Mathematics Commons^™

Learning Motion Primitive Selection And Environment Abstraction, Edison Alberto Martinez Samaniego, Natalie Alexander, Kaelyn Weddle

Discovery Day - Daytona Beach

Learning Motion Primitive Selection and Environment Abstraction Advanced Air Mobility (AAM) is emerging as a transformative solution for short and medium range transportation; however, it introduces an operational model that differs significantly from conventional aviation. AAM vehicles are expected to operate closer to populated areas, with increased autonomy, in dense urban and suburban environments. These settings present constrained maneuvering conditions which highlights the importance of maintaining safe operation under degraded flight conditions. Abnormal conditions may endanger onboard passengers, people on the ground, and surrounding infrastructure, making rapid detection and mitigation essential to prevent loss of control. Recent research has explored …

Advancements In Spacecraft Trajectory Generation Through Matrix Decomposition Techniques, David Stoev, Oshani Jayawardane, Kaitlyn Cavanaugh, Kristiyan Stefanov, Andrew Murphy

Discovery Day - Daytona Beach

The Circular Restricted Three-Body Problem (CR3BP) is renowned for its intricate and chaotic dynamics, leaving it without a closed-form solution. In this poster, we introduce an innovative approach to determine spacecraft trajectories within the CR3BP framework using matrix factorization techniques. We formulate a matrix equation where the right-hand side vector is constructed from the spacecraft's position and velocity data, while the coefficient matrix is derived from the spacecraft's temporal data. Subsequently, we apply several matrix decomposition techniques, including modified Gram-Schmidt, the Householder technique, and Givens Rotation, to analyze the coefficient matrix and derive the spacecraft trajectories. Finally, we evaluate the …

Determinants And Invertibility In Finite Modular Systems, Osasu Omobude

Discovery Day - Daytona Beach

This project investigates determinants and matrix invertibility in finite modular systems, focusing on matrices over Zn. Using the Hill cipher as context, it examines the algebraic conditions under which a matrix is invertible in modular arithmetic. In particular, the project studies how the determinant determines invertibility, showing that a matrix over Zn is invertible if and only if its determinant is coprime with n.   The project further compares invertibility over the real numbers with invertibility over modular systems, highlighting the distinction between prime moduli Zp and composite moduli. In the prime case, matrices behave similarly to those over fields, where …

Low-Complexity Polynomial Ring Learning For Quantum Space Assets, Lola Torres

Discovery Day - Daytona Beach

Secure communication for space-based systems requires cryptographic methods that remain both reliable and efficient under strict computational constraints. This work investigates a low-complexity polynomial ring learning algorithm designed for quantum space assets, including satellite–ground communication systems. The project focuses on post-quantum cryptographic principles, where encryption and decryption rely heavily on repeated polynomial operations; this can be computationally expensive with constrained platforms. This is addressed with reformulating polynomial multiplication as a structured linear transformation on coefficient vectors. By representing these operations as matrices with a cyclic structure, the structure allows the use of the discrete Fourier transform (DFT); this will simplify …

The General Solution Analysis Of Homogeneous Linear Equations, Jacob Schwamb, Edward Whipple

Discovery Day - Daytona Beach

The general solution analysis of homogeneous linear equations are any systems of equations in which all constant terms are equal to zero is classified as a homogeneous linear equation. Some key characteristics of homogeneous linear equations are that there are “zero” solutions, where every system has at least a single solution where all variables are zero, also all solutions to any homogenous linear equation is linearly independent, along with having preserved homogeneity, where if any variable (x) may be added to the system, then any scalar multiple of the variable is also a solution. The General solution of any homogeneous …

A Compact Representation Of Oscillatory Limits Via Asymptotic Value Distributions, Ibrahim Arnous, Eric M. Rodarte, Chirag Kumar

Discovery Day - Daytona Beach

Classical limits describe asymptotic behavior through convergence to a single value, but many important oscillatory functions do not converge in this sense. Standard examples such as sin(𝑥) as 𝑥→∞ and sin(1/x) as x→0 instead display stable distributions of values over time. This project examines how such behavior can be described using a measure-theoretic framework, particularly through occupation measures and, in sequence-based settings, Young measures. The objective is to present this perspective in a clear and accessible way while introducing the Ansatz representation, a compact notation for recording the support and density of an asymptotic value distribution when the limiting measure …

Differentiating The Impossible: Feynman's Trick In Applications Of Modern Physics, Yaohua Zhao

Discovery Day - Daytona Beach

We discuss Feynman’s method of differentiating with respect to a parameter inside an integral and explore its significance on selective topics of modern physics. This powerful technique allows us to integrate functions that may seem impossible. Depending on the underlying parameters, the Feynman method becomes a unifying framework that connects mathematical concepts to many parameter-dependent equations in modern physics. In statistical mechanics, this appears directly in the partition function, where derivatives with respect to temperature-related parameters yield thermodynamic quantities such as internal energy and heat capacity; this demonstrates how parameter dependence gives rise to macroscopic behaviors observable at a larger …

Bridging Discrete And Continuous Systems: Fibonacci Sequences And Exponential Growth From Odes, Martyna Wojcik

Discovery Day - Daytona Beach

Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations       It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions …

Engineering Path Trajectories With Gravitational Fields, Eliane Dean, Aidan Hart, Isabel Noot, Nathan Browning, Valeria Villazon Fito

Discovery Day - Daytona Beach

This project explores how vector calculus concepts play a role in aerospace engineering though spacecraft trajectory design. In particular, the notion of vector fields is used to model the gravitational force, whose work done is expressed through line integrals. By taking the curl of the gravitational field and showing it is zero, the field is recognised as conservative, implying that the work done by gravity is path independent. This property is conceptually linked to gravitational potential energy and the principle of energy conservation. The results are then applied to spacecraft motion, where engineers use energy-base methods to determine efficient trajectories …

Visualizing The Invisible: Simulating Electromagnetic Field In 3d Space With Matlab, Aashman Gupta, Eliane Dean, Riley Esperanca, Hailey Grabinski, Jacob Summerhays, Ameya Sute

Discovery Day - Daytona Beach

Electromagnetic field behaviours in free space are defined by Maxwell’s Equations, which couple the temporal and spatial variations of electric and magnetic fields through partial derivatives. These derivatives quantify the rate of change of each field’s vector component with respect to position and time in a 3D lattice, forming the basis for numerical field analysis. This research will develop a mathematical and computational framework using multivariable calculus to model, simulate, and visualize electromagnetic wave propagation in free space using MATLAB. Gradient, divergence, and curl operations are implemented to compute local field variations and energy transfer. The resulting data are used …

Modelling Quadcopter Thrust And Torque Using Lu-Decomposition, Diya Patil

Discovery Day - Daytona Beach

3 & 4 motor systems have been in the world of aviation in many different forms as the field grows and evolves. To understand the complexities of an Unmanned Aerial Vehicle (UAV) and its stability, assessing the amount of thrust put into each motor can help generate the torque produced despite factors such as multidirectional movement. While a UAV does this multiple times a second, producing a simplified version of this calculation can aid in simpler models simulating UAV movement. Due to the popularity of the quadcopter drone, a simple algorithm depicting thrust through each spinning motor can aid in …

Sums Of Consecutive Integers And Sequence Of Divisors, Mushrat Fatema, Abdulkadir Hassen

STEM Student Research Symposium Posters

This work is concerned with two problems in number theory. The first part is to find the maximum number of sequence of integers such that two consecutive numbers in the list are multiple or factor of the other, and integers that can be expressed as a sum of consecutive integers.

On The Fractional Laplacian Type Operator, Maysam Abdulnaser Zain

Thesis/ Dissertation Defenses

In this thesis, we study analytical structures arising from Dunkl theory and their applications to harmonic analysis and fractional Laplacian operators. Dunkl operators are differential-difference operators associated with finite reflection groups, providing a natural generalization of the classical Fourier analysis through the introduction of root systems and multiplicity functions. Within this framework, several classical transforms appear as special cases of the (k,a)-generalized Fourier transform. We study the generalized Fourier transform, its kernel, and the associated translation operator and convolution structures. Using these tools, we construct the corresponding heat kernel and analyze the associated heat semigroup. Our main contribution concerns the …

Dynamic Homeostasis In Relaxation And Bursting Oscillations, Christopher J. Ryzowicz

Biology and Medicine Through Mathematics Conference

No abstract provided.

Effectiveness Of The Guided Discovery Method In Teaching The Surface Area Of A Cylinder, Paul Ahortu

2026 Symposium

This study investigates the impact of the guided discovery instructional method on students’ understanding of the surface area of a cylinder. A quasi-experimental pre-test–post-test design was conducted with 100 senior high school students in Cape Coast, Ghana, divided into experimental and comparison groups..

Results showed a substantial improvement in performance for students exposed to guided discovery, with mean scores increasing from 1.25 (pre-test) to 9.43 (post-test) and a large effect size (Cohen’s d = 2.70). Statistical analysis also revealed significant gender differences in achievement.

These findings indicate strong improvement following the guided discovery intervention and suggest its potential to enhance …

Dynamical Systems Modeling To Determine The Role Of Crosstalk In Shaping Stat Signaling Profiles, Laura F. Strube, Anamarie Martinez, Neha Cheemalavagu, Karsen Shoger, James Faeder, Rachel Gottschalk

Biology and Medicine Through Mathematics Conference

No abstract provided.

Detection, Mapping, And Spraying Of Carolina Redroots In Cranberry Bogs Using Ai And Autonomous Drones, Duwon Ham, Bishal Neupane, Thien Ba Nguyen, Thanh Nguyen, Hieu D. Nguyen, Thierry Besancon

STEM Student Research Symposium Posters

Use artificial intelligent and autonomous drones to automatically detect Carolina Redroots in cranberry bogs, create density maps of the weed, and perform spot spraying.

Area Approximation Of Jordan Curves Via The Isoperimetric Inequality And Cauchy-Crofton Formula, Nikita Veselkin, Rashad Kaiyal

Mathematics Colloquium Series

We combine the Isoperimetric Inequality (Dido's Problem) and the Cauchy-Crofton formula to approximate the area enclosed by Jordan curves in the 2D plane. The Cauchy-Crofton formula provides a consistent estimate of a curve's length, which the Isoperimetric Inequality then uses to approximate the enclosed area. We additionally present software that automates the Cauchy-Crofton length computation, making the method practical for real-world use. Empirical testing validates the accuracy of this combined approach, with potential applications in tumor segmentation from MRI scans.

Normal Matrices, Fuzhen Zhang

Mathematics Colloquium Series

Normal matrices form a central class in matrix analysis, including Hermitian, skew-Hermitian, and unitary, positive semidefinite, permutation matrices and so on. This presentation surveys fundamental properties of normal matrices, including spectral characterization, unitary diagonalization, and trace (in)equality through majorization. It highlights equivalent conditions for normality, with discussions extending to matrix exponentials and polynomials. Examples and counterexamples are provided to clarify certain subtle points about matrix normality. The talk is based on a recent paper published in JMC (joint work with Y.-J. Hu)

Counting Hamiltonian Cycles In Quartic Circulant Graphs, Allison Hilliard

Seaver College Research And Scholarly Achievement Symposium

We consider the problem of counting Hamiltonian cycles in circulant graphs $C^K_n$ where $n$ is the number of vertices and $K$ is a set containing elements that correspond to the allowed edges in the circulant graphs. After sorting the cycles by a topological invariant called the winding number, we use a modified transfer matrix method to convert local data into global structures. The result is a generating function that counts the number of Hamiltonian cycles in a circulant graph with $n$ vertices. The results for $K=\{1,2\}$ and $K=\{1,3\}$ have been found by previous authors. We focus on the case where …

Tipping Points In Crayfish Management: Exploring Population Dynamics With The Bifurcations Activity In Slopes, Evan Inrig, Benjamin Lucas

Seaver College Research And Scholarly Achievement Symposium

Slopes is an interactive environment for exploring numerical methods and graphical solutions to ordinary differential equations. The app launched with five activities for exploration: slopefields, phase planes, oscillations, solutions to systems, and numerical methods for approximation. Bifurcations is a new sixth activity that we designed to investigate changes in the long term behavior of solutions to autonomous differential equations. This activity displays a slopefield and implements the ability to add solutions, but also introduces two new views that show how varying a single parameter impacts the values and stability of equilibrium solutions. We demonstrate the value of the new bifurcations …

From Counting To Advanced Math: How Five Simple Axioms Shape The Mathematical Landscape, Joy Skaggs

ATU Scholars Symposium

In the late nineteenth and early twentieth centuries, mathematics faced a foundational crisis: Greog Cantor’s set theory led to an interesting self-referencing paradox in math and questions about the logical consistency of mathematics. From this crisis, two opposing viewpoints emerged: the Formalists and the Intuitionists. The Formalists praised Cantor’s work as a way to place math on a secure logical foundation, ensuring the discipline’s purity, however the Intuitionists despised Cantor’s work and heralded Cantor as a charlatan and corrupter of the youth. The leader of the Formalists, David Hilbert, proposed a formal system of rigorous proofs to build a complete …

Ideals And Lattices In Number Fields, Sarah Ali Saeed Alyammahi

Thesis/ Dissertation Defenses

This thesis investigates algebraic number fields and their rings of integers, which generalize the ring of integers Z in Q. The study focuses on ideals, units, and ideal class groups, which describe the arithmetic structure of number fields and the failure of unique factorization. Key invariants such as the norm, trace, and discriminant are developed and applied, with particular emphasis on quadratic number fields and classical examples such as the Gaussian and Eisenstein integers. Some explicit computations of ideal class groups are carried out. The thesis also explores connections with lattice theory by interpreting rings of integers as lattices and …

On Topological And Algebraic K-Theories, Amar Yasser Aldakheel

Thesis/ Dissertation Defenses

This thesis explores topological K-theory, a powerful framework for studying invariants of topological spaces via algebraic structures. We investigate the construction of K-groups, with particular emphasis on the categorical formulation. The exposition is structured pedagogically, with careful development of the main concepts supported by detailed proofs and examples. The categories of vector bundles and projective modules are thoroughly defined and investigated. A central result studied in this thesis is the Serre–Swan theorem, which provides a bridge between the algebraic context of projective modules and the geometric context of vector bundles, enabling the use of algebraic techniques to solve geometric problems, …

Local-Nonlocal Dispersal, Behavioral Responses, And Intervention Strategies In Infectious Disease And Addiction Dynamics, Ghilmana Sarmad

Thesis/ Dissertation Defenses

This dissertation explores how epidemiological and behavioral processes interact across spatial and temporal scales to shape the dynamics of infectious diseases and addiction. Bringing together mathematical rigor and biological realism, it develops and analyzes a series of nonlinear models that capture the effects of spatial heterogeneity, mobility patterns, behavioral adaptation, and intervention strategies. Using tools from semigroup theory, functional analysis, stability analysis, and numerical simulation, the study provides a unified framework for understanding how awareness programs, fear-driven protection, vaccination, and social reinforcement influence transmission thresholds, persistence, and long-term population outcomes.

awareness-based interventions, where a basic reproduction number is derived and …

Indexed Concatenation (I-Cat): A Method For Representing And Operating On Repeating Numbers*, Abigail Touma, Isaac Wolford

Campus Research Month

Our research presents Indexed Concatenation (I-Cat) notation as a structured way to represent numbers with repeating patterns, including both decimals and whole numbers. Instead of treating expressions like 0.333... or 735735735 as unstructured expansions, they are rewritten as compact repeating objects called I-Cats. The presentation demonstrates how arithmetic operations, including addition and multiplication, can be performed using hypothesized rules such as the "U = M/C" method, unpacking, and carry propagation. Examples progress from simple conversions and multiplication by integers to the multiplication of two I-Cats. A live visual demonstration will show how standard numerics transform into I-Cat form and how …

Variational Data Assimilation With Steepest Descent Method For Coupled Time-Dependent Stokes-Darcy Model With Bjsj Interface Condition, Yafang Hei

Miners Solving for Tomorrow Research Conference

Variational data assimilation (VDA) determines the initial condition of a dynamical system by minimizing the mismatch between model predictions and observed data. This work studies VDA for the time-dependent Stokes–Darcy system with the BJSJ interface condition. The problem is formulated as a PDE-constrained optimization problem, and the first-order optimality system is derived using the Gâteaux derivative and adjoint variables. A steepest descent method is applied for efficient computation. Spatial and temporal discretizations are carried out using the finite element method and backward Euler scheme, respectively. Numerical results confirm accuracy and convergence.

Application Of The Tridiagonal Representation Approach And The J Matrix Method Of Scattering In Theoretical Physics, Tunde Joseph Osunmusanmi

Thesis/ Dissertation Defenses

This dissertation investigates the application of the Tridiagonal Representation Approach (TRA) to linear systems and introduces, for the first time, an extension of the J-matrix method of scattering to nonlinear phenomena. The TRA provides an algebraic framework for solving second-order linear ordinary differential equations by combining algebraic structure with the analytical properties of orthogonal polynomials and special functions. Computationally, the method benefits from efficient numerical techniques for tridiagonal matrices. Within the TRA, solutions of differential equations, including the Schrödinger equation, are expressed as convergent expansions in square-integrable basis functions chosen to yield a tridiagonal representation of the differential operator. This …

Lipscomb University Parking Garage Analysis, Morgan Hayes, Reagan Maxwell, Daniel Diaz Tortolero, Dominick Dingus, Michael Ent

Student Scholar Symposium

Many students struggle to easily find parking during the school day at Lipscomb University, and this problem will only be emphasized with the potential removal of the nearby Stokes parking lot (containing 255 spaces). To combat this need and help provide additional room for university growth, we propose the addition of a parking garage west of the Fields engineering building, between Belmont Boulevard and Grandview Drive. This garage would be separated from the current garage behind Fields and be slightly larger, containing about 500 spaces over five levels. We evaluate the current number of parking spaces needed at Lipscomb based …

Mathematical Developments In American Spacecraft Technology, Tylah Grace Lynn, Lauren Reece Terry

Student Scholar Symposium

No abstract provided.