Doctoral Dissertation Defense

Digraphs and Homomorphisms: Cores, Colorings, and Constructions

Mike Severino, University of Montana

Wednesday, May 14, 2014 at 9:10 am in Math 312

Abstract (PDF)

Presentation of Master’s Project

The Impact of ‘Tipi Geometry’, A Lesson Combining Elementary Geometry Concepts and Blackfeet Tipi Building, on Preservice K-8 Teachers

Elizabeth Lask, University of Montana

Monday, May 12, 2014 at 1:10 pm in Math 108

Abstract (PDF)

Presentation of Master's Project

Progress on the 123-Conjecture

Cody Fouts, University of Montana

Wednesday, May 7 at 3:10 pm in Math 108

Abstract (PDF)

Presentation of Master's Project

The Character Table of PGL(2,q)

Daisy Matthews, University of Montana

Tuesday, May 6 at 4:10 pm in Math 211

Presentation of Master's Project

Mathematics Anxiety and the Nontraditional Student

Jack Lelko, University of Montana

Tuesday, May 6 at 2:10 pm in Math 108

Abstract (PDF)

Colloquium

A short construction of highly chromatic digraphs without short cycles

Mike Severino, University of Montana

Monday, May 5, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

A natural digraph analogue of the graph-theoretic concept of an 'independent set' is that of an 'acyclic set', namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets. In the spirit of a famous theorem of P. Erdős [Graph theory and probability, Canad. J. Math., 11:34-38, (1959)], it was shown probabilistically in [D. Bokal et al., The circular chromatic number of a digraph, J. Graph Theory, 46(3): 227-240, 2004] that there exist digraphs with arbitrarily large girth and chromatic number. Here I give a construction of such digraphs.

Colloquium

Number Theory in Several Complex Variables

Dusty Grundmeyer, University of Michigan

Wednesday, April 30, 2014 at 4:10 p.m. in Math 103
5:00 p.m. Refreshments in Math Lounge 109

We will discuss mapping problems in complex analysis with a focus on developing connections to other fields of mathematics. In particular we will study a family of canonical CR mappings that have many intriguing number-theoretic and combinatorial properties.

Presentation of Master's Project

Investigating and Assessing Student Interest in a Math Seminar Class

Adam Clinch, University of Montana

Tuesday, April 29 at 2:10 pm in Math 305

Abstract (PDF)

Colloquium:

Minerva: Big Exoplanet Science with Small Telescopes

Nate McCrady, Department of Physics & Astronomy, University of Montana

Monday, April 21 at 3:10 p.m. in Math 103

The Kepler mission has identified over 3000 candidate planets in the past three years, adding to the over 800 confirmed planets from radial velocity (RV) surveys. One of the most striking results of these surveys is that the number of planets increases rapidly with decreasing size. It is apparent that there are more small, rocky planets in the Galaxy than stars. These planets must be common around nearby stars, though few have yet been discovered. Finding these planets requires high precision RV measurements and high cadence observing to densely sample the orbital phase. Project Minerva is a robotic observatory dedicated to detection of rocky planets in the habitable zone around nearby stars. The observatory will consist of four 0.7-m telescopes that will use fiber optics to simultaneously feed a stable spectrometer to perform an intense campaign of precise velocimetry on the brightest, nearest, Sun-like stars. I will present simulated Minerva observations to estimate our expected exoplanet yield and habitable zone planet detections.

Colloquium:

Some Questions on Hex

Ryan B Hayward, Department of Computing Science, University of Alberta

Monday, April 14 at 3:10 p.m. in Math 103

Hex is the classic connection game invented by Piet Hein and, independently, John Nash. For years, Hex has intrigued gamers, mathematicians, and computer scientists. For nxn boards there exists a winning first-player strategy, but (other than for small boards) the win/loss value of particular opening moves is not known. Recently, the winning value of two 10x10 opening moves was found. I will describe how, and then discuss some questions:

Hein designed Hex to be fair. Did he succeed? How close are computers to solving 11x11 Hex?

An expert offers you 10-1 odds to play 11x11 Hex. To make it interesting, she offers you first move, plus a 1-stone handicap (so you can play two stones on your first move). How much should you wager?

You have 8 hours to write a program to play Hex. What algorithm should you use?

Colloquium & Doctoral Dissertation Defense

Bayesian Estimator Assessment Methods for Minimizing Costs in Multivariate Driving Performance Studies

Clark Kogan, University of Montana

Monday, March 24 at 3:10 p.m. in Math 103

Reduced alertness and high levels of cognitive fatigue due to sleep loss bring forth substantial risks in today’s 24/7 society. Biomathematical models can be used to help mitigate such risks by predicting quantitative levels of fatigue under sleep loss. These models help manage risk by providing information on the timing at which high levels of fatigue will occur; countermeasures can then be taken to reduce accident risk at such critical times.

Biomathematical models of fatigue predict cognitive performance based on homeostatic and circadian processes. Such models have typically been fitted to group average data. Due to large individual variation, group-average predictions are often inaccurate for any given individual. However, individual differences are trait-like. Between-subjects variation can therefore be captured by individualizing model parameters. These parameters may be estimated using the technique of Bayesian forecasting to combine new individual data with prior distributions that have been pre-specified using population data. In many cases the amount of data collected on the individual at hand, and consequently, the prediction accuracy, will be limited by factors such as the availability of data and cost of collecting it. However, prediction accuracy may be improved by including information from alternative, correlated performance measures in a multivariate Bayesian forecasting framework. Investigation of this latter technique is the topic of this thesis.

When collecting data from two performance measures, we consider how to minimize the cost of data collection while meeting a desired average level of prediction accuracy. We extend a commonly used measure of prediction accuracy, the mean squared error (MSE), by integrating over observed data values to create a uniquely determinable accuracy measure for specific parameterized Bayesian models with fixed data collection strategies. We call this new measure the marginal MSE. We derive the marginal MSE of the prediction accuracy for a general Bayesian linear model.

To understand how the marginal MSE depends on the number of measurements from primary and secondary tasks in the simplest case, we specify the accuracy for the bivariate Bayesian linear model of subject means. For this simple model, we further assume that observations from each performance measure have a fixed cost per data point, and use this assumption to determine the number of measurements of each variable needed to minimize the cost while still obtaining no less than the desired level of accuracy.

To aid the extension of the findings from the linear case to state-of-the-art nonlinear biomathematical fatigue models, we focus on obtaining our extended measure of accuracy for the nonlinear case. Computing this accuracy analytically is often infeasible without reliance on model approximations. Model simulations can be used to compute the accuracy; however, such simulations can be time consuming, especially for models that lack analytic solutions and require that a system of differential equations be solved to produce model dynamics.

Much of this computational burden in assessing estimator accuracy, however, is produced by using the Bayesian minimum mean squared error (MMSE) estimator, and could be reduced by taking advantage of the more rapidly computable Bayesian maximum a posteriori (MAP) estimator. We show for a nonlinear biomathematical model that the accuracy assessment using repeated simulation with the MAP estimator yields a reasonable estimate of the accuracy obtained using the MMSE estimator.

Still, for any given case, determination of whether the MMSE accuracy can be approximated with the MAP accuracy requires these time-consuming simulations. We begin to analytically identify classes of models where the MMSE accuracy can be approximated by the MAP accuracy. We consider a class of quadratic Bayesian models, and show by analytic approximation that for this class, the MAP has twice the marginal MSE of the MMSE.

Colloquium

Students' Reasoning about Variation

Rachel Chaphalkar, University of Montana

Monday, March 17 at 3:10 p.m. in Math 103

The study of statistics is becoming increasingly important in both K-12 education and at the college level which can be seen in curricular documents including the Common Core State Standards (2010) and the Guidelines for Assessment and Instruction in Statistics Education (2005), as well as substantial increases in the number of students taking the Advanced Placement Statistics Exam (College Board, 2011) and college statistics course enrollment (Scheaffer & Stasny, 2000). One of the main components for statistical thinking is consideration of variation (Wild & Pfannkuch, 1999). Previous studies show that students have misconceptions about variation (e.g., Reading, 2004; Torok & Watson, 1999) and often lack students who are able to give sophisticated answer (Shaughnessy, 2007). In this talk, I will provide examples of research in the area of student reasoning about variation, frameworks used for assessing the quality of student responses, and some examples of data from my dissertation study, which is aimed to explore how do students' reasoning about variation in a distributional context change as they progress through an introductory college-level statistics course.

Colloquium

Statistical Tests for Regularization in Ill-posed Inverse Problems

Dr. Jodi Mead, Boisse State University

Monday, February 10 at 3:10 p.m. in Math 103

Most inverse problems are ill-posed due to the fact that inputs such as parameters, physics and data are missing or inconsistent. This results in solution estimates that are not unique or unstable, i.e. small changes in the inputs result in large changes in the estimates. One common approach to resolving ill-posedness is to use regularization methods whereby information is added to the problem so that data are not over-fitted. Alternatively, one could take the Bayesian point of view and assign a probability distribution to the unknowns and estimate it by exploiting Monte Carlo techniques.

In this work we take the regularization approach and use uncertainties to weight added information and data in an optimization problem. This allows us to apply statistical tests with the null hypothesis that inputs are combined within their uncertainty ranges to produce estimates of the unknowns. For example, the Discrepancy Principle can be viewed as using a chi-squared test to determine the regularization parameter.

Colloquium

Turán numbers for forests

Cory Palmer, University of Montana

Monday, February 3 at 3:10 p.m. in Math 103

The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not contain H as a subgraph. The Erdős-Stone-Simonovits Theorem establishes (essentially) ex(n,H) for graphs H of chromatic number 3 or greater. For bipartite graphs much is still unknown. Of particular interest is the Turán number for trees (this is the Erdős-Sós conjecture). We will concentrate our attention on the Turán number of forests. Bushaw and Kettle determined the Turán number of a forest made up of copies of a path of a fixed length. We generalize their result by finding the Turán number for a forest made of up arbitrary length paths. We also determine the Turán number for a forest made up of arbitrary size stars. In both cases we characterize the extremal graphs.

(joint work with Hong Liu and Bernard Lidický)

Doctoral Dissertation Defense

A Longitudinal Study of Students' Reasoning About Variation in Distributions in an Introductory College Statistics Course

Rachel Chaphalkar
Department of Mathematical Sciences

Monday, December 8, 2014 at 10:30 am in Math 103

Current curricular documents including the Common Core State Standards (2010) and the Guidelines for Assessment and Instruction in Statistics Education (2005) have increased the need for students’ understanding and reasoning about statistics at both the K-12 and college levels. In addition, an increasing number of students are taking the Advanced Placement Statistics Exam (College Board, 2011) or a college-level introductory statistics course (Scheaffer & Stasny, 2000). One of the main components for statistical thinking is consideration of variation (Wild & Pfannkuch, 1999). Previous studies have shown that students have misconceptions about variation (e.g. Reading, 2004; Torok & Watson, 1999) and often lack students who are able to give sophisticated answers (Shaughnessy, 2007). The goal of this study was to better understand how students' reasoning about variation in a distributional context change as they progress through an introductory college-level statistics course. In order to better understand the longitudinal nature of this process during a semester-long introductory statistics course, both quantitative and qualitative data were collected at three different times (beginning, middle, end of the course) in surveys and interviews. The Structure of Observed Learning Outcomes (SOLO) Taxonomy (Biggs & Collis, 1982) was used to understand and assess the quality of their reasoning. Qualitative data came from two sources: three interviews from each of the ten interviewees and three survey questions on each of three surveys from all participants. The interviews were transcribed and responses were sorted into appropriate locations in the SOLO Taxonomy. After coding responses to each question in each interview, themes of progress were then identified. These themes showed that students progressed through four different paths of reasoning including: improved, maintained, decreased, and inconsistent. Quantitative data showed that while students were good at reasoning about situations involving bar graphs and dot plots with regards to comparing variability in distributions, they struggled with reasoning about histograms. Overall, this study found that there was no statistically significant improvement in reasoning about variability when comparing distributions as students progressed through a college-level introductory statistics course. This lack of improvement showed that college students needed to have direct intervention or cognitive conflict in order to make more progress in reasoning about variability when comparing distributions.

Dissertation Committee
Dr. Ke Wu, Chair (Mathematical Sciences), Dr. Bharath Sriraman (Mathematical Sciences),
Dr. James Hirstein (Mathematical Sciences), Dr. David Patterson (Mathematical Sciences),
Dr. David Erickson (Curriculum & Instruction)

Master’s Thesis Defense

SOME MATHEMATICAL MODELS IN NEUROSCIENCE APPLICATIONS

Denis Shchepakin
Department of Mathematical Sciences

Thursday, December 4th, 2014 at 3:00 p.m. in Skaggs 117

A neuron is an electrically excitable cell that processes and transmits information through electrical and chemical signals. Neurons connect and pass signals to other cells through the structure called synapse. We focus on synapses through which the signals are transferred by signaling molecules called neurotransmitters. One of the predominant excitatory neurotransmitters in the central nervous system of the mammals, including humans, is glutamate. It is directly or indirectly involved in most brain functions. However, the excessive stimulation of the glutamate receptors is toxic to neurons, therefore it is important to rapidly clear the glutamate from the extracellular space and keep its concentration low. Glutamate transporters play a crucial role in regulating glutamate concentration in synaptic clefts. Thus, it is important to understand the mechanisms underlying this process.

We describe measurement of the glutamate concentration in the extracellular space. It is important to estimate the baseline glutamate concentration to use it in future models and studies. However, two existing methods of measuring the glutamate concentration in the extracellular space give inconsistent results with about 100—fold difference. We construct the model of the process of the glutamate concentration measurement in order to explain that discrepancy.

Also, we consider an experiment, derive a model for it, and use the data from the experiment for estimation of some important constants of glutamate dynamic process in a brain.

Thesis Committee
Dr. Leonid Kalachev (Mathematical Sciences), Dr. Emily Stone (Mathematical Sciences),
Dr. Michael Kavanaugh (Center for Structural and Functional Neuroscience)

Colloquium

A mechanism of mode locking in slow-fast delayed systems

Dr. Dmitry Rachinskiy
The University of Texas at Dallas

Monday, November 24, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

I will discuss one bifurcation scenario which is deemed responsible for generating periodic solutions of special type in delay differential models of semiconductor mode-locked lasers. The solution of interest is a periodic train of short pulses with the period close to the delay time. I would like to explore how general this bifurcation scenario is for delay differential systems with a slow-fast structure. To this end, I will test a number of population dynamics models and formulate some conjecture regarding the conditions that ensure the formation of a stable pulse train.

Colloquium

Regularity of the Bergman Projection on Finite Type and Infinite Type Domains

Andy Raich
University of Arkansas

Monday, November 10, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

In this talk, I will introduce the Bergman projection, discuss its importance in complex analysis, and conclude with a discussion of recent, joint work with Khanh Tran in which we prove regularity for the Bergman projection in L^p-Sobolev and Holder spaces on finite type and a large class of infinite type domains.

Colloquium

Ma-Kap-Pii: Historical Trauma Perceptions from Diverse Cultures

Lydia Silva
Missoula India Center

Monday, November 3, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Within the Native American culture there are traumas that have not been acknowledged. For this reason, I created a film to help with the understanding of historical trauma. It is encouraged for non natives who work with Native people to understand historical trauma. In this discussion I hope to shed light on historical trauma and help with the understanding of the cultural differences.

Colloquium

Distance between chaotic trajectories by fractal dimension concepts

Heikki Haario
Lappeenranta University of Technology (Finland)

Monday, October 20, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Several concepts of dimension have been developed to characterize properties of chaotic trajectories. To estimate parameters of chaotic dynamical systems  a  measure to quantify the likelihood function of chaotic variability (the 'distance' between different trajectories) is needed. We review problems encountered by previously used method and propose a method related to the correlation dimension concept. The major advantage of the new construct is its insensitivity with respect to varying initial values, to the choice of a solver, numeric tolerances, etc.  A way to create the statistical likelihood for model parameters is presented, together with a sound framework for Markov chain Monte Carlo sampling. The methodology is illustrated using computational examples for the Lorenz~63 and Lorenz~95 systems.

Colloquium

An Introduction to Research in the Department of Mathematical Sciences

John Bardsley, Brian Steele, and Ke Wu

Monday, October 13, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

In this colloquium members of the faculty give short talks on their research.  If you are a graduate student who has not yet picked an MA project/thesis advisor, this is an excellent opportunity to learn a bit about what research questions interest us.

Colloquium

An Introduction to Research in the Department of Mathematical Sciences

Jen Kacmarcik, Emily Stone, Eric Chesebro, Cory Palmer

Monday, October 6, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

In this colloquium and the next, members of the faculty give short talks on their research.  If you are a graduate student who has not yet picked an MA project/thesis advisor, this is an excellent opportunity to learn a bit about what research questions interest us.

Update and Discussion regarding Electronic & IT Accessibility Activities

Janet Sedgley, EITA Coordinator, and Lucy France, Legal Counsel

Monday, September 22, 2014 at 3:10 p.m. in Math 103

Colloquium

Partitioning edge-2-colored graphs into monochromatic paths and cycles

Dániel Gerbner, Alfréd Rényi Institute of Mathematics

Monday, September 15, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

We are given a graph G and its edges are colored with two colors. How many vertex-disjoint monochromatic paths/cycles are needed to cover its vertices? We prove several theorems. We show that two times the independence number many cycles are always enough to cover almost all vertices of G. Furthermore, if the minimum degree is at least 3n/4, then 2 cycles can cover almost all vertices of G. Finally, if the graph does not contain a forbidden bipartite subgraph, then 2 paths can cover almost all the vertices of G. (joint work with J. Balogh, J. Barát, A. Gyárfás, G. Sárközy)

Colloquium

MINERVA: Small Planets from Small Telescopes

Nate McCrady, Department of Physics and Astronomy

Monday, September 8, 2014 at 3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Planet occurrence studies such as NASA's Kepler mission are clearly demonstrating that small rocky planets are common in our Galaxy. A significant portion are located in the Habitable Zone (HZ) of their host star, where surface liquid water is possible. While these results are primarily based on stars typically several hundred light years distant, such planets are presumably common around nearby stars as well. These planets would be extremely valuable targets for follow-up studies with the next generation of telescopes, particularly if any are also transiting. Finding small HZ planets around nearby stars requires high precision RV measurements and high cadence observing to densely sample the orbital phase and beat down stellar noise sources. Project MINERVA is a robotic observatory dedicated to detection of rocky planets in the HZ around nearby stars. I will discuss our approach to dispatch scheduling and present simulated Minerva observations to estimate our expected exoplanet yield.

Colloquium

The Bayesian Approach to Inverse Problems

Andrew Stuart, Warwick University

Monday, July 14, 2014 at 3:10 p.m. in Math 103

Many problems in the physical sciences require the determination of an unknown function from a finite set of indirect measurements. Examples include oceanography, oil recovery, water resource management and weather forecasting. The Bayesian approach to these problems is natural for many reasons, including the under-determined and ill-posed nature of the inversion, the noise in the data and the uncertainty in the differential equation models used to describe complex mutiscale physics. The object of interest in the Bayesian approach is the posterior probability distribution on the unknown field [1].

In its purest form the Bayesian approach presents a computationally formidable task as it results in the need to probe a probability measure on an infinite dimensional space; furthermore the likelihood is defined through the solution of a partial differential equation. In this talk I will discuss three computational approaches designed to make this computational task feasible. I will describe Monte Carlo-Markov chain methods tailored to scale well under mesh refinement in the computational model [2]. I will discuss ensemble Kalman filter methods which may be viewed as derivative free optimization methods [3]. And I will describe approximation of the posterior probability distribution by a Gaussian measure, looking for the closest approximation with respect to the Kullback-Leibler divergence [4]; furthermore I will show how the approximate Gaussians can be used to speed-up MCMC sampling of the posterior distribution [5], linking up with ideas from [2].

[1] A.M. Stuart. "Inverse problems: a Bayesian perspective." Acta Numerica 19(2010) and http://arxiv.org/abs/1302.6989

[2] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White, "MCMC methods for functions: modifying old algorithms to make them faster". Statistical Science 28(2013). http://arxiv.org/abs/1202.0709

[3] M.A. Iglesias, K.J.H. Law and A.M. Stuart, "Ensemble Kalman Methods for Inverse Problems." Inverse Problems, 29(2013) 045001. http://homepages.warwick.ac.uk/~masdr/JOURNALPUBS/stuart99.pdf

[4] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Kullback-Leibler Approximations for measures on infinite dimensional spaces." http://arxiv.org/abs/1310.7845

[5] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Algorithms for Kullback-Leibler approximation of probability measures in infinite dimensions." In preparation.

Colloquium

Diophantine approximation: automatic numbers and their generalisations

Michael Coons, University of Newcastle

Wednesday, May 21, 2014 at 11:10 a.m. in Math 103
12:00 p.m. Refreshments in Math Lounge 109

In 1844, Liouville gave the first examples of transcendental numbers, providing a criterion for transcendence based on how well a number can be approximated by rationals. This criterion has been refined and generalised, and it may well be the basis for what is now called Diophantine approximation. The first major refinement of Liouville's criterion was made by Thue in 1909, around the time that he was investigating patterns in binary sequences. Thue noted that any binary sequence of length at least four must contain a square. He then asked, is it possible to find an infinite binary sequence that contains no cube, or even no overlap? Thue's questions began an area now called combinatorics on words.

In this presentation, I will discuss the strong relationship between Diophantine approximation and combinatorics on words. This relationship includes the work of Mahler on functional equations, a connection to finite automata, and very recent results on the rational-transcendental dichotomy of associated classes numbers.