Physical Sciences and Mathematics Commons^™

Using Circle Packings To Approximate Harmonic Measure Distribution Functions, Ella Wilson

Undergraduate Mathematics Day: Past Content

Harmonic measure distribution functions, h-functions, encode information about the geometry of domains in the plane. Specifically, given a domain and a basepoint in the domain, for a fixed radius, r, the value h(r) is the probability that a Brownian particle first exits the domain within distance r of the basepoint. There are many domains for which we can compute h-functions, such as the disk and the inside and outside of a wedge. However, exact computation is often difficult or impossible for more complicated domains, so we need methods to approximate these h-functions. In this paper, we develop two methods for …

Finding An Effective Shape Parameter Strategy To Obtain The Optimal Shape Parameter Of The Oscillatory Radial Basis Function Collocation In 3d, Quinnlan Aiken, Annika Murray, Ar Lamichhane

Undergraduate Mathematics Day: Past Content

Recent research into using the Method of Approximate Particular Solutions to numerically solve partial differential equations, has shown promising results. High levels of accuracy can be obtained when implementing this method, however the success of this collocation method is dependent on a shape parameter that is found in nearly all radial basis functions. If the shape parameter is not appropriately chosen, then it can provide an unacceptable result. Two shape parameter strategies are considered, a random variable shape parameter strategy and a leave-one-out cross validation strategy. The main objective of this work is to assess the viability of using these …

Efficient Conformal Binary Classification Under Nearest Neighbor, Maxwell Lovig

Undergraduate Mathematics Day: Past Content

There are many types of statistical inferences that can be used today: Frequentist, Bayesian, Fiducial, and others. However, Vovk introduced a new version of statistical inference known as Conformal Predictions. Conformal Predictions were designed to reduce the assumptions of standard prediction methods. Instead of assuming all observations are drawn independently and identically distributed, we instead assume exchangeability. Meaning, all N! possible orderings of our N observations are equally likely. This is more applicable to fields such as machine learning where assumptions may not be easily satisfied. In the case of binary classification, Vovk provided the nearest neighbors (NN) measure which …

Fixed Points Of Functions Below The Line Y = X, Grace Fryling, Harrison Rouse

Undergraduate Mathematics Day: Past Content

This paper concerns fixed points of functions whose graphs lie on or below the line y = x. Using the Monotone Convergence Theorem, we show that positive fixed points of such functions are “attracting on the right” so long as we include a couple of further assumptions about these functions near their fixed points. As an illustrative example, we confirm that this is the case for the function y = x sin x; the positive fixed points of this function “attract on the right” and “repel on the left.” Further, we generalize by showing that differentiability is in fact not …

Program: 2021 Undergraduate Mathematics Day, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Past Content

Schedule and general information about the event.

21st Annual Kenneth C. Schraut Memorial Lecture: "One Health: Connecting Humans, Animals and the Environment" (Suzanne Lenhart, University of Tennessee)

Plenary talk: "The Crossings of Art, History, and Mathematics" (Jennifer White, St. Vincent College)

Analysis Of Weights In Central Difference Formulas For Approximation Of The First Derivative, Preston R. Boorsma

Undergraduate Mathematics Day: Past Content

Manipulations of Taylor series expansions of increasing numbers of terms yield finite difference approximations of derivatives with increasing rates of convergence. In this paper, we consider central difference approximations of arbitrary order of accuracy. We derive explicit formulas for the weights of terms and explore their limits for increasing orders of accuracy.

Climbing The Branches Of The Graceful Tree Conjecture, Rachelle Bouchat, Patrick Cone

Undergraduate Mathematics Day: Past Content

This paper presents new ways to look at proving the Graceful Tree Conjecture, which was first posed by Kotzig, Ringel, and Rosa in 1967. In this paper, we will define an adjacency diagram for a graph, and we will use this diagram to show that several classes of trees are graceful.

Derivation Of The (Closed-Form) Particular Solution Of The Poisson’S Equation In 3d Using Oscillatory Radial Basis Function, Anup R. Lamichhane, Steven Manns

Undergraduate Mathematics Day: Past Content

Partial differential equations (PDEs) are useful for describing a wide variety of natural phenomena, but analytical solutions of these PDEs can often be difficult to obtain. As a result, many numerical approaches have been developed. Some of these numerical approaches are based on the particular solutions. Derivation of these particular solutions are challenging. This work is about how the Laplace operator can be written in a more convenient form when it is applied to radial basis functions and then use this form to derive the (closed-form) particular solution of the Poisson’s equation in 3D with the oscillatory radial function in …

Baseball: Defense Or No?, Jacob D. Stemmerich

Undergraduate Mathematics Day: Past Content

Defense wins championships, or so they say. How do baseball organizations find the right defenders to win games? FanGraphs has published a series of metrics that teams throughout Major Leage Baseball use to quantify players’ fielding prowess. Baseball analysts use Wins Above Replacement, WAR, to predict who should be the league most valuable player, MVP. This uses defensive metrics to quantify how many runs the player produces when the team wins. The paper will discuss the metrics that already exist, and the technology that has been developed to analyze these metrics and other measurements of a player’s defensive skills.

Magic Polygons And Their Properties, Victoria Jakicic, Rachelle Bouchat

Undergraduate Mathematics Day: Past Content

Magic squares are arrangements of natural numbers into square arrays, where the sum of each row, each column, and both diagonals is the same. In this paper, the concept of a magic square with 3 rows and 3 columns is generalized to define magic polygons. Furthermore, this paper will examine the existence of magic polygons, along with several other properties inherent to magic polygons.

Finite Sum Representations Of Elements In R And R2, Lewis T. Dominguez, Rachelle R. Bouchat

Undergraduate Mathematics Day: Past Content

In February 2017, a number theoretic problem was posed in Mathematics Magazine by Souvik Dey, a master’s student in India. The problem asked whether it was possible to represent a real number by a finite sum of elements in an open subset of the real numbers that contained one positive and one negative number. This paper not only provides a solutionto the original problem, but proves an analogous statement for elements of R2.

Alcoholism: A Mathematical Model With Media Awareness Campaigns, Erik H. Ander, Zeynep Teymuroglu

Undergraduate Mathematics Day: Past Content

In this paper, we study how media awareness campaigns influence the spread and persistence of drinking behavior in a community. Here, we present a compartmental population model with an additional differential equation to describe the dynamics of media awareness campaigns in combating problem drinking ([10], [12], [21]). Our model indicates a basic reproductive number, R0, where there exists an asymptotically stable drinking-free equilibrium if R0 < 1, and a unique endemic state, which appears to be stable when R0 > 1. We found that the following two components affect the basic reproductive number: the strength of peer influence of problem drinkers on susceptibles and the average overall time spent in the problem drinking environment. Furthermore, …

How One’S Risk Preferences Affect Their Investment Decisions, Kari Hayes, Anna Petrick

Undergraduate Mathematics Day: Past Content

The purpose of our project was to display how our personal risk preferences affect our investment decisions, if we invested on two assets: one risky asset (stock) and one risk-free asset (bank account). We considered the problem in both discrete and continuous case. In particular, the stock price follows a multinomial tree in the discrete case; and follows a Geometric Brownian motion in the continuous case. We then found the expected value of the stocks at varying times. By setting what we expect our bank account to be at those times equal to these expected values, we solved for the …

Mathematics With Only Rods, Jianqiao Mao, Zheng Yang

Undergraduate Mathematics Day: Past Content

We discuss in this expository paper the rod system used in ancient China based on the mathematical classic work of Sun Zi, with a focus on application to solving systems of linear equations. The mathematics involved is authentic and beautiful, and we believe it is also of interest from historical, cultural, and pedagogical perspectives.

Generalized Catalan Numbers And Objects: X; Y Equivalence Classes And Polyominoes, Emily S. Dautenhahn, Hannah E. Pieper

Undergraduate Mathematics Day: Past Content

No abstract provided.

Distance Functions And Attribute Weighting In A K-Nearest Neighbors Classifier, Alyssa C. Frazee, Matthew A. Hathcock, Samantha C. Bates Prins

Undergraduate Mathematics Day: Past Content

To assess environmental health of a stream, field, or other ecological object, characteristics of that object should be compared to a set of reference objects known to be healthy. Using streams as objects, we propose a k-nearest neighbors algorithm (Bates Prins and Smith, 2006) to find the appropriate set of reference streams to use as a comparison set for any given test stream. Previously, investigations of the k-nearest neighbors algorithm have utilized a variety of distance functions, the best of which has been the Interpolated Value Difference Metric (IVDM), proposed by Wilson and Martinez (1997). We propose two alternatives to …

The Marshall Differential Analyzer: A Visual Interpretation Of Mathematics, Bonita A. Lawrence, Richard P. Merritt, Devon A. Tivener

Undergraduate Mathematics Day: Past Content

Mechanical integration is an idea dating back to the late 1800's discovered by James Thomson, brother of Lord Kelvin. This idea was then expanded to build a calculating machine, called a differential analyzer, by Vannevar Bush (M.I.T) in 1929. The Marshall University Differential Analyzer Team has followed in the footsteps of Dr. Bush and a gentleman named Dr. Arthur Porter, who was the first to build a differential analyzer in England when he was a student of Dr. Douglas Hartree. He built his machine of Meccano components, the British version of Erector Set. In the early days of Arthur Porter's …

Uniqueness Of Solutions Implies Existence And Uniqueness Of Solutions Of Boundary Value Problems For Third Order Differential Equations, Veronica Respress

Undergraduate Mathematics Day: Past Content

In this paper we are concerned with uniqueness implies uniqueness and uniqueness implies existence questions for solutions of a class of boundary value problems for the third order ordinary differential equation (ODE). First we show uniqueness of solutions of a class of two-point problems implies the uniqueness of solutions of an associated class of three-point problems. Then we establish uniqueness of solutions of the class of two-point problems implies the existence of solutions of the class of two point problems and the associated class of three-point problems.

On The Construction Of Order Six Multilevel Hadamard Matrices, Keli Parker

Undergraduate Mathematics Day: Past Content

The existence of multilevel Hadamard matrices (MHMs) of all orders as well as a construction for full-rate circulant MHMs of all orders n 6= 4 is known. We use computer search methods to look for previously unknown full-rate circulant MHMs of orders 5, 6, and 7 and find solutions that potentially do not follow from the known construction. We then give an alternate construction to explain some order six MHMs.

Just Sit Back And Let The Girth Model Make Money For You, Ellham Negahdary

Undergraduate Mathematics Day: Past Content

The Girth Model will use the 10-period exponential moving average (EMA) and the 20-period EMA as their proxy for market trend. The Girth Model is a trend following model incorporating volatility, momentum and velocity. We will use girth as an early close indication to both long and short positions. Typically, early exit due to decreasing girth results in a more favorable profit position than that taken if the trader simply waited for an exit on the EMA cross to the downside.

2010 Vol. 4 Table Of Contents, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Past Content

No abstract provided.

2008 Vol. 3 Table Of Contents, University Of Dayton. Department Of Mathematics

Undergraduate Mathematics Day: Past Content

No abstract provided.

Breaking The Curve, Amanda Dahlman, Jesse Depinto, Kyle Kremer, Joe Plattenburg

Undergraduate Mathematics Day: Past Content

Imagine you are up to bat in a major league baseball game. Would you rather face a pitch with smaller curvature or smaller break? Would you know the difference? In this paper we will derive a model for the path of a pitch based on actual data from MLB.com's GameDay™ feature. Then, employing our model, we shall analyze the curvature and break of the pitch.

An Application Of Analytic Geometry To Designing Machine Parts--And Dresses, Karl Hess

Undergraduate Mathematics Day: Past Content

This paper presents the solution of an engineering problem that the author was asked to solve. The problem involves creating a flat pattern that could be cut from a piece of sheet metal and rolled to form a tube whose top edge would be contained in a plane that is not perpendicular to the central axis of the tube. A piece of this nature needs to be fabricated whenever two sheet metal tubes must be joined at any angle other than a straight angle.

Rearrangement On Conditionally Convergent Integrals In Analogy To Series, Edward J. Timko

Undergraduate Mathematics Day: Past Content

Rearrangements on conditionally convergent series suggests the existence of a similar process for integrals, here also referred to as rearrangement. In this document, a general theorem concerning rearrangement for conditionally convergent integrals is presented, as well as supporting theorems and a corollary to the general theorem.

A New Spin On Baseball, Allison Horney, Taylor Lowry, Eric Schwenker, Evan Wray

Undergraduate Mathematics Day: Past Content

All baseball fans know what a curveball is physically, but what is curveball mathematically, and how does it differ from a fastball? The secret of a pitch lies in its spin. In this paper we shall define the spin of a baseball and investigate the effects of its magnitude and direction by employing data collected by MLB.com Gameday from the league's best pitchers. We shall then employ this model to differentiate between the spin of a curveball and that of a fastball.

Acknowledgements: We would like to thank our teacher Scott Mitter for all that he has done for us. …

Pricing The Asian Call Option, Vinh Xuan Dang, Scott Glasgow, Harrison Potter, Stephen Taylor

Undergraduate Mathematics Day: Past Content

Background material on measure-theoretic probability theory and stochastic calculus is provided in order to clarify notation and inform the reader unfamiliar with these concepts. These fields are then employed in exploring two distinct but related approaches to fair option pricing: developing a partial differential equation whose solution, given specified boundary conditions, is the desired fair option price and evaluating a riskneutral conditional expectation whose value is the fair option price. Both approaches are illustrated by example before being applied to the Asian call option. .

Two results are obtained by applying the latter option pricing approach to the Asian call …

Determining The Statistical Significance Of Observed Frequencies Of Short Dna Motifs In A Genome, Philip E. Pfeiffer, Peter W. Hovey, Sudhindra R. Gadagkar

Undergraduate Mathematics Day: Past Content

Until recently over 90 percent of the DNA in the human genome was considered junk DNA, with no known function. However, this non-coding DNA is now known to harbor elements that perform important functions in gene regulation. In particular, there is currently much interest in the search for short DNA motifs collectively known as cis-regulatory elements. Most studies attempt to identify these elements by means of cross-species comparisons. We have approached the problem of finding cis-regulatory elements by searching for conserved DNA motifs within genomes. This requires searching for DNA motifs that are repeated in the genomes either more …

Adventures With Rubik's Ufo, Bill Higgins

Undergraduate Mathematics Day: Past Content

Enro Rubik invented the puzzle which is now known as Rubik’s Cube in the 1970’s. More than 100 million cubes have been sold worldwide. The mathematics behind solutions to the cube have been extensively studied by mathematicians and puzzle enthusiasts alike. In this article the mathematics behind a solution of the related puzzle known as Rubik’s UFO is analyzed.

Brownian Motion And Its Applications In The Stock Market, Angeliki Ermogenous

Undergraduate Mathematics Day: Past Content

Wilfrid Kendall notes on the complexity of the paths of Brownian motion: If you run Brownian motion in two dimensions for a positive amount of time, it will write your name. The oddness and complexity of Brownian motion reveal a really deep subject in the field of mathematics that cannot be fully understood and explained even until now. The purpose of this paper is to introduce the Brownian motion with its properties and to explain how it is applied in an everyday but totally unpredictable environment like the stock market.