TY - INPR
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 - INPR
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 - INPR
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 - INPR
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 - INPR
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 - INPR
A1 - Pousttchi, Key
A1 - Tilson, David
A1 - Lyytinen, Kalle
A1 - Hufenbach, Yvonne
T1 - Introduction to the Special Issue on Mobile Commerce: Mobile Commerce Research Yesterday, Today, TomorrowWhat Remains to Be Done?
T2 - International journal of electronic commerce
N2 - Mobile commerce (m-commerce) in the smartphone age is revolutionizing established value networks and transforming the wider economy. In this introduction we strive to build a bridge from the past of m-commerce research to its future. We examine more than a decade of research and conduct a Delphi study among leading scholars in the field. The review reveals significant changes in m-commerce topics as time goes on, and provides initial insights into what the future may hold for us. The most sobering finding is that the m-commerce field has still to establish a strong theoretical foundation. This has been reflected in less than overwhelming success in publishing on the subject in the most prestigious journals of the Information Systems discipline. At the same time, m-commerce forms one of the epicenters of the ongoing digitalization of our life. Therefore, we look forward to m-commerce research rising to the challenge and making significant contributions to understanding one of the important phenomena of our time.
KW - Delphi study
KW - literature review
KW - m-commerce
KW - mobile commerce research
Y1 - 2015
U6 - http://dx.doi.org/10.1080/10864415.2015.1029351
SN - 1086-4415 (print)
SN - 1557-9301 (online)
VL - 19
IS - 4
SP - 1
EP - 20
PB - Routledge, Taylor & Francis Group
CY - Abingdon
ER -
TY - INPR
A1 - Daviter, Falk
T1 - The political use of knowledge in the policy process
T2 - Policy sciences : integrating knowledge and practice to advance human dignity
N2 - The role of knowledge in the policy process remains a central theoretical puzzle in policy analysis and political science. This article argues that an important yet missing piece of this puzzle is the systematic exploration of the political use of policy knowledge. While much of the recent debate has focused on the question of how the substantive use of knowledge can improve the quality of policy choices, our understanding of the political use of knowledge and its effects in the policy process has remained deficient in key respects. A revised conceptualization of the political use of knowledge is introduced that emphasizes how conflicting knowledge can be used to contest given structures of policy authority. This allows the analysis to differentiate between knowledge creep and knowledge shifts as two distinct types of knowledge effects in the policy process. While knowledge creep is associated with incremental policy change within existing policy structures, knowledge shifts are linked to more fundamental policy change in situations when the structures of policy authority undergo some level of transformation. The article concludes by identifying characteristics of the administrative structure of policy systems or sectors that make knowledge shifts more or less likely.
KW - Evidence-based policy making
KW - Knowledge creep
KW - Knowledge utilization
KW - Organizational epistemology
KW - Punctuated equilibrium theory
Y1 - 2015
U6 - http://dx.doi.org/10.1007/s11077-015-9232-y
SN - 0032-2687 (print)
SN - 1573-0891 (online)
VL - 48
IS - 4
SP - 491
EP - 505
PB - Springer
CY - Dordrecht
ER -
TY - INPR
A1 - Caliendo, Marco
A1 - Kritikos, Alexander S.
T1 - Searching for the entrepreneurial personality New evidence and avenues for further research
T2 - Journal of economic psychology : research in economic psychology and behavioral economics
Y1 - 2012
U6 - http://dx.doi.org/10.1016/j.joep.2011.06.001
SN - 0167-4870 (print)
VL - 33
IS - 2
SP - 319
EP - 324
PB - Elsevier
CY - Amsterdam
ER -