Wikipedia:Reference desk/Archives/Mathematics/2010 June 20

= June 20 =

Fields medal
How come no women has ever got the fields medal? —Preceding unsigned comment added by 58.109.119.6 (talk) 08:16, 20 June 2010 (UTC)
 * Many people haven't got the Fields medal. Some of them were not qualified. The rules discriminate against age, but not against gender. Bo Jacoby (talk) 11:09, 20 June 2010 (UTC).
 * Also, mathematics is very much a male dominated subject. In my faculty there was a single female member of the academic staff amongst 50 or so male members. Why haven't any women won Fields Medals? Well, it's just a statistical thing more than anything else. •• Fly by Night (talk) 20:48, 22 June 2010 (UTC)

A-invariant subspaces
Let V a finite-dimensional vector space over a field K, A ∈ HomK(V,V), V = ⊕undefinedk Wi with A-cyclic subspaces Wi ≠ {0}, and Ai := Aundefined for 1 ≤ i ≤ k. Furthermore, the characteristic polynomials fA i are pairwise coprime. My question: Is v = ∑undefinedk vi with vi ∈ Wi for all 1 ≤ i ≤ k an A-generator of V iff vi is an A-generator of Wi for all 1 ≤ i ≤ k? --84.62.192.52 (talk) 13:26, 20 June 2010 (UTC)
 * If you don't get an answer here, you might try taking this question to www.mathoverflow.com . However if you post it there, you're going to need a little more motivation for the question.  Read the FAQ - questions that aren't presented so as to interest mathematicians will be closed very quickly. 24.6.2.115 (talk) 22:37, 23 June 2010 (UTC)

Right-angled triangles in a quadrant
The "number of right triangles with nonnegative integer coordinates less than or equal to N and one corner at the origin." (A155154) seems to have a very regular graph- is there an explicit formula for this sequence? 70.162.12.102 (talk) 19:40, 20 June 2010 (UTC)
 * It seems very unlikely to me. There are n2 triangles with the right angle at the origin, but then for the ones with the right angle elsewhere it gets pretty messy pretty fast.  Any right triangle on the lattice is going to have its legs in a rational ratio so we can think of all the triangles as scaled up versions of the triangles with leg lengths a, b, with a, b positive and relatively prime.  There are two possible orientations, so just consider the one where the right angle is closer to the x-axis than the other vertex.  For given a, b there is the triangle with right angle at (a, 0) and the other vertex at (a, b) but we can also replace our "x unit" (0, 1) with any (c, d) where c, d are non-negative and not both zero.  Then the "y unit" becomes (-d, c), so the vertices of the triangle become (ac, ad) and (ac - bd, ad + bc).  For these points to be inside the region we want they have to satisfy bd ≤ ac ≤ n and ad + bc ≤ n.  So the problem is now counting all the values of a, b, c, d that satisfy these conditions for a given n, which is not straight forward. Rckrone (talk) 21:31, 20 June 2010 (UTC)
 * I agree, there probably isn't a nice formula (it is, of course, almost impossible to prove that). There may, however, be an asymptotic formula that is very accurate even for small values of n. There must be some explanation for the graph looking so regular. --Tango (talk) 00:27, 21 June 2010 (UTC)
 * an ≈ 3.1735n2.14937±7.65 for 0≤n≤44. Bo Jacoby (talk) 12:44, 23 June 2010 (UTC).
 * Or perhaps an ≈ 3.1356n2.152891.00303±1. Bo Jacoby (talk) 08:33, 24 June 2010 (UTC).