Compressive Sensing

Compressive sensing is a novel paradigm for acquiring signals and has a wide range of applications. The basic assumption is that one can recover a sparse or compressible signal from far fewer measurements than traditional methods. The difficulty lies in the construction of efficient recovery algorithms. In this thesis, we review two main approaches for solving the sparse recovery problem in compressive sensing: l1-minimization methods and greedy methods. Our contribution is that we look at compressive sensing from a different point of view by connecting it with sparse interpolation. We introduce a new algorithm for compressive sensing called generalized eigenvalues (GE). GE uses the first m consecutive rows of discrete Fourier matrix as its measurement matrix. GE solves for the support of a sparse signal directly by considering generalized eigenvalues of Hankel systems. Under Fourier measurements, we compare GE with iterated hard thresholding (IHT) that is one of the state-of-the-art greedy algorithms. Our experiment shows that GE has a much higher probability of success than IHT when the number of measurements is small while GE is a bit more sensitive for signals with clustered entries. To address this problem, we give some observations from the experiment that suggests GE can be potentially improved by taking adaptive Fourier measurements. In addition, most greedy algorithms assume that the sparsity k is known. As sparsity depends on the signal and we may not be able to know the sparsity unless we have some prior information about the signal. However, GE doesn\u27t need any prior information on the sparsity and can determine the sparsity by simply computing the rank of the Hankel system

Neutron star matter in the quark-meson coupling model in strong magnetic fields

The effects of strong magnetic fields on neutron star matter are investigated in the quark-meson coupling (QMC) model. The QMC model describes a nuclear many-body system as nonoverlapping MIT bags in which quarks interact through self-consistent exchange of scalar and vector mesons in the mean-field approximation. The results of the QMC model are compared with those obtained in a relativistic mean-field (RMF) model. It is found that quantitative differences exist between the QMC and RMF models, while qualitative trends of the magnetic field effects on the equation of state and composition of neutron star matter are very similar.Comment: 16 pages, 4 figure

Delay-dependent robust stability of stochastic delay systems with Markovian switching

In recent years, stability of hybrid stochastic delay systems, one of the important issues in the study of stochastic systems, has received considerable attention. However, the existing results do not deal with the structure of the diffusion but estimate its upper bound, which induces conservatism. This paper studies delay-dependent robust stability of hybrid stochastic delay systems. A delay-dependent criterion for robust exponential stability of hybrid stochastic delay systems is presented in terms of linear matrix inequalities (LMIs), which exploits the structure of the diffusion. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method

Almost sure exponential stability of stochastic differential delay equations

This paper is concerned with the almost sure exponential stability of the multidimensional nonlinear stochastic differential delay equation (SDDE) with variable delays of the form dx(t) = f(x(t−δ1(t)), t)dt+g(x(t−δ2(t)), t)dB(t), where δ1, δ2 : R+ → [0, τ ] stand for variable delays. We show that if the corresponding (nondelay) stochastic differential equation (SDE) dy(t) = f(y(t), t)dt + g(y(t), t)dB(t) admits a Lyapunov function (which in particular implies the almost sure exponential stability of the SDE) then there exists a positive number τ ∗ such that the SDDE is also almost sure exponentially stable as long as the delay is bounded by τ ∗ . We provide an implicit lower bound for τ ∗ which can be computed numerically. Moreover, our new theory enables us to design stochastic delay feedback controls in order to stabilize unstable differential equation