Graphs, Surfaces and Homology

Author: Peter Giblin

Publisher: Cambridge University Press

ISBN: 1139491172

Category: Mathematics

Page: N.A

View: 9757

Homology theory is a powerful algebraic tool that is at the centre of current research in topology and its applications. This accessible textbook will appeal to mathematics students interested in the application of algebra to geometrical problems, specifically the study of surfaces (sphere, torus, Mobius band, Klein bottle). In this introduction to simplicial homology - the most easily digested version of homology theory - the author studies interesting geometrical problems, such as the structure of two-dimensional surfaces and the embedding of graphs in surfaces, using the minimum of algebraic machinery and including a version of Lefschetz duality. Assuming very little mathematical knowledge, the book provides a complete account of the algebra needed (abelian groups and presentations), and the development of the material is always carefully explained with proofs given in full detail. Numerous examples and exercises are also included, making this an ideal text for undergraduate courses or for self-study.

Graphs, Surfaces and Homology

An Introduction to Algebraic Topology

Author: P. Giblin

Publisher: Springer Science & Business Media

ISBN: 9400959532

Category: Juvenile Nonfiction

Page: 329

View: 6500

viii homology groups. A weaker result, sufficient nevertheless for our purposes, is proved in Chapter 5, where the reader will also find some discussion of the need for a more powerful in variance theorem and a summary of the proof of such a theorem. Secondly the emphasis in this book is on low-dimensional examples the graphs and surfaces of the title since it is there that geometrical intuition has its roots. The goal of the book is the investigation in Chapter 9 of the properties of graphs in surfaces; some of the problems studied there are mentioned briefly in the Introduction, which contains an in formal survey of the material of the book. Many of the results of Chapter 9 do indeed generalize to higher dimensions (and the general machinery of simplicial homology theory is avai1able from earlier chapters) but I have confined myself to one example, namely the theorem that non-orientable closed surfaces do not embed in three-dimensional space. One of the principal results of Chapter 9, a version of Lefschetz duality, certainly generalizes, but for an effective presentation such a gener- ization needs cohomology theory. Apart from a brief mention in connexion with Kirchhoff's laws for an electrical network I do not use any cohomology here. Thirdly there are a number of digressions, whose purpose is rather to illuminate the central argument from a slight dis tance, than to contribute materially to its exposition.

Real and Complex Singularities

Author: Ana Claudia Nabarro,Juan J. Nuño-Ballesteros,Raúl Oset Sinha,Maria Aparecida Soares Ruas

Publisher: American Mathematical Soc.

ISBN: 1470422050

Category: Differential geometry -- Classical differential geometry -- Classical differential geometry

Page: 355

View: 9205

This volume is a collection of papers presented at the XIII International Workshop on Real and Complex Singularities, held from July 27–August 8, 2014, in São Carlos, Brazil, in honor of María del Carmen Romero Fuster's 60th birthday. The volume contains the notes from two mini-courses taught during the workshop: on intersection homology by J.-P. Brasselet, and on non-isolated hypersurface singularities and Lê cycles by D. Massey. The remaining contributions are research articles which cover topics from the foundations of singularity theory (including classification theory and invariants) to topology of singular spaces (links of singularities and semi-algebraic sets), as well as applications to topology (cobordism and Lefschetz fibrations), dynamical systems (Morse-Bott functions) and differential geometry (affine geometry, Gauss-maps, caustics, frontals and non-Euclidean geometries). This book is published in cooperation with Real Sociedad Matemática Española (RSME)

A First Course in Topology

Continuity and Dimension

Author: John McCleary

Publisher: American Mathematical Soc.

ISBN: 0821838849

Category: Mathematics

Page: 211

View: 5167

How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincare argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century. The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time.The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension. This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study.

Electromagnetic Theory and Computation

A Topological Approach

Author: Paul Wolfgang Gross,P. Robert Kotiuga

Publisher: Cambridge University Press

ISBN: 9780521801607

Category: Mathematics

Page: 278

View: 1243

Although topology was recognized by Gauss and Maxwell to play a pivotal role in the formulation of electromagnetic boundary value problems, it is a largely unexploited tool for field computation. The development of algebraic topology since Maxwell provides a framework for linking data structures, algorithms, and computation to topological aspects of three-dimensional electromagnetic boundary value problems. This book attempts to expose the link between Maxwell and a modern approach to algorithms. The first chapters lay out the relevant facts about homology and cohomology, stressing their interpretations in electromagnetism. These topological structures are subsequently tied to variational formulations in electromagnetics, the finite element method, algorithms, and certain aspects of numerical linear algebra. A recurring theme is the formulation of and algorithms for the problem of making branch cuts for computing magnetic scalar potentials and eddy currents. Appendices bridge the gap between the material presented and standard expositions of differential forms, Hodge decompositions, and tools for realizing representatives of homology classes as embedded manifolds.

3D Topography

A Simplicial Complex-based Solution in a Spatial DBMS

Author: Friso Penninga

Publisher: N.A

ISBN: N.A

Category: Geomorphology

Page: 192

View: 5915

Physiological Basis of Aging and Geriatrics, Third Edition

Author: S. Timiras Paola

Publisher: CRC Press

ISBN: N.A

Category: Medical

Page: 326

View: 4700

Aging is an inevitable aspect of living. This book covers the aging process from a physiological viewpoint, examining all systems of the body and describing changes that occur with normal aging and in disease. Its primary focus is on humans, but some animal and in vitro research is included to explain mechanisms and theories of aging. Integrated aspects of aging (e.g., demography of aging) are considered, and preventive and interventive measures (pharmacology, diet, exercise) for ensuring healthful aging are discussed. The Second Edition of Physiological Basis of Aging and Geriatrics considers many new approaches and methods for studying the aging process. New and updated information on molecular and cellular mechanisms, pathology, and aging-related diseases is provided to promote a better understanding of basic biological mechanisms. Theories, epidemiology, demographics, and comparative and differential aging are also presented.

British Book News

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Best books

Page: N.A

View: 415

Maß und Kategorie

Author: J.C. Oxtoby

Publisher: Springer-Verlag

ISBN: 364296074X

Category: Mathematics

Page: 112

View: 9263

Dieses Buch behandelt hauptsächlich zwei Themenkreise: Der Bairesche Kategorie-Satz als Hilfsmittel für Existenzbeweise sowie Die "Dualität" zwischen Maß und Kategorie. Die Kategorie-Methode wird durch viele typische Anwendungen erläutert; die Analogie, die zwischen Maß und Kategorie besteht, wird nach den verschiedensten Richtungen hin genauer untersucht. Hierzu findet der Leser eine kurze Einführung in die Grundlagen der metrischen Topologie; außerdem werden grundlegende Eigenschaften des Lebesgue schen Maßes hergeleitet. Es zeigt sich, daß die Lebesguesche Integrationstheorie für unsere Zwecke nicht erforderlich ist, sondern daß das Riemannsche Integral ausreicht. Weiter werden einige Begriffe aus der allgemeinen Maßtheorie und Topologie eingeführt; dies geschieht jedoch nicht nur der größeren Allgemeinheit wegen. Es erübrigt sich fast zu erwähnen, daß sich die Bezeichnung "Kategorie" stets auf "Bairesche Kategorie" be zieht; sie hat nichts zu tun mit dem in der homologischen Algebra verwendeten Begriff der Kategorie. Beim Leser werden lediglich grundlegende Kenntnisse aus der Analysis und eine gewisse Vertrautheit mit der Mengenlehre vorausgesetzt. Für die hier untersuchten Probleme bietet sich in natürlicher Weise die mengentheoretische Formulierung an. Das vorlie gende Buch ist als Einführung in dieses Gebiet der Analysis gedacht. Man könnte es als Ergänzung zur üblichen Grundvorlesung über reelle Analysis, als Grundlage für ein Se minar oder auch zum selbständigen Studium verwenden. Bei diesem Buch handelt es sich vorwiegend um eine zusammenfassende Darstellung; jedoch finden sich in ihm auch einige Verfeinerungen bekannter Resultate, namentlich Satz 15.6 und Aussage 20.4. Das Literaturverzeichnis erhebt keinen Anspruch auf Vollständigkeit. Häufig werden Werke zitiert, die weitere Literaturangaben enthalten.

Gentechnologie für Einsteiger

Author: Terence A. Brown

Publisher: Springer

ISBN: 9783827413024

Category:

Page: 409

View: 3571

T. A. Browns "Gentechnologie fA1/4r Einsteiger" hat sich in mehreren Auflagen international als StandardeinfA1/4hrung in die Genklonierung und DNA-Analyse etabliert. Das Grundlagenlehrbuch setzt nur wenig Kenntnisse in Molekularbiologie und Genetik voraus und erlAutert in klarer Sprache sowohl die fundamentalen Prinzipien der wichtigsten gentechnologischen Methoden als auch die immer breiter werdende Palette ihrer Anwendungen. Die groAe Bedeutung der Gentechnologie fA1/4r die Grundlagenforschung und fA1/4r neue Entwicklungen in Biotechnologie und Medizin wird klar herausgearbeitet. Aoeber 250 zweifarbige Illustrationen erleichtern das VerstAndnis des Stoffes und machen die Konzepte und Arbeitstechniken der Gentechnologen anschaulich. Die neue Auflage ist vollstAndig A1/4berarbeitet und enthAlt zwei neue Kapitel. Eines prAsentiert die Methoden, mit denen man ganze Genome sequenziert, und beschreibt, was man aus den gewonnenen Sequenzinformationen ablesen kann. Das zweite widmet sich den Anwendungen der Gentechnologie in der Gerichtsmedizin - ein spannendes und aufschlussreiches Beispiel des Einsatzpotenzials der DNA-Analyse. Der groAen Bedeutung der Polymerasekettenreaktion (PCR) entsprechend wird diese Technik jetzt in einem frA1/4hen Abschnitt des Buches behandelt und durchgehend stArker berA1/4cksichtigt.

Differentialgeometrie von Kurven und Flächen

Author: Manfredo P. do Carmo

Publisher: Springer-Verlag

ISBN: 3322850722

Category: Technology & Engineering

Page: 263

View: 2673

Inhalt: Kurven - Reguläre Flächen - Die Geometrie der Gauß-Abbildung - Die innere Geometrie von Flächen - Anhang

Groups Acting on Graphs

Author: Warren Dicks,M. J. Dunwoody

Publisher: Cambridge University Press

ISBN: 9780521230339

Category: Mathematics

Page: 283

View: 5820

Originally published in 1989, this is an advanced text and research monograph on groups acting on low-dimensional topological spaces, and for the most part the viewpoint is algebraic. Much of the book occurs at the one-dimensional level, where the topology becomes graph theory. Two-dimensional topics include the characterization of Poincare duality groups and accessibility of almost finitely presented groups. The main three-dimensional topics are the equivariant loop and sphere theorems. The prerequisites grow as the book progresses up the dimensions. A familiarity with group theory is sufficient background for at least the first third of the book, while the later chapters occasionally state without proof and then apply various facts which require knowledge of homological algebra and algebraic topology. This book is essential reading for anyone contemplating working in the subject.

Kategorien und Funktoren

Author: Bodo Pareigis

Publisher: N.A

ISBN: N.A

Category: Categories (Mathematics)

Page: 192

View: 5725

Lineare Funktionalanalysis

Eine anwendungsorientierte Einführung

Author: Hans Wilhelm Alt

Publisher: Springer-Verlag

ISBN: 3662083868

Category: Mathematics

Page: 294

View: 6544

Computational Topology

An Introduction

Author: Herbert Edelsbrunner,John Harer

Publisher: American Mathematical Soc.

ISBN: 0821849255

Category: Mathematics

Page: 241

View: 1383

Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

Differentialgeometrie, Topologie und Physik

Author: Mikio Nakahara

Publisher: Springer-Verlag

ISBN: 3662453002

Category: Science

Page: 597

View: 482

Differentialgeometrie und Topologie sind wichtige Werkzeuge für die Theoretische Physik. Insbesondere finden sie Anwendung in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das vorliegende beliebte Buch, das nun erstmals ins Deutsche übersetzt wurde, ist eine ideale Einführung für Masterstudenten und Forscher im Bereich der theoretischen und mathematischen Physik. - Im ersten Kapitel bietet das Buch einen Überblick über die Pfadintegralmethode und Eichtheorien. - Kapitel 2 beschäftigt sich mit den mathematischen Grundlagen von Abbildungen, Vektorräumen und der Topologie. - Die folgenden Kapitel beschäftigen sich mit fortgeschritteneren Konzepten der Geometrie und Topologie und diskutieren auch deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie. - Daran anschließend findet eine Zusammenführung von Geometrie und Topologie statt: es geht um Faserbündel, characteristische Klassen und Indextheoreme (u.a. in Anwendung auf die supersymmetrische Quantenmechanik). - Die letzten beiden Kapitel widmen sich der spannendsten Anwendung von Geometrie und Topologie in der modernen Physik, nämlich den Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer gemetrischen Perspektive. Mikio Nakahara studierte an der Universität Kyoto und am King’s in London Physik sowie klassische und Quantengravitationstheorie. Heute ist er Physikprofessor an der Kinki-Universität in Osaka (Japan), wo er u. a. über topologische Quantencomputer forscht. Diese Buch entstand aus einer Vorlesung, die er während Forschungsaufenthalten an der University of Sussex und an der Helsinki University of Sussex gehalten hat.

ABPR

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 3438