Mathematics Commons^™

English Translation Of Einige Gesetze Ueber Die Theilung Der Ebene Und Des Raumes, Justin Heavilin

Mathematics and Statistics Faculty Publications

The article “Einige Gesetze ueber die Theilung der Ebene und des Raumes.” was published by J. Steiner in the very first volumn of the Journal fuer die reine und angewandte Mathematik in 1826. Journal fuer die reine und angewandte Mathematik is the oldest mathematics periodical in existence.† . My translation is meant to convey the ideas published by Steiner, and when presented with the choice between translating faithfully to the original text or clarity of his ideas, I admit to choosing the later. There are two footnotes original to the text, which appear with asterisks. Where helpful I included additional …

Econophysics And Fractional Calculus: Einstein's Evolution Equation, The Fractal Market Hypothesis, Trend Analysis And Future Price Prediction, Jonathan Blackledge, Derek Kearney, Marc Lamphiere, Raja Rani, Paddy Walsh

Articles

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial …

Conflict Free Connectivity And The Conflict-Free-Connection Number Of Graphs, Travis D. Wehmeier

College of Graduate Studies: Theses & Dissertations

We explore a relatively new concept in edge-colored graphs called conflict-free connectivity. A conflict-free path is a (edge-) colored path that has an edge with a color that appears only once. Conflict-free connectivity is the maximal number of internally disjoint conflict-free paths between all pairs of vertices in a graph. We also define the c-conflict-free-connection of a graph G. This is the maximum conflict-free connectivity of G over all c-colorings of the edges of G. In this paper we will briefly survey the works related to conflict-free connectivity. In addition, we will use the probabilistic method to achieve a bound …

On The Regularity Radius Of Delone Sets In R3, Nikolay Dolbilin, Alexey Garber, Undine Leopold, Egon Schulte

School of Mathematical & Statistical Sciences Faculty Publications

We complete the proof of the upper bound ρ^3≤10R for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all 10R-clusters of a Delone set X with parameters (r,R) are equivalent, then X is a regular system.

On A Class Of Parameter Dependent Series Generalizing Euler’S Constant, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Series which depend on a parameter and generalize the constant discovered by Euler are introduced and studied. Convergence results are established. An infinite series expansion is obtained from these generalized formulas which can be used to evaluate the generalized constant. Euler’s constant can be obtained as a special case. Some asymptotic results are formulated and limits of some closely related sequences are given.

On The Determination Of The Number Of Positive And Negative Polynomial Zeros And Their Isolation, Emil Prodanov

Articles

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real co-efficients and degree $n$ can be restricted with significantly better determinacy than that provided by the Descartes' rule of signs and also isolate quite successfully the zeros of the polynomial. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically and …

Van Der Waals Universe With Adiabatic Matter Creation, Emil Prodanov, Rossen Ivanov

Articles

A FRWL cosmological model with perfect fluid comprising of van der Waals gas and dust has been studied in the context of dynamical analysis of a three-component autonomous non-linear dynamical system for the particle number density $n$, the Hubble parameter $H$, and the temperature $T$. Perfect fluid isentropic particle creation at rate proportional to an integer power $\alpha$ of $H$ has been incorporated. The existence of a global first integral allows the determination of the temperature evolution law and hence the reduction of the dynamical system to a two-component one. Special attention is paid to the cases of $\alpha = …

Injective Tensor Products Of Tree Spaces, Milena Venkova, Christopher Boyd, Costas Poulios

Articles

We study tensor products on tree spaces; in particular, we give necessary and sufficient conditions for the n-fold injective tensor product of tree spaces to contain a copy of l_1.

Equatorial Wave–Current Interactions, Adrian Constantin, Rossen Ivanov

Articles

We study the nonlinear equations of motion for equatorial wave–current interactions in the physically realistic setting of azimuthal two-dimensional inviscid flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface. We derive a Hamiltonian formulation for the nonlinear governing equations that is adequate for structure-preserving perturbations, at the linear and at the nonlinear level. Linear theory reveals some important features of the dynamics, highlighting differences between the short- and long-wave regimes. The fact that ocean energy is concentrated in the long-wave propagation modes motivates the pursuit of in-depth nonlinear analysis in the long-wave …

Current Source Density Approaches Improve Spatial Resolution In Event Related Potential Analysis In People With Parkinson's Disease, Conor Fearon, John Butler, Saskia M. Waechter, Isabelle Killane, Richard B. Reilly, Timothy Lynch

Articles

No abstract provided.

On The Reidemeister Spectrum Of An Abelian Group, Brendan Goldsmith, Fatemeh Karimi, Noel White

Articles

The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group G/(ϕ − 1G)(G), and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G. In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then …

The Sons Report (1989-1994, Mathematical Association Of America): The Way It Was, Linda R. Sons

Numeracy

Recollections and commentary by Linda R. Sons on a 1994 national report entitled Quantitative Reasoning for College Graduates: A Complement to the Standards. Professor Sons chaired the committee which wrote the report and championed its use.

This paper traces the development of the 1994 MAA report Quantitative Reasoning for College Graduates: A Complement to the Standards--a report which is still surprisingly relevant. The paper highlights some major parts of the report, describing the context in which its conclusions were made. Additionally, it challenges and encourages those continuing the quest for QL in the nation by noting progress made …

Developing Mathematics Teachers’ Attention To Quantitative Reasoning In Task Design: A Modeling Approach, David Glassmeyer

Numeracy

This study examines how a models-and-modeling perspective affected teachers' attention to quantitative reasoning in task design. A Model-Eliciting Activity (MEA) was implemented with 21 teachers over four weeks, challenging teachers to design a quantitative reasoning task for their students. Teachers’ initial quantitative reasoning tasks did not incorporate quantities or quantitative relationships, two essential components of quantitative reasoning. As teachers revised their tasks through the MEA, most teachers began attending to these components. This article details how a modeling approach to teacher education provided a method to describe and support teachers to incorporate quantitative reasoning in their classroom tasks, though attending …

The Second Decade Of Numeracy: Entering The Seas Of Literacy, H. L. Vacher

Numeracy

This multipurpose editorial explores and tries to count the many types of literacy that are referred to by name in Wikipedia and Numeracy. Wikipedia’s Category:Literacy page identifies 44 kinds of literacy that are the subject of articles, ranging from numeracy and graphicacy to braille literacy and diaspora literacy. In addition, searching Google finds more than 30 adjective-literacy or noun-literacy collocations, including quantitative literacy, adult literacy, and document literacy, that do not have Wikipedia pages of their own but are mentioned on other Wikipedia pages. The sum puts this modest literacy count in line with the more than 70 bodies …

The Predictive Performance Of Objective Measures Of Physical Activity Derived From Accelerometry Data For 5-Year All-Cause Mortality In Older Adults: National Health And Nutritional Examination Survey 2003-2006, Ekaterina Smirnova, Andrew Leroux, Quy Cao, Lucia Tabacu, Vadim Zipunnikov, Ciprian Crainiceanu, Jacek Urbanek

Mathematics & Statistics Faculty Publications

Background: Declining physical activity (PA) is a hallmark of aging. Wearable technology provides reliable measures of the frequency, duration, intensity, and timing of PA. Accelerometry-derived measures of PA are compared to established predictors of 5-year all-cause mortality in older adults in terms of individual, relative, and combined predictive performance.

Methods: Participants between 50 and 85 years old from the 2003-2006 National Health and Nutritional Examination Survey (NHANES, n = 2978) wore a hip-worn accelerometer in the free-living environment for up to 7 days. A total of 33 predictors of 5-year all-cause mortality (number of events = 297), including 20 measures …

On Hybrid Temporal Basis Functions For Stable Numerical Solution Of Time Domain Boundary Integral Equations, Fang Q. Hu

Mathematics & Statistics Faculty Publications

Problems in unsteady aerodynamics and aeroacoustics can sometimes be formulated as integral equations, such as the boundary integral equations. Numerical discretization of integral equations in the time domain often leads to so-called March-On-in-Time (MOT) schemes. In the literature, the temporal basis functions used in MOT schemes have been largely limited to low-order shifted Lagrange basis functions. In order to evaluate the accuracy and effectiveness of the temporal basis functions, a Fourier analysis of the temporal interpolation schemes is carried out. Based on the Fourier analysis, the spectral resolutions of various temporal basis functions are quantified. It is argued that hybrid …

Conceptualizing And Interpreting Mean And Median With Future Teachers, Eryn Stehr Maher, Ha Nguyen, Gregory Chamblee, Sharon Taylor

Mathematical Sciences: Faculty Publications

Mathematical Education of Teachers II (METII), echoed by the American Statistical Association publication, Statistical Education of Teachers, recommended teacher preparation programs support future teachers in developing deep understandings of mean and median, such that middle grades teachers may use them to “summarize, describe, and compare distributions” (Conference Board of Mathematical Sciences, 2012, p. 44; Franklin et al., 2015). Georgia Standards of Excellence require statistical reasoning from students beginning as early as 6-7 years old, including interpretation of measures of center and statistical reasoning about best measures of center (Georgia Department of Education, 2015). This level of understanding and interpretation of …

Cronbach’S Alpha Under Insufficient Effort Responding: An Analytic Approach, Stephen W. Carden, Trevor R. Camper, Nicholas S. Holtzman

Psychology: Faculty Publications

Surveys commonly suffer from insufficient effort responding (IER). If not accounted for, IER can cause biases and lead to false conclusions. In particular, Cronbach’s alpha has been empirically observed to either deflate or inflate due to IER. This paper will elucidate how IER impacts Cronbach’s alpha in a variety of situations. Previous results concerning internal consistency under mixture models are extended to obtain a characterization of Cronbach’s alpha in terms of item validities, average variances, and average covariances. The characterization is then applied to contaminating distributions representing various types of IER. The discussion will provide commentary on previous simulation-based investigations, …

Proceedings Of Thirteenth 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 13th Annual GAMTE Proceedings Front Matter:

• Officers of GAMTE
• Reviewers
• Copyright & Licensing Terms
• Purposes and Goals of GAMTE
• Conference Schedule
• Dedication
• Table of Contents

Conceptualizing And Interpreting Mean And Median With Future Teachers, Eryn M. Stehr, Ha Nguyen, Gregory Chamblee, Sharon Taylor

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

Mathematical Education of Teachers II (METII), echoed by the American Statistical Association publication, Statistical Education of Teachers, recommended teacher preparation programs support future teachers in developing deep understandings of mean and median, such that middle grades teachers may use them to “summarize, describe, and compare distributions” (Conference Board of Mathematical Sciences, 2012, p. 44; Franklin et al., 2015). Georgia Standards of Excellence require statistical reasoning from students beginning as early as 6-7 years old, including interpretation of measures of center and statistical reasoning about best measures of center (Georgia Department of Education, 2015). This level of understanding and interpretation of …

An Alternative Approach To The Traditional Internship, Basil M. Conway, David Erikson, Christopher Parrish, Marilyn Strutchens, Jennifer Whitfield

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

This paper reports the benefits and challenges of incorporating a paired-placement model at four different post-secondary teacher preparation programs in secondary mathematics education. The paired-placement model places two secondary mathematics clinical teachers with one mentor (or cooperating) teacher during their internship experience. Benefits exhibited were increased collaboration, more knowledgeable cooperating teachers, increased sense of community, teaming, pedagogical risk-taking, increased reflective practice, established natural professional learning communities, Plan-Do-Study-Act Cycle (PDSA), and increased accountability. Challenges found through the PDSA cycle include personnel issues, number of days teaching, perceived classroom management preparation, preparing university supervisors, mentors, and teacher candidates, and support for collaboration …

K-2 Mathematicians & Writers: Professional Learning Communities For Developing Conceptual Understanding, Doris Santarone, Angel R. Abney, Sandra M. Webb

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

National Council of Teachers of Mathematics (NCTM) has long supported the use of children’s literature, writing, and manipulatives to improve conceptual understanding of mathematics (2000). In a professional learning community for K-2 teachers, professional development was designed and implemented on ways to incorporate literacy and manipulatives into a mathematics lesson. The teachers were charged with collaboratively planning lessons that included multiple components: the standard(s), a mathematics activity, manipulatives, a writing task, and children’s literature. As the data were analyzed, it became apparent that while most of the lessons were well connected, this did not happen for all of the lessons. …

Preparing Pre-Service Teachers To Present At A State Conference, Heidi Eisenreich

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

As mathematics teacher educators, we have a responsibility to prepare as many people as we can, to teach mathematics in ways that foster a deeper understanding of the content. We do this by teaching current and future teachers in college programs and providing professional development to in-service teachers. A less explored way is to prepare these “students” to present ideas they have learned to colleagues at their school, other schools in their district, and conferences. In this paper, I share my experience of helping students go through the process of preparing to present over the last two years at our …

Creative Writing In The Mathematics Classroom, William Lacefield, Laura Markert

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

Creative writing in the mathematics classroom promotes mathematical applications in the real world, constructivist learning, embodied learning, transfer of mathematical ideas, and student engagement. When students are allowed to write about mathematical concepts creatively, they are able to take concepts that they have learned and put them into their world or even create a situation where the mathematical concept applies. Applying mathematical concepts to other environments helps learners transfer mathematical concepts. Learners are able to take the mathematics content and contextualize it outside of the classroom. Writing in mathematics also is a way for students to embody learning. Because writing …

The Near-Peer Mathematical Mentoring Cycle: Studying The Impact Of Outreach On High School Students’ Attitudes Toward Mathematics, Aaron T. Wilson, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

College students may be seen as near-peers to high school students and high school students are often able to see themselves in the college students who are but one step ahead. This nearness in maturity and educational level may place college students in a particularly powerful position when it comes to reaching out to high school students to promote higher education in math and science. In this study college students gave dynamic mathematics outreach presentations, MathShows, to minority and low-income high school students in a mid-sized public school district on the U.S. border with Mexico. The study investigated the impacts …

Totally Acyclic Complexes, Holly M. Zolt

College of Graduate Studies: Theses & Dissertations

We consider the following question: when is every exact complex of injective modules a totally acyclic one? It is known, for example, that over a commutative Noetherian ring of finite Krull dimension this condition is equivalent with the ring being Iwanaga-Gorenstein. We give equivalent characterizations of the condition that every exact complex of injective modules (over arbitrary rings) is totally acyclic. We also give a dual result giving equivalent characterizations of the condition that every exact complex of flat modules is F-totally acyclic over an arbitrary ring.

The Relationship Between Housing Affordability And Demographic Factors: Case Study For The Atlanta Beltline, Chapman T. Lindstrom

College of Graduate Studies: Theses & Dissertations

Housing affordability has been a widely examined subject for populations residing in major metropolitan regions around the world. The relationship between housing affordability and the city’s demographics and its volume of urban development are important to take into consideration. In the past two decades there has been an increasing volume of literature detailing Atlanta Georgia’s large-scale redevelopment project, the Atlanta BeltLine (ABL), and its relationship with Atlanta’s Metropolitan population and housing affordability. The first objective of this paper is to study the relationship between housing affordability at two scales within the Atlanta Metropolitan Area (AMA) for both renters and homeowners. …

Taking A Canon To The Adjunction Formula, Paul M. Harrelson

College of Graduate Studies: Theses & Dissertations

In this paper, we show how the canonical divisor of a graph is related to the canonical divisor of its subgraph. The use of chip firing and the adjunction formula for graphs ex- plains said relation and even completes it. We go on to show the difference between the formula for full subgraphs and that of non-full subgraphs. Examples are used to simplify these results and to see the adjunction formula in action. Finally, we show that though the adjunction formula seems simple at first glance, it is somewhat complex and rather useful.

Analyzing Mathematicians' Concept Images Of Differentials, Timothy Shawn Mccarty

Graduate Theses, Dissertations, and Problem Reports (ETD)

The differential is a symbol that is common in first- and second-year calculus. It is perhaps expected that a common mathematical symbol would be interpreted universally. However, recent literature that addresses student interpretations of differentials, usually in the context of definite integration, suggests that this is not the case, and that many interpretations are possible. Reviews of textbooks showed that there was not a lot of discussion about differentials, and what interpretations there were depended upon the context in which the differentials were presented. This dissertation explores some of these issues. Since students may not have the experience necessary to …

A Doubly Nonlocal Laplace Operator And Its Connection To The Classical Laplacian, Petronela Radu, Kelseys Wells

Department of Mathematics: Faculty Publications

In this paper, motivated by the state-based peridynamic frame- work, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces addi- tional degrees of exibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connec- tions with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero.