Five recent book reviews

Publisher:

Princeton University Press

Year:

2008

ISBN:

978-0-691-13131-3

Price (tentative):

USD 27.95

MSC main category:

00 General

Review:

The second popular-mathematics book of the author is in some sense a free sequel to the first one Nonplussed. This book, just like its predecessor, contains a collection of baffling paradoxes and mathematical occurrences that are in sharp conflict with a common, or even scientific, intuition. The book shows how deceiving the intuition can occur and where it can lead us. The author illustrates this unsettling thought on several examples of truly mind-boggling and delightfully amusing paradoxes. After a warm-up of “common-knowledge” classical puzzles and paradoxes based on elementary logic, the serious business starts. The reader will find, among many other topics, Simpson’s famous paradox, a nightmare of statisticians and a destructive weapon of demagogues, Braess’ paradox, disguised in a less common form with a rather surprising positive effect of a temporary closure of a road on the smoothness of traffic in a big city, and finally, at the end of the book, the king of all paradoxes, the Banach-Tarski theorem. This paradox, for instance, is a deep and serious result that lies on the crossroads of measure theory, the theory of sets and mathematical logic, a consequence of the existence of sets that have no volume at all, combined with the highly disputed axiom of choice. It states that a three-dimensional ball can be cut into just five pieces from which one can assemble two such balls. This result, whilst true, is so highly counterintuitive that the author himself comments on it: ‘If nothing else in this book was considered Impossible by the reader, it is hoped that this result might just have saved the author’s day.’

Reviewer:

lp

Publisher:

American Mathematical Society

Year:

2008

ISBN:

978-0-8218-4760-2

Price (tentative):

USD 39

MSC main category:

00 General

Review:

The main objective of these lecture notes is to describe local and global duality, primarily focusing on irreducible algebraic varieties over an algebraically closed field. Both local and global duality theorems are based on two linear operators - we have the residue map defined on the top local cohomology of the canonical sheaf, and the integral as a linear form on the top global sheaf cohomology of algebraic variety. There is then a comparison given by the residue theorem relating the integral operator to a sum of residues or, when specializing to projective algebraic curves, yielding the Serre duality theorem expressed in terms of differentials and their residues. Possible applications include the generalization of residue calculus to toric residues.

Reviewer:

pso

Publisher:

Oxford University Press

Year:

2003

ISBN:

ISBN 0-19-852531-1

Price (tentative):

£65

MSC main category:

62 Statistics

Review:

It is not easy to describe the contents of this book in a few words. Generally speaking, the author describes the course of development of statistical concepts from the perspective of the present state of the subject. Some chapters of the book emphasize the philosophical context and the others are devoted to the history of selected statistical and probabilistic methods. Anyway, the reader is assumed to have a sufficient knowledge of mathematical statistics. The book is divided into two parts. Part I called Perspective is oriented to a philosophical background of statistical thought and an interpretation of probability. Part II called History describes the birth of probability theory, fundamental ideas (including the central limit theorem, maximum likelihood, information criteria, outliers, robustness, and many others), and the role of outstanding scientists like Pascal, Bernoulli, Bayes, Laplace, Gauss, Poisson, Galton, Pearson, Student, and Fisher. However, this volume is neither a book on the history nor on the philosophy of statistics. To illustrate the subject of the book, I would like to mention some details about sufficiency described in it. All statisticians know that this concept was introduced by R.A. Fisher in 1922. It is less known (see p. 255) that Fisher discovered the principle of sufficiency when he solved a problem raised by the astronomer and physicist Eddington in a 1914 book on astronomy (Which of the two given estimators of standard errors has a better performance?). But in fact, sufficiency was used already in 1860 by the American statistician Simon Newcomb (see p. 254), who observed that in a sample X1,…,Xn with replacement from {1,…,N}, the statistic max Xi in some sense summarizes information from the complete sample. The book can be recommended to teachers and students who are interested in philosophical principles of statistics and in the history of probability and statistics.

Reviewer:

ja

Publisher:

Oxford University Press

Year:

2002

ISBN:

0-19-851623-1

Price (tentative):

£12.99

MSC main category:

00 General

Review:

Think of a three-figure number such that its first and last digit differ by two or more. Now, reverse the number, and subtract the smaller of the resulting two numbers from the larger: for example, 782 – 287 = 495. Finally, reverse the new three-digit number, and add: 495 + 594 = 1089. Whatever your initial choice, the result is always 1089. This wonderful and rather surprising trick serves as an opening of the lovely book by David Acheson, a professor at Jesus College, Oxford, and a jazz guitar player, according to whom this ‘1089-trick’ had been the first piece of mathematics that had really impressed him when he learnt it back in 1956. The trick even made the title of the book.

This simple algebraic idea is a well-chosen appetiser, giving the reader an idea of what to expect from this nice little book. The author pinpoints, in a witty and reader-friendly style, certain interesting facts from mathematics and related subjects. The book is obviously aimed at an audience far wider than just professional mathematicians, but each reader, whether mathematician or keen layman, will be delighted. David Acheson writes in his own style, the main feature of which is the fascinating fact of how much he is able to communicate in a rather small amount of words. Naturally, he has something to say about many of the notorious and often publicised pieces from mathematics, history, applications and, of course, famous open problems (or famous ex-open problems that had been open for a long time and have been solved recently). Well-known topics are not missing in the book, but even though typical readers will have at least some knowledge of all these, they will always find something new and interesting in this book.

Several chapters of the book are outstanding and deserve extra mention here. First, from the past, is mathematical analysis (calculus) with its great explanation of the Leibniz dy/dt notation, and one of the subsequent chapters in which the origin of the leopard’s spots is claimed to be governed by a certain differential equation. Another is the chapter on the magical Indian rope trick [see David Acheson’s article in this issue: ed]: a length of a rope is thrown up in the air and stays there, defying gravity. A small child then climbs the rope. The author obtained a rather sophisticated mathematical description of this remarkable phenomenon and, to demonstrate it, persuaded a friend to construct an upside-down double pendulum. Equipped for both theory and experiment, they then proved that the trick really works. They demonstrated unbelievable stability in the pendulum: after being pushed over by as much as 40 degrees, it would gradually wobble back to the upward vertical.

There are more fascinating things in the book that cannot be described here. So, here is the message to all potential readers of this type of mathematical writing: even though you have doubtless read everything by Keith Devlin, Simon Singh, Martin Gardner, Raymond Smullyan, Lewis Carroll and you-name-it, this wonderful work is yet another ‘must’ for your bookshelf!

Reviewer:

lp

Publisher:

Cambridge University Press

Year:

2002

ISBN:

0-521-01678-9

Price (tentative):

£21.95

MSC main category:

00 General

Review:

This is an excellent collection of mathematical problems and puzzles at all levels of mathematical sophistication. The tasks are presented to little Dorothy by Dr. Oz, a mathematically obsessed alien. Dorothy tries to solve problems, each described via a nice story: this motivates the reader to solve the challenge. Full solutions are presented in the ‘Further Exploring’ part of the book.

The problems cover various topics: geometry and mazes, sequences, series, sets, arrangements, probability and misdirection, number theory, arithmetic and the physical world. As an example, consider the following problem: ‘Can you divide a triangle into finitely many triangles to produce all acute internal angles?’ The reader is guided to think about many interesting areas, and various methods are exploited and presented in a non-traditional way. The author has written many interesting books, and his website www.pickover.com is well known.

Reviewer:

pp