Georgia Institute of Technology College of Sciences

Given F, finite set of square matrices of dimension n, it is possible to define the Joint Spectral Radius or simply JSR as a generalization of the well known spectral radius of a matrix. The JSR evaluation proves to be useful for instance in the analysis of the asymptotic behavior of solutions of linear difference equations with variable coefficients, in the construction of compactly supported wavelets of and many others contexts. This quantity proves, however, to be hard to compute in general. Gripenberg in 1996 proposed an algorithm for the computation of lower and upper bounds to the JSR based on a four member inequality and a branch and bound technique. In this talk we describe a generalization of Gripenberg's method based on ellipsoidal norms that achieve a tighter upper bound, speeding up the approximation of the JSR. We show the performance of this new algorithm compared with Gripenberg's one. This talk is based on joint work with V.Y.Protasov.

In this talk, I will present two examples of the application of wavelet analysis to the understanding of mild Traumatic Brain Injury (mTBI). First, the talk will focus on how wavelet-based features can be used to define important characteristics of blast-related acceleration and pressure signatures, and how these can be used to drive a Naïve Bayes classifier using wavelet packets. Later, some recent progress on the use of wavelets for data-driven clustering of brain regions and the characterization of functional network dynamics related to mTBI will be discussed. In particular, because neurological time series such as the ones obtained from an fMRI scan belong to the class of long term memory processes (also referred to as 1/f-like processes), the use of wavelet analysis guarantees optimal theoretical whitening properties and leads to better clusters compared to classical seed-based approaches.

In this era of "big data", Mathematics as it applies to human behavior is becoming a much more relevant and penetrable topic of research. This holds true even for some of the less desirable forms of human behavior, such as crime. In this talk, I will discuss the mathematical modeling of crime on two different "scales", as well as the results of experiments that are being performed to test the usefulness and accuracy of these models. First, I will present a data-driven model of crime hotspots at the scale of neighborhoods -- adapted from literature on earthquake predictions -- along with the results of this model's application within the LAPD. Second, I will describe a game-theoretic model of crime and punishment at the scale of a society, and compare the model to results of lab-based economic experiments performed by myself and collaborators.

We discuss various exponential time differencing (ETD) schemes designed to handle nonlinear parabolic systems. The ETD schemes use certain Pade approximations of the matrix exponential function. These ETD schemes have potential to be implemented in parallel and their performance is very robust with respect to the type of PDE. They are unconditionally stable and computationally very fast due to the technique of computing the nonlinear part explicitly. To handle the problem of irregular initial or boundary data an adaptive ETD scheme is utilized, which adds sufficient damping of spurious oscillations. We discuss algorithm development, theory and applications.