In (l) Hunter proved that the complete symmetric functions of even order are positive definite by obtaining the inequality

where ht denotes the complete symmetric function of order t. In this note we show that the inequality can be strengthened, which, in turn, enables theorem 2 of (l) to be sharpened. We also obtain a special case of an inequality conjectured by McLeod(2).

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

Introduction and statement of results. It is well known that if V is a vector space over a field k then its dual vector space Homk (V, k) has dimension not less than that of V, and that the dimensions are equal precisely when V is finite-dimensional. In this note we introduce a ‘dual module’ for a module over a principal ideal domain, and show that some though not all of the properties of the field case carry over.

Let F be a field, L a commutative F-algebra and K an extension field of F. An important area of commutative algebra is the study of the passage from L to the k-algebra K ⊗FL, i.e. the investigation of the behaviour of the ideals of L under ‘extension of scalars’. In most problems of this kind one finds that the problem is reduced to the case when the algebra L is itself an extension field of F. It is for this reason that tensor products of fields play an important role (see, for example, (2), chap, viii, (3), (5), (9) and (12), vol. I).

All rings considered in this paper will be commutative and Noetherian, will have identities, and will be assumed to be non-trivial unless otherwise specified. The letter A will always denote such a ring. It is to be assumed that ring homomorphisms respect identity elements.

Let N be a von Neumann subalgebra of a von Neumann algebra M. A linear mapping π: M → N is called a retraction if it is idempotent and has norm one. By a result of Tomiyama(15) a retraction is a positive mapping and is a module homo-morphism over N. A retraction is normal if it is ultraweakly continuous, and faithful if it does not annihilate any nonzero positive element of M. Suppose that (Nn)n≥1 is an increasing sequence of von Neumann subalgebras of M whose union is weakly dense in M and that, for each n, πn: M → Nn is a faithful normal retraction. The sequence (πn) is called a martingale if, whenever m ≥ n,

Recall that a categorical covering of a space B is a covering by closed sets each of which is contractible in B. Suppose that B admits a finite categorical covering, and hence one where the number of sets is minimal. The category of B is then defined to be one less than that minimum number. Category is generally associated with nilpotency, in homotopy theory. In this note we describe a further illustration of this, from the theory of fibre spaces.

In their paper (l) on the general theory of connectednesses and disconnectednesses, Arhangel'skiĭ and Wiegandt conclude by proposing several problems. We shall show in this note, by elementary constructions (assuming of course the axiom of choice), that the answer to their questions (d) and (e) is negative in both cases; we also consider a modified form of question (d) to which the answer is positive. Our notation is that of (l) and references, unless otherwise stated, are to that paper; thus ‘(1·3)’ means ‘Proposition 1·3 of (l)’, but ‘(2·1 above)’ means ‘Proposition 2·1 of the present note.’

This paper will study certain rational BP operations. We will work in the category of spaces having the homotopy type of CW complexes. Recall that for − 1 ≤ n ≤ ∞ there is defined in this category the homology theory BP(n)*(X). (See (7).) In particular, BP(∞)*(X) = BP*(X) is the usual Brown Peterson homology of X, BP(1)*(X) = l*(X) is the canonical direct summand of connective K-theory with Qp coefficients (Qp are the integers localized at the prime p), BP(0)*(X) = Hℤ*(X) is ordinary homology with Qp coefficients, and BP(−1)*(X) = Hℤ/p*(X) is ordinary homology with ℤp coefficients (ℤp are the integers reduced mod p). For 1 ≤ n ≤ ∞, BP(n)*(X) is a module over BP(n)*(pt) = Qp[v1, …, vn] (deg vs = 2ps − 2). The BP(n) theories are all related via canonical maps p(m, n):BP(m)*(X) → BP(n)*(X) whose performance is described in (7).

In (1), J. F. Adams showed that for p odd, the Adams E2-term for a sphere, , is zero for s ≤ t ≤ (2p − 1)s − 1, while is the one-dimensional vector space generated by where corresponds to the Bockstein.

Although examples have been known since 1969 [Si], which reveal a quite similar failure of the differentiable and piecewise-linear s-cobordism theorems, these did not immediately suggest any topological failure.

If E and F are ordered spaces, a map T: E → F is positive if

The map T is bi-positive if f ≥ 0 ⇔ Tf ≥ 0. It is a natural problem to seek information about the positive and bi-positive linear maps between pairs of ordered vector spaces. The present study is an attempt to elucidate one facet of this general problem (see, for example, (2), vol. II, § 16·7·1, p. 278).

In statistical problems, use is often made of the variance s2 of a finite sequence of numbers x1, x2, …, xn, where

x¯ being the arithmetic mean of x1, x2, …, xn. The object of this note is to introduce a generalized variance for an infinite sequence x = (xk) and to examine some of its properties.

1. Let X be a normed linear space of periodic functions (of period 2π), which includes the class of trigonometric polynomials. We restrict ourselves throughout to functions f ∈ X. Let D) = (dn, k) be the matrix of a regular sequence-to-sequence transformation. (We could also consider sequence-to-function transformations, in which case n is replaced by a continuous variable.) We suppose that, for all n,

Let {Ln(x)} be the D transform of the Fourier series of f(x). If f(x) is a constant, it follows from (1) that, for all n, Ln(x) = f(x). However, it is often found that, roughly speaking, except in this trivial case, Ln(x) cannot tend to f(x) (in the topology given by the norm) with more than a certain degree of rapidity. This leads to the concept of ‘saturation’, which was first introduced by Favard (1) and Zamanski (8) and which has since been investigated by many authors (see (2), (3), (4), (5), (6), (7)). This is defined as follows.

Let Xnj, 1 ≤ j ≤ kn, be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the Xnj, the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of ΣjXnj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

Let X, Y be real Banach spaces, and let = {f(λ):λ ∈ m} be an m-parameter family of bounded linear operators from X to Y, with f(λ) depending continuously on λ. The cases m = 1 and m = 2 are considered, and conditions on are found which determine the null space of f(λ) for all λ near a given λ0 such that f(λ0): X → Y is a Fredholm operator. The results obtained are shown to be of particular interest in perturbed bifurcation theory.

Let Ω ⊆ n be open and bounded, with ∂Ω smooth. Sufficient conditions on f are given in order that for p > n, the equation

has a solution for every h ∈ Lp; L is a second order symmetric elliptic operator and B represents either the first, second or third boundary value problems. These conditions are in terms of the asymptotic behaviour of f, in its second variable, in relation to those λ for which there is a nontrivial solution of

We consider globally hyperbolic spacetimes (M, g) of dimension ≥ 3 satisfying the curvature condition Ric (g) (v, v) ≥ 0 for all non-spacelike tangent vectors v in TM. This curvature condition arises naturally as an energy condition in cosmology. Suppose (M, g) admits a smooth globally hyperbolic time function h: M → such that for some t0, the Cauchy surface h−1(t0) satisfies the strict curvature condition Ric (g) (v, v) > 0 for all non-spacelike v attached to h−1(t0). Then M admits a metric g′ conformal to g satisfying the strict curvature condition Ric (g′) (v, v) > 0 for all non-spacelike v in TM. If the curvature and strict curvature conditions are restricted to null vectors, the analogous result may be obtained. Similar results may also be obtained for the scalar curvature in dimension ≥ 2 and for non-positive Ricci curvature.