{"product_id":"the-elements-of-cantor-sets-9781118405710","title":"The Elements of Cantor Sets","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eA systematic and integrated approach to Cantor Sets and their applications to various branches of mathematics\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eThe Elements of Cantor Sets: With Applications\u003c\/i\u003e features a thorough introduction to Cantor Sets and applies these sets as a bridge between real analysis, probability, topology, and algebra.\u003c\/p\u003e \u003cp\u003eThe author fills a gap in the current literature by providing an introductory and integrated perspective, thereby preparing readers for further study and building a deeper understanding of analysis, topology, set theory, number theory, and algebra.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eThe Elements of Cantor Sets\u003c\/i\u003e provides coverage of:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eBasic definitions and background theorems as well as comprehensive mathematical details\u003c\/li\u003e \u003cli\u003eA biography of Georg Ferdinand Ludwig Philipp Cantor, one of the most significant mathematicians of the last century\u003c\/li\u003e \u003cli\u003eChapter coverage of fractals and self-similar sets, sums of Cantor Sets, the role of Cantor Sets in creating pathological\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“This book could be used as a text for an undergraduate survey course of higher mathematics.  It is an excellent reference for a graduate student, researcher or university instructor.”  (\u003ci\u003eAmerican Mathematical Society\u003c\/i\u003e, 1 March 2015)\u003c\/p\u003e \u003cp\u003e“Summing Up: Recommended.  Upper-division undergraduates through researchers\/faculty.”  (\u003ci\u003eChoice\u003c\/i\u003e, 1 March 2014)\u003c\/p\u003e \u003cp\u003e \u003c\/p\u003e\n\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eForeword xiii\u003c\/p\u003e \u003cp\u003ePreface xv\u003c\/p\u003e \u003cp\u003eAcknowledgments xvii\u003c\/p\u003e \u003cp\u003eIntroduction xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 A Quick Biography of Cantor 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Basics 5\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Review 5\u003c\/p\u003e \u003cp\u003eExercises 14\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Introducing the Cantor Set 17\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Some Definitions and Basics 17\u003c\/p\u003e \u003cp\u003e3.2 Size of a Cantor Set 21\u003c\/p\u003e \u003cp\u003e3.3 Large and Small 46\u003c\/p\u003e \u003cp\u003eExercises 48\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Cantor Sets and Continued Fractions 51\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Introducing Continued Fractions 52\u003c\/p\u003e \u003cp\u003e4.2 Constructing a Cantor Set 59\u003c\/p\u003e \u003cp\u003e4.3 Diophantine Equations 60\u003c\/p\u003e \u003cp\u003e4.4 Miscellaneous 63\u003c\/p\u003e \u003cp\u003eExercises 65\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 p-adic Numbers and Valuations 67\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Some Abstract Algebra 67\u003c\/p\u003e \u003cp\u003e5.2 p-adic Numbers 72\u003c\/p\u003e \u003cp\u003e5.3 p-adic Integers and Cantor Sets 80\u003c\/p\u003e \u003cp\u003e5.4 p-adic Rational Numbers 82\u003c\/p\u003e \u003cp\u003eExercises 88\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Self-Similar Objects 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 The Meaning of Self-Similar 91\u003c\/p\u003e \u003cp\u003e6.2 Metric Spaces 92\u003c\/p\u003e \u003cp\u003e6.3 Sequences in (\u003ci\u003eS; d\u003c\/i\u003e) 97\u003c\/p\u003e \u003cp\u003e6.4 Affine Transformations 106\u003c\/p\u003e \u003cp\u003e6.5 An Application for an IFS 112\u003c\/p\u003e \u003cp\u003eExercises 115\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Various Notions of Dimension 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Limit Supremum and Limit Infimum 117\u003c\/p\u003e \u003cp\u003e7.2 Topological Dimension 121\u003c\/p\u003e \u003cp\u003e7.3 Similarity Dimension 125\u003c\/p\u003e \u003cp\u003e7.4 Box-Counting Dimension 126\u003c\/p\u003e \u003cp\u003e7.5 Hausdorff Measure and Dimension 129\u003c\/p\u003e \u003cp\u003e7.6 Miscellaneous Notions of Dimension 134\u003c\/p\u003e \u003cp\u003eExercises 138\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Porosity and Thickness Looking\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eat the Gaps 141\u003c\/p\u003e \u003cp\u003e8.1 The Porosity of a Set 141\u003c\/p\u003e \u003cp\u003e8.2 Symmetric Sets and Symmetric Porosity 144\u003c\/p\u003e \u003cp\u003e8.3 A New and Different Definition of Cantor Set 147\u003c\/p\u003e \u003cp\u003e8.4 Thickness of a Cantor Set 148\u003c\/p\u003e \u003cp\u003e8.5 Applying Thickness 149\u003c\/p\u003e \u003cp\u003e8.6 A Bit More on Thickness 151\u003c\/p\u003e \u003cp\u003e8.7 Porosity in a Metric Space 152\u003c\/p\u003e \u003cp\u003eExercises 154\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Creating Pathological Functions via\u003c\/b\u003e C \u003cb\u003e155\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Sequences of Functions 155\u003c\/p\u003e \u003cp\u003e9.2 The Cantor Function 159\u003c\/p\u003e \u003cp\u003e9.3 Space-Filling Curves 165\u003c\/p\u003e \u003cp\u003e9.4 Baire Class One Functions 169\u003c\/p\u003e \u003cp\u003e9.5 Darboux Functions 171\u003c\/p\u003e \u003cp\u003e9.6 Linearly Continuous Functions 175\u003c\/p\u003e \u003cp\u003eExercises 178\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Generalizations and Applications 179\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Generalizing Cantor Sets 179\u003c\/p\u003e \u003cp\u003e10.2 Fat Cantor Sets 183\u003c\/p\u003e \u003cp\u003e10.3 Sums of Cantor Sets 184\u003c\/p\u003e \u003cp\u003e10.4 Differences of Cantor Sets 191\u003c\/p\u003e \u003cp\u003e10.5 Products of Cantor Sets 193\u003c\/p\u003e \u003cp\u003e10.6 Cantor Target 195\u003c\/p\u003e \u003cp\u003e10.7 Ana Sets 196\u003c\/p\u003e \u003cp\u003e10.8 Average Distance 199\u003c\/p\u003e \u003cp\u003e10.9 Non-Averaging Sets 201\u003c\/p\u003e \u003cp\u003e10.10 Cantor Series and Cantor Sets 203\u003c\/p\u003e \u003cp\u003e10.11 Liouville Numbers and Irrationality Exponents 205\u003c\/p\u003e \u003cp\u003e10.12 Sets of Sums of Convergent Alternating Series 207\u003c\/p\u003e \u003cp\u003e10.13 The Monty Hall Problem 209\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Epilogue 215\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eReferences 217\u003c\/p\u003e\n\u003c\/li\u003e\n\u003c\/ul\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49406865080663,"sku":"9781118405710","price":75.56,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118405710.jpg?v=1730497379","url":"https:\/\/bookcurl.com\/products\/the-elements-of-cantor-sets-9781118405710","provider":"Book Curl","version":"1.0","type":"link"}