Physical Sciences and Mathematics Commons^™

Generalized Periodicity And Applications To Logistic Growth, Martin Bohner, Jaqueline Mesquita, Sabrina Streipert

Mathematics and Statistics Faculty Research & Creative Works

Classically, a continuous function f:R→R is periodic if there exists an ω>0 such that f(t+ω)=f(t) for all t∈R. The extension of this precise definition to functions f:Z→R is straightforward. However, in the so-called quantum case, where f:qN0→R (q>1), or more general isolated time scales, a different definition of periodicity is needed. A recently introduced definition of periodicity for such general isolated time scales, including the quantum calculus, not only addressed this gap but also inspired this work. We now return to the continuous case and present the concept of ν-periodicity that connects these different formulations of periodicity for …

The Cubic-Quintic Nonlinear Schrödinger Equation With Inverse-Square Potential, Alex H. Ardila, Jason Murphy

Mathematics and Statistics Faculty Research & Creative Works

We consider the nonlinear Schrödinger equation in three space dimensions with a focusing cubic nonlinearity and defocusing quintic nonlinearity and in the presence of an external inverse-square potential. We establish scattering in the region of the mass-energy plane where the virial functional is guaranteed to be positive. Our result parallels the scattering result of [11] in the setting of the standard cubic-quintic NLS.

Exact Solutions Of Stochastic Burgers–Korteweg De Vries Type Equation With Variable Coefficients, Kolade Adjibi, Allan Martinez, Miguel Mascorro, Carlos Montes, Tamer Oraby, Rita Sandoval, Erwin Suazo

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations.

A New Fractional Derivative Extending Classical Concepts: Theory And Applications, Mutaz Mohammad, Mohamed Saadaoui

All Works

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle's theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

Categorical Chain Conditions For Étale Groupoid Algebras, Sunil Philip

Dissertations, Theses, and Capstone Projects

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G, Steinberg defined an étale groupoid algebra RG. These étale groupoid algebras generalize various algebras, including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. In this work, we characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, thereby generalizing existing results for Leavitt path algebras and introducing new results for inverse …

Twisted Alexander Polynomials And Ptolemy Varieties Of Knots And Surface Bundles, Michael R. Marinelli

Dissertations, Theses, and Capstone Projects

The first focus of this dissertation is to compute Ptolemy varieties for triangulations of two infinite families of manifolds. Given an ideal triangulation of a cusped manifold, one can compute the Ptolemy variety and using it, obtain parabolic representations of the fundamental group. We compute certain obstruction classes for these manifolds, which are necessary to obtain the discrete faithful representation. This leads to our second focus of the dissertation, the twisted Alexander polynomial. The twisted Alexander polynomial (TAP) is a variation of the classical Alexander polynomial twisted by a representation of the fundamental group into a linear group. It was …

Oer Textbook Review For Calculus - Openstax Calculus, Jing Hu Ph.D.

Open Educational Resources Publications

This OER textbook review provides a comprehensive evaluation of the "Calculus" textbook series published by OpenStax. The reviewer, Jing Hu, an adjunct lecturer at Bentley University, highlights the textbook's strengths, including its thorough coverage of essential calculus topics, accurate and well-established mathematical principles, practical relevance, and user-friendly design. The open-access nature of the resource is seen as a significant advantage, contributing to its long-term utility and accessibility for both students and educators. Overall, the review concludes that the OpenStax Calculus textbook is a high-quality, comprehensive, and freely available resource that effectively supports the learning and teaching of calculus.

Graph And Group Theoretic Properties Of The Soma Cube And Somap, Kyle Asbury, Ben Glancy

Mathematical Sciences Technical Reports (MSTR)

The SOMA Cube is a puzzle toy in which seven irregularly shaped blocks must be fit together to build a cube. There are 240 distinct solutions to the SOMA Cube. One rainy afternoon, Conway and Guy created a graph of all the solutions by manually building each solution. They called their graph the SOMAP. We studied how the geometric structure of the SOMA Cube pieces informs the graph theoretic properties of the SOMAP, such as subgraphs that can or cannot appear and vertex centrality. We have also used permutation group theory to decipher notation used by Knuth in previous work …

New Class Function In Dual Soft Topological Space, Maryam Adnan Al-Ethary, Maryam Sabbeh Al-Rubaiea, Mohammed H. O. Ajam

Al-Bahir Journal for Engineering and Pure Sciences

In this paper we introduce a new class of maps in the dual Soft topological space and study some of its basic properties and relations among them, then we study and mapping.

Dynamic Optimization With Timing Risk, Erin Cottle Hunt, Frank N. Caliendo

Economics and Finance Faculty Publications

Timing risk refers to a situation in which the timing of an economically important event is unknown (risky) from the perspective of an economic decision maker. While this special class of dynamic stochastic control problems has many applications in economics, the methods used to solve them are not easily accessible within a single, comprehensive survey. We provide a survey of dynamic optimization methods under comprehensive assumptions about the nature of timing risk. We also relax the assumption of full information and summarize optimization with limited information, ambiguity, imperfect hedging, and dynamic inconsistency. Our goal is to provide a concise user …

Math Developmental Models Examined: Pass Rate, Duration For Completion, Enrollment Consistency And Racial Disparity, Xixi Wang, Annie Childers, Lianfang Lu

Journal of Access, Retention, and Inclusion in Higher Education

No abstract provided.

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.

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 …

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 …

The Fundamental Groupoid In Discrete Homotopy Theory, Udit Ajit Mavinkurve

Electronic Thesis and Dissertation Repository

Discrete homotopy theory is a homotopy theory designed for studying graphs and for detecting combinatorial, rather than topological, “holes”. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis.

In this thesis, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us …

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

Journal of Stochastic Analysis

No abstract provided.

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

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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.

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 …

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.

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 …

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 from 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 …

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 …

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 …

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, …