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.     

拟对称packing极小Moran集

本文研究了一维Moran集的拟对称packing极小性的问题.利用质量分布原理的方法,获得了直线上一类packing维数为1的Moran集为拟对称packing极小集的结果,推广了参考文献中关于拟对称packing极小性的已知结果.     

关于Subordinator的指数与维数

设X(t)是R^d(d为正整数)中的Levy过程，本文首先对前人所定义的X(t)的各种指数给出了另外一种刻划，事实上它们可以用极限的形式表达．这种刻划使得这些指数的几何意义非常明确．且适合于构造各种各样的例子，若X(t)是一个Subordinator，本文证明了X(t)的象集的填充维数就是X(t)的上指数β，并且举例说明存在Subordinator，它的象集的Hausdorff维数为0而填充维数为1.     

一类d-维齐次Moran集的维数结果

本文构造了一类特殊的d-维齐次Moran集:{m_k^d}-拟齐次完全集,并在一定的条件下得到它们的Hausdorff维数和上盒维数.     

Zd中离散分形指标的线性不变性质

研究欧几里得格 Zd 内离散分形指标的线性不变性质 ,即证明了上、下离散质量维数的线性不变性质 ,离散 Hausdorff维数的线性不变性质以及离散填充维数的线性不变性质 .     

一类复值函数图象的fractal维数

本文讨论一类连续而无处可微的复值函数,给出其图象的box维数与packing维数的表达式。     

齐次Moran集的维数

设μ(|n_k|_k≥1,|c_k|_k≥1)为正整数序列|n_k|_k≥1与正数序列|c_k|_k≥1确定的齐次Moran集的集类.确定了μ中元素的Hausdorff维数的最大值与最小值,并证明对任一介于上述最大值和最小值之间的数s,存在μ中的元素,其Hausdorff维数等于s,前述结论对于填充维数亦成立。还讨论了齐次Moran集的其他性质。     

关于可数点集的盒维数的一些探讨

本文研究了一类可数点集的盒维数的计算问题.通过构造双Lipschitz映射,把原可数点集的盒维数的求解问题转化为求解一类相对简单的可数点集的盒维数.获得了两个单调的可数点集在具有同阶间隔时具有相同的上盒维数和下盒维数的结论.该结论为计算一类可数点集的盒维数提供了方便.     

自相似测度的联合发散点和填充维数(英文)

近年来,发散点引起了数学界的广泛关注.对单测度的发散点,前人已经作了很完备的研究.对有限多个自相似测度的联合发散点的集合,仅其豪斯多夫维数被L.Olsen研究并给出了,然而,我们对其填充维数仍一无所知.因此,本文主要研究联合发散点集合的填充维数.     

概率空间（Ω,f,μ）中关于μ的packing维数与经典的实直线上的packing维...

本文将概率空间（Ω，ｆ，μ）中ｐａｃｋｉｎｇ维数的定义与经典的实直线上的ｐａｃｋｉｎｇ维数的定义相联系，证明了在Ｌｅｂｅｓｇｕｅ情形，对所有的Ａ∈ｆ，关于μ的ｐａｃｋｉｎｇ维数Ｄｉｍμ（Ａ）与被Ｔａｙｌｏｒ和Ｔｒｉｃｏｔ所定义的ｐａｃｋｉｎｇ维数Ｄｉｍ（Ａ）是一致的。Ｂｉｌｌｉｎｇｓｌｅｙ的结果与我们的结果相结合，表明在Ｌｅｂｅｓｇｕｅ情形，关于μ的分形与被Ｔａｙｌｏｒ所定义的分形是一致的。     

Category and dimensions for cut-out sets

In this paper, we study the modified box dimensions of cut-out sets that belong to a positive, nonincreasing and summable sequence. Noting that the family of such sets is a compact metric space under the Hausdorff metric, we prove that the lower modified box dimension equals zero and the upper modified box dimension equals the upper box dimension for almost all cut-out set in the sense of Baire category.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fractal dimensions of fractional integral of continuous functions

In this paper,we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals.Riemann–Liouville integral of a continuous function f(x) of order v(v0) which is written as D~(-v) f(x) has been proved to still be continuous and bounded.Furthermore,upper box dimension of D~(-v) f(x) is no more than 2 and lower box dimension of D~(-v) f(x) is no less than 1.If f(x) is a Lipshciz function,D~(-v) f(x) also is a Lipshciz function.While f(x) is differentiable on [0,1],D~(-v) f(x) is differentiable on [0,1] too.With definition of upper box dimension and further calculation,we get upper bound of upper box dimension of Riemann–Liouville fractional integral of any continuous functions including fractal functions.If a continuous function f(x) satisfying H?lder condition,upper box dimension of Riemann–Liouville fractional integral of f(x) seems no more than upper box dimension of f(x).Appeal to auxiliary functions,we have proved an important conclusion that upper box dimension of Riemann–Liouville integral of a continuous function satisfying H?lder condition of order v(v0) is strictly less than 2-v.Riemann–Liouville fractional derivative of certain continuous functions have been discussed elementary.Fractional dimensions of Weyl–Marchaud fractional derivative of certain continuous functions have been estimated.     

Dimensions for random self–conformal sets

A set is called regular if its Hausdorff dimension and upper box–counting dimension coincide. In this paper, we prove that the random self–conformal set is regular almost surely. Also we determine the dimensions for a class of random self–conformal sets.     

DIMENSION AND DIFFERENTIABILIIY OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, a new iterated function system consisting of non-linear affinemaps is constructed. We investigate the fractal interpolation functions generated bysuch a system and get its differentiability, its box dimension, its packing dimension,and a lower bound of its Hansdorff dimension.     

Dimensions of supercritical branching processes in varying environments

Let ∂Γ be the boundary of a family tree Γ associated with a supercritical branching process in varying environments. In this paper, the Hausdorff dimension, the upper box dimension and the packing dimension of ∂Γ are computed explicitly. In contrast to the (fixed environment) Galton–Watson case, the Hausdorff and upper box dimension may take different values.     

THE REGULARITY OF RANDOM GRAPH DIRECTED SELF-SIMILAR SETS

A set in R^d is called regular if its Hausdorff dimension coincides with its upper box counting dimension. It is proved that a random graph-directed self-similar set is regular a.e..     

Squares and their centers

We study the relationship between the size of two sets B, S ? R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.     

ON THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.