Georgia Institute of Technology College of Sciences

Many physical models without dissipation can be written in a Hamiltonian form. For example, nonlinear Schrodinger equation for superfluids and Bose-Einstein condensate, water waves and their model equations (KDV, BBM, KP, Boussinesq systems...), Euler equations for inviscid fluids, ideal MHD for plasmas in fusion devices, Vlasov models for collisionless plasmas and galaxies, Yang-Mills equation in gauge field theory etc. There exist coherent structures (solitons, steady states, traveling waves, standing waves etc) which play an important role on the long time dynamics of these models. First, I will describe a general framework to study linear stability (instability) when the energy functional is bounded from below. For the models with indefinite energy functional (such as full water waves), approaches to find instability criteria will be mentioned. The implication of linear instability (stability) for nonlinear dynamics will be also briefly discussed.

Given a finite set of matrices F, the Markovian Joint Spectral Radius represents the maximal rate of growth of products of matrices in F when the matrices are multiplied each other following some Markovian law. This quantity is important, for instance, in the study of the so called zero stability of variable stepsize BDF methods for the numerical integration of ordinary differential equations. Recently Kozyakin, based on a work by Dai, showed that, given a set F of N matrices of dimension d and a graph G, which represents the admissible products, it is possibile to compute the Markovian Joint Spectral Radius of the couple (F,G) as the classical Joint Spectral Radius of a new set of N matrices of dimension N*d, which are produced as a particular lifting of the matrices in F. Clearly by this approach the exact evaluation or the simple approximation of the Markovian Joint Spectral Radius becomes a challenge even for reasonably small values of N and d. In this talk we briefly review the theory of the Joint Spectral Radius, and we introduce the Markovian Joint Spectral Radius. Furthermore we address the question whether it is possible to reduce the exact calculation computational complexity of the Markovian Joint Spectral Radius. We show that the problem can be recast as the computation of N polytope norms in dimension d. We conclude the presentation with some numerical examples. This talk is based on a joint work with Nicola Guglielmi from the University of L'Aquila, Italy, and Vladimir Yu. Protasov from the Moscow State University, Russia.

The Rehabilitation Engineering and Applied Research Lab (REARLab) performs both experimental research and product development activities focused on persons with disabilities. The REARLab seeks collaboration from the School of Mathematics on 2 current projects. This session will introduce wheelchair seating with respect to pressure ulcer formation and present two projects whose data analysis would benefit from applied mathematics. 3D Tissue Deformation- Sitting induces deformation of the buttocks tissues. Tissue deformation has been identified as the underlying cause of tissue damage resulting from external loading. The REARLab has been collecting multi-planar images of the seated buttocks using MRI. This data clearly shows marked differences between persons, as expected. We are interested in characterizing tissue deformation as a combination of displacement and distortion. Some tissues- such as muscle- displace (translate within the sagittal, coronal and transverse planes) and distort (change shape). Other tissue such as skin and subcutaneous fat, simple distorts. We seek a mathematical means to characterize tissue deformation that reflects its multi-planar nature. Categorizing Weight-shifting behaviors - many wheelchair users have limitations to their motor and/or sensory systems resulting in a risk of pressure ulcers. Pressure ulcers occur when localized loading on the skin causes ischemia and necrosis. In an attempt to reduce risk of pressure ulcer occurrence, wheelchair users are taught to perform weight-shifts. Weight shifts are movements that re-distribute loads off the buttocks for short periods of time. The REARLab is measuring weight shifting behaviors of wheelchair users during their everyday lives. We seek a means to classify patterns of behavior and relate certain patterns to healthy outcomes versus other patterns that result in unhealthy outcomes.