harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J(j,n,\epsilon)=\bigl\{k\mid 1\le k\le n,\ \min_{m\in\mathbb Z}\{|k\theta-c_j-m|\}<\epsilon\bigr\}\intertext{and}I_\epsilon=\bigl\{x\in[0,1]\mid \min_{m\in\mathbb Z}|x-c_j-m|\ge\epsilon j=1,\ldots,p\bigr\}.
0 =(-d^*f)(ae_xe_y)=f(d(ae_xe_y))=f(d(a)e_xe_y)+f(ad(e_x)e_y)-f(ae_xd(e_y)) &=f(a(1-x)e_y)-f(ae_x(1-y))=f(ae_y)-f(axe_y)-f(ae_x)+f(ae_xy),
{A_{4}^{Y M}= \langle T_{\mu_{1}\nu_{1}}( 1 ) T_{\mu_{2}\nu_{2}}( 2 ) T_{\mu_{3}\nu_{3}}( 3 ) T_{\mu_{4}\nu_{4}}( 4 ) \rangle = N^{2}A_{4}^{( 1 )}+ \tilde{k}N^{1 / 2}f^{( 0 , 0 )}( S , \bar{S}) A_{4}^{( 2 )}+ \dots ,}
0 \leq \int_{\Omega}|\nabla u + F_\epsilon u|^2 dx = \int_{\Omega}|\nabla u|^2 dx + \int_{\Omega}|F_\epsilon|^2 |u|^2 dx + \int_{\Omega}F_\epsilon \cdot \nabla |u|^2 dx = \int_{\Omega}|\nabla u|^2 dx- \int_{\Omega}\Big(F_\epsilon - |F_\epsilon|^2\Big)|u|^2 dx.
\frac{1}{2}(c(s)+\overline{c(s)})= \int_{0}^{\infty}(x(x+1))^{-h(s)}\cos\Big( \log\Big( \frac{x}{x+1}\Big)^{-\tilde{h}(s)}\Big) dx \leq \int_{0}^{\infty}(x(x+1))^{-h(s)}dx=:c_{h}(s).
B^C = \{(P_f)_{f\in G}\in\wp(G)^G\; |\;{\rm there\; exists}\; g\in P_f \;{\rm and}\; h\in P_{gf}\;{\rm such \; that}\; hg\notin P_{f}\;{\rm for \;some\;}f \in G\}&= \bigcup_{f,g, h\in G}(W_{(f, g)}\cap W_{(gf,h)}\cap W_{(f, hg)}^C).
X \Box ( Y \wedge Z) = (X \Box Y) \wedge Z + (-1)^{\bar{p}(X)p(Y)}Y \wedge (X \Box Z) - (X \Box 1)\wedge Y \wedge Z, (X \wedge Y) \Box Z = X \wedge (Y \Box Z) + (-1)^{p(Y)\bar{p}(Z)}(X \Box Z) \wedge Y.
\mu_k = \left( \frac{\pi}{\Omega_\beta}k \right)^\gamma(1+o(1)), k \to \infty, \mu_{k+1}- \mu_k = \frac{2\pi}{\Omega_\beta'}\left(\frac{\pi}{\Omega_\beta}k\right)^{\gamma-1}(1+o(1)), k \to \infty,
F_{3}(a_{1},a_{2},b_{1},b_{2};c;x,y)=\sum_{m,n=0}^{\infty}\frac{(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}m!n!}x^{m}y^{n},
x^{(N)}= \frac{1}{N}\sum_{i=1}^N x_i,y^{(N)}= \frac{1}{N}\sum_{i=1}^N y_i, m^{(N)}=\frac{1}{N}\sum_{i=1}^N m_i ,p^{(N)}=\frac{1}{N}\sum_{i=1}^N p_i.
\langle \Delta_r u,\varphi_k \rangle =\Delta_r \langle u, \varphi_k \rangle, \langle \Delta_\theta u,\varphi_k \rangle = \langle u, \Delta_\theta \varphi_k \rangle =-\lambda_k\langle u,\varphi_k\rangle, \langle \frac{\partial u}{\partial r},\varphi_k\rangle =\frac{\partial \langle u,\varphi_k\rangle}{\partial r}.
\frac{s^{(2\ell)}_n}{s^{(2m)}_{n+1}}= \dfrac{(2\ell-1)!!}{(2m-1)!!}\cdot (1-2\alpha)^{m-\ell}\cdot \dfrac{n^{\ell}}{(n+1)^m}&\sim \dfrac{(2\ell-1)!!}{(2m-1)!!}\cdot (1-2\alpha)^{m-\ell}\cdot n^{\ell-m}(n \to \infty),
\lVert f\rVert_{L^q_H L^p_z(U)}:=\begin{cases}\left(\int_{U'}\lVert f(x',\cdot)\rVert^q_{L^p(U_3)}\,dx'\right)^{1/q},q\in [1,\infty), &_{x'\in U'}\lVert f(x',\cdot)\rVert_{L^p(U_3)}, q=\infty.\end{cases}
I = \lim_{k\to\infty}\int_\Omega\Big\{|\phi_k(x)\nabla u_k(2\pi(x) - x)|^2- |\phi_k(x)\nabla u_k(x)|^2\Big\}~\mbox{d}x & = \lim_{k\to\infty}\int_{\Omega\cap\{\mathrm{dist}(x,\partial\Omega)<1/k\}}\Big\{|\nabla u_k(2\pi(x) - x)|^2 - |\nabla u_k(x)|^2\Big\}~\mbox{d}x.
{\frac{d n_{s}( \omega | \epsilon )}{d \omega}}={\frac{\cal A}{4 \pi^{2}\kappa^{3}}}\left[{\frac{\omega^{2}}{\epsilon^{2}}}+{\frac{m^{2}\kappa^{2}}{2}}\ln{\frac{\epsilon^{2}m^{2}}{4}}-{\frac{m^{2}\kappa^{2}}{2}}\left( 1 + 2 \mathrm{R e}~ \psi ( i \omega / \kappa ) \right) \right] .
\begin{aligned}\langle dN_{\phi}(e_{\phi}^{i}),d\big(\mathrm{li}_{\phi}(f_{\phi}^{j})\big)\rangle_{L^{2}\Lambda^{1}(\Omega)}=\int_{\partial\Omega}e_{\phi}^{i}\wedge f_{\phi}^{j}. \end{aligned}
y\sqrt{y}\sqrt{a}-x^2 ~=~ x \sqrt{y}\sqrt{a}- x^2 - x \sqrt{y}\sqrt{a}+ y \sqrt{y}\sqrt{a}begin{align*}.5em] = ~ x(\sqrt{y}\sqrt{a}- x) + \sqrt{y}\sqrt{a}(y-x) begin{align*}.5em] \leq ~ x(y- x) + \sqrt{y}\sqrt{a}(y-x) begin{align*}.5em] =~ (x + \sqrt{y}\sqrt{a})(y-x),
\mu_j^{-\frac{1}{2}}\,U_j(\zeta_i+\mu_i z)=4\pi\alpha_3\,G_{\lambda}(\zeta_i+\mu_i z,\zeta_j)+\mu_j^{2-\sigma}\hat{\theta}(\mu_j,\zeta_i+\mu_i z,\zeta_j).
\forall v, \ \tilde{\lambda}_v = \frac{\rho_v \tilde{p}_v \chi\left( \gamma \left( \pmb{\rho}\right) \right)e^{- \gamma \left( \pmb{\rho}\right)}}{e^{- \gamma\left( \pmb{\rho}\right)}+ \tau\left( 1 - e^{- \gamma\left( \pmb{\rho}\right)}\right)}.
\check{\Gamma}_{\alpha \beta}^{\mu \nu}\epsilon_{\mu \nu}^{3 , 4}= \left( k^{2}+ \Pi_{B}\right) \epsilon_{\mu \nu}^{3 , 4}+ 2 \left( 1 - \frac{k^{2}}{( k \cdot u )^{2}}\right) \Pi_{6}\epsilon_{\mu \nu}^{5 , 6}
{\mathcal{L}_{S D}}= \beta^{2}g \left( m^{2}A^{2}/ 2 \beta^{2}\right) - \frac{\beta^{2}}2 h \left( m^{2}B^{2}/ 2 \beta^{3}\right) - \frac{\chi \theta}{2}\varepsilon^{\mu \nu \rho \lambda}\partial_{\mu}B_{\nu \rho}A_{\lambda}\; .
\int_{0}^{\infty}\int_{I}|F_k(z)|^2 h(y) \frac{dx dy}{y^2}= \left(|I|+O\left( \frac{1}{(\log k)^3}\right) \right) \langle E(z|h) F_k, F_k \rangle +O\left( (\log k)^{\varepsilon}\prod_{p \le k}\left(1-\frac{(|\lambda_f(p)|-1)^2}{p}\right) \right),
<\psi^{l}(x,y)H(x_{1},y_{1})\psi^{l}(x,y)>={\cal A}(l)\sum_{n_{1}=0}^{r}\sum_{n_{2}=n_{1}}^{r}\sum_{m_{1}=0}^{s}\sum_{m_{2}=0}^{s}\sum_{m_{3}=0}^{s-m_{1}}c^{n_{1}n_{2}}_{m_{1},m_{2},m_{3}}a^{n_{1}n_{2}}_{m_{1},m_{2},m_{3}}(l).
\psi(1) = 1-\bigl(1+\psi(0)-\mathbb{E}S\bigr)/y_0, \psi(u) = 1+\frac{1}{s_0}\Biggl(\psi(u-2)-1+\sum\limits_{k=1}^{u-1}s_k\bigl(1-\psi(u-k)\bigr) \Biggr) &-\frac{x_{u-1}(1-\psi(1))}{x_0},u\in\{2,3,\ldots\}.
X_n(t)=\int_{\{\tau_{{E'}}(\xi)\in(d_{n-1},d_n]\}}\bigl(\mathrm{e}^{\mathrm{i}\langle t,\xi\rangle}-1 \bigr) \frac{\widetilde{\mathcal{M}}(\mathrm{d}\xi)}{\psi(\xi)^{1 + Q/2}}.
\sum_{n = 0}^{\infty}x^{2 n}\, \frac{e^{- | z |^{2}}}{( n ! )^{2}}\, \left\{\sum_{\mu , \nu = 1}^{M}\left( e^{i \omega_{\mu}}| z_{\mu +}^{+}|^{2}+ e^{- i \omega_{\mu}}| z_{\mu -}^{+}|^{2}\right) \left( e^{i \omega_{\nu}}| z_{\nu +}^{-}|^{2}+ e^{- i \omega_{\nu}}| z_{\nu -}^{-}|^{2}\right) \right\}^{n}\ \ \ .
D_{i j}( t - t^{\prime}) = T_{i j}f_{i}( t ) f_{j}( t^{\prime}) = \hbar \left[ A_{i}B_{j}\theta ( t - t^{\prime}) e^{- i \omega ( t - t^{\prime})}+ A_{j}B_{i}\theta ( t^{\prime}- t ) e^{i \omega ( t - t^{\prime})}\right] ,
{\cal W}_4={1\over 8N}\oint\oint\oint\oint{\rm Tr}[{\cal P}(T^a_xT^a_yT^b_z T^b_w)] D_{\mu\nu}(x-y)D_{\rho\sigma}(z-w)\; dx^{\mu}dy^{\nu}dz^{\rho}dw^{\sigma}\;\; ,
\begin{cases}\bar p_\tau = \bar p_{xx}, (x,\tau)\in (0,1)\times (0,\infty), \bar p_x(0,\tau) = c \bar p(0,\tau), \tau\in(0,\infty), \bar p_x(1,\tau) = 0, \tau\in(0,\infty), \bar p(x,0) = \bar p_0(x), x\in(0,1),\end{cases}
S^{p}= \int \sqrt{g}\delta_{[ \alpha_{1}\cdots \alpha_{2 p}]}^{[ \beta_{1}\cdots \beta_{2 p}]}R_{\beta_{1}\beta_{2}}^{\alpha_{1}\alpha_{2}}\cdots R_{\beta_{2 p - 1}\beta_{2 p}}^{\alpha_{2 p - 1}\alpha_{2 p}}d^{d - 2}x \ ,
\delta^N(t)\leq CN\left[e^{-\frac{\lambda}{4}N^2}\sum_{i=1}^{k}\frac{C^iN^{(3\alpha+2)i}T^{2i}}{(2i)!}+\frac{C^kN^{(3\alpha+2)k}T^{2k}}{(2k)!}\right].
\mathcal{N}(\omega) := \inf_{U \in H^1_{\rm C}(\mathcal{T}) \backslash \{0\}}\left\{\frac{3}{2}S_{\omega}(U) : \;\; \| U \|^6_{L^6(\mathcal{T})}=\frac{1}{3}B_{\omega}(U) \right\}
|\Phi_{v}^{\nu =1/k}\rangle ={\cal N}\varepsilon^{\alpha_{1}\dots\alpha_{N}}({\bf A}{_{\alpha_{1}}^{+}})\Big({\bf A}_{\alpha_{2}}^{+}\Big)^{2}\dots\Big({\bf A}_{\alpha_{N}}^{+}\Big)^{N}\{\bf I}\;|v\rangle
\limsup_{n \to \infty}\frac{1}{n}\sum_{k = 0}^n V_k\leq \limsup_{n \to \infty}\frac{i_n}{\sigma_{i_n}}\cdot \left[ \frac{1}{i_n}\sum_{j = 0}^{i_n + 1}Q_j \right]= \limsup_{i \to \infty}\frac{i}{\sigma_i}\cdot \left[ \frac{1}{i}\sum_{j = 0}^{i + 1}Q_j \right]= \frac{\mu_{Q}}{\mu_{\sigma}}~,~ P_{0,\pi}^V ~ \mbox{a.s.}
2\ell\left( s,\chi\right) +2 =-\chi\left( -1\right) \sum_{j=1}^{l}\frac{s\left( s+1\right) ...\left( s+j-1\right)}{j!}\overline{E}_{j,\overline{\chi}}\left( 1\right) & +\chi\left( -1\right) \frac{s\left( s+1\right) ...\left( s+l\right)}{l!}\int\limits_{1}^{\infty}\frac{\overline{E}_{l,\overline{\chi}}\left(x\right)}{x^{s+l+1}}dx,
\langle \hat{u}(\lambda), Z_{s}(\epsilon_{s}) \rangle = \langle \hat{u}(\lambda), Z_{s}+ \epsilon_{s}V_M \rangle &= \langle u , Z_{s}\rangle + \lambda \langle u' , Z_{s}\rangle + \epsilon_{s}(\langle u, V_M \rangle + \lambda \langle u', V_M \rangle).
M(t,x) := \frac{\langle w \rangle_{f_t,\varphi}}{\langle 1 \rangle_{f_t,\varphi}}= \frac{\int_{\mathbb{T}}\!\mathrm{d}y \int_{\mathbb{R}}\!\mathrm{d}w\, f_t(y,w)\,\varphi(x-y)\, w}{\int_{\mathbb{T}}\!\mathrm{d}y \int_{\mathbb{R}}\!\mathrm{d}w\, f_t(y,w)\,\varphi(x-y)}\,.
\mathfrak{B}_d((\partial_k + \eth_k^*)(a + a^*), b + b^*) = \mathfrak{B}_d(\partial_k(a) + \eth_k^*(a^*), b + b^*) = \langle \partial_k(a), b^* \rangle + \langle \eth_k^*(a^*), b \rangle = \langle a, \partial_k^*(b^*) \rangle + \langle a^*, \eth_k(b) \rangle = \mathfrak{B}_d(a + a^*, (\eth_k + \partial_k^*)(b + b^*)).
{\cal A}\propto \int d^{J - 1}\theta \int_{[ p ]}d z \oint d w \frac{1}{w^{n}}\eta_{\mu \nu}\langle \V_{- 1}^{\mu}( 0 ) \V_{- 1}^{\nu}( \infty ) \V_{+ 1 / 2}[ u_{a}] ( w ) \V_{- 1 / 2}[ \lambda_{a}] ( z ) \ \rangle
\begin{aligned}|N_1(\mu_1,\vartheta_1)| \leq c_1|\mu_1|^2+c_2|\vartheta_1^\prime|^2\,, |N_2(\mu_1,\vartheta_1)| \leq c_3|\mu_1|^2+c_4|\vartheta_1|^2\,, \end{aligned}
K_{\Omega}(z,z^{\prime};\bar\zeta,\bar{\zeta^{\prime}})=\sum_{\mathbf a,\mathbf b}\frac{(z\bar \zeta)^{\mathbf a}\phi_{\mathbf a,\mathbf b}(z^{\prime})\overline{\phi_{\mathbf a,\mathbf b}(\zeta^{\prime})}}{\|z^{\mathbf a}\phi_{\mathbf a,\mathbf b}(z^{\prime})\|^2_{L^2(\Omega)}}.
{\lefteqn{\sum_{j=1}^{n+m}B_{i_1}\cdots B_{i_{j-1}}D_{i_j}C_{i_{j+1}}\cdots C_{i_{n+m}}}}&=\begin{aligned}[t] \sum_{j=1}^{m}B_{i_{m+1}}\cdots B_{i_{j-1}}D_{i_j}C_{i_{j+1}}\cdots C_{i_{n+m}}&+ B_{i_1}\cdots B_{i_m}\sum_{j=m+1}^{n+m}B_{i_{m+1}}\cdots B_{i_{j-1}}D_{i_j}C_{i_{j+1}}\cdots C_{i_{n+m}}\end{aligned}
M =\left(\begin{array}{cc|c|cc}m_{1,1}m_{1,2}m_{1,3}m_{1,4}m_{1,5}0 m_{2,2}m_{2,3}m_{2,4}m_{2,5}\hline 0 0 m_{3,3}m_{3,4}m_{3,5}\hline 0 0 0 m_{4,4}m_{4,5}0 0 0 0 m_{5,5}\end{array}\right).
\partial_{x}f^{p}\left( x_{0}\right) ={\displaystyle\prod\limits_{j=0}^{p-1}}f_{x}\left( x_{j}\right) =\pm1 & =\left( -1\right)^{p}{\displaystyle\prod\limits_{j=0}^{p-1}}\frac{F_{x}\left( x_{j},x_{j+1\left( \operatorname{mod}p\right)}\right)}{F_{y}\left( x_{j},x_{j+1\left( \operatorname{mod}p\right)}\right)}=\pm1.
E[X^{m}]=\frac{(2\kappa\beta)^{-m/\alpha}}{1+\kappa\frac{m}{2\alpha}}\frac{\Gamma(\frac{1}{\kappa}+\frac{m}{\alpha})\Gamma(1-\frac{m}{2\alpha})}{\Gamma(\frac{1}{\kappa}+\frac{m}{2\alpha})}
c_p(n) =\frac{1}{p2^p}\Bigl[ \bigl(\zeta(p,1+\lambda_1)+\zeta(p,1+\lambda_2)\bigr)\Bigl(\frac{B_{p+1}(q)-B_{p+1}}{p+1}\Bigr)-\zeta(p)n + \zeta(p)\times \times\Bigl(\frac{B_{p+1}(n+1)-B_{p+1}}{p+1}\Bigr) - \zeta(p,2+\lambda_1+\lambda_2)\Bigl(\frac{B_{p+1}(2q-1)-B_{p+1}(q-1)}{p+1}\Bigr)\Bigr],
D_{\downarrow}V(t,\overline{x}(t))(\alpha,\theta) =\limsup_{\tau \rightarrow 0^+}\frac{V(t+\alpha \tau,x_{\tau}(t+\alpha \tau))-V(t,\overline{x}(t))}{\tau}&\leq \limsup_{\tau\rightarrow 0^+}\frac{\phi(x_{\tau}(T)) - \phi(\overline{x}(T))}{\tau}\leq 0.
\kappa_W (A,r,s):= \frac{\max\limits_{1\leq i \leq r}\sigma_i(X^T\!AX)/\sigma_i(A)}{\min\limits_{1\leq j \leq r}\sigma_j(X^T\!AX)/\sigma_j(A)}= \max\limits_{1\leq i \leq r}\max\limits_{1\leq j \leq r}\frac{\sigma_i(X^T\!AX)}{\sigma_j(X^T\!AX)}\frac{\sigma_j(A)}{\sigma_i(A)}
X_1X_\delta= X(1,0)(X(-1,-1)+hX(-1,0)+hX(0,-1)+X(-1,1)+X(1,-1)) = q^{-\frac{1}{2}}X(0,-1)+h+q^{-\frac{1}{2}}hX(1,-1)+q^{\frac{1}{2}}X(0,1)+q^{-\frac{1}{2}}X(2,-1).
\frac{z_{t}}{z_{k}}\le\frac{2C_{1}\left(\sqrt{\Delta_{1}}+2\sqrt{A}C_{1}\right)+4C_{1}^{2}\sqrt{A}}{2C_{1}\sqrt{\Delta_{1}}+4C_{1}^{2}\sqrt{A}}=\frac{\sqrt{\Delta_{1}}+4\sqrt{A}C_{1}}{\sqrt{\Delta_{1}}+2\sqrt{A}C_{1}}\le2
\|\delta^{k+1}(\tau_V^t(V))\| \leq \|\delta_V^{k+1}(V)\| + \sum_{l=0}^k \sum_{j=0}^l{l \choose j}2\|\delta_V^j(V)\| \| \delta_V^{l-j}\delta^{k-l}(\tau_V^t(V)) \|.
\rho_k = \frac{\phi(x_k) - \phi(x_k+s_k) + 2\epsilon_f+r}{m(x_k) - m(x_k+s_k)}= \frac{L_1\|x_k\|^2/2 - L_1\|x_k-\delta_k g_k / \|g_k\|\|^2/2 + 2\epsilon_f+r}{\|g_k\| \delta_k}.
\begin{cases}D_{r_k}h_3(z_*)=\cos \frac{2\pi(k-1)}{n}, D_{x_k}h_3(z_*)=\frac{\cos \frac{\pi}{n}-\cos \frac{\pi(2k-1)}{n}}{2 \sin \frac{\pi}{n}}, \k=1,2,\ldots,n, \end{cases}
\frac{d^{2}\overline{{{\phi}}}( x )}{d x^{2}}= V^{\prime}( \overline{{{\phi}}}) \quad \longrightarrow \quad \frac{d}{d x}\left[ \frac{1}{2}\left( \frac{d \overline{{{\phi}}}}{d x}\right)^{2}\right] = \frac{d}{d x}V ( \overline{{{\phi}}}) .
(\begin{matrix}w&x&y&z\\x&\frac{x^{2}}{w+1}+1&\frac{yx}{w+1}&\frac{zx}{w+1}\\y&\frac{xy}{w+1}&\frac{y^{2}}{w+1}+1&\frac{zy}{w+1}\\z&\frac{xz}{w+1}&\frac{yz}{w+1}&\frac{z^{2}}{w+1}+1\\ \end{matrix})
(0,\, b) S^{-1}A(\varepsilon_0) S\left(\begin{matrix}\mathfrak{a}\mathfrak{b}\end{matrix}\right)= (\mathfrak a'\,,\mathfrak b') S^{-1}A(\varepsilon_0) S\left(\begin{matrix}0 -c\alpha+a\beta\end{matrix}\right).
\P_x(\tau_\vartheta > n) = \P_0(\tau_\vartheta > n) + \sum_{y\in E\setminus(E\star x)}\sum_{k=1}^n \P_x(X(t_1) = y, \; t_1 = k) \P_y(\tau_\vartheta > n-k) \geq \P_0(\tau_\vartheta > n) + \sum_{y\in E\setminus(E\star x)}\sum_{k=1}^n \P_x(X(t_1) = y, \; t_1 = k) \P_y(\tau_\vartheta > n)
\psi_{\frac{1}{2}\frac{1}{2}}= \psi_{- \frac{1}{2}- \frac{1}{2}}^{*}= \frac{i}{\sqrt{2}}f_{2}e^{- i \phi}, \quad \psi_{\frac{1}{2}- \frac{1}{2}}= - \psi_{\frac{1}{2}- \frac{1}{2}}= - \frac{i}{\sqrt{2}}f_{1}.
\frac{1}{F_{ax}}=1.0+(\frac{4}{5})(\frac{\xi^{2}}{1+\xi^{2}})+(\frac{4\cdot6}{5\cdot7})(\frac{\xi^{2}}{1+\xi^{2}})^{2}+(\frac{4\cdot6\cdot8}{5\cdot7\cdot9})(\frac{\xi^{2}}{1+\xi^{2}})^{3}+...
S_{b o s}\, = \, \int d^{3}x \, \left( \pm \frac{i}{2}A_{\mu}\epsilon_{\mu \nu \lambda}\partial_{\nu}A_{\lambda}\, - \, \frac{i}{\sqrt{4 \pi}}s_{\mu}\epsilon_{\mu \nu \lambda}\partial_{\nu}A_{\lambda}\right) ,
\left[\begin{matrix}|A|A^* I_k \end{matrix}\right]\left[\begin{matrix}A C B \end{matrix}\right]\left[\begin{matrix}I_m -f(|A|)A^*C I_l \end{matrix}\right]=\left[\begin{matrix}|A|^3 (|A|-|A|^3f(|A|))A^*C B \end{matrix}\right].
r(q_1)=\left( \begin{array}{cccc}1 0 0 0 0 0 \frac{1+\beta t_{j}}{1+\beta t_{j^{\prime}}}0 0 1 q_1^{-1}\frac{t_{j}-t_{j^{\prime}}}{1+\beta t_{j^{\prime}}}0 0 0 0 1\end{array}\right)\in\operatorname{End}[\mathbb{Z}[\![q_1,q_{1}^{-1}]\!]\otimes V(t_j)\otimes V(t_{j\rq{}})] \;
(A_L u \mid v) = (u \mid A_0^*v) - \bigl\langle \bigl(u(0-),u'(0-),u''(0-)\bigr) \mid \bigl(v(0+),v'(0+),v''(0+)\bigr)\bigr\rangle_- + \bigl\langle L\bigl(u(0-),u'(0-),u''(0-)\bigr),\bigl(v(0+),v'(0+),v''(0+)\bigr)\bigr\rangle_+.
Q=\frac{2}{3}[(n_{u}-n_{\overline{u}})+(n_{c}-n_{\overline{c}})+(n_{t}-n_{\overline{t}})]-\frac{1}{3}[(n_{d}-n_{\overline{d}})+(n_{s}-n_{\overline{s}})+(n_{b}-n_{\overline{b}})].
S_{j,k}^{(1,n)}:=i({\bf C}^{(1)}_j(n); Y^n|{\bf C}^{(2)}_k (n)), S_{j,k}^{(2,n)}:=i({\bf C}^{(2)}_k(n); Y^n|{\bf C}^{(1)}_j (n)), S_{j,k}^{(3,n)}:=i({\bf C}^{(1)}_j(n),{\bf C}^{(2)}_k(n);Y^n),
\Gamma_{k+1}^{\ell,p+1}= \Gamma_{k+1}^{\ell,p}- \big(\nabla^2 \tilde D_{k+1}^{\ell}(\Gamma_{k+1}^{\ell,p}) \big)^{-1}\nabla \tilde D_{k+1}^{\ell}( \Gamma_{k+1}^{\ell,p}), p\ge 1,
S_{eff}=NS_{0}(\sigma^{0})+S_{1}(\sigma^{0})+{i\over 2}B_{2}+{1\over 2}A_{2}+\sum_{n=1}^{\infty}{1\over\sqrt{N^{n}}}\big[-i{(-1)^{n+1}\over (n+2)}B_{n+2}+iL^{d}\delta^{d}(0){(-1)^{n+1}\over n}B_{n}+{1\over (n+2)!}A_{n+2}\big]
\mathcal A f(x,s) = \int_0^\infty \int_{S^{d-1}}(f(x+y,s)-f(x,s))\frac{d|y|}{|y|^{1+\alpha}}S(x,s,\bar y)\, d_S \bar y &+ \frac{w(x,s)}{\Gamma(-\beta)}\int_0^\infty (f(x,s+v)-f(x,s))\frac{1}{v^{1+\beta}}\,dv &=: \mathcal L f(x,s)+ w(x,s) \partial_{-s}^\beta f(x,s)
\|I^{+}f\|_{\mathcal{B}_{\alpha}^{2}}\leq \left(\sup_{k\in\mathbb{Z}^{n}\setminus\{0\}}\frac{1}{1-\alpha/|k|^{3}}\right)\|f\|_{\mathcal{B}_{\alpha}^{0}}= \frac{1}{1-\alpha}\|f\|_{\mathcal{B}_{\alpha}^{0}}.
\mathbb N_{\varepsilon_0,\delta_0}=\{n\in\mathbb N \ :\ \|X\varphi_n(T)x\|\geq\delta_0\|\varphi_n(T)x\| \ \ 1-\varepsilon_0\leq\|\varphi_n(T)x\|\leq 1+\varepsilon_0\}
\left[ - \nabla_{\widehat{\mathrm{\boldmath ~ \scriptstyle{\x i}~}}, D}^{2}- \widehat{{\mathcal W}}^{({D})}( \widehat{\mathrm{\boldmath ~ \x i ~}}) \right] \, \widehat{\Phi}( \widehat{\mathrm{\boldmath ~ \x i ~}}) = \widehat{\eta}\, \widehat{\Phi}( \widehat{\mathrm{\boldmath ~ \x i ~}}) \; .
\phi_{i}(x_1,x_2,\ldots,x_{i+1}) = +\{(-1)^{(|x_1|+ \ldots+ |x_{i-2}|)|x_{i}|}-(-1)^{|x_{i-1}||x_{i+1}|}\}(-1)^{(|x_1|+ \ldots+ |x_{i-2}|)|x_{i+1}|}\overline{[[x_1,\ldots}\overline{\ldots, x_{i-1}]_l,[x_{i},x_{i+1}]]}& \in \ker (\lambda_{i}).
G_{2}(\theta,0)=-\frac{1}{d^{1/2}}\left\{\frac{\gamma^{3/2}\sinh \theta + \cosh \theta}{\sinh \theta \cosh \theta}-\frac{\alpha \gamma (\sinh \theta -\gamma^{1/2}\cosh \theta)}{(\cosh \theta-\gamma^{1/2}\sinh \theta)^{2}}\right\}<0,
\frac{d}{d\beta}\Big(\frac{\psi(\beta p)}{\psi(\beta q)}\Big) = \sqrt{\frac{q}{p}}\frac{1}{2}\sqrt{\frac{1-\beta q}{1-\beta p}}\Big(\frac{-p(1-\beta q)-(-q)(1-\beta p)}{(1-\beta q)^{2}}\Big) &=\frac{1}{2}\cdot \sqrt{\frac{q}{p}}\sqrt{\frac{1-\beta q}{1-\beta p}}\Big(\frac{q-p}{(1-\beta q)^{2}}\Big)<0\; (p>q).
\ell =\frac{1}{k_1}N_1(\delta)+\frac{1}{k_2}N_2(\delta), &=\frac{1}{k_1}N_1(\delta)+\frac{1}{k_2}N_1(-\delta). \intertext{Replacing $\delta$ by $-\delta$ in the above argument gives}\ell =\frac{1}{k_1}N_1(-\delta)+\frac{1}{k_2}N_1(\delta), \intertext{so}\left(\frac{1}{k_1}-\frac{1}{k_2}\right)N_1(\delta) =\left(\frac{1}{k_1}-\frac{1}{k_2}\right)N_1(-\delta)
m_{D,S}(P') =\frac{1}{d}\sum_{j=1}^rn_jm_{Q_j,S}(\phi(P'))+O(1)\leq \frac{n_1+n_2+\epsilon}{d}h(\phi(P'))+O(1) &\leq(n_1+n_2+\epsilon)h(P')+O(1).
|\rho_h(x,u)| = \left|\int_0^1 (1-t)u^\top\nabla^2 g(x+t\cdot hu)u\cdot dt\right| &\leq h^2|u|^2 \cdot \int_0^1 (1-t)\|\nabla^2 g(x+t\cdot hu)\|_{}dt
J_{\Gamma}( x ) = \bar{\psi}( x ) \Gamma \psi ( x ) \equiv \frac{1}{8 \pi}\int d \hat{\epsilon}e^{i \chi_{\Gamma}( \hat{\epsilon})}\left\{\left[ \psi^{\dagger}( x + \epsilon ) , \gamma^{0}\Gamma \psi ( x - \epsilon ) \right] , e^{i e \int_{x - \epsilon}^{x + \epsilon}d x_{i}A_{i}^{T}}\right\}_{| \epsilon | , | \eta | \propto 1 / \Lambda}
d_m(\pi_m(x),\pi_m(x')) = d(x,x')-d(\rho_m(x),\rho_m(x')) = \begin{cases}d(x,\rho_m(x)) + d(x',\rho_m(x')) \rho_m(x)\ne \rho_m(x'), d(x,x') \rho_m(x)= \rho_m(x'), \end{cases}
\sum_{\substack{k_1\ge0,\,k_2\ge0,\,\ldots,\,k_m\ge0, k_1+k_2+\cdots+k_m=k}}\Biggl(\prod_{i=1}^r b(t,j+1,k_i)\Biggr)\Biggl(\prod_{i=r+1}^m b(t,j,k_i)\Biggr)
\! \! \! \! - \frac{1}{2}\sum_{n_{1}, n_{2}}\, \log \left( \frac{\mid \tau \mid^{2}n_{1}^{2}- 2 \tau_{1}n_{1}n_{2}+ n_{2}^{2}+ \lambda^{2}}{\mid \tau \mid^{2}n_{1}^{2}- 2 \tau_{1}n_{1}n_{2}+ n_{2}^{2}}\right) \; .
\mu_T^{\varepsilon_i,\delta_j}(\mathbb{R}^n)+ \int_0^T \int_{\mathbb{R}^n}\varepsilon_i \Big( -\Delta \varphi^{\varepsilon_i ,\delta_j}+\frac{F'_{\delta_j}(\varphi^{\varepsilon_i ,\delta_j})}{\varepsilon_i^2}\Big)^2 \, dxdt = \mu_0^{\varepsilon_i,\delta_j}(\mathbb{R}^n) \leq D_1.
|M^N_t(x) - M_s^N(x)| \leq \left| \frac{1}{N}\sum_{i=1}^N \int_s^t \nabla e^{(t-u)\Delta}V^N(X_u^{i,N}-x) \cdot dW^i_u \right| &+ \left| \frac{1}{N}\sum_{i=1}^N \int_0^s \nabla e^{(s-u)\Delta}\left[e^{(t-s)\Delta}V^N(X_u^{i,N}-x)- V^N(X_u^{i,N}-x)\right] \cdot dW^i_u \right| &=: |I^N_{s,t}(x)| + |II^N_{s,t}(x)| .
\mathbf{H}(t,f) =\sum_{p=1}^{P}\mathbf{H}_p(t)\exp(-j2\pi f \tau_p) =\sum_{p=1}^{P}\sum_{r=1}^{R_p}\beta_{r,p}\mathbf{a}_{\mathrm{r}}(\theta_{r,p})\mathbf{a}^T_{\mathrm{t}}(\phi_{r,p})\exp(j\nu_{r,p}t-j2\pi f \tau_p)
\Theta(z) =I-(1-z\overline{\mu})C(I-zT)^{-1}P^{-1}(I-\overline{\mu}T^*)^{-1}C^*J &=I+(z-\mu)C(I-zT)^{-1}P^{-1}(\mu I-T^*)^{-1}C^*J,
h_m(z) = \lambda_m^{-1}\left( f_m\circ\eta_{v_m}(z) - f_m\circ\eta_{v_m}(0) \right) k_m(z) = \lambda_m^{-1}\left( f_m\circ\eta_{\frac{b_m}{2}+ v_m}(z) - f_m\circ\eta_{\frac{b_m}{2}+ v_m}(0) \right).
<BR>B=3D-{{\sqrt{-2\nabla^2}}\over=202}(Q_1+Q_2);~~<BR>E_i=3D-{1\over{\sqrt{-2\nabla^2}}}[\epsilon =_{ij}\partial=20_j(P_1+P_2)+(\mu + m)\partial_iQ_1+(\mu -m)\partial=20_iQ_2],<BR><BR>
\hat F_{[A] [B]}^{[\dot A][\dot B]}= \sum_{s=0}^{\min (2a,2b)}\sum_{\dot t=0}^{\min (2\dot a,2\dot b)}S[A]S[B]S[\dot A]S[\dot B] \epsilon_{[s]}\epsilon^{[\dot t]}F_{[C(s)]}^{[\dot C(\dot t)]}\ \ \ .
Lx = A(t) R(\lambda, A(t)) (A(t)^{-1}- A(s)^{-1})x & = R(\lambda, A_{-1}(t)) A_{-1}(t) (A(t)^{-1}- A(s)^{-1})x & = R(\lambda, A_{-1}(t)) (A(s) - A_{-1}(t)) A(s)^{-1}x
\|\bar{Z}\bar{Z}^\top\| \leq \sum_{i \in \widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}}\sum_{t=0}^{T-1}\|\Sigma^{(i)}_t\| \left\| \sum_{l=1}^{N_i}\chi^{(i)}_{l,t}\chi^{(i),\top}_{l,t}\right\|,
\tilde x^{t}= \alpha_t (x^{t-1}- x^{t-2}) + x^{t-1}. \underline x^{t}= \left( \tilde x^t + \tau_t \underline x^{t-1}\right) / (1+\tau_t). g^{t}= \nabla f(\underline x^{t}). x^{t}= \Pr_{X}(g^t, x^{t-1}, \eta_t).
\frac1N\sum_{i=1}^N |X^{i,N}_t|^{2(1+\rho)}\leq \frac{c_\rho^{1-\rho}}{N}\sum_{i=1}^Na(X^{i,N}_t,M^N_t)^{2(1-\rho)}\leq \frac{c_\rho^{1-\rho}}{N^\rho}\,\big[\langle\mu_N(t),a^2\rangle\big]^{1-\rho},
[|D_0|,|D_0|^{\frac{1-q}3}a^4|D_0|^{-\frac13}] = |D_0|^{\frac{1-q}3}\delta_0(a^4)|D_0|^{-\frac13}= |D_0|^{\frac{1-q}3}\delta_0(a^2)a^2|D_0|^{-\frac13}+|D_0|^{\frac{1-q}3}a^2\delta_0(a^2)|D_0|^{-\frac13}.
\eta_{p}(2s)=\frac{1}{1+\frac{1}{\pi p}}\cdot\frac{1}{2s-1}+C_{p}^{(2)}-\frac{2e^{2\pi p}}{1+\frac{1}{\pi p}}Q_{2\pi p}(0)+O\left(s-\frac{1}{2}\right),
f_\varepsilon(t)= (\|\nabla u\|_{L^2}^2+\varepsilon\|\Delta u\|_{L^2}^2+\| \sqrt\rho\partial_tu\|_{L^2}^2+\|\Delta d\|_{L^2}^2+ \|\nabla\partial_td\|_{L^2}^2)(t) +\frac12\int_0^t(\|\sqrt\rho\partial_tu\|_{L^2}^2+\|\nabla\partial_tu\|_{L^2}^2+\varepsilon \| \Delta\partial_tu\|_{L^2}^2+\|\nabla\Delta d\|_{L^2}^2+\|\Delta\partial_td\|_{L^2}^2)ds.
k_\gamma = k_{0^r}k_{\gamma_1}^{(0)}\cdots k_{\gamma_s}^{(s-1)}= k_{0^r}\left( \sum_{i_1=0}^r (-1)^{i_1}\binom{r}{i_1}k_{\gamma_1-i_1}^{(r)}\right)\cdots\left( \sum_{i_s=0}^r (-1)^{i_s}\binom{r}{i_s}k_{\gamma_s-i_s}^{(r+s-1)}\right) = \prod_{j=r+1}^{r+s}(1-L_j)^r k_{(0^r,\gamma)}\,.
\widehat{b_{\lambda}^{n}}(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}b_{\lambda}^{n}(e^{it})e^{-ikt}{\rm d}t=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i(nf_{\lambda}(t)-kt)}{\rm d}t=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ig(t)}{\rm d}t=\frac{1}{\pi}\Re\left\{\int_{0}^{\pi}e^{ig(t)}{\rm d}t\right\},
\int\limits_0^{\pi}Cf (0,e_n,\psi)\sin\psi d\psi &=\frac{n-2}{|S^{n-3}|}\int\limits_0^{\pi/2}\int\limits_{S^{n-2}}Rf(((\cos\psi_0)\omega,\sin \psi_0),0)d\omega \sin\psi_0(\cos\psi_0)^{n-2}d\psi_0 &=\frac{n-2}{|S^{n-3}|}\int\limits_{\pi/2}^\pi \int\limits_{S^{n-2}}Rf(((\sin \phi) \omega,-\cos \phi),0) (-\cos\phi)(\sin\phi)^{n-2}d\omega d\phi.
g_j(f_{jk}(z_k,s),s)=g_{jk}(g_k(z_k,s),h(s)) \,\,\,\,\,\,\,\,U_j\cap U_k\ne \emptyset \Lambda_{M_j}^{r,s}(g_j(z_j,s),h(s))=\sum_{\alpha,\beta=1}^n \Lambda_{N_j}^{\alpha,\beta}(z_j,s)\frac{\partial g_j^r}{\partial z_j^{\alpha}}\frac{\partial g_j^s}{\partial z_j^{\beta}}\,\,\,\,\,\,\,\,U_j.
P_{\tilde g}u= \Delta^2 u+\frac{1}{2(n-1)}R_{,kl}(0)x^kx^l (c_1^*\frac{u'}{r}+c_2^*u'')-\frac{4}{9(n-2)r^2}\sum_{kl}(W_{ikjl}(0)x^i x^j)^2(u''-\frac{u'}{r}) &+\frac{n-4}{24(n-1)}|W_g(0)|^2 u+(\frac{\psi_5(x)}{r^2}+\psi_3(x))u'' -(\frac{\psi_5(x)}{r^3}+\frac{\psi_3'(x)}{r}) u'+\psi_1(x)u &+O(r^4)u''+O(r^3) u'+O(r^2) u,
\left\{\delta S_{0}\left[{\frac{\delta}{\delta \eta_{a}}},{\frac{\delta}{\delta \bar{\eta}_{a}}},{\frac{\delta}{\delta J_{a}^{\mu}}}\right] + \delta S_{T}\left[{\frac{\delta}{\delta \eta_{a}}},{\frac{\delta}{\delta \bar{\eta}_{a}}},{\frac{\delta}{\delta J_{a}^{\mu}}}\right] \right\}Z \left[ J_{\mu}, \eta , \bar{\eta}\right] = 0 .
V ( u ) = E \left\{1 - \frac{1}{2}\left( \frac{u}{u_{*}}\right)^{7 - p}\left[ 1 - \frac{1}{\left( 1 + ( u_{*}/ u )^{7 - p}\right)^{2}}\right] \right\}+ \frac{E u_{* *}^{2}}{2 u^{2}\left( 1 + ( u_{*}/ u )^{7 - p}\right)^{2}},
t_{0}\leqslant \frac{1}{\left(\frac{1-\theta}{\theta}\right)\left(\left(\frac{2\theta}{1-\theta}2^{\frac{\theta+1}{\theta}}C_{1}^{\frac{1}{\theta}}\right)^{\frac{\theta}{1-\theta}}K^{\frac{2}{1-\theta}}\right)^{\frac{1-\theta}{\theta}}2^{-\frac{\theta+1}{\theta}}C_{1}^{-\frac{1}{\theta}}K^{-\frac{2}{\theta}}}=\frac{1}{2}.
ds^2 = e^{-2k|y|}\left[ -\left( 1 - \frac{U_T^4}{k^4}e^{4 k |y|}\right) dt^2 +\sum^3_{i=1}dx^i dx^i \right] + \frac{dy^2}{1 - \frac{U_T^4}{k^4}e^{4 k |y|}}
\#\left\{(A_i,B_i)_i(X_j)_j\in G^{2g}\times \prod_{j=1}^{k}C_j\Big\arrowvert A_1\sigma(B_1)A_1^{-1}B_1^{-1}\prod_{i=2}^{g}[A_i,B_i]\prod_{j=1}^{k}X_j=1\right\}= (n'\ast n^{g-1}\ast 1_{C_1}\ast\cdots\ast 1_{C_{k}})(1).
\begin{array}{ccl}\Delta_m = (-1)^mq^{A_m+k+m-1/2}(1-q^{1/2})((1+q^{k+m+3/2})(1-q^{k+m}) -q^{k+m+1}(1+q^{k+m-1/2})(1-q^{k+m+1}))/T_m = U_m(1-q^{1/2})(1-q^{k+m})((1+q^{k+m+3/2})-q^{k+m+1}(1+q^{k+m-1/2})) -U_m(1-q^{1/2})q^{2k+2m+1}(1+q^{k+m-1/2})(1-q) = U_m(1-q^{1/2})(K_m+L_m+M_m)~,\end{array}