harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_R \equiv \langle \mathcal{H}\rangle_R=\frac{1}{\mathcal{N}_R}\sum_{2k-n=R}E_{n,k}(1,\lambda) &=2R \sqrt{1+\lambda^2}+N\left (\sqrt{1+\lambda^2}-\frac{1}{1+\lambda^2}\right)\left [1-\frac{4}{\mathcal{N}_R}\sum_{2k-n=R}C_{n-1}^{k-1}C_{N-n-1}^{k-1}\right ]\ .
\begin{cases}\Phi''-{\Delta}\Phi +A^T\Phi=0\hskip2.1cm~ \hbox{in}(0,+\infty) \times \Omega, \partial_\nu \Phi + B^T\Phi = 0 \hskip3cm~ \hbox{on}(0,+\infty) \times \Gamma \end{cases}
\chi^*(\Phi(D)) = \chi_\infty(\Phi(D)) + d_\infty(\Phi(D)) = d_\infty(\Phi(D)) \le (\ell + \zeta)Md_\infty(D) & \le (\ell + \zeta)M [d_\infty(D)+\chi_\infty(D)] = (\ell + \zeta)M\chi^*(D).
-\sum_{k=1}^{p}\frac1k \frac{x^{p-k}}{(x-1)^{p-k+1}}=\sum_{\ell=0}^{p-1}\binom{p}\ell \frac{1}{(x-1)^{\ell+1}}(H_\ell-H_p) &= \sum_{\ell=0}^{p}\binom{p}\ell \frac{1}{(x-1)^{\ell+1}}H_\ell- H_p\frac{x^p}{(x-1)^{p+1}}.
w_n(t) = \begin{cases}w_{n-1}(t) + \chi_{-}(t) t \leq x_n - R_n, m(t - (x_n - R_n)) + w_{n-1}(x_n - R_n) + c_{-}x_n - R_n < t <x_n - \tau_n, w_{n-1}(x_n) + \tilde{w}_n (t) x_n - \tau_n \leq t \leq x_n + \tau_n, m(t - (x_n + R_n)) + w_{n-1}(x_n + R_n) + c_{+}x_n + \tau_n < t < x_n + R_n, w_{n-1}(t) + \chi_+ (t) x_n + R_n \leq t.\end{cases}
N_{2 2}( x , y , a , b ) = \frac{1}{4 a b}\sum_{n = 1}^{\infty}\cos ( \frac{2 n \pi x}{a}) \int_{\frac{n \pi}{a}b}^{\infty}d q \left( \coth q - 1 \right) \cosh ( \frac{2 q y}{b}) .
\delta_t U^1 + \sum\limits_{q=1}^{2}\widetilde{Q}_{1}^{(q)}(\mathbf{B}_qU) = f^{1}, \delta_t^{(2)}U^n + \sum\limits_{q=1}^{2}\widetilde{Q}_{n}^{(q)}(\mathbf{B}_qU) = f^{n}, n\geq 2, U^0=u_0,
\begin{aligned}E_{x_3}\big[E_{x_2}\big[ E_{x_1}\big[F\big(\mathcal Z_{h_1}(x_1,\cdot)+ \mathcal Z_{h_2}(x_2,\cdot)\big)G\big(\mathcal Z_{h_3}(x_1,\cdot)+ \mathcal Z_{h_4}(x_3,\cdot)\big) \big]\big]\big] &=E_x\big[F\big(\mathcal Z_{\mathbf{s}(h_1,h_2)}(x,\cdot)\big)\big]E_x\big[G\big(\mathcal Z_{\mathbf{s}(h_3,h_4)}(x,\cdot)\big)\big]\end{aligned}
\frac{d}{d\psi_0}\int\limits_{\psi_0}^{\pi-\psi_0}Cf(0,e_n,\psi)h(\psi_0,\psi)d\psi &= (2-n)\cos \psi_0 \sin\psi_0 \int\limits_{\psi_0}^{\pi-\psi_0}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi.
Z_0 :=X_1\cdots X_N, Z_1 :=Z_0L_1\left(\frac{Y_1}{X_1},\dots,\frac{Y_N}{X_N}\right), &\vdots Z_r :=Z_0L_r\left(\frac{Y_1}{X_1},\dots,\frac{Y_N}{X_N}\right),
T_{\mu\nu}=e^{-2\phi}[F_{\mu\rho}F_{\nu}{}^{\rho}-{}^{\star}F_{\mu\rho}{}^{\star}F_{\nu}{}^{\rho}]-2[\nabla_{\mu}\phi\nabla_{\nu}\phi-{\textstyle\frac{1}{2}}g_{\mu\nu}(\nabla\phi)^{2}]\, ,
\nu_1^{(1)}\circledast \rho_{i}+ \nu_{-1}^{(1)}\circledast \rho_{-i}= \bigl( c_1\|\rho_{i}\|_\alpha^\alpha +c_{-1}\|\rho_{-i}\|_\alpha^\alpha\bigr)\;\nu^\alpha, \ \i=\pm 1\,.
\Gamma^{\theta}:\,= \big(\Gamma^{\theta}(-1),\, \Gamma^{\theta}(0),\, \Gamma^{\theta}(1)\big) = \bigg( \frac{b\, \exp(-\theta)}{Z(\theta)},\,\frac{1 - 2b}{Z(\theta)},\,\frac{b\, \exp(\theta)}{Z(\theta)}\bigg),
\left| \|u\|^2_{\dot H^1(\Omega)}- \|\chi u\|^2_{\dot H^1(\Omega)}- \|\eta u\|^2_{\dot H^1(\Omega)}\right|=\int_\Omega(|\nabla \chi|^2 + |\nabla \eta|^2)|u|^2 \le C \int_\Omega |u|^2.
X(\theta) \log X(\theta)= \theta X^0\log X^0-\theta (1-\theta)^2\frac{X^0}{(\tilde X^0)^2}(X^1-X^0)^2 + (1-\theta)X^1\log X^1 - \theta^2 (1-\theta)\frac{X^1}{(\tilde X^1)^2}(X^1-X^0)^2,
\Lambda \partial_{\Lambda}{\cal{G}}_{\rm ir}^{\Lambda , \Lambda_0}\{\varphi_{\bf{k}}\}= - \partial_t{\cal{G}}_t \{\tilde{\varphi}_{\bf{q}}\}+ \hat{\cal{R}}_{s}{\cal{G}}_t \{\tilde{\varphi}_{\bf{q}}\}+ \frac{N_t}{2}\left[ \eta_t + D \ln Z_t \right] \; ,
\int_{\Delta(0,2)}|\zeta|^{2k}|f(\zeta)|^2\,dm(\zeta) =2\pi\sum_{j=k}^\infty\frac{2^{2j+2}}{2j+2}\,|a_{j-k}|^2=2\pi\sum_{j=0}^\infty\frac{2^{2j+2+2k}}{2j+2+2k}\,|a_{j}|^2 &\geq \frac{2^{2k}}{k+1}\,2\pi\sum_{j=0}^\infty\frac{2^{2j+2}}{2j+2}\,|a_{j}|^2=\frac{2^{2k}}{k+1}\,\int_{\Delta(0,2)}|f(\zeta)|^2\,dm(\zeta)\,.
[g,h]^{n+1}= [g,h][g,h]^n=ghg^{-1}h^{-1}[g,h]^n=ghg^{-1}[g,h]^nh^{-1}= &=[g,h^2]h[g,h]^{n-1}h^{-1}=[g,h^2][hgh^{-1},h].
p_{\ell}^{(N)}= \frac{1}{2^{\ell}}\sum_{\begin{stackrel}{t = \tfrac{1}{2}\left( \frac{2 - \ell}{N}-1 \right)}{\ell - 2r = (2t+1)N}\end{stackrel}}^{\tfrac{1}{2}\left( \frac{\ell}{N}-1 \right)}(-1)^{t}A(\ell-1,r).
b_{n,1}= \tau (p_{r(0^{(n)}0^{(n)})}) = \tau (p_{r(1^{(n)}0^{(n)}0^{(n)}1^{(n)})}) = \tau (p_{r(1^{(n+1)}0^{(n+1)})}) = b_{n+1, 3}\, , b_{n,4}= \tau (p_{r(1^{(n)}1^{(n)})}) = \tau (p_{r(0^{(n)}1^{(n)}1^{(n)}0^{(n)})}) = \tau (p_{r(0^{(n+1)}1^{(n+1)})}) = b_{n+1, 2}\, ,
g_V\left(At,t\right)= \sum_{i=1}^n g_V\left( H_{e_i}K_{e_i}t,t\right)=-\sum_{i=1}^n g_U\left(K_{e_i}t,K_{e_i}t\right) = \sum_{i=1}^n g_V\left(t, H_{e_i}K_{e_i}t\right)=g_V(t,At)
X(a,\zeta)X(b,\delta) = X(ab,\zeta\delta), X(a,\zeta)Y(b,\delta) =Y(ab,\zeta\delta), Y(b,\delta)X(a,\zeta) = Y(\zeta^2a^{-1}b,\zeta\delta), Y(a,\zeta)Y(b,\delta) = X(-\delta^2ab^{-1},\zeta\delta).
= 8 m^{2}\mathrm{T r}\Big [ \frac{1}{- \triangle + m^{2}}\not{\! \! B}\frac{1}{- \triangle + m^{2}}\not{\! \! B}\frac{1}{- \triangle + m^{2}}\not{\! \partial}\not{\! \! B}\frac{1}{- \triangle + m^{2}}\not{\! \partial}\not{\! \! B}\Big ] ,
{\bf G}= U ( x ) \left( \mathrm{{1}\k e r n - . 2 5 e m \mathrm{l}}+ i ~ \overline{{{\theta}}}\chi \cdot \tau +{\frac{\overline{{{\theta}}}\theta}{2}}( i Z \cdot \tau +{\frac{\overline{{{\chi}}}\chi}{2}}) \right) ~ ,
-\frac{1}{2}\int_{\mathbb{R}}|P_n\varphi|^2~dx +\frac{1}{2\pi}\int_0^{\infty}\left(\left(\frac{\psi_n^2}{2}\right)' \widehat{u\varphi}\bar{\hat\varphi}+\psi_n^2\widehat{u\varphi}'\bar{\hat\varphi}\right)~d\xi=0.
c_p= \frac{1}{2^{p^2}p^2(p^2-1)}+ \frac{1}{2^{p}p^2}+ \frac{1}{2^{p^2}(p^2-1)}\cdot\sum_{1\neq d\mid p^2-1}2^{(p^2-1)(d-1)/d}\phi(d) ,
c_L(\tau) =\hat\mu+\mathbf{r}_L(\tau)'\mathbf{R}_L^{-1}(\mathbf \Upsilon-\hat\mu \mathbf 1), \nabla_\tau c_L(\tau) =\hat\mu+\mathbf{Q}_L(\tau)\mathbf{R}_L^{-1}(\mathbf \Upsilon-\hat\mu \mathbf 1),
X^{(\epsilon)}(t) =\eta(0) + \int_0^t f(s,X_s^{\epsilon})ds + \int_0^t g(s, X_{s}^{\varepsilon})dW(s) &\quad+ \int_0^t h_0(s,X_{s}^{\epsilon})\Lambda(\epsilon) dB(s)+ \int_0^t \int_{\mathbb{R}_0}h_0(s,X_{s}^{\epsilon})(\lambda(z)-\lambda_{\varepsilon}(s)) \tilde{N}(ds,dz) X_0^{(\epsilon)}=\eta
\frac{\mathrm{d}}{\mathrm{d}\sigma}\ell= -\frac{1}{\sigma}-\frac{\alpha-1}{\sigma^2}(x-\mu)\frac{\phi_s\left(\frac{x-\mu}{\sigma}\right)}{\Phi_s\left(\frac{x-\mu}{\sigma}\right)}+\frac{\beta-1}{\sigma^2}(x-\mu)\frac{\phi_s\left(\frac{x-\mu}{\sigma}\right)}{1-\Phi_s\left(\frac{x-\mu}{\sigma}\right)}-\frac{s}{\sigma}\left|\frac{x-\mu}{\sigma}\right|^s,
\begin{array}{l}{\inf}\; \mathbb{E}_{\xi_2,\ldots,\xi_{T_{\max}}}\Big[{\overline{f}}_{T_{\max}}(x_{T_{\max}}, x_{T_{\max}-1}, \xi_{T_{\max}}) \Big] x_t \in X_t( x_{t-1}, \xi_t )\;\mbox{a.s.}, x_{t}\;\mathcal{F}_t\mbox{-measurable,}t=1,\ldots,T_{\max}.\end{array}
\mathcal{K}_j:=\left(\Psi^{Ai}_{0}\right)^{-1}\Psi^{Ai}_{j}\left(\frac{2}{3}\right)^{-j}=\frac{3^j}{2^{j+1}}e^{\frac{i\pi}{4}\sigma_3}\begin{pmatrix}(-1)^{j}(s_j+\nu_j) s_j-\nu_j (-1)^{j}(s_j-\nu_j) s_j+\nu_j \end{pmatrix}e^{-\frac{i\pi}{4}\sigma_3}.
\begin{array}{ll}\| | \partial_{x_{i}}\tilde{\omega}_{R^{'}}|^{\frac{1}{p-1}}\|^{p-1}_{L^{Q}}\lesssim \mu_{4}|\alpha| = k: \, \| \partial^{\alpha}\tilde{\omega}_{R^{'}}\|_{L^{\infty}}\lesssim \frac{\mu_{4}^{k}}{R'^{k}}\end{array}
| \mathcal{T}( T ) \rangle = U_{h ( z ) \circ z ( v )}^{\dagger}( h ( z ) \circ z ( \rho ) \circ \frac{1}{2 \pi i}\int d \rho b ( \rho ) ) \left( h ( z ) \circ z ( w ) \circ \mathcal{B}\right) | 0 \rangle ,
z_{1}= z_{0}+ 2^{-\frac{4}{3}}N^{-\frac{1}{3}}\eta_{-}^{-1}(\pi_{*}- 2), w_{1}= z_{0}+ 2^{-\frac{4}{3}}N^{-\frac{1}{3}}\eta_{-}^{-1}(\pi_{*}- 1),
\chi_{T}= 2^{4 / 3}\pi^{2 / 3}\kappa^{- 2 / 3}N m r_{+}^{- 2}\frac{r_{+}^{2}+ 2 l^{2}}{3 r_{+}^{4}+ 5 l^{2}r_{+}^{2}+ 2 l^{4}}\frac{3 r_{+}^{2}+ 2 l^{2}}{3 r_{+}^{2}- 2 l^{2}}.
Z_I(u_0) = \sum_{\substack{u_1, \ldots, u_{2t-1}: d(u_0, u_i) \,=\, d(u_0, u_{2t-i}) \,=\, i \,\, \forall i \in [t], u_i \,=\, u_{2t-i}\,\, \forall i \in I.}}W\!\left(u_0, \ldots, u_{2t-1}, u_0\right),
{\cal{Z}}^{P F}( X_{1}, \cdots , X_{M}; p ) ={\frac{\det ( X_{j}^{2 M + p + 1 - i}- X_{j}^{i})}{\det ( X_{j}^{2 M + 1 - i}- X_{j}^{i})}}; 1 \le i , j \le M .
d{\mathcal V}_m(t) =\sigma \,dW_1(t) d{\mathcal X}_m(t) =\sqrt{m}(f(V^{(m)}(t),X^{(m)}(t))-f(v_t,x_t))\,dt &+\sqrt{\lambda\delta+\delta f(V^{(m)}(t),X^{(m)}(t))}\,dW_2(t),
\int\limits_{\mathbb{R}_+}\left||v_{j,n}|^2\log|v_{j,n}|^2-|v_{j}|^2\log|v_{j}|^2\right|dx\leq \int\limits_{\{|v_{j,n}|\geq |v_{j}|\}}\left(1+|\log|v_{j,n}|^2|\right)\left||v_{j,n}|^2-|v_{j}|^2\right|dx &+ \int\limits_{\{|v_{j,n}|\leq |v_{j}|\}}\left(1+|\log|v_{j}|^2|\right)\left||v_{j,n}|^2-|v_{j}|^2\right|dx.
A =(-d)[[\cdots [F_{3,1},F_{2,0}]_{q},\cdots,F_{n-2,0}]_q,F_{n-1,0}]_{q}&=(-d)[[[\cdots [F_{3,1},F_{4,0}]_{q},\cdots,F_{n-2,0}]_q,F_{n-1,0}]_{q},F_{2,0}]_{q}\,,
\omega\big[\kappa(\theta)\big]\phi=\nu\begin{bmatrix}\kappa(\theta)\otimes I 0 0 \kappa(\theta)\otimes I \end{bmatrix}\phi=(z_{11}\cos\theta -z_{21}\sin\theta )^r(z_{11}z_{22}-z_{21}z_{12})^{\mu_2}z_{33}^\nu.
S_n\bigl(\lambda, \lambda_1,\lambda_2\bigr) \triangleq \int\limits_{0<t_1<\cdots<t_n<1}\prod\limits_{i=1}^n t_i^{\lambda_1 d(i)}(1-t_i)^{\lambda_2 d(n-i+1)}\prod\limits_{k<p}^n |t_p-t_k|^{2\lambda\,d(p-k)}\,dt
\sum_{i\neq l}^k\int_{\mathcal{A}_l}U_{\mu_i,\xi_i}\bigg|\ln\ln \Big(e+(-1)^lU_{\mu_l,\xi_l}+\sum_{i\neq l}^k(-1)^iU_{\mu_i,\xi_i}\Big)\bigg|^{\frac{2n}{n+2}}dx=O\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big).
\begin{array}{r c l}{\omega^{x}}{=}{2 \, f_{t t}\, d f_{t x}\wedge d f_{t t}+ \frac{3}{2}\, d f_{x}\wedge d f_{t x},}{\omega^{t}}{=}{2 \, f_{t x}\, d f_{t t}\wedge d f_{t x}- \frac{1}{2}\, d f_{x}\wedge d f_{x x}}\end{array}
f\left( t \right) +{I_{{0^+}}^\alpha \,I_{{b^-}}^\alpha \left(\lambda + q \left(t\right) \right)f\left( t \right) -{{\left({\frac{t}{b}}\right)}^\alpha}{{\left.{I_{{0^+}}^\alpha \,I_{{b^-}}^\alpha \left(\lambda + q \left(t\right) \right)f\left( t \right)}\right|}_{t = b}}}={\frac{L}{b^\alpha}}t^\alpha .
( \Delta \Gamma_{\mu}\Delta )^{M}( p^{\prime}, p ; k ) = \int \bar{d}^{4}\! k^{\prime}\frac{( p + p^{\prime}- 2 k^{\prime})_{\mu}E^{M}( k^{\prime})}{[ ( p^{\prime}- k^{\prime}) - m^{2}] [ ( p - k^{\prime})^{2}- m^{2}]}.
\mu(R_\lambda \cap R_\nu)= \mu(\tau_{\lambda_1}( R_{\lambda_2}\cap R_{\nu_2})) = \int_{R_{\lambda_2}\cap R_{\nu_2}}1 \, d(\mu \circ \tau_{\lambda_1}) = \int_{R_{\lambda_2}\cap R_{\nu_2}}\Phi_{\lambda_1}\, d\mu = 0.
\dot{\widetilde{\delta p}}+ \left( 3{\dot{a}_b \over a_b}+{\dot{n}_b \over n_b}\right) \widetilde{\delta p}= k^2 \widetilde{\delta \pi}+ 6 \left({\rho+P\over\lambda}\right) \left({\dot{a}_b \over a_b}\delta p -{k^2 \over2}\delta\pi \right) \,.
0 \mkern1.5mu\ge\mkern1.5muy'_-\mkern-1.2mu(1) \mkern1.5mu+\mkern1.5mu (s_{\mkern-1mu+}\mkern-3mu-\mkern-1.5mu1)\mkern.3mu y'_-\mkern-1.2mu(s_{\mkern-1mu+}\mkern-3mu-\mkern-1.5mu1) &\mkern1.5mu=\mkern1.5mu([y_1^{\textsc{d}}] + [-y_1^{\textsc{d}}])-([-y_{1+}^{\textsc{g}}] + [y_{1+}^{\textsc{g}}]) \mkern1.5mu\ge\mkern1.5mu0.
K_{i\bar j}= \frac 12 \left( -\mu_{\bar ki}^r\mu_{k\bar j}^{\bar r}+\mu_{k\bar r}^{\bar i}\mu_{\bar k r}^j -\mu_{i\bar r}^{\bar k}\mu_{\bar j r}^{k}+\mu_{k\bar k}^{\bar i}\mu_{\bar r r}^j-\mu_{k \bar k}^{\bar r}\mu_{\bar ji}^{r}- \mu_{\bar kk}^{r}\mu_{i \bar j}^{\bar r}\right) \,.
t^{d}\sum_{r=0}^\infty h_r(a_1, \ldots , a_d) t^r= t\sum_{i=1}^d \sum_{k=0}^\infty a_i^k t^k \prod^d_{\substack{j=1 j\neq i}}\frac{1}{a_i-a_j}& \sum_{r=0}^\infty h_r(a_1, \ldots , a_d) t^{r+d-1}= \sum_{k=0}^\infty\Big( \sum_{i=1}^d a_i^k \prod^d_{\substack{j=1 j\neq i}}\frac{1}{a_i-a_j}\Big) t^k.
\mathbb{P}_0\left(\widetilde{\tau}^{(n)}_{M_n(b)}>(1+\epsilon)t_n\right) =\int_0^\infty\lambda_n(b)e^{-\lambda_n(b)s}\mathbb{P}_0(S_n(b)>(1-\epsilon)t_n-s)ds &\ge\mathbb{P}_0(S_n(b)>(1+\epsilon)t_n-r)e^{-\lambda_n(b)r}.
w_k =\left(\sum_{k \in \mathcal{K}}y_k^2\right)|(\boldsymbol{v}_{k})^{\mathrm{H}}\boldsymbol{g}_{k}|^{2}{E}_{k}>0, o_k =y_k\sqrt{(1+\eta_k)\left|\left(\boldsymbol{v}_{k}\right)^{\mathrm{H}}\!\boldsymbol{g}_{k}\right|^{2}{E}_{k}}>0, s_k = -\sqrt{\frac{z_k^2}{4C_k}}(\frac{E_k}{L\kappa_k})^{\frac{1}{4}}<0.
{\mathrm{\boldmath ~ e ~}}( u = 0 , t ) = \operatorname*{l i m}_{\epsilon \rightarrow + 0}{\mathrm{\boldmath ~ e ~}}( \epsilon , t ) ={\mathrm{\boldmath ~ e ~}}( t ) \equiv \frac{\dot{\mathrm{\boldmath ~ x ~}}( t )}{| \dot{\mathrm{\boldmath ~ x ~}}|}\; ;
\makebox [ 2 . 8 c m ]{}\times \frac{( 2 x_{1}+ x_{2})^{2}( x_{1}+ 2 x_{2})^{2}}{\Big [ ( 2 x_{1}+ x_{2})^{2}- E^{2}\Big ] \Big [ ( x_{1}+ 2 x_{2})^{2}- E^{2}\Big ]}
(D-\mu)\Phi_n= \left[E, y c_n(\log y)\,y^{\alpha}\sum_{i=1}^k(\partial_i b)\, e_i\cdot \psi_0 \pm b(x) c_n'(\log y) y^{\alpha}\i \psi_0\right. &\left. + b(x)c_n(\log y) (\pm \i\alpha \mp \i\frac{k}{2}- \mu) y^\alpha \psi_0\right].
\mathcal H_0(\rho,S)=\mathcal K(S,\rho)+\mathcal F(\rho)-\alpha L(\rho),\; &\mathcal H_1(\rho,S)=\eta_1 \mathcal K(S,\rho)+\eta_2 I(\rho)+\eta_3\mathcal V(\rho)+\eta_4\mathcal W(\rho)-\eta_5 L(\rho)
\pi_{\widehat{\alpha}}^{\P}\pi_{\widehat{\beta}}^{\P}= \sum_{\widehat{\gamma}}c_{\widehat{\alpha}\widehat{\beta}}^{\widehat{\gamma}}\pi_{\widehat{\gamma}}^{\P}c_{\widehat{\alpha}\widehat{\beta}}^{\widehat{\gamma}}= c_{\alpha,\beta}^{\gamma}.
\sum_{i=j-1}^{j+1}I_i(x)\lesssim\left(u_{p\wedge q,\widetilde{\lambda}}^*\right)_{Q^*}+\left(u_{p\wedge q,\widetilde{\lambda}}^*\right)_Q+\left(u_{p\wedge q,\widetilde{\lambda}}^*\right)_{Q^{**}},
G_{2 \mu \nu}^{h}( x , \ldots ) = \partial_{\mu}G_{\nu}^{h}( x , \ldots ) + \partial_{\mu}G_{\mu}^{h}( x , \ldots ) - \frac{2}{D}\delta_{\mu \nu}\partial_{\lambda}G_{\lambda}^{h}( x , \ldots ) ,
\left( \sum_{j_{1},\dots,j_{m}=1}^{\infty}\left\vert A(e_{j_{1}},\dots,e_{j_{m}})\right\vert^{\frac{1}{1-\left\vert 1/\mathbf{p}\right\vert}}\right)^{1-\left\vert 1/\mathbf{p}\right\vert}\leq D_{m,\mathbf{p}}^{\mathbb{K}}\Vert A\Vert
\Delta_{\mathbf{L}(\mathbf{t})}(\mathcal{I})\cdot\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I})=\mathbf{t}^{\mathbf{m}_{0}}\cdot\Delta_{\mathbf{L}(\mathbf{1})}(\mathcal{I})\cdot\Delta_{\mathbf{R}(\mathbf{1})}(\mathcal{I})\cdot\prod_{\alpha\in\mathcal{I}}\mathbf{t}^{\psi(\alpha)},\quad\mathcal{I}\in\wp_{k}[n].
\pi^*{\rm vol}_{g(q_0)}(v_1 ,\ldots ,v_k)={\rm vol}_{g(q_0)}(\pi_* v_1 ,\ldots ,\pi_* v_k) \leq |{\rm vol}_{g(q)}(\pi_* v_1 ,\ldots ,\pi_* v_k) | \leq{\rm vol}_{V}(v_1 ,\ldots ,v_k).
\frac{\partial^{2}\varphi_{L}( t )}{\partial t^{2}}+ 3 \Bigl ( \frac{\partial a ( t ) / \partial t}{a ( t )}\Bigr ) \frac{\partial \varphi_{L}( t )}{\partial t}+ \frac{\delta^{2}V ( \hat{\phi})}{\delta \hat{\phi}^{2}}\varphi_{L}( t ) = 0 .
\sum_{j = 0}^{i}q^{j r}\chi_{1 , k ( r + 1 ) + 1 - i + 2 j}^{( p , k p + 1 )}( q ) = \displaystyle \sum_{n_{1}\geq \cdots \geq n_{k}\geq 0}\frac{q^{n_{1}^{2}+ \cdots + n_{k - 1}^{2}+ 2 n_{k}^{2}+ r ( n_{1}+ \cdots + n_{k - 1}+ 2 n_{k}) - ( n_{1}+ \cdots + n_{i})}}{( q )_{n_{1}- n_{2}}\cdots ( q )_{n_{k - 1}- n_{k}}( q )_{2 n_{k}+ r}}\overline{{{F}}}_{p - 1 ; r}^{( 2 n_{k}+ r )}( q ) .
\sum_{l=j}^{i}a_{l}\frac{a_{i}}{a_{N}}\prod_{k=l+1}^{N}\left(1-\frac{a_{k}}{a_{j}}\right)=a_{j}\frac{a_{i}}{a_{N}}\prod_{k=i+1}^{N}\left(1-\frac{a_{k}}{a_{j}}\right),
H_{(X,K3)}(y_1,y_2)=e^{\frac{H\log y_1+P\log y_2}{2\pi i}}\sum_{d_1,d_2\geq 0}y^d \frac{\Gamma(1+\frac{4H}{2\pi i}+4d_1)\Gamma(1+\frac{H+P}{2\pi i}+d_1+d_2)}{\Gamma (1+\frac{H}{2\pi i}+d_1)^5\Gamma(1+\frac{P}{2\pi i}+d_2)}[\textbf{1}]_{(d_1,0)},
\mathcal{J}_1 := \{j \in \{2, \dots, J \}\mid \forall n, \s_{j,n}=0 \}\mathcal{J}_2 := \{j \in \{2, \dots, J \}\mid \ \lim_{n \to +\infty}- \frac{s_{j,n}}{\mu_{j,n}}= \pm \infty \}.
\lim_{r \rightarrow \infty}\frac{\mu \left( B^T_{\xi, r}(z) \cap R_{(\omega_1 , \ldots , \omega_N )}\right)}{\mu \left( B^T_{\xi, r}(z) \right)}= \rho_{\xi, (\omega_1 , \ldots , \omega_N )}( z )
\begin{array}{l}\ddot M(x,\lambda)=-\int_0^x M(x,\lambda)M(t,\lambda)^{-1}R\,2\dot M(t,\lambda)dt = -\int_0^x \big(E_\lambda(x)+ \hat M(x,\lambda) \big)\big(E_\lambda(-t)+ (M(t,\lambda)^{-1})\hat{}\big) R\,2\big(\dot E_\lambda(t)+ \hat{\dot M}(t,\lambda) \big)dt\end{array}
\sum^{\lfloor p^{l}/e\rfloor}_{\substack{r=1 p\nmid r}}\frac{1}{r^{2}}\equiv -J_{e}(p^{l})\frac{B_{\phi(p^{l})-1}(\frac{1}{e})}{\phi(p^{l})-1}\pmod{p^{l}}.
[\begin{matrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{matrix}]=[\begin{matrix}14&11&13&9\\ 9&14&11&13\\ 13&9&14&11\\ 11&13&9&14\end{matrix}][\begin{matrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{matrix}]
d_+ f = T_1 T_2 \ldots T_{\ell}f[X + (t-1)y_{\ell+1}] d_- f = \left. -y_{\ell}f[X - (t-1)y_{\ell}] \sum_{i \geq 0}h_i[-X/y_{\ell}] \right|_{y_{\ell}^0}
{P^{\textrm{proposed}}_{{\mathop{\textrm{int}}}}}= \prod\limits_{i = 1}^N{\left[{\frac{{\zeta ({m_i},{k_i},{\alpha_{ie}})}}{{{m_i}\Gamma ({m_i})}}{{\left({\frac{{{m_i}}}{{{\alpha_{is}}}}}\right)}^{{m_i}}}}\right]}&\cdot{\left({\frac{1}{{{\lambda_{me}}}}}\right)^{\sum\limits_{i = 1}^N{{m_i}}}},
y_1^{j_1}y_2^{j_2}\cdots y_{n-1}^{j_{n-1}-1}y_iy_{n-1}=(k_1'Y_1'a_1'+k_2'Y_2'a_2'+\cdots+k_s'Y_s'a_s')y_{n-1}&=\sum_{i=1}^s(-1)^{|x_{n-1}||a_i'|}k_i'Y_i'(y_{n-1}a_i'-\{x_{n-1}, a_i'\})
V_{i j}^{a}\ = \ \frac{1}{2 \sqrt 2}\Sigma^{2}f_{x i j}^{a}K^{x}, \qquad T_{i j}^{a}\ = \ \frac 1 2 \Sigma^{- 1}L_{i}^{I}{}^{k}f_{x k j}^{a}K_{I}^{x},
\begin{aligned}\eta_i^\varepsilon(t) =\eta^\varepsilon(\tau_i, u_i| \bar \tau_i, \bar u_i)(t) =\dfrac{\varepsilon^2}{2}(u_i(t)-\bar u_i(t))^2 - P(\tau_i(t)|\bar \tau_i(t)). \end{aligned}
A_{0}\bar{\partial}g_{0}^{f}({g_{0}^{f}})^{- 1}+ \bar{A}_{0}({g_{0}^{f}})^{- 1}\partial g_{0}^{f}+ A_{0}g_{0}^{f}\bar{A}_{0}({g_{0}^{f}})^{- 1}+ A_{0}\bar{A}_{0}
\sigma^{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \qquad \sigma^{2}= \left( \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right) , \qquad \sigma^{3}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) .
\|\frac{g_1+g_2}{2}\|_{\partial M,\rho}^2+\|\frac{g_1-g_2}{2}\|_{\partial M,\rho}^2=\frac{\|g_1\|_{\partial M,\rho}^2+\|g_2\|_{\partial M,\rho}^2}{2}=M_H(Z_0,J,\rho),
\exp\Big\{t\mathrm{E}\sum_{i=1}^k Y_{[i]}\Big\}\le\mathrm{E}\exp\Big\{t\sum_{i=1}^k Y_{[i]}\Big\}\le \sum_{I: |I|=k}\mathrm{E}\exp\Big\{t\sum_{i\in I}Y_i\Big\}\le \tbinom{n}{k}A_k(t).
\int_0^t\int_0^1 \int (v_1-u_1)^2 B{\chi}_jff_* F_j(f')F_j(f'_*)(s,x,v,v_*,\theta )dvdv_*d\theta dxds &\leq c\int_0^t \int_0^1 \int (v_1-u_1)^2 ff_*(s,x,v,v_*)dvdv_* dxds &= \frac{c}{2}\int_0^t \int_0^1 \int (v_1-v_{*1})^2 ff_*(s,x,v,v_*)dvdv_*dxds &\leq c(1+t).
\Big| \int_{t_1}^{t_2}e^{iL(\xi,s)}I_{k_1,k_2,k_3}^{++-}(\xi,s) \, ds \Big| \lesssim | N_1 (\xi,t_1) | + | N_1 (\xi,t_2) | + \int_{t_1}^{t_2}| N_2(\xi,s)|+ | N_3(\xi,s) | + | N_4(\xi,s)| + |(\partial_sL)(\xi,s)|| N_1(\xi,s) | \, ds,
\mu^{\prime \prime}+ \biggl \{\frac{4 \pi^{2}}{l_{\mathrm{C}}^{2}}a^{2}\operatorname{t a n h}^{2 / p}\biggl [ \biggl ( \frac{l_{\mathrm{C}}}{\lambda ( \eta )}\biggr )^{p}\biggl ] - \frac{a^{\prime \prime}}{a}\biggr \}\mu = 0 .
\int\limits_0^T\int\limits_0^{\ell}\Big[-B(x,t,v(x,t))\psi_t+v_x\psi_x-f\psi\Big]\,dxdt - \int\limits_0^{\ell}B_0(x,0,\Phi(x))\psi(x,0)\,dx - - \int\limits_0^Tp(t)\psi(\ell,t)\,dt+\int\limits_0^Tg(t)\psi(0,t)\,dt=0, \forall \psi\in W_2^{1,1}(D),\quad\psi(x,T)=0.
\Big\{\frac{\|V\|(B_\rho(x)) + \|V\|(\tilde{B}_\rho(x)) + 2\sigma\mathcal{H}^n\lfloor_{B^+}(B_\rho(x))}{\omega_n\rho^n}\Big\}^{\frac1p}\Big(1+C\varkappa\rho\Big(1+\dfrac{1}{p-n}\Big)\Big) &+ \dfrac{\Gamma \rho^{1-\frac{n}{p}}}{p-n}
\tau_{g\circ f}=F(g\circ f,C'')\circ \tau_{C''}= F(C,g\circ f)\circ \tau_C =F(f,C'')\circ F(g,C'')\circ \tau_{C''}=F(f,C'')\circ \tau_g =F(C,g)\circ F(C,f)\circ\tau_C = F(C,g)\circ \tau_f.
\mathrm{var}[\widehat{K}_{\mathrm{a-cpt-d}}^\mathbb{R}]\!=\!\frac{\psi}{2}\Big(\frac{1}{N_2\bar{\gamma}_c'}\!-\!\frac{\mathrm{li}(1\!-\!\xi)}{1-\xi}\sum_{i\in\mathcal{K}'}\frac{1}{1\!+\!N_1\bar{\gamma}_i}\Big)
S_{ij}= g^{AB}{\cal R}_{\mu\nu}\eta^{\mu}_{A}\eta^{\nu}_{B}g_{ij}- \frac{1}{2}g^{AB}{\cal R}_{\mu\nu\rho\sigma}\eta^{\mu}_{A}\eta^{\sigma}_{B}Z^{\nu}_{,i}Z^{\rho}_{,j}
\sum_{n<X}\mathbf{1}_{\mathcal{R}}(n)=\frac{X}{\log{X}}\idotsint\limits_{(e_1,\dots,e_r)\in\mathcal{R}}\frac{d e_1\dots d e_{r-1}}{\prod_{i=1}^{r}e_i}+O_{\mathcal{R}}\Bigl(\frac{X}{(\log{X})^2}\Bigr).
Z ( t ) ={\frac{1}{\sqrt{2 \pi}}}{\frac{1}{g \hbar^{2}t^{3 / 2}}}\, \operatorname{e x p}\! \left({\frac{t v^{4}}{4 g^{2}}}\right) K_{0}\! \left({\frac{t v^{4}}{4 g^{2}}}\right) .
r^2G(r)\rightarrow 1+\frac a4\ln (\mu r)+\frac{a^2}{64}\ln (\mu r)\left(1+2\ln (\mu r)\right) +\frac{a^3}{1536}\ln (\mu r)\left( 9+6\ln (\mu r)+4\ln^2(\mu r)\right) +O(a^4).
-\Delta_p \overline{u}_\lambda-\Delta_q \overline{u}_\lambda = \lambda\tilde{u}_\lambda^{-\eta}+\lambda a(x)\overline{u}_\lambda^{\tau-1}+f(x,\overline{u}_\lambda) \Omega, \overline{u}_\lambda\big|_{\partial \Omega}=0.
\frac{1}{i \hbar}[ W_{1}, W_{2}] = \{W_{1}, W_{2}\}_{M}= \frac{2}{\hbar}\sin \frac{\hbar}{2}( \frac{\partial}{\partial p_{2}}\frac{\partial}{\partial \lambda_{1}}- \frac{\partial}{\partial p_{1}}\frac{\partial}{\partial \lambda_{2}}) W_{1}W_{2}
\left(\frac{1}{p_2}, \frac{1}{q_2}\right) = \left(\frac{(n-1)(1-\sigma)}{n+1},\frac{n-3 + 4\sigma}{2(n+1)}\right) = \left(\frac{n-1}{n+1}(1-\sigma), \frac{1}{2}- \frac{2 (1-\sigma)}{n+1}\right).
|M_m(x)| \leq\exp\left(-\int_{[|x|]}^{|x|}\frac{B(u)-m^6+1}{u}du\right)\leq \exp\left(-\int_{[|x|]}^{|x|}\frac{|x|-1-m^6+1}{u}du\right) =\exp\left((m^6-|x|)\ln\frac{|x|}{[|x|]}\right) \leq C\exp(m^6-|x|),
B(\vec\gamma) := \Theta^{-1}(\vec\gamma) \cong\bigl\{f \in R \; \big| \; T_{(\alpha, 2\mu_\alpha)}^-(f) = \gamma_\alpha \cdot T_{(\alpha, 2\mu_\alpha)}^+(f) \;\; \mbox{\rm for all}\;\;\alpha \in \Pi\bigr\}.
\sum_{u\le p< z}\frac{1}{p}=\sum_{p<z}\frac{1}{p}-\sum_{p<u}\frac{1}{p}\le\log\log z-\log\log u+\frac{1.445\cdot 10^{-2}}{\log z}+\frac{2.964\cdot 10^{-6}}{\log u}<\log\log z-\log\log u+1.6236\cdot 10^{-3}+\frac{2.964\cdot 10^{-6}}{\log u}
H ={\cal H}+ ( i \hbar )^{- 1}[ \Omega , \bar{\cal P}_{\alpha}] \lambda^{\alpha}+ \bar{\cal P}_{\alpha}{\cal P}^{\alpha}+ \pi_{\alpha}\chi^{\alpha}+ \bar{C}_{\alpha}( i \hbar )^{- 1}[ \chi^{\alpha}, \Omega ] .
\tilde{j}_{q}^{( 2 )}( \tilde{x}, t ) = 2 j_{q 2}( \tilde{x}, t ) - 4 j_{q 1}( \tilde{x}, t ) - 4 j_{q 0}( \tilde{x}, t ) - \frac{1}{6}j_{\mathrm{u n}}( \tilde{x}, t ) .
V ={\frac{3}{4 f}}\left( f^{\prime \prime}-{\frac{1}{4 f}}f^{\prime}f^{\prime}\right) + m^{2}f ={\frac{3}{4}}\left( 5 f - 3 -{\frac{c_{0}^{2}}{8 f^{3}}}\right) + m^{2}f \ ,
\omega_{n'}(y_i - y_i',y_j-y_j') = \omega_{n'}(y_i,y_j) - \omega_{n'}(y_i',y_j) - \omega_{n'}(y_i,y_j') + \omega_{n'}(y_i',y_j') &= 2 \omega_{n'}(y_i',y_j') - 2 \omega_{n'}(y_i',y_j') = 0,
E_{n \ell}\approx \hbar \omega \left[ \left( n + \frac{D}{2}\right) + \frac{1}{2}\left\{( k^{2}+{k^{\prime}}^{2}) \left( n + \frac{D}{2}\right)^{2}+ ( k^{2}-{k^{\prime}}^{2}) \left( L^{2}+ \frac{D^{2}}{4}\right) +{k^{\prime}}^{2}\frac{D}{2}\right\}\right] \; .
\partial_{\iota}\tilde{h}-\tilde{a}\left(r,\,\iota\right)\partial_{rr}^{2}\tilde{h}-\tilde{b}\left(r,\,\iota\right)\partial_{r}\tilde{h}-\tilde{c}\left(r,\,\iota\right)\tilde{h}=\tilde{f}\left(r,\,\iota\right)
|L(v)|/n^2 \le \binom{t}{2}\left(\frac{1}{t+2}+\epsilon\right)^2 + 2\times 102t^2\epsilon^{1/4}+ 2\epsilon^{1/4}& = \frac{t(t+1)}{2(t+2)^2}- \frac{t}{(t+2)^2}+\frac{t(t-1)}{t+2}\epsilon + \frac{t(t-1)}{2}\epsilon^2+204t^2\epsilon^{1/4}+ 2\epsilon^{1/4}& < 3(\lambda-\delta),
\frac{k}{2\pi}\partial_{+}gg^{-1}\equiv a^{I}G^{I}\equiv \left( J^{3}\frac{\sigma^{3}}{2}+J^{+}\sigma^{+}+J^{-}\sigma^{-}\right) +\left( \frac{d-1}{2C_{\rho}}B^{a}T^{a}\right)+\left( F_{+\alpha}R^{+\alpha}+F_{-\alpha}R^{-\alpha}\right)
\begin{aligned}\frac{dS(t)}{dt}= \mu \cdot (1-S(t)) -\beta \cdot S(t) \cdot V(t) =g_S \frac{dI(t)}{dt}=\beta \cdot S(t) \cdot V(t) -(\mu+\gamma)\cdot I(t) =g_I \frac{dV(t)}{dt}=\vartheta \cdot (1-V(t)) \cdot I(t) -\nu \cdot V(t) =g_V \end{aligned}
\sum_{i=1}^{a-1}\epsilon_i = 2\eta L \sum_{i=1}^{a-1}\left( \eta KL \right)^{i-1}< 2\eta L \sum_{i=0}^\infty \left( \eta KL \right)^i = 2\eta L \left( \frac{1}{1-\eta KL}\right),
\begin{aligned}{f_k}\left({{x_k}}\right) ={\log_2}\left({{1 +{{\bar C}_k}\left({1 -{2^{- 1 - \left({{x_k}+ 1}\right)}}}\right)}}\right) -{\log_2}\left({{1 +{{\bar C}_k}\left({1 -{2^{- 1 -{x_k}}}}\right)}}\right) ={\log_2}\left({1 + \frac{1}{{{2^{2 +{x_k}}}\left({1/{{\bar C}_k}+ 1}\right) - 2}}}\right).\end{aligned}
\mathrm{tr}(B_q^{2j-1}) = \mathrm{tr}(B_q^{2(j-1)}B_q)=\mathrm{tr}((q A_q)^{j-1}B_q) &= \mathrm{tr}(q^{j-1}q^{j-2}A_q B_q)= q^{2j-3}\mathrm{tr}(A_q B_q).
P_{R}^{\frac{1}{2}}( k ) = 2^{\mu - \frac{1}{2}}\frac{\Gamma ( \mu )}{\Gamma ( 3 / 2 )}( \mu - \frac{1}{2})^{\frac{1}{2}- \mu}\frac{1}{m_{p l}^{2}}\frac{H^{2}}{| H^{\prime}|}|_{k = a H}