The ninth meeting of the Carolina Mathematics Seminar took place at University of South Carolina Sumter, Sumter, SC. This is a list of our speakers, titles and abstracts.
Shannon Michaels (Student at USC Sumter)
Title: A Relation Between Fibonacci Numbers and Lucas Numbers.
Abstract: In this talk I discuss some elementary properties of Fibonacci numbers. I will prove an identity that shows that the difference of two Fibonacci squares is a Lucas Number.
Thomas L. Fitzkee (Francis Marion University)
Title: Topology Explains Why Automobile Sunshades Fold Oddly.
Abstract: The article uses topology and abstract algebra to examine "automatic folding" sunshades that coil up when not in use. From the authors' experience, it seems impossible simply to fold such a sunshade in half (i.e. coil it into exactly two loops). The object here is to figure out how many loops can appear in the coil and to understand why.
Kevin Milans (USC Columbia)
Title: Subtrees with Few Labeled Paths.
Abstract: Consider a {0,1}-edge-labeling of the complete rooted ternary tree T of depth n. The edge labels along a path from the root to a leaf produce a bitstring of length n; such a bitstring is called a path label. For each complete binary subtree S of depth n, let L(S) be the set of path labels that occur along paths in S. We study the problem of finding a subtree S such that |L(S)| is small. The problem originated from a question in computability theory. This is joint work with Rod Downey, Noam Greenberg, and Carl Jockusch.
Jason Burns (USC Sumter)
Title: Extending Hadwiger's characterization theorem.
Abstract: In 1957, Hugo Hadwiger classified the continuous, rigid-motion invariant, finitely additive measures on convex sets in n-dimensional Euclidean space. Let me explain those buzzwords and persuade you this is important, then prove a 'baby' version and persuade you this is true, then talk about extensions to non-Euclidean space and persuade you there is more to be discovered.
Leandro Junes (USC Sumter)
Title: Convolutions on the Geometry of Fibonacci numbers.
Abstract: We define a discrete convolution C using Fibonacci numbers that acts on the Hosoya’s triangle. We prove that this convolution give rise to some counting theorems. Those theorems are used to count some words’ patterns in formal languages. In particular, we have found that that those theorems can be used to count the weight of non-decreasing Dyck paths. Work in progress in collaboraton with Rigo Florez.