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It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto not available. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. The analysis of this tree will provide new insights into the structure of Collatz sequences. The findings are of special interest to possible cycles within a sequence. Next, we describe the conditions which must be fulfilled by a cycle. Finally, we demonstrate how these conditions could be used to prove that the only possible cycle within a Collatz sequence is the trivial cycle, starting with the number one, as conjectured by Lothar Collatz.
It is well known that the inverted Collatz sequence can be represented as a graph or a tree. Similarly, it is acknowledged that in order to prove the Collatz conjecture, one must demonstrate that this tree covers all odd natural numbers. A structured reachability analysis is hitherto unavailable. This paper investigates the problem from a graph theory perspective. We define a tree that consists of nodes labeled with Collatz sequence numbers. This tree will be transformed into a sub-tree that only contains odd labeled nodes. Furthermore, we derive and prove several formulas that can be used to traverse the graph. The analysis covers the Collatz problem both in it’s original form 3x + 1 as well as in the generalized variant kx + 1. Finally, we transform the Collatz graph into a binary tree, following the approach of Kleinnijenhuis, which could form the basis for a comprehensive proof of the conjecture.
Künstliche Intelligenz ist in aller Munde. Immer mehr Anwendungsbereiche werden durch die Auswertung von vorliegenden Daten mit Algorithmen und Frameworks z.B. des Maschinellen Lernens erschlossen. Dieses Buch hat das Ziel, einen Überblick über gegenwärtig vorhandene Lösungen zu geben und darüber hinaus konkrete Hilfestellung bei der Auswahl von Algorithmen oder Tools bei spezifischen Problemstellungen zu bieten. Um diesem Anspruch gerecht zu werden, wurden 90 Lösungen mittels einer systematischen Literaturrecherche und Praxissuche identifiziert sowie anschließend klassifiziert. Mit Hilfe dieses Buches gelingt es, schnell die notwendigen Grundlagen zu verstehen, gängige Anwendungsgebiete zu identifizieren und den Prozess zur Auswahl eines passenden ML-Tools für das eigene Projekt systematisch zu meistern.
ERP und MES : Teil 3
(2008)