A Geometric Definition Of Schubert Polynomials and Dual Schubert Polynomials For Classical Lie Groups

In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its corresponding infinite dimensional projective Stiefel manifold(which is homotopic to the product of infinite dimensional complex projective space $\mathbb{C}P^{\infty}$), we define the Schubert polynomials and dual Schubert polynomials. Finally we discuss the property and the computation of these polynomials.Comment: This paper has been withdrawn by the author due to a crucial This paper have a vital error in Lemma 2.1. So the definition for Schubert polynomials are not valid for Lie groups of type B,C,

Dynamics and correlation length scales of a glass-forming liquid in quiescent and sheared conditions

We numerically study dynamics and correlation length scales of a colloidal liquid in both quiescent and sheared conditions to further understand the origin of slow dynamics and dynamic heterogeneity in glass-forming systems. The simulation is performed in a weakly frustrated two-dimensional liquid, where locally preferred order is allowed to develop with increasing density. The four-point density correlations and bond-orientation correlations, which have been frequently used to capture dynamic and static length scales $\xi$ in a quiescent condition, can be readily extended to a system under steady shear in this case. In the absence of shear, we confirmed the previous findings that the dynamic slowing down accompanies the development of dynamic heterogeneity. The dynamic and static length scales increase with $\alpha$-relaxation time $\tau_{\alpha}$ as power-law $\xi\sim\tau_{\alpha}^{\mu}$ with $\mu>0$. In the presence of shear, both viscosity and $\tau_{\alpha}$ have power-law dependence on shear rate in the marked shear thinning regime. However, dependence of correlation lengths cannot be described by power laws in the same regime. Furthermore, the relation $\xi\sim\tau_{\alpha}^{\mu}$ between length scales and dynamics holds for not too strong shear where thermal fluctuations and external forces are both important in determining the properties of dense liquids. Thus, our results demonstrate a link between slow dynamics and structure in glass-forming liquids even under nonequilibrium conditions.Comment: 9 pages, 17 figures. Accepted by J. Phys.: Condens. Matte