Georgia Institute of Technology College of Sciences

We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a convex domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda. Further, the Hardy term in this inequality is stronger than the one in the classical case. The result can be extended as well to more general domains

Nonnegative Matrix Factorization (NMF) has attracted much attention during the past decade as a dimension reduction method in machine learning and data analysis. NMF provides a lower rank approximation of a nonnegative high dimensional matrix by factors whose elements are also nonnegative. Numerous success stories were reported in application areas including text clustering, computer vision, and cancer class discovery. In this talk, we present novel algorithms for NMF and NTF (nonnegative tensor factorization) based on the alternating non-negativity constrained least squares (ANLS) framework. Our new algorithm for NMF is built upon the block principal pivoting method for the non-negativity constrained least squares problem that overcomes some limitations of the classical active set method. The proposed NMF algorithm can naturally be extended to obtain highly efficient NTF algorithm for PARAFAC (PARAllel FACtor) model. Our algorithms inherit the convergence theory of the ANLS framework and can easily be extended to other NMF formulations such as sparse NMF and NTF with L1 norm constraints. Comparisons of algorithms using various data sets show that the proposed new algorithms outperform existing ones in computational speed as well as the solution quality. This is a joint work with Jingu Kim and Krishnakumar Balabusramanian.

The purpose of the Georgia Scientific Computing Symposium (GSC 2010) is to provide an opportunity for professors, postdocs and graduate students in the Atlanta area to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The one-day symposium is open to the whole research community. The event is free but registration is required.

I will introduce the AJ conjecture (by Garoufalidis) which relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. Then I will verify it for the trefoil and the figure 8 knots (due to Garoufalidis) and torus knots (due to Hikami) by explicit calculations.