分形数集自相似新异类型

按照集合特征把分形自相似划分为4类,以前分形数集中出现的自相似可归为第Ⅰ、Ⅱ、Ⅲ类,未见第Ⅳ类。发现一种新的三元数集,具有Ⅳ类自相似性。数集整体外观像3叶片叶轮,叶边有序排列着无数芽枝。芽枝上结有两种不同形状的自相似子集,结在枝梢段的子集具有全集形,结在枝干段的子集具有Mandelbrot集形,呈现一棵树结两种果的现象。两种子集各带有自己的自相似后代,自相似传递方式迥异。Mandelbrot集形子集的后代都是Mandelbrot集形;全集形子集的后代也分两种形,像全集那样遵循枝干段与枝梢段不同的规则。芽枝干梢两段有可辨的交接处,却是无穷精致的极限点。比较多种分形数集的运算规则和形象,得出数算规则小差别可以导致数集形态和自相似特性大差别的结论,也推测分形数集或有自相似的趋向性。文中给出了该三元数集的定义、样集参数和空间位置,附图21幅展示了整体三维形象、两种子集的剖面形态以及局部复杂的自相似结构。

双函复合迭代下分形数集成形试验

此前分形数集已经出现了动物形模样和三维自相似形态,那还只是单个函数的运算结果。这里进行的是两个函数复合迭代运算共同确定一个数集的试验。这种双函数集的形体模样既不是两单函数集的集合交并,也不是它们形体的几何合成,更像是与生物现象类似的数集杂交。双函数集的形态会传承其两单函数集的某些特征,又不同其单函数集而有自己的独特性。发现不仅两函数是而且两函数的复合迭代顺序也是双函数集形态的决定因素,表明分形数集存在同函异序双生子。高级双函数集的函数还有配型条件,对称双函数集的产生须要两函数的单函数集是同类对称数集,自相似双函数集的产生须要两函数的单函数集是同类自相似数集。文中提供了5个具有代表性的迭代函数,31幅附图展示了11个数集及自相似子集的三维形象。通过比对发现双函数集成形与动物杂交现象在多方面都有惊人的一致。推测动物基因结合有类似的顺序原理,生物基因信息不仅记录在基因中而且会记录在基因顺序中。

Generalization of 3D Mandelbrot and Julia sets

In order to further enrich the form of 3D Mandelbrot and Julia sets, this paper first presents two methods of generating 3D fractal sets by utilizing discrete modifications of the standard quaternion algebra and analyzes the limitations in them. To overcome these limitations, a novel method for generating 3D fractal sets based on a 3D number system named ternary algebra is proposed. Both theoretical analyses and experimental results demonstrate that the ternary-algebra-based method is superior to any one of the quad-algebra-based methods, including the first two methods presented in this paper, because it is more intuitive, less time consuming and can completely control the geometric structure of the resulting sets. A ray-casting algorithm based on period checking is developed with the goal of obtaining high-quality fractal images and is used to render all the fractal sets generated in our experiments. It is hoped that the investigations conducted in this paper would result in new perspectives for the generalization of 3D Mandelbrot and Julia sets and for the generation of other deterministic 3D fractals as well.

Vertical derivative conversion of irregular-range potential fi eld data based on improved projection onto convex sets method

Gravity and magnetic exploration areas are usually irregular,and there is some data defi ciency.Missing data must be interpolated before the vertical derivative conversion in the wavenumber domain.Meanwhile,for improved processing precision,the data need to be edge-padded to the length required by the fast Fourier transform algorithm.For conventional vertical derivative conversion of potential fi eld data(PFD),only vertical derivative conversion is considered,or interpolation,border padding,and vertical derivative conversion are executed independently.In this paper,these three steps are considered uniformly,and a vertical derivative conversion method for irregular-range PFD based on an improved projection onto convex sets method is proposed.The cutoff wavenumber of the filter used in the proposed method is determined by fractal model fi tting of the radial average power spectrum(RAPS)of the potential fi eld.Theoretical gravity models and real aeromagnetic data show the following:(1)The fitting of the RAPS with a fractal model can separate useful signals and noise reasonably.(2)The proposed iterative method has a clear physical sense,and its interpolation,border padding error,and running time are much smaller than those of the conventional kriging and minimum curvature methods.

Kinetic Simulation of Fractal Aggregation on Liquid Surfaces

A modified fractal growth model based on the deposition, diffusion, and aggregation (DDA) with cluster rotation is presented to simulate two-dimensional fractal aggregation on liquid surfaces. The mobility (including diffusion, and rotation) of clusters is related to its mass, which is given by D-m = D-0s(-gamma D) and theta(m) = theta(0)s (-gamma theta,) respectively. We concentrate on revealing the details of the influence of deposition flux F, cluster diffusion factor gamma(D) and cluster rotation factor gamma(B) on the dynamics of fractal aggregation on liquid surfaces. It is shown that the morphologies of clusters and values of cluster density and fractal dimension depend dramatically on the deposition flux and migration factors of clusters.

Morphological Variation of Coilia Mystus （Clupeiformes： Engraulidae） in Three Chinese Estuaries

Truss network data were collected and investigated in order to clarify the morphological variation in populations of Coilia mystus from three Chinese estuaries. Nineteen morphometric measurements were made for each individual, and Burnaby＇s multivariate method was used to obtain size-adjusted shape data. The cluster analysis and discriminant analysis were used to discriminate morphological differences among populations. The results indicated that 1） the three populations were clustered into two distinct groups： the first group included Changiiang C. mystus and Zhujiang C. mystus, the last one included Minjiang C. mystus, and 2） discriminant analysis with selected 5 morphological parameters showed that the identification accuracy was between 98.7952% and 100%, and global identification accuracy was 99.2933%. Reproductive isolation and adaption to environmental condition are determinant factors for morphological variation between populations of C. mystus.

The Dense Fractal Sets with the Hausdorff Dimension

In this paper, for any given s (0 ≤s≤1), we construct a Cantor-type set E, such that dimH Es- s and Es is dense in [0,1].

I.I.D.STATISTICAL CONTRACTION OPERATORS AND STATISTICALLY SELF-SIMILAR SETS

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.

Lipschitz equivalence of fractal sets in R

Let T（q, D） be a self-similar （fractal） set generated by {fi（x） = 1/q（（x ＋ di）}^Ni=1 where integer q 〉 1and D = {d1, d2 dN} C R. To show the Lipschitz equivalence of T（q, D） and a dust-iik-e T（q, C）, one general restriction is 79 C Q by Peres et al. [Israel] Math, 2000, 117： 353-379]. In this paper, we obtain several sufficient criterions for the Lipschitz equivalence of two self-similar sets by using dust-like graph-directed iterating function systems and combinatorial techniques. Several examples are given to illustrate our theory.

Self-similar fractals: An algorithmic point of view

This paper studies the self-similar fractals with overlaps from an algorithmic point of view.A decidable problem is a question such that there is an algorithm to answer"yes"or"no"to the question for every possible input.For a classical class of self-similar sets{E b.d}b,d where E b.d=Sn i=1(E b,d/d+b i)with b=(b1,...,b n)∈Qn and d∈N∩[n,∞),we prove that the following problems on the class are decidable:To test if the Hausdorff dimension of a given self-similar set is equal to its similarity dimension,and to test if a given self-similar set satisfies the open set condition(or the strong separation condition).In fact,based on graph algorithm,there are polynomial time algorithms for the above decidable problem.