How must n equal non-overlapping circles be packed on a sphere so that the angular diameter of the circles will be as great as possible? In the paper, the conjectured solutions of this problem for n = 18, 27, 34, 35, 40 are improved on the basis of an idea of Danzer. Using the theory of bar structures it is ascertained that, in these cases, the edge-length of the graphs of the circle-packings can be increased till, in the graphs, additional edges appear which prevent further motions apart from rigid motions. The cases of n = 17 and 32 are also dealt with and there are references to the possibilities of further applications of the method applied in this paper (n = 59, 80, 110, 122).

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.

Suppose that (ni) = (n1, n2,…) and (mi) are infinite sequences of positive integers with ni < mi < ni+1. It is well-known and easily proved that, if a series σak is (C, 1) summable to s and has lacunae (ni, mi) such that ak = 0 (ni < k < mi) with

then where

This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).

Let ℒ be the class of those groups G which can be presented in the form

where u and v are positive words in the given generators. Here a word w is termed positive if only non-negative powers of a, b,…, c occur in w. If each generator occurs with exponent sum zero in uv-1, we term the ℒ-group G a -group. This class contains, in particular, the class X of those groups G which can be presented in the form

where u and v are positive words, and where [u, v] is the commutator uvu-1v-1.

Let R be a noetherian local ring and x = x1, …, xn a system of parameters for R. If R is an equicharacteristic local ring then Hochster(3) proved there is a big Cohen-Macaulay module with respect to x, i.e. an R-module M, not necessarily noetherian, with x1, …, xn a regular sequence on M and M/(x) M ≠ 0. Such modules are important for the study of the homological conjectures in commutative algebra(3). Nevertheless, for mixed characteristic local rings virtually nothing is known about their existence.

Every oriented knot or link in S3 can be represented in many ways as the closure of a braid β ∈ Bn, the braid group on n strings, for some n. Braids β ∈ Bn, γ ∈ Bm are called closure-equivalent if and are equivalent as oriented knots. It is a well-known result of Markov, see, for example, (l), that β and γ are closure equivalent if and only if there is a sequence of elementary (Markov) moves in which β ∈ Bn is replaced by

(a) a conjugate in Bn, or , or

(c) β1 ∈ Bn−1, where ,

and the process repeated until γ is reached.

In (1) it was shown that PL characteristic numbers determine (unoriented) PL bordism. In §§ 1 and 2 of this paper we use D. Sullivan's obstruction theory (9), (5) to provide some geometric insight into this result, as follows. We take an unoriented PL manifold M and let νM: M → BPL classify its stable normal bundle, and assume that all its PL characteristic numbers vanish, i.e. that (νM)* [M] = 0, where [M] is the fundamental class of M and . To show that M bounds, we first ‘resolve the homology to zero’ given by (νM)* [M] = 0. This is done by defining obstructions in whose vanishing ensures that the homology can be ‘resolved’, i.e. can be replaced by a bordism F: W → BPL between νM and a constant map, g: N → BPL, say. Then it is shown that these obstructions all vanish in this case.

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

Exact sequences, which were subsequently used for calculating the group ℰ(X) of homotopy classes of self-homotopy equivalences of a space X, were given by Barcus and Barratt (§5 of (1)) in the case where X is obtained from a simply-connected (q + 1 > 1)-dimensional complex by adding one (q + l)-cell (q ≥ 3): these were later extended by Kudo and Tsuchida (theorems 2·2 and 2·8 of (6)) and by the author (theorem 3·1* of (15)), who also obtained a related sequence (theorem 2·3* of (15)). In the case of a two-cell complex, one or more of these sequences has been shown to split by Oka, Sawashita and Sugawara (theorems 3·9, 3·13 and 3·15 of (11)). The sequences have been used to calculate ℰ (X) for a number of complexes having two, three or more cells by various authors, including Oka (8), Oka, Sawashita and Sugawara (11), Rutter (17) and Sawashita (18). However the aforementioned sequences are only applicable to the addition of top-dimensional cells if the complex has no cells in its penultimate dimension. In this article I obtain sequences which are applicable without this latter restriction, show that one of them is generally split, and in special cases where there is only one top-dimensional cell obtain a further splitting: sequences are given which are valid without the assumption that A is simply connected. Also I give a new formula for calculating ℰ(X)in the case where X is not 2-connected.

V. P. Snaith has shown that a number of interesting homology theories (e.g. K-theory, complex cobordism) arise when one localizes the stable homotopy type of certain H-spaces (BU(1), BU, etc.). In this paper we sketch a construction which, given an H-space X and , produces a ring spectrum IΣ−nX which lies between Xand its localization with respect to b, and in many cases has a tractable structure. We apply this when X is an Eilenberg-MacLane space K(ℤ/, n) and is a generator (p = 0 or a prime), and get complete results on the localizations in this case. The results confirm one's suspicions that localization is a fairly drastic manoeuvre, cutting down the large and complicated ring to something more or less trivial, except when p = 0 and n = 2, in which case the rich geometry of K(ℤ, 2) = BU(1) gives complex K-theory.

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

Let X be a compact connected homogeneous complex manifold, which is Kāhlerian and has the second Betti number equal to one: b2(X) = 1; dimcX ≥ 3.

It is known that these conditions imply the following: X is a projective-rational homogeneous manifold (see (3)); X has an ‘algebraic cell-decomposition’: the 2s-dimensional closed cells are s-dimensional irreducible algebraic sets in X and they form a basis for the 2s-homology group of X, s = 1, 2, …, dimcX (see (1)); there are no holomorphic maps of X on lower dimensional (normal) analytic spaces except constants (see (9)).

Given a smooth plane curve or surface in ℝ3 its parallels consist of those curves or surfaces a fixed distance d down the normals in a fixed direction. Generically they have Legendre singularities. We are concerned here with the way in which these parallels change as we alter the distance d. (Alternatively the manner in which wave-fronts change as they evolve from an initial smooth wavefront.)

This problem was considered in (1) by V. I. Arnold. In a very beautiful paper he describes the generic evolution of wavefronts but does not prove that for a generic initial wavefront in ℝ2 or ℝ3 the evolution is of the type described there. This we do here, using the tool of transversality. A more positive outcome of our investigation is that some of Arnold's generic forms do not occur (those corresponding to A2 singularities).

Criteria for singularity and absolute continuity analogous to Zygmund's for Riesz products, are demonstrated for those generalized products introduced by Modica and Mortola.

Consider a sequence of i.i.d. random variables attached to the points of an independent point process . The random record process is the process of epochs of successive maxima in this sequence. Various limit theorems are proved for the distribution of times to successive records and interrecord times. Some new results on the tail-behaviour of subordinated distributions are needed.

The method used by Gage and Reid(10) to investigate hydrodynamic stability of thermally stratified fluid is extended to hydromagnetic stability to study the effect of aligned magnetic field on the stability of unstable thermal stratification under the assumption of small magnetic Reynolds number. The interaction between the Tollmien–Schlichting–Stuart mechanism of instability due to shear and magnetic field and Rayleigh–Bénard–Thompson mechanism of instability due to thermally unstable stratification and magnetic field is brought out in detail. It is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, Squire's theorem is not valid. This conclusion follows from the fact that in our analysis the Richardson number Ri ( < 0) will not be greater than the value −0·92 × 10−6. In particular, it is shown that for the values of stratification parameter n ≤ 0·6 the effect of magnetic field for small values of Stuart number N is to augment instability and impose the restriction on the validity of our numerical procedure. However, for η = 0·8 a sharp transition from unstable to stable flow takes place at N = 0·3. A physical explanation for this based on eddies is given.

Alan Rahilly has pointed out to me that the proof of the Theorem in my paper (2) is incomplete. This correction will now complete it. I would also like to acknowledge here that the results which are actually established in (2) can also be found in the paper (5) by Praeger and Rahilly.

The trouble was in the proof of Proposition 2. Although the group G has a subgroup H1 which intersects each of its conjugates trivially, the same is not necessarily true of the image H1N/N of H1 in the 2-transitive representation G/N of G referred to in the paper. A theorem of M. E. O'Nan from (4) was used along with Proposition 2 to establish my theorem. What should have been done was to look at O'Nan's results more deeply and combine them with other results. Here is the way that it is done.

There is nothing actually wrong with the argument in my paper; it is just that it is not complete. The proof here will continue the argument of (2) using the notations established there.