矩阵广义对角化的探讨

利用子空间关于矩阵的最小多项式研究了矩阵可广义对角化的充要条件,给出了矩阵可广义对角化的一种算法.     

建立模糊诊断矩阵的一种新方法

本文首先提出了一个新的模糊逻辑算子——模糊加权综合算子“#”,进而给出建立模糊诊断矩阵的新方法——网络法 ,并给出了相应的算法     

Synthesis of achiral PEG‐PANI rod‐coil block copolymers and their helical superstructures

Poly(ethylene glycol) (PEG) was modified with aniline groups at both the end, and then PEG‐PANI rod‐coil block polymers have been synthesized by polymerization of the aniline with the aniline‐modified PEG. FTIR, NMR, and elemental analysis provided the chemical strucutre of the as‐prepared polymers. The achiral rod‐coil copolymer could form different superstructures by means of self‐assembly when adding diethyl ether into its THF solution and the length of PANI segments is a key factor to the superstructures. AFM measurements revealed that they form spring‐like helical superstructures from the short PANI‐containing copolymers while these form fibrous helical superstructures from the longer PANI‐containing copolymer. A possible mechanism of the helical superstructures is suggested in this article and the driving force is believed the π–π stacking of the rigid segment of the copolymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 12–20, 2008     

Shifted Jacobi multiplier sequences and transplantation between sine and cosine series

It is shown that the multiplier norm of a shifted Jacobi multiplier sequence can be estimated by the (same) multiplier norm of the original sequence uniformly with respect to the shift. Muckenhoupt’s transplantation theorem for Jacobi series is used essentially, for which also a functional analytic understanding is given in terms of the minimality of the Jacobi system in weighted L ^p -spaces.      

Adjacency matrices and chemical transformation graphs

The vast numbers of molecular objects and chemical transformations of a substrate cannot be evaluated even intuitively, which necessitates the development of simple reaction monitoring methods. Here this problem is formulated for the first time, and possible solutions are substantiated.     

对称广义中心对称矩阵模型修正的矩阵逼近法及其扰动性

X,B是实测的位移矩阵和载荷矩阵,C是有限元方法得到的估计矩阵,给出了AX=B的对称广义中心对称矩阵解集合ζ的表达式,对于逼近问题||C-A||F=min A∈ζ||C-A||F的解A,给出了它的表达式并分析了解A的扰动性,数值结果表明方法是行之有效的.     

A comparison of preconditioners for incompressible Navier–Stokes solvers

We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the so‐called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium‐sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd.     

A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem

The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471074, 10771116) and the Doctoral Program of the Ministry of Education of China (Grant No. 20060003003)     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.