Set Theory

Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780821838358

Category: Mathematics

Page: 352

View: 3700

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.

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: 7792

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.

Set Theory and Metric Spaces

Author: Irving Kaplansky

Publisher: American Mathematical Soc.

ISBN: 0821826948

Category: Mathematics

Page: 140

View: 315

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.

Topology with Applications

Topological Spaces via Near and Far

Author: Somashekhar A Naimpally,James F Peters

Publisher: World Scientific

ISBN: 9814407674

Category: Mathematics

Page: 296

View: 9142

The principal aim of this book is to introduce topology and its many applications viewed within a framework that includes a consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces. This book provides a complete framework for the study of topology with a variety of applications in science and engineering that include camouflage filters, classification, digital image processing, forgery detection, Hausdorff raster spaces, image analysis, microscopy, paleontology, pattern recognition, population dynamics, stem cell biology, topological psychology, and visual merchandising. It is the first complete presentation on topology with applications considered in the context of proximity spaces, and the nearness and remoteness of sets of objects. A novel feature throughout this book is the use of near and far, discovered by F Riesz over 100 years ago. In addition, it is the first time that this form of topology is presented in the context of a number of new applications. Contents:Basic FrameworkWhat is Topology?Symmetric ProximityContinuity and Proximal ContinuitySeparation AxiomsUniform Spaces, Filters and NetsCompactness and Higher Separation AxiomsInitial and Final Structures, EmbeddingGrills, Clusters, Bunches and Proximal Wallman CompactificationExtensions of Continuous Functions: Taimanov TheoremMetrisationFunction Space TopologiesHyperspace TopologiesSelected Topics: Uniformity and Metrisation Readership: 3rd year undergraduate students, graduate students and researchers in topology; professional and practitioners who are interested in applying topology and its applications especially in science and engineering. Keywords:Applications;Close;Far;Near;Nearness;Remoteness;Proximity;Set Theory;TopologyKey Features:Complete overview of famous results in topologyFirst topology textbook to link proximity space theory (nearness and remoteness) with well-known results in topologyPresentation of a collection of new applications in a variety of areas such as digital image analysis, stem cell biology, visual merchandising, forgery and paleontologyThe materials in the book have been class-tested over the past thirty years by the authorsReviews: “The book contains a lot of mathematical material from different fields that can complement and enrich a more standard brief introduction into the field of general topology.” Zentralblatt MATH

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: 1120

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.

Leopold Kronecker's Werke

Author: K. Hensel

Publisher: American Mathematical Soc.

ISBN: 0821849824

Category: Mathematics

Page: 509

View: 9701

Part 4: Uber die algebraisch auflosbaren Gleichungen I; Note sur les fonctions semblables des racines d'une equation; Sur quelques fonctions symetriques et sur les nombres de Bernoulli; Uber die algebraisch auflosbaren Gleichungen II; Uber Gleichungen des siebenten Grades; Sur la resolution de l'equation du cinquieme degree; Sur la theorie des substitutions; Mitteilung uber algebraische Arbeiten; Uber Abelsche Gleichungen; Einige entwickelungen aus der Theorie der algebraischen Gleichungen; Uber die symmetrischen Funktionen; Die Composition Abelscher Gleichungen; Die kubischen Abelschen Gleichungen des Bereichs $(\sqrt{-31})$; Zur Theorie der Abelschen Gleichungen; Uber eine Stelle in Jacobi's Aufsatz ""Observatiunculae ad theoriam aequationum pertinentes""; Sur une formule de Gauss; Uber die elliptischen Functionen, fur welche complexe Multiplication stattfindet; Uber die Anzahl der verschiedenen Classen quadratischer Formen von negativer Determinante; Uber eine neue Eigenschaft der quadratischen Formen von negativer Determinante; Uber die complexe Multiplication der elliptischen Functionen; Uber die Auflosung der Pell'schen Gleichung mittels elliptischer Functionen; Uber den Gebrauch der Dirichlet'schen Methoden in der Theorie der quadratischen Formen; Uber quadratische Formen von negativer Determinante; Uber die algebraischen Gleichungen, von denen die Theilung der elliptischen Functionen abhangt; Uber den vierten Gauss'schen Beweis des Reciprocitatsgesetzes fur die quadratischen Reste; Summirung der Gauss'schen Reihen $\sum^{h=n-1}_{h=0}e^{2h^2\pi i/n}$; Uber die Dirichlet'sche Methode der Werthbestimmung der Gauss'schen Reihen; Zur Theorie der elliptischen Functionen; Bemerkungen uber die Multiplication der elliptischen Functionen; Weitere Bemerkungen uber der elliptischen Functionen; Zur Theorie der elliptischen Functionen. (CHEL/224.4.H)

Hausdorff on Ordered Sets

Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780821890516

Category: Mathematics

Page: 322

View: 5184

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.


Author: Karl Menger,Georg Nöbeling

Publisher: American Mathematical Soc.

ISBN: 9780828401722

Category: Curves

Page: 374

View: 3184

This classic book is a treatise on the topology of curves. The class of curves considered is quite broad, including smooth curves, rational curves, trees, Cantor curves and so on. It was one of a small handful of landmark books on topology, in particular point-set topology, that were published during the important period of the 1930s. Many of the properties of curves explored by Menger are of renewed importance today in various contexts, notably the topology of dynamics.

Grundzüge der Mengenlehre

Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780828400619

Category: Mathematics

Page: 476

View: 9169

This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.


Aus dem Englischen übersetzt von Annette A’Campo

Author: Michael Artin

Publisher: Birkhäuser

ISBN: 9783764359386

Category: Mathematics

Page: 705

View: 802

Important though the general concepts and propositions may be with which the modem and industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics, and that to master their difficulties requires on the whole the harder labor. HERMANN WEYL Die Arbeit an diesem Buch begann vor etwa zwanzig Jahren mit Aufzeichnungen zur Ergänzung meiner Algebravorlesungen. Ich wollte einige konkrete Themen, wie Symmetrie, lineare Gruppen und quadratische Zahlkörper, ausführlicher be­ handeln als dies im vorgesehenen Text der Fall war, und darüberhinaus wollte ich den Schwerpunkt in der Gruppentheorie von den Permutationsgruppen auf Matrixgruppen verlagern. Ein anderes ständig wiederkehrendes Thema, nämlich Gitter, sind spontan aufgetaucht. Ich hoffte, der konkrete Stoff könne das Interesse der Studenten wecken und gleichzeitig die Abstraktionen verständlicher machen, kurz gesagt, sie sollten weiter kommen, indem sie beides gleichzeitig lernten. Das bewährte sich gut. Es dauerte einige Zeit, bis ich entschieden hatte, welche Themen ich behandeln wollte, und allmählich verteilte ich mehr und mehr Aufzeichnungen und ging schließlich dazu über, die ganze Vorlesung mit diesem Skript zu bestrei­ ten. Auf diese Weise ist ein Buch entstanden, das, wie ich meine, etwas anders ist als die existierenden Bücher. Allerdings haben mir die Probleme, die ich damit hatte, die einzelnen Teile des Buches zu einem Ganzen zusammenzufügen, einige Kopfschmerzen bereitet; ich kann also nicht empfehlen, auf diese Art anzufangen, ein Buch zu schreiben.

Gesammelte Abhandlungen

Author: Oswald Teichmüller,Lars Valerian Ahlfors,Frederick W. Gehring

Publisher: N.A


Category: Mathematics

Page: 751

View: 2310


Geometry and the Imagination

Author: David Hilbert,Stephan Cohn-Vossen

Publisher: University of Pennsylvania Press

ISBN: 9780821819982

Category: Mathematics

Page: 357

View: 960

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer - even after more than half a century has passed! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. 'Hilbert and Cohn-Vossen' is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces.The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 1/5 - 1/7 - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.One of the most remarkable chapters is 'Projective Configurations'. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Gottingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the 'pantheon' of great mathematics books.

Lehrbuch Der Algebra

Author: Heinrich Weber

Publisher: American Mathematical Soc.

ISBN: 0821832581

Category: Mathematics

Page: 703

View: 6587

Weber's three-volume set on algebra was for many years the standard text on algebra. Published at the end of the nineteenth century, it helped usher group theory to a central place in twentieth century mathematics. Volume 1 covers foundational material. Volume 2 covers group theory and its applications, plus the theory of algebraic numbers. Volume 3 covers advanced topics, such as algebraic functions, elliptic functions and class field theory. Although notations have changed somewhat and algebra has become more abstract that it was in Weber's day, many of the same themes and ideas important today are central topics in Weber's book, which may be why it has become a classic.

Vorlesungen Über Die Theorie Der Elliptischen Modulfunctionen: Bd. Fortbildung Und Anwendung Der Theorie

Author: Felix Klein

Publisher: Wentworth Press

ISBN: 9780270545777

Category: History

Page: 736

View: 4859

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Seminar of mathematical analysis

proceedings, Universities of Malaga and Seville (Spain), september 2003-february 2004

Author: Daniel Girela Álvarez,Genaro López Acedo,Rafael Villa Caro

Publisher: Universidad de Sevilla

ISBN: 9788447208579

Category: Mathematics

Page: 320

View: 6616

This volume consists of the lecture notes of the Seminar on Mathematical Analysis which was held at the Universities of Malaga and Seville, Septembre 2002-February 2003.

Felix Hausdorff-Gesammelte Werke

Band IV: Analysis, Algebra und Zahlentheorie

Author: Felix Hausdorff

Publisher: Springer

ISBN: 9783642639920

Category: Mathematics

Page: 554

View: 7979

Felix Hausdorff gehört zu den herausragenden Mathematikern der ersten Hälfte des 20. Jahrhunderts. Eine Gesamtausgabe seiner Werke galt lange als Desideratum. Die auf 8 Bände veranschlagte Edition wird Hausdorffs gesamtes publiziertes Opus enthalten, ferner eine Reihe bemerkenswerter Stücke aus dem umfangreichen wissenschaftlichen Nachlaß. Alle Texte werden von Fachleuten auf den einzelnen Gebieten sorgfältig kommentiert; an dieser Arbeit sind mehr als 20 Mathematiker, Mathematikhistoriker, Astronomen, Philosophen und Literaturwissenschaftler aus vier Staaten beteiligt. Der vorliegende Band IV enthält Hausdorffs Arbeiten zur Analysis, Algebra und Zahlentheorie, darunter die klassischen auch heute noch vielzitierten Texte zu Hausdorff-Maß und Hausdorff-Dimension und zum Hausdorffschen Kugelparadoxon. Aus dem Nachlaß werden 19 Faszikel publiziert, ferner einige interessante Briefe.


Author: Felix R. Gantmacher

Publisher: Springer-Verlag

ISBN: 3642712436

Category: Mathematics

Page: 654

View: 5819