harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{n=1}^{\infty}\frac{1}{e^{n^{2}x}-1}=\frac{1}{4}+\frac{\pi^{2}}{6x}+\frac{1}{2}\sqrt{\frac{\pi}{x}}\,\zeta\left(\frac{1}{2}\right)+\frac{1}{2}\,\sqrt{\frac{\pi}{x}}\,\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}\left(\frac{\cos(2\pi\sqrt{\pi n/x})-\sin(2\pi\sqrt{\pi n/x})-e^{-2\pi\sqrt{\pi n/x}}}{\cosh(2\pi\sqrt{\pi n/x})-\cos(2\pi\sqrt{\pi n/x})}\right),\,\,\,\,x>0.
\eta_{t}\lambda_{t}^{p}=\frac{\sqrt{\Delta_{1}}T^{\frac{1-p}{3p-2}}}{8\sqrt{L}\gamma}\lambda_{t}^{p-1}\ge\frac{\sqrt{\Delta_{1}}T^{\frac{1-p}{3p-2}}}{8\sqrt{L}\gamma}\frac{8\gamma}{\sqrt{L\Delta_{1}}}T^{\frac{p-1}{3p-2}}\sigma^{p}=\frac{\sigma^{p}}{L}
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=(2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m-1}}(2^{2i}-1)}{|M_2^{K_{\ell}}|})(2^{(\dim M_2^L-m)/2}\cdot\frac{\displaystyle{\prod_{i=1}^{m}}(2^{2i}-1)}{|M_2^L|})^{-1}\leq 2\cdot\frac{2^{n_{2,\nu_2}}-1}{2^{n+1}-1}
\left( 1+j^{-2}\sum_{k=1}^{\ell-1}2^{n}C(i+k)^{-2}\right)\left(1+C\left(\frac{i+\ell+2}{i+\ell+1}\right)^{n'-m_2'}j^{-2}(i+\ell)^{-2}\right) &\leq 1+j^{-2}\sum_{k=1}^{\ell-1}2^{n}C(i+k)^{-2}+2C2^{n-1}j^{-2}(i+\ell)^{-2}\leq 1+j^{-2}\sum_{k=1}^{\ell}2^{n}C(i+k)^{-2}.
\Lambda = \{(s_1,\dots,s_{n-1},p)\in \mathbb G_n \,: & \; (s_1,\dots,s_{n-1}) \in \sigma_T(F_1^*+pF_{n-1}\,,\,F_2^*+pF_{n-2}\,,\,\dots\,, F_{n-1}^*+pF_1) \},
\frac{1}{\rho_0(\eta)}=O\left(\sqrt\eta\right), \rho_0(\eta) = O\left(\eta^{-\frac34}\right), \quad begin{align*}2pt] \frac{\varphi(z_0) - \varphi(\overline z_0)}{(z_0 - \overline z_0)\rho_0^2(\eta)}=-1 + O\left(\eta\right).
\sum_{n = 1}^{\infty}f \left( n \right) = - \frac{1}{2}f \left( 0 \right) + \int_{0}^{\infty}d x f \left( x \right) + i \int_{0}^{\infty}d x \frac{f \left( i x \right) - f \left( - i x \right)}{e^{2 \pi x}- 1}.
\tilde{S}_{r s}= \oint \frac{d \xi}{2 \pi \imath}\frac{d \xi^{\prime}}{2 \pi \imath}\xi^{- r - \frac{1}{2}}\xi^{\prime}{}^{- s - \frac{1}{2}}\frac{2 \imath}{\sqrt{1 + \xi^{2}}\sqrt{1 + \xi^{\prime}{}^{2}}}\ln \left( \frac{( 1 + i \xi ) ( 1 - i \xi^{\prime})}{( 1 - i \xi ) ( 1 + i \xi^{\prime})}\right) .
\left.\begin{array}{rcl}0 = \displaystyle{\inf_{v \in U}\left[ \mathcal L \bar u^k_{\alpha}(\theta,x,v) + \theta (r_k(x,v)-g^k_{\alpha}) \bar u^k_{\alpha}\right]}\displaystyle{\bar \nabla u^k_{\alpha}\cdot \gamma(x)}= 0 \mbox{on}\partial D, \ \bar u^k_{\alpha}(\theta,x_0)=1 \, .\end{array}\right.
L[f'](t) = \frac{2}{\pi^2}\int_{1/r}^1 \frac{\sqrt{1-t^2}\,ds}{\sqrt{1-s^2}(s^2-t^2)}= - \frac{2}{\pi^2}\int_0^{1/r}\frac{\sqrt{1-t^2}\,ds}{\sqrt{1-s^2}(s^2-t^2)}= F_t(0) - F_t(1/r) = - F_t(1/r),t\in(-1,-1/r]\cup[1/r,1).
\| u\|_{L^p_{k,\nu}}= \left( \sum_{j=0}^k{\| r^{-\frac{n}{p}-\nu+j}\nabla^j u\|^p_{L^p}}\right)^{\frac{1}{p}}, \| u\|_{C^{k,\alpha}_\nu}= \sum_{j=0}^k{\|r^{-\nu+j}\nabla^j u\|_{C^0}+ [r^{-\nu+k}\nabla^k u]_\alpha}.
-\,\frac{m}{2\mu_*\left(m+2\right)}\le \gamma < \frac 1{m+2}-\frac{2(n+1)}{\mu_\ast (m+2)(n-1)}=\frac 3{m+2}-\frac{n\left(2\mu_\ast -m\right)}{\mu_\ast (m+2)(n-1)}.
\Big \langle e^{- i e \sum_{b}( j_{\mu}^{( b )}, a_{\mu}^{( b )})}\Big \rangle = e^{- \frac{e^{2}}{2 \pi}\sum_{b , c}( \varepsilon_{\mu \nu}\partial_{\mu}a_{\nu}^{( b )}, M^{( b , c )}\varepsilon_{\rho \sigma}\partial_{\rho}a_{\sigma}^{( c )})},
v =(v_1,v_2,\dots,...,v_k):=G_{R_k(z)}^{-1}(R_k(u)) &=\left(u_{0}-z_{0}\frac{u_k}{z_k}, u_1-z_1\frac{u_k}{z_k},\dots,u_{k-1}-z_{k-1}\frac{u_k}{z_k},u_{k+1}-z_{k+1}\frac{u_k}{z_k},\dots, u_{d}-z_{d}\frac{u_k}{z_k}\right).
\delta_{1}=-a_1a_{n-2k}, b_{2}=\frac{(-1)^{k-1}(a_2+1)^k\theta_{n}}{a_1}, \theta_{n-k+i}=(-1)^{k-i}(a_2+1)^{k-i}\theta_{n}, 1\leq i\leq k-1.
e_{1}=\left\{v, a_{1},\cdots, a_{k-2}, a \right\},\quad~e_{2}=\left\{v, b_{1}, \cdots, b_{k-2},b\right\}, e_{1}^{\prime}=\left\{v,a,b,a_{2},\cdots,a_{k-2}\right\},~~e_{2}^{\prime}=\left\{v,a_{1},b_{1}, \cdots, b_{k-2}\right\}.
\omega_{n_{1}n_{2}}= \biggl ( \biggl ( \frac{n_{1}\pi}{L_{1}}\biggr )^{2}+ \biggl ( \frac{n_{2}\pi}{L_{2}}\biggr )^{2}\biggr )^{\frac{1}{2}}n_{1}, n_{2}= 1 , 2 , \dots ,
\nabla^{{\rm SO}(M)}_{X^h}Y^h =\left( \nabla_XY+\frac{1}{2}R_{\xi_X}(Y)+\frac{1}{2}R_{\xi_Y}(X) \right)^h \hphantom{\nabla^{{\rm SO}(M)}_{X^h}Y^h =}{}+\frac{1}{2}\big( \nabla_X\xi_Y+\nabla_Y\xi_X+\xi_{[X,Y]}\big)^{\ast}+\textrm{$\mathfrak{g}_P$-component}.
\Lambda_{g,k}^{W_1\oplus W_2}(\lceil m_1n_1 \; ; \; \gamma_1\delta_1 \rfloor, \; \ldots, \; \lceil m_kn_k \; ; \; \gamma_k\delta_k \rfloor) &=\Lambda_{g,k}^{W_1}(\lceil m_1 \; ; \; \gamma_1 \rfloor, \, \ldots, \,\lceil m_k \; ; \; \gamma_k \rfloor)\, \, \Lambda_{g,k}^{W_2}(\lceil n_1 \; ; \; \delta_1 \rfloor, \, \ldots,\, \lceil n_k \; ; \; \delta_k \rfloor).
\frac{1}{\sqrt{2}k}\left[ 1 - \kappa \left( \frac{d}{d r}+ \frac{\kappa + 1}{r}\right) \right] R_{E \ell ( \kappa )}( r ) = \left( E + \frac{1}{2 k^{2}}\right)^{\mathrm{\frac{1}{2}}}R_{E \ell ( - \kappa )}( r ) .
q^{\alpha^{2}/2}\left(-zq^{\alpha+1/2};q\right)_{\infty}=\int_{-\infty}^{\infty}\frac{\exp\left(x^{2}/\log q^{2}+i\alpha x\right)dx}{\left(ze^{ix};q\right)_{\infty}\sqrt{\pi\log q^{-2}}},
\tilde{E}(u) =\Pi^\perp \bigg [ E(u)- \left( H(u^T)\right)'' +m\left( H(u^T)\right)' - \mathcal{L}_{\Sigma}H(u^T) \bigg] &= \Pi^\perp \bigg [ -\mathcal{M}_{\Sigma}(u) +\mathcal{L}_{\Sigma}u +N_1(u) - \left( H(u^T)\right)'' +m\left( H(u^T)\right)' - \mathcal{L}_{\Sigma}H(u^T) \bigg] &=:\textsc{I+II},
d_1 \le \sqrt{\sum_{n=N_T +1}^{+\infty}\frac{\phi^2(u) H_{n-1}^2(u)}{n! T^d}\int_{[-T,T]^d}\rho^n(t) \prod_{j=1}^d (T- |t_j|)\,dt}\cr\le K \sqrt{\phi(u)}\sqrt{\int_{\mathbb R^d}|\rho(t)|\,dt}\sqrt{\sum_{n=N_T +1}^{+\infty}\frac{1}{n (n-1)^{1/6}}}\cr= O(N_T^{-1/12}),
\begin{aligned}(fg)x = (fx)g + f(gx) + \alpha_1 \big( f(xg) + f(gx) \big) + \alpha_2 \big( g(xf) + g(fx) \big) x(fg) = (xf)g - (xg)f + \beta_1 \big( f(xg) + f(gx) \big) + \beta_2 \big( g(xf) + g(fx) \big)\end{aligned}
\forall 1 \leq i,j,k \leq n, \: i \neq j, \: k \neq p, \:\:\:\: t_0(w_{i,j},w_{k,p})=w_{(i+k-1) mod(n) +1, (j+p-2) mod(n) +1}&t_1(w_{i,j},w_{k,p})=w_{(i+k-2) mod(n) +1, (j+p-2) mod(n) +1}&t_2(w_{i,j},w_{k,p})=w_{(i+k-1) mod(n) +1, (j+p-1) mod(n) +1}&t_3(w_{i,j},w_{k,p})=w_{(i+k-2) mod(n) +1, (j+p-1) mod(n) +1}.
I_{cc}^{2}= \left.\frac{1}{26}\left(\frac{-7 + 10 x}{26x^2 - 70x + 49}\right)\right|_{x=1}^{x=2}= \frac{1}{65}I_{cc}^{3}= \left.3\frac{1}{9+x^2}\right|_{x=2}^{x=5}= -\frac{63}{442}I_{cc}^{4}= \left.\frac{48 (7 - 2 x)}{338 x^2 - 3276x + 11466}\right|_{x=5}^{x=7}= -\frac{3}{119}.
\sin \rho_n x - \rho_n x = O \left( \left(\rho_n x\right)^3 \right) = O \left( \cfrac{1}{n^3}\right), \sin^2 \cfrac{\rho_n x}{2}= O \left( \left(\cfrac{\rho_n x}{2}\right)^2 \right) = O \left( \cfrac{1}{n^2}\right).
S\sp c={1\over 32\pi\sp 2}\int\sb 0\sp\infty de\sb 0 \left(\det\frac{\sinh\frac{e\sb0gF}{2}}{gF}\right)\sp{-\frac{1}{2}}\Psi(x\sb{out},\,x\sb{in},\,e\sb0)e\sp{{i}I\lbrack x\sb{cl},e\sb0\rbrack}\,,
\left[ ( L_{+}+ L_{-}) + ( R_{+}+ R_{-}) \right] \rightarrow \left[ ( L_{+}^{\prime}+ L_{-}^{\prime}) + ( R_{+}^{\prime}+ R_{-}^{\prime}) \right] = \left[ ( L_{+}+ L_{-}) + ( R_{+}+ R_{-}) \right]
I_{r}= \frac{c_{b}}{2}\int_{0}^{\infty}d\xi \sin(t\xi) \xi \chi_{\leq 1}(r\xi) \left(J_{0}(r\xi)-J_{2}(r\xi)\right) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}
\Vert J_{44}^{\left( 5\right)}\Vert_{p}=\mathcal{O}\left( \frac{1}{n}\right) (v(|z_{1}|)+(v(|z_{2}|))\left( \frac{\omega (h_{1})}{v(h_{1})}+\frac{\omega (h_{2})}{v(h_{2})}\right) .
\begin{aligned}\tilde{b}_{s,s+k+1}\left[ Z_s, t ;\left\{t_j,v_{s+j},\omega_j,i_j \right\}_{j=1}^{k+1}\right] = &= \tilde{b}_{s,s+k}\left[ Z_s,t;\left\{t_j,v_{s+j},\omega_j,i_j\right\}_{j=1}^k \right] \times \omega_{k+1}\cdot \left( v_{s+k+1}-v_{i_{k+1}}^\prime\right) \end{aligned}
\tilde{\mathcal{E}}_{z_1}=\Big\{\tilde x\geq \frac{z_1}{\sqrt{n}}\Big\},\qquad\tilde{\mathcal{E}}_{z_1,z_2}=\Big\{\frac{z_1}{\sqrt{n}}\leq \tilde x\leq \frac{z_2}{\sqrt{n}}\Big\},\tilde{\mathcal{E}}=\Big\{\frac{1}{n}\sum_{i=1}^nf_i^2\leq \frac{1}{r}\Big\}.
f\biggr(\frac{F_{k_0+2}}{F_{k_0+1}},\dots,\frac{F_{k_0+n+1}}{F_{k_0+n}}\biggr)=\biggr(\frac{F_{k_0+3}}{F_{k_0+2}},\dots,\frac{F_{k_0+n+2}}{F_{k_0+n+1}}\biggr).
(r(X)\otimes 1)\alpha(f\otimes w)(g) = \left[\frac{d}{dt}f(g\exp(tX)) \otimes \pi(\exp(-tX)g^{-1})w\right]_{t=0}&= r(X)f(g)\otimes g^{-1}w - f(g)\otimes \pi(X)\pi(g)^{-1}w &= \alpha(r(X)f\otimes w)(g) -(1\otimes (I_{\wedge^\bullet\overline{\mathfrak{u}}}\otimes I_E\otimes \pi(X))) \alpha(f\otimes w)(g)
\{A , D \}_{( 3 , 1 )}= - 4 d{\partial}_{1}a \cdot{\gamma}^{2}- 4{\gamma}^{2}d \cdot{\sum_{k = 1}^{\infty}}{\gamma}^{2 k}{\partial}_{1}a_{2 k}- 2 ( 1 - 4{\gamma}^{2}){\partial}_{1}d \cdot{\sum_{k = 1}^{\infty}}k{\gamma}^{2 k}a_{2 k},
\|u\|_{(\kappa+1)p^*}= \|u^{\kappa+1}\|_{p^*}^{{\frac{1}{\kappa+1}}}\leq M_{29}^{\frac{1}{\kappa+1}}\left((\kappa+1)^{M_{28}}\right)^{{\frac{1}{\kappa+1}}}\left[\|u^{\kappa+1}\|_{\tilde{q}_1}^p+1\right]^{\frac{1}{(\kappa+1)p}}.
\Pr\left[\exists X,Y e_{R_2}(X,Y)=0\right]\le \binom{n}{s}^2(1-pq')^{(1-o(1))s^2}&\le \left(\frac{en}{s}\right)^{2s}e^{-(1-o(1))pq's^2}\le e^{O\left(n(\ln n)^{-0.45}\ln \ln n\right)}e^{-\Omega\left(n(\ln n)^{-0.05}\right)}=o(1).
0 =F\nabla_{j}\frac{1}{\varphi}+\frac{1}{\varphi}\nabla_{j}F-\frac{\beta-1}{\beta}\lambda_{j}^{-1}\nabla_{j}F &=-F\lambda_{1}^{-2}h_{11j}+(\lambda_{1}^{-1}-\frac{\beta-1}{\beta}\lambda_{j}^{-1})\nabla_{j}F.
Q/Q(2) \downarrow X = T(8) \otimes 1^{[1]}+ 5 + 1^{[1]}, Q(2)/Q(3) \downarrow X = T(8) + 5 \otimes 1^{[1]}+ 0, Q(3)/Q(4) \downarrow X = T(9) + 5 + 1^{[1]}, Q(4)/Q(5) \downarrow X = 5 \otimes 1^{[1]}, Q(5)/Q(6) \downarrow X = 5, Q(6)/Q(7) \downarrow X = 0.
\begin{array}{c}u(\tau )=e^{\sqrt{\gamma^2+m^2/2}\,\tau}\left[ u_0\cosh (\gamma \tau)-\left( u_0\sqrt{\gamma^2+m^2/2}-\dot u_0\right) \frac 1\gamma \sinh(\gamma \tau )\right] v(\tau )=e^{-\sqrt{\gamma^2+m^2/2}\,\tau}\left[ v_0\cosh (\gamma \tau)+\left( v_0\sqrt{\gamma^2+m^2/2}+\dot v_0\right) \frac 1\gamma \sinh(\gamma \tau )\right] ,\end{array}
0\leq\frac{1}{\|h\|}_{\!H^1}\!\bigg|\int_{[0,1]}(h')^2\bigg|\leq\frac{1}{\|h\|}_{\!H^1}\!\bigg|\int_{[0,1]}\big(h^2+(h')^2\big)\bigg|=\|h\|_{H^1}\,,
\lim_{x\nearrow x^*_t}\rho_t(x)= \psi_0 \left(\varphi_{0, t}^{- 1}(x^*_t) \right)\exp \left\{- \int_0^t [ f - \lambda ] ( \varphi_{s, t}^{- 1}(x^*_t) ) ds \right\}
\omega(v_\alpha \diamond v_{\beta+n})=\omega(-ie_{\alpha,\beta}+ie_{\beta+n,\alpha+n})=-i\sum_{k=1}^{r}(u_{\alpha+n,k}\frac{\partial}{\partial u_{\beta+n,k}}+u_{\alpha,k}\frac{\partial}{\partial u_{\beta,k}})-i r \delta_{\alpha \beta}.
\int f^*\,d\nu + \int f\,d\eta = \int f^*(\nabla V(x)) + f(x)\,\eta(dx) & \geq \int \nabla V(x)\cdot x\,\eta(dx) & = \int V^*(\nabla V (x))+V(x)\,\eta(dx) & = \int V^*\,d\nu + \int V\,d\eta.
Q[u_n]=-\lambda\,\lVert G^\lambda\mspace{1.5mu}\eta_n\rVert_{L^2_{\mathrm{sym}}(\mathrm{R}^6)}+\Phi^\lambda[\eta_n]\leq-\lambda\,\lVert G^\lambda f\rVert_{L^2_{\mathrm{sym}}(\mathrm{R}^6)}+\Phi^\lambda[\eta_n]
=\exp\left\{-\frac{g^2}{2}\left[\frac{M^2}{2}\int\limits_{\Sigma}^{}d\sigma_{\mu\nu}({\bf x})\int\limits_{\Sigma}^{}d\sigma_{\mu\nu}({\bf y})+\oint\limits_{{\cal C}}^{}dx_\mu \oint\limits_{{\cal C}}^{}dy_\mu\right]D_M({\bf x}-{\bf y})\right\}.
N= \frac{1}{4 \beta \left(\frac{2}{\sqrt{d_{\perp}}}\right)^{d_{\perp}}}~\int_{y_{N}}^{y_{end}}\frac{(1-y)^{d_{\perp}-1}dy}{y^3 \left( 1 + (\frac{d_{\perp}}{4}-\frac{3}{2}) y \right)}
E=\frac{1}{(m|g|)^2}\sum_{\substack{\ell=1 (\ell\neq k)}}^n\sum_{i,h,h'=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{j,\ell}[\tilde{v}_{i-h-1}\tilde{w}_{i-h-1}]_{\ell,k}[\tilde{v}_{i-h'-1}\tilde{w}_{i-h'-1}]_{k,j}\frac{\lambda_\ell^{m-h-1}\lambda_k^{m-h'-1+h}\lambda_j^{h'}}{(\lambda_\ell^m-\lambda_k^m)(\lambda_k^m-\lambda_j^m)}.
S_{NS}^{\mathrm{quad}}=\int A*Q_BA=\langle A^{\prime}|b_0Q_Bb_0|A^{\prime}\rangle=\langle A^{\prime}|\{b_0,Q_B\}b_0|A^{\prime}\rangle=\langle A^{\prime}|L_0^{\mathrm{tot}}b_0|A^{\prime}\rangle .
\sum_{\gamma': \gamma'\sim \gamma}\prod_{e\in E_{\tilde \gamma}}|M_e|^{\tilde m_e}\leq \left( \sqrt\frac{K \rho}{mn}\right)^{4sk-2a+2h}\sum_{\gamma': \gamma'\sim \gamma}\prod_{e\in E_{\tilde\gamma}\setminus H}\frac{Q_e}{\sqrt{mn}}, =(K\rho)^{2sk+h-a}(\sqrt{mn})^{a-h-4sk}\sum_{\gamma': \gamma'\sim \gamma}\prod_{e\in E_{\tilde\gamma}\setminus H}{Q_e},
\begin{aligned}\left|\nabla\left(\frac{mt^2}{2}-\varphi\right)\right|^2= m^2t^2|\nabla t|^2-2mt\nabla\varphi\cdot \nabla t+|\nabla\varphi|^2 \ge m^2t^2|\nabla t|^2-2m |t| |\nabla\varphi| |\nabla t|+|\nabla\varphi|^2. \end{aligned}
\ll S_{|f|}(N;\widetilde W,A) \sum_{d|a',\,d>1}\sum_{\substack{b < a: \gcd(\widetilde{W}b + A,a')=d~}}\frac{1}{a}\ll S_{|f|}(N;\widetilde W,A) \sum_{d|a',\,d>1}\frac{1}{a}\frac{a}{a'}\phi\Big(\frac{a'}{d}\Big) \ll S_{|f|}(N;\widetilde W,A) \sum_{d|a',\,d>1}\frac{1}{d},
N_{i}^{- 1}{\tilde{L}}( \lambda | x_{i}, \partial_{x_{i}}) N_{i - 1}= \left( \begin{array}{c c}{{- ( y_{i - 1}- x_{i})}}&{{1}}\\{{0}}&{{\frac{\lambda + \beta_{i}}{y_{i - 1}- x_{i}}}}\\ \end{array}\right) ~ .
\langle \delta Y \rangle = \delta Y \bigg ( \frac{\delta{\cal W}}{\delta{\cal J}_{i}}+ \frac{\hbar}{i}\frac{\delta}{\delta{\cal J}_{i}}, \, \psi^{\alpha}, \, \phi_{A a}^{*}, \, \bar{\phi}_{A}\, ; ( - 1 )^{\varepsilon_{i}}\frac{1}{i \hbar}{\cal J}_{i}, \,
\int_M[e_{\epsilon}(u_{\epsilon})-\frac{1}{2}|ju_{\epsilon}|^2]=\int_M[\frac{1}{2}(1-|u_{\epsilon}|^2)|du_{\epsilon}|^2+\frac{1}{2}|d|u_{\epsilon}||^2+\frac{W(u_{\epsilon})}{\epsilon^2}\leq C;
x_i^{\prime}(t) \leq x_i(t) \biggl(-\mu_i x_i(t)-\sum_{j=1}^{p}a_{ij}\int_0^{\infty}K_{ij}(s)(x_j(t-s)-x_j^*)\, ds & -c_i \int_0^{\infty}G_{i}(s)u_i(t-s)\, ds-\sum_{j=p+1}^n a_{ij}\int_0^{\infty}K_{ij}(s)x_j(t-s)\, ds \biggr).
\begin{cases}\dot{\bf x}_N= A{\bf x}_N +BR^{-1}B^T{\bf y}_N +G{\bf m}_N +\gamma{\bf p}_N, \dot{\bf m}_N= (A + G){\bf m}_N +B\bar u^* +\gamma{\bf p}_N, \dot{\bf p}_N = -(A+G)^T{\bf p}_N -(I-\Gamma)^T Q [{\bf x}_N- (\Gamma{\bf m}_N+\eta)] , \dot{\bf y}_N=-A^T{\bf y}_N +Q[{\bf x}_N-(\Gamma{\bf m}_N+\eta)],\end{cases}
F_{s+1}\left( \xi \right) =-{\dfrac{1}{2}}\frac{dF_{s}\left( \xi \right)}{d\xi}-{\dfrac{1}{2}}\sum\limits_{j=1}^{s-1}{F_{j}\left( \xi \right)F_{s-j}\left( \xi \right)}\ \left({s=2,3,4\cdots}\right) .
\Omega_2(x)=\frac{\Gamma(1-a-ax-\gamma)\Gamma(1-a-ax-\delta)\Gamma(1-a-ax+b+bx-\gamma-\delta)}{\Gamma(1-a-ax+b+bx-\gamma)\Gamma(1-a-ax+b+bx-\delta)\Gamma(1-a-ax-\gamma-\delta)},
\kappa_{0}= - k^{2}/ 4 \ , \quad \kappa_{1}/ \kappa_{0}= a + 1 / 2 \ , \quad{\kappa_{2}/ \kappa_{0}= \ln ( \sqrt{2 \pi}/ G_{+}( i ) ) - 1}+ \ln k + a ( \gamma_{\mathrm{E}}- 1 + \ln 8 ) - b \ .
G(x)={{2{B_0}-{C_1}+ 2({B_1}-{C_2}){\omega}+ (2{B_2}- 3{C_3}){{\omega}^2}- 4{C_4}n{{\omega}^3}}\over{4{\sqrt{{C_0}+{C_1}{\omega}+{C_2}{{\omega}^2}+{C_3}{{\omega}^3}+{C_4}{{\omega}^4}}}}}.
\begin{aligned}\dot{\phi}- R K \sin \delta \leq (\varepsilon R(0) - R K) \sin \delta \leq (\varepsilon - K) R(0) \sin \delta < 0, & \dot{\phi}+ R K \sin \delta > (-\varepsilon R(0) + R K ) \sin \delta > (K- \varepsilon) R(0) \sin \delta > 0, & \int_{I_\delta^-}\cos(\theta-\phi)\rho^{2}~ d\theta \leq -(\cos \delta) \Lambda(t). \end{aligned}
q^{{n}\choose{2}}C_n\left(\frac{1}{q},t\right)=\sum_{\lambda \in \mathcal{P}, \atop n_\lambda \leq n}q^{|\lambda|}t^{h_{1,1}(\lambda)}= \sum_{\lambda \in \mathcal{P},\atop \lambda_1+l(\lambda) \leq n}\prod_{i \geq 1}\left[\lambda_i-\lambda_{i+2}+1 \atop \lambda_{i+1}-\lambda_{i+2}\right]_t q^{{\lambda_1\choose 2}+|\lambda|}.
q^{{n}\choose{2}}C_n\left(\frac{1}{q}\right)=\sum_{\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k),\atop \lambda_1+k\leq n}\prod_{i \geq 1}{\lambda_i-\lambda_{i+2}+1 \choose \lambda_{i+1}-\lambda_{i+2}}q^{{\lambda_1\choose 2}+|\lambda|},
\widehat{{\cal M}}^{(B=0)}=\hat{V}^{(B=0)}(x,t)(\hat{V}^{B=0)})^{T}(x,{\frac{1}{t}})=\left(\begin{array}{cc}{\cal M}(\omega ) 0 0{\cal M}^{-1}(\omega )\end{array}\right)
\frac{\Sigma_n}{a^{m+1}}\,g_a^{m+1}=D_{\lambda+m+2}\left[B_0^{m+1}+\sum_{k=1}^m(B_k^{m+1}+C_k^{m+1})\cos^k\theta+C_{m+1}^{m+1}\cos^{m+1}\theta\right],
= \lim_{K\to \infty}\frac{1}{M}\sum_{n=1}^{N}{\beta_{ni}\textrm{tr}(\boldsymbol\Lambda)}&= \lim_{K\to \infty}\frac{1}{M}\sum_{n=1}^{N}{\beta_{ni}\left( \frac{M}{N}\right)}&= \overline{\beta_{i}},
2\varepsilon = \sum_{i=1}^N (t_y^{(i)}\cdot 1) \leq \left(\sum_{i=1}^N (t_y^{(i)})^q\right)^{1/q}\cdot N^{1 - \frac{1}{q}}\leq \left(\sum_{i=1}^N (t_y^{(i)})^{1+2p}\right)^{1/q}\cdot P^{1 - \frac{1}{q}},
d_{M,}= \liminf_{n\rightarrow \infty}E(v_n) \geq \liminf_{n\rightarrow \infty}\left(E(V) + E(r_n) \right) \geq E(V) + \liminf_{n\rightarrow \infty}E(r_n) &\geq E(V) \geq d_{M,}.
g_{-}^{- 1}( u ) E_{j j}^{( n )}g_{-}( u ) = g_{-}^{- 1}( u ) \left( \frac{\partial}{\partial u_{j}^{( n )}}g_{-}( u ) \right) + \left( \frac{\partial}{\partial u_{j}^{( n )}}g_{+}( u ) \right) g_{+}^{- 1}( u ) \, .
v'(T) =V(T) T^{1/(n-1)}-V(T)^{(n-1)/n}\left( \int_0^{T}V(t)^{(n-1)/n}\,dt \right)^{1/(n-1)}&\le V(T) T^{1/(n-1)}-V(T)^{(n-1)/n}\left( TV(T)^{(n-1)/n}\right)^{1/(n-1)}=0,
A_{( 0 )}= \frac{1}{R_{T}^{2}}\frac{( d - 3 )^{2}}{k_{d}^{\frac{2}{d - 3}}}\frac{F_{d - 2}^{2 \frac{d - 2}{d - 3}}}{( \partial_{r}F_{d - 2})^{2}+ ( \partial_{z}F_{d - 2})^{2}}
\bar x_{i,j-1}^+ :=-\bar{A}(a,\Delta t)+\sqrt{\bar{A}(a,\Delta t )^2+B(a,\Delta t)}\; =\Delta t \Bigg(-\frac{d_{i,j-1}}{4}+\sqrt{\frac{d_{i,j-1}^{2}}{16}+\bar c_{+}(i,a) \bar c_{-}(i,a)}\Bigg) &=:\Delta t\,\bar y_{i,j-1}(a).
(dL_{p})_{Id}\bigr(W_{1}\bigr)=[\tilde{Y}^{0},\,\tilde{Y}^{\bf i}]_{p}, &(dL_{p})_{Id}\bigr(W_{2}\bigr)=[\tilde{Y}^{0},\,\tilde{Y}^{\bf j}]_{p}, &(dL_{p})_{Id}\bigr(W_{3}\bigr)=[\tilde{Y}^{0},\,\tilde{Y}^{\bf k}]_{p},
T_2(g) :=(n-2)\sigma_1(A_g) g-8A_g, T_4(g) :=-\frac{3 n^2-12 n-4}{4}\sigma_1(A_g)^2 g+4(n-4)\|A_g\|^2 g+8(n-2) \sigma_1(A_g) A_g &+(n-6)\Delta_g \sigma_1(A_g) g+48A_g^2-\frac{16}{n-4}B_g, v_6(g) :=-\frac{1}{8}\sigma_3(A_g)-\frac{1}{24(n-4)}\langle B_g, A_g\rangle_g,
X_{N \times N}^{\mu}\ni \left( \begin{matrix}{. \qquad . \qquad .}&{.}\\{. \qquad . \qquad .}&{.}\\{. \quad X^{\mu}\quad .}&{| \phi^{\mu}\rangle}\\{. \qquad . \qquad .}&{.}\\{. \qquad . \qquad .}&{.}\\{. \quad \langle \phi^{\mu}| \quad .}&{x^{\mu}}\\ \end{matrix}\right)_{( N + 1 ) \times ( N + 1 )},
\sum_{k_3\in \mathbb{N}}\left(\hat{u}_3(k)\frac{k_3\pi}{h}\cos(\frac{\pi k_3x_3}{h})+2\pi i [\hat{u}_1(k)\frac{k_1}{\Pi_1}+\hat{u}_2(k)\frac{k_2}{\Pi_2}] \sin(\frac{\pi k_3x_3}{h})\right)=0,
h_{n-i}^n(t)= \frac{(-1)^n\sqrt{2n-2i+1}\,(n+1)}{(-n)_i}\prod_{m=0}^{\frac{i-3}{2}}(n-1-2m)^2 &\qquad\times\lim_{\epsilon\rightarrow0}{}_{i+2}F_{i+1}\left(\begin{matrix}-n,\; \alpha_1,\; \beta_1 1, \;\alpha_0,\; \beta_0 \end{matrix}\,; t\right),
\tilde{\mathcal Z}_Q = \mbox{span}\Bigl \{(2-N)x_j Q + |x|^2 \partial_{x_j}Q - 2 x_j x \cdot \nabla Q, \partial_{x_j}Q, 1 \leq j\leq N, x_j \partial_{x_k}Q - x_k \partial_{x_j}Q, 1 \leq j < k \leq N, \frac{N-2}{2}Q + x \cdot \nabla Q \Bigr \},
\begin{aligned}D^{t-1}-D^{t}= -(M(\breve{v}^{t}-v^*)+M(\breve{v}^{t-1}-v^*) )^THM(\breve{v}^{t}-\breve{v}^{t-1}) \leq \|M(\breve{v}^{t}-v^*)+M(\breve{v}^{t-1}-v^*)\|_H\|M(\breve{v}^{t}-\breve{v}^{t-1})\|_H \leq 2\sqrt{N_2}\sqrt{E^t}.\end{aligned}
\frac{{\lambda}^{2}}{1 6 \pi^{2}}\, \left({{\sigma}_{c}}\,{{{\phi}_{c}}}^{2}+ \frac{{{\sigma}_{c}}\,{{{\phi}_{c}}}^{2}}{2 \, \epsilon}+ \frac{{{{\phi}_{c}}}^{4}}{4}+ \frac{{{{\phi}_{c}}}^{4}}{4 \, \epsilon}+{{{\sigma}_{c}}}^{2}+ \frac{{{{\sigma}_{c}}}^{2}}{4 \, \epsilon}\right)
\begin{array}{c c c}{{\hat{g}_{I J}= \left( \begin{array}{c c}{{g_{\mu \nu}}}&{{g_{\mu \lambda}a_{j}^{\lambda}}}\\{{a_{i}^{\kappa}g_{\kappa \nu}}}&{{g_{i j}+ a_{i}^{\kappa}a_{j}^{\lambda}g_{\kappa \lambda}}}\\ \end{array}\right)}}&{{\ \ \mathrm{a n d}\ \}}&{{\hat{\omega}_{I J}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{\varepsilon_{i j}}}\\ \end{array}\right)}}\\ \end{array}
\bigg|\int fd\nu_N-\int fd\nu_{\infty}\bigg|\le \mathbb{E}||X_N-X_{\infty}|| \le \mathbb{E}|X|\bigg(\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k^{-\tau}}{n^{1/\gamma}}-\sum_{k=1}^{n}\frac{k^{-\tau}}{n^{1/\gamma}}\bigg),
\zeta_{ij}\left(\frac{z}{w}\right)= \left(\frac{zq^{-1}-w}{z-w}\right)^{\delta_{ij}}\prod_{e=\vec{ij}\in E}\left(\frac{1}{t_e}-\frac{z}{w}\right) \prod_{e=\vec{ji}\in E}\left(1-\frac{wt_e}{zq}\right) \,.
\delta \chi^{j}= i / \kern - . 5 e m J^{j}\epsilon - \frac 1 2 \varepsilon^{j k l}( \gamma_{p}\epsilon ~ \bar{\chi}^{k}\gamma_{p}\chi^{l}+ \gamma_{\mu}\epsilon ~ \bar{\chi}^{k}\gamma^{\mu}\chi^{l}) , \qquad \delta J_{\mu}^{j}= - \bar{\epsilon}( \partial_{\mu}\chi^{j}+ \varepsilon^{j k l}J_{\mu}^{k}\chi^{l}) ~ .
L^{\xi}= p\cdot\hat{H}+X^{\xi},\quad X^{\xi}=i\sum_{\rho\in\Delta_{+}}g_{|\rho|}\,(\rho\cdot\hat{H})\left(x(\rho\cdot q)-x(\rho^\vee\!\cdot\hat{H}\xi)\right)\hat{s}_{\rho},
g_{k,\ell}:=\begin{cases}\max(e^{\delta_n q_n}, |\ell|, 1) e^{-(\beta_n-6\varepsilon)q_n}\ \beta_n\geq \delta_n+200\varepsilon e^{2\varepsilon k}\beta_n<\delta_n+200\varepsilon\end{cases}
\mathcal D^{t}:= \Delta w^{t}= \frac 1 t \frac{\partial_N\,v}{N\cdot \nu^0}\, \mathcal{H}^{n-1}\restriction_{\Gamma^0}- \left(\frac{1}{t^2}\Delta h^0 + \frac{1}{t}\Delta \dot h^0\right) \chi_{\Omega^0 \setminus \Omega^t}- \frac 1 2 \Delta \delta_t^2 h^0 \chi_{\Omega^t}.
S^{env}_{s,t}= -\int_{s}^t 2\partial_{p_l}H (r,x_r) \gamma_{jl}(r,q_r)(\Sigma^{-1})^{jk}(r,q_r)\circ d(p_r)_k &+\int_{s}^t 2\partial_{p_l}H (r,x_r) \gamma_{jl}(r,q_r)(\Sigma^{-1})^{jk}(r,q_r)(- \nabla_q H+\tilde F)_k(r,x_r) -\nabla_p\cdot \tilde F(r,x_r)dr.
U_{b}^{a}= e_{\mu}^{a}( x ) \, \frac{\partial x^{\mu}}{\partial q^{\alpha}}\, e_{b}^{\alpha}( q ) = \left( \begin{array}{c c c}{{\cos \theta \cos \phi}}&{{- \sin \phi}}&{{\sin \theta \cos \phi}}\\{{\cos \theta \sin \phi}}&{{\cos \phi}}&{{\sin \theta \sin \phi}}\\{{- \sin \theta}}&{{0}}&{{\cos \theta}}\\ \end{array}\right) .
B_T: =(\hat \phi_{\max T \min T})_\ast(C_{\min T}|_{(\phi_{\max T\min T})^{-1}(\pi_{\max T \min T}(U_T)})) &\equiv (\tilde\pi_{\max T\min T}\circ\hat\pi_{\max T\min T})_\ast(C_T|_{U_T}).
D^{BS}_{2N}(\lambda) =(-2)^{N}(\lambda-\frac n2-2N)_N (2N-1)!!D_{2N}(n-\lambda), D^{BS}_{2N+1}(\lambda) = (-2)^{N+1}(\lambda-\frac n2-2N-1)_{N+1}(2N+1)!!D_{2N+1}(n-\lambda).
I(X^n;W,Z^n,F) &\; \overset{(a)}{\leq}n\big(I(X;Z|A,U)+I(X;A,V,Y)-I(X;Y|A,U) \big) &+n\big(I(V;Z|A,X,U)\!-\!I(V;Y|A,X,U)\big) &+n(I(U;Z|A) -I(U;Y|A)+\epsilon^{\prime\prime}) &\; \overset{(b)}{=}n\big(I(V,X;Z|A)+I(X;A,V,Y)-I(V,X;Y|A) \big) &+n\epsilon^{\prime\prime}
w(T) = D_{\hat n}= \frac{\sum_{j=1}^{n-1}D_{\hat j}}{n-2}= \frac{(n-1)w(T) - \sum_{1}^{n-1}w(\mbox{twig}_j)}{n-2}\geq w(T) + \frac{x}{n-2}> w(T)
\begin{array}{l l}{\partial X_{c l}( z ) = a \omega ( z ) ,}&{\partial \bar{X}_{c l}( z ) = \bar{a}\omega^{\prime}( z ) ,}\\{\bar{\partial}X_{c l}( \bar{z}) = b \bar{\omega}^{\prime}( \bar{z}) ,}&{\bar{\partial}\bar{X}_{c l}( \bar{z}) = \bar{b}\bar{\omega}( \bar{z}) ,}\\ \end{array}
\int_0^t \left(1+\tau\right)^{-d}e^{-\left(\beta(\tau)-\beta(t)\right) E_s (Z_s)}e^{-\left(\mu(\tau)-\mu(t)\right) s}d\tau \leq\frac{\sum_{k=0}^\infty r_k^{-1}\left(1+k\right)^{-d}}{s+\frac{\beta_0}{2}E_s(Z_s)}
u_{A,q}^s(x,y) =\frac{1}{4\pi}\frac{e^{ik\vert x\vert}}{\vert x\vert}\frac{e^{ik\vert y\vert}}{\vert y\vert}u^\infty_{-A,q}\left(\widehat{y}, -\widehat{x}\right)+ \frac{O(1)}{|x|^2}\|u_{A,q}(\cdot,y)\|_{L^2(D)}\cr + \frac{O(1)}{|x||y|^2}\|u_{-A,q}(\cdot,-\hat x)\|_{L^2(D)}.
\Phi u =e, \|u\|_{l_1^d}\le C_3 \|e\|_J\le C_3\,c\,\varepsilon\sqrt{m}, \|u\|_{l_2^d}\le C_4\sqrt{\log(d/m)/m}\|e\|_J\le C_4\,c\,\varepsilon\sqrt{\log(d/m)}.
r^{\prime}:=(E_{i_1^{\prime},j_1^{\prime}}\otimes d_1)\cdot\dots\cdot (E_{i_m^{\prime},j_m^{\prime}}\otimes d_m) =E_{i_1^{\prime}j_1^{\prime}}\cdots E_{i_m^{\prime}j_m^{\prime}}\otimes d_1\cdots d_m.
\sum_{i}q^{-(2\rho,\lambda_{i})}(\mathsf{M}_{m}^{n})_{i}^{i}=q^{-(2\rho,\lambda_{n})}\sum_{i}q^{(2\rho,\lambda_{n}-\lambda_{i})}u_{i}^{m*}u_{i}^{n}=\delta_{m}^{n}q^{-(2\rho,\lambda_{m})}.
\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{|\xi|\leq \lambda}d_\xi\smallint\limits_G\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx= \lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{|\xi|\leq \lambda}d_\xi^2 \smallint\limits_{G}\varkappa(x)dx=\smallint\limits_{G}\varkappa(x)dx.
\mathcal{I}_3(\bar X, t)= \int_{S^N(t)\setminus S^{N+1}(t)}dy\int dw\,\frac{f_0^N(y,w)}{\left|\bar{X}-Y^{N+1}(t)\right|^2}\leq &\frac{1}{[\delta^N(t)]^2}\int_{\left(S^{N+1}(t)\right)^c}dy\int dw\,f_0^N(y,w).
g(h,h') := \, \left(\partial_xf(h)-\frac{y-2y'}{2}\partial_zf(h)\right)^2 +\left(\partial_yf(h)+\frac{x-2x'}{2}\partial_zf(h)\right)^2 +\beta^2\left(\partial_z f(h)\right)^2 = \, ((X + y' Z)f(h))^{2}+ ((Y - x' Z)f(h))^{2}+ \beta^{2}(Zf(h))^{2}.