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Symmetry 2011, 3(3), 402-442; doi:10.3390/sym3030402

Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content

Brain Products, Freiburg, Germany, and Simion Stoilow Institute of Mathematics of the Romanian Academy Research Unit 5, P.O. Box 1-764, RO-014700 Bucharest, Romania
Received: 11 January 2011 / Revised: 15 June 2011 / Accepted: 30 June 2011 / Published: 7 July 2011
(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)
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Abstract

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.
Keywords: recurrent double sequence; expansive system of context-free substitutions; automatic proof procedure; homomorphisms of finite Abelian groups; Klein’s Vierergruppe; M2(F2); geometric content recurrent double sequence; expansive system of context-free substitutions; automatic proof procedure; homomorphisms of finite Abelian groups; Klein’s Vierergruppe; M2(F2); geometric content
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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Prunescu, M. Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content. Symmetry 2011, 3, 402-442.

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