Mathematics Commons^™

Spectral Properties And The Behavior Of The Current-Current Correlation Measure For The Luttinger-Sy Model, Jonathan Benoit

Theses and Dissertations--Mathematics

The Luttinger-Sy Model, sometimes referred to as the Pieces Model, is a Random Schrodinger Operator on L²(R) which is characterized in part by "pieces" whose endpoints are chosen by a Poisson Point Process. The Hamiltonian in this setting is then given as a direct sum of Laplacians with Dirchlet boundary conditions on each piece. In this work, we show several spectral properties of the Luttinger-Sy Model, including proving the deterministic spectrum is [0,infinity) and that a Wegner-type and Minami-type estimate both hold. Additionally, we show that the finite-volume Current-Current Correlation Measure is singular continuous with respect …

Student Conceptions Of Summation And Limits In The Definite Integral Following Quantitatively Focused Instruction, Caleb Daniel Holloway

2026 Scholarly Teaching Conference: Concurrent Session Papers

Recent studies have examined how students form productive conceptions of the definite integral and discussed techniques for promoting such conceptions. In this paper I present findings from interviews held with six students enrolled in second-semester calculus, four of whom had received quantitatively focused instruction on the definite integral. All six were chosen for their observed use of summation conceptions on definite integral problems, and here their conceptions are explored further. Additionally, we gain insight on their thinking regarding limits as related to the definite integral. The findings presented here add to our understanding of student thinking regarding the integral and …

Test Retakes In Introductory Math Courses, John Hird, Chris Mcclain

2026 Scholarly Teaching Conference: Poster Session Papers

In this poster, we describe the implementation of an exam retake model using specifications grading for math courses taken by non-STEM majors. This system was implemented by two faculty members over three years in two sequential courses. During that time, we tried several versions of allowed retakes, with varying restrictions on partial credit. Some of the challenges that we faced were scaling the system for use by different faculty members and with different courses, managing faculty workload on writing and grading multiple exams, and managing student expectations.

Plumbed 3-Manifolds And Neumann Moves, Noah J. Pope

Theses and Dissertations

We give a constructive proof that every weakly negative definite plumbing tree can be transformed into a negative definite one by a finite sequence of Neumann moves. The argument combines Neumann’s plumbing calculus with the diagonalization algorithm of Duchon, Eisenbud, and Neumann, which extracts the eigenvalues of the framing matrix directly from the combinatorics of the tree. We show that any positive eigenvalues are supported on linear branches and can be eliminated systematically via controlled applications of Neumann moves. This provides an explicit algorithm reducing weakly negative definite plumbing trees to negative definite ones.

Traveling Waves For A Diffusive Sir Epidemic Model With Delay In The Diffusion Term, William Barker

Faculty Scholarship

This paper investigates the existence of traveling waves in a diffusive SIR model with delay incorporated in the diffusion terms and a nonlinear incidence rate with delay. By employing a cross-iteration scheme and partial monotonicity conditions, we establish that the existence of quasi-upper and lower solutions, along with suitable super and sub-solutions, provides sufficient conditions for the existence of a traveling wavefront. This existence result is obtained via Schauder’s fixed-point theorem. Furthermore, given an appropriate basic reproduction number, the traveling wavefront transitions from the disease-free steady state to the endemic steady state. To illustrate our approach, we explicitly construct super- …

Mg-Spair: Multi-Grade Sparse-Guided Implicit Representation For Training-Data-Free Image Restoration, Jianmin Liao, Lei Huang, Ronglong Fang, Ashley Prater-Bennette, Lixin Shen, Yuesheng Xu

Mathematics & Statistics Faculty Publications

MG-SpaIR is a training-data-free framework for restoring a clean image from a single observation corrupted by a mixture of blur, downsampling, noise, and missing pixels. Building on implicit neural representations (INRs), we introduce a multi-grade residual hierarchy that progressively refines the reconstruction from low to high spatial frequencies across grades, improving representational fidelity and mitigating spectral limitations. To stabilize reconstruction optimization and suppress INR-induced artifacts, we further propose an explicit sparse proximal regularization (e.g., ℓ0 type) applied directly in the high-resolution image domain, which discourages spurious high-frequency patterns while preserving sharp structures. The resulting optimization is solved efficiently via a …

Machine Learning-Based Intrusion Detection System For Iot Networks Using The Rt-Iot 2022 Dataset, Bukunmi Ebenezer Afolabi

Theses, Dissertations and Capstones

The rapid expansion of the Internet of Things (IoT) has transformed modern computing by enabling seamless connectivity among heterogeneous devices across diverse application domains. However, this increased interconnectivity has significantly enlarged the attack surface of IoT networks, exposing them to a wide range of sophisticated cyber threats. Conventional security mechanisms often lack the capability to detect emerging attacks in real time, thereby necessitating the development of intelligent Intrusion Detection Systems (IDS) capable of accurately identifying malicious network activities. This study developed and evaluated a machine learning-based intrusion detection framework for multiclass IoT attack detection using the RT-IoT2022 dataset. The dataset …

On Values Taken By Characters Of Finite Groups, Christopher William Herbig

Graduate Research Theses & Dissertations

The characters of finite groups are a fundamental tool for analyzing the structure of finite groups and are interesting in their own right. In particular, we are concerned with how certain assumptions on the values taken by characters impact the structure of finite groups and vice versa. After establishing notation and stating some fundamental results in Chapter 2, this work begins with a presentation of results in relation to a conjecture of N. N. Hung and P. H. Tiep on fields generated by character values. In particular, we have found a large family of counterexamples to the conjecture by taking …

The Effects Of Problem-Based Learning On Mathematical Creativity And Self-Efficacy Of High School Students, Margaret Remus

Graduate Research Theses & Dissertations

Mathematical creativity is recognized as an important component of mathematics education; however, there is limited understanding of how instructional approaches can effectively support this skill among high school students, particularly through problem-based learning (PBL). This study examined the effects of problem-based learning on mathematical creativity among high school students.

This study employed a mixed-methods approach using an alternating treatment design with two groups of participants. Group 1 received the PBL intervention followed by a control condition, while Group 2 received the control condition followed by the PBL intervention. Quantitative data were collected through a mathematical creativity test measuring flexibility, fluency, …

Quantum Mechanics As A Framework For Data Assimilation And Its Application To Atmospheric Parameterization, David Freeman

Dartmouth College Ph.D Dissertations

Quantum mechanics, as a mathematical system, can be understood as a generalization of classical probability theory. Quantum Mechanical Data Assimilation (QMDA) is a method in which classical dynamical systems are embedded into a quantum mechanical setting, with an associated data assimilation scheme leveraging the operator algebraic setting. In this dissertation, the algebraic structure underlying the operator theoretic formulation of QMDA is discussed. A procedure for closure of dynamical systems based on QMDA, known as Quantum Mechanical Closure (QMCl), is then constructed, and the procedures for constructing the quantum embeddings and implementing QMCl in practice are laid out and implemented for …

Delaying Cancer Progression By Integrating Toxicity Constraints In A Model Of Adaptive Therapy, Jana L. Gevertz, Harsh Vardhan Jain, Irina Kareva, Kathleen P. Wilkie, Joel Brown, Yitong Pepper Huang, Eduardo Sontag, Vladimir Vinogradov, Mark Davies

Mathematics Sciences: Faculty Publications

Cancer therapies often fail when intolerable toxicity or drug-resistant cancer cells undermine otherwise effective treatment strategies. Over the past decade, adaptive therapy has emerged as a promising approach to postpone emergence of resistance by altering dose timing based on tumor burden thresholds. Despite encouraging results, these protocols often overlook the crucial role of toxicity-induced treatment breaks, which may permit tumor regrowth. Herein, we explore the following question: would incorporating toxicity feedback improve or hinder the efficacy of adaptive therapy? To address this question, we propose a mathematical framework for incorporating toxic feedback into treatment design. We and that the degree …

Categories, Homology And Sheaves For Hypergraphs, Robert Green

Electronic Theses & Dissertations (2024 - present)

Hypergraphs are a prominent tool for representing networks with connections among three or more entities. There is an inherent flexibility that allows hypergraphs to more naturally represent certain types of networks than graphs or simplicial complexes can on their own. This flexibility, however, comes at a cost, as there is a zoo of various categories and homology theories that are applicable to hypergraphs. The first chapter of this dissertation explores various categorical perspectives on hypergraphs, focusing on what the natural notion of morphism between hypergraphs should be. It also contains an exploration of the functoriality of vertex-edge duality in these …

On Quantum Processes And The Epistemic Constraints, Varun Immanuel Premkumar Immanuel

Electronic Theses & Dissertations (2024 - present)

This doctoral dissertation on the foundations of quantum theory tells the story of a conceptual protagonist I have called “Epistemic Constraint.” Here, epistemic constraints are the definite, intersubjectively agreeable, ordinary-language conditions under which experiments are described.

The usual formulation of the quantum measurement problem, which I call the Schrodingerian measurement problem, has the structure of an anomaly: if we take quantum theory at face value, we expect no definite values, and yet we see definite values in experiments. The responses to this problem have been either to solve it or to dissolve it. These responses, which have taken the form …

Developing Collaboration And Community Through An Online Virtual Modality: Participatory Action Research, Jennifer Reed

Doctor of Education Dissertations

This action research study examined educators’ perceptions of collaboration and community within a virtual professional learning community and investigated how participation influenced collaborative actions over time. The study also compared the needs and goals of secondary and postsecondary educators participating in a shared VPLC model. Grounded in the Community of Inquiry and Online Collaborative Learning theoretical frameworks, this study addressed a growing need for effective, flexible professional learning structures that support collaboration across educational contexts. Data were collected from six educators, including three secondary and three postsecondary instructors, through pre- and post-administration of the Professional Learning Community Assessment–Revised, recorded virtual …

Pursuer Evader Surveillance Game, Zijie Mu

Honors Theses

This thesis studies the pursuer evader surveillance game with a triangular obstacle in the short-term. In the game, the pursuer aims to maintain surveillance of the evader as long as possible while the evader aims to break surveillance in a finite time. We classify the player strategies into  ideal ones and best admissible ones. The outcome of the game is determined by line of sight. We reduce the 4D game to a 3D game with a boundary separating two different local regimes. When the evader starts inside the threshold, we show that there exists an admissible evader that maintains an …

Spillover Effects Of Medicare Advantage On Fee-For-Service Post-Acute Care Spending, Nyel Bangash

Honors Theses

Does the growth of Medicare Advantage reduce fee-for-service post-acute care spending through practice-pattern spillovers, or do observed spending differences primarily reflect favorable selection? Using a county-level panel of roughly 2,700 counties (2014–2023) and a two-way fixed effects specification, I find that a one percentage-point increase in MA penetration is associated with $9.54 less per-capita standardized FFS spending. Spending per episode falls while participation rates remain stable, consistent with practice-pattern spillovers rather than compositional changes from selection. Welfare indicators from County Health Rankings, CDC PLACES, and CMS Care Compare show no evidence that spending reductions harm health or care quality. The …

Education And The Maternal Childcare Gap: Evidence From The Uk Covid-19 Pandemic, Lara Bakhaya

Honors Theses

This paper examines whether college education shapes working mothers’ access to remote work, flexible working, and childcare hours in the United Kingdom, and whether COVID-19 school closures amplified these inequalities. Using data from the UK Time Use Survey (2016–2021), a repeated cross-sectional diary dataset spanning the pre-pandemic period and five COVID-19 waves, this paper estimates weighted logistic and ordinary least squares regressions on a sample of married or cohabiting, employed mothers. School closures serve as a natural experiment, providing an exogenous shock to caregiving demands that affected all mothers simultaneously regardless of education level. College education significantly predicted working from …

Mechanisms Driving Disparities In Income Mobility Across The Income Distribution, Joe Larkins

Honors Theses

This study examines intergenerational income persistence across the income distribution, testing whether mechanisms driving inequality differ between families in the top and bottom halves of the income distribution. Using data from the National Education Longitudinal Study of 1988 (NELS:88), a nationally representative longitudinal survey of 8th grade students and their parents, this research estimates an interaction model comparing parental income effects for children in advantaged versus disadvantaged economic circumstances. The analysis reveals that a $1,000 increase in parental income yields eight times greater income gains for children in the bottom half of the distribution compared to those in the top …

From Shock To Routine: The Evolving Impact Of Shutdown-Related Sentiment On Stock Markets, Yiran Shao

Honors Theses

To address gaps in existing research, this paper selects two U.S. government shutdown periods, 2018-2019 and 2025, as research samples to explore the effect of policy uncertainty on sentiment. This paper primarily analyzes the following two research questions.

First, what is the correlation between government shutdown-related sentiment during the shutdown period and daily market fluctuations? Specifically, can the sentiment index constructed from shutdown-related news effectively predict the next-day stock return during the event period?

Second, does the market have a learning effect? That is, between 2018-2019 and 2025, has the relationship between shutdown-related emotions and market outcomes weakened, shortened the …

Exponentially Many Correspondence Colourings Of Planar And Locally Planar Graphs, Luke Postle, Evelyne Smith-Roberge

Faculty Publications – Mathematics

We show that there exists a constant c>0 such that if G is a planar graph with a 5-correspondence assignment (L,M), then G has at least 2c⋅v(G) distinct (L,M)-colourings. This confirms a conjecture of Langhede and Thomassen. More broadly, we introduce a general method showing how hyperbolicity theorems for certain families of critical graphs can be used to derive lower bounds on the number of colourings of the associated class of planar graphs. Hence our main result follows from this method plus a technical theorem (that we proved in a previous paper) involving the hyperbolicity …