Preface

Fractal analysis has proven its ability to resolve many problems and ambiguities in the full spectrum of sciences such as physics, chemistry, human biology, and geosciences. The aim of this book is to show some applications of fractal analysis in the fields of sciences authored by Dr. Sid-Ali Ouadfeul, an Associate Professor of geophysics at Khemis Miliana University. The first chapter introduces the readers to the book, while the second chapter shows the methods and challenges of fractal analysis of time-series data sets. The third chapter demonstrates fractal geometry as an attractive choice for miniaturized planar microwave filter design. The fourth chapter presents fractal antennas for wearable applications. The objective of the fifth chapter is to show some Parrondian games in discrete dynamic systems, while the last chapter reveals fractal structures of carbon nanotube system arrays.

> **Sid-Ali Ouadfeul** University of Khemis Miliana, Algeria

**1**

**Chapter 1**

Sciences

*Sid-Ali Ouadfeul*

**1. Introduction**

rotations [1].

cardiograms [5].

sequences.

series data sets.

Introductory Chapter: Fractal in

The notion of fractal was introduced for the first time in 1975 by the mathematician Benoit Mandelbrot in his book entitled *Fractal Objects* which marked the beginning of his fame. The first definitions of the adjective fractal (from the Latin

The irregularities of nature, of chaotic appearance, such as the irregularities of the seacoasts and the shape of the clouds, a tree, and a fern leaf, are in fact the expression of a very complex geometry of the sea. 'infinitely small. It can be said, however, that a fractal object is an invariant object by dilations, translations, and

The fractal analysis has been widely used in sciences, for example, in physics, the fractal analysis is used in thermodynamics, particularly for the study of fully developed turbulence [1], in image segmentation and processing [2, 3], in astrophysics for the study of hydrogen distribution [4], in physical medicine for tumor localization from mammograms [3], and in cardiology, for the study of the electro-

In geoscience, the fractal analysis has been used in petrophysics for the segmentation or classification of geological formations [6–9]. It has also been used in geomagnetism to characterize the outer part of the geomagnetic field [10–14]. In environmental sciences, Burrough [15] used the semivariogram method to estimate the fractal dimension D for various environmental transects (e.g., soil factors, vegetation cover, iron ore content in rocks, rainfall levels, crop yields). In medicine and human biology, the fractal analysis has been applied in cell, protein, and chromosome structures, for example, Takahashi [16] supposed that the basic design of a chromosome has a tree-like pattern. Xu et al. [17] assumed that the twistings of DNA-binding proteins have fractal properties. Self-similarity has recently demonstrated in DNA sequences (see Stanley [18]; see also papers in Nonnenmacher et al. [19]). Glazier et al. [20] used the multifractal spectrum approach to rebuild the evolutionary history of organisms from mDNA

The aim of this book is to gather advance researches in the field of fractal analysis; the book contains seven chapters: one chapter is discussing the Parrondian games in discrete dynamic systems, two chapters are debating the application of the fractal analysis in microwave and antennas, and another chapter is showing some applications in medicine, while another one is talking about the fractal structures of the carbon nanotube system arrays and another chapter discuss the methods and challenges of the fractal analysis of the time-

adjective fractus) come from the word "frangere" which means to break.
