Zuzana Masáková

Factor versus palindromic complexity of uniformly recurrent infinite words

We study the relation between the palindromic and factor complexity of infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1)@[emailxa0protected](n)+2, for all [emailxa0protected]?N. For a large class of words this is a better estimate of the palindromic complexity in terms of the factor complexity than the one presented in [J.-P. Allouche, M. Baake, J. Cassaigne, D. Damanik, Palindrome complexity, Theoret. Comput. Sci. 292 (2003) 9-31]. We provide several examples of infinite words for which our estimate reaches its upper bound. In particular, we derive an explicit prescription for the palindromic complexity of infinite words coding r-interval exchange transformations. If the permutation @p connected with the transformation is given by @p(k)=r+1-k for all k, then there is exactly one palindrome of every even length, and exactly r palindromes of every odd length.

Minimal distances in quasicrystals

A general expression is derived for the minimal distance between points of a cut and project quasicrystal in with a convex acceptance window . The study of minimal distances amounts to the study of one-dimensional quasicrystals and their rescalings which occur in . For an n-dimensional ball as , the exact value of is calculated for any radius; for `close to a ball, a simple formula is given; for all upper and lower bounds for are found. The latter are easy to use even when is of a complicated shape.

Complexity of infinite words associated with beta-expansions

We study the complexity of the infinite word uβ associated with the Renyi expansion of 1 in an irrational base β > 1 . When β is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity C(n) = n + 1 . For β such that dβ (1) = t1 t2 ...tm is finite we provide a simple description of the structure of special factors of the word uβ . When tm =1 we show that C(n) = (m - 1)n + 1 . In the cases when t1 = t2 = ... tm-1 or t1 > max{t2 ,...,tm-1 } we show that the first difference of the complexity function C(n + 1) - C(n ) takes value in {m - 1,m} for every n , and consequently we determine the complexity of uβ . We show that uβ is an Arnoux-Rauzy sequence if and only if dβ (1) = tt...t1 . On the example of β = 1 + 2 cos(2π/7), solution of X3 = 2X2 + X - 1 , we illustrate that the structure of special factors is more complicated for dβ (1) infinite eventually periodic. The complexity for this word is equal to 2n+1 .

Classification of Voronoi and Delone tiles of quasicrystals: III. Decagonal acceptance window of any size

This paper is the last of a series of three articles presenting a classification of Vornoi and Delone tilings determined by point sets ?(?) (quasicrystals), built by the standard projection of the root lattice of type A4 to a two-dimensional plane spanned by the roots of the Coxeter group H2 (dihedral group of order 10). The acceptance window ? for ?(?) in the present paper is a regular decagon of any radius 0 < r < ?. There are 14 distinct VT sets of Voronoi tiles and 6 sets DT of Delone tiles, up to a uniform scaling by the factor and . The number of Voronoi tiles in different quasicrystal tilings varies between 3 and 12. Similarly, the number of Delone tiles is varying between 4 and 6. There are 7 VT sets of the generic type and 7 of the singular type. The latter occur for seven precise values of the radius of the acceptance window. Quasicrystals with acceptance windows with radii in between these values have constant VT sets, only the relative densities and arrangement of the tiles in the tilings change. Similarly, we distinguish singular and generic sets DT of Delone tiles.

Arithmetics in number systems with a negative base

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set

Self-similar Delone sets and quasicrystals

{rm Fin}(-beta)

Inflation centres of the cut and project quasicrystals

and

Number representation using generalized (-β)-transformation

Z_{-beta}

Substitution rules for aperiodic sequences of the cut and project type

of numbers having finite resp. integer

Proceedings 8th International Conference Words 2011

(-beta)