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An Analysis Of A Family Of Root Finding Methods, Arabella Fuzak

Celebrating Scholarship and Creativity Day (2018-)

This thesis analyzes the behavior of a family of iterative root-finding methods, the Hansen-Patrick Family, which has a parameter, alpha, to create methods such as Newton’s methods, Halley’s method, and Euler’s method. By varying the parameter alpha, with both real and complex values, this project examines how the parameter can change convergence, divergence, and stability for different functions. This is shown by basin maps and Mandelbrot-like sets to visualize this behavior and classify points based on whether they converge to a root, diverge to infinity, or remain bounded without converging.

Redefining Certainty: Non-Euclidean Geometry And Theology Transformation Throughout The Intellectual Unrest Of The Early 1800s, Luke Bensinger

Honors Theses

To bridge the gap between mathematics and theology, it is necessary to explore their intersection and challenge the notion that these fields are incompatible. This study focuses on the 19th century, a period when non-Euclidean geometries emerged and disrupted mathematical certainty, while Protestant theologians such as Barton W. Stone and Alexander Campbell grappled with Calvinism and shifting theological perspectives. By analyzing mathematicians studying geometry, such as Gauss, Lobachevsky, and Riemann, this research examines how both disciplines balance change with enduring truths.

Bibliography Of Religious Faith And The Mathematical Sciences, Calvin Jongsma

Faculty Work Comprehensive List

This Bibliography is offered as a helpful resource for anyone who wishes to thoughtfully explore the connections between religious faith and the mathematical sciences. Searching the database for a topic of interest will bring up items with that focus. An entry’s attached PDF (when publicly available) can be opened and read while in the database. Alternatively, each item contains a URL/web link to a location where one can either read the item or retrieve information about how to obtain a copy of it.

Lebesgue’S Theory Of Integration: An Introduction To Modern Integration Theory, Nathanael C. Krug

Senior Honors Theses

The depth and richness of integration theory far surpasses the introductory material presented in an elementary calculus classroom. The work of Bernhard Riemann and Henri Lebesgue demonstrates just a small sample of the richness of the field of analysis. In formulating and contrasting the Riemann and Lebesgue integrals, students can gain an enriched and well-rounded introduction to integration theory. This not only deepens understanding and love for previously learned material, but also enables further study within the fields of analysis and measure theory. An introductory primer to integration theory equips students with the tools needed to continue their exploration of …

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.

Rosenzweig's Elements And Universal History, Martin Zwick

Complex Systems Faculty Publications and Presentations

This paper uses Rosenzweig’s conception of the three elements of God, World, and Human from The Star of Redemption in a model of universal history. The model also has some relation to Rosenzweig’s geopolitical essay, Globus, which is very different from the meta-historical Star. Based on a systems-theoretic schema of events and processes, the model views human history in terms of three linked processes: a primary process labeled “World” – the origin and development of human society embedded in nature, a secondary process labeled “God” – the origin and development of the Axial religions and philosophies, and a …

Boolean Rank Via Monomial Ideals And Neural Ideals, Juliann Marie Geraci

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

This dissertation develops new connections between Boolean matrix factorization, combinatorial neural codes, and commutative algebra. The central goal is to understand how structural and algebraic invariants can be used to measure and bound complexity in discrete data. We begin by studying the Boolean matrix factorization (BMF) problem, which seeks to express a binary matrix as a product over the Boolean semiring with minimal inner dimension, known as the Boolean rank. By interpreting binary matrices as bipartite graphs, we relate Boolean rank to biclique covers and introduce algebraic techniques to study this quantity. We associate to a matrix its edge ideal …

Complexity Of The Zero Set Of A Matrix Schubert Ideal, Cesar Julian Meza

Arts & Sciences Graduate Student Theses and Dissertations

T-varieties are normal varieties equipped with an action of an algebraic torus T. When the action is effective, the complexity of a T-variety X is dim(X)−dim(T). Matrix Schubert varieties, introduced by Fulton in 1992, are T-varieties consisting of n×n matrices satisfying certain constraints on the ranks of their submatrices. In this dissertation, we focus on the complexity of certain torus-fixed affine subvarieties of matrix Schubert varieties. Concretely, given a matrix Schubert variety X_w where w∈S_n, we study the complexity of Y_w obtained by the decomposition X_w = Y_w ×C^k with k as large as possible. Building on results by Escobar–Mészáros …

Finiteness Of Torsion Loci And Normal Functions Arising From Cycles On Hadamard Products, Devin Akman

Arts & Sciences Graduate Student Theses and Dissertations

The unifying theme of this dissertation is normal functions. In the first chapter, we study an invariant of knot exteriors and similar manifolds called the A-polynomial through the lens of vanishing loci of normal functions. Using a special case of the Zilber-Pink conjecture, proven in the second chapter, we show that only finitely many irreducible Laurent polynomials of bounded overgenus appear as factors of A-polynomials. In the third chapter, we construct normal functions from hypergeometric variations of Hodge structure and compute their regulators. Finally, we determine under which conditions incomplete motivic cohomology cycles on families complete to cycles on their …

Halo: High Autonomous Low-Swap Operations, Sloan Hatter, Blake Gisclair

Mathematics and System Engineering Student Publications

Orbital object detection is a vital aspect of space operations, particularly for identifying satellite components. Convolutional Neural Networks (CNNs) are typically used for such operations by running onboard models directly on satellite systems. However, a new neural network architecture, known as Vision Transformers (ViTs), have shown greater effectiveness due to their ability to capture global context. One main issue of deploying systems with such capabilities is resource allocation. One solution is to run models on a Low-SWaP system; however, this results in inefficient performance. To enable efficient ViT operations on Low-SWaP systems, the model must be scaled down through quantization, …

Stochastic Network Resilience Under Random Failures, Blake Gisclair

Mathematics and System Engineering Student Publications

This project investigates threshold-crossing probabilities in stochastic networks. Using probabilistic modeling and transform-based analytical methods, the work derives expressions that characterize when cumulative losses exceed prescribed limits with the goal of providing insight into the relationship between local random behavior and global network risk.

Determining Material Transport Method For Moon Colony Using Differential Equations, Scott Meeson, Anthonie Page, Matas Vaitkevicius

Mathematics and System Engineering Student Publications

NASA Artemis Missions: NASA’s Artemis program marks a fundamental shift from short-term lunar exploration to sustained, permanent colonization.

The Invisible Shield: Why Our Stomach Doesn’T Digest Itself, Kevon Findley

Mathematics and System Engineering Student Publications

The stomach contains highly acidic gastric fluid with a very low pH.  A thin mucus lining protects the stomach from self-digestion and keeps nearby epithelial cells near a pH 7.  A weakened mucus layer is associated with conditions such as Gastritis and Peptic Ulcer Disease.

How can the protective mucus barrier be maintained and be of stable thickness?

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.

Theory And Simulations Of Delayed Stochastic And Deterministic Models Of Prion Diseases, Gangadhara Boregowda, Omar Sharif, Daniel Gutierrez Iii, Allegra Simmons, Laurent Pujo-Menjouet, Tamer Oraby, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Neurodegenerative diseases (NDs), such as Alzheimer’s, Parkinson’s, and prion diseases, are characterized by the dynamical spread of toxic proteins through the brain. In prion diseases, cellular prion protein (PrPC), produced by neurons, misfolds into a toxic form, known as scrapie prion protein (PrPSc). PrPSc induces neuronal stress which ultimately leads to cell death. In this paper, we develop mathematical models for the progression of prion diseases, incorporating a cellular defense mechanism that introduces a delay term affecting protein translation and a volatility term accounting for unaccounted biological factors influencing the system. We also extend the model to capture the spatial …

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.

Massively Parallel Kalman Filtering: Scaling State Estimation Via Cuda Kernels, Joseph Campione, Peter Chim, Hannah Depuydt, William Johnston, Jackson Phillips, Brian White

Mathematics & Computer Science Student Scholarship

This project addresses the challenge of state estimation in real-world conditions by fusing data from multiple sensors using the Kalman Filter. To ensure numerical stability, we use the Joseph Form covariance update, which guarantees valid results but introduces significant computational overhead due to its complexity. To overcome this limitation, we implement a parallelized solution using custom CUDA kernels on a GPU, distributing matrix operations across thousands of threads rather than relying on sequential CPU execution. Through systematic benchmarking across matrix sizes ranging from 4×4 to 4096×4096, we identify a crossover region where GPU performance surpasses CPU efficiency. This work shows …

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.

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 …

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

Electronic Theses and Dissertations

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 …

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. …

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.