Titles are hyperlinked to pdf copies of the final project write-up. Course coordinator: Patrick Keef

• TITLE: THE USE OF LINEAR ALGEBRA IN MODELING THE PROBABILITIES OF PREDICTED FUTURE OCCURRENCES (selected outstanding senior project)

AUTHOR: Gabrielle Boisrame
ABSTRACT: Singular Value Decomposition (SVD) and similar methods can be used to factor matrices into subspaces which describe their behavior. In this paper we review the SVD and generalized singular value decomposition (GSVD) and some of their applications. We give particular attention to how these tools can be used to isolate important patterns in a dataset and provide predictions of future behavior of these patterns. A major focus of this project is the examination of a component resampling method described by Michael Dettinger which provides estimates of probability distributions for small sets of data. We tested the results of using both the SVD and the GSVD for Dettinger's method. Similarly to Dettinger, we found that the method had a tendency to give probability distributions a Gaussian shape even when this did not seem to be represented in the original data. For some data sets, however, both using the SVD and GSVD provided what appear to be reasonable probability distributions. There was not a significant difference in how well original probability distributions were estimated when using Dettinger's original method or the modifications with the reduced SVD or the GSVD. Using Dettinger's method rather than a simple histogram always provided a higher resolution of information, and was sometimes capable of matching the shape of the original probability distributions more closely.

• TITLE: COMMUTATIVITY IN NON-ABELIAN GROUPS
AUTHOR: Cody Clifton
ABSTRACT: Let $P_2(G)$ be defined as the probability that any two elements selected at random from the group $G$, commute with one another. If $G$ is an Abelian group, $P_2(G) = 1$, so our interest lies in the properties of the commutativity of non-Abelian groups.

Particular results include that the maximum commutativity of a non-Abelian group is 5/8, and this degree of commutativity only occurs when the order of the center of the group is equal to one-fourth the order of the group. Explicit examples will be provided of arbitrarily large non-Abelian groups that exhibit this maximum commutativity, along with a proof that there are no 5/8 commutative groups of order 4 mod 8.

Further, we prove that no group exhibits commutativity 0, there exist examples of groups whose commutativity is arbitrarily close to 0. Then, we show that for every positive integer $n$ there exists a group $G$ such that $P_2(G) = 1/n$. Finally we prove that the commutativity of a factor group $G/N$ of a group $G$ is always greater than or equal to the commutativity of $G$.

• TITLE: THE EUCLIDEAN ALGORITHM AND A GENERALIZATION OF THE FIBONACCI SEQUENCE
AUTHOR: Ian Cooper
ABSTRACT: This paper will explore the relationship between the Fibonacci numbers and the Euclidean Algorithm in addition to generating a generalization of the Fibonacci Numbers. It will also look at the ratio of adjacent Fibonacci numbers and adjacent generalized Fibonacci numbers. Finally it will explore some fun applications and properties of the Fibonacci numbers.

• TITLE: THE GAME SET AS ${\bf F}^4_3$
AUTHOR: Hillary Fairbanks
ABSTRACT: In this paper we construct an isomorphism between the card game Set and the four-dimensional vector space over the three element field, ${\bf F}_3$, to draw various results about the game. By creating a one-to-one and onto correspondence between the cards and points in ${\bf F}_3^4$, we find that a collectable 3 set is in fact a line in the vector space. Using this, we are able to determine the total number of collectable sets that exist in the game, as well as find the maximum number of cards that can be played without having a collectable set. Furthermore, we can simulate the game using the Monte Carlo method to find the probability of having a collectable set in a random selection of cards from the deck.

• TITLE: CURVES OF CONSTANT WIDTH AND THEIR SHADOWS
AUTHOR: Lucia Paciotti
ABSTRACT: In this paper we will investigate curves of constant width and the shadows that they cast. We will compute shadow functions for the circle, Reuleaux Triangle, and the curves of constant width described by Stanley Rabinowitz. From these functions we will prove that you can distinguish the different curves from their shadows. A result about the perimeter and area of these curves is also presented.

• TITLE: MULTIPLICATIVE GROUPS IN ${\bf Z}_m$
AUTHOR: Brian Sloan
ABSTRACT: Our goal will be to find subsets of ${\bf Z}_m$ that form groups under the operation of multiplication modulo $m$. By utilizing the isomorphism ${\bf Z}_m = {\bf Z}_n \oplus {\bf Z}_k$ , we will find multiplicative groups contained in ${\bf Z}_n \oplus {\bf Z}_k$ and then map these back to ${\bf Z}_m$. In particular, if $m = nk$ with $\gcd(n, k) = 1$, our objective is to find particular multiplicative subsets of ${\bf Z}_n \times {\bf Z}_k$ that are groups and whose first coordinate is a projection onto $U(n)$. We will give a method to calculate the total number of these subsets, and identify the elements of which they are composed.

• TITLE: SYSTEMS OF PYTHAGOREAN TRIPLES
AUTHOR: Chris Tobin-Campbell
ABSTRACT: This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines which numbers can be the sides of primitive right triangles and how many primitive right triangles those numbers can be a side of, and finally explores systems of three and four right triangles that fit together in three dimensions.
ABSTRACT: This paper examines two methods of determining whether a positive integer can correspond to the semiperimeter of a number of Pythagorean triangles. For all positive integers $k$, using Bertrand's Postulate, we can find semiperimeters corresponding to $k$ or more isoperimetrical triangles, and using the Prime Number Theorem, we can find exactly $k$ generator pairs which correspond to a semiperimeter.