| Talk Date: | 20080118 |
| Speaker Name: | Ming-Jun Lai |
| E-mail: | mjlai@math.uga.edu |
| webpage: | www.math.uga.edu/~mjlai |
| Talk Title: | Tight Wavelet Frames on the Sphere |
| Talk Abstract: | I will first explain how to construct tight wavelet frames on bounded domains. Then I will introduce trigonometric B-splines which are necessary for constructing C^1 smooth surfaces on the unit sphere among certain class of functions. Then we explain how to find Fourier transform of trigonometric B-splines and we use them to construct tight wavelet frames. |
| Talk Date: | 20080201 |
| Speaker Name: | Sergiy Borodachov |
| E-mail: | borodasv@math.gatech.edu |
| webpage: | |
| Talk Title: | Optimal cubature formulas related to computer tomography |
| Talk Abstract: | We consider the problem of optimal recovery of the integral of the function f defined on the cube C. A priori, we know that the function has a given majorant for the modulus of continuity (i.e. belongs to the corresponding class of functions). Assume also that we know or can easily compute mean values of f along intersections of C with n arbitrary planes or n arbitrary straight lines. The problem is to choose best positions of these n planes (lines) and the best way to assign to the n-tuple of sampled means the approximate value of the integral of f along the whole cube C in order to minimize the maximal error over this class of functions. This problem is a generalization of the classical Kolmogorov-Nikol'skii problem about optimal quadratures. |
| Talk Date: | 20080215 |
| Speaker Name: | Tom Lyche |
| E-mail: | tom@ifi.uio.no |
| webpage: | |
| Talk Title: | New Formulas for Divided Differences and Partitions of a Convex Polygon |
| Talk Abstract: | Divided differences are a basic tool in approximation theory and numerical analysis: they play an important role in interpolation and approximation by polynomials and in spline theory. So it is worth while to look for identities that are analogous to identities for derivatives. An example is the Leibniz rule for differentiating products of functions. This rule was generalized to divided differences by Popoviciu and Steffensen 70 years ago. To our surprise it was discovered that there were no analog of a 150 year old formula for differentiating composite functions (Faa di Bruno's formula) and for differentiating the inverse of a function. In this talk I will discuss chain rules and inverse rules for divided differences. The inverse rule turns out to have a surprising and beautiful structure: it is a sum over partitions of a convex polygon into smaller polygons using only non-intersecting diagonals. This provides a new way of enumerating all partitions of a convex polygon with a specified number of triangles, quadrilaterals, and so on. The talk is based on joint work with Michael Floater. 1 |
| Talk Date: | 20080222 |
| Speaker Name: | David Prager |
| E-mail: | pragerdj@uga.edu |
| webpage: | www.math.uga.edu/~pragerdj/ |
| Talk Title: | Stock Loan Pricing Under Geometric Brownian Motion and Mean-Reverting Stock Models |
| Talk Abstract: | A stock loan is a debt instrument in which the borrower uses a share of stock as collateral. When the loan matures, the borrower may regain the share of stock by repaying the loan principle plus interest, or if the borrower fails to repay the loan, the lender may retain the collateral. Unlike traditional debt instruments, the collateral of a stock loan is subject to wide and frequent price fluctuations, creating difficulties in determining a fair price, interest rate, or service fee for the loan. This talk will introduce stock loans and give some examples to illustrate why such instruments are desirable. We will explore some of the difficulties associated with pricing these instruments. Closed-form solutions will be given for the cases when 1) the loan is American, perpetual, and the price of the stock behaves a geometric Brownian motion and 2) the loan is European and the price of the stock behaves a mean-reverting price model. |
| Talk Date: | 20080229 |
| Speaker Name: | Kun Zhao |
| E-mail: | kzhao@math.gatech.edu |
| webpage: | |
| Talk Title: | Initial Boundary Value Problem for 2D Viscous Boussinesq System |
| Talk Abstract: | The 2D Boussinesq systems share similar vortex stretching effects as that of 3D incompressible Euler or Navier-Stokes equations. In this talk, I will present some recent result on 2D viscous Boussinesq equations in bounded domains. The global existence of smooth solutions is obtained by the method of energy estimate. Also, the kinetic energy is found to be uniformly bounded. This is a joint work with Mingjun Lai and Ronghua Pan. |
| Talk Date: | 20080321 |
| Speaker Name: | Alex Petukhov |
| E-mail: | petukhov@math.uga.edu |
| webpage: | |
| Talk Title: | Hausdorff metric of battle fields. |
| Talk Abstract: | Hausdorff metric is widely used in the computer graphics for approximation (measuring the closeness) of smooth surfaces. At the same time, the Hausdorff metric is a very natural to measure the distance between discontinuous functions or functions with low smoothness. Such representation is currently very actual for many applied problems. Say, for the representation of mountain (urban) landscape is very important in 3D navigators currently used for military applications, especially for helicopters and unmanned planes. The Hausdorff distance is essentially non-linear and cannot be generated by any norm. This fact was an obstacle for theoretical studies and development of the approximation methods. We will discuss how this obstacle can be overcome. In particular, we give criteria for the Hausdorff convergence of convolution on (quasi)Banach spaces. |
| Talk Date: | 20080328 |
| Speaker Name: | Malcolm Adams |
| E-mail: | adams@math.uga.edu |
| webpage: | |
| Talk Title: | Hopf Bifurcations in a Parvo virus model |
| Talk Abstract: | Any dog owner is familiar with the Parvo virus. This causes a severe illness in canines that can be spread by animal to animal contact or by contact with the feces of an infected animal. Not only is it a threat to pets, but antibodies to this virus have also been found in wild carnivores. I will develop a simple modification of the SIR model in epidemiology that incorporates the delay mechanism of infection through fecal contact. I will give an analytic proof of a Hopf bifurcation in this model. |
| Talk Date: | April 4, 2008 |
| Speaker Name: | Ettinger Bree |
| E-mail: | bree@math.uga.edu |
| webpage: | |
| Talk Title: | Ozone Prediction using Bivariate Splines to Approximate Functional Regression Models |
| Talk Abstract: | Functional linear regression models are models where the explanatory variable is a random surface and the response is a real random variable. We will discuss the resent results for bounded or normal real random responses. We use bivariate splines to represent the random surfaces then we use this representation to construct least squares estimators of the regression function. We will discuss the two cases of the least squares estimators, one with a penalty term and one without. Finally we will explore the functional regression model’s application to predicting ground level ozone from 2006 EPA data over the continental United States. |