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Proceedings Of The Seventh Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Introductory Texts

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

Contents of 7th Annual GAMTE Proceedings Front Matter:

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

How Differing Interpretations Of Making Mathematics Fun Influence Teaching Practice, Amanda Sawyer

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

I investigated an experienced teacher and a beginning teacher who held similar beliefs about making mathematics learning fun yet held different interpretations of implementation. I found when a teacher equated fun with problem solving, her classroom practice included activities with higher-level thinking skills. In contrast, a teacher who defined fun as students’ enjoyment layered manipulatives and group work on top of procedures. Therefore, teachers need opportunities to reflect on the nature of student understanding as a precursor to shaping their views of fun.

Negative phrases and attitudes toward mathematics have become commonplace in popular culture. Perhaps in response to this …

Applications Of The Lopsided Lovász Local Lemma Regarding Hypergraphs, Austin Tyler Mohr

Theses and Dissertations

The Lovász local lemma is a powerful and well-studied probabilistic technique useful in establishing the possibility of simultaneously avoiding every event in some collection. A principle limitation of the lemma's application is that it requires most events to be independent of one another. The lopsided local lemma relaxes the requirement of independence to negative dependence, which is more general but also more difficult to identify. We will examine general classes of negative dependent events involving maximal matchings of uniform hypergraphs, partitions of sets, and spanning trees of complete graphs. The results on hypergraph matchings (together with the configuration model of …

Study On Covolume-Upwind Finite Volume Approximations For Linear Parabolic Partial Differential Equations, Rosalia Tatano

Theses and Dissertations

In this thesis we solve two-dimensional linear parabolic partial differential equations with pure Dirichelet boundary conditions, using the bilinear covolume-upwind finite volume method on rectangular grids to discretize the spatial variables and the Crank-Nicholson method for the time variable. These PDEs provide a model for problems from various fields of engineering and applied sciences, such as unsteady viscous flow problems, the simulation of oil extraction from underground reservoirs, transport of air and ground water pollutants and modeling of semiconductor devices. Finite volume method has the important advantage of allowing the conversion of integrations over the control volume to integrations over …

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 Processing Of Irregularly Distributed Point Clouds, Kamala Hunt Diefenthaler

Theses and Dissertations

We address critical issues arising in the practical implementation of processing real point cloud data that exhibits irregularities. We develop an adaptive algorithm based on Learning Theory for processing point clouds from a stationary sensor that standard algorithms have difficulty approximating. Moreover, we build the theory of distribution-dependent subdivision schemes targeted at representing curves and surfaces with gaps in the data. The algorithms analyze aggregate quantities of the point cloud over subdomains and predict these quantities at the finer level from the ones at the coarser level.

The Weierstrass Approximation Theorem, Larita Barnwell Hipp

Theses and Dissertations

In this thesis we will consider the work began by Weierstrass in 1855 and several generalization of his approximation theorem since. Weierstrass began by proving the density of algebraic polynomials in the space of continuous real-valued functions on a finite interval in the uniform norm. His theorem has been generalized to an arbitrary compact Hausdorff space and the approximation with elements from more general algebras of continuous real-valued functions. We will consider proofs that use brute force and proofs based on convolutions and approximate identities, trudge through probability and the use of the Bernstein polynomials, and become intimately close to …

The Mc-Dagum Distribution And Its Statistical Properties With Applications, Broderick O. Oluyede, Sasith Rajasooriya

Mathematical Sciences: Faculty Publications

In this paper, a new class of distributions called Mc-Dagum distribution is proposed. This class of distributions contains several distributions such as beta-Dagum, beta-Burr III, beta-Fisk, Dagum, Burr III and Fisk distributions as special cases. The hazard function, reverse hazard function, moments, mean residual life function, Renyi entropy and Fisher information are obtained. Lorenz, Bonferroni and Zenga curves are derived. Maximum likelihood estimates of the model parameters and numerical examples are given to illustrate the usefulness of the proposed class of distributions.

Rademacher's Infinite Partial Fractions Conjecture Is (Almost Certainly) False, Andrew Sills, Doron Zeilberger

Mathematical Sciences: Faculty Publications

In his book Topics in Analytic Number Theory, Hans Rademacher conjectured that the limits of certain sequences of coefficients that arise in the ordinary partial fraction decomposition of the generating function for partitions of integers into at most N parts exist and equal particular values that he specified. Despite being open for nearly four decades, little progress has been made towards proving or disproving the conjecture, perhaps in part due to the difficulty in actually computing the coefficients in question. In this paper, we present a recurrence (alias difference equation) which provides a fast algorithm for calculating the Rademacher …

On Q-Analogs Of Wostenholme Type Congruences For Multiple Harmonic Sums, Jianqiang Zhao

Mathematical Sciences: Faculty Publications

Multiple harmonic sums are iterated generalizations of harmonic sums. Recently Dilcher has considered congruences involving q-analogs of these sums in depth one. In this paper we shall study the homogeneous case for arbitrary depth by using generating functions and shuffle relations of the q-analog of multiple harmonic sums. At the end, we also consider some non-homogeneous cases.

Number Representation And Calculation: An Overview Of Chapter 4 Of Thinking Mathematically, Emily Pawlicki

A with Honors Projects

A presentation providing an overview of numeration systems and using different bases in calculations.

Students' Development And Use Of Internal Representations When Solving Algebraic Tasks, Laban J. Cross

Theses and Dissertations

The difficulty in observing, recording, and examining internal representations has been well documented (Goldin & Shteingold, 2001). However, the important role that these internal representations play in the learning and understanding of mathematical concepts has been noted (Yackel, 2000). This study sought to develop a framework for examining the internalization patterns of school-aged children, gain a deeper understanding of how these internal representations are developed and used, and search for internalization patterns associated with successful generalizations.

Semi-structured interviews were conducted with six sixth-grade students and six tenth-grade students. During these interviews the students were asked to solve a series of …

Equivalences Of Dessins D'Enfants, Rachel M. Volkert

Honors Program Theses

Dessins d'enfants are bipartite graphs with a cyclic ordering given to the set of edges that meet at each vertex. Merling and Perlis presented a method by which to construct pairs of dessins d'enfants using the permutations induced by the action of a finite group on the cosets of two locally conjugate subgroups of that group. They called these pairs of dessins Gassmann equivalent and investigated some of their properties. First, we discuss several properties of pairs of dessins that imply Gassmann equivalence. Then, using elementwise conjugate subgroups, we introduce and investigate a weaker type of equivalence of dessins, which …

Developing Crochet Patterns For Surfaces Of Non-Constant Curvature, Katherine Lea Pearce

Honors Program Theses

The purpose of this project is to develop an algorithm to create crochet patterns for a variety of surfaces. I start with surfaces of constant curvature: the Euclidean surface and the sphere. Then, I generate patterns for surfaces of revolution by calculating the change in circumference for each row of stitches. My methods suggest an approach to crochet more surfaces such as surfaces whose cross section is not a circle. This research demonstrates how crochet can act as a discrete model of differential geometry. Producing these patterns allows for further research into the surfaces themselves by providing accurate models as …

Simplicial Complexes Obtained From Qualitative Probability Orders, Paul H. Edelman, Tatiana Gvozdeva, Arkadii Slinko

Vanderbilt Law School Faculty Publications

The goal of this paper is to introduce a new class of simplicial complexes that naturally generalize the threshold complexes. These will be derived from qualitative probability orders on subsets of a finite set that generalize subset orders induced by probability measures. We show that this new class strictly contains the threshold complexes and is strictly contained in the shifted complexes. We conjecture that this class of complexes is exactly the set of strongly acyclic complexes, a class that has previously appeared in the context of cooperative games. Beyond the results themselves, this new class of complexes allows us to …

Analytic Matrix Elements Of The Schrödinger Equation, Muhammad I. Bhatti

School of Mathematical & Statistical Sciences Faculty Publications

A previously defined analytic technique of constructing matrix elements from the Bernstein-polynomials (B-poly) has been applied to Schr¨odinger equation. This method after solving generalized eigenvalue problem yields very accurate eigenenergies and eigenvectors. The numerical eigenvectors and eigenvalues obtained from this process agree well with exact results of the hydrogen-like systems. Furthermore, accuracy of the numerical spectrum of hydrogen equation depends on the number of B-polys being used to construct the analytical matrix elements. Validity of eigenvalues and quality of the constructed wavefunctions is verified by evaluating the Thomas-Reiche-Kuhn (TRK) sum rules. Excellent numerical agreement is seen with exact results of …

Zero-Bounded Limits As A Special Case Of The Squeeze Theorem For Evaluating Single-Variable And Multivariable Limits, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

Many limits, typically taught as examples of applying the ‘squeeze’ theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful for both single-variable limits and multidimensional limits. A comprehensive treatment of multidimensional limits and continuity is also outlined.

On Equivalent Characterizations Of Convexity Of Functions, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

A detailed development of the theory of convex functions, not often found in complete form in most textbooks, is given. We adopt the strict secant line definition as the definitive definition of convexity. We then show that for differentiable functions, this definition becomes logically equivalent with the first derivative monotonicity definition and the tangent line definition. Consequently, for differentiable functions, all three characterizations are logically equivalent.

Optimization Problem In Single Period Markets, Tian Jiang

Electronic Theses and Dissertations

There had been a number of researches that investigated on the security market without transaction costs. The focus of this research is in the area that when the security market with transaction costs is fair and in such fair market how one chooses a suitable portfolio to optimize the financial goal. The research approach adopted in this thesis includes linear algebra and elementary probability. The thesis provides evidence that we can maximize expected utility function to achieve our goal (maximize expected return under certain risk tolerance). The main conclusions drawn from this study are under certain conditions the security market …

Nonparametric And Empirical Bayes Estimation Methods, Rida Benhaddou

Electronic Theses and Dissertations

In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski’s method and show that the estimator attains …

Extensions Of S-Spaces, Bernd Losert

Electronic Theses and Dissertations

Given a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y . Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y . We answer the first question for some …

Characterizations Of Exponential Distribution Based On Sample Of Size Three, George Yanev, Santanu Chakraborty

School of Mathematical & Statistical Sciences Faculty Publications

Two characterizations of the exponential distribution based on equalities among order statistics in a random sample of size three are proved. This proves two conjectures stated recently in Arnold and Villasenor [4].

The Two-Phase Arterial Blood Flow With Or Without A Catheter And In The Presence Of A Single Or Multi Stenosis, Ani E. Garcia, Daniel N. Riahi

School of Mathematical & Statistical Sciences Faculty Publications

We consider the problem of blood flow in an artery with or without a catheter and in the presence of single or multi stenosis whose shape is based on the available experimental data for the stenosis in a human’s artery. The presence of stenosis in the artery, which locally narrows portion of the artery, can be a result of fatty materials such as cholesterol in the blood. The use of catheter is important as a standard tool for diagnosis and treatment in patience whose blood flow passage in the artery is affected adversely by the presence of the stenosis within …

Accelerated Life Model With Various Types Of Censored Data, Kathryn Pridemore

Electronic Theses and Dissertations

The Accelerated Life Model is one of the most commonly used tools in the analysis of survival data which are frequently encountered in medical research and reliability studies. In these types of studies we often deal with complicated data sets for which we cannot observe the complete data set in practical situations due to censoring. Such difficulties are particularly apparent by the fact that there is little work in statistical literature on the Accelerated Life Model for complicated types of censored data sets, such as doubly censored data, interval censored data, and partly interval censored data. In this work, we …

Spectrally Uniform Frames And Spectrally Optimal Dual Frames, Saliha Pehlivan

Electronic Theses and Dissertations

Frames have been useful in signal transmission due to the built in redundancy. In recent years, the erasure problem in data transmission has been the focus of considerable research in the case the error estimate is measured by operator (or matrix) norm. Sample results include the characterization of one-erasure optimal Parseval frames, the connection between two-erasure optimal Parseval frames and equiangular frames, and some characterization of optimal dual frames. If iterations are allowed in the reconstruction process of the signal vector, then spectral radius measurement for the error operators is more appropriate then the operator norm measurement. We obtain a …

Numerical Simulations For The Flow Of Rocket Exhaust Through A Granular Medium, Kristina Kraakmo

Electronic Theses and Dissertations

Physical lab experiments have shown that the pressure caused by an impinging jet on a granular bed has the potential to form craters. This poses a danger to landing success and nearby spacecraft for future rocket missions. Current numerical simulations for this process do not accurately reproduce experimental results. Our goal is to produce improved simulations to more accurately and effi- ciently model the changes in pressure as gas flows through a porous medium. A two-dimensional model in space known as the nonlinear Porous Medium Equation as it is derived from Darcy’s law is used. An Alternating-Direction Implicit (ADI) temporal …

Creating An Interdisciplinary Research Course In Mathematical Ecology, Glenn Ledder, Brigitte Tenhumberg

School of Biological Sciences: Faculty Publications

An integrated interdisciplinary research course in biology and mathematics is useful for recruiting students to interdisciplinary research careers, but there are difficulties involved in creating and implementing it. We describe the genesis, objectives, design policies, and structure of the Research Skills in Theoretical Ecology course at the University of Nebraska–Lincoln and discuss the difficulties that can arise in designing and implementing interdisciplinary courses.

An Interdisciplinary Research Course In Theoretical Ecology For Young Undergraduates, Glenn Ledder, Brigitte Tenhumberg, G. Travis Adams

School of Biological Sciences: Faculty Publications

As part of an interdepartmental effort to attract promising young students to research at the interface between mathematics and biology, we created a course in which groups of recent high school graduates and first-year college students conducted a research project in insect population dynamics. The students set up experiments, collected data, used the data to develop mathematical models, tested their models against further experiments, and prepared their results for dissemination. The course was self-contained in that the lecture portion developed the mathematical, statistical, and biological background needed for the research. A special writing component helped students learn the principles of …

A Generalized White Noise Space Approach To Stochastic Integration For A Class Of Gaussian Stationary Increment Processes, Daniel Alpay, Alon Kipnis

Mathematics, Physics, and Computer Science Faculty Articles and Research

Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.

Non-Commutative Stochastic Distributions And Applications To Linear Systems Theory, Daniel Alpay, Guy Salomon

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we introduce a non-commutative space of stochastic distributions, which contains the non-commutative white noise space, and forms, together with a natural multiplication, a topological algebra. Special inequalities which hold in this space allow to characterize its invertible elements and to develop an appropriate framework of non-commutative stochastic linear systems.