We fix a number field L and a finite group G, and write Cl (ℤL[G]) for the reduced Grothendieck group of the category of finitely generated projective ℤL[G]-modules. We let RG denote the ring of complex characters of G, with SG the additive subgroup which is generated by the irreducible symplectic characters. We shall say that an element c ∈ Cl (ℤL[G]) is ‘(arithmetically) realizable’ if there exists a tamely ramified Galois extension N/K of number fields with L ⊆ K and an identification Gal (N/K) →˜ G via which c is the class of some Gal (N/K)-stble ℤN-ideal. We let RL(G) denote the subgroup of Cl (ℤL[G]) which is generated by the realizable elements for varying N/K. Our interest in RL(G) arises from the fact that it is the largest subset of Cl (ℤL[G]) upon which the results of Chinburg and the author in [Bu, Ch] can be used to give an explicit module theoretic description of the action of the integral semi-group ring AL, G of the Adams-Cassou-Noguès-Taylor operators (ΨL, k): k ∈ ℤ, 2 × k if SG ≠ {0}}. Whilst the results of [Bu, Ch] can (at least partially) be understood ‘geometrically’ via the action of Bott cannibalistic elements on suitable Grothendieck groups (cf. [Ch, E, P, T], [Bu]), the underlying problem of finding an explicit module theoretic interpretation of the action of AL, G on all elements of Cl(ℤL[G]) is of course essentially algebraic in nature. It is in this context that we were originally motivated to investigate RL(G).

Tricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

Let X be a surface in Euclidean 3-space, hereafter denoted by ℝ3. In the paper [13] Montaldi considered the contact of the surface X with circles, and obtained some very attractive results. In this piece of work we want to address some more detailed questions concerning such contact. In keeping with a general theme within singularity theory we shall bundle the circles up into fibres of certain maps and consider the restriction of these mappings to our surface X. In other words we shall be interested in the simultaneous contact of the surface X with special families of circles. The particular families we shall consider are parameterized by the set K of all lines in ℝ3; associated to such a line we have the family of all circles lying in planes orthogonal to the line, and centred on the line. The line will be referred to as the axis of the circle. Suppose, for example, the line in question is given by x1 = x2 = 0. We can consider the map ℝ3 → ℝ2 given by . The fibres of this mapping are clearly the set of circles with the properties described above together, of course, with single points on the line itself. So the family of oriented lines parameterizes a family of mappings ℝ3 → ℝ2, and by restriction a family of mappings X → ℝ2. It is of interest to relate the singularities of this mapping to the differential geometry of X. The key geometric invariant of any smooth family is its bifurcation set, that is the set of parameter values for which the corresponding map fails to be stable. We shall see that for the family of circle maps the bifurcation set is of some interest.

Using potential theoretic methods and exploiting the connection with eigenvalues of Toeplitz matrices, we investigate the limiting behaviour of zeros of Faber polynomials generated by a Laurent series. Our results build upon fundamental work of J. L. Ullman. For example, we show that if E is a compact set with simply connected complement and connected interior whose boundary is either (i) not a piecewise analytic curve or (ii) a piecewise analytic curve but with a singularity other than an outward cusp, then the equilibrium distribution for E is a limit measure of the sequence of normalized zero counting measures for the Faber polynomials associated with E.

In this paper we study the homology of groups with coefficients in metabelian Lie powers, and apply the results to obtain information about elements of finite order in certain free central extensions of groups. Perhaps the most prominent example to which our results apply is the relatively free group

where Fd is the (absolutely) free group of rank d. Thus Fd(Bc) is the free group of rank d in the variety Bc of all groups which are both centre-by-(nilpotent of class ≤ c − 1)-by-abelian and soluble of derived length ≤ 3. It was pointed out in [1] that the order of any torsion element in Fd(Bc) divides c if c is odd and 2c if c is even. This, however, is a conditional result as it does not answer the question of whether or not there are any torsion elements in (1·1). Up to now, this question had only been answered in case when c is a prime number [1] or c = 4 [8]. In these cases Fd (Bc) is torsion-free if d ≤ 3, and elements of finite order do occur in Fd(Bc) if d ≥ 4. Moreover, the torsion elements in Fd(Bc) form a subgroup, and the precise structure of this torsion subgroup was exhibited in [1] in the case when c is a prime and in [8] for c = 4. In the present paper we add to this knowledge. On the one hand, we show that for any prime p dividing c the group Fd(Bc) has no elements of order p for all d up to a certain upper bound, which takes arbitrarily large values as c varies over all multiples of p. On the other hand, we show that for prime powers does contain elements of order p whenever d ≥ 4. Finally, we exhibit the precise structure of the p-torsion subgroup of when p ≠ 2. Precise statements are given below (Corollaries 1 and 2). Our results on (1·1) are a special case of more general results (Theorems 1′−3′) which refer to a much wider class of groups, and which are, in their turn, a consequence of our main results on the homology of metabelian Lie powers (Theorems 1–3).

Let IΩn, k denote the bordism group of immersions of closed oriented n-manifolds into (n + k)-space. The object of this paper is to study certain group extension problems arising from Pastor's calculations of IΩn, k.

The bordism group of immersions was first studied by Wells [12] who calculated the unoriented bordism group I Rn, k for k = n and k = n − 1 ≡ 3(4). Later these unoriented bordism groups were completely determined by Koschorke and Olk for k ≥ n − 2 with the help of an exact sequence measuring the difference between IRn, k and Rn (see [4]). A similar program has been carried out by Pastor [7] to determine the oriented bordism group I Ωn, k for k ≥ n − 2 except for certain group extension problems and some low dimensional cases.

The purpose of this note is to present a few facts about the Julia set of a rational function that are well known to the experts in the subject of complex dynamics but whose documented exposition in the literature seems to need a little clarification. For example, under conditions set out in Theorem 1, the Julia set of a rational function can be expressed as the limit of a sequence of finite sets. In particular, for certain choices of a point α, the Julia set is the limit as n increases of the inverse image sets R−n(α). This formulation is widely exploited in the backwards iteration algorithm to produce computer illustrations of Julia sets (see for example Section 5·4 of [14]).

We show that if a rearrangement invariant Banach function space E on the positive semi-axis satisfies a non-trivial lower q-estimate with constant 1 then the corresponding space E(M) of τ-measurable operators, affiliated with an arbitrary semi-finite von Neumann algebra M equipped with a distinguished faithful, normal, semi-finite trace τ, has the uniform Kadec-Klee property for the topology of local convergence in measure. In particular, the Lorentz function spaces Lq, p and the Lorentz-Schatten classes Cg, p have the UKK property for convergence locally in measure and for the weak-operator topology, respectively. As a partial converse, we show that if E has the UKK property with respect to local convergence in measure then E must satisfy some non-trivial lower q-estimate. We also prove a uniform Kadec-Klee result for local convergence in any Banach lattice satisfying a lower q-estimate.

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined.

The existence and nature of singularities in locally spatially homogeneous solutions of the Einstein equations coupled to various phenoraenological matter models is investigated. It is shown that, under certain reasonable assumptions on the matter, there are no singularities in an expanding phase of the evolution and that unless the spacetime is empty a contracting phase always ends in a singularity where at least one scalar invariant of the curvature diverges uniformly. The class of matter models treated includes perfect fluids, mixtures of non-interacting perfect fluids and collisionless matter.

This is a study of entire functions whose coefficients are independent random variables. When the space of such functions is symmetric it is shown that independence of the coefficients alone is sufficient to ensure that almost all such functions will, for large z, be large except in certain small neighbourhoods of the zeros called pits. In each pit the function takes every not too large value and these pits have a certain uniform distribution.

The notion of a Gaussian as the exponential of a quadratic is rather familiar. Such functions are of considerable importance in a number of contexts, for example within quantum theory. Thus, in the Schrödinger representation of the canonical commutation relations they alone minimize uncertainty and they appear as ground states for harmonic oscillators. Also in the complex-wave representation of a free boson field they arise as transforms of the Fock vacuum under certain Bogoliubov automorphisms.

Solvable model of a quantum dot as a resonator with semitransparent boundary is constructed in the framework of the operator extensions theory. It is proved that the model operator is a limit of the short range Hamiltonians in the norm resolvent sense.