Reconstruction of the shape of object with near field measurements in a half-plane

We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem.Some numerical examples to show the feasibility of the method are presented.     

A unification of some matrix factorization results

A unification of some factorization results regarding products of positive-definite matrices, commutators, and products of involutions is presented. This topic was first presented by Sourour for fields with, in the latter two classes, sufficiently many elements in terms of the order of the matrix being factored. The current presentation is valid for matrices over any field with at least four elements, and is independent of the order of the matrix being factored.     

Majorization, range inclusion, and factorization for bounded linear operators

In this paper, relationships among the concepts, majorization, range inclusion, and factorization, are studied in a general setting for bounded linear operators. Some applications of these concepts are given.

     

Rank reduction, factorization and conjugation

We investigate the rank reduction procedure, related factorizations and conjugation algorithms. An exact characterization of the full rank factorization produced by the rank reduction algorithm is given. This result is then used to derive matrix decompositions and conjugation procedures.     

A note on 3‐factorizations of K10*

In this note, we enumerate all nonisomorphic 3‐factorizations of K₁₀. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 379–383, 2001     

Fast solution of unsymmetric banded Toeplitz systems by means of spectral factorizations and Woodbury's formula

A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a companion matrix pencil constructed from the coefficients of the generating function. The factorization is followed by the Woodbury inversion formula and solution of several banded triangular systems. Stability of the algorithm is discussed and its performance is demonstrated by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Imposing symmetry in augmented linear systems

This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least‐squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero‐by‐definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose‐built’ examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical solution of an extended White–Metzner model for eccentric Taylor–Couette flow

In this study, we have developed a new numerical approach to solve differential-type viscoelastic fluid models for a commonly used benchmark problem, namely, the steady Taylor—Couette flow between eccentric cylinders. The proposed numerical approach is special in that the nonlinear system of discretized algebraic flow equations is solved iteratively using a Newton–Krylov method along with an inverse-based incomplete lower-upper preconditioner. The numerical approach has been validated by solving the benchmark problem for the upper-convected Maxwell model at a large Deborah number. Excellent agreement with the numerical data reported in the literature has been found. In addition, a parameter study was performed for an extended White–Metzner model. A large eccentricity ratio was chosen for the cylinder system in order to allow flow recirculation to occur. We detected several interesting phenomena caused by the large eccentricity ratio of the cylinder system and by the viscoelastic nature of the fluid. Encouraged by the results of this study, we intend to investigate other polymeric fluids having a more complex microstructure in an eccentric annular flow field.     

关于整系数多项式有理根求法的注记

现行高等代数教材给出了求整系数多项式有理根的经典方法 ,周仲旺近日撰文又给出了一个新方法 ,称其“要比经典的方法有趣简捷”,但没有给出两个方法运算量的定量分析与比较 .本文先对经典方法从数学原理和算法设计两个方面作较详细明确的描述 ;再给出经典算法与周方法运算量的定量分析 ,比较的结果是周方法运算量比经典算法运算量多得多 .     

On almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X^c-I, where X^c denotes the complement of X and I a perfect matching (1-factor) in X^c. Almost self-complementary circulant graphs were first studied by Dobson and Šajna [Almost self-complementary circulant graphs, Discrete Math. 278 (2004) 23-44]. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost self-complementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.