Mathematics Commons^™

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 …

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 …

In Defense Of Frivolous Questions, Gizem Karaali

Pomona Faculty Publications and Research

Is there any reason for today's academic institutions to encourage the pursuit of answers to seemingly frivolous questions? The opinionated business leader who does not give a darn about your typical liberal arts classes "because they do not prepare today’s students for tomorrow's work force" might snicker knowingly here: Have you seen some of the ridiculous titles of the courses offered by the English / literature / history / (fill in the blank) studies department in the University of So-And-So? Why should any student take "Basketweaving in the Andes during the Peloponnesian Wars"? Just what would anyone gain from …

Unitary Equivalence To A Complex Symmetric Matrix: Low Dimensions, Stephan Ramon Garcia, Daniel E. Poore '11, James E. Tener '08

Pomona Faculty Publications and Research

A matrix T∈M_n(C) is UECSM if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. We develop several techniques for studying this property in dimensions three and four. Among other things, we completely characterize 4×4 nilpotent matrices which are UECSM and we settle an open problem which has lingered in the 3×3 case. We conclude with a discussion concerning a crucial difference which makes dimension three so different from dimensions four and above.

On The Matrix Equation Xa + Ax_T = 0, Stephan Ramon Garcia, Amy L. Shoemaker '14

Pomona Faculty Publications and Research

The matrix equation XA+AX^T=0, which has relevance to the study of Lie algebras, was recently studied by De Terán and Dopico (Linear Algebra Appl. 434 (2011), 44–67). They reduced the study of this equation to several special cases and produced explicit solutions in most instances. In this note we obtain an explicit solution in one of the difficult cases, for which only the dimension of the solution space and an algorithm to find a basis of this space were known previously

On The Closure Of The Complex Symmetric Operators: Compact Operators And Weighted Shifts, Stephan Ramon Garcia, Daniel E. Poore '11

Pomona Faculty Publications and Research

We study the closure $\bar{CSO}$ of the set $CSO$ of all complex symmetric operators on a separable, infinite-dimensional, complex Hilbert space. Among other things, we prove that every compact operator in $\bar{CSO}$ is complex symmetric. Using a construction of Kakutani as motivation, we also describe many properties of weighted shifts in $\bar{CSO} \backslash CSO$. In particular, we show that weighted shifts which demonstrate a type of approximate self-similarity belong to $\bar{CSO}\backslash CSO$. As a byproduct of our treatment of weighted shifts, we explain several ways in which our result on compact operators is optimal.

Review: Classification Of Four And Six Dimensional Drinfel'd Superdoubles, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Lusztig Symmetries And Automorphisms Of Quantum Superalgebras, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.