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.