harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma^{\mu}\gamma^{\lambda}= g^{\mu \lambda}+ \sigma^{\mu \lambda}, \ \left( g^{\mu \lambda}= \frac{1}{2}\left\{\gamma^{\mu}, \gamma^{\lambda}\right\}, \ \sigma^{\mu \lambda}= \frac{1}{2}\left[ \gamma^{\mu}, \gamma^{\lambda}\right] \right) ,
S_{eff}= \int{d^4 x\left({\Pi^i \stackrel{\bf{\cdot}}{A_i}+{\cal P}\stackrel{\bf{\cdot}}{C}+{\overline{\cal P}}\stackrel{\bf{\cdot}}{\overline C}+ \Pi^0 \stackrel{\bf{\cdot}}{A_0}-H_0 - \left[{\Omega ,\Psi}\right]}\right)},
\max_{t\in[0,\infty)}E(tw_k) = E(t_kw_k) \geq \frac{1}{n}M\left(\left( \frac{2n-\mu}{2n}\alpha_n\right)^{n-1}\right) & \; \frac{d}{dt}(E(tw_k))|_{t=t_k}=0.
\int_\Omega g(u(x))dx= \int_{\Omega\setminus\Omega_0}g(u(x))dx= \int_{\Omega\setminus \Omega_0}\lim_{n \rightarrow \infty}g( u_n(x))dx \leq \lim_{n \rightarrow \infty}\inf \int_{\Omega\setminus\Omega_0}g(u_n(x))dx.
\delta x = \dot{x}\xi \,\,,\,\,\,\,\,\,\delta e = \frac{d}{dt}(e\xi) \,\,,\,\,\,\,\,\,\delta \psi^n = \dot{\psi}^n\xi \, \,,\,\,\,\,\,\,\delta \chi = \frac{d}{dt}(\chi\xi)\, ,
\Psi'(t)=\frac{1}{2}\sum_{i,j=1}^dE\left[\frac{\partial^2\varphi}{\partial x_i\partial x_j}(\sqrt{1-t}F+\sqrt{t}Z)(\mathfrak{C}(i,j)-\langle DF_i,-DL^{-1}F_j\rangle_H)\right]
I_3(\kappa)^{-1}=\big(I_3(\kappa)+S_3\big)^{-1}+\big(I_3(\kappa)+S_3\big)^{-1}S_3\big\{S_3-S_3\big(I_3(\kappa)+S_3\big)^{-1}S_3\big\}^{-1}S_3\big(I_3(\kappa)+S_3\big)^{-1}&=\big(I_3(\kappa)+S_3\big)^{-1}+\big(I_3(\kappa)+S_3\big)^{-1}S_3\big\{S_3-S_3\big(I_3(0)+S_3\big)^{-1}S_3+\O(\kappa)\big\}^{-1}S_3\big(I_3(\kappa)+S_3\big)^{-1}.
\mathrm{T r}K^{( j )}( s ) = \mathrm{T r}\left[ e^{- s \bigtriangleup^{( j )}}\right] ={\frac{1}{( 4 \pi s )^{d / 2}}}\left( A_{0}^{( j )}+ s A_{1}^{( j )}+ s^{2}A_{2}^{( j )}+ . . . . \right) ~
\xi_n \in A^{\epsilon_n}(y_n) \eta_n =c_nM_n\xi_n+y_n-x_n, \|\eta_n\|^2_{M_n^{-1}}+2c_n\epsilon_n \leq\sigma_n^2(\|c_nM_n\xi_n\|^2_{M_n^{-1}}+\|y_n-x_n\|^2_{M_n^{-1}}).
\begin{array}{l}H (y) = \left( \frac{n-2}{n}-1 \right) y^{\frac{n}{n-2}}\tilde{g}(y)- \frac{n-2}{n}\int_{0}^{y}s^{\frac{n}{n-2}}\tilde{g}^{'}(s) \; ds, \; \; H(y) \approx - y^{\frac{n}{n-2}}\tilde{g}(y) \cdot\end{array}
\mathrm{det}\left(\mathbf{A}\right)=\begin{vmatrix}\begin{array}{ccc}A_{11}A_{12}A_{13}A_{21}A_{22}A_{23}A_{31}A_{32}A_{33}\end{array}\end{vmatrix}=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}=\epsilon_{ijk}A_{i1}A_{j2}A_{k3}
Bi^{\prime}(z)\sim\frac{z^{1/4}}{\sqrt{\pi}}exp(\frac{2}{3}z^{3/2})[\sum_{n=0}^{\infty}\frac{1+6n}{1-6n}\frac{\Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})(\frac{3}{4})^{n}}{2\pi n!z^{3n/2}}].
\int_u^v \frac{x^n(qx/u,qx/v;q)_{\infty}}{(bx,cx;q)_{\infty}}d_qx=\frac{(1-q)v (q,u/v,qv/u,bcuv;q)_{\infty}}{(bu,bv,cu,cv;q)_{\infty}}W_n(b,c,u,v).
\begin{cases}u_{tt}-\Delta u +\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=|u|^p, x\in \mathbb{R}^n, \t>0, u(0,x)=\varepsilon u_0(x), x\in \mathbb{R}^n, u_t(0,x)=\varepsilon u_1(x), x\in \mathbb{R}^n,\end{cases}
\begin{array}{l}{{\langle \Phi ( x ) \Phi^{*}( y ) \rangle}}\\{{\simeq \int_{0}^{\infty}d \tau E [ \langle \delta \left( q \left( \tau , x \right) - y \right) \rangle \exp \Big ( - \frac{1}{2}\langle \int_{0}^{\tau}d q^{B}( s ) d q_{C}( s^{\prime}){\cal D}_{B}^{C}\left( q \left( s \right) - q \left( s^{\prime}\right) \right) \rangle \Big ) ]}}\\ \end{array}
\lim_{t\to 0^+}\frac{1}{t}\left[ \int_0^1( e^{-tv(1-v) \Delta_0}V, V) \,dv - \| V \|_{L^2}^2 \right] = \frac 16 \, \lim_{t\to 0^+}\frac{1}{t}\left[ \int_0^1( e^{-t\Delta_0}V, V) \,dv - \| V \|_{L^2}^2 \right]
\left( \sum_{j_{1}=1}^{N}\left( \sum_{j_{2}=1}^{N}\left( \cdots \left(\sum_{j_{m}=1}^{N}\left\vert T(e_{j_{1}},...,e_{j_{m}})\right\vert^{q_{m}}\right)^{\frac{q_{m-1}}{q_{m}}}\cdots \right)^{\frac{q_{2}}{q_{3}}}\right)^{\frac{q_{1}}{q_{2}}}\right)^{\frac{1}{q_{1}}}\leq C_{m,\infty ,\mathbf{q}}\left\Vert T\right\Vert
\frac{d}{d r}\left( e^{( \Gamma - \Lambda ) / 2}( \Gamma^{\prime}- \phi^{\prime}) \, r^{2}- \frac{\alpha^{\prime}e^{\phi}}{g^{2}}e^{( \Gamma - \Lambda ) / 2}[ ( \phi^{\prime}- \Gamma^{\prime}) \, ( 1 - e^{- \Lambda}) + e^{- \Lambda}\phi^{\prime}\Gamma^{\prime}r ] \right) = 0
{\cal S}= \left( \begin{matrix}{1 + 3 \epsilon}&{\mu}&{0}&{0}\\{\lambda}&{1 + \epsilon}&{0}&{2 \mu / 9}\\{0}&{0}&{1 - 3 \epsilon}&{- \lambda}\\{0}&{- 6 \lambda}&{- \mu}&{1 - \epsilon}\\ \end{matrix}\right)
\partial_t q_t(x,y)=-\partial_y \int_{-\infty}^{x}\partial_tp_t(y,z)dz =-\partial_y\int_{-\infty}^{x}L_yp_t(y,z)dz &=-\hat{L}_y^*\partial_y\int_{-\infty}^{x}p_t(y,z)dz &=\hat{L}_y^*q_t(x,y).
\widehat{T}_{\varphi}\widehat{T}_{\psi}=\begin{bmatrix}T_{\varphi}T_{\psi}T_{\varphi}{\Gamma^{*}_{\bar{\psi}}}+{\Gamma^{*}_{\bar{\varphi}}}{S_{\psi}}0 S_{\varphi}S_{\psi}\end{bmatrix}~~~\widehat{T}_{\varphi \psi}=\begin{bmatrix}T_{\varphi \psi}{\Gamma^{*}_{\overline{\varphi \psi}}}0 S_{\varphi \psi}\end{bmatrix}.
\left( \Gamma_{1}\otimes \Gamma_{2}\right) \left( \Gamma_{3}\otimes \Gamma_{4}\right) = ( - 1 )^{\mathrm{d e g}\left( \Gamma_{2}\right) \mathrm{d e g}\left( \Gamma_{3}\right)}\Gamma_{1}\Gamma_{3}\otimes \Gamma_{2}\Gamma_{4}\, ,
\delta \tilde{\lambda}(t,s, \{\vec{k}\})=[i\delta (0)]~\frac{3}{2}a(t)^{-1}\tilde{\lambda}^2(t)(2\pi)^{-3}\int d^3\vec{q}[\vec{q}^2+ \tilde{m}^2(t) - i\epsilon]^{-1}
a=\lim_{t\to1}(t-1)^{2}\Psi(t)=\frac{1}{\left(q;q\right)_{\infty}}\,_{0}\phi_{0}\left(-;-;q,q\xi^{-1}\right) =\frac{\left(q\xi^{-1};q\right)_{\infty}}{\left(q;q\right)_{\infty}}
{\mathrm{P}}_{N}\left(d_H\left(\iota,I\right)>\delta, I_{(N,k)}=\iota\right) \leq\displaystyle\sum_{j=0}^{\ell-1}\displaystyle\sum_{x\in U_{j,\delta}}{\mathrm{P}}_{N}\left(x\in I , I_{(N,k)}=\iota\right) \leq\displaystyle\sum_{j=0}^{\ell-1}\displaystyle\sum_{x\in U_{j,\delta}}{\mathrm{P}}_{N}\left(x\in I, \iota\subset I\right).
- \int_{Q_T}\rho u\eta(p)\log(1+\frac{1}{p})^{r}= \int_{Q_T}\rho \eta(p)\log(1+\frac{1}{p})^{r}\frac{\partial_t p-|\nabla p|^2}{p}= \int_{Q_T}-\rho (\frac{d}{dt}h(p)-\nabla h(p)\cdot \nabla p)
(|D^m w|^2)^{\frac{1}{2}}_{Q^+_r(X_0)}\le N\kappa^{m+d/2}\sum_{|\alpha|\le m}\lambda^{\frac{|\alpha|}{2m}-\frac{1}{2}}(|f_\alpha|^2)_{Q^+_{\kappa r}(X_0)}^{\frac{1}{2}}.
\kappa_{\omega}=\frac{2\Gamma (\Delta_{\omega})}{\pi \Gamma(1-\Delta_{\omega})}\left( \frac{\sqrt{\pi}\Gamma \left( \frac{1}{2-2\Delta_{\omega}}\right)}{2 \Gamma \left( \frac{\Delta_{\omega}}{2-2\Delta_{\omega}}\right)}\right)^{2-2\Delta_{\omega}}\, .
B_k=\left(\begin{matrix}0 \dots 0 0 0 \dots 0 \vdots \vdots \vdots \vdots \vdots A_1 \dots A_{k-1}A_k+\mu I_p A_{k+1}\dots A_n \vdots \vdots \vdots \vdots \vdots 0 \dots 0 0 0 \dots 0\end{matrix}\right)\in\mathbb{C}^{np\times np}
\|\frac{1}{V_F}e^{ik\frac{2\pi}{V_F}v}p_k(t,g)\|_{L^1((0,V_F)\times\mathbb{R})}\leq \int_{0}^{V_F}\frac{1}{V_F}|e^{ik\frac{2\pi}{V_F}v}|dv\int_{\mathbb{R}}|p_k(t,g)|dg &=\|p_k(t,g)\|_{{L^1(\mathbb{R})}}.
-\frac{1}{\sin^2\phi_{-L}(n,\omega,E)}\frac{\partial}{\partial E}\phi_{-L}(n,\omega,E)=-\frac{1}{\cos^2\phi_{-L}(n-1,\omega,E)}\frac{\partial}{\partial E}\phi_{-L}(n-1,\omega,E)+1
\|u^n-u_h^{n}\|+\gamma^{\frac12}\Big{(}\Delta t\sum_{k=1}^n\|(\sigma-\sigma_h)^{k-\theta}\|^2\Big{)}^{\frac12}+\Big{(}\Delta t\sum_{k=1}^n\|\nabla (u-u_{h})^{k-\theta}\|^2\Big{)}^{\frac12}\leq Ch^{m+1},
R(t) =i\int_0^t e^{i(t-s)(-\Delta)^{\sigma}}\Big\{\mu\big(|u|^{p-1}u-|\tilde{u}|^{p-1}\tilde{u}\big)(s) &\quad\quad\quad\quad\quad\quad\quad\quad+e^{is|v|^{2\sigma}}(\mathcal{E}\phi^{(\nu)}\big(s,\nu(x-2\sigma s|v|^{2(\sigma-1)}v)\big)\Big\}ds.
\delta_{\epsilon}\lambda = \left(\not\!\partial\phi+{\textstyle\frac{1}{2\cdot 3!}}\Gamma_{11}\not\!\! H\right)\epsilon +{\textstyle\frac{5i}{4}}m e^{\phi}\epsilon-{\textstyle\frac{3i}{8}}e^{\phi}\not\! G^{(2)}\Gamma_{11}\epsilon\, ,
\sum_{m,n\in\mathbb{Z}}\mu(T_m(\phi_1) \cap T_n(\phi_2)) x^my^n = \frac{3}{64}xy^{-1}+ \frac{15}{64}x^{-1}y + \frac{35}{192}x^{-2}+ \frac{7}{192}y^{-2}+ \frac{7}{32}x^{-1}y^{-1}+ \frac{9}{32}.
D F(p^{\pm})=\left(\begin{array}{cccc}f_{1,u_1}(p^{\pm}) f_{1,u_2}(p^{\pm}) \cdots f_{1,u_m}(p^{\pm}) f_{2,u_1}(p^{\pm}) f_{2,u_2}(p^{\pm}) \cdots f_{2,u_m}(p^{\pm}) \vdots \vdots \ddots \vdots f_{m,u_1}(p^{\pm}) f_{m,u_2}(p^{\pm}) \cdots f_{m,u_m}(p^{\pm})\end{array}\right).
f^{}+f^{}=\frac{\chi}{12}+\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{|m\tau+n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}
T^{+q}_{-\dot p+2}=\frac12\gamma^i_{q\dot q}\Omega^{+2i}_{-\dot p}\psi^{1-}_{+2\dot q}+\frac{\chi}{2}\gamma^i_{q\dot p}\Omega^{+2i}_{+2},\T^{+q}_{-\dot p-2}=\frac{\chi}{2}\gamma^i_{q\dot p}\Omega^{+2i}_{-2},
\left( \binom{kl}{3}-k\binom{l}{3}\right) \frac{1}{k^{3}}=\frac{l}{6}\cdot\frac{\left( lk-1\right) \left( lk-2\right) -\left( l-1\right)\left( l-2\right)}{k^{2}}& =\frac{l^{2}}{6}\left( l\left( 1-\frac{1}{k^{2}}\right) -\frac{3\left(k-1\right)}{k^{2}}\right) ,
g_\varepsilon^{(s)}(t) =\tilde{T}_s (t) g_\varepsilon^{(s)}(0) +\ell^{-1}\int_0^t \tilde{T}_s (t-\tau)\tilde{C}_{s+1}g_\varepsilon^{(s+1)}(\tau) d\tau (s\geq m-1)
\begin{aligned}(\tilde{\Delta}_E\tilde{h},\tilde{h})_{L^2(\tilde{g})}=\int_{0}^{\infty}(\varphi')^2r^n dr +\kappa_i\int_{0}^{\infty}\varphi^2r^{n-2}dr \geq\left[\frac{(n-1)^2}{4}+\kappa_i\right]\int_{0}^{\infty}\varphi^2r^{n-2}dr . \end{aligned}
T_e N = \sum_{\lambda \in \Lambda_0}(\mathfrak{g}_0 \cap S_\lambda) + \sum_{\alpha \in \Delta^+}\sum_{\lambda \in \Lambda_\alpha}(\mathfrak{g}_\alpha \cap S_\lambda), T^\perp_e N = \mathfrak{g}_0 \cap T^\perp_e N + \sum_{\alpha \in \Delta^+}(\mathfrak{g}_\alpha \cap T^\perp_e N),
\Lambda_{m,n}(t) =-i m \sum_{(m',n') \in \mathbb{Z}^2_{0,n}}\left ( \theta^{(1)}_{m',n'}(t) + \theta^{(2)}_{m',n'}(t) \right ) \left ( \theta^{(1)}_{m-m',n-n'}(t)- \theta^{(2)}_{m-m',n-n'}(t) \right )
-\int_\Sigma \nabla R^2 \cdot \nabla (\zeta_\sigma ( R)) \, dx_1dx_2 \geq 2 \int_\Sigma \zeta_\sigma ( R) \, (1-O(R^2)) |\nabla u|^2 \, dx_1dx_2 = 4 \int_\Sigma \zeta_\sigma( R) \, (1-O(R^2)) \, \lambda \, dx_1dx_2.
\sum_{j = 0}^k{\sum_{s = 0}^j{( - 1)^{nj + s}\binom kj\binom js\frac{{G_{n + r}^{j - s}G_n^s}}{{G_0^j}}H_{m + rk + (n - r)j + rs}}}= \left({\frac{{G_r}}{{G_0}}}\right)^k H_m,G_0\ne 0\,,
\begin{array}{ccc}\{U^1_{\alpha}\}\stackrel{\rho_{V, \{U^1_{\alpha}\}}}{\longleftarrow}\{U^1_{\alpha}|_V \}\makebox[0pt][r]{$\scriptstyle{\rho_{12}}$}\uparrow \uparrow \makebox[0pt][l]{$\scriptstyle{\rho_{V, 12}}$}\{U^2_{\alpha}\}\stackrel{\rho_{V, \{U^2_{\alpha}\}}}{\longleftarrow}\{U^2_{\alpha}|_V \}\end{array}
\theta^{1 , \alpha}= \left( \begin{array}{c}{{\theta^{1 , \alpha_{1}}}}\\{{0}}\\ \end{array}\right) \, , \hspace{1 c m}\theta^{2 , \alpha}= \left( \begin{array}{c}{{\theta^{2 , \alpha_{1}}}}\\{{0}}\\ \end{array}\right) \, .
\begin{cases}\partial_{tt}u - \Delta u + b_{1}\partial_{t}u -{\rm div}(b_{2}\nabla \partial_{t}u)= 0 \mbox{in}(0, \infty) \times \Omega, u(t,\sigma) = 0 \mbox{on}(0, \infty)\times \partial\Omega, u(0,x) = u_{0}(x) \mbox{in}\Omega u_{t}(0,x) = u_{1}(x) \mbox{in}\Omega,\end{cases}
\forall s \in \textrm{MT}(\mathbf{k}),\,\, \eta_(s)=s, \forall s = \left(\mathbf{k}^{(n)},\mu^{(n)},\mathbf{k}^{(n-1)},\mu^{(n-1)},\ldots,\mathbf{k}^{(1)},\mu^{(1)}\right) \in \textrm{GMT}(\mathbf{k}),\,\, \eta_(s)=(\mathbf{k}_1,\mathbf{k}_2,\ldots,\mathbf{k}_n).
\begin{aligned}f(u_{n+1}) \leq f(\bar{u}_{n+1}) - \langle u_{n+1}-\bar{u}_{n+1}, w_n\rangle + \left( \frac{L}{2}- \frac{1}{2t}\right) \lVert u_{n+1}- u_n \rVert^2 & + \left( \frac{L}{2}+ \frac{1}{2t}\right) \lVert \bar{u}_{n+1}- u_n \rVert^2 - \frac{1}{2t}\lVert u_{n+1}-\bar{u}_{n+1}\rVert^2.\end{aligned}
+\sum_{r=2}^{n}\sum_{l=1}^{r-1}\sum_{i=0}^{l-1}\left(S_{irl}+ S_{ilr}\right)\left(\frac{H_{i}}{(i-r)(i-l)}+\frac{H_{r}}{(r-i)(r-l)}+ \frac{H_{l}}{(l-i)(l-r)}\right),
\lim_{n \to +\infty}\frac{\sum_{t=1}^{n}n_t}{n}= \lim_{n \to +\infty}\frac{0\cdot\sum_{t=1}^n \mathbf{1}_{N_{t+1}=0}\!+\! \sum_{t=1}^n \mathbf{1}_{N_{t+1}>0}}{n}&= \Psi_S,
2g(y^*)-2G(\xi(-T^*)) \leqslant 2\langle\nabla G(y^*),y^*-\xi(-T^*)\rangle+C|y^*-\xi(-T^*)|^2 &=2\Big\langle\nabla G(y^*),(\frac{y^*-y^-}{2}+\frac{y^*-y^+}{2})\Big\rangle+C|y^*-\xi(-T^*)|^2 &\leqslant C|y^*-y^-|^2+C|y^*-y^+|^2+C|y^*-\xi(-T^*)|^2 &=C|h|^2,
\int_{\partial D_R}\Big(e^{-\frac i2 \theta_0}(i\nabla+A_0)z_R\cdot\nu-e^{-\frac i2 \theta_{\mathbf e}}(i\nabla+A_{\mathbf e})\Psi_k\cdot\nu\Big) \psi_k\,ds=ik\sqrt\pi\big(\xi(1)-\sqrt \pi \big),
|\mathcal{F}|=\sum_{i=0}^n |\mathcal{F}^i|=|\mathcal{F}^{\frac{n+l-3}{2}}|+\sum_{i=0}^{\lfloor\frac{n+l}{2}-2\rfloor}(|\mathcal{F}^i|+|\mathcal{F}^{n+l-3-i}|)+\sum_{i=n+l-2}^n |\mathcal{F}^i| \leq |\mathcal{F}^{\frac{n+l-3}{2}}|+\sum_{i=\lceil\frac{n+l}{2}-1\rceil}^n{n \choose i}.
i \varepsilon \partial_t \, \mathcal{U}^*(t;0) \, \omega_{N,t}^k \, \mathcal{U}(t;0)= - \mathcal{U}^*(t;0) \left[ V * ( \rho_t -\rho_t^k), \omega_{N,t}^k\right] \mathcal{U}(t;0),
v_i(\tau, \xi) = \int_t^\tau \sum_{j=1}^n \Big( C_{i j}\big(x_i(s, \tau, \xi) \big) v_j \big(s, x_i(s, \tau, \xi)\big) + D_{i j}\big(x_i(s, \tau, \xi) \big) v_j(s, 0) + f_i\big(s, x_i(s, \tau, \xi) \big) \Big) \, ds begin{align*}6pt] + \sum_{r=1}^R \sum_{j =1}^n A_{r, ij}(t) v_j\big(t, x_r \big) + \int_{0}^1 \sum_{j =1}^n M_{ij}(t, x) v_j(t, x) \, dx + h(t),
\frac{1}{n^2}\frac{2l^2}{\pi(xy)^{3/2}}\sum\limits_{i=0}^{k}\frac{{k\choose i}p^{i}q^{k-i}i(k-i)}{k^2}e^{-\frac{l^2}{2}\left(\frac{i^2}{k^2x}+\frac{1}{y}\left(1-\frac{i}{k}\right)^2\right)},
R_{a_{2}}^{c_{2}}( \lambda \theta_{2}) S_{a_{1}c_{2}}^{c_{1}d_{2}}( \theta_{1}+ \theta_{2}) R_{c_{1}}^{d_{1}}( \lambda \theta_{1}) S_{d_{2}d_{1}}^{b_{2}b_{1}}( \theta_{1}- \theta_{2}) = S_{a_{1}a_{2}}^{d_{1}d_{2}}( \theta_{1}- \theta_{2}) R_{d_{1}}^{c_{1}}( \lambda \theta_{1}) S_{d_{2}c_{1}}^{c_{2}b_{1}}( \theta_{1}+ \theta_{2}) R_{c_{2}}^{b_{1}}( \lambda \theta_{2})
\mathsf{all}(\lambda) = \left[ \begin{array}{rrrrrrrrrrrrrr}1 \; . \; . \; . \; . \; . \; . \; . \; -1 \; . \; . \; . \; . \; . begin{align*}-1pt] . \; . \; . \; . \; . \; . \; . \; 1 \; . \; . \; . \; . \; . \; -1 begin{align*}-1pt] . \; . \; . \; . \; . \; . \; . \; . \; . \; 1 \; . \; . \; -1 \; 0 \end{array}\right],
\int\limits_{-\infty}^{\infty}dz_{1}\int\limits_{-\infty}^{\infty}dz_{2}\;\psi_{z_{1},z_{2}}^{\ast \;\nu ,\mu}(u^{0},x^{\prime},y^{\prime})\psi_{z_{1},z_{2}}^{\nu ,\mu}(u^{0},x,y)=\delta \left( x-x^{\prime}\right) \delta \left( y-y^{\prime}\right) \;.
\mathbb{D}^{X,Y}= \underline{\mathrm{Nat}}(-\,{\otimes_{A}}\,X,-\,{\otimes_{A}}\,Y) := \underline{\mathrm{Hom}}_{\mathcal Fun_{A,B}}(-\,{\otimes_{A}}\,X,-\,{\otimes_{A}}\,Y) \,\in \mathcal{Z}(\mathcal{C}) \,.
|\hat{\rho}^n_{\hat{f}_n^{*}}- \rho_{f^{*}}| = \max (\hat{\rho}^n_{\hat{f}_n^{*}}- \rho_{f^{*}}, \rho_{f^{*}}- \hat{\rho}^n_{\hat{f}_n^{*}}) &\leq \max (\hat{\rho}^n_{f^{*}}- \rho_{f^{*}}, \rho_{\hat{f}^{*}_n}- \hat{\rho}^n_{\hat{f}_n^{*}}) &\leq \sup_f |\hat{\rho}^n_f - \rho_f|.
\frac{d^2}{dr^2}U^+_{k,l}+\frac{1}{r}\frac{d}{dr}U^+_{k,l}-\Bigg(4l^2+\frac{(k-\frac{1}{2})^2}{r^2}\Bigg)U^+_{k,l}=0, \frac{d^2}{dr^2}U^-_{k,l}+\frac{1}{r}\frac{d}{dr}U^-_{k,l}-\Bigg(4l^2+\frac{(k+\frac{1}{2})^2}{r^2}\Bigg)U_{k,l}^-=0.
\real \left( \frac{zv_{{\nu}}^{\prime}(z)}{v_{{\nu}}(z)}\right) \geq1-\sum_{n\geq1}\frac{4\left\vert z\right\vert^{4}}{j_{{\nu},n}^{4}-\left\vert z\right\vert^{4}}=\frac{\left\vert z\right\vert v_{{\nu}}^{\prime}(\left\vert z\right\vert )}{v_{{\nu}}(\left\vert z\right\vert )}
|((V_1\cup W_1^H)\times(V_2\cup W_2^H))\cup ((W_1\cup V_1^H)\times(W_2\cup V_2^H))| &=(|V_1|+ |W_1|)(|V_2|+|W_2|)+ (|W_1|+ |V_1|)(|W_2|+|V_2|) &=2(|V_1|+ |W_1|)(|V_2|+|W_2|),
{V_{T}}^{G}( \psi , z ) = : \left( \prod_{a = 1}^{M}{\frac{i}{( n_{a}- 1 ) !}}{\frac{d^{n_{a}}}{d z^{n_{a}}}}C^{i_{a}}( z ) \right) e^{i \lambda \cdot C ( z )}: \gamma_{\lambda},
G_0(\bar{d},\bar{\sigma},\xi) = \alpha_n a_1 d_1^{n-2}\varphi(\xi) + a_3 \sum\limits_{i=1}^{k-1}\Big(\frac{d_{i+1}}{d_i}\Big)^{\frac{n-2}{2}}g(\sigma_i) - a_4\sum\limits_{i=1}^k \frac{2}{2i-1}|\ln d_i|, & G_h(d_1,\xi)=\frac{\alpha_n}{2}a_2\partial_{\xi_h}\varphi(\xi)d_1^{n-2},\quad\mbox{for}\ \h=1,\cdots,n.
f \circ \psi \circ f^{-1}(\xi_1, \xi_2, w_{12}) = (\xi_1^{L}, \xi_2^{L}, w_{12}) f \circ \phi \circ f^{-1}(\xi_1, \xi_2, w_{12}) = \left(w_{12}, w_{12}^{-1}, - \frac{p +q\xi_1\xi_2 - \xi_1}{p +q\xi_1\xi_2 - \xi_2}\right).
\Box_{s,y}v = 5 I^{-2}\overline{Q_l}^4 v + 10 I^{-1}\overline{Q_l}^3 v^2 + 10 \overline{Q_l}^2 v^3 + 5 I \overline{Q_l}v^4 + I^2 v^5 &=:\sum_{i=1}^5 G_j^{(3)}v^j.
\begin{aligned}\star_7 \beta = (-1)^k 2^{(-2+k)}|\Omega|_\omega^{(\frac{4}{3}- \frac{2}{3}k)}dr \wedge \star_6 \beta, \star_7 (dr \wedge \beta) = 2^{(-4+k)}|\Omega|_\omega^{(\frac{8}{3}- \frac{2}{3}k)}\star_6 \beta,\end{aligned}
{\cal I}_{U ( 1 )}- \delta{\cal I}_{U ( 1 )}= \lim_{\beta \rightarrow 0}2^{N / 2 - 1}\mathrm{T r^{\prime}}\; ( - 1 )^{P}e^{( \beta / 2 ) \nabla^{2}}= \lim_{\beta \rightarrow 0}2^{N / 2 - 1}\; \int [ d X ] \; \langle - X \vert \, e^{( \beta / 2 ) \nabla^{2}}\, \vert X \rangle .
\mathcal{C}_{1}^{n}=\frac{\mathcal{B}_{1}^{n}}{n},\thinspace\thinspace\mathcal{C}_{1,2}^{n}=\frac{\left(\mathcal{C}_{1}+\mathcal{B}_{2}\right)^{n}}{n},\dots\,\,\,\,\,\mathcal{C}_{1,2,\dots,k+1}^{n}=\frac{\left(\mathcal{C}_{1,2,\dots,k}+\mathcal{B}_{k+1}\right)^{n}}{n}
2\bigl(m-d_\epsilon-\tilde{d}_{\tilde{\epsilon}}\bigr)\xi + \sum_{1\leq r\leq d}\int_0^\xi u_{a_r}(\theta)\theta + \sum_{1\leq r\leq \tilde{d}}\int_0^\xi u_{\tilde{a}_r}(\theta)\theta =\pi (2l+\epsilon_- +\tilde{\epsilon}_-) ,
\sum_{r=1}^k \sum_{(i_1,j_1),\dots,(i_r,j_r)}\sum_{\gamma,\gamma' \in \Gamma}w_\gamma w_{\gamma'}\prod_{l=1}^r 1_{p_{i_l}=p'_{j_l}}1_{p_{i_l}|p_1+\dots+p_{i_l-1}- p'_1 - \dots - p'_{j_l-1}}&\frac{p_{i_1}\dots p_{i_r}}{p_1 \dots p_k p'_1 \dots p'_k}= o(1),
c_N = \int_{\mathbb{R}^{2N}}\delta(E_N-H_N(\vec{p},\vec{u})) d\vec{p}d\vec{u}\equiv\int_{H_N(\vec{p},\vec{u})= E_N}\frac{d\sigma(\vec{p},\vec{u})}{|\nabla H_N(\vec{p},\vec{u})|};
\phi_{1 2}^{+}\widetilde T_{1 2 , 2 3}^{3 D}\phi_{2 3}= ( - 2 i \pi )^{2}\left[ \overline{\Phi}_{1 2}T_{1 2 , 2 3}\Phi_{2 3}\right]_{3 1}^{\delta \delta}\beta_{1}
|\sum_{l}\sum_{T\in\mathbf{T}_{k,l}}\Lambda^{\epsilon}_{T}(\tilde F_{0},\tilde F_{1}, \tilde G)|\lesssim\sum_{l=-\infty}^{\lceil pk \rceil}2^{2l}2^{-k}\min(1,2^{-l}) 2^{-l}\lesssim2^{-k}(1+pk).
d(p_i,x)+d(q_i,x)-d(p_i,q_i) \leq d(p_i,\tilde{\xi}_i)+d(q_i,\tilde{\xi}_i)+2d(x,\tilde{\xi}_i)-d(p_i,q_i) =2d(x,\tilde{\xi}_i)<r_i\epsilon_i.
X^N_E(t) =X^N_E(t^N_{k-1})+\sigma(X^N_E(t^N_{k-1}))(B(t)-B(t^N_{k-1}))+\tilde{b}(X^N_E(t^N_{k-1}))(t-t^N_{k-1}) & \quad+\Phi^N_E(t)-\Phi^N(t^N_{k-1})t^N_{k-1}\le t\le t^N_k,
Z = \beta^{N / 2}\Gamma ( ( N - 1 ) / 2 )^{- 1}{\LARGE \int_{0}^{\infty}}d \sigma{\LARGE \int_{- \infty}^{\infty}}d \sigma_{1}( 4 \beta^{2}\sigma^{2}+ ( 2 \beta \sigma_{1}+ a )^{2}+ m^{2})^{M}\sigma^{N - 2}e x p [ - \beta ( \sigma^{2}+ \sigma_{1}^{2}) ]
D(0, \varepsilon) = \sigma_{\varepsilon}(T\diamond S\circ_{\ast}I)=\sigma_{\varepsilon}(\varphi(T)\diamond \varphi(S)\circ_{\ast}\varphi(I)) &=\sigma_{\varepsilon}(2\varphi(T)\varphi(I)^{\ast}-2\varphi(I)\varphi(T)).
\frac{1}{n}\sum_{k\in N}e_{k,g}^n +\sum_{\tau=0}^{n-s-2}\frac{1}{n}e^{n-\tau-1}_{i,g+\tau+1}&+\left(1-\frac{1}{n}\right)\left[\sum_{\tau=0}^{n-s-2}\frac{1}{n^{n-\tau-s-1}}e^{s}_{i,g+\tau+1}\right]+\frac{1}{n}\sum_{\tau=1}^{s-1}e_{i,g+n-1-s+\tau}^{s-\tau}.
\frac{d(p)}{c(p)}=\alpha(p), \ \frac{b(-p)}{a(-p)}=\gamma(p)=2\cosh(\beta p^0/2)-\alpha(p), \ \frac{b(-p)}{a(-p)}\cdot \frac{d(p)}{c(p)}=\gamma(p)\cdot \alpha(p)=1,
\widetilde{\psi}(d_{1}a^{\ast}d_{2}w) =\widetilde{\psi}\left(d_{1}c_{k}^{-1}\pi_{e_{k}a}i \Pi_{2}(d_{2}w)+d_{1}c_{k}^{-1}\pi_{e_{k}a}j'\sigma_{2}\Pi_{3}(d_{2}w)\right) &=d_{1}c_{k}^{-1}\psi\left(\pi_{e_{k}a}i\Pi_{2}(d_{2}w)\right)+d_{1}c_{k}^{-1}\psi \left(\pi_{e_{k}a}j'\sigma_{2}\Pi_{3}(d_{2}w)\right)
\exp \left( - i e \Bigg [ \frac{1}{\sqrt{\pi \! + \! g N}}\Big ( A_{\mu}^{( 1 )}, \varepsilon_{\nu \mu}\partial_{\nu}\Phi^{( 1 )}\Big ) + \sum_{I = 2}^{N}\frac{1}{\sqrt{\pi}}\Big ( A_{\mu}^{( I )}, \varepsilon_{\nu \mu}\partial_{\nu}\Phi^{( I )}\Big ) \Bigg ] \right) \Bigg \rangle_{\{K^{( I )}\}}\; =
\sum_{i=1}^N{\mathsf{R}}^{\lambda}_{\bar{\mathcal{N}}_i}=\sum_{s\in [0:1]^N}\lambda_s\left [\sum_{j=1}^N \left(\max_{i \in \mathcal{A}^{\star}_j}\ell_{i,s}^{\prime}+\max_{i \in ([1:N]\backslash \{j\}) \backslash \mathcal{A}^{\star}_j}r_{i,s}^{\prime}\right)\right],
\left( \boxtimes_{i = 1}^{l}\left(U_i^{\boxtimes |\tau^{(i)}|}\otimes \mathcal{S}^{\tau^{(i)}}\right) \right) \boxtimes \left(\mathbf{1}^{\boxtimes (n - |\sigma|)}\otimes \mathcal{S}^{(n - |\tau^{(0)}| - |\sigma|, \tau^{(0)})}\right)
C^*(\pi^*) \Big( \sum_{e\in X}ee^* \Big) = \sum_{e\in X}\sum_{g\in (\pi^1)^{-1}(e)}gg^* = \sum_{g\in (\pi^1)^{-1}(X)}gg^* & = \sum_{w\in (\pi^0)^{-1}(v)}\Big( \sum_{g\in Y_w}gg^*\Big) = \sum_{w\in (\pi^0)^{-1}(v)}w= C^*(\pi^*) (v) \, ,
\bar{\rho_{\mathrm{s}}}( \zeta ) \; \approx \; 4 \beta \left[ 1 + k^{2}\left( \ln{\frac{4}{k^{\prime}}}+{\frac{1}{2}}\right) + O ( k^{2}) \right] \left[ \zeta \ln{\frac{1 + \zeta}{1 - \zeta}}+ k^{2}\left( \ln{\frac{4}{k^{\prime}}}+{\frac{1}{2}}\right){\frac{\zeta^{2}}{1 - \zeta^{2}}}+ O ( k^{2}) \right] \; ,
\int_{\exp(58)}^x\frac{|\theta(t) - t|}{t\log^2 t}t \le \int_{\exp(58)}^{x}\frac{A_1 \log^B t\exp(-Cu(t))}{\log^2 t}t \le \int_{\exp(58)}^{x}h'(t)t \leq A_1x\log^{-\alpha}x\exp(-Cu(x)) = \log^{1 - B - \alpha}x \cdot A_1x\log^{B-1}x\exp(-Cu(x)) \le \log^{1 - B - \alpha}x_0 \cdot A_1x\log^{B-1}x\exp(-Cu(x)),
\eta_+^A=e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}\hat{\eta}_R -\hat{\eta}_L\end{array}\right)^A\ ,\qquad \eta_-^A=\epsilon(y)e^{-\frac{1}{2}k|y|}\left(\begin{array}{c}\hat{\eta}_L \hat{\eta}_R\end{array}\right)^A\ ,
\frac{d}{dt}\phi_X(t) = \frac{d}{dt}\left(\int_\mathbb{R}f_X(x)e^{\mu tx}dx\right) &= \int_\mathbb{R}f_X(x)\frac{d}{dt}\left(e^{\mu tx}\right)dx &= \int_\mathbb{R}f_X(x)e^{\mu tx}xdx\mu .
\int_{\mathcal{H}}(\mathfrak{Re}\langle e_k, T^nx \rangle)^2\,(\mathfrak{Im}\langle e_\ell, x \rangle)^2\, dm(x) =\frac{1}{2}(\mathfrak{Re}\langle RT^{*n}e_k, ie_\ell \rangle)^2+\sigma_k^2\sigma_\ell^2 &=\frac{1}{2}(\mathfrak{Im}\langle RT^{*n}e_k, e_\ell \rangle)^2+\sigma_k^2\sigma_\ell^2.
\lim\limits_{A,B\rightarrow0}\int_{x_1^{AB}(t)}^{x_2^{AB}(t)}\rho_*^{AB}dx==\lim\limits_{A,B\rightarrow0}\int_{0}^{t}\rho_*^{AB}(\sigma_2^{AB}-\sigma_1^{AB})dt=\sqrt{\rho_{+}\rho_{-}}(u_--u_+)t.
\frac{d^{2}x^{\beta}}{d\tau^{2}}+\frac{1}{2}g^{\mu\beta}(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_{\mu}g_{\alpha\nu})\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}=0
A := \left(\lambda +\frac{2a}{\lambda}\left(1-\frac{1}{n}\right)\right)\left(1-\frac{1}{n}\right)>0, B := \lambda\theta\left(2-\frac{1}{n}\right)-\varepsilon \left(1-\frac{1}{n}\right)^2+2a, C := \lambda(\theta^2-\varepsilon)<0.
\mathbb{S}^{3}_{1}(c^2)_{+}= \{(u^0,u^1,u^2,u^3)\in\mathbb{S}^{3}_1(c^2): \c(u^{0}+u^{1})>0\}; \mathbb{S}^{3}_{1}(c^2)_{0}=\{(u^0,u^1,u^2,u^3)\in\mathbb{S}^{3}_1(c^2): \u^{0}+u^{1}=0\}; \mathbb{S}^{3}_{1}(c^2)_{-}=\{(u^0,u^1,u^2,u^3)\in\mathbb{S}^{3}_1(c^2): \c(u^{0}+u^{1})<0\}.
\Lambda(f,g,h) := \textup{p.v.}\int_{(\mathbb{R}^2)^3}f(x)g(y)h(z)\delta(x+y+z)\det\begin{pmatrix}1 1 1 x y z \end{pmatrix}^{-1}dxdydz & \,= \frac{1}{3}\,\textup{p.v.}\int_{(\mathbb{R}^2)^3}\widehat{f}(\xi)\widehat{g}(\eta)\widehat{h}(\zeta)\det\begin{pmatrix}1 1 1 \xi \eta \zeta \end{pmatrix}^{-1}d\xi d\eta d\zeta.
M(\widetilde \varphi ')_* K(\widetilde \varphi ')^* = (M(\widetilde \varphi) \circ M(k_y ) )_* (c_{k_y}\circ K(\widetilde \varphi ) )^*=M(\widetilde \varphi)_* M(k_y)_* K(\widetilde \varphi )^* c_{k_y}^* &=M(\widetilde \varphi)_* K(\widetilde \varphi )^* M(k_y)_* c_{k_y}^*.
4 \partial_{+}\partial_{-}\rho + 2 \partial_{+}\psi \partial_{-}\psi + \frac{1}{2}\partial_{+}\psi_{1}\partial_{-}\psi_{1}+ 2 e^{- \psi_{1}- 2 \rho}\Omega F_{2 + -}^{2}+ 2 e^{\psi_{1}- 2 \rho}\Omega F_{+ -}^{2}= 0
\epsilon (\|y_1\|_2^2 + \|u_1\|_2^2) \leq \langle u_1, y_1 \rangle + \langle u_2, y_2 \rangle & = \langle e_1 - y_2, y_1 \rangle + \langle e_2 + y_1, y_2 \rangle & = \langle e_1, y_1 \rangle + \langle e_2, y_2 \rangle.
t_\infty(y)h =\lim_{\lambda\in \Lambda}\rho_\infty(e_\lambda)t_\infty(y)h=\lim_{\lambda\in \Lambda}t_\infty(e_\lambda \cdot y)h &=\lim_{\lambda\in \Lambda}t_\infty(\gamma_F(e_\lambda) \cdot y)h=\lim_{\lambda\in \Lambda}\rho_\infty(\gamma_F(e_\lambda)) t_\infty(y)h
[fX,Y] =[f\wedge X,Y] &=-(-1)^{(k-1)(l-1)}[Y,f\wedge X] &=-(-1)^{(k-1)(l-1)}([Y,f]\wedge X+f\wedge[Y,X]) &=-(-1)^{(k-1)(l-1)}((-1)^{(l-1)}i_fY\wedge X+f[Y,X]) &=f[X,Y]-(-1)^{(l-1)k}i_fY \wedge X &=f[X,Y]-X\wedge i_fY,