harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\operatorname{Ad}(\tilde{g}) c = c, & \operatorname{Ad}(\tilde{g})d = d - g'g^{-1}- \frac{1}{2}\| g'g^{-1}\|^2 c, &\operatorname{Ad}(\tilde{g}) u = gug^{-1}+ \langle g' g^{-1}, gug^{-1}\rangle c
\hskip-20mm c(n)\equiv \Gamma(-n)e^{-{n(n+1)T\over 2}}\prod_{i=1}^{n}\int d^2z_i\, \prod_{i<j}|z_i-z_j|^2\prod_{i,j}{1\over |1-z_i\bar{z}_je^{-T}|^{{2 \over Q^2}+1}}.
{\cal L}= - 2 f_{\phi}^{2}\ln ( S + S^{+}) |_{\theta^{2}\bar{\theta^{2}}}+ \frac{1}{2}S W_{j}^{\alpha}W_{\alpha}^{j}|_{\theta^{2}}+ \frac{1}{2}S^{+}\overline{{{W}}}{}_{j}^{\dot{\alpha}}\overline{{{W}}}{}_{\dot{\alpha}}^{j}|_{\bar{\theta}^{2}}
\sum_{k=0}^n \binom{p+c+k}{p}\binom{q+n+d-k}{q}\binom{c+k}{c}\binom{n+d-k}{d}H_{p+c+k}H_{q+n+d-k}& ={}\binom{n+p+q+c+d+1}{n}\binom{p+c}{c}\binom{q+d}{d}[H_{p+q+c+d+1}^{(2)}-H_{n+p+q+c+d+1}^{(2)}+(H_{n+p+q+c+d+1}-H_{p+q+c+d+1}+H_{p+c})(H_{n+p+q+c+d+1}-H_{p+q+c+d+1}+H_{q+d})]
\begin{cases}\partial_tg^{n+1}+v\cdot \nabla_{x}g^{n+1}+\mathcal{L}_1g^{n+1}=\Gamma(g^n,g^{n+1}) -\mathcal{L}_2g^n , g^{n+1}|_{t=0}=g_0, \hskip1.5cm ( n = 0,1,2,\cdots, \enskip g^0 = 0)\,.\end{cases}
\hat{\mathbf{h}}= \frac{1}{\sqrt{N_c}}\left(\check{\mathbf{F}}^{H}\mathbf{S}^H\mathbf{S}\check{\mathbf{F}}\right)^{-1}\check{\mathbf{F}}^{H}\mathbf{S}^H\mathbf{F}\mathrm{Diag}\left(e^{j\boldsymbol{\theta}}\right)^H\mathbf{y}.
\hat{S}= \int d^{1 0}\hat{x}\sqrt{| \hat{g}|}\, \left\{e^{- 2 \hat{\phi}}\left[ \hat{R}- 4 ( \partial \hat{\phi})^{2}\right] +{\textstyle \frac{( - 1 )^{7 - p}}{2 \cdot ( 8 - p ) !}}\left( \hat{G}^{( 8 - p )}\right)^{2}\right\}\, .
J(t,j)=\int_{-\infty}^{\infty}\,x^{j}\,\exp\left(-t x-\frac{x^2}{2}\right)\mathrm{d}x=(-1)^j\,\sqrt{2\pi}\,\,\,\frac{\partial^j}{\partial t^j}\left\{\exp\left(\frac{t^2}{2}\right)\right\}
\gamma \frac{d f}{d \gamma}\; + \; \sum_{n = 1}^{\infty}\Big [ \lambda_{n}\frac{d f}{d \lambda_{n}}+ \kappa_{n}\frac{d f}{d \kappa_{n}}+ \bar{\kappa}_{n}\frac{d f}{d \bar{\kappa}_{n}}\Big ] \; \; = \; \; f \; .
(\Phi\Psi)' =\Phi'\Psi+\Phi\Psi' (\Phi\Psi)'' =\Phi''\Psi+2\Phi'\Psi'+\Phi\Psi'' (\Phi\Psi)''' =\Phi'''\Psi+3\Phi''\Psi'+3\Psi'\Phi''+\Psi\Phi''' (\Phi\Psi)'''' =\Phi''''\Psi+4\Phi'''\Psi'+6\Phi''\Psi''+4\Phi'\Psi'''+\Psi\Phi''''
J_{\mathrm{Alg}}^{s}\epsilon_{\mathrm{L}}=\epsilon_{\mathrm{BrL}}^{s}J_{\mathrm{Alg}}^{s}J_{\mathrm{Lie}}^{s}\eta_{\mathrm{L}}=\eta_{\mathrm{BrL}}^{s}J_{\mathrm{Lie}}^{s},
\sum_{n=0}^{\infty}q^{n^{2}/2}S_{n}\left(xq^{n};q\right)t^{n}=\sum_{k=0}^{\infty}\frac{q^{5k^{2}/2}\left(-xt\right)^{k}}{\left(q;q\right)_{k}}\left(-tq^{2k+1/2};q\right)_{\infty}.
\int_{-1}^{1}\frac{(1-x^2)^{k-i}-(1-t^2)^{k-i}}{x-t}dt =-x\sum_{j=0}^{k-i-1}(1-x^2)^{k-i-j-1}2^{2j+1}\frac{(j!)^2}{(2j+1)!}.
F(x) ={E_P^2\over 8\pi}\int^\infty_{-\infty}{dk\over 2\pi}e^{-ikx}\ln \left( 1 +{8\pi k^2\over E_P^2}\right) = -{E_P^3\over 8\pi}\left({L_P\over |x|}\right) \exp \left( -{|x|\over \sqrt{8\pi}L_P}\right)
\begin{aligned}\, \|\partial_t^j D^l_x D_v^{m+2}u\|_{L_{\infty}(Q_{1/2})}\leq N (d, j, l, m) \delta^{-\theta}\big( \|D_v^2 u - (D_v^2 u)_{Q_1}\|_{L_2 (Q_1)}+ \|D_x u\|_{L_2 (Q_1)}\big). \end{aligned}
[f'(0),a]\cap \sigma_{ess}(A) \subset \left[\inf_{s\in \mathbb{R}\setminus\{0\}}\dfrac{f(s)}{s},\sup_{s\in \mathbb{R}\setminus\{0\}}\dfrac{f(s)}{s}\right]\cap\sigma_{ess}(A) = \emptyset a\notin \sigma(A).
V_m=\tilde{\Phi}_5(\alpha_5;\mathcal{F}(\sigma_3,\sigma_2;\tilde{\Phi}_1(\alpha_1;V_a))),Z_m=Z_a+\alpha_4, &V_b=\tilde{\Phi}(\beta_5,\beta_3,\beta_2,\beta_1;V_m),Z_b=Z_m+\beta_4, &V_b=\tilde{\Phi}_5(\gamma_5;\mathcal{F}(\sigma'_3,\sigma'_2;\tilde{\Phi}_1(\gamma_1;V_a))),Z_b=Z_a+\gamma_4,
\begin{cases}h_P^{\infty}(v) :=h'_P(v) - \max\mathrm{Im\,}h'_P\mbox{for}P \neq \infty, h_\infty^\infty(v) := h'_{\infty}(v) + \sum_{P \neq \infty}\max\mathrm{Im\,}h'_P. \end{cases}
N_{1,\varphi}^{*-}(x) = \left(\begin{array}{cc}0 - \frac{i \left(e^{-i \varphi x}- e^{- 4 \varphi}\right)}{x - 4 i}+ \frac{i \left(e^{-i \varphi x}- e^{- 2 \varphi}\right)}{x + 2 i}\frac{i \left(e^{-i \varphi x}- e^{- 2 \varphi}\right)}{x - 2 i}- \frac{i \left(e^{-i \varphi x}- e^{- 4 \varphi}\right)}{x + 4 i}0\end{array}\right),
\bar{G}( M^{2}) ={\frac{\lambda}{2}}\{{\frac{T^{2}}{1 2}}-{\frac{M T}{4 \pi}}+{\frac{M^{2}}{8 \pi^{2}}}( \ln{\frac{T}{\mu}}+ c o n s t . ) + O ({\frac{M^{4}}{T^{2}}}) \}
\tilde S(\eta,\mu/\sqrt{\eta}) = e^{-ia\ln4}e^{ia\ln|\eta^2-\mu^2/\eta|}\left(A + B e^{2ia\ln|(\eta+\mu/\sqrt{|\eta|})/2|}\frac{e^{i\beta (\eta+\mu/\sqrt{\eta})^3}}{(\eta+\mu/\sqrt{\eta})^3}\right) &\times\left(A + B e^{2ia\ln|(\eta-\mu/\sqrt{|\eta|})/2|}\frac{e^{i\beta (\eta-\mu/\sqrt{\eta})^3}}{(\eta-\mu/\sqrt{\eta})^3}\right)
\int_{C_{h}}{\frac{Y_{\nu}( \lambda z ) J_{\nu + m}( \sigma z ) - J_{\nu}( \lambda z ) Y_{\nu + m}( \sigma z )}{J_{\nu}( z ) Y_{\nu}( \lambda z ) - J_{\nu}( \lambda z ) Y_{\nu}( z )}F ( z ) d z}
d s_{4}^{2}= - \frac{1}{\sqrt{-{\frac{1}{6}}H^{0}\, C_{A B C}H^{A}H^{B}H^{C}}}\, d t^{2}+ \sqrt{-{\textstyle{\frac{1}{6}}}H_{0}\, C_{A B C}H^{A}H^{B}H^{C}}\, d \vec{x}^{2}\; .
U_{K G}( z ) = U ( z ) +{\frac{a M_{K G}^{2}/ 2}{({\cal Z}- z )^{2}}}={\frac{K + a M_{K G}^{2}/ 2}{({\cal Z}- z )^{2}}}\equiv{\frac{K_{K G}}{({\cal Z}- z )^{2}}}\, .
\pi_{TM, X_{1,V}, Y_{1,V}, X_{2,V}, Y_{2,V},c}({\bf x},{\bf y})=\left( \begin{array}{ccc|ccc}0 0 0 0 0 0 0 0 0 0 0 x^1 0 0 0 0 -x^1 0 \hline 0 0 0 0 0 -x^2 0 0 -x^1 0 0 y^1+x^1 0 -x^1 0 x^2 -y^1-x^1 0 \end{array}\right).
\langle{\cal O}\rangle_A=\langle{\cal O}\rangle_L+\int_0^1 d\kappa\int D\phi \biggl(\tilde\delta_1[\phi]+\kappa\tilde\delta_2[\phi]\biggr)\Theta^\prime[\phi]{\delta{\cal O}\over{\delta\phi}}e^{iS^M_{\rm eff}}.
\bigl(\mathcal{S}_{M-1}B^{(f)}_{M}(x)\bigr)(q\,|\,b) - \bigl(\mathcal{S}_{M-1}B^{(f)}_{M}(x)\bigr)(0\,|\,b) = M! \prod\limits_{j=1}^{M-1}b_j \Bigl( B^{(f)}_{1}(q+b_0)- B^{(f)}_{1}(b_0)\Bigr).
\max_{\boldsymbol{\rho}}\sum_{t}R_{t}(\boldsymbol{\rho})+\sum_{r}R_{r}(\boldsymbol{\rho}), \rho_{m}^{\mathrm{t}}, \rho_{m}^{\mathrm{r}}\in\{0,1\},\forall m \in \mathcal{M}, &\rho_{m}^{\mathrm{t}}+\rho_{m}^{\mathrm{r}}=1,\forall m \in \mathcal{M},
\begin{array}{lll}\mathcal{P}_{FD}(\mathcal{R}) = 1- \Pr \{\mathcal{I}_{1,fd}>\mathcal{R},\dots,\mathcal{I}_{K,fd}>\mathcal{R}\}&= 1-\prod_{k=1}^K [1-\mathcal{P}_{\mathcal{I}_{k,fd}}(\mathcal{R})]\end{array}
\widehat{\alpha}_{t,\varepsilon}:=\widehat{\alpha}_{t,\varepsilon}(0)=\int_0^t\int_0^sf_\varepsilon(X^H_s-X^H_r)drds &=\frac{1}{2\pi}\int_{D}\int_{\mathbb{R}}e^{-\frac{\varepsilon}{2}p^2}e^{\iota p(X^H_s-X^H_r)}dpdrds,
\mathbf{1}_0 = p^X_0 c_{w_0}^{X^{-1}}= p^\lambda_X p^X_\lambda (w^{-1}_\lambda c_{v_\lambda}^{X^{-1}}) c_{w_\lambda}^{X^{-1}}= p^\lambda_X c_{v_\lambda}^{X^{-1}}p^X_\lambda c_{w_\lambda}^{X^{-1}}= p^\lambda_X c_{v_\lambda}^{X^{-1}}\mathbf{1}_\lambda.
{\cal D}_{g_{\mu \nu}(w)}\nabla^{g_{\kappa \lambda}(z)}_{\sigma(x)}\equiv \delta(z-w) \left[t_{\kappa \lambda\ \ \ , \sigma \tau}^{\ \ \ \mu \nu}\nabla^{\tau (x)}\right] \delta (w-x) +{\cal O}[l^{-1}] \; ,
\begin{aligned}\partial_tH= \alpha H^{-\alpha-1}\Delta H-\alpha(\alpha+1)H^{-\alpha-2}|\nabla H|^2-|A|^2H^{-\alpha}+2H^{-\alpha}= -\Delta H^{-\alpha}-|A|^2H^{-\alpha}+2H^{-\alpha}. \end{aligned}
\begin{aligned}\big(T_{q, k_1}(F)*T_{q, k_2}(G)\big)_{q}^{(g_1,g_2;h_3,h_4)}(y) &=T_{q,\mathbf{s}(h_3,k_1)}(F)\big(\mathcal Z_{g_1}(y,\cdot)\big) T_{q,\mathbf{s}(h_4,k_2)}(G)\big(\mathcal Z_{g_2}(y,\cdot)\big). \end{aligned}
\sum_{\beta\in B^*}\sum_{k:\ \beta_k \le \beta}w^{-(\beta-\beta_k)}t^{2(\beta-\beta_k)}=\sum_{k=1}^N\sum_{\beta\in B^*:\, \beta\ge\beta_k}w^{-(\beta-\beta_k)}t^{2(\beta-\beta_k)}.
\begin{aligned}\mbox{there are}\,\, \beta>2\,\, \mbox{and}\,\, r_0>0\,\, \mbox{such that}& tf(x,t)\geq \beta F(x,t)>0\,\, \mbox{for any}\,\, t\geq r_0\,\, \mbox{and every}\,\,\, x\in D.\end{aligned}
x \, \frac{\theta_2'(x)}{\theta_2(x)}+ \tfrac{1}{x}\, \frac{\theta_4' \left( \tfrac{1}{x}\right)}{\theta_4 \left( \tfrac{1}{x}\right)}= -\frac{1}{2}x \, \frac{\theta_4'(x)}{\theta_4(x)}+ \tfrac{1}{x}\, \frac{\theta_2' \left( \tfrac{1}{x}\right)}{\theta_2 \left( \tfrac{1}{x}\right)}= -\frac{1}{2},
{\cal L}_{\mathrm{b u l k}}( t , y ) ={\frac{M_{3}^{2}}{2}}\sqrt g \ \left( R - \frac{1}{2}( \partial \phi )^{2}-{\frac{3}{2}}\frac{e^{2 \phi}{\cal F}^{2}}{5 !}\right) = -{\frac{3 M_{3}^{2}\tilde{A}^{3}B}{\tilde{N}}}\left[ \left({\frac{\dot{\tilde{A}}}{\tilde{A}}}\right)^{2}-{\frac{7}{4}}\left({\frac{\dot{D}}{D}}\right)^{2}\right] .
(U_i^*U_{i+1}L) = (U_i^*)^{\otimes r_{i+1}}\otimes(U_{i+1}L)^{\otimes r_i}&=(U_i^*)^{\otimes r_{i+1}}\otimes(U_{i+1})^{\otimes r_i}\otimes(L)^{\otimes r_ir_{i+1}}
\rho_{i \pm}( \tau_{i \pm}( \tau , \vec{\sigma}) , \vec{\sigma}) = r_{i \pm}( \tau_{i \pm}( \tau , \vec{\sigma}) , \vec{\sigma}) [ 1 \mp \vec{\beta}_{i}( \tau_{i \pm}( \tau , \vec{\sigma}) ) \cdot \hat{r}_{i \pm}( \tau_{i \pm}( \tau , \vec{\sigma}
\eta(t,u) = \frac{1}{(1 - \gamma t (e^{\gamma u}-1))^{\lambda/\gamma}}\Big( \frac{e^{\gamma u}- \gamma t (e^{\gamma u}- 1)}{1 - \gamma t (e^{\gamma u}-1)}\Big)^{X(0)/\gamma}= \frac{(e^{\gamma u}- \gamma t (e^{\gamma u}- 1) )^{X(0)/\gamma}}{(1 - \gamma t (e^{\gamma u}-1))^{Y(0)/\gamma}}
|V_{2m}(x)|\leq\sum_{n,n'=1}^N\sum_{\substack{1\leq ||j||_{\infty},||j'||_{\infty}\leq G, ||M^T_nj\pm M^T_{n'}j'||_{\infty}<||M^T_{m'}||_{\infty}}}(|a_j|+|b_j|)(|a_{j'}|+|b_{j'}|)\leq C\sqrt{d}||p||_2N
\delta L=\sum_{j=1}^{n}(\frac{\partial L}{\partial q_{j}}\delta q_{j}+\frac{\partial L}{\partial\dot{q}_{j}}\delta\dot{q}_{j}),\delta\dot{q}_{j}\equiv\delta\frac{dq_{j}}{dt}\equiv\frac{d(\delta q_{j})}{dt},
\mathrm{d}x\left( t\right) =f\left( t,x\left( t\right) ,r\left(t\right) \right) \mathrm{d}t+g\left( t,x\left( t-d\left( t,r(t)\right)\right) ,r\left( t\right) \right) \mathrm{d}w\left( t\right) ,
\lim_{\xi_{1}\rightarrow 0}\langle 0_{D}| T_{i}^{k}| 0_{D}\rangle = \langle 0_{R}| T_{i}^{k}| 0_{R}\rangle + \langle T_{i}^{k}\rangle_{D}^{( 1 b )}( \xi_{2}, \xi ) , \quad \xi < \xi_{2},
\vec{\phi}( \tau , x ) = \frac{1}{\sqrt{1 + \left( f^{( \perp )}( \tau , x ) \right)^{2}}}\left( \begin{array}{c}{{\sin f^{( \parallel )}( \tau , x )}}\\{{f^{( \perp )}( \tau , x )}}\\{{\cos f^{( \parallel )}( \tau , x )}}\\ \end{array}\right)
{}^{*}( d x^{\alpha_{1}}\wedge d x^{\alpha_{2}}\wedge . . . \wedge d x^{\alpha_{m}}) = \frac{\sqrt{| g |}}{( n - m ) !}\varepsilon_{\ \ \ \beta_{m + 1}. . . \beta_{n}}^{\alpha_{1}. . . \alpha_{m}}d x^{\beta_{m + 1}}\wedge . . . \wedge d x^{\beta_{n}},
0< \int_0^{x_0}\frac{\tilde{\theta}(y)}{y-x}\, dy = \int_0^{x_0}\frac{\tilde{\theta}(y) - \tilde{\theta}(x)}{y-x}\, dy + \tilde{\theta}(x) \left(\log|x_0-x| - \log|x| \right) \leq C+ \log \frac{1}{|x|}.
h_2^2(P) = (v_1+v_2)^{2}+ (v_2+v_3)^{2}+ (v_3+v_4)^{2}+ (v_1+v_4)^{2}+\tfrac{1}{2}(v_1+v_3)^{2}+\tfrac{1}{2}(v_2+v_4)^{2}- v_1^{2}- v_2^{2}- v_3^{2}- v_4^{2}=\tfrac{1}{2}(v_1+v_2+v_3+v_4)^{2}+\tfrac{1}{2}(v_1+v_2)^{2}+\tfrac{1}{2}(v_2+v_3)^{2}+\tfrac{1}{2}(v_3+v_4)^{2}+\tfrac{1}{2}(v_1+v_4)^{2}.
0 \le E [V_{0,a,a+b}] = \varphi_a + \varphi_b - \varphi_{a+b}\le 2E \sum_{i=1}^a \sum_{\ell=1}^b g_{a,\alpha}(i) G(S_i,\tilde{S}_{\ell})\tilde{g}_{b,\alpha}(\ell) = 2 E Y_{a,b}\,.
\epsilon \frac{dc}{dt}=\mu h\frac{b+c}{1+c}-\frac{\Gamma}{K_1}\frac{c}{K+c}+\frac{\lambda}{K_1}\theta=R_1/K_1=F_1, \frac{d\theta}{dt}=-\theta+T(c)=R_2, \frac{dh}{dt}=\frac{1}{1+c^2}-h=R_3.
\left( \begin{array}{c}{{Z_{1}}}\\{{Z_{2}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\cos ( \mu x^{+})}}&{{\sin ( \mu x^{+})}}\\{{- \sin ( \mu x^{+})}}&{{\cos ( \mu x^{+})}}\\ \end{array}\right) \left( \begin{array}{c}{{x_{1}}}\\{{x_{2}}}\\ \end{array}\right) .
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X,E)=\lim_{t\rightarrow0^+}\int_X\sum^n_{\ell=0}t^{-\ell}\Bigr(\mathrm{Tr\,}a^+_\ell(t,x,x)-\mathrm{Tr\,}a^-_\ell(t,x,x)\Bigr)dv_X(x).
\frac{(-1)^{n-3}\det\Big( l_{i_1}, \ldots , l_{i_{n-2}}, n_j, \widetilde{n}\Big)}{(n-2)! \sqrt{1 - \langle n_j,\widetilde{n}\rangle^2}}\sum_{\omega =1}^{n-2}\left( \langle u_{i_\omega}, \widetilde{n}\rangle^{-1}\prod_{\gamma = 1}^{n-2}\frac{\langle u_{i_\gamma}, \widetilde{n}\rangle}{\langle l_{i_\gamma}, \xi_0 \rangle}\right) .
\int_{\mathbb{R}^N\times \mathbb{R}^N}\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}\, dx\, dy &=\int_{E\times E}\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}+2\int_{\Omega\times E^c}\frac{J_p(u(x)-u(y))\,\varphi(x)}{|x-y|^{N+s\,p}}\, dx\, dy.
\Pi(1) = \{(1)\}\Pi(2) = \{(2), (1,1)\}\Pi(3) = \{(3), (2,1), (1,1,1)\}\Pi(4) = \{(4), (3,1), (2,2), (2,1,1), (1,1,1,1)\}\Pi(6) = \left\{(6), (5,1), (4,2), (4,1,1), (3,3), (3,2,1), (3,1,1,1),\right. \ \ \ \ \ \ \left.(2,2,2), (2,2,1,1), (2,1,1,1,1), (1,1,1,1,1,1)\right\}.
\varphi \big( g^{j}_1 g_3 \big) \varphi \big( g^{i}_1 g_2 \big) = \varphi (g_3) \varphi (g^{n}_1) \varphi (g_2) \overset{(\beta_2)}{=}\varphi (g^{n}_1 g_3) \varphi (g_2) \,,
\alpha_n^{(d)}(i,j,k_1,\dots,k_{d-2}) = \alpha_n^{(3)}(i,j,k_1+\cdots+k_{d-2}) \beta_n^{(d)}(i,j,k_1,\dots,k_{d-2}) = \beta_n^{(3)}(i,j,k_1+\cdots+k_{d-2})
a_{\mathrm{pw}}(\phi_{\mathrm{nc}}({j}),v_{\mathrm{nc}})=\lambda_h(j) b(\phi_{\mathrm{pw}}({j}),v_{\mathrm{nc}})\phi_{\mathrm{pw}}({j})-\phi_{\mathrm{nc}}({j})=\lambda_h({j})\kappa_{m}^2h_{\mathcal{T}}^{2m}\phi_{\mathrm{pw}}({j}).
N(n) = \left\|\frac{1}{\lambda_{1,n}^{(d-2)/2}}\tilde{V}_1 \left(\frac{x - x_{1,n}}{\lambda_{1,n}}\right)\cdot \frac{1}{\lambda_{2,n}^{(d-2)/2}}\tilde{V}_{2}\left(\frac{x - x_{2,n}}{\lambda_{2,n}}\right)\right\|_{L^{2^\ast/2}}\rightarrow 0.
\begin{aligned}b_4 b_4 = b_0 b_8 -\dfrac{b_1^1}{b_0^0}b_7 + 2 \dfrac{b_1^3}{b_0^2}b_5 - \dfrac{3}{2}\dfrac{b_1^5}{b_0^4}b_3 + \dfrac{1}{2}\dfrac{b_1^7}{b_0^6}b_1.\end{aligned}
\Omega_{\rho}( z_{1}, z_{2}) = \sum_{n_{1}= 0}^{\infty}\sum_{n_{2}= 0}^{\infty}A ( n_{1}\! + \! \rho_{1}, n_{2}\! + \! \rho_{2}) z_{1}^{n_{1}+ \rho_{1}}z_{2}^{n_{2}+ \rho_{2}},
(X_{a^{\ast}}(\widetilde{P}))_{\overline{N}}=\biggl(c_{k}\displaystyle \sum_{b \in T_{k}, r \in L(k)}r^{-1}(^{\ast}b)\rho(bra)\biggr)_{\overline{N}}&=c_{k}\displaystyle \sum_{b \in T_{k},r \in L(k)}(r^{-1}(^{\ast}b)\rho(bra))_{\overline{N}}
S_{\mathrm{o d d}}( a ; b_{1}, \ldots , b_{k - 1}) = \sum_{n = 1}^{\infty}\frac{1}{( 2 n - 1 )^{a}}\prod_{i = 1}^{k - 1}\sum_{m_{i}= 1}^{n - 1}\frac{1}{( 2 m_{i}- 1 )^{b_{i}}}\, ,
[\begin{matrix}I\\P\\T\end{matrix}]_{PQC2}=(\frac{1}{8192}[\begin{matrix}8192&799&1681\\ 8192&-933&1091\\ 8192&267&-5545\end{matrix}])^{-1}[\begin{matrix}L^{\prime}\\M^{\prime}\\S^{\prime}\end{matrix}]
u_{lip}\nu^l = [c^{pq}+ \nu^p \nu^q ] u_{liq}\nu^l = c^{pq}[D_q (u_{li}\nu^l)- u_{li}D_q \nu^l] + \nu^p \nu^q u_{liq}\nu^l = c^{pq}D_q (c^{ij}D_j\varphi - c^{ij}u_l D_j \nu^l + \nu^i \nu^j \nu^l u_{lj}) -c^{pq}u_{li}D_q \nu^l + \nu^p \nu^q \nu^l u_{liq}.
\begin{array}{ll}U_{x^{\sigma}}V_{y^{-\sigma},x^{\sigma}}=V_{x^{\sigma},y^{-\sigma}}U_{x^{\sigma}}V_{U_{x^{\sigma}}y^{-\sigma},y^{-\sigma}}=V_{x^{\sigma},U_{y^{-\sigma}}x^{\sigma}}U_{U_{x^{\sigma}}y^{-\sigma}}=U_{x^{\sigma}}U_{y^{-\sigma}}U_{x^{\sigma}}\end{array}
H=\sum_{i=1}^{N}\left(x_i \frac{\partial}{\partial x_i}\right)^2+\beta\sum_{i<j}\frac{x_i+x_j}{x_i-x_j}\left(x_i\frac{\partial}{\partial x_i}-x_j\frac{\partial}{\partial x_j}\right)-2\beta\sum_{i<j}\frac{x_i x_j}{\left(x_i-x_j\right)^2}\left(1-K_{ij}\right),
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{\ell_1}c_1 \,a^{\ell_1}\, b^{p^{e_1}}\, d^{p^{e_1}}c_2 \,a^{\ell_1}\, b^{p^{e_2}}\, d^{p^{e_2}}0 a^{\ell_2}0 0 0 d^{- \ell_3}\end{array}\right) .
y(x) =\sum_{n=0}^{\infty}\frac{(l_k+\frac{m}{2}+\lambda)\cdots(l_k+\frac{m}{2}+\lambda+n-1)}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+n+1+\lambda-i)n!}x^n &=\sum_{n=0}^{\infty}\frac{(l_k+\frac{m}{2}+\lambda-i+1+n-1)\cdots(l_k+\frac{m}{2}+\lambda+n-1)}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)n!}x^n
\begin{aligned}\langle \mathcal{M}^n (\varphi ) \rangle_t =\int_0^t \big( L_n ( \mathcal{X}^n_s (\varphi ) )^2 - 2 \mathcal{X}^n_s (\varphi ) L_n \mathcal{X}^n_s (\varphi ) \big) ds & = \frac{1}{n}\int_0^t \sum_{j \in \mathbb{Z}}g_n (\eta^n_j (s) ) \big[ n ( \varphi^n_{j+1}(s) - \varphi^n_j (s) ) \big]^2 ds ,\end{aligned}
S_{3}:=\{\begin{bmatrix}1 2\end{bmatrix},\begin{bmatrix}-1 -2\end{bmatrix},\begin{bmatrix}1 -2\end{bmatrix},\begin{bmatrix}-1 2\end{bmatrix},\begin{bmatrix}2 1\end{bmatrix},\begin{bmatrix}-2 -1\end{bmatrix},\begin{bmatrix}2 -1\end{bmatrix},\begin{bmatrix}-2 1\end{bmatrix}\}.
\mathbb{Q}'_{\mathbb{I}}:=\{Q'\in\mathbb{Q}': Q'\subseteq\{\xi^{1}_{1}<\xi^{1}_{2}\}\}, \,\,\,\,\,\,\,\,\, \mathbb{Q}'_{\mathbb{II}}:=\{Q'\in\mathbb{Q}': Q'\subseteq\{\xi^{1}_{1}>\xi^{1}_{2}\}\}.
n^2\pi^2\int_{S_2}\vert \varphi \vert^{r'}d\mu - \dfrac{1}{r'}\int_{S_2}\mathcal{G}\cdot \dfrac{\partial (\vert \varphi^{r'}\vert)}{\partial u}d\mu = \int_{S_2}h\vert \varphi \vert^{r'-1}sgn(\varphi)d\mu
\beta_1 \coloneqq x_{1,n}, \beta_2 \coloneqq \begin{vmatrix}x_{1,n-1}x_{1,n}x_{2,n-1}x_{2,n}\end{vmatrix}, \beta_3 \coloneqq \begin{vmatrix}x_{1,n-2}x_{1,n-1}x_{1,n}x_{2,n-2}x_{2,n-1}x_{2,n}x_{3,n-2}x_{3,n-1}x_{3,n}\end{vmatrix}, \cdots
( \gamma^{\mu}{\cal C}) P_{\mu}= \left( \begin{array}{}{{c ]{c c c c}i \partial_{+}}}&{{\ast}}&{{\ast}}&{{\ast}}\\{{\ast}}&{{i \partial_{+}}}&{{\ast}}&{{\ast}}\\{{\ast}}&{{\ast}}&{{i \partial_{-}}}&{{\ast}}\\{{\ast}}&{{\ast}}&{{\ast}}&{{i \partial_{-}}}\\ \end{array}\right) \, ,
{\cal Z}( G , \lambda A , N ) = \int_{\Sigma ({\cal M})}d \nu e^{- \frac{( n + \tilde{n}) \lambda A}{2}}\frac{( - 1 )^{i + \tilde{t}}N^{2 - 2 g}}{| S_{\nu}|}\prod_{j = 1}^{\max ( 0 , - I )}( - 1 )^{x_{j}},
R_{\hat{\frac 1 2}\otimes j}( u ) = \left( \begin{array}{c c c c}{{\sin ( u + ( \frac{1}{2}+ \hat{H}) \eta )}}&{{\sin ( \eta ) \hat{F}}}\\{{\sin ( \eta ) \hat{E}}}&{{\sin ( u + ( \frac{1}{2}- \hat{H}) \eta )}}\\ \end{array}\right) ,
S_n^{[0, \infty)}(t) ={1\over n}\left( D_{n,2\lfloor nt \rfloor}(\vec{m}_{2n}) - \mathbb{E}[ D_{n,2\lfloor nt \rfloor}(\vec{m}_{2n}) ] \right) R_n^{[0, \infty)}(t) = \frac{1}{n}\mathbb{E}[ D_{n,2\lfloor nt \rfloor}(\vec{m}_{2n}) ]
c(x',y,t) = \sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x) \vert}\left[c(x,y,t)+\frac{1}{4}h(x)(2x^{2}-1)\frac{\partial^{2}}{\partial y^{2}}\alpha(y,t)-\frac{1}{6}x^{3}h(x)\frac{\partial^{3}}{\partial y^{3}}p_{0}(y,t)\right]
(u_1,u_2) = \left({\phi_2 \over \phi_1},{\phi_3 \over \phi_1}\right), \quad (v_1,v_2) = \left({\phi_1 \over \phi_2},{\phi_3 \over \phi_2}\right), \quad (w_1,w_2) = \left({\phi_1 \over \phi_3},{\phi_2 \over \phi_3}\right) .
E(x,y,T)\; =\; \mathcal{O}\Big(T^{-1}(|x|+|y|)(T^{1/6}+|x|^{1/6}+|y|^{1/6})^2\Big)=\mathcal{O}\Big( T^{-1}(|x|+|y|)T^{1/3}\Big) =\; \mathcal{O}\Big( T^{-1/12}\big(1+|x|^{5/12}+|y|^{5/12})\Big) =: \;\mathcal{O}_1(x,y,T),
\int_{\{-t_1\leq\psi<-t_2\}}|F|^2e^{-\varphi}a(-\psi)=\frac{G(T_1;c)}{\int_{T_1}^{+\infty}c(l)e^{-l}dl}\int_{t_2}^{t_1}a(t)e^{-t}dt
\mathbf{E}\left[\|\zeta(s,\mathbb{X}_\alpha)-\zeta_L(s,\mathbb{X}_\alpha)\|^2\right] = \sum_{m,n>L}\mathbf{E}[\mathbb{X}_\alpha(m) \overline{\mathbb{X}_\alpha(n)}]\left\langle (m+\alpha)^{-s}, (n+\alpha)^{-s}\right\rangle &= \sum_{n>L}\|(n+\alpha)^{-s}\|^2
\mathfrak{h}^0_{i+1}= \left( \begin{array}{c}\frac{h_{i+1}^+}{\sqrt{z}}\frac{h^-_{i+1}}{\sqrt{\bar{z}}}\end{array}\right)+\mathfrak{h}_{\mathfrak{R},i+1};\mbox{}\mathfrak{c}^0_{i+1}= \left( \begin{array}{c}\frac{c_{i+1}}{2\sqrt{z}}\frac{c^{aps}_{i+1}}{2\sqrt{\bar{z}}}\end{array}\right)+\mathfrak{c}_{\mathfrak{R},i+1}.
f\big( a_1(i) e_1(i) \big) = \sum_{k\in I_2}f_i^k \, \frac{a_1(i)}{a_2(k)}\, \big(a_2(k) e_2(k) \big) f\left ( \sqrt{a_1(i)}e_1(i) \right) = \sum_{k\in I_2}f_i^k \, \sqrt{\frac{a_1(i)}{a_2(k)}}\, \left( \sqrt{a_2(k)}\, e_2(k) \right).
(D_XR')(Y,Z) =\nabla '_XR'(Y,Z)-R'(\tilde{\nabla}_XY,Z)-R'(Y,\tilde{\nabla}_XZ), (D_XQ)_{\alpha}(Y) =\tilde{\nabla}_XQ_{\alpha}(Y)-Q_{\nabla '_X\alpha}(Y)-Q(\tilde{\nabla}_XY).
d_W\theta^\alpha +{1\over 2}C^{\alpha}_{\beta\gamma}\theta^\beta\wedge \theta^\gamma = u^\alpha\ ,\{}d_W u^\alpha = - C^{\alpha}_{\beta\gamma}\theta^\beta \wedge u^\gamma\ ;\{}i_{W\,X_\alpha}\theta^\beta=\delta_\alpha^\beta\ ,\{}i_{W\,X_\alpha}u^\beta=0\ ;
\psi^{\otimes n}(\xi)^* \rho(x) \psi^{\otimes n}(\xi) = \psi^{\otimes n}(\xi)^* \left( \sum_{i=1}^N \psi^{\otimes n_i}(\xi_i) \psi^{\otimes m_i}(\eta_i)^* \right) \psi^{\otimes n}(\xi) &= \pi (g) + \sum_{n_i > m_i}\psi^{\otimes n'_i}(\xi'_i) + \sum_{n_i < m_i}\psi^{\otimes m'_i}(\eta'_i)^*.
{}_{r+1}\phi_r^{(m)}\left(\begin{matrix}a_1,\dots,a_{r+1}b_1,\dots,b_r\end{matrix};q,x\right)=\sum_{0\leq k_1<k_2<\dots<k_m}\Delta(q^k)^2\prod_{j=1}^m\frac{(a_1,\dots,a_{r+1};q)_{k_j}x^{k_j}}{(q,b_1,\dots,b_r;q)_{k_j}}.
\bar{H}(\bar{x}, \bar{p})= \lim_{\delta \rightarrow 0}\sup_{\beta(\cdot)}\delta E \left[ \int_0^\infty \left (|\sigma(\bar{x}, Z_t)^T \bar{p}|^2 - |\beta(t)|^2\right) e^{-\d t}dt \, |\, Z_0=z\right]
0 =\widehat{\varphi}\circ\widehat{f}-\widehat{\varphi}-\beta =\widehat{\varphi}_{n}\circ\widehat{f}_{n}-\widehat{\varphi}_{n}+\left(\widehat{\varphi}-\widehat{\varphi}_{n}\right)\circ\widehat{f}_{n}-\left(\widehat{\varphi}-\widehat{\varphi}_{n}\right)+\sum_{i\geq1}\frac{\widehat{\varphi}^{(i)}\circ\widehat{f}_{n}}{i!}\widehat{g}_{n}^{i}- \beta ,
\frac{1}{w_{1}}-\frac{1}{2}\coth\left(\frac{w_{1}}{2}\right)= -\sum_{k = 0}^{+ \infty}\frac{w_{1}^{2k+1}}{\left(2k+2\right)!}B_{2k+2}\thinspace \thinspace \thinspace \thinspace\frac{1}{w_1+w_2}= \frac{1}{w_2}\sum_{l \ge 0}\left(- \frac{w_1}{w_2}\right)
e^{t_0}\lim_{j\rightarrow+\infty}\frac{G(t_0)-G(t_0+B_j)}{B_j}=\lim_{j\to+\infty}\frac{G(t_0)-G(t_0+B_j)}{e^{-t_0}-e^{-t_0-B_j}}&=\liminf_{B\to0+0}\frac{G(t_0)-G(t_0+B)}{e^{-t_0}-e^{-t_0-B}}
J_q(L,r)=\sum q^{X_r(L)}\prod_{i=1}^{\nu-2}\left[\begin{array}{c}n_i+m_i n_i \end{array}\right]_q=\sum q^{X_r(L)}\prod_{i=1}^{\nu-2}\left[\begin{array}{c}n_i+\tilde{n}_i-V_{i,r}n_i \end{array}\right]_q ;\;\; L+r\equiv even \;\;
H_\kappa(G) \leq I(V;W^*) = I(V,A;W^*) = I(A;W^*) + \sum_{a \in \mathcal{A}}P_A(a) I(V;W^*|A = a) = \sum_{a \in \mathcal{A}}P_A(a) I(V_a;W_a^*) = \sum_{a \in \mathcal{A}}P_A(a) H_\kappa(G_a);
u_t(X_t^{h^i})-u_t(X_t) = F(t,L_t X_t^{h^i},X^{h^i}(t)) -F(t,L_t X_t,X(t)) = &=F(t,X(t)+h^i,L_t X_t^{h^i}) -F(t,X(t),L_t X_t)\, .
\frac{E_{0}\left( \ell , \mu , d \right)}{L^{d - 1}}\approx - \frac{C \left( d \right) \left( 2^{d}- 1 \right) \Gamma \left( d \right) \zeta_{R}\left( d + 1 \right)}{2^{3 d - 1}\pi^{d / 2}\Gamma \left( \frac{d}{2}\right) \ell^{d}}.
S_{E}\,=\,M_{0}^{2}\,\int\limits_{0}^{T}\!\! dt\int\limits_{0}^{R}\!\!dr\left[1\,+\,\frac{1}{2}{\bf u}\,\partial^{2}{\bf u}\right]\,+\,\sum_{a=1}^{2}\,\frac{m_a}{2}\,\int\limits_{t_1}^{t_2}\!\!dt\,{\dot{\bf u}}^2(t,r_a)\,{,}
det(AB)=|\begin{matrix}1&1\\ 3&1\end{matrix}|\cdot|\begin{matrix}1&1\\ 3&1\end{matrix}|+|\begin{matrix}1&2\\ 1&-1\end{matrix}|\cdot|\begin{matrix}3&1\\ 0&2\end{matrix}|+|\begin{matrix}1&2\\ 3&-1\end{matrix}|\cdot|\begin{matrix}1&1\\ 0&2\end{matrix}|.
h(x) = \sum_{\substack{|k| \geq n+1 n \, \nmid \, k}}e^{-\pi \alpha k^2}\left( \sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}\right)e^{2 \pi i k x}.
\begin{aligned}\left(\frac{3 \boldsymbol{u}_h^n-3\widetilde{\boldsymbol{u}}_h^n}{2 \Delta t}+\nabla (p_h^{n}-p_h^{n-1}), \boldsymbol{y}_h \right) =0 (\nabla\cdot \boldsymbol{u}_h^n, q_h) =0 \boldsymbol{u}_h^n|_{\partial\Omega}=0\end{aligned}
\max_{p_i\in\mathbb{R},~i=1,\dots,K}~ \sum_{i=1}^K\alpha_i\log_2(1+\zeta_i(\{p_k\}_{k{\ne}i})Q_{ii}p_i){\rm s.t.}~ 0~{\le}~ p_i~{\le}~ P_i,~i=1,\dots,K,
\int\limits_{Y_{3T_1}}(u_s^f\overline{v^g}-u^f\overline{v_s^g})dsdy' =2 \int\limits_{Y_{3T_1}}u_s^f\overline{v^g}dsdy'-\int\limits_{\partial Y_{3T_1}\cap\{y_n=0\}}u^f(T_2-T_1,0,y')\overline{v^g}(T_2-T_1,0,y')dy'.
\frac{d}{dt}\int \mathcal{E}dx+\frac{d}{dt}\int \left( \frac{1}{2}|\psi_y(t,y)|^2 + \frac{1}{4}|\psi(t,y)|^4 + \alpha g(v(t,y)) h(|\psi(t,y)|^2) \right)dy=0.
\tilde{\rho}_{B^{2}}( s ) = \frac{\beta}{2 \pi}\ \ \rho_{B^{2}}( s ) = \frac{\beta}{2 \pi}\sqrt{\frac{b^{2}- s}{s - c^{2}}}\ \ \rho_{B}( \eta ) = \frac{\beta \eta}{\pi}\sqrt{\frac{b^{2}- \eta^{2}}{\eta^{2}- c^{2}}}
\Phi^{(s+1)}_{\alpha}:=d\Phi^{(s)}_{\alpha}(X_{T}):\approx0\ \ \ (s=2,3,\cdots,m_{\alpha}-1), &\Phi^{(m_{\alpha}+1)}_{\alpha}:=d\Phi^{(m_{\alpha})}_{\alpha}(X_{T})=C_{\alpha s}^{\beta}\Phi^{(s)}_{\beta},
h = \frac{8 \pi}{15}l_p^{\prime 6}\sum_{n=1}^{N}\sum_{n_0 = -\infty}^{\infty}\frac{Q_n}{\left( |\vec{r}- \vec{r}_n |^2 + (x_{11}- x_{n11}+ n_0 R_{11}^{\prime})^2 \right)^3},
VT_0 = T_0,VT_{-1}= - T_{-1}, VT_i = \zeta_i T_i, V\overline T_i = \overline\zeta_i \overline T_i, i=1,2,\dots, n, S_0V = S_0,S_{-1}V = - S_{-1}, S_iV_i = \zeta_i S_i, \overline S_i V= \overline\zeta_i \overline S_i, i=1,2,\dots, n,