Colloquium Schedule - Spring 2012
Friday, May 11, 2012, 11am-1pm, DMS 102
Masters of Science in Mathematics Thesis Defense
Student Heidi Tan
Department of Mathematics and Statistics
University of Nevada, Reno
Asymptotic Distribution of the Estimated CDF of the Bivariate Distribution with Truncated Logistic and Geometric marginals
Abstract: In this work, we derive the limiting distribution of the cumulative distribution function for the bivariate model with truncated logistic and geometric marginals. We discuss the limiting behavior of the asymptotic variance and illustrate the convergence of the cumulative distribution function to the limiting normal law using Monte Carlo simulations. Last, we present an example of applying these results to estimate error probabilities for hydrological and financial data.
Tuesday, May 1, 2012, 2:30-3:45pm, DMS 102
Honors Thesis Defense Student Jessica Reynolds
Department of Mathematics and Statistics
University of Nevada, Reno
The Mathematical Signature of a Composer: A Preliminary Analysis
Abstract: The possibility of analyzing and comparing works of musical composers using statistical methods is explored. Statistical methods are already used to analyze literary authors, examining word lengths, frequencies, and idiosyncratic tendencies. These methods are studied and related to the field of music by analyzing musical patterns of different composers. In this research, information from individual notes from solo-instrument musical compositions is coded into five variables. This new coding system allows for observation of note frequencies, average note durations, note repetitions, sequences, and the composer’s tendency to adhere to a specific key. Several works by Bach and Stravinsky are analyzed and compared based on these criteria. If these composers are successfully shown to be statistically different, it could be the starting point to the development of an identification system for works of unknown or uncertain authorship.
Thursday, April 12, 2012, 2:30-3:45pm, DMS 102
The University of Texas at El Paso
Department of Mathematical Sciences & Computational Science Ph. D. Program
Automated Solution of Equations with Uncertain Parameters
Abstract: Mathematical modeling allows prediction of the future characteristics of engineering structures without performing expensive experiments. In many cases it is hard to get exact values of the parameters necessary to specify mathematical model. If only limited information is available sometimes it is possible to obtain upper and lower bound of the parameter (i.e. ). In this talk an efficient method for solution of equations with uncertain parameters (interval, random, and fuzzy parameters) will be presented. The method allows adaptive error estimation.
Many numerical results, visualizations as well as mathematical theorems which are related to this presentation were developed automatically by the SelfNet system. The system is capable to perform many typical scientific tasks (e.g. prove theorems, prepare visualizations, get numerical/symbolic solution) automatically. SelfNet is capable to develop selected scientific ideas automatically and improve it itself. The system engenders not only the final results of the calculations, but also shows all intermediate steps which lead to its solution. New knowledge generated by the system is saved and can be used automatically in the future problems calculations. Once the new idea is added to the system it will never be forgotten and the system is capable to apply it for processing of future problems. SelfNet was developed by the author of this presentation.
Thursday, April 5, 2012, 2:30-3:45pm, DMS 102
Wayne State University
Localized Modes and Energy Trapping Phenomena in Coupled Nonlinear Oscillators
Abstract:
High energy dynamics or resonance interactions between system subcomponents are usually accompanied by most interesting physical effects. At the same time, conventional analyses of such effects face challenging mathematical problems due to strong nonlinearities or dimension increase. The main idea of the present talk is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear however still simple enough to play the role of basis. Such a stand point will be illustrated on practically reasonable mechanical models, whose dynamics may include both resonance interaction and nonlinear localization effects. The nonlinear mode localization has been known for a long time as both micro- and macro-level phenomenon. However, in the area of nonlinear dynamics, the mode localization recently became of a growing interest due to the idea of dynamic energy absorption. In contrast to stochastic localization in disordered linear systems discovered in quantum physics about one half of a century ago, nonlinear local modes may occur even in perfectly symmetric systems with symmetry braking. In general terms, localization means that one or few interacting particles become dynamically isolated from the rest of the system due to specific initial conditions and/or variation of physical parameters. As a result, the energy, while oscillating between two oscillators on low energy levels, does not oscillate any more and becomes trapped on one of the two oscillators.
Thursday, March 1, 2012, 2:30-3:45pm, DMS 102
University of Nevada, Las Vegas
Department of Mathematical Sciences
Recent advanced in time-domain simulation of electromagnetic wave propagation in metamaterials
Abstract: Since 2000, there is a growing interest in the study of metamaterials due to their potential applications in areas such as design of invisibility cloak and sub-wavelength imaging. In this talk, I'll first give a brief introduction to the short history of metamaterials. Then I'll focus on mathematical modeling of metamaterials, and discuss some numerical schemes we developed in recent years. Finally, I'll conclude the talk with our cloak simulation and some open issues for further exploration.
A short bio: Dr. Jichun Li is Professor of Mathematics and Director of Center for Applied Mathematics and Statistics (CAMS). He has worked at the Institute for Computational Engineering and Sciences (ICES) in Univ of Texas at Austin, the Institute for Pure and Applied Mathematics (IPAM) at UCLA, and U.S. Air Force Research Laboratory (AFRL). His research interest is in scientific computing, imaging processing, inverse problems, computational electromagnetics. He has published over 60 refereed journal papers and two books. He currently serves on editorial boards of three international journals.
Thursday,February 23, 2012, 2:30-3:45pm, DMS 102
University of Nevada, Las Vegas
Department of Mathematical Sciences
Sequential Confidence Limits for the Ratio of Two Binomial Proportions with Unequal Sample Sizes
Abstract: We propose the approximate confidence limits for the ratio of two independent binomial variates. Due to the nonexistence of an unbiased estimator for the ratio, we develop the procedure based on a modified maximum likelihood estimator. When sample sizes are not equal, by defining the sample-ratio we can generalize results of Cho and Govindarajulu (2008). We investigate the large-sample properties of the proposed estimator and its finite sample behavior through numerical studies. In addition, we make comparisons from the sample information view points.
Monday, January 23 2012, 11am-12pm, DMS 102
Mathematics and Statistics
Dalhousie University, Halifax, Nova Scotia
Moduli Spaces and Orbifolds: What is Half a Point?
Abstract: A moduli space is a geometric object whose points parametrize (encode) all structures of a particular algebraic or geometric type. For instance, there is a moduli space of annuli: it is 2-dimensional if you want to vary the diameter and the width independently and if you only keep track of the proportion of the two, you would obtain a 1-dimensional parameter space.
Moduli spaces are used in the study of elliptic curves and Riemann surfaces of higher genus, and other areas of mathematics that are related to mathematical physics.
When the family of objects we are parametrizing comes with a symmetry
structure this needs to be reflected in the moduli space. This leads us to moduli spaces for which the local structure consists of an open subset of Euclidean space together with the action of a finite group. Such gadgets are called orbifolds and form a natural generalization of manifolds. In this talk we will see how orbifolds arise in a natural way from a moduli problem and then we will discuss some techniques for modeling and studying them and in particular we will look at the question what smooth maps between orbifolds should be.
Thursday, December 8, 2011, 2:30-3:45pm, AB 102
Biochemistry and Molecular Biology
University of Nevada, Reno
Using Boolean Networks to Model Gene Interactions, Karen Schlauch and Bryson Wheeler
Abstract: The analysis of large amounts of microarray data is a significant challenge for the researcher. The parallel assay of thousands of data points, not all of which are independent, across a number of temporal states, provides an interesting platform for statistical analyses and the construction of models. Complex networks are often used to model genetic regulatory systems or protein interaction networks, both represented by these types of data.
As a first approximation, Boolean networks can provide useful models of the underlying regulatory networks of a collection of genes (or proteins) over time (e.g., a pathway). Gene signals can be typically characterized as being on or off, lending perfectly to the Boolean model. Interactions among individual genes are inferred by examining gene expression levels measured over a series of temporal states. Often, more than one regulatory model may exist to represent the genes’ interactions.
In this approach, we generate all possible Boolean network models that correctly represent the system under observation. Using published experimental data about the genes within the given network, as well as simple entropy measures, many of these network models can be quickly eliminated. The resulting network space includes biologically meaningful models that characterize possible structure and interaction patterns of the regulatory system.
Thursday, December 1, 2011, 2:30-3:45pm, AB 102
Department of Statistics
Stanford University
Inequalites: Theory of Majorization and its Applications
Abstract:There are many theories of "equations": linear equations, differential equations, functional equations, and more, However, there is no central theory of "inequations" There are several general themes that lead to many inequalities. One such theme is convexity. Another theme is majorization, which is a particular partial order. What us important in this context is that the partial order have lots of examples, and that teh order-preserving functions be a rich class. In this case majorization arises in many fields: in mathematics:geometry, numerical analysis, graph theory; in other fields: physics, chemistry, political science, economics. In this talk we describe the origins of majorization and many examples of majorization and its consequences.
Thursday, November 17, 2011, 2:30-3:45pm, AB 102
Visiting Post-Doc
Penn State
Montesinos knots and the slice-ribbon conjecture
Abstract:The slice-ribbon conjecture states that a knot in the three sphere is the boundary of an embedded disc in the four ball if and only if it bounds a disc in the sphere which has only ribbon singularities. This conjecture was proposed by Fox in the early 70s. There doesn't seem to be any conceptual reason for it to be true, but large families of knots (i.e. pretzel knots, two bridge knots) satisfy it. In this colloquium we will show different ways to approach this conjecture with special emphasis in the case of Montesinos knots.
Friday, November 4, 2011, 2:30-3:45pm, DMS 102
Applied Mathematics, U of Arizona
Cofounder, Acunum Algorithms and Simulations, LLC
Numerical Laplace Transform Inversion Methods with Selected Applications
Abstract:Mathematical methods based on the use of the Laplace transform are a standard component of undergraduate engineering, mathematics, and physics education. Outside of the classroom however, real world problems often yield Laplace space solutions which are too complex to be analytically inverted to expressions in physically meaningful variables. A robust numerical inversion approach is thus desirable for these nontrivial cases. In this talk, I will present a few of the more common methods that have been developed to compute an approximate inverse. I will also discuss the inherent difficulties in performing numerical Laplace transform inversion. Finally, I will show through a selection of applications that these numerical inversion methods can be utilized to efficiently produce accurate results.
Thursday, November 3, 2011, 2:30-3:45pm, AB 102
American University
Washington, D.C.
Title: A Gentle Introduction to Stable Distributions
Abstract: Stable distributions are a class of heavy tailed
probability distributions that generalize the Gaussian distribution and that can be used to model a variety of problems. An overview of univariate stable laws is given, with emphasis on the practical aspects of working with stable distributions. Then a range of statistical applications will be explored. If there is time, a brief introduction to multivariate stable distributions will be given.
Thursday, September 22, 2011, 2:30-3:45pm, PE 208
Institute of Chemical Physics
Moscow, Russia
Energy exchange, localization, and transfer in finite oscillatory chains: Weak coupling approximation
Abstract: I will present a general approach to non-stationary dynamics of weakly coupled nonlinear oscillators. Two significant limiting cases can be distinguished. (i) Infinite (or very long) oscillatory chains can be considered in continuum approximation. Hence, they present appropriate objects for application of the ideas and methods of classical nonlinear field theory. Depending on the initial conditions, both wave-like (normal vibrations and waves) and particle-like (solitons) excitations may take place. However, for (ii) relatively short chains a different approach is required. The reason is that the formation of localized excitations and irreversible energy transfer is preceded by the stage of intensive energy exchange between groups of particles (“effective particles”). Maximal possible energy exchange occurs on the Limiting Phase Trajectory, which is a novel concept alternative to the quantum stationary state and the classical nonlinear normal mode. We obtain the description of the short chain in terms of “effective particles” when we reduce the oscillatory chain (in a certain frequency range) to a system of weakly coupled oscillators. Mathematically, the latter is similar to a multi-level quantum system. Then we determine the threshold for the transition from thestate of intensive energy exchange to the state of energy localization on an effective particle (or, possibly, energy transfer along the chain).
Special attention will be paid to classical linear and nonlinear systems with variable parameters, whose mathematical description is similar to that for a quantumsystem in an external field. An outstanding example of such a quantum system is given by the Landau–Zener Tunneling. The very possibility of a unified description of quantum and classical systems clearly shows the asymptotic nature of the wave-particle duality.
Download a PowerPoint presentation of the talk here.
Thursday, September 15, 2011, 2:30-3:45pm, AB 102
Department of Mathematics
Oregon State University
High-Order Staggered Finite Difference Methods for Maxwell's
Equations in Dispersive Media
Abstract: We consider high order (in space) staggered finite difference
schemes for Maxwell's equations coupled with a Debye or Lorentz
polarization model. A novel expansion of the symbol of arbitrary (even)
order finite difference approximations of the first order spatial
derivative operator allows us to derive a concise formula for the
numerical dispersion relation for all (even) order schemes applied to each
model, including the limiting (infinite order) case. We further derive a
closed-form analytical stability condition for these schemes as a function
of the order of the method. Using representative numerical values for the
physical parameters, we validate the stability criterion while quantifying
numerical dissipation. Lastly, we demonstrate the effect that the spatial
discretization order, and the corresponding stability constraint, has on
the dispersion error.
Tuesday, September 6, 2011, 2:30-3:45pm, AB 102
Department of Mechanical Engineering
University of South Florida
An Open Courseware for Numerical Methods
Abstract: Funded by National Science Foundation, since 2001, an innovative open courseware (http://numericalmethods.eng.usf.edu) has been developed for a comprehensive undergraduate course in Numerical Methods. The topics include 1) Introduction to Scientific Computing, 2) Differentiation, 3) Nonlinear Equations, 4) Simultaneous Linear Equations, 5) Interpolation, 6) Regression, 7) Integration, 8) Ordinary Differential Equations, 9) Partial DifferentialEquations, 10) Optimization, and 11) Fast Fourier Transforms.
The open courseware resources enhance instructor preparation and development as well as the student educational experience by facilitating a hybrid educational approach to theteaching of Numerical Methods, a pivotal STEM course, via a) customized textbooks, b) adapted course websites, c) social networking via blogs and YouTube, d) YouTube and iTunes digital audiovisual lectures, e) concept inventory, f) self-assessment of the level of learning via online multiple-choice question tests and algorithm-based unlimited attempt quizzes, g) worksheets in a computational system of choice, and h) real-life applications based on the choice of one’s STEM major.
The popularity of the open courseware is unprecedented. If you conduct a web search for “numerical methods”, you will find that the courseware is ranked #2 on Google, #4 on Yahoo, and #4 on Bing. In 2010, there were
· 1,016,206 page views (330,597 visits) to http://numericalmethods.eng.usf.edu,
· 532,675 views of the audiovisual lectures on YouTube (http://youtube.com/numericalmethodsguy), and
· 79,034 visits to the “Numerical Methods Guy” blog (http://autarkaw.wordpress.com).
In this talk, the speaker will discuss the development, refinement, and assessment process of the open courseware. The assessment results will include those of comparing several instructional modalities, measuring student learning, effect of collecting homework for a grade, using online quizzes as a substitute for grading homework, and interpreting summative ratings of the courseware, student satisfaction, and Google analytics.