Notes: 4/29/04 We had picked a notetaker and snackbringer for tuesday. We picked it on Tuesday and I forgot to ask who it was. But we do have one! Jim asks whether or not we can call 2^n by N. This will not work because we call the Newton operator = N. Could we use script N for operator and regular N for 2^n? This may work. People writing the paper can fight this out amongst themselves. We wanted to see what the notation is in the interpolation section. Brendan has not updated his site since Tuesday, so that section is not there yet. We also need references for the noncommuting q case. The reference Sam found was by Gasper and Rahman called "Basic Hypergeometric Functions". Sam was going to go to the library and check out the book to see what they say about the noncommuting q case. Hal and Carl talk about how to modify the current Newton proof to prove the non-commuting q case. Carl thinks we can just modify the algebra in the current proof. Hal is skeptical (I think I heard that conversation right). Jim asks if anybody has read Bressoud's book. He realizes that Sam got his reference from that book, so Bressoud's book would probably not be a good reference. Sam got back from the library. He found that the above book was a bad reference. The good reference is Schutzenberger. The exact reference is: [12] M. P. Schutzenberger, Une interpretation de certaines solutions de l'equation fonctionnelle: F(x + y) = F(x)F(y), C. R. Acad. Sci. Paris 236 (1953), 352-353. Jim asks what restaurant we go to: Everybody seems to be fine either way Takara: Hal, Sam Tutto: Carl, Martin There seems to be a dead heat. We'll vote again on Tuesday. Jim wants to have exit interviews with each of us with Melania, Steven, John, Jim. Jim wants at least all the undergrads and Hal and Martin. Jim will decide either Tuesday or Thursday for exit interviews. He'll find out schedules for Melania, Steven, and John and pick a day. Jim's idea: Link from SSL homepage to articles as they currently exist. We think this is a good idea. Vigre may look at the SSL site to decide who will get the grant, and if we have papers being written then that could be a tie breaker! Jim is doing this right now (putting links onto the main site). Sam and Hal want to change the title. Suggestions: Newton's method as a rational recurrence Newton's method as a recurrence Newton's recurrence Newton's method applied formally to quadratics A formal application of Newton's method applied to quadratics Newton's method as a formal recurrence Newton's method considered as a formal recurrence. Jim likes the last two best. We also don't like the title of Emilie and Paul's paper. Suggestions: Sequences Similar to Somos Sequences Similar to Dana Scott New Quadratic Recurrences Yielding Integers we can keep our title and call it tentative...with an "a" in front: "A New Family of Somos-like Recurrences" Jim has updated the website with the articles now linked to: jamespropp.org/SSL Paul said that he finished the proof of the cube snake weighted conjecture. He'll update the paper to have theorem instead of conjecture. Jim said that he should update the authors list to include everybody that worked on the cube snake problem. Jim is adding that writeup to the SSL site as well. Sam asks whether or not they should put the Fibonacci special case of the Newton recurrence. Jim thinks that is a good idea. Jim will email the domino forum and ask whether or not the Newton result is new. A debate occurs about chocolate ice cream. Is it really chocolate? Jim says no, Hal says yes. Hal says that chocolate ice cream is related to chocolate milk. We're taking our break now! Sam asks why Mobius transformation is not the same as fractional linear transformation. The Mobius transformation is a fractional linear transformation that preserves the unit disk. Note: Rick Kenyon needs recognition in the Newton paper!!!! Jim: is there any other way to see if the Newton polynomial sequence is new. You could look in the integer sequence handbook. The sequence would be: the number of terms in the numerator/denominator, or the max of the coefficients in either numerator or denominator. We enter the recurrences into maple in order to calculate some of these sequences. We put some of them into the sequence handbook and nothing came back. That's a good sign that the Newton paper is a new result. Another way to see if the result is new is to to a good old fashioned Google search. Search for things like: Newton recurrence, Newton's method, Newton 2^n... Sam asks what is umbral calculus. Jim says do a Google search with "Finite Operator Calculus". The main people to look for are Rota. There's a book published in 1975 by Rota. Umbral calculus is: Take the set of all sequences and treat it like a vector space. T: xvector -> yvector y_n = x_{n+1} f_{n+2}-f_{n+1}-f_{n} = 0 /\ || \/ (T^2-T-I)(the f vector) = the 0 vector It is 5:30 now. Time to adjourn for the day.