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Mathematics & Statistics

# Discrete Mathematics (Leader: Prof. Greg McColm)

## Monday, April 15, 2002

Title
Speaker
Time
Place

Monotonicity of Eigenvalues of Hermitian Matrices
9:00am-10:00am
PHY 120

Abstract

We discuss various discrete techniques to describe how the eigenvalues of a parameter-dependent Hermitian matrix change as a function of the parameter.

## Monday, April 8, 2002

Title
Speaker
Time
Place

Cocycle Knot Invariants, Alexander and Burau Matrices, Part II
Marina Appiou-Nikiforou
9:00am-10:00am
PHY 120

## Monday, April 1, 2002

Title
Speaker
Time
Place

Cocycle Knot Invariants, Alexander and Burau Matrices
Marina Appiou-Nikiforou
9:00am-10:00am
PHY 120

Abstract

We will talk about extensions of quandles by $$2$$-cocycles in relation to knot colorings. We will also demonstrate relations between cocycle invariants and Alexander and Burau matrices. Examples of the Whitehead link and Borromean rings will be given.

## Monday, March 25, 2002

Title
Speaker
Time
Place

Using Circulant Matrices to Solve Low-Level Polynomials
Karol McIntosh
9:00am-10:00am
PHY 120

Abstract

The idea is to construct a circulant matrix with a specified characteristic polynomial. Roots of the polynomial become eigenvalues which are trivially found for circulant matrices. This approach provides a beautiful unity to the solutions of quadratic, cubic, and quartic polynomials. This is a talk on a paper by D. Kalman & J. E. White.

## Monday, March 18, 2002

Title
Speaker
Time
Place

Distance-Regular Graphs, Part II
Brian Curtin
9:00am-10:00am
PHY 120

Abstract

We continue with a gentle introduction to distance-regular graphs. This second part will focus on the Bose-Mesner algebra of a distance-regular graph and its representations. We shall also discuss the “$$Q$$-polynomial” property (which is defined algebraically) and some its characterizations.

The script of the first part is available on my web page at http://www.math.usf.edu/~bcurtin/DRG1.pdf.

## Monday, March 4, 2002

Title
Speaker
Time
Place

Distance-Regular Graphs
Brian Curtin
9:00am-10:00am
PHY 120

Abstract

I will discuss some basic combinatorial and algebraic features of distance-regular graphs, objects central to my research. This talk is intended to be a gentle introduction to the subject. I will give some examples and briefly discuss some connections to other topics as well as presenting some basic facts.

## Monday, February 25, 2002

Title
Speaker
Time
Place

Software to Generate Codes for DNA Computations
David Kephart
9:00am-10:00am
PHY 120

Abstract

In DNA nontechnology and DNA based computations the design of DNA sequences that are error resistant is of essential importance. The set of all sequences that are generated by a biomolecular protocol forms a language over the four letter alphabet $$\{A,G,C,T\}$$. This alphabet is associated with a natural involution mapping $$h$$, $$h(A)=T$$ and $$h(G)=C$$. In order to avoid undesirable Watson-Crick bonds between the words (undesirable hybridization), the language has to satisfy certain variations of coding properties such as: being a prefix (suffix) code, comma-free code, and more particular for DNA, no involution of a word is a subword of another word, or no ivolution of a word is a subword of a composition of two words.

We will demonstrate the alpha-test version of software that can either (1) test whether a given set of code words satisfy any of these properties or (2) design a set of code words that satisfy certain coding properties defined by the user. Though the default alphabet for the program is the DNA alphabet of $$\{A,G,C,T\}$$ it is designed to accept (or generate) words over arbitrary alphabet and as such it can be used for a variety of applications.

Note: This work is joint with K. Mahalingam and N. Jonoska.

## Monday, February 18, 2002

Title
Speaker
Time
Place

Coloring Knots With a $$4$$-Color Palette
Masahiko Saito
9:00am-10:00am
PHY 120

Abstract

First I review a coloring scheme of knots and links. For a given $$4$$-colored link, I give a rule of changing the (underlying) colors by a yellow marker. Whether or not you can consistently change colors this way depends on a given link, and can be used to distinguish links.

Then I will discuss how to come up with such marker rules, and mention a relation to the quandle cocycle knot invariants.

## Monday, February 11, 2002

Title
Speaker
Time
Place

Games, Logics, and Complexity, Part III
Greg McColm
9:00am-10:00am
PHY 120

Abstract

We will conclude with a look at second order logic, the polynomial hierarchy, and other hallucinatory nonsense.

## Monday, February 4, 2002

Title
Speaker
Time
Place

Games, Logics, and Complexity, Part II
Greg McColm
9:00am-10:00am
PHY 120

Abstract

We continue with the game for PTIME, and move on to NLOGSPACE.

## Monday, January 28, 2002

Title
Speaker
Time
Place

Games, Logics, and Complexity
Greg McColm
9:00am-10:00am
PHY 120

Abstract

There are a number of logics that are associated with famous complexity classes. And there are a number of games that are associated with famous logics. This gives us an opportunity for some representation theorems that may enable to understand the complexity classes better.