Consider actions of a reductive complex Lie group G on an analytic space germ (X, 0). In a previous paper [16] we proved that such an action is determined uniquely (up to conjugation with an automorphism of (X, 0)) by the induced action of G on the tangent space of (X, 0). Here it will be shown that every deformation of such an action, parametrized holomorphically by a reduced analytic space germ, is trivial, i.e. can be obtained from the given action by conjugation with a family of automorphisms of (X, 0) depending holomorphically on the parameter. (For a more precise formulation in terms of actions on analytic ℂ-algebras, see Theorem 2 below. An analogue for actions on formal ℂ-algebras is given there too.)

We introduce a new method for proving ‘witness lemmas’ in BQO theory. As an application, we obtain witness-lemmas for arbitrary trees and for σ-scattered linear orderings. A positive answer to a question by Van Engelen, Miller and Steel follows.

The aim of this paper is to make some remarks about the theory of parallel immersions in ℝn. We show that the existence of parallels to a curve in ℝn depends on the values of the integrals of the n − 1 curvature functions along the curve. We then give examples of immersions of S1 in ℝ3 and of immersions of codimension 2 with local parallel rank zero, thereby answering certain questions in [1].

If R is a Noetherian local ring and I = (x1, …, xn)R is an ideal of R then the Rees algebra R[It] can be represented as a homomorphic image of the polynomial ring R[Z1, …, Zn]. The kernel is a homogeneous ideal, and the smallest of the degree bounds among all generating sets, called the relation type of I, is independent of the representation. We derive formulae connecting the relation type of I with the reduction number of I when the analytic spread of I exceeds height(I) by one. In the process we define complete d-sequences with respect to I and use them to help achieve our results. In addition some results on the behaviour of the relation type modulo an element are proved, and examples where the relation type is explicitly computed are presented.

If D is an acyclic digraph, define the height h = h(D) to be the length of the longest directed path in D. We prove that the values of h(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Cn and variance C′n, where C ≈ 0·764334 and C′ ≈ 0·145210. Furthermore, define V0(D) to be the set of sinks (vertices of out-degree 0) and, for r ≥ 1, define Vr(D) to be the set of vertices ν such that the longest directed path from ν to V0(D) has length r. For each k ≥ 1, let nk(D) be the number of sets Vt(D) which have size k. We prove that, for fixed k, the values of nk(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Ckn and variance C′kn, for positive constants Ck and C′k. Results of Bender and Robinson imply that our claim holds also for unlabelled acyclic digraphs.

In recent years, much work in algebraic topology has been devoted to stable splitting phenomena. Often the existence of these splittings has first been detected at the cohomological level in terms of modules over the Steenrod algebra.

For example, W. Richter has exhibited a decomposition of ΩSU(n) of the form

(see [7]). Not only were cohomology calculations the initial evidence for this situation, but they further suggested that each summand Gk might be the Thom complex of a suitable k-plane complex vector bundle. This possibility was also verified by Mitchell.

Let K be a knotted theta-curve with exterior X, and let ∂_X be one of the two pieces into which ∂X is divided by the meridians of the edges of K. Let X be the universal abelian cover of X. Then is a module over the group ring of H1(X); i.e. over . We call this the Alexander module of K, and denote it by A(K). This, rather than H1(X), seems to be the analogue of the Alexander module of a classical knot; it is a torsion module of deficiency 0. Moreover, it is not an invariant of X alone.

In [7], the authors proposed the problem of giving a geometric characterization of those Banach spaces which admit an algebraic structure. Motivated by the geometry imposed by measuring processes on the set of observables of a quantum mechanical system, they introduced the category of facially symmetric spaces. A discrete spectral theorem for an arbitrary element in the dual of a reflexive facially symmetric space was obtained by using the basic notions of orthogonality, protective unit, norm exposed face, symmetric face, generalized tripotent and generalized Peirce projection, which were introduced and developed in this purely geometric setting.

If PL(v,z) = Σbi(v)zi is the skein polynomial of a link L, and D = D1 * D2 is the diagram which is a planar star (Murasugi) product of D1 and D2 then bϕ(D)(v) = bϕ(D1)·bϕ(D2)(v) where ϕ(D) = n(D)– (s(D) – 1) and n(D) denotes the number of crossings of D, and s(D) is the number of Seifert circles of D.

In [2], B. Schweizer and A. Sklar proposed the problem of whether every triangle function is causal. In [1], T. B. M. McMaster solved the problem negatively by giving an example of a non-causal triangle function. In this note we provide a class of simpler non-causal triangle functions which are continuous on a dense subset of Δ+ but in general not continuous on Δ+. Since McMaster's triangle function is also not continuous on Δ+, the problem of determining whether or not every continuous triangle function is causal remains open.

Let X be an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Let denote the ring of regular functions on X ([6], chapter 3). (If X is an algebraic subset of ℝn then is comprised of all functions of the form f/g, where g, f: X → ℝ are polynomial functions with g−1(O) = Ø.) In this paper, assuming that X is compact, non-singular, and that dim X ≤ 3, we compute the Grothendieck group of projective modules over (cf. Section 1), and the Grothendieck group and the Witt group of symplectic spaces over (cf. Section 2), in terms of the algebraic cohomology groups and generated by the cohomology classes associated with the algebraic subvarieties of X. We also relate the group to the Grothendieck group KO(X) of continuous real vector bundles over X, and the groups and to the Grothendieck group K(X) of continuous complex vector bundles over X.

When considering *-algebras of unbounded operators, the usual approach is perhaps the one to be expected. One starts with a Hilbert space , and then defines a common dense domain X associated with some *-algebra of unbounded operators on . The algebra is then used to define a locally convex topology (the graph topology) on X, with respect to which the inner product on X is continuous, and this in turn defines a variety of topologies on , which are then studied.

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

In 1953, Calabi proved a rigidity theorem for Kählerian submanifolds in complex space forms [3]. The Calabi rigidity theorem, since then, has been successfully applied to various areas in geometry. Among them is the study of minimal surfaces in real space forms; see [4] for example.

A tempered distribution on ℝ whose Fourier transform is supported in an interval [−Ω,Ω], where Ω>0, can be characterized by the behaviour of its successive derivatives. On the other hand, a tempered distribution on ℝ whose Fourier transform vanishes in an interval (−Ω,Ω), where Ω>0, can be characterized by the behaviour of a particular sequence of successive antiderivatives. Similar considerations apply to general convolution operators acting on J′(ℝn) and yield characterizations for tempered distributions having their Fourier transforms supported in sets of the form or , where and Ω>0.

A n.c. JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky for C*-algebras. Different characterizations of associativity are given.

Let (rj) be a Rademacher sequence of random variables – that is, a sequence of independent random variables, with , for each j. A biorthogonal system in a Banach space X is called an RUC-system[l] if for every x in [ej] (the closed linear span of the vectors ej), the series

converges for almost every ω. A basis which, together with its coefficient functionals, forms an RUC-system is called an RUC-basis. A biorthogonal system is an RLTC-svstem if and only if there exists 1 ≤ K < ∞ such that

for each x in [ej]: the RUC-constant of the system is the smallest constant K satisfying (1) (see [1], proposition 1.1).

Singularity theory is concerned with the study of smooth mappings between smooth manifolds. Given two such manifolds X and Y and a pair of smooth mappings f1,f2: X→Y we say that f1 and f2 are -equivalent if there are diffeomorphisms α: X→X and β: Y→Y with βof1oα = f2. Clearly -equivalence is an equivalence relation, and one aims to classify smooth mappings f: X→Y up to this equivalence.

Group algebras and crossed products have always played an important role in the theory of C*-algebras, and there has also been considerable interest in various twisted analogues, where the multiplication is twisted by a two-cocycle. Here we shall discuss a very general family of twisted actions of locally compact groups on C*-algebras, and the corresponding twisted crossed product C*-algebras. We shall then establish some of the basic properties of these algebras, motivated by the requirements of some applications we have in mind [2, 9, 10]. Some of our results will be known to others, at least in principle, but we feel that a coherent account might be useful.

Let BP denote the localization at υn, of the Brown-Peterson spectrum (associated to the prime p). There is a related ring spectrum E(n) with homotopy ring

(as a quotient ring of in fact the cohomology theory E(n)*( ) is determined via a Conner-Floyd type isomorphism from on finite complexes, and moreover E(n) and BP are in the same Bousfield class (see [2, 14]). Although it is known (essentially from [17]) that BP cannot be a product of suspensions of E(n) in a multiplicative sense, D. Ravenel conjectured that such a splitting might occur after suitable completion of these spectra (see the introduction to [14]). This question was the original motivation of the present paper; however in proving Ravenel's conjecture we were naturally led to the consideration of some fundamental results in the theory of liftings of formal group laws and ‘change of ring’ results for Ext groups occurring in connection with the work of [10, 11, 12].