Let X be a linear space over the field of real or complex numbers. We characterize solutions f : X → K and M : K → K of the equation f(x + M(f(x))y) = f(x)f(y) under assumption that the set {x ∈ X : f(x) 6= 0} has an algebraically interior point. As a consequence we give solutions of the equation such that f is bounded on a set having an algebraically interior point.

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincaré-Hopf formula is then a consequence of the smooth Poincaré-Hopf theorem and of additivity of the Euler-Poincaré characteristic with compact supports, once we have a suitable definition of index. AMS classification: 55N33 57R25

In this paper, we investigate the concept of topological stationary for locally compact semigroups. In [4], T. Mitchell proved that a semigroup S is right stationary if and only if m(S) has a left Invariant mean. In this case, the set of values ?(f) where ? runs over all left invariant means on m(S) coincides with the set of constants in the weak* closed convex hull of right translates of f. Th...

in this paper, we investigate the concept of topological stationary for locally compact semigroups. in [4], t. mitchell proved that a semigroup s is right stationary if and only if m(s) has a left invariant mean. in this case, the set of values ?(f) where ? runs over all left invariant means on m(s) coincides with the set of constants in the weak* closed convex hull of right translates of f. th...

In this paper, we investigate some properties of inverse limits of linear algebraic groups. For example, we show that if G = lim ←−Gi, where (Gi, πji)i,j∈I is an inverse system of algebraic groups over an algebraically closed field. Then each canonical projection πi : G → Gi maps closed subgroups of G onto closed subgroups of Gi. Furthermore, we prove directly that the inverse limit of an inver...

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Diffalg(X) the group of algebraic diffeomorphisms of X into itself. The group Diffalg(X) acts diagonally on X n, for any natural integer n. We show that this action is transitive, for all n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebra...

Let T * C* x C* act meromorphically on a compact Kahler manifold X, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if .Vis projective. PROBLEM. Classify all T-invariant open subsets U of X such that the geometricquotient U —> U/Texists with U/Ta compact complex space (necessarily algebraic if Xis). In t...

We define a motivic analogue of the Haar measure for groups of the form G(k((t))), where k is an algebraically closed field of characteristic zero, and G is a reductive algebraic group defined over k. A classical Haar measure on such groups does not exist since they are not locally compact. We use the theory of motivic integration introduced by M. Kontsevich to define an additive function on a ...