The theory described in this paper is directed towards obtaining a general expression for the development of the free surface of a fluid, subsequent to a given initial state and prescribed boundary conditions, as a power series in g, the gravitational acceleration. In an earlier paper [4], a result, applicable to the particular case of the entry of a thin wedge into an incompressible fluid, was obtained and gave the shape of the free surface as such a power series. This series was valid for values of the ratio ut/x < 1, where u, was the (constant) velocity of entyr of the wedge; x the horizontal distance from the vertex of the wedge; and t the times elapsed after entry. This particular problem was first investigated by Mackie [6] who derived an asymptotic solution.

Euclid's scheme for proving the infinitude of the primes generates, amongst others, the following sequence defined by p1 = 2 and pn+1 is the highest prime factor of p1p2…pn+1.

This paper is concerned with a class of singular elliptic partial differential equations related to the operator of Weinstein's generalized axially symmetric potential theory (GASPT) [1, 2] which has numerous applications.

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.

Throughout this paper G denotes an infinite compact connected Hausdorff Abelian group with character group X. Given a map of α X into itself, we are concerned with the set of a, ∈ G such that the function ϕa ∈ l∞ (X) defined by is a multiplier of type (þ, þ), where it can be assumed without loss of generality that 1 ≤ þ < ∞.

In a previous paper [4] we showed that Γ3,1 = 16/. For the definition of Γr, s for an indefinite quadratic form in n = r + s variables of the type (r, s) see the above paper. Here we shall show that Γ2,2 = 16. More precisely we prove: Theorem. Let Q (x, y, z, t) be an indefinite quaternary quadratic form with determinant D > 0 and signature (2, 2). Then given any real numbers x0, y0, z0, t0 we can find integers x, y, z, t such thatEquality is necessary if and only if either where ρ ≠ 0. For Q1 equality occurs if and only if

In the present paper we shall consider some subgroups of (increasing) autohomeomorphisms of the closed real interval <0, 1>; mainly because of brevity, we shall defer discussing possible generalizations of our results to more general ordered fields.

On the basis of physical principles having a very general nature, A. Landé [1] has demonstrated that the mathematical structure of quantum mechanics can be derived without having recourse to the introduction of special assumptions of an ad hoc type (such as commutation rules governing canonical observables) which are not immediately suggested by our knowledge of the physical world, but which have simply originated as rules which mathematical physicists have discovered by past experience to yield conclusions in conformity with experiment.

In the terminology of J. R. Isbell [5], an element d of a semigroup S is dominated by a subsemigroup U of S if, for an arbitrary semigroup X and arbitrary homomorphisms α β, from S into X, α(u) = β(u) for every u in U implies α(d) = β(d). The set of elements of S dominated by U is a subsemigroup of S containing U and is called the dominion of U. It was shown by Isbell that if one takes two disjoint isomorphic copies S+, S− of S and forms their amalgamated free product S+ * US− that is to say, the quotient of the free product S+ * S− by the congruence p generated by , (u+ and u− being the images of u in S+, S− respectively) then the homomorphisms: μ+: S→ S+ * uS−, μ−: S→ S+ * uS− defined by are one-one. Moreover, μ+(s) = μ−(t) only if s = t, and μ+(s)= μ−(s) if and only if s is in the dominion of U. In other words, the two natural copies of S in S+ * uS− intersect precisely in the dominion of U. Thus, in particular, and in the terminology of [3], the amalgam [S+, S−; U] is embeddable if and only if U is self-dominating (that is to say, its own dominion) in S.

Let (X, T, π) be a topological transformation group, where X is a Hausedorff continuum. We will say that X is irreducibly T-invariant if no proper subcontinuum of X is T-invariant. Wallace, [6], has shown that if T is abelian and X is irreducibly T-invariant, then X has no cut point; he then asked if this statement remains true if “abelian” is replaced by “compact”. In this paper we answer this question in the affirmative, and prove a related result when T satisfies a recursive property.

It is a consequence of the Kurosh subgroup theorem for free products that if a group has two decompositions where each Ai and each Bj is indecomposable, then I and J can be placed in one-to-one correspondence so that corresponding groups if not conjugate are infinite cycles. We prove here a corresponding result for free products with a normal amalgamation.

Let α be an arbitrary positive number. For every integer n ≦ 0 we can write where is the largest integer not greater than, i.e the integral part of, and rn is its fractional part and so satisfies the inequality

Let ℜk be the variety of all nilpotent groups of class at most k. The purpose of this note is to prove the following Theorem 1. Letbe a variety of groups containing ℜ2, let A and B be torsion-free abelian hopfian groups and let P be the free-product2 of A and B. If P is residually torsion-free nilpotent, then P is hopfian.

A study is made of the aerodynamic damping in a cascade of oscillating aerofoils in subsonic compressible flow with the system mode described by a constant interblade phasing predicted by Lane (9). The integral downwash equation is obtained as an extension of Possio's equation for an isolated aerofoil. Procedures for a practical solution of the equation have allowed the aerodynamic reactions and damping derivatives to be evaluated with a digital computer after minimisation of the critical flutter velocity with respect to interbiade phase angle. The effect of aerodynamic lag in separation conditions as a function of reduced frequency and chordwise location is compared with results without separation.

The concept of a compactly generated lattice has been studied extensively in connection with decomposition theory (see [1]). This paper investigates the order topology in a lattice which is, along with its dual, compactly generated (hence, bicompactly generated). We show that order convergence is topological and that the order topology is Hausdorff, totally disconnected, and has an open subbase of ideals and dual ideals in any bicompactly generated lattice; furthermore, with an additional restriction, the lattice operations are continuous in the order topology. Next we consider the order topology in certain special types of compactly generated lattices, namely atomic Boolean algebras and sub-complete lattices of atomic Boolean algebras in the former structures the order topology is uniformizable, in the latter, compact.

Consideration of the primitive homomorphic images of semigroups arises naturally from their matrix representation theory. (See, for example, papers [1] and [2].) This is basically because a semigroup of matrices, in which all the matrices have the same rank, is necessarily primitive. (The set of idempotents in the semigroup may be empty, however).