Displaying similar documents to “Tilings associated with non-Pisot matrices”

The higher transvectants are redundant

Abdelmalek Abdesselam, Jaydeep Chipalkatti (2009)

Annales de l’institut Fourier

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Let A , B denote generic binary forms, and let 𝔲 r = ( A , B ) r denote their r -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the { 𝔲 r } . As a consequence, we show that each of the higher transvectants { 𝔲 r : r ≥ 2 } is redundant in the sense that it can be completely recovered from 𝔲 0 and 𝔲 1 . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of S L 2 -representations,...

Billiard complexity in the hypercube

Nicolas Bedaride, Pascal Hubert (2007)

Annales de l’institut Fourier

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We consider the billiard map in the hypercube of ℝ d . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that n 3 d - 3 is the order of magnitude of the complexity.

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to β -shifts

Veronica Baker, Marcy Barge, Jaroslaw Kwapisz (2006)

Annales de l’institut Fourier

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This article is devoted to the study of the translation flow on self-similar tilings associated with a substitution of Pisot type. We construct a geometric representation and give necessary and sufficient conditions for the flow to have pure discrete spectrum. As an application we demonstrate that, for certain beta-shifts, the natural extension is naturally isomorphic to a toral automorphism.

Substitutions, abstract number systems and the space filling property

Clemens Fuchs, Robert Tijdeman (2006)

Annales de l’institut Fourier

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In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo 1 and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

On the Number of Partitions of an Integer in the m -bonacci Base

Marcia Edson, Luca Q. Zamboni (2006)

Annales de l’institut Fourier

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For each m ≥ 2 , we consider the m -bonacci numbers defined by F k = 2 k for 0 ≤ k ≤ m - 1 and F k = F k - 1 + F k - 2 + ⋯ + F k - m for k ≥ m . When m = 2 , these are the usual Fibonacci numbers. Every positive integer n may be expressed as a sum of distinct m -bonacci numbers in one or more different ways. Let R m ( n ) be the number of partitions of n as a sum of distinct m -bonacci numbers. Using a theorem of Fine and Wilf, we obtain a formula for R m ( n ) involving sums of binomial coefficients modulo 2 . In addition we show that this formula may be used to determine the...