Mathematics Commons^™

Missing Data Imputation With Longitudinal Data, Joseph Omar Alanis

Theses and Dissertations

In the repeated measures longitudinal datasets where missing data is a relatively common issue, we explored different imputation methods, including machine learning (ML) methods in order to examine potential efficiencies for using traditional versus newer computational methods. In order to accomplish said comparison, we used a Monte Carlo simulation experiment of a population of size N=70000 to mimic a clinical trial, with different scenarios of missing at random (MAR) data in the response variable. To compare the behavior of each method resulting from the difference between population and sample dataset, a real dataset was used from the Boston College on …

Recognizing Triangles Using Weighted Ehrhart Polynomials, Rafael Sarabia

Theses and Dissertations

Ehrhart theory is a classical topic for polygons or polytopes with vertices with integer coordinates. One of the main results of the theory claims that for every convex polygon P with integer vertices, the number of integer points in its nth dilation is a polynomial in n, the Ehrhart polynomial of P. This property can be extended if for a polynomial weight w(x, y), we sum values of w(x, y) over all points with integer coordinates in the nth dilation of P. The resulted polynomial is called the weighted Ehrhart polynomial of P associated with the weight w. In this …

A Survey On Homomorphic Encryption Based On The Learning With Rounding Problem, Jose Manuel Montoya

Boise State University Theses and Dissertations

Much of modern day cryptography deals with the notion of security and efficiency. The main concern with any cryptosystem deals with the practicality of being able to implement any encryption algorithm over large data sets. In this paper, we will look at the various properties of the Learning With Rounding problem (LWR) which comes from the Learning With Errors problem (LWE) first introduced by Oded Regev. We will also present the topic of Homomorphic Encryption (HE) as well as its three variants: Partially Homomorphic Encryption (PHE), Somewhat Homomorphic Encryption (SHE), and Fully Homomorphic Encryption (FHE). This paper also highlights SABER, …

Probabilistic Frames And Concepts From Optimal Transport, Dongwei Chen

All Dissertations

As the generalization of frames in the Euclidean space $\mathbb{R}^n$, a probabilistic frame is a probability measure on $\mathbb{R}^n$ that has a finite second moment and whose support spans $\mathbb{R}^n$. The p-Wasserstein distance with $p \geq 1$ from optimal transport is often used to compare probabilistic frames. It is particularly useful to compare frames of various cardinalities in the context of probabilistic frames. We show that the 2-Wasserstein distance appears naturally in the fundamental objects of frame theory and draws consequences leading to a geometric viewpoint of probabilistic frames.

We convert the classic lower bound estimates of 2-Wasserstein distance \cite{Gelbrich90, …

Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles

All Dissertations

This dissertation will explore factorization within orders in a number ring. By far the most well-understood of these orders are rings of algebraic integers. We will begin by examining how certain types of subrings may relate to the larger rings in which they are contained. We will then apply this knowledge, along with additional techniques, to determine how the elasticity in an order relates to the elasticity of the full ring of algebraic integers. Using many of the same strategies, we will develop a corresponding result in the rings of formal power series. Finally, we will explore a number of …

Gan With Skip Patch Discriminator For Biological Electron Microscopy Image Generation, Nishith Ranjon Roy

Graduate Theses and Dissertations

GAN models have been successfully used for image generation in various sections such as real-life objects like human faces, cars, animal faces, landscapes, etc. This work focuses on biological electron microscopy (EM) image generation. Unlike other real-life objects, biological EM images are obtained through electron microscopy techniques to study biological specimens. Electron microscopy offers high resolution and magnification capabilities, making it a powerful tool for visualizing biological structures at the nanoscale. However, using GAN models for biological EM image generation poses challenges due to the complex and unique arrangements of biological structures and the sparse and asymmetrical patterns in EM …

A Uniformly Most Powerful Test For The Mean Of A Beta Distribution, Richard Ntiamoah Kyei

Electronic Theses and Dissertations

The beta distribution is used in numerous real-world applications, including areas such as manufacturing (quality control) and analyzing patient outcomes in health care. It also plays a key role in statistical theory, including multivariate analysis of variance (MANOVA) and Bayesian statistics. It is a flexible distribution that can account for many different characteristics of real data. To our surprise, there has been very little work or discussion on performing statistical hypothesis testing for the mean when it is reasonable to assume that the population is beta distributed. Many analysts conduct traditional analyses using a t-test or nonparametric approach, try transformations, …

On Neumann Boundary Conditions For Nonlocal Models With Finite Horizon, Scott Alex Hootman-Ng

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

Nonlocal models are have recently seen an explosive interest and development in the context of fracture mechanics, diffusion, image processing, population dynamics due to their ability to approximate differential-like operators with integral operators for inherently discontinuous solutions. Much of the work in the field focuses on how concepts from partial differential equations (PDEs) can be extended to the nonlocal domain. Boundary conditions for PDEs are crucial components for applications to physical problems, prescribing data on the domain boundary to capture the behavior of physical phenomena accurately with the underlying model. In this thesis we specifically examine a Neumann-type boundary condition …

Some Experiments In Additive Number Theory, Yunan Wang

All Dissertations

This dissertation explores fundamental conjectures in number theory, focusing on the distribution patterns of representation functions in prime pairs. The work concentrates on twin primes, cousin primes, and primes separated by six units, offering a fresh heuristic interpretation of the Hardy-Littlewood correction factor. The analysis progresses to investigate the partition function for prime pairs in the form $(p, p+k)$, specifically for $k = 2, 4, 6$. The study culminates in the derivation of a general formula for prime pairs $(p, p+d)$, where $d$ is an even integer. Drawing on the insights gleaned from examining the correction factor, this dissertation proposes …

Mitochondrial Function In Lungs Of Rats With Different Susceptibilities To Hyperoxia-Induced Acute Lung Injury, Pardis Taheri, Devanshi D. Dave, Ranjan K. Dash, Guru P. Sharma, Anne V. Clough, Elizabeth R. Jacobs, Said H. Audi

Biomedical Engineering Faculty Research and Publications

Adult rats exposed to hyperoxia (>95% O₂) die from respiratory failure in 60–72 h. However, rats preconditioned with >95% O₂ for 48 h followed by 24 h in room air acquire tolerance of hyperoxia (H-T), whereas rats preconditioned with 60% O₂ for 7 days become more susceptible (H-S). Our objective was to evaluate lung tissue mitochondrial bioenergetics in H-T and H-S rats. Bioenergetics was assessed in mitochondria isolated from lung tissue of H-T, H-S, and control rats. Expressions of complexes involved in oxidative phosphorylation (OxPhos) were measured in lung tissue homogenate. Pulmonary endothelial filtration coefficient …

Virtual Unknotting Numbers For Families Of Virtual Torus Knots, Kaitlin R. Tademy

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

A virtual torus knot T(p,q,VC) sits in the intersection of the well-understood torus knot and the not-so-well-understood virtual knot, making it an intriguing object to study.

The unknotting number of a classical knot K is defined unambiguously. However, "the" unknotting number when K is a virtual knot is not as clear to define, since virtual knots have both classical and virtual crossings. We will define virtual unknotting number vu(K) as the minimum number of (classical) crossing changes required to unknot K. Under this definition of virtual unknotting, not all …

Gevrey Class Estimates Towards Null Controllability Of A Fluid Structure Interaction System, Dylan Mcknight

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

Fluid-Structure Interaction concerns the interaction of parabolic fluids and hyperbolic elastic structures via numerous mechanisms such as boundary coupling and pressure. These models find application in blood flow, fluid flow in the eye, and air flow over plane wings. Parabolic equations are well known for “infinite speed of propagation,” which manifests itself via a uniform bound on the resolvent of the infinitesimal generator of the associated strongly continuous semigroup. Qualitatively, a solution of a parabolic pde with rough initial data is immediately smooth for any positive time. A priori, it is not clear whether a fluid structure interaction inherits any …

On Regularity Of Graph C*-Algebras, Gregory Joseph Faurot

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

We prove that for any countable directed graph E with Condition (K), the corresponding graph C*-algebra C*(E) has nuclear dimension at most two. We also prove that the nuclear dimension of certain extensions is at most one, which can be applied to certain graphs to achieve the optimal upper bound of one. Finally, we generalize some previous results for O_∞ -stability of graph algebras, and prove some partial results for Z-stability.

Advisor: Christopher Schafhauser

Making Sandwiches: A Novel Invariant In D-Module Theory, David Lieberman

Department of Mathematics: Dissertations, Theses, and Student Research

Say I hand you a shape, any shape. It could be a line, it could be a crinkled sheet, it could even be a the intersection of a cone with a 6-dimensional hypersurface embedded in a 7-dimensional space. Your job is to tell me about the pointy bits. This task is easier when you can draw the shape; you can you just point at them. When things get more complicated, we need a bigger hammer.

In a sense, that “bigger hammer” is what the ring of differential operators is to an algebraist. Then we will say some things and stuff …

Robust Multivariate Estimation And Inference With The Minimum Density Power Divergence Estimator, Ebenezer Nkum

Open Access Theses & Dissertations

The estimation of the location vector and scatter matrix plays a crucial role in many multivariate statistical methods. However, the classical likelihood-based estimation is greatly influenced by outliers, potentially leading to unreliable decisions. Hence, a fundamental challenge in multivariate statistics is to develop robust alternatives that can maintain performancein the presence of outliers and deviations from the assumed data distribution. Unfortunately, methods with good global robustness often substantially sacrifice efficiency. To address this, we propose the adoption of Minimum Density Power Divergence (MDPD) estimation, a well-established robust technique known for its efficiency and statistical robustness to outliers and model violations. …

First Diffusion Course: "The Structure Of Language As A Connection Between Artificial Intelligence, Information And Ethics", Dioneia Monte-Serrat

Journal of Humanistic Mathematics

From August 22nd to November 21st, the first Diffusion Course on “The structure of language as a connection between artificial intelligence, information and ethics” will take place on a Thursday of each month, in Ribeirao Preto, Brazil. You are cordially invited to sign up for in-person classes and join other researchers and students on this course.

7th International Conference On Creative Mathematical Sciences Communication (Cmsc`24), Frances A. Rosamond

Journal of Humanistic Mathematics

The 7th International Creative Mathematical Sciences Communication (CMSC) conference is scheduled for October 2024 in Trier, Germany. Initiated in Darwin, Australia in 2013, CMSC aims to explore novel methods of imparting computational thinking to diverse audiences including non-specialists, modern citizens, and children. Participants from around the world and from various and interdisciplinary disciplines such as science, education, dance, drama, and visual arts convene to exchange ideas, present experimental approaches, and collaborate on engaging children in the exploration of ongoing, unresolved research challenges.

Oh Statistics!, Heather L. Cook

Journal of Humanistic Mathematics

This poem was written about statistics and the usefulness thereof.

Discordium Mathematica - A Symphony In Aleph Minor, Vijay Fafat

Journal of Humanistic Mathematics

How did Mathematics arise? Who created it? Why is it subject to Godel’s Incompleteness Theorems? And what does all this have to do with Coleridge’s poem, “Kubla Khan”, and “The Person from Porlock”? Here is a complete mythology of Mathematics set in an epic poetry format, fusing thoughts and verses from Western religions and Eastern mysticism… Those with immense patience and careful reading shall reap the fruit… (best read on a large screen or in printed form)

Pi: A Perpetual Journey, Ravindra K. Bisht

Journal of Humanistic Mathematics

A brief history of the constant pi is presented with a poetic flavor.