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Full-Text Articles in Mathematics

Math Department 2022-2023 Newsletter, Mathematics Department Jan 2023

Math Department 2022-2023 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. We had an exciting year with many activities and accomplishments. Our students have received awards, attended conferences, conducted research, and have really contributed to the life of the department. Our faculty have had many accomplishments in teaching and scholarship. We conducted several events such as Alumni Night, hosted a guest speaker for the university community, had our Pi Mu Epsilon induction and awards ceremony, and much more. We have heard from several alumni and are happy to report what they are doing. Our Math Club has been becoming more active. On the curriculum …

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A Distinguished Subgroup Of Compact Abelian Groups, Dikran Dikranjan, Wayne Lewis, Peter Loth, Adolf Mader Jan 2022

A Distinguished Subgroup Of Compact Abelian Groups, Dikran Dikranjan, Wayne Lewis, Peter Loth, Adolf Mader

Mathematics Faculty Publications

Here “group” means additive abelian group. A compact group G contains δ" role="presentation" style="box-sizing: border-box; max-height: none; display: inline; line-height: normal; font-size: 13.2px; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">δ–subgroups, that is, compact totally disconnected subgroups Δ" role="presentation" style="box-sizing: border-box; max-height: none; display: inline; line-height: normal; font-size: 13.2px; text-align: left; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">Δ such that G/Δ" role="presentation" style="box-sizing: …

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Evolutionary Kuramoto Dynamics, Elizabeth A. Tripp, Feng Fu, Scott D. Pauls Jan 2022

Evolutionary Kuramoto Dynamics, Elizabeth A. Tripp, Feng Fu, Scott D. Pauls

Mathematics Faculty Publications

Biological systems have a variety of time-keeping mechanisms ranging from molecular clocks within cells to a complex interconnected unit across an entire organism. The suprachiasmatic nucleus, comprising interconnected oscillatory neurons, serves as a master-clock in mammals. The ubiquity of such systems indicates an evolutionary benefit that outweighs the cost of establishing and maintaining them, but little is known about the process of evolutionary development. To begin to address this shortfall, we introduce and analyse a new evolutionary game theoretic framework modelling the behaviour and evolution of systems of coupled oscillators. Each oscillator is characterized by a pair of dynamic behavioural …

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Math Department 2021-2022 Newsletter, Mathematics Department Jan 2022

Math Department 2021-2022 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. As we are slowly coming out of the pandemic, we were able to resume many of our normal events.

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Upward-Closed Hereditary Families In The Dominance Order, Michael D. Barrus, Jean A. Guillaume Jan 2021

Upward-Closed Hereditary Families In The Dominance Order, Michael D. Barrus, Jean A. Guillaume

Mathematics Faculty Publications

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris (2002), the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class F of graphs to be dominance monotone if whenever no realization of e contains an element F as …

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Math Department 2020-2021 Newsletter, Mathematics Department Jan 2021

Math Department 2020-2021 Newsletter, Mathematics Department

Mathematics Newsletter

This was an unusual year for our department as the coronavirus pandemic continued. However, we made the best of the situation and did as many activities as we could -- In the Fall, we inaugurated our new math major with three tracks: the traditional track, actuarial science track, and data science track -- Faculty changes -- Graduating seniors -- Pi Mu Epsilon Induction Ceremony and Awards Night -- Alumni Page -- Faculty spotlight -- Noyce grant -- Alumni night -- Coming attractions.

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Global Dynamics Of Generalized Second-Order Beverton–Holt Equations Of Linear And Quadratic Type, Elliott J. Bertrand, Mustafa R. S. Kulenović Jan 2021

Global Dynamics Of Generalized Second-Order Beverton–Holt Equations Of Linear And Quadratic Type, Elliott J. Bertrand, Mustafa R. S. Kulenović

Mathematics Faculty Publications

We investigate second-order generalized Beverton–Holt difference equations ... In particular, we will investigate the local and global dynamics in the event f is a certain type of linear or quadratic polynomial, and we explore the existence problem of period-two solutions.

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Free Subgroups With Torsion Quotients And Profinite Subgroups With Torus Quotients, Wayne Lewis, Peter Loth, Adolf Mader Jan 2020

Free Subgroups With Torsion Quotients And Profinite Subgroups With Torus Quotients, Wayne Lewis, Peter Loth, Adolf Mader

Mathematics Faculty Publications

Here “group” means abelian group. Compact connected groups contain δ-subgroups, that is, compact totally disconnected subgroups with torus quotients, which are essential ingredients in the important Resolution Theorem, a description of compact groups. Dually, full free subgroups of discrete torsion-free groups of finite rank are studied in order to obtain a comprehensive picture of the abundance of δ-subgroups of a protorus. Associated concepts are also considered.

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Math Department 2019-2020 Newsletter, Mathematics Department Jan 2020

Math Department 2019-2020 Newsletter, Mathematics Department

Mathematics Newsletter

Welcome to our annual Mathematics Department Newsletter. This was an unusual year for our department as the coronavirus cut short our time on campus this year. Hence we were unable to have our usual events such as the Annual Mathematics Lecture, the Pi Mu Epsilon Induction and Awards Night, and there was no in-person commencement ceremony. But despite that, many great things happened in our department.

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Math Department Newsletter, 2018-2019, Mathematics Department Jan 2019

Math Department Newsletter, 2018-2019, Mathematics Department

Mathematics Newsletter

No abstract provided.

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Number Theory: Niven Numbers, Factorial Triangle, And Erdos' Conjecture, Sarah Riccio Apr 2018

Number Theory: Niven Numbers, Factorial Triangle, And Erdos' Conjecture, Sarah Riccio

Mathematics Undergraduate Publications

In this paper, three topics in number theory will be explored: Niven Numbers, the Factorial Triangle, and Erdos's Conjecture . For each of these topics, the goal is for us to find patterns within the numbers which help us determine all possible values in each category. We will look at two digit Niven Numbers and the set that they belong to, the alternating summation of the rows of the Factorial Triangle, and the unit fractions whose sum is the basis of Erdos' Conjecture.

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Global Dynamic Scenarios For Competitive Maps In The Plane, Elliott J. Bertrand, M.R.S. Kulenovic Jan 2018

Global Dynamic Scenarios For Competitive Maps In The Plane, Elliott J. Bertrand, M.R.S. Kulenovic

Mathematics Faculty Publications

In this paper we present some global dynamic scenarios for general competitive maps in the plane. We apply these results to the class of second-order autonomous difference equations whose transition functions are decreasing in the variable \(x_{n}\) and increasing in the variable \(x_{n-1}\).We illustrate our results with the application to the difference equation ...

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Global Dynamics Of Higher-Order Transcendental-Type Generalized Beverton–Holt Equations, Elliott J. Bertrand, M.R.S. Kulenovic Jan 2018

Global Dynamics Of Higher-Order Transcendental-Type Generalized Beverton–Holt Equations, Elliott J. Bertrand, M.R.S. Kulenovic

Mathematics Faculty Publications

We investigate generalized Beverton–Holt difference equations of order k.

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Math Department Newsletter, 2017-2018, Mathematics Department Jan 2018

Math Department Newsletter, 2017-2018, Mathematics Department

Mathematics Newsletter

No abstract provided.

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The Classification Of Infinite Abelian Groups With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth Nov 2017

The Classification Of Infinite Abelian Groups With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider the class of abelian groups with partial decomposition bases, which includes groups classified by Ulm, Warfield, Stanton and others. We define an invariant and classify these groups in the language L∞ω, or equivalently, up to partial isomorphism. This generalizes a result of Barwise and Eklof and builds on Jacoby's classification of local groups with partial decomposition bases in L∞ω.

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Math Department Newsletter, 2016-2017, Mathematics Department Jun 2017

Math Department Newsletter, 2016-2017, Mathematics Department

Mathematics Newsletter

No abstract provided.

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The Classification Of Zp -Modules With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth Nov 2016

The Classification Of Zp -Modules With Partial Decomposition Bases In L∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω-equivalence. In this paper, we extend this classification to a class of mixed Zp-modules which includes all Warfield modules and is closed under L∞ω-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in L∞ω using invariants deduced from the classical Ulm and Warfield invariants.

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Musical Sound: A Mathematical Approach To Timbre, Timothy Weiss (Class Of 2016) Oct 2016

Musical Sound: A Mathematical Approach To Timbre, Timothy Weiss (Class Of 2016)

Writing Across the Curriculum

What is the mathematical reasoning behind the ear’s ability to distinguish two completely different musical sounds? In answering this question, one must call to mind a fundamental term with regards to music: timbre.

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Math Department Newsletter, 2015-2016, Mathematics Department Jun 2016

Math Department Newsletter, 2015-2016, Mathematics Department

Mathematics Newsletter

No abstract provided.

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To The Memory Of R. Freivalds, Efim Kinber Jan 2016

To The Memory Of R. Freivalds, Efim Kinber

School of Computer Science & Engineering Faculty Publications

The paper contains author’s memories of his mentor and teacher R. M. Freivalds.

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Math Department Newsletter, 2014-2015, Mathematics Department Jun 2015

Math Department Newsletter, 2014-2015, Mathematics Department

Mathematics Newsletter

No abstract provided.

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The Unimodality Of Pure O-Sequences Of Type Two In Four Variables, Bernadette Boyle Jan 2015

The Unimodality Of Pure O-Sequences Of Type Two In Four Variables, Bernadette Boyle

Mathematics Faculty Publications

Since the 1970's, great interest has been taken in the study of pure O-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into …

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Pure Injective And *-Pure Injective Lca Groups, Peter Loth Jan 2015

Pure Injective And *-Pure Injective Lca Groups, Peter Loth

Mathematics Faculty Publications

No abstract provided.

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The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle Oct 2014

The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle

Mathematics Faculty Publications

Since the 1970’s, great interest has been taken in the study of pure O-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.

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Zp-Modules With Partial Decomposition Bases In L Δ ∞Ω, Carol Jacoby, Peter Loth Jan 2014

Zp-Modules With Partial Decomposition Bases In L Δ ∞Ω, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider the class of mixed Zp-modules with partial decomposition bases. This class includes those modules classified by Ulm and Warfield and is closed under L∞ω-equivalence. In the context of L∞ω- equivalence, Jacoby defined invariants for this class and proved a classification theorem. Here we examine this class relative to Lδ∞ω, those formulas of quantifier rank ≤ some ordinal δ, defining invariants and proving a classification theorem. This generalizes a result of Barwise and Eklof.

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Math Department Newsletter, 2013-2014, Mathematics Department Jan 2014

Math Department Newsletter, 2013-2014, Mathematics Department

Mathematics Newsletter

No abstract provided.

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An Explicit Construction Of Kleinian Groups With Small Limit Sets, Andrew Lazowski Jan 2014

An Explicit Construction Of Kleinian Groups With Small Limit Sets, Andrew Lazowski

Mathematics Faculty Publications

This paper provides an explicit construction of Kleinian groups that have small Hausdorff dimension of their limit sets. It is known that such groups exist and they can be constructed by results of Patterson. The purpose here is to work out the methods of calculation.

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Hyperbolic Geometry And God, Andrew Lazowski Sep 2013

Hyperbolic Geometry And God, Andrew Lazowski

Presidential Seminar on the Catholic Intellectual Tradition

Development of hyperbolic geometry --Fifth axiom is a problem --Model of the divine --Infinite power --Infinite knowledge --Infinite benevolence.

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Horizon Content Knowledge In The Work Of Teaching: A Focus On Planning, Nick Wasserman, Julianna Connelly Stockton Jan 2013

Horizon Content Knowledge In The Work Of Teaching: A Focus On Planning, Nick Wasserman, Julianna Connelly Stockton

Mathematics Faculty Publications

Horizon content knowledge, one component of Ball et al.’s mathematical knowledge for teaching framework (e.g., Ball, Thames, & Phelps, 2008), has yet to reach adequate clarity and consensus in the field. Recently, various scholars have worked to further conceptualize and describe the mathematical horizon (e.g., Jakobsen, Thames & Ribeiro, 2013; Figueiras et al., 2011; Zazkis & Mamolo, 2011). In this communication, we identify some limitations in the ways such knowledge has thus far been described and offer an additional form of potential impact of horizon content knowledge on the work of teaching.

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Math Department Newsletter, 2012-2013 (Year In Review), Mathematics Department Jan 2013

Math Department Newsletter, 2012-2013 (Year In Review), Mathematics Department

Mathematics Newsletter

No abstract provided.

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