Symmetric and balanced numbers

Joint work with Josh Zelinsky.

Combinatorics/Algebra Seminar, ISU, April 2, 2018

In a recent talk on "integer complexity", Josh Zelinsky looked at the binary expansion of simple fractions 1/n, where n is odd. Depending on whether the number of 1's in the period of that expansion is less than, greater than, or equal to the number of 0's, he calls n an efficient, inefficient, or balanced number. While not central to the topic of integer complexity, these concepts raise interesting questions themselves:

  • What is the density of these sets of numbers?
  • What is the density of prime numbers with these properties?
  • Are there easy criteria to determine whether a number is efficient/inefficient or balanced?

One easy criterion is that an odd number n is balanced if -1 equals a power of 2 modulo n. The converse is true for the case of n prime, but false in general. This motivates us to call an odd number n symmetric if - 1 equals a power of 2 modulo n. Then we can state that every symmetric number is balanced. Again, we can ask the questions above about symmetric numbers and primes. We have some answers to these questions, relying on some all-time classics (Fermat's Little Theorem, Quadratic Reciprocity, and the structure of the multiplicative group modulo n).

Galois Theory and the Ring of Sauron

Joint work with John Gillespie.

Combinatorics/Algebra Seminar, ISU, April 24, 2017 

Consider a polynomial f(x) over a field k with degree n and discriminant D. The Galois group G of f(x) can be viewed a a subgroup of the symmetric group Sn via its action on the roots of f(x). It is a well-known application of Galois Theory that G is contained in the alternating group An exactly if D is a square in the ground field k. Moreover, if this is not the case, then sqrt(D) generates the fixed field of the intersection of G with An. We are looking for formulas that generalize this - i.e., for any transitive subgroup H of Sn, find expressions in the coefficients of a generic polynomial of degree n which are in the ground field exactly if G is contained in H, and which generate the fixed field of the intersection of G and H otherwise. At first, it is not even clear that such expressions have to exist. But they do, and they in principle provide a way to determine G. Practical computations are lengthy and best done by computer, but the theory is very appealing and natural. In particular, we use multivariate polynomial rings and classical representation theory of finite groups (in a very basic way).

Rashomon - Reconstructing Reality from inexact measurements

Joint work with Heike Hofmann, Dianne Cook.

MAA section meeting, Graceland University, Lamoni, IA, Oct 3, 2015

Here are slides.
The art installation 'Rashomon' was displayed on the Iowa State University campus during summer 2015. It consists of 15 identical, abstract sculptures. Artist Chuck Ginnever posed the challenge whether it is possible to display the sculptures so that no two of them are in the same position. We investigated the related question of reconstructing such a sculpture from (ordinary tape-measure) inexact measurements. Mathematics involved are the Cayley-Menger determinant, and the gradient method / Steepest Descent. We'll explain the mathematics with some simple examples and then show the results of our reconstruction. We will only assume elementary linear algebra (matrix - vector multiplication, determinants).