Project Advisor: Dr. Michael J. Barry
This senior project looks at the probability function in terms of a linear recurrence relation for an experiment where an arbitrary coin is tossed until the first occurrence of a pattern consisting of a number of heads followed by a tail. The moment-generating functions for three cases are derived, and the mean and variance for those cases are then calculated using the respective moment-generating function. A general approach to finding the moment-generating function for a recurrence relation is then briefly explored. Several conjectures about the case involving an arbitrary number of heads are then stated, and the mean and variance for the general case are shown and proved.
Stephanie Bruggeman
Project Advisor: Dr. Caryn Werner
This paper focuses on basic properties of eigenvalues and eigenvectors within matrix algebra. In addition, advanced topics such as complex matrices are introduced, as well as Hessenberg matrices.
Jonathon Hall
Project Advisor: Dr. Michael Barry
I will combine the techniques used in Linear and Abstract Algebra to determine the “nice” froms to which a matrix is conjugate in GL(2,C), SL(2,C), GL(2, R), and SL(2,R), where GL(2,C) and GL(2,R) are groups of 2 x 2 invertible matrices, and SL(2,C) and SL(2,R) are groups of 2 x 2 invertible matrices having a determinant of 1. A. Seibel worked out the case for GL(2,F) where F is a finite field. I will adapt this approach to GL(2,C).
Tyler Lowry
Project Advisor: Dr. Michael Barry
In this paper we will calculate the smallest number of elements that will generate groups of order less than or equal to fifteen. We will also look at minimal generating sets for some groups of permutations.
Joseph Stephen Ribaudo
Project Advisor: Dr. Michael Barry
Currently at Allegheny College there is no taught course covering the subject of differential geometry. For those motivated to pursue a career in theoretical physics, this is rather unfortunate. Differential geometry is a base on which much of theoretical physics is based, and it is because of this that this project was undertaken. This paper has been developed as a complementary text to aide future persons pursuing the study of differential geometry. We have relied heavily on Elementary Topics in Differential Geometry by J.A. Thorpe as our main reference while expanding on and including numerous examples of concepts and their applications into other fields. It is hoped that this paper will serve as a more in depth examination of the subject of differential geometry thus allowing readers a more fundamental understanding the mathematics behind the subject.
Sarah R. Shoemaker
Project Advisor: Dr. Tamara J. Lakins
Change ringing is the traditional English method of ringing church bells. A team of ringers attempts to produce a peal, or all the permutations on a set of bells, according to rules. For example, each possible way of ringing the bells (permutation) must be rung exactly once, and no bell may switch more than one position between changes. This project will use group theory to analyze change ringing, showing that many methods of ringing give the symmetric group broken down into cosets, often of the dehedral group of alternating group. An understanding of these groups and their generators, as well as ideas about cosets and group actions, will be used to do the analysis.
Project Advisor: Dr. Tamara J. Lakins
Some counting problems are simple enough to solve by observation, but many require a more sophisticated approach. Burnside’s Theorem is a result of group theory that is often used to calculate the number of nonequivalent arrangements of colorings of objects in a set under a group of permutations. In this project, we will discuss the notion of an arbitrary group acting on a set, the analysis and several applications of Burnside’s Theorem, and a generalization of the theorem.
Melanie Wilson
Project Advisor: Dr. Caryn Werner
In this paper, we will investigate polynomial invariants for knots and links. We will then look at how we can simplify the calculations of the Jones Polynomials of knots or links by using their corresponding braid representations and braid words. Theses calculations lead to a closed formula for the Jones Polynomial’s of knots or links represented by braids with 2-strands. We also begin to find general formulas that can relate the Jones Polynomials of knots or links represented by braids with 2-strands. We also begin to find general formulas that can relate the Jones Polynomials of knots or links represented by braid with 3 strands to those represented by braids with 2 strands.