函数系矩阵生成的分形完备簇的豪斯夺夫维数和盒维数

本文给出了函数系矩阵的定义及运算,用函数系矩阵构造了分形完备簇,并确定了相似分形完备簇在开集条件下的Hausdorff维数和Box维数.     

Structural and stability characteristics of agglomerated clusters

We investigated the cohesion of agglomerates formed by sticking two fractal clusters, each cluster having been previously generated by particle aggregation on a 3D lattice. The degree of cohesion of an agglomerate of a given configuration was defined by the number of connections simultaneously established on the two stuck clusters. All the possible nonoverlapping configurations were investigated and the corresponding porosity and brittleness as well as the pore volume and connection frequencies were determined. The numerical study showed the greater internal cohesion of agglomerates issued from sticking of reaction-limited aggregation (RLA) clusters compared to that of diffusion-limited-aggregation (DLA) clusters. DLA and RLA agglomerates presented continuously decreasing pore volume frequency curves, the latter agglomerates being characterised by a greater frequency of large pores. Comparison with typical controlled fragmentation experiments showed the number of connections to be the prevailing factor in the cohesion of aggregates formed in aqueous suspensions under various conditions. Received: 25 January 2001 Accepted: 16 May 2001     

On the size of the algebraic difference of two random Cantor sets

In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Parabolic scaling of tree-shaped constructal network

We investigate the multi-scale structure of a tree network obtained by constructal theory and we propose a new geometrical framework to quantify deviations from scale invariance observed in many fields of physics and life sciences. We compare a constructally deduced fluid distribution network and one based on an assumed fractal algorithm. We show that: (i) the fractal network offers lower performance than the constructal object, and (ii) the constructal object exhibits a parabolic scaling explained in the context of the entropic skins geometry based on a scale diffusion equation in the scale space. Constructal optimization is equivalent to an equipartition of scale entropy production over scale space in the context of entropic skins theory. The association of constructal theory with entropic skins theory promises a deterministic theory to explain and build optimal arborescent structures.     

The upper and lower quantization coefficient for Markov‐type measures

We study the asymptotic quantization error for Markov‐type measures μ on a class of ratio‐specified graph directed fractals E . Assuming a separation condition for E , we show that the quantization dimension for μ of order r exists and determine its exact value in terms of spectral radius of a related matrix. We prove that the ‐dimensional lower quantization coefficient for μ is always positive. Moreover, we establish a necessary and sufficient condition for the ‐dimensional upper quantization coefficient for μ to be finite.     

Talbot effect with Cantor transmittances

We show the Talbot effect for Cantor transmittances, which are obtained as a product superposition of periodic components. The self-images for each periodic component can be superimposed with the self-images or the optical noise corresponding to the remaining components. Due to the integer scaling factor among periodic components, there are also self-images positions for the complete fractal structure.     

potential theory and analytic properties of self-similar fractal and multifractal distributions

By the use of recursion relations and analytic techniques we deduce general analytic results pertaining to the electrostatic potential, moments, and Fourier transform of exactly self-similar fractal and multifractal charge distributions. Three specific examples are given: the binomial distribution on the middle-third Cantor set, which is a multifractal distribution, the uniform distribution on the Menger sponge, which illustrates the added complication of higher dimensionality, and the uniform distribution on the von Koch snowflake, which illustrates the effect of rotations in the defining transformations.     

The fractal interpretation of the weak scattering of elastic waves

Nondestructive methods, in particular the measurement of elastic waves, have become increasingly important in determining the microstructure of many materials in recent years. One of these methods is observing the attenuation of ultrasonic waves of known amplitude and direction, e.g., in granular metals. The waves are exponentially attenuated with distance with a frequency-dependent attenuation fractor. The attenuation factor can be decomposed into two parts: absorption and scattering. Experimentally, the absorption part varies linearly with frequency, while the scattering part has a noninteger power law behavior, the exponent of which is related to the strength of the material. Theoretically, at long wavelengths the exponent is 4 (Rayleigh scattering) while for grain-sized wavelengths it is 2 (diffusive scattering). We relate the attenuation factor to the forward scattering amplitude which is related to the frequency dependence of the scatterers and their cross sections. We attribute the noninteger attenuation exponent to a fractal distribution of grain shapes and sizes.Supported by the Defense Advanced Research Projects Agency.     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.