(x_{1},x_{j})}}\Big| N_{(0,x_{1})}= 0 \Bigg] =\frac{F_{\alpha}(\vec{x},\vec{s})}{F_{\alpha}(x_{1},0)},
\Phi (w,z)=\bigl(\Gamma(w,z),\Gamma(-w,z)\bigr) \quad\mbox{where}\Gamma(w,z) = \left(\frac{\sqrt{8-|w|^2}\left((8-2|w|^2-|z|^2)w+\bar{w}z^2\right)}{8(4-|w|^2)}+\frac{iz}{4}\sqrt{8-2|w|^2-|z|^2},\right. \left. 1-\frac{|w|^2+|z|^2}{4}-\frac{\sqrt{(8-|w|^2)(8-2|w|^2-|z|^2)}}{4(4-|w|^2)}Im(w\bar{z})\right).
- \int_{S}x(-\Psi_y\theta_x+\Psi_x\theta_y)dz=-\int_{S}\Psi_y\theta dz=\int_{S\times S}\frac{-\sin(y-\xi_2)}{2(\cosh(x-\xi_1)-\cos(y-\xi_2))}\theta(z,t)\theta(\xi,t)dzd\xi=0\,,
U_{k_l}(z)=\frac{\sum_{j=1}^{d_1\cdots d_{k_l}}\log{|z-z_{j,k_l}|}}{-d_1\cdots d_{k_l}}=\sum_{j=1}^{d_1\cdots d_{k_l}}\frac{\log{|a_{j,d_j}|}}{d_1\cdots d_j}-\frac{\log{|F_{k_l}(z)-a|}}{d_1\cdots d_{k_l}}.
\lim_{y \to 0}\left[ \phi_{n}^{1}( y ) \chi_{n}^{1}( y ) - \rho \phi_{n}^{2}( y ) \chi_{n}^{2}( y ) \right] = \phi_{n}^{1}( 0 ) \chi_{n}^{1}( 0 ) - \rho \phi_{n}^{2}( 0 ) \chi_{n}^{2}( 0 )
i_{z}( \varphi^{\ast}\omega^{\prime}) ( z_{2},\cdots ,z_{q})=\omega^{\prime}( \varphi ( z) ,\varphi ( z_{2}) ,\cdots ,\varphi ( z_{q}) )=i_{\varphi ( z)}\omega^{\prime}( \varphi ( z_{2}) ,\cdots ,\varphi (z_{q}) ) & \displaystyle=\varphi^{\ast}( i_{\varphi ( z)}\omega^{\prime}) (z_{2},\cdots ,z_{q}) ,
(\mathcal{G}_{D}u_{0})_t (x) =\int_{B_R(0)}u_0(y)p_D(t,x,y)\,\d y &\geq \int_{B_{R-\epsilon}(0)}u_0(y)p_D(t,x,y)\,\d y &\geq c_1e^{-\mu_1t},
\overline U_{M_n,n}(x,t) ={\bf\Phi}_{M_n,n}(\widetilde u_T )(x,t)-\widetilde \Phi_{M_n,n}( \widetilde g)(x,t)-\sum_{p=0}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}F_p(\overline U_{M_n,n})(s)ds \Bigg]\phi_p(x).
A = \left[ \begin{array}{ccccc}a_1 a_2 \hdots a_{d-1}a_d 0 a_1 a_2 \hdots a_{d-1}\ddots \ddots \vdots 0 \hdots 0 a_1 a_2 0 \hdots 0 0 a_1 \end{array}\right].
\lambda_{N,k}=\mathcal{R}_{s}(2^{n_{k}})+2^{-n_{k}+1}\,(2^{s}\,\mathcal{R}_{s}(2^{n_{k}+1})-\mathcal{R}_{s}(2^{n_{k}}))\,\sum_{j=k+1}^{\tau(N)}2^{n_{j}}=\mathcal{R}_{s}(2^{n_{k}})+2^{-n_{k}+1}\,(2^{s}\,\mathcal{R}_{s}(2^{n_{k}+1})-\mathcal{R}_{s}(2^{n_{k}}))\,\left(\sum_{j=k+1}^{M}2^{n_{j}}+\sum_{j=M+1}^{\tau(N)}2^{n_{j}}\right).
I ( p ; m_{0}, r ) = \left( \prod_{\mu = 1}^{d}c_{\mu}\right) \left[ m_{0}+ r \sum_{\rho}( c_{\rho}- 1 ) + r \sum_{\rho}\frac{s_{\rho}^{2}}{c_{\rho}}\right] ( H_{0}^{2})^{- n - \frac{1}{2}}.
\begin{aligned}\|\bar{u}\partial_xu_1 + u_2\bar{u}_x\|_{L^2([0,T]\times I)}\leq \|\bar{u}\|_{L^\infty([0,T]\times I)}\|\partial_xu_1\|_{L^2([0,T]\times I)}&+ \|u_2\|_{L^2([0,T];L^\infty(I))}\|\bar{u}\|_{L^\infty([0,T];H^1(I))}&\leq T^\frac{1}{2}\left(\|u_1\|_{X}+ \|u_2\|_X\right)\|\bar{u}\|_{X}.\end{aligned}
F_{s}\left( \begin{matrix}{{{\bf a}_{1}{\bf a}_{+}}}\\{{{\bf a}_{3}{\bf a}_{4}}}\\ \end{matrix}\right) ( z , \bar{z}) = z^{- 2 \Delta_{+}}\sum_{p = 0 , \pm 1}M_{p s}F_{p}\left( \begin{matrix}{{{\bf a}_{4}{\bf a}_{+}}}\\{{{\bf a}_{3}{\bf a}_{1}}}\\ \end{matrix}\right) ( 1 / z , 1 / \bar{z}) ~ ,
\sum_{z \leq p}\left (\frac{1}{p}\sum_{\gcd(k,p-1)=1}1\right )\left (\frac{1}{p}\sum_{\gcd(n,p-1)=1}1\right )=\left (\frac{\varphi(p-1)}{p-1}\right )^2p+O\left(1\right) .
\sum_{j=0}^{\infty}F_j''(t)\varepsilon^j=\sum_{j=0}^{\infty}\left(F_j(t)\varepsilon^{j+1}+F'_j(t)\varepsilon^j(\varepsilon+1-e^{-\varepsilon t}\sum_{k=0}^{\infty}\varepsilon^{k+1}F_k)\right),
\frac{\omega_1}{2\pi}\int_{t-\pi/\omega_1}^{t+\pi/\omega_1}\left(\frac{A(x)^2}{2}+ \frac{B(x)^2}{2}+ A(x)B(x) \cos(\eta s) \right)ds= \frac{A(x)^2}{2}+ \frac{B(x)^2}{2}+ A(x)B(x) \cos(\eta t) + O\left(\frac{\vert A(x) B(x) \eta\vert}{\omega_1}\right),
\mathbb{E}\big\{{{\rm s}^{\rm T}(t)}\boldsymbol{\phi}^{\rm T}\boldsymbol{\phi}^{\ast}{{\rm s}^{\ast}(t)}\big\}\!=\!\sum_{i=1}^{N}{\beta_{i}^{2}}\mathbb{E}\big\{{{\rm s}^{\rm T}(t)}{{\rm s}^{\ast}(t)}\big\}=N,
\int_{(\Omega_h\setminus\Omega) \cap \pi(S,\delta)}|f(y)|^p\,dy = \int_{S'}\int_0^{\max\{0, t^*(z')\}}|f(\Psi(z', t))|^p \operatorname{det}J \, dt \, dz' &\le C \int_{S'}\int_0^{\max\{0, t^*(z')\}}\Big( |f(\Phi_h(z'))|^p + |f(\Psi(z', t)) - f(\Phi_h(z'))|^p \Big) \, dt \, dz' &=: I_1 + I_2,
\tilde \psi'_\epsilon(b)-\tilde \psi'_\epsilon(a)+(n-1)\frac{\tilde \psi_\epsilon(b)}{b}-(n-1)\frac{\tilde \psi_\epsilon(a)}{a}+(n-1)\int_a^b\frac{\tilde \psi_\epsilon(r)dr}{r^2}=\epsilon^2\int_a^b W'(\tilde\psi_{\epsilon}(r))dr.
\tilde L_{k, a, k_\rho}({\mathbf x}) =\frac{\hat L_k({\mathbf x}) \hat\Delta_{k_\rho,a}(a{\mathbf x})-\hat L_k(a{\mathbf x}) \hat\Delta_{k_\rho,a}({\mathbf x})}{\hat\Delta_{k_\rho,a}(a{\mathbf x}) -a \hat\Delta_{k_\rho,a}({\mathbf x})},
H(S_0)\leq\kappa\min\{I(V_0;Y_1),I(V_0;Y_2)\}, &H(S_i)\leq\kappa[\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_i;Y_i|V_0)],i=1,2, &H(S_1,S_2)\leq\kappa[\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_1;Y_1|V_0)+I(X;Y_2|V_0,V_1)], &H(S_1,S_2)\leq\kappa[\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_2;Y_2|V_0)+I(X;Y_1|V_0,V_2)].
\sum_{w(h)=1}^rw(h)^{2k}=1^{2k}+2^{2k}+\cdots+r^{2k}=\frac{1}{2k+1}\sum_{i=0}^{2k}\frac{(2k+1)!}{i!(2k+1-i)!}B_ir^{2k+1-i},
\sup_{0 \le t \le \tau_3}\left|\int_0^t S(t-s)\begin{pmatrix}0 \sigma_1 dW_s^{(1)}\end{pmatrix}\right| \le R \sqrt{\tau_3}, \int_0^{\tau_3}\sum_{j=3}^n |x_{t,j}|^2 dt \le \sqrt{\delta}\tau_3 K^{2r}, \sup_{0 \le t \le \tau_3}|W_t| \le R \sqrt{\tau_3}, \sup_{0 \le t \le \tau_3}|x_{t,1}| \le (\delta_1^{1/4}+ \delta^{1/4})K^{4r}+ 1.
S=\,{8\pi^2\over g^2}+2\pi^2\rho^2(|v_+|^2+|v_-|^2)-{i\over \sqrt{2}}\left(\begin{array}{llll}\bar{v}_+ & \bar{v}_-\end{array}\right)_{f}^{\,\,\dot\beta}\mu_{\dot\beta}\, ({\cal K}_f+\tilde{\cal K}_f) \ ,
\left\|( u^s_{yy}+\partial_y\tilde{w}_1)\partial^{\alpha_1}_x \bar{v}\right\|_{L^2_{k+\ell}}=\left\|( u^s_{yy}+\partial_y\tilde{w}_1)\partial^{-1}_y\partial^{\alpha_1}_x \bar{u}_x\right\|_{L^2_{k+\ell}}&\qquad\qquad\le C (1+\| \tilde{w}_1\|_{H^{m-1}_{k+\ell}(\mathbb{R}^2_+)})\left\|\partial^{\alpha_1+1}_x\bar{w}\right\|_{L^2_{\frac 32+\delta}(\mathbb{R}^2_+)}.
r \phi_d^{-1}* a + r \phi_d^{-1}* b = r * (r \phi_d^{-1}* a + r \phi_d^{-1}* b) = r\phi_d^{-1}* r\phi_d * (r \phi_d^{-1}* a + r \phi_d^{-1}* b) = r\phi_d^{-1}* (r\phi_d *r \phi_d^{-1}* a + r\phi_d *r \phi_d^{-1}* b) = r\phi_d^{-1}* (r * a + r * b) = r\phi_d^{-1}* (a + b)
\| \tilde{f}\|_{\dot H^s(\mathbb R^n)}\le C \| \tilde{f}\|_{(L^2(\mathbb R^n), \dot H^2(\mathbb R^n))_{\frac{s}{2},2}}= C\left\{\int_0^\infty (\lambda^{-\frac{s}{2}}K(\lambda, \tilde{f}))^2\frac{d\lambda}{\lambda}\right\}^\frac12,
V(B,\varepsilon,\gamma)= \{x\in LS(\mathcal M):~\mathrm{there~exist}~p\in\mathcal P(\mathcal M)~\mathrm{and}~z\in\mathcal P(Z(\mathcal M)~\mathrm{such~that}&xp\in\mathcal M, \|xp\|_{\mathcal M}\leq\varepsilon,\varphi(z^\bot)\in W(B,\varepsilon,\gamma)~\mathrm{and}~\Delta(zp^\bot)\leq\varepsilon\varphi(z)\},
\textup{smoothness term}= 4\biggl(\sum_{i=1}^n(ac_{ii}+ b_i^2)+ \frac{1}{4}\sum_{i,j=1}^n \bigl(ac_{ij}^2+b_i b_j c_{ij}+ 4b_i c_{ij}\bigr)\biggr)^{1/2}&+ \frac{1}{4}\Big(\sum_{i=1}^n b_i^2\Big)^{1/2}\Big(\sum_{i=1}^n c_{ii}^2\Big)^{1/2}+ 3\sum_{i=1}^n c_{ii}+ \log 2.
\Delta^{\alpha \beta}_{\pi}(y_{1}- y_{2}) = \delta^{\alpha \beta}\int_{0}^{\infty}\frac{d T}{T}\exp \{-m^{2}\frac{T}{2}\}<V_{5}(y_{1}) V_{5}(y_{2}) >_{x, \psi_{\mu}, \psi_{5}, \psi_{6}}
\exists\ \{\nu_{i}\}_{i\in I(\bar{x})}\geq 0\ \forall_{k\in I(\bar{x})}|\bar{x}_k-x_k|^{p-2}{(\bar{x}_k-x_k)}=- \delta_k \nu_k \forall i \in J\backslash I(\bar{x}) \bar{x}_i=x_i.
f(\mathbf{x})=\sum_{i=1}^{m-1}a_{i}x_{\pi(i)}x_{\pi(i+1)}+ \sum_{k=1}^{m}d_{k}x_{k}^{2}+ \sum_{k=1}^{m}c_{k}x_{k}=\mathbf{x}\mathbf{A}\mathbf{x}^{T}+\mathbf{c}\mathbf{x}^{T},
\frac{1}{2 \pi i}\int_{(c_1)}\mathcal{B}(s_1,s_2) \Phi(s_1) \zeta(s_1+1)^{k}X^{s_1}ds_1 = \mathrm{Res}_{s_1=0}\Big( \mathcal{B}(s_1,s_2) \Phi(s_1) \zeta(s_1+1)^{k}X^{s_1}\Big) + g(s_2)
\mathrm{K e r}Q_{n}^{[ s ]}/ \mathrm{I m}Q_{n}^{[ s - 1 ]}= \Bigl \lbrace \begin{matrix}{{0}}&{{f o r}}&{{s \not = 0}}\\{{{\cal H}_{n , n^{\prime}}}}&{{f o r}}&{{s = 0}}\\ \end{matrix},
l_1^{'}(a)=-2(\kappa-1)a^{2\kappa-3}-\kappa(\kappa-1)a^{\kappa-1}+2\kappa(\kappa-1)a^{\kappa-2}-(\kappa-1)(\kappa-2)a^{\kappa-3}&=(\kappa-1)a^{\kappa-3}[-2a^\kappa-\kappa a^2+2\kappa a-\kappa+2]=(\kappa-1)a^{\kappa-3}[-\kappa(a-1)^2-2(a^\kappa-1)]\ge 0.
\begin{array}{rl}\tilde v_1 : = \!\! \!\!\displaystyle v_1 - \zeta^2[ g(0)\nu_1 + C_1(1 + M_2(R))|x|^2] = \!\! \!\!\displaystyle \zeta^2 [w_1 - g(0)\nu_1 - C_1(1 + M_2(R))|x|^2]\end{array}
M^2_t-M^2_{t-1}=(h-R_{\tau_h+t})^2-(h-R_{\tau_h+t-1})^2 = (\eta_{\tau_h+t}-1)^2+2(h-R_{\tau_h+t-1})(1-\eta_{\tau_h+t}) &=(\eta_{\tau_h+t}-1)^2+2M_{t-1}(1-\eta_{\tau_h+t})
I =\Big((I\cap S')S\Big)\oplus \bigoplus_{\tau\subset [s]}\bigoplus_{w\in \mathcal{M}_{\tau}}\Bigg(\Bigg(\Big(\bigcap_{j\in [s]\setminus \tau}Q_j\cap wS_{\tau}\Big)S_{\tau}[x_{r+1}, \ldots, x_n]\Bigg) \cap \Bigg(\Big(\bigcap_{j\in \tau}Q_j\cap wS''\Big)S_{\tau}[x_{r+1}, \ldots, x_n]\Bigg)\Bigg),
| f |_{\mathcal{H}^\beta}= \max_{1 \leq j <\beta}\left( \sup_{x\in[0,1]}\frac{|f^{(j)}(x)|^\beta}{|f(x)|^{\beta-j}}\right)^{1/j}= \max_{1 \leq j <\beta}\Big\| |f^{(j)}|^\beta/|f|^{\beta-j}\Big\|_\infty^{1/j}
y^{(n)}=u_{1}y_{1}^{(n)}+\cdot\cdot\cdot+u_{n}y_{n}^{(n)}+u_{1}^{\prime}y_{1}^{(n-1)}+u_{2}^{\prime}y_{2}^{(n-1)}+\cdot\cdot\cdot+u_{n}^{\prime}y_{n}^{(n-1)}.
{\cal L}= - \frac{1}{4}F_{i j}^{a}F_{i j}^{a}+ \frac{1}{2}F_{i 0}^{a}F_{i 0}^{a}+ j_{i}^{a}A_{i}^{a}- j_{0}^{a}A_{0}^{a}+{\cal L}_{\mathrm{g a u g e}}-{\bar{\psi}}\left( m - i{\gamma}\cdot \partial \right) \psi
f^{(5)}(T)={512\pi^2 \over (T-{\bar T})^3}\sum_{\ell=1}^6\int\frac{d^2\tau}{\tau_{2}^{\; 3/2}}\bar{C}_{\ell}(\bar{\tau})\,\partial_{\bar{\tau}}\left[ \tau_2^2 \partial_{\bar{\tau}}(\tau_{2}^{\; 3/2}\sum_{p_L,p_R\in \Gamma_{\ell}}p_L^3{p}_{R}\, e^{\pi i\tau\, |p_{L}|^2}\,e^{-\pi i \bar{\tau}\, p_{R}^2})\right]\, .
\texttt{> l := [seq(binomial(2n,n)\^{}2,n=0..10)]:}\texttt{> listtodiffeq(l, y(x));}4y(x) + (32x - 1)\frac{\mathrm{d}}{\mathrm{d}x}y(x) + (16x^2 - x)\frac{\mathrm{d}^2}{\mathrm{d}x^2}y(x) = 0.
\frac{\partial \mathbf H^{(2)}}{\partial Z_{jk}^{(2)}}=\mathbf L_t\mathbf e_{k}\mathbf e_{j}^T - \mathbf L_t\mathbf e_{k}\mathbf e_{j}^T\mathbf M_t\mathbf L_t\mathbf M_t^* - \mathbf L_t\mathbf M_t^*\mathbf e_{j}\mathbf e_{k}^T\mathbf L_t\mathbf M_t^*.
\tilde{J}= \frac{6 4}{3}\frac{1}{({\cal V}_{2 , 2}^{( 1 )}+{\cal V}_{2 , 2}^{( 2 )})^{2}}\, \frac{{\cal V}_{2 , 2}^{( 2 )}}{{\cal V}_{2 , 1}^{( 1 )}}.
\left|u_\Lambda(x)-\bar{u}\right|\cdot\left|u_\Lambda(x)-\bar{v}\right|+\left|u_\Lambda'(x) \right|=O(1)e^{-\left(\sqrt{2}\sqrt{1-4c_0^2}+O\left(\frac{1}{\sqrt{\Lambda}}\right)\right)|x|},
\max_{x_{k_j}}\;\; \sum_{k=1}^K\sum_{j=1}^J{x_{k_j}v_{k_j}}\mbox{s.t.}\;\; \sum_k x_{k_j}\leq L_j,~j=1,2,\cdots,J \;\; \sum_j x_{k_j}\leq 1,~k=1,2,\cdots,K \;\; \mbox{Constraints (\ref{eqn:mmhn95}),~(\ref{eqn:mmhn69})}.
\partial_{r_{|_{(x,y)}}}=\frac{r}{2R_{+}^{2}}\left(\sum_{j=1}^{5}x^{j}\partial_{x^{j}_{|_{(x,y)}}}\right)+\frac{r}{2R_{-}^{2}}\left(\sum_{j=1}^{5}y^{j}\partial_{y^{j}_{|_{(x,y)}}}\right).
kn < e(S_1(x), L_1(x) \setminus \{z\}) + |N_1(x)\cap N_1(z)|\mathbf{\mathrm{v}}_z + \frac{\epsilon^2 n}{2}&\leq (k+\epsilon)n - |N_1(x)\cap N_1(z)| + \left(1-\frac{1}{16k^3}\right)|N_1(x)\cap N_1(z)| + \frac{\epsilon^2 n}{2}& = kn + \epsilon n + \frac{\epsilon^2 n}{2}- \frac{|N_1(x)\cap N_1(z)|}{16k^3},
\Gamma_{x}^{n_{x}}( k_{x}^{n_{x}}, M ) ={\frac{\gamma_{x}^{n_{x}}}{M}}\sqrt{[ M^{2}- ( m_{x}+ \mu_{x}^{n_{x}})^{2}] [ M^{2}- ( m_{x}- \mu_{x}^{n_{x}})^{2}]}
W^{2}\equiv \epsilon_{i j}\epsilon_{k l}W_{\mu \nu}^{i j}W_{\mu \nu}^{k l}= T_{\mu \nu}T_{\mu \nu}- 2 ( \epsilon_{i j}\theta^{i}\sigma_{\mu \nu}\theta^{j}) R_{\mu \nu \lambda \rho}T_{\lambda \rho}- ( \theta^{i})^{2}( \theta^{j})^{2}R_{\mu \nu \lambda \rho}R_{\mu \nu \lambda \rho}+ \dots
\begin{array}{l}\phi_k(\chi)=\displaystyle\frac{1}{\omega_{n-1}}\int_{uu'=1}a_k(u)P_{\alpha_k(m_0)}(\chi,u)\dot{u}~, \psi_k(\chi)=\displaystyle\frac{1}{2\pi}\int_{uu'=1}b_k(u)P_{\alpha_k(m_0)}(\chi,u)\dot{u}~.\end{array}
\Phi_h\bigg( \bigg( \sum_{i=0}^\ell r_i x_i \bigg)^h \bigg) = \frac1{h!}\sum_{\sigma\in\Sigma_h}\sum_{m_1=0}^\ell \cdots \sum_{m_h=0}^\ell r_{m_{\sigma1}}\cdots r_{m_{\sigma h}}z_{m_{\sigma1}m_{\sigma2}}\cdots z_{m_{\sigma(h-1)}m_{\sigma h}}.
R_{12}(\lambda )=\left( \begin{array}{cccc}\sinh (\lambda +\eta ) 0 0 0 0 \sinh \lambda \sinh \eta 0 0 \sinh \eta \sinh \lambda 0 0 0 0 \sinh (\lambda +\eta )\end{array}\right) \in (\mathcal{H}_{1}\otimes \mathcal{H}_{2}),
\pi_v(G^+ \cap G^-) = \pi_v((G^+ \cap Q_v^-) \cap (G^- \cap Q_v^+)) = \pi_v(G^+ \cap Q_v^-) \cap \pi_v(G^- \cap Q_v^+), \pi_v(G^+ \cap Q_v^-) = \pi_v(G^+ \cap (Q_v^+ \cap Q_v^-)) = \pi_v(G^+) \cap \partial \pi_v(Q), \pi_v(G^- \cap Q_v^+) = \pi_v(G^- \cap (Q_v^+ \cap Q_v^-)) = \pi_v(G^-) \cap \partial \pi_v(Q).
\sum_{\kappa\in \mathbb{Z}}e^{-4\pi^2s( \kappa+a)^2}= \sum_{p\in \mathbb{Z}}(4\pi s)^{-1/2}e^{-p^2/4s}e^{2\pi ipa}, \sum_{\kappa\in \mathbb{Z}}e^{-4\pi^2s( \kappa+a)^2}4\pi^2s( \kappa+a)^2 = \sum_{p\in \mathbb{Z}}(4\pi s)^{-1/2}e^{-p^2/4s}e^{2\pi ipa}\left(\frac{1}{2}-\frac{p^2}{4s}\right).
Z_{G}[ \overline{{\xi}}, \xi ] = \exp \left\{i \sum_{a , b = 1 , . . 8}\int d^{4}x d^{4}y \overline{{\xi}}^{a}\left( x \right) \widetilde{D}_{G}^{a b}( x - y ) \xi^{b}\left( y \right) \right\},
\begin{array}{l l}{{D_{\alpha}= \partial_{\alpha}+ i \sigma_{\alpha \dot{\alpha}}^{\mu}\bar{\theta}^{\dot{\alpha}}\partial_{\mu},}}\\{{D_{\dot{\alpha}}= - \bar{\partial}_{\alpha}- i \theta^{\alpha}\sigma_{\alpha \dot{\alpha}}^{\mu}\partial_{\mu}.}}\\ \end{array}
k_{x_i}(t,p,z) \leq H w^T(A_{r_0}(O)) k(t+4r_0^2,p, A_{r_0}(O)) \prod_{\stackrel{j=1}{j\neq i}}^{n_0}x_j,\quad\forall\, (t,z)\in\Delta_{2r}(t,O), k_{x_i}(t,p,z) \geq H w^T(A_{r_0}(O)) k(t-2r_0^2,p, A_{r_0}(O)) \prod_{\stackrel{j=1}{j\neq i}}^{n_0}x_j,\quad\forall\, (t,z)\in\Delta_{r}(t,O),
...\to\mathrm{Ext}^1(L(2),L(2))\overset{\delta}{\to}\mathrm{Ext}^2(L(2),E) &\to\mathrm{Ext}^2(L(2),F)\to\mathrm{Ext}^2(L(2),L(2))\to\mathrm{Ext}^3(L(2),E) &\to\mathrm{Ext}^3(L(2),F)\to...
a_{f}= \frac{1 0 9}{4}- \frac{1 5}{2 \lambda^{( 1 )}}\left( \lambda^{( 1 ) 2}- \frac{1}{4}\right) - \left( \lambda^{( 2 ) 2}- \frac{1}{4}\right) \eta^{( 2 )}( \eta^{( 2 )}- 2 c_{1}( B ) ) ,
Z^{\beta}_{i}=\big\{x^{(\beta,s)}_{i,t}\, \big| \, 1\leq s\leq d_{\beta}, 1\leq t\leq \nu_{\beta,i}\big\}\forall\ \beta\in\Delta^{+}, i\in \beta, Z^{>\beta}_{i}=\big\{x^{(\alpha,s)}_{i,t}\, \big| \, \beta<\alpha, 1\leq s\leq d_{\alpha}, i\in\alpha, 1\leq t\leq \nu_{\alpha,i}\}\forall\ \beta\in\Delta^{+}, 1\leq i\leq 2.
K= \overline{K\cap U_{1}}\cup \ldots \cup \overline{K\cap U_l}= \left( \bigcap_{k=1}^{\infty}V^1_{k}\right) \cup \ldots \cup \left(\bigcap_{k=1}^{\infty}V^l_{k}\right)\subset \bigcap_{k=1}^{\infty}\left( V^1_{k}\cup \ldots \cup V^l_{k}\right).
\Psi^{c}( x ) \, = \, \sum_{\sigma = 0 , \pm 1}\int{\frac{d^{3}p}{( 2 \pi )^{3}}}{\frac{1}{2 \, \omega_{\vec{p}}}}\Big [ S ( C ) \, u_{\sigma}^{\ast}( \vec{p}\, ) \, a_{\sigma}^{\dagger}( \vec{p}\, ) \, e^{i p \cdot x}+ S ( C ) \, v_{\sigma}^{\ast}( \vec{p}\, ) \, b_{\sigma}( \vec{p}\, ) \, e^{- i p \cdot x}\Bigr ] \quad .
\boldsymbol{A}(S^{\varepsilon_1}_{\alpha_1}S^{\varepsilon_2}_{\alpha_2}\cdots S^{\varepsilon_m}_{\alpha_m})=\boldsymbol{A}(S_{\alpha_1})^{\varepsilon_1}\boldsymbol{A}(S_{\alpha_2})^{\varepsilon_2}\cdots\boldsymbol{A}(S_{\alpha_m})^{\varepsilon_m}
\frac{\operatorname*{d e t}g_{\mu \nu}}{\operatorname*{d e t}\left( g - \Omega^{( 2 ) T}g^{- 1}\Omega^{( 1 )}\right)_{\mu \nu}}= \left( \frac{1}{1 - q_{c}^{n}}\right)^{p + 1}~ .
\sum_{\substack{W \in \underline{k}^{-1}(\mathcal{U}) \vee\mathcal{V}W= \underline{k}^{-1}(U) \cap V}}e^{t S_{\underline{g}}\varphi(W)}\leq \Big( \sum_{V\in \mathcal{V}}e^{t S_{\underline{k}}\varphi(V)}\Big)\Big(\sum_{U\in \mathcal{U}}e^{t S_{\underline{h}}\varphi(U )}\Big)
B_2(W_1\times W_2, L^\infty(\Omega)) = B(W_1\overset{\wedge}{\otimes}W_2,L^\infty(\Omega)) & = B(L^1(\Omega), (W_1\overset{\wedge}{\otimes}W_2)^*) & = B(L^1(\Omega), B(W_1,W_2^*)) & = L^\infty_\sigma\bigl(\Omega; B(W_1,W_2^*)\bigr).
I_P= \int_{D_P}\chi(ac(f(y_1,\cdots, y_n)))|f(y_1,\cdots, y_n)|^s|\pi^{-n}||dy_1\cdots dy_n| = q^n\int_{D_P}\chi(ac(f(y_1,\cdots, y_n))|f(y_1,\cdots, y_n)|^s|dy_1\cdots dy_n| = q^nZ_f(s, \chi, D_P).
\begin{cases}\begin{aligned}\Delta^m u=-e^u, &u^{(2i+1)}(0)= \Delta^k u(0) = 0, \forall\; i = 0\ldots m-1;\; k = 1, \ldots m-2, &\Delta^{m-1}u(0) = c_0, &u(0)=-b < 0. \end{aligned}\end{cases}
A^{-1}\mathcal{L}_g = I_{n+1}\frac{\partial^2}{\partial x_n^2}+ B \frac{\partial}{\partial x_n}+ C = \Bigl(I_{n+1}\frac{\partial}{\partial x_n}+ B - Q\Bigr)\Bigl(I_{n+1}\frac{\partial}{\partial x_n}+ Q\Bigr),
(p'\theta-p'\theta^*_L)_+\Big(1-\max_{j=1,\dots,J}\Phi\Big(\frac{g_j(\theta)-c_L(\theta)}{\hat\varsigma s_L(\theta)}\Big)\Big)=\min_{j=1,\dots,J}(p'\theta-p'\theta^*_L)_+\Big(1-\Phi\Big(\frac{g_j(\theta)-c_L(\theta)}{\hat\varsigma s_L(\theta)}\Big)\Big).
\Sigma_0(\delta_0) = \left\{\gamma \in \mathcal{P}(\mathcal{S}): \boldsymbol{K}(\mu, \gamma) \le \delta_0 \right\}\Sigma(\delta) = \left \{\gamma \in \mathcal{P}(\mathcal{S}): \boldsymbol{K}_\ell(\mu_{\ell,3}, \gamma_{\ell,3}) \le \delta_\ell, \ \forall \ell=1,2\right\}.
\begin{array}{c c}{{\mathrm{s d e t}\: F ( z_{1 2}) = - x_{\bar{1}2}^{2}\, , ~}}&{{~ \det v ( z_{2 1}) = \displaystyle{\frac{x_{\bar{2}1}^{2}}{x_{\bar{1}2}^{2}}}\, ,}}\\ \end{array}
\lambda(q(c(z_1z_2...z_{m-1})), q(c(z_1z_2...z_{m-1}))l_m) = z_m &\lambda(q(c(z_1z_2...z_{n-2})), q(c(z_1z_2...z_{n-2}))a) = z_{n-1}' &\lambda(q(c(z_1z_2...z_{n-2}z_{n-1}' )), q(c(z_1z_2...z_{n-2}z_{n-1}' ))b)= z_n' &\lambda(q(c(z_1z_2...z_{n-2}z_{n-1}'z_n' )), q(c(z_1z_2...z_{n-2}z_{n-1}'z_n' ))b)= z_{n+1}'
\lim_{x \to 0}\sup_{y \in H}\, \| \nabla^K \nabla_{\xi}^G R_t[\Phi](x +y) - \nabla \nabla_{\xi}^G R_t[\Phi](y) \|_{L(K,J)}= \lim_{x \to 0}\sup_{y \in H}\sup_{|k|_K = 1}\, | \nabla_k \nabla_{\xi}^G R_t[\Phi](x +y) - \nabla_k \nabla_{\xi}^G R_t[\Phi](y) |_{J}=0,\;\; \xi \in U.
\| (P_k( \sum_{j=-L}^L P_j^{\psi}f_j) )_{l_k^2}\|_p \lesssim \sum_{|a|\le n_0+10}\| (P_k P_{k+a}^{\psi}f_{k+a})_{l_k^2}\|_p &\lesssim \| (f_j)_{l_j^2}\|_p,
\tilde{D}^{L}( X )_{i j}= \sum_{k}[ Q , P_{I}( F_{i k}^{I}) ]^{L}( X_{k j}) , \ \tilde{D}^{R}( X )_{i j}= \sum_{k}[ Q , P_{I}( F_{i k}^{I}) ]^{R}( X_{k j}) \ .
-245592 - 295812 s - 124830 s^2 - 9975 s^3 + 625 s^4 + 15120 a_1 + 9216 s a_1 - 6180 s^2 a_1 &\quad- 400 s^3 a_1 + 54648 a_2 + 5016 s a_2 + 110 s^2 a_2=0.
d ( l_{0}+ 1 , k_{1}; l_{0}, k_{1}) c ( l_{0}, k_{1}; l_{0}+ 1 , k_{1}) = \frac{( k_{1}+ k_{0}) a - ( k_{1}- k_{0}) b}{2 k_{1}( k_{1}- l_{0}) ( k_{1}+ l_{0}+ 1 )},