Mathematics Commons^™

Maximal Estimates For The ∂-Neumann Problem On Non-Pseudoconvex Domains, Phillip S. Harrington, Andrew Raich

Mathematical Sciences Faculty Publications and Presentations

It is well known that elliptic estimates fail for the ∂-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the ∂-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for (0, 1)-forms) and Ben Moussa (for (0, q)-forms when q ≥ 1). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also …

Mesenchymal Stem Cells In Autoimmune Disease: A Systematic Review And Meta-Analysis Of Pre-Clinical Studies, Hailey N. Swain, Parker D. Boyce, Bradley A. Bromet, Kaiden Barozinksy, Lacy Hance, Dakota Shields, Gayla R. Olbricht, Julie A. Semon

Mathematics and Statistics Faculty Research & Creative Works

Mesenchymal Stem Cells (MSCs) Are of Interest in the Clinic Because of their Immunomodulation Capabilities, Capacity to Act Upstream of Inflammation, and Ability to Sense Metabolic Environments. in Standard Physiologic Conditions, They Play a Role in Maintaining the Homeostasis of Tissues and Organs; However, there is Evidence that They Can Contribute to Some Autoimmune Diseases. Gaining a Deeper Understanding of the Factors that Transition MSCs from their Physiological Function to a Pathological Role in their Native Environment, and Elucidating Mechanisms that Reduce their Therapeutic Relevance in Regenerative Medicine, is Essential. We Conducted a Systematic Review and Meta-Analysis of Human MSCs …

Shapley Value Under Interval Uncertainty And Partial Information, Kittawit Autchariyapanikul, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

In the 1950s, the future Nobelist Lloyd Shapley solved the problem of how to fairly divide the common gain. Namely, he showed that some reasonable requirements determine a unique division -- which is now known as the Shapley value. The main limitation of Shapley's solution is that it assumes that for each subgroup of the original group of participants, we know exactly how much this group could gain if it acted by itself, without involving others. In practice, we rarely know these exact values. At best, we know the bounds on each such value -- i.e., in other words, an …

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, …

Evolving Teacher Pedagogy Through Implementation Of Mathematical Modeling And Participation In Responsively Designed Professional Development, Geena Taite

Theses, Dissertations and Culminating Projects

Modeling with mathematics (MP4) is a mathematical practice standard that can be used to engage students in mathematical content standards as well as additional mathematical practice standards. Research has shown the benefits of mathematical modeling (e.g., Aguirre et al., 2019; Cirillo et al., 2016a), but also the challenges teachers can face implementing modeling (e.g., Asempapa & Sturgill, 2019; Manouchehri, 2017). This dissertation aimed to explore how mathematical modeling professional development was responsively designed and facilitated to support teachers’ pedagogical goals (RQ1). This dissertation also aimed to explore how my role as the designer and facilitator of professional development evolved (RQ2) …

Experimental And Computational Investigation Of Green Laser Propagation Through Turbulent Medium In A Water Tank, Richard Owusu Adansi

Open Access Theses & Dissertations

Turbulence medium poses significant challenges to the propagation of laser beams, impacting applications such as free-space optical communication, remote sensing, and laser weapons systems. Therefore, comprehending and mitigating turbulence effects are vital for enhancing the performance of these systems. Analysis of a strong turbulence impact on electromagnetic wave propagation requires extensive knowledge of the Earth's atmospheric and Oceanic phenomena and Maxwell's electromagnetic theory of light as a propagating wave of electric and magnetic fields. Optical systems' propagation often encounters Strong Turbulence, causing interference and diffraction. Nonetheless, light propagation through randomly fluctuating media has been an exciting and active area of …

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. …

A Generalization Of The Graham-Pollak Tree Theorem To Even-Order Steiner Distance, Joshua N. Cooper, Gabrielle Tauscheck

Faculty Publications

Graham and Pollak showed in 1971 that the determinant of a tree’s distance matrix depends only on its number of vertices, and, in particular, it is always nonzero. The Steiner distance of a collection of 𝑘 vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for 𝑘 =2 , this reduces to the ordinary definition of graphical distance. Here we show that the hyperdeterminant of the 𝑘 th order Steiner distance hypermatrix is always nonzero if 𝑘 is even, extending their result beyond 𝑘 =2 . Previously, the authors showed that the …

Enumeration Of Level-K Hypertriangulations, Patrick J. Kaylor

Theses and Dissertations

Triangulations are a classical object in discrete and computational geometry that finds it uses in many other fields, including numerous applications. In this thesis we approach the question of enumerating all hypertriangulations of a point set, the family of tilings introduced by Olarte and Santos in 2022 as induced projections of hypersimplices. Unfortunately, the usual approach to enumeration of trinagulations using flips may not work for hypertriangulations as flip-connectivity is only conjectured for that family even in the two-dimensional case. As a result, we develop a DFS-based algorithm and present its Python implementation that enumerates all hypertriangulations of small point …

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 …

Exploring Intraplate Seismicity In The Midwest, Alexa Fernández

Department of Earth and Atmospheric Sciences: Dissertations, Theses, and Student Research

Intraplate seismicity represents a notable occurrence within the stable North American Craton. This research explores the potential sources of stresses that could reactivate older faults and influence seismic activity within this region. Among these sources, the enduring impact of the last glacial period is considered, which includes continued glacial isostatic adjustments (GIA). During GIA the lithosphere rebounds due to the retreating ice, and the forebulge caused by far-field flexure in response to the glacial load, collapses. This results in significant faulting, fracturing, and seismic activity associated with the deglaciation phase. The adjustment of the lithosphere manifests as both near surface …

A Study On The Vanishing Of Ext, Andrew J. Soto Levins

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

This thesis has two goals. The first is to study an Ext analog of the rigidity of Tor, and the second is to study Auslander bounds.

In Chapter 2 we show that if R is an unramified hypersurface, if M and N are finitely generated R-modules, and if the nth Ext modules of M against N is zero for some n less than or equal to the grade of M, then the ith Ext module of M against N is zero for all i less than or equal to n. A corollary of this says that if …

Semigroup Well-Posedness And Finite Element Analysis Of A Biot-Stokes Interactive System, Sara Mcknight

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

The coupling of a porous medium modeled by the Biot equations and a fluid has many biological applications. There are numerous ways by which to model the fluid and to couple the porous medium with the fluid. This particular model couples the Biot equations to Stokes flow along the boundary, through the Beavers-Joseph-Saffman conditions. We address semigroup well-posedness of the system via an inf-sup approach, which along the way requires consideration of a related but uncoupled static Biot system. We also present the results of finite element analysis on both the uncoupled Biot system and the coupled system.

Advisor: Sara …

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 …

Spreads And Transversals And Their Connection To Geproci Sets, Allison Joan Ganger

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

Spreads of [set of prime numbers]³ over finite fields can yield geproci sets. We study the existence of transversals to such spreads, proving that spreads with two transversals exist for all finite fields, before further considering the groupoids coming from spreads when transversals do or do not exist. This is further considered for spreads of higher dimensional projective spaces. We also consider how certain spreads might generalize to characteristic zero and the connection to the previously known geproci sets coming from the root systems D₄ and F₄.

Advisor: Brian Harbourne

Perturbations Of Representations Of Cartan Inclusions, Catherine Zimmitti

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

A free semigroup algebra is the unital, weak operator topology closed algebra generated by a collection of Cuntz-Toeplitz isometries in B(H). Ken Davidson and David Pitts asked in [9] if a self-adjoint free semigroup algebra exists; Charles Read answered this question in [28] by constructing such an example, which Ken Davidson later simplified in [8]. The construction takes a standard representation of O₂ and multiplies it by a unitary operator in the diagonal MASA of the representation. This results in a new "perturbed" representation of O₂ generating a self-adjoint free semigroup algebra.

In this thesis, …

Leveraging International Large-Scale Assessments: Insights From An Item-Writing Professional Development Program For High School Mathematics Teachers, Mark Lester B. Garcia, Catherine P. Vistro-Yu

Mathematics Faculty Publications

The Philippine Department of Education, despite seeing minimal improvements in the country’s score and rankings in the recent cycles of international large-scale assessments (ILSAs) such as TIMSS and PISA, maintained that it shall continue to participate in ILSAs to monitor the progress of educational effectiveness and benchmark with other countries by observing and adopting commendable practices. While legislators and other stakeholders push for increased financial resources in the education sector as part of the country’s preparations for upcoming ILSAs, one crucial aspect that is largely overlooked is teacher training. This paper presents a viable yet contentious way of upskilling teachers …

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 …

Cohen-Macaulay Type Of Open Neighborhood Ideals Of Unmixed Trees, Jounglag Lim

All Theses

Given a tree T and a field k, we define the open neighborhood ideal N(T) of T in k[V] to be the ideal generated by the open neighborhoods of all vertices in the graph. If T is unmixed with respect to the total domination problem, then it is known that N(T) is Cohen-Macaulay. Our goal is to compute the (Cohen-Macaulay) type of k[V]/N(T) using graph theoretical properties of T. We achieve this by using homological algebra and properties of monomial ideals. Along the way, we also provide a different characterization of unmixed trees and a generalization of the total dominating …