Clifford分析中k-正则函数的一些性质和高阶Cauchy型积分的边值特性

本文首先给出了定义于R~n取值于Clifford代数C(V_(n,0))中k-正则函数的若干性质,如唯一性定理,Cauchy-Pompeiu公式,高阶Cauchy积分公式,平均值定理等,然后在k-正则函数的高阶Cauchy积分公式的基础上,相应的定义了r次连续可微函数的高阶Cauchy型积分,并给出了它的Cauchy主值,Plemelj公式,边值的Ho|¨lder连续性及其Privalov定理.     

利用调和分析方法证明柯西积分公式

柯西积分公式是复变函数中的重要公式之一,它的证明在一般的教材中是利用柯西积分定理以及函数的连续性来证明的.而在该论文中提供了另一种的柯西积分公式证明方法,主要是利用调和函数和数学分析中的格林公式来证明.     

积分第一中值定理的进一步推广

给出了积分第一中值定理的几种推广形式,在更弱的条件下得到更好的结果,同时推广了柯西型积分中值定理,得到了柯西型积分第一中值定理.     

Clifford分析中双正则函数的Taylor展式及其性质

首先借助实Clifford分析中双正则函数的累次积分的换序公式,给出了双正则函数的Cauchy积分公式,然后由特征边界的Cauchy积分公式,得到了双正则函数的Taylor展式,并由此给出了双正则函数的唯一性定理,柯西不等式和Weierstrass定理.     

柯西积分定理

<正> 柯西积分定理是解析函数中最重要的基础定理,解析函数的很多重要性质,都是由这个定理派生出来的.柯西原始的积分定理创立于1825年,当时要求导函数f′(z)在积分围线上是连续的.1900年古尔莎(E.Goursat)证明的柯西积分定理改进为只要求     

取值于Banach空间上的向量值函数的柯西型积分的一些性质

应用初等的方法讨论了取值于Banach空间上的向量值函数的柯西型积分的连续性、可导性、解析性和高阶导数公式,以及柯西型积分在具有可数基的Banach空间中的性质.     

复变函数中值定理的改进和推广

改进了复变函数柯西中值定理的表达形式,将实函数加权型中值定理推广到复变函数上,同时推广复变函数积分中值定理到一般光滑曲线上.     

基于微分中值定理的积分中值定理

运用微分中值定理,讨论并导出相应拉格朗日型或柯西型积分中值定理,在吏弱的条件下,得出比通常积分中值定理更强的结论．     

圆周上带希尔伯特核的柯西奇异积分方程的数值解(英文)

本文研究了圆周上带希尔伯特核的柯西奇异积分的复合梯型公式.利用连续的分片线性函数逼近被积函数,得到积分公式的误差估计;然后用积分公式构造求解对应奇异积分方程的两种格式;最后给出数值实验验证理论分析结果.     

Stokes公式的一个注记

利用Stokes公式证明了一个对满足散度为零的向量场的第二型曲面积分可化为其边界封闭曲线的第二型曲线积分来计算的定理.该定理对于满足上述条件向量场的曲面积分,给出了具体转化为曲线积分进行计算的公式,最后利用该公式计算了一个例子.     

Integral Formula of Isotonic Functions over Unbounded Domain in Clifford Analysis

In this paper we mainly study the so-called isotonic Dirac system over more general types of unbounded domains in Euclidean space of even dimension. In such systems different Dirac operators in the half dimension act from the left and from the right on the functions considered. We obtain the integral representation of isotonic functions satisfying the decay condition over the unbounded domains, and show that the integral representation formula over the unbounded domains for holomorphic functions of several complex variables and for Hermitean monogenic functions may be derived from it.     

Martinelli-Bochner formula using Clifford analysis

In the even dimensional case the Dirac equation may be reduced to the so-called isotonic Dirac system, in which different Dirac operators appear from both sides in half the dimension. This system is then used to derive the classical Martinelli-Bochner formula for several complex variables. Frank Sommen: Supported by FWO-Krediet aan Navorsers 1.5.065.04. Dixan Pe?a Pe?a: Supported by a Doctoral Grant of the Special Research Fund of Ghent University. Received: 8 March 2006     

A Discrete Bochner�CMartinelli Formula

In the even dimensional case the discrete Dirac equation may be reduced to the so-called discrete isotonic Dirac system in which suitable Dirac operators appear from both sides in half the dimension. This is an appropriated framework for the development of a discrete Martinelli–Bochner formula for discrete holomorphic functions on the simplest of all graphs, the rectangular \mathbbZ^m{\mathbb{Z}^m} one. Two lower-dimensional cases are considered explicitly to illustrate the closed analogy with the theory of continuous variables and the developed discrete scheme.     

Hartogs Extension Theorem for Functions with Values in Complex Clifford Algebras

A regular extension phenomenon of functions defined on Euclidean space with values in a Clifford algebra was studied by Le Hung Son in the 90’s using methods of Clifford analysis, a function theory which, is centred around the notion of a monogenic function, i.e. a null solution of the firstorder, vector-valued Dirac operator in . The isotonic Clifford analysis is a refinement of the latter, which arises for even dimension. As such it also may be regarded as an elegant generalization to complex Clifford algebra-valued functions of both holomorphic functions of several complex variables and two-sided biregular function theories. The aim of this article is to present a Hartogs theorem on isotonic extendability of functions on a suitable domain of . As an application, the extension problem for holomorphic functions and so for the two-sided biregular ones is discussed.      

偶数维复Clifford代数中的Dirac旋量空间

本文引入了偶数维欧氏空间的复结构及Witt基,在此基础上讨论了偶数维复Clifford代数中的Dirac旋量空间.由Fock空间的结果我们得到了Dirac旋量空间视为复Clifford代数中极小左理想,最后我们研究了Dirac旋量空间的对偶空间.     

Explicit solutions of the inhomogeneous dirac equation

In this paper we establish a general principle which may be used to construct many explicit solutions to special inhomogeneous Dirac equations with distributional right-hand side. These solutions are presented as series of products of Clifford algebra valued functions which themselves satisfy Dirac equations in a lower dimension. We also present several special examples, including plane waves, zonal functions, Cauchy kernels and electromagnetic fields.     

The Second Dirac Eigenvalue of a Nearly Parallel G2-Manifold

We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension n = 7.     

Dirac operators and spectral triples for some fractal sets built on curves

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log3/log2.     

Complexes of Dirac operators in Clifford algebras

In this paper we study complexes of k Dirac operators (or variations of Dirac operators) in the real or complex Clifford algebras i.e. complexes in which the first map is induced by the matrix where is the Dirac operator with respect to the variable . In particular we prove that, if , the complex in the case of 3 operators can be described in terms of relations coming from the so called radial algebra. Moreover we show that if the dimension m is less than 2k-1, then the Fischer decomposition does not hold. Received: 2 February 2000; in final form: 20 June 2000 / Published online: 25 June 2001     

A Resolution for the Dirac Operator in Four Variables in Dimension 6

The Dirac operator in several operators is an analogue of the - operator in theory of several complex variables. The Hartog’s type phenomena are encoded in a complex of invariant differential operators starting with the Dirac operator, which is an analogue of the Dolbeault complex. In the paper, a construction of the complex is given for the Dirac operator in 4 variables in dimension 6 (i.e. in the non-stable range). A peculiar feature of the complex is that it contains a third order operator. The methods used in the construction are based on the Penrose transform developed by R. Baston and M. Eastwood. The work presented here is a part of the research project MSM 0021620839 and was supported also by the grant GA ČR 201/05/2117.