一类随机Moran集的分形维数

本文研究了随机压缩向量满足一定条件下的随机Moran集的分形维数.利用计算上盒维数的上界和分形维数之间的性质,得到Moran集各种分形维数. 并在一般情形下,给出随机Moran集的上盒维数的上界.     

f.f.p.维数

本文对每个环R定义了同调维数l.f.f.p.D(R),并讨论了该维数与环的弱维数及整体维数之间的关系。同时刻画了l.f.f.p.D(R)为有限的环。此外还计算了可换凝聚局部环的维数f.f.p.D(R)。H.Bass的一个早期结果是本文一主要结果之推论。     

一类分形曲面的精细计盒维数公式

本文研究由一个二变元四阶差分方程边值问题生成的分形曲面的精细计盒维数问题,给出了一个自然的维数公式,若该边值问题的边界上的连续函数的图象的精细计盒维数为γ,则该解曲面的精细计盒维数为（1＋γ）。     

螺线的Steinhaus维数

Ｄｕｐａｉｎ Ｙ,Ｆｒａｎｃｅ Ｍ．Ｍ．和 Ｔｒｉｃｏｔ Ｃ．［１］利用积分几何中的经典的Ｓｔｅｉｎｈａｕｓ定理,引入 Ｓｔｅｉｎｈａｕｓ维数,并研究了螺线的 Ｓｔｅｉｎｈａｕｓ维数与盒维数的关系．本文深入这一研究,对Ｓｔｅｉｎｈａｕｓ维数的值域,单调性等基本性质作了进一步的考察．     

R^d中Hausdorff维数和packing维数的确定

本文目的在于建立确定Ｒ＾ｄ中Ｈａｕｓｄｏｒｆｆ维数ｄｉｍ和ｐａｃｋｉｎｇ维数Ｄｉｍ的两个命题，进而寻求Ｒ＾ｄ中Ｈａｕｓｄｏｒｆｆ维数ｄｉｍ与ｐａｃｋｉｎｇ维数Ｄｉｍ相等的条件；这使得我们能够引入分形测度的测度论定义。     

具有有限X-分解维数的模的同调性质

引入了左R-模M关于可解模类X以及内射余生成子W的同调维数.给出了M的X-分解维数有限的几种刻画,进而讨论了M的这两种维数之间的关系.研究了相对于有限W-分解维数的模的稳定性以及相对于模类X的模的稳定性.     

粗糙面分形计算理论研究进展

为提出一种工程上适用可靠的粗糙面分形维数计算方法,在分形曲线的维数计算方法(码尺法,盒维法)基础上,先后提出了星积分形曲面的维数计算方法、三角形棱柱表面积法、投影覆盖法、立方体覆盖法、改进的立方体覆盖法、分形的增变量描述法等曲面分形维数理论.鉴于上述方法的共有缺陷——获取三维坐标的激光表面仪器的扫描尺度限制,研究者提出了粗糙面图像维数计算理论,包括二值化图像维数、灰度图像维数、RGB图像维数计算理论.最后,本文展望了分形维数计算理论领域内亟待解决的三大问题.     

SG(2,2)上布朗运动的维数性质

本文研究分形集合ＳＧ（２,２）上布朗运动的维数性质,得到了ＳＧ（２,２）上布朗运动的样本图以及象集的Ｈａｕｓｄｏｒｆｆ维数与盒维数。     

有限余生成维数

引入模的有限余生成维数的概念，研究了它的性质．同时，我们探讨了模的有限余生成维数、有限余表现维数和内射维数三之间的关系。     

d维平稳高斯过程极集的充分条件及维数

本文研究了d维平稳高斯过程极集的性质,给出了d维平稳高斯过程广义极性的充分条件,并通过一个特殊的Cantor型集的构造将极集的维数与容度巧妙地结合起来,得到了d维平稳高斯过程非极集的Hausdorff维数的下确界.     

具有无界变差的连续函数研究进展

讨论了具有无界变差的连续函数的结构.首先按照局部结构和分形维数对连续函数进行了分类,给出了相应的例子.对这些具有无界变差的函数的性质进行了初步的讨论.对于新定义的奇异连续函数,给出了一个等价判别定理.基于奇异连续函数,又给出了局部分形函数和分形函数的定义.同时,分形函数又由奇异分形函数、非正则分形函数和正则分形函数组成.相应于不连续函数的情形也进行了简单的讨论.     

Bush型函数的分形维数及其奇异性

本文给出了一类无处可微的连续函数──Ｂｕｓｈ型函数的Ｂｏｘ维数的精确值及其Ｈａｕｓｄｏｒｆｆ维数的下界估计值，同时讨论了Ｂｕｓｈ型函数的奇异性特征．     

一个分形函数的分数阶微积分函数

Based on the combination of fractional calculus with fractal functions, a new type of is introduced； the definition, graph, property and dimension of this function are discussed.     

Dimensions of the coordinate functions of space-filling curves

The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.     

一类分形函数的Weyl-Marchaud分数阶导数的Hausdorff维数

首先介绍广义Weierstrass型函数的Weyl-Marchaud分数阶导数,得到了带随机相位的广义Weierstrass型函数的Weyl-Marchaud分数阶导数图像的Hausdorff维数,证明了该分形函数图像的Hausdorff维数与Weyl-Marchaud分数阶导数的阶之间的线性关系.     

The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus

The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus has been investigated. On some special conditions, the linear connection between them has been proved, and the other case has also been discussed.     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.      

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.

     

On a class of Besicovitch functions to have exact box dimension: A necessary and sufficient condition

Let 1 < s < 2, λ_k > 0 with λ_k → ∞ satisfy λ_k+1/λ_k ≥ λ > 1. For a class of Besicovich functions B(t) = sin λ_kt, the present paper investigates the intrinsic relationship between box dimension of their graphs and the asymptotic behavior of {λ_k}. We show that the upper box dimension does not exceed s in general, and equals to s while the increasing rate is sufficiently large. An estimate of the lower box dimension is also established. Then a necessary and sufficient condition is given for this type of Besicovitch functions to have exact box dimensions: for sufficiently large λ, dim _BΓ(B) = dim _BΓ(B) = s holds if and only if lim_n→∞ = 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.