harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\overline{{{g}}}_{\mu \nu}= \left[ \begin{array}{c c}{{1 + l ( x^{0})^{2}\overline{{{\lambda}}}( x^{0}, x^{1})^{2}}}&{{l ( x^{0})^{2}}}\\{{l ( x^{0})^{2}\overline{{{\lambda}}}( x^{0}, x^{1})^{2}}}&{{l ( x^{0})^{2}}}\\ \end{array}\right] .
log\gamma_{\pm}=\frac{-A_{\gamma}|z_{+}z_{-}|I^{1/2}}{1+\rho I^{1/2}}+\frac{(0.06+0.6B|z_{+}z_{-}|)I}{(1+\frac{1.5}{|z_{+}z_{-}|}I)^{2}}+BI
\lim_{q\rightarrow\infty}\frac{\left\|\frac{f}{|x|^{\frac{\beta}{q}}}\right\|_{L^{q}(B(x_{0},r))}}{(\Gamma(q/p'+2))^{\frac{1}{q}}\|f\|_{L^{p}_{Q/p}(B(x_{0},r))}}\leq \lim_{q\rightarrow\infty}C_{\varepsilon}^{\frac{1}{q}}(\widehat{\alpha}-\varepsilon)^{-\frac{1}{p'}}=(\widehat{\alpha}-\varepsilon)^{-\frac{1}{p'}}.
\left( \sum_{n = 0}^{\infty}\lambda_{n}P_{n}\right)^{m}= \sum_{n = 0}^{\infty}\lambda_{n}^{m}P_{n}\quad \mathrm{s o ~ t h a t}\quad \frac{\partial V}{\partial \phi}\left( \sum_{n}\lambda_{n}P_{n}\right) = \sum_{n}P_{n}\frac{\partial V}{\partial \phi}( \lambda_{n}) = 0 .
z \, ( U + g z ) \, \omega_{1 - 1}- ( U + g z ) \, t_{\Phi}- g \, t_{1 - 1 \Phi}- \omega \, \omega_{1 - 1}+ c \, \omega_{1 \ 0 - 1}+ c \, \omega_{- 1 - 1 - 1}= 0
\Pr_{r}'[T_{\mathrm{Bad}_i}\le 2^{18 n-2}\mid T_{\mathrm{Bad}_i}<T_{\mathrm{Nice}}] \le \Pr_{o_i}'[T_{\mathrm{BL}_i}\le 2^{18 n-2}\mid T_{\mathrm{BL}_i}<T_{\mathrm{GMP}_i}]=o(1/n),
D_{N,M,2}=\left(\sum_{j=M+1}^{\tau(N)}\frac{2^{n_{j}}}{N}\right)\sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{N}\right)^{s}(2^{s+1}\,\mathcal{R}_{s}(2^{n_{k}+1})-2\,\mathcal{R}_{s}(2^{n_{k}})).
H_m^2(f) := \int_I\Big(\sqrt{f(x)}-\sqrt{\hat f_m(x)}\Big)^2 \nu_0(dx), A_m^2(f) := \int_I\Big(\widehat{\sqrt{f}}_m(y)-\sqrt{f(y)}\Big)^2\nu_0(dy), B_m^2(f) := \sum_{j=1}^m\bigg(\int_{J_j}\frac{\sqrt{f(y)}}{\sqrt{\nu_0(J_j)}}\nu_0(dy)-\sqrt{\nu(J_j)}\bigg)^2.
C \leq \frac{1}{1-r_0^{-2}}e^{(\frac{1}{r_0-r_0^{-1}})^6}=\frac{1}{1-r_0^{-2}}e^{\frac{1}{(N^2-4)^3}}\leq \frac{1}{1-\frac{4}{N^2}}e^{\frac{1}{125}}\leq \frac{9}{5}e^{\frac{1}{125}}\leq 2.
{\rm{I}}_1\leqslant O(1) \int_\mathbb{R}\sum_{i,j=1}^n\bigg(\delta \sum_{k=1 \atop k \neq p}^n |u^k_x|+|u^p_x| \chi_{\{i,j \neq p\}}+ |u^p_x|^2 + |\rho_{xxt}|+ e\bigg) | w_iw_j | dx.
q_{11}^k(t,k,l,\xi,\eta)=\frac{\varepsilon_0e^{-5(|k-l|+|\xi-\eta|)}}{t^2}\cdot\frac{l\cdot A_k(t,\xi)-k\cdot A_l(t,\eta)}{\sqrt{A_k(t,\xi)A_l(t,\eta)}},
\langle \phi\, | \psi \rangle = \int d^2z\,\overline{\phi(z,\bar z)}\,\psi(z,\bar z)\,, \langle\, T^{(s,\bar s)}(g)\phi\, |\, T^{(s,\bar s)}(g)\psi\, \rangle = \langle \phi\, | \psi \rangle.
K(\theta) = \widetilde{E}_x\left[\int_{0}^{\infty}\frac{1}{1-\alpha}\theta^{1-\alpha}_{t}e^{X_t-Y_t}e^{-\widetilde{r}t}\,dt -\int_{0}^{\infty}e^{-\widetilde{r}t}\,d\theta_{t}\right] .
\frac{u_{m-2}'}{u_m'}= \frac{u_{m-2}}{u_m}+ t\,\frac{u_{m-1}}{u_m}+ \frac12\,t^2 = \frac{u_{m-2}}{u_m}+ t\,\left(\frac{u_{m-1}}{u_m}+\frac{t}{2}\right)
(e_nx^2+ f_nx-f_n)\mathcal{L}_{n+1}(x)=(p_nx^3+q_nx^2+r_nx+s_n)\mathcal{L}_n(x) +(t_nx^4+u_nx^3+(-u_n+t_n\omega^2)x^2+\omega^2u_nx-u_n\omega^2)\mathcal{L}_{n-1}(x), n\geq 1.
P^n_\tau : =n^{\alpha+\frac12}\int^{\tau}_0(t-s)^{\alpha}\sigma'(X_s)\, \left(\int^s_{\eta_n(s)}(s-\eta_n(u))^{\alpha}\sigma(X_{\eta_n(u)})\,dW_u\right)ds = n^{\alpha+\frac12}\int^{\tau}_0(t-s)^{\alpha}\sigma'(X_{s}) \sigma(X_{\eta_n(s)}) \Xi^{n,2}_s ds,
{\bf r}_{i+1}^{T}\mathcal{M}(\gamma){\bf r}_{i}={\bf r}_{i}^{T}\mathcal{M}(\gamma){\bf r}_{i}-\frac{{\bf r}_{i}^{T}\mathcal{M}(\gamma){\bf r}_{i}}{{\bf p}_{i}^{T}M^{T}\mathcal{M}(\gamma){\bf p}_{i}}({\bf p}_{i}^{T}M^{T}\mathcal{M}(\gamma){\bf p}_{i}-\beta_{i}{\bf p}_{i-1}^{T}M^{T}\mathcal{M}(\gamma){\bf p}_{i})
\frac{\mathbf{Q}(L_w[0,j]=\gamma [0,j])}{\mathbf{Q}(L_{v_0}[0,j]=\gamma [0,j])}=\frac{\mathbf{Q}(\widehat{S}^w(\tau_{D_j})=\gamma_j)\mathbf{Q}(L_{\gamma_j}[0,j]=\gamma[0,j])}{\mathbf{Q}(\widehat{S}^{v_0}(\tau_{D_j})=\gamma_j)\mathbf{Q}(L_{\gamma_j}[0,j]=\gamma[0,j])}&=\frac{H_j(w,\gamma_j)}{H_j(v_0, \gamma_j)}.
S_{M\otimes I_d}(\{0,d\}) = S_{M\otimes I_d}(S_{M\otimes I_d}(N);\{0,d\}) S_{M\otimes I_d}(\{1,d+1\}) = S_{M\otimes I_d}(S_{M\otimes I_d}(N);\{1,d+1\}) &\vdots S_{M\otimes I_d}(\{d-1,d+(d-1)\}) = S_{M\otimes I_d}(S_{M\otimes I_d}(N);\{d-1,d+(d-1)\})
\log \Phi\big(z-\tfrac{1}{2}; \lambda\big)= \sum_{k=0}^\infty \frac{2p_{2k+1}(\lambda)}{2k+1}\big(z-\tfrac{1}{2}\big)^{-(2k+1)}\hphantom{\log \Phi\big(z-\tfrac{1}{2}; \lambda\big)}{}= \sum_{k=0}^\infty\frac{2p_{2k+1}(\lambda)}{2k+1}\frac{1}{z^{2k+1}}\sum_{j=0}^\infty \frac{(2k+1)(2k+2) \cdots(2k+j)}{j!}\frac{1}{(2z)^j}.
\rho_{0}( x , x ; \beta ) = \int_{y ( 0 ) = x}^{y ( \beta ) = x}{\cal D}\eta \, \exp \left\{- \int_{0}^{\beta}\frac{1}{2}\, \dot{\eta}^{2}\, d \tau \right\}= \frac{1}{\sqrt{2 \pi \beta}}
\frac{1}{i \partial_{-}- e A_{-}}= \frac{1}{i \partial_{-}}+ \left( \frac{1}{i \partial_{-}}e A_{-}\right) \ast \frac{1}{i \partial_{-}- e A_{-}}= \frac{1}{i \partial_{-}}+ \frac{1}{i \partial_{-}- e A_{-}}\ast \left( e A_{-}\frac{1}{i \partial_{-}}\right) .
l_{2}(x)=\frac{x-x_{0}}{x_{2}-x_{0}}\cdot\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{x-2}{5-2}\cdot\frac{x-4}{5-4}=\frac{1}{3}x^{2}-2x+\frac{8}{3}
\frac{\int_J \frac{|w'|^2 V_{\beta_0}}{|\beta_0'|}+ \frac{k^2 |w|^2 | \beta_0'|}{V_{\beta_0}}\,dt}{\int_J |w|^2 V_{\beta_0}|\beta_0'| \,dt}= \frac{\int_0^L \frac{|w'|^2 V_{\gamma_0}}{|\gamma_0'|}+ \frac{k^2 |w|^2 | \gamma_0'|}{V_{\gamma_0}}\,dt}{\int_0^L |w|^2 V_{\gamma_0}|\gamma_0'| \,dt}< \lambda_{k,n}(\beta_0)
\rho_{n-j}(\omega'_1,\ldots,\omega'_{2n})=a^jb^j\rho_{n-j}(\omega_1,\ldots,\omega_{2n}), \rho_{n-j}(\omega'_1,\ldots,\omega'_{2n+1})=a^{j+1}b^j\rho_{n-j}(\omega_1,\ldots,\omega_{2n+1}).
(S \circ \rho)(I_{132,312}) = I_{213,231}(S \circ \rho)(I_{231,132}) = I_{312,213}(S \circ \rho)(I_{231,213}) = I_{312,132}(S \circ \rho)(I_{213,312}) = I_{132,231}S(I_{231,123}) = I_{312,123}
\left|X(t)-\Pi_nX(t)\right| = \left|X(t) - X\left(t^n_{i-1}\right)-{n \over T}\left(t-t^n_{i-1}\right)\left(X\left(t^n_{i}\right)- X\left(t^n_{i-1}\right)\right)\right| \leq 2 \sup_{t\in \left[t^n_{i-1},t^n_{i}\right]}\left| X(t)- X\left(t^n_{i-1}\right)\right|,
\tfrac{d}{dP_1}\left(\tfrac{1}{a_1}\right) =-\tfrac{(h_1-h_2)h_1h_2}{(h_2+h_1h_2P_1)^2}\left[\gamma(h_2P)-\gamma(h_2P_1)\right]<0 P_1\neq P \tfrac{d}{dP_1}\left(\tfrac{1}{a_2}\right) =\tfrac{(h_1-h_2)h_1h_2}{(h_1+h_1h_2P_1)^2}\gamma(h_1P_1)>0 P_1\neq 0.
-H_y|y|^{-2}> - \tfrac{1}{2}\Delta_y|y|^{-2}= (n-4) |y|^{-4}|y|>{\sf m}_1, -H_y |y|^{-(n-4)}> - \tfrac{1}{2}\Delta_y|y|^{-(n-4)}= (n-4) |y|^{-(n-2)}|y|>{\sf m}_1.
D=s_{1}^{2}+(N-1)\frac{Ns_{2}d^{2}-s_{1}^{2}(1+d^{2})}{N-1-d^{2}}=\frac{N(N-1)^{2}d^{2}}{N-1-d^{2}}((N-1)s_{2}-s_{1}^{2})
\theta^\star_{xy}\, q_{ij}(x,y)= \frac{n(r-3)i}{k}q_{ij}(x,y) + \frac{n(i+1)(k-i-j)}{k(n-i-2j)}q_{i+1,j}(x,y) +\frac{n(i+1)(n-k-j+1)}{k(n-i-2j+2)}q_{i+1,j-1}(x,y) +\frac{n(n-j-k)(j+1)(r-2)}{k(n-i-2j)}q_{i-1,j+1}(x,y)+\frac{n(k-i-j+1)(n-i-j+2)(r-2)}{k(n-i-2j+2)}q_{i-1,j}(x,y),
\tilde \varphi_t^* \tilde \Omega_N = (\eta \circ \varphi_t \circ \eta^{-1})^* \eta_* \Omega = (\eta^{-1})^*\varphi_t^*\eta^*(\eta^{-1})^*\Omega=\eta_*\varphi_t^*\Omega = \tilde \Omega_N,
{\cal H}_{k+1}^{(s,t)}(z)=(z-Q_{k+1}^{(s,t)}-E_{k}^{(s,t)}-\mu^{(t)}){\cal H}_{k}^{(s,t)}(z)-Q_{k}^{(s,t)}E_{k}^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z),k=0,1,\dots,m-1.
\Delta^{\mu\nu,\lambda\sigma}_{N,2}=\frac{1}{2(p^2-i\epsilon)}\left ( g^{\mu\nu}g^{\lambda\sigma}+(g^{\mu\nu}n^{\lambda}n^{\sigma}+n^{\mu}n^{\nu}g^{\lambda\sigma})\frac{p^2}{|{\bf p}|^2}+n^{\mu}n^{\nu}n^{\lambda}n^{\sigma}\frac{(p^2)^2}{|{\bf p}|^4}\right ).
{d{e}_{1}(s)\over ds}= -{e}^*_{2}(s) e_{3}(s), \quad{d{e}_{2}(s)\over ds}= -{e}^*_{1}(s){e}_{3}(s), \quad{d{e}_{3}(s)\over ds}= -{e}_{1}(s)e_2(s).
\tilde{\rm E}_0(x_0,y_0;b) \simeq \frac{(eq)^2}{m_0}(1 - e^{-m_0 |x_0 - y_0|}) + \hbar q \frac{e^2}{2\rm L}[\frac{eb\rm L}{2\pi}] (x_0^2 - y_0^2) +\hbar q^2 \frac{e^4}{8\pi m_0}(x_0 - y_0)^2.
\theta_n(T) =\frac{\beta_n r_{n,n}(T)T}{2 \alpha_{n+1}}+ \frac{c \beta_{n+1}}{2T}+\frac{c}{\alpha_{n+1}}\Big(2 d_{n.n}(T) +\frac{T \beta_n r_{n.n}(T)}{2}&+ \frac{k_{n,n}(T)}{T}\Big)+ \frac{c}{\alpha_{n+1}}\Big(2 d_{n-1.n}(T) \Big)\,.
A_{S,R}= \frac{4}{\pi}\left(1 - \frac{1}{\sqrt{M}}\right) \int_{0}^{\pi{/}2}\sin^{2 m_{S,R}}{\rm d}\theta - \frac{4}{\pi}\left(1 - \frac{1}{\sqrt{M}}\right)^{2}\int_{0}^{\pi{/}4}\sin^{2 m_{S,R}}{\rm d}\theta.
\begin{array}{c}{{\overline{{{A}}}_{\hspace{0 . 1 c m}\nu}^{1}\longrightarrow W_{\nu}^{-},}}\\{{\overline{{{A}}}_{\hspace{0 . 1 c m}\nu}^{2}\longrightarrow W_{\nu}^{+},}}\\{{\overline{{{A}}}_{\hspace{0 . 1 c m}\nu}^{3}\longrightarrow Z_{\nu},}}\\{{\overline{{{A}}}_{\hspace{0 . 1 c m}\nu}^{0}\longrightarrow A_{\nu}.}}\\ \end{array}
\begin{array}{c l}{{}}&{{A_{\lambda}^{( m )}( x | z^{( n )}| \cdots | z^{( 1 )}) |_{z_{m}^{( k )}= a \tau^{2 n - 2 k}~ ~ ( k = 1 , \cdots , n )}= ( x - a \tau^{n - 1})}}\\{{\times}}&{{\displaystyle \sum_{\rho = 0}^{n - 1}( - 1 )^{\rho}\sigma_{\rho}( a \tau , a \tau^{3}, \cdots , a \tau^{2 n - 3}) A_{\lambda - \rho}^{( m - 1 )}( x | z^{( n )}| \cdots | z^{( 1 )}) .}}\\ \end{array}
P_N(\gamma_1,\gamma_2)= 2 \int_{0<s_1<s_2<1}ds_1ds_2 \int_{0}^{\infty}(u+3/N)^{\gamma_1}(s_2-s_1+u+3/N)^{\gamma_1}du &\qquad\qquad\qquad\qquad\times \int_{0}^{\infty}(u+3/N)^{\gamma_2}(s_2-s_1+u+3/N)^{\gamma_2}du.
\int F_1(\lambda r, \lambda^{-2}z) b_0(r,z) dr dz = \int F_1 (\lambda r, \lambda^{-2}z) \frac 1 r \bigl(-\partial_{zz}a_0 - \partial_{rr}a_0 +\frac 1 r \partial_r a_0\bigr) dr dz.
\int_{S^n}{\tilde{\phi}}_{\epsilon^2}(x)~d\sigma(x) ={\frac 1{|{\hat S}^l|}}\sum_{y\in{\hat S}^lx_0}\phi(y)\int_{S^n}H_t(y,x)~d\sigma(x) =\int_{S^n}\phi(x)~d\nu(x).
W=\int\frac{2d^{10}\tilde p_1}{\tilde p_1^2\tilde p_2^2}\sum_{(\lambda,\lambda')}\hat A_{(0)}(\{p,\epsilon\}_{(i)},\tilde p_1\lambda,\lambda')\hat A_{(n-1)}(\lambda,\lambda';\tilde p_1,\{p,\epsilon\}_{(f)})\,,
M^{\mu_{n}( i )}{}_{( j )}\left( p \right) f_{\left[ k \right]}^{\mu_{n}( j )}\left( p \right) = \lambda_{\left[ k \right]}^{\mu_{n}}\left( p \right) f_{\left[ k \right]}^{\mu_{n}( i )}\left( p \right) , p \in V_{\mu_{n}}^{+},
= \;\Bigg|{\; \atop{\mbox{det}\atop{\scriptstyle i,j=1,...,2n_a}}}\Bigg(\frac{1}{w^{(a)}_i - z^{(a)}_j}\Bigg) \Bigg| \; \times \;\Bigg|{\; \atop{\mbox{det}\atop{\scriptstyle i,j=1,...,2n_b}}}\Bigg(\frac{1}{w^{(b)}_i - z^{(b)}_j}\Bigg) \Bigg|
\begin{aligned}\mu_{1,2}^T(u)= \|u\|_{L_T^\infty H^s}+\|D_x^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}+\|D_y^{s}\,\partial_x u\|_{L_x^\infty L_{yT}^2}&+\|u\|_{L_T^3 L_{xy}^\infty}+\|\partial_x u\|_{L_T^{\frac94}L_{xy}^\infty}+\| u\|_{L_x^{2}L_{yT}^{\infty}}.\end{aligned}
H(N,N^{a},\Lambda)=\frac{1}{2}\int_{\Sigma}d^{3}xN\frac{q_{ab}}{\sqrt{det(q)}}[\tilde{E}^{a}\tilde{E}^{b}+B^{a}B^{b}]+N^{a}F_{ab}\tilde{E}^{a}+\Lambda\nabla_{a}\tilde{E}^{a}
(x_{m_1}\cdots x_{m_n},x_{k_1}\cdots x_{k_l})= (x_{m_2}\cdots x_{m_n},\Omega_\psi(-m_1)x_{k_1}\cdots x_{k_l}) =\delta_{m_1,k_1}(x_{m_2}\cdots x_{m_n}, x_{k_2}\cdots x_{k_n}) \hskip 75pt+\sum_{r\geq 0}g_{q^{-1}}(r) (x_{m_2}\cdots x_{m_n},x_{k_1+r}\Omega_\psi(-m_1-r)x_{k_2}\cdots x_{k_l}).
L_{e f f}= - R = \frac{1}{2}A_{i j}^{\mu \nu}\partial_{\mu}{\delta y}^{i}\partial_{\nu}{\delta y}^{j}+ B_{i j}^{\mu}{\delta y}^{i}\partial_{\mu}\delta y^{j}+ \frac{1}{2}C_{i j}\delta y^{i}\delta y^{j},
\sup_{z,w\in Y_{0}(N)_{\varepsilon}}\big|g_{Y_{0}(N),\mathrm{can}}(z,w)-g_{Y_{0}(N),\mathrm{hyp}}(z,w)\big|= &O_{\varepsilon,\delta}\bigg(\frac{\big(|\mathcal{P}_{Y_{0}(N)}|+|\mathcal{E}_{Y_{0}(N)}|\big)}{g_{Y_{0}(N)}}\bigg(1+\frac{1}{\lambda_{Y_{0}(N),1}}\bigg)\bigg)=O_{\varepsilon,\delta}(1);
{\mathrm{Re}(e^{-i\alpha}w)}= \zeta_1\left(||f_1||^2-2||f_2||||f_3||\delta_1\cos{(\theta_1-\frac{\pi}{3})}\right) +\zeta_2\left(||f_2||^2-2||f_1||||f_3||\delta_2\cos{(\theta_2-\frac{\pi}{3})}\right) +\zeta_3\left(||f_3||^2-2||f_1||||f_2||\delta_3\cos{(\theta_3-\frac{\pi}{3})}\right),
L(\mu,\lambda)=(\frac{\lambda}{2\pi})^{\frac{n}{2}}(\prod_{i=1}^{n}\frac{w_{i}}{X_{i}^{3}})^{\frac{1}{2}}exp(\frac{\lambda}{\mu}\sum_{i=1}^{n}w_{i}-\frac{\lambda}{2\mu^{2}}\sum_{i=1}^{n}w_{i}X_{i}-\frac{\lambda}{2}\sum_{i=1}^{n}w_{i}\frac{1}{X_{i}}).
\limsup_{n \rightarrow \infty}\| F_\alpha-R_{n,\textup{\textbf{m}},\alpha}\|_{K(\varepsilon)}^{1/n}\leq \frac{\|\Phi\|_K}{\rho_{|\textup{\textbf{m}}|}(\textup{\textbf{F}})},\alpha=1,2,\ldots,d,
S_{i_\ell}=\{i_1,i_2,\ldots,i_{\ell-1},i_\ell-1,i_{\ell+1}-1,i_{\ell+2}-1,\ldots,i_{s}-1\}, \widehat{S}_{i_\ell}=\{i_1,i_2,\ldots,i_{\ell-1},\widehat{i_\ell},i_{\ell+1}-1,i_{\ell+2}-1,\ldots,i_{s}-1\},
V_{i}[t+1]=\{\begin{matrix}V_{i}^{R}&ifX_{i}[t]=1\\ \mu_{i}V_{i}[t]&ifX_{i}[t]=0\end{matrix}\}+E_{i}[t]+\sum_{j\in I\backslash\{i\}}w_{j\rightarrow i}X_{j}[t]
\Vert f\Vert_*:=\max\Bigg(\Vert f\Vert,\sup_{j\geqslant 0}2^{-j-1}\sup_{k_0,\dots,k_j\geqslant 0}\Bigg\Vert \prod_{i=0}^{j}\big(\rho(g_{k_i})-\mathrm{Id}\big)f\Bigg\Vert\Bigg)
\forall x \in X^*, \ \langle a_{\mu},x \rangle = \int_X \langle x, z\rangle\mu(dz) \forall x,y \in X^{*}, \langle y, K_{\mu}x\rangle = \int_X \langle x - a_{\mu},z\rangle \ \langle y - a_{\mu},z\rangle\mu(dz)
\frac{d\tilde{u}_k}{dt}= - u_0\sum_{i\in F\cup G}\sum_{j\in F}\kappa_i \tilde{u}_je_{ijk}&+tu_0^2\sum_{i\in F\cup G}\sum_{j\in G}\sum_{s\in F\cup G}\sum_{t\in F}\kappa_i\kappa_s e_{ijk}e_{stj}\tilde{u}_{t}, k \in F
\sum_{x\in\Bbb F_{q^2}}f(x)^{1+(q-2)q}=-a^{1+(q-2)q}\biggl[ \sum_{-j-l-1=-q}\binom{q-2}j\binom jl a^{-qj}b^{q(j-l)}\cr +\sum_{-j-l=-q}\binom{q-2}j\binom jl a^{-1-qj}b^{1+q(j-l)}\cr +\sum_{-j-l+1=-q,1}\binom{q-2}j\binom jl a^{-1-qj}b^{q(j-l)}\biggr].
M_{0}(v) \coloneqq v-\frac{1}{v^2}\int_0^vv'^2dv'=\frac{2}{3}v, M_{1}(v) \coloneqq\frac{1}{v^2}\int_0^v\left(1-e^{-\int_{v'}^vA(v'')dv''}\right)v'^2dv', M_{2}(v) \coloneqq -\frac{A(v)}{v}\int_0^ve^{-\int_{v'}^vA(v'')dv''}v'^2dv'
G_{m}( x_{1}, x_{2}, x_{3}, x_{4}) = [ \prod_{i = 1}^{m}\Gamma ( \alpha_{i}) ]^{- 1}r_{1 2}^{\alpha_{4}- \frac{1}{2}D}r_{1 4}^{\frac{1}{2}D - \alpha_{1}- \alpha_{4}}r_{2 4}^{\frac{1}{2}D - \alpha_{2}- \alpha_{4}}r_{3 4}^{- \alpha_{3}}F ( u , v )
{\cal K}_{{\hat{\imath}}_{1}\dots{\hat{\imath}_{4}}}^{( 4 )}={\frac{1}{2 ! \sqrt{\mathrm{d e t \Pi^{( A )}}}\sqrt{1 + e^{2 \phi}k^{- 2}( i_{k}C^{( 1 )})^{2}}}}\Pi_{{\hat{\imath}}_{1}{\hat{\jmath}}_{1}}^{( A )}\dots \Pi_{{\hat{\imath}}_{4}{\hat{\jmath}}_{4}}^{( A )}\epsilon^{{\hat{\jmath}}_{1}\dots{\hat{\jmath}}_{6}}{\cal K}_{{\hat{\jmath}}_{5}{\hat{\jmath}}_{6}}^{( 2 )}\, ,
Q_1(g,\varphi) = \tfrac{1}{2}\cos^2(\theta) (\alpha_1\otimes\alpha_1)_0 + \tfrac{1}{2}\sin^2(\theta) (\alpha_2 \otimes \alpha_2)_0, Q_2(g,\varphi) = -|\alpha_1|^2 \cos(\theta) e^{\alpha_1(x)\omega}\psi_1 - |\alpha_2|^2 \sin(\theta) e^{\alpha_2(x)\omega}\psi_2 &+\big( \cos^2(\theta) |\alpha_1|^2 + \sin^2(\theta) |\alpha_2|^2\big)\varphi.
P_{\mathrm{c},\mathrm{L}}(T) =\int_{0}^{\infty}\mathbb{P}\left[C_\mathrm{L}h_0M_\mathrm{r}M_\mathrm{t}x^{-\alpha_\mathrm{L}}>T\left(\sigma^2+I_\mathrm{L}+I_\mathrm{N}\right)\right] &\hat{f}_\mathrm{L}(x)\mathrm{d}x,
f= \sum_jm_j\phi_{11}^*(e_j^*)y_1^*+n_j\phi_{21}^*(e_j^*)y_1^* &+\sum_jm_j\phi_{12}^*(e_j^*)y_2^*+n_j\phi_{22}^*(e_j^*)y_2^* &+\sum_i m_i y_1^*e_i^* + n_i y_2^*e_i^* = \sum_i m_i(y_1^*e_i^*+\phi_{11}^*(e_i^*)y_1^*+\phi_{12}^*(e_i^*)y_2^*) &+n_i(y_2^*e_i^*+\phi_{21}^*(e_i^*)y_1^*+\phi_{22}^*(e_i^*)y_2^*),
S_{n}:=\sum_{i=0}^{n-1}\alpha_{n-1-i}\big\{(-1)^{i}\beta_{i}-\sigma\beta_{i}\sigma\big\}, \beta_{i}:=\sum_{j=0}^{N-2}\frac{\tau((j,N))}{(u-x_{j})^{i+1}}, \alpha_{2n}(x(u,0)) =\frac{\kappa}{2n}S_{2n}, \alpha_{2n+1}(x(u,0)) =\rho\left(\frac{\kappa}{2n+1-2\kappa},\frac{\kappa}{2n+1+2\kappa}\right) S_{2n+1}.
h_{2i}^{2j}=\frac{(-1)^{j+i}(2j)!}{\prod\limits_{d=0,\;d\neq i}^{\lceil n/2\rceil-1}((2i)^2-(2d)^2)}\frac{\prod\limits_{d=0,\;d\neq i}^{\lceil n/2\rceil-1}((2i)^2-(2d)^2)}{\prod\limits_{d=0,\;d\neq i}^{j}((2i)^2-(2d)^2)}=\frac{(-1)^{j+i}(2j)!}{(2i)!!(-1)^{j-i}(2j-2i)!!\frac{(2j+2i)!!}{(2i-2)!!4i}}& =\frac{2^{1-2j}(2j)!}{(j-i)!(j+i)!}=2^{1-2j}\binom{2j}{j-i},
\begin{array}{l l}{{J_{0}^{3}: \lambda , \omega > = \omega : \lambda , \omega >}}&{{}}\\{{\overline{{{J}}}_{0}^{2}: \lambda , \omega > = ( - \frac{1}{2}+ i \lambda ) : \lambda , \omega > ,}}&{{\omega , \lambda \r e a l .}}\\ \end{array}
H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}e^{-x^{2}}=(-1)^{n}e^{x^{2}}\frac{n!}{2\pi i}\oint_{\gamma}\frac{e^{-z^{2}}}{(z-x)^{n+1}}dz.
a(r)\le\left( \frac1{|B_{\eta^{k_r+1}}|}\int_{B_{\eta^{k_r}}}|\nabla u(x)|^p\,dx\right)^{\frac1p}=\eta^{-n/p}a(\eta^{k_r}) &\qquad\qquad\le\eta^{-n/p}\big( C(\eta)M+2^{-k_r}a(1)\big)\le C(M,\eta)(1+a(1)).
\begin{aligned}\Delta(x) = 1 \otimes x + x \otimes 1 -\frac{3}{2}x \otimes x + \frac{1}{2}y \otimes y, \Delta(y) = 1 \otimes y -\frac{3}{2}x \otimes y + y \otimes 1 -\frac{3}{2}y \otimes x.\end{aligned}
\eqref{Gfourier5}=\int_{x_0}^x \frac{c_0}{2}\frac{\sin (2\tilde{\theta}(y,{E})-2\tilde{\theta}(y,\hat{E}))}{(1+y-b)}dx+\sum_{k=1}^\infty c_k \cos(2\pi ky)\frac{\sin (2\tilde{\theta}(y,{E})-2\tilde{\theta}(y,\hat{E}))}{(1+y-b)}dy +\sum_{k=1}^\infty d_k \sin(2\pi ky)\frac{\sin (2\tilde{\theta}(y,{E})-2\tilde{\theta}(y,\hat{E}))}{(1+y-b)}dy.
\begin{array}{l r}{{F_{3 4}( x_{3}, x_{4}) = 3 D | \phi_{2}( x_{3}, x_{4}) |^{2}}}&{{}}\\{{( D_{3}- i D_{4}) \phi_{2}( x_{3}, x_{4}) = 3 D 0}}&{{\hspace{6 . 4 c m}( 9 )}}\\ \end{array}
\sum_{i=1}^{s_{n}}\mbox{Var}(\delta_{i,n})=\sum_{i\in \Upsilon_{3}}\frac{\sigma_{n}^{2}(I_{i,n})}{\sigma_{n}^{2}(C_{n})}\leq 4\sum_{i\in \Upsilon_{3}}\frac{{\bf P}(I_{i,n})\,\sigma^{2}}{\sigma^{2}}\leq 4.
|\partial_\sigma c(s,\sigma)| \leq C\frac{|\partial_\sigma(r_1-r_2)|}{(\sqrt{s}+\alpha |\sigma|)^2}&+C\frac{(|\partial_\sigma r_1|+|\partial_\sigma r_2|+\alpha)|r_1-r_2|}{(\sqrt{s}+\alpha |\sigma|)^3}.
I = \frac{1}{2}\frac{\mu}{g^{2}}tr\int d^{3}x [-\mu(v_{m}- U^{-1}\partial_{m}U) (v^{m}- U^{-1}\partial^{m}U) + \epsilon^{mnp}v_{m}(\partial_{n}v_{p}+ \frac{2}{3}v_{n}v_{p})] ,
\mbox{\mathversion{bold}$X$}(\xi ,t)={}^{t}(R_{1}(t)\cos (\xi), R_{1}(t)\sin(\xi), z_{1}(t)), \mbox{\mathversion{bold}$Y$}(\xi ,t)={}^{t}(R_{2}(t)\cos (\xi), R_{2}(t)\sin (\xi), z_{2}(t)),
\Vert J_{44}^{\left( 7\right)}\Vert_{p}=\mathcal{O}\left( 1\right)(v(|z_{1}|)+(v(|z_{2}|))\left( \frac{\omega (h_{1})}{v(h_{1})}+\frac{\omega(h_{2})}{v(h_{2})}\right) .
(R_{s,t}^{n+1})^{1,1}=\Gamma_{s,t}^{n+1,1}+\int_{t}^s \Gamma_{s,u}^{n+1,1}\left\{\sum_{j=2}^{n+1}a_u^{1,j}(R_{u,t}^{n+1})^{j,1}\right\}du,
S_{b h}~ \equiv ~ \log{\cal N}( p ) ~ \approx ~{\frac{A_{S}}{4 l_{P}^{2}}}~ - ~{\frac{3}{2}}~ \log \left({\frac{A_{S}}{4 l_{P}^{2}}}\right) ~ + ~ c o n s t . ~ + ~ O ( A_{S}^{- 1}) .
|k(t_j)| \le Var(\mathbb{K}_h(x -X_0))^\frac{1}{2}Var(\mathbb{K}_h(x -X_{t_j}))^\frac{1}{2}\le \int_{\mathbb{R}^d}(\mathbb{K}_h (x -y))^2 \pi(y) dy \le \frac{c}{\prod_{l = 1}^d h_l}.
\vec{v}_{1}-\vec{v}_{2}=-D_{12}(\frac{n^{2}}{n_{1}n_{2}}\nabla(\frac{n_{1}}{n})+\frac{m_{2}-m_{1}}{m}\nabla(lnP)+\alpha_{T}\nabla(lnT)-\frac{m_{1}m_{2}}{mkT}(\vec{a}_{1}-\vec{a}_{2}))
\begin{array}{lcl}a_2 = \frac{-1}{6}u_2 a_3 = \frac{1}{3}\theta u'_2 a_4 = -\frac{4}{9}{\theta}^{2}u''_2+\frac{7}{72}u_{2}^{2}a'_5 = \frac{4}{9}{\theta}^{3}u^{(4)}_2 - \theta(\frac{17}{18}u''_{2}u_{2}+ \frac{7}{9}{u'_2}^{2})+\frac{1}{9}u'_{3}u_2 \end{array}
\begin{cases}Cov_N(\delta_i, y_i) = \frac{1}{N}\sum\limits_{i\in U}\delta_i y_i - \big( \frac{1}{N}\sum\limits_{i\in U}\delta_i \big) \big( \frac{1}{N}\sum\limits_{i\in U}y_i \big) = 0 E_N(\delta_i) = \bar{p}_N \equiv \sum\limits_{i\in U}\delta_i/N > 0 \end{cases}
(|(-\Delta_x)^{1/6}\widetilde u_0|^p)^{1/p}_{Q_{1, R}}\le N (d, p, \delta) ( 2 \nu r) \sum_{k = 0}^{\infty}2^{-k}\big(\sum_{i = 1}^d(|\widetilde f_i|^p)^{1/p}_{Q_{1, 2^{k+ 1}R}}+ \lambda^{-1/2}(|\widetilde g|^p)^{1/p}_{Q_{1, 2^{k+ 1}R}}\big) .
(x_{1}+x_{2}+x_{3})^{n}=\sum_{k_{1}+k_{2}+k_{3}=n}(\begin{matrix}n\\k_{1},k_{2},k_{3}\end{matrix})x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}};k_{1},k_{2},k_{3},n\in\mathbb{N}_{0}
\begin{bmatrix}1 a \sum_i (a_i,b_i)\otimes (x_i,y_i) 1 b 1 \end{bmatrix}\mapsto\begin{bmatrix}1 aA+bC \sum_i (a_i,b_i)T\otimes (x_i,y_i)T 1 aB+bD 1 \end{bmatrix}.
\langle \sigma_{n_{1}, i_{1}}. . . \sigma_{n_{s}, i_{s}}\rangle_{g}= \int_{{\cal M}_{g , s}}\int e^{- S}\prod_{k}e^{\alpha_{k}{\tilde{\pi}}( z_{k})}\prod_{a ,{\bar{a}}= 1}^{3 g - 3}G ( \chi_{a}){\bar{G}}({\bar{\chi}}_{\bar{a}}) \prod_{j = 1}^{s}\int_{\Sigma}\sigma_{n_{j}, i_{j}}^{( 2 )}
H = -\frac{e}{2}\pi^2+\imath\pi_\mu\psi^\mu\chi-\left(\frac{1}{8!}\epsilon_{\nu\mu\rho_2\ldots\rho_9}\pi^\mu\psi^{\rho_2}\psi^{\rho_9}+\frac{\alpha}{2^8}\pi_\nu\right) b^\nu\;.
| z , \gamma , n_{1}\rangle = e^{- \frac{1}{2}| z |^{2}}T^{\dagger}e^{- \frac{1}{4}\operatorname{l n}\gamma ( a_{2}^{2}- ( a_{2}^{\dagger})^{2})}e^{z a_{2}^{\dagger}}| 0 , n_{1}\rangle
I_{h,k,n}(x) = \frac{1}{2 \pi n^{1/3}}\,\int_{- \frac{2 \pi n^{1/3}}{m+m'}}^{\frac{2 \pi n^{1/3}}{m+m''}}\, \exp\left[n^{2/3}\left(\frac{L_m(x)^3}{2 ( \Re L_m(x) - iz)^2}+ ( \Re L_m(x) - iz)\right)\right] \, & \times\exp\left[g_{h,k}\left( x, \frac{1}{n^{1/3}}( \Re L_m(x) - iz) \right)\right]\,dz .
\widehat{\left(i\ \epsilon\right)}\boxtimes\widehat{\left(j\ \epsilon_{1}\right)}=\widetilde{\left(\frac{i+j}{2}\ \epsilon+\epsilon_{1}\right)}+\widetilde{\left(k+\frac{i+j}{2}\ \epsilon+\epsilon_{1}+1\right)}+\sum_{1\le r\le2k-1,r\ \mbox{even}}\left(i+r\j-r\right)
m_{i,j}={\small \det \begin{pmatrix}a_i 1 0 1 a_{i+1}1 \ddots \ddots \ddots 1 a_{j-1}1 0 1 a_{j}\end{pmatrix}}, begin{align*}0.5em]m_{i,j}=a_j m_{i,j-1}-m_{i,j-2}, m_{i,j}=a_i m_{i+1,j}-m_{i+2,j}, begin{align*}0.5em] m_{i,j}=m_{i,k-1}m_{k,j}-m_{i,k-2}m_{k+1,j}.
\delta\left(\mathrm{dvol}_ge^{-2\phi}\left(R+4|d\phi|^2-\frac{1}{2}|H_3|^2\right) \right) =\delta (e^{-2\phi})\left(\mathrm{dvol}_g\left(R+4|d\phi|^2-\frac{1}{2}|H_3|^2\right)\right) &\quad+4e^{-2\phi}\delta (d\phi\wedge*d\phi) &=-2e^{-2\phi}\delta\phi\left(\mathrm{dvol}_g\left(R+4|d\phi|^2-\frac{1}{2}|H_3|^2\right)\right) &+ 8e^{-2\phi}d\delta\phi\wedge*d\phi.
\delta_1:=(\gamma_1,\gamma_1,\gamma_0,\ldots,\gamma_0), \delta_2:=\delta_1^h=(\gamma_2,\gamma_2,\gamma_0,\ldots,\gamma_0), &\delta_3:=\delta_2^h=(\gamma_3,\gamma_1,\gamma_0,\ldots,\gamma_0), \delta_4:=\delta_3^h=(\gamma_4,\gamma_2,\gamma_0,\ldots,\gamma_0).
\left\Vert \tilde{J_{1}}\right\Vert_{\mathbb{L}^{2}(B)}= \mathcal{O}\left(a^{5}\; \left\Vert \overset{1}{\mathbb{P}}\left(\tilde{E}_{j}^{T}\right) \right\Vert_{\mathbb{L}^{2}(B)}\right).
=(x_{2}-x_{1})+F_{1}\left(\frac{T^{2}(x_{2})}{2}-\frac{T^{2}(x_{1})}{2}+T(x_{2})s\right)+F_{2}\frac{s^{2}}{2}=(x_{2}-x_{1})-F_{1}\frac{s^{2}}{2}+F_{2}\frac{s^{2}}{2}=(x_{2}-x_{1})+\frac{s}{2}V(x_{1},x_{2}).
\partial^2_\mu D^\prime(p||p_{\mu,\sigma}) = \frac{1}{\mu_0^2}+ \frac{3\mu^2 + \sigma^2 - \mathbb{E}_p[x^2]}{\sigma_0^4}, \partial_\mu \partial_\sigma D^\prime(p||p_{\mu,\sigma}) = \frac{2\mu\sigma}{\sigma_0^4}, \partial^2_\sigma D^\prime(p||p_{\mu,\sigma}) = \frac{\mu^2 + 3\sigma^2 - \mathbb{E}_p[x^2]}{\sigma_0^4}.
D_{1}A^{\prime \prime}+ D_{2}B^{\prime \prime}+{\frac{D_{1}}{2}}{A^{\prime}}^{2}+{\frac{D_{2}}{2}}{B^{\prime}}^{2}={\frac{4 \kappa^{2}}{D - 2}}\left( - \Lambda +{\frac{F^{2}}{4 g^{2}}}\mathrm{e}^{- 2 B}\right) ,
I\phi(\cdot,x):= \sum_{j=1}^{l}\int_{\mathbb{R}^p}\{\phi(\cdot,x+ \gamma^{j}(x,z_j))-\phi(\cdot,x)-\nabla\phi(\cdot,x) \gamma^{j}(x,z_{j})\}\nu_{j}(dz_{j}),\;{\forall x\in \mathbb{R}^p.}
\alpha_N^\dagger =\alpha^\dagger\Big(e^{-2\pi \sqrt{N/3}}\Big),\intertext{and}C_N({q}) = \frac{1}{a^2(q)a^2(q^N)}\,\operatorname{det}\!\left(\begin{matrix}a(q) a(q^N){q}\dfrac{da(q)}{d{q}}q\dfrac{da(q^N)}{d{q}}\end{matrix}\right).
g_\gamma \,=\,\prod_{1\leq i < j\leq\ell}(1-R_{ij}) k_\gamma\,,\;\,\text{where\( k_\gamma := k_{\gamma_1}^{(0)}k_{\gamma_2}^{(1)}\cdots k_{\gamma_\ell}^{(\ell-1)}\,.\)}
R^{(0)}_{i\bar{j}}=R^{(0)}{}^{k}_{\phantom{1}k i\bar{j}}=-R^{(0)}{}^{k}_{\phantom{1}k\bar{j}i}&=-\partial_{\bar{j}}\partial_{i}(\log{g_{m\bar{n}}})^{k}_{\phantom{1}k}=-\partial_{\bar{j}}\partial_{i}tr(\log{g_{m\bar{n}}}) &=-\partial_{\bar{j}}\partial_{i}\log{\mathcal{G}}
\Lambda = 1, M = 32, \Lambda_r = \begin{cases}0.286+0.001r 1 \leq r \leq 14, 0.300 + 0.025(r-14) 15 \leq r \leq 22, 0.5 + 0.05(r-22) 23 \leq r \leq 32, \end{cases}
V_1|_{\phi=0}=\frac{\partial V_1}{\partial \phi}|_{\phi=0}=\frac{\partial^2 V_1}{\partial \phi^2}|_{\phi=0}=\frac{\partial^3 V_1}{\partial \phi^3}|_{\phi=0}=\frac{\partial^4 V_1}{\partial \phi^4}|_{\phi=0}=0
\Delta^{(2)}p^- = \frac{g^2}{4\pi}\int_0^{p^+}\frac{dk^+}{p^+-k^+}\frac{\left( \frac{m}{p^+}+\frac{m}{k^+}\right)^2}{p^--\frac{m^2}{k^+}- \frac{\lambda^2}{p^+-k^+}}.
\left( \begin{array}{c}{{\mathrm{p e r t u r b a t i v e \; \; c o n t r i b u t i o n}}}\\{{\mathrm{f r o m \; \; t h e \; \;}U \; \; \mathrm{f i e l d}}}\\ \end{array}\right) = - \frac{N_{c}^{2}- 1}{8 g^{2}}\! \! \int \! \! \frac{d^{3}p}{( 2 \pi )^{3}}S ( p ) \frac{2 g^{2}}{L ( p )}