Other Mathematics Commons^™

Splittings Of Relatively Hyperbolic Groups And Classifications Of 1-Dimensional Boundaries, Matthew Haulmark

Theses and Dissertations

In the first part of this dissertation, we show that the existence of non-parabolic local cut point in the relative (or Bowditch) boundary, $\relbndry$, of a relatively hyperbolic group $(\Gamma,\bbp)$ implies that $\Gamma$ splits over a $2$-ended subgroup. As a consequence we classify the homeomorphism type of the Bowditch boundary for the special case when the Bowditch boundary $\relbndry$ is one-dimensional and has no global cut points.

In the second part of this dissertation, We study local cut points in the boundary of CAT(0) groups with isolated flats. In particular the relationship between local cut points in $\bndry X$ and …

On Some One-Complex Dimensional Slices Of The Boundedness Locus Of A Multi-Parameter Rational Family, Matthew Hoeppner

Theses and Dissertations

Complex dynamics involves the study of the behavior of complex-valued functions when they are composed with themselves repeatedly. We observe the orbits of a function by passing starting values through the function iteratively. Of particular interest are the orbits of any critical points of the function, called critical orbits. The behavior of a family of functions can be determined by examining the change in the critical orbit(s) of the functions as the values of the associated parameters vary. These behaviors are often separated into two categories: parameter values where one or more critical orbits remain bounded, and parameter values where …

Cocompact Cubulations Of Mixed 3-Manifolds, Joseph Dixon Tidmore

Theses and Dissertations

In this dissertation, we complete the classification of which compact 3-manifolds have a virtually compact special fundamental group by addressing the case of mixed 3-manifolds. A compact aspherical 3-manifold M is mixed if its JSJ decomposition has at least one JSJ torus and at least one hyperbolic block. We show the fundamental group of M is virtually compact special iff M is chargeless, i.e. each interior Seifert fibered block has a trivial Euler number relative to the fibers of adjacent blocks.

Asymptotic Expansion Of The L^2-Norm Of A Solution Of The Strongly Damped Wave Equation, Joseph Silvio Barrera

Theses and Dissertations

The Fourier transform, F, on R^N (N≥1) transforms the Cauchy problem for the strongly damped wave equation u_tt(t,x) - Δu_t(t,x) - Δu(t,x) = 0 to an ordinary differential equation in time t. We let u(t,x) be the solution of the problem given by the Fourier transform, and v(t,ƺ) be the asymptotic profile of F(u)(t,ƺ) = û(t,ƺ) found by Ikehata in [4].

In this thesis we study the asymptotic expansions of the squared L^2-norms of u(t,x), û(t,ƺ) - v(t,ƺ), and v(t,ƺ) as t → ∞. With suitable initial data u(0,x) and u_t(0,x), we establish the rate of growth or decay of …