The number (up to isomorphism) of positive-definite, even, unimodular lattices of rank 8r grows rapidly with r. However, Bannai [1] has shown that, when counted according to weight, those with non-trivial automorphisms make up a fraction of the whole, which goes rapidly to zero as r→∞. Therefore it is of some interest to produce families of positive-definite, even, unimodular lattices with large automorphism groups and unbounded ranks.

Suppose that G is a finite group and V is an irreducible ℚ[G]-module such that V[otimes ]ℝ is still irreducible. Then, as observed by Gross [8], the space of G-invariant symmetric bilinear forms on V is one-dimensional and is necessarily generated by a positive-definite form, unique up to scaling by non-zero positive rationals. Thompson [23] showed that, if V is also irreducible modp for all primes p, then it contains an invariant lattice (unique up to scaling) which is even and unimodular with appropriate scaling of the quadratic form. Examples arising in this manner are the E8-lattice of rank 8, the Leech lattice of rank 24 and the Thompson–Smith lattice of rank 248. Gow [6] has also constructed some examples associated with the basic spin representations of 2An and 2Sn.

Suppose that [Pscr ] is a distribution of N points in the unit square U=[0, 1]2. For every x=(x1, x2)∈U, let B(x)=[0, x1]×[0, x2] denote the aligned rectangle containing all points y=(y1, y2)∈U satisfying 0[les ]y1[les ]x1 and 0[les ]y2[les ]x2. Denote by Z[[Pscr ]; B(x)] the number of points of [Pscr ] that lie in B(x), and consider the discrepancy function

D[[Pscr ]; B(x)]=Z[[Pscr ]; B(x)]−Nμ(B(x)),

where μ denotes the usual area measure.

One of the principal impulses for this paper was the consideration of the question, first raised in 1973 by G. Baumslag [2], of whether every finitely generated Abelian by polycyclic group can be embedded in a finitely presented Abelian by polycyclic group. That this cannot be done follows at once from

THEOREM A. Every finitely presented Abelian by polycyclic group has a metanilpotent normal subgroup of finite index.

All rings in this paper are commutative, with identity. If the ring R is either finitely generated, or semi-local (by which I mean Noetherian and possessing only finitely many maximal ideals), then R has only a finite number of ideals of each finite index. Thus it makes sense to study the function n[map ]an(R), where an(R) denotes the number of ideals of index n in R. In an earlier note [10], I determined an(R) for certain 2-dimensional domains R. The results to be discussed here are less precise, but more general; in particular, they deal with submodules of a finitely generated module. This makes it possible to translate them into results about the growth of normal subgroups in certain metabelian groups, the original motivation for this work: for example, we determine the finitely generated metabelian groups with ‘polynomial normal subgroup growth’ (Theorem 6.1).

Let G be a group, and let Fn[G] be the free G-group of rank n. Then Fn[G] is just the natural non-abelian analogue of the free ℤG-module of rank n, and correspondingly the group Φn(G) of equivariant automorphisms of Fn[G] is a natural analogue of the general linear group GLn(ℤG). The groups Φn(G) have been studied recently in [4, 8, 5]. In particular, in [5] it was shown that if G is not finitely presentable (f.p.) then neither is Φn(G), and conversely, that Φn(G) is f.p. if G is f.p. and n≠2. It is a common phenomenon that for small ranks the automorphism groups of free objects may behave unstably (see the survey article [2]), and the main aim of the present paper is to show that this turns out to be the case for the groups Φ2(G).

We develop a theory of harmonic analysis and duality for finite commutative hypergroups by considering somewhat more general objects called signed hypergroups. A notion of entropy is defined, and a Second Law of Thermodynamics is established. Applications to group theory and to the fusion rule algebras of conformal field theory are given.

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of:

(1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equations);

(2) elements from the polynomial envelope of a linear differential equation with an irreducible monodromy group (in terms of the degree of the envelope);

(3) successive derivatives of a function defined by a linear irreducible equation (in terms of the order of the derivative).

These results generalize the bounds from [2, 5, 6] that were previously obtained for the number of real isolated zeros.

In this paper we consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. self-similar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set K the function ρ(x)=O(xds/2) as x→∞, for some ds>0. We show that if there exist localized eigenfunctions (that is, a non-zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions (‘the lattice case’) then ρ(x)x−ds/2 does not converge as x→∞. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K. In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.

We consider operators on the matrix-valued disc algebra. We show that any bounded linear operator T: A[otimes ][Kscr ]→X to a Banach space X of cotype 2 induces a bounded operator Gν→X defined on some weighted Bergman space Gν on the unit disc. We give sufficient conditions on the weight ν for the formal inclusion A[otimes ][Kscr ]→Gν to be 2−C*-summing. Decompositions of T with respect to operator-valued measures are obtained.

In the last decade much research has been done on questions related to circle packings, which are collections of circles in the plane with prescribed patterns of tangencies encoded by simplicial 2-complexes. The interest in circle packings and their connections with analytic functions was initiated by W. Thurston in 1985 when he suggested a method for approximation of conformal mappings using circle packings [20]. This method was shown to work by B. Rodin & D. Sullivan [15] later that year (see also [17]). Since then several results have been published (for example, [3, 4, 5, 10, 11, 14]) which seem to indicate the possibility of creating a discrete analog of the theory of analytic functions via circle packings. However, most of these results deal with univalent circle packings, that is, genuine packings of circles. Influenced by the idea of [3] that circle packings might be used to construct discrete parallels of analytic functions, we started in [7] to investigate non-univalent circle packings, particularly branched packings. We introduced there the notion of discrete Blaschke products and proved a branched version of the finite Riemann mapping theorem (see [20, 15]). We feel though that to have a solid foundation for a circle packing analog of the theory of analytic functions one needs something more. Since complex polynomials form a fundamental and very important class of analytic functions, it is natural to try to establish circle packing counterparts of these mappings. This is exactly the main theme of our paper.

We render the cut locus more accessible to analysis by showing that each non-conjugate cut point has a neighbourhood on which the distance function is the minimum of finitely many smooth ‘radial functions’. This enables us to generalise to an arbitrary complete manifold a number of recent results, mainly from stochastic analysis, which were either limited or not valid on the cut locus because of the lack of differentiability there of the distance function.

On sait associer à certaines structures de Poisson sur ℝn, de 1-jet nul en 0, des actions de ℝ2 sur ℝn, données par le ‘rotationnel’ de leur partie quadratique et un autre champ de vecteurs. Lorsque ces actions sont ‘non résonantes’ et ‘hyperboliques’, on montre que ces structures sont ‘quadratisables’, en ce sens qu'il existe des coordonnées dans lesquelles, elles sont quadratiques. Dans le cas de la dimension 3, nos résultats mènent à la ‘non-dégénérescence’ générique des structures de Poisson quadratiques à rotationnels inversibles.

We can associate with some Poisson structures defined on ℝn with a zero 1-jet at zero, actions from ℝ2 on ℝn, given by the ‘curl’ of their quadratic part and another vector field. Assuming that those actions are ‘hyperbolics’ and without ‘resonances’, we give a normal form for those structures. On ℝ3, we prove that every quadratic Poisson structure with invertible curl, is generically ‘non degenerate’.

We consider aperiodic shifts of finite type σ with an equilibrium state m and associated skew-products σf where f: X→G is Hölder and G is a compact Lie group. We show that generically σf is weak-mixing and give constructive methods for achieving weak-mixing by perturbing an arbitrary f at a finite number of small neighbourhoods. When σf is not ergodic we describe the (closed) ergodic decomposition precisely. The key result shows that certain measurable eigenfunctions are essentially Hölder continuous. This leads to conditions for weak-mixing or ergodicity in terms of a functional equation which involves Hölder rather than measurable functions.

In this article we extend well-known results of Livsic on the regularity of measurable solutions to the cocycle equation. Livsic proved a regularity result for real valued cocycles, but for compact Lie group valued cocycles his result was restricted to coboundaries (that is, trivial or constant cocycles). In this article we prove the complete result.

We present several useful applications of these results.