Fractals

Database Tomography (DT) is a textual database analysis system consisting of two major components: (1) algorithms for extracting multi-word phrase frequencies and phrase proximities (physical closeness of the multi-word technical phrases) from any type of large textual database, to augment (2) interpretative capabilities of the expert human analyst. DT was used to obtain technical intelligence from a Fractals database derived from the Science Citation Index/Social Science Citation Index (SCI). Phrase frequency analysis by the technical domain experts provided the pervasive technical themes of the Fractals database, and the phrase proximity analysis provided the relationships among the pervasive technical themes. Bibliometric analysis of the Fractals literature supplemented the DT results with author/journal/institution publication and citation data.

The analytical expressions of the fractal dimensions for wetting and non-wetting phases for unsaturated porous media are derived and are found to be a function of porosity, maximum and minimum pore sizes as well as saturation. There is no empirical constant in the proposed fractal dimensions. It is also found that the fractal dimensions increase with porosity of a medium and are meaningful only in a certain range of saturation S_w, i.e. S_w>S_min for wetting phase and S_w<S_max for non-wetting phase at a given porosity, based on real porous media for requirements from both fractal theory and experimental observations. The present analysis of the fractal dimensions is verified to be consistent with the existing experimental observations and it makes possible to analyze the transport properties such as permeability, thermal dispersion in unsaturated porous media by fractal theory and technique.

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.

Reprocessing of single-use medical devices is an issue of concern and discussion due to infection risk and operation failure. Cleaning procedures and sterilization processes can produce structural changes and relevant topographical alteration of the biomedical device surfaces which may activate mechanisms able to cause unwanted biological response. Atomic Force Microscopy analysis (AFM) to measure Poly(vinyl chloride) (PVC) catheters surface roughness variation along successive sterilization cycles was applied. In this work, the relation between the number of reprocessing cycles and the fractal mass dimension of reprocessed catheter surface microscopic images is studied.

A Neogene transcurrent fault system affecting the Apenninic-Maghrebian thrust belt, located in northwestern Sicily, has been investigated using fractal analysis. The present-day structural setting of the sector between the Palermo and Madonie Mountains is the result of the superimposition, in space and time, of two distinct deformational events, a Miocene southeast-verging thrusting followed by Messinian to Pliocene strike-slip faulting, which cuts obliquely through the compressive fronts. The spatial distribution properties of the fault array were investigated by means of a fractal analysis. Fractal dimension was computed by adopting the correlation integral method. Fractal ranges have been evidenced between 350 and 3000 m. The fractal dimension obtained for the whole array is D=1.66. The scaling spatial distribution property of the fault array has been analyzed by calculating the fractal dimension with a moving window.

We show power-scaling behaviors for fluctuations in share volume, which no other studies have done so far. After analyzing a database of the daily transactions for all securities listed on the Tokyo Stock Exchange, we selected 1050 large companies that each had an unbroken series of daily trading activity from January 1975 to January 2002. We found that the cumulative distributions of daily fluctuations in share volumes can be well described by a power-law decay, and that the cumulative distributions for almost all of the companies can be characterized by an exponent within the stable Lévy domain 0 < α < 2. Furthermore, more than 35% of the cumulative distributions can be well approximated by Zipf's law, i.e. the cumulative distributions have an exponent close to unity.

During the last few decades, fractal geometries have found numerous applications in several fields of science and engineering such as geology, atmospheric sciences, forest sciences, physiology and electromagnetics. Although the very fractal nature of these geometries have been the impetus for their application in many of these areas, a direct quantifiable link between a fractal property such as dimension and antenna characteristics has been elusive thus far. In this paper, the variations in the input characteristics of multi-resonant antennas based on generalizations of Koch curves and fractal trees are examined by numerical simulations. Schemes for such generalizations of these geometries to vary their fractal dimensions are presented. These variations are found to have a direct influence on the primary resonant frequency, the input resistance at this resonance, and ratios resonant frequencies of these antennas. It is expected that these findings would further enhance the popularity of the study of fractals.

The prevalence of a Patent Foramen Ovale is described in merely 30% of the asymptomatic population. This patency has been shown to be an increasing risk factor for paradoxical cerebral embolization. Some desaturation or decompression situations in human activities such as scuba diving or altitude flight are prone to provoke embolisations. The association with the presence of a patent Foramen Ovale and the onset of cerebral decompression sickness seems to be presenting an odds ratio value of about 5.1.¹ The presence of asymptomatic brain lesion-like "spots" has been investigated in a randomized population of diving individuals (n=42 randomized out of 200). The inclusion criteria were drastic and included: age (less than 41 years of age); diving experience (more than 200 logged dives); no decompression sickness episodes; no contraindications for the MRI examination; and no known central nervous system conditions. Data of the magnetic resonance investigation of the brain has been performed in 42 (diving) volunteers fully informed on the experimental procedures. The statistical comparison (Anova test after Kolmogorov-Smirnov compatibility testing and Neuman–Keuls discriminant post-test) of the fractal dimension obtained by means of the box counting method with the slope analysis (Harfa fractal analysis program). The comparison was performed with known pathological images such as multiple sclerosis (a pathology not emerging from vascular problems), ischemic thrombotic lesions (vascular problem), diver's asymptomatic brain spots, and the arteriography of the internal carotid in non-pathological humans (clearly vascular). The statistical difference (p<0.001) between the vascular related images, as well as the absence of statistical difference (p>0.05) with the non-vascular spots images advocates with a non-vascular origin of the diver's asymptomatic spots and thus the link between the patency of the cardiac Foramen Ovale and the brain "spots" seems not to be as clear as it was believed.

This paper deals with several of the possible uses of Zipf and inverse Zipf laws in the field of audio signal analysis. We show that these laws are powerful analysis tools allowing the extraction of information not available by standard methods. The adaptation of Zipf and inverse Zipf laws to audio signals requires a coding of these signals into text-like data, considered as sequences of patterns. Because these codings are of first importance since they have to bring to the fore relevant information within signals, three types of codings have been developed, depending on the representation of the audio signal it is based on: temporal, frequential and time-scale representations. Once audio signal has been coded, features linked to Zipf and inverse Zipf approaches are computed. Finally, the classification step aims at the identification of signals. Four classification methods have been considered as well as a fusion method that combines these classifiers. In order to evaluate our method, we apply it on medical acoustical signals. They occur when swallowing and contain xiphoidal sounds. The problem is to extract and characterize xiphoidal sounds according to the gastro-oesophageal reflux pathological state. The aim is to help medical doctors to characterize and diagnose this pathology, and to give, in the end, a decision help tool as efficient as possible.

Malignant melanoma is nowadays one of the most frequent type of skin cancer among white-skinned populations and one of the most malignant tumors. In this work, we investigate fractal properties of color images of dermal tissue microscopic samples as a basis for detection of malignant melanoma lesions. The enhancement image process is applied to hematoxylin-eosin stained tissue samples microscopic images. A segmented image based on a color model attribute is processed with a local Hurst transformation added with edge detection algorithm. The result is used to modulate the intensity information of the original image in order to support the expert analysis essentially in low quality images. Our findings show that fractal measures such as the Hurst exponent can be used to detect regions with a high density of melanocytes nests included in dermal tissue allowing early diagnosis of malignant melanoma tumors.

We present efficient methods for simulation, using the Fast Fourier Transform (FFT) algorithm, of two classes of processes with symmetric α-stable (SαS) distributions. Namely, (i) the linear fractional stable motion (LFSM) process and (ii) the fractional autoregressive moving average (FARIMA) time series with SαS innovations. These two types of heavy-tailed processes have infinite variances and long-range dependence and they can be used in modeling the traffic of modern computer telecommunication networks.

We generate paths of the LFSM process by using Riemann-sum approximations of its SαS stochastic integral representation and paths of the FARIMA time series by truncating their moving average representation. In both the LFSM and FARIMA cases, we compute the involved sums efficiently by using the Fast Fourier Transform algorithm and provide bounds and/or estimates of the approximation error.

We discuss different choices of the discretization and truncation parameters involved in our algorithms and illustrate our method. We include MATLAB implementations of these simulation algorithms and indicate how the practitioner can use them.

An algorithm for the visualization of n-dimensional binary spaces is considered. The algorithm is put in metric terms, and reformulated in terms of branching structures. The difference between metric and structural visualizations is discussed. Then, a hybrid method is proposed. Tree structures are grown in accordance with a metric error function. The method is illustrated with an example in the context of microbial taxonomy, and compared with a Kohonen map.

An irregular Sierpinski triangle is a non-empty compact set F satisfying where each T_i is a contracting affine transformation. The irregular Sierpinski triangle is not generated with similarity transformations. Therefore, the calculation of its fractal dimension is not straightforward. We associate with each irregular Sierpinski triangle a self-similar regularization. The dimension of the regularization dictates the shape of the irregular Sierpinski triangle. We also describe properties of the regularizations and show their relationship to the classic Sierpinski triangle.