While physics makes heavy use of mathematics compared to other disciplines, most of what a physicist does is closer to advanced arithmetic than pure mathematics. We frequently abuse mathematical concepts and teach our students to do the same in the interest of developing their physical intuition. Ironically, the most useful parts of mathematics are often the ones that students perceive […]
Lamé’s Theorem is a little known result that describes a beautiful connection between the Fibonacci sequence and the Euclidean Algorithm on the integers. We will describe what this connection is, beginning with what how the Euclidean Algorithm works on the integers.
In part I of this series, we introduced formal language theory, focusing on languages that are recognized by simple computational machines called Finite State Automata. In part II, we turn our attention to how Finite State Automata (FSA) can be used to solve computational problems in groups.
What is needed in order to form a language? In their most basic form, languages require an alphabet (a set of letters) and a collection of finite strings of letters called words. In order to recognize a language, one must be able to distinguish strings of letters that form words from those that do not form words.
In the universe of mathematics there are many worlds. Of these, there’s the world where we apply our theory, the world where the theory is interesting for its own sake, and the world where mathematics is entertaining to the uninitiated. Some mathematical objects manage to live in all of these worlds at once.