August 31:
Speaker: Andras Kroo, Renyi Institute of Hungarian Academy of Sciences and Sam Houston State University
Title: The Density of Homogeneous Polynomials in the Space of Continuous Functions
Abstract: The classical Weierstrass Approximation Theorem on density of polynomials in the space of continuous functions is one of the cornerstones of Analysis. Numerous papers are devoted to its various extensions for ''smaller'' classes of polynomials.
We shall consider the density problem for homogeneous polynomials on convex and star-like surfaces and discuss how the geometry of the domain effects this question.
September 14:
Speaker: Yuliya Babenko, Sam Houston State Univesity
Title: On asymptotically optimal methods of adaptive spline interpolation
Abstract: Spline functions are widely used in applications which involve geometric design: aircraft and automobile manufacturing, pharmacology, animation etc. Given a function which may describe a surface of an object to be designed one can consider two major types of splines: approximating or interpolating splines. Unfortunately, an algorithm which would provide us with the best possible interpolating or approximating spline for any given function would be NP-hard. However, an algorithm for providing an interpolating spline for any function which is asymptotically the best can be constructed.
In this talk we shall present the exact asymptotics of the optimal error in the weighted $L_p$-norm, $1\leq p \leq \infty$, of linear spline interpolation of an arbitrary function $f \in C^2([0,1]^2)$. Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations for linear interpolation. The connections with the problem of approximating the convex bodies by polytopes and the problem of adaptive mesh generation for finite element methods will also be discussed.
Similar results are obtained for some other classes of splines. We shall also discuss the analogous multivariate results as well.
September 20:
Speaker: TBA,
Title: TBA
Abstract: TBA
September 27: Special Day and Time: Wednesday, 2pm !!!
Speaker: Jianzhong Wang, Sam Houston State University
Title: Bayesian Approach to Total Variation Denoising
Abstract: In data processing, an important problem is how to preserve data feature in noise removal processing. Total Variation (TV) methods are
well-developed methods for solving the problem. In this talk, a Bayesian
approach TV denoising is presented and the relation between the TV
method and the Bayesian MAP (maximum a posterior estimate) method is
revealed. The algorithm based on geometric diffusion kernel for the TV
denoising is also developed.
October 5:
Speaker: Fumiko Futamura, Vanderbilt University
Title: Localized Operators and the Construction of Localized Frames
Abstract: A frame for a Hilbert space is a kind of generalized orthonormal
basis which is useful in signal processing. A localized frame is a frame
whose elements are "well-localized", in the sense that the inner products
of their elements decay as the differences of their indices increase.
Grochenig in 2004 proved that localized frames for Hilbert spaces extend
to frames for a family of associated Banach spaces. We generalize
localized frames to the operator setting, and say an operator is localized
with respect to given frames if there is an off-diagonal decay of the
matrix representation of an operator with respect to the frames. We prove
that operators localized with respect to localized frames are bounded on
the same family of Banach spaces, and that they can be used in the
construction of new localized frames. We also consider the special case
where the frames are unitary shifts of a single atom function.
October 12:
Speaker: Joe Ward, Dept. of Mathematics, Texas A&M
Title: Interpolation of Scattered Data on Spheres Using SBFs:
Direct and Inverse Error Estimates and a Bernstein Inequality
Abstract: In this talk, we will discuss SBFs - spherical basis functions - which provide a powerful tool for
interpolating scattered data on the n-sphere. In particular, interpolation error estimates - both direct and inverse -
will be presented along with a Bernstein-type inequality where the smallest separation between data sites plays the
role of a Nyquist frequency.
October 17. Special day and time: Tuesday 3:30 - 4:30 pm !!!
Speaker: Francis Narcowich, Dept. of Mathematics, Texas A&M
Title: Positive-weight quadrature and localized tight frames on
Euclidean spheres
Abstract:
In this talk we will first discuss a positive-weight
quadrature formula for the sphere that the authors have recently
developed. In the remaining time, we will present a new class of frames
on the sphere. The novel features here are that these frames are tight,
and that they have excellent localization properties. We will discuss
what these new frames are and how they can be implemented using the
quadrature formula mentioned above.
October 25: NO MEETING THIS WEEK
November 3. Special day and time: Friday, 1:30 - 2:30 pm !!!
Speaker: Vasilis Zafiris, Dept. of Computer and Mathematical Sciences, University of Houston - Downtown
Title: Geometric Characteristics of Trivariate Maps
Abstract: Abstract:
Volume grid cells are usually constructed using a trivariate polynomial
map defined on a reference domain. The simplest and most popular
trivariate is the trilinear. The map and its Jacobian are represented in
Bezier form and a pyramid algorithm is utilized to simultaneously
compute points and geometric characteristics associated with the map. In
addition, sufficient conditions are given for the positivity of the
Jacobian determinant and an iterative algorithm for solving the
inversion problem is derived. The convergence and the accuracy of
numerical solutions to partial differential equations strongly depend on
the quality of the grids on which these solutions are computed. First
and second order geometric characteristics for hexahedral volume grids
cells are formulated and applied to evaluate the quality of
three-dimensional grid structures. Examples measuring the Jacobian and
the orthogonality of geologic grids are given.
The extended abstract and additional information about Professor Zafiris can be found
here.
November 10. Special day and time: Friday, 1:30 - 2:30 pm !!!
Speaker: Edwin Tecarro , Dept. of Computer and Mathematical Sciences, University
of Houston-Downtown
Title: Interdisciplinary Research in Biology and Mathematics
Abstract: The application of mathematical theories and computational
schemes in helping understand various biological phenomena accentuates a new
trend in life sciences research. The study of biology depends increasingly
on data and models; advances in the computer sciences have made it possible
to handle a lot of information with relative ease. The mechanics of
interdisciplinary research in biological and quantitative sciences will be
described in terms of actual studies conducted in the mathematical modeling
of the mammalian cell cycle protein network.
November 17: SPECIAL DAY AND TIME : FRIDAY 1:30 - 2:30
Speaker: John Alford, Sam Houston State University
Title: A Reaction-Diffusion Model for Eradication
of the Screwworm Fly By Sterile Fly Release Method.
Abstract: The screwworm fly is a parasite that causes myiasis (larval
infestations in tissues) in wounded mammals. It was eradicated
from the United States, Mexico and parts of Central America by the sterile
insect release method (SIRM). A permanent sterile barrier
zone is now maintained in Panama to prevent renewed invasion into
eradicated territory. We have modeled SIRM control of the
screwworm fly with a system of reaction-diffusion equations. Our
results suggest that the barrier zone could be shortened
substantially, reducing costs without risk of screwworm
reinvasion.
December 7:
Speaker: Nira Dyn, Tel-Aviv University, Israel
Title: Subdivision schemes in Geometric Modelling: from discrete data to smooth shapes.
Abstract: Subdivision schemes are efficient computational methods for the design,
representation and approximation of surfaces of arbitrary topology in 3D.
Subdivision schemes generate curves/surfaces from discrete data by repeated
refinements. This talk reviews some of the theory of linear stationary subdivision schemes and their applications in geometric modeling. The first part is
concerned with ''classical'' schemes refining control points. The second part
reviews subdivision schemes refining other ob jects, such as compact sets and
nets of curves. Examples of various schemes are presented.