22 January 2009
Algebras and Related Rings
Efraim Armendariz
Department of Mathematics
University of Texas at Austin

Let A be an ring which is also an algebra over a field K; A is said to be algebraic over K if every element in A is a root of a polynomial with coefficients in K. These algebras have been extensively studied, yet many interesting open problems persist. I will discuss these problems as well as possible extensions of known classes of rings that encompass algebraic algebras.

29 January 2009
Persistence in Discrete-time Dynamical Systems
Paul Leonard Salceanu
Department of Mathematics and Statistics
Arizona State University

The concept of persistence, or permanence, emerged in the late seventies as a mathematical tool to describe the long term survival, or coexistence, of some, or all interacting species in an ecosystem. The typical mathematical framework for persistence is that of dynamical systems, or semiflows, generated, for example, by differential or difference equations. In this framework, persistence requires that the set of all extinction states, which is usually a closed subset of the boundary of the state space, is a repeller for the dynamics on the complementary set. Thus, persistence allows a more general characterization of coexistence, compared to the previously used global convergence to an equilibrium. Epidemiological models are especially rich sources of issues that can be decided by the theory: Can a disease drive the host population to extinction? This is a question of host persistence. Does the disease become endemic in a population? This is the question of disease persistence. However, to-date, the theory has been difficult to apply to dynamical systems arising in epidemiology and population dynamics because of the complexity of the boundary attractor. First, I will present an age structured (juvenile-adult), SI (susceptible and infected) model of Emmert and Allen for which we show that the host persists despite the presence of the disease. Moreover, working under the assumption of convergent boundary dynamics, we provide sufficient conditions for the disease to persist in the population. Then, using the notion of Lyapunov exponents, I will construct a framework which will decide (disease) persistence in the more general case that the boundary dynamics is not convergent.

3 February 2009 (TUESDAY)
Plant-herbivore Interactions Mediated by Plant Toxicity
Rongsong Liu
Dept. of Mathematics
Purdue University

We explore the impact of plant toxicity on the dynamics of a plant-herbivore interaction, such as that of a mammalian browser and its plant forage species, by studying a mathematical model that includes a toxin-determined functional response. In this functional response, the traditional Holling Type 2 response is modified to include the negative effect of toxin on herbivore growth, which can overwhelm the positive effect of biomass ingestion at sufficiently high plant toxicant concentrations. A detailed bifurcation analysis of the system reveals a rich array of possible behaviors including cyclical dynamics through Hopf bifurcations and homoclinic bifurcation.

5 February 2009
High Order Irregular Singularities in Differential Equations and Applications in Mesoscopic Systems
Andrei Ludu
Dept. of Chemistry and Physics
Northwestern State University
Natchitoches

Mesoscopic superconductors (quantum wires and dots, etc.) are microscopic systems of Cooper-pairs described by the Ginzburg-Landau field theory model, and their corresponding dynamical partial differential equation (PDE) can be reduced to a 3-dimensional Gros-Pitaevski equation (nonlinear Schrödinger equation) with boundary conditions. In general, the external magnetic field symmetry and the sample geometry are not compatible, so exact solutions for this system are very difficult to be found. The most interesting solutions are related to superconducting vortices structures that can be a future candidate for ultra-fast reliable memory devices.
 
We investigate cylindrical and spherical samples (solid and hollow) in uniform external magnetic field and in the internal field of a magnetic dipole and find that the PDE can be reduced in appropriate coordinates to a linear differential equation with higher order singularities of Heun type. We study such analytic solutions, the Sturm-Liouville associate problem, and their corresponding vortex structures.
 
The nonlinear terms of the model are taken into account by minimizing the free energy functional of the field theory on the space of the linearized independent solutions.

12 February 2009
Degree Bounds in Invariant Theory
Mara D. Neusel
Department of Mathematics and Statistics
Texas Tech University
Lubbock

Invariant Theory of Finite Groups studies linear actions of (finite) groups on polynomials. Typical questions are: (a) How do we construct invariant polynomials? (b) How do we find all of them? (c) Which (algebraic, homological, geometric, combinatorial, ...) properties does the resulting ring of invariant polynomials have? (d) How do we determine those? (e) How do they depend on the input data? In order to approach these problems adequately, we use methods and results from a wide range of mathematics, like combinatorics, algebraic topology, commutative algebra, group theory, homological algebra, and algebraic geometry. On the other hand, invariant theoretical results have not only applications in the above mentioned fields, but also to areas like graph theory, numerical analysis, control theory, coding theory, and even to physics and engineering. I will give an introductory survey on Invariant Theory by focusing on the particular problem of degree bounds.

19 February 2009
Field theoretical approach to dynamics of deformation and fracture
Dr. Sanichiro Yoshida
Department of Chemistry and Physics
Southeastern Louisiana University
Hammond

26 February 2009
Mathematical Models of T-cell Development
Stanca M. Ciupe
Laboratory of Computational Immunology
Duke University Medical Center

The immune response to infectious agents involves the presence and maintenance of a large number of T cells with highly variable antigen receptors and functional diversity. We develop a stochastic population-dynamic model that studies the mechanisms responsible for the establishment of T cell receptor diversity. We fit the model to human data from immunocompromised DiGeorge anomaly patients undergoing thymus transplantation. The dynamics we see in the evolution of T cells gives valuable information about the characteristics of the healthy immune system.

26 March 2009 CANCELED
Mathematical Modeling of Dominant or Recessive Transgenic Mosquitoes and Allee Effects
Jia Li
Department of Mathematical Sciences
University of Alabama in Huntsville

To prevent the transmission of malaria and other mosquito-borne diseases, transgenic (genetically-altered) mosquitoes, that are resistant to malaria infection, become an effective weapon. To study the impact of releasing transgenic mosquitoes into the field of wild mosquitoes, we formulate mathematical models of interacting mosquito populations, with dominant or recessive transgenes, respectively. Dynamics of these models are explored by investigating existence and stability of boundary and positive equilibria. The models exhibit richer dynamics which are demonstrated by numerical simulations.

31 March 2009 TUESDAY
A Fundamental Bifurcation Theorem for Darwinian Matrix Models
J. M. Cushing
Department of Mathematics & Interdisciplinary Program in Applied Mathematics
University of Arizona
Tucson

Matrix models (systems of difference equations or maps) are commonly used to model the (discrete time) dynamics of biological populations structured according to some classification scheme (age, size, life cycle stage, etc.). A fundamental problem in theoretical population dynamics and ecology is to determine under what conditions a model predicts that a population will go extinct or will survive. The Fundamental Bifurcation Theorem for population dynamic matrix models deals with this problem from a bifurcation theory point-of-view. Under general conditions, this theorem asserts the loss of stability of the extinction equilibrium, and the resulting bifurcation of non-extinction equilibria (whose stability depends on the direction of bifurcation), as a basic demographic parameter increases through a critical value. Using methods of evolutionary game theory, one can extend population dynamic matrix models to so-called Darwinian matrix models. These evolutionary models include the dynamics of phenotypic traits (that have a heritable component and are subject to natural selection) and describe how the dynamics (evolution) of these traits affects the population dynamics and vice versa. I will show how the Fundamental Bifurcation Theorem can be generalized to Darwinian matrix models. I will also give some applications and discuss a few open problems.

30 April 2009
On Commuting Automorphisms of Groups
Gary L. Walls
Southeastern Louisiana University

Let $G$ be a finite group. An automorphism $\alpha \in Aut(G)$ is said to be a commuting automorphism provided $\textrm{ for all } x \in G \ xx^{\alpha}=x^{\alpha}x$. In this talk we will investigate the consequences for a group of the existence of non-trivial commuting automorphisms. We also consider the relationship between commuting automorphisms and central automorphisms and briefly discuss $A$-groups, groups in which all automorphisms are commuting.

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.

2 December 2008 (TUESDAY)
Stoichiometry of Daphnia, Algae and Bacteria and Species Competition
Hao Wang
Georgia Tech

We carried out a microcosm experiment evaluating competition of an invasive species Daphnia lumholtzi with a widespread native species, Daphnia pulex. We applied two light treatments to these two different microcosms and found strong context-dependent competitive exclusion in both treatments. To better understand these results we developed and tested a mechanistically formulated stoichiometric model. This model exhibits chaotic coexistence of the competing species of Daphnia. The rich dynamics of this model as well as the experiment allow us to suggest some plausible strategies to control the invasive species D. lumholtzi.
 
We modeled bacteria-algae interactions in the epilimnion with the explicit consideration of carbon (energy) and phosphorus (nutrient). We hypothesized that there are three dynamical scenarios determined by the basic reproductive numbers of bacteria and algae. Effects of key environmental conditions were examined through these scenarios. Bifurcation diagrams for the depth of epilimnion mimic the profile of Lake Biwa, Japan. Competition of bacterial strains were modeled to examine Nishimura's hypothesis that in severely P-limited environments such as Lake Biwa, P-limitation exerts more severe constraints on the growth of bacterial groups with higher nucleic acid contents, which allows low nucleic acid bacteria to be competitive.

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.

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.

20 September 2007
The Axiom of choice: Origins, Equivalences, and Some Applications
Henry Heatherly
Mathematics Department
University of Louisiana at Lafayettte

Origins and easy uses of the Axiom of Choice are discussed. Applications of the axiom in various areas of mathematics are given, with emphasis on applications in abstract algebra. Various equivalent conditions to the Axiom of Choice in Zermelo-Fraenkel set theory are considered. Some recent results, arising in ring theory, are examined.

27 September 2007
Solving equations in algebra
Gunter F. Pilz
Vice Rektor and Dept. of Algebra
Johannes Kepler Univ. Linz

What is an equation? Few people know that. What is an algebraic equation? When is such an equation or system of equations solvable? For instance, xyz=0 and zyx=8 is clearly unsolvable in the class of commutative rings with identity, but there exists a solution in the set of 2x2-matrices over the reals. We might fix an algebraic structure A and look if a system of equations has a solution in A or in an extension B of A. We get a criterion for solvability in some extension by a generalization of Hilbert's Nullstellensatz. But very strange things can happen. For instance, an equation can be solvable in an extension C of A, but not in an extension of B (as above). We also touch the theory of algebraically closed groups and other structures, and also algorithmic aspects like Groebner bases.

18 October 2007
Nonlinear Wave Phenomena in Continuum Mechanics: Some Recent Findings
Pedro Jordan
Naval Research Laboratory
Stennis Space Center, Mississippi

Traveling wave solutions (TWS) are explored in the context of nonlinear acoustics. Exact solutions are given, including one involving the recently introduced Lambert W-function, along with asymptotic and stability results. Poroacoustic propagation under Darcy's law is examined, as well as acoustic phenomena in thermoviscous fluids. Additionally, a connection between discontinuity formation in the TWS and the associated singular surface, which is known as an acceleration wave, is pointed out. Lastly, if time permits, applications to nonlinear kinematic wave phenomena (e.g., second-sound and traffic flow) are briefly noted.

25 October 2007
Quenching for Degenerate Parabolic Problems with Nonlocal Boundary Conditions
H. Terrence Liu
Department of Applied Mathematics, Tatung University
Taipei, Taiwan 104, Republic of China
Department of Mathematics, University of Southern Mississippi

Let $q$ be a nonnegative real number, $a$ and $T$ be positive constants, $G$ be a nonnegative function in the form of either $f(u(x,t))$, or $\int_{0}^{a}h(x,t)f(u(x,t))dx$ for some positive, bounded and continuous function $h$ with $f>0$, $f'>0$, $f''\geq0$, and $\lim_{u\rightarrow1^{-}}f(u)=\infty$. We study the following degenerate parabolic equation, \[x^{q}u_{t}-u_{xx}=G(u)\text{\ in\ }(0,a)\times(0,T),\] subject to the initial condition, \[u(x,0)=0\text{\ on\ }[0,a],\] and the nonlocal boundary conditions, \[u(0,t)=\int_{0}^{a}M(x)\left| u\left( x,t\right) \right| ^{p}dx\text{, }u\left( a,t\right) =\int_{0}^{a}N\left( x\right) \left| u\left( x,t\right) \right| ^{r}dx\text{, }t>0,\] where $p$ and $r$ are constants greater than or equal to $1$, and $M$ and $N$ are given functions. A solution $u$ is said to quench at $T$ if $\lim _{t\rightarrow T^{-}}\max_{0\leq x\leq a}u\left( x,t\right) =1$. Existence, uniqueness and criteria for quenching and non-quenching of a solution $u$ are discussed.

20 November 2007 (TUESDAY)
The Schur Horn Theorem In Infinite Dimensions
Victor Kaftal
Department of Mathematical Sciences
University of Cincinnati

In this talk we will explore the connections between (finite dimensional) majorization theory, stochastic matrices, diagonals of selfadjoint matrices and the extension of these notions to infinite dimensions. The main result of the joint work of the speaker and Gary Weiss is that a positive nonsummable sequence x decreasing to zero is the diagonal of a positive compact Hilbert space operator with eigenvalue list y if and only if y majorizes x.

27 November 2007 (TUESDAY)
Effects Of Concentrated Nonlinear Sources On Blow-Up And Quenching Phenomena
C. Y. Chan
Mathematics Department
University of Louisiana at Lafayettte

Blow-up and quenching phenomena modeled by parabolic first initial-boundary value problems due to concentrated nonlinear sources are investigated. For the one-dimensional problems, existence, uniqueness, and behavior of solutions are given. A correct formulation of such problems to multi-dimensions is discussed. The talk should be of interest to a general audience.

30 November 2007 (FRIDAY: 10:00 room 312)
Measuring objects and holes, I
Nell Sedransk
National Institute of Statistical Sciences

Objects and holes exist in 3-space. Unstable axes or centroids distort 3-dimensional shapes, while dents, bulges, creases, ridges, pits and nubs perturb smooth shapes locally, regionally and/or globally. Making inference depends on the measurement process; designing an efficient measurement process is a 3-dimensional challenge and often simultaneously a 2-dimensional and 1-dimensional challenge as well. The problem of experimental design is examined for two very different (engineering) contexts and from two different vantage points; the first is centered inside, and the second is centered outside.
 
Design and Analysis when the Center is Internal
 
The range of controllable arm motion for a patient with a spinal cord injury exists as an imaginary object, often with a vacuous center. Expansion of this "controllable space" is a metric for assessing success of a therapeutic intervention or progressing degradation of function through stricture or muscle weakness and failure. Inference about changes in the controllable space depends upon the measurement protocol, i.e., the experimental design. Patient fatigue during the measurement process severely limits the number of trials. Bayesian methods for inference as well as design offer some solutions for making inferences about the shape of the controllable space and that of its vacuous center.

30 November 2007 (FRIDAY: 2:00 room 208)
Measuring objects and holes, II
Nell Sedransk
National Institute of Statistical Sciences

Objects and holes exist in 3-space. Unstable axes or centroids distort 3-dimensional shapes, while dents, bulges, creases, ridges, pits and nubs perturb smooth shapes locally, regionally and/or globally. Making inference depends on the measurement process; designing an efficient measurement process is a 3-dimensional challenge and often simultaneously a 2-dimensional and 1-dimensional challenge as well. These examples illustrate the transfer classical experimental design principles to modern, non-standard experimental situations.
 
Design and Analysis when the Vantage Point is External
 
High-precision is required for reflective and optical surfaces that are used from the megamacro- to the nano-scale in state-of-the-art scientific equipment from telescopes to scanning tunneling electron microscopes. Standard objects for calibration of such equipment must be ultra-smooth and as perfectly formed as is possible with current technology. The canonical, but real, problem is the manufacturing of a perfect sphere. Local curvature can be measured, pointwise from the surface, by refraction. The design strategy is to proceed from the global form down to the surface details.

18 January 2007
Geometrical Operations on Hierarchical Structures with Result Verification
Eva Dyllong
Institute of Informatics and Interactive Systems
University of Duisburg-Essen, Germany

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Efficient and reliable computational routines to realize geometrical operations are essential for many applications. In this talk, we focus on the problem of a reliable construction and geometrical operations on interval-based hierarchical structures with result verification.
Hierarchical object representations are the most frequently used data structures for the reconstruction of a real scene. This object modeling does not depend on the nature of the real object but only on the maximum subdivision level of the tree. This is a useful property for objects with a complex shape that are difficult to describe via exact mathematical formulas.
Techniques of reliable computing like interval arithmetic can be used to guarantee a reliable solution even in the presence of numerical round-off errors. The need to trace bounds for the error function separately can be eliminated using these techniques.
In this talk, we show how the techniques and algorithms of reliable computing can be applied to operations which are utilized for the construction and further processing of hierarchical object representations. We conclude the talk with some examples of implementations.

25 January 2007
Allee effects and pulsed invasion in the gypsy moth
Derek Johnson
Department of Biology
University of Louisiana at Lafayette

Biological invasions impose considerable threats to the world’s ecosystems and incur substantial economic losses. A prime example is the invasion of the gypsy moth in the United States for which over $194 million were spent on management and monitoring between 1985-2004 alone. The spread of the gypsy moth across eastern North America is, perhaps, the most thoroughly studied biological invasion in the world, providing a unique opportunity to explore spatiotemporal variability in rates of spread. Herein we report evidence of periodic pulsed invasions, defined as regularly punctuated range expansions interspersed among periods of range stasis, based on a second-order density-dependent theoretical model. The model was parameterized from long-term monitoring data and shows how an interaction between strong Allee effects (negative population growth at low densities) and stratified diffusion (most individuals disperse locally, but a few seed new colonies by long-range movement) can explain the invasion pulses. Our results suggest that suppressing population peaks along range borders may greatly slow invasion.

1 February 2007
On sets with convex shadows
Jan J. Dijkstra
Vrije Universiteit Amsterdam

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We investigate topological properties of objects that appear convex when viewed from different directions. This research project in geometric tomography was carried out jointly with Stoyu Barov (Bulg. Acad. Sci.). If P is a set of projection directions in Rⁿ then two subsets of Rⁿ are called weak P-imitations of each other if in every direction from P the shadows of the two sets have identical closures. The following result is representative.
Theorem. Let B be a closed convex subset of Rⁿ that contains no hyperplane and let P be an open set of projection directions that contains a direction in which the shadow of B is not a full hyperplane. If C is a closed weak P-imitation of B then C = B or C contains a closed subset that is an (n-2)-manifold.
In addition to results of this type we also present examples of `minimal imitations' of convex bodies that show that our theorems are sharp.

7 February 2007 (WEDNESDAY)
Availability of a Multi-unit Repairable System
Jyoti Sarkar
Department of Mathematical Sciences
Indiana University Purdue University Indianapolis

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The availability of a repairable system is an important aspect of reliability theory. While the reliability of a multi-unit coherent system can be calculated easily as a function of the reliability of the individual units (assuming that the units operate independently), the same function is rarely valid in calculating the instantaneous availability or the steady-state availability. Practical considerations make availability calculation a whole lot more complicated. We demonstrate this feature in some simple models, and expose a host of unsolved problems.

8 February 2007
Algorithms in Discrete Morse Theory
Kevin Knudson
Mississippi State University

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Discrete Morse theory was developed by Robin Forman to provide a combinatorial analogue, for simplicial complexes, of classical smooth Morse theory on manifolds. Constructing efficient discrete Morse functions is a nontrivial task. In this talk, I will present an algorithm that begins with a function h defined on the vertices of a complex K and extends it to a discrete Morse function on the entire complex so that the resulting discrete gradient field mirrors the large scale behavior of h. This has applications to the analysis of point cloud data sets and several examples will be given. No prior knowledge of Morse theory (discrete or smooth) will be assumed.

12 February 2007 (MONDAY)
Apparent paradoxes in disease models with horizontal and vertical transmission
Horst Thieme
Department of Mathematics and Statistics
Arizona State University

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The question as to how the ratio of horizontal to vertical transmission depends on the coefficient of horizontal transmission is investigated in host-parasite models with one or two parasite strains. In an apparent paradox, this ratio decreases as the coefficient is increased provided that the ratio is taken at the equilibrium at which both host and parasite persist. Moreover, a completely vertically transmitted parasite strain that would go extinct on its own can coexist with a more harmful horizontally transmitted strain by protecting the host against it. Several stability results are presented for the coexistence equilibrium (host and two parasite strains). Under standard incidence, undamped oscillations may occur.

15 February 2007
A Bayesian Hierarchical Non-Overlapping Random Disc Growth Model
Athanasios Micheas
Department of Statistics
University of Missouri Columbia

A methodology is proposed to efficiently model a random set via a multistage hierarchical Bayesian model. We define a Non-Overlapping Random Disk Model (NORDM), which includes the well-known Poisson-Boolean model as a special case. This model is formulated in a conditional setting that facilitates Bayesian sampling of important parameters in the model. Utilizing a transformation to the disc, this framework can accommodate any object, not just those with disk shapes, although the model can be easily extended to include any known compact convex set instead of the disc, e.g., polygons or ellipses. We further propose a growth model that is conceptually simple and allows straightforward estimation of parameters, without the need for tedious calculations of hitting or inclusion probabilities. The model is applied to severe storm cell development as obtained from weather radar.

23 February 2007 (FRIDAY 3:00)
Free Subgroups of Uniform Lattices
Lewis Bowen
Department of Mathematics
Indiana University

A discrete group F acting by isometries on hyperbolic space Hⁿ has a natural "shape" or modulus. This is the isometry class of the quotient manifold Hⁿ/F. For a given a uniform lattice Gamma of the isometry group of Hⁿ there are two natural problems.
1. For a given compact set of moduli, count the number of conjugacy classes of subgroups of Gamma with modulus in that set.
2. Describe the set of all moduli of subgroups of Gamma.
If we restrict attention to the case of subgroups isomorphic to the integers, then these are much-studied problems concerning the length spectrum. On the other hand, if we restrict to the case of subgroups isomorphic to the fundamental group of a closed surface of high genus then these problems are wide open and answers are much desired. The talk will focus on the intermediate case, when the subgroups are free groups.

8 March 2007
Instantaneous Availability of a Repairable System
Jyoti Sarkar
Department of Mathematical Sciences
Indiana University Purdue University Indianapolis

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A repairable system undergoes cycles of operation, breakdown, repair and re-installation. What is the probability that at any particular instant the system is operational? We answer this question for a continuously monitored system under a perfect repair policy when there is no time lost for commencement of repair or installation to operation. Also we answer the question if a spare unit takes over operation when the main unit is undergoing repair. What if we further allow recall of a nonfailed unit when preventive maintenance on it is likely to be quicker than repair of a failed unit? We present partial answer to this question, and to questions arising in some other repair models.

12 March 2007 (MONDAY 3:30)
Traveling Waves in Epidemic Models
Shigui Ruan
Department of Mathematics
University of Miami

In this talk, we first review some classical epidemic models, such as the Ross-Macdonald model, the Kermack-McKendrik model, the Kendall model, etc. The existence of traveling waves in some epidemic models is demonstrated. Then we propose a host-vector model for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modeled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting convex sets technique. When the spatial variable is one-dimensional and the delay kernel assumes some special form, we establish the existence of traveling wave solutions by using the linear chain trick and the geometric singular perturbation method.

15 March 2007
Fixed points in group cohomology, topological algebra, and homotopy theory
Daniel Davis
Department of Mathematics
Wesleyan University

Let G be a group. The study of Z[G]-modules and their G-fixed points leads naturally to group cohomology. Similarly, if G is a topological group and one requires continuous group actions, consideration of pro-discrete G-modules leads to continuous cochain cohomology. A review of these topics, after a natural change of context, allows us to understand some theorems in chromatic homotopy theory about G-homotopy fixed points for continuous G-spectra. For example, the iterated homotopy fixed points of the Lubin-Tate spectrum behave simply, just like the iterated fixed points of a G-module in algebra.

22 March 2007
Residuated (and related) mappings revisited
Richard J. Greechie
Mathematics and Statistics
Louisiana Tech University

Our setting is that of partially ordered sets and lattices, sometimes specified to Boolean algebras or orthomodular lattices. We survey some known results on residuated mappings and discuss some recent results on the calculation of approximations of isotone mappings by residuated mappings.

29 March 2007
GlobSol -- Present State and Future Developments
Baker Kearfott
Mathematics Department
University of Louisiana at Lafayette

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There has been increasing interest in automatically verified global optimization in recent years. A consequence is that the community is learning more about the strengths and pitfalls of techniques in the associated software. We will discuss these strengths and pitfalls, then describe ongoing work to use this knowledge to improve our GlobSol package.

3 April 2007 (TUESDAY 2:30)
Quandles and their homology with applications in knot theory
Maciej Niebrzydowski
George Washington University

16 April 2007 (MONDAY)
A class of C*-algebras arising from minimal shift spaces
Efren Ruiz
University of Toronto

In recent years, there have been mutual interactions between operator algebras and dynamical systems. In particular, dynamical systems have provided many interesting and important examples of C*-algebras and C*-algebras have provided interesting equivalence relations between two dynamical systems. One of the most famous result is due to Giordano, Putnam, and Skau. Using C*-algebras, they showed that minimal Cantor systems can be classified up to orbit equivalence (in a strong sense) via their naturally associated pointed ordered groups.
In this talk, I will talk about the work of Matsumoto in which he constructed a C*-algebra from a shift space. I will show a certain class of these Matsumoto algebras can be classified using K-theoretical invariants. If we say that two shift spaces are equivalent if the associated Matsumoto algebras are stably isomorphic, then a consequence of this classification result is that this relation is a coarser relation than flow equivalence. Also, this relation is determined by computable objects.
This is joint work with Soren Eilers and Gunnar Restorff.

17 April 2007 (TUESDAY 2:30)
Biased bootstrap methods for semiparametric models
Mihai Guircanu
Department of Statistics
University of Florida

19 April 2007
Topological orbit equivalence of free, minimal actions on the Cantor set
Thierry Giordano
University of Ottawa

In 1959, H. Dye introduced the notion of orbit equivalence and proved that any two ergodic finite measure preserving transformations on a Lebesgue space are orbit equivalent. He also conjectured that an arbitrary action of a discrete amenable group is orbit equivalent to a Z-action. This conjecture was proved by Ornstein and Weiss and its most general case by Connes, Feldman and Weiss by establishing that an amenable non-singular countable equivalence relation R can be generated by a single transformation, or equivalently is hyperfinite, i.e., R is up to a null set, a countable increasing union of finite equivalence relations. In the Borel case, Weiss proved that actions of Zⁿ are (orbit equivalent to) hyperfinite Borel equivalence relations, whose classification was obtained by Dougherty, Jackson and Kechris. In this talk, after having reviewed these results, I will describe the topological counterpart of the orbit equivalence and the classification up to orbit equivalence of minimal, free actions of Z and Z² on the Cantor set.

23 April 2007 (MONDAY)
Effect of Nonlinear Coupling on the Particle-Like Behaviour of the Localized Waves in Vector NLSE
M. D. Todorov
Dept. of Differential Equations
Faculty of Applied Mathematics and Informatics
Technical University of Sofia
Sofia, Bulgaria

For the Coupled Nonlinear Schr\"odinger Equations (CNLSE)
\psi_t = \beta\psi_{xx} + \bigl[\alpha_1|\psi|^2 + (\alpha_1+2\alpha_2)|\phi|^2\bigr]\psi = 0 ,
\phi_t = \beta\phi_{xx} + \bigl[\alpha_1|\phi|^2 + (\alpha_1+2\alpha_2)|\psi|^2\bigr]\phi = 0,
we construct a conservative fully implicit scheme with internal iterations (in the vein of the proposed in (Christov et al. 1994) difference is that our scheme utilizes complex arithmetic which makes it four time more efficient in the sense of the computational resources used. The scheme conserves the ``mass'', momentum, and energy within the round-off error. We study the dynamical behaviour of the solitary waves/quasi-particles (QP) upon varying the coefficient of the nonlinear coupling. For small values of $\alpha_2$, the initially linear polarization changes and becomes elliptic. When the coefficient increases along with the main two, additional new solitons are born. The cross-modulation enhances the excitability of the system which causes a phase shift after the collision and change of the carrier frequencies. In consequence of the radiation the initial masses of individual QPs decrease slightly, their energies transform into negative but the full energy of the system is preserved as result of the used conservative numerical scheme.

26 April 2007
The Concept of Quasi-Particle and the Non-probabilistic Interpretation of Wave Mechanics
Christo I. Christov
Department of Mathematics
University of Louisiana at Lafayette

In recent author's works (Christov 2006, Christov 2007) the argument has been made that the Hertz equations of electrodynamics reflect the material invariance (indifference) of the latter. Then the principle of material invariance was postulated in lieu of Lorentz covariance. The absolute medium was called the metacontinuum and assumed to be Maxwell viscoelastic liquid. Then it was shown that the Maxwell-Hertz electrodynamics is a straightforward corollary from the governing equations of metacontinuum.
Here we assume that the metacontinuum is a thin 3D hypershell in the 4D space. Then the deflection along the fourth dimension is a new independent variable (on top of the electromagnetic phenomena in the 3D middle surface of the hypershell). The ``master'' equation for the deflection \zeta of very thin but very stiff shells is
\mu [(\partial^2 \zeta) / \partial t^2] = F^{(4)} + D [-\Delta \Delta \zeta - (\Delta \zeta)^3] + \sigma \Delta \zeta.
where \mu is the density of the material, D is the stiffness, and \sigma is the membrane tension. The linearized part is nothing else but the Schrodinger wave equation written for the real or imaginary part of the wave function (fact acknowledged by Schrodinger himself (Schrodinger, 1926)
A dispersive nonlinear equation of type of above admits solitary wave solutions (solitons) that behave as particles upon collisions. Such waves are called Quasi-Particles (QPs). We stipulated that the material particles are our perception schaumkommen in Schrodinger's own words) of the QPs of the equation above. The wave function has a clear non-probabilistic interpretation as the actual amplitude of the flexural deformation. We show the passage from the continuous Lagrangian to the discrete Lagrangian of the centers of QPs and introduce the concept of (pseudo)mass. We interpret the membrane tension as an attractive (gravitational?) force acting between the QPs and proportional to the inverse square of the distance.

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3 May 2007
A new numerical method for the simulation of 2-D axisymmetrical viscous incompressible flow
N.P. Moshkin
School of Mathematics
University of Technology
Thailand

A novel finite-difference method for simulating 2D axisymmetric incompressible fluid flow is introduced. It is based on a representation of the Navier-Stokes equations in a form of new functions which have been proposed in Aristov and Pukhnachev (2004). New functions are stream function, axial velocity component and new function which we called \Phi. Physical sense of \Phi is unclear. The axisymmetric Navier-Stokes equations in term of new functions contain two transport equations for the stream function and the axial velocity component and one elliptic equation which coupled function \Phi with other. System of the equations are suffering from lack off boundary conditions for function \Phi and two boundary conditions for the stream function. Situation very similar to the case of the stream function and vorticity case. Developed finite-difference method solve the stream function-\Phi equations as a system. Therefore, the two boundary conditions for the stream function are imposed directly and the 'correct' boundary condition for \Phi come out as a part of the coupled solutions.
Bifurcation phenomena in the Taylor-Couette problem of flow between two concentric cylinder has provided a good test problem for quantitative comparison between numerical studies of the axisymetric Navier-Stokes equations set on physical boundary conditions and experimental observation.
To validate our numerical scheme we choose a novel variant of standard Taylor-Couette flow when the end plates can be rotated independently from the inner cylinder.
Good agreement between our numerical computations and numerical and experimental results of Abshagen et al. (2004) has been demonstrated.