harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sqrt{\langle \hat{\Phi}^{\dagger}( p ) \hat{\Phi}( p ) \rangle \; \langle \hat{\Pi}^{\dagger}( p ) \hat{\Pi}( p ) \rangle}={\frac{1}{2 \; \mathrm{t h}{\frac{\beta E ( p )}{2}}}}\ge{\frac{1}{2}}\; ,
\sum_{\substack{x<n\le x+H \gcd(n,\Gamma M)=1}}1 \hskip 0.1cm= \hskip 0.1cmH \sum\limits_{u\mid \Gamma M}\dfrac{\mu(u)}{u}-\sum\limits_{u\mid \Gamma M}\mu(u)\left(\left\{\dfrac{x}{u}\right\}+\left\{\dfrac{x+H}{u}\right\}\right) &\hskip 0.1cm\ge \hskip 0.1cm H\dfrac{\phi(\Gamma M)}{\Gamma M}-2^{\omega(\Gamma M)+1},
T_{\pm}(e_{\pm})=\int_{0}^{P_{\beta}}\frac{dx^{\prime}}{\sqrt{2(e_{\pm}\mp\beta(x^{\prime}))}}=\frac{P_{\beta}}{\sqrt{2e_{\pm}}}+O(\frac{\varepsilon}{\left( e_{\pm}\right)^{\frac{3}{2}}}),
S \approx \left\{2\pi^2 Q^3 e^{\pi Q^2}T \left[\exp{\left({1\over 2\pi^2 Q^3 e^{\pi Q^2}T}\right)}-1\right]\right\}^{-1}-\ln{\left[1-\exp{\left(-{1\over 2\pi^2 Q^3 e^{\pi Q^2}T}\right)}\right]}.
H_{0}:= \begin{pmatrix}\mathbb{H}_{0,\perp}- bd 0 0 \mathbb{H}_{0,\perp}+ bd\end{pmatrix}=: \begin{pmatrix}\mathbb{H}_{0,\perp}^{-}0 0 \mathbb{H}_{0,\perp}^{+}\end{pmatrix}.
D_{a \alpha}\mathrm{v}^{\beta}{}_{\gamma}( \theta ) = \textstyle{\frac{2}{3}}\delta_{\alpha}{}^{\beta}D_{a \delta}\mathrm{v}^{\delta}{}_{\gamma}( \theta ) - \textstyle{\frac{1}{3}}\delta^{\beta}{}_{\gamma}D_{a \delta}\mathrm{v}^{\delta}{}_{\alpha}( \theta ) ,
\times \Bigg\{\sum_{\sigma \in Z_N}\frac{\omega^{\sigma k_2}w(x_3,x_{13},x_1|\sigma)w(x_4,x_{24},x_2|k_4+\sigma)}{w(x_4,x_{14},x_1|k_3+\sigma)w(x_3/\omega,x_{23},x_2|k_1+\sigma)}\Bigg\}_0
\begin{gathered}r_{13}r_{12}= \sum\limits_{p}s_{ij}^{kl}s_{pi}^{st}e_{pj}\otimes e_{kl}\otimes e_{st}, r_{12}r_{23}= \sum\limits_{p}s_{ij}^{kl}s_{lp}^{st}e_{ij}\otimes e_{kp}\otimes e_{st}, r_{23}r_{13}= \sum\limits_{p}s_{ij}^{kl}s_{pr}^{lt}e_{pr}\otimes e_{ij}\otimes e_{kt}.\end{gathered}
B_{k,X}=\left\{\sum_{i=T_k}^{T_{k+1}-1}\max_{s\in [0,1]}|X_i(s)| >\frac{k^\alpha}{3}\right\}, \ \B_{k,Y}=\left\{\sum_{i=T_k}^{T_{k+1}-1}\max_{s\in [0,1]}|Y_i(s)| >\frac{k^\alpha}{3}\right\}.
H_n'(t) = \Bigg(\frac{(n-1)^3 (n^2 +6n -12)}{n^2}+ \frac{(n-2)(2n^2 -7n +7)}{\sinh(t)^2}+ \frac{(n-2)^2(n-4)}{\sinh(t)^4}\Bigg) &\qquad\times J_n(t)\, \sinh(t)^{n-1},
\frac{m}{2 J}s \mathrm{T r}_{N}[{\hat{\bar{\Psi}}}{\hat{\Psi}}] \ + \ \frac{1}{2 J}s \mathrm{T r}_{N}[ \mu ({\hat{\bar{\Psi}}}{\hat{\Psi}})^{2}+ \nu ({\hat{\bar{\Psi}}}\Gamma_{0}{\hat{\Psi}})^{2}] \ .
\bar{\Phi}_{A}( \theta ) = \bar{\phi}_{A}- \theta^{a}\phi_{A a}^{*}- \theta^{2}\eta_{A}, \qquad \frac{\delta}{\delta \bar{\Phi}_{A}( \theta )}= \theta^{2}\frac{\delta}{\delta \bar{\phi}_{A}}+ \frac{\delta}{\delta \phi_{A a}^{*}}\theta_{a}- \frac{\delta}{\delta \eta_{A}}.
\int_{\{z\in M:-\Psi(z)\in S_{i,n}\backslash U_k\}}|F_{t_0,\tilde{c}}|^2_hc(-\Psi) = \int_{\{z\in M:-\Psi(z)\in S_{i,n}\backslash U_k\}}|\tilde F|^2_hc(-\Psi)+\int_{\{z\in M:-\Psi(z)\in S_{i,n}\backslash U_k\}}|F_{t_0,\tilde{c}}-\tilde F|^2_hc(-\Psi).
\frac{\partial l}{\partial\sigma_{B}^{(k)^{2}}}=\sum_{i=1}^{m}\frac{\partial l}{\partial y_{i}^{(k)}}(x_{i}^{(k)}-\mu_{B}^{(k)})(-\frac{\gamma^{(k)}}{2}(\sigma_{B}^{(k)^{2}}+\epsilon)^{-3/2})
\langle [ A_{2}]_{N}({\bf x}) \rangle = \frac{1}{\sqrt{2 k \omega}}\left\{\frac{1}{2}( 1 - \frac{m}{\omega}) J_{N + 1}( k r ) e^{i ( N + 1 ) \varphi}- \frac{1}{2}( 1 + \frac{m}{\omega}) J_{N - 1}( k r ) e^{i ( N - 1 ) \varphi}\right\}.
e^{\int_{\tau_{\ell+1}}^{\tau_{j+1}}G_{\ell}(v) \: dv}\leq \prod_{h=\ell+1}^j e^{\int_{\tau_h}^{\tau_{h+1}}G_h(v) \: dv}\leq \bigg(\frac{2s}{\mu}\bigg)^{j-\ell}.
\sum_{\tiny\begin{matrix}\beta+\gamma =\alpha, b+c=a |\gamma|+|c|\leq[(\alpha+|a|)/2]\end{matrix}}\|\nabla U^{(\beta,b)}\|^2_{L^2}\| \langle t \rangle\nabla U^{(\gamma,c)}\|^2_{L^\infty(r\leq \langle t \rangle/2)}&\lesssim E_{|\alpha|+|a|+1}X_{[(|\alpha|+|a|)/2]+3}.
W_{n}(T,x)= 2\pi\lambda\sum_{k=1}^{4}b_k \int_{\psi_\mathrm{N}(x)}^{\infty}F\left(N_\mathrm{L},\frac{nC_\mathrm{L}\eta_\mathrm{N}\bar{a}_kTx^{\alpha_\mathrm{N}}}{C_\mathrm{N}N_\mathrm{L}t^{\alpha_\mathrm{L}}}\right) &p(t)t\mathrm{d}t,
4 A r^{3}\left( \sqrt{1 - r_{H}^{4}/ r^{4}}- 1 \right) \ + \ \frac{2 A r_{H}^{4}}{r \sqrt{1 - r_{H}^{4}/ r^{4}}}\ + \ 2 ( r^{2}- r r_{0}) ( 2 r - r_{0}) ,
Z ( \beta , \Omega ) = | Z ( \tau ) |^{2}= \exp \left( \frac{\pi^{2}l^{2}}{2 \hbar^{2}G \beta ( 1 + l^{2}\Omega^{2})}+ \frac{\pi^{2}l}{\hbar \beta ( 1 + l^{2}\Omega^{2})}\right) .
{\frac{g ( \tau , \vec{\sigma})}{\gamma ( \tau , \vec{\sigma})}}= g_{\tau \tau}( \tau , \vec{\sigma}) - \gamma^{{\check{r}}{\check{s}}}( \tau , \vec{\sigma}) g_{\tau{\check{r}}}( \tau , \vec{\sigma}) g_{\tau{\check{s}}}( \tau , \vec{\sigma}) .
\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}
U_p^{(\kappa)}(u) :=U(u)\cap\bigcap_{i=1}^\kappa\left\{w_{U(u)}^i\leq p\right\}U_p^{(\infty)}(u):=U(u)\cap\bigcap_{i=1}^\infty\left\{w_{U(u)}^i\leq p\right\}=\bigcap_{\kappa=0}^\infty U_p^{(\kappa)}(u) Q_{p,i}^\kappa(u) :=f^{-i}\left(U_p^{(\kappa)}(u)\setminus U_p^{(\kappa+1)}(u)\right)=f^{-i}\left(U_p^{(\kappa)}(u)\cap\{w_{U(u)}^{\kappa+1}>p\}\right)
0\leq \frac{\sqrt{\pi t}}{2}- \int_0^\delta e^{-s^2/t}\, ds = \int_\delta^\infty e^{-s^2/t}\, dt = \frac{\sqrt{t}}{2}\int_{\delta^2/t}^\infty \frac{e^{-y}}{\sqrt{y}}\, dy \leq \frac{t}{2\delta}e^{-\delta^2/t}\, .
\exp{\left\{- i ( \xi F_{1}+ \eta F_{2}) \right\}}= \left( \begin{matrix}{1}&{0}&{0}&{\xi}\\{0}&{1}&{0}&{\eta}\\{\xi}&{\eta}&{1}&{( \xi^{2}+ \eta^{2}) / 2}\\{0}&{0}&{0}&{1}\\ \end{matrix}\right) .
\left(I_{-}^{\alpha,\beta,\gamma}f\right)(x)=\frac{1}{\Gamma(\alpha)}\int_x^{\infty}(t-x)^{\alpha-1}t^{-\alpha-\beta}{}_{2}F_1\left(\alpha+\beta,-\gamma;\alpha;1-\frac{x}{t}\right)f(t)\,dt,
g_{1}= \frac{2 \phi}{R}\eta_{\mu \nu}\Big ( ( \tilde{E}^{\mu}- E^{\mu}) \otimes E^{\nu}+ E^{\mu}\otimes ( \tilde{E}^{\nu}- E^{\nu}) \Big ) + \frac{R^{2}}{4 \phi^{2}}\eta_{i j}\tilde{E}^{i}\otimes \tilde{E}^{j}.
\int [ d \delta \phi ]_{e^{\phi}{\hat{g}}}e^{- S_{L}[ \phi ,{\hat{g}}] + S_{\mathrm{b o u n d a r y}}[ \phi ,{\hat{e}}]}= \int [ d \delta \phi ]_{{\hat{g}}}^{\prime}\int_{- \infty}^{\infty}d \phi_{0}e^{- S [ \phi_{R},{\hat{g}}]}\quad ,
\mathrm{I m}{\cal N}_{\Lambda \Sigma , \Gamma \Delta}f_{A B}^{\Gamma \Delta}= -{\frac{1}{2 \sqrt{2}}}\left( e^{- \Phi_{+}}\tau^{\left( + \right) \Lambda \Sigma}\tau^{\left( + \right) A B}+ e^{- \Phi_{-}}\tau^{\left( - \right) \Lambda \Sigma}\tau^{\left( - \right) A B}\right) .
\times \prod^{r_1 + l_2 -1}_{j= r_1}d_j (\lambda_1) \prod^{r_1+l_2-1}_{j=l_1 +l_2}d_j (\lambda_1) \prod^{r_2- 1}_{j=r_2 - l_2}d_j (\lambda_2) \prod^{r_1+r_2-l-1}_{j=r_2 -l_2}d_j(\lambda_2)
\begin{aligned}\frac{d}{dt}\int |\nabla^2 B|^2dx+\frac{4}{1+t}\int |\nabla^2 B|^2 dx &\le \frac{16}{(1+t)^2}\int |\nabla B|^2 dx+C(1+t)^{-5}&\lesssim (1+t)^{-2}(1+t)^{-\frac{5}{2}}+(1+t)^{-5}&\le C(1+t)^{-\frac{9}{2}},\end{aligned}
\left(\begin{array}{ccc}1 0 0 \lambda 1+\lambda 0 0 0 1+\lambda\end{array}\right)\left(\begin{array}{ccc}1 0 0 \mu 1+\mu 0 0 0 1+\mu\end{array}\right)= \left(\begin{array}{ccc}1 0 0 (1+\lambda)(1+\mu)-1 (1+\lambda)(1+\mu) 0 0 0 (1+\lambda)(1+\mu)\end{array}\right)
\left\{a \, \middle|\, 1 \leq a \leq l(\lambda) + 1,\j(\lambda)_a = i, \begin{array}{l}\bullet\, m(\lambda)_{a-1}< m(\lambda)_{a}; \bullet\, m(\lambda)_{a-1}= m(\lambda)_{a}j(\lambda)_{a-1}> i+1\end{array}\right\}
L_{0}^{\beta}\left( N\right) =\frac{1}{2}\beta_{0}^{2}+\sum_{e\geq2}^{2N}\beta_{-e}^{x}\cdot \beta_{e}^{x}+\sum_{o\geq 1}^{2N-1}\beta_{-o}^{p}\cdot \beta_{o}^{p}.
\frac{\partial^{2}{Z_{0}}_{l}( r )}{\partial r_{*}^{2}}- V ( r ){Z_{0}}_{l}( r ) = - \frac{4 \pi q_{s}m_{0}f ( r ) Y_{l}( 0 )}{r^{( d - 2 ) / 2}}( \frac{d t}{d \tau})_{r_{0}}^{- 1}\delta ( r - r_{0}) ,
\lim_{n\to \infty}n \vec{c}^T\mathsf{MSE}_{\vec{\theta}}[\frac{\vec{g}^n(X^{n+1})}{n}]\vec{c}= \lim_{n\to \infty}n \vec{c}^T\mathsf{Cov}_{\vec{\theta}}[\frac{\vec{g}^n(X^{n+1})}{n}] \vec{c}= \vec{c}^T\mathsf{H}_{\vec{\theta}}[\phi]\vec{c},
| I + t A |_{i i}= \mathrm{d e t}( I + t A ) \left( \mathrm{d e t}( I + t A^{i i}) \right)^{- 1}= \ \sum_{k = 0}^{+ \infty}\, \Lambda_{k}({\bf \alpha}-{\bf \alpha}^{i}) \, t^{k}
\varepsilon_{\mu_1\ldots \mu_d}=e^{-1}e_{\mu_1}^{a_1}\ldots e_{\mu_d}^{a_d}\varepsilon_{a_1\ldots a_d}\,,\qquad\varepsilon^{\mu_1\ldots \mu_d}=e e^{\mu_1}_{a_1}\ldots e^{\mu_d}_{a_d}\varepsilon^{a_1\ldots a_d}\,.
V = \frac{1}{8}\int_{a}^{R}s^{3}\, d s \, d \psi \wedge \sin \theta \, d \theta \wedge d \phi = \pi^{2}\int_{a}^{R}s^{3}\, d s = \frac{\pi^{2}}{4}\left( R^{4}- a^{4}\right) ,
\frac{\partial \chi^{\varepsilon}}{\partial t}+\displaystyle\sum_{j=1}^{d}f_{j}^{\prime}(c)\,\frac{\partial\chi^{\varepsilon}}{\partial x_{j}}=\left(-\Delta_{x,t}+1\right)^{\frac{1}{2}}\left(-\Delta_{c}+1\right)^{\frac{r}{2}}g^{\varepsilon}\,\,\mbox{in}\,\,\mathcal{D}^{\prime}\left(\Omega\times\mathbb{R}\times (0,T)\right).
A_{i}(u) = T_{(1, \ldots, i),(1, \ldots, i)}(u), B_{i}(u) = T_{(1, \ldots, i),(1, \ldots, i-1, i+1)}(u), C_{i}(u) = T_{(1, \ldots, i-1, i+1),(1, \ldots, i)}(u).
\int_{\{\Psi<-t_1\}}|F_{\delta}-(1-b_{t_0,B}(\Psi))f|^2e^{v_{t_0,B}(\Psi)-\Psi}\le \left(\frac{1}{\delta}e^{-t_1}+e^{-t_1}-e^{-t_0-B}\right)\int_D\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|f|^2e^{-\Psi}.
\Phi(g, \pi) - \left( \chi \Phi(g_1,\pi_1) + (1-\chi) \Phi(g_2,\pi_2) \right) &=\chi D\Phi|_{(g_1, \pi_1)}((1-\chi)(g_2 - g_1, \pi_2-\pi_1)) + \chi Q_{(g_1, \pi_1)}((1-\chi)(g_2 - g_1, \pi_2-\pi_1)) &\quad+(1- \chi) D\Phi|_{(g_2, \pi_2)}(\chi(g_1 - g_2, \pi_1-\pi_2)) + (1-\chi) Q_{(g_2, \pi_2)}(\chi(g_1 - g_2, \pi_1-\pi_2)).
P_{\gamma}=\lambda^{\gamma}\sum_{\hat{\alpha}\in \hat{\mathcal{A}}}b_{\gamma \hat{\alpha}}\delta^{\left\vert \gamma \right\vert -\left\vert \hat{\alpha}\right\vert}\left[ S_{\hat{\alpha}}\odot_{0}\bar{P}_{\chi\left( \hat{\alpha}\right)}\right] \gamma \in \hat{\mathcal{A}}
\overline{Q}_{i}^{a,b}(t) = (b-t)\det\begin{bmatrix}bs_{i,-1}-1 1 b-\lambda_{i,2}^{2}t\end{bmatrix}, i\in\mathbb{N}\cap[1,\eta],\a,b\in(0,\infty),\a<b \overline{P}_{i}^{a,b}(0) = \sigma\left(\frac{\overline{Q}_{i}^{a,b}(t)-\overline{Q}_{i}^{a,b}(0)}{t}\right), i\in\mathbb{N}\cap[1,\eta],\a,b\in(0,\infty),\a<b.
\zeta_{\bar{\cal V}}( s ) = \frac{m L}{\sqrt{8 \pi}}\frac{1}{\Gamma ( s )}\int_{0}^{\infty}d \beta \beta^{s - \frac{3}{2}}e^{- 4 \beta}= \frac{m L}{\sqrt{8 \pi}}\frac{1}{4^{s - \frac{1}{2}}}\frac{\Gamma ( s - \frac{1}{2})}{\Gamma ( s )}
\omega_{\mathbb{A}}\left[\begin{pmatrix}0 T S 0\end{pmatrix}\right]\geq\frac{1}{2}\sqrt{\left\|\big(S^{\sharp_A}(S^{\sharp_A})^{\sharp_A}+ (T^{\sharp_A})^{\sharp_A}T^{\sharp_A}\big)\right\|_A+ 2c_A(TS)}=\Phi(T,S),
\gamma(s,\pi,r_0 \boxtimes \eta,\psi) = \gamma(s,(r_0 \circ \sigma) \otimes \eta,\psi), L(s,\pi,r_0 \boxtimes \eta) = L(s,(r_0 \circ \sigma) \otimes \eta) \varepsilon(s,\pi,r_0 \boxtimes \eta,\psi) = \varepsilon(s,(r_0 \circ \sigma) \otimes \eta,\psi).
Z^{(2)}(J,B) = \{\det \tilde{K}_{ik})\}^{-1}\exp \left\{-\frac{ig^2}{2}\int d^4 x (L_i (J) \tilde{K}^{-1}_{ik}L_k(J) +J_{+}(K_{++})^{-1}J_{+}) \right\}
\begin{gathered}a_{ij}^\varepsilon(x,t) = a_{ji}^\varepsilon(x,t) \quad(x,t)\in Q_{\varepsilon,T}, \, i,j=1,\dots,n, \sum_{i,j=1}^na_{ij}^\varepsilon(x,t)\xi_i\xi_j \geq 0 \quad(x,t)\in Q_{\varepsilon,T}, \, \xi=(\xi_1,\dots,\xi_n)^T\in\mathbb{R}^n \end{gathered}
e^{(N+1)t}R_{N+1}v_N(t) &=R_{N+1}v_N(0)-\sum_{m+j=N+1}\int_0^tR_{N+1}B(q_m(\tau), q_j(\tau))\d\tau+\int_0^te^{(N+1)\tau}h_N(\tau)\d\tau.
{\rm Hilb}\big(3T_5^{0}(3,2)^{ab},t\big)=(1,6,15,17,7) \hphantom{{\rm Hilb}\big(3T_5^{0}(3,2)^{ab},t\big)}{}=t^3 (1+t)+3 t^2~(1+t)^2+2t (1+t)^3+(1+t)^4,
\sum_{p\in V_1}\frac{f(p)}{c_p\prod_{e\in E^p}\alpha(e)}=\frac{1}{2}\sum_{\substack{pq\in E_\Gamma \alpha(pq)=\lambda\alpha_i}}\left(\frac{f(p)}{c_p\prod_{e\in E^p}\alpha(e)}+\frac{f(q)}{c_q\prod_{e\in E^q}\alpha(e)}\right).
\begin{aligned}X_{11}X_{12}- qX_{12}X_{11}, X_{12}X_{22}- qX_{22}X_{12}, X_{11}X_{21}- qX_{21}X_{11}, X_{21}X_{22}- qX_{22}X_{21}, X_{12}X_{21}-X_{21}X_{12}, X_{11}X_{22}-X_{22}X_{11}-(q-q^{-1})X_{12}X_{21};\end{aligned}
\mu_\Gamma(\gamma)= \sum_{\substack{e_0 \in{\rm Edges}^\pm(\Gamma) \smallsetminus \{\bar e_1\}\tau(e_0) = \tau(\bar \gamma)}}\mu_\Gamma(e_0 \gamma) = \sum_{\substack{e_{q+1}\in{\rm Edges}^\pm(\Gamma) \smallsetminus \{\bar e_q\}\tau(\gamma) = \tau(\bar e_{q+1})}}\mu_\Gamma(\gamma e_{q+1})
\langle \varphi h e_i , \varphi h e_j \rangle = (c \varphi h). \langle \varphi h e_i , \varphi h e_j \rangle_0 = \langle (c \varphi h)^{-1}\varphi h e_i , (c \varphi h)^{-1}\varphi h e_j \rangle_0 = c^{-2}\delta_{i j}.
\int_{G_d}\Phi^{(j)}(P \cap gQ,A\cap gB) \mu (dg) &=\Phi^{(j)}(P,A) \lambda_Q(B) + \sum_{m=j+1}^{d}\Phi^{(j)}_m(P,A) \sum_{G \in{\cal F}_{d+j-m}(Q)}\sigma^{(F^\bot)}(n(Q,G)) \lambda_G(B)\cr = \Phi^{(j)}(P,A) \lambda_Q(B) + \sum_{m=j+1}^{d}\Phi^{(j)}_m(P,A) \Phi_{d+j-m}(Q,B)
\det(T_N)(\tau)=\det\left(\operatorname{diag}\left(\frac jNq^{\frac{j^2}{2N}}\right)_{1\leq j\leq\frac N2-1}\left(\left(\frac{m^2}{2N}\right)^{k-1}+O(q)\right)_{1\leq m,k\leq\frac N2-1}\right).
\psi_{1 a}( x^{\mu}) = \left( \begin{matrix}{\nu_{e}}\\{e^{-}}\\{u}\\{d}\\ \end{matrix}\right) , \psi_{2 a}( x^{\mu}) = \left( \begin{matrix}{\nu_{\mu}}\\{\mu^{-}}\\{c}\\{s}\\ \end{matrix}\right) ,
r\begin{pmatrix}B_{1}D_{1}0 C_{1}^{\eta^{\ast}}\end{pmatrix}=r\left(B_{1}\right)+r\left(C_{1}\right), & r\begin{pmatrix}B_{1}C_{1}D_{1}\end{pmatrix}=r\begin{pmatrix}B_{1}C_{1}\end{pmatrix}.
\check{F}_{l,x}+2u\check{G}_{l+1}=2w\check{F}_{l+1}, \check{H}_{l,x}-2v\check{G}_{l+1}=-2w\check{H}_{l+1}, \check{G}_{l,x}=u\check{H}_{l+1}-v\check{F}_{l+1},
\begin{array}{@{}l@{}}{\displaystyle \mathcal{E}(t) = \int_{\Omega}\big(\frac{1}{2}\big\vert \left(i\nabla + q\mathbf{A}\right)\Psi(t,\mathbf{x})\big\vert^{2}+ V|\Psi(t,\mathbf{x})|^{2}}begin{align*}2mm]{\displaystyle + \frac{\epsilon}{2}|\mathbf{E}(t,\mathbf{x})|^{2}+ \frac{1}{2\mu}|\mathbf{B}(t,\mathbf{x})|^{2}\big)\mathrm{d}\mathbf{x}.}\end{array}
S_{2}: = \sum_{n \ge 0}\frac{2 n + 1}{( 3 n + 1 )^{2}( 3 n + 2 )^{2}}= \frac{2}{3^{3 / 2}}\mathrm{C l}_{2}( 2 \pi / 3 ) = \frac{4}{3^{5 / 2}}\mathrm{C l}_{2}( \pi / 3 )
\psi_{\kappa}(\theta, x) \ = \ \inf_{v_2 \in{\mathcal M}_2}E^{\hat v_1, v_2}_x \Big[e^{\frac{\kappa \|r_2\|_{\infty}}{\alpha}}e^{\theta \int^{T_\kappa}_0 e^{-\alpha t}r_2(X(t), \hat v_1(\theta e^{-\alpha t},X(t)), v_2(t, X(t)))dt}\Big],
\bar{\partial}_{\bar{A}}F_{g}= \bar{C}_{\bar{A}\bar{B}\bar{C}}e^{2 K}G^{B \bar{B}}G^{C \bar{C}}\left( D_{B}D_{C}F_{g - 1}+ \frac{1}{2}\sum_{r}D_{B}F_{r}\cdot D_{C}F_{g - r}\right) ,
L^{i n t}= - e ( \overline{\psi}_{1}\gamma^{\mu}\gamma^{5}\psi_{1}- \overline{\psi}_{2}\gamma^{\mu}\gamma^{5}\psi_{2}) C_{\mu}+ i e ( \overline{\psi}_{1}\gamma^{\mu}\psi_{1}+ \overline{\psi}_{2}\gamma^{\mu}\psi_{2}) B_{\mu}
C^{ML}_{( i j )( k l )}\! =\!C^{ML}_{( ji )( kl )}\! = \!C^{ML}_{( ij )( lk )}\! = \!C^{ML}_{( ji )( lk )}\! = \!C^{ML}_{( kl )( ij )}\! = \!C^{ML}_{( lk )( ij )}\! = \!C^{ML}_{( kl )( ji )}\! = \!C^{ML}_{( lk )( ji )}\,.
\begin{array}{cc}\mu_{X\otimes Y}= (\mu_X \otimes \mu_Y)\circ(1_X \otimes \sigma_{YX}\otimes 1_Y) \qquad \eta_{X\otimes Y}= \eta_X \otimes \eta_Y \delta_{X\otimes Y}= (1_X \otimes \sigma_{XY}\otimes 1_Y)\circ(\delta_X \otimes \delta_Y) \qquad \epsilon_{X\otimes Y}= \epsilon_X \otimes \epsilon_Y. \end{array}
I_{G}B = (1-\rho)B \subseteq \bigl((1+\sigma)\rho\bigr)B+ (1+\sigma\rho)B=\bigl(1+\sigma\bigr)B+ (1+\sigma\rho)B=N_\Sigma B+N_{\Sigma'}B\subseteq B^{\Sigma}+B^{\Sigma'}.
\mathcal{H}_{i+1}(\vec{r}+\vec{e_i},h_1) - \mathcal{H}_{i+1}(\vec{r},h_1) = \big( r_i+1 \big) h_i + \frac{(r_i+1)r_i}{2}- \Big( r_i h_i + \frac{r_i(r_i-1)}{2}\Big) = h_i + r_i < \Big( \frac{1}{r_i}+ \frac{1}{h_i}\Big) h_{i+1}
X_{\alpha}=\left[\begin{array}{ccc}0 1 0 0 0 0 0 0 0 \end{array}\right], X_{\beta}=\left[\begin{array}{ccc}0 0 0 0 0 1 0 0 0 \end{array}\right]\quad\mbox{and}X_{\alpha+\beta}=\left[\begin{array}{ccc}0 0 1 0 0 0 0 0 0 \end{array}\right].
({\widehat \varphi}^{A B})_{~ v}^{u}= f ( x , \rho ) \, \Big ( \mu^{[ A u}\bar{\mu}_{~ \, v}^{B ]}- \frac{1}{2}\, \mu^{[ A p}\, \bar{\mu}_{~ \, p}^{B ]}\, \tilde{\delta}_{~ v}^{u}\Big )
\partial^{\mu}( \sqrt{- g}R_{\mu}) = \frac{1}{1 9 2 \pi^{2}}\left[ ( N_{c}^{2}- 1 ) - \frac{1}{3}\sum_{i}( - 1 + c_{i}^{( 0 )}) \right] \epsilon_{\mu \nu \lambda \delta}R^{\mu \nu \sigma \rho}R_{\sigma \rho}^{\lambda \delta},
\gamma^0=\left(\begin{array}{lr}0 \hspace*{2ex}1 1 0 \end{array}\right)\; , \;\;\;\;\; \gamma^1=\left(\begin{array}{lr}0 -1 1 0 \end{array}\right)\; , \;\;\;\;\; \gamma^5=\gamma^0\gamma^1=\left( \begin{array}{lr}1 0 0 -1 \end{array}\right) \; ,
C=i\gamma^{2}\gamma^{0}=(\begin{matrix}-i\sigma^{2}&0\\ 0&i\sigma^{2}\end{matrix})=(\begin{matrix}0&-1&0&0\\ 1&0&0&0\\ 0&0&0&1\\ 0&0&-1&0\\ \end{matrix})=-i\sigma^{3}\otimes\sigma^{2}.
2\ell\left( s,a,\chi\right) =E_{0,\overline{\chi}}\left( 0\right)a^{-s}-sE_{1,\overline{\chi}}\left( 0\right) a^{-s-1}+s\left( s+1\right)\chi\left( -1\right) \int\limits_{0}^{\infty}\frac{\overline{E}_{1,\overline{\chi}}\left( t\right)}{\left( t+a\right)^{2+s}}dt.
\widetilde{F}_{\eta,\widetilde{m}}(v_g,\omega): =W(\omega^{-1}P_{\widetilde{m}}v_g,1)F_{\eta}(v_g,\omega)W(\omega^{-1}P_{\widetilde{m}}v_g,1)^*+\lVert \omega^{-1/2}P_{\widetilde{m}}v_g \lVert^2 &=\eta W(2\omega^{-1}P_{\widetilde{m}}v_g,-1)+d\Gamma(\omega)+\varphi(\overline{P}_{\widetilde{m}}v_g)
\nabla \mathrm{H}(\mathrm{x}) - \alpha \ \nabla\int_\Omega\nabla\mathbb{G}^{(0)}(\mathrm{x},\mathrm{y})\cdot\nabla \mathrm{H}(\mathrm{y}) d\mathrm{y}= \nabla\mathrm{H}^{\textbf{in}}(\mathrm{x}) + \alpha \mathbb{T}\Big[\nabla\mathrm{H}\Big].
{\rm grad}\, \tilde{J}(A_r,B_r) = (-2A_r{\rm sym}(P^2 -X^T X) A_r, &\quad\quad\quad\quad\quad4PB_r-4X^TB ), &{\rm Hess}\, \tilde{J}(A_r,B_r) [ (A'_r,B'_r)] = ( -2A_r{\rm sym}( P'P+PP'-X'^TX-X^TX') A_r &\, -2{\rm sym}(A'_r{\rm sym}(P^2-X^TX) A_r), &\,\, 4(P'B_r+PB'_r) -4 X'^TB ),
\int_{0}^{\infty}\left(\frac{\cos(2Q_{1}(R))-1}{R^{2}\lambda(t)^{2}}\right)v_{3}(t,R\lambda(t)) \phi_{0}(R) R dR &= \frac{16}{\lambda(t)}\int_{t}^{\infty}K_{3}(s-t,\lambda(t)) \lambda''(s) ds + \int_{0}^{\infty}\left(\frac{\cos(2Q_{1}(R))-1}{R^{2}\lambda(t)^{2}}\right)E_{5}(t,R\lambda(t)) \phi_{0}(R) R dR
\partial_{\bar{\nu}_{\bar{\xi}_i}}u^{(i)}_{p\alpha +k}= \big( w^{(i)}_{p\alpha +k -1}+ u^{(i)}_{p\alpha +k -1}\big) \, \overline{V}^{(i)}\boldsymbol{\boldsymbol{\cdot}}\bar{\nu}_{\bar{\xi}_i}\, - \, \delta_{p\alpha +k -1, \, \alpha -1}\, \varphi^{(i)}\big(\bar{\xi}_i, x_i, t\big), \bar{\xi}_i \in \partial \Upsilon_i,
\begin{array}{l}\int\limits_{\mathcal{P}(I)\times E}T_s^*F d\nu^*=\lim\limits_{t\rightarrow \infty}\int\limits_{\mathcal{P}(I)\times E}T^*_{t+s}Fd\tilde{m}_0=\int\limits_{\mathcal{P}(I)\times E}Fd\nu^*.\end{array}
W(p, q) = W_4(p, q; 0) = \left( \begin{array}{cccc}1+ \frac{p^2 + q^2}{2}p q -\frac{p^2 + q^2}{2}p 1 0 -p q 0 1 -q \frac{p^2 + q^2}{2}p q 1 -\frac{p^2 + q^2}{2}\end{array}\right)
C(z; \tau) = \prod_{n \geq 1}\frac{(1-q^n)^2 (1-\zeta^{\pm 2}q^n)(1-\zeta^{\pm 3}q^n)}{(1-q^n)(1-\zeta^{\pm 1}q^n)(1-\zeta^{\pm 2}q^n)(1-\zeta^{\pm 3}q^n)}&= \prod_{n \geq 1}\frac{1-q^n}{(1-\zeta^{1}q^n)(1-\zeta^{-1}q^n)}.
d(\tau)-d(\nu)+d((\eta\alpha)(e_i,d(\eta\alpha)))-d(\tau\beta) =d(\eta)+d(\alpha)-e_i-d(\nu)-d(\beta) &=d(\eta)+d(\rho)\vee d(\lambda\nu)-d(\rho)-e_i &\qquad-d(\nu)-d(\rho)\vee d(\lambda\nu)+ d(\lambda\nu) &=d(\eta)-d(\rho)-e_i+ d(\lambda) &=d(\eta)-d(\rho).
\begin{array}{rcl}\Psi(t_1) = \sqrt[4]{\Delta}X_1+\frac{\sqrt{\gamma}}{\sqrt[4]{\Delta}}X_3+\frac{\sqrt{\alpha}}{\sqrt[4]{\Delta}}X_4, \Psi(t_2) = \sqrt{\alpha}X_1+X_3, \Psi(t_3) = \sqrt{\gamma}X_1+X_4.\end{array}
F_N(a,b;t)=\frac{(1-b)(1-tq^N)}{(1-t)(1-bq^N)}\sum_{n=0}^{N}\begin{bmatrix}N n\end{bmatrix}\frac{(atq/b)_n(q)_n(b)_{N-n}b^n}{(tq)_n(b)_N}.
W = \prod_{a = 1}^{p - 1}\left( \frac{n^{\frac{1}{l}}}{L_{a}}\right) \int_{0}^{\infty}\frac{d s}{s}\left( \frac{\pi}{2 s}\right)^{\frac{p - 1}{2}}e^{- r^{2}s}\frac{4 \sinh^{2}( \omega_{l}s \sin{\frac{\phi}{2}}) - \sinh^{2}( 2 \omega_{l}s \sin{\frac{\phi}{2}})}{\cos{\frac{\phi}{2}}~ \sinh ( 2 \omega_{l}s \sin{\frac{\phi}{2}})}\, .
(-1)^{2j+m_{3}}\sqrt{\frac{(2j+1)(2l_{3}+1)}{(2l_{1}+1)(2l_{2}+1)}}\left[\begin{array}[c]{ccc}l_{1}l_{2}l_{3}m_{1}m_{2}-m_{3}\end{array}\right] \!\!{[j]}& =C_{0,0,0}^{l_{1},l_{2},l_{3}}C_{m_{1},m_{2},m_{3}}^{l_{1},l_{2},l_{3}}+\frac{1}{n+1}C_{m_{1},m_{2},m_{3}}^{l_{1},l_{2},l_{3}}P({l_{1},l_{2},l_{3}})+O((n+1)^{-2})
\left|\psi(y, s)+e^{- s\lambda_l}\phi_l(y)\right| \le \left|(e^{-A(s- s_0)}\widetilde\phi_{l})(y)\right| +\int_{s_0}^{s}\left |\left(e^{-A(s-\tau)}F(\psi(\tau))\right)(y)\right |d\tau =S_1+S_2.
\left[ \Gamma_{01}f,\left( A_{1}-z \right)^{-1}\Gamma_{01}h \right]+\left[ \Gamma_{02}f,\left( A_{2}-z \right)^{-1}\Gamma_{02}h \right]=0,\thinspace \forall z\in \rho \left( A \right),\thinspace \forall h\in H
\frac{1}{\left(1-a_{j}z\right)\cdots\left(1-a_{i}z\right)}=\sum_{t=j}^{i}\frac{1}{1-a_{t}z}\left(\prod_{\genfrac{}{}{0pt}{}{k=j}{k\neq t}}^{i}\frac{a_{t}}{a_{t}-a_{l}}\right).
\lim_{c\to +\infty}e^{-c |J|}V^{}_{\varepsilon J}(\xi ;g^{(c)}) =V^{}_{\varepsilon J}(\xi ;g), \lim_{c\to +\infty}e^{-cp}U^{}_{K,p}(\xi ;g^{(c)}) = U^{}_{K,p}(\xi ;g) , \lim_{c\to +\infty}e^{-c\ell}E^{}_\ell (x+c\rho_L) = E^{}_\ell (x) ,
\hat{P}(C_{1}= C_{2}) =\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\! \hat{f}_{XYZS}(x,y,z,s) \, \mathrm{d}u_{1}\mathrm{d}u_{2}\Big|_{x=y=u_{1}, z=s=u_{2}}= \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\! \hat{f}_{XYZS}(u_{1},u_{1},u_{2},u_{2}) \, \mathrm{d}u_{1}\mathrm{d}u_{2}
E_-(s,t)=|A_-'(n^{2/3}\zeta)|n^{-s}+|A_-(n^{2/3}\zeta)|n^{-t}Ei(s,t)=|Ai'(n^{2/3}\zeta)|n^{-s}+|Ai(n^{-2/3}\zeta)|n^{-t}
\ddot{\rho}\left( r + \frac{\rho^2}{r}h_0''(d) \right) = \frac{1}{r}\left( \rho + \frac{\omega^2}{\rho^3}\right) - \dot{\rho}+ \frac{\rho}{r}\left( d - \dot{\rho}^2 \right)h_0''(d).
\langle \hat{E}_{0}| \hat{W}_{0}( f ) | \hat{E}_{0}\rangle = e^{- i \pi q ( \widetilde{f_{+}}( 0 ) - \widetilde{f_{-}}( 0 ) )}\, \langle \hat{V}_{0}| \hat{W}_{0}( f ) | \hat{V}_{0}\rangle .
\mathcal{F}_{_{v_{1},...,v_{m},m,\mu}}(z)=\Bigg[(m\ \mu+1)\int_{0}^{z}\prod_{j=1}^{m}\big(D_{\lambda}^{n,\gamma}\mathcal{J}_{\nu_{j}}(t)\big)^{\mu}dt\Bigg]^{1/(m\ \mu +1)},
\Psi_{0}^{( 1 )}= N e^{- \pi x_{4}^{2}/ V}\theta_{2}( \tau | \bar{z}) \quad ; \quad \Psi_{0}^{( 2 )}= N e^{- \pi x_{4}^{2}/ V}\theta_{3}( \tau | \bar{z}) \quad ; \quad \tau \equiv 2 i{\frac{R_{4}}{R_{5}}}\ ,
index (M_k \cap B^{(N,g)}_{R r^{y,\ell}_k}(p^{y,\ell}_k))= index (\tilde{M}^{y,\ell}_k \cap B^{(\tilde{N},\tilde{g})}_R(0)) \geq index(\Sigma^y_\ell\cap B_R(0)) = index(\Sigma^y_\ell)
\begin{array}{lccllc}\mathcal{B}= \{\mathcal{B}^0 = \{210201202120102012, 210201021202102012\}, \mathcal{B}^1 = \{021012010201210120, 021012102010210120\}, \mathcal{B}^2 = \{102120121012021201, 102120210121021201\}\}.\end{array}
L^\bullet_{YM}= \mathrm{Tr}\left[ \frac12 F_A \star F_A + A^\dag d_Ac + \frac12c^\dag[c,c] + cd_A\star F_A + \frac12 A^\dag[c,c] + \frac12[c,c] \star F_A\right] \theta^\bullet_{YM}= \mathrm{Tr}\left[A^\dag \delta A + c^\dag \delta c + \delta A \star F_A + A^\dag \delta c + c \delta(\star F_A)\right],
\Vert \Gamma u\Vert_{L^{r}(\mathbb{R};M^s_{p,q})}\leq \Vert S(t)u_{0}(x)\Vert_{L^{r}(\mathbb{R};M^s_{p,q})}+C\Vert u\Vert^{\lambda+1}_{L^{r}(\mathbb{R}; M^s_{p,q})}& \leq \epsilon+C(2\epsilon)^{\lambda+1}=\epsilon+C2^{\lambda+1}\epsilon^\lambda\epsilon.
A_+^{(i)}ds+A_-^{(i)}d\tau -\sum_{j=1}^{n-1}A_j^{(i)}dy_j = \tilde A_+^{(i)}d\hat s+\tilde A_-^{(i)}d\hat\tau-\sum_{j=1}^{n-1}\tilde A_j^{(i)}d\hat y_j,
\begin{aligned}\bar{\boldsymbol{\psi}}(t)\!=\!\bar{\boldsymbol{\psi}}(t_k)\!+\!\frac{(t\!-\!t_k)}{b_{S,k+1}}\left[\bar{\boldsymbol{\psi}}(t_{k+1})\!-\!\bar{\boldsymbol{\psi}}(t_k)\right], t\!\in\![t_k, t_{k+1}].\end{aligned}
K(s_\nu^\Lambda)=\sum_{\substack{(\lambda,a)\in E\times G, (\mu,b)\in F\times H}}\Theta_{s_\lambda^\Lambda \phi(a), s_\mu^\Lambda \phi(b)}(s_\nu^\Lambda)=\sum_{\substack{(\lambda,a)\in E\times G, (\mu,b)\in F\times H}}s_\lambda^\Lambda \phi(ab^*){s_\mu^\Lambda}^*s_\nu^\Lambda=0.
\alpha(h_0 + v) - \alpha(h_0) = Av - T \big(\tilde{\alpha}(t_0 + s, \langle \zeta, h_0 - \phi(t_0) + v \rangle) - \tilde{\alpha}(t_0 + s, \langle \zeta, h_0 - \phi(t_0) \rangle ) \big) \in V,
{\mathfrak{A}}_L:=\Big\{A=\bigl(a_{rs}^{\alpha\beta}\bigr)_{\substack{1\leq r,s\leq n begin{align*}1pt] 1\leq\alpha,\beta\leq M}}\in{\mathbb{C}}^{\,nM}\times{\mathbb{C}}^{\,nM}:\,L=L_A\Big\}.