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Zuzana Masáková

Publications and source records attributed to Zuzana Masáková.

36 records · Page 2Linked to original sources
arXiv2013

Greedy and lazy representations in negative base systems

We consider positional numeration systems with negative real base $-β$, where $β>1$, and study the extremal representations in these systems, called here the greedy and lazy representations. We give algorithms for determination of minimal and maximal $(-β)$-representation with respect to the alternate order. We also show that both extremal representations can be obtained as representations in the positive base $β^2$ and a non-integer alphabet. This enables us to characterize digit sequences admissible as greedy and lazy $(-β)$-representation. Such a characterization allows us to study the set of uniquely representable numbers. In case that $β$ is the golden ratio and the Tribonacci constant, we give the characterization of digit sequences admissible as greedy and lazy $(-β)$-representation using a set of forbidden strings.

math.DS↗
arXiv2013

Enumerating Abelian Returns to Prefixes of Sturmian Words

We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set $\mathcal{APR}_u$ of abelian returns of all prefixes of a Sturmian word $u$ in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set $\mathcal{APR}_u$ and we determine it for the characteristic Sturmian words.

cs.FL↗
arXiv2013

Integers in number systems with positive and negative quadratic Pisot base

We consider numeration systems with base $β$ and $-β$, for quadratic Pisot numbers $β$ and focus on comparing the combinatorial structure of the sets $\Z_β$ and $\Z_{-β}$ of numbers with integer expansion in base $β$, resp. $-β$. Our main result is the comparison of languages of infinite words $u_β$ and $u_{-β}$ coding the ordering of distances between consecutive $β$- and $(-β)$-integers. It turns out that for a class of roots $β$ of $x^2-mx-m$, the languages coincide, while for other quadratic Pisot numbers the language of $u_β$ can be identified only with the language of a morphic image of $u_{-β}$. We also study the group structure of $(-β)$-integers.

math.NT↗
arXiv2012

Optimal number representations in negative base

For a given base $γ$ and a digit set ${\mathcal B}$ we consider optimal representations of a number $x$, as defined by Dajani at al. in 2012. For a non-integer negative base $γ=-β<-1$ and the digit set ${\mathcal A}_β:={0,1,...,\lceilβ\rceil-1}$ we derive the transformation which generates the optimal representation, if it exists. We show that -- unlike the case of negative integer base -- almost no $x$ has an optimal representation. For a positive base $γ=β>1$ and the alphabet ${\mathcal A}_β$ we provide an alternative proof of statements obtained by Dajani et al.

math.NT↗
arXiv2012

Purely periodic expansions in systems with negative base

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base $β$, we find a sufficient condition so that there exist an interval $J$ containing the origin such that the $(-β)$-expansion of every rational number from $J$ is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property (${\rm Fin}(β)=\Z[β]$) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that ${\rm Fin}(-β)=\Z[β]$ is not necessary in the case of negative bases.

math.NT↗
arXiv2011

Proceedings 8th International Conference Words 2011

WORDS is the main international event in Combinatorics on Words. It is a biannual conference devoted to research of words (i.e., finite or infinite sequences of symbols over a finite alphabet) from combinatorial, algebraic and algorithmic points of view. The emphasis of the conference is on mathematical theory of words but the conference is also open to applications, mainly in computer science, biology, linguistics and physics, gaining from the fact that words arise as a natural object in many areas. The eighth edition of the conference was organized in Prague from 12th to 16th September 2011 as a joint undertaking of the Czech Technical University and the Charles University. This volume consists of contributed papers accepted for presentation at the conference and summaries of invited lectures.

cs.FL↗
arXiv2011

Number representation using generalized $(-β)$-transformation

We study non-standard number systems with negative base $-β$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-β}$ of the interval $\big[-\fracβ{β+1},\frac{1}{β+1}\big)$ into itself, we suggest a generalization using an interval $[l,l+1)$ with $l\in(-1,0]$. Such generalization may eliminate certain disadvantages of the Ito-Sadahiro system. We focus on the description of admissible digit strings and their periodicity.

cs.DM↗
arXiv2010

Ito-Sadahiro numbers vs. Parry numbers

We consider positional numeration system with negative base, as introduced by Ito and Sadahiro. In particular, we focus on algebraic properties of negative bases $-β$ for which the corresponding dynamical system is sofic, which happens, according to Ito and Sadahiro, if and only if the $(-β)$-expansion of $-\fracβ{β+1}$ is eventually periodic. We call such numbers $β$ Ito-Sadahiro numbers and we compare their properties with Parry numbers, occurring in the same context for Rényi positive base numeration system.

math.NT↗
arXiv2009

On the number of factors in codings of three interval exchange

We consider exchange of three intervals with permutation $(3,2,1)$. The aim of this paper is to count the cardinality of the set $3\iet(N)$ of all words of length $N$ which appear as factors in infinite words coding such transformations. We use the strong relation of 3iet words and words coding exchange of two intervals, i.e., Sturmian words. The known asymptotic formula $# 2\iet(N)/N^3\sim\frac1{π^2}$ for the number of Sturmian factors allows us to find bounds $\frac1{3π^2} + o(1) \leq # 3\iet(N)/N^4 \leq \frac2{π^2} + o(1)$.

math.CO↗
arXiv2008

Morphisms fixing words associated with exchange of three intervals

We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.

math.CO↗
arXiv2008

Relation between powers of factors and recurrence function characterizing Sturmian words

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.

math.CO↗
arXiv2007

Sturm numbers and substitution invariance of 3iet words

In this paper, we give a necessary condition for an infinite word defined by a non-degenerate interval exchange on three intervals (3iet word) to be invariant by a substitution: a natural parameter associated to this word must be a Sturm number. We deduce some algebraic consequences from this condition concerning the incidence matrix of the associated substitution. As a by-product of our proof, we give a combinatorial characterization of 3iet words.

math.CO↗
arXiv2006

Self-Matching Properties of Beatty Sequences

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function $\lfloor jβ\rfloor $ against $j$ for every quadratic unit $β\in(0,1)$. We show that translation in the argument by an element $G_i$ of generalized Fibonacci sequence causes almost always the translation of the value of function by $G_{i-1}$. More precisely, for fixed $i\in\N$, we have $\bigl\lfloor β(j+G_i)\bigr\rfloor = \lfloor βj\rfloor +G_{i-1}$, where $j\notin U_i$. We determine the set $U_i$ of mismatches and show that it has a low frequency, namely $β^i$.

math.CO↗
arXiv2006

Quasicrystals: algebraic, combinatorial and geometrical aspects

The paper presents mathematical models of quasicrystals with particular attention given to cut-and-project sets. We summarize the properties of higher-dimensional quasicrystal models and then focus on the one-dimensional ones. For the description of their properties we use the methods of combinatorics on words.

math-ph↗
arXiv2006

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) \leq ΔC(n) + 2, for all n \in N. For a large class of words it is a better estimate of the palindromic complexity in terms of the factor complexity then the one presented by Allouche et al. 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 πconnected with the transformation is given by π(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.

math.CO↗
arXiv2006

Palindromic complexity of infinite words associated with simple Parry numbers

A simple Parry number is a real number β>1 such that the Rényi expansion of 1 is finite, of the form d_β(1)=t_1...t_m. We study the palindromic structure of infinite aperiodic words u_βthat are the fixed point of a substitution associated with a simple Parry number β. It is shown that the word u_βcontains infinitely many palindromes if and only if t_1=t_2= ... =t_{m-1} \geq t_m. Numbers βsatisfying this condition are the so-called confluent Pisot numbers. If t_m=1 then u_βis an Arnoux-Rauzy word. We show that if βis a confluent Pisot number then P(n+1)+ P(n) = C(n+1) - C(n)+ 2, where P(n) is the number of palindromes and C(n) is the number of factors of length n in u_β. We then give a complete description of the set of palindromes, its structure and properties.

math.CO↗
arXiv1999

s-convexity, model sets and their relation

The relation of s-convexity and sets modeling physical quasicrystals is explained for quasicrystals related to quadratic unitary Pisot numbers. We show that 1-dimensional model sets may be characterized by s-convexity for finite set of parameters s. It is shown that the three Pisot numbers $\frac12(1+\sqrt5)$, $1+\sqrt2$, and $2+\sqrt3$ related to experimentally observed non-crystallographic symmetries are exceptional with respect to s-convexity.

math.RA↗