harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
p_{\alpha}A^{\alpha}\rightarrow \frac{1}{2}\left[ \hat{p}_{\alpha}, A^{\alpha}( \hat{x}) \right]_{+}\, , \, \, \, \, \, \, \, \, p_{\alpha}F^{\alpha \beta}\rightarrow \frac{1}{2}\left[ \hat{p}_{\alpha}, F^{\alpha \beta}( \hat{x}) \right]_{+}\, ,
h(x) \le \Bigl(g_3(x)+\frac{2}{6^4}(3^4x-33)(7-2^4x)+\frac{2}{6^5}(2^5x-13)(101-3^5x) &+\frac{2}{6^6}(2^6x-26)(303-3^6x)\Bigr)\Bigr|_{x=\frac{27}{65}}+ \frac1{10\cdot 6^6}- \eta &=-\frac{1906349521544493873261549971920349}{425623699615692261466802423622110256000}<0,
\cdots{\buildrel Q_{n}\over\longrightarrow}W_{-n+2P,n'}{\buildrel Q_{P-n}\over\longrightarrow}W_{n,n'}{\buildrel Q_{n}\over\longrightarrow}W_{-n,n'}{\buildrel Q_{P-n}\over\longrightarrow}W_{n-2P,n'}{\buildrel Q_{n}\over\longrightarrow}W_{-n-2P,n'}{\buildrel Q_{P-n}\over\longrightarrow}\cdots
\begin{pmatrix}a_{11}a_{12}a_{13}a_{21}a_{22}a_{23}a_{31}a_{32}a_{33}\end{pmatrix}\cdot (z_1, z_2) = \frac{\begin{pmatrix}a_{21}+a_{22}z_1 +a_{23}z_2 a_{31}+a_{32}z_1 + a_{33}z_2\end{pmatrix}}{a_{11}+a_{12}z_1+ a_{13}z_2}
{\frac{\bar{t}}{\bar{t}_{*}}}= \left({\frac{9 - 4 r^{2}}{1 2}}\right) \left[{\frac{2}{3 - 2 r}}\left|{\frac{\eta}{\eta_{*}}}\right|^{( 3 / 2 ) - r}+{\frac{2}{3 + 2 r}}\left|{\frac{\eta}{\eta_{*}}}\right|^{( 3 / 2 ) + r}\right]
\Gamma(A) := \exp\left(L_{\rm uo}(A)\right) \; \exp\left(L_{\rm pf}(A)\right) \; \left( \gamma_0+L_{\rm pp}(A)\right) \; \exp\left(-L_{\rm pf}(A)\right) \; \exp\left(-L_{\rm uo}(A)\right),
\deg{E_0}=(1,0,1,0), \deg{E_1}=(0,1,-1,0), \deg{E_2}=(0,-1,0,-1), \deg{E_3}=(-1,0,0,1), \deg{F_0}=(-1,0,-1,0), \deg{F_1}=(0,-1,1,0), \deg{F_2}=(0,1,0,1), \deg{F_3}=(1,0,0,-1), \deg H_i =(0,0,0,0).
\bar{u}'(t) = \sum_{n=1}^{\infty}u_n'(t)=\sum_{n=2}^{\infty}\frac{-(n-1)\xi_{n-1}}{t^{n}}G_{\alpha+1,\sigma}, A\bar{u}(t) =\sum_{n=1}^{\infty}\frac{A\xi_n}{t^{n}}G_{\alpha,\sigma}.
\mathfrak{R}_{\sigma_{3,7,2,0}}(q;\beta)=(1,6,21,36,51,48,{\bf 26})_{\beta}+ q\beta (6,36,126,216,306,288,156)_{\beta}\hphantom{\mathfrak{R}_{\sigma_{3,7,2,0}}(q;\beta)=}{}+ q^2 \beta^3 (20,125,242,403,460,289)_{\beta}+ q^3 \beta^5 (6,46,114,204,{\bf 170})_{\beta}, \mathfrak{R}_{\sigma_{3,7,2,0}}(q;1) = (189,1134,1539,540)_{q}=27 (7,42,57,20)_{q}.
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) ,
\frac{1}{k^2(\xi)+\alpha^2(\xi)}=\left\langle \nu^H, \xi^H\right\rangle^2 + \left\langle \eta, \xi^H\right\rangle^2 &=\left\langle \nu^H, \xi^H\right\rangle^2 + \left\langle \eta, \xi^H\right\rangle^2 + \sum_{j=1}^{n-1}\left\langle V_j, \xi^H\right\rangle^2 + \left\langle W_j, \xi^H\right\rangle^2=|\xi^H|^2,
\begin{cases}(1) \mathfrak{m}_\lambda^{(k)}\neq 0,0\leq k\leq m+n (2) E_{i,i+1}^{(M)}\, \mathfrak{m}_\lambda^{(k)}=0 , 1\leq i\leq m+n-1,\; i \neq m, (3) E_{-\beta_i}\, \mathfrak{m}_\lambda^{(k)}=0=E_{\beta_j}\, \mathfrak{m}_\lambda^{(k)}, 1\leq i\leq k< j\leq mn.\end{cases}
~ \begin{aligned}\phi\Bigl( \Pi\bigl((a_{-i},s_i); p^{\tau_{ij}}\bigr) \Bigr) - \phi\Bigl( \Pi\bigl((a_{-i},u_i); p^{\tau_{ij}}\bigr) \Bigr) \leq \phi\Bigl( \Pi\bigl((a_{-i},s_i); p^{\tau_{ij+1}-1}\bigr) \Bigr) - \phi\Bigl( \Pi\bigl((a_{-i},u_i); p^{\tau_{ij+1}-1}\bigr) \Bigr), \end{aligned}
{S_L}\left[\varphi,\hat{g}\right]=-{{d-26}\over{48\pi}}\int{d^2}\xi\sqrt{\hat{g}}\left({1\over{2}}\varphi\hat{\Delta}\varphi+\hat{R}\varphi\right)+{\mu_1^2}\int{d^2}\sqrt{\hat{g}}{e^{\varphi}},
\begin{array}{rcl}E = E_0+E_{q(J)-{\rm vib}}(0) = E_0+\displaystyle hc\nu_{\rm vib}\frac{1}{2\sinh(\gamma(J)/2)}\sinh\left(\gamma \left(J\right)\left({1\over 2}+c(J)\right)\right) begin{align*}2mm] +\displaystyle \frac{h^2}{8\pi^2I}J(J+1)~,\end{array}
\frac{d^2q}{dt^2}=\frac 12\left(\frac 1q+\frac 1{q-1}+\frac 1{q-t}\right)\left(\frac{dq}{dt}\right)^2-\left(\frac 1t+\frac 1{t-1}+\frac 1{q-t}\right)\frac{dq}{dt}+\frac{q(q-1)(q-t)}{t^2(t-1)^2}\left(\alpha+\beta\frac t{q^2}+\gamma\frac{t-1}{(q-1)^2}+\delta\frac{t(t-1)}{(q-t)^2}\right).
J(u)=\frac{\sum \alpha_{i}^{2}\int L_{g_{0}}\varphi_{i}\varphi_{i}}{(\sum_{i}\alpha_{i}^{\frac{2n}{n-2}}\int K\varphi_{i}^{\frac{2n}{n-2}})^{\frac{n-2}{n}}}+o_{\varepsilon}(1)=c_{0}\frac{\sum_{i}\alpha_{i}^{2}}{(\sum_{i}\alpha_{i}^{\frac{2n}{n-2}}K_{i})^{\frac{n-2}{n}}}+o_{\varepsilon}(1)
[v_1v_2] =[v_1]+[v_2]+\eta.[v_1].[v_2] &= [v_1]+[v_2]+\eta.[c_1^2s].[c_2^2(1-s)] &= [v_1]+[v_2]+\eta.([c_1^2]+[s]).([c_2^2] + [1-s]) &= [v_1]+[v_2],
W ( C )_{\mathrm{m o n}}^{\cal N}= \frac{1}{N}{\,}\mathrm{t r}{\,}\exp \left( \frac{i g}{2}\vec{H}\int d^{3}x \vec{F}_{\mu \nu}^{\cal N}\Sigma_{\mu \nu}\right) = \frac{1}{N}\sum_{\alpha = 1}^{N}W_{\alpha}^{\cal N}.
[L^{p_0}, L^{p_1}]_{\theta,r}=[L^{p_0,r_0}, L^{p_1,r_1}]_{\theta,r}=L^{p,r}, \frac1p=\frac{1-\theta}{p_0}+\frac{\theta}{p_1}, ~~1\leq r\leq \infty;
\frac{(h^t)^{n-p}}{(n-p)!}\circ (\ast\frac{h^p}{p!})=\det h\frac{g^{n-p}}{(n-p)!}\,{\rm and}\,\, \big( \ast \frac{(h^t)^p}{(p)!}\big)\circ \frac{h^{n-p}}{(n-p)!}=\det h\frac{g^{n-p}}{(n-p)!}.
\begin{array}{ll}\nabla_{e_1}e_1 = \nabla_{X}X = e^{-2x_2}e_2, \nabla_{e_2}e_2 = \nabla_{U}U = 0, \nabla_{e_1}e_2 = \nabla_{X}U = -e_1, \nabla_{e_2}e_1 = \nabla_{U}X = -e_1.\end{array}
I_{\mathrm{g}}( \tau ) = ( \det P^{+}P )^{1 / 2}( \det \Delta_{\mathrm{g}})^{- 1 3}= e^{c ( 2 \mathrm{g}- 2 )}\left( \frac{d}{d s}Z ( s ) |_{s = 1}\right)^{- 1 3}Z ( 2 ) ,
\Gamma_{\underline{\alpha}\underline{\beta}}^{a}= \left( \begin{array}{c c}{- \gamma_{\alpha \beta}^{a}C_{q r}}&{0}\\{0}&{\tilde{\gamma}^{a \alpha \beta}C^{q r}}\\ \end{array}\right) , \quad a = 0 , 1 , . . . , 5 ,
\frac{d}{dx}\frac{1}{2}\rho^2(x)= h(x)h'(x)+(m-p)\tanh x\frac{h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}h'^2(x) &-\frac{m-1}{2}\frac{(3-p)h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}\sin 2h(x)h'(x) \leq C\rho^2(x)
{\bf Per}_k = \{(- t^{x_1}, \dots,- t^{x_n}, t^{y_1}, \dots, t^{y_n}) \cup ( t^{x_1}, \dots,t^{x_n}, -t^{y_1}, \dots, -t^{y_n}) : & (t^{x_1}, \dots, t^{x_n}, t^{y_1}, \dots, t^{y_n}) \in \tilde{Per}_k\}
\log\varrho\left(L_0^{\phi},L_1^{\phi}\right)+2^{-n}N\mu([\omega]) \leq \log\varrho\left(L_0^{\phi_{n}},L_1^{\phi_n}\right) &=\Lambda(\phi_{n},\nu_{n}) &\leq \Lambda(\phi,\nu_{n}) +\Lambda(2^{-n}\psi,\nu_{n}) & \leq \log\varrho\left(L_0^\phi,L_1^\phi\right) + 2^{-n}N\nu_{n}([\omega]),
\sigma^{(1,1,0)}\star \sigma^{(4, 2, 1)}= (\alpha_1+\alpha_2+\alpha_3)(\alpha_1+\cdots+\alpha_6)\sigma^{(4,2,1)}+ (\alpha_1+2\alpha_2+2\alpha_3+\alpha_4+\alpha_5+\alpha_6)\sigma^{(4, 2,2)}+(\alpha_1+\cdots+\alpha_6)\sigma^{(4, 3,1)}+\sigma^{(4, 3,2)}+ q\sigma^{(1,1,0)}+q\sigma^{(2, 0,0)}+(\alpha_1+\alpha_2+\alpha_3)q\sigma^{(1,0 ,0)}
\begin{aligned}\Bigl|\int_{t_{i-1}}^{t_i}(\mu(W_s)-\mu(W_{t_{i-1}}))\, ds \Bigr| \le c\,\Bigl(\int_{t_{i-1}}^{t_i}|W_s-W_{t_{i-1}}|\, ds +\int_{t_{i-1}}^{t_i}\sum_{\ell=1}^k 1_{D_\ell}(W_s,W_{t_{i-1}})\, ds \Bigr)\end{aligned}
\phi(-q^{{\frac{j-i}{2}}}x_{j-1}x_{j-2}\cdots x_{i})=\displaystyle \sum_{m\geq 0}\frac{q^{\frac{(2-(j-i))m^{2}}{2}}x_{i}^{m}x_{i+1}^{m}\cdots x_{j-1}^{m}}{(q)_{m}}.
\mathcal{A}^{(2)}= -\big((n+1-|v_n|)\c(v_n),\psi_k'(w_n)\big) \leq C(\gamma,\delta,n) \int_{0}^{1}1\cdot \psi_k'(w_n)\d x & \leq C(\gamma,\delta,n)\int_{0}^{1}|w_n(x)|_+^2\d x.
{\widetilde Q}^T \begin{pmatrix}c -c \end{pmatrix}=\begin{pmatrix}(Q_+ + Q_-)^Tc -(Q_- + Q_+)^Tc \end{pmatrix}=\begin{pmatrix}Q^Tc -Q^Tc \end{pmatrix}=\begin{pmatrix}c -c \end{pmatrix},
val(1-fr) =val(1-fg_{\nu}-(1-fr))=val(f(r-g_{\nu}))= &=val(f)+val(r-g_{\nu})<val(f)+val(r-g_{\mu})= &=val(f(r-g_{\mu}))=val(1-fg_{\mu}-(1-fr))=val(1-fr),
|F(\psi( s))(y)|\lesssim e^{-\lambda_{l}s}y^{-\gamma}\begin{cases}y^{-2}\Gamma^{\gamma}y<\Gamma e^{-\omega_l s}y^{-2}\Gamma^{-2\gamma}+y^{2l}e^{-(1-2\sigma)\lambda_{l}s}y\ge\Gamma e^{-\omega_{l}s}\end{cases}
\sum_{n=0}^\infty x^{2n}\,\frac{e^{-|z|^2}}{(n!)^2}\,\left\{\sum_{\mu,\nu=1}^M\left(e^{i\omega_\mu}|z^+_{\mu+}|^2+e^{-i\omega_\mu}|z^+_{\mu-}|^2\right)\left(e^{i\omega_\nu}|z^-_{\nu+}|^2+e^{-i\omega_\nu}|z^-_{\nu-}|^2\right)\right\}^n\ \ \ .
\int_0^1 (\{ax\}- \{\nu a x\})^2 \mathrm{d}x = \frac{1}{a}\int_0^a (\{x\}- \{\nu x\})^2 \mathrm{d}x = \int f \mathrm{d}\mu_a \underset{a\to\infty}{\longrightarrow}\frac{1}{6}.
\phi_i(\tau) = \big(\pi_i^\mu (\tau) + \frac i2 \xi_i^\mu(\tau)\big)\int{d^3\sigma \rho_\mu (\tau, \vec{\sigma})\equiv \big(\pi_i^\mu(\tau) + \frac i2 \xi_i^\mu(\tau)\big){p_s}_\mu}\approx 0,
\mathbb{P}(h^{D_{N}}(x)\geq a_{N}+b) =(1+o(1))\frac{1}{\sqrt{2\pi}}\frac{\sqrt{\gamma\log N+s_{D}(x)+o(1)}}{a_{N}+b}e^{-\frac{(a_{N}+b)^{2}}{2(\gamma\log N+s_{D}(x)+o(1))^{2}}}&=(1+o(1))\frac{1}{4\lambda\sqrt{\pi}\sqrt{\log N}}e^{-\frac{a_{N}^{2}}{2\gamma\log N}}e^{-\pi\lambda b}e^{\frac{4\lambda^{2}s_{D}(x)}{\gamma}}.
\frac{H}{p}w w^{\prime}\Bigg |_{r_{1}}^{r_{2}}= \int_{r_{1}}^{r_{2}}d r \Bigg \{\frac{H}{p}( w^{\prime})^{2}- \frac{p}{H}u^{2}w^{2}- \frac{1}{p r^{2}}w^{2}( 1 - w^{2}) \Bigg \}.
K^g_t =\sum_{s \leq t}\mathbf{1}_A \left(\omega_{s-}\right)g(\omega_{s}) +\int_0^t \mathbf{1}_A(\omega_{s-})\bigl(-\Delta g + b\cdot\nabla g \bigr)(\omega_s)ds &=\sum_{s \leq t}\mathbf{1}_A \left(\omega_{s-}\right)g(\omega_s).
\bar \zeta (s) =\int^{\infty}_{-\infty}\frac{dk}{2\pi}\int_{0}^{\infty}\frac{\lambda\,d\lambda}{(k^2+\lambda^2)^s}=\frac{1}{2\sqrt{\pi}}\frac{\Gamma(s-1/2)}{\Gamma(s)}\int_{0}^{\infty}\lambda^{2-2s}d\lambda\,{.}
K_{L , n}^{( 2 )}= \frac 3 4 \frac{\sqrt{\pi}\Gamma ( L + \frac 1 2 ) \Gamma ( N + L + 1 )}{\Gamma ( n + 1 ) \Gamma ( L + 1 ) \Gamma ( 2 L + 2 )}\frac{1 - 4 N^{2}}{( 2 L - 1 ) ( 2 L + 3 )}
\varphi(u) =\langle J^{'}(u),u\rangle =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\left( \frac{u(x)-u(y)}{\vert x-y\vert^{\alpha}}\right)\frac{u(x)-u(y)}{\vert x-y\vert^{\alpha}}\frac{dxdy}{\vert x-y\vert^{d}}& + \int_{\mathbb{R}^{d}}g(u)u dx - \int_{\mathbb{R}^{d}}K(x)f(x,u)u dx.
S_{4}= ( 2 \pi )^{4}\int \left[ \prod_{j = 1}^{2}\frac{d^{\, 4}P_{j}}{( 2 \pi )^{4}}\frac{d^{\, 4}Q_{j}}{( 2 \pi )^{4}}\right] G_{4}\, H \, \delta^{4}\left( \sum_{l}P_{l}+ \sum_{j = 1}^{2}( Q_{j}- P_{j}) \right) .
S_{\mathrm{G F}}= \int \, d^{4}x \{B_{\mu}D_{\nu}B^{\nu \mu}+ \varepsilon_{a b}B^{a}D_{\mu}C^{\mu b}- \frac{1}{2}B_{\mu}B^{\mu}- \frac{1}{2}\varepsilon_{a b}B^{a}B^{b}\},
\sum_{\alpha=1}^t\sum_{\beta=1}^\phi\left(z^{\alpha+r-1}\sum_{i=0}^{\beta-2}z^i+x^{\alpha-1}\sum_{i=\beta-1}^{\phi-1}z^i\right)= \sum_{\alpha=1}^tz^{\alpha-1}\left(\sum_{i=0}^{\phi-1}(i+1)z^i+x^{r+\phi-1}\sum_{i=0}^{\phi-1}\left(\frac{1}{z}\right)^i\right).
S_{t+s}=Ue^{-(t+s)L_1}U^{-1}=Ue^{-t L_1}U^{-1}Ue^{-sL_1}U^{-1}=S_t S_s \lim_{t\to 0}Ue^{-tL_1}U^{-1}f =U\lim_{t\to 0}e^{-tL_1}U^{-1}f=UU^{-1}f=f
\mathbb E\|N_{\infty}-N^n_{\infty}\|^p \eqsim_p \mathbb E[N-N^n]_{\infty}^{\frac p2}= \mathbb E \bigl([N-N^n]^c_{\infty}+ [N-N^n]^q_{\infty}+ [N-N^n]^a_{\infty}\bigr)^{\frac p2}=\mathbb E \bigl([N]^c_{\infty}+ [N-N^n]^q_{\infty}+ [N]^a_{\infty}\bigr)^{\frac p2}\geq \mathbb E \bigl([N]^c_{\infty}+ [N]^a_{\infty}\bigr)^{\frac p2},
\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )}Q(\lambda)=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh2\lambda_{a}\right) .
|-4 b \int_{t+\log^{(\alpha-1)b}(t)+\frac{1}{2}}^{\infty}\frac{1}{x^{2}\log^{b+1}(x)}\frac{1}{(\log^{(\alpha-1)b}(t)+x-t)}\frac{dx}{(1+x-t)^{3}}| & \leq \frac{C}{t^{2}\log^{b+1}(t)}\int_{t+\log^{(\alpha-1)b}(t)+\frac{1}{2}}^{\infty}\frac{dx}{(x-t)^{4}}&\leq \frac{C}{t^{2}\log^{b+1}(t)}
s_{n+2}=3s_{n+1}+24\sum_{j=0}^{n}(\begin{matrix}2n+2\\ 2j+2\end{matrix})s_{n-j}\sum_{k=0}^{j}(\begin{matrix}2j+1\\ 2k+1\end{matrix})s_{k}s_{j-k},s_{0}=1,s_{1}=3,
\left< T_{U U}\right> ={\frac{\mu}{6 \pi U^{2}}}\{\frac{1 - 2 \mu}{( 1 - 4 \mu )^{2}}}; \quad \left< T_{V V}\right> ={\frac{\mu}{6 \pi V^{2}}}\{\frac{1 - 2 \mu}{( 1 - 4 \mu )^{2}}}.
<\sigma_\gamma,\phi>=\int_{\Omega \times \Omega}\mathrm{d}\gamma(x,y)\int_{0}^{1}\phi(\omega_{x,y}(t))\,||{\omega}^\prime_{x,y}(t)||\,\mathrm{d}t\;\;\; \mbox{for all}\;\;\phi\;\in\;C(\Omega)
-C_1(1-\eta)^{\alpha_0}\leq \partial_\xi w+\partial_\tau w\leq \frac{\delta}{2}win [0,T_1]\times[0,X]\times[0,1] , -C_1(1-\eta)^{\alpha_0}\leq\partial_\tau w \leq b\delta(1-\eta)^{\alpha_0}in [0,T_1]\times[0,X]\times[0,1].
\frac{d}{dt}|\rho|^q_{D^{2,q}}\leq C|\nabla u|_\infty|\rho|^q_{D^{2,q}}+C|\rho|_{\infty}|u|_{D^{3,q}}|\rho|^{q-1}_{D^{2,q}}+|\nabla \rho|_\infty |u|_{D^{2,q}}|\rho|^{q-1}_{D^{2,q}},
L_{\omega,\mathbf{c}}(V) =L_{\omega,\mathbf{c}}(V_n)-L_{\omega,\mathbf{c}}(V_n-V)-(L_{\omega,\mathbf{c}}(V_n)-L_{\omega,\mathbf{c}}(V_n-V)-L_{\omega,\mathbf{c}}(V)) &<L_{\omega,\mathbf{c}}(V_n)-6\mu_{\omega,\mathbf{c}}-(L_{\omega,\mathbf{c}}(V_n)-L_{\omega,\mathbf{c}}(V_n-V)-L_{\omega,\mathbf{c}}(V)) &\rightarrow 0
\begin{aligned}v = (x_1 + x_2, S_1 x_1 + S_2 x_2) = (x_1 + x_2, S_1 (x_1 + x_2) + U x_2) & = (x, S_1 x + U_1 x) = (x, (S_1 + U_1)x). \end{aligned}
M_Rf(x,y) :=\frac{1}{y}\int_{x}^{x+y}f(t)dt M_Lf(x,y) :=\frac{1}{y}\int_{x-y}^{x}f(t)dt \Delta Mf(x,y) :=\frac{1}{y}\int_{x}^{x+y}f(t)dt-\frac{1}{y}\int_{x-y}^{x}f(t)dt \Delta m_f(x) :=\sup_{y>0}\vert \Delta Mf(x,y)\vert m_f^{\delta}(x) :=\sup_{0<y<\delta}\vert Mf(x,y)\vert
( \Gamma^{\underline{m}})_{\underline{\beta}}^{\underline{\alpha}}( C \Gamma_{\underline{m}})_{\underline{{\gamma \delta}}}+ ( \Gamma^{\underline{m}})_{\underline{\gamma}}^{\underline{\alpha}}( C \Gamma_{\underline{m}})_{\underline{{\delta \beta}}}+ ( \Gamma^{\underline{m}})_{\underline{\delta}}^{\underline{\alpha}}( C \Gamma_{\underline{m}})_{\underline{{\delta \beta}}}= 0 ,
r=\frac{1}{3Ns^{2}}\sum_{n=1}^{N}\{(x_{n,1}-\overline{x})(x_{n,2}-\overline{x})+(x_{n,1}-\overline{x})(x_{n,3}-\overline{x})+(x_{n,2}-\overline{x})(x_{n,3}-\overline{x})\},
A\left( x\right) =\int_{\mathbb{R}^{d}}K_{R}\left( y\right)\left( \chi_{D}\left( x-y\right) -H_{R}\left( x-y\right) \right)dy B\left( x\right) =\int_{\mathbb{R}^{d}}K_{R}\left( y\right)\left( \chi_{D}\left( x-y\right) +H_{R}\left( x-y\right) \right)dy,
\mathcal{T}_D^k(\langle h_k, P_r(\theta - \cdot)+iQ_r(\theta - \cdot)\rangle) = \langle h_k, (P_r(\theta - \cdot)+iQ_r(\theta - \cdot))(e^{i(\cdot)}- re^{i\theta}+\overline{e^{i(\cdot)}- re^{i\theta}})^k\rangle,
\begin{aligned}\int_0^T \int_{\tau}^{1}\Big( s \Theta (v_x)^2 + s^3 \Theta^3\left(\frac{x-x_0}{a}\right)^2 v^2 \Big) e^{2s \varphi}\, dx dt\le C \int_{0}^T \int_{\omega}\frac{v^2}{a}dxdt,\end{aligned}
\begin{aligned}\rho_t +\nabla \cdot (\rho Q\ast \mathbf{u})=0,\; x\in \mathbb{R}^N, \; t>0, & \mathbf{u}_t + (\mathbf{u}\cdot \nabla)\mathbf{u}=\rho(Q\ast \mathbf{u}-\mathbf{u}), \end{aligned}
\psi_1(u,v,w)= \beta uw+ \alpha v(1+u), \psi_2(u,v,w)=\beta vw+ \alpha u(1+v), \psi_3(u,v,w)= \beta vw+\alpha v\left(1+u\right)\psi_4(v,w)=\beta uw+\alpha u(1+v).
A=\frac{y_2}{W(y_1,y_2)}\frac{1}{R^7u'\sqrt{{u'}^6-(1+\theta){u'}^2+\theta}},B=\frac{y_1}{W(y_1,y_2)}\frac{1}{R^7u'\sqrt{{u'}^6-(1+\theta){u'}^2+\theta}}.
W ={\frac{1}{6 4 \pi^{{\frac{1}{2}}}}}{\frac{\tau}{a}}( \Phi_{0}^{2}a^{2})^{2}L o g ({\frac{1}{M^{2}a^{2}}}) [ 1 + m^{2}a^{2}+ . . . ] -{\frac{c}{1 9 2 ( 2 \pi )^{\frac{1}{2}}}}({\frac{\tau}{a}}( \Phi_{0}^{2}a^{2})^{3})
\hat{R}=\frac{\hat{Y}}{\hat{Z}}=\frac{\sum_{i=1}^{n}w_{i}y_{i}^{\prime}}{\sum_{i=1}^{n}w_{i}z_{i}^{\prime}}\approx R+\frac{1}{Z}\sum_{i=1}^{n}(\frac{y_{i}^{\prime}}{\pi_{i}}-R\frac{z_{i}^{\prime}}{\pi_{i}})
|E[f(F)]-E[f(Z^*)]| \leq E\left[\max_{1\leq i,j\leq d}|C^{ij}-\langle DF_i,-DL^{-1}F_j\rangle_H|\sum_{i,j=1}^p|\partial^2_{i,j}U_0f(F)|\right] &\lesssim\beta\delta^{-1}E\left[\max_{1\leq i,j\leq d}|C^{ij}-\langle DF_i,-DL^{-1}F_j\rangle_H|\right].
\kappa^{\mathrm{mu}}=\frac{1}{\mathcal{E}_\mathrm{s}}(\mathbf{R}_{\mathbf{r}x}^{\mathrm{mu}})^{H}(\mathbf{R}_{\mathbf{rr}}^{\mathrm{mu}})^{-1}\mathbf{R}_{\mathbf{r}x}^{\mathrm{mu}}.
\Delta V = V''\big(\rho\big) + \Delta \rho V'\big(\rho\big) \le V''(\rho) + (n-1)\frac{f_a^\prime(\rho)}{f_a(\rho)}V'(\rho) = (n-1) \frac{f_a^\prime(\rho)}{f_a^n(\rho)}\int_0^\rho a_0(t) f_a^{n-1}(t) dt - a_0(\rho) - (n-1)\frac{f_a^\prime(\rho)}{f_a^n(\rho)}\int_0^\rho a_0(t) f_a^{n-1}(t) dt = -a_0(\rho),
r_{s,p}=\frac{\overline{\alpha'(a_s)}(\overline{b}_1-\overline{a}_s)r_{s,1}+\overline{\alpha'(a_1)}(\overline{a}_1-\overline{b}_1)r_{1,1}+\overline{\alpha'(a_1})(\overline{b}_p-\overline{a}_1)r_{1,p}}{\overline{\alpha'(a_s)}(\overline{b}_p-\overline{a}_s)}
\big\langle C_1\times 0,\,C_1\times 0 \big\rangle_{(F\!H)^2}=0, \big\langle C_1\times 0,\, \widehat C_{12}\big\rangle_{(F\!H)^2}=0; \big\langle 0\times C_2,\, 0\times C_2 \big\rangle_{(F\!H)^2}=0, \big\langle 0\times C_2,\,\widehat C_{12}\big\rangle_{(F\!H)^2}=0.
1 -{\frac{g}{6}}( 1 +{\frac{2}{9}}g \pi \sqrt{3}) \ln [ g ] +{\frac{1}{2 7}}g^{2}\pi \sqrt{3}\ln [ 1 +{\frac{1}{9}}g \pi \sqrt{3}( 1 +{\frac{g}{1 2}}) ] = 0 .
g_d:=-2[\bar{p}, \mathfrak{H}]\frac{\partial_{\alpha}\bar{f}}{z_{\alpha}}-2[\bar{f},\mathfrak{H}]\frac{\partial_{\alpha}\bar{p}}{z_{\alpha}}-2[\bar{p},\mathfrak{H}]\frac{\partial_{\alpha}\bar{p}}{z_{\alpha}}-4p_t.
\left\{\frac{d^{2}}{d \varrho^{2}}+\alpha^{2}\varrho^{2(\alpha-1)}\left[E - V( \varrho^{\alpha}) \right]- \frac{\alpha^{2}(l + \nu)^{2}- 1/4}{\varrho^{2}}\right\}\widetilde{u}(\varrho) = 0\; ,
\P\biggl( \sup_{x\in Q}Y(t,x) > R\mathrel{\Big|}A\biggr) \geq C\P\biggl(\sup_{x\in Q}g(t-\tau,x-\eta) >4 R\mathrel{\Big|}A\biggr) &=C\P ((2\pi (t-\tau))^{-\frac d2}>4R\mid A).
H_{p}^{G}=\{~U\in T_{p}(P)~|~\theta^{A}_{p}(U)=0~~{^\forall}A\in\mathfrak{g}~\}, &H_{p}^{K}=\{~U\in T_{p}(P)~|~\theta^{A}_{p}(U)=0~~{^\forall}A\in\mathfrak{k}~\}.
\sum_{n = 0}^\infty{{\|{{f_n}-{g_n}}\|}^2}= \sum_{n = 0}^{{n_0}}{{{\|{{f_n}-{g_n}}\|}^2}}+ \sum_{n ={n_0}+ 1}^\infty{{{\|{{f_n}-{g_n}}\|}^2}}&\le{M_0}+{\pi^3}\sum_{n ={n_0}+ 1}^\infty{{|{{\lambda_n}-{\nu_n}}|}^2}< \infty ;
C_{2}= \sup_{\tau \in \bar{D}(0,\rho) \cup U_{d}}\frac{1 + |\tau|^{2k}}{|\tau|}\exp( -\nu |\tau|^{\kappa}) \int_{0}^{|\tau|^k}\frac{\exp( \nu h^{\kappa/k})}{1+h^2}h^{\frac{1}{k}}(|\tau|^{k}-h)^{\chi_2}h^{\nu_2}dh= \sup_{x \geq 0}B(x)
(\begin{matrix}1&1&1&1&1&1&1&1\\ 1&0&1&0&1&0&1&0\\ 1&1&0&0&1&1&0&0\\ 1&1&1&1&0&0&0&0\\ 1&0&0&0&1&0&0&0\\ 1&0&1&0&0&0&0&0\\ 1&1&0&0&0&0&0&0\\ \end{matrix})
\theta \rightarrow{\frac{\pi}{2}}, c = - \cos{\theta}\rightarrow 0 , s = \sin{\theta}\rightarrow 1 , \qquad x^{+}={\frac{x^{0}+ x^{1}}{\sqrt 2}}, \x^{-}={\frac{x^{0}- x^{1}}{\sqrt 2}},
M_k=M_k(\beta):=\left(\begin{array}{cccc}M[0,0](\beta) M[0,1](\beta) \cdots M[0,k](\beta) M[1,0](\beta) M[1,1](\beta) \cdots M[1,k](\beta) \vdots \vdots \ddots \vdots M[k,0](\beta) M[k,1](\beta) \cdots M[k,k](\beta)\end{array}\right),
\frac{dA}{dD}=\left(\frac{-A^4D^2}{\lambda_1^2\lambda_2\lambda_4^2\lambda_5}+\frac{-A^3B}{\lambda_1\lambda_2\lambda_3D}\right)\left(\frac{\lambda_1\lambda_2\lambda_3}{A^2B}\right) &=\frac{-\lambda_3A^2D^2}{\lambda_1\lambda_4^2\lambda_5B}+\frac{-A}{D}
\left( 1+\delta\right) c_{+}\left\Vert x\right\Vert \geq\sum_{i\geq1}\sum_{j=r_{i}^{+}}^{m_{i+1}^{+}-1}\left\vert \mathbf{S}_{j}^{t}\right\vert\geq\sum_{i\geq1}\left( m_{i+1}^{+}-1-r_{i}^{+}\right) MN
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^{\vec{r}}(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^{\vec{r}}(\mathbb{R}^n)}.
\lim_{M \to \infty}y = \lim_{M \to \infty}\frac{M - d -tN}{\sqrt{N}}= \lim_{M \to \infty}\frac{-d + \frac{z}{\sqrt{t}}\sqrt{M}}{\sqrt{\frac{M}{t}+ \frac{z}{t^{3/2}}\sqrt{M}}}= z.
\mathbb{E}\| g - \tilde{f}_{m_0,m_1}\|^2 - \| g - f \|^2 = \mathbb{E}\| \tilde{f}_{m_0,m_1}- \mathbb{E}\tilde{f}_{m_0,m_1}\|^2 + \| g - \mathbb{E}\tilde{f}_{m_0,m_1}\|^2 - \| g - f \|^2 & = \mathbb{E}\| \tilde{f}_{m_0,m_1}- \mathbb{E}\tilde{f}_{m_0,m_1}\|^2 + \langle f - \mathbb{E}\tilde{f}_{m_0,m_1}, 2g - f - \mathbb{E}\tilde{f}_{m_0,m_1}\rangle.
R_{n}(s)-R_{n-1}(s) =\sum_{k=n+1}^\infty \frac{(-1)^{k-1}}{k^s}-\sum_{k=n}^\infty \frac{(-1)^{k-1}}{k^s}=\frac{(-1)^{n}}{n^s}R_{n}(s)-R_{n+1}(s) =\sum_{k=n+1}^\infty \frac{(-1)^{k-1}}{k^s}-\sum_{k=n+2}^\infty \frac{(-1)^{k-1}}{k^s}=\frac{(-1)^{n}}{(n+1)^s}
{\cal M}_{1 0}= \sum_{\sigma}\int d \Omega d^{n - 2}p \sigma \Omega [ d_{p}^{( \sigma ) \dagger}d_{p}^{( \sigma )}+ \bar{d}_{\tilde{p}}^{( \sigma )}\bar{d}_{\tilde{p}}^{( \sigma ) \dagger}]{.}
\sum_{l_{1}}\frac{\bar{g}_{1 l_{1}}}{2 I + 1}C_{\sigma_{1}\sigma_{1}^{\prime}}C_{\sigma_{2}\sigma_{2}^{\prime}}\delta_{k m_{I}}\left\{\begin{array}{l l l}{{j_{1}}}&{{j_{2}}}&{{I}}\\{{j_{1}}}&{{j_{2}}}&{{l_{1}}}\\ \end{array}\right\}\left( \begin{array}{l l l}{{j_{1}}}&{{j_{2}}}&{{I}}\\{{\sigma_{1}^{\prime}}}&{{\sigma_{2}^{\prime}}}&{{m}}\\ \end{array}\right) .
\bar{n}= \sum_{n = 0}^{\infty}n p_{2 n}= p_{0}\lambda \frac{\partial}{\partial \lambda}\sum_{n = 0}^{\infty}\lambda^{n}q_{2 n}( M ) \vert_{\lambda = 1}= \lambda \frac{\partial}{\partial \lambda}2 \, \mathrm{I m}\, W ( \lambda M ) \vert_{\lambda = 1}= \frac 1 2 \mathrm{t r}\, \frac{M}{1 \mp M}.
A_0 \le (1+\epsilon)\frac{|R_0|^a}{|R_1|^a}A_0^{1-a}A_1^{a}+C(M_0+M_1)^{as}N_0^{(1-a)s/q}\le (1+\epsilon)2^{-an}((1-a)A_0+aA_1)+C(M_0+M_1)^{as}N_0^{(1-a)s/q}.
\|F\|_1= |\nabla F|_2+|F|_2\leq |\nabla F|_2+|F-\overline{F}|_2+\overline{F}|\Omega|^{\frac{1}{2}}\leq C\big( |\sqrt{\rho}F|_{2}+(1+|\rho|_2) |\nabla F|_{2}\big),
\Phi_S(u,w) = e^{it\Delta}u_0 + \frac{1}{2}\mathcal{J}_S[(w + \overline{w})u](t), \Phi_{W}(u,w) = e^{-it \langle \nabla \rangle}w_0 + \mathcal{J}_{W_{+}}\Bigl[\frac{\Delta}{\langle \nabla \rangle}(u \overline{u}) + \frac{1}{2 \langle \nabla \rangle}(w + \overline{w}) \Bigr](t).
h[n+1](x,y) := \big(-\Delta_x + V_\mathrm{trap}(x)\big)\delta(x-y) + (\upsilon_N\ast|\phi_{n+1}|^2)(x)\delta(x-y) &+\frac{1}{N}(\upsilon_N \Big(\frac{k_{n+1}\circ\overline{k_{n+1}}}{\delta - k_{n+1}\circ \overline{k_{n+1}}}\Big))(x,y) + \frac{1}{N}(\upsilon_N \ast\rho^\mathrm{pair}[n+1])(x)\delta(x-y).
\nabla_{\partial_t}\nabla_{\partial_t}Y-\nabla_{\nabla_{\partial_t}\partial_t}Y =\nabla_{\nabla_{Y^T}\partial_t}\partial_t-u\nabla_{\nabla u}Y^T+\langle u\nabla u,\nabla \frac{Y^{\perp}}{u}\rangle\cdot\partial_t
X_{i}(t) = X_{i}(0) + \int_{\Gamma\times [0,t]}b\left(s,X_{i}(s),\frac{1}{N}\sum_{j=1}^{N}\delta_{X^{N}_{j}(s)},\gamma\right) \rho^{i}(d\gamma,ds) &+ \int_{0}^{t}\sigma\left(s,X_{i}(s),\frac{1}{N}\sum_{j=1}^{N}\delta_{X^{N}_{j}(s)}\right)dW_{i}(s),t\in [0,T],
h-\tilde{h}=\sum_{j=-\infty}^{\infty}h_j\left(1-\prod_{j'>j}(1-U_{j'})\right) &= \sum_j h_j\sum_{j^{\prime}>j}U_{j^{\prime}}\prod_{j<j^{\prime\prime}<j^{\prime}}\left(1-U_{j^{\prime\prime}}\right) &=\sum_{j^{\prime}}U_{j^{\prime}}\sum_{j<j^{\prime}}h_j\prod_{j<j^{\prime\prime}<j^{\prime}}\left(1-U_{j^{\prime\prime}}\right)= \sum_j U_jV_j.
\P^{\theta}(\tilde{\pi}_{0:n-1},e_{1:n}) = \prod_{t=0}^{n-1}\P^{\theta}(\tilde{\pi}_t,e_{t+1}|\tilde{\pi}_{0:t-1},e_{1:t}) &=\prod_{t=0}^{n-1}1_{\theta_{t+1}[\tilde{\pi}_t]}(e_{t+1})\times &\sum_{y_t,m^1,m^2}\pi_t(m^1,m^2) Q(y_t|e^1_t(m^1),e^2_t(m^2))1_{\tilde{F}(\tilde{\pi}_{t-1},e_t,y_t)}(\tilde{\pi}_t)
{n \choose k}E[(U_1+\cdots+U_k)^{n-k}] S_{1,\lambda}(n,n) &\quad-{n-1 \choose k-1}S_{1,\lambda}(n-1,n-1) E[(U_1+\cdots+U_{k-1}+1)^{n-k}&=-\sum_{m=k-1}^{n-1}E[(U_1+\cdots+U_k)^{m-k}]{m \choose k}S_{1,\lambda}(n-1,m-1) &+ \sum_{m=k-1}^{n-2}{m \choose k-1}S_{1,\lambda}(n-1,m) E[(U_1+\cdots +U_{k-1}+1)^{m-k+1}].
Q_{-}(t_{0},x_{0},r) = (t_{0},t_{0}+\delta\tau r^{2\beta/\alpha})\times B(x_{0},\delta r), Q_{+}(t_{0},x_{0},r) = (t_{0}+(2-\delta)\tau r^{2\beta/\alpha},t_{0}+2\tau r^{2\beta/\alpha})\times B(x_{0},\delta r).
D^k\big(N^{4-2n}\langle(\nabla R^P)^\sharp(\psi,\psi)\psi,\psi\rangle\big) &=\sum_{\sum l_i+\sum{m_j}=k}D^{l_1}N^{4-2n}\star \nabla^{l_2+1}R^P\star\underbrace{D^{m_1+1}\phi\star\ldots\star D^{m_{l_2}+1}\phi}_{l_2-\textrm{times}}\star D^{l_3}\psi\star D^{l_4}\psi\star D^{l_5}\psi\star D^{l_6}\psi.
\begin{array}{c}{\widetilde{\tau}= \operatorname{c o s h}\omega \, \tau - \operatorname{s i n h}\omega \, \sigma , \hfill}{\widetilde{\sigma}= \operatorname{c o s h}\omega \, \sigma - \operatorname{s i n h}\omega \, \tau . \hfill}\end{array}
\mathcal{R}_{n,\alpha}(\varepsilon) = \inf_{Q \in \mathcal{Q}_\alpha}\inf_{\phi\in \Phi_Q}\bigg\{\sup_{P \in \mathcal{P}_\varepsilon(P_0)}\mathbb{E}_{P,Q}(\phi) + \sup_{P' \in \mathcal{P}_\varepsilon(P_1)}\mathbb{E}_{P',Q}(1-\phi) \bigg\},
\gamma_m =\int_{I^n}\Big[\prod_{k=1}^{n}r_k^{2m_k}(1-r_k^2)^\alpha r_k\Big ]\omega(\mathbf{r})d\mathbf{r}&\le \int_{I^n}\Big[\prod_{k=1}^{n}(1-r_k^2)^\alpha r_k\Big ]\omega(\mathbf{r})d\mathbf{r}<\infty.