Mathematics Commons^™

Operator Theoretic Methods For Coupled Pde In General Geometries, Yuhao Mu

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

This thesis develops and applies operator theoretic methods applied to coupled PDE analysis in general geometries, in which the coupling involves interchange across a boundary. The general geometries refer to domains with merely Lipschitz continuous boundaries (e.g. any non-convex polygon), as opposed to those with any further smoothness conditions on the boundary. The key results include a new pressure elimination method for coupled fluid–structure interaction (FSI) PDEs with boundary interchange, which allows an explicit semigroup representation of the PDE in terms of just the fluid and structure variables. This novel technique, valid over arbitrary bounded Lipschitz domains, leads to a …

Course Portfolio: Elements Of Physics Phys 151, Evan A. Rich

UNL Faculty Course Portfolios

This course portfolio documents the instructional design, teaching methods, and ongoing assessment efforts for PHYS 151: Elements of Physics, an algebra-based introductory physics course at the University of Nebraska-Lincoln. The course serves a broad undergraduate population, including architecture, construction management, and life science majors. The portfolio describes the teaching framework that integrates pre-lecture video preparation, active in-class engagement through iClicker questions, and collaborative weekly recitation sections, all unified around a structured six-step problem-solving approach. A central concern of the course is building students’ self-efficacy in physics, particularly among those with math anxiety or limited preparation. Two assessments are reported: 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 …

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 …

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 …

Traffic Prediction For Research And Education Networks: Anomaly-Aware Deep Learning And Benchmarking, Mohammad Arafath Uddin Shariff

School of Computing: Dissertations, Theses, and Student Research

Research and Education Networks (RENs) and High-Performance Computing (HPC) environments are critical infrastructures for modern scientific discovery, demanding sustained high-throughput and low-latency data transfers. Unlike commercial networks, RENs exhibit unique traffic characteristics, including predominant “elephant flows,” inherent burstiness, and complex temporal-spatial dynamics often decoupled from human-driven cycles. Traditional traffic forecasting methods, tailored for commercial Wide Area Networks (WANs), consistently fail to capture these distinct REN dynamics, leading to inefficient resource management and potential impediments to scientific progress.

This thesis addresses this critical gap by developing and validating a robust, scalable, and anomaly-aware traffic forecasting framework specifically tailored for REN/HPC networks. …

Analysis Of Graph-Based Decoders For Quantum Low Density Parity Check Codes, Kirsten Morris

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

Quantum computing has the potential for radically increased computational ability. However, the physical realization of quantum states are fragile and susceptible to noise and decoherence. For this reason, robust quantum error correction is imperative to achieve quantum computation at scale.

Of particular interest in realizing effective quantum error correction are quantum low density parity check (QLDPC) codes. Classical LDPC codes were invented by Robert Gallager in the 1960s and came in to prominence in the 1990s. Due to Daniel Gottesman’s stabilizer formalism and the invention of Calderbank-Shor-Steane (CSS) codes, we can apply LDPC codes to the quantum setting.

As in …

Interpolation In Weighted Projective Spaces, Shahriyar Roshan Zamir

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

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree $d$ singular at a given set of points.

After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this thesis, we use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space.

A main contribution of this work is the careful adaption of several classical algebro-geometric techniques to the …

On Kernels And Antiderivatives Of Nonlocal Derivatives, Alex John Heitzman

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

Nonlocal operators are mathematical operators taking functions to other functions f → Df , where to evaluate the operator Df at a point x, one must know the value of f in some region around x, and that region cannot be arbitrarily small. Nonlocal derivatives are like derivatives in that they measure the deviation of a function f(z) from f(x) when z is close to x. In this thesis, we will study nonlocal operators of the form

D_kf(x) = [integral]_Ω [f(x) …

A Zariski-Nagata Theorem For Smooth Toric Surfaces, Jordan Vincent Barrett

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

The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the n^th symbolic power of a radical ideal I in a polynomial ring over a field in terms of the n^th ordinary powers of the maximal ideals in [affine variety]_max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with …

Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors, Johan Benedict A. Cristobal

Department of Mathematics: Dissertations, Theses, and Student Research

Why do instructors teach the way that they do? This question is key in understanding how and why asset-based or deficit-based practices are enacted in the classroom. In consequence, this question also gives insight into how the environment where students learn is shaped. Mathematics education research, in particular, has been concerned with how instructors respond to students’ contributions in the classroom and how these ways of responding can (dis)empower students.

One way to answer this question is to understand how instructors frame their own teaching and their students’ learning. Frames are mental constructs which help individuals filter details, interpret information, …

Modules Of Finite Projective Dimension And Singularities, Nawaj Kc

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

In the first part of this thesis, we study liftings of modules of finite projective dimension. We introduce a notion of “Serre liftable” modules and deduce applications to multiplicity conjectures in local algebra. In the second part, we introduce and study a module of mixed K\"ahler differentials for finite algebras over ramified discrete valuation rings of mixed characteristic. We state and prove a Jacobian criterion for computing the singular loci of such algebras.

Advisors: Jack Jeffries and Mark Walker

Using Permutation Groups To Identify Family Of Capacity Achieving Codes, Daniel Joseph Welchons

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

When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.

This this thesis seeks to …

Using Permutation Groups To Identify Families Of Capacity Achieving Codes, Daniel Welchons

Department of Mathematics: Dissertations, Theses, and Student Research

When communicating over a noisy channel, the probability of message interference sets a maximum possible transmission rate known as the channel capacity. Any family of codes which have rates converging to the channel capacity and arbitrarily low probability of decoding failure is called capacity achieving. Such codes have been known to exist since the birth of information theory, but are difficult to find explicitly. It has recently been shown that the permutation groups of a family of codes can be used to show that the family is capacity achieving on the q-ary erasure channel.

This thesis seeks to apply the …

Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors, Johan Benedict Arroyo Cristobal

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

Why do instructors teach the way that they do? This question is key in understanding how and why asset-based or deficit-based practices are enacted in the classroom. In consequence, this question also gives insight into how the environment where students learn is shaped. Mathematics education research, in particular, has been concerned with how instructors respond to students’ contributions in the classroom and how these ways of responding can (dis)empower students.

One way to answer this question is to understand how instructors frame their own teaching and their students’ learning. Frames are mental constructs which help individuals filter details, interpret information, …

Equity Vs. Excellence: Dichotomy Or Duo?, Nadjia I. Logans

Honors Program: Senior Projects (Public)

In this thesis, I investigate recent changes in mathematics education and how these developments connect to the principles of equity and excellence. I place a special focus on “active learning” methods. My methodology primarily consisted of interviews with educators, along with analyzing literature from math education researchers. I first define and explain the motivations behind active learning instruction methods. Next, I explore not only how instructors define equity and excellence but how educators perceive the role of the two principles within education. Finally, I challenge the pervasive media portrayal of equity and excellence as being mutually exclusive. Instead, I find …

Algorithms For Order Statistics In Farey Sequences: A Computational Study, Connor Weyers

School of Computing: Dissertations, Theses, and Student Research

Farey sequences are the sets of irreducible fractions in increasing order with denominator less or equal to some integer n. They are a well-known concept in number theory problems and are related to many other concepts in number theory including integer factoring, Fibonacci sequences, and Riemann’s Zeta function. In this paper, we investigate some known algorithms to solve certain problems in Farey sequences from a computational perspective. In particular, we implement established algorithms that have not been previously implemented with the goal of creating a package that can be used more broadly. We also develop a new algorithm for rational …

Navigating Geometry With Interactive Lessons: A Collection Of Resources In Geogebra That Address Frequently Held Misconceptions, Sara Vance

Honors Program: Senior Projects (Public)

Nearly every high school student in the United States is required to take a geometry class, and for many, it’s unlike any math class they’ve taken before. Additionally, there are certain topics with which past and present geometry students alike tend to struggle. This project was created to address these issues; my goal is that these materials will help students better understand these topics and clear up their frequently held misconceptions about geometry. This project was supported by interviews I conducted with veteran high school geometry teachers with over 75 years of teaching math between them.

This collection of interactive …

The Tate Resolution Over A Complete Intersection Ring, Kolton O'Neal

Honors Program: Senior Projects (Public)

Given a Noetherian commutative ring R and an ideal I ⊆ R, Tate provided a construction in [5] to produce a DG R-algebra that is also a free resolution for R/I. In this work, we review free resolutions and DG algebras, describe Tate’s construction, and present a proof of a result from Tate’s paper about his construction when a regular sequence is involved. Specifically, this result is that it only takes two steps of Tate’s construction to resolve a characteristic 0 field k over k[[x1, . . . , xn]]/(f1, . . . , fc), where f1, . . . …

Analysis Of Impulsive Differential Equation Models Of Cell Populations Undergoing Radiation Therapy, Abigail D'Ovidio Long

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

Radiation therapy is a mode of treatment which is implemented for approximately 50% of cancer patients. Treatment needs to be able to kill cancer cells, but also do minimal damage to surrounding healthy tissue. We propose two main impulsive differential equation models of radiation therapy to capture the periodic nature of the treatment. These models build off of previous studies using clinical data to ensure biological relevance. The first model incorporates only cancer cell populations, and we provide parameter relationships which theoretically ensure treatment outcomes of cancer eradication, cancer approaching a carrying capacity, and cancer approaching a periodic solution. We …

A Measurement Of The Differential Drell-Yan Cross Section As A Function Of Invariant Mass In Proton–Proton Collisions At √ S = 13 Tev, William Robert Tabb

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

The Drell-Yan process, a crucial mechanism for producing lepton pairs in highenergy hadron collisions, serves as an essential probe for testing the Standard Model of particle physics. This dissertation presents a comprehensive measurement of the differential cross section with respect to the invariant mass of the lepton pairs, utilizing data collected by the CMS experiment at CERN from 2016 to 2018. Cross sections are essential for refining our understanding of parton distribution functions and the underlying quantum chromodynamics processes, thereby providing constraints on theoretical predictions. In this analysis, the cross sections are compared to theoretical models and simulations, offering new …

Exploring Intraplate Seismicity In The Midwest, Alexa Fernández

Department of Earth and Atmospheric Sciences: Dissertations, Theses, and Student Research

Intraplate seismicity represents a notable occurrence within the stable North American Craton. This research explores the potential sources of stresses that could reactivate older faults and influence seismic activity within this region. Among these sources, the enduring impact of the last glacial period is considered, which includes continued glacial isostatic adjustments (GIA). During GIA the lithosphere rebounds due to the retreating ice, and the forebulge caused by far-field flexure in response to the glacial load, collapses. This results in significant faulting, fracturing, and seismic activity associated with the deglaciation phase. The adjustment of the lithosphere manifests as both near surface …

Making Sandwiches: A Novel Invariant In D-Module Theory, David Lieberman

Department of Mathematics: Dissertations, Theses, and Student Research

Say I hand you a shape, any shape. It could be a line, it could be a crinkled sheet, it could even be a the intersection of a cone with a 6-dimensional hypersurface embedded in a 7-dimensional space. Your job is to tell me about the pointy bits. This task is easier when you can draw the shape; you can you just point at them. When things get more complicated, we need a bigger hammer.

In a sense, that “bigger hammer” is what the ring of differential operators is to an algebraist. Then we will say some things and stuff …

Spreads And Transversals And Their Connection To Geproci Sets, Allison Joan Ganger

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

Spreads of [set of prime numbers]³ over finite fields can yield geproci sets. We study the existence of transversals to such spreads, proving that spreads with two transversals exist for all finite fields, before further considering the groupoids coming from spreads when transversals do or do not exist. This is further considered for spreads of higher dimensional projective spaces. We also consider how certain spreads might generalize to characteristic zero and the connection to the previously known geproci sets coming from the root systems D₄ and F₄.

Advisor: Brian Harbourne

Semigroup Well-Posedness And Finite Element Analysis Of A Biot-Stokes Interactive System, Sara Mcknight

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

The coupling of a porous medium modeled by the Biot equations and a fluid has many biological applications. There are numerous ways by which to model the fluid and to couple the porous medium with the fluid. This particular model couples the Biot equations to Stokes flow along the boundary, through the Beavers-Joseph-Saffman conditions. We address semigroup well-posedness of the system via an inf-sup approach, which along the way requires consideration of a related but uncoupled static Biot system. We also present the results of finite element analysis on both the uncoupled Biot system and the coupled system.

Advisor: Sara …

On Regularity Of Graph C*-Algebras, Gregory Joseph Faurot

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

We prove that for any countable directed graph E with Condition (K), the corresponding graph C*-algebra C*(E) has nuclear dimension at most two. We also prove that the nuclear dimension of certain extensions is at most one, which can be applied to certain graphs to achieve the optimal upper bound of one. Finally, we generalize some previous results for O_∞ -stability of graph algebras, and prove some partial results for Z-stability.

Advisor: Christopher Schafhauser

Perturbations Of Representations Of Cartan Inclusions, Catherine Zimmitti

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

A free semigroup algebra is the unital, weak operator topology closed algebra generated by a collection of Cuntz-Toeplitz isometries in B(H). Ken Davidson and David Pitts asked in [9] if a self-adjoint free semigroup algebra exists; Charles Read answered this question in [28] by constructing such an example, which Ken Davidson later simplified in [8]. The construction takes a standard representation of O₂ and multiplies it by a unitary operator in the diagonal MASA of the representation. This results in a new "perturbed" representation of O₂ generating a self-adjoint free semigroup algebra.

In this thesis, …

On Neumann Boundary Conditions For Nonlocal Models With Finite Horizon, Scott Alex Hootman-Ng

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

Nonlocal models are have recently seen an explosive interest and development in the context of fracture mechanics, diffusion, image processing, population dynamics due to their ability to approximate differential-like operators with integral operators for inherently discontinuous solutions. Much of the work in the field focuses on how concepts from partial differential equations (PDEs) can be extended to the nonlocal domain. Boundary conditions for PDEs are crucial components for applications to physical problems, prescribing data on the domain boundary to capture the behavior of physical phenomena accurately with the underlying model. In this thesis we specifically examine a Neumann-type boundary condition …

Virtual Unknotting Numbers For Families Of Virtual Torus Knots, Kaitlin R. Tademy

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

A virtual torus knot T(p,q,VC) sits in the intersection of the well-understood torus knot and the not-so-well-understood virtual knot, making it an intriguing object to study.

The unknotting number of a classical knot K is defined unambiguously. However, "the" unknotting number when K is a virtual knot is not as clear to define, since virtual knots have both classical and virtual crossings. We will define virtual unknotting number vu(K) as the minimum number of (classical) crossing changes required to unknot K. Under this definition of virtual unknotting, not all …

A Study On The Vanishing Of Ext, Andrew J. Soto Levins

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

This thesis has two goals. The first is to study an Ext analog of the rigidity of Tor, and the second is to study Auslander bounds.

In Chapter 2 we show that if R is an unramified hypersurface, if M and N are finitely generated R-modules, and if the nth Ext modules of M against N is zero for some n less than or equal to the grade of M, then the ith Ext module of M against N is zero for all i less than or equal to n. A corollary of this says that if …