Volume 10, Issue 2

We give a computer-free proof for Weiss' theorem of the uniqueness of the Rudvalis simple group acting flag transitively on an extended generalized octagon with empty triangles.

This paper presents an intrinsic differential algebraic framework for considering feedback in nonlinear control systems. In particular, filtrations of differential field extensions are shown to be useful for the definition of state feedback and the interpretation of two algorithms, namely the Structure Algorithm and the Dynamic Extension Algorithm, wellknown in the context of control theory. This difierential algebraic approach allows defining quasi-static state feedback.

Let be σ Galois automorphism of a number field E . We determine the cuspidal representations π of GL 2 () such that σ(π) ⋍ π ⊗ ω for some Hecke character ω. Our result is applied to extend the descent criterion of the base change theorem for GL(2) from cyclic extensions of prime degree to arbitrary cyclic extensions.

Let G : ℝ → ℝ be a continuous function. Denote by T G the corresponding composition operator which sends ƒ to G ( ƒ ). Then we investigate necessary and sufficient conditions on the parameters s , p , q , r and on the function G such that an inclusion like is true. Here F s p, q denotes a space of Triebel-Lizorkin type and W m p denotes a Sobolev space, respectively. Necessary and sufficient conditions for such an inclusion to hold will be given in cases G ( t ) = t k , k ∈ ℕ, G ( t ) = | t | μ , G ( t ) = t | t | μ - 1 , μ > 1, G ∈ C 0 ∞ , and G a periodic C ∞ -function.

If is the union of a smooth strictly ascending chain of B 2 -subgroups G α , then, in the case when κ < ℵ ω , a criterion is established under which G becomes a B 2 -group. This criterion is dependent on a new class of torsion-free groups and generalizes earlier criteria for freeness established by Paul Hill. The result of S. Shelah and others establishing that, for a weakly compact cardinal κ “ κ -free” implies “free”, is extended to the case of B 2 -groups.

Let R be an henselian valuation domain; then the Krull-Schmidt property holds for finite rank torsion-free R -modules, namely direct decompositions of such modules into indecomposable summands are unique up to isomorphism. Vámos gave examples of nonhenselian valution domains such that the Krull-Schmidt property holds. We show that the Krull-Schmidt property fails for finite rank torsion-free R -modules, whenever the valuation domain R contains a prime ideal P such that R/P is nonhenselian and [ K ( R/P )ˆ: K ( R/P )] ≥ 6; K ( R/P ) is the field of fractions of R/P , and K ( R/P )ˆ denotes its completion in the topology of the valuation. We examine nonhenselian valuation domains to which the preceding result is not applicable. If [ K ( R/P )ˆ: K ( R/P )] < ∞ for every prime ideal P of R , and Spec ( R ) is well-ordered by the reverse inclusion, we prove that R has characteristic zero and contains a prime ideal J such that [ K ( R/P )ˆ : K ( R/P )] is 2 if P = J , and 1 otherwise. Moreover, K ( R/J )ˆ is algebraically closed, and K ( R/P ) is algebraically closed if and only if P ⊃ J .