## Recent book reviews

Five recent book reviews

## Why cats land on their feet. And 76 other paradoxes and puzzles

Author(s):
Mark Levi
Publisher:
Princeton University Press
Year:
2012
ISBN:
978-0-691-14854-0
Price (tentative):
£13.95
Short description:

The author explains with elementary physical laws many phenomena observed in real life, but also gives answers to more academic puzzles. It's fun to read. It is addressed to a general intrigued and curious audience. Some elementary physical principles are added in an appendix.

URL for publisher, author, or book:
http://press.princeton.edu/titles/9666.html
MSC main category:
00 General
MSC category:
00A79
Other MSC categories:
70-01, 74-01, 76-01, 80-01
Review:

The title (plus subtitle) says it all. Although the author is a mathematics professor, the book is about physics rather than mathematics. All the reader is supposed to know are some elementary physical principles that are collected in an appendix. Things like Newton's laws, energy, center of mass, angular momentum, etc. Of course this involves some formulas and hence some mathematics, but secondary school knowledge suffices.

The example of the cat comes only in chapter 13 and explains why the cat manages to flip in a split second. Its trick is to squirm before making the roll movement, creating opposite spins. A surprising connection is made with winds that slow down the rotation of the earth. This phenomenon has extended a day, which is estimated to have originally lasted only 6.5 hours, to its current duration. Ant it is is basically a consequence of the same physical laws. These two items make it only a short chapter, but the other 12 chapters just keep going on applying the same thematic physical principle of the chapter in quite different situations. All of them are really surprising and keep you reading on. It is almost standard that the correct answer is opposite to what intuition predicts. Sometimes the wrong answer is explained according to apparently correct arguments and the reader is asked to find the flaw. In any case, the right arguments with the right answer are always given.

Some puzzles are practical like how to sail against the wind?' or why is it harder to suck up water through a straw than to blow it out?'. Others are academic like if a water surface spinning in a bowl takes a parabolic shape, how should one steer a boat to navigate on the slope of this inclined sea?'.

Most chapters are centered around some physical principle or law like for example coriolis force: why goes fair weather with anticyclones and why do they rotate clockwise in the northern hemisphere and how do they link with trade winds at the equator? Or some topics related to a gyroscope: how does it manage to seemingly deny gravity and how can it be used to find out where is north? Or thermodynamics: how to mix two liquids of different temperature to end up with a mixture that has a temperature higher than the average?

Each chapter can be read in a few minutes time, say while you are drinking a cup of tea or coffee. It will give you a lot of inspiration to challenge or entertain your friends during a reception or another get-together with some different kind of beverages. Of course you will impress them only when they haven't read the book themselves already. Hence make sure that you are the first.

Reviewer:
Affiliation:
KU Leuven

## Raising public awareness of mathematics

Author(s):
Ehrhard Behrends, Nuno Crato, José Francisco Rodrigues (eds.)
Publisher:
Springer Verlag
Year:
2012
ISBN:
978-3-642-15709-4
Price (tentative):
49.95 € (net)
Short description:

This book contains the proceedings of an international conference held in Óbidos (PT) in 2010, ten years after the World Mathematical Year 2000. It was jointly organized by the Centro Internacional de Matematicá and the Raising Public Awareness (RPA) committee of the EMS. In 28 contributions is reported on many different initiatives concerning RPA (websites, exhibitions, real or virtual museums, national initiatives, etc.) from all over the world but mainly from within Europe.

URL for publisher, author, or book:
http://www.springer.com/mathematics/book/978-3-642-25709-4
MSC main category:
00 General
MSC category:
00A79
Review:

The World Mathematical Year 2000 boosted many initiatives intended to Raise Public Awareness in Mathematics (RPAM). Ten years later in 2010 an international workshop was organized by the Centro Internacional de Matematicá and the Raising Public Awareness committee of the EMS. This book contains the (post factum) proceedings of this conference held in Óbidos (PT). It contains 28 contributions which report on many different RPAM initiatives.

The harvest is abundant and the formats are very diverse, but the editors organized the texts in four parts: National experiences, Exhibitions and mathematical museums, Popularisation activities and Popularisation - why and how.

John D. Barrow and Robin Wilson give some snapshots' of what is happening in the UK, Jean-Pierre Bourgain does something similar for France, and Wolfram Koepf explains the German website mathematik.de complemented with other "do's and don'ts" given by Günter M. Ziegler and Thomas Vogt raised from experience of the Mathematical Media Office of the DMV. The account by Reinhard Laubenbacher about the USA includes arguments for convining politicians to fund mathematics. Renata Ramalho and Nuno Crato highlight debate and results for RPAM and Mathematical education in Portugal. The many initiatives in Spain are reported by Raúl Ináñez Torres.

The second part about exhibitions and mathematical museums shows different examples of real and virtual exhibitions. We give a selection. Mathema is a large and most successful exhibition organized in Berlin in 2008; Mathematikum is a mathematical science center in Giessen (DE); The Italian exhibition Il Giardino di Archimedes pays attention to mathematics in history, play, and everyday's life; In Portugal, the Atractor Association is involved in both physical exhibitions but also in virtual aids like GeCla, a DVD and website with interactive possibilities to play many mathematical puzzles, generate tilings, and experiment with symmetry; IMAGINARY is another very extensive open source math exhibition platform.

Part three has more of the same. The website www.mathematics-in-europe.eu is an initiative of the EMS that was initiated at the Óbidos workshop in 2010 (the text was written in January 2011). Furthermore the reader can find guidelines for resorting maximal effect from minimal cost, a retrospective of the period WMY2000-2010, mathematics used in magic trics, how to exploit RPAM from an interdisciplinary item or from any event that raised public awareness, linking mathematics (visualisation) and art, etc.

In part 4, collects some papers of a more reflective nature. Again a selection: About the usefulness of mathematics and rigour used in communicating about it; the role played by Martin Gardner and his books about recreational mathematics and his columns in Scientific American; research-application-communication are three different aspects of our mathematical experience; how do we keep all current attention for mathematics alive.

The authors are all responsible for or working on the project they report on, or they are really knowledgeable about the survey given. If you want to start up another initiative in this context, you may learn many things from this book: a white paper for setting up an exhibition, topics and the material needed for organizing a workshop, links to interactive websites you want to add to your own creation, they are all there. Or maybe you are just generally interested and then there is much fun exposed left for you to grasp. I could not resist to hop around on the cited websites and play with many of the most amazing interactive possibilities offered by them.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven

## Mathematics of Approximation

Author(s):
Johan de Villiers
Publisher:
Atlantis Press/Springer Verlag
Year:
2012
ISBN:
978-94-91216-49-7 (P), 978-94-91216-50-3 (E)
Price (tentative):
59.95 € (net)
Short description:

This is the first volume of a new series Mathematical Textbooks for Science and Engineering. It builds up standard approximation theory from scratch (requiring only some advanced calculus and linear algebra) up to a reasonably advanced level. No complex analysis or advanced functional analysis is needed. As a textbook it includes all the proofs and has many exercises following each chapter.

URL for publisher, author, or book:
<a href="http://www.springer.com/mathematics/book/978-94-91216-49-7">www.springer.com/mathematics/book/978-94-91216-49-7</a>
MSC main category:
65 Numerical analysis
MSC category:
65-01
Other MSC categories:
42A10, 42A16, 42C10, 65D05, 65D07, 65D10, 65D15, 65D32, 65T40
Review:

The idea of the new series Mathematical Textbooks for Science and Engineering (MSTE) is to publish self contained textbooks on applied or applicable mathematics at undergraduate and graduate level of mathematics science and engineering. It should be easy reading, even for self study. This book is volume 1 and it is devoted to the standard basics of approximation in the broad sense.

The subjects are not a surprise and line up as one would expect: Polynomial interpolation, best approximation (in $L_2$ and $L_\infty$-sense by polynomials), elementary orthogonal polynomials (Chebyshev, Legendre), numerical quadrature, Fourier series, and spline approximation. A more detailed table of contents can be found on the publisher's website. Note that it involves only approximation by polynomials, trigonometric polynomials or piecewise polynomials (no rationals or other function systems) and only real approximation (complex analysis is avoided completely).

All the material is standard and fully proved in the book, and no historical notes are given about the origin of these results. Hence no references to research literature are needed and thus there is no list of references. As with most textbooks, it is an endeavour resulting from many years of experience while teaching the course to many students and it is not different in this case. Such a ripening process is needed to make all the things explicit that a teacher considers obvious but may actually be a source of wild and unexpected speculations by students. So it is a remarkable achievement to introduce the right amount of rigour to avoid any misunderstanding from the reader's side and yet not to overload the text and keep it very readable also for self studying. In that respect de Villiers has made some specific choices that allowed to deliver a thoroughly thought over product that serves these characteristics very well.

Some of these choices have their consequences. Although the series aims at an audience from science and engineering', this book is in my opinion more oriented towards science than towards engineering. It is practical in the sense that a lot of attention goes to error estimates and that it handles approximation techniques that are important and actually used in practical applications, but it is not so practical that proper attention is paid to the pure numerical and algorithmic aspects. Many of the subjects discussed will also appear in a textbook on numerical analysis, but there much more attention will be given to rounding errors and efficient implementation. This particular choice also shows by a total absence of graphs. A simple graph of an orthogonal polynomial, the location of the Chebyshev points, a spline, or whatever could relax the reader a bit from the rigorous and sometimes complicated formulas to explain a relatively simple idea. Another exponent of this choice of approach is a lack of numerical examples. There are some examples included, but mostly quite simple and of an academic nature. So there are also no graphs or tables that illustrate the error or the convergence behaviour of some of these methods. Of course a particular phenomenon might require a numerical explanation that has been deliberately avoided in this book. Another example is the incidental mentioning of FFT in the chapter on Fourier series: [...] indeed (9.3.73)-(9.3.76) form the basis of the widely used Fast Fourier Transform (FFT), a detailed presentation of which is beyond the scope of this book.'. In many computational issues, the linear algebra aspect takes an important role when it comes down to implementation (e.g. the computation of Gaussian quadrature formulas by solving an eigenvalue problem). This is another aspect that has not been included in this book.

But even, if the contents has a strong theoretical component, the scrupulous attention paid to error estimates, estimates of Lebesgue constants, and the different ways in which a solution can be represented and computed have a definite `applicable' component as well. So, I am somewhat surprised that, besides Lagrange and Newton forms of interpolation polynomials, there is no mentioning of the barycentric representation which allows some nice theory and has definitely great practical importance. Another nice piece of theory could have been devoted to wavelets, which are absent as well (except for a brief note in an exercise about splines). Let me make it clear that I do not consider my enumeration of what is not in this book as a critique. After all, any course that is actually taught has to transfer knowledge in a finite number of teaching hours, and there is enough material left to teach a major course on approximation. My only intention is to inform potential buyers of what is and what is not to be expected.

A completely different approach is for example taken in the matlab based package Chebfun developed by Nick Trefethen and his team. This is less broad, less theoretical, and obviously much more hands-on. Learn by experience, not by formulas. A book on Chebfun is announced as a SIAM publication for early 2013. That book will be in many aspects different from the present book under review: probably not the opposite but certainly a complement.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven

## Probability tales

Author(s):
Charles M. Grinstead, William P. Peterson, J. Laurie Snell
Publisher:
AMS
Year:
2011
ISBN:
978-0-8218-5261-3
Short description:

Probability tales explores four real-world topics through the lens of probability theory. The authors assume as a prerrequisite the basic concepts in probability theory and focus, in an expository style, on four specific subjects: streaks, the stock market, lotteries and fingerprinting.

MSC main category:
60 Probability theory and stochastic processes
MSC category:
60-01
Review:

Probability is a subject well-considered by most people outside the mathematical community. As mathematicians, we know that research in algebra or geometry leads us to a better understanding of the physical world, and that sometimes amazing connections or applications appear for mathematical theories which have been already developed. Probability arises in many places in our daily life: sports, economy, design.. and yet many textbooks usually introduce the same collection of examples when they try to explain the basic notions in probability.

Probability tales tries to solve this problem by explaning probability theory applied to four different real-world topics, one per chapter. More concretely, the first chapter is devoted to streakiness: cold or hot streaks occur in different sports and the authors explain how different mathematical models (Bernouilli, Markov chains...) are used to fit the existent data and to explain these success and failure runs in baseball, basketball, tennis... The second chapter is focused on the stock market, and on how the variations in the prize of a stock can be modelled, from the basics to more complicated models. A nice exposition on Powerball lottery and the funny behaviour of lottery players occupies chapter 3, and a historical evolution of fingerprinting and its relationship with probability is the topic of chapter four.

The style is informal and the topics are not presented as in an standard textbook: the book is a a collection of four independent short essays on probability, which can be read easily avoiding technicalities. However, the details in each topic are gathered in short appendices after each chapter, and several references to research articles are provided.

To get a good grip on the text, the prerrequisites are probability and calculus at the level covered in a basic undergraduate course. The authors point out the possible use of the text as a good complement to an ordinary undergraduate textbook: Probability tales explains some basic notions (hypothesis tests, p-values, distributions)put into context, so it might serve well to this purpose.

Reviewer:
Javier Martinez
Affiliation:

## Interpolation, Schur Functions and Moment Problems II

Author(s):
Daniel Alpay, Bernd Kirstein (eds.)
Publisher:
Birkhäuser Basel
Year:
2012
ISBN:
978-3-0348-0427-1
Price (tentative):
94.95 € (net)
Short description:

Like the previous volume with the same title, this is a collection of papers in the field of Schur analysis, a mathematical research fiels involving moment problems and more generally interpolation in the complex plane by bounded analytic (Schur) functions or positive real (Carathéodry) functions. The papers are produced by the team of the U. Leipzig and one coworker who are well known ambassadors of Schur analysis. This volume contains a coherent set of papers related to matrix valued moment problems on the complex unit circle and on real line, a half line or a finite interval.

URL for publisher, author, or book:
www.springer.com/mathematics/book/978-3-0348-0427-1
MSC main category:
30 Functions of a complex variable
MSC category:
30E05
Other MSC categories:
42C05
Review:

Interpolation, Schur Functions and Moment Problems (Part I) was edited by D. Alpay and I. Gohberg and published by Birkhäuser in 2006 as volume 165 in the same series on Operator Theory: Advances and Applications. This book is volume 226 in the same series. Many (in fact all) other volumes in this series are also related to topics in operator theory and analysis that play a role in the interaction between pure mathematics and applications in systems theory, signal processing, linear algebra, etc. By the number of volumes in the series, founded in 1979 by I. Gohberg, it is seen that this is a very productive and active research area. That is why the series has now two subseries: Linear Operators and Linear Systems (to which the present book belongs) and Advances in partial Differential Equations.

The topics of this volume, as well as the ones in volume 1 dwell in the realm of Schur analysis. The name stems from I. Schur who published two papers in 1917/18 where he proposes an algorithm to solve a coefficient problem (i.e. the trigonometric moment problem), which is a kind of continued fraction-like decomposition of a function analytic in the complex unit disk that is bounded by 1 (now called a function in the Schur class). This involves a recurrence that coincides with the recurrence relation of polynomials orthogonal on the unit circle (studied by Ya. Geronimus and G. Szegő) and can be interpreted as a discretized transmission line in circuit theory (V. Belevich) or a digital prediction filter in signal processing (N. Wiener and P. Masani). It were the applications that revived the interest in Schur analysis in the 1960s and it hasn't stopped since. This has been generalized much further in many different directions and linked to other work by some great mathematicians like R. Nevanlinna, G. Pick, G. Herglotz, T. Stieltjes, H. Weyl, etc. A translation of Schur's original papers into English can be found e.g. in volume 18 of the OT series ( I. Schur methods in operator theory and signal processing (I. Gohberg (ed.), 1986) and many other of the basic papers in Schur analysis are collected in Ausgewählte Arbeiten zu den Urspüngen de Schur-Analysis Band 16 of Teubner Archiv zur Mathematik, (B. Fritzsche, B. Kirstein (eds.), 1991).

The six papers of this volume are mainly focussing on block generalizations of moment problems and Nevanlinna's theory. Moment theory asks for the existence and characterization of a measure when a sequence of moments are given. Nevanlinna's analogue is a generalization when the moments are not given at one point, but at more than one point, i.e. instead of Taylor series coefficients at just one point, a number of Taylor coefficients are given at different points, leading to a form of Hermite interpolation at these points. The block generalization refers to the fact that the functions and the moments are matrix valued, and hence also the measure one is looking for. As a consequence, the theory does not involve (matrix valued) orthogonal polynomials but more general matrix valued orthogonal rational functions.

Bernd Fritzsche and Bernd Kirstein from Leibniz University have been passionate ambassadors of Schur analysis. The papers in this volume are all written by them and their coworkers at their institute (C. Mädler, T. Schwartz, A. Lasarow, A. Rahn) with one exception, the paper by A.E. Choque Rivero (Mexico) is a continuation of his earlier collaboration with the Leipzig team.
Here is a short summary.
Papers 1-3 involve reciprocal sequences and their applications, papers 4-5 discuss power moment problems on the real line and the last papers is about orthogonal rational functions.

1. B. Fritzsche, B. Kirstein, C. Mädler and T. Schwarz. On the Concept of Invertibility for Sequences of Complex p × q-matrices Application to Holomorphic p×q-matrix-valued Functions
This involves the construction the a q× p-sequence which is the reciprocal of a p×q-matrix sequence f(z). This is a matrix version of constructing the scalar coefficients in a series 1/f(z), given the coefficients of the series f(z). It applies to investigating the holomorphicity of the Moore-Penrose inverse of a matrix valued holomorphic function.
2. B. Fritzsche, B. Kirstein, A. Lasarow and A. Rahn. On Reciprocal Sequences of Matricial Carathéodory Sequences and Associated Matrix Functions
This studies a particular case of the previous paper. It concentrates on the case p = q for Carathéodory functions (functions holomorphic is the disc and whose Hermitian part is nonnegative definite). The reciprocal of such Carathéodory function is again a Carathéodory function.
3. B. Fritzsche, B. Kirstein, C. Mädler and T. Schwarz. On a Schur-type Algorithm for Sequences of Complex p×q-matrices and its Interrelations with the Canonical Hankel Parametrization
This is another application of reciprocal sequences. The Hamburger moment problem involves power moments for a (matrix-valued) measure on the real line, and the measure has to be recovered. The moments are coefficients in a Taylor series and a second sequence of moments can be defined as the Taylor coefficients in the reciprocal series. On the other hand partially defined (block) Hankel matrices that have a positive definite extension play an important role in the existence of a solution. By recursively constructing Schur complements, a sequence of matrices can be defined characterizing them. This allows to formulate a Schur-type algorithm to find this so called canonical characterization of Hankel matrices and hence finding solutions of truncated Hamburger moment problems.
4. A.E. Choque Rivero. Multiplicative Structure of the Resolvent Matrix for the Truncated Hausdorff Matrix Moment Problem
The Hausdorff moment problem has to find a measure, supported on a finite interval, given the power moments. This involves two sequences of nested (block) Hankel matrices corresponding to an even or an odd number of prescribed moments. The solution of the moment problem is given in terms of a resolvent matrix.
In this paper, the resolvent matrix is constructed and its factorization in elementary factors (Blaschke-Potapov factors). Again this corresponds to some recurrence relation for the (matrix-valued) orthogonal polynomials on a finite interval. This factorization of the resolvent has been treated before in the case of an even number of moments, here the odd case is analysed.
5. B. Fritzsche, B. Kirstein and C. Mädler. On a Special Parametrization of Matricial α-Stieltjes One-sided Non-negative Definite Sequences
The Stieltjes moment problem looks for a measure supported on the positive real line, given its power moment sequence. Like in the Hamburger case, the existence of a solution involves (block) Hankel matrices containing the moments that have to be in a certain class. Being a member or not can be easily checked in the Hausdorff case by their canonical parametrization that can be constructed by a recursive Schur-like algorithm. Here a similar characterization is derived for the Stieltjes case and the relation between both is established.
6. B. Fritzsche, B. Kirstein and A. Lasarow. On Maximal Weight Solutions of a Moment Problem for Rational Matrix-valued Functions
Truncated moment problems on the unit circle with moments given at several points involves rational interpolation (rather than matching the initial terms in series expansions). For a finite number of moments, a measure solving the moment problem can be described as a quadrature formula, i.e. it is a discrete measure with mass at a finite number of mass points. In this paper such particular solutions are investigated that have extremal mass at a prescribed point on the unit circle.

The contributors of this volume are very productive and they have published a large number of papers in many journals and books. Bringing a number of papers together and publish them as a book is a good idea. They have a particular style of writing, involving very precise formulations that also require complex notation. It may take a while for the reader to get used to it, but once familiar with these habits and the constructs involved, it is good reading. It will be of high interest to anyone who is involved from far or near with Schur analysis and the whole universe of related topics that I sketched in the beginning. Given the special character of this book (sub)series, it is clear that whoever was interested in volumes of this OT subseries before will be interested in practically all of them, a fortiori in this one. Be warned though that this book is at an advanced level, treating particular and specialized results in the domain, so this is not the right place to start learning the subject.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven