arXiv subjects

Stanley Orlando Juriaans

Publications and source records attributed to Stanley Orlando Juriaans.

5 recordsLinked to original sources

arXiv2016

Dirichlet-Ford domains and Double Dirichlet domains

We continue investigations started by Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic $2$- and $3$-space, or which have a Dirichlet domain with multiple centers. Such domains are called DF-domains and Double Dirichlet domains respectively. Making use of earlier obtained concrete formulas for the bisectors defining the Dirichlet domain of center $i \in \HQ^2$ or center $j \in \HQ^3$, we obtain a simple condition on the matrix entries of the side-pairing transformations of the fundamental domain of a Fuchsian or Kleinian group to be a DF-domain. Using the same methods, we also complement a result of Lakeland stating that a cofinite Fuchsian group has a DF domain (or a Dirichlet domain with multiple centers) if and only if it is an index $2$ subgroup of the discrete group G of reflections in a hyperbolic polygon.

math.GR↗

arXiv2014

Poincaré Bisectors in Hyperbolic Spaces

We determine explicit formulas for the bisectors used in constructing a Dirichlet fundamental domain in hyperbolic two and three space. They are compared with the isometric spheres employed in the construction of a Ford domain and used to find a finite set of generators for discrete groups of finite covolume. Applications are given to Fuchsian groups, Kleinian groups, including the Bianchi groups, and for the construction of a finite set of generators of the unit group of the integral group ring of a finite nilpotent group. An easy implementable algorithm, DAFC, is also given and used in the search for generators of discrete groups.

math.GR↗

arXiv2013

From the Poincaré Theorem to generators of the unit group of integral group rings of finite groups

We give an algorithm to determine finitely many generators for a subgroup of finite index in the unit group of an integral group ring $\mathbb{Z} G$ of a finite nilpotent group $G$, this provided the rational group algebra $\mathbb{Q} G$ does not have simple components that are division classical quaternion algebras or two-by-two matrices over a classical quaternion algebra with centre $\mathbb{Q}$. The main difficulty is to deal with orders in quaternion algebras over the rationals or a quadratic imaginary extension of the rationals. In order to deal with these we give a finite and easy implementable algorithm to compute a fundamental domain in the hyperbolic three space $\mathbb{H}^3$ (respectively hyperbolic two space $\mathbb{H}^2$) for a discrete subgroup of ${\rm PSL}_2(\mathbb{C})$ (respectively ${\rm PSL}_2(\mathbb{R})$) of finite covolume. Our results on group rings are a continuation of earlier work of Ritter and Sehgal, Jespers and Leal.

math.GR↗

arXiv2009

Colombeau's Algebra of full Generalized Numbers

Let $\bar{\Kset}_f$ denote the commutative unital ring of Colombeau's full generalized numbers. This ring can be endowed with an ultra-metric in such a way that it becomes a topological ring. There are many interesting question about $\bar{\Kset}_f$ in the framework of Commutative Algebra and General Topology as well as of the superposition of these two subjects. The purpose of this paper aims to give an initial step toward the study of this ring.

math.AC↗

arXiv2008

Generalized Solutions of a Nonlinear Parabolic Equation with Generalized Functions as Initial Data

In \cite{bf} Brézis and Friedman prove that certain nonlinear parabolic equations, with the $δ$-measure as initial data, have no solution. However in \cite{cl} Colombeau and Langlais prove that these equations have a unique solution even if the $δ$-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais their result proving that we may take any generalized function as the initial data. Our approach relies on resent algebraic and topological developments of the theory of Colombeau generalized functions and results from \cite{A}.

math.AP↗