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Â鶹ÊÓƵ University

Mathematics

Mathematics at Â鶹ÊÓƵ

Student Scholarship

UUSS - Spring 2025

Solving "Lights Out" and Variations with Linear Algebra (O)

Presenter: Joseph Carman
Faculty Project Advisor: Matt Lunsford

Using linear algebra, we model the game Lights Out. The game is played using a 5x5 square grid of lights in which some lights are turned on and some are off. The objective of the game is to turn off all the lights from a configuration, using the rules of the game. We construct a matrix that represents the rules of the game and a matrix equation whose solution yields a strategy for winning. The matrix equation models how to obtain a solution of the game (if it exists) from the scrambled configuration. We expand the model to grids ranging from a 1x1 to a 9x9 representations of the game. Then we explore the same process but for a variation of the game called "Knight's Move". The talk will end with results and observations for further research.

Symplectic Algorithms for Hamiltonian Dynamical Systems (O)

Presenter: Josiah Hayes
Faculty Project Advisor: Troy Riggs

In a Hamiltonian dynamical system, ‘area’ in the phase space of the system is conserved over time. Symplectic algorithms are numerical algorithms that solve for the time evolution of a Hamiltonian system and are explicitly designed to preserve areas in phase space. Due to their area-preserving properties, symplectic algorithms often demonstrate much higher accuracy over long time intervals than the equivalent non-symplectic numerical algorithms for Hamiltonian systems. However, symplectic algorithms are still susceptible to rounding error as they use finite-precision floating-point numbers in their computations. We describe a modified symplectic method due to Robert Skeel that takes rounding errors into account, and we mathematically verify the accuracy of this method for the case of the simple harmonic oscillator.

UUSS - Spring 2023

Lost in the Divergence Zone (O)

Presenter: Abigail Branson
Faculty Project Advisor: Bryan Dawson

In this talk, we will consider the level comparison test for series, based on infinitesimal calculus, which outlines the fate of the infinite series if the reciprocal of the Ω term in a series is either always in the convergence zone or always in the divergence zone. Will a series still be able to converge if it has an infinite number of terms in the divergence zone? After finding a few such series on divergence levels consecutively further from the convergence zone, I propose that the answer is yes — if there is a sufficiently large proportion of terms in the convergence zone to those in the divergence zone. In order to understand this, we will cover proofs that show the sum of the reciprocals of squares and fourth powers. We will also define convergence and divergence zones. During this we will consider a story to enhance comprehension of the topic.

UUSS - Spring 2022

That's Impossible! An Exploration of Three Famous Greek Constructions (O)

Presenter: Lisa Reed
Faculty Project Advisor: Matt Lunsford

The three famous ancient Greek construction problems involve using only a straightedge and compass to double the cube, trisect an angle, and square the circle. Attempts at these constructions captivated geometers for centuries. It was not until the nineteenth century that Wantzel proved the impossibility of doubling the cube and trisecting an angle and Lindemann completed the proof of the impossibility of squaring the circle. While the problems seem geometric in nature, proving the impossibility of these constructions requires abstract algebra. This talk will introduce the idea of constructible numbers, i.e., lengths that can be constructed using only a compass and straightedge and will introduce two important theorems concerning constructible numbers. In addition, a proof of the transcendence of pi will be outlined. Using these concepts from abstract algebra and fact that pi is a transcendental number, proofs of the three impossibilities will be presented.

UUSS - Spring 2020

Math and Marriage: How Graphs Can Model Polygamy

Student Scholarship: Josephine Carrier
Faculty Advisor: Matt Lunsford

With plenty of men and women in the world who are searching for the perfect spouse, is there a way to know if everyone can be paired with someone they like? Hall's Marriage Theorem is an important graph theory result that addresses this very question. A proof will be sketched of a generalization of Hall's Theorem, which creates the polygamous case by increasing the capacities of the nodes in the second set of a bipartite graph, analogous to the number of women a man can marry. Using the Ford-Fulkerson Maximum Flow Minimum Cut Theorem, one can show why the generalized Hall condition is not only necessary, but also sufficient for ensuring a matching in which all the women are matched with a man whom they like.

UUSS - Spring 2018

Determining Unique Local Minima in Complex Systems (O)

Presenter: Amy Murdaugh
Faculty Advisor: George Moss

When fitting models of complex systems, local minima of the cost function, which measures the fit of the model to the data, pose problems both for fitting algorithms and for model interpretation. The surface of the cost function is usually assumed to be rough, but this characteristic may be overstated. Local minima produced by a fitting algorithm may actually lie in the same basin of attraction, due to differences in the algorithm's stopping criteria and numerical resolution. To examine this, we construct geodesic paths between local minima produced by a fitting algorithm. We solve the geodesic equation numerically in Julia as a boundary value problem, using the Shooting and Multiple Shooting Methods. Convergence is improved by interpolating between parameter space and the model manifold using a Levenberg-Marquardt parameter, λ. Evaluating the cost function along the geodesic paths allows us to differentiate between distinct minima and find unique basins of attraction. This work has implications for characterizing models based on the number of distinct local minima present and technical results for algorithm development. It also may provide insight into the existence of low-dimensional effective theories in the complex system.

Senior Seminar Project Archives

2017

  • Lydia Black - "Using History of Mathematics in the Classroom"
  • Chandler Clark - "Game-Based Learning in Mathematics Education"
  • Andrew Edmiston - "The Algorithm Heard Around the World: Google and the Wonder of PageRank"
  • Graham Gardner - "Using KL Divergence to Create an R-Squared Measure of Goodness of Fit for the Exponential Family of Regression Models"
  • Michayla Kramer - "Congressional Apportionment, LaGrange's Identity, and the 2016 Election"
  • Amy Murdaugh - "Determining Unique Local Minima in Complex Systems"
  • Matthew Owen - "Geometry of nonlinear least squares with applications to sloppy models and optimization"
  • Renee Seavey - "Using Maximum Likelihood Estimators in the Analysis of Censored and Truncated Data Sets"

2016

  • Caleb Dahl - "Fermat's Last Theorem: n=3 Case"
  • John Gollihugh - "Developing Algebraic Reasoning through Problem Solving"
  • Najla Qaadan - "Nice Matrices"
  • Joshua Stucky - "Approximation of Irrational Numbers by Rationals"
  • Jessica Vinyard - "Students' Difficulties with Proof"

2009-2010

2008-2009

2007-2008

2006-2007

2004-2005

2003-2004

2001-2002

2000-2001

1999-2000

1998-1999

Off-campus Presentations
  • Amy Murdaugh won first place for her presentation at MAA Southeastern Section Meeting at Clemson University, 2018
  • Joshua Stucky won first place for his presentation at the Kappa Mu Epsilon regional convention in Nebraska, 2016
Publications
  • Joshua Stucky, "" The Pentagon, Spring 2017
  • Joshua Stucky, "" The Pentagon, Spring 2016
  • Kimberly Lukens, "" The Pentagon, Fall 2012
  • Emilie Huffman, "" The Pentagon, Fall 2012
  • David Clark, et. al., "Discrete Calculus on a Scaled Number Line," PanAmerican Mathematical Journal, 2016