Fractals

For any continuous probability measure μ on ℝ we construct an IFS with probabilities having μ as its unique measure-attractor.

The multifractal structure of one dimensional charged particle density distribution in ²⁸Si-Ag/Br interactions at 14.5 GeV per nucleon is investigated by using two different techniques. The experimental measurements are compared with a microscopic transport model of particle production based on the Ultra-relativistic Quantum Molecular Dynamics (UrQMD). Various parameters related to multifractality, for example the Lévy's index, are obtained. Our analysis shows that multifractal structure is present both in the experiment as well as in the simulation. As far as the self-similar nature of the density fluctuation is concerned, there exists, however, a small but definite quantitative difference between the two.

In this paper, we compute the Hausdorff dimension of a graph-directed set when the underlying multigraph is a Cartesian product or a tensor product of several multigraphs. We give explicit formulas in terms of the eigenvalues of the graph and the similarity ratios used with each graph.

A human-sized microwave invisibility cloak has been realized. The invisibility cloak uses fractal metamaterials with two cloaking layers to achieve a high fidelity re-attainment of the intensity of an unobstructed direct path over a better than 50% bandwidth. A human subject was cloaked demonstrating a new milestone in diverted imaging capabilities: electrically large; high fidelity; and broad bandwidth. Transformational optics must now be considered less limiting in the guidance of practical applications.

In this paper, the complexity-entropy causality plane approach is applied to analyze traffic data. The R/S analysis and detrended fluctuation analysis (DFA) methods are also used to compare with this approach. Moreover, based on the concept of entropy, we propose to use permutation to calculate the probability distribution of the time series when applying the representation plane. The empirical results indicate that traffic dynamics exhibit different levels of traffic congestion and demonstrate that this statistical method can give a more refined classification of traffic states than the R/S analysis and DFA.

The study of d dimensional traces of functions of m several variables leads to directional behaviors. The purpose of this paper is two-fold. Firstly, we extend the notion of one direction pointwise Hölder regularity introduced by Jaffard to multi-directions. Secondly, we characterize multi-directional pointwise regularity by Triebel anisotropic wavelet coefficients (resp. leaders), and also by Calderón anisotropic continuous wavelet transform.

The object under study is a particular closed and simple curve on the square lattice ℤ² related with the Fibonacci sequence F_n. It belongs to a class of curves whose length is 4F_3n+1, and whose interiors tile the plane by translation. The limit object, when conveniently normalized, is a fractal line for which we compute first the fractal dimension, and then give a complexity measure.

Natural data sets, such as precipitation records, often contain geometries that are too complex to model in their totality with classical stochastic methods. In the past years, we have developed a promising deterministic geometric procedure, the fractal-multifractal (FM) method, capable of generating patterns as projections that share textures and other fine details of individual data sets, in addition to the usual statistics of interest. In this paper, we formulate an extension of the FM method around the concept of "closing the loop" by linking ends of two fractal interpolating functions and then test it on four geometrically distinct rainfall data sets to show that this generalization can provide excellent results.

We introduce a new method, multifractal cross-correlation analysis based on statistical moments (MFSMXA), to investigate the long-term cross-correlations and cross-multifractality between time series generated from complex system. Efficiency of this method is shown on multifractal series, comparing with the well-known multifractal detrended cross-correlation analysis (MFXDFA) and multifractal detrending moving average cross-correlation analysis (MFXDMA). We further apply this method on volatility time series of DJIA and NASDAQ indices, and find some interesting results. The MFSMXA has comparative performance with MFXDMA and sometimes perform slightly better than MFXDFA. Multifractal nature exists in volatility series. In addition, we find that the cross-multifractality of volatility series is mainly due to their cross-correlations, via comparing the MFSMXA results for original series with those for shuffled series.

The fractal dimensions of real world objects are commonly investigated using digital images. Unfortunately, these images are unable to represent an infinitesimal range of scales. In addition, a proper evaluation of the applied methods that encompass the image processing techniques is often missing. Several mathematical well-defined fractals with theoretically known fractal dimensions, represented by digital images, were investigated in this work. The very popular Box counting method was compared to a new image pyramid approach as well as to the Minkowski dilation method. Effects from noise and altered aspect ratios were also considered. The new Pyramid method is quite identical to the Box counting method, but it is easier to implement. Additionally, the calculation times are much shorter and memory requirements are almost comparable.