Labib Haddad

On uniqueness of p-adic entire functions

Abstract Let K be a complete ultrametric algebraically closed field, let A (K) be the ring of entire functions in K. Unique Range Sets ( urss ) were defined in [5], and studied in [6] for complex entire or meromorphic functions. Here, we characterize the urss for polynomials, in any algebraically closed field, and we prove that in non archimedean analysis, there exist urss of n elements, for entire functions, for any n ≥ 3. When n = 3, we can characterize the sets of three elements that are urss for entire functions.

Les cogroupes et les D-hypergroupes

Resume Parmi dautres resultats, on montre en particulier que les D-hypergroupes ne forment pas une classe elementaire, autrement dit, que la structure de D-hypergroupe ne peut pas etre caracterisee par un systeme daxiomes du premier ordre, dans le langage des hypergroupes.

ANALYTIC ERDÖS-TURÁN CONJECTURES AND ERDÖS-FUCHS THEOREM

We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdos-Turan conjectures in terms of supported series and extending to them a version of Erdos-Fuchs theorem.

On the Additive Group Structure of the Nonstandard Models of the Theory of Integers

Let denote the inverse limit of all finite cyclic groups. Let F, G and H be abelian groups with H ≤ G. Let FβH denote the abelian group (F × H, +β), where +βis defined by (a, x) +β (b, y) = (a + b, x + y + β(a) + β(b) — β(a + b)) for a certain β : F G linear mod H meaning that β(0) = 0 and β(a) + β(b) — β(a + b) ∈ H for all a, b in F. In this paper we show that the following hold: (1) The additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to (ℤ*+/H)βH for a certain β : ℤ*+/H linear mod H. (2) is isomorphic to (ℤ+/H )βH for some β : /H ℚ linear mod H, though is not the additive group of any model of Th(ℤ, +, ×) and the exact sequence H /H is not splitting.

Les groupes, les hypergroupes et l'énigme des Murngin. The Murngin case

Abstract This is a contribution to the algebraic study of an Australian kinship system known as the ‘Murngin system’. It is shown how the introduction of hypergroups and semi-direct products of groups can shed new light on some aspects of this particular system and further enhance the abstract theory of kinship.