Mathematics Commons^™

Disturbing The Q-Dyson Conjecture, Andrew Sills

Mathematical Sciences: Faculty Publications

I discuss the computational methods behind the formulation of some conjectures related to variants on Andrews’ q-Dyson conjecture.

On The Ordinary And Signed Göllnitz-Gordon Partitions, Andrew Sills

Mathematical Sciences: Faculty Publications

A partition of an integer n is a representation of n as an unordered sum of positive integers. In a recent paper [1], Andrews introduced the notion of a "signed partition," that is, a representation of a positive integer as an unordered sum of integers, some possibly negative.

Preservice Teachers’ Disposition Self-Appraisals: Is There A Connection To Mathematics Instructional Practices?, Shonda Lemons-Smith

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

This paper explores preservice teachers’ dispositions and the connection to mathematics instructional practices. The Theory of Culturally Relevant Pedagogy served as the theoretical grounding for the study. Hence, its three broad propositions: conceptions of self and others, social relations, and conceptions of knowledge structure the argument presented. The participants in the study are elementary preservice teachers enrolled in a post-baccalaureate alternative certification program. The full-time, accelerated program focuses on and is specifically designed for individuals committed to teaching in urban, high-poverty schools. A dispositions survey, open-ended narratives, and teaching observation rubric served as data sources for the qualitative study. Findings …

Becoming Critical Mathematics Pedagogues: A Journey, David W. Stinson

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

This session will report the findings of a study that explored the beginning transformations in the pedagogical philosophies and practices of three mathematics teachers (middle, high school, and 2-year college) who completed a graduate-level mathematics education course that focused on critical theory and teaching for social justice, and how these transformations are compatible (or not) with reform mathematics education as suggested by the National Council of Teachers of Mathematics (NCTM), and in turn, the new Georgia Performance Standards (GPS). The study

employed Freirian participatory research methodology; in fact, the participants were not only co- researchers, but also co-authors of the …

Inequalities And Exponential Approximations For Residual Life Reliability Functions, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

Given that a unit is of age t, the remaining life after time t is random. The expected value of this random residual life is called the mean residual life at time t. Specifically, if T is the life of a component with distribution function F, then δF (t) = E(T −t|T > t) is called the mean residual life function (MRLF). It is well known that the class of distributions with decreasing mean residual life (DMR) contains the class of distributions with increasing hazard rate (IHR). In this note, …

Harmonic Analysis Related To Schrödinger Operators, Gestur Olafsson, Shijun Zheng

Mathematical Sciences: Faculty Publications

In this article, we give an overview of some recent developments in Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in our previous work [J. Fourier Anal. Appl. 12 (2006), no. 6, 653–674; MR2275390]. We elaborate our approach by considering a potential in C^∞₀ or the Schwartz class in one dimension. In particular, the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give a maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with H. Then we …

Proceedings Of The Second Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Front Matter

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 2nd Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents
• Letter from President

On The Simplification Of Certain Q-Multisums, Andrew Sills

Mathematical Sciences: Faculty Publications

Some examples of naturally arising multisum q-series which turn out to have representations as fermionic single sums are presented. The resulting identities are proved using transformation formulas from the theory of basic hypergeometric series.

Proceedings Of The First Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Front Matter

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 1st Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents
• Abstracts of Papers

The Kennesaw State University Mathematics Methods Model, Angela L. Teachey

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Kennesaw State University’s comprehensive, nine-credit-hour, methods course integrates general and mathematics-specific pedagogical training with a structured four-week field experience prior to student teaching. This course blends essential units on conceptual understanding of mathematics, lesson planning, assessment, classroom management, and diversity with mathematics-specific methods. All topics are aligned with National Council of Teachers of Mathematics standards and Georgia Performance Standards. Throughout the course, students complete a variety of assignments that require them to practice the skills highlighted in class readings and discussions, and they adapt and generalize those skills during their field experiences. Students have numerous opportunities in class and in …

A Note On Relations For Reliability Measures In Zero-Adjusted Models, Broderick O. Oluyede, Mavis Pararai

Mathematical Sciences: Faculty Publications

In this note we examine and study relations in zero-adjusted models. Relations for reliability measures in the adjusted and unadjusted models are established and appropriate comparisons including the relative error are presented. The relative error is shown to be a decreasing function of the counts. Some inequalities and comparisons for weighted zero-adjusted models are established.

Closure Under Transfinite Extensions, Edgar Enochs, Alina Iacob, Overtoun Jenda

Mathematical Sciences: Faculty Publications

The closure under extensions of a class of objects in an abelian category is often an important property of that class. Recently the closure of such classes under transfinite extensions (both direct and inverse) has begun to play an important role in several areas of mathematics, for example, in Quillen's theory of model categories and in the theory of cotorsion pairs. In this paper we prove that several important classes are closed under transfinite extensions.

Bounds And Comparisons For Weighted Renewal-Type Integral Equations, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this note, inequalities and bounds for weighted renewal-type integral equations are presented. Some upper and lower bounds for the weighted renewal-type integral equations with monotone weight functions are derived. Some upper and lower bounds for the weighted renewal-type equations with monotone weight functions are derived. Bounds for the difference between two weighted renewal functions as well between the parent and weighted renewal functions are obtained in terms of the parent renewal reliability functions and their first and second moments. Relations for renewal-type integrals of the ruin probability are presented. Some inequalities, bounds and convergence results are also established.

On Energy And Expected Uncertainty Measures In Weighted Distributions, Broderick O. Oluyede, Mekki Terbeche

Mathematical Sciences: Faculty Publications

In this note, bounds and inequalities for the comparisons of weighted energy functions, entropy, and discrimination information measures and their unweighted counterparts are presented. Inequalities for weighted expected uncertainty, cross-entropy or discrimination information measures are also presented. A useful result on the convergence of the weighted kernel density informational energy estimates is given and some informational energy applications presented.

Preservice Teachers’ Mapping Structures Acting On Representational Quantities, Gunhan Caglayan

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

In this article, I write about my research on five preservice secondary teachers’ (PST) understanding and sense making of representational quantities associated with magnetic color cubes and tiles. Data came from individual interviews during which I asked PST problems guided by five main tasks: prime and composite numbers, summation of counting numbers, odd numbers, even numbers, and polynomial expressions in x and y. My work drew upon an analysis framework (Behr et. al, 1994) supported by a unit coordination construct (Steffe, 1988) associated with linear and areal quantities inherent in the nature of figures produced by these PST. Linear quantities …

Assessing Understanding Of Multiplication Through Words, Pictures, And Numbers, Stephanie Z. Smith

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

The objective of this session is to engage mathematics teacher educators in a discussion of how to assess an understanding of the concept of multiplication as an operation and its relationships to other operations. The session will begin with a presentation of a previously published study assessing children’s understanding of multiplication as grouping and the relationship between multiplication and addition. The assessment asked a series of problems involving words, pictures, and numbers. The results of the study indicate that the types of problems asked were successful in providing evidence of children’s understanding of multiplication. The study also found that a …

The Mathematical Preparation Of Secondary School Teachers, Kelly W. Edenfield

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

In the summer of 2007, a group of doctoral students at the University of Georgia gathered to discuss the mathematical preparation of secondary teachers. The group used Mathematics for High School Teachers: An Advanced Perspective by Usiskin, Peressini, Marchisotto, and Stanley (2003) as the catalyst for the discussion. Participants agreed that future teachers need opportunities to examine high school and college mathematics differently from the way they had as students, with specific emphasis on connections, representations, and history. Features of this text that were highlighted in the discussions were the attention topics with commonly held misconceptions, the historical rationales and …

Using Technology To Design Teaching Modules In Mathematics And Science, Ollie I. Manley

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Technology is changing the way in which mathematics and science are taught, and this radical transformation in teaching is causing teachers to take a closer look at how lessons are designed. In an effort to demonstrate how to design instructional modules using technology, this paper will include the following: 1)A review of the National Educational Technology Standards for teachers to establish a framework for the development of the teaching modules; 2)instructional designs and techniques with special emphasis on multiple intelligence and critical thinking skills; 3) strategies and techniques for infusing technology into a standard based curriculum; and 4) an analysis …

Change And Relationships In Elementary Preservice Teachers’ Mathematics Pedagogical Beliefs, Teaching Efficacy Beliefs, And Content Knowledge, Susan L. Swars, Lynn C. Hart , Ed., Stephanie Z. Smith, Marvin E. Smith

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

This study investigated the mathematics beliefs and content knowledge of 103 elementary preservice teachers in a developmental teacher preparation program that included a two course mathematics methods sequence. Preservice teachers’ pedagogical beliefs became more cognitively-oriented during the teacher preparation program with these changes occurring during the two methods courses. Pedagogical beliefs remained stable during student teaching. The preservice teachers also significantly increased their personal efficacy for teaching mathematics throughout the program with these shifts occurring across both methods courses and into student teaching. Pedagogical beliefs and teaching efficacy beliefs were not related at the beginning of the program, but, in …

Some General Notions Of Stochastic Orderings For Weighted Reliability And Uncertainty Measures With Applications, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this note, stochastic comparisons of reliability measures and related functions are presented. Inequalities for uncertainty of a residual life distribution and certain modified cross-entropy or discrimination information measures under weighted models are established. Comparisons of the expected uncertainty about the remaining lifetime of a component for weighted conditional distributions and unweighted conditional distributions are presented.

On Sequential And Fixed Designs For Estimation With Comparisons And Applications, Mekki Terbeche, Broderick O. Oluyede, Ahmed Barbour

Mathematical Sciences: Faculty Publications

A fully sequential approach to the estimation of the difference of two population means for distributions belonging to the exponential family of distributions is adopted and compared with the best fixed design. Results on the lower bound for the Bayes risk due to estimation and expected cost are presented and shown to be of first order efficiency. Applications involving the Poisson and exponential distributions with gamma priors as well as the Bernoulli distribution with beta priors are given. Finally, some numerical results are presented.

Vertices Of Self-Similar Tiles, Da-Wen Deng, Sze-Man Ngai

Mathematical Sciences: Faculty Publications

The set V_n of n-vertices of a tile T in \R^d is the common intersection of T with at least n of its neighbors in a tiling determined by T. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of V_n for general and strictly self-similar tiles. We show that if T is a general self-similar tile in \R² whose interior consists of finitely many components, then any tile in any self-similar tiling generated by T has a finite …

Self-Similar Measures Associated To {Ifs} With Non-Uniform Contraction Ratios, Sze-Man Ngai, Yang Wang

Mathematical Sciences: Faculty Publications

In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, Peres, et al.

Julian Cecil Stanley Papers, Zach S. Henderson Library, Special Collections

Finding Aids

This collection consists of the research materials of Johns Hopkins professor of psychology, Julian C. Stanley. Materials range from 1950-2005 and include his published reprints, abstracts, reports, letters, and seminar papers. The collection also includes reviews of research materials and articles by prominent psychologists in similar fields.

Find this collection in the University Libraries' catalog.

Long-Step Homogeneous Interior-Point Method For P*-Nonlinear Complementarity Problem, Goran Lesaja

Mathematical Sciences: Faculty Publications

A P*-Nonlinear Complementarity Problem as a generalization of the P*Linear Complementarity Problem is considered. We show that the long-step version of the homogeneous self-dual interior-point algorithm could be used to solve such a problem. The algorithm achieves linear global convergence and quadratic local convergence under the following assumptions: the function satisfies a modified scaled Lipschitz condition, the problem has a strictly complementary solution, and certain submatrix of the Jacobian is nonsingular on some compact set.

A Note On Computing The Generalized Inverse A^(2)_{T,S} Of A Matrix A, Xiezhang Li, Yimin Wei

Mathematical Sciences: Faculty Publications

The generalized inverse A T,S (2) of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T,S (2) has been recently developed with the condition σ (GA| T)⊂(0,∞), where G is a matrix with R(G)=T andN(G)=S. In this note, we remove the above condition. Three types of iterative methods for A T,S (2) are presented if σ(GA|T) is a subset of the open right half-plane and they are extensions of existing computational procedures of A T,S (2), including special cases such as the weighted Moore-Penrose …

On Inequalities And Partial Orderings For Weighted Reliability Measures, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

Inequalities, relations and partial ordering for weighted reliability measures are presented. Inequalities for Lévy distance measure for weighted distributions are obtained in terms of the parent distributions. Reliability inequalities and stability results are established for weighted distributions with monotone hazard and mean residual life functions.

On Stochastic Inequalities And Comparisons Of Reliability Measures For Weighted Distributions, Broderick O. Oluyede, E. Olusegun George

Mathematical Sciences: Faculty Publications

Inequalities, relations and stochastic orderings, as well as useful ageing notions for weighted distributions are established. Also presented are preservation and stability results and comparisons for weighted and length-biased distributions. Relations for length-biased and equilibrium distributions as examples of weighted distributions are also presented.

On Moment Inequalities And Stochastic Ordering For Weighted Reliability Measures, Broderick O. Oluyede, Mekki Terbeche

Mathematical Sciences: Faculty Publications

We obtain stochastic inequalities, error bounds, and classification probability for a general class of distributions. We introduce the notion of variability ordering via the probability functional and comparisons made for the weighted and the original distributions. We present moment inequalities, comparisons, and applications.

On Stochastic Comparisons Of Energy Functions With Applications, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

We develop simple methods for the stochastic comparisons of informational energy functions. We introduce modified informational energy functions and uncertainty of parameter functions are introduced for models with realistic parameter spaces. We present inequalities, comparisons, and applications including test procedures for testing the equality of informational energy functions. Some illustrative examples are also presented.