Fractal Analysis - Basic Concepts and Applications

✍ Scribed by Carlo Cattani, Anouar Ben Mabrouk, Sabrine Arfaoui

• Publisher: World Scientific
• Year: 2022
• Tongue: English
• Leaves: 244
• Series: Series on Advances in Mathematics for Applied Sciences 91
• Edition: 1
• Category: Library

✦ Synopsis

The aim of this book is to provide a basic and self-contained introduction to the ideas underpinning fractal analysis. The book illustrates some important applications issued from real data sets, real physical and natural phenomena as well as real applications in different fields, and consequently, presents to the readers the opportunity to implement fractal analysis in their specialties according to the step-by-step guide found in the book.

Besides advanced undergraduate students, graduate students and senior researchers, this book may also serve scientists and research workers from industrial settings, where fractals and multifractals are required for modeling real-world phenomena and data, such as finance, medicine, engineering, transport, images, signals, among others.

For the theorists, rigorous mathematical developments are established with necessary prerequisites that make the book self-containing. For the practitioner often interested in model building and analysis, we provide the cornerstone ideas.

✦ Table of Contents

Contents
Preface
About the Authors
List of Figures
List of Table
1. Introduction
2. Basics of Measure Theory
2.1 σ-algebras
2.2 Some topological concepts
2.3 Outer measures
2.4 Regular outer measures
2.5 Metric outer measures
2.6 Lebesgue measure on Rd
2.7 Convergence of measures on metric spaces
2.8 Exercises for Chapter 2
3. Martingales with Discrete Time
3.1 Generalities
3.2 Conditional expectation
3.3 Convergence and regularity of martingales
3.4 Regularity of integrable martingales
3.5 Positive and upper martingales
3.5.1 Stopping time
3.5.2 Positive upper martingales
3.6 Exercises for Chapter 3
4. Hausdor Measure and Dimension
4.1 Hausdor measure
4.2 Hausdor dimension of Cantor-type sets
4.3 Other variants of Hausdor dimension
4.4 Upper and lower bounds of the Hausdor dimension
4.5 Frostman's Lemma
4.6 Application
4.7 Exercises for Chapter 4
5. Capacity Dimension of Sets
5.1 Generalities
5.2 Self-similar sets
5.3 Billingsley dimension
5.4 Eggleston theorem
5.5 Exercises for Chapter 5
6. Packing Measure and Dimension
6.1 Bouligand–Minkowski dimension
6.2 Packing measure
6.3 Packing dimension
6.4 Exercises for Chapter 6
7. Multifractal Analysis of Gibbs Type Measures
7.1 The multifractal formalism
7.2 Existence of Gibbs measures
7.3 Exercises for Chapter 7
8. Extensions to Multifractal Cases
8.1 Generalized multifractal versions of the Hausdor , and packing measures, and dimensions
8.2 Generalized Bouligand-Minkowski dimension
8.3 The multifractal spectrum
8.4 Exercises for Chapter 8
9. Some Applications
9.1 Introduction
9.2 Fractals in plants' nature
9.3 Fractals in human body anatomy
9.4 Fractals for time series
9.5 Fractals for signals/images: The case of nano images
9.6 A classical fractal self-similar set
9.7 A case of self-similar type measures
9.8 Exercises for Chapter 9
Bibliography
Index

✦ Subjects

Fractals, Measure Theory, Martingales, Hausdorff Measure, Capacities, Packing Measures, Multifractal Analysis