On the Fundamental Group of self-affine plane Tiles

Jun Luo; Jörg M. Thuswaldner

Displaying similar documents to “On the Fundamental Group of self-affine plane Tiles”

The first to $(k + 1)$ -th smallest Wiener (hyper-Wiener) indices of connected graphs

Liu Mu-huo, Xuezhong Tan (2009)

Kragujevac Journal of Mathematics

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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 ${𝔲_{r} : r \geq 2}$ 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.

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 $m \geq 2,$ we consider the $m$ -bonacci numbers defined by $F_{k} = 2^{k}$ for $0 \leq k \leq m - 1$ and $F_{k} = F_{k - 1} + F_{k - 2} + \dots + F_{k - m}$ for $k \geq m .$ 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...

Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

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Suppose $A \in G l_{d} (ℤ)$ has a 2-dimensional expanding subspace $E^{u}$ , satisfies a regularity condition, called “good star”, and has $A^{*} \geq 0$ , where $A^{*}$ is an of $A$ . A morphism $θ$ of the free group on ${1, 2, \dots, d}$ is called a of $A$ if it has structure matrix $A$ . We show that there is a $Θ$ whose “boundary substitution” $θ = \partial Θ$ is a non-abelianization of $A$ . Such a tiling substitution $Θ$ leads to a self-affine tiling of $E^{u} \sim ℝ^{2}$ with $A_{u} {: = A |}_{E_{u}} \in G L_{2} (ℝ)$ as its expansion. In the last section we find conditions on $A$ so that $A^{*}$ has no negative entries. ...

Displaying similar documents to “On the Fundamental Group of self-affine plane Tiles”

The first to ( k + 1 ) -th smallest Wiener (hyper-Wiener) indices of connected graphs

The higher transvectants are redundant

Billiard complexity in the hypercube

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

Tilings associated with non-Pisot matrices

The first to $(k + 1)$ -th smallest Wiener (hyper-Wiener) indices of connected graphs

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