Related papers: Modeling rational numbers by Cantor series
We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen…
The paper deals with a construction of a separating system of rational invariants for finite dimensional generic algebras. In the process of dealing an approach to a rough classification of finite dimensional algebras is offered by…
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…
A quantum theoretic representation of real and complex numbers is described here as equivalence classes of Cauchy sequences of quantum states of finite strings of qubits. There are 4 types of qubits each with associated single qubit…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
We give another proof of a result of Adamczewski and Bell concerning Mahler equations: A formal power series satisfying a $p-$ and a $q-$Mahler equation over ${\mathbb C}(x)$ with multiplicatively independent positive integers $p$ and $q$…
While recent work has begun to uncover the internal strategies that Large Language Models (LLMs) employ for simple arithmetic tasks, a unified understanding of their underlying mechanisms is still lacking. We extend recent findings showing…
We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in…
We introduce two modal natural deduction systems that are suitable to represent and reason about transformations of quantum registers in an abstract, qualitative, way. Quantum registers represent quantum systems, and can be viewed as the…
Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~…
We present a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in $\mathbb{Q}$ (where the Cauchy condition is defined…
In this paper, we consider a problem of counting rational points near self-similar sets. Let $n\geq 1$ be an integer. We shall show that for some self-similar measures on $\mathbb{R}^n$, the set of rational points $\mathbb{Q}^n$ is…
We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams:…
We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a…
It is shown that there is an absolute constant $C$ such that any rational $\frac bq\in]0, 1[, (b, q)=1$, admits a representation as a finite sum $\frac bq=\sum_\alpha\frac {b_\alpha}{q_\alpha}$ where $\sum_\alpha\sum_ia_i(\frac…
This article is written with the hope to draw attention to a method that uses integral transforms to find exact values for a large class of convergent series (and, in particular, series of rational terms). We apply the method to some series…
Criteria are given for determining whether an irreducible sextic equation with rational coefficients is algebraically solvable over the complex numbers.
A new definition of a real number is that it is a rule which says Yes or No based on whether the real number ought to be in a given rational interval. This is a teaser paper for formalizing, exploring, and generalizing this definition. The…