Related papers: An iteration based on prime and composite factors
Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example,…
The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…
In this paper, we study a class of functions defined recursively on the set of natural numbers in terms of the greatest common divisor algorithm of two numbers and requiring a minimality condition. These functions are permutations, products…
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed…
Incremental computation aims to compute more efficiently on changed input by reusing previously computed results. We give a high-level overview of works on incremental computation, and highlight the essence underlying all of them, which we…
I develop a function that, for any integer $n \geq 2$, takes a value of 1 if $n$ is prime, 0 if $n$ is composite. I also discuss two applications: First, the characteristic function provides a new expression for the prime counting function.…
We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes,…
We present a simple iteration for the Lebesgue identity on partitions, which leads to a refinement involving the alternating sums of partitions.
Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other are prime number indexed basis entities taken…
In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.
We show one possible dynamical approach to the study of the distribution of prime numbers. Our approach is based on two complexity methods, the Computable Information Content and the Entropy Information Gain, looking for analogies between…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…
Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…
We examine the notion of anticonfinement and the role it has to play in the singularity analysis of discrete systems. A singularity is said to be anticonfined if singular values continue to arise indefinitely for the forward and backward…
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…
The exact computation of orbits of discrete dynamical systems on the interval is considered. Therefore, a multiple-precision floating point approach based on error analysis is chosen and a general algorithm is presented. The correctness 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…