**Emily Stone**

Email

# 1999 Colloquia

## Spring 1999

### February 4 -Isometries and Biholomorphy in Several Complex Variables

**Cancelled**

**Isometries and Biholomorphy in Several Complex Variables**Characterization of domains in Cn up to biholomorphic equivalence is one of the most active areas of research in SCV. From a functional analytic point of view, this problem may be converted to a question about the linear isometries of subspaces of certain Banach spaces. In this talk, we will discuss the connection between these two problems and give a characterization of isometries on certain important subspaces of the above Banach spaces.

Thursday, 4 February 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 1 - Exploiting Spatial Information for Improving Statistical Classification Methods and Error Analysis: Applications to Satellite Remote Sensing and Vegetation Mapping

**Exploiting Spatial Information for Improving Statistical Classification Methods and Error Analysis: Applications to Satellite Remote Sensing and Vegetation Mapping****and candidate for the Statistics position**

Satellite remote sensing has become an important tool in natural resource assessment and management during the past 10 years. However, the sheer quantity of data contained satellite imagery poses substantial difficulties for extracting usable and accurate information. Mathematics, and statistics in particular, have not been utilized to their full extent for analyzing satellite imagery, and so the area is rich in opportunities for statistical research. In this talk, new approaches to statistical classification and analysis of classification errors based on exploiting spatial information are discussed. Ongoing efforts aimed at assessing habitat suitability of the Salmon-Selway Wilderness for grizzly bears will be mentioned. The talk is intended to be accessible to a broad audience

Monday, 1 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 4 - Computing Geodetic Coordinates

**Computing Geodetic Coordinates**Much of this talk is intended to be accessible to anyone with about a year of any calculus including at least some proofs. The talk will explain the design of an algorithm to compute the geodetic latitude and altitude of a point (aircraft, spacecraft, or submarine) above or slightly under the surface of an oblate-spheroidal planet.

The algorithm specifications include the requirement of a mathematical proof that the algorithm will deliver a specified accuracy within a specified number of computer-arithmetic operations, taking into account the mathematical approximation of the algorithm and the rounding errors from the computer. This means that for each tolerance of accuracy epsilon, and for each tolerance of computer rounding delta, the proof must supply an integral number of operations, which guarantees results within epsilon of the exact value, even if each operations suffers from a perturbation of relative size at most delta.

With IEEE double-precision floating-point arithmetic, the current proof of the current version of the algorithm guarantees a millionth of a degree for the latitude and one centimeter in the altitude, for any point from the deepest ocean to the edge of the galaxy.

There is an "exact" solution by solving a quartic equation, but apparently no tractable upper bounds on the rounding errors.

The lesson is that practical projects for which accuracy is crucial may require not only calculus but also epsilon-delta proofs .

Thursday, 4 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 5 - How quickly do we generate the matrix algebra?

**How quickly do we generate the matrix algebra?****and candidate for the Algebra position**

*F*be a field and let

*M*be the full ring of

_{n}(F)*n*by

*n*matrices over

*F*. Let

*S*be a finite set of matrices generating

*M*as an algebra. In this talk, we look at how quickly such a set

_{n}(F)*S*generates

*M*. By this we mean, how many multiplications (among elements of

_{n}(F)*S*) are required to pick up a basis for the algebra?

Friday, 5 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 8 - Exact Bootstrap Moments of an L-estimato

**Exact Bootstrap Moments of an L-estimator****and candidate for the Statistics position**

Because many bootstrap problems are analytically intractable, the bootstrap is commonly viewed solely as a resampling technique. We show that for the broad class of statistics that are linear combinations of order statistics (L-estimators) exact analytic expressions for the bootstrap mean and variance can be obtained, eliminating the error due to bootstrap resampling. The expressions follow from direct calculation of the bootstrap mean vector and covariance matrix of the whole set of order statistics. We examine the non-negligible error of the resampling approach for estimating the bootstrap variance using some classical L-estimators such as the trimmed mean and the median on some real data. We also consider exact percentiles and moments of more general functions of order statistics.

Monday, 8 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 9 - Superalgebras and their uses

**Superalgebras and their uses****and candidate for the Algebra position**

I will demystify the term "superalgebra". The name has origins in physics (which will be described), and the objects themselves are of great importance to mathematics. Several fundamental questions about their algebraic structure are still unsolved.

I shall spend some time on familiarization by means of basic examples. After short excursions to outline the necessary historical background in mathematics and physics, the talk will turn to algebraic questions, my work in the area, and open problems.

Tuesday, 9 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 22 - Dual Model Robust Regression

**Dual Model Robust Regression****and candidate for the Statistics position**

In typical normal theory regression, the assumption of homogeneity of variances is often not appropriate. Instead of treating the variances as a nuisance and transforming away the heterogeneity, the structure of the variances may be of interest and it is desirable to model the variances. Aitkin (1987) proposes a parametric dual model in which a log linear dependence of the variances on a set of explanatory variables is assumed. Aitkin's parametric approach is an iterative one providing estimates for the parameters in the mean and variance models through joint maximum likelihood. Estimation of the mean and variance parameters are interrelated as the responses in the variance model are the squared residuals from the fit to the means model. When one or both of the models (the mean or variance model) are misspecified, parametric dual modeling can lead to faulty inferences. An alternative to parametric dual modeling is to let the data completely determine the form of the true underlying mean and variance functions (nonparametric dual modeling). However, nonparametric techniques often result in estimates which are characterized by high variability and they ignore important knowledge that the user may have regarding the process. Mays and Birch (1997) have demonstrated an effective semi-parametric method in the one regressor, single-model regression setting which is a "hybrid" of parametric and nonparametric fits. Using their techniques, we develop a dual modeling approach which is robust to misspecification in either or both of the two models. Examples will be presented to illustrate the new technique, termed Dual Model Robust Regression.

Monday, 22 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 23 - On the numerical solution of differential equations with algebraic constraints

**On the numerical solution of differential equations with algebraic constraints****and Numerical Analysis candidate for Position 3**

The numerical solution of differential equations subject to algebraic state constraints and invariants has become increasingly important in several fields of application. Research focused on solving such problems usually centres around the following question: How do you most efficiently utilize the explicit constraint in the discretization scheme of the differential equation? There are two usual lines of attack. The first attempts to build a discretization scheme that embodies the constraint in a direct way. The second augments a generic numerical scheme with additional techniques to enforce the constraints. However, there are significant differences between the corresponding numerical treatments of equalities and inequalities.

In this talk, I will take you from the sometimes tenuous safety of equality constraints to the wild frontiers of inequality constraints by means of several diverse and innovative applications.

Tuesday, 23 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 25 - Pattern formation in the Extended Fisher Kolmogorov equation

**Pattern formation in the Extended Fisher Kolmogorov equation****Remark**: Dr. Troy will also present a talk in the Chemistry Colloquium on Monday, March 29 at 4 p.m. in Chemistry 109 titled "Solutions of the one dimensional Ginzburg-Landau model of superconductivity".

Thursday, 25 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### March 26 - A Non-commutative Version of Polynomial Ring

**A Non-commutative Version of Polynomial Rings****and Algebra candidate for Position 3**

The Artin-Schelter regular (AS regular) algebras are the non-commutative analog to the polynomial ring *k[x _{1},...,x_{i}]* over a field

*k*. Because AS regular algebras exhibit many of the same features as their commutative cousin, they are a good starting place for non-commutative algebraic geometry. The classification of the 3-dimensional AS regular algebras has been the inspiration for much interesting mathematics. This talk will focus on a few examples to illustrate how a large family of 4-dimensional AS regular algebras has now been classified.

Friday, 26 March 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### April 1 - Connectivity for Binary Matroids and Graphs

**Connectivity for Binary Matroids and Graphs**Connectivity theory is one of the most important subjects in graph theory and matroid theory. There has been much interest in generalizing graph results to matroids, especially to binary matroids. In this talk, we will present several binary matroid connectivity results which generalize certain graph results. A very useful graph result of Mader states that if *C* is a cycle of a 3-connected graph *G* such that for all elements *x* of *C*, the deletion of *x* from *G* is not 3-connected, then *C* meets at least two vertices of degree three. We prove that this result is a special case of a connectivity result for binary matroids. Another well-known graph result of Halin states that a minimally 3-connected graph with *n* vertices has at least 2*n*+6 / 5 vertices of degree three. We prove a binary matroid generalization of this result. We use the concept of non-separating cocircuits in matroids.

Thursday, 1 April 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### April 22 - Invertible Polynomial Transformations

**Invertible Polynomial Transformations****and algebra candidate for position 3**

The *general affine group* *GA _{n}*(C) generalizes the more familiar general linear group

*GL*(C). In particular,

_{n}*GA*(C) is the set of one-to-one functions

_{n}*F*= (

*F*

_{1}, …,

*F*

_{n}) : C

*à C*

^{n}*such that each*

^{n}*F*is a polynomial in

_{i}*n*variables over C. Remarkably, such

*F*are also onto, and

*F*

^{-1}is an element of

*GA*(C). So

_{n}*GA*(C) is indeed a group, consisting of the invertible polynomial transformations of C

_{n}*, with*

^{n}*GL*(C) as a subgroup. The aim of this talk is to give an overview of what is known about this important and much-studied group.

_{n}The *Structure Theorem* for *GA _{2}*(C) gives a fairly complete understanding in this case. For

*n*>= 3, relatively little is known about the structure of

*GA*(C), except that it is amazingly complicated. For example, the

_{n}*tame*subgroup

*T*contained in

_{n}*GA*(C) is easy to define, and the Structure Theorem implies

_{n}*T*

_{2}=

*GA*(C), but it remains an open question whether

_{2}*T*=

_{n}*GA*(C) for any

_{n}*n*>= 3.

Naturally, one wishes to study certain kinds of group actions in which an "algebraic" group *G* acts "algebraically" on C* ^{n}*, since these give rise to embeddings of

*G*as a subgroup of

*GA*(C). Classically, the case in which

_{n}*G*is a

*reductive*group (like

*G*=

*SL*(C)) has been studied since the Nineteenth Century, and many positive results are known. The case in which

_{n}*G*is a

*unipotent*group (like

*G*= C

^{+}, the additive group of C) is not as well understood, though the importance of this case is widely recognized. Much of my own work has focused on actions of C

^{+}on C

*.*

^{n}Thursday, 22 April 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### April 27 - Introduction to Cellular Automata (CA)

**Introduction to Cellular Automata (CA)**Tuesday, 27 April 1999

4:10 p.m. in LA 106

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### April 30 - Controllability of Excitable Cells

**Controllability of Excitable Cells**Mathematical models of cell electrical activity typically consist of a current balance equation, channel activation (or inactivation) variables and concentrations of regulatory agents. These models can be thought of as nonlinear filters whose input is some applied current I (possibly zero) and output is a membrane potential V.

A natural question to ask is if the applied current I can be deduced from the potential V. For a surprisingly large class of models the answer to this question is yes.

Friday, 30 April 1999

4:10 p.m. in FOR 106

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### May 13 - The mathematics of problem solving in sport and nature

**The mathematics of problem solving in sport and nature**This talk is about the topics in a problem solving course which considers aspects of mathematics that encourage students to explore and create. The course covers magic squares, Russian multiplication, cube building, pattern exploration, the use of Excel to solve problems, basic operations with Mathematica, a casino visit, sport modeling, task creation, finance fun problems, oral math presentations, age charts, matrices and secret messages, 100! and other big numbers, counting irregular patterns, train shuntings, optimization, frog jumping, Pythagoras.

Thursday, 13 May 1999

4:10 p.m. in MA 109

Coffee/Tea/Treats 3:30 p.m. in MA 104 (Lounge)

### May 25 - Topological Structure of the Space of Composition Operators on H ^infinity

**Topological Structure of the Space of Composition Operators on H ^{infinity}**

Dr. Ruhan Zhao

Kyoto University, Japan

Components and isolated points of the topological space of composition operators on *H ^{infinity}* in the uniform operator topology are characterized. Compact differences of two composition operators are also characterized. With the aid of these results, we show that a component in

*C(H*is not in general the set of all composition operators that differ from the given one by a compact operator.

^{infinity})Tuesday, 25 May 1999

3:10 p.m. in MA 109

Coffee/Tea/Treats 2:30 p.m. in MA 104 (Lounge)

## Fall 1999

### September 9 - Mathematics, Computer Games and Gende

President's Lecture Series

**Mathematics, Computer Games and Gender**This talk is sponsored by The University of Montana President's Lecture Series and is the keynote address for the Big Sky Conference.

Thursday, September 9, 1999, 8:00pm

### September 9 - The Impact of User Interfaces in Educational Software

**The Impact of User Interfaces in Educational Software**This talk is sponsored by and presented as an element of The University of Montana President's Lecture Series. The Big Sky Conference is also sponsored by the National Science Foundation and the Department of Mathematical Sciences.

### September 10 - An Algorithmic Approach to Multicolourin

**An Algorithmic Approach to Multicolouring**Multicolouring is the assignment of sets of colours to the vertices of a graph, so that sets on adjacent vertices are disjoint. Weights on the vertices prescribe the cardinality of the colour sets. Multicolouring was first studied in the context of the polyhedral approach to graph colouring. The application of graph colouring to frequency assignment in cellular networks has given a new impulse to the study of multicolouring for its own sake.

While graph colouring is known to be hard in general, the situation can be different when we restrict ourselves to special classes of graphs. In the case of multicolouring, we can use the structure of the underlying graph to guide our algorithm. We will show a number of approximation algorithms with constant performance ratio for multicolouring on certain types of graphs. We focus on graphs derived from subgraphs of the triangular lattice, the so-called hexagon graphs, since such graphs arise naturally in the context of cellular networks. We will also discuss variations of the graph colouring problem that arise from the frequency assignment problem

The Big Sky Conference is sponsored by the National Science Foundation and the Department of Mathematical Sciences.

Friday, 10 September 1999

4:10 p.m. in Continuing Education 203/204

Treats at 3:30 p.m. in the same

### September 23 - Mineral Leaching, and Clean Green Applied Mathematics

**Mineral Leaching, and Clean Green Applied Mathematics**This naturally raises the question of where the acid goes once it has been injected into the rock, and what percentage of it can actually be recovered. This talk will present a new design strategy, that can in principle recover all the injected acid. This is expected to be of considerable economic and environmental benefit. The solution makes use of a rather novel Green function combined with a numerical solution, and shows how powerful applied mathematics can be as a design tool.

Thursday, 23 September 1999

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (lounge)

### October 14 - Using Interdisciplinary Applications to Integrate Mathematical Topics in the Undergraduate Curriculum

**Using Interdisciplinary Applications to Integrate Mathematical Topics in the Undergraduate Curriculum**In the first two years of our undergraduate mathematics program, single and multivariable calculus are integrated with discrete dynamical systems, linear algebra, probability and statistics, ordinary and partial differential equations, and systems of differential equations. This program is offered as a sequence of four 5-semester-credit courses. There are four class meetings and one computer lab session per week.

Thursday, 14 October 1999

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (lounge)

### November 18 - Random Ramblings on Graph Pebbling

**Random Ramblings on Graph Pebbling**Given a connected graph *G*, and a distribution of pebbles to the vertices of *G*, a pebbling step consists of removing 2 pebbles from a vertex *v* and placing one pebble on a neighbor of *v*. The “lost pebble” could represent the cost of a computation. For a particular root vertex *r*, the distribution is *r-solvable* if it is possible to place a pebble on *r* after a finite number of pebbling steps. The distribution is *solvable* if it is *r*-solvable for every* **r*. The pebbling number *f(G)* is the least integer *t *so that every distribution of *t* pebbles onto the vertices of *G* is solvable. Thus, starting with *f(G)* pebbles --- even if placed by the devil --- guarantees solvability. What if we place the pebbles at random and ask only for an almost sure guarantee? This introductory talk will explore these ideas and questions, revealing their connections with familiar mathematical ideas.

Thursday, 18 November 1999

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (lounge)

### November 23 - Market Offering Strategies for Hydro-electric Generators

**Market Offering Strategies for Hydro-electric Generators**In the recent years, many countries have switched to a market mechanism for determining the price of electricity. Such markets have also been instituted in parts of the US including the Pacific northwest. We'll consider the problem of offering electricity, produced by a price-taker hydro generator operating a river chain, into a central market.

The market model is a simplified version of the New Zealand wholesale electricity market. The prices are modeled by an independent sequence or a Markov process.

Tuesday, 23 November 1999

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (lounge)