# Spring 2011

### The geometry of automorphisms of free groups

#### Mladen Bestvina

#### Department of Mathematics, University of Utah

#### Thursday, April 14, 2011, 2:30-3:45pm, DMS 103

**Abstract:** The modern study of automorphisms of free groups of finite rank is modeled on the study of SL_n(Z) and homeomorphisms of surfaces. The growth rate of an individual automorphism is understood in terms of its train track representative (analogous to the Jordan normal form), and groups of automorphisms are studied by their action on Outer space (analogous to a symmetric space or the Teichmuller space). In the talk I will define the basic objects in this theory and survey the results.

### Khovanov homology of certain symmetric unions

#### T.J. Gaffney

UNR Mathematics Student

#### Wednesday, April 13, 2011, 1:00-2:15pm, MS 321

**Abstract:** Symmetric unions have been studied due to their failure to be distinguished by some invariants. A new and powerful invariant, Khovanov Homology, has been the subject of recent study. In our talk, we will discuss Khovanov Homology's ability to distinguish symmetric unions in some scenarios.

### Application of Adomian Method to Model Concentration near the Surface of a Rotating Disk

#### Masakazu Gesho, Mathematics Student

#### University of Nevada, Reno

#### Wednesday, April 13, 2011, 9:00-11:00am, DMS 105

**Abstract:** The Adomian decomposition method is a recurrent procedure to obtain approximate solutions in the form of series. We solve an equation of a diffusion type with the Adomian decomposition method. We construct the velocity field in fluid due to the rotating disk and analyze the concentration of a specie present in the fluid that undergoes irreversible chemical reaction. We represent the nonlinear chemical reaction term with Adomian polynomials, and we calculate the concentration with respect to distance from the rotating disk.

### Sequence Comparison Without Alignment

#### Michael Waterman, Professor of Biological Sciences, Mathematics, and Computer Science

#### University of Southern California

#### Tuesday, April 7, 2011, 2:30-3:45pm, TBD

**Abstract:** The sum of the products of the k-word counts from each of two sequences has been used to test if the sequences have a significant similarity. The statistic can be rapidly computed which is important for large scale applications. When the sequence alphabet is not uniformly distributed, the statistic has approximately a normal distribution. However this property arises from the individual sequences themselves, not their cross relationship. In this talk the known results will be summarized and it will be shown that in the case just mentioned the statistical power of the statistic behaves very poorly and some superior statistics are available.

### Low-dimensional Quaternionic Matrix Groups and the Exponential Map

#### Casey Machen, Mathematics Student

#### Tuesday, April 5, 2011, 2:30-3:45pm, DMS 105

**Abstract:** We study low dimensional matrix groups over the quaternions H and the exponential map. We focus in this talk on the properties of the Lie groups Sp(2) and SL2(H) and their Lie algebras sp(2) and sl2(H). We compute the Weyl group of Sp(2) and show that the exponential map from sp(2) to Sp(2) is surjective.

### Aperiodic Order with a little topology and dynamical systems

#### Ian Putnam

#### Department of Mathematics and Statistics, University of Victoria

#### Thursday, March 31, 2011, 2:30-3:45pm, DMS 105

**Abstract:** In the 1960's, mathematicians discovered various geometric structures, particularly tilings, in Euclidean space which displayed a high degree of order, but which were not periodic. The most famous examples were given by Roger Penrose. Later, physicists discovered physical materials which displayed the same properties and are now known as quasicrystals. Since, this subject has grown substantially and the mathematics is notable for the wide variety of fields from which techniques are used in the study of these objects. In the first part of the talk, I will present a basic introduction to the subject. In the latter part, I will indicate how some ideas from topology and dynamical systems have been effective tools.

### On bootstrap principle and its application in statistics, engineering, biology and economics

#### Jacek Leskow

#### Department of Quantitative Methods in Management

#### The Polish-American Graduate School of ManagementWSB-NLU Nowy Sacz, Poland

#### Thursday, March 22, 2011, 2:30-3:45pm, DMS 103

**Abstract:** In the talk, the principle of bootstrap approach to building confidence intervals and tests will be presented. We will show how bootstrap-derived procedures provide better tools for statistics and applications then those based on normal distributions. The bootstrap perspective will be then applied to various applications ranging from telecommunications signal processing, vibromechanics, biology and economics. In telecommunications, we will show how to classify signals according to their characteristic frequencies and how bootstrap principle can be applied there. For those interested in engineering, we will show how bootstrap can help identifying problems in transmissions and jet engines. Folks interested in economics should start using bootstrap to get better forecasts of their risk assessments - we will show how bootstrap can be helpful in analyzing Value-at-Risk. Recent advances in time series analysis were making possible application of bootstrap to gene expression data and we will be showing how to do it as well.

### Resampling methods for nonstationary almost periodic stochastic models

#### Jacek Leskow

#### Department of Quantitative Methods in Management

#### The Polish-American Graduate School of Management

#### WSB-NLU Nowy Sacz, Poland

#### Thursday, March 24, 2011, 2:30-3:45pm, DMS 103

**Abstract:** In the talk, the nonstationary almost periodic models for time series and stochastic processes will be presented. The applications of such models are very wide and include signal processing, climatology and finance. One of the most fundamental problems for such models is estimation of covariance and identification of frequencies. We will start from presentation of classical asymptotic results and then move to resampling methods that work for time series and continuous-time models as well. The consistency results will be illustrated with simulations and real data applications.

### Propensity Scores and Estimators of the Odds Ratio

#### Christiana Drake

#### Department of Statistics, UC Davis

#### Thursday, March 10, 2011, 2:30-3:45pm, DMS 103

**Abstract:** The odds ratio is a commonly used measure of disease-exposure association. The odds ratio can be estimated from cohort as well as from case-control studies. The regression coefficients obtained from fitting logistic regression models to binary outcomes are log odds ratios. Thus one could be tempted to view non-parametric estimators of the odds ratio and logistic regression models as the binary equivalent of linear regression models and exposure effects estimated as mean differences. However, the conditions for collapsibility are different for estimators of the odds ratio and additive effects. Marginal and conditional effects differ, even in the absence of confounding (Gail, Wieand and Piantadosi, 1983). Senn, Graf and Caputo (2007) compared the properties of marginal estimators and conditional regression estimators in the linear model. They showed that adjustment by the propensity score does not necessarily lead to a less efficient marginal estimator of an additive exposure effect as compared to the conditional estimator from a regression model.

Epidemiological studies often use logistic regression to estimate exposure effects adjusted for other risk factors and confounders. The Mantel-Haenszel estimator of the odds ratio is seen as an alternative to logistic regression. It has the desirable property of being consistent under both regular asymptotics ( when the number of strata is fixed and the sample size in each stratum increases proportionately to the total sample) and sparse data asymptotics when the stratum size remains fixed and finite and the number of strata increases with the sample size. However, the Mantel-Haenszel estimator is consistent for the conditional odds ratio only if the confounders are constant over the strata. Thus stratifying on a continuous covariate will lead to an inconsistent estimate while matching will lead to a consistent estimate of the conditional odds ratio. We show that matching on the propensity score suffers from similar problems as does forming strata for a continuous confounder. We also investigate weighting by propensity, an approach that Lunceford and Davidian (2004) show to be efficient for additive exposure effects.

### The Discontinuous Galerkin Method and its Application to the Maxwell Time Domain Problem

#### Sascha Schnepp

#### Computational Electromagnetics Group (CEM)Technische Universitaet Darmstadt

#### Thursday, February 24, 2011, 2:30-3:45pm, DMS 103

**Abstract:** The discontinuous Galerkin method (DGM) has received considerable attention in recent years. It is a high order method for the solution of partial differential equations. Its advantage over other high order methods is the preservation of a highly localized formulation. The DGM contains conceptual elements from Finite Element and Spectral Element Methods as well as the Finite Volume Method. The talk will introduce key ideas of the method by examining the semi-discrete formulation of Maxwell's equations in 1D (space-discrete, time-continuous). A short analysis of the semi-discrete formulation touching dispersion, dissipation and accuracy will be presented before an insight to the formulation in higher space dimensions is given. The talk will then proceed with presenting the fully discrete formulation (discretization of time) and its analysis. Applications to examples in one- and three-dimensional space will be presented including examples involving hp-adaptivity.

### Testing Isotropy and a related Random Walk problem

#### S. Rao Jammalamadaka

#### Department of Statistics, UC Santa Barbara

#### Thursday, February 3, 2011, 2:30-3:45pm, DMS 103

**Abstract:** One comes across directions of the observations in a number of situations. The first inferential question that one should answer when dealing with such data is, "Are they isotropic or uniformly distributed?" The answer to this question goes back in history which we shall retrace a bit and provide an exact and approximate solution to this so-called "Pearson's Random Walk" problem.

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