Infinity is where things happen that don't. — S. Knight Ever since Benoit Mandelbrot published his book on fractals [1], we've become accustomed to seeing fantastic and beautiful fractal images such as the ones at the bottom of the pages of this article. A fractal is a mathematical object that exhibits self similarity — it looks the same at any scale. If you zoom in on an image of a fractal, yo...

A novel fractal antenna-array type is proposed. The design is based on the Sierpinski rectangular carpet concept. However, the generator is a circular ring area, filled with radiating elements, so the higher stages of the fractal development produce large arrays of circular rings which, besides the high directivity, have the advantage of the almost uniform azimuthal radiation pattern, attribute...

a major concern of farmers is deterioration of the soil and water resources. soil management is the most direct way for mankind to interfere with natural processes that may either intensify or hinder soil degradation. fractal geometry has been widely used to describe heterogeneity in a wide range of natural processes, including physical systems of soil. in this study, the effects of different m...

Purpose – Fractal geometry can be used to model natural objects which cannot be easily represented by the euclidean geometry. However, contemporary computer-aided design (CAD) and computer-aided manufacturing (CAM) systems cannot be used to model a fractal object efficiently. In a general layer manufacturing (LM) workflow, a model described by the euclidean geometry is required in order to gene...

Keywords: fractal geometry, fractional derivative, fractional Fourier transform, fractional power of a matrix, self similarity, complex partial differential equation, broadband ultrasound, frequency-dependent attenuation, time domain. 1. Backgrounds The rational behind this model is schematically illustrated below: Fractal geometry (irregular soft tissues) → Fractional Fourier transform (freque...

The irregularity has been very attractive to the antenna and packaging engineers. More specifically, the self-similarity has inspired many researchers to use them for broadband designs, due to the fact that a fractal antenna is a repeated shape in different scales, which can be interpreted as a multiscale (“multi-wavelength”) radiator. In addition to this, the space filling property seems to be...

In this report, we present a simple geometric generation principle for the fractal that is obtained when applying Newton's method to nd the roots of a general complex polynomial with real coee-cients. For the case of symmetric polynomials z ? 1, the generation mechanism is derived from rst principles. We discuss the case of a general cubic and are able to give a description of the arising fract...

Abstract. We study the almost sure convergence of the normalized columns in an infinite product of nonnegative matrices, and the almost sure rank one property of its limit points. Given a probability on the set of 2 × 2 nonnegative matrices, with finite support A = {A(0), . . . , A(s − 1)}, and assuming that at least one of the A(k) is not diagonal, the normalized columns of the product matrix ...