## Recent book reviews

Five recent book reviews

## AN INTRODUCTION TO QUADRILATERAL GEOMETRY

Author(s):
Ovidiu T. Pop, Nicusor Minculete, and Mihaly Bencze
Publisher:
Editura Didactica Si Pedagogica, R. A.
Year:
2013
ISBN:
978-973-30-3324-0
Short description:

The book consists of seven chapters written by three eminent mathematicians in the subject of Euclidean Geometry. It is especially useful for students and teachers who are preparing for National and International Mathematical Olympiads.

MSC main category:
51 Geometry
Other MSC categories:
14-xx
Reviewer:
Themistocles M. Rassias
Affiliation:
National Technical University of Athens

## X and the City: Modeling Aspects of Urban Life

Author(s):
Publisher:
Princeton University Press
Year:
2012
ISBN:
9780691154640
Price (tentative):
$30 Short description: This book is a leisure reading of different aspects of urban life which are modeled mathematically. It tries to make accessible to an average reader a collection of arguments for analysing daily questions with elementary mathematical tools. The book may be recommended to undergraduate students who are yet to be convinced that mathematics appear in more places than a non-math inclined person may expect. URL for publisher, author, or book: http://press.princeton.edu/titles/9663.html MSC main category: 00 General MSC category: 00A08 Other MSC categories: 00A09 Review: This book is a leisure reading of different aspects of urban life which can be modeled mathematically. It tries to make accessible to an average reader a collection of arguments for analysing daily questions with elementary mathematical tools. Examples of questions appearing on the book are: how many restaurants are in a city of a given size? should we walk or run under the rain? how much a car in a dense traffic slows down the other cars? how fast cities grow depending on the available resources? how many people have ever lived in London through time? among many others. It is very interesting how the author makes inferences without statistical real data, just gessing. And the good thing is that at least the order of magnitude of the results obtained are very likely to be correct. These mathematical problems are solved with elementary computations, accessible to undergraduate students, and even many of them to high school mathematics. The book focuses in estimations, rounding off numbers very often. This point of view is very interesting for a young reader, and it is the main strength of the book. In general terms, the problems analysed by the book are treated in a shallow way, with very simple answers, and not entering into very sophisticated mathematical models. Some of the later chapters (e.g. those on analysis of traffic flows) are of higher mathematical level and require some basic knowledge of differential equations. The book may be recommended to undergraduate students who are yet to be convinced that mathematics appear in more places than a non-math inclined person may expect. However, anyone acquainted with higher mathematical background may find it a bit boring. Reviewer: Vicente Muñoz Affiliation: UCM ## Geometry of Mobius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL_2(R) Author(s): Vladimir V. Kisil Publisher: Imperial College Press Year: 2012 ISBN: 978-1848168589 Price (tentative):$78
Short description:

This book is an exposition of geometries associated with Möbius transformations of the plane, based on properties of the group $SL_2({\mathbb R})$. The presentation is self-contained, starting from elementary facts in group theory, and unveiling surprising new results about the geometry of circles, parabolas and hyperbolas. The treatment of elliptic, parabolic and hyperbolic Möbius transformations is provided in a uniform way. This is possible due to an appropriate usage of complex, dual and double numbers, which represent all non-isomorphic commutative associative two-dimensional algebras over the real numbers.

MSC main category:
51 Geometry
Review:

This book is a deep analysis of Möbius transformations from an unusual point of view. The approach is based on the Erlangen programme of Felix Klein, who defined geometry as a study of invariants under a transitive group action. The book focuses on the group $SL_2({\mathbb R})$ and its action by Möbius transformations: $x \mapsto \frac{ax+b}{cx+d}$. This acts on the complex plane, but it also acts on the plane of dual numbers and on the plane of double numbers. Actually, these are the three possible non-isomorphic commutative associative two-dimensional algebras over the real numbers, which are ${\mathbb R}[\sigma]$, with $\sigma^2=-1,0,+1$. The corresponding actions are called elliptic, parabolic and hyperbolic Möbius transformations. The three geometries correspond to the homogeneous spaces with group $SL_2({\mathbb R})$ for the three possible one-dimensional subgroups.

The book studies in depth the geometry associated to the "cycles" in these spaces (circles in the first case, parabolas with horizontal directrix in the second, and equilateral hyperbolas in the third). There is a three dimensional real projective space parametrising such cycles, and a corresponding action of $SL_2({\mathbb R})$ on it. Moreover, there is a naturally defined (indefinite) quadratic form on the space of cycles which serves to recover the initial geometric space, and the usual geometric transformations on it. Then the books moves on to analyse many geometric properties of cycles. This is completed with several aside considerations: the relationship with the physics of Minkowski and Galilean space-time, the more classical point of view of (semi)riemannian geometry, questions on conformal geometry, and more far away subjects like optics or tropical algebra.

The book is accompanied by a DVD with a program which runs under linux (also freely available in internet) which serves the reader to perform computations that appear along the book. This is used often in the book to complete some proofs, which are done by brute force calculation. However, the use of the program requires some knowledge of programming, as the interface is not very user-friendly. This is useful to the reader to complete the arguments and get convinced that the results are true. However, I would have preferred at some points to read a concise and theoretic proof, much more appealing than checking a calculation.

The book is addressed to undergraduate and graduate students in the areas of geometry and algebra. The presentation is basically self-contained. There are many exercises scattered along the book for the interested reader. On the one hand, the point of view is not classical, so a student trying to learn basic properties of Möbius transformations and the relation with complex/Kähler geometry may not get totally satisfied. On the other hand, I think that the author has been successful in transmitting the idea (as he confess in the epilogue that this was his intention) that the three geometries: elliptic, parabolic and hyperbolic deserve to be treated on an equal footing, and that all of them are very rich.

Reviewer:
Vicente Muñoz
Affiliation:
UCM

## The Tower of Hanoi - Myths and Maths

Author(s):
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, Ciril Petr
Publisher:
Birkhäuser Basel
Year:
2013
ISBN:
978-3-0348-0236-9 (hbk)
Price (tentative):
47.65 € (hbk)
Short description:

The mathematics of the classical tower of Hanoi puzzle with 3 pegs and n disks is by now well known and optimal solutions have been described in the form of algorithms that need a minimal number of moves. Besides some historical notes and an introduction to the Chinese ring puzzle, also called Baguenaudier puzzle, the graphs, the number sequences, and in general, the mathematics and algorithms are explained for the tower of Hanoi puzzle. As far as they are currently known, also the methods for its possible generalizations and variants are given. The latter leaves many challenging open problems. Exercises are provide after each chapter.

URL for publisher, author, or book:
www.springer.com/978-3-0348-0236-9
MSC main category:
05 Combinatorics
MSC category:
05C90
Other MSC categories:
00A08, 11B37, 20B25, 68T20
Review:

The Tower of Hanoi (TH) is a classic puzzle that was invented by Éduard Lucas (1842-1891) using the alias N. Claus (de Siam), an anagram of Lucas d'Amiens. Given three pegs, two of them empty, and a pile of disks of decreasing size is stacked on the third one. The problem is to move the stack to another peg, moving one disk at a time from one peg to another, but never putting a larger disk on top of a smaller one (the divine rule). The Chinese ring puzzle (CR) and the topologically equivalent Cardano rings are related. In the latter two the problem is to get all the rings off the upper bar or to remove the string from the linked pillar structure. (See image attached.) These puzzles are older and CR, which Lucas called Baguenaudier puzzle, has probably inspired him to formulate the TH puzzle.

So this is a book about recreational mathematics, but do not be mistaken, the origin of the questions may be recreational, the problems discussed are undeniable mathematics. There is a lot of graph theory, automata, group actions, complexity analysis and algorithmic aspects around with definitions, theorems and proofs.
Chapter 0 gives an introduction to the problem and the history of the TH puzzle and other related puzzles, but it also introduces definitions and properties of graphs that are used in solving these problems.
Then chapter 1 discusses the CR puzzle as a prototype to set up a model for approaching this kind of problems. The classical TH problem is then much more thoroughly discussed in chapter 2. Algorithms and proofs are given for optimal (w.r.t. the number of steps) solutions. The simplest problem is to move a perfect pile from one peg to another one following the divine rule. All possible intermediate states are called regular. It is more difficult to transform an arbitrary regular state into another one. To solve this, a good understanding and analysis of the associated graph becomes more important.
Lucas considered also another problem where the disks are placed randomly on the three pegs, and the goal is still to transform this irregular state to a perfect or regular state still following the divine rule. That's a short chapter 3.
The next chapter introduces another type of graphs to analyse the TH problem. They are called Sierpinski graphs because of their relation with the Sierpinski triangle.
Generalizing TH from 3 pegs to 4 or more pegs, still looking for an optimal solution (known as Reve's problem), goes way beyond a trivial step. So there is much intuition, conjectures and open problems to be found in chapter 5.
Some further variations are briefly discussed in the next 3 chapters. like more towers (of different color) and more pegs, and/or relaxing the rules of the game (chapter 6) or Towers of London: colored balls on a number of pegs to be brought form one state to another in a minimum number of steps (chapter 7) or with oriented disk moves (chapter 8).
Chapter 9 is just a list of open problems that were encountered in the previous chapters.

Chapters 0-8 are followed by a list of exercises (hints and solutions are provided in an appendix). In principle the mathematics needed are all introduced in the text, but the introduction to the necessary concepts is kept to a minimum and some basis is tacitly assumed known. For example, the notion of group is used without much ado. Hence a certain mathematical background and some familiarity with graphs is advisable and even required if you want to solve the exercises. Besides the usual subject and name indexes, the glossary of terms with one-liner definitions is quite useful to recall a concept of a previous chapter if you are not so familiar with graphs.
The authors use a very pleasant and amusing style, but they keep the discussion to the point, and leave much more to be explored using many pointers to the extensive reference list. The many illustrations make the technicalities more easy to digest.

Thus if you love puzzles, and more in particular the mathematics behind it, this is a book for you. That holds for mathematicians but for computer scientists as well. Also if you are looking for a lifelasting occupation, then you may find here a list of open problems that will keep you busy for a while.

Reviewer:
A. Bultheel
Affiliation:
KU Leuven

## Probability and Stochastics

Author(s):
Erhan Çinlar
Publisher:
Springer. Graduate Texts in Mathematics 261
Year:
2011
ISBN:
978-0-387-87858-4;e-ISBN: 978-0-387-87859-1; DOI 10.1007/978-0-387-87859-1; ISSN 0072-5285
Price (tentative):
Between 51.16€ (ebook)-62.35€
Short description:

The book is an introduction to the modern theory of probability and stochastic processes. It based on the lecture-notes for two-semester, fairly popular, which Prof. Çinlar, a well-known professor and researcher, at Princeton University,ecinlar$princeton.edu, has offered for many years. The course attracks graduate students in Business-Banking,computer sciences, engineering, eonomics, physics, finance, and mathematics and others related disciplines, and may be recommended to undergraduate students who are yet convinced that the mathematics appear in more places or in differents fields of knowledge, and also suitable for individual study. The style and coverage is directed toward the theory of stochastic processes and its applications. In many instances, the gist of the problems is introduced in every-day language and then all written in a precise matematical form. The first four chapters are in probability theory: measure and integration, probability theory, convergence and conditioning. There follows chapters on martingales and stochastics, Poisson random measures, Brownian motion and Markov Processes. The book is full of insights and observations that only a lifetime researcher in Probability and Stochastic Processes can have. URL for publisher, author, or book: http://w.w.w.springer.com/series/136;ecinlar$princeton.edu.
MSC main category:
60 Probability theory and stochastic processes
MSC category:
60-01;60A10;60B10;60C05;60F05;60G05;60J05;60H05
Other MSC categories:
60J65;60G45;60G46;60G48;60G51;60J75;60.60
Review:

The book is an introduction to the modern theory of probability and stochastic processes. It based on the lecture-notes for two-semester, fairly popular, which Prof. Çinlar, a well-known professor and researcher, at Princeton University,ecinlar@princeton.edu, has offered for many years.
The first four chapters are in probability theory: measure and integration, probability theory, convergence and conditioning.
The first chapter is a review of measure and integration in the context of the modern literatura on probability and stochastic processes. The second introduces probability spaces as espeila measure spaces.The chapter three is on convergence, routinely classical, and the chapter four in on conditional expectations included Radon-Nikodyn derivatives.
There follows chapters on martingales and stochastics, Poisson random measures, Brownian motion and Markov Processes.
Martingales are introduced in chapter V, the treatment of continuous martingales contain an improvement, achieved through the introductiion of a Doob martingale, a stopped martingale that is uniformly integrable. Two great theorems are considered:martingale characterization of Brownian motion due to Lévy and the characterization of Poisson process due to Watanabe.
Poisson random measures are in Chapter VI, the treatment is from the point of vue of their uses,specially,of Lévy processes. This chapter pays some attention to processes with jumps. The chapter VIII on Brownian motion is mostly on the standard material. Finally, the Chapter IX, on Markov processes, Itô diffusions and jump-diffusions are introduced via stochastic integral equations, as an integral path in a field of Lévy Proceses.
In short, the book is higher recommended. It provides new simple proofs of important results on Probability Theory and Stochastics Processes. In my opinion, it is a stimulating textbook will be for the teaching and research of the materia. A well written text with excellent tools for many instances, in every day language, and then all written precise in mathematical form.

Reviewer:
Francisco José Cano Sevilla
Affiliation: