This book is a thorough introduction to the Cantor (Ternary) Set and its applications and brings together many of the topics (advanced calculus, probability, topology, and algebra) that mathematics students are required to study, but unfortunately are treated as separate ideas. This book successfully bridges the gap between how several mathematical fields interact using Cantor Sets as the common theme. While the book is mathematically self–contained, readers should be comfortable with mathematical formalism and have some experience in reading and writing mathematical proofs. Chapter coverage includes: a biography of Cantor; an introduction to the Cantor (Ternary) Set; Self–Similar Sets and Fractal Dimensions; sums of Cantor Sets; the role of Cantor Sets to create pathological functions; and additional topics such as continued fractions, Ana Sets, and p–adic numbers. INDICE: Foreword xiii Preface xv Acknowledgments xvii Introduction xix 1 A Quick Biography of Cantor 1 2 Basics 5 2.1 Review 5 Exercises 14 3 Introducing the Cantor Set 17 3.1 Some Definitions and Basics 17 3.2 Size of a Cantor Set 21 3.3 Large and Small 46 Exercises 48 4 Cantor Sets and Continued Fractions 51 4.1 Introducing Continued Fractions 52 4.2 Constructing a Cantor Set 59 4.3 Diophantine Equations 60 4.4 Miscellaneous 63 Exercises 65 5 p–adic Numbers and Valuations 67 5.1 Some Abstract Algebra 67 5.2 p–adic Numbers 72 5.3 p–adic Integers and Cantor Sets 80 5.4 p–adic Rational Numbers 82 Exercises 88 6 Self–Similar Objects 91 6.1 The Meaning of Self–Similar 91 6.2 Metric Spaces 92 6.3 Sequences in ( S; d ) 97 6.4 Affine Transformations 106 6.5 An Application for an IFS 112 Exercises 115 7 Various Notions of Dimension 117 7.1 Limit Supremum and Limit Infimum 117 7.2 Topological Dimension 121 7.3 Similarity Dimension 125 7.4 Box–Counting Dimension 126 7.5 Hausdorff Measure and Dimension 129 7.6 Miscellaneous Notions of Dimension 134 Exercises 138 8 Porosity and Thickness Looking at the Gaps 141 8.1 The Porosity of a Set 141 8.2 Symmetric Sets and Symmetric Porosity 144 8.3 A New and Different Definition of Cantor Set 147 8.4 Thickness of a Cantor Set 148 8.5 Applying Thickness 149 8.6 A Bit More on Thickness 151 8.7 Porosity in a Metric Space 152 Exercises 154 9 Creating Pathological Functions via C 155 9.1 Sequences of Functions 155 9.2 The Cantor Function 159 9.3 Space–Filling Curves 165 9.4 Baire Class One Functions 169 9.5 Darboux Functions 171 9.6 Linearly Continuous Functions 175 Exercises 178 10 Generalizations and Applications 179 10.1 Generalizing Cantor Sets 179 10.2 Fat Cantor Sets 183 10.3 Sums of Cantor Sets 184 10.4 Differences of Cantor Sets 191 10.5 Products of Cantor Sets 193 10.6 Cantor Target 195 10.7 Ana Sets 196 10.8 Average Distance 199 10.9 Non–Averaging Sets 201 10.10 Cantor Series and Cantor Sets 203 10.11 Liouville Numbers and Irrationality Exponents 205 10.12 Sets of Sums of Convergent Alternating Series 207 10.13 The Monty Hall Problem 209 11 Epilogue 215 References 217
- ISBN: 978-1-118-40571-0
- Editorial: Wiley–Blackwell
- Encuadernacion: Cartoné
- Páginas: 246
- Fecha Publicación: 16/08/2013
- Nº Volúmenes: 1
- Idioma: Inglés