Selected Master’s/UROP projects

Continued Fractions:

Sam Judnick
, finishing in May, 2013 did a project related to simple continued fractions.  The basic question:  Given the expansion of x, what happens to that expansion  if you add a very small quantity to x?

Jesse Schmeig, finishing in March, 2015 did a UROP on continued fractions of the form a + z/(b + z/(c + …)).  He focused mostly on expansions of rational numbers when z is also rational.

Pablo Mello, finishing in August, 2017 did a UROP on continued fractions of the form a + z/(b + z/(c + …)) where z = sqrt(n).  His major focus was on patterns when rational numbers are expanded with z = sqrt(n), investigating cases where expansions were finite, and cases with periodic expansions.

Products of 2x2 matrices:

Kate Niedzielski, May 2008, investigated recurrence relations that had a nonlinear twist on the Fibonacci sequence:  for some fixed rational xm if the some of two successive terms has the property that x(a_(n-1) + a_(n-2)) is an integer, let a_n be this product.  Otherwise a_n = a_(n-1) + a_(n-2).  One of the main tools in this investigation was products of (many copies) of two matrices A and B.

Brittany Fanning, January 2015, extended the ideas in Kate’s paper to more general second order recurrence relations.  That is, she investigated the recurrence  a_n = c a_(n-1) + d a_(n-2), or  a_n = x(c a_(n-1) + d a_(n-2)).

Zhanwen Huang, December 2009, investigated products of two 2x2 matrices A, B, asking questions about how often ABAB has a larger trace that AABB, among other questions.

Andrew Schneider, May 2015, extended Zhanwen’s work, getting explicit counts for questions of the following type:  Given a product of n copies of A and 2 copies of B, in how many ways can the traces of such products be arranged?  For example, if x_1 = Tr(ABBABB), x_2 = Tr(ABABBB) and x_3 = Tr(AABBBB) then 5 of the 6 possible orderings of the x’s can occur:  x_2 > x_3 > x_1 can not happen.

Some combinatorial projects:

Kyle Krueger, August 2012, investigated generating functions for cyclic permutations of a multi set of 0’s and 1’s, weighted by the inversion number or by the Major index.

Ondrej Zjevik, July 2014, used a max flow algorithm to find symmetric chain decompositions in various partially ordered sets, most notably, for the weak Bruhat order on the symmetric group up to S_11, and for Young’s Lattice up to L(12, 12).

Other projects (harder to classify):

Melissa Larson, May 2008, used an integral to verify and construct first order BBP-type formulas for π.

Amy Schmidt, May 2011, investigated the numerical stability of Dodgson’s condensation method for calculating determinants.

Trevor Brennan, March 2012, investigated how efficient Erods’s scheme is for constructing Carmichael numbers.  In particular, he shows that of m = LCM(1, 2, …, 17) then there 141 primes p with p - 1 dividing m, and these lead to at least 1.26 x 10^36 Carmichael numbers as products of these p’s.

Mark Broderius, August 2019, With Lucas numbers defined via U_0 = 0, U_1 = 1, U_n = P U_(n-1) - Q U_(n - 2), Mark looked into patterns for when a Lucas sequence can not contain infinitely many primes.