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Full-Text Articles in Physical Sciences and Mathematics
Cellular Automata And Music: A New Representation, Richard French
Cellular Automata And Music: A New Representation, Richard French
Honors Theses
For millenia, we’ve thought of musical composition as a purely human activity. However, we once also thought of an activity like chess to be purely human, but Deep Blue was able to defeat Kasparov in 1995 all the same. Could there perhaps be some tool or algorithm for musical composition that can replicate to some extent what human beings can do with music? This project explores this idea through the use of a tool called a cellular automaton. A cellular automaton is a grid space with a finite number of states for each of the ”cells” or ”squares” where a …
An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams
An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams
Honors Theses
It is well known that polynomials commute under addition and multiplication. It turns out that certain polynomials also commute under composition. In this paper, we examine polynomials with coefficients in the field of complex numbers that commute under composition (also referred to as “commuting polynomials”). We begin this examination by defining what it means for polynomials to commute under composition. We then introduce sequences of commuting polynomials and observe how the polynomials in these sequences (later defined as chains) along with other commuting polynomials relate to a concept called similarity. These observations allow us to better understand the qualities and …