Related papers: Numeration systems without a dominant root and reg…
Let $\mathcal{O}$ be an order, that is a commutative ring with $1$ whose additive structure is a free $\mathbb{Z}$-module of finite rank. A generalized number system (GNS for short) over $\mathcal{O}$ is a pair $(p,\mathcal{D} )$ where…
To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well…
Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
We show how to represent an interval of real numbers in an abstract numeration system built on a language that is not necessarily regular. As an application, we consider representations of real numbers using the Dyck language. We also show…
The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration…
The family of lattice simplices in $\mathbb{R}^n$ formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative…
Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which…
A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor,…
In this paper will be introduced large, probably complete family of complex base systems, which are 'proper' - for each point of the space there is a representation which is unique for all but some zero measure set. The condition defining…
Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated…
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
Number systems with a rational number $a/b > 1$ as base have gained interest in recent years. In particular, relations to Mahler's 3/2-problem as well as the Josephus problem have been established. In the present paper we show that the…
Alignments, i.e., position-wise comparisons of two or more strings or ordered lists are of utmost practical importance in computational biology and a host of other fields, including historical linguistics and emerging areas of research in…
When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared…
Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in R^d, consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the…
Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over $\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the language of…
A new construction of the real number system, that is built directly upon the additive group of integers and has its roots in the definition due to Henri Poincar\'e of the rotation number of an orientation preserving homeomorphism of the…
Ordering the collection of states of a given automaton starting from an order of the underlying alphabet is a natural move towards a computational treatment of the language accepted by the automaton. Along this path, Wheeler \emph{graphs}…
In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition…
In this paper we discourse basises of representable algebras. This question lead to arithmetic problems. We prove algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive…