Five recent book reviews

Publisher:

Princeton University Press

Year:

2013

ISBN:

978-0691-1-5991-1 (hbk)

Price (tentative):

$24.95 / £16.95 (net)

Short description:

Martin Gardner's boost in popularity for the broad public came about when he started in 1956 his *Mathematical Games* column in *Scientific American* that he continued for the next 25 years. This book is an autobiography that he wrote shortly before he passed away in 2010 at the age of 95. It reveals his prolific and diverse career and his strong opinions on things that matter to him such as art, poetry, philosophy, religion, pseudoscience, etc. and it has many anecdotes about people from his broad circle of acquaintances and friends.

URL for publisher, author, or book:

press.princeton.edu/titles/10066.html

MSC main category:

01 History and biography

MSC category:

01A70

Other MSC categories:

00A08

Review:

Martin Gardner (1914-2010) hardly needs any introduction in a review intended for mathematicians. His popularity, even among a broader public, has grown to a legendary level. His interests, reflected in his numerous publications, have created a loyal circle of followers among the species of *homo ludens* collecting geeks that love recreational mathematics, card tricks, and other magic hocus-pocus. Some may know about his admiration for Lewis Caroll via his edition of *The Annotated Alice* (1960) and for L. Frank Baum's Wizard of Oz via his book *Visitors From Oz* (1998). These were inspirational for many other of his writings as well. Perhaps a bit less known is that he was also a big fan of G. K. Chesterton (the author of the Father Brown detective stories).

If this is about as far as your familiarity with Martin Gardner's work goes, then this autobiography will bring some surprises. It is not revealing unexpected issues about his private life, but it will be an eye-opener knowing that he was active on so many diverse fields outside the ones listed above. Clearly he has been interested in magic, chess, card games, and recreational mathematical issues since he was a young boy. But the opening chapter is about colours, and Gardner immediately connects this with colours in *The Wizard of Oz* and in Chesterton's novels. This is very typical. Whatever topic or period in his live is covered in the different chapters, there are always numerous references to and citations from books by others and of course also by himself. His opinion about some poetry and more in general about other art forms is outspoken and clearly put on display.

Another issue that mattered a lot to him was religion. After going through several stages in his life, he finally became a believer in God and in an afterlife, although not on a rational basis. This is the subject of the preface, but is recurrent throughout the book. It flares when he is discussing the views of his philosophy professors M.J. Adler and R. McKeon at the University of Chicago, and when he has a chapter on loosing his faith, and it is very explicitly in his penultimate chapter entitled `God' and in the last one where he summarizes his philosophy as a kind of testament for posterity.

Another of his pet subjects is his aversion for pseudoscience. There is a separate chapter on his rejection of what he calls `bad science'. There is no mercy for Dianetics and Scientology, orgonomy, UFOs, homeopathy, chiropractors, phrenology, palmistry etc. or for frauds like Uri Geller. On this topic, his book *Fads and Fallacies in the Name of Science* (1958) originally published in 1952 with a different title has become a classic.

Almost `between the lines' we learn about his life: his schools, the University of Chicago, his service in the Navy during WW II and later his career as a contributor to *Esquire*, editor of *Humpy Dumpty* and how it really was taking off when he published his first contribution to *Scientific American* on flexagons. Obviously there is also a chapter about his parents, one about his wife and family, and one about good friends. However, these are not digging very deeply into the lives of these people. Even in these chapters, he finds hooks to his views and convictions and to his or somebody else's publications. The chapter on his math and magic friends has a lot of anecdotes, but there is a constant stream throughout the chapters with amazing details and funny stories about an endless number of people he has known or worked with and who often became friends. The list of names and the list of works compiled in the index at the end of the book is 19 pages long. That is a lot of people on only 200 pages. That Salvador Dali was one of his fans will be a surprise for many. A `photo essay' has 24 pages of photographs and illustrations including some of the caricatures he made, showing another of his skills as yet undiscussed.

So lovers of mathematical games and recreations should not look for more of this stuff here. Nevertheless, with the many references and citations, they can consider it as an annotated (although incomplete) guide to the work of Gardner. Moreover they will be surprised that there are so many probably unexpected facets to this man. Martin Gardner will live on in the biannual G4G (Gatherings for Gardner) that started in 1993. His broad mathematical impact may be explained just because he never got a formal degree in mathematics. As he confesses that he sometimes had to work hard to understand the subject himself before he could write about it, implying that if he understood, also interested but non-mathematical readers would understand what he wanted to communicate.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

Birkhäuser

Year:

2013

ISBN:

978-1-4614-7033-5, 978-1-4614-7048-9, 978-1-4614-7131-8 (hbk)

Price (tentative):

116.59 €, 148.99 €, 148.99 €

Short description:

The major part of these three thick volumes consists of reprints from a large part of the work of Walter Gautschi. He is a renowned numerical analyst whose main contributions are in the domain of numerical integration and orthogonal polynomials. The reprints are organized in 13 different topics. Each topic is commented and summarized by different specialists.

URL for publisher, author, or book:

www.springer.com/birkhauser/mathematics/book/[978-1-4614-7033-5 | 978-1-4614-7048-9 | 978-1-4614-7131-8]

MSC main category:

65 Numerical analysis

MSC category:

65-06

Other MSC categories:

65Dxx, 65Lxx, 65Axx, 65Yxx, 01Axx

Review:

Walter Gautschi got a Ph.D in 1953 under A.M. Ostrowski in Basel. So, being active in the early years of numerical analysis, he has helped shaping the field of wat is now a broad subject. With his three authored books, several others edited, and around 200 journal papers and book chapters he is now globally recognized as a respected numerical analyst. He is best known for his major work about quadrature formulas and orthogonal polynomials.

The larger part of these three thick volumes (2375 pages in total) consist of reprints of his own selection of his papers. The first volume starts with a very short biography by the editors and an original contribution by Gautschi himself in which he summarizes his work and cathegorizes it in 13 different domains, and this survey is followed by a complete bibliography.

This subdivision in 13 subjects is maintained in the rest of the volumes: topics 1-3 go into volume 1, 4-8 in volume 2 and 8-13 in volume 3. In each topic it goes in two steps: first there is a part in which well known specialists comment and summarize the contributions of Gautschi each of the topics in the volume and in a second part the papers classified in this topic follow. The layout in which they were originally published is maintained. Here is the list of the topics and in brackets are the authors who wrote the commentaries.

- numerical conditioning (N.J. Higham)
- special functions (J. Segura)
- interpolation and approximation (M.M. Spalević)
- orthogonal polynomials on the real line (G.V. Milovanović)
- orthogonal polynomilas on the semicircle (L. Reichel)
- Chebyshev quadrature (J. Korevaar)
- Kronrod and other quadratures (G. Monegato)
- Gauss quadrature (W. Van Assche)
- linear difference equations (L. Lorentzen)
- ordinary differential equation (J. Butcher)
- computer algorthms and softwarepackages (G.V. Milovanović)
- history and biography (G. Wanner)
- miscelenea (M.J. Gander)

Walter Gautschi is a very kind person and the commentators are not only world leading specialists, but they are also friends of Walter. So some of the comments also contain some personal reminiscences. Since his official retirement from Purdue University in 2000 to which he has been associated since 1963, Gautschi has remained active ever since. He published his book on *Orthogonal Polynomials* with Oxford University Press in 2004 and his book on *Numerical Analysis* published by Birkhäuser Boston got a second edition in 2012.

Of course the topics listed above are not disjunct and there is obvious overlap. The conditioning for example is about Vandermonde matrices and polynomials, important for interpolation and orthogonal polynomials for quadrature nodes. The recursive computation of these polynomials and special functions links to the stability in using recurrence relations, and special functions often have an integral representation that can be evaluated using quadratures. His work is mainly constructive and practical which resulted in two chapters in Abramowitz and Stegun's *Handbook of Mathematical Functions* and in software packages such as ORTHOPOL (orthogonal polynomias) and several other routines available on his website.

Somewhat separate from the rest is the section on history and biography. There we find his work about Euler (2007 was the Euler year and Gautschi gave a main lecture about Euler at the ICIAM Congress in Zürich). More historical work is about his thesis adviser Ostrowski, and about the Bieberbach conjecture. Also a survey paper is devoted to the work of E. Christoffel, while others are written as a tribute to colleagues he has known and that have passed away: Y. Luke, P. Rabinowitz, L. Gatteschi, G. Golub, and one for his friend G. Milovanović on the occasion of his 60th birthday in 2008.

The commentaties are usually rather short and do not extend or give additional results. They just summarize what the papers are about with little personal comments. The more important sections like Orthogonal Polynomials and Quadrature have more papers and hence also longer introductions.

Volume 3 has an extra third part. Walter Gautschi had a twin brother Werner, also a mathematician, who died in 1959. This part reprints 5 of Werner's papers, and two orbituaries, and a link to the Springer website where one may find a recording of 35 minutes in which Werner performs Schubert's Trout Quitet.

With this series *Contemporary Mathematicians*, Birkhäuser has started to publish the collected works (or at least a selection) of contemporary mathematicians whose publications are sometimes scattered and/or may be difficult to find. Bringing them together in one, or as in this case, several volumes is an excellent service to the scientific community. In the e-book version, the chapters are downloadable separately which is of course a highly desirable feature. Libraries will find these hard copies on their shelves to be of lasting significance.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

A K Peters/CRC Press

Year:

2013

ISBN:

978-14-6650-976-4 (hbk)

Price (tentative):

£19.99 (hbk)

Short description:

This is a collection of mathemagical card effects with an underlying mathematical explanation. Colm Mulcahy realizes with this book a suggestion that Martin Gardner made to him many years ago. Many of the effects have been described in his blog *Card Colm* sponsored by the MAA. The mathematics are based on counting, combinatorics and sometimes a bit of probability that applies to the particular constellation of a (sometimes carefully chosen subset of a) card deck. All aspects are enlightened: how to perform the trick, why and how does it work, what mathematics are involved, what are the variations and generalizations, what mnemotechnical tricks can be used, what is the origin, etc.

URL for publisher, author, or book:

www.crcpress.com/product/isbn/9781466509764

MSC main category:

00 General

MSC category:

00A08

Other MSC categories:

11Z05

Review:

Colm Mulcahy has Irish roots, with a MSc from the University College Dublin, but he got a PhD from Cornell, and he is teaching at the Math Department of Spelman College, Atlanta (GA) since 1988. He regularly contributes a math column to The Huffington Post, but he is foremost an enthusiastic user of internet media. He has a Math Colm blog, and on his *Card Colm website* he has many interesting links, for example to material related to his much admired hero Martin Gardner, but most of all to the items of his bimonthly *Card Colm blog*, that is sponsored by the MAA. In the latter he discusses mathematically based effects (he avoids using the word 'tricks' for some reason) you may obtain by manipulating a card deck. His website has extra videos illustrating the manipulation of the cards, and of course a link to the book under review. This book is the eventual realization of a suggestion made to him by Martin Gardner, many years ago.

So here it is: a book presenting 52 effects arranged in 13 chapters, with 4 sections each, a numerological predestination for a book about card tricks. The actual contents is however a bit liberal with these numbers. There is for example an unnumbered appetizing chapter that also introduces some terminology and card shuffling techniques, and there is a coda that gives some additional information. Also each chapter may have the four `suit'able variations, but they often consist of several introductory sections as well. The effects get some inspiring names like `Full House Blues', `Easy as Pi', or `Twisting the Knight Away' etc. The effects are usually introduced in several steps: 'How it looks' is just describing what an uninformed observer will experience, then `How it works' is explaining what the mathemagician actually is doing, later a section `Why it works' is explaining in more detail what is going on. This is often followed with different options on how to present the magic to the public. Also the origin of the method is referred to, which, no wonder, is often one of his Card Colms blogs. The chapters end with some `Parting Thoughts' which are further elaborations of the previously introduced principles, sometimes in the form of exercises: `prove that...', `what if...', `how to...'.

Mulcahy also gives his effects some Michelin-like ratings in the margins. It can have one ♣ (easy) up to four ♣♣♣♣ (difficult) clubs for mathematical sophistication, similarly hearts grade entertaining value, spades are used to rate the preparatory work needed, and diamonds refers to the concentration and counting that is needed during the performance.

The effects are based on underlying mathematical principles. There are many of them, and most are believed to be original in their application to card magic. Since these principles are introduced but also re-used at several instances, they get some mnemotechnic names of which some recurrent ones are the COAT (Count Out And Transfer) or a generalized overCOAT or underCOAT, and TACO is some kind of inverse, not to be confused with a CATO (Cut And Turn Over). These give rise to neologisms like `minimal underCOATing' of `Fibbing' when it concerns Fibonacci numbers. TOFUH stands for Turn Over And Flip Under Half, sounds healthier than the alternative FAT (Flip And Transfer). In many cases, the order of the suits is important. So the deck can be in cyclic CHaSeD (Clubs, Hearts, Spades, Diamonds) order. And there are many other mnemotechnical tricks to recall certain orders of cards or operations. These witty namings and word plays make the text fun to read.

Since Mulcahy stresses at many places that the mathemagician should not reveal the mathematics underlying the magic, so neither will I uncover them here in this review. As he writes: The best answer to the question `How did you do that?' is to say `Reasonably well, I think'. Unless the audience is really interested in the mathematics, it is unwise to explain what is going on. Otherwise comments like `So is this all you did? It is just mathematics', will kill all the magic. It is absolutely rewarding however to convince young people that mathematics is everywhere and can be fun to play with. So it is perfectly all right if a teacher explains the mathematics to his pupils.

In most cases, the mathematics are not that advanced. Just counting will do, but it is not just plain combinatorics or modulo calculus. The fact that there are 13 different faces and 4 different suits make the counting special. Fibonacci numbers sometimes play a role, occasionally there is some probability involved. The effect may be for example to bring by some seemingly random shuffling a card from the bottom of the deck to the top. Or if the mathemagician is given the sum of the values of 2 or 3 cards that a spectator has randomly selected from a prepared deck, he will be able to name the faces of these cards. As one reads along, the effects become more involved, hence more complicated to perform, but they will have a higher magical alloy and thus be more rewarding. Some involve an accomplice to assist the magician.

One word about the typography. The numbering of the chapters is with cards (A for the first, 2 for the second,..., K for the 13th) and the effects within the chapters are numbered like 6♣ for the first one in chapter 6, or K♠ for the third one in chapter 13 (CHaSeD order). Printed on glossy paper with many colour illustrations, it is not only fun, but also a pleasure to read. The apprentice magician will have a lot to practice on but even the professional magician will find many things to think about while mastering this wonderful calculus of the card deck.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

Princeton University Press

Year:

2012

ISBN:

978-0691153483

Price (tentative):

$75

Short description:

This book proves a conjecture of Deser and Schwimmer regarding the algebraic structure of global conformal invariants (conformally invariant integrals of scalar functionals on the Riemannian metric). The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand.

URL for publisher, author, or book:

http://press.princeton.edu/titles/9768.html

MSC main category:

53 Differential geometry

MSC category:

53C

Review:

Given a Riemannian manifold $(M,g)$, consider a formal polynomial expression $L(g)$ on the coefficients of the metric $g_{ij}$, its partial derivatives, and $(\det g)^{-1}$. If $L(g)$ is invariant by changes of charts, then it can be written as a linear combination $\sum_{l\in L} a_l C^l(g)$ of some complete contractions of covariant derivatives of the Riemannian curvature tensor. A global conformal invariant is a functional of the form $\int_M L(g) dV_g$ which remains invariant under conformal rescalings of the metric. A conjecture or the physicists Stanley Deser and Adam Schwimmer in 1993 asserts that

$$

L(g)=W(g)+ \mathrm{div}_i T^i(g)+C \cdot \mathrm{Pfaff}(R_{ijkl}),

$$

where $W(g)$ is a local conformal invariant, $T^i(g)$ is a vector field, $C$ is a constant, and $\mathrm{Pfaff}(R_{ijkl})$ is the Pfaffian of the curvature tensor. The Chern-Gauss-Bonnet theorem says that this last term integrates to a multiple of the Euler-Poincaré characteristic of $M$.

This book provides a proof of this conjecture. The method consists on an iterative procedure to reduce the expression $\sum_{l\in L} a_l C^l(g)$ to another expression of a similar type for a reduced subset $L'\subset L$ of "good" complete contractions, trading them by local conformal invariants and divergences. The second step is a lengthy algebraic proof that the expression $\sum_{l\in L'} a_l C^l(g)$ satisfies the required property.

Reviewer:

Vicente Muñoz

Affiliation:

UCM

Publisher:

Springer

Year:

2013

ISBN:

978-1-4614-6761-8

Short description:

This book presents a collection of methods and techniques through problem-solving in fundamentals of Mathematical Analysis. In particular, emphasis is given to the computation and/or existence of limits of sequences, series and fractional part integrals.

It is the first book of its kind to give emphasis on the fractional part of functions which are involved in convergence, differentiation and integration problems.

MSC main category:

26 Real functions

MSC category:

11B73

Other MSC categories:

33B15, 26A06, 40B05

Review:

The book provides an extensive presentation of a wide collection of problems on three selected topics of Mathematical Analysis: limits, series, and fractional part integrals. Emphasis is given to a variety of methods and techniques which apply to the solution of special computational problems. These problems are mainly oriented to undergraduate and beginning graduate students, who wish to be prepared for various problems-solving mathematical competition including the Putnam exam. The book will be particularly useful to teachers of Calculus and Real Analysis courses who wish to enrich their class with challenging problems, some of which require special skill and ingenuity.

Some of the problems appear for the first time in this book and have been introduced by the author or other mathematicians.

The book is highly recommended for libraries as well as for personal use.

Reviewer:

Th. M. Rassias

Affiliation:

National Technical University of Athens