Browsing by Author "Farlow, Kasie Geralyn"
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- Max-Plus AlgebraFarlow, Kasie Geralyn (Virginia Tech, 2009-04-27)In max-plus algebra we work with the max-plus semi-ring which is the set โmax=[-โ)โชโ together with operations ๐โ๐ = max(๐,๐) and ๐โ๐= ๐+๐. The additive and multiplicative identities are taken to be ฮต=-โ and ฮต=0 respectively. Max-plus algebra is one of many idempotent semi-rings which have been considered in various fields of mathematics. Max-plus algebra is becoming more popular not only because its operations are associative, commutative and distributive as in conventional algebra but because it takes systems that are non-linear in conventional algebra and makes them linear. Max-plus algebra also arises as the algebra of asymptotic growth rates of functions in conventional algebra which will play a significant role in several aspects of this thesis. This thesis is a survey of max-plus algebra that will concentrate on max-plus linear algebra results. We will then consider from a max-plus perspective several results by Wentzell and Freidlin for finite state Markov chains with an asymptotic dependence.
- The Reflected Quasipotential: Characterization and ExplorationFarlow, Kasie Geralyn (Virginia Tech, 2013-05-06)The Reflected Quasipotential V(x) is the solution to a variational problem that arises in the study of reflective Brownian motion. Specifically, the stationary distributions of reflected Brownian motion satisfy a large deviation principle (with respect to a spatial scaling parameter) with V(x) as the rate function. The Skorokhod Problem is an essential device in the construction and analysis of reflected Brownian motion and our value function V(x). Here we characterize V(x) as a solution to a partial differential equation H(DV(x))=0 in the positive n-dimensional orthant with appropriate boundary conditions. H(p) is the Hamiltonian and DV(x) is the gradient of V(x). V(x) is continuous but not differentiable in general. The characterization will need to be in terms of viscosity solutions. Solutions are not unique, thus additional qualifications will be needed for uniqueness. In order to prove our uniqueness result we consider a discounted version of V(x) in a truncated region and pass to the limit. In addition to this characterization of V(x) we explore the possibility of cyclic optimal paths in 3 dimensions.