Pseudodifferential Operators on Prehomogeneous Vector Spaces

ABSTRACT

Let G be a connected, linear algebraic group defined over ?, acting regularly on a finite dimensional vector space V over ? with ?-structure V _?. Assume that V possesses a Zariski-dense orbit, so that (G, ?, V) becomes a prehomogeneous vector space over ?. We consider the left regular representation π of the group of ?-rational points G _? on the Banach space C₀(V _?) of continuous functions on V _? vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G _?. Denote the complement of the dense orbit by S, and put S _? = S ∩ V _?. It turns out that, on V _? ? S _?, π(f) is a smooth operator. If S _? = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V _? ? {0}. Its nonunique extension to V _? can then be regarded as a trace of π(f). If G is reductive, and S and S _? are irreducible hypersurfaces, π(f) corresponds, on each connected component of V _? ? S _?, to a totally characteristic pseudodifferential operator. In this case, the restriction of the Schwartz kernel of π(f) to the diagonal defines a distribution on V _? ? S _? given by some power |p(m)| ^s of a relative invariant p(m) of (G, ?, V) and, as a consequence of the Fundamental Theorem of Prehomogeneous Vector Spaces, its extension to V _?, and the complex s-plane, satisfies functional equations similar to those for local zeta functions. A trace of π(f) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.     

Theories of rings with divisibility

We consider several different theories of commutative rings with a new predicate symbol | together with the additional axiom a|b ? ?c (a . c = b) and show that none of these has a model companion.     

Pure elements and intertwining classes of simple strata in local central simple algebras

Let S be a semigroup of words over an alphabet ∑ . Suppose tliar every two words u and e over ∑ are equal in S if (1) the sets of subwords of length k of the words a and b coincide and are non-empty. (2) the prefix (suffix) of u of length k1 is equal to the prefix (suffix) of e. Then S is called k-testable. A semigroup is locally testable if it is k-testable for some k > 0.

We present a finite basis of identities of the variety of A'-testable semigroups. The structure of k-testable semigroup is studied. Necessarv and sufficient conditions for local testability will be given. A solution to one problem from the survey of Shevrin and Sukhanov (1985) will be presented.     

On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.     

Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups

We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in with “degree” ∂(a)=m satisfies

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