The Princeton Companion to Applied Mathematics
Edited by Nicholas J. Higham
PRINCETON UNIVERSITY PRESS 2015, 1016 PAGES
PRICE (HARDBACK) £69.95 ISBN 978 0691150390
This collection of almost 200 authoritative, but accessible, articles on applied mathematics (spread over a thousand or so pages) is arranged thematically with the following themes:
Part I  Introduction to Applied Mathematics
Part II  Concepts
Part III  Equations, Laws, and Functions of Applied Mathematics
Part IV  Areas of Applied Mathematics
Part V  Modeling
Part VI  Example Problems
Part VII  Application Areas
Part VIII  Final Perspectives
In the Preface the editor, Professor Higham (Richardson Professor of Applied Mathematics, University of Manchester), writes “The Companion differs from an encyclopedia in that it is not an exhaustive treatment of the subject, and it differs from a handbook in that it does not cover all relevant methods and techniques.” It does however provide a flavour of a wide range of applied mathematics including such aspects as how to read and write it in addition to descriptions of the mathematics itself.
The target audience is stated to be mathematicians at undergraduate level or above; students, researchers, and professionals in other subjects who use mathematics; and mathematically interested lay readers. In other words the background knowledge required to understand the articles varies considerably from article to article! There are some articles easily understandable to sixthform students (for example, Part VI.2 Bubbles, by Andrea Prosperetti; Part VI.6 The Flight of a Golf Ball, by Douglas N. Arnold) and some where a mathematically interested lay reader might struggle without suitable background knowledge (for example, Part IV.13 Numerical Solution of Partial Differential Equations, by Endre Süli; Part IV.29 Magnetohydrodynamics by David W. Hughes).
I was pleased to see there are plenty of diagrams throughout the book; most of these are monochrome, but there is a centrally placed collection of 23 colour plates.
I can’t claim to have read all the articles in The Companion (yet!), but here are a few that I stopped to read as I flicked through the pages (I’ll list one from each theme).
Part I.6 The History of Applied Mathematics, by June BarrowGreen and Reinhard SiegmundSchultze. This is probably the longest article in the book (about 25 pages), but is easy to read and accessible to all.
Part II.21 Invariants and Conservation Laws, by Mark R. Dennis. I like this because it explained Noether’s theorem in a way I could understand!
Part III.17 The Lambert W Function, by Robert M. Corless and David J. Jeffrey. This implicit elementary function seems strange when one first comes across it, but the authors clarify it nicely here.
Part IV.7 Special Functions, by Nico M. Temme. This covers a raft of interesting functions used by applied mathematicians, physicists, engineers and statisticians. It includes Bernoulli numbers, Euler numbers, Stirling numbers, the Gamma function, Gauss Hypergeometric function, Bessel functions, Legendre functions and others.
Part V.5 Mathematical Physiology, by Anita T. Layton. An application far from my area of expertise, but with a surprisingly familiar set of equations!
Part VI.5 Insect Flight, by Z. Jane Wang. I was attracted by a couple of the section titles; namely: “How Do Insects Fly?” and “How Do Insects Turn?”
Part VII.10 Compressed Sensing, by Yonina C. Eldar. A relatively young topic I think I should know more about.
Part VIII.3 How to Write a General Interest Mathematics Book, by Ian Stewart. Who could possibly resist such an article from Ian Stewart?!
The readers of this magazine are likely to be interested in many of the articles in The Companion. There are too many individual topics to list here, but a pdf file of the Contents may be downloaded from: http://press.princeton.edu/titles/10592.html. Files containing the Preface and a list of the contributors may also be downloaded from the same site.
Safe to say there is something for everyone in The Princeton Companion to Applied Mathematics.
Alan Stevens CMath FIMA
Review first published in Mathematics Today (December 2015)
Advanced General Relativity: Gravity Waves, Spinning Particles, and Black Holes
Claude Barrabès and Peter A. Hogan
OXFORD UNIVERSITY PRESS 2013, 152 PAGES
PRICE (HARDBACK) £60.00 ISBN 9780199680696
This book is a research monograph which contains 6 chapters, each of which is intended to serve as a platform from which a beginning graduate student could launch a research project. The expected readership is people who have either completed a taught MSc in General Relativity or have reached this level through self study after completing an introductory book on this subject, e.g. Introduction to Black Hole Physics by Frolov and Zelnikov.
The six chapters cover the following material:
Chapter 1 – is concerned with various kinds of Lorentz transformations, ending with a section on gravitational waves.
Chapter 2 – considers waves, both gravitational and electromagnetic, plane, shock and high frequency. The chapter starts with the approach used in initial courses on general relativity, that of presenting gravitational waves as solutions of the vacuum field equations in linear approximation. It then continues via a sequence of gauge transformations to the metric tensor that is the exact solution of the vacuum field equations. The approach is then to specialise to waves with a Dirac delta function profile.
Chapter 3 – deals with an approach to calculating the equations of motion of various types of particles, e.g. Kerr particles moving in external fields using the appropriate vacuum field equations, that was developed starting in 2008. The chapter ends with a section on spinning test particles. This approach results in avoiding divergent integrals.
Chapter 4 – starts by considering OzsvathRobinsonTozga planefronted gravitational waves with a cosmological constant, this eventually leads to gravitational waves appearing in isotropic models as perturbations. Covariant and gauge invariant perturbation theory are covered and used to demonstrate how gravitational waves are introduced into isotropic models.
Chapter 5 – introduces black holes, initially covering basic properties, followed by some classical and quantum aspects of them. The formation of trapped surfaces during gravitational collapse, scattering properties of a high speed Kerr black hole and creation of de Sitter universes inside a black hole are considered.
Chapter 6 – is a short chapter covering black holes in more than 4D spacetimes. This is usually investigated as part of attempts to unify gravity with the other fundamental forces. The chapter starts with a brief outline of higher dimensional black holes and goes on to consider other aspects including the loop conjecture in D dimensions.
I would recommend this book for people planning a career in General Relativity research or who are considering embarking on a PhD concerning some aspect of this subject.
John Bartlett CMath MIMA
Book review published directly onto IMA website (December 2015)
Professor Stewart’s Casebook of Mathematical Mysteries
Ian Stewart
Profile Books Ltd 2014, 307 pages
PRICE (HARDBACK) £12.99 ISBN 9781846683473
Professor Stewart’s Casebook of Mathematical Mysteries is the third in a series of popular mathematics books. It contains some 125 short articles some featuring two fictional characters: Victorian detective Hemlock Soames and Dr John Watsup who may bear some resemblance to the creations of Sir Arthur Conan Doyle. This actually works quite well as a device to make the stories readable. There is a mix of serious mathematics, anecdotes, mnemonics and jokes.
A few merit particular mention:
Prime Number Mysteries gives an interesting analysis of unsolved problems in Number Theory. There is discussion of computer verification for large numbers of specific cases including computer produced graphs, but not of computer aided proofs.
Sign of One is a series of four articles on the problem of expressing integers using only the digit 1 a fixed number of times (akin to the more familiar ‘four fours’ puzzle). Which integers can be expressed in this way depends on which symbols are allowed. By making use of the exotic ‘floor’ and ‘ceiling’ functions, familiar to spreadsheet users, along with a combination of factorials and square roots expressions, many integers can be derived. A missing square root sign in two of the expressions (pp. 116) makes the reasoning a bit tricky to follow. There is also use of the ‘double factorial’ symbol with which readers may not be familiar although this is explained on pp. 108. Eventually it is shown that every integer can be expressed using just a single 1 – but only by repeatedly using the natural log, exponential (ex) and ceiling functions.
The Wave of Translation gives an introduction to solitons (solitary waves), recounting the experience of the Scottish civil engineer John Scott Russell, who observed an instance of the phenomena on a canal in 1834 and chased it for several miles on horseback.
A Tiling That Is Not Periodic starts with considering shapes that can be used to tile the plane periodically (repeating indefinitely) or nonperiodically. But questions about nonperiodic tiling lead into deep areas of Mathematical Logic. In particular it has been proved that there is no general algorithm that can determine whether a given set of shapes can tile the plane (the domino problem), making it undecidable in the sense of Gödel’s Theorem.
How to Write Very Big Numbers explores the names and symbols used for expressing large numbers starting with some historical background, from Roman numerals to the different British and American ‘billion’. Eventually we get to Knuth’s arrow notation with applications in string theory and cosmology.
There is a large section The Mysteries Demystified (58 pages) of detailed answers and explanations to many of the puzzles posed. There is no index or formal bibliography, but there are intext references including web sites and the work of other mathematicians is acknowledged throughout. Some problems are solvable with the help of simple programming tools or spreadsheets. The articles are wellillustrated in blackandwhite.
This book is first and foremost entertaining, and will appeal to the mathematicallyinclined reader at all levels. Professor Stewart uses mysteries and puzzles as a vehicle on a journey into deeper understanding and further enquiry.
Francis McGonigal CMath MIMA
Birmingham City University
Book review published directly onto IMA website (December 2015)

50 Visions of Mathematics
Murder, medicine, gambling, sport…; the list of topics to which mathematics contributes is endless. The 50 Visions of Mathematics contains articles on, well, 50 of them! The book celebrates the 50th anniversary (in 2014) of the Institute of Mathematics and its Applications. It comprises fifty short articles on various aspects of mathematics, together with colour plates containing fifty mathematical images. The contributors come from all over the world and include professors of mathematics, teachers, journalists, students and others.

A Guide to Groups, Rings, and Fields
As the author of this book notes in his very ﬁrst sentence, ‘[a]lgebra has come to play a central role in mathematics’. There are a vast number of textbooks available on ‘algebra’ in its broadest sense, each with its own particular focus. In the book under review, the author has endeavoured to provide a summary of the major topics of current abstract algebra, and thereby supply students with a handbook where they might quickly check deﬁnitions or the statements of theorems. To this end, the author has omitted all proofs, apart from some sketches (‘shadows of proofs’) here and there. Another goal of the book is to provide a uniﬁed picture of algebra, so that students might see ‘how it all hangs together’: the author intends to furnish insight, rather than focus on formal structure. This is billed as a practical text, aimed at users of algebra, and so need not be (and probably shouldn’t be) read in the order in which it is presented. Indeed, the author frequently, and knowingly, employs concepts that have not yet been deﬁned.

A Guide to MonteCarlo Simulations in Statistical Physics (Third Edition)
This is a graduate level text that deals primarily with MonteCarlo simulation of complex physical systems encountered in condensedmatter physics and statistical mechanics. It also provides a very brief overview of some alternative computer simulation methods of use in statistical physics, such as molecular dynamics, quasiclassical spin dynamics, dissipative particle dynamics, lattice gas cellular automata and others. The emphasis however, is firmly on various MonteCarlo simulation approaches to lattice based systems, especially Ising spin type models, and offlattice systems, exemplified by binary fluids and polymer mixtures.

A Mathematical Orchard: Problems and Solutions
This is a collection of 208 challenging problems, with carefully crafted instructional solutions, offering the reader a feast of mathematical entertainment. One hundred and thirty of the problems in the book were previously published in 1993 by the MAA in The Wohascum County Problem Book. The problems are not trivial to solve and given consideration, often offer scope for extension.

A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics
A Mathematical Tapestry combines some practical recreational mathematics, in particular paper folding constructions, with deep theoretical ideas in geometry, number theory and group theory. This more than justifies the subtitle Demonstrating the Beautiful Unity of Mathematics.

A Student's Guide to Vectors and Tensors
Before starting my undergraduate studies, all I was told of tensors was that they are ‘matrices on steroids’, which in hindsight seems an apt description. Tensors are described in the preface as ‘the facts of the Universe’, given their allpervading nature in applied mathematics and physics. Tensors changed our understanding of the fundamental structure of the universe when Albert Einstein succeeded in expressing his theory of gravity in terms of tensors in 1915, which can quite rightfully be seen as the highlight of this book in the final chapter.

A Transition to Advanced Mathematics: A Survey Course
This text is intended to aid students in the move from a calculational, or operational, approach to mathematics to a more conceptual, proof based approach, and also to provide a broad overview of undergraduate mathematics. Chapters cover, in turn, the basics of mathematical logic, an introduction to abstract algebra via group theory, number theory, real analysis, probability and statistics, graph theory and complex analysis.

Advanced Topics in Quantum Field Theory: A Lecture Course
There are essentially three to four stages of studying the subject of Quantum Field Theory (QFT), as covered in American universities. In the first two of these stages there are usually a standard set of topics. In brief, these are: QFT 1  Relativistic Quantum Mechanics, spinors, Dirac's Equation, Hamiltonian QFT, Canonical Quantization, etc., and QFT II  Path Integral formalism, Perturbation Theory beyond tree level, Renormalization is thoroughly discussed, QCD, etc. These are covered in the first and second years of postgraduate study.

Algebraic Shift Register Sequences
There are many situations that require sequences with a given set of properties. For example, if the sequences are being used to manage multiple, simultaneous communications within the same medium then we would like a family of sequences with low crosscorrelation (so that the individual communications do not ‘bump into each other’): one application of this technique is Code Division Multiple Access (CDMA), which is widely used for mobile phones. Additionally, if the sequence is being used to protect information then we would like it to be in some sense difficult to predict.

Algorithmic Puzzles
This book is a collection of one hundred and fifty puzzles that can be solved by using clearly defined algorithmic procedures. The authors state that solving algorithmic puzzles is the most productive and definitely the most enjoyable way to develop one’s algorithmic thinking skills. If I had any doubts about this before I started working my way through this book I assure you that they are now completely gone.

An Equation for Every Occasion: FiftyTwo Formulas and Why They Matter
This book is written for a general audience. The preface opens with the words: ‘This is not a math book. It’s a storybook’. There are fiftytwo stories each inspired by an equation from mathematics or science. ‘Equation’ is interpreted quite widely to include the statement of a constant (π or the speed of light) or a conversion factor. About half of the equations are derived from physics, with others from biology, finance, geometry, statistics and pure mathematics. Each chapter is given a title which relates more to the story than the formula.

An Illustrated Guide to Relativity
An Illustrated Guide to Relativity explains Einstein’s Theory of Relativity but largely without the equations found in traditional texts. It does this mainly by means of spacetime diagrams and cartoons. To an extent it succeeds although the diagrams themselves become quite complex in the more advanced parts of the book, particularly on Lorenz transformation.

An Introduction to Metric and Topological Spaces (Second Edition)
Second editions of maths textbooks occupy a strange place in the literary universe. They are not really equivalent to a ‘Greatest Hits’ album, on which only the best examples from a long career survive. Nor can they be considered as being analogous to a Director’s Cut of a movie, in which creativity is given free reign over commerciallydictated constraints on the maximum time of the film.

Analytic Perturbation Theory and its Applications
This book contains nine chapters which are organised into three sections. The first part is concerned with finite dimensional perturbations. The next part deals with applications of these to optimisation and Markov processes, the final part discusses infinite dimensional perturbations

Beautiful Geometry
Beautiful Geometry by Maor and Jost, a mathematician and artist respectively, links art and mathematics through the common search for pattern. The wider perceived disconnect between the two is explored, of art expressing feelings and emotions and mathematics as rational and detached. However the authors set out from the Renaissance tradition where both ‘not only were practised together, they were regarded as complementary aspects of the human mind’.

Calendrical Calculations (Third Edition)
Reingold and Dershowitz present a comprehensive review of calendars, and it is suitable for all people who have an appreciation for mathematics and/or history.

Computation and its Limits
More than most people mathematicians (and physicists) are good with limits. Understanding the behaviour of a function, f(x), as x tends to zero (or infinity) is part of our early training. We also appreciate how an infinite sequence can tend towards a value, but never quite reach it. Given these insights it is perhaps surprising how little I had thought about limits in other contexts. As this text demonstrates limits can be subtle, important and interesting.

Count Like an Egyptian: A Handson Introduction to Ancient Mathematics
Asserting the shortcomings of existing writings on ancient Egyptian mathematics (at least for the popular audience), David Reimer has produced this stepbystep guide, from which the principles of Egyptian arithmetic may be learnt in a handson manner through their application to specific problems. The target audience is quite wide – the book is, I believe, accessible to secondaryschool pupils, and yet contains much that will be of interest to people with a broader mathematical background. Count like an Egyptian is a beautifully glossy and colourful book; the presentation of hieroglyphs is particularly well done, and fully integrated into the surrounding text.

Cows in the Maze – And Other Mathematical Explorations
The book is an eclectic mix of mathematics selected from Ian Stewart’s columns in Scientific American. It is the third such collection he has written. The book offers a wonderful insight into the range of diverse topics that mathematics as a subject has to offer.

Data Analysis and Graphics Using R: An Examplebased Approach (Third Edition)
Much has been made over recent years of the need for academic research to contribute to the wider economy. Likewise, there have also been significant debates about the merits and effectiveness of peer review, and what information researchers should provide to facilitate this.

Decoding Reality: The Universe as Quantum Information
If a poll was held to identify the most significant written phrase what do you think would win it? Perhaps something from a famous poem about how one should treat triumph and adversity? Or maybe something from Shakespeare about which is the noblest path of action?

Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation
This book is part of the series of books 'Frontiers in Applied Mathematics'. It is aimed mostly at Mathematicians working on solving various Partial Differential Equations numerically. The first chapter is accessible to undergraduates but the rest is at a much higher level.

Essential Math Skills for Engineers
Mathematics is at the heart of engineering design. So states the author, Clayton R. Paul, a professor of electrical and computer engineering at Mercer University, in Macon, Georgia. I expect readers of this magazine are likely to agree, though I’ve known a few engineers who wouldn’t!

Famous Puzzles of Great Mathematicians
This book essentially looks at puzzles from recreational mathematics that have been tackled by leading professional mathematicians. The criteria for classifying problems as ‘puzzles’ is that the questions are easy to understand but often require advanced mathematical techniques to solve. Some of these were posed in antiquity but the complete solution is accessible only with modern mathematical techniques and computing resources.

Feature Extraction and Image Processing for Computer Vision, 3rd edition
This book is the third edition of Feature Extraction and Image Processing for Computer Vision. The first observation of this book is that the page number has considerably increased over the first two editions. The main changes are as follows: a) A completely new chapter on moving object detection, tracking and analysis, b) Updated material on ViolaJones, SURF, SIFT and Haar wavelets, c) Vastly updated literature review, (d) Development of symmetry operators and (e) Expanded coverage of distance measures and classification.

Fluid Dynamics Part 1: Classical Fluid Dynamics
This is the first book of a four part series. The aim over the four books is to provide a full and coherent description of fluid dynamics from classical theory for undergraduates in Part 1, providing increasingly advanced material until the books reach the level of current research in the field by the end of book 4: Hydrodynamic Stability Theory.

Fractals: A Very Short Introduction
126 pages at A6, it is both short and small, but within it is contained a multitude of detail, explanation, background and relevance.
The book is very well written and is accessible to the interested middleschooler, and definitely to those with some basic knowledge of geometric series and post GCSE mathematics. Whatever ‘special’ maths that is required, is carefully and succinctly explained; such as the basics behind complex numbers, squaring complex numbers, and a simple overview of the log laws.

From Vector Spaces to Function Spaces — Introduction to Functional Analysis with Applications
This book is aimed at almost a beginning undergraduate in Mathematics as it starts with the simplest view of a vector and proceeds to introduce the axioms of a vector space and all related concepts, as usually seen in a first year Algebra course. The book is probably easier for a second year to understand, since these ideas will then be well assimilated, however a wellmotivated first year could find the book very interesting.

GameChanger: Game Theory and the Art of Transforming Strategic Situations
Perhaps more so than any other mathematical field, game theory has entered the public consciousness. The Prisoner’s Dilemma has been particularly influential, with its (apparently) dismal implications for the possibility of cooperation between two parties, even when the situation of both would be improved by working together. In this book, David McAdams considers different qualitative responses to the Dilemma in great detail, starting from the premise that ‘the game can always be changed’.

Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry
This book aims to introduce its readers to a mathematical topic that was once widely taught, but that has vanished from textbooks and syllabuses over the past 60 years: spherical trigonometry. The book is intended as a teaching resource; indeed, it appears to have grown out of the author’s summer workshops for teenagers. The author clearly believes that the principles of spherical trigonometry deserve to be betterknown, and has set out to impress the reader not only with its great practicality, but also with its elegance.

How to Fold It: The Mathematics of Linkages, Origami and Polyhedra
This book is a triptych, unsurprisingly consisting of parts on linkages, origami and polyhedra.

How to Guard an Art Gallery and Other Discrete Mathematical Adventures
‘How to Guard an Art Gallery and Other Discrete Mathematical Adventures’ models solutions to a variety of problems  what is the largest number of pizza slices that we can make with n straight lines, how does a computer configure the best arrangement of pixels to represent a straight line, what is the minimum number of guards needed to guard an art gallery?

In the Dark on the Sunny Side: A Memoir of an OutofSight Mathematician
Many will know Professor Lawrence (Larry) Baggett for his work on harmonic analysis, wavelet theory and unitary representations of groups. He is a world leading mathematician and has been for many decades. However, there is something you might not know about Larry, he is blind. This book serves as Larry’s memoirs, describing everything from his musical exploits and mathematical pursuits, to his personal life. Many of these moments, feelings and activities have been experienced by each of us, and often he describes a professional environment that lots of us are familiar with. However, Larry talks about them from a completely different, and in many cases, unique position.

In Pursuit of the Traveling Salesman – Mathematics at the Limits of Computation
In Pursuit of the Traveling Salesman gives a very readable account of progress in tackling one of the most important problems in Applied Mathematics: the Travelling Salesman Problem. The problem of finding the shortest route through a number of cities is finite and therefore solvable in principle but becomes disproportionately complex as this number grows. Theoretical, practical and historical aspects of the problem are thoroughly explained.

Introduction to Differential Equations Using Sage
Sage is a free opensource mathematical software system which combines the power of many mathematical software packages into a common PYTHONbased interface. David Joyner, who has worked on the development of Sage, has coauthored with Marshall Hampton this book, the purpose of which is to teach the theory of the solution of differential equations with the aid of Sage as a valuable learning tool.

Introduction to Mathematical Physics: Methods and Concepts (Second Edition)
This book consists of eight chapters. Some chapters are very detailed and relatively simple to follow; other chapters are much more advanced and terse in their presentation. A description of the chapters follows. Chapter 1 – This chapter is a detailed introduction to vectors and vector fields, reaching vector differential operators in curvilinear coordinates. Chapter 2 – Is very similar to Chapter 1 in level and covers matrices, eigenvalues and into operators and matrix groups. Chapter 3 – This is the first of the more advanced chapters covering special relativity, spinors of various kinds and tensors. Chapter 4 – This chapter covers Fourier series, Fourier transforms, Green’s functions and generalised Fourier series, amongst other topics.

Linear and Nonlinear Functional Analysis with Applications
This book is a selfcontained volume containing the fundamentals of linear and nonlinear functional analysis. It consists of nine chapters. The first is a very brief description of real analysis and functions of several real variables, it consists of theorems, no proofs are given. Chapters 2–5 cover all of linear functional analysis, Chapter 6 applies this knowledge to linear PDE, and Chapters 7–9 cover nonlinear functional analysis.

Loving + Hating Mathematics: Challenging the Myths of Mathematical Life
Mathematicians are different from other people, lacking emotional complexity. Mathematics is a solitary pursuit. Mathematics is a young man’s game. Mathematics is an effective filter for higher education.

Math Bytes: Google Bombs, Chocolatecovered Pi, and Other Cool Bits in Computing
We know that mathematics (or, as the Americans would have it, math) underpins a large number of disciplines, but, perhaps, Computer Science is the one that depends upon it most. Hence, there ought to be a wealth of material for this book, which sits at the crossroads of these two subjects, as indicated by the punning title.

Mathematical Underpinnings of Analytics: Theory and Applications
Have you ever kept a tally of the running cost of the items in your trolley while shopping at the supermarket? Well, the simple arithmetic required for this task is as nothing compared to the complicated mathematical calculations performed on the contents of your trolley by the supermarket itself! Correlating pairs of items from hundreds of thousands of products across multiple customers using similarity matrices and dendrograms is but one example. The detailed analysis of this and other activities in customerfacing sectors such as supermarkets is one aspect of what Professor Peter Grindrod, of the Mathematical Institute at the University of Oxford, calls Analytics.

Mathematics in Victorian Britain
Mathematics in Victorian Britain is a collection of essays edited by Raymond Flood, Adrian Rice and Robin Wilson.

Mathematics of Life: Unlocking the Secrets of Existence
‘Biology will be the great mathematical frontier of the twentyfirst century.’ So says Ian Stewart, Emeritus Professor of Mathematics at Warwick University (and undoubtedly well known to all readers of this magazine). In the Mathematics of Life he suggests that, historically, there have been five revolutions that have changed the way scientists think about life and that a sixth – the mathematical solution of biological problems – is on its way.

Mathematics of Social Choice: Voting, Compensation, and Division
This book is an introduction to, as the title suggests, the mathematics of voting, compensation and division. Each section is selfcontained, consisting of short and snappy chapters without being overconcise, and accompanied by mercifully doable exercises which not only tie in well with the text, but are mostly provided with that rare luxury of worked solutions in the back.

Modelling and Reasoning with Bayesian Networks
One of the key themes underlying mathematics, and especially mathematical proof, is that of bringing together separate elements and combining them so that they tell a whole new story. Whether it’s in the common theme that unites two apparently disparate branches of work, or in the combination of approaches that together create a complicated proof, amalgamating pieces of evidence (or knowledge) in a reasoned fashion is a critical aspect of the continued progress of mathematics.

Multiagent Systems: Algorithmic, GameTheoretic and Logical Foundations
One of the characteristics of society is interaction, with many different people coming together either to achieve a common goal, for example electing a government, or to compete with each other, for example in a business context.

Naked Statistics (Stripping the Dread from the Data)
Charles Wheelan is also the author of the bestselling Naked Economics – so I suspected I was definitely in for a good read. I have taught Alevel Statistics, and more recently Advanced Placement Statistics, so I am not really new to the subject. I confess, reading the earlier chapters, I was a little hungry for more mathematics and more detail. However, I gradually learned to suspend my own prior knowledge, and read this from the perspective of a reader approaching the subject (in any depth) for the first time  which is of course, the likely background of the intended audience.

Networks, Crowds, and Markets  Reasoning about a Highly Connected World
The growth of connectedness in modern society in recent years seems to have escalated at a spectacular rate. The rapid technological growth of the internet and global communications, as well as the spread of epidemics and financial crises, affects the wholeworld with surprising speed and intensity.

Networks: A Very Short Introduction
This short introduction to Networks is a good way to introduce the ideas of Networks and how widespread they are in our lives. The text is part of a series of ‘a very short introduction’ books, however it is independent in terms of the topic  Networks. It is written in a formal but relaxed tone, talking through each point in present tense allowing the reader to follow along as if in a conversation.

NIST Handbook of Mathematical Functions
How do you improve on perfection?The Handbook of Mathematical Functions edited by Abramowitz and Stegun (A&S) and produced for NIST (then known as the National Bureau of Standards) in 1964 has been the most highly cited of NIST’s publications. Indeed it was one of the standard references at my workplace and I personally found it invaluable.

NumberCrunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction
In 1841 Augustus De Morgan challenged his students to calculate the real root of a particular cubic equation to more than thirty decimal places – which they did! Richard Feynman once said, “For my money Fermat’s theorem is true” after having produced a probabilistic ‘proof’ of the theorem!

Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present
There are inherent problems and paradoxes when the choices of a number of individuals need to be combined to make one collective decision. With the recent referendum on voting reform such issues are topical today, but they actually go back to antiquity. Szpiro writes from a historical and mathematical perspective. He also looks at the principal characters and gives a short biography of each, putting their ideas into a social context.

Numbers: A Very Short Introduction
This beautiful 230 page paperback is one of a series of 260 Very Short Introductions to a variety of topics, from Aristocracy, through Quantum Theory to Architecture.

Numerical Notation: A Comparative History
Comprehensive, encyclopaedic and scholarly are the first three words that spring to mind when reading Chrisomalis’ mammoth work on numerical notation. Not for the fainthearted, this thoroughly researched academic tome, with an extensive bibliography stretching to some 30 pages, covers over 100 different numerical notation systems spanning over 5,000 years.

Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek and Arabic Roots
Readers expecting a dictionary in the purely descriptive style of the OED are in for a shock with this book, which instead takes the prescriptive approach of Samuel Johnson, with frequent condemnation of supposedly ‘bad’ words. Indeed, the language employed suggests that this was a very deliberate choice; many words are, for example, dismissed as ‘low’, a favourite epithet of Johnson, meaning simply ‘vulgar’. Moreover, this dictionary frequently exhorts us to adhere to the usages of the nebulouslydefined ‘best authors’, to whom Johnson also appealed.

Partial Differential Equations in General Relativity
This book is written with two main audiences in mind: physicists with a good knowledge of General Relativity (GR) who wish to gain an understanding of how Partial Differential Equation (PDE theory can be applied to GR; and mathematicians with a good knowledge of PDE theory who want to understand how it is applied to GR.

Principles of Multiscale Modeling
Children delight in responding to an explanation with ‘Why?’, and it takes few iterations to get from candles to quarks. Their natural curiosity vividly reveals a hierarchy of models and explanations. Multiscale modelling is the formal version of this child's game, in which through analysis and computation the levels in this hierarchy of physical models are connected, and information is rigorously passed between them. Principles of Multiscale Modeling is a lucid, wellwritten, and broad introduction to the field aimed at mathematical scientists and engineers. It is attractively illustrated in colour and contains many useful and relevant references for further study.

Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks
Advances in genetic sequencing in the late twentieth century have been accompanied by an increased emphasis on the factors influencing how and when genes are expressed. For example, a skin cell and a liver cell contain the same DNA but regulate it differently, leading to divergence in gene expression and thus distinct forms and functions for the cells. This timely book discusses the use of probabilistic Boolean networks as a systemslevel model of such interactions.

Professor Stewart’s Hoard of Mathematical Treasures
Professor Stewart’s Hoard of Mathematical Treasures is the latest collection of puzzles, jokes and mathematical snippets by Ian Stewart, FRS, CMath FIMA, Professor of Digital Media at Warwick University. This collection is a sequel to his very successful Cabinet of Mathematical Curiosities and there is no drop in quality from the previous work. Over many collections (some from his column in Scientific American) Professor Stewart has developed an engaging and irreverent style featuring lots of puns and a series of recurring themes.

Quantum Field Theory
This book provides a complete introduction to quantum field theory and the elementary particles. It is set at the level of graduate students who have covered special relativity and quantum mechanics. It covers, amongst other areas, the unification of forces, super symmetry, the renormalization group, quark and gluon scattering, and nonpeturbative physics (magnetic monopoles and instantons).

Riot at the Calc Exam and Other Mathematically Bent Stories
The intention of the book is given by the blurb on the back which runs as follows:
“What’s so funny about math? Lots! Especially if you’re mathematically bent. In the world of Colin Adams, differential equations bring on tears of laughter. Hollywood producers hire algebraic geometers to punch up a script. In this world, math and humor are synonymous. ‘Riot at the Calc Exam’ is a proof of this fact." 
Routes of Learning: Highways, Pathways and Byways in the History of Mathematics
In this volume Ivor GrattanGuinness collects together a range of papers spanning forty years of his distinguished career as a historian of mathematics. The central theme linking chapters in the first two sections (the ‘Highways’ and ‘Pathways’ of the subtitle) is the place of history in mathematical education. The author criticises accounts which conflate ‘history’ (the development of a particular piece of work) with ‘heritage’ (the impact of that work on future mathematics).

Secret Days: Codebreaking in Bletchley Park
Asa Briggs, the eminent historian, has written this book about his memories of his time as a codebreaker at Bletchley Park, or BP as it was known to those who worked there, during the years 1943 to 1945.

Security and Game Theory: Algorithms, Deployed Systems, Lessons Learned
The need for security of transport around the world has never been so high, because of the increase in the number of threats from terrorism and other sources. Security and Game Theory provides an insight into how game theory can be used to allocate the limited resources of the governing bodies of different companies, which will decrease the likelihood of such events happening.

Soliton Equations and Their AlgebroGeometric Solutions (Volume II: (1 + 1)  Dimensional Discrete Models)
One definition of the soliton is “a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed”. More informally it is a “selfreinforcing solitary wave that maintains its shape while it travels at constant speed”. The name is derived from ‘solitary wave solutions’. The phenomenon was first described by John Scott Russell who observed such a wave on a Scottish canal in 1834.

Spectral Theory and its Applications
The aim of the book is to introduce various aspects of spectral analysis and to apply the theory to examples from different branches of physics, including Schrödinger operators and statistical physics. The book contains sixteen chapters, covering various aspects of the theory and an introduction. Every chapter, except the last, describes some aspect of spectral theory and includes a number of problems at the end to test the reader's understanding. The final chapter contains many problems intended to challenge the reader's understanding of the previous material.

Structure and Randomness: Pages from Year One of a Mathematical Blog
Terence Tao is a distinguished mathematician, perhaps best known for his work in combinatorics and number theory, linked especially to the theory of arithmetic progressions of prime numbers. In 2007, he turned his homepage into a weblog, and this book collects some of his online writings which first appeared there. In the book’s collection of some of these blogs, it sketches out unusual proofs for classical theorems, the texts of three of his invited lectures, a selection of discussions of open problems, and a few number curiosities.

Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature (Second Edition)
Symmetry and chaos seem unlikely bedfellows; yet Field, from the University of Houston, and Golubitsky, from Ohio State University, have produced a book full of beautiful pictures by combining the two. To quote from the Introduction: “Our pictures are created by merging symmetry and chaos. At first sight this seems paradoxical: a merging of order and disorder or yin and yang”.

The Best Writing on Mathematics 2010
After reading this wonderfully crafted book, I put pen to paper somewhat apprehensively to write a review of the ‘best writing’. The book is divided into six parts; Mathematics Alive, Mathematics and the Practice of Mathematics, Mathematics and its Applications, Mathematics Education, History and Philosophy of Mathematics and Mathematics in the Media, offering a selection of articles by various authors publish during 2009. With such a great diversity of writing I found myself loving some of the articles, reading others with interest and skipping swiftly over a few, purely due to personal interest and nothing to do with the quality of the article itself. That’s the beauty of the book, you can pick out what appeals to you.

The Blind Spot: Science and the Crisis of Uncertainty
A fair number of authors have the ability to explain complicated ideas in simple terms, but William Byers also has the rarer gift of taking the almost banally familiar and revealing its hidden depths and complications. Mathematicians manipulate fractions on a daily basis and can quickly forget how bizarre the idea of writing one number above another may appear to a learner encountering them for the first time.

The History of Mathematics: A Very Short Introduction
In The History of Mathematics: A Very Short Introduction, Jacqueline Stedall considers a variety of themes in seven chapters. Mathematics is clearly portrayed as a human endeavour evolving from dead ends, failure and the capacity of a mathematician to use, refine and finesse the work of another. The book is a manual on how to approach a historical study, detailing the constraints and possible motivations. Stedall rejects the linearity of progress stepping stone approach right from the start. She takes a wider view that also includes the most written about period of post15th century Western European mathematics, considering the input from earlier times across the globe.

The Num8er My5teries  A Mathematical Odyssey through Everyday Life
Marcus du Sautoy’s latest book The Num8er My5teries is written for the general reader. The book bursts with creativity, analysis and explanation in a clear nonspecialist style. Complex issues in the subject become accessible as he exposes readers to the big ideas in mathematics.

What's Luck Got to Do with It? The History, Mathematics, and Psychology of the Gambler's Illusion
Joseph Mazur’s book takes us on a fascinating journey looking at the misconceptions involved in gambling. The reader learns the history of gambling, the way mathematicians analyse luck and the psychology that affects gamblers.

Zombies and Calculus
Generally speaking, Mathematics Today doesn’t review fiction. There are many, more mainstream publications that cater for such works and, whilst they are appropriate for specialist mathematics texts, our review and publication cycles are not well matched to the faster timelines associated with fiction. So, it was with some trepidation that I approached this book. The trepidation was increased by the subject matter – I’m not scared of calculus, but zombies, well, they’re a different matter.