Fractals

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Conscious activity is generally consistent with a deterministic mode of action, however some activities demand randomness. The goal of this study was to find whether mind can produce randomness. We asked various human subjects, aged between 21 and 57, to produce sequences of random integers. The non-repetition of the terms in the series was the "algorithm" generally used by all the subjects. The series were analyzed by using spectral analysis (SA), detrended fluctuation analysis (DFA), information (I₂) and approximate entropy (AE) and correlation dimension analysis (CDA). SA of the series revealed 1/f^β(-0.4<β<1) types of spectra. The α scaling exponent resulting from DFA ranged between 0.2 and 1.1. The maximum of the distribution of α was centered at about 0.62. This is equivalent to a quarter power law in spectral terms. A significant percentage of subjects were able to produce random series. I₂ of mental series revealed short-range correlation for pairs of successive numbers in the series. The correlation decreased gradually and it disappeared at a delay step of about 7 which is the characteristic number of short memory. AE had a minimum value at α=1 and a maximum value at α=0.5 for mental and random series, respectively. CDA showed no presence of attractors in the series, therefore none of the randomness produced by the subjects was of deterministic origin. A comparison of the scaling exponents revealed that the mental series have a certain long-range characteristic which is common to the human language as evident in the literature writings.

The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number π while exploring a seemingly unrelated property of the Mandelbrot set.¹ Boll's finding is easy to describe and understand, and yet it is not widely known — possibly because the result has not been rigorously shown. The purpose of this paper is to present and prove Boll's result.

Fractal behavior in the quasi-steady states of the two types of speech signals, namely, quasi-periodic and quasi-random, are studied. The signals of the first group consist of seven vowels and those for the second group consist of three variants each of three sibilants. Ten rendering of each of these signals by one native Bengali male speaker of 45 years of age are used as the signal database. Standard box-counting method is used for generating ln (p^q) versus ln (1/r) curves. D₀, D_q and the knee expanse (KE) of the curves, their interrelations and the projections on the source characteristics are the objects of analysis. D₂>D₀ is found to indicate locally dense nature of the map and are found to be associated mainly with some sibilants. D_q are found to be a family of simple polynomial functions of q for all signals. D₀, D_q and KE are related to the nature of different signals and character of the sources generating the signals. The study of fractal dimensions and the generalized dimensions for these signals reveal intermittency behavior and multifractality, which indicate basically, turbulent source or sources for speech generation. That the quasi-periodic and the quasi-random sounds are fundamentally different is reflected in the behavior of generalized fractal dimensions.

We generalize the notion of fractal interpolation functions (FIFs) to stochastic processes. We prove that the Minkowski dimension of trajectories of such interpolations for self-similar processes with stationary increments converges to 2-α.

We generalize the notion of vector-valued FIFs to stochastic processes. Trajectories of such interpolations based on an equally spaced sample of size n on the interval [0,1] converge to the trajectory of the original process. Moreover, for fractional Brownian motion and, more generally, for self-similar processes with stationary increments (α-sssi) processes, upper bounds of the Minkowski dimensions of the image and the graph converge to the Hausdorff dimension of the image and the graph of the original process, respectively.

Fractal-skeletal based channel network (F-SCN) model, which describes how the boundary of the basin constrains the channel network evolution, produces a channel network pattern that obeys Horton's laws. The statistical features of this model conform well with real networks. For this F-SCN that depends on general shape of the initiator-basin, generating mechanism and rule, and the nature of skeletonization process, the estimation of fractal dimension (D) is Log R_BLog R_L. The estimated D, 1.76 is approximated to the observed value, 1.8.

We study time series concerning rare events. The occurrence of a rare event is depicted as a jump of constant intensity always occurring in the same direction, thereby generating an asymmetric diffusion process. We consider the case where the waiting time distribution is an inverse power law with index μ. We focus our attention on μ<3, and we evaluate the scaling exponent δ of the time in the resulting diffusion process. We prove that δ gets its maximum value, δ=1, corresponding to the ballistic motion, at μ=2. We study the resulting diffusion process by means of joint use of the continuous time random walk and of the generalized central limit theorem (CLT), as well as adopting a numerical treatment. We show that rendering the diffusion process to be asymmetric yields the significant benefit of enhancing the value of the scaling parameter δ. Furthermore, this scaling parameter becomes sensitive to the power index μ in the whole region 1<μ<3. Finally, we show our method in action on real data concerning human heartbeat sequences.

This paper seeks to check the validity of the apparent fractal conjecture,^1,2 which states that the observed power-law behavior for the average density of large-scale distribution of galaxies arises when some observational quantities, selected by their relevance in average density profile determination, are calculated along the past light cone. Since general relativity states that astronomical observations are carried out in this spacetime hypersurface, observables necessary for direct comparison with astronomical data must be calculated along it. Implementing this condition in the proposed set of observational relations profoundly changes the behavior of many observables in the standard cosmological models. In particular, the average density becomes inhomogeneous, even in the spatially homogeneous spacetime of standard cosmology, change which was already analyzed by Ribeiro (1992b, 1993, 1994, 1995)^3–6 for a non-perturbed model. Here we derive observational relations in a perturbed Einstein-de Sitter cosmology by means of the perturbation scheme proposed by Abdalla and Mohayaee (1999),⁷ where the scale factor is expanded in power series to yield perturbative terms. The differential equations derived in this perturbative context, and other observables necessary in our analysis, are solved numerically. The results show that our perturbed Einstein-de Sitter cosmology can be approximately described by a decaying power-law like average density profile, meaning that the dust distribution of this cosmology has a scaling behavior compatible with the power-law profile of the density-distance correlation observed in the galaxy catalogues. These results show that, in the context of this work, the apparent fractal conjecture is correct.

We investigate the Japanese personal income distribution in the high income range over the 112 years (1887–1998), and that in the middle income range over the 44 years (1955–1998). It is observed that the distribution pattern of the log-normal with power law tail is the universal structure. However, the indexes specifying the distribution differ from year to year. One of the index characterizing the distribution is the mean value of the log-normal distribution; the mean income in the middle income range. It is found that this value correlates linearly with the gross domestic product (GDP). To clarify the temporal change of the equality or inequality of the distribution, we analyze Pareto and Gibrat indexes, which characterize the distribution in the high income range and that in the middle income range, respectively. It is found for some years that there is no correlation between the high income and the middle income. It is also shown that the mean value of Pareto index equals to 2, and the change of this index is effected by the change of the asset price. From these analysis, we derive four constraints that must be satisfied by mathematical models.

The interaction between the Earth's magnetic field and the solar wind plasma results in a natural plasma confinement system which stores energy. Dissipation of this energy through Joule heating in the ionosphere can be studied via the Auroral Electrojet (AE) index. The apparent broken power law form of the frequency spectrum of this index has motivated investigation of whether it can be described as fractal coloured noise. One frequently-applied test for self-affinity is to demonstrate linear scaling of the logarithm of the structure function of a time series with the logarithm of the dilation factor λ. We point out that, while this is conclusive when applied to signals that are self-affine over many decades in λ, such as Brownian motion, the slope deviates from exact linearity and the conclusions become ambiguous when the test is used over shorter ranges of λ. We demonstrate that non self-affine time series made up of random pulses can show near-linear scaling over a finite dynamic range such that they could be misinterpreted as being self-affine. In particular, we show that pulses with functional forms such as those identified by Weimer within the AL index, from which AE is partly derived, will exhibit nearly linear scaling over ranges similar to those previously shown for AE and AL. The value of the slope, related to the Hurst exponent for a self-affine fractal, seems to be a more robust discriminator for fractality, if other information is available.