8:45-9:00 Alexander Engau,
Clemson University
Simultaneous Flows in Multiple Networks
The development of network flow programming was originally motivated from classical operations research tasks such as communication, transportation, production or scheduling. However, it has also been found that a large number of other combinatorial problems can frequently be formulated in terms of network flows. While such problems can generally be embedded into the theory of linear programming, a number of benefits arises from a separate treatment and by making use of the special network structure. In particular, many solution algorithms allow for a significant improvement with respect to complexity, running time and required computing resources.
We study an integer program whose constraint matrix can be partitioned into a collection of submatrices that are consecutive one in rows. Based on linear programming relaxation and duality techniques, this integer program is transformed into a simultaneous flow problem in several underlying networks that are related through a bijection on subsets of their respective arcs. Similar to simultaneous flows that have identical values on corresponding arcs in different networks, one can study simultaneous tree problems, matching problems, etc. This new area coined "simultaneous graph theory" will be subject of forthcoming University of Kaiserslautern working papers.
Keywords: network flow programming, simultaneous network flows.
9:05-9:20 Nicholas Orlowski, North Carolina State University
Statistical Rank Estimation: Observations and Insights
Factor analysis has been an invaluable tool for rank reduction of data matrices. Singular value and spectral decompositions provide a ranking of the importance of different factors. By choosing a suitable number of factors, a good low-rank approximation to these matrices can be obtained. However, deciding on an adequate number of factors is a difficult and subjective task. Often the chosen rank is decided by choosing a cutoff arbitrarily. Statistical methods are discussed that provide a confidence measurement for deciding the number of factors. Also, an adaptation of estimates for linear regression coefficients is shown to apply to factor analysis.
9:25-9:40 David M. Schlorff,
North Carolina State University
Generalized Proof of Convergence for "k-means like" Clustering Method
The k-means method is perhaps the best known example of a class of Hard (non-fuzzy) clustering algorithms which improve iteratively upon some set of initial clusters by means of reassigning data objects and assigning new representatives to existing clusters. This class of methods is defined, and a generalized proof of convergence is given. The requirements for membership in this class are examined, and a framework is briefly presented to allow comparisons between methods.
9:45-10:00 Necibe Tuncer,
Auburn University
Finite Element Method on the Sphere
The numerical approximation of partial differential equations on the sphere has been an attractive subject of late. We study the model problem -Δ u = f(x,y) where u (x,y) = g(x,y) on the boundary of the domain. Here the domain is the unit disc. In our research, we generate the FEM on the unit disc, by mapping the unit square to the domain with the radial projection. We introduce the mapped finite elements on the unit disc as "radially projected finite elements". We discuss the approximation properties of radially projected finite elements on the unit disc and the sphere