harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
n_{k}( t ) = \int \delta ( t_{1}+ t_{2}+ \ldots + t_{k}+ t_{k + 1}- t ) \frac{d^{3}r_{1}}{\ell_{0}^{3}}\ldots \frac{d^{3}r_{k}}{\ell_{0}^{3}}\frac{d^{3}r_{k + 1}}{\ell_{0}^{3}}
{\cal C}_{k}( \eta ) \simeq c_{1}( k ) \biggl [ - \frac{\eta}{\eta_{1}}\biggr ]^{\sqrt{3}+ \frac{1}{2}}+ c_{2}( k ) \biggl [ - \frac{\eta}{\eta_{1}}\biggr ]^{- \sqrt{3}+ \frac{1}{2}}
\Phi(\O_F(1)) = \O_F(1) - \O_F(2) = -\O_p -\Phi(\O(1,1)) = -\O(1,1) + \O(3,2) + \O(2,2) + \O(1,2) - \O(0,1) - \O(-1,1) -\Phi(\O(2,1)(-e)) = -\O + \O(4,2) + \O(3,2) - \O(1,1)
B_{\infty,\infty}^s=\{f\in L_2[0,1]:\, \|f\|_{B_{\infty,\infty}^s}<\infty \}\quad &B_{\infty,\infty}^s(L)=\{f\in L_2[0,1]:\, \|f\|_{B_{\infty,\infty}^s}<L \},
A(t)=\cosh(s(t)) \begin{pmatrix}\cos(\mu(t)) -\sin(\mu(t)) \sin(\mu(t)) \cos(\mu(t)) \end{pmatrix}+\sinh(s(t)) \begin{pmatrix}\cos(\theta(t)) \sin(\theta(t)) \sin(\theta(t)) -\cos(\theta(t)) \end{pmatrix}\,.
\tilde{Z}_{i j}= i \sigma_{2}\left( \begin{matrix}{| Z |}&{0}\\{0}&{| Z |}\\ \end{matrix}\right) \ , \qquad \tilde{Z}_{i j}= i \sigma_{2}\left( \begin{matrix}{| Z |}&{0}&{0}&{0}\\{0}&{| Z |}&{0}&{0}\\{0}&{0}&{| Z |}&{0}\\{0}&{0}&{0}&{| Z |}\\ \end{matrix}\right) \ ,
G_{0}(y) = d{\pi}^{3}y^{2/3}\, \int_{0}^\infty V_1\left( \frac{z}{N}\right) e\left(\frac{zu}{p_1qQ}\right)\, z^{-1/3+i\nu}\, \left(e(3 z^{1/3}y^{1/3}) -e(-3 z^{1/3}y^{1/3}) \right)dz + .
\Phi_2^{(3)}\left(b_1,b_2,b_3;c;x,y,z\right):=\sum_{m,n,p=0}^\infty\frac{(b_1)_{m}\,(b_2)_{n}\,(b_3)_{p}}{(c)_{m+n+p}}\,\frac{x^{m}}{m!}\frac{y^{n}}{n!}\frac{z^{p}}{p!}.
I = \int d^{n}x{\cal L}= - \int d^{n}x \sqrt{- g}\left( \frac{1}{2}g^{\mu \nu}\partial_{\mu}\Phi \partial_{\nu}\Phi + \frac{\mu^{2}}{2}\Phi^{2}+ \frac{\lambda}{4 !}\Phi^{4}\right) ,
M = \mu \left( \begin{array}{c c c c c c}{{0}}&{{- a_{1}}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{a_{1}}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- a_{2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{a_{2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{- a_{3}}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{a_{3}}}&{{0}}\\ \end{array}\right)
\phi_{T+1}^{(i)}(\hat{k}^{(i)}) = \begin{cases}\frac{1}{\mu}R_T(k^{(i)},\hat{k}^{(i)}), \hat{k}^{(i)}\neq k^{(i)}, 1- \sum_{l\neq k^{(i)}}\frac{1}{\mu}R_T(k^{(i)},l), \hat{k}^{(i)}= k^{(i)},\end{cases}
\phi=\sum_{r+s=p+1, s\geq 1}\phi_{j \alpha_1\cdots\alpha_r\beta_1\cdots\beta_s}(z)d\bar{z}_j^{\alpha_1}\wedge \cdots \wedge d\bar{z}_j^{\alpha_r}\wedge \frac{\partial}{\partial z_j^{\beta_1}}\wedge\cdots \wedge\frac{\partial}{\partial z_j^{\beta_s}}
\frac{1}{(2\pi)^{\dim M}}\int_{T^*M}\chi(x){\rm tr}_s(\sigma_{t^{-1}}(\Pi(t)))(x,\xi) dxd\xi =\frac{(-1)^p}{(2\pi i)^p}\frac{p !}{2p!}\,\int_M \chi\hat{A}(M)f_0^\varphi df^\varphi_1\wedge df^\varphi_k &=\frac{(-1)^p}{(2\pi i)^p}\frac{p !}{2p!}\,\int_M \chi\hat{A}(M)\wedge \omega_\varphi.
\bigg(\sum_{\ell=1}^p(\varphi_{I_\ell\cap J_i}^{w_{i+1}})\bigg)\cap(\varphi_{I_0\cap J_i}^{w_{i+1}}) = \sum_{\ell=1}^p(\varphi_{I_\ell\cap I_0\cap J_i}^{w_{i+1}}).
Q_n(x) = 2^{7/6}\, \sqrt[12]{1-x}\, \frac{1}{\sqrt{6 \pi n^{4/3}}}\, | Li_3(x )|^{1/6}{\rm exp}\left( \frac{3}{4}\sqrt[3]{2}n^{2/3}|Li_3(x)|^{1/3}\right) &\times\, \, \left( \cos\left(\frac{3 \sqrt{3}}{4}\sqrt[3]{2}n^{2/3}|Li_3(x)|^{1/3}+ \frac{\pi}{6}\right)+ O_X( n^{-1/3}) \right) .
\left(\begin{array}{l}f_0 g_0 \end{array}\right)=\left(\begin{array}{l}0 1 \end{array}\right)\,\,\,\,\left(\begin{array}{l}f_1 g_1 \end{array}\right)=\left(\begin{array}{l}v^{-1}-v^{-2}u+cv^{-3}v''-\frac32cv^{-4}(v')^2 \end{array}\right)\,.
\begin{aligned}J_\nu(z) =\frac{(z/2)^{\nu}}{\Gamma(1+\nu)}(1+\mathcal{O}(z^2)), \nu \geq 0 Y_\nu(z) = -\frac{\Gamma(\nu)\,(z/2)^{-\nu}}{\pi}(1+\mathcal{O}(z^2)) + \frac{\cot(\nu \pi)\, (z/2)^{\nu}}{\Gamma(1+\nu)}(1+\mathcal{O}(z^2)), \ \ \nu>0,\end{aligned}
\int_{a_i}^{a_{i+1}}|V(x,t)|^2=\left|\begin{pmatrix}S_{d,1}(t,t_{1,ex}(a_i))S_c(t_{1,ex}(a_i),t_{1,en}(a_{i+1})))S_d(t_{1,en}(a_{i+1}),0)v_{1,0}(x)+R_{1,i}S_{d,2}(t,t_{2,ex}(a_i))S_c(t_{2,ex}(a_i),t_{2,en}(a_{i+1})))S_d(t_{2,en}(a_{i+1}),0)v_{1,0}(x)+R_{2,i}\end{pmatrix}\right|^2
\mathrm{div}_Y(dT)= \prod_{i=1}^{s}\prod_{\mathfrak{P}|P_i}{\mathfrak{P}_i}^S \prod_{Q \mid \mathfrak{p}_\infty}Q^{q-2}\cdot_{\mathbb{F}_q(T)/\mathbb{F}_{q^d}K_{q,P^\alpha}}(\mathfrak{p}_\infty^{-2}),
\omega((a_1,x_1),(a_2,x_2),(a_3,x_3)) := \widehat{F}(x_1,x_2)(a_3) \ \epsilon(x_1,x_2,x_3) \mu(x_1, (a_2,x_2),(a_3,x_3)) = \left(\widehat{F}(x_1,x_2)(a_3)\ \epsilon(x_1,x_2,x_3) \right)^{-1}\gamma(y)(x_1,(a_2,x_2))= \widehat{F}(y,x_1)(a_2)\ \epsilon(y,x_1,x_2,).
\mathcal{C}_{\mathrm{\tiny F P}}c = \bar{c}, \mathcal{C}_{\mathrm{\tiny F P}}\bar{c}= - c , \mathcal{C}_{\mathrm{\tiny F P}}B = B + \widetilde Z_{1}( 1 - \alpha ) ( \bar{c}\times c ) , \mathcal{C}_{\mathrm{\tiny F P}}A = A .
2(f_t+f^*_t)+(f+f^*)(f_z+f^*_{z^*})+i(g+g^*)(f_z-f^*_{z^*})=0, &2(g_t+g^*_t)+(f+f^*)(g_z+g^*_{z^*})+i(g+g^*)(g_z-g^*_{z^*})=0.
{\cal D}= z^{1}z^{5}z^{9}+ q^{2}z^{2}z^{6}z^{7}+ q^{2}z^{3}z^{4}z^{8}- q z^{1}z^{6}z^{8}- q^{3}z^{3}z^{5}z^{7}- q z^{2}z^{4}z^{9}\ ,
H_\gamma(\phi) = 2E_\gamma(\phi) +\frac{4}{p+1}\|\phi\|^{p+1}_{L^{p+1}}&\leq 2I^m_\gamma(c) + B m^{\frac{N(p-1)}{4}}c^{\frac{4-(N-2)(p-1)}{4}}&\leq 2\omega^0_\gamma c+ B m^{\frac{N(p-1)}{4}}c^{\frac{4-(N-2)(p-1)}{4}}
K(L,r)=\int_{\frac{1-r^2}{1+r^2}}^1 \int_0^{\alpha(s,r)}\frac{s^{2L-n}}{1-s^{2L}}\left(2\cos\theta-2\cos\alpha(s,r)\right)^{n-2}\left(s+\frac 1s-2\cos\theta\right) d\theta\; ds\ .
\left( (A_i-\overline{X}_i)^T\otimes( \Phi^{s}_i(X)) \right) (\overline{M}^s -I_4) - \left( (A_i-\overline{X}'_i)^T\otimes( \Phi^{s}_i(X)) \right) (\overline{M}^s -I_4) = \| \overline{X}_i - \overline{X}'_i \| \left[ \Phi^{s}_i, \Phi^{s}_i \right] ~ \left[\alpha_1,\alpha_2\right] (\overline{M}^s -I_4) = \mathbf{0}, ~~~ s=\pm,
v_s-e^{i(t-s)H}E_+(t-s)v_s= \gamma(H)v_s-e^{i(t-s)H}E_+(t-s)v_s = (\gamma(H)-\gamma(H_0))v_s &+(\gamma(H_0)-P_+-P_-)v_s &+(P_+-e^{i(t-s)H}E_+(t-s))v_s+P_-v_s .
\sum_{1-n \leq 2r \leq n}\frac{1}{4r^2 - 1}\left(1 +\frac{r}{j n}\right)^{-s-1}& = -1 + \sum_{k=1}^{n/2}\frac{1}{4k^2 - 1}\left[ \left( 1 +\frac{k}{jn}\right)^{-s-1}+ \left( 1 - \frac{k}{jn}\right)^{-s-1}\right] ~~~~~~~~~~~ - \frac{1}{n^2 - 1}\left( 1 - \frac{1}{2j}\right)^{-s-1}.
\|\mathcal{H}_2\| \leq \Delta_{\max}|\widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}| \frac{9\sum_{i \in \widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}}N_i \sum_{t=0}^{T-1}\|\Sigma^{(i)}_t\|}{2\lambda_{\min}\left(\sum_{i \in \widehat{\mathcal{C}}_1}N_i \sum_{t=0}^{T-1}{\Sigma}_{t}^{(i)}\right)},
\mathcal{Q}^{(2)}_1 \frac{\partial \bar{\mathcal{F}}_2}{\partial x_1}+ \mathcal{Q}^{(2)}_2 \frac{\partial \bar{\mathcal{F}}_2}{\partial x_2}= (\Delta_{L-1}- \mathcal{V}^{(2)}) \bar{\mathcal{F}}_2 \; .
\Psi_n\left(\frac{t+1}{2}\right) =D^n_{-1}\Phi_n(t+1)+D^n_1\Phi_n(t) &=D^n_{-1}P_n(2t+1)\chi_{[-1,0)}(t)+D^n_1P_n(2t-1)\chi_{[0,1)}(t)
\sum_{\substack{\sigma\in BNC(\hat{\chi}) \pi\leq\sigma\leq \hat{\epsilon}}}\mu_{BNC}(\pi, \sigma) = \sum_{\substack{\sigma\in BNC(\overline{\chi}) \overline{\pi}\leq\overline{\sigma}\leq \hat{\epsilon}}}\mu_{BNC}(\overline{\pi}, \overline{\sigma}).
r \widetilde{F}\left( \beta \right) \approx - \frac{\pi^{4}}{4 5}\frac{r^{4}}{\beta^{4}}- \frac{1}{2 4 0}- \frac{\pi^{2}\left| \mu_{e f f}^{2}\right|}{1 2}\frac{r^{2}}{\beta^{2}}- \frac{3 \left| \mu_{e f f}^{2}\right|}{1 6},
E[(1+t)^U] = E[e^{U \log(1+t)}] = \sum_{k=0}^\infty E[U^k] \frac{1}{k!}\log^k(1+t) &= \sum_{k=0}^\infty E[U^k] \sum_{n=k}^\infty S_1(n,k) \frac{t^n}{n!}= \sum_{n=0}^\infty \left( \sum_{k=0}^n E[U^k] S_1(n,k) \right) \frac{t^n}{n!}.
\Lambda_{-}(X) = \sum_{i\in\mathcal{I}}\left( (A_i-X)^T\otimes( \Phi^{-}_{i,T}(X)) \right)+ \sum_{i\in\mathcal{I}^+}\left( (A_i-\overline{X}_i)^T\otimes(\Phi^{-}_{i,T}(X)) \right) (\overline{M}^- -I_4),
\mathbf{q}_p^{\mathrm{H}}(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}\mathbf{q}_p\leq\lambda_p^{-1}\cos^2\theta_{p,p}+\sum_{i\neq p}\lambda_i^{-1}\sin^2\theta_{p,p}&=\lambda_p^{-1}+\{\mathrm{Tr}[(\mathbf{H}_{\mathrm{e}}^{\mathrm{H}}\mathbf{H}_{\mathrm{e}})^{-1}]-2\lambda_p^{-1}\}\sin^2\theta_{p,p},
\tilde{E}^a_{m,n}\left[\sum_{i=1}^{t_1(x_{\cdot})\wedge r}G(a,\tilde{R}^1_{i,0})\right]=\sum_{x_{\cdot}\in \Pi_{m,n}}\left(\sum_{i=1}^{t_1(x_{\cdot})\wedge r}G(a,\tilde{R}^1_{i,0})\right)\tilde{Q}_{m,n}^a(x_{\cdot}).
ss(\eta_x)= s(F(x)) \star_0 \eta_{t_0(x)}\star_1 \cdots \star_{n-2}\eta_{t_{n-2}(x)}= s(\eta_{s(x)}) = st(\eta_x) begin{align*}0.5ex] ts(\eta_x) = t(\eta_{t(x)}) = \eta_{s_{n-2}(x)}\star_{n-2}\cdots \star_1 \eta_{s_0(x)}\star_0 t(G(x))= tt(\eta_x).
\Box \tilde \phi= \left(|\tilde \phi_{t}|^2- |\tilde \phi_{x}|^2\right) \tilde \phi+ \mathbf{1}_{\omega}\tilde f^{\tilde \phi^{\perp}}, \tilde \phi[0]= (a, b) \textrm{and}\tilde \phi[T]= (p, 0).
\frac{4}{\sqrt{\pi}}\left( \, \theta(\tau_{0})-\pi\right) \geq \sqrt{\frac{2 \pi M}{\tau_{0}^{\alpha}}}\, \sum_{j=0}^{\infty}\mu^{j/2}\exp\left(-\frac{2 \pi M\mu^j}{\tau_{0}^{\alpha}}\right).
\sum_{k=r}^\infty k(k-1)...(k-r+1)(1-p)^{k-r}= (-1)^r\frac{d^r}{dp^r}\sum_{k=0}^\infty (1-p)^k= (-1)^r\frac{d^r}{dp^r}\frac{1}{p}= \frac{r!}{p^{r+1}}
e^{- \lambda^2/2}\mathbf{E}\left( e^{- \lambda\mathcal{N}}\mathbf{1}_{\{\mathcal{N}>0\}}\right) =\frac{1}{\sqrt{2\pi}}\int_0^{\infty}e^{-(y+\lambda)^2/2}dy=1-\Phi \left( \lambda\right)
\delta_{\mathsf{P,Ach}}(\mu,r) = \lim_{B\rightarrow \infty}\lim_{P\rightarrow \infty}\limsup_{L\rightarrow \infty}\frac{(B+1)}{B}\frac{\max\left(T_F,T_E\right)}{L/\log(P)}& = \max\left(\delta_F,\delta_E\right),
\mathcal{F}^{-1}[|\xi|\hat{f}(\xi)] =\mathcal{F}^{-1}[\mathrm{sign}(\xi)]\ast\mathcal{F}^{-1}[\xi\hat{f}(\xi)], &=\Big( -\frac{i}{\pi x_1}\Big)\ast i\frac{\partial f}{\partial x_1}, &=\Big( \frac{1}{\pi x_1}\Big)\ast \frac{\partial f}{\partial x_1}.
\mathbf{a}+\mathbf{b}=(\alpha_1+\beta_1,\alpha_2+\beta_2,\alpha_3+b_1,a_1+b_2,a_2+b_3,\cdots,a_{2k-2}+b_{2k-1},a_{2k-1}+\beta_3,\alpha_4+\beta_4),
\frac{c_8}{c_6}=-u\frac{e^{\frac{i}{2}\Theta\pi-\frac{i}{4}\pi}+e^{-\frac{i}{2}\Theta\pi+\frac{i}{4}\pi}D_0C_0^{-1}}{{e^{-\frac{i}{2}\Theta\pi-\frac{i}{4}\pi}+e^{\frac{i}{2}\Theta\pi+\frac{i}{4}\pi}D_0C_0^{-1}}},
A(z)\frac{zA(z)+c+\alpha_{-1}}{A(z)-h(z)}-z\left(A(z)+\frac{\alpha_{-1}}{z}\right)= cA(z), \frac{1}{z^2}\left(A(z)\frac{zA(z)+\delta_0}{1+h(z)}-(d\alpha_{-1}+1)z-\delta_0\right)-\frac{1}{z}(A(z)-1)= dA(z).
A_2:= - \max(s_-^2,s_+^2) \sup_{\substack{\psi \in \mathcal V_0 \|\psi\|_{\ell^2(\square)}= 1}}\sup_{\substack{\varphi \in \mathcal V_0^\perp \|\varphi\|_{\ell^2(\square)}= 1}}\frac{\left| \left\langle \psi, V^\square \varphi \right\rangle \right|^2}{\left\langle H^\square_0(\theta) \varphi, \varphi \right\rangle}\leq 0.
d s_{E}^{2}\approx - e^{- 2 \phi ( r_{H})}( 1 - b^{\prime}( r_{H}) ) \left({\frac{r - r_{H}}{r_{H}}}\right) d t^{2}+{\frac{1}{( 1 - b^{\prime}( r_{H}) )}}\left({\frac{r_{H}}{r - r_{H}}}\right) d r^{2}+ r_{H}^{2}( d \theta^{2}+ \sin^{2}\theta \; d \varphi^{2}) .
Z_{n}=\frac{X_{1}+\cdot\cdot\cdot+X_{n}-n\mu}{\sqrt{n\sigma^{2}}}=\sum_{i=1}^{n}\frac{X_{i}-\mu}{\sqrt{n\sigma^{2}}}=\sum_{i=1}^{n}\frac{1}{\sqrt{n}}Y_{i},
D^\alpha_{a^+}f (t) = \frac{\mathrm d^m}{\mathrm dt^m}I_{a^+}^{m-\alpha}f(t) = \frac{1}{\Gamma(m-\alpha)}\frac{\mathrm d^m}{\mathrm dt^m}\int_{a}^t (t-s)^{m-1-\alpha}f(s) \mathrm ds.
\begin{aligned}\bigg[-\hat{L}^2_i+g_i(\Omega_i)+\frac{1}{4}(d_i-1)(d_i-3)\bigg]Y_i(\Omega_i)=\lambda_i Y_i(\Omega_i),~~~i=1,2,\cdots,N-1, \bigg[-\hat{L}^2_N+\frac{1}{4}(d_N-1)(d_N-3)\bigg]Y_N(\Omega_N)=\lambda_NY_N(\Omega_N),\end{aligned}
\Xi(s)= -W_{s,1}(\mathcal{P}_{1,s}f)-W_{s,3}(\mathcal{P}_{2,s}f) &-\int_{\mathbb{T}^2}\lambda(u,v)\theta_s^S(u)\theta_s^I(v)f(u)\left(\tilde{G}_s(u)-\tilde{F}_s(u)\right)dudv.
\int_{\mathcal{T}^d}\prod_{i=1}^d u_i^{\lambda_i+n_i-1}\, du = \int_{T^{d-1}}u_1^{\lambda_1+n_1-1}\cdots u_{d-1}^{\lambda_{d-1}+n_{d-1}-1}(1-|u|)^{\lambda_d+n_d-1}\, du_1\cdots du_{d-1}= \frac{\Gamma(\lambda_1+n_1)\cdots \Gamma(\lambda_d+n_d)}{\Gamma(|\boldsymbol{\lambda}|+|\mathbf{n}|)},
C = \frac{1}{2}(D + D (-\frac{1}{2}\otimes (E_1 F_1 + E_2 F_2)) - (-\frac{1}{2}\otimes (E_1 F_1 + E_2 F_2)) D) \in \mathcal{I}C^{-}= D - C \in \mathcal{I}.
\sum\limits_{\substack{(A,B)\in \mathbb{Z}^2}}\Gamma\left(\frac{A}{X^4}\right)\Gamma\left(\frac{B}{X^6}\right) \mu^2\left(\Delta_{A,B}\right)=X^{10}\cdot \frac{1}{3}\cdot \prod\limits_{p>3}\left(1-\frac{2p-1}{p^3}\right) + O\left(X^{7-5/27+\varepsilon}\right).
\Gamma [ \sigma ] = \int d^{D}\! x \sqrt{- g}( - \frac{\sigma^{2}}{2 \lambda_{0}}) - i \mathrm{T r}\mathrm{L n}[ - \sqrt{- g}( \gamma^{\mu}\nabla_{\mu}+ \sigma ) ] +{\cal O}( \frac{1}{N}) .
{\cal{G}}_{2}^{( d - \tilde{\eta}_{o})}( x_{1}, x_{2}, x_{3}, x_{4}) = \left[ C ( \tilde{\eta}_{o}) \right]^{2}{\cal{G}}_{2}^{( \tilde{\eta}_{o})}( x_{1}, x_{2}, x_{3}, x_{4}) ,
Z(f, L; s, \phi) = \int_{G_+/\Gamma}\chi(g)^s\phi\left(g\right)\sum_{x\in L \setminus L_0}f(g\cdot x)dg Z(f, \hat{L}; s, \phi) = \int_{G_+/\Gamma}\chi(g)^s\phi\left(g\right)\sum_{x\in \hat{L}\setminus \hat{L}_0}f(g\cdot x)dg.
\sum_{\substack{y_1 > y_2 > \ldots > y_N\geq 0 y_1, \ldots, y_N \in E}}{\det_{i, j}[P_1(t; x_i, y_j)]\Delta_N(\mathbf{y})}= e^{c_Nt}\Delta_N(\mathbf{x}), \textrm{for all}\mathbf{x}\in E_N, t\geq 0.
\left[\Bigl(\overline{V_\Gamma}+\varepsilon\,\overline{\partial^\circ g_i}+\varepsilon^2\bar{g}_i\bar{\tau}_\varepsilon^i\cdot\overline{\nabla_\Gamma V_\Gamma}\Bigr)\rho^\varepsilon\right](x,t) = V_\Gamma\eta_0(r)+\varepsilon\{(\partial^\circ g_i)\eta_0(r)+V_\Gamma\eta_1(r)\}+O(\varepsilon^2),
V_i(x_i) = e^{-L_i \tau_i}W_i(z_i) \geq e^{-L_i N_{0i}}\max_{j=1}^{n}\tilde\chi_{ij}(W_j(z_j)) = \max_{j=1}^{n}\chi_{ij}(W_j(z_j)) \geq \max_{j=1}^{n}\chi_{ij}(V_j(x_j)),
\begin{array}{r c l}{{{p^{\prime}}_{\alpha}}}&{{=}}&{{{p^{\prime}}_{\alpha}( p ) , \qquad \alpha = 1 , \ldots , q - 1 \, ,}}\\{{}}&{{}}&{{}}\\{{p_{q}^{\prime}}}&{{=}}&{{p_{q}\, ,}}\\ \end{array}
\begin{array}{rcl}g(s;y,z) = \frac{2^{1-s}}{\Gamma (s)}\sum_{k=0}^N \frac{{\bf B}_{2k}}{(2k)!}y^{2k}\Gamma (s-1+2k)\Bigl \{2\zeta_R \left(s-1+2k;1+\frac{z}{2}\right)+ &+ \left(\frac{z}{2}\right)^{1-2k-s}\Bigr \}+O\left(y^{2N+2}\right),\end{array}
\int\displaylimits_{0}^{+\infty}e^{-\frac{i}{2c}(r-n-h)g}\mathrm{d}g = \frac12\int\displaylimits_{0}^{+\infty}e^{-\frac{i}{2c}(r-n-h)g}\mathrm{d}g+\frac12\int\displaylimits_{-\infty}^{0}e^{\frac{i}{2c}(r-n-h)g}\mathrm{d}g
\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(2i-1)(i+j+1)}\equiv 1-4p(-1)^{\frac{p-1}{2}}-2p^2-p^2\sum_{k=1}^{\frac{p-3}{2}}\frac{(-1)^k}{k^2}\pmod{p^3}.
\acute{o}^{A}\bar{o}^{A^{\prime}}+ o^{A}\acute{\bar{o}}{}^{A^{\prime}}= o^{A}\dot{\bar{\iota}}{}^{A^{\prime}}+ \dot{o}^{A}\bar{\iota}{}^{A^{\prime}}+ \iota^{A}\dot{\bar{o}}{}^{A^{\prime}}+ \dot{\iota}^{A}\bar{o}{}^{A^{\prime}}.
\frac{\sup_{y \in J_\alpha}\P (S^y_\delta (\tau_D)=a_\delta \ |\S^y_\delta (\tau_D) \in A)}{\inf_{y \not\in J_\alpha}\P (S^y_\delta (\tau_D)=a_\delta \ |\S^y_\delta (\tau_D) \in A)}\leq C.
A^{1,0}a_0\Big|_0 = n^m 2^{m}A^{1,0}a_0\Big|_1 = - n^{m+1}2^{m+1}P_1(m) A^{1,0}a_0\Big|_2 = n^{m+2}2^{m+2}P_2(m) + n^m 2^{m}m \ \alpha_p (3),
I \neq J \{i_1 , \ldots , i_n\}\neq \{j_1 ,\ldots , j_n \}\Rightarrow Z^{I;\vec{i}}_n \cap Z^{J;\vec{j}}_n = \emptyset I = J \{i_1 , \ldots , i_n\}= \{j_1 ,\ldots , j_n \}\Rightarrow Z^{I;\vec{i}}_n = Z^{J;\vec{j}}_n
\mathrm{dvol}_{g_E}\left(R_E-\frac{1}{2}|d\phi|^2_E\right) =\mathrm{dvol}_ge^{-2\phi}\left(R+\frac{9}{2}\Delta\phi+\frac{9}{2}|d\phi|^2-\frac{1}{2}|d\phi|^2 \right) &=\mathrm{dvol}_ge^{-2\phi}\left(R+\frac{9}{2}\Delta\phi-5|d\phi|^2 \right).
\langle \Sigma_{f}, J_{\phi}^{f}\vert \Sigma_{i}, J_{\phi}^{i}\rangle = \delta ( Q_{f}- Q_{i}) \int D [ g , \phi ]{e^{- S + i \int_{t_{i}}^{t_{f}}d^{3}x \sqrt{g}J_{\phi}^{\mu}\partial_{\mu}\phi}}
\hat{X}_t^{J, \kappa, \delta}= e^{A( \delta + t )}P_J(\xi) + \int_0^t e^{A( \kappa + t - s )}P_J F( \hat{X}_s^{J, \kappa, \delta}) \, ds + \int_0^t e^{A( \kappa + t - s )}P_J B( \hat{X}_s^{J, \kappa, \delta}) \, dW_s .
\mathcal{R}_{t}:= \overline{\left\{(p, [\ell_1], [\ell_2], [F])\in{\Theta}^t_{h}\left| \begin{array}{l}\ell_1\neq \ell_2 \exists D\subset \Lambda_{p,X_F}^h \ell_1\cap \Lambda_{p,X_F}^h \ell_2\cap \Lambda_{p,X_F}^h\end{array}\right.\right\}}.
\vert T_{F,G,N}^L(f_1-1,1,\dots,1)\vert = \vert T_{F,G,N}^L(f_1,1,\dots,1) - T_{F,G,N}^L(1,1,\dots,1)\vert & = \vert 0 - T_{F,G,N}^L(1,1,\dots,1)\vert\gg_{c,C,\varepsilon}1
( \bar{S}\bar{R})_{o o^{\prime}}={\frac{1 6 i^{o + o^{\prime}+ 2}}{\pi^{2}o^{\prime}}}\sum_{e = 2}^{\infty}{\frac{e^{3}}{( e^{2}- ( o^{\prime})^{2}) ( e^{2}- o^{2})}}= \pm \infty .
\Psi_{\sigma}^{( n )}( w^{\prime}, x^{4}w_{2 n}| \zeta^{\prime}, x^{2}\zeta_{2 n}) J ( w^{\prime}x^{4}w_{2 n}| \zeta^{\prime}, x^{2}\zeta_{2 n}) = \sigma \Psi_{\sigma}^{( n )}( w_{2 n}, w^{\prime}| \zeta ) J ( w_{2 n}, w^{\prime}| \zeta ) \prod_{j = 1}^{2 n - 1}\left( \frac{\zeta_{j}}{\zeta_{2 n}}\right)^{\frac{1}{r}},
\begin{array}{ll}z=\displaystyle\frac{1}{\sqrt{m^*}}\left( \sqrt{B}\big(x_{1}+ix_{2}\big) + \frac{1}{\sqrt{B}}\big(-ip_{1}+p_{2}\big)\right), \hfill begin{align*}8pt] w=\displaystyle\frac{1}{\sqrt{B}}\big(-ip_{1}-p_{2}\big),\hfill \end{array}
\begin{aligned}\partial_u \nabla (H_{\geq 5}\circ \Phi_B)(T_{\delta})[h] =(\partial_u \nabla H_{\geq 5})(\Phi_B(T_{\delta}))[h]+\mathcal{R}_{H_{\geq 5}}(T_{\delta})[h]= &=\partial_x (r_1 (T_{\delta})\,\partial_x h)+r_0(T_{\delta}) h+\mathcal{R}_{H_{\geq 5}}(T_{\delta})[h]\end{aligned}
{M_{Pl}\over m}=\left({3n^3P^S_k\over 128\pi}\right)^{{1\over n-4}}\left({\phi\over M_{Pl}}\right)^{-{n+2\over n-4}}=\left({3n^3P^S_k\over 128\pi}\right)^{{1\over n-4}}\left({4\pi\over nL}\right)^{{n+2\over 2(n-4)}},
c={(4\pi)}^{4/3}\int_0^{\infty}du \sum_{l=0}^{\infty}\Big(\frac{u^{3l}{(-1)}^l l!}{32\pi^2(4l+2)!}-\frac{u^{3l+3/2}{(-1)}^l (2l+1)!!}{32\pi^{3/2}2^l(4l+4)!}\Big)
\mu \Lambda = \begin{pmatrix}\mu \Lambda_1 \mu \Lambda_0 0 \end{pmatrix}, \nu_{m-n}\mu \Lambda = \begin{pmatrix}\nu_{m-n}\mu \Lambda_1 \nu_{m-n}\mu \Lambda_0 0 \end{pmatrix}= \begin{pmatrix}\nu_{m-n}\mu \Lambda_1{?}0 C \end{pmatrix}.
W = 2 \sqrt{2}m e^{{\frac{5}{\sqrt{1 5}}}\tilde{\varphi}}- e^{{\frac{2}{\sqrt{1 5}}}\tilde{\varphi}}( \sqrt{2}\mu e^{{\frac{3}{\sqrt{1 0}}}\tilde{f}}+{\frac{R_{4}}{2 \sqrt{2}\mu}}e^{-{\frac{2}{\sqrt{1 0}}}\tilde{f}}) .
\begin{aligned}n_0^0 + n_0^1 = \frac{q'l^0 - a}{q}+ \frac{q'l^1 - b}{p}&= \frac{q'(pl^0+ql^1) - pa - qb}{pq}&< \frac{2q'pq - p'q' + pqk}{pq}< k. \end{aligned}
\frac14\bar\nabla \log f \cdot U = -\frac14 \theta(f)(-\bar\nabla f)B\frac{U}{\theta(f)B}\geq - \frac14 \Psi\big(\frac{U}{\theta(f)B}\big)\theta(f)B -\frac14\Psi^{*}\big(-\bar\nabla f\big)\theta(f)B\;.
T^*_pg_{h(Y)}(T_p^*)^* = T^*_pg_{\Gamma^p\cap \Gamma_q}T_p = g_{\Gamma^{p\cdot r}\cap \Gamma_{r_1}}= g_{h(\Delta_{rp}\cap \Delta^{r_1})}= g_{h((\Delta_p\cap \Delta^q)_r)}&= g_{h(Y_r)}.
S = \int \frac{1}{2}\epsilon^{\mu \nu \lambda \rho}\left( B_{\mu \nu}^{a}( x ) R_{\lambda \rho}^{a}( x ) + \underbrace{\phi^{a b}( x ) B_{\mu \nu}^{a}( x ) B_{\lambda \rho}^{b}( x )}_{\mathrm{c o n s t r a i n t \: t e r m}}\right) .
1 - g_{m_n}+ \lambda G_{m_n}= 1 - (\varphi^d_{m_n}- c^d_{m_n}\psi_{m_n}) + \lambda G_{m_n}= - \Phi^d_{m_n}+ c^d_{m_n}\psi_{m_n}-\sum_{k=1}^{d}\lambda^k G^k_{m_n},
\mathcal{R}=\Bigl\{(x_1,\dots,x_r)\in \mathcal{Q}_r(\eta):\, -\gamma\le L(x_{i_1},\dots,x_{i_\ell})\le \gamma,\,\sum_{i\in\mathcal{I}'}x_i\in [\theta_1,\theta_2+\epsilon]\Bigr\}.
\mathfrak{B}_J=\begin{cases}\mathfrak{S}_{p_2}\times\cdots\times \mathfrak{S}_{n-p_k}, p_1=0, \mathfrak{B}_{p_1}\times \mathfrak{S}_{p_2-p_1}\cdots\times \mathfrak{S}_{n-p_k}, p_1>0.\end{cases}
\chi_{x}^{(\mathfrak{L})}\left( \omega_{1}\right) \left( y\right) :=\omega_{1}\left( y+x\right) \ ,\ \chi_{x}^{(\mathfrak{b})}\left( \omega_{2}\right) \left( \{y,y^{\prime}\}\right) :=\omega_{2}\left(\{y+x,y^{\prime}+x\}\right) \ .
\begin{array}{l l}{{\{\phi_{a}^{i}, \phi_{b}^{j}\}}}&{{= \Omega_{a b}^{i j}}}\\{{\{\phi_{a}^{i}, G_{b}\}}}&{{= C_{a b}^{c}\phi_{c}^{i}}}\\{{\{G_{a}, G_{b}\}}}&{{= C_{a b}^{c}G_{c}}}\\ \end{array},
E_1(H,r,w):= \left\{\exists \, i\in\left\{0,\dots, m \right\}:\y_i\big( \tfrac{H}{r}, r \big)\right\}, E_2(H,r,w):= \bigcup_{\substack{0 \leq i \leq m 0 \leq j \leq r-1}}\left\{X_{(j + 1)\frac{H}{r}}^{y_i}- X_{j\frac{H}{r}}^{y_i}\geq \left( v_+ + \frac{\theta}{2 r}\right)\frac{H}{r}\right\}.
\mu_i(j)=\begin{cases}\dfrac{1-2^{-i}}{i}\textrm{for $j\in \{0,\ldots,i-1\}$}begin{align*}0.2in]\phantom{aa}\dfrac{2^{-i}}{i}\textrm{for $j\in \{i,\ldots,2i-1\}$}\end{cases},\textrm{if $i$ is odd.}
\mathbf{V}_{\alpha, p}\mu(x) \le c_1\, \mathbf{W^{c}}_{\alpha, p}\mu(x) =c_1 \, \int_0^{\infty}\left(\frac{\mu(B_r(x))}{r^{n-\alpha p}}\right)^{\frac{1}{p-1}}e^{-c r}\frac{dr}{r}, x \in \mathbb{R}^n.
\int{{[d\delta \phi]}\over{{\rm Vol}({\rm Weyl})}}\int_0^{\infty}dl ~ J_b(l;{\hat{g}}) e^{-{{10-d}\over{48\pi}}S_{SL}[\phi,\zeta,g]}({\rm det}^{\prime}\Delta )_{{\hat{g}}}^{-5}\quad ,
( D_{j})_{i}^{l}= \left( \begin{array}{c c c}{{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}{c c c}{{a_{j ( 1 )}\cdot{\bf 1}}}&{{\dots}}&{{0}}\\{{0}}&{{\ddots}}&{{0}}\\{{0}}&{{\dots}}&{{a_{j ( N )}\cdot{\bf 1}}}\\ \end{array}\right)
\tilde{\mathbf{P}}^2= \mathbf{A}^{-1}\mathbf{Q}\mathbf{P}^{-1}_f \mathbf{P}_2 \mathbf{A}^{-1}\mathbf{Q}\mathbf{P}^{-1}_f \mathbf{P}_2=\mathbf{A}^{-1}\mathbf{Q}\mathbf{P}^{-1}_f \mathbf{P}_f \mathbf{P}^{-1}_f \mathbf{P}_2=\tilde{\mathbf{P}},
R = \begin{bmatrix}1 -q^-_2 0 0 \ldots 0 0 -q^+_2 1 -q^-_3 0 \ldots 0 0 0 -q^+_3 1 -q^-_4 \ldots 0 0 \vdots \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 0 \ldots 1 -q^-_{N-1}0 0 0 0 \ldots -q^+_{N-1}1\end{bmatrix},
V ( a ) = - \frac{\kappa_{( 4 )}^{2}}{6}\rho_{X 0}a^{- 3 w_{X}+ 2}- \frac{\kappa_{( 4 )}^{2}}{1 2 \lambda}\rho_{X 0}^{2}a^{- 6 w_{X}+ 2}- \frac{\sigma_{0}^{2}}{6}a^{- 4}- \frac{\Lambda_{( 4 )}}{6}a^{2}- \frac{{\cal U}_{0}}{\lambda \kappa_{( 4 )}^{2}}a^{- 2}+ \frac{k}{2}
\begin{array}{c}\bar{a}_\alpha(x)=\tilde{a}_\alpha(r_\alpha x),\bar{h}_\alpha(x)=\tilde{h}_\alpha(r_\alpha x),\bar{f}_\alpha(x)=\tilde{f}_\alpha(r_\alpha x), \bar{c}_\alpha(x)=\tilde{c}_\alpha(r_\alpha x),\bar{d}_\alpha(x)=\tilde{d}_\alpha(r_\alpha x),\bar{Y}_\alpha(x)=\tilde{Y}_\alpha(r_\alpha x).\end{array}
p^\intercal = \lim_{t_j\to 0}\frac{\big(h(t_j)-h\big)^\intercal}{t_j}= 0 H'(p-L_Xh) =\lim_{t_j\to 0}\frac{1}{t_j}H'|_{g_{t_j}}\big( h(t_j)- \psi^*_t(h) \big) &=-\lim_{t_j\to 0}\frac{1}{t_j}\psi_t^*\big(H'|_{g}(h)\big)=-X(H'(h)),
\prod_{m = 1}^{M}( Z^{1 \dagger})_{j_{m}}^{i_{m}}\prod_{m^{\prime}= 1}^{M^{\prime}}( Z^{2 \dagger})_{j_{m^{\prime}}^{\prime}}^{i_{m^{\prime}}^{\prime}}\prod_{n = 1}^{N k}( \Psi^{1 \dagger})_{l_{n}}| 0 \rangle ,
h_{\rm top}(T) = \sup_{\mu \in M (X,T)}h_\mu (T) = \sup_{\mu \in M (X,T)}\mu (U) h_{\mu_{U}}(T_U) & \leq \epsilon \sup_{\mu \in M (X,T)}h_{\mu_{U}}(T_U) = \epsilon h_{\rm top}(T_U) ,
\mathbf e \approx \sum_{j=m+1}^M r_j \begin{bmatrix}\mathbf 0 [\mathbf L_{22}^{-1}]_{:,j}\end{bmatrix}\approx \sum_{j=m+1}^M r_j \frac{h}{2}\begin{bmatrix}\mathbf 0 G(\mathbf S_2, S_j) \end{bmatrix},
Q_{\Sigma\Delta}^r : A_m(\mathcal{X}) \rightarrow \mathcal{A}^m y \rightarrow q, \textrm{where}\ \ q_i = Q_s\left( \sum\limits_{j=1}^r (-1)^{j-1}\binom{r}{j}u_{i-j}+y_i\right) & u_i = y_i - q_i + \sum^{r}_{j=1}{r \choose j}(-1)^j u_{i-j}\ \i=1,...,m & u_0 =c_0,
d_X(F^2(x_0,y_0),F(x_0,y_0))= d_X(F(F(x_0,y_0),G(y_0,x_0)),F(x_0,y_0)) \leq k\d_X(F(x_0,y_0),x_0)+ l\d_Y(G(y_0,x_0),y_0) \leq (k+l)\ [d_X(x_1,x_0)+ d_Y(y_1,y_0)]
\mathbb{P}(Y^2- B_n^2 > t ) = \mathbb{P}(|Y|^2 > t+ B_n^2) & \leq \mathbb{P}(|\varepsilon|^2 > (1/2)(t+ B_n^2) - B^2) & \leq e^{-\frac{t}{2\nu}}e^{-\frac{1}{v}(\frac{B_n^2}{2}-B^2)}\mathbb{E}e^{|\varepsilon|^2/\nu}.
\partial_s I(1,\alpha,s)=-\int_0^\alpha\left[\int_0^{+\infty}\frac{\log \left({r}^{2}+2\,r\,\cos t+1\right) r\,dr}{{\left({r}^{2}+2\,r\,\cos t+1\right)}^{\frac{2+s}{2}}}\right]\,dt.
\nu\left(d_{j',k,i}Q_{i+1}^j\right)-\nu_{i+1}(h)=\nu\left(d_{j',k,i}\right)-\nu\left(d_{\delta,i+1}\right)+(j-\delta)\beta_{i+1}\ge &\ge(j'-j)\beta_i+\nu\left(d_{j',i}\right)-\nu\left(d_{\delta,i+1}\right)+(j-\delta)\beta_i=\nu\left(d_{j',i}Q_i^{j'}\right)-\nu\left(d_{\delta,i}Q_i^\delta\right)\ge\nu_i^+(h)-\nu_i(h),
\lambda^\star(\hat{x},x^{\star}) = \left(\frac{z}{b}\right)\frac{l_2^{-1}z^{-l_2}-l_1^{-1}z^{-l_1}}{l_1^{-1}z^{-l_1}+l_2^{-1}z^{-l_2}}-1 \kappa^\star(\hat{x},x^{\star}) =z\left[l_1^{-1}+l_2^{-1}-1\right]-z\frac{l_1^{-1}z^{-l_1}-l_2^{-1}z^{-l_2}}{l_1^{-1}z^{-l_1}+l_2^{-1}z^{-l_2}}\left[l_1^{-1}-l_2^{-1}+\ln{\hat{x}}-\ln{x^\star}\right],