Digital Commons Network^™

Extensions Between Modules Defined By Lattice Paths In The Preprojective Algebra, Chloe Napier

Theses and Dissertations--Mathematics

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of …

A Character Theory For Loop Representations Of Symmetric Monoidal Bicategories, Jordan Sawdy

Theses and Dissertations--Mathematics

Character theory arises in many distinct fields of mathematics, but its many instantiations often share a few key features: they arise in contexts where one object is acted on or parametrized by another, and they are often computed via "trace-like" formulas. Focusing on these properties, we present a categorical formalism for constructing such characters. We first define a notion of "loop representation" for symmetric monoidal bicategories, then build a character for such representations via the canonical symmetric monoidal trace. We then show that this character defines a symmetric monoidal functor which satisfies commutativity properties with respect to both restriction- and …

Combinatorial Models For Nonnegativity In Flag Varieties, Williem L. Rizer

Theses and Dissertations--Mathematics

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh, and Carolina Benedetti has further developed the combinatorics and geometry of these structures. In this dissertation, we extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. While cell decompositions in this setting were previously obtained, notably in work of Konstanze Rietsch, our focus is on providing new combinatorial models that make this structure more explicit and computationally tractable. We introduce flag positroid pipe …

The Interaction Between Additive And Multiplicative Structures In Arithmetic Settings, Ali Alsetri

Theses and Dissertations--Mathematics

The first part of this thesis is concerned with Goldbach-type problems. In recent years, there has been an interest in developing density versions of Goldbach-type results. Namely, given a relatively dense subset A of the primes, one may study representations of integers as sums of primes belonging to the subset A. These density Goldbach-type results have been facilitated by the development of new tools from additive combinatorics, in particular the Fourier-analytic transference principle due to Green. We apply the transference principle to obtain a variant of Vinogradov’s theorem involving subsets of primes confined to the residue class 1 (mod 3). …

Local Energy Decay For Non-Stationary Damped Wave Operators, Nicholas Dj Arsenault

Theses and Dissertations--Mathematics

This work establishes integrated local energy decay (ILED) estimates for the damped wave equation on certain non-stationary spacetimes. The main technical result is a high frequency estimate that holds in great generality, provided that null geodesics trapped in a compact region are sufficiently damped. This is combined with low- and medium-frequency estimates to establish full local energy decay. We conclude by providing a counterexample where the damping assumption fails and local energy decay does not hold.

Understanding Möbius Inversion As Composite Of Dual Pairs, Isaac B. Kochmaan

Theses and Dissertations--Mathematics

Möbius inversion is a well-studied computational tool in number theory and combinatorics, but it exhibits a pattern which may be familiar to those who study categorical traces. In 2016, Kate Ponto and Michael Shulman characterized duality and traces in the bicategory of profunctors for a particularly nice class of one-cells. We extend these results. Consequently, we construct, for a given poset, a profunctor whose Euler characteristic is the Möbius function of said poset. In light of this, we propose a categorical notion of Möbius inversion.

Barycentric Subdivision And Hyperbolic Geometry, Hannah Elisabeth Steger

Dissertations and Theses

Barycentric subdivision of a triangle is the geometrical process of repeatedly subdividing a triangle by connecting the midpoints of the sides to the opposite vertices. The transformations which determine this subdivision form a group acting on the hyperbolic plane, action which we will show is topologically transitive. We find specific cases when the barycentric subdivision process leads to flat triangles (all vertices on the x-axis) and, on the contrary, situations when shapes are positioned on an orbit that is a circle, hence never becoming flat. We will also analyse this process when the starting triangle is already flat and we …

All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

Spectral Decimation On The Schreier Graphs Of The Basilica Group: A Thesis In Fractal Analysis, Anne D. Bannon

Scripps Senior Theses

This thesis is intended to provide a comprehensive overview of the literature required to fully understand research conducted during the University of Connecticut's Fractals & Stochastics REU in the summer of 2025. The literature review includes a description of Robert Strichartz's seminal work pertaining to the Laplacian spectrum of the Sierpiński Gasket, which provides a framework for how we approach studying the spectrum of the basilica Julia set. Defining the basilica Julia set and the closely-related Basilica group involves graph theory, automata theory, iterated monodromy group theory, and amenable group theory. Further time is dedicated to defining the graph Laplacian …

An Introduction To Modern Conversations On Knot Invariants, Stella Shah

Scripps Senior Theses

Knot Theory is a vast and diverse subfield of modern mathematics involving the classification and abstraction of knots and links. In this thesis, we wish to provide the necessary background for and an explanation of two papers in different subfields of knot theory, The Forbidden Quiver of a Link, and Biquandle Fares and Link Invariants.

In Chapter I, we begin with an introduction to knot theory and knot invariants. We continue to present the example of Fox Colorings, and conclude the chapter with an example of the Fox Coloring Number Invariant.

In Chapter II, we explore the derivation and utilization …

Real Interpolation: An Approximate Introduction, Madeline Anderson

Scripps Senior Theses

This thesis provides an introduction to real interpolation. We establish

relevant notions in functional analysis first, and use these concepts to study

real interpolation using J. Peetre’s 𝐾-functional in some detail. We also

explore the basics of approximation theory, in particular the connection

between approximation and interpolation results.

Catalan And Hyper-Catalan Numbers: Combinatorial Applications To Polynomial Equations, Leilani Natale

Williams Honors College, Honors Research Projects

In this paper, we study Catalan numbers and their generalization, hyper-Catalan numbers, and explore how these sequences arise naturally in the context of solving polynomial equations using infinite power series. We begin by introducing the Catalan numbers through their combinatorial interpretation as triangulations of convex polygons. Using this geometric definition, we derive a relation whose recursive structure leads to a quadratic functional equation. Interpreting this relation as a formal power series equation allows us to express solutions to quadratic equations as infinite power series whose coefficients are given by the Catalan numbers. This framework is then extended by allowing polygon …

Representations Of Finite Groups And Diagrammatic Algebras, Hudson Yeend

CMC Senior Theses

Representation theory allows mathematicians to study abstract mathematical objects using the powerful and concrete tools of linear algebra. This thesis aims to present some foundational concepts in representation theory and apply these concepts to specific groups and algebras. We begin by examining representations of finite groups, culminating with a proof of Maschke's theorem. We then use the correspondence between a group and its group algebra to segue into a study of representations of diagrammatic algebras, where we introduce analogous notions of decomposition. We end with a study of quiver representations, noting that Gabriel's theorem and the kQ-modular structure transcend …

Mat 1500 Calculus I Syllabus, Tian Cai

Open Educational Resources

No abstract provided.

Mat 1600 Syllabus Ii Syllabus, Tian Cai

Open Educational Resources

No abstract provided.

Mat 2100 Calculus Iii Syllabus, Mei Xing

Open Educational Resources

No abstract provided.

Alexander Duals Of Symmetric Simplicial Complexes And Stanley-Reisner Ideals, Ayah Almousa, Kaitlin Bruegge, Martina Juhnke-Kubitzke, Uwe Nagel, Alexandra Pevzner

Mathematics Faculty Publications

Given an ascending chain (I_n)_n∈N of Sym-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals (I_n^∨)_n∈N. Using a novel combinatorial tool, which we call avoidance up to symmetry, we provide an explicit description of the minimal generating set up to symmetry in terms of the original generators. Combining this result with methods from discrete geometry, this enables us to show that the number of orbit generators of I_n^∨ is given by a polynomial in n for sufficiently large n. The same is true for …

Small Antiperfect Steiner Triple Systems, Justin Z. Schroeder, Joshua Ganschow

Research & Publications

The cycle structure of Steiner triple systems (STS) has been well studied with regard to uniform STS and cycle switching. Of particular interest among uniform STS are perfect STS, in which every cycle graph consists of a single cycle. In this paper, we initiate the study of antiperfect STS, in which every cycle graph consists of a union of at least two cycles. We prove that an antiperfect STS(n) exists for all admissible n ≥ 15 and provide a complete listing of all antiperfect STS(n) for n ≤ 19 and all antiperfect STS(21) with a non-trivial automorphism. Furthermore, it is …

Special Issue: Innovative Numerical Approaches For Problems In Science And Engineering, Xiaoming He, Shuhao Cao, Qiao Zhuang

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

Equilibrium Stability Under Nuclear Confrontation, Martin Bohner, A. A. Martynyuk

Mathematics and Statistics Faculty Research & Creative Works

This article proposes and analyzes mathematical models of confrontation between two and n countries, including countries with nuclear weapons. The proposed models are based on a generalization of Richardson's well-known mathematical model of the arms race. Namely, the factor of hostility is filled with expanded content, including public opinion and the armed forces of the opposing countries. Qualitative analysis of confrontation models is carried out by the method of Lyapunov functions and by applying nonlinear integral inequalities. As a result of the analysis, the conditions for the stability of the equilibrium state of the opposing countries are established, and the …

Threshold Asymmetric Conditional Autoregressive Range (Tacarr) Model, Isuru Ratnayake, V. A. Samaranayake

Mathematics and Statistics Faculty Research & Creative Works

This paper introduces a Threshold Asymmetric Conditional Autoregressive Range (TACARR) model for analyzing the daily price ranges of financial assets. The proposed formulation assumes that the conditional expected range switches between two regimes, representing upward and downward market states, with the disturbance distribution also allowed to vary across regimes. A self-adjusting threshold component, determined by past values of the series, is used to identify the prevailing market regime. In this way, the model is able to capture asymmetric and heteroscedastic volatility behavior in financial markets. The TACARR model is designed to address several limitations of existing price range models, including …

Group-Based Integer Factorization: Theory And Performance, Phuong Cao

Honors Theses

Integer factorization, the problem of finding a nontrivial factor of a composite integer N=pq for large primes p,q, particularly at the size of RSA moduli, is a notoriously difficult challenge that takes classical methods such as Trial Division and Fermat’s Algorithm trillions of years to solve. This thesis studies four probabilistic algorithms that exploit algebraic group structures to achieve significantly better, subexponential efficiency for certain classes of N: Pollard’s p-1, Williams’ p+1, Lenstra’s Elliptic Curve Method, and Pell’s Conic Method. In each case, the algorithm operates on a group over ℤ/Nℤ that decomposes, via the Chinese Remainder Theorem, into corresponding …

A Nonstandard Exploration Of Approximate Identity And Unitization, Tong (Nicole) Wu

HMC Senior Theses

The goal of this senior thesis is to explore general nonstandard analysis and some possible applications to 𝐶^*-algebras in functional analysis. More specifically, we shall define an approximate identity of a 𝐶^*-algebra using nonstandard analysis and study nonstandard hulls of internal 𝐶^*-algebra in the context of different unitizations. We shall also prove a few results for ideals in 𝐶^*-algebra using nonstandard definitions of approximate identities. We shall also briefly discuss the history and developments of nonstandard analysis.

Digital Technologies And Calculus: Students’ Peaks And Pits, Ricela Feliciano-Semidei, Kevin A. Palencia Infante, Alcibiades Bustillo-Zarate

Faculty Articles, Papers, and Other Scholarship

Understanding calculus students’ perspectives can provide valuable insights into their learning needs and help develop strategies to enhance persistence in STEM programs. Numerous studies have shown that the use of digital technologies (DT) influences student engagement, motivation, and mathematics achievement. In this project, we explored students’ perspectives on using DT in calculus. We interviewed eight calculus students from a midwestern doctorate-granting institution in the United States and used thematic analysis informed by the didactical tetrahedron, accounting for internal and external factors that may influence the teaching and learning process. Students reflected on their use of technology, identifying various benefits, including …

Soft-Constrained Variants Of T-Distributed Stochastic Neighbor Embedding For Global Structure Preservation, Joseph A. Balderas

Mathematics Dissertations

Dimensionality reduction (DR) is a fundamental tool in data science and machine learning that transforms high-dimensional data into a low-dimensional representation while preserving important structural properties of the original data. Among modern DR methods, t-distributed stochastic neighbor embedding (t-SNE) has become one of the most widely used techniques for visualization due to its strong ability to preserve local neighborhood structure and produce visually separated clusters. However, despite its popularity, t-SNE is well known to struggle with preserving global structure of data, often producing embeddings in which distances between clusters and neighborhoods do not accurately reflect relationships in the high-dimensional space. …

Global Weak Solutions Of Optical Variational Wave System, Shahrazad Hamed Alnafie

Graduate Theses, Dissertations, and Problem Reports (ETD)

The coupling of a variational wave equation with Maxwell’s equations gives rise to the optical variational wave system, a hyperbolic PDE system that models the director field of the nematic liquid crystals. This system presents unique analytical challenges that have not been addressed in the existing literature. In this dissertation, we study the one-dimensional case of this system.

We establish the global existence of conservative weak solutions to the associated Cauchy problem. The hyperbolic system is derived using the energy variational method. Through a sequence of suitable coordinate transformations, we reformulate the system while preserving its hyperbolicity. Due to degeneracy …

I Think, Therefore I Can: Self-Efficacy, Self-Regulation, And Sense Of Belonging In Introductory Physics, Danielle Christine Maldonado

Graduate Theses, Dissertations, and Problem Reports (ETD)

Cognitive and affective beliefs in physics classrooms are important indicators of student success beyond traditional measurements of academic performance. This body of work explores major beliefs and processes, including self-efficacy, self-regulation, and sense of belonging, as they relate to academic achievement among introductory undergraduate physics students. First, this dissertation explores the development of physics self-efficacy and physics sense of belonging, two highly correlated variables. This work identifies unique sources of physics self-efficacy and sense of belonging development from current and prior academic experiences, personality factors, and demographic variables. Then, relationships between early- semester self-efficacy and sense of belonging, mid-semester …

Classifying Surfaces With Handle Decomposition, Elizabeth Sipes

Murray State Theses and Dissertations

Among topological spaces, manifolds draw a lot of interest. An
n-manifold is a space that is locally like R^n. Manifolds of dimension 2
are called surfaces. Using handle decomposition, we decompose surfaces
into k-handles, where 0< =k< =2. Techniques such as handle sliding and
handle cancellation allow us to get a more favorable representation of
our surface. We use these tools and calculation of the fundamental group
to classify all compact surfaces.

Infimum Dimension Nash Embeddings For 2d Projective Shape Analysis, Robert L. Paige, Vic Patrangenaru

Mathematics and Statistics Faculty Research & Creative Works

Vector embeddings make complicated data extracted from networks, words and images, more amendable to data science applications. At the present time, the Veronese-Whitney (VW) matrix embedding of the real projective space is the state of the art for making inference about digital images from an uncalibrated camera, such as a cell phone or security camera. In this work we consider vector embeddings for the projective shape data and in particular determine the minimum dimension isometric (distance-preserving or Nash) vector embedding for a projective space. We determine such an embedding for the projective plane in closed-form. From this embedding we determine …

A Fully Discrete Semi-Implicit Numerical Scheme And Its Optimal Error Estimates For Cahn-Hilliard-Mhd Model With Variable Density, Dongmei Duan, Fuzheng Gao, Xiaoming He, Yanping Lin

Mathematics and Statistics Faculty Research & Creative Works

This paper proposes and analyzes a fully discrete semi-implicit unconditionally energy stable numerical scheme to solve the Cahn-Hilliard Magnetohydrodynamics (Cahn-Hilliard-MHD) model with variable density. The unconditional energy stability and optimal L² error estimates are established for the fully discrete scheme. Major challenges in error estimation arise from the variable density, the strong nonlinearities, and the multi-physics coupling of the model. Under the mathematical induction framework, the Ritz quasi-projection and the Stokes quasi-projection, proposed in [SIAM J. Numer. Anal., 61(3):1218-1245, 2023], are utilized to avoid the gradient terms of the projection errors. The H⁻¹ superconvergence error estimates of Ritz …