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University of Alaska Fairbanks
Department of Mathematics and Statistics

2005-2006 Colloquium Series

Colloquiums are usually held on Thursdays at 1:00 in Chapman 106.

Spring Schedule

Fall Schedule

Abstracts

Perigees and distributive structure in the real-valued cycles of the 3x + 1 problem

Dixon Jones

University of Alaska Fairbanks

The 3x+1 problem goes like this: (1) Pick a positive integer. (2) If it's odd, multiply it by 3 and add 1; if it's even, divide it by 2. (3) Iterate step 2. The 3x+1 conjecture is that, for any initial positive integer, the iterates eventually reach the value 1, at which point the cycle 1 -> 4 -> 2 ensues. A further conjecture is that if we're allowed to start with a negative integer, the iterates eventually produce one of the values -17, -5, -1, 0, or 1, which are the perigees, or points of least magnitude, of the 3x+1 problem's five known integer cycles. Now more than 70 years old, this unsolved problem has generated a small industry of research---an annotated bibliography of 200 relevant papers is available on the arxiv.

In this talk we move the problem of 3x + 1 cycles from the integers to the real numbers. We look at a simple one-dimensional iterated function system that extends the 3x + 1 function to the real line, and exhibit a structure in which cycles are represented by their perigees. We show several interesting combinatorial aspects of this structure, involving staircase paths, words on two symbols, and the distribution of those symbols within a word. We conclude with some thoughts on how these results might be used to classify 3x + 1 cycles and thereby determine if other integer cycles exist.

The talk should be accessible to anyone who has taken undergraduate discrete math or abstract algebra.

A cycle.

Variational Analysis and Generalized Differentiation: New Trends and Developments

Boris Mordukhovich
Department of Mathematics
Wayne State University

Nonsmooth functions, sets with nonsmooth boundaries, and set-valued mappings naturally and frequently appear in various aspects of analysis. Constrained optimization, calculus of variations and its modern form of optimal control, stochastic and statistical problems, mathematical economics, etc., are among those areas of mathematics and its applications, where appropriate tools of generalized differentiation lead to essential achievements. New constructions of generalized differentiations have been recently developed in the framework of the so-called variational analysis, which has been recognized as a fruitful area in mathematics that, on one hand, concerns with the study of optimization-related problems and, on the other hand, applies variational methods to a broad spectrum of non-variational problems. Nonlinear systems and variational principles in physics, economics, and other applied sciences give rise to nonsmooth structures, and these are some of the prime motivations for the development of new forms of analysis.

This talk provides an overview of the basic principles, new trends and developments on the generalized differentiation theory with its various applications. It does not require any preliminary knowledge on the subject.

Some Statistical Issues with No Child Left Behind

Ellis Ott

Iowa State University

The No Child Left Behind Act mandates states to hold schools accountable for the performance of their students. All students are to reach proficiency in Reading and Math as measured by a state-administered test by the 2013-2014 school year. Specifically, Iowa has defined this proficiency as a student scoring at or above the 41st national percentile rank on the Iowa Test of Basic Skills (relative to a 2000 standardization group). To achieve Adequate Yearly Progress (AYP), schools must have a given percentage of proficient students in Math and Reading within every subgroup (by race, poverty status, disability, and 1st language). A school failing to meet AYP in two consecutive years is labeled as a "School in Need of Improvement."

Originally, AYP was determined using a single cutoff percentage that increased periodically. Recently, the Department of Education in the State of Iowa has obtained permission from the federal Department of Education to judge AYP for a single school's percentage of proficient students using a confidence interval. However, this confidence interval method has assumptions which are not met. Using Item Response Theory, the measurement source of error will be discussed and a research question for determining a more reasonable confidence interval for a school's proficiency percentage will be posed.

Schools that fail AYP are assumed to be schools of poor quality ("School in Need of Improvement"). However, the exam performance of a group of students in a school may not be solely due to the quality of the instruction and the school. External factors that are beyond the control of teachers, staff, and school officials could also impact student performance. A data set of 1400 schools in Iowa including demographics of the school, population demographics of the district, urban status, proportion of economically disadvantaged students in the school, teacher information for the district (experience, salary, etc,), and finance data for the district will be introduced and used to determine important factors in whether a school fails AYP.

Gaussian Process Models for the Sphere, with Application to the Rotation Measures of the near Galactic Sky

Margaret Short

Los Alamos National Laboratory

Our primary goal is to obtain a smoothed summary estimate of the magnetic field generated in and near to the Milky Way by using Faraday rotation measures (RM's). The ability to estimate the magnetic field generated locally by our galaxy and its environs will help astronomers distinguish local versus distant properties of the universe. Each RM in our data set provides an integrated measure of the effect of the magnetic field along the entire line of sight to an extragalactic radio source. RM's can be considered prototypical of geostatistical data on a sphere. In order to model such data, we employ a Bayesian process convolution approach which uses Markov chain Monte Carlo (MCMC) for estimation and prediction. Complications arise due to contamination in the RM measurements, and we resolve these by means of a mixture prior on the errors.

This represents joint work with Dave Higdon and Philipp Kronberg.

Ice Sheets and Obstacle Problems

Ed Bueler
Department of Mathematics and Statistics
University of Alaska Fairbanks

Accurate numerical simulation of large ice sheets, like those currently occupying Greenland and Antarctica but also including recently-departed sheets covering parts of North American and Eurasia, is prerequisite to understanding of long-term climate and dynamical earth processes. For problems of ice flow over long times (at least longer than days) and of large scale (but not mountain glaciers) the continuum description of ice flow within ice sheets is now reasonably standard. It is a "shallow" model in the sense that certain simplifications can be made using the small aspect ratio of the sheet.

Surprisingly, there is no existing well-posed formulation of the steady state problem: What is the configuration of a grounded ice sheet given a steady climate, that is, a steady spatially-dependent distribution of accumulation or ablation of snow/ice? Note that everyone in the field "knows how to get there", that is, to steady state, but not how to describe what you get or how to get there in one step!

This talk will describe some significant recent steps in the direction of such a well-posed formulation. It will turn out to be not just a PDE, but rather an "obstacle problem" with surprising mathematical properties. (The classical obstacle problem will indeed be described.)

The first half of the talk will be suitable for a general scientific audience.

On eigenfrequencies and zero sets of eigenmodes of the 2D sloshing problem

Vladimir Kozlov
Department of Mathematics
Linkoping University

I'll discuss properties of eigenfrequences and eigenmodes, in particiular a topology of their zero sets, of the 2D sloshing problem, which describes the free wave motion in a canal.

An Easy Solution to a Hard Problem that Looks Easy

Walt Tape
Department of Mathematics and Statistics
University of Alaska Fairbanks

It does not matter so much what the problem is; the fun is in seeing the easy solution. But here is the problem anyway: The point A and the surface S are fixed, with S being the part of a sphere that is visible from A. With ∠PAQ fixed, how do you position the points P and Q on S so as maximize the distance between them?

I care because the problem is equivalent to finding the maximum deviation of light produced by a given prism. And I care about that because ...

The Connected Mathematics Project and Multicultural Education

Anthony Rickard
Department of Mathematics and Statistics
University of Alaska Fairbanks

Multicultural education is an important issue in K-12 mathematics education. This presentation examines how and to what extent a problem-centered middle school mathematics curriculum addresses multicultural education and then discusses the effects of the curriculum on the mathematics achievement of diverse groups of students. Results include that the curriculum incorporates several categories of multicultural elements to address multicultural education. Moreover, the effects of the curriculum on the mathematics achievement of all students, especially diverse groups of students, are positive and well documented. Implcations of findings for future research will also be discussed.

Maximum Likelihood in Phylogenetic Inference

Elizabeth Allman
Department of Mathematics and Statistics
University of Alaska Fairbanks

Inferring evolutionary relationships between species is an important problem in evolutionary and systematic biology. One approach to this problem involves sequencing the DNA of a particular gene for each of the species of interest, and then using this DNA data to construct a 'phylogenetic tree' that describes evolutionary relationships encoded in these genes. In practice, this constructs a 'gene tree' and, only by extrapolation, perhaps a species tree. This talk will begin with an introduction and overview to some techniques of phylogenetic tree construction from aligned DNA sequences. Then, assuming a probabilistic model describing the mutation process, we describe some of the challenges, difficulties, and benefits of using the approach of maximum likelihood to the tree reconstruction problem.

Linear Forests, k-Ordered, and Pancyclic Groups

Ralph J. Faudree
Department of Mathematical Sciences
University of Memphis

Given integers k, s, t with 0 &le s &le t and k &ge 0, a (k, t, s)-linear forest F is a graph that is the vertex disjoint union of t paths with a total of k edges and with s of the paths being single vertices. Given integers m and n with k + t &le m &le n, a graph G of order n is (k, t, s, m)-pancyclic if for any (k, t, s)-linear forest F and for each integer r with m &le r &le n, there is a cycle of length r containing the linear forest F. If the paths of the forest F are required to appear on the cycle in a specified order, then the graph is said to be (k, t, s, m)-pancyclic ordered. If, in addition, each path in the system is oriented and must be traversed in the order of the orientation, then the graph is said to be strongly (k, t, s, m)-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k, t, s, m)-pancylic, as well as degree conditions that imply a graph is (strongly) (k, t, s, m)-pancylic ordered will be given. Examples showing the sharpness of the conditions will be described. Problems and open questions related to these conditions will be presented.
Please contact John Gimbel or David Maxwell if you are interested in giving a colloquium talk.

'04 - '05 Colloquium Schedule