The tenth meeting of the Carolina Mathematics Seminar took place at The Citadel, Charleston, SC. This is a list of our speakers, titles and abstracts.
Chris Rufty (Student at USC Salkehatchie)
Title: A Japanese Ladder Game, a simple version.
Abstract: In MAA FOCUS newsmagazine Vol.31, No.3 June/July 2011, Dr. Dougherty and Dr. Vasquez introduced a puzzle, known as Japanese Ladder Game. It is known that every Japanese ladder will provide a 1-1 mapping, and for any sequence at the bottom of the ladder, one can always find a minimum solution. By studying a simpler version, we were able to use our method to prove some of the old results. In this talk, we will introduce the way to play the game, and the basic Math related to this game. Finally, we will give two simple proofs of our theorems.
Kristopher Liggins (Student at Benedict College)
Title: On Generalization of Fibonacci Numbers.
Ralph Howard (USC Columbia)
Title: Integral Geometry with Applications to Geometric Inequalities.
Abstract: We outline a few of the basic results in integral geometry, and their application to geometric inequalities such as the Bonnesen inequality (a refinement of the isoperimetric inequality) and Fray-Milnor inequality on the total curvature of knots. If time permits some recent work on knot energy will be given. The talk should be accessible to undergraduates with a knowledge of vector calculus.
Chuck Groetsch (The Citadel)
Title: Inverse problems, von Neumann’s theorem, and stable approximate evaluation of unbounded operators.
Abstract: The mathematical framework for many linear inverse problems in the mathematical sciences requires the application of an unbounded operator to a vector which is not in its domain – an impossible task! We present a brief introduction to a general approach, based on a classical theorem of von Neumann, for addressing this difficulty.
Oleg Smirnov (College of Charleston)
Title: Lie Algebras, Groups Triple Systems: the Truth .
Abstract: In the talk we consider functorial connections between the categories of the Lie algebras, Lie groups, Lie triple systems, and Symmetric Spaces.
We present a full subcategory of graded Lie algebras which is equivalent to the category of triple systems and discuss a similar connection between Lie groups and symmetric spaces.
Mei Chen (The Citadel)
Title: Eigenpairs of Adjacency Matrices of Balanced Signed Graphs.
Abstract: In this talk, we present results on eigenvalues λ and their associated eigenvectors x of an adjacency matrix A of a balanced signed graph. A graph G =(V,E) consists of a set V of vertices and a set E of edges between two adjoined vertices. A signed graph is a graph for
which each edge is labeled with either + or -. A signed graph is said to be balanced if there
are an even number of negative signs in each cycle (a simple closed path).
Signed graphs were first introduced and studied by F. Harary to handle a problem in social psychology. It was shown by Harary in 1953 that a signed graph is balanced if and only if its vertex set V can be divided into two sets (either of which may be empty), X and Y, so that each edge between the sets is - and each within a set is +. Based on this fundamental theorem for balanced signed graphs, vertices of a balanced signed graph can be labeled in a way so that its adjacency matrix is well structured. Using this special structure, we find exactly all eigenvalues and their associated eigenvectors of the adjacency matrix A of a given balanced signed graph. We will present eigenpairs ( λ, x) of adjacency matrices of three types of balanced signed graphs: (1) graphs that are complete; (2) graphs with t vertices in X or in Y that are not connected; and (3) graphs that are bipartite
Joint work with Spencer P. Hurd of The Citadel.