**Emily Stone**

Email

# 2004 Colloquia

## Spring 2004

### May 7 - Mathematical Analysis of Facilitated Diffusion in Membrane Transport

**Mathematical Analysis of Facilitated Diffusion in Membrane Transport**Many nutrients that a cell needs are large and composed of lipid-insoluble molecules. These do not easily cross cell walls. In order to transport these large hydrophilic permeants, cells have specialized membrane-embedded proteins called carriers that by a conformational change, transport a specific permeant from one side of the membrane to the other. This mechanism is called facilitated diffusion.

In this talk we will study the mechanisms of facilitated diffusion from a mathematical perspective. Our main goal is to categorize each possible physiological situation with the appropriate mathematical model that fits the available data and known physiological constraints.

Friday, 7 May 2004

4:10 p.m. in Math 109

### May 6 - Mathematical Quantization and Operator Spaces

*Mathematical Quantization and Operator Spaces*After a brief discussion of quantization in physics, we will discuss the analogous program in mathematics---mathematical quantization. We will then explain how mathematical quantization applied to the theory of Banach spaces leads to the theory of operator spaces. In the second half of the talk, we will discuss three areas of operator space theory to which we have made contributions:

- One-sided multipliers and M-ideals of operator spaces.
- Algorithms for deciding complete positivity.
- Arveson's non-commutative Choquet boundary.

Thursday, 6 May 2004

4:10 p.m. in Math 109

### May 4 - Non-Local Dispersal and the Spatial Spread of Disease

**Non-Local Dispersal and the Spatial Spread of Disease**Invasion of diseases into new territory is a worldwide problem. Examples include West Nile fever in the US, HIV in Africa and Asia, and dengue in Latin America. Traditionally, the spatial spread of disease has been modeled using a local process, diffusion, to model dispersal. However, if dispersal is non-local, diffusion can greatly underestimate speeds of invasion. In this talk, I will discuss integrodifferential-equation models that incorporate knowledge about the dispersal of disease propagules and infected hosts to describe disease infection.

Tuesday, 4 May 2004

4:10 p.m. in Math 109

### April 29 - Multiplicatively spectrum preserving maps of function algebras II

**Multiplicatively spectrum preserving maps of function algebras II**Let *cl A* be a closed, point-separating sub-algebra of *C _{û}(X)C* , where X is a locally compact Hausdorff space. Assume that X is the maximal ideal space of

*clA*. If ƒ∈

*clA*, the set

*ƒ*(

*X*)∪{0} is denoted by σ(ƒ) . After characterizing the points of the Choquet boundary as strong boundary points this equivalence is used to complete the discussion initiated in a previous paper proving the

Main Theorem: If Φ : *cl A*→*cl A* is a surjective map with the property that σ(ƒ*g*)=σ(Φ(ƒ)Φ(g)) for every pair of functions ƒ, *g*∈*cl A* , then there is an onto homeomorphism Λ: *X*→*X* and a signum function *g*(*x*) on X such that Φ(ƒ)(Λ(*x*)) = *g*(*x*)ƒ(*x*) , for all *x* ∈ *X* and ƒ∈*cl A*.

Thursday, 29 April 2004

4:10 p.m. in Jour 304

### April 22 - Cauchy Regression and Confidence Intervals for the Slope

**Cauchy Regression and Confidence Intervals for the Slope**This paper uses computer simulations to verify several features of the Greatest Deviation (GD) nonparametric correlation coefficient. First its asymptotic distribution is used in a simple linear regression setting where both variables are bivariate. Second, the distribution free property of GD is demonstrated by using both the bivariate normal and bivariate Cauchy distributions. Third, the robustness of the method is shown by estimating parameters in the Cauchy case. Fourth, a general geometric method is used to estimate a ratio of standard deviations used in the confidence interval. The methods in this paper are an outgrowth of general research on the use of nonparametric correlation coefficients in statistical estimations. The results in this paper are not specific to GD and are appropriate for other rank based correlation coefficients. Part of this far-reaching research on correlation coefficient methods is available on Professor Gideon's website.

Thursday, 22 April 2004

4:10 p.m. in Jour 304

### April 15 - Discovering the structure of real-valued functions on binary strings

*Discovering the structure of real-valued functions on binary strings*This work addresses the problem of discovering the structure of a function from fixed length binary strings to the nonnegative reals when the function is given as a black box. The function is assumed to be a sum of component functions, where each component function depends on at most k bits (where k is less than the string length). An algorithm is given that finds the complete structure of the given function by sampling function values. Under the assumption that k is constant and that the number of component functions grows linearly with the string length, the complexity of this algorithm is shown to be *O(L ^{2}*log

*L)*function evaluations where L is the string length.

(This is joint work with Robert Heckendorn, Department of Computer Science, University of Idaho).

Thursday, 15 April 2004

4:10 p.m. in Jour 304

### March 19 - A Study of Post-Calculus II Students' Understandings of Indirect Proofs

**A Study of Post-Calculus II Students' Understandings of Indirect Proofs**Proving is an integral and indispensable part of mathematical activities. However, the ability to understand a mathematical proof involves many complex cognitive constructs-from interpreting the arguments and identifying the assumptions to recognizing the method.

I will discuss the methodology used and the objectives of some of the tools I developed in order to explore students' perceptions and understandings of certain aspects of indirect mathematical processes. Then I will elaborate on the findings in this research that showed students' lack of understanding of the deep structure of the method of contradiction, as well as many other aspects of indirect processes including finding counterexamples. what is taught in the calculus.

Friday, 19 March 2004

4:10 p.m. in Math 109

### March 18 - Approximating Irrationals by Means of Calculator Technology: Some Classical Examples.

*Approximating Irrationals by Means of Calculator Technology: Some Classical Examples.*This is a presentation that follows on the colloquium delivered at The University of Montana on November 7, 2003. Prepare to use a very powerful calculator to investigate multi-digit arithmetic (up to 600 significant digits) to generate approximations of irrational numbers. We will look at several classical algorithms in the history of mathematics and use graphing to study their convergence properties. In each case, the level of mathematics will not exceed what is taught in the calculus.

Note: Some TI 92s and TI Voyage 200s will be available for you to use during the talk.

Thursday, 18 March 2004

4:10 p.m. in Jour 304

### March 11 - Pollinators synchronize chaotic flowering in a perennial plant: Mathematical and statistical approaches to understanding ecological dynamics

**Pollinators synchronize chaotic flowering in a perennial plant:**

**Mathematical and statistical approaches to understanding ecological dynamics**In most plant species, reproduction is distributed bimodally over time, with high or low flowering years more common than "average" years. These fluctuations affect plant population dynamics, as well as populations of numerous animal species. In the past, ecologists have attempted to explain these patterns through correlations with weather or climate variables, with mixed success. Recently, theoretical ecologists have demonstrated that nonlinear dynamics of resource allocation to flowering at the individual level can make individuals flower at regular or erratic intervals. Given nonlinear dynamics within individuals, only small amounts of external forcing are necessary to generate population-level synchrony. I compared the ability of a number of resource models to explain alternate-year flowering in a perennial wildflower, Astragalus scaphoides (Bitterroot milkvetch). Pollen limitation in low-flowering years and differences among individuals in resource gain were necessary and sufficient to generate patterns seen in natural populations. More obvious drivers of synchrony, such as variation in temperature and precipitation, were not sufficient to generate synchrony. I compare these modeling results to statistical and experimental approaches to understanding flowering dynamics, and discuss implications for predicting flowering and fruiting in a broad array of plant species.

Thursday, 11 March 2004 **4:10 p.m. in Jour 304**

### March 9 - Integer Partitions and Composite Fermions

**Integer Partitions and Composite Fermions**Combinatorial mathematics is not frequently associated with quantum physics. However, work in one discipline can motivate investigations in the other and vice versa. A recent conjecture regarding allowed multiplets in the composite fermion model led to a proof of the unimodality of restricted partitions with duplicate or consecutive parts. This in turn, allowed the original physics conjecture to be verified. Using generating functions and the KOH theorem, this talk will follow the harmonic development of these two fields, show how to generalize the original physics results and make connections to recent breakthroughs investigating the fractional quantum hall effect.

Tuesday, 9 March 2004

4:10 p.m. in Math 109

### February 19 - The Voter Model with Confidence Levels

**The Voter Model with Confidence Levels**The voter model on a finite connected graph G is a stochastic process where each vertex has an opinion, 0 or 1. As time progresses, each voter's opinion is influenced by its neighbors. The voter model has been used to model the spread of opinions, as well as the spread of cultural ideas, geographic species dominance, consensus in computer networks, and spin states of atoms. We introduce a modification of the voter model that changes how quickly a voter will change its opinion based on its confidence in its opinion. We show that the voter model with confidence levels always results in a uniform opinion, and we determine the probability of each outcome (uniform 1 or 0) based on the initial opinions and the structure of the graph.

Thursday, 19 February 2004

4:10 p.m. in Math 109

### February 12 - Valuations in Algebraic Geometry

**Valuations in Algebraic Geometry**Let *k* be a field of characteristic zero, *K* an algebraic function field over *k*, and *V* a *k*-valuation ring of *K*. Zariski's theorem of local uniformization shows that there exist algebraic regular local rings *R*_{i } with quotient field *K* which are dominated by *V*, and such that the direct limit ∪*R*_{i}=*V*.

A fter giving the necessary background and definitions we discuss generalizations of Zariski's theorem and we give examples of valuations that arise naturally in Algebraic Geometry.

Thursday, 12 February 2004

4:10 p.m. in CP 109

## Fall 2004

### December 9 - Causal Discovery of Latent-Variable Models from a Mixture of Experimental and Observational Data

**Causal Discovery of Latent-Variable Models from a Mixture of Experimental and Observational Data**I will describe a Bayesian method for learning causal Bayesian networks through networks that contain latent variables from an arbitrary mixture of observational and experimental data. Observational data are passively observed. Experimental data, such as those produced by randomized controlled trials, result from the experimenter's manipulating, typically randomly, one or more variables and observing the states of other variables. I will also present Bayesian methods (including two new methods) for learning the causal structure and parameters of the underlying causal process that is generating the data, given that (1) the data contain a mixture of observational and experimental case records, and (2) the causal process is modeled as a causal Bayesian network. These learning methods were applied using as input various mixtures of experimental and observational data that were generated from the ALARM causal Bayesian network. In these experiments, the absolute and relative quantities of experimental and observational data were varied systematically. For each of these training datasets, the learning method was applied to predict the causal structure and to estimate the causal parameters that exist among randomly selected pairs of nodes in ALARM. I report how these structure predictions and parameter estimates compare with the true causal structures and parameters as given by the ALARM network. I show that one of the new methods for learning Bayesian network structure from a mixture of data, the implicit latent variable modeling method, is asymptotically correct and efficient.

Thursday, 9 December 2004

4:10 p.m. in Math 109

### December 2 - Topological Measures and Quasi-linear Maps

**Topological Measures and Quasi-linear Maps**A quasi-linear map on a space of continuous functions C(X) is one that is linear on each singly generated subalgebra of C(X). An analog of the Riesz Representation Theorem associates set functions called topological measures to such maps. These generalize the usual regular Borel measures and are linked to much deeper aspects of the topology of the underlying space X. I will outline the theory as it now stands, provide some standard examples, and mention several open questions.

Thursday, 2 December 2004

4:10 p.m. in Math 109

### November 15 - The Truth behind the Data: PISA 2003

**The Truth behind the Data: PISA 2003**In December of 2004 the new data for PISA mathematics will be released. A new horse race with some surprising results, maybe. The US will be again 'mathematically challenged', and other countries will be proud of their 'mathematical standing'.

But do the data tell the truth? Are the excelling countries really good? The data from studies like TIMSS and PISA do give that indication. But if one looks at the items and scores in the successful countries, and especially at the items that did not reach the study because they were too 'difficult' a very somber picture emerges.

Monday, 15 November 2004

4:10 p.m. in Skaggs 114

### November 4 - Hybrid (those taught partially on-line and partially in-class) Mathematics Courses-Feasibility and Samples

**Hybrid (those taught partially on-line and partially in-class) Mathematics Courses-Feasibility and Samples**Our last presentation addressed the feasibility (and inevitability) of mathematics courses taught completely on-line. This session we would like to talk about (present ideas and show actual examples) of a less than total on-line mathematics offering at the post-secondary level-that of the hybrid (part classroom, part on-line) mathematics course offering. Hybrid courses may address the needs of a more diversified student audience, as well as allow the possibility of offering higher level mathematics courses with internet components. Ryan has been included in our group because he can give information and answer questions of a more general nature about the present online course movement both in Montana and nationally in higher education.

Thursday, 4 November 2004

4:10 p.m. in Math 109

### October 21 - A Science & Math Curriculum Structure for Effective Student Learning

**A Science & Math Curriculum Structure for Effective Student Learning**State standards and benchmarks are identified and essential questions that serve as advance organizers for the instructional unit are developed, and used as the controlling device for the development and flow of the unit, lesson by lesson. We developed a novel formulation of a unit plan through a template that is based on essential questions and offers a range of teaching and learning strategies which promotes different types of thinking required for effective learning. The unit design and its lessons will be presented together with a full layout of examples in each of the following two applications:

Modernizing Montana Science Curriculum: Extremophiles mark the boundaries and robustness of life, and they are inherently an interesting part of Montana environments. In order to modernize science curriculum, supplement older textbooks, and customize curriculum to the natural environment of Montana, NASA materials related to the Extremophiles theme are used to design grades 5-8 science materials. A demonstration will be given of the electronically-disseminated materials which will be available to educators and students.

Augmenting Precalculus and Calculus textbooks: The same unit design is also used to construct precalculus and calculus materials. A demonstration of precalculus examples will be given.

Thursday, 21 October 2004

4:10 p.m. in Jour 304

### October 14 - Statistical Tools for Chemical Kinetics

**Statistical Tools for Chemical Kinetics**Modelling of chemical kinetics often leads to complex dynamical systems whose parameters are to be identified from experimental data. The accuracy of the model predictions should be estimated by statistical methods that take into account the noise in the data as well as possible limitations in modelling. While the models typically are strongly nonlinear, classical statistics is restricted to linear theory and may thus lead to misleading results. New MCMC (Markov chain Monte Carlo) methods allow a proper reliability analysis even for nonlinear models. Here we present applications of this Bayesian methodology to parameter estimation and optimal design of experiments. Examples are given from chemical kinetics as well as from biological modelling of algal mass occurrences in lake systems.

Thursday, 14 October 2004

4:10 p.m. in Jour 304

### October 7 - Use of Network Algorithms in Forestry Applications

**Use of Network Algorithms in Forestry Applications**This presentation will describe several forestry applications of network algorithms that can be used to optimize forest road locations, transportation routes, and timber harvesting layouts. Most forest transportation problems dealing with road construction and timber transportation can be solved as non-linear cost minimization problems. Approximate solution methods have been developed and implemented into several computer programs to efficiently solve the non-linear transportation problems. These computer programs include NETWORK2000, NETWORK2001, and CPLAN. NETWORK2000 solves variable and fixed cost, multiple period transportation problems, while NETWORK2001 has the ability to constraint road system length by using weighted objective function components. CPLAN is designed for simultaneously optimizing landing location, harvest equipment, logging profile, and road location using information from a GIS. Both the solution methods and computer programs will be introduced in this presentation.

Thursday, 7 October 2004

4:10 p.m. in Math 109

### September 30 - Mathematical Modeling of Heterogeneous Bio-Switches

**Mathematical Modeling of Heterogeneous Bio-Switches**Various biological systems that exhibit transitions between different possible stable steady states under influence of internal and/or external perturbations are usually modeled in terms of nonlinear differential equations with multiple equilibria. Ordinary differential equation models describe cases where fast mixing of isolated species (biological, chemical, etc.) occurs so that spatial dependence of species population/concentration changing in time can be neglected. In these systems perturbation of species population/concentration above a certain threshold level leads to a transition from one spatially uniform steady state to a new spatially uniform state. Such systems can be interpreted as homogeneous (spatially independent) bio-switches. When spatial dependence in the models is important we arrive at heterogeneous switches where the initiation of a transition from one stable equilibrium to another will depend on the type of boundary conditions imposed on a system (no flux conditions vs. fixed species population/concentration conditions), on the presence/absence of convection, as well as on the shape of the initial perturbation. The ideas behind mathematical modeling of heterogeneous bio-switches (i.e., the discussion of why transitions between various steady states occur, how the transitions are initiated, how the tune-ups of switches can be done to change the transition threshold values and to make transitions asymmetric, etc.) are going to be addressed in the presentation.

Thursday, 30 September 2004

3:50 p.m. in UC Theatre

This talk is a part of the UM-Toyo U Symposium on Bio-Nano Technology & Sciences.

### September 23 - Teaching Applied Mathematics Through Open-Ended Projects

**Teaching Applied Mathematics Through Open-Ended Projects**At Carroll College we have redesigned several upper division courses in applied mathematics, such as numerical methods, statistics, and optimization, to focus on open-ended projects. These projects can present students with many of the ambiguities and complexities that appear whenever we use mathematics in the real world, thus offering the students a variety of possible approaches, each with its own advantages and disadvantages. We have found that these projects teach students to become systematic in their explorations, more aware of the different methods that they are forced to choose from at each stage in the project, and that regular peer review helps them to view their own work more critically.

Thursday, 23 September 2004

4:10 p.m. in Math 109

### September 16 - Polytopal Graphs and Digraphs

*Polytopal Graphs and Digraphs*In the first half, I'll present some basic definitions and interesting facts about polytopes and their edge graphs. In the second half, I'll introduce the definition of a polytopal digraph and characterize them for certain special classes of polytopes. The talk should be reasonably self-contained and I'll do my best to convey why I think polytopes are such interesting objects to study.

Thursday, 16 September 2004

4:10 p.m. in Math 109

### September 10 - The Structure of Clawfree Graphs

**The Structure of Clawfree Graphs**A graphis said to be clawfree if it has no induced subgraph isomorphic to K1, 3. Line graphs are one well-known class of clawfree graphs, but there others, such as circular arc graphs and subgraphs of the Schläfli graph. It has been an open question to describe the structure of all clawfree graphs. Recently, in joint work with Paul Seymour, we were able to prove that all clawfree graphs can be constructed from basic pieces (which include the graphs mentioned above, as well as a few other ones) by gluing them together in prescribed ways. In this talk we will survey some ideas of the proof, and present examples of clawfree graphs that turned out to be of importance in the description of the general structure. We will also describe some new properties of clawfree graphs that we learned while working on the subject.

Friday, 10 September 2004

4:10 p.m. in Skaggs 117

Refreshments at 3:30 p.m. in Lobby

This talk is part of The Big Sky Conference, and is sponsored in part by the National Science Foundation & the Department of Mathematical Sciences.

### September 9 - Mathematics and Poker

**Mathematics and Poker**Television and on-line poker have brought about an incredible growth in poker. There is a spectrum of opinion ranging from the belief that poker is nothing more than gambling to the belief that poker is highly amenable to accurate mathematical analysis. This talk examines these beliefs and looks at some of the mathematical aspects of this fascinating game.

Thursday, 9 September 2004

8:00 p.m. in the Music Recital Hall

Refreshments following lecture in Lobby

This talk is part of The Big Sky Conference, and is sponsored in part by the National Science Foundation, The Presidents Office & the Department of Mathematical Sciences.

### September 9 - Groups and Graphs

**Groups and Graphs**Like most areas of mathematics, graph theory has an interaction with group theory. In this talk, we examine some of this interaction. We look at old results and new directions of research.

Thursday, 9 September 2004

4:10 p.m. in the UC Theatre

Refreshments at 3:30 p.m. in Lobby

This talk is part of The Big Sky Conference, and is sponsored in part by the National Science Foundation & the Department of Mathematical Sciences.