Related papers: Representing Real Numbers in a Generalized Numerat…
Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to…
I propose a class of non-positional numeral systems where numbers are represented by Dyck words, with the systems arising from a recursive extension of prime factorization. After describing two proper subsets of the Dyck language capable of…
In this article, for modelling numeral systems, the operator approach, which is introduced in [25], is generalized for a certain case. An example of such numeral systems is introduced and considered.
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
It is known that any rational abstract numeration system is faithfully, and effectively, represented by an N-rational series. A simple proof of this result is given which yields a representation of this series which in turn allows a simple…
I present the most fundamental features of an implemented system designed to manipulate representations of regular languages. The system is structured into two layers, allowing regular languages to be represented in an increasingly compact,…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…
A real number is a rule that, when provided with a rational interval, answers Yes or No depending on if the real number ought to be considered to be in the given interval. Since the goal is to define the real numbers, this can only motivate…
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…
The article is devoted to introduce and investigation of new numeral systems. These representations are generalizations of classical representations of real numbers. The main properties of investigating representations are described. The…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.
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 describe the design of a quantifier elimination framework for the complex numbers in the language of ordered rings supplemented with symbols for the imaginary unit, real parts, imaginary parts, and conjugates. Technically, we use a…
In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number…
We provide a setting-independent definition of reals by introducing the notion of a streak. We show that various standard constructions of reals satisfy our definition. We study the structure of reals by noting that its pieces correspond to…
Researchers are often perplexed when their machine learning algorithms are required to deal with complex number. Various strategies are commonly employed to project complex number into real number, although it is frequently sacrificing the…
System identification is an important area of science, which aims to describe the characteristics of the system, representing them by mathematical models. Since many of these models can be seen as recursive functions, it is extremely…
In this work, we establish a direct connection between supplemented Dyck language and the signed expectation value of chains of second quantization operators relatively to the physical vacuum and relatively to a one-determinant state.…
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…