Physical Sciences and Mathematics Commons^™

Modeling, Analysis, And Simulation Of Discrete-Continuum Models Of Step-Flow Epitaxy: Bunching Instabilities And Continuum Limits, Nicholas O. Kirby

University of Kentucky Doctoral Dissertations

Vicinal surfaces consist of terraces separated by atomic steps. In the step-flow regime, deposited atoms (adatoms) diffuse on terraces, eventually reaching steps where they attach to the crystal, thereby causing the steps to move. There are two main objectives of this work. First, we analyze rigorously the differences in qualitative behavior between vicinal surfaces consisting of infinitely many steps and nanowires whose top surface consists of a small number of steps bounded by a reflecting wall. Second, we derive the continuum model that describes the macroscopic behavior of vicinal surfaces from detailed microscopic models of step dynamics.

We use the …

Convergence Of Eigenvalues For Elliptic Systems On Domains With Thin Tubes And The Green Function For The Mixed Problem, Justin L. Taylor

University of Kentucky Doctoral Dissertations

I consider Dirichlet eigenvalues for an elliptic system in a region that consists of two domains joined by a thin tube. Under quite general conditions, I am able to give a rate on the convergence of the eigenvalues as the tube shrinks away. I make no assumption on the smoothness of the coefficients and only mild assumptions on the boundary of the domain.

Also, I consider the Green function associated with the mixed problem on a Lipschitz domain with a general decomposition of the boundary. I show that the Green function is Hölder continuous, which shows how a solution to …

Combinatorial Aspects Of Excedances And The Frobenius Complex, Eric Logan Clark

University of Kentucky Doctoral Dissertations

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that …

Topological And Combinatorial Properties Of Neighborhood And Chessboard Complexes, Matthew Zeckner

University of Kentucky Doctoral Dissertations

This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_n,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SG_n,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation …