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.     

Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

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.     

Are Operads Algebraic Theories?

I exhibit a pair of non-symmetric operads that, while not themselvesisomorphic, induce isomorphic monads. The existence of sucha pair implies that if ‘algebraic theory’ is understoodas meaning ‘monad’ then operads cannot be regardedas algebraic theories of a special kind. 2000 Mathematics SubjectClassification 18D50 (primary), 18C10, 18C15 (secondary).     

The Kissing Numbers of Convex Bodies -- A Brief Survey

Nearly four hundred years ago, the cubic close-packing of equalspheres in R³ was discovered by Kepler [21], in which each spheretouches 12 others. In 1694, Gregory and Newton discussed thefollowing thirteen spheres problem. Can a rigid material spherebe brought into contact with 13 other such spheres of the samesize? Gregory believed ‘yes’, while Newton thought‘no’. 1991 Mathematics Subject Classication 11H31,52C17.     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

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.     

Nathan Jacobson (1910-1999)

Nathan Jacobson, who died on 5 December 1999, was an outstandingalgebraist, whose work on almost all aspects of algebra wasof fundamental importance, and whose writings will exercisea lasting influence. He had been an honorary member of the Societysince 1972. Nathan Jacobson (later known as ‘Jake’ to his friends)was born in Warsaw (in what he describes as the ‘Jewishghetto’) on 5 October 1910 (through an error some documentshave the date 8 September); he was the second son of CharlesJacobson (as he would be known later) and his wife Pauline,née Rosenberg. His family emigrated to the USA duringthe First World War, first to Nashville, Tennessee, where hisfather owned a small grocery store, but they then settled inBirmingham, Alabama, where Nathan received most of his schooling.Later the family moved to Columbus, Mississippi, but the youngNathan entered the University of Alabama in 1926 and graduatedin 1930. His initial aim was to follow an uncle and obtain adegree in law, but at the same time he took all the (not verynumerous) mathematics courses, in which he did so well thathe was offered a teaching assistantship in mathematics in hisjunior (3rd) year. This marked a turning point; he now decidedto major in mathematics and pursue this study beyond College.During his final year at Alabama he applied for admission andfinancial aid to three top graduate schools in the country:Princeton, Harvard and Chicago. He was awarded a research assistantshipat Princeton; after the first year he was appointed a part-timeinstructor for two years, and during his fourth year he wasappointed a Procter Fellow. The stipend was enough to enablehim to make a grand tour of Europe by car in 1935, in the companyof two Princeton fellow-students at the time: H. F. Bohnenblustand Robert J. Walker.     

Transformations Between Sturm-Liouville Problems with Eigenvalue Dependent and Independent Boundary Conditions

Explicit relationships are given connecting ‘almost’isospectral Sturm-Liouville problems with eigen-value dependent,and independent, boundary conditions, respectively. Applicationis made to various direct and inverse spectral questions. 2000Mathematics Subject Classification 34B24, 34L05, 47E05.     

Wild Spectral Behaviour of Anharmonic Oscillators

We show that the norms of the sequence of spectral projectionsof certain non-self-adjoint ordinary differential operatorscannot be polynomially bounded. The proof involves the investigationof ‘tame’ bi-orthogonal systems in Banach spaces.2000 Mathematics Subject Classification 34L10, 46B15.     

Existence of Solutions for Asymptotically 'Linear' p-Laplacian Equations

Under very general conditions, a proof is given, via Morse theory,of the existence of solutions for asymptotically ‘linear’p-Laplacian equations, where the asymptotic limit may be greaterthan the second eigenvalue. The existence of nonzero solutionsis also considered. 2000 Mathematics Subject Classification35J65, 58E05.     

On the Tightness of Capacities Associated with Sub-Markovian Resolvents

This paper investigates the tightness property of the capacityinduced by the reduction operator with respect to the resolventof a right Markov process. Tightness is verified in two particularsituations: under the ‘weak duality hypothesis’,and if a substitute for ‘the axiom of polarity for thedual theory’ holds. In the second context, the quasi-continuityproperty for the excessive functions is derived. These are extensionsand improvements of results of Lyons and Röckner, Ma andRöckner, Le Jan, and Fitzsimmons, mainly obtained in thecontext of Dirichlet forms. 2000 Mathematics Subject Classification31C25, 60J45 (primary), 31C15, 60J40 (secondary).     

Monstrous Moonshine: The First Twenty-Five Years

Twenty-five years ago, Conway and Norton published in this journaltheir remarkable paper ‘Monstrous Moonshine’, proposinga completely unexpected relationship between finite simple groupsand modular functions. We review the progress made in broadeningand understanding that relationship. 2000 Mathematics SubjectClassification 11F22, 17B69, 17B67, 81T40.     

Generic Projections of Stable Mappings

Mather has shown that almost all linear projections of a smoothsubmanifold of a vector space to a subspace yield a genericmapping (so in the nice dimensions, most linear projectionsgive rise to ‘stable’ mappings). In this paper weshow that the same is true if we replace the submanifold bythe image of a stable mapping. 2000 Mathematics Subject Classifiation58C.     

Un Theoreme De Finitude Dans Le Spectre Automorphe Pour Les Formes Interieures De GLn Sur Un Corps Global

A proof is given to show that for an inner form of GL_n overa global field of zero characteristic, there exist only a finitenumber of automorphic representations with fixed local factor(up to equivalence) at almost every place. What is new in comparisonto earlier work (see A. I. Badulescu and P. Broussous, ‘Unthéorème de finitude’, Compositio Math.132 (2002) 177–190) is the case when the local factorsare not fixed at the infinite places, as well as the statementof the result for the automorphic spectrum, rather than thecuspidal one. 2000 Mathematics Subject Classification 11F70.     

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.     

Cubic Forms Modulo One

By using a more efficient way of ‘almost diagonalizing’,known results on small fractional parts of general real cubicforms in a small number of variables, are improved. In particular,for ternary forms the method gives a bound superior to thatarising from simply reducing to a cubic form in one variable.2000 Mathematics Subject Classification 11D75, 11E76, 11J54.     

On the Hecke Equivariance of the Borcherds Isomorphism

The Borcherds isomorphism is proved to be Hecke equivariantif one considers multiplicative Hecke operators acting on theintegral weight meromorphic modular forms. This answers a partof a question of Borcherds (see ‘Automorphic forms onO_{s+2, 2}(R) and infinite products’, Invent. Math. 120 (1995)161–213, 17.10), using his suggestion to define the multiplicativeHecke operators. 2000 Mathematics Subject Classification 11F37.     

How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.