Characterization of classical groups by orbit sizes on the natural module

We show that if is a finite vector space, and is a subgroup of having the same orbit sizes on 1-spaces as an orthogonal or unitary group on , then, with a few exceptions, is itself an orthogonal or unitary group on .

     

Weak separation properties for self-similar sets

We develop a theory for self-similar sets in that fulfil the weak separation property of Lau and Ngai, which is weaker than the open set condition of Hutchinson.

     

Multiplicative -quotients

Let be the Dedekind -function. In this work we exhibit all modular forms of integral weight , for positive integers and and arbitrary integers , such that both and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group via a generalized McKay-Thompson series.

     

A Large Deviation Principle for Polynomial Hypergroups

Let S_n, n 1, be a random walk on a polynomial hypergroup (N₀,*), that is, a Markov chain on the nonnegative integers withstationary transition probabilities P_ij = _i* µ({j}), whereµ is a fixed probability measureon N₀. Under certain conditionson this measure, the principle of large deviations is shownfor the distributions of S_n/n. This result comprises the largedeviation principle for birth and death random walks associatedwith the polynomials generating the polynomial hypergroup.     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.