Special Topics in Mathematics for Computer Scientists

Sets, Categories, Topologies and Measures

Author: Ernst-Erich Doberkat

Publisher: Springer

ISBN:

Category: Mathematics

Page: 719

View: 657

This textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems. Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Gödel’s Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics. Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.

An Introduction to Algebraic Topology

Author: Joseph J. Rotman

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 437

View: 831

A clear exposition, with exercises, of the basic ideas of algebraic topology. Suitable for a two-semester course at the beginning graduate level, it assumes a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.

Basic Topology

Author: M.A. Armstrong

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 251

View: 524

In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for their calculating. Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over 139 illustrations and more than 350 problems of various difficulties help students gain a thorough understanding of the subject.

Combinatorial Topology

Author: Pavel S. Aleksandrov

Publisher: Courier Corporation

ISBN:

Category: Mathematics

Page: 148

View: 841

Clearly written, well-organized, 3-part text begins by dealing with certain classic problems without using the formal techniques of homology theory and advances to the central concept, the Betti groups. Numerous detailed examples.

Urbane Topologie

Author: Joachim Huber

Publisher:

ISBN:

Category: Architecture and society

Page: 574

View: 349

Basic topology

Author: Mark Anthony Armstrong

Publisher: McGraw-Hill Companies

ISBN:

Category: Mathematics

Page: 251

View: 739