Mathematics Commons^™

Orthogonal Polynomials, George Gevork Antashyan

Theses Digitization Project

This thesis will show work on Orthogonal Polynomials. In mathematics, the type of polynomials that are orthogonal to each other under inner product are called orthogonal polynomials. Jacobi polynomials, Laguerre polynomials, and Hermite polynomials are examples of classical orthogonal polynomials that was invented in the nineteenth century. The theory of rational approximations is one of the most important applications of orthogonal polynomials.

Solutions To A Generalized Pell Equation, Kyle Christopher Castro

Theses Digitization Project

This study aims to extend the notion of continued fractions to a new field Q (x)*, in order to find solutions to generalized Pell's Equations in Q [x] . The investigation of these new solutions to Pell's Equation will begin with the necessary extensions of theorems as they apply to polynomials with rational coefficients and fractions of such polynomials in order to describe each "family" of solutions.

Monomial And Permutation Representation Of Groups, Rebeca Maria Blanquet

Theses Digitization Project

The purpose of this project is to introduce another method of working with groups, that is more efficient when the groups we wish to work with are of a significantly large finite order. When we wish to work with small finite groups, we use permutations and matrices. Although these two methods are the general methods of working with groups, they are not always efficient.

Cassini Ovals As Elliptic Curves, Nozomi Arakaki

Theses Digitization Project

The purpose of this project is to show that Cassini curves that are not lemniscates, when b does not equal 1, represent elliptic curves. It is also shown that the cross-ratios of these elliptic curves are either real numbers or represented by complex numbers on the unit circle on the conplex plane.

Non-Genera Of Curves With Automorphisms In Characteristic P, Darren B. Glass

Math Faculty Publications

We consider which integers g and r can occur respectively as the genus and p-rank of a curve defined over a field of odd characteristics p which admits an automorphism of degree p.

Prouhet-Tarry-Escott Problem, Juan Manuel Gutierrez

Theses Digitization Project

The purpose of this research paper is to gain a deeper understanding of a famous unsolved mathematical problem known as the Prouhet-Terry-Escott Problem. The Prouhet-Terry-Escott Problem is a complex problem that still has much to be discovered. This fascinating problem shows up in many areas of mathematics such as the study of polynomials, graph theory, and the theory of integral quadratic forms.

Ua94/6/9 Student / Alumni Personal Papers Wku Lena Annis, Wku Archives

WKU Archives Collection Inventories

Records created by and about Lena Annis. Series includes term papers and subject notebooks for English, Mathematics, Geography and Education. Also included are Class of 1931 reunion materials.

The Role Of Concavity In Applications Of Avery Type Fixed Point Theorems To Higher Order Differential Equations, Abdulmalik A. Altwaty, Paul W. Eloe

Mathematics Faculty Publications

In this article we apply an extension of an Avery type fixed point theorem to a family of boundary value problems for higher order ordinary differential equations. The theorem employs concave and convex functionals defined on a cone in a Banach space. We begin by extending a known application to a right focal boundary value problem for a second order problem to a conjugate boundary value problem for a second order problem. We then extend inductively to a two point boundary value problem for a higher order equation. Concavity of differentiable functions plays a key role in the application to …

Visualizations And Intuitive Reasoning In Mathematics, Kajsa Bråting

The Mathematics Enthusiast

On the basis of historical and didactical examples we consider the role of visualizations and intuitive thinking in mathematics. Examples from the 17th and 19th century have been used as well as smaller empirical studies at upper secondary school level and university level. We emphasize that a mathematical visualization does not reveal its intended meaning. With experience we can learn to interpret the visualization in different ways, depending on what is asked for.

If Not, Then What If As Well- Unexpected Trigonometric Insights, Stanley Barkan

The Mathematics Enthusiast

In performing an exercise of “What if not”, one can end up with a paucity of structure. Adding alternative structure can be a rich source of discovery, as we present here. The framework of this presentation is the original voyage of discovery, from a trivial geometric problem to the derivation of some unexpected trigonometric formulae based on regular polygons. The original “voyage” has been changed only sufficiently to make the text readable.

Reflections On Teaching With A Standards-Based Curriculum: A Conversation Among Mathematics Educators, Jill Newton, Ricki Geller, Lindsay Umbeck, Lisa Kasmer

The Mathematics Enthusiast

Many teachers and researchers have written about the challenges inherent in adopting new teaching practices in mathematics classrooms (e.g., Chazan, 2000; Clarke, 1997; Heaton, 2000). The authors of this article, all with secondary mathematics teaching experience, are convinced by research suggesting that Standards-based mathematics curricula are beneficial for student learning.1 However, the first three authors had not used such curriculum materials in their own classrooms, and we desired experience using a Standards-based mathematics curriculum with secondary students. To this end, we taught a week-long summer course with a focus on linear functions to high school students who had previously struggled …

Stability And Clustering Of Self-Similar Solutions Of Aggregation Equations, Hui Sun, David Uminsky, Andrea L. Bertozzi

Mathematics

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρ _t = ∇ · (ρ∇K * ρ) in Rd , d ⩾ 2, where K(r) = r ^γ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math.70, 2582–2603 (Year: 2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the …

The Generalised Zakharov-Shabat System And The Gauge Group Action, Georgi Grahovski

Articles

The generalized Zakharov–Shabat systems with complex-valued non-regular Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studied. This study includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary equations, solvable by the inverse scattering method, and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures. The results are illustrated on the example of the multi-component nonlinear Schr¨odinger (MNLS) equations and the corresponding gauge-equivalent multi-component Heisenberg ferromagnetic (MHF) type …

Ramanujan Sums As Supercharacters, Christopher F. Fowler '12, Stephan Ramon Garcia, Gizem Karaali

Pomona Faculty Publications and Research

The theory of supercharacters, recently developed by Diaconis-Isaacs and Andre, can be used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.

Review: The Dirichlet Space: A Survey, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Absolute Differentiation In Metric Spaces, W. J. Charatonik, Matt Insall

Mathematics and Statistics Faculty Research & Creative Works

In this article, we introduce a new notion of (strong) absolute derivative, for functions derived between metric spaces, and we investigate various properties and uses of this concept, especially regarding the geometry of abstract metric spaces carrying no other structure.

Coverings And Matchings In R-Partite Hypergraphs, Douglas S. Altner, J. Paul Brooks

Statistical Sciences and Operations Research Publications

Ryser's conjecture postulates that for r -partite hypergraphs, τ ≤ (r - 1)ν where τ is the covering number of the hypergraph and ν is the matching number. Although this conjecture has been open since the 1960s, researchers have resolved it for special cases such as for intersecting hypergraphs where r ≤ 5. In this article, we prove several results pertaining to matchings and coverings in r -partite intersecting hypergraphs. First, we prove that finding a minimum cardinality vertex cover for an r -partite intersecting hypergraph is NP-hard. Second, we note Ryser's conjecture for intersecting hypergraphs is easily resolved …

Visualizing Chaos, Andrew Nicklawsky

Mathematics Student Work

An important piece of information when dealing with a polynomial in the complex plane is its roots, the value or values of x for a given function f such that f(x)=0. Iterative root finding methods, such as Newton’s method, are utilized to discover an approximate value when these values cannot be explicitly solved. This process can be graphically represented for complex-valued functions and has been achieved with relative ease on a 2-Dimensional plane. However, this process can also be embodied on a sphere through the method of stereographic projection, which has not been attempted. In this research, I worked with …

Non-Abelian Groups With Perfect Order Subsets, Hongying Zhao

Mathematics Student Work

The purpose of this paper is to explore non-abelian finite groups with perfect order subsets. A finite group is said to have perfect order subsets (POS) if the number of elements of each given order can divide the order of the group. The study of such groups was initiated by Carrie E. Finch and Lenny Jones. In this paper, we construct POS-groups by considering semi-direct products of cyclic groups (and sometimes quaternions).

On The Quantization Of Zero-Weight Super Dynamical R-Matrices, Gizem Karaali

Pomona Faculty Publications and Research

Solutions of the classical dynamical Yang-Baxter equation on a Lie superalgebra are called super dynamical r-matrices. A super dynamical r-matrix r satisfies the zero weight condition if

[h ⊗ 1 + 1 ⊗ h, r(λ)] = 0 for all h ∈ ɧ, λ ∈ ɧ ∗ .

In this paper we explicitly quantize zero-weight super dynamical r-matrices with zero coupling constant for the Lie superalgebra gl(m, n) . We also answer some questions about super dynamical R-matrices. In particular, we prove a classification theorem and offer some support for one particular …