941 research outputs found
Collisions and their Catenations: Ultimately Periodic Tilings of the Plane
Motivated by the study of cellular automata algorithmics and dynamics, we investigate an extension of ultimately periodic words to two-dimensional infinite words: collisions. A natural composition operation on tilings leads to a catenation operation on collisions. By existence of aperiodic tile sets, ultimately periodic tilings of the plane cannot generate all possible tilings but exhibit some useful properties of their one-dimensional counterparts: ultimately periodic tilings are recursive, very regular, and tiling constraints are easy to preserve by catenation. We show that, for a given catenation scheme of finitely many collisions, the generated set of collisions is semi-linear
Geometrical regular languages and linear Diophantine equations: The strongly connected case
AbstractGiven an arbitrarily large alphabet Σ, we consider the family of regular languages over Σ for which the deterministic minimal automaton has a strongly connected state diagram. We present a new method for checking whether such a language is semi-geometrical or not and whether it is geometrical or not. This method makes use of the enumeration of the simple cycles of the state diagram. It is based on the construction of systems of linear Diophantine equations, where the coefficients are deduced from the set of simple cycles
The homeomorphism problem for closed 3-manifolds
We give a more geometric approach to an algorithm for deciding whether two
hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach
to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.Comment: first version: 12 pages. Replacement: 14 pages. Includes minor
improvements to exposition in response to referee's comment
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