Brian Hartley began his algebraic career as one of Philip Hall's research students in Cambridge. He obtained his Ph.D. in 1964, spent two post-doctoral years in the USA and, on his return to the United Kingdom, accepted a lectureship in the newly established Mathematics Department at Warwick University; there he was promoted to a readership in 1973. He was appointed to a chair of pure mathematics at the University of Manchester in 1977 and was Head of the Mathematics Department there from 1982–4. He was elected to the London Mathematical Society in 1968 and served on Council from 1987–9. He won an EPSRC Senior Research Fellowship, but died on 8 October 1994, a few days after taking it up. He travelled widely and took a lively interest in other cultures and languages. His intellectual energy, enthusiasm for algebra, direct manner and dry sense of humour endeared him to the many mathematical friends he made around the world. He was devoted to mathematics and gave generously of his time and energy in support of younger colleagues.

The quantum stochastic differential equation dkt = kt ∘ θαβdΛβα(t) is considered on a unital C*-algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non-commutative topological space.

It is proved that the cone length or strong category of a product of two co-H-spaces is less than or equal to two. This yields the following positive solution to a problem of Ganea. Let α ∈ π2p(S3) be an element of order p, p a prime [ges ] 3, and let X(p) = S3∪αe2p+1. Then X(p) × X(p) is the mapping cone of some map φ : Y → Z where Z is a suspension.

The local Caratheodory conjecture on the index of an isolated singularity of the principal foliations in surface theory is equivalent to a conjecture of Loewner on the index of the isolated singularities of the Hessian operator of a smooth function in the plane. Here we prove the latter conjecture for a special class of functions.

The paper considers the asymptotic behaviour of the heat content of a riemannian manifold M, when Dirichlet conditions are imposed on a small compact subset of M.

Nathan Jacobson, who died on 5 December 1999, was an outstanding algebraist, whose work on almost all aspects of algebra was of fundamental importance, and whose writings will exercise a lasting influence. He had been an honorary member of the Society since 1972.

Nathan Jacobson (later known as ‘Jake’ to his friends) was born in Warsaw (in what he describes as the ‘Jewish ghetto’) on 5 October 1910 (through an error some documents have the date 8 September); he was the second son of Charles Jacobson (as he would be known later) and his wife Pauline, née Rosenberg. His family emigrated to the USA during the First World War, first to Nashville, Tennessee, where his father owned a small grocery store, but they then settled in Birmingham, Alabama, where Nathan received most of his schooling. Later the family moved to Columbus, Mississippi, but the young Nathan entered the University of Alabama in 1926 and graduated in 1930. His initial aim was to follow an uncle and obtain a degree in law, but at the same time he took all the (not very numerous) mathematics courses, in which he did so well that he was offered a teaching assistantship in mathematics in his junior (3rd) year. This marked a turning point; he now decided to major in mathematics and pursue this study beyond College. During his final year at Alabama he applied for admission and financial aid to three top graduate schools in the country: Princeton, Harvard and Chicago. He was awarded a research assistantship at Princeton; after the first year he was appointed a part-time instructor for two years, and during his fourth year he was appointed a Procter Fellow. The stipend was enough to enable him to make a grand tour of Europe by car in 1935, in the company of two Princeton fellow-students at the time: H. F. Bohnenblust and Robert J. Walker.

Dr Jack Howlett, who died on 5 May 1999 at the age of 86, a Founder Fellow of the Institute of Mathematics and its Applications, was a mathematician with a special interest in numerical analysis who early recognised the power of computing methods and who strongly influenced the development of some mechanical computing machines and electronic computers as we know them today. The widespread advance of the use of computer models in all scientific disciplines was made possible thanks to the efforts of a small number of mathematicians who laid the foundations of modern numerical analysis during the late 1930s to early 1960s, a period which covered the greater part of Jack's working life and in which he was a star player.

Explicit relationships are given connecting ‘almost’ isospectral Sturm–Liouville problems with eigen-value dependent, and independent, boundary conditions, respectively. Application is made to various direct and inverse spectral questions.