Mathematics Commons^™

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.

Quantifying The Sensitivity Of Land Use Land Cover Metrics Through Simulation Techniques, Haley Burger

All Graduate Theses and Dissertations, Fall 2023 to Present

As human activities and climate change continue to reshape our landscape, understanding how land use changes over time is becoming increasingly important. Accurate ways to track and analyze these changes are essential for governments, businesses, and communities to make informed decisions. Monitoring agricultural land is particularly critical, as shifts in land use can impact food production and environmental pollutants. One of the primary tools used in the United States to monitor agricultural land is the Cropland Data Layer (CDL), an annual map created by the United States Department of Agriculture (USDA) from satellite images. While the CDL is highly accurate, …

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 Graph-Based Decoders For Quantum Low Density Parity Check Codes, Kirsten Morris

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

Quantum computing has the potential for radically increased computational ability. However, the physical realization of quantum states are fragile and susceptible to noise and decoherence. For this reason, robust quantum error correction is imperative to achieve quantum computation at scale.

Of particular interest in realizing effective quantum error correction are quantum low density parity check (QLDPC) codes. Classical LDPC codes were invented by Robert Gallager in the 1960s and came in to prominence in the 1990s. Due to Daniel Gottesman’s stabilizer formalism and the invention of Calderbank-Shor-Steane (CSS) codes, we can apply LDPC codes to the quantum setting.

As in …

Interpolation In Weighted Projective Spaces, Shahriyar Roshan Zamir

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

Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree $d$ singular at a given set of points.

After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this thesis, we use commutative algebra to prove analogous statements in the weighted projective space, a natural generalization of the projective space.

A main contribution of this work is the careful adaption of several classical algebro-geometric techniques to the …

On Kernels And Antiderivatives Of Nonlocal Derivatives, Alex John Heitzman

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

Nonlocal operators are mathematical operators taking functions to other functions f → Df , where to evaluate the operator Df at a point x, one must know the value of f in some region around x, and that region cannot be arbitrarily small. Nonlocal derivatives are like derivatives in that they measure the deviation of a function f(z) from f(x) when z is close to x. In this thesis, we will study nonlocal operators of the form

D_kf(x) = [integral]_Ω [f(x) …

Properties Of A Class Of Analytic Functions Associated With Exponentially Convex Functions, K. R. Karthikeyan, Elangho Umadevi, G. Thirupathi, Dharmaraj Mohankumar

All Works

Studies in univalent function theory comprising the exponential of differential characterizations are rarely considered. The prominent study in this direction is the study of so-called α-exponentially convex functions. Here we study a class of analytic functions which satisfy an analytic characterization influenced by the definition of the multiplicative derivative and α-exponentially convex functions. Integral representation and coefficient inequalities of the defined function class are the main results of the paper.

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 …

Constructing Code-Based Zero-Knowledge Proofs Leveraging Generic Errors And Bounded Vectors, Freeman Slaughter

All Dissertations

Quantum computing is developing at an expeditious rate, and once fully scalable quantum computers become realized, classical cryptographic systems face obsolescence. This approaching peril has prompted a paradigm shift away from pre-quantum cryptography and towards post-quantum primitives, such as those that arise from the field of coding theory. Among these, zero-knowledge proofs have emerged as a dynamic tool instrumental in constructing quantum-resilient digital signature schemes.

We being by introducing HammR, a pre-quantum zero-knowledge proof protocol designed to verify Hamming weight and entry constraints of error vectors, and comprehensively establish its security. Subsequently, we extend HammR to the multi-party computation setting, …

Online Multiobjective Optimization, Kristen Joyce

All Dissertations

Online optimization (OO) is an iterative process of decision making under uncertainty. At every step, a decision is made before the outcome of this decision is known. For the online optimization model, the objective function is unknown at the time the decision is being made. It is very likely that the taken decision is not optimal, so the decision maker incurs a loss, called regret, in every iteration. The goal of the online optimization algorithm is to compute a decision at every step so that the overall regret cost is minimized. In particular, the average regret produced by an ideal …

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 …

Utilizing The Horseshoe Prior In Exploratory Factor Analysis And Gaussian Graphical Networks, James Thomas Roddy

Graduate Theses and Dissertations

High-dimensional data analysis frequently involves extracting meaningful structure from noisy, sparse signals. In recent years, Bayesian shrinkage priors—particularly global-local shrinkage priors—have emerged as powerful tools for inducing sparsity while preserving signal fidelity. Among these, the Horseshoe prior has gained notable attention for its capacity to simultaneously shrink irrelevant parameters and retain substantial signals. This dissertation explores the Horseshoe prior as a unified framework for sparse Bayesian inference across theory, simulation, and real-world application. The first component develops new theoretical results establishing the asymptotic Bayes optimality of the Horseshoe prior in Gaussian graphical models (GGMs). We consider sparse precision matrix estimation …

Preservation Of The Bernstein Property For Sums Of Independent Random Variables, Iosif Pinelis

Michigan Tech Publications

It is shown that Bernstein-type conditions on independent random variables are preserved by their sum. Some optimality properties of such preservation are proved.

Rethinking Iterative Proportional Fitting: Scalable And Hybrid Approaches To Joint Distribution Fitting, William Ofosu Agyapong

Open Access Theses & Dissertations

The Iterative Proportional Fitting (IPF) algorithm is widely used in contingency table estimation, survey weighting, and synthetic population generation due to its simplicity and strong theoretical foundation for matching observed marginal distributions. However, in high-dimensional settings, IPF faces substantial computational and memory demands, as well as statistical instability caused by sparse contingency tables. Moreover, IPF is less useful in modern population synthesis tasks that require both scalability and realism because, despite its superiority in matching known marginal distributions, it cannot produce realistic out-of-sample data points. To address these limitations, we first propose a blockwise IPF framework, in which the feature …

Laser Scan Path Design For Controlled Microstructure In Additive Manufacturing With Integrated Reduced-Order Phase-Field Modeling And Deep Reinforcement Learning, Augustine Twumasi

Open Access Theses & Dissertations

Laser Powder Bed Fusion (L-PBF) is a well-established additive manufacturing technique for fabricating intricate metal components with exceptional precision. A significant challenge in L-PBF is the formation of complex microstructures that influence final material properties. We propose a physics-guided, machine learning-aided approach to optimize scan paths for desired microstructure outcomes, such as equiaxed grains. We employed a phase-field method (PFM) to model the evolution of the crystalline grain structure. To reduce computational costs, we trained a surrogate machine learning model, a 3D U-Net convolutional neural network, using single-track phase-field simulations with varying laser powers to predict crystalline grain orientations based …

2-Adic Quantum Mechanics, Continuous-Time Quantum Walks, And The Space Discreteness, Wilson A. Zuniga-Galindo

School of Mathematical & Statistical Sciences Faculty Publications

The authors show that a large class of 2-adic Schrödinger equations is the scaling limit of certain continuous-time quantum Markov chains (CTQMCs). Practically, a discretization of such an equation gives a CTQMC. As a practical result, new types of continuous-time quantum walks (CTQWs) on graphs using two symmetric matrices are constructed. The transport between nodes in one direction is described by one matrix, while the transport between nodes in the opposite direction. This construction includes, as a particular case, the CTQWs constructed using adjacency matrices. The final goal of this work is to contribute to the understanding of the foundations …

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 …

Mathematical Modeling Of Effects Of Tumor Location On Lung Function., Lamargaret Temukisa Johnson

Master of Engineering Theses

Lung cancer has the highest rates of incidence and mortality of all cancers. Most lung cancer tumors are Non-Small Cell Lung Cancer (NSCLC). NSCLC patients with lesions in the upper lobes are found to have better prognosis compared to those with lesions in the middle and lower lobes. Previous studies have suggested various causes for this discrepancy at both the organ-scale and tissue-scale. To model NSCLC growth in different locations within the lung, an organ scale lung model and tissue scale tumor model were coupled through the tissue pressure, and oxygen and carbon dioxide partial pressures. The coupling was used …