本原代换的非周期不动点的Hausdorff维数函数

设 Ａ＝｛０, １,…, Ｎ－１｝,ξ是 Ａ上的一个本原代换, ｕ是ξ的非周期不动点．记 Ｘ（ξ）＝Ｏｒｂ（ｕ） 本文证明了：Ｘ（ξ）的Ｈａｕｓｄｏｒｆｆ维数函数是ｈ（ｘ）＝－ｌｏｇＮ／ｌｏｇｘ     

二维带形无界区域中Navier－Stokes方程整体吸引子及其维数估计

该文讨论二维无界带形区域中Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程其中Ω＝（０，ｄ）×Ｒ，ｄ＞０为一常数，ｕ与ｐ为未知量，其中ｕ＝（ｕ１，ｕ２）为速度场，ｐ表示压力．我们证明了当ｕ０∈Ｈ，ｆ∈Ｖ且ｆ［ｌｏｇ（ｅ＋｜ｘ｜２）］１／２∈Ｌ２（Ω）时，问题（Ｉ）在Ｈ中存在整体吸引子Ａ，它是的一个子集．对Ａ的Ｈａｕｓｄｏｒｆｆ维数与Ｆｒａｃｔａｌ维数我们也给出了估计．     

关于自相似集的一个维数定理

本文对严格自相似集，提出了一个比“开集”条件更弱的“可解”条件，并且证明：在可解条件下，自相似集的Ｈａｕｓｄｏｒｆｆ维数及Ｂｏｕｌｉｇａｎｄ维数与其相似维数一致．     

一类递归集Hausdorff维数及Bouligand维数

本语言给出了递归集的Ｈａｕｓｄｏｒｆｆ维数的下界估计，并由此确定了一类递归集的维数，所获结果包含并推广了Ｂｅｄｆｏｒｄ，Ｄｅｋｋｉｎｇ及文志英，钟红柳等人的有关结果。     

一类递归集的Hausdorff维数及Bouligand维数

本文给出递归集的Ｈａｕｓｄｏｒｆｆ维数的下界估计，并由此确定了一类递归集的维数，所获结果包含并推广了Ｂｅｄｆｏｒｄ，Ｄｅｋｋｉｎｇ及文志英、钟红柳等人的有关结果。     

Belousov－Zhabotinskii化学反应Field－Noyes模型的整体吸引子和惯性流形

本文讨论Ｂｅｌｏｕｓｏｖ—Ｚｈａｂｏｔｉｎｓｋｉｉ化学反应Ｆｉｅｌｄ—Ｎｏｙｅｓ模型（三维的方程组）整体吸引子的存在性、维数估计以及惯性流形的存在性．     

一类随机级数图像的Hausdorff和Bouligand维数

本文确定了一类取复值的随机级数的图像的Ｈａｕｓｄｏｒｆｆ和Ｂｏｕｌｉｇａｎｄ维数．     

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．     

Canto型集合C^m（i1,i2,…ik）的Hausdorff维数和Hausdorf…

证明了较一般的Ｃａｎｔｏｒ型集合Ｃ＾ｍ（ｉｄ１，ｉ２，…ｉｋ）的Ｈａｕｓｄｏｒｆｆ维数和它的Ｂｏｕｌｉｑａｎｄ维数，自要上似维数一致，并且给出了估计Ｃ＾ｍ（ｉ１，…ｉｋ）的Ｈａｕｓｄｏｒｆｆ测度上界的方法。     

一类准自相似集的研究

本文引入并研究了准自相似集，利用动力系统技巧讨论了其Ｈａｕｓｄｏｒｆｆ维数的上、下界，得到了一类严格准自相似集的Ｈａｕｓｄｏｒｆｆ维数公式并确定了一类由共形映射族所确定的准自相似集的Ｈａｕｓｄｏｒｆｆ维数．     

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.     

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.      

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.     

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.     

Box dimension and fractional integral of linear fractal interpolation functions

In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs.     

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.     

Box Dimension of Weyl Fractional Integral of Continuous Functions with Bounded Variation

We know that the Box dimension of f(x) ∈ C~1[0,1] is 1. In this paper, we prove that the Box dimension of continuous functions with bounded variation is still1. Furthermore, Box dimension of Weyl fractional integral of above function is also 1.     

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.