24 January 2008
Principally Quasi-Baer Rings
Jae Keol Park
Department of Mathematics
Busan National University
South Korea

A ring R with identity is called right principally quasi-Baer (simply, right p.q.-Baer) if the right annihilator of every principal right ideal of R is generated by an idempotent of R. We discuss right p.q.-Baer rings with finite triangulating dimension. Also we introduce various properties of right p.q.-Baer rings emphasizing on the possibility of applications to C^*-algebras.

29 January 2008 (TUESDAY)
Prime Ideals in Mixed Polynomial/Power Series rings
Christina Eubanks-Turner
Department of Mathematics
University of Nebraska-Lincoln

This talk will present research on the partially ordered set of prime ideals of rings involving power series. In particular, we characterize the partially ordered set of prime ideals of R[[x]][g/f], where R is a one-dimensional Noetherian UFD and (f,g) is an R[[x]]-sequence.

31 January 2008
Asymptotic behaviour of models of structured population dynamics
Jozsef Farkas
Department of Computing Science and Mathematics
University of Stirling
Stirling, Scotland

In this talk we are going to discuss some recent results on the asymptotic behaviour of solutions of certain physiologically structured scramble and contest competition models. In particular, we employ semigroup and spectral methods to investigate the local asymptotic stability/instability of equilibrium solutions.

12 February 2008 (TUESDAY)
Spatial problems in mathematical ecology
Department of Mathematics
Andrew Nevai
Ohio State University

In this talk, I will introduce two spatial problems in theoretical ecology together with their mathematical solutions. The first part of the talk concerns competition between plants for sunlight. In it, I use a mechanistic Kolmogorov-type competition model to connect plant population vertical leaf profiles (or VLPs) to the asymptotic behavior of the resulting dynamical system. For different VLPs, conditions can be obtained for either competitive exclusion to occur or stable coexistence at one or more equilibrium points. The second part of the talk concerns the spatial spread of infectious diseases. Here, I use a family of SI-type models to examine the ability of a disease, such as rabies, to invade or persist in a spatially heterogeneous habitat. I will discuss properties of the disease-free equilibrium and the behavior of the endemic equilibrium as the mobility of healthy individuals becomes very small relative to that of infected. The family of disease models consists variously of systems of difference equations (which I will emphasize), ODEs, and reaction-diffusion equations.

21 February 2008
Sensitivity functions and their uses in parameter estimation problems
Sava Dediu
Center for Research in Scientific Computation
North Carolina State University

One of the most important questions in parameter estimation problems for dynamical systems is: How do we choose the length T of the sampling interval, such that to obtain more accurate parameter estimates when sampling data points from [0,T]? Another extremely important question is: Given a fixed number of measurements to be taken, what is their optimal time sampling in the interval [0,T] such that to obtain the most accurate estimates, once a time limit T was chosen? In this talk we will present our latest efforts to answer these questions based on the information provided by the traditional sensitivity functions (TSF) and the generalized sensitivity functions (GSF) from the perspective of least squares estimation problems for a Logistic Growth Population Model and a recently developed Agricultural Production Network Model. We argue that TSF and GSF provide the basis for new tools for investigators in design of inverse problem studies.

28 February 2008
Multi-scale stress analysis in random media
Robert Lipton
Department of Mathematics
Louisiana State University

A method for upscaling the local field concentrations inside random composite and polycrystalline media is presented. The talk focuses on gradient or strain fields associated with solutions of second order elliptic PDE used in the description of thermal transport and elasticity inside random media. We develop a method for assessing the L^p integrability of gradient and strain fields inside microstructured media. The results are described in terms of the p^th order moments of the solution of two-scale corrector problems. Examples are provided that illustrate the theory and its application.
In the second part of this talk we present new lower bounds on the L^p norms of local gradient or strain fields inside random media. The bounds are given in terms of the available statistical information describing the microstructure. We show that these bounds are the best possible as they are realized by several different classes of microstructures including coated confocal ellipsoids, coated spheres, and layered microstructures.

6 March 2008
Operator algebras - A sampler
Richard Kadison
Department of Mathematics
University of Pennsylvania

This will be a view of some of the basics of the theory of operator algebras, with a look at some intriguing questions - as time allows. My emphasis is on making what I say understandable, rather than on mentioning a great many topics.

13 March 2008
How to Define Multiplication on an Additive Structure
Gary R. Birkenmeier
Department of Mathematics
University of Louisiana at Lafayette

Let (G, +) denote an arbitrary (not necessarily commutative) group. We discuss how to construct all "multiplications" on G which are associative and/or left or right distributive over +. Also we consider the implementation of these concepts on a computer for finite groups. Elementary examples are provided.

20 March 2008
Numerical simulation of small scale coastal coherent structures in microtidal sea
Philippe Fraunie
Laboratoire de Sondages Electomagnetiques de l'Environnement Terrestre, CNRS and University of Toulon, France
(Visiting Professor at Arizona University)

Coherent structures on the mircotidal continental shelf are mainly driven by the wind forcing as observed from satellite images and drifted buoys and HF radar surface currents measurements. Such flows are induced by the non linear interaction of upwellings and density fronts in complex bathymetry. Specific numerical techniques have been recently developed including Eulerian and Lagrangian data assimilation, nested models and front capture TVD type numerical schemes. The sensitivity of the coastal flows to wind forcing both at mesoscale (continental and offshore winds) and sub mesoscale has been investigated using process oriented, realistic and climate scale (10 years) modeling. As a result, scales of oceanic response to the wind forcing are shown to lock on the local external and internal Rossby radius as well as bathymetric undulations in the case of microtidal coastal flows, contributing in small scale turbulent energy.

27 March 2008
Eigenfunction expansion method for the damped Boussinesq equation in a disc
Vladimir Varlamov
Department of Mathematics
University of Texas - Pan American

Solutions of semi-linear evolution equations in bounded domains can be constructed by the method of eigenfunction expansions. In contrast to Galerkin's method, the projection is made onto the infinite-dimensional space spanned by the set of eigenfunctions of the main elliptic operator. Of particular interest is a problem of excitation of a circular elastic membrane by an incident acoustic wave. Membrane oscillations are governed by the 2D damped Boussinesq equation. The solution in question is represented by a series of eigenfunctions of the Laplace operator in a disc. Proving decay of the eigenfunction expansion coefficients leads to an appearance of a new family of special functions, convolutions of Rayleigh functions with respect to the Bessel index. Rayleigh functions appear in classical linear problems of vibrating drumheads, heat conduction in cylinders and Fraunhofer diffraction through circular apertures. They are defined as a series \sigma_{l}(m)=\sum_{n=1}^{\infty}{\lambda_{m,n}^{2l}}, where \lambda_{m,n} are positive zeros of the Bessel function J_{m}(x), m=\pm1,\pm2,... and l,n=1,\,2,\,3,... Convolutions of such sums with respect to the Bessel function index form a new family of special functions. A general representation for this family is obtained and asymptotic expansions as |m|\to\infty are computed for practically important cases.

10 April 2008
Constructing morphic and quasi-morphic rings
Yiqiang Zhou
Memorial University of Newfoundland
St. John's, Canada

This talk is about the two notions of a morphic ring and a quasi-morphic ring, introduced by Nicholson and Sanchez Campos (2004) and by Camillo and Nicholson (2007) respectively. We will introduce some basic properties and examples of these rings as well as their natural connections with von Neumann regular rings and unit regular rings. The emphasis is to discuss how new examples of these rings are constructed and how these examples answer some existing questions and generate new questions. The talk is accessible to graduate students.

17 April 2008
An Introduction to the Matrix Theory of Field and Motion in the n-dimensional Riemannian space. Applications to the Motion in the Electromagnetic-Gravitational Field
Alexander D. Dymnikov
Louisiana Accelerator Center
University of Louisiana at Lafayette

A new matrix theory of motion in the n-dimensional Riemann space is developed. Two spaces are considered: the n-dimensional Riemannian space and the n-dimensional Euclidean space of non-integrable vectors and matrices. We call the last space as the absolute space. The absolute field matrix is introduced. The matrix field equations are derived. It is shown that the matrix metric equation is another form of the matrix field equation. Applications to the four-dimensional Riemannian spacetime are considered. It is shown that in this case the absolute field matrix is the absolute electromagnetic-gravitational field matrix. The symmetric part of this matrix is the gravitational field matrix. The antisymmetric part is the electromagnetic field matrix. The differential equations for these two matrices and for the metric matrix were obtained. The partial case of these equations is the Maxwell equations for the electromagnetic field. The mathematical metric matrix equation differs from the Einsteins metric equation.
Different solutions of the obtained metric equations in the Riemannian 4-spacetime are considered, for example, Friedmann-Lobachevsky model and the mathematical model of the Big Bang. The general solution for the static spherically symmetric gravitational field is found. This solution has no an apparent singularity at the Schwarzschild radius which the Schwarzschild metric has and it corresponds to the constant invariant density which is not zero (the Schwarzschild metric has zero density). For these solutions the determinant of the metric matrix is the same as for the flat spacetime. The approximate formulae for the minimum and maximum orbital velocity and for the perihelion precession of planets through perihelion and aphelion distances are given. A new solution for the static spherically symmetric gravitational field has been found. For this solution the determinant of the metric matrix differs from the determinant of the metric of the flat spacetime.

24 April 2008
Galois theory, commutative rings, and chromatic homotopy theory
Daniel Davis
Department of Mathematics
University of Louisiana at Lafayette

Some time ago, Galois theory for fields was extended to Galois theory for commutative rings. More recently, Rognes developed a Galois theory for commutative ring objects in stable homotopy theory, with an emphasis on finite group actions. Mark Behrens and I have extended a piece of Rognes's work to the case where the group is profinite. In particular, we have obtained a result about the "module of homomorphisms" between certain types of these commutative ring objects - a result which Behrens used to obtain an interesting result in chromatic theory about a finite resolution of the K(n)-local sphere. We will discuss these results and another theorem (due to myself and others) that Behrens and Lawson have generalized (in a beautiful and sprawling work that does many other things), for the purpose of using the arithmetic of Shimura varieties to understand the homotopy groups of the sphere.

1 May 2008 (SPECIAL TIME: 2:30)
Some Collapsibility Results for Multi-Dimensional Contingency Tables
P. Vellaisamy
Department of Statistics
Michigan State University
(On leave from Department of Mathematics, Indian Institute of Technology, Bombay)

Analysis of a large dimensional contingency table is difficult. Most often, the problems at hand can be analyzed using marginal (collapsed) tables. For a multidimensional contingency table, we discuss several necessary and sufficient conditions for collapsibility and strict collapsibility. The results are obtained using the technique of Mobius inversion formula. As a consequence, the results of Whittemore (1978, Journal of the Royal Statistical Society B, 40, 328-340) are stated in a form which are easy to understand and implement in practice. Our proofs are much simpler and straightforward. Several new results on collapsibility and strict collapsibility with respect to a set of interaction parameters and their relationships to conditional independence will be discussed. Some typical examples on collapsibility, strict collapsibility and conditional independence will be addressed. It will be shown that Bishop et al.[1975, Discrete Multivariate Analysis: Theory and Practice] conditions are necessary and sufficient for strict collapsibility with respect to a set of interaction factors.

1 May 2008 (SPECIAL TIME: 3:45)
Within-host virus dynamics: modeling, analysis and treatment
Patrick De Leenheer
Department of Mathematics
University of Florida

We revisit a standard model describing the infection cycle of a virus in an individual. One example of this model is furnished by HIV, a retrovirus which infects a particular class of immune cells, the CD4+ T cells, giving rise to infected T cells that spawn off new virus. The model is amenable to global analysis: A Lyapunov function can be found under certain conditions, establishing global stability of the infection steady state. Moreover, the system is a 3 dimensional competitive dynamical system. Consequently, it shares the Poincare-Bendixson property with planar system, and thus a fairly complete picture of its dynamical behavior can be obtained. This behavior is richer than thought previously. For instance, sustained oscillations are possible, at least theoretically. We will go on to discuss the effect of periodic treatment on the system, and give tight bounds for the drug efficiencies required to eradicate the infection. Finally, if time permits, we will modify the model to account for mutations. It turns out that for small mutation rates, many results of the single strain model carry over.

8 May 2008
DAETS: a Differential-Algebraic Equation code in C++ for high index and high accuracy
John D. Pryce
(recently retired)
Royal Military Academy
Shrivenham

Ned Nedialkov (McMaster University, Canada) and John Pryce (Cranfield University, UK) are the authors of DAETS, a C++ code for solving differential-algebraic equations (DAEs), version 1.0 of which has just been released. It uses Pryce's structural analysis theory, and expands the solution in Taylor series using automatic differentiation. DAETS is very effective when high accuracy is required, and at solving problems of high index---we have solved artificial DAEs of index up to 47. It is versatile: higher-order systems do not have to be cast in first-order form; it can solve explicit and implicit ODEs; it can solve purely algebraic problems, by simple or by arc-length continuation.