Set Theory

Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780821838358

Category: Mathematics

Page: 352

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In the early twentieth century, Hausdorff developed an axiomatic approach to topology, which continues to be the foundation of modern topology. The present book, the English translation of the third edition of Hausdorff's Mengenlehre, is a thorough introduction to his theory of point-set topology. The treatment begins with topics in the foundations of mathematics, including the basics of abstract set theory, sums and products of sets, cardinal and ordinal numbers, and Hausdorff's well-ordering theorem. The exposition then specializes to point sets, where major topics such as Borel systems, first and second category, and connectedness are considered in detail. Next, mappings between spaces are introduced. This leads naturally to a discussion of topological spaces and continuous mappings between them. Finally, the theory is applied to the study of real functions and their properties. The book does not presuppose any mathematical knowledge beyond calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first-year graduate students should have no difficulty in making the material their own.
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Topology of Digital Images

Visual Pattern Discovery in Proximity Spaces

Author: James F. Peters

Publisher: Springer Science & Business Media

ISBN: 3642538452

Category: Computers

Page: 411

View: 8831

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This book carries forward recent work on visual patterns and structures in digital images and introduces a near set-based a topology of digital images. Visual patterns arise naturally in digital images viewed as sets of non-abstract points endowed with some form of proximity (nearness) relation. Proximity relations make it possible to construct uniform topologies on the sets of points that constitute a digital image. In keeping with an interest in gaining an understanding of digital images themselves as a rich source of patterns, this book introduces the basics of digital images from a computer vision perspective. In parallel with a computer vision perspective on digital images, this book also introduces the basics of proximity spaces. Not only the traditional view of spatial proximity relations but also the more recent descriptive proximity relations are considered. The beauty of the descriptive proximity approach is that it is possible to discover visual set patterns among sets that are non-overlapping and non-adjacent spatially. By combining the spatial proximity and descriptive proximity approaches, the search for salient visual patterns in digital images is enriched, deepened and broadened. A generous provision of Matlab and Mathematica scripts are used in this book to lay bare the fabric and essential features of digital images for those who are interested in finding visual patterns in images. The combination of computer vision techniques and topological methods lead to a deep understanding of images.
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Parallel Computing Technologies

15th International Conference, PaCT 2019, Almaty, Kazakhstan, August 19–23, 2019, Proceedings

Author: Victor Malyshkin

Publisher: Springer

ISBN: 3030256367

Category:

Page: N.A

View: 1189

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Innovations in Intelligent Image Analysis

Author: Halina Kwasnicka,Lakhmi C. Jain

Publisher: Springer Science & Business Media

ISBN: 3642179339

Category: Computers

Page: 312

View: 5858

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This book presents an introduction to new and important research in the images processing and analysis area. It is hoped that this book will be useful for scientists and students involved in many aspects of image analysis. The book does not attempt to cover all of the aspects of Computer Vision, but the chapters do present some state of the art examples.
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Computing and Combinatorics

18th Annual International Conference, COCOON 2012, Sydney, Australia, August 20-22, 2012, Proceedings

Author: Joachim Gudmundsson,Julián Mestre,Taso Viglas

Publisher: Springer

ISBN: 3642322417

Category: Computers

Page: 606

View: 9812

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This book constitutes the refereed proceedings of the 18th Annual International Conference on Computing and Combinatorics, held in Sydney, Australia, in August 2012. The 50 revised full papers presented were carefully reviewed and selected from 121 submissions. Topics covered are algorithms and data structures; algorithmic game theory and online algorithms; automata, languages, logic, and computability; combinatorics related to algorithms and complexity; complexity theory; computational learning theory and knowledge discovery; cryptography, reliability and security, and database theory; computational biology and bioinformatics; computational algebra, geometry, and number theory; graph drawing and information visualization; graph theory, communication networks, and optimization.
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Hausdorff on Ordered Sets

Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780821890516

Category: Mathematics

Page: 322

View: 3192

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Georg Cantor, the founder of set theory, published his last paper on sets in 1897. In 1900, David Hilbert made Cantor's Continuum Problem and the challenge of well-ordering the real numbers the first problem of his famous lecture at the international congress in Paris. Thus, as the nineteenth century came to a close and the twentieth century began, Cantor's work was finally receiving its due and Hilbert had made one of Cantor's most important conjectures his number one problem. It was time for the second generation of Cantorians to emerge. Foremost among this group were Ernst Zermelo and Felix Hausdorff. Zermelo isolated the Choice Principle, proved that every set could be well-ordered, and axiomatized the concept of set. He became the father of abstract set theory. Hausdorff eschewed foundations and developed set theory as a branch of mathematics worthy of study in its own right, capable of supporting both general topology and measure theory. He is recognized as the era's leading Cantorian. Hausdorff published seven articles in set theory during the period 1901-1909, mostly about ordered sets. This volume contains translations of these papers with accompanying introductory essays. They are highly accessible, historically significant works, important not only for set theory, but also for model theory, analysis and algebra. This book is suitable for graduate students and researchers interested in set theory and the history of mathematics. Also available from the AMS by Felix Hausdorff are the classic work, Grundzuge der Mengenlehre, and its English translation, Set Theory, as Volume 69 and Volume 119 in the AMS Chelsea Publishing series. Information for our distributors: Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners.
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Set Theory and Metric Spaces

Author: Irving Kaplansky

Publisher: American Mathematical Soc.

ISBN: 0821826948

Category: Mathematics

Page: 140

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This is a book that could profitably be read by many graduate students or by seniors in strong major programs ... has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. ... There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem ... The presentation of metric spaces before topological spaces ... should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar. --Canadian Mathematical Bulletin Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent. -- Lance Small, UC San Diego This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.
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Matters Mathematical

Author: I. N. Herstein,Irving Kaplansky

Publisher: American Mathematical Soc.

ISBN: 9780828403009

Category: Mathematics

Page: 246

View: 5172

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From the Preface: ``This book is based on notes prepared for a course at the University of Chicago. The course was intended for nonmajors whose mathematical training was somewhat limited ... Mastery of the material requires nothing beyond algebra and geometry normally covered in high school ... [I]t could be used in courses designed for students who intend to teach mathematics ... We want the reader to see mathematics as a living subject in which new results are constantly being obtained.''
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Lebesgue's Theory of Integration

Its Origins and Development

Author: Thomas Hawkins

Publisher: American Mathematical Soc.

ISBN: 9780821829639

Category: Mathematics

Page: 227

View: 7486

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In this book, Hawkins elegantly places Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries. Hawkins was awarded the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for notable exposition and exceptional scholarship in the history of mathematics.
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