Five recent book reviews
In the early 1990s, Tom Rodgers conceived the idea of hosting a weekend gathering in honour of Martin Gardner, to bring together people interested in problems and puzzles. Members of the three communities of mathematicians, magicians and puzzlers met to honour their forefront man and nexus, and so far four ‘Gatherings for Gardner’ (known as G4G1 through G4G4) have been organised. At these gatherings, held in 1993, 1994, 1998 and 2002, participants share problems and puzzles, knowledge and ideas. In 1999, ‘The mathemagician and pied puzzler’ was published as a tribute to Martin Gardner, based on contributions to G4G1. This is a second volume.
In the preface, the work of Martin Gardner and his contribution to the popularisation of mathematics is briefly reviewed: in the words of John Conway, ‘Martin Gardner has brought more mathematics, to more millions, than anybody else’. The book continues with a section dedicated to the memory of Mel Stover, Harry Eng and David Klarner, three G4G participants who have passed away. Called ‘The toast tributes’, this section contains several short moving articles commemorating their lives and ideas. The rest of the book contains 56 contributions of G4G participants, divided into six sections. This presents an incredible range of beautiful material, witty ideas, tantalising puzzles, intriguing problems, and surprising ingenious solutions. Anyone interested in mind-boggling problems will find plenty to interest them here. The famous authors in the area, such as Barry Cipra, Roger Penrose, Raymond Smullyan, Keith Devlin, John Conway, and many more, are among the contributors. The material covers all kinds of problems that one can possibly think of: card problems, labyrinth problems, topological games, knots, floor-tiling problems, spider-and-fly problems, pentacube towers, chess, magic squares, pure arithmetic problems, geometry, trigonometry, Japanese puzzles and much, much more.
Opening a book on projective geometry, we expect an investigation of objects occurring in projective space. We expect to meet subspaces, quadrics, algebraic subvarieties, differential submanifolds, and many other objects. The book under review is not of this type, and this explains perhaps, why it carries the title Modern Projective Geometry. The main aim of the book is to introduce the category of projective geometries. This means that the authors’ goal is to look at projective geometries not only from inside, but also from outside. They adopt a synthetic definition of a projective geometry, and this definition has a fundamental influence on the style of the book. We find deep relations between projective geometries and other mathematical structures. First of all, a relation to lattice theory, i.e., the category of projective spaces is equivalent with the category of projective lattices. Secondly, a relation to closure spaces (and to matroids, in particular), i.e., an equivalence between the category of projective geometries and a category of certain closure spaces. The general approach leads us also to other geometries, e. g. affine geometries, hyperbolic geometries, and Möbius geometries. But this look at projective geometries from outside does not mean that we do not find information about these particular geometries. The book is written according to an excellent plan, and it surely represents a milestone in the development of projective geometry. The text is organized very carefully and each chapter is followed by many exercises. It is a great advantage of the book that it requires very modest prerequisites. Hence, it can be recommended already to undergraduate students in the first year of their study. On the other hand, I expect that also professional mathematicians will appreciate it.
Books containing interesting problems from probability theory and/or mathematical statistics are very popular among teachers and students. This kind of recreational mathematics offers supplementary material for use in classes. The formulation of problems gives an opportunity of discussing how to construct models in real life. Solutions often lead to seemingly paradoxical results. Sometimes calculations are based on theorems from quite different branches of mathematics, which makes it possible to present the beauty of mathematical thinking. Collections of such problems have been published for many years. Contemporary books containing such material include those by G. J. Székely (Akadémiai Kiadó, Budapest, 1986), P. J. Nahin (Princeton Univ. Press, 2000), H. Tijms (Cambridge Univ. Press, 2004) and also my book (J. Anděl, Wiley, New York, 2001).
In this book the author includes problems that come from some aspect of everyday real life. However, even simply formulated problems may be too hard to be solved analytically. On the other hand, numerical answers can be reached using simulations. This is the main difference between this book and the others. The author first presents a MATLAB code and the results of simulations and only then an analytic solution is given (or outlined) – if it is known. The book is divided into three parts. The first part contains the formulation of the problems with some historical remarks, the second part describes the MATLAB solutions to the problems and the third part is composed of nine appendices. In the appendices, we find some theoretical complements to presented problems. Of course, some additional references to problems presented in the book can be given. For example, an interesting motivation to the material on page 22 can be found in the book by J. Swift (The Math. Teacher, 76, 268-269,1983). I have two critical remarks. In all computations, where a random number generator is used, I strongly recommend fixing the seed at the beginning of calculations. Only in this way can everybody repeat the whole process with the same results. Unfortunately, the author does not use this approach. Secondly, the author presents remarks as a special subsection at the end of each section. I find this annoying; it would be much better to print them as footnotes. In conclusion, though, I believe that this book will find many readers and that the problems presented here will refresh introductory courses in probability theory.
The goal of these lecture notes is to introduce the reader to the central concepts related to wavelets and their applications as quickly as possible. By focusing on essential ideas and arguments, the authors indicate appropriate places in the literature for detailed proofs and real applications. The book is organised as follows. A preliminary chapter containing some of the required concepts and definitions is included for reference. In chapter 2, time-frequency analysis is reviewed. The authors start with Fourier series and the Fourier transform. The windowed Fourier transform, the Gabor basis and local trigonometric bases are discussed. Finally the authors gain localization at the scales with the wavelet transform. In chapter 3, the notion of multiresolution analysis (MRA) is introduced. The Haar wavelets and MRA is carefully described. This example contains most of the important ideas behind MRA. Daubechies compactly supported wavelets are described briefly and pointers to the literature are given.
In chapter 4, the authors discuss variants of the classical orthogonal wavelets and MRA. The MRA associated to biorthogonal and multiwavelets are also carefully described. The authors also explain how to construct wavelets in two dimensions by tensor products. The wavelets confined to an interval or a domain in the space are very briefly mentioned. The authors also discuss wavelet packets and some relatives and mutations of wavelets that have been constructed to tackle more specialized problems. Finally the prolate spheroidal wave functions are studied. In chapter 5, a few applications are described without attempting to be systematic or comprehensive. The choice of sample applications is dictated by the authors’ experiences in this area. A description is made in some detail of how to calculate derivatives using biorthogonal wavelets, and how one can construct wavelets with more fancy differential properties. The authors also describe how wavelets can characterize a variety of function spaces and how well adapted wavelets are able to identify very fine local regularity properties of functions. Finally the authors very briefly describe how wavelets could be used for the study of differential equations. The authors expect the reader to have been exposed to some real and complex analysis, calculus and linear algebra and some concepts of orthonormal bases, orthogonal projections and orthogonal complements on Hilbert space. The book is intended for beginning graduate students and beyond.
This interesting book is the English translation of the first French edition published in 2001 by Jean-Pierre Luminet, an astrophysicist at the Paris-Meudon Observatory in France. The main aim of the book is to describe a particular approach to cosmology. The author discusses the topology of the universe, which is not a simple theme for popularisation. To understand the main topics, there is a requirement for a deeper knowledge of mathematics (e.g. non-Euclidean geometry, the classification of surfaces, the classification of three-dimensional spaces and symmetries), physics (the theory of relativity, the curvature of the universe and cosmic crystallography), astrophysics (astronomical distances, cosmic repulsions, etc.) and the history of cosmological ideas and philosophical concepts of the cosmos and spaces.
The central notion is the theory of “the wraparound universe”. The author discusses a model of wrapped universes, their experimental ramifications, problems of their sophisticated observations and their consequences (e.g. size of space, fraction of distant galaxies, folds in the universe, expansion and the infinite, the rate of expansion and the age of the universe). The author also tries to explain the most important parts of general relativity, big bang models and cosmic topology. The book is very well written and nicely illustrated; the author uses a simple graphical representation to solve and explain complex problems. The author combines his understanding of history, scientific knowledge and expository skill to produce the book, which can be recommended to all readers interested in mathematical and astronomical ideas and cosmology, as well as the topology of the universe.