Related papers: Curious Properties of Iterative Sequences
We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which…
A two-dimensional arrangement of toothpicks is constructed by the following iterative procedure. At stage 1, place a single toothpick of length 1 on a square grid, aligned with the y-axis. At each subsequent stage, for every exposed…
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.
We consider iterations of integer-valued functions $\phi$, which have no fixed points in the domain of positive integers. We define a local function $\phi_n$, which is a sub-function of $\phi$ being restricted to the subdomain $\{0, ..., n…
In this paper we develop a classification of real functions based on growth rates of repeated iteration. We show how functions are naturally distinguishable when considering inverses of repeated iterations. For example, $n+2\to 2n\to 2^n\to…
We study the $k$-Bonacci word over the infinite alphabet $\mathbb{N}$. Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in…
Within the study of uncertain dynamical systems, iterated random functions are a key tool. There, one samples a family of functions according to a stationary distribution. Here, we introduce an extension, where one sample functions…
Let $f(n)$ denote the number of distinct unordered factorisations of the natural number $n$ into factors larger than 1.In this paper, we address some aspects of the function $f(n)$.
A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
We provide an overview of iterated function systems (IFS), where randomly chosen state-to-state maps are applied iteratively to a state. We aim to summarize the state of art and, where possible, identify fundamental challenges and…
The Kaprekar transformation is uniquely determined by parameters based on differences between symmetric values in the ordered numeric sequence. The parametric analysis of the iteration of the process requires developing Ki functions that…
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…
Sequences whose terms are equal to the number of functions with specified properties are considered. Properties are based on the notion of derangements in a more general sense. Several sequences which generalize the standard notion of…
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of…
We explore the possibility of using machine learning to identify interesting mathematical structures by using certain quantities that serve as fingerprints. In particular, we extract features from integer sequences using two empirical laws:…
The notion of transducer integer sequences is considered through a series of examples. By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite automata encoding tree morphisms…
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…
We investigate solutions to the functional equation $f(f(x)) = e^x$, which can be interpreted as the problem of finding a half iterate of the exponential map. While no elementary solution exists, we construct and analyze non-elementary…
Developing a system of parallel non-linear iterations, we establish the consistency of $\mathfrak{b}<\mathfrak{s}<\mathfrak{d}<\mathfrak{c}$ where $\mathfrak{b}, \mathfrak{d}, \mathfrak{c}$ are arbitrary subject to the known ZFC…