Quantization dimension for infinite self-similar probabilities

The quantization dimension function for a probability measure induced by a set of infinite contractive similarity mappings and a given probability vector is determined. A relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established. The result in this paper is an infinite extension of Graf and Luschgy [S. Graf, H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002) 103-109].     

中国突发性地质灾害分布的统计规律

本文根据地域间突发性地质灾害发生的相似性,将全国区划为9个自然区域;然后依据统计学的大数原理,建立了概率模型分析地质灾害在9个自然区域的分布特点以及隶属于各自然区域的所有省、市和自治区的地质灾害分布特征。结果显示不同类型与不同的区域之间存在着不同程度的强相依性,分布格局明显。     

Induced polarization forward modelling using finite element method and the fractal model

The modeling of induced polarization data from 2D and 3D geological medium was developed. We applied the finite element method and the fractal model to complex resistivity of rocks as an intrinsic electrical property of geological medium. The LU factorization and pre-conditioned complex bi-conjugated gradient method were applied to solve the complex linear equation system. The modeling implemented in this work was efficient and prompt. In addition, the induced polarization survey using the fractal model as intrinsic property of the medium is a promising method for environmental investigation.     

The global dimensions of crossed products and crossed coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.     

A topological phase transition between small-worlds and fractal scaling in urban railway transportation networks?

Fractal and small-worlds scaling laws are applied to study the growth of urban railway transportation networks using total length and total population as observational parameters. In spite of the variety of populations and urban structures, the variation of the total length of the railway network with the total population of conurbations follows similar patterns for large and middle metropolis. Diachronous analysis of data for urban transportation networks suggests that there is second-order phase transition from small-worlds behaviour to fractal scaling during their early stages of development.     

Precise (m,k)-Zipf diagram analysis of mathematical and financial time series when m=6, k=2

For studying short-range time correlations in financial signals, we have envisaged to combine the Zipf method and the i-variability diagrams (VD) as useful tools. The 2-VD describes the local curvature short-range correlations. We have resulted into ranking the 2-VD data according to their frequency of occurrence. After having tested the ideas and estimated the error bars on a Brownian motion signal, we have examined two stocks, i.e. SGP and OXHP closing price and volume of transaction long series. A precise (m,k)-Zipf diagram analysis when m=6, k=2 has been shown to lead to a non-immediate information on the signal behavior, even taking into account error bars. The set of curvatures (translated into “words”) indicates a Brownian motion-like set for the closing price local curvature of such signals over a 6 day span. Moreover, it has been shown that the conjecture about a simple relationship between the Hurst exponent H and the ζ exponent of Zipf plots does not seem to be substantiated here.     

Examples of entropy generating sequence

We introduce the notion of entropy generating sequence for infinite words and define its dimension when it exists. We construct an entropy generating sequence for each symbolic example constructed by Cassaigne such that the dimension of the sequence is the same as its topological entropy dimension. Hence the complexity can be measured via the dimension of an entropy generating sequence. Moreover, we construct a weakly mixing example with subexponential growth rate.     

Orthonormal bases generated by Cuntz algebras

We show how some orthonormal bases can be generated by representations of the Cuntz algebra; these include Fourier bases on fractal measures, generalized Walsh bases on the unit interval and piecewise exponential bases on the middle third Cantor set.