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).

A family of κ-sized sets is said to be almost disjoint if each pair of sets from the family intersect in a set of power less than κ. Such an almost disjoint family ℋ is defined to be κ-maximally almost disjoint (κ-MAD) if |∪ℋ| = κ and each κ-sized subset of ∪ ℋ intersects some member of ℋ in a set of cardinality κ. A set T is called a representing set of a family if T ⊆ ∪ and T has non-empty intersection with each member of .

In (12), Takeuchi characterized those Fuchsian groups which are arithmetic in terms of the traces of the matrices involved in the group. Further, he used these criteria (13) to show that there are only finitely many arithmetic Fuchsian triangle groups and indeed to determine the signatures of all such groups. We note that, if two Fuchsian triangle groups have the same signature, then they are conjugate in PGL(2, ) and hence either all Fuchsian triangle groups of a given signature are arithmetic or none of them are. This will not, in general, be the case for non-triangle Fuchsian groups and in this paper, we examine Fuchsian groups with signature of one of the forms (1; n; 0), n ≽ 2 or (0; 2,2, 2, n; 0), n ≽ 3 and odd in which case the space of conjugacy classes has dimension two. Our principal results state that for each n, there are only finitely many conjugacy classes of arithmetic Fuchsian groups of given signature (1; n; 0) or (0; 2, 2, 2, n; 0) and for large enough n there are no arithmetic Fuchsian groups of that given signature (Theorems 3 and 4). Together with the results of (13) and known results for non-cocompact Fuchsian groups, these results show that there are only finitely many conjugacy classes of two-generator arithmetic Fuchsian groups (see §7).

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.

The thickness of a graph is a measure of its nonplanarity and has applications in the theory of printed circuits. To determine the thickness of an arbitrary graph is a seemingly intractable problem. This is made precise in this paper where we answer an open problem of Garey and Johnson (2) by proving that it is NP-complete to decide whether a graph has thickness two.

Let unity and the irrational components ri of a d-dimensional vector r be linearly independent, and consider the integers n for which

where {x} denotes the fractional part of x. It is a trivial restriction to assume that ri and φi lie between 0 and 1. This note is concerned with the question: what are the gaps between the successive n?

We let K1 and K2 be two convex bodies in Ed (d ≥ 3) and consider translates of K2 for which ∂K1 ∩ ∂ ≠ ø and K1 ≠ . We show that if, for all such translates,∂K1 ∩ is contained in a hyperplane then K1 and K2 are homothetic ellipsoids.

If K1 and K2 are two planar non-coincident convex sets whose interiors intersect we define α(K1, K2) to be the number of connected components of ∂K1 ∩ ∂K2. The starting point for our investigations is the result of Fujiwara(S) and Bol(2) to the effect that K is a circular disc if and only if α(K, K′) = 2 for all congruent copies K′ of K. An easy extension of this result is the observation that K1 and K2 are congruent circular discs if and only if for all congruent copies of K2. When working with such results it is natural to look for analogues in the situation where only translates are allowed rather than all possible congruences. The appropriate analogue was obtained in (7) and we state it here for completeness.

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

My first paper (5), submitted for publication by my supervisor Philip Hall, dealt with iteration of a continuous map of an interval. This subject has recently become fashionable and many interesting results are discussed in the book of Collet and Eckmann (4). Several of these results hold for special types of map, for example those with negative Schwarzian derivative. However here, as in my original paper, we will be concerned with arbitrary continuous maps.

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).

In (5), Peter Freyd recently raised the question of whether every Grothendieck topos could be obtained from the topos of sets by means of the two constructions of taking sheaves on a locale and of taking continuous actions of a topological group (i.e. the topos-theoretic analogues of the set-theorists' techniques of forcing extensions and permutation models). He showed that these two constructions do suffice to within epsilon; provided we allow ourselves the freedom to take exponential varieties (4) (which do not change the internal logic of the topos) we can obtain every Grothendieck topos in this way.

Let A be a local ring with maximal ideal m. Let M be a finitely generated module over A. Let a1 …, ar be a sequence of elements of m. Let qi denote the ideal (a1,…, ai), i = 1, …, r, and set q0 = 0A (the zero ideal of A), q = qr.

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.

This paper studies the Coxeter–Todd lattice its automorphism group (which is Mitchell's reflection group 6·PSU(4, 3)·2), and the associated 12-dimensional real lattice K12. We give several constructions for , which is a Z[ω]-lattice where ω = e2πi/3; enumerate the congruence classes of and where θ = ω − ω¯; prove the lattice is unique; determine its covering radius and deep holes; and study its connections with the lattice E6 and the Leech lattice. A number of new dense lattices in dimensions up to about 107 are constructed. We also give an explicit basis for the invariants of the Mitchell group. The paper concludes with an extensive bibliography.

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).

In considering the geometric invariant theory of linear systems (pencils, nets, …) of geometrical objects (quadrics, cubic curves, binary forms,…) there are several ways to apply the basic theory. The object of this note is to point out the equivalence of different approaches and to apply this remark to a number of special cases.

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

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.

Let g be a semi-simple complex Lie algebra with enveloping algebra U(g). It is shown that the Krull dimension of U(g) is bounded above by dim g − r, where r is half the minimal dimension of a non-trivial G orbit in g (G is the adjoint group of g).

Kirby and Lickorish have introduced the idea of prime tangle (5) where the term ‘tangle’ is borrowed from Conway (3). While exploring this notion, Lickorish has shown in ((6), theorem 3): A tangle, which is the Conway sum of two tangles, one of which is prime and connected to both arcs of the other one which itself is prime or untangle, is prime.