11 September 2008
What Is A Semigroup?
Henry E. Heatherly
Department of Mathematics
University of Louisiana at Lafayette

The study of abstract semigroups was motivated by work with various concrete types of semigroups: in algebra, topology, and analysis (including differential equations). We look at some of these motivational examples, especially those from algebra. Basic concepts and results of semigroup theory are introduced. The importance of von Neumann regularity in modern semigroup theory is discussed. Some recent results on semigroups of endomorphisms are given, illustrating one direction semigroup theory research is now taking.

25 September 2008
Non-Oscillatory Central Schemes -- a Powerful Black-Box-Solver for Hyperbolic PDEs
Alexander Kurganov
Department of Mathematics
Tulane University

I will first give a brief description of finite-volume, Godunov-type methods for hyperbolic systems of conservation laws. These methods consist of two types of schemes: upwind and central. My lecture will focus on the second type -- non-oscillatory central schemes.
 
Godunov-type schemes are projection-evolution methods. In these methods, the solution, at each time step, is interpolated by a (discontinuous) piecewise polynomial interpolant, which is then evolved to the next time level using the integral form of conservation laws. Therefore, in order to design an upwind scheme, (generalized) Riemann problems have to be (approximately) solved at each cell interface. This however may be hard or even impossible.
 
The main idea in the derivation of central schemes is to avoid solving Riemann problems by averaging over the wave fans generated at cell interfaces. This strategy leads to a family of universal numerical methods that can be applied as a black-box-solver to a wide variety of hyperbolic PDEs and related problems. At the same time, central schemes suffer from (relatively) high numerical viscosity, which can be reduced by incorporating of some upwinding information into the scheme derivation -- this leads to central-upwind schemes, which will be presented in the lecture.
 
During the talk, I will show a number of recent applications of the central schemes.

9 October 2008
Quasi-Stationary Optical Solitons with non-Kerr Law Nonlinearity
Anjan Biswas
Applied Mathematics and Theoretical Physics Department
Delaware State University

This talk is going to be on optical solitons and its perturbations that is governed by the generalized nonlinear Schrödinger's equation with non-Kerr law nonlinearities. The multiple scale perturbation analysis is applied to study the perturbed nonlinear Schrodinger's equation. A new definition of the phase is going to be introduced that will capture the variation of the soliton parameters up to order epsilon, which otherwise is a failure by soliton perturbation theory. The perturbation terms that are going to be considered are nonlinear damping and saturable amplifiers.

6 November 2008
Numerical Implementation of the Asymptotic Boundary Conditions for Steadily Propagating 2D Solitons of Boussinesq Equation
Christo I. Christov
Department of Mathematics
University of Louisiana at Lafayette

The Boussinesq equation (BE) was the first equation that was derived for surface waves in shallow fluid layer when both nonlinearity and dispersion were taken into account. BE appears also in a modeling elastic rods and shells. In a coordinate frame moving with the center of the propagating wave, BE reduces to the Korteweg-De Vries equation (KdVE) which is widely studied in 1D, especially in connection with solitary waves and solitons. At the same time, results for the 2D localized solutions of BE (or KdVE) cannot be found, which justifies a deeper look into the problem.
 
Unlike the 1D case, where analytical one- and two-soliton solutions can be obtained for some of the limiting cases of BE, none of the well known techniques (such as Hirota bilinear transformation, Backlund transformation, inverse scattering) are available in 2D which leaves numerical and semi-analytical techniques as the only possible tools for attacking the problem. In the present talk, the issues to be overcome for obtaining accurate difference solution are discussed: the bifurcation nature of the localized solution; and the proper implementation of the asymptotic boundary conditions. In 2D, the decay of the profile at infinity is second order algebraic which is much slower in comparison with the exponential decay in 1D. This imposes more demanding requirements on the approximation on top of the fact that a.b.c.'s are nonlocal as well: involving the two different partial derivatives of the solution.
 
Results for the 2D shapes are presented for different values of the governing parameters and for different phase speeds of the solitons. For validation, the obtained shapes are compared to the results of an asymptotic semi-numerical solution for small phase speeds, and the results are in excellent agreement.

13 November 2008
Mainstream contributions of Interval Computations in Engineering and Scientific Computing
R. Baker Kearfott
Department of Mathematics
University of Louisiana at Lafayette

Interval arithmetic, visible in its present form in the scientific computing literature for at least 46 years, has had a consistent strong following among experts in the field. Giving mathematical rigor to machine computations based on rounded approximations to real numbers, interval arithmetic has held enticing promise. Here is an outline:
1. We first present the basic mathematical questions that interval arithmetic can possibly answer.
2. We then briefly review the elements of interval arithmetic.
3. We point out pitfalls in the naive use of interval arithmetic.
4. We mention subjects in which interval arithmetic has already had a significant impact in commercial software and applications.
5. We briefly outline current research and promising areas for future impact.

20 November 2008
Compatible Discretizations for Continuous Dynamical Systems
Hristo Kojouharov
Department of Mathematics
University of Texas at Arlington

A new class of one-step nonstandard finite difference methods is developed for first-order ordinary differential equations. The proposed numerical techniques are based on a nonlocal modeling of the right-hand side function and a nonstandard discretization of the time-derivative. This approach leads to significant qualitative improvements in the behavior of the numerical solution. For multi-dimensional autonomous dynamical systems, positive and elementary-stable nonstandard finite difference methods are formulated and analyzed, based on an extension of the nonstandard discretization rules. Applications of the nonstandard finite difference methods to specific biological systems are also presented.

25 November 2008 (TUESDAY)
Estimating species phylogenies under the coalescence model Liang Liu
Department of Organismic and Evolutionary Biology
Harvard University

Estimating the evolutionary history of species is one of the most important problems in evolutionary biology and recently there has been greater appreciation of the need to estimate species trees directly, other than using gene trees as a surrogate. In this talk, I will introduce three approaches for estimating species phylogenies from multilocus data under the coalescence model. The Bayesian approach, known as BEST, uses the full dataset to infer the species phylogeny, while the other two approaches (STAR and STEAC) use only partial information of the dataset. All three approaches can consistently estimate species phylogenies. Since the Bayesian approach involves intensive computation, it is impossible to use it to analyze large-scale genomic data which may include thousands of genes. By contrast, STAR and STEAC are based on summary statistics of coalescence times which are easy to compute and thus are suitable for the phylogenetic analysis of large-scale genomic data.

4 December 2008
Minimum Hellinger Distance Estimation: Strategies for Improved Efficiency
Ayanendranath Basu
Department of Statistics
Pennsylvania State University

The minimum Hellinger distance estimator and related methods are popular tools in statistical inference, primarily because they combine full asymptotic efficiency with attractive robustness properties (e.g. Beran, 1977; Simpson 1987; Lindsay 1994). The first half of the talk will give a general introduction to the philosophy and application domains of minimum distance procedures in general and the procedures based on the minimized Hellinger distance in particular. The popularity of many of these methods is partially tempered in practice by the poor small sample properties of these estimators compared to maximum likelihood estimator. Empirical evidence appears to suggest that this deficiency is at least partially due to the improper treatment of inliers by these procedures. In the second half of the talk general strategies to improve the small sample performance of these procedures will be described, and the performance of the resulting estimators will be discussed.