Digital Commons Network^™

Existence And Uniqueness Of The Solution Of A Traffic Flow Partial Differential Equation On Multi-Lane Freeways, Sina Zareian

CGU Theses & Dissertations

In this dissertation, we shall prove the existence of a solution of the stochastic partial differential equation describing the density of cars on a multi-lane freeway using the operator splitting method. Furthermore, we shall prove the uniqueness of the solution of the stochastic differential equation which forms when we apply the operator splitting method to the traffic flow stochastic partial differential equation.

Diophantine Avoidance, Number Fields, And Quadratic Forms, Sehun Jeong

CGU Theses & Dissertations

Diophantine avoidance has been studied by several authors in recent years. This term refers to effective results on existence of points of bounded size (where size is measured by norm or height, depending on the context) in a given algebraic set avoiding some specified subsets. The application of avoidance conditions allows to understand how ``well distributed" are points of bounded size in a given set. If it is possible to find them outside of some prescribed collection of subsets of the set in question, then it suggests that they are evenly distributed, in some appropriate sense. Our first result investigates …

Classical And Quantum Computational Methods For Predicting Fluid Transport In Fracture Networks, John Kath

CGU Theses & Dissertations

This dissertation addresses the challenge of modeling complex geophysical systems by developing efficient surrogate models and scalable quantum algorithms. Our approaches provide uncertainty quantification, assessing confidence in estimates while accounting for subsurface heterogeneity. These innovations are designed to replace costly solvers with parsimonious emulators. In combination with multi-fidelity and quantum techniques, they make physics-informed modeling computationally feasible. One of our studies involves the use of Gaussian process regression to generate Bayesian predictions for gas transport in 3D discrete fracture networks. This provides accurate estimates while offering substantial savings over computationally intensive high-fidelity simulations. Additionally, we study multi-fidelity modeling through the …

Characterizations Of Stability Via Morse Limit Sets, Jacob D. Garcia

Mathematics Sciences: Faculty Publications

Subgroup stability is a strong notion of quasiconvexity that generalizes convex cocompactness in a variety of settings. In this paper, we characterize stability of a subgroup by properties of its limit set on the Morse boundary. Given H < G, both finitely generated, H is stable exactly when all the limit points of H are conical, or equivalently when all the limit points of H are horospherical, as long as the limit set of H is a compact subset of the Morse boundary for G We also demonstrate an application of these results in the settings of the mapping class …

Topological Symmetry Groups Of The Generalized Petersen Graphs, Angelynn Álvarez, Erica Flapan, Mark Hunnell, John Hutchens, Emille Lawrence, Paul Lewis, Candice Price, Ruth Vanderpool

Mathematics Sciences: Faculty Publications

The topological symmetry group TSG(Γ) of an embedded graph Γ in S³ is the subgroup of the automorphism group of the graph which is induced by homeomorphisms of (S³,Γ). If we restrict to orientation-preserving homeomorphisms then we obtain the orientation-preserving topological symmetry group TSG₊(Γ). In this paper, we determine all groups that can beTSG(Γ) or TSG₊(Γ) for some embedding Γ of a generalized Petersen graph other than the exceptional graphs P(12,5) and P(24,5).

Optimization Of Free-Time Controlled Constrained Sweeping Processes And Applications, Thi Dai Trang Nguyen

Wayne State University Dissertations

Optimization and optimal control have a wide range of applications across fields like engineering, biology, data science, and economics. This dissertation is devoted to advanced optimal control problems discontinuous constrained differential inclusions of the sweeping type involving the duration of the dynamic process into optimization, which are truly challenging and underinvestigated in control theory while being highly important for various applications. To attack such problems, we use the method of discrete approximation while establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. This approach, married to advanced tools of variational analysis, enables this research to …

Examining Student Understanding Of Series Convergence Using Script Writing, Eduardo Torres Manzanarez

Mathematics Dissertations - Archive

This exploratory study investigates second-semester calculus students’ understanding of series convergence using both nonscripting and scripting-based tasks. Researchers recognize that students encounter difficulties with learning series, including series convergence, and that there is a need for more meaningful tasks that aid students’ learning of series. Script writing, used mainly with prospective mathematics teachers, can be used to explore mathematical understandings. Thus, this study examines how script writing, in the form of a scripting task, elicits students’ understanding of series convergence in contrast to nonscripting-based tasks to determine the potential of script writing in assessing this student understanding. We examine students’ …

Golden Spirals Everywhere? Solutions For Fermi Questions, January 2025, John Adam

Mathematics & Statistics Faculty Publications

The article discusses different types of spirals, including Archimedean, hyperbolic, and logarithmic spirals, with a focus on the golden ratio and golden spirals. It addresses the misconception that golden rectangles and spirals can be found in various natural and man-made structures, emphasizing the importance of understanding the properties of logarithmic spirals. The article provides mathematical explanations and solutions for questions related to pitch angles and self-similarity in logarithmic spirals, using examples like the nautilus shell and an ammonite-like stone. It concludes by referencing additional sources for further exploration of the golden ratio and golden spiral myths.

Enhancing The Accuracy Of Image Classification For Degenerative Brain Diseases With Cnn Ensemble Models Using Mel-Spectrograms, Sang-Ha Sung, Michael Pokojovy, Do-Young Kang, Woo-Yong Bae, Yeon-Jae Hong, Sangjin Kim

Mathematics & Statistics Faculty Publications

Alzheimer’s disease (AD) and Parkinson’s disease (PD) are prevalent neurodegenerative disorders among the elderly, leading to cognitive decline and motor impairments. As the population ages, the prevalence of these neurodegenerative disorders is increasing, providing motivation for active research in this area. However, most studies are conducted using brain imaging, with relatively few studies utilizing voice data. Using voice data offers advantages in accessibility compared to brain imaging analysis. This study introduces a novel ensemble-based classification model that utilizes Mel spectrograms and Convolutional Neural Networks (CNNs) to distinguish between healthy individuals (NM), AD, and PD patients. A total of 700 voice …

Construction Techniques For Linear Realizations Of Multisets With Small Support, M. A. Ollis

Emerson Authors, Researchers, & Creators

The Buratti-Horak Rosa Conjecture is a problem in graph theory asking when we can find a Hamiltonian path in the complete graph that has various properties to do with edge lengths. In this paper, we use constructive methods to show that the conjecture is true for various parameter sets where it was previously unknown.

Random Graph Models For Dual Graphs, Anne Friedman

Scripps Senior Theses

This paper aims to better characterize dual graphs derived from state districting maps by developing random graph models that replicate their structural properties. Dual graphs provide a simplified way to represent districting maps, making it computationally feasible to analyze their structure. These representations enable researchers, legislators, and courts to assess district compactness, detect signs of gerrymandering, and generate alternative districting plans. A deeper understanding of the structural patterns of these dual graphs can help researchers choose or design more effective algorithms for redistricting analysis. The random graph models developed in this study serve as testbeds for evaluating algorithmic approaches to …

Analyzing Patterns In Chicago Motor Vehicle Crashes Using Time-Series Techniques, Christina Trotta

Senior Honors Theses and Projects

This project explores time series forecasting of daily traffic crash rates in Chicago from 2018 to 2024, with a focus on understanding how past crash patterns and external conditions influence future risk. The primary research question asks: To what extent does yesterday’s crash rate help predict today’s? Using a combination of Holt-Winters exponential smoothing, Prophet forecasting, and SARIMAX models, we assess the role of autoregression, seasonality, and exogenous variables such as weather and roadway conditions. Daily crash data was cleaned, aggregated, and enriched with engineered features including holiday indicators, weather metrics from O’Hare and Midway airports, and binary flags for …

A New Formulation Of Hardy-Type Dynamic Inequalities On Time Scales, Martin Bohner, Irena Jadlovská, Ahmed I. Saied

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we introduce a novel formulation of dynamic Hardy-type inequalities on a time scale, motivated by a recently established convexity approach in the Haar measure. The classical Hardy inequality is refined so that the classical Lebesgue-measure constant is replaced by the sharp constant 1. We obtain time-scale analogues on finite intervals with best constants, and, for nonincreasing and nondecreasing functions, reversed inequalities with explicit weights described by incomplete β-functions. To establish our results, we employ two distinct time scales and apply the chain rule, together with the substitution rule, the derivative of inverse functions, and Fubini's theorem for …

Self-Tor Persistence Of Modules Over Determinantal Rings, Tatheer F. Ajani

Mathematics Dissertations - Archive

Tor-persistence is the claim that Tor of a module with itself is only zero if the module has finite projective dimension. Work done by Avramov, Iyengar, Nasseh, Sather-Wagstaff, and various other authors have proved Tor-persistence of modules over certain rings. In this work, we will prove Tor-persistence for certain modules over determinantal rings, specifically for the hypersurface defined by the determinant of a generic matrix. We will then give an explicit proof that Tor^R_2(M,M) is never zero, that Tor^R_1(M,M)=0, and due to the periodicity of the given free resolution, our result can be extended to the entire complex, showing that …

Hyper B-Ary Representations Of A Number, Mingway Wang

Masters Theses

In this work, we summarize combinatorial, analytical, and computational properties of Stern's diatomic sequence, and do the same for a generalization of a combinatorial property of this sequence to higher bases. We then construct special ``bit conversion'' functions $g_{b}$ that allows us to find a semi-explicit formula for the generalized sequences. Lastly, we analyze some of the properties of these $g_{b}$ functions.

Adding Libraries To The Equation: Mathematical Sciences’ Underutilization Of Academic Librarians, Jennifer Lc Burke, Elizabeth C. Novosel, Daniel G. Kipnis, Rasitha Jayesekere

Scholarship and Professional Work

Academic librarians do not engage with all disciplinary departments equally. Despite equal or even greater efforts, some departments are less responsive to librarian outreach. One such department is mathematics. To understand mathematics departments’ relationships with their academic librarians, three mathematics librarians created a 20-question survey that was disseminated to mathematics faculty, instructors, and instructional staff in the United States and Canada. Of the 188 survey participants, more than a third reported that they never engage with their librarians, approximately half only do so occasionally, and a mere eight percent of participants collaborated with librarians to provide information literacy instruction (IL) …

Modeling Deductive Inference: A Historico-Philosophical Introduction To First-Order Logic, David J. Buller

Faculty Books & Book Chapters

This book is a companion text for lectures on first-order logic and its elementary metatheory (used in Intermediate Logic at Northern Illinois University). It covers the basic concepts of set theory necessary for a mathematical development of first-order logic; develops a formal language of first-order logic; presents a classical Tarskian semantics for the language and the “semantic” conception of logical consequence; presents a Gentzenian proof system and the “syntactic” conception of logical consequence; develops a partial decision procedure for logical consequence in the language; demonstrates applications of the formal system to modeling deductive inference expressed in natural language; and extends …

Ms-Yolo: Infrared Object Detection For Edge Deployment Via Mobilenetv4 And Slideloss, Jiali Zhang, Thomas S. White, Haoliang Zhang, Wenqing Hu, Donald C. Wunsch, Jian Liu

Mathematics and Statistics Faculty Research & Creative Works

Infrared imaging has emerged as a robust solution for urban object detection under low-light and adverse weather conditions, offering significant advantages over traditional visible-light cameras. However, challenges such as class imbalance, thermal noise, and computational constraints can significantly hinder model performance in practical settings. To address these issues, we evaluate multiple YOLO variants on the FLIR ADAS V2 dataset, ultimately selecting YOLOv8 as our baseline due to its balanced accuracy and efficiency. Building on this foundation, we present MS-YOLO (MobileNetv4 and SlideLoss based on YOLO), which replaces YOLOv8's CSPDarknet backbone with the more efficient MobileNetV4, reducing computational overhead by 1.5% …

Optimizing Decision-Making In A Cerebral Palsy Model Using Reinforcement Learning, Richard Ampah

Pitzer Senior Theses

This study presents an original interdisciplinary investigation into how reinforcement learning (RL) can model motor and cognitive defects and potentially improve motor and cognitive functions in individuals with cerebral palsy (CP), a non-progressive neurological disorder that impairs movement and adaptability. Integrating computational neuroscience and machine learning, the research applies policy gradient methods and Markov Decision Processes (MDPs) to simulate adaptive learning in agents with and without CP-related constraints.

The central aim is to compare the cumulative rewards of optimal policies, derived from value iteration, and human-like learning policies using the REINFORCE algorithm, both with and without the Bellman baseline. The …

The Anatomy Of A Reconstruction: From Fourier Space To Image Recovery In Computed Tomography, Charlotte P. Maurer

Honors Theses

This thesis develops the mathematical foundations of computed tomography (CT) reconstruction through the lens of harmonic analysis. Beginning with the Schwartz class, we introduce the Fourier transform and its role in expressing the Radon transform and its inversion via a fractional Laplacian. After constructing the Radon transform in general dimension R^d, we specialize to the cases d = 2 and d = 3, demonstrating explicit inversion formulas and the associated instability in lower dimensions. For its computational advantages, we study filtered back-projection using classical low-pass filters (Ram-Lak, Shepp–Logan, Cosine, Gaussian) and formulate a discrete reconstruction algorithm grounded in …

Local Limit Theorems On Finitely Generated Abelian Groups, Yutong Yan

Honors Theses

In this thesis, we classify the pointwise behavior of finite-range random walks on finitely generated abelian groups in terms of local limit theorems. Random walks are central objects of research in probability theory, and the theory has found applications in statistics, physics, and even card shuffling. One significant topic in this line of study is random walks on finitely generated groups. Starting from the pioneering work of G. Pólya and H. Kesten, random walks on finitely generated groups have been studied extensively. However, many notable results on the subject (local limit theorems, for example) make assumptions about periodicity and irreducibility …

Affine Groups: A Functorial Perspective, Vladimir Khabaev

Honors Theses

A familiar construction associated to any commutative ringRwith1is its group of units, traditionally denoted by R^x= {u in R | uv = 1 for some v in R}. This is but one out of many ways to get a group from a ring. To see at least one other way, we need a mild change in perspective: units may instead be characterized as elements for which the linear transformation f(r) = u ⋅ r is an isomorphism of R as a module over itself. That is to say, R^x = GL(1, R …

On Higher Level Zhu Algebras Of N-Graded Vertex Algebras Associated With Simple Leibniz Algebras That Contain Sl2, Christian Soltermann

Theses and Dissertations

In this thesis, we study how the higher-level Zhu algebras of a vertex algebra reflect the structure of associated simple Leibniz algebras. In particular, we construct a vertex algebra from a vertex algebroid containing the simple Lie algebra sl₂ and analyze its higher level Zhu algebras. The irreducible modules of this vertex algebra were completely classified in [JY20b], but the structure of its indecomposable modules remains an open problem. Since modules for higher level Zhu algebras correspond to modules of vertex algebras, studying these algebras provides a method for understanding their broader representation theory.

A Comparative Analysis Of Data-Driven And Model-Based Neutrosophication Methods: Advancing True Neutrosophic Logic In Medical Data Transformation, Maikel Yelandi Leyva Vázquez, Lorenzo Cevallos-Torres, Omar Mar Cornelio, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Neutrosophic logic extends fuzzy logic by explicitly modeling indeterminacy (I), offering a robust framework for uncertainty representation. The transformation of crisp data into neutrosophic triplets {T, I, F}—known as neutrosophication—is crucial for applying neutrosophic models in real-world analysis. However, comparative evaluations of existing neutrosophication methods remain limited. This study presents a systematic comparison of five approaches: three model-based methods (Parabolic, Threshold Distance, Fuzzy Membership), one density-based method (Kernel Density Estimation), and a proposed data-driven K-Means clustering method integrating sigmoid membership functions. Using a medical dataset of 299 patients and six continuous clinical variables, we assessed statistical behavior, consistency, and alignment …

Strong Neighborhood-Prime Labelings Under Ordinary & Gaussian Integers, Micheal Arnal-Brown

Murray State Theses and Dissertations

This thesis introduces and studies the notion of a strong neighborhood-prime labeling, a strengthening of the neighborhood-prime labeling where the label 1 can be assigned to any vertex in a graph. We prove that several graph families—including paths, cycles (excluding those congruent to 2 modulo 4), caterpillars, helm graphs, closed helm graphs, gear graphs, and graphs with universal vertices—admit such labelings, and also provide results to more general classes of graphs. We extend this new labeling concept to the Gaussian integers using a spiral order- ing on Z[i] and define a Gaussian analogue of strongly neighborhood-primeness. To support this extension, …

The Local Deep Galerkin Method Applied To The (2+1)-D Cahn-Hilliard Equation, Caden Fischer

Schultz-Werth Award Papers

Physics-informed Neural Networks (PINNs) are an alternative approach to solving Partial Differential Equations. This study examines the Local Deep Galerkin Method (LDGM), and its application to the (2+1)-Dimensional Cahn-Hilliard Equation (2D-CH). The 2D-CH models phase separation. One application of this is to modeling biofilm. The LDGM is trained on a loss function that minimizes the sum of the squares of the residuals of the system of equations. The LDGM is compared to a numerical simulation based on the Finite Element Method. The results of this study show that LDGM does not produce accurate results for the 2D-CH equation for large …

Empirical Analysis Of Political Districting Splitability Via Uniform Spanning Trees In Polynomial Time, Brooke C. Feinberg

Scripps Senior Theses

This work expands a recently proven conjecture that a polynomial fraction of all uniform spanning trees (USTs) are splittable into k balanced partitions on grid graphs to real-world political districting plans. We investigate whether similar structural properties hold for the planar dual graphs of U.S. counties (cnty) and tracts (t), using Wilson’s algorithm to generate uniform random spanning trees and Breadth- First Search (BFS) to check for splitability into balanced partitions. Our empirical findings suggest that real-world districting plans can be split into 2-balanced, connected partitions in a fraction of polynomial time. This result highlights the potential for scalable redistricting …

Quantification Of Changes With Combined Shape Mode Analysis And Swimming Simulations, Kelli E. Gutierrez, Becca Thomases, Paulo E. Arratia, Robert D. Guy

Mathematics Sciences: Faculty Publications

Many different microswimmers propel themselves using flagella that beat periodically. The shape of the flagellar beat and swimming speed have been observed to change with fluid rheology. We quantify changes in the flagellar waveforms of Chlamydomonas reinhardtii in response to changes in fluid viscosity using (i) shape mode analysis and (ii) a full swimmer simulation to analyse how shape changes affect the swimming speed and to explore the dimensionality of the shape space. By decomposing the gait into the time‑independent mean shape and the time‑varying stroke, we find that the flagellar mean shape substantially changes in response to viscosity, while …

Circadian Variation In Mgmt Promoter Methylation And Expression Predicts Sensitivity To Temozolomide In Glioblastoma, Maria F. Gonzalez-Aponte, Yitong Huang, William A. Leidig, Tatiana Simon, Omar H. Butt, Marc D. Ruben, Albert H. Kim, Joshua B. Rubin, Erik D. Herzog, Olivia J. Walch

Mathematics Sciences: Faculty Publications

Purpose Recent studies show that glioblastoma (GBM) is more sensitive to temozolomide (TMZ) in the morning. In cells, inhibiting O6-Methylguanine-DNA-Methyltransferase (MGMT) abolished time-dependent TMZ efficacy, suggesting that circadian regulation of this DNA repair enzyme underlies daily TMZ sensitivity. Here, we tested the hypotheses that MGMT promoter methylation and protein abundance vary with time-of-day in GBM, resulting in daily rhythms in TMZ efficacy.

Methods We assessed daily rhythms in MGMT promoter methylation in GBM in vitro and retrospectively analyzed MGMT methylation status in human GBM biopsies collected at different times of day. Next, we measured MGMT and BMAL1 protein …

Reasoning Through Change: Exploring Covariational Thinking In Student Exponential Graph Interpretation, Samantha Lynn Pearson

Honors Theses and Capstones

This study explores how life science students reason about exponential functions in the context of an introductory physics course. While exponential functions are essential in both physics and biology—appearing in scenarios like population growth and temperature cooling—students often encounter challenges in interpreting them. Using a resources framework, we examined the cognitive resources these students bring to exponential reasoning, focusing on what ideas and strategies they already possess. Our methodology involved conducting think-aloud interviews with nine students from PHYS 402. Interviews were coded using a framework informed by prior research on covariational reasoning and exponential understanding. Results show that students generally …