一种求吸引子（IFS）分维的有效方法

众所周知,分形集的维数在分形几何中是相当重要的一个概念。如何精确求得分形集的维数,已有很多学者进行了研究。本文所讨论的是如何求迭代函数系(IFS)吸引子的分形集的维数。一大类分形集可定义如下:(X,d)是一个完备的距离空间,若X上的N个     

由复映射z←zα+c(α<0)所构造的广义M-集的研究

阐述了构造复映射z←z^α c(α<0)所构造的广义Mandelbrot集（简单称M－集或M－分形图）的逃逸时间算法。通过改变参数α,作出了一系列有趣的分形图,这些分形图为若干卫星群环绕中央恒星的星群结构,定量地研究了恒星和卫星群的几何结构,并对α取非整数时分形图的结构特点和卫星群胚胎出现的原因进行了分析,最后给出几点结论。     

正实数维分形集的构造方法

给出了测量一类分形集维数的简单方法. 根据这种测量方法, 可以构造出任意实数维分形集,并且分形集可以不是自相似的.     

分形集的拟一致不连通性

一致不连通性在分形几何的研究中起重要作用.本文讨论了拟一致不连通性:对于一类由数列定义的集合,得到其(拟)一致不连通性成立的充要条件;还引入了一类缓变莫朗集,得到其拟一致不连通性成立的一个充分条件.     

由复映射z←z~α十c(α<0)所构造的广义M-集的研究

阐述了构造复映射z←zα＋c（α＜0）所产生的广义Mandelbrot-集（简称M-集或M-分形图）的逃逸时间算法．通过改变参数a，作出了一系列有趣的分形图．这些分形图为若干卫星群环绕中央恒星的星群结构．定量地研究了恒星和卫星群的几何结构，并对a取非整数时分形图的结构特点和卫星群胚胎出现的原因进行了分析．最后给出几点结论．     

广义Kuramoto—Sivashinsky方程吸引子的分形结构

首先给出广义Ｋｕｒａｍｏｔｏ＿Ｓｉｖａｓｈｉｎｓｋｙ（ＧＫＳ）方程周期初边值问题在Ｈ２空间惯性集的构造，进而给出并证明ＧＫＳ方程吸引子的分形结构，同时发现吸引子的一个分形局部化指数型逼近序列·上述结果精细和推进了［１，３，５，７］关于惯性集和吸引子的结论，刻划了吸引子的一种几何结构     

由几何反演导出的分形几何

本文介绍了由几何反演而导出的自反分形概念.构造了一个对称的,分叉自反分形曲线和一个自反分形尘集.推广了分形密切的概念而且提出了一个新的概念——分形包络.最后给出了“肥皂”和“鸡蛋”的两个自反分形的实例.     

随机康托集的分形性质及分形渗流过程

本文综合介绍了两种随机康托集的分形性质和与之相关的随机过程的一些研究成果.     

随机康箍集的分形性质及分形渗流过程

本文综合介绍了两种随机康托集的分形性质和与之相关的随机过程的一些研究成果。     

弱分离条件下的自共形迭代函数系统

开集条件是分形几何的一个重要概念,弱分离条件(WSC)在研究有重叠的迭代函数系统(IFS)中扮演着重要角色.本文考虑满足弱分离条件的自共形迭代函数系统,并给出确定其不变集的Hausdorff维数的一种方式.     

分形统计测度构建及其在投资组合中的应用研究

准确测量证券的风险和收益无论是对投资管理,还是对金融理论研究,甚至对理论成果向实践应用转化都至关重要。本文在证券价格具有分形特征的现实背景下,基于分形理论构建了分形期望和分形方差两个分形统计测度,以克服非分形统计测度在风险收益方面测不准或不可测的缺陷。在此基础上,应用分形统计测度构建了投资组合模型,给出了分形组合模型的解析解;随后,利用实证分析验证了分形统计测度在投资组合应用中的有效性。本文创新之处在于针对证券价格具有分形特征的现实背景构建了分形期望和分形方差两个分形统计测度;并基于分形统计测度构建了投资组合模型,将证券价格普遍存在的分形特征纳入投资组合的研究框架。     

分形插值曲面理论及其应用

本文叙述了分形曲面的生成原理，给出了分形插值曲面的计算公式，证明了分形插值曲面迭代函数系唯一性定理，导出了分形插值曲面的维数定理，并应用实际数据进行了分形插值曲面的实例研究·     

Design and analysis of fractal detector for high resolution radars

Recently, fractal geometry has been used as a tool for improving the detection of targets in radar systems. The fractal dimension is utilized as a feature to distinguish between target and clutter in fractal detectors. In this paper, a general model is proposed for the target and clutter signals in high resolution radar (HRR). The fractal dimensions of the clutter and the target plus clutter are evaluated. Performing statistical tests on the distribution of the fractal dimension, it is proved that a gaussian distribution can approximately model the distribution of the fractal dimension for HRR signals. The fractal detector is designed based on the gaussian distribution of the fractal dimension and its performance is compared with a semi-optimum detector. It is demonstrated that the fractal detector has great capabilities in the rejection of colored clutter. Moreover, we show that the fractal detector is CFAR, i.e., the probability of false alarm remains approximately constant in different interference powers.     

Structure function and spectral density of fractal profiles

Spectral density and structure function for fractal profile are analyzed. It is found that the fractal dimension obtained from spectral density is not exactly the same as that obtained from structure function. The fractal dimension of structure function is larger than that of spectral density for small fractal dimension, and is smaller than that of spectral density for larger fractal dimension. The fractal dimension of structure function strongly depends on the spectral density at low and high wave numbers. The spectral density at low wave number affects the structure function at long distance, especially for small fractal dimension. The spectral density at high wave number affects the structure function at short distance, especially for large fractal dimension. This problem is more serious for bifractal profiles. Therefore, in order to obtain a correct fractal dimension, both spectral density and structure function should be checked.     

Construction of fractal surfaces by recurrent fractal interpolation curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system (RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.     

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.     

Closed contour fractal dimension estimation by the Fourier transform

This work proposes a novel technique for the numerical calculus of the fractal dimension of fractal objects which can be represented as a closed contour. The proposed method maps the fractal contour onto a complex signal and calculates its fractal dimension using the Fourier transform. The Fourier power spectrum is obtained and an exponential relation is verified between the power and the frequency. From the parameter (exponent) of the relation, is obtained the fractal dimension. The method is compared to other classical fractal dimension estimation methods in the literature, e.g., Bouligand–Minkowski, box-counting and classical Fourier. The comparison is achieved by the calculus of the fractal dimension of fractal contours whose dimensions are well-known analytically. The results showed the high precision and robustness of the proposed technique.     

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.     

铜基复合材料组织形态分形特征的统计分析与研究

通过对铜基复合材料显微组织结构相图的分析和研究,根据分形理论,计算了不同实验条件下铜基复合材料横截面和平行压制力面的显微组织结构相图的分形维数,同时结合统计方法分析了铜基复合材料分形维数的一些统计特性,结果表明,分形维数反映了石墨在样品中的分布规律,分形维数越大,组织结构相图越复杂,石墨分布越不规则,故石墨分布的不规则性可用分形维数来刻画,分形维数可作为材料组织形态分析的一个表征参数,通过统计分析可知,铜基复合材料横截面和平行压制力面的组织结构相图的分形维数服从正态分布,且横截面和平行压制力面的分形维数随石墨含量变化的情况互不影响。     

Describing some characters of serine proteinase using fractal analysis

In this paper we calculated the fractal dimensions of four proteins, chymotrypsin, elastase, trypsin and subtilisin, which are made up of about 220–275 amino acids and belong to the family of serine proteinase by using three definitions of fractal dimension i.e. the chain fractal dimension (D_L), the mass fractal dimension (D_m) and the correlation fractal dimension (D_c). We also analyzed the relationship between fractal dimension and space structure or secondary structure contents of proteins. The results showed that the values of fractal dimensions are almost same for the global mammalian enzymes (chymotrypsin, elastase and trypsin), but different for the global subtilisin. This demonstrated that the more similar structures, the more equal fractal dimensions, and if the fractal dimensions of proteins are different from each other, the three dimensional structures should not be similar. On the other hand, the detailed structures and fractal dimensions of the active sites of four enzymes are extraordinarily similar. Therefore, the fractal method can be applied to the elucidation of the proteins evolution.