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

Sur Les Densités de Certaines Suites D'Entiers

Abstract: Soit n un nombre entier, w(n) le nombre de ses facteurs premiers distincts et r(n) le nombre de ses diviseurs . Nous désignons par { pj(n) : 1 ,j , (u (n) } la suite croissante des facteurs premiers et par {dj(n) : 1 --j < 7r(n)} celle des diviseurs . Le comportement normal de ces deux suites a été abondamment étudié dans la littérature-avec bien entendu des résultats plus précis pour la première . Les fonctions de répartition globales sont notamment assez bien connues : Erdős P montré en 1946 que l'on a, pour toute fonction ~(n) et presque tout n,

Classification of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries

Abstract: We use Elie Cartan's method of equivalence to give a complete classification, in terms of differential invariants, of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries. We then apply our results to the determination of all second-order equations which are equivalent, under fibre-preserving transformations, to the free particle equation. In addition we present those equations of Painleve' type which admit a transitive symmetry group. Finally we determine the symmetry group of some equations of physical interest, such as the Duffing and Holmes-Rand equations, which arise as models of non-linear oscillators.

Primitive Subgroups of Wreath Products in Product Action

Abstract: This paper is concerned with finite primitive permutation groups G which are subgroups of wreath products W in product action and are such that the socles of G and W are the same. The aim is to explore how the study of such groups may be reduced to the study of smaller groups. The O'Nan-Scott Theorem (see Liebeck, Praeger, Saxl [12] for the most recent and detailed treatment) sorts finite primitive permutation groups into several types, the groups in any one type admitting a common discussion. One of the types (III(b) in [12]) consists of groups G which are contained in, and contain the socle of, a suitable wreath product W in product action. It is easy to see that in this case the socles of G and W actually coincide. Thus the aim here amounts to pursuing the discussion of primitive groups of this type beyond the conclusions reached, say, in [12]. It has not proved possible to make direct use of those conclusions here. Instead, it seems necessary to repeat, elaborate, extend, and recombine arguments from various proofs of the O'Nan-Scott Theorem. No attempt will be made here to trace the origins of the ideas so used. For a sketch of the main results, some terminology is needed. By a wreath product A Wr Sn we mean the usual semidirect product W of the symmetric group Sn and the n-fold direct power A\" of the (abstract) group A. The projection of W onto Sn corresponding to this semidirect decomposition will be denoted by n. Consider n a permutation representation of W and take a point stabilizer Wo: this has an obvious direct factorization

ℵ0-Categorical Structures Smoothly Approximated by Finite Substructures

Abstract: A classification is given of primitive K0-categorical structures which are smoothly approximated by a chain of finite homogeneous substructures. The proof uses the classification of finite simple groups and some representation theory. The main theorems give information about a class of structures more general than the X0-categorical, co-stable structures examined by Cherlin, Harrington, and Lachlan.

Embedding Topological Median Algebras in Products of Dendrons

Abstract: Dendrons and their products admit a natural, continuous median operator We prove that there exists a two-dimensional metric continuum with a continuous median operator, for which there is no median-preserving embedding in a product of finitely many dendrons Our method involves ideas and results concerning graph colouring and abstract convexity The main result answers a question in [16] negatively, and is sharply contrasting with a result of Stralka [15] on embeddings of compact lattices

Finiteness Conditions on Subgroups and Formal Language Theory

Abstract: We show in this article that the most usual finiteness conditions on a subgroup of a finitely generated group all have equivalent formulations in terms of formal language theory. This correspondence gives simple proofs of various theorems concerning intersections of subgroups and the preservation of finiteness conditions in a uniform manner. We then establish easily the theorems of Greibach and of Griffiths by considering free reductions of languages that describe the computations of pushdown automata in one case and of Turing machines in the other, thus making clear that they are essentially the same.