{"id":"https://openalex.org/W2033061148","doi":"https://doi.org/10.1080/10586458.2008.10129021","title":"On the Convex Closure of the Graph of Modular Inversions","display_name":"On the Convex Closure of the Graph of Modular Inversions","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W2033061148","doi":"https://doi.org/10.1080/10586458.2008.10129021","mag":"2033061148"},"language":"en","primary_location":{"id":"doi:10.1080/10586458.2008.10129021","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10586458.2008.10129021","pdf_url":null,"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"],"host_organization_lineage_names":["Taylor & Francis"],"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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060419914","display_name":"Mizan R. Khan","orcid":null},"institutions":[{"id":"https://openalex.org/I203331390","display_name":"Eastern Connecticut State University","ror":"https://ror.org/01mhgwt57","country_code":"US","type":"education","lineage":["https://openalex.org/I203331390","https://openalex.org/I4210132746"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Mizan R. Khan","raw_affiliation_strings":["Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, CT 06226","#N#Eastern Connecticut State University#N#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, CT 06226","institution_ids":["https://openalex.org/I203331390"]},{"raw_affiliation_string":"#N#Eastern Connecticut State University#N#","institution_ids":["https://openalex.org/I203331390"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030033231","display_name":"Igor E. Shparlinski","orcid":"https://orcid.org/0000-0002-5246-9391"},"institutions":[{"id":"https://openalex.org/I99043593","display_name":"Macquarie University","ror":"https://ror.org/01sf06y89","country_code":"AU","type":"education","lineage":["https://openalex.org/I99043593"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Igor E. Shparlinski","raw_affiliation_strings":["Department of Computing, Macquarie University, Sydney, NSW 2109, Australia","Macquarie University"],"affiliations":[{"raw_affiliation_string":"Department of Computing, Macquarie University, Sydney, NSW 2109, Australia","institution_ids":["https://openalex.org/I99043593"]},{"raw_affiliation_string":"Macquarie University","institution_ids":["https://openalex.org/I99043593"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067788139","display_name":"Christian L. Yankov","orcid":"https://orcid.org/0000-0002-9472-7002"},"institutions":[{"id":"https://openalex.org/I203331390","display_name":"Eastern Connecticut State University","ror":"https://ror.org/01mhgwt57","country_code":"US","type":"education","lineage":["https://openalex.org/I203331390","https://openalex.org/I4210132746"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Christian L. Yankov","raw_affiliation_strings":["Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, CT 06226","#N#Eastern Connecticut State University#N#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, CT 06226","institution_ids":["https://openalex.org/I203331390"]},{"raw_affiliation_string":"#N#Eastern Connecticut State University#N#","institution_ids":["https://openalex.org/I203331390"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5060419914"],"corresponding_institution_ids":["https://openalex.org/I203331390"],"apc_list":null,"apc_paid":null,"fwci":2.6693,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.88304392,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"17","issue":"1","first_page":"91","last_page":"104"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11166","display_name":"Analytic Number Theory Research","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9975000023841858,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9969000220298767,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8511583805084229},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6376259326934814},{"id":"https://openalex.org/keywords/heuristic","display_name":"Heuristic","score":0.6257066130638123},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.6029288172721863},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.5749043226242065},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4840042293071747},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.459891140460968},{"id":"https://openalex.org/keywords/convex-combination","display_name":"Convex combination","score":0.4125415086746216},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3844911456108093},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.32281070947647095},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.1772489845752716},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06465628743171692}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8511583805084229},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6376259326934814},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.6257066130638123},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.6029288172721863},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.5749043226242065},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4840042293071747},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.459891140460968},{"id":"https://openalex.org/C111110010","wikidata":"https://www.wikidata.org/wiki/Q2627315","display_name":"Convex combination","level":4,"score":0.4125415086746216},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3844911456108093},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.32281070947647095},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.1772489845752716},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06465628743171692},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1080/10586458.2008.10129021","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10586458.2008.10129021","pdf_url":null,"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"],"host_organization_lineage_names":["Taylor & Francis"],"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:CULeuclid:euclid.em/1227031900","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.em/1227031900","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.315.3492","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.315.3492","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0711.4100v1.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W1562455717","https://openalex.org/W1607203414","https://openalex.org/W1965911928","https://openalex.org/W1972106664","https://openalex.org/W1973971481","https://openalex.org/W1976003589","https://openalex.org/W1979832469","https://openalex.org/W1983455472","https://openalex.org/W1989620204","https://openalex.org/W2005936791","https://openalex.org/W2012987871","https://openalex.org/W2019387672","https://openalex.org/W2025299767","https://openalex.org/W2035439008","https://openalex.org/W2037357508","https://openalex.org/W2046541871","https://openalex.org/W2069469525","https://openalex.org/W2076597612","https://openalex.org/W2078645706","https://openalex.org/W2146421406","https://openalex.org/W2159774990","https://openalex.org/W2326291957","https://openalex.org/W2334612034","https://openalex.org/W2572248928","https://openalex.org/W2916007620","https://openalex.org/W3009825533","https://openalex.org/W3175367423"],"related_works":["https://openalex.org/W2391247224","https://openalex.org/W2363904277","https://openalex.org/W868123585","https://openalex.org/W2300145982","https://openalex.org/W2376997122","https://openalex.org/W4296997005","https://openalex.org/W2365830408","https://openalex.org/W2580512045","https://openalex.org/W981962503","https://openalex.org/W2040676043"],"abstract_inverted_index":{"In":[0,122],"this":[1,123],"paper":[2],"we":[3,72,94],"give":[4],"upper":[5],"and":[6,61,81,115],"lower":[7],"bounds":[8],"as":[9,11],"well":[10],"a":[12,96],"heuristic":[13,52,79,97],"estimate":[14],"on":[15,55,105],"the":[16,21,25,69,74,78,106,110,125,134,140],"number":[17],"of":[18,20,24,59,109,112],"vertices":[19],"convex":[22,70,107],"closure":[23,108],"set":[26],"Gn":[27],"=":[28,143],"{(a,":[29],"b)":[30],":":[31],"a,":[32,44],"b":[33,45],"\u2208":[34],"\u2124,":[35],"ab":[36],"\u2261":[37],"1":[38,42],"(mod":[39],"n)":[40,145],",":[41],"\u2264":[43,46],"n":[47,120],"-":[48],"1}":[49],".":[50,121],"The":[51],"is":[53,146],"based":[54],"an":[56],"asymptotic":[57],"formula":[58],"Renyi":[60],"Sulanke.":[62],"After":[63],"describing":[64],"two":[65],"algorithms":[66],"to":[67],"determine":[68],"closure,":[71],"compare":[73],"numeric":[75,126],"results":[76,127],"with":[77,133],"estimate,":[80],"find":[82],"that":[83,139],"they":[84],"do":[85],"not":[86],"agree\u2014there":[87],"are":[88,128],"some":[89,102],"interesting":[90],"peculiarities,":[91],"for":[92],"which":[93,136],"provide":[95],"explanation.":[98],"We":[99],"then":[100],"describe":[101],"numerical":[103],"work":[104],"graph":[111],"random":[113],"quadratic":[114],"cubic":[116],"polynomials":[117],"over":[118],"\u2124":[119],"case":[124],"in":[129],"much":[130],"closer":[131],"agreement":[132],"heuristic,":[135],"strongly":[137],"suggests":[138],"curve":[141],"xy":[142],"1(mod":[144],"\"atypical.\"":[147]},"counts_by_year":[{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}