实对称矩阵对角化中正交矩阵的初等变换求法

对实对称矩阵正交对角化过程中正交矩阵的求解方法进行了研究,给出了利用初等变换求解正交矩阵的方法,该方法不需要通过特征方程求解特征值与特征向量,仅仅使用初等变换和Schmidt正交化方法.     

几何对称法在求取某些区域格林函数中的应用

格林函数法是数学物理方程中一种常用的方法,适用于求解狄利克雷问题.针对几种特殊区域上的上狄利克雷问题,采用几何对称法求取这些区域对应的格林函数,该方法对于该区域上格林函数的求解是有效的.     

RBF-PU方法求解二维非局部扩散问题和近场动力学问题

采用单位分解径向基函数(radial basis function partition of unity,RBF-PU)方法,数值求解了二维非局部扩散问题和近场动力学问题。主要思想是对求解区域进行局部划分,在局部子区域上分别进行函数逼近,然后加权得到未知函数的全局逼近。这种基于方程强形式的径向基函数方法在求解非局部问题时,不需要处理网格与球形邻域求交的问题,避免了额外的一层积分计算,实施简便,计算量小。数值实验显示计算结果与解析解吻合较好,RBF-PU方法可以准确有效地求解非局部扩散方程和近场动力学方程。     

高维紧支撑正交对称的小波

基于仿酉矩阵的对称扩充方法,该文提出了一种尺度因子为3的紧支撑高维正交对称小波构造算法.即设φ(x)∈L~2(R~d)是尺度因子为3的紧支撑d维正交对称尺度函数,P(ξ)是它的两尺度符号,p_(0,v)(ξ)为P(ξ)的相位符号.首先提出一种向量的对称正交变换,应用对称正交变换对3~d维向量(p_(0,v)(ξ))_v,v∈E_d的分量进行对称化.通过仿酉矩阵的对称扩充,给出了3~d-1个紧支撑高维正交对称小波构造.这种方法构造的小波支撑不超过尺度函数的支撑.最后给出一个构造算例.     

对称反对称紧支撑正交多小波的构造

对于给定的对称反对称紧支撑正交r重尺度函数,给出一种构造对称反对称紧支撑正交多小波的方法.通过此方法构造的多小波与尺度函数有相同的对称性与反对称性,并且给出算例.     

Cn空间中有界域上一种积分表示

本文应用单位分解的观点及积分表示中核函数的构造理论,得到ln空间中有界域上积分表示的一种抽象的一般形式,根据这种一般形式,可以得到至今许多区域上光滑函数和全纯函数种种已有的抽象公式和具体的积分公式.     

三进制双正交对称小波的设计

本文给出了一种三进制双正交对称小波的设计方法.在给定插值紧支撑对称尺度函数的情况下,指出了如果对偶尺度函数同为紧支撑插值的,则它们同为1-型对称.并且给出了对偶尺度函数为紧支撑插值和非插值情况下的通解计算公式.还提出了频率优化方法设计对偶尺度函数和小波函数,把双正交条件归结为线性约束的二次规划问题,最后通过线性方程组来求解.对于小波函数本文也给出了一组特解公式.     

求Hermite型插值的广义Lagrange基函数方法

在求解各类插值问题时候,各种不同方法的本质是基函数的构造不同.本文提出了每个节点上的广义La-grange基函数概念,从数据表列的角度出发给出了一种求解各类Hermite插值问题的新方法,并通过各种算例进行了验证.     

取值在复Clifford代数上的函数的结构

1.前言在[1,第13章]中证明过,以{e_j}_(j-1)~n为正交基的空间R~n,可以构造一个2~n维实Clifford代数A_n,以e_0为其单位元素且基底元素为e_0,e_1,…,e_n,…,e_1…e_n。在最近的书[2]中证明了古典的复变函数论的许多观点都能够推广到这样的函数论上去,其函数取值于A_n上,这种函数论叫实Clifford分析。     

矩阵恒等式及应用

利用矩阵函数的性质得到了一类矩阵行列式的恒等式,作为应用,得到了一类无限维矩阵的行列式和迹.     

Frequency domain Walsh functions and sequences: An introduction

In this letter, a new set of orthogonal band-limited basis functions is introduced. This set of basis functions is derived from the inverse Fourier transform of the frequency domain Walsh functions. The Fourier transforms of the Walsh functions were calculated by Siemens and Kitai in 1973 but they have been overlooked in the literature. Some of the properties of these functions are studied in this paper. Moreover, the orthogonal discrete version of these functions is obtained by truncation, sampling and orthogonalization utilizing the orthogonal Procrustes problem.     

A basis function for approximation and the solutions of partial differential equations

In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well‐known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz‐type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein-Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.     

ORTHOGONAL PIECE-WISE POLYNOMIALS BASIS ON AN ARBITRARY TRIANGULAR DOMAIN AND ITS APPLICATIONS

1. IntroductionIt is well known that the concept of orthogonal polynomials plaes a key role in numericalanalysis. How to generalize orthogonal polynomials iDto higher dimension, without using tensorproduct, is still an open problem. As we know, the original orthogonal polynomial has beenstudied in univariable case. Strictlyl the tensor product approach is still staying in the onedimension level via decreasing dimension. The result only can use in rectangular domain.' There are some different…     

The canonical structure of a pencil of degenerate matrix functions

In this paper we study global properties of a pencil of identically degenerate matrix functions with a compact domain of definition. Matrix functions are assumed to have a constant rank and all roots of the characteristic equation of the matrix pencil are assumed to have a constant multiplicity at each point in the domain of definition. We obtain sufficient conditions for the smooth orthogonal similarity of matrix functions to the upper triangular form and sufficient conditions for the smooth equivalence of the pencil of matrix functions to its canonical form. We illustrate the obtained results with simple examples.     

A Practical Method for Generating Trigonometric Polynomial Surfaces over Triangular Domains

Mediterranean Journal of Mathematics - A class of trigonometric polynomial basis functions over triangular domain with three shape parameters is constructed in this paper. Based on these new basis...     

基于正交规范化的小波的有限差分表示

本文研究了在时域内小波的一种表达形式.利用正交规范化,获得了小波的有限差分表示.不仅该形式构造了任意次B样条正交小波.而且在时域中用来直接获得小波滤波器是有效的.     

〈I〉型三角剖分下非张量积连续小波基的构造

多维非张量积小波是近年小波研究领域中的热点问题之一 ,它们与多维张量积小波相比具有更多的优势 .关于高维张量积、非张量积小波 ,目前已有一些很好的工作 (见文[2 ] [3 ] [4 ] ) ,但关于样条小波 ,还有许多问题有待于研究 .本文针对〈I〉型三角剖分下的二维线性元空间 ,讨论其具有紧支集和对称性的半正交样条小波基 .给定 x1 x2 平面上的〈I〉型三角剖分 (图 1 ( a)所示 ) ,记 j=( j1 ,j2 ) ,| j| =j1 + j2 ,πm= { 0≤ |j|≤ mCj1j2 xj11 xj22 ,Cj1,j2 是任意实数 }为次数不超过 m的代数多项式全体 .引入剖分尺度为 1的线性元空间 V0…     

三角域上带两个形状参数的Bézier曲面的扩展

给出了三角域上带双参数λ1,λ2的类三次Bernstein基函数,它是三角域上三次Bernstein基函数的扩展.分析了该组基的性质并定义了三角域上带有两个形状参数λ1,λ2的类三次Bernstein-Bézier(B-B)参数曲面.该基函数及参数曲面分别具有与三次Bernstein基函数及三次B-B参数曲面类似的性质.当λ1,λ2取特殊的值时,可分别得到三次Bernstein基函数及三次B-B参数曲面以及参考文献中所定义的类三次Bernstein基函数及类三次B-B参数曲面.由实例可知,通过改变形状参数的取值,可以调整曲面的形状.     

Triangular domain extension of algebraic trigonometric Bézier-like basis

In computer aided geometric design (CAGD), Bézier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bézier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bézier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bézier-like surfaces. Finally, some applications of the proposed basis are shown.