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# Discrete Mathematics (Leader: Prof. Greg McColm <mccolm (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'mccolm' + '&#64;' + 'usf.edu' + '">Prof. Greg McColm</a>');)

## Monday, November 29, 2010

No seminar this week.

## Monday, November 22, 2010

No seminar this week.

## Monday, November 15, 2010

Title

Speaker
Time
Place

Hierarchical Self Assembly: Physical Implementation of Substitution Rules to Produce Quasicrystalline Patterns
3:05pm-4:05pm
LIF 263

Abstract

Substitution rules efficiently produce quasicrystalline patterns such as those found in the Penrose and Robinson tilings. A few simple modifications to the Tile Assembly Model allow tiles to follow a hierarchical self assembly pathway that can effect substitution rules in a physical assembly process. I will introduce these modifications along with a tile set that can self assemble into Robinson patterns via a hierarchical process. I will also discuss how these tiles could be built using DNA origami and DNA strand exchange mechanisms.

## Monday, November 8, 2010

Title
Speaker
Time
Place

Characterizing the Dynamics of an Asynchronous Automaton on a Family of Graphs
Apurva Bhatty
3:05pm-4:05pm
LIF 263

Abstract

Parallel distributed algorithms can be modeled as the application of automata (with $$k$$ states) on a single component undirected graph structure, $$G$$ with $$n$$ vertices. The dynamics of parallel asynchronous processing can be described as the transitions of the global state of the automata on the graph, defined by a directed graph $$G_d$$ with $$k^n$$ vertices. I will describe and define a polynomial which can encode the stability of automata over a family of graphs, along with a combinatorics technique for calculating (in polynomial time) the sizes of the congruence classes defined by the number of unstable vertices over the graph. This allows the characterization of the directed graph $$G_d$$ by the number of directed edges in it.

## Monday, November 1, 2010

Title
Speaker

Time
Place

Statistical Models for Social Networks: Biased Nets vs. Exponential Random Graphs
John Skvoretz
USF Department of Sociology
3:05pm-4:05pm
LIF 263

Abstract

I review two types of statistical models for social networks: ones rooted in the random and biased net theory first proposed by Rapoport in the 1950s and ones in the exponential family based on the Hammersley-Clifford Theorem applied to network data. In the context of social science, models of the first type are theoretical models while models of the second are methodological models. In either case the basic challenge raised by network data is massive interdependency among the random variables defining the presence or absence of a tie. The two families of models approach the problem differently but seek to incorporate similar effects determining the outcome of tie formation processes. Several specific models are set out to illustrate the types of effects commonly assumed to govern the formation of social ties.

## Monday, October 25, 2010

The seminar was cancelled this week.

## Monday, October 18, 2010

Title
Speaker
Time
Place

Assembly Graphs and Assembly Polynomials
Egor Dolzhenko
3:05pm-4:05pm
LIF 263

Abstract

Assembly graphs are used to model gene recombination in ciliates. We associate an assembly polynomial to each assembly graph. In the talk, I will discuss a few basic properties of the assembly polynomials and show their relationship to circle graphs.

## Monday, October 11, 2010

No seminar this week.

## Monday, October 4, 2010

Title
Speaker
Time
Place

Negative binomials: properties and applications
3:05pm-4:05pm
LIF 263

Abstract

We will discuss the recursive, asymptotic, and other properties of the negative binomial coefficients. The recent pure binomial results and some applications involving general binomial and exponential inequalities, hypergeometric series, special functions and conformal maps will be presented.

## Monday, September 27, 2010

Title
Speaker

Time
Place

Modeling Enforcement Mechanisms with Security Automata
Jay Ligatti
USF Computer Science Engineering Department
3:05pm-4:05pm
LIF 263

Abstract

This talk will present recent work on modeling security mechanisms as automata. The mechanisms monitor the runtime behavior of software and modify that behavior when it violates a mechanism's policy. We will discuss the mechanisms' operational semantics and the transition function it implies. After defining a model for security automata, which includes definitions of automaton traces, policies, and policy enforcement, we will briefly analyze the sorts of policies these security automata can enforce.

## Monday, September 20, 2010

Title
Speaker
Time
Place

A Manifesto for Reticular Geometry, Part II
Greg McColm
3:05pm-4:05pm
LIF 263

Abstract

In our previous episode, we decided that a good place to start devising a “reticular geometry” was with CW complexes. After a brief review, we will look at how the tools of geometric group theory and other even more sinister areas can be adapted to address the objects of reticular geometry.

## Monday, September 13, 2010

Title
Speaker
Time
Place

A Manifesto for Reticular Geometry
Greg McColm
3:05pm-4:05pm
LIF 263

Abstract

By “Reticular Geometry”, we mean the study of the articulation of many geometric components and their assembly or arrangement into larger structures. Reticular geometry, like “prose”, is something many people (mathematicians and otherwise) have been doing for a long time without treating it as a field in itself. However, some polemical chemists suggest that there is a demand for reticular geometry as a coherently organized body of knowledge developed by some kind of scholarly community. We will look at what this demand consists of, and what resources mathematicians may offer in the near future.

## Tuesday, September 7, 2010

Title
Speaker

Time
Place

Notes on unknotting operations
Takuji Nakamura
Department of Engineering Science
Osaka Electro-Communication University
Osaka, Japan
3:30pm-4:00pm
NES 102

Abstract

A knot is a knotted loop in a $$3$$-dimensional space. For a given knot, When we want to know how much it is being knotted we consider the number of times to apply a certain operation to untie it. We call such an operation an unknotting operation. In this talk, we will introduce several unknotting operations and show recent results about “sharp unknotting operation”.

Title
Speaker

Time
Place

Open book decomposition of $$3$$-manifold and intersection numbers of arc system
Ryosuke Yamamoto
College of Science and Engineering
Ritsumeikan University
Kyoto, Japan
4:00pm-4:30pm
NES 102

Abstract

Every closed orientable $$3$$-manifold has a structure called open book decomposition, which is determined by an orientable surface with boundary and an automorphism of the surface fixing the boundary pointwise. Such an automorphism is completely described by an arc system on the surface and its image of the automorphism. We will focus on (geometric and algebraic) intersection numbers of arcs with their images and discuss relations between such intersection numbers and some topological properties of open book decomposition.
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