# Research

## Discrete Mathematics

### Monday, December 4, 2006

Title: Hurwitz Equivalence in Tuples of Generalized Quaternion Groups and Dihedral Groups
Speaker: Xiang-dong Hou
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

Let $$Q_2m$$ be the generalized quaternion group of order $$2^m$$ and $$D_N$$ the dihedral group of order $$2N$$. We classify the orbits in $$\left(Q_2m\right)^n$$ and $$(D_pm)^n$$ $$\left(p'\right)$$ under the Hurwitz action.

### Monday, November 27, 2006

Title: Questions About Dynamics of Membrane Systems
Speaker: Giuditta Franco
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

Membrane systems were introduced in 1998 as a distributed computational model inspired by the structure and the functioning of the living cell. Their computational power has been extensively investigated, while their feasibility as models of cellular and biochemical processes is lately receiving an increasing interest. In this context, it is still an open problem to find a suitable mathematical setup to describe membrane systems as (discrete) dynamical systems. Two possible approaches will be suggested, one based on linear operators (so called “stoichiometric matrices”) and the other one based on symbolic dynamics.

### Monday, November 20, 2006

Title: Subconstituent algebras of Latin square
Speaker: Ibtisam Daqqa
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

In this talk we are going to define a subconstituent algebra $$T(p)$$ of a Latin square $$L$$ with respect to a base point $$p$$. We will introduce the cycle structure of $$L$$ with respect to $$p$$. And see how one can span a $$T$$-module using a given cycle of order $$k$$. This cycle structure will play an important role in determining the isomorphism classes of $$T(p)$$.

### Monday, November 13, 2006

Title: String Pointer Reduction System: Formalization of gene assembly in ciliates
Speaker: Angela Angeleska
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

In this talk we give a short overview of a combinatorial model for DNA recombination in ciliates which are the unicellular organisms characterized by the presence of two nuclei in a single cell (macronucleus MAC and micronucleus MIC).

The assembly of MIC-gene into MAC-gene in ciliates might be viewed as a composition of three molecular operations that can be formalized through string rewriting rules. The string rewriting rules define a String Pointer Reduction System, which describes every posible gene recombination observed in rearrangements from MIC into MAC genes.

### Monday, November 6, 2006

Title: Building Block Approach to Porous Materials
Speaker: Mohamed Eddauodi, Chemistry Department, USF
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, October 30, 2006

Title: The Spectrum of a Pot With DNA Molecules and Related Problems
Speaker: Ana Staninska
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

A theoretical model of DNA self-assembly will be presented. For this model a problem is encoded in the molecules in the pot and a solution is represented by a complete complex (a complex that does not contain free sticky ends) of appropriate size.

In most experiments, a lot of useless material (non-complete complexes) also appears. To optimize the initial solution so as to minimize the amount of useless material at the end one needs to use proper proportion of molecule types. The set of vectors representing these proper proportions is called the “spectrum” of the pot.

The spectrum reveals much more information about the pot with DNA molecules, than just giving the proper proportion. It helps to classify the pots and to determine the minimal complete complexes.

I will present some already proved facts as well as problems that I am currently working on.

### Monday, October 23, 2006

Title: Minimal Generators of Zero-Dimensional Ideals
Speaker: Boris Shekhtman
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

Let $$F[x]$$ be the ring of polynomials in $$d$$ variables over the real or complex field $$F$$. A zero-dimensional ideal is an ideal in $$F[x]$$ of finite codimension (colength). I present bounds for the minimal number of generators for such ideals and “extreme cases”, that is the cases where bounds are actually archived. These questions popped up naturally (believe it or not) in Analysis.

### Monday, October 16, 2006

Title: Coloring Random Knots
Speaker: Enver Kardayi
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

I will discuss creating a random knot and computing its Determinant by using Maple and the distribution of non-trivial and $$p$$-colorable random knots for different stick numbers.

### Monday, October 9, 2006

Title: TBA
Speaker: Joni Piernot
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, October 2, 2006

Title: TBA
Speaker: Dr. Brian Curtin
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, September 25, 2006

Title: Isomorphisms and homeomorphisms of graphs
Speaker: Dr. Brian Curtin
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

We show that the isomorphism class of a graph $$G$$ is determined by the set $$\{(H,n)\mid H\text{ is a graph, }n\text{ is the number of homomorphic images of }H\text{ in }G\}$$. We use partition functions to encode the computation of $$n$$ into a polynomial, and then use some elementary invariant theory to study these polynomials.

### Monday, September 18, 2006

Title: Blueprints for Very Tiny Structures, Part II
Speaker: Dr. Greg McColm
Time: 3:00pm‐4:00pm
Place: PHY 108

### Monday, September 11, 2006

Title: Blueprints for Very Tiny Structures
Speaker: Dr. Greg McColm
Time: 3:00pm‐4:00pm
Place: PHY 108

#### Abstract

With chemists designing crystals, computer scientists carrying out DNA computations, and pharmacists creating new proteins, lots of scientists are now building nanostructures. Presumably, such architectural planning would involve blueprints of the final building. We present an algebraic system for such blueprints, and look at examples.