Comprehensive Mathematics for Computer Scientist 2.pdf

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

123

Guerino Mazzola

Gérard Milmeister

Jody Weissmann

Department of Informatics

University of Zurich

Winterthurerstr. 190

8057 Zurich, Switzerland

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Library of Congress Control Number: 2004102307

Mathematics Subject Classiﬁcation (1998): 00A06

ISBN 3-540-20861-5 Springer Berlin Heidelberg New York

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Preface

This second volume of a comprehensive tour through mathematical core

subjects for computer scientists completes the ﬁrst volume in two re-

gards:

Part III ﬁrst adds topology, diﬀerential, and integral calculus to the top-

ics of sets, graphs, algebra, formal logic, machines, and linear geometry,

of volume 1. With this spectrum of fundamentals in mathematical edu-

cation, 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 part IV add a selection of more

advanced topics. In view of the overwhelming variety of mathematical

approaches in the computational sciences, any selection, even the most

empirical, requires a methodological justiﬁcation. Our primary criterion

has been the search for harmonization and optimization of thematic di-

versity and logical coherence. This is why we have, for instance, bundled

such seemingly distant subjects as recursive constructions, ordinary dif-

ferential equations, and fractals under the unifying perspective of con-

traction theory.

For the same reason, the entry point to part IV is category theory. The

reader will recognize that a huge number of classical results presented

in volume 1 are perfect illustrations of the categorical point of view,

which will deﬁnitely dominate the language of mathematics and theo-

retical computer science of the decades to come. Categories are advan-

tageous or even mandatory for a thorough understanding of higher sub-

jects, such as splines, fractals, neural networks, and λ -calculus. Even for

the specialist, our presentation may here and there oﬀer a fresh view on

classical subjects. For example, the systematic usage of categorical limits

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