General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ + Mδc, in terms of those of the measures dμ and (x − c)2dμ. In particular, these relations allow us to show that Nevai's class M(0, 1) is closed under adding a mass point, as well as obtain several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

Assuming a conjecture on distinct zeros of Dirichlet $L$-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

A formal solution of the coupled linear thermoelastic equations representing oscillatory motion of an elastic body is obtained, and the decay of the normal modes of vibration examined in detail. The main results of Zener's theory of the thermal damping of a vibrating body are then derived, and somewhat amplified, and consistency with the foregoing normal mode analysis established.

This paper is concerned with marginal convection in a self-gravitating sphere of uniform incompressible fluid containing a uniform distribution of heat sources. Its purpose is twofold. The first aim is to present the mathematical argument in a form which, the author believes, is more succinct than that which has been given heretofore. The second aim is to determine the effect of the convective motions upon the moments of inertia of the body and, in the light of the results obtained, examine briefly the hypothesis that the moon is in a state of convection.

We construct realizations of Dyck's regular map of genus three as polyhedra in ℝ3. One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dyck's regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.

Let S be an ordered set, i.e. a set with a transitive irreflexive binary relation “<” such that, for any a, bεS, either a = b or a < b or b < a. By an order automorphism of S we mean a one-one mapping α of S onto itself such that

In this note I settle a question which arose out of my first paper under the above title (cf. [1]), where I considered the classgroup C(Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C() of the maximal order of the rational groupring Q(Γ), and C() is the product of the ideal classgroups of the algebraic number fields which occur as components of Q(Γ) and is thus in a sense known. One is then interested in the kernel D(Z(Γ)) of C(Z(Γ)) → C() and in its order k(Γ). In [1] I proved that, for Γ a p-group, k(Γ) is a power of p. I also computed k(Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q(n) with n4 = 1 or n6 = 1. The numerical results obtained led me to the question whether in fact k(Γ) tends to infinity with the order of Γ.

Let h1(x1, …, xn), …, hs(x1, …, xn) be polynomials with integer coefficients. We give conditions on these polynomials which guarantee the existence, for all sufficiently large primes p, of small solutions to the system of congruences

Previous investigations of this problem include those of Mordell [10], Chalk and Williams [5], and Smith [14]. Smith's main result, which encompasses the other results, can be stated as follows.

Let Q(x1 …, xn) be an indefinite quadratic form in n variables with real coefficients. Suppose that when Q is expressed as a sum of squares of real linear forms, with positive and negative signs, there are r positive signs and n—r negative signs. It was proved recently by Birch and Davenport that, if

then for any ε > 0 the inequality

is soluble in integers x1, …, xn, not all 0.

It was proved by Heilbronn that if ε > 0, N > 1 and ϑ is any real number then there exists an integer n satisfying 1 ≤ n ≤ N such that

where C depends only on ε. Here ║α║ denotes the difference between α and the nearest integer, taken positively. Professor Heilbronn has remarked (in conversation) that the exponent of N cannot be decreased beyond -1, since if p is an odd prime and a is not divisible by p then

for 1 ≤ n ≤ p—1. He has also remarked that if one could improve the exponent of N to -1 + η, say, it would follow that the absolutely least quadratic non-residue (mod p) is less than Cpn. For if a is a quadratic non-residue (mod p) then so is each of the numbers an2 (1 ≤ n ≤ p—1) and ║an2/p║<Cp-1+η implies that an2 is congruent (mod p) to a number of absolute value less than Cpη.

Let a1, …, am and b1, … bm be non-negative real numbers. The well-known inequality of Minkowski states that

if n ≥ 1. If n is a positive integer, this inequality asserts a property of a particular symmetric form (i.e. homogeneous polynomial) in m variables, namely the sum of the n-th powers of the variables. Some time ago, Prof. A. C. Aitken conjectured that similar properties are possessed by certain other symmetric forms. In particular, let E(n)(a) denote the n-th elementary symmetric function of a1, …, am and let C(n)(a) denote the n-th complete symmetric function of a1, …, am, the formal definitions being

Then Prof. Aitken conjectured that

Integral geometry is the study of measures of sets of geometric figures. Commonly a measure of this sort is an integral of a density or differential form; the density is determined by the type of figure, but is independent of the particular set of such figures to which the measure is assigned. As one of the simplest examples, the area of a plane convex point set K is the integral over K of the density dx dy for points with Cartesian co-ordinates x, y. But when we assign a Hausdorff linear measure to the set of boundary points of K, we obtain a measure of quite another sort. This is representable as a Stieltjes integral of arc length density; here the density depends on the choice of K. The examples suggest examining measures for other sets of figures, where each such set is made up of all those figures from a certain class which support, in some sense, a convex body. Further, the examples lead us to expect that measures of this kind will appear as integrals of densities which may depend on the choices of . Here we treat a question of the type just described: to determine a measure for sets of q–flats which support a convex body.

Let d≥2 and let K⊂ℝd be a convex body containing the origin 0 in its interior. For each direction ω, let the (d−l)-volume of the intersection of K and an arbitrary hyperplane with normal ω attain its maximum when the hyperplane contains 0. Then K is symmetric about 0. The proof uses a linear integro-differential operator on Sd−1, whose null-space needs to be, and will be determined.

For a primitive root g modulo a prime p≥1 we obtain upper bounds on the gaps between the residues modulo p of the N consecutive powers agn, n=1,…,N, which is uniform over all integers a with gcd (a,p)=1.

In this paper I show that complex-valued multiplicative functions g which satisfy |g(n)|≤1 for all positive integers n, are generally well distributed in residue classes to small moduli.

If K is a set in n-dimensional Euclidean space En, n ≥ 2, with non-empty interior, then a point p of En is called a pseudo-centre of K provided tha each two dimensional flat through p intersects K in a section, which is either empt or centrally symmetric about some point, not necessarily coinciding with p. A pseudo centre p is called a false centre of K, if K is not centrally symmetric about p. A dee result of Aitchison, Petty and Rogers [1] asserts that, if K is a convex body in E and p is a false centre of K, with p in the interior, int K, of K, then K is an ellipsoid Recently J. Höbinger [2] extended this theorem to any smooth convex body K witl a false centre p anywhere in En. At a recent meeting in Oberwolfach, he asked if the condition of smoothness can be omitted and the purpose of this note is to prove sucl a result.

Let p be an odd prime and denote by [p], the finite field of residue classes, mod p. In Euclidean n-space, let n denote the lattice of points x = (x1, …, xn) with integral coordinates and C = C(n, p), the set of points of n satisfying

A polygon, with all sides and all diagonals of rational length, will be called a rational polygon.

Prof. I. Schoenberg has set a problem whether rational polygons are everywhere dense in the class of all polygons, that is, given an arbitrary polygon whether there exists a rational polygon whose sides and diagonals are of length arbitrarily near to those of the given one. Once set, the problem becomes very interesting both for its simplicity and for its fundamental nature. For obvious reasons the likely answer to the problem is in the negative. In this note I consider two problems simply related to the above problem.

A Borel isomorphism that, together with its inverse, maps ℱσ-sets to ℱσ-sets will be said to be a Borel isomorphism at the first level. Such a Borel isomorphism will be called a first level isomorphism, for short. We study such first level isomorphisms between Polish spaces and between their Borel and analytic subsets.

How many digraphs are isomorphic with their own converses? Our object is to derive a formula for the counting polynomial dp′(x) which has as the coefficient of xq, the number of “self-converse” digraphs with p points and q lines. Such a digraph D has the property that its converse digraph D′ (obtained from D by reversing the orientation of all lines) is isomorphic to D. The derivation uses the classical enumeration theorem of Pólya [9[ as applied to a restriction of the power group [6] wherein the permutations act only on 1–1 functions.