WILLIAM J RYAN

Fun with Random Walks

7 July 2021

I coded a "chaos game" based on a Numberphile Video on Youtube linked here. The game uses the three vertices of a triangle, each weighted with equal proability on a dice, and a single starting point. A dice is thrown and a new point is placed midway between the associated vertex and the starting point. Using the new point as the new starting point, this process is repeated N times. With enough points a Sierpiński triangle forms. Starting with hexagon rather than a triangle results in the mesmerizing image in the thumbnail. This random walk process is studied in chaos theory, a field I am interested in and will continue to study and write about here.

The blazar Markarian 421 is a supermassive black hole with a jet, accelerating particles to near lightspeed, pointed towards Earth. This paper presents a short-term and a long-term variability analysis, using the discrete correlation function, a series of least-squared fits, and the Lomb-Scargle periodogram, that probes Markarian 421's jet structure and possible long term periodicity in Markarian 421's average emmisions.

Steiner Systems

5 June 2021

This report introduces Steiner systems, a type of block design, and how such systems can be used to geometrically solve the Kirkman student problem, a famous question in combinatorics.