The two approaches (the hydrodynnmical-numerical and the harmonic methods) to the solution of the long wave equations are compared, enabling an analysis of the limits imposed by time stepping. The minimum time scale of motions, which may be modelled by the time stepping scheme, is determined from the stability condition derived for the harmonic method.

An algorithm is presented for computing the unique stationary distribution of an infinite stochastic matrix possessing at least one column whose elements are bounded away from zero. Elementwise convergence rate is discussed by means of two examples.

It is shown that if E is a barreled locally convex lattice and F is a quasi-complete and order complete locally convex lattice then E′ ⊗ F equipped with the cone of positive continuous linear maps of finite rank is a lattice if and only if E′ or F has finite dimensional order intervals.

Let A = {ai} be a finite set of integers and let p, m denote the orders of A + A = {ai+aj} and A – A = {ai–aj} respectively. J.H. Conway conjectured that p ≤ m always and that p = m only if A is symmetric about 0. This conjecture has since been disproved; here we make several other observations on the values of p and m.

Let M (N) be a σ–algebra of subsets of a set S (T) and let X, Y be Banach spaces with (,) a continuous bilinear map from X × Y into the scalar field. If μ: M → X (v: N → Y) is a vector measure and λ is the scalar measure defined on the measurable rectangles A × B, A ∈ M, B ∈ N, by λ(A×B) = 〈μ(A), v(B)〉, it is known that λ is generally not countably additive on the algebra generated by the measurable rectangles and therefore has no countably additive extension to the σ-algebra generated by the measurable rectangles. If μ (v) is an indefinite Pettis integral it is shown that a necessary and sufficient condition that λ have a countable additive extension to the σ-algebra generated by the measurable rectangles is that the function F: (s, t) → 〈f(s), g(t)〉 is integrable with respect to α × β.

A polyhedral decomposition of a finite trivalent graph G is defined as a set of circuits = {C1, C2, … Cm} with the property that every edge of G occurs exactly twice as an edge of some Ck. The decomposition is called even if every Ck is a simple circuit of even length. If G has a Tait colouring by three colours a, b, c then the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: if G has an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grünbaum) in terms of polyhedral decompositions alone.

It is shown that:

(i) a continuous function from a compact quasi-uniform space into a quasi-uniform space is not necessarily quasiuniformly continuous;

(ii) if the range of the function is a uniform space, the function will be necessarily quasi-uniformly continuous.

(This contradicts an example in the literature and generalizes a classical result.) Finally, a generalization of (ii) is given by means of a suitable boundedness notion.

For a nonconstant L2 (−π, π) function f, we prove that and that the inequalities are sharp.

The purpose of this paper is to study the propagation of cylindrical shear waves in nonhomogeneous four-parameter viscoelastic plates of arbitrary thickness. The plates have a transverse cylindrical hole and their material properties are functions of the radial distance from the center of this opening. They are initially unstressed and at rest. A suddenly rising shearing traction is applied uniformly over the boundary of the opening and parallel to the faces of the plates and thereafter steadily maintained; they are otherwise free from loading. We consider both the case of a finite plate with a stress-free cylindrical outer boundary, and an infinite plate composed of two media in welded contact along a cylindrical surface symmetrical with respect to the center of the opening. We find that a reflected pulse is produced at the outer boundary of the finite plate while reflected and transmitted pulses are produced at the interface in the infinite bi-viscoelastic plate. Ray techniques are used throughout, and formal asymptotic wavefront expansions of the solution functions are obtained.

One problem in set theory without the axiom of choice is to find a reasonable way of estimating the size of a non-well-orderable set; in this paper we present evidence which suggests that this may be very hard. Given an arbitrary fixed aleph κ we construct a model of set theory which contains a set X such that if Y ⊆ X then either Y or X - Y is finite, but such that κ can be mapped into S(S(S(X))). So in one sense X is large and in another X is one of the smallest possible infinite sets. (Here S(X) is the power set of X.)

This paper presents a theoretical investigation into the forced oscillations produced in an elongated lake by wind stresses varying in time. Analysis of the appropriate hydrodynamical equations of motion, in the absence of friction, and the equation of continuity give an estimate of the response function of the longitudinal component of the wind stress onto water level. Two mathematical models are used, one giving an analytical solution and the other requiring numerical methods for solution. The first model assumes that the lake is a homogeneous rectangular body of water and the second uses the mean depth h(x) and area of cross section A(x), considered as functions of distance x directed along the longitudinal axis of the lake.

Let R be a ring of nonzero characteristic and let G be a finite group with subgroup H. It is shown that H is a normal subgroup of the group of units of the group ring RG if and only if H is contained in the centre of G or R is the field with 2 elements, G is the symmetric group on 3 letters and H is normal in G.

The category of solid convergence spaces is introduced, and shown to lie strictly between the category of all convergence spaces and that of pseudo-topological spaces. A wide class of convergence spaces, including the c-embedded spaces of Binz, is then characterized in terms of this concept. Finally, several illustrative examples are given.

The group G is called trifactorisable if G has three subgroups, A, B, and C such that G = AB = BC = CA. Obviously the structure of the group G will be restricted by the structure of these subgroups. In this paper it will be shown that a finite group G is π-separable if and only if it satisfies Dπ and has a trifactorisation with two factors π closed and the third, C say, π-separable. In this case we show that the π- and π′-lengths of G can be at most one more than those of C, and so it is this factor which “controls” the structure of G. Similar results are proved for π-solubility and solubility.

It is shown that rings in which the projective cover and injective hull of cyclic modules are isomorphic are equivalent to uniserial rings. Further, it is shown that rings for which the top and bottom of finitely generated modules are isomorphic also are equivalent to uniserial rings.