Elements of Cantor Sets : With Applications.
Title:
Elements of Cantor Sets : With Applications.
Author:
Vallin, Robert W.
ISBN:
9781118548707
Personal Author:
Edition:
1st ed.
Physical Description:
1 online resource (248 pages)
Contents:
Cover -- Title Page -- Copyright Page -- CONTENTS -- Foreword -- Preface -- Acknowledgments -- Introduction -- 1 A Quick Biography of Cantor -- 2 Basics -- 2.1 Review -- Exercises -- 3 Introducing the Cantor Set -- 3.1 Some Definitions and Basics -- 3.2 Size of a Cantor Set -- 3.2.1 Cardinality -- 3.2.2 Category -- 3.2.3 Measure -- 3.3 Large and Small -- Exercises -- 4 Cantor Sets and Continued Fractions -- 4.1 Introducing Continued Fractions -- 4.2 Constructing a Cantor Set -- 4.3 Diophantine Equations -- 4.4 Miscellaneous -- Exercises -- 5 p-adic Numbers and Valuations -- 5.1 Some Abstract Algebra -- 5.2 p-adic Numbers -- 5.2.1 An Analysis Point of View -- 5.2.2 An Algebra Point of View -- 5.3 p-adic Integers and Cantor Sets -- 5.4 p-adic Rational Numbers -- Exercises -- 6 Self-Similar Objects -- 6.1 The Meaning of Self-Similar -- 6.2 Metric Spaces -- 6.3 Sequences in (S, d) -- 6.4 Affine Transformations -- 6.5 An Application for an IFS -- Exercises -- 7 Various Notions of Dimension -- 7.1 Limit Supremum and Limit Infimum -- 7.2 Topological Dimension -- 7.3 Similarity Dimension -- 7.4 Box-Counting Dimension -- 7.5 Hausdorff Measure and Dimension -- 7.6 Miscellaneous Notions of Dimension -- Exercises -- 8 Porosity and Thickness-Looking at the Gaps -- 8.1 The Porosity of a Set -- 8.2 Symmetric Sets and Symmetric Porosity -- 8.3 A New and Different Definition of Cantor Set -- 8.4 Thickness of a Cantor Set -- 8.5 Applying Thickness -- 8.6 A Bit More on Thickness -- 8.7 Porosity in a Metric Space -- Exercises -- 9 Creating Pathological Functions via C -- 9.1 Sequences of Functions -- 9.2 The Cantor Function -- 9.3 Space-Filling Curves -- 9.4 Baire Class One Functions -- 9.5 Darboux Functions -- 9.6 Linearly Continuous Functions -- Exercises -- 10 Generalizations and Applications -- 10.1 Generalizing Cantor Sets -- 10.2 Fat Cantor Sets.
10.3 Sums of Cantor Sets -- 10.4 Differences of Cantor Sets -- 10.5 Products of Cantor Sets -- 10.6 Cantor Target -- 10.7 Ana Sets -- 10.8 Average Distance -- 10.9 Non-Averaging Sets -- 10.10 Cantor Series and Cantor Sets -- 10.11 Liouville Numbers and Irrationality Exponents -- 10.12 Sets of Sums of Convergent Alternating Series -- 10.13 The Monty Hall Problem -- 11 Epilogue -- References -- Index.
Abstract:
A systematic and integrated approach to Cantor Sets and their applications to various branches of mathematics The Elements of Cantor Sets: With Applications features a thorough introduction to Cantor Sets and applies these sets as a bridge between real analysis, probability, topology, and algebra. The 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. The Elements of Cantor Sets provides coverage of: Basic definitions and background theorems as well as comprehensive mathematical details A biography of Georg Ferdinand Ludwig Philipp Cantor, one of the most significant mathematicians of the last century Chapter coverage of fractals and self-similar sets, sums of Cantor Sets, the role of Cantor Sets in creating pathological functions, p-adic numbers, and several generalizations of Cantor Sets A wide spectrum of topics from measure theory to the Monty Hall Problem An ideal text for courses in real analysis, topology, algebra, and set theory for undergraduate and graduate-level courses within mathematics, computer science, engineering, and physics departments, The Elements of Cantor Sets is also appropriate as a useful reference for researchers and secondary mathematics education majors..
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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Electronic Access:
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