harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal G(k,d)= \left\{F\in\binom{[n]}{k}:[d-1]\subseteq F,\, F\cap[d,k-1]\neq\emptyset\right\}\cup\{[2,k]\cup\{i\}:i\in[k+2,n]\}&\cup\left\{F\in\binom{[n]}{k}:[d-1]\cup[k,k+1]\subseteq F\right\}\cup\{[2,k-1]\cup\{k+1,i\}:i\in[k+2,n]\}&\cup\left\{F\in\binom{[n]}{k}:|[d-1]\cap F|=d-2,\, [d,k+1]\subseteq F\right\}.
\widetilde{G}_{m,n}\ge \frac{a_{n-1}^2}{b_{n-1}^2}\frac{a_{n-2}^2}{b_{n-2}^2}\cdots \frac{a_{n-m}^2}{b_{n-m}^2}\sim n^{2m}, n=0\,(\operatorname{mod}q), m<q,
d s_{4 + N}^{2}= \frac{1}{( k \sum_{i = 1}^{N}\, | z_{i}| + 1 )^{2}}\, \Big ( \eta_{\mu \nu}d x^{\mu}d x^{\nu}+ \sum_{j = 1}^{N}\, ( d z^{j})^{2}\Big ) \, .
|P_1^{2^*-2}\bigl( u_0 +\sum_{j=2}^k P_j)|\le C U_1^{2^*-2}\le\lambda^{-\frac{N+2}4+\frac\tau2}\frac{\lambda^{\frac{N+2}2}}{(1+ \lambda|y-\xi_1|)^{\frac{N+2}2+\tau}},y\in S.
R_{t,s}(x) = \frac12 DV_{ts}(x) V_{ts}(x) + \frac s 2 e^{i f(x) s}f(x) \int_s^t \Big( \nabla f(x) - \overline{\nabla f(x)}e^{2 i f(x) r}\Big) \,dr + f(x) \bar{\nabla f(x)}\frac 12 \int_s^t r e^{2i f(x) r}\,dr.
\mathcal{Q}_{v}(\mathcal{U}, \mathcal{W}) = \int_{\Omega_{\kappa}}\nabla \mathcal{U}\nabla \mathcal{W}dy - \left( \frac{2}{\alpha+ 2}\right)^2 \int_{\Omega_{\kappa}}f'(v)\mathcal{U}\mathcal{W}dy, \ \ \mathcal{U}, \mathcal{W}\in H^1_0(\Omega_{\kappa})
(\Omega_{\tau})_{k,l}=u_{l-k}(\tau)X_k X_{l-k}-\frac{\frac{\partial x_0}{\partial z}(0,\tau)}{n-2}\cdot \left( \sum_{1\leq i \leq n-1, i \neq k-l}\frac{\alpha_{l-k}(\tau)}{\alpha_i(\tau) \alpha_{i+l-k}(\tau)}X_{i+k}X_{i+l-k}\right) ,
{\cal E}=\frac{1}{2}\mid D_{0}\Phi \mid^{2}+\frac{1}{2}\mid D_{i}\Phi \mid^{2}+\frac{1}{4}F_{ij}^{2}+\frac{1}{2}F_{0i}^{2}+\frac{\lambda}{8}( \mid \Phi \mid^{2}- 1)^{2},
z'_R (t) = \frac{y''_R (t) y_R (t) - [y'_R (t)]^2}{y_R^2 (t)}\geq \frac{1}{d-2}\left[\frac{y'_R (t)}{y_R (t)}\right]^2 = \frac{1}{d-2}z_R^2 (t).
p\left(x_1,x_2,\ldots,x_d\right) = \sum_{i_1 =1}^k \cdots \sum_{i_d =1}^k W_{i_1,\ldots,i_d}p_{1,i_1}(x_1)p_{2,i_2}(x_2)\cdots p_{d,i_d}(x_d).
2^{3}u^{3}\varOmega_3(u)={}\det \left(\begin{array}{rrr}\mu^1_{2,1}(u) \mu^1_{2,2}(u) \mu^1_{2,3}(u) \sqrt{u}\acute\mu^1_{2,1}(u) \sqrt{u}\acute\mu^1_{2,2}(u) \sqrt{u}\acute\mu^1_{2,3}(u) u\mu^2_{2,1}(u) u\mu^2_{2,2}(u) u\mu^2_{2,3}(u) \end{array}\right).
E\left[ \sigma'(X_{\eta_n(s_1)-\delta}) M_{s_1}^{n,3}\sigma'(X_{\eta_n(s_2)-\delta}) M_{s_2}^{n,3}\right] =E\left[ \sigma'(X_{\eta_n(s_1)-\delta}) M_{s_1}^{n,3}\sigma'(X_{\eta_n(s_2)-\delta}) E[M_{s_2}^{n,3}| \mathcal{F}_{\eta_n(s_2)-\delta}] \right] =0.
\prod_{i = 1}^{n - 1}\left[ \int d^{D}x | f_{i}( x ) |^{q q_{1}/ q}\right]^{\frac{1}{q q_{1}/ q}}\left[ \int d^{D}x | f_{n}( x ) |^{p}\right]^{\frac{1}{p}}= \prod_{i = 1}^{n}\| f_{i}\|_{q_{i}}.
L_{m}(\alpha) \equiv \left (\begin{array}{cc}e^{-iV_m^-}e^{\alpha}e^{iV_m^+}e^{\alpha}e^{-iV_m^+}e^{iV_m^-}\end{array}\right ) \, ,\quad \bar L_{m}(\alpha)\equiv \left ( \begin{array}{cc}e^{-iV_m^+}e^{-\alpha}e^{iV_m^-}e^{-\alpha}e^{-iV_m^-}e^{iV_m^+}\end{array}\right ) \, ,
S_{B H}={\frac{2 \pi}{\kappa_{2}^{2}}}\Phi_{H}={\frac{2 \pi}{\kappa_{2}^{2}}}\left[ 2 \ell M \left( 2 -{\frac{\gamma}{\delta^{2}}}\right) \right]^{\frac{\delta^{2}}{2 \delta^{2}- \gamma}}.
d{\cal L}_2 \vert_{\Psi_-^I=0}= -2 i E^-_{\dot{q}}\wedge E^-_{\dot{q}}\wedge E^{++}+{1 \over 2}E^i \wedge ( E^{\mp\mp}\wedge \Omega^{\pm\pm i}- 4i E^+_q \wedge E^-_{\dot{q}}\gamma^i_{q\dot{q}})
\Big( \sum_{\vert \alpha \vert = m}\Vert x_{\alpha}\Vert^{q}\Big)^{\frac{1}{q}}\leq C^{m}\Big( \int_{\mathbb{T}^{n}}\Big\Vert \sum_{\vert \alpha \vert = m}x_{\alpha}z^{\alpha}\Big\Vert^{2}dz \Big)^{\frac{1}{2}}\,.
\mathrm{Tr}(\Omega(c)B^zA^z) = \mathrm{Tr}(a^{4z}\Omega(c)(1+D^2)^{-\frac{z}{6}}) = \mathrm{Tr}(\Omega(c)(1+D^2)^{-\frac{z}{6}}),\Re(z) > 3p.
\hat{H}^{\prime}= \frac{1}{4 \Omega_{G}^{2}}\Bigl ( \frac{1}{4}\hat{a}_{0}^{\dagger 4}+ \hat{a}_{0}^{\dagger 3}\hat{a}_{0}+ \frac{3}{2}\hat{a}_{0}^{\dagger 2}\hat{a}_{0}^{2}+ \hat{a}_{0}^{\dagger}\hat{a}_{0}^{3}+ \frac{1}{4}\hat{a}_{0}^{4}\Bigr ) .
F_{i_{1}i_{2}, i_{3}}+ F_{i_{2}i_{3}, i_{1}}+ F_{i_{3}i_{1}, i_{2}}= - \frac{1}{( m - 3 ) !}\delta \left( \begin{array}{c c c c c c}{{1}}&{{2}}&{{. . .}}&{{. . .}}&{{. . .}}&{{m}}\\{{j_{1}}}&{{. . .}}&{{j_{m - 3}}}&{{i_{1}}}&{{i_{2}}}&{{i_{3}}}\\ \end{array}\right) \hspace{- 1 m m}\sqrt{\bar{g}}K^{j_{1}. . . j_{m - 3}},
\begin{aligned}|\partial_{\alpha}\bar{V}_i \, \Phi_{,i\alpha}^A(\bar{F}) \,J_A| \leqslant C\Bigl(|\Xi\!-\!\bar{\Xi}|^2\!+\!|F\!-\!\bar{F}|^2 \!\!\int_{0}^{1}\!\!\!\int_{0}^{1}\!\!|\bar{F}\! + \!\tau s(F \!-\!\bar{F})|^{p-3}ds\,d\tau \Bigr) &\leqslant C\Bigl(|\Xi\!-\!\bar{\Xi}|^2\!+\!|F\!-\!\bar{F}|^2(1\!+\!|F|^{p-3})\Bigr).\end{aligned}
u =c_1-c_2y-zc_3-xc_4-c_5yz-c_6xy-c_7xz-c_8(y^2-z^2)-c_9yz+xyz+\frac{1}{12}[2xy^3+6xy^2z &+6xy^2-6xyz^2-2xz^3+6y^2z-2z^3-9x^2+c_5(2z^3+6yz^2-6y^2-6y^2z-2y^3+18x) +(6c_5^2-12xc_5+6x^2)\log(x-c_5)].
\chi_1+\chi_2+\rho = (\chi_1+\rho_1)+(\chi_2+\rho_2)+(\rho-\rho_1-\rho_2) \in \textbf{W}(d_1)_{w_1}+\textbf{W}(d_2)_{w_2}+\frac{1}{2}\mathfrak{g}^{\nu<0}=\textbf{W}(d_1)_{v_1}+\textbf{W}(d_2)_{v_2}+\frac{3}{2}\mathfrak{g}^{\nu<0}.
\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2};1)
\upsilon^{\textrm{F}}_{i,(k,m)}=\prod\limits_{p\in\varphi\setminus\{i\}}\Big(\sum\limits_{u\in[1,M-1]}\sum\limits_{q\in[0,K]}\pi_{p,(q,u)}&+ \sum\limits_{l\in\mathcal{G}^\textrm{F}_{p,k}}\pi_{p,(l,M)}\Big), &~\textrm{if}~k \geq 0,~m = M,
\epsilon^{\mu \nu}{\partial}_{\mu}B_{\nu}^{a b}= - \epsilon^{\mu \nu}B_{\mu}^{a c}B_{\nu}^{c b}-{\textstyle \frac 1 2}\epsilon^{\mu \nu}( A_{\mu}^{a c}B_{\nu}^{c b}+ B_{\mu}^{a c}A_{\nu}^{c b}) .
\sum_{n=2}^{\infty}\left|\frac{h}{N(\beta_1)}T_0^{l,(n)}(\beta_1)-\frac{h}{N(\beta_2)}T_0^{l,(n)}(\beta_2)\right|\le c \beta_1^{-\frac{1}{2}}M^{(a_1+a_2+2a_4-a_3)(l+1)}\alpha^{-8}.
N(n,-1)=q^{\binom{n-1}{2}}\prod^{n-1}_{l=1}(q^{l}-(-1)^{l})=q^{(n-1)^{2}-\binom{n-(-1)-1}{2}}\prod^{n-(-1)-2}_{j=1}(q^{j}-(-1)^{j}),
\delta{\cal E}_{0}= - i \frac{3 \hbar c}{2}\int d^{3}\vec{r}\varphi^{\dagger}\epsilon_{0}\left[ 1 - \frac{\hbar^{2}}{4 m^{2}c^{2}}( \vec{\sigma}\cdot \vec{\nabla})^{2}\right] \varphi{.}
{\cal A}= 6 4 \cdot 9 6 \pi^{3}\lim_{\rho \rightarrow 0}\rho^{4}\int_{0}^{r}x_{0}^{3}d x_{0}\int_{0}^{\pi}{\frac{\sin^{2}\theta d \theta}{( x^{2}+ x_{0}^{2}+ \rho^{2}- 2 x x_{0}\cos \theta )^{4}}}.
\longrightarrow H^{g + 1}( F ) \stackrel{p r}{\longrightarrow}H^{g}( F_{r e l}) \stackrel{\{Q , c_{0}\}}{\longrightarrow}H^{g + 2}( F_{r e l}) \stackrel{i}\longrightarrow H^{g + 2}( F ) \stackrel{p r}{\longrightarrow}H^{g + 1}( F_{r e l}) \longrightarrow .
\bar{y}_s^{t,\bar{A}_t^\epsilon;Z_t \otimes u^{\epsilon^\prime}, W_t^\otimes v}:= \phi(X_s^{t,\bar{A}_t^\epsilon; Z_t \otimes u^{\epsilon^\prime}, W_t \otimes v}; (Z_t \otimes u^\epsilon)_s,(W_t \otimes v)_s) &~~~ - y_s^{t,\bar{A}_t^\epsilon; Z_t \otimes u^{\epsilon^\prime}, W_t \otimes v}
\{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}= W_{~ 1 2}^{0}+ \left\{\theta_{1}[ \theta_{2}, r ] \right\}+ \mathrm{T r}_{3 4}( W_{~ 1 2 3 4}^{2}\theta_{3}\theta_{4}) + \mathrm{T r}_{3 4 5 6}( W_{~ 1 2 3 4 5 6}^{4}\theta_{3}\theta_{4}\theta_{5}\theta_{6}) \ldots .
\nabla^{(i)}\cdot\frac{\partial f}{\partial(\nabla^{(i)}\rho)}=\sum_{\alpha_{1},\alpha_{2},\cdot\cdot\cdot,\alpha_{i}=1}^{n}\frac{\partial^{i}}{\partial r_{\alpha_{1}}\partial r_{\alpha_{2}}\cdot\cdot\cdot\partial r_{\alpha_{i}}}\frac{\partial f}{\partial\rho_{\alpha_{1}\alpha_{2}\cdot\cdot\cdot\alpha_{i}}}.
\rho_j(i_j)={\displaystyle \sum_{n_j=1}^{i_j}\prod_{r=1}^{n_j-1}\left({q_j(r)\over p_j(r)}\right) \over \displaystyle \sum_{n_j=1}^{N_j}\prod_{r=1}^{n_j-1}\left({q_j(r)\over p_j(r)}\right)}.
L_{M}= - \frac{1}{4}e^{2 \kappa \Psi}F_{\mu \nu}^{a}F^{\mu \nu , a}- \frac{1}{2}\partial_{\mu}\Psi \partial^{\mu}\Psi - \frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}- e^{- 2 \kappa \Psi}V ( \Phi^{a}) - \frac{1}{2}m^{2}\Psi^{2}
\nabla^2\Phi_{(S)}- (D-2)\partial^c\Phi_{(S)}\partial_c\ln r - \frac{2(D-4)\Lambda}{(D-1)(D-2)}\Phi_{(S)}-\frac{k^2-(D-2)K}{r^2}\Phi_{(S)}+ \frac{\Delta_{(S)}}{r^2}= 0.
W(F)=e^{k\,\|{\rm dev}_2 \log U\|^2}=e^{k\,\|\log \frac{U}{\det U^{1/2}}\|^2}=g(\lambda_1,\lambda_2),\ \ \g:\mathbb{R}^2_+\rightarrow\mathbb{R},\ \g(\lambda_1,\lambda_2):=e^{\frac{k}{2}\,\left(\log \frac{\lambda_1}{\lambda_2}\right)^2}.
\prod_{\substack{1\leq b \leq k d_b + d_{b - 1}= 1}}\cos\left(\dfrac{\pi d^{(b - 1)}}{2^{b}}\right)\prod_{\substack{1 \leq b \leq k d_b + d_{b - 1}= 0}}\sin\left(\dfrac{\pi d^{(b - 1)}}{2^{b}}\right) \leq \left(\frac{\sqrt{3}}{2}\right)^k
\begin{array}{l}\Theta_1 \equiv \pi_0 \approx 0 \hspace{1.5cm}\Theta_2\equiv E' + \phi' + \pi + A_1 \approx 0 \Theta_3 \equiv E\approx 0 \hspace{1.5cm}\Theta_4 \equiv -\pi - \phi' - 2A_1 + A_0 \approx 0 \end{array}
F'=\prod^{i-2}_{l=1}\overline{W}_{l-1}T^{\varepsilon_l}_{\alpha_l}{\overline{W}\,}^{-1}_l\cdot\overline{W}_{i-2}T^{\varepsilon_i-1}_{\alpha_i-1}{\overline{W}\,}^{-1}_{i-1}\cdot\overline{W}_{i-1}T^{\varepsilon_i}_{\alpha_i}{\overline{W}\,}^{-1}_{i}\cdot\prod^{m}_{l=i+1}\overline{W}_{l-1}T^{\varepsilon_l}_{\alpha_l}{\overline{W}\,}^{-1}_l
|A+A| = \left|\left(\bigcup_{i=1}^{n}A_{i}\right)+\left(\bigcup_{i=1}^{n}A_{i}\right)\right| & \leq \sum_{k=1}^{n}|A_{k}+A_{k}|+\sum_{i=1}^{n-1}\sum_{j>i}|A_{i}+A_{j}| & = \frac{5n(n+1)}{2}.
\|H(f)\|_{L_{|x|_h}^pL_{\theta}^{\bar{p}_1}(\mathbb H^n)\rightarrow L_{|x|_h}^pL_{\theta}^{\bar{p}_2}(\mathbb H^n)}=\omega_Q^{1/\bar{p}_2-1/\bar{p}_1}\frac{\omega_QQ}{(Q-Q/p)Q/p}.
\{\omega^{i_0,\alpha_0}\,;\,\omega^{i_1,\alpha_1}\}= \{\omega^{i_{0}, 0}, \omega^{i_{0}, 1}, \ldots, \omega^{i_{0},N_{0}}; \omega^{i_{1}, 0}, \omega^{i_{1}, 1}, \ldots,\omega^{i_{1}, N_{1}}\}
\left(\sum_{0\le t<n}\frac{\exp\left\{i\frac{2\pi}{n}\left(u\,\sqrt{t}-\sqrt{t}\,w\right)^{2}\right\}}{\sqrt[3]{n}}\,\frac{\exp\left\{i\frac{2\pi}{n}\left(u\,\sqrt{t}-v\,\sqrt{t}\right)^{2}\right\}}{\sqrt[3]{n}}\,\frac{\exp\left\{i\frac{2\pi}{n}\left(\sqrt{t}\,v-\sqrt{t}\,w\right)^{2}\right\}}{\sqrt[3]{n}}\right)=
s_n(\Gamma_f)\,z^{-1}\overline{b_m(1/\bar z)}\overline{j(1/\bar z)}\overline{w_n(1/\bar z)}=\frac{1}{2i\pi}\int_{G}\frac{v_n(\xi)h(\xi)}{z-\xi}d\xi,|z|\geq1.
\dot{x}^{A A^{\prime}}= o^{A}\bar{o}^{A^{\prime}}- \frac{\rho^{\sigma}}{\rho^{\tau}}\acute{x}^{A A^{\prime}}, \acute{x}^{A A^{\prime}}= o^{A}\bar{\iota}^{A^{\prime}}+ \iota^{A}\bar{o}^{A^{\prime}}.
m = \int_{\Gamma(t)}c(t) \, \mathrm{d}\mathcal{H}^d= c(t) \mathcal{H}^d\big( \Gamma(t) \big)= \alpha_d c(t) R(t)^d \phantom{xx}\Leftrightarrow \phantom{xx}c(t) = \frac{m}{\alpha_d}R(t)^{-d}
\begin{pmatrix}Q_{t_1}Q_{t_2}\end{pmatrix}\widehat Q_t \widehat Q_t^* \begin{pmatrix}Q_{t_1}^* Q_{t_2}^* \end{pmatrix}\widetilde V_t \widetilde B_t^* = Q_t Q_t^* \widetilde V_t \widetilde B_t^* \approx \widetilde V_t \widetilde B_t^*,
\tfrac{1}{6}r^{3}+ \tfrac{1}{2}r|v|^{2}+ P(v) = \hat{P}(v, r) = \hat{Q}(\phi(v) + ru, |v|^{2}) &= \hat{Q}(ru, |v|^{2}) = \tfrac{1}{6}|v|^{6}+ \tfrac{1}{2}r^{2}|v|^{2}|u|^{2}+ r^{3}Q(u),
V(\lambda)_{<_{\mathrm{or}}\underline{\mathbf m}}=(U(\mathfrak n^-_P)_{<_{\mathrm{or}}\underline{\mathbf m}})\cdot v_\lambda,V(\lambda)_{\le_{\mathrm{or}}\underline{\mathbf m}}=(U(\mathfrak n^-_P)_{\le_{\mathrm{or}}\underline{\mathbf m}})\cdot v_\lambda.
{\cal H}_{can}= \Pi^{-}\partial_{+}A_{-}+\Pi^i\partial_{+}A_i -{\cal L}_{Weyl}= \frac 1 2 \Pi^2 + \frac 1 2 (\partial_i A_j)^2 + \Pi\partial_i A_i - A_i J^i - A_{-}J^{-}
F_2^{\rm class}\left( \phi_0, \phi_1, \phi_2 \right) =\left[ a_0 \left( \phi_1 \right) - a_0 \left( \phi_0 \right) \right]\left[ a_1 \left( \phi_2 \right) - a_1 \left( \phi_1 \right) \right]
\left| \left. a,1/2\right| n\right\rangle =\frac{1}{g^{\left| 1/2\right\rangle}_{\left| \left. a,1/2\right| n\right\rangle}}K_{a\frac{1}{2}}\left( i\, \nu_{n}\right) \left| \frac{1}{2}\right\rangle \; ,\quad \mathrm{where}\; \left| \frac{1}{2}\right\rangle \equiv \left| B_{\frac{1}{2}}\right\rangle \; ,
{\rm d}\|V(t)\|^2_{L^2}=2 \left[-\langle AV(t),V(t)\rangle - \langle B(V(t), v_1(t)), V(t)\rangle-\langle B(v_2(t),V(t)),V(t)\rangle \right]{\rm d}t &+2\langle \left[G(v_1(t))-G(v_2(t)) \right]{\rm d}W(t),V(t)\rangle &+\|G(v_1(t))-G(v_2(t))\|^2_{L_{}(\mathcal{H};\mathbb{L}^2)}.
|A + \{a_{1,1}, \dots, a_{r_0,3}\}| \geq \sum_{i=1}^{r_0}|p_1 + \{a_{i,1}, \dots, a_{i,3}\}| \geq \sum_{i=1}^{r_0}(|p_1| + |p_i| - 1) \geq r_0 |p_1| > (d+1)^2 |A|(1 - 100^{-d^2})/d \geq (d+2)|A|,
\sigma(X^g,Y^g,Z^g) =\chi(\mu_g^{d(X,Y)}\lambda_{g,X}\mskip1mu \lambda_{g,Y})\chi(\mu_g^{d(Y,Z)}\lambda_{g,Y}\mskip1mu\lambda_{g,Z}) &\phantom{{}={}}\times\chi(\mu_g^{d(Z,X)}\lambda_{g,Z}\mskip1mu \lambda_{g,X})\sigma(X,Y,Z) \phantom{\sigma(X^g,Y^g,Z^g)}=\chi(\mu_g)^{d(X,Y)+d(Y,Z)+d(Z,X)}\sigma(X,Y,Z).
+\int_{\partial B_R}\sum_{j=1}^3\, \partial x_m \mathfrak Z_{}(x,y)(u_{j}\,u) (y)\, \frac{y_{j}}{R}\hbox{d}o_{y}={\mathcal B_1}+{\mathcal S}^{\mathfrak D}+\mathcal S_R^{\mathcal B},
\int_{-a}^{a}\int_{-\sqrt{b^{2}-x^{2}/a^{2}}}^{\sqrt{b^{2}-x^{2}/a^{2}}}\int_{-\sqrt{c^{2}-x^{2}/a^{2}-y^{2}/b^{2}}}^{\sqrt{c^{2}-x^{2}/a^{2}-y^{2}/b^{2}}}dzdydx.
c_{3a}=-3\nu^6+12\nu^5+632\nu^4+1794\nu^3-37624\nu^2+65244\nu+64877>0 c_{3b}=2(1-\nu)(-3\nu^7+12\nu^6+652\nu^5+2417\nu^4-42561\nu^3+73864\nu^2+41336\nu+91323) +(1-\mu) (3\nu^6(220-\nu^2+4\nu)+2\nu(1490\nu^4-22993\nu^3+39898\nu^2+890\nu+109262)+8477)\ge 0
[ \hat{p}_{\alpha}+ \beta_{\alpha \beta}\hat{\epsilon}^{\beta}, \hat{p}_{\alpha^{\prime}}+ \beta_{\alpha^{\prime}\beta^{\prime}}\hat{\epsilon}^{\beta^{\prime}}] = i ( \beta_{\alpha \alpha^{\prime}}+ \beta_{\beta \alpha^{\prime}}\partial_{\alpha}\epsilon^{\beta}+ \beta_{\alpha \beta}\partial_{\alpha^{\prime}}\epsilon^{\beta}) .
s_{i,k}= - \sum_{\ell = 1}^k x_{\ell, k}s_{i+\ell,k}, \sigma_{i,k+1}= - \sum_{\ell = 1}^k (x_{\ell, k}+ \xi_{k+1}x_{\ell-1,k}) \sigma_{i+\ell,k+1}.
\langle \frac{\partial}{\partial\theta_i},\frac{\partial}{\partial r}\rangle = 0,\ \ \langle \frac{\partial}{\partial r},\frac{\partial}{\partial r}\rangle = 1, \ \ \langle \frac{\partial}{\partial\theta_i},\frac{\partial}{\partial \theta_j}\rangle = \phi^2(r) s_{ij}.
\mathcal{T}_\delta^{n}(P)\mspace{-1mu}=\mspace{-1mu}\Big\{\mathbf{x}\in\mathcal{X}^n\Big|\mspace{5mu}\big|\nu_{\mathbf{x}}(x)-P(x)\big|\leq\delta P(x),\ \forall x\mspace{-1mu}\in\mspace{-1mu}\mathcal{X}\Big\}.
\sum_{k=1}^m a_k + \sum_{k=1}^m b_k + \sum_{k=1}^{m+1}c_k &= \sum_{k=1}^m\sum_{i=0}^{\ell -1}(g_{k+im}-h_{k+im}) + \sum_{k=1}^m \psi(p_k) + \sum_{k=1}^{m+1}\psi(g_k) &= \sum_{k=1}^{\ell m}(g_k - h_k) + \psi(1) &= \sum_{k=1}^{\ell m}g_k - \sum_{k=1}^{\ell m}\alpha^{k-1}(vv^*) + \sum_{k= 1}^{\ell m + 1}x_k^*x_k.
y_{n_{r}}= p \left\{1 + C_{n_{r}}p^{- 2 / 3}+ \left( \lambda_{l}^{( \ast )}\right)^{1 / 2}\left[ - c + \ln \Theta_{l}\right] p^{- 1}+ O \left( p^{- 4 / 3}\right) \right\}.
d(\mathcal{G}_1, \mathcal{G}_2) = \left|n_1 - n_2 \right| + \sum_{k\geq 0}\left|[t^k]\left((-t)^{n_1}\chi_{\mathcal{G}_1}(-t^{-1}) - (-t)^{n_2}\chi_{\mathcal{G}_2}(-t^{-1}) \right)\right|.
d^2\!f(\overline{x})(w|\overline{z}) =\lim_{k\to\infty}\frac{2[f(\overline{x}+\tau_kw+\frac{1}{2}\tau_k^2z^k)-f(\overline{x})-\tau_kdf(\overline{x})(w)]}{\tau_k^2}=\lim_{k\to\infty}\Delta_{\tau_k}^2f(\overline{x}|\overline{v})(w^k)+\langle\overline{v},z^k\rangle\ \{\rm with}\ \w^k=w+\frac{1}{2}\tau_kz^k.
G=\begin{pmatrix}{-{p}_{1,5}}{-{p}_{2,5}}{-{p}_{3,5}}{-{p}_{4,5}}0{p}_{1,4}{p}_{2,4}{p}_{3,4}0{-{p}_{4,5}}{-{p}_{1,3}}{-{p}_{2,3}}0{p}_{3,4}{p}_{3,5}{p}_{1,2}0{-{p}_{2,3}}{-{p}_{2,4}}{-{p}_{2,5}}0{p}_{1,2}{p}_{1,3}{p}_{1,4}{p}_{1,5}\end{pmatrix},
= \frac{1}{2}\int_{0}^{L}( \vec{l}- i \partial_{x}\vec{n})^{2}\; d x \, = \frac{1}{2}\int_{0}^{L}( \vec{l}^{2}- ( \partial_{x}\vec{n})^{2}) \; d x \, - i \int_{0}^{L}( \vec{l}\cdot \partial_{x}\vec{n}) \; d x
\begin{array}{c c l}{L_{a - b}- \! L_{a + b}= \displaystyle 2 \sqrt{k + 2}\exp \left( \! - \frac{i \pi ( a^{2}+ b^{2})}{4 ( k + 2 )}\right) \sin \left( \frac{\pi a b}{2 ( k + 2 )}\right) E_{k + a + b}\ .}\\ \end{array}
g = \frac{1}{2}u_{b}^{2}b_{\mathrm{e x t}}- L \ \ , \ \{\cal E}\simeq 2 \pi \frac{\Delta u}{u_{b}}\left[ f^{2}g^{2}+ \frac{1}{2}\kappa^{2}u_{b}^{2}f^{4}- \kappa^{2}u_{b}^{2}f^{2}\right] .
{d^{\dagger}}( \eta^{n}\cdot \tau^{n}{\cal{G}}) \; \; = \; \; \eta^{n}\cdot \tau^{n}\,{d^{\dagger}}{\cal{G}}+{\textsf{P}}_{s}( n ) \; \eta^{n - 1}\cdot \tau^{n - 1}\,{d^{\dagger}}{\cal{G}}
\partial_t\frac{1}{2}\|u\|_{L^2(\mathbb{T})}^2 = \frac{1}{2}\int_\mathbb{T}\overline{u}\left(iu_xx - i|u|^{p-1}u-\gamma u +if\right) + u\left(-i\overline{u}_xx+i|u|^{p-1}\overline{u}-\gamma \overline{u}-i\overline{f}\right)\,dx = -\gamma \|u\|_{L^2_x(\mathbb{T})}^2 + \int_\mathbb{T}\Im(u\overline{f})\,dx
F^{\prime \prime}- 3 A^{\prime}F^{\prime}- 4 A^{\prime \prime}F - 2 \frac{\phi_{0}^{\prime \prime}}{\phi_{0}^{\prime}}F^{\prime}+ 4 \frac{\phi_{0}^{\prime \prime}}{\phi_{0}^{\prime}}A^{\prime}F = - m^{2}F .
CR(\mathbf{A}) =\frac{\left( \lambda_{\max}(\mathbf{A}) - n \right) / (n-1)}{\left( \overline{\lambda_{\max}^{n \times n}}- n \right) / (n-1)}=\frac{\lambda_{\max}(\mathbf{A}) - n}{\overline{\lambda_{\max}^{n \times n}}- n},
\frac{(r-1)!!}{(r-2)!!}= 1+\frac{1}{r-2}+\frac{r-1}{(r-2)(r-4)}+\frac{(r-1)(r-3)}{(r-2)(r-4)(r-6)}+ & + \dots + \frac{(r-1)(r-3)\cdots 4}{(r-2)(r-4)\cdots 3},
|d_{p_{j+1}-1}|=|a_{p_{j+1}-1}^2 \cdot \beta_{{p_{j+1}}}^{2,0}+b_{p_{j+1}-1}s_{p_{j+1}-1}^{2,0}+b_{p_{j+1}-1}y_{p_{j+1}-1}^{2,0}|\leq \frac{4D^2}{C^2}(6YX+1).
f_{k,u_l}=\partial_{u_l}f_k\geq0\ \ (k\not=l)\ \{\rm in}\ \ (p^-,p^+):=(p_1^{-},p_1^{+})\times(p_2^{-},p_2^{+})\times\cdots\times(p_m^{-},p_m^{+}),
hP_{\pi}=[\begin{matrix}h_{1}&h_{2}&\cdot\cdot\cdot&h_{n}\end{matrix}][\begin{matrix}e_{\pi(1)}\\e_{\pi(2)}\\ \vdots\\e_{\pi(n)}\end{matrix}]=[\begin{matrix}h_{\pi^{-1}(1)}&h_{\pi^{-1}(2)}&\cdot\cdot\cdot&h_{\pi^{-1}(n)}\end{matrix}]
\pi_1(e_{n+p}) = \ \frac{-\lambda_p a_p}{\sqrt{1+{\lambda_p}^2}}\; , \pi_2(e_{n+p}) = \ \frac{a_{n+p}}{\sqrt{1+{\lambda_p}^2}}\,\,\,\, 1 \le p \le r\;,\,\, \,\, \,\, \pi_1(e_{n+p}) = 0 \; , \pi_2(e_{n+p}) =a_{n+p}\,\, \,\,r+1 \le p \le m\; .
\int_0^1(\theta (1-\theta))^{s-2}\exp((n+1-q)\theta t) d\theta = \sum_{j=0}^\infty \frac{(n+1-q)^j}{j!}\bigl(\int_0^1\theta^{s+j-2}(1-\theta)^{s-2}d\theta\bigr) t^j &=\sum_{j=0}^\infty (n+1-q)^j \frac{\Gamma(s-1)\Gamma(s+j-1)}{\Gamma(j+1)\Gamma(2s-2+j)}t^j.
V_{0}(q)=-1+2\sum_{n\ge0}\frac{q^{n^{2}}(-q;q^{2})_{n}}{(q;q^{2})_{n}}=-1+2\sum_{n\ge0}\frac{q^{2n^{2}}(-q^{2};q^{4})_{n}}{(q;q^{2})_{2n+1}}
\tau_{m e}= \frac{1}{1 + b \chi + \chi^{2}}\left( \begin{array}{c c c}{( b + \chi )^{2}}&{- 2}&{b + \chi}\\{- \frac{1}{2}}&{\chi^{2}}&{\frac{\chi}{2}}\\{- 2 ( b + \chi )}&{- 4 \chi}&{- 1 + b \chi + \chi^{2}}\\ \end{array}\right)
\frac{1}{|W^{\mathrm{abs}}_{M_S}|}\sum\limits_{w \in W^{\mathrm{abs}}_{M_S}}\sum\limits_{\gamma \in \Gamma}w(\gamma(\mu))=\frac{1}{|W^{\mathrm{abs}}_{M_S}|}\sum\limits_{w \in W^{\mathrm{abs}}_{M_S}}\sum\limits_{\gamma \in \Gamma}\gamma(w)(\gamma(\mu))
\mathfrak{b}_{\ell^2}(N)=\Gamma\mathcal{O}_N\ \Longrightarrow\N_{N/L}(\Gamma)\mathcal{O}_L=N_{N/L}(\mathfrak{L}_1\mathfrak{L}_2)=\mathcal{L}^4=\mathfrak{b}_{\ell}(L)^4.
\left( -\frac{d^{2}}{dr^{2}}+\omega^{2}r^{2}+\frac{\mu +A_{\ell,j,m,n_1,n_2}-\frac{1}{4}}{r^{2}}-E_{k,\ell,j,m,n_1,n_2}\right) F_{k,\ell,j,m,n_1,n_2}(r)=0.
|s_t(x,y)| \lesssim \frac{\ell(R)^{\alpha}}{d(Q,R)^{m+\alpha}}\lesssim \frac{\ell(Q)^{\alpha/2}\ell(R)^{\alpha/2}}{D(Q,R)^{m+\alpha}}= 2^{-\alpha \ell/2}2^{-(m+\alpha)j}\ell(Q)^{-m}.
\aligned\det(F)= \Big\{(p-1)^2+(\nu^2\alpha\beta(\frac{\alpha}{2}-1)(\frac{\beta}{2}-1)-\frac{1}{4}\nu^2\alpha^2\beta^2)(\frac{x_1}{x_2})^{\alpha-p}&+(p-1)\nu\beta(\frac{\beta}{2}-1)(\frac{x_1}{x_2})^{\frac{\alpha}{2}}+(p-1)\nu\alpha(\frac{\alpha}{2}-1)(\frac{x_2}{x_1})^{p-\frac{\alpha}{2}}\Big\}x_1^{p-2}x_2^{p-2}.\endaligned
E_{0,l}= \lambda [2a(2l+d) + 2b_1(4a-2l-d+3) - l(l+d-1)], \psi_{0,l}(r) \propto r^l (1+\lambda r^2)^{-a}\exp\left(- \frac{b_1}{1+\lambda r^2}- \frac{b_2}{(1+\lambda r^2)^2}- \frac{b_3}{(1+\lambda r^2)^3}\right),
[\begin{matrix}\sigma_{xx}-\sigma_{hydrostatic}&\tau_{xz}\\ \tau_{zx}&\sigma_{zz}-\sigma_{hydrostatic}\\ \end{matrix}]+[\begin{matrix}\sigma_{hydrostatic}&0\\ 0&\sigma_{hydrostatic}\\ \end{matrix}]+[\begin{matrix}0&0\\ 0&\sigma_{z}\\ \end{matrix}]
\begin{array}{rcl}\frac{\partial^2}{\partial \rho^2}x(t,\rho) = \frac{\partial^2}{\partial \rho^2}(H(t,\rho)\rho^2) + \frac{\partial^2}{\partial \rho^2}(H^2(t,\rho)\rho^3) + \dots &= 2H(t,\rho) + R_1(t,\rho),\end{array}
{{\cal G}^{- 1}}_{a b}^{c d}\eta_{c d}^{\mathrm{L T T}}= \left( k^{2}- \Lambda_{l}+ \frac{1}{2 r^{2}}N ( N - 3 ) - \frac{N + d}{N + d - 2}\Lambda \right) \eta_{a b}^{\mathrm{L T T}},
u_x = \frac{u_s}{x_s},\quad\; u_y=\frac{u_t}{k},\quad\; u_{xx}= \frac{x_s\, u_{ss}- u_s\, x_{ss}}{x_s^3},\quad\; u_{xy}=\frac{u_{st}}{k\, x_s},\quad\; u_{yy}=\frac{u_{tt}}{k^2},\quad\; \ldots.
F(D^2\phi)=\log \det \begin{pmatrix}\ddot \phi \nabla \dot \phi (\nabla \dot \phi)^t g_{i\bar j}^T+\phi_{i\bar j}\end{pmatrix}-\log \det(g^T_{i\bar j})=\log \left(\ddot{\phi}_t-|\nabla \dot{\phi}_t|^2_{\omega_{\phi_t}}\right)+\log\frac{\det(g_{i\bar j}^T+\phi_{i\bar j})}{\det(g_{i\bar j}^T)}
\begin{array}{ll}\forall x\in\mathbb{R}^2 \langle \mathcal{F}(g)(x), x^\perp\rangle = \langle \mathcal{F}(f)(x), x\rangle & \langle \mathcal{F}(g)(x), x\rangle = - \langle \mathcal{F}(f)(x), x^\perp\rangle\end{array}
\limsup_{\lambda\to\infty}A_{12}(\lambda,\varepsilon) \leq C(n,\varepsilon,\zeta) \sum_{k=0}^{d-2}\sum_{y_{\tau}\in S_{\tau}}\int_{\mathbb{R}^d}\mathbf{1}_{B_{20\tau}(y_{\tau})}(x)|\langle \eta_{d-k}^{\kappa}(y_{\tau}), \eta(x)\rangle| d|\mu|^{s}(x).
|\langle S^{\nu,\mu^{n}_{\nu}}_{f}(h),g\rangle_{A^{2}_{\mu^{n}_{\nu}}}| =|\langle h_{f}^{\nu}(g),h\rangle_{A^{2}_{\nu}}|\le\|h_{f}^{\nu}(g)\|_{A^{1}_{\nu}}\|h\|_{\mathcal{B}}&\lesssim\|h_{f}^{\nu}\|_{A^{p}_{\omega}\to A^{1}_{\nu}}\|g\|_{A^{p}_{\widetilde{\omega}}}\|h\|_{\mathcal{B}}, g\in H^\infty,h\in H^\infty.
\sigma_{\hat{g}}^{2}\approx(\frac{\partial\hat{g}}{\partial T})^{2}\sigma_{T}^{2}=(\frac{-8L\pi^{2}}{T^{3}}\alpha(\theta))^{2}\sigma_{T}^{2}\Rightarrow(\frac{-8\overline{L}\pi^{2}}{\overline{T}^{3}}\alpha(\overline{\theta}))^{2}\frac{\sigma_{T}^{2}}{n_{T}}
\begin{array}{l}{\displaystyle \frac{d^{2}}{dt^{2}}\, y_{1}+ \frac{1}{t}\frac{d}{dt}\, y_{1}+ \mbox{e}^{-4y_{1}-2y_{3}}- \mbox{e}^{2y_{1}-2y_{3}}=0}\,,{\displaystyle \frac{d^{2}}{dt^{2}}\, y_{3}+ \frac{1}{t}\frac{d}{dt}\, y_{3}+ \mbox{e}^{2y_{1}-2y_{3}}- \mbox{e}^{4y_{3}+2y_{1}}=0}\,.\end{array}
(n-2)^3-(n+1)(n-3)^2=-n^2 + 9n -17<0, &(48n^2-2)^2(n-3)-2^{11}n^3(n-1)^2=256 n^5 - 2816 n^4 - 2240 n^3 + 576 n^2 + 4 n - 12>0.
A'_{1}+B_{1}= A'+A+B+e+\sigma \leq \Big[l(t)+m(t)\ln(A+e)+n(t)\big(\ln(A+B+e)\big)^{\alpha}\Big](A+e)+A+e+\sigma+f(t) = \Big[l(t)+m(t)\ln (A_{1}-\sigma)+n(t)\big(\ln (B_{1}-\sigma)\big)^{\alpha}\Big](A_{1}-\sigma)+A_{1}+f(t) \leq \Big[1+l(t)+m(t)\ln A_{1}+n(t)\big(\ln B_{1}\big)^{\alpha}\Big]A_{1}+f(t).
\gamma_4 = \alpha_2^2 - 2\beta_4 = \frac{1}{2}(q^5s+q^6-8q^6s+2q^6s^2+18q^7s-8q^7s^2-8q^8-8q^8s+8q^8s^2+8q^9-7q^9s+q^{10}) \overset{(a)}{\le}\frac{1}{2}(q^5s+q^6-8q^6s+2q^6s^2+18q^7s) \overset{(b)}{\le}2q^5 \, ,
(\begin{matrix}e_{i}^{0}\\ \vdots\\e_{i}^{m-1}\end{matrix})\oplus^{*}(\begin{matrix}e_{j}^{0}\\ \vdots\\e_{j}^{m-1}\end{matrix})=(\begin{matrix}e_{i}^{0}\oplus e_{j}^{0}\\ \vdots\\e_{i}^{m-1}\oplus e_{j}^{m-1}\end{matrix})
slnp-tlnq=ln(1+\frac{d}{q^{t}})=\sum_{m=1}^{\infty}(-1)^{m+1}\frac{(\frac{d}{q^{t}})^{m}}{m}=\sum_{n=0}^{\infty}\frac{2}{2n+1}{(\frac{d}{2q^{t}+d})}^{2n+1}.
R^a{}_{nbn}=W^a{}_{nbn}+p_{nn}\delta^a{}_b,R^a{}_{nbc}=W^a{}_{nbc}+\delta^a{}_cp_{nb}+\delta^a{}_bp_{nc}, R^a{}_{n0b}=-p^a{}_b+\delta^a{}_bp_{n0},R^a{}_{n0n}=p^a{}_n.
0 \geq 2 \left[e^{2u}+\left(\frac{1}{2}+\epsilon\right)\frac{1}{t}\right]^2-2e^{4u}-4\left(\frac{1}{2}+\epsilon\right)\frac{1}{t}e^{2u}-\left(\frac{1}{2}+\epsilon\right)\frac{1}{t^2}= \left[2\left(\frac{1}{2}+\epsilon\right)^2-\left(\frac{1}{2}+\epsilon\right)\right]\frac{1}{t^2}=(2\epsilon^2+\epsilon)\frac{1}{t^2}
{\rm Hess}\,f(x) +\frac{K}{n-m}f^{-3}(x)E_n &\geq{\rm Hess}\,f(x)+\frac{K}{n-m}f(x)E_n &=(1+|x|^2)^{-\frac94}\left(\frac54 x_ix_j-\frac12(1+|x|^2)\delta_{ij}+\frac{K}{n-m}(1+|x|^2)^2\delta_{ij}\right)_{ij}&\geq \frac54(1+|x|^2)^{-\frac94}(x_ix_j)_{ij}\geq O
\sum_{i = 1}^l \frac{\alpha_s(y_i)}{\alpha_s}(1 - s)(x_i) = \frac{1}{\alpha_s}\left(\sum_{i = 1}^l \alpha_s(y_i)(x_i - s(x_i)s)\right) = \frac{1}{\alpha_s}(\alpha_s - s(\alpha_s)s) = 1 + s
\|\sigma\|_\infty = \max_{1 \leq j \leq d}(\Sigma_{j j})^{1/2}= \max_{1 \leq j \leq d}\Big(\big(\Sigma_1^{1/2}- \Sigma_2^{1/2}\big)^2\Big)_{j j}^{1/2}\leq \big\|\Sigma_1^{1/2}- \Sigma_2^{1/2}\big\|_2.