Related papers: Defining arithmetical operations on infinite decim…

We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…

A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number…

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By "accurate" we mean that the computed…

A simple mathematical expression for the universal map for cellular automata is found in closed form with the help of a digit function, whose most basic properties are established. This result is found after proving a theorem on the…

Rota-Baxter operators on algebras, which appeared in 1960, have connections with different versions of the Yang-Baxter equation, pre- and postalgebras, double Poisson algebras, etc. In 2020, the notion of Rota-Baxter operator on a group was…

Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical…

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a sum, product and quotient of some series of positive rational…

There has been always an ambiguity in division when zero is present in the denominator. So far this ambiguity has been neglected by assuming that division by zero as a non-allowed operation. In this paper, I have derived the new set of…

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions.…

We determine all pairs of positive integers $(a,b)$ such that $a+b$ and $a \times b$ have the same decimal digits in reverse order: \[ (2,2), (9,9), (3,24), (2,47), (2,497), (2,4997), (2,49997), \ldots \] We use deterministic finite…

Differential operators usually result in derivatives expressed as a ratio of differentials. For all but the simplest derivatives, these ratios are typically not algebraically manipulable, but must be held together as a unit in order to…

We start by presenting a theory of finite sets using the approach which is essentially that taken by Whitehead and Russell in Principia Mathematica}, and which does not involve the natural numbers (or any other infinite set). This theory is…

Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, 'division by zero' is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in…

The present work includes some of the author's original researches on integer solutions of Diophantine liner equations and systems. The notion of "general integer solution" of a Diophantine linear equation with two unknowns is extended to…

We analyze the convergence order of an algorithm producing the digits of an absolutely normal number. Furthermore, we introduce a stronger concept of absolute normality by allowing Pisot numbers as bases, which leads to expansions with…

We present methods that provide all zeroes and extrema of a function that do not require differentiation. Using point process theory, we are able to describe the locations of zeroes or maxima, their number, as well as their distribution…

All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained through a systematic scheme of proof and not trial and error. The proof is also…

We introduce the continued logarithm representation of real numbers and prove results on the occurrence and frequency of digits with respect to this representation

Unlike some other formal systems, the proof system Metamath has no built-in concept of "decimal number" in the sense that arbitrary digit strings are not recognized by the system without prior definition. We present a system of theorems and…

The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…