Georgia Institute of Technology College of Sciences

A rectangulation is a tiling of a rectangle by rectangles. The rectangulation is called generic if no four of its rectangles share a corner. We will consider the problem of counting generic rectangulations (with n rectangles) up to combinatorial equivalence. This talk will present and explain an initial step in the enumeration: the fact that generic rectangulations are in bijection with permutations that avoid a certain set of patterns. I'll give background information on rectangulations and pattern avoidance. Then I'll make the connection between generic rectangulations and pattern avoiding permutations, which draws on earlier work with Shirley Law on "diagonal" rectangulations. I'll also comment on two theories that led to this result and its proof: the lattice theory of the weak order on permutations and the theory of combinatorial Hopf algebras.

Cone Beam Computer tomography (CBCT) has been broadly applied incancer radiation therapy, mainly for positioning patients to align withtreatment radiation beams. As opposed to tomography reconstruction problemsfor diagnostic purposes, CBCT reconstruction in radiotherapy requires a highcomputational efficiency, since it is performed while patient is lying on acouch, waiting for the treatment. Moreover, the excessive radiation dosefrom frequent scans has become a clinical concern. It is therefore desirableto develop new techniques to reconstruct CBCT images from low dose scans. Inthis talk, I will present our recent work on an iterative low dose CBCTreconstruction technique via total variation regularization and tight frameregularization. It is found that 40~60 x-ray projections are sufficient toreconstruct a volumetric image with satisfactory quality in about 2min. Wehave also studied 4 dimensional CBCT (4DCBCT) reconstruction problem viatemporal non-local means (TNLM) and high quality 4DCBCT images can beobtained. Our algorithms have been fully implemented on a graphicsprocessing unit. Detailed implementation techniques will also be addressed.

We study four discrete time stochastic systems on $\bbN$ modelingprocesses of rumour spreading. The involved individuals can eitherhave an active ora passive role, speaking up or asking for the rumour. The appetite inspreading or hearing the rumour is represented by a set of randomvariables whose distributionsmay depend on the individuals. Our goal is to understand - based on those randomvariables distribution - whether the probability of having an infiniteset of individuals knowing the rumour is positive or not.

Motivated by a problem in the synthesis of nanowires, a sequential space filling design, called Sequential Minimum Energy Design (SMED), is proposed for exploring and searching for the optimal conditions in complex black-box functions. The SMED is a novel approach to generate designs that are model independent, can quickly carve out regions with no observable nanostructure morphology, allow for the exploration of complex response surfaces, and can be used for sequential experimentation. It can be viewed as a sequential design procedure for stochastic functions and a global optimization procedure for deterministic functions. The basic idea has been developed into an implementable algorithm, and guidelines for choosing the parameters of SMED have been proposed. Convergence of the algorithm has been established under certain regularity conditions. Performance of the algorithm has been studied using experimental data on nanowire synthesis as well as standard test functions.(Joint work with V. R. Joseph, Georgia Tech and T. Dasgupta, Harvard U.)