Mathematics Commons^™

Solvability Of Stochastic Linear-Quadratic Optimal Control Problems Under Partial Stabilizability Conditions, Al-Sadh Rahman Imadh

Honors Undergraduate Theses

Optimal Control Theory, a branch of Control Theory, is applicable in fields such as engineering, operations research, and economics. Stochastic Optimal Control deals with noisy systems and data using Ito’s formulation. Given a noisy system and a cost functional, the goal is to find a control that will minimize the cost. This thesis focuses on linear quadratic stochastic optimal control, and we explore state equations that are not stabilizable. We first address measurability concerns arising from the semigroup property of the state trajectory. The notions of partial stability and partial stabilizability are introduced, and we formulate their corresponding Lyapunov and …

Quantized Average Agreement Algorithms With Error Correction For Digraphs, Shuaib A. Mughal

Honors Undergraduate Theses

Multi-agent systems have become more and more prevalent as technology increasingly gets integrated into our daily lives. Some of these technological systems are large in size; for example, the smart grid where multiple devices are used to monitor and control different aspects of the energy grid. Another example is a team of autonomous systems deployed for a specific task. When these systems are spatially distributed, an important component of distributed algorithms is the ability for the agents to reach consensus on the global state of the system. Reaching agreement enables the spatially distributed agent make decisions or determine the next …

Automorphism Groups Of N-Graded Vertex Algebras Associated With Cyclic Leibniz Algebras With Small Dimensions, Alexander M. Keene

Theses and Dissertations

A fundamental problem in the study of vertex (operator) algebras V is the determination of the group of (grading-preserving) N-graded vertex algebras associated with cyclic Leibniz algebras of dimensions 2 and 3 that were classified by C. Barnes, E. Martin, J. Service, and G. Yamskulna in [1].

In each case examined, investigation of the automorphism group relies on the key fact that the action of an automorphism σ is determined solely by its value at a single basis element b. Furthermore, we employ a result in [19] by H. Li and G. Yamskulna which states that we can determine the …

Centralized Deep Reinforcement Learning For Homogeneous Multi-Component Maintenance Optimization, Joseph W. Wittrock

Theses and Dissertations

This thesis explores an application of reinforcement learning (RL) in maintenance optimization. Recent advances in hardware-accelerated computation and deep learning have made RL a powerful tool for solving optimization problems which are too complex for traditional methods. Maintenance optimization involves improving the efficiency and effectiveness of maintenance activities through data-driven approaches, ultimately reducing costs and increasing asset availability. Making informed maintenance decisions is crucial to long-term sustainability.

A desirable maintenance policy maximizes a utility signal while minimizing the cost of maintenance. Techniques in sequential decision making such as dynamic programming (DP) and RL have found success in optimizing these maintenance …

Editorial Insights: Reflections On The Volume And Charge To The Field, Heather Howell, Meghan Shaughnessy, Barbara Stengal, Carrie Wilkerson Lee, Liza Bondurant, Bima Sapkota, Gregory Benoit, Yvonne Lai

School of Mathematical & Statistical Sciences Faculty Publications

In this, the final chapter of Section 3, the authors document an interview that took place in June of 2024, as the editorial team wrapped up reviews of the chapters submitted to this volume. The interview was led by two of the advisory board members, Meghan Shaughnessy and Barbara Stengal, and included the editorial group, Heather Howell, Carrie Wilkerson Lee, Liza Bondurant, and Bima Sapkota, and other members of the advisory board, Greg Benoit and Yvonne Lai. The motivation to create this interview-based chapter was to capture some of the authors' own learning and reflection as they brought the volume …

Oer Ancient Egyptian Numerals Activity, Cynthia Huffman Ph.D.

Open Educational Resources - Math

Ancient Egypt captures our imagination not just because of its pyramids, but because of its incredible staying power. This civilization thrived for over 3,000 years, influencing cultures across the globe—including the very roots of mathematics. As Egyptologist Bob Brier puts it, “No civilization lasted so long, contributed so much, or repeatedly amazed as did ancient Egypt.”

Ready to crack the code of Ancient Egypt and the pharoahs? In this activity, you will learn to decipher Egyptian numerals by examining real photos from temples and museums. Master this skill, and next time you visit a museum exhibit on ancient Egypt, you …

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.

Cost-Effective Strategies For Feral Swine Control: Exploring Trapping Equipment Cooperatives, Phil Kenkel, Riza Radmehr, Rodney Holcomb, Mckenzie Boyce

Human–Wildlife Interactions

Feral swine (Sus scrofa) in the United States have become a growing concern, with a population of ≥6 million and causing annual damages of ≥$1.5 billion USD. Because of their high reproductive capacity, effective control methods are crucial to reducing or slowing their population growth. While remotely monitored and triggered traps have proven to be an effective control strategy, the cost of sophisticated trapping equipment, which can cost ≥$7,000 USD, is often cost prohibitive for many landowners. Despite the pressing need to address the challenge of making effective feral swine control methods more affordable, there are few studies …

Machine Learning Methods For Intrusion Detection And Response In Network Security, Ayomide Oyemaja

College of Graduate Studies: Theses & Dissertations

Intrusion Detection Systems (IDS) play a crucial role in computer network security by identifying malicious activities and potential cyberattacks. This thesis combines machine learning and cybersecurity by applying Reinforcement Learning (RL) in intrusion detection and response using the NSL-KDD dataset.

We designed and implemented a Q-learning framework where an agent learns to classify network traffic over time by interacting with the environment and receiving rewards based on detection accuracy. We also look at the importance of feature selection and classification techniques and how effective they are in improving model performance, reducing the complexity of computation, and producing more desirable results. …

Towards A Baker-Campbell-Hausdorff Theorem In Positive Characteristic, Nikolas A. Koutroulakis

College of Graduate Studies: Theses & Dissertations

We investigate whether there is an analog of the Baker-Campbell-Hausdorff (BCH) theorem for Lie algebras over fields of positive characteristic. We begin by introducing the proof of the BCH formula in characteristic zero. We then introduce the Artin-Hasse exponential and show that it is $p$-integral. Our main result provides sufficient conditions under which a BCH-type formula exists for the Artin-Hasse exponential in positive characteristic. Additionally, we derive a formula for computing an inverse of the Artin-Hasse exponential.

Bayesian Networks For Safety-Critical Systems, Joseph Mietkiewicz

Theses

This thesis addresses a operational challenge in modern industrial operations: the increasing complexity of systems and the consequent cognitive burden on operators. As industrial technologies advance, the human-computer interface has become the primary conduit for information flow, playing a pivotal role in operational decision-making. However, the proliferation of data often leads to information overload, potentially compromising rather than enhancing operator performance. This research explores an approach to this pressing issue through the application of Bayesian networks as decision support systems in safety- critical scenarios. Our study employs a multi-faceted approach, combining theoretical modeling with empirical testing. Through collaboration with industry …

Theoretical And Applied Considerations In Dynamical Sampling, Brendan Miller

Graduate Research Theses & Dissertations

Dynamical sampling is an area of mathematics which studies the theory and applications of signals evolving through time, as well as the stable recovery of said signals from spatiotemporal measurements. This manuscript is devoted to problems which arise as theoretical considerations in signal recovery and as applications of using space-time measurements as opposed to static measurements.

For the former, we consider the problem of recovering an evolving signal via space-time measurements at irregular time intervals. To do so, we generalize the classic Kadec $1/4$ theorem using techniques of Banach $L^1(\R)$-modules. For the latter, we show how dynamical samples can be …

Methods For Finding High Quality And Optimal Solutions Of Binary Quadratic Optimization Problems, James Haas

Graduate Research Theses & Dissertations

BiqAlps is an extension of the BiqCrunch project that leverages its bounding procedures within a computer cluster environment. The primary goal of this work was to investigate whether distributing the branch-and-bound tree search across multiple CPUs could improve solve times. While some problem instances demonstrated speedups of up to fivefold, others showed no measurable improvement, revealing that parallelization benefits are problem-dependent.

Beyond parallelization, BiqAlps introduces enhanced heuristics for generating high-quality initial solutions and refining existing ones. Additionally, the framework explores the impact of propagating triangle inequality cuts to child nodes during search, with the aim of accelerating bounding processes and …

Cubic Blaschke Products, Alexandra Hill

Graduate Research Theses & Dissertations

It is well known that Blaschke products can be classified as elliptic, hyperbolic, or parabolic. Based on the classification of the Blaschke product, we know that the Julia set is either the entire unit circle or a Cantor subset of the unit circle. For unicritical Blaschke products of degree d, it has been shown that the parabolic functions are parameterized by an epicycloid with d-1 cusps, with the interior of the epicycloid giving rise to the elliptic parameters, and the relative complement of the epicycloid within the unit disk providing the hyperbolic parameters. In this dissertation, we study in more …

Green's Functions And Fixed Points Of Quasiregular Maps, Mark Broderius

Graduate Research Theses & Dissertations

This dissertation is on the dynamics of planar quasiregular maps of degree 2. These maps have three parameters. One of these parameters determines the distortion, one of them determines the orientation, and the last one is a shift parameter. Our first objective is to construct a new dynamical version of Green's functions for these maps. These functions provide information about the escaping sets of our quasiregular maps and are used to prove topological results about them. We show that the boundary of the escaping set is either connected or consists of infinitely many components. Additionally, we prove that if one …

Facial Reduction Of Semidefinite Relaxations Of Binary Quadratic Problems, Ian Sugrue

Graduate Research Theses & Dissertations

A standard technique for bounding a combinatorial optimization problem is to form a semidefinite relaxation of the problem by "lifting" the problem into a higher dimensional space of symmetric matrices. For some combinatorial problems such as maximum k-cluster, the resulting semidefinite relaxation is not strictly feasible. In the context of the maximum k-cluster problem, we present an equivalent, strictly feasible semidefinite relaxation by way of the dimensionality reduction technique known as facial reduction. We provide a recipe to form this semidefinite relaxation directly, bypassing the lifting and facial reduction steps entirely. We then present our work on generalizing this result, …

Frobenius And Commutative Pseudomonoids In The Bicategory Of Spans, Ivan Contreras, Rajan Amit Mehta, Walker H. Stern

Mathematics Sciences: Faculty Publications

In previous work by the first two authors, Frobenius and commutative algebra objects in the category of spans of sets were characterized in terms of simplicial sets satisfying certain properties. In this paper, we find a similar characterization for the analogous coherent structures in the bicategory of spans of sets. We show that commutative and Frobenius pseudomonoids in Span correspond, respectively, to paracyclic sets and Γ-sets satisfying the 2-Segal conditions. These results connect closely with work of the third author on A∞ algebras in ∞-categories of spans, as well as the growing body of work on higher Segal objects. Because …

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 …

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).