Mathematics Commons^™

Generalized Heineken--Mohamed Type Groups, Orest Artemovych

Turkish Journal of Mathematics

We prove that a torsion group G with all subgroups subnormal is a nilpotent group or G=N(A_1 \times \cdots \times A_n) is a product of a normal nilpotent subgroup N and p_i-subgroups A_i, where A_i=A_1^{(i)} \cdots A_{m_i}^{(i)} \lhd G, A_j^{(i)} is a Heineken--Mohamed type group, and p_1, \ldots, p_n are pairwise distinct primes (n\geq 1; i=1, ... ,n; j=1, ... ,m_i and m_i are positive integers).

The Fundamental Theorems Of Algebroid Functions On Annuli, Yang Tan, Qingcai Zhang

Turkish Journal of Mathematics

An extension of Nevanlinna value distribution theory for algebroid functions on annuli is proposed. The main characteristics are one-parameter and possess the same properties as in the classical case. Analogs of the Cartan theorem, the first fundamental theorem, the second fundamental theorem, deficient values, and the uniqueness of algebroid functions on annuli are proved.

Real Hypersurfaces In Complex Two-Plane Grassmannians Whose Shape Operator Is Recurrent For The Generalized Tanaka-Webster Connection, Juan De Dios Perez, Young Jin Suh, Changhwa Woo

Turkish Journal of Mathematics

We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose shape operator $A$ is generalized Tanaka-Webster recurrent if the principal curvature of the structure vector field is not equal to trace(A).

Analysis And Simulation For Homogeneous And Heterogeneous Sir Models, Joseph Wilda

Electronic Theses and Dissertations

In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous Susceptible-Infectious-Removed (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented …

Positive Fragments Of Coalgebraic Logics, Adriana Balan, Alexander Kurz, Jirí Velebil

Engineering Faculty Articles and Research

Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on …

Presenting Distributive Laws, Marcello M. Bonsangue, Helle H. Hansen, Alexander Kurz, Jurriaan Rot

Engineering Faculty Articles and Research

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation …

Narrowing The Mathematical Achievement Gap Among African American And Latino Students, Muriel Eileen Smith

Walden Dissertations and Doctoral Studies

This study focused on the continued mathematical achievement gap between African American and Latino students in an urban elementary school. An illustrative case study design was used to examine the teachers' perception of factors contributing to this gap in mathematical performance, and what instructional math strategies can narrow the achievement gap. Socioconstructivism and culturally relevant pedagogy were the learning theories used to form the conceptual framework in this study. Qualitative data were obtained from 6 individual interviews with 4th grade math educators, classroom observations, and teacher artifacts. Data analysis in this study included data triangulation and coding, as well as …

Extensions Of Functors From Set To V-Cat, Adriana Balan, Alexander Kurz, Jirí Velebil

Engineering Faculty Articles and Research

We show that for a commutative quantale V every functor Set --> V-cat has an enriched left- Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric.

Approximation Of Nested Fixpoints, Alexander Kurz

Engineering Faculty Articles and Research

The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible nontermination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekic lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is …

Coalgebraic Semantics Of Reflexive Economics (Dagstuhl Seminar 15042), Samson Abramsky, Alexander Kurz, Pierre Lescanne, Viktor Winschel

Engineering Faculty Articles and Research

This report documents the program and the outcomes of Dagstuhl Seminar 15042 “Coalgebraic Semantics of Reflexive Economics”.

Combinatorial Game Theory: An Introduction To Tree Topplers, John S. Ryals Jr.

College of Graduate Studies: Theses & Dissertations

The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing.

Engaging In Lesson Study At Georgia College, Angel R. Abney, Brandon Samples, Doris Santarone

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

A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught.

Using Ti-Nspire To Engage Preservice Mathematics Teachers In An Exploratory Geometry Module, Alesia D. Mickle, Pier A. Junor Clarke

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

In the mathematics classroom, most preservice mathematics teachers possess basic skills to use technology as an instructional strategy in communicating content standards. However, today’s demands for preservice teachers to engage in a variety of “best teaching practices” in their preservice teaching and edTPA requirements can oftentimes place the acquisition of technical skills and integration of new technology in content curriculum far from the forefront of their minds. Ertmer, Conklin, Lewandowski, Osika, Selo, and Wignall (2003) acknowledged preservice teachers’ desires to gain the adequate technical skills necessary to use technology in teachers’ daily tasks of facilitating and managing their classrooms. They …

Volterra Difference Equations, Nasrin Sultana

Doctoral Dissertations

"This dissertation consists of five papers in which discrete Volterra equations of different types and orders are studied and results regarding the behavior of their solutions are established. The first paper presents some fundamental results about subexponential sequences. It also illustrates the subexponential solutions of scalar linear Volterra sum-difference equations are asymptotically stable. The exact value of the rate of convergence of asymptotically stable solutions is found by determining the asymptotic behavior of the transient renewal equations. The study of subexponential solutions is also continued in the second and third articles. The second paper investigates the same equation using the …

Essays On Unit Root Testing In Time Series, Xiao Zhong

Doctoral Dissertations

"Unit root tests are frequently employed by applied time series analysts to determine if the underlying model that generates an empirical process has a component that can be well-described by a random walk. More specifically, when the time series can be modeled using an autoregressive moving average (ARMA) process, such tests aim to determine if the autoregressive (AR) polynomial has one or more unit roots. The effect of economic shocks do not diminish with time when there is one or more unit roots in the AR polynomial, whereas the contribution of shocks decay geometrically when all the roots are outside …

On Testing Common Indices For Several Multi-Index Models: A Link-Free Approach, Xuejing Liu

Doctoral Dissertations

"To avoid the curse of dimensionality, and to help us better understand the structure of the high dimensional data, methods for dimension reduction are clearly called for. The common linear dimension reduction techniques for single population include principal component analysis (PCA) which is unsupervised in regression and supervised Partial Least Squares (PLS). Modern sufficient dimension reduction techniques, like the ones we consider, constitute a form of supervised linear dimension reduction which outperform PCA and PLS without the underlying model assumptions.

In practice, we often deal with situations where the same variables are being measured on objects from different groups, and …

Discrete Analogues Of Some Classical Special Functions, Thomas Joseph Cuchta

Doctoral Dissertations

"Analogues of special functions on time scales are studied with special focus on the time scale 𝕋 = hℤ. Functions investigated in particular include complex monomials, new trigonometric functions, Gaussian bell, Hermite and Laguerre polynomials, Bessel functions, and hypergeometric series"--Abstract, page iii.

Small Sample Saddlepoint Confidence Intervals In Epidemiology, Pasan Manuranga Edirisinghe

Doctoral Dissertations

"In section 1, we develop a novel method of confidence interval construction for directly standardized rates. These intervals involve saddlepoint approximations to the intractable distribution of a weighted sum of Poisson random variables and the determination of hypothetical Poisson mean values for each of the age groups. Simulation studies show that, in terms of coverage probability and length, the saddlepoint confidence interval (SP) outperforms four competing confidence intervals obtained from the moment matching (M8), gamma-based (G1,G4) and ABC bootstrap (ABC) methods.

In section 2, we first consider Brillinger's classical model for a vital rate estimate with a random denominator. We …

Small Sample Umpu Equivalence Testing Based On Saddlepoint Approximations, Renren Zhao

Doctoral Dissertations

"In the first section, we consider small sample equivalence tests for exponentiality. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require a much larger sample size, in comparison to classical "difference tests", to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. …

Teacher Influence On Elementary School Students’ Participation In Science, Technology, Engineering, And Mathematics, Courtney Hartman

Honors College Theses

The purpose of this study is to explore the influence of elementary school teachers on encouraging students’ interest and participation in Science, Technology, Engineering, and Mathematics. The researcher sought to understand what methods teachers use in their classrooms to encourage students to participate in STEM subjects and programs. This mixed methods study consisted of a questionnaire to collect quantitative data, as well as an interview of selected teachers who participated in the questionnaire to collect qualitative data. The data was analyzed to determine the overall perceptions of teachers regarding the importance of encouraging students to participate in STEM. The qualitative …