We construct several examples of positive definite functions, and use the positive definite matrices arising from them to derive several inequalities for norms of operators.

Eric Charles Milner was born on 17 May 1928 and brought up in London. His father was an engineer, but times were hard and work was often difficult to obtain. So his mother had to help out by working as a seamstress, and Eric was often looked after by his grandmother. At the age of 11, he won a scholarship to the Haberdashers' Aske's Boys' School, but never attended it in its permanent London buildings because the outbreak of the Second World War caused all London schools and their pupils to 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, after unsuccessful attempts to find him another billet, he roamed the streets and missed school. After some time, he was eventually found another billet where he received kindness and was much happier. Despite these disruptions and the other inevitable shortcomings of a war-time education, Eric's intelligence more than sufficed to surmount such hurdles, and in later life he could speak and write better than most of us.

From 1946 to 1951, Eric attended King's College, London. He graduated with First Class Honours in 1949, when he was awarded the Drew Gold Medal as the most distinguished Mathematics student in that year, and a Research Studentship. He then studied for an MSc degree, taking ‘Modern algebra’ and ‘Quantum mechanics (Wave mechanics)’ as his selected subjects, his supervisors being Richard Rado (then a Reader at King's College) and Professor Charles Coulson. He received the MSc degree, with distinction, in 1950. This was followed by a year's research in quantum mechanics under the supervision of Professor Coulson.

We study the behaviour of non-negative solutions to the equation

formula here

This equation is an ‘explicitly’ soluble special case of a class of non-local evolution equations, of which the behaviour of solutions has been studied by Uchiyama [9]. In this paper, certain fine properties of solutions such as a local relaxation within their supports are obtained for the present special case. The asymptotic forms (for large time) of the solutions whose initial measures have compact supports are identified within the error of the order O(t−3/2). It is also shown that the comparison principle does not hold. With a simple change of variables, the equation ([midast ]) is transformed to the equation arising in Dyson's approach to the Wigner semi-circle law of eigenvalues of random matrices, and the present results have immediate consequences on the latter.

We show that operators on n × n matrices which are representable in the form T(X) = [sum ][lscr ]i=1aiXbi (where ai and bi are n × n matrices) and are k-positive for k = [√[lscr ]] must be completely positive. As a consequence, elementary operators on a C*-algebra with minimal length [lscr ] which are k-positive for k = [√[lscr ]] must be completely positive.

Let X be a separable infinite dimensional Banach space. There exist a closed set A ⊂ X which contains a translate of any compact set in the unit ball of X, and a bi-Lipschitz homeomorphism f of X onto X so that every line in X intersects f(A) in a set of Lebesgue measure zero.

Given an oriented manifold and its immersion in a euclidean space, we compute the oriented cobordism class of the manifold of [sum ]1r singular points of the projection of the immersion to a hyperplane. For immersions of non-oriented manifolds, we show that the cobordism class of the domain manifold determines those of all [sum ]1r singularity manifolds of the hyperplane projection. Finally, we investigate the possible (algebraic) number of cusps (that is, [sum ]1,1 singular points) of generic maps of oriented 4t-manifolds in R6t−1.