Mathematics Commons^™

Creating A Random Number Generator Harnessing: Lava Lamps As A Source Of Randomness, Rachel Barter, Nicolas Guerra, Sarah O'Connor

Mathematics & Computer Science Student Scholarship

No abstract provided.

The Euler Characteristic, Cara Admiraal

SPARK Symposium Presentations

The Euler characteristic is an example of a topological invariant most famously Leonard Euler proved that for any convex polyhedron with $v$ vertices, $f$ faces, and $e$ edges, $v-e+f=2$. In this presentation, we will extend his ideas to define the Euler characteristic for surfaces.

Graph Theoretical Modeling Of Self-Assembling Dna Of The Double Cone Graph, Philiffe Tebalan, Evan Burns

Rose-Hulman Undergraduate Mathematics Journal

The unique properties of double-stranded DNA molecules make DNA a valuable structural material with which to form nanostructures, and the field of DNA nanotechnology is largely based on this premise. By modeling nanostructures with discrete graphs, efficient DNA self-assembly becomes a mathematical puzzle. These nanostructures have wide-ranging applications, such as containers for the transport and release of nano-cargos, templates for the controlled growth of nano-objects, and in drug-delivery methods. This research centers around exploring graph theoretical and combinatorial properties of DNA self-assembly to optimize the nanostructure construction for the Double Cone Graph.

Objections To The Use Of The Axiom Of Choice To Model The Physical World, Kensey Doughtie

Rose-Hulman Undergraduate Mathematics Journal

We look at platonistic mathematics and the application of this perspective in the physical world. We recognize paradoxes within Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that conflict with physical reality, giving us reason to question if the axiom of choice should be so freely applied in theories of the physical world especially since it appears to enable a deterministic perspective. In theories of quantum physics, the axiom of choice is used to assume noncomputable numbers as initial conditions. This is equivalent to assuming a finite system contains an infinite amount of information at an instant in time; …

Bass Numbers Of Veronese Submodules And Lifting The Frobenius Trace, Taylor Jeffrey Murray

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

The first part of this thesis is inspired by the works of G. Lyubeznik, C. Hunkeke, R. Sharp, and L. Núñez-Betancourt on Bass numbers, associated primes, and injective dimension of local cohomology modules. In particular, we study the question: how do Bass numbers behave under the Veronese functors? We show for a positive integer n, a reasonably nice, graded, finitely generated algebra over field R, and a graded module M, that if the Bass numbers of M are finite over R, then so are the Bass numbers of Mⁿ/Rⁿ; this recovers …

Cohomological Support Varieties Along Ring Maps, Ryan Watson

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

Inspired by Quillen’s geometric approach to study group cohomology, Avramov introduced the theory of support varieties to study finite modules over local complete intersection rings. Geometric properties of the variety encode important homological information, and this theory has been a useful tool in studying complete intersection rings leading to many advances in local commutative algebra. By the work of several authors, this theory has now been expanded to encompass any noetherian local ring. Notably, Pollitz developed the theory of cohomological support varieties over Koszul complexes and used them to answer a question of Dwyer, Greenlees, and Iyengar regarding the structure …

Analyzing Label Structure And Regional Similarity In Watershed Data Via Spectral Clustering, Yifan Luo

25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026)

This project focuses on identifying patterns of seasonal transitions and nutrient salt fluctuations within the watershed environment. To capture the complex relationships between multiple sampling sites and environmental variables, we represent the watershed dataset as a weighted graph, where nodes correspond to water samples and edge weights reflect similarity in environmental conditions or nutrient concentrations. Using the Gaussian kernel function, we encode the connectivity structure of this network and quantify how similar different nodes are. We then perform spectral embedding by projecting the high-dimensional graph into a lower-dimensional space using the eigenvectors of the Laplacian matrix. This approach preserves the …

The “How Many” Routine As A Catalyst For Computational Fluency And Student Participation, Lillian Iden, Ella Williams, Jen Munson, Sarah Larison, Leslie Yuqui

25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026)

It is not uncommon to hear adults claim they are bad at math or always disliked the subject. This negativity often stems from early mathematical experiences which ranged from boring to highly discouraging and embarrassing. Thus a new emphasis in mathematics education recommendations is to change the narrative and help students develop joy, wonder, and curiosity about mathematics (MAISA & GELN, 2023). This is reflected in defining computational fluency (skill in carrying out arithmetic procedures like addition or multiplication) as comprised of flexibility, accuracy, efficiency, and appropriate strategy use (NRC, 2001). The “How Many” Routine, in which a carefully-designed image …

Investigations In Bertrand’S Paradox, Mary Moore, Hope Weeda, Annika Cunill Krones

25th Annual A. Paul and Carol C. Schaap Celebration of Undergraduate Research and Creative Activity (2026)

Bertrand’s paradox is a classic problem that highlights how different notions of randomness can lead to different outcomes, even in a simple geometric setting. It concerns the lengths of chords chosen “at random” in a circle. In this talk, we begin by reviewing the three original methods Bertrand proposed for generating random chords, along with several related distributions that have been studied since. We then turn to a geometric application, examining triangles formed by two random chords that share a common endpoint. By joining the remaining endpoints, we obtain a random triangle and compute the probability that it is acute. …

A Computer Vision Approach To Analyzing Taxane Effects On Prostate Cancer Cells, Diana Elizabeth Dancea

Electronic Theses and Dissertations 2020 - Present

Actin is a family of proteins that help create the structure of the cytoskeleton, which gives shape to the cell. In many chemotherapy treatments, researchers target actin because it controls the cell division process. Therefore, if they are able to understand the actin fibers, that may help in formulating methods to stop or slow down cancer cells from reproducing. Another important protein is PAK6, which regulates actin. In our research, a collaborative effort with Prof. Michael Lu’s lab at Florida Atlantic University, we use machine learning techniques to analyze cells which had their PAK6 protein knocked out, and compare them …

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)

Emergent Storylines That Influence Positions And Mathematical Status In Collaborative Small-Group Proof Activity, Brittney M. Ellis, Tenchita Alzaga Elizondo

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we used positioning theory to examine storylines that emerged in students’ discourse as they collaborated on a proof construction task. We purposefully selected a case of group work from an inquiry-oriented introduction to proof course as prior analyses showed it was highly collaborative (Alzaga Elizondo, 2022), yet power dynamics seemed unbalanced. We hypothesized that positioning theory could provide a useful lens to interrogate such power dynamics. Through this analysis, we identified several implicit storylines that influenced the interaction related to the nature of proofs, the nature of mathematics, writing proofs, the role of an external authority, …

Modeling Bitcoin Dynamics Using Differential Equations, Boone M. Fleenor

Departmental Honors & Graduate Capstone Projects

In this thesis, we develop and analyze two nonlinear systems of ordinary differential equations to model Bitcoin price dynamics. Analytical techniques are used to obtain exact or approximate solutions where possible. Then, numerical simulations using a fourth-order Runge–Kutta method are employed to explore system behavior beyond analytically tractable regimes. Finally, model outputs are compared to historical Bitcoin price data using normalized and resampled time series. These results suggest that deterministic models can provide meaningful insight into the structural behavior of Bitcoin markets, while highlighting the need for stochastic or time-dependent extensions for more realistic modeling.

Fourier Analysis Of Electronic Synthesizer Waveforms, Jiayan Ling, Jack W. Maseberg

SACAD: Scholarly Activities

We compute Fourier series coefficients for some standard electronic synthesizer waveforms and provide plots of our results.

Unique Combinations Of Packing Integer Squares, Keith M. Dreiling, Austin Leanna, William Mooney

SACAD: Scholarly Activities

This research investigates a function, informally named WAK(x), that describes the number of ways to divide an integer square into integer subsquares counting only the list of parts. Previous research has shown values up to 28, though finding these values is computationally complex and requires a long runtime using computer algorithms. We attempt to find patterns in the values and many aspects of the values, hoping to find a general solution. We are unsure if a solution exists, but we have ideas for how to move forward in finding a solution.

Mathematical Modeling, Analysis And Numerical Simulation Of Air Quality Using The Advection-Diffusion-Reaction Equation, Md Timorul Islam

Math Theses

In this thesis, we study an air quality model by using a nonlinear one-dimensional advection-diffusion-reaction system of partial differential equations for NO-NO2-O3 chemical cycle. The model incorporates advection, diffusion, chemical reactions, and source terms, and is formulated based on standard atmospheric chemistry.

We first develop numerical methods to approximate the solutions of the governing equations. Both explicit and implicit finite difference schemes are considered, and the implicit scheme provides approximations without any restrictions on the stability.

The model is then nondimensionalized to identify key parameter combinations governing the system. This leads us to a regime in which the transport and …

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 …

Irreversible K-Threshold Dynamics On Corona And Base-B Corona Product Graphs, Eric J. Moon, Soumya Bhoumik, Paul Flesher

SACAD: Scholarly Activities

This poster studies the irreversible k-threshold process on corona-type graph products, where a vertex becomes colored once at least k of its neighbors are colored and then remains colored permanently. We focus on corona, double corona, and base-b corona product graphs built from cycles and complete graphs, with particular attention to how graph structure affects complete activation from a minimum seed set.

A generalized reduction lemma is used to relate threshold dynamics on layered corona graphs to smaller residual graphs, yielding explicit formulas for the irreversible k-threshold conversion number on both corona and double corona families. The …

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 …