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Fractals and Universal Spaces in Dimension Theory / by Stephen Lipscomb
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Monographs in Mathematics | SpringerLink BücherPublisher: New York, NY : Springer-Verlag New York, 2009Description: Online-Ressource (XVIII, 242p. 91 illus., 15 illus. in color, digital)ISBN:- 9780387854946
- 1281850950
- 9781281850959
- 514 23
- 514.742
- QA614.86
Contents:
Summary: Construction of = -- Self-Similarity and for Finite -- No-Carry Property of -- Imbedding in Hilbert Space -- Infinite IFS with Attractor -- Dimension Zero -- Decompositions -- The Imbedding Theorem -- Minimal-Exponent Question -- The Imbedding Theorem -- 1992#x2013;2007 -Related Research -- Isotopy Moves into 3-Space -- From 2-Web IFS to 2-Simplex IFS 2-Space and the 1-Sphere -- From 3-Web IFS to 3-Simplex IFS 3-Space and the 2-Sphere.Summary: For metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods — the classical (separable metric) and the modern (not necessarily separable metric). While the classical theory is now well documented in several books, this is the first book to unify the modern theory (1960 – 2007). Like the classical theory, the modern theory fundamentally involves the unit interval. By the 1970s, the author of this monograph generalized Cantor’s 1883 construction (identify adjacent-endpoints in Cantor’s set) of the unit interval, obtaining — for any given weight — a one-dimensional metric space that contains rationals and irrationals as counterparts to those in the unit interval. Following the development of fractal geometry during the 1980s, these new spaces turned out to be the first examples of attractors of infinite iterated function systems — “generalized fractals.” The use of graphics to illustrate the fractal view of these spaces is a unique feature of this monograph. In addition, this book provides historical context for related research that includes imbedding theorems, graph theory, and closed imbeddings. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. It can also serve as a text for graduate seminars or self-study — the interested reader will find many relevant open problems that will motivate further research into these topics.PPN: PPN: 1647563925Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Contents; Preface; Introduction; CHAPTER 1 Construction of JA = Ja; CHAPTER 2 Self-Similarity and Jn+1 for Finite n; CHAPTER 3 No-Carry Property of ?A; CHAPTER 4 Imbedding JA in Hilbert Space; CHAPTER 5 Infinite IFS with Attractor ?A; CHAPTER 6 Dimension Zero; CHAPTER 7 Decompositions1; CHAPTER 8 The Jn+1A Imbedding Theorem; CHAPTER 9 Minimal-Exponent Question; CHAPTER 10 The J8A Imbedding Theorem; CHAPTER 11 1992-2007 JA-Related Research; Illustration of Chapter 12 IsotopyThat Moves J5 into 3-space1; CHAPTER 12 Isotopy Moves J5 into 3-Space
CHAPTER 13 From 2-Web IFS to 2-Simplex IFS2-Space and the 1-SphereCHAPTER 14 From 3-Web IFS to 3-Simplex IFS3-Space and the 2-Sphere; APPENDIX 1 Background Basics; APPENDIX 2 The Standard Simplex ?A in l2(A); APPENDIX 3 Measures and Fractal Dimension; Bibliography; Index;
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