{"id":"https://openalex.org/W2803821225","doi":"https://doi.org/10.3390/e20070504","title":"Diffusion on Middle-\u03be Cantor Sets","display_name":"Diffusion on Middle-\u03be Cantor Sets","publication_year":2018,"publication_date":"2018-07-02","ids":{"openalex":"https://openalex.org/W2803821225","doi":"https://doi.org/10.3390/e20070504","mag":"2803821225","pmid":"https://pubmed.ncbi.nlm.nih.gov/33265594"},"language":"en","primary_location":{"id":"doi:10.3390/e20070504","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20070504","pdf_url":"https://www.mdpi.com/1099-4300/20/7/504/pdf?version=1530497100","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/20/7/504/pdf?version=1530497100","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Alireza Khalili Golmankhaneh","orcid":"https://orcid.org/0000-0002-5008-0163"},"institutions":[{"id":"https://openalex.org/I4210163840","display_name":"Islamic Azad University of Urmia","ror":"https://ror.org/05km8ys10","country_code":"IR","type":"education","lineage":["https://openalex.org/I110525433","https://openalex.org/I4210163840"]}],"countries":["IR"],"is_corresponding":true,"raw_author_name":"Alireza Khalili Golmankhaneh","raw_affiliation_strings":["Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran","Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran"],"raw_orcid":"https://orcid.org/0000-0002-5008-0163","affiliations":[{"raw_affiliation_string":"Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran","institution_ids":["https://openalex.org/I4210163840"]},{"raw_affiliation_string":"Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran","institution_ids":["https://openalex.org/I4210163840"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Arran Fernandez","orcid":"https://orcid.org/0000-0002-1491-1820"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Arran Fernandez","raw_affiliation_strings":["Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK"],"raw_orcid":"https://orcid.org/0000-0002-1491-1820","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Ali Khalili Golmankhaneh","orcid":null},"institutions":[{"id":"https://openalex.org/I4210163840","display_name":"Islamic Azad University of Urmia","ror":"https://ror.org/05km8ys10","country_code":"IR","type":"education","lineage":["https://openalex.org/I110525433","https://openalex.org/I4210163840"]}],"countries":["IR"],"is_corresponding":false,"raw_author_name":"Ali Khalili Golmankhaneh","raw_affiliation_strings":["Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran","Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics, Urmia Branch, Islamic Azad University, Urmia, Iran","institution_ids":["https://openalex.org/I4210163840"]},{"raw_affiliation_string":"Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran","institution_ids":["https://openalex.org/I4210163840"]}]},{"author_position":"last","author":{"id":null,"display_name":"Dumitru Baleanu","orcid":"https://orcid.org/0000-0002-0286-7244"},"institutions":[{"id":"https://openalex.org/I106245021","display_name":"\u00c7ankaya University","ror":"https://ror.org/056wqre19","country_code":"TR","type":"education","lineage":["https://openalex.org/I106245021"]},{"id":"https://openalex.org/I4210142834","display_name":"Institute of Space Science - INFLPR Subsidiary","ror":"https://ror.org/054a6wv56","country_code":"RO","type":"facility","lineage":["https://openalex.org/I4210142834"]}],"countries":["RO","TR"],"is_corresponding":false,"raw_author_name":"Dumitru Baleanu","raw_affiliation_strings":["Department of Mathematics, Cankaya University, Ankara 06530, Turkey","Institute of Space Sciences, P.O. Box, MG-23, R 76900 Magurele-Bucharest, Romania"],"raw_orcid":"https://orcid.org/0000-0002-0286-7244","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Cankaya University, Ankara 06530, Turkey","institution_ids":["https://openalex.org/I106245021"]},{"raw_affiliation_string":"Institute of Space Sciences, P.O. Box, MG-23, R 76900 Magurele-Bucharest, Romania","institution_ids":["https://openalex.org/I4210142834"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210163840"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":1185,"currency":"EUR","value_usd":1278},"fwci":6.5666,"has_fulltext":true,"cited_by_count":45,"citation_normalized_percentile":{"value":0.97756352,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"20","issue":"7","first_page":"504","last_page":"504"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.5444999933242798,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.5444999933242798,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12037","display_name":"Algebraic and Geometric Analysis","score":0.06310000270605087,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.0471000000834465,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/cantor-function","display_name":"Cantor function","score":0.9070000052452087},{"id":"https://openalex.org/keywords/cantor-set","display_name":"Cantor set","score":0.8507000207901001},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.6190000176429749},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.6092000007629395},{"id":"https://openalex.org/keywords/cantors-diagonal-argument","display_name":"Cantor's diagonal argument","score":0.5982999801635742},{"id":"https://openalex.org/keywords/integrable-system","display_name":"Integrable system","score":0.5583000183105469},{"id":"https://openalex.org/keywords/multifractal-system","display_name":"Multifractal system","score":0.512499988079071},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.392300009727478}],"concepts":[{"id":"https://openalex.org/C149701218","wikidata":"https://www.wikidata.org/wiki/Q938883","display_name":"Cantor function","level":3,"score":0.9070000052452087},{"id":"https://openalex.org/C36605576","wikidata":"https://www.wikidata.org/wiki/Q273188","display_name":"Cantor set","level":2,"score":0.8507000207901001},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.791700005531311},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.6190000176429749},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.6092000007629395},{"id":"https://openalex.org/C70654966","wikidata":"https://www.wikidata.org/wiki/Q729471","display_name":"Cantor's diagonal argument","level":3,"score":0.5982999801635742},{"id":"https://openalex.org/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.5583000183105469},{"id":"https://openalex.org/C133905733","wikidata":"https://www.wikidata.org/wiki/Q2629238","display_name":"Multifractal system","level":3,"score":0.512499988079071},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4837000072002411},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.399399995803833},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.392300009727478},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.37290000915527344},{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.3662000000476837},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.3303000032901764},{"id":"https://openalex.org/C164602753","wikidata":"https://www.wikidata.org/wiki/Q567544","display_name":"Anomalous diffusion","level":3,"score":0.3244999945163727},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.28780001401901245},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.2867000102996826},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C116006050","wikidata":"https://www.wikidata.org/wiki/Q280081","display_name":"Sierpinski carpet","level":4,"score":0.27970001101493835},{"id":"https://openalex.org/C42357961","wikidata":"https://www.wikidata.org/wiki/Q213363","display_name":"Open set","level":2,"score":0.2667999863624573},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.2639999985694885},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2630999982357025},{"id":"https://openalex.org/C75413324","wikidata":"https://www.wikidata.org/wiki/Q1530791","display_name":"Square-integrable function","level":2,"score":0.2515999972820282}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.3390/e20070504","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20070504","pdf_url":"https://www.mdpi.com/1099-4300/20/7/504/pdf?version=1530497100","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:33265594","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33265594","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:arXiv.org:1805.01536","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1805.01536","pdf_url":"https://arxiv.org/pdf/1805.01536","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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:7528a501-6f9c-4e03-989e-8b575740a538","is_oa":true,"landing_page_url":"https://avesis.yyu.edu.tr/publication/details/7528a501-6f9c-4e03-989e-8b575740a538/oai","pdf_url":"https://avesis.yyu.edu.tr/publication/details/7528a501-6f9c-4e03-989e-8b575740a538/oai/document.pdf","source":{"id":"https://openalex.org/S7407055328","display_name":"Van Y\u00fcz\u00fcnc\u00fc Y\u0131l University Academic Data Management System","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":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:080177e720234ee8bbc60013bc695e01","is_oa":true,"landing_page_url":"https://doaj.org/article/080177e720234ee8bbc60013bc695e01","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 20, Iss 7, p 504 (2018)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/20/7/504/","is_oa":true,"landing_page_url":"http://dx.doi.org/10.3390/e20070504","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:7513040","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7513040","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e20070504","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e20070504","pdf_url":"https://www.mdpi.com/1099-4300/20/7/504/pdf?version=1530497100","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2803821225.pdf","grobid_xml":"https://content.openalex.org/works/W2803821225.grobid-xml"},"referenced_works_count":44,"referenced_works":["https://openalex.org/W433794350","https://openalex.org/W636061291","https://openalex.org/W850594699","https://openalex.org/W1530342450","https://openalex.org/W1592886334","https://openalex.org/W1970366402","https://openalex.org/W1972716504","https://openalex.org/W1972793885","https://openalex.org/W1974220233","https://openalex.org/W1977782935","https://openalex.org/W1978851592","https://openalex.org/W1986372629","https://openalex.org/W1996160582","https://openalex.org/W2001966464","https://openalex.org/W2009873874","https://openalex.org/W2040509711","https://openalex.org/W2065049775","https://openalex.org/W2086219623","https://openalex.org/W2087875149","https://openalex.org/W2092529649","https://openalex.org/W2118676499","https://openalex.org/W2131864324","https://openalex.org/W2140741294","https://openalex.org/W2156264000","https://openalex.org/W2169401034","https://openalex.org/W2260062985","https://openalex.org/W2307990652","https://openalex.org/W2313148625","https://openalex.org/W2317903225","https://openalex.org/W2333597439","https://openalex.org/W2414935967","https://openalex.org/W2465752070","https://openalex.org/W2467729541","https://openalex.org/W2489930379","https://openalex.org/W2558706252","https://openalex.org/W2567351708","https://openalex.org/W2571654160","https://openalex.org/W2578008795","https://openalex.org/W2606484856","https://openalex.org/W2793674850","https://openalex.org/W2793914237","https://openalex.org/W2795912180","https://openalex.org/W6681386923","https://openalex.org/W6697460734"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,37,86],"study":[4],"C\u03b6-calculus":[5,47],"on":[6,48,63,77,101],"generalized":[7,45,50],"Cantor":[8,51,56,66,80],"sets,":[9],"which":[10],"have":[11,44,59,82,87],"self-similar":[12],"properties":[13],"and":[14,85,99],"fractional":[15,41],"dimensions":[16],"that":[17],"exceed":[18],"their":[19],"topological":[20],"dimensions.":[21],"Functions":[22],"with":[23,73],"fractal":[24,102],"support":[25],"are":[26,104],"not":[27],"differentiable":[28],"or":[29],"integrable":[30],"in":[31],"terms":[32],"of":[33,71],"standard":[34],"calculus,":[35],"so":[36],"must":[38],"involve":[39],"local":[40],"derivatives.":[42],"We":[43,58],"the":[46,49,64,78,89],"sets":[52,67,81,103],"known":[53],"as":[54],"middle-\u03be":[55,65,79],"sets.":[57],"suggested":[60],"a":[61],"calculus":[62],"for":[68,96],"different":[69],"values":[70],"\u03be":[72],"0<\u03be<1.":[74],"Differential":[75],"equations":[76],"been":[83],"solved,":[84],"presented":[88],"results":[90],"using":[91],"illustrative":[92],"examples.":[93],"The":[94],"conditions":[95],"super-,":[97],"normal,":[98],"sub-diffusion":[100],"given.":[105]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":13},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2018-06-01T00:00:00"}
