Related papers: Real numbers having ultimately periodic representa…
We define a class of quantum systems called regular quantum graphs. Although their dynamics is chaotic in the classical limit with positive topological entropy, the spectrum of regular quantum graphs is explicitly computable analytically…
We present here the notion of breadth-first signature and its relationship with numeration system theory. It is the serialisation into an infinite word of an ordered infinite tree of finite degree. We study which class of languages…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…
The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical…
We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study…
For any irrational $\alpha > 0$ and any initial value $z_{-1} \in \mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{\infty}$ as follows: $z_n$ is $z_{n-1} + e^{2 \pi i \alpha n}$ or $z_{n-1} - e^{2 \pi i \alpha n}$,…
We present the model theoretic concepts that allow mathematics to be developed with the notion of the potential infinite instead of the actual infinite. The potential infinite is understood as a dynamic notion, being an indefinitely…
The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our…
This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
A numeration system originally implies a digitization of real numbers, but in this paper it rather implies a compactification of real numbers as a result of the digitization. By definition, a numeration system with $G$, where $G$ is a…
Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several…
We extend the well-known Dumont--Thomas numeration systems to $\mathbb{Z}$ using an approach inspired by the two's complement numeration system. Integers in $\mathbb{Z}$ are canonically represented by a finite word (starting with…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
This work is concerned with regular languages defined over large alphabets, either infinite or just too large to be expressed enumeratively. We define a generic model where transitions are labeled by elements of a finite partition of the…
The present article is devoted to the investigation of some properties of the generalized shift operator of numbers represented in terms of numeral systems with a variable alphabet.
We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$, which we call $\boldsymbol{\beta}$-representations. In doing so, we generalize both representations…
We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily…
We prove the existence of an effective universal upper bound for the order of any integral periodic orbit of any integral algebraic dynamical system in a fixed ambient space. Using this, we demonstrate the decidability of periodicity in…