{"id":"https://openalex.org/W2143716195","doi":"https://doi.org/10.1142/s1230161212500072","title":"Process Dimension of Classical and Non-Commutative Processes","display_name":"Process Dimension of Classical and Non-Commutative Processes","publication_year":2012,"publication_date":"2012-03-01","ids":{"openalex":"https://openalex.org/W2143716195","doi":"https://doi.org/10.1142/s1230161212500072","mag":"2143716195"},"language":"en","primary_location":{"id":"doi:10.1142/s1230161212500072","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s1230161212500072","pdf_url":null,"source":{"id":"https://openalex.org/S27332288","display_name":"Open Systems & Information Dynamics","issn_l":"1230-1612","issn":["1230-1612","1573-1324","1793-7191"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Open Systems & Information Dynamics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1108.3984","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Wolfgang L\u00f6hr","orcid":null},"institutions":[{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]},{"id":"https://openalex.org/I62318514","display_name":"University of Duisburg-Essen","ror":"https://ror.org/04mz5ra38","country_code":"DE","type":"education","lineage":["https://openalex.org/I62318514"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Wolfgang L\u00f6hr","raw_affiliation_strings":["Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany","Universit\u00e4t Duisburg-Essen, Universit\u00e4tsstra\u00dfe 2, 45141 Essen, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327"]},{"raw_affiliation_string":"Universit\u00e4t Duisburg-Essen, Universit\u00e4tsstra\u00dfe 2, 45141 Essen, Germany","institution_ids":["https://openalex.org/I62318514"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Arleta Szko\u0142a","orcid":null},"institutions":[{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Arleta Szko\u0142a","raw_affiliation_strings":["Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327"]}]},{"author_position":"last","author":{"id":null,"display_name":"Nihat Ay","orcid":null},"institutions":[{"id":"https://openalex.org/I1308548392","display_name":"Santa Fe Institute","ror":"https://ror.org/01arysc35","country_code":"US","type":"funder","lineage":["https://openalex.org/I1308548392"]},{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]}],"countries":["DE","US"],"is_corresponding":false,"raw_author_name":"Nihat Ay","raw_affiliation_strings":["Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany","Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Inselstra\u00dfe 22, 04103 Leipzig, Germany","institution_ids":["https://openalex.org/I4210091327"]},{"raw_affiliation_string":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA","institution_ids":["https://openalex.org/I1308548392"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.12837883,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"19","issue":"01","first_page":"1250007","last_page":"1250007"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.3407000005245209,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.3407000005245209,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11303","display_name":"Bayesian Modeling and Causal Inference","score":0.11089999973773956,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.06449999660253525,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.6567000150680542},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.5792999863624573},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.515999972820282},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5105999708175659},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.5077000260353088},{"id":"https://openalex.org/keywords/observable","display_name":"Observable","score":0.5056999921798706},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5026000142097473},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.46650001406669617}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8055999875068665},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6567000150680542},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.5792999863624573},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.515999972820282},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5105999708175659},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.5077000260353088},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.5056999921798706},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5026000142097473},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.46650001406669617},{"id":"https://openalex.org/C151424130","wikidata":"https://www.wikidata.org/wiki/Q17123911","display_name":"Stationary ergodic process","level":4,"score":0.44190001487731934},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.4092999994754791},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.3978999853134155},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3790000081062317},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3675000071525574},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3395000100135803},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.3337000012397766},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3158999979496002},{"id":"https://openalex.org/C48057537","wikidata":"https://www.wikidata.org/wiki/Q3551145","display_name":"Statistical theory","level":2,"score":0.30219998955726624},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.2791999876499176},{"id":"https://openalex.org/C122203268","wikidata":"https://www.wikidata.org/wiki/Q5862903","display_name":"Probability theory","level":2,"score":0.2782000005245209},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.25099998712539673}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s1230161212500072","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s1230161212500072","pdf_url":null,"source":{"id":"https://openalex.org/S27332288","display_name":"Open Systems & Information Dynamics","issn_l":"1230-1612","issn":["1230-1612","1573-1324","1793-7191"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Open Systems & Information Dynamics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1108.3984","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1108.3984","pdf_url":"https://arxiv.org/pdf/1108.3984","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:edoc.mpg.de:645655","is_oa":false,"landing_page_url":"http://edoc.mpg.de/645655","pdf_url":null,"source":{"id":"https://openalex.org/S4406922265","display_name":"Max Planck Institute for Plasma Physics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Open Systems and Information Dynamics, v.19 (2012)","raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1108.3984","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1108.3984","pdf_url":"https://arxiv.org/pdf/1108.3984","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W2036613247","https://openalex.org/W2046755574","https://openalex.org/W2065890766","https://openalex.org/W2074426935","https://openalex.org/W2075694080","https://openalex.org/W2149960632","https://openalex.org/W4300128422"],"related_works":[],"abstract_inverted_index":{"We":[0,32,56],"treat":[1],"observable":[2],"operator":[3],"models":[4],"(OOM)":[5],"and":[6,60,78],"their":[7],"non-commutative":[8,54],"generalisation,":[9],"which":[10,41,84],"we":[11,67],"call":[12],"NC-OOMs.":[13],"A":[14],"natural":[15],"characteristic":[16],"of":[17,24,51,53,81,88,100,103,116],"a":[18,49],"stochastic":[19],"process":[20,30,46,117],"in":[21],"the":[22,29,37,71,75,79,86,93,98,101,114],"context":[23],"classical":[25],"OOM":[26,77],"theory":[27],"is":[28],"dimension.":[31,118],"investigate":[33],"its":[34],"properties":[35],"within":[36],"more":[38],"general":[39],"formulation,":[40],"allows":[42],"one":[43],"to":[44,108,113],"consider":[45],"dimension":[47],"as":[48],"measure":[50],"complexity":[52],"processes:":[55],"prove":[57],"lower":[58],"semi-continuity,":[59],"derive":[61],"an":[62,110],"ergodic":[63],"decomposition":[64],"formula.":[65],"Further,":[66],"obtain":[68],"results":[69],"on":[70],"close":[72],"relationship":[73],"between":[74],"canonical":[76],"concept":[80],"causal":[82,104],"states":[83],"underlies":[85],"definition":[87],"statistical":[89,95],"complexity.":[90],"In":[91],"particular,":[92],"topological":[94],"complexity,":[96],"i.e.":[97],"logarithm":[99,115],"number":[102],"states,":[105],"turns":[106],"out":[107],"be":[109],"upper":[111],"bound":[112]},"counts_by_year":[],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2016-06-24T00:00:00"}