contact info:

Siddique Khan

siddique@mit.edu

410 Memorial Drive

Cambridge, MA 02139

**I am working with the following on P-adic properties of enumeration of domino tilings : **
John

Kezia

Amanda

and

Trevor

**
I am getting academic credit from MIT for REACH
**

- 11/19 REACH meeting, 3pm-5pm
- 11/21 missed REACH meeting
- I finished up reading
- Met with Kezia to get up to date with what was discussed at the last meeting
- Spent some time working with the cube recurrence to see how it works under Kuo condensation to give the number of tilings of the 2n by 2n grid embedded in an Aztec diamond.
- Thought about a possible Maple program that could compute and display each level of the cube recurrence given an initial matrix. This could possibly help the group with computations for large n. I'll talk to the group more about it at the next meeting
- Total time spent this week: 7 hours
- Total hours spent on REACH to date: 65 hours

- 11/12 missed REACH meeting
- 11/14 REACH meeting, 3pm-5pm.
- I began reading
- Started learning how to use vaxmaple
- I began reading
- Total time spent this week: 5 hours
- Total hours spent on REACH to date: 58 hours

- 11/5 REACH meeting, 3pm-5pm.
- 11/7 REACH meeting, 3pm-5pm.
- I wrote a maple procedure with Kezia's help called quadrecurr that , for a given k and a sequence of integers, proves that no quadratic recurrence of the form C[k]*a[m-k]*a[m+k] + C[k-1]*a[m-k+1]*a[m+k-1] + C[k-2]*a[m-k+2]*a[m+k-2]+ ... + C[0]*a[m]^2 = 0 exists for that sequence.
- Here is the
- I tried using this program to see if there might be a quadratic recurrence for the Pachter regions. The program gave non-zero determinants for k from 1 through 7. I believe that this proves that no quadratic recurrence of the above form exists for Pachter regions for k up to 7. For k higher than 7, I couldn't get Maple to handle the large integers properly.
- Total time spent this week: 10 hours
- Total hours spent on REACH to date: 53 hours

- 10/29 Missed REACH meeting
- 10/31 REACH meeting, 3pm-5pm.
- Read
- Tried using Pachter's result that N(2n,2n)=2^n(H_n)^2 to simplify our problem of proving that N(2n,2n) is divisible by 3 when n is congruent to 2 (mod 5). I attempted to find a recurrence for the Pachter regions with no success. I think that we might be able to use induction on such a recurrence and the recurrence for N(2,m) to prove divisibility by 3.
- Total time spent this week: 6 hours
- Total hours spent on REACH to date: 43 hours

- 10/22 REACH meeting, 3pm-5pm
- 10/24 REACH meeting, 3pm-5pm.
- Chose P-adic properties of enumeration of domino tilings project
- Studied Lindstrom's lemma and uses for this project
- Started trying to read Cohn's paper: http://front.math.ucdavis.edu/math.CO/0008222
- Total time spent this week: 5 hours
- Total hours spent on REACH to date: 37 hours

- 10/15 REACH meeting, 3pm-5pm
- 10/17 REACH meeting, 3pm-5pm.
- Continued reading of the eight previews and the notes Jim added
- Read more on the Gale-Robinson Polynomial project from the Fomin and Zelevinsky paper
- I decided to focus on the P-adic properties of domino tilings project
- Read the link to Jim's data on the 3-adic behavior of N(n)
- Used the maple code for msq to investigate N(n) modulo 9 (one of Jim's questions). From the little data I generated, i think that possibly N(n) is divisible by 9 for n congruent to 2 or 7 modulo 10 (i.e., Jim's conjecture for 3-adic behavior may also hold for 9-adic behavior) 2, 0, 5, 7, 5, 1, 0, 4, 2, 7, 2, 0, 2, 7, 5, 7, 0, 7, 2, 0, 8, 0, 8, 4, 2, 7, 0, 7, 5, 1, 2, 0, 2, 7, 8, 0, 0, 7, 5, 7
- Note to self: Try to find out if there is any way to improve the running time of path and msq - Perhaps Maple's Determinant algorithm is not ideal?
- Total time spent this week: 7 hours
- Total hours spent on REACH to date: 32 hours

- 10/8 Missed REACH meeting
- 10/10 REACH meeting, 3pm-5pm.
- Caught up with work discussed in the two meetings missed by reading the minutes.
- Read the Reciprocity Theorem for Domino Tilings paper at front.math.ucdavis.edu/math.CO/0104011
- Read the proof of the quadratic recurrence for Fibonnaci recurrence at http://www.math.harvard.edu/~propp/reach/integrality.html and the combinatorial interpretation of running the Fibonnaci sequence backwards at http://www.math.harvard.edu/~propp/reach/symmetry.html
- Read http://jamespropp.org/bilinear/domino
- Brushed up on some Maple programming by working a bit on the 3n+1 problem (a favorite number theory problem I've been looking at recently)
- Read the eight 1-pagers
- Total time spent this week: 14 hours
- Total hours spent on REACH to date: 25 hours

- 10/1 REACH meeting, 3pm-5pm.
- 10/3 Missed REACH meeting
- No significant work done to report during this week ( really bad week due to midterms, papers and labs)
- Total time spent this week: 2 hours
- Total hours spent on REACH to date: 11 hours

- 9/24 Attended REACH first meeting, 3pm-5pm
- 9/26 REACH meeting, 3pm-5pm.
- Found solution of ladder matchings for 1 by n case. Completed generating function solution for 2 by n case. Worked on finding a recurrence for 3 by n case. (Total of about 5 hrs spent).
- Total time spent this week: 9 hours
- Total hours spent on REACH to date: 9 hours