Related papers: Functional Equations Associated to Collatz-Type Ma…
We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they…
The Collatz conjecture (also known as the $3x+1$ problem) concerns the behavior of the discrete dynamical system on the positive integers defined by iteration of the so-called $3x + 1$ function. We investigate analogous dynamical systems in…
The Collatz map (or the $3n{+}1$-map) $f$ is defined on positive integers by setting $f(n)$ equal to $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The Collatz conjecture states that starting from any positive integer $n$, some iterate…
We study an analogue of the Collatz map in the polynomial ring $R[x]$, where $R$ is an arbitrary commutative ring. We prove that if $R$ is of positive characteristic, then every polynomial in $R[x]$ is eventually periodic with respect to…
Colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on Schwartz distributions. Many variants and modifications of these algebras exist for various applications. We present a…
This paper details a generalization of the formalism presented in the author's 2024 paper, "The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory",…
The present work focuses on the study of the renowned Collatz conjecture, also known as the $3x +1$ problem. The distinguished analysis approach lies on the dynamics of an iterative map in binary form. A new estimation of the enlargement of…
We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under the topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key…
We investigate the problems related to the Collatz map $T$ from the point of view of functional analysis. We associate with $T$ certain linear operator $\mathcal{T}$ and show that cycles and (hypothetical) diverging trajectory (generated by…
In this paper a new conjecture equivalent to Collatz conjecture is presented. In particural, showing that (all) the solution(s) of newly introduced iterative functional equation(s) have a given property is equivalent to prove Collatz…
Let A be the integral closure of the ring of polynomials CC[t], within the field of algebraic functions in one variable. We show that A interprets the ring of integers. This contrasts with the analogue for finite fields, proved to have a…
Collatz Conjecture is one of the most famous, for its simple form, proposed more than eighty years ago. This paper presents a full attempt to prove the affirmative answer to the question proposed by the conjecture. In the first section, we…
In this article, we define a very important sequence of functions, all the functions of this sequence present behaviors very close to that of the Collatz function. The study of such functions allows us to obtain very interesting results…
The axiomatic formulation of quantum field theory (QFT) of the 1950's in terms of fields defined as operator valued Schwartz distributions is re-examined in the light of subsequent developments. These include, on the physical side, the…
We address the conjectures left by the recent article by Ferreira et al. titled ``Commuting maps and identities with inverses on alternative division rings.'' We also present an example showing the necessity of the conditions of the results…
We define generalized Collatz mappings on free abelian groups of finite rank and study their iteration trajectories. Using geometric arguments we describe cones of points having a divergent trajectory and we deduce lower bounds for the…
We introduce an explicit logarithmic transformation $T(x) = \{\log_6(x + 1/5)\}$ under which the Collatz map becomes a rigid circle rotation by the irrational angle \(\alpha = \log_6 3\), perturbed by a uniformly bounded error term. We…
We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case…
The works of R. Descartes, I. M. Gelfand and A. Grothendieck have convinced us that commutative rings should be thought of as rings of functions on some appropriate (commutative) spaces. If we try to push this notion forward we reach the…
We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be identified with pointwise maps from A into the ring of…