Fractals

We demonstrate that it is possible to distinguish with a complete certainty between healthy subjects and patients with various dysfunctions of the cardiac nervous system by way of multiresolutional wavelet transform of RR intervals. We repeated the study of Thurner et al. on different ensemble of subjects. We show that reconstructed series using a filter which discards wavelet coefficient related with higher scales enables one to classify individuals for which the method otherwise is inconclusive. We suggest a delimiting diagnostic value of the standard deviation of the filtered, reconstructed RR interval time series in the range of ~ 0.035 (for the above mentioned filter), below which individuals are at risk.

We propose the application of a recent vector space technique for the analysis of DNA sequences. In this approach the similarity of two series of symbols is measured by the dot products of vectors corresponding to the given sequences. In this paper we demonstrate that this measure, depending on the short-range properties of the sequences, may be used for identifying different elements of the genome.

Darwin's theory of evolution by natural selection revolutionized science in the nineteenth century. Not only did it provide a new paradigm for biology, the theory formed the basis for analogous interpretations of complex systems studied by other disciplines, such as sociology and psychology. With the subsequent linking of macroscopic phenomena to microscopic processes, the Darwinian interpretation was adopted to patterns observed in molecular evolution by assuming that natural selection operate fundamentally at the level of DNA. Thus, patterns of molecular evolution have important implications in many fields of science. Although the evolution rate of a given gene seems to be of approximately the same order of magnitude in all species, genes appear to differ in rate from one another by orders of magnitude, a fact which standard theory does not adequately explain. An understanding of the statistics of rates across different genes may shed light on this problem. The evolution rates of mammalian DNA, based on recent estimates of numbers of nonsynonymous substitutions in 49 genes of human, rodents, and artiodactyls, are studied. We find that the rate variations are better described by lognormal statistics, as would be the case for a multiplicative process, than by Gaussian statistics, which would correspond to a linear, additive process. Thus, we introduce a multiplicative evolution statistical hypothesis (MESH), in which the theoretical explanation of these statistics requires the evolution of different substitution rates in different genes to be a multiplicative process in that each rate results from the interaction of a number of interdependent contingency processes.

We study the spatial distribution in aggregating systems of mixtures of magnetic and non-magnetic particles using Monte-Carlo simulations together with scaling arguments. In particular, we show that (a) as the system size grows, the fractal dimension of the composite system is dominated by the smaller fractal dimension, (b) the system is realized as a back-bone consisting of magnetic particles (lower fractal dimension) with denser regions of non-magnetic particles attached to it at random positions. Using simple connectivity features observed in pure magnetic and non-magnetic clusters and self-similarity arguments we predict, via Real-Space-Renormalization, fractal exponents D_m = 1.25 ± 0.05 for the magnetic clusters and D_nm = 1.4 ± 0.1 for the non-magnetic clusters.

Two fundamental empirical laws have been established in the analysis of galaxy space distribution. First, recent analyses have revealed that the three-dimensional distribution of galaxies and clusters is characterized by large-scale structures and huge voids: such a distribution shows fractal correlations up to the limits of the available samples. This has confirmed the earlier de Vaucouleurs power-law density — distance relation, now corresponding to a fractal structure with dimension D ≈ 2, at least, in the range of scales ~1 ÷ 200 Mpc (H₀=55 km/sec/Mpc). An eventual cut-off towards homogenization has not been yet identified. Second, since Huble's discovery, the linear redshift-distance law has been well established within 200 Mpc and also much deeper. The co-existence of these laws within the same scales is a challenge for the standard cosmology, where the linear Hubble law is a strict consequence of homogeneity of the expanding universe. This puzzle is now sufficiently strong to raise doubts for the standard cosmology.

This paper shows that the dendritic patterns formed by low-resistance channels in a river drainage basin are reproducible and can be deduced from a single principle that acts at every step in the development of the pattern: the constrained minimization of global resistance in area-to-point flow. The river basin is modeled as a two-dimensional territory with Darcy flow through a saturated heterogeneous porous medium with uniform flow addition per unit area. From one step to the next, small elements of the porous medium are dislodged and removed in ways that minimize the global flow resistance. The removed elements are replaced by channels with lower flow resistance. The channels form a dendritic pattern that is deterministic, not random. The finest details of this structure are sensitive to internal properties and external forcing, i.e. variations in the local properties of the flow medium, and the manner in which the total area-to-point flow rate varies as the structure develops. Remarkably insensitive to such effects are the basic type and rough size of the flow structure (channels versus no channels, dendrite, number of branches) and the minimized global resistance to flow.

Entropy spectrum of self-affine fractal interfaces created by contraction maps is investigated. Interfaces are created by a single (or multi-) generator(s) all of whose segments have the same anisotropy of scaling and different scaling factors. The whole interface is decomposed into many subsets and h (topological entropy), λ (decay exponent), D_D (divider dimension) and D_B (box dimension) of each subset are calculated. Entropy spectrum is obtained from the relation between h and λ. We find that H (Hurst or roughness exponent), D_D and D_B for only the remaning subset at n =∞ are related as and D_B = 2 - H.

In this paper, a special kind of FIF (fractal interpolation function) is studied. The authors prove the stability of the fractal interpolations. When the interpolation data have a small perturbation, the corresponding FIFs also have a small perturbation.

An image texture measure based on the box counting algorithm is evaluated for its potential to characterize human trabecular bone structure in medical images. Although bone images lack the self-similarity of theoretical fractals, bone images are candidates for characterization using fractal analysis because of their highly complex structure. The fractal based measure, herein called the box counting dimension (BCD), is an effective dimension, and does not imply an underlying fractal geometry. The importance of resolution in quantifying bone characteristics using the BCD is addressed. The relationship of BCD to standard measures of trabecular bone structure is also analyzed. To evaluate the variability of the BCD with change in resolution, the BCD is determined for two sections from each of seven 3D X-ray Tomographic Microscopy (XTM) images of human radius bone specimens, while the resolution is varied using lowpass filtering. An automated method of choosing the range of scales for the fractal analysis curve regression is used. The relationship of BCD to trabecular bone width and spacing is analyzed both for the XTM images and for simulated images representing idealized structures. The range of BCD values is 1.21–1.54. Variation in BCD over a range of resolutions is found to be small compared to the variation in BCD between different bone specimens. Maximum change in BCD over a large range of resolutions (17.60–176 microns per pixel) is 0.08. BCD decreases as space between trabeculae increases. Fractal based texture measures may potentially allow clinical monitoring of changes in bone structure — for example, using Magnetic Resonance Imaging at 150–200 micron resolution.

In this paper we present a study of the variation of morphology of polyethylene oxide (PEO) films complexed with ammonium perchlorate (NH₄ClO₄) formed by evaporating a methanolic solution. Films with salt concentration x = 0 - 0.35 (x is the weight fraction of salt) were prepared for the morphology study. The wide varitey of structures obtained were systematically studied by photographs and polarizing microscopy. Some preliminary studies of x-ray diffraction, differential scanning calorimetry and variable temperature polarizing microscopy have also been done to identify the various phases present in the films.

We have extended the work of Lei Tan on the similarity between the Mandelbrot set and the Julia sets. We show that the fractal structures of dynamic and parameter spaces are asymtotically similar at Misiurewicz points for the cubic mappings.