Some Borel measures associated with the generalized Collatz mapping
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In this article, the authors extended T to a mapping of polyadic numbers and constructed finitely many ergodic Borel measures on the polyadic number Ẑ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.Abstract:
1. Abstract. This paper is a continuation of a recent paper [2], in which the authors studied some Markov matrices arising from a mapping T : Z → Z, which generalizes the famous 3x + 1 mapping of Collatz. We extended T to a mapping of the polyadic numbers Ẑ and construct finitely many ergodic Borel measures on Ẑ which heuristically explain the limiting frequencies in congruence classes, observed for integer trajectories.read more
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The 3x+1 problem: An annotated bibliography
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The 3x+1 Problem: An Annotated Bibliography, II (2000-2009)
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The 3x+1 problem: An annotated bibliography (1963--1999)
TL;DR: The 3x+1 Conjecture as mentioned in this paper states that for every positive integer n > 1, the forward orbit of n under iteration by T includes the integer 1. At present, the problem remains unsolved.
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The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)
TL;DR: The 3x+1 Conjecture as discussed by the authors states that for every positive integer n > 1, the forward orbit of n under iteration by T includes the integer 1. At present, the problem remains unsolved.
Journal ArticleDOI
Functional Equations Connected With The Collatz Problem
Lothar Berg,Günter Meinardus +1 more
TL;DR: In this paper, some aspects of the famous 3n + 1 problem, due to L. Coilatz, are studied, and two equivalent analytic versions, concerning the rational character of some generating functions, and the set of solutions of a linear functional equation, respectively.
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Probability measures on metric spaces
TL;DR: The Borel subsets of a metric space Probability measures in the metric space and probability measures in a metric group Probability measure in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability probabilistic probability measures on $C[0, 1]$ and $D[0-1]$ Bibliographical notes Bibliography List of symbols Author index Subject index as mentioned in this paper