平面双线性映射的伸缩率分析

拟共形映射能较好地保持角度,在形状编辑等几何处理领域有着广泛应用.但该类映射不易构造,特别是复杂区域之间的拟共形映射构造,是一个困难且重要的问题.本文研究了一类简单的拟共形映射,即双线性映射,讨论了其伸缩率的分布情况,证明了伸缩率的最大值一定在四边形区域的顶点上取得.相关结论为复杂区域之间拟共形映射的构造提供了良好的理论基础.数值实验验证了结论的正确性.     

调和映射的可积拟共形延拓

本文研究了平面调和映射的可积拟共形延拓问题.利用经典的共形映射的拟共形延拓方法和调和映射的性质,获得了一些条件使得调和映射可以拟共形延拓至整个平面且其复伸缩商关于双曲度量是p次可积的,推广了解析单叶函数的相关结果.     

拟共形映射的极值问题

研究了拟共形映射的极值问题.通过对一类具有边界对应的拟共形扩张函数的伸缩商的上界估计,得到了一些新的方法和新的结果.     

塌陷型极值拟共形映射的性质

证明了如果f~μ是单位圆△上塌陷型极值拟共形映射,则存在与f~μ等价的极值拟共形映射f~(?)以及单位圆的子区域G,使得面(z)=0,(?)z∈G.     

模和拟共形映射

研究了模和拟共形映射，得到了一个同胚为拟共形映射的几个条件．     

强John域中的Lipschitz扩张和拟双曲度量

利用拟双曲度量刻划了强John域,并且获得了强John域中拟共形映射的Hardy-Littlewood性质.     

具有不可缩伸缩商的拟共形映射

在研究拟共形映射的唯一极值性时,Reich引入了具有不可缩伸缩商的拟共形映射的概念.本文将继续讨论这类映射,并给出无穷小情形下的一些类似结果.     

同胚映射的代数指数、H?lder指数和拟对称指数

给定实轴上的同胚映射,定义了拟对称指数、代数指数和H?lder指数.上述指数刻画了同胚映射的局部特征,同时在拟对称映射和拟共形映射的研究中具有重要作用.探索了三个指数的相互关系并给出了几个实例.     

具有不可缩伸缩商的拟共形映射

在研究拟共形映射的唯一极值性时,Reich引入了具有不可缩伸缩商的拟共形映射的概念.本文将继续讨论这类映射,并给出无穷小情形下的一些类似结果     

Apollonian度量与Apollonian边界条件

证明了(1)(-R)n中真子域D上的Apollonian度量aD是拟共形映射的拟不变量;(2)(-R)n中严格一致域是拟共形不变的;(3)(-R)2中的Jordan域D是拟圆当且仅当D是严格一致域.作为应用,进一步得到了Apollonian边界条件,拟共形映射和局部Lipschitz映射之间的关系.     

Grunsky inequalities and Fredholm spectrum in general domains

We discuss the Fredholm spectrum for general domains and study its applications to conformal and quasi-conformal mappings. In particular, we establish an improvement of the Grunsky inequalities which is valid for general domains. This improvement constitutes an extension of a recent result of Schiffer concerning the sharpening of Grunsky inequalities for the unit disk by a factor smaller than 1, and which is the reciprocal of the least Fredholm eigenvalue of the smooth simply-connected image domain.     

Noncommutative,quasi-conformal integral transforms over quaternions and octonions

This paper is devoted to the study of holomorphic and meromorphic functions of quaternion and octonion variables. New classes of quasi-conformal and quasi-meromorphic mappings are defined and studied. Different properties of these functions such as residues and the argument principle are studied. We prove that the family of all quasi-conformal diffeomorphisms of a domain is a topological group relative to composition of mappings. In particular, cases where it is a finite-dimensional Lie group over R are studied. Relations between quasi-conformal functions and integral transformations of functions over quaternions and octonions are investigated. Noncommutative analogs of the Mellin transformations are studied and used. In addition, examples of such functions are given. Finally, applications to problems of complex analysis are discussed. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional Analysis, 2008.     

QCMC: quasi-conformal parameterizations for multiply-connected domains

This paper presents a method to compute the quasi-conformal parameterization (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain S onto a punctured disk D _S associated with a given Beltrami differential. The Beltrami differential, which measures the conformality distortion, is a complex-valued function \(\mu :S\to \mathbb {C}\) with supremum norm strictly less than 1. Every Beltrami differential gives a conformal structure of S. Hence, the conformal module of D _S , which are the radii and centers of the inner circles, can be fully determined by μ, up to a Möbius transformation. In this paper, we propose an iterative algorithm to simultaneously search for the conformal module and the optimal quasi-conformal parameterization. The key idea is to minimize the Beltrami energy with the conformal module of the parameter domain incorporated. The optimal solution is our desired quasi-conformal parameterization onto a punctured disk. The parameterization of the multiply-connected domain simplifies numerical computations and has important applications in various fields, such as in computer graphics and vision. Experiments have been carried out on synthetic data together with real multiply-connected Riemann surfaces. Results show that our proposed method can efficiently compute quasi-conformal parameterizations of multiply-connected domains and outperforms other state-of-the-art algorithms. Applications of the proposed parameterization technique have also been explored.     

单位圆内拟亚纯映射的重值

单位圆内充满圆序列是一个复杂的问题,该文改进了现有一些证法,使单位圆内充满圆证法类似于平面上充满圆证法.同时得到单位圆内K-拟亚纯映射涉及重值的充满圆序列及相应的奇异半径     

Removability of certain sets in the class of three-dimensional quasiconformal mappings

The removability is established, in the class of three-dimensional quasi-conformal mappings, of closed sets on sufficiently smooth curves.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 717–722, June, 1970.     

DECIMAL EXTENSIONS OF THE ADDITIVE GROUP OF INTEGERS

We introduce the concept of decimal extension of the additive group of integers. We characterize these extensions as the linearly ordered abelian groups containing at least one Archimedean element. Finally, we study the additive group of rationals and its subgroups as particular cases of decimal extension.     

一维p—Laplace方程多个正解的存在性

在非线性项ｆ呈Ｓ形的条件下，讨论了ｐ－Ｌａｐｌａｃｅ方程多个正解的存在性，所用工具为锥拉伸与锥压给不动点定理。     

一个特征标环素谱的连通性

本文确定了一个有限群特征标环通过代数整数环扩张后素谱的结构，在此基础之上，利用与[1]中类似的方法证明了这个扩张后的特征标环素谱的连通性。同时还计算了有限群的复类函数空间的幂等元．