The knowledge of the metric entropy of precompact subsets of operators on finite dimensional Euclidean space is important in particular in the probabilistic methods developped by E. D. Gluskin and S. Szarek for constructing certain random Banach spaces. We give a new argument for estimating the metric entropy of some subsets such as the Grassmann manifold equipped with natural metrics. Here, th...

In this chapter, we introduce a unjfied rugh-dimensional geometric framework for analyzing the phase transition phenomenon of (1 minimization in compressive sensing. This framework connects srudying the phase transitions of ( 1 minimization with computing the Grassmann angles in high-dimensional convex geometry. We demonstrate the broad applications of this Grassmann angle framework by giving s...

A generalized definition of a deformation of the fermionic oscillator (k-fermionic oscillators) is proposed. Two prescriptions for the construction of generalized Grassmann coherent states for this kind of oscillators are derived. The two prescriptions differs in the nature of the generalized Grassmann variables used. While we use Majid’s definition for such variables in the first case, Kerner’...