44th Annual WKU Mathematics Symposium Schedule and Abstracts
Schedule Overview
3:00pm (Friday)
Free registration is available throughout the whole event and begins at 3:00pm. It is located at Snell Hall First Floor. There will be free refreshments.
3:30 - 3:45pm
Dr. David Brown, Ogden College of Science and Engineering, Dean
Dr. Kanita DuCloux, Mathematics Department, Chair
3:45 - 4:45pm
Snell Hall 1108
5:00 - 6:25pm
Sessions will be in Snell Hall. For specific information see below.
6:30 - 7:00pm
Snell Hall First Floor near Da Vinci's.
7:00 - 8:25pm
Sessions will be in Snell Hall. For specific information see below.
8:30 - 9:15pm
Caroline Boone, Jerry Shaw, DJ Price
Snell Hall 1108
8:00 - 8:30am (Saturday)
Snell Hall First Floor
8:30 - 10:25am
Sessions will be in Snell Hall. For specific information see below.
10:45 - 11:45am
Snell Hall 1108
Detailed Schedule
Click on the Title and Presenter(s) to read the abstract of the presentation, and see a photo (when available).
Notes: (GA) Gatton Academy High School Student, (U) Undergraduate Student, (G) Graduate Student, (P) Postdoctoral Fellow, (F) Faculty, (I) Industry, * denotes presenter
Each 15-minute talk follows with a 5- to 10-minute Q&A. The next presenter should set up the presentation towards the end of the Q&A.
Friday, November 8
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Time |
Room |
Title and Presenter(s) |
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5:00-5:25pm |
SH 1101 |
Columnar and one-rowed Weyl symmetric functions by Robert G. Donnelly* (F) |
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SH 1102 |
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SH 1103 |
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SH 1108 |
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5:30-5:55pm |
SH 1101 |
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SH 1102 |
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SH 1103 |
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|
SH 1108 |
Trigonometric Functions in Discrete Fractional Calculus by Asa Ashley* (GA), Ferhan Atici (F) |
|
|
6:00-6:25pm |
SH 1101 |
Semiorders Induced by Uniform Random Points by Caroline Boone* (G) |
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SH 1102 |
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SH 1103 |
Numerical Solution of Differential Equations by Mark P. Robinson* (F) |
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SH 1108 |
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7:00-7:25pm |
SH 1101 |
Knight-free Capacities of Rectangular Chessboards by Doug Chatham* (F) |
|
SH 1102 |
Learned Dependency in Students: Origins and Solutions? by Ed Thome* (F) |
|
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SH 1103 |
Structural Types in Typescript with Higher Order Functions by DJ Price* (G) |
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SH 1108 |
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|
7:30-7:55pm |
SH 1101 |
|
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SH 1102 |
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SH 1103 |
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SH 1108 |
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8:00-8:25pm |
SH 1101 |
Containment Lattices for Interval Posets of Permutations by Asa Ashley* (GA), Thomas Richmond (F) |
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SH 1102 |
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SH 1103 |
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SH 1108 |
Saturday, November 9
|
Time |
Room |
Title and Presenter(s) |
|
8:30-8:55am |
SH 1101 |
|
|
SH 1102 |
A survey of Linear programming problems by Olusegun Adebayo* (G) |
|
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SH 1103 |
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|
9:00-9:25am |
SH 1101 |
|
|
SH 1102 |
Review on Integer Linear Programming and Some Applications by Andrew Adegoju Oluwashijibomi* (G) |
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SH 1103 |
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9:30-9:55am |
SH 1101 |
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SH 1102 |
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SH 1103 |
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10:00-10:25am |
SH 1101 |
|
|
SH 1102 |
Fractional Dimensions of Compactly Supported Refinable Functions by Jay Coughlon* (U) |
|
|
SH 1103 |
Abstracts for Presentations
Title: Columnar and one-rowed Weyl symmetric functions
Presenters: Robert G. Donnelly* (F)
Abstract: Two families of distinguished symmetric polynomials are the elementary symmetric polynomials
and the complete homogeneous symmetric polynomials.
Elementary symmetric polynomials are sums of monomials of some fixed degree whose
variable factors are distinct. Complete homogeneous symmetric polynomials are sums
of monomials of some fixed degree whose variable factors are not required to be distinct.
Elementary (respectively, complete homogeneous) symmetric polynomials arise in representation
theory as invariants of the exterior (respectively, symmetric) powers of the defining
representations of the special linear Lie algebras. The latter Lie algebras are identified
as type A in the famous classification of the finite-dimensional simple Lie algebras
over C and have the finite symmetric groups as their associated Weyl groups. In this
talk, what we call the columnar (respectively, one-rowed) Weyl symmetric functions
are analogs of elementary (respectively, complete homogeneous) symmetric polynomials
for the odd orthogonal (type B), symplectic (type C), and even orthogonal (type D)
Lie algebras; in the type B and C cases, the associated Weyl group is a hyperoctahedral
group of symmetries of a hyperoctahedron or of the related hypercube. We discuss some
extant and potential combinatorial models for columnar and one-rowed Weyl symmetric
functions in types A, B, C, and D.
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Title: Distributive lattice models for one-rowed representations of the classical Lie algebras
Presenters: Goran Omerdic* (G)
Abstract: Lie algebras are canonically used to describe the symmetries of continuous functions. Such algebras are rich in enumerative properties. We consider the classical Lie algebras – comprised of the ‘special linear’, ‘symplectic’, and ‘orthogonal’ Lie algebras through a lens of algebraic combinatorics, using colored modular and distributive lattice models to describe said properties. Research related to special linear, symplectic, and odd orthogonal Lie algebras has been fruitful, yielding families of lattice models and related coefficients to generate direct graphs. We use these methods to explore the properties of even orthogonal lie algebras, their lattice and one-rowed tableau representations.
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Title: Semiorders Induced by Uniform Random Points
Presenters: Caroline Boone* (G)
Abstract: In this talk we will discuss semiorders induced by uniform random points. We will define an algorithm for the calculation of the probability of generating a specific semiorder on a given uniform interval. More specifically, we will explore the equivalance classes created by the probabilities of generating these semiorders as we increase the length of the uniform interval. The equivalence classes can largely be determined by a lexicographic decomposition of the semiorder over a chain. Continuing research on the probabilities of the semiorders which cannot be decomposed over a chain, referred to as bricks, will also be addressed in this talk.
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Title: Knight-free Capacities of Rectangular Chessboards
Presenters: Doug Chatham* (F)
Abstract: How many black and white chess pieces -- excluding knights -- can we place on the squares of an mxn chess board so that none of those pieces attack any other piece? We answer this question for all rectangular boards. If n>1, the answer is⌈2m/3⌉n plus either 0, 1, or 2. We also present partial answers for the question with knights included.
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Title: The Impact of the Fibonacci Sequence and Golden Ratio on the Mathematical and Aesthetic Perception of Universal Beauty
Presenters: Andrew Adegoju Oluwashijibomi (G)
Abstract: This talk explores the intrinsic connection between the Fibonacci Sequence and the Golden Ratio in defining beauty in nature, art, and modern applications. Mathematical properties such as symmetry, proportionality, and harmonization are fundamental to the perception of beauty, and these are exemplified by the Golden Ratio. The paper establishes the step-by-step derivation of the Golden Ratio and its mathematical relationship with the Fibonacci Sequence. Historical applications, such as in the Mona Lisa and the Pyramids of Giza, demonstrate the longstanding influence of these concepts on art and architecture. In contemporary times, the Golden Ratio continues to shape design in fields such as logo creation, magazine layout, and even medical fields like facial surgery and neuro-analysis. The study highlights the enduring relevance of these mathematical principles and calls for educators to bridge the gap between theoretical mathematics and practical applications in everyday life to foster greater student engagement.
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Title: Containment Lattices for Interval Posets of Permutations
Presenters: Asa Ashley*(GA), Thomas Richmond (F)
Abstract: Interval posets of permutations register the intervals appearing in permutations according to set inclusion. The interval posets of all permutations for an element size can be ordered by a containment lattice defined by a group action from certain functors, with the quotients being the colimits of these functors. We study these lattices, focusing on their informational properties, structural representations in higher dimensional space, and symmetries. As an enumerative result from a categorical construction, lower bounds are found for the number of minimal interval posets for a given permutation size by defining minimal interval posets in terms of lower-dimensional minimals.
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Title: Universal/ubiquitous hyper-sequences in Rd
Presenters: Attila Pór* (F)
Abstract: A k-uniform hyper-sequence of of length 
. We say that a hyper-sequence is universal if any two set of d+1 points of the same
type have the same orientation. Here two point sets have the same type if there is
an order preserving bijection on their indices. We describe all one dimensional universal
hyper-sequences.
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Title: Relativistic Fermion Dynamics in Fuzzy Space Time: The Fuzzy Dirac Equation
Presenters: Asa Ashley* (GA), Valentino Simpao (F)
Abstract: We present the derivation of a wave equation for the relativistic description of massive spin 1/2 particles in finite-noncommutative spacetime. We first construct a complex projective space CP^n and define a Kähler structure, extending the assumptions of Affine Quantum Mechanics to a complex coordinate space and establish the isomorphic embedding of CP^n to the relativistic spinning top. A worldsheet effective non-abelian action in the 11-D configuration space is described then made conformally invariant by replacing particle mass with scalar pseudo-Weyl curvature. As a beautiful result, the Lagrangians are shown to be sheet fibrations between the complex and real space. Characterizing separable coordinates in terms of the first homology group, we obtain the bundles of extremals belonging to the family of equidistant hypersurfaces, yielding a parity-invariant semilinear eigenvalue equation. We then formalize the transition to Hydrodynamical Quantum Mechanics then modern Quantum Field Theory via the facile transformation of complex Poisson structures to Dirac brackets and quantization of antiparticles via a Foldy-Wouthuysen transformation of the Newton-Wigner representation of the inhomogeneous coordinates, permitting the discussion of CPT symmetry, respectively. Finally, we show complete physical equivalence with the ordinary Dirac equation at sufficiently large scales beyond the Planck length. This theory is compared with those of quantum loop gravity and the noncommutative Yang Mills framework of M-theory in respect to gauge invariances. Holistically, the presented theory leverages insights from contemporary superstring theories to provide a direct analytical framework for the dynamics of fermions at minimal scales, a unique perspective in the ongoing pursuit of unification.
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Title: Uniform Exponential Stability in Model Reduction of a Magnetizable Piezoelectric Beam with Non-Collocated Observers
Presenters: Uthman Adeniran Rasaq* (G), Ahmet Ozer (F)
Abstract: This study, part of my thesis work, delves into the mathematical modeling and stability analysis of a magnetizable piezoelectric beam with free boundary conditions, governed by partial di!erential equations (PDEs) that encapsulate the dynamics of longitudinal vibrations and electrode charge accumulation. Advancing beyond traditional collocated control methodologies, I propose a non-collocated control and observer framework that facilitates state recovery and boundary output feedback control at one end of the beam, utilizing estimates derived from an observer positioned at the opposite end. Recent studies have established exponentially stable solutions within this non-collocated model, prompting a focused investigation into model reduction strategies that preserve this stability. In response, I develop an order-reduction-based Finite Difference method tailored specifically to this beam model. This approach incorporates midpoints in a uniform discretization mesh and leverages average operators to construct a discrete Lyapunov function. Through comprehensive analysis, I establish that both the observer and observer error dynamics maintain uniformly exponentially stable solutions as the discretization parameter tends toward zero. Crucially, the stability decay rate is shown to be independent of the discretization parameter, aligning with the inherent rhythmic properties of the original PDE syste .
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Title: Uniformly Exponentially Stable Finite-Difference Model Reduction of a Heat and Piezoelectric Beam Interactions with Static or Hybrid Feedback Controllers
Presenters: Ibrahim Khalilullah* (G), Ahmet Ozer (F)
Abstract: This research presents a model reduction technique for a system of partial di!erential equations (PDEs) modeling heat transfer, mechanical vibrations, and charge distributions in a magnetizable piezoelectric beam within a transmission line framework. Leveraging recent results in designing static and dynamic feedback controllers, we reduce the system to a lower-dimensional model while preserving uniform exponential stability. The proposed method incorporates thermal effects and mitigates high-frequency eigenmode issues using Finite Differences with average operators. Importantly, the decay rate of the discrete model is identical to that of the original PDE model and is independent of the discretization parameters h1 and h2, ensuring uniform exponential stability. Stability is achieved through a discrete Lyapunov function, which provides an explicit decay rate for selecting feedback gains and ensures robust performance across different scales. This framework e!ectively applies Finite Difference techniques to complex systems with coupled thermal and electromagnetic dynamics, ensuring stability without requiring direct spectral methods or observability results.
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Title: Exploring the concept of Null space in the Hallways task: An Inquiry-Oriented Linear Algebra task sequence
Presenters: Zac Bettersworth* (F)
Abstract: As part of an NSF-funded research project, our research group has developed four instructional task sequences related to important linear algebra concepts. In this talk, I will demonstrate the mathematics underlying the design of the hallways task which is an instructional task sequence designed to support student understanding of null space in an experientially real setting. This presentation will highlight mathematical ideas that are approachable to any undergraduate STEM major who has taken Linear Algebra.
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Title: The Disconnect Between Undergraduate Standard Mathematics and Modern Engineering Mathematics
Presenters: Cory Wang* (G)
Abstract: As technology increasingly becomes a focus in modern society, there is a growing demand for engineering graduates around the world. Recently, there have been concerns about the amount of engineering graduates from universities due to the high drop-out rates of engineering programs (Flegg et al., 2012, Faulkner et al., 2019). Various studies have attributed this to undergraduate engineering students’ struggles with mathematics in their engineering coursework, and the transfer of mathematical skills and knowledge from the standard mathematics curriculum (Calculus I-III, Linear Algebra, Differential Equations) to engineering contexts. The goal of this literature review is to study the disconnect between the standard mathematics curriculum and modern engineering mathematics. Due to the breadth of research in the topic, this review will focus on the case of education of integral transforms (such as the Laplace and Fourier Transforms), as this topic provides a bridge between the standard mathematics curriculum (Differential Equations) and modern engineering mathematics (Fourier analysis). Implications and potential future studies informed by this review will also be discussed.
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Title: Reinventing the Salty Tank Through Guided Inquiry
Presenters: Nick Fortune* (F), Lucy Oremland (F), Justin Dunmyre (F)
Abstract: In this presentation, we discuss mathematical modeling opportunities that can be included in an introductory Differential Equations course. In particular, we focus on the development of and extensions to the single salty tank model. Typically, salty tank models are included in course materials with matter-of-fact explanations. These explanations miss the opportunity for students to develop rate of change equations for themselves. We highlight an open-ended single salty tank task that provides the foundation for more complicated salty tank models that we also detail.
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Title: Learned Dependency in Students: Origins and Solutions?
Presenters: Ed Thome* (F)
Abstract:
Many of us teaching Freshman and Sophomore courses are noticing an increasing number of this generation who have difficulty investing in much at all. This is also true of teachers in K-12, and of many employers as well. We have always had students who we called "unmotivated", but unmotivated assumed that the student had the capability to succeed (and knew that they had it) but chose not to invest in our courses. I contend that many of these students are not aware that they have the capability to succeed and are afraid that they do not. Also, they are unaware of the tools necessary to access this capability. Why is this, and how can our universities help the students learn how to invest of themselves? The question mark which closes the title is because there are multiple contributors to this problem and I don't know that my suggestions are really solutions. However, they are a starting point from which we can learn more about the students and how to help them succeed on their own.
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Title: Hiding in Plain Sight - Uncovering Statistical Properties of Feasible Traveling Salesman Problem Solutions
Presenters: Jerry Shaw*
Abstract: The Traveling Salesman Problem (TSP) is the canonical combinatorial optimization problem famous throughout the engineering and computer science literature. Given a set of locations on a plane we seek to identify the “shortest” Hamiltonian cycle through the locations. That is, you start and end at the same location visiting each location one time. The problem has been shown to be NP Hard. Most of the work on the problem has sought to identify the shortest path through a given graph. We show that “lengths” of the cycles (in the sense of Lebesgue measure) tend to a Normal distribution as the number of vertices grows large as a consequence of the Central Limit Theorem. This allows a user to estimate the likelihood that a lower objective function value exists without having to solve a problem instance to optimality. There is potentially high economic value to knowledge since it allows practitioners to determine acceptable stopping criteria for immensely large problems. This work links Graph Theory, Combinatorics, Optimization, and Probability Theory.
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Title: The Use of Binomial Index of Dispersion and Centralized Cosine Measure in Related Records Searching
Presenters: Ngoc Nguyen* (F)
Abstract: In related records searching within a bibliographic database, an individual article is linked to other articles on the basis of the number of references cited they share in common, the theory being that two articles that cite many of the same sources are likely to be highly similar in subject content. Results of the search are usually displayed in the order of each item’s actual number of commonly-shared references. In this project, we suggest two indexes to measure the association between articles: the Binomial Index of Dispersion and the Centralized Cosine Measure. Both of them are equipped with statistical test for the significance of the association.
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Title: A survey of Linear programming problems
Presenters: Olusegun Adebayo* (G)
Abstract: Linear programming, a subset of a mathematical programming and an operations research technique is widely used in finding solutions to complex managerial decision problems; it consists of objective function which represents profit or loss and a set of constraints which are linear equations or inequalities. This study was carried out to survey and arrive at the optimal solution of linear programming models arising from production and transportation problems.
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Title: Review on Integer Linear Programming and Some Applications
Presenters: Andrew Adegoju Oluwashijibomi* (G)
Abstract: In linear programming, common solution methods, such as the Simplex and Dual Simplex Methods, are highly e!ective across a range of optimization problems. However, these methods fall short when applied to integer programming problems, where optimal solutions must meet strict integer constraints. Such cases arise frequently in operations research and other fields requiring feasible, integer-constrained solutions. This study investigates the limitations of the Simplex and Dual Simplex Methods in handling integer constraints and explores the effcacy of alternative techniques: the Branch-and-Bound and Cutting Plane algorithms. These advanced algorithms are specifically tailored to address integer requirements, iteratively refining feasible solutions until integer-optimal outcomes are achieved. An initial feasible and optimal solution is obtained through Simplex or Dual Simplex, followed by Branch-and-Bound or Cutting Plane algorithms if integer conditions are unmet. This study provides insights into applying these methodologies to meet integer constraints, demonstrating their importance for optimization in complex, integer-dependent applications. This study thereby applies Branch-and-Bound algorithm to Urban and Regional Planning and Cutting Plane algorithm to Production Management.
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Title: An Introduction to Wavelets
Presenters: David Roach* (F)
Abstract: A wavelet is a function that is perpendicular to all dilated and shifted versions of itself (compressed and stretched by powers of two along with all of the integer shifts). The first wavelet was discovered in 1910 by Haar and is the "step" function defined as one on [0,1) and zero otherwise. No other wavelets were found until the 1980's with the seminal work of Daubechies who characterized a construction method to find the "nicest" wavelets on a given support length. In this talk, we will explore this class of functions and demonstrate how they are used to solve the image compression problem.
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Title: Fractional Dimensions of Compactly Supported Refinable Functions
Presenters: Jay Coughlon* (U)
Abstract: Herein we explore refinable functions with compact support, including the “step” and “hat” functions, as well as functions with fractional dimensions. A function is refinable whenever it can be written as a linear combination of dilated versions of itself. In this talk, we consider refinable functions with compact support. The step function, which is nonzero on the interval [0,1), and the hat function, which can be defined as nonzero on the interval [0,2), are classic examples of refinable functions with compact support. Although these functions are the best-known refinable functions within these compact supports, refinable functions with fractional dimensions also exist. The fractional (or “fractal”) dimension of a function can be determined either theoretically or experimentally. We will explore fractal dimensions for the class of refinable functions that includes the hat function.
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Title: Providing Stability to Quantum Computers: Möbius Transformations, Algebras, and Topological Invariants
Presenters: Emma Bunch* (U)
Abstract: In my honors thesis, guided under Dr. Molly Dunkum and Dr. Tilak Bhattacharya, I am exploring the relationship among linear fractional transformations, topological invariants, various algebras, and stability in quantum computers. Quantum computers are composed of quantum bits (qubits) that can be in a superposition of states (rather than just 0 or 1 like classical bits), which makes quantum computers very powerful for certain types of calculations. However, this superposition is very unstable, and providing stability to the quantum system can enable us to accelerate our work with quantum computers and its future applications, including drug discovery, biological modeling, cryptography, and many more. Therefore, I am exploring the potential of linear fractional transformations, specifically the Möbius transformation, acting as a topological invariant of-sorts in a quantum system, exploring the algebras behind it. I will also be utilizing IBM's Qiskit language to experimentally test this hypothesis.
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Title: Qualitative Behavior of Solutions to Differential Equations in Abstract Spaces
Presenters: Austin Anderson* (GA), Lan Nguyen (F)
Abstract: The qualitative behavior of solutions to differential equations mainly addresses the various questions arising from the study of the long run behavior of solutions. The contents of this study are related to three of the major problems of the qualitative theory developed by Henri Poincare and Aleksandr Lyapunov in the late 19th century. These problems are the stability, boundness, and the periodicity of a differential equation solution. The definitions are as follows: stability means how a system responds to a slight disturbance, boundness implies there are restrictions in a system, and periodicity means there is an interval or pattern a solution follows. Learning the qualitative behavior of such solutions is an important part of the theory of differential equations. As applications, we model the population of a bacteria type in a patient's body as a solution to a particular differential equation subject to a new form of treatment represented by a function. We look for conditions such that the number of bacteria in this patient is curbed. Finding such conditions will prevent the outbreak of an illness in future patients.
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Title: Numerical Solution of Differential Equations
Presenters: Mark P. Robinson* (F)
Abstract: Several methods for the numerical solution of differential equations are examined. This presentation will touch on topics in the numerical solution of initial value problems (including for systems of ordinary differential equations and differential-algebraic systems) and numerical methods for the solution of partial differential equations.
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Title: Structural Types in Typescript with Higher Order Functions
Presenters: DJ Price* (G)
Abstract: Typescript (TS) is a programming language that extends the functionality of Javascript (JS) by adding a structural typing system on top of it. Any valid TS program can be transpiled into a valid JS program. The structural typing system of Typescript is very reminiscent of set theory and allows a programmer to write extremely tight, auto-verified code when used appropriately. This talk will provide an overview of the TS type system, show how it relates to set theory, and give some interesting, industry relevant examples.
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Title: One-Way High-Dimensional ANOVA
Presenters: Lukun Zheng* (F)
Abstract: ANOVA is one of the most important tools in comparing the treatment means among different groups in repeated measurements. The classical F test is routinely used to test if the treatment means are the same across different groups. However, it is inefficient when the number of groups or dimension gets large. We propose a smoothing truncation test to deal with this problem. It is shown theoretically and empirically that the proposed test works regardless of the dimension. The limiting null and alternative distributions of our test statistic are established for fixed and diverging number of treatments. Simulations demonstrate superior performance of the proposed test over the F test in different settings.
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Title: Military Expenditures and Economic Growth: An Analysis Through Quantile Regression
Presenters: Kole Ingram* (U), Lukun Zheng (F)
Abstract: The relationship between military expenditures and economic growth in developing countries has been studied continuously with varying results. We analyze the relationship through a fixed panel quantile regression model examining various factors to model this economic growth, including GDP (total and per capita), literacy rate, inflation rate, population growth, and foreign direct investment inflow. This analysis will be done across the period from 1990-2023. The finding come as a contribution to the somewhat limited usage of quantile regression in examining the relationship between military spending and economic growth. The theoretical benefits of this model come from it’s benefits in analyzing heteroskedastic and skewed data like that of military expenditures. The ultimate goal of the project will be to understand what this relation is for developing countries as a whole, but also how countries will differ and what factors will influence those different effects of military expenditures on their respective economies.
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Title: Prey-Predator Model of One Plant species with two Inter-competing herbivore species
Presenters: Sukurat Kofoworola* (G)
Abstract: In this project, a Prey-Predator model describing the interaction between a plant species and two inter-competing herbivore species was formulated and analyzed. Five different equilibrium points of the model, namely: trivial, axial, boundary points and interior points were obtained, and their stability conditions were determined through the analysis of Jacobian of the eigenvalues of the matrix.
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Title: Master Theorems in Integration
Presenters: Dominic Lanphier* (F)
Abstract: A master theorem in integration is a general formula used to solve a variety of cases of definite integrals. We introduce some famous master theorems, and in particular Ramanujan's master theorem and Glasser's master theorem. We show how these extremely useful theorems give several interesting integral formulas. We then briefly survey some more obscure master theorems. We introduce a new master theorem and give some interesting new (and old) integral formulas as a result.
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Title: Measuring and Modeling the Discharge of a Stream in Great Onyx Cave, Mammoth Cave National Park
Presenters: Ava Lich* (U)
Abstract: Atmospheric carbon dioxide removal can be measured through limestone dissolution in caves. This project aims to measure and model the discharge of a subterranean stream in Mammoth Cave National Park through utilization of a barrel weir. By comparing the modeled data with the measured data, refinements to the overall process of calculating discharge will be made.
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Title: Naive RAG - Augmenting LLMs with Data
Presenters: DJ Price* (G)
Abstract: After the release of ChatGPT in Nov 2022, Large Language Models (LLMs) have started to become a ubiquitously known machine learning tool. LLMs possess strong capabilities for giving coherent and human understandable textual output when prompted with textual input. However, LLMs tend to "drift" or "hallucinate" in their answers. They also don't have access to many forms of private information, resulting in limited capabilities when it comes to providing textual output that requires references to that private information. Retrieval Augmented Generation (RAG) helps address this. This talk will provide an overview of Naive RAG, the most basic form.
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Presenters: Claus Ernst* (F)
Abstract: The pretzel knot family is simple well-known family in knot theory. After an introduction to this family using lots of pictures, we present some recent result about pretzel knots.
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Title: Trigonometric Functions in Discrete Fractional Calculus
Presenters: Asa Ashley* (GA), Ferhan Atici (F)
Abstract: In this talk, our goal is to introduce and study the linear fractional h-difference equation with order between 1 and 2 in hN space. Our approach is grounded in the formalization of Picard iteration via the construction of fixed points and the definition of quasi-nonexpansive operators on hN. This method also addresses the complexities arising from the variable metric of the space in the limit to the reals via the employment of weak contractions. Through this framework, we derive a series expansion of the Mittag-Leffler type. This expansion converges to functions analogous to the generalized R-function in continuous fractional calculus and the discrete Sine function. Our findings attempt to bridge the gap between continuous and discrete fractional calculus, offering new insights into the behavior of fractional difference equations with possible physical applications such as in chaos theory and electrical modeling.
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Title: Analytical and Numerical Methods for Solving Ordinary Differential Equations with Fractional Caputo Derivative
Presenters: Emmanuel Tomiwa Siyanbola* (G)
Abstract: This study explores analytical and numerical methods for solving ordinary differential
equations (ODEs) with fractional Caputo derivatives. Fractional calculus,
extending classical calculus to non-integer orders, is increasingly applied in science
and engineering due to its capacity to model memory and inheritance properties of
various processes. The main focus of this study is to develop and analyse numerical
methods that efficiently and accurately solve these equations. The study provides
a foundation in fractional calculus, covering essential concepts and special functions,
such as the Gamma and Mittag-Leffler functions, which support the numerical methods
used. Various numerical techniques are implemented and compared, including adaptations
of the Euler method, the Adams-Bashforth-Moulton method, and a modified version of
the Adams-Bashforth-Moulton method. These methods are tailored to handle fractional
derivatives in the Caputo sense, making them particularly useful for real-world initial
value problems. Additionally, numerical approaches for sequential ordinary differential
equations of fractional order are examined, where Caputo derivatives are applied
sequentially. The stability and convergence of these methods are analysed through
examples and case studies, with results validated to demonstrate their effectiveness
in solving fractional ODEs. Finally, the findings show the possibilities for advancing
numerical solutions for fractional di!erential equations.
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Title: Discrete Forest Growth Model
Presenters: Jakob Barker* (U)
Abstract: Quercus alba L., more commonly known as the Eastern White Oak, is one of the most common species of trees found across Kentucky. It grows with resilience, only surrendering to the driest of soils. With extended and strong root systems, the White Oak is an adaptable species that provides structural stability to the soil surrounding each tree. In an age of climate instability, the strength of the White Oak is vital to the survival and prosperity of our environment. Aside from its structural integrity, it also produces some of the highest-quality lumber in the country. Most commonly, the White Oak’s lumber is found in wooden barrels, a staple of Kentucky’s barrel Bourbon Industry. This study aims to utilize the Matrix Model of Forest Dynamics, as documented in the literature, to model the growth of White Oak in Southern Kentucky. By applying the principles of discrete calculus, we will evaluate and predict growth patterns. Additionally, we will use this expanded model to analyze the impacts of various harvest and artificial reproduction rates on the White Oak.
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Title: Various Student Research Problems and the Results of a Mixed-Effects Model on Wound Healing
Presenters: Richard Schugart* (F)
Abstract: This talk will have two parts. One part will be to outline research problems that students might be able to work on. The other will be to highlight some previous work on a structurally identifiable mixed-effects model with data from patients with diabetic foot ulcers.
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Title: Analyzing A Structurally Identifiable Model for Tuberculosis Transmission Using Data from Bangladesh
Presenters: Anika Tahsin* (G)
Abstract: We developed a structurally identifiable differential-equation model for tuberculosis (TB) transmission using data from Bangladesh. We will show that the model is structurally identifiable and derive a formula for the basic reproduction number. In the future, we will curve fit the model to data and analyze the practical identifiability of the model.
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