Driss Abouabdillah

Driss Abouabdillah ( _ar. ادريس أبو عبدالله) (or Driss Bouabdillah) is a Moroccan mathematician born in Meknès on April 1, 1948. He is a former teacher of mathematics at the ENS (higher teacher training school) of Rabat.

Contributions

*In geometry, he gave several characterizations of similarities as mappings that preserve circles or spheres [http://www.geocities.com/bwakeel/] .
*He proposed an alternative axiomatic approach to Euclidean geometry by giving an axiomatic definition of equipollence. [http://www.geocities.com/bwakeel/]
*He studied maximal sub-groups of the group of positive isometries and gave several examples of such subgroups.
*He also contributes to abelian group theory and to Commutative Algebra where he gave the definition of topologically prüferian rings.
*In number theory, he discovered the following theorem on antichains of N. ( An antichain of N, for divisibility, is a set of nonzero integers such that no one is divisible by another. It is not difficult to prove that the maximal cardinality of an antichain of $E\_\{2n\}$ = {1,2,...,2n} is n).

Abouabdillah's Theorem

Let c∈$E\_\{2n\}$ , c = 2^kc', c' odd. Then $E\_\{2n\}$ contains an antichain of cardinality n containing c if and only if 2n < 3^k+1c'.

This result constitutes a complete solution of a problem the mathematician Paul Erdős posed in 1937 in "Amer. Math. Monthly" 44 (1937), p.179.

Publications

* Topologies Linéaires Minimales Sur Un Groupe Abélien. Lecture Notes in Mathematics .1006. (Springer-Verlag). Abelian group Theory. Proceedings Honolulu 1982-83.pp 582-588.

* Topologies de corps A-linéaires . Pacific journal of Mathématics. Vol.107,No.2,1983.

* Sur les similitudes d'un espace euclidien, Revue de Mathématiques Spéciales (RMS) No. 7 Mars 1991.(Vuibert).

* Sur les sous-groupes maximaux du groupe des isométries positives. Revue de Mathématiques de l'enseignement supérieur (RMS) Numéro 9-10. Mai-Juin 1999.(Vuibert).

* Contributions en Géométrie. Monographie de cinq articles de recherche en Géométrie. Avril 2001, ISBN 978-9954-8096-0-0, Avril 2001.

* Notions fondamentales de géométrie ; Ouvrage destiné à la préparation de l'Agrégation mathématique ; ISBN 978-9981-1697-2-2, Editions Dar Es-Salam ,1997.

* K. Benabdallah and D. Bouabdillah. Straight and Strongly Straight Primary Modules over Principal Ideal Domains. Abelien Groups and Modules. Proceeding of The Udine Conference . April 1984.CISM Courses And Lectures No.287 Springer Verlag pp.115-130.

* D. Abouabdillah & J.Turgeon, On a 1937 problem of Paul Erdős concerning certain finite sequences of integers none divisible by another. Congressus Numerantium. A conference journal on numerical themes. Vol.43, December,1984, (Winnipeg, Canada), pp.19-22.