arXiv subjects

Edita Pelantová

Publications and source records attributed to Edita Pelantová.

At least 73 records · Page 4Linked to original sources

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

Languages invariant under more symmetries: overlapping factors versus palindromic richness

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.

math.CO↗

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

Parallel addition in non-standard numeration systems

We consider numeration systems where digits are integers and the base is an algebraic number $β$ such that $|β|>1$ and $β$ satisfies a polynomial where one coefficient is dominant in a certain sense. For this class of bases $β$, we can find an alphabet of signed-digits on which addition is realizable by a parallel algorithm in constant time. This algorithm is a kind of generalization of the one of Avizienis. We also discuss the question of cardinality of the used alphabet, and we are able to modify our algorithm in order to work with a smaller alphabet. We then prove that $β$ satisfies this dominance condition if and only if it has no conjugate of modulus 1. When the base $β$ is the Golden Mean, we further refine the construction to obtain a parallel algorithm on the alphabet $\{-1,0,1\}$. This alphabet cannot be reduced any more.

math.NT↗

arXiv2011

Infinite words rich and almost rich in generalized palindromes

We focus on $Θ$-rich and almost $Θ$-rich words over a finite alphabet $\mathcal{A}$, where $Θ$ is an involutive antimorphism over $\mathcal{A}^*$. We show that any recurrent almost $Θ$-rich word $\uu$ is an image of a recurrent $Θ'$-rich word under a suitable morphism, where $Θ'$ is again an involutive antimorphism. Moreover, if the word $\uu$ is uniformly recurrent, we show that $Θ'$ can be set to the reversal mapping. We also treat one special case of almost $Θ$-rich words. We show that every $Θ$-standard words with seed is an image of an Arnoux-Rauzy word.

math.CO↗

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↗

arXiv2011

Infinite Words with Finite Defect

In this paper, we provide a new characterization of uniformly recurrent words with finite defect based on a relation between the palindromic and factor complexity. Furthermore, we introduce a class of morphisms P_ret closed under composition and we show that a uniformly recurrent word with finite defect is an image of a rich (also called full) word under a morphism of class P_ret. This class is closely related to the well-known class P defined by Hof, Knill, and Simon; every morphism from P_ret is conjugate to a morphism of class P.

math.CO↗

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↗

arXiv2009

On representations of Lie algebras compatible with a grading

The paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is defined and applied to finite-dimensional representations of $sl(n,\mathbb{C})$ in relation with its $\mathbb{Z}_2$-gradings. For representation theory of $sl(n,\mathbb{C})$ the Gel'fand-Tseitlin method turned out very effective.

math-ph↗

arXiv2008

Factor complexity of infinite words associated with non-simple Parry numbers

The factor complexity of the infinite word $\ubeta$ canonically associated to a non-simple Parry number $β$ is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words $\ubeta$ only. To complete the description of its special factors, we define and study $(a,b)$-maximal left special factors. This enables us to characterize non-simple Parry numbers $β$ for which the word $\ubeta$ has affine complexity.

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↗

arXiv2007

Fine group gradings of the real forms of $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$

We present an explicit description of the 'fine group gradings' (i.e. group gradings which cannot be further refined) of the real forms of the semisimple Lie algebras $sl(4,\C)$, $sp(4,\C)$, and $o(4,\C)$. All together 12 real Lie algebras are considered, and the total of 44 of their fine group gradings are listed. The inclusions $sl(4,\C)\supset sp(4,\C)\supset o(4,\C)$ are an important tool in our presentation. Systematic use is made of the faithful representations of the three Lie algebras by $4\times 4$ matrices.

math-ph↗

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

Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers

We study arithmetical and combinatorial properties of $β$-integers for $β$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $β$-fractional positions, which may arise as a result of addition of two $β$-integers. For the infinite word $u_β$ coding distances between consecutive $β$-integers, we determine precisely also the balance. The word $u_β$ is the fixed point of the morphism $A \to A^{m-1}B$ and $B\to A^{m-n-1}B$. In the case $n=1$ the corresponding infinite word $u_β$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $β$-integers are related.

cs.DM↗

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.

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