44185
2020
2020
eng
29
3rd version
preprint
1
2020-01-06
2020-01-06
--
Collatz Sequences in the Light of Graph Theory
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.
10.25932/publishup-44185
urn:nbn:de:kobv:517-opus4-441859
false
true
Creative Commons - Namensnennung, 4.0 International
Eldar Sultanow
Christian Koch
Sean Cox
eng
uncontrolled
Collatz
eng
uncontrolled
Cayley Graph
eng
uncontrolled
Free Group
eng
uncontrolled
Reachability
Sozialwissenschaften
NUMBER THEORY
open_access
Wirtschaftswissenschaften
Nicht referiert
Third Version
Universität Potsdam
https://publishup.uni-potsdam.de/opus4-ubp/files/44185/algebrabook.pdf
43741
2019
2019
eng
21
2nd version
preprint
1
2019-11-05
2019-11-05
--
Collatz Sequences in the Light of Graph Theory
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.
10.25932/publishup-43741
urn:nbn:de:kobv:517-opus4-437416
Creative Commons - Namensnennung, 4.0 International
Eldar Sultanow
Christian Koch
Sean Cox
eng
uncontrolled
Collatz
eng
uncontrolled
Cayley Graph
eng
uncontrolled
Free Group
eng
uncontrolled
Reachability
Sozialwissenschaften
NUMBER THEORY
open_access
Wirtschaftswissenschaften
Second Version
Universität Potsdam
https://publishup.uni-potsdam.de/opus4-ubp/files/43741/collatz_sequences.pdf
43008
2019
2019
eng
15
preprint
1
2019-06-12
2019-06-12
--
Collatz Sequences in the Light of Graph Theory
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.
10.25932/publishup-43008
urn:nbn:de:kobv:517-opus4-430089
Creative Commons - Namensnennung, 4.0 International
Eldar Sultanow
Christian Koch
Sean Cox
eng
uncontrolled
Collatz
eng
uncontrolled
Cayley Graph
eng
uncontrolled
Free Group
eng
uncontrolled
Reachability
Sozialwissenschaften
NUMBER THEORY
open_access
Wirtschaftswissenschaften
First Version
Universität Potsdam
https://publishup.uni-potsdam.de/opus4-ubp/files/43008/collatz_sequences.pdf
40473
2017
2017
eng
17
2nd version
preprint
1
2017-12-08
2017-12-08
--
Introducing a Finite State Machine for processing Collatz Sequences
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.
urn:nbn:de:kobv:517-opus4-404738
<a href="http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-399223">Prior version of the Paper</a>
Keine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Eldar Sultanow
Denis Volkov
Sean Cox
eng
uncontrolled
Collatz Conjecture
eng
uncontrolled
State Machine
eng
uncontrolled
Graph
eng
uncontrolled
Double Colored Edges
Sozialwissenschaften
NUMBER THEORY
open_access
Wirtschaftswissenschaften
Second version
Universität Potsdam
https://publishup.uni-potsdam.de/opus4-ubp/files/40473/preprint_sultanow-volkov-cox.pdf
39922
2017
2017
eng
1st version
preprint
1
2017-09-06
2017-09-06
--
Introducing a Finite State Machine for processing Collatz Sequences
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.
urn:nbn:de:kobv:517-opus4-399223
online registration
<a href="http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-404738">Subsequent version of the paper</a>
Keine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Eldar Sultanow
Denis Volkov
Sean Cox
eng
uncontrolled
Collatz Conjecture
eng
uncontrolled
State Machine
eng
uncontrolled
Graph
eng
uncontrolled
Double Colored Edges
Mathematik
NUMBER THEORY
open_access
Wirtschaftswissenschaften
First version
Universität Potsdam
https://publishup.uni-potsdam.de/opus4-ubp/files/39922/preprint_sultanow-volkov-cox.pdf
38871
2015
2015
eng
1
20
20
4
19
preprint
Routledge, Taylor & Francis Group
Abingdon
1
--
--
--
Introduction to the Special Issue on Mobile Commerce: Mobile Commerce Research Yesterday, Today, TomorrowWhat Remains to Be Done?
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.
International journal of electronic commerce
10.1080/10864415.2015.1029351
1086-4415 (print)
1557-9301 (online)
wos:2015
WOS:000356301300001
Pousttchi, K (reprint author), Univ Potsdam, Business Informat & Digitalizat, Potsdam, Germany., key.pousttchi@uni-potsdam.de; david.tilson@simon.rochester.edu; kalle.lyytinen@case.edu; yvonne.hufenbach@wi-mobile.de
Key Pousttchi
David Tilson
Kalle Lyytinen
Yvonne Hufenbach
eng
uncontrolled
Delphi study
eng
uncontrolled
literature review
eng
uncontrolled
m-commerce
eng
uncontrolled
mobile commerce research
Wirtschaftswissenschaften
Referiert
38427
2015
2015
eng
491
505
15
4
48
preprint
Springer
Dordrecht
1
--
--
--
The political use of knowledge in the policy process
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.
Policy sciences : integrating knowledge and practice to advance human dignity
10.1007/s11077-015-9232-y
0032-2687 (print)
1573-0891 (online)
wos:2015
WOS:000364573800005
Daviter, F (reprint author), Univ Potsdam, Dept Econ & Social Sci, August Bebel Str 89, D-14482 Potsdam, Germany., daviter@uni-potsdam.de
Falk Daviter
eng
uncontrolled
Evidence-based policy making
eng
uncontrolled
Knowledge creep
eng
uncontrolled
Knowledge utilization
eng
uncontrolled
Organizational epistemology
eng
uncontrolled
Punctuated equilibrium theory
Wirtschaftswissenschaften
Referiert
36007
2012
2012
eng
319
324
6
2
33
preprint
Elsevier
Amsterdam
1
--
--
--
Searching for the entrepreneurial personality New evidence and avenues for further research
Journal of economic psychology : research in economic psychology and behavioral economics
10.1016/j.joep.2011.06.001
0167-4870 (print)
wos:2011-2013
WOS:000301829900001
Caliendo, M (reprint author), Inst Study Labor IZA, Bonn, Germany., caliendo@iza.org; akritikos@diw.de
Marco Caliendo
Alexander S. Kritikos
Wirtschaftswissenschaften
Referiert