Let G be a locally compact Abelian non-discrete group. Let M(G) be the convolution algebra of Radon measures on G. Let µ be an element of M(G) with its Lebesgue decomposition [1]

into absolutely continuous, purely discontinuous and continuous singular parts. The chief problem one encounters in the study of the invertibility of µ is with the case µs ≠ 0. As observed by Wiener and Pitt [2], the problem can be handled provided µs be “not too large”. In fact, Wiener and Pitt (loc. cit.) proved the following:

Let µ be a Radon measure on R (the real line) such that

whereare the Fourier transforms of µ, µd, and ≑µs≑ is the variational norm of µs. Then, µ has an inverse in M (R).

A transversely isotropic elastic material can transmit three body waves in each direction, a quasi-longitudinal (QL) wave, a quasi-transverse (QT) wave, and a purely transverse wave. When the material is able to conduct heat the properties of small amplitude QL and QT waves are modified and we consider here the analysis of such thermo-elastic interactions in plane harmonic disturbances. The modified QL and QT waves are both found to exhibit frequency-dependent dispersion and damping of the kind known to affect dilatational waves in isotropic heat-conducting elastic materials, and in addition we show that the particle paths in the associated motions are ellipses with their axes inclined to the wave normal. This latter effect is peculiar to body waves travelling in anisotropic heat-conducting elastic materials and seems not to have been studied in detail hitherto. Numerical results referring to the propagation of plane harmonic body waves in a single crystal of zinc are presented and discussed.

Erdös and Straus have conjectured that for every integer n > 1,

is soluble in positive integers x, y, z. Schinzel has conjectured that for every a > 0 if n > no(a),

is soluble in positive integers x, y, z.

The fluid mechanical problem with which we are concerned is the behaviour of fluid occupying the half-space x > 0 above a rotating disc which is coincident with the horizontal plane x = 0 and rotating about its axis which remains fixed. Studying rotationally symmetric solutions of this problem, von Kármán [1] (see also [2; p. 93]; [3; p. 133]) reduced it to the solution of two simultaneous equations in functions f(x), g(x) which may, with suitable normalisation, be written in the form

with the boundary conditions

Generating functions for the number of linearly independent invariants of a set of tensors under a given group of transformations are given by the theory of group representations. For the full and proper orthogonal groups these generating functions are in the form of definite integrals. The classical theory of algebraic invariants gives generating functions for the number of invariants of tensors under two-dimensional unimodular transformations, these generating functions being algebraic expressions. Because of a correspondence between the two-dimensional unimodular group and the three-dimensional proper orthogonal group, the corresponding generating functions are equivalent. The main result of this paper is an explicit demonstration of this equivalence. In addition, algebraic generating functions for the three-dimensional full orthogonal group are obtained and the use of the algebraic generating functions illustrated by applying them to a third order symmetric tensor.

In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible number of j-faces of a d-polytope with v vertices is achieved by a cyclic polytope C(v, d).

It is natural to say that a set S in a metric space has infinite generalized Hausdorff dimension if there is no Hausdorff measure Λh with Λh(S) = 0. In this note we study such sets. We first need some definitions.

We say that h(x) is a Hausdorff measure function if it satisfies the conditions:

A method of “inner and outer expansions” employed by Cox and Brenner (1967) is used to calculate the hydrodynamic force experienced by either of two identical small solid spheres, approaching each other, at low Reynolds number, with the same velocity along their line of centres, for the limiting case when the gap width between them tends to zero. Numerical results are compared with those furnished by the exact solution and an improved expression is then obtained for a similar problem arising from the motion of a small sphere towards an identical stationary sphere, in the limit of small gap widths.

All of our work takes place in Ed, d-dimensional Euclidean space, with unit ball B. Unless specifically noted to the contrary, all sets will be presumed to be closed and convex. If X is a subset of a sphere S with centre p, we will say X is spherically convex if X is contained in some open hemisphere of S and if the cone generated by X with vertex p is convex. The distance between two points x, y ε Ed will be denoted |x − y|. If K1K2 are two convex sets, ρ(K1, K2) will mean the usual Hausdorff distance between them.

It is well known that Cantor's ternary set C, constructed on [0, 1], has a difference set, D(C), equal to [0, 1]. If E ⊂ R we define Dk(E)(⊂ Rk-1), by Dk(E) = {(d1, d2, …, dk-1); di ≥ 0, and there is x ∈ E such that x + di ∈ E for all i, 1 ≤ i < k}. Thus D2(E) ≡ D(E) and Dk(E) tells us whether or not a particular set of k real numbers can be translated into E. We call Dk(E) the k-difference set of E. In this work we seek criteria for finding the Besicovitch dimension of Dk(E) (written dim Dk(E))) and in particular, conditions on certain classes of linear sets E that ensure that Dk(E) should contain an open interval in Rk-1.

Closed form solutions are obtained for a class of singular integral equations of the first kind with difference kernels. The kernel function is the sum of a polynomial and a second polynomial multiplied by a logarithm, with the possible addition of a strong singularity. A wider class of kernels have approximate representations in one of the above forms, where, for a specified accuracy, the polynomial orders will depend on the range of any parameter present. For example, the modified Bessel function kernels K0(|γx|), K1(|γx|)sgn x in the interval |x| ≤ 2 can be so expressed using 16th order polynomials with a maximum error 10-5 for real γ in the range 0 < γ ≤ 4.

The assertions about the congruence properties of the coefficients Ci, Di, in formula (3) of the paper (Mathematika, 16 (1969), 101–105) referred to above are patently false. They are, moreover, irrelevant, though the succeeding argument is so condensed and badly expressed that the true and the false in the paper cannot easily be distinguished. In short, the paper ought to be re-written. My thanks are due to the critic who first complained of the mistakes to the Editors and who thereby took upon himself the arduous task of refereeing this Corrigendum. His patient criticisms led to many improvements of the arguments alluded to in my paper, as well as of those which are presented here, and I am most grateful to him for his help.

A classical problem in the theory of convex polytopes is the enumeration of the distinct combinatorial types of d-polytopes with υ vertices (υ ≥ d + 1). Following Grünbaum [1] c(υ, d) will denote the number of such types. Apart from the general results c(d + l, d)= 1 and c(d + 2, d) = [¼d2], which may be established by elementary arguments, the only other known values of c(υ, d) are for small values of υ and d. These have been determined empirically; details of the most recent results are contained in [2].

Various methods have been developed for solutions of boundary value problems involving discs of finite radius and spherical caps. A recent account of this work is described in the book by Sneddon [1]. In the present paper a simple method is presented for the solutions of potential problems for the electrified disc and spherical cap by reducing the axially symmetric boundary value problems to a corresponding problems for the two-dimensional Laplace equation. The essence of the method is to employ integral operators which map two-dimensional harmonic functions into axially symmetric potentials and are closely related to the integral transformations given in [3]. In particular it is shown how the mixed boundary value problems for the disc and spherical cap are mapped into Dirichlet problems for the two-dimensional Laplace equation in the half plane and interior of the unit circle respectively. In both cases a standard Green's function approach is applied to determine the solution of the two-dimensional problem. Williams [2] demonstrated how the potential problem for the lens can be found using a similar method. It is noted that Rostovtsev [5] Mossakovskii [4] and Heins [7] have used techniques similar to that presented in this paper.

The study of linear stability of a layer of stratified fluid in horizontal shearing motion leads, in the absence of diffusive effects, to a second order differential equation, often called the Taylor-Goldstein equation. This equation possesses a singularity at any critical point, i.e. at any point at which the flow speed, U, is equal to the wave speed, c. If c is complex, a similar singularity arises at any point at which the analytic extension of U into the complex plane is equal to c. Assuming the stratification is thermal in origin, the introduction of a small viscosity and heat conductivity removes this singularity, but leads to a governing equation of sixth order, four solutions being of rapidly varying WKBJ form. The circumstances in which the remaining two solutions can be uniformly represented in the limit of small viscosity and conductivity by the solutions of the Taylor-Goldstein equation are examined in this paper.

§1. Preliminaries. A Cauchy process in d-dimensional Euclidean space, Rd, is a stochastic process, Xt(ω), with stationary independent increments and with a continuous transition density, p(t, y − x) defined by

and

where m, the isotropic measure, is a probability measure on Sd, the unit sphere in Rd, such that when d > 1 the support of m is not contained in any d − 1 dimensional subspace. In (2) w is given by

where . It follows that for each t > 0 and y we have p(t, y) > 0 and that for each t > 0 p(t, y) is a bounded and continuous function of y. Xt(ω) can be considered as being a standard Markov process (for a full description of the definition of such a process see Chapter 1 of [1]) and in particular we can assume that the sample functions of Xt(ω) are right continuous and have left limits. We can also assume that Xt(ω) enjoys the strong Markov property. We write Px and Ex for probabilities and expectations conditional on X0(ω) = x, and we write P for P0.

Suppose K is a convex body in Euclidean n-space En and that all the orthogonal projections of K onto p-dimensional linear subspaces have the same p-dimensional volume, that is K has constant outer p-measure. If p = 1, this means that K is of constant width; if p = n − 1, this means that K has constant brightness. It is known that, when the boundary of K is smooth enough to admit principal radii of curvature R1, …, Rn-1 as functions of the outer normal u, and if we define Fp(u) to be the p-th elementary symmetric function of these radii, then, when K has constant width,

Let f(z) be an entire function. The definition of a Phragmén–Lindelöf indicator of f(z) requires the preliminary construction of a fairly regular comparison function V(r).

If f(z) is of order λ (0 < λ < + ∞), and of mean type, one takes

and associates with f(z) the indicator

In [1] Fröhlich considers the kernel D(Z(Γ)) of the map of class-groups C(Z(Γ)) → C(), Γ a finite abelian group, the maximal order in the rational group ring Q(Γ). We obtain under mild hypotheses a non-trivial lower bound for the cardinality k(Γ) of the finite group D(Z(Γ)) when Γ is the cyclic group of order 2pn, p an odd prime. In fact, let f be the smallest positive integer such that 2f ≡ 1 mod.pn, If 2|f then k(Γ) > 1 for pn ≠ 3, 32, 5 and k(Γ) is divisible by primes ≠ 2, p except possibly when pn is a Fermat prime or when pn = 32. The latter result contrasts with the fact that D(Z(Γ)) is a p-group if Γ is a p-group [1; Theorem 5].