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Mathematics & Statistics

# Colloquia — Fall 2000

## Friday, September 8, 2000

Title
Speaker
Time
Place

The Elementary Solution of Hilbert's Third Problem
David Benko
3:00pm-4:00pm
PHY 013
Masahiko Saito

Abstract

If two polygons have the same area, it is always possible to decompose one of them into a finite number of parts which can be rearranged to form the second polygon. This is the Bolyai-Gerwien theorem. One might ask whether this is still true for polyhedra; in fact, Farkas Bolyai and Gauss already asked this around 1830.

Hilbert raised this question again; it was the third problem of his celebrated list of 23 problems in 1900. The negative answer was first given by Max Dehn in the same year. Even today (after 170 years), the simplest proof of the problem uses Dehn's idea.

Not any more! In my talk I will show that the solution of this seemingly elementary question is indeed elementary. My proof does not use vector spaces or additive functions. In fact, the proof finally boils down to the pigeon hole principle...

## Monday, September 18, 2000

Title
Speaker

Time
Place

Image Modeling and Its Impact on Image/Video Compression, Metrics, and Indexing
Anastase Nakassis
National Bureau of Standards & Technology
3:00pm-4:00pm
PHY 013
Arunava Mukherjea

Abstract

There is a multiplicity of techniques (in use, former and actual wannabes) for image compression, metrics, and indexing. These techniques reflect modeling choices (most often those underlying signal processing) and exhibit the strengths and weaknesses of the underlying model.

The talk will be a survey of the state of the practice and of the technological challengers with emphasis on the underlying models and on the consequences of the modeling choices.

Examples (images,simulated clips, other) will be included as often as possible. Underlying mathematical theories (Fourier, Wavelets, Catastrophe theory/Topology, Fractals) will be briefly mentioned and explained in apopthegmatic/telegraphic style with emphasis on the properties that are relevant to image modeling. Proofs, details, and alternate interpretations will be left as an exercise to the audience.

Sparse references to the literature will be included.

## Friday, October 6, 2000

Title
Speaker

Time
Place

Reflections on Mirrors
Anthony Quas
University of Memphis
3:00pm-4:00pm
PHY 013
Nataša Jonoska

Abstract

We consider a random spatial process in which mirrors are placed randomly on the lattice at $$\pm 45^\circ$$ to the axes. A light is then shone from a point on the lattice and the resulting light beam is studied. The talk uses techniques from ergodic theory and percolation theory and these will be explained as needed. The results of computer experimentation will be presented.

## Friday, October 13, 2000

Title
Speaker

Time
Place

Biodiversity and Statistics: We Can't Save the Planet if we Don't Know What's Going On
A. K. Dewdney
University of Waterloo
3:00pm-4:00pm
PHY 013
W. Edwin Clark

Abstract

The term “biodiversity” has many meanings, mathematically speaking, no two the same. The outcome of biodiversity measures depends on what measure is being used, yet the preservation of natural areas may well hinge on such measurements. An unacknowledged crisis in theoretical biology has resulted from the lack of sound research in natural communities, particularly the patterns (i.e., distributions) of species abundance that prevail in them. I demonstrate the presence of a uniform $$J$$-shaped abundance distribution which I call the Logistic-$$J$$ Distribution. I show how I discovered it by computer, how I tested it against 100 randomly-selected biosurveys, and what it tells us about “biodiversity”, as well as the twin phenomena of speciation and extinction.

## Friday, October 27, 2000

Title
Speaker

Time
Place

On Hierarchical Bayesian Neural Networks: An Application to Prostate Cancer Study
Malay Ghosh
Distinguished Professor of Statistics
University of Florida
3:00pm-4:00pm
PHY 013
Arunava Mukherjea

Abstract

Prostate cancer is one of the most common cancers in American men. Management depends on the staging of prostate cancer. Only cancers that are confined to organs of origin are potentially curable. The paper considers a hierarchical Bayesian neural network approach for posterior prediction probabilities of certain features indicative of non-organ confined prostate cancer. The Bayesian procedure is implemented by an application of the Markov Chain Monte Carlo numerical integration technique. For the problem at hand, the hierarchical Bayesian neural network approach is shown to be superior to the one based on hierarchical Bayesian logistic regression model.

## Friday, November 3, 2000

Title
Speaker

Time
Place

Combinatorial Dynamics: Directed Graphs and Maps of the Interval
Ethan M. Coven
Wesleyan University
3:00pm-4:00pm
PHY 013
Nataša Jonoska

Abstract

Combinatorial Dynamics deals with periodic orbits of self-maps of one-dimensional spaces. The language and concepts of directed graphs are particularly useful for stating and proving results about periodic orbits of continuous self-maps of the interval. We will illustrate this by discussing “minimal combinatorial models” for some periodic point properties of continuous self-maps of the interval.

## Wednesday, November 8, 2000

Title
Speaker

Time
Place

A. W. Goodman
Professor Emeritus
3:00pm-4:00pm
PHY 013

Abstract

The talk and the topic uses only the elementary theory of numbers of the positive integers. If you wish to learn what a pure multigrade is you must attend the lecture for at least 5 minutes. The lecture will mention a number of open problems that may be attractive for future research.

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Speaker

Time
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