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Fractal Analysis: Basic Concepts And Applications [Kõva köide]

(Univ Of Kairouan, Tunisia & Univ Of Monastir, Tunisia & Univ Of Tabuk, Saudi Arabia), (Univ Of Monastir, Tunisia & Univ Of Tabuk, Saudi Arabia), (Univ Of Tuscia, Italy)
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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.



"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 andsenior 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"--
Preface vii
About the Authors ix
List of Figures
xv
List of Table
xvii
1 Introduction
1(5)
2 Basics of Measure Theory
6(21)
2.1 σ-algebras
6(2)
2.2 Some topological concepts
8(1)
2.3 Outer measures
9(1)
2.4 Regular outer measures
10(1)
2.5 Metric outer measures
11(2)
2.6 Lebesgue measure on Rd
13(1)
2.7 Convergence of measures on metric spaces
14(9)
2.8 Exercises for
Chapter 2
23(4)
3 Martingales with Discrete Time
27(23)
3.1 Generalities
27(4)
3.2 Conditional expectation
31(2)
3.3 Convergence and regularity of martingales
33(2)
3.4 Regularity of integrable martingales
35(3)
3.5 Positive and upper martingales
38(9)
3.5.1 Stopping time
38(3)
3.5.2 Positive upper martingales
41(6)
3.6 Exercises for
Chapter 3
47(3)
4 Hausdorff Measure and Dimension
50(27)
4.1 Hausdorff measure
50(7)
4.2 Hausdorff dimension of Cantor-type sets
57(6)
4.3 Other variants of Hausdorff dimension
63(3)
4.4 Upper and lower bounds of the Hausdorff dimension
66(2)
4.5 Frostman's Lemma
68(4)
4.6 Application
72(2)
4.7 Exercises for
Chapter 4
74(3)
5 Capacity Dimension of Sets
77(23)
5.1 Generalities
77(3)
5.2 Self-similar sets
80(11)
5.3 Billingsley dimension
91(4)
5.4 Eggleston theorem
95(2)
5.5 Exercises for
Chapter 5
97(3)
6 Packing Measure and Dimension
100(16)
6.1 Bouligand--Minkowski dimension
101(3)
6.2 Packing measure
104(6)
6.3 Packing dimension
110(3)
6.4 Exercises for
Chapter 6
113(3)
7 Multifractal Analysis of Gibbs Type Measures
116(18)
7.1 The multifractal formalism
118(5)
7.2 Existence of Gibbs measures
123(6)
7.3 Exercises for
Chapter 7
129(5)
8 Extensions to Multifractal Cases
134(45)
8.1 Generalized multifractal versions of the Hausdorff, and packing measures, and dimensions
135(14)
8.2 Generalized Bouligand-Minkowski dimension
149(10)
8.3 The multifractal spectrum
159(15)
8.4 Exercises for
Chapter 8
174(5)
9 Some Applications
179(24)
9.1 Introduction
179(1)
9.2 Fractals in plants' nature
180(3)
9.3 Fractals in human body anatomy
183(3)
9.4 Fractals for time series
186(3)
9.5 Fractals for signals/images: The case of nano images
189(5)
9.6 A classical fractal self-similar set
194(2)
9.7 A case of self-similar type measures
196(3)
9.8 Exercises for
Chapter 9
199(4)
Bibliography 203(12)
Index 215