TY - RPRT A1 - Sultanow, Eldar A1 - Koch, Christian A1 - Cox, Sean T1 - Collatz Sequences in the Light of Graph Theory N2 - 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. KW - Collatz KW - Cayley Graph KW - Free Group KW - Reachability Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-437416 ET - 2nd version ER - TY - RPRT A1 - Sultanow, Eldar A1 - Koch, Christian A1 - Cox, Sean T1 - Collatz Sequences in the Light of Graph Theory N2 - 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. KW - Collatz KW - Cayley Graph KW - Free Group KW - Reachability Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-430089 ER - TY - RPRT A1 - Sultanow, Eldar A1 - Koch, Christian A1 - Cox, Sean T1 - Collatz Sequences in the Light of Graph Theory N2 - 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. KW - Collatz KW - Cayley Graph KW - Free Group KW - Reachability Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-441859 ET - 3rd version ER - TY - RPRT A1 - Sultanow, Eldar A1 - Koch, Christian A1 - Cox, Sean T1 - Collatz Sequences in the Light of Graph Theory N2 - 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. KW - Collatz KW - Cayley Graph KW - Free Group KW - Reachability Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-443254 ET - 4th version ER - TY - RPRT A1 - Sultanow, Eldar A1 - Koch, Christian A1 - Cox, Sean T1 - Collatz Sequences in the Light of Graph Theory N2 - 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. KW - Collatz Conjecture KW - Free Group KW - Multiplicative Group KW - Cyclic Group KW - Cayley Graph KW - Cycle KW - Tree KW - Binary Tree Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-482140 ET - Fifth version ER - TY - JOUR A1 - Brockmann, Carsten A1 - Gronau, Norbert A1 - Sultanow, Eldar T1 - ERP und MES : Teil 3 Y1 - 2008 SN - 1617-948X ER - TY - RPRT A1 - Sultanow, Eldar A1 - Volkov, Denis A1 - Cox, Sean T1 - Introducing a Finite State Machine for processing Collatz Sequences N2 - 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. KW - Collatz Conjecture KW - State Machine KW - Graph KW - Double Colored Edges Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-399223 ET - 1st version ER - TY - RPRT A1 - Sultanow, Eldar A1 - Volkov, Denis A1 - Cox, Sean T1 - Introducing a Finite State Machine for processing Collatz Sequences N2 - 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. KW - Collatz Conjecture KW - State Machine KW - Graph KW - Double Colored Edges Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-404738 ET - 2nd version ER - TY - JOUR A1 - Sultanow, Eldar A1 - Weber, Edzard T1 - Klassifikation und Identifikation von Kommunikationsbarrieren in Unternehmen Y1 - 2009 SN - 978-3-88579-239-0 ER - TY - JOUR A1 - Sultanow, Eldar A1 - Weber, Edzard T1 - Management-Leitstände 2.0 : Kollaboration, Semantic Web und Web 3D Y1 - 2009 SN - 0945-0491 ER -