Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2007

Last Update: April 29, 2007.

Feel free to propose your own project!

Untaken Projects

1. Write a package and invesigate the "super-C-finite ansatz", which are solutions of linear recurrences whose coefficients are C-finite functions.
2. Write a Maple package that empirically finds "all" hypergeometric identities of bounded complexity (ask me for more details).
3. Write a Maple package to find lower bounds for the minimal number of "monochromatic" entities generalizing the lower bounds in Aaron Robertson, Pablo Parillo and Dan Saracino's article On the asymptotic minimum number of monochromatic 3-term arithmetic progressions and Aaron Robertson, and Doron Zeilberger's great article on Schur Triples. [ask me for details].
4. Use the holonomic ansatz to investigate solid partitions and 3-dim. Young Tableaux.

Taken Projects

• Develop a Maple package called QuasiPolynomial to handle and quasi-polynomials. Recall that a sequence f(n) is quasi-polynomial if there exists an A, such that the sequences f_c(m):=f(A*m+c) (for c=0,1, ..., A-1) are polynomials in m. Famous sequences that are known to be quasi-polynomials are those whose generating functions are of the form

  1/((1-x^a1)^b1*(1-x^a2)^b2*... *(1-x^ak)^bk)

  for finite positive integers a1, ..., ak, b1, ..., bk .

  [Claimed by Andrew Baxter]

• Develop, in great detail, the rudimentary ideas of experimental-yet-rigorous findings, develped in class, of statistical quantities (moments and generalized covariances) for permutation statistics like des, maj, inv, and generalizations of them (that you are welcome to make up), as well as for multi-set permutations. [Claimed by Jason Chu]

• Study, via experiments unavoidable words.

  [Suggested and Claimed by Justin P.]

• Write a Maple package CFinite that handles solutions of linear recurrence equations with constant coefficients, and proves identities amongst them, using a purely empirical approach. Also, have the computer discover elegant identities in the style of Hugh Montgomery's problem described in my article What is Experimental Mathematics .

  [Claimed by Justin Bush]

• Study, experimentally (and, if possible, rigorously) non-linear recurrences of the kind in Raghib M. Abu-Saris's paper (given in class), "A note on the attractivity of period-four solutions of a third-order rational difference equation" (J. of Difference Eqations and Applications vol. 12, no. 2, February 2006, 233-235).

  [Claimed by Humberto Montalvan Gamez]

• Develop extensions of hyper-exponential functions (ask me for more details). [Claimed by Mangesh Gupte]
• Investigate Non-Linear (quadratic) Somos-type recurrences .

  [Proposed and claimed by Emilie Hogan]

• Do the analogs of my articles Symbolic Moment Calculus I and Symbolic Moment Calculus II to study higher moments for the r.v. "number of monochromatic van-der-Waerden triples" (or more generally "k-tuples" for numeric, specified k) in a random r-coloring of the integers from 1 to n, and the r.v. "number of Schur triples".

  [Claimed by Paul Raff]

• Find computer-assisted bijections between Dyck words avoiding some subwords and binary trees avoiding some subtrees.

  [Suggested and Claimed by Eric Rowland]

• Develop a package for "bi-basic" identities as hinted in A=B section 9.8

  [Claimed by Ke Wang]

Challenges

• ($25) Solve Amitai Regev's problem.