Mathematics Commons^™

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 …

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. …