October 26-27, 2007
Location: Lee Drain Building (see the map), Room 418
NOTE: files of some presentations are available now (see below)!
FRIDAY, October 26
2-2:50 pm, Colloquium
Vladislav Babenko, Dnepropetrovsk National University, Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine
Landau-Kolmogorov type inequalities and their applications.
We shall present a survey of known results on sharp Landau-Kolmogorov-type inequalities for univariate and multivariate functions, as well as a series of new results of this type. We shall also discuss the connections between the problem on sharp Kolmogorov-type inequalities with other extremal problems of Analysis and, in particular, Approximation Theory. We shall present recent results in Approximation Theory obtained exploring this connection.
Full presentation
Bio information:
Vladislav Babenko is Professor of Mathematics and Chair of the Department
of Mathematical Analysis at DNU, as well as Senior Research Fellow at the
Institute of Applied Mathematics and Mechanics of National Academy of
Sciences of Ukraine. He has more than 200 publications and 3 monographs,
he has suprvised 8 PhD students and currently has 5. He has been awarded the
State Prize in the area of sciences -- the highest governmental award of
Ukraine, and has been named Honored Worker of Science and Technology of
Ukraine. His areas of research include Multivariate Approximation Theory,
Optimization of Algorithms, Complexity, Extremal Problems of Analysis
among others.
2:50-3:10 Coffee break
3:10-3:35
Sergiy Borodachov, Georgia Tech
Optimal approximate integration of functions with bounded Laplace operator on a ball.
We consider cubature formulas for integration along a d-dimensional ball
which use mean values of the function along concentric hyperspheres. The cubature formula with the least worst case error over the class of
functions whose Laplace operator is bounded in L- or C-norm on the ball is described.
3:35-4:00
Tatyana Sorokina, Towson University
Geometric constraints for finite elements: why they arise and how to
deal with them
The talk is based on the following papers:
1. A multivariate Powell-Sabin interpolant, T. Sorokina and A.J. Worsey, Advances in Comp. Math.,
to appear
2. A $C^1$ multivariate Clough-Tocher interpolant, T. Sorokina, submitted
3. A $C^1$ quadratic trivariate macro-element space defined over arbitrary tetrahedral partitions, L. L. Schumaker, T. Sorokina
and A. J. Worsey, submitted
4. Smooth macro-elements over $k$-plus splits of simplices, T. Sorokina, in preparation
5. A modified trivariate Clough-Toucher interpolant, P. Alfeld and T. Sorokina, in preparation
4:10-5:00
Francis Narcowich, Texas A&M
Divergence-free SBFs on spheres and other surfaces,
In this talk, we present a new tool, based on radial basis functions,
for fitting a divergence-free vector field tangent to a two dimensional
surface, orientable surface $\mathcal P \subset R^3$ to samples of such
a field taken on scattered sites on $\mathcal P$. When the surface is a
sphere, there are important physical applications. This joint work with
J. Ward and G. Wright.
5:10-6:00
Jianzhong Wang, Sam Houston State University
Hyper-spectral image data dimensionality reduction
Hyper-spectral sensors typically collect hundreds of narrow (5-20 nm)
contiguous bands. The data of each band can be represented as a single
channel image. The hyper-spectral images are used for many purposes such
as area segmentation, object detection, and classification. Because the
image size is huge, it is necessary to reduce it to a few bands (5-10)
so that the data processing can be taken. The talk introduces the
structure of hyper-spectral images, the popular methods used for its
dimensionality reduction and explain how to apply diffusion wavelets in
their dimensionality reduction.
6:30-9:30 WINE AND BEER PARTY AT KEN SMITH'S HOUSE (address: 335 Parkhill St., Huntsville, TX)
SATURDAY, October 27
9:30-10
Coffee at LDB 419
10-10:50
Joe Ward, Texas A&M
Norming Sets in Multivariate Approximation: An Overview
This talk will focus on the idea of norming sets and their utility in multivariate
approximation theory. We will describe how this notion was useful
for establishing error estimates via radial basis interpolation. Subsequently many
other applications of this notion were found in problems concerning positive
quadrature
formulas for the sphere, the extention of error estimates to larger classes of
functions by means of ``dual norming sets'', and, more recently, in the derivation
of Bernstein inequalities and inverse estimates for radial basis approximation. We
plan to discuss these other developments as well.
Full presentation
11-11:25
Jian-ao Lian Prairie View A&M University
Subdivision schemes induced from refinable scaling functions
We revisit and recover some classical curve and surface subdivision
schemes by univariate and bivariate refinable scaling functions. Some new schemes
will be established and demonstrated.
11:30-11:55
Luis Garcia-Puente Sam Houston State University
Linear precision for toric patches
In 2002, Krasauskas generalized the standard Bezier
and tensor product patches of geometric modeling to
multi-sided toric patches. While these offer the
promise of greater design flexibility, it is not clear
whether they possess the desirable properties of
the standard patches. One such property is linear
precision, which is the ability to replicate a linear
function.
I will discuss work with Frank Sottile on linear precision.
We show that every patch has a reparametrization having
linear precision. The reparametrization is not rational
unless the patch has a very singular geometry. For toric
patches, the existence of such rational reparametrizations
has an appealing mathematical reformulation in terms of
Cremona transformations. Moreover, this reparametrization
can be numerically computed using a standard method in
statistical inference known as iterative proportional scaling.
Full presentation
12 noon - 1 pm LUNCH
1-1:25
Plamen Simeonov University of Houston, Downtown
$q$-Difference Operators and Orthogonal Polynomials
In this work we apply a $q$-ladder operator approach to
orthogonal polynomials arising from a class of indeterminate moments
problems. We derive general representation of first and second order
$q$-difference operators and we study the solution basis of the
corresponding
second order $q$-difference equations and its properties. The results
are applied to the Stieltjes-Wigert and the $q$-Laguerre polynomials.
1:30-1:55
Dmytro Skorokhodov, Dnepropetrovsk National University, Ukraine
On linear approximation of quadratic function of two variables in Lp-metric and applications
In this talk we shall present the following extremal
property of triangles: for given $p$, $1\le p<\infty$,
equilateral triangles realize the minimum
$$
\displaystyle \min_{T} \frac{L_p-\hbox{error of linear interpolation of} \; x^2 + y^2 \;
\hbox {on} \; T }{ |T| ^{1+\frac 1p}},
$$
in which min is taken over all triangles $T$ and $|T|$
denotes the area of the triangle $T$.
As applications we have obtained the exact asymptotics
of the optimal $L_p$-error of linear spline
interpolation of an arbitrary function $f\in C^2([0,1]^2)$.
In addition, we shall discuss the algorithm of construction of
asymptotically optimal triangulation of $[0,1]^2$.
Full presentation
2-2:25
Nataliya Parfinovych, Dnepropetrovsk National University, Ukraine
On the class preserving approximation of functional classes by splines
The problems of the class preserving and shape preserving
approximations of functional classes are studied. The exact order
of relative widths of classes of the periodic differentiable
functions, exact values and exact asymptotic of the best relative
approximations of such classes by splines will be discussed.
2:25-2:45 Coffee break
2:45-3:10
John Alford, Sam Houston State University
Deterministic Models of Initiation and Propagation of Unidirectional
Excitations (Action Potentials) In Excitable Media
This talk will present the computations and analysis of some
differential equation models which simulate initiation of unidirectional
excitations (action potentials) in excitable media. The models depend on a
heterogeneous distribution of stimulus and coupling (or diffusion)
parameters. In the case of circular spatial domains, unidirectional
propagation may result in a rotating wave, which is referred to as
re-entrant arrhythmia in cardiac tissue. The equilibria and steady-states
are computed in order to determine the critical parameter values and ranges
over which unidirectional propagation may occur.
3:15-3:40
Yuliya Babenko, Sam Houston State Univesity
On existence of functions with prescribed norms of the derivatives
In this talk we shall discuss the following problem which was posed by Kolmogorov:
for given integer d, function space X, and r positive numbers, find necessary and sufficient conditions
for existence of a function from the class X that has these given numbers as values of its subsequent derivatives.
We shall give a short review of known results and present new ones. In particular, we will give a complete
characterization of sets of four numbers such that there exists l-monotone function with prescribed
smoothness that has these numbers as values of sup-norms of its corresponding derivatives. As for the arbitrary number of prescribed derivatives we shall give a sufficient condition on the existence of the function.
Full presentation
