TITLE:
FROM POSETS TO DERANGEMENTS:
AN EXPLORATION OF THE MÖBIUS INVERSION FORMULA
AUTHOR: Allison Beemer
ABSTRACT: This paper works through an independently developed proof of
the Möbius inversion formula. Beginning with a simple seed
of an idea, that of a partially ordered set, we proceed by considering
properties of compatible matrices, walking through a proof of the
existence and uniqueness of what is known as the Möbius
function, $\mu$, and subsequently offering a proof of the
Möbius inversion formula. The development of the
Möbius function and the proof of the
Möbius inversion formula lead to results in several
branches in mathematics, including combinatorics and number theory. In
particular, we will prove the principle of inclusion and exclusion and
several number theoretic results.
ADVISOR: David Guichard
TITLE: MODEL SELECTION AND SHRINKAGE: A
PROJECT IN LINEAR MODELING
AUTHOR: David DeVine
ABSTRACT: Linear modeling is a statistical tool which fleshes predictive
information from large and unweildy data sets. The most familiar linear
modeling technique is the Ordinary Least Squares (OLS) technique which
minimizes the sum of the squares of residual errors from a predictor
curve. As this paper discusses, OLS has significant shortcomings in
regression scenarios with many predictors and with correlation between
predictors. This paper provides an introduction to modeling techniques
that attempt to improve upon these pitfalls. It studies the mathematical
machinery behind Ridge Regression, Subset Selection, Forward Stepwise
Selection, Lasso, and Adaptive Lasso including a look at the asymptotic
behavior of the Lasso estimates and their limiting distribution. It then
moves on to apply the theory in a variety of regression scenarios and
investigates the relative performances of each technique using the
software package R. Under the criteria of Mean Square Error it
concludes that Lasso is the most widely applicable and reliable modeling
technique of the ones studied.
ADVISOR: Kelly McConville
TITLE: AN INTRODUCTION TO SURREAL NUMBERS
AUTHOR: Gretchen Grimm
ABSTRACT: In this paper, we investigate John H. Conway's surreal numbers.
Surreal numbers are defined by two sets of numbers, which differentiates
them from the real numbers. Based on two axioms, we map out all of the
surreal numbers, finding infinities greater than any real number and
infinitesimal numbers smaller in absolute value than any real number. We
investigate addition and multiplication of surreal numbers, and show that
they form a totally ordered field, $S$, which contains the real numbers.
We also give an introduction to an application of surreal numbers in game
theory.
ADVISOR: Barry Balof
TITLE: PATHOLOGICAL: APPLICATIONS OF LEBESGUE
MEASURE TO THE CANTOR SET AND NON-INTEGRABLE DERIVATIVES
AUTHOR: Price Hardman
ABSTRACT: The Lebesgue integral is a hallmark of modern analysis, and the
theoretical foundation of the Lebesgue integral is measure theory. Here,
we develop the basics of Lebesgue measure theory on the real line, a
theory which is concerned with generalizing the notion of length to sets
that are not intervals. We then investigate several fascinating
"pathological" examples -- particularly counterintuitive and insightful
results -- and use the tools of measure theory to explore both their
bizarre features as well as their historical importance. These examples
include the Cantor set, an uncountable set of measure zero; the
Smith-Volterra-Cantor set, a set of positive measure yet one that contains
no intervals; and Volterra's function, a function with a bounded
derivative that is not Riemann integrable.
ADVISOR: Russ Gordon
TITLE: PÓLYA'S COUNTING THEORY
AUTHOR: Mollee Huisinga
ABSTRACT: P$\acute{\text{o}}$lya's Counting Theory is a spectacular tool
that allows us to count the number of distinct items given a certain
number of colors or other characteristics. We will count two objects as
'the same' if they can be permuted to produce the same configuration. We
can use Burnside's Lemma to enumerate the number of distinct objects.
However, sometimes we will also want to know more information about the
characteristics of these distinct objects. P$\acute{\text{o}}$lya's
Counting Theory is uniquely useful because it will act as a picture
function - actually producing a polynomial that demonstrates what the
different configurations are, and how many of each exist. As such, it has
numerous applications. Some that will be explored here include chemical
isomer enumeration, graph theory and music theory.
ADVISOR: Pat Keef
TITLE: A STUDY OF CONVEX FUNCTIONS WITH
APPLICATIONS
AUTHOR: Matthew Liedtke
ABSTRACT: This paper is a rigorous analysis of convex functions and their
applications to selected topics in real analysis and economics. In our
theoretical survey we explore the continuity and differentiability of
convex functions. In addition we state a simplified criteria of convexity
for continuous functions. Our topics from real analysis include
right-endpoint approximation for integrals and pointwise convergence. In
our discussion of economics, we will use the theory of consumer behavior
under uncertainty and Jensen's Inequality to show the relation between
convex functions and the modeling of risk loving behavior.
ADVISOR: Russ Gordon
TITLE: ON PROOFS WITHOUT WORDS
AUTHOR: Robin Miller
ABSTRACT: Proofs Without Words (PWWs) are pictures or diagrams which claim
to prove mathematical theorems without the aid of formal written logic.
PWWs have been included in MAA journals since 1975 and portray results
from a wide variety of mathematical fields, however there is not a
consensus regarding whether they can truly be called "proofs". In this
paper, we explore a collection of PWWs from geometry and calculus, as well
as relationships between the integers. We also present philosophical
arguments for and against the classification of PWWs as proofs.
ADVISOR: Albert Schueller
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