SSL Notes for meeting #3 (9/16)
Today's note-taker: Stephen (Hal next time)
Today's snack-provider: Sam (Martin next time)
Go around with names, and ask everyone how they heard about SSL.
Carl Edquist (guest): Heard about SSL thru ?
Hal: from Scott Simon
Josh: from Jim Propp directly
Stephen: from Jim Propp
Jim: from Jim Propp, in a prophetic dream involving a turtle and six mongooses.
Sam: From Gloria
Paul: from Gloria
Emily: from Gloria
Abby: from Gloria
Martin: from random papers; Jim's home page
What's the deal with making Maple available?
Everyone thinks they have access now; Sam still has to make sure.
Hal's question: Where to get public methods/algorithms for doing computations
in Maple?
Hal says the URL for the Maple Web site is http://www.maplesoft.com/ .
Wolfram will be in town to plug his book "A New Kind of Science" Oct. 7.
Martin says that Maple is not installed in UPL (undergrad projects lab); he
will look into the suitability of that place.
What's the deal with the Math 192 DVDs?
Jim would like someone to find out what format it is
Martin agrees to take 192 DVD to someplace that can identify its format/type.
It turns out that no one is receiving SSL email Jim has been sending out
to ssl@math.wisc.edu . Jim will explore this issue.
Jim wants to know how he can make the SSL web-site more helpful.
(Links to your SSL sites? Links to minutes?)
Many believe more math, less logistics would be good (we spent nearly an hour
on logistics this time).
Work done by group members since last time:
Hal wrote a nice perl code to compute #'s of perfect matchings, and compared
these #'s to Markoff #'s. Hal will email a link explaining how to get ?perl?
(spelling?)
Paul found how to quickly count perfect matchings of small snakes, and counted
matchings of some rectangles, and modified rectangles. Verifies in examples
that removing any exterior vertex from an odd by odd rectangle results same #
of matchings; removing an interior vertex results in fewer.
Sam played with arithmetic progression/Fibonacci progression rule for
recursively computing #M(G) for snakes G, also found that "staircase" snakes
with n boxes have n+1 matchings. (Thus every positive integer occurs as #M(G)
for some G)
Jim:
Here is a formula for counting perfect matchings: In the kth box of a snake,
write the number of perfect matchings of the snake formed by the first k boxes:
E.g.
_ _
|2|3|
- -
Then the rules are:
_ _ ___
|a|b|a+b|, and
- - ---
____
|2b-a|
_ ____
|a| b |.
_ ____
Notice: the rule is compatible with reading the snake backwards, in a generic
example.
Can anyone verify these? Homework: Proof of these rules.
Emily concentrated on a proof of 3F_n=F_{n+2}+F_{n-2}, also found a pattern
agreeing w/ the rule just given by Jim
Abby: Gave an (algebraic) proof of the identity above.
Martin: Played with the rule Jim gave.
Josh: Encoded snakes as sequences of integers, where the kth integer is the
number of boxes in the kth row of the snake.
Carl: Encoded snakes as sequences of zeros and ones, with the ones marking a
turn.
Jim:
Here's a symmetric, non-bijective proof of the Fibonacci identity:
_ _ _ _ _ _
Every perfect matching of |_|_|_|_|_|_| is of one of the types:
_ _ _ _ _ _ _ _ _ _ _ _ _ _
|_|_|_|_| _, _ |_|_|_|_|, or | |_|_|_|_| |.
But there are some matchings which occur as both of the first two types; these
are of the type
_ _ _ _
_ |_|_| _, and this gives the formula.
Moral: Sometimes bijections aren't the best way to go.
Stephen will post notes giving closed formulas for #M(G) for snakes G as soon
as he figures out how to draw nice pictures on the computer.
For Thursday: Find a combinatorial proof that F_n^2 = F_{n-1} F_{n+1} +- 1.
Also, everyone should make a SSL web page as soon as possible.