In boundary wave problems, when there are two or more boundaries where conditions have to be satisfied, it is often necessary to set up a process of successive approximation in order to gain the solution. Invariably the process cannot be continued beyond a few stages, and the error incurred by halting the process cannot be satisfactorily determined. Such seems to be the case in the surface wave problem when there is a streaming motion of constant velocity past a submerged circular cylinder, which is set in liquid of infinite depth with its axis perpendicular to the stream.

In a recent paper I. J. Schoenberg [1] considered relations

where the av are rational integers and the ζv are roots of unity. We may in (1) replace all negative coefficients av by −av replacing at the same time ζv by −ζv so that we may, if it is convenient, assume that all av are positive. If we do this and arrange the ζr so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all av are non-negative. We shall call a polygon (1) k-sided if all av are positive. The polygon is called degenerate if two of the ζv are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p-gons where p is a prime.

Our purpose in this note is to present a natural geometrical definition of the dimension of a graph and to explore some of its ramifications. In §1 we determine the dimension of some special graphs. We observe in §2 that several results in the literature are unified by the concept of the dimension of a graph, and state some related unsolved problems.

Let L be a lattice in euclidean n-space Rn of determinant d(L). A well-known problem in the geometry of numbers is whether

(1) if d(L) = l and there exists a sphere centred at the origin containing n linearly independent points of L on its boundary and none other than the origin inside, then any sphere in Rn of radius ½√ contains a point of L.

If an is a sequence of numbers between 0 and 1, then

has infinitely many integral solutions n, l either for almost all real x or for almost no real x[1,4]. Duffin and Schaeffer [2], improving on an earlier theorem of Khintchine [7], proved that for decreasing sequences an, (1) has infinitely many solutions a.e. if and only if Σan diverges. They also gave an example of a sequence an for which Σan diverges, but for which (1) has only finitely many solutions a.e. No general necessary and sufficient condition for (1) to have infinitely many solutions a.e. is known.

Some of the elastic properties of liquids in shear can be detected and measured by observing suitable types of oscillatory motion. Oscillating systems have proved to be fairly simple to design and to control in practice, and can lead, in the case of purely viscous liquids, to accurate measurements of the viscosity [see, for example, 1, 2, 3]. One such system, used for the purpose of investigating the rheological properties of dilute polymer solutions, is the coaxial-cylinder elastoviscometer of Oldroyd, Strawbridge and Toms [4]; in this experimental arrangement, the liquid is contained between two cylinders with a common, vertical axis, the inner cylinder being suspended by a vertical torsion wire. The theory of the motion of such an instrument in the case when the outer cylinder is forced to oscillate about its axis, which is fixed, with a known constant frequency, and the resulting motion of the inner cylinder is constrained by the torsion wire, has been considered by Oldroyd [5], who shows that the experimental results available for some typical polymer solutions can be interpreted in terms of an idealised elastico-viscous liquid characterised by three constants (a viscosity and two relaxation times). A new representation of the relaxation spectrum of a liquid has subsequently been used by Walters [6] in order to develop the theory of oscillatory flow of the most general (linear) visco-elastic liquid. Walters has shown that the experimental results for dilute polymer solutions (previously interpreted in terms of a discrete relaxation spectrum by Oldroyd [5]) can, equally, be interpreted in terms of a simple continuous relaxation spectrum characterised by three constants.

The flows induced by an oscillating cylinder and by a torsionally oscillating disk are considered. For the case of the cylinder attention is restricted to the high frequency case whilst for the disk both the high and low frequency cases are discussed.

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

The purpose of this paper is to prove that the altitudes of an n-simplex (a simplexin an n-space) S form an associated set of n+1 lines (see Baker, [4] for n = 4) such that any (n–2)-space meeting n of them meets the (n+1)th too. As an immediate consequence 2 quadrics are associated with S, one touching its primes at the respective feet of its altitudes and the other touching n(n+1) primes, n parallel to each of its altitudes and 2 through each of its (n−2)-spaces. Certain special cases are also mentioned.

Let ν be a discrete random variable taking on nonnegative integer values and set P{ν = κ} = Pk, κ = 0, 1, hellip. Suppose that the binomial moments are finite. Frequently the problem arises under what conditions the probabilities Pk, k = 0, 1,…, can be determined uniquely by the sequence of moments Br, r = 0, 1,…, and how it can be done.

The following paper is a sequel to the author's earlier paper [2]. In that paper some general results were obtained which described the motion of a fluid with a free surface subsequent to a given initial state and prescribed boundary conditions of a certain type. The analysis was based on a linearized theory but gravity effects were included. Viscosity, compressibility and surface tension effects were neglected. Among the problems treated was that of the normal symmetric entry of a thin wedge into water at rest. This Water entry problem has attracted a considerable amount of attention since the pioneer paper by Wagner [5]. Both linear and non-linear approximations have been used but all papers apart from [2] neglect gravity on the assumption that in the early stages of the penetration this is unimportant. One of the objects of [2] was to determine the solution with the gravity terms retained. A formal solution was obtained but no attempt was made to analyse this quantitatively. In the present paper we examine the extent of this effect in some detail. It will be of help to the reader to have some familiarity with the first three or four sections of [2] but in order to make the present paper self-contained we shall first reintroduce the notation used there and quote the necessary results from that paper without proof.

The mechanics of a system of packed spheres has relevance to several physical disciplines. A particular case has been a recent trend among engineers to use a close-packed sphere model to aid research into the strength of cohesionless granular masses, such as sand.

Let G be a locally compact group with left invariant Haar measure m. Le H be a closed subgroup of G and K a compact group of G. Let R be the equivalence relation in G defined by (a, b)∈R if and if a = kbh for some k in K and h in H. We call E =G/R the double coset space of G modulo K and H. Donote by a the canonical mapping of G onto E. It can be shown that E is a locally compact space and α is continous and open Let N be the normalizer of K in G, i. e. .

The groups whose 2-generator subgroups are all nilpotent of class at most 2 are nilpotent of class at most 3 (see Levi [6]). Heineken [3] generalized Levi's result by proving that for n ≧ 3, if the n-generator subgroups of a group are all nilpotent of class at most n, then the group itself is nilpotent of class at most n. Other related problems have been considered by Bruck [1].

Let X be an non-empty set and denote by PX the set algebra consisting of all sunsets of X. An operator i: PX → PX is said to be Pre-interior if (i) iX = X; (ii) i(A∩B) = iA ∩ iB for all A, B in PX.

In a recent paper by P. D. Finch and myself [1], the solution for the limiting distribution of a moving average queueing system was obtained. In this paper the system is generalised to the case of batch arrivals in batches of size ρ > 1.

In [2] we defined an irreducible B(J)-cartesian membrane, and used this to obtain a characterization of an n-sphere by generalizing the definition of simple closed curve given by Theorem 1.2 below. There B(J) is a class A(n) of (n−1)-spheres, but here it is a class of mutually homeomorphic continua. In Theorem 1.1 we give a definition of hereditarily unicoherent continua and generalize this in Section 3 by means of B(J)-cartesian membranes. To do this we paraphrase by a translation some of Wilder's work in [7]. In his Unified Topology [8: p. 674] he gives a principle: “The connectedness of a domain is a special case of the bounding properties of its i-cycles”. We substitute the element J of B(J) for the i-cycle and for “bound” we substitute that “J membrane-bases an irreducible B(J)-cartesian membrane. The very nature of an i-cycle seems to limit the complexity of the point set studied, although the restriction to “nice” manifolds is due partly to the difficulty of the subject matter treated. There are similar difficulties here, but also advantages, in the very general set-theoretic approach by means of B(J)-membranes.