Digital Commons Network^™

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: …

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 …

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 …

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.

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.

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 ...

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.

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.

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.

Mathematics Without Calculations – It’S A Beautiful Thing!, Jason J. Molitierno

Mathematics Faculty Publications

All students should have the opportunity to do mathematics in a meaningful way for the sheer fun of it. Such experiences, if well designed, improve students’ effective thinking skills, increase their appreciation of the beauty and utility of mathematics, and prepare them to be mathematically-literate members of society. This session invites talks on how we can engage the liberal arts student through courses specifically designed for them. We welcome presentations on innovative course design, pedagogy, projects, or activities, as well as talks on tools used to assess such courses. Presentations should include a research basis for the design or pedagogical …

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 …

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

Mathematics Faculty Publications

No abstract provided.

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.

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.

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.

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.

Instructor Use Of Tablet Pcs In A College Pre-Calculus Course: Implementation & Assessment, Julianna Connelly Stockton, Peter Gregory

Mathematics Faculty Publications

A group of six math instructors used tablet PCs to teach their individual sections of a high enrollment gateway Pre-Calculus course in a diverse urban four-year college. Student performance in the experimental sections were compared to those in 31 other sections in terms of student retention, pass rates, and score on the department-wide standardized final exam. Student performance was higher in Tablet PC sections across all three measures, although in some cases the improvement was not substantial enough to improve students’ overall course grades. Surveys of students and faculty in classes using a Tablet PC reflected overall positive impressions of …

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann

Mathematics Faculty Publications

We consider the class of abelian groups possessing partial decomposition bases in L^δ_∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.

Finite Factors Of Bernoulli Schemes And Distinguishing Labelings Of Directed Graphs, Andrew Lazowski, Stephen M. Shea

Mathematics Faculty Publications

A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let G be a directed graph associated to the k -block presentation of a Bernoulli scheme X . We determine the automorphism group of G , and thus the distinguishing labelings of G . A labeling of G defines a finite factor of X . We define demarcating labelings and prove that demarcating labelings define finitarily Markovian finite factors of X . We use …

Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth

Mathematics Faculty Publications

We consider abelian groups with partial decomposition bases in L^δ_∞ω for ordinals δ. Jacoby, Leistner, Loth and Str¨ungmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Warfield invariants and Ulm-Kaplansky invariants up to ωδ for some ordinal δ, then they are equivalent in L^δ_∞ω. Here we prove that the modified Warfield invariant is expressible in L^δ_∞ω and hence the converse is true for appropriate δ.

Mathematical Competitions In Hungary: Promoting A Tradition Of Excellence & Creativity, Julianna Connelly Stockton

Mathematics Faculty Publications

Hungary has long been known for its outstanding production of mathematical talent. Extracurricular programs such as camps and competitions form a strong foundation within the Hungarian tradition. New types of competitions in recent years include team competitions, multiple choice competitions, and some exclusively for students who are not in a special mathematics class. This study explores some of the recent developments in Hungarian mathematics competitions and the potential implications these changes have for the very competition-driven system that currently exists. The founding of so many new competitions reflects a possible shift in the focus and purpose of competitions away from …

Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann

Mathematics Faculty Publications

Warfield modules are direct summands of simply presented Z_p - modules, or, alternatively, are Z_p - modules possessing a nice decomposition basis with simply presented cokernel. They have been classified up to isomorphism by theor Ilm-Kaplansky and Warfield invariants. Taking a model theoretic point of view and using infinitary languages we give here a complete theoretic characterization of a large class of Z_p - modules having a nice decomposition basis. As a corollary, we obtain the classical classification of countable Warfield modules. This generalizes results by Barwise and Eklof.

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

Mathematics Faculty Publications

We will give a positive answer for the unimodality of the Hilbert functions in the smallest open case, that of Artinian level monomial algebras of type three in three variables.

Education Of Mathematically Talented Students In Hungary, Julianna Connelly Stockton

Mathematics Faculty Publications

Hungary is famous for its production of large numbers of highly talented mathematicians and physicists. This study explores the Hungarian system for educating mathematically talented secondary school students with the goal of identifying successful features that may be applicable to education in the United States. Highlights of the Hungarian approach include an emphasis on problem solving, problem posing, detailed explanation or proof for solutions, and development of mathematical creativity through the search for multiple solution paths.

Hungary And The United States: A Comparison Of Gifted Education, Julianna Connelly Stockton

Mathematics Faculty Publications

There is a lot that can be learned about a country based on the programs and provisions it has for mathematically talented students. While it is difficult to identify a single U.S. "program" or "approach" for gifted education, in general the trend is to put mathematically talented students through the standard mathematics sequence, just starting at an earlier age. In Hungary, on the other hand, the focus is on enrichment over acceleration. This paper explores how some very different historical, cultural, and political forces have shaped these two countries’ different approaches to educating mathematically talented students.

An A1 Function That Is Not In Lipα For Any Positive Α, Ryan Mullen

Mathematics Faculty Publications

Let A1 be the Banach algebra of all continuous functions on the torus whose Fourier coefficients are in 1, and let Lipα be the Banach algebra of all continuous function f on the torus such that ...

Hungary, Slovakia, And Romania: International Relations Examined Through Minority Language Education, Julianna Connelly Stockton

Mathematics Faculty Publications

The Hungarians in Romania and Slovakia are not just ethnic minorities but also linguistic minorities whose language has almost no linguistic similarity to their host nations’ language. One aspect of life in which linguistic minority issues arise most frequently is that of education. Should minority linguistic groups be forced to learn the national language in order to attend school or participate in official functions or should the government provide or allow for education in their native language?

The ethnic Hungarian minorities did not, for the most part, emigrate from Hungary to these other countries by choice the way most immigrants …

On Determining Paint By Numbers Puzzles With Nonunique Solutions, Ryan Mullen

Mathematics Faculty Publications

Paint by Numbers is a classic logic puzzle in which the squares of a p×n grid are to be colored in such a way as to display a picture. The decision on which squares to color is determined by sequences of numbers above each column and to the left of each row. The numbers describe how many consecutive squares are to be colored in that row or column, and multiple numbers represent multiple blocks of colored in squares (with at least one uncolored square in between blocks). Certain natural questions arise. For a given p × n grid, how many …

On T-Pure And Almost Pure Exact Sequences Of Lca Groups, Peter Loth

Mathematics Faculty Publications

A proper short exact sequence in the category of locally compact abelian groups is said to be t-pure if φ(A) is a topologically pure subgroup of B, that is, if for all positive integers n. We establish conditions under which t-pure exact sequences split and determine those locally compact abelian groups K ⊕ D (where K is compactly generated and D is discrete) which are t-pure injective or t-pure projective. Calling the extension (*) almost pure if for all positive integers n, we obtain a complete description of the almost pure injectives and almost pure projectives in the category of …

Pure Extensions Of Locally Compact Abelian Groups, Peter Loth

Mathematics Faculty Publications

In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every …