Tuesday, October 20, 2009

The Axiom of Choice

Three useful links dealing with the Axiom of Choice: The Axiom of Choice - Stanford Encyclopedia of Philosophy entry by John L. Bell, the University of Western Ontario; The Axiom of Choice (a short paper by Prof. John L. Bell) ; The Relative Consistency of the Axiom of Choice - Mechanized Using Isabelle/ZF by Lawrence C. Paulson, Computer Laboratory Univ. of Cambridge. An important result, in a categorical setting linked to intutionistic set theory, was obtained by Radu Diaconescu (Axiom of choice and complementation, Proc. Amer. Math. Soc. 51, 1975, 176-178): a topos satisfying the axiom of choice must be boolean (in short... the axiom of choice implies the law of the excluded middle).

Tuesday, October 13, 2009

A conjecture on primes

The Feit-Thompson conjecture: there are no distinct prime numbers p and q for which (p^q-1)/(p-1) divides (q^p-1)/(q-1). Note that a stronger statement stating that (p^q-1)/(p-1) and (q^p-1)/(q-1) are relatively prime whenever p and q are distinct primes does not hold.
Indeed, 112643 = GCD((3313^17 -1)/3312, (17^3313-1)/16) - see Stevens (1971).
Note that (3313^17 - 1)/3312 factors as
78115430278873040084455537747447422887 * 23946003637421 * 112643,
while (17^3313-1)/16 modulo (3313^17 -1)/3312 equals...
149073454345008273252753518779212742886488244343395482423
The 1970 Fields medalist and 2008 Abel prize winner John Griggs Thompson was born on Ottawa, KS on October 13.

Sunday, October 4, 2009

Dimitrie Pompeiu (1873–1954)

Dimitrie Pompeiu (October 4, 1873, Broscǎuţi, Botoşani – October 8, 1954, Bucharest) - orthodox christian, and one of the greatest Romanian mathematicians. He is remembered in the mathematical world for numerous contributions such as: the set distance [1] that he introduced in his 1905 Université de Paris dissertation published in the same year in the Annales de la faculté des sciences de Toulouse (a set distance was introduced in a slightly different form in 1914 by Hausdorff, who credited Pompeiu's definition though), his contributions in complex analysis, including the areolar derivative [2] and the seminal Cauchy-Pompeiu's formula (higher dimensional analogues of the Cauchy-Pompeiu formula are topics of current research, while the formula was used in the theory of functions of several complex variables by Dolbeault and Grothendieck [5]), and for the celebrated Pompeiu's Conjecture that he formulated in his 1929 C. R. Acad. Sci. Paris article [4], a conjecture not fuly proved yet. Still, elegant analogues of Pompeiu's Conjecture continue to be proved in other areas [6] - this is an indicator of the fertility of the idea.

REFERENCES

[1] T. Bârsan and D. Tiba. One hundred years since the introduction of the set distance by Dimitrie Pompeiu. Institute of Mathematics of the Romanian Academy.
[2] D. Pompeiu. Sur une classe de fonctions d'une variable complexe. Rendiconti del Circolo Matematico di Palermo, t. XXXIII, Ist sem. 1912, pp. 108-113.
[3] Pompeiu's biography from the The MacTutor History of Mathematics archive.
[4] D. Pompeiu. Sur certains systèmes d'équations linéaires et sur une propriété intégrale des fonctions de plusieurs variables, Comptes Rendus de l'Académie des Sciences Paris Série I. Mathématique, 188, 1138 –1139 (1929).
[5] R. Remmert. Theory of Complex Functions. Graduate Texts in Mathematics, Springer Verlag, 2nd Edition (1989).
[6] D. Zeilberger. Pompeiu's problem on Discrete Space. Proc. Natl. Acad. Sci. USA, Vol. 75 (8), 3555-3556 (1978).