The Mathematical Gazette Sir Wifred H Cockcroft 1923-1999 Volume 84: Number 499 THE MATHEMATICAL ASSOCIATION March 2000 £16.00 CONTENTS (continued) THE MATHEMATICAL ASSOCIATION Notes 84.01 to 84.28 (continued) AN ASSOCIATION OF TEACHERS AND STUDENTS OF ELEMENTARY MATHEMATICS Unexpected symmetry in a derived Fibonacci sequence Alexander J Gray 87 A recurrence relation among Fibonacci sums Alexander J Gray 89 Some unusual iterations Mark Thornber 90 When the sum equals the product Leo Kurlandchik and Andrzej Nowicki 91 Never say never: some mistaken identities Mark J Cooker 94 A curious property of the integer 24 M H Eggar 96 What cycles of a given length generate? Mowaffaq Hajja 97 A game with positive and negative numbers M H Eggar 98 An inductive proof of the arithmetic mean − geometric mean inequality Zbigniew Urmanin 101 Weighted mean in a trapezium Larry Hoehn 102 (Chair) A formula for integrating inverse functions S Schnell and C Mendoza 103 (Chair) Mathematician versus machine P Glaister 105 On a conjecture of Paul Thompson Tim Jameson 107 (Chair) Maximal volume of curved folding boxes Kenzi Odani 110 (Chair) More on a sine product formula Walther Janous and Jeremy King 113 On a limit for prime numbers J A Scott 115 SHM and projections P Glaister 116 Another cautionary chi-square calculation Nick Lord 119 More on dual Van Aubel generalisations Michael de Villiers 121 Obituary Sir Wilfred Cockcroft 1923-1999 Peter Reynolds 123 'I hold every man a debtor to his profession, from the which as men of course seek to receive countenance and profit, so ought they of duty to endeavour themselves by way of amends to be a help and an ornament thereunto.' BACON THE COUNCIL PRESIDENT Professor John Berry IMMEDIATE PAST PRESIDENT PRESIDENT DESIGNATE CHAIR OF COUNCIL SECRETARY TREASURER REPRESENTATIVES BRANCHES OF COMMITTEES CONFERENCES PROFESSIONAL DEVELOPMENT PUBLICATIONS PUBLICITY & MEMBERSHIP TEACHING COMMITTEE EDITOR IN CHIEF Mr Bill Richardson MEMBERS WITHOUT OFFICE Mr Robert Barbour Mr Neil Bibby Mr David Carter Miss Susie Jameson Dr Jim Message OFFICE MANAGER Professor Chris Robson Mr Steve Abbott Dr Sue Sanders Mr Roy Ashley Mr Paul Metcalf Mr Bob Francis Mr Martin Bailey Ms Sue Jennings Mr Peter Bailey Mr David Hodgson Mr Doug French (Chair) (Chair) Ms Trish Morgan Mr Michael Mudge Ms Robyn Pickles Mr Tony Robin Mrs Marcia Murray EDITORIAL COMMITTEE OF THE MATHEMATICAL GAZETTE Editor Mr Steve Abbott Production Editor Mr Bill Richardson Reviews Editors Mr Bud Winteridge Mrs Rosalie McCrossan Problems Editors Mr Graham Hoare Mr Tim Cross Assistant Editor Mr Gerry Leversha Correspondence 125 Notices 127 Problem corner G T Q Hoare 128 Student problems Tim Cross 135 Other Journals Anne C Baker 139 Book Reviews 140 © The Mathematical Association 2000 CONTENTS Editorial The Mathematical Gazette One hundred years on Graham T Q Hoare Lewis Carroll − mathematician and teacher of children Canon D B Eperson Snubbing with and without eta H Martyn Cundy 14 The Fermat-Torricelli points of n lines Roy Barbara 24 Continued fractions Robert Macmillan 30 A construction of magic cubes Marián Trenkler 36 The factorial function: Stirling's formula David Fowler 42 A simple energy-conserving model Richard Bridges 51 The Hale-Bopp comet explored with A level mathematics H R Corbishley 58 THE MATHEMATICAL GAZETTE Articles Notes 84.01 to 84.28 Circumradius of a cyclic quadrilateral Larry Hoehn 69 A neglected Pythagorean-like formula Larry Hoehn 71 An unexpected reduced cubic equation J A Scott 74 Touching hyperspheres D F Lawden 75 Comments on note 82.53—a generalised test for divisibility Andrejs Dunkels 79 A matrix method for a system of linear Diophantine equations A J B Ward 81 On the application of Whittaker's theorem J A Scott 84 Digital roots and reciprocals of primes Alexander J Gray 86 (The contents are continued inside the back cover.) Printed in Great Britain by J W Arrowsmith Ltd ISSN 0025-5572 Vol 84 No 499 66 MARCH 2000 A portrayal of right-angled triangles which I Grattan-Guinness generate rectangles with sides in integral ratio Sir Wifred H Cockcroft 1923-1999 Volume 84: Number 499 THE MATHEMATICAL ASSOCIATION March 2000 £16.00 The Mathematical Gazette A JOURNAL OF THE MATHEMATICAL ASSOCIATION Vol 84 March 2000 No 499 Editorial: It's voting time again! The time has come to vote for the Fifth Annual Mathematical Gazette Writing Awards Please use the address carrier from this issue of the Gazette to identify the articles and notes of 1999 that impressed you most The Index for 1999 will remind you of the many good submissions There will again be a prize draw among those who respond The prize, worth about £30, will be a copy of the book Mathematics: frontiers and perspectives, edited by Vladimir Arnold, Michael Atiyah, Peter Lax and Barry Mazur (AMS, 2000) Previous Annual Mathematical Gazette Writing Awards Year Best Article Best Note 1996 Colin Fletcher Two prime centenaries David Fowler A simple approach to the factorial function Ann Hirst and Keith Lloyd Colin Dixon Cassini, his ovals and a space probe to Saturn Geometry and the cosine rule Robert M Young Robert J Clarke Probability, pi, and the primes The quadratic equation formula 1997 1998 Please indicate, in the spaces provided on the voting form, the titles of your favourite Articles and your favourite Notes Note that Matters for Debate count as Articles Alternatively, you can just write your choices in a letter or on a postcard Each vote will be given equal weighting The results will be announced in the July 2000 issue Return the form as soon as you can, and definitely by 31st May 2000 to: Gazette Poll, 91 High Road West, Felixstowe IP11 9AB, UK STEVE ABBOTT THE MATHEMATICAL GAZETTE One hundred years on GRAHAM T Q HOARE David Hilbert, one of the giants of mathematics, delivered a lecture at the International Congress of Mathematics at Paris in 1900 The first part of the lecture, a preamble to his announcement of the now-famous 23 problems, began with the words: ‘Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will thrive? What new methods and new facts in the wide and rich fields of mathematical thought will the new centuries disclose? History teaches the continuity of the development of science We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development Just as every human undertaking pursues certain objects, so also mathematical research requires its problems It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.’ Later we find the oft-quoted passage: ‘This conviction of the solvability of every mathematical problem is a powerful incentive to the worker We hear within us the perpetual call: There is the problem Seek its solution You can find it by pure reason, for in mathematics there is no ignorabimus.’ Hilbert considered that the 23 problems he had chosen were those most likely to stimulate important new advances in mathematics It redounds to his perspicacity that much fruitful mathematical activity resulted in addressing these problems in the twentieth century As we shall see from the list below, which we give together with short commentaries and notes, ONE HUNDRED YEARS ON the so-called problems vary from specific mathematical questions to programmes of research Some have been reformulated or extended without losing their identity We note the importance Hilbert attached to algebraic number theory, since the 8th problem, partly, and the 9th, 11th and 12th, entirely, are devoted to it Problems 1, and 10 belong to mathematical logic, whereas 6, 19, 20 and 23 fall within the provinces of applications Observe too that topology, then at an early stage of its development, features strongly Readers will appreciate that we cannot justice to Hilbert's vision in a short article such as this Hilbert's Problems Cantor's continuum hypothesis (CH) and well-ordering 1(a) Is 2¼0 = 1? Undecidable Assuming the consistency of the Zermelo-Fraenkel axioms for set theory (ZF), the work of K Gödel (1938) and P Cohen (1963) established that both the statement of the hypothesis (that 2¼0 = 1) and its negation are consistent with ZF Thus the hypothesis is completely independent of the axioms of set theory 1(b) Hilbert also asked whether the continuum of numbers can be wellordered This problem is related to the Axiom of Choice (AC), but in 1963 P Cohen proved the independence of AC from the other axioms of set theory, so the problem remains unresolved Note: Gödel believed that the AC and the CH were either true or false and that ZF did not encapsulate what was ‘obviously’ true about set theory The task was to think of some new axiom which would determine AC and CH He did not succeed in devising such an axiom (the existence of measurable cardinals was proposed as such, but was not in any sense ‘obvious’) so this remains an unresolved consequence of the Hilbert challenge ¼ ¼ To establish the consistency of the axioms of arithmetic Gödel's two theorems shattered the Hilbert programme The second of these proves that the consistency of a theory at least as strong as arithmetic cannot be proved within the theory To show, using only the congruence axioms, whether two tetrahedra having the same altitude and base area have the same volume Proved false by M Dehn (1900) To investigate geometries (metrics) in which the line segment between any pair of points gives the shortest path between the pair (geodesic) Considered too vague Can the assumption of differentiability for the functions defining a continuous transformation group be avoided? Reformulated to encompass a larger domain of topological groups, the problem was solved in the form that a locally Euclidean topological group is THE MATHEMATICAL GAZETTE a Lie group by A Gleason (1952) and by D Montgomery and L Zippin (1955) Note: If each point of a topological group G has a neighbourhood homeomorphic to an open set of a given Euclidean space, then G is called a locally Euclidean group If the underlying topological space of a topological group has the structure of a real analytic manifold, where the group operations (x, y) → xy, x → x−1 are real analytic mappings, then G is a Lie group S Lie envisaged an approach to solving partial differential equations analogous to Galois' group-theoretic resolution of algebraic equations The mathematical axiomatisation of physics Hilbert considered that physics was too difficult to be left to physicists Progress has been minimal, not least because the meaning of is unclear Again, Hilbert could not have foreseen the many developments in 20th century physics We can record, however, that the axiomatisation of probability theory was accomplished by A Kolmogorov and that of quantum physics by A Wightman To establish the transcendence of certain numbers The following generalisation of Lindemann's theorem was conjectured by A O Gelfond (1929) and proved by A Baker (1966) If α1, α2, … , αr, β1, β2, … βr are non-zero algebraic numbers such that ln α1, ln α2, … , ln αr are linearly independent over the rationals then β1 ln α1 + β2 ln α2 + … +βr ln αr ≠ A special case of this, found independently by Gelfond and T Schneider (1934), which answers Hilbert's enquiry about the nature of 2, states that if α is an algebraic number ≠ 0, and β is an irrational number, then αβ is a transcendental number To investigate problems concerning the distribution of prime numbers; in particular, to show the correctness of the Riemann hypothesis Tantalisingly, the Riemann hypothesis evades resolution ∞ Note: The Riemann zeta function is defined by ζ (s) = ∑1 n−s for s = σ + iτ ∈ c and σ > This converges when σ > 1, and can be continued to all of c by a formula giving ζ (1 − s) in terms of ζ (s) The Riemann hypothesis states that the non-trivial roots of the Riemann zeta function all lie on the line σ = 12 Riemann had already noted that, if ζ (s) = 0, then ≤ Re (s) ≤ He believed, for example, that a proof of the hypothesis might establish the existence of an infinity of twin primes To find the most general law of reciprocity in an algebraic number field Hilbert contributed to this, but it was E Artin (1927) who established it for Abelian extensions of q; the non-Abelian case is still open ONE HUNDRED YEARS ON Note: The quadratic reciprocity law state that if p, q are different odd primes then p q = (−1)(p − 1)(q − 1)/4 , q p a where , Legendre's symbol, is defined for any integer a and any odd p prime p as ()() () if x2 ≡ a (mod p) is solvable for x a = −1 if x2 ≡ a (mod p) is not solvable for x p if a ≡ (mod p) Gauss was the first to solve the quadratic and cubic reciprocity laws () 10 To find an algorithm for deciding whether any given Diophantine equation has a solution Following pioneering work by M Davis, H Putnam and J Robinson, the problem was finally solved, negatively, by Y (1970) Matijaseviè 11 To investigate the theory of quadratic forms over an arbitrary algebraic number field of finite degree H Hasse (1929) and C L Siegel (1936, 1951) obtained important results A Weil and T Ono (1964-1965) demonstrated a connexion between the problem and algebraic groups Generally, still incomplete 12 Extension of Kronecker's theorem on Abelian fields to an arbitrary algebraic field Poorly posed by Hilbert, the problem was corrected and solved by T Takagi In 1922 he proved the following fundamental theorem: every Abelian extension of an algebraic number field F is a class field for the field (corresponding to a congruence class group in F) and, conversely, every class field E of F is an Abelian extension of F Note: Given a group G of automorphisms of a given field L, and K a subfield of L, the group consisting of all automorphisms of L leaving every element of K invariant is denoted by G (L / K) A Galois extension is called an Abelian extension when G (L / K) is Abelian Kronecker's theorem states that cyclotomic fields are Abelian extensions of q and, conversely, every Abelian extension of q is a subfield of a cyclotomic field The problem is related to finding functions which, for an arbitrary field, play the same role as the exponential function for the rational field and elliptic modular functions for imaginary quadratic fields 13 To show the impossibility of the solution of the general algebraic equation of the 7th degree by compositions of continuous functions of two variables THE MATHEMATICAL GAZETTE Solved by V I Arnol'd (1957) for continuous functions; still unsolved if analyticity is required Note: We mention, in passing, the beautiful results of Kolmogorov and Arnol'd that arbitrary real-valued continuous functions of any number of variables can be represented exactly as compositions of a finite number of such functions of only two variables 14 To consider invariants which arise when only the transformations of a subgroup of the totality of linear transformations, the projective linear group, are permitted By producing a counter-example, M Nagata (1958) showed that the invariants need not be finitely generated Note: An invariant is a mathematical object which remains unchanged under certain kinds of transformation Recently there has been renewed activity in invariant theory; it has widened its scope and has entered the realm of abstract algebra Indeed Problem 14, in algebraic language, can be rendered as: Given fields k , k (x1, … , xn), and K , where k ⊆ K ⊆ k (x1, … , xn), the problem is to determine whether the ring K ∩ k [ x1, … , xn] , is finitely generated over K Here k (x1, … , xn) is the field of rational functions in (x1, … , xn) with coefficients in k, and k [ x1, … , xn] is the ring of polynomials with coefficients in k 15 To establish the foundations of algebraic geometry, in particular, H Schubert's enumerative calculus Solved by B L van der Waerden (1938-1940), A Weil (1950) and others In the late 1950s and 1960s, A Grothendieck rewrote the foundations of algebraic geometry after Weil Note: Algebraic geometry is the study of algebraic curves, algebraic varieties and their generalisations to n dimensions Suppose V is an ndimensional vector space with scalars in some field F If W is a subset of V composed of all points (x1, … , xn) which satisfy each of a set of polynomial equations {pi (x1, … , xn) = 0}, i ∈ z+, with coefficients in F, then W is an algebraic variety Originally, enumerative calculus was developed for counting the number of curves touching a given set of curves, and enumerative geometry refers to Schubert's application of the conservation of number principle [1] 16 To study the topology of real algebraic curves and surfaces Sporadic results 17 Suppose f (x1, … , xn) is a rational function with real coefficients that takes a positive value for any n-tuple (x1, … , xn) The problem is to determine whether the function f can be written as a sum of squares of rational functions ONE HUNDRED YEARS ON Solved, affirmatively, by Artin (1926-1927) for real closed fields In 1967 DuBois gave a negative solution to the general case In the same year Pfister gave the number of squares required 18 To investigate the existence of non-regular space-filling polyhedra K Reinhardt (1928), a student of Hilbert, showed that such a ‘tiling’ exists In 1910, L Bieberbach proved that, up to equivalence, there are only finitely many n-dimensional crystallographic groups 19 To determine whether the solution of regular problems in the calculus of variations are necessarily analytic Solved by S Bernstein, I G Petrovskii, and others 20 To investigate the existence of solutions of partial differential equations with prescribed boundary conditions Hilbert contributed here by resurrecting Dirichlet's problem; a vast amount of work has been done in this area pre- and post-Hilbert Note: This ‘problem’ is closely linked to the 19th ∂ 2u A typical boundary problem takes the form − ∇ u = f in some ∂t ∂u region R, with u (0, t) = u1 and (0, t) = u2 on the boundary of R ∂t An elliptic partial differential equation, for example, is a real 2nd order partial differential equation of the form: n ∂ 2u ∑ aij ∂ xi∂ xj ( + F x1, … , xn, u, i,j = ) ∂u ∂u ,…, = ∂ x1 ∂ xn n such that the quadratic form ∑ aijxixj is non-singular and positive definite i, j = Typical examples are the Laplace (Dirichlet's problem) and Poisson equations We might, in passing, mention the link with potential theory 21 To show that there always exists a linear differential equation of the Fuchsian class with given singular points and monodromy group Several special cases have been solved, for example by H Röhrl (1957) and P Deligne (1970), but a negative solution was found by B Bolibruch (1989) Note: The first indication of a deep relationship between groups and differential equations emerged in Riemann's investigation of the hypergeometric differential equation, which belongs to class of equations of Fuschian type As it is linear, and of second order, its solutions are expressible as a sum of basic solutions, the analytic continuation of which around each singular point gives rise to more branches of the solution that 178 THE MATHEMATICAL GAZETTE Sadly Rota died on 19 April 1999, but he would have seen this work under review written by one of his former students at MIT Indeed Peter Olver paid due respect to his former teacher and the ‘wonderful lectures [which] opened my vistas’ (p xxi) Invariant theory aims to bring out intrinsic properties In the simple case of polynomials of one variable there would be no point in treating say, x3 + 2x − and 8x3 − 12x2 + 10x − separately, since one is a linear transformation of the other: they are therefore equivalent in an algebraic sense In the language of invariant theory both have the same set of invariants and covariants In practice invariant theory does not deal with numerical polynomials but with symbolic ones The simplest example of an invariant is the discriminant b2 − 4ac for the ordinary quadratic polynomial ax2 + bx + c In projective geometry, invariant theory is concerned with the intrinsic properties which not depend on choice of coordinate axes The history of invariant theory reminds us that mathematics is subject to the vagaries of fads and fashions just as any other human enterprise While a topic in mathematics comes back into fashion at intervals it invariably comes back wearing different clothes Modern invariant theory hardly follows the lines of Cayley’s and Sylvester’s research programme Rota erected an abstract scheme for the theory in a way never dreamed of by the pioneers of the 1840s and 1850s though his work built on theirs In any explanation of invariant theory, it is significant that the common discriminant b2 − 4ac of the quadratic remains the standard exemplar for all the algebraic versions of the subject which have arisen The newness of present day modern invariant theory is due to the central role which group theory plays The present book brings much of classical invariant theory up-to-date Group theory is introduced early, and there is the recognition of the part David Hilbert played in the 1880s However, the modern theory is far from being a continuation from the point at which Hilbert left off To be sure, Hilbert’s Basis Theorem, the Syzygy theorem and the Nullstellensatz are still central planks of the theory, but there has been a reorganisation too It might be well to compare the present text with a text which appeared a century ago that was designed for the same purpose as the book under review: i.e to bring invariant theory within the compass of students and to bring it up to date The algebra of quantics was written by Edwin Bailey Elliott, who taught the subject at Oxford when Sylvester was there in the 1880s The work is highly derivative of the English approach to invariant theory, as we might expect, with only a brief nod in the direction of the German mathematicians Paul Gordan and Alfred Clebsch, though it does contain a summary of Hilbert’s work [2] A good portion of Elliott’s text is bound up with combinatorial techniques in the shape of Eulerian generating functions This was quite natural for Elliott since generating functions could be used not only to count the number of irreducible invariants and covariants but to discover them and the linear dependencies (syzygies) between them Elliott, influenced very heavily by the English school, used his introductory chapters for an amplification of material found in George Salmon’s texts It adopts the English terminology and it contains a thorough discussion of Cayley’s twin differential operators Ω and Ο Elliott gives the complete listing of the irreducible invariants and covariants for the polynomials of degree five and six (already established by the ‘King of the invariants’ Paul Gordan, using a compressed notation) and goes on to show the connection between invariant theory and analytical geometry (Olver decided to leave out the combinatorial aspect of invariant theory in his book because of pressure of space, so he refers the reader elsewhere [3]) REVIEWS 179 As mentioned above, the modern view of invariant theory makes transformation groups as the unifying idea From this there is an easy extension from polynomials to the study of differential invariants of Lie groups It also lends itself to an easier appreciation of Sophus Lie’s attempt to construct a ‘Galois theory’ for differential equations Olver’s book gains added interest since his own background is in differential equations and mathematical physics An interesting feature of the book is the reappearance of Sylvester’s ‘chemical’ viewpoint, a way of representing invariants and covariants by formulae expressed graphically Invariant theory is a highly technical subject To study it will never be easy but the present text presents the leading modern ideas in a highly cogent and understandable form References Gian-Carlo Rota, Two turning points in invariant theory, The Mathematical Intelligencer 21 (1999), pp 20-27 E B Elliott, An introduction to the algebra of quantics, 2nd edn Reprint (1913) Bernd Sturmfels, Algorithms in invariant theory, Springer (1993) TONY CRILLY Middlesex Business School, The Burroughs, Hendon, London NW4 4BT e-mail: t.crilly@mdx.ac.uk Abelian groups and modules, edited by Paul Eklof and Rüdiger Göbel Pp 373 SFr168 1999 ISBN 7643 6172 (Birkhäuser) Analysis and geometry in several complex variables, edited by Gen Komatsu and Masatake Kuranishi Pp 314 SFr158 1999 ISBN 7643 4067 (Birkhäuser) These books, published in the Trends in Mathematics series, contain conference proceedings Naturally, the papers deal with matters at the forefront of research The first book arises from the International Conference on Abelian Groups and Modules that was held in Dublin during August 1998 Some of the papers deal with methods borrowed from other areas of mathematics and applied to abelian groups and modules, including model theory, category theory, infinite combinatorics, classical algebra and geometry Other papers use abelian group theory in the study of module theory and non-commutative groups The second book is a collection of papers from the 40th Taiguchi Symposium Analysis and Geometry in Several Complex Variables, held in Kataka, Japan in June 1997 Several of the papers cover recent applications of complex analysis to other areas, such as partial differential equations, differential geometry, quantum mechanics and algebraic geometry These books will have limited appeal beyond the respective research communities, but both contain papers which may interest workers in other fields STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Combinatorics: a problem oriented approach, by Daniel A Marcus Pp 136 £16.95 1999 ISBN 883 85710 (Mathematical Association of America) As the title might suggest, the greater part of this attractive little book consists of problems The four sections that make up Part I cover Strings, Combinations, Distributions and Partitions Part II covers more advanced methods of counting, with 180 THE MATHEMATICAL GAZETTE sections on Inclusion and Exclusion, Recurrence Relations, Generating Functions and the Pólya-Redfield Method Within each section there are introductory problems that build towards one of the nineteen Standard Problems (for example, #9 is ‘Find the number of distributions of a given set of identical balls into a given set of distinct boxes’) These are generally followed by further problems that can be solved by suitable adaptation of the Standard Problem The problems are connected by fairly short sections of text which include examples and any definitions that are required The first sections are found in most elementary books on combinatorics, and the treatment of generating functions is quite short, so as far as content is concerned, it is the final section that distinguishes the book from its competitors The PólyaRedfield Method is useful for solving counting problems where there is an element of symmetry One example of this type of problem is to find the number of ways of colouring the squares of a by grid using two colours, two colourings being considered the same if one can be obtained by rotating the other Not surprisingly, this final section contains a dose of group theory The book is based on the author's problem-led course on combinatorics to ‘mathematics and computer science majors … generally third and fourth year’ at California State Polytechnic University The prerequisites are few however, and the book could form the basis of a first-year undergraduate course It would also be suitable for independent study, for example by a student preparing for the Olympiad STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Discrete mathematics using latin squares, by Charles F Laywine and Gary L Mullen Pp 305 £51.95 1998 ISBN 471 24064 (Wiley-Interscience) A latin square of order n is an n × n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column Readers may have come across such objects in statistical designs used to determine whether significant differences in some variable exist between various samples The subject itself is rich with unsolved problems, and methods employed on obtaining general results touch on a variety of other mathematical areas, especially in combinatorics, finite geometry and coding theory The book introduces many basic properties and results of latin squares together with diverse applications The sixteen chapters are divided into four parts, with the first two parts devoted to the introduction to latin squares and generalisations such as permutation cubes, orthogonal hypercubes and frequency squares Related mathematics, such as the sieve principle, groups and graphs, are dealt with in the third part The nine chapters on applications are given in the last part, which constitutes half of the book There is a useful chapter on ‘nets’, which are point sets with a very uniform distribution in a high dimensional cube, and can be used to overcome the problem of generating truly random sequences in numerical techniques such as Monte Carlo methods Other topics include affine designs, statistics, errorcorrecting codes and cryptology, with some topics being discussed in quite considerable detail Duplicate bridge players may be interested in a short chapter on ‘Room squares’, a topic based on the article [1] in the Gazette by T G Room, which is related to the construction of Howell movements The last chapter gives short introductions to more applications including conflict-free access to parallel memories, broadcast squares and tournaments Thus, although many readers will be able to construct solutions to round-robin tournaments, perhaps few will be able to tackle a mixed-doubles tournament with a spouse-avoiding condition REVIEWS 181 The well-written book, which can be read by undergraduates, contains very useful notes and references at the end of each chapter Many of the exercises have hints or partial solutions given in an appendix Unfortunately, I am afraid that the very high price for such a text means that it may only be bought by libraries Reference T G Room, A new type of magic square, Math Gaz 39 (1955), p 307 P SHIU Department of Mathematical Sciences, Loughborough University LE11 3TU Graph theory as I have known it, by W T Tutte Pp 156 £27.50 1998 ISBN 19 850251 (Oxford University Press) W T Tutte is one of the principal pioneers in the field of graph theory, where he has exerted a major influence for over 60 years This book is an account of a preretirement series of lectures that Tutte gave in 1984 in which he reflected on his life's work and disclosed many of his lines of thought, creative processes, triumphs and frustrations As he has also memorably described elsewhere in [1], Tutte's introduction to graph theory occurred while he was an undergraduate at Cambridge in the 1930s through his involvement with the Trinity College ‘Team of Four’ (Brooks, Smith, Stone, Tutte) Initially fired by one of H E Dudeney's Canterbury puzzles, they eventually disproved Lusin's conjecture, that there is no dissection of a square into a finite number of unequal smaller squares, and Tait's conjecture (which implies the Four Colour Theorem, FCT), that there is a Hamiltonian circuit on the edges of any convex polyhedron ‘Squaring the square’ led to generalisations, involving triangulations of triangles and parallelograms, and to work on rotational symmetries of graphs (including the search for highly symmetrical graphs) and on graphs on spheres: here Tutte whisks us from Brooks' Theorem via Hadwiger's Conjecture (which generalises the FCT and is still unproved) to the theory of bridges of bonds, ‘a beautiful theory needing applications’ Apart from the FCT, two other nineteenth century precursors of twentieth century graph theoretical concerns were Cayley's famous enumeration formula nn − for the number of labelled trees on n vertices and Kirchhoff's ‘Matrix-Tree’ Theorem which asserts that the singular n × n matrix K = (cij) associated to the graph G = {v1, … , vn} by: cii = valency of vi, cij = −1 if vi, vj are adjacent, = otherwise, has constant cofactors, the constant being the number of spanning trees of G Tutte describes his own work on the enumeration of various types of triangulations: this is a fiendishly difficult task which put me in mind of Piet Hein's ‘grook’, ‘Problems worthy of attack prove their worth by hitting back.’ Kirchhoff's ideas spurred Tutte's interest in subgraphs which culminated in his f-Factor Theorem Another important enumerative tool is the chromatic polynomial, P (G, x), of a (loopless) graph G This has degree equal to the number of vertices of G and, for any whole number m, P (G, m) is the number of colourings of the vertices of G using m colours in which no edge has both its ends the same colour Much of Tutte's later work has dealt with specific properties of the roots of P (G, x) or chromatic eigenvalues, including reasons for the ubiquitous appearance of the golden ratio and other Beraha numbers of the form cos2 (π / k), and also with generalisations such as 182 THE MATHEMATICAL GAZETTE the 2-variable Tutte polynomial which mimics the behaviour of the chromatic polynomial with respect to contraction and formation of subgraphs His paper with the catchy title, ‘All the King's horses’ demonstrated, among other things, that P (G, x) can be reconstructed from a knowledge of P (G − {vi } , x) for i = 1, … , n, although Ulam's full reconstruction conjecture − that G is determined up to isomorphism by the isomorphism classes of the subgraphs G − {vi} − remains stubbornly unresolved More abstract algebraic techniques also feature: Tutte traces the impact of some homological ideas from combinatorial topology, and the route that led to his 1959 characterisation of which abstract matroids are graphic This is unusual and enchanting book which will be accessible to anyone who is comfortable with the contents of [2] Tutte modestly shows us ‘mathematics in the making’ and portrays in a delightfully whimsical, personal manner the multidimensional warps and wefts, cul de sacs, wishful thinking, pleasures and satisfaction associated with a fulfilled life in mathematics References Martin Gardner, More mathematical puzzles and diversions Penguin 1980 Chapter 17 Robin J Wilson, Introduction to graph theory Longman 1975 NICK LORD Tonbridge School, Kent TN9 1JP Graph theory and its applications, by Jonathan Gross and Jay Yellen Pp 585 £47.50 1999 ISBN 8493 3982 (CRC Press)* Graph theory is a relatively new area of mathematics with many applications in optimisation, scheduling, communication networks, computer architecture and even biology This comprehensive text assumes little background and the authors claim it can be used as the basis of advanced undergraduate or beginning graduate courses in general graph theory, data structures and algorithms, or operations research and optimisation The first few chapters cover the fundamentals of graph theory such as types of graph, matrix representation, spanning trees, Eulerian trails, Hamiltonian cycles and travelling salesman problems Later chapters introduce more advanced ideas and applications such as drawing graphs on various surfaces, planarity, graph colourings, digraph applications, network flows, enumeration, voltage graphs and non-planar layouts The variety of courses is possible by being selective in the coverage of the later chapters in particular Although the book caters in theory for readers with minimal background, in practice such readers would have to absorb an awful lot Taking Chapter (Drawing Graphs and Maps) as a typical example, there are 72 definitions, 17 remarks and 20 propositions, theorems and corollaries The first section is particularly tough for anyone without any knowledge of topology: there are 20 definitions to be mastered in order to state (but not prove) the Jordan Curve Theorem To be fair though, an advanced undergraduate who has not studied the topology of surfaces, and therefore has to work hard in this chapter, will probably have studied topics which make other chapters more straightforward As well as being comprehensive, the book has several other useful features There are hundreds of illustrations ‘to strengthen intuition’ and over 1600 exercises, * CRC Press has become part of the same group as Springer-Verlag REVIEWS 183 some to secure understanding and others to challenge In many sections explicit algorithms are given for those who wish to use computers to solve particular problems Finally, a large number of applications are described, some in considerable detail The Chinese postman problem (in Chapter 6) provides one example of the ‘algorithm and applications’ approach The problem, proposed in 1962 by the Chinese mathematician Meigu Guan, is to find the shortest closed walk that traverses every edge of a graph at least once It corresponds to the postman (or woman) seeking the shortest route that allows all the mail to be delivered, starting and finishing at the sorting office The authors describe an algorithm for solving the problem and give five examples of applications: street sweeping, mechanical graph plotters, arranging a sequence of two-person meetings, determining an RNA chain from its fragments, and information encoding For the RNA example, they devote three pages to providing enough information about RNA fragmentation for readers to appreciate the value of the solution This book succeeds in its aim of comprehensive coverage It will be useful for anyone needing to learn about algorithms, applications, specific topics or about graph theory in general STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Theory of differentiation: a unified theory of differentiation via new derivate theorems and new derivatives, by Krishna M Garg Pp 525 £80.95 1998 ISBN 471 25387 (Wiley) This imposing research monograph is the outcome of a life-time's research into the minutiae of the theory of differentiation In it, Garg heroically strives to collate, unify, systematize and, in several instances, improve the various results about generalised derivatives that have sporadically appeared in the literature since the days of the early pioneers such as Dini, Peano, Denjoy, Perron, G C and W H Young, Lusin, Banach and Saks The archetypal unilateral ‘derivates’ are the upper and lower Dini derivates: f (x + h) − f (x) f (x + h) − f (x) D+f (x) = limsup , D−f (x) = liminf h h h↓0 h↑0 (with analogous definitions for D+f (x) and D−f (x)) Notable early theorems were those of Denjoy-Young (Almost everywhere, either f is differentiable or one of the upper derivates is +∞ and one of the lower derivates −∞.) and of Denjoy-Young-Saks (An arbitrary function is differentiable at almost every point at which it has a unilateral derivative.) Garg bases his development on what he calls upper and lower new derivatives defined − where they exist − as the − set-valued functions x [ D+f (x) , D−f (x)] and x [ D f (x) , D+ f (x)] ; the sets involved are, in fact, singletons nearly everywhere (These link with the subgradients of convex analysis and the notions of sub/super/semi-differentiability in non-smooth analysis.) What might be hoped for of a generalised derivative? • Versions of the familiar manipulative devices such as the product, quotient, chain and l'Hơpital's rules • Versions of characteristic results such as the mean value theorem and the ‘monotonicity theorem’, guaranteeing that a function is nondecreasing if its derivative is non-negative 184 THE MATHEMATICAL GAZETTE • Versions of familiar structural properties of derivatives such as the Darboux (intermediate value) property, the Denjoy property (that (f ′)−1 (a, b) is either empty or non-null), and the Lusin property (f (E) is null whenever E is null) • Reconstructibility of a function from its derivative by a suitably generalised Denjoy-Perron integration process • Analogues of the Banach-Saks theorem, which characterises those continuous functions f that are differentiable (everywhere) on almost all of their level sets f −1 (y) • Although since Weierstrass it has been known that continuous, nowhere differentiable functions exist, a continuous function might have a generalised derivative on, hopefully, a rich set of points Again, the set of somewhere differentiable functions is meagre in the space of continuous functions: does the same hold for generalised derivatives? These constitute some of the recurring themes explored in this book which culminates in an axiomatic framework which, building on work with new derivatives, explains the differences in behaviour of the various notions of generalised derivative that have appeared in the literature (Garg lists over a dozen such!) Throughout, the author's canvas is that of real-valued functions defined on subsets of r − indeed, he is able to use the symbol C to denote the spaces of continuous functions on [0, 1]! This is a dauntingly technical work (the index of symbols runs to pages!) which demands close concentration on the part of the reader But, as a not inconsiderable feat of presentation of a frustratingly diffuse area of analysis, it will repay such attention by real-variable aficionados NICK LORD Tonbridge School, Kent TN9 1JP Functional analysis and differential equations in abstract spaces, by S Zaidman Pp 226 £36.00 1999 ISBN 58488 011 (Chapman & Hall/CRC) In form and content, this is very much a book of two contrasting halves The first half consists of a carefully written, smoothly organised introduction to classical linear functional analysis The route chosen to the three big theorems (uniform boundedness, closed graph, Hahn-Banach) is a very familiar one, but there are some nice pedagogical flourishes (such as the easy proof of Hahn-Banach for Hilbert spaces: extend f defined on Y to Y by uniform continuity and thence to the whole space by setting f = on Y ⊥) After that, the topics presented reflect the less standard prerequisites for the second half of the text with material on unbounded/ closed/closable operators, operator semigroups and their infinitesimal generators, compact operators, symmetric operators (with their square roots obtained by a pretty iterative method) and a soupỗon of spectral theory In the more technical second half of the book, these tools are shown in action in the context of results taken from the recent research literature on differential equations in Hilbert and Banach spaces As ever, the devil is in the details, but we can glimpse something of what is involved from the simplest first order problem: solve u′ = Au with u0 given Here, u : [0, ∞) → X is a Banach space-valued function, u′ is the strong derivative (obtained by replacing | | by in the usual definition of derivative) and A is a closed densely defined operator If the problem is well-posed in a strong enough sense, the prescription u (t) = T (t) u0 gives rise to an operator semigroup T (t) : X → X with infinitesimal generator A, i.e REVIEWS 185 Ax = limt ↓ (T (t) x − x) / t , and the solution of the related non-homogeneous equation u′ = Au + f (u0 given) is u (t) = T (t) u0 + t ∫0 T (t − s) f (s) ds The majority of Zaidman's later theorems relate to the question of uniqueness of solutions He introduces the concepts of weak and ultraweak solutions (which rely on test functions in much the same way that these are used in the theory of distributions) and, via delicate differential inequalities, proves restricted uniqueness results in this situation for u″ = Mu (M symmetric on Hilbert space) and u′ = Au (A closed and densely defined on a reflexive Banach space) He is also able to transfer well-known results such as the resolvent representation formula ∞ ∫0 e−λtT (t) x dt = (λI − A)−1 x (T, A as above, Re λ large) to the ultraweak setting Finally, on a different tack, he discusses the striking theorem that, for solutions of equations such as u′ = Au + f (with A the infinitesimal generator of an almost periodic (a.p.) group of operators and f a.p.) the concepts of weakly a.p and strongly a.p coincide with a simple connection between the spectrums of f and u: here, u is weakly a.p if x′u is a.p in the classical sense for all x in X′ and strongly a.p if u mimics the classical definition but with replacing | | There are no exercises and, to my mind, a dearth of illustrative and motivational examples in the text The juxtaposition of the elementary and rather standard contents of the first half with the specialised and much more rarefied concerns of the second half certainly give the book an unusual and very distinctive flavour but may, I fear, serve to split its potential readership NICK LORD Tonbridge School, Kent TN9 1JP Ordinary differential equations and applications: mathematical methods for applied mathematicians, physicists, engineers, bioscientists, by Werner S Weigelhofer and Kenneth A Lindsay Pp 215 £14 1999 ISBN 898563 57 (Horwood Publishers) This book, according to the authors' preface, is based on lecture notes for a third year course on mathematical methods at Glasgow University The book opens with three chapters entitled respectively: Differential Equations of First Order; Modelling Applications; and Linear Differential Equations of Second Order The content of the first and third is fairly conventional, but the second deals with a variety of modelling topics coming from fields not usually touched on in differential equation textbooks, such as Newton's law of cooling, the Gompertz population law and pursuit curves The fourth chapter contains work on a variety of methods which have proved useful in the solution of linear second order equations The fifth chapter on oscillatory motion includes a discussion on the concept of resonance The later chapters deal with more advanced work The sixth chapter gives an introduction to the use of Laplace transforms for solving linear differential equations with constant coefficients The seventh chapter deals with higher order initial value problems (the marching problems of numerical analysis) and introduces the idea of the Wronskian The eighth chapter discusses systems of first order linear equations and shows how such systems can be dealt with using the aid of matrices The two final chapters deal with more sophisticated material The ninth introduces the concept of eigenvalues and eigenfunctions and Sturm-Liouville theory, and the tenth gives an introduction to the calculus of variations and considers some elementary examples An appendix gives a number of self-study projects from a variety of 186 THE MATHEMATICAL GAZETTE fields; for example, rockets, bridges and snowploughs Examples with solutions are scattered throughout each chapter and at the end of each chapter there are a number of tutorial examples, the answers to which (taking up about a quarter of the book) are given in a second appendix This is an interesting book and reads easily The topic mix may not suit every course, but many will find it a useful textbook Parts of chapter and chapters 2, and could in fact be useful to A level students (Might one suggest a pamphlet on these lines by the authors?) There are one or two places where I feel the book could be improved In the first chapter there is a mention of singular solutions but no mention of envelopes, and some discussion could have been given in the ninth and tenth chapters of the connection between eigenvalues and the calculus of variations However, the appearance of the book is pleasing and the price is, for these days, reasonable I have no hesitation in recommending it Ll G CHAMBERS School of Mathematics, University of Bangor LL57 1UT Elementary Lie group analysis and ordinary differential equations, by Nail H Ibragimov (Mathematical Methods in Practice 4) Pp 347 £55 1999 ISBN 471 97430 (Wiley) The series Mathematical Methods in Practice is intended to provide a one-stopshop for applied scientists who wish to use mathematics in their work The idea is to combine some of the theory traditionally taught in pure mathematics courses with applications that are sometimes taught in the absence of rigour Lie originally developed his theory as a way of unifying the treatment of many types of differential equations, analogous to Galois's theory for algebraic equations Many books on Lie groups develop the theory in the most general form, requiring the reader to be acquainted with topological groups and manifolds, which are difficult ideas for beginners This book is intended for students who wish to understand Lie group analysis specifically in the context of differential equations In order that the book be self-contained, the first part gives a brief but fairly comprehensive treatment of the classical approach to differential equations The first chapter gives many excellent examples of the use of differential equations in mathematical modelling The next three cover methods for various ordinary equations, general properties of solutions and first order partial differential equations The second part develops the fundamental ideas of Lie group analysis, beginning with an interesting historical survey of Lie theory from Lie's original work, to the resurrection of applied group analysis led by L V Ovsyannikov in the sixties and seventies The other chapters cover transformation groups, infinitesimal transformations and local groups, differential algebra, symmetry of differential equations and invariants Having established the machinery of Lie group analysis in part two, the remaining section is devoted to showing how it brings unity to the ad hoc methods for differential equations presented in the first part This book presents Lie groups in a way that will appeal to two groups: those who want an accessible introduction to the theory and those who primary focus is on differential equations Each chapter is supplemented with numerous historical notes and references and a collection of problems, graded in difficulty Answers or hints are collected in an appendix STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG REVIEWS 187 Beginning partial differential equations, by Peter V O’Neil Pp 500 £51.95 1999 ISBN 471 23887 (Wiley) This is a self-confessed bread-and-butter book on partial differential equations Since the subject is a difficult one for students, the appearance of this book is very welcome, although the price will not be conducive to large sales In fact, its exorbitant price will ensure its exclusion from all but the most well-heeled libraries The topics chosen are first and second order differential equations, Fourier analysis, the treatment of the wave and heat equations (the latter now used as a tool in financial mathematics) and finally, a long section on the Dirichlet and Neumann problems To his credit the author makes certain prerequisites clear at the beginning, and it is not one of those books with lofty ideas (not to mention blurbs written by the marketing department) about teaching the latest research to students without GCSE The reader is not asked to plunge into partial differential equations without a facility with the standard properties of real-valued functions of n real variables, vector calculus (theorems of Green and Gauss), a post-calculus course on ordinary differential equations and the convergence of series and improper integrals Access to the computer software MAPLE would also be handy, but is not essential for a proper study of the mathematical contents of this book The very sparse historical comments are too incomplete to be of much help but their inclusion is a nod in the right direction A few afternoons in a reasonable library would have offered much more to bolster the meagre comments made here Why not give us a bit more (than nothing) on say, Jean Marie Constant Duhamel (1797-1872), who crops up in various places in the book And which Neumann of the Neumann problem is the author not talking about? A curious diversion in a technical book is the long historical digression on ‘The Great Debate Over the Age of the Earth’ (pp 317-320) One can only guess that this is still a popular topic in Alabama, where the author is based In this portion, which is presented well, the author summarises Joe Burchfield’s Lord Kelvin and the age of the Earth (1875) and shows the way William Thomson used the heat equation to gain his estimate of the Earth’s age as between 100 and 400 million years The book will appeal mostly to applied mathematicians and those engineers who are relatively strong mathematically The presentation of surfaces in three dimensions in these days of computer graphics leaves something to be desired It is a useful book but the same information can be found in books which are more reasonably priced TONY CRILLY Middlesex Business School, The Burroughs, Hendon, London NW4 4BT e-mail: t.crilly@mdx.ac.uk Numerical solution of partial differential equations in science and engineering, by Leon Lapidus and George F Pinder Pp 677 £41.95 1999 ISBN 471 35944 (paperback) (Wiley) Originating as a support text for courses given by both authors at Princeton University, this book is ideally suited to students from a variety of academic disciplines It is virtually free from jargon or other nomenclature which may deter certain students; however it provides references to applications in a diversity of subject areas It is very sad to report the sudden death (5 May 1977) of co-author, Leon Lapidus, whilst at work in the Department of Chemical Engineering at Princeton University This tragedy undoubtedly contributed to the delay in publication 188 THE MATHEMATICAL GAZETTE which in no way devalues this work The objectives of providing a balanced treatment of finite differences and finite element methods which can be read either by equation-type or by numerical approximation have been fully achieved The first quarter of the text consists of introductory material relating to partial differential equations of the first and second orders including their classification, together with the method of characteristics Basic concepts of finite differences and finite elements follow, the latter being extended to include triangular, isoparametric and three-dimensional elements There are limited references here covering both standard texts such as R Courant and D Hilbert Methods of mathematical physics (Interscience 1962) and some less readily available publications!! For example B G Galerkin, Vestn Inzh Tekh (USSR), 19, pp 897-908 (1915) As expected the majority of the book is trisected to cover parabolic, elliptic and hyperbolic partial differential equations Each section concludes with a very substantial list of references The absence of sets of exercises for the reader is only partially offset by the inclusion of selected ‘Example Problems’ within the body of the text However the style of presentation is such that almost all readers would be expected to go away and practice what they have assimilated; either in ‘real world’ applications or in sets of exercises taken from elsewhere The three types of PDE each receive a thorough treatment, carefully balanced between finite differences and finite elements, as well as between one, two and three spatial dimensions Authors of the twenty-first century attempting to supersede this work, possibly by the inclusion of extensive material relating to digital computer software, will well to ask themselves ‘Which came first, the chicken or the egg?’ Whilst the publication price of £41.95 for a paperback edition of approximately 700 pages may be considered substantial it must be emphasised that − despite its origins − the work is lecturer independent Thus the reader has a readily accessible self-contained reference work for the subject of the numerical solution of PDEs − itself central to all applicable mathematics This book is strongly recommended for all students with significant involvement in this subject area MICHAEL R MUDGE 23, Gors Fach, Pwll-Trap, St Clears SA33 4AQ Evaluation and optimisation of electoral systems, by Pietro Grilli di Cortona, Cecilia Manzi, Aline Pennisi, Federica Ricca and Bruno Simeone Pp 230 $53 (SIAM members $42.40) 1999 ISBN 89871 422 (Society for Industrial and Applied Mathematics) The problem of determining the best electoral system has occupied many minds in this country recently New assemblies in Scotland and Wales and the currently suspended assembly for Northern Ireland have been established and the system for electing Members of the European Parliament has been changed Reform of the House of Lords and the possibility of proportional representation for the Commons will keep the subject of electoral systems in the public mind This book is the result of a collaboration involving experts from operational research, statistics, decision sciences and political sciences It has four sections, three presenting mathematical treatments of various aspects of electoral systems and a final section taking the political view The mathematical treatment has been kept as accessible as possible in the hope of attracting readers from beyond the mathematical community Though the book is essentially mathematical, the consequences of the various equations and formulas have been interpreted in words for those uncomfortable with mathematical notation REVIEWS 189 Previous analyses of electoral systems have taken various points of view; for example, axiomatic, statistical, game-theoretic and geometric approaches have all been used The present authors consider the optimisation of various criteria In Chapter 2, Cecilia Manzi proposes a general set-theoretic model for electoral systems that encompasses quotient methods and first-past-the-post, double ballot, alternative transferable vote and single transferable vote systems This model enables a classification of the electoral systems of many countries These systems can be compared using a variety of criteria and indicators that are dealt with in the following chapter The next five chapters, by Aline Pennisi, consider the design of electoral systems Different electoral formulas are treated as algorithms that minimise certain measures of unfairness, with the surprising result that the same formula may minimise more than one measure The third section, by Fedrica Ricca, considers the problem of designing electoral districts An artificial example is given of a territory consisting of 45 wards to be grouped into equal constituencies Although 24 wards favour party C and 21 favour party P, it is possible to assign wards to constituencies to achieve seats to victories for either party The political sensitivity of deciding constituency boundaries could not be better illustrated In the final section, Pietro Grilli di Cortona analyses the benefits, limitations and political implications of the methodology presented in the first three parts This is a fascinating study that deserves to be widely read STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Perplexing problems in probability: Festschrift in honor of Harry Kesten, ed Maury Bramson and Rick Durrett Pp 398 DM158 1999 ISBN 7643 4093 (Birkhauser) It has become fashionable recently for academics to honour a distinguished colleague on reaching a certain age by publishing a Festschrift, being a collection, in book form, of papers, each on a topic on which the person honoured has worked Often the contributors are former students Cynics say this is a device for the authors to get another paper to add to their c.v Others would say that it provides an opportunity both to honour a major contributor and to present a view of the state of knowledge in a particular field The book under review is such a Festschrift in honour of Harry Kesten, consisting of a paper by Rick Durrett, which summarizes his work, and twenty papers of original material on topics on which Kesten has worked The honorand has made major contributions to mathematical probability and has gained a deserved reputation as a superb solver of problems and, as such, his work is not confined to a single topic The papers therefore range widely and all require substantial background for their appreciation One class of problems concerns percolation processes Take a square, integer lattice and add edges connecting adjacent sites Imagine the edges to be channels conveying a fluid which are open (or closed) with probability p (or − p) independently of other channels A question of physical interest is the set of sites that can be reached from the origin by the use of open channels It turns out that there is a critical value at p = 12 and the number of such sites is infinite with positive probability only if p exceeds that value Kesten has extended enormously our knowledge of the behaviour for values of p near to 12 Another problem, again on the lattice, is the self-avoiding random walk, which is the usual walk except that a walk can never visit the same site twice This is an especially difficult scenario 190 THE MATHEMATICAL GAZETTE because the process has to remember all its past and cannot forget most of it, unlike a Markov process These are difficult papers of limited appeal When a writer says ‘well-known’ he likely means ‘well-known to 500 people’ But nevertheless they represent real advances in difficult problems and form a fine testimonial to one of the world's leading probabilists D V LINDLEY Woodstock, Quay Lane, Minehead TA24 5QU Resampling methods: a practical guide to data analysis, by Phillip I Good Pp 269 SFr118 1999 ISBN 7643 4091 (Birkhäuser) This book has been written for a diverse audience: medical students, health workers, business people, biologists, social scientists, industrial statisticians and statistical consultants are all mentioned in the preface as potential beneficiaries As the title suggests, it is not a traditional statistics text, preferring to emphasise ‘tablefree’ resampling methods (bootstrap, density estimation and permutations) rather than methods based on tables of standard distributions Much of the first part of the book is aimed at those with minimal experience of statistics The first two chapters include such material as types of data, measures of average, statistical diagrams and the binomial distribution However, the resampling approach is introduced as early as chapter one, where the bootstrap method is used to establish the precision of a sample median This involves repeatedly sampling with replacement from the original sample of size 22, each case yielding another sample of size 22, usually involving repetitions The sample medians of 50 ‘resamples’ are computed These are illustrated on a number line to ‘provide a feel for what might have been had we sampled repeatedly from the original population’ Clearly such a method has drawbacks, and these are discussed in a section headed ‘Caveats’ Permutation tests are introduced in the third chapter, Testing Hypotheses Two samples are compared: one has been subjected to a treatment that the other has not The results of the treated sample are 121, 118, 110; those of the untreated sample results are 34, 22, 12 The total of the treated sample is compared with the totals of every possible choice of results Since it is the highest of the 20 possible results one can assert at the 5% significance level that the treatment was effective Density estimation is not introduced until chapter 10, Classification and Discrimination, which considers the problem of deciding how many distinct populations are represented in a sample Suppose that a sample containing data from n distinct populations is used to construct a histogram Whether or not the histogram shows the n sub-populations depends on the number of intervals used and the extent to which the populations ‘overlap’ The histogram can be smoothed by replacing each of its blocks by a normal distribution curve of corresponding area, centred on the block, and then summing the results The resulting curve is likely to have a number of modes (maxima) The process can be repeated for many different interval widths to determine the smallest interval width hk that gives k modes For each of these critical interval widths, bootstrap resampling can be used with the smoothing process to estimate the proportion of times that more than k modes appear This proportion will be close to if k < n and close to if k ≥ n, thus allowing us to estimate the value of n This book illustrates many applications of resampling methods In each case, the author discusses the assumptions that have been used and the corresponding limitations He writes in a conversational style and makes effective use of concrete REVIEWS 191 examples to introduce the various methods Each chapter has a chapter summary, a section called ‘To Learn More’ guiding readers to appropriate references, and a set of exercises Overall this is a helpful introduction, but one that will stretch many of the target groups mentioned in the preface His claim that readers need only ‘highschool algebra’ is stretching the truth: an expression like n B ˆ ] = ∑ ∑ [ yi, η [ xi, w∗b]] eff [ w∗, F B b=1 n i=1 would frighten the life out of most ‘Physicians and physicians in training, nurses and nursing students’ of this reviewer's acquaintance! STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG ( ) Epidemiology: study design and data analysis, by Mark Woodward Pp 699 £39.95 1999 ISBN 584 88009 (Chapman & Hall/CRC) This book has been aimed at statisticians who want to see how their subject can be applied to epidemiology and to medical researchers who need a better understanding of statistics The first two chapters reflect this dual aim: the statisticians need Chapter to get a basic understanding of the fundamental issues of epidemiology and the researchers need Chapter to remind them of the basic tools of statistics Epidemiologists investigate the causes of disease as well as modelling the spread of disease The emphasis in this book is on the first category of questions, including the estimation of risk, the effect of confounding variables and the interaction among risk factors A good example of a confounding variable is the presence of grey hair among stroke victims: the important risk factor is age, but grey hair increases with age An example of interaction between two risk factors occurs in the study of lung disease among porcelain painters: lung function is affected by exposure to cobalt, but this is much worse among painters who smoke Necessarily Woodward pays a good deal of attention to the design of studies to investigate risk factors, with chapters on cohort studies, case-control studies and intervention studies The last four chapters present statistical methods such as hypothesis testing, analysis of variance for one or more explanatory variables, various forms of regression analysis and probability models for hazard and survival functions, such as the exponential and Weibull distributions The book contains many examples and exercises based on real epidemiological data In some cases, where the data set is large, the information is also provided electronically on an associated web site (http://www.reading.ac.uk/AcaDepts/sn/ wsn1/publications99.html) The use of real data sets to illustrate ideas, the emphasis on practicality and the omission of proofs are all features that will appeal to those interested in applying statistical methods to the study of disease The book looks particularly suitable for students STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG 192 THE MATHEMATICAL GAZETTE THE MATHEMATICAL ASSOCIATION The fundamental aim of the Mathematical Association is to promote good methods of mathematical teaching A member receives each issue of the Mathematical Gazette and/or Mathematics in School (according to the class of membership chosen), together with Newsletters Reports are published from time to time and these are normally available to members at a reduced rate Those interested in becoming members should contact the Executive Secretary for information and application forms The address of the Association Headquarters is 259 London Road, Leicester LE2 3BE, UK (telephone 0116 221 0013) The Association should be notified of any change of address If copies of the Association periodicals fail to reach a member through lack of such notification, duplicate copies can only be supplied at the published price If change of address is due to a change of appointment, the Association will be glad to be informed Subscriptions should be submitted to the Treasurer via Headquarters Correspondence relating to Teaching Committee should be addressed to Mr Doug French The Association's Library is housed in the University Library, Leicester THE MATHEMATICAL GAZETTE Editor: Mr Steve Abbott, 91 High Road West, Felixstowe IP11 9AB Production Editor: Mr Bill Richardson, Kintail, Longmorn, Elgin IV30 8RJ Problem Corner: Mr Graham Hoare, Russett Hill, Chalfont St Peter, Bucks SL9 8JY Reviews Editors: 'Bud' Winteridge, The University of Birmingham, Westhill, Weoley Park Road, Birmingham B29 6LL e-mails Mrs Rosalie McCrossan, Bill Richardson Garnock Academy, wpr3@tutor.open.ac.uk School Road, Kilbirnie, Bud Winteridge KA25 7AX d.j.winteridge@bham.ac.uk Material for publication should be sent to the Editor Suggestions for improvements will be welcome Books for review should be sent to Bud Winteridge Advice to authors of notes and articles Study the format of articles in the Gazette Please note the format for references MSS should be typed (two copies please) and formulae that cannot be typed should be clearly hand written If, in addition, files on disk are submitted the disks should be 3½ inch TEX files are welcome This edition of the Gazette was produced on an Acorn machine using TechWriter and Draw