具有移动底边界的水波问题的仿线性化（英文）

主要研究了带表面张力的无旋不可压缩重力水波问题,该水波的流动区域除了有自由上边界外,还具有给定的移动底边界.主要目的是利用仿微分方法对非线性水波问题的Zakharov表示进行仿线性化,关键在于处理Dirichlet-Neumann算子.借助Possion核定义正则映射来拉平边界会使仿线性化过程更加精细.这一仿线性化结果使非线性的水波方程成为线性系统,为研究具有移动底边界的水波方程适定性奠定了基础.     

多分支自相似集的自仿嵌入

本文研究了一类多分支自相似集的自仿嵌入.利用压缩映射的不动点,在一定条件下,证明了若一个自相似集能自仿嵌入到另一个自相似集中,则它们对应的迭代函数系的压缩比满足某种代数性质.     

多分支自相似集的自仿嵌入（英文）

本文研究了一类多分支自相似集的自仿嵌入.利用压缩映射的不动点,在一定条件下,证明了若一个自相似集能自仿嵌入到另一个自相似集中,则它们对应的迭代函数系的压缩比满足某种代数性质.     

格L上点式仿度量的研究

本文通过对经典度量的比较,提出了格上点式仿度量概念且研究了它的拓扑与余拓扑的关系及其它的一些性质,并给出了它的一个应用。     

数字集的谱性与谱自仿测度

本文主要确定与谱自仿测度μ_(M,D)相关的数字集D的谱性.研究所涉及的谱性问题与Dutkay、Han和Jorgensen的一个猜测密切相关.此猜测表明,在一维情形下,如果μ_(M,D)是谱自仿测度,则相应的数字集D总是一个谱集.对于一个自仿测度μ_(M,D),本文获得使数字集D具有谱性的一些条件,为这个猜测的成立提供了依据.另外,本文所得的结果在某些情形下也推广许多已知的相应结果.     

Locale的仿紧完全正则反射

1972年, Isbell利用locale中由正规覆盖构成的一致结构的完备化, 证明了仿紧完全正则locale范畴是locale范畴的满反射子范畴. 本文通过在locale的补零元理想格上做核映射的方法, 给出locale的仿紧完全正则反射的明确构造, 并证明locale的仿紧完全正则反射是locale的 Stone-Čech紧化的子locale.     

格上点式仿紧性

本文在拓扑分子格（ＴＭＬ）上提出一种仿紧性，它以格上点式紧性［３］为特款，讨论了它在ＴＭＬ上的几种不同的表现形式以及它与正规性的关系．     

空间自仿测度下无限正交指数系存在的条件

自仿测度μ_M,D的谱与非谱问题是自仿测度谱理论研究的主要内容之一,而μ_M,D-正交指数系的有限性或无限性问题在研究自仿测度是否为谱测度中起着重要的作用.本文主要探讨空间自仿测度下无限正交指数系存在的条件.通过利用函数m_D(x)零点集Z(m_D)中的非零中间点(即坐标为0或1/2的点)的性质,得到存在无限μ_(M,D)-正交指数系的许多条件,为进一步研究自仿测度μ_(M,D)的谱性质奠定基础.     

自仿函数分数阶导数的分形维数

研究一类自仿函数的分数阶导数,获得了自仿函数的Weyl-Marchaud分数阶导数的图像盒维数,证明了分数阶导数的阶与分形维数之间的线性关系.     

椭圆型边值问题的边界积分-微分方程和它们的数值解

本文给出了椭圆型边值问题边界归化的一个新方法,它将原边值问题归化为边界积分一微分方程。它的优点是能够保持原边值问题的自伴性。基于边界积分一微分方程给出了一个新的边界元方法并得到了数值解的误差估计。本文中的数值例子说明这个方法是很有效的。     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

On a Theorem of Bandt and Wang and Its Extension to <Emphasis Type="Italic">p</Emphasis>2-tiles

We study tilings of the plane by a single prototile with respect to the lattice and to the crystallographic group p2. We are interested in the connection between the neighbors of a tile in the tiling and its topology. We show that lattice and p2-tiles always have at least six neighbors. We characterize self-affine tiles that are homeomorphic to a disk in a rather easy way by the set and number of neighbors of the central tile in the tiling. This extends the work of Bandt and Wang devoted to lattice self-affine disk-like tiles of the plane.     

A Survey on Topological Properties of Tiles Related to Number Systems

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its ‘neighbors’ is discussed.     

Singularity of certain self-affine measures

The self-affine measure associated with an iterated function system and a weight is uniquely determined. The problem of determining whether a self-affine measure is absolutely continuous or singular has been studied extensively in recent years. In the present paper we consider the singularity of certain self-affine measures. We obtain a sufficient condition for such measures being singular. Two applications of this result are given, which extend several known results in a simple manner.     

Porosity of Self-affine Sets

In this paper, it is proved that any self-affine set satisfying the strong separation condition is uniformly porous. The author constructs a self-affine set which is not porous, although the open set condition holds. Besides, the author also gives a C^1 iterated function system such that its invariant set is not porous.     

Spectral analysis of generalized Volterra equation

A generalized Volterra lattice with a nonzero boundary condition is considered by virtue of the inverse scattering transform. The two-sheeted Riemann surface associated with the boundary problem is transformed into the Riemann sphere by introducing a suitable variable transformation. The associated spectral properties of the lattice in single-valued variable was discussed. The constraint condition about the nonzero boundary condition and the scattering data is found.     

Self-affine Lattice Reptiles with Two Pieces in IRn

We prove that for every number n ≥ 1, there is a finite number of affine types of self-affine lattice tilings in ?ⁿ, such that the expansion carries each tile onto the union of two tiles.     

Natural Tiling, Lattice Tiling and Lebesgue Measure of Integral Self-Affine Tiles

In the existing theory of self-affine tiles, one knows thatthe Lebesgue measure of any integral self-affine tile correspondingto a standard digit set must be a positive integer and everyintegral self-affine tile admits some lattice Zⁿ as a translationtiling set of Rⁿ. In this paper, we give algorithms to evaluatethe Lebesgue measure of any such integral self-affine tile Kand to determine all of the lattice tilings of Rⁿ by K. Moreover,we also propose and determine algorithmically another type oftranslation tiling of Rⁿ by K, which we call natural tiling.We also provide an algorithm to decide whether or not Lebesguemeasure of the set K (K+j), j–Zⁿ, is strictly positive.     

Spectra of a class of self-affine measures

In the present paper we will study the spectral property of a class of self-affine measures under the condition of compatible pair. We first answer a question of Dutkay and Jorgensen concerning the relationship between spectral self-affine measure and compatible pair. We then consider the spectra of Bernoulli convolutions and obtain a sharp result which extends the corresponding result of Jorgensen, Kornelson and Shuman. Finally, we provide a structural property for the integer spectrum of a spectral self-affine measure.     

Analysis of lattice Boltzmann boundary conditions

The correct implementation of Navier‐Stokes boundary conditions in the framework of lattice Boltzmann schemes is complicated by the non‐availability of analytical methods to assess the consistency of such discretizations. To close this gap, we propose a simple direct asymptotic analysis which is readily applicable to finite difference discretizations of initial boundary value problems in general and to lattice Boltzmann methods in particular. Results of the analysis applied to the classical lattice Boltzmann scheme with bounce back boundary condition are reported.