Let G be an arbitrary group and Zn(G) denote the group algebra of G over the integers modulo n. If δi(G) denotes ith power of the augmentation ideal δ(G) of Zn(G), then

is easily seen to be a normal subgroup of G. It is denoted by Di, n(G) and is called ith dimension subgroup of G modulo n. It can be shown that these dimension subgroups are determined by the dimension subgroups modulo a power of a prime p. Hence we shall restrict our attention to these dimension subgroups.

Let D be an integral domain with identity. In (3), Cohn has shown that if D is a Schreier ring, then D can be inertly embedded in a pre-Bezout domain B(D), which Cohn calls the pre-Bezout hull of D. Further, D is an HCF-ring if and only if B(D) is a Bezout domain, and D is a UFD if and only if B(D) is a PID. Using a modification of Cohn's techniques, Samuel in (8) has shown that a Krull domain can be inertly embedded in a Dedekind domain. In this paper, we show that these embedding theorems are also obtained by considering the embedding of a v-ring in its Kronecker function ring with respect to the v-operation. To wit, if D is a v-ring and if Dv is the Kronecker function ring of D with respect to the v-operation, then Dv is a Bezout domain, and we show that D is inertly embedded in Dv if an only if D is an HCF-ring. Moreover, D is a UFD if and only if Dv is a PID, and if D is a Krull domain, D is inertly embedded (in Samuel's sense) in Dv, a Dedekind domain.

From a commutative ring A, Lazard(8) has made a flat injective epimorphism: A → B of commutative rings, such that if A → C is another flat injective epimorphism of commutative rings, then there is one and only one ring morphism: B → C such that the diagram

commutes; and he shows too that B → C is a flat injective epimorphism. The main aim of the present paper is to make a similar object for not necessarily commutative rings: this is achieved thanks to the notion of an A-prering, intermediate between that of an A-bimodule and that of an A-ring. In passing, prerings are also used to construct a kind of non-commutative ring of fractions.

Let A = (ank) denote an infinite matrix. Then A is said to be of type M if the conditions

imply αn = 0(n = 0,1,2,…).

Braid groups on the plane were denned by Artin(1) in 1925. More recently Fox(8) defined braid groups on arbitrary topological spaces, the situation being particularly interesting if the space is a 2-manifold. Presentations of the braid groups on R2, S2 and P2 can be found in (2), (6) and (13) respectively, and some general results on braid groups of compact 2-manifolds can be found in (7). In section 1 of this paper, we give finite presentations for the braid groups on all closed 2-manifolds except S2 and P2.

By a multi-valued map from a space X to a space Y we mean a map which assigns to each point x in X a non-empty subset F(x) of Y. When X = Y, a point x in X is a fized point for F if x is in F(x).

Let X be a CW-complex with finite skeletons and with H*(X, Z) free of p-torsion, where p is some odd prime. We shall prove two theorems on the action of the Steenrod powers

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectra

inducing a morphism

of homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequence

where Z is viewed as a Z[t] module via the augmentation

and, is induced by η*in the natural way.

Let M and Q be closed manifolds, and write ℝ+ = [0, ∞). Our main result is that, in both the PL and topological categories, M × ℝ+ unknots in Q × ℝ+ in codimension ≥ 3 (see Theorem 1 for a precise statement). The proof is essentially a generalization of Stallings's topological unknotting of spheres(13), the main tool being engulfing.

All spaces considered in this paper are Hausdorff. θX is a compactification of a completely regular space X means that X is identified with a dense subspace of θX. We shall thus always regard X to be contained in θX. A compactification θX of X is zero-dimension (countable) if its ‘outgrowth” θX ~ X is zero-dimensional (countable).

A function f( , ) of two variables is said to be separately continuous if for each fixed y and each fixed x, the functions f(, y) and f(x,) are continuous functions of one variable. Separately continuous functions seem to be playing a larger rôle in analysis than formerly. In (1) there is an account of the theory of semigroups in which the multiplication is separately continuous, and in (9) various analytic theorems are obtained by the study of separately continuous functions f(s, t), where t ranges over a set T, and s ranges over a set of functions on T.

Sawon and Warzecha established in (2) the general form of a subalgebra of codimension 1 in a commutative Banach algebra with identity. A short and elegant proof of their result was later given by Detraz in(1). This note is intended to show that an even more elementary proof is possible and which has the advantage that no use is made of the existence of maximal ideals.

Let A(Z) be the Banach algebra of Fourier-Lebesgue coefficients: a function f is an element of A(Z) if it is the Fourier transform

of some function fεL1(T), where T, as usual, denotes the circle group. We write

In this paper, by extending the results of Yoshikawa (8), we obtain local a priori inequalities for hypoelliptic pseudo-differential operators. Using these inequalities we then show how the results of Hormander ((3), Theorem 8·7·2), on the solvability of the adjoint operator of a principally normal operator can be extended to the adjoint operator of a hypoelliptic pseudo-differential operator. Finally, we consider a class of operators which satisfy more particular a priori inequalities and we show that these operators are hypoelliptic. This class of operators was studied by Egorov (1), and he shows them to be ‘of principal type’. They include elliptic operators and also the subelliptic operators of Hormander (4).

One of the classical models of applied probability is that which may be described as the covering of a line with a random collection of intervals of random length. It appears as such in ((9), pp. 23–25), but also as the problem of Type II counters with random dead time (1, 4, 10), as a model for a pedestrian crossing a busy road(5), and as the queue M/G/∞. If an excuse is required for returning to a problem which has been the object of so much research in the past, it is to be found in the fact, noted by Kendall (5), that the model gives rise to the general stable infinitely divisible p-function, and that the theory of the semigroup P of standard p-functions (6) requires deeper information than previous, practically motivated, studies have provided.

Let M1 denote the set of Borel probability measures on the real line and M1 the space of their Fourier transforms, or ‘characteristic functions’. It is well known that the natural correspondence between M1 and M1 is in fact a homeomorphism, provided M1 is endowed with the usual weak topology relative to bounded continuous functions (which topology can be metrized by the construction of Lévy) and that M1 is given the topology (also metric) appropriate to locally uniform convergence, which we call the ‘compact-open’ topology. In particular, if

where μn ∈ M1 (and so φn ∈ M1), then φn(λ)→φ0(λ) uniformly on all compact sets if and only if μn ⇒ μ0 (weak convergence). However, the continuity theorem for characteristic functions implies that even if it is only known that φn(λ)→φ0 pointwise (for all λ), the conclusion that μn ⇒ μ0 is still valid. This raises a question as to whether the pointwise (i.e. product) topology for M1 might not, in fact, be equivalent to the compact-open one, since the corresponding notions of sequential convergence do coincide. The purpose of this note is to show that the answer is negative, even in a much more general setting.

The generalized function (x + iO)λ is defined by

for λ ╪ -1, -2, … and (x + iO)λ is denned by

for n = 1, 2,…, see (l).

In order to study the convergence of a series arising from the use of Martin's method in potential scattering a framework consisting of a modifidation to Schwartz's formulation of distribution theory is set up. This modification makes essential use of norms on the spaces of test functions and linear functionals. By making the norm on the space of test functions satisfy extra axioms a bound is obtained on the norm of a generalized convolution of two linear functionals, and this bound is used to prove the convergence of the series. The proof of convergence is generalized to cover the multi-channel problem. The method used may also be applied to many other physical problems.

We consider here the supersonic inviscid flow past a wing-body combination consisting of a semi-infinite plane wing symmetrically placed about an infinite convex cylinder. When the cylinder is circular in cross-section a formal solution for the Laplace transform of the velocity potential exists (Neilson(l), and Stewartson, 1951, unpublished). This solution is given in the form of a series which is only slowly convergent near the boundary surface separating the disturbed and the undisturbed regions. However, this slow convergence has been overcome by Waechter(4) using the Poisson Summation formula on the series solution. This enables the solution to be expressed in terms of integrals which can be evaluated as a convergent residue series, taking on different forms in the different regions of the interaction.

In (general) elastico-viscous liquids the response to stress at any instant will depend on previous rheological history, the equations of state needed to describe the rheological properties of a typical material element at any instant t being expressible in the form of a (properly invariant†) set of integro-differential equations relating its deformation, stress and temperature histories (as defined by a metric tensor (of a convected frame of reference), a stress tensor and the temperature measured in the element as functions of previous time t'( < t)) together with the time lag (t – t') and physical constant tensors associated with the element (1). Thus in any type of oscillatory motion a rheological history will necessarily be a function of the frequency of the forcing agent, the rheological history of a number of different types of elastico-viscous liquids in some simple shearing oscillatory flows being a rather simple oscillatory history (see, for example, (2–4)). It is, therefore, to be expected that a liquid with elastic properties will behave somewhat differently from any inelastic viscous liquid when subjected to any kind of oscillatory motion, and it is for this reason that oscillatory motions have been used extensively to detect and measure the elastic properties of liquids (see, for example, (2–5)).