Today was the last day of my independent study at BGSU.

To start off, I talked with Dr. Zirbel about finding a good problem to work on once this intensive period worked on. In addition to the one I described in the last post, I found a more well known problem having to do with the Kolakoski sequence, which is a sequence of 1s and 2s that encodes its own run length. Each 1 in the sequence corresponds to a single number later on, and each 2 corresponds to a pair of the same number later on.

In addition to this problem, there is a problem that he got from a professor who retired which essentially involves determining the number of points of an n-dimensional cube that an n-dimensional sphere with radius r < sqrt(n) can contain. We figured out the lower dimensionality cases, but the higher dimensions are more difficult once sqrt(n) > 2, because then the sphere can contain more lower-dimension edges before it reaches the upper bound of the radius. I plan to work on this first by determining a general algorithm for finding which points must be evaluated for finding this number of vertices. Once I have a computer program that can plot it, then I’ll be able to analyze that data and try to create an analytic solution.

In addition to this, I also attended the probability class and the pedagogy class. In probability, we finished the proof of the martingale convergence theorem, which was really interesting, because it made me realize that we often alter the way we look at specific cases in proofs without actually altering the statements. In pedagogy, we talked about academic honesty. It was good to see that academic honesty is taken very seriously and that the system is designed to prevent people from being dishonest.