# William Andrew Coppels' books

We list below eight books by W A Coppel. For each of these books we give some further information, mostly extracts from Prefaces and reviews.

Stability and asymptotic behaviour of differential equations (1965)

Disconjugacy (1971)

Linear systems (1972)

Dichotomies in stability theory (1978)

A Chinese-English mathematics primer (1989)

Dynamics in one dimension (1992) with Louis Block

Foundations of convex geometry (1998)

Number Theory: An Introduction to Mathematics: 2nd edition (2009)

**Click on a link below to go to that book**Stability and asymptotic behaviour of differential equations (1965)

Disconjugacy (1971)

Linear systems (1972)

Dichotomies in stability theory (1978)

A Chinese-English mathematics primer (1989)

Dynamics in one dimension (1992) with Louis Block

Foundations of convex geometry (1998)

Number Theory: An Introduction to Mathematics: 2nd edition (2009)

**1. Stability and asymptotic behaviour of differential equations (1965), by W A Coppel.**

**1.1. Review by: F S Van Vleck.**

*The American Mathematical Monthly*

**75**(3) (1968), 319-320.

This book is a monograph on stability and asymptotic behaviour of solutions of ordinary differential equations whose independent variable is real. With the addition of two topics, this book would furnish a good background for prospective research workers in the area of stability theory and would furnish those interested primarily in applications with a very readable account of the basics of stability theory. Liapunov's direct method is not treated; however, there are several books which deal with this topic so this is no real hardship. The topic which this reviewer would have liked to have seen in this book is the problem of global stability. Liapunov's direct method is frequently used to study this problem, but there are some interesting theorems on this subject that are analogous to results presented here.

The five chapters of the book are titled Initial Value Problems, Linear Differential Equations, Stability, Asymptotic Behaviour, and Boundedness. In addition, there is a nice appendix on the Routh-Hurwitz Problem. The first two chapters are a (good and somewhat different) treatment of the standard material on ordinary differential equations which is necessary to discuss stability and asymptotic behaviour. The Schauder-Tychonov theorem and the Contraction Principle are introduced early (Chapter 1) and are used throughout the book. Chapter III is especially noteworthy; it develops carefully the parallelism which exists among five types of stability-stability, asymptotic stability, uniform stability, uniform asymptotic stability, and strong stability. A good treatment is also given of conditional and orbital stability.

This is a very well written book which contains few errors (although, since it is a monograph it contains no exercises). This reviewer heartily recommends this book to any person who is interested in learning about stability and asymptotic behaviour of solutions of ordinary differential equations. While it may not always contain the most general (and most complicated) theorems known, it does provide a firm and readily understandable foundation on which to build.

**1.2. Review by: Jack H Hale.**

*Mathematical Reviews*MR0190463 (

**32 #7875**).

This book is a welcome addition to the existing literature on ordinary differential equations. It is a concise and clear presentation of those aspects of stability and asymptotic behaviour which can be easily discussed from the variation of constants formula and a few elementary concepts of functional analysis. Fortunately, results are not always presented in their most general form, yet the methods are exposed in such a clear fashion as to allow the reader to consult current literature without difficulty. This book could be used very easily as part of a graduate course in differential equations. The table of contents by chapter is (1) Initial value problems; (2) Linear differential equations; (3) Stability; (4) Asymptotic behavior; (5) Boundedness; Appendix: The Routh-Hurwitz problem.

**2. Disconjugacy (1971), by W A Coppel.**

**2.1. From the Preface.**

Disconjugacy has assumed growing significance in recent years and for this reason seemed a worthwhile topic for the study group in differential equations at the Australian National University. The group met almost once a week throughout 1970 and covered most aspects of the subject. I am grateful to the other regular participants. Dr A Howe. Mr G C O'Brien and Mr A N Stokes. for their collaboration and support. The notes were prepared to provide a permanent record of the lectures. They follow a logical rather than a chronological order. For their preparation I accept full responsibility. In particular the decision to omit special results for third and fourth order equations was mine. I hope that by making the notes available in their present form they may prove useful to a wider audience. I thank Professor P Hartman for his comments on the first draft of these notes. I also thank Mrs Barbara Geary for her careful typing of the manuscript. and Professor B H Neumann and Dr M F Newman for their assistance with the proof-reading.

**2.2. Review by: D V V Wend.**

*Mathematical Reviews*MR0460785 (

**57 #778**).

This book is an excellent introduction to disconjugacy of ordinary linear differential equations in the real domain. It is essentially self-contained: all definitions are given and the lemmas, theorems, and propositions are completely proved.

...

The development in Chapters 1 and 2 is somewhat parallel, and the author introduces the reader to a variety of techniques for studying disconjugacy such as the associated Riccati equation, Green's functions, quadratic functionals, and differential and integral inequalities. Many specific tests for disconjugacy and examples are included. The book ends with a good selection of about 80 relevant references and an index of terms and symbols. Especially valuable are the Notes, tucked away between the end of Chapter 3 and the References, which give the sources for the various results presented, including comments and historical remarks.

**3. Linear systems (1972), by W A Coppel.**

**3.1. Review by: Junji Kato.**

*Mathematical Reviews*MR0437142 (

**55 #10075**).

This is a course of lectures given at the 12th Summer Research Institute of the Australian Mathematical Society (January/February 1972). The course consists of eight lectures. The object of the author is to give a systematical view of a subject which electrical engineers call "mathematical systems theory". Owing to the limited time most of the proofs are omitted (the exact sources are quoted), but precise statements are given for the main results. By a linear system the author means a finite-dimensional system of linear differential equations $\dot{x}= A(t)x + B(t)u$ together with a finite system of linear algebraic equations $y = C(t)x + D(t)u$. Here $u \in \mathbb{R}^{m}$ is called the input, $x \in \mathbb{R}^{n}$ is called the state, and $y \in \mathbb{R}^{p}$ is called the output. The author is mainly interested in the input-output relations such as "controllability (Lecture 2)", "realizations (Lecture 3)", "linear-quadratic optimal control (Lecture 5)" and related topics: "matrix Riccati equations (Lecture 4)", "the Popov problem (Lecture 6)" and "the results of Jakubovič (Lectures 7, 8)".

**4. Dichotomies in stability theory (1978), by W A Coppel.**

**4.1. From the Preface.**

Several years ago I formed the view that dichotomies, rather than Lyapunov's characteristic exponents, are the key to questions of asymptotic behaviour for non- autonomous differential equations. I still hold that view, in spite of the fact that since then there have appeared many more papers and a book on characteristic exponents. On the other hand, there has recently been an important new development in the theory of dichotomies. Thus it seemed to me an appropriate time to give an accessible account of this attractive theory.

The present lecture notes are the basis for a course given at the University of Florence in May, 1977. I am grateful to Professor R. Conti for the invitation to visit there and for providing the incentive to put my thoughts in order. I am also grateful to Mrs Helen Daish and Mrs Linda Southwell for cheerfully and carefully typing the manuscript.

**4.2. Review by: Bucur B Ionescu.**

*Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série*

**24**(72) (1) (1980), 101.

The Author of the book has formed the view that dichotomies, rather than Lyapunov's characteristic exponents, are the key to questions of asymptotic behaviour for nonautonomous differential equations. The present volume is constituted from the following nine lectures : Stability, Exponential and Ordinary Dichotomy, Dichotomies and Functional Analysis, Roughness, Dichotomies and Reductibility, Criteria for an Exponential Dichotomy, Dichotomies and Lyapunov Functions, Equations on R and Almost Periodic Equations, Dichotomies and the Hull of an Equation. Finally, in an appendix, is presented The Method of Perron.

Tersely, the Author succeded to give an accessible account of this attractive theory.

**4.3. Review by: George R Sell.**

*Mathematical Reviews*MR0481196 (

**58 #1332**).

These notes are based on a series of lectures given by the author in 1977 at the University of Florence. The subject matter is concerned with the theory of linear differential equations (1) $x' = A(t)x$, which are defined for $t \in \mathbb{R}^{+} = [0, ∞]$, or $t \in \mathbb{R}$. The author concentrates on the interconnections between various uniform stability concepts and the theory of exponential and ordinary dichotomies. The underlying theme of these notes is based upon recent work of the author.

**5. A Chinese-English mathematics primer (1989), by W A Coppel.**

**5.1. From the Preface.**

My own interest in the work of Chinese mathematicians arises from their significant contributions to the qualitative theory of ordinary differential equations and, in particular, of plane quadratic systems. However, Chinese mathematics covers a very wide range. This is hardly surprising, since a quarter of the world's population is Chinese. It is predictable that during the next quarter century the importance of Chinese mathematics will increase greatly.

Although a number of Chinese mathematicians read English they may prefer to write in Chinese, either because writing is more difficult than reading or because they give the home market priority. Thus an ability to read Chinese mathematical papers will become increasingly useful.

Unfortunately the Chinese language presents special difficulties to a Western reader - even consulting a dictionary is a problem! The accompanying Word Lists and Index of characters are intended to alleviate these difficulties for those, like myself, who have neither the time nor the capacity to undertake a thorough study of the language.

The Word Lists contain around 1800 compound characters, together with their pinyin transcriptions (which determine their pronunciation) and English translations. These differ widely in their meanings (e.g. at, always, theorem, holomorphic), but could all be encountered in mathematical papers. They are listed according to some common feature (resp. prepositions, time, logical notions, function). It is believed that they will be more quickly memorised in this way than with the essentially random order of a dictionary. Moreover, in the case of technical terms, those in the same list will more frequently occur in the same paper. It will be observed that I have given some preference to my own mathematical interests. Those with different interests can easily supplement the lists here with their own specialised vocabulary by picking out the appropriate words from the English-Chinese Mathematics Vocabulary (Scientific Publishing House, Beijing, 1974). My intention has been to show the way, rather than to be encyclopaedic.

The compound characters have been chosen to introduce a broad range of simple characters in typical or significant situations. There is no attempt to be exhaustive, and there may be other ways of expressing the same English meaning. Compound characters are grouped, as far as possible, according to a common simple character. However, the simple character need not be the initial element of the compound character. This offers some advantage over the arrangement in a dictionary, since quite often the less common character appears first in a compound. Simple characters usually have several different meanings, even if they cannot express these meanings on their own. For those simple characters which head a group of compound characters the most relevant meanings are given here. For other meanings and other simple characters a dictionary should be consulted. Our aim is to be a supplement, rather than a substitute, for a dictionary.

English words also may have several meanings. In general the meaning of the English translation should be understood from the context. For example, in List 78 right means the opposite of left, not correct, and left does not mean remaining. In some cases additional help is provided. Thus in List 23 we have both opposite (facing), and opposite (contrary).

The Index contains all those simple characters (about 950) which appear in the lists of compound characters. It gives also their pinyin transcriptions and the numbers of the lists in which they appear. The number is in italics when the simple character appears on its own in the corresponding list. The simple characters are ordered according to the total number of strokes and, for characters with the same number of strokes, according to their radicals. The meaning of this is explained in the Introduction.

It will often be possible to bypass the Index by consulting instead the Key which precedes the word lists. For each list, except the first, this indicates most of the simple characters which head a group of compound characters in that list (and occasionally a simple character which does not). Although the key may look formidable at first sight, it gives an overview of the more important characters and will become more useful as the reader becomes familiar with them.

In addition, an Introduction to the Chinese Language has been provided. It gives not only an account of the Chinese script and its pinyin transliteration, but also a brief outline of Chinese grammar. It is hoped that, in spite of the inevitable omissions and oversimplifications, this will be of help to the prospective reader of Chinese mathematical articles. The illustrative examples have been chosen to equip such a reader with a useful starting vocabulary of characters. The material here could form the basis for a short course, with the students constructing additional examples drawn from their own areas of specialization.

The preceding remarks should make clear our purpose. Advice on errors and omissions, and other suggestions for improvement, will be welcome from those whose knowledge of Chinese is much greater than my own.

Acknowledgements. I have been helped in different ways and at different times by a number of people, including Hilary Chappell, Hongli Jia, Jong Li, David Kelly, Tzee-Char Kuo, Neville Smythe and the late Kurt Mahler. Neil Trudinger made available to me the facilities of the Centre for Mathematical Analysis. Special thanks are due to Joyce Heinz, who produced a remarkably accurate typescript without previous acquaintance with Chinese or a Macintosh. Yousong Luo and Xiaoji Wang assisted me with the proofreading.

**5.2. Review by: Perry Smith.**

*Mathematical Reviews*MR1019558 (

**91d:00001**).

This book, in 142 pages, does for Chinese what S H Gould's Russian for the mathematician (1972) did for Russian: it provides enough information on grammar, mathematical vocabulary, and example sentences to enable English-speaking mathematicians to tackle Chinese mathematical papers. ... About 950 characters are included; this is a large number for a beginning book, but falls well short of the 3000 or so needed for a basic Chinese vocabulary.

**6. Dynamics in one dimension (1992), by Louis Block and W A Coppel.**

**6.1. From the Publisher.**

The behaviour under iteration of unimodal maps of an interval, such as the logistic map, has recently attracted considerable attention. It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. The purpose of the book is to give a clear account of this subject, with complete proofs of many strong, general properties. In a number of cases these have previously been difficult of access. The analogous theory for maps of a circle is also surveyed. Although most of the results were unknown thirty years ago, the book will be intelligible to anyone who has mastered a first course in real analysis. Thus the book will be of use not only to students and researchers, but will also provide mathematicians generally with an understanding of how simple systems can exhibit chaotic behaviour.

**6.2. From the Preface.**

There has recently been an explosion of interest in one-dimensional dynamics. The extremely complicated - and yet orderly - behaviour exhibited by the logistic map, and by unimodal maps in general, has attracted particular attention. The ease with which such maps can be explored with a personal computer, or even with a pocket calculator, has certainly been a contributing factor. The unimodal case is extensively studied in the book of Collet and Eckmann, for example.

It is not so widely known that a substantial theory has by now been built up for arbitrary continuous maps of an interval. It is quite remarkable how many strong, general properties can be established, considering that such maps may be either real-analytic or nowhere differentiable. The purpose of the present book is to give a clear, connected account of this subject. Thus it updates and extends the survey article of Nitecki. The two books by Sarkovskii and his collaborators contain material on the same subject. However, they are at present available only in Russian and in general omit proofs. Here complete proofs are given. In many cases these have previously been difficult of access, and in some cases no complete proof has hitherto appeared in print.

Our standpoint is topological. We do not discuss questions of a measure-theoretical nature or connections with ergodic theory. This is not to imply that such matters are without interest, merely that they are outside our scope. [A forthcoming book by de Melo and van Strien discusses these matters, and also the theory of smooth maps.] The material here could indeed form the basis for a course in topological dynamics, with many of the general concepts of that subject appearing in a concrete situation and with much greater effect.

Several of the results included here were first established for piecewise monotone maps. There exist also other results which are valid for piecewise monotone maps, but which do not hold for arbitrary continuous maps. Although we include some results of this nature, we do not attempt to give a full account of the theory of piecewise monotone maps.

The final chapter of the book deals with extensions to maps of a circle of the preceding results for maps of an interval. In contrast to the earlier chapters, the results here are merely stated, with references to the literature for the proofs. [Complete proofs are given in a forthcoming book by Alsedà, Llibre and Misiurewicz, which also discusses the material in our Chapters 1.7 and 8.] We do not discuss at all some results which have been established for other one-dimensional structures. The pre-eminent importance of the interval and the circle appears to us adequate justification for our title. The list of references at the end of the book, although extensive, has no pretence to completeness.

This book has its origin in a course of lectures which the older author gave at the Australian National University in 1984. The first four chapters are based on the xeroxed notes for that course. However, the older author acknowledges that without the assistance of the younger author the book could never have reached its present greatly expanded form. We accept responsibility equally for the final product.

Our manuscript was originally submitted as a whole volume for the series Dynamics Reported. After its submission responsibility for publication of this series passed from Wiley and Teubner to Springer-Verlag. The resulting changes in format would not have presented insurmountable difficulties if the authors had been experts with TEX or LATEX. Since we were not, we decided instead to produce a good camera-ready manuscript, following the instructions to authors provided by Springer-Verlag for its Lecture Notes in Mathematics series. We are extremely grateful to the Managing Editors of Dynamics Reported, Professors U Kirchgraber and H O Walther, for the time and care they devoted to our manuscript, for obtaining valuable referees' reports, and finally for generously agreeing to its appearance in the Lecture Notes in Mathematics rather than in Dynamics Reported.

We thank Professor Xiong Jincheng for contributing some unpublished results (Propositions VI.53 and VI.54), and the referees for several useful suggestions. We take this opportunity to thank also the numerous typists who have assisted us over a period of eight years. W.A.C. is grateful to the University of Florida for support during a visit to Gainesville in 1987. L.S.B. would like to thank the Australian National University for its hospitality during visits in 1988 and 1990. These visits considerably accelerated progress on the book. L.S.B. also thanks the University of Göttingen for its hospitality during a visit in 1988, and Zbigniew Nitecki for helpful conversations during that visit. Finally he thanks Ethan Coven for many helpful conversations over the past few years.

We dedicate this book to our families, in gratitude for their support.

**6.3. Review by: Frederick R Marotto.**

*Mathematical Reviews*MR1176513 (

**93g:58091**).

This monograph brings together major developments of the past several decades concerning the dynamics of iterated continuous maps in one dimension. Unlike P Collet's and J-P Eckmann's book [Iterated maps on the interval as dynamical systems (1980)] in which smooth unimodal maps of the interval are the primary focus, this work investigates general one-dimensional endomorphisms that are not necessarily differentiable.

The approach is topological in nature, intentionally avoiding measure-theoretic and ergodic considerations. Even so, this work provides a useful summary of many dynamical properties that have been established for such maps in recent years by these and numerous other authors. Results are developed systematically, with all proofs included for the case of continuous maps of the interval, which comprises most of the book.

...

As the authors observe, this monograph grew out of and could be used as a text for a lecture course for which the prerequisite is real analysis. Results are for the most part coherently described and concisely proven. The work is more likely to be of use to researchers of one-dimensional dynamical systems. It is at present unique as a unified treatment of an area for which no comparable English language summary has heretofore appeared.

**7. Foundations of convex geometry (1998), by W A Coppel.**

**7.1. From the Publisher.**

This book on the foundations of Euclidean geometry aims to present the subject from the point of view of present day mathematics, taking advantage of all the developments since the appearance of Hilbert's classic work. Here real affine space is characterised by a small number of axioms involving points and line segments making the treatment self-contained and thorough, many results being established under weaker hypotheses than usual. The treatment should be totally accessible for final year undergraduates and graduate students, and can also serve as an introduction to other areas of mathematics such as matroids and antimatroids, combinatorial convexity, the theory of polytopes, projective geometry and functional analysis.

**7.2. From the Preface.**

Euclid's Elements held sway in the mathematical world for more than two thousand years. In the nineteenth century, however, the growing demand for rigour led to a re-examination of the Euclidean edifice and the realisation that there were cracks in it. The critical reappraisal which followed was synthesized by Hilbert (1899), in his

*Grundlagen der Geometrie*, which has in turn held sway for almost a century.

In what ways may Hilbert's treatment be improved upon today? Apart from technical improvements, a number of which were included in later editions of his book, it may be argued that Hilbert followed Euclid too closely. The number of undefined concepts is unnecessarily large and the axioms of congruence sit uneasily with the other axioms. In addition, the restriction to three-dimensional space conceals the generality of the results and seems artificial now.

For these reasons Hilbert's approach is often replaced today by a purely algebraic one - the axioms for a vector space over an arbitrary field, followed by the axioms for a vector space over the real field with a positive definite scalar product. This permits a rapid development, but it simply begs the question of why it is possible to introduce coordinates on a line. Only when one attempts to answer this question does one realise that many important results in no way depend on the introduction of coordinates. It is curious that some powerful advocates of 'coordinate-free' linear algebra have used real analysis to prove purely geometric results, such as the Hahn-Banach theorem, without any apparent twinge of conscience.

A system of axioms for Euclidean geometry in which the only undefined concepts are

*point*and

*segment*was already given before Hilbert by Peano. We follow his example in the present work, although some of our axioms differ. The choice of a system of axioms is inherently arbitrary, since there will be many equivalent systems. However, the purpose of an axiom system is not only to provide a basis for rigorous proof, but also to reveal the structure of a subject. From this point of view one axiom system may seem preferable to another which is equivalent to it. The development should seem natural, almost inevitable.

Considerations of this nature have led us to isolate a basic structure, here called a

*convex geometry*, which is defined by two axioms only. Additional axioms are chosen so that each individually, in conjunction with these, guarantees some important property. Four such additional axioms define another basic structure, here called a

*linear geometry*, for which a rich theory may be developed. Examples of linear geometries, in addition to Euclidean space, are hyperbolic space and (hemi)spherical space, where a 'segment' is the geodesic arc joining two points.

Apart from a dimensionality axiom, only seven axioms in all are needed to characterise ordinary Euclidean or, more strictly,

*real affine*space. However, dimensionality plays a role in achieving this number, by eliminating some other possibilities in one or two dimensions. Although the characterisation of Euclidean space may be regarded as our ultimate goal, it would be contrary to our purpose to impose all the axioms from the outset. Instead we adjoin axioms successively, so that results are proved under minimal hypotheses and may be applied in other situations, which are of interest in their own right. For example. Proposition III.17 establishes the theorems of Helly and Radon in any linear geometry, and Chapter IV similarly extends the facial theory of polytopes. In this way each result appears, so to speak, in its 'proper place'. When there is a branching of paths, we choose the one which leads to our ultimate goal. This approach, I believe, has not previously been pursued in such a systematic manner. I have found it illuminating myself and hope that the illumination succeeds in shining through the present account.

We have spoken of the characterisation of Euclidean space as our ultimate goal, but we are actually concerned with characterising only a convex subset of Euclidean space. The greater freedom of the whole Euclidean space is mathematically desirable, but it is reasonable to require only that our physical world may be embedded in such a space. This type of non-Euclidean geometry was already mentioned by Klein (1873) and was further considered by Schur (1909). In view of the great historical importance of the parallel axiom, it is of interest that it can play no role here, since it fails to hold in any proper convex subset of Euclidean space (except a lower-dimensional Euclidean space).

The foundations of geometry have been studied for thousands of years. and thousands of papers have been written on the subject. Since it is impossible to do justice to all previous contributions in such a situation, we have deliberately restricted the number of references. Thus the absence of a reference does not necessarily imply ignorance of its existence and is not a judgement on its quality. Some references are included simply to provide a time scale and others for their useful bibliographies. Our task has been to select and organise, and sometimes extend. Some open problems are mentioned at me ends of Chapters III, IV and VIII.

The topic of axiomatic convexity has been omitted from the recent extremely valuable

*Handbook of convex geometry*(P M Gruber and J M Wills (eds.), 1993), and ordered geometry receives only passing reference (on p. 1311) in the equally valuable

*Handbook of incidence geometry*(F Buekenhout (ed.), 1995). It is hoped that the present work may in some measure repair these omissions. Our aim has been to give a connected account of these subjects. which may be read without reference to other sources. Since many results are established under weaker hypotheses than usual, rather detailed proofs have been given in some cases.

The work is not arranged as a textbook, with starred sections and exercises, and is perhaps more difficult than the usual final-year undergraduate or first-year graduate course. However, the mathematical prerequisites are no greater and I believe that an interesting course could be constructed from the material here. which would acquaint students with a cross-section of mathematics in contrast to the usual compartmentalised course.

Familiarity is assumed with the usual language and notation of set theory, with such algebraic concepts as group, field and vector space, and with Dedekind's construction of the real numbers from the rationals. Concepts from other areas, such as partially ordered sets and lattices, projective geometry, topology and metric spaces, are defined in the text. A few elementary properties of metric spaces are stated without proof. On the two or three occasions when appeal is made to some other result the subsequent development is not at stake. Although the work is essentially self-contained, the reader is still encouraged to consult other treatments, both in the references cited in the notes at the end of each chapter and in the introductory chapters of more specialised works on convexity theory, such as Bonnesen and Fencbel (1934), Valentine (1964), Leichtweiss (1980) and Schneider (1993). The remaining prerequisites for the present work are more substantial - the ability and resolve to follow a detailed logical argument and a love of mathematics.

For assistance in various ways I thank M Albert, B Davey, V Klee, J Reay, J Schäffer, V. Soltan and H Tverberg. I am especially grateful to H Tverberg for the detection of errors, misprints and obscurities in the original manuscript. The exposition and index have been improved by suggestions from the referees, and the appearance by suggestions from the copy-editor.

As. I write these lines on the eve of my retirement from paid employment in the Institute of Advanced Studies at the Australian National University, I take the opportunity to acknowledge that this book could not have been written without the privileged working conditions which I have enjoyed.

**7.3. Review by: Zbynek Nádenik.**

*Mathematical Reviews*MR1629043 (

**99f:52001**).

This very interesting and, for graduate students, very useful book can be characterised as an axiomatic approach to convex geometry. Chapter I, Alignments, contains the definition of this notion as a collection of subsets with certain properties. Chapter II, Convexity, issues from the definition of the convex geometry on a set X if two axioms for the elements of X are satisfied. Chapters III and IV, Linearity, introduce the notion of a linear geometry and establish the theorems of Helly, Radon and Carathéodory in a linear geometry and the generalization of the theorem of Carathéodory by I. Bárány. Also, polytopes are studied. Chapter V, Density and unendingness, contains two new axioms concerning these notions. Their study is applied to cones and polyhedra. Chapter VI, Desargues, is devoted to projective geometry and the Desargues property. Chapter VII, Vector spaces, introduces coordinates, defines isomorphism of linear geometries and indicates the result of J P Doignon concerning the embedding of a dense linear geometry of dimension ≥ 2 with the Desargues property in a projective space. Chapter VIII, Completeness, begins with the axiom of completeness as an analogy for linear geometries to the Dedekind property for the real line. Chapter IX is devoted to spaces of convex sets. The chapters are provided with historical remarks.by

**8. Number Theory: An Introduction to Mathematics: 2nd edition (2009), by W A Coppel.**

**8.1. From the Publisher.**

Undergraduate courses in mathematics are commonly of two types. On the one hand are courses in subjects - such as linear algebra or real analysis - with which it is considered that every student of mathematics should be acquainted. On the other hand are courses given by lecturers in their own areas of specialization, which are intended to serve as a preparation for research. But after taking courses of only these two types, students might not perceive the sometimes surprising interrelationships and analogies between different branches of mathematics, and students who do not go on to become professional mathematicians might never gain a clear understanding of the nature and extent of mathematics. The two-volume "Number Theory: An Introduction to Mathematics" attempts to provide such an understanding of the nature and extent of mathematics. It is a modern introduction to the theory of numbers, emphasizing its connections with other branches of mathematics. Part A, which should be accessible to a first-year undergraduate, deals with elementary number theory. Part B is more advanced than the first and should give the reader some idea of the scope of mathematics today. The connecting theme is the theory of numbers. By exploring its many connections with other branches, we may obtain a broad picture.

**8.2. From the Preface.**

Undergraduate courses in mathematics are commonly of two types. On the one hand there are courses in subjects, such as linear algebra or real analysis, with which it is considered that every student of mathematics should be acquainted. On the other hand there are courses given by lecturers in their own areas of specialisation, which are intended to serve as a preparation for research. There are, I believe, several reasons why students need more than this.

First, although the vast extent of mathematics today makes it impossible for any individual to have a deep knowledge of more than a small part, it is important to have some understanding and appreciation of the work of others. Indeed the sometimes surprising interrelationships and analogies between different branches of mathematics are both the basis for many of its applications and the stimulus for further development. Secondly, different branches of mathematics appeal in different ways and require different talents. It is unlikely that all students at one university will have the same interests and aptitudes as their lecturers. Rather, they will only discover what their own interests and aptitudes are by being exposed to a broader range. Thirdly, many students of mathematics will become, not professional mathematicians, but scientists, engineers or schoolteachers. It is useful for them to have a clear understanding of the nature and extent of mathematics, and it is in the interests of mathematicians that there should be a body of people in the community who have this understanding.

The present book attempts to provide such an understanding of the nature and extent of mathematics. The connecting theme is the theory of numbers, at first sight one of the most abstruse and irrelevant branches of mathematics. Yet by exploring its many connections with other branches, we may obtain a broad picture. The topics chosen are not trivial and demand some effort on the part of the reader. As Euclid already said, there is no royal road. In general I have concentrated attention on those hard-won results which illuminate a wide area. If I am accused of picking the eyes out of some subjects, I have no defence except to say "But what beautiful eyes!"

The book is divided into two parts. Part A, which deals with elementary number theory, should be accessible to a first-year undergraduate. To provide a foundation for subsequent work, Chapter I contains the definitions and basic properties of various mathematical structures. However, the reader may simply skim through this chapter and refer back to it later as required. Chapter V, on Hadamard's determinant problem, shows that elementary number theory may have unexpected applications.

Part B, which is more advanced. is intended to provide an undergraduate with some idea of the scope of mathematics today. The chapters in this part are largely independent, except that Chapter X depends on Chapter IX and Chapter XIII on Chapter XII.

Although much of the content of the book is common to any introductory work on number theory, I wish to draw attention to the discussion here of quadratic fields and elliptic curves. These are quite special cases of algebraic number fields and algebraic curves, and it may be asked why one should restrict attention to these special cases when the general cases are now well understood and may even be developed in parallel. My answers are as follows. First, to treat the general cases in full rigour requires a commitment of time which many will be unable to afford. Secondly, these special cases are those most commonly encountered and more constructive methods are available for them than for the general cases. There is yet another reason. Sometimes in mathematics a generalisation is so simple and far-reaching that the special case is more fully understood as an instance of the generalisation. For the topics mentioned, however, the generalisation is more complex and is, in my view, more fully understood as a development from the special case.

At the end of each chapter of the book I have added a list of selected references, which will enable readers to travel further in their own chosen directions. Since the literature is voluminous, any such selection must be somewhat arbitrary, but l hope that mine may be found interesting and useful.

The computer revolution has made possible calculations on a scale and with a speed undreamt of a century ago. One consequence has been a considerable increase in 'experimental mathematics' - the search for patterns. This book, on the other hand, is devoted to 'theoretical mathematics ' - the explanation of patterns. I do not wish to conceal the fact that the former usually precedes the latter. Nor do I wish to conceal the fact that some of the results here have been proved by the greatest minds of the past only after years of labour, and that their proofs have later been improved and simplified by many other mathematicians. Once obtained, however, a good proof organises and provides understanding for a mass of computational data. Often it also suggests further developments.

The present book may indeed be viewed as a 'treasury of proofs'. We concentrate attention on this aspect of mathematics, not only because it is a distinctive feature of the subject, but also because we consider its exposition is better suited to a book than to a blackboard or a computer screen. In keeping with this approach, the proofs themselves have been chosen with some care and I hope that a few may be of interest even to those who are no longer students. Proofs which depend on general principles have been given preference over proofs which offer no particular insight.

Mathematics is a part of civilisation and an achievement in which human beings may take some pride. It is not the possession of any one national, political or religious group and any attempt to make it so is ultimately destructive. At the present time there are strong pressures to make academic studies more 'relevant'. At the same time, however, staff at some universities are assessed by 'citation counts' and people are paid for giving lectures on chaos, for example, that are demonstrably rubbish.

The theory of numbers provides ample evidence that topics pursued for their own intrinsic interest can later find significant applications. I do not contend that curiosity has been the only driving force. More mundane motives, such as ambition or the necessity of earning a living, have also played a role. It is also true that mathematics pursued for the sake of applications has been of benefit to subjects such as number theory: there is a two-way trade. However, it shows a dangerous ignorance of history and of human nature to promote utility at the expense of spirit. This book has its origin in a course of lectures which I gave at the Victoria University of Wellington, New Zealand, in 1975. The demands of my own research have hitherto prevented me from completing it, although I have continued to collect material. If it succeeds at all in conveying some idea of the power and beauty of mathematics, the labour of writing it will have been well worthwhile.

As with a previous book, I have to thank Helge Tverberg, who has read most of the manuscript and made many useful suggestions.

In this revised edition of my book, the original edition of which appeared in 2002, I have removed an error in the statement and proof of Proposition 11.1 2 and filled a gap in the proof of Proposition II.12. The statements of the Weil conjectures in Chapter IX and of a result of Heath-Brown in Chapter X have been modified, following comments by J-P Serre. I have also corrected a few misprints, made many small expository changes and expanded the index.

**8.3. Review by: Władysław Narkiewicz.**

*Mathematical Reviews*MR1923251 (

**2003f:11001**).

The author writes: "Many students of mathematics will become, not professional mathematicians, but scientists, engineers or schoolteachers. It is useful for them to have a clear understanding of the nature and extent of mathematics. ...The present book attempts to provide such an understanding. ... The connecting theme is the theory of numbers ..." The book starts with a quick introduction to basic mathematical structures, like sets, mappings, integers, rationals and reals, complex numbers, metric spaces, groups, rings, fields and vector spaces. Then the main notions and results of elementary number theory (divisibility, congruences, quadratic reciprocity, linear Diophantine equations) and quadratic fields are presented. There follows a chapter on continued fractions with applications to quadratic Diophantine equations. The modular group is introduced and its relation to non-Euclidean geometry pointed out. In the next chapter the Hadamard matrices are considered, with applications to the study of designs and codes. The final chapter of part A is devoted to $p$-adic numbers.

Part B starts with an introduction to the theory of quadratic spaces, culminating with the proof of the Hasse-Minkowski theorem. The main results of the geometry of numbers follow, with Minkowski's convex body theorem, and Mahler's compactness theorem. The chapter on prime numbers contains Chebyshev's inequalities, the prime number theorem (proved via Ikehara's theorem) and a discussion of Riemann's Hypothesis. The proof of Dirichlet's theorem on primes in progression follows, and then linear representations of finite groups are introduced and their main properties established. A chapter on uniform distribution and ergodic theory follows, containing Weyl's criterion, Birkhoff's ergodic theorem and Poincaré's recurrence theorem. As an application van der Waerden's theorem on partitions of the set of integers is proved. The penultimate chapter brings an introduction to elliptic functions and the book is concluded with several arithmetical results: We find there Jacobi's four squares theorem, Rogers-Ramanujan identities and elliptic curves with a proof of the theorem of Mordell. The main text is accompanied by comments in which further development of the theory is presented and references to further reading are provided. The book is a welcome addition to the literature, but it is difficult to agree with the author when he writes that it is suitable for first-year undergraduates, as in several places it presupposes the knowledge of calculus.

Last Updated April 2024