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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, April 25, 2011

Title
Speaker
Time
Place

Short Circuits in the Brain
Apurva Bhatty
3:05pm-3:55pm
PHY 108

Abstract

The brain's connectivity diagram is a directed graph with cell bodies as vertices and axons and dendrites as directed edges. While the complete directed graph of the human brain's circuits has not been fully defined or understood, short characteristic circuits (subgraphs) have been identified and partially understood in humans, insects, and other animals. I will describe three key circuits in asynchronous neural information processing (detection and discrimination):

1. coincidence detection and delay lines;
2. lateral inhibition;
3. grouping into generalization and specialization,

and how they are connected together in visual and auditory perception and motor control, with examples of sound localization and sound recognition, stereo-depth-perception, spatial perception and looming (obstacle avoidance) response in human and insect circuits.

## Monday, April 18, 2011

Title
Speaker
Time
Place

A Report on a Knot Quandle Coloring Project, Part II
Masahiko Saito
3:05pm-3:55pm
PHY 108

Abstract

This is a continuation of Edwin's talk. I would like to go over some of the actual outputs obtained, and make observations and comments. The actual data are intriguing in many aspects. For example, the distributions of certain types of quandles are polarized depending on the order of quandles. The data I plan to go over include characterizing the 431 connected quandles in the list, which quandles give the same coloring numbers for all knots in the list, and relations to extensions of quandles by cocycles. More open questions and conjectures from outputs will be discussed.

## Monday, April 11, 2011

Title
Speaker
Time
Place

A Report on a Knot Quandle Coloring Project
W. Edwin Clark
3:05pm-3:55pm
PHY 108

Abstract

This concerns a project that Masahico Saito, Mohamed Elhamdadi, Tim Yeatman and I are working on. As is known, given a quandle $$Q$$ then the number $$N(Q,K)$$ of “colorings” of a knot $$K$$ by $$Q$$ is a knot invariant. If $$Q$$ is generated by one of the colorings then $$Q$$ must be a connected quandle. Now, thanks to Leandro Vendramin, we have a complete list of the 431 finite connected quandles of orders $$< 36$$. Also we have available from various places descriptions of 12965 knots with up to 13 crossings. Our goal is in part to find and analyze a $$431\times 12965$$ matrix $$M$$ with $$M(i,j)=N(\text{quandle}(i),\text{knot}(j))$$. I will discuss our progress so far and some apparently open questions that this project provokes. The talk will be more or less self-contained.

## Monday, April 4, 2011

No seminar this week.

## Monday, March 28, 2011

Title
Speaker
Time
Place

Discrete Wavelets and Image Compression
Catherine Bénéteau
3:05pm-3:55pm
PHY 108

Abstract

Wavelet theory was an immensely popular research area in the 1990’s that brought together ideas from people working in completely different areas such as electrical engineering, physics, mathematics, and computer science. In mathematics, the subject attracted researchers from real and harmonic analysis, statistics, and approximation theory. Applications of wavelets turn up in lots of different places such as identifying art or handwriting forgeries, JPEG2000 image compression, and FBI fingerprint storage algorithms, among others. In this talk, I will discuss Discrete Wavelet transforms, how they can be used in image compression, and their connections to the famous breakthroughs of Ingrid Daubechies in the late 1980s and early 1990s.

## Monday, March 21, 2011

Title
Speaker

Time
Place

On charts with three crossings representing $$2$$-knots
Teruo Nagase and Akiko Shima
Tokai University
Japan
3:05pm-3:55pm
PHY 108

Abstract

A chart is an oriented labeled graph in a disk satisfying some conditions. A chart represents an embedded oriented surface in $$4$$-space. We introduce results about charts with three crossings. And we prove the first useful lemma and introduce properties of $$k$$-minimal charts.

## Monday, March 7, 2011

Title
Speaker
Time
Place

Deterministic Acoustic Models
Apurva Bhatty
3:05pm-3:55pm
PHY 108

Abstract

Reflections occur in a deterministic fashion, and for small values of $$n$$ ($$n$$ reflections), the deterministic model is an accurate representation of the physical interaction of sound and media. Simulation of pressure waves and sound propagation in complex environments and inhomogeneous media can model the effects of TBI (trauma to the brain and skull) and changes in diffusion (DTI) activity in the brain. Discrete and graph theory mathematical algorithms and techniques can help to segment and divide the work of these medical simulations onto scalable modular software that can take advantage of parallel processing in multi-core CPUs and in many-core GPU Chips for speed-up in numerical modeling. I will describe discrete approaches to real-time acoustic simulation and of perceptual modeling in synthetic soundscapes and environments for 3D audio.

## Monday, February 28, 2011

Title
Speaker
Time
Place

Automorphism Groups of Alexander Quandles
Xiang-dong Hou
3:05pm-3:55pm
PHY 108

Abstract

An Alexander quandle is a module $$M$$ over $$\mathbb{Z}\left[t,t^{-1}\right]$$ whose quandle operation is defined by $$x*y=tx+(1-t)y$$, $$x,y\in M$$. The automorphism group of such a quandle, previous unknown, is now determined.

## Monday, February 21, 2011

Title
Speaker
Time
Place

Synchronizing words and splicing
Nataša Jonoska
3:05pm-3:55pm
PHY 108

Abstract

Contextual cross-over operation on words, called splicing, is motivated by certain biomolecular processes and has been of interest in formal language theory since 1987 when it was introduced by T. Head. Splicing of words models the recombination obtained by restriction enzyme cleavage of DNA and ligation.

It was observed in 1991 by K. Culik and T. Harju that the set of words generated by a finite set of splicing rules (finite set of enzymes) applied to a finite set of initial words (molecules) belongs to the class of regular languages, however, the general characterization of such generated languages, called splicing languages, remains elusive. It has been conjectured that the minimal deterministic automaton of every splicing language must have a synchronizing word (i.e., it must have a word such that all paths in the automaton labeled with this word end at a unique vertex). We prove that this conjecture is true.

This is joint work with Paola Bonizzoni, U. of Milano-Bicocca.

## Monday, February 14, 2011

Title
Speaker
Time
Place

Completeness and Representability in Boolean Algebras
Joseph Van Name
3:05pm-3:55pm
PHY 108

Abstract

Given a cardinal $$k$$, a Boolean algebra is $$k$$-complete if every subset with cardinality less than $$k$$ has a least upper bound. $$k$$-complete algebras of sets are defined similarly. We shall discuss a canonical way to represent $$k$$-complete Boolean algebras as $$k$$-complete algebras of sets.

## Monday, February 7, 2011

Title
Speaker
Time
Place

On the Structure of Finite Latin Quandles
W. Edwin Clark
3:05pm-3:55pm
PHY 108

Abstract

A quandle is a pair $$Q=(X,*)$$ where $$X$$ is a set and $$*$$ is a binary operation on $$X$$ satisfying for all $$x,y,z$$ in $$X$$:

1. $$x*x=x$$ [$$X$$ is idempotent]
2. $$x\to x*y$$ is a bijection for all $$y$$ [$$X$$ is a right quasigroup]
3. $$(x*y)*z=(x*z)*(y*z)$$ [$$X$$ is right distributive]

If we take $$X=\{1,\dotsc,n\}$$ then $$(X,*)$$ may be represented as an $$n\times n$$ matrix $$A$$ where $$A(i,j)=i*j$$. Thanks to Ricardo Restrepo and James McCarron, we now have files of all quandles (up to isomorphism) of orders 1 through 9. For example, there are 11,079 isomorphism classes of quandles of order 9. Looking at these leads to lots of conjectures. For example, we note that the matrix $$A$$ of some quandles is a Latin square, that is, each element appears exactly once in each row and each column. Some people have called such quandles, Latin quandles.

One way to construct a Latin quandle is to find an automorphism $$f$$ of an abelian group $$(G,+)$$ such that $$1-f$$ is also an automorphism and define $$*$$ on $$G$$ by $$g*h=f(g)+(1-f)(h)\text{ for }g,h\text{ in }G.$$ Sam Nelson conjectured that all Latin quandles can be constructed in this way.

Using results known to the community of non-associative algebraists for over half of a century, I will show that Nelson's conjecture is false and discuss what is known about this problem as time permits. It is closely related to commutative Moufang loops which are generalizations of abelian groups. In fact the smallest commutative Moufang loop that is not an abelian group has order 81.

In particular, we show how to construct all commutative quandles up to order 728 and how to fix Nelson's conjecture in the case where $$Q$$ and $$Q^{(op)}$$ are both quandles. [$$Q^{op}$$ is $$Q$$ with the multiplication reversed.]

Thanks to Mohamed Elhamdadi and Jennifer MacQuarrie for stirring up my interest in this subject and to Larry Dunning, Apurva Bhatty, and James McCarron for stimulating discussions on these matters. And, special thanks to Michael Kinyon for telling me about Belousov's work and the relationship to commutative Moufang loops.

## Monday, January 31, 2011

Title
Speaker
Time
Place

Thresholds for Stopping Times
Greg McColm
3:05pm-3:55pm
PHY 108

Abstract

Given a monotone increasing random process, a “stopping time” is the time that a property is satisfied. For example, if a process consists of a graph randomly acquiring edges, one stopping time would be the number of edges acquired when it finally becomes connected. A “threshold” for a given property is a device for predicting the appearance of a property; and some properties are far more predictable than others. We present a hierarchy of types of thresholds for such properties.