harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{F_nF_{n-2}}{F_{n-1}^2}=\frac{\alpha^2\left(\alpha(n-2)+\beta\right)^2(n-1)^2}{\left(\alpha(2n-2)+\beta\right)\left(\alpha(2n-3)+\beta\right)^2\left(\alpha(2n-4)+\beta\right)},
C_0 = \int_2^{x_1}\frac{720 - a(t)}{\log^7 t}\, t - 2\sum_{k=1}^5 \frac{k!}{\log^{k+1}2}, C_1 = \int_2^{x_1}\frac{720 + a(t)}{\log^7 t}\, t - 2\sum_{k=1}^5 \frac{k!}{\log^{k+1}2}.
\bar{d}^{k+1}_t-\Delta \bar{d}^{k+1}=\nabla \bar{d}^k \cdot (\nabla d^k+\nabla d^{k-1})\cdot d^k+|\nabla d^{k-1}|^2 \bar{d}^k -(\bar{u}^k \cdot \nabla) d^k-(u^{k-1}\cdot \nabla)\bar{d}^k.
i \frac{\partial}{\partial t}\langle \vec{q}^{\prime}, t \mid \vec{q}, 0 \rangle = - \frac{1}{2}\sum_{i = 1}^{3}\frac{\partial^{2}}{\partial q_{i}^{2}}\langle \vec{q}^{\prime}, t \mid \vec{q}, 0 \rangle + V ( \vec{q}^{\prime}) \langle \vec{q}^{\prime}, t \mid \vec{q}, 0 \rangle
\int_0^{\infty}t^{-1-\alpha/2}\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}f\|^2_{L^2(\mathbb{R}^n, \mu_2)}dt \leq C'\int_0^{\infty}t^{-1-\alpha/2}\sum_{j\geq 0}\|g_0^{j,t}\|^2_{L^2(C^{j,t}_0, \mu_2)}dt + &C'\int_0^{\infty}t^{-1-\alpha/2}\sum_{k\geq 1}e^{-c 2^k}\sum_{j\geq 0}\|g_k^{j,t}\|^2_{L^2(C^{j,t}_k, \mu_2)}dt.
X_{I}^{\mu}= \frac{i}{2 \sqrt{\pi T_{I}}}\sum_{n \neq 0}\frac{1}{n}\left[ \alpha_{n}^{\mu}( s ) e^{i ( 1 + s ) n ( \sigma - \tau )}+ \tilde{\alpha}_{n}^{\mu}( s ) e^{- i ( 1 + s ) n ( \sigma + \tau )}\right] ,
\begin{aligned}\big (\mathbf{S}( \mathbf{D}_1,\mathbf{R}_1,\mathbf{E})- \mathbf{S}(\mathbf{D}_2,\mathbf{R}_2,\mathbf{E})\big ): \big (\mathbf{D}_1-\mathbf{D}_2+\mathbf{R}_1-\mathbf{R}_2\big )>0\,. \end{aligned}
{_aD_x^{\alpha}}f(x) = \frac{d^m}{d x^m}\left[{_aD_x^{-(m-\alpha)}}f(x) \right] &=\frac{1}{\Gamma(m - \alpha)}\frac{d^m}{d x^m}\int_a^x f(t) (x - t)^{m-\alpha-1}dt.
{{\cal R}^{\hat{t}\hat{\imath}}}_{\hat{t}\hat{\imath}}= -{\frac{1}{G^{2}}}{( \ln B )^{\prime}}^{2}= -{\frac{( d - 2 )^{2}Q^{2}}{2 \Delta ( D - 2 )}}\left[ r^{d - 2}+{\frac{1}{d - 2}}\sqrt{\frac{\Delta}{2 ( D - 2 )}}| Q | \right]^{- 2 \left( 1 +{\frac{p + 1}{\Delta}}\right)}r^{- a^{2}{\frac{D - 2}{\Delta}}}.
\omega((x,y),(u,v)\bigr) := \Bigg( \begin{pmatrix}B_- 0 0 -B_+\end{pmatrix}\begin{pmatrix}x y\end{pmatrix}\Bigg| \begin{pmatrix}u v\end{pmatrix}\Bigg)_{\mathcal{G}_-\oplus \mathcal{G}_+},
\sum_{k_1,k_2=-S}^S \bigl\{S,S,k_1,k_2 \vert J_1,m_1 \bigr\}_q \,\bigl\{S,S,k_1,k_2 \vert J_2,m_2 \bigr\}_q= \delta_{J_1,J_2}\, \delta_{m_1,m_2}\,.
\nu =\begin{cases}(-1)^{2d}=1=(-1)^{n}(-1)^{2d-k}=(-1)^{-k}=-(-1)^{n}(-1)^{2d-k+1}=(-1)^{-k+1}=-(-1)^{n}(-1)^{0}=1=(-1)^{n}\end{cases}
[\rho(f \bigotimes v) \varphi]_{n}({\bf{x_{1}}}\sigma_{1},{\bf{x_{2}}}\sigma_{2}, .... ,{\bf{x_{n}}}\sigma_{n}) = \sum_{i=1}^{n}f({\bf{x_{i}}}) a(\sigma_{i}) \varphi_{n}({\bf{x_{1}}}\sigma_{1},{\bf{x_{2}}}\sigma_{2}, .... ,{\bf{x_{n}}}\sigma_{n})
\lambda^*\left(s_{0}\right) = &\inf_{\pi \in \Pi}\limsup_{n \rightarrow \infty}\mathrm{E}_\nu\left[\frac{1}{n}\sum_{t=1}^n \ell\left(s_t, w_t \right) \mid s_{1}^{t-1}, w_{1}^{t-1}\right],
P'_{R}=(\;\;\;{\rho}\frac{\breve m}{2}\;\;\; +(\;\;G_{\bot}\;\;-\;\;B_{\bot}+ \frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\breve n}- \;\;\frac{1}{2}A_{\bot}^{t}C_{\bot}{\breve l}, 0 ) = (p'_{R},\;0 ).
\frac{w}{z(w+1)}\frac{1}{F_{12}}= -\frac{1}{F_1F_2}- \frac{1}{z^2(w+1)}+ \frac{1}{z}\cdot \frac{1}{F_1}+ \frac{1}{z}\cdot \frac{1}{F_2}&= -\left( \frac{1}{F_1}- \frac{1}{z}\right)\left( \frac{1}{F_2}- \frac{1}{z}\right) + \frac{w}{z^2(w+1)}\,.
d_i\Delta^{n-i}e_i =\Delta^{n-i}\phi^{n-i}(d_i)e_i (\;d_i\Delta=\Delta\phi(d_i)) &=\Delta^{n-i}\phi^{n-i}(d_i)\left(\phi^{n-i}(d_i)\backslash \Delta\right) (\;e_i=\phi^{n-i}(d_i)\backslash \Delta) &=\Delta^{n-i+1}(\;\phi^{n-i}(d_i)\left(\phi^{n-i}(d_i)\backslash \Delta\right) =\Delta).
G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|wz)=w^{b_{q}}\sum_{h=0}^{\infty}\frac{(w-1)^{h}}{h!}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{1},...,b_{q-1},b_{q}+h\end{matrix}|z),m<q,
- \int_1^\infty\frac{1-x^{-2N-2}}{1-x^{-2}}x^{-a+ib}(g(x)+g^*(x)-\frac{2g(1)}{x^{2-a+ib}}\big)\frac{dx}{x}+(\log N-\sum_{m=1}^{N+1}\frac{1}{m})f(1) + O\big(\frac{1}{N}\big).
\Gamma_{inv}= \int \left\{-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}(D_{\mu}\underline \varphi ) (D^{\mu}\underline \varphi) - \frac{1}{8}\frac{m_H^2}{m^2}e^2 (\varphi_1^2 + 2\frac me \varphi_1 +\varphi_2^2 )^2 \right\}
2 \delta \sqrt{2m}\geq \|H^\dagger \Phi (x-\hat x)\|_2 = \|V^T \Sigma^{-1}U\Phi (x-\hat x)\|_2\geq \|\Sigma^{-1}U\Phi (x-\hat x))\|_2 \geq \|\Sigma^{-1}_\ell U_\ell \Phi_\ell (x-\hat x)\|_2 \geq \sigma_\ell(H^\dagger) \| U_\ell \Phi_\ell (x-\hat x)\|_2 .
H = \frac{1}{2 M}: \left( \hat{P}- \sum_{n , m = 1}^{\infty}\hbar g_{m n}\hat{b}_{n}^{\dag}\hat{b}_{n}\right)^{2}: + \sum_{n = 1}^{\infty}\hbar \Omega_{n}\hat{b}_{n}^{\dag}\hat{b}_{n}
\int_M \nabla_gu(x)\nabla_gw(x){\rm d}v_g -\mu\int_M\frac{u(x)w(x)}{d^2_g(x_0,x)}{\rm d}v_g+\int_Mu(x)w(x){\rm d}v_g=\lambda\int_M \alpha(x)\xi_x w(x){\rm d}v_g.
N(\sigma_k) = \{e_i - e_j : \underline x_k \leq \overline x_k + 1 - \mu \leq i \leq \overline x_k < j < \underline y_k\}\cup \{e_i - e_j : \overline x_k < i < \underline y_k \leq j \leq \underline y_k + \mu - 1 \leq \overline y_k\}.
\delta_{l s}\tilde{E}^{a}= - 2 i \tilde{E}^{\alpha}\Gamma_{\alpha \beta}^{a}\epsilon^{\beta}( x , \tilde{\theta}( x ) ) , \quad \delta_{l s}\tilde{E}^{\alpha}={\cal D}\epsilon^{\alpha}( x , \tilde{\theta}( x ) ) + \ldots \quad \ldots
\sum_{n=1}^2 \frac{1}{2^n}(I_2 \otimes \cdots \otimes I_2 \otimes A \otimes I_2 \otimes \cdots) = \mathrm{diag}\left(\frac{3}{4}, \frac{1}{4}, -\frac{1}{4}, -\frac{3}{4}\right).
g_{ij}=\begin{pmatrix}1+\lambda(u^2)^2+\mu(u^3)^2 -\lambda(u^3+u^1 u^2) -\lambda u^2-\mu u^1u^3 -\lambda(u^3+u^1 u^2) \lambda (u^1)^2+\mu 2\lambda u^1 -\lambda u^2-\mu u^1 u^3 2\lambda u^1 \lambda+\mu(u^1)^2\end{pmatrix}, w_{12}=\sqrt{\frac{\lambda(\mu^2-\lambda^2)}{\det g}}u^1,w_{23}=0,w_{31}=\sqrt{\frac{\lambda(\mu^2-\lambda^2)}{\det g}},
\Big|\frac{1}{2 \varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega - \Omega_1(x)}\overleftarrow{\eta}^{\ell}(x) [\eta(x+1)-\eta(x+1+r) ] [f(\eta)-f(\eta^{x+1,x+1+r})] d\nu_{b}\Big|
\xi_1 = \pi_u(\zeta_1)\pi_v(\zeta_1), \ \xi_2 = \pi_u(\zeta_1)\pi_v(\zeta_2), \xi_3 = \pi_u(\zeta_2)\pi_v(\zeta_1), \ \xi_4 = \pi_u(\zeta_2)\pi_v(\zeta_2).
a^{11}=N^2,a^{12}=-MN, a^{22}=LN, &{b}^2=2B^2K_t+3BB_tK-\frac{5B_t}{B}M^2+\frac{B_s}{B}MN+\frac12MN^3\partial_s\left(\frac{1}{N^2(s,0)}\right),
\begin{aligned}{\rm Bias}(\hat D_l) = \sum_{j\in J}\gamma_j\varphi_j(l)\varGamma^{-j/2d}+ o\left(\varGamma^{-1/2}\right), \end{aligned}\begin{aligned}{\rm Var}(\hat D_l) =O\left(\varGamma^{-1}\right). \end{aligned}
P_x=\frac{2}{\pi\alpha'\sqrt{-K}(1-KL^2)}\left\{-\frac{2}{3}+\frac{1}{KL^2}-\frac{1-KL^2}{KL^3\sqrt{-K}}[\frac{\pi}{2}-\arcsin\frac{1}{\sqrt{1-KL^2}}]\right\},
\phi^{-1}(\boldsymbol{e}_{12}(a)) =\mathbf{f}(a\oplus0), \phi^{-1}(\boldsymbol{e}_{13}(a)) =-\mathbf{g}(0\oplus\rho(a)), \phi^{-1}(\boldsymbol{e}_{21}(a)) =\mathbf{g}(a\oplus0), \phi^{-1}(\boldsymbol{e}_{23}(a)) =\mathbf{v}(a\oplus0), \phi^{-1}(\boldsymbol{e}_{31}(a)) =\mathbf{f}(0\oplus\rho(a)), \phi^{-1}(\boldsymbol{e}_{32}(a)) =\mathbf{w}(a\oplus0),
-\overline{R}_n = \overline{\Delta}_{n,k}- \sum_{j=1}^k \overline{U}_j \,, \Delta_{n,k}=\sum_{u=1}^p \sum_{j=1}^{2^{u-1}}V_{(2j-2) n_u', (2j-1) n_u' ,2j n_u'}\,,
L(g, s) = L\big(\psi^2, s-1\big) = \prod_{v \ne \mathfrak P, \infty}\frac{1}{1 - (a-b{\rm i})^2 Nv^{-s}}= \sum_{\mathfrak I \in S_1}\tilde \psi(\mathfrak I)^2 (N \mathfrak I)^{-s}
\mathrm{IV}=\sum_{(j,k,n)\in \pi_3}\frac{1}{8}(R_jR_k\Gamma_{jk}^nR_n+R_k\Gamma_{jk}^nR_nR_j) &=\frac{1}{4}\sum_{(j,k,n)\in \pi_3}\Gamma_{jk}^nR_jR_kR_n &=\frac{1}{2}\sum_{(j,k,n)\in \pi_3\ \mathrm{is\even}}\Gamma_{jk}^nR_jR_kR_n &=-\frac{\Gamma_{12}^3+\Gamma_{23}^1+\Gamma_{31}^2}{2},
y\binom{r-1}{2}+ x \left(\binom{r}{2}-k\right) = \frac{2m-xr}{r-1}(r-1)(r-2)/2 - x\left(k - \frac{r(r-1)}{2}\right) &= (r-2)m - \frac{x}{2}(2k-r) &\le (r-2)m
| \mathcal{T}( T ) \rangle_{\alpha_{- 1}\cdot \tilde{\alpha}_{- 1}\tilde{c}_{1}c_{1}| 0 \rangle}\sim \frac{1}{T^{2}}\, \, U_{f ( v )}^{\dagger}c ( i ) \partial X^{\mu}( i ) \, \tilde{c}( i ) \bar{\partial}X_{\mu}( i ) | 0 \rangle ,
\prod_{i=1}^d (1 - \lambda_i x_i)^{-k_i}= \sum_{n_1,\ldots,n_d=0}^\infty{\binom{k_1+n_1-1}{n_1}}\cdots{\binom{k_d+n_d-1}{n_d}}\lambda_1^{n_1}\cdots \lambda_d^{n_d}x_1^{n_1}\cdots x_d^{n_d},
y(t) := \int (\partial_x^lu)^2\chi(x+\nu t) \; dx+ \int_0^t\int (\partial_x^{l+2}u)^2 \chi'(x+\nu\tau) \; dxd\tau &\leq c(\nu;8(l-5)) + \int_0^t(c_0+c_1\|\partial_x^3u(\tau)\|_{L_x^\infty})\int(\partial_x^lu)^2\chi(x+\nu\tau) \; dxd\tau &\leq c(\nu;8(l-5)) + \int_0^t(c_0+c_1\|\partial_x^3u(\tau)\|_{L_x^\infty})y(\tau)\;d\tau.
\overline{u}e_{i(m)}e_{-i(m)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}=\overline{s_{i(1)}\dots s_{i(m-1)}}e_{i(m)}^{-1}e_{i(m-1)}e_{-i(m-1)}^{-1}\dots e_{i(1)}e_{-i(1)}^{-1}.
\prod_{j=1}^{i-1}(1-\beta t_j) \left( \frac{1-\beta T}{2-\beta T}(1-\beta T)^{-\lambda_i}\left[ c(i)\prod_{j=1}^{i-1}\frac{1-t_j}{1+t_j-\beta t_j}\right]_{\lambda_i}+ \frac{1}{2}(-1)^i e(i) \right).
<\exp\{i\sqrt{\pi}[\phi (z_1)+\phi (z_2)-\phi(z_3^*)-\phi (z_4^*)]\}> \propto \left[{(z_1-z_2)(z_3^*-z_4^*)\over(z_1-z_3^*)(z_1-z_4^*)(z_2-z_3^*)(z_2-z_4^*)}\right]^{1/4}
-2\lambda^{2m+1}mT_{-m,0}-3m\frac{\partial T_{1,0}}{\partial M_0}\lambda^mM_0-3m\frac{\partial T_{-1,0}}{\partial M_0}\lambda^{m+2}M_0+\lambda^mT_{1,0}-\lambda^{m+2}T_{-1,0}=(-m+1)\lambda^{2m}T_{-m+1,0}+(-m-1)T_{-m-1,0}\lambda^{2m+2}.
-\dfrac{Q^{(n-3)}(0)}{\left( n+1\right) (n-3)!}=(n-2)\left(n^{2}-13n+6\right) +\dfrac{1}{2}(n+1)n(n-1)(n-2) =\dfrac{1}{2}\left( n-2\right) \left( n-4\right) \left( n^{2}+6n-3\right) >0
(\lambda^k+\epsilon_2-\delta_n,\lambda^k+\epsilon_2-\delta_n+2\rho) = 2k(k+m-n-1)+2(m-n-2), (\lambda^k+\epsilon_1-\epsilon_{m-1},\lambda^k+\epsilon_1-\epsilon_{m-1}+2\rho) = 2k(k+m-n)+2(m-1), (\lambda^k+\epsilon_2-\epsilon_m,\lambda^k+\epsilon_2-\epsilon_m+2\rho) = 2k(k+m-n)+2(m-n-1).
\sum_{k=1-\epsilon_0(\Gamma)}^{d-\ell(\Gamma)+\epsilon_1(\Gamma)}\Phi_{s(c,m,d)}(\Gamma_{(k)}) = \sum_{k=1-\epsilon_0(\Gamma)}^{d-\ell(\Gamma)+\epsilon_1(\Gamma)}\big(\alpha+\beta (c+ km) +\gamma m\big) = M_1(\alpha+\beta c+\gamma m)+\frac{M_1M_2}{2}\beta m,
\begin{aligned}F(z) \geq F_t(z;x)-\tfrac{\mu + t^{-1}}{2}\|z-x\|^2\geq F_{t}({\bar x};x)+\tfrac{1}{2t}\|{\bar x}-z\|^2-\tfrac{\mu + t^{-1}}{2}\|z-x\|^2 &\geq F({\bar x})+\frac{1}{2t}\|{\bar x}-z\|^2-\tfrac{\mu + t^{-1}}{2}\|z-x\|^2 + \tfrac{t^{-1}-\mu}{2}\|{\bar x}-x \|^2,\end{aligned}
\begin{bmatrix}1 \alpha' \gamma' 1 \beta' 1 \end{bmatrix}\mapsto\begin{bmatrix}1 \overline{\alpha'\alpha+\beta'\gamma}\overline{(\alpha\delta-\beta\gamma)\gamma'+\tau(\alpha',\beta')}1 \overline{\alpha'\beta+\beta'\delta}1 \end{bmatrix}.
N^Q_{J,K}(\varepsilon):= \#\Big\{k\in\mathfrak{Z}^{Q}_{J,K}~:~ |a_{j,\ell,k}|>\varepsilon \Big\}\lesssim \min\Big\{\ell_J , \varepsilon^{-2}L_K^{-2d}2^{-js(1+\alpha)}\ell_J^{-2\beta-1}\Big\}.
\delta\dfrac{dx}{dt}-\dfrac{d}{dt}\delta\,x=0,\quad\delta\dfrac{dy}{dt}-\dfrac{d}{dt}\delta\,y=0,\delta\dfrac{dz}{dt}-\dfrac{d}{dt}\delta\,z=q\left(\dot{y}\delta\,x-\dot{x}\delta\,y\right).
{\bf w}_t^{(i)}={\bf w}_{\Delta}^{(i)}\frac{t}{\Delta}+\frac{1}{2}a_{i,0}+\sum_{r=1}^{\infty}\left(a_{i,r}{\rm cos}\frac{2\pi rt}{\Delta}+b_{i,r}{\rm sin}\frac{2\pi rt}{\Delta}\right),
\begin{gathered}\sum_{i,j}(\partial_i \partial_j g_{\nu (t-s)}* ) (\eta_i (s) - \eta_i (t) ) \sigma_{jk}(s), \sum_{i,j}\eta_i (t) \left ( \partial_i \partial_j g_{\nu (t-s)}\right )* \sigma_{jk}(s) , \,\, \mathrm{and \;respectively}\sum_{i,j}\left ( \partial_i \partial_j g_{\nu (t-s)}\right ) * (\eta_i (s) \sigma_{jk}(s) ).\end{gathered}
\{((k-2)e+1,(k-1)e+1)\cdots ((k-1)e,ke)\}\{(-e+1, (k-1)e+1)\cdots (0,(k-1)e)\}& \{((k-2)e+1,(k-1)e+1)\cdots ((k-1)e,ke)\}= (-e+1, ke+1)\cdots (0,ke),
\varphi^{2,1}(A_{0,0}) = (0,0), \varphi^{2,1}(B_{0,0}) = \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right), \varphi^{2,1}(C_{0,0}) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right), \varphi^{2,1}(A_{0,1}) = (0,0), \varphi^{2,1}(B_{0,1}) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right), \varphi^{2,1}(C_{0,1}) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right).
(h\boxtimes T)(\lambda u) =\sum_{i=1}^n\delta_{(h(x_i),h(y_i))}\boxtimes T(\lambda e_i) &=\sum_{i=1}^n\delta_{(h(x_i),h(y_i))}\boxtimes \lambda T(e_i) &=\lambda\sum_{i=1}^n\delta_{(h(x_i),h(y_i))}\boxtimes T (e_i) &=\lambda(h\boxtimes T)(u).
T_{i}T_{i+1}\cdots T_{j-1}\big(x_{j}^+\big) = E_{i,j+1},T_{i}T_{i+1}\cdots T_{j-1}\big(x_{j}^-\big) = E_{j+1,i}, T_{j}T_{j-1}\cdots T_{i+1}\big(x_{i}^+\big) = (-1)^{j-i}E_{i,j+1},T_{j}T_{j-1}\cdots T_{i+1}\big(x_{i}^-\big) = (-1)^{j-i}E_{j+1,i}.
E_{\mathbb M_{m_1}(M)\overline{\otimes}\psi_2(A)}(a\psi(u_n)b) = (\psi_1\otimes\tau)((1\otimes\xi)u_n(1\otimes\zeta))\psi_2(p) &=((\psi_1\otimes\tau)\circ E_{P_1\overline{\otimes}P_2})((1\otimes\xi)u_n(1\otimes\zeta))\psi_2(p) &=\psi_1(E_{P_1}(u_nE_{P_2}(\zeta\xi)))\psi_2(p),
\sum_{j = 0}^k{j^5 G_j}= 2671G_0 + 4322G_1 + (k^5 - 5k^4 + 50k^3 - 310k^2 + 1285k - 2671)G_k &\qquad+ (k^5 - 10k^4 + 80k^3 - 500k^2 + 2080k - 4322)G_{k + 1}\,,
\frac{1}{N + 1}{L_{N}^{1}}( x ) = \frac{1}{2 \pi i}\int_{{\cal C}_{0}}\frac{d z}{z^{N + 1}}\exp \Biggl [ \sum_{k = 1}^{+ \infty}L_{k}^{0}( \frac{k x}{N + 1}) \frac{z^{k}}{k}\Biggr ] ,
\bar\sigma^1(x)=\overline{\mathcal{A}}^1(x)+TR_{\bar{s}(x)^{-1}}\dot{\bar{s}}(x),\bar\sigma^2(x)=\overline{\mathcal{A}}^2(x)+TR_{\bar{s}(x)^{-1}}\mathbf{d}\bar{s}(x)
-\frac{\varphi(z_n) - \varphi(\overline z_n)}{z_n-\overline z_n}= -\frac{\varphi_n(z_n) - \varphi_n(\overline z_n)}{z_n-\overline z_n}-\sum_{m \ne n}\frac{\varphi_m(z_n) - \varphi_m(\overline z_n)}{z_n-\overline z_n}=\rho^2_n(\eta)+O(1)
\int_{\bar{a}_n}^{\bar{b}_n}\left(\frac{d\varphi}{d\bar{\ss}_n}\right)^2d\bar{\ss}_n=\sum_{j}\int_{I_{k_j}}\left(\frac{d\varphi}{d\ss_{k_j}}\right)^2d\ss_{k_j}=\sum_{j}\int_{I_{k_j}}\left(\frac{d\varphi}{d\bar{\ss}_n}\right)^2d\bar{\ss}_n.
\|w\|_{L^\infty(\Omega)}\leq \delta\int_\Omega \gamma(v)u^2dx+C_{\delta}\left(\int_\Omega \gamma^{-\frac{p}{2-p}}(v)dx\right)^{\frac{2-p}{p}}\leq \delta\int_\Omega \gamma(v)u^2dx+C_{\delta}\left(\int_\Omega w^{\frac{pk}{2-p}}dx\right)^{\frac{2-p}{p}}+C_{\delta}^\prime.
d s^{2}= \exp^{- 2 A_{0}( y )}\left( 1 + 2 f ( t , y ) \right) d t^{2}- a ( t )^{2}\exp^{- 2 A_{0}( y )}\left( 1 + 2 g ( t , y ) \right) d \vec{x}^{2}- b_{0}^{2}d y^{2},
\begin{cases}u''-\Delta u=0 \hbox{in}(0, +\infty)\times \Omega, v''-\Delta v +u=0 \hbox{in}(0, +\infty)\times \Omega, u=v=0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu u=\partial_\nu v=0 \hbox{on}(0, +\infty)\times \Gamma_1\end{cases}
Q ( \Phi_{0})_{i j}= Q \left( \sum_{k}( e^{- \Psi_{0}})_{i k}\star Q ( e^{\Psi_{0}})_{k j}\right) = - \sum_{k , l , m}( e^{- \Psi_{0}})_{i k}\star Q ( e^{\Psi_{0}})_{k l}\star ( e^{- \Psi_{0}})_{l m}\star Q ( e^{- \Psi_{0}})_{m j}
= \frac{1}{2}+ \frac{1}{3}- ( \frac{1}{3}+ \frac{1}{4}) + \frac{1}{4}+ \frac{1}{5}- ( \frac{1}{5}+ \frac{1}{6}) + \frac{1}{6}+ \frac{1}{7}- ( \frac{1}{7}+ \frac{1}{8})
\mathbf{P}_{\beta, T}(d\omega)=\frac{\exp\{-\beta H_T(\omega)\}}{Z_{\beta,T}}\mathbf{P}_T(d\omega) =\frac{\exp\{\beta \int_0^Tv(\omega(t))dt\}}{Z_{\beta,T}}\mathbf{P}_T(d\omega),
\frac{e^{- 4 \pi \tau}}{1 - e^{- 4 \pi \tau}}\sim - \frac 1 4 \frac{q^{2}}{p^{( 2 ) 2}}, \frac{e^{- 4 \pi l^{\prime}}}{1 - e^{- 4 \pi l^{\prime}}}\sim - \frac 1 4 \frac{k^{2}}{p^{( 1 ) 2}}.
I(s,f^{(n)},\chi):=\int\limits_{O_v^{\times 2}}\chi(ac ((y^3-x^2)^2+\pi^{8n}x^4y^4))\ |(y^3-x^2)^2+\pi^{8n}x^4y^4|^s\ |dxdy| =\int\limits_{O_v^{\times 2}}\chi(ac (x^{12}y^4\widetilde{f^{(n)}}(x,y)))\ |\widetilde{f^{(n)}}(x,y)|^s\ |dxdy|.
{\mathbb E}( \|{\mathbb E}_0(Y_k)\|_H) ={\mathbb E}\Big (\Big(\int_0^1 (F_{X_k|X_0}(t)-F(t))^2 dt \Big)^{1/2}\Big) &\leq{\mathbb E}\Big( \sup_{t \in [0,1]}|F_{X_k|X_0}(t)-F(t)|\Big):=\beta(k).
-\int_{Y^m}\left({\bf R}{\bf R}^T \nabla_y \right) \cdot \vec{\phi}(\vec{y}) \;{\theta}(\vec{y}) \;\mathrm{d}\vec{y}= \int_{Y^m}\vec{\phi}(\vec{y}) \cdot \left({\bf R}{\bf R}^T \nabla_y \right) \;{\theta}(\vec{y}) \;\mathrm{d}\vec{y}
\boldsymbol{\Phi}(\mathbf{a},t)=\mathbf{a}, \,\mathbf{U}(\mathbf{a},t)=\mathbf{0}, \, \mathbf{R}(\mathbf{a},t)\mathbf{=I}, \, \mathbf{W}(\mathbf{a},t)\mathbf{=I,\}, \, \mathbf{Z}(\mathbf{a},t)=\mathbf{0.}
{\frac{1}{\Theta}}\left({\frac{\partial^{2}\Theta}{\partial \theta^{2}}}+{\frac{\cos \theta}{\sin \theta}}{\frac{\partial \Theta}{\partial \theta}}\right) +{\frac{1}{\sin^{2}\theta}}{\frac{1}{\Phi}}{\frac{\partial^{2}\Phi}{\partial \varphi^{2}}}\equiv - l ( l + 1 ) ,
F(\tau,x) :=\frac{1}{4\pi}\Big(\lvert x-e\rvert V(x,2\tau-\lvert x\rvert)+\lvert x\rvert v_{2}(x,2\tau-\lvert x-e\rvert) &\ \ \ \ \ \ +4\pi\lvert x\rvert\lvert x-e\rvert\int\limits_{\lvert x\rvert}^{2\tau-\lvert x-e\rvert}V(x,2\tau-t)v_{2}(x,t)dt\Big)
\begin{aligned}\sum_{j=2}^{k}\left\{m_{l}\hat{\theta}_{j}\right\}=\sum_{j=2}^{k_1}\left\{m_{l}\hat{\theta}_{j}\right\}+\sum_{j=k_{1}+1}^{k}\left\{m_{l}\hat{\theta}_{j}\right\}&=\sum_{j=2}^{k_1}\left\{p_{j}\{m_{l}\theta\}+\xi_{j}\right\}+\sum_{j=k_{1}+1}^{k}\left\{p_{j}\{m_{l}\theta\}\right\}. \end{aligned}
\delta x^{\mu}={\dot{x}}^{\mu}\xi\;,\; \delta e = \frac{d}{d\tau}\left(e \xi\right)\;,\; \delta f_{ab}= \frac{d}{d\tau}\left(f_{ab}\xi\right)\;,\;\; \delta \psi_{a}^{n}= \dot{\psi}_{a}^{n}\xi\;,\;\; \delta{\chi}_{a}= \frac{d}{d\tau}\left(\chi_a\xi\right)\;,
\|Y_0^T\| \leq (L_1 T+L_1 T \|W_0^T\| + \|B_0^T\|) e^{\int_0^T L_1 dt}& = e^{L_1 T}(L_1 T+L_1 T \|W_0^T\| + \|B_0^T\|) & = L_2 + L_2 \|W_0^T\| + L_2 \|B_0^T\|.
\xi^{h}_{\ell,t}=\begin{cases}\frac{1}{t^{h}_{\ell,s^*}-\tau(t^h_{\ell,s^*})}, \tau(t^{h}_{\ell,s^*}) \leq t < t^{h}_{\ell,s^*}t^{h}_{\ell,s^*}=\min\limits_{\begin{subarray}~k''\in\mathbb{N}_+\end{subarray}}\{t^{h}_{\ell,k''}|t^{h}_{\ell,k''}> t\}, 0, ,\end{cases}
\langle \prod_{i = 1}^{N - 2 + ( N - k ) d}{\cal O}_{e}( z_{i}) \rangle = \int_{{\cal M}_{0 , d}^{M_{N}^{k}}}\bigwedge_{i = 1}^{N - 2 + ( N - k ) d}\phi_{i}^{*}( e ) = \int_{{\cal M}_{0 , d}^{C P^{N - 1}}}c_{( N , k )}\wedge ( \bigwedge_{i = 1}^{N - 2 + ( N - k ) d}\varphi_{i}^{*}( e ) )
\langle 0 p_{1}p_{2}p_{3}p_{4}\rangle ={\frac{\xi_{1 2 3}^{2}\xi_{1 2 4}^{2}}{( 1 2 )^{2}}}{\cal F}^{p_{1}- 2 \vert p_{2}- 2 \vert p_{3}- 2 \vert p_{4}- 2}( x , u ) + O ( \theta^{5}\bar{\theta}) \; .
{\mathfrak g}_{\ell}^+ = \langle H, P_0, P_1, \dots, P_{\ell-1}\rangle,{\mathfrak g}_{\ell}^0 = \langle D, P_{\ell}\rangle,{\mathfrak g}_{\ell}^- = \langle C, P_{\ell+1}, P_{\ell+2}, \dots, P_{2\ell}\rangle.
Z_- = q^{2\theta(i)}\cdot X_1X_2 \dots X_{2\theta(i)+1}\cdot Y_1, &Z_0 = q^{-2n}\cdot X_{2\theta(i)}^{-1}\dots X_2^{-1}X_1^{-1}\cdot Y_{2i}^{-1}\dots Y_2^{-1}Y_1^{-1}, &Z_+ = q^{2i}\cdot X_1 \cdot Y_1Y_2 \dots Y_{2i+1}.
p_k(s,a;r) =\frac{1}{g(s,a;r)}\frac{\partial}{\partial a}\big(g(s,a;r)p_{k-1}(s,a;r)\big) =... = \frac{1}{g(s,a;r)}\frac{\partial^l}{\partial a^l}(g(s,a;r) p_{k-l}(s,a;r)).
\pi_{TM}({\bf x},{\bf y})=\left(\begin{array}{ccc|ccc}0 0 0 0 0 0 0 0 0 0 0 x^1 0 0 0 0 -x^1 0 \hline 0 0 0 0 0 0 0 0 x^1 0 0 y^1 0 -x^1 0 0 -y^1 0\end{array}\right),
\underline{\underline{\epsilon}}=[\begin{matrix}\epsilon_{1}&0&0\\ 0&\epsilon_{2}&0\\ 0&0&\epsilon_{3}\end{matrix}]\Rightarrow\epsilon=\epsilon_{1}n_{1}\otimes n_{1}+\epsilon_{2}n_{2}\otimes n_{2}+\epsilon_{3}n_{3}\otimes n_{3}
-\mathfrak{d}h=\omega^{\sharp}\circ d^{G\backslash}\left( -h\right) :\mathfrak{\mathfrak{g}}^{\ast}\mathfrak{\rightarrow z}_{d}:\mu\rightarrow\left( ad_{\frac{\delta h}{\delta \mu}}^{\ast}\mu ,\mu ,\frac{\delta h}{\delta \mu}\right) .
\mathcal{Q}_{t}(x_{t-1}, \xi_{[t-1]}, D_{t-1})=\mathbb{E}_{\xi_t, D_t}\Big[ \mathfrak{Q}_t ( x_{t-1}, \xi_{[t-1]}, D_{t-1}, \xi_t , D_t ) | D_{t-1}, \xi_{[t-1]}\Big],
\delta E = \sum_{i}\tilde{\epsilon}_i \tilde{N}_i + \frac{2\pi}{L}\sum_i \frac{\tilde{v}_i}{2}\left[\left(\sum_j [Z^{-1}]_{ij}\tilde{N}_j\right)^2 +\left( \sum_j Z_{ji}\tilde{D}_j\right)^2\right]
\phi_1\big(\tfrac13,y\big) = \ \phi_1\big(\tfrac23,y\big) = \frac{\sin(2\pi y)}{\sin(\pi y)}, \phi_2\big(\tfrac13,y\big) = - \phi_2\big(\tfrac23,y\big) = \frac{\sin(2\pi y)}{\sin(\pi y)}, \phi_3\big(\tfrac13,y\big) = \ \phi_3\big(\tfrac23,y\big) = \frac{\sin(3\pi y)}{\sin(\pi y)}.
\sqrt{\gamma_{1}}\, \sqrt{\gamma_{2}}\C_{\mathrm{t o t}}( X_{1}, X_{2}) = 3 2 \, G_{N}\, T^{2}\, ( \partial_{+}\, X_{1}^{\mu}\ \partial_{+}\, X_{2 \mu}) ( \partial_{-}\, X_{1}^{\nu}\ \partial_{-}\, X_{2 \nu}) \, .
\lim_{N\to \infty}\frac{1}{N}\#\{1\le n \le N \mid \widetilde{T}^n (x;a) \in C_s \times A\}=\lim_{N\to \infty}\frac{1}{N}\#\{1\le n \le N \mid T^n x \in C_s \}&= \mu(C_s) .
\mathbb{E}^{0}(\varphi(\tau_{x}){\bf 1}_{\tau_{x>t}}L_{T}|\mathcal{F}_{t}^{Q})= \mathbb{E}^{0}[\varphi(\tau_{x}){\bf 1}_{\tau_{x>t}}] + \mathbb{E}^{0}\left(\int_{0}^{t}S_{u}\rho_{u}\mathbb{E}^{0}(L_u h(X_{u}){\bf 1}_{\tau_{x}> u}\varphi(\tau_{x-X_u})|\mathcal{F}_u^Q)du \right).
|b^{(j)}(S_\varepsilon(\rho))|\leq h^j=\begin{cases}\max_{[0,\bar\rho-\alpha_0]}|b^{(j)}| \rho \leq\bar\rho-\alpha_0 |b^{(j)}(\rho)| \rho>\bar\rho-\alpha_0.\end{cases}
\begin{gathered}\inf_{n\geq 0}\alpha_{2n+1}>0, \ \ \inf_{n\geq 0}\alpha'_{2n}>0, \inf_{n\geq 0}(1+c_n^2)^{1/2}\alpha_{2n}>0, \ \ \inf_{n\geq 0}(1+c_n^2)^{1/2}\alpha'_{2n+1}>0.\end{gathered}
(Y+x_0^3+y_0^3-z_0^3)=3D_1, (Y+x_0^3-y_0^3+z_0^3)=3D_2, (Y+x_1^3+y_1^3-z_1^3)=3D_3, (Y-x_2^3+y_2^3+z_2^3)=3D_4.
l'_{i,j}=\min \{l'_{i,j-1},l_{i-1,j}\}+l'_{i+1}+l_{i,j+1}-\min \{l'_{i+1,j},l_{i,j+1}\}-l_{i,j}& =\min \{l'_{i,j-1},l_{i-1,j}\}+\max \{l'_{i+1,j},l_{i,j+1}\}-l_{i,j}.
\begin{array}{cc}a_{0}=\int_{0}^{1}tw_{0}\left( t\right) dt=\frac{1}{2}a_{1}=\int_{0}^{1}tw_{1}\left( t\right) dt=-\frac{1}{4}a_{2}=\int_{0}^{1}tw_{2}\left( t\right) dt=-\frac{1}{8}a_{3}=\int_{0}^{1}tw_{3}\left( t\right) dt=0\end{array}
\deg\left(_\infty\left(\frac{\Delta}{\Delta_{N,12}}\right)\right)=p^{n-1}q^{m-1}(p+1)(q+1)-1 &\deg\left(_\infty\left(\frac{\Delta_N}{\Delta_{N,12}}\right)\right)=p^{n-1}q^{m-1}((p+1)(q+1)-pq)=p^{n-1}q^{m-1}(p+q+1).
\langle \nabla^{\Sigma,L}_{\dot{\gamma}}{\dot{\gamma}},\dot{\gamma}\rangle_{\Sigma,L}=(\overline{q}\frac{\dot{\gamma}_1}{\gamma_1}-\overline{p}\dot{\gamma}_3)\cdot\left(\overline{q}\frac{\ddot{\gamma}_1\gamma_1-(\dot{\gamma}_1)^2}{\gamma^2_1}-\overline{p}\ddot{\gamma}_3\right).
h^{\beta}(a^2)={1\over{2 \pi{\hat{\beta}}^2}}\left\{\frac{\pi}{2a}+{1\over 2}\ln a +{1\over 2}(\gamma - \ln 4 \pi) - \frac{\zeta(3)}{16 \pi^2}a^2 + \frac{3 \, \zeta(5)}{256 \, \pi^4}a^4 \right\}+{\cal O}(a^6) \; .
\Phi_\infty^* = \begin{pmatrix}0 |f|^{-1/2}\kappa^{-1}|f|^{-1/2}\kappa \bar f 0 \end{pmatrix}dz , \varphi_\infty = \begin{pmatrix}0 \frac 12 |f|^{-1/2}\kappa^{-1}\dot f \frac 12 |f|^{1/2}\kappa \dot f/f 0 \end{pmatrix}dz
\mathbb{E}_{N}^{eq}\left[\exp\left\{\sum_{i=1}^{N}\alpha_{i}\left(\widetilde{A}_{i}^{N}(t_{i})-\widetilde{A}_{i}^{N}(s_{i})\right)\right\}\right]\le\exp\left\{C\sum_{i=1}^{N}(\alpha_{i}^{2}+\alpha_{i}^{4})(t_{i}-s_{i})\right\}
\rho\circ X^{R-Q}\circ\rho =\frac{(X+\omega)^{R-Q}+\omega (\omega X+1)^{R-Q}}{\omega (X+\omega)^{R-Q}+(\omega X+1)^{R-Q}}&=\frac{(X+\omega)^R(\omega X+1)^Q+\omega (\omega X+1)^R(X+\omega)^Q}{\omega (X+\omega)^R)(\omega X+1)^Q+(\omega X+1)^R (X+\omega)^Q}&=X^{(-1)^\ell}\circ g(X).
\frac{d E}{d \Omega}= \frac{2{\cal G}}{( 2 \pi )^{D - 2}}\frac{1}{D - 2}\frac{\gamma_{1}^{2}m_{1}^{2}v_{1}^{2}( v_{1}+ v_{2})^{2}\sin{\theta_{1}}^{4}}{( 1 - v_{1}\cos \theta_{1})^{2}( 1 + v_{2}\cos \theta_{1})^{2}}\times \omega_{c}^{D - 3}\, .
{\Pi}= S +{\bar{S}}- \frac{\delta_{G S}}{8 \pi^{2}}\log \left( ( T +{\bar{T}}) ( U^{\prime}+{\bar{U}}^{\prime}) - ( B +{\bar{C}}) ( C +{\bar{B}}) \right) + ( m o d u l a r f u n c t i o n ) .
\dot{\gamma}(t)=\left[\overline{q}\dot{\gamma}_3-\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)\right]e_1-\frac{l_L}{l}L^{\frac{1}{2}}\omega(\dot{\gamma}(t))e_2.
\frac{\pi^{d/2}}{\Gamma(d/2+1)}=\frac{2^{2m+1}\pi^mm!}{(2m+1)!}\le \frac{2^{d}\pi^{(d-1)/2}m^{m+1/2}e^{d-m}e^{{1}/{12}}}{d^{d+1/2}}\le \bigg(\frac{e^{1/6}}{3\pi}\bigg)^{1/2}\frac{(2\pi e)^{d/2}}{d^{d/2}}\le\frac{(2\pi e)^{d/2}}{2d^{d/2}}.
\sum_{\substack{w A, B j As, i Bs}}\mathrm{ad}_{w}=\sum_{\substack{w A, B j-1 As, i Bs}}\mathrm{ad}_{w}\mathrm{ad}_{A}+\sum_{\substack{w A, B j As, i-1 Bs}}\mathrm{ad}_{w}\mathrm{ad}_{B},
{3-2s \over 2}\int_0^{\infty}t^{2-2s}\phi^2 = -(1-2s) \int_0^{\infty}t^{2-2s}(\phi')^2 - \int_0^{\infty}t^{3-2s}\phi'\phi'' &= -(1-2s) \int_0^{\infty}t^{2-2s}(\phi')^2 +{3-2s \over 2}\int_0^{\infty}t^{2-2s}(\phi')^2 -{1 \over 2}\left.t^{3-2s}(\phi')^2\right]_{t=0}^{\infty}&={1+2s \over 2}\int_0^{\infty}t^{2-2s}(\phi')^2.