34 research outputs found
Sturmian numeration systems and decompositions to palindromes
We extend the classical Ostrowski numeration systems, closely related to
Sturmian words, by allowing a wider range of coefficients, so that possible
representations of a number better reflect the structure of the associated
Sturmian word. In particular, this extended numeration system helps to catch
occurrences of palindromes in a characteristic Sturmian word and thus to prove
for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni
and the author: If a word is not periodic, then for every it has a prefix
which cannot be decomposed to a concatenation of at most palindromes.Comment: Submitted to European Journal of Combinatoric
Nested quasicrystalline discretisations of the line
One-dimensional cut-and-project point sets obtained from the square lattice
in the plane are considered from a unifying point of view and in the
perspective of aperiodic wavelet constructions. We successively examine their
geometrical aspects, combinatorial properties from the point of view of the
theory of languages, and self-similarity with algebraic scaling factor
. We explain the relation of the cut-and-project sets to non-standard
numeration systems based on . We finally examine the substitutivity, a
weakened version of substitution invariance, which provides us with an
algorithm for symbolic generation of cut-and-project sequences
Decidability Problems for Self-induced Systems Generated by a Substitution
International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems
Analysis of the Student Numerical Literacy in Completing the Division Count and Multiplication Operation of the 6th Grade Elementary School Round Number
The background of this study is that students are unable to solve the problems and problems of daily life and which of the students associated with literacy, are unable to think critically and be able to reason in understanding the concept of division operations and multiplication integers. The aim of this study was to describe numerical literacy in the division and multiplication operations of integers in 6th grade elementary school. Research type use descriptive with a qualitative approach. Research instrument consist of interview instrument, observation instrument, and test subject. Research has found no significant obstacle, the study subject can complete most numerical literacy components on the test issue. Research subjects has not been able to dominate numerical literacy of patterns and statistic intact, but some of the components appearing on the subject have not been fully resolved
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference