Zhengfang Zhou

Peter G. Casazza, Matthew Fickus, Andreas Heinecke et al.

Spectral tetris is a fexible and elementary method to construct unit norm frames with a given frame operator, having all of its eigenvalues greater than or equal to two. One important application of spectral tetris is the construction of fusion frames. We first show how the assumption on the spectrum of the frame operator can be dropped and extend the spectral tetris algorithm to construct unit norm frames with any given spectrum of the frame operator. We then provide a suffcient condition for using this generalization of spectral tetris to construct fusion frames with prescribed spectrum for the fusion frame operator and with prescribed dimensions for the subspaces. This condition is shown to be necessary in the tight case of redundancy greater than two.

Peter Casazza, Andreas Heinecke, Keri Kornelson et al.

Spectral Tetris has proved to be a powerful tool for constructing sparse equal norm Hilbert space frames. We introduce a new form of Spectral Tetris which works for non-equal norm frames. It is known that this method cannot construct all frames --- even in the new case introduced here. Until now, it has been a mystery as to why Spectral Tetris sometimes works and sometimes fails. We will give a complete answer to this mystery by giving necessary and sufficient conditions for Spectral Tetris to construct frames in all cases including equal norm frames, prescribed norm frames, frames with constant spectrum of the frame operator, and frames with prescribed spectrum for the frame operator. We present a variety of examples as well as special cases where Spectral Tetris always works.

Lianzhang Bao, Zhengfang Zhou

The process by which one may take a discrete model of a biophysical process and construct a continuous model based on it is of mathematical interest as well as being of practical use. In this paper, we first study the singular limit of a class of reinforced random walks on a lattice for which a complete analysis of the existence and stability of solutions are possible. In the continuous scenario, we obtain the regularity estimate of this aggregation diffusion model. As a by-product, nonexistence of solution of the continuous model with pure aggregation initial data is proved. When the initial is purely in diffusion region, asymptotic behavior of the solution is obtained. In contrast to continuous model, bounded-ness of the lattice solution, asymptotic behavior of solution in diffusion region with monotone initial date and the interface behaviors of the aggregation, diffusion regions are obtained. Finally we discuss the asymptotic behaviors of the solution under more general initial data with non-flux when the lattice points $N\leq 4$.