Physical Sciences and Mathematics Commons^™

Economics And Game Theory, Jeremiah Patrick Prenn

Mathematics Senior Showcase 2020

Game theory is one of the major fields of mathematics. Game theory is the study of how games, their players, and players’ strategies are defined, and how the games might play out. The outcomes of games are ultimately based on decisions, much like in the science of economics. Economics analyzes how scarce resources are to be allocated to suit unlimited needs. Every decision has an economic cost, and every decision has a utility value (utility being a quantitative measure of usefulness). Economics and game theory go hand in hand: Both analyze the effects of decisions and the rules imposed on …

The History And Application Of Benford's Law, Hunter Clark

Mathematics Senior Showcase 2020

My Poster is on the history and application of Benford’s law. This is a law that states that the leading digit of a set of numbers will be the number 1 approximately 30% of the time. This is a natural phenomenon and what I mean by that is that in order for this law to hold the numbers cannot be assigned. They must be random as in financial statements or logs. This law does not work on sets that are assigned such as time sheets and addresses. You will see in my poster that the original person to discover this …

Using The Chi-Square Test To Analyze Voter Behavior, Bailey Fadden

Mathematics Senior Showcase 2020

We explain the Chi-Square Test and how to use it to analyze voter behavior. Specifically we look at the behavior of U.S citizens and whether or not they voted in the 2016 U.S presidential election, and how this relates to income.

Morse-Code Encoded Eye Blinking As A Source Of Biometric Authentication Via Eeg, Ben Adams, Meghan Edgerton, Gabe Miles, Callum Young

Mathematics Senior Showcase 2020

Brain-Computer Interfaces (BCIs) have historically provided many uses in the medical field, including mobility for individuals with differing levels of paralysis. Present day research is focused around testing the efficacy of such devices on mental diseases such as Alzheimer's, Dementia, and Parkinson's. Leading companies that are spearheading the research of such devices, are looking at BCI's as a tool for solving many of the problems that these diseases produce, with the end goal of generalizing BCIs to appeal to the healthy layperson by providing an additional interface between them and the technological world. If such devices were present in society …

Internal Sorting Methods, Rebekah Marie Bitikofer

Mathematics Senior Showcase 2020

Internal sorting methods are possible when all of the items to be accessed fit in a computer's high-speed internal memory. There are quite a few (Knuth's third volume of The Art of Computer Programming covers 14 in total) but I will go over the four I found to be most versatile and useful. Each algorithm that I cover has a specific benefit that merits its' use in computer science. Some have faster run times (Heapsort), simpler code (Straight Insertion), run with a smaller memory space (Quicksort), or work well with large sets (Radix Sorting). Different sorting tasks lead users to …

Cybersecurity Of The Artificial Pancreas, D. J. Cooke, Andres Guzman, Robert Kinney, Christine Patterson, Josh Stone

Mathematics Senior Showcase 2020

We live in a world of cyber-enabled devices that enhance many aspects of life, including the treatment of diabetes. Type I Diabetes is a chronic autoimmune disorder characterized by destruction of pancreatic cells and subsequent deficiency of insulin - a crucial hormone in regulating blood glucose levels. The development of an Artificial Pancreas System is automating the maintenance of this disease by integrating wireless devices to continuously balance blood glucose levels without patient interaction. An integral part of this system is the Continuous Glucose Monitor (CGM) which wirelessly transmits blood glucose measurements every 5 minutes. CGMs and other Implantable Medical …

Construction Of A First Order Logic Theorem Prover, Luke Philip Tyler

Mathematics Senior Showcase 2020

There are many systems that have been researched in the past on automating the process of theorem proving in first-order logic. This research explores one of these systems, the tableau method. A point of interest within the tableau method is whether or not the method is sound and complete. This research was done in tandem with a computer implementation of the tableau method written in Haskell. The basic design of the implementation was to construct a fair rule for tableau expansion and expand the tableau until it was found to be closed, open, or infinite, thereby proving or disproving of …

A Fast And Accurate Algorithm For Spherical Harmonic Analysis On Healpix Grids With Applications To The Cosmic Microwave Background Radiation, Kathryn P. Drake, Grady B. Wright

Mathematics Faculty Publications and Presentations

The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) scheme is used extensively in astrophysics for data collection and analysis on the sphere. The scheme was originally designed for studying the Cosmic Microwave Background (CMB) radiation, which represents the first light to travel during the early stages of the universe's development and gives the strongest evidence for the Big Bang theory to date. Refined analysis of the CMB angular power spectrum can lead to revolutionary developments in understanding the nature of dark matter and dark energy. In this paper, we present a new method for performing spherical harmonic analysis for HEALPix data, …

Meager Sets, Games And Singular Cardinals, Liljana Babinkostova, Marion Scheepers

Mathematics Faculty Publications and Presentations

We show that a statement concerning the existence of winning strategies of limited memory in an infinite two-person topological game is equivalent to a weak version of the Singular Cardinals Hypothesis.

The Classification Of Countable Models Of Set Theory, John Clemens, Samuel Coskey, Samuel Dworetzky

Mathematics Faculty Publications and Presentations

We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of ZFC.

What Is The Derivative Of Music?, Thad B. Welch, Cameron H.G. Wright, Michael G. Morrow

Electrical and Computer Engineering Faculty Publications and Presentations

In our continuing effort to prove to students that Signals & Systems is not just another mathematics course taught by the ECE Department, we ask the question, “What is the Derivative of Music?”

The first-order difference (or first-difference) is an incredibly simple algorithm that very accurately approximates the numeric derivative operator, especially for oversampled signals. Its inverse also accurately approximates the numeric integration operator, but not without numeric difficulty.

Given a real-time demonstration using winDSK8, we can now show students that these mathematical operators provide powerful signal processing filtering tools for real-world signals.

During this ASEE session, we will include …

A Bound For The Waring Rank Of The Determinant Via Syzygies, Mats Boij, Zach Teitler

Mathematics Faculty Publications and Presentations

We show that the Waring rank of the 3 × 3 determinant, previously known to be between 14 and 18, is at least 15. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the symmetric cactus rank of the 3 × 3 permanent is at least 14.

Quantifying Cds Sortability Of Permutations By Strategic Pile Size, Marisa Gaetz, Bethany Flanagan, Marion Scheepers, Meghan Shanks

Mathematics Faculty Publications and Presentations

The special purpose sorting operation, context directed swap (CDS), is an example of the block interchange sorting operation studied in prior work on permutation sorting. CDS has been postulated to model certain molecular sorting events that occur in the genome maintenance program of some species of ciliates. We investigate the mathematical structure of permutations not sortable by the CDS sorting operation. In particular, we present substantial progress towards quantifying permutations with a given strategic pile size, which can be understood as a measure of CDS non-sortability. Our main results include formulas for the number of permutations in S_n with …

A Robust Hyperviscosity Formulation For Stable Rbf-Fd Discretizations Of Advection-Diffusion-Reaction Equations On Manifolds, Varun Shankar, Grady B. Wright, Akil Narayan

Mathematics Faculty Publications and Presentations

We present a new hyperviscosity formulation for stabilizing radial basis function-finite difference (RBF-FD) discretizations of advection-diffusion-reaction equations on manifolds �� ⊂ ℝ³ of codimension 1. Our technique involves automatic addition of artificial hyperviscosity to damp out spurious modes in the differentiation matrices corresponding to surface gradients, in the process overcoming a technical limitation of a recently developed Euclidean formulation. Like the Euclidean formulation, the manifold formulation relies on von Neumann stability analysis performed on auxiliary differential operators that mimic the spurious solution growth induced by RBF-FD differentiation matrices. We demonstrate high-order convergence rates on problems involving surface advection and …