Author:
Vallin, Robert W.
Edition Statement:First edition.
Imprint:Hoboken, New Jersey : John Wiley & Sons, Inc., [2013]
Description225 pages : illustrations ; 24 cm.
Note:1. 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. Analysis Point of View -- 5.2.2. 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. Meaning of Self-Similar -- 6.2. Metric Spaces -- 6.3. Sequences in (S, d) -- 6.4. Affine Transformations -- 6.5. 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. Porosity of a Set -- 8.2. Symmetric Sets and Symmetric Porosity -- 8.3. New and Different Definition of Cantor Set -- 8.4. Thickness of a Cantor Set -- 8.5. Applying Thickness -- 8.6. 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. 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. Monty Hall Problem -- 11. Epilogue.
Bibliography Note:Includes bibliographical references (p. 219-222) and index.
Note:"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"-- Provided by publisher.
Note:Recommended in Resources for College Libraries
