en>LA2: link journal title

2010-08-17T10:46:00Z

link journal title

New page

'''Abouabdillah's theorem''' refers to two distinct [[theorem]]s in [[mathematics]], proven by Moroccan mathematician Driss Abouabdillah: one in [[geometry]] and one in [[number theory]].

==Geometry==
In geometry, similarities of a [[Euclidean space]] preserve circles and spheres. Conversely, Abouabdillah's theorem states that every injective or surjective transformation of a Euclidean space that preserves circles or spheres is a [[Similarity (geometry)|similarity]].

More precisely:

'''Theorem.''' Let <math>E</math> be a Euclidean [[affine space]] of dimension at least 2. Then:

1. Every surjective mapping <math>f: E \rightarrow E</math> that transforms any four [[concyclic points]] into four concyclic points is a similarity.

2. Every injective mapping <math>f: E \rightarrow E</math> that transforms any circle into a circle is a similarity.

==Number theory==
The number-theoretic theorem of Abouabdillah is about [[antichain]]s in the [[partially ordered set]] ''EN'' consisting of the [[positive integer]]s in the interval [1,''N''], partially ordered by [[divisibility]]. With this partial order, an antichain is a set of integers within this interval, such that no member of this set is a divisor of any other member. It possible to prove using ideas related to [[Dilworth's theorem]] that the maximum number of elements in an antichain of ''E''2''n'' is exactly ''n'': there exists an antichain of this size consisting of all the numbers in the subinterval [''n'' + 1,2''n''], so the maximum size of an antichain is at least ''n''. However, there are only ''n'' odd numbers within the interval [1,2''n''], for each odd number ''c'' in this interval at most one number of the form 2''k''''c'' may belong to any antichain, and every number in the interval has this form for some ''c'', so the maximum size of an antichain is also at most ''n''.

Abouabdillah's theorem characterizes more precisely the numbers that may belong to an antichain of maximum size in ''E''2''n''. Specifically, if ''x'' is any integer in the interval [1,2''n''], decompose ''x'' as the product of a [[power of two]] and an odd number: ''x'' = 2''k''''c'', where ''c'' is odd. Then, according to Abouabdillah's theorem, there exists an antichain of cardinality ''n'' in ''E''2''n'' that contains ''x'' if and only if 2''n''  < 3''k'' + 1''c''.

The smallest value in any maximum antichain of ''E''2''n'' is at least 2''k'', where 3''k'' + 1 is the first power of three that is greater than 2''n'', as had been posed as a problem by {{harvs|authorlink=Paul Erdős|first=Paul|last=Erdős|year=1937|txt}} and solved by {{harvs|authorlink=Emma Lehmer|first=Emma|last=Lehmer|year=1939|txt}}. Lehmer's solution immediately implies the special case of Abouabdillah's theorem for ''c'' = 1. Abouabdillah's theorem generalizes this solution to all values within the given interval.

==References==
*{{citation|first=D.|last=Abouabdillah|title=Sur les similitudes d'un espace euclidien|journal=Revue de Mathématiques Spéciales|volume=7|year=1991}}.
*{{citation|first1=D.|last1=Abouabdillah|first2=J.|last2=Turgeon|contribution=On a 1937 problem of Paul Erdős concerning certain finite sequences of integers none divisible by another|title=Proc. 15th Southeastern Conf. Combinatorics, Graph Theory and Computing (Baton Rouge, La., 1984)|series=Congressus Numerantium|volume=43|year=1984|location=Winnipeg, Canada|publisher=Util. Math.|pages=19–22|mr=0777348}}.
*{{citation|authorlink=Paul Erdős|first=Paul|last=Erdős|title=Advanced Problem 3820|journal=[[American Mathematical Monthly]]|volume=44|issue=3|year=1937|jstor=2301675|author2=Thébault, V|author3=Goormaghtigh, R|author4=Musselman, J. R|publisher=Mathematical Association of America|pages=179–180}}.
*{{citation|authorlink=Emma Lehmer|first=Emma|last=Lehmer|title=Solution to Advanced Problem 3820|journal=[[American Mathematical Monthly]]|volume=46|issue=4|year=1939|pages=240–241|jstor=2303086|author2=Lehmer, Emma|publisher=Mathematical Association of America}}.

== External links ==
* [http://sites.google.com/site/contgeom/ Contributions in Geometry]
* [http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1017.51003&format=complete Contributions en géométrie]

{{DEFAULTSORT:Abouabdillah's Theorem}}
[[Category:Theorems in geometry]]
[[Category:Euclidean geometry]]
[[Category:Theorems in number theory]]

Fluent calculus - Revision history

en>LA2: link journal title