Some fractal properties of a class of self-similar Markov processes

LetX=(Ω,F,F _t ,X _t , θ _t ,P ^x be a self-similar Markov process on (0, ∞). The exact Hausdorff measure function of the level sets are obtained. An appropriate condition is given under which the self-similar Markov process corresponds to a stable process, and some more fractal properties of the sample path ofX are obtained in this case. Supported by the National Natural Science Foundation of China Huang Lihu: born in March 1972, Ph. D graduate student     

一类长方形sierpinski垫片的Hausdorff测度与上凸密度

在平面上,由长为1,宽为a[1/2≤a≤1]的长方形生成的一类自相似集,也就是一个Sierpinski垫片。在满足强分离条件及维数小于1的条件下,证明了自然覆盖为其实现上凸密度1计算的最好形状,自然覆盖即是最好的覆盖。作为它的直接推论,可以得到该类自相似集的Hausdorff测度的精确值。     

自相似集的一类维函数

设E是由满足开集条件的IFS{fi}in=1生成的自相似集,其中fi为相似映射,相似比为ci,0ci1.已经知道,E的Hausdorff维数,填充维数,盒维数及相似维数相等,而且E具有正有限s维Hausdorff测度及预填充测度.将要证明若g为维函数且满足条件:(1)∑ni=1g(ci)=1;(2)对于由数字{1,2,…,n}生成的任意一个k项序列σ=i1…ik,有g(ci1…cik)=g(ci1)…g(cik),则E具有正有限g-Hausdorff测度及g-预填充测度.     

The convergence of statistically self-similar sets and the upper bound and lower bound of Hausdorff measure

We constructed a class of self-similar sets and proved the convergence in this paper. Besides these, the upper bound and lower bound of Hausdorff measures of them were given too. Supported by the National Natural Science Fundation and the Doctral Programme Fundation of China Hu Dihe: born in May 1935, Professor     

The construction of statistically self-similar measures

We have studied statistically self-similar measures together with statistically self-similar sets in this paper. A special kind of statistically self-similar measures has been constructed and a class of statistically self-similar sets as well. Supported by the National Natural Science Fundation and the Doctral Programme Fundation of China Hu Dihe: born in May 1935, Professor     

The pointwise dimension of self-similar measures

The problem about the existence of pointwise dimension of self-similar measure has been discussed. The Hausdorff dimension of the set of those points for which the pointwise dimension does not exist equals the dimention of support of the self-similar measure.     

Some Results about the Sample Path Properties of Markov Processes with Independent Self-Similar Components

This paper considers a special class of operator self-similar processes Markov processes {X（t）, t≥0} with independent self-similar components, that is, X （ t ） =（X^1（t）,…,X^d（t））, where {X^i（t）,t≥0}, i=1,2,…,d are d independent real valued self-similar Markov processes. By means of Brel-Cantelli lemma, we give two results about asymptotic property as t→∞ of sample paths for two special classes of Markov processes with independent self-similar components.     

自相似测度对一些参数的连续依赖性

证明自相似测度对迭代函数系统的压缩因子、平移向量以及概率向量在Hutchinson度量意义下是连续依赖的.     

一个特殊自相似分形集的Hausdorff测度的下界估计

利用自相似分形的结构性质和质量分布原理,通过定义支撑在分形集上恰当的质量分布,具体地分析了直径在不同的分区内的可测集的直径大小与分布在其上的质量多少之间的关系,得到了一个由Falconer提出的特殊分形集Hausdorff测度的下界估计,HS(F)≥0.807 758 0.     

Hausdorff measure of Sierpinski carpet

The estimate of Hausdorff measure H' (F) of Sierpinski carpet F with Hausdorff dimension s =logS/log3 is derived as Hs(F)≤55102s--864855992=1.089147….     

一类算子稳定过程图集的Hausdorff测度

设X={X（t）,t∈R＋}是取值R^d上的,指数为对角矩阵的算子稳定过程．文章将探讨X的图集的Hausdorff测度问题．更确切地,利用逗留时得到了图集G（[0,1]）={（t,X（t））：0≤t≤1}的确切Hausdorff测度函数．这一结果推广了已有文献中类似的结果．     

局部自相似过程的小波估计

首先, 针对时变自相似参数提出了一种基于最大重叠离散小波变换的估计方法; 对此进行了蒙特卡洛模拟研究, 发现提高了估计的精度; 最后, 将研究结果应用于海洋垂直切变序列.     

关于一类自相似集的上密度

研究了自相似集上的Hausdorff测度H的上s密度,尤其对于E是一个自相似集,且满足条件c^1＋1-3c＋1=0,l∈N,c是相似集E上的比例常数,获得了一类自相似集E上的Hausdorff测度的精确上s密度．     

The structure and the sufficint and necessary conditions for generalized statistically self-similar sets

We constructed a class of generalized statistically self-similar sets and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson, Falconer, Graf are the special cases of ours. Foundation item: Supported by the National Natural Science Foundation of China Biography: DENG Ai-jiao (1974-), female, Ph.D. candidate, research interes: is in stochastic process and random ractal     

Kock曲线的Hausdorff测度

对 Kock曲线的 Hausdorff测度进行了估计 ,并给出了一个公式 .由此公式 ,得到了 Kock曲线的Hausdorff测度的上界估计 ,并推翻了关于它的一个猜测 .     

一个特殊分形集的Hausdorff测度估计

利用满足开集条件的自相似分形的性质，得到一个特殊分形Hausdorff测度的上界估计公式。由此公式，对它的Hausdorff测度的上界进行了估计，并用两种方法估计了它的Hausdorff测度的下界。     

Exact Hausdorff measure of the lever sets of a multi-parameter stable process

Let X(t)be a(N,d,α)stable process of type A with Nα＞ d.The exact Hausdorff measure of the lever sets of X - 1(x)is evaluated.