{"id":"https://openalex.org/W2011080089","doi":"https://doi.org/10.1080/10586458.2001.10504441","title":"On the Random Character of Fundamental Constant Expansions","display_name":"On the Random Character of Fundamental Constant Expansions","publication_year":2001,"publication_date":"2001-01-01","ids":{"openalex":"https://openalex.org/W2011080089","doi":"https://doi.org/10.1080/10586458.2001.10504441","mag":"2011080089"},"language":"en","primary_location":{"id":"doi:10.1080/10586458.2001.10504441","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2001.10504441","pdf_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2001.10504441","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2001.10504441","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081580336","display_name":"David H. Bailey","orcid":"https://orcid.org/0000-0002-7574-8342"},"institutions":[{"id":"https://openalex.org/I148283060","display_name":"Lawrence Berkeley National Laboratory","ror":"https://ror.org/02jbv0t02","country_code":"US","type":"facility","lineage":["https://openalex.org/I1330989302","https://openalex.org/I148283060","https://openalex.org/I39565521"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David H. Bailey","raw_affiliation_strings":["Lawr ence Berkeley National Laboratory","Lawr ence Berkeley National Laboratory , Berkeley , CA , 94720 , United States E-mail: dhbailey@lbl.gov"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Lawr ence Berkeley National Laboratory","institution_ids":["https://openalex.org/I148283060"]},{"raw_affiliation_string":"Lawr ence Berkeley National Laboratory , Berkeley , CA , 94720 , United States E-mail: dhbailey@lbl.gov","institution_ids":["https://openalex.org/I148283060"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111508743","display_name":"Richard E. Crandall","orcid":null},"institutions":[{"id":"https://openalex.org/I55486353","display_name":"Reed College","ror":"https://ror.org/00a6ram87","country_code":"US","type":"education","lineage":["https://openalex.org/I55486353"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Richard E. Crandall","raw_affiliation_strings":["Center for Advanced Computation, Reed College","Center for Advanced Computation, Reed College , Portland , OR , 97202 , United States E-mail: crandall@reed.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Advanced Computation, Reed College","institution_ids":["https://openalex.org/I55486353"]},{"raw_affiliation_string":"Center for Advanced Computation, Reed College , Portland , OR , 97202 , United States E-mail: crandall@reed.edu","institution_ids":["https://openalex.org/I55486353"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":12.004,"has_fulltext":false,"cited_by_count":112,"citation_normalized_percentile":{"value":0.98574571,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"10","issue":"2","first_page":"175","last_page":"190"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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":0.9962000250816345,"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"}},{"id":"https://openalex.org/T13720","display_name":"Benford\u2019s Law and Fraud Detection","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8590892553329468},{"id":"https://openalex.org/keywords/randomness","display_name":"Randomness","score":0.8345077037811279},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.6590831279754639},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.6416041851043701},{"id":"https://openalex.org/keywords/pseudorandom-number-generator","display_name":"Pseudorandom number generator","score":0.5934672951698303},{"id":"https://openalex.org/keywords/character","display_name":"Character (mathematics)","score":0.5878115892410278},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.5532472133636475},{"id":"https://openalex.org/keywords/pseudorandomness","display_name":"Pseudorandomness","score":0.5422810316085815},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.48104050755500793},{"id":"https://openalex.org/keywords/base","display_name":"Base (topology)","score":0.4780697822570801},{"id":"https://openalex.org/keywords/normality","display_name":"Normality","score":0.4598437547683716},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3562045097351074},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20213952660560608},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1240379810333252},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.1129559576511383},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08283022046089172}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8590892553329468},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.8345077037811279},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.6590831279754639},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.6416041851043701},{"id":"https://openalex.org/C140642157","wikidata":"https://www.wikidata.org/wiki/Q1623338","display_name":"Pseudorandom number generator","level":2,"score":0.5934672951698303},{"id":"https://openalex.org/C2780861071","wikidata":"https://www.wikidata.org/wiki/Q1062934","display_name":"Character (mathematics)","level":2,"score":0.5878115892410278},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5532472133636475},{"id":"https://openalex.org/C19086982","wikidata":"https://www.wikidata.org/wiki/Q2115856","display_name":"Pseudorandomness","level":3,"score":0.5422810316085815},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.48104050755500793},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.4780697822570801},{"id":"https://openalex.org/C2776157432","wikidata":"https://www.wikidata.org/wiki/Q1375683","display_name":"Normality","level":2,"score":0.4598437547683716},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3562045097351074},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20213952660560608},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1240379810333252},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.1129559576511383},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08283022046089172},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1080/10586458.2001.10504441","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2001.10504441","pdf_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2001.10504441","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.137.9407","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.137.9407","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://crd.lbl.gov/~dhbailey/dhbpapers/baicran.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.23.7720","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.7720","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.nersc.gov/~dhb/dhbpapers/baicran.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.32.4113","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.4113","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.nersc.gov/~dhb/dhbpapers/bcrandom.ps","raw_type":"text"},{"id":"pmh:oai:CULeuclid:euclid.em/999188630","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.em/999188630","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"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":"","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1080/10586458.2001.10504441","is_oa":true,"landing_page_url":"https://doi.org/10.1080/10586458.2001.10504441","pdf_url":"https://www.tandfonline.com/doi/abs/10.1080/10586458.2001.10504441","source":{"id":"https://openalex.org/S119636414","display_name":"Experimental Mathematics","issn_l":"1058-6458","issn":["1058-6458","1944-950X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Experimental Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2011080089.pdf","grobid_xml":"https://content.openalex.org/works/W2011080089.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W199708800","https://openalex.org/W753951602","https://openalex.org/W1531131534","https://openalex.org/W1560933130","https://openalex.org/W1607458528","https://openalex.org/W1976927746","https://openalex.org/W2026833808","https://openalex.org/W2046937767","https://openalex.org/W2073085121","https://openalex.org/W2095709299","https://openalex.org/W2103236565","https://openalex.org/W2145880011","https://openalex.org/W2332689278","https://openalex.org/W2752853835","https://openalex.org/W2952193010","https://openalex.org/W2970621922","https://openalex.org/W3102232858","https://openalex.org/W4239648988","https://openalex.org/W4300914388"],"related_works":["https://openalex.org/W2082419723","https://openalex.org/W3203585077","https://openalex.org/W4241950541","https://openalex.org/W2176553175","https://openalex.org/W2185337847","https://openalex.org/W2143898630","https://openalex.org/W1994207320","https://openalex.org/W2343905204","https://openalex.org/W2795569938","https://openalex.org/W95325373"],"abstract_inverted_index":{"We":[0,82],"propose":[1],"a":[2,25,51,55,84],"theory":[3,93],"to":[4,79],"explain":[5],"random":[6],"behavior":[7],"for":[8],"the":[9,12,19,29,32,69,71,92],"digits":[10],"in":[11,50],"expansions":[13],"of":[14,21,31,45,57,94],"fundamental":[15],"mathematical":[16],"constants.":[17],"At":[18],"core":[20],"our":[22,88],"approach":[23],"is":[24,74],"general":[26],"hypothesis":[27],"concerning":[28],"distribution":[30],"iterates":[33],"generated":[34],"by":[35],"dynamical":[36,89],"maps.":[37],"On":[38],"this":[39],"main":[40],"hypothesis,":[41,70],"one":[42],"obtains":[43],"proofs":[44],"base-2":[46],"normality\u2014namely":[47],"bit":[48],"randomness":[49],"specific":[52],"technical":[53],"sense\u2014for":[54],"collection":[56],"celebrated":[58],"constants,":[59],"including":[60],"\u03c0,":[61],"log":[62],"2,":[63],"\u03b6(3),":[64],"and":[65,91],"others.":[66],"Also":[67],"on":[68],"number":[72,96],"\u03b6(5)":[73],"either":[75],"rational":[76],"or":[77],"normal":[78],"base":[80],"2.":[81],"indicate":[83],"research":[85],"connection":[86],"between":[87],"model":[90],"pseudorandom":[95],"generators.":[97]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":8},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":4}],"updated_date":"2026-08-26T07:47:46.906454","created_date":"2025-10-10T00:00:00"}
