Related papers: Expansions in multiple bases
We construct explicit exponential bases on finite unions of disjoint rectangles of $\mathbb{R}^d$ with rational vertices.
Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
In this paper we extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups…
Given two real numbers $q_0,q_1$ with $q_0, q_1 > 1$ satisfying $q_0+q_1 \ge q_0q_1$, we call a sequence $(d_i)$ with $d_i\in \{0,1\}$ a $(q_0,q_1)$-expansion or a double-base expansion of a real number $x$ if \[…
It is shown how Dedekind cuts can be used to introduce the extended real numbers along with sound arithmetic laws via one simple rule for the addition of sets. The crucial idea is that the use of the lower and the upper part of the cuts,…
This paper draws on diverse areas of computer science to develop a unified view of computation: (1) Optimization in operations research, where a numerical objective function is maximized under constraints, is generalized from the numerical…
We develop a theory of real numbers as rational Cauchy sequences, in which any two of them, $(a_n)$ and $(b_n)$, are equal iff $\lim\,(a_n-b_n)=0$. We need such reals in the Countable Mathematical Analysis ([4]) which allows to use only…
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…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
The aim of this article is to present a topological tool for the study of additive basis in additive number theory. It will be proposal a metric for the set of all additive basis, in which it will be possible to study properties of some…
In this article, we define the notion of slim (normal) bases and show their existence for various fields. As an application, an algorithm will be given that computes the spectrum of a basefield transform by merely using O(n) additions.
Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a…
We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the R\'enyi system with one real base. We focus on the so-called…
We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number $r\geq 3$, we say a set $C$ is a generalized Cantor set in base $r$ if there is a non-empty…
From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different…
We define a notion which contains numerous basic notions of Analysis as special cases, for example limit, continuity, differential, Riemann and Lebesgue integral, root and exponential functions. Properties like additivity or linearity of…
We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over…
The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.
The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic…