Showing papers in "Bulletin of The London Mathematical Society in 1989"

A Survey on Spectra of infinite Graphs

Abstract: Introduction. Operateurs lineaires associes a un graphe. Resultats fondamentaux. Rayon spectral, fonctions generatrices de marche et mesures spectrales. Croissance et nombre isoperimetrique d'un graphe. Fonctions propres positives. Graphes de groupes, arbres et graphes reguliers en distance. Quelques remarques sur les applications en chimie et en physique

Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle

Abstract: Etude des relations entre la theorie des moments, les polynomes orthogonaux, la quadrature et les fractions continues associees au cercle-unite. Revue d'ensemble des resultats generaux

REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples

Abstract: Page 55, proof of Lemma 3.13. This proof is incorrect as it stands because it involves an interchange of limits that has not been justified. A naive attempt to fix the proof might involve assuming that the given representation is continuous into the uniform operator topology, but this assumption is not necessarily valid. Instead, a different approach is needed, and such an approach may be found in the paper by M. Welleda Baldoni, “General representation theory of real reductive groups,” Proc. Sumposia in Pure Math. 61 (1997), 61–72. The relevant proof is the proof of Proposition 1 on page 63. The argument is straightforward enough, but it makes use of the fact that g → π(g)v is of class C1 if and only if for each w in the Hilbert space, g → hπ(g)v, wi is of class C1. She attributes this result to Grothendieck and points to J. B. Neto, Trans. Amer. Math. Soc. 111 (1964), 381–391, for “discussion and results.” Reading Proposition 1 and its proof, one sees that by approaching the development in a slightly different order, one gets around the temptation to use the uniform operator topology and obtains a valid proof of Proposition 3.14.

A Guided Tour through Excursions

Abstract: (l.i) what happens after Tn is independent of what happened before; (l.ii) the evolution of the game from Tn on, {ST +k'.k^ 0}, is just like the evolution of the original game {Sk: k ^ 0}. The whole game starts afresh at time Tn! Thus if we define the nth excursion ^ " b y Zn = {Sk:Tn_x^k^Tn}, nsN, (1 .i) can be re-expressed as saying £15..., £n are independent of£n+1, £n+2,...; because of (l.ii), £n+1 has the same distribution as £v Hence

D.R.E. Degrees and the Nondiamond Theorem

Abstract: If we study the boolean algebra generated by the recursively enumerable sets we are naturally led to the difference hierarchy of Ershov [6,7,8]. If a set A can be represented as (A1 — A2) U ... U (An_x — An) for r.e. sets An c An_x £ ... £ Ax we say that A is «-r.e. In particular A is r.e. if A is 1-r.e. A set A is called d.r.e. if it is 2-r.e. (so that A = B— C with B and C r.e.). The reader should note that a set A is n-r.e. if and only if there is a recursive function / such that for all x, \\ims f[x,s) — A(x), J{x,0) = 0 and \\{s:J{x,s + 1) =Ax,s)}\\ ^ n. In the obvious way, we call a degree a A?-r.e. (d.r.e.) if it contains an n-r.e. (respectively, d.r.e.) set. It is natural to ask the extent to which the w-r.e.—and in particular the d.r.e.—degrees (Dn) resemble the r.e. degrees (R). Early results (such as the fact that the «-r.e. degrees have no minimal elements, and—as Jockusch observed—the d.r.e. degrees are not complemented) stressed the differences between the d.r.e. degrees and the A\" degrees and the similarities between the d.r.e. degrees and the r.e. degrees. In particular, it was open whether the d.r.e. degrees and the r.e. degrees were elementarily equivalent (see, for example, [5]). Answering a question of Sacks (see [13]), both Harrington (unpublished) and Lachlan and Soare (unpublished) show that D2 and R are not elementarily equivalent by showing that 0' is a minimal cover in D2. Cooper, Lempp and Watson [3] verified a conjecture of Arslanov [1] to produce another elementary difference at the two quantifier level (in the language L( U , n ,0,1)). Specifically, they showed that the following sentence holds in D2: