harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\max_{j<\beta N}\left\{\Pr_{v\sim \mu_t}[|v H|=j]\cdot 2^{2\epsilon N\log(1-\frac{2j}{N})}\right\}\leq\max_{\delta< \beta}\left\{2^{-(1-h(\delta))(1-\eta) N}\cdot 2^{2\epsilon N\log(1-2\delta)}\right\}.
\Omega f(\zeta) =\sum_{x\in \mathbb{Z}^d}\big[f(\zeta^{x,0})-f(\zeta)\big]+\lambda\sum_{x\in \mathbb{Z}^d}\big[f(\zeta^{x,U(\zeta,x)})-f(\zeta)\big] &+(1-2\lambda d)\sum_{x\in \mathbb{Z}^d}f_x^{\prime}(\zeta)\zeta(x)
\begin{array}{c}{( \dot{x}, \dot{y}) = i ( a \alpha \zeta^{\alpha - 1}, b \beta \zeta^{\beta - 1}) = ( A \zeta^{\alpha - 1}, B \zeta^{\beta - 1})}{( i \dot{y}^{*}, - i \dot{x}^{*}) = ( i B^{*}( \zeta^{*})^{\beta - 1}, - i A^{*}( \zeta^{*})^{\alpha - 1})}\end{array}
\|I_\mathcal{H}-S_{B,\Psi}S_{A,\Psi}^{-1}\| =\left\|\sum_{j\in \mathbb{J}}A_j^*\Psi_j S_{A,\Psi}^{-1}-\sum_{j\in \mathbb{J}}B_j^*\Psi_j S_{A,\Psi}^{-1}\right\|=\left\|\sum_{j\in \mathbb{J}}(A_j^*-B_j^*)\Psi_j S_{A,\Psi}^{-1}\right\| &\leq \sum_{j\in \mathbb{J}}\|A_j-B_j\|\|\Psi_j S_{A,\Psi}^{-1}\| =\beta<1.
\lim_{\sigma_{1 2}\rightarrow \infty}N_{S p ( 4 )_{2}}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1 / 2}}&{{- 1}}&{{- 1 / 2}}\\{{0}}&{{- 1}}&{{- 1}}&{{- 1}}\\{{0}}&{{- 1 / 2}}&{{- 1}}&{{- 1 / 2}}\\ \end{array}\right) .
\|f\|_{L^{2}([0,1])}^{2}\leq C_{H}\int_{0}^{1}dt\int_{t}^{1}(s-t)^{-\frac{1}{2}-H}|\varphi(s)|^{2}ds =C_{H}\int_{0}^{1}|\varphi(s)|^{2}ds\int_{0}^{s}(s-t)^{-\frac{1}{2}-H}dt =C_{H}\int_{0}^{1}s^{\frac{1}{2}-H}|\varphi(s)|^{2}ds \leq C_{H}\|\varphi\|_{L^{2}([0,1])}^{2}=C_{H}\|f\|_{\mathcal{H}}^{2},
\mathcal{C}_{-1}(\alpha, \nu, q + w', \lbrace 1, \ldots, q-1 \rbrace, \lbrace q, \ldots, \ell \rbrace \setminus \lbrace q+w' \rbrace) < \mathcal{C}_{-1}(\alpha, \nu, q, \lbrace 1, \ldots, q-1 \rbrace, \lbrace q+1, \ldots, \ell \rbrace) = \nu_q + \ell - 2q + 1 = \mu^{\circ}+ \ell - 2q + 1,
\left\langle H^\square_0(\theta) (\psi^* + \varphi^*), (\psi^* + \varphi^*) \right\rangle = \left\langle H^\square_0(\theta) \varphi^*, \varphi^* \right\rangle \geq g \|\varphi^*\|_{\ell^2(\square)}^2,
B_t^{d,p}=\{x\in T^d(r):\eta_t^{d,p}(x)=0\}, &C_t^{d,p}=\{x\in T^d(r):\sum_{y:y\sim x}\eta_t^{d,p}(y)\geq d\}, &D_t^{d,p}=\{x\in T^d(r):\sum_{y:y\sim x}\eta_t^{d,p}(y)\leq d\}, &E_t^{d,p}=\bigcup_{s\leq t}C_s^{d,p}.
\left(h_{i_{1}\,0\,i_{3}\cdots i_{m}}\,h_{i_{2}\,0\,i_{4}\cdots i_{m}i_{1}}\cdots h_{i_{m}\,0\,i_{2}\cdots i_{m-1}}\right)+\left(h_{i_{1}\,1\,i_{3}\cdots i_{m}}\,h_{i_{2}\,1\,i_{4}\cdots i_{m}i_{1}}\cdots h_{i_{m}\,1\,i_{2}\cdots i_{m-1}}\right)=0
a_{0}z_{1}z_{2}z_{3}z_{4}z_{5}+ a_{1}z_{1}^{4}+ a_{2}z_{2}^{4}+ a_{3}z_{3}^{4}+ a_{4}z_{4}^{6}+ a_{5}z_{5}^{1 2}+ a_{6}z_{4}^{2}z_{5}^{8}+ a_{7}z_{4}^{4}z_{5}^{4},
T_{{\bar m}{\bar n}{\bar s}}\rightarrow T_{{\bar m}{\bar n}{\bar s}}+\nabla_{\bar m}\nabla_{\bar n}\Lambda_{\bar s}\hskip 1cmH_{\alpha\nu}\rightarrow H_{\alpha\nu}+W_{\alpha\nu s}G^{s{\bar s}}\Lambda_{\bar s}\ ,
C^{\Lambda_{0}}= ( I \otimes \mathrm{t r}) \{[ I \otimes \pi_{\Lambda_{0}}( q^{2 h_{\rho}}) ] \Gamma \}= \sum_{s , t}\mathrm{t r}( \pi_{\Lambda_{0}}( q^{2 h_{\rho}}a_{s}b_{t}) ) b_{s}a_{t}
b=\left(\frac{N}{A}\right)\frac{\sinh(N\tau) +\sqrt{1-\frac{A^2}{N^2}}\cosh(N\tau)}{\cosh(N\tau) +\sqrt{1-\frac{A^2}{N^2}}\sinh(N\tau)},\;\;e^{-\phi}=\cosh(N\tau) +\sqrt{1-\frac{A^2}{N^2}}\sinh(N\tau),
\det\left(\begin{matrix}C I_{i_{1},i_{1}+1,\ldots,\widehat{i_{j}+\epsilon},\ldots,i_{r},i_{r}+1}\end{matrix}\right)=\left(-1\right)^{k+i_{j}-\epsilon}\det\left(C_{[n]\setminus\{i_{1},i_{1}+1,\ldots,\widehat{i_{j}+\epsilon},\ldots,i_{r},i_{r}+1\}}\right).
\phi(h_1,(F_{2})_{N}) &= (h_{1}|\mathcal{F}((F_{2})_{N}))-(\mathcal{F}^{-1}(h_{1})|(F_{2})_{N}) &= (\mathcal{F}((f_{1})_{N})|\mathcal{F}((F_{2})_{N}))-((f_{1})_{N}|(F_{2})_{N}).
\tau(q) =f(e^{2\pi i y},e^{2 \pi i (x-y)})=(1-e^{2 \pi i (x-y)})e^{2\pi i y}, \kappa(\tau)(p_0) =f_0=f(e^{2\pi i (y+w_0)},e^{2 \pi i (x-y)})=(1-e^{2 \pi i (x-y)})e^{2\pi i (y+w_0)}, \kappa(\tau)(p_1) =f_1=f(1,e^{2 \pi i (x-y)})=1-e^{2 \pi i (x-y)}.
\lim_{|z| \rightarrow 0}\Psi^{\pm}(x;z) =\begin{pmatrix}1 -u_x(x) -\bar{u}_x(x) 1\end{pmatrix}, begin{align*}4pt] \lim_{|z| \rightarrow \infty}\Psi^{\pm}(x;z)=e^{-ic_\pm\sigma_3}.
\eta < -(m+p)(\partial_t-\tilde X_{j_l}\cdot\nabla_Y)\varphi_{j_l}(\tilde X_{j_l},\tilde Y_{j_l},\tilde t_{j_l})+(m+p)(\partial_t-X_{j_l}\cdot\nabla_Y)\psi_{j_l}(X_{j_l},Y_{j_l},t_{j_l}) &={{j_l^4}}(\tilde X_{j_l}-X_{j_l})\cdot (\tilde Y_{j_l}-Y_{j_l})|\tilde Y_{j_l}-Y_{j_l}|^2 =0,
\sum_{\substack{X^{\theta_2-\theta_1}\le p_1\le \dots\le p_\ell X^{\theta_1}\le\prod_{i\in\mathcal{I}}p_i\le X^{\theta_2}p_1\cdots p_\ell\le X/p_j}}^*S_{p_1\cdots p_\ell}(p_j)=o_{\mathcal{L}}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr),
\begin{bmatrix}q -\alpha J(T^{-i-r},T^{-3r}) 0 -\overline\alpha J(T^{-i-3r},T^{-r}) -\overline\alpha J(T^{-i},T^{-r}) q -\alpha J(T^{-i-2r},T^{-3r}) 0 0 -\overline\alpha J(T^{-i-r},T^{-r}) q -\alpha J(T^{-i-3r},T^{-3r}) -\alpha J(T^{-i},T^{-3r}) 0 -\overline\alpha J(T^{-i-2r},T^{-r}) q \end{bmatrix}
\frac{1}{2}<\sum_{i=b_n}^{n}\frac{1}{i-2}=\sum_{i=b_n-2}^{n-2}\frac{1}{i}<\int_{b_n-\frac{5}{2}}^{n-\frac{3}{2}}\frac{dx}{x}=\log\left(\frac{n-\frac{3}{2}}{b_n-\frac{5}{2}}\right)
\mathbb{P}_1=\mathbb{P}\left[^{+}_{\omega \ge \omega^{*}}S(\omega)+r^{h}(\omega) \in d\omega_2, \max\limits_{\omega \ge \omega^{*}}S(\omega)+r^h(\omega) \in dx^{*}, \right. \left. S(\omega) \le 0 \omega \ge \omega^{*}\vert S(\omega^{*})=y-x^{*}\right]
\mathbf{P}_{\rm w}\mathbf{A}^{-1}\tilde{\mathbf{y}}_i = \mathbf{P}_{\rm w}\mathbf{A}^{-1}\mathbf{Q}\mathbf{P}^{-1}_f \mathbf{P}_1 \bar{\mathbf{d}}_{i-1}+ \mathbf{P}_{\rm w}\mathbf{A}^{-1}\mathbf{H}^{-1}_i \mathbf{n}_i.
a_1 \cdot ((\alpha_{-1}- \beta_{-1}) \cdot (\alpha_0 + \beta_0 + \frac{5}{3}\alpha_2)) = \frac{1}{2^2}(\alpha_{-1}\cdot \beta_0 - (\alpha_0 + \frac{5}{3}\alpha_2) \cdot \beta_{-1}) + \frac{1}{2^2}\langle \beta_0, \beta_{-1}\rangle .
\hat{G}^{\xi = 0}[a, b; E] = \frac{e^{-\sqrt{2E}|a - b|}}{\sqrt{2E}}+ \frac{\frac{\sqrt{2E}}{v^{ren}}+ 1}{\frac{\sqrt{2E}}{v^{ren}}- 1}\frac{e^{-\sqrt{2E}(|a| + |b|)}}{\sqrt{2E}}
\int^{T}_0\int_{\Omega}(\rho_{N}\partial_t\psi+\rho_{N}u_{N}\cdot\nabla\psi-\varepsilon\nabla\rho_{N}\cdot \nabla\psi)\,dxdt+\int_{\Omega}\rho_0\psi(0,x)\,dx &\qquad\qquad\qquad=\int_0^T\int_{\Gamma_{\rm{in}}}\rho_B u_B\cdot \nu(x)\psi\,d\sigma(x)dt;
\lim_{s\to \infty}\P ( Y_s \in B_r(0)) \leq \Gamma\left(\frac{1+r}{2}\right)-\Gamma\left(\frac{1-r}{2}\right), \lim_{s\to \infty}\P ( Y_s \in B_r(0)) \leq a(r) :=( 1/2 + \pi^{-1/2}- 2 \pi^{-3/2}) r + 1/2 - \pi^{-1/2}+ 2 \pi^{-3/2}.
\rho(0,0)=(\begin{matrix}1&0\\ 0&1\end{matrix}),\rho(1,0)=(\begin{matrix}-1&0\\ 0&-1\end{matrix}),\rho(0,1)=(\begin{matrix}0&1\\ 1&0\end{matrix}),\rho(1,1)=(\begin{matrix}0&-1\\ -1&0\end{matrix}).
\sigma_1 = (1\a_2\a_3\ \cdots\a_{n-1}\a_{n}), \sigma_2 = (1\a_3\a_4\ \cdots\a_{n}\a_{2}), &\ \ \ \ \ \vdots \sigma_{n-1}= (1\a_n\a_2\ \cdots\a_{n-2}\a_{n-1}).
[a_l,a_m,a_k,a_{\infty}]= \exp(4 \pi i (\eta_{m}-\eta_{l})_1(\eta_k)_2)\left( \frac{\theta[\eta_{S_l \circ U_\eta}+2\eta_{S_l \cap U_\eta}+n](\Omega)\theta[\eta_{T_m \circ U_\eta}+2\eta_{T_m \cap U_\eta}+n](\Omega)}{\theta[\eta_{S_m \circ U_\eta}+2\eta_{S_m \cap U_\eta}+n](\Omega)\theta[\eta_{T_l \circ U_\eta}+2\eta_{T_l \cap U_\eta}+n](\Omega)}\right)^2.
\log\prod_p\big(1+p^{\alpha}r(p)^2\omega(p)\big)-\log\prod_p\big(1+r(p)^2\omega(p)\big) &\qquad=\sum_p\log\left(1+\frac{(p^{\alpha}-1)r(p)^2\omega(p)}{1+r(p)^2\omega(p)}\right) &\qquad\leqslant\alpha\sum_p\log p\cdot r(p)^2\omega(p)+O\bigg(\alpha^2\sum_p\log^2p\cdot r(p)^2\omega(p)\bigg)
\begin{cases}e^{i\theta\infty}(w,\phi_i)_{L^2(\mathbb{R}_+^d)}+(\sigma_{i\infty}\nabla w_i,\nabla\phi_i)_{L^2(\mathbb{R}_+^d)}=0, e^{i\theta\infty}(w,\phi_e)_{L^2(\mathbb{R}_+^d)}-(\sigma_{e\infty}\nabla w_e,\nabla\phi_e)_{L^2(\mathbb{R}_+^d)}=0, w=w_i-w_e, \end{cases}
X^a{}_b=\left( \begin{array}{cccc}-k_1^{\,2}A+k^2m^2 -k_0k_1A k_0k_2\,k^2 k_0k_3\,k^2 k_0k_1A k_0^{\,2}A+k^2m^2 -k_1k_2\,k^2 -k_1k_3\,k^2 -k_0k_2\,k^2 -k_1k_2\,k^2 k_3^{\,2}B-k^2l^2 -k_2k_3 B -k_0k_3\,k^2 -k_1k_3\,k^2 -k_2k_3 B k_2^{\,2}B-k^2l^2\end{array}\right)
\#_{w_X}A \ll XW(\frac{m}{X^2}) L(1,\chi_{D}) \prod_{\beta(p)\geq2}\Big(1+\frac{2}{p}\Big) \ll XW(\frac{m}{X^2}) L(1,\chi_{D}) \big(\frac{v_0}{\varphi(v_0)}\big)^2.
\begin{cases}\gamma \leq{(\alpha-1)}\min\{1 ,{1}/{L_1}\}\left[{\alpha B^2\left( C_0 + 1 \right)}\right]^{-1}; \gamma \leq B^{-2}\Big[\alpha^2 + (e-1) \Big(L_{0}+ L_1 C_0\Big) \Big]^{-1}. \end{cases}
\left\lbrace\begin{tabular}{l l l l}$u(0,t) = 0$ $u(L,t) = 0$ $u_{x}(L,t) = 0$, in $(0,T)$, $v(0,t) =0,$ $v(L,t) = 0,$ $v_{x}(L,t) =g_2(t)$, in $(0,T)$.\end{tabular}\right.
\lim_{s\rightarrow0}\Gamma(s)\left( \frac{1}{\left( dz-c\right)^{s}}G(Tz,s;A,B;r_{1},r_{2})-G(z,s;A_{-c},B_{b};R_{2},-R_{1})\right) & =2\pi is\left( -c,d;A_{c},B_{-b};-R_{1},-R_{2}\right) -\frac{2\pi i}{d}\frac{1}{dz-c}B_{0}(B)P_{2}(cR_{2}-dR_{1},A)-\frac{2\pi i}{d}(dz-c)B_{0}(B)P_{2}(-R_{2},A_{-c}).
i(I-\mathcal{H})A=i-[D_t\zeta,\mathcal{H}]\frac{\partial_{\alpha}\mathfrak{F}}{\zeta_{\alpha}}-[D_t^2\zeta,\mathcal{H}]\frac{\bar{\zeta}_{\alpha}-1}{\zeta_{\alpha}}-(I-\mathcal{H})\frac{i}{2\pi}\sum_{j=1}^2 \frac{\lambda_j(D_t\zeta(\alpha,t)-\dot{z}_j(t))}{(\zeta(\alpha,t)-z_j(t))^2}
\frac{dt^i}{dv}=(\int_{N^i_t \cap \{|z|\leq \frac12\}}H^{-1}_i dS)^{-1}&\leq (\int_{N^i_t \cap \{|z|\leq \frac12\}}H^2_i dS)^\frac12 (Area (N^i_t \cap \{|z|\leq \frac12\}))^{-\frac32}
\left\Vert T_{\delta}(t)\right\Vert_{\mathfrak{d}\to \mathfrak{d}}\leqslant C_{\circ}|t|,|t|\leqslant t^{0}, \left\Vert T_{\delta}^{(j)}\right\Vert_{\mathfrak{d}\to \mathfrak{d}}\leqslant \widetilde{B},j=1,\dots,2p,
P(\lambda_{1}<\lambda_{2}|n_{1},n_{2})=\int_{\lambda_{1}=0}^{\infty}\frac{\lambda_{1}^{n_{1}}d\lambda_{1}}{e^{\lambda_{1}}n_{1}!}\int_{\lambda_{2}=\lambda_{1}}^{\infty}\frac{\lambda_{2}^{n_{2}}d\lambda_{2}}{e^{\lambda_{2}}n_{2}!}
(\delta_{c_2}\circ \delta_{c_3}) \circ \delta_{c_4}|_{c_0}= \delta_{c_2}\circ (\delta_{c_3}\circ \delta_{c_4})|_{c_0},~~~ (\delta_{c_2}\circ \delta_{c_4}) \circ \delta_{c_3}|_{c_0}=\delta_{c_2}\circ (\delta_{c_4}\circ \delta_{c_3})|_{c_0},
\displaystyle{\sum_{i=1}^n e_{1+i-n}(U_i)(0)}+ \sum_{j=1}^q e_1(U_{n+j})(0)+ \sum_{j=q+1}^p e_0(U_{n+j})(0) \le C_2 \Big( \int_0^T \sum_{l=0}^q || \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}||_{G_{n+l}}^2 + \sum_{l=q+1}^p || \mathcal{\mathbf{B}}_{n+l}^{\ast}W_{n+l}||_{G_{n+l}}^2\,dt\Big) \,.
\lim_{t\to\infty}\prod_{\kappa=s}^{t-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta\leq\lim_{t\to\infty}\hat{\theta}^{t-\hat{\tau}}\prod_{\kappa=s}^{\hat{\tau}-1}\frac{\beta(\kappa)}{\beta(\kappa+1)}\theta=0.
\mathrm{P}_{(\Phi ,-\hat{f}\left( 0\right) \delta_{\Psi})}^{\sharp}=\mathrm{P}_{(\Phi ,-\hat{f}\left( 0\right) \delta_{\Psi})}^{\flat}\leq\inf_{\gamma \in \left( 0,1\right)}\mathrm{P}_{\Phi +\mathcal{K}_{\gamma}\left( \Psi ,-f\right)}^{\sharp}\ ,\Psi \in \mathbb{S},\ \Phi \in \mathcal{W}_{1},\f\in \mathfrak{D}_{0,+}\ ,
F_{B}={\frac{1}{3}}H_{i j l}^{1}( k ) H_{i j l}^{2}( p ){\frac{\Gamma ( 1 - 2 k_{0}^{2}) \Gamma ( \vec{k}\cdot \vec{p}+ k_{0}^{2})}{\Gamma ( \vec{k}\cdot \vec{p}- k_{0}^{2}+ 1 )}}\ .
\phi_{1}'''(\kappa;u)-\kappa\big(\phi_{1}''(\kappa;u)-u\phi_{1}(\kappa;u)\big) -u\phi_{1}'(\kappa;u) -\frac{1}{2}\phi_{1}(\kappa;u) =0, \phi_{2}'''(\kappa;v)+\kappa\big(\phi_{2}''(\kappa;v)-v\phi_{2}(\kappa;v)\big) -v\phi_{2}'(\kappa;v) -\frac{1}{2}\phi_{2}(\kappa;v) =0.
K\left(\mathbb{T}_T(d_1)_{v_1}\boxtimes \cdots \boxtimes \mathbb{T}_T(d_k)_{v_k}\right)\otimes_{\mathbb{K}}\mathbb{F}=\bigotimes_{i=1}^k K_T(\mathbb{T}(d_i)_{v_i}) \otimes_{\mathbb{K}}\mathbb{F},
\tilde{E}_{i}(h,\theta)= \int\limits_{y_{i}+2n_{i-1}s}^{y_{i}}\big(W(U_{h,\theta}(ih-,y))-W(U_{h,\theta_i}(ih-,y))\big)dy-\int\limits_{y_{i}+2n_{i-1}s}^{y_{i}+(2n_{i-1}+d_i)s}W(U_{h,\theta}(ih-,y))dy.
V_nV_{n(d-1)}\int_0^{aR}(R^2 - r^2)^{n(d-1)/2}r^{n-1}dr &= V_nV_{n(d-1)}R^{nd}\int_0^{aR}(1 - \frac{r^2}{R^2})^{n(d-1)/2}\frac{r^{n-1}}{R^{n}}dr &\leq \frac{V_nV_{n(d-1)}}{n}a^nR^{nd}&\leq \frac{V_nV_{n(d-1)}}{nV_{nd}}a^n \alpha^n(B).
\limsup_{\varepsilon \rightarrow 0^{+}}\limsup_{n \rightarrow \infty}\mathbb{E}_{\mu_n}\Big[ \Big| \int_{0}^{t}\frac{1}{n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) [ \overleftarrow{\eta}_{s}^{\ell}(x) - \overleftarrow{\eta}_{s}^{\varepsilon n}(x)] \overrightarrow{\eta}_{s}^{\varepsilon n}(x+1) ds \Big| \Big] =0.
\mathbb{P}(|\mathcal{C}(V_n)|>k)=\mathbb{P}(|\mathcal{A}_t|>0\forall t\leq k) =\mathbb{P}(1+\sum_{i=1}^{t}(\eta_i-1)>0\forall t\leq k) &\leq \mathbb{P}(1+\sum_{i=1}^{t}(X_i-1)>0\forall t\leq k).
\dfrac{d}{dt}\eta_i^\varepsilon = \varepsilon^2 (u_i -\bar u_i) \dfrac{d}{dt}(u_i-\bar u_i) -(p(\tau_i) - p(\bar \tau_i))\dfrac{d}{dt}\tau_i + (\tau_i-\bar \tau_i)p'(\bar\tau_i) \dfrac{d}{dt}\bar \tau_i .
T_{c}= 2^{- 3 / 2}\pi^{1 / 4}\kappa^{- 1 / 2}N^{- 1 / 2}l x^{- 1 / 4}( 8 + x )^{1 / 2}( ( 8 + x )^{1 / 2}- x^{1 / 2})^{1 / 2},
\mu_T(e_1,e_1)=ae_1,\ \mu_T(e_1,e_2)= \frac{\lambda_1a}{1-\lambda_2}e_1, &\mu_T(e_2,e_1)=\lambda_1 a e_1+ \lambda_2 a e_2,\ \mu_T(e_2,e_2)=\frac{\lambda_1a}{1-\lambda_2}(\lambda_1e_1+ \lambda_2 e_2).
\begin{aligned}\bar\Lambda^n(A\times B) \doteq \int_B \bar\Lambda^n(A\mid t) dt, &\bar \xi^n(A\times C \times B) \doteq \int_B \bar \xi^n(A \times C \mid t) dt, \;\;\bar \Xi^n(A\times C \times B) \doteq \int_B \bar \Xi^n(A \times C \mid t) dt\end{aligned}
\partial{}_{\mu}\overline{{F}}^{\prime a \mu \nu}- \overline{{f}}^{a}{}_{c b}\overline{{F}}^{\prime c \mu \nu}\overline{{A}}^{\prime b}{}_{\mu}{}+ \overline{{A}}^{\prime b}{}_{\mu}\overline{{R}}^{\prime a}{}_{b}{}^{\mu \nu}- \overline{{C}}^{a}{}_{d \mu}\overline{{F}}^{\prime d \mu \nu}= 0 ,
S = - M_{0}^{2}\mathop{\int \! \! \! \int}_{\Sigma \hspace *{0 . 2 5 c m}}d \Sigma \sqrt{- g}\left[ 1 - \frac{\alpha}{2}r_{s}^{2}\Delta x^{\mu}\Delta x_{\mu}\right] - \sum_{a = 1}^{2}m_{a}\int_{C_{a}}^{}\! \! d s_{a}{,}
\int_{\O}|\eta^n_h|^{p(x)-2}\eta^n_h \Phi\,dx +r\int_{\O}(\eta^n_h-Bu^n_h)\Phi \, dx=\int_{\O}\lambda^n_h \Phi\, dx, \lambda^{n+1}_h =\lambda^n_h +r(Bu^n_h- \eta^n_h),
\mathbf{P}_n^{\lambda,\theta,T}(\Gamma(x_i : i \in \mathcal{B}\cup \mathcal{O}\cup \mathcal{U}))= \left( \det \limits_{i,j \in \mathcal{O}\sqcup \mathcal{U}\sqcup \mathcal{B}}K^*_{i,j}\right) \prod_{i \in \mathcal{B}}\mathrm{d}t_i
g = U_1 + U_2, - \frac{f'}{f}= \frac{V[g]}g =\frac{V[U_1] + V[U_2]}{U_1+U_2}= \frac{V[U_1]/U_2 + V[U_2]/U_2}{e^\Phi + 1}, e^\Phi = \frac{U_1}{U_2}.
\rho = q|\Psi|^{2}, \mathbf{J}= -\frac{\mathrm{i}q\hbar}{2m}\big(\Psi^{\ast}\nabla{\Psi}-\Psi\nabla{\Psi}^{\ast}\big)-\frac{\vert q\vert^{2}}{m}\vert\Psi\vert^{2}\mathbf{A},
S=\frac{1}{2\kappa^{2}}\int d^{D}x\sqrt{-g}[-\frac{2(D-26)}{3\alpha^{\prime}}e^{\frac{4\tilde{\Phi}}{D-2}}+\tilde{R}-\frac{4}{D-2}\partial_{\mu}\tilde{\Phi}\partial^{\mu}\tilde{\Phi}+O(\alpha^{\prime})],
u_2^3x_0^3x_1^3x_2^3x_9^3x_{10}^3x_{11}^3u_2^3x_0^3x_1^3x_2^3x_9^3x_{10}^3x_{11}^3=(u_2x_0x_1x_2x_9x_{10}x_{11})^6\in\langle\partial w\rangle
\frac{d^2}{ds^2}h(\mu_{st}) =-\int\Delta(\mathbb{E}[U^2|Z_{st}]\mu_{st})\ln\mu_{st}-\int[\nabla(\mathbb{E}[U|Z_{st}]\mu_{st})]^2\frac1{\mu_{st}}&=-\int\mathbb{E}[U^2|Z_{st}]\mu_{st}\Delta\ln\mu_{st}-\int[\nabla(\mathbb{E}[U|Z_{st}]\mu_{st})]^2\frac1{\mu_{st}},
K_\alpha(x,y) = \sum_{j=-1}^{j_0(R,\lambda)}K_\alpha(x,y)\mathcal{X}_{A\bigl(x,\tfrac{R}{\nu^{j+1}},\tfrac{R}{\nu^j}\bigr)}(y) &\leq\frac{\alpha}{R^{1+s}}\sum_{j=-1}^{j_0(R,\lambda)}H^{j+1}&= \frac{\alpha}{R^{1+s}}C(R,\lambda).
\begin{pmatrix}A B_1 C_2 C_1 D_{11}D_{12}B_2 D_{21}D_{22}\end{pmatrix}^{\mathrm{osp}}:=\begin{pmatrix}\overline{A}^t -\overline{\rho(B_2)}^t \overline{\rho(C_1)}^t \overline{\rho(C_2)}^t \overline{D}_{22}^t -\overline{D}_{12}^t -\overline{\rho(B_1)}^t -\overline{D}_{21}^t \overline{D}_{11}^t\end{pmatrix},
f ( T , U ) = \int_{( T_{0}, U_{0})}^{( T , U )}\{d T^{\prime}Q ( U , U^{\prime}) ( T - T^{\prime})^{2}\partial_{T^{\prime}}^{3}f ( T^{\prime}, U^{\prime}) + ( T \leftrightarrow U , T^{\prime}\leftrightarrow U^{\prime}) \}\ ,
\mathbb{N}_{x}\left(X_{t_{1}}\in{\rm d}\nu_{1},\X_{t_{2}}\in{\rm d}\nu_{2},\cdots, X_{t_{n}}\in{\rm d}\nu_{n}\right) =L_{t_{1}}(x,{\rm d}\nu_{1})Q^{0}_{t_{2}-t_{1}}(\nu_{1},{\rm d}\nu_{2})\cdots Q^{0}_{t_{n}-t_{n-1}}(\nu_{n-1},{\rm d}\nu_{n}).
{\mathrm A d}_{w}( P_{\mu}) = \epsilon_{\mu}P_{\mu}; \hspace{. 2 c m}\epsilon_{0}= \epsilon_{n - 1}= 1 ; \hspace{. 2 c m}\epsilon_{j}= - 1 \mathrm{~ i f ~}1 \leq j \leq n - 2 .
\sum_{n = 1}^{\infty}\beta_{n}\, \left[ ( g \bar{\theta})^{n}\right]^{a b}+ \Omega^{a b}+ \Omega^{a c}\sum_{n = 1}^{\infty}\beta_{n}\, \left[ ( g \bar{\theta})^{n}\right]^{c b}= 0 \, .
C\,T^{(\kappa \lambda )\sigma}=-e\left( 2m+n-4d\right)D^{\lambda \sigma \kappa}+\frac{e}{2}\left( 2a+c\right) \epsilon_{\mu \nu}{}^{\kappa \lambda}D^{\nu \sigma \mu}-\frac{e}{2}E\,\epsilon_{\mu}{}^{\sigma \kappa \lambda}D^{\mu}-\frac{e}{2}F\,G^{\sigma \kappa \lambda}\,.
\begin{aligned}\max\left\{u(y_j, t), u(z_j, t)\right\}\leq \left({1\over \min\{\lambda, 1-\lambda\}}\right)^{1\over q}(u_{q, \lambda}(x, t)+1) \leq \left({1\over \min\{\lambda, 1-\lambda\}}\right)^{1\over q}(u_{\star, \lambda}(x, t)+1) \end{aligned}
{k\choose t}{n-k+t+2\choose t+2}= \ \frac{k!(n-k+t+2)!}{(k-t)!t!(t+2)!(n-k)!}\le \ \Big(\frac{n-k-2}{n-k-t-4}\Big)^{t+4}\cdot \frac{k!(n-k-2)!}{(k-t)!(2t+2)!(n-k-t-4)!}\cdot{2t+2\choose t}\le \ \Big(\frac 54\Big)^{t+4}\frac{(n-k-2)!}{(2t+2)!(n-k-2t-4)!}\cdot 2^{2t}\le \ 5^{t+1}{n-k-2\choose 2t+2}.
\alpha M+R \geq (n-\gamma)\left\lfloor\alpha\right\rfloor +\frac{\left\lfloor\alpha\right\rfloor(3(n-\gamma)^2-(n-\gamma)+1)-\left\lfloor\alpha\right\rfloor((n-\gamma)^2-(n-\gamma)+1)}{2(n-\gamma)}= 2(n-\gamma)\left\lfloor\alpha\right\rfloor.
F_{C}( a , T ) = \frac{5 \xi^{2}}{3 2 a \pi}- \frac{8 a^{3}\pi^{3}T^{4}\xi^{2}}{1 3 5}- \frac{5 1 2 a^{5}\pi^{5}T^{6}\xi^{2}}{4 7 2 5}+ O ( T^{7}) .
I_{1}= \int d x \left[ \alpha \Omega^{1 / 2}e^{- \psi_{1}}\alpha^{\prime}\Omega^{\prime}- \frac{3}{4}\alpha^{2}\Omega^{3 / 2}e^{- \psi_{1}}\psi_{1}^{\prime 2}- \frac{\mu l^{2}}{2}\Omega^{- 1}e^{\psi_{1}}\right] ,
J^{R}( 0 , x^{-}) \equiv \sqrt{2}\lim_{\epsilon^{-}\rightarrow 0}\left( e^{- i e \int_{x}^{x + \epsilon^{-}}v d x^{-}}\psi_{R}^{\dagger}( 0 , x^{-}+ \epsilon^{-}) \psi_{R}( 0 , x^{-}) - \mathrm{V E V}\right)
\begin{cases}\partial_t u= \partial_{xx}u + g(u)(\partial_x u)^2 + h(u)(\partial_x u)+k(u) +f(u)\xi\,, u(0,x)=u_0(x) u(t,0)=u(t,1) \partial_xu(t,0)=\partial_x u(t,1)\,\end{cases}
{J_{{N_y},{N_t}}}= \sum\limits_{l = 0}^{{N_t}}{_{{t_f}}p_{B,{N_t},{N_t}+ 2,l}^{(1)}\,\sum\limits_{k = 0}^{{N_y}}{_Lp_{B,{N_y},{N_y}+ 2,k}^{(1)}}}\left({{r_1}\,{x_{k,l}^2}+{r_2}{\mkern 1mu}{u_{k,l}^2}}\right),
\beta_1:=\alpha_3|_{\frak{t}^\sigma},\beta_2:=\alpha_4|_{\frak{t}^\sigma},\beta_3:=\alpha_5|_{\frak{t}^\sigma},\beta_4:=(\alpha_1+\alpha_3+\alpha_4+\alpha_5+\alpha_6)|_{\frak{t}^\sigma},
{{\cal L}^1}^{\rm eff}(H,\mu,T)= \frac{eH}{2\pi^2}\sum_{n=0}^\infty b_n \int_{-\infty}^\infty d\! p_{z}{}~\frac{p_{z}^2}{\varepsilon_n(p_{z})}~(f_+(T) + f_-(T)).
v(a)=e^{-\gamma \frac{4}{3}\pi E r^3}\Big(e^{\gamma \frac{4}{3}\pi E r^3},\, 1,\, -a \gamma Er^2 \pi ,\, -a^2 \gamma 2 Er ,\, a^2\frac{\gamma^2\pi^2 (Er^2)^2}{2}, &\;\; -a^3\gamma ,\, a^3 \gamma^2 2\pi E r Er^2,-a^3 \frac{\gamma^3}{6}\pi^3 (E r^2)^3 \Big).
X_1=\begin{bmatrix}a 0 ab(c+d)-a-b b\end{bmatrix}, X_2 = \begin{bmatrix}\frac{1}{a}-\frac{1}{a}-c-d+\frac{1}{a}+\frac{1}{b}c+d-\frac{1}{a}\end{bmatrix}, X_3=\begin{bmatrix}1 1 0 1\end{bmatrix}.
\sup_{n}\mathbb{E}_{\mathbb{P}}\left[\exp \left\{\kappa \sum_{i=1}^{n}\int_{T_{0}}^{T_{0}+\delta}\triangle K_{t}^{i, n}\cdot d W_{t}^{i}-\frac{\kappa}{2}\sum_{i=1}^n \int_{T_{0}}^{T_{0}+\delta}\left|\triangle K_{t}^{i, n}\right|^{2}d t\right\}\right]
\frac{\partial \xi}{\partial u}(t,s)-\frac{\partial \eta}{\partial u}+ \int_s^t \frac{\partial^2}{\partial x\partial x}V(\tau,x(\tau,s)) \frac{\partial y}{\partial u}\, d\tau &=-\int_s^t \int_s^\tau \frac{\partial^2}{\partial x\partial x}V(\tau,x(\tau,s)) \frac{\partial \xi}{\partial u}(\sigma,s)\, d\sigma.
( \mathcal Q_0 +\ell R(\ell, \omega) )^{-1}= ( I + \ell \mathcal Q_0^{-1}R(\ell, \omega))^{-1}\mathcal Q_0^{-1}&= \sum_{k=0}^\infty \ell^k \bigr( - \mathcal Q_0^{-1}R(\ell, \omega) \bigr)^{k}\mathcal Q_0^{-1}= \mathcal Q_0^{-1}+ \mathcal O(\ell) ,
m_t = \frac 1{u_3}\hat u_t - \frac{(u_3)_t}{u_3^2}\hat u = \frac1{u_3}\left((A^{-1}-\hat u\xi^T)\begin{pmatrix}r_1 r_2\end{pmatrix}+ r_3 \hat u\right) + \frac 1{u_3^2}\left(u_3\xi^T \begin{pmatrix}r_1 r_2\end{pmatrix}- u_3 r_3\right) \hat u
U \left({{\bf y},{\bf 0}}\right) \equiv P \exp \left({i g \int_{\bf 0}^{\bf y}{d z^{i}}A_{i}^{a}\left( z \right) T^{a}}\right) = P \left({1 + i g \int_{\bf 0}^{\bf y}{d z^{i}}A_{i}^{a}\left( z \right) T^{a}+ . . .}\right) .
\begin{array}{r l}{{}}&{{\hat{\Gamma}( k_{1}, k_{2}, k_{3})}}\\{{=}}&{{\displaystyle ( 2 \pi )^{4}\delta^{4}( \sum_{i}k_{i}) \sum_{\{\lambda_{j}\}}\prod_{j = 1}^{3}\left[ \eta_{\lambda_{j}}( k_{j}) \right]}}\\{{}}&{{\displaystyle \times \sum_{\pi_{3}}\exp [ - i ( k_{\pi ( 1 ) \lambda_{\pi ( 1 )}}, k_{\pi ( 2 ) \lambda_{\pi ( 2 )}}, k_{\pi ( 3 ) \lambda_{\pi ( 3 )}}) ] .}}\\ \end{array}
(q_{13}^{m_0}-1)c_1 \rho_{I_0+e_3}\delta_{J_0}\sigma_{e_2+e_3+m_0e_1}+ \textit{rest}= (q_{13}^{m_0}-1)c_2 \rho_{P_0+e_2}\delta_{Q_0}\sigma_{e_2+e_3+m_0e_1}+ \textit{rest}.
\quad\sum_{j=1}^{d_k}k\langle (k+\Delta')^{-1}a_N\square_{N+1}i_{\mu}u_j,i_{\mu}u_j\rangle &\leq \sum_{j=1}^{d_k}k|a_N|\|(k+\Delta')^{-1}\square_{N+1}i_{\mu}u_j\|\|i_{\mu}u_j\| &\leq \sum_{j=1}^{d_k}|a_N|\|\square_{N+1}i_{\mu}u_j\|\|i_{\mu}u_j\|,
F_\mu=\delta_a^{b'}\delta_{a'}^b\int \bigg\lbrack e^{iq\cdot (z-y)}-e^{iq\cdot(z-x)}\bigg\rbrack{(x_\mu-y_\mu)\over q\cdot (x-y)}\; \;{d^4q\over (2\pi)^4}+F_\mu^T\; ,
|R^{(-)}_1(\nu,k)|\leq\frac{2(1.1)}{\sqrt{\pi}}\sum_{\ell=1}^{5}(\frac{4}{3})^{\ell}|\sum_{j=0}^{5-\ell}|C^{(2)}(\ell,j)|(-1)^j\Gamma(j+\frac{1}{2};\pi)|\equiv C'_2,
(K-i)\prod_{j=i+1}^K(1-p^j(o_i))\left[1-\frac{\prod_{j=1}^i(1-p^j(o_i))}{\prod_{j=1}^i(1-p^j(g_j))}\right] &\leq i\left[1-\prod_{j=i+1}^K(1-p^j(o_i))\right].
\left\{\prod_{X_{i}\in ({\it G}_{1})_{X_{i^{\prime}}^{\prime}}}\; e^{\eta_{1}^{\prime}\sum_{\stackrel{i_{\alpha_{1}}< j_{\alpha_{1}}< k_{\alpha_{1}}}{\left\{i_{\alpha_{1}}, j_{\alpha_{1}}, k_{\alpha_{1}}\right\}\in J_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}t_{k_{\alpha_{1}}}{\bf 1}_{( w ( t_{i_{\alpha_{1}}}) = w ( t_{j_{\alpha_{1}}}) = w ( t_{k_{\alpha_{1}}}) )}}\right\}.
G_f(z,z')=G_{(\sigma)}(z,z')-\sum_{s=1}^n\sum_{p=a,b}\sum_{N_s}\int\limits_{C_p^{(s)}}G_{(\sigma)}(z,z'')\frac{d\theta'' dz''}{2\pi i}Y_{p,N_s}^{(0)}(t'')\chi_{N_s}^{(0)}(z')
\Big| |\chi(x) D^\alpha u (x) - \chi(y) D^\alpha u (y)|^2 + |\eta(x) D^\alpha u (x) - \eta(y) D^\alpha u (y)|^2 &\quad\qquad- |D^\alpha u(x)- D^\alpha u(y)|^2 \Big| &= \left| \Big( (\chi(x)-\chi(y))^2 + (\eta(x)-\eta(y))^2 \Big) \Re D^\alpha \overline{u}(x)D^\alpha u(y) \right| &\leq C |x-y|^2 \Big( |D^\alpha u(x)|^2 + |D^\alpha u(y)|^2 \Big).
X^{m}= X^{m}( \xi^{m}, \eta^{\mu}( \xi^{m}) ) , \qquad \Theta^{1 \mu}= \Theta^{1 \mu}( \xi^{m}, \eta^{\mu}( \xi^{m}) ) \qquad \Theta^{2 \mu}= \Theta^{2 \mu}( \xi^{m}, \eta^{\mu}( \xi^{m}) )
A = (2a+2)(2a+3), B_1 = - 2(4a+3)z_1^2 + 2(4a-4b_1-2l-d+3)z_1 + 2b_1(4a-2b_1-2l-d+3){}- 4b_2(4a+1), B_2 = - 2(4a+3)z_1^3 + 2(4a-4b_1-2l-d+3)z_1^2 + 8(b_1-2b_2)z_1 + 4b_1^2{}+ 4b_2(4a-4b_1-2l-d+1) - 6b_3(4a-1),
\sum_{\sigma\in\{1,2\}^m}\prod_{j=1}^m (-\gamma_{\sigma_j}- \gamma_{\sigma_{j-1}'}-1)^{-1}=\sum_{0\le r\le m/2}2{m \choose 2r}(-2\gamma_1-1)^{-r}(-2\gamma_2-1)^{-r}(-\gamma_1-\gamma_2-1)^{-(m-2r)}.
\left(\begin{array}{c}f_0^{(t)}f_1^{(t)}\vdots f_{m-1}^{(t)}\end{array}\right)=\left(\begin{array}{cccc}1 1 \cdots 1 \lambda_1 \lambda_2 \cdots \lambda_m \vdots \vdots \ddots \vdots \lambda_1^{m-1}\lambda_{2}^{m-1}\cdots \lambda_{m}^{m-1}\end{array}\right)\left(\begin{array}{c}c_{1}^{(t)}c_{2}^{(t)}\vdots c_{m}^{(t)}\end{array}\right).
g_n =(3-n)2^{n-3}-1+\sum_{a=1}^{n-2}2^{a-1}e_{n-1-a}+\sum_{a=2}^{n-1}\left(2^{n-1-a}-1-\binom{n-a}{2}\right)(a-1)2^{a-2}&\quad+\sum_{i=0}^{n-2}\binom{n-1+i}{2i+1}+\sum_{a=2}^{n-3}\sum_{\ell=1}^{n-2-a}\sum_{i=0}^{a-1}2^{n-2-a-\ell}\binom{a-1}{i}\binom{\ell+i}{i}, n \geq 3.
:\underset{\{x[i],y[i]\}}{\max}~ \underset{i=1}{\overset{N}{\sum}}\underset{j=i}{\overset{N}{\sum}}~w[i][j] f_3(x[i],x[j],y[i],y[j]) \mathrm{s.t.}~~ (\ref{maxV1}), (\ref{maxV2}), (\ref{maxV3}).
w_{n+1}(s) = \int_0^s e^{(s-\tau)L}\Big [Q( w_{n}, w_{n}) - \Big ( Q(u_\mathrm{app},w_{n}) + Q(w_{n},u_\mathrm{app}) \Big) & + \varepsilon \Big (R_1(w_n)+ R_2(w_n,w_n) - R_2(u_\mathrm{app},w_n) - R_2(w_n,u_\mathrm{app}) \Big) - R_\mathrm{app}\Big] \, d\tau &=: I_1(s) + I_2(s) + I_3(s) + I_4(s).