1. Introduction. All spaces in this paper are Hausdorff. We recall that a space X is sequentially compact, if every countable subset of X contains a convergent sequence. Let us consider the three statements:

(1) Every compact space of cardinality ≤ ʗ contains a point of countable character.

(2) Every compact space of cardinality ≤ ʗ is sequentially compact.

(3) Every infinite compact space of cardinality ≤ ʗ contains a convergent sequence.

Segre's classification of pencils of quadrics is well known, and appears in standard texts such as (1), (2). By contrast, other linear systems of quadrics do not lend them-selves to a similar exhaustive treatment. Here we discuss the simplest case, that of nets of conics. The result is intrinsically interesting, and involves some pleasant geometry. As well as deriving a list of types, and enumerating their properties, we study the elementary geometrical properties of the partition of the space Ω of nets into equivalence classes (or strata). We work over the field ℝ of real numbers, after performing preliminary calculations over ℂ.

1. This paper is a sequel to (2). The notation is the same and we assume the reader to be familiar with that paper. We there proved various theorems which established conditions under which ‘almost all’ labelled (n, q) graphs contained a k-cycle, i.e. under which the proportion of all labelled (n, q) graphs which contained a k-cycle tended to 1 as n → ∞. We here consider a different but related problem. If a graph contains a cycle of length k for every k such that 3 ≤ k ≤ k0, we say that the graph is pancyclic up to k0. We find conditions on k0, n and q such that almost all (n, q) graphs are pancyclic up to k0. If k0 = 0(1), the result follows trivially from Theorem 3a of (1) (Theorem 1 of (2)), but if k0 → ∞with n the matter is not so simple. We shall prove the following theorem.

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:

Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

The Vidav–Palmer theorem [(11), (5), (2) (p. 65)] characterizes C*-algebras among Banach algebras in terms of the algebra and norm structure alone, without reference to an involution, in the following way. Let B denote a complex unital Banach algebra, and let Her (B) denote the set of Hermitian elements of B, that is the elements of B with real numerical ranges. In this notation, the Vidav–Palmer theorem tells us that if

then B is isometrically isomorphic to a C*-algebra of operators on a Hilbert space, with the Hermitian elements corresponding to the self-adjoint operators in the C*-algebra.

Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variable

where ω is any primitive mth root of unity (e.g. ω = e2πi/m). Then

where 〈.〉 denotes expectation.

In this note, we discuss some questions on the arithmetic properties of recurrence sequences. Our primary purpose is to mention some methods which originated as lemmas in certain transcendence studies, but in order to give context to these ideas, we briefly review other techniques applicable to the problems considered. These problems concern the distribution of zeros in a recurrence, the rate of growth of its terms and the size of the greatest prime factor of the terms.

Given any embedding f:Sr × Bn−r → Sn, f ∣ Sr × 0 can be extended to a homeomorphism of Sn, except when r = n − 2 and f is actually knotted (10). Is there any sense in which this extension can be chosen to vary continuously with respect to f? If r = n − 1, there certainly is: this is simply the canonical Schoenflies theorem, proved in varying (but equivalent) forms in (6), (5) and (1). One of the principal results of this paper provides an affirmative answer to this question when r < n − 1. In the usual fashion of such canonical extension theorems, we look for a local version; that is attempt to find a neighbourhood of the inclusion in E(Sr × Bn−r, Sn) on which a map F can be defined with each F(f) a homeomorphism of Sn which extends f and F (inclusion) = identity. Now if the neighbourhood is sufficiently small, any element of it can be viewed as an embedding Sr × Bn−r → Sr × Rn−r, still close to the inclusion, by removing a standard Sn−r−1 from the image space. There are known methods for extending such embeddings. It would not be hard to adapt Kister's methods (8) for extending embeddings of Bk in Rk canonically to homeomorphisms (approximately the method that is sketched for the r = n − 1 case in (5)). Alternatively the existence of a canonical extension in this situation follows explicitly from the more general Theorem 5 of (1). Unfortunately both methods extend the embedding in a ‘radial’ fashion, with the result that there is little control over the extension towards infinity, and the final homeomorphism cannot necessarily be further extended over the missing Sn−r−1 to yield a homeomorphism of Sn. In fact this failure is due to the extension map F, derived in either of these ways, being continuous with respect to the compact-open topology on the image space, H(Sr × Rn−r). The extension map to be constructed here is continuous with respect to the uniform topology on the image space, and so the homeomorphisms F(f) of Sr × Rn−r are bounded and can thus be extended over all of Sn by setting F(f) ∣ Sn−r−1 equal to the identity.

In this paper, which assumes a familiarity with (1), we produce a short proof of the following theorem. The proof uses MOG techniques to extend a powerful observation of Conway (2) on intransitive subgroups of M24.

In this paper, the duality theory for vector lattices developed by the author in (2) is applied to the study of vector lattices freely generated by distributive lattices.

If L is a finite distributive lattice defined by a set of generators X, subject to a finite set of relations R, then the vector lattice FVL(L) freely generated by the elements of L subject to the relations governing intersection and union in L is a finitely generated projective vector lattice, being a quotient of the vector lattice freely generated by elements of X by the principal ideal generated by the relations R. Thus, there is a polyhedron Q, such that FVL(L) is isomorphic with the vector lattice of real valued polyhedral functions on Q, under pointwise operations, and the isomorphism class of FVL(L) determines and is determined by the polyhedral type of Q. The principal result of this paper shows that the polyhedral class to which Q belongs can be simply characterised in terms of the nerve of the poset P of join-irreducible elements of L. Explicitly, up to polyhedral equivalence, Q is a suspension of the nerve of P.

Two collections of locally convex space topologies are shown to have the intertwined completeness property. This is done by relating their completions with sets of sequentially continuous functionals on the dual.

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We write

If α is a real number, ∥α∥ denotes the distance from α to the nearest integer.

Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).

for all X ≥ 1. We say that θ is singular if

Singular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed that

for almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.

This paper considers perturbations in locally convex spaces of semi Fredholm operators T by a sequence of operators {Kn} converging compactly to zero in a sense extending that from (2). It is shown that, under suitable conditions, α(T + Kn) ≤ α(T), β(T + Kn) ≤ β;(T) and κ(T + Kn) = κ(T) for large n.

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.

A Riesz space E has the (sequential) order-continuity property if every positive linear map from E to an Archimedean Riesz space is (sequentially) order-continuous. This is the case if and only if the canonical maps from E to its Archimedean quotient spaces are all (sequentially) order-continuous. I relate these properties to others that have been described elsewhere.

Using Pitt's Tauberian classes and a theorem of Pitt (3), Krishnan(2) has recently obtained a gap Tauberian theorem for (Aα) summability. We recall briefly the notation introduced in (2). Let

and assume that the series a(t) and A(t) converge for all t > 0.

We prove the following theorem:

Theorem A. Let G be a finite simple group of order divisible by 3. Suppose

(i) G contains no element of order 6;

(ii) G has cyclic Sylow 3-subgroups;

(iii) G does not involve A4;

(iv) Every 3′-simple section of G is a Suzuki group.

Then G ≅ SL(2, 2n) for some odd n ≥ 3.

Some explicit formulae for the right coaction of A* on H*BO are given. As corollaries, closed formulae for χ(ξk) and other elements in A* are given.

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

Suppose that f is analytic in the unit disk D. If its range f(D) is contained in a simply connected proper subdomain of the plane, then the principle of subordination and the distortion theorem for univalent functions show that

where M(r, f) denotes the maximum modulus of f. Cartwright (2) studied functions which, instead of omitting all values on a continuum stretching to infinity, omit only a sequence of values. She assumed that the sequence {wn} satisfies

and

and proved that if f(D) contains none of the points {wn} thenm

for every ε > 0. Cartwright's proof was based on the Ahlfors Distortion Theorem, and is quite complicated. A much simpler proof was given by Pommerenke in (10). The key idea in his proof will also be used in the present paper.