harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x_{i_{1},i_{2}}x_{i_{3},i_{4}}=x_{i_{\sigma_{1}},i_{\sigma_{2}}}x_{i_{\sigma_{3}},i_{\sigma_{4}}}, y_{i_{1},i_{2}}y_{i_{3},i_{4}}=y_{i_{\sigma_{1}},i_{\sigma_{2}}}y_{i_{\sigma_{3}},i_{\sigma_{4}}},
T(z) = O_n\left(\begin{matrix}1 \log(\vert z + 1\vert) 1 \log(\vert z + 1\vert)\end{matrix}\right) , T(z) = O_n\left(\begin{matrix}\log^{1/2}(\vert z-1\vert) \log^{3/2}(\vert z - 1\vert) \log^{1/2}(\vert z-1\vert) \log^{3/2}(\vert z - 1\vert)\end{matrix}\right) .
e T_{\mu \nu}= e \Theta_{\mu \nu}+ \frac{1}{2}g_{\mu \nu}\left({\tilde{\overline{\Psi}}}_{L_{0}}\frac{\delta{\tilde{S}}_{F}}{\delta{\tilde{\overline{\Psi}}}_{L_{0}}}+{\tilde{\Psi}}_{L_{0}}\frac{\delta{\tilde{S}}_{F}}{\delta{\tilde{\Psi}}_{L_{0}}}\right) .
{\cal B}^\pm_+ = B^\pm b^{(0)\pm}, \quad{\cal B}^+_+ = ({\cal B}^-_+)^* \;\;{\rm for}\; q = 2, \quad{\rm or}\quad{\cal B}^\pm_- = B^\pm b^{(0)\mp},\quad{\cal B}^+_- = ({\cal B}^-_-)^* \;\;{\rm for}\; q = - 2,
B_{\mu \nu}( t ) = \sum_{\rho = 0}^{N}\frac{\eta_{\mu \rho}\eta_{\nu \rho}}{\sqrt{8 \omega_{\mu}}}\left[ \left( 1 + \frac{\omega_{\mu}}{\Omega_{\rho}}\right) \mathrm{e}^{i \Omega_{\rho}t}+ \left( 1 - \frac{\omega_{\mu}}{\Omega_{\rho}}\right) \mathrm{e}^{- i \Omega_{\rho}t}\right] .
\hat \Delta_\mu ={{g^2}\over{8\pi^2}}\epsilon_{\mu\nu\rho\sigma}[\hat A^\nu\hat F^{\rho\sigma}-g\theta_{\alpha\beta}(\hat A^\alpha\hat F^{\beta\nu}\hat F^{\rho\sigma}-\hat A^\nu\hat F^{\rho\alpha}\hat F^{\sigma\beta}+{1\over 2}\hat A^\alpha\hat F^{\rho\sigma}\partial^\nu \hat A^\beta +\hat A^\nu \hat A^\alpha\partial^\beta\hat F^{\rho\sigma})].
\frac{q \widetilde{W}(x)}{|I|}\sum_{\substack{m \in I m \equiv A ~(q \widetilde{W}(x))}}f_x(m)- S_{f_x}(x;\widetilde{W}(x),A) &= O_{E,H,\kappa}\bigg(\varphi'(x) \frac{1}{\log x}\frac{\widetilde W(x)}{\phi(\widetilde W(x))}\prod_{\substack{p<x p\nmid \widetilde W(x)}}\left(1 + \frac{|f(p)|}{p}\right) \bigg)
\Xi X = \left( \begin{array}{c c}{{m}}&{{b}}\\{{\bar{b}}}&{{n}}\\ \end{array}\right) \left( \begin{array}{c c}{{M}}&{{B}}\\{{\bar{B}}}&{{N}}\\ \end{array}\right) = \left( \begin{array}{c c}{{m M + b \bar{B}}}&{{m B + b N}}\\{{\bar{b}M + n \bar{B}}}&{{\bar{b}B + n N}}\\ \end{array}\right) ,
\mu_{\alpha, \infty}= \prod_{k=1}^d \left(\int_0^\infty \left(\frac{s}{\pi}\right)^{\alpha_k}e^{-s}\, ds \right) = \prod_{k=1}^d \frac{\Gamma(\alpha_k)}{\pi^{\alpha_k}}= \prod_{k=1}^d \frac{\alpha_k!}{\pi^{\alpha_k}}= \frac{\alpha!}{\pi^{|\alpha|}}.
\mathcal{H}^n =\!\left(-1+\frac{\tau}{2}\eta^n-\frac{\tau^2}{2}\vartheta^n\!\right)\!\phi^{n-1}+\frac{\tau^2}{4}\Delta\phi^{n-1}+2\phi^n+\tau^2\left(\!(\vartheta^n+\lambda^n)\phi^n-\frac{1}{2}\Delta\phi^n-G(\phi^n)\!\right)
\eta_{E}^{\prime}= \left( \begin{array}{c}{{\eta_{E}^{+}}}\\{{\eta_{E}^{-}}}\\ \end{array}\right) , \quad \eta_{M}^{\prime}= \left( \begin{array}{c}{{\eta_{M}^{+}}}\\{{\eta_{M}^{-}}}\\ \end{array}\right) ,
\partial_{m}\hat{X}^{\underline{{{m}}}}( \xi ) \partial_{n}\hat{X}_{\underline{{{n}}}}( \xi ) ={\frac{1}{2}}g_{m n}( \xi ) g^{p q}( \xi ) \partial_{p}\hat{X}^{\underline{{{m}}}}( \xi ) \partial_{q}\hat{X}_{\underline{{{n}}}}( \xi ) .
\begin{aligned}\alpha_s^{(N)}= (-1)^{N+s}\, (q^s - q^{-s}) \, N \, a_s, \beta_s^{(N)}= [s]^2 \, (b_s^+ - b_s^-), \gamma_s^{(N)}= [s]^2 \,\left(\dfrac{s}{N}b_s^+ + \dfrac{N-s}{N}\, b_s^-\right) +(q^s + q^{-s}) \, a_s, \qquad(1 \leq s \leq N-1) \gamma_0^{(N)}=a_0, \qquad\qquad\qquad\ \gamma_N^{(N)}= -a_N. \end{aligned}
\left\Vert \frac{\partial}{\partial y_{h}}Z_{\varepsilon,\xi(y)}^{l}\right\Vert_{\varepsilon}=O\left( \frac{1}{\varepsilon}\right),\qquad\left\Vert \frac{\partial}{\partial y_{h}}W_{\varepsilon,\xi(y)}\right\Vert_{\varepsilon}=O\left( \frac{1}{\varepsilon}\right) .
\tilde{a}_{X}| 0 \rangle_{X Y \bar{c}c}= \tilde{a}_{Y}| 0 \rangle_{X Y \bar{c}c}= \hat{c}| 0 \rangle_{X Y \bar{c}c}= \hat{\bar{c}}| 0 \rangle_{X Y \bar{c}c}= 0
I_{0}\left[ A_{\alpha}^{\;\;(\lambda )}\right] =\int d^{3}x\varepsilon^{\alpha \beta \gamma}A_{\alpha (\lambda )}\partial_{\beta}A_{\gamma}^{\;\;(\lambda )},\;\delta_{\epsilon}A_{\alpha}^{\;\;(\lambda)}=\partial_{\alpha}\epsilon^{(\lambda )},
\int_{B_r(x_0)}\Big(\left|\nabla u(x)\right|^p-\left|\nabla v(x)\right|^p\Big)\,dx =\; p\int_0^1\frac{1}{s}\bigg(\int_{B_r(x_0)}\left(\left|\nabla u_s(x)\right|^{p-2}\nabla u_s(x)-\left|\nabla v(x)\right|^{p-2}\nabla v(x)\right)\cdot\nabla(u_s-v)(x)\,dx\bigg)\,ds.
\begin{aligned}\frac{d}{dt}\Big|_{t=1}\|t\phi+R_n\|_{L^2}^2 =\frac{d}{dt}\Big|_{t=1}\Big\{t^2 \|\phi\|_{L^2}^2+2t\Re \int_{\mathbb{R}}\phi\overline{R_n}dx+\|R_n\|_{L^2}^2\Big\}&=2 \|\phi\|_{L^2}^2+2\Re \int_{\mathbb{R}}\phi\overline{R_n}dx &=2\|\phi\|_{L^2}^2+o_n(1)\quad\textup{(by Proposition \ref{bubble decomposition}$(ii)$)}\end{aligned}
\Delta_0^{(\widetilde{1})}=\frac1{2m^2}ip^{(\widetilde{1})}\left( ip^{(\widetilde{1})}-\varepsilon m\right) \left( 1-\sigma_p^{(\widetilde{1})2}\right) =\Psi_0^{(\widetilde{1})}\cdot\overline{\Psi}_0^{(\widetilde{1})}
E_{1}(\mbox{\mathversion{bold}$\varphi$}(t))=E_{1}(\mbox{\mathversion{bold}$\varphi$}_{0})-R_{1}(\mbox{\mathversion{bold}$v$}^{R},\mbox{\mathversion{bold}$\varphi$})+R_{1}(\mbox{\mathversion{bold}$v$}^{R},\mbox{\mathversion{bold}$\varphi$}_{0}).
S=\sum\limits_{\mu\nu}\frac{\beta_{\mu\nu}}{2N}\sum\limits_{\{p_{\sigma}\mu\nu\}}ReTr(U_{p_{\sigma}\mu\nu}+U^{+}_{p_{\sigma}\mu\nu})=\sum\limits_{\mu\nu}\frac{\beta_{\mu\nu}}{2N}\sum\limits_{\{p_{\sigma}\mu\nu\}}Tr(U_{p_{\sigma}\mu\nu}+U^{+}_{p_{\sigma}\mu\nu}),
\sqrt[6]{32(\tfrac{146410001}{48400})^3-6(\tfrac{146410001}{48400})+\sqrt{(32(\tfrac{146410001}{48400})^3-6(\tfrac{146410001}{48400}))^2-1}}= &\sqrt[6]{\tfrac{\psi^{1,-1}_{3}\left((110)^2+\tfrac{1}{(110)^2}\right)}{2}+\sqrt{\tfrac{\left(\psi^{1,-1}_{3}\left((110)^2+\tfrac{1}{(110)^2}\right)\right)^2}{4}-1}}=\sqrt{(110)^2}=110.
s_{n}=-\frac{1}{H_{n-1}(c_{k+1})}\left(\frac{H_{n-1}(c_{k})H_{n}(c_{k+1})}{H_{n}(c_{k})}+\frac{H_{n}(c_{k})H_{n-2}(c_{k+1})}{H_{n-1}(c_{k})}\right).
\partial_{z}W ( z ) \vert_{z = z_{0}}= - e^{W ( z_{0}) - \sqrt{2}b \kappa \Xi ( z_{0})}\frac{\sqrt{2}}{6}\lambda_{0}, \quad \partial_{z}\Xi ( z ) \vert_{z = z_{0}}= - e^{W ( z_{0}) - \sqrt{2}b \kappa \Xi ( z_{0})}b \kappa^{- 1}\lambda_{0}.
\big( [\gamma'] \in P_{n,m}^{(d)}, (f_{1},\ldots,f_{d}) \in \mathcal{F}(\mathbb{N},\mathbb{Q})^{d}\big)\overset{S}{\longmapsto}\sum_{\substack{\gamma=(n_{i}) \in [\gamma']}}\prod_{i}f_{(i,[\gamma'])}(n_{i})
\sup_{x}\left\vert \hat{\gamma}_{2\ell}(x)-\gamma_{20}(x)\right\vert =O_{p}(n^{-d_{1}(2\xi_{1}-1)/(2\xi_{1}+1)}\ln(n)),\left\vert\hat{\gamma}_{3\ell}-\gamma_{30}\right\vert =O_{p}(n^{-d_{1}}), \left\Vert \hat{\gamma}_{2\ell}-\gamma_{20}\right\Vert =O_{p}(n^{-d_{1}2\xi_{1}/(2\xi_{1}+1)}\ln(n)),(\ell=1,...,L).
\omega_x =\frac{i}{2}\sum_{\ell=1}^n dz_\ell\wedge d\bar{z}_\ell\,, c_1(L,\widetilde{h})_x =dd^c \varphi_x(0)=dd^c \psi_x(0)=dd^c \psi^\prime_x(0)=\frac{i}{\pi}\sum_{\ell=1}^n\lambda_\ell\, dz_\ell\wedge d\bar{z}_\ell\,.
\langle T_{a_{1}b_{1}}({\vec{x}}_{1}) T_{a_{2}b_{2}}({\vec{x}}_{2}) T_{a_{3}b_{3}}({\vec{x}}_{3}) T_{a_{4}b_{4}}({\vec{x}}_{4}) \rangle_{O}\propto \langle T_{a_{1}b_{1}}({\vec{x}}_{1}) T_{a_{2}b_{2}}({\vec{x}}_{2}) \rangle \, \langle T_{a_{3}b_{3}}({\vec{x}}_{3}) T_{a_{4}b_{4}}({\vec{x}}_{4}) \rangle \, .
u_2(z) =\sum_{n=0}^a\left( \int_{D^2}\frac{\mu(\zeta)f(\zeta)}{\zeta^{n+1}}d\zeta\wedge d\bar{\zeta}\right)z^n-\sum_{n=0}^{a}\left(\int_{D(0,2|z|)}\frac{\mu(\zeta)f(\zeta)}{\zeta^{n+1}}d\zeta\wedge d\bar{\zeta}\right)z^n &+\sum_{n=a+1}^{\infty}\left(\int_{D\setminus D(0,2|z|)}\frac{\mu(\zeta)f(\zeta)}{\zeta^{n+1}}d\zeta\wedge d\bar{\zeta}\right)z^n.
\hat{q}_{\beta}(n,t)=\sum_{k=1}^{n}\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\prod_{j=1}^{k}\frac{(1-p)^{m_{j}}}{m_{j}!}\left(\frac{-\lambda t^{\beta}}{\ln p}\right)^{k}E_{\beta,k\beta+1}^{k+1}(-\lambda t^{\beta}).
(ge_\ell,e_{\ell'}) =(e_\ell+\mu\varphi (e_\ell),e_{\ell'})=\mu(\varphi (e_\ell),e_{\ell'}) & =\mu(e_\ell,\varphi^*(e_{\ell'}))=(e_\ell,e_{\ell'}+\mu\varphi^*(e_{\ell'}))=(e_\ell,g^{-1}e_{\ell'}).
k'_{\pm}=\frac{1}{\sqrt{4\beta \hslash^{2}/3}}\left( \sqrt{\frac{\sqrt{(1-|a|)^{2}+|b|^{2}}+(1-|a|)}{2}}\mp i \sqrt{\frac{\sqrt{(1-|a|)^{2}+|b|^{2}}-(1-|a|)}{2}}\right).
S_{d u a l i t y}~ = ~ \left( \begin{matrix}{{0}}&{{0}}&{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{- 1}}\\{{- 1}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- 1}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}&{{0}}\\ \end{matrix}\right)
\mathbf{\hat{d}}_{m_\textbf{i}}[n_{\textbf{i}}] = \mathbf{{d}}_{m_\textbf{i}}[n_{\textbf{i}}] + \sum\limits_{m_\textbf{s}\in \mathcal{M}_\textbf{s}}\mathbf{\eta}_{m_\textbf{s}\rightarrow m_\textbf{i}}[n_\textbf{i}] + \mathbf{w}_{\textbf{i}}[n_\textbf{i}],
\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{\frac{3(i+2j)(i+2j-1)}{2}+2i+\frac{i(i-1)}{2}+3j}}{(q;q)_i\left(q^3;q^3\right)_j}=\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{2i^2+6ij+6j^2}}{(q;q)_i\left(q^3;q^3\right)_j}.
h_{2,m}\partial_{t}v^{(2,m)}_0 =h_{2,m}\partial^2_{x_2 x_2}v^{(2,m)}_0 - h_{2,m}k_2\big(v^{(2,m)}_0\big) - 2 \delta_{\alpha_2,1}\kappa_2\big(v^{(2,m)}_0\big) + 2 \delta_{\beta_2,1}g^{(2)}_0
y^{(2)}_{1,2}(\theta) =\frac{t^{(2)}_{1,2}(\theta+i\frac{\pi}{3}) t^{(2)}_{1,2}(\theta-i\frac{\pi}{3})}{t^{(1)}_{1,2}(\theta)}y^{(1)}_{1,2}(\theta) =\frac{t^{(1)}_{1,2}(\theta+i\frac{\pi}{3}) t^{(1)}_{1,2}(\theta-i\frac{\pi}{3})}{t^{(2)}_{1,2}(\theta)}
\int_{0}^{1}\int_{0}^{1}\frac{dxdy}{1+x^{2}y^{2}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=\frac{1}{1^{2}}-\frac{1}{3^{2}}+\cdot\cdot\cdot
m \simeq \frac{\alpha}{\pi^{2}}\int d q_{3}d q_{4}\int_{0}^{\infty}d r^{2}\mathrm{e}^{- r^{2}}\frac{1}{2 r^{2}+ L^{2}( q_{3}^{2}+ q_{4}^{2})}~ \frac{m}{m^{2}+ ( q_{3}^{2}+ q_{4}^{2})}
g(\tau,r) = \frac{r}{t_+}\left[(u_r^2 +u_t^2)(t_+^2 + r^2) + 4 u_r u_t t_+ r + 2\rho u u_r r + 2\rho u u_t t_+ + \frac{1}{3}e^{2\tau}|u|^6\right].
\frac{1}{i}Zu(t,y) = \left(h^{\frac{1}{2}}\left[ h^{-1}Op_h^w(xp(\xi) + \xi) - \frac{1}{2i}Op_h^w(p'(\xi)) \right] v + h^{\frac{1}{2}}x \widetilde{P}\right)|_{x=\frac{y}{t}}& = \left(h^{\frac{1}{2}}\left[Op_h^w(p(\xi))\mathcal{L}v - \frac{1}{i}Op_h^w(p'(\xi))v + x\widetilde{P}\right] \right)|_{x=\frac{y}{t}}\,,
\dfrac{h(y)}{h(y+1)}-1 =\dfrac{y+1}{y}\exp\left\{\dfrac{\log^2(y+1)-\log^2(y)}{2\sigma^2}\right\}-1 &\le \exp\left\{\dfrac{1}{y}+\dfrac{\log(y+1)}{\sigma^2y}\right\}-1 &\le c\left(\dfrac{1}{y}+\dfrac{\log(y+1)}{\sigma^2y}\right), \c>0\ , &\le \dfrac{c_2}{(y+1)^{\beta}}.
\widehat{A}_{j l}^{( s )}( \omega ) = \frac{2 \operatorname{s i n h}( ( s - j ) \omega ) \operatorname{c o s h}( \omega ) \operatorname{s i n h}( l \omega )}{\operatorname{s i n h}( \omega ) \operatorname{s i n h}( s \omega )}
S=\int d^{6}x\left[{\frac 1 6}F_{{\mu}{\nu}{\sigma}}(A)F^{{\mu}{\nu}{\sigma}}(A)+{\frac 1 2}{\lambda}_{{\mu}{\nu}}{\cal F}^{{\mu}{\rho}{\sigma}}(A){{\cal F}^{\nu}}_{{\rho}{\sigma}}(A)\right],
P(f)=\biggl\{a(t,x)+b(t,x)\cdot\Big(1+\frac{\mathcal{I}}{mc^2}\Big) p+c(t,x)\biggl(cp^0\Big(1+\frac{\mathcal{I}}{mc^2}\Big)-\widetilde{e}(T_0)\biggl)\biggl\}\sqrt{F_E^0}
F(h) =\lim_n F(h_n)=\lim_n F\left(\sum_{i=1}^nh\chi_{[i-1,i)}\right)=\lim_n\sum_{i=1}^n F(h\chi_{[i-1,i)}) &=\lim_n\sum_{i=1}^n F(\chi_{[i-1,i)})h=\left(\sum_{i=1}^\infty F(\chi_{[i-1,i)})\right)h.
f'(t) =-\int_{-\infty}^t \frac{1}{4}e^{2x}\bigg(\int_x^{\infty}F(y)e^{-2y}dy\bigg)dx+\int_t^{\infty}\frac{1}{4}e^{-2x}\bigg(\int_{-\infty}^x F(y)e^{2y}dy\bigg)dx &=-\frac{1}{8}e^{2t}\int_t^{\infty}F(x)e^{-2x}dx -\frac{1}{8}\int_{-\infty}^t F(x)dx &+\frac{1}{8}e^{-2t}\int_{-\infty}^t F(x)e^{2x}dx+\frac{1}{8}\int_t^{\infty}F(x)dx.
D_\mathcal{G}(Tu,Tw) =D_{\mathcal{G}}((u,F(u)),(w,F(w))) = \lvert u-w \rvert+ H_d(Fu,Fw) \leq \lvert u-w \rvert+l\lvert u-w \rvert \leq (1+l) \lvert u-w \rvert, D_{\mathcal{G}}(Tu,Tw) \leq (1+l)\lvert u-w \rvert
\int_\Omega\widetilde{\theta}^2dx+ \iint_{Q_t}\widetilde{\theta}_x^2dxds \leq C\sqrt{\mu}+\epsilon\iint_{Q_t}\big(\widetilde{u}_x^2+|\widetilde{\mathbf{b}}_x|^2\big)dxds +\frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,~~\forall\epsilon\in(0, 1).
\mathcal{H}^{(1)}\psi^{(1)}(x) =\mathcal{E}\psi^{(1)}(x), \mathcal{H}^{(1)}\varphi^{-1}_1(x)=\tilde{\mathcal{E}}_1\varphi^{-1}_1(x), \mathcal{H}^{(1)}\varphi^{(1)}_{1,l}(x) =\tilde{\mathcal{E}}_l\varphi^{(1)}_{1,l}(x)\ \ (l\neq 1).
E^+(z) = \sum_{n \geq 0}E_n z^n,E^-(z) = - \sum_{n \leq -1}E_n z^n,F^+(z) = - \sum_{n \geq 1}F_n z^n,F^-(z) = \sum_{n\leq 0}F_n z^n, & K_1^{\pm}(z) = (s_{11}^{(0)})^{\pm 1}\exp (\pm (q-q^{-1}) \sum_{s > 0}h_{\pm s}z^{\pm s}), K_2^{\pm}(z) = K_1^{\pm}(z) \phi^{\pm}(z).
M(s,x_0,y_0,z_0) :={1 \over (1 + s)}\chi_{|\tau| \le 5}\Vert \partial \phi \Vert_{H^{10}(\Sigma_0)}+ \sup_{|\alpha| \le 10}\sup_{\psi_1}\int_0^s \int_{\Sigma_t}|\partial^\alpha (\Box \phi)| |\partial \psi| d x d t.
\mu(h(q,a))f(x) =\mu\left(\bmatrix I 0 \sigma_q I\endbmatrix\right)\mu\left(\bmatrix a 0 0 \;^t\!a^{-1}\endbmatrix\right) f(x) =\pm e^{\pi i\langle\sigma_q x,x\rangle}\mu\left(\bmatrix a 0 0 \;^t\!a^{-1}\endbmatrix\right) f(x) =\pm e^{\pi i\langle\sigma_q x,x\rangle}(\det a)^{-1/2}f(a^{-1}x).
\begin{cases}\|\dot{U}^{a,\pm}+V^{\pm}_k+\tau\delta V^{\pm}_k\|_{s,X}\le C\delta(1+\theta_k^{(s-\alpha)_+}+\theta_k^{s-\alpha-2}), \forall s_0\le s\le s_1, s\neq \alpha begin{align*}2mm]\|\dot{U}^{a,\pm}+V^{\pm}_k+\tau\delta V^{\pm}_k\|_{\alpha,X}\le C\delta(1+\log \theta_k+\theta_k^{-2})\end{cases}
\sum_{\nu}\,^{\prime}\, \sum_{\mu}\sum_{\kappa}\int d^{2}x \int d^{2}y \int d^{2}z ( \Psi_{\nu}^{*}( x ) \delta \lambda_{\mu}\Psi_{\mu}( x ) ) ( \Psi_{\mu}^{*}( y ) e^{- \epsilon \lambda_{\kappa}}\Psi_{\kappa}( y ) ) ( \Psi_{\kappa}^{*}( z ){\frac{1}{\lambda_{\nu}}}\Psi_{\nu}( z ) ) \, ,
G(m,2, a, \epsilon,s) \leq 2 \sum_{k=1}^{({m(1+\epsilon) \over 2})^{{1 \over a}}}k^{-ds}({m\over 3})^{-{ds\over a}}\cdot 2\varepsilon m^{{1\over a}}\cdot C_4(a) \leq 4 \cdot \zeta(ds) \cdot 3^{\frac{ds}a}\cdot C_4(a) \cdot \varepsilon m^{\frac{1-ds}a}.
X=(\begin{matrix}X_{1(p\times p)}&-\overline{X}_{2}\\X_{2}&\overline{X}_{1}\end{matrix}),Y=(\begin{matrix}Y_{1(q\times q)}&-\overline{Y}_{2}\\Y_{2}&\overline{Y}_{1}\end{matrix}),Z=(\begin{matrix}Z_{1(p\times q)}&-\overline{Z}_{2}\\Z_{2}&\overline{Z}_{1}\end{matrix}),
\frac{d}{ds}\left(-\zeta_v(s)^{-1}Z_{r_2}(f,s)+\epsilon_v(1,\psi_v)\zeta_v(s+1)^{-1}Z_{r_1}(d_2(f), s+1)\right)\Big|_{s=0}=(\log q_v)\bigg((e(\psi_v)+1)\tilde{c}_{2,v}(I(f))-\tilde{a}_{2,v}(I(f))\bigg).
R_2(g_1g_2,g_3)-R_2(g_1,g_2g_3)=R_1(g_1)\circ R_2(g_2,g_3)-R_2(g_1,g_2)\circ R_1(g_3) 0=R_2(g_1,g_2)\circ R_2(g_3,g_4)
\mathbb{E}_q(x_1) \star \xi_0 = \mathbb{E}_q(x_{-1}) \cdot \left( \mathbb{E}_{q^2}(q^{\frac{1}{2}}x_{(1,1)}) \star \xi_0 + \mathbb{E}_{q^2}(q^{-\frac{1}{2}}x_{(1,1)}) \star \xi_{(1,1)}\right).
\tau_{k}\left( n\right) =\max_{v\left( G\right) =n}\lambda_{1}\left(G\right) +\cdots+\lambda_{k}\left( G\right) , \xi_{k}\left( n\right) =\max_{v\left( G\right) =n}\lambda_{1}^{\ast}\left( G\right) +\cdots+\lambda_{k}^{\ast}\left( G\right) .
{{v}_{x}}+{{u}_{y}}+v\left({{\alpha}_{x}}-{{\beta}_{y}}\right)+u\left({{\beta}_{x}}+{{\alpha}_{y}}\right)=0{{u}_{x}}-{{v}_{y}}+u\left({{\alpha}_{x}}-{{\beta}_{y}}\right)-v\left({{\beta}_{x}}+{{\alpha}_{y}}\right)=0
u|_{x=\xi_j(t)}=u^j, 0<t\leq T, j=\overline{1,J}, begin{align*}3mm] [u]|_{x=\xi_j(t)}=0, 0<t\leq T, j=\overline{1,J}, begin{align*}3mm] [k(u)u_x]|_{x=\xi_j(t)}= \gamma_j\frac{d\xi_j}{dt}, 0<t\leq T, j=\overline{1,J}, begin{align*}3mm] u(x,0) = \phi(x), 0\leq x \leq \ell, begin{align*}3mm] k(u)u_x|_{x=0}=g(t),k(u)u_x|_{x=\ell}= p(t), 0<t\leq T,
\frac{\Gamma}{V_{3}}\simeq 2 \left( \frac{\Delta S_{E}}{2 \pi}\right)^{3 / 2}\exp \left[ \frac{4}{3}\pi R^{3}\Delta U_{e f f}- 4 \pi R^{2}\left({\cal S}_{0}+{\cal S}_{1}\right) \right] \; ,
h_i(u) [\lambda] &= \prod_{(x,y) \in A_{\lambda,i}}\dfrac{u - (\varepsilon_1(x - 1) + \varepsilon_2(y - 1))}{u - (\varepsilon_1 x + \varepsilon_2 y)}\prod_{(x,y) \in R_{\lambda,i}}\dfrac{u - (\varepsilon_1 (x + 1) + \varepsilon_2 (y + 1))}{u - (\varepsilon_1 x + \varepsilon_2 y)}[\lambda].
\displaystyle\frac{d\lambda}{d\tau_\beta}= \frac{\lambda\left(\alpha e^{-\lambda\tau_\alpha}-s-1\right)}{\left(1+\alpha \tau_\alpha e^{-\lambda\tau_\alpha}\right)\left(\frac{\lambda\tau_\beta}{m}+1\right)+\tau_\beta\left(\lambda+1-\alpha e^{-\lambda\tau_\alpha}\right)}\;.
G(x,y)=\frac{\langle{\bf A}_y^{-1}(x-y),(x-y)\rangle^{\frac{2-n}{2}}}{(n-2)|S^{n-1}|\sqrt{\det{\bf A}_y}}\,e^{I_y(\sqrt{\langle{\bf A}_y^{-1}(x-y),(x-y)\rangle})-I_y(0)}(1+\xi_y(x))
-\frac{\partial L_{d}DAHI(p\|q)}{\partial q_{j}}=\sum_{j}p_{j}\left[\underbrace{\frac{a-1}{a-b}\left(\overline{MH}\right)^{a-2}-\frac{b-1}{a-b}\left(\overline{MH}\right)^{b-2}}_{Z>0}\right]\frac{\partial \overline{MH}}{\partial q_{j}}
\lim_{n\rightarrow\infty}\frac{1}{N(n)}\sum_{[\xi_j]:\langle \xi_j\rangle\leq n}A_j=(2\pi)^{-n}= \lim_{n\rightarrow\infty}\frac{1}{N(n)}S_{N(n)}=\smallint\limits_{T^*\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi')d\mu_L(x,\xi').
\frac{E_0( g_{\vec{\tau}}, s ,\varphi)}{\sqrt{N(\vec{v})}}= \varphi(0) \cdot N(\vec{v})^{s/2}- N(\vec{v})^{-s/2}\frac{\Lambda(s , \chi )}{\Lambda(s+1,\chi)}\cdot M (s,\varphi) .
\displaystyle{\mathcal{G}_{K,L}\left( A, B; \omega^m \right) \equiv\frac{1}{\sqrt{(K+L)!K!L!}}\partial_x^K \partial_y^L \prod_{s=0}^{K+L-1}\left( \omega^{-s/2}x \mathbf{A}+ \omega^{s/2}y \mathbf{B}\right) \vert_{x=y=0}.}
S(\omega_{{\sf f}_1 \cdots{\sf f}_n}\| \omega_{{\sf g}_1 \cdots{\sf g}_m}) = S(\omega \circ \alpha_U \| \omega \circ \alpha_V) = S(\omega \circ \alpha_{V^*U}\| \omega) = S(\omega_{{\sf g}_m \cdots{\sf g}_1{\sf f}_1 \cdots{\sf f}_n}\| \omega)\,.
( 4 . 1 3 ) = \frac{2 C_{1}C_{2}}{\pi}\int_{0}^{\infty}d w_{1}\; w_{1}^{- \frac{1}{2}}\left( w_{1}+{\sigma}_{1}^{2}\right)^{\frac{\lambda + m}{2}}\int_{0}^{\infty}d w_{2}\; w_{2}^{- \frac{1}{2}}\left( w_{2}+{\sigma}_{2}^{2}\right)^{\frac{\lambda + m}{2}}=
\mathcal{T}_{0}\left( \dot{Q}\right) =\mathcal{V}_{0}\left( Q\right)-\operatorname{Re}\left( \dot{Q},\theta Q\right) -\left( \frac{\operatorname{Im}\zeta}{\operatorname{Re}\zeta}\right)^{2}\operatorname{Re}\left( \dot{Q},\theta Q\right) .
A = \frac{1}{8 \pi^{D / 2}}\Gamma ( \frac{D}{2}- 1 ) \frac{( 1 - ( - 1 )^{2 - D})}{3 - D}( 1 - \frac{\lambda}{2}( 3 - D ) ) = A_{0}+ \epsilon A_{1}+ \ldots ,
D_{\sigma,n-1}\Big(\frac{1}{1-z\overline{w}}\Big) =(n-1)|w|^{2n-2}\int_{\mathbb D}\frac{(1-|z|^2)^{n-2}}{|1-z\overline{w}|^{2n}}dA(z) &=\frac{|w|^{2n-2}}{(1-|w|^2)^n}.
\begin{cases}\displaystyle\frac{dy(t)}{dt}=Ay(t) + A_1 y(t-r)+ f \dot B(t),\,\,\,\,t\ge 0, y(0)=\phi_0,\,\,\,\,y_0=\phi_1,\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X},\end{cases}
\mathcal{I}((f_i), (f_j))(x) = e^{-\mathcal{Q_+}(B_0 x)}e^{\mathcal{Q_-}(B_{m+1}x)}\prod_{i=1}^{m^+}f_i^{c_i}(B_i x) \prod_{j=1+m^+}^m f_j^{c_j}(B_j x) \quad\textup{for $x \in H$.}
\Vert u_{\rho}-{\check{u}}_{n}\Vert_{{H^{1}}({{\mathbb{R}}})}+\Vert{\mathcal{L}}u_{\rho}-{\check{v}}_{n}\Vert_{{H^{1}}({{\mathbb{R}}})}<\rho \; \; J(u_{\rho})<c_{n}=J({\check{u}}_{n})\;.
\begin{cases}k_{0,j}= a_{0}j(j-1) + b_{0}j - \lambda_{n}, k_{1,j}= a_{0}j(j-1)(f_{j-1}+ f_{j-2}) + b_{0}j f_{j-1}+ a_{1}j(j-1) + b_{0}j g_{j-1}+ b_{1}j, k_{2,j}= a_{0}j(j-1) f_{j-2}^2 + a_{1}j(j-1) f_{j-2}+ b_{0}j g_{j-1}f_{j-2}+ a_{2}j (j - 1) + b_{1}j g_{j-1}.\end{cases}
f-\underbrace{(f^*-\varepsilon)}_{t}\,=\,\sum_{\ell=1}^pf^\ell\quad\mbox{with}f^\ell\,=\,\sum_{(\alpha,\beta)\in N^{\ell}}c_{\alpha\beta}^\ell\,h_{\alpha\beta},\quad\ell=1,\ldots,p,
( 2 6 ) - \omega_{\alpha \bar{\beta}}( \kappa_{a}^{\alpha}\overline{{{\kappa_{b}^{\beta}}}}- \kappa_{b}^{\alpha}\overline{{{\kappa_{a}^{\beta}}}}) = \nabla_{a}T_{b}- \nabla_{b}T_{a}= f_{a b}^{c}T_{c}.
\lim_{\lambda\to\infty}2^{n\lambda}\int_{\mathbb{R}^{n}}R^{\alpha,\beta}(2^{\lambda}\mathbf{x}+\mathbf{z}, 2^{\lambda}\mathbf{y}+\mathbf{z})[f^{(\alpha)}(\mathbf{y})-f^{(\alpha)}(\mathbf{x})]d\mathbf{y}=0.
\gamma \cdot \left( \begin{array}{c c}{\Psi^{3}}\\{\Psi^{4}}\\ \end{array}\right) = R_{D_{2}}^{r e g}( \gamma )^{- 1}\cdot \left( \begin{array}{c c}{\Psi^{3}}\\{\Psi^{4}}\\ \end{array}\right) \cdot R_{D_{2}}^{r e g}( \gamma ) .
\|{\mathcal E}(t,\tau) \| \le{C}_{18}+ C_{19}|\tau| t^2,\ 0< |t|\le t^{00},\N_0=0;C_{18}={C}_{14}+ c_*^{-1/2}C_{12}, \C_{19}= c_*^{-1/2}C_{13}.
\overline{u}(x) = g(x,y,v_\rho(y)) \leq g(x,y_0,z_0)+|g_y||y-y_0|+|g_z||v_\rho(y)-z_0| \leq g_0(x) + 2\rho\Vert g \Vert_{C^1(\Gamma)}.
\langle b\delta_t,\lambda(x)b\delta_t\rangle=\big\langle b\delta_t,\sum_{s\in G}\lambda_s(b_s)b\delta_t\big\rangle=\sum_{s\in G}\langle b\delta_t,b_sb\delta_{st}\rangle=b^*b_eb=b^*\mathbb{E}(x)b.
A_{123}=\arg k_{12}k_{23}k_{31}& =\arg\left( \frac{1}{\left( 1-r^{2}\omega^{1-2}\right) \left(1-r^{2}\omega^{2-3}\right) \left( 1-r^{2}\omega^{3-1}\right)}\right)^{\lambda}& =3\lambda\arg\left( 1-r^{2}\omega\right)
{\tilde{R}}( z ) = R ( z ) + \frac{i}{q ( z )^{2}}\left( \frac{\beta}{2}\left( p ( z )^{2}- q ( z )^{2}\right) + \alpha z \left( p^{\prime}( z ) q ( z ) - p ( z ) q^{\prime}( z ) \right) \right) + O ( \alpha , \beta ) ,
T =\frac{\Big(\int_{\mathbb{R}^3}fdv\Big)\Big(\int_{\mathbb{R}^3}f|v|^2dv\Big)-\Big|\int_{\mathbb{R}^3}fvdv\Big|^2}{3\Big(\int_{\mathbb{R}^3}fdv\Big)^2}\geq \frac{\gamma_{\ell}}{3a_{u}^2}.
r^{j(3)}_\alpha = -\eta^{j(4)}(\hat{X}_\alpha) &= \frac{1}{8}R_{\alpha\beta i}^{j}(q)z^i x^\beta + \frac{1}{24}J_{\beta\gamma}^j(q) R_{\alpha\beta^\prime\gamma^\prime}^\gamma(q) x^\beta x^{\beta^\prime}x^{\gamma^\prime}+ \frac{1}{4}x^\beta J_{\alpha\beta}^{j(2)}.
a(n,n) =\frac{(-1)^{n}}{n!}, a(n-1,n) =\frac{(-1)^{n+1}}{2!(n-2)!}3,\;n\geq 2, a(n-2,n) =\frac{(-1)^{n}}{4!(n-3)!}(27n-73),\;n \geq 3, a(1,n) =\frac{1-2^{\nu_{2}(n)+1}}{n},\;n\geq 1.
\varphi_1(x, x,s)=0,\ \d_x\varphi_1(x, x,s)=d_x\varphi(x, x,s),\ \d_y\varphi_1(x, x,s)=d_y\varphi(x, x,s),\ \forall (x,x,s)\in\Omega, &{\rm Im\,}\varphi_1(x,y,s)\geq0,\ \ \forall (x,y,s)\in\Omega,
C_p(\nu_1) - C_p(\nu_2) \geq \frac{C_p(\nu_2) (C_p(\nu_2) - 1)^2}{3(C_p(\nu_2) - 1)^2 + 4 C_p(\nu_2)}&\geq\begin{cases}\frac{1}{6}(C_p(\nu_2) - 1)^2 1 \leq C_p(\nu_2)< 2 \frac{1}{3}(C_p(\nu_2) - 1) - 1/6 C_p(\nu_2)\geq 1,\end{cases}
\frac{\partial}{\partial t}\frac{\delta\bar\ell}{\delta\bar\sigma^1}+\operatorname{div}\frac{\delta\bar\ell}{\delta\bar\sigma^2}=-\operatorname{ad}^*_{\bar\sigma^1}\frac{\delta\bar\ell}{\delta\bar\sigma^1}-\operatorname{Tr}\left(\operatorname{ad}^*_{\bar\sigma^2}\frac{\delta\bar\ell}{\delta\bar\sigma^2}\right),
DF^{\omega}(\varphi,T\varphi)=\underbrace{D[\gamma^{\partial B_0}(\nu^{\partial B_1})Q^{\partial B_0}\zeta^{\partial B_1}\gamma^{X}(\nu^{\partial B_0})Q^{X}\zeta^{\partial B_0}]}_{=:a(\varphi,T\varphi)}\omega+\underbrace{D[-\mu_{\partial B_1}]}_{=0\ref{D0}}
\begin{aligned}\mathcal{C}:\mathcal{M}_{p,n}\longrightarrow\mathcal{M}_{p,p-n}\ , W^{(n)}_\lambda \longmapsto \mathcal{C}[W^{(n)}_\lambda]=W^{(p-n)}_{\mathcal{C}(\lambda)}\ ,\end{aligned}
\left\|C^{-1}\right\|_{X,X}\le\frac{|c_0|^{-1}}{1-\left\|I-c_0^{-1}C\right\|_{X,X}}\le\frac{|c_0|^{-1}}{1-\frac{\sum_{m\not=0}|c_m|}{|c_0|}}.
|f(x) - s(x) | \le \frac{h_i}{4c_i}\max \Big \{|y_i^{(1)}-d_i|, |y_{i+1}^{(1)}- d_{i+1}|\Big \}+ \frac{1}{384 c_i}\Big\{h_i^4 \| f^{(4)}\| (1 + \frac{|r_i -3|}{4}) + 4 |r_i - 3| \big ( h_i^3 \| f^{(3)}\| + 3 h_i^2 \| f^{(2)}\|\big ) \Big \},
\xi^{\langle k' \rangle}\cdot \xi^{\langle k'' \rangle}=\sum_{max\{k',k''\}\leq l\leq k'+k''}\frac{q_{k'}!\,q_{k''}!\,l!\,}{q_l!\,(k'+k''-l)!\,(l-k')!\,(l-k'')!\,}\xi^{\langle l\rangle}.
( \mathrm{C o n t r i b u t i o n f r o m ( \ref{4 3}) t o ( \ref{3 4})}) = \prod_{{C_{\alpha}\cap C_{\beta}\neq \emptyset}\atop{\alpha \neq \beta \alpha , \beta \in E d g e ( \Gamma )}}\frac{1}{\frac{\lambda_{i}- \lambda_{j}}{d_{\alpha}}+ \frac{\lambda_{i}- \lambda_{l}}{d_{\beta}}}
\frac{1}{2}\,\sum_{1\le i\le k\le N}\partial_{x^{(k)}_i}^2 -\sum_{1\le i\le k\le N}\sum_{j\neq i}\frac{\beta/2-1}{x^{(k)}_i-x^{(k)}_j}\,\partial_{x^{(k)}_i}+\sum_{1\le i\le k+1\le N}\sum_{j=1}^k \frac{\beta/2-1}{x^{(k+1)}_i-x^{(k)}_j}\,\partial_{x^{(k+1)}_i},
\|z(t)\|^2 \leq e^{-L(t-t_0)+\frac{2}{N}\sum\limits_{j=1}^{N}\int_{t_0}^{t}O_{\tau}^{(j)}d\tau}\|z(t_0)\|^2 +\frac{1}{N}\sum\limits_{j=1}^{N}\frac{\|f^{(j)}(0)\|^2}{L}\int_{t_0}^{t}e^{-2O_{u}^{(j)}}e^{-L(t-u)+\frac{2}{N}\sum\limits_{k=1}^{N}\int_{u}^{t}O_{\tau}^{(k)}d\tau}du.
\begin{array}{ll}l_{k}= \Bigl[ |U_*|^{2^{*}-2}U_*-u_0^{2^*-1}+\sum_{j=1}^{k}P_j^{2^{*}-1}\Bigr] + \sum_{j=1}^{k}\bigl( U_j^{2^*-1}- P_j^{2^{*}-1}\bigr) &=: J_1+ J_2.\end{array}
\int_{\Omega \times \Omega^c}J \bigg( \frac{x-y}{t}\bigg) \dxdy = \sum_{i=1}^{12}\int_{U_i \times V_i}J \bigg( \frac{x-y}{t}\bigg) \dxdy + \sum_{i \neq j}\int_{U_i \times V_j}J \bigg( \frac{x-y}{t}\bigg) \dxdy.