Physical Sciences and Mathematics Commons^™

The Word Problem For The Automorphism Groups Of Right-Angled Artin Groups Is In P, Carrie Anne Whittle

Graduate Theses and Dissertations

We provide an algorithm which takes any given automorphism f of any given right-angled Artin group G and determines whether or not f is the identity automorphism, thereby solving the word problem for the automorphism groups of right-angled Artin groups. We do this by solving the compressed word problem for right-angled Artin groups, a more general result. A key piece of this solution is the use of Plandowski's algorithm. We also demonstrate that our algorithm runs in polynomial time in the size of the given automorphism, written as a word in Laurence's generators of the automorphism group of the given …

Hardy Space Properties Of The Cauchy Kernel Function For A Strictly Convex Planar Domain, Belen Espinosa Lucio

Graduate Theses and Dissertations

This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\mathbb{C}^N$, the Henkin-Ram\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\in(0,N)$.

The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\subset\mathbb{C}$ of class $C^2$ if we fix $\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function

\mathcal{K}(\zeta,z)=\frac{1}{2\pi i}\frac{1}{\zeta-z}

as a …

The Effect Of Symmetry On The Riemann Map, Jeanine Louise Myers

Graduate Theses and Dissertations

The Riemann mapping theorem guarantees the existence of a conformal mapping or Riemann map in the complex plane from the open unit disk onto an open simply-connected domain, which is not all of the complex plane. Although its existence is guaranteed, the Riemann map is rarely known except for special domains like half-planes, strips, etc. Therefore, any information we can determine about the Riemann map for any class of domains is interesting and useful.

This research investigates how symmetry affects the Riemann map. In particular, we define domains with symmetries called Rectangular Domains or RDs. The Riemann map of an …