A new lower bound of the Hausdorff measure of the Sierpinski gasket

For the Sierpinski gasket, by using a sort of cover consisting of special regular hexagons, we define a new measure that is equivalent to the Hausdorff measure and obtain a lower bound of this measure. Moreover, the following lower bound of the Hausdroff measure of the Sierpinski gasket has been achieved H^s（S）≥0.670432,where S denotes the Sierpinski gasket, s = dimn（S） = log23, and H^s（S） denotes the s-dimensional Hausdorff measure of S. The above result improves that developed in .     

三分Sierpinski垫上的Cauchy变换所涉及的辅助函数的一些性质

三分Sierpinski垫上的α(=1+(log_2)/(log_3))维Hausdorff测度的Cauchy变换F(z)和二分的情形一样,表现出一些不平常的几何性质．本文考虑联系到F(z)的几个辅助函数,利用Laplace变换和Dong X．H．和Lau K．S．(2003)中的结果,得到了它们的一些重要性质．这些性质对研究F(z)的几何性质和渐近性质起关键作用．     

Bounds of Hausdorff measure of the Sierpinski gasket

By a new method, we obtain the lower and upper bounds of the Hausdorff measure of the Sierpinski gasket, which can approach the Hausdorff measure of the Sierpinski gasket infinitely.     

Maximum density for the Sierpinski gasket

In the paper we show that there exists a closed convex set attaining the maximum density for the Sierpinski gasket.     

一类Sierpinski地毯的Hausdorff测度的估计

设Sr是压缩比为r(0.250≤r≤0.292)的Sierpinski地毯,该文证明了Sr的Hausdorff测度满足公式:21-s/2≤Hs(Sr)≤2s/2,其中s=-logr4.     

Some results on the upper convex densities for the self-similar sets

In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439-3444].     

关于Sierpinski垫片的Hausdorff测度

本文给出了 Sierpinski垫片的另一构造方法 ,并给出了它的 Hausdorff测度的精确值     

关于两篇有关自相似分形集的Hausdorff测度的论文的注记

指出了发表在《数学的实践与认识》上的两篇关于Sierpinski垫片和Sierpinski锥的Hausdorff测度的论文中主要结论的错误所在,并从若干方面讨论了相关的几个问题.     

经典分形集测度上估的计算机搜索Ⅰ──对典型例子Sierpinski垫片编码技术的剖析

分形集的Hausdorff测度是一个非线性科学的理论课题,至今结果甚少,即使对于一些生成很有规则的经典分形集亦是如此间．SISrpinski垫片就是这样一个经典分形集,但因其预分形的图形尚未被研究者解析地认识,所以对其Hausdorff测度的研究进展很慢．文献问猜测1十年之后,这个猜测才被文献问以估值否定．接着,文献同又得到更好的估值出由于采取了对Sierpinski垫片发生规则的编码技术,从而方便地得到H”（S）的一个上方估值函数o（x）．上面提到的一系列猜测值或上方估值,依次由0（1／2）,o（l／4）,以司对给出,而则给出TH”（S）…     

Haudorff测度与等径不等式

对于:Hausdorff维数为s>0的满足开集条件的自相似集E(?)Rn(n>1),我们引入等径不等式Hs|E(X)≤|X|s,以及使该不等式等号成立而直径大于0的极限集U(?)Rn.这里,Hs|E(·)是限制到集合E上的s维Hausdorff测度,而|X|指集合X在欧氏度量下的直径.当s=n时,n维球是唯一的极限集;当s∈(1,n)时,除去一些反面例子以外,我们对上述等径不等式的极限集的基本性质所知甚少.可以看出,这些不等式与Hs(E)的准确值的计算有密切联系.作为特例,我们将考虑Sierpinski垫片,指出计算这一典型自相似集的In2/In3维Hausdorff测度准确值的困难何在.由此可以大致推想,为什么除去平凡情形以外,至今还没有一个具体的满足开集条件而维数大于1的自相似集的:Hausdorff测度准确值被计算出来.     

一类Sierpinski垫的Hausdorff测度

设S_λ为压缩比为λ(λ≤1/3)的一类Sierpinski垫,s=-log_λ3为S_λ的Hausdorff维数,N为产生S_λ的所有基本三角形的集合.本文使用网测度方法,获得了S_λ的s-维Hausdorff测度的精确值H~s(S_λ)=1,同时证明了H~s(S_λ)可由S_λ关于网N的s-维Hausdorff测度H_N~s(S_λ)确定,获得了S_λ的非平凡的最佳覆盖.     

Backward problems for stochastic differential equations on the Sierpinski gasket

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section 2) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman–Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.     

SOME GEOMETRIC PROPERTIES OF BROWNIAN MOTION ON SIERPINSKI GASKET

Let {X(t),t≥0} be Brownian motion on Sierpinski gasket,The Hausdorff and packing dimensions of the image of a ompact set are studied,The uniform Hausdorff and packing dimensions of the inverse image are also discussed.     

Sierpinski gasket上Brown运动的重点

设（Ｘ（ｔ），ｔ≥０）为Ｓｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ上的Ｂｒｏｗｎ运动，Ｅ为（０，＋∞）中闭子集，本文考虑了何时Ｘ（Ｅ）具有Ｋ重点，给出了一个充分条件和一个必要条件。     

Sierpinski地毯Hausdorff测度的一个初等证明

设Sm为压缩比为1/m（m≥4）的Sierpinski地毯,Sn为产生Sm的第n级基本正方形集合,U为平面点集,U的直径|U|＞0,αn（U）表示Sn中与U相交的基本正方形的个数,本文用初等方法证明了对充分大的n有,从而证明了Sm的s-维Hausdorff测度Hs（Sm）＝2s/2．     

The Hausdorff measure of a class of Sierpinski carpets

In this paper we obtain the exact value of the Hausdorff measure of a class of Sierpinski carpets with Hausdorff dimension no more than 1 and show the fact that the Hausdorff measure of such Sierpinski carpets can be determined by coverings which only consist of basic squares.     

Sierpinski Gasket上Brownian运动的局部时

本文证明了Ｓｉｅｒｐｉｎｓｋｉｇａｓｋｅｔ上Ｂｒｏｗｎ运动局部时的联合连续性。     

Spectral Self-Affine Measures on the Generalized Three Sierpinski Gasket

The self-affine measure μM,Dassociated with an iterated function system{φd(x)=M~(-1)(x + d)}_(d∈D) is uniquely determined. It only depends upon an expanding matrix M and a finite digit set D. In the present paper we give some sufficient conditions for finite and infinite families of orthogonal exponentials. Such research is necessary to further understanding the non-spectral and spectral of μM,D. As an application,we show that the L~2(μM,D) space has infinite families of orthogonal exponentials on the generalized three Sierpinski gasket. We then consider the spectra of a class of self-affine measures which extends several known conclusions in a simple manner.     

经典分形集测度上估的计算机搜索Ⅱ──对典型例子Sierpinski垫片计数技术和格点跟踪技术的剖析

A upper estimate function ν(x) of Hausdorff measure Hs (S) ofSierpinski Gasket is given. A mathematical representation of the upperapproximate value νN (x) to ν(x) and a simple algorithm ofνN (x) based on lattice tracing technique are also derived. As asimple corollary, the estimation Hs(S)≤ min ν15 (n·10-5)=ν15 (0.50783)=0.81794 …is obtained.     

经典分形集测度上估的计算机搜索Ⅱ──对典型例子Sierpinski垫片计数技术和格点跟踪技术的剖析

相同生标题的文献山,通过编码技术导出了义献问给出的Sierninski垫片Hausdorff测度＊”（S的L方估值函数…X）．本文的第一个目的,是利用计数技术,推导出更好的上方估值函数“一及其可计算的过剩近似值。N（）的数学表达式其中l二0．lllZ…2n…,t二k·2－N二0．tltZ…iN,（二0f122··＜N是二进制小数,ZI—…二打．1二0,。三2满足！1＋…十认．1二！。l＝1,＊表示l的整数部分,unitx二1,若xZ地否则unitx二队尽管op00的上述表达式把人们从看图作业和手工计算的状态中解脱出来,但其计算复杂性却仍然是指数爆炸的,对大如10的N…