Knof, Albrun (2024)
Characterizations of Synchronizability - in terms of road-colored graphs, Markov chains and quantum Markov processes.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00026533
Ph.D. Thesis, Primary publication, Publisher's Version
Abstract
The thesis is dedicated to the investigation of the property of synchronizability of road-colored directed graphs, which can also be understood as deterministic finite automata. In this context, Markov chains, a special class of stochastic processes, are considered. They have a canonical representation as directed graphs and, conversely, a directed graph with a probability distribution on the outgoing edges of its vertices induces a Markov chain. Such an identification is also possible between Markov chains and road-colored directed graphs. Especially with regard to the property of synchronizability, this connection has already led to interesting results in both areas, the deepening and expansion of which is one of the main goals of this work. Moreover, it has already been shown in the field of quantum probability theory that the notion of asymptotic completeness of certain quantum Markov processes, motivated by scattering theory, is closely related to the classical idea of synchronizability. Indeed, when considering commutative systems, the synchronizability of a road-colored graph provides an equivalent characterization of the asymptotic completeness of the associated quantum Markov process. In this framework, the present thesis deals with possible generalizations of the classical concept of synchronizability in a quantum mechanical context.
Item Type: | Ph.D. Thesis | ||||
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Erschienen: | 2024 | ||||
Creators: | Knof, Albrun | ||||
Type of entry: | Primary publication | ||||
Title: | Characterizations of Synchronizability - in terms of road-colored graphs, Markov chains and quantum Markov processes | ||||
Language: | English | ||||
Referees: | Kümmerer, Prof. Dr. Burkhard ; Gohm, Dr. Rolf | ||||
Date: | 22 March 2024 | ||||
Place of Publication: | Darmstadt | ||||
Collation: | vi, 162 Seiten | ||||
Refereed: | 20 December 2023 | ||||
DOI: | 10.26083/tuprints-00026533 | ||||
URL / URN: | https://tuprints.ulb.tu-darmstadt.de/26533 | ||||
Abstract: | The thesis is dedicated to the investigation of the property of synchronizability of road-colored directed graphs, which can also be understood as deterministic finite automata. In this context, Markov chains, a special class of stochastic processes, are considered. They have a canonical representation as directed graphs and, conversely, a directed graph with a probability distribution on the outgoing edges of its vertices induces a Markov chain. Such an identification is also possible between Markov chains and road-colored directed graphs. Especially with regard to the property of synchronizability, this connection has already led to interesting results in both areas, the deepening and expansion of which is one of the main goals of this work. Moreover, it has already been shown in the field of quantum probability theory that the notion of asymptotic completeness of certain quantum Markov processes, motivated by scattering theory, is closely related to the classical idea of synchronizability. Indeed, when considering commutative systems, the synchronizability of a road-colored graph provides an equivalent characterization of the asymptotic completeness of the associated quantum Markov process. In this framework, the present thesis deals with possible generalizations of the classical concept of synchronizability in a quantum mechanical context. |
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Alternative Abstract: |
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Uncontrolled Keywords: | Markovketten, Quantenmechanik, Quantenwahrscheinlichkeitstheorie, fast gleichmäßige Konvergenz, Synchronisierbarkeit, synchronisierendes Wort, asymptotische Vollständigkeit, straßengefärbte Graphen, Quantenmarkovprozess, Markov chain, quantum mechanics, quantum probability theory, almost uniform convergence, synchronizability, synchronizing word, asymptotic completeness, road-colores graph, road-coloring, quantum markov process, von Neumann algebra, dual extended tranistion operator, Moller operator, Coupling from the past | ||||
Status: | Publisher's Version | ||||
URN: | urn:nbn:de:tuda-tuprints-265335 | ||||
Classification DDC: | 500 Science and mathematics > 510 Mathematics | ||||
Divisions: | 04 Department of Mathematics 04 Department of Mathematics > Didactics and Pedagogy of Mathematics |
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Date Deposited: | 22 Mar 2024 13:10 | ||||
Last Modified: | 28 Mar 2024 08:44 | ||||
PPN: | |||||
Referees: | Kümmerer, Prof. Dr. Burkhard ; Gohm, Dr. Rolf | ||||
Refereed / Verteidigung / mdl. Prüfung: | 20 December 2023 | ||||
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