Mathematics Commons^™

Enestr¨Om-Kakeya Type Results For Complex And Quaternionic Polynomials, Matthew Gladin

Electronic Theses and Dissertations

The well known Eneström-Kakeya Theorem states that: for P(z)=∑_i=0ⁿ a_i zⁱ, a polynomial of degree n with real coefficients satisfying 0 ≤ a₀ ≤ a₁ ≤ ⋯≤ a_n, all zeros of P(z) lie in |z|≤1 in the complex plane. In this thesis, we will find inner and outer bounds in which the zeros of complex and quaternionic polynomials lie. We will do this by imposing restrictions on the real and imaginary parts, and on the moduli, of the complex and quaternionic coefficients. We also apply similar restrictions on complex polynomials with …

The Italian Domatic Number On Varying Graph Families, Keith Gallegos

Scholar Week 2016 - present

See file download option for presentation.

.pdf of abstract, which contains mathematical formulas, is available as a "Supplemental File" below.

It Application Administrator Internship, Payton Simmons

Research & Creative Achievement Day

This presentation is a focus on my internship with Fastenal as an IT Application Administrator. I'll talk about some of my daily tasks as well as the CMDB installation that will take place in the future. I am incredibly grateful to everyone who helped make this position possible because it gave me a wealth of knowledge that I will use in the near future.

Optimal Control Problems For Stochastic Processes With Absorbing Regime, Yaacov Kopeliovich

Journal of Stochastic Analysis

No abstract provided.

Bounds For The Augmented Zagreb Index, Ren Qingcuo, Li Wen, Suonan Renqian, Yang Chenxu

Theory and Applications of Graphs

The augmented Zagreb index (\rm{AZI} for short) of a graph $G$, introduced by Furtula et al. in 2010, is defined as ${\rm AZI}(G)=\sum\limits_{v_iv_j\in E(G)}{\left(\frac{d(v_i)d(v_j)}{d(v_i)+d(v_j)-2}\right)}^3$, where $E(G)$ is the edge set of $G$, and $d(v_i)$ denotes the degree of the vertex $v_i$. In this paper, we give some new bounds on general connected graphs, molecular trees and triangle-free graphs. \\[2mm] {\bf Keywords:} Agumented Zagreb index; molecular trees; Triangle-free graph

New Diagonal Graph Ramsey Numbers Of Unicyclic Graphs, Richard M. Low, Ardak Kapbasov

Theory and Applications of Graphs

Grossman conjectured that $R(G, G) = 2 \cdot |V(G)| - 1$, for all simple connected unicyclic graphs $G$ of odd girth and $|V(G)| \geq 4$. In this note, we prove his conjecture for various classes of $G$ containing a triangle. In addition, new diagonal graph Ramsey numbers are calculated for some classes of simple connected unicyclic graphs of even girth.

Ts-Reconfiguration Of $K$-Path Vertex Covers In Caterpillars For $K \Geq 4$, Duc A. Hoang

Theory and Applications of Graphs

A $k$-path vertex cover ($k$-PVC) of a graph $G$ is a vertex subset $I$ such that each path on $k$ vertices in $G$ contains at least one member of $I$. Imagine that a token is placed on each vertex of a $k$-PVC. Given two $k$-PVCs $I, J$ of a graph $G$, the \textsc{$k$-Path Vertex Cover Reconfiguration ($k$-PVCR)} under Token Sliding ($\mathsf{TS}$) problem asks if there is a sequence of $k$-PVCs between $I$ and $J$ where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known …

Ramsey Numbers For Connected 2-Colorings Of Complete Graphs, Mark Budden

Theory and Applications of Graphs

In 1978, David Sumner introduced a variation of Ramsey numbers by restricting to $2$-colorings in which the subgraphs spanned by edges in each color are connected. This paper continues the study of connected Ramsey numbers, including the evaluation of several cases of trees versus complete graphs.

Over-Determined Control Systems On Time Scales, Danian Castillo, Drew Barnes, Nick Wintz

2023 Student Academic Showcase

Control theory is a branch of mathematics focused on observing or controlling a process governed by a dynamic equation. We use state-space notation to represent all meaningful information about our process. This means our processes are expressed in vector form. Typically, the state matrix is square. In this project, we consider a control system where the corresponding state is over-determined, meaning there are more rows than columns. In addition, our state equation is on a time scale T, which allows us to consider discrete, continuous, or hybrid measurements. Here, we offer two methods to solve the dynamic system. Finally, we …

A New Approach To Proper Orthogonal Decomposition With Difference Quotients, Sarah Locke Eskew, John R. Singler

Mathematics and Statistics Faculty Research & Creative Works

In a Recent Work (Koc Et Al., SIAM J. Numer. Anal. 59(4), 2163–2196, 2021), the Authors Showed that Including Difference Quotients (DQs) is Necessary in Order to Prove Optimal Pointwise in Time Error Bounds for Proper Orthogonal Decomposition (POD) Reduced Order Models of the Heat Equation. in This Work, We Introduce a New Approach to Including DQs in the POD Procedure. Instead of Computing the POD Modes using All of the Snapshot Data and DQs, We Only Use the First Snapshot Along with All of the DQs and Special POD Weights. We Show that This Approach Retains All of the …

Numerical Range Of Strictly Triangular Matrices Over Finite Fields, Ariel Russell

Mathematics Student Scholarship

In this paper, we investigate the numerical range of matrices over finite fields, particularly triangular matrices. We conjecture that all strictly triangular matrices over finite fields of dimension 3 or greater have a numerical range encompassing the entirety of the finite field. We use both algebraic and computational methods to support this claim, making some concrete progress towards the algebraic proof. Further, we conjecture that all matrices over finite fields have a numerical range falling into one of five potential categories, providing an extensive appendix of randomly generated computational examples which seems to support this conjecture.

Geometry And Coding: Introducing An Interactive And Integrated Mathematics-Computer Science Unit, Kimberly Beck, Jessica F. Shumway

Publications

As part of a collaborative project between Utah State University, the Cache County School District, and Stanford, instructional units were designed for fifth-grade students. These units integrated math concepts of geometrical shapes and computer science concepts of sequences, conditionals, and loops. One component of the unit was implemented in math classrooms by math teachers, and the other component was implemented in computer labs. This presentation will focus on the math unit as presented at the National Council of Teachers of Mathematics (NCTM-V).

Ramanujan–Sato Series For 1/Π, Timothy Huber, Daniel Schultz, Dongxi Ye

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We compute Ramanujan–Sato series systematically in terms of Thompson series and their modular equations. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.

Discovering Dune: Essays On Frank Herbert’S Epic Saga., Edited By Dominic J. Nardi And N. Trevor Brierly, G. Connor Salter

Mythlore: A Journal of J.R.R. Tolkien, C.S. Lewis, Charles Williams, and Mythopoeic Literature

G. Connor Salter reviews Discovering Dune: Essays on Frank Herbert’s Epic Saga, edited by Dominic J. Nardi and N. Trevor Brierly, considering its new contributions to studies of Frank Herbert's work. Essays included fit into four categories (Politics and Power, History and Religion, Biology and Ecology, and Philosophy, Choice and Ethics) and range from Herbert's use of ecology in Dune to how game theory may help explain certain characters' apparent ability to see the future. Discovering Dune also includes an appendix which contains the only up-to-date bibliography of Herbert's work (primary and secondary sources).

Euler Archive Spotlight, Erik R. Tou

Euleriana

A survey of two translations posted to the Euler Archive in 2022.

Euler And The Duplication Formula For The Gamma-Function, Alexander Aycock

Euleriana

We show how the formulas in Euler’s paper "Variae considerationes circa series
hypergeometricas" [ 4] imply Legendre’s duplication formula for the Γ-function. This
paper can be seen as an Addendum to [2].

Euler Found The First Binary Digit Extraction Formula For Π In 1779, Nick Craig-Wood

Euleriana

In 1779 Euler discovered two formulas for π which can be used to calculate any binary digit of π without calculating the previous digits. Up until now it was believed that the first formula with the correct properties (known as a BBP-type formula) for this calculation was published by Bailey, Borwein and Plouffe in 1997.

Analytical Observations (Translation Of E326), Cynthia Huffman Ph.D.

Euleriana

Euler, in this publication with Eneström number E326, provides an induction fallacy which arises from analyzing a particular sequence. Euler wrote this work in 1763, one of only two papers he wrote on sequences and/or series in the 1760’s, out of a total of 79 papers on series during his career. His goal in E326 is to investigate the middle terms in the expansion of powers of quadratic trinomial expressions, beginning with the specific simple quadratic , before considering the general quadratic .

The induction fallacy shows up during the analysis of the simple case when Euler first finds an …

Euler's Anticipations, Christopher Goff, Erik Tou

Euleriana

Welcome to Volume 3 of Euleriana. This issue highlights occasions where Euler's work anticipated future results from other others, sometimes by decades or even centuries!

Problem Of The Week: A Student-Led Initiative To Bring Mathematics To A Broader Audience, Jordan M. Sahs, Brad Horner

UNO Student Research and Creative Activity Fair

Problem of the Week (POW!) is a weekly undergraduate mathematics competition hosted by two graduate students from the UNO Math Department. It started with the goal to showcase variety, creativity, and intrigue in math to those who normally feel math is dry, rote, and formulaic. Problems shine light on both hidden gems and popular recreational math, both math history and contemporary research, both iconic topics and nontraditional ones, both pure abstraction and real-world application. Now POW! aims to increase availability and visibility in Omaha and beyond. Select problems from Fall 2021 to Spring 2023 are highlighted here: these received noteworthy …