Five recent book reviews

Publisher:

Springer

Year:

2011

ISBN:

978-1-4614-1498-8

Short description:

This book gives the basic definitions and results as well as some physical instances where the theory is applied, of Hamiltonian systems in terms of Lagrangian submanifolds and their generating families.

MSC main category:

70 Mechanics of particles and systems

Review:

Among the brilliant developments achieved by Differential Geometry along the XX century, the geometric formulation of Classical Mechanics has proved to be one of the most relevant. In this context, the Hamiltonian formulation has attired much attention of both mathematicians and physicists. For instance, from a mathematical point of view, symplectic Geometry finds essential inspiration in Hamiltonian Mechanics whereas physicists formalize many plausible models by means of the symplectic tools studied in Geometry.

Given a manifold equipped with a symplectic structure, the submanifolds that are isotropic and coisotropic with respect to the symplectic structure are called Lagrangian. Many important concepts and definitions (or, according to many specialists, all the concepts and definitions!) in symplectic Geometry or in Hamiltonian Mechanics can be written in terms of these special submanifolds. Furthermore, if the manifold is a cotangent bundle with the canonical symplectic structure, a Lagrangian submanifold is locally generated by a function. This construction can be generalized by the so-called generating families to describe more general Lagrangian sets that are connected with some physical meaningful phenomena. The main topic of this book covers the definition, description and applications of generating families. For this purpose, the introduction of the notion of symplectic relation plays an important role.

This reference is an enhanced version of a previous book edited in Russian. This new work gives an improved presentation of the theoretical part and deeper developments of its applications. In fact, the applications of the Lagrangian submanifolds and symplectic relations given in chapters 6, 7 and 8 are especially motivating. They present the Hamilton-Jacobi theory in geometric Optics, Hamiltonian Optics in Euclidean space and control of thermostatic systems respectively. There are many other scenarios where Hamiltonian systems play a key role. A comprehensive study of them would simply overwhelm the length of a single book. From this point of view, the choice done by this book could have been different, though the importance and elegance of applications in these chapters need not further justification.

This book is aimed at both undergraduate and graduate students with just some initial knowledge in Algebra and Geometry. For this reason, the chapters try to include enough preliminaries to provide a gentle introduction to the topics covered by them. Specialists will also find a nice reference in this book specially, I think, with respect to the classical applications to Optics and thermostatics.

Reviewer:

Marco CASTRILLON LOPEZ

Affiliation:

Universidad Complutense de Madrid, Spain

Publisher:

Springer Verlag

Year:

2013

ISBN:

978-88-470-2888-3 (hbk)

Price (tentative):

59,99 € (net)

Short description:

This is a sequel to the first volume of Imagine Math and it contains again a collection of papers that make a connection between mathematics and cultural activities like music, poetry, literature, film, and/or scientific disciplines like philosophy, sociology, biology, polemology, myrmecology, etc.

URL for publisher, author, or book:

www.springer.com/mathematics/book/978-88-470-2888-3

MSC main category:

00 General

MSC category:

00A99

Other MSC categories:

00A65; 00A66; 00A67

Review:

Like the previous volume *Imagine Math* and the parallel series of *Mathematics and Culture*, (see e.g. vol I and vol VI) we again find here a collection of 26 very diverse essays, often translated from Italian, that deal with the mathematical aspects of arts, culture and scientific applications. The volumes in both series are edited by M. Emmer, but whereas the *Mathematics and Culture* provides translations of the Italian proceedings of recurrent conferences on the topic, this series is not directly connected to a conference. Since there are too many contributions to review each paper in detail here, I just a few examples below.

There is mathematics, poetry and literature which take up about a quarter of the book. Some examples: a paper by *V. Della Mea* who wrote a collection of poems *L'Infanzia di Gödel* (Gödel's childhood) and one by *Ph. Schlogt* who wrote a novel *The wild numbers* about a mathematician who proved some great theorem about a crazy fictitious sequence of numbers. That sequence got an entry in the on-line encyclopedia of integer sequences afterwards. Many classics from world literature appear in a survey analysing the attitude of the authors towards mathematics (or science). The contribution by *C. Ambrosini* deals with two operas he wrote (*Big bang circus* and *Il killer di parole*) in which the libretto, especially in the latter, deals with numbers, language and mathematics.

Mathematics and film is also a subject that shows up in all volumes of these series. Many types of links are possible. It can be a biographical film of some mathematician, or a mathematical element can play an essential dramatic role,

or structural elements may be defined by mathematical procedures, or geometric elements may be used as in an abstract painting using light as a paint, etc. Of course there are also instructive movies like *Flatland* or *Donald in Mathmagic Land* or the more formal computer generated *Dimensions*. This volume has a paper by *E. Frenkel* about his erotic short film *Rites of love and math*, a remake of the Japanese cult classic *Rites of love and death* (Y. Mishima, 1966) in which two lovers commit suicide. The harakiri at the end in the original is replaced by the tattooing of a math formula on the belly of the woman in the remake. *Arithmétique* is a short animation video by *G. Munari* and *D. Rovazzani*

Painting, music and math are also mixed in the papers by the musician *D. Amodio* and the engineer *C. de Fabritiis*. They `mapped' the well known painting *Summertime 9A* by *J. Pollock* into a piece of music, which they called *Jacksontime*.

The homage to *Alan Turing* contains a short biography and the script of the stage play *Alan Turing and the poisoned apple* by *M.Vicenzi*. More performance art, relating the real and the virtual is described in the paper by *A. Mondot* on the company *Adrien M / Claire B* that he has set up together with *C. Bardainne*.

Somewhat more technical are contributions on dynamical systems and morphology, weather prediction, on modeling the social behaviour of ant colonies and more generally modelling swarms, and cellular automata for the game of life.

All this and much more can be found in here. Mathematics is everywhere as this book illustrates and yet hardly a formula to be found in it. Good reads for all math lovers with a broad and open mind.

Reviewer:

A. Bultheel

Affiliation:

KU Leuven

Publisher:

American Mathematical Society

Year:

2010

ISBN:

978-0-8218-5280-4

Short description:

This book is a collection of articles originating from Terence Tao’s mathematical blog (terrytao.wordpress.com) which were posted during the year 2009. The material has been updated and improved by the author to reach a publishable form. It contains introductory articles on topics of broad interest, as well as more technical articles mostly related to the research interests of the author.

MSC main category:

00 General

Review:

This book is a continuation of the series of books by Terence Tao which derive from his mathematical blog. The preceding books were Structure and Randomness and Poincaré’s legacies, both published under the American Mathematical Society. The current review corresponds to An Epsilon of Room, volume 2. The content of this volume has been divided into two large chapters: the first one is devoted to expository articles covering a wide variety of topics, while the second one deals with more technical articles closer to the research interests of the author.

The reader will appreciate from the very first page the exquisite expository style of the author and the beauty of the proofs and ideas that are presented in the text. Though elaborated at certain steps, the technicalities in the arguments are kept to a minimum, while the important ideas are clearly highlighted, in an attempt to make them available in different contexts. The genuine insight of the author provides a perfect approach to each of the topics that are considered in the book.

The articles deal with different aspects of analysis, probability theory, mathematical physics, logic, complexity, combinatorics, number theory… The subjects covered in this volume include, for instance, Talagrand’s concentration inequality and its connection with random matrices, the Agrawal-Kayal-Saxena primality test, Grothendieck’s definition of a group, or the prime number theorem in arithmetic progressions and dueling conspiracies. This is just to mention a few of the subjects corresponding to the first part of the book. Besides, among the topics included in the second half, we find articles devoted to Szemerédi’s regularity lemma via random partitions and the correspondence principle, the Kakeya maximal function conjecture, the prevalence of determinantal point processes, and approximate bases, sunflowers, and nonstandard analysis in additive combinatorics.

Each of the topics is introduced in a very elegant way, including the appropriate motivation, but immediately, the author gets to the point of each problem, revealing where the difficulties lie. He recalls the main contributions to every issue and frequently provides examples and toy versions of the theorems that are considered. Informal overviews of some of the important arguments and/or heuristic proofs are often given in order to clarify the ideas without getting extremely technical.

All these ingredients make the book a wonderful reading to anyone with an interest in current approaches to both classical problems and emerging areas of mathematics. The author really succeeds in transmitting his passion for mathematics, and I am pretty sure that every reader will feel the necessity of broadening his/her area of interest after reading this excellent compilation of mathematical ideas.

Reviewer:

Pedro Tradacete

Affiliation:

Universidad Carlos III de Madrid

Publisher:

World Scientific Publishing Co. Pte. Ltd. Singapore, River Edge, N. J. 2013. Second Edition, XII, 209 p. Illus., 23cm. Hardcover.

Year:

2013

ISBN:

978-981-4401-90-6 (Hardcover); 978-981-4401-91-3 (Softcover, pbk.)

Price (tentative):

From 2l.30 € (18.00 GBP) (eBook) to 44.95€ (38.00GBP) (Hardcover).

Short description:

The book begins with the basics of the combinatorial analysis and topics, as a branch of mathematics dealing with discretely structured problems. Its scope of study includes selections and arrangements of objects with established conditions, configurations, and designs of experimental schemes according to determined rules. In short, the counting problem is one of the basic problems in combinatorics, with applications in various branches of mathematics and other disciplines such as engineering, computer science, operational research and life sciences.

This book is a concise, interesting, self-contained, useful, clear, attractive introduction to basic in counting, thinking skills and techniques in general problem solving. It contains numerous exercises, and their solutions, and includes updates on new tools of combinatorics.

In this second edition new chapters are included: the Principle of Inclusion and Exclusion, The Pigeonhole Principle, Recurrence Relations, The Stirling and the Catalan Numbers, with interesting counting arguments, this is why a new edition, (the second edition), seemed appropriate. A good collection of examples and exercises, have also been added to covering a range of difficulties, and a recommended bibliography for further reading, answers to exercises, and index. The book is a beautiful systematic collection of examples and solved exercises, related to 20 different chapters on the basics of the combinatorial theory.

The book helps to give to readers, to all people who appreciate mathematics, to avid puzzle-solvers, to undergraduate students as well as teachers, an early start to learning problem, solving heuristics and ways of counting, and so on.

URL for publisher, author, or book:

www.worldscientific.com

MSC main category:

05 Combinatorics

MSC category:

05AXX, 05CXX, 97KXX, 97MXX

Review:

The book begins with the basics of the combinatorial analysis and topics, as a branch of mathematics dealing with discretely structured problems. Its scope of study includes selections and arrangements of objects with established conditions, configurations, and designs of experimental schemes according to determined rules. In short, the counting problem is one of the basic problems in combinatorics, with applications in various branches of mathematics and other disciplines such as engineering, computer science, operational research and life sciences.

The book Counting, is self-contained, good, admirably clear, and a stimulating and very well written. This concise introductory text-book is an early start to learning problem solving heuristics and ways of counting. It contains numerous exercises, and their solutions, and includes updates on new tools and techniques in practical combinatorial theory. Not many mathematical prerequisites are needed to read this book because the mathematical material is sufficiently closed.

The aim of the present book is intended not only as an introduction to basic counting techniques for upper secondary to undergraduate students, and teachers, but is a rich source of ideas to describe the main tools and techniques in practical combinatorial theory. For the reader is suitable solving some of the many proposed and solved exercises. ¡Good experience for share seriously the examples and exercises!

I feel sure that it will be of great use both to students of practical combinatorial theory, in particular graph theory, and mathematics, in general, and captivate to students, instructors, puzzle devotees, and teachers in high/secondary schools and colleges.

The book contains preface, table of contents, list de figures (pp. i-xii), and twenty chapters, and a recommended bibliography for further reading, answers to exercises, and index, in 209 p.

The first two chapters cover two basic principles, the addition and the multiplication, both principles are wide and commonly uses in counting. The first chapter, (p. 1-7), introduces the basic, definitions and notions, devoted to present the addition principle and illustrate how it is applied.

This chapter considered four examples and nine exercises, very quite easy. The proposed exercises are solved with their respective solutions, in one appendix on answers to exercises.

In the chapter two, on introduce the multiplication principle, (p. 9-16), with four detailed examples and six exercises, for stimulating to reader and the beginner, even the most requires very little work.

In the chapter three, (p.17-23), Subsets and arrangements, provides the concepts of combinations and permutations by considering as subsets and arrangements of a set of objects. This chapter analyzed how the multiplication principle, by incorporating the addition principle, enables us to solve more complicated problem, in particular four exercises are presented, and solved with their respective solutions.

The chapter four, Applications, (p. 25-33), is completely devoted to various applications of the concepts learnt and now gives some examples to illustrate how the concepts of r-permutations and r-combinations of an n-element set can be applied. On introduce the third basic principle for counting, the principle of complementation. Moreover, six examples and fourteen exercises are proposed.

In the chapter five, the Bijection principle, (p. 35-47), on introduce the fourth basic principle for counting, named the bijection principle, and discuss some of its applications. On presented here three examples and nine interesting exercises for illustrating the mainly concepts.

The chapter six, distribution of balls into boxes, (p. 49-53), introduces a very useful perspective to which many counting problems can be converted to the distribution, for example balls into boxes. Moreover one example is discussed joint four exercises.

The following next three chapters flesh out the Bijection Principle with a number of applications and variations. In particular, the chapter seven, More Applications of Bijection Principle (p. 55-64), on analyzed with details, six examples and fifteen elaborated exercises.

The chapter eight, Distribution of distinct objects into distinct boxes, (p. 65-68), presented two examples and four exercises, related to the distribution problem, that is, the counting of ways of distributing objects into boxes. This is a basic model for many counting problems, because objects can be identical or distinct, and boxes too can be identical or distinct. Thus, there are, in general, four cases to be considered, namely. The first case: objects identical and boxes distinct have been studied in chapters 6 and 7. The second case: objects distinct and boxes distinct on analyzed in this chapter. The third case: objects distinct and boxes identical will be discussed in chapter 18 as Stirling numbers of the second type, and finally the fourth case will not touched upon in this book.

The chapter nine, (p. 69-74) deals on other variations of the distribution problem in their cases one and two mentioned above. In this chapter on considered the ordering of objects in each box. Two examples and eight exercises are also presented.

From chapter ten to chapter twelve put this family of combinatorial numbers in the context of the binomial expansions and Pascal´s triangle. A great number of useful identities are proven and problems are posed where theses identities surprisingly appear.

The chapter ten, the binomial expansion, (p.75-78), deals with the binomial theorem and four simple identities involving binomial coefficients, derived easily equivalent formula for the combinatorial number. Moreover, on presented here four proposed exercises, and analyzed their solutions in the answers to exercises.

In the chapter eleven, (p. 79-85), on derived some more identities, involving binomial coefficients from the binomial expansion. These identities are useful in simplifying certain algebraic expressions. Three worked examples and five proposed exercises are presented here.

The chapter twelve, (p. 87-96), is completely devoted to the Pascal´s triangle. An interesting and worked example and eight elaborated exercises are done.

Chapter 13 and Chapter 14 cover the Principle of Inclusion and Exclusion, and its general statement. Many situations of counting are complicated by the possibility of double counting.

In chapter thirteen, the principle of inclusion and exclusion (p. 97-108), added as new in this second edition, express the cardinal of the union of two sets in terms of cardinal of each one regardless of both are disjoint. The resulting identity is a simplest form of a principle called the Principle of Inclusion and Exclusion, a powerful tool in enumeration. Four examples and ten proposed exercises illustrating, this important and surprisingly principle appear.

The chapter fourteen, (p. 109-119), corrected and added in this second edition, extends the principle of inclusion and exclusion to three finite sets, and then to four sets and in general to n≥ 2, finite sets, and if so, what identity would one get in general, for the general statement of the principle of inclusion and exclusion.

The Pigeonhole Principle is studied in chapter fifteen (p. 121-132), the principle not actually count the number ways for a particular situation, it is used to check the existence for a determined setting. The principle deals to transform the particular problem, partly into one of distributing a number of objects into a number of boxes. The principle is very interesting to prove the contra positive statement of any question, and so proves the principle. It also known as the Dirichlet Drawer Principle, used to prove some results in Number Theory. In this chapter on introduces the Ramsey number and the problem of n-coloring edges in complete graph, and so on, as an application of the Pigeonhole Principle. Six examples and nine proposed exercises, and a worked problem with proof illustrating this chapter. This chapter is included in this second edition.

The next four chapters, chapter 16 to chapter 19, also added in this second edition, are on recurrence relations, in a new defiant aspect about the techniques, tools and principles learnt.

The chapter sixteen, (p. 133-146), introduces the technique of using recurrence relations. These represent algebraically the situation where the solution of a counting problem of bigger size can be obtained from the solutions of the same problem but of smaller size one, and so that the original counting problem can be solved. On consider in this chapter, the sequence of Fibonacci numbers, the rabbit problem, the tower of Hanoi, the theory about the last largest prime number of Edouard Lucas, and so on. Two well worked examples and ten proposed exercises, illustrating this chapter

Chapter seventeen to chapter nineteen study three series of numbers which are derived from special recurrence relations. These are the Stirling numbers of the firs kind, the Stirling numbers of the second kind, and Catalan numbers.

In the chapter seventeen, (p. 147-157), devoted completely to the study of the Stirling numbers of the first kind, that is, the coefficient of xk in the expansion of the polynomial in x of degree m, [x]m = {x(x-1)(x-2)…(x-(m-1))}, where 0≤k≤m. This coefficient is designed by s (m, k), because depends on both m and k, and it is called the Stirling number of the first kind, which is purely algebraic in nature. The chapter studied the combinatorial interpretation of the Stirling number of first kind and the corresponding relation for the number of ways of arranging m distinct objects around k identical circles with at least one object at each circle, denoted by s* (m,k), the absolute value of the real number s(m,k). The sequence of numbers s (m, 1), s (m, 2), s (m, m) alternate in sign with s (m, 1) positive when and only when m is odd. These give the mainly combinatorial argument and explain the nature of Stirling numbers. Five interesting exercises, illustrating this chapter

The chapter eighteen, (p. 159-166), introduces the other sequence of Stirling number of the second kind. The Stirling number of the second kind, denoted by S(n,k) and given two positive integers n and k, with k≤n, is defined as the number of ways of dividing n distinct objects into k(nonempty) groups, that is, the number of ways of partitioning an n-element set into k nonempty subsets. The chapter is dedicated to these numbers and their combinatorial interpretations. Moreover, five exercises are proposed.

In the chapter nineteen, (p. 167-178), introduces the Catalan numbers, that just been obtained as the number of shortest routes f(n) from O(0,0) to A(n,n) which do not cross the diagonal y=x in the rectangular coordinate system is given by f(n) = Cn2n/(n+1), combinatorial number 2n over n. The numbers 1, 2, 5, 14, 42, 132, 429 …, Cn2n/ (n+1), ..., are called Catalan numbers after the Belgium Mathematician Eugene Catalan. In particular, the 1-1 correspondences among the routes, the ways of parenthesising x1. x2 … .xn , and the binary sequences, are examples of the Catalan numbers.

It is interesting some more general problem, known as the Ballot Problem as an extension of some problems involves Catalan numbers; the Euler´s Polygon Division Problem; the solution to how many ways are there to pair, 2n (n≥ 1) distinct fixed points on the circumference of a circle using n nonintersecting chords, an others similar problems considered in the specialized literature. Six interesting exercises are proposed in this chapter.

The chapter twenty, miscellaneous problems, (p. 179-195), closes the book with a collection of sixty interesting problems in which the approaches to solving them appear as applications of one or more concepts learnt in all the earlier chapters. The exercises and problems are designed to aid understanding. Some are quite easy, a few are quite difficult, and the other for works adequately. All problems are solved in answers to exercises.

Books recommended for further reading, (p. 197). There is an explicit recommended bibliography with nine references of basic books.

Answers to Exercises, (p. 199-208). There are worked solutions, to quite all of them.

Index (p. 209), of relevant subjects considered.

In this second edition new chapters are included: the Principle of Inclusion and Exclusion, The Pigeonhole Principle, Recurrence Relations, The Stirling and the Catalan Numbers, with interesting counting arguments, this is why a new edition seemed appropriate. A good collection of examples and exercises, have also been added to covering a range of difficulties, and a recommended bibliography for further reading, answers to exercises, and index. The book is a beautiful systematic collection of examples and solved exercises, related to different 20 chapters on the basics of the combinatorial theory.

This book is a concise, interesting, self-contained, useful, clear, attractive introduction to basic in counting, thinking skills and techniques in general problem solving. It contains numerous exercises, and their solutions, and includes updates on new tools of practical combinatorial theory.

Definitely, practical combinatorial theory is considered difficult by students, and by this the book make more accessible with the numerous interesting exercises and applications, to captivate the attention towards an book that helps to give to readers, to all people who appreciate mathematics, to avid puzzle-solvers, to undergraduate students as well as teachers, an early start to learning problem, solving heuristics and ways of counting, and so on.

Reviewer:

Francisco José Cano Sevilla

Affiliation:

Profesor Universidad Complutense

Publisher:

World Scientific Publishing Co. Pte. Ltd.5 Toh Tuck, Singapore 596224. 2013, (Jan).i-ix, 221 p.; 23cm. Hardcover.

Year:

2013

ISBN:

976-981-4412-57-5 (Hardcover); 978-981-4412-59-9 (Softcover, pbk.)

Price (tentative):

From 58.00 € (49.00 GBP) (eBook) to 76.89€ (65.00GBP) (Hardcover).

Short description:

The book is organized according to the main concepts and topics in this area and it provides new inequalities for sums of random variables, their maximum, martingales, Brownian motions, diffusion processes, point processes and their maximum.

The emphasis on the inequalities is aimed at graduate students and researchers having the basic knowledge in Analysis, Integration Theory and Probability. The book gives a survey of classical inequalities in several fields with the main ideas of their proofs, as well as an account of inequalities in vector and functional spaces with applications to probability and to complete and extend on them. The inequalities presented are not an exhaustive list of them.

This book contains many proofs about basic inequalities for martingales with discrete or continuous parameters, and the proofs of the new results are detailed, accessible to the readers, and finally are presented in great detail. All illustrated by applications in probability theory. Its scope of study includes inequalities in vector spaces and functional Hilbert spaces in order to simplify the approach of uniform bounds for stochastic processes in functional classes. It also develops new extensions of the analytical inequalities, with sharper bounds and generalization to the sum on the maximum of random variables, to martingales and the transformed Brownian motions.

This book is an attractive introduction and extension to applications of the analytic inequalities to probability, random variables, martingales, stochastic processes with values in Banach spaces, complex spaces, tail behavior of processes, and so on.

The book helps to give to readers, to graduate students and beginners and senior researchers, a fundamental background and compendium in probability, stochastic processes, martingales, Brownian motions, and integration theory, with widely applications.

URL for publisher, author, or book:

www.worldscientific.com

MSC main category:

60 Probability theory and stochastic processes

MSC category:

26DXX, 32CCC, 60GXX, 60JXX

Other MSC categories:

60G16

Review:

The book is organized according to the main concepts and topics in this area and it provides new inequalities for sums of random variables, their maximum, martingales, Brownian motions, diffusion processes, point processes and their maximum.

The emphasis on the inequalities is aimed at graduate students and researchers having the basic knowledge in Analysis, Integration Theory and Probability. The book gives a survey of classical inequalities in several fields with the main ideas of their proofs, as well as an account of inequalities in vector and functional spaces with applications to probability and to complete and extend on them. The inequalities presented are not an exhaustive list of them.

This book contains many proofs about basic inequalities for martingales with discrete or continuous parameters, and the proofs of the new results are detailed, accessible to the readers, and finally are presented in great detail. All illustrated by applications in probability theory. Its scope of study includes inequalities in vector spaces and functional Hilbert spaces in order to simplify the approach of uniform bounds for stochastic processes in functional classes. It also develops new extensions of the analytical inequalities, with sharper bounds and generalization to the sum on the maximum of random variables, to martingales and the transformed Brownian motions.

This book is an attractive introduction and extension to applications of the analytic inequalities to probability, random variables, martingales, stochastic processes with values in Banach spaces, complex spaces, tail behavior of processes, and so on.

The book helps to give to readers, to graduate students and beginners and senior researchers, a fundamental background and compendium in probability, stochastic processes, martingales, Brownian motions, and integration theory, with widely applications. I feel sure that it will be of great use both to graduate students, and researchers, in analysis and probability theory.

The book contains preface, table of contents, list de figures (p. i-ix), seven chapters, and an appendix A with the basics used in Probability Theory, as definitions and convergences in probability spaces; boundary-crossing probabilities; distances between distributions, and expansion in L2 ( R) an interesting updated bibliography with 110 references, and index, in 221 p.

The first chapter, preliminaries, (p. 1-30), introduces the basic, definitions and notions, devoted to present the origin of the inequalities for convex functions and its generalizations to inequalities for the tail distribution of sums of independent or dependent variables, and to martingales indexed by discrete or continuous sets. These inequalities are the decisive arguments for bounding series, integrals, and for proving other inequalities.

The chapter introduces the following items: Cauchy and Hölder inequalities; inequalities for transformed series and functions; applications in Probability Theory; Hardy´s inequality; inequalities for discrete martingales; martingales indexed by continuous parameters; large deviations and exponential inequalities, and finally functional inequalities. Five remarkable theorems and two propositions are presented here.

The next chapters develop extension and applications of the classical results presented in above chapter.

In the chapter two, inequalities for means and integrals, (p. 31-58), on extends, the Cauchy, Hölder and Hilbert inequalities for arithmetic and integral means in real spaces. Hardy´s inequality is presented with new versions of weighted inequalities in real analysis. This chapter deals basically with inequalities for means in real vector spaces; Hölder and Hilbert inequalities; generalizations of Hardy´s inequality; Carleman´s inequality and generalizations; Minkowski´s inequality and generalizations; inequalities for the Laplace transform, and inequalities for multivariate functions.

This chapter works on ten theorems, nineteen propositions and eight examples that illustrating the different theoretical lines proposed. Some are quite difficult for the beginner.

The chapter three, (p.59-90), analytic inequalities, presents some of the most important analytic inequalities for the arithmetic and geometric means, in particular functional means for the power and the logarithm functions. The upper and lower bounds for the logarithm mean function on R+, Carlson´s inequality, is improved and is applied to other functions. On the other hand, inequalities for the arithmetic and the geometric means extend Cauchy´s results, and also established results for the median, the mode, mean residual time, and the mean of density or distributions functions. Finally, functional equations, Young´s integral inequality and several results about the entropy are proved.

The following subjects are considered: bounds for series; Cauchy´s inequalities and convex mappings: inequalities for the mode and the median; mean residual time; functional equations; Carlson´s inequality; functional means; Young´s inequalities, and th3e concepts of entropy and information.

Three theorems, twenty six propositions and six examples, with their respective proofs are deals with great detail, and it shows the excellent work on them.

In the chapter four, (p. 91-126), inequalities for martingales, on deal inequalities for sums and maximum of n independent random variables, extended to discrete and continuous martingales an to their maximum variables. The Bürkholder- Davis-Gundy inequalities are improving. Moreover, the Chernov and Bennet theorems and other exponential inequalities are generalized to local martingales, applied to Brownian motions and Poisson processes, and turn out generalized in several forms to dependent and bounded variables and to local martingales, using Lenglart´s inequality. Finally another question is also considered, the solutions of diffusion equations are explicitly established, and then, some level crossing problems are studied.

This chapter presents the following statements inequalities for sums of independent random variables; inequalities for discrete martingales and for martingales indexed by R+; Poisson processes; Brownian motion; Diffusion processes; Level crossing probabilities, and martingales in the plane.

The chapter is illustrated with nine theorems, three lemmas, one corollary, and twenty four propositions elaborated, all them intensively and adequately proved.

The chapter five, Functional inequalities, (p. 127-152), concerns inequalities in functional spaces, for sums of real functions of random variables and their supremum on class of the different kinds defining the transformed variables. Functional of discrete or continuous martingales are extended, using the Chernov´s, Hoeffding´s and Bennet´s theorems. Uniform versions of the Bürkholder- Davis-Gundy inequalities, allows extended the above functional of martingales. Moreover some applications to the weak convergence are considered.

Chapter 5 analyzed exponential inequalities for functional empirical processes; exponential inequalities for functional martingales; weak convergence of functional processes; differentiable functional of empirical processes; regression functions and biased length, and finally regression functions for processes.

On presented here six very interesting theorems, one lemma, three corollaries, sixteen propositions and two examples, for illustrating the mainly concepts in this chapter.

The chapter six, inequalities for processes, (p. 153- 178), introduces a very useful perspective about inequalities for stochastic processes, in particular, new results for Gaussian processes, and the distribution of the ruin time of the Sparre Anderson ruin model and some more optimistic stochastic models with a diffusion term. By end, some spatial stationary measures, and also their weak convergence deduced from their tail behavior.

Chapter 6, concerns stationary processes; ruin models; comparison of models; moments of the processes at Tu; empirical process in mixture distributions; integral inequalities in the plane and spatial point processes.

The chapter is illustrated with four theorems, five lemmas, one corollary, and seven propositions elaborated, all them intensively and adequately proved, some are a little hard.

The chapter seven, Inequalities in complex spaces, (p. 179-200), focuses on complex spaces and on the Fourier transform. The classical theory is extended to generalize the expansions of analytic functions of several variables in series. On study the orthonormal basis of the Hermite polynomials and their properties, with the orders of the approximations. The isometry between R2 and C, extended to the same between R3 and a complex space where the Fourier transform is defined. Conditions for the differentiability of complex functions, as Cauchy conditions, are established, as well as in higher dimensions.

.

Chapter 7 presents some interesting questions as polynomials; Fourier and Hermite transforms; inequalities for the transforms; inequalities in C; complex spaces of higher dimensions and stochastic integrals.

The chapter included one special theorem, one lemma, two corollaries, eight propositions and two examples, with their proofs. A great detailed study about above them, are done.

An appendix A closed the book, with the basics of probability, definitions kinds or convergences in probability spaces, distances between probabilities, and other questions basic of interest.

In detailed form:

Appendix A. Probability (p. 201)

1. Definitions and convergences in probability spaces (p. 201-206).

2. Boundary-crossing probabilities (p. 206-207)

3. Distances between probabilities (p.207-209)

4. Expansions in

Reviewer:

Francisco José Cano Sevilla

Affiliation:

Profesor Universidad Complutense