Related papers: Maximum distances in the four-digit Kaprekar proce…

A base-b junction number u has the property that there are at least two ways to write it as u = v + s(v), where s(v) is the sum of the digits in the expansion of the number v in base b. For the base-10 case, Kaprekar in the 1950's and…

In this study, several interesting iterative sequences were investigated. First, we define the iterative sequences. We fix function f(n). An iterative sequence starts with a natural number n, and calculates the sequence f(n),f(f(n)),…

For every non-negative integer $a$ and positive integer $b$, the congruence speed of the tetration $^{b}a$ is the difference between the number of the rightmost digits of $^{b}a$ that are the same as those of $^{b+1}a$ and the number of the…

In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove…

An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. For $b\geq 2$ and $k \in \mathbb{Z}^+$, a $k$-desert base $b$ is a set of $k$ consecutive non-negative…

Given a continuous function $f:[a,b]\to\mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $|x-y|$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide…

Let $k \geq 2$ and $b \geq 3$ be integers, and suppose that $d_1, d_2 \in \{0,1,\dots, b - 1\}$ are distinct and coprime. Let $\mathcal{S}$ be the set of non-negative integers, all of whose digits in base $b$ are either $d_1$ or $d_2$. Then…

The critical exponent of an infinite word $\bf x$ is the supremum, over all finite nonempty factors $f$, of the exponent of $f$. In this note we show that for all integers $k\geq 2,$ there is a binary infinite $k$-automatic sequence with…

Two sums over the inverse of the product of an integer n and its greatest prime factor G(n), are computed to first 13 decimal digits. These sums converge, but converge very slowly. They are transformed into sums involving Mertens' prime…

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent…

The main aim of this article is to prove that for any continuous function $f \colon X \to X$, where $X$ is metrizable (or, more generally, for any family $\mathcal{F}$ of such functions, satisfying an additional condition), there exists a…

Recently, a new distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this distance…

This paper focuses on regularisation methods using models up to the third order to search for up to second-order critical points of a finite-sum minimisation problem. The variant presented belongs to the framework of [3]: it employs random…

We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations where n is the number of polynomials (as well as the dimension of the ambient space), D…

Given an integer base $b>1$, a set of integers is represented in base $b$ by a language over $\{0,1,...,b-1\}$. The set is said to be $b$-recognisable if its representation is a regular language. It is known that eventually periodic sets…

We consider the Keplerian distance $d$ in the case of two elliptic orbits, i.e. the distance between one point on the first ellipse and one point on the second one, assuming they have a common focus. The absolute minimum $d_{\rm min}$ of…

We devise an analytically simple as well as invertible approximate expression, which describes the relation between the minimum distance of a binary code and the corresponding maximum attainable code-rate. For example, for a rate-(1/4),…

Let $X$ be a projective nested product of fields and let $\delta_X(d)$ be the minimum distance in degree $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(\delta_X)$ of the minimum distance function…

The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for…

In the present paper we provide a formula that allows to compute the number of stable digits of any integer tetration base $a\in\mathbb{N}_0$. The number of stable digits, at the given height of the power tower, indicates how many of the…