## Recent book reviews

Five recent book reviews

## Turbulent Times in Mathematics. The Life of J.C. Fields and the History of the Fields Medal.

Author(s):
E. McKinnon, F. Hoffman
Publisher:
American Mathematical Society, The Fields Institute
Year:
2011
ISBN:
9780821869147
Short description:

The book is about the life of J.C. Fields (1863 – 1932), from his beginning as a student at Hamilton (Canada) until his death as a Research Professor at the University of Toronto. Basically, it deals with his postdoctoral work at Europe where he gets introduced into the mathematical society of the moment, and the breaking out of the World War I and the influence made by Fields upon the mathematical community now confronted.

MSC main category:
01 History and biography
MSC category:
01A05, 01A55, 01A60, 01A70, 01A73, 01A99
Other MSC categories:
97A30, 97A80, 97A40
Review:

The book begins with the Fields’ parents and describes their life at Hamilton. Then it shows his life at the Toronto University, how he graduates with the gold medal in 1884, and enhance his four year sojourn at the Johns Hopkins University. There he gets his PhD. under the supervision of T. Craig. In 1892 he went to Paris, in 1894 stayed in Göttingen where he met F. Klein, and Berlin where he attended several courses and seminars and become friend of Takagi, Caratheodory, and Wilczynski among others. Finally in 1901 he went to Chicago where he knew to Maschke, Bolza, and Moore.
Back into his homeland, we read about his teacher career at the Toronto University: 1902 special lecturer, 1905 associate prof. 1914 prof. and 1923 research prof. Meanwhile he wrote his only book “Theory of the Algebraic Functions of a Complex Variable” in 1906. During those still years, he liked to travel to Europe and almost every summer met with some acquaintance as Mittag-Leffler etc.
The book now turns into the break out of the World War I and makes a thorough description of the events from the history point of view and also from the mathematical one. It gives a detailed account of the 93 pamphlet and the bitterness and confusion it created among mathematicians. Once the conflict ended, we read that Fields got the support to realize the International Mathematical Congress (the book explains perfectly well the difference between IMC and ICM) at Toronto in 1924 and as he couldn’t invited the mathematicians of Central Europe he realized the world needed some kind of solution. At the same time, the Nobel Prize, which was well established at that time, lacked of a Mathematical prize. So, Fields got enough support to create his counterpart: the medal that he didn’t wanted to bear his name or any other name of any institution or country, to promote the union that the world needed.
The reader can use the book also to learn about the history of the ICMs as he will find in the appendix all the winners with a brief account of their mathematical work from the one at Oslo (1936) up to and include the one at Hyderabad (2010).

Reviewer:
José L. Guijarro
Affiliation:

## Affine Maps, Euclidean Motions and Quadrics

Author(s):
Agustí Reventós Tarrida
Publisher:
Springer-Verlag
Year:
2011
ISBN:
978-0-85729-709-9
Short description:

This is a textbook on Affine and Euclidean Geometry, with emphasis on classification problems: classification of affinities, of Euclidean motions, of affine quadrics, of Euclidean quadrics. It can be used in a basic course and still to present full results on the topic at a reasonable cost.

MSC main category:
51 Geometry
MSC category:
51N10, 51N20
Review:

The assertion that this is a textbook may need some nuance, the book being xi+411 pages long. Since quite often readers loose impetus after 150 pages, such a thick book could discourage many. However those 400 pages can be estimated as 200 of real course. Indeed, the last 100 pages include appendices that are themselves mini-courses in Linear Algebra and Quadratic Forms, and maybe almost that many pages could be counted for the lists of exercises suggested at the end of every chapter, and the many examples fully worked out along the lessons either as illustrations of previous results or as motivation for proofs that follow. For instance, there is a 7 page section titled Clarifying Examples prior to the classification of affinities in arbitrary dimension. In addition, there are some sections that could be marked as optional, either because they include material better understood with some extra knowledge (as Projective Geometry), or because they deal with the recovering of classical results. Here one cannot help praising the succinct 5 page introduction where the author summarizes Euclid’s axiomatic of Affine Geometry and explains how it motivates the text. All these are of course added bonuses, but it is good to separate them from the main content of the text. Thus we are left with say 180 pages of Affine and Euclidean Geometry, which cover the usual: Affine spaces, subspaces and their operations; frames; affine maps and invariants, classification; Euclidean affine spaces and Euclidean motions, classification; real affine and Euclidean quadrics, classifications. Very little need to explain further this list, but even running the usual path, several qualities distinguish this text from others.

First, concerning the global view, there is a neat scheme of action: to give the definitions of the objects to study and discuss their classification problems. Here, although the approach is algebraic, the geometric meaning of the corresponding classification results (canonical forms, canonical equations, tables and lists) is always stressed, with careful choices of terminology. Second, the job is done thoroughly without fault. It is not that common to find: (1) a full classification of affinities as given here, (2) the clear distinction between a quadric and its equations and the exact description of their relationship. Third, there are separate sections for every classification question in the low dimensional cases, sometimes preceding as preparation the general result. Fourth, it is quite clear that this is a real life course, that is, a text written for and from true teaching. Thus any professor will easily find the way to adapt the text to particular whims, discarding technicalities or lightening some lessons. Also, students will find a self-contained book containing all they need to catch the matter: full details and many solved and proposed examples.

All in all the text is a highly recommendable choice for a course on Affine Geometry, and fills some gaps in the existing literature.

Reviewer:
Jesús M. Ruiz

## Numerical Analysis (second edition)

Author(s):
Walter Gautschi
Publisher:
Birkhäuser
Year:
2012
ISBN:
978-0-8176-8258-3
Price (tentative):
64,95 &euro; (net)
Short description:

This is the second updated edition of the 1997 version. It's a classical book on numerical analysis that can be used as course notes. Numerical linear algebra is not included and it stays away from real life applications. There are however many exercises and matlab assignments.

URL for publisher, author, or book:
http://www.springer.com/birkhauser/mathematics/book/978-0-8176-8258-3
MSC main category:
65 Numerical analysis
MSC category:
65-XX
Other MSC categories:
65Dxx, 65G50, 65H05, 65H10, 65L05, 65L06, 65L07, 65L10, 65L20
Review:

The classical subjects are treated in a classical way. Here is a list: rounding errors, approximation and interpolation, differentiation and integration, zero finding for (systems of) nonlinear equations, ODE's: initial, as well as boundary value problems. Note however that numerical linear algebra, optimisation and PDE's are absent.

The author concentrates on the analysis and basic principals of the methods and the algorithms. The text is never very abstract, i.e., there are some theorems and proofs, but they certainly do not dominate. For the more difficult parts, the reader is referred to the literature and/or some comments are given at the end of a chapter. This makes the theory easily digestible and never dull.

There is a lot of matab code included to play with and so one can gain some feeling of what the theory means in practice'. However, there are no real life examples from applied fields like engineering, economics, etc. Even the academic ones are rare. On the other hand, there is an amazingly extensive list of theoretical' exercises (i.e., proving or deriving things with paper and pencil) but also some machine-assignments per chapter.

The second edition has updated references and notes, and matlab has now been adapted as the sole computer language. Some exercises are added and the book's subtitle `an introduction' in the previous edition has been dropped. The major change is however that now solutions to all exercises and computer assignments are available. A selected number is included after each chapter, but all of them are available as a pdf to instructors upon request from the publisher on the book's webpage. It's a pity that it is not possible to download the matlab code as well. On that webpage one can also find a list of typo's that should be corrected.

Conclusion: one of the better handbooks on the market today, based on several decades of teaching experience of the author. It is an excellent tool for teaching a classical numerical analysis course.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven

## Morse Theoretic Aspects of p-Laplacian Type Operators

Author(s):
K. Perera, R.P. Agarwal, D. O'Regan
Publisher:
AMS
Year:
2011
ISBN:
978-082184968
Price (tentative):
$69 Short description: The purpose of this book is to present a Morse theoretic study of a very general class of homogeneous operators that includes the p-Laplacian as a special case. The p-Laplacian operator is a quasilinear differential operator that arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media. URL for publisher, author, or book: http://www.ams.org/bookstore-getitem/item=SURV-161 MSC main category: 47 Operator theory MSC category: 47J05 Other MSC categories: 58E05, 47J10, 35J60 Review: Standard Morse theory is a useful tool to compute homology groups of finite dimensional manifolds from the critical points of a sufficiently generic function. The tools of Morse theory can be developed in an infinite dimensional setting to some extent. This point of view can be applied to solve differential equations, by means of writing a Morse functional from a Banach vector space whose critical points are the solutions to the original equation. Each critical point has a degree and some cohomological information. This together with some standard tools of algebraic topology may allow finding restrictions to the number and properties of critical points. The current book is devoted to apply infinite dimensional Morse theory to study the p-Laplacian and other quasilinear operators where the Euler functional is not defined on a Hilbert space or is not$C^2$or where there are no eigenspaces to work with. The p-Laplacian operator arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media. The p-Laplacian problem consists on finding the eigenfunctions of$\Delta_p u= \lambda |u|^{p-2} u$, where$\Delta_p u= div( |\nabla u|^{p-2} \nabla u)\$.

The book is very technical, and it is only accessible to experts in the field. Even the introduction and overview is written in a technical language, assuming some knowledge with the notation. Chapters 2, 3 and 4 are more readable, containing background material. A large part of the rest of the book is of research level. Researchers in the area will find the book of interest, containing many results, and with a large bibliography.

Reviewer:
Vicente Muñoz
Affiliation:
UCM

## Statistics for Non-Statisticians

Author(s):
Publisher:
Springer
Year:
2011
ISBN:
978-3-642-17655-5
Short description:

A quick and applied introduction to Statistics

MSC main category:
62 Statistics
MSC category:
62-01
Review:

As the author claims in the preface of the book, everything is nowadays supported by numbers. Hence, if numbers are such an important source of information, the art of dealing and assessing with statistical data material is becoming more and more essential in our society.
The spirit of this book is a first course for practitioners. The word “first” means that the basic concepts of classical Statistics are all motivated from their fundamentals whereas “practitioner” (or non-statistician, as indicated in the own title) means that the presentation does not cover the mathematical theory behind those concepts but practical information and useful details thought, v.g., for planning and interpreting surveys.
The book is organized in nine chapters. They successively introduce the basics in descriptive Statistics (that is, presentation, collection and description of data) and the main notions in classical analytical Statistics (confidence interval, linear regression, hypothesis testing, samplings,...). The emphasis is put on practical situations where all these ideas can be applied. This emphasis is intensified by a frequent (excessive?) use of exclamation marks found in those sentences containing the essential information. Additionally, every chapter shows an implementation of its topics within the framework of the statistical freeware Calc. The final chapter contains a summary of Probability and Statistics, a glossary of terms and, like most books in this field, tables of useful distributions.
The presentation of the book, both visually and mathematically, is good and well organized and may provide a pleasant (and useful) reading for either statisticians and non-statisticians.

Reviewer:
Marco CASTRILLON LOPEZ
Affiliation: