Let Z(Γ) be the integral groupring of a finite Abelian group Γ. There is some interest in the study of its class group (Picard group) C(Z(Γ)) (cf. e.g. [1] and [5]). One knows that this group is mapped surjectively onto the class group of the maximal order in the rational groupring Q(Γ). Now is known in the sense that it is the product of the classgroups of the algebraic integer rings, whose quotient fields appear in the decomposition of Q(Γ). One is thus also interested in the kernel D(Z(Γ)) of the map , and it is this which concerns us here. I shall show that it can become very big.

Let be a (finite) non-empty sequence of integers, and let K be a positive integer. The aim of the Brun-Selberg sieve is to obtain bounds for the “sifting function”

where z ≥ 2 is a real number and

For any fixed integer a and real variables x, y with y < x let Na(x, y) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y. As an elementary application of the following deep theorem of Bombieri on arithmetic progressions,

Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that ifwith l = log x, then for x > 1

where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m;

we shall prove the following theorem:

Theorem 1. Let a be any fixed integer and let x > e. We then have

where the double sum is taken over primes p and q.

In general the terminology and notation of [1] is used throughout. A correspondence for topological spaces is a triple f: P → Q where P and Q are topological spaces and f is a subset of P × Q, the graph of f: P → Q. A correspondence f: P → Q will be called graph-compact, or graph-closed, or graph-Souslin, or graph-analytic if f is, respectively, compact or closed or Souslin or analytic in P × Q.

For a 3-connected graph G imbedded in the plane let νk(G) denote the number of k-valent vertices, and let pk(G) denote the number of k-gonal faces (countries) of the map defined by G. (We consider only graphs without loops or multiple edges; hence, νk(G) and pk(G) are defined only for k ≥ 3.) It is a wellknown consequence of Euler's formula that

Let G be a plane domain with ∞ ∊ G. Let E be the compact complement of G and cap E the logarithmic capacity. We shall assume that cap E = 1 and E ⊂ {|Z| ≤ R. Then R ≥ 1, with equality if and only if E is a closed disc.

The uniform approach of a solid sphere of radius a towards a stationary solid sphere through a viscous fluid is considered when the fluid motion may be regarded as essentially steady with Reynolds number sufficiently small to permit the linearization of the Navier–Stokes equations by neglect of the inertia terms. Discussion is given when the minimum clearance εa between the spheres is arbitrary and a detailed analysis is made of the asymptotic behaviour of the solution as the minimum clearance tends to zero by constructing matched asymptotic expansions. The forces on the spheres are shown to be of the form as a0ε−1 + b0logε + c0 + O(εlogε) when ε is small where a0, b0 and c0 are constants. The values of a0 and b0 are determined explicitly in the general case of arbitrary sized spheres and c0 is also determined explicitly when the stationary sphere degenerates into a plane. For this case, it is shown that the asymptotic expression for the force is useful in practice even when ε is not very small, the error being less than 3% when ε ∼ 0·5.

An old theorem of Pólya and Carlson [2] states that, if the power series has rational integer coefficients, positive radius of convergence, and can be continued analytically to a region that contains points outside the closed unit disc, then the function that the power series represents is rational. This result has been extended in a number of ways (cf. e.g. Petersson [4]). The present note gives a new extension based on a recent theorem of Güting [3]. My thanks to Professors Henry Helson and Raphael Robinson for introducing me to this subject.

The object of this paper is to prove the following:

Theorem. Every perfect septenary quadratic form assumes its minimum value at a set of 7 points with integer co-ordinates whose determinant is 1.

This is true also, as shown by Rankin [1], with n ≤ 6 in place of 7. The proof will be shortened considerably by using the weaker result obtained in [1] for n = 7, and we shall also use the following classical results, see, e.g., [2], for Hermite's constant γn:

Recently there has been much interest in theories describing the behaviour of a mixture of two or more continua, which are allowed to diffuse through each other and which interact mechanically and thermally. In the present note we consider such a mixture consisting of two isotropic incompressible non-linear elastic solids; physically this may describe a composite of two rubber-like materials. Such a mixture possesses internal friction between the constituents when deformed (called diffusive resistance or diffusive force), which acts as a damping mechanism. This damping mechanism has a static part and a dynamic part, and we here show that the exact static solutions which exist for an isotropic incompressible non-linear elastic solid are such that the damping mechanism is expressible as a gradient, and hence are also controllable for such a mixture. We also discuss the corresponding dynamic solutions.

From a practical viewpoint, some experiments carried out on vulcanized synthetic rubber-like materials may involve a mixture rather than a pure solid.

Let D be a compact set in the plane, t a real number, and Dt the linear set {x + ty|x + iy ε D}. We are interested in the Hausdorff dimensions of D and Dt, and assume that dim D = d ≤ 1. A number t is “exceptional” if dim Dt < d; the exceptional numbers form a Borel set of d-dimensional measure zero [3]. (Marstrand [4, p. 268] proves a similar conclusion for the Lebesgue measure of the exceptional set.) In this note we exhibit a planar set D of Hausdorff dimension d and a linear set E for which dim Dt ≤ r < d for every t in E, but dim E > 0. The method does not come very close to the probable truth, that the set E can have dimension r. But perhaps the result can be improved by more subtle calculation, for example, as in Jarník [2].

Artin, in 1927, conjectured that for any given non-zero integer a other than — 1 or a perfect square there exist infinitely many primes for which a is a primitive root. He also conjectured that the number of primes not exceeding x, denoted by Na(x), for which a is a primitive root is given by the asymptotic formula

where A(a) is a constant depending on a.

Necessary and sufficient conditions for Abel's integral equation to have a solution have been given by Tamarkin [18]. The form of the solution was obtained by Abel [1]. The corresponding integral equation for an infinite range of integration was introduced by Liouville [12], who found a solution in a restricted class of cases. In the present paper, we find necessary and sufficient conditions for Liouville's equation to have a solution, and also give the form of the solution.

The objectjof this paper is to extend to algebraic number fields some of the recent results proved by the large sieve method. In particular we prove generalizations of Bombieri's form of the large sieve inequality [1; Theorem 1] and of the theorems of Davenport and Halberstam [2] and Bombieri [1, 4], on the average distribution of primes in arithmetic progressions.

Let be the class of non-constant functions f(z), holomorphic in |z| < 1, which have asymptotic values at a dense set of points on |z| = 1. MacLane [4; p. 18] asked whether the sum and product of two functions in had to be either constant or in . Recently Barth and Ryan [2] have shown that this was not necessarily so; in this note we will demonstrate, in a totally elementary way, why not. Our principal result is:

Theorem 1. Any non-constant function R(z), holomorphic in |z| < 1, may be represented both as a sum and as a product of pairs of functions in.

In this paper the response of an Euler-Bernoulli viscoelastic beam to impulsive excitation is obtained using Volterra's model for the stress-strain relationship. In order to achieve a better approximation of actual materials a series of parallel connections of one or more basic models must be used. However, the analytic solutions to most problems then become very difficult. Therefore an alternative approach is to formulate such a problem in terms of the hereditary integral as proposed by Vito Volterra [1].

It is well known [1; pp. 333–339, 2; pp. 181–183] that the quadrics of S4 through a normal elliptic quintic curve 1C5 form a homaloidal system, and that this gives rise to an interesting Cremona transformation T2, 3 of S4 into S4′ for which the reverse system consists of the (Segre) cubic primals of S4′ that contain an elliptic quintic scroll 1R5. Less well known, though the fact was noted by one of us many years ago, is that the quadrics of S6 through the most general type of normal elliptic septimic scroll 1R7 likewise form a homaloidal system giving rise to a T2, 4 of S6 into S6′. We show in this paper that the latter transformation has properties no less remarkable than those of the above mentioned T2, 3. We propose also to show in a subsequent note that T2, 3 and T2, 4 are the initial members of a sequence of Cremona transformations T2, k+2 of S2k+2 (k = 1,2, …) of which the general member is generated by the homaloidal system of quadrics of S2k+2 that contain an elliptic ruled variety 1Rk2k+3, locus of an elliptic pencil of [k − 1]'s. Since the T2, 4 of S6, however, has certain cyclic properties peculiar to itself, we have thought it best to describe it separately in the present paper.

One of the earliest investigations of the acoustic disturbance created in a fluid of semi-infinite extent by the movement of a piston surrounded by an infinite plane rigid baffle was undertaken in 1878 by Rayleigh [1], who studied the harmonic oscillations of a piston of arbitrary shape. In subsequent years, a number of authors have presented methods for studying the “baffled piston problem”. A large portion of the literature has been concerned with a circular piston and particular attention has been devoted to the case in which the small amplitude oscillations of the piston are harmonic. References to these earlier works are given by Chadwick and Tupholme [2]. A solution to the more general problem in which the normal velocity of a circular piston is an arbitrary function of time was derived from the corresponding time-harmonic solution by Oberhettinger [3] and obtained more directly by Miles [4, see also 5]. Kozina and Makarov [6] considered the acoustic fields produced by the time-dependent motion of an arbitrarily-shaped piston.

We present here the eigenfunction expansions of the bi-harmonic Love strain function for a cylinder whose plane faces are stress-free. The convergence for the expansions of arbitrary functions in terms of these eigenfunctions is studied. As an example, the symmetrical deformations in a cylinder whose plane faces are stress-free and on whose curved edge arbitrary radial displacement and transverse shear force are prescribed is worked out.

The reviewer [Math. Reviews, 32 (1966), 7526] has stated that he could not follow case (iii) of the proof of Lemma 7 and upon examination we find that the argument is incomplete (on p. 183, the degree of d1 may not be smaller than that of g). The following version of the proof (which is given with full details), circumvents the difficulty and has been designed to show, further, that the constant implied in the O-symbol is independent of the number of variables. This minor refinement may be of some interest as the corresponding estimates of Lang and Weil [Amer. J. Math., 76 (1954), 819–827; cf., Lemmas 1, 2 and Theorem 1] for general varieties over a finite field lack this feature. Throughout, we use the symbol deg F to denote the total degree of an element F of a polynomial domain , where k = [p] is the field of residue classes mod p and . Nr(†) will denote the number of r-tuples (xn−r+1, …, xn) ε kr with some specific property †. We prove that, if n ≥ 1 and f, g are elements of with no non-constant common factor in , i.e. (f, g)p = 1, then

where the constant in the O-symbol depends only upon d = max (deg f, deg g). The argument is entirely elementary and uses a method of descent in which the number of variables n and the minimum degree, defined by

are simultaneously diminished at each step of the descent. It makes use of a corresponding (trivial) estimate for the case of one polynomial, i.e. if and ∂ = deg F, then Nn(F = 0) = O∂(pn−1). Thus, if f, g are disjoint in the sense that f ε k[x1, …, xr], g ε k[xr+1, …, xn] after a suitable permutation of x1, …, xn, then (f, g)p = 1 implies that