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Top Papers in Collatz iteration

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Almost all orbits of the Collatz map attain almost bounded values

Define the \emph{Collatz map} $\mathrm{Col} : \mathbb{N}+1 \to \mathbb{N}+1$
on the positive integers $\mathbb{N}+1 = \{1,2,3,\dots\}$ by setting
$\mathrm{Col}(N)$ equal to $3N+1$ when $N$ is odd and

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Collatz meets Fibonacci

The Collatz map is defined for a positive even integer as half that integer,
and for a positive odd integer as that integer threefold, plus one. The Collatz
conjecture states that when the map is iter

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An Engineering and Statistical Look at the Collatz (3n + 1) Conjecture

The famous (3n + 1) or Collatz conjecture has admitted some progress over the
last several decades towards the conclusion that the conjecture is true (i.e.
that all Collatz sequences will eventually r

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Binary expression of ancestors in the Collatz graph

The Collatz graph is a directed graph with natural number nodes and where
there is an edge from node $x$ to node $T(x)=T_0(x)=x/2$ if $x$ is even, or to
node $T(x)=T_1(x)=\frac{3x+1}{2}$ if $x$ is odd

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On the convergence of infinite Collatz sequences

Collatz convergence is a Hydra game

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Termination of string rewriting

An Automated Approach to the Collatz Conjecture

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On Cycles of Generalized Collatz Sequences

We explore the cycles and convergence of Generalized Collatz Sequence, where
$3n+1$ in original collatz function is replaced with $3n+k$. We present a
generating function for cycles of GCS and show a

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Proof of Collatz Theorem

In this article, we will show that Collatz is theorem and we proof it by
method that we made in section 2 and 3. In section 1, first we introduction
Collatz problem and idea of mathematician about thi

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The Collatz Conjecture

From a conjecture of Collatz to Thompson's group F, via a conjunction of Girard

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Pruning the binary tree, proving the Collatz conjecture

The yet unproven Collatz conjecture maintains that repeatedly connecting even
numbers n to n/2, and odd n to 3n+1, connects all natural numbers to a single
tree with 1 as its root. Pruning ever more n

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The Collatz Problem generalized to 3x+k

The Collatz problem with $3x+k$ is revisited. Positive and negative limit
cycles are given up to k=9997 starting with $x_0=-2\cdot10^7...+2\cdot10^7$. A
simple relation between the probability distrib

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Small tile sets that compute while solving mazes

We ask the question of how small a self-assembling set of tiles can be yet
have interesting computational behaviour. We study this question in a model
where supporting walls are provided as an input s

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Complete Proof of the Collatz Conjecture

The \textit{Collatz's conjecture} is an unsolved problem in mathematics. It
is named after Lothar Collatz in 1973. The conjecture also known as Syrucuse
conjecture or problem.
Take any positive intege

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Actor-Critic or Critic-Actor? A Tale of Two Time Scales

We revisit the standard formulation of tabular actor-critic algorithm as a
two time-scale stochastic approximation with value function computed on a
faster time-scale and policy computed on a slower t

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In-Context Policy Iteration

In-Context Policy Iteration

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Distributed Picard Iteration

The Picard iteration is widely used to find fixed points of locally
contractive (LC) maps. This paper extends the Picard iteration to distributed
settings; specifically, we assume the map of which the

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The 3x+1 Problem: An Overview

This paper is an overview and survey of work on the 3x+1 problem, also called
the Collatz problem, and generalizations of it. It gives a history of the
problem. It addresses two questions: (1) What ca

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Collatz map as a power bounded nonsingular transformation

Let $T$ be the map defined on $\N$ by $T(n) = \frac{n}{2} $ if $n$ is even
and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system
$(\N, 2^{\N}, \nu, T)$ where $\nu$ is a finite me

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Problems in number theory from busy beaver competition

By introducing the busy beaver competition of Turing machines, in 1962, Rado
defined noncomputable functions on positive integers. The study of these
functions and variants leads to many mathematical

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