一类准自相似集的研究

本文引入并研究了准自相似集，利用动力系统技巧讨论了其Ｈａｕｓｄｏｒｆｆ维数的上、下界，得到了一类严格准自相似集的Ｈａｕｓｄｏｒｆｆ维数公式并确定了一类由共形映射族所确定的准自相似集的Ｈａｕｓｄｏｒｆｆ维数．     

关于齐次Cantor集的一个注记

设Ｃ是［０,１］上Ｈａｕｓｄｏｒｆｆ测度为正有限的齐次Ｃａｎｔｏｒ集类,本文证明了,这里ｓ是Ｅ的Ｈａｕｓｄｏｒｆｆ维数,Ｈｓ（Ｅ）是Ｅ的ｓ维Ｈａｕｓｄｏｒｆｆ测度,Ｈｓ（Ｅ）的定义见引言,     

算子自相似马氏过程的Hausdorff维数定理

对各向同性,算子自相似马氏过程,本文在恰当的条件下得到了其象集与图集的Ｈａｕｓｄｏｒｆｆ维数。其证明表明该维数的估计与其算子特征值的实部有关。     

保形图递归集的Hausdorff维数和测度

本文确定了保形图递归集的 Ｈａｕｓｄｏｒｆｆ维数,证明了相应的 Ｈａｕｓｄｏｒｆｆ度是正σ－有限的,并且我们给出了 Ｈａｕｓｄｏｒｆｆ测度为正有限的充分必要条件．     

关于自相似集的一个维数定理

本文对严格自相似集，提出了一个比“开集”条件更弱的“可解”条件，并且证明：在可解条件下，自相似集的Ｈａｕｓｄｏｒｆｆ维数及Ｂｏｕｌｉｇａｎｄ维数与其相似维数一致．     

关于自相似集的Hausdorff测度

得到了 Ｈａｕｓｄｏｒｆｆ容度与 Ｈａｕｓｄｏｒｆｆ测度相等的集的充分必要条件．对于满足开集条件的自相似集,验证了它的Ｈａｕｓｄｏｒｆｆ容度与Ｈａｕｓｄｏｒｆ测度相等并给出了它的Ｈａｕｓｄｏｒｆｆ测度的一个便于应用的公式．作为例子,给出了均匀康托集的Ｈａｕｓｄｏｒｆｆ测度的一种新的计算方法,对于Ｋｏｃｈ曲线的Ｈａｕｓｄｏｒｆｆ测度的上限也作了讨论．     

两条相互独立的非对称Cauchy过程轨道的乘积集的分形性质

设Ｘ１,Ｘ２是Ｒ＾ｄ上两条相互独立的非对称Ｃａｕｃｈｙ过程,我们求出了,两条轨道的乘积集的确切Ｈａｕｓｄｏｒｆｆ测度函数ψ（ｈ）＝ｈ＾２／ｌｏｇ＾ｈ,同时ψ（ｈ）,同时ψ（ｈ）也是图集的乘积集的确切Ｈａｕｓｄｏｒｆｆ测度函数,另外,我们还求出了乘积集的Ｈａｕｓｄｏｒｆｆ维数和Ｐａｃｋｉｎｇ维数均为２,从而证明了乘 集仍然是分形集。     

关于一类Ornstein—Uhlenbeck型马氏过程Range的维数

本文我们考虑一类Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ型马氏过程Ｒａｎｇｅ的分形性质，给出了它们的Ｈａｕｓｄｏｒｆｆ维数的上界和下界，此外在文末我们对这类过程的水平集的维数给了估计。     

R~n上分形集的多重维数

本文推广Ｈａｕｓｄｏｒｆｆ测度和维数的概念，引入了被称作为多重维测度和多重维数的概念．文中证明了关于多重维测度的Ｆｒｏｓｔｍａｎ定理，构造了一个例子说明存在一类点集，其Ｈａｕｓｄｏｒｆｆ测度是零或十∞，但其多重维测度是一个正数，并说明了多重维数除第一个分量是正数外，其它分量可以取到任何实数．     

Ornstein—Uhlenbeck过程的重点问题

本文主要讨论了单参数ｄ维Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ过程的重点的存在性和多重时的Ｈａｕｓｄｏｒｆｆ维数，得到了：当ｄ≤３时，以正概率样本轨道具有两重点，且两重时具有Ｈａｕｓｄｏｒｆｆ维数２－ｄ２；当ｄ≤２时，以正概率样本轨道具有三重点，且三重时具有Ｈａｕｓｄｏｒｆｆ维数３－ｄ。     

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

广义向量拟平衡问题解集映射的稳定性

我们在局部凸Hausdorff拓扑向量空间中,讨论了广义向量拟平衡问题解集映射的上半连续性以及闭性,并利用扰动间隙函数证明解集的Hausdorff下半连续性.     

THE DIMENSION FOR RANDOM SUB-SELF-SIMILAR SET

In this article,the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions.Several examples are given at the end.     

The distributional dimension of fractals

In the book [1] H.Triebel introduces the distributional dimension of fractals in an analytical form and proves that： for Г as a non-empty set in R^n with Lebesgue measure ｜Г｜ = 0, one has dimH Г = dimD Г, where dimD Г and dimH Г are the Hausdorff dimension and distributional dimension, respectively. Thus we might say that the distributional dimension is an analytical definition for Hausdorff dimension. Therefore we can study Hausdorff dimension through the distributional dimension analytically. By discussing the distributional dimension, this paper intends to set up a criterion for estimating the upper and lower bounds of Hausdorff dimension analytically. Examples illustrating the criterion are included in the end.     

一个广义Siperpinski地毯的Hausdorff测度

§ 1 .　Introduction　　Infractalgeometry ,thestudyingonthetheoryandcalculationofHausdorffdimensionhasmadegreatprogress.ButasfarasHausdorffmeasureisconcerned ,onlyafewresultsareknown .ZHOUZuo linghasobtainedtheexactvalueofHausdorffmeasureforaSierpinskicar pet(see [3]) .MADong kuihasdevelopedthetechniquesin [3](see [4 ]) .ButineverystepofconstructingtheSierpinskicarpetsin [4 ],theremaininglittlesquaressatisfytheconditionthatfourofthemareatthefourvertexanglesoftheirimmediatepredecessorrespe…     

The gonality of smooth curves with plane models

The Hausdorff measure with fractional index is used in order to define a functional on measurable sets of the plane. A fractal set, constructed using the well-known Von Koch set, is involved in the definition. This functional is proved to arise as the limit of a sequence of classical functionals defined on sets of finite perimeter. Thus it is shown that a natural extension of the ordinary functionals of the calculus of variations leads both to fractal sets and to the fractional Hausdorff measure.     

The Hausdorff dimension of the level sets of Takagi’s function

Recently, Maddock (2006) [12] has conjectured that the Hausdorff dimension of each level set of Takagi’s function is at most 1/2. We prove this conjecture using the self-affinity of the function of Takagi and the existing relationship between the Hausdorff and box-counting dimensions.     

The necessary and sufficient conditions for various self-similar sets and their dimension

We have given several necessary and sufficient conditions for statistically self-similar sets and a.s. self-similar sets and have got the Hausdorff dimension and exact Hausdorff measure function of any a.s. self-similar set in this paper. It is useful in the study of probability properties and fractal properties and structure of statistically recursive sets.