Let µ be Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is the size condition µ(B(x, r)) ≤ Crn for some fixed n є (0, d). Recently, Tolsa introduced the spaces RBMO(µ) and Hatb1.∞ (µ), which, in some ways, play the role of the classical spaces BMO and H1 in case u is a doubling measure. In this paper, the author considers the local versions of the spaces RBMO(µ) and Hatb1.∞ (µ) in the sense of Goldberg and establishes the connections between the spaces RBMO(µ) and Hatb1.∞ (µ) with their local versions. An interpolation result of linear operators is also given.

We define a notion of conjugacy in singular Artin moniods, and solve the corresponding conjugacy problem for finite types. We sgiw that this definition is appropriate to describe type (1) singular Markov moves on singular braids. Parabolic submonoids of singular Artin monoids are defined and, in finite type, are shown to be singular Artin monoids. Solutions to conjugacy-type problems of parabolic submonoids are described. Geometric objects defined by Fenn, Rolfsen and Zhu, called (j, k)-bands, are algebraically characterised, and a procedure is given which determines when a word represents a (j, k)-band.

For a permutation group H on an infinite set X and a transformation f of X, let 〈f: H〉 = 〈{hfh-1:h є; H}〉 be a group closure of f. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation f of X to generate distinct group closures of f. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

Let F′ be the commutator subgroup of F and let Γ0 be the cyclic group generated by the first generator of F. We continue the study of the central sequences of the factor L(F′), and we prove that the abelian von Neumann algebra L(Γ0) is a strongly singular MASA in L(F). We also prove that the natural action of F on [0, 1] is ergodic and that its ratio set is {0} ∪ {2k; k ∞ Z}.

Hypercentrally embedded subgroups of finite groups can be characterized in terms of permutability as those subgroups which permute with all pronormal subgroups of the group. Despite that, in general, hypercentrally embedded subgroups do not permute with the intersection of pronormal subgroups, in this paper we prove that they permute with certain relevant types of subgroups which can be described as intersections of pronormal subgroups. We prove that hypercentrally embedded subgroups permute with subgroups of prefrattini type, which are intersections of maximal subgroups, and with F-normalizers, for a saturated formation F. In the soluble universe, F-normalizers can be described as intersection of some pronormal subgroups of the group.

Suppose that the finite group G acts faithfully and irreducibly on the finite G-module V of characteristic p not dividing |G|. The well-known k(GV)-problem states that in this situation, if k(G V) is the number of conjugacy classes of the semidirect product GV, then k(GV) ≤ |V|. For p—solvable groups, this is equivalent to Brauer's famous k(B)-problem. In 1996, Robinson and Thompson proved the k(GV) problem for large p. This ultimately led to a complete proof of the k(GV)-problem. In this paper, we present a new proof of the k(G V)-problem for large p.

Kallenberg has introduced the concept of conditional spreadability for random sequences and has developed characterizations of this property in terms of one dimensional martingales and optional times, and as well has proven a predictable sampling theorem. This paper investigates the relationship between planar martingale structures and the natural analogues of conditional spreadability when extended to arrays of random elements. Analogues of the predictable sampling theorem are also established for spreadable arrays.