If D is an integral domain with quotient field K, then by an overring of D we shall mean a ring D′ such that D ⊂ D′ ⊂ K. Gilmer and Ohm in [GO], Davis in [D] and Pendleton in [P] have studied the class of integral domains D having the property that each overring D′ of D is of the form Ds for some multiplicative system S of D. In [D] and [P] a domain with this property is called a Q-domain and in [GO] such a domain is said to have the QR-property. Slightly altering the terminology of [D] and [P], we shall say “QR-domain” instead of “Q-domain”. Noetherian QR-domains are precisely the Dedekind domains having torsion class group [GO; p. 97] or [D; p. 200]. Pendleton in [P; p. 500] classifies QR-domains as Prüfer domains satisfying the additional condition that the radical of each finitely generated ideal is the radical of a principal ideal. It is pointed out in [P; p. 500] that this characterization of QR-domains still leaves unresolved the question of whether the class group of a QR-domain is necessarily a torsion group. We show in §1 that a QR-domain need not have torsion class group. Our construction is direct; however, the problem can be viewed in terms of the realization of certain ordered abelian groups as divisibility groups of Prüfer domains, and we conclude §1 with a brief discussion of this approach to the problem.

Let E be a real normed linear space. Let K be a closed convex set containing 0, the origin, as an extreme point. Let A be a linear operator with AK ⊆ K. Stated below are theorems concerning eigenvectors and spectral (partial spectral) radius of A which generalize the well-known theorems of Bonsall [3] and Krein and Rutman [7] on positive operators. Proofs are given in §2.

In a recent article [3] L. Mirsky proved a theorem which gives necessary and sufficient conditions for the existence of a finite integral matrix whose elements, row sums, and column sums all lie within prescribed bounds. Mirsky suggested to me the problem of extending his theorem to infinite matrices, and it is the solution of this problem that is presented in this note. To allow for extra generality, instead of prescribing upper and lower bounds for the row and column sums we shall prescribe upper and lower bounds for the row and column deficiencies (a term to be explained later). The theorem when upper and lower bounds for the row and column sums are prescribed is then a special case of the deficiency theorem. The solution of the problem depends on a construction of Mirsky [3] and a theorem of mine [1] concerning the existence of a partial transversal of a family of sets satisfying certain properties. As will be seen, we shall take a rather broad view of the notion of a matrix.

When the Helmholz equation

is separated in ellipsoidal coordinates [1; §1.6] and the technique of separation of variables applied, there results the ordinary differential equation

known as the ellipsoidal wave equation or Lamé wave equation. In this equation k is the modulus of the Jacobian elliptic function sn z, and is related to the eccentricity of the fundamental ellipse of the ellipsoidal coordinates; a, b are separation constants, and the parameter q is connected with the wave number χ by

l being a real constant, the dimensional parameter of the coordinate system.

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].

A hemisphere, resting on a horizontal plane, is initially at rest relative to an incompressible, inviscid, non-diffusive fluid whose density is vertically stratified. The hemisphere is then given, impulsively, a small constant horizontal velocity which is maintained thereafter. Assuming that the Froude number is small, and using the Boussinesq approximation, the equations of motion are linearised and solved using a Laplace transform. The disturbance in the fluid is analysed for large times and is found to contain a steady component of purely horizontal flow, an internal wave field and internal oscillations at the Brunt-Väisälä frequency, together with their various interactions. The effects of viscosity and diffusivity are discussed qualitatively by considering their effects on an internal wave.

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.

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.

Weber proves in §§114-124 of his Algebra [19] that if w is complex quadratic and ℤ[ω] is the ring of integers of the field ℚ(ω) then the absolute class field of ℚ(ω) is generated by the modular invariant j(w); he calls j(w) a class invariant. He goes on in §§125-144 to consider the values f(w) of other modular functions f(z); he shows that in certain cases the degree of the extension ℚ(ω, f(ω)) of ℚ(ω, j(ω)) is much less than that of ℚ(f(z)) over ℚ(f(z)); indeed, f(ω) is often in ℚ(ω, j(ω)), and in such circumstances Weber calls f(ω) a class invariant too. Using such results, Weber computes many class invariants—an end in itself, since the numbers are so beautiful. More recently, results of this type have been applied to determine all the complex quadratic fields with class number 1, and to prove that elliptic curves of certain families always have infinitely many rational points-see [9, 2, 5].

Let X be a finite set of points in Ed. Then a partition of X into two non-empty subsets X1 and X2 (X1 ∪ X2 = X, X1 ∩ X2 = ∅) will be called a Radon partition if

.

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 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.

An analogue of a problem of Littlewood. The arguments of this paper are not wholly correct; a detailed corrigendum will be published in due course.

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.

Irrotational gravity waves of finite height: a numerical study. The sentence on page 146 in which equation (30) occurs should read:

The method consists of expanding the equation as

In an earlier paper [Shail, 1] one of the present authors considered the effect of a magnetic field on the frictional couple experienced by an axisymmetric solid insulator which rotates slowly in a bounded viscous conducting fluid, the applied magnetic field being parallel to the axis of rotation. Results for the couple in various geometrical configurations were obtained in the form of power series expansions in the Hartmann number M, and hence are valid only for M ≪ 1. However, because of the increased rigidity given to the fluid by the applied field, it is physically evident that the magnetic field will have a more dramatic effect on the flow pattern for large values of M. Thus one object of this paper is to investigate the frictional couple on a solid insulator rotating in an unbounded fluid for values of M ≫ 1.

We investigate the steady pressure driven flow of conducting fluid through an insulating circular pipe in the presence of a transverse applied magnetic field. It is well known that an exact solution is available, and the series expressions for the fluid velocity and induced magnetic field have often appeared in the literature, Uflyand (1960), Uhlenbusch and Fischer (1961), Gold (1962); however, no progress has been made in deriving asymptotic expansions for large Hartmann number directly from these series. Boundary layer methods have been used by Shercliff (1953. 1962) to determine the core structure, the Hartmann layer structure, the current distribution and the flow rate; however, these analyses do not describe the tangential layers (called obscure layers by Shercliff), which are located in the vicinity of the points at which the applied magnetic field is tangential to the pipe.