Abstracts
Dr. Nicholas Baeth - University of Central Missouri
Matrices Behaving Badly: Examples of Non-unique Factorization
within Matrix Semigroups
ABSTRACT: In the perfect world of the positive integers every element can be
uniquely factored (up to order) as a product of prime numbers. However, this
nice property of "unique factorization" does not hold in every algebraic
setting. In this talk we will investigate unique and non-unique factorizations
of elements in various categories of integer-valued matrices. Through these
examples, we will explore the good, the bad, and the ugly in the world of of
factorization theory and introduce several important factorization-theoretic
invariants.
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Lane Bloome - Millikin University
Compressed Zero-divisor Graphs of Finite Commutative Rings
In abstract algebra, we learned that in certain rings it is
possible to have that xy = 0 when neither x nor y =0.
(We call x and y zero-divisors.) This led algebraists to
develop the notion of a zero-divisor graph. Recently, this idea has been
expanded upon to take sets of vertices in zero-divisor graphs that serve the
same function and compress them down to one vertex, yielding a new graph: the
compressed zero-divisor graph. We will explore this structure and some of
its properties.
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Trey Brock - University of Central Missouri
Monoids Defined by Second Order Recurrence Relations
Born out of the study of non-unique
factorization in integral domains, the study of factorization properties in
commutative monoids has generated much interest over the past few decades. In
this talk, we investigate certain properties of submonoids, defined by linear
equations whose coefficients come consecutive entries in a sequence defined by a
second order recursive relation. In particular, we classify all irreducible
elements of the monoids and we give a measure of how they are from being free;
i.e., having unique factorization.
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Kaitlyn Cassity - University of
Central Missouri
Factorization in k-furcus Semigroups
Every positive integer can be factored uniquely
into a product of primes. However, not all algebraic systems have this nice
property of unique factorization. In this talk certain types of nonunique
factorization are studied in semigroups. A bifurcus semigroup, recently defined
by Adams et al., is a semigroup in which every nonunit nonatom can be factored
into the product of two atoms. In this talk we generalize the idea of bifurcus
to
k-furcus;
a semigroup S
is k-furcus
if whenever an element of S
can be factored into at least
k elements, then it
can be factored as the product of exactly k
atoms of S.
Examples of k-furcus
semigroups are given and the results about bifurcus semigroups are generalized.
Two variations of k-furcus
- quasi k-furcus
and strongly k-furcus
- are given and these properties are compared and contrasted.
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Kelly Collins - University of Evansville
Minimum Energy Solutions for the Shape of a Slice of French Bread
If you have ever wondered why baked goods obtain
specific shapes, you are certainly not alone. The answer can be summed up in two
words: energy minimization. A heuristic model for the shape of a slice of French
bread, derived from the principle of least action, has been formulated by Dr. D.
Finn and confirmed by past REU groups. Having inherited this mathematical model,
we seek to understand under what conditions it provides minimum energy–and,
therefore, physically possible–solutions.
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James Gossell - University of
Central Missouri
Creating Irreducible Divisor Graphs for Numerical Monoids
Given a generating set N = <e1, e2,
… , en> where e1,e2,…,en
are in Z+, a numerical monoid N is a set of all
non-negative integers x that can be written as x = c1e1
+ c2e2 + … + cnen
where c1,c2,…,cn
are non-negative integers. An irreducible divisor graph G(x)
is a visual representation of the factorization of an element x in a
numerical monoid. The graph G(X) contains all elements in the
generating set that appear in some combination of x. We connect two
vertices if and only if they appear in the same combination of x. In
this talk we will examine how to construct certain types of graphs such path
graphs, cycle graphs, and bipartite graphs.
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Chelsey
Johnson -
University of Central Missouri
Cliques and Numerical Monoids
A numerical monoid N is the set of all sums of elements of a
relatively prime set of natural numbers.
An irreducible divisor graph G(x) where x is in N, can be constructed
where the generators form the vertex set.
In this talk, we will discuss examples of numerical monoids related to
irreducible divisor graphs.
Furthermore, we will prove a theorem that enables us to create an arbitrary
number of disjoint 3-cliques.
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Gina Luciano - Millikin University
Using Data Mining to Analyze Admissions Data
Data mining is the process of finding useful patterns in
data. A data mining program called Rattle was used to analyze admission data for
Millikin University. The purpose of analyzing this set of data was to find
patterns that could potentially explain why prospective students do not fully
complete the admission process that ends with registration. An original data set
of 42 variables was narrowed down and analyzed to find the characteristics of
students that do register and explanations as to why some students do not
complete this process. The findings will be relayed to the Office of Admission
to help them more effectively reach out to students.
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Joe Nance - University of Illinois
A Method for Recursively Generating
Sequential Rational Approximations to
We intend to derive
an elementary recursion that generates a sequence of fractions approximating
with increasing accuracy.
The recursion is then defined in terms of a series of first-ordernon-linear
difference equations and then analyzed as a discrete dynamical system.
Convergence behavior of initial trajectories is then discussed in the language
of finite dimensional vector spaces and eigenvectors, effectively proving
convergence without the notions of standard analysis.
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Darrin Weber
- Millikin University
Zero-divisor Lattices on Commutative Rings
A zero-divisor graph is a graph whose vertices are the nonzero
zero-divisors of a ring and two vertices are connected if and only if their
product is zero. We look to expand on the idea of zero-divisor graphs of rings
into lattices on those rings. We take the zero-divisors of a ring, examine their
annihilator sets, and assign an order to them. We then place these annihilators
into a lattice structure and observe its properties.
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