Mathematics Commons^™

The Bicomplex Tensor Product And A Bicomplex Choi Theorem, Daniel Alpay, Antonino De Martino, Kamal Diki, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we extend the concept of tensor product to the bicomplex case and use it to prove the bicomplex counterpart of the classical Choi theorem in the theory of complex matrices and operators. The concept of hyperbolic tensor product is also discussed, and we link these results to the theory of quantum channels in the bicomplex and hyperbolic case.

Free Energy Differences In Nonequilibrium Thermodynamic Processes, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Systems which may be allowed to go out of equilibrium have been of interest recently. A quantity is formulated whose average over an ensemble of microscopic realizations of the process depends only on the initial and final states. This is so even though the system may not be in equilibrium during the process. A generalization to the case where the initial and final states are not equilibrium states is developed here. Quantum analogues of these relations are derived, and an indication of how this might be applied to study entropy increase in thermodynamics is presented.

A Second Homotopy Group For Digital Images, Gregory Lupton, Oleg R. Musin, Nicholas A. Scoville, P. Christopher Staecker, Jonathan Treviño-Marroquín

School of Mathematical & Statistical Sciences Faculty Publications

We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second homotopy group from algebraic topology. We illustrate that our approach can be effective by computing this (digital) second homotopy group for a digital 2-sphere.

Modeling, Analysis, Approximation, And Application Of Viscoelastic Structures And Anomalous Transport, Yiqun Li

Theses and Dissertations

(Variable-order) fractional partial differential equations are emerging as a competitive means to integer-order PDEs in characterizing the memory and hereditary properties of physical processes, e.g., anomalously diffusive transport, viscoelastic mechanics and financial mathematics, and thus have attracted widespread attention. In particular, optimal control problems governed by fractional partial differential equations are attracting increasing attentions since they are shown to provide competitive descriptions of challenging physical phenomena. Nevertheless, variable-order fractional models exhibit salient features compared with their constant-order analogues and introduce mathematical difficulties that are not typical encountered in the context of integer-order and constant-order fractional partial differential equations.

This dissertation …

Generalizations Of The Graham-Pollak Tree Theorem, Gabrielle Anne Tauscheck

Theses and Dissertations

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. This dissertation will generalize their result via two different directions: Steiner distance k-matrices and distance critical graphs. 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 Steiner distance k-matrix is always zero if …

Representation Dimensions Of Algebraic Tori And Symmetric Ranks Of G-Lattices, Jason Bailey Heath

Theses and Dissertations

Algebraic tori over a field k are special examples of affine group schemes over k, such as the multiplicative group of the field or the unit circle. Any algebraic torus can be embedded into the group of invertible n x n matrices with entries in k for some n, and the smallest such n is called the representation dimension of that torus. Representation dimensions of algebraic tori can be studied via symmetric ranks of G-lattices. A G-lattice L is a group isomorphic to the additive group Zⁿ for some n, along with an action …

Erlang-Distributed Seir Epidemic Models With Cross-Diffusion, Victoria Chebotaeva

Theses and Dissertations

We examine the effects of cross-diffusion dynamics in epidemiological models. Using reaction-diffusion dynamics to model the spread of infectious diseases, we focus on situations in which the movement of individuals is affected by the concentration of individuals of other categories. In particular, we present a model where susceptible individuals move away from large concentrations of infected and infectious individuals.

Our results show that accounting for this cross-diffusion dynamics leads to a noticeable effect on epidemic dynamics. It is noteworthy that this leads to a delay in the onset of epidemics and an increase in the total number of people infected. …

Global Well-Posedness Of Nonlocal Differential Equations Arising From Traffic Flow, Thomas Joseph Hamori

Theses and Dissertations

Macroscopic traffic flow models describe the evolution of a function ρ(t, x), which represents the traffic density at time t and location x according to a differential equation (typically a conservation law). Numerous models have been introduced over the years which capture the phenomenon of shock formation in which the solution develops a discontinuity. This presents difficulties from the standpoint of mathematical analysis, necessitating the consideration of weak solutions. At the same time, this undesirable mathematical behavior corresponds to unsafe driving conditions on real roadways, in which the heaviness of traffic may vary abruptly and dramatically. This thesis introduces and …

Traveling Wave Phenomena Of Inhomogeneous Half-Wave Equation, Zhaosheng Feng, Yu Su

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we are concerned with traveling wave phenomena of the inhomogeneous half-wave equation, which models the energy of a spin zero particle in the Coulomb field. We study the Gagliardo-Nirenberg and critical Hardy-Sobolev inequalities with velocity 0 < | v | < 1 and obtain the estimates for the best constants and optimizers of inequalities. Moreover, we establish the non-scattering results with small traveling wave for energy subcritical and critical cases.

An Introduction To Category Theory, Joseph Kopp

Electronic Proceedings of Undergraduate Mathematics Day

Category theory is a relatively new field of mathematics that has grown much in popularity in recent years. It is a general theory of mathematical structure that lends itself to making overarching, yet deep, connections between many branches of mathematics. This power to make such wide-reaching statements is what has drawn many to study it. However, category theory has also been criticized for being "abstract nonsense," in that some believe the theory to be too abstract to carry meaning, much less be applied to the real world. The goal of this paper is to introduce the main ideas of category …

Derivation Of The Sliding Catenary Curve Via Calculus Of Variations, Ethan Shade

Electronic Proceedings of Undergraduate Mathematics Day

Using the calculus of variations this paper derives the general equation for the "sliding catenary curve" — a hanging chain with terminal links free to slide along two poles, one tilted and one vertical. By applying physical assumptions along with the Euler-Lagrange equation, the Beltrami identity, the Legendre-Clebsch condition, the transversality condition, Lagrange multipliers, and the isoperimetric constraint, we derive the general equation for the sliding catenary curve through a functional that measures the potential energy of the hanging chain. This general equation is then compared to a real-life construction of a sliding catenary curve. Additionally the paper explores a …

Mathematical Modeling, Analysis, And Simulation Of Patient Addiction Journey, Adan Baca, Diego Gonzalez, Alonso G. Ogueda, Holly C. Matto, Padmanabhan Seshaiyer

CODEE Journal

This paper aims to develop a mathematical model to study the dynamics of addiction as individuals go through their detox journey. The motivation for this work is three fold. First, there has been a significant increase in drug overdose and drug addiction following the COVID-19 pandemic, and addiction may be interpreted as a infectious disease. Secondly, the dynamics of infectious disease could be modeled via compartmental models described by differential equations and one can therefore leverage the existing analytical and numerical methods to model addiction as a disease. Finally, the work helps to inform how mathematical models governed by differential …

Bernoulli Convolution Of The Depth Of Nodes In Recursive Trees With General Affinities, Toshio Nakata, Hosam Mahmoud

Journal of Stochastic Analysis

No abstract provided.

An Em-Based Likelihood Inference For Degradation Data Analysis Using Gamma Process, Lochana Palayangoda, N. Balakrishnan

Mathematics Faculty Publications

The gamma process is widely used for the lifetime estimation of highly reliable products that degrade over time. Typically, incomplete likelihood is used to estimate the model parameters and the reliability estimates for the first passage time distribution of the gamma process; however, it (i.e., pseudo method) does not consider interval censoring and right censoring information of the degradation data. In this work, the expectation-maximization algorithm-based method (EM method) is developed for the estimation of the gamma process model parameters and the reliability estimates incorporating interval censoring and right censoring. The asymptotic variance–covariance matrix and the asymptotic confidence intervals for …

Frames And Spaces For Distributive Quasi Relation Algebras And Distributive Involutive Fl-Algebras, Andrew Craig, Peter Jipsen, Claudette Robinson

Mathematics, Physics, and Computer Science Faculty Articles and Research

Analogous to atom structures for relation algebras, we define partially ordered frames and prove they are duals for complete perfect distributive quasi relation algebras and distributive involutive FL-algebras. We then extend this dual representation to all algebras and their corresponding frames with a Priestley topology.

For relation algebras up to size 16 it has been determined which algebras are representable by binary relations. We compute all finite distributive quasi relation algebras up to 8 elements and provide representations for some of them.

On The Structure Of Balanced Residuated Partially Ordered Monoids, Stefano Bonzio, José Gil-Férez, Peter Jipsen, Adam Přenosil, Melissa Sugimoto

Mathematics, Physics, and Computer Science Faculty Articles and Research

A residuated poset is a structure ⟨A,⩽, ·, \, /, 1⟩ where ⟨A,⩽⟩ is a poset and ⟨A, ·, 1⟩ is a monoid such that the residuation law x · y ⩽ z ⇐⇒ x ⩽ z/y ⇐⇒ y ⩽ x\z holds. A residuated poset is balanced if it satisfies the identity x\x ≈ x/x. By generalizing the well-known construction of Płonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two …

Waves In Cosmological Background With Static Schwarzschild Radius In The Expanding Universe, Karen Yagdjian

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we prove the existence of global in time small data solutions of semilinear Klein–Gordon equations in space-time with a static Schwarzschild radius in the expanding universe.

On The Colorability Of The Sphere Complex, Bennett Haffner

Master's Theses

One of the most prominently studied groups in geometric group theory is the outer automorphism group of the free group Out(F). The sphere complex provides a topological model for Out(F). We demonstrate the chromatic number of the sphere complex is finite.

Examining The Lived Experiences Of Educators Using Different Levels Of Support For Teaching Math To Students With Learning Disabilities In Math Computation And Problem-Solving For Teachers At Public Cyber Charter High Schools In The Northeastern United States: A Transcendental Phenomenological Study, Leeann E. Mccullough

Doctoral Dissertations and Projects

The purpose of this transcendental phenomenological study was to describe the lived experiences of educators using different levels of support for teaching math to students with learning disabilities in math computation and problem-solving for teachers at public cyber charter high schools in the Northeastern United States. The theory guiding this study was Sweller’s cognitive load theory, as it explained the learning process of students with learning disabilities and how educators developed instructional methods that complement the learner’s needs. The central research question was, “What is the lived experience of 9-12th-grade mathematics teachers in supporting students with differing learning abilities in …

A Measure Of Interactive Complexity In Network Models, Will Deter

Northeast Journal of Complex Systems (NEJCS)

This work presents an innovative approach to understanding and measuring complexity in network models. We revisit several classic characterizations of complexity and propose a novel measure that represents complexity as an interactive process. This measure incorporates transfer entropy and Jensen-Shannon divergence to quantify both the information transfer within a system and the dynamism of its constituents’ state changes. To validate our measure, we apply it to several well-known simulation models implemented in Python, including: two models of residential segregation, Conway’s Game of Life, and the Susceptible-Infected-Susceptible (SIS) model. Our results reveal varied trajectories of complexity, demonstrating the efficacy and sensitivity …

Bootstrap Methods For Bias-Correcting Probability Distribution Parameters Characterizing Extreme Snow Accumulations, Kenneth Pomeyie, Brennan Bean

Mathematics and Statistics Student Research and Class Projects

Accurately quantifying the threat of collapse due to the weight of settled snow on the roof of a structure is crucial for ensuring structural safety. This quantification relies upon direct measurements of the snow water equivalent (SWE) of settled snow, though most weather stations in the United States only measure snow depth. The absence of direct load measurements necessitates the use of modeled estimates of SWE, which often results in the underestimation of the scale/variance parameter of the distribution of annual maximum SWE. This paper introduces a novel bias correction method that employs a bootstrap technique with regression-based models to …

A Measurement Of The Differential Drell-Yan Cross Section As A Function Of Invariant Mass In Proton–Proton Collisions At √ S = 13 Tev, William Robert Tabb

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

The Drell-Yan process, a crucial mechanism for producing lepton pairs in highenergy hadron collisions, serves as an essential probe for testing the Standard Model of particle physics. This dissertation presents a comprehensive measurement of the differential cross section with respect to the invariant mass of the lepton pairs, utilizing data collected by the CMS experiment at CERN from 2016 to 2018. Cross sections are essential for refining our understanding of parton distribution functions and the underlying quantum chromodynamics processes, thereby providing constraints on theoretical predictions. In this analysis, the cross sections are compared to theoretical models and simulations, offering new …

Hardy Spaces And Canonical Kernels On Quadric Cr Manifolds, Albert Boggess, Jennifer Brooks, Andrew Raich

Mathematical Sciences Faculty Publications and Presentations

R functions on an embedded quadric M always extend holomorphically to M + iΓ_M where Γ_M is the closure of the convex hull of the image of the Levi form. When Γ_M is a closed polygonal cone, we show that the Bergman kernel on the interior of M + iΓ_M is a derivative of the Szegö kernel. Moreover, we develop the L^p Hardy space theory which turns out to be particularly robust. We provide examples that show that it is unclear how to formulate a corresponding relationship between the Bergman and Szegö kernels …

What If The Resulting Interval Is Too Wide: From A Heuristic Fuzzy-Technique Idea To A Mathematically Justified Approach, Marc Fina, Vladik Kreinovich

Departmental Technical Reports (CS)

In engineering designs, we usually need to make sure that the values of some characteristics y do not exceed a certain threshold y0 – e.g., that the stress at each location does not exceed a certain critical value. Usually, we know how each of these characteristics y depends on the design parameters x1, . . . ,xn, i.e., we know the function y= f (x1, . . . ,xn). However, it is not enough to use the nominal values of the design parameters in our analysis, since the actual values are, in general, somewhat different from the nominal values. Often, …

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 …

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 …

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) …

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 …