Mathematics Commons^™

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 …

Reduced Product Type Monoid-Module Extensions, Darryl Jent

Dissertations

In 1955, I. M. James introduced the James Construction, a free topological monoid that models the loops on the suspension of a given space. In 1969, S. Y. Husseini generalized this idea to RPT monoids: topological monoids with a free-like monoid structure that can be used to model a broader class of loop spaces. In order to prove that these topological monoids are models of loop spaces, both I. M. James and S. Y. Husseini constructed contractible spaces on which these topological monoids act. We define a topological module as a space equipped with an action by a topological monoid. …

From Total Domination To Graph Coloring, Sawyer Isaac Osborn

Dissertations

A question involving a chess piece called a prince on the 8×8 chessboard leads to a concept in graph theory involving total domination. We say a vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A subset S of the vertex set of a graph G is a total dominating set for G if every vertex in G is totally dominated by at least one vertex of S. If S is a total dominating set of G, then σ_S(v) denotes the number of …

Differential-Geometric Methods For Neural Signed Distance Fields: Parameterized Surface Extraction And Curvature Regularization For Cad Models, Haotian Yin

Dissertations

Neural signed distance fields have emerged as a powerful framework for representing three-dimensional geometry through continuous and differentiable neural functions. Their flexibility, resolution independence, and compatibility with gradient-based optimization make them especially attractive for surface reconstruction and geometric learning. However, despite these advantages, two fundamental challenges remain for engineering-grade applications. First, higher-order geometric properties such as curvature are difficult to model reliably during training and often require computationally expensive second-order differentiation. Second, while neural signed distance fields provide implicit surface representations, they do not directly yield a globally consistent forward map or parameterization for downstream geometric processing.

This dissertation addresses …

Math Anxiety, Math Self-Concept And Math Self-Efficacy: A Study Of The Jingle-Jangle Fallacies, Marsha Natasha Durrant-Walker

Dissertations

Problem

The overlap and lack of clear distinction among the constructs of math anxiety, math self-concept, and math self-efficacy presents issues for research and practice. The literature reveals that math anxiety is closely linked to math self-concept (Klee et al., 2022). Additionally, math self-concept and math self-efficacy often overlap and are not easily distinguishable (Kranzler & Pajares, 1997; Pajares & Miller, 1994; Pajares & Urdan, 1996). Each of these constructs has been shown to play a critical role in student math achievement (Timmerman et al., 2016). -- When constructs are not defined or measured distinctly, inconsistencies may emerge in research …

Problems In Extremal Graph Theory And Spectral Graph Theory, Fareeha Jamal

Dissertations

Spectral graph theory is a subfield of algebraic graph theory that studies the matrices associated with graphs. It lives at the nexus of Linear Algebra and Combinatorics. Many intriguing results in the domains of Matrix Theory and Combinatorics have come from studying the eigenvalues of graph matrices; in fact, several open problems in both areas have been resolved. Beyond its theoretical appeal, spectral graph theory has found meaningful applications in theoretical chemistry, particularly in the mathematical classification of chemical graphs. These classifications underpin quantitative structure–property relationships (QSPRs), facilitating the prediction of physicochemical properties such as enthalpy of vaporization, molar refractivity, …

The Global Phase Space Of The Three-Vortex Interaction System And Its Application To Vortex-Dipole Scattering, Atul Anurag

Dissertations

This dissertation presents a global reduction of the classical three-vortex problem that is free from coordinate singularities, enabling a comprehensive analysis of the system's dynamics across all circulation regimes.

To achieve this, a two-step symplectic reduction procedure is developed. The first step introduces Jacobi coordinates adapted to the symplectic structure of the vortex system, and the second applies a Lie-Poisson reduction to the resulting system. This formulation eliminates the non-physical singularities associated with collinear vortex configurations and facilitates a global phase space analysis, including a detailed and novel investigation of bifurcations.

Within this reduced framework, all relative fixed points are …

Reduced Order Models Of Hydrodynamically Interacting Flapping Wings, Jose Pabon

Dissertations

Fish schools exhibit a collective behavior and self-organization that is mediated by hydrodynamic interactions between individual fish. However, the long-time evolution of hydrodynamically interacting collectives is challenging to investigate due to the persistent influence of long-lived vortical structures, and the high-resolution requirements of direct numerical simulation at large Reynolds numbers. Reduced-order models have therefore played an important role in theoretical investigations of collectives of swimming bodies. The main results detailed herein are several new reduced-order models of swimmers that self-propel by flapping, i.e., by executing a prescribed periodic rigid body motion. The models are extensions of a discrete-time dynamical system …

An Alternative Approach To Non-Relativistic Quantum Mechanics In Curved Space, Robert A. Hulsey

Dissertations

In the research presented in this dissertation, we propose an alternative formulation of non-relativistic quantum mechanics in curved spaces (Riemannian manifolds). Some toy quantum models (2D quantum harmonic oscillator in Poincaré half-plane model and the flat chart model of hyperbolic 2-space) are studied to understand the physical implications of this alternative formulation.

Multiple Monochromatic Subgraphs In Edge-Colored Graphs, Emma Felicity Jent

Dissertations

Ramsey theory, though a relatively young branch of mathematics, has captivated the attention of graph theorists, combinatorialists, and theoretical computer scientists alike through its raw beauty, versatility, and powerful applications. Before it emerged as a branch of mathematics, the central idea of Ramsey theory appeared in the form of three lemmas in three separate papers by three different mathematicians working on three distinct areas of research. The first such lemma was published by David Hilbert in 1892, followed by the second lemma published by Issai Schur in 1916. However, Frank Ramsey’s renowned lemma, published in 1930, compelled mathematicians to establish …

On The Design Of A Framework For Large-Scale Exploratory Graph Analytics, Oliver Andres Alvarado Rodriguez

Dissertations

Large-scale exploratory graph analytics merges data science with high-performance computing to extract critical insights from network-representable data. Data scientists routinely analyze data from the natural, social, and computing sciences by representing it as networks, or graphs, where objects become vertices and their relationships become edges. This representation allows data scientists to add graph analytics to their toolbox. However, designing tools for large-scale exploratory graph analytics is challenging due to the complexities of graph algorithms, such as high communication in distributed systems and large memory demands. These challenges can lead to overly complex software, which limits usability and development to a …

From Neural Networks To Large Language Models: Innovations In Financial Ai, Mathematical Reasoning, And Structured Data Representation, Junyi Ye

Dissertations

This dissertation explores the evolution and application of artificial intelligence techniques across three critical domains: financial modeling, mathematical reasoning, and structured data analysis. The dissertation presents seven research projects that chart a progression from specialized neural architectures to sophisticated large language models (LLMs), contributing novel methodologies and frameworks at each stage.

In the financial domain, the research first introduces TS-Mixer, a MLP-based architecture for time-series forecasting that captures both feature relationships and temporal dependencies through a simple yet effective design, outperforming more complex models in S&P500 index prediction. The dissertation then presents DySTAGE, a dynamic graph representation learning framework that …

Quadratic Stochastic Processes: Algebraic Structures And Their Applications, Taimun Saleh Qaisar

Dissertations

This research focuses on the algebraic structures of the Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠). In this work, we first study 𝜉𝑎- Quadratic Stochastic Operators (𝑄𝑆𝑂𝑠) linked to the partition P3. We simultaneously discuss the dynamics of the obtained 𝑄𝑆𝑂𝑠. Moreover, algebraic structure of the associated genetic algebra is studied. Further, we build Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠) using the given Markov processes. Consequently, we obtain an ordinary differential equation for the resultant Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠). Besides, we apply the solution of this ordinary differential equation for the option pricing problem. Thereafter, we construct Quadratic Stochastic Processes (𝑄𝑆𝑃𝑠) in three-dimensional space by …

The Implementation Of Stream In Mathematics Classrooms In Abu Dhabi Primary Schools: Prospects, Priorities, Processes, And Problems, Nadeia Rashed Alalawi

Dissertations

This study shed light on implementing science, technology, reading and writing, engineering, art, and mathematics (STREAM) in mathematics classrooms in Abu Dhabi primary schools. The study aimed to explore mathematics teachers’ views on the prospects, priorities, processes, and problems of the implementation of STREAM in their classrooms. The study employed qualitative methods to collect and analyze data to support the findings using interviews, classroom observations, and document analysis. Fifteen in-service mathematics teachers were interviewed to explore their views about the application of STREAM in their classrooms regarding prospects, priorities, processes, and problems (4 Ps). Then, three of them were observed …

A Micromagnetic Study Of Skyrmions In Thin-Film Multilayered Ferromagnetic Materials, Nicholas J. Dubicki

Dissertations

Magnetic skyrmions are topologically protected, localized, nanoscale spin textures in non-centrosymmetric thin ferromagnetic materials and heterostructures. At present they are of great interest to physicists for potential applications in information technology due to their particle-like properties and stability. In a system of multiple thin ferromagnetic layers, the stray field interaction was typically treated with various simplifications and approximations. It is shown that extensive analysis of the micromagnetic equations leads to an exact representation of the stray field interaction energy in the form of layer interaction kernels, a so-called 'finite thickness' representation. This formulation reveals the competition between perpendicular magnetic anisotropy …

A Microgenetic Learning Analysis Of Contextuality In Reasoning About Exponential Modeling, Elahe Allahyari

Dissertations

This work explores the complex cognitive processes students engage in when addressing contextual tasks requiring linear and exponential models. Grounded within Piagetian constructivism and the Knowledge in Pieces (KiP) epistemological perspective (diSessa, 1993, 2018), this empirical study in a clinical setting develops a Microgenetic Learning Analysis (MLA) of the reasoning of 14 students from an Algebra II course. It reveals the critical role of cognitive disequilibrium as an essential cognitive state for conceptual development and the process of reorganizing knowledge systems. The study uncovers the fluctuations in students’ reasoning patterns and the significant impact on students’ reasoning patterns of task-specific …

On Near-Linear Cellular Automata Over Near Spaces, Abdul-Rahman M. Nasser

Dissertations

Cellular Automata can be considered as examples of massively parallel machines. They are computational mathematical objects consisting of a grid of cells, each of which can exist in a finite number of states. These cells evolve over discrete time steps according to a set of predefined rules based on the states of neighboring cells. The notion of cellular automata was first introduced by Ulam and von Neumann and then popularized by John H. Conway in the 1970s with one of the most famous examples being The Game of Life.

This research builds on and generalizes the work of Tullio Ceccherini-Silberstein …

Mathematical Modeling For Dental Decay Prevention In Children And Adolescents, Mahdiyeh Soltaninejad

Dissertations

The high prevalence of dental caries among children and adolescents, especially those from lower socio-economic backgrounds, is a significant nationwide health concern. Early prevention, such as dental sealants and fluoride varnish (FV), is essential, but access to this care remains limited and disparate. In this research, a national dataset is utilized to assess sealants' reach and effectiveness in preventing tooth decay, particularly focusing on 2nd molars that emerge during early adolescence, a current gap in the knowledge base. FV is recommended to be delivered during medical well-child visits to children who are not seeing a dentist. Challenges and facilitators in …

Finding Combinatorial Patterns In Real Valued Omics Data, Kenneth Smith

Dissertations

Precision medicine is a healthcare approach which tailors disease prevention and treatment to an individual, based on their genetics, environment, lifestyle, and physiological state. These factors interact to produce biological changes that can be measured to produce data called omics, and include genomics, lipidomics, and proteomics. Despite the abundance of omics data and analysis techniques, researchers still struggle to identify biological findings that replicate across data sets and translate into clinical applications. In this dissertation, we employ combinatorial optimization techniques to improve upon three steps in the precision medicine analysis pipeline: 1) data cleaning, 2) community detection, and 3) feature …

On The Projections And Unitary Groups Of Unital C*-Algebras, Fouzia Shaheen

Dissertations

H. Dye proved that the unitary group in a factor determines the algebraic type of that factor. Al-Rawashdeh, Booth and Giordano established that, for a large class of simple unital C*-algebras, an isomorphism between the unitary groups induces an isomorphism between their K₀-ordered group and 1-groups. Then using the results of Dadarlat-Elliot-Gong and Kirchberg-Phillips, the C*-algebras are isomorphic. Dye introduced special projections P_i,j (a) of the matrix algebra M_n(A), and he used it as a main tool to establish his results in the case of von Neumann factors. Precisely, in case of von Neumann algebra, …

Exploring Topological Phonons In Different Length Scales: Microtubules And Acoustic Metamaterials, Ssu-Ying Chen

Dissertations

The topological concepts of electronic states have been extended to phononic systems, leading to the prediction of topological phonons in a variety of materials. These phonons play a crucial role in determining material properties such as thermal conductivity, thermoelectricity, superconductivity, and specific heat. The objective of this dissertation is to investigate the role of topological phonons at different length scales.

Firstly, the acoustic resonator properties of tubulin proteins, which form microtubules, will be explored The microtubule has been proposed as an analog of a topological phononic insulator due to its unique properties. One key characteristic of topological materials is the …

Topological Data Analysis Of Convolutional Neural Networks Using Depthwise Separable Convolutions, Eliot Courtois

Dissertations

In this dissertation, we present our contribution to a growing body of work combining the fields of Topological Data Analysis (TDA) and machine learning. The object of our analysis is the Convolutional Neural Network, or CNN, a predictive model with a large number of parameters organized using a grid-like geometry. This geometry is engineered to resemble patches of pixels in an image, and thus CNNs are a conventional choice for an image-classifying model.

CNNs belong to a larger class of neural network models, which, starting at a random initialization state, undergo a gradual fitting (or training) process, often a …

Topics On Asymmetric Classification, Subhrasish Chakraborty

Dissertations

Asymmetric classification refers to a situation where the cost of misclassifying one class is significantly higher than the cost of misclassifying the other class. This problem is common in many real-world scenarios, such as medical diagnosis or fraud detection. In this dissertation two of the common types of asymmetric classification problems have been dealt with — imbalanced classification and ordinal classification. An example of imbalanced classification is to detect fraudulent credit card transactions where the distribution of the normal and fraud transactions are extremely skewed. On the other hand, ordinal classification, also known as ordinal regression, is widely used in …

Topological Data Analysis Of Weight Spaces In Convolutional Neural Networks, Adam Wagenknecht

Dissertations

Convolutional Neural Networks (CNNs) have become one of the most commonly used tools for performing image classification. Unfortunately, as with most machine learning algorithms, CNNs suffer from a lack of interpretability. CNNs are trained by using a training data set and a loss function to tune a set of parameters known as the layer weights. This tuning process is based on the classical method of gradient descent, but it relies on a strong stochastic component, which makes the weight behavior during training difficult to understand. However, since CNNs are governed largely by the weights that make up each of the …

Irregular Domination In Graphs, Caryn Mays

Dissertations

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters …

Zonality In Graphs, Andrew Bowling

Dissertations

Graph labeling and coloring are among the most popular areas of graph theory due to both the mathematical beauty of these subjects as well as their fascinating applications. While the topic of labeling vertices and edges of graphs has existed for over a century, it was not until 1966 when Alexander Rosa introduced a labeling, later called a graceful labeling, that brought the area of graph labeling to the forefront in graph theory. The subject of graph colorings, on the other hand, goes back to 1852 when the young British mathematician Francis Guthrie observed that the countries in a map …

Using Visual Imagery To Develop Multiplication Fact Strategies, Gina Kling

Dissertations

The learning of basic facts, or the sums and products of numbers 0–10 and their related differences and quotients, has always been a high priority for elementary school teachers. While memorization of basic facts has been a hallmark of elementary school, current recommendations focus on a more nuanced development of fluency with these facts. Fluency is characterized by the ability to demonstrate flexibility, accuracy, efficiency, and appropriate strategy use. Despite recommendations to focus on strategy use, there is insufficient information on instructional approaches that are effective for developing strategies, particularly for multiplication facts. Using visual imagery with dot patterns has …

The Relationships Between Flow, Mathematics Self-Efficacy, And Mathematics Anxiety Among International Undergraduate Students In The United States, Samah Abduljabbar

Dissertations

Problem

A worldwide problem, math anxiety is defined as an anxious state with an unpleasant feeling of tension characterized by fear of failing to achieve mathematics targets. Psychologically, math anxiety involves anxiety, tension, discomfort, nervousness, fear, shock, and insecurity. Math anxiety has been perceived as a key influencer of reduced math achievement, and avoidance of math-related careers. On the other hand, abilities, flow, interests, and psychological conditions contribute to student mathematics success. Belief in one's ability to perform a specific task boosts self-efficacy, which has been studied widely as a predictor of student academic performance. When students are interested in, …

Computation Of Risk Measures In Finance And Parallel Real-Time Scheduling, Yajuan Li

Dissertations

Many application areas employ various risk measures, such as a quantile, to assess risks. For example, in finance, risk managers employ a quantile to help determine appropriate levels of capital needed to be able to absorb (with high probability) large unexpected losses in credit portfolios comprising loans, bonds, and other financial instruments subject to default. This dissertation discusses the computation of risk measures in finance and parallel real-time scheduling.

Firstly, two estimation approaches are compared for one risk measure, a quantile, via randomized quasi-Monte Carlo (RQMC) in an asymptotic setting where the number of randomizations for RQMC grows large, but …

Low-Reynolds-Number Locomotion Via Reinforcement Learning, Yuexin Liu

Dissertations

This dissertation summarizes computational results from applying reinforcement learning and deep neural network to the designs of artificial microswimmers in the inertialess regime, where the viscous dissipation in the surrounding fluid environment dominates and the swimmer’s inertia is completely negligible. In particular, works in this dissertation consist of four interrelated studies of the design of microswimmers for different tasks: (1) a one-dimensional microswimmer in free-space that moves towards the target via translation, (2) a one-dimensional microswimmer in a periodic domain that rotates to reach the target, (3) a two-dimensional microswimmer that switches gaits to navigate to the designated targets in …