Mathematics Commons^™

Coloring Pythagorean Triples And A Problem Concerning Cyclotomic Polynomials, Daniel White

Theses and Dissertations

One may easily show that there exist $O( \log n)$-colorings of $\{1,2, \ldots, n\}$ such that no Pythagorean triple with elements $\le n$ is monochromatic. In Chapter~\ref{CH:triples}, we investigate two analogous ideas. First, we find an asymptotic bound for the number of colors required to color $\{1,2,\ldots ,n\}$ so that every Pythagorean triple with elements $\le n$ is $3$-colored. Afterwards, we examine the case where we allow a vanishing proportion of Pythagorean triples with elements $\le n$ to fail to have this property.

Unrelated, in 1908, Schur raised the question of the irreducibility over $\Q$ of polynomials of the form …

Analysis And Simulation Of Kinetic Model For Active Suspensions, Panon Phuworawong

Mathematics & Statistics Theses & Dissertations

In this research, we study the recently proposed kinetic model for active suspensions, where the active particles are assumed to be rigid rod and are driven in the suspension either by their own biological/chemical forces or external electric/magnetic fields. We first study the stability of the isotropic suspension in quiescent flow. Then we investigate the weak shear perturbation of the isotropic state and study some rheological properties of the suspension by explicit analytic formulas derived directly from the model. For imposed shear, we give some bifurcation diagrams of the stable states in some parametric spaces through numerical simulations. Some rheological …

The Eichler-Selberg Trace Formula For Level-One Hecke Operators, Alex Barron

Honors Theses

This paper explains the steps involved in the proof of the Eichler-Selberg Trace Formula for Hecke operators of level one. It is based on an appendix in Serge Lang's Introduction to Modular Forms written by Don Zagier, though I also draw heavily from sections of Toshitsune Miyake's Modular Forms and Xueli Wang's and Dingyi Pei's Modular Forms with Integral and Half-Integral Weights.

Section 2 summarizes the necessary background in the theory of modular forms. The material covered here is standard, so I've left out most of the details and proofs. Most of the section is based on Chapter VII of …

Shimura Images Of A Family Of Half-Integral Weight Modular Forms, Kenneth Allan Brown

Theses and Dissertations

In 1973, Shimura introduced a family of maps between modular forms of half-integral weight and modular forms of even integral weight. We will give explicit formulas for the images of two different classes of modular forms under these maps. In contrast to Shimura's difficult analytic construction, our formulas will fall out of relatively simple combinatorial derivations. Using the Shimura correspondence, we will prove congruences for the eigenvalues of a family of eigenforms introduced by Garvan. Using deep results of Eichler and Shimura, we state these congruences in terms of the number of points on associated elliptic curves, and we provide …

Generalizations Of Sperner's Theorem: Packing Posets, Families Forbidding Posets, And Supersaturation, Andrew Philip Dove

Theses and Dissertations

Sperner's Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an $n$-set, $[n]:=\{1,2,\dots,n\}$, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset $P$, we are interested in maximizing the size of a family $\mathcal{F}$ of subsets of $[n]$, where each maximally connected component of $\mathcal{F}$ is a copy of $P$, and finding the extreme configurations that achieve this value. For instance, Sperner showed …

Ricean Over Gaussian Modelling In Magnitude Fmri Analysis—Added Complexity With Negligible Practical Benefits, Daniel W. Adrian, Ranjan Maitra, Daniel B. Rowe

Mathematics, Statistics and Computer Science Faculty Research and Publications

It is well known that Gaussian modelling of functional magnetic resonance imaging (fMRI) magnitude time-course data, which are truly Rice distributed, constitutes an approximation, especially at low signal-to-noise ratios (SNRs). Based on this fact, previous work has argued that Rice-based activation tests show superior performance over their Gaussian-based counterparts at low SNRs and should be preferred in spite of the attendant additional computational and estimation burden. Here, we revisit these past studies and, after identifying and removing their underlying limiting assumptions and approximations, provide a more comprehensive comparison. Our experimental evaluations using Receiver Operating Characteristic (ROC) curve methodology show that …

Considerations In Designing Human-Computer Interfaces For Elderly People, Drew Williams, Mohammad Arif Ul Alam, Sheikh Iqbal Ahamed, William Chu

Mathematics, Statistics and Computer Science Faculty Research and Publications

As computing devices continue to become more heavily integrated into our lives, proper design of human-computer interfaces becomes a more important topic of discussion. Efficient and useful human-computer interfaces need to take into account the abilities of the humans who will be using such interfaces, and adapt to difficulties that different users may face – such as the difficulties that elderly users must deal with. Interfaces that allow for user-specific customization, while taking into account the multiple difficulties that older users might face, can assist the elderly in properly using these newer computing devices, and in doing so possibly achieving …

(P,\\Lambda)-Koszul Algebras And Modules, Ii, Jiafeng Lu

Turkish Journal of Mathematics

This paper is a continuous work of [14], where the notions of (p,\lambda)-Koszul algebra and (p,\lambda)-Koszul module were first introduced. More precisely, some new criteria for a positively graded algebra to be (p,\lambda)-Koszul are provided. We also generalize (p,\lambda)-Koszul objects to the nongraded case and define the so-called quasi-(p,\lambda)-Koszul objects. Further, the relationships between (quasi-) (p,\lambda)-Koszul modules and minimal Horseshoe Lemma are established.

On Some Results On Ip-Graphs, Roghayeh Hafezieh, Moohammadali Iranmanesh

Turkish Journal of Mathematics

The IP-graph of a naturally valenced association scheme and some of its properties have been studied recently. In this paper we introduce the bipartite version of this graph for a naturally valenced association scheme (X,S), denoted by BIP(S). We also investigate some of its properties.

On The Centroid Of Prime Semirings, Hasret Yazarli, Mehmet Ali̇ Öztürk

Turkish Journal of Mathematics

We define and study the extended centroid of a prime semiring. We show that the extended centroid is a semifield and give some properties of the centroid of a right multiplicatively cancellable prime semiring.

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

Mathematics Newsletter

No abstract provided.

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.

Eigenfunction Expansions Associated With The One-Dimensional Schrödinger Operator, Daphne Gilbert

Articles

We consider the form of eigenfunction expansions associated with the time-independent Schrödinger operator on the line, under the assumption that the limit point case holds at both of the infinite endpoints. It is well known that in this situation the multiplicity of the operator may be one or two, depending on properties of the potential function. Moreover, for values of the spectral parameter in the upper half complex plane, there exist Weyl solutions associated with the restrictions of the operator to the negative and positive half-lines respectively, together with corresponding Titchmarsh-Weyl functions.

Second Order Approximations In Sequential Point Estimation Of The Probability Of Zero In Poisson Distribution, Eisa Mahmoudi, Mohammad Hatami Kamin

Turkish Journal of Mathematics

In the analysis of the count data, the Poisson model becomes overtly restrictive in the case of over-dispersed or under-dispersed data. When count data are under-dispersed, specific models such as generalized linear models (GLM) are proposed. Other examples are the zero-inflated Poisson model (ZIP) and zero-truncated Poisson model (ZTP), which have been used in literature to deal with an excess or absence of zeros in count data. Thus having a knowledge of the probability of zeros and its estimation in Poisson distribution can be significant and useful. Some estimation problems with unknown parameter cannot attain minimum risk where the sample …

The Wright Message, 2013-2014, University Of Northern Iowa. Department Of Mathematics.

The Wright Message

Inside this issue:

-- Dear Department Alumni and Friends
-- Around Wright Hall
-- New Faculty: C. Adam Feldhaus
-- 2013 - 2014 Tenure-Stream Faculty
-- Faculty Spotlight: Syed Kirmani
-- UNI on the March
-- The Litwiller Bequest: A Gift that Keeps on Giving
-- Retiring Faculty Member: Glenn Nelson
-- The Center for Teaching and Learning Mathematics
-- David & Lois Kail: Donor Spotlight
-- Addresses by Mathematics Faculty
-- Department of Mathematics Goes Digital
-- Alumni Spotlight: Slade Hovick & Sarah Miesner
-- Student Spotlight: Jesse Moeller, Katherine Pearce, & Adam Feller
-- The 2013 Math Shindig …

Two-Step Arithmetic Word Problems, Enrique Castro-Martínez, Antonio Frías-Zorilla

The Mathematics Enthusiast

This study uses the perspective of schemes to analyze characteristics of arithmetic word problems that can influence the process of translation from the verbal statement to an arithmetical representation. One characteristic that we have detected in the two-step word problems is the presence of one or two connections (nodes) in schemes that represent them, and this paper explores whether the number of nodes affects the activation of the associated schemas. With students from the 5th and 6th grades of elementary school (11 and 12 years of age), we analyze the written productions and would stress that the number of connections …

Trajectory Of A Problem: A Study In Teacher Training, Alain Kuzniak, Bernard Parzysz, Laurent Vivier

The Mathematics Enthusiast

Problems are frequently used in mathematics to introduce and convey new notions and skills. Hence, teachers transform and adjust those problems to their students' level. The present study focuses on this transformation process on the particular case of a geometric problem posed by two teacher educators in one French Institute for Teacher Training. The whole process is described as a trajectory of the problem through various institutions from training center to secondary school and back. Before presenting the notion of trajectory of the problem, some elements about a general theoretical frame which refers to didactics of mathematics are presented.

Primary School Teacher’S Noticing Of Students’ Mathematical Thinking In Problem Solving, Ceneida Fernández, Salvador Llinares, Julia Valls

The Mathematics Enthusiast

Professional noticing of students’ mathematical thinking in problem solving involves the identification of noteworthy mathematical ideas of students’ mathematical thinking and its interpretation to make decisions in the teaching of mathematics. The goal of this study is to begin to characterize pre-service primary school teachers’ noticing of students’ mathematical thinking when students solve tasks that involve proportional and non-proportional reasoning. From the analysis of how pre-service primary school teachers notice students’ mathematical thinking, we have identified an initial framework with four levels of development. This framework indicates a possible trajectory in the development of primary teachers’ professional noticing.

Tme Volume 10, Numbers 1 And 2

The Mathematics Enthusiast

No abstract provided.

On Discrete Analytic Functions: Products, Rational Functions, And Reproducing Kernels, Daniel Alpay, Palle Jorgensen, Ron Seager, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We introduce a family of discrete analytic functions, called expandable discrete analytic functions, which includes discrete analytic polynomials, and define two products in this family. The first one is defined in a way similar to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us to define rational discrete analytic functions. To define the second product we need a new space of entire functions which is contractively included in the Fock space. We study in this space some counterparts of Schur analysis.