Narrow Results

In this paper, we proposed a unified formulation to explain the reason why spurious eigensolution occurs in the eigenproblem of interior acoustics using the real-part and imaginary-part BEMs and why fictitious frequency occurs in exterior acoustics using the complex-valued BEM. Both the two problems stem from the rank deficiency of the influence matrix. Based on the circulant properties and degenerate kernels, an analytical study in a discrete system for a circular cavity is conducted. The Fredholm alternative theorem is employed to study the rank-deficiency problems in conjunction with singular value decomposition updating technique. The spurious and fictitious boundary modes are found to locate in the column vectors of left unitary matrix. Also, the effects of different types of boundary condition on the spurious and fictitious solutions using direct and indirect methods are discussed. The mathematical and physical meanings for the nontrivial boundary solution in spurious eigensolution and fictitious frequency are explained. Numerical experiments are found to agree with the analytical predictions.

A classical lazy random walk on cycles is known to mix with the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibits strong non-uniform mixing properties. First, we prove that the instantaneous distribution of a quantum walk on most even-length cycles is never uniform. More specifically, we prove that a quantum walk on a cycle C_n is not instantaneous uniform mixing, whenever n satisfies either: (a) n = 2^u, for u ≥ 3; or (b) n = 2^uq, for u ≥ 1 and q ≡ 3 (mod 4). Second, we prove that the average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform. Nevertheless, we show that the average distribution of a quantum walk on the cycle C_n is O(1/n)-uniform.

The numerical solution of large scale 3D elliptic boundary value problems is discussed in this article. We analyze the parallel complexity of the circulant block-factorization preconditioners when solving elliptic problems by Preconditioned Conjugate Gradient (PCG) method. Both, the conjugate gradient method and the inversion of circulant matrices using Fast Fourier Transform (FFT), are highly parallelizable. Estimates for the parallel time, the speed-up and the parallel efficiency are derived. The numerical tests have been executed on a symmetric multiprocessor systems.

We present a unified approach to limiting spectral distribution (LSD) of patterned matrices via the moment method. We demonstrate relatively short proofs for the LSD of common matrices and provide insight into the nature of different LSD and their interrelations. The method is flexible enough to be applicable to matrices with appropriate dependent entries, banded matrices, and matrices of the form where X is a p × n matrix with real entries and p → ∞ with n = n(p)→ ∞ and p/n → y with 0 ≤ y < ∞.

This approach raises interesting questions about the class of patterns for which LSD exists and the nature of the possible limits. In many cases the LSD are not known in any explicit forms and so deriving probabilistic properties of the limit are also interesting issues.