Eric Primrose was appointed to a Lectureship in Pure Mathematics at the University College, Leicester, in 1947 and promoted to a Senior Lectureship in 1954. He came to Leicester direct from Oxford where he had spent the 1946–47 year completing his degree, which had been interrupted by war service. Eric was awarded his PhD in 1957 by the University of London.

Eric's secondary education was at Chigwell School and in 1939 he won an Open Scholarship in Mathematics and went up to St John'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 various locations throughout Europe.

The nonexistence of a submersion from the 23-sphere to the Cayley projective plane is proved.

It is proved that a ring $R$ is right Artinian if and only if, for each countably generated right $R$-module $M$, there exists a finite subset $F$ of $M$ such that the annihilator of $M$ in $R$ equals the annihilator of $F$ in $R$.

In this note we wish to prove a purely characteristic p > 0 variant of the Kodaira–Akizuki–Nakano vanishing for smooth complete intersections of dimension at least two in projective space. This has some interesting applications; in particular, we show that all Frobenius pull-backs of the tangent bundle of any complete intersection of general type and of dimension at least three in Pn are stable. We also show (see Remark 3.4) that a small modification of the techniques of [5] and a theorem of Mehta and Ramanathan (see [3]) together allow us to extend this stability result to smooth projective hypersurfaces of degree d, where (n + 1)/2 < d < n + 1 (that is, to some Fano hypersurfaces).

It is well known that behaviour of stability under Frobenius pull-backs is a subtle problem of the theory of vector bundles in characteristic p > 0, and hence this result is not without interest. We end with an obvious conjectural form of our variant for a general class of varieties.

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).

Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

1. Introduction

The classical circle method of Hardy and Littlewood (see Baker [1]) ensures that the diophantine equation

a1xk1+[ctdot ]+ asxks =0 (1.1)

has a nontrivial solution in nonnegative integers x1, [ctdot ], xs, provided s[ges ]C(k) and the coefficients a1, [ctdot ], as are all integers and not all the same sign.

This paper introduces the concept of orbit-homogeneity of permutation groups: a group $G$ is orbit-$t$-homogeneous if two sets of cardinality $t$ lie in the same orbit of $G$ whenever their intersections with each $G$-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of $t$-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan.

Further, this paper shows that any group generated by orbit-$t$-homogeneous subgroups is orbit-$t$-homogeneous, and that the condition becomes stronger as $t$ increases up to $\lfloor n/2\rfloor$, where $n$ is the degree. So any group $G$ has a unique maximal orbit-$t$-homogeneous subgroup $\Omega_t(G)$, and $\Omega_t(G)\le\Omega_{t-1}(G)$. Some structural results for orbit-$t$-homogeneous groups, and a number of examples, are also given.

A description is given of serial rings whose maximal quotient rings are quasi-Frobenius (QF). Every serial ring is a factor of a serial ring whose maximal quotient ring is a QF-ring. This result is used to give a new, conceptual proof for the selfduality of serial rings, emphasising the importance of weakly symmetric rings.

A C*-algebra A is said to be monotone (respectively monotone σ-) complete if every increasing net (respectively increasing sequence) of elements in the ordered space Ah of all hermitian elements of A has a supremum in Ah. It is straightforward to verify that every monotone complete C*-algebra is an AW*-algebra. For type I AW*-algebras, the converse is known to be true. However, for general AW*-algebras, this question is still open, although an impressive attack on the problem was made by E. Christensen and G. K. Pedersen, who showed that properly infinite AW*-algebras are monotone σ-complete [4].

The most natural candidates to the definition of equisingularity of germs of complex space curves are discussed, showing that they coincide in the case of Lagrangean curves (projectivizations of conic Lagrangean varieties of a symplectic manifold of dimension four).

Well over two decades have now passed since the publication of the classic books Singular integrals and differentiability properties of functions by E. M. Stein [32] and An introduction to Fourier analysis on Euclidean spaces by E. M. Stein and G. Weiss [37]. These two texts would, I am sure, be universally regarded as defining the ‘common core’ of harmonic analysis in the Calderón–Zygmund tradition in the early 1970s. What has been going on in the subject since then?

It is shown that known solvability conditions and several new existence results for semilinear elliptic boundary value problems at resonance can be considered from a unified point of view. The proofs apply Leray–Schauder degree arguments.

The purpose of this note is initially to present an elementary but surprising connectedness principle pertaining to the intersection of a fixed subvariety X of some ambient space Z with another subvariety Y which is ‘mobile’ (in the sense of being movable, rather than actually moving). It is via this mobility that monodromy enters the picture, permitting the crucial passage from ‘relative’ or total-space irreducibility to ‘absolute’ or fibrewise connectedness (and sometimes irreducibility). A general form of this principle is given in Theorem 2 below.

Oscillation criteria for the first-order linear delay differential equation

formula here

are established. Our results improve the known results in the literature.

For a semigroup $S$, the finitary power semigroup of $S$, denoted $P_f(S)$, consists of all finite subsets of $S$ under the usual multiplication. The main result of this paper asserts that $P_f(G)$ is not finitely generated for any infinite group $G$.

We show that the analytic rank, as defined by Murphy, of a unital graph C*-algebra is either 1 or 0, depending on whether or not the underlying graph possesses an initial loop. As a corollary, we show that the analytic rank of certain quantum spaces (including some quantum spheres, projective spaces and lens spaces) is 1.

We consider the equation

a1x1 + ··· + anxn = 1, (1.1)

where the coefficients a1, …, an are non-zero elements of a field K, to be solved for (x1, …, xn) in a finitely generated multiplicative subgroup G of (K*)n. Evertse [2] and van der Poorten and Schlickewei [6] showed, independently, for the most important case in which K is a number field and G is a group of S-units in K, that (1.1) has only finitely many solutions (x1, …, xn) for which no subsum in (1.1) vanishes. Such solutions are called non-degenerate.

In 1990, Schlickewei [7] was the first to obtain an explicit upper bound for the number of non-degenerate solutions of (1.1) when G is a group of S-units of K. His result went through various successive improvements, and the best result to date is the bound (235n2)n3s by Evertse [3, Theorem 3], where s is the cardinality of S. Very recently, Evertse, Schlickewei and Schmidt [4, 5] obtained a remarkable result. They showed, again for K a number field, that the number of non-degenerate solutions of (1.1) with (x1, …, xn) in a finitely generated subgroup of (K*)n of rank r is at most c(n)r+2, with c(n) = exp((6n)4n). The importance of this result lies with its uniformity with respect to the rank r of the group and its independence of the field K. The proofs of the above-mentioned results are all quite difficult and depend on deep tools from Schmidt's Subspace Theorem and diophantine approximation. In this paper we consider equation (1.1) in the rational function field K = k(t) where k is an algebraically closed field of characteristic 0. Of course, results of this type will follow rather easily from the above-mentioned results by a specialization argument. However, our object is to show that our approach to the function field case is quite elementary and certainly very different, being a simple and direct consequence of the powerful abc-theorem.

Before stating our result, we first define proportional solutions of (1.1). We say that (x1, …, xn) and (x′1, …, x′n) are proportional if xi, /, x′i ∈ k, 1 [les ] i [les ] n. This determines equivalence classes of solutions.