Let Un be a U-statistic whose kernel depends on the size n of the sample under consideration. It is shown that when Un is suitably normalised its distribution function differs in Lp norm from the distribution function of a standard normal variable by a term of O(n-½).

Let K be a bounded closed convex set in the plane containing no points of the integral lattice in its interior and having width w, area A, perimeter p and circumradius R. The following best possible inequalities are established:

In this paper some nonlinear analogues of Gronwall's integral inequality are established and an application to differential equations is given.

In this note it is shown that any ribbon link is a sublink of a ribbon link for which surgery on the longitudes gives a connected sum of copies of S1 x S2. In particular there are many links for which the analogue of the knot theoretic Property R fails, and sublinks of homology boundary links need not be homology boundary links. Higher dimensional analogues of these results are also given and it is shown that if n ≥ 2 the group of a ν-component ribbon n-link has a presentation of deficiency ν. Hence there are high dimensional slice knots which are not ribbon knots.

This paper establishes the radii of convexity of the following two classes of regular functions,

where

.

The following theorem is proved: Let D be a division ring such that for all x, y in D there exists a positive integer n = n(x, y) for which (xy)n − (yx)n is in the center of D. Then D is commutative. This theorem also holds for semisimple rings.

In this paper we investigate the spectrum of a variety of quasigroups satisfying the 2-variable identity x(xy) = yx, called Stein quasigroups. Stein quasigroups are known to be self-orthogonal and have been given a considerable amount of attention because of this property. It is known that there are no Stein quasigroups of order 2, 3, 6, 7, 8, 10, 12, 14. The object of this paper is to show that for all but 36 values of n ≥ 15 there exists a Stein quasigroup of order n. In particular, the spectrum of Stein quasigroups contains all n ≥ 191.

In this paper, a class of systems governed by second order linear parabolic partial delay-differential equations in “divergence form” with Cauchy conditions is considered. Existence and uniqueness of a weak solution is proved and its a priori estimate is established.

In this paper we prove a G-invariant implicit function theorem and indicate how it can be used to improve an earlier result of the author on the bifurcation of solutions of nonlinear equations in the presence of continuous groups of symmetries. We also use our theorem to show that, under reasonable hypotheses, the method of looking for solutions in invariant subspaces yields all solutions. This can be used to answer a question raised by Sattinger [J. Math. Phys. 19 (1978), 1729]. The abstract result is also of interest because it provides a theorem which should be of use in other symmetric situations.

We prove, in normed linear spaces, the existence of extensions of continuous linear functionals from linear subspaces to the whole space, with arbitrarily prescribed larger norm. Also, we prove that under an additional boundedness assumption, in the known separation theorems for convex sets, there exist hyperplanes which separate and support both sets.

Convergence notions are used extensively in the areas of probability and statistics. Many times proofs can be simplified by considering an appropriate convergence structure on the space and using well-known results from the theory of convergence spaces; for example, compactness arguments are sometimes simplified by using a generalized Ascoli theorem in the convergence space setting. The theory of convergence spaces is also used to generalize some results in probability and statistics.

In this note sufficient conditions for non-compactness of composition operators on two different functional Hilbert spaces have been obtained.

It is shown that if I is a finitely generated ideal of a commutative ring R such that the multiplication map I ⊗RI → I is an injection, then I is locally principal. As a corollary, one obtains a new homological characterization of Prüfer domains.

The familiar notion of the determinant is generalised to include rectangular matrices. An expression for a normalised generalised inverse of a matrix is given in terms of its determinant and a possible generalisation of the Schur complement is discussed as a simple application.