OF THE EUROPEAN MATHEMATICAL SOCIETY

conjecture concerning Weyl’s problem on spectra of the sum of
Hermitean matrices. While elementary parts of the book can
be presented even in a basic course of analysis, the book goes
rather deeply into the subject. A unifi ed approach based on an
application of means of different types is combined with many
classical and recent results to show to the reader convexity as
one of central notions of mathematics. The book well documents
sources of ideas and the origins of results and will be of
interest even for specialists in the fi eld; it may also be used as a
reference book on the subject. (jive)
D. Pei: Authentication Codes and Combinatorial Designs, Discrete
Mathematics and its Applications, Chapman & Hall/CRC,
Boca Raton, 2006, 244 pp., USD 89,95, ISBN 1-58488-473-8
This book is concerned with two authentication schemes. In
both of them there is a sender, a receiver and an opponent. The
scheme consists of a set of source states, a set of messages and
a set of keys. The goal is to transmit a source state by means
of sending a message. Each key is a one-to-one mapping from
a subset of the message set into the set of source states. The
sender and receiver agree on some key and the opponent tries
to mislead the receiver with a fraudulent message. If the sender
and the receiver are supposed to trust each other, one speaks
about an authentication scheme with three participants. The
scheme with four participants involves an arbiter between the
sender and receiver in the case when they are not regarded as
trustworthy. The main line of the book consists of considering
the designs that appear when a key is associated with a set
of admissible messages (i.e. with the range of a mapping with
which we identify a given key).
In the beginning the author uses probability and entropy
computations to show that the perfect 3-party authentication
schemes with uniform probability distributions correspond to
strong partially balanced t-designs. The defi nition of a partially
balanced design differs from that of a t-design with parameters
(v, b, k, λ) by allowing the existence of k-sets that do not extend
to any t-set. Such a design is called strong when it is also a
partial balanced r-design for every r smaller than t. The conditions
for perfect 4-party authentications are of a similar type
but much more technically complicated. Much of the content of
the book is a survey of various designs that can be used for the
studied authentication schemes. A lot of material is taken from
other sources and the exposition is very careful, often in an elementary
manner. Even some well known topics are covered in
detail, like Steiner triples or mutually orthogonal Latin squares.
There is an overview of constructions of orthogonal arrays and
a description of interesting families of authentication schemes
based on curves and on fi nite geometries. (ad)
V. Runde: A Taste of Topology, Universitext, Springer, Berlin,
2005, 176 pp., EUR 32,95, ISBN 0-387-25790-X
This book is an introductory text for a study of topology. In
general, the book is oriented to second-year undergraduates.
It presents the basic language used in various fi elds of modern
mathematics. The book covers classical topics of topology.
Chapter 1 is about set theory. Chapter 2 covers metric spaces
including continuity, compactness and completeness. These notions
are studied in chapter 3 from the point of view of topological
spaces. Separation properties are included as well. Chapter
4 is devoted to systems of continuous functions, covering
Recent books
the Urysohn lemma, the Stone-Čech compactifi cation and the
Stone-Weierstrass theorems. Chapter 5 presents basics of algebraic
topology, containing a study of homotopy, the fundamental
group and covering spaces. The text also presents several
non-typical approaches to various topics. For instance, the Baire
theorem is derived from Bourbaki’s Mittag-Leffl er theorem,
the Tychonoff theorem is proved intuitively using nets, and the
complex Stone-Weierstrass theorem is obtained using a short
and elegant approach. The book is also a source of exercises on
basic topological notions. (pp)
M. Sanz-Solé: Malliavin Calculus with Applications to Stochastic
Partial Differential Equations, Fundamental Sciences:
Mathematics, Chapman & Hall/CRC (EPFL Press), Boca Raton,
2005, 162 pp., USD 84,95, ISBN 0-8493-4030-6
Professor Sanz-Solé (University of Barcelona) has written an
inspiring text in an alert style, which can be used for many different
purposes from individual reading to graduate level seminars;
the interest of potential users is continuously kept alive.
She demonstrates how Malliavin calculus can provide penetrating
insights into deep aspects of probability theory. The book
gives a self-contained introduction to the background material
(including the Ornstein-Uhlenbeck operator and the representation
of Wiener functionals). However, the choice of topics is
highly selective, with emphasis on those frequently used in research
and inspired by special problems discussed in the later
chapters. These include questions related to stochastic partial
differential equations. Several results are published here for the
fi rst time in book form. The quick progress from fundamentals
to applicable topics ensures that both graduate students and
young researchers will benefi t from the book. (kzit)
K.-G. Steffens: The History of Approximation Theory – From
Euler to Bernstein, Birkhäuser, Boston, 2006, 219 pp., EUR 68,
ISBN 0-8176-4353-2
This nice little book is devoted to the early development of approximation
theory. It mainly concentrates on the role of Russian
mathematicians but it confronts it with the development of
approximation theory in the West. It fi rst explains the approximation
results of Euler connected with geodesy (he constructed
a best possible approximation for a map of the Russian Empire
satisfying certain side conditions) and Laplace’s contribution
to the determination of the best ellipsoidal approximation of
the planets. Then more than a quarter of the book is devoted to
Chebyshev and his results and another part of about the same
size to Saint Petersburg mathematical school (among his direct
successors one fi nds the names Bessel, Sochocki, Zolotarev,
A. A. and V. A. Markov, Lyapunov, Voronoy and others). The
list of Western mathematicians who contributed to approximation
theory includes Klein, Blichfeldt, Kirchberg, Weierstrass,
Runge, Borel, Young, Fréchet, Fejér and Jackson. The author
also tries to present a closer look at the personalities of those
he mentions; we learn many facts about their lives and their
philosophical attitudes to different approximation results. He
quite often quotes Russian sources (with translations), which
will help most readers gain understanding of their personalities.
The last chapter describes the work of Pscheborski and Bernstein
and their role in the creation of the Kharkov School. The
book will be of interest for historians of mathematics and for
specialists in approximation theory. (jive)
EMS Newsletter September 2006 55