Related papers: Generalized Collatz Maps with Almost Bounded Orbit…

We study the stopping time of the Collatz map for a polynomial $f \in \mathbb{F}_2[x]$, and bound it by $O({\rm deg} (f)^{1.5})$, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence…

The present paper is devoted to the study of classes of mappings with non--bounded characteristics of quasiconformality. It is proved that the normal families of mappings distorting the families of mappings in ${\Bbb R}^n$ by special way,…

Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to…

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 document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

We prove several results concerning the theory of Toeplitz algebras over $p$-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm…

We consider a sequence of complex rational maps (f_n) of a fixed degree d at least 2. Building on the seminal work of Kiwi, we introduce the notion of generalized rescaling limits. These are rational maps possibly defined over a…

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian…

In the present paper, it was studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions…

Let $f:X\times Y\times Z\longrightarrow W $ be a bounded tri-linear map on normed spaces. We say that $f$ is close-to-regular when $f^{t****s}=f^{s****t}$ and we say that $f$ is completely regular when all natural extensions are equal. In…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

Given a finite set T of maps on a finite ring R, we look at the finite simple graph G=(V,E) with vertex set V=R and edge set E={(a,b) | exists t in T, b=t(a), b not equal to a}. An example is when R=Z_n and T consists of a finite set of…

Consider $q_n$ a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with $n$ faces. In this paper we show that, when $n$ goes to $+\infty$, $q_n$ suitably normalized converges weakly in a certain sense…

It is well known that the following Collatz Conjecture is one of the unsolved problems in mathematics. Collatz Conjecture: For any positive integer $n>1$, the following recursive algorithm will convergent to 1 by a finite number of steps.…

The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we…

Let $F : \mathrm{End}_{\mathbb{F_p}}(\mathbb{G}_{a/K}^d)$ be an additive polynomial mapping over a global function field $K/\mathbb{F}_q$, and let $P \in \mathbb{G}_a^d(K)$. Following Silverman, consider $\delta := \lim_{n \in \mathbb{N}}…

The traditional approach to analyzing mean motion resonances is through canonical perturbation theory. While this is a powerful method, its generality leads to complicated combinations of variables that are challenging to interpret and…

In this work the generalized Collatz problem $qn+1$ ($q$ odd) is studied. As a natural generalization of the original $3n+1$ problem, it consists of a discrete dynamical system of an arithmetical kind. Using standard methods of number…

Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$…

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…