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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 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.
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 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 1, as conjectured by Lothar Collatz.
The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequence’s "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.
Auf dem Gelände der Landesgartenschau 2018 in Würzburg untersuchte unsere Forschungsgruppe das Anpassungsverhalten der BesucherInnen an Hitze. Ziel war es herauszufinden, wie BesucherInnen von Großveranstaltungen Hitzetage erleben und wie sie sich während unterschiedlicher Wetterbedingungen verhalten. Auf Grundlage der Ergebnisse sollen Empfehlungen zur Förderung individuellen Anpassungsverhaltens bei Hitzebelastung an Veranstalter ausgesprochen werden. An sechs aufeinanderfolgenden Wochenenden im Juli und August führten wir Temperaturmessungen, Verhaltensbeobachtungen und Befragungen unter den BesucherInnen durch. Die Wetterlage an den zwölf Erhebungstagen fiel unterschiedlich aus: Es gab sechs Hitzetage mit Temperaturen über 30 °C, vier warme Sommertage und zwei kühle Regentage.
Es ließen sich unterschiedliche Anpassungsmaßnahmen bei den 2741 beobachteten BesucherInnen identifizieren. Hierzu gehören das Tragen von leichter oder kurzer Kleidung und von Kopfbedeckungen, das Mitführen von Getränken oder Schirmen sowie das Aufhalten im Schatten oder Abkühlen in einer Wasserfläche. Dabei fanden sich Unterschiede zwischen den verschiedenen Altersgruppen: Jüngere und Ältere hatten unterschiedliche Präferenzen für einzelne Anpassungsmaßnahmen. So suchten BesucherInnen über 60 Jahren bevorzugt Sitzplätze im Schatten auf, wohingegen sich Kinder zum Abkühlen in Wasserflächen aufhielten.
Die Befragung von 306 BesucherInnen ergab, dass Hitzetage als stärker belastend wahrgenommen wurden als Sommer- oder Regentage. Die Mehrheit zeigte zudem ein hohes Bewusstsein für die Thematik Hitzebelastung und Anpassung. Dies spiegelte sich aber nur bei einem Teil der Befragten in ihrem tatsächlich gezeigten Anpassungsmaßnahmen wider. Offizielle Hitzewarnungen des DWD waren den meisten BesucherInnen an Tagen mit ebendiesen nicht bekannt.
Auf Grundlage unserer Untersuchungsergebnisse empfehlen wir eine verbesserte Risikokommunikation in Bezug auf Hitze. Veranstalter und Behörden müssen zielgruppenspezifisch denken, wenn es um die Förderung von Hitzeanpassung geht. Angeraten werden u. a. die Schaffung von schattigen Sitzplätzen besonders für ältere BesucherInnen und Wasserstellen, an denen Kinder und Jugendliche spielen und sich erfrischen können. Da sich Hitzewellen in Zukunft häufen werden, dienen die Erkenntnisse dieser Untersuchung der Planung und Durchführung weiterer Open-Air-Veranstaltungen.
The Collatz conjecture is a number theoretical problem, which has puzzled countless researchers using myriad approaches. Presently, there are scarcely any methodologies to describe and treat the problem from the perspective of the Algebraic Theory of Automata. Such an approach is promising with respect to facilitating the comprehension of the Collatz sequences "mechanics". The systematic technique of a state machine is both simpler and can fully be described by the use of algebraic means.
The current gap in research forms the motivation behind the present contribution. The present authors are convinced that exploring the Collatz conjecture in an algebraic manner, relying on findings and fundamentals of Graph Theory and Automata Theory, will simplify the problem as a whole.
The present work will introduce a Finite State Machine (FSM) that processes any Collatz Sequence; further, we will endeavor to investigate its behavior in relationship to transformations of a special infinite input. Moreover, we will prove that the machine’s word transformation is equivalent to the standard Collatz number transformation and subsequently discuss the possibilities for use of this approach at solving similar problems. The benefit of this approach is that the investigation of the word transformation performed by the Finite State Machine is less complicated than the traditional number-theoretical transformation.