Mathematics Commons^™

Dg Algebra Techniques For Tor Algebras And Edge Ideals, Desiree Martin

Dissertations - ALL

This thesis is comprised of two projects. The first is individual work where I use semifree Gamma-extensions to show, for an ideal I with infinite projective dimension over a Noetherian local ring R, there exists a splitting morphism from a Tor algebra Tor(R/I,R/I) to the exterior algebra of the R/I-free summands of the conormal module. In particular, when the conormal module is free over R/I, we see that the exterior algebra of the conormal module splits completely from Tor(R/I,R/I) as algebras. For the second project my collaborators, Hugh Geller and Henry Potts-Rubin, and I completely classify all trees and cycle …

On Compact Quasi-Einstein Metrics Of Constant Scalar Curvature, Eric Morgan Cochran

Dissertations - ALL

This thesis concerns a generalization of Einstein manifolds called quasi-Einstein manifolds. These manifolds are a triple $(M,g,X)$ where $(M,g)$ is a Riemannian manifold, and $X$ is a smooth vector field on $M$. Quasi-Einstein manifolds are of interest in general relativity, where a special class of quasi-Einstein manifolds are known as near horizon geometries. They are also related to Ricci solitons, which are self-similar solutions of the Ricci flow. In the first set of results in this thesis, we study the existence of Killing fields on compact quasi-Einstein manifolds. In particular, we prove an equivalence between compact quasi-Einstein manifolds of constant …

Quasi-Einstein Metrics And Homogeneous Conformally Einstein Manifolds, Nazia Valiyakath

Dissertations - ALL

This thesis investigates a classic question in Riemannian geometry: What is the best metric to put on a Riemannian manifold? It combines two projects that explore different aspects of this question. The first project studies nilpotent and unimodular solvable Lie groups that admit m-quasi-Einstein metrics (M,g,X) with X a left-invariant vector field, which we call totally left-invariant quasi-Einstein metrics. We give a complete classification of nilpotent Lie groups admitting such metrics, showing that this occurs if and only if the group is isomorphic to Heisenberg Lie group. For unimodular solvable Lie groups S, we prove that the existence of a …

Greedy Algorithms And Matroids, Kiri M. Strack

LSU Master's Theses

In a connected graph with weights on the edges, a minimum-weight spanning tree can be obtained by repeatedly choosing minimum-weight edges while avoiding choosing the edge set of any cycle. This algorithm is known as Kruskal’s Algorithm, although it was first introduced by Boruvka in 1926. Prim introduced an alternative algorithm in which, at each step, the chosen set of edges forms a connected graph. Both of these algorithms make locally optimal choices that eventually yield a global optimum. This thesis considers how these algorithms can be extended to matroids. In particular, it is shown that matroids are exactly the …

The Isoperimetric Inequality And Wirtinger’S Inequality, Mason Neal

Honors Theses

In this thesis, we present Hurwitz’s proof of the Isoperimetric Inequality, which roughly states that the area enclosed by a simple closed curve is always less than or equal to the area of a circle with the same perimeter. Hurwitz’s proof relies on Wirtinger’s Inequality. We survey results about periodic functions and Fourier series, and we use them to provide a proof of Wirtinger’s Inequality. We then give a new proof of a variant of Wirtinger’s Inequality due to Alzer and generalize this variant to higher powers.

Monotone Traveling Waves In A General Discrete Model For Populations With K-Generation Long-Term Memory, Austin Kernel Sanders

Theses and Dissertations

We investigate the spatial dynamics of a population modeled by a general discrete integrodifference equation incorporating k-generation long-term memory. While classical models rely solely on the im- mediately preceding state to determine population growth and dispersal, introducing multiple past states causes the associated evolution operator to lose compactness. Consequently, standard fixed- point theorems are insufficient to prove the existence of traveling wave solutions. We overcome this difficulty by constructing a time-independent moving frame operator and employing the monotone iteration method. By assuming the fecundity function is locally Lipschitz, bounded, and nondecreasing on a specified interval, and by relaxing continuity requirements …

Toward The Salmon Prize: A Computational Certificate For The M5, M6, And M9 Equation Families Of Σ4(P3 × P3 × P3), John Trevino

Theatre Faculty Publications

We present a computational certificate for three families of explicit polyno- mial equations whose zero locus contains the fourth secant variety σ4(P3×P3×P3) inside P63. The three families are: 192 degree-5 Strassen commutation equa- tions (M5); 160 degree-6 equations lifted from the Bates–Oeding generators of σ4(P2 × P2 × P3) via the Landsberg–Manivel–Friedland theorem (M6); and 64 degree-9 Ottaviani 9 × 9 determinant equations (M9). All 416 generators are explicit polynomials in the coordinate ring Z[Zijk | i, j, k ∈ {0, 1, 2, 3}]. We verify by exact integer arithmetic that every generator vanishes on rank-4 test tensors (six independent …

Largest 2-Regular Subgraphs In Complete S-Partite Graphs, Yiyang Jiang

McKelvey School of Engineering Graduate Student Theses & Dissertations

In this thesis, we focus on the class of complete $S$-partite graphs, for $S$ an undirected graph possibly with self-loops, and address the problem of finding largest $2$-regular subgraphs of these graphs, which can be formulated as an integer linear program. Roughly speaking, a complete $S$-partite graph is obtained by replacing every single node of $S$ with a number of nodes, preserving the edge/non-edge relations of $S$. Our motivation in studying largest $2$-regular subgraphs is rooted in the structural systems theory, particularly in the problem of finding largest subnetworks that can sustain controllability or asymptotic stability of the corresponding subsystems. …

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

Towards Improving The Performance Of The Adcirc Storm Surge Modeling Software, Nick Weldner, Tim Stitt, Jijun Tang

Caravel Undergraduate Research Journal

Accurately predicting storms and hurricanes is critical to saving lives and reducing economic loss. Therefore, it is necessary to use the most efficient software and hardware technology available in order to improve the performance and fidelity of these predictive mathematical models. For over ten years, the Computational Hydraulics Lab (CHL) at the University of Notre Dame has been involved in developing the high-resolution ADvanced CIRCulation (ADCIRC) storm surge model to predict storm surges in coastal areas. The objective of the work reported here was to port a novel adaption of the parallel ADCIRC code to the state-of-the-art Intel Xeon Phi …

Proofs Of A Trigonometric Inequality, Cameron Butler, Wei-Kai Lai

Caravel Undergraduate Research Journal

A trigonometric inequality is introduced and proved using Hölder’s inequality, Cauchy-Schwarz inequality, and Chebyshev’s order inequality.

Leveraging Information Theory And Ecological Network Analysis To Monitor Communication Networks In A Student Aerospace Team, Christine Sessions

Doctoral Dissertations and Master's Theses

Effective communication is a critical component of successful collaboration in group projects and team settings. However, systematically tracking and analyzing team communications can be challenging, especially in complex, multi-member teams such as those found in the aerospace industry. This paper explores an information-theoretic approach that leverages encoding techniques and graph theory to analyze communication networks within a university’s multi-year, student-led cubesat design project (Project COMET). By utilizing Shannon Entropy as a measure of information flow, encoding communication patterns, and analyzing Ecological Network Analysis parameters, this research aims to understand how an Embry-Riddle Aeronautical University student project team evolves over the …

Primitive Pythagorean Triples In Lean And Reduction Modulo Odd Prime Powers, Luke Biddle

Mathematical Sciences Undergraduate Honors Theses

Primitive Pythagorean triples (PPTs) are (a,b,c) triples that satisfy the Pythagorean theorem and share no other common factors outside of 1. This project examines these PPTs reduction modulo odd prime powers by combining proof writing and number-theoretical analysis with the process of verification and formalization in the Lean proof coding language. Using the parameterization of PPTs generated by using the unit circle with additional conditions, we investigate how these triples behave modulo  for odd primes , with emphasis on counting the number of elements in the set of PPTs (a,b,c) modulo pⁿ . By using cases based on initial …

From The Hopf Fibration To Instantons: Geometry In Gauge Theory, Emily D. Wessman

Undergraduate Honors Capstone Projects

This paper explores the relationship between topology, differential geometry, and gauge theory through the study of Yang-Mills theory and its solutions, known as instantons. Beginning with the Hopf fibration, we show how principal fiber bundles encode topological information and appear in physical contexts such as electromagnetism. In particular, we consider how the fibration of S³ over CP¹ ≅ S² represents the Dirac magnetic monopole, and how the Chern number associated with the bundle is exactly the winding number for the monopole.

We then develop the framework of gauge theory, focusing on connections on principal SU(2) bundles …

Rainbow Dominating Sets Of Graphs, Samuel L. Powell

All Graduate Theses and Dissertations, Fall 2023 to Present

The content of this thesis may be compared to a stack of plates with a common design that are broken, one at a time. If the plates are broken into large enough pieces you would be able to choose one fragment from each broken plate to discover the entire design that the plates share. For example, if you were still missing the center of the design after taking a piece from some plates, you could look for the piece of the next broken plate with the region in question.

We study the question of how many broken plates might guarantee …

Course Insight Portfolio: Math 1060 Calculus Of One Variable I, Minh Ha, Dinaniaina Florence Rafidimanantsoa, Preston Sessoms, Rishabh Shukla

Publications

The following portfolio is a representation of the course MATH 1060, Calculus I, taught at Clemson University. The contents can be broken down into 3 major components.

• The first being an analysis of the students who may typically be in the course, the prerequisites required to take the course, and what paths students have after completing the course. This analysis is based upon the course curriculum at Clemson, and as a result will likely not align with that of other universities

. • The second major component is a map of the concepts discussed in MATH 1060 as well …

Course Insight Portfolio Math 1040: Precalculus And Introductory Differential Calculus, Jessica Ayers, Morgan Hayes, Meghan Kerrick, Uthman Rasaq, Drishty Singh

Publications

This portfolio includes four primary elements. First is a document titled “Where did they come from? Where are they going?” that provides a description of the types of students who are likely to take the course, followed by the “Course Concept Map” that visually depicts the flow of topics and possible misconceptions in MATH 1040. Additionally, the “Misconceptions Overlay” document lists articles that offer interesting discussion and further insight into the ideas students may bring to MATH 1040 or develop throughout the course. Two lesson plans are included that cover “Continuity and Introduction to Classifying Discontinuities” as well as an …

Course Insight Portfolio: Math 1080, A Gentle Introduction To Infinite Series, Igor Luis Aureliano, Leyanis Falcon Hernandez, Ross George, Fnu Monika

Publications

This lesson aims to introduce the concept of an infinite series through geometric representations and partial sums to help students develop an intuitive understanding of convergence. A common misconception students may have when first encountering infinite series is the belief that “a sum of infinitely many things must be infinite”. Through the study of a convergent geometric series, students will investigate how the limit of partial sums can produce a finite value, thereby addressing this misconception and developing a more accurate understanding of infinite series.

A Low-Rank Solver For The Stokes–Darcy Model With Random Hydraulic Conductivity And Beavers–Joseph Condition, Yujun Zhu, Yulan Ning, Zhipeng Yang, Xiaoming He, Ju Ming

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes, analyzes, and demonstrates an efficient low-rank solver for the stochastic Stokes-Darcy interface model with a random hydraulic conductivity both in the porous media domain and on the interface. We consider three interface conditions with randomness, including the Beavers–Joseph interface condition with the random hydraulic conductivity, on the interface between the free flow and the porous media flow. Our solver employs a novel generalized low-rank approximation of the large-scale stiffness matrices, which can significantly cut down the computational costs and memory requirements associated with matrix inversion without losing accuracy. Therefore, by adopting a suitable data compression ratio, the …

A Computational And Experimental Study Of Energy Extraction From Vertical Axis Wind Turbines In Realistic Environments, Kevin Vargas

Theses, Dissertations and Culminating Projects

This thesis presents preliminary analyses on the performance of small scale vertical axis wind turbines (VAWTs) conducted through both experimental and computational methods. Laboratory experiments were carried out in the Complex Fluids Laboratory at Montclair State University to evaluate turbine behavior under controlled conditions. The key focus of this work was to investigate the impact of inlet blockage on turbine performance, simulating real-world environmental constraints such as flow obstruction and urban interference and measuring power generated via laboratory tools. Computational fluid dynamic (CFD) simulations were performed to further analyze flow characteristics and validate experimental observations and trends. Turbine geometries were …