Mathematics Commons^™

The Concept Of T-Basis And Vector-Valued Hardy Classes, Bilal Bilalov, Sabina Rahib Sadigova

Turkish Journal of Mathematics

This paper introduces the concept of a t-basis generated by some bilinear mapping t(·;·). It is considered the vector-valued class L_p(X) := L_p(J; X), 1 ≤ p < +∞, where J = [−π, π] and X is a Banach space with the UMD property, and it is proven that the classical system of exponents {e^int}_n∈ℤ forms a t-basis for L_p(X), 1 < p < +∞. Using this fact, the Hardy vector classes _nH_p^±(X), 1 < p < +∞, different from the classical ones, are defined, and an equivalent definition of these classes is given and some of their properties are studied. In addition, the concept of t-Riesz property of a system of exponentials is introduced in L_p(X), 1 < p < +∞, and it is proved that this system has the t-Riesz property. A new method is given for establishing the Plemelj–Sokhotski formulas for X-valued Cauchy type integrals when X has the UMD property. An abstract analogue of the “1/4-Kadets” theorem is obtained for …

Independent Vs. Signed Musicians: Does Thier Status Affect Their Success?, Zachary Vincent

Honors Projects in Mathematics and Economics

As technology has advanced and become more widely available, and as the public learns more about the recording industry and its workings, more musicians are choosing not to sign with record labels, becoming independent musicians. This now raises the question: does a musician’s status of independent or signed determine their level of success? It has been previously found that independent musicians have a more difficult time competing with signed musicians due to the resources provided by record labels and the way labels negotiate with large streaming platforms, giving signed labels more opportunities to be seen in the public eye, but …

Algorithms To Estimate Contours: Two Applications Of Analytical Tools In Differential Geometry And Topology, Mohammad Abirul Islam

Computer Science ETDs

We develop distributed robotics algorithms with analytical tools needed to define and analyze angle turned and distance traversed by robots executing geometric algorithms. We then use these analytical tools to obtain information, via sensor measurements, about an a priori unknown surface. Our contributions are threefold. First, we develop the Sketch Algorithm, which estimates the boundary of any unknown contour and is asymptotically optimal in terms of distance traversed and angle turned. Second, we present experimental field work that validates the Sketch Algorithm. Finally, we propose an approach to find multiple sources of a surface with potential applications to approximate that …

Analyticity, Superoscillations And Supershifts In Several Variables, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Alain Yger

Mathematics, Physics, and Computer Science Faculty Articles and Research

Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A different way of thinking about superoscillations consists in realizing that it is possible to reproduce the exponential function far away from the origin by only knowing its value in a countable set of points near the origin. By using this perspective, one can extend the idea of superoscillations to functions that are not a sum of exponential functions, namely to the notion of supershift. The study …

Applications Of The Mathieu Groups And Information Theory In Dna Encoding Functions, Juan C. Nava Jr

Theses and Dissertations

A foundational idea in mathematics lies in breaking down existing components into their bare fundamentals. As evidenced by prime numbers and composites, we learn this idea at an early age. Categorizing these broken-down components into their simplest form allows mathematicians to construct proofs from emergent patterns. John Conway’s Atlas of Finite Groups in the 1990s was particularly concerned with the categorization of structures known as groups. There are certain axioms a group must adhere to, which amount to the retention of symmetry; ultimately a group helps us to better understand symmetric actions performed on a set with a binary operation. …

Maximal Simple Regular Matroids, Elana Joy Israel

Dissertations - ALL

In 1980, Seymour completely classified regular matroids. In this thesis, we use this classification to provide a complete list of maximal simple regular matroids up through rank six. In each rank, exactly one of these is graphic. Additionally, there are one, two, and eight cographic, but not graphic, maximal simple regular matroids in ranks four, five, and six, respectively. In rank five, we also have the 10-element sporadic matroid, and in rank six we have a 12-element maximal simple regular matroid and 16-element maximal simple regular matroid, both of which are decomposable matroids that are not graphic, cographic, or sporadic. …

Asymptotics Of Fubini-Study Currents For Sequences Of Line Bundles, Melody Ann Wolff

Dissertations - ALL

Let $(X,\omega)$ be a Kähler manifold, and $L_p$ be a sequence of line bundles on $X$ equipped with continuous hermitian metrics $h_p$. Let $\gamma_p$ be the Fubini-Study current and $h_p^{eq}$ the equilibrium metric corresponding to the metric $h_p$. Our goal is to generalize a theorem of Tian in Complex Geometry to a setting using arbitrary line bundles equipped with metrics with non-positive curvature. This amounts to finding sufficient conditions for $\gamma_p-c_1(L_p,h_p^{eq})$, when appropriately scaled, to converge to $0$ in the sense of currents. To achieve this goal, we imply methods of Demailly and the Oshawa-Takegoshi Extension Theorem to bound a …

Encircling Chains Around 4- And 6-Clusters In Fullerenes, Hannah Melissa Kimbrell

Dissertations - ALL

Fullerenes are purely carbon molecules which can be represented by a 3-regular planar graph consisting of only hexagonal and (exactly 12) pentagonal faces. Since carbon atoms have valence 4 but our graphs are trivalent we need to double the edges in a perfect matching to bring the valence up to 4. The edges in a perfect matching form a Kekulé structure and the hexagonal faces bound by three Kekulé edges are called benzene rings. A maximal independent set of benzene rings for a given Kekulé structure is called a Clar set, and the maximum possible size of a Clar set …

On The Embeddability Of Small Cubic Graphs Into The Torus, Marie Kramer

Dissertations - ALL

The embeddability of graphs into different surfaces has been studied for nearly a century. In 1930, Kuratowski famously proved that a graph is planar if and only if it does not contain the complete graph K5 or the complete bipartite graph K3,3 as a topological minor. In particular, K3,3 is the only cubic obstructions for the plane. This sparked interest in two different directions: What are analogous results for other surfaces, and what can be said about embeddings of K5 and K3,3 into other surfaces? Regarding the first question, it was shown in the 1970s and 80s by Glover, Huneke, …

Pluricomplex Green Functions And Transform Methods For The Laplacian, Jesse Joel Hulse

Dissertations - ALL

There are two research themes within complex analysis presented. The first is developing generalizations of the Unified Transform Method as a novel way to compute fluid flows and numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian and complex Helmholtz equation. The second is finding a formula for the pluricomplex Green function with two poles of equal weights for the bidisk. The Unified Transform Method (also sometimes known as the Fokas Method) is a technique for numerically solving boundary value problems for linear and integrable nonlinear PDEs. Crowdy reformulated the Unified Transform Method in the complex …

Exploration Into Homotopy Equivalence And Vietoris-Rips Complexes, Alexander J. Mobley

Honors Theses

An important question in discrete topology and geometry is how to recover the structure and characteristics of a manifold when only given a finite set of points sampled from that manifold. Thus, if mathematicians have a point cloud of data which induces a discrete metric space, they look for some structure to these points. Understanding the structure of these points often gives needed insight to solve this problem. One such way to determine structure to these points is to use these points to construct a Vietoris-Rips complex. In abstract, Latschev shows that this allows us to recover the homotopy type …

Understanding Student And Teacher Perceptions Of A Checking For Accuracy Method In Algebra Class, Abigail A. Jameson

Masters of Education in Teaching and Learning

Mathematics encompasses learning from one’s mistakes and developing accuracy with problem-solving. This action research study analyzed the classroom teacher’s and students' perceptions of a checking for accuracy math method that was implemented in an eighth-grade algebra classroom. Additionally, the researcher wanted to understand how the participants felt about the accuracy method and its influence on students’ attitudes toward math and mastery of math concepts. Student surveys, individual teacher (with student artifacts) and student interviews and focus groups (with individual artifacts) were the qualitative data collected using the constant comparative method. Descriptive statistics was used to collect quantitative data with calculated …

Crystallization Of The Quantized Function Algebras Of Suq(N + 1), Manabendra Giri

Doctoral Theses

The $q$-deformation of a connected, simply connected Lie group $G$ is typically studied through two Hopf algebras associated with it: the quantized universal enveloping algebra $\mathcal{U}_q(\mathfrak{g})$ and the quantized function algebra $\mathcal{O}(G_q)$. If $G$ has a compact real form $K$, one can use the Cartan involution to give a $*$-structure on $\mathcal{O}(G_q)$. The QFA $\mathcal{O}(G_q)$ with this $*$ structure is denoted by $\mathcal{O}(K_q)$ and its $C^*$-completion by $C(K_q)$. Here we study the crystal limits of $\mathcal{O}(SU_q(n+1))$ and $C(SU_q(n+1))$ and classify all irreducible representations of the crystallized algebras. We also prove that the crystallized algebra carries a natural bialgebra structure.

Derived $P$-Adic Heights And The Leading Coefficient Of The Bertolini–Darmon–Prasanna $P$-Adic $L$-Function, Francesc Castella, Chi-Yun Hsu, Debanjana Kundu, Yu-Sheng Lee, Zheng Liu

School of Mathematical & Statistical Sciences Faculty Publications

Let E/Q be an elliptic curve and let p be an odd prime of good reduction for E. Let K be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which p splits. The goal of this paper is two-fold: (1) we formulate a p-adic BSD conjecture for the p-adic L-function LBDP p introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue F BDP p of LBDP p , we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected …

Statistics - What Does My Data Say About Me?, Taylor Gadsden-Deterville

Student Scholar Symposium Abstracts and Posters

For my Introduction to Statistics Class, I have been tasked with collecting unique, personal data to give insight into my daily routine. I decided to record nine different outcomes (two qualitative and seven quantitative). On February 6, 2025, I began with a blank Excel sheet, and so far, I have 57 full days of data collected. I will continue monitoring my findings for the remainder of the Spring 2025 Semester. Per my project instructions, I must include tables and graphs for my qualitative and quantitative outcomes. So far, I have collected daily quantitative data on my screen time (Instagram and …

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

Libraries Scholarship

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) …

A Zariski-Nagata Theorem For Smooth Toric Surfaces, Jordan Vincent Barrett

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

The Affine Zariski-Nagata theorem is a classical result in commutative algebra that gives an expression for the n^th symbolic power of a radical ideal I in a polynomial ring over a field in terms of the n^th ordinary powers of the maximal ideals in [affine variety]_max(I). In this thesis we discuss a well-known projective analog of Zariski-Nagata and provide the necessary background on toric varieties to present a generalization of this result to toric surfaces. We conclude with a brief discussion about work toward characterizing which abstract toric varieties have smooth point fibers with …

Transformations Of Bound Quivers, Ari Pincus-Kazmar

Mathematics, Statistics, and Computer Science Honors Projects

A quiver Q is a directed multigraph. Representations of Q are assignments of vector spaces and linear maps to the vertices and arrows of Q; these are categorically equivalent to the modules over the path algebra kQ. The indecomposable representations of a given quiver and the irreducible morphisms between them are summarized combinatorially in the Auslander-Reiten quiver which, in the case of algebras of finite representation type, gives almost complete information about the category of representations. We introduce an intuitive transformation from a family of bound quivers to a family of A-type quivers which preserves properties of the Auslander-Reiten quiver.

Modeling Literary Connections: Exploring Transregional Resistance In Dalit Poetry, Antara Bhattacharyay

Mathematics, Statistics, and Computer Science Honors Projects

Structuring socio-political identities, the caste system (a graded form of hierarchy) remains entrenched in contemporary Indian society. Dalits, marginalized by the caste system, have expressed their resistance through literature, envisioning substantive equality and social change. In this thesis, I draw on digital humanities methods to examine regional variation in translated Dalit poetry from Bengali, Hindi/Urdu, Marathi, and Tamil languages. I utilize topic modeling—a machine learning algorithm that detects latent semantic structures in a text—as a point of departure for poetry analysis. I observe how topic modeling enables newer readings of the poems, revealing regionally resonant and broader Dalit themes.

Oriented Matroid Circuit Polytopes, Jodi Mcwhirter

Arts & Sciences Graduate Student Theses and Dissertations

Matroids give rise to several natural constructions of polytopes. Inspired by this, we examine polytopes that arise from the signed circuits of an oriented matroid. We give the dimensions of these polytopes arising from graphical oriented matroids and their duals. Moreover, we consider polytopes constructed from cocircuits of oriented matroids generated by the positive roots in any type A root system. We give an explicit description of their face structure and determine the Ehrhart series. We also study an action of the symmetric group on these polytopes, giving a full description the subpolytopes fixed by each permutation. These type A …

A Preliminary Study Of Hilbert–Kunz Functions: Coefficient Behavior In A Normal Affine Semigroup Ring, Jesus A. Mendiola Herrera

Theses and Dissertations

In 1890, David Hilbert published a set of notes on what now constitutes one of the bases of Commutative Algebra; his work would eventually influence the efforts of mathematicians like Ernst Kunz. In 1969, Ernst Kunz introduced a particular mapping regarding modules of regular local rings. His goal was to characterize Noetherian local rings of prime characteristic by computing the length of the composition series under Frobenius power transformations. In this thesis, the focus will be on stating the initial steps on finding the coefficients of the Hilbert-Kunz function of the normal affine semigroup ring of the form R = …

How Much Should Consumers With Mild To Moderate Hearing Loss Spend On Hearing Devices?, Vinaya Manchaiah, Steve Taddei, Abram Bailey, De Wet Swanepoel, Hansapani Rodrigo, Andrew Sabin

School of Mathematical & Statistical Sciences Faculty Publications

Background: This study examined the relationship between hearing device price and sound quality. Method: A novel consumer-centric metric of sound quality (“SoundScore”) was used to assess hearing devices’ audio performance. Each hearing device is tested with two fittings. The “Initial Fit” is designed to approximate the most likely fitting for an individual with a mild-to-moderate sloping sensorineural hearing loss. The “Tuned Fit” includes adjusting parameters optimized to hit prescriptive fitting targets (NAL NL2) on an acoustic manikin. Each fitting is evaluated across five dimensions. Both fittings are combined using a weighted average to create a single number from 0 to …

Studies In Algebraic Cycles: Hodge Theory Of Degenerations, Regulators, And Cluster Varieties, Ricardo Jaime Acuna

Arts & Sciences Graduate Student Theses and Dissertations

This dissertation brings together results in Hodge theory and the theory of cluster varieties. The second chapter, based on a paper written in collaboration with Devin Akman and Matt Kerr, uses admissible normal functions to establish the first finiteness result for zero-dimensional components of the Hodge locus. The third chapter, based on joint work with Matt Kerr, shows that weak polar- ized relations constrain possible adjacencies of mixed Hodge structures across boundary strata in geometric compactifications. The fourth chapter gives a geometric construction of a 6-dimensional cluster variety with a disconnected mutation graph, as discovered by Yan Zhou. These three …

Some Interpolation Problems In The Projective Plane, Lilah Estes

Mathematical Sciences Undergraduate Honors Theses

Given some set of r general points in the projective plane, we want to better understand: what is the smallest degree of any polynomial passing through the points m times? How many linearly independent equations of this degree pass through the points m times? The investigation of these questions, particularly for the case of m=3 and r< 16, motivates the development of several results. We translate Terracini's inductive argument, a tool for evaluating the expectedness of certain sets of double points, into a version which can be used for triple points, and prove that the argument holds. We compute the minimal graded free resolutions for the ideals corresponding to up to 15 points, for m up to 6, and we conjecture a connection between the expectedness of these ideals and what their resolutions look like. Further, we prove that this conjecture holds when m=1, and we either fully or partially prove that these ideals are …

On The Hexgame, Corwin Jones

Mathematical Sciences Technical Reports (MSTR)

The SOMA Cube has been studied by mathematicians for a number of decades, but so far methods for solving three-dimensional space-filling puzzles like the SOMA cube remain numerical; we do not have a means to predict the number of solutions to SOMA-like puzzles. We present a two-dimensional puzzle that shares certain features of the SOMA Cube, with the hope that it will be a more convenient object of study for future research into space-filling/space-covering puzzles.

The Stem Gender Gap: Can It Be Closed?, Makenna Roberts

Undergraduate Honors Capstone Projects

Despite increasing gender parity in many professional fields, women remain underrepresented in STEM industries, particularly in math-intensive areas such as engineering, computer science, and physics. This report explores the persistent gender gap in STEM through analysis of two data sources: gendered differences in the US national Scholastic Aptitude Test (SAT) performance and post-graduate employment patterns among scientists and engineers.

The first portion of the analysis examines SAT score data from 2018 to 2024, with a focus on high-performing test-takers. Using Python and R, data was extracted and visualized to compare male and female performance on the Math and Evidence-Based Reading …

Data Science For Engineers, Heidi Moulton

Undergraduate Honors Capstone Projects

Undergraduate research is a core pillar of Utah State University’s College of Engineering. Many students become involved with research during their Junior and Senior years and begin to generate various forms of data. Most students, however, have received little formal education on how to process data, and there are currently no readily available resources within the College of Engineering. As a Mechanical Engineering and Data Science double major, I found the data processing techniques I learned in my Data Science courses invaluable as an undergraduate researcher, and now as a Mechanical Engineer at Apogee Instruments, I frequently draw upon these …

Geometric Algebra For Field Theory In Curved Spacetime, Kevin Rhine

All Graduate Reports and Creative Projects, Fall 2023 to Present

Physics seeks to understand the universe by uncovering the fundamental laws that govern matter, energy, space, and time. At its heart lies the challenge of unification: finding a mathematical framework that consistently describes these interactions across all scales, from the subatomic to the cosmological.

This thesis explores geometric algebra, a mathematical language that unifies algebra and geometry, as a tool for advancing this understanding. By extending this framework to curved spacetimes, where gravity influences the structure of space and time, we investigate its ability to describe physical phenomena such as electromagnetism and general relativity. A notable contribution includes the geometric …

Decision Space Decomposition For Multiobjective Programs, Emma Soriano

All Dissertations

Being inspired by the parametric decomposition theorem for multiobjective optimization problems (MOPs) of Cuenca and Miguel (2017), and by the block- coordinate descent for single objective optimization problems, we present a decom- position theorem for computing the set of minimal elements of a partially ordered set. This set is decomposed into subsets whose minimal elements are used to retrieve the overall minimal elements. We apply this approach to strictly convex MOPs de- composing their decision space into lines. The line decomposition benefits from the fact that a multiobjective line search problem is equivalent to solving a collection of single objective …

The Spectral Asymptotics Of Toeplitz Operators On Hilbert Spaces Of Analytic Functions, Trevor Camper

All Dissertations

Many physical systems, whether they are ocean waves or particles moving through space, can be described using the mathematical language of “partial differ- ential equations.” In many circumstances, it is useful to study how these equations amplify an input to the equation, in which case the amplification factor is called an “eigenvalue.” The usefulness of these amplification factors is that they can be used to describe properties of the physical system. In this dissertation, I have studied this amplification factor for a related set of equations called “Toeplitz operators.” In par- ticular, I have studied eigenvalues using statistical techniques. The …