1. Introduction. In a principal ideal domain R, any two-sided ideal is of the form Rc = cR, i.e. it has an invariant element as generator, and the customary development of ideal theory in a principal ideal domain (cf. e.g. (10), ch. III) takes on a more transparent form when expressed in terms of invariant elements. Likewise, the one-sided bounded ideals may be studied in terms of their generators.

In 1934 Kurosh(9) proved that ‘a subgroup of a free product of groups is again a free product’. Many other proofs of this, and attempts to generalize it to amalgamated free products, have appeared (e.g. (7), (1), (10) and (8)). Recently the theory of groupoids has been applied to this area with increasing success. In 1966 Higgins (6) used groupoids to prove the generalization of Grushko's Theorem (3) due to Wagner (14).

It is pointed out that many axiomatic systems include a condition of closure under binary composition. In the present development a study is made of the consequences of the weaker requirement of closure under ternary, or three-fold composition. As a preliminary step in the investigation, subsets of groups closed under n-fold multiplication are examined and characterized abstractly. Attention is then focused upon ternary algebras defined as linear subspaces of ℬ (ℋ) (bounded operators on some Hilbert space ℋ) closed under three-fold multiplication. These are shown to satisfy, under suitable conditions, a secondanticommutanttheorem, the analogue of the second commutant theorem which holds for (binary) algebras. It is a result which provides a useful technique for studying such ternary algebras, the structure of which are examined in some detail.

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.

1. Introduction. The principle of the Conservation of Number is concerned with the following situation. One starts with a system of algebraic equations having only a finite number of solutions and then applies a homomorphism whose domain contains the coefficients of those equations. This produces a new system. Let us suppose that the new system of equations also has only a finite number of solutions. The question then arises as to how the number of solutions before specialization compares with the number present afterwards. In a typical geometrical situation, one usually wishes to assert that the two systems have equally many solutions. However, it is easy to construct algebraic situations where the number changes,† and where the change is not to be explained away through the confluence of solutions or by their slipping off to infinity. At first sight this represents a breakdown of the conservation principle, but this principle has proved so useful in the past that one has a natural reluctance to discard it. The alternative is to attempt a reformulation and in (l) the present author gave such a reformulation for the case in which the specialization consisted in mapping a regular local ring on to its residue field. The modified theory requires that we take account of systems of equations which arise in connexion with the homology modules of a certain complex. The system associated with the homology module of degree zero is found to be the same as the one that arises in the naive theory, and usually this is the only one that makes a contribution. However, in cases where the number of solutions appears to change, the other systems become active and act in such a way that the balance is restored. For an amplification of these remarks we must refer the reader to (l). They are made here to indicate how the relevance of homological concepts first became clear in any detail. In the present paper these ideas are taken further, the principal gain being that it is no longer necessary to restrict the type of specialization to that which consists in mapping a regular local ring on to its residue field. Indeed one can use very general specializations provided that one transfers the homological requirement from the ring to the system of equations under consideration. In this way, one obtains a theory which is more general and, in some of its aspects, simpler as well.

In this paper we shall consider contractible 2-dimensional cell complexes K with one 0-cell v, one l-cell a, and one 2-cell D, where the boundary of D is attached to a by a word in a and a−1. In order for such a complex K to be contractible, the attaching word in a and a−1 must have total degree ± 1. The simplest example occurs when the attaching word is a; in this case K is a 2-cell, and is of no interest since K is collapsible. If any other word of degree ± 1 is used, the resulting K is contractible but not collapsible.

Let Mm be a compact connected PL-manifold, we say M is metastably connected if πi(M) = 0 for i ≤ [⅓m] + 1, and m ≥ 6. Our notation will follow (3).

1. Introduction. Let X be a complex Banach space and let (X) denote the Banach algebra of all bounded linear operators on X. In this paper we study subalgebras of (X) that contain non-zero compact operators and that are Banach algebras with respect to some norm dominating the operator norm. Our main aim is to give conditions for the existence of minimal one-sided ideals in . Barnes ((l) Theorem 2·2) has shown that if every element of a semi-simple Banach algebra has countable spectrum then the algebra has minimal one-sided ideals. If, in particular, is a uniformly closed sub-algebra of the compact operators on X, then every element of has countable spectrum by the Riesz–Schauder theory and so has minimal one-sided ideals. We extend this latter result by showing that if is a semi-simple uniformly closed subalgebra of (X) that contains some non-zero compact operator, then has minimal one-sided ideals. The proof depends heavily on the fact (see e.g. Bonsall (3)) that the spectral projection at a non-zero eigenvalue of a compact operator T belongs to the least closed subalgebra of (X) that contains T. The uniform norm is quite critical for Bonsall's result, but we are able to give a mild generalization which leads to conditions for the existence of minimal one-sided ideals in subalgebras of (X, Y,〈,〉) where (X, Y,〈,〉) are Banach spaces in normed duality.

1. Introduction. Let P be a collection of functions mapping some set into the complex plane C and let δ be a subset of C. A function F: δ → C is said to operate on P provided the composition F o φ belongs to P whenever φ ∈ P and the range of φ is contained in δ. The basic problem is to characterize the functions which operate.

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

1. Definitions and notations. Let be a given infinite series with the sequence of its partial sums {sn}. Let {pn} be a sequence of constants, real or complex, and let us write

1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-strip

and if there exists an x0 in (a, b) such that

as y → ∞, then

as y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and if

as y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such that

as y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)

asy → ∞, then

as y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker condition

as n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.

If u is subharmonic in some open subset of Euclidean space, RN (N ≥ 1), which contains the ball |x| ≤ ρ and if S(r) and B(r) denote the mean values of u taken over the sphere |x| = r and the ball |x| ≤ r respectively (0 ≤ r ≤ ρ) then, ((2), p. 196),

Let p = 1 + 5n be a rational prime congruent to 1 (mod 5). Let ζ = e2πi/p and let g be a primitive root mod p. Let the non-zero residues g, g2, …, gp-1 (mod) p be divided into five classes , ℬ, , , ℰ, where gν ∈ , ℬ, , , ℰ according as ν ≡ 0, 1, 2, 3, 4 (mod 5). Let

be the 5-nomial periods. Then it is well known (see (3)) that they are the roots of a monic polynomial with integral coefficients. Our object is to determine these coefficients in terms of the quantities A, B, C, D, E, Y, Z considered in a previous paper (2), p. 65. A large number of relations connecting these quantities have been obtained in the above-mentioned paper and we shall use these relations to simplify the coefficients and get them in a reasonably compact and symmetrical form.

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfying

where H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

The differential equation

defining the function of the amplitude sL is integrated by an iterative procedure. An expression for the phase

is obtained in terms of integrals of ωL and its derivatives. The method is applied to the numerical calculation of Coulomb functions and the accuracy of the results is discussed. Recurrence relations are used to obtain øL, L ≽ 1.

A technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.

1. Introduction. Flow of viscous fluid at low Reynolds number has attracted a great deal of attention over the last hundred years or so and is still an area of active research at the present time. For a general background on this field, we refer the reader to Langois(4). The present paper is concerned with steady flow past a finite flat plate which spans a long rectangular channel, of large aspect-ratio, at the centre ‘line’ (see Fig. 1). The flow is motivated by a constant pressure difference which is applied

between the ends of the channel. We examine such flows when the Reynolds number based on the semi-channel height, a, is much less than 1. Our main task is to show that Stokes approximation, i.e. neglection of the inertia terms, is uniformly valid † over the whole flow field. (†Such a result contrasts with the non-uniformity of the approximation when the channel walls are removed and the external flow is a uniform stream.) Harper & Chang (1) who treat the corresponding problem for a circular cylinder, give a full account of recent theoretical and experimental work on this class of fluid flows. We shall therefore defer comments on related papers until the appropriate part of the text.

1. Introduction. The historical basis for the work in this paper lies in a remarkable fact discovered by Dean in 1948. He found that time-periodic surface waves in an ideal fluid experience no reflexion when they encounter normally a fixed, submerged, right-circular cylinder. We might reasonably ask if a similar non-reflective property carries over to different geometrical configurations of submerged objects, for example, two or more cylinders. This question motivates the investigation which follows.