Related papers: A Family of Iterated Maps on Natural Numbers
We introduce the novel concept of hypercomplex iterated function system (IFS) on the complete metric space $(\mathbb{A}_{n+1}^k,d)$ and define its hypercomplex attractor. Systems of hypercomplex function systems arising from hypercomplex…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset $X\subset\mathbb{R}^m$ is a sequence $\Phi=(\{\phi^{(j)}_{i}\}_{i\in I^{(j)}})_{j=1}^{\infty}$ of collections of uniformly contracting maps…
We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such…
We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability" : at each…
This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained…
We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard…
We investigate properties of families $F$ of subsets of a finite set in a situation where subsets are incomparable by the binary inclusion relation and a) for any $A\notin F$, there is such set $A'\in F$ that either $A\subset A'$ or…
We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a…
This paper investigates the \textbf{graphical $r$-Stirling numbers of the first kind}, denoted by $\str{G}{k}$, which enumerate partitions of a vertex set $V(G)$ into $k$ disjoint cycles such that $r$ specified vertices occupy distinct…
In this paper we present a systematic study of continuous local iterated function systems. We prove local iterated function systems admit compact attractors and, under a contractivity assumption, construct their code space and present an…
Let $K$ be a finitely generated field of characteristic zero. We study, for fixed $m \geq 2$, the rational functions $\phi$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$th powers. For $m \geq 5$ we show the…
In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other…
We construct a converging geometric iterated function system on the moduli space of ordered triangles, for which the involved functions have geometric meanings and contain a non-contraction map under the natural metric.
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
We show, that the Complex Step approximation to the Fr\'echet derivative of real matrix functions is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the…
Questions concerning small fractional parts of polynomials and pseudo-polynomials have a long history in analytic number theory. In this paper, we improve on earlier work by Madritsch and Tichy. In particular, let $f=P+\phi$ where $P$ is a…
Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose…
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…
We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family includes smooth numbers, almost primes and practical numbers.