harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[V] = \frac{[\mathbb C_x]}{D}D = \prod_{n=1}^\infty (1- e^{-n\delta})^{r}\prod_{n=1}^\infty \prod_{\alpha \in \Delta_+^0}(1-e^{-\alpha}e^{(1-n)\delta}) (1- e^{\alpha}e^{-n\delta}).
-\bar{R}_{\mu\nu}F^{\rho\mu}F_{\rho}~^\nu+\frac 14 F^{\lambda\nu}F^{\mu\rho}F_{\lambda\mu}F_{\nu\rho}-{1 \over 16}(F^{\mu\nu}F_{\mu\nu})^2+\frac 12 F_{\mu\rho}~^{;\rho}F^{\mu\lambda}~_{;\lambda}~~.
\sigma_L(H_n) = \sigma_L(H_1) = \sigma_L\left(\begin{bmatrix}0 Q^Te_1 (Q^Te_1)^T 0\end{bmatrix}\right) = \sigma_L\left(\begin{bmatrix}0 e_1 e_1^T 0\end{bmatrix}\right) = \{\pm1\},
e^{S_{I_{k;q}}}\,V^{(k,q-1)}_{I_{l;i}}\,e^{-S_{I_{k;q}}}=V^{(k,q-1)}_{I_{l;i}}+\sum_{n=1}^{\infty}\frac{1}{n!}\,ad^{n}S_{I_{k;q}}(V^{(k,q-1)}_{I_{l;i}}) \,.
\tilde{\lambda}_v = \frac{\rho_v^{(\infty)}\tilde{p}_v \chi\left( \gamma \left( \pmb{\rho}^{(\infty)}\right) \right)e^{- \gamma \left( \pmb{\rho}^{(\infty)}\right)}}{e^{- \gamma\left( \pmb{\rho}^{(\infty)}\right)}+ \tau\left( 1 - e^{- \gamma\left( \pmb{\rho}^{(\infty)}\right)}\right)}\,.
\xi = ([\xi_0^i: \xi_1^i])_{i=1}^N{\xi}' = ([{\xi'}_0^i :{\xi'}_1^i])_{i=1}^N \omega = ([\omega_0^{(k,l)}: \omega_1^{(k,l)}])_{1 \leq k<l\leq N}{\omega}' = ([{\omega'}_0^{(k,l)}:{\omega'}_1^{(k,l)}])_{1 \leq k<l\leq N}
H_{i,j}(v)=\sum_{k\notin\{i,j\}}v_k^\alpha\;, H_i(v)=\sum_{j,k;\,i\neq j\neq k\neq i}v_j^\alpha v_k^\alpha = \sum_{j\neq i}v_j^\alpha H_{i,j}(v)\;, H(v)=\sum_{i,j,k;\, i\neq j\neq k\neq i}v_i^\alpha v_j^\alpha v_k^\alpha=\sum_{i}v_i^\alpha H_i(v)\;.
\begin{aligned}h_j =\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}\tilde U_{aj}(t)^*U_a(t)\tilde \Omega_j f &=\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}\tilde U_{aj}(t)^*\varphi(H^a)(\psi_{aj}(H^a)-P^a_j)P_a^{{\varepsilon}_a}(t)e^{-itH}f.\end{aligned}
d_{\max}(b_1,b_2/C) = \inf_{c\in C}(d(b_1,c)+d(b_2,c)), d_{\min}(b_1,b_2/C) = \max\left\{\sup_{c\in C}|d(b_1,c)-d(b_2,c)|,\textstyle\frac{1}{3}\displaystyle d(b_1,b_2)\right\}.
[f(s|D|),x]-sf'(s|D|)\delta(x) = s\int_{-\infty}^{\infty}\Big(\int_0^1\widehat{f'}(u)\Big(e^{ius(1-v)|D|}\delta(x)e^{iusv|D|}-e^{ius|D|}\delta(x)\Big)dv\Big)du = s\int_{-\infty}^{\infty}\Big(\int_0^1\widehat{f'}(u)\Big(e^{ius(1-v)|D|}[\delta(x),e^{iusv|D|}]\Big)dv\Big)du.
\sum_{u\in V}x_{u}^{2}d\left( u\right) =\sum_{u\in V\backslash V_{k}}d\left( u\right) +\sum_{u\in V_{k}}\left( \chi-1\right)^{2}d\left(u\right) & =2m+\chi\left( \chi-2\right) e_{k}.
c(u_1\otimes u_1) = \xi^{mi+tj}u_1\otimes u_1, c(u_1\otimes u_2) = \xi^{it+mj}u_2\otimes u_1, c(u_2\otimes u_1) = \xi^{it+mj + 2(i^{2}-j^{2})}u_1\otimes u_2, c(u_2\otimes u_2) = \xi^{mi+tj}u_2\otimes u_2.
-\frac{2Ric'_{ij}f_{i}f_{j}}{(1-f)^{2}}+\frac{2f_{j}f_{jji}}{(1-f)^{2}}-\frac{2f_{i}f_{ijj}}{(1-f)^{2}}=-\frac{4Ric'_{ij}f_{i}f_{j}}{(1-f)^{2}},
\frac{M_{\beta}(s)}{M_{\beta}(s+1)}= \frac{\Gamma(1+a+\gamma s)\Gamma(1+b+\gamma s)\Gamma(1+(s-1)\gamma)\Gamma(2+a+b+(s+1)\gamma)}{\Gamma(2+a+b+2s\gamma)\Gamma(2+a+b+(2s+1)\gamma)}
\langle u^+,v^+\rangle_{*\mathcal R^+}=\int_0^\infty \Big[\,A_+\big({t^+}^2-{U^+}^2\big)-B\big({t^-}^2-{U^-}^2\big)\,\Big](u^+_1v^+_1+u^+_2v^+_2)r\,{\mathrm d}r,
s\left( c,d;A_{c},B_{b}\right) +s\left( d,c;B_{b},A_{c}\right) & =-P_{1}\left( 0,B_{-b}\right) P_{1}\left( 0,A_{-c}\right) +\frac{1}{2dc}B_{0}(B)B_{2}(A)+\frac{d}{c}B_{0}(A)P_{2}\left( 0,B_{b}\right)+\frac{c}{d}B_{0}\left( B\right) P_{2}\left( 0,A_{c}\right) -f(0)P_{1}\left( 0,B\right) .
\begin{array}{rll}\Pi^{\mu\nu,ab}(x,y) = \int \frac{d^4p}{(2\pi )^4}\Pi^{\mu\nu,ab}(p)\exp (-ip(x-y)) = -ig^2C_A\frac 1{(4\pi )^2}\frac{11}{3}\delta^{ab}(g^{\mu\nu}\partial^2-\partial^{\mu}\partial^{\nu}) H(x-y).\end{array}
\tilde{Q}_{m,n}^a(x_{\cdot}):=\frac{1}{Z_{m,n}(a)}\prod_{i=1}^{m+n}\tilde{\omega}_{(x_{i-1},x_i)}=\frac{1}{Z_{m,n}(a)}\prod_{i=1}^{t_1(x_{\cdot})\wedge r}\tilde{R}^1_{i,0}\prod_{i=t_1(x_{\cdot})\wedge r+1}^{m+n}\omega_{(x_{i-1},x_i)}.
\frac{\partial}{\partial{t}}U_{i(k)}^{2,2}(x,t)= -\int\limits^t_{A}\int\limits_{\mathbb{R}^{3}_{+}}\frac{\partial}{\partial{t}}G_{ij}^{2}(x,y,t-\tau)\Big[\frac{\partial}{\partial y_{l}}F^{k}_{jl}(y,\tau)+\frac{\partial}{\partial{y_j}}p^{1(k)}(y,\tau)\big]dyd\tau.
\; b \neq 0 v_t(u) = \frac{u e^{-bt}}{\left(1+\frac{\delta^\alpha}{\alpha b}u^{\alpha-1}\left(1-e^{-(\alpha-1)bt}\right)\right)^{\frac{1}{\alpha-1}}}\; b=0 v_t(u) = u \left(\frac{\alpha}{\alpha + (\alpha-1) \delta^\alpha u^{\alpha-1}t}\right)^\frac{1}{\alpha-1}.
S_{\mathrm{I I B}}= \frac{1}{2 \kappa^{2}}\int d^{1 0}x \sqrt{- g}\left[ R + \frac{1}{4}\mathrm{T r}\left( \partial{\cal{M}}\partial{\cal{M}}^{- 1}\right) - \frac{1}{1 2}{\bf F}^{T}{\cal{M}}{\bf F}\right]
\widetilde{\kappa}_2(x) > x^{4n}- (4m+6)x^{3n}= x^{3n}(x^n - (4m+6)), \widetilde{\kappa}_1(x) > (7m+9)x^{n}, \widetilde{\kappa}_0(x) = x^{4n}- 2x^{2n-1}+ 1 \le x^{4n}- 1 < x^{4n}.
\lim_{n \rightarrow \infty}\int \int |\lambda_{n}^{j}v^{j}(\lambda_{n}^{j}t, \lambda_{n}^{j}x)|^{2}|\lambda_{n}^{k}v^{k}(\lambda_{n}^{k}t, \lambda_{n}^{k}x)|^{2}dx dt = 0.
\begin{cases*}\rho_1:= Q_0(\partial_x(\ln \alpha), \tilde{\Lambda})+Q_0(\ln \alpha, \partial_x \tilde{\Lambda}), \rho_2:= -2\Big[\sinh(2\lambda +2\tilde{\Lambda})\left(Q_0(\partial_x\phi,\phi)+Q_0(\phi,\partial_x\phi)\right)+2\partial_x\tilde{\Lambda}\cosh(2\lambda+2\tilde{\Lambda})Q_0(\phi,\phi)\Big].\end{cases*}
\lim_{n \rightarrow \infty}E\int_{t_0}^{t_1}I_{B(R)}\sup_{|x| \leq R}\{|b^n_t(x)-b_t(x)|^2+|\sigma^n_t(x)-\sigma_t(x)|^2\}dt =0 \lim_{n \rightarrow \infty}E \int_{t_0}^{t_1}I_{B(R)}\sup_{|x| \leq R}\int_{Z}|\gamma^n_t(x,z)-\gamma_t(x,z)|^2\nu(dz) dt =0.
\mu_n\left(\overline{\bigcup_{j\in E}C_{p_j}\setminus C_{p_{j-1}}}\right) =\mu_n\left(\bigcup_{j\in E_1}C_{p_j}\setminus C_{p_{j-1}}\right)+\mu_n\left(\overline{\bigcup_{j\in E_2}C_{p_j}\setminus C_{p_{j-1}}}\right) &\leq\sigma_n(D_1)+\sigma_n(D_2)
\Delta_{\nu}:=4(1+\langle z,z\rangle )\left( \sum_{i,j=1}^{n}(\delta_{ij}+z_{i}\bar{z_{j}})\frac{\partial^{2}}{\partial z_{i}\partial \bar{z_{j}}}-\nu \sum_{j=1}^{n}\left( z_{j}\frac{\partial}{\partial z_{j}}-\bar{z_{j}}\frac{\partial}{\partial \bar{z_{j}}}\right) -\nu^{2}\right) +4\nu^{2}
F(t) = \dfrac{d_R^2}{144 \pi D_\theta t}\exp \bigg(\dfrac{- (G_z - W_z t)^2}{4 D_\theta t}\bigg) \cdot \bigg( 4 \sum_{k = 0}^{3}\lambda(k,t) +2 \sum_{k=1}^{3}\xi(k,t) + 2 \omega(t_s) \bigg),
\int_0^t\!\!(\lambda \overline{b}_h^i(t)+(1-\lambda)\hat{b}_h^i(t))dt\leq\lambda\int_0^T\!\!\overline{b}_h^i(t)dt+\int_0^T\!\!\hat{b}_h^i(t)dt<\int_0^T\!\!\sum_{i=1}^r\xi_h^i(t)dt+R
\mathcal{P}(3) = \bigg\{\Big\{\{1\},\{2\},\{3\}\Big\}, \Big\{\{1,2\},\{3\}\Big\}, \Big\{\{1,3\},\{2\}\Big\}, \Big\{\{1\},\{2,3\}\Big\}, \Big\{1,2,3\Big\}\bigg \},
\mathbb{E}[G_{n}^{i}(z)Z^{i}(n,1)] ~~~ = \left( \prod_{j=i-1}^{n-1}\frac{2j+2+z}{2j}\right) \mathbb{E}[G_{i-1}^{i}(z)] + \sum_{k=i-1}^{n-1}\left( \prod_{j=k+1}^{n-1}\frac{2j+2+z}{2j}\right) \frac{1}{2k}\mathbb{E}[Z^{i}(k,1)],
\overline{\mu}_{-n}= \mu_{n}= \int_{\mathcal{C}}\zeta^{-n}d \hat{\mu}(\zeta) = \frac{1+c_1^2}{4d_1}[\nu_n - \nu_{n+1}] = \frac{(-b-1)_n}{(\overline{b}+2)_n}, n = 0, 1, 2, \ldots \ .
F_{\mu \nu}^{a}\left( x \right) = \partial_{\mu}A_{\nu}^{a}\left( x \right) - \partial_{\nu}A_{\mu}^{a}\left( x \right) + g f^{a b c}A_{\mu}^{b}\left( x \right) A_{\nu}^{c}\left( x \right) .
H\left(\theta^{(2)}_{1}(l) ~|~ \theta(l) = c_{j}\right) = - \sum_{k = 1}^{2^{m}}\mbox{Prob}\left(\theta^{(2)}_{1}(l) = b_{k}~| ~\theta(l) = c_{j}\right)\mbox{log}_{2}\left(\mbox{Prob}\left(\theta^{(2)}_{1}(l) = b_{k}~| ~\theta(l) = c_{j}\right)\right)
d_Y(Rf_\ast(\mathcal G))\overset{\eqref{eq:defofdecomphomoofschemes}}{=}(Rf_\ast\mathcal F)\otimes^L_{\mathcal O_E}\mathcal O_E/\mathfrak m_E= Rf_\ast(\mathcal F\otimes^L_{\mathcal O_E}\mathcal O_E/\mathfrak m_E)\overset{\eqref{eq:defofdecomphomoofschemes}}{=}Rf_\ast (d_X(G)).
F(0) = \left(\log \left(\cosh 1 + \frac{k (\eta \sinh 2 + 2 \gamma \sinh 1)}{4 (\gamma + \eta \cosh 1)^2}\right)^2\right)^2-4 \intertext{and}F\left(\frac{\pi}{2}\right) = \left(\log \left(\cos 1 - \frac{k(\eta \sin 2 + 2 \gamma \sin 1)}{4 ( \gamma +\eta \cos 1 )^{2}}\right)^2\right)^2-4
\begin{aligned}g_{1}=x_{j-1,1}^{s_{j-1}+2}(x_{j,1}x_{j,2})^{s_{j}}\prod_{\ell=i}^{j-2}x_{\ell,1}^{s_{\ell}+1}\prod_{\ell=j+1}^{n}(x_{\ell,1}x_{\ell,2})^{s_{\ell}+1}, g_{2}=x_{i,1}^{s_{i}}\prod_{\ell=i+1}^{j-1}x_{\ell,1}^{s_{\ell}+1}\prod_{\ell=j}^{n}(x_{\ell,1}x_{\ell,2})^{s_{\ell}+1}.\end{aligned}
N_{K(\beta)/K}\left(\beta-\frac{1}{c}\right) =\prod_{~\alpha \{\rm of}\g(f^{m-1}(z))}\left(\alpha-\frac{1}{c}\right)=(-1)^tg(\left(f^{m-1}\left(\frac{1}{c}\right)\right)=(-1)^tg(f^{(m)}(0))
\dim(E_2(\Gamma_0(40),\chi_1))=8,~\dim(S_2(\Gamma_0(40),\chi_1))=2, &\dim(E_2(\Gamma_0(40),\chi_2))=4,~\dim(S_2(\Gamma_0(40),\chi_2))=4, &\dim(E_2(\Gamma_0(40),\chi_3))=4,~\dim(S_2(\Gamma_0(40),\chi_3))=4.
V_{+}=\frac{-e^{2}}{4\pi\epsilon_{0}}\frac{1}{r}+\frac{h^{2}(l+1)(l+2)}{2m}\frac{1}{r^{2}}+\frac{e^{4}m}{32\pi^{2}h^{2}\epsilon_{0}^{2}(l+1)^{2}}
\frac{\mathsf{sgn}(x_j-x_i)\cdot d(x_j,x_i)}{h(x_j)-h(x_i)}= \frac{\mathsf{sgn}(x_j-x_i)\cdot d(x_j,x_i)}{\mathsf{sgn}(x_j-x_i) |h(x_j)-h(x_i)|}= \frac{d(x_i,x_j)}{|h(x_i)-h(x_j)|}>0,
p_{\mathrm{S}_1}(m | n) = \sum_{q=0}^{\infty}p(m|n,q)p(q) &= \sum_{q=0}^{\infty}\left(\int_0^\infty p_{\mathrm{S}_1}(t_\mathrm{p}|n,q)\frac{(\lambda_\mathrm{B}t_\mathrm{p})^{m}}{m!}e^{-\lambda_\mathrm{B}t_\mathrm{p}}dt_\mathrm{p}\right)\frac{(\lambda_\mathrm{B}t_\mathrm{s})^{q}}{q!}e^{-\lambda_\mathrm{B}t_\mathrm{s}}.
T_L(t_2,t_1)^A_C ={\cal P}\exp [\int_{t_2}^{t_1}dt Omega\circ H]^A_C, \qquad T_R(t_2,t_1)^B_C ={\cal P}\exp [\int_{t_2}^{t_1}dt \H \circ \Omega]^B_C
L(t,0) = \ \lim_{s\to 0}L(t,s) = \lim_{s\to 0}\ell(t,\zeta(t,s)) = \lim_{s\to T^{-1}(t)}\ell(t,s) = \L_0(T^{-1}(t)) - k \ln{(1+t L_1(T^{-1}(t)))}=1\ ,
\begin{array}{r c l}{{S}}&{{=}}&{{{\frac{1}{\kappa}}\int d^{1 1}x \sqrt{| g |}[ R ]}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{-{\frac{T_{K K 1 1}}{2}}\int d^{7}\xi \sqrt{| \gamma |}\left[ k^{4 / 7}\gamma^{i j}\partial_{i}X^{\mu}\partial_{j}X^{\nu}\Pi_{\mu \nu}- 5 \right] \, ,}}\\ \end{array}
\lim_{\varepsilon \rightarrow 0}\left\Vert M\left( t,\varepsilon \right)\right\Vert_{B\left( E\right)}=0,\lim_{\varepsilon \rightarrow0}\left\Vert N\left( t,\varepsilon \right) \right\Vert_{B\left( E\right)}\leq C_{0}.
\begin{array}{ll}\delta\Psi_\alpha=\displaystyle\frac 18\omega^{mn}(\Gamma_{mn})_\alpha{}^\beta\Psi_\beta, \Gamma_{mn}\equiv\Gamma_m\tilde\Gamma_n-\Gamma_n\tilde\Gamma_m, \delta\chi^\alpha=\displaystyle\frac 18\omega^{mn}(\tilde\Gamma_{mn})^\alpha{}_\beta\chi^\beta, \tilde\Gamma_{mn}\equiv\tilde\Gamma_m\Gamma_n-\tilde\Gamma_n\Gamma_m,\end{array}
E(M_{i})=\frac{1}{16\pi}lim_{r\rightarrow\infty}\int_{|x|=r}\sum_{p=1}^{3}\sum_{q=1}^{3}(\partial_{q}(\Phi_{i}^{*}g)_{pq}-\partial_{p}(\Phi_{i}^{*}g)_{qq})\frac{x^{p}}{|x|}dH^{2}(x),
\int_{W_3}| \tilde{k}_\gamma(\zeta,z) - \tilde{k}_{\gamma}(\zeta,w) | dV_X(\zeta) \lesssim \sum_{k=0}^{2n-1}\frac{1}{r^{2n-k-1}}\int_{B_r(0) \cap X}\frac{1}{\|\zeta\|^k}dV_X(\zeta) \lesssim r,
\sum_{j=1}^m \frac 1{nF_j}= \sum_{j=1}^m \Delta_j \frac{1}{n \Delta_j F_j}\asymp n^{\frac{1-2\beta}{2\beta+1}}\int_0^1 f_0(x)^{-\frac{2\beta+3}{2\beta+1}}dx.
(\mathrm{Det}( \mathrm{A}_1\ast \mathrm{A}_2))^{-\alpha}_{\lambda}=(\mathrm{Det}\mathrm{A}_1\ast \mathrm{Det}\mathrm{A}_2)^{-\alpha}_{\lambda}=\mathrm{det}\,{A_1}_{\lambda}^{-\alpha}\,\mathrm{det}\,{A_2}^{-\alpha}_{\lambda}\;.
C^{-}_{p}= \tau^{-1}\{(j_{r},l_{r})\in p: j_{r}\not \in \{0,2n-1,4n-2\}, \l_{r-1}<l_{r},l_{r+1}<l_{r}\}& \bigsqcup \{(n,l)\in \mathcal{D}: (2n-1,l-\epsilon)\not\in p (2n-1,l+\epsilon)\in p\},
{\cal D}(g) = \left(\begin{array}{ccc}z_1 ' - \bar{z}_2 ' - \chi z_2 ' \bar{z}_1 ' - \bar{\chi}\theta ' -\bar{\theta}' \lambda\end{array}\right) \,, \quad \quad \sum_{i}|z_i '|^2 + \bar{\theta}' \theta ' = 1
I(U_i; Y|U_{1:i-1}\!=\!u_{1:i-1}) \le \!I(U_i;Z|U_{1:i-1}\!=\!u_{1:i-1}) I(U_i; X|U_{1:i-1}\!=\!u_{1:i-1})\le \!I(U_i;Z|U_{1:i-1}\!=\!u_{1:i-1}).
R_{0}(z) =\frac{1}{z-\sum_{\ell=0}^{p}a_{-\ell}\,S_{p-\ell-1}^{(1)}(z)\,T_{\ell-1}^{(1)}(z)}R_{j}(z) =R_{0}(z)\,S_{j-1}^{(1)}(z)1\leq j\leq p.
c_g=c_r=1; &a_k=a_{g+1-k}^{-1}\ \ \mathrm{and}\ \b_k=a_{g+1-k}b_{g+1-k}^{-1}a_{g+1-k}^{-1}, \ \ 1\leq k\leq r; a_{g+1-k}=a_k^{-1}\ \ \mathrm{and}\ \b_{g+1-k}=a_{g+1-k}^{-1}b_k^{-1}a_{g+1-k}, \ \ 1\leq k\leq r.
(D_{j})_{i}^{l}= \left( \begin{array}{ccc}d_{j(1)}\dots 0 0 \ddots 0 0 \dots d_{j(k)}\end{array}\right) \, , \qquad (\hat D_{j})_{\alpha}^{\beta}= \left( \begin{array}{ccc}a_{j(1)}\cdot{\bf 1}\dots 0 0 \ddots 0 0 \dots a_{j(N)}\cdot{\bf 1}\end{array}\right)
b_1 = \beta_1 + 3\varphi\gamma_1 + \psi\gamma_2\;, b_2 = \beta_2 + 2\varphi\gamma_2 + 2\psi\gamma_3\;, b_3 = \beta_3 + \varphi\gamma_3 + 3\psi\gamma_4\;,
{\cal{G}}( t , t^{\prime}) = -{\frac{\Gamma [ - \nu ]}{\sqrt{4 \pi i E}}}D_{\nu}\left( \sqrt{\frac{2 i}{E}}( E t - k_{x}) \right) D_{\nu}\left( - \sqrt{\frac{2 i}{E}}( E t^{\prime}- k_{x}) \right)
\tau(xe_i^m,yf_i^n)= \delta_{mn}\tau(x,y)\frac{q_i^{n(n-1)/2}}{(q_i^{-1}-q_i)^n}[n]!_{q_i}&(x\in U^+\cap T_i(U^+), y\in U^-\cap T_i(U^-)), \tau(e_i^mx',f_i^ny')= \delta_{mn}\tau(x',y')\frac{q_i^{n(n-1)/2}}{(q_i^{-1}-q_i)^n}[n]!_{q_i}&(x'\in U^+\cap T_i^{-1}(U^+), y'\in U^-\cap T_i^{-1}(U^-)).
-\cos\psi\cdot\left(\frac{F[q_{0}](\frac{i}{\mu_{0}})w_{1}[q_{0}](\frac{i}{\mu_{0}})w_{3}[q_{0}](\frac{i}{\mu_{0}})}{w_{2}[q_{0}](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{2}\gamma^{0}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\langle a\cos\theta, \bar{a}\cos \bar{\theta}\rangle=\frac1{2}\left[ \int_{0}^{1}a(t)\bar{a}(t)\cos (\bar{\theta}(t)+\theta(t)) dt+\int_{0}^{1}a(t)\bar{a}(t)\cos (\bar{\theta}(t)-\theta(t)) dt\right] .
- \frac{i^{3}}{3 !}\int d x_{1}d x_{2}d x_{3}( \partial_{\tau}h_{2}) ( x_{1}) h_{2}( x_{2}) h_{2}( x_{3}) [ . . . R_{3}^{\tau}( x_{1}; x_{2}, x_{3}; x ) . . . , \psi_{4}( y_{4}) ] .
S=-2^k(0.606)^{2k+1}1.2T_p\int_{R^{1,8-2k}}d^{p+1}x \left[1+\frac{(2\pi\alpha')^2}{4}(F_{\mu\nu}F^{\mu\nu}+2\partial_{\mu}X^i\partial^{\mu}X^i+2i\overline{\theta}\Gamma^{\mu}\partial_{\mu}\theta )\right]
E^{N_1', N_2'}_{\gamma', \gamma, I}(\mathbf{t})[\alpha] = \left|f_{\gamma \setminus \gamma', I, g^{N_1'}_{\gamma', \gamma, I}}(\mathbf{t})[\alpha] - f^{N_2'}_{\gamma \setminus \gamma', I, g^{N_1'}_{\gamma', \gamma, I}}(\mathbf{t})[\alpha]\right|
{\frac{1}{{g_{grav}}^2 (p^2)}}={\frac{1}{{g_{grav}}^2 ({M_I}^2)}}+{\frac{{\tilde b}_{grav}}{16 \pi^2}}\log{\frac{M_{I}^2}{p^2}}- \frac{{\tilde a}_{grav}}{16 \pi^2}\log \left((T +{\bar T})(U +{\bar U})\right).
M^n =\begin{pmatrix}\Gamma_{1,n}\Gamma_{1,n-1}\ldots \Gamma_{1,n-(N-1)}\Gamma_{2,n}~ ~ \Gamma_{2,n-(N-1)}\vdots ~ ~ \vdots \Gamma_{N,n}\Gamma_{N,n-1}\ldots \Gamma_{N,n-(N-1)}\end{pmatrix}
\varphi_1\circ wX^{q^s}\circ \varphi_2 = (cg^{q^{s+l}}w^{q^l}+ dh^{q^{s+l+n}}w^{q^{l+n}})X^{q^{j+s+l}}+ (ch^{q^{s+l}}w^{q^l}+ dg^{q^{s+l+n}}w^{q^{l+n}}) X^{q^{j+s+l+n}},
(*) =\prod_{i=1}^{|A|}\int_{[0,a]^{A_i}}P_N(\phi_{x_i}+\tilde{\psi}_{x_i}\in[0,a] |\phi=0 \, on \, A_i)g_i(\psi)d\psi &=\prod_{i=1}^{|A|}\int_{[0,a]^{A_i}}P_{A_i^c}(\phi_{x_i}+\tilde{\psi}_{x_i}\in[0,a])g_i(\psi)d\psi &\geq \prod_{i=1}^{|A|}P_{A_i^c}(\phi_{x_i}\in[0,a] ) &\geq [c(1/2\wedge a)]^{|A|}
J_5=D-\sum_{\substack{i,j,k=1, i\not=j}}^n\frac{\mu_k\partial_y[\tau_i(\mathcal{L}\mathfrak{B}_{ijk})(\varphi_{ij}-1)]}{x^3_{ij}}-\sum_{\substack{i,j,k=1, i\not=j}}^n\partial_y\bigg(\frac{\mu_k\tau_i([|D|,\varphi_{ij}]\mathfrak{B}_{ijk})}{x_{ij}^3}\bigg),
\frac{a_{7}x^{7}+a_{6}x^{6}+a_{5}x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}}{b_{4}x^{4}-b_{3}x^{3}-b_{2}x^{2}-b_{1}x-b_{0}}
\frac{1 + \lambda}{\sigma}(\partial_t \widetilde{U}- \mathcal{L}_{\frac{\sigma}{1+\lambda}}\widetilde{U}) \geq \lambda \sum_{j=1}^m \frac{\partial_j B(U)}{U_j}|\nabla U_j|^2 - \sum_{j,k=1}^m \partial_{j,k}B(U)\langle \nabla U_j, \nabla U_k \rangle,
S^F_{lc}= \frac{i}{4 \pi l_s^2}\int d \tau \int_{0}^{2 \pi l_s^2 p^+}d \sigma\left[ \left( \eta^{\alpha \beta}\delta_{AB}- \epsilon^{\alpha \beta}\sigma^3_{AB}\right) \partial_\alpha x^+ \bar{\theta}^A \Gamma_+ \left( \mathcal{D}_\beta \theta \right)^B \right],
U_{\varepsilon,z_\varepsilon}(y)^2=\gamma_\varepsilon^2-2\tilde{t}_\varepsilon+\frac{\tilde{t}_\varepsilon^2}{\gamma_\varepsilon^2}+\frac{2\tilde{S}_{0,\varepsilon}}{\gamma_\varepsilon^2}+O\left((|A(\gamma_\varepsilon)|+\gamma_\varepsilon^{-4})(1+\tilde{t}_\varepsilon(y)^2)+|y-z_\varepsilon| \right)
\overline{{{\chi}}}^{( + ) m}\chi_{n}^{( + )}\: \overline{{{\eta}}}_{( s )}^{n}\gamma^{i}\eta_{( s ) m}+ \overline{{{\chi}}}^{( - ) m}\chi_{n}^{( - )}\: \overline{{{\eta}}}_{( t )}^{n}\gamma^{i}\eta_{( t ) m}
\begin{array}{l}{{\displaystyle \sum_{i , j}\int \! \mathrm{d}^{4}x_{1}\dots \mathrm{d}^{4}x_{i}\, \mathrm{d}^{4}y_{1}\dots \mathrm{d}^{4}y_{j}\enspace f_{i}^{\ast}( x_{1}, \dots , x_{i}) \, f_{j}( y_{1}, \dots , y_{j})}}\\{{\qquad \times{\cal S}( \theta x_{1}, \dots , \theta x_{i}, y_{1}, \dots , y_{j}) \ge 0 \, ,}}\\ \end{array}
(\tau_z\otimes \mathrm{id})(u^{(\pi)})=Q_\pi^{\mathrm{i}z}u^{(\pi)}Q_\pi^{-\mathrm{i}z},(\sigma_z\otimes \mathrm{id})(u^{(\pi)})=Q_\pi^{\mathrm{i}z}u^{(\pi)}Q_\pi^{\mathrm{i}z}.
\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)K'}{(1+N_1\bar{\gamma}')(1-\xi)}\Big).
C_{\mu \nu}= \nabla^{2}\left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{- 1}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{- 1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \delta^{2}( x - y )
\nu_{-}= \left( \begin{array}{c c}{I_{n}}&{0}\\{\nu_{- 2 1}}&{I_{k}}\\ \end{array}\right) , \qquad \nu_{+}= \left( \begin{array}{c c}{I_{n}}&{\nu_{+ 1 2}}\\{0}&{I_{k}}\\ \end{array}\right) , \qquad \eta = \left( \begin{array}{c c}{\eta_{1 1}}&{0}\\{0}&{\eta_{2 2}}\\ \end{array}\right) ,
[ \Delta_{\perp}]^{- 1}( x_{\perp}) \rightarrow [ \Delta_{\perp}- m^{2}]^{- 1}( x_{\perp}) = - \int \frac{d^{2}k_{\perp}}{( 2 \pi )^{2}}\frac{e^{i k_{\perp}\cdot x_{\perp}}}{k_{\perp}^{2}+ m^{2}}= - \frac{1}{2 \pi}K_{0}( m \sqrt{x_{\perp}^{2}}) ,
\frac{z\;{}_{a}\mathtt{B}'_{2a-1, a \nu-a+1,1}(a^{a/2}z)}{{}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(a^{a/2}z)}=\frac{z\mathtt{J}_{\nu-1}\left(z\right)}{\mathtt{J}_{\nu}\left(z\right)}-(2-a)\nu +1-a.
\mathbb{E}_{U,\xi}\left[\left\| \nabla \phi_{\mu}^{1/\bar{\rho}}(x_{t^*})\right\|_2^2 \right] \leq \frac{\bar{\rho}}{\bar{\rho}-\rho}\frac{\left(\phi_{\mu}^{1/\bar{\rho}}(x_0) - \underset{x}{\min}\ \phi_{\mu}(x) \right)+ 2(n^2 + 2n)\bar{\rho}L_{f,0}^2 \sum_{t=0}^{T}\alpha_t^2}{\sum_{t=0}^T \alpha_t}.
n_{{\mathfrak{a}_{i+1}}}(\mathfrak{b})=n'_{{\mathfrak{a}_{i+1}}}(\mathfrak{b}) -\; \; \sum_{\mathclap{\{\mathfrak{d}\mid \ell(\mathfrak{d}) < \ell({\mathfrak{a}_{i+1}})\}}}\; \;(n'_{{\mathfrak{a}_{i+1}}}(\mathfrak{d})|_{t=0})n_\mathfrak{d}(\mathfrak{b}).
{\mathcal A}^{(<)}=\frac{\hbar^{2}}{2M}\frac{\hat{\lambda}B^{2}}{2}\, \xi\,I_{0}( \xi)\,I'_{0}( \xi)\stackrel{(a \rightarrow 0)}{=}E_{{\rm (gs)}}\,\times O\left( \xi^{2}\ln \xi\right) \; ;
a_n^{-1}l\sum_{i\in [n]}d_i^2\Big(\frac{1}{\ell_n(T)}-\frac{1}{\ell_n}\Big)=O(n^{2\rho+1-\alpha-2}/L(n)^3)=O(n^{(\tau-4)/(\tau-1)}/L(n)^3)=o(1),
d ( x , y ; e^{2}) = d ( t , y ; e^{2}) d \left( \frac{x}{t}, \frac{y}{t}; e^{2}d ( t , y ; e^{2}) \right) , \; s ( x , y ; e^{2}) = s ( t , y ; e^{2}) s \left( \frac{x}{t}, \frac{y}{t}; e^{2}d ( t , y ; e^{2}) \right)
S_{j i}^{({\widehat G}/ H )}( \theta , q ) = \left( x^{- \rho_{H}\cdot h}\otimes x^{- \rho_{H}\cdot h}\right) S_{j i}^{({\widehat G})}\left( x^{\rho_{H}\cdot h}\otimes x^{\rho_{H}\cdot h}\right) \ ,
\begin{cases}\underbar s_1^{k+1}-\epsilon \le u(x,t;t_0,u_0,v_0) \le \bar s_1^{k+1}+\epsilon\cr\underbar s_2^{k+1}-\epsilon \le v(x,t;t_0,u_0,v_0) \le \bar s_2^{k+1}+\epsilon\end{cases}
S_{J,j}V^{,j}-\lambda_{1}S_{J}=0 \left( S_{J,k}\delta_{j}^{i}+2S_{J},_{j}\delta_{k}^{i}\right) V^{,k}+a_{7}\left( 2Y_{J}^{;i}{}_{;~j}-a_{0}S_{J}\delta_{j}^{i}\right) =0
\mathcal{C}_{\pm}^{\mathrm{local}}=\left\{z_{1}+ r e^{\pm i\frac{2}{3}\pi}: r \in [0, 2^{\frac{5}{3}}N^{-\frac{1}{3}}\kappa]\right\}, \mathcal{C}_{\pm}^{1}=\left\{t \pm i 2^{\frac{2}{3}}\sqrt{3}N^{-\frac{1}{3}}\kappa: t \in [z_{1}- 2^{\frac{2}{3}}N^{-\frac{1}{3}}\kappa, 1]\right\},
{\rm I}_\pm(\varepsilon) \,=\, \frac{\varepsilon}{4\pi |1-\hbar|}\int_{S^1_\pm\cap[T^2 - \Delta(\varepsilon)]_+}\mskip-20mu d\phi\,\,\frac{\dot{\gamma}\times\ddot{\gamma}}{|\gamma\times\dot{\gamma}|}\, \exp{\!\left(-\frac{\Lambda\,||\dot{\gamma}||^2\,\varepsilon}{2\,|1-\hbar|\,|\gamma\times\dot{\gamma}|}\right)}\,.
\begin{bmatrix}\alpha_{11}\cdots \alpha_{1n}0 0 \cdots 1 \vdots \vdots \vdots \vdots \vdots \alpha_{(m-1)1}\cdots \alpha_{(m-1)n}0 1 \cdots 0 \alpha_{m1}\cdots \alpha_{mn}1 0 \cdots 1 \end{bmatrix},
\frac{1}{4}\,W^{+}_{ijkl}W^{+}_{ipkq,t}W^{+}_{jplq,t}= \,\, \lambda |d\lambda|^{2}+\mu|d\mu|^{2}+\nu|d\nu|^{2}&-\lambda(\mu-\nu)^{2}|a|^{2}-\mu(\nu-\lambda)^{2}|b|^{2}-\nu(\lambda-\mu)^{2}|c|^{2}\,,
z^3 y^3 = x^6 + ax^4 y^2 + bx^2 y^4 + y^6 \, ; z^3 y^2 = x(x^4 + ax^2 y^2 + y^4) \, ; z^3 y^3 = x^6 + a x^3 y^3 + y^6 \, ; z^3 y^2 = x^5 + y^5 \, ; z^3 y^3 = x^6 + y^6 \, ; z^3 y^2 = x(x^4 + y^4) \, .
\chi_{i}(G\circ H)\leq|\mathbb{P}|=\chi_{i}(G)i_{H}+\chi_{2}(G)\big{(}|V(H)|-i_{H}\big{)}=\chi_{2}(G)|V(H)|-i_{H}\big{(}\chi_{2}(G)-\chi_{i}(G)\big{)}.
= \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{{\psi_{l R}}}\\{{\psi_{l}+ \psi_{u}}}\\ \end{array}\right) = \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{{\psi_{1 R}}}\\{{\sqrt{2}\psi_{1 L}+ \psi_{1 R}}}\\ \end{array}\right) .
\begin{aligned}(\nabla_{\bar{b}}\eta^{s\bar{q}}) = - \eta^{s\bar{k}}\eta^{j\bar{q}}\nabla_{\bar{b}}\eta_{j\bar{k}}&= - \eta^{s\bar{k}}\eta^{j\bar{q}}\left(\alpha^{p\bar{m}}\omega_{p\bar{k}}\nabla_{\bar{b}}\omega_{j\bar{m}}+ \alpha^{p\bar{m}}\omega_{j\bar{m}}\nabla_{\bar{b}}\omega_{p\bar{k}}\right),\end{aligned}
= \frac{1}{2}(\partial_{t}A_{0}(\textbf{w}+\bar{\textbf{u}}){\textbf{w}},{\textbf{w}}) +\frac{1}{2}(\partial_{x}A_{1}(\textbf{w}+\bar{\textbf{u}}){\textbf{w}},{\textbf{w}}) + (D_{1}(\textbf{u}, \bar{\textbf{u}}, \textbf{w}){\textbf{w}},{\textbf{w}}).
S \varrho(a) \sum_i \varrho(a_i)Vx_i \ =\ \sum_i S \varrho(aa_i) Vx_i \ =\ \sum_i \varrho'(aa_i) V'x_i \ =\ \varrho'(a) \sum_i \varrho'(a_i)V'x_i \ =\ \varrho'(a) S \sum_i \varrho(a_i)V x_i.
\rho^2_s \asymp \begin{cases}\left(\frac{\sqrt{b}n}{\sqrt{\log(n)}}\right)^{-\frac{2s}{2s+1}}, \; m \geq n^{\frac{1}{2s+1}}, \left(\frac{\sqrt{b}n}{\sqrt{m \log(n)}}\right)^{-\frac{2s}{2s+1/2}}, \; m < n^{\frac{1}{2s+1}}, \end{cases}
D_1=\sum_{\substack{-B<b_1\le B -B<b_2\le B -B<b_3\le B}}\sum_{|n|\not=0}\Big(\prod_{j=1}^3E(b_j,q,\beta)\Big)\Delta(n,q,-\beta)T(q;b_1,b_2,b_3,n-v),
\widetilde{B}=\{(\mu_1,\mu_2)|\mu_1=\zeta_{m^2}^{a\alpha}\zeta_m^x,\ \mu_2=\zeta_{m^2}^{a\beta}\zeta_m^y;\ \mu_1^{m+\alpha}\neq 1, \ \mu_2^{m+\beta}\neq 1; (x,y)\in B \}
Y_t =g(X_{\tau_D})+\int^{\tau_D}_{t\wedge\tau_D}f(X_s,u(X_s))\,ds &\quad+A^{\mu}_{\tau_D}-A^{\mu}_{t\wedge\tau_D}-(M^x_{\tau_D}-M^x_{t\wedge\tau_D}), t\ge0,P_x\mbox{a.s.}
q_{c}( a , r ) = \frac{1}{4 \pi^{2}a^{4}}\sum_{m = 0}^{\infty}{'}\int_{0}^{\infty}{d z \chi_{\mu}( z / a ) F_{c m}^{( q )}( z , x )}, \quad r > a , \quad q = \varepsilon , p_{i},
V_{2,1}(\tau) = e^{\Delta \tau}(V_{2,1}(0)) + \int_0^\tau e^{(\tau - \tau')\Delta}\left( - 2 (V_2 \nabla \psi_1) + V_2 \Delta\psi_1 + |\ln(T-t_0(x_0))| \psi_1 F_2(U_1, U_2))(\tau') \right) d \tau'.
\int_{-1}^1 (y'(x,\lambda)-y'(\xi,\lambda))\log\left(\frac{(x+\xi)^2+(y(x,\lambda)+y(\xi,\lambda))^2}{(x-\xi)^2+(y(x,\lambda)-y(\xi,\lambda))^2}\right)d\xi=0, x\in [-1,1]
H^{\phi}_{1}(f) = \lim_{\nu \to 1}H_{\nu}(f) = \frac{1}{2}{\rm log}\frac{2 \pi e [\alpha (1-\alpha)]}{n}+ \frac{(\alpha-\gamma)^2}{2\alpha (1-\alpha)}+ O\left(\frac{1}{n}\right)
\leq \ \frac{1}{\mathcal{T}}\sum_{l=0}^{\lfloor \frac{\mathcal{T}}{\Delta(\epsilon)}\rfloor}\int_{t_l}^{t_{l+1}}|\tilde G(\mu^{N,M}(t),z^{N,M}(t)) - \tilde G( \mu^{N,M}(t_l), z^{N,M}(t))|dt \ + \L \kappa_2(\epsilon) \ + \ 2M\Delta(\epsilon) \ \ \ \ \ \ \forall \mathcal{T}\geq 1,
\mathcal{D}_{N}(f)=\frac{1}{8}\int_{G_{N}}\frac{|\nabla f(x)|^{2}}{f(x)}dx+\frac{\lambda N}{2}\sum_{i\neq j}\int_{F_{ij}}\left[\sqrt{f_{ij}(x)}-\sqrt{f_{ji}(x)}\right]^{2}dS_{ij}(x).
( 2 6 ) - \omega_{\alpha \bar{\beta}}( \kappa_{a}^{\alpha}\overline{{\kappa_{b}^{\beta}}}- \kappa_{b}^{\alpha}\overline{{\kappa_{a}^{\beta}}}) = \nabla_{a}T_{b}- \nabla_{b}T_{a}= f_{a b}^{c}T_{c}.
\|G(v_1)-G(v_2)\|^2_{L_{HS}(\mathcal{H};\mathbb{L}^2)}\le \left( \sigma(v_1)-\sigma(v_2)\right)^2\left(\sum_{k=1}^{\infty}c_k^2 \|e_k\|^2_{H^{1-g,2}}\right)\le L^2 \left( \sum_{k=1}^{\infty}c_k^2\right) \|v_1-v_2\|^2_{\mathbb{L}^2}.