Mathematics Commons^™

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.

Wavelet Representation Of Singular Integral Operators, Jeremy Cummings

Arts & Sciences Graduate Student Theses and Dissertations

The idea of representing singular integral operators as averages dyadic shifts has proven fruitful since Petermichl's representation of the Hilbert transform, and its generalization by Hyt\"onen to prove the $A_2$ conjecture. These results employ a random dyadic decomposition of the operator in terms of Haar shifts of all complexities. An alternate approach to wavelet representation was provided by Di Plinio, Wick, and Williams (2022) in which the random dyadic grids are replaced by zero-complexity wavelet projections, providing finer control of smooth operators and a more efficiently computable representation. The goal of this thesis is to provide two main generalizations of …

A Mathematical Theory Of Epitaxial Growth, Brock C. Price

Theses and Dissertations

In this dissertation we investigate several PDE models of Epitaxial growth. These models are fourth-order PDE’s featuring exponential nonlinearities as well the p-Laplacian and 1-Laplacian. The presence of the exponential nonlinearity is what provides the main mathematical difficulty. Theexponent in particular does not have enough estimates to guarantee any compactness. Because of this, one has to allow the inclusion of a singular portion to the exponent in the sense of the Lebesgue decomposition theorem. In thefirst chapter weinvestigate a related epitaxial growth, with transition rates of the Metropo lis variety and a linear exponent. The metropolis rates induce an extra …

Differentiation And Certain Operators On Scaled Sectional Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we study certain sectional structures of the t-scaled hypercomplex numbers H_t for a scale t ∈ R, including the quaternions H_-1, and the split quaternions H₁. For a fixed scale t ∈ R, by defining the collection S_t of certain pureimaginary t-scaled hypercomplex number in H_t , we sectionize H_t from the imaginaries of S_t. We concentrate on a section SH_It for an arbitrarily fixed imaginary I_t ∈ S_t , called the t-scaled section for I_t. Differentiation theory on the …

Skolem Number Of Kagome Lattice Graphs, Braxton Carrigan, Max Martone

Theory & Applications of Graphs

A proper Skolem labelling of a graph $G$ is a function assigning a positive integer to each vertex of $G$ such that any two vertices assigned the same integer are that distance apart in the graph. The Skolem number of a graph is smallest number $n$ such that there exists a proper Skolem labelling only using the positive integers less than or equal to $n$. In this paper, we will begin by proving the Skolem number for another family of subgraphs of the hexagonal lattice and then prove the Skolem number for two families of subgraphs of the Kagome Lattice.

Two Problems: Hankel Operators And Dyadic Paraproducts, Ana Čolović

Arts & Sciences Graduate Student Theses and Dissertations

Paraproducts can be thought of "parts of a product" of two functions, that isolate particular properties of each of the functions. They have played an essential role in the study of commutators in harmonic analysis, in particular commutators of multiplication by a function and Calder\'{o}n-Zygmund operators. In complex analysis, Hankel and Toeplitz operators can be used to decompose a product of two functions. They play the same role as paraproducts in analyzing commutators of certain operators, so they can be thought of as complex analytic analogues of paraproduct operators. The thesis consists of two parts. In the first part, we …

Math 122: Precalculus Instructor Guide, Seth Lehman

Open Educational Resources

OER Instructor guide for Math 122: Precalculus at Queens College.

The Characteristic Function Of The Cube Of A Gaussian Random Variable, Andreas Boukas

Journal of Stochastic Analysis

Using the spectral resolution of the multiplication operator on the Schwartz class of L2(R,C), we compute the characteristic function of the cube of a Gaussian random variable.

Energy-Stable And Efficient Finite Element Schemes For The Shliomis Model Of Ferrofluid Flows, Guo Dong Zhang, Kejia Pan, Xiaoming He, Xiaofeng Yang

Mathematics and Statistics Faculty Research & Creative Works

In this paper, we aim to design two energy-stable and efficient finite element schemes for simulating the ferrofluid flows based on the well-known Shliomis model. The model is a highly nonlinear, coupled, multi-physics system, consisting of the Navier–Stokes equations, magnetostatic equation, and magnetization field equation. We propose two reliable numerical algorithms with the following desired features: linearity and unconditional energy stability. Several key techniques are used to achieve the required features, including the auxiliary variable method, consistent terms method, prediction-correction method, and semi-implicit stabilization method. The first scheme is based on a hybrid continuous/discontinuous finite elements spatial approximation, and the …

An Alternative Approach To Non-Relativistic Quantum Mechanics In Curved Space, Robert A. Hulsey

Dissertations

In the research presented in this dissertation, we propose an alternative formulation of non-relativistic quantum mechanics in curved spaces (Riemannian manifolds). Some toy quantum models (2D quantum harmonic oscillator in Poincaré half-plane model and the flat chart model of hyperbolic 2-space) are studied to understand the physical implications of this alternative formulation.

Multiple Monochromatic Subgraphs In Edge-Colored Graphs, Emma Felicity Jent

Dissertations

Ramsey theory, though a relatively young branch of mathematics, has captivated the attention of graph theorists, combinatorialists, and theoretical computer scientists alike through its raw beauty, versatility, and powerful applications. Before it emerged as a branch of mathematics, the central idea of Ramsey theory appeared in the form of three lemmas in three separate papers by three different mathematicians working on three distinct areas of research. The first such lemma was published by David Hilbert in 1892, followed by the second lemma published by Issai Schur in 1916. However, Frank Ramsey’s renowned lemma, published in 1930, compelled mathematicians to establish …

Analysis Of Multi Grade Deep Learning, Ronglong Fang

Mathematics & Statistics Theses & Dissertations

Multi-Grade Deep Learning (MGDL) is a training framework that incrementally builds deep neural networks. It does this by dividing the training process into multiple “grades,” where each grade sequentially trains a shallow neural network to learn the residue from the previous one, using the outputs of prior grades as input. This approach progresses from shallow to deep architectures. This dissertation offers a comprehensive theoretical and numerical analysis of the MGDL methodology.

We first demonstrate that MGDL can effectively learn target functions within the sum-composition learning format. In this context, MGDL approximates high-frequency components by composing multiple low-frequency functions. This unique …

Μ-Distributions And Μ-Distributed Sequences, Noah Graham Giddings

Undergraduate Honors Thesis Collection

The project attempts to generalize the notion of a uniform distribution for a broader class of probability measures and illustrate one construction of a sequence that satisfies these properties. A sequence ⟨a_n⟩ is uniformly distributed if the limiting relative frequency of sequence elements in any interval I ⊆ [0,1] corresponds with length(I). We generalize the notion of “uniform distribution” for an atomless Borel probability measure µ. We say that a sequence ⟨a_n⟩ is µ-distributed if the limiting relative frequency of sequence elements in any interval I ⊆ [0,1] equals µ( …

A Study Of Complex Analysis After Whittaker And Watson, Crystal Steed

All Graduate Reports and Creative Projects, Fall 2023 to Present

The goal of this report is to provide solutions to the exercises found in chapter five of the book titled, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions with an Account of the Principal Transcendental Functions by E.T. Whittaker and G.N. Watson. The fifth chapter is titled, "The Fundamental Properties of Analytic Functions; Taylor's, Laurent's and Liouville's Theorems." This report solves the end-of-chapter exercises in addition to providing details for some in-chapter exercises, which are left to the reader. Many of these exercises are results from famous mathematicians.

Traffic Prediction For Research And Education Networks: Anomaly-Aware Deep Learning And Benchmarking, Mohammad Arafath Uddin Shariff

School of Computing: Dissertations, Theses, and Student Research

Research and Education Networks (RENs) and High-Performance Computing (HPC) environments are critical infrastructures for modern scientific discovery, demanding sustained high-throughput and low-latency data transfers. Unlike commercial networks, RENs exhibit unique traffic characteristics, including predominant “elephant flows,” inherent burstiness, and complex temporal-spatial dynamics often decoupled from human-driven cycles. Traditional traffic forecasting methods, tailored for commercial Wide Area Networks (WANs), consistently fail to capture these distinct REN dynamics, leading to inefficient resource management and potential impediments to scientific progress.

This thesis addresses this critical gap by developing and validating a robust, scalable, and anomaly-aware traffic forecasting framework specifically tailored for REN/HPC networks. …

Analysis Of Popular Songs In The Us Market, Michael Myers

Theses, Dissertations and Culminating Projects

In today’s digitally driven music landscape, understanding what drive’s a song’s popularity requires insight not only into its acoustic and lyrical content, but also into patterns of listener engagement across platforms. This thesis explores the predictive and descriptive dimensions of song popularity by applying supervised and unsupervised machine learning models to a multi-source dataset integrating audio features, sentiment analysis, and temporal consumption behavior. Drawing from a novel, multi-platform dataset that includes Billboard Hot 100 rankings, Spotify acoustic features and popularity scores, streaming, airplay, and sales metrics as reported on Luminate’s Music Connect, and lyrics from AZLyrics, the study investigates the …

Certified Approximation Algorithms Of Algebraic Curves, Michael Byrd Jr.

All Dissertations

One of the fundamental problems in mathematics is to determine the set of solutions to a system of equations. In algebraic geometry, the equations studied are polynomials, and the solution set is called an algebraic variety. For single variable polynomials of degree less than five, the roots can be determined exactly using algebraic methods, but for polynomials of degree five or higher, numerical methods are required. When using numerical methods, it is important to know when the computed approximation is indeed a correct solution, which leads to the idea of a certified algorithm. An algorithm is said to be …

Active Calculus: Single Variable, 2nd Edition, Matthew Boelkins, David Austin, Christina Safranski, Steven Schlicker

Open Textbooks

Active Calculus is different from most existing calculus texts in at least the following ways: the text is freely readable online in HTML format and is also freely available for in PDF; in the electronic formats, graphics are in full color and there are live links to java applets; there are live WeBWorK exercises in each chapter, which are fully interactive in the HTML format and included in print in the PDF; the text is open source, and interested users can gain access to the original source files on GitHub; the style of the text requires students to be active …

Homomesies And Toggleability Spaces, Alec Mertin

All Dissertations

We study the homomesy phenomenon under the rowmotion operator acting on order ideals of posets. We provide details for the extensions of several results in the literature concerning homomesies of toggleability statistics from finite to infinite orbits, which allows us to obtain homomesies for piecewise-linear and birational rowmotion, even in the case of infinite orbits. Integral to this is a novel generalization of a result in the literature, which relaxes the conditions needed to lift a statistic.

We completely describe the order ideal (resp. antichain) toggleability space for general fences: the space of statistics which are linear combinations of …

An Introduction To Reverse Mathematics Through The Weakened Base System Rca_0^*., Kaden Dvorak

Boise State University Theses and Dissertations

The purpose of reverse mathematics, a field of mathematical logic, is to determine which axioms are required to prove particular mathematical theorems. Gödel's first incompleteness theorem states that within any standard consistent formal system of mathematics, there are statements for which neither themselves nor their negations can be proven. Thus, the goal of reverse mathematics cannot be the discovery of some formal system which underlies all mathematics, which was that of Hilbert's program. Instead, the questions lie in how much mathematical reasoning can be represented and how strong the formal systems are required to be to conduct said reasoning. In …