Related papers: Computable Absolutely Pisot Normal Numbers
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…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which…
In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression.…
A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not…
Champernowne famously proved that the number $0.(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)...$ formed by concatenating all the integers one after another is normal base 10. We give a generalization of Champernowne's construction to various…
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.
Let $b \geq 3$ be a positive integer. A natural number is said to be a base-$b$ Zuckerman number if it is divisible by the product of its base-$b$ digits. Let $\mathcal{Z}_b(x)$ be the set of base-$b$ Zuckerman numbers that do not exceed…
Let s be an integer greater than or equal to 2. A real number is simply normal to base s if in its base-s expansion every digit 0, 1, ..., s-1 occurs with the same frequency 1/s. Let X be the set of positive integers that are not perfect…
Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this…
Inspired by computer assisted proofs in analysis, we present an interval approach to real-number computations.
Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with…
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…
A. Renyi \cite{Renyi} made a definition that gives one generalization of simple normality in the context of $Q$-Cantor series. Similarly, in this paper we give a definition which generalizes the notion of normality in the context of…
When describing images, humans tend not to talk about the obvious, but rather mention what they find interesting. We argue that abnormalities and deviations from typicalities are among the most important components that form what is worth…
Evaluating performance across optimization algorithms on many problems presents a complex challenge due to the diversity of numerical scales involved. Traditional data processing methods, such as hypothesis testing and Bayesian inference,…
This paper presents two algorithms on certain computations about Pisot numbers. Firstly, we develop an algorithm that finds a Pisot number $\alpha$ such that $\Q[\alpha] = \F$ given a real Galois extension $\F$ of $\Q$ by its integral…
In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.
We create a simple test for distinguishing between sets of primes and random numbers using just the sum-of-digits function. We find that the sum-of-the-digits of prime numbers does not have an equal probability of being odd or even. The…
In a previous paper, an implementable algorithm was introduced to compute discrete solutions of sweeping processes (i.e. specific first order differential inclusions). The convergence of this numerical scheme was proved thanks to…