E-Book Overview
This two-volume textbook Comprehensive Mathematics for the Working Computer Scientist is a self-contained comprehensive presentation of mathematics including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. Despite the theoretical soundness, the material stresses a large number of core computer science subjects, such as, for example, a discussion of floating point arithmetic, Backus-Naur normal forms, L-systems, Chomsky hierarchies, algorithms for data encoding, e.g., the Reed-Solomon code. The numerous course examples are motivated by computer science and bear a generic scientific meaning. This text is complemented by an online university course which covers the same theoretical content, however, in a totally different presentation. The student or working scientist who once gets involved in this text may at any time consult the online interface which comprises applets and other interactive tools.
E-Book Content
Guerino Mazzola · Gérard Milmeister Jody Weissmann
Comprehensive Mathematics for Computer Scientists 2 Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda Calculus With 114 Figures
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Guerino Mazzola Gérard Milmeister Jody Weissmann Department of Informatics University of Zurich Winterthurerstr. 190 8057 Zurich, Switzerland
The text has been created using LATEX 2ε . The graphics were drawn using the open source illustrating software Dia and Inkscape, with a little help from Mathematica. The main text has been set in the Y&Y Lucida Bright type family, the heading in Bitstream Zapf Humanist 601. Library of Congress Control Number: 2004102307
Mathematics Subject Classification (1998): 00A06 ISBN 3-540-20861-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg Typesetting: Camera ready by the authors Production: LE-TEX Jelonek, Schmidt & V¨ ockler GbR, Leipzig Printed on acid-free paper 40/3142YL - 5 4 3 2 1 0
Preface This second volume of a comprehensive tour through mathematical core subjects for computer scientists completes the first volume in two regards: Part III first adds topology, differential, and integral calculus to the topics of sets, graphs, algebra, formal logic, machines, and linear geometry, of volume 1. With this spectrum of fundamentals in mathematical education, young professionals should be able to successfully attack more involved subjects, which may be relevant to the computational sciences. In a second regard, the end of part III and par