Mathematics Commons^™

Matching The Stars To A Game Of Aggression, E. Chambers, Kristen Barnard

Electronic Proceedings of Undergraduate Mathematics Day

Region against region and army against army, we chart and battle among the stars! This paper studies the combinatorial game Aggression using graph-theoretic models. We focus on how winning strategies depend on the structure of the game graph, with particular attention to star graphs, matchings, and their disjoint union. The results presented here come from research conducted under the guidance of Dr. Kristen Barnard during the summer of 2025. We begin by introducing the game, describing the rule set used, and explaining how maps can be represented as graphs. From there, we examine how adjacency, placement order, and tie-breaking rules …

Strict Steiner Symmetrization For Polygons, Joseph Chai, Sun-Yung Alice Chang

Electronic Proceedings of Undergraduate Mathematics Day

In this note, we introduce a method called Strict Steiner Symmetrization (see Section 2.2) that is an altered form of Steiner Symmetrization that fixes the number of vertices. The context of this paper will be to prove the Isoperimetric Inequality in R2 via approximation by polygons. Specifically, to establish that among all n-gon domains in the plane, the regular m-gon, m ≥ n, uniquely minimizes the isoperimetric ratio and that the limit of this regular polygon (that is the circle) achieves equality in the isoperimetric inequality. This note is an extracted portion of my junior thesis (A Polygonal Proof of …

Predator-Prey Dynamics: Exploration And Logistic Extension Of The Lotka-Volterra System, Amanda Maylath, Samuel Brensinger

Electronic Proceedings of Undergraduate Mathematics Day

The Lotka-Volterra system is a classic model of predator–prey interactions, capturing the interdependent growth and decline of two species. Using a dynamical systems approach, we identify fixed points, linearize the system via the Jacobian, and analyze phase space trajectories to determine stability and long-term behavior. This analysis provides insight into predator-prey dynamics without requiring explicit solutions. We further extend the model by incorporating a carrying capacity for the prey, reflecting more realistic population dynamics.

Huffman Tree Uniqueness, Elijah Diller, Adam Volk

Electronic Proceedings of Undergraduate Mathematics Day

The Huffman algorithm is a standard algorithm used to generate an optimized binary encoding of a string. Using this algorithm, it may be possible to come up with more than one possible optimized binary encoding for a given string, so the following question arises: what strings have a unique optimized binary encoding? By unique encoding, we mean a unique vertex-labeled binary tree resulting from the Huffman algorithm. We do not worry about edge labels as every binary tree corresponding to a string with n distinct characters has 2n−1 ways of assigning the edge labels and determining the code. This investigation …

An Introduction To Category Theory, Joseph Kopp

Electronic Proceedings of Undergraduate Mathematics Day

Category theory is a relatively new field of mathematics that has grown much in popularity in recent years. It is a general theory of mathematical structure that lends itself to making overarching, yet deep, connections between many branches of mathematics. This power to make such wide-reaching statements is what has drawn many to study it. However, category theory has also been criticized for being "abstract nonsense," in that some believe the theory to be too abstract to carry meaning, much less be applied to the real world. The goal of this paper is to introduce the main ideas of category …

Derivation Of The Sliding Catenary Curve Via Calculus Of Variations, Ethan Shade

Electronic Proceedings of Undergraduate Mathematics Day

Using the calculus of variations this paper derives the general equation for the "sliding catenary curve" — a hanging chain with terminal links free to slide along two poles, one tilted and one vertical. By applying physical assumptions along with the Euler-Lagrange equation, the Beltrami identity, the Legendre-Clebsch condition, the transversality condition, Lagrange multipliers, and the isoperimetric constraint, we derive the general equation for the sliding catenary curve through a functional that measures the potential energy of the hanging chain. This general equation is then compared to a real-life construction of a sliding catenary curve. Additionally the paper explores a …

How Generative Ai Models Such As Chatgpt Can Be (Mis)Used In Spc Practice, Education, And Research? An Exploratory Study, Fadel M. Megahed, Ying-Ju (Tessa) Chen, Joshua A. Ferris, Sven Knoth, L. Allison Jones-Farmer

Mathematics Faculty Publications

Generative Artificial Intelligence (AI) models such as OpenAI's ChatGPT have the potential to revolutionize Statistical Process Control (SPC) practice, learning, and research. However, these tools are in the early stages of development and can be easily misused or misunderstood. In this paper, we give an overview of the development of Generative AI. Specifically, we explore ChatGPT's ability to provide code, explain basic concepts, and create knowledge related to SPC practice, learning, and research. By investigating responses to structured prompts, we highlight the benefits and limitations of the results. Our study indicates that the current version of ChatGPT performs well for …

Program: 2023 Undergraduate Mathematics Day, University Of Dayton

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

No abstract provided.

A Cryptographic Tour Of The Enigma, Frank Casabianca

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

Cryptography is the study of hidden information. This includes techniques to create and read secure secret messages (encryption/decryption) as well as methods to break the security of hidden messages, regardless of the intended recipient (cryptanalysis). This talk will explore basic cryptographic principles using manual cryptosystems, progress to the concepts behind - and the mechanical inner workings of - the well-known Enigma Cipher deployed in WWII, and discuss how cryptography surrounds us in the modern day.

Poster: 2023 Undergraduate Mathematics Day, University Of Dayton

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

No abstract provided.

23rd Annual Kenneth C. Schraut Memorial Lecture: Languages, Alphabets, And Group Theory, Gizem Karaali

Undergraduate Mathematics Day: Programs, Lectures, Promotional Materials

In 1960 Eugene Wigner wrote a now-famous article titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In 1993, the homophonic quotient groups for French and English (the quotient of the free group generated by the French (respectively English) alphabet determined by relations representing standard pronunciation rules) were explicitly characterized. In this talk, we will explore the mathematical and philosophical connections between these two works. Mathematically I will describe how my colleagues Herbert Gangl, Woohyung Lee (PO’15) and I, native speakers of three quite different languages, applied the methodology proposed in 1993 to our three language systems: German, Korean, …

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 …

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 …

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 …

Dpp: Deep Predictor For Price Movement From Candlestick Charts, Chih-Chieh Hung, Ying-Ju (Tessa) Chen

Mathematics Faculty Publications

Forecasting the stock market prices is complicated and challenging since the price movement is affected by many factors such as releasing market news about earnings and profits, international and domestic economic situation, political events, monetary policy, major abrupt affairs, etc. In this work, a novel framework: deep predictor for price movement (DPP) using candlestick charts in the stock historical data is proposed. This framework comprises three steps: 1. decomposing a given candlestick chart into sub-charts; 2. using CNN-autoencoder to acquire the best representation of sub-charts; 3. applying RNN to predict the price movements from a collection of sub-chart representations. An …

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)

Quasilinearization Applied To Boundary Value Problems At Resonance For Riemann-Liouville Fractional Differential Equations, Paul W. Eloe, Jaganmohan Jonnalagadda

Mathematics Faculty Publications

The quasilinearization method is applied to a boundary value problem at resonance for a Riemann-Liouville fractional differential equation. Under suitable hypotheses, the method of upper and lower solutions is employed to establish uniqueness of solutions. A shift method, coupled with the method of upper and lower solutions, is applied to establish existence of solutions. The quasilinearization algorithm is then applied to obtain sequences of lower and upper solutions that converge monotonically and quadratically to the unique solution of the boundary value problem at resonance.

A Data Analytic Framework For Physical Fatigue Management Using Wearable Sensors, Zahra Sedighi Maman, Ying-Ju Chen, Amir Baghdadi, Seamus Lombardo, Lora A. Cavuoto, Fadel M. Megahed

Mathematics Faculty Publications

The use of expert systems in optimizing and transforming human performance has been limited in practice due to the lack of understanding of how an individual's performance deteriorates with fatigue accumulation, which can vary based on both the worker and the workplace conditions. As a first step toward realizing the human-centered approach to artificial intelligence and expert systems, this paper lays the foundation for a data analytic approach to managing fatigue in physically-demanding workplaces. The proposed framework capitalizes on continuously collected human performance data from wearable sensor technologies, and is centered around four distinct phases of fatigue: (a) detection, where …

A Two-Stage Machine Learning Framework To Predict Heart Transplantation Survival Probabilities Over Time With A Monotonic Probability Constraint, Hamidreza Ahady Dolatsaraa, Ying-Ju (Tessa) Chen, Christy Evans, Ashish Gupta, Fadel M. Megahed

Mathematics Faculty Publications

The overarching goal of this paper is to develop a modeling framework that can be used to obtain personalized, data-driven and monotonically constrained probability curves. This research is motivated by the important problem of improving the predictions for organ transplantation outcomes, which can inform updates made to organ allocation protocols, post-transplantation care pathways, and clinical resource utilization. In pursuit of our overarching goal and motivating problem, we propose a novel two-stage machine learning-based framework for obtaining monotonic probabilities over time. The first stage uses the standard approach of using independent machine learning models to predict transplantation outcomes for each time-period …

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 …

Stochastic Technique For Solutions Of Non-Linear Fin Equation Arising In Thermal Equilibrium Model, Iftikhar Ahmad, Hina Qureshi, Muhammad Bilal, Muhammad Usman

Mathematics Faculty Publications

In this study, a stochastic numerical technique is used to investigate the numerical solution of heat transfer temperature distribution system using feed forward artificial neural networks. Mathematical model of fin equation is formulated with the help of artificial neural networks. The effect of the heat on a rectangular fin with thermal conductivity and temperature de-pendent internal heat generation is calculated through neural networks optimization with optimizers like active set technique, interior point technique, pattern search, genetic algorithm and a hybrid approach of pattern search - interior point technique, genetic algorithm - active set technique, genetic algorithm - interior point technique, …

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.

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.

Three Point Boundary Value Problems For Ordinary Differential Equations, Uniqueness Implies Existence, Paul W. Eloe, Johnny Henderson, Jeffrey T. Neugebauer

Mathematics Faculty Publications

We consider a family of three point n − 2, 1, 1 conjugate boundary value problems for nth order nonlinear ordinary differential equations and obtain conditions in terms of uniqueness of solutions imply existence of solutions. A standard hypothesis that has proved effective in uniqueness implies existence type results is to assume uniqueness of solutions of a large family of n−point boundary value problems. Here, we replace that standard hypothesis with one in which we assume uniqueness of solutions of large families of two and three point boundary value problems. We then close the paper with verifiable conditions on the …

Quasilinearization And Boundary Value Problems At Resonance, Kareem Alanazi, Meshal Alshammari, Paul W. Eloe

Mathematics Faculty Publications

A quasilinearization algorithm is developed for boundary value problems at resonance. To do so, a standard monotonicity condition is assumed to obtain the uniqueness of solutions for the boundary value problem at resonance. Then the method of upper and lower solutions and the shift method are applied to obtain the existence of solutions. A quasilinearization algorithm is developed and sequences of approximate solutions are constructed, which converge monotonically and quadratically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

Mittag–Leffler Stability Of Systems Of Fractional Nabla Difference Equations, Paul W. Eloe, Jaganmohan Jonnalagadda

Mathematics Faculty Publications

Mittag-Leffler stability of nonlinear fractional nabla difference systems is defined and the Lyapunov direct method is employed to provide sufficient conditions for Mittag-Leffler stability of, and in some cases the stability of, the zero solution of a system nonlinear fractional nabla difference equations. For this purpose, we obtain several properties of the exponential and one parameter Mittag-Leffler functions of fractional nabla calculus. Two examples are provided to illustrate the applicability of established results.

The Number Of Fixed Points Of And-Or Networks With Chain Topology, Lauren Geiser

Honors Theses

Boolean networks are sets of Boolean functions, which are functions that contain Boolean variables and the logical operators AND, OR, and NOT. In the simple case, the variables can be in one of two states—either 1 or 0, which can be interpreted in different ways such as ON or OFF, or TRUE or FALSE, depending on the application. Arranging model systems into Boolean functions, we can study steady states of these networks. This refers to the overall state of the dynamical system given an initial condition and another theoretical condition such as a subsequent point in time. Boolean networks have …

Topology Of Fractals, Amelia Pompilio

Honors Theses

No abstract provided.

Comparison Of Green's Functions For A Family Of Boundary Value Problems For Fractional Difference Equations, Paul W. Eloe, Catherine Kublik, Jeffrey T. Neugebauer

Mathematics Faculty Publications

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.