1 September 2011
Rings and their semigroups
H.E. Heatherly
Department of Mathematics
University of Louisiana at Lafayette

There are many semigroups naturally associated with any ring. These include (but are not limited to) the additive group and multiplicative semigroup of the ring, multiplicative semigroups of ideals (left, right, two-sided), and semigroups of morphisms on the ring. Some connections between a ring and such semigroups are given, with emphasis on recent results obtained jointly with E.P. Armendariz concerning the multiplicative semigroup of all two-sided ideals when this semigroup is commutative.

8 September 2011
A structured population model with diffusion and dynamic boundary condition for Wolbachia dynamics
Jozsef Z. Farkas
Department of Mathematics
University of Louisiana at Lafayette
and
Institute of Computing Science and Mathematics
University of Stirling

Physiologically structured population models attracted a vast amount of interest since Gurtin and MacCamy introduced and analyzed a general nonlinear age-structured model in a seminal paper in 1974. Quasilinear models still pose difficult mathematical challenges, but the theory of semilinear equations is well understood. In this talk we are going to consider a structured population model with diffusion. Individuals are structured with respect to pathogen load. The class of uninfected individuals constitutes a special compartment which carries mass; hence the model is equipped with generalized Wentzell (or dynamic) boundary condition. Our model is intended to describe the spread of infection of a vertically transmitted disease, such as Wolbachia in mosquito populations, and hence the nonlinearity arises in the recruitment term. Well-posedness of the model and the Principle of Linearised Stability follow from standard semilinear theory. In our main result we establish existence of positive steady states in the model. Our method utilizes an operator theoretic framework with a fixed point approach.

15 September 2011
The Kervaire Invariant One Problem in Topology
Duane Randall
Department of Mathematical Sciences
Loyola University New Orleans

This expository lecture on the Kervaire Invariant One Problem begins with the work of M.Kervaire and J.Milnor in differential topology, followed by fundamental theorems of W.Browder and also M.Mahowald and co-authors, and ending with the surprising recent solution by M.Hill, M.Hopkins, and D.Ravenel. We review briefly the origins of the Kervaire invariant as an obstruction to framed surgery, the identification of Kervaire invariant one elements in the stable homotopy of spheres with the survival of certain elements in the Adams spectral sequence, and the birth of Kervaire invariant one elements conjectured through the vanishing of certain Whitehead products involving Im J generators. We enumerate known examples of KI one elements and then briefly state the remarkable nonexistence theorem of HHR, except for the open dimension 126.
 
Since the lecture will be an explanation and overview of the KI One problem without proofs, hopefully this subject will not be too technical for a general audience, but also informative for those with interests in topology.

22 September 2011
Rings whose semigroup of right ideals is J-trivial
Ralph Tucci
Department of Mathematical Sciences
Loyola University New Orleans

This is joint work with Henry E. Heatherly. A semigroup S is J-trivial if any two distinct elements of S must generate distinct ideals of S. We investigate this condition for the semigroup of all right ideals of a ring under right ideal multiplication. There is a rich interplay between the underlying ring and the semigroup of all of its right ideals.

29 September 2011
Low-degree cohomology for finite groups of Lie type
Niles Johnson
Department of Mathematics
University of Georgia

In this talk we describe joint work with the UGA VIGRE Algebra Group determining first and second degree cohomology for finite groups of Lie type, with coefficients in certain simple modules. This is a large-scale joint project, and involves ideas from algebraic groups, Lie algebra cohomology, algebraic topology, and some interesting combinatorics. We'll give a survey of the various techniques and describe how they're applied to these low-degree cohomology computations.
In slightly more detail, we work over an algebraically closed field k of characteristic p, and let G be a simple, simply-connected algebraic group over F_p. Our calculations significantly extend -- and provide new proofs for -- earlier results of Cline, Parshall, Scott, and Jones. We also have a range of vanishing results for the second cohomology group, together with a number of non-zero (one-dimensional) results.

6 October 2011
Homology of quandles
Maciej Niebrzydowski
Department of Mathematics
University of Louisiana at Lafayette

Knot theory is a rapidly developing part of topology with applications to physics, chemistry, biology, and quantum computing. For example, mitochondrial DNA often forms knots, and some of the questions concerning mechanisms of DNA recombination can be answered using knot-theoretical models. Quandles are algebraic structures introduced in 1982 by Joyce and Matveev as a universal tool for classifying knots. Quandle homology, defined in 1999, is a source of powerful knot invariants. In this talk we will describe the attempts to understand the structure and patterns appearing in these homology groups.

13 October 2011
Classification of Nonsimple graph C*-algebras
Mark Tomforde
Department of Mathematics
University of Houston

I will discuss how K-theory can be used to provide complete stable isomorphism invariants for certain classes of nonsimple graph C*-algebras. Moreover, I will show how these invariants can be calculated from data determined by the graph, and describe the range of the invariants.

27 October 2011
Robust Uniform Persistence and Competitive Exclusion in a Nonautonomous SIR Epidemic Model with Multiple Infection Strains
Paul Salceanu
Department of Mathematics
University of Louisiana at Lafayette

A nonautonomous version of the SIR epidemic model of Ackleh and Allen (2003) is considered, for competition of n infection strains in a host population. Sufficient conditions for the robust uniform persistence of the total population, as well as of the susceptible and infected subpopulations, are given. The first two forms of persistence depend entirely on the rate at which the population grows from the extinction state, respectively, the rate at which the disease is vertically transmitted to offspring. We also discuss the competitive exclusion among the n infection strains, namely when a single infection strain survives and all the others go extinct. Numerical simulations are also presented, to account for the situations not covered by the analytical results.

8 November 2011 (TUESDAY)
Homology of semi-lattices and distributive lattices
Józef H. Przytycki
Department of Mathematics
George Washington University

Motivated by rack homology defined by Fenn, Rourke and Sanderson, we introduce homology theory of finite distributive monoids. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term, 2-term, and 3-term homology, and then discussing 4-term homology for Boolean algebras and distributive lattices.

10 November 2011
On Skew-Normal Distribution (SND)
Nabendu Pal
Department of Mathematics
University of Louisiana at Lafayette

Three-parameter Skew-Normal distribution (SND) is a useful generalization of the usual two-parameter normal distribution, and has applications in modelling data over the real line. SND can take positively skewed or negatively skewed shape depending on the positive or negative value of its shape parameter. If the shape parameter takes the value zero then SND reduces to the usual normal distribution. In this talk we will review some interesting characterization properties of SND, and we will also see how it provides a better approximation for a binomial distribution compared to the usual normal approximation based on the Central Limit Theorem. [This talk is geared mainly for the graduate students.]

18 November 2011 (FRIDAY 1:00)
Mathematical Concepts with Image Analysis Applications
Nikolay Metodiev Sirakov
Department of Mathematics and Department of Computer Science and Info Systems
Texas A&M University Commerce

In his talk the speaker will promote the idea that fundamental mathematical concepts are an incubator for methods and algorithms for image analysis and computer vision. He will present his active contour models based on the Heat PDE and will prove that the convex hull of an object in a gray level or color image has no level set presentation. Two kinds of models will be discussed: automatic boundary extraction of multiple image objects; the automatic convex hull definition for single and multiple image objects. Experimental results will be presented to validate the models' capabilities.

1 December 2011
Vanishing Viscosity Limit of the Navier-Stokes Equations to the Euler Equations for Compressible Fluid Flow
Mikhail Perepelitsa
Department of Mathematics
University of Houston

In this talk we will discuss the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. We will follow the approach of R.DiPerna (1983) and reduce the problem to the study of a measure-valued solution of the Euler equations, obtained as a limit of a sequence of the vanishing viscosity solutions. For a fixed pair (x,t), the (Young) measure representing the solution encodes the oscillations of the vanishing viscosity solutions near (x,t). The Tartar-Murat commutator relation with respect to two pairs of weak entropy-entropy flux kernels is used to show that the solution takes only Dirac mass values and thus it is a weak solution of the Euler equations in the usual sense. In DiPerna's paper and the follow-up papers by other authors this approach was implemented for the system of the Euler equations with the artificial viscosity. The extension of this technique to the system of the Navier-Stokes equations is complicated because of the lack of uniform (with respect to the vanishing viscosity), pointwise estimates for the solutions. We will discuss how to obtain the Tartar-Murat commutator relation and how to work out the reduction argument using only the standard energy estimates. This is a joint work with Gui-Qiang Chen (Oxford University).

18 January 2011 (TUESDAY)
Analysis of a size-structured cannibalism model with infinite dimensional environmental feedback
Jozsef Farkas
Department of Computing Science and Mathematics
University of Stirling

First I will give a brief introduction to structured population dynamics. Then I will consider a size-structured cannibalism model with the model ingredients depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system. We show how the point spectrum of the linearised semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.

20 January 2011
Thermal Blow-up in Media with Anomalous Diffusion Effects
W. Edward Olmstead
Department of Engineering Sciences and Applied Mathematics
Northwestern University

Media with anomalous diffusion effects exhibit thermal transport properties that are either faster or slower than classical diffusion. Superdiffusive media are characterized by the influence of Levy flights, which enhance any transport process including the diffusion of thermal energy. Subdiffusive media have a restricted capability to conduct heat as is observed in some porous materials. The mathematical modeling of anomalous diffusion is accomplished through the use of fractional differential operators. In the case of superdiffusion, the classical heat operator is modified by introducing a fractional version of the Laplacian, whereas in the case of subdiffusion, a fractional time derivative is applied to the standard Laplacian. To gain insight into the influence of these fractional operators, some nonlinear problems with the possibility of a blow-up solution are examined.

10 February 2011
Arbitrary Morava modules, their Adams spectral sequence, and continuous group cohomology
Daniel Davis
Department of Mathematics
University of Louisiana at Lafayette

In chromatic homotopy theory, a helpful strategy for being able to compute the stable homotopy groups of the sphere and other spaces is to do this computation for all K(n)-local spectra. After gaining an understanding of this statement by considering fundamental objects like pointed spaces and quotient rings, we consider the Adams spectral sequence, one of the main tools for doing the aforementioned K(n)-local computations. We discuss some new results about the relationship between the Adams spectral sequence and the continuous cohomology of continuous cochains of a certain profinite group. Part of the work behind these results is joint with Takeshi Torii.

17 February 2011
A mixed integer linear model for optimal placement of points of distribution for disaster relief supplies
R. Baker Kearfott
Department of Mathematics
University of Louisiana at Lafayette

A common method of distribution of relief supplies post-disaster is by transporting supplies to points of distribution (such as churches, schools, large department stores, or special locations set up by the government); disaster victims then travel to these locations to pick up needed supplies. Entities distributing supplies operate under limited budgets. However, an additional consideration is convenience of the selected locations. For example, a point of distribution in Opelousas will probably not be of maximal use to disaster victims in Morgan City, or placing all supplies at one point would not be appropriate if most of the affected population were nearer to another point.
 
Previous models have considered cost. In contrast, we have proposed and implemented a multi-objective model that considers both cost and convenience. The dual-objective model can be dealt with directly, using goal programming techniques, or can be simplified by formulating one objective or the other as constraints. Our model is posed as a mixed integer linear program. We have run both simplified models, subject to various budget or convenience constraints, for actual census data and site location data for Lafayette Parish. The results are interesting, and we present them in map form.
 
Model solution, using a public open-source mixed linear solver, runs in a modest amount of time on a single processor. However, we have proposed and are implementing multiprocessing solutions for more extensive data, such as the New Orleans metropolitan area or the southern third of Louisiana, and model refinements such as multiple kinds of relief supplies.
 
This is joint work, with much of the work having been done by my student Haochun Zhang and by Dr. Raju Gottumukkala at the Center for Business Information Technologies.

24 February 2011
A Mathematical Model of Terrorism
Jairo Santanilla
Department of Mathematics
The University of New Orleans

In this talk we give a short account of the history of mathematics in warfare and provide some evidence of the use of mathematics in preventing a terrorist attack, saving tens of thousands of lives. We discuss a dynamical system which models terrorism and obtain conditions under which a terrorist organization will collapse.

3 March 2011
Uniform Persistence in Discrete and Continuous Non-Autonomous Dynamical Systems With Application to Epidemic Models
Paul Salceanu
Department of Mathematics
University of Louisiana at Lafayette

This is an extension of the work of Salceanu and Smith (2009), where boundary attractors for autonomous dynamical systems on the positive orthant of R^m, generated by maps, were characterized as uniformly weak repellers, in order to obtain conditions for uniform persistence. Here we take a unified approach, for both discrete and continuous time non-autonomous systems. We show that when a compact subset of the invariant boundary that attracts all orbits of the boundary, has certain repelling properties, robust uniform persistence for the complementary dynamics is obtained. We also discuss some particular cases for this boundary attracting set that often occur in applications, and conclude by giving sufficient conditions for robust uniform persistence of the disease in two epidemic models.

15 March 2011 (TUESDAY)
Hulls of Semiprime Rings with Applications to C*-Algebras
Gary F. Birkenmeier
Department of Mathematics
University of Louisiana at Lafayette

We shall show that every semiprime ring (not necessarily with unity) has a smallest essential overring that is quasi-Baer (i.e., the right annihilator of every ideal is generated by an idempotent). Various connections and applications to C*-algebras will be discussed, including a characterization of the C*-algebras which are C*-direct products of prime C*-algebras and the characterization of the PI C*-algebras (i.e., satisfy a polynomial identity) which have only finitely many minimal prime ideals.

24 March 2011
Robust estimators for Type I censored samples under non-normality
Evrim Oral
Biostatistics Section
School of Public Health
LSU Health Sciences Center
New Orleans

Non-normal, particularly skewed distributions frequently occur in practice, especially in environmental data sets and HIV studies, which are subject to non-detects. In such situations, the maximum likelihood estimation can be problematic. It is also very common that left censored samples contain outliers on the right tail. In this paper, we use the modified maximum likelihood (MML) methodology to estimate parameters in a Type I censored sample when the data is non-normal. The resulting estimators, called modified maximum likelihood estimators (MMLEs) are explicit functions of sample observations and are easy to compute. We show that the resulting MMLEs are robust. We compare the MMLEs with the traditional naive approaches and develop hypothesis testing procedures for the population mean.

31 March 2011
Presentations of monoidal homotopy theories and the homotopy theory of cubical sets
Samuel B. Isaacson
Department of Mathematics
University of Texas at Austin

What is a topological space? In order to study homotopy-invariant properties of spaces---i.e., those captured by the "homotopy theory of spaces"---we can replace spaces with more combinatorial models such as simplicial sets. In this talk I'll discuss alternative presentations of the homotopy theory of spaces and some applications using the framework of Quillen model categories. In particular, I'll discuss cubical models for spaces and applications to the study of presentations of symmetric monoidal homotopy theories.

7 April 2011
Fixed points imply chaos for a class of differential inclusions that arise in economic models
Brian Raines
Department of Mathematics
Baylor University

We consider multi-valued dynamical systems with continuous time of the form dot(x) in F(x), where F(x) is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, omega-chaos and topological entropy for these differential inclusions that is in terms of the natural R-action on the space of all solutions of the model. By considering this more complicated topological space and its R-action we show that chaos is the 'typical' behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, omega-chaotic, and has infinite topological entropy. This work is joint work with David Stockman (University of Delaware).

11 April 2011 (MONDAY 3:30)
Parameter Estimation on Nonlinear Mixed-Effects Pharmacokinetic (PK) Models
Seongho Kim
Department of Bioinformatics & Biostatistics
University of Louisville
Louisville, KY

We study three challenges, which are the speed of convergence, statistical identifiability, and global optimization, in parameter estimation of nonlinear mixed-effects pharmacokinetic (PK) models. First, various Bayesian Monte Carlo Markov Chain (MCMC) methods and the proposed algorithm, Gibbs maximum a posteriori (GMAP) algorithm, are compared for implementing the nonlinear mixed-effects model in PK studies. The three-stage hierarchical nonlinear mixed model is constructed. Data analysis and model performance comparisons show that GMAP converges the fastest, and provides reliable results. Second, we study the convergence rate of MCMC on the statistically unidentifiable nonlinear model involving the Micaelis-Menten kinetic equation. We demonstrate that a single component MCMC (SCM) scheme is faster than the group component MCMC (GCM) scheme on unidentifiable models, while GCM is faster than SCM when the model is statistically identifiable. A novel MCMC method is then developed using both SCM and GCM schemes, which is called the Switching MCMC (SWM) method. The proposed SWM possesses the advantage of not having to know the identifiability of a model and, as a result, of being able to estimate parameters regardless of the statistically identifiable situations. Lastly, we propose an efficient global search algorithm called NONMEP. It combines the global search strategy by particle swarm optimization (PSO) and the local estimation strategy of NONMEM, which is one of the most popular approaches to a population PK analysis. In the proposed algorithm, initial values (particles) are generated randomly by PSO, and NONMEM is implemented for each particle to find a local optimum for fixed effects and variance parameters. NONMEP guarantees the global optimization for fixed effect and variance parameters. Under certain regularity conditions, it also leads to global optimization for random effects. In the simulation studies, we have shown that NONMEP has much improved convergence performance when compared with NONMEM. Even when the initial values were far away from the global optimal, NONMEP converged nicely for all fixed effects, random effects, and variance components.

14 April 2011
Vector Schroedinger Equation: Coupling, Polarization, Phase Difference, Quasi-Particle Dynamics
Michail D. Todorov
Department of Applied Mathematics and Computer Science
Technical University of Sofia (Bulgaria)

For a system of nonlinear Schroedinger equations (SCNLSE) coupled both through linear and nonlinear terms, we investigate numerically the head-on and taking-over collision dynamics of polarized solitons. In the case of general elliptic polarization, analytical solutions for the shapes of steadily propagating solitons are not available, and we develop an auxiliary numerical algorithm for finding the initial shape. We use a superposition of polarized solitons as the initial condition for investigating the soliton dynamics (sech-like as well as general form). We consider the interactions with and without cross-modulation, with stationary shapes and with breathers. For general nontrivial cross-modulation, a jump in the polarization angles of the solitons takes place after the collision (`polarization shock'). For moderate and large values of the nonlinear coupling parameter, additional solitons are created during the collision of the initial ones.
We also find that, depending on the initial phases of the solitons, the polarizations of the system of solitons after the collision change, even for trivial cross-modulation. This sets the limits of practical validity of the celebrated Manakov solution. In the majority of cases the solitons survive the interaction, preserving approximately their phase speeds and the main effect is the change of individual polarization but the total net polarization of the system is conserved. However, in some intervals for the initial phase difference, the interaction is ostensibly inelastic: either one of the solitons virtually disappears, or additional solitons are born after the interaction.
In the case of pure linear coupling the Manakov system is enriched by a linear coupling term with complex-valued coefficient and the individual solitons are actually either blowing or dispersing breathers. The momentum of the individual quasi-particles (QPs) is conserved while the masses of the individual QPs oscillate, but the sum of the masses for the two QPs is constant. Respectively, the total energy oscillates during one period of the breathing, but the average over the period is conserved. The individual and total polarization angles for the two QPs oscillate with different periods before and after the interaction. This extends to the case of breathing our earlier results about the conservation of the total polarization for the interaction of non-breathing solitons.
The results of this work elucidate the role of the linear and nonlinear couplings, the initial phase, and the initial polarization on the interaction dynamics of soliton systems in SCNLSE. Since the Manakov system loses its full integrability when the nontrivial nonlinear coupling is present, the approach for its study is numerical. All the numerical experiments and computer simulations are implemented by using a fully conservative difference scheme in complex arithmetic.

15 April 2011 (FRIDAY 1:00)
Mathematical Modeling and Inverse Problems in Understanding Mechanisms of Behavior Change
Karyn Sutton
Center for Research in Scientific Computation and
Center for Quantitative Studies in Biomedicine
North Carolina State University
Raleigh, NC

Modeling in the behavioral sciences presents unique challenges as compared to the physical and biological sciences and as a result, the field is considerably less developed. Recent advances in data collection methods have resulted in intensive longitudinal data sets, which are particularly well suited for dynamic modeling. Approaches commonly taken thus far have largely been restricted to statistical analysis relying on static methods, which are limited in their ability to account for relationships that may change over time - i.e., dynamic relationships. There is much to be gained from the development of dynamic mathematical models, such as the formulation and testing of hypotheses of mechanisms leading to behavior change. We discuss here developing efforts in understanding why some problem drinkers are able to successfully reduce their drinking long-term and some are not, guided by such an intensive longitudinal data set. These studies have indicated a need for the incorporation of nontrivial features such as nonlinearities, delayed and cumulative effects, and thresholds. Such issues will likely arise in other applications in modeling of the behavioral sciences as the field grows and connections to data are pursued. Thus, the development of mathematical foundations of aspects of inverse problems for nonlinear nonautonomous dynamical systems with delays is clear. We include a discussion of some results and future directions. (This is joint work with H. T. Banks and Keri L. Rehm at N.C. State University, and Jon Morgenstern, Alexis Kuerbis, Lisa Hail, Christine Davis at Columbia University.)

25 April 2011 (MONDAY)
Lexical Ambiguity in Statistics: The Cases of Random and Spread
Diane Fisher
Department of Mathematics
University of Louisiana at Lafayette

Language plays a crucial role in the classroom. The use of specialized language in a domain can cause a subject to seem more difficult to students than it actually is. When words that are part of everyday English are used differently in a domain, these words are said to have lexical ambiguity. Studies in other fields, such as mathematics and chemistry education suggest that in order to help students learn vocabulary instructors should exploit the lexical ambiguity of the words. This presentation is part of a sequence of studies designed to understand the effects of and develop techniques for exploiting lexical ambiguities in the statistics classroom. This session will focus on research results from pre- and post-testing students' definitions, both everyday and statistical, of the words random and spread and will present suggestions and activities that instructors can use to exploit the lexical ambiguity associated with the words random and spread.

28 April 2011
Construction of Explicit Solutions to the Matrix Equation X²AX = AXA
Aihua Li
Department of Mathematical Sciences
Montclair State University

Consider the matrix equation AXA = X²AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. The solution space is explicitly constructed for all 2×2 complex matrices using Gröbner basis techniques. When A is a 2×2 matrix, the equation AXA = X²AX is equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In the procedure for solving these equations, Gröbner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classifying all solutions for 2×2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Gröbner bases are extraordinarily computationally demanding, and so a different approach is taken. This technique can be applied to more general matrix equations, and the focus here is placed on solutions coming from a particular class of matrices.

5 May 2011
Estimation of delta=P(X < Y) for Burr XII distribution for progressively first failure-censored samples
Tzong-Ru Tsai
Department of Industrial and Manufacturing Systems Engineering
Kansas State University
(On leave from Tamkang University, Tamsui, Taiwan)

Let X and Y have two-parameter Burr XII distributions. The maximum likelihood estimator of delta=P(X < Y) is studied under the progressively first failure-censored samples. Three confidence intervals of delta are constructed by using an asymptotic distribution of the maximum likelihood estimator of delta and two bootstrapping procedures, respectively. Some computational results from intensive simulations are presented. An illustrative example is provided to demonstrate the application of the proposed method.

9 May 2011 (MONDAY)
Differential-Equation-Based Statistical Models with Application to Biomedical Research
Tao Lu
Department of Biostatistics and Computational Biology
University of Rochester School of Medicine and Dentistry

A dynamical system in engineering and physics, specified by a set of differential equations, is usually used to describe a dynamic process which follows physical laws or engineering principles. The idea of dynamical systems has been introduced into biomedical fields in recent years because of the rapid development of computational power and deep understanding of biological processes at a cellular level with the assistance of modern biotechnologies. Examples of biological dynamical systems include gene regulatory networks, tumor cell kinetics and viral dynamics. In this talk, I will give two examples of its application. One is modeling the long-term dynamics of HIV viral load by incorporating clinical factors such as pharmacokinetics, compliance to treatment and drug susceptibility. The other is modeling a complicated interactive network, such as a gene regulatory network based on time course microarray data. The proposed models not only help us understand the underlying mechanism of disease processes but also provide guidance for disease treatment. We are currently applying these models to other disease-related areas, such as oncology.

2 June 2011
The classification of C*-algebras and the Cuntz semigroup
Leonel Robert
Department of Mathematical Sciences
University of Copenhagen

I will first recall the definition of C*-algebra and give some examples. I will then talk about the classification program for C*-algebras, the main goal of which is to distinguish C*-algebras from each other by means of data (so called, "classifying invariants") extracted from them. Next, I will talk about the use of the Cuntz semigroup as a classifying invariant for certain classes of C*-algebras. Finally, I will give some applications of the classification by the Cuntz semigroup.

9 September 2010
Simple Rings: A Survey
Henry Heatherly
Department of Mathematics
University of Louisiana at Lafayette

A ring is said to be simple if it has no non-zero, proper ideals. In the 1920's simple rings were found to be basic building blocks in describing the structure of a wide class of non-commutative rings. They continued to play a key role in the development of structure theory for rings, and do so to this day. We give some highlights of this development. Examples of simple rings are given to illustrate the theory and to exhibit their exotic and diverse nature. Finally, a challenging open problem concerning simple rings is discussed.

16 September 2010 (SPECIAL TIME 4:30/4:45)
Accurate interval arithmetic with rounding to nearest
Naoya Yamanaka
Department of Applied Mathematics
Waseda University (Tokyo)

In this joint work with Shin'ichi Oishi, we give simple and accurate methods to compute basic operations of interval arithmetic using only floating-point operations in rounding to nearest. Proposed methods are based on "Error Free Transformations". Since proposed methods work on rounding to nearest mode, these algorithms improve the portability of interval arithmetic. Furthermore, for some operations including dot product, proposed methods are more accurate than methods with changing rounding mode.

23 September 2010
What good is period three?
Tom Ingram
Emeritus
Missouri University of Science and Technology

Any satisfactory answer to this question surely depends on who poses it. To a dynamicist familiar with the Sarkovskii theorem and considering a real-valued map of the interval, period three is a goldmine. To others, it is a path to chaos. To a continuum theorist, perhaps a different answer altogether is more satisfying. In this talk aimed at a general audience, we will explore some continuum theoretic consequences of period three.

7 October 2010
On Bootstrap Inconsistency and an Oracle Bootstrap
Mihai C. Giurcanu
Department of Mathematics
University of Louisiana at Lafayette

In this talk, I will first make a short overview on LASSO-type estimation and model selection of regression models. Then, I will present some results on the standard bootstrap distribution estimation for LASSO-type estimators. Then, I will describe a new bootstrap procedure, called the oracle bootstrap, which provides consistent bootstrap estimation in model selection setting. I will end my talk with some simulation studies to illustrate the behavior of the bootstrap methods for various estimation problems, including the regularization parameters estimation.

14 October 2010
Rings of Bounded Index: A Survey
Efraim Armendariz
Department of Mathematics
University of Texas at Austin

A ring R is said to have bounded index (of nilpotency) if there is a positive integer k such that x^k = 0 whenever x is a nilpotent element of R. Recent developments related to radical and radical-type properties of such rings will be discussed along with the historical background for the study of such rings. The effect of bounded index on special classes of rings will be discussed and open problems will be identified.

21 October 2010
Optimal Treatment Strategies for Malaria Infection
Jeremy Thibodeaux
Department of Mathematical Sciences
Loyola University New Orleans

Malaria has long been a devastating illness affecting millions of people in many parts of the world. In recent years, mathematical models have been used to gain insight into different aspects of the disease. In this presentation, a new mathematical model will be presented that accounts for aspects of the disease that have not been previously considered. A numerical method for simulating the model will then be presented along with an existence-uniqueness result for solutions of the model. Through several simulations, it will be demonstrated that the model predicts a high sensitivity to the number of parasites released per infected host cell. With this knowledge, an optimization problem can be formulated. The techniques for solving this problem will then be presented along with the results.

28 October 2010
Homology of distributive structures
Józef H. Przytycki
Department of Mathematics
George Washington University

Homology theory of associative structures like groups and rings has been zealously studied throughout the past starting from the work of Hopf, Eilenberg, and Hochschild, but non-associative structures, like distributive lattices or racks, were neglected till recently. The condition which can often replace associativity is distributivity and my talk will be devoted to homology of distributive structures with an eye on a hypothetical connection to Khovanov homology.

11 November 2010
The many faces of free groups
Arturo Magidin
Department of Mathematics
University of Louisiana at Lafayette

In this expository talk, I will give several different ways of thinking about free groups, depending on whether one wants to approach them from a group-theoretical, a topological, or a categorical point of view. I will also discuss some of their nice properties (for example, that a subgroup of a free group is free), and some of their weird ones (for example, that the free group on two generators contains as a subgroup the free group on infinitely many generators). I will conclude the talk by mentioning some still open problems about free groups. No previous knowledge of free groups will be assumed.

18 November 2010
Quantum cohomology: a machine to count curves
Leonardo Mihalcea
Department of Mathematics
Baylor University/University of Louisiana at Lafayette

In 1993 Kontsevich gave a seemingly simple solution to a problem that eluded algebraic geometers for more than a century: what is the number of rational curves in the plane which pass through some number of random points. In the process he introduced quantum cohomology - a tool which started a whole mathematical industry which is flourishing nowadays. My goal is to present the main definitions and idea behind Kontsevich's computation. If time allows, I will point out some of my own contributions to this subject, which gives a way to "count" infinitely many curves.

2 December 2010
Broken Bracelets, Molien Series, Paraffin Wax and an Elliptic Curve of Conductor 48
Mahir Can
Mathematics Department
Tulane University

A jeweler is asked to design a necklace consisting of a chain with n placements for k pieces of diamond. The client asks for one group of r diamonds to be placed next to each other and the remaining diamonds are to be isolated, that is, each one is mounted so that the two adjacent places are left empty. These special diamonds are called the medallion of the necklace. A configuration is a broken necklace resulting from one of the r+1 cuts to the left, right or in between the medallion.
Problem: Determine the number of configurations up to symmetry.
In this talk we will show amusing connections of this enumerative problem to the other fields of mathematics.
This is joint work with Tewodros Amdeberhan and Victor Moll.

21 January 2010
Quantum K-Theory of Grassmannians and the Geometry of Spaces of Curves
Leonardo Mihalcea
Department of Mathematics
Baylor University

The 3-point, genus 0, Gromov-Witten invariants of a Grassmannian count rational curves of degree d satisfying certain incidence conditions - if the number of curves is expected to be finite. Recently, Givental and Lee defined the K-theoretic Gromov-Witten invariants, which associate an integer to spaces of rational curves in question, regardless of whether it is finite or not. The resulting quantum cohomology theory - the quantum K-theory - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants. I will show how the Gromov-Witten invariants and quantum K-theory algebra can be computed explicitly. The key is a "quantum=classical" phenomenon: the K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary K-theory of certain two-step flag manifolds.

11 February 2010
GasDay: Forecasting Customer Demand for Natural Gas
George F. Corliss
Electrical and Computer Engineering
Marquette University
Milwaukee, Wi

Natural gas for heating takes 2 - 3 days to get to Milwaukee from Louisiana. If our local utility does not order enough, they buy it on the spot market at premium prices. If they order too much, the pipeline companies charge them penalties. The GasDay lab at Marquette University licenses to local natural gas utilities software to forecast the gas demands of their customers hours, days, and months in the future. We license to 23 utilities across the US. Each day, software written by our students helps forecast 20% of the gas used by residential, commercial, and industrial customers in the US. Our customers tell us our forecasts save their customers tens of millions of dollars each winter. We also offer analysis consulting services including peak day studies and data cleaning. We will discuss some of the mathematical modeling challenges, as well as the entrepreneurial challenges of running a $500K business with students within the university.

18 February 2010
Max-Plus Algebra and the Computation of Nonlinear Controls
Ben G. Fitzpatrick
Loyola Marymount University

Mathematical methods for linear systems have had enormous impact on engineering applications, especially in control systems. Nonlinear approaches, however, suffer from many drawbacks, not the least of which are computational difficulties. The recent successes of max-plus and more general idempotent structures for attacking nonlinear control problems offer the potential for revolutionary improvements in our ability to design and implement nonlinear controls for real applications. In this talk, we describe the basic concepts behind max-plus methods. The key observation is that Bellman's dynamic programming principle yields a linear equation in the max-plus arithmetic. We will outline the max-plus approach to nonlinear control and provide some preliminary results in the extension from deterministic to stochastic problems. We will discuss numerical approximation schemes and examples problems in counterinsurgency and UAV sensing planning.

16 March 2010 (TUESDAY)
Leavitt path algebras and graph C*-algebras: at the crossroads of algebra and functional analysis
Mark Tomforde
Department of Mathematics
University of Houston

In the past 15 years Functional Analysts have considered a method for constructing C*-algebras from directed graphs. These graph C*-algebras include many previously considered classes of C*-algebras, and provide a unified framework in which to consider many examples. Furthermore, numerous properties of the graph C*-algebra can be translated into properties of the graph, allowing for a very satisfying theory in which C*-algebraic questions can be reformulated as graph questions. In the past 5 years Algebraists have used similar methods to construct a K-algebra from a directed graph. These K- algebras are called Leavitt path algebras, and they generalize the Leavitt algebras (which are fundamental examples of algebras without the Invariant Basis Number property). To many researchers' surprise, it has been found that when properties of a graph correspond to properties of the associated C*-algebra, these same graph properties correspond to the algebraic properties of the associated Leavitt path algebra. What is even more astonishing is that neither the graph C*- algebra results nor the Leavitt path algebra results are obviously logical consequences of the other. Often different methods are used in the proofs, and moreover, neither set of results can easily be seen to imply the other. This has left many researchers wondering exactly what the relationship is between the Leavitt path algebras and the graph C*- algebras. In this talk I will describe the many similarities (and a few of the differences) between these two classes.

18 March 2010
Global stability in a multi-species periodic Leslie-Gower model
Robert J. Sacker
University of Southern California
Los Angeles, California

The d-species Leslie--Gower competition model is studied in which all the parameters are p-periodic. It is shown that whenever the coupling parameters are small, there is a positive p-periodic state that is exponentially asymptotically stable and globally attracts all initial states having positive coordinates. The theory extends to systems with delays and to certain systems having fractional quadratic form.

22 April 2010
A Skew-Normal Approximation to the Distribution of Aggregate Claim
Kumer P. Das
Department of Mathematics
Lamar University
Beaumont, Texas

In collective risk model, an insurance portfolio is regarded as a process that produces claims over time. The size of these claims are taken to be independent, identically distributed random variables, independent also of the number of claims generated. This makes the total claims the sum of a random number of iid individual claim amounts. The third central moment of aggregate claims under both the compound Poisson and compound negative binomial distribution is not zero. In fact, for positive claim amount distributions, the third central moment of aggregate claims is positive in each case. This skewed nature of the distribution of aggregate claims assures that a normal approximation may not be the best approximation to the aggregate claims distribution. In this study, a more general approximation to the distribution of aggregate claims using the skew normal distribution has been sought. In many occasions the obtained results are sharper than that of translated gamma approximation technique.

29 April 2010
Stabilization of hydrodynamic instabilities in Hele-Shaw flows
Prabir Daripa
Department of Mathematics
Texas A&M University

Motivated by applications and numerical simulation of fronts, multi-layered Hele-Shaw flow will be considered as a way to control growth of instabilities. Upper bound results on the growth rates will be discussed in cases of constant viscosity layers and variable viscosity layers. The upper bound provides a way to assess cumulative effects of many layers and many interfaces on the growth rates of unstable waves. As an application of the bound, we obtain some necessary conditions for suppressing instability of two-layer flows by introducing arbitrary number of constant viscosity fluid layers in between. This necessary condition has very practical relevance because it narrows the choice of internal layer fluids based on surface tensions of all interfaces and viscosity of fluids. Importance of this condition which has been hitherto unknown is also discussed. Other consequences of these upper bounds and necessary conditions are discussed. The case of internal fluid layers having unstable viscous profiles will be treated. The effect of diffusion on the instability will also be discussed.

10 September 2009
A Pressure Boundary Integral Method for Direct Numerical Simulations of Particle-laden Flows
Julian Simeonov
Marine Geosciences Division
Naval Research Laboratory
Stennis Space Center

We consider the coupling of Direct Numerical Simulations (DNS) and Discrete Particle Simulations (DPS) for the purpose of modeling particle-laden turbulent flows with strongly interacting particles at finite particle Reynolds numbers. The flow and the particle evolution are determined respectively from the Navier-Stokes and Newton's equations of motion where the hydrodynamic force on a particle is obtained by integrating the resolved pressure and viscous stress on the particle surface. For colliding particles, the normal and tangential particle-contact forces are modeled with springs and friction, respectively. The primary focus of this talk will be the numerical method for solving the Boundary Value Navier-Stokes problem in domains with moving boundaries. The numerical method is based on a fictitious domain formulation where the pressure Poisson equation is extended discontinuously across particle boundaries such that the pressure is continuous but its normal derivative has a jump (essentially the density of a surface layer) on the particle surface. A boundary integral equation is then solved to determine the unknown jumps for a given pressure Neumann boundary condition. Assuming that the fictitious pressure inside particles is harmonic, we can also determine the discontinuities of the pressure second derivatives. With known derivative discontinuities we can regularize the second order finite-difference Laplacian and solve the pressure Poisson equation efficiently using Fast Poisson Solvers on a regular Cartesian grid. Once the pressure field is obtained, the fluid momentum equations are discretized with finite-differences and solved only outside the particles. The hydrodynamics solver is tested against theoretical and experimental results for flows around spheres. A preliminary simulation of visco-elastic particle collision is also discussed.

16 September 2009 WEDNESDAY
Mathematica in Education and Research
Sean McDonald
Academic Account Executive
Wolfram Corporation

This seminar will highlight the latest directions in technical computing with Mathematica, and the impact of these new technologies on education and research. Participants will come away with a comprehensive understanding of Mathematica's key capabilities and core design principles. A wide variety of practical and theoretical applications will be discussed, and no Mathematica experience is required.

24 September 2009
Foundations of Interval Computation
Trong Wu
Department of Computer Science
Southern Illinois University Edwardsville

This talk reports a study of numerical computation problems from a theoretical viewpoint. It shows that computer systems are not capable of computing of real numbers correctly due to the differences between the algebraic structures of real numbers and model numbers. These two classes of numbers are not isomorphic. From this study, we have learned that there are no machine errors or computation errors. If fact, one can view it is a human mistake by putting real valued problem onto a model number platform for computation. This paper proposes use of the concept of computer model numbers to approximate rough numbers for computation. Moreover, we revise an arbitrary initial compact interval to a shortest initial closed-open model interval for ordinary interval computation. This way, we can assure that the final resulting interval will be the shortest interval and that the computation will result in the greatest precision.

15 October 2009
Adjoint groups of rings
Henry E. Heatherly
Mathematics Department
University of Louisiana at Lafayette

A survey is given of the theory of adjoint groups of rings, from the seminal work of Jacobson in 1945 up through some very recent results. The focus is on certain aspects which are less technical and more understandable to a general mathematical audience. Examples are given to motivate and illustrate the theory developed, and several open problems are discussed.

29 October 2009
An Introduction to Inverse Limits with Set-valued Functions
Tom Ingram
Emeritus, Missouri University of Science and Technology

Since the first paper on the subject just over five years ago, the study of inverse limits with set-valued functions has flourished. We shall provide an introduction to this topic intended for a general audience. We shall include many examples chosen to illustrate various aspects of subject and to give some sense of the rich variety of inverse limits that result even in the simple case that the domain of the set-valued function is the interval [0,1].

5 November 2009
Modeling antibody response during HIV infection
Stanca Ciupe
Mathematics Department
University of Louisiana at Lafayette

One of the first anti-HIV immunologic responses recognized is the presence of neutralizing antibodies that seem able to inactivate several HIV strains. In vitro studies have shown the existence of monoclonal antibodies that exhibit broad neutralizing potential against the constantly mutating virus. Yet their number is low and slow to develop in vivo. In this study, we investigate the hypothesis that broadly neutralizing antibodies develop alongside strain specific neutralizing antibodies. We develop a mathematical model for the interaction between families of B lymphocytes producing broad and strain-specific antibodies following infection with several HIV variants. Using local and global stability analyses, we explore how cross-reactive immune mechanisms may help the virus escape, gaining insight into the failure to ultimately control the HIV infection.

12 November 2009
Strong seasonality produces spatial asynchrony in the outbreak of infectious diseases
Scott M. Duke-Sylvester
Department of Biology
University of Louisiana at Lafayette

Models for infectious diseases usually assume a fixed demographic structure. Yet a disease can spread over a region encountering different local demographic variations that may significantly alter local dynamics. Spatial heterogeneity in the resulting dynamics can lead to important differences in the design of surveillance and control strategies. We illustrate this by exploiting a north-south gradient in the seasonal demography of raccoon rabies over the eastern United States. We find that the greater variance in the timing of spring births characteristic of southern populations can lead to the spatial synchronization of southern epidemics while a narrow birth pulse associated with northern populations can lead to an irregular patchwork of epidemics. These results indicate that surveillance in the southern states can be reduced relative to northern locations without loss of detection ability. The importance of seasonality in many widely distributed diseases indicates that our findings will find applications beyond raccoon rabies.

19 November 2009
An approach to persistence: Lyapunov exponents and uniform weak repellers
Paul Salceanu
Mathematics Department
University of Louisiana at Lafayette

The concept of mathematical persistence is relatively new in mathematics, as it has been around only for a few decades now. Its development has been motivated especially by the need for predicting the long term survival of some species in an ecosystem. Of a particular interest there is the question of whether or not (and if yes, under what circumstances) the disease can drive the host population to extinction, or else, the disease can persist in the population. Lyapunov exponents are used to characterize the attractor in a bounding hyperplane as uniform weak repeller and then to derive sufficient conditions for uniform persistence, for a large class of both discrete and continuous dissipative dynamical systems on the positive cone. This is done under the assumption that all so-called normal Lyapunov exponents are positive on such attractor. Two applications to both discrete and continuous models are presented.

3 December 2009
Part 1 (3:30-4:00): An Introduction to Survival Analysis
Part 2 (4:10-5:00): Parameter Estimations under Progressive Type-I Interval Censoring
Yuh-Long Lio
Department of Mathematical Sciences
University of South Dakota

The estimates, via maximum likelihood, moment method and probability plot, will be discussed under progressive type-I interval censoring. A simulation study and a model selection process for fitting a real data set which contains with plasma cell myeloma are discussed. Finally, a possible further research will be presented.

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

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

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

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

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

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

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

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.

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.