Comprehensive Mathematics for Computer Scientists 2nd ed [Vol 2] - G. Mazzola, et al., (Springer, 2006) WW.pdf

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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
The text has been created using A T E X2 ε . The graphics were drawn using the open
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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
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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
VI
Preface
in neural networks has enabled an original formal restatement of Hebbian
learning, perceptron convergence, and the back-propagation algorithm.
However, a secondary, but no less relevant selection criterion has been
applied. It concerns the delimitation from subjects which may be very
important for certain computational sciences, but which seem to be nei-
ther mathematically nor conceptually of germinal power. In this spirit, we
have also refrained from writing a proper course in theoretical computer
science or in statistics. Such an enterprise would anyway have exceeded
by far the volume of such a work and should be the subject of a specific
education in computer science or applied mathematics. Nonetheless, the
reader will find some interfaces to these topics not only in volume 1, but
also in volume 2, e.g., in the chapters on probability theory, in spline the-
ory, and in the final chapter on λ -calculus, which also relates to partial
recursive functions and to λ -calculus as a programming language.
We should not conclude this preface without recalling the insight that
there is no valid science without a thorough mathematical culture .One
of the most intriguing illustrations of this universal, but often surprising
presence of mathematics is the theory of Lie derivatives and Lie brack-
ets, which the beginner might reject as “abstract nonsense”: It turns out
(using the main theorem of ordinary differential equations) that the Lie
bracket of two vector fields is directly responsible for the control of com-
plex robot motion, or, still more down to earth: to everyday’s sideward
parking problem. We wish that the reader may always keep in mind these
universal tools of thought while guiding the universal machine, which is
the computer, to intelligent and successful applications.
Zurich,
Guerino Mazzola
August 2004
Gérard Milmeister
Jody Weissmann
Contents
III Topology and Calculus
1
27 Limits and Topology
3
27.1Introduction.............................................
3
27.2 Topologies on Real Vector Spaces . . . . . . . . ................
4
27.3 Continuity . . . . ...........................................
14
27.4Series ...................................................
21
27.5 Euler’s Formula for Polyhedra and Kuratowski’s Theorem
30
28 Differentiability
37
28.1Introduction.............................................
37
28.2Differentiation ..........................................
39
28.3 Taylor’s Formula . . . . . . . .................................
53
29 Inverse and Implicit Functions
59
29.1Introduction.............................................
59
29.2TheInverseFunctionTheorem...........................
60
29.3TheImplicitFunctionTheorem ..........................
64
30 Integration
73
30.1Introduction.............................................
73
30.2 Partitions and the Integral . . . . . ..........................
74
30.3 Measure and Integrability . . . . . . ..........................
81
31 The Fundamental Theorem of Calculus and Fubini’s Theorem 87
31.1Introduction.............................................
87
31.2 The Fundamental Theorem of Calculus . . ................
88
31.3 Fubini’s Theorem on Iterated Integration . ................
92
32 Vector Fields
97
32.1Introduction.............................................
97
32.2VectorFields ............................................
98

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