Clemson Analysis SeminarFridays, 3:30--4:30pm Martin M-102 Fall 2017 September 8 Walton Green (Clemson University) September 15 Jeong-rock Yoon (Clemson University) September 22 Oleg Yordanov (Clemson University & Bulgarian Academy of Sciences) September 26 (Tuesday) James Melbourne (University of Minnesota) October 6 October 13 October 20 October 27 November 3 November 10 November 17 December 1 December 8Title & AbstractSeptember 26 James Melbourne Title: A R\'enyi Entropy TrilogyAbstract: In part of an effort to properly axiomatically characterize Shannon entropy, Alfred R\'enyi put forth a family of "information measures", parameterized $r \in [0,\infty]$. The Shannon entropy corresponded to $r=1$, and his famed entropy power inequality (EPI), fully proved by Stam some years later, could be written $N_1(X+Y) \geq N_1(X)+N_1(Y)$ for independent random variables $X,Y$. This provides an archetype for exploring further convolution inequalities under other Renyi entropy parameters. In particular, when $r=0$ one can interpret the Brunn-Minkowski inequality of Convex Geometry as a Renyi EPI of a nearly identical form, while setting r=\infty allows one to cleanly formulate some new projection inequalities important in Random Matrix Theory. We will properly define the terminology and notation used above in order to discuss this background and motivation before describing some recent progress in the understanding of $r=\infty$ case. Time permitting some general superadditivity properties.will be explained as well.September 22 Oleg Yordanov Title: Approximate, Saturated and Blurred Scaling of Random Fields: ApplicationsAbstract: Scaling (homogeneous, power-law) functions are empirically identified in a variety of natural phenomena and structures. An important class of irregular structures and processes, modeled as random fields, exhibit scaling of their second order, two-point correlation functions. Among these, referred to also as random fractals, are the morphology of rough surfaces, the fully developed turbulence, star and galaxy clusters, and many others. In all these cases, the scaling is accounted for by using power-law functions, which are singular and have limited range of validity. In this talk, I present examples of random fields whose correlation functions are defined over the entire real line and are analytic: yet they exhibit scaling properties albeit not exact. The fields are constructed over a finite band of wavenumbers/frequencies in the Fourier space. The scaling arises as an asymptotic behavior and therefore is only approximate. I also present applications of the above fields and discuss certain technical subtleties involved in these applications.September 15 Jong-rock YoonTitle: Various models of viscoelasticity including fractional derivative modelSeptember 8 Walton Green Title: Brownian Rotation of Magnetized Particles in Magnetic Particle ImagingAbstract: Magnetic Particle Imaging (MPI) is a medical imaging technique which is implemented by measuring the voltage emitted by magnetized particles in a domain. I will derive the current model (equilibrium model) and propose a more sophisticated one (relaxation model) and compare the two in both simulation and reconstruction. |