harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\partial M}\left(\left(W_{\delta,q}+\delta V_{\delta,q}+\theta\Phi\right)^{+}\right)^{\frac{2}{n-2}}\Phi^{2}d\sigma \le C\|\Phi\|_{H}^{2}\left(\left\Vert W_{\delta,q}+\delta V_{\delta,q}+\theta\Phi\right\Vert_{L^{\frac{2(n-1)}{n-2}}(\partial M)}^{\frac{2}{n-2}}\right) \le C\|\Phi\|_{H}^{2}=o(\varepsilon^{2});
I_2 \le \omega_d a_2 4^{\delta_2}a_1^{-1}f(s^{-1}) \int_s^{r_0}(2^{\eta}(t/s)^{\eta}-1)(t/s)^{-\delta_1}t^{-1}dt &=\omega_d a_2 4^{\delta_2}a_1^{-1}f(s^{-1}) \int_{1}^{r_0/s}( (2t)^{\eta}-1) t^{-\delta_1-1}dt &\le \omega_d a_2 4^{\delta_2}a_1^{-1}f(s^{-1})\int_{1}^{\infty}( (2t)^{\eta}-1) t^{-\delta_1-1}dt
\sum_{K\in\mathcal{T}_h}J_2+J_4 \leq Ch\sum_{K\in\mathcal{T}_h}\Big|\Big|\nabla\Big(\frac{\partial^2 u}{\partial x_1^2}\Big|_{e_2}-\frac{\partial^2 u}{\partial x_1^2}\Big|_{e_4}\Big)\Big|\Big|_{L^2(K)}|w_h|_h &\leq Ch^2|u|_{H^4(\Omega)}|w_h|_h.
U_k= \frac{Hp_{k-1}p_{k-1}^TH}{p_{k-1}^T Hp_{k-1}}-\frac{B_{k-1}p_{k-1}p_{k-1}^TB_{k-1}}{p_{k-1}^T B_{k-1}p_{k-1}}+\phi_k p_{k-1}^TB_{k-1}p_{k-1}ww^T,
f(x) \cos(\kappa_j t) \stackrel{?}{=}\frac{1}{2}\left( \frac{\xi L}{2}\left\{\frac{1}{Z_j^2}+ 2 \sum_{n=1}^\infty \frac{\cos[k_n (x-t)]}{Z_j^2-(\pi n)^2}\right\}+\frac{\xi L}{2}\left\{\frac{1}{Z_j^2}+ 2 \sum_{n=1}^\infty \frac{\cos[k_n (x+t)]}{Z_j^2-(\pi n)^2}\right\}\right),
\int \frac{x^{n-2}}{(-x^2;q^2)_\infty}\widetilde{h}_n(x;q) d_qx= \frac{x^n}{[n-1]_q(-x^2;q^2)_\infty}\left( \frac{\widetilde{h}_n(x;q)}{x}- \widetilde{h}_{n-1}(x;q)\right).
x_{m}=\frac{\psi(\tau x',\tau^{2}y_{2},\cdots,\tau^{k}y_{k})}{\tau}~~~~~\overline{x}_{m}=\frac{\psi(\tau \overline{x}',\tau^{2}\overline{y}_{2},\cdots\tau^{k}\overline{y}_{k})}{\tau}.
\Phi^{\overline{e}}_{wv}= (\Phi^e_{vw})^{-1}= |\Phi^e_{vw}|^{-1}(V^e_{vw})^{-1}= \underbrace{(V^e_{vw})^*}_{}\underbrace{V^e_{vw}|\Phi^e_{vw}|^{-1}(V^e_{vw})^*}_{}
\|P e_\lambda\|_{L^p(\partial M)}=\|P(I+\sqrt[3]{\Theta})^{-k}(I+\sqrt[3]{\Theta})^k e_\lambda\|_{L^p(\partial M)}&=(1+\lambda)^k\|P(I+\sqrt[3]{\Theta})^{-k}e_\lambda\|_{L^p(\partial M)}\lesssim\lambda^k\|e_\lambda\|_{L^p(\partial M)}.
\frac{C_m(\lambda)}{W_m(\lambda)}= -\frac{\Gamma(\frac 12 +m+|m|)}{\Gamma(1+2 |m|)}\, \frac{b'_m+|m+\alpha|\, b_m}{a'_m+|m+\alpha|\, a_m}\ (1+\mathcal{O}(\lambda)) & \quad\ + \lambda^{|m+\alpha|}\ \frac{2 \pi\, (i-\cot(|m+\alpha|\, \pi))}{\Gamma^2(|m+\alpha|)\, (a'_m+|m+\alpha|\, a_m)^2}\ (1+\mathcal{O}(\lambda)+\mathcal{O}(\lambda^{|m+\alpha|}))
\Phi_{L}\left( p^{\mu}\right) = \Lambda_{L}\left( \dot{p}^{\mu}\to p^{\mu}\right) \Phi_{L}\left( \dot{p}^{\mu}\right) = e^{i \vec{J}\cdot ( \vec{\vartheta}+ i \vec{\varphi})}\Phi_{L}\left( \dot{p}^{\mu}\right) .
\P(\widehat{\tau}_0<\infty,\widehat{X}(\widehat{\tau}_0-)\in dy,\widehat{X}(\widehat{\tau}_0)\in dx,\Delta C(\widehat{\tau}_0)=\Delta C_i(\widehat{\tau}_0))=\frac{1}{c}\Lambda_i(-y+dx)dy~,
\alpha_\pm = \frac{\theta^2}{2}\left( 1 \pm \sqrt{1 -4/\theta^2}\right), \ \ \ \ \ \frac{\lambda}{\gamma^2}=\frac{2\pi^2}{\theta^2}\left( 1 + \frac{4\alpha^2}{\theta^2}+ \frac{6\alpha^4}{\theta^4}\right).
\tau^\mathrm{an}(M/S,F) =\sum_k \bigg\{\frac{2^{2k}\big(k!\big)^2}{(2k+1)!}\tau^\mathrm{BL}(M/S,F) \bigg\}^{[2k]}\;, \tau^\mathrm{top}(M/S,F) =\sum_k \bigg\{\! - \frac{k!}{(2\pi)^k}\tau^\mathrm{IK}(M/S,F) + \frac{\zeta'(-k)\mathrm{rk}F}{2}\int_Z \mathrm{e}(TZ)\mathrm{ch}(TZ) \bigg\}^{[2k]}\;,
\frac{d}{dx}\left(\log \frac{\Gamma(x)}{\Gamma(x-a)}\right) = \frac{\Gamma'(x)}{\Gamma(x)}-\frac{\Gamma'(x-a)}{\Gamma(x-a)}= \sum_{k\geq0}\left(\frac{1}{x-a+k}-\frac{1}{x+k}\right) > 0,
\int{\cal D}V = \int \prod_{n}\left( d V_{n}^{1}d V_{n}^{2}d V_{n}^{3}d \lambda_{n}^{1}d \lambda_{n}^{2}d \bar{\lambda}_{n}^{1}d \bar{\lambda}_{n}^{2}d D_{n}\right) \prod_{i}^{2 n_{0}}d V_{0}^{i}\prod_{i}^{n_{0}}d \lambda_{0}^{i}
\sum_{y<p\le \sqrt{x}}\frac{x}{p\log(x/p)}\ge \frac{x}{\log y}\left(\frac{\log(u-1)}{u}-\frac{2c_1}{u\log^2y}\right) &=\frac{x}{\log y}\left(\omega(u)-\frac{2c_1}{u\log^2y}\right)-\frac{x}{\log x}.
\langle \prod_{j=1}^m A_j^{\alpha_j}B_j^{1- \alpha_j}\rangle_{\rho}= (-1)^{\frac{|J(\alpha)|}{2}+ \sum_{j \in J(\alpha)}j}\cdot \langle \prod_{j\in J(\alpha)}C^{(1)}(f_j^{(1)}, g_j^{(1)}) \rangle_{\rho_1}\cdot\langle \prod_{j\in J(\alpha)^c}C^{(2)}(f_j^{(2)}, g_j^{(2)}) \rangle_{\rho_2}.
F_1(0):=\lim_{x\downarrow 0}F_1(x)=e^{-\int_{0}^{\infty}\frac{1}{t}e^{-qt}\left(1-e^{-p t}\right)\mathbb P\left(X_t > 0\right)dt}-1,\ \ \F_1(\infty):=\lim_{x \uparrow \infty}F_1(x)=0.
\bigg|\int_a^b f(y)E\,dy\bigg| =\bigg|\int_a^b \bigg(f(b)+\int_y^b |f'(z)|\,dz\bigg)E\,dy\bigg| &\le f(b)\bigg|\int_a^b E\,dy\bigg|+\int_a^b \bigg|\int_a^z E\,dy\bigg| |f'(z)|\,dz &\le 24b^{1/2}f(b)+24\int_a^b z^{1/2}|f'(z)|\,dz &= 24b^{1/2}f(b)+24\bigg( a^{1/2}f(a)-b^{1/2}f(b)+\frac12\int_a^b y^{-1/2}f(y)\,dy\bigg) &= 24a^{1/2}f(a)+12\int_a^b y^{-1/2}f(y)\,dy.
i \left( \begin{matrix}{\langle U_{z}, \bar{U}_{\bar{z}}\rangle}&{\langle U_{z}, V \rangle}\\{\langle \bar{V}, \bar{U}_{\bar{z}}\rangle}&{\langle \bar{V}, V \rangle}\\ \end{matrix}\right) = \left( \begin{matrix}{( z + \bar{z})^{- 2}}&{\frac{a - 1}{a + 1}( z + \bar{z})^{- 1}}\\{\frac{a - 1}{a + 1}( z + \bar{z})^{- 1}}&{1}\\ \end{matrix}\right)
\sum_{i=1}^{n+1}b_i = \sum_{i=1}^{n+1}\log v_i = \log\prod_{i=1}^{n+1}v_i = \log\prod_{i=1}^{n+1}u_i = \sum_{i=1}^{n+1}\log u_i = \sum_{i=1}^{n+1}a_i.
\pi^3_{\rm co}:N\otimes_{\rm max}N^\circ\rightarrow \mathcal{B}(L^2(N,\tau)\otimes L^2(N,\tau)) \pi^3_{\rm co}(x\otimes y^o)(\xi\otimes\eta) :=x\xi\otimes\eta yx, y\in N\, ,\xi, \eta\in L^2(N,\tau)\, ,
\left\langle e_j , \widetilde{\mathcal{L}_{\star}}\left( \widetilde{\varrho}\right) e_j \right\rangle=- 2 \kappa j \left\langle e_{j}, \widetilde{\varrho}\, e_{j}\right\rangle+ 2 \kappa \left(j+1\right) \left\langle e_{j+1}, \widetilde{\varrho}\, e_{j+1}\right\rangle \forall j \geq 0
\begin{cases}\rho(x) w_{tt}- \operatorname{div}(K(x) \nabla w) + 2vuz+v^2w =0 ~\hbox{in}~ \Omega \times (0,T), \rho(x) z_{tt}- \operatorname{div}(K(x) \nabla z) + 2uvw+u^2z=0 ~\hbox{in}~ \Omega\times (0,T), w=z=0 ~\hbox{in}~ \omega\times(0,T),\end{cases}
\Phi^t_\varepsilon=\mathcal P\exp\!\left(\int_{\frac 1 2}^t a_3^s(B_\varepsilon)ds\right)\ \Phi^\frac 1 2_\varepsilon\ \mathcal P\exp\!\left(-\int_{\frac 1 2}^t a_3^s(A_\varepsilon)ds\right).
\nabla\left(\Phi_{\ast}(\omega)\right)(y)= \nabla\left((\nabla \Phi\circ\Psi) (\omega\circ\Psi)\right)(y) \smallskip = \sum_{k=1}^n S^k(\Phi,y)\omega_k(\Psi(y))+\nabla \Phi(\Psi(y))\,\nabla\omega(\Psi(y))\,\nabla \Psi(y),
\frac{px}{(1-p^2x)(1-px)}\sum_{r=1}^{\infty}\bigg((px^2)^r - (p^2x^3)^r\bigg) - \frac{px}{(1-p^2x)(1-p^{-1})}\sum_{r=1}^{\infty}\bigg((p^3x^3)^r - (p^2x^3)^r\bigg)
u(x,y) = \frac{P y}{6 EI}\left[ (9L-3x)x + (2+\nu) \left( y^2 - \frac{D^2}{4}\right) \right], v(x,y) = -\frac{P}{6 EI}\left[ 3 \nu y^2(L-x) + (4+5\nu) \frac{D^2x}{4}+ (3L-x)x^2 \right].
\overset{2}{R}(X,Y,AZ,AW) = \overset{2}{R}(AX,AY,Z,W), \overset{2}{R}(X,AY,AZ,W) = \overset{2}{R}(AX,Y,Z,AW), \overset{2}{R}(AX,AY,AZ,AW) = \overset{2}{R}(X,Y,Z,W).
\begin{array}{l c l}{{\phi_{a}}}&{{=}}&{{( \phi_{+}, \phi_{-}, H , \chi )}}\\{{Y_{a}}}&{{=}}&{{( Y_{+}, Y_{-}, Y_{H}, Y_{\chi})}}\\ \end{array}\quad \begin{array}{l c l}{{\hat{\phi}_{a}}}&{{=}}&{{( \hat{\phi}_{+}, \hat{\phi}_{-}, \hat{H}, \hat{\chi})}}\\{{q_{a}}}&{{=}}&{{( q_{+}, q_{-}, q_{H}, q_{\chi})}}\\ \end{array}
{\widehat{A}(TX,\nabla^{TX})}{\rm ch}(\triangle(X)){\rm ch}(\Theta_1(T_{C}X))=\prod_{j=1}^{k}\frac{2 x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\frac{\theta_1(x_j,\tau)}{\theta_1(0,\tau)},
\sum_{a+b=n}\sum_{0\le i\le a-1}(-1)^{\deg x_1+\dots+\deg x_i+i+S}{m}_a(x_1,\dots,x_i,{m}_b(x_{i+1},\dots,x_{i+b}),x_{i+b+1},\dots,x_{a+b-1})=0
\left. \frac{d}{ds}\left( \frac{1}{\Gamma(s)}\int_1^\infty t^{s-1}K\left(\frac{t}{m}, \tau(m)\right) dt \right) \right|_{s=0}= \int_1^\infty t^{-1}K\left(\frac{t}{m}, \tau(m)\right) dt.
n = \sum_{i=1}^{\ell}b_i \sum_{j =1}^{m_i}n_{i,j}= \sum_{i \in I}b_i \sum_{j=1}^{m_i}(m_i - j +1) d_{i,j}= \sum_{i=1}^{\ell}\left( b_i m_i d_{i,1}+ \sum_{j=2}^{m_i}b_i (m_i - j +1) d_{i,j}\right)
\displaystyle \int_{\mathbb{R}^{N}}\phi(|\nabla (t_{1}u)|)|\nabla(t_{1}u)|^{2}dx + \mu \displaystyle\int_{\mathbb{R}^{N}}\phi(|t_{1}u|)|t_{1}u|^{2}dx = \displaystyle\int_{[u>0]}f(t_{1}u)t_{1}udx
E\left[E\left[\exp\left(-\eta_{j}^{i_{j'}}\min(W_{j}, t-T_{j}')| W_{j}\right)\right]\right] = \frac{1}{m(t)}\left(m(t) - \eta_{j'}^{i_{j'}}\int_{0}^{t}\exp(-\eta_{j'}^{i_{j'}}w)m(t-w)\bar{G}_{W}(w)dw\right)
\langle{\cal O}_{\mu_{1}\cdots \mu_{s}}( x ){\cal O}_{\nu_{1}\cdots \nu_{s}}( 0 ) \rangle = c_{s}\frac{1}{( | x | \mu )^{2 h_{s}}}{\prod}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}\left( \frac{1}{| x |^{4}}\right) ,
x_\lambda \mapsto M_{\lambda}:=M_{(0^{\lambda_0}, \ldots, r+1^{\lambda_{r+1}})}= v_0^{\otimes \lambda_0}\otimes \cdots \otimes v_{r+1}^{\otimes \lambda_{r+1}}\in \mathbb{V}^{\otimes d},\forall \lambda \in \Lambda_{n,d}.
\biggl\| \sum_{l=1}^{n}\left( \cos(\varepsilon^{-1}\tau \sqrt{\mathstrut \lambda_l(t)}) - \cos(\varepsilon^{-1}\tau |t| \sqrt{\mathstrut \gamma_l}) \right) (\cdot,\omega_l)\omega_l \biggr\| \varepsilon^{s}(t^2 + \varepsilon^2)^{-s/2}\le \widehat{C}(\tau) \varepsilon
(\tilde{n}-1)! M^{{\tilde{n}}-1}\frac{vol(I_{N,M}^{\tilde{n}}\cap\hat{\mathfrak{D}}_{\tilde{n}})}{vol(\hat{\mathfrak{D}}_{\tilde{n}})}=\, \sum_{S}(-1)^{|S|}\sum_{0\le K_i\le M-1\atop(1\le i \le\tilde{n})}\sum_{l=0}^{\tilde{n}}\tilde{M}\left(\sum_{i\not\in S}K_i +\sum_{j\in S}(K_j-\tau_j)-Ml\right)^{\tilde{n}-1},
e'_{\delta, \, \gamma}(u) = \sum_{n = 1}^\infty e'_{n \delta, \, \gamma}\, u^n, e_{\delta, \, \gamma}(u) = \sum_{n = 1}^\infty e_{n \delta, \, \gamma}\, u^n, & f'_{\delta, \, \gamma}(u^{-1}) = \sum_{n = 1}^\infty f'_{n \delta, \, \gamma}\, u^{- n}, f_{\delta, \, \gamma}(u^{-1}) = \sum_{n = 1}^\infty f_{n \delta, \, \gamma}\, u^{- n}
d\left(x_1^{a_1}\cdots x_n^{a_n}((a_i+1)x_1 dx_2\wedge\cdots \wedge dx_n-(-1)^{i-1}(a_1+1)x_idx_1\wedge \cdots\wedge\widehat{dx_i}\wedge\cdots\wedge dx_n\right) ={}x_1^{a_1}\cdots x_n^{a_n}((a_1+1)(a_i+1)-(a_i+1)(a_1+1))dx_1\wedge\cdots \wedge dx_n ={}0.
\lambda(\{xg,yt \})= \,\,\lambda(\{x, y\}gt - (t*x)ygt+x(g*y)gt) = \,\,\{x, y\}h_{gt}+gt\beta_{\{x, y\}}- (t*x)yh_{gt}-gt\beta_{(t*x)y}\,\,+ x(g*y)h_{gt}+gt\beta_{x(g*y)}.
\bar{T}_U:=(\wedge^2 U\wedge (U_1\oplus U_2))/\wedge^3U_1\subset (\wedge^3 U_1\oplus (\wedge^2 U_1\otimes U_2)\oplus ( U_1\otimes \wedge^2 U_2))/\wedge^3U_1 &\cong(\wedge^2 U_1\otimes U_2)\oplus ( U_1\otimes \wedge^2 U_2).
X_{n,i}=U_{n,i}-\Lambda_n Z_{n,i}\hskip1cm \mathrm{where}\ \ \Lambda_n =\left(\sum_{i=1}^n E[U_{n,i}]Z_{n,i}^{\prime}\right)\left(\sum_{i=1}^nZ_{n,i}Z_{n,i}^{\prime}\right)^{-1}.
[-u+w,u+w]=\left[\tfrac{\widetilde{\theta}+\eta \tfrac{\alpha_t}{1-q}-2\sqrt{\tfrac{\alpha_t}{1-q}}\sqrt{\eta\widetilde{\theta}+1-q}}{1-q},\tfrac{\widetilde{\theta}+\eta \tfrac{\alpha_t}{1-q}+2\sqrt{\tfrac{\alpha_t}{1-q}}\sqrt{\eta\widetilde{\theta}+1-q}}{1-q}\right].
\frac{1}{\lambda_{i}}-\frac{1}{m}\sum_{j=1}^{m}\frac{1}{\lambda^{ij}_{\epsilon}}= \frac{1}{m}\sum_{j=1}^{m}<\frac{1}{\lambda_{i}}u^{ij}-\upsilon_{\epsilon}^{ij},u^{ij}>+\theta(\frac{\epsilon^{3}}{\lambda^{i}}).
\ell_1'=-\frac{c_{16}}{b_{11}^2}\quad, \ell_2'=-\frac{c_{24}}{b_{22}^2}c_{35}\ell_{1}^{2}\ell_{2}^{2}b_{11}^{4}b_{22}^{4}+ c_{16}b_{22}^{2}\ell_{2}+ c_{24}b_{11}^{2}\ell_{1}=0 b_{12}=b_{13}=b_{23}=0\ .
\sum_{k\ne i}^{}\frac{\lambda_k^2(t)}{\lambda_i(t)-\lambda_k(t)}=\sum_{k\ne i}^{}\frac{\lambda_i^2(t)}{\lambda_i(t)-\lambda_k(t)}-(N-2)\lambda_i(t)-\sum_{k=1}^{N}\lambda_k(t),
\textbf{e}^n=-\sum_{k=1}^{n-1}\omega^\alpha_k(\textbf{I}+\tau^\alpha\textbf{K})^{-1}\textbf{e}^{n-k}+\sum_{k=0}^{n-1}\omega^\alpha_k(\textbf{I}+\tau^\alpha\textbf{K})^{-1}\textbf{e}^0,
A(b)^2 = 4 \left| Z_{b^{-1}}\phi \left( \tfrac{1}{4}, \tfrac{1}{2}\right) \right|^4 =8 \pi^2 \, \frac{b^{3}\vartheta_1'(0, i b)^4}{2^4}\, e^{-\pi \frac{b}{4}}\frac{\vartheta_3 (\frac{1}{2}- \frac{i b}{4}, i b)^4}{\vartheta_4 (\frac{1}{2}, i b)^4 \vartheta_4 (\frac{1}{4}, \frac{i}{b})^4}.
\begin{aligned}\sum_{k\geq 1}\int_{\mathcal{H}_{-s}}\Big|\lambda_k\frac{\partial^2\psi(X)}{\partial z_k^2}\Big|\mu(dX) = \int_{\mathcal{H}_{-s}}\Big|\sum_{k\geq 1}\lambda_kk^{-2s}k^{2s}\frac{\partial^2\psi(X)}{\partial z_k^2}\Big|\mu(dX) &\leq \| D^2\psi\|_\infty \int_{\mathcal{H}_{-s}}\sum_{k\geq 1}\lambda_kk^{-2s}\mu(dX)<\infty.\end{aligned}
\mathrm{tr}(XY)= \sum_{m=1}^x \sum_{q=1}^Q \sum_{r=1}^Q \sum_{a=1}^x c_q(m-a) S(a,m;r) = \sum_{q=1}^Q \sum_{r=1}^Q \sum_{m=1}^x \sum_{a=1}^x c_q(m-a) S(a,m;r) = \sum_{q=1}^Q \sum_{r=1}^Q \delta (q,r) x^2 \widetilde{\varphi}(q)= x^2 \sum_{q=1}^q \widetilde{\varphi}(q)=x^2 \widetilde{\Phi}(Q)
K(u,w,\|\mathbf{x}-\mathbf{y}\|)=\frac{p(u,w,\left\Vert \mathbf{x}-\mathbf{y}\right\Vert )}{p(u) p(w)}= 1+\sum_{k=1}^{\infty}\gamma^{k}(\|\mathbf{x}-\mathbf{y}\|)e_{k}(u)e_{k}(w)
\frac{[m]_q}{[n]_q}q^{m-n}-1 = \frac{q^{2m}-1}{q^{2n}-1}- \frac{q^{2n}-1}{q^{2n}-1}= \frac{q^{2m}-q^{2n}}{q^{2n}-1}= O(q^{2m}) + O(q^{2n}),
\left\{\begin{bmatrix}N_1 (\lambda)^T -\widehat{N}_2 (\lambda) M(\lambda) N_1 (\lambda)^T\end{bmatrix}h_1(\lambda), \ldots , \begin{bmatrix}N_1 (\lambda)^T -\widehat{N}_2 (\lambda) M(\lambda) N_1 (\lambda)^T\end{bmatrix}h_p(\lambda) \right\}
t(e_{x}|f_{y})\leftarrow\frac{t(e_{x}|f_{y})}{\lambda_{y}}\sum_{k,i,j}\frac{\delta(e_{x},e_{i}^{(k)})\delta(f_{y},f_{j}^{(k)})}{\sum_{j^{\prime}}t(e_{i}^{(k)}|f_{j^{\prime}}^{(k)})}
\frac{A_{nh}5\pi\sqrt{n}}{2\cdot 5^n\exp\bigl(-\frac{5h^2}{4n}\bigr)}= \frac{25h\sqrt{5\pi}\exp(\alpha^2)}{4n^2}G(\alpha)\Bigl(1+ O\Bigl(\frac{\log n}{n^{1/2-2\varepsilon}}\Bigr)\Bigr).
\mathbb{P}\left[A^c_{\rho,S, \frac{C}{2}}(p)\right] \leq 2\exp\left( -\frac{C^2}{4}\frac{c_0^2k^2n \log n/2}{\left(k(n-k)+\binom{k}{2}\right)p+Cc_0k \sqrt{n \log n}/6}\right).
Y_{1}=(\begin{matrix}0&0&0\\ 0&0&y_{1}\\ 0&y_{1}^{*}&0\end{matrix}),Y_{2}=(\begin{matrix}0&0&y_{2}^{*}\\ 0&0&0\\y_{2}&0&0\end{matrix}),Y_{3}=(\begin{matrix}0&y_{3}&0\\y_{3}^{*}&0&0\\ 0&0&0\end{matrix}).
\chi_1(z)=-\frac{1}{2}\delta^{i_1i_2}(\nabla_q\beta)^{i_3}G_{i_1i_2i_3}^{j_1j_2j_3}&\times \left(z_{j_1}z_{j_2}z_{j_3}+2\beta^{-1}\left(\gamma_{j_1j_2}\delta_{j_3l}(\tilde\gamma^{-1})^{lk}+\gamma_{j_1j_3}\delta_{j_2l}( \tilde\gamma^{-1})^{lk}+\gamma_{j_2j_3}\delta_{j_1l}(\tilde\gamma^{-1})^{lk}\right)z_k \right),
y^2 + 10154960719 x y - 66798078951809458114391930400 y = x^3 17 y^2 + 8412073331 x y + 7384158420201525518270114400 y = x^3 13 y^2 + 19223749711 x y - 435665346791890005577936749600 y = x^3 13 y^2 + 8589423667 x y - 30679410326232604531989794400 y = x^3 17 y^2 + 35429815349 x y - 169064164426703584254124708800 y = x^3 15
\frac{\partial\rho^{*}u^{*}}{\partial t^{*}}+\nabla^{*}\cdot(\rho^{*}u^{*}\otimes u^{*}+\frac{p_{0}}{\rho_{0}u_{0}^{2}}p^{*})=-\frac{\tau_{0}}{\rho_{0}u_{0}^{2}}\nabla^{*}\cdot\tau^{*}+\frac{f_{0}r_{0}}{u_{0}^{2}}f^{*}
\frac{f_9}{f_1}= \frac{f^{3}_{12}}{f_{36}}\left(\frac{f_{18}}{f_{2}}\right)\left(\frac{1}{f_{2}f_{6}}\right) +q\frac{f^{2}_4f_{36}}{f_{12}}\left(\frac{f_{6}}{f^{3}_{2}}\right).
I_\sigma \le C^n \frac{t^{n(2-\beta_0)}}{n!}\sum_{m=0}^n \binom{n}{m}\frac{1}{\Gamma (n\varepsilon_0+1)}\le C^n \frac{t^{n(2-\beta_0)}}{n!}(n+1) 2^n \frac{1}{(n\varepsilon_0/3)^{n\varepsilon_0}},
F_{\mu\nu}=-\frac{i}{g}[D_{\mu},D_{\nu}]=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+ig[A_{\mu},A_{\nu}]=(\partial_{\mu}A_{\nu}^{i}-\partial_{\nu}A_{\mu}^{i}+gf^{ijk}A_{\mu}^{j}A_{\nu}^{k})T^{i}
\mathcal{A}^{t}\varphi(\rho_{-})=-\frac{\varphi'(\rho_{-})}{tH''(\rho_{-})}+\int_{\rho_{-}}^{\infty}(\varphi(\rho_{+})-\varphi(\rho_{-})) n(\rho_{-},\rho_{+},t)d\rho_{+}
\sum_{\epsilon_{l}< 0}u_{\epsilon_{l}l}\left( r \right) u_{\epsilon_{l}l}\left( r^{\prime}\right) + \int_{0}^{\infty}\frac{\mathrm{d}k}{2 \pi}u_{k l}\left( r \right) u_{k l}\left( r^{\prime}\right) = \delta \left( r - r^{\prime}\right) ,
B(r)^2 \rightarrow - \frac{4q^2 r}{(r^2 + q^2)^n}\lbrace\int^{r}\frac{(s^2 + q^2)^n}{s^2}ds - \int^{iq}\frac{(s^2 + q^2)^n}{s^2}ds- \bar{\alpha}\rbrace,
\frac{1}{n}\sum_{i=1}^n(y_i-T\widetilde{f}_m(x_i))^2-\frac{1}{n}\sum_{i=1}^n(y_i-f(x_i))^2 + \frac{1}{n}\sum_{i=1}^n(T\widetilde{f}_m(x^{\prime}_i))^2-Tf(x^{\prime}_i))^2 & \leq \frac{2v^2\epsilon^2_1(1+M_1/m_0)}{m_0}+ \frac{v^2M_1}{2m^2_0}+ 8B_nv\left(1+M_1/m_0\right)\epsilon_2+ \frac{2}{n}\sum_{i=1}^n(|y_i|^2-B^2_n)\mathbb{I}\{|y_i| > B_n \}.
p_{l}^{\eta}(t,x,y) \geq\int_{M_{x,y}\cap W}\eta(u)K(u,x)\rho_{t}(u)m_{x,y}(du) =\int_{M_{x,x}\cap V_{x,h}}\eta(S_{x,y}^{-1}v)K(S_{x,y}^{-1}v,x)\rho_{t}(S_{x,y}^{-1}v) \, m_{x,y}\circ S_{x,y}^{-1}(dv).
S^{( N )}\left[{\bf r}_{1}, \dots{\bf r}_{N - 1}\right] ({\bf r}^{\prime \prime},{\bf r}^{\prime}; t^{\prime \prime}, t^{\prime}) \equiv S^{( N )}\left[{\bf r}_{1}, \dots{\bf r}_{N - 1}\right] ({\bf r}^{\prime \prime},{\bf r}^{\prime}; t^{\prime \prime}- t^{\prime}) \; ,
M = \frac{g_{Y M}^{2}}{2 V{\cal N}}( w^{i}+ \Theta^{i j}n_{j})^{2}+ \frac{V}{4 g_{Y M}^{2}{\cal N}}( m_{i j}+{\cal N}\Phi{i j})^{2}+ \frac{1}{\cal N}\sqrt{\frac{k_{i}^{2}}{\Sigma_{i}^{2}}},
w_{\alpha-1,\beta}'=\left(\beta(1-x)-(\alpha-1)(1+x)\right)\cdot w_{\alpha-2,\beta-1}>0 \quad \lim_{x\to-1+}|f|^2 w_{\alpha-1,\beta}=\lim_{x\to1-}|f|^2 w_{\alpha-1,\beta}=0.
\int_{0}^1 \frac{\big(\beta_{\mu,2}^{(k)}(x,r)\big)^2}{r}\, \d r =\sum_{q=0}^{\infty}\int_{\lambda^{q+1}}^{\lambda^q}\frac{\big(\beta_{\mu,2}^{(k)}(x,r)\big)^2}{r}\,\d r\leq C\,\sum_{q=0}^{\infty}\big(\beta_{\mu,2}^{(k)}(x,\lambda^q)\big)^2.
a:\left(\begin{pmatrix}u u_{|\partial \Omega}\end{pmatrix},\begin{pmatrix}v v_{|\partial \Omega}\end{pmatrix}\right)\mapsto \int_\Omega \nabla u\cdot \overline{\nabla v}\dx+\eta_1 \int_{\partial \Omega}\nabla u\cdot \overline{\nabla v}\d\sigma-\eta_2 \int_{\partial \Omega}u \overline{v}\d\sigma,
\mu_t \big( \mu_t (a, b), c \big) =~ \mu_t \big( a, \mu_t(b,c) \big), \mu_t \big( R_t (a), R_t (b) \big) =~ R_t \big( \mu_t (R_t (a), b) + \mu_t (a, R_t(b)) \big) + \kappa ~ \mu_t (a, b),
E_{\mu}= z^{\mu}+ \sum_{\nu \prec \mu}c_{\mu,\nu}(q,t) z^{\nu},c_{\mu,\nu}(q,t)\in\mathbb{Q}(q,t), Y_iE_{\mu}= y_i(\mu;q,t)E_{\mu},\forall\ 1 \leq i \leq n, \mu \in \mathbb{Z}_{\geq 0}^n,
(D\pi(a)\xi)(s) =(D\pi(a)\xi)(s)=\frac{-1}{2\pi i}\frac{d}{ds}\big(\pi(a)\xi\big)(s)=\frac{-1}{2\pi i}\frac{d}{ds}\big(\alpha_s^{-1}(a)\xi\big)(s) =\frac{-1}{2\pi i}\Big(-\alpha_s^{-1}\big(\partial(a)\big)\xi(s)+\alpha_s^{-1}(a)\xi^\prime(s)\Big).
\left[\begin{smallmatrix}\bar a(z) \bar b(z) \bar c(z) \bar d(z)\end{smallmatrix}\right]=\left[\begin{smallmatrix}0 1-z^2 1 0\end{smallmatrix}\right]\left[\begin{smallmatrix}a(z) b(z) c(z) d(z)\end{smallmatrix}\right]\left[\begin{smallmatrix}0 1-z^2 1 0\end{smallmatrix}\right].
(\tilde{u}_R(t),\varphi(t))_{\mathcal{H}}-(\tilde{u}_{R0},\varphi(0))_{\mathcal{H}}= \int_0^t \Big[ (\tilde{u}_R,\partial_t \varphi)_{\mathcal{H}}+2\nu a(\tilde{u}_R,\varphi)+2\nu a(H,\varphi) -b_R(\tilde{u}_R ,\varphi,\tilde{u}_R)-b(H,\varphi,\tilde{u}_R)-b(\tilde{u}_R,\varphi, H) \Big]dt.
L_1(y) = \sum^m_{i=1}y_i (V^T A_i V), \;\; L_1^*(R) = \left( (V^T A_1 V) \cdot R, \cdots, (V^T A_m V) \cdot R \right)^T L_2(y) = \sum^m_{i=1}y_i (U^T A_i V) \;\; L_2^*(Z) = \left( (U^T A_1 V) \cdot Z , \cdots, (U^T A_m V) \cdot Z \right)^T,
\Pi ~ = ~ \mathrm{\O}\int \left[{\frac{e_{1}^{2}}{( p \! \! \! / - m_{1}) ( q \! \! \! / - m_{1})}}~ - ~{\frac{e_{2}^{2}}{( p \! \! \! / - m_{2}) ( q \! \! \! / - m_{2})}}\right] ~ d^{4}p
\sigma_{k}^{(n)}= \begin{pmatrix}1 2 \ldots{k-1}k{k+1}{k+2}\ldots n+k 1 2 \ldots{k-1}{n+k}{k}{k+1}\ldots n+k-1 \end{pmatrix}\hphantom{\sigma_{k}^{(n)}}{}=s_k s_{k+1}\cdots s_{n+k-1}\in \mathbb{S}_{n+k}.
\sigma^{\rm reg}(\xi) = -\,{|\xi|\over 2}\, -\,{|\xi| \over \pi}\sum_{n=1}^\infty{K_1(2 n \pi\,|\xi|)\over n}= -\,{|\xi|\over 2}\, -\,{|\xi| \over \pi}G(2\pi\,|\xi|) \ .
\delta^{1}u^{n}_{j}=2 \Delta x (u_x)^n_j+O(\Delta x^3), &\delta^{2}u^{n}_{j}=4 \Delta x^2 (u_{xx})^n_j+O(\Delta x^4), &\delta^{3}u^{n}_{j}=O(\Delta x^3).
\begin{array}{c}{\left[ A^{-}, A^{+}\right]_{q \left( x \right)}= - 2 \alpha \left( x \right) \sinh \frac{i}{2}\frac{d}{d x}\left[ \alpha \left( x \right) \sinh 2 \rho_{\frac{s}{2}}\left( x \right) \right] \rightarrow}\\{\rightarrow \frac{d^{2}\rho \left( x \right)}{d x^{2}}}\\ \end{array}
a_n^{(\theta)}= \frac{1}{x_1 - x_2}\big( T_{q, x_2}a_n^{(\theta - 1)}- T_{q, x_1}a_{n-1}^{(\theta - 1)}\big), 1\leq n\leq \theta-1, a_0^{(\theta)}= \frac{1}{\prod\limits_{i=0}^{\theta - 1}{(x_1 - q^i x_2)}}, a_{\theta}^{(\theta)}= \frac{1}{\prod\limits_{i=0}^{\theta - 1}{(x_2 - q^i x_1)}}.
\begin{aligned}\delta(P) =\frac12 \widehat{g}(z_P,z_P)=\frac12 p^t\cdot G\cdot p, \widehat{g}(z_P,z_Q) =4\delta\left( \frac{P+Q}{2}\right)-\delta(P)-\delta(Q)=p^t\cdot G\cdot q, G_{ij}=\widehat{g}(z_{R_i},z_{R_j})=4\delta\left( \frac{R_i+R_j}{2}\right)-\delta(R_i)-\delta(R_j),\end{aligned}
\sum_{|\alpha| = k_{1}+ 2}\int | \partial^{\alpha}\psi|^{2}\xi^{2}\leq C \bigg( \int \sum_{|\beta_{1}| = k_{1}}|\partial^{\beta_{1}}\Delta \psi|^{2}\xi^{2}+ \int \sum_{|\beta_{2}| = k_{1}+1}|\partial^{\beta_{2}}\psi|^{2}\xi^{2}\bigg),
{\rm Tr}e^{-\beta\,P \bar{\cal K}}=-1+\left[\frac{mL}{\sqrt{8\pi \beta}}+\frac{1}{\sqrt{\pi}}\sum_{n=1}^\infty \frac{2^{n}(1+2^{2n-1})}{(2n-1)!!}\beta^{n-\frac{1}{2}}\right]e^{-4 \beta}
\begin{cases}|x|^{-s}u_{t}-\mathrm{div}(|\nabla u|^{p-2}\nabla u)=|u|^{q-2}u\ln|u| ~\Omega \times(0,T), u(x,t)=0 ~\partial\Omega \times(0,T), u(x,0)=u_0(x) ~x\in\Omega,\end{cases}
w_r(\tau) = e^{\tau \Delta}w_r(0) + \int_0^\tau e^{(\tau - s)\Delta}\big(e^u \Delta \phi_r - 2\nabla \cdot (e^u \nabla \phi_r)\big) ds &+ \int_0^\tau e^{(\tau - s)\Delta}e^u\phi_r (\partial_\tau u - \Delta u - |\nabla u|^2)ds,
\frac{z\mathtt{g}_{a, \nu}'(z)}{\mathtt{g}_{a, \nu}(z)}= a (1-\nu)+ a^{a/2}\left(\frac{z\mathtt{J}'_{\nu}\left(z\right)}{\mathtt{J}_{\nu}\left(z\right)}-(\nu-1)(1-a)\right) &= a (1-\nu)+ a^{a/2}\left(a\nu-a+1-\sum_{n=1}^\infty \frac{2 z^2}{\mathtt{j}_{\nu, n}^2-z^2}\right),
P_{M}\left(\Pi,V\right)=\sum_{j\in J}e^{-\hat{r}T_{j}}\left[\sum_{i\in I_{j}}\Pi\left(t_{i},T_{j}\right)\left(\sum_{c\in C}V_{c}\left(t_{i},T_{j}\right)+\sum_{p\in P}V_{p}\left(t_{i},T_{j}\right)\right)\right]
\left( 1-i\varepsilon \int d^{4}z\frac{\partial F^{\beta}}{\partial \alpha}\mid_{_{\alpha_{0}}}\delta \alpha (\partial .A-F)^{\alpha}(x)i\int d^{4}xM^{'-1\alpha \beta}(x,z;A;\alpha_{0})+O[\left( \delta \alpha \right)^{2}]+O[g]\right)
\theta_{0}(q,q^{-2}u) =1+q^3u^3+q^{15}u^{-3}+q^{24}u^{6}+\cdots, \theta_{1}(q,q^{-2}u) =q^{-1}u+q^{12}u^{-2}+q^{8}u^4+\cdots, \theta_{2}(q,q^{-2}u) =u^2+q^3u^{-1}+q^{15}u^{5}+q^{24}u^{-4}+\cdots, \lim_{q\to 0}\ [\theta_0,\theta_1,\theta_2] (q,q^{-2}u)=\lim_{q\to 0}\ [1,q^{-1}u,u^2]=[0,1,0]
V \rho^X_{\gamma^1}(-)=\Gamma^X_2 \widehat{V}\Gamma^X_1 \rho^X_{\gamma^1}(-)=\Gamma^X_2 \widehat{V}\rho^X_{\widehat{\gamma}^1}(-) \Gamma^X_1=\Gamma^X_2 \rho^X_{\widehat{\gamma}^2}(-) \widehat{V}\Gamma^X_1=\rho^X_{\gamma^2}(-) \Gamma^X_2 \widehat{V}\Gamma^X_1=\rho^X_{\gamma^2}(-) V.
\vartheta(z)=2\varphi_{zz}-\varphi_z^2= \begin{cases}2\mathcal{S}\left(J_1^{-1}\right)(z),\quad z\in \Omega_1 2\mathcal{S}\left(J_2^{-1}\right)(z),\quad z\in \Omega_2.\end{cases}
\overline{{{\Psi}}}( x ) = ( 2 \pi )^{- 3 / 2}\int \left[ \overline{{{\Psi}}}^{+}( \mathbf{p}) e^{i p x}+ \overline{{{\Psi}}}^{-}( \mathbf{p}) e^{- i p x}\right] d^{3}p
S^{env,0}_{s,t}= \int_{s}^t 2\hat b_+^j(r,q_r) (\tilde \Sigma^{-1})_{jk}(r,q_r)\circ dq^k_r &-\int_{s}^t 2\hat b_+^j(r,q_r) (\tilde \Sigma^{-1})_{jk}(r,q_r)b^k_-(r,q_r) +\nabla\cdot b_-(r,q_r)dr,
\times \prod_{b = 1}^{N}\prod_{j = 1}^{n_{b}}\exp \left( + 2 e \sum_{I = 2}^{N}U_{I b}\Big ( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, Q^{( I )}\delta ( x_{j}^{( b )}) \Big ) \right)
\mathcal{D}_{z}= \left( \begin{array}{c c}{\mathbf{a}+ \mathbf{b}\mathbf{x}}{\mathbf{0}}{\mathbf{0}}{\mathbf{a}^{\prime}+ \mathbf{b}^{\prime}\mathbf{x}^{\prime}}\end{array}\right) \equiv \left( \begin{array}{c c}{D_{z}}{\mathbf{0}}{\mathbf{0}}{D_{z}^{\prime}}\end{array}\right) \, ,
{\cal T}_{S}\, = \, \left[ \begin{array}{c c}{{1}}&{{\qquad 0}}\\{{}}&{{}}\\{{1}}&{{\qquad 1}}\\ \end{array}\right] \qquad{\cal S}_{S}\, = \, q \gamma_{q}\left[ \begin{array}{c c}{{0}}&{{\qquad - 1}}\\{{}}&{{}}\\{{1}}&{{\qquad 0}}\\ \end{array}\right]
T_{- \dot{p}+ 2}^{+ q}= \frac 1 2 \gamma_{q \dot{q}}^{i}\Omega_{- \dot{p}}^{+ 2 i}\psi_{+ 2 \dot{q}}^{1 -}+ \frac{\chi}{2}\gamma_{q \dot{p}}^{i}\Omega_{+ 2}^{+ 2 i}, \T_{- \dot{p}- 2}^{+ q}= \frac{\chi}{2}\gamma_{q \dot{p}}^{i}\Omega_{- 2}^{+ 2 i},
G_t(s) = \big\langle \exp{\big[-\mathrm{e}^{(\eta(0,t) - s )}\big]}\big\rangle = \sum_{N=0}^\infty \frac{(-1)^N}{N!}\mathrm{e}^{-N(s -(t/24))}\langle Z(0,t)^N\rangle\,.