harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\underbrace{|\alpha|^{4}}_{\in\mathbb{Q}(\sqrt{-D})}=\underbrace{A^{2}+C^{2}\alpha^{2}+E^{2}\overline{\alpha}^{2}}_{\in\mathbb{Q}(\sqrt{-D})}+2AC\alpha+2AE\overline{\alpha}+2CE\underbrace{(A+C\alpha+E\overline{\alpha})}_{|\alpha|^{2}},
\begin{array}{c}{\displaystyle <\prod_{<ij>\in \omega}U_{ij}>=\hspace{6cm}}{\displaystyle \frac{1}{Z_D}\sum^{\alpha-1}_{k=0}\sum_{\{\tau_m,\tau^{'}_m\}}\exp \sum_{<mn>\in L_D^1}\left(J_D{\tilde V}_{mn}(\gamma_k) \tau_m \tau_n +J_D^{'}{\tilde V}_{mn}(\gamma_k+\omega)\tau^{'}_m \tau^{'}_n\right)}\end{array}
\left\langle \left[ \partial_{\mu y}A^m_\nu (y) \right] \left[ g \epsilon^{3mn}a^\mu (x) A^{n \nu}(x) \right] \right\rangle \Bigr |_{y=x}= - 8 i g a^\mu(x) \partial_{\mu y}\mathcal P (y,x) \Bigr |_{y=x}.
T = \mu[U]\left(a^{(1)}\otimes \ldots \otimes \widetilde{t}^{(i)}\otimes \ldots \otimes a^{(m)}\right) = \sum_{k = 1}^l\beta_k\mu[U]\left(a^{(1)}\right. \left.\otimes \ldots \otimes b^{(i)}_k\otimes \ldots \otimes a^{(m)}\right).
L_{k,p,q}\triangleq\int_{J_{p}}\left|\sum_{l=1}^{k-1}\int_{J_{q}}\frac{\left|\dot{\bar{h}}_{k}(u)\right|+\left(\frac{|I_{1}|+\cdots+|I_{k}|}{|I_{1}|+\cdots+|I_{l-1}|}\right)^{H-\frac{1}{2}}\cdot\left|\dot{\bar{h}}_{l}(v)\right|}{|q_{k,l}(u,v)|^{H+\frac{1}{2}}}|I_{l}|dv\right|^{2}|I_{k}|du,
P_{(a)}^{(b)}=\frac{1}{2\eta}\{2C^{bd}C_{ad}+C^{db}C_{ad}+C^{bd}C_{da}-C_{d}^{d}(C_{a}^{b}+C_{a}^{b})+\delta_{a}^{b}[(C_{d}^{d})^{2}-2C^{df}C_{df}]\}.
\psi_{k}(x,\bar{x},E, \bar{E}, \mathbf{g}) = \omega^{k}\psi(\omega^{-k}x,\omega^{k}\bar{x},\omega^{-3kM}E,\omega^{3kM}\bar{E},\mathbf{g}) \,, \ \ \omega = \mbox{e}^{\frac{2\pi i}{3(M+1)}}\,,
\left\lbrace\begin{array}{ll}1-\theta_1^{\lambda}\leq 1-\theta_1 \leq (1-\theta_1)^{\lambda},\,\, \beta-a.e., begin{align*}5mm]\sum_{k \geq 1}\theta_k^{\lambda}- 1 = \sum_{k \geq 2}\theta_k^{\lambda}- (1 - \theta_1^{\lambda}) \leq \sum_{k \geq 2}\theta_k^{\lambda},\,\, \beta-a.e.\end{array}\right.
\| q - \tilde q\|_{L_2(0, \pi)}\le C_r \Xi + \sqrt{\frac{2}{\pi}}\big\| \{\xi_n - \tilde \xi_n\}_{n \in \Omega}\big\|, \Xi := \big\| \{n(\rho_n - \tilde\rho_n)\}_{n \in \overline{\Omega}}\big\|_{\bf a},
\biggl(\frac{c_{a_0}}{a_0!}\biggr)^{m-1}\frac{c_{_{\sum_{k=0}^ma_k}}}{\bigl(\sum_{k=0}^ma_k\bigr)!}\ge \prod_{k=1}^m\frac{c_{a_0+a_k}}{(a_0+a_k)!}
{\Gamma}_{\mu}=-\frac{n_{\mu}}{{\partial}_{\bot}^2+n^2{\partial}_+^2}B-\frac{{\partial}_{\mu}}{{\partial}_{\bot}^2+n^2{\partial}_+^2}\left( C-n^2x^-B-\frac{n^2n_+}{{\partial}_{\bot}^2+n^2{\partial}_+^2}{\partial}_+B \right)
\alpha_1^n = ||\nabla \cdot \mathbf{w}_{h}(t^{n})||_{L_{\infty}(\Omega_{t^{n}})}, \alpha_2^n = \sup_{t \in (t^{n},t^{n+1})}~ ||J_{\mathcal A_{{t_{n}},t_{n+1}}}\nabla \cdot \mathbf{w}_{h}||_{L_{\infty}(\Omega_{t})},
{\cal L}_{SYM}\ =\ -{1\over 4}\, G_{\mu\nu}^a G^{\mu\nu \, a}\ +\{\theta g^2 \over 32 \pi^2}\, G_{\mu\nu}^a \widetilde G^{\mu\nu \, a}\ +\{1\over 2}\, \bar\lambda^a i \slash \!\!\!\! D \lambda^a \ +\{\rm gauge\fixing\ +\ghosts\ ,}
\Vert Y_{j+1}\Vert_* \le \Vert Y''_j \Vert_* +\Vert W_{j+1}\Vert_* - \Vert H^{\lambda_{j-1}}(W_j) \Vert_* = \Vert W_{j+1}\Vert_* + \sum\nolimits^{j}_{i=1}\beta^d_i.
\begin{array}{lll}e_1^2=\omega_0 e_1,\ \ \ \e_1s_1e_1=e_1=e_1\bar s_1e_1, s_ie_1=e_1s_i,\ \ \bar s_je_1=e_1\bar s_j\ \ (i,j\ne1), e_1s_1\bar s_{1}e_1s_{1}= e_1s_1\bar s_1 e_1\bar s_{1}, s_1e_1s_1\bar s_1 e_1 = \bar s_1 e_1s_1 \bar s_1 e_1.\end{array}
A\overset{1}{T}(AX,AY) =A\overset{1}{T}(X,Y) - \overset{1}{T}(AX,Y) - \overset{1}{T}(X,AY), \overset{1}{T}(X,Y,Z) = \overset{1}{T}(AX,AY,Z) + \overset{1}{T}(AX,Y,AZ) + \overset{1}{T}(X,AY,AZ), \underset{XYZ}{\sigma}\overset{1}{T}(X,Y,Z) = \underset{XYZ}{\sigma}(\overset{1}{T}(AX,Y,AZ) + \overset{1}{T}(X,AY,AZ)),
\|u_{22}\| =\|\rho_{m,j}(x_j^*x_j+a_j^*a_j)-\rho_{m,n}\big(\rho_{n,j}(x_j)^*\rho_{n,j}(x_j)+\rho_{n,j}(a_j)^*\rho_{n,j}(a_j)\big)\|\leq 4.
R_{\mu\nu}-2\nabla_{\mu}\nabla_{\nu}\phi +{\textstyle\frac{1}{4}}H_{\mu}{}^{\rho\sigma}H_{\nu\rho\sigma}-{\textstyle\frac{1}{4}}e^{2\phi}\sum_{n}{\textstyle\frac{(-1)^{n}}{(n-1)!}}T^{(n)}{}_{\mu\nu}\, ,
\log (\gamma_M^{-1}\cdot\gamma_N)+\frac 12\Big(\sum_{M<n\leq N}\log(1-|F_n|^2) \Big)I =\log\Big(\prod_{M<n\leq N}\Big(I+(0 , F_n z^n)\Big)\Big) &=\log(I+(\sigma_N-\sigma_M)+H_M^N),
\left.\left<\left<F_{\mu_1\nu_1}(x_1)\cdots F_{\mu_n\nu_n}(x_n)\right>\right>=(ig)^{-n}\frac{\delta^n\ln{\cal Z}\left[J_{\mu\nu}\right]}{\delta J_{\mu_1\nu_1}(x_1)\cdots \delta J_{\mu_n\nu_n}(x_n)}\right|_{J=0}.
\pmb e\partial_{\lambda}(\overline{\pmb e}m_e(\lambda\pm))=-\frac{1}{2\pi\lambda}\int_{\mathbb{R}}u(y)m_e(y,\lambda\pm)~dy+G_{\lambda \pm 0i}*(u\pmb e\partial_{\lambda}(\overline{\pmb e}m_e(\lambda\pm))).
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}=1.
=\sum_{i,\,j=1}^{n}\left(\delta_{ij}-\frac{\partial_{\boldsymbol{z}_{i}}\hat{\boldsymbol{\psi}}_{k}\,\partial_{\boldsymbol{z}_{j}}\hat{\boldsymbol{\psi}}_{k}}{1+\left|\nabla\hat{\boldsymbol{\psi}}_{k}\right|^{2}}\right)\partial_{\boldsymbol{z}_{i}\boldsymbol{z}_{j}}^{2}\hat{\boldsymbol{\psi}}_{k}\,-\,\frac{n-1}{\hat{\boldsymbol{\psi}}_{k}}
L_{s,\mathsf{k}}(\theta_\mathsf{k}) = \begin{bmatrix}l_{s,\mathsf{k}}0 0 l_{s,\mathsf{k}}\end{bmatrix}+ \mathrm{R}(2\theta_\mathsf{k}) \begin{bmatrix}l_{sa,\mathsf{k}}0 0 -l_{sa,\mathsf{k}}\end{bmatrix}.
y_x = -\Re\hat h(x,A)/\Re\hat g(x,A),y_A = 1/(A\Re\hat g(x,A)) &\Phi_x = \Im\hat h(x,A) - \Re\hat h(x,A)\Im\hat g(x,A)/\Re\hat g(x,A),\Phi_A = \Im\hat g(x,A)/(A\Re\hat g(x,A))
\begin{aligned}\pi[\tau_2<\infty |X_{\tau_1},Y_{\tau_1}]=\pi\Big[\int_{\tau_1}^\infty 1_{\{|x+\omega_{X_0}(s)-y-\omega_{Y_0}(s)|\leq 1\}}ds>K |X_{\tau_1},Y_{\tau_1}\Big],\end{aligned}
\frac{1}{4\tan \frac{\pi lb^{-1}}{p}\tan\frac{\pi l}{p}}\geq \frac{-1}{\left|4\tan \frac{\pi lb^{-1}}{p}\right|\left|\tan\frac{\pi l}{p}\right|}\geq \frac{-p}{4\tan\frac{\pi}{4b}l}=-\frac{c_2(b)|p|}{|l|}.
-\int_{\Sigma}\varphi (L\varphi)e^{\frac{|x|^2}{4}}d\sigma=\int_{\Sigma}\left(|\nabla\varphi|^{2}-\bigr(|A|^{2}-\frac12)\varphi^2\right)e^{\frac{|x|^2}{4}}d\sigma\geq 0.
S_{F}^{- 1}= \not \! \! p_{L}\left[ 1 - \Sigma_{L}({\vec{p}}_{L}^{2},{\vec{p}}_{\perp}^{2}) \right] + \not \! \! p_{\perp}- \left[ m + \Sigma_{0}({\vec{p}}_{L}^{2},{\vec{p}}_{\perp}^{2}) \right] .
\Gamma_{\mu_1\mu_2\mu_3}^{a_1 a_2 a_3}\left(k_1,k_2,k_3\right)=\frac{g^3}{\left(2\pi\right)^3}\int\frac{{\rm d}^3{\bf Q}}{2|{\bf Q}|}N\left(|{\bf Q}|\right)\left[{\cal S}_{\mu_1\mu_2\mu_3}^{a_1 a_2 a_3}\left(k_1,k_2,k_3,Q\right) + Q\rightarrow -Q \right],
\Delta_{\omega_{\phi_{\epsilon}}}\Psi_{\epsilon,\rho}\geq -Ctr_{\omega_{\phi_{\epsilon}}}\omega_{\epsilon}+C\Sigma_{i=1}^{n}\frac{1}{(|S|^2+\epsilon)^{1-\rho}}|\frac{\partial z}{\partial w_{i}}|^2\frac{1}{\lambda_i}.
Y_n E_\delta = t^{\rho(\delta)_n}q^{\delta_n}E_\delta = t^{(n+1)/2-d_r}q^{\delta_n}E_\delta &= \omega T_1^{-1}\cdots T_{n-1}^{-1}E_\delta = t^{(-d_r+1}t^{d_0+d_1+\ldots + d_{r-1})/2}\omega f_{\delta_n\delta_1\ldots\delta_{n-1}}&= t^{(n+1)/2-d_r}\omega f_{\delta_n\delta_1\ldots\delta_{n-1}},
f(x)=\frac{1}{\sigma\mathrm{B}(\alpha,\beta)}\,\left[\Phi_s\left(\frac{x-\mu}{\sigma}\right)\right]^{\alpha-1}\left[1-\Phi_s\left(\frac{x-\mu}{\sigma}\right)\right]^{\beta-1}\phi_s\left(\frac{x-\mu}{\sigma}\right).
\sigma_{k-1}(\mu|i)\ge \frac{1}{\sigma_{k-1}(\mu|n)}\bigg\{\left[1-\frac{(k-1)(n-k-1)}{k(n-k)}\right]\big[\sigma_{k-1}(\mu|i,n)\big]^2 +\sigma_{k-2}(\mu|i,n)M \bigg\}.
\frac{\alpha}{2\gamma}\frac{d}{dt}\int_{0}^{+\infty}v^2dx+ \frac{\alpha}{\gamma}\int_{0}^{+\infty}v_{xxx}vdx+\frac{\alpha}{\gamma}\int_{0}^{+\infty}v^2v_xdx=\alpha\int_{0}^{+\infty}(|u|^2)_xvdx
\begin{array}{l}{\rm E}\left[ \left( \sum\limits_{k = 1}^\beta{n_{N,k}x_{(\mathcal{A},\mathcal{B}),i\beta + k}^{}}\right)^2 \right] = \beta{\rm E}[ x_{(\mathcal{A},\mathcal{B}),i\beta + k}^{2}] N_0/2 = E_b^{(i)}N_0/2 \end{array}
\mu_{p}(A)\leq \mu_{p}\bigg(\bigcup_{n=N}^{\infty}\{x\mid v_{\phi}(g_{n}^{-1}x)>n\}\bigg)\leq \sum_{n=N}^{\infty}\mu_{p}(\{x\mid v_{\phi}(g_{n}^{-1}x)>n\})<\epsilon.
E\left[T_1\right] = \frac{\nu_1}{\alpha_1}\, M H_{\nu_1 M}\, + \, O\left(e^{-\varepsilon M}\right)= (\nu_1 + \lambda \nu_2) M H_{\nu_1 M}\, + \, O\left(e^{-\varepsilon M}\right),M \to \infty,
\begin{aligned}\left [{x r (x - a) - x^2 r' \over 2 r^2 r'}+{r' y^2 - r y^2 \over 2 r^2 r'}+ 1 \right ] \underline{L}={r (x - a) - x r' \over r r'}\left [{x \over 2 r}L -{y \over r}e_{x y}\right ] - L' +{r y - r' y \over r r'}\left [{y \over 2 r}L +{x \over r}e_{x y}\right ]. \end{aligned}
R ( u_{1}- u_{2}) t_{b}^{c}( u_{1}) \otimes t_{a}^{b}( u_{2}) = \sum_{d}W \left( \left. \begin{array}{c c}{c}&{d}\\{b}&{a}\\ \end{array}\right| u_{1}- u_{2}\right) t_{a}^{d}( u_{1}) \otimes t_{d}^{c}( u_{2}) .
\; + \; \frac{1}{2 \pi}m c \int_{\Lambda}d^{2}x \; \left[ e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) + i \theta}\; + \; e^{- 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) - i \theta}\right]
\int_G f(x)\rho(x)dx =\int_G f(x) \rho(x)Q(g)(\ddot{x})dx &=\int_K\int_H\int_G f(x)\rho(x) g(k^{-1}xh) dxdhdk &=\int_G \int_K\int_H f(kxh^{-1}) \Delta_H(h^{-1})\Delta_K(k) \rho(x) g(x) dh dk dx &=\int_Gg(x)\rho(x)\Big(\int_K\int_H f(k^{-1}xh)dhdk\Big) dx.
\int_{X}e^{f}\Delta\theta_{v}\omega^{n}&=-\int_{X}e^{f}\sqrt{-1}\partial f\wedge n \bar{\partial}\theta_{v}\wedge\omega^{n-1}&=-\int_{X}e^{f}\sqrt{-1}\partial f\wedge i_{v}(\omega)^{n}&=-\int_{X}e^{f}v(f) \omega^{n}.
\int_{B_\rho(\zeta_i)}U_i^5\,U_j= 2\,a_1\,\mu_i^{\frac{1}{2}}\mu_j^{\frac{1}{2}}G_\lambda(\zeta_i,\zeta_j)-2\,a_3\,\mu_i^{\frac{3}{2}}\,\mu_j^{\frac{1}{2}}g_\lambda(\zeta_i)\,G_\lambda(\zeta_i,\zeta_j) &+\mathcal{R}_{i,j}^1+\mathcal{R}_{i,j}^2+5\,\mathcal{R}_{i,j}^3,\mathcal{R}_{i,j}^{5,1}.
\mathrm{E}[X^{n}|x_{p}<X<x_{q}]=\frac{\int_{x_{p}}^{x_{q}}x^{n}\frac{c_{1}}{\sigma}g_{1}\left(\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)\mathrm{d}x}{F_{X}(x_{p},x_{q})}.
\overline{W}_{1}^{0}(x)=\frac{1}{x},\overline{W}_{1}^{1}(x)=\frac{\kappa p+iq}{\kappa x(1-x)},\overline{W}_{1}^{2}(x)=\frac{(\kappa p-iq-\kappa+1)(\kappa p+iq)}{\kappa^2(x-1)^2},
\begin{aligned}\int_{t_{0}}^{t_{0}+h}e^{-s \omega}s^{-3 (\frac{1}{p}-\frac{1}{q})}\ds \leq \int_{t_{0}}^{t_{0}+h}s^{-3 (\frac{1}{p}-\frac{1}{q})}\ds = \frac{1}{1-3(\frac{1}{p}-\frac{1}{q})}\cdot \Big[(t_{0}+h)^{1-3(\frac{1}{p}-\frac{1}{q})}- t_{0}^{1-3(\frac{1}{p}-\frac{1}{q})}\Big]\end{aligned}
\begin{array}{ccc}T[0]=\left [\begin{array}{ccccc}1 0 0 0 0 -1 1 0 0 0 1 -1 1 0 0 0 0 -1 1 0 0 0 1 -1 1\end{array}\right ] ~~~ T[\infty]=\left [\begin{array}{ccccc}-5 4 -11 3 -6 -1 -1 0 0 0 1 -1 1 0 0 0 0 -1 1 0 0 0 1 -1 1\end{array}\right ]\end{array}
\begin{array}{r c l}{{v_{\mu}A_{\nu}^{a}= i \, \mathrm{s g n}( k ){\epsilon}_{\nu \mu \rho}{\partial}^{\rho}c^{a}}}&{{,}}&{{v_{\mu}c^{a}= 0 \, ,}}\\{{v_{\mu}{\bar{c}^{a}}= A_{\mu}^{a}}}&{{,}}&{{v_{\mu}B^{a}= - D_{\mu}c^{a}\, ,}}\\ \end{array}
A(u^*-u_h^*,\varphi_h) + B(u^*,\varphi_h)- B(u_h^*,\varphi_h)+ D(u^*- u_h^*,\varphi_h) + E(u^*-u_h^*,\varphi_h)+ J(u^*-u_h^*,\varphi_h)=0.
\langle x, \pi(k,\ell)g \rangle \overline{\langle x, \pi(k+q, \ell + p)g \rangle}= \frac{e^{2\pi i k p/M}}{3}\sum_{t = 0}^{2}e^{2\pi i t/3}\vert \langle x, \pi(k, \ell)g_{qpt}\rangle \vert^2
\ell_{\hat \mu + x}= \ell_{\hat \mu}\prod_{i=0}^{x - 1}\frac{\ell_{\hat \mu + i + 1}}{\ell_{\hat \mu + i}}= \ell_{\hat \mu}\prod_{i=0}^{x - 1}(1 + o_n(1)) d(\hat \mu + i).
\renewcommand{\arraystretch}{1.5}\begin{array}{rcl}f_1^{(1)}={}_2F_1(\lambda_3, 1-\lambda_1; \lambda_2+\lambda_3; 1-x) \,, * f_2^{(1)}= x^{1-\lambda_3-\lambda_4}(1-x)^{1-\lambda_2-\lambda_3}{}_2F_1(\lambda_1, 1-\lambda_3; \lambda_1+\lambda_4; 1-x) \,, \end{array}
F( x,y,p,q,s) := \left( \left( \sum_{(1,\beta) \in P}(p_{1,\beta}- y_1 q_{1,\beta}) s^\beta, \ldots, \sum_{(n,\beta) \in P}(p_{n,\beta}- y_n q_{n,\beta}) s^\beta \right) \right. ,y,p,q, \left. \vphantom{\sum_{(1,\beta) \in P}}\{s_{j,\alpha}- \partial^\alpha f_j(x)\}_{(j,\alpha) \in S}\right).
\mathbb{E}[Y] = d_1 = \log(k) - \frac{1}{2k}+ O(k^{-2}) \operatorname{Var}(Y) = d_2 - d_1^2 = \frac{1}{k}+ \frac{1}{2k^2}+ O(k^{-3}) \mathbb{E}[(Y-\mathbb{E}[Y])^4] = d_4 - 4 d_1d_3 + 6d_1^2d_2 - 3d_1^4 = 3(d_2 - d_1^2)^2 + O(k^{-3}) = \frac{3}{k^2}+ O(k^{-3})
\omega_0^{(\Lambda,N)}\big(H_0^{(\Lambda^+)}(\phi);\phi\big) =\frac{1}{q}{\rm Tr}\; H_{0,-}^{(\Lambda^+)}P_0^{(\Lambda,N)}(\phi) &=\frac{1}{q}\frac{1}{2\pi i}\oint dz\;{\rm Tr}\;H_{0,-}^{(\Lambda^+)}\frac{1}{z-H_{0,-}^{(\Lambda)}(\phi)}.
\underbrace{\left(5a_0-\dfrac{2}{3}M_SL^\frac{1}{2}M_P(M_P+1)(E(0))^\frac{1}{2}-a_1M_P^2-2\epsilon\right)}_{:=D}\|\partial_x^2 u\|^2\leq -\partial_t\left(\int_0^Lxu^2 dx\right) &+2C_\epsilon\int_0^\infty g(s)\|\partial_x^{k}\eta^t\|^2ds.
\sum_{i=1}^n \lambda_i -2\lambda_n =\sum_{i=1}^{n-1}\lambda_i -\lambda_n = \sum_{i=1}^{n-1}\sum_{j=1}^n R_{ijji}- \sum_{i=1}^{n-1}R_{inni}=\sum_{i,j=1}^{n-1}R_{ijji}.
(f_h\circ \pi_x)(y) =\int_D{\bf 1}_{\{r\}}\> d(\delta_{\pi(x,y)}*\delta_h)= (\delta_{\pi(x,y)}*\delta_h)(\{r\}) &= \frac{\omega_r}{\omega_h \omega_{\pi(x,y)}}p_{\pi(x,y),h}^r=\frac{1}{\omega_h}p_{r,\bar h}^{\pi(x,y)},
\mathcal{M}\left( p^{2}+ p_{\perp}^{2}+ M^{2}\right) G ( p , p_{\perp}) + \frac{p^{2}}{2 \pi}\int_{- \infty}^{\infty}d p_{\perp}^{\prime}\, G ( p , p_{\perp}^{\prime}) = 1 ,
f(k,\tau) =f(k,\alpha m_0 m_1\beta_1\dots m_r\beta_r) &=f(k,\alpha m_0 m_1\beta_1\dots m_j\beta_j)m_{j+1}\beta_{j+1}\dots m_r\beta_r &=k\alpha m_0 m_1\beta_1\dots m_j\beta_j m_{j+1}\beta_{j+1}\dots m_r\beta_r &=k\tau.
d s^{2}= - 4 d x^{+}d x^{-}- \bigg [ \sum_{a = 1}^{4}x_{a}^{2}+ ( 1 +{\frac{\tilde{m}}{{\tilde{r}}^{4}}}) \sum_{b = 5}^{7}x_{b}^{2}+ ( 1 - 3{\frac{\tilde{m}}{{\tilde{r}}^{4}}}) x_{8}^{2}\bigg ] ( d{\frac{\mu x^{+}}{R^{2}}})^{2}+ \sum_{i = 1}^{8}d x_{i}^{2}.
v_{T}\left( x\right) =u_{T}^{\left( 1\right)}\left( x\right) -u_{T}^{\left(2\right)}\left( x\right) ,p_{T}\left( x\right) =m_{T}^{\left(1\right)}\left( x\right) -m_{T}^{\left( 2\right)}\left( x\right) ,x\in \Omega ,
{\cal L}_{\mathrm{K}_{\mu}}= \frac{- 1}{8 g^{2}}\left( \mathrm{T r}\{{\cal F}_{\ast}^{\mu \nu}{\cal F}_{\mu \nu}\}- i \, \mathrm{T r}\{{\cal F}_{\ast}^{\mu \nu}{\cal F}_{\mu \nu}\}\, i \right) ,
(1+E_\ell)^{1-1/p}\leq (1+E_{0})^{1-1/p}+ C_p\tau \sum_{k=1}^\ell\left(1 + \frac{H^\tau[u^{k-1}] - H^\tau[u^{k}]}{\tau}\right)^{1/p}&\leq (1+E_{0})^{1-1/p}+ C_p\tau\ell^{1-1/p}\left(\ell + \frac{H^\tau[u_0] - H^\tau[u^{\ell}]}{\tau}\right)^{1/p}.
\begin{cases}\|(S_{\theta_{n}}-I)f_a^\pm\|_{s,X}\le C\theta_n^{s-\alpha-1}\|f_a^\pm\|_{\alpha+1, X}, s\le \alpha+1\cr\|(S_{\theta_{n}}-I)f_a^\pm\|_{s,X}\le \|S_{\theta_{n}}f_a^\pm\|_{s,X}+\|f_a^\pm\|_{s,X}\le C\theta_n^{s-\alpha-1}\|f_a^\pm\|_{\alpha+1, T}+C\delta, \alpha+2\le s\le s_1+1.\end{cases}
E \| L_n^{-1/2}\xi_k^{(n)}\|_{\ell_2}^{2+\delta}= E \left( \frac{1}{L_n}\sum_{j=1}^{L_n}[ \xi_k^{(n)}( j ) ]^2 \right)^{1+\delta/2}\le L_n^{-1}\sum_{j=1}^{L_n}E | \xi_k^{(n)}(j) |^{2+\delta}< \infty,
\big|D^{(m_{e_2(j,\ell)})}(T^{\psi}_df_{j+\ell})(x)\big| = \Big|D^{(m_{e_2(j,\ell)})}\Big( \big[\bigotimes_{i=1}^d\varphi\big] f_{j+\ell}(\psi ,...,\psi )\Big)(x)\Big| &\lesssim \sum_{\alpha}\Big| D^{(\alpha)}\Big[ \bigotimes_{i=1}^d\varphi\Big](x) D^{(\beta)}[f_{j+\ell}(\psi ,...,\psi)](x)\Big|\,.
\frac{\ell+1}{\sin^6 \phi}\ell^2 \sum_{v=0}^{3}{_3}\omega_{2,v}(\cos \phi) p_{1,\ell+v}(\phi) = \frac{\ell^3}{\sin^6 \phi}\big[- \sin^3 \phi \sin \psi_{1,\ell}h_{1}(0) s_{1,0}(\ell,\phi) \big] &\;\;+ \phi^{-6-1/2}O(\ell^{-1/2})+\phi^{-4}O(\ell),
L_{ij}=\left(\begin{array}{ccccc}0 k_{1}-k_{3}-k_{1}+k_{2}k_{3}-k_{2}0 -k_{1}+k_{3}0 k_{1}-k_{3}0 k_{1}-k_{2}-k_{1}0 k_{2}0 -k_{3}+k_{2}k_{3}-k_{2}0 0 0 0 0 0 0\end{array}\right) .
\pi_E(U^*\widetilde{F}U)e =U^*\widetilde{F}Ue=\sum_{g\in \Gamma}\epsilon_{g^{-1}}(c(\widetilde{F}Ue)(g))=\sum_{g\in \Gamma}\epsilon_{g^{-1}}(cF(Ue)(g)) =\sum_{g\in \Gamma}\epsilon_{g^{-1}}(cFc\epsilon_g(e))=\sum_{g\in \Gamma}\gamma_{g^{-1}}(c)F\gamma_{g^{-1}}(c)e,
\begin{array}{l}{\bf M}_{{\cal N}}(E) \equiv \hspace*{-1pt}\left(\renewcommand{\arraystretch}{1.3}\begin{array}{cc}\ddots \vdots \cdots \begin{array}{cccc}2E-{\cal N}+ 7 0 ({\cal N}-2)({\cal N}-3) 0 -6g 2E-{\cal N}+5 0 ({\cal N}-1)({\cal N}-2) 0 -4g 2E-{\cal N}+3 0 0 0 -2g 2 E-{\cal N}+1 \end{array}\end{array}\right) \end{array}
f^{(0)}_m = C m^2 |m|^{1/2}\frac{1}{g_{\rm YM}^{12}}\left( \frac{g_{\rm YM}^2}{\phi}\right)^{5/2}K_{5/2}(|m| \phi / g_{\rm YM}^2) e^{i m \phi^8 /g_{\rm YM}^2}
P(q^2)=1-24\sum^{\infty}_{n=1}\frac{nq^{2n}}{1-q^{2n}}=\frac{3}{\pi\sqrt{r}}+\left(1+k^2_r-\frac{3\alpha(r)}{\sqrt{r}}\right)\frac{4}{\pi^2}K^2[r].
X_{L}^{\mu}( \sigma_{+}) = \frac{1}{2}X^{\mu}+ \frac{1}{2}P_{L}^{\mu}\sigma_{+}+ \frac{i}{2}\sum_{k \neq 0}\frac{1}{k}\alpha_{k}^{\mu}\mathrm{e}^{- 2 i k \sigma_{+}}\, .
r(p)\lambda_1 =\sum_{i\neq1}K_{max}\lambda_i-\sum_{i\neq1}(K_{max}-K_{i1})\lambda_i \leq-\lambda_1 K_{max}+\lambda_1\sum_{i\neq1}(K_{max}-K_{i1}) =((n-2)K_{max}-\mu)\lambda_1.
\int_{t-h}^t Ce^{As}x \, ds - \int_t^{t+h}Ce^{As}x \, ds = \int_{t-h}^t \left( Ce^{At}x-\int_s^t CAe^{Ar}x \, dr \right)ds-\int_t^{t+h}\left(Ce^{At}x+\int_t^s CAe^{Ar}x \, dr \right)ds = -\int_{t-h}^t \int_s^t CAe^{Ar}x \, dr \, ds - \int_t^{t+h}\int_t^s CAe^{Ar}x \, dr \, ds.
(\partial_s+\Delta)w = v(\partial_s+\Delta)\varphi+\varphi(\partial_s+\Delta)v-2g(\nabla\varphi,\nabla v) &=v(\partial_s+\Delta)\varphi+\varphi(\partial_s+\Delta)v-2\rho\lambda e^{\lambda u}\varphi g(\nabla u,\nabla v).
0 \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{E}xt^1_{\mathcal{O}_X}(\mathcal{O}_X \to L,\mathcal{O}_X) \longrightarrow \mathcal{E}xt^1_{\mathcal{O}_X}(L,\mathcal{O}_X) \longrightarrow 0,
\mathcal{B}^+_I = \left\{\begin{aligned}\left(\tau,v_{s+k+1},\omega_{k+1}\right)\in\mathcal{A}^+ \textnormal{such that}& \inf_{i\in \left\{1,\dots,s,s+1,\dots,s+k\right\}\backslash \left\{i_{k+1}\right\}}\left| \left(x_{i_{k+1}}^\prime - x_i^\prime\right) - \tau \left(v_{i_{k+1}}^\prime - v_i^\prime\right)\right| \leq y\end{aligned}\right\}
\bar{g}(s)=\frac{g s^{\frac{y}{2}}}{1-\frac{\pi A(y)b(y)g}{y}(s^{\frac{y}{2}}-1)}=\frac{g_{IR}^* |\tilde{g}|s^{y/2}}{1+|\tilde{g}|s^{y/2}},\ \ \ \ \ \ \tilde{g}=\frac{g}{g-g_{IR}^*}\ \in \ \ ( -\infty,0).
F(r,R) = \frac{r}{(1+r)^2}\left(\frac{r^2-1}{2}\sin(2x) + R'(x)(\cos^2(x) + r \sin^{2}(x)) + (r-1)\sin(2x)R(x) + R(x)R'(x)\right) & + \frac{1}{4\pi}\sum_{i=1}^{2}F_{i}(r,R),
[W, \lambda](x, y)=\begin{cases}W\left(\frac{x}{1-\lambda}, \frac{y}{1-\lambda}\right) 0\le x\le 1-\lambda 0\le y\le 1-\lambda\,, 1 1-\lambda< x\le 1 1-\lambda< y\le 1\,. \end{cases}
{\cal L}_{\rm daughter}=\frac{1}{2}\sum_{i=1}^{2N}\dot{x}_{i}^{2}-\frac{1}{2}\sum_{i=1}^{2N}\left\{-2Ax_{i}^{2}+x_{i}^{6}\right\}-\frac{1}{2}\sum_{\stackrel{i,j=1}{j\neq i}}^{2N}\frac{1}{(x_{i}-x_{j})^{2}}\,,
\sigma^{n}( \lambda ) = \sum_{j = q_{+}, q_{-}}Z_{n j}( \lambda -{\frac{( - )^{j}}{\alpha_{0}}}) - \sum_{j = - , +}( A_{n j}* \sigma^{j}) ( \lambda ) -{\frac{1}{N}}A_{n l}( \lambda - \lambda_{0}) ,
A_{\lambda, \alpha}(\boldsymbol{n}) = \sum_{k \in \mathbb{Z}}\alpha^k \prod_{j = 1}^{\ell}\binom{n_{s (j) + 1}+ \ldots +_{}n_{s (j) + \lambda_j}- (\lambda_j - 1) k}{n_{s (j) + 1}- k, \ldots, n_{s (j) + \lambda_j}- k, k}.
M_2(p,H)={\pi^2\over 8}\left (1+\frac{N_2(p,H)}{p}\right )={\pi^2\over 8}\left (1+O(p^{-1/2})\right )\hbox{and}h_K^-\leq 2\left (\frac{p+o(p)}{32}\right )^{(p-1)/12},
\sum_{m=1}^\infty\frac 1{m^2}\int\limits_{\stackrel{|z|^2<1-(\frac{\varepsilon_L}2)^{1/L}}{\rho_L(z,0)\leq \varepsilon_L}}(1-|z|^2)^{mL}d\mu(z) = \textrm{O}\left(\frac{\log^{n-1}L}{L^{2n+1}}\right)\ .
\rho\frac{\partial^{2}\vec{\zeta}}{\partial t^{2}}=-\nabla(-\gamma p\nabla\cdot\vec{\zeta}-\vec{\zeta}\cdot\nabla p)-\vec{B}\times(\nabla\times(\nabla\times(\vec{\zeta}\times\vec{B})))+(\nabla\times\vec{B})\times(\nabla\times(\vec{\zeta}\times\vec{B}))+(\nabla\Phi)\nabla\cdot(\rho\vec{\zeta})
\mathfrak{P}_{tdo}=\{\mathfrak{p}=(T_m, \lambda_1, \lambda_2, \dots , \lambda_k) \ |\T_m-, \ \lambda_1>\lambda_2>\dots >\lambda_k, \lambda_i \in \frac{1}{2}+\mathbb{Z}_{\geq 0}, \i=1, \dots , k \}.
\forall x_0\in A^+ : \ \lim_{\Omega\ni x\to x_0}u(x)=+\infty .\footnote{In view of \eqref{bc3}, this solution `connecting' $+\infty$ to $-\infty$, may be compared to the \emph{heteroclinic}orbits that connect in a similar way two distinct equilibrium points (cf. for instance \cite{indiana}).}
\mathfrak{F}(u)= \max_{\phi\in K'}\int_{\Gamma_u}\langle \phi , \nu_{\Gamma_u}\rangle \, d \mathcal{H}^{n}= \max_{\phi\in K'}\int_{\Omega\times \mathbb{R}}\langle \phi ,D\textbf{1}_{\{u>t\}}\rangle,
\xi E_i+ \xi^2 \delta E_i + \xi^3 E_i = \xi E_{i+1}+\xi^2 \delta E_{i+1}+ \xi^3 E_{i+1}(\xi + \xi^2 \delta + \xi^3 )E_i = (\xi +\xi^2 \delta + \xi^3 )E_{i+1}
Q_1 = \big\{(x, t) \in \Omega \times (0,T) \,|\, \big|\dot{u}(x, t)\big| > 1 \big\}, Q_2 = \big\{(x, t) \in \Omega \times (0,T) \,|\, \big|\dot{u}(x, t)\big| \le 1 \big\}.
\nabla\cdot[(x_k+\phi_k)(a\nabla u+g)-ua(e_k+ \nabla\phi_k)] =(e_k+\nabla\phi_k)\cdot g, \nabla\cdot[x_k(a_h\nabla v+g_i(e_i+\partial_i\phi))- v a_he_k] =e_k\cdot [g_i(e_i+\partial_i\phi)],
\prod_{j=1}^md^{k(n_j+nq)}h\circ r_n\cdot (r_n)^*{\mathrm{Leb}}= \prod_{j=1}^m d^{k(n_j+nq)}\cdot(r_n)^*\mu & \underset{n\rightarrow\infty}{\longrightarrow}(\phi_0)^*\left(\bigwedge_{j=1}^m(\pi_j)^*\mu_{f_0}\right).
M - N\overline{M}^{-1}\overline{N}=\left(\begin{array}{rrr}0.5756 - 0.0000i 0.3906 + 0.2060i -0.1576 - 0.5196i 0.3906 - 0.2060i 0.8131 - 0.0000i -0.5220 - 0.6089i -0.1576 + 0.5196i -0.5220 + 0.6089i 0.9137 - 0.0000i\end{array}\right)
\Phi_{+}^{[ 0 ]}( n , p ) = \left( \begin{matrix}{{| \eta_{+}( n ) >}}\\{{0}}\\ \end{matrix}\right) \otimes | p > , \Phi_{-}^{[ 0 ]}( n , p ) = \left( \begin{matrix}{{0}}\\{{| \eta_{-}( n ) >}}\\ \end{matrix}\right) \otimes | p > ,
2\Re(p_\mu^2\Psi,V\Psi) =2\Re\{(p_\mu\Psi,Vp_\mu \Psi)-(p_\mu\Psi,[V,p_\mu]\Psi)\}&\geq -2 \| \partial_\mu V \| \|p_\mu\Psi\| \|\Psi\|\geq -\epsilon\|p_\mu\Psi\|^2 -C_\epsilon\|\Psi\|^2.
|\nabla \rho| ={}\tfrac1{\rho}\left|\Re{\textstyle \sum_{j=1}^N}\overline{u_j}\nabla u_j \right| \le{}\tfrac1{\rho}\left|{\textstyle \sum_{j=1}^N}\overline{u_j}\nabla u_j \right|= \sqrt{{\textstyle \sum_{j=1}^N}|\nabla u_j|^2 -{(2\rho^2)}^{-1}{\textstyle \sum_{j_1\neq j_2}}|u_{j_1}\nabla u_{j_2}- u_{j_2}\nabla u_{j_1}|^2}.
n^{-2}g u^{k\bar l}u_{k\bar l \bar j}u^{p\bar q}u_{p\bar q j}+ n^{-1}g (u^{p\bar q})_{\bar j}u_{p\bar q j}+ n^{-1}g u^{p\bar q}u_{p \bar q j \bar j}=2 Re( g_{vj}v_{\bar j}) + g_v v_{j\bar j}+ g_{vv}|v_j|^2+ g_{j\bar j}.
|\{g\in\mathbb H^n: |[b,R_j](f)(g)|>\lambda \}| &\leq|\{g\in\mathbb H^n: \big|b(g)R_j(f)(g) \big|>\lambda/2 \}|+|\{g\in\mathbb H^n: \big| R_j(bf)(g) \big|>\lambda/2 \}| &\leq C\|b\|_{L^\infty(\mathbb H^n)}{\|f\|_{L^1(\mathbb H^n)}\over\lambda},
\big(g\odot f\big)(x) := f(x)\cdot \sum_{m=r}^{x}\frac{g(m)}{f(m-1)}\textrm{for}x\geq r \big(f\otimes g\big)(x) := g(x)\cdot \Big(\sum_{m=r}^{x}\frac{g(m)}{f(m-1)}\Big)^{-1}\textrm{for}x\geq r+1
c_1(Z)^2 = (c_1(M)-c_1)^2c_2-2(c_1(M)-c_1)c_3+c_4 c_2(Z) = (c_2(M)-c_1(M)c_1+c_2(A)-c_2(B)+c_1(B)^2-c_1(A)c_1(B))c_2+ +(-c_1(M)+2c_1)c_3+c_4.
\prod_{v \in S}\frac{1}{\beta_{M_v}}= 2^{|S|}\prod_{v \in S}\begin{cases}\frac{1}{1 - q_v^{-1}}\frac{1}{1 +q_v^{-1}}\end{cases}\leq 2^{|S|}\prod_{v \in S}\frac{1}{1 - q_v^{-1}}\leq 2^{|S|}\left( \frac{1}{1-\frac{1}{2}}\right)^{|S|}= \exp(O([K:\mathbb{Q}]^2)) n = 2.
\begin{aligned}\iota\bigl(f\bigr)(s) = \operatorname{Ad}(v_s) \Bigl( \operatorname{diag}\bigl[\overbrace{c(f(0)), \dots, c(f(0))}^a, f(t_1(s)), \dots, f(t_k(s)), \underbrace{c(f(1)), \dots, c(f(1))}_b \bigr] \Bigr)\end{aligned}
-2u\cdot G_h\ast u^{n-1}= - u \cdot G_h\ast u + ( u-u^{n-1}) \cdot G_h\ast( u-u^{n-1}) - u^{n-1}\cdot G_h \ast u^{n-1}+ u^{n-1}\cdot G_h \ast u - u\cdot G_h \ast u^{n-1},
{\bf{V}}_1^{[2]}(n) = \left[{\begin{array}{*{20}{c}}{\frac{{h_1^{[22]}(1)}}{{h_1^{[22]}(n - 2)}}}0 0{\frac{{h_2^{[22]}(1)}}{{h_2^{[22]}(n - 2)}}}\end{array}}\right],{\bf{V}}_2^{[2]}(n) = \left[{\begin{array}{*{20}{c}}{\frac{{h_1^{[12]}(6)}}{{h_1^{[12]}(n - 2)}}}0 0{\frac{{h_2^{[12]}(6)}}{{h_2^{[12]}(n - 2)}}}\end{array}}\right].
\sum_{i=1}^n{}_N\langle \xi, a_i\rangle a_i =J_\phi\sum_{i=1}^n J_\phi ({}_N\langle \xi, a_i\Omega\rangle a_i\Omega)=J_\phi\sum_{i=1}^n (J_\phi a_i\Omega){}_N\langle a_i\Omega, \xi\rangle &=J_\phi\sum_{i=1}^n (J_\phi a_i\Omega) \langle J_\phi a_i\Omega| J_\phi\xi\rangle_N=J_\phi\sum_{i=1}^n b_i \langle b_i| J_\phi\xi\rangle_N=J_\phi J_\phi\xi=\xi.