Mathematics Commons^™

Theoretical Properties Of The Weighted Feller-Pareto Distributions, Oluseyi Odubote, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

In this paper, for the first time, a new six-parameter class of distributions called weighted Feller-Pareto (WFP) and related family of distributions is proposed. This new class of distributions contains several other Pareto-type distributions such as length-biased (LB) Pareto, weighted Pareto (WP I, II, III, and IV), and Pareto (P I, II, III, and IV) distributions as special cases. The pdf, cdf, hazard and reverse hazard functions, monotonicity properties, moments, entropy measures including Renyi, Shannon and s-entropies are derived.

How We Got From There To Here: A Story Of Real Analysis, Eugene Boman, Robert Rogers

Milne Open Textbooks

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) …

Fully Inert Subgroups Of Abelian P-Groups, Brendan Goldsmith, Luigi Salce, Paolo Zanardo

Articles

A subgroup H of an Abelian group G is said to be fully inert in G, if for every endomorphism ϕ of G, the factor group (H+ϕ(H))/H" role="presentation"> is finite. This notion arises in the study of the dynamical properties of endomorphisms (entropy). The principal result of this work is that fully inert subgroups of direct sums of cyclic p-groups are commensurable with fully invariant subgroups of the direct sum.

Note On Rainbow Connection In Oriented Graphs With Diameter 2, Rebecca Holliday, Colton Magnant, Pouria Salehi Nowbandegani

Mathematical Sciences: Faculty Publications

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter $2$. For a tournament $T$ of diameter $2$, we show $2 \leq \overrightarrow{rc}(T) \leq 3$. Furthermore, we provide a general upper bound on the rainbow $k$-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of $k^{th}$ diameter $2$ has rainbow $k$-connection number at most approximately $k^{2}$.

Rainbow Generalizations Of Ramsey Theory - A Dynamic Survey, Shinya Fujita, Colton Magnant, Kenta Ozeki

Mathematical Sciences: Faculty Publications

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.

Building A Better Bijection Between Classes Of Compositions, James D. Diffenderfer

Mathematical Sciences: Faculty Publications

A bijective proof is given for the following theorem: The number of compositions of n into parts congruent to a (mod b) equals the number of compositions of n + b - a into parts congruent to b (mod a) that are greater than or equal to b. The bijection is then shown to preserve palindromicity.

A Decomposition Of Gallai Multigraphs, Alexander Halperin, Colton Magnant, Kyle Pula

Mathematical Sciences: Faculty Publications

An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees.

A New Class Of Generalized Dagum Distribution With Applications To Income And Lifetime Data, Broderick O. Oluyede, Shujiao Huang, Mavis Pararai

Mathematical Sciences: Faculty Publications

The generalized beta distribution of the second kind (GB2), McDonald [11], McDonald and Xu [12] is an important distribution with applications in finance and actuarial sciences, as well as economics, where Dagum distribution which is a sub-model of GB2 distribution plays an important role in size distribution of personal income. In this note, a new class of generalized Dagum distribution called gamma-Dagum distribution is presented. The gamma-Dagum (GD) distribution which includes the gamma-Burr III (GB III), gamma-Fisk or gamma-log logistic (GF of GLLog), Zografos and Balakrishnan-Dagum (ZB-D), ZB-Burr III (ZB-B III), ZB-Fisk of ZB-Log logistic (ZB-F or ZB-LLog), Burr III …

Discrete Fourier Restriction Associated With Schrödinger Equations, Yi Hu, Xiaochun Li

Mathematical Sciences: Faculty Publications

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain’s level set result on Strichartz estimates associated with Schrödinger equations on torus. Some sharp estimates on L 2(d+2) d norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on L 2 (Z).

Determinants Of Incidence And Hessian Matrices Arising From The Vector Space Lattice, Saeed Nasseh, Alexandra Seceleanu, Junzo Watanabe

Mathematical Sciences: Faculty Publications

We give explicit formulas for the determinants of the incidence and Hessian matrices arising from the interaction between the rank 1 and rank n−1 level sets of the subspace lattice of an n-dimensional finite vector space. Our exploration is motivated by the fact that both of these matrices arise naturally in the study of the combinatorial and algebraic Lefschetz properties for the vector space lattice and the graded Artinian Gorenstein algebra associated to it, respectively.

Proceedings Of The Eighth 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

• GAMTE 2014 Officers
• GAMTE 2014 Conference Committee
• GAMTE 2014 Proceedings Committee
• Purposes and Goals of GAMTE
• Letter from the President

A Proposed Pedagogical Approach For Preparing Teacher Candidates To Incorporate (Academic) Language, Woong Lim, Lynn Stallings

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

The edTPA is a performance-based assessment that aims to measure teacher candidates’ readiness for teaching. Beginning in the fall of 2015, this assessment will be a mandatory requirement for those seeking certification in Georgia. General agreement exists in the field of education about the basic knowledge and skills essential for beginning teachers to demonstrate in classroom teaching. Does edTPA measure the knowledge and skills essential for beginning mathematics teachers in particular? Assuming that edTPA can successfully measure that knowledge and skills for beginning teachers, the use of the assessment could be valuable.

One of the critical components of edTPA is …

Design Considerations For Visually-Aided Discussion Prompts: Emphasizing Mathematical Reasoning In Teacher Education, Anne Marie S. Marshall, Kadian M. Callahan

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

The availability and familiarity of online discussion tools create new instructional options that teacher educators can use to foster prospective teachers’ understanding of mathematics. In particular, online discussion blogs provide an avenue through which teacher educators can press prospective teachers to explore mathematical concepts and share their mathematical reasoning with peers. Furthermore, by incorporating visual stimulations as a design component of these discussion blogs, prospective teachers can make sense of and respond to others’ ideas about mathematical concepts with greater clarity. This paper shares preliminary findings of a research study that examined the extent to which the design of a …

Differences In Beliefs Across A Series Of Four Mathematics Content Courses, Susanna Molitoris Miller, Caitlin Walkey

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

College students often ask questions such as, “Why do I have to take this class? Is there a point to it?” For Early Childhood Education (ECE) majors these questions may often take on a slightly different form, wondering, “How can I incorporate this information into my classroom?” or “Do I understand this well enough to teach this to my students?” It is especially important for pre- service teachers to feel confident working with the mathematical content that they are learning and for them to believe that they can successfully teach that same information to a group of students. Swackhamer, Koellner, …

Do You See What I See? Deepening Teachers’ Understanding Of Linear Equations Through Student Interviews, Tamara Pearson, Kelli L. Nipper, Catherine Matos

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

Many teachers have trouble transitioning their students between natural recursive thinking about the data and algebraic notation for representing linear functions (Zazkis & Liljedahl, 2002).

In this study, we interviewed eighteen middle school students to see how they used prior instruction to think about a geometric pattern and construct its corresponding linear equation. All students

were given the same task to complete and were questioned about their thinking during the interview.

We found that the recording of pattern recognition plays a substantial part in helping students recognize and write explicit patterns. By having students decompose the total perimeter into how …

Caep, Nctm, And Secondary Mathematics Program Revisions, Dianna J. Spence

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

Eight assessments were developed for CAEP (formerly NCATE) and NCTM recognition of our secondary mathematics program. These assessments include internship work samples, field evaluations, and candidate portfolios addressing content knowledge, pedagogical methods, and mathematics technology. Based on data collected from these assessments, alongside ongoing evaluation of the program, several curriculum and program revisions were implemented, including: 1) development of mathematics content-specific courses in classroom management, assessment, and secondary curriculum; 2) restructuring of a senior seminar course in mathematics education; and 3) an increased content focus in probability and statistics. The adoption of new NCTM standards in the CAEP review process …

A Mathematics Teacher’S Journey Of Identity Construction And Change, Anthony B. Stinson

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

Despite some gains, improving mathematics instruction remains an area of concern in the United States. The implementation of the Common Core Standards and the challenge of teaching the 21st Century student require mathematics teachers to examine their pedagogy to determine if they need to change or improve their practices. This paper provides a personal account of my journey when determining my identity as a mathematics teacher and how constructing my identity helped in changing and improving my practices as a mathematics teacher. The study was done using autoethnography, a burgeoning research method, and identity theory. This study has the goals …

Safety Effect Of Missouri's Strategic Highway Safety Plan: Missouri's Blueprint For Safer Roadways, Mojtaba A. Mohammadi, V. A. Samaranayake, Ghulam H. Bham

Mathematics and Statistics Faculty Research & Creative Works

This study systematically evaluated the changes in motor vehicle crashes that occurred on the Missouri Interstate highway system following the implementation of the Missouri Strategic Highway Safety Plan (MSHSP) between 2004 and 2007. The MSHSP implemented crash injury reduction strategies in enforcement, education, engineering, and public policy. Empirical Bayesian methods were commonly used to evaluate the effects of any change in safety as a result of countermeasures. This paper presents a simple new approach to evaluating the effects of Missouri's safety plans on roadway crashes. For crash data associated with traffic and roadway characteristics, negative binomial regression models were developed …

W.R.I.T.E: Writing’S Role In Thoughtful Endeavors, Catherine M. Miller

Faculty Work

“Mathematicians actually spend a great deal of time writing. If a mathematician wants to contribute to the greater body of mathematical knowledge, she must be able communicate her ideas in a way which is comprehensible to others.” Lee, 2014, pg. 1

Why should our students be different from the mathematicians Lee (2014) refers to in the quote above? Learning mathematics requires a confluence of spoken, written and mathematically abstract languages. After all, who thinks entirely in mathematical symbols? No one, except a few mathematical thinkers who are live in the abstract worlds they construct. Thinking about mathematics in the language …

An Introduction To Set Theory And Topology, Ronald C. Freiwald

Books and Monographs

These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. …