2013 Colloquia

Spring

Presentation of Master’s Project

“Hamilton-Stable Edge-Labellings of Graphs”
Takehiko Yamaguchi
Department of Mathematical Sciences

Tuesday, May 14, 2013
1:10 pm in Math 103

We call an edge-labelling of a graph Hamilton-stable if it has constant weight on Hamilton cycles. In this presentation, we survey most of the existing knowledge about Hamilton-stable edge-labellings, consisting of the early papers by Chandrasekan (1986) and Krynski (1994), the study of SC-Hamiltonian graphs by Punnen and coauthors (2003−), the focused study of labellings of complete graphs by Kayll and coauthors (2003−), and the generalized study to abelian group labellings by Gimbel (2004).

Masters Committee
Mark Kayll, Chair (Mathematical Sciences),
George McRae (Mathematical Sciences),
Mary Riegel (Mathematical Sciences)

Dissertation Defense:
"Abstract Primal-Dual Affine Programming"
Tien Chih
University of Montana

Doctoral Dissertation Defense

Jeffrey Johnson
Department of Mathematical Sciences

Wednesday, May 8, 2013
3:10 pm in Math 103

Dissertation Committee
Thomas Tonev, Chair (Mathematical Sciences),
Eric Chesebro (Mathematical Sciences),
Jennifer Halfpap (Mathematical Sciences)
Karel Stroethoff (Mathematical Sciences),
Eijiro Uchimoto (Physics and Astronomy)

Spectrally Arbitrary Zero Nonzero Patterns 
Timothy Melvin
Washington State University & Carroll College 

A zero-nonzero pattern ℒ is a matrix whose entries are from the set {*,0}, where * denotes a nonzero entry. An n × n zero-nonzero pattern is called a spectrally arbitrary pattern (SAP) over the field F if for every monic polynomial p(t) with coefficients from F of degree n, there exists a matrix A over F with zero-nonzero pattern ℒ such that the characteristic polynomial of A is p(t).

The Nilpotent-Jacobian Method is a powerful tool developed to determine if a pattern is a SAP over ℝ (and ℂ). We will explore this method to determine what information can be gleaned when we look at a pattern over finite fields, ℚ, and extensions of ℚ.

Monday, 6 May 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Doctoral Dissertation Defense

“Computational Methods for Support Vector Machine Classification and Large-Scale Kalman Filtering”
Marylesa Howard
Department of Mathematical Sciences

Monday, April 29, 2013
3:10 pm in Math 103

The first half of this talk focuses on computational methods for solving the bound and equality constrained quadratic program within the support vector machine classifier. An augmented Lagrangian approach will be presented, in which all the constraints are incorporated into the objective function to yield an unconstrained quadratic program, allowing us to apply the conjugate gradient method. This method outperforms other state-of-the-art methods on three image test cases.

The second half of this talk focuses on computational methods for large-scale Kalman filtering applications. The Kalman filter (KF) is a method for solving a dynamic, coupled system of equations. Standard KF is often infeasible in large-scale implementations due to the storage requirements and inverse calculations of large, dense covariance matrices. The use of the conjugate gradient (CG) method within various forms of the Kalman filter will be discussed for low-rank approximations of the covariance matrices, with lowstorage requirements. In test cases, the CG-based KF methods perform similarly in root-mean-square error when compared to the standard KF methods, when these implementations are feasible.

Dissertation Committee
John Bardsley, Chair (Mathematical Sciences),
Jon Graham (Mathematical Sciences),
Jesse Johnson (Computer Science),
Albert Parker (Mathematical Sciences, MSU),
David Patterson (Mathematical Sciences)

(Jointly with the Analysis Seminar)

A Principal Function Problem — Finding a meromorphic function with a given boundary behavior or singularity
Mikihiro Hayashi
Hokkaido University, Sapporo, Japan

For a given meromorphic function s on an open subset A of an (open) Riemann surface R, we want to find a meromorphic function q on R such that both q/s and s/q are bounded on A. Incidentally, we also mention an associated easier problem, finding a meromorphic function p on R such that p−s is bounded on A. Roughly speaking, these functions p and q behave like s on A and are meromorphically extended to R.

Monday, 22 April 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

(Jointly with the Analysis Seminar)

On the multiplier algebra of certain locally m-convex algebras.
Lourdes Palacios
Universidad Autónoma Metropolitana - Iztapalapa, Mexico City

If A is a topological algebra, a bounded mapping T : AA is called a left (right) multiplier on E if T(xy) = T(x)y (resp. T(xy) = xT(y)) for all x; yA; it is called a two-sided multiplier on E if it is both a left and a right multiplier. Denote by ℳl(A), ℳr(A) and ℳ(A) the sets of all left, right and two-sided multipliers of A, respectively. Multipliers play an important role in different areas of mathematics with an algebra structure, due to important applications of non-normed topological algebras in other fields. In this talk, we describe the multiplier algebra of a certain locally m-convex algebra with involution and a perfect projective system of decomposition. We give conditions under which ℳ(A) is isomorphic to the inverse limit of the multiplier algebras of its normed factors; this happens, for instance, in locally m-convex H*-algebras. Moreover, we describe the multiplier algebra of a locally m-convex algebra under certain conditions. Suitable examples will be given.

Wednesday, 17 April 2013
4:10 p.m. in Math 311
3:30 p.m. Refreshments in Math Lounge 109

Incorporating Academic Service Learning Projects and Model-Eliciting Activities into Introduction to Statistics
Rachel Chaphalkar
The University of Montana

The Guidelines for Assessment and Instruction in Statistics Education College Report (Aliaga, et. al., 2010) encourages the use of conceptual understanding, active learning, and the use of technology for analysis of real data in introductory statistics courses. Academic Service Learning (ASL) projects combine what students are learning in class through a project that serves their community (Hadlock, 2005). Model-Eliciting Activities (MEA) present students with data and ask them to construct a shareable model (Lesh, et. al., 2000). In this quasi-experimental study ASL and MEA projects were implemented in STAT 216 during the summer of 2011. Comparisons of the effectiveness of project on student learning of statistics and their attitudes toward the relevance and usefulness of statistics were conducted. This talk will present the findings of this research project and discuss its implications to the teaching and research of college level introductory statistics courses.

Tuesday, 16 April 2013
10:10 a.m. in Math 103
11:00 a.m. Refreshments in Math Lounge 109

(Jointly with the Analysis Seminar)

On Banach Algebras of Bounded Continuous Functions with Values in a Banach Algebra 
Hugo Arizmendi
National Autonomous University of Mexico, Mexico City 

Let X be a completely regular Hausdorff space and A be a complex commutative unital Banach algebra with norm ∥∥. We denote by C (X, A) the unital algebra of all A-valued continuous functions on X with pointwise operations and unit element e (x)≡e, where e is the unit element of A. We denote by (Cb (X, A), ∥∥) the subalgebra of C (X, A) of all bounded continuous functions, provided with the sup-norm ∥∥ on X given by

formula

for every ƒ ∈Cb(X, A), and by (Cp(X, A), ∥∥) the subalgebra of all functions ƒ ∈Cb(X, A) such that ̅ƒ ̅(̅) is compact in A. It is easy to see that both are Banach algebras.

We study the maximal ideal space M ((Cb(X, A), ∥∥)), invertibility in (Cb(X, A), ∥∥) and establish necessary and suficient conditions in order the set X × M (A) to be dense in M((Cb(X, A), ∥∥)) where M(A) is the maximal ideal space of A.

Monday, 15 April 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Colloquium:

"Giving a MOOC – a Survival Guide"
Stanford University Mathematician, Author, and the NPR Math Guy
The game of Cops and Robbers 
William Kinnersley
Ryerson Univesity
C & O Candidate

We discuss the classic Cops and Robbers game on graphs, in which a team of cops attempts to capture a robber. The game has been extensively studied, with applications ranging from video game AI to counterterrorism. It has also spawned many variants: cops in helicopters, cops with tasers, drunk robbers, and so on.

When studying the game, mathematicians have traditionally asked how many cops are needed to catch the robber. In this talk, we focus on a different question that has recently gained interest: given that there are enough cops to catch the robber, how quickly can they do so? We answer this question for the n-dimensional hypercube. Along the way, we encounter a situation in which drunkenness is asymptotically optimal.

This is joint work with Anthony Bonato, Przemysław Gordinowicz, and Paweł Prałat.

Monday, 18 March 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

On the tree packing conjecture 
Cory T. Palmer
University of Illinois at Urbana-Champaign
C & O Candidate

A set of graphs is said to pack into the complete graph, Kn , if the graphs can be found as edge-disjoint subgraphs of Kn. In 1978, Gyarfas conjectured that any set of n-1 trees T1, T2, ..., Tn-1 such that Ti has n-i edges packs into Kn. Even when we weaken the statement to claim that the largest t trees T1, T2, ..., Tt pack into Kn the conjecture remains open. Among others we will discuss our recent result that any t=(1/10)n¼ trees T1, T2, ..., Tt such that Ti has n-i edges packs into Kn+1 (for n large enough). (This is joint work with J. Balogh.)

Thursday, 14 March 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

An Exploration of Ordered Rings 
Tien Chih
PhD Candidate, The University of Montana

The usual study of linear (affine) programming is done over the real numbers or the integers. However, the notion of a maximum or minimum value is sensible over any ring with a well-defined order. However, in general, ordered rings can be very peculiar objects, and may not exhibit all the properties one associates with the real numbers and its sub-rings.

In this talk, we will present some of the properties of ordered and orderable rings, including some historical examples. We will give examples of ordered rings that are potentially non-Archimedean and even non-commutative. We will then discuss how these may present challenges when attempting to do linear (affine) programming over these rings. 

Monday, 11 March 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109  

Boundaries in Functional Analysis 
Jeff Johnson
PhD Candidate, The University of Montana

The notion of boundary is ubiquitous throughout mathematics. The most familiar definitions of the concept come from geometry and point-set topology, but boundary has somewhat different definitions in other areas. In functional analysis, and the theory of commutative Banach algebras in particular, the boundary is defined in terms of the functions on a carrier space and not just the space itself. The talk will give an introduction to two of the most common boundaries found in commutative Banach algebras, the Shilov boundary and Choquet boundary.

Monday, 25 February 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Relative Likelihood Comparisons: Heuristics, Biases and Fallacies
Egan J Chernoff
University of Saskatchewan

In this lecture, the heuristics, biases and fallacies associated with relative likelihood comparisons (e.g., which sequence of coin flips is less likely to occur: HHTHTT or TTTTTH) will be presented. The lecture begins, first, with an overview of Tversky and Kahneman’s heuristics and biases program (of the late 1960s and early 1970s) — with a particular emphasis on the representativeness heuristic. Second, the alternative theories, frameworks and models that mathematics education researchers used to account for incorrect, inconsistent, incomprehensible relative likelihood comparisons (during the 1990s and late 2000s) will be discussed. Third, the more recent developments to the heuristics and biases program, which have largely and inexplicably been ignored by the mathematics education community, will shed new light on unintended comparisons of relative likelihood. Lastly, the lecture will focus on an emerging area of research that uses logical fallacies (e.g., the fallacy of composition, the appeal to ignorance and others) instead of heuristics and biases to account for prospective math teachers' relative likelihood comparisons. Ultimately, this lecture is designed to provide the background for a lively discussion on whether ACCBDCAADB or CCCBBBBBBB is least likely to be the answer key (with four possible answers: A, B, C and D) for a 10 question multiple choice math quiz.

Monday, 11 February 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109

Non-Beatles’ perspectives on non-Edmonds graphs
Mark Kayll
University of Montana

In 1965, at the height of Beatlemania, Jack Edmonds published his groundbreaking characterization of the perfect matching polytope of a graph G = (V,E), i.e., the convex hull P of the characteristic vectors of the perfect matchings in G. Edmonds described P polyhedrally as the set of nonnegative vectors in ℝE satisfying two families of constraints: 'saturation' and 'blossom'. Graphs for which the latter constraints are implied by the former are now called non-Edmonds graphs. As it turns out, this graph class interacts interestingly with more familiar classes. For example, bipartite graphs are non-Edmonds, and this assertion is equivalent to the Birkhoff–von Neumann Theorem on doubly-stochastic matrices. This talk will explore several connections of this nature and will be accessible to non-experts.

Monday, 4 February 2013
3:10 p.m. in Math 103
4:00 p.m. Refreshments in Math Lounge 109 

Fall 2013

Doctoral Dissertation Defense

Abstracted Primal-Dual Affine Programming

Tien Chih, University of Montana

Monday, December 9 at 11:10 am in Math 108

Abstract (PDF)

Colloquium

Physics based interpolation techniques

Jesse Johnson, Department of Computer Science, University of Montana

Monday, December 2 at 3:10 p.m. in Math 103

Increasingly, the demands of computational models challenge the limits of observational data. For instance, models require first and higher derivatives of velocity and thickness measurements, however numerical derivatives of data are often characterized by noise that makes their interpretation difficult. Specific examples include strain-rates and flux divergences computed from observations of velocity and thickness. Two approaches to physics based interpolation of observational data are presented here. The first is known as the "mass conserving bed", and entails using the continuity equation to interpolate between measurements of ice thickness. Our favored approach utilizes least squares rather than Lagrange multipliers, and is shown to be accurate, robust, and scalable to large problems. The second application is to InSAR surface velocity observations. In order to smooth these frequently discontinuous data we again look to the continuity equation, this time solving for vertically averaged velocity. Attaching a Lagrange multiplier to the forward model, and adding misfit over the domain, we find adjoint, control, and objective equations allowing minimization of differences between model and observed surface velocity. Bounds set in the minimization algorithm ensure optimal velocities are consistent with reported errors in thickness, surface mass balance, surface velocity, and surface rate of change. The resulting velocity field is in excellent agreement with observation, provides complete coverage, and satisfies stronger requirements for continuity. Both bed and velocity fields produced by these techniques are of use to the community for; initialization of ice sheet models, calculation of the force budget, inversion for parameter estimation, assessment of ice sheet sensitivity to perturbation, and mission planning.

Colloquium

Linear Preserver Problems and Algebraic Groups

Hernando Bermudez, Emory University

Monday, November 25 at 3:10 p.m. in Math 103

In 1897 Frobenius posed and solved the following problem: Determine the set of all endomorphisms of the space of complex n by n matrices that preserve the values of the determinant. This is the first example of a Linear Preserver Problem. The work I will discuss in this talk began with an attempt to find a solution to an interesting class of such problems. Our methods of attack come from the theory of Linear Algebraic Groups; I will give an overview of the basic concepts and then apply them to describe the solution of some LPP's. In an interesting recent development it turns out that understanding the solution of the LPP's in turn gives a great deal of information about the general structure of Linear Algebraic Groups.

Colloquium

Modeling Cognitive Performance under Sleep Restriction

Clark Kogan, PhD Candidate, University of Montana

Monday, November 18 at 3:10 p.m. in Math 103

Mathematical modeling of performance impairment due to sleep restriction produces the results that can be used to optimize work scheduling which, e.g., involve night shifts, and which, in turn, can reduce the risk of automotive, flight, and other accidents. Models’ parameters may be estimated using the data related to driver centered metrics such as lane deviation and eye movement. The fitted models can be used to predict driving performance and make drivers aware of cognitive impairments. Population information for the models can be collected in laboratory based studies on high-fidelity driving simulators. This information may be employed to construct prior distributions for model parameters; it can also be combined with real time driving data to predict driving performance. Individual metrics differ with respect to cost, ease of implementation, correlations, and noise, and it is often of interest to determine the most cost effective combination of metrics. In the linear setting, we analytically determine a simple relationship between the improvement in prediction accuracy due to a secondary task and the sample size for that task.

In the nonlinear case, we may assess the improvement in prediction MSE using simulation; however, for models stated in the form of differential equations with no analytic solution, such repeated evaluation of the Bayesian minimum mean squared error estimator (MMSE) can be time consuming. An alternative estimator, the Bayesian maximum a posteriori (MAP), can reduce these computations by one or more orders of magnitude; however, it does not guarantee minimum mean squared error. Simulations show that in some important nonlinear cases there is virtually no difference in accuracy between these two estimators.

We hypothesize that the reduction in estimator accuracy (owing to the substitution of the MAP for the MMSE) can be approximated without the need for numerical integration. We begin by considering a limited class of nonlinear modeling scenarios, and analytically approximate the increase in MSE due to using the MAP estimator for this class. Our work builds a foundation for further construction of methodology to quickly determine modeling scenarios (i.e., model functions, covariate values and prior parameters) for which the MAP estimator will result in no significant loss of accuracy.

Colloquium

Guided Reinvention: The Case for Technology in the Mathematics Education Classroom

Matt Roscoe, University of Montana

Monday, October 28 at 3:10 p.m. in Math 103

Well-known Mathematics Education theorist Hans Freudenthal argues in favor of a “guided reinvention” approach to mathematics education. I will summarize Fruedenthal’s argument and then demonstrate how the technologies of Smartboard, GeoGebra, Desmos and Elmo make Freudenthal’s vision for mathematics teaching and learning more accessible to teachers of mathematics in the modern era.

Colloquium

Large-deviations limits for non-equilibrium Markov processes, path entropies, and applications

D. Eric Smith, George Mason University

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

A variety of methods now exist, some developed over the past 40 years in statistical physics, to compute large-deviation functions for non- equilibrium Markovian stochastic processes. More important than technical methods is the understanding that these functions define the relevant notions of entropy and thermodynamic limits for systems both at and away from equilibrium, in domains ranging from mechanics to inference. Much of the richness of non-equilibrium large- deviations theory is being developed in systems chemistry, cell and molecular biology, and population processes, because these combine features of intermediate numbers of degrees of freedom and complex state-space structure that make fluctuations both important and challenging to compute. I will sketch key concepts and some methods for modern non-equilibrium large-deviations theory, and present a collection of examples from cell and population biology showing the use of symmetry and collective fluctuations as organizing concepts, and the appropriate role of entropy corrections in fundamentally non- equilibrium systems. I will close with some future areas of opportunity particularly in molecular biology and possibly microbial ecology.

Colloquium

Mathematical Modeling and Computational Methods for Breast Tomosynthesis Image Reconstruction

Jim Nagy, Dept. of Mathematics & Computer Science, Emory University

Thursday, October 3 at 3:10 p.m. in Math 103

In digital tomosynthesis imaging, multiple projections of an object are obtained along a small range of different incident angles in order to reconstruct a pseudo-3D representation of the object. In this talk we describe a mathematical model for polyenergetic digital breast tomosynthesis image reconstruction that explicitly takes into account various materials composing the object and the polyenergetic nature of the x-ray beam. This model allows for computing weight fractions of the individual materials that make up the object, which can then be used to reconstruct pseudo-3D images. The reconstruction process requires solving a large-scale inverse problem. The mathematical model and computational approaches, including an efficient GPU implementation, are described in detail. The effectiveness of our approach is illustrated with real data taken of an object with known materials that simulates an actual breast.

Colloquium

Math for Fun and (Small) Profit: An Eclectic Tour of Prize Problems in Mathematics

Katherine St. John, City University of New York

Monday, September 23 at 3:10 p.m. in Math 103

Many open problems in mathematics have small monetary or consumable prizes associated with them. The Hungarian mathematician Paul Erdős was known for this tradition, with prizes ranging from $25 to many thousands. In honor of the 100th anniversary of his birth (26 March 1913), we will survey some of the prize problems including some from Erdős and the million dollar Millennium Prize Problems from the Clay Mathematics Institute. The second half of the talk will focus on prize problems at the intersection of biology, computing, mathematics, and statistics.

Colloquium

Son of Dog & Pony Show*

Ke Wu, Math Education
Cory Palmer, C & O
Kelly McKinnie, Algebra
University of Montana

Monday, September 16 at 3:10 p.m. in Math 103

*1. an elaborately staged activity, performance, presentation, or event designed to sway or convince people (from a derisive term for a small circus)

2. Pretending to be something you're not or doing something that you want people to think is important when it really isn't.

3. A demonstration or proof of concept (often done for a client) to prove the product works as sold or to show how a design on paper looks once actually built. May or may not be a waste of time. Used in the special effects industry among many others.

From the urbandictionary.com

Colloquium

Dog & Pony Show*

Emily Stone, Applied Math
Jennifer Halfpap, Analysis
Matt Roscoe, Math Ed
Eric Chesebro, Topology
University of Montana

Monday, September 9 at 3:10 p.m. in Math 103

*1. an elaborately staged activity, performance, presentation, or event designed to sway or convince people (from a derisive term for a small circus)

2. Pretending to be something you're not or doing something that you want people to think is important when it really isn't.

3. A demonstration or proof of concept (often done for a client) to prove the product works as sold or to show how a design on paper looks once actually built. May or may not be a waste of time. Used in the special effects industry among many others.

From the urbandictionary.com