Given a variety X over a field k and a dense open subset U of X, the related generalized albanese problem has two parts. First we want to classify rational maps with domain U into commutative algebraic groups, into reasonable categories, and then in each category we want to find an object α which is universal in the sense that any β in this category factors through α.

In the lattice theory that underlies topology, that is, in the study of frames, a class of frames arising naturally is that of the stably continuous frames (see §0 for definitions). On the one hand, they correspond to the most reasonable not necessarily Hausdorff compact spaces, and on the other, they are precisely the retracts of coherent frames. Moreover, an important special case of stably continuous frames are the compact regular frames which correspond to compact Hausdorff spaces.

Pregroups were defined by Stallings[7] who showed that the elements of the group they define have a normal form up to an equivalence called interleaving. Recently Rimlinger[5] has shown that subject to a discreteness and a boundedness condition any pregroup P defines a graph of groups. We show here that closer analysis of P makes the boundedness condition superfluous. In § 1 we give results of Stallings and Rimlinger and prove some key lemmas. In §2 we show that the discreteness condition gives an integer-valued length function in the sense of Lyndon [4]. It follows from the work of Chiswell [2] and Serre [6] that this defines a graph of groups. I would like to thank the referee for his careful reading and useful comments on this paper.

Let ƒ = {In}n ≽ 0 be a filtration on a ring R, let

(In)w = {x ε R; x satisfies an equation xk + i1xk − 1 + … + ik = 0, where ij ε Inj} be the weak integral closure of In and let ƒw = {(In)w}n ≽ 0. Then it is shown that ƒ ↦ ƒw is a closure operation on the set of all filtrations ƒ of R, and if R is Noetherian, then ƒw is a semi-prime operation that satisfies the cancellation law: if ƒh ≤ (gh)w and Rad (ƒ) ⊆ Rad (h), then ƒw ≤ gw. These results are then used to show that if R and ƒ are Noetherian, then the sets Ass (R/(In)w) are equal for all large n. Then these results are abstracted, and it is shown that if I ↦ Ix is a closure (resp.. semi-prime, prime) operation on the set of ideals I of R, then ƒ ↦ ƒx = {(In)x}n ≤ 0 is a closure (resp., semi-prime, prime) operation on the set of filtrations ƒ of R. In particular, if Δ is a multiplicatively closed set of finitely generated non-zero ideals of R and (In)Δ = ∪KεΔ(In, K: K), then ƒ ↦ ƒΔ is a semi-prime operation that satisfies a cancellation law, and if R and ƒ are Noetherian, then the sets Ass (R/(In)Δ) are quite well behaved.

If G is a locally compact group such that G/G0 contains a uniform compactly generated nilpotent subgroup, then G has a maximal compact normal subgroup K such that G/G is a Lie group. A topological group G is an N-group if, for each neighbourhood U of the identity and each compact set C ⊂G, there is a neighbourhood V of the identity such that for each g∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

Consider K⊂G, a proper pair of equal rank compact connected Lie groups. It is known (see [6], proof of theorem 1·1) that the group of self-homotopy equivalences of the rationalization of G/K is (anti)isomorphic, in a natural way, to the group of graded algebra automorphisms of H*(G/K; ℚ). On one side of the matter there is the fact that AutH*(G/K; ℚ) almost gives the integral picture of the group of homotopy classes of the self-homotopy equivalences of G/K, that is, up to a finite ambiguity and up to grading automorphisms of H*(G/K; ℚ) (i.e. those acting on H2i as λi.id, for some non-zero λ∈ℚ): see also [6]. On the other side there is the classical description by Borel[2] of H*(G/K; ℚ) in terms of invariants of Weyl groups, which gives hope for a satisfactory understanding of AutH*(G/K; ℚ).

Whenever a covariant (resp. contravariant) functor F from a category of groups is given, it is natural to ask the following question: if a homomorphismof groups induces the isomorphism (resp., ), is f itself an isomorphism?

In [6] and [9] the second author and Bill Richter showed that the natural ‘degree’ filtration on the homology of ΩSU(n) has a geometric realization, and that this filtration stably splits (as conjectured by M. Hopkins and M. Mahowald). The purpose of the present paper is to prove the real and quaternionic analogues of these results. To explain what this means, consider the following two ways of viewing the filtration and splitting for ΩSU(n). When n = ∞, ΩSU = BU. The filtration is BU(1)⊆BU(2)⊆… and the splitting BU≅ V1≤<∞is a theorem of Snaith[14]. The result for ΩSU(n) may then be viewed as a ‘restriction’ of the result for BU. On the other hand there is a well-known inclusion ℂPn−1. This extends to a map ΩΣℂPn−1→ΩSU(n), and the filtration (or splitting) may be viewed, at least algebraically, as a ‘quotient’ of the James filtration (or splitting) of ΩΣℂPn−1. It is now clear what is meant by the ‘real and quaternionic analogues’. In the quaternionic case, we replace BU by BSp=Ω(SU/SP), ΩSU(n) by Ω(SU(2n)/SP(n))and ℂPn−1 by ℍPn−1. The integral homology of Ω(SU(2n)/SP(n)) is the symmetric algebra on the homology of ℍPn−1, and may be filtered by the various symmetric powers. We show that this filtration can be realized geometrically, and that the spaces of the filtration are certain (singular) real algebraic varieties (exactly as in the complex case). The strata of the filtration are vector bundles over filtrations of Ω(SU(2n−2)/SP(n−1)), and the filtration stably splits. See Theorems (1·7) and (2·1) for the precise statement. In the real case we replace BU by Ω(SU/SO), Ω(SU(n)/SO(n)) and ℂPn−1 by ℝPn−1. Here integral homology must be replaced by mod 2 homology, and splitting is only obtained after localization at 2. (Snaith's splitting of BO in [14] can be refined [2, 8] so as to be exactly analogous to the splitting of BU:BO≅V1≤<∞MO(k).)

The Jones polynomial VL(t) of a link L in S3 contains certain information on the homology of the 2-fold branched covering D(L) of S3 branched along L. The following formulae are proved by Jones[3] and Lickorish and Millett[6] respectively:

1. Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixed n ≥ 1. We define (Γ, μ) to be a Fatou pair in Ω if

(a) Γ is a continuous family of boundary curves γw in Ω, one ending at each w ∈ ∂Ω,

(b) μ is a positive finite Borel measure on ∂Ω, and

(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:

(a) The map (w, t) → γw(t) is continuous, from ∂Ω × [0, 1) into Ω, and

for every w in the boundary ∂Ω of Ω.

(c) For every f ∈ H∞(Ω) (the class of all bounded holomorphic functions in Ω), the limit

exists a.e. [μ].

When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).

Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.

The notion of statistical convergence was introduced by Fast[1] and has been investigated in a number of papers[2, 5, 6]. Recently, Fridy [2] has shown that k(xk–xk+l) = O(1) is a Tauberian condition for the statistical convergence of (xk). Existing work on statistical convergence appears to have been restricted to real or complex sequences, but in the present note we extend the idea to apply to sequences in any locally convex Hausdorif topological linear space. Also we obtain a representation of statistical convergence in terms of strong summability given by a modulus function, an idea recently introduced in Maddox [3, 4]. Moreover Fridy's Tauberian result is extended so as to apply to sequences of slow oscillation in a locally convex space, and we also examine the local convexity of w(f) spaces.

For a time in the mid-1970s probabilists were tantalized by a seemingly simple problem posed by Araki and Kakutani[3]. An interval is repeatedly divided by points chosen successively at random, the nth point being uniformly distributed over the largest of the n intervals formed by the first n − 1 points. Is this sequence of points asymptotically uniformly distributed?

The connection problems associated with the one-dimensional Schrödinger equation in the presence of a general isolated cluster containing an unspecified number of complex transition points in unspecified positions can be studied by means of the phase-integral method developed by Fröman and Fröman. Any anti-Stokes line, i.e. any line in the complex z-plane on which the solutions behave as travelling waves with constant flow, must asymptotically (i.e. in the limit of large values of |z|) point in one of m +2 possible directions, which divide the region around the cluster into m +2 sectors, where m is the degree of the cluster. The tracing of these waves from an anti-Stokes line, bounding a sector, to an anti-Stokes line constituting the other boundary of the same sector is expressed by means of the Stokes constant for the sector in question. This paper examines the relation between these m + 2 Stokes constants in the general case when the transition points in the cluster may also be close-lying in the sense that it is impossible to treat them individually, when the solutions are traced. Under the assumption that the effective potential in the Schrodinger equation is a regular analytic function in a sufficiently large region containing the cluster, it is shown that the m + 2 Stokes constants are in general constrained by three algebraic relations, which are obtained for arbitrary m. The cases m = 1, 2, 3 and 4 are worked out in detail.

In many cases of practical interest there is some kind of symmetry associated with an ordinary second-order differential equation of the Schrödinger type. This symmetry is reflected in certain relations between different F-matrices or different elements of the same F-matrix of the phase-integral method. These symmetry relations imply in turn further relations between the Stokes constants, in addition to those which exist for a general cluster of transition points, and thereby reduce the number of independent Stokes constants. The present paper deals with two important kinds of symmetry relations.