Driss Abouabdillah

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Driss Abouabdillah (Arabic: 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.

[edit] Contributions

In geometry, he gave several characterizations of similarities as mappings that preserve circles or spheres [1].
He proposed an alternative axiomatic approach to Euclidean geometry by giving an axiomatic definition of equipollence. [2]
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 $E 2 n$ = {1,2,...,2n} is n).

[edit] Abouabdillah's Theorem

Let c∈ $E 2 n$ , c = 2^kc', c' odd. Then $E 2 n$ 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.

[edit] 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.