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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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