Selected Master’s/UROP projects
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.
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
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
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.