Minutes for Reach 13 February 2003 Members Present (I can't remember if we usually include this but from force of habit from other clubs): Gabe, Jen, Reid, Ilya, Ian, Ananda, Dave, Inna, Nick, Cecilia, Trevor, Andy, Jim, Neil Notetaker for next time: Ian Cecilia, Reid, and Neil need math 192 dvds. Nick will make them. Next *THURSDAY* is demo day in the computer room. We went around w/names again, also naming thms/branches of math that sounded interesting (name-wise). -- We talked about what demos/computer things should be high-priority for Thursday's demos and in general. Here's what we came up with: - small-group sessions on maple/mathematica - Jim doesn't want to go thru the same web tour he did last term, since it'll be old news for most people - we should have a list of links (and short descriptions of what they point to) somewhere: CECILIA WILL COMPILE THIS, everyone should send ideas for useful online references to her. - important web sites: the online encyclopedia of integer sequences, the los alamos archive (front.math.ucdavis.edu) - we should compile a list of papers we've looked at ("reading list") - Superseeker: send an email w/a seq, sends you back generating function, etc. different from the encyclopedia of integer sequences because it does more than just prove the database. little-known fact: while there's a limit of one query per hour per user, if you ask for unlimited use, you'll probably get it. Other tangential matters: - We have a REACH library, which currently has 0 books. If people buy things with REACH money for the REACH library, then REACH owns it for as long as REACH exists (i.e. until the end of this term). We don't know what happens after that. Anyway, Ilya might buy Stanley's Enumerative Combinatorics for the REACH library. - TREVOR will be compiling a list/document of resources NOT available online (and how to get at them). -- We talked about the (1,4) recurrence. (Well, mostly Gabe talked about it.) (NOTE: TT is a capital pi. :-) d|a,b s(n) = (s(n)^a + 1)/s(n-1), n odd (s(n)^b + 1)/s(n-1), n even define d ( TT (f(i_1,..,i_j-1,..,i_d,n))^(a/d) ) + 1 j=1 f(i1,i2,..,id,n+1) = ____________________________________________, n odd f(i_1,i_2,..,i_d,n-1) replace i_j-1 with i_j+1 and a/d with b/d, n even seems to give laurent polynomials. It appears to be d-dim'l but really it's (d-1)-dim'l, because, if you evaluate apply the formula recursively to compute f(j_1,j_2,...j_d,n+1) in terms of other values of f, you find that all the index-combinations i_1,i_2,...,i_d,n+1 that arise have the property that, for some fixed k, i_1 + ... + i_d = k (n odd) or k+1 (n even). Someone should try to check laurentness. Can all the coefficients be 1? Jim gave kind of a proof (that it couldn't?). I didn't write it down because Gabe said partway through that it would fail and Jim agreed. Then they changed their minds. :-[ [I think we agreed the proof would work: By inspecting the recurrence, we find that f_(i_1,...,i_d,n) only involves initial variables of the form x_(j_1,...,j_d,m) (where we assume, say, m <= 1 is initial) for which |i_1-j_1| + ... + |i_d-j_d| <= n. The number of possible such initial variables is bounded by a polynomial P(n). Since the recurrence also is of fixed degree, say a, each monomial in a polynomial at height n has degree roughly a^n, so the number of such possible monomials is bounded by, say, (a^n)^(P(n)) = a^(nP(n)). So if every monomial has coefficient 1, then s_n (the value of the Laurent polynomial when all initial conditions equal 1) has at most polynomial-exponential growth in n. However, we know that the sequence s_n should have double-exponential growth, a contradiction. --- Gabriel] - might look at orders of growth - this is like the octahedron recurrence, but only half a turn at a time maybe we can throw other recurrences "off-balance"? Ananda will look at using cluster algebra methods. -- Let's do a session on trying to prove Laurentness: Jim on TEAM MAPLE *versus* Gabriel on TEAM MATHEMATICA Computer demos will be postposed til THURSDAY (since lots of people can't make it tuesday) -- Look at Jim's document about projects (on the private web site) so we won't have to spend meeting time talking about it. READ SECTION B BY TUESDAY jim talked about section B a little. i won't transcribe it, since we're all going to read the section :-) -- Dylan Thurston came to talk about cluster algebras and "exchange algebras" I'll transcribe it later, I promise... or more likely, I'll scan my notes and put them somewhere useful... the end. -------------------- Jennifer Krishnan 320 Memorial Dr. Cambridge, MA 02139