Department Colloquium
 
Fall 2004:
 

1.  Friday, September 3, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Frederic Mynard, Georgia Southern

Title: Compactoidness: compactness for families of sets.

Abstract: Motivations for the introduction of compactness properties of families of sets are presented. Compactoid families and in particular compactoid filters are studied. It is the intent of this talk to show that general results on such objects have a wide range of consequences,
leading to results concerning both spaces and  maps.

2.  Friday, September 10, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Scott Kersey, Georgia Southern

Title: Subdivision curves: An introduction.

Abstract:

Subdivision curves are generated by successively refining a broken line (piecewise linear curve), producing a sequence of broken lines that converge to a limit.  The main issue is whether or not such limit curves are smooth.  In this talk,
we give an introduction to the area of subdivision, including those schemes that generate cardinal spline
curves, interpolatory schemes and variational schemes.

3.  Friday, September 17, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Francis Jordan, Georgia Southern

Title: Selection Problems in Continuum Theory.

Abstract:
For our purposes a selection will be a function f from a collection C of compact subsets of a continuum X into X such that f(A)
is an element A for every A in C.  In particular, we will talk how the existance of continuous selections for certain types of collections of
compact sets determines the structure of a continuum X and vice versa.

4.  Friday, September 24, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Billur Kaymakcalan, Georgia Southern

Title: Langenhop's Inequality and Applications for Dynamic Equations

Abstract:
Langenhop type inequality is given for dynamic equations on time scales.  This result is further employed to obtain lower bounds of solutions of certain dynamic equations. As an application, usage of the derived Langenhop's inequality in determining the oscillatory behavior of
a damped second order delay dynamic equation is illustrated. The results obtained are important in the qualitative sense.
 

5.  Friday, October 1, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Xiezhang Li, Georgia Southern

Title:  Perturbation Theory of Invariant Subspaces and a Perturbation Bound of the Drazin Inverse.

Abstract: The perturbation theory of invariant subspaces of matrices proposed by G. W. Stewart is introduced. A constructive perturbation
bound of the Drazin inverse of a square matrix is derived with an application of the perturbation theory. It is a totally new  approach to
develop perturbation bounds for the Drazin inverse of a matrix. A  numerical example which indicates the
sharpness of the perturbation bound is presented.

6.  Friday, October 8, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Francois Ziegler, Georgia Southern

Title: Bohr Density of Simple Linear Group Orbits

Abstract: As is well known, an irrational line in Rⁿ has dense image in the torus Rⁿ/Zⁿ. In this talk, I will discuss subsets that
enjoy a much stronger property: to have dense projection in Rⁿ/L for every lattice L. In particular I will explain this latest result: if G is a noncompact, simple Lie group acting irreducibly on Rⁿ, then all nonzero G-orbits have the strong density property. (This is rather
unexpected, as such orbits can have arbitrarily large codimension.)
 

7.  Friday, October 15, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Rheinhard Piltner, Georgia Southern

Title: Selected PDEs in Elasticity and Finite Element Approximations for Biomedical Applications.

Abstract:  Two- and three-dimensional elasticity problems are described by coupled partial differential equations. For several model
equations it is possible to give solutions for the differential equations in terms of complex valued functions. For the construction of
finite element functions the possibility of using complex valued functions is discussed. Finally a brief overview of the use
of finite element approximations for biomedical problems will be given.
 

8.  Friday, October 22, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Broderick Oluyede, Georgia Southern

Title: Modified Chi-square tests for a class of ordered alternatives.

Abstract:  Some basic notions of local dependence for multinomial populations are defined. Relation between these notions of local
dependence and stochastic ordering is established. Modified chi-square test procedures for locally ordered alternatives are developed.
Numerical results on power comparisons are also presented. Furthermore, extensions to likelihood ratio and quadrant dependence in ordinal contingency tables are discussed.
 

9.  Friday, October 29, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker:  Sze-Man Ngai, Georgia Southern

Title: Fractal geometry and Hausdorff dimension of self-similar sets.

Abstract: Computing the Hausdorff dimension of a fractal set is a central problem in fractal geometry. However, even for the most
basic class of fractal sets known as self-similar sets, this problem is far from being  solved. In this talk, we describe several results on
computing the Hausdorff dimension of self-similar sets, including the open set condition and the finite type condition. The main
purpose of this talk is to introduce the generalized finite type condition.
 

10.  Friday, November 5, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 .

Speaker: Steven Damelin, Georgia Southern

Title: Approximation methods and stability of singular integral equations on the line: A tale of several countries.

Abstract: In the subjects of target recognition and earthquake predication, there arise classes of integral equations with Cauchy
singular kernels. An important subclass of such equations, which are of present interest to geoscientists and mathematical physicists are classes
where the underlying data is defined over an unbounded domain, such as the real line. Up until recently, the study of such classes of equations has been hampered due to the lack of natural tools to deal with such problems.

In this paper, we show that there exist positive, finite numbers $\mu$ which allow us to approximate singular integral equations on the line of the form
\[
\mu w^2 f - K[f] = g w^{2+\delta}.
\]
Here $w$ is a fixed even exponential weight of smooth
polynomial decay at $\pm \infty$, $\delta>0$, $K[\cdot]:=H[\cdot w^2]/\pi$
is a weighted Hilbert transform and $g$ is a fixed real valued
function in a weighted locally Lipschitz space of order $0<\lambda<1$.

I will discuss the problem and the interesting tale which allowed us to deal with it. The talk will be easy to follow, both for faculty and
graduate/undergraduate students.

This is joint work with K. Diethelm  and is supported, in part, by a EPSRC Fellowship (with the University of Leicester).

PDF file of abstract.

11.  Friday, November 12, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314 . (Visiting Lecture).
 

Speaker:Piotr Minc, Aurburn University.

Title: Embedding of simplicial arcs into the plane

Abstract: Suppose f: H->G is a simplicial map between graphs and g is an embedding of G in the plane.
It is not always true that H can be embedded in the plane with an embedding arbitrarily close to gof.
We will talk about combinatorial obstructions preventing such embeddings and give a full characterization in the case
when H is an arc.

12.  Friday, November 19, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314  (Visiting Lecture).

Speaker: Phillip Zenor, Auburn University.

Title: Continuously extending continuous functions

Abstract: Recall that if H is a subset of X and f:H --> Y, then ef:X --> Y is an extension of f over X if ef(x)=f(x) for all x in H.
We will give a brief history of results concerning continuous extensions of continuous functions and a review of basic
topological notions involved.  Of particular interest is C_K(X), the set of all real valued functions with domains compact
subsets of the Hausdorff space X and the topology on C_K(X).  Finally, we finish with necessary and sufficient conditions
on X so that there is a continuous function e:C_K(X) --> C(X) such that if f is in C_K(X), then ef is an extension
of f over X.

This talk will be mostly self contained and will be accessible to anyone who has had an introductory course in topology.
 

13.  Friday, November 26. No colloquium this week. Thanksgiving Holiday.
 

14.  Friday, December 3, 2004, 3:00 p.m. - 4:00 p.m.  Room: MP 3314.

Speaker:  Goran Lesaja and Verlynda Slaughter, Georgia Southern, Georgia Southern

Title: Interior point methods for a class of conic quadratic programming problems.

Abstract: This talk is based on the master thesis that Verlynda Slaughter completed in the Department of Mathematical Sciences at
Georgia Southern University under direction of Dr. Goran Lesaja. The goal of the thesis was to solve Stratified Sampling Problem (SSP)
using two different interior point methods and perform numerical tests to compare which one is more effective on this particular problem.
The stratified sampling problem is an important problem that arises in different applications and very often in statistics.  First, a homogeneous conic quadratic interior point method was used to solve the SSP as a reformulated conic quadratic problem.  Then, a nonlinear homogeneous interior point method was used to solve the stratified sampling problem as a reformulated monotone complemetarity problem.  Finally, the
methods were compared by CPU time and the number of iterations showing that the first approach works better for problems with
higher dimensions.
 

If you have any question regarding the Colloquium, please send e-mail to Dr. Sze-Man Ngai: ngai@gsu.mat.GeorgiaSouthern.edu.
 

( Last updated: January 12, 2004.)