Fractals

The heart beat data recorded from samples before and during meditation are analyzed using two different scaling analysis methods. These analyses revealed that meditation severely affects the long range correlation of heart beat of a normal heart. Moreover, it is found that meditation induces periodic behavior in the heart beat. The complexity of the heart rate variability is quantified using multiscale entropy analysis and recurrence analysis. The heart beat during meditation is found to be more complex.

Many important definitions in the theory of multifractal measures on ℝ^d, such as the L^q-spectrum, L^∞-dimensions, and the Hausdorff dimension of a measure, cannot be applied to non-compactly supported or infinite measures. We propose definitions that extend the original definitions to positive Borel measures on ℝ^d which are finite on bounded sets, and recover many important results that hold for compactly supported finite measures. In particular, we prove that if the L^q-spectrum is differentiable at q = 1, then the derivative is equal to the Hausdorff dimension of the measure.

The Horton laws of stream numbers and magnitudes are proved in the limit of large network order for the broad class of Tokunaga model of river networks. Tokunaga model is built on the assumption of mean self-similarity in the side tributary structure, and an additional assumption of Tokunaga self-similarity, which is supported by data from real networks. Tokunaga model is gaining increasing recognition in the recent literature, because data supports several predictions of the model.

The fractal dimension of signals represents a powerful tool for analyzing the irregular behavior and state of the dynamical systems. Analysis of waveforms has been used to identify and distinguish specific complex patterns. A variety of algorithms are available for the computation of fractal dimension of waveforms. In this paper we evaluate the performance of our algorithm based on size measure relationship method, quantifying the synthetic waveforms and electroencephalographic signals. Compared to Katz's, Higuchi's and Petrosian's algorithm advantages of this method include greater speed and not affected by noise. The computation time for the algorithm suggested in this paper is much less than the other methods.

We show how to calculate the spectrum of the Laplacian operator on fully symmetric, finitely ramified fractals. We consider well-known examples, such as the unit interval and the Sierpiński gasket, and much more complicated ones, such as the hexagasket and a non-post critically finite self-similar fractal. We emphasize the low computational demands of our method. As a conclusion, we give exact formulas for the limiting distribution of eigenvalues (the integrated density of states), which is a purely atomic measure (except in the classical case of the interval), with atoms accumulating to the Julia set of a rational function. This paper is the continuation of the work published by the same authors in Ref. 1.

The record of the sunspot number visible on the sun is regularly collected over the centuries by various observatories for studying the different factors influencing the sunspot cycle and solar activity. Sunspots appear in cycles, and last several years. These cycles follow a certain pattern which is well known. We analyzed monthly and yearly averages of sunspot data observed from year 1818 to 2002 using rescaled range analysis. The Hurst exponent calculated for monthly data sets are 0.8899, 0.8800 and 0.8597 and for yearly data set is 0.7187. Fractal dimensions¹ calculated are 1.1100, 1.1200, 1.1403 and 1.2813. From the study of Hurst exponent and fractal dimension, we conclude that time series of sunspots show persistent behavior.

The fundamental tool of signal processing is the fast Fourier transform technique (FFT). The sunspot data is also analyzed using FFT. The power spectrum of monthly and yearly averages of sunspot shows distinct peaks at 11 years confirming the well known 11-year cycle. The monthly sunspot data is also analyzed using FFT to filter the noise in the data.

We present an iterative method to define a two-parameter family of continuous functions f_a,θ: I → ℂ such that f_1/3,π/3 is the Koch curve. We consider the two-cases θ = π/3 and θ = π/4 of these generalized Koch curves f_a,θ(I). In each case we determine the pivotal value of a, the largest value of a for which the corresponding curve is not simple. We give characterizations of the double points of the curve (points on the curve that have two inverse images). In the case where θ = π/3 double points are vertices of equilateral triangles. When θ = π/4 the double points form Cantor sets in the plane.

We conclude with a more general result that proves that if the fixed set (attractor) of an iterated function system is connected, then it is a curve.

We demonstrate that videofeedback augmented with one or two mirrors can produce stationary fractal patterns obtained by an iterated function system (IFS) with two transformations (one with a reflection, one without), and with four transformations (two with reflections, one without, one with a 180° rotation). Camera placement yielding only a partial image in one or both mirrors can be achieved using IFS with memory. The IFS rules are obtained from the images of three non-collinear points in each of the principal pieces of the image.