{"id":"https://openalex.org/W2145549086","doi":"https://doi.org/10.1016/0166-218x(84)90116-1","title":"A duality theorem for linear congruences","display_name":"A duality theorem for linear congruences","publication_year":1984,"publication_date":"1984-01-01","ids":{"openalex":"https://openalex.org/W2145549086","doi":"https://doi.org/10.1016/0166-218x(84)90116-1","mag":"2145549086"},"language":"en","primary_location":{"id":"doi:10.1016/0166-218x(84)90116-1","is_oa":false,"landing_page_url":"https://doi.org/10.1016/0166-218x(84)90116-1","pdf_url":null,"source":{"id":"https://openalex.org/S171741597","display_name":"Discrete Applied Mathematics","issn_l":"0166-218X","issn":["0166-218X","1872-6771"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Applied 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/A5110611221","display_name":"Hugh Williams","orcid":"https://orcid.org/0009-0007-5086-5319"},"institutions":[{"id":"https://openalex.org/I98677209","display_name":"University of Edinburgh","ror":"https://ror.org/01nrxwf90","country_code":"GB","type":"education","lineage":["https://openalex.org/I98677209"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"H.P. Williams","raw_affiliation_strings":["Department of Business Studies, University of Edinburgh, U.K"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Business Studies, University of Edinburgh, U.K","institution_ids":["https://openalex.org/I98677209"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110611221"],"corresponding_institution_ids":["https://openalex.org/I98677209"],"apc_list":{"value":2750,"currency":"USD","value_usd":2750},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.27738482,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"7","issue":"1","first_page":"93","last_page":"103"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.8543000221252441,"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"}},"topics":[{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.8543000221252441,"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/mathematics","display_name":"Mathematics","score":0.8413604497909546},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.8067052960395813},{"id":"https://openalex.org/keywords/congruence-relation","display_name":"Congruence relation","score":0.7988203763961792},{"id":"https://openalex.org/keywords/strong-duality","display_name":"Strong duality","score":0.5742824077606201},{"id":"https://openalex.org/keywords/linear-programming","display_name":"Linear programming","score":0.5619715452194214},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.5353757739067078},{"id":"https://openalex.org/keywords/integer-programming","display_name":"Integer programming","score":0.4966192841529846},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.4963696599006653},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4628784656524658},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45262712240219116},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.44151216745376587},{"id":"https://openalex.org/keywords/wolfe-duality","display_name":"Wolfe duality","score":0.436576783657074},{"id":"https://openalex.org/keywords/unimodular-matrix","display_name":"Unimodular matrix","score":0.4263901710510254},{"id":"https://openalex.org/keywords/weak-duality","display_name":"Weak duality","score":0.41458219289779663},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3722163736820221},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.21674534678459167},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.1350114345550537},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06856423616409302}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8413604497909546},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.8067052960395813},{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.7988203763961792},{"id":"https://openalex.org/C163863214","wikidata":"https://www.wikidata.org/wiki/Q7624553","display_name":"Strong duality","level":3,"score":0.5742824077606201},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.5619715452194214},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.5353757739067078},{"id":"https://openalex.org/C56086750","wikidata":"https://www.wikidata.org/wiki/Q6042592","display_name":"Integer programming","level":2,"score":0.4966192841529846},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.4963696599006653},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4628784656524658},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45262712240219116},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.44151216745376587},{"id":"https://openalex.org/C109456813","wikidata":"https://www.wikidata.org/wiki/Q8029991","display_name":"Wolfe duality","level":5,"score":0.436576783657074},{"id":"https://openalex.org/C201958917","wikidata":"https://www.wikidata.org/wiki/Q1367055","display_name":"Unimodular matrix","level":2,"score":0.4263901710510254},{"id":"https://openalex.org/C80682859","wikidata":"https://www.wikidata.org/wiki/Q7977943","display_name":"Weak duality","level":4,"score":0.41458219289779663},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3722163736820221},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.21674534678459167},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.1350114345550537},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06856423616409302},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.0},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","level":1,"score":0.0},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1016/0166-218x(84)90116-1","is_oa":false,"landing_page_url":"https://doi.org/10.1016/0166-218x(84)90116-1","pdf_url":null,"source":{"id":"https://openalex.org/S171741597","display_name":"Discrete Applied Mathematics","issn_l":"0166-218X","issn":["0166-218X","1872-6771"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:researchonline.lse.ac.uk:31659","is_oa":false,"landing_page_url":"https://doi.org/10.1016/0166-218X(84)90116-1","pdf_url":null,"source":{"id":"https://openalex.org/S4306401593","display_name":"London School of Economics and Political Science Research Online (London School of Economics and Political Science)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I909854389","host_organization_name":"London School of Economics and Political Science","host_organization_lineage":["https://openalex.org/I909854389"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},{"id":"pmh:oai:eprints.lse.ac.uk:31659","is_oa":false,"landing_page_url":"http://eprints.lse.ac.uk/31659/","pdf_url":null,"source":{"id":"https://openalex.org/S4306401593","display_name":"London School of Economics and Political Science Research Online (London School of Economics and Political Science)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I909854389","host_organization_name":"London School of Economics and Political Science","host_organization_lineage":["https://openalex.org/I909854389"],"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":"Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":4,"referenced_works":["https://openalex.org/W1517845546","https://openalex.org/W2027875058","https://openalex.org/W2127470768","https://openalex.org/W2796604629"],"related_works":["https://openalex.org/W3213165575","https://openalex.org/W2029896594","https://openalex.org/W4210245775","https://openalex.org/W4283709143","https://openalex.org/W2345066843","https://openalex.org/W4220928172","https://openalex.org/W2061589448","https://openalex.org/W2056178476","https://openalex.org/W2384039353","https://openalex.org/W2136200239"],"abstract_inverted_index":null,"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-07-02T09:51:11.867554","created_date":"2025-10-10T00:00:00"}
