Mathematics Commons^™

Symmetric Conditions For A Weighted Fourier Transform Inequality, Ryan Berndt

Mathematics Faculty Scholarship

We discuss conditions on weight functions, necessary or sufficient, so that the Fourier transform is bounded from one weighted Lebesgue space to another. The sufficient condition and the primary necessary condition presented are similar, one being phrased is terms of arbitrary measurable sets and the other in terms of cubes. We believe that the symmetry amongst the two conditions helps frame how a single condition, necessary and sufficient, might appear.

Operation Comics: The Story Continues, Bruce Kessler, Janet Tassell, Tressa Tullis

Mathematics Faculty Publications

During the 2008-2009 academic year, the author K. wrote three issues of Operation Comics, a comic book with embedded mathematics content appropriate for 4th through 6th grade students. Several printed comics were placed in Cumberland Trace Elementary in the Warren County School System in Bowling Green, Kentucky, US. The author Ta. was enlisted to measure the impact of the comics on the attitudes and motivation of the students using the comics. A preliminary report was given by K. at the 2009 Bridges Banff Conference, and the written report appeared in the proceedings. Since then, data has been collected on the …

Quantitative Reasoning In The Contemporary World, 3: Assessing Student Learning, Caren Diefenderfer

Mathematics & Statistics Faculty Scholarship

In this third paper in a series describing the Quantitative Reasoning in the Contemporary World course, the authors provide an adaptation of the Association of American Colleges and Universities quantitative literacy VALUE rubric. Describing achievement levels in six core competencies (interpretation, representation, calculation, analysis/synthesis, and communication), the resulting Quantitative Literacy Assessment Rubric (QLAR) is applicable to grading student work and has exhibited a high degree of reliability in two separate scoring tests (97% and 88% respectively). The distribution of the six core competencies across the 24 case studies in the authors’ quantitative reasoning casebook shows that interpretation, calculation, and analysis/ …

Modeling And Control For Heave Dynamics Of A Flexible Wing Micro Aerial Vehicle Distributed Parameter System, Lisa M. Kuhn

Doctoral Dissertations

In recent years, much research has been motivated by the idea of biologically-inspired flight. It is a conjecture of the United States Air Force that incorporating characteristics of biological flight into air vehicles will significantly improve the maneuverability and performance of modern aircraft. Although there are studies which involve the aerodynamics, structural dynamics, modeling, and control of flexible wing micro aerial vehicles (MAVs), issues of control and vehicular modeling as a whole are largely unexplored. Modeling with such dynamics lends itself to systems of partial differential equations (PDEs) with nonlinearities, and limited control theory is available for such systems.

In …

A Characterization Of Ramsey Graphs For R(3,4), Nicholas M. Richardson

Doctoral Dissertations

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4).

The process used to …

Research On Practical Rationality: Studying The Justification Of Actions In Mathematics Teaching, Patricio Herbst, Daniel Chazan

The Mathematics Enthusiast

Building on our earlier work conceptualizing teaching as the management of instructional exchanges, we lay out a theory of the practical rationality of mathematics teaching—that is, a theory of the grounds upon which instructional actions specific to mathematics can be justified or rebuffed. We do that from a perspective informed by what experienced practitioners consider viable but also in ways that suggest operational avenues for the study of instructional improvement, in particular for improvements that enable students to do more authentic mathematical work. We show how different kinds of experiments can be used to engage in theory building and provide …

Complex Learning Through Cognitively Demanding Tasks, Lyn D. English

The Mathematics Enthusiast

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s …

Conceptualizations And Issues Related To Learning Progressions, Learning Trajectories, And Levels Of Sophistication, Michael T. Battista

The Mathematics Enthusiast

In this paper the nature of learning progressions and related concepts are discussed. The notions of learning progressions and learning trajectories are conceptualized and their usage is illustrated with the help of examples. In particular the nuances of instructional interventions utilizing these concepts are also discussed with implications for the teaching and learning of mathematics.

On The Idea Of Learning Trajectories: Promises And Pitfalls, Susan B. Empson

The Mathematics Enthusiast

Learning mathematics is a complex and multidimensional if not an inherently indeterminate process. A necessary goal of research on learning is to simplify this complexity without sacrificing the ability of research to inform teaching. This goal has been addressed in part by researchers focusing on how to represent research on learning for teachers and on how to support teachers to use and generate models of students’ learning (e.g., Franke, Carpenter, Levi, & Fennema, et al., 2001; Hammer & Schifter, 2001; Simon & Tzur, 2004; Steffe, 2004). Recently, the idea of learning trajectories has gained attention as a way to focus …

Can Dual Processing Theories Of Thinking Inform Conceptual Learning In Mathematics?, Ron Tzur

The Mathematics Enthusiast

Concurring with Uri Leron’s (2010) cross-disciplinary approach to two distinct modes of mathematical thinking, intuitive and analytic, I discuss his elaboration and adaptation to mathematics education of the cognitive psychology dual-processing theory (DPT) in terms of (a) the problem significance and (b) features of the theory he adapts. Then, I discuss DPT in light of a constructivist stance on the inseparability between thinking and learning. In particular, I propose a brain-based account of conceptual learning—the Reflection on Activity-Effect Relationship (Ref*AER) framework—as a plausible alternative to DPT. I discuss advantages of the Ref*AER framework over DPT for mathematics education.

Tme Volume 8, Number 3

The Mathematics Enthusiast

No abstract provided.

A Numerical Method For Studying Thermal Deformation In 3d Double-Layered Thin Films With Imperfect Interfacial Thermal Contact Exposed To Ultrashort-Pulsed Lasers, Runzhou Liu

Doctoral Dissertations

Micro heat transfer induced by Ultrashort-pulsed lasers is an important research topic in mechanical engineering and material science. In order to apply ultrashort-pulsed lasers successfully, studying the thermal deformation in double-layered thin films with imperfect thermal interfacial contact induced by ultrashort-pulsed lasers is important for preventing thermal damage. For the ultrashort-pulsed laser, the thermal damage is different from that caused by the long-pulsed lasers, and ultrafast cracks occur after heating.

This dissertation presents a new finite difference method for investigating the thermal deformation in a 3D gold-chromium thin film with imperfect interfacial thermal contact exposed to ultrashort-pulsed lasers. The method …

A Characterization Of Ramsey Graphs For R(3,4), Nicholas M. Richardson

Doctoral Dissertations

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4).

The process used to …

Applications Of Variational Analysis To A Generalized Heron Problem, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr

Mathematics Research Reports

This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point …

Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and …

Practical Rationality, The Disciplinary Obligation, And Authentic Mathematical Work: A Look At Geometry, Deborah Moore-Russo, Michael Weiss

The Mathematics Enthusiast

Grossman and McDonald (2008) recently argued that the research community needs to move its “attention beyond the cognitive demands of teaching … to an expanded view of teaching that focuses on teaching as a practice (p. 185).” Building on the work of Bourdieu (Bourdieu and Wacquent, 1992; Bourdieu, 1985, 1998), Herbst and Chazan (2003, 2006) have written about mathematics teaching as a practice, just as law and medicine are considered practices, in an attempt to better understand the rationality that produces, regulates, and sustains mathematics instruction. This practical rationality is the commonly held system of dispositions or the “feel for …

Editorial: “Glocal”, “Glocavores”: Good Gadgetry?, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Mathematical Modeling And Dynamical Analysis Of The Operation Of The Hypothalamus - Pituitary - Thyroid (Hpt) Axis In Autoimmune (Hashimoto's) Thyroiditis, Balamurugan Pandiyan

Dissertations (1934 -)

This thesis is a mathematical modeling study of the operation of the negative feedback control through the hypothalamus-pituitary- thyroid (HPT) axis in autoimmune (Hashimoto's) thyroiditis. Negative feedback control through the HPT axis is a mechanism in which the high levels of thyroid hormone; free thyroxine (FT4) in the blood inhibits the secretion of the pituitary hormone, thyroid stimulating hormone (TSH) into the blood. Similarly, the low levels of free thyroxine (FT4) sensed by the pituitary gland and then it secretes thyroid stimulating hormone (TSH) into the blood. Autoimmune (Hashimoto's) thyroiditis is a disease in which the immune system turns against …

Springer Representations On The Khovanov Springer Varieties, Heather M. Russell, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H*(Xn). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H*(Xn) is irreducible in each degree. …

Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov

Articles

The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T^∗(T^N) to the flow momentum (X^∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The …

Using Performance Assessments To Connect Fractions And Rational Expressions: Noyce Scholars As Mentors To Pre-Service Elementary Teachers, Joy W. Darley, Michelle Cawthorn, Marlynn Griffin, Brian P. Koehler, James Lobue

Mathematical Sciences: Faculty Presentations (1991-2022)

At Georgia Southern University, Noyce scholars are not only being mentored, but they are also serving as mentors to preservice elementary teachers. One topic that proves to be problematic for many students is the conceptual understanding of fractions and rational expressions. Since our Noyce scholars with mathematics degrees will be teaching algebra, it is important that they are fluent with the arithmetic to algebra connection. In addition, it is crucial that these mathematics majors become stakeholders in mathematics education at the elementary school level.

Performance assessments can provide the structure necessary for assisting pre-service elementary teachers in firmly establishing the …

The Time At Which A Lévy Process Creeps, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We show that if a Levy process creeps then, as a function of u, the renewal function V (t, u) of the bivariate ascending ladder process (L⁻¹,H) is absolutely continuous on [0,∞) and left differentiable on (0,∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d⁻¹_H, the reciprocal of the (positive) drift of H. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As …

Path Decomposition Of Ruinous Behaviour For A General Lévy Insurance Risk Process, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We analyse the general Levy insurance risk process for Levy measures in the convolution equivalence class S^(alpha), alpha > 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u -> ∞, conditional on ruin occurring, under our assumptions. Existing asymptotic results under the S^(alpha) assumption are synthesised and extended, and proofs …

Path Decomposition Of Ruinous Behaviour For A General Lévy Insurance Risk Process, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We analyse the general Levy insurance risk process for Levy measures in the convolution equivalence class

S^(alpha), alpha> 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u → ∞, conditional on ruin occurring, under our assumptions. Existing asymptotic results under the S^(alpha)

assumption are synthesised and extended, and …

A Positive Monk Formula In The S1-Equivariant Cohomology Of Type A Peterson Varieties, Megumi Harada, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A_n−1, with respect to a natural S¹ -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of …

The Time At Which A Lévy Process Creeps, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We show that if a Levy process creeps then, as a function of u, the renewal function V (t, u) of the bivariate ascending ladder process (L⁻¹,H) is absolutely continuous on [0,∞) and left differentiable on (0,∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d⁻¹_H , the reciprocal of the (positive) drift of H. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As …

Income Inequality And Spatial Distribution : Firm Location, Product Quality And Welfare Of Poor., Namrata Gulati Dr.

Doctoral Theses

The key idea explored in this thesis is the following: though being poor in itself is a huge disadvantage, the situation might be influenced considerably by the type of neighbourhood the poor lives in as private establishments like educational institutions, health care facilities or credit institutions take both the location and income mix of people into account while making strategic decisions like whether to enter into the neighbourhood at all, and, upon entry, what price and quality to choose for their products and services. Is staying with the rich a virtue for the poor or a source of resentment? Are …

Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, James Mclaughlin, Andrew V. Sills

Mathematics Faculty Publications

No abstract provided.

Hamilton-Waterloo Problem With Triangle And C9 Factors, David C. Kamin

Dissertations, Master's Theses and Master's Reports - Open

The Hamilton-Waterloo problem and its spouse-avoiding variant for uniform cycle sizes asks if Kv, where v is odd (or Kv - F, if v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. This thesis examines the case in which r of the factors are made up of cycles of length 3 and s of the factors are made up of cycles of length 9, for any r and s. We also discuss a constructive solution to the general (m,n) case which fixes r and s.

Nonlocal Conditions For Differential Inclusions In The Space Of Functions Of Bounded Variations, Ravi P. Agarwal, Abdelkader Boucherif

Mathematics and System Engineering Faculty Publications

We discuss the existence of solutions of an abstract differential inclusion, with a right-hand side of bounded variation and subject to a nonlocal initial condition of integral type.