Volume 10, Issue 1

This article is motivated by the study begun in [8] of modular invariants of finite groups using as tools, the Steenrod algebra and the Dickson algebra. The ring of invariants of a representation q : G GL(h, F) of a finite group G over a Galois field F of characteristic p is an unstable graded connected commutative Noetherian algebra over the Steenrod algebra P*. We adopt this more general point of view and study P*-invariant ideals in unstable graded connected commutative Noetherian algebras H* over a Galois field F. (An ideal I= H* is called ,P*-invariant if it is closed under the action of the Steenrod algebra.) Our goal is to show that .P*-invarlant ideals have a .P*-invariant primary decomposition.

We give the atomic decomposition of Triebel-Lizorkin and Besov scales on Riemannian manifolds of bounded geometry. In order to do it we modify the moment condition for atoms.

Interpretations of several existing results on spectral controllability of a linear delay system are given. A general characterization in a recently proposed module theoretic framework is presented. This setting enables a clear and precise comparison of the various examined notions.

Given a regular symmetric and isotropic bilinear form on a finite-dimensional vector space V (dim(C) > 3) over a commutative field K of characteristic +- 2. The commutator- group Ω(V) of the orthogonal group O(V) is generated by the set of Eichler-transformations. We will compute for each Ω(V) the length I(n), i.e. the minimal number of Eichler- transformations in a product that equals n.

For arbitrary N and for all even Dirichlet characters z (mod N) we calculate in an elegant closed form the entry (φ)ai,(s) in φ(S) for all singular cusps a, b that have the property that [y,(N/y)], = 1 When z(mod N) is primitive or N is squarefree, all singular cusps have this property. We then present an argument to support our contention that it is unlikely that one would be able to obtain an elegant, closed form evaluation for every term in the scattering matrix when z(mod N) is an arbitrary even character and N is an arbitrary modulus.

The volume contraction in dissipative reversible transitive Anosov flows obeys a large deviation rule (fluctuation theorem).

We introduce the notions of weak and strong sense recurrence of time inhomogeneous Markov processes and give general criteria for them. These notions of recurrence are invariant under certain time changes but the Dirichlet forms are not stable under time changes m general. We shall apply the general criterion for the strong recurrence to time inhomogeneous diffusion processes obtained by certain drift transformations of recurrent diffusion processes as well as the processes obtained by time change of Brownian motions.