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].

1. The purpose of this paper is to give simple proofs for some recent versions of Linnik's large sieve, and some applications.

The first theme of the large sieve is that an arbitrary set of Z integers in an interval of length N must be well distributed among most of the residue classes modulo p, for most small primes p, unless Z is small compared with N. Following improvements on Linnik's original result [1] by Rényi [2] and by Roth [3], Bombieri [4] recently proved the following inequality: Denote by Z(a, p) the number of integers in the set which are congruent to a modulo p.

In 1953 Erdős† proved in a characteristically ingenious manner that an irreducible‡; integral polynomial f(n) of degree r ≥ 3 represents (r – 1)-free integers (that is to say integers not divisible by an (r – 1)th power other than 1) infinitely often, provided that the obvious necessary condition be given that f(n) have no fixed (r – 1)th power divisors other than 1. His method did not, however, give a means for determining an asymptotic formula for N(x), the number of positive integers n not exceeding x for which f(n) is (r – 1)-free, nor did it even show that the positive integers n for which f(n) is (r – 1)-free had a positive lower density.

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.

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.

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.

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

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.

The velocity potential near the boundary of the disturbance is determined for steady supersonic flow past a simple body-and-wing model. It is found that the disturbance decays exponentially as it spreads round the body; the alteration caused by changing the radius of curvature is discussed. A universal formula for the potential away from the fuselage is also derived.

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.

A well-known theorem of Schauder asserts that a continuous self-map T of a closed convex subset K of a normed linear space X, such that the closure of T(K) is compact, has at least one fixed point. No general method is known for a computation of the fixed point(s) of T.

It was proved in a recent paper† that if au α1, …, αn denote non-zero algebraic numbers and if‡ logαn, …, log αn and and 2πi are linearly independent over the rationals then log α1, …, log αn are linearly independent over the field of all algebraic numbers. Further it was shown that if α1 …, αn are positive real algebraic numbers other than 1 and if β1, …, βn denote real algebraic numbers with 1, β1 …, βn linearly independent over the rationals then is transcendental.

Introduction and result. Let

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