Preface. In order to make this paper intelligible to the reader, I repeat the following definitions and results from my paper ‘On lattice points in star domains’, which is to appear in the Proceedings of the London Mathematical Society.

In this paper, a method is given for solving certain types of integral equations. It involves a technique of building up the solution of the given integral equation from that of a much simpler auxiliary equation and a consideration of this auxiliary equation forms a large section of the work. The method of solution is applied to a particular problem in semi-circular electron focusing in metals.

In this paper, we consider the family of the p-groups of maximal class and study some invariants related to the conjugacy classes of these groups. This subject was first taken up in [7] and [6], whose notation and terminology we maintain and give after this introduction. In [7], the invariants σG and k(G) are found for the p-groups of maximal class satisfying |G| ≤ p7. Here, apart from extending those results to the case when G has order p8 or p9, we consider the problem in all its generality and propose to determine their values for as large a family of p-groups of maximal class as possible. In this direction, we get some general results in the first paragraph, from which we want to point out the following ones.

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequality

has infinitely many solutions if and only if

diverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

1. I appreciate the careful reply which Dirac, Peierls and Pryce (hereafter referred to as DPP) have made to my criticism of the usual theory of the eigen-functions of a hydrogen atom. Their paper will be generally welcomed because there has not, I think, previously been available an authoritative and clear statement of what is assumed and the grounds for assuming it. Their defence, which is partly on lines I had not expected, has required very serious consideration. But the result of making the arguments more explicit is two-fold. The theory is perhaps more self-consistent than it had appeared to be; but on the other hand, the pressing need for amendment becomes too plain to be overlooked. I have endeavoured to show in the last twelve years that this amendment opens out very fertile extensions.

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.

Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

A Banach space is c0-saturated if all of its closed infinite-dimensional subspaces contain an isomorph of c0. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that c0-sums of c0-saturated spaces are c0-saturated; (2) C(K, E) is c0-saturated if both C(K) and E are; (3) the tensor product is c0-saturated, where JH is the James Hagler space

An amenable semigroup of positive linear unital mappings on a W*-algebra is considered. Two main questions are dealt with: the existence of a normal faithful state invariant with respect to this semigroup and the description of ergodicity conditions. An explicit form of the ergodic projection, useful in treating the ergodicity problems, is also derived.

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equation

where u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.

We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have non-vanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.

This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

Consider a sphere of radius $\sqrt{n}$ in $n$ dimensions, and consider $\vc{X}$, a random variable uniformly distributed on its surface. Poincaré's Observation states that for large $n$, the distribution of the first $k$ coordinates of $\vc{X}$ is close in total variation distance to the standard normal $N(\vc{0},I_k)$. In this paper we consider a larger family of manifolds, and $\vc{X}$ taking a more general distribution on the surfaces. We establish a bound in the stronger Kullback–Leibler sense of relative entropy, and discuss its sharpness, providing a necessary condition for convergence in this sense. We show how our results imply the equivalence of ensembles for a wider class of test functions than is standard. We also deduce results of de Finetti type, concerning a generalisation of the idea of orthogonal invariance.

Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integer n as the sum of s r-free integers. For s ≥ 4 their formula was subsequently sharpened by Barham and Estermann, and for s = 3 recently by me.

About twenty years ago a number of results were given expressing the sum of n terms of an ordinary hypergeometric series with unit argument in terms of an infinite series of the type 3F2. The interest in the subject was aroused by a theorem due to Ramanujan, who stated that

We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).

Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?

It is well known that a path-connected Hausdorff space is arc-connected. Indeed, given a path P, that is, a continuous map. f of the unit interval I0(0 ≤ t ≤ 1) into the Hausdorff space X, there is an arc joining the points f(0), f(1) (supposed distinct) which is obtained by ‘cutting out loops in P’. More precisely, there exist a continuous increasing map α of I0 onto itself and a homeomorphism φ of I0 onto φ(I0) ⊂ X, such that if [t0, t1] is any maximal interval of constancy of α (including the case t0 = t1) then f(t0) = f(t1) = φοα(t0). The function φ, α, can be defined by a constructive process.

Abelian theorems for the H-transform of functions and generalized functions are obtained as the complex variable of the transform approaches zero or infinity in a wedge domain in the right half plane. Quasi-asymptotic behaviour (q.a.b.) of the H-transformable generalized functions is defined. A structure theorem for generalized functions possessing q.a.b. is proved and is applied to obtain the asymptotic behaviour of the H-transform of generalized functions having q.a.b. The theorems are illustrated by examples.

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined as

and the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).