harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D^2_{x'}v_i({{x}}'):D^2_{p'}\Phi({x},\nu_{E_{t_i}}({x})) =\left({D^2_{x'}v_i}^TD^2_{p'}\Phi({x},\nu_{E_{t_i}}({x}))\right) &=\sum_{k,l=1}^{N-1}\partial_{x_kx_l}v_i({{x}}')\ \partial_{p_kp_l}\Phi({x},\nu_{E_{t_i}}({x})).
\lim_{k\rightarrow\infty}J_{n}(f^*_{n,k},x) = J_n(f_n^*,x) \coloneqq J_n^*(x) \lim_{k\rightarrow\infty}V_{n}(f^*_{n,k},x) = V_n(f_n^*,x), \coloneqq V_n^*(x)
{\rm Dom}(\widehat{A}_\varepsilon^*) =\bigl\{k\in \mathcal{K}_\alpha: \exists\tilde{k}\in \mathcal{K}_\alpha \ \ \forall G\in{\rm Dom}(\widehat{A}_\varepsilon) \ \ \langle\! \langle \widehat{A}_\varepsilon G, k\rangle \!\rangle =\langle\! \langle G, \tilde{k}\rangle\!\rangle\bigr\}.
\limsup_{n\rightarrow +\infty}(\mathbb E[Z_n^*])^{-1}\mathbb E\left[(Z_{n+\lfloor \varepsilon n\rfloor}^*-Z_{n+\lfloor \varepsilon n\rfloor}) \mathbf 1_{\{\sup_{k=1,\ldots,\lfloor \varepsilon n\rfloor}(Z_{k}-Z_{k+1})> b_{dn}\}}\right]<\rho-1,
={1 \over (d-2)!}\sum_{k=0}^{\infty}C_k(z)\sigma^k\sum_{n=0}^{\infty}\left(n +{d+1 \over 2}\right)^{-2z-2k}\left[d^2+d(n-3)+4-n\right]\sum_{p=0}^{d-3}n^pw'_p~~~,
\beta_2(y)=-\int_0^y \frac{2 e^u}{e^u u^{\prime}+2i \mathrm{Im}(\lambda^{-3}\psi)}ds &=-2\int_0^y \frac{e^{2u}u^{\prime}}{(e^u u^{\prime})^2+4[\mathrm{Im}(\lambda^{-3}\psi)]^2}ds+4i\mathrm{Im}(\lambda^{-3}\psi)\int_0^y\frac{e^u}{(e^u u^{\prime})^2 + 4[\mathrm{Im}(\lambda^{-3}\psi)]^2}ds,
\rho_{i-\frac{1}{2}}^{L}=\rho_{i-1}+0.5\phi(r_{i-1})(\rho_{i}-\rho_{i-1}),\rho_{i-\frac{1}{2}}^{R}=\rho_{i}-0.5\phi(r_{i})(\rho_{i+1}-\rho_{i}).
\cos \theta = \frac{\vert \overrightarrow{A_{1}D}\cdot \overrightarrow{C_{1}E}\vert}{\vert \overrightarrow{A_{1}D}\vert \cdot \vert \overrightarrow{C_{1}E}\vert}= \frac{\vert - 4 + 0 + 4 . 5 \vert}{\sqrt{4 + 9}\cdot \sqrt{4 + \frac{9}{4}}}= \frac{\frac{1}{2}}{\sqrt{1 3}\times \sqrt{\frac{2 5}{4}}}= \frac{\frac{1}{2}}{\frac{5 \sqrt{1 3}}{2}}
\mathcal{A}^x_{h_L, \, F}(f_0) = \, c \, h_L^y, \mathcal{A}^x_{h_L, \, F}(f_0) = \, p'\circ(K^x+ f_0^x) \cdot h_L^x + (K^y + f_0^y) \cdot q'\circ(K^x+ f_0^x) \, h_L^x & + q\circ(K^x + f_0^x) \cdot h_L^y + (Du+D g) \circ (K + f_0) \cdot h_L,
\begin{aligned}\left|m_u\left(B\left(z, \frac{R}{2^j}\right)\right)-m_u\left(B\left(z, \frac{R}{2^i}\right)\right)\right| & \leq A [u]_{W^{\alpha, G}_{s}(X,d,\mu)}\sum_{l=i}^{j-1}G^{-1}\left( \frac{R^{\alpha p_{0}-s}}{2^{l(\alpha p_{0}-s)}}\right),\end{aligned}
\rho_{m_{0}\rightarrow m_{1}+ \cdots + m_{N}}^{D}= \left( \prod_{i = 1}^{N}\int \frac{d^{n}\! \vec{\mu_{i}}}{( 2 \pi )^{n}}\right) \int d^{n}\vec{k}\mathrm{e}^{i \vec{k}. \sum_{i}\vec{\mu_{i}}}\rho_{m_{0}\rightarrow m_{1 \mu_{1}}+ \cdots + m_{N \mu_{N}}}^{D - n}
Q_0[h] = \frac{1}{2}(\delta + h)^{ab}(\delta + h)^{pq}\Big(\nabla_i h_{pa}\nabla_j h_{qb}+ 2 \nabla_a h_{jp}\nabla_q h_{ib}- 2\nabla_a h_{jp}\nabla_b h_{iq}- 2\nabla_j h_{pa}\nabla_b h_{iq}- 2\nabla_i h_{pa}\nabla_b h_{jq}\Big) - \nabla_a(\delta +h)^{ab}\nabla_b h_{ij}
\frac{1}{2\pi i}\int_{\substack{\vert s\vert\leq T \Re(s)=c}}C_1 \mathbb{E}(L(1,\mathbb{X}_D)^{-s})\left(\int_{1}^{X}y'(x)\ell(x)^{-s}dx\right)H^{s}\left(\frac{e^{\lambda s}-1}{\lambda s}\right)^{N}\frac{ds}{s}+\mathcal{E}
\left[ (\bar{\sigma}_\omega\circ E^\omega)(x) \right](\lambda) = \lim_{n\to \omega}[E(x_n)](\sigma(\lambda_n)) &= \lim_{n\to \omega}[(\bar{\sigma}\circ E )(x_n)](\lambda_n) &= \lim_{n\to\omega}[(E\circ \beta)(x_n)](\lambda_n) &= [(E^\omega\circ \beta_\omega)(x)](\lambda).
\begin{array}{l c r}{( | X_{1}|^{2}- | X_{2}|^{2}) - ( | Y_{1}|^{2}- | Y_{2}|^{2}) = R_{2}}{X_{1}\overline{{Y}}_{1}- X_{2}\overline{{Y}}_{2}= \overline{{X}}_{1}Y_{1}- \overline{{X}}_{2}Y_{2}= 0 .}\end{array}
\frac{( \eta^{\omega}+ \eta^{- \omega})^{3 / 2}}{1 - \eta^{2}}\left( \frac{3 \omega}2 \frac{\eta^{\omega}- \eta^{- \omega}}{\eta^{\omega}+ \eta^{- \omega}}+ \frac{1 + \eta^{2}}{1 - \eta^{2}}\right) = 0 .
\alpha=\beta=\frac{\sqrt{p_1p_2}}{2}, \sum_{x\sim y}\gamma_{x,y}E_{x,y}=\eta\Delta_1A_{1,2}, Z=\begin{bmatrix}\varepsilon_1\Delta_1A_{1,1}0 0 \varepsilon_2\Delta_2A_{2,2}\end{bmatrix}.
\langle \Phi_{1}^{\ell}\Phi_{2}^{\ell^{\prime}}\rangle = \frac{\sqrt{( 2 \ell + 1 ) ( 2 \ell^{\prime}+ 1 )}}{4 \pi^{2}}\int_{| t | < 1}\frac{d t d \bar{t}}{( 1 - | t |^{2})^{2}}\int_{- 2 \pi}^{2 \pi}d \psi \ \Phi_{1}^{\ell}\bar{\Phi}_{2}^{\ell^{\prime}}\, ,
-\dfrac{d}{dx}\left[ (1-x^2)\dfrac{d}{dx}\varphi^{\alpha}_{n,c}(x) \right] + \left(c^2x^2- \dfrac{\dfrac{1}{4}-\alpha^2}{x^2}\right)\varphi^{\alpha}_{n,c}(x)=\chi_{n,\alpha}(c)\varphi^{\alpha}_{n,c}(x),x\in [0,1].
\frac 1 2 \Sigma_{\mu \nu}^{( + )}\widetilde{F}_{\mu \nu}= \mathbf{S}\widetilde{\mathbf{X}}, \hspace{1 . 0 i n}\frac 1 2 \Sigma_{\mu \nu}^{( - )}\widetilde{F}_{\mu \nu}= \mathbf{S}\widetilde{\mathbf{X}}^{*},
\hat P_{1,q}= \frac{i\lambda_2\sqrt{K_1}}{2(\lambda_1 +\lambda_2)}\left(\tilde A_1 -\tilde A_1^\dagger\right) -\frac{i\lambda_1\sqrt{K_2}}{2(\lambda_1 +\lambda_2)}\left(\tilde A_2 -\tilde A^\dagger_2\right);
R_{T}^{( r )}( E , R_{0}) = 2 \pi \sum_{m = 1}^{\infty}\int_{0}^{\infty}d q \int_{- \infty}^{\infty}d k \, N_{2}^{2}\, J_{m}^{2}( q R_{0}) \, \delta \left( E - \frac{m}{R_{0}}\sinh \Omega R_{0}+ \omega \cosh \Omega R_{0}\right) .
D_L^u(P) =D_L^u(\mathfrak{uce}^{\phi}_A(f)(Ker(u_K))) &=(D_L^u\circ\mathfrak{uce}^{\phi}_A(f))(Ker(u_K)) &=(\mathfrak{uce}^{\phi}_A(f)\circ D_K^u)(Ker(u_K)) &\subset \mathfrak{uce}^{\phi}_A(f)(Ker(u_K))=P.
\Delta_{q_1}(f, <q_1>_f) \Delta_{p_1}(f, <p_1>_f) = \frac{1}{2}\left( \frac{1}{2}\mu^2 E^2 a + \frac{\mu^2 \theta^2}{\lambda^2 ab}+ \frac{\eta^2 E^2 a^2 b}{32 \mu^2}+ \frac{\theta^2 \eta^2}{16 \mu^2 \lambda^2}\right)^{\frac{1}{2}}
c_1(t) = c_{1,0}- 2t x_1^{(i)}(t) = \gamma_{1,i}+ \frac{\gamma_{2,i}}{c_{1,0}+c_{2,0}}\cdot \sqrt{\frac{c_{2,0}+2t}{c_{1,0}-2t}}c_2(t) = c_{2,0}+ 2t x_2^{(i)}(t) = \gamma_{2,i}- \frac{\gamma_{2,i}}{c_{1,0}+c_{2,0}}\cdot \sqrt{\frac{c_{1,0}-2t}{c_{2,0}+2t}}
U_{N 1}\; = \; \frac{1}{\sqrt{2}}\; \; \; , \; \; \; U_{N 2}\; = \; \frac{-1}{\sqrt{2}}\; \; \; , \; \; \;U_{N b}\; = \; 0 \; \; \; \; \mbox{for}\; \; 3 \leq b \leq N \; \; ,
\sum_{r , s}e^{- i ( r - s ) \Omega_{G}t}\frac{\partial}{\partial t}\Bigl ( b_{( n )}^{( r , s )}\Bigr ) \hat{a}_{0}^{\dagger r}\hat{a}_{0}^{s}= i \sum_{r , s}e^{- i ( r - s ) \Omega_{G}t}b_{( n - 1 )}^{( r , s )}[ \hat{a}_{0}^{\dagger r}\hat{a}_{0}^{s}, \hat{H}^{\prime}] .
\int_{I^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k \leq \int_{I^k}n^k(2\ln n)^{\frac{k}{2}}\prod_{j=1}^k\mathbb{P}(\xi^{(n)}([y_j,y_j+G_n(x_j)/S(I)])=0)dy_1\cdots dy_k = \prod_{j=1}^k\left[n(2\ln n)^{\frac{1}{2}}\int_{I}\mathbb{P}(\xi^{(n)}([y_j,y_j+G_n(x_j)/S(I)])=0)dy_j\right].
\mathcal{R}(|\widehat{gd\sigma}|^2)(e_n,t) = \int_{\mathbb{R}^{n-1}}|\widehat{gd\sigma}(x',t)|^2\, dx' = \int_{\mathbb{S}^{n-1}}\frac{|g(\xi)|^2}{|\xi_n|}\, d\sigma(\xi)=T_0(|g|^2)(e_n).
| 0 ; g \rangle = A d T ( g ) | 0 ; e \rangle = \sum_{I , I_{3}, Y}| ( 3 , 0 ) ; ( I , I_{3}, Y ) ; e \rangle D_{( I , I_{3}, Y ) ; ( 0 , 0 , - 2 )}^{( 3 , 0 )}( g )
\stackrel[i]{}{\overline{\lambda}}^{\,\mu}=e^{-\rho(x)}\stackrel[i]{}{{\lambda}}^{\mu}\,\,( \, \stackrel[i]{}{\overline{\lambda}}\,_{\!\!\!\mu}=e^{\rho(x)}\stackrel[i]{}{{\lambda}}\,_{\!\!\!\mu}),
\begin{array}{rr}\begin{cases}\eta_{i,j}=(1-(-1)^{j-i})(\lambda+i),\i<j \eta_{i,j}=0,\i\geq j \end{cases}\begin{cases}\theta_{j\pm 1,j}={\displaystyle \frac{\pm 1}{2(\lambda+j)}}\theta_{i,j}=0,\i\neq j\pm 1\end{cases}\end{array}, j\geq 0
n(x_{1})-f(x_{1},y_{1},v_{1})v_{1}=0, f(x_{1},y_{1},v_{1})v_{1}=\frac{a\varphi_{1}(y_{1})}{G_{1}}, \frac{u}{kG_{2}}=\frac{\varphi_{1}(y_{1})}{v_{1}}.
\bar{\boldsymbol k}_k(i)=\frac{\lambda^{-1}\bar{\boldsymbol \Phi}_k(i-1)\bar{\boldsymbol x}_k(i)}{1+\lambda^{-1}\bar{\boldsymbol x}_k^H(i)\bar{\boldsymbol \Phi}_k(i-1)\bar{\boldsymbol x}_k(i)}.
e_{\pm}(x,\xi) = \frac{x + p'(\pm\xi)}{\xi \mp d\varphi(x)}\gamma\left(\langle\xi\rangle^2 (x+p'(\pm\xi))\right) \emph{and}\tilde{e}_{\pm}(x,\xi) = \frac{\xi \mp d\varphi(x)}{x + p'(\pm\xi)}\gamma\left(\langle\xi\rangle^2(x+p'(\pm\xi))\right)
\Psi = \left( \begin{array}{c c c c}{{\psi_{1}}}&{{0}}&{{0}}&{{0}}\\{{\psi_{2}}}&{{0}}&{{0}}&{{0}}\\{{\psi_{3}}}&{{0}}&{{0}}&{{0}}\\{{\psi_{4}}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \mathrm{i n s t e a d ~ o f}\psi = \left( \begin{array}{c}{{\psi_{1}}}\\{{\psi_{2}}}\\{{\psi_{3}}}\\{{\psi_{4}}}\\ \end{array}\right) ~ .
\mathcal{R}^{\nu}_{kij}(\alpha_{1,\nu})=\{\xi\in\mathcal{O}_{\nu}:|\langle k,\omega_{\nu}(\xi)\rangle+ \bar{\Omega}_{\nu,i}(\xi)-\bar{\Omega}_{\nu,j}(\xi)|<\alpha_{1,\nu}\frac{\max\{|i|,|j|\}}{\langle k\rangle^{\tau}}\},i\neq \pm j.
\hat{G}= \begin{bmatrix}e_1 0 \end{bmatrix}+ (Z-I) T^{-1}(Z-I)^{-1}\begin{bmatrix}-g e_1 \end{bmatrix}, \hat{B}= \begin{bmatrix}0 e_1 \end{bmatrix}+ (Z-I) T^{-*}(Z-I)^{-1}\begin{bmatrix}e_1 -b \end{bmatrix}
h_{\mathcal{M}}^{}=\lim_{\tau\rightarrow\infty}\left\{\lim_{\Delta\tau\rightarrow0}\left[ \frac{\mathcal{S}_{\mathcal{M}}\left(\tau+\Delta\tau\right) -\mathcal{S}_{\mathcal{M}}\left( \tau\right)}{\Delta\tau}\right] \right\}.
I_2 \le c_3 x_d^p \begin{cases}y_d^{2\alpha-p+\beta_1+\beta_2}\left(\log(1/{y_d})\right)^{\beta_4}2\alpha-2p+\beta_1+\beta_2<0; y_d^p \left(\log (1/y_d)\right)^{\beta_4+1}2\alpha-2p+\beta_1+\beta_2=0. \end{cases}
\begin{array}{rl}\left[ \Gamma^{\hspace{0.25cm}\mu}_{GV{\overline{G}}(c)}(p_{1}, k,-p_{2})\right]^{[r,0]}= \tilde{B}^{\mu}_{(c)0,r}\left(p^{2}_{1},p^{2}_{2};k^{2}\right) \mbox{}+ \sum \limits^{r}_{s=1}\, \tilde{B}^{\mu}_{(c)s,r}\left(p^{2}_{1}, p^{2}_{2}\right) \left( \frac{\Lambda^{2}}{k^{2}+\tilde{u}'_{r,2s}\Lambda^{2}}\right) \, ,\end{array}
\sup_{m\in \mathbb Z_M}\|e_{m}\|^2 \le C\exp\left(-(a-2\epsilon)T+2 \int_0^{T}\|u(t)\|_{L_{\infty}}\|u_{\tau,m}^D(t)\|_{L_{\infty}}dt\right) & \cdot \Bigg(\sum_{k=0}^{M-1}R_0^k+R_1^k+R_2^k+R_3^k+R_4^k\Bigg).
\Big|\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{a_mb_n}{m+n}\Big| \leq \pi\Big(\sum_{m=1}^\infty |a_m|^2\Big)^\frac{1}{2}\Big(\sum_{n=1}^\infty |b_n|^2\Big)^\frac{1}{2}.
\int_m^\infty e^{-it\lambda}\chi_j(\lambda) [\mathcal R_0^+(\lambda)-\mathcal R_0^-(\lambda)] \, d\lambda = \int_0^\infty e^{-it\sqrt{z^2+m^2}}\frac{z \chi_j(z) [\mathcal R_0^+(\lambda)-\mathcal R_0^-(\lambda)]}{\sqrt{z^2+m^2}}\, dz.
\int_{K}v^{\frac{N+1}{N}}\,dx \leq \left[\int_{K}v^{\frac{N}{N-1}}\,dx\right]^{\frac{N-1}{N}}\cdot \left[\int_{K}v \,dx\right]^{\frac{1}{N}}\leq C(N) \int_{K}|Dv| \,dx \cdot \left[\int_{K}v \,dx\right]^{\frac{1}{N}}.
\sum_{\alpha<n<\beta}\left( -1\right)^{n}\chi\left( n\right) f(n) =\sum_{\alpha<n<\beta}\chi\left( 2n\right) f(2n)-\sum_{\alpha<2n+1<\beta}\chi\left( 2n+1\right) f(2n+1) & =2\chi\left( 2\right) \sum_{\alpha/2<n<\beta/2}\chi\left( n\right)f(2n)-\sum_{\alpha<n<\beta}\chi\left( n\right) f(n).
\frac{p_r^2}{g_{rr}}+ \frac{p_\theta^2}{g_{\theta \theta}}+ \frac{-{g'}_{tt}}{- \cal D}\left( p_\phi + \frac{g_{t \phi}+ \Omega_0 g_{\phi \phi}}{{g'}_{tt}}E \right)^2 = \left( \frac{E^2}{-{g'}_{tt}}- \mu^2 \right),
\aligned \widehat{\mathcal U^+}({\rm mor};3,1;\alpha_1,\alpha_3) = \bigcup_{\alpha_2 \in \frak A_2}\widehat{\mathcal U^+}({\rm mor};2,1;\alpha_1,\alpha_2)\times_{R_{\alpha_2}}\widehat{\mathcal U^+}({\rm mor};3,2;\alpha_2,\alpha_3)\endaligned
\| \phi\|_{\infty}\leq \| \tilde{\phi}_0\|_{\infty}+ \sum_{i=1}^{\infty}\|\tilde{\phi}_i\|_{\infty}\leq \|\tilde{\phi}_0\|_{\infty}+ \sum_{i=1}^{\infty}2^{-i}= \|\tilde{\phi}_0\|_{\infty}+ 1 = C;\ \textrm{and}\|\phi- \phi_n\|_{\infty}\leq \sum_{i=n+1}^{\infty}\|\tilde{\phi}_i\|_{\infty}\leq \sum_{i = n+1}^{\infty}2^{-i}= 2^{-n}.
\lefteqn{\theta_S(H(n, d))}&= \max\left\{{\hat f(0)\over f(0)}: \hat f \ge 0, f(x)=0, x\in[1,d]\cap\mathbb{Z}, f(x) \ge 0,x\in[0,n]\cap \mathbb{Z}\right\}= \min\left\{2^n{g(0)\over \hat g(0)}: \hat g \ge 0, g(x)\le 0, x \in[d+1, n]\cap \mathbb{Z}, \hat g(0)>0 \right\}
\dot{\xi}_{\underline{{c}}}-{h}_{\ \underline{{c}}}^{i}\,{h}_{\ \underline{{b}}}^{j}\, T_{i j}^{\underline{{a}}}\, \xi_{\underline{{a}}}\, \xi^{\underline{{b}}}-{h}_{\ \underline{{c}}}^{i}\, T_{i 0}^{\underline{{a}}}\, \xi_{{a}}= 0 ,
\langle\partial_{i}z,\partial_{j}z\rangle =\sum_{\alpha,\beta=1}^{m}\frac{\partial x^{\alpha}}{\partial z^{i}}\frac{\partial x^{\beta}}{\partial z^{j}}\langle\partial_{\alpha}x,\partial_{\beta}x\rangle =\sum_{\alpha=1}^{m}\frac{\partial x^{\alpha}}{\partial z^{i}}\frac{\partial x^{\alpha}}{\partial z^{j}}.
f^{n}(x)=a+(x-a)f^{\prime}(a)^{n}+\frac{(x-a)^{2}}{2}(f^{\prime\prime}(a)f^{\prime}(a)^{n-1})(1+f^{\prime}(a)+\cdot\cdot\cdot+f^{\prime}(a)^{n-1})+\cdot\cdot\cdot
\bar{H}(y) = - \frac{1}{N}\log \int_{Px = y}\exp \left(- H(x) \right)\mathcal{L}(dx) &= \frac{1}{2}\langle y, (Id + P(M_{ij}) NP^* y \rangle_Y + \langle P^* y , s \rangle - \frac{1}{N}\log \int_{Pz=0}\exp \left( -H_{(M_{ij})}(z,y) \right) \mathcal{L}(dz).
P_1 = \frac{1}{\epsilon(\mu_1-1)}\begin{pmatrix}-\epsilon 0 1 0 \end{pmatrix}, P_2 = \frac{1}{\mu_1\epsilon}\begin{pmatrix}0 0 1 0 \end{pmatrix}, P_3 =\frac{1}{\mu_2}\begin{pmatrix}0 0 \epsilon 1 \end{pmatrix}, P_4 = \frac{1}{\mu_3 \epsilon}\begin{pmatrix}-\epsilon^2 -\epsilon \epsilon 1 \end{pmatrix},
X = \; x_1+x_2+\ldots+x_{n-2}w_{n-2}X^{n-2}= \;(u_{n-2})^{n-2}x_1x_2\ldots x_{n-2}{u_{n-2}v_{n-2}\over w_{n-2}X}= \;{1\over x_1}+{1\over x_2}+\ldots+{1\over x_{n-2}}{1\over X^2}-1= \;{4w_{n-2}\over(1-u_{n-2})v_{n-2}}
\bar u'(t)+A\bar u(t) +B(\bar u(t),\bar u(t)) =\frac{A\xi_1}t + \sum_{n=2}^{\infty}\frac1{t^{n}}\Big\{-(n-1)\xi_{n-1}+A\xi_n+\sum_{k=1}^{n-1}B(\xi_k,\xi_{n-k})\Big\}&=\sum_{n=1}^\infty \phi_n t^{-n}=f(t).
\mathcal{L}= \sum_{i = 1}^{k}\sum_{j = 1}^{N_{i}}V ( \lambda_{i}, N_{i}; \lambda , N ; \varepsilon ) \bar{\psi}_{( i )}^{j}\left( \partial \! \! \! \slash + \lambda_{i}U ( \lambda_{i}, N_{i}; \lambda , N ; \varepsilon ) \sigma \right) \psi_{( i )}^{j},
\sum_{s_1\geq\dots\geq s_{r}\geq-\left\lfloor m/2\right\rfloor}\frac{q^{s_1^2+\dots+s_{r}^2+m(s_1+\dots+s_{r-1})-s_1-\dots-s_{i}}(-q)_{m+2s_r-1}}{(q)_{s_1-s_2}\dots(q)_{s_{r-2}-s_{r-1}}(q^2;q^2)_{s_{r-1}-s_{r}}}(-1)^{s_r}\left[{m+s_r\atop m+2s_r}\right]_{q^2}=a_m,
P_0 = 2 \frac{1}{n}\sum_{i=1}^n \psi_i \psi_i^T\Omega^{-1}\bar{\psi}\Pi^T\Omega^{-1}\psi_i - \frac{1}{n}\sum_{i=1}^n \psi_i \psi_i^T\Omega^{-1}\bar{\psi}\bar{\psi^T}\Omega^{-1}\psi_i \psi_i^T\Omega^{-1}\bar{\psi}\,.
\begin{aligned}Q_{\psi}(T) = Q_{A_{\psi}}(T) \cdot Q_{B_{\psi}}(T), \; \psi \neq \psi_0 Q_{\psi_0}(T) = Q_0(T) \cdot Q_{A_{\psi_0}}(T) \cdot Q_{B_{\psi_0}}(T).\end{aligned}
\sum_{i=0}^{b-2}\frac 1{\sqrt{2w_i^-}}= \sum_{i=0}^b\frac 1{\sqrt{2w(v_i)}}-\frac 1{\sqrt{2w(v_0)}}+\frac 1{\sqrt{2w(v_{\lfloor\frac b2\rfloor})}}-\frac 1{\sqrt{2w(v_{b-1})}}&-\frac 1{\sqrt{2w(v_b)}}\sim \sum_{i=0}^b\frac 1{\sqrt{i^2+(b-i)^2}}.
\phi_{\chi}^o(s) = \int_{\frac{1}{P}}^{\infty}y^{\frac{s}{2}-1}\tilde{\Psi}_{\chi}(y) dy + \frac{-jP^{1-s}}{G\left(\overline{\chi},P\right)}\int_{\frac{1}{P}}^{\infty}y^{\frac{1-s}{2}-1}\tilde{\Psi}_{\overline{\chi}}(y) dy
\cdots{\frac{\buildrel Q}{\longrightarrow}}{\cal F}_{l_{- 1}, k}{\frac{\buildrel Q}{\longrightarrow}}{\cal F}_{l_{0}, k}{\frac{\buildrel Q}{\longrightarrow}}{\cal F}_{l_{1}, k}{\frac{\buildrel Q}{\longrightarrow}}\cdots .
\mathbf{D}= \left(\begin{array}{cccccc}2(i-j) c_{1}c_{1}2(i-j) c_{2}& c_{2}2(i-j) c_{3}& \ddots \ddots \ddots & c_{m-2}2(i-j) c_{m-1}& c_{m-1}2(i-j)\end{array}\right),
Z_{i, -1,0}^* := n\sqrt{\Delta_i}\int \phi_i(t) d\widetilde Y_t, \i=1,\ldots,m, Z_{i,j,k}^* := \sqrt{\frac{n}{f_0(x_{i-1})}}\int \psi_{i,j,k}(t) d\widetilde Y_t , \ (i,j,k)\in \Lambda, \ 0\leq j\leq \overline{J},
\underline{h}^{N}(t,x):= h_0(x)\mathbf{1}_{]-\tau,0]}(t) + \sum_{k=1}^{N}h_k(x) \mathbf{1}_{](k-1)\tau,k\tau]}(t) + h_N(x) \mathbf{1}_{]N\tau,+\infty[}(t).
|\mathfrak m_N(\xi)-\lambda_N^2(\xi)|\le 17\min\Big\{e^{-\frac{c\kappa(d, N)^2}{400}\sum_{i=1}^d\cos^2(\pi\xi_i)}, \kappa(d, N)^2\sum_{i=1}^d\cos^2(\pi\xi_i)\Big\}.
F(z) = -\frac{1}{40}\cdot\frac{E_{4}(z) + 4E_{4}(2z) - 9E_{4}(3z) - 36E_{4}(6z)}{(\eta(z)\eta(2z)\eta(3z)\eta(6z))^{2}}= q^{-1}-4-83q-296q^{2}+ \dots
\begin{aligned}\sup_{Q_s \in \mathfrak{Q}_s, s \in \lbrace 1,\ldots,k \rbrace}\min_{G \in \mathcal{G}(Q_0,\ldots,Q_k)}\det(G^\intercal G)\geq \sup_{Q_s \in \mathfrak{Q}_s', s \in \lbrace 1,\ldots,k \rbrace}\min_{G \in \mathcal{G}(Q_0,\ldots,Q_k)}\det(G^\intercal G),\end{aligned}
_{-\infty}D_{x}^{\alpha}f(x) =\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{\infty}\frac{f(x)-f(x-x^{\prime})}{x^{\prime\alpha+1}}dx^{\prime},_{\infty}D_{x}^{\alpha}f(x) =-\frac{\alpha}{\Gamma(1-\alpha)}\int_{0}^{\infty}\frac{f(x+x^{\prime})-f(x)}{x^{\prime\alpha+1}}dx^{\prime}.
\begin{aligned}(\xi_1+\xi_2)^2 + (\xi_3 +\xi_4)^2 - 2(\xi_1+\xi_2)(\xi_3+\xi_4)= \ (\xi_1+\xi_2-\xi_3-\xi_4)^2 = \ 4( \xi_1 - \xi_4)(\xi_2-\xi_3) + (\Delta^4 \xi)^2.\end{aligned}
I_K(\tilde{P}^*_\lambda(;q,t,T;t_0);q,t,T;t_0,t_1,t_2,t_3){}=\frac{(-t_0T/t)^{-|\lambda|}q^{-n(\lambda')}t^{2n(\lambda)}C^0_\lambda(T,Tt_0t_1/t,Tt_0t_2/t,Tt_0t_3/t;q,t)}{C^0_\lambda\big(T^2t_0t_1t_2t_3/t^2;q,t\big)C^-_\lambda(q,t;q,t)}.
\lambda(u, s) = \mu(s) \cdot \lim_{n\to +\infty}\left(\frac{u_0^{(n)}}{s_0^{(n)}}\right)^{2^n}= \mu(s) \cdot \lim_{n\to +\infty}\left(\frac{u_0^{(n)}}{\mu(s)}\right)^{2^n}.
(\begin{matrix}1&1&0&0&1\\ 1&1&1&0&0\\ 0&1&1&1&0\\ 0&0&1&1&1\\ 1&0&0&1&1\end{matrix})(\begin{matrix}\alpha\\ \beta\\ \gamma\\ \delta\\ \epsilon\end{matrix})=(\begin{matrix}2A\\ 2B\\ 2C\\ 2D\\ 2E\end{matrix})
S_{[ 1 ]}^{c}\, = \, e^{i \vartheta_{[ 1 ]}^{c}}\, \left( \begin{array}{c c}{{0}}&{{\Theta_{[ 1 ]}}}\\{{- \, \Theta_{[ 1 ]}}}&{{0}}\\ \end{array}\right) \,{\cal K}\, \equiv \,{\cal C}_{[ 1 ]}\,{\cal K}\quad .
h^\theta(x,u)\le \sup_{\rho\in \Theta_{n-1}}E_{\xi_1^\theta(u)}\biggl[\varphi\biggl(x+u+\sum_{i=1}^{n-1}Z_i^{\rho}\biggr)\biggr]=V\left(\xi_1^\theta(u),n-1,x+u\right).
s\left( -c,d;A_{c},B_{-b};-R_{1},-R_{2}\right) -s\left( d,c;B_{b},A_{c};R_{2},-R_{1}\right) & =P_{1}\left( R_{2},B_{-b}\right) P_{1}\left( -R_{1},A_{-c}\right)-\frac{1}{cd}B_{0}(B)P_{2}(cR_{2}-dR_{1},A)-\frac{d}{c}B_{0}(A)P_{2}\left(R_{2},B_{-b}\right) -\frac{c}{d}B_{0}\left( B\right) P_{2}\left(R_{1},A_{c}\right) .
T(n_i)+T(n_{i+1})+T(n_{i+2}) <\left( \frac{85}{27}+ \frac{1216}{729}\right)\cdot \frac{1}{X_0}= \frac{3511}{729}\cdot \frac{1}{X_0}&<7 \cdot \frac{3}{4}\cdot \frac{1}{X_0}= (k_i+k_{i+1}+k_{i+2}) \cdot \frac{3}{4}\cdot \frac{1}{X_0}.
k_{1}= \left( \begin{array}{c}{p}\\{p}\\ \end{array}\right) k_{2}= \left( \begin{array}{c}{e}\\{- p}\\ \end{array}\right) k_{3}= \left( \begin{array}{c}{- p}\\{- p}\\ \end{array}\right) k_{4}= \left( \begin{array}{c}{- e}\\{p}\\ \end{array}\right) .
|\tilde u_j(x)| =|\frac{u_j((x^j)'+d_jx)-u_j((x^j)')}{d_j^2}| &\le \frac{|\nabla u_j(r_{x^j})||x|}{d_j}=\frac{|\nabla u_j(r_{x^j})-\nabla u_j(x^j)||x|}{d_j}\le T_1,
m(f_{n+1}(C_{f_{n+1}, \delta}\cap I)) \leq\sum_{i=1}^N \sum_{i' = 1}^{\tilde N_i}\big( (1-K) \mu(\tilde I_i^{i'}) + \epsilon_2 \big) \leq (1-K)^2 \mu(I) + \tfrac{1}{2}\epsilon_1,
mp'q-npq' = m \sum_{i=0}^n i p_i x^{i-1}\sum_{j=0}^m q_j x^j - n\sum_{i=0}^n p_i x^i \sum_{j=0}^m j q_j x^{j-1}& = \sum_{i,j}p_i q_j mi x^{i+j-1}- \sum_{i,j}p_i q_j nj x^{i+j-1}&= \sum_{i,j}p_i q_j (mi-nj) x^{i+j-1}.
\tilde{S}=\int d^2{x}\left\{g^2(1+\varphi_{1}^2)^2(\partial\tilde{\varphi_{0}})^2+8\theta g^2\partial_{a}\tilde{\varphi_{0}}\partial^{a}\varphi_{1}+\frac{(1+16\theta^2 g^4)}{g^2(1+\varphi_{1}^{2})^2}(\partial\varphi_{1})^2\right\}
\dfrac{\hat{k}_2{(\Delta)}}{\hat{k}_2(\Delta\setminus\sigma)}= \dfrac{D_{1,2}D_{2,2}D_{3,2}}{D_{1,1}D_{2,1}D_{3,1}}=\dfrac{(x_{1,1}+x_{1,2})(x_{2,1}+x_{2,2})(x_{3,1}+x_{3,2})}{x_{1,1}x_{2,1}x_{3,1}}.
g(k,i,n+1) =g(k-1,1,n)+g(k,i+1,n) &=g(k-1,1,n)+g(k-1,1,n-1)+g(k,i+2,n-1) &\dots =[g(k-1,1,n) +g(k-1,1,n-1)+\dots+g(k-1,1,0)]+g(k,i+n+1,0).
H_{(Y_+,D_{Y,+})}(y)=e^{\frac{t_+}{2\pi i}}\sum_{d\in\mathbb K_{+}}y^d\left(\frac{\prod_{i=1}^k \Gamma(1+\frac{v_i}{z}+v_i\cdot d)}{\prod_{i\in M_0}\Gamma (1+\frac{\bar D_i}{z}+D_i\cdot d)}\right)\textbf{1}_{[d]}[\textbf{1}]_{(D_i\cdot d)_{i\in I_+}}.
2\pi x \sum_{m=1}^\infty \int_0^1{\pi^2 B_2(t)\over (x^2+(2\pi (t+m))^2)^{{3\over 2}}}dt =2\pi^3 x \sum_{k=0}^\infty{-{3\over 2}\choose k}{x^{2k}\over(2\pi)^{3+2k}}\sum_{m=1}^\infty \int_0^1{B_2(t)\over (t+m)^{3+2k}}dt \ .
l_f(0,r) =\inf\big\{d(f(0),f(y)):d(0,y)= r\big\}&\geq \big(1-O(r)\big)\inf\big\{d^0(0,Df(0)y):d(0,y)=(1-O(r))r\big\}-2o(r) &\geq \big(1-O(r)\big)^2\|Df(0)\|_sr-2o(r).
\begin{aligned}a_{S}(\kappa_x,\kappa_y,\bold{s})=e^{-\jmath \boldsymbol{\kappa}^{T}\bold{s}}=e^{-\jmath(\kappa_x s_x+\kappa_y s_y +\kappa_z s_z)}, &a_{R}(k_x,k_y,\bold{r})=e^{\jmath \bold{k}^{T}\bold{r}}=e^{-\jmath(k_x r_x+k_y r_y +k_z r_z)},\end{aligned}
2\lambda mT_{m,0}+2\lambda^{2m+1}mT_{-m,0}+6m\frac{\partial T_{1,0}}{\partial M_0}\lambda^mM_0+6m\frac{\partial T_{-1,0}}{\partial M_0}\lambda^{m+2}M_0 &=(m+1)T_{m+1,0}-(-m+1)T_{-m+1,0}\lambda^{2m}+(m-1)T_{m-1,0}\lambda^2-(-m-1)T_{-m-1,0}\lambda^{2m+2}.
c_{n}=\frac{\langle f,(J_{\alpha})_{n}\rangle}{\langle(J_{\alpha})_{n},(J_{\alpha})_{n}\rangle}=\frac{\int_{0}^{b}xf(x)(J_{\alpha})_{n}(x)dx}{\frac{1}{2}(bJ_{\alpha\pm1}(u_{\alpha,n}))^{2}}
S ={\frac{1}{4}}( 4 \pi a^{2}) ={\frac{1}{4}}A_{\mathrm{h o r i z o n}}; \quad E ={\frac{1}{2}}a = \left({\frac{A_{\mathrm{h o r i z o n}}}{1 6 \pi}}\right)^{1 / 2}
\langle P[\{\Phi^{(I)}\}] \rangle \; = \; \frac{1}{Z}\sum_{n=0}^\infty \frac{(-1)^n}{n!}\; \langle \; P[\{\Phi^{(I)}\}] \; S_{int}[ \{\Phi^{(I)}\}]^n\; \rangle_Q \; .
\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) .
\hat{f}^\lambda_x:\mathbb{R}^3\longrightarrow\mathbb{R}_+,x,\lambda\in\mathbb{R}_+, \mspace{27mu}\vec{\tau}\longmapsto \frac{e^{-x\mspace{1.5mu}\sqrt{\frac{3}{4}\tau^2\,+\,\lambda}}}{x}
\begin{dcases}\sum_{e\in E(G_i)}w_i(e)=1, \sum_{e\in E(G_i):\,v\in e}B_i(v,e)=1,\ \v\in V(G_i), w_i(e)^{p-r}\prod_{v\in e}B_i(v,e)=\alpha_i,\ \e\in E(G_i).\end{dcases}
\beta_{s}\left( a,b\right) :=\sum_{l=0}^{\left[ p\right] -1}\frac{1}{l!}\left( D^{l}f\right) \left( x_{s}\right) \pi_{l+1}\left( g_{s}^{-1}a\left(b-1\right) \right) \forall a,b\in \mathcal{G}_{\left[ p\right]}\forall s\in \left[ 0,T\right]
\frac{1 - \Phi(y_1)}{1 - \Phi(y)}= 1 + \left( \int_{y - \frac{C_1}{\sigma \sqrt{n}}}^{y}e^{- \frac{t^2}{2}}dt \right) \left( \int_y^{\infty}e^{- \frac{t^2}{2}}dt \right)^{-1}.
\gamma^{\textbf{int}}_{1,\mathrm{x}}\mathcal{V}_{\Omega \times\mathbb{R}}\Big[\psi\Big](\mathrm{x},t) = \delta \Big(\gamma^{\textbf{int}}_{1,\xi}\overline{\mathcal{V}}_{B\times\mathbb{R}}\Big[\hat{\psi}\Big]\Big)^\vee
{\mathcal W}^{(n)}_{\mu_1\cdots\mu_n}(x_1,\cdots,x_n)\;=\; \left. \frac{\delta^n{\mathcal W}[b_\mu]}{\delta b_{\mu_1}(x_1) \cdots \delta b_{\mu_n}(x_n)}\right|_{b_\mu = 0}\;.
K_{-}(\lambda ;p)=\left( \begin{array}{cc}\lambda -\eta /2+p 0 0 p-\lambda +\eta /2\end{array}\right) ,K_{+}(\lambda ;q,\xi )=\left( \begin{array}{cc}\lambda +\eta /2+q \xi (\lambda +\eta /2) \xi (\lambda +\eta /2) q-(\lambda +\eta /2)\end{array}\right) ,
= \;\mbox{Tr}(\gamma_\mu \gamma_\alpha \gamma_\nu \gamma_\beta)\int_{x=0}^1 dx\frac{1}{(4\pi)^\omega \Gamma(2)}\Bigg( \frac{1}{2}\delta_{\alpha \beta}\frac{\Gamma(1-\omega)}{[m^2 +k^2x(1-x)]^{1-\omega}}
-\frac{\partial L_{d}KL(q\|p)}{\partial q_{j}}=1-\left[\frac{a}{\left(a-b\right)}\frac{q_{i}^{a-1}}{p_{i}^{a-1}}-\frac{b}{\left(a-b\right)}\frac{q_{i}^{b-1}}{p_{i}^{b-1}}\right]
\mathrm{Solv}\left(\frac{\mathrm{USp}(3,3)}{\mathrm{SU}(3)\times{U}(1)}\right) \subset \mathrm{Solv}\left(\frac{\mathrm{SU}(3,3)}{\mathrm{SU}(3) \times \mathrm{SU}(3)\times \mathrm{U}(1)}\right)~.
y_1=x_2, \, y_2=\frac{1}{\sqrt{2}}(x_1+x_3), \, y_3=\frac{1}{\sqrt{2}}(x_1-x_3); [y_1,y_2]=[y_3,y_1]=\frac{w-1}{\sqrt{2}}y_1, \, [y_2,y_3]=\frac{1+w}{\sqrt{2}}(y_2+y_3).
\begin{aligned}\nu \int_0^\infty \partial_y^2 D^\alpha u\cdot \langle y \rangle^{2(l+k)}D^\alpha u\mathrm{d}y\leq -\frac{\nu}{2}\Vert \langle y\rangle^{l+k}\partial_yD^\alpha u\Vert_{L^2_y}^2 &+\delta_1\Vert \partial_y (u,h)\Vert_{H^m_0}^2+C\delta_1^{-1}E_{u,h}^2(1+E_{u,h}^2)+C(u_b).\end{aligned}
\frac{1}{2}\frac{d}{dt}\int_{\mathbb T^2}\varphi^2(\theta)\,dx+\int_{\mathbb T^2}\nabla\cdot\left(\frac{1}{2}\varphi^2(\theta)u\right)\,dx +\nu\int_{\mathbb T^2}\Lambda^\alpha\theta\varphi(\theta)\,dx=\int_{\mathbb T^2}f\varphi(\theta)\,dx.
\left( \begin{array}{c c c}{- 1}&{0}&{0}\\{0}&{0}&{- 1}\\{0}&{- 1}&{0}\\ \end{array}\right) , \quad \left( \begin{array}{c c c}{0}&{- 1}&{0}\\{- 1}&{0}&{0}\\{0}&{0}&{- 1}\\ \end{array}\right) , \quad \left( \begin{array}{c c c}{0}&{0}&{- 1}\\{0}&{- 1}&{0}\\{- 1}&{0}&{0}\\ \end{array}\right)