harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat f_t=\Biggl(\sum_{i=1}^N\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\widehat\Gamma_i'\Biggr)^{-1}\Biggl(\sum_{i=1}^N\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\widehat B\dot z_{it}\Biggr).
\hat{\mathbf{x}}_{\mathcal{S}}=\left(\mathbf{A}_{\mathcal{S}}^t\mathbf{A}_{\mathcal{S}}\right)^{-1}\mathbf{A}_{\mathcal{S}}^t\mathbf{y_s}&=\mathbf{x}_{\mathcal{S}}+\mathbf{e}_{LS},
W(\rho)^d =\lambda_{K/F}\cdot W(\chi_K) =\lambda_{K/F}\cdot q_{K}^{-\frac{1}{2}}\sum_{x\in U_K/U_{K}^{1}}\chi_{K}^{-1}(x/c)\psi_K(x/c) =\lambda_{K/F}\cdot q_{K}^{-\frac{1}{2}}\tau(\chi_K),
|\psi\rangle = \psi_{g}|\psi_{g}\rangle + \psi_{q\bar{q}}|\psi_{q\bar{q}}\rangle + \psi_{q\bar{q}\gamma}|\psi_{q\bar{q}\gamma}\rangle + \psi_{q\bar{q}gg}|\psi_{q\bar{q}gg}\rangle + \ldots .
f(t) = \prod_{j=1}^{\frac{n+1}{2}}(t - \lambda_{2j+1}) + (t - \lambda_{1}) \sum_{j=1}^{\frac{n+1}{2}}\frac{\sin^{2}\left[\frac{(2j + 1)\pi}{n + 3}\right]}{\sin^{2}\left(\frac{\pi}{n + 3}\right)}\prod_{\substack{m=1 m \neq j}}^{\frac{n+1}{2}}(t - \lambda_{2m+1})
a_n^{(\ell)}=\left(\left|\begin{array}{cc}\Lambda_{n-1}^{(\ell)}(\theta_{n+\ell}^n)^\top \theta_{n+\ell+1}^n \boxed{m_{2n+\ell+1}}\end{array}\right|+\left|\begin{array}{cc}\Lambda^{(\ell)}_{n-1}e_{n-1}^\top \theta_{n+\ell}^n \boxed{0}\end{array}\right|H_n^{(\ell)}\right)(H_n^{(\ell)})^{-1},b_n^{(\ell)}=H_n^{(\ell)}(H_{n-1}^{(\ell)})^{-1}.
\left(I + \frac{Y_1}{z}+ O_n\big(z^{-2}\big)\right)z^{n\sigma_3}= \left(\frac{F_\infty}{2^n}\right)^{\sigma_3}\left(I + \frac{Q_1}{z}+ O_n\big(z^{-2}\big)\right)\left(\frac{\phi^n(z)}{F(z)}\right)^{\sigma_3}.
M_{|g|}(x) \ll_B B\frac{x}{\log x}\int_1^x \frac{M_{|g|}(u)}{u^2}du = B\frac{x}{\log x}\left(\sum_{n \leq x}\frac{|g(n)|}{n}- \frac{M_{|g|}(x)}{x}\right) \ll_B B\frac{x}{\log x}\prod_{p \leq x}\left(1+\frac{|g(p)|}{p}\right).
f_1(x) =\sum_{l=0}^n\frac{1}{l!}\left(\frac{d}{d y}\right)^lf_1(y)\Big|_{y=x_0}(x-x_0)^l &+\frac{1}{n!}\int_{x_0}^x(x-y)^n\left(\frac{d}{d y}\right)^{n+1}f_1(y)dy.
\sum_{k=n-4}^nc_k\leq 9\prod_{j=1}^{\frac{n-6}{2}}\left(\frac{n-2j}{2}\right)^2 =9\cdot \frac{\left(\left(\frac{n-2}{2}\right)! \right)^2}{(2!)^2}=\frac{9}{4}\left(\left(\frac{n-2}{2}\right)! \right)^2.
{\hat{T}}_{\ \nu}^{\mu}[ C ] ={\frac{1}{2 ( p + 1 ) !}}e^{2 \Phi}G^{\mu \lambda \ldots \sigma}G_{\nu \lambda \ldots \sigma}-{\frac{( p + 1 )}{1 6}}e^{2 \Phi}{\frac{G^{2}}{( p + 2 ) !}}g_{\, n u}^{\mu}\, .
\widetilde{R}_4(z):=\frac{r^{(2)}(z)}{1+|r(z)|^2}, \widetilde{R}_3(z):=\frac{\overline{r^{(2)}}(z)}{1+|r(z)|^2}, \widetilde{R}_6(z):=\overline{r^{(2)}}(z), \widetilde{R}_1(z):=r^{(2)}(z).
\frac{\partial^{2}\tilde{A}}{\partial p_{i}\partial \tilde{p}_{j}}\, dp_{i}\, d\tilde{p}_{j}= \frac{\partial^{2}\tilde{A}}{\partial p_{k}' \partial \tilde{p}_{l}'}\left( \frac{\partial p_{k}'}{\partial p_{i}}\right) \left( \frac{\partial \tilde{p}_{l}'}{\partial \tilde{p}_{j}}\right) \, dp_{i}\, d\tilde{p}_{j}
\Psi_{*,F_2''}^{\alpha,\beta}f (x) \simeq\sup_{t \ge 3x}t^{-\beta}\int_{(t-x)/2}^{t-x}(t+x - z)^{-\alpha - 1/2}(t - x - z)^{\alpha + \beta - 1/2}f(z) \, dz & \simeq\sup_{t \ge 2x}t^{-\beta}\int_{t/2}^{t}(t - z + x)^{-\alpha - 1/2}(t - z)^{\alpha + \beta - 1/2}f(z) \, dz & =: J_1 f (x) + J_2 f(x),
(I+GV)X_1\psi =(I+(G_\Delta+G_B)V)X_1\psi +\O{}(h^\infty)\psi &=(I+G_\Delta^{1/2}(I+G_{\Delta}^{-1/2}G_BG_{\Delta}^{-1/2})G_{\Delta}^{1/2}V)X_1\psi+\O{}(h^\infty)\psi=f.
\frac{\partial}{\partial \beta}f_{d,\beta}(\rho)=-\frac{d}{2}\frac{\frac{2}{k}-\|\rho\|_2^2\frac{2c_\beta}{k^2}e^{-\beta}}{1-\frac{2}{k}c_\beta+\frac{\|\rho\|_2^2}{k^2}c_\beta^2}<0.
L_0(s) = \int_{\mathbb{R}}\left( \frac{1}{2}F_y^2 + G^2 \right)dy, L_1(s) = \int_{\mathbb{R}}\left( f_y^2 + g^2 \right) dy- \int_{\mathbb{R}}f^2 dy, L_2(s) = \int_{\mathbb{R}}y^2 \left( \frac{1}{2}f_y^2 + g^2 \right)dy+ 2\int_{\mathbb{R}}y f_y \left( f+ g \right) dy, L_3(s) = \left( \frac{d\alpha}{ds}(s) \right)^2, L_4(s) = 0
0 = B_\phi 0 &= B_\phi (I-Q) F^q(u + \psi(u,p),p) &= (I-Q) B_\phi F^q(u + \psi(u,p),p) &= (I-Q) F^q(B_\phi[u+\psi(u,p)], p) &= (I-Q) F^q(B_\phi u + B_\phi \psi(u,p), p),
\pi(e) = \prod_{h \in X_e}\pi(e) = \prod_{h \in X_e}(\pi(e) - \varepsilon(h) + \varepsilon(h)) = \sum_{S \subseteq X_e}\left ( \left ( \prod_{h \in S}\varepsilon(h) \right ) \cdot \left ( \prod_{h \in X_g \setminus S}(\pi(e) - \varepsilon(h)) \right ) \right ).
R_{1,N}(C,\epsilon_m) := \sum_{r = N+1}^{\infty}b_re^{\beta_m r}= \sum_{r=N+1}^{\infty}\frac{T_re^{-r}}{(r-1)!}e^{-(\delta-\beta_m) r}\leq \sum_{r=N+1}^{\infty}\frac{T_re^{-r}}{(r-1)!}
\Delta^{\prime}\lambda = \left( \frac{d^{4}G_{E}}{d \varphi_{0}^{4}}\right)_{s = 1}= - \frac{3 \lambda^{2}}{3 2 \pi^{3}}\, \sqrt{a_{3}a_{4}}\, \lim_{s \rightarrow 1}A_{2}^{c^{2}}( s , a_{3}, a_{4}) \, .
{\cal K}\, = \, - \mathrm{l o g}\left( \mathrm{i}\langle \Omega \, \vert \, \bar{\Omega}\rangle \right) \, = \, - \mathrm{l o g}\left[ \mathrm{i}\left({\bar{X}}^{\Lambda}F_{\Lambda}-{\bar{F}}_{\Sigma}X^{\Sigma}\right) \right]
\int\limits_0^\infty \frac{dt}{t}\Bigl[\frac{t^2\,\bigl(e^{-t/2(1+\tau(1+\lambda_1+\lambda_2))}-1\bigr)}{(e^{t/2}-1)(e^{t\tau/2}-1)}\Bigr]= \int\limits_0^\infty \frac{dt}{t}\Bigl[\frac{(2t)^2\,\bigl(e^{-t(1+\tau(1+\lambda_1+\lambda_2))}-1\bigr)}{{(e^t-1)(e^{t\tau}-1)}}\Bigr],
P_{G,Y}(v_1,\dots,v_n,u_1,\dots,u_n) = \begin{cases}\sum_{k=0}^{n}\bar{Y}_{1k}\, v_1 u_k - S_1 = 0 \vdots \sum_{k=0}^{n}\bar{Y}_{nk}v_n u_k - S_n = 0 begin{align*}1.5ex] \sum_{k=0}^{n}Y_{1k}\,\, u_1 v_k - \bar{S}_1 = 0 \vdots \sum_{k=0}^{n}Y_{nk}\, u_n v_k - \bar{S}_n = 0 \end{cases}
\tilde{M}_{F_{s}, F_{r}^{\prime}}( \{\sigma_{q}\}) = \int_{C_{v_{r}}}\Phi_{\sigma , F_{s}}^{( 0 )}( z ) \frac{d z}{2 \pi i}P_{F_{r}^{\prime}}( z ) - \frac{[ 1 - ( - 1 )^{2 l_{1_{r}}}]}{2}\int_{\tilde{C}_{r}}\Phi_{\sigma , F_{s}}^{( 0 )}( z ) \frac{d z}{2 \pi i}P_{F_{r}^{\prime}}^{( a )}( z ) .
f \star g = \exp \Bigg [ i \frac{\hbar}{2}\Bigg ( \frac{\partial}{\partial q}\frac{\partial}{\partial \tilde{p}}- \frac{\partial}{\partial p}\frac{\partial}{\partial \tilde{q}}\Bigg ) \Bigg ] f ({\bf x}) g ({\bf \tilde{x}}) \vert_{{\bf x}={\bf \tilde{x}}},
\left\lbrace \begin{array}{lll}^{C}D_{t}^{q}x(t)=Ax(t)+f(t, x(t), (Hx)(t))+\mathcal{B}u(t), t\in (0, b]\setminus\lbrace t_1, t_2,\ldots, t_m\rbrace, x(0)+g(x)=x_{0}\in X, \triangle x(t_{i})=I_{i}(x(t_{i}^{-}))+\mathcal{D}v(t_{i}^{-}), i=1,2,\ldots,m,(u, v)\in U_{ad},\end{array}\right.
e^{2 \phi}= \sqrt{\frac{C^{2}}{8 \kappa \phi_{0}}}\left( a^{2 \sqrt{\phi_{0}/ K}}+ a^{- 2 \sqrt{\phi_{0}/ K}}\right) \approx \sqrt{\frac{C^{2}}{8 \kappa \phi_{0}}}a^{- 2 \sqrt{\phi_{0}/ K}}.
\left\Vert \nabla u_{p}\right\Vert_{p}^{p}=\Lambda_{p}(\Omega)=\min\left\{\left\Vert \nabla u\right\Vert_{p}^{p}:u\in W_{0}^{1,p}(\Omega)\text{\and\}\left\Vert u\right\Vert_{\infty}=1\right\}.
g^\alpha_{D^*}\mu^*(x^*) =\kappa_\alpha\mu^*(x^*)-\kappa_\alpha(\mu^*)^{A^*}(x^*){}=|x-y|^{n-\alpha}\bigl(\kappa_\alpha\mu(x)-\kappa_\alpha\mu^A(x)\bigr)=|x-y|^{n-\alpha}g_D^\alpha\mu(x)
\textbf{x}_{k}-\textbf{x}_{0}\in \textit{F}_{k}(\textit{A}, \textbf{r}_{0})=\texttt{span}(\textbf{r}_{0}, \textit{A}\textbf{r}_{0},\ldots,\textit{A}^{k-1}\textbf{r}_{0}),
A = \bigoplus_{i= 1}^\infty A_{0,i}\oplus \bigoplus_{i=1}^\infty A_{1,i}\oplus \bigoplus_{i=1}^\infty A_{2,i}B=\bigoplus_{j= 1}^\infty B_{0,j}\oplus \bigoplus_{j=1}^\infty B_{1,j}\oplus \bigoplus_{j=1}^\infty B_{2,j}
\Phi(x_1, x_2; z_1, z_2) := \frac{1}{2^\mu \cdot \mu!}\Bigl(H_{(x_1, x_2)}-z_1^2 - z_2^2\Bigr)^\mu \diamond \bigl(\delta(x_1, x_2) \cdot \exp(x_1 z_1 + x_2 z_2)\bigr).
{\cal A}_{z^{\prime}= z}F = \frac{1}{2}\sum_{i = 1}^{\infty}\frac{1}{k !}\partial_{z}^{i - 1}\mathrm{R e s}_{z^{\prime}= z}( z^{\prime}- z )^{i}F , \hspace{1 e m}{\cal B}_{z^{\prime}= z}F = 0 .
V_{\beta}(\phi)=V_0(\phi)+\frac{N}{2}LF_\beta(\phi)_R-\frac{N}{4!}(\lambda_0+\frac{\eta_0 \phi^2}{10N})(F_{\beta}(\phi)_R)^2 -\frac{2N\eta (F_{\beta}(\phi)_R)^3}{6!}.
F^\alpha_j{\frac{\partial C^i_{kl}}{\partial u^\alpha}}+F^\alpha_k{\frac{\partial C^i_{lj}}{\partial u^\alpha}}+F^\alpha_l{\frac{\partial C^i_{jk}}{\partial u^\alpha}}=\bigl(C^i_{mj}C^m_{kl}+C^i_{mk}C^m_{lj}+C^i_{ml}C^m_{jk}\bigr).
p_t (1+\rho) (x) = 1+\frac{Z e^{-\frac c\nu x}}{\sqrt{4\pi\nu t}}\int e^{-\frac{y^2}{4\nu t}-\frac{c}{\nu}y}\, dy = 1+\frac{Z e^{-\frac c\nu x + \frac{c^2}{\nu}t}}{\sqrt{4\pi \nu t}}\int e^{-\frac{(y+ 2c t)^2}{4\nu t}}\, dy = 1+Z e^{-\frac c\nu x + \frac{c^2}{\nu}t}\, .
H= \left(\begin{matrix}1 \lambda \mu 0 1 0 0 0 1\end{matrix}\right), \T =\left(\begin{matrix}1 0 0 0 1 1 0 0 1\end{matrix}\right), \S = \left(\begin{matrix}1 0 0 0 0 -1 0 1 0 \end{matrix}\right).
=\sum_{a^{n}\in T_{\delta}^{p^{n}}}Pr\{E_{a^{n}}\}Pr_{S}\{\bigcup_{b^{n}\in T_{\delta}^{p^{n}},b^{n}\ne a^{n}}E_{a^{n}}^{\dagger}E_{b^{n}}\in N(S)\}
{d^{\dagger}}^{( i )}( \eta^{n}\cdot \tau^{n}{\cal{G}}) = \eta^{n}\cdot \tau^{n}{d^{\dagger}}^{( i )}{\cal{G}}+{\textsf{P}}_{N}( n ) \eta^{n - 1}\cdot \tau^{n - 1}{d^{\dagger}}^{( i )}{\cal{G}}
r(\tau)=r_o+\frac{4(6H^2r^2_o-1)e^{-\sqrt{6H^2r^2_o-1}\;(\tau-\tau_o)}}{H(e^{-\sqrt{6H^2r^2_o-1}\;(\tau-\tau_o)}-4H^2r^2_o)^2-4H(6H^2r^2_o-1)}.
\partial^\nu_x(g\circ f)=\nu!\cdot \sum_{m=1}^{\|\nu\|_1}g^{(m)}\bigl(f(x)\bigr) \sum_{j =1}^{\|\nu\|_1}\sum_{\substack{(k_1,\ldots,k_j,\ell_1,\ldots,\ell_j) \in \mathcal{Q}_j(\nu,m)}}\prod_{q=1}^j\frac{\{\partial^{\ell_q}_x(f)\}^{k_q}}{k_q! \cdot (\ell_q!)^{k_q}}.
\mu_{1}^{\Delta}= \left( \mu_{1}^{A}, m_{1}^{I}\right) , \; \nu_{2}^{\Delta}= \left( \nu_{2}^{A}, n_{2}^{I}\right) , \; \nu_{3}^{\Delta}= \left( \nu_{3}^{A}, n_{3}^{I}\right) , \; \omega^{\Delta}= \left( \omega^{A}, o^{I}\right) ,
B(\Psi_m, \Phi_k; u)-(1-\alpha)D(\Psi_m, \Phi_k) =\sum_{l=1}^{m}e^{kl}(t) d_m^l(t), D(\Psi_m, \Phi_k) =\sum_{l=1}^m \tilde{e}^{kl}\Re(d_m^l(t)),
U_{t}^{p h}( h , h^{\prime}) = \left( \frac{l}{i \hbar t}\right)^{r / 2}\sum_{\hat{s}\in W_{A}}( \kappa ( h ) \kappa ( \hat{s}h^{\prime}) )^{- 1}\exp \left( \frac{i \pi l ( h - \hat{s}h^{\prime})^{2}}{\hbar t}+ i t E_{0}\right) \ ,
\sum_{j \in J}p_n(0,x_j) = \sum_{i \in I}\sum_{j \colon x_j \in C_i}p_n(0,x_j)\le \sum_{i \in I}\rho L^d p_n(0,z_i) & \le \rho \sum_{i \in I}\sum_{x \in C_i}|p_n(0,x) -p_n(0,z_i)| + \rho.
({\cal T}_{\alpha},{\cal T}_{\beta})_{\Omega}= 0 , \quad ( \bar{\cal P}_{\alpha}, \bar{\cal P}_{\beta})_{\Omega}= ( i \hbar )^{2}U_{\alpha \beta}^{\gamma}\bar{\cal P}_{\gamma}( - 1 )^{\varepsilon_{\alpha}+ \varepsilon_{\beta}+ \varepsilon_{\gamma}},
\omega = \log |{\frac{q^{2}| c_{2}|}{c_{3}| \Lambda |}}{\frac{\cosh^{2}\sqrt{\frac{c_{3}}{2}}( z - z_{3})}{\{-{\frac{c_{1}}{2}}\sin \sqrt{\frac{c_{2}}{2}}( z - z_{2}) -{\frac{| c_{2}|}{2}}\cos \sqrt{\frac{c_{2}}{2}}( z - z_{2}) \}^{2}}}| .
\Psi(0)={}_3F_2\bigg[\begin{matrix}-a \beta \gamma &1 \delta\end{matrix}\bigg|\,1\bigg] = \frac{\Gamma(\delta)\Gamma(1+\delta+a-\beta-\gamma)}{\Gamma(\delta+a)\Gamma(1+\delta-\beta-\gamma)}\cdot{}_3F_2\bigg[\begin{matrix}-a 1-\beta 1-\gamma &1 1+\delta-\beta-\gamma\end{matrix}\bigg|\,1\bigg].
M_y (s,x_0) :={1 \over (1 + s)^2}\chi_{|\tau| \le 5}\Vert \phi \Vert_{H^{11}(\Sigma_0)}+ \sup_{|\alpha| \le 10}\sup_{\psi_1}\int_0^s \int_{\Sigma_t}|\partial^\alpha (\Box \phi)| |\partial_y \psi| d x d t.
E_\epsilon= \;O(h^{2/3-\epsilon}): L^2(D(h^\epsilon))\to H_h^2(\mathbb{R}^n\setminus\mathcal{O}), K_\epsilon= \;O(e^{-C^{-1}h^{-1+\frac{3\epsilon}{2}}}): L^2(D(h^{\epsilon}))\to H_h^k(\mathbb{R}^n\setminus\mathcal{O}),\;\; \forall k\in\mathbb{R}.
\begin{array}{lll}r_1 := uv - qvu , r_2 := u'v' - qv'u', r_3 := uu'- q u'u, r_4 := vu' - q^2 u'v - (q - q^3) v'u, r_5 := uv' - q^2 v'u, r_6 := vv' - q v'v .\end{array}
\partial_y(d_X) = \begin{pmatrix}0 0 0 j [y^{j-1}] 0 0 (N-j) [y^{N-j-1}] 0 0 j [y^{j-1}] 0 0 (N-j) [y^{N-j-1}] 0 0 0\end{pmatrix}\,.
e^{-\lambda_k t}\geq \frac{\sqrt{m(A_j)m(A_l)}}{m(V)}+\frac{1}{\sqrt{m(A_j) m(A_l)}}\sum_{i=2}^{k-1}e^{-\lambda_i t}a^i_ja^i_l-e^{-\frac12 \zeta(D_m t, d(A_j,A_l))}.
[d,f](x_n,x_m,1)=0\Rightarrow -[\alpha (x_n),g(x_m)]+[\alpha (x_m),g(x_n)]+g([x_n,x_m])+\bar{f}(1,\varphi (x_n,x_m)) &-\varphi (\alpha (x_n),g(x_m))+\varphi (\alpha (x_m),g(x_n))+\hat{v}(1,[x_n,x_m])+\bar{v}(1,\varphi (x_n,x_m)).
a_{\{2,8\}}(\hat{n})=\sqrt{5^{\hat{n}}\cdot c_{a_{\{2,8\}}}(\hat{n})-1}=\sqrt{5^{\hat{n}-n}\cdot 5^n \cdot c_{a_{\{2,8\}}}(\hat{n})-1}\Rightarrow V\left(a_{\{2,8\}}(n)\right) = n \leq \hat{n}.
dX_i(t)=\sqrt{2(X_i^2(t)+1)}dW_i(t)+\left[\left(2-2N-2\Re(s)\right)X_i(t)+2\Im(s)+\sum_{j\ne i}^{}\frac{2(X^2_i(t)+1)}{X_i(t)-X_j(t)}\right]dt.
\Delta_G =\frac1{\sin \theta}\frac{\partial}{\partial \theta}\biggl( \sin \theta \frac{\partial}{\partial \theta}\biggr)+\frac1{\sin^2 \theta}\biggl( \frac{\partial}{\partial \phi}- \cos \theta \frac{\partial}{\partial \psi}\biggr)^2+\frac{\partial^2}{\partial \psi^2}.
\varepsilon = \{\varepsilon^{i j}\}= \left( \begin{array}{c c c c c}{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\varepsilon^{1 2}}}&{{\varepsilon^{1 3}}}&{{0}}\\{{0}}&{{- \varepsilon^{1 2}}}&{{0}}&{{\varepsilon^{2 3}}}&{{0}}\\{{0}}&{{- \varepsilon^{1 3}}}&{{- \varepsilon^{2 3}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
(X)_{ll'}= \int_{0}^{\infty}\int_{0}^{\pi}\int_{0}^{2\pi}\Psi_{nlm}^*(r,\theta,\varphi;\delta_1,\delta_2)X \Psi_{nl'm}(r,\theta,\varphi;\delta_1,\delta_2) \,r^2 \sin\theta dr d\theta d\varphi.
L^\lambda (z^2)= \sum_{n\in \mathbb{Z}}L_n (z^2)^{-n-2}= -\sum_{n\in \mathbb{Z}}\Big(\sum_{k+l=n}\left(\lambda (k+1) +(\lambda +1)l\right) :\chi_{2k+\frac{1}{2}}\chi_{2l-\frac{1}{2}}:\Big)(z^2)^{-n-2},
f_{\mathbf{M^{\ast\ast}}_{-\xi_{\boldsymbol{s}k},\xi_{\boldsymbol{t}l}}}(\boldsymbol{u})=c_{n-2,\xi_{\boldsymbol{s}k},\xi_{\boldsymbol{t}l}}^{\ast\ast}\overline{\mathcal{G}}_{n}\left\{\frac{1}{2}\boldsymbol{u}^{T}\boldsymbol{u}+\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}+\frac{1}{2}\xi_{\boldsymbol{t}l}^{2}\right\},~\boldsymbol{u}\in \mathbb{R}^{n-2},
X_0(z) = \frac{z^g \sum_{j=0}^{g-1}\P(G=j) \left(\prod_{i=g+1}^c Y_i(z) - z^{j-g}\,\prod_{i=j+1}^c Y_i(z)\right)}{z^g-\prod_{i=1}^c Y_i(z)}&= \frac{(1-\P(G=g)) z^g \prod_{i=g+1}^c Y_i(z) - \sum_{j=0}^{g-1}\P(G=j) z^{j}\,\prod_{i=j+1}^c Y_i(z)}{z^g-\prod_{i=1}^c Y_i(z)}.
R\big(\pi^* x^*,\pi^* (\sigma^*)^{-1}B(y, \delta) \big) = R\big(\tau^* \pi^* x^*, \tau^*\pi^* (\sigma^*)^{-1}B(y, \delta) \big) = R\big( \pi x, \pi B(y, \delta) \big) \supseteq R\big( \pi x, B(\pi y,\delta') \big).
x^{m_1}y^{n_1}x^{m_2}y^{n_2}x^{m_3}y^{n_3}\cdots x^{m_k}y^{n_k}x^{m_{k+1}}= [x^{m_1},y^{n_1}]\cdot [y^{n_1},x^{m_1+m_2}]\cdot [x^{m_1+m_2},y^{n_1+n_2}]\cdot [y^{n_1+n_2},x^{m_1+m_2+m_3}] &\cdot [x^{m_1+m_2+m_3},y^{n_1+n_2+n_3}]\cdots [y^{n_1+\cdots n_k},x^{m_1+\cdots + m_k+m_{k+1}}].
\mathcal{E}(f_{nm}) =\frac 12\sum_{i,j}\pi_iq_{ij}(f_{nm}(j)-f_{nm}(i))^2-\sum_{i}\pi_i\gamma_if_{nm}(i)^2 =\frac 12\sum_{i,j\in G_{nm}}\pi_iq_{ij}(f_j-f_i)^2+\sum_{i}\pi_i q(i, G_{nm}^c)f_i^2-\sum_{i\in G_{nm}}\pi_i\gamma_if_i^2.
{\frac{2}{g^{2}}}\sin \alpha_{j}+{\frac{4 \sin \alpha_{j}}{m^{4}+ 4 m^{2}+ 4 \cos^{2}({\frac{\alpha_{j}}{2}})}}\; = \; \sum_{k \neq j}\cot \left|{\frac{\alpha_{j}- \alpha_{k}}{2}}\right| .
d v_{\underline{{\mu}}}^{~ \underline{{\alpha}}}= 1 / 4 v_{\underline{{\mu}}}^{~ \underline{{\beta}}}( \Gamma_{\underline{{a}}\underline{{b}}})_{\underline{{\beta}}}^{~ \underline{{\alpha}}}\Omega^{~ \underline{{a}}\underline{{b}}}( d ) \qquad
R_{b}=x_0 \mathbb{E}\left[\sum_{t=t_b}^{t'_b}u(t,n)-\mu_b^*\right] &=x_0 \mathbb{E}\left[\sum_{n\in \mathcal{N}_b}k_{n,b}u_m\delta_{n,b}\right]=x_0u_m\sum_{n\neq a_b^*}\delta_{n,b}\mathbb{E}[k_{n,b}],
\widetilde{t}_{j,k}^{\left[\widetilde{p}-1,q-1,L1a\right]}= \widetilde{t}_{j,k}^{\left[\widetilde{p}-1,q-1\right]}+{q-1 \over \widetilde{p}-2}\,\widetilde{t}_{j,k+1}^{\left[\widetilde{p}-2,q-2\right]}\,\left(t_{j,k}-t_{j,k+1}\right).
\phi_{1}( x + y ) ={\frac{\biggl | \begin{matrix}{{\phi_{2}( x )}}&{{\phi_{2}( y )}}\\{{\phi_{3}( x )}}&{{\phi_{3}( y )}}\\ \end{matrix}\biggr |}{\biggl | \begin{matrix}{{\phi_{4}( x )}}&{{\phi_{4}( y )}}\\{{\phi_{5}( x )}}&{{\phi_{5}( y )}}\\ \end{matrix}\biggr |}}
\begin{array}{c}{< j_{1}- m_{1}\j_{2}- m_{2}| J - M >_{q}= < j_{2}m_{2}\j_{1}m_{1}| J M >_{q}=}\\{= ( - )^{j_{1}+ j_{2}- J}< j_{1}m_{1}\j_{2}m_{2}| J M >_{q^{- 1}}}\\ \end{array}
\mathcal{E}_{n,\lambda}\bigg(\frac{1}{2}\bigg) =\sum_{k=0}^{n}k!S_{2,\lambda}(n,k)\sum_{j=0}^{k}\binom{\frac{1}{2}}{j}\bigg(-\frac{1}{2}\bigg)^{k-j}&=\sum_{k=0}^{n}k!S_{2,\lambda}(n,k)(-1)^{k}\sum_{j=0}^{k}\binom{2j}{j}\frac{1}{(1-2j)2^{k+j}}.
\|(1-\nu)AX-\nu XB \|_{2}^{2}\leqslant \| A^{1-\nu}XB^{\nu}\|_{2}^{2}+(1- r_{0})^{2}\|AX-XB\|_{2}^{2}& -\sum_{k=1}^{\infty}r_{k}\|A^{1-\frac{m_{k}}{2^{k}}}XB^{\frac{m_{k}}{2^{k}}}- A^{1-\frac{m_{k}+1}{2^{k}}}XB^{\frac{m_{k}+1}{2^{k}}}\|_{2}^{2}.
a_{j-1}w' =a_{n-k}(a_{n-k}a_k)^{i_k-1}(a_{n-k-1}a_{k+1})^{i_{k+1}}\dotsm(a_{s+1}a_{s-1})^{i_{s-1}}a_s^{i_s}a_{j-1}a_kv, a_jw' =a_{n-k}(a_{n-k}a_k)^{i_k-1}(a_{n-k-1}a_{k+1})^{i_{k+1}}\dotsm(a_{s+1}a_{s-1})^{i_{s-1}}a_s^{i_s}a_ja_kv.
\ddot{\Phi}- \frac{2}{a}\, \dot{\Phi}_{, r}+ \frac{1}{a^{2}}\, \Phi_{, r r}+ H \dot{\Phi}+ \frac{2}{a^{2}}\left( \frac{1}{r}- \dot{a}\right) \Phi_{, r}= 8 \pi G \, \dot{\varphi}\left( \delta \dot{\varphi}- \frac{1}{a}\, \delta \varphi_{, r}\right) \ .
[D^*D,D^k]\phi =\sum_{l=0}^{k-1}D^{l}R^M\star D^{k-l}\phi &\quad+\sum_{\sum l_i+\sum{m_j}=k-1}{}^{G}\nabla^{l_1}R^P\star\underbrace{D^{m_1+1}\phi\star\ldots\star D^{m_{l_1}+1}\phi}_{l_1-}\star D^{l_2+1}\phi\star D^{l_3+1}\phi\star D^{l_4+1}\phi.
4\tau^{2}\langle\Delta'u,|\nabla\phi|^{2}u\rangle = \tau^{2}\langle\Delta'u,|x'|^{2}u\rangle+16\tau^{2}\langle\Delta'u,(x_{n+1}^{1-2s}-x_{n+1})^{2}u\rangle \ge -\tau^{2}(1+\epsilon_{0})\|\nabla'u\|^{2}-\tau^{2}C\epsilon_{0}^{-1}\|u\|^{2}-16\tau^{2}\|(x_{n+1}^{1-2s}-x_{n+1})\nabla'u\|^{2}.
(4+t+2u)p_{i,j}+(3+2t+2u)p_{j,i}& =(2+t)p_{i,j}+(1+2t)p_{j,i}+p_{i+1,j}+p_{j,i+1}+p_{i,j+1}+p_{j+1,i}& +up_{i-1,j}+up_{j,i-1}+up_{i,j-1}+up_{j-1,i}
\biggl(\sum_{u,v=1}^m \sum_{v=1}^m b_{i,u}(T_N)b_{i,v}(T_N)V_N(W^u,W^v) + \sum_{u,v=1}^l c_{i,u}(T_N)c_{i,v}(T_N)V_N(B^{H,u},B^{H,v}) + \sum_{u=1}^m \sum_{v=1}^l b_{i,u}(T_N)c_{i,v}(T_N)V_N(W^u,B^{H,v}) \biggr)^{1/2}\leq C \Delta^{1/2}\delta^{-1/2}\mathit{S}.
|f(T) - f(T^M)| \le \log \frac{\rho_{0,\inf}^M(T)e^{\tau_0^M(T)}+ \rho_{*,\inf}^M(T)e^{\tau_*^M(T)}}{\rho_{0,\inf}^M(T) + \rho_{*,\inf}^M(T)}\le \log e^{\eta^M(T)}= \eta^M(T),
\int_X f_n\, d\mu = \int_{A_n}f_n\, d\mu +\int_{X\setminus A_n}f_n\, d\mu \leq \|f_n\|_\infty \mu (A_n) + \frac{\varepsilon}{2\mu ({\rm supp}f_1)}\mu ({\rm supp}f_n) < \varepsilon.
1-|x_1-y_1|+(1- |x_1-y_1|)\left|x-y\right|+\left|x-y\right|^2 = (1-|x_1-y_1|+|x-y|)-|x_1-y_1|\left|x-y\right|+\left|x-y\right|^2 \approx 1+|x-y|(1-|x_1-y_1|+\left|x-y\right|) \approx 1+|x-y|^2.
\left[\mathbf{J}(\Theta)\right] =\frac{2}{\sigma^2}\mathfrak{R}\left[(\mathbf{G}_5^H\mathbf{G}_5)\odot(\mathbf{G}_4^H\mathbf{G}_4)\odot(\mathbf{G}_3^H\mathbf{G}_3) \right. \quad\quad\quad\left. \odot(\mathbf{G}_2^H\mathbf{G}_2) \odot(\mathbf{G}_1^H\mathbf{G}_1)\right]
\,\phi \leq 2 \int_{\rho B_{1}}\int_{\rho B_{1}}(\psi(x)-\psi(y))^{2}\phi w^{2}k(x,y) dxdy & + 4 \int_{\rho B_{1}}\int_{\mathbb{R}^{n}\backslash(\rho B_{1})}(\psi(x)-\psi(y))^{2}\phi w^{2}k(x,y)dydx \leq C_{1}(n,\Lambda,\delta)(\rho - \rho')^{-2\beta}\int_{\rho B_{1}}\phi w^{2}dx,
I(H,\varnothing) = ~ \mathrm{span}(\{\mu \nu^* \colon \mu \nu r(\mu) = r(\nu) \in H \}) = ~ \mathrm{span}\left( \left\{\big( \widehat{s(\mu)}\cdot \mu \cdot \widehat{r(\mu)}) ( \widehat{r(\nu)}\cdot \nu^* \cdot \widehat{s(\nu)}\big) \colon r(\mu) = r(\nu) \in H \right\}\right) \; ,
R_1= -8F-4s+8t_1+4C_1+8C_2+13C_3+18C_4+15C_5+12C_6+10C_7+8C_8+ &+6C_9+4C_{10}+3C_{11}+2C_{12}+C_{13}+3B_1+6B_2+3B_3, R_2= -4F-2s+2t_1+2C_1+4C_2+5C_3+6C_4+5C_5+4C_6+4C_7+4C_8+ &+4C_9+4C_{10}+3C_{11}+2C_{12}+C_{13}+B_1+2B_2+B_3,
B ( R ) = \exp \left( \begin{matrix}{{- i \rho K_{3}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{R}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1 / R}}\\ \end{matrix}\right) ,
2 \epsilon_1-\epsilon_m-\delta_n,\; \epsilon_1+\epsilon_2-\epsilon_m-\delta_n,\; \epsilon_1-\epsilon_m,\; \delta_1-\delta_n,\; \epsilon_1+\delta_1-2\delta_n,\; \epsilon_1+\delta_1-\delta_{n-1}-\delta_n,\; \epsilon_1-\epsilon_m+ \delta_1- \delta_n.
A_1^{\frac{1}{k}}(z) =(A+h)^{\frac{1}{k}}=A^{\frac{1}{k}}\left(1+ \frac{h}{A}\right)^{\frac{1}{k}}= A^{\frac{1}{k}}\left(1+ O\left({\frac{|h|}{|A|}}\right)\right),
\boldsymbol{\alpha}_t^*=(\alpha_t^{*,i})_{1\leq i\leq N}=\left(-R^{-1}By_t^{*,i}-h\big(\rho_i^N(\Delta^{N,1}_{*,t},\ldots,\Delta^{N,N}_{*,t})\big)\right)_{1\leq i\leq N}.
\begin{array}[c]{c}\sqrt{-1}\partial_{B}\overline{\partial}_{B}\log\Omega=\frac{\partial}{\partial t}\widehat{\omega}_{t}=-\frac{1}{T_{0}}(\omega_{0}+\psi^{\ast}\omega_{N})\in c_{1}^{B}(M),\int_{M}\Omega\wedge\eta_{0}=1.\end{array}
\Biggl | \frac{\Delta T_{1}}{{( T_{1})}_{0}}\Biggr | \; , \; \Biggl | \frac{\Delta T_{2}}{{( T_{1})}_{0}}\Biggr | \geq \frac{( 2 q_{0})^{- 1 / 2}}{( T_{1})_{0}}1 0^{- 4}
\varphi(x)=-\int_{V}G(x,x^{\prime})\rho(x^{\prime})d^{3}x^{\prime}+\int_{S}[\varphi(x^{\prime})\nabla^{\prime}G(x,x^{\prime})-G(x,x^{\prime})\nabla^{\prime}\varphi(x^{\prime})]\cdot d\hat{\sigma}^{\prime}.
\alpha_2(m+1)=\alpha_2(m)^{\frac{2^{m-1}-1}{2^m-1}}\Bigg(\bigg(\frac{2^{m-1}-1}{2^{m-1}}\bigg)^{-\frac{2^{m-1}-1}{2^m-1}}+\bigg(\frac{2^{m-1}-1}{2^{m-1}}\bigg)^{\frac{2^{m-1}}{2^m-1}}\Bigg).
\aligned -(\psi_2^{i+1 i}\circ \frak h_{ab}^i )\vert_{E^i}+ (\frak h_{ab}^{i+1}\circ \psi_1^{i+1 i})\vert_{E^i}- (\frak h_{a}^{i+1 i})\vert_{E^i}+ (\frak h_{b}^{i+1 i})\vert_{E^i}&=\left(\hat d_{2}^{i+1}\circ \frak H_{ab}^{i+1 i}-\frak H_{ab}^{i+1 i}\circ \hat d_{1}^{i}\right)\vert_{E^i}.\endaligned
\sum_{n=0}^{\infty}\frac{q^{3n^{2}}\left(-w\right)^{n}}{\left(q^{2};q^{2}\right)_{n}\left(-wq^{3};q^{4}\right)_{n}}=\frac{1}{\left(1+wq^{3}\right)}\sum_{n=0}^{\infty}\frac{q^{3n^{2}}\left(-q^{2}w\right)^{n}}{\left(q^{2};q^{2}\right)_{n}\left(-wq^{7};q^{4}\right)_{n}}.
\lim_{\varepsilon \to 0}\frac{\| f_\varepsilon -\mu \|_{L^2([0,t_\varepsilon] \times \Omega_\varepsilon)}}{\| \langle v\rangle^m (f_\varepsilon|_{t=0}- \mu) \|^\alpha_{H^s(\mathbb{T}^3 \times \mathbb{R}^3)}}=+\infty
\sqrt{n}(\hat{\theta}-\theta_{n}) =\sqrt{n}(\hat{\theta}-\theta_{0})+\sqrt{n}(\theta_{0}-\theta_{n})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\zeta(W_{i})+o_{p}(1)+\mu\bar{\sigma}^{2}& =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\{\zeta(W_{i})-E_{F_{n}}[\zeta(W)]\}+\mu\bar{\sigma}^{2}\overset{d}{\longrightarrow}N(\mu\bar{\sigma}^{2},V).Q.E.D.
I^{R}( 0 ) = - 2 \sum_{n = 1}^{\infty}n^{2}\ln ( n ) +{\frac{1}{6 0}}\ln ( r_{+}/ r_{-}) + 2 \sum_{n = 1}^{\infty}n^{2}\ln \Bigr [ 1 - ( r_{-}/ r_{+})^{2 n}\Bigr ] .
\mathbf{w}_S^S = \mathbf{g}_S^S - \beta \mathbf{g}_S^S = (1-\beta)\mathbf{g}_S^S \mathbf{w}_{S^c}^S = \mathbf{1}_{S^c}+ \beta \mathbf{P}_{S^c N}\mathbf{g}^S - \mathbf{g}_{S^c}^S = \mathbf{1}_{S^c}+ \beta \mathbf{P}_{S^c S}\mathbf{g}_S^S.
y^{(m)}_{r_1,\ldots,r_k}=\sum_{a_m}\frac{y_{r_1,\ldots,r_{m-1},a_m,r_{m+1},\ldots,r_k}}{\varphi(a_m)}+O\left(\frac{y_{max}\varphi(VP_f)\log X}{VP_fD_0}\right).
\frac{\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (s_k)_{k=0}^{m}\right)}{a\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (1,0,\ldots,0)\right)}\stackrel{a\to 0+}{\longrightarrow}\infty.
\bar{\cal M}_{\alpha \beta}^{\zeta}= \left( \begin{array}{l l}{- i \alpha}{\gamma \partial_{t}+{\textstyle{\frac{1}{2}}}\dot{\gamma}+ i \alpha}{- \gamma \partial_{t}-{\textstyle{\frac{1}{2}}}\dot{\gamma}+ i \alpha}{- i \alpha}\end{array}\right)
\|\nabla^2\mathcal{H}\|^2 = g^{ik}g^{jl}\nabla^2\mathcal{H}(\partial_i,\partial_j)\nabla^2\mathcal{H}(\partial_k,\partial_l), \nabla^2\mathcal{H}(\partial_i,\partial_j) = \partial_i \partial_j \mathcal{H}-\nabla_{\nabla_{\partial_i}\partial_j}\mathcal{H}=\mathcal{H}_{ij}-\Gamma_{ij}^k\mathcal{H}_k.
y^\ell+hf^\ell(y^\ell,u) =\Psi^\top \mathrm My+h\Psi^\top\mathrm Mf(\Psi\Psi^\top\mathrm My,u) &=\Psi^\top \mathrm M\big(y+hf(y,u)\big)+h\Psi^\top\mathrm M\big(f(\Psi\Psi^\top\mathrm M y,u)-f(y,u)\big).
H_{S u s y}= p_{t A}p_{t A}+ \frac{1}{2}\f_{A B C}f_{A B^{\prime}C^{\prime}}q_{s B}q_{t C}q_{s B^{\prime}}q_{t C^{\prime}}+ i q_{t C}f_{A B C}\gamma_{\alpha \beta}^{t}\Theta_{\alpha A}\Theta_{\beta B}\ ,
K^{(2r+1,r_L)}_0 (x)=\sum_{i=-r}^{r}E^{(ii)}+\frac{q-x^2}{h_0(a,c,x)}\Big(\sum_{r_L < i \leq r}t_0 E^{(-i,-i)}+E^{(r+1-i,r+1-i)}{}-\sum_{r_L < i \leq r}q^{-i}E^{(-i,r+1-i)}+q^{r+1-i}t_0E^{(r+1-i,-i)}\Big),
\begin{cases}-\Delta \psi_i-\frac{\l}{|x|^2}\psi_i-pu_{\l}^{p-1}\psi_i=\frac{\nu_i}{|x|^2}\psi_i, \ \ \B_1\setminus\{0\}\psi_i\in H^1_0(B_1)\end{cases}
r=\frac{cov(X,Y)}{\sigma_{X}\sigma_{Y}}=\frac{\sum_{i=1}^{N}(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{\sqrt{\sum_{i=1}^{N}(X_{i}-\overline{X})^{2}}\sqrt{\sum_{i=1}^{N}(Y_{i}-\overline{Y})^{2}}}.