Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:

Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

There is a pairing between two Borel subalgebras of a quantized Kac–Moody algebra, which plays the rôle of R-matrix. Over the field ℚ(q) this pairing is non-degenerate. We show the existence of a braiding in some categories of representations of a quantized Kac-Moody algebra.

A natural number n is said to be squarefull if p|n implies p2|n for primes p. The set of all squarefull numbers is not much more dense in the natural numbers than the set of perfect squares but their additive properties may be rather different. We are more precise only in the case of sums of two such integers as this is the problem with which we are concerned here. Let U(x) be the number of integers not exceeding x and representable as the sum of two integer squares. Then, according to a theorem of Landau [4],

as x tends to infinity.

In this paper I give an algorithm to find all ‘small’ S-integral points on an elliptic curve. I would like to thank N. Stephens for suggesting I consider such equations and the Wingate Foundation for supporting me whilst carrying out the research. As is usual c1, c2, …, will denote positive real constants which are effectively computable.

A profinite topological group is compact and therefore possesses a unique invariant probability measure (Haar measure). We shall see that it is possible to define a fractional dimension on such a group in a canonical way, making use of Haar measure and a natural choice of invariant metric. This fractional dimension is analogous to Hausdorff dimension in ℝ.

It is therefore natural to ask to what extent known results concerning Hausdorff dimension in ℝ carry over to the profinite setting. In this paper, following a line of thought initiated by B. Volkmann in [12], we consider rings of a-adic integers and investigate the possible dimensions of their subgroups and subrings. We will find that for each prime p the ring of p-adic integers possesses subgroups of arbitrary dimension. This should cause little surprise since a similar result is known to hold in ℝ. However, we will also find that there exists a ring of a-adic integers possessing Borel subrings of arbitrary dimension. This is in contrast with the situation in ℝ, where the analogous statement is known to be false.

Let q be a prime power, which will be fixed throughout the paper, let k be a field, and let be the field with q elements. Let Gn(k) be the general linear group GL(n, k), and Sn(k) the special linear group SL(n, k). The corresponding groups over will be denoted simply by Gn and Sn. We may embed Gn(k) in Gn+1(k) via the map

Forming the direct limit of the resulting system, we obtain the stable general linear groupG∞(k) over k.

Let p be an odd prime, and h*( − ) a multiplicative mod p cohomology theory with a complex orientation satisfying the 2(p − 1)-sparseness condition. A formula concerning the formal group law for such h*( − ) is used to investigate the structure of h*(BG), G being a finite group whose p-Sylow subgroups are cyclic.

We consider coupled sets of identical cells and address the problem of which symmetries are permissible in such networks. For example, n linearly coupled cells with one independent variable in each cell cannot be constructed with the symmetry group An, the alternating group on n symbols. Using a graphical technique, we show that it is possible to construct cell networks with any desired finite group of symmetries. In particular, we show that any subgroup of Sn can be realized as the symmetries of a group of n cells. Special forms of coupling (especially low order polynomial coupling) are shown to restrict the possible symmetries. We give some upper and lower bounds for the degree of polynomial required to realize several classes of subgroups of Sn.

In [R] D. Rees introduced the notions of reduction and integral closure for modules over a commutative Noetherian ring and proved the following remarkable result. Let R be a locally quasi-unmixed Noetherian ring and I an ideal generated by n elements. Suppose that height (I) = h. Then the ith module of cycles in the Koszul complex on a set of n generators for I is contained in the integral closure of the ith module of boundaries for i > n − h. This result should be considered a dimension-theoretic analogue of the famous depth sensitivity property of the Koszul complex demonstrated by Serre and Auslander-Buschsbaum in the 1950s. At roughly the same time, Hoschster and Huneke introduced the notion of tight closure and thereafter gave a number of theorems in the same (though considerably broader) vein for tight closure. In particular, in [HH] they showed that if R is an equidimensional local ring of characteristic p > 0, which is a homomorphic image of a Gorenstein ring, then for all i > 0, the ith module of cycles is contained in the tight closure of the ith module of boundaries for any complex satisfying the so-called standard rank and height conditions (see the definitions below). Since the tight closure is contained in the integral closure for such rings, the result of Hochster and Huneke extends (in characteristic p) considerably the result of Rees. In fact, their result could be considered a dimension-theoretic analogue of the Buchsbaum-Eisenbud exactness theorem ([BE]), which in a certain sense is the ultimate depth sensitivity theorem. Moreover, using the technique of reduction to characteristic p, Hochster and Huneke have shown that their results hold in equicharacteristic zero as well, whenever the tight closure is defined.

Let A be an artin algebra over a commutative artin ring R, mod A be the category of finitely generated right A-modules, and rad∞ (modA) be the infinite power of the Jacobson radical rad(modA) of modA. Recall that A is said to be representation-finite if mod A admits only finitely many non-isomorphic indecomposable modules. It is known that A is representation-finite if and only if rad∞ (mod A) = 0. Moreover, from the validity of the First Brauer–Thrall Conjecture [26, 2] we know that A is representation-finite if and only if there is a common bound on the length of indecomposable modules in mod A.

Let A = k ⊕ ⊕n ≥ 1An connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.

Let G be a reductive group over an algebraically closed field K. In [8] and [9] we defined and studied certain finite dimensional K-algebras SK(π), associated to G via a finite saturated set π of dominant weights. The algebras are defined over ℤ, i.e. SK(π) = K ⊗ℤSℤ(π) for an order Sℤ(π) of Sℚ(π), and if G is a general linear group or a Chevalley group then the order Sℤ(π) arises naturally from the corresponding group scheme G over ℤ (or Kostant ℤ-form Uℤ). These algebras may be regarded as (and were obtained as) direct generalizations of the Schur algebras S(n, r) studied by Green in [10].

Full and reduced C*-coactions are shown to be essentially equivalent as far as the representations and cocrossed products are concerned, at least in the presence of non-degeneracy. This is shown to be particularly true for a special class of full coactions which are given the name normal.

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context of kG-modules where k is a field of finite characteristic p and G is a finite group whose order is divisible by p. Aside from some general considerations, the main results of [2] related the existence of kG-modules M with H*(G, M) = 0 to the structure of the centralizers of the p-elements in G. Specifically it was shown that there exists a non-projective module M in the principal block of kG with H*(G, M) = 0 whenever the centralizer of some p-element of G is not p-nilpotent. The converse was proved in the special case that the prime p is an odd integer (p > 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.

The δ-homogeneity of the Patterson measure is used for a closer study of the limit sets of Kleinian groups. A combination of the properties of this measure with concepts of diophantine approximations is shown to lead to a more detailed understanding of these limit sets. In particular, it is seen to how great an extent the studies of these sets, in terms of Hausdorff measure or Hausdorff dimension, are limited in a natural way.

Let d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.

The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.

For a locally compact group G with left Haar measure and modular function δ the conjugation representation γG of G on L2(G) is defined by

f ∈ L2(G), x, y ∈ G. γG has been investigated recently (see [19, 20, 21, 24, 32, 35]). For semi-simple Lie groups, a related representation has been studied in [25]. γG is of interest not least because of its connection to questions on inner invariant means on L∞(G). In what follows suppγG denotes the support of γG in the dual space Ĝ, that is the closed subset of all equivalence classes of irreducible representations which are weakly contained in γG. The purpose of this paper is to establish relations between properties such as a variant of Kazhdan's property and discreteness or countability of supp γG and the structure of G.

Let N be a compact normal subgroup of a locally compact group G. One of our goals here is to determine when and how a given compactification Y of G/N can be realized as a quotient of the analogous compactification (ψ, X) of G by Nψ = ψ(N) ⊂ X; this is achieved in a number of cases for which we can establish that μNψ ⊂ Nψ μ for all μ ∈ X A question arises naturally, ‘Can the latter containment be proper?’ With an example, we give a positive answer to this question.

The group G is an extension of N by GN and can be identified algebraically with Nx GN when this product is given the Schreier multiplication, and for our further results we assume that we can also identify G topologically with N x GN. When GN is discrete and X is the compactification of G coming from the left uniformly continuous functions, we are able to show that X is an extension of N by (GN)(X≅N x (G/N)) even when G is not a semidirect product. Examples are given to illustrate the theory, and also to show its limitations.

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

1. Let be a Banach algebra. We say that homomorphisms from are continuous if every homomorphism from into a Banach algebra is automatically continuous, and that derivations from are continuous if every derivation from into a Banach -bimodule is automatically continuous.