A Character Theory For Loop Representations Of Symmetric Monoidal Bicategories,
2026
University of Kentucky
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,
2026
University of Kentucky
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,
2026
University of Kentucky
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,
2026
University of Kentucky
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,
2026
University of Kentucky
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,
2026
University of South Dakota
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 …
Representations Of Finite Groups And Diagrammatic Algebras,
2026
Claremont McKenna College
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,
2026
CUNY Kingsborough Community College
Mat 1500 Calculus I Syllabus, Tian Cai
Open Educational Resources
No abstract provided.
Mat 1600 Syllabus Ii Syllabus,
2026
CUNY Kingsborough Community College
Mat 1600 Syllabus Ii Syllabus, Tian Cai
Open Educational Resources
No abstract provided.
Mat 2100 Calculus Iii Syllabus,
2026
CUNY Kingsborough Community College
Mat 2100 Calculus Iii Syllabus, Mei Xing
Open Educational Resources
No abstract provided.
Alexander Duals Of Symmetric Simplicial Complexes And Stanley-Reisner Ideals,
2026
University of Minnesota - Twin Cities
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 (In)n∈N of Sym-invariant squarefree monomial ideals, we study the corresponding chain of Alexander duals (In∨)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 In∨ is given by a polynomial in n for sufficiently large n. The same is true for …
Mechanisms Driving Disparities In Income Mobility Across The Income Distribution,
2026
University of Richmond
Mechanisms Driving Disparities In Income Mobility Across The Income Distribution, Joe Larkins
Honors Theses
This study examines intergenerational income persistence across the income distribution, testing whether mechanisms driving inequality differ between families in the top and bottom halves of the income distribution. Using data from the National Education Longitudinal Study of 1988 (NELS:88), a nationally representative longitudinal survey of 8th grade students and their parents, this research estimates an interaction model comparing parental income effects for children in advantaged versus disadvantaged economic circumstances. The analysis reveals that a $1,000 increase in parental income yields eight times greater income gains for children in the bottom half of the distribution compared to those in the top …
Small Antiperfect Steiner Triple Systems,
2026
Dakota State University
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,
2026
Missouri University of Science and Technology
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,
2026
Missouri University of Science and Technology
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,
2026
Missouri University of Science and Technology
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,
2026
Bucknell University
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,
2026
Harvey Mudd College
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.
Pursuer Evader Surveillance Game,
2026
University of Richmond
Pursuer Evader Surveillance Game, Zijie Mu
Honors Theses
This thesis studies the pursuer evader surveillance game with a triangular obstacle in the short-term. In the game, the pursuer aims to maintain surveillance of the evader as long as possible while the evader aims to break surveillance in a finite time. We classify the player strategies into ideal ones and best admissible ones. The outcome of the game is determined by line of sight. We reduce the 4D game to a 3D game with a boundary separating two different local regimes. When the evader starts inside the threshold, we show that there exists an admissible evader that maintains an …
Education And The Maternal Childcare Gap: Evidence From The Uk Covid-19 Pandemic,
2026
University of Richmond
Education And The Maternal Childcare Gap: Evidence From The Uk Covid-19 Pandemic, Lara Bakhaya
Honors Theses
This paper examines whether college education shapes working mothers’ access to remote work, flexible working, and childcare hours in the United Kingdom, and whether COVID-19 school closures amplified these inequalities. Using data from the UK Time Use Survey (2016–2021), a repeated cross-sectional diary dataset spanning the pre-pandemic period and five COVID-19 waves, this paper estimates weighted logistic and ordinary least squares regressions on a sample of married or cohabiting, employed mothers. School closures serve as a natural experiment, providing an exogenous shock to caregiving demands that affected all mothers simultaneously regardless of education level. College education significantly predicted working from …
