Clifford分析中无界域上正则函数的边值问题

在引入修正Cauchy核的基础上,讨论了无界域上正则函数的带共轭值的边值问题：a(t)Φ (t) b(t)Φ (t) c(t)Φ-(t) d(t)Φ=g(t)．首先给出了无界域上正则函数的Plemelj公式,然后利用积分方程方法和压缩不动点原理证明了问题解的存在唯一性．     

Norms and Generating Functions in Clifford Algebras

Generating functions are commonly used in combinatorics to recover sequences from power series expansions. Convergence of formal power series in Clifford algebras of arbitrary signature is discussed. Given , powers of u are recovered by expanding (1 − tu)⁻¹ as a polynomial in t with Clifford-algebraic coefficients. It is clear that (1 − tu)(1 + tu + t ² u ² + ...) = 1, provided the sum (1 + tu + t ² u ² + ...) exists, in which case u ^m is the Cliffordalgebraic coefficient of t ^m in the series expansion of (1 − tu)⁻¹. In this paper, conditions on for the existence of (1 − tu)⁻¹ are given, and an explicit formulation of the generating function is obtained. Allowing A to be an m × m matrix with entries in , a “Clifford-Frobenius” norm of A is defined. Norm inequalities are then considered, and conditions for the existence of (I − tA)⁻¹ are determined. As an application, adjacency matrices for graphs are defined with vectors of as entries. For positive odd integer k > 3, k-cycles based at a fixed vertex of a graph are enumerated by considering the appropriate entry of A ^k . Moreover, k-cycles in finite graphs are enumerated and expected numbers of k-cycles in random graphs are obtained from the norm of the degree-2k part of tr(1 − tu)⁻¹. Unlike earlier work using commutative subalgebras of , this approach represents a “true” application of Clifford algebras to graph theory.      

Riemann Boundary Value Problems for Triharmonic Functions in Clifford Analysis

In this paper, we study the R _m (m > 0) Riemann boundary value problem for triharmonic functions with values in a universal Clifford algebra Cl(V _n,n ). By using the Plemelj formula and generalized Liouville theorem for triharmonic functions, the explicit representation of solution of this problem is given.     

Clifford 分析中无界域上正则函数带Haseman位移的边值问题

该文在引入修正的Cauchy核的基础上,讨论了Clifford 分析中无界域上正则函数带 Haseman 位移的边值问题. 首先给出了无界域上Cauchy 型积分的Plemelj公式,再利用积分方程方法和压缩不动点定理证明了问题解的存在唯一性.     

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.      

The Additive Cousin Problem and Related Problems for Regular Functions with Parameter Taking Values in a Clifford Algebra

This article deals with the Runge type approximations problem, the solution of inhomogeneous system and the additive Cousin problem for Clifford-algebra-valued functions ?(x, y) which are regular with respect to x and real-analytic in y. Some main results concerning these three fundamental problems from complex analysis are proved for the class of functions mentioned above.     

分裂的正则Hom-Leibniz-Rinehart代数

作为Hom-Leibniz代数胚的代数类比, 本文引入Hom-Leibniz-Rinehart代数的概念. 证明了分裂的正则Hom-Leibniz-Rinehart代数$L$写成$L=U+\sum_{\gamma}I_\gamma$, 其中$U$为极大交换子代数$H$的子空间和$I_\gamma$为$L$的理想, 若$[\gamma]\neq[d]$, 满足$[I_\gamma, I_d]=0$. 随后分别发展了分裂Hom-Leibniz-Rinehart代数的根和权的连通技术.最后研究了紧致的正则Hom-Leibniz-Rinehart代数的结构.     

On the Riemann Hilbert type problems in Clifford analysis

In this paper boundary value problems combining Jump — Riemann and Hilbert problems for monogenic functions in Ahlfors-David regular surfaces and in the upper half space respectively are investigated. The explicit formula of the solution is obtained.     

分裂的正则Hom-Poisson 着色代数结构

We introduce the class of split regular Hom-Poisson color algebras as the natural generalization of split regular Hom-Poisson algebras and the one of split regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show∑that such a split regular Hom-Poisson color algebras A is of the form A = U +αIα with U a subspace of a maximal abelian subalgebra H and any Iα, a well described ideal of A, satisfying[Iα, Iβ] + IαIβ = 0 if [α]≠[β]. Under certain conditions, in the case of A being of maximal length, the simplicity of the algebra is characterized.     

On Serre's Problem for RIT Algebras

Abstract

We construct a counterexample to Serre's problem for over a RIT algebra, which is a noncommutative quadratic algebra, given by quadratic relations, more general than the relations of the Weyl algebra and of a universal enveloping of finite dimensional Lie algebra.     

具有奇性的非线性椭圆方程的边值问题

本文研究奇异椭圆方程的边值问题.利用变分方法和锥理论中的混合单调方法,证明了奇异方程正解的存在性、唯一性.     

On a Frankl-Type Boundary-Value Problem for a Mixed-Type Degenerating Equation

Ukrainian Mathematical Journal - We study the existence and uniqueness of solutions for an analog of the Frankl-type boundary-value problem for a parabolic-hyperbolic-type equation. The uniqueness...     

On a Two-Point Boundary-Value Problem with Spurious Solutions

The Carrier–Pearson equation     with boundary conditions   u (−1) = u (1) = 0  is studied from a rigorous point of view. Known solutions obtained from the method of matched asymptotics are shown to approximate true solutions within an exponentially small error estimate. The so-called spurious solutions turn out to be approximations of true solutions, when the locations of their "spikes" are properly assigned. An estimate is also given for the maximum number of spikes that these solutions can have.     

解析函数的加权代数中的闭理想问题

该文讨论了单位圆盘上解析函数的加权代数犃犘中的闭理想问题,并且利用加权系给出了一个闭理想成为犃犘中的补子空间的判断准则     

Clifford分析中广义超正则函数的一个非线性边值问题

讨论了Cliffrd分析中广义超正则函数的一个非线性边值问题.首先将广义超正则函数分解为两个奇异积分算子,然后给出了广义超正则函数的Plemelj公式及相关奇异积分算子的性质,最后利用Schauder不动点原理证明了广义超正则函数的一个非线性边值问题的解的存在性及积分表达式.     

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.