Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2009

First posted: March 18, 2009; Last Update: April 14, 2009.

Feel free to propose your own project!

The suggestions below are very brief. Feel free to ask me more. Once you pick a project, I'll tell you more about it. You are welcome to collaborate. At most two people per project.

Untaken Projects

1. In my seminal paper Symbol-Crunching With the Gambler's Ruin Problem, I describe how to compute the moments for the random variable "expected duration" of the game for any fixed moment. In an even more seminal paper I describe how to compute asymptotic moments for the symbolic 2r-th (and (2r+1)-th) moment for certain combinatorial random variables. Apply (or extend) this method to handle the random variable "duration of the game" in a fair gambler's ruin problem (you start with xA dolalrs, (0 < x < 1) and you keep playing until you either broke or have A dollars. (Note: I don't know anything about this limiting distribution, except that it it not asymptotically normal.

2. Milk the generating function F(z,t1,t2,t2,t4) this paper as much as possible. Using and if possible, extending the Maple package FELLER.

3. Using the Maple package TILINGS accompanying my paper figure out the first ten or whatever terms in the sequence : number of n by n (empty) KenKen puzzles.

4. Xavier Viennot and his school study directed animals, that can ve viewed as triangles consisting of 0's and 1's such that every entry is weakly between the two right above it (in the NE and NW directions). Experimentally (and who knows, even rigorously) study such triangles where the entries can be (i) any non-neg. integer ≤ k (k=2,3) (ii) any non-neg. integer

5. Write a Maple program (better than Maple's simplify(%,GAMMA) that would simplify quotients of products of powers of GAMMA evaluated at rational arguments. For example, the output of
   (GAMMA(1/14)*GAMMA(9/14)*GAMMA(11/14))/
   (GAMMA(3/14)*GAMMA(5/14)*GAMMA(13/14))
   should be 2. [Added April 13, 2009: it turns out this is probably the only one, and thanks to Dennis Hou's brilliant solution, this project is really closed, and hence removed].

Taken Projects

• Fancy Encoding and Decoding, by Michael Ratner [suggested by him]. Comming up later in the summer.

• The Gambler's Ruin problem can be viewed as random walk in the fixed interval on the discrete line segment [0,A]. Study, both by simulation, and by symbol-crunching walks in the 2D lattice with unit steps in the four fundamental directions where you start at (x,y) travel in [0,A]x[0,B] and get kicked out as soon as you hit one of the walls x=0,x=A,y=0,y=B. This is very hard, so how about fix B to be numeric (B=1,2,3, etc.) but keep A symbolic. [Claimed by Liyang Diao].

• Studying the Cayley graph of Thompson's Group F. [Claimed by Andrew Baxter and Dan Staley, and suggested by them].
  [(last update May 19, 2009): Here is Dan Staley and Andrew Baxter's Maple package, and here is some auxiliary file, and Maple worksheet.

• Study experimentally/symbolically Percolation in a strip. [Kellen Myers and Robert DeMarco]. Here is a preliminary version of Kellen and Bobby's project. It would hopefully be replaced by a final version.

• Study expermintally/symbolically the Ising Model with a magnetic field in a finite-width infinite strip. [Claimed by Dennis Hou]

• Make the program gSAW(n,a,b,c,d,PAT) from Mar9.txt much more efficient, and experiment with many a,b,c,d, PAT, to get sequences with θ that are not zero. Possibly using the method of this paper, or more systematically that paper, teach the computer how to find closed form rational generating functions with finite a and b but c=-infinity, d=infinity. [Claimed by Brian Nakamura, and (possibly) Jeffrey Amos]