harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal A}_{\rm IR}={\cal A}( \infty ) -{\cal A}( \Lambda ) \simeq \int_0^{\infty}\frac{dS}{S}S^{n/2 - 12}( e^{- p \circ p S / 2 \pi \alpha^{\prime}}- e^{- ( p \circ p + 1/ \Lambda^2 ) S / 2 \pi \alpha^{\prime}}) ~,
\left( \prod_{i=0}^{ka-1}\binom{mk/n+i}{bk/n}\binom{bk/n+i}{bk/n}^{-1}\right)^{\frac{n}{k}}\!\!\!\!\!\! - \sum_{\substack{j|k j < k}}\left( \prod_{i=0}^{ja-1}\binom{mj/n+i}{bj/n}\binom{bj/n+i}{bj/n}^{-1}\right)^{\frac{n}{j}}.
4 \pi{\cal L}^{\mathrm{g a u g e}}= -{\textstyle{\frac{i}{8}}}({\cal N}_{I J}\, F_{\mu \nu}^{+ I}F^{+ \mu \nu J}- \bar{\cal N}_{I J}\, F_{\mu \nu}^{- I}F^{- \mu \nu J}) ,
G ( x , x^{\prime}) = i \int \frac{d^{3}q}{( 2 \pi )^{3}}e^{i{\bf q}. ({\bf x}-{\bf x}^{\prime})}\left( \begin{array}{l l}{{G^{+ +}({\bf q}, t - t^{\prime})}}&{{\: \: G^{+ -}({\bf q}, t - t^{\prime})}}\\{{G^{- +}({\bf q}, t - t^{\prime})}}&{{\: \: G^{--}({\bf q}, t - t^{\prime})}}\\ \end{array}\right) \: ,
m=l^B_A(m) r^B_A(m) =l^B_A(m) q(r^B_A(m))q(r^B_A(m))^{-1}r^B_A(m) &\stackrel{(L2)}=q(l^B_A(m) r^B_A(m)) q(r^B_A(m))^{-1}r^B_A(m) &=q(m)q(r^B_A(m))^{-1}r^B_A(m)
(q-1)[z^n]\left(h(q,z)-\frac12 h(q^2,z^2)\right)=\sum_{(2l-1)m=n}\frac{q-1}{q^{2l-1}-1}\frac{(-1)^{m-1}Q_m(q^{2l-1})}{(2l-1)q^{(2l-1)\binom{d}{2}}}.
\begin{cases}\frac{\partial}{\partial t}u_j(t,x)=c_j(t)\frac{\partial}{\partial x}u_j(t,x)+q_j(t,x)u_j(t,x), x\in\left(0,1\right),\t\geq 0, u_j(t,0)= f_j(x), x\in\left(0,1\right), \phi^-_{ij}c_j(t)u_j(t,1)=\omega_{ij}\sum_{k\in\mathbb{N}}{\phi^+_{ik}c_k(t)u_k(t,1)}, t\geq0.\end{cases}
\frac{d\rho}{d t}=\frac{\partial}{\partial S}\mathcal H_0(\rho,S)+\frac{\partial}{\partial S}\mathcal H_1(\rho,S)\circ dW(t), &\frac{d S}{dt}=-\frac{\partial}{\partial \rho}\mathcal H_0(\rho,S)-\frac{\partial}{\partial \rho}\mathcal H_1(\rho,S)\circ dW(t).
|g(x) - x^2| = |2^{-n}(2k+1) (x-k2^{-n}) + k^2 4^{-n}- x^2| = |2^{-n}(2k+1) \delta + k^2 4^{-n}- (k2^{-n}+ \delta)^2| = |\delta (2^{-n}- \delta)| \leq 4^{-n}.
\Psi_{M-1,2}^\alpha(t) =\int_t^{t_M}\Big[\int_{t_{M-1}}^t e^{-\lambda_\alpha(t-\tau)}d\tau \Big]^2ds =\frac{[1-e^{-2\lambda_\alpha(t-t_{M-1})}]^2}{\lambda_\alpha}k &\leq \frac{1-e^{-2\lambda_\alpha(t-t_{M-1})}}{\lambda_\alpha}k^2\leq \frac{1-e^{2\lambda_\alpha k}}{\lambda_\alpha}k^2.
\sum_{k = 1}^{\infty}\frac{T_{\nu}( \lambda_{\nu , k})}{\lambda_{\nu , k}^{\sigma}}J_{\sigma + \nu + 1}( \lambda_{\nu , k}) J_{\nu}( \lambda_{\nu , k}t ) = \frac{1}{2}\int_{0}^{\infty}{J_{\sigma + \nu + 1}( z ) J_{\nu}( z t ) \frac{d z}{z^{\sigma}}}= \frac{\left( 1 - t^{2}\right)^{\sigma}t^{\nu}}{2^{\sigma + 1}\Gamma ( \sigma + 1 )}
E_{R}={\frac{( \epsilon - 1 )^{2}}{4 \pi a}}\sum_{l = 1}^{\infty}\nu^{2}\int_{0}^{\infty}d z \left[ F_{l}( \nu z ) -{\frac{t^{4}}{4}}+{\frac{t^{1 0}}{8 \nu^{2}}}( 1 + 8 z^{2}- 5 z^{4}+ z^{6}) \right] ,
s=\frac{3\alpha}{4}\prod_{k=2}^{\infty}{\frac{(\frac{k^2}{1+k})[\frac{s^2(k-1)^2}{(1+s(k-1))}+\alpha]}{(\frac{(1+k)^2}{2+k})[\frac{s^2(k-2)^2}{(1+s(k-2))}+\alpha]}}
\frac{[G_n:B_{n}^{-}]}{|U_n|}\Psi_{\underline{\chi}_{n}}^{GL_{n}}(g_{1,\cdots, 1}(a_1,\cdots,a_n))=\sum_{t\in T_n}f_{T_n}(e_{\psi_n}g_{1,\cdots, 1}(a_1,\cdots,a_n)e_{\psi_n})(t)\underline{\chi}_n(t).
\hat{A}_{1}\mathcal{D}\tilde{\mathcal{Z}}_{p}\hat{A}_{1}^{\top}=-\tilde{A}_{1}\hat{Q}^{-1}\hat{A}_{12}^{\top}\left(\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}\right)^{-1}\hat{A}_{p}\frac{\partial v''_{p}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi\right]}\tilde{A}_{1}^{\top}.
\begin{aligned}Lu = \sum_{i=1}^n \left(x_ia_{ii}(z)u_{x_ix_i}+ b_i(z)u_{x_i}\right) + \sum_{i,j=1}^n x_ix_j \tilde a_{ij}(z)u_{x_ix_j}&\quad+\sum_{i=1}^n\sum_{l=1}^m x_i c_{il}(z)u_{x_iy_l}+\sum_{k,l=1}^m d_{kl}(z)u_{y_ky_l}+ \sum_{l=1}^m e_l(z)u_{y_l},\end{aligned}
\hat{\mathbb{E}}[\sum_{i=1}^{n}\hat{\mathbb{E}}_{t_i}[\xi_i]{\varphi^{i,m}_{l}}\prod_{j=1}^{i-1}{\widetilde{\varphi}^{j,m}_l}]=\hat{\mathbb{E}}[\hat{\mathbb{E}}_{t_1}[\sum_{i=1}^{n}\xi_i{\varphi^{i,m}_{l}}\prod_{j=1}^{i-1}{\widetilde{\varphi}^{j,m}_l}]]=\hat{\mathbb{E}}[\sum_{i=1}^{n}\xi_i{\varphi^{i,m}_{l}}\prod_{j=1}^{i-1}{\widetilde{\varphi}^{j,m}_l}].
A \le\sum_{x \in\Gamma}\left| \langle\mathbf{1}_{\{y\}}, \lambda_{G}(x)\varphi\rangle\right|^{2}& = \sum_{x \in y^{-1}\Gamma}|\varphi(x)|^{2}& \le\sum_{x \not \in U}|\varphi(x)|^{2}< A
\overline{Q}_{k,a}(x;q)=\sum_{i=1}^a (xq)^i \left[\sum_{h=1}^{k-i}(xq^2)^h\overline{Q}_{k,k-h}(xq^2;q) +\sum_{h=0}^{k-i-1}(xq^2)^h\overline{Q}_{k,k-h}(xq^2;q)\right] &\quad+ \sum_{i=0}^{a-1}(xq)^i\left[\sum_{h=1}^{k-i}(xq^2)\overline{Q}_{k,k-h}(xq^2;q)+\sum_{h=0}^{k-i-1}\overline{Q}_{k,k-h}(xq^2;q)\right].
G^{J J^{\prime}}( x ) = \frac{\delta^{J J^{\prime}}}{| x |^{2 J}}\left( \frac{g_{\mathrm{Y M}}^{2}N}{8 \pi^{2}}\right)^{J}\cdot J \cdot \frac{\operatorname{s i n h}\left( \frac{1}{2}\frac{J^{2}}{N}\right)}{\frac{1}{2}\frac{J^{2}}{N}}
15 AB(A^2-B^2)(2A^2+5AB+2B^2)P_5(A,B,t) = 3(A^2-B^2)t{1+(t(A+B))^2\over(1-(t(A+B))^2)^2}+3(B^2-(A+B)^2)t{1+(tA)^2\over(1-(tA)^2)^2}+3((A+B)^2-A^2)t{1+(tB)^2\over(1-(tB)^2)^2},
\mathbb{M}^s(\mathbb{R}^d) =\mathbb{H}^{s}(\mathbb{R}^d),s\geq 0, \mathbb{M}^s_{\mathrm{e}}(\mathbb{R}^d) =\mathbb{E}^{\frac{1}{2}}_q(\mathbb{R}^d),q=\frac{s}{\sqrt{\pi}},s\geq 0, \mathbb{M}^s_{\mathrm{e}^2}(\mathbb{R}^d) =\mathbb{E}^{1}_q(\mathbb{R}^d),q=\ln(\tfrac{\pi}{\pi-s}),\pi>s\geq 0.
M_j:=\phi_{j-1}\circ\cdots \circ\phi_1\left(\underbrace{\bigcap_{0\leq k\leq j-1}\left\{\frac{f}{\det \nabla (\phi_{k}\circ \ldots \circ \phi_1)}\geq \frac{1}{2}\right\}}_{=:A_j}\right)
\sum_{3r_0+2r_1+r_2=d_1+1}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}{d_1+1-r_0-r_1 \choose r_2}& \frac{(d_2+3)!}{2^{r_2}6^{r_3}(d_2-2d_1+1)!}{n_1 \choose d_1+1}{n_2 \choose d_2+3},
\int_{B_{\frac{R+r}{2}}}\int_{\mathbb{R}^N\setminus B_R}\left|\frac{\delta_h u(y)}{|h|^t}\right|^{p}\,\frac{1}{|x-y|^{N+s\,p}}\,dx\,dy =\frac{C}{R^{s\,p}}\,\int_{B_\frac{R+r}{2}}\mathrm{Snail}\left(\frac{\delta_h u}{|h|^t};x,B_R\right)^p\,dx &\le\frac{C}{R^{s\,p}}\,\langle u\rangle^p_{\mathcal{Y}^{t,p}_s(B_\frac{R+r}{2};B_R)}.
||S^X_{Y,Z}(U)(y)||_{M_K(M_n(X\otimes_hZ))}=||\sum_k (A_k\otimes Id_K) \odot (B_k(y))|| &\leq \sum_k||A_k\otimes Id_K||_{M_K(M_{n,n_k}(X))}||B_k(y)||_{M_K(M_{n_k,1}(Z))}&\leq \sum_k||A_k||_{M_{n,n_k}(X)}||B_k||_{CB(M_{n,1}(Y),M_{n_k,1}(Z))}||y||_{M_K(M_{n,1}(Y))}
\left( d_{c o c h a i n}\omega^{1}\right) ( x , y , z ) \! : = \! \omega^{1}( y , z ) \! - \! \omega^{1}( x y , z ) \! + \! \omega^{1}( x , y z ) \! - \! \omega^{1}( x , y ) \! = \! 0 ,
\sum_{\substack{m\in \mathbb{Z}3m^2+2m+1\leq n}}\left(3m+1\right) c\left(\omega(-q); n-3m^2-2m-1\right) =6\tilde{R}_n+\sum_{\substack{a,b\in\mathbb{Z}^+ ab=n a>3b a\equiv b\pmod{2}}}a+\sum_{\substack{a,b\in\mathbb{Z}^+ ab=n a>3b-2 a\equiv b\pmod{2}}}3b.
\begin{array}{lll}\ \left(\sigma_i,\sum_{j=1}^{\infty}\sigma_j\,Sa_{in}\left(\overline{\sigma}_j\right)S^{-1}+\overline{\sigma}_j\,Sa_{in}^{\dagger}\left(\sigma_j\right)S^{-1}\right)_{K.G.}&= Sa_{in}\left(\overline{\sigma}_i\right)S^{-1}\;.\end{array}
\begin{aligned}D^{\beta}(|x|^{r_1}W(t) u_0)(x,y) &= W(t) (D^{\beta}|x|^{r_1}u_0)(x,y) + W(t)(D^{\beta}(\{\Phi_{1,t,r_1}(\widehat u_0)(\xi,\eta)\}^{\lor}))(x,y)\end{aligned}
\frac12\int_{s_0}^{s_1}[\frac12\left(\frac{dA_1}{ds}-\frac{dA_2}{ds}\right)^*\left(\frac{dA_1}{ds}-\frac{dA_2}{ds}\right)+\sum_{i=1}^2\left(\frac{dZ}{ds}-ieA_iZ\right)^*\left(\frac{dZ}{ds}-ieA_iZ\right)]ds
\partial_tf =-y\cdot \partial_zf-\nabla^{\perp}\nu\cdot\nabla\omega+v\cdot\partial_zf-\theta(\partial_v-t\partial_z)f &=-\theta\cdot\partial_vf-(h+1)\nabla^{\perp}\phi\cdot\nabla f+(v-y+t\theta+\partial_y^{-1}\mathbb{P}_0\omega(t,y))\cdot \partial_zf,
M_{\pm,1}(x;z)-e_1=\mathcal{P}^{\pm}\left(M_-(x ;\cdot) R(x;\cdot)\right)_{11}, z \in \mathbb{R}, &(2ik)^{-1}(M_{\pm,1}(x;z)-e_1)=\mathcal{P}^{\pm}\left((2ik^{-1}) M_-(x ;\cdot) R(x;\cdot)\right)_{11}, z \in \mathbb{R},
{\frac{d \bar{w}}{d t}}\, \Pi^{T}\,{\frac{d w}{d t}}={\frac{1}{\kappa^{2}}}\sum_{r = 3}^{N}\left[{\frac{t_{1 r}\bar{t}_{1 r}}{b + r}}+{\frac{t_{2 r}\bar{t}_{2 r}}{b - r}}\right] \, \, .
A = \dfrac{ab}{2}\left(r^k-\dfrac{1}{r^k}\right)^3\left(r^k +\dfrac{1}{r^k}\right)^2 &=\dfrac{(s^2+s-t^2)}{2\sqrt{5}^2}\left(r^k+\dfrac{(-1)^{k+1}}{r^k}\right)^2\left(r^k+\dfrac{-1(-1)^{k+1}}{r^k}\right)^3.
J^{2}(h_{+}\allowbreak )-M\left( h_{+}\right) =\dfrac{4p_{1}\left(s-v\right)^{2}(s-2h_{+})^{2}}{s^{4}}\times \left( p_{1}-\dfrac{{\large (}2wst-(t^{2}-s^{2})v{\large )}^{2}}{s^{2}+t^{2}}\right)
\begin{matrix}0\leftrightarrow1,&1\leftrightarrow2,&2\leftrightarrow3,&3\leftrightarrow4,&4\leftrightarrow5,&...\\ 0\leftrightarrow0,&1\leftrightarrow2,&2\leftrightarrow4,&3\leftrightarrow6,&4\leftrightarrow8,&...\end{matrix}
\Lambda^{\gamma}\theta(x_t,t)= c_{\gamma,n}\int_{\mathbb{T}^n}\frac{\theta(x_t)-\theta(x_t-y)}{|y|^{n+\gamma}}dy+c_{\gamma,n}\sum_{k\in\mathbb{Z}^n\setminus\{0\}}\int_{\mathbb{T}^n}\frac{\theta(x_t)-\theta(x_t-y)}{|y-2k\pi|^{n+\gamma}}dy\leq 0.
Z_f(s, \chi, D) =q^{-ls}Z_{h_2}(s, \chi, D) &=q^{-ls}\Big(v(\bar{h}_2, D, \chi)+\sigma(\bar{h}_2, D, \chi)\dfrac{(1-q^{-1})q^{-s}}{1-q^{-1-s}}+Z_{h_2}(s, \chi, D_{S(h_2, D)})\Big) &=\dfrac{H_{2,3}(q^{-s}, \chi)}{1-q^{-1-s}}+q^{-ls}Z_{h_2}(s, \chi, D_{S(h_2, D)}),
G_{( \sigma )}( z , z^{\prime}) = \sum_{\Gamma}\frac{\exp \pi i [ \Omega_{\Gamma}( \{\sigma_{s}\}) + \sum_{s}2 l_{1 s}\sigma_{s}( J_{( o ) s}( z ) - J_{( o ) s}( z^{\prime}) ) ]}{[ z - g_{\Gamma}( z^{\prime}) ] [ c_{\Gamma}z^{\prime}+ d_{\Gamma}]^{3}}
\left| \frac{\xi_q(\varepsilon_b z + m_b)}{\xi(z+m_b +j_b)}\right| \leq C \begin{cases}(1+|t|)^{-0.03}, \mbox{unless $j_b=0$}, (1+|t|)^{0.01/n^4}, \mbox{$j_b$}= 0, m_b \neq 1 (1+|t|)^{0.01/n^4}\log(5+|t|), \mbox{$j_b$}= 0, m_b = 1 \end{cases}
\mathbf{D}(r,\rho,ru)=\frac{\rho\,np'(n)}{r^2}\begin{pmatrix}0 0 0 -1 1 0 0 0 0 \end{pmatrix}+\theta \begin{pmatrix}0 0 0 0 n^2/r^2 0 -{\rho u}/{r}0{\rho}/{r}\end{pmatrix}
\lambda_a^{(s)}(p(x))=2\,\sum_{r\in{\cal R}_{+,s}}\frac{\nabla p_a(x)\cdot r}{l_r(x)}\,, \lambda_a^{(l)}(p(x))=2\,\sum_{r\in{\cal R}_{+,l}}\frac{\nabla p_a(x)\cdot r}{l_r(x)}\,,\forall\,a=1,\ldots,n\,,
A_0^{j_1j_2j_3}= -B^{i_1i_2i_3}_1\int_0^\infty (e^{-t\tilde \gamma})_{i_1j_1}(e^{-t\tilde \gamma})_{i_2j_2}(e^{-t\tilde \gamma})_{i_3j_3}dt = -\frac{1}{2}\delta^{i_1i_2}(\nabla_q\beta)^{i_3}\int_0^\infty (e^{-t\tilde \gamma})_{i_1j_1}(e^{-t\tilde \gamma})_{i_2j_2}(e^{-t\tilde \gamma})_{i_3j_3}dt
\int_{X_j}|F_{\epsilon,j}-(1-v'_{\epsilon}(\Psi))fF^{1+\delta}|^2_h e^{v_\epsilon(\Psi)-\delta \max\{\psi+T,2\log|F|\}}c(-v_\epsilon(\Psi)) \le \int_{X_j}v''_{\epsilon}(\Psi)|fF|^2_h e^{-u(-v_{\epsilon}(\Psi))}<+\infty.
P\big(\widetilde{C}_T^{d,p}\geq\widetilde{C}_0^{d,p}e^{\frac{4T}{1-2p_0}}\big)\leq P\big(t^d_{K_d}\leq T\big)=P\big(\sum_{l=1}^{K_d}\tau^d_l\leq T\big).
\mathbb{P}_{\epsilon}\left(\epsilon>t \right)= \int_{t}^{\infty}\frac{1}{\sqrt{2 \pi}}\exp \left(\frac{-x^2}{2}\right) dx < \int_{t}^{\infty}\frac{x}{t}\exp \left(\frac{-x^2}{2}\right) dx = \frac{1}{t}\exp \left(\frac{-t^2}{2}\right)
\Big\|\sum_{n = N_1}^{N_2-1}\big| m_{n+1}- m_n \big|\Big\|_{\ell^1}\leq\bigg(\frac{1}{\vartheta_B(N_1)}- \frac{1}{\vartheta_B(N_2)}\bigg) \vartheta_B(N_1)+ \frac{2}{\vartheta_B(N_1)}\big(\vartheta_B(N_2) - \vartheta_B(N_1) \big),
u_2^* (0)= \lim_{t \to T}u_2(x_0, t) = \frac{(T- t_0 (x_0))^{- \frac{1}{p-1}}}{|\ln (T- t_0 (x_0))|}\lim_{\tau \to 1}V_2 (x_0, 0, \tau) \sim \frac{(T- t_0 (x_0))^{- \frac{1}{p-1}}}{|\ln (T- t_0 (x_0))|}\left(\frac{(p - 1)^2}{4 p}\right)^{- \frac{p}{p-1}}(K_0^2)^{- \frac{1}{p-1}}.
r_{+}= R_{0}\sqrt{1 + \frac{1}{3 ( 1 - b ) \omega}}, \qquad r_{-}= R_{0}\left( - \frac{1}{6}\kappa_{5}^{2}\rho R_{0}\sqrt{\frac{3 \omega}{( 1 - b )}}\right)^{1 / ( 1 - b )}.
Lv_a= -a(a-1) \rho^{-2}\bigl(\textrm{log}\rho\bigl)^{-2}v_a - \bigl(N-k-a\bigl) \rho^{-2}(\textrm{log}\rho)^{-1}v_a + \l \eta \rho^{-2}v_a+ O\bigl(\textrm{log}\rho) \rho^{-3/2}\bigl)v_a.
\mathbb{C}_x^{f}\Biggl( \sum_{k=1}^K \sum_{j=1}^{J_k}\sum_{a=0}^{1}C_{k}(j,a)X_{j,k}^{\pi,a}(\cdot) \Biggr) \qquad\mbox{with}f\in\{\mathrm{av}, \beta\},
\prod_{r=1}^{2^q}(s_{r,e}2^{-q-1})^{s_{r,e}}((q-s_{r,e})2^{-q-1})^{q-s_{r,e}}> \prod_{r=1}^{2^q}(s_{r,e}2^{-q-1})^{s_{r,f}}((q-s_{r,e})2^{-q-1})^{q-s_{r,f}}
\tilde{E}f=\int_{B_1^{d-n}}\int_{B_1^{n}}f(\xi_1,\xi_2) e^{i(x\cdot \xi_1+y\cdot \xi_2+t(\frac{\mathfrak{m}(N\xi_1)}{N^2 \mathfrak{m}(N)}+\frac{\mathfrak{m}(N\xi_2)}{N^2 \mathfrak{m}(N)}))}\, d\xi_1 d\xi_2.
\beta_{1,\infty}:=\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_r}\frac{1}{\left|x\right|^{4-a\beta_k}}e^{a\hat{u}_{k}}=\frac{a\beta_{\infty}\left[2\beta-4(N+1)-\beta_{\infty}\right]}{4\left[1-a(N+1)\right]}
\lim_{n\to\infty}\frac{1}{-f_4(n)\log(b_c^{(n)}/f_4(n))}\log P(B_1^{(n)}) =\lim_{n\to\infty}\frac{a_c^{(n)}}{-f_4(n)\log(b_c^{(n)}/f_4(n))}\lim_{n\to\infty}\frac{1}{a_c^{(n)}}\log P(B_1^{(n)}) &=-J(x_0)\lim_{n\to\infty}\frac{a_c^{(n)}}{-f_4(n)\log(b_c^{(n)}/f_4(n))}=-\infty.
\Delta W^{(k+1)}= M_1(-\tilde{f}_1+\rho A^Tc -\rho A^TB\Delta x^{(k)}-\rho A^T \Delta \bar{u}^{(k)}), \Delta x^{(k+1)}=M_2(-\tilde{f}_2+\!\rho B^Tc\! -\!\rho B^T\!A\Delta W^{(k+1)}\!-\!\rho B^T\! \Delta \bar{u}^{(k)}), \Delta \bar{u}^{(k+1)}= \Delta \bar{u}^{(k)}+ A \Delta W^{(k+1)}+ B \Delta x^{(k+1)}- c,
[ \Delta_{1}( \epsilon_{1}) , \Delta_{- 1}( \epsilon_{2}) ] g = [ \int_{x_{0}}^{x}\partial_{\mu}( g \epsilon_{1}g^{- 1}) d y^{\mu}, \epsilon_{2}] g + g [ \int_{x_{0}}^{x}\partial_{\mu}( g^{- 1}\epsilon_{2}g ) d y^{\mu}, \epsilon_{1}] = [ g , \epsilon_{1 2}]
( n - 2 p + 1 ) \left[ ( B^{+})_{k , i_{1}, . . . , i_{n - 2 p}}^{m}+ ( B^{-})_{k , i_{1}, . . . , i_{n - 2 p}}^{m}\right] + ( B_{\{k})_{i_{1}, . . . , i_{n - 2 p}\}}^{m}= 0 ,
\Delta_H \colon \H_\gg\to H_\gg\hat\otimes_{U(\gg)}H_\gg,u\sharp P\stackrel{\Delta_H}\mapsto u \sharp P_{(1)}\otimes_{U(\gg)}P_{(2)}, \Delta_{\hat{S}(\gg^*)}(P) = P_{(1)}\otimes P_{(2)},u\in U(\gg), P\in\hat{S}(\gg^*).
1-r^{2}= (1-r)(1+r) &= \left[ \frac{\alpha}{2}(t_{2}-t_{1})^{2}+O\left(m^{3/2}(t_{2}-t_{1})^{3}\right)\right]\left[2-\frac{\alpha}{2}(t_{2}-t_{1})^{2}+ O\left(m^{3/2}(t_{2}-t_{1})^{3}\right)\right] &= \alpha(t_{2}-t_{1})^{2}\left( 1+O(\sqrt{m}(t_{2}-t_{1}))\right),
\chi_{({\cal V}_{m_1,n_1}\otimes{\cal V}_{m_2,n_2})_{\mathrm f}}(\tau) = \sum_{m_3=|m_1-m_2|+1}^{m_1+m_2-1}\sum_{n_3=|n_1-n_2|+1}^{n_1+n_2-1}\chi_{m_3,n_3}(\tau)\,,
\{\{\{\Theta,\tilde{e}\}, e\},f\}= \pounds_{\rho(\{\tilde{e},e\})}f +\{\{\Theta,e\}, \{\tilde{e}, f\}\}- \{\tilde{e},\{\{\Theta,e\},f\}\}&= \pounds_{\rho(\{\tilde{e},e\})}f + \pounds_{\rho(e)}(\pounds_X f) -\pounds_{X}(\pounds_{\rho(e)}f).
\|\widehat{G}\|^2_{L^2(C,d\sigma_c)}=\sum_{\overline{m}\in \mathbb F_q^{d+1}}G(\overline{m})~ \overline{(G\ast (d\sigma_c)^\vee)(\overline{m})}&\leq\|G\|_{L^{\frac{2d+4}{d+4}}(\mathbb F_q^{d+1}, d\overline{m})}\|G\ast (d\sigma_c)^\vee \|_{L^{\frac{2d+4}{d}}(\mathbb F_q^{d+1}, d\overline{m})}.
\left.H=t^{(2s)}_{2s}(u)^{-1}\frac{d}{du}t^{(2s)}_{2s}(u)\right|_{u=-(2s-1)\lambda}= \prod_{j=1}^{2s}\sinh(j \lambda)^{-1}\left( \sum_{j=1}^{N-1}h_{j,j+1}+ h^{F}_{N,1}\right),
J_{\mathrm{F}}= \exp \{i S [ g ] + \frac{i}{4 \pi}\int \! d^{2}x \, [ \mathrm{T r}( B_{+}( \partial_{-}g ) g^{- 1}) + \frac{N}{2 \pi}\partial_{-}\theta \partial_{+}\theta + \frac{N}{\pi}C_{+}\partial_{-}\theta ] \}\quad ,
\nabla_{\mu}{\cal J}^{\mu}= 0 \, , \hspace{1 c m}{\cal J}_{\mu}= 2 \partial_{\mu}{\cal M}{\cal M}^{- 1}= 2 \left( \begin{array}{c c}{{\frac{1}{2}j_{\mu}^{( \varphi )}}}&{{j_{\mu}}}\\{{}}&{{}}\\{{j_{\mu}^{( 0 )}}}&{{- \frac{1}{2}j_{\mu}^{( \varphi )}}}\\ \end{array}\right) \, ,
\sum^{m}_{h=0}a_{1,h}= -n^m 2^{m-n}\sum^{m}_{h=0}(1 + 2h)a_{1,h}= -n^m 2^{m+1-n}P_1(m) \sum^{m}_{h=0}(1+2h)^2 a_{1,h}=\begin{aligned}[t] -n^m 2^{m+3-n}P_2(m) &+ n^{m-1}2^{m+1-n}\alpha_p (2) - n^{m-2}2^{m+1-n}m \alpha_p (3).\end{aligned}
q_{\beta}(n,t)=\sum_{k=1}^{n}\sum_{\Lambda_{n}^{k}}k!\prod_{j=1}^{n-k+1}\frac{(\theta^{j}/j!)^{x_{j}}}{x_{j}!}\left(\alpha t^{\beta}\right)^{k}E_{\beta,k\beta+1}^{k+1}\left(-\alpha(e^{\theta}-1)t^{\beta}\right).
\langle T_{\ \mu}^{\mu}\rangle = \frac{1}{4 8 \pi}\Biggl [ \left( \partial_{\mu}\partial_{\nu}h^{\mu \nu}- \partial_{\mu}\partial^{\mu}h_{\ \nu}^{\nu}\right) \mp \frac{1}{2}\varepsilon_{\mu \lambda}\partial^{\lambda}\partial_{\nu}h^{\mu \nu}\Biggr ] \, .
V_x(G) = V(G) \cap \{x_i \mid i \in \mathbb{Z}_n\}V_y(G) = V(G) \cap \{y_i \mid i \in \mathbb{Z}_n\}V_u(G) = V(G) \cap \{u_i \mid i \in \mathbb{Z}_n\}V_v(G) = V(G) \cap \{v_i \mid i \in \mathbb{Z}_n\}
\frac1X \int_X^{2X}\Big(\Delta_k\Big(x+\frac{x^{1-\frac1k}}{L}\Big)-\Delta_k(x) \Big)^2\, dx= C_k \cdot \frac{X^{1-\frac1k}}{L}(\log L)^{k^2-1}+O\Big(\frac{X^{1-\frac1k}}{L}(\log L)^{k^2-2}\Big).
CI=\frac{(D)_{1}}{(D_{x})_{1}}+\frac{(D)_{2}}{(D_{x})_{2}}=\frac{(D)_{1}}{(D_{m})_{1}[f_{a}/(1-f_{a})]^{1/m_{1}}}+\frac{(D)_{2}}{(D_{m})_{2}[f_{a}/(1-f_{a})]^{1/m_{2}}}
q^{N_{a}}| m_{1}, \ldots , m_{r}\rangle =q^{m_{a}}| m_{1}, \ldots , m_{r}\rangle, & \beta_{a}^{*}| m_{1}, \ldots , m_{r}\rangle =| m_{1}, \ldots , m_{a}+1 , \ldots , m_{r}\rangle, \beta_{a}| m_{1}, \ldots , m_{r}\rangle =(1-q^{2 m_{a}})| m_{1}, \ldots , m_{a}-1 , \ldots , m_{r}\rangle.
\sum_{a_i,b_i}H^{(l_0)}_{a_0 b_0}H^{(l_1)}_{a_1 b_1}.....H^{(l_n)}_{a_n b_n}\sum_{\alpha_i,\beta_i}(\gamma_{\mu_0})_{\alpha_0 \beta_0}(\gamma_{\mu_1})_{\alpha_1 \beta_1}.....(\gamma_{\mu_n})_{\alpha_n \beta_n}
\begin{array}{l l l}{{\Lambda_{2 4}^{2}}}&{{=}}&{{\{< A , B , 0 > , \; < 0 , A , B > , \; < B , 0 , A > : \; \; A , B \in \frac{1}{\sqrt{2}}{\cal E}_{8}^{o d d},}}\\{{}}&{{}}&{{A B^{\dagger}= \pm \frac{1}{2}e_{a}, \; \; a \in \{0 , . . . , 7 \}\},}}\\ \end{array}
\mathbb{E}(\tau) L^k(f,\mu) \geq \int_{\tilde \Omega^k}d(f(e),f(\gamma_{\tau(\tilde\omega)}^k(\tilde\omega))) d\mathbb{P}(\tilde\omega) = \sum_{i=1}^{\infty}b_i \left( \int_{\Gamma}d(f(e),f(\gamma)) d\mu^{ik}(\gamma)\right)
5995,\, 555,\, 5635,\, -357,\, 1243,\, -477\stackrel{4}{=}-245,\, 1035,\, -605, \,5883,\, 763,\, 5763, 5995.555.5635.(-357).1243.(-477)=(-245).1035.(-605).5883.763.5763 .
W_3' = O\biggl( \max \biggl\{\frac{\log n}{nk^{1/2}}\, , \, \frac{k^{\frac{1}{2}+\frac{2\beta}{d}}}{n^{1+ \frac{2\beta}{d}}}\, , \,\frac{k^{\frac{2\alpha}{\alpha+d}- \epsilon}}{n^{\frac{2\alpha}{\alpha+d}- \epsilon}}\biggr\}\biggr).
E=\dfrac{1}{2M}\int{\left\{\sum_{\sigma=\uparrow,\downarrow}|(D_1^{\sigma}-iD_2^{\sigma})\psi_{\sigma}|^2+eB\left(|\psi_{\uparrow}|^2+|\psi_{\downarrow}|^2\right)-eB\left(|\psi_{\uparrow,0}|^2+|\psi_{\downarrow,0}|^2\right)\right\}}dx,
\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^r q^{r(Lr+L-2i)/2}=\displaystyle\sum_{n_1 \geq \cdots \geq n_{k-1}\geq 0}\frac{q^{n_1^2 +\cdots + n_{k-1}^2 +n_{i}+\cdots +n_{k-1}}}{(q)_{n_1 -n_2}\cdots (q)_{n_{k-2}-n_{k-1}}(q^a ; q^a )_{n_{k-1}}},
S = \int d^{4}x \, \sqrt{- \tilde{g}}\left\{\tilde{R}- 2 \tilde{g}^{a b}\nabla_{a}\Phi \nabla_{b}\Phi -{\frac{1}{2}}e^{- 2 \Phi}\tilde{g}^{a e}\tilde{g}^{b f}F_{a b}F_{e f}\right\}\; .
\sigma_{\mu}^{A \dot{B}}( x^{\mu}+ i p^{\mu}) = 4 i \left\{\left( \begin{array}{c}{\psi_{l}^{A}}\\{0}\\ \end{array}\right) , \left( \begin{array}{c}{0}\\{\psi_{r}^{\dot{B}}}\\ \end{array}\right) \right\}
( F \circ F^{\dag})_{\mu}{}^{\nu}= \frac{1}{2}\ \epsilon \delta_{\mu}{}^{\nu}+ \frac{C ( d - 2 ) + 2}{2 ( d C + 2 )}\g_{\mu 0}\delta_{0}{}^{\nu}= \frac{\epsilon}{2}\left( \begin{matrix}{\frac{2 ( d - 1 ) C + 4}{d C + 2}}&{}\\{}&{1_{d - 1}}\\ \end{matrix}\right) ,
(\Psi_{\gamma+t},e^{i\left\langle \gamma+\delta+t,x\right\rangle})=\frac{1}{d(\gamma,\delta)}\sum_{\gamma_{1}\in\Gamma(k+)}q_{\gamma_{1}}(\Psi_{\gamma+t},e^{i\left\langle \gamma+\delta-\gamma_{1}+t,x\right\rangle})
\hat{X}^n_2(t) = X^n_2(0) + \hat{V}_1^n(t) - \hat{M}_2^{n}(t) - \int_0^t\left(\hat{X}^n_2(s)-\hat{X}_3^n(s)\right)ds - \hat{V}_2^n(t), \hat{X}^n_i(t) = X^n_i(0) - \hat{M}_i^{n}(t) - \int_0^t\left(\hat{X}^n_i(s)-\hat{X}_{i+1}^n(s)\right)ds, 3 \le i \le k-1, \hat{X}^n_k(t) = X^n_k(0) - \hat{M}_k^{n}(t) - \int_0^t\hat{X}^n_k(s)ds.
D_kD_lh^{11}=h^{11}_{mn}D_kD_lh_{mn}+h^{11}_{mn,rs}D_lh_{mn}D_kh_{rs}&=-(h^{11})^2D_kD_lh_{11}+h^{ms}(h^{11})^2D_lh_{1m}D_kh_{1s}+h^{ns}(h^{11})^2D_lh_{1n}D_kh_{1s}&=-(h^{11})^2D_kD_lh_{11}+2(h^{11})^2h^{rs}D_1h_{lr}D_1h_{ks}.
\overset{5}{R}(X,Y,AZ,AW) = \overset{5}{R}(AX,AY,Z,W), \overset{5}{R}(X,AY,AZ,W) = \overset{5}{R}(AX,Y,Z,AW), \overset{5}{R}(AX,AY,AZ,AW) = \overset{5}{R}(X,Y,Z,W).
\mathrm{H}(\mathrm{x}) - \alpha \int_\Omega \nabla\mathbb{G}^{(\mathrm{k})}(\mathrm{x},\mathrm{y})\cdot\nabla \mathrm{H}(\mathrm{y}) d\mathrm{y}= \mathrm{H}^{\textbf{in}}(\mathrm{x}),
\sigma_{(a,i)}\sigma^{-1}_{(0,0)}((b,j))=\sigma_{(a,i)}((\alpha^{h}(b)+g,j+h))=(b+\alpha^{-h}(g)-\alpha^{i-h}(g),j)=(b,j)+(\alpha^{-h}(g)-\alpha^{i-h}(g),0).
\mathcal I_s^q(\mu)=\int_{\underline a \in \Sigma}\left(\sum_{n=0}^{\infty}\int_{\underline b' \in\Sigma:\underline b' \wedge \sigma^n\underline a = \phi}\frac{\mu[a_1\cdots a_n]}{d(\underline a,a_1\cdots a_n\underline b')^s}d\mu(\underline b')\right)^{q-1}d\mu(\underline a)
Dh_m(\rho) =D^2T \left( \operatorname{Id}, (\rho \circ T)(DT)^{\otimes m}\right) + DT \left( D\rho(T) \right)\left(DT, (DT)^{\otimes m}\right) &+ DT (\rho \circ T) \sum_{i=0}^{m-1}\left( (DT)^{\otimes i}, D^2T, (DT)^{\otimes m-1-i}\right).
\int_{\Sigma}\left(|A|^2+|\nabla \log w|^2\right)\phi^2e^{\frac{|x|^2}{4}}d\sigma =\int_{\Sigma}(\dfrac12-\mu_1)\phi^2e^{\frac{|x|^2}{4}}d\sigma-\int_{\Sigma}\phi^2(\mathcal{L}\log w)e^{\frac{|x|^2}{4}}d\sigma &=\int_{\Sigma}(\dfrac12-\mu_1)\phi^2e^{\frac{|x|^2}{4}}d\sigma+\int_{\Sigma}\left<\nabla \phi^2, \nabla \log w\right>e^{\frac{|x|^2}{4}}d\sigma
m_{\mu_\alpha\boxplus\mu_\beta}(z)- m_{\mu_\alpha\boxplus\mu_\beta}(E_-)= m_{\mu_\alpha}'(\omega_\beta(E_-))\sqrt{\frac{-2 (\omega_\beta(E_-))}{\widetilde z''(\omega_\beta(E_-))}}\sqrt{E_--z}+O(|z-E_-|)\,,
\int_{0}^\infty \frac{1}{\rho(a)}L^a_t da = \int_0^t 1_{X_s > X_s'}\frac{1}{\rho(X_s - X_s')}d\langle X - X', X - X'\rangle_s= \int_0^t 1_{X_s > X_s'}\frac{(\sigma(X_s) - \sigma(X_s'))^2}{\rho(X_s - X_s')}ds \leq t.
\int_0^t (t-\tau)^{\frac{m-\frac52}{\alpha}}\tau^{\frac{\frac52-\alpha-m}{\alpha}}\, d\tau=\int_0^1 (1-\tau')^{\frac{m-\frac52}{\alpha}}(\tau')^{\frac{\frac52-\alpha-m}{\alpha}}\, d\tau'\leq C
\displaystyle K^{Q,<\epsilon}(f,a)= \int_{H_L\left[<\epsilon \sigma(a)\right]}\int_{aU_Q\left[<\epsilon \sigma(a) \right]a^{-1}}f\left(a^{-1}h_L u_Q a\right) \xi_L(h_{L}) du_Q dh_{L}
\mathrm{Res}_{D_i\cap D_j}^{(1)}:\mathbf D_i^1=\sum_{j\in I-\{i\}}\Gamma(D_i,\Omega^1_{D_i}(\mathrm{log}D_i\cap D_j))\to \Gamma(D_i\cap D_j,\mathcal O_{D_i\cap D_j})=:\mathbf D_{ij}^0.
\frac{\partial z}{\partial b}(t,x;b,y) =\frac{4(1+t)\big[(y-x)^2-(t-b)(t+b+2)\big]}{\big[(t+b+2)^2-(y-x)^2\big]^2}, \frac{\partial z}{\partial y}(t,x;b,y) =-\frac{8(y-x)(1+t)(1+b)}{\big[(t+b+2)^2-(y-x)^2\big]^2},
\max\{|S|:S\in\bigcap_{i=1}^m\mathcal{F}_i\}=\min\{r_1(X)+r_2(X_1\backslash X)+\dots+r_{m-1}(X_{m-2}\backslash X_{m-3})+r_m(E\backslash X_{m-2}): X\subseteq X_1\subseteq X_2\subseteq \dots\subseteq X_{m-2}\subseteq E\}
\|S^{1/2}U(t)\|+\gamma\int_0^te^{\gamma(t-s)}\|S^{1/2}U(s)\|ds \leq e^{\gamma t}\,\|S^{1/2}U(0)\|+\int_0^te^{\gamma(t-s)}\|S^{1/2}{\tilde L}_aU(s)\|ds.
\alpha_{A}( \varphi ) = - \left[ \ln{\frac{\widehat \Lambda_{s}}{m_{A}}}+{\frac{1}{2}}\right]{\frac{\partial \ln B ( \varphi )}{\partial \varphi}}= + \ln{\frac{\widehat \Lambda_{s}^{\prime}}{m_{A}}}{\frac{\partial \ln B^{- 1}( \varphi )}{\partial \varphi}}\ ,
\xi_n= \varrho_{i(n)}\ast u_0 \hbox{on}\partial Q, \;\;\;\xi_n\to u_0 \hbox{in}L^1(Q;\mathbb R^d), \;\;\;\overline{v}_n\rightharpoonup v \hbox{in}L^p(Q;\mathbb R^m), \int_Q \overline{v}_n dx = b, \;\liminf_{n \to +\infty}\int_Q f ( w_n, v_n,\nabla w_n)dx\geq \limsup_{n\to +\infty}\int_Q f( \xi_n,{\bar v}_n, \nabla \xi_n)dx.
E_2' \ll \widehat{P}^{n-5}\widehat{Y}^{3+-n/2}\widehat{Z}^{3-n/2}\widehat{V}^{n(1-1/n-1))}& = \widehat{P}^{n/3-5}\widehat{Y}^{2+n/3}\widehat{Z}^{3+n/6}& \ll \widehat{P}^{11n/18}\widehat{Y}^{n/6-1}&\ll \widehat{P}^{8n/9 - 5/3}
M_{p l}^{2}={M_{X}^{3}}\left({\frac{k_{0}}{\displaystyle{k_{0}^{2}- ( k_{1}- k_{c})^{2}}}}e^{- 2 \sigma ( y_{0})}+{\frac{k_{1}}{\displaystyle{k_{1}^{2}- ( k_{0}+ k_{c})^{2}}}}e^{- 2 \sigma ( y_{1})}\right) . \,
\begin{aligned}-dy_{(n,t)}=\bigg[G\big(y_{(n,t)},z_{(n,t)},\tilde{z}_{(n,t)},\gamma_{(n,t,e)}\big)+\varphi_n(t)\bigg]dt-z_{(n,t)}dW_t-\tilde{z}_{(n,t)}d\xi_t &\quad-\int_{\mathcal{E}}\gamma_{(n,t,e)}\tilde{N}(de,dt),\t\in[0,\infty).\end{aligned}
\ell_1:=P \left(1-\left(\frac{1}{P_1}+\frac{1}{P_2}+\frac{1}{P_3}\right) \right), \ell_2:=P\left(1-\left(-\frac{1}{P_1}-\frac{1}{P_2}+\frac{1}{P_3}\right)\right), \ell_3:=P \left(1-\left(\frac{1}{P_1}-\frac{1}{P_2}-\frac{1}{P_3}\right) \right) , \ell_4:=P\left(1-\left(-\frac{1}{P_1}+\frac{1}{P_2}-\frac{1}{P_3}\right)\right) .