# Research

## Discrete Mathematics

### Monday, April 24, 2006

Title: On the Number of Inequivalent Binary Self-Orthogonal Codes
Speaker: Xiang-Dong Hou
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

Let $$\Psi k,n$$ denote the number of inequivalent binary self-orthogonal $$[n,k]$$ codes. We present a method which allows us to compute $$\Psi k,n$$ explicitly for a moderate $$k$$ and an arbitrary $$n$$. Included in this talk are explicit formulas for $$\Psi k,n$$ with $$k\le 5$$.

### Monday, April 17, 2006

Title: Spatial Graphs and Chemistrys
Speaker: Enver Kardayi
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

We will talk about spatial graphs and chromatic polynomials. We will go over Bing's conjecture about complete graphs and Yamada Polynomials. Finaly, we will discuss the chirality of spatial graphs.

### Monday, April 10, 2006

Title: Forbidding-enforcing graphs
Speaker: Daniela Genova
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

We propose a new way of defining classes of graphs based on boundary conditions. Forbidding conditions state that certain combinations of subgraphs are forbidden in a graph and enforcing conditions state that certain subgraphs induce larger subgraphs in the graph structure. A “forbidding-enforcing family” of graphs is specified as the set of graphs that satisfy such forbidding and enforcing conditions. The talk will include examples and some properties of these families.

### Monday, April 3, 2006

Title: Bose-Mesner algebra from Latin square
Speaker: Ibtisam Daqqa
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

We recall Latin square and a construction of a Bose-Mesner algebra from a Latin square. We show that this Bose-Mesner algebra has the so-called amorphous property.

### Monday, March 27, 2006

Title: A graphic representation of a pot with DNA molecules
Speaker: Ana Staninska
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

Junction DNA molecules with flexible branches self-assemble into larger complexes using weak hydrogen bonds. We approach this self-assembly process form a graph theoretical point-of-view. Given a pot of molecules, we assign a starlike graph to every molecule, a labeled multigraph to the complexes that can arise from the pot, and a labeled multigraph to the pot itself. This representation is used to determine what complexes can assemble from the molecules in the given pot.

### Monday, March 20, 2006

Title: Breaking Highgrade Ciphers in World War II: Working With Alan Turing
Speaker: Peter Hilton, Distinguished Professor Emeritus
Oxford University
Time: 10:00am‐10:50am
Place: LIF 268

#### Abstract

I will reminisce about the experience of working on the German Naval Enigma and Geheimschreiber (“Secret Writer”) during World War II. Concentrating on the Patter — the most sophisticated German coding machine — I will describe how the Germans made very serious mistakes which enormously facilitated our work.

I will also talk about the great logician Alan Turing, whose contribution to breaking Enigma was unique and decisive.

### Monday, March 6, 2006

Title: Quandle Cocycle Invariants and Tangle Embeddings
Speaker: Kheira Ameur
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

Quandles are sets with self-distributive binary operations that generalize the Fox-$$n$$ colorings. A quandle coloring along with a quandle cocycle can be used to define invariants for knots and knotted surfaces.

For some Alexander quandles we contruct polynomial type cocycles, we then use them to compute invariants for certain families of knots and their twist-spins. An interesting application is tangle embedding, where the cocycle invariant can be used as obstructions to embedding tangles in knots. We will define the cocycle invariant for tangles, and then compute the invariant for some tangles. By comparing the invariant values, informations can be obtained in which knots a given tangle can be embedded.

### Monday, February 27, 2006

Title: A Generalized Urn Model, Part II
Speaker: Kevin Wagner
Time: 10:00am‐10:50am
Place: PHY 109

### Wednesday, February 22, 2006

Title: A Generalized Urn Model
Speaker: Kevin Wagner
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

$$m$$ “$$-1$$” balls and $$p$$ “$$+t$$” balls are placed in an urn and drawn out randomly without replacement. Before any ball is drawn, a player decides whether to place a bet on the ball, the payoff being the value of the ball that is then drawn. The process continues until all balls are removed from the urn.

In part one, we will find an optimal betting strategy, and determine the expected gain $$G(m,p)$$ (or give suitable bounds) as a sum involving binomial coefficients. In the case where an exact formula is present, we will then transform the sum into one involving a binomial distribution.

In part two, we determine what $$G(m,p)$$ is asymptotically under various circumstances, via Stirling's formula, normal approximation, and such.

### Monday, February 13, 2006

Title: Numerical Calculation of Growth Rates for Some Random Fibonacci Sequences
Speaker: Edgardo Cureg
Time: 10:00am‐10:50am
Place: PHY 109

#### Abstract

We will consider the numerical determination of the growth rate of some random Fibonacci sequences.