Eric Charles Milner

Eric Charles Milner was born on 17 May 1928 and brought up inLondon. His father was an engineer, but times were hard andwork was often difficult to obtain. So his mother had to helpout by working as a seamstress, and Eric was often looked afterby his grandmother. At the age of 11, he won a scholarship tothe Haberdashers' Aske's Boys' School, but never attended itin its permanent London buildings because the outbreak of theSecond World War caused all London schools and their pupilsto be evacuated to safer parts of the country. As a result,Eric, an only child and knowing none of his new schoolfellows,was billeted at a home near Reading where he was extremely unhappy.In despair, he ran away and returned to London, where, afterunsuccessful attempts to find him another billet, he roamedthe streets and missed school. After some time, he was eventuallyfound another billet where he received kindness and was muchhappier. Despite these disruptions and the other inevitableshortcomings of a war-time education, Eric's intelligence morethan sufficed to surmount such hurdles, and in later life hecould speak and write better than most of us. From 1946 to 1951, Eric attended King's College, London. Hegraduated with First Class Honours in 1949, when he was awardedthe Drew Gold Medal as the most distinguished Mathematics studentin that year, and a Research Studentship. He then studied foran MSc degree, taking ‘Modern algebra’ and ‘Quantummechanics (Wave mechanics)’ as his selected subjects,his supervisors being Richard Rado (then a Reader at King'sCollege) and Professor Charles Coulson. He received the MScdegree, with distinction, in 1950. This was followed by a year'sresearch in quantum mechanics under the supervision of ProfessorCoulson.     

Alan Breach Tayler

Alan Breach Tayler, CBE, Director of the Oxford Centre for Industrialand Applied Mathematics, died on 28 January 1995, aged 63. Alan went up to Oxford in 1951 to read Mathematics at BrasenoseCollege. He obtained a first, and after a brief excursion tothe Bristol Aircraft Company, he returned to work for a DPhilwith George Temple. His thesis, completed in 1959 and entitled‘Problems in compressible flow’, contained a mixtureof analytic, approximate and numerical solutions which foreshadowedthe new practical applied mathematics that he embraced later.He became University Lecturer and Tutorial Fellow of St Catherine'sSociety in 1959. During the next twenty-five years, Alan Tayler brought a newethos to applied mathematics. This change came about throughhis recognition that the status quo in the 1960s, which compriseda delicate balance between theory and practice in the area ofapplied mechanics, was capable of far-reaching generalisation;indeed, he saw that such a development was essential since thefollowing decades were to be dominated by computers and an ever-increasingneed for mathematical modelling. In 1967, with Leslie Fox, heinitiated the mathematical Study Groups with Industry, whereinacademic and industrial researchers interact in week-long workshops.These were an immediate success: (1) with industry, who foundnew insights into their problems and new recruiting possibilities;(2) with students, whose enthusiasm to use their theoreticalknowledge soon led to the highly popular MSc in MathematicalModelling and Numerical Analysis; and (3) with faculty, bothpure and applied, who found an undreamed of source of fascinatingnew theoretical problems. For example, one intellectual consequencewas the use of industrial case studies to uncover the new fieldof ‘free boundary problems’, on which several thousandlearned articles have appeared since 1970.     

Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’by A. R. D. Mathias defends a so-called full-blooded set theorywithout full detail [3]. He again objects to a weak set theorywhich he calls ‘Mac’, in which the usual Zermelo–Fraenkelseparation scheme is required only for formulas with suitably‘restricted’ quantifiers. I had proposed that suchseparation is adequate for all standard uses of set theory inmathematics. But Mathias has not produced any counter examplesof actual mathematics which requires the use of a stronger separation.     

Eric John Fyfe Primrose (1920-1998)

Eric Primrose was appointed to a Lectureship in Pure Mathematicsat the University College, Leicester, in 1947 and promoted toa Senior Lectureship in 1954. He came to Leicester direct fromOxford where he had spent the 1946–47 year completinghis degree, which had been interrupted by war service. Ericwas awarded his PhD in 1957 by the University of London. Eric's secondary education was at Chigwell School and in 1939he won an Open Scholarship in Mathematics and went up to StJohn's College, Oxford. He took ‘shortened finals’in 1941 and then entered the RAF as a Technical Officer (Radar),reaching the rank of Flight Lieutenant, having served in variouslocations throughout Europe.     

Richard Rado

Richard Rado was born in Berlin; he was the second son of LeopoldRado, from Budapest. At one stage of his education he had todecide whether to become a concert pianist or a mathematician.He chose the latter in the belief that he could continue withmusic as a hobby, but that he could never treat mathematicsin that way. He studied at the University of Berlin, but alsospent some time in Göttingen. He took a DPh at Berlin withhis thesis ‘Studien zur Kombinatorik’ [3] underIssai Schur in 1933. During this period he was also influencedby Erhard Schmidt. On 16 March 1933, he married Luise Zadek, the elder daughterof Hermann Zadek, whom he had earlier come to know when he neededa partner to play piano duets. It was indeed a remarkable partnership. As Hitler came to power in 1933, the Rados, being Jewish, madetheir way to England, Richard having obtained a scholarshipof £300 p.a. from Sir Robert Mond through the recommendationof Professor Lindemann (later Lord Cherwell), who had interviewedhim in Berlin, to enable him to study at Cambridge.     

Brian Hartley, 1939-94

Brian Hartley began his algebraic career as one of Philip Hall'sresearch students in Cambridge. He obtained his Ph.D. in 1964,spent two post-doctoral years in the USA and, on his returnto the United Kingdom, accepted a lectureship in the newly establishedMathematics Department at Warwick University; there he was promotedto a readership in 1973. He was appointed to a chair of puremathematics at the University of Manchester in 1977 and wasHead of the Mathematics Department there from 1982–4.He was elected to the London Mathematical Society in 1968 andserved on Council from 1987–9. He won an EPSRC SeniorResearch Fellowship, but died on 8 October 1994, a few daysafter taking it up. He travelled widely and took a lively interestin other cultures and languages. His intellectual energy, enthusiasmfor algebra, direct manner and dry sense of humour endearedhim to the many mathematical friends he made around the world.He was devoted to mathematics and gave generously of his timeand energy in support of younger colleagues.     

Kunihiko Kodaira

Kunihiko Kodaira, who died on 26 July 1997, was the outstandingJapanese mathematician of the post-war period, his fame establishedby the award of the Fields Medal at the Amsterdam Congress in1954. He was born on 16 March 1915, the son of an agricultural scientistwho at one time was Vice Minister of Agriculture in the JapaneseGovernment and had also played an active role in agriculturaldevelopments in South America. Kodaira studied at Tokyo University,taking degrees in both mathematics and physics. From 1944 to1951 he was an associate professor of physics at the University.His PhD thesis was published in the Annals of Mathematics [18],and it immediately attracted international attention. Essentiallythis filled a significant lacuna in the basic theorem of W.V. D. Hodge on harmonic integrals. Kodaira had worked on thisfor many years but, because of the war, his research was carriedout in isolation from the international community and did notbecome known until much later. Hermann Weyl, who had been a keen supporter of Hodge's work,realised the importance of Kodaira's thesis, and arranged forhim to come to the Institute for Advanced Study in Princetonin 1949. This was the start of Kodaira's 18-year residence inthe United States, a fruitful period which saw the full blossomingof his research, much of it in collaboration with Donald Spencer.Kodaira spent many years at Princeton, divided between the Instituteand the University, but the years 1961–67 were more unsettled,seeing him successively at Harvard, Johns Hopkins and finallyStanford. In 1967 he returned to a professorship at the Universityof Tokyo, where he remained until the normal retiring age. From1975 to 1985 he worked at Gakushuin University, where retirementrestrictions did not apply.     

Kenneth O. May, 1915–1977 : His early life to 1946

Kenneth Ownsworth May graduated from the University of California at Berkeley in 1936 with highest honors in mathematics. The following year he received his Masters degree and became a fellow of the Institute of Current World Affairs, and during the next two years he traveled to England, Europe, and Russia. On his return to the United States he became active in the Communist Party, the consequences of which would plague him for years. He joined the United States Army in 1942, serving with distinction, and after the war returned to Berkeley, where he obtained his Ph.D. in 1946. He immediately accepted an assistant professorship at Carleton College in Northfield, Minnesota, later moving to the University of Toronto.This part of May's biography focuses on the events up to his accepting a position at Carleton College. In this early phase his openness, his emphasis on good communications in the process of education, and his interest in practical procedures emerge which later set the background for his successful career as a leading historian of mathematics and the founding editor of Historia Mathematica.     

The contributions of W.A. harris Jr. to difference equations

William Ashton Harris Jr. was born December 18, 1930 in New Orleans, Louisiana, USA. He studied mathematics with minors in physics and appl ied mechanics (elasticity) at the University of Minnesota receiving his Ph.D. in mathematics in 1955 with the thesis "A boundary value problem for a system of ordinary linear differential equations involving powers of a parameter". His thesis advisor was Professor H.L. Turrittin.He remained at the University of Minnesota and became professor in 1968. In 1970 he moved to the University of Southern California where he stayed until his untimely death on January 8, 1998. He has held visiting appointments at several other universities.     

The sound of music and its link with mathematics

This article discusses a relatively unnoticed application ofmathematics by describing its connection with an aspect of music,in particular, the musical scales. Stemming from a problem foundin a Year 9 mathematics textbook commonly used in Singapore,the article illustrates the role of mathematics in musical scalesby first considering the frequency ratios of consecutive musicalnotes in the ‘just scale’ and secondly explaininghow an anomaly in the ‘just scale’ caused by theuneven frequency ratios is resolved with the help of mathematics,thereby leading to the development of the well-tempered scale.The article ends with an exploration of the frequency ratiosof consecutive musical notes in the well-tempered scale. Notonly does the article aim to broaden the teachers’ horizonswith such an introduction to the mathematical aspect of music,it also hopes to enrich their mathematical experiences as well.     

Philip Holgate (1934-1993)

Philip Holgate was born in Chesterfield on 8 December 1934.His family moved from Derbyshire to Devon in 1945, and he waseducated at Newton Abbot Grammar School from 1945 to 1952, andthe University College of the South-West (now Exeter University,but which then awarded London degrees) from 1952 to 1955. Hequalified as both a teacher (King's College, London, 1955–56)and a statistician (University College, London, 1956–57). After teaching mathematics and physics at Burgess Hill School,in Hampstead and then in Borehamwood, Philip joined the StatisticsSection at Rothamsted Experimental Station (1961–62).He then spent five years in the Biometrics Section of the NatureConservancy. His first publications date from this period, andthe interests he acquired then were to develop into what becamehis most enduring, and distinctive, scientific interests. Philip joined the Department of Statistics, Birkbeck College,University of London, in 1967, and remained at Birkbeck forthe rest of his career, until his death from a heart attackon 13 April 1993.     

Foreword

Steve Smale set the agenda for FoCM in his call for the 1995 conference in Park City, Utah. No stranger he to ambitious agendas and extraordinary accomplishments. He is one of the dominant figures in the mathematics of the second half of the twentieth century. Smale’s theory of immersions, the generalized Poincare conjecture, and H-cobordism theorems with their far-reaching consequences are the bedrock of differential topology. His horseshoe is the hallmark of chaos, and his hyperbolic dynamics the rejuvenation of the geometric theory of dynamical systems. He is one of the pioneers in the introduction of infinite-dimensional manifolds for the study of nonlinear problems in the calculus of variations and partial differential equations. The list goes on: the systematic use of differential techniques in microeconomics, electrical circuit theory, chaos in predator–prey equations and, finally, for the twentieth century, the foundations of computational mathematics and complexity theory, and now, in the twenty-first century, the theory of learning. It has been our privilege to be among his collaborators and students in the broadest sense of the word. With these issues (Volume 5 Number 4 and Volume 6 Number 1, as well as an earlier article appearing in Volume 5 Number 2, are dedicated to Steve Smale’s 75th Birthday) we celebrate Steve’s 75th birthday and continuing vitality. He sets the bar high. We do our best.     

Topologically Irreducible Representations and Radicals in Banach Algebras

It is shown that the topologically irreducible representationsof a normed algebra define a certain topological radical inthe same way that the strictly irreducible representations definethe Jacobson radical and that this radical can be strictly smallerthan the Jacobson radical. An abstract theory of ‘topologicalradicals’ in topological algebras is developed and usedto relate this radical to the Baer radical (prime radical).The relations with topologically transitive representationsand standard representations in the sense of Meyer are alsoexplored. 1991 Mathematics Subject Classification: 46H15, 46H25,16Nxx.     

Bibhutibhusan datta (1888–1958), historian of Indian mathematics

Born to a poor Bengali family, Bibhutibhusan Datta (1888–1958) was indifferent to wordly pleasures and gains. He never married. His doctoral thesis was on hydrodynamics, but he is best known for his work on the history of mathematics. He retired voluntarily from the University of Calcutta at the age of 45 and in 1938 took sany$\text{a}$sa (literally, renunciation) to become known as Swami Vidyaranya. He also wrote on Indian religion and philosophy.     

Promoting student engagement with mathematics support

This article reports the findings of qualitative research undertakento seek to identify the key reasons why some students are notengaging with mathematics support provided by Loughborough University.The research involved a number of focus groups and ‘onthe spot’ interviews with ‘non-users’ fromacross the campus. Barriers identified include a lack of awarenessof the location of support and a fear of embarrassment. Furtherinterviews were conducted with regular users of the supportin an attempt to understand how some of these barriers to usagemight be overcome. The article will discuss actions that maybe taken to improve student engagement with mathematics supportand the issue of how student motivation may affect such action.     

N.G. de Bruijn’s contribution to the formalization of mathematics

N.G. (Dick) de Bruijn was the first to develop a formal language suitable for the complete expression of a mathematical subject matter. His formalization does not only regard the usual mathematical expressions, but also all sorts of meta-notions such as definitions, substitutions, theorems and even complete proofs. He envisaged (and demonstrated) that a full formalization enables one to check proofs automatically by means of a computer program. He started developing his ideas about a suitable formal language for mathematics in the end of the 1960s, when computers were still in their infancy. De Bruijn was ahead of his time and much of his work only became known to a wider audience much later. In the present paper we highlight de Bruijn’s contributions to the formalization of mathematics, directed towards verification by a computer, by placing these in their time and by relating them to parallel and later developments. We aim to explain some of the more technical aspects of de Bruijn’s work to a wider audience of interested mathematicians and computer scientists.     

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

Toward a scientific and personal biography of Tullio Levi-Civita (1873–1941)

Tullio Levi-Civita was one of the most important Italian mathematicians of the first part of the 20th century, contributing significantly to a number of research fields in mathematics and physics. In addition, he was involved in the social and political life of his time and suffered severe political and racial persecution during the period of Fascism. He tried repeatedly and in several cases successfully to help colleagues and students who were victims of anti-Semitism in Italy and Germany. His scientific and private life is well documented in the letters and documents contained in his Archive. The authors' aim is to illustrate the events of his life by means of his large and remarkable correspondence.     

Why mechanics should be integral to secondary school mathematics

Mechanics has never been the most popular subject in A-levelmathematics, the UK’s public examination for 16–18-yearolds, either with students, teachers or educators. The attemptsto popularize mechanics have failed and it is conceivable thatthe subject will be dropped from the A-level syllabus in theforeseeable future. This article argues the importance of mechanicsand why it should be integral to secondary school mathematics:Mechanics is the exemplar of mathematical modelling, is thelogical point of entry for the enculturation into scientificthinking and provides the means to develop an understandingof the relationship between mathematics, the theoretical objectsof science and the way science and mathematics speak of theworld. It enables learners across the ‘ability range’to think in the abstract and as such should be taught priorto the 6th form, that is, prior to the UK’s post-compulsorylevel of education.