Mathematics Colloquia

Spring 2003

• Fall 2001 Talks
• Spring 2002 Talks
• Fall 2002 Talks

Feb. 7 (12 p.m., LA5-265) Victor Acosta, CSULB.

The Random Schrodinger Operators On a Lattice.

We discuss certain discrete random Schrodinger operators and their spectra. Specifically, the Hamiltonian H_l = -D + lV, where V is a random potential and l is the disorder, is investigated. Such a Hamiltonian was introduced by P. Anderson to model the motion of an electron in a crystal with impurities. For certain lattice structures, results are obtained pertaining to the density of states, exponential localization, and the existence of extended states.

Feb. 21 (12 p.m., LA5-265) Janet Myhre, Claremont McKenna College.

To Believe or Not To Believe Test Equivalancy: A Bayesian Solution.

A problem facing many engineers is how to estimate the failure probability of a system during its intended use, e.g. missile flight, based on test data. One solution is to use Bayesian priors on the ratios of different failure probabilities for the different test types. Use of these priors allows one to obtain a maximum likelihood estimate of the desired failure probability. Asymptotic unbiasedness of these estimates is obtained if one of the test types in the ratio is the desired use and if the domain of positivity of the prior includes the true ratio. The maximum likelihood estimates are proven to behave reasonably well in the sense that the estimate lies in the range of the estimates from the single data sources and in fact behaves like a weighted average of those estimates. Mean square error studies show that in practice the engineer’s prior does not have to be very accurate if the number of failures is small. There is a high degree of robustness to the prior used. On the other hand, if there are failures in the primary data source and the number of failures for the alternate data source is large the theorem shows that a maximum likelihood estimator should not be used. Other results relevant to a small number of failures will be presented during the talk.

Feb. 28 (12 p.m., LA5-265) Ko Honda, USC.

Farey Tessellation and Contact Topology.

Bart Simposon’s rule of adding two fractions is a/b + c/d = (a+c)/(b+d). In this talk, we will describe a setting where the Bart Simpson addition rule makes sense, namely the Farey tessellation of the hyperbolic disk H ². We will then go on to describe the relationship with continued fractions and my subject of research, namely contact topology.

Feb. 28 (2:10 p.m., LA5-263) James Stewart, McMaster University.

Three Centuries of Calculus Wars.

(Dr. James Stewart is the author of the best selling Calculus textbook.)

Mar. 7 (12 p.m., LA5-265) Gwen Fisher, CSU San Louis Obispo.

Employing a Theoretical Framework for Students' Understanding of the Derivative: Comparing Daily Writing Tasks with Individual Interviews and Traditional Exams.

This study compares the use of open-ended writing tasks with interviews and traditional exams to assess students' understanding of the concept of the derivative. The theoretical framework for the data analysis depicts the derivative as a ratio, as a limit, and as a function, each of which can be conceptualized in any of six representations: numeric-tabular, numeric-computational, symbolic-definition, symbolic-abstract, graphical, and contextual. The results show that daily writing tasks are reliable assessments for some forms of students' understanding that are not assessed by traditional exams. Of practical concern, this study also showed that, for a group of students taken as a whole, the range of responses in the group’s writing is usually very similar to the range of responses found during interviews.

Mar. 14 (12 p.m., LA5-265) Robert Mena, CSULB.

The Fibonacci Numbers — Exposed.

The Fibonacci sequence has been referred to as one of two shining stars in the vast array of integer sequences. But as the talk will exemplify, it may be difficult to find properties of the sequence, as attractive as it may be, that are not specifications of properties of the much more general two-term linear recurrences. The talk is based on joint work with Dan Kalman, of American University, and should be accessible to an advanced undergraduate student.

Mar. 17 (Monday, 12 p.m., FO3-200A) Doug Dunham, U. Minnesota.

Mathematical Art and M. C. Escher.

For hundreds, if not thousands of years, artists in many cultures have created artistic designs based on mathematical principles. The Dutch graphic artist M. C. Escher (1898-1972) certainly employed more mathematics in his works than any artist. Escher made use of the mathematical concepts of infinity, Mobius bands, deformations, reflections, Platonic solids, spirals, and especially repeating patterns. In addition to his many Euclidean plane patterns, Escher also created spherical and hyperbolic patterns, thus becoming the first artist to utilize all three of the "classical" geometries. Escher has inspired many other artists, including myself, to produce much more mathematical art. I will show many examples of art that is based on mathematics. This fruitful collaboration between mathematics and art is being celebrated as the theme for Mathematics Awareness Month, April 2003.

Mar. 21 (12 p.m., LA5-265) Sergey Lototsky, USC.

Estimating the Population Mean.

In a Gaussian population, the best estimator of the population mean is the sample mean. This is no longer the case when the distribution of the population is not normal. For example, for the Cauchy population, while there is a natural analog of the population mean, the sample mean is not even a convergent estimator. In my talk, I will describe the construction of the best estimator of the population mean for various population distributions and observation models.

Mar. 28 (12 p.m., LA5-265) Giulio Della Rocca, CSULB.

From Mackey to Takesaki, the New Approach to Classification.

In his work on group representation, G. Mackey established the so-called "Mackey doctrine" on conditions for a good classification. Despite his enormous successes, the paradigm has limited applications and results. Fromthe early work of M. Takesaki and A. Connes a new approach is coming to life. The need for a new paradigm has only become evident in the last decade. We will present an application to the case of UHF algebrasof a so-called non-smooth classification.

Apr. 4 (12 p.m., LA5-265) Mahmood Ghamsary, Loma Linda University.

Effect of Air Pollution on CHD, Coronary Heart Disease.

Numerous studies have shown a positive association between daily mortality, particularly CHD, and particulate air pollution, even at concentrations below regulatory limits. These findings have motivated interest in the shape of the exposure-response relationship. We have developed flexible modeling strategies including spline, log-linear model, Poisson regression, etc..

Apr. 11 (1 p.m., LA5-265) Burkhard Englert, UCLA.

Applications of Ramsey's Theorem in Distributed Algorithms.

Recently, several different long-lived adaptive algorithms whose operation step complexity is a function of the actual number of processors running concurrently with the operation execution, have been presented. Regardless of the type of adaptiveness and the type of object implemented, they require W(N) MWMR registers in their implementation, where N is the total number of processors that may participate in the implementation. We show that, even if the maximum allowed concurrency is a constant, the number of MWMR registers used must depend on N. The proof is based on an application of Ramsey's Theorem.

Apr. 25 (12 p.m., LA5-265) Maya Chhetri, University of North Carolina at Greensboro.

Effects of Harvesting and Diffusion on Population Dynamics.

We will first discuss the logistic growth model introduced by Verhulst in 1863. Then we will notice the change in dynamics when constant effort harvesting is introduced in the model. We will see that the mathematical analysis of these models becomes very challenging if the species are allowed to disperse in the domain.