On the wavelet transformation of fractal objects

The wavelet transformation is briefly presented. It is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half-plane near the boundary of its domain of analyticity. As an example, a Weierstrass-like fractal function is considered, for which the wavelet transform is related to a Jacobi theta function. Some of the scalings of this theta function are analyzed, and give some information about the scaling behavior of this fractal.     

Dirichlet Spaces Connected by Points and Application to Diffusions on Finitely Ramified Fractals

We illustrate the results of [13] by giving some computation methods and some explicite computations of the renormalization map involved in the construction of a diffusion on a finitely ramified self-similar set. Using the inverse of the Dirichlet form (defined on the set of finite 0-energy measures) we show that in some cases the map is linear and given by a nonnegative matrice. We also use the representations of the group of symmetries of the fractal to compute T in the coordinates associated with the eigenvalues of the Dirichlet form. In this way we give the meaning in terms of representations of groups of the change of variable made in [1]. We explicitely compute T for the snowflake and we are able to give an approximated value of the fixed point, and so of the probabilities of transition of the diffusion.     

Sierpinski Carpet格点图上边渗流及其临界现象

本文讨论了一种非平移不变格点图──格点分形图上的边渗流及其临界现象。     

Fractal approach of nucleation in liquid and gas phases

Summary The authors present a generalised fractal treatment of the nucleation from the liquid solution or gas-phase reactant.     

相应于随机自相似分形的记忆函数和分数次积分

For a physics system which exhibits memory, if memory is preserved only at points of random self-similar fractals, we define random memory functions and give the connection between the expectation of flux and the fractional integral. In particular, when memory sets degenerate to Cantor type fractals or non-random self-similar fractals our results coincide with that of Nigmatullin and Ren et al. .     

The strong open set condition for self-conformal random fractals

We prove that the open set condition and the strong open set condition are equivalent for self-conformal random fractals.

     

Completeness of the Space of Separable Measures in the Kantorovich-Rubinshtein Metric

We consider the space M(X) of separable measures on the Borel σ-algebra ?(X) of a metric space X. The space M(X) is furnished with the Kantorovich-Rubinshtein metric known also as the “Hutchinson distance” (see [1]). We prove that M(X) is complete if and only if X is complete. We consider applications of this theorem in the theory of selfsimilar fractals.     

Theory of self-avoiding walks on percolation fractals

A phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self-avoiding walks (SAW) is used to derive a new expression for the Flory exponent describing the average radius of gyration of SAWs on fractals. We focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of SAWs. The mean-field version of our new formula corresponds to the Aharony and Harris expression, who used the standard Flory approach for its derivation.On leave from Santipur College, Nadia 741404, India.     

谢宾斯基分形结构声散射特性与视觉偏好评价*

声扩散界面设计应兼顾声学和美学需求。结合参数化设计方法,谢宾斯基分形的模块化结构具有灵活生成个性化界面的潜力。然而,针对分形结构声学特性与视觉效益尚缺乏综合的量化研究。本文根据谢宾斯基分形的构造规则提出了一种宽频的扩散结构设计方法,分别从声学性能与视觉偏好的角度,研究模块形状、模块高度和随机排布对性能的影响。采用边界元素法的数值仿真与实测验证、结合虚拟仿真技术的主观调查、以及多目标综合分析,探究分形设计要素的作用机制和最佳阈值范围。结果表明,与相同体量的传统二次残差扩散体相比,具有不同平面尺寸和高度模块的分形结构改善了扩散性能,特别是在1 kHz以下的中低频范围。此外,发现在室内使用中高复杂度（迭代三次）、低随机排布、谢宾斯基三角截锥分形结构,有助于提高居住者对环境的偏好度。