随机网分形的多重分形分解

本文讨论了随机网分形的多重分形分解．计算了随机网分形上任意随机自相似测度的多重分形谱,     

一类自相似集重分形分支的量纲

本文首先定义具有量纲函数的重分形测度,然后证明当Euclid空间中的两个重分形测度具有等价的量纲函数时,它们也等价.进一步,对于直线上满足强分离条件(SSC)的自相似集,在某些加倍条件下,本文给出判断其重分形分支的量纲函数的充要条件.     

关于自相似集的Hausdorff测度的一个判据及其应用

讨论了满足开集条件的自相似集。对于此类分形，用自然覆盖类估计它的Hausdorff测度只能得到一个上限，因而如何判断某一个上限就是它的Hausdorff测度的准确值是一个重要的问题。本文给出了一个判据。作为应用，统一处理了一类自相似集，得到了平面上的一个Cantor集－Cantor尘的Hausdorff测度的准确值，并重新计算了直线上的Cantor集以及一个Sierpinski地毯的Hausdorff测度。     

控制系统中的分形

本文将整数维与分形的Hausdorff测度引入并应用于控制系统,同时也介绍了准自相似集这个新概念,证明了这种集合的存在性与唯一性.并将计算自相似集维数的公式推广到准自相似集,在此基础上,说明了控制系统的可达集可以具有分数维.表明在分析非线性系统可控性与可观性时,分形几何学也将是一种有意义的工具.     

Sierpinski地毯上自相似测度以及Markov测度的加倍性质

众所周知,当自相似集满足强分离条件时,其上的自相似测度都是加倍的.本文进一步证明在强分离条件下,自相似集上的Markov测度都是加倍的.随后,本文讨论了Sierpinski地毯S上自相似测度及Markov测度的加倍性质.当S不满足强分离条件时,将S分为不同的类型,完全刻画了S上加倍的自相似测度及加倍的Markov测度.     

均匀三部分康托集的Hausdorff中心测度

均匀三部分康托集K(λ,3)是满足开集条件的自相似分形集．本文通过一个概率测度μ在点x的上球密度的计算给出了K(λ,3)的s维Hausdorff中心测度的精确值,其中s=logλ1/3是K(λ,3)的Hausdorff维数．     

随机分形的测度性质

本文主要介绍随机过程样本轨道、Ｈａｗｋｅｓ模型、统计自相似集、统计自仿射集的测度性质，同时也将介绍一些离散分形的结果．文中还列出一些尚未解决的问题．     

相伴于随机自相似分形的随机测度的强测度的收敛性(英文)

构造了随机自相似分形及其上的记忆函数,并得出了有关结论,在此基础上,我们可以定义一个随机概率测度dΦn(τ)=Kn(τ)dτ,Φn(τ)弱收敛于Φ,进一步可得到强测度序列Ψn(.)=EΦn(.),则{Ψn}弱收敛于Ψ=EΦ.     

关于自保形测度与Lebesgue测度的等价性

本文我们研究了自保形测度与Lebesgu测度的关系,对Yuvla Peres等的结果进行了推广,证明了自相似测度要么是奇异的,要么关于Lebesgue测度 绝对连续的,并且若将Lebesgue测度限制在自相似测度的紧支撑上,则其关于非奇异的自相似测度是绝对连续的。     

区间值直观模糊集的相似测度和距离测度

对现有的模糊集和直观模糊集的相似测度和距离测度的公理化定义进行分析,并做出改进;然后提出区间值直观模糊集的相似测度和距离测度的公理化定义,并各引入它们的一种计算方法;最后给出区间值直观模糊集的相似测度和距离测度在模式识别中的一个应用实例.     

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.     

A Local Property of Hausdorff Centered Measure of Self-Similar Sets

We analyze the local behavior of the Hausdorff centered measure for selfsimilar sets. If E is a self-similar set satisfying the open set condition, then Cs(E∩B(x,r)) ≤(2r)s for all x ∈ E and r 0, where Csdenotes the s-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.     

A lower bound for the symbolic multifractal spectrum of a self-similar multifractal with arbitrary overlaps

By now the multifractal structure of self-similar measures satisfying the so-called Open Set Condition is well understood. However, if the Open Set Condition is not satisfied, then almost nothing is known. In this paper we prove a nontrivial lower bound for the symbolic multifractal spectrum of an arbitrary self-similar measure. We emphasize that we are considering arbitrary self-similar measures (and sets) which are not assumed to satisfy the Open Set Condition or similar separation conditions. Our results also have applications to self-similar sets which do not satisfy the Open Set Condition (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the exact Hausdorff measure of a class of self-similar sets satisfying open set condition

In this paper,we provide a new effective method for computing the exact value of Hausdorff measures of a class of self-similar sets satisfying the open set condition(OSC).As applications,we discuss a self-similar Cantor set satisfying OSC and give a simple method for computing its exact Hausdorff measure.     

A construction of singular overlapping asymmetric self-similar measures

Summary In [8] we found a class of overlapping asymmetric self-similar measures on the real line, which are generically absolutely continuous with respect to the Lebesgue measure. Here we construct exceptional measures in this class being singular.     

自相似集的质量分布原理与Hausdorff测度及其应用

本文提出了满足开集条件的自相似集的质量分布原理.作为应用,得到了计算一类满足开集条件的自相似集的Hausdorff测度的准确值的方法,并举例说明了此方法对于计算一类满足开集条件的自相似集的Hausdorff测度的准确值是行之有效的.     

关于广义齐次自相似集上Hausdorff测度的一个特征

设E是Hausdorff测度正有限的广义齐次自相似集,本文证明了s维Hausdorff测度是E上唯一的非扩张概率测度.     

Quantization dimension of random self-similar measures

In this paper, we study the quantization dimension of a random self-similar measure μ supported on the random self-similar set K(ω). We establish a relationship between the quantization dimension of μ and its distribution. At last we give a simple example to show that how to use the formula of the quantization dimension.     

ON THE SELF-SIMILAR SOLUTIONS OF THE MAGNETO-HYDRO-DYNAMIC EQUATIONS

In this paper, we show that, for the three dimensional incompressible magnetohydro-dynamic equations, there exists only trivial backward self-similar solution in L^p（R^3） for p ≥ 3, under some smallness assumption on either the kinetic energy of the self-similar solution related to the velocity field, or the magnetic field. Second, we construct a class of global unique forward self-similar solutions to the three-dimensional MHD equations with small initial data in some sense, being homogeneous of degree -1 and belonging to some Besov space, or the Lorentz space or pseudo-measure space, as motivated by the work in [5].