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.
ADVISOR: Douglas Hundley
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 $G$.
ADVISOR: David Guichard
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.
ADVISOR: Patrick Keef
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.
ADVISOR: Barry Balof
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.
ADVISOR: Albert Schueller
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.
ADVISOR: David Guichard
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.
ADVISOR: Laura Schueller
TITLE: PERIMETERS
OF PRIMITIVE PYTHAGOREAN TRIANGLES
AUTHOR: Lindsey Witcosky
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.
ADVISOR: Russell Gordon
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