Fractals

Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).

This paper gives a method to obtain Hausdorff dimensions of sub-self-similar sets, where the ratios of similitudes are different. By using Gibbs measure and pressure in the sense of thermodynamic formalism, we obtain the Hausdorff dimensions of sub-self-similar sets in the space of infinite sequences. We then consider sub-self-similar sets in R^d and show how to calculate their Hausdorff dimensions under an open set condition.

It is shown that metric representation of DNA sequences is one-to-one. By using the metric representation method, suppression of nucleotide strings in the DNA sequences is determined. For a DNA sequence, an optimal string length to display genomic signature in chaos game representation is obtained by eliminating effects of the finite sequence. The optimal string length is further shown as a self-similarity limit in computing information dimension. By using the method, self-similarity limits of bacteria complete genomic signatures are further determined.

This paper considers four geoelectrical time series, measured in a seismic area of Southern Italy. Lomb Periodogram method, Higuchi analysis, Detrended Fluctuation Analysis (DFA) and the mean distance spanned within time L are used to discuss the correlation properties of these signals. The values of the scaling exponents from these methods of the geoelectrical data indicate that the long-range correlations are present. Furthermore, it is found that these correlations are all linear.

Universal constructions of univariate and bivariate Gaussian distributions, as transformations of diffuse probability distributions via, respectively, plane- and space-filling fractal interpolating functions and the central limit theorems that they imply, are reviewed. It is illustrated that the construction of the bivariate Gaussian distribution yields exotic kaleidoscopic decompositions of the bell in terms of exquisite geometric structures which include non-trivial crystalline patterns having arbitrary n-fold symmetry, for any n > 2. It is shown that these results also hold when fractal interpolating functions are replaced by a more general class of attractors that are not functions.

The scaling properties of the portal vein and the hepatic vein are examined, based on three-dimensional computed tomography images of casts of human livers. In particular, a quantitative analysis of the branching ratio based on the Strahler ordering scheme as well as the diameter and length ratios are performed. To quantify the segment anatomy of the liver, the volume of supplied liver parenchyma is measured and related to the corresponding vessel radius. The implications of these findings for segment-oriented liver surgery are discussed. We also investigate the 3D interdependence of the intertwined portal and hepatic veins based on a concept of tree distance that will be introduced.

The fractal component in the long-term heart rate variability (HRV) in health and in certain heart disease conditions were studied in the framework of dyadic random bounded cascade. The physiology of the fractal component was also proposed and tested in the simulation of HRV in autonomic nervous system blockade. Numerical results suggest the intrinsic mechanism behind HRV is of multiplicative nature and a "failure mechanism" due to the change of the fractal generating mechanism in certain heart diseases and in autonomic blockade.

The nature of the zeolite 4A coatings prepared by using the substrate heating method was investigated. It was determined that the fractal dimensions of the surfaces of the coatings varied between about 1.14 and 1.19 with respect to the coating thickness in a mesh size range of about 1.5 μm - 48.5 μm. The fractal dimension value increased to some extent for the thicker zeolite coatings. It is hypothesized that the significance of nucleation is relatively high in the substrate heating method when compared to conventional synthesis methods and this leads to zeolite coatings with fractal surfaces and intercrystal porosity.

Functional digraphs are sometimes fractal sets. As a special kind of fractal sets, the dimension properties of the functional digraph are studied in this paper. Firstly, the proof of a Minkowski dimension theorem is discussed and a new proof is given. Secondly, according to this dimension theorem, the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions are discussed. And the relations between these Minkowski dimensions and the Minkowski dimensions of the digraphs of the two functions are established. In the conclusion, the maximum Minkowski dimension of the two functional digraphs plays a decisive part in the Minkowski dimensions of the digraphs of the sum, deviation, product and quotient of two functions.

We present here an overview of the history, applications and important properties of a function which we refer to as the Lévy integral. For certain values of its characteristic parameter, the Lévy integral defines the symmetric Lévy stable probability density function. As we discuss however, the Lévy integral has applications to a number of other fields besides probability, including random matrix theory, number theory and asymptotics beyond all orders. We exhibit a direct relationship between the Lévy integral and a number theoretic series which we refer to as the generalized Euler-Jacobi series. The complete asymptotic expansions for all natural values of its parameter are presented, and in particular it is pointed out that the intricate exponentially small series become dominant for certain parameter values.

Let the self-similar set C in R be defined by with a disjoint union, where the h_j's are similitude mappings with ratios 0 < a_j < 1. Let μ on C be the self-similar probability measure corresponding to the probability vector , where ξ = dim_HC is the Hausdorff dimension of C. Let S be the set of points at which the probability distribution function F of μ has no derivative, finite or infinite. We prove that dim_H S = (dim_HC)² and dim_P S = dim_B S = dim_HC.

The study of basic fractal geometry can help build students' enthusiasm for learning early in their undergraduate careers. To most undergraduate students, fractals are new, visually appealing, useful, and mathematically accessible. As a result, fractals can be an effective vehicle for introducing and reinforcing multiple modes of learning, which at many institutions is one of the main goals of general first-year undergraduate education. This article describes how fractals are used in one institution's "Freshman Seminar" program to help accomplish these goals.