In the present paper the researches initiated in the two earlier papers of this series are continued, and, by suitable generalizations of the techniques employed therein, solutions are obtained to some further well known problems from the theory of transcendental numbers. It will be proved, for example, that a non-vanishing linear form, with algebraic coefficients, in the logarithms of algebraic numbers, cannot be algebraic. This implies, in particular, that π + log α is transcendental for any algebraic number α ≠ 0, and also eα π+ß is transcendental for all algebraic numbers α, β with β ≠ 0.

Let be the class of normalised polynomials of the form:

of degree n which are univalent in |z| < 1, and let be the class of “polynomials” of the form

of degree n which are univalent in 0 < |z| < 1. In this note we discuss the coefficients of polynomials in (k = 2, 3, 4) and (k = 2, 3).

Let p be a prime number, Qp the field of p-adic numbers and Ωp the completion of the algebraic closure of Qp with its valuation normed by setting |p| = 1/p. We shall designate by log the p-adic logarithm defined by the usual series

Let Λ be a lattice in 3-dimensional space which provides a double covering for spheres of unit radius. By this we mean that if X is any point of space, there are at least two distinct lattice points P, Q such that XP ≤ 1, XQ ≤ 1. Let d(Λ) be the determinant of Λ. We shall prove that

where

In the author's paper [1] in which an exact solution is given to the Stokes equations for the steady motion without rotation of a rigid sphere through a viscous fluid in a direction parallel to and at an arbitrary distance d from a fixed plane, expressions were obtained for the forces and couples which are exerted on the sphere and the plane by the fluid. The forces were shown to be equal and opposite. The couple acting on the sphere was found to have Cartesian components (0, Gy, 0), when moments of the surface stress are taken about the centre of the sphere, where

and the couple acting on the plane was found to be (0, Gy′, 0) where

The notation of (1) and (2) follows that as explained in [1] but a misplaced minus sign is corrected.

In 1933, K. Borsuk [1] established the well-known result that if n closed sets cover Sn−1 then at least one set contains antipodal points, where Sn−1 is the surface of the ball Tn of centre O and unit diameter in Rn. This result prompted H. Hadwiger [2] to make a still unresolved conjecture which, in the spirit of B. Griinbaum's survey [3], we state as follows:Let r be the largest integer such that whenever r closed sets cover Sn−1 at least one set realizes all distances between 0 and 1. Then r = n.

In many physical problems it is necessary to express a solution of Laplace's equation relative to one set of coordinates in terms of harmonics relative to another set. We term such a relationship an addition or shift formula. Well-known examples are the formulae for spherical harmonics [Hobson 1, p. 139] and cylindrical harmonics [Watson 2, p. 360]. In this paper we shall consider the problem for oblate spheroidal coordinate systems and obtain some addition theorems for the corresponding harmonics. These addition theorems are used to write down “two-centred” expansions of the Coulomb Green's function, and by a limiting process a new form of the “one-centred” expansion is obtained in a non-orthogonal coordinate system, closely related to oblate spheroidal coordinates. This expansion is applied to the evaluation of the Coulomb (or gravitational) energy of spheroidal distributions of charge (or mass) in which the surfaces of constant density are concentric similar spheroids, a situation which occurs in both nuclear physics and cosmology [Carlson 3]

A steady two-dimensional motion of viscous liquid resulting from a maintained tangential velocity on a part of the bounding surface is considered. It is assumed that the liquid is contained in a long circular cylinder of radius a, and that a constant tangential velocity V is maintained over an arcual length 2b of the boundary. It should be possible to approximate to these conditions experimentally.

This paper deals with diophantine equations and inequalities in three variables x, y, z. Write

Given an algebraic number field K of degree n and an element ξ of K, denote the conjugates of ξ by ξ(1) = ξ, ξ(2), …, ξ(n), and its norm ξ(1) … ξ(n) by N(ξ). Recent results [3] in diophantine approximations enable us to prove the following theorems.

Estimates involving polynomials can often naturally be given in terms of the discriminants of these polynomials or of the resultants of pairs of polynomials. Since the discriminant of a polynomial with multiple zeros vanishes as does the resultant of two polynomials with common zeros these results become trivial when applied to such polynomials or pairs of polynomials. Therefore it is often necessary to exclude polynomials with multiple zeros from a given investigation. In theoretical studies of a measure-theoretical nature this often does not affect the results; however for the purpose of constructing polynomials with specified properties it can be an advantage if it is not necessary to restrict the attention to polynomials without multiple zeros.

It has been pointed out to us by Professor L. Schoenfeld that there is a fallacy in the proof of Theorem 3 of our paper “The values of a trigonometrical polynomial at well spaced points” [ Mathematika, 13 (1966), 91–96]. The fallacy occurs in the appeal to Theorem 1 at the end of the proof. If this is to be made explicitly, we must not only put n = dn′ but also put q = dq′ but then the sum over m goes from 1 to q instead of from 1 to q′, and if one allows for this the final result becomes much weakened.

Let the real constants qij (i, j = 0, 1, 2, …) satisfy the conditions

Introduction and result. Let

be analytic in |z| < 1. The Hankel determinants are defined by

We shall establish here a new relation between the classnumber h of an algebraic number field K and the signature of its group of units Y.

Write dim2(A) for the number of even invariants of a finite Abelian group A. Denote by C the absolute classgroup of K (ideals modulo principal ideals), and by P the quotient group of principal ideals modulo totally positive principal ideals.

Let Q(x, y) = ax2 + bxy + cy2 be a positive definite quadratic form with discriminant d = b2 – 4ac. The Epstein zeta function associated with Q is given by

where Σ′ means the sum is over all pairs (x, y) of integers not both zero, and as usual, s = σ + it.

1. G. B. Jeffery [1] investigated the axisymmetrical flow of an incompressible viscous fluid caused by two spheres rotating slowly and steadily in the liquid about their line of centres. The numerical results he gave were for spheres of equal radii.

The present problem is to investigate what happens when the spheres touch each other either externally or internally and when they are unequal in size. This problem could be approached by taking a limit of Jeffery's solution, but in fact it will be more convenient to use a co-ordinate system different from Jeffery's and, of course, his results would not yield information about unequal spheres.

A little over a hundred years ago E. B. Christoffel in [6] asserted a proposition concerning the determination of a surface in Euclidean 3-space from a specification of its mean radius of curvature as a function of the outer normal direction. In that paper an assumption was made which limited the class of surfaces considered to be boundaries of convex bodies. The argument rested on the construction of functions describing the co-ordinates of surface points corresponding to outer normal directions as solutions of certain partial differential equations involving the mean radius of curvature. However, it was pointed out by A. D. Alexandrov [1], [2], that the conditions laid down by Christoffel on the preassigned mean radius of curvature function were not sufficient to ensure that that function actually be a mean radius of curvature function of a closed convex surface. Hence Christofrel's discussion is incomplete. Different and similarly incomplete treatments of Christoffelés problem were given by A. Hurwitz [9], D. Hilbert [8], T. Kubota [11], J. Favard [7], and W. Suss [13]. A succinct discussion of the question is in Busemann [5] but the footnote on p. 68, intended to rectify the discussion in Bonnesen and Fenchel [4], is also not correct. A sufficient, but not necessary condition was found by A. V. Pogorelov in [12].

Recent work on the mechanics of interacting continua has led to the formulation of linearized theories governing thermo-mechanical disturbances of small amplitude in mixtures of an elastic solid and a viscous fluid and of two elastic solids. These theories are well posed in the crude sense that the number of field and constitutive equations equals the number of field quantities to be determined, but the proper posing of initial and initial-boundary-value problems has not been studied. In this paper we prove, for both types of mixtures, the uniqueness of sufficiently smooth solutions of the field and constitutive equations subject to initial and boundary data which include conditions of direct physical significance. Sufficient conditions for uniqueness are given in the form of inequalities on the material constants appearing in the constitutive equations. These restrictions fall into two categories, one arising from the application of the entropy-production inequality, and therefore intrinsic to the theory, and the other representing constraints on the Helmholtz free energy of the mixture. Our results include as special cases uniqueness theorems for unsteady linearized compressible flow of a heat-conducting viscous fluid and for the linear theory of thermoelasticity.

Throughout this note X will denote a completely regular Hausdorff space and ℬ the σ-algebra of Borel sets in X (see §2 for terminology). For x in X we may define the atomic Borel measure δx to be the unit mass placed at the point x. Observe, however, that the example of Dieudonné [3; §52, example 10] shows that not all atomic measures need have this form. The problem investigated in this note is the existence of finite Borel measures other than the atomic measures. In §3 we show that such a measure necessarily exists on a compact space without isolated points, though we are not able to add to the very meagre supply of examples which are known at the moment. A converse to our result has been given by Rudin [7] and, for completeness, we include a new proof of his result.