Mathematics Commons^™

Model-Theoretic Arguments In Philosophy, Peter Susanszky

Dissertations, Theses, and Capstone Projects

This dissertation is on model-theoretic arguments in philosophy, especially those of Quine, Davidson, and Putnam. In the first part, to ground the debate, I give a rigorous introduction to the salient parts of first-order model theory. I start the second part by giving an introduction to Quine's philosophy, and how the model-theoretic arguments fit into it. After considering how Donald Davidson adopted the Quinean lesson, I move on to Putnam's model-theoretic arguments. Putnam's spin on these model-theoretic considerations significantly departs from Quine and Davidson, while retaining many of the core ideas. Most importantly, I argue that the target of Putnam's …

Machine Learning For Predictive Energy And Emissions Modeling Of Vehicles And Power Grids In The United States, S M Tanvir Faysal Alam Chowdhoury

Dissertations

The environmental benefits of electric vehicle (EV) adoption depend on more than replacing internal combustion engine vehicles with electric powertrains. EV adoption reshapes electricity demand, interacts with regional generation mixes, and influences travel behavior and congestion, creating a coupled transportation-energy system in which vehicle and power-plant emissions must be evaluated together. This dissertation develops machine-learning frameworks for predicting energy consumption and emissions from vehicles and power grids under rising EV adoption. The first component forecasts grid emissions from EV charging. Using simulation data from NREL's Cambium database, a Prophet-based time-series framework predicts carbon dioxide, nitrous oxide, and methane emission rates …

Spatial Prediction Under Uncertainty: Methodological And Computational Advances In Bayesian Maximum Entropy, Kinspride K. Duah

All Graduate Theses and Dissertations, Fall 2023 to Present

Environmental decisions such as infrastructure design, water management, and snow load estimation depend on spatial data that are often incomplete or uncertain. In many cases, measurements are not exact values but ranges, reflecting limitations in data collection methods. Traditional mapping techniques typically simplify these uncertain measurements, which can lead to less accurate predictions. This dissertation introduces improved statistical tools for making spatial predictions when data are uncertain or partially known. By utilizing a framework called Bayesian Maximum Entropy (BME), this research demonstrates how exact measurements and range-based data can be combined in a mathematically consistent way. The work demonstrates that …

Learning Latent Structure In High-Dimensional Data Via Geometry And Graphs, Haozhe Chen

All Graduate Theses and Dissertations, Fall 2023 to Present

Modern datasets often contain many measured variables for each observation, such as gene-expression levels, brain activity signals, or features in tabular data. These data are also often noisy, meaning that useful patterns are mixed with measurement error or irrelevant variation. Although such datasets can appear complex, they are frequently represented by simpler hidden structures, such as trajectories, clusters, or relationships between observations. This dissertation develops methods for uncovering these hidden structures by learning geometric and graph-based representations directly from data. The first part introduces Functional Information Geometry, which represents local patterns in high-dimensional data using functional features and constructs a …

Unifying And Expanding Global And Local Variable Importance Methods For Explainable Machine Learning, Kelvyn K. Bladen

All Graduate Theses and Dissertations, Fall 2023 to Present

Machine learning methods are powerful analytical tools used across all scientific disciplines and many other fields of investigation for prediction and inference from diverse data sources. Despite their broad applicability, machine learning methods are often highly complex and difficult to interpret. Developing a greater understanding of which variables most influence a response is essential for increasing the interpretability of these models and supporting informed decision-making. This research focuses on improving how we evaluate the importance of these variables.

One common approach is to shuffle the values of a variable and see how much the model accuracy gets worse. Another approach …

Applications Of Machine Learning To Gas Plume Analysis In Longwave Infrared Hyperspectral Images, Scout C. Jarman

All Graduate Theses and Dissertations, Fall 2023 to Present

Each pixel from a hyperspectral camera measures the intensity of light over a continuous range of wavelengths, which is in contrast to traditional color cameras, which just measure the intensity of red, green, and blue wavelengths of light. Longwave infrared hyperspectral images can be used to detect gases from a distance by measuring how different materials emit and absorb heat. This makes them useful for applications such as monitoring industrial emissions or locating hazardous gas leaks. In practice, however, gas signatures in these hyperspectral images are often weak and easily obscured by variations in the background scene, making reliable identification …

Mapler: An R Package For Estimating The Impact Of Climate Change On Maple Syrup Production, Matthew T. White

All Graduate Theses and Dissertations, Fall 2023 to Present

Successful maple sap tapping depends on the freeze/thaw cycle (i.e., temperatures fluctuating above/below freezing) during the winter and spring. Climate change threatens to alter the timing and duration of the tapping season. This necessitates research into how maple sap tapping will be impacted by climate change in order to help maple syrup producers prepare for the future. We define a sap day as a day where the freeze/thaw cycle occurred. Using information climate scientists use to predict future temperatures, we calculate how many sap days could occur each year. We develop software to analyze these sap day calculations to determine …

Advantages Of Dynamic Representation For Related Rates Problems In Calculus, Eri Osuna

Electronic Theses, Projects, and Dissertations

Related-rates problems are a standard yet persistently difficult topic in first-semester calculus. Research increasingly recommends dynamic visualization tools such as GeoGebra, but direct comparisons of static and dynamic representations in related-rates settings remain scarce. This qualitative study examines how representation type shapes the quality of students' reasoning and their perceived experience during related-rates problem solving. Six mathematics students who had completed Calculus —a group of four undergraduates and a pair of graduate students—completed a static sliding-ladder task and a dynamic airplane-and-camera task supported by an interactive GeoGebra applet, followed by an interview. Findings indicate that the two representations supported reasoning …

Geodesic Completeness And The Hopf-Rinow Theorem, Christopher Farias

Electronic Theses, Projects, and Dissertations

Differential geometry is concerned with the properties of calculus and geometry on curved n-dimensional manifolds. As a result, thinking about such a space often runs counter to the Euclidean geometer's intuition of distances, angles, and transformations. This thesis aims to build up to proving an important result in the study of Riemannian manifolds: the Hopf-Rinow theorem.

In Chapter 2, we begin by defining what a manifold is and showing that the collection of directional derivatives at a point on the manifold spans a tangent vector space. After defining a basis and a metric for this space, in Chapter 3, we …

Comparing 3-Connectedness And Roundness In Matroid Theory, Blanca Delia Larios

Electronic Theses, Projects, and Dissertations

A matroid is a discrete mathematical object that abstracts and connects the various notions of independence found throughout mathematics. Such notions of independence include linear independence, algebraic independence, as well as notions of independence that arise in graph theory. There are many broad classes of matroids. Important examples include binary matroids, graphic matroids, regular matroids, uniform matroids, and various levels of connected matroids. Some of the most important problems in matroid theory involve characterizing classes of matroids so that such characterizations can be used to prove results concerning these matroid classes. This thesis is a study of two important classes …

Student Attitudes And Perceptions Of Proof In Mathematics, Emelin G. Sibrian Marquez

Electronic Theses, Projects, and Dissertations

Traditionally, mathematical proof is viewed primarily as a tool for validation or verification. However, proof holds many other important roles such as discovery, reasoning, explanation, and justification. For many students, these other roles are not always obvious. Those encountering rigorous proof for the first time often find the process abstract, intimidating or disconnected from their previous learning. This disconnect can lead to negative attitudes as students transition from computational mathematics to advanced proof-based mathematics. Utilizing a mixed-methods approach, this study examined undergraduate and graduate mathematics students at a Hispanic-Serving Institution (HSI) in Southern California. We investigated what students perceive the …

Error-Tolerant Metric Dimension, Leander Ten Hoff

Master's Theses

Metric dimension is a graph parameter that measures the smallest distance-based unique coordinate system on a graph. Fault-tolerant metric dimension is a variant that requires redundancy. Inspired by this idea, we propose and investigate a new variant we call error-tolerant metric dimension. We first prove some fundamental results to gain familiarity with this more complex variant. Then, we use these tools to study relative behaviors of error-tolerant metric dimension. We prove explicit computations for simple graph families, including bipartite complete graphs and paths. Finally, we extend extremal results from previous papers on metric dimension, including a full correction of an …

Probabilistic Metric Dimension, Kyle Worley

Master's Theses

The metric dimension of a graph G is the smallest number of unique vertices (often called “marked vertices” or “landmarks”) such that each vertex has a unique distance to the marked vertices. If a set of vertices S ⊆ V (G) when marked gives all vertices unique distances, it is called a resolving set. Hence if G is a graph with a resolving set S, then for all v, u ∈ V (G) there exists s ∈ S such that d(v, s) ̸= d(u, s). The metric dimension dim(G) = min(|S|). Two widely studied variants exist, one where edges are …

The Relationship Between Foliations Of The Plane, Kaplan Diagrams, And Non-Hausdorff 1-Manifolds, Monique Justine Howe

Master's Theses

In this paper we seek to explain the relationship between foliations of the plane, Kaplan diagrams, and simply connected non-Hausdorff 1-manifolds with countable basis that are orientable with an ordering on branch points. We will walk the reader through the definitions of all of these objects, and provide examples with a focus on the motivating example of the Reeb foliation. This paper will describe and define the known bijection from the set of foliations of the plane, F, to the set of Kaplan diagrams, K, its inverse, and the one-to-one map from F to the set of simply connected non-Hausdorff …

Monoids With K-Strictly Local Geodesic Languages, William M. Hong

Master's Theses

A paper by Gilman, Hermiller, Holt and Rees (2011) prove that a group G finitely generated by X is virtually free if and only if the language of geodesics words in G over X is k-strictly local if and only if the word problem is context-free. If M is a monoid finitely generated by X, we define Γu(M,X) to be the underlying, undirected graph of Γ(M,X). We prove that if the language of geodesics in Γu(M,X) based at the identity is k-strictly local, then the monoid and semi-group word problems are context-free.

Rewriting The Calculus 3 Workshop At San José State University, Moorea Lippert

Master's Theses

Calculus 3 at San José State University is a very important course for any student pursuing a STEM degree. However, the course is very challenging, so the university offers a supplemental workshop course for Calculus 3 students. Unfortunately, this workshop has not aligned with modern research in math education. This thesis documents efforts to recreate the workshop to better align with this modern research.

Fermat's Last Theorem Over Quadratic Number Fields, Richie Tay

Master's Theses

Fermat’s Last Theorem states that the equation xn + yn = zn has no nontrivial integer solutions for n ≥ 3. While the rational case is completely solved, the situation over quadratic number fields is not as straightforward. For some exponents nontrivial solutions exist, while for others they do not. This thesis studies Fermat-type equations over quadratic number fields, combining classical methods with more modern techniques from the theory of elliptic curves. The thesis first reviews the rational cases n = 2, n = 3, n = 4, then develops the necessary background on quadratic number fields, including rings of …

Rational Non-Euclidean Triangles And Elliptic Curves, Tyler Morales

Master's Theses

We say a triangle △ is rational if its side lengths are rational. For a geometry of constant curvature κ, we say △ is a rational triangle if the generalized tangents of its side lengths are rational. We are able to parameterize rational triangles having the same inradius and semiperimeter on an plane curve, write a bijection between the rational points on this curve and triples of side lengths, and show that this curve is an elliptic curve. Then, we write a general transformation for our plane curve to short Weierstrass form to compute ranks and add points easily using …

Fundamental Solutions To The Fractional Heat Operator, Jacob Flores

Math Theses

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …

Pseudodifferential Absorbing Boundary Conditions For Waves, Lauren Taylor

Mechanical Engineering Theses

Absorbing boundary conditions (ABCs) are required to truncate the computational domain when performing Finite Element Analyses of exterior acoustic problems where physical domain is unbounded. ABC is applied on a fictitious boundary containing the scatterer and ideally allows outgoing waves to leave without non-physical reflections. Preventing artificial reflections is essential to benefit from the accuracy of the numerical method used. Otherwise, ABC acts as a reflective surface, and artificial reflections distort the solution in the entire domain which is not recoverable by any type of refinement. Pseudodifferential ABCs were used to describe the Dirichlet-to-Neumann map which maps known boundary values …

Analytic Solutions To Oceanographic Kinematic Property Equations, James Turbett

Theses and Dissertations

With the recent advent of large submesoscale drifter deployments, interpreting the timeseries of kinematic properties (KP) - divergence, vorticity, shear, and normal strain rates observed therein is a pressing issue. The evolution equations for the four KP are derived from the two-dimensional momentum conservation equations. The resulting equations, a nonlinear coupled system of ordinary differential equations, capture the submesoscale motion of drifters along fixed ocean surfaces, with time and space scales on the order of days and kilometers. This thesis builds theory around KP timeseries behavior by solving for the analytic solutions in the particular cases that one KP is …

Cdt-1d Cnn Integration With Simpson-Sobolev Regularization For High-Frequency Options Trading: With Fem-Based Heston Option Pricing, Daniel M. Margolis, Johannes Tausch, Arthur K. Selender

Mathematics Theses and Dissertations

This dissertation presents a computational framework for high-frequency options trading that combines Cross-Data-Type 1-D Convolutional Neural Networks (CDT-1D CNN) with Simpson-Sobolev regularization for directional prediction, and finite element methods (FEM) for realistic option pricing during backtesting. The core innovation lies in developing a mathematically rigorous regularization approach that maintains the adaptability of modern deep learning while enabling accurate evaluation through stochastic volatility models. The primary contribution is the Simpson-Sobolev regularization scheme, which extends traditional Sobolev regularization by incorporating Simpson’s rule for numerical integration. This approach achieves higher-order accuracy in approximating the Sobolev norms that control function smoothness. Simpson’s rule attains …

On The Extension Conjecture For Koszul Algebras, Benjamin Kaufman

Dissertations - ALL

We investigate questions related to the extension conjecture for finite-dimensional algebras. Suppose $\Lambda = \kk Q/I$ is a finite-dimensional algebra given by a quiver with relations. The extension conjecture asserts that if $S$ is a simple right $\Lambda$-module corresponding to a vertex with a loop, that is, $\Ext_\Lambda^1(S,S)\neq 0$, then $\Ext^n_\Lambda(S,S)\neq 0$ for infinitely many $n$. We show that one can obtain information about the non-vanishing of the extensions $\Ext^n_\Lambda(S,S)$ by looking at the graded Cartan matrix and its inverse. We use this connection to prove the extension conjecture for standardly graded $\kk$-algebras on two vertices with Loewy length at …

Image Restoration Regularized By Structured Sparsity Promoting Functions: Theory, Algorithms, And Applications, Jianchen Wei

Dissertations - ALL

Image restoration aims to recover an unknown clean image from observations degraded by blur, subsampling, missing pixels, or noise. This dissertation develops and analyzes variational image restoration models based on structured sparsity promoting functions (SSPFs) and their Moreau-enveloped regularization. The main objective is to design nonconvex regularization models that promote structured sparsity in transformed image representations while remaining analytically tractable and computationally effective. The first part of the dissertation studies an SSPF regularized image restoration model. Under the identifiability condition $\ker(A)\cap \ker(B)=\{\bm0\}$, where $A$ is a linear forward degradation operator and $B$ is a transform operator used to extract image …

Bias, Structure, And Inference In Applied Network Analysis, Anna Vasenina

Dartmouth College Ph.D Dissertations

This dissertation develops mathematical and statistical methods for extracting reliable information from network data across biological applications, with an emphasis on understanding what observed network structure can and cannot resolve. The first study leverages protein–protein interaction network topology in the c-di-GMP signaling system of Pseudomonas fluorescens, showing that node centrality measures accurately classify protein domain types and that physical interaction structure contributes statistically significant predictive power for biofilm formation phenotypes across nearly 200 environments, while gene expression does not. The second study examines sampling bias in lemur-plant trophic interaction networks in Madagascar, demonstrating that differential detection of diurnal versus …

Uniform Stability Of Katyusha In Strongly-Convex Settings, Don Li

University Honors Theses

Acceleration of convergence and reduction of variance constitute a trade-off in the design of stochastic optimization machine learning algorithms. Katyusha was introduced to address this trade-off, synthesizing Nesterov Accelerated Gradient (NAG) and Stochastic Variance-Reduced Gradient (SVRG) into a single first-order optimizer with promising empirical performance. However, the generalization properties of Katyusha remain largely unexplored. We conjecture that, in the smooth quadratic regime (i.e., under assumptions of strong convexity and smoothness of the loss function, and boundedness of gradients), Katyusha is uniformly stable in the sense of Bousquet and Elisseeff. Instantiating our framework for NAG, we extend the use of Lyapunov …

The Method Of Periodic Averaging Applied To Reduced Coupled Mode Theory Models For Fiber Laser Amplifiers, Rebecca Nicole Bryant

Dissertations and Theses

Fiber laser amplifier (FLA) models are often implemented without rigorous mathematical justification or thorough numerical validation. Without a proper theoretical basis for assumptions and approximations, or a technical analysis of model performance, there is significant uncertainty about the limitations of any given reduced model and its suitability for an application. This research aims to address the lack of comprehensive assessment of FLA models by directly comparing distinct models and recommending a mathematical alternative to replace heuristic model-reduction techniques. The work in this dissertation is divided into two projects: a comparative study that uses existing FLA models to assess the validity …

The Malaugh Operations And The Quadratic Invariant Gaps Of $\Si_4^C$, Forrest M. Hilton

ETDs from 2020-2029

Connected Julia sets of polynomials generally correspond to laminations, sets of chords of the unit disc that reflect the dynamics of the Julia set. If the circle is measured in revolutions and the polynomials studied are of degree $d$, then the dynamics on the lamination is given by the covering map $\sigma_d(t) := td \pmod 1$ where chords are mapped by their end points. Every lamination has at least one laminational invariant set, which is loosely an invariant complementary component of the lamination. That set has a significant impact on the shape of the Julia set. James Malaugh showed how …

The Impact Of Evolution And Strong Allee Effects On The Dynamics Of A Discrete-Time Predator-Prey System, Neerob Basak

Doctoral Dissertations

This dissertation investigates the dynamics of discrete-time predator-prey systems, focusing on both evolutionary responses and ecological interactions. The first part of this dissertation, in Chapters 2 and 3, explores how evolutionary processes, particularly the development of resistance to toxicants in predators, influence the persistence and stability of predator-prey populations. In this part, we extend the predator-prey model developed in Ackleh et al., 2019 to incorporate the evolution of a predator's resistance to toxicant effects. We consider three cases: (1) lethal effects, where the toxicant directly influences the predator's survival; (2) sublethal effects, where the toxicant impacts the predator's fecundity, and …

The Interplay Of Seasonality, Evolution, And Density-Dependence In Discrete-Time Predator-Prey Dynamics, Narendra Pant

Doctoral Dissertations

We extend the discrete-time mathematical models developed in (Ackleh et al., 2019) and (Ackleh et al., 2024) to account for seasonal prey reproduction and build a class of discrete-time predator-prey seasonal models. Each model distinguishes between breeding and non-breeding seasons, representing prey reproduction as a periodic function of period 2. Altogether, three different models are analyzed. In the first part, we extend the predator-prey model from (Ackleh et al., 2019) to incorporate seasonality. We study the resulting dynamics and show that when the inherent reproduction number of the prey and the invasion reproduction number of the predator are larger than …