Friday, December 2, 2005
| Title |
On the Asymptotic Number of Non-Equivalent q-Ary Linear Codes |
| Speaker |
Xiang-Dong Hou |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
Two subspaces (linear codes) of Fq
n are called equivalent if one can be obtained form the
other through the action of a monomial matrix (a product of a permutation matrix
and a diagonal matrix). We confirmed a recent conjecture about the asymptotic
number of non-equivalent codes as n tends to infinity. If time permits,
we will also mention some related results.
Friday, November 18, 2005
| Title |
Graphs and Marriages |
| Speaker |
David
Nezelek |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
The marriage game concerns two disjoint sets of players, men and women, along
with each player's preferences among the acceptable members of the opposite set.
The object is to form stable marriages among the players, such that no two players
would prefer to leave their mates for each other. We use the directed-graph
approach proposed by Balinski and Ratier to demonstrate the existence of stable
marriages, and to explore other items of interest.
Friday, November 4, 2005
| Title |
Thresholds and the Creation of Random Structures |
| Speaker |
Greg McColm |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
Quite by accident, the models of random structures mathematicians looked at
first were models of random assembly. But there is another kind of model: the
model in which all the random elements appear at the beginning, and in which all
subsequent development is deterministic. We will look at a class of these, the
random geometric graphs — which can be problematic with weak thresholds, and
surprise us with strong ones.
Friday, October 28, 2005
| Title |
Weak Thresholds and the Simulation of Random Structures |
| Speaker |
Greg McColm |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
There are many models of random graphs, and the model of random strings
presented last week can simulate many of these. We will see some models of random
accretion, and see that by using the model of random strings, we can prove that
these models have weak thresholds. We will also look at some open questions.
Friday, October 21, 2005
| Title |
Weak Thresholds and the Evolution of Random Structures |
| Speaker |
Greg McColm |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
There are many extant models of random structures, and it is about time that we
start organizing them. Towards this end, we present a general model that subsumes
many other models of the random assembly of complex structures. For a theme, we
look at the question “at what point of the development does the partially
assembled structure satisfy criterion θ?” and we find that in dealing with at
least ballpark answers, this model provides some readily accessible results.
Friday, October 14, 2005
| Title |
Twist Numbers of Links from the Jones Polynomial |
| Speaker |
Matt Williamson |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
In this talk, I will present some basic information about knot theory, which I
learned over the summer for a Research Experience for Undergraduate program. My
research deals with twist numbers of links, and their relation to the Jones
Polynomial. In a paper written by Dasbach and Lin, the twist number of any
alternating knot was found to be encoded inside the Jones Polynomial. I extended
this theorem to include alternating links, and I also found a way to read two of
the Jones Polynomial coefficients from only using the diagram of the alternating
link.
Friday, October 7, 2005
| Title |
The Spectrum of the DNA Self-Assembly Model with Flexible
Tiles |
| Speaker |
Ana Staninska |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
I’ll talk about the spectrum of a given pot with DNA junction molecules, which
is another aspect of the model described by Dr. Jonoska in the previous two
meetings. In the final pot of an experiment, only complete complexes represent the
solution to a given problem while incomplete complexes are wasted material. With
careful selection of the proportion of each proto-tile added in the pot we can
annul the amount of incomplete complexes. The space of all the distribution
vectors of the proto-tile proportions is called the Spectrum of the pot type. The
properties of the spectrum will be the main topic of the talk.
Friday, September 30, 2005
| Title |
A Computational Model for Self-Assembling Flexible Tiles,
Part II |
| Speaker |
Natasa Jonoska |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Friday, September 23, 2005
| Title |
A Computational Model for Self-Assembling Flexible Tiles |
| Speaker |
Natasa Jonoska |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
A theoretical model for self-assembling tiles with flexible branches will be
introduced. An instance of a “problem” is encoded as a pot of such
tiles, and a “solution” is encoded as an assembled complete complex
without any free sticky ends (called ports), whose number of tiles is within
predefined bounds.
We prove that this model of self-assembly precisely captures NP-computability
when the number of tiles in the minimal complete complexes is bounded by a
polynomial.
Friday, September 16, 2005
| Title |
Leonard pairs and Leonard triples
|
| Speaker |
Brian Curtin |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
We begin by introducing the notion of a Leonard pair. We briefly discuss their
connections to the orthogonal polynomials in the Askey-scheme and examples arising
in representation theory.
We shall also discuss our work on the extension of Leonard pairs to Leonard
triples.
Friday, September 9, 2005
| Title |
Salient Properties of Scale-free Random Graph Processes |
| Speaker |
Dr. Zoran Nikoluski
Charles University
Prague |
| Time |
10:00-11:00 a.m. |
| Place |
PHY 120 |
Abstract
Recent empirical studies of the Internet and the WWW have shown statistical
similarities between these and other networks, as diverse as the network of
phone-calls, power network, citation, and the network of sexual contacts. The
extent to which these, so-called, scale-free (scale-invariant)
networks have pervaded, influenced, and conditioned the modern society, have
prompted the study of scale-free random graph processes. In this talk, we review
results pertinent to the three salient characteristics: scale-free degree
distribution, small average path length, and large clustering coefficient. We
define a fourth characteristic — degree-degree correlation — and
obtain a rigorous result using static description of the scale-free random graph
process per Barabasi and Albert. Implications of these salient properties to
reliable searching, fast data-communication, and strategies for quarantining a
disease, will also be addressed.