Let g(y) denote an arbitrary real-valued function satisfying g(y) → ∞ as y → ∞. One expects, and, subject to certain conjectures, Heath-Brown(3) has proved, that for all y in (1, X], apart from a set of measure o(X), the interval (y, y + g(y) log y] contains primes.

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

In (2) Lennox and Stonehewer proved: Let G be a group with subnormal subgroups H and K, and let J = 〈H, K〉. Suppose that J/J′ has finite rank. Then

(i) given positive integers a and b, there exists c such that γc(J) ≤ γa(H). γb(K) and

(ii) J is subnormal in G.

The object of this paper is to prove a necessary and sufficient condition on two groups H, K for their join always to be subnormal in a group G whenever they are embedded subnormally in G.

Let G be a simply primitive permutation group on a set Ω of order p2, where p is a prime (necessarily odd). In theorem 27·2 of (9), Wielandt states without proof:

Theorem A. (i) ¦G¦ is not divisible by p3;

(ii) if G has a pair of Sylow p-subgroups with nontrivial intersection, then G has an imprimitive subgroup of index 2 which is the direct product of two intransitive groups.

The braid group on n + 1 strings has the presentation

where xi corresponds to the single crossing of the i string over the i + 1 string, and rij is a commutator when i < j − 1, and a braid relator when i = j − 1. B1 is the trivial group and B2 is infinite cyclic; we shall assume n ≥ 2 throughout.

A wild orbit is an orbit of a transformation group which has the property that its dimension appears to increase in a paradoxical way. The existence of such orbits was noted in a previous paper (1).

Let A be a uniform algebra with maximal ideal space M and Shilov boundary Σ; see (8), (18) or (21) for the basic definitions. If Σ is different from M, there is often analytic structure in M/Σ. However this is not always the case, as is shown by the classical example of Stolzenberg in (16). Hence much of the considerable amount of research on this topic has been devoted to finding conditions which ensure the presence of analytic structure in M/Σ One particularly fruitful line of development has been concerned with one-dimensional analytic structure; in particular we have in mind the classical theorem of Bishop (see (2), chapter 11) and the more recent result of Aupetit and Wermer(2).

This paper investigates the ‘general position’ properties which ANR's may possess. The most important of these is the disjoint discs property of Cannon (5), which plays a vital role in recent striking characterizations of manifolds (5, 9, 12, 18, 19, 22). Also considered are the property Δ of Borsuk(2) (which ensures an abundance of dimension-preserving maps), and the vanishing of local homology groups up to a given dimension (cf. (9)). Our main results give relations between these properties, and clarify their behaviour under the stabilization operation of taking cartesian product with the real line. In the last section these results are applied to give partial solutions to questions about homogeneous ANR's.

In this paper we simultaneously prove a conjecture of G. W. Whitehead concerning symmetric product spectra and extend the epimorphism of the Kahn-Priddy Theorem to a long exact sequence. This sequence can be interpreted as a minimal resolution of the integral Eilenberg-MacLane spectrum by suspension spectra.

Let M, N be smooth manifolds and ℋ ⊂ TN a smooth sub-bundle. A smooth map φ:M → N will be called horizontal if

At points where φ(M) is a submanifold of N, the tangent spaces to φ(M) will be sub-spaces of the fibres of ℋ. In other words where φ(M) is a submanifold it is an integral submanifold of ℋ. If ℋ is not integrable it will follow that the rank of Æ must be less than dim ℋx. But we can often place much stricter bounds on the rank of φ by examining the integrability tensor of ℋ. This we shall do in this note.

This paper discusses the local and global geometry of coisotropic foliations and complex polarizations on symplectic manifolds and draws attention to an analogy between coisotropic foliations and Hermitian vector bundles, in which connections and characteristic classes are modelled by objects in symplectic geometry.

A class of f-driven algorithms due to Chvátal for finding the stability number of a graph are studied. It is shown that, for almost all graphs, the computation time grows subexponentially with n, the number of vertices. If, however, each edge exists with probability δ/n independently on other edges then asymptotically almost certainly the computation time is exponential. Still, for all large enough δ's, these algorithms perform noticeably better than a naive algorithm. The results are extended to random graphs with a fixed number of edges.

Suppose that {Xn} is a countable non-homogeneous Markov chain and

If converges for any i, l, m, j with , then

whenever lim , whereas if converges, then

where and . The behaviour of transition probabilities between various groups of states is studied and criteria for recurrence and transience are given.

Let Xi be iidrv's and Sn = X1 + X2 + … + Xn. Let (r)Sn be Sn minus the r terms of largest absolute value. If (Sn/Bn)− An converges to normality for some Bn, An, then so does ((r)Sn/Bn) − An. We show that the converse is true if Xi have a continuous symmetric distribution, and give some related results.

Let F be a distribution function with support confined to ( − π, π), W a Wiener process on (0, 1), and W0 a Brownian bridge. We examine the order of the approximation to W(F) and W0(F) by their Fourier series. The results we obtain are of theoretical interest since the Wiener process possesses the unusual property of satisfying a Lipschitz condition of order ∝ for each 0 < ∝ < ½, but not one of order ∝. = ½. The classical results on the rate of convergence of Fourier series are not sufficiently specific to provide an exact rate in the case of the Wiener process. The work also has practical significance because many statistical procedures can be closely approximated by functions of a Wiener process, and several procedures rely heavily on Fourier series methods. Thus, some of our results may be applied to obtain limit theorems for quantities such as trigonometric series estimates of density and distribution functions.