Kristen M. Cheman
Project Advisor: Dr. Caryn Werner
This project will study the Zariski topology on the prime spectrum Spec(R) of a commutative ring R. The notion of a topology will be introduced and extended to construct the Zariski topology on the topological space Spec(R). In particular, the Zariski topology on the prime spectrum of R and C, as well as the polynomial rings over each of these fields in one and two variables, will be examined. Several properties of the prime spectrum and its topology will be discussed, including the spectrum as a T0 topological space and a Hausdorff space, and the compactness of the Zariski topology. Finally, Spec(R) will be considered in terms of geometry based on the correlation between the affine variety of an ideal of a ring and the prime spectrum of its coordinate ring.
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Laura R. Greenhow
Project Advisor: Dr. Michael Barry
This senior project will explore strange functions in real analysis. Throughout this project, we will first investigate a nowhere differentiable function which is everywhere continuous functions with varying differentiability. We will then use various properties of continuity and differentiability to solve an advanced integral.
Elizabeth V. McLaughlin
Project Advisor: Dr. Tamara J. Lakins
It is known that the polynomial ring F[X] is a unique factorization domain when F is a field. However when we take away some of the units from this domain we can get a half factorial domain K + XF[X] where K is a proper fubfield of F (Note that a half factorial domain is a domain where every non-zero, non-unit element can be factored into a product of irreducibles, and each such factorization has the same length, but the factorization may not be unique up to associates). In this paper we will count the number of factorizations of elements in K + XF[X] where K = Z2 and F = F4 and conjecture about the number of factorizations for general finite fields K Í F. Also we will look at the asymptotic behavior of the number of numbers of factorizations of length n for Z2 + X F4[X], and make conjectures about general domains of the form K + XF[X].
Project Advisor: Dr. Michael Barry
The purpose of this senior project is to study and answer the following question: Starting with a given arrangement of the fifteen Puzzle, how many of these 15! Possible arrangements can be reached by moving the tablets according to the rules of the puzzle. I will also extend this question to the Lucky Seven Puzzle in figuring out how many of the 7! Possible arrangements can be reached by moving the tablets according to the rules of this puzzle.
Lolly Beth Roscher
Project Advisor: Dr. Anthony Lo Bello
I show that the winning Powerball, Hot Lotto, and Wild Card combinations all pass a new test for randomness.
Steven W. Rusnica
Project Advisor: Dr. Anthony Lo Bello
The purpose of this Comprehensive Senior Project is to present an alternative proof to a theorem of Fabrizio Polo on the expected number of sequences of various lengths in Euler’s Genoan Lottery.
August Seibel
Project Advisor: Dr. Michael Barry
I will use a mix of linear and abstract algebra methods to derive a formula for the number of elements of GL(2,Zp) and GL(3, Zp), corresponding to the group of 2 X 2 and 3 X 3 invertible matrices respectively, taking entries in a finite field of p elements, all of whose eigenvalues lie in the field. The 2 X 2 case has been worked out by Olsavsky and Barry, I will supply the missing details.
Dan Slowey
Project Advisor: Dr. Anthony Lo Bello
The following is a presentation of the mathematical principles upon which the design and execution of national election polls is based.
John Tarr
Project Advisor: Dr. Tamara J. Lakins
The purpose of this project is to combine the disciplines of mathematics and art. The mathematics I am working with are isometries of the plane as well as symmetry groups and strip patterns. The study of isometries and products of isometries aid in the study of symmetry groups. Products of isometries allow us to eliminate combinations of symmetries that are not possible in a strip pattern. The sculptures made have an underlying concept of the mathematics studied. More specifically I fuse artistic ideas such as form, composition, and materials, with mathematical ideas of isometries, mathematical notation, and infinity. The body of work presented gives visual examples of this fusion, while the document here further explains the underlying mathematics as well as the conceptual background of the work.
Zachary S. Webber
Project Advisor: Dr. Anthony Lo Bello
The purpose of this Comprehensive Senior Project is to expand on the findings of Fabrizio Polo and Steven Rusnica involving Euler’s Genoan Lottery. Specifically, this paper will deal with finding the expected value, variance, and factorial moments of the number of sequences in an outcome of this lottery.
Project Advisor: Dr. Caryn Werner
This paper looks at magic squares. Specifically, it looks at the square-palindromic property. It also looks at other properties of matrices, such as persymmetric matrices, centro-symmetric matrices, and show how they exhibit the square-palindromic property.