## Recent book reviews

Five recent book reviews

## Famous Puzzles of Great Mathematicians

Author(s):
Miodrag S. Petkoviç
Publisher:
American Mathematical Society
Year:
2009
ISBN:
ISBN-10: 0-8218-4814-2; ISBN-13: 978-0-8218-4814-2
Price (tentative):
US36$(List Price); US28.80$ (Member Price)
Short description:

This book is an interesting work on recreational problems on mathematics in the history. These problems have survived, not because they were fostered, by the textbook writers, but because of their inherent appeal to our love of mystery. This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. About 65 intriguing problems, marked by * in the book, are given as exercises, to the readers. The selected problems do not require advanced mathematics, making this excellent book accessible to a variety of readers.
The history of mathematics is replete with examples of puzzles, games and entertaining problems that have fostered the development of new and emergent disciplines and sparked further research. The book is intended principally to amuse and entertain, incidentally to introduce the general reader to other intriguing mathematical topics and ideas. With this book, many stories and famous puzzles can be very useful to prepare teaching or lecture notes, to inspire and amuse students, and to instill affection for mathematics. In my opinion, it is a stimulating and excellent text will be for amuse and entertaining, and the teaching in recreational mathematics.

URL for publisher, author, or book:
http://w.w.w.ams.org; http://w.w.w.ams.org/bookpages/mbk-63; order code: MBK/63.
MSC main category:
00 General
MSC category:
2000 MSC: 00A08, 97A20, 01A05, 01A70, 05A05, 05C45, 05C90
Other MSC categories:
2000 MSC: 11D04, 11D09, 51E10, 51M16, 52C15, 52C22, 97D4O
Review:

This book is an interesting work on recreational problems on mathematics in the history. This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. About 65 intriguing problems, marked by *, are given as exercises, to the readers. The selected problems do not require advanced mathematics, making this excellent book accessible to a variety of readers.
The book is intended principally to amuse and entertain, incidentally to introduce the general reader to other intriguing mathematical topics and ideas. Important relations and connections exist between those problems originally meant to amuse and entertain and mathematical concepts critical to combinatorial and chess, geometrical, and arithmetical puzzles, geometry, graph theory, optimization theory, probability, number theory, and related areas.
The book contains eleven chapters and four appendices.
The first six chapters are on: recreational mathematics (a brief and concise history of mathematics); arithmetics; number theory; geometry; tiling and packing and physics.

The chapter one is on Recreational Mathematics contains, before taking up the noteworthy mathematical thinkers and their memorable problems, a brief overview of the history of mathematical recreations. Perhaps the oldest known example is the magic square. Known as lo-shu to Chinese mathematicians around 2200 B.C., the magic square was supposedly constructed during the reign of the Emperor Yii. Chinese myth holds that Emperor Yii saw a tortoise of divine creation swimming in the Yellow River with the lo-shu, or magic square figure, adorning its shell. The Rhind (or Ahmes) papyrus dating to around 1650 B.C., suggests that the early Egyptians based their mathematics problems in puzzle form. Perhaps their main purpose was to provide intellectual pleasure. The ancient Greeks also delighted in the creation of problems strictly for amusement. The cattle problem is one of the most famous problems in number theory, whose complete solution was not found until 1965 by a digital computer. The Dido´s problem, cited by Virgil, and the elegant solution, established by Jacob Steiner, regarded as first problem in a new mathematical discipline, established 17 centuries later, as calculus of variations. Others interesting problems are included in this book, as Josephus problem; Alcuin of York´s problems and variants; Fibonacci´s amusing problems; IbnKallikan´s problem about the number of wheat grains on a standard 8x8 chessboard; and many instances interesting.
In chapter two, named Arithmetics, are related many instances of famous puzzles: Diophantus´ age; Mahavira: number of arrows; Fibonacci´s: square numbers problem, money in a pile, sequence, how many rabbits?; triangle with integral sides (Bachet); sides of two cubes (Viète), and others puzzles.
Chapter three on Number Theory deal with the following famous: cattle problem; dividing the square; wine problem; amicable numbers, Qorra formula; how many soldiers?; horses and bulls; the sailors, the coconuts and the monkey; stamp combinations, with a generalized problem of Frobenius and Sylvester.
In chapter four, on Geometry, are related some instances: Arbelos problem of Arquimedes, archimedean circles, perpendicular distance and two touching circles; minimal distance of Heron, a fly and a drop of honey, peninsula problem; dissection of three squares; dissection of four triangles; the minimal sum of distances in a triangle; volumes of cylinders and spheres of Kepler; Dido´s problem; the shortest bisecting arc of area de Polya, and other interesting problems.
Chapter five on Tiling and Packing deal interesting and amusing problems in the history of mathematics as: mosaics; non-periodic tiling; maximum area by pentaminoes; kissing spheres; the densest sphere packing, and the cube-packing puzzles.
The chapter six is related to famous problems on Physics as the gold crown of King Hiero; the length of traveled trip; meeting of ships; a girl and the bird, and the lion and the man.

There follows chapters on combinatorics; probability; graphs; chess and miscellany which contains problems from Alcuin de York, Abu´lWafa, Fibonacci, Bachet, Huygens, Newton and Euler.

The chapter seven on Combinatorics, deal the Josephus problem; rings puzzle; the problem of the misaddressed letters; eulerian squares, and the famous Kirkman´s schoolgirls problem. Others interesting problems as counting problem, the tower of Hanoi, the tree planting problem, etc., are included in this chapter.
In the chapter 8 on Probability are considered, the famous problem of the points, gambling game with dice, gambler´s ruin, Petersburg paradox, the probability problem with the misaddressed letters, and the match problem are all treated of elegant form.
The chapter nine deal on Graphs. Contains the famous problem of Königsberg´ bridges, Hamilton´game on a dodecahedron, some problems of Alcuin of York, Erdös, Poinsot, Poisson, Listing, and others problems of interest.
In the chapter ten on chess, many instances and problems are considered: classical knight, queen, rooks and the longest uncrossed knight´s tour.
The chapter eleven, titled Miscellany, contains problems from Alcuin de York, Abu´lWafa, Fibonacci, Bachet, Huygens, Newton and Euler.
Finally, the four appendices, for to help readers, on refer: method of continued fractions for solving Pell´s equation; geometrical inversion; some basic facts from graph theory; linear differences equations with constant coefficients.
The author includes bibliographical references and index, and sometimes amusing anecdotal material, with the objective that to underscore the informal and recreational character of the book.
The book is also high recommended also for individual study. In my opinion, it is a stimulating and excellent text will be for amuse and entertaining, and the teaching in recreational mathematics.

Reviewer:
Francisco José Cano Sevilla
Affiliation:

## Virtual knots. The State of the Art

Author(s):
Vassily Olegovich Manturov and Denis Petrovich Ilyutko.
Publisher:
New Jersey. World Scientific Publishing Co. Pte. Ltd. SERIES ON KNOTS AND EVERYTHING: Vol. 51.
Year:
2012
ISBN:
ISBN-13: 978-981-4401-12-8; ISBN 10:9814401129; ISSN 0219-9769
Price (tentative):
168$(Hardcover: alk. paper), 91.05$ (kindle Edit)
Short description:

This book is the first systematic and full-length book on the theory of virtual knots and links, devoted to an intriguing and comprehensive study of (virtual and classical) knots as integral part. The book is self-contained. The mathematical material is sufficiently closed, and contains up-to-date exposition of the key aspect of virtual (and classical) knot theory. The book is quite accessible for undergraduate students of low courses, thus it can be used as a basic course book on virtual and classical knot theory.
The aim of the present book is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. A large part of the present title is devoted to rapidly developing areas of modern knot theory: such as virtual knot theory and Legendrian knot theory.
The book contains nine chapters, bibliography and an index, in 520 pp. It goes from the basics to the frontiers of research. The book is volume 51 of Series on Knots and Everything, polarized around the theory of Knot Theory. The questions treated reach out beyond theory itself into physics, mathematics, logic, linguistics, philosophy, biology and practical experienced. The book is dedicated to the memory of Oleg Vassilievich Manturov (1936-2011), father of Vassily Olegovich Manturov, first-named author of the book.

Definitely the book is high recommended to undergraduate, graduate, professionals and amateur mathematicians, because it goes from the basics to the frontiers of research. Finally, as L.H. Kauffman says: “this book self-contained is motivated, to delve, into the adventure proposed, by this intriguing and remarkable book”.

URL for publisher, author, or book:
www.worldscientific.com
MSC main category:
57 Manifolds and cell complexes
MSC category:
57M27, 57M25, 57Q45
Other MSC categories:
57M15, 08Ag2, 08C05
Review:

The study of knots and their properties is known as knot theory. Classical theory counts more than two hundred and fifty years. Knots appear in Chinese knotting, Tibetan Buddhism, intricate Celtic knot-work, in the 1200 year old Book of Kells, and so on. As a mathematical theory appeared in 1771 by French mathematician Vandermonde, when discussing the properties of knots related to the geometry of position. Mathematical studies of knots began in the 19th century with Gauss, who defined the linking integral. This theory was given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations, P.G. Tait then cataloged possible loops with different knots by trial and error, the first knot tables with up to ten crossings, known as the Tait conjectures, this record motivated that knot theory became part of the emerging subject of topology. Over six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century. Much progress has been made in the intervening years, for example, Alexander, Dehn, Klein, Reidemeister, and other outstanding mathematicians.

A knot is defined as a closed non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., unknot). To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand. A knot can be generalized to a link which is simply a knotted collection of one o more closed strands. A virtual knot represents a natural combinatorial generalization of a classical knot, simply introduced a new type of a crossing and extended new moves to the list of the Reidemester moves, the new crossing (virtual) should be treated as a diagrammatic picture of two part of a knot (a link) on the plane which are far from each other, and the intersection of these parts is artifact of such a drawing. In short, a virtual diagram (or a diagram of a virtual link) is the image of an inmersion of a framed 4-valent graph in R2 with a finite number of intersections of edges. Moreover, each intersection is a transverse double point which named a virtual crossing and mark by a small circle, and each vertex of the graph is endowed, with the classical crossing structure. The theory of knots in three-dimensional Euclidean space or in the three-sphere (the classical theory) is an integral part of a much larger theory, knots in 3-manifolds.

This remarkable book is the first systematic and full-length book on the theory of virtual knots and links, devoted to an intriguing and comprehensive study of (virtual and classical) knots as integral part. The book is self-contained. The mathematical material is sufficiently closed, and contains up-to-date exposition of the key aspect of virtual (and classical) knot theory. The book is quite accessible for undergraduate students of low courses, thus it can be used as a basic course book on virtual and classical knot theory. This book can also be useful for professionals and amateur mathematicians because it contains the newest and the most significant scientific developments in knot theory. The book was written using knots.tex fonts containing special symbols from knot Theory.

The aim of the present book is to describe the main concepts of modern knot theory together with full proofs that would be both accessible to beginners and useful for professionals. A large part of the present title is devoted to rapidly developing areas of modern knot theory: such as virtual knot theory and Legendrian knot theory.

Over the last decades, knot theory was enriched by numerous methods and subtle invariants, which today constitute a powerful tool in knot theory and low-dimensional topology. A breakthrough in knot theory is due to discoveries of Conway, Jones and Vassiliev (Conway and Jones polynomials, Vassiliev´s finite type invariants). For their impact to knot theory which related knot theory to various branches of mathematics and physics, Jones, Witten, Drinfeld (1990) y Kontsevich(1998) were awarded the highest honor of mathematics, The Fields medals.

Virtual knots were discovered by Louis H. Kauffman, in 1996, and independently by Goussarov, Polyak and Viro in 2000, about finite type invariants of classical and virtual knots. The first paper on virtual knot theory appeared in 1999, by Louis H. Kauffman in European Journal of Combinatory. The virtual knot theory helped one to understand better some aspects of the classical knot theory. By means of virtual knot theory, the problem of existence of combinatorial formulae for finite type invariants for classical knots was solved.

A common point of view allows us to treat classical and virtual knots uniformly.

A classical knot (link) can be given by a planar diagram. There are classical crossings and curves connecting crossings to each other. In the classical knot is possible that the curves connecting crossings can be chosen to be non-intersecting and in some cases it is impossible to situate these curves without additional intersections. This new intersections are marked as virtual (encircled) and we get a virtual diagram. Virtual crossings appear every time, when a 4-valent graph defined by classical crossings and ways of connection, is not planar, which happens quite often.

Thus, virtual knots are related to classical knots approximately in the same way as graphs are related to planar graphs. Herewith, the equivalence (isotopy) of classical knot diagrams is defined by means of formal combinatorial transformations (Reidemeister moves), which are applied to crossing lying close to each other.

From the topological point of view, the virtual knots are knots in thickened surfaces (products of sphere with handles with an interval) considered up to isotopy and stabilizations.

The book contains nine chapters, bibliography and an index, in 520 pp. It goes from the basics to the frontiers of research. The book is volume 51 of Series on Knots and Everything, polarized around the theory of Knot Theory. The questions treated reach out beyond theory itself into physics, mathematics, logic, linguistics, philosophy, biology and practical experienced. The book is dedicated to the memory of Oleg Vassilievich Manturov (1936-2011), father of Vassily Olegovich Manturov, first-named author of the book.

The first chapter, untitled basic, definitions and notions, devoted to the diagrammatic and combinatorial definitions and to a discovery of the self-linking number of a virtual knot. This chapter is a compendium, remarkable and fascinating encyclopedic treatment of basics.

In Chapter 2, is dedicated to virtual knots and the three dimensional topology. The authors present a discussion of Kuperberger´s Theorem, and the surface genus of virtual knots. Kuperberger proved that if a virtual knot (link) is represented in its minimal genus surface, then this embedding type is unique. The result is interesting because is fruitful for getting deeper invariants, a stronger version of the Jones polynomial for virtual knots and links. This chapter also proves that virtual knots are algorithmically recognizable by generalizing the technique of Haken and Hemion for classical knots. The recognition problem for classical knots was one of the central problems in low-dimensional topology. Its first solution is related to Haken´s normal surface theory, and the final steps belong to Matveev. The result of algorithmic recognizability of a certain object in low-dimensional topology is important, because, in low-dimensional topology, the algorithmic non-realizability takes place for many objects. When virtual knot theory appeared, the problem of algorithmic realizability of virtual knots arose. This problem is resolved positively, in this chapter. This result relies not only on Haken´s theory, but also on Kuperberg´s theorem.

In chapter 3 on consider quandles (distributive groupoids) and their generalizations to virtual knots. Generalizations of invariants that emerged, from virtual knot theory, is the use of the biquandle, and its powerful applications to the theory. Biquandle is a generalization of the quandle, which in turn generalizes the fundamental group of a knot (link). It worth the Lie algebraic techniques for invariants, long virtual knots, and the hierarchy of virtual knots, where have ordered by some choice of an ordinal, in terms of their ability to move across one another. Some invariants are found here by the hierarchy of virtual knots.

Chapter 4 does the basics of the Jones-Kauffman polynomials via the bracket state sum and introduces the concept of atoms (surfaces, orientable or not, bearing the virtual knot diagram). It also treated the chord diagrams and the passage from atoms to chord diagrams. The spanning tree, the leading and lowest terms of the Kauffman bracket polynomial, Kauffman bracket for rigid knots, and so on. It ends with the study of minimal diagrams of long virtual knots.

In Chapter 5, in the present book, on dealing with many ideas related to virtual theory can be directed to the study of Khovanov homology, as the homology of an algebraic complex which is constructed with a diagram of a knot (link), this homology detects the unknot, which seriously enlarges horizons of the theory. The Khovanov theory, associates with each knot diagram a chain complex, whose homology is a knot invariant, and the Euler characteristic of this complex coincides with the Jones polynomial. These chains constructed with a knot diagram, correspond to formal smoothing of this diagram at all classical crossings. It is interesting that the differentials of the Khovanov complex are defined combinatorially, and the homology is invariant under Reidemeister moves. To extend Khovanov homology theory to virtual knots, Manturov had to revisit completely the theory and construct a complex homotopically equivalent to the usual Khovanov complex. The problem, in the case of virtual knots, is related to the well definedness in order for the square of the differential to be equal to zero. This procedure needs to be computed, and compared with the Khovanov Rozansky Categorified Link Homology. A key role in this construction was played by the notion of atom and played a crucial role in the proof of Vassiliev´s conjecture. Such construction is very nice because required a number of new ideas: orientation and enumeration of the state curves; twisted coefficients in the Frobenius algebra representing, and the usage of exterior products instead of usual products. In short, the authors gave a first combinatorial solution to the question of constructing integral Khovanov homology for the virtual knots and links.

Chapter 6 deals with virtual braids and the work of Kamada (2007), and Kauffman and Lambropoulou(2006), and the considerations of the invariants of virtual braids, in the following cases, the construction of the main invariant, then the representation of virtual braid group. On the other hand, studies on completeness in the classical case and the case of two-strand braids.

Chapter 7 treats combinatorial aspects of the Vassieliev invariant theory and the work of Goussarov, Polyak and Viro who used virtual knots in the guise of general Gauss diagrams to construct a theory of Gauss diagram formulas for virtual knots. Virtual knot theory, its constructions and methods are closely related to various branches of classical knot theory, in particular, to Vassiliev invariants. These occupy a special position in classical knot theory; it turned out just, when this theory appeared, that all polynomial and quantum invariants were expressible in terms of Vassiliev invariants. In the case of virtual knots, the theory of Vassiliev knot invariants is much more complicated; even the space of order zero invariants is infinite-dimensional. In this chapter, by using atoms and d-diagrams, on proved Vassiliev´s conjecture about planarity of framed 4.valent graphs (graphs where at each vertex four half-edges are split into two pairs of opposite ones); this conjecture solved positively, plays a key role in Vassiliev´s work on the existence of integer-valued combinatorial formulae for invariants of finite order.

The chapter 8 is devoted to parity in knot theory. In virtual knot theory, there are many unexpected invariants which do not take place in the classical case. This chapter includes work on the Goldman bracket, and the Turaev cobracter and on cobordism of free knots. At first it was thought (a conjecture of Turaev) that free knots were trivial, but the most striking example of such theory is the parity theory conceived by Vassily Olegovich Manturov, where all classical crossings are either even or odd, herewith the property of being even is naturally preserve by Reidemeister moves. By parity, we mean any such natural way of labeling of all classical crossing which is defines for all knots from this theory. Manturov showed, using parity that this is not the case and that there are non-trivial cobordism classes of free knots. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams. The analogous virtual knot theory is to study virtual knots up to change of orientation of the crossings as we describe of the beginnings, to study virtual knots up to virtualization equivalence. The existence of different parities and different projections (from knots to knots) allows one to establish various filtrations on the space of knots. Besides that, such projections allows, one to lift invariants, from classical knots to virtual knots.

The theories mentioned above are related to different problems of combinatorics, three-dimensional topology, and four-dimensional topology, representation theory for Lie groups and algebras. Representation theory is the starting point for constructing quantum invariants of knots and 3- manifolds.

The book is the result of the research for over 10 years, different questions of virtual knot theory were discussed in the seminar “Knots and the representation theory” and Seminar on “Tensor and Vector Analysis” (the latter exists since 1920s) in the Moscow State University.

Definitely the book is high recommended to undergraduate, graduate, professionals and amateur mathematicians, because it goes from the basics to the frontiers of research. Finally, as L.H. Kauffman says: “this book self-contained is motivated, to delve, into the adventure proposed, by this intriguing and remarkable book”.

Reviewer:
Francisco José Cano Sevilla
Affiliation:

## Dynamic Mixed Models for Familial Longitudinal Data

Author(s):
Publisher:
Springer
Year:
2011
ISBN:
1441983414, 9781441983411
Short description:

The book focuses on the analysis of familial and longitudinal data by means of dynamic mixed models.

MSC main category:
62 Statistics
MSC category:
62J12
Other MSC categories:
62P10
Review:

The book focuses on the analysis of familial and longitudinal data by means of dynamic mixed models. These data are correlated as (a) the responses of the members of a particular family share a common random family effect and (b) the repeated responses of the same individual are not independent. These familial and longitudinal correlation structures play a crucial role in the analysis of this type of data and greatly condition the model estimates and the rest of the statistical inferences. The book consists of eleven chapters. The first exhaustively lists the most relevant antecedents on familiar and longitudinal models in the last three decades. Chapters 2 and 3 give an overview of the analysis of longitudinal data by linear models (fixed and mixed, respectively). The rest of the chapters are divided in four pairs and each pair focusses on the study of count data and binary data, respectively, using distinct kind of models. In particular, the following: familial models (chapters 4 and 5), longitudinal models (chapters 6 and 7), longitudinal mixed models (in the following pair of chapters), and finally familial longitudinal models (in the last pair). The different models are discussed in depth with a special care for the technicalities behind the countless amounts of different correlation models. This means that the statistical background of the potential readers of the book needs to be rather advanced and the reading becomes occasionally arduous. This drawback is alleviated by the fact that the book also broaches the analysis of a considerable number of real life data sets, mainly within the context of biostatistics and econometrics. Researchers of these two areas along with applied statistics undoubtedly constitute the main target of the book.

Reviewer:
Teófilo Valdés Sánchez
Affiliation:

## Geometry of crystallographic groups

Author(s):
Andrzej Szczepa ́ski
Publisher:
World Scientific
Year:
2012
ISBN:
978-981-4412-25-4
Short description:

An $n$-dimensional crystallographic group $\Gamma$ is a discrete subgroup of the group $O(n)\ltimes{\mathbb R}$ of isometries of ${\mathbb R}^n$ having a compact fundamental domain. If $\Gamma$ is torsion free the quotient $M:={\mathbb R}^n/\Gamma$ is a manifold whose fundamental group is $\Gamma$, and since this group acts on ${\mathbb R}^n$ as a group of isometries, $M$ inherits a Riemannian structure making it into a flat manifold, i.e. a manifold with sectional curvature zero. Conversely, any compact flat manifold is obtained in this way, and many parts of this excellent book can be understood as a dictionary explaining the relationship between the geometric properties of $M$ and the algebraic properties of $\Gamma$.

MSC main category:
20 Group theory and generalizations
MSC category:
20H15
Other MSC categories:
53C55
Reviewer:
José Manuel Gamboa Mutuberria
Affiliation:

## The Real Numbers and Real Analysis

Author(s):
Ethan D. Bloch
Publisher:
Springer
Year:
2011
ISBN:
978-0-387-72177-4
Short description:

This book is devoted to an introduction to the real numbers and real analysis. The main goal of the book is to provide to secondary school teachers of a solid background on analysis. The book may be used also as an introduction to one variable analysis for undergraduates majoring in mathematics.

URL for publisher, author, or book:
www.springer.com
MSC main category:
26 Real functions
MSC category:
2601
Review:

The author's purpose is to cover with this book the necessary mathematical background for secondary school teachers. The book is also useful for an introductory one real variable analysis course.

The book basic contents, which correspond with that introductory course, consists of seven chapters, namely the numbers 2, 3, 4, 5, 8, 9 and 10. The first chapter devoted to the real numbers, which are introduced axiomatically, and their properties. This chapter contains also a rigorous presentation of mathematical induction and recursively defined functions. The chapter concludes with a deep study of the Least Upper Bound Property and several of its consequences.

The second chapter is devoted to the limits of functions and continuity. It includes the proofs of the Extreme Value and Intermediate Value Theorems. Moreover, it is proved that these theorems are equivalent to the Least Upper Bound Property. This result is obviously very interesting, but its proof might be omitted in a basic course.

The treatment of differentiation is the standard one. The chapter includes the Rolle's Theorem and the Mean Value Theorem, as well as increasing and decreasing functions, local extrema and convex - called concave up in the text - functions.

In the chapter devoted to integration the author presents the Riemann integral, proving the Fundamental Theorem of Calculus and Lebesgue's Theorem, characterizing Riemann integrable functions as those which have a zero measure set of discontinuity points. In my opinion, introducing Riemann sums with respect to a partition and a representative set of the partition contributes nothing new with respect to the more usual way throughout upper and lower sums.

The study of sequences and series of real numbers is presented in two chapters that include the Bolzano Weierstrass Theorem and the completeness of the real numbers field. A proof of the equivalence between these results and the Least Upper Bound Theorem is given. Other results as Cantor's Lemma and the usual series convergence criteria are presented too. The author's option choice is to delay the study of sequences after the development of the course main core. It is a personal decision which has some advantages as the fact that one may immediately present some applications. But the early introduction of the sequences is, in my opinion more adequate. The ε-n arguments appearing while studying sequences, are a good introduction of the ε-δ arguments, essential for the theoretic study of continuity, differentiation and integration. On the other hand, using Bolzano Weierstrass Theorem in order to prove classical results instead of the Least Upper Bound Property has the advantage of an easier generalization of the results to higher dimensions.

This basic part ends with a chapter devoted to the study of sequences and series of functions. These concepts had been suggested before, throughout the study of power series. The chapter includes pointwise and uniform convergence, Taylor series, and ends with the appealing and interesting example of a continuous but nowhere differentiable function.

In order to provide a solid background in analysis to future secondary school teachers, the book includes the following topics. A first chapter introducing natural, integer, and rational numbers in a standard way, and the real numbers throughout Dedeking Cuts. This way is necessary once the study of sequences has been delayed. Both methods, Dedeking Cuts and Cauchy Sequences, contribute interesting ideas for the understanding of the real numbers and therefore for analysis.

I consider very interesting the chapter devoted to the trigonometric functions, to the exponential and the logarithm. The logarithm is defined by the integral, defining later the exponential as its inverse. In order to introduce the trigonometric functions, the starting point is the arcsine, which again is introduced throughout the integral. The main interest of this chapter is that it closes the gap existing between the elementary definitions and the analytic treatment of such functions.

Another good decision is the inclusion of the study of areas and lengths, which sometimes is delayed to a several variables course, or even to a Measure Theory course.

Two last remarks. The book has an interesting and useful collection of exercises; they are in general achievable for the students, but on the other hand, they also go deeply into the theory, completing the results introduced previously. Last but not least, the historic notes are excellent.

One more word, I consider this book of great interest for the academic training of the future secondary school teachers, so the author's purpose is greatly fulfilled.

Reviewer:
Juan Ferrera
Affiliation: