{"id":"https://openalex.org/W1145565286","doi":"https://doi.org/10.2478/forma-2015-0003","title":"Matrix of \u2124-module <sup>1</sup>","display_name":"Matrix of \u2124-module <sup>1</sup>","publication_year":2015,"publication_date":"2015-03-01","ids":{"openalex":"https://openalex.org/W1145565286","doi":"https://doi.org/10.2478/forma-2015-0003","mag":"1145565286"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2015-0003","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2015-0003","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/23/1/article-p29.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://content.sciendo.com/downloadpdf/journals/forma/23/1/article-p29.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076839736","display_name":"Yuichi Futa","orcid":null},"institutions":[{"id":"https://openalex.org/I177738480","display_name":"Japan Advanced Institute of Science and Technology","ror":"https://ror.org/03frj4r98","country_code":"JP","type":"education","lineage":["https://openalex.org/I177738480"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yuichi Futa","raw_affiliation_strings":["Japan Advanced Institute of Science and Technology, Ishikawa, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Japan Advanced Institute of Science and Technology, Ishikawa, Japan","institution_ids":["https://openalex.org/I177738480"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048322538","display_name":"Hiroyuki Okazaki","orcid":"https://orcid.org/0009-0000-1022-361X"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hiroyuki Okazaki","raw_affiliation_strings":["Shinshu University, Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University, Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["Shinshu University, Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University, Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.7805,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.88006791,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"23","issue":"1","first_page":"29","last_page":"49"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.5672000050544739,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.5672000050544739,"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/T11797","display_name":"graph theory and CDMA systems","score":0.06639999896287918,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.06260000169277191,"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":0.8258881568908691},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.531265914440155},{"id":"https://openalex.org/keywords/gramian-matrix","display_name":"Gramian matrix","score":0.5118183493614197},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.49947500228881836},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36224275827407837},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33939066529273987},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.16521331667900085}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8258881568908691},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.531265914440155},{"id":"https://openalex.org/C77246614","wikidata":"https://www.wikidata.org/wiki/Q1409400","display_name":"Gramian matrix","level":3,"score":0.5118183493614197},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.49947500228881836},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36224275827407837},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33939066529273987},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.16521331667900085},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2015-0003","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2015-0003","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/23/1/article-p29.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:e41342a79bd74b29820438cae3fad69b","is_oa":false,"landing_page_url":"https://doaj.org/article/e41342a79bd74b29820438cae3fad69b","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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":"Formalized Mathematics, Vol 23, Iss 1, Pp 29-49 (2015)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/4836","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/4836","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"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":{"id":"doi:10.2478/forma-2015-0003","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2015-0003","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/23/1/article-p29.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1453757988","display_name":"Study on secure and safety technology for sensor networks","funder_award_id":"21240001","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G5886570192","display_name":"Content Development for Distance Education in Advanced University Mathematics Using Mizar","funder_award_id":"22300285","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1145565286.pdf","grobid_xml":"https://content.openalex.org/works/W1145565286.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W113552438","https://openalex.org/W116725595","https://openalex.org/W148282774","https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W1485280854","https://openalex.org/W2026904929","https://openalex.org/W2098464720","https://openalex.org/W2100168951","https://openalex.org/W2104238184","https://openalex.org/W2105559527","https://openalex.org/W2114186533","https://openalex.org/W2134386295","https://openalex.org/W2138253682","https://openalex.org/W2145706675","https://openalex.org/W2147599721","https://openalex.org/W2150348213","https://openalex.org/W2159772409","https://openalex.org/W2163641194","https://openalex.org/W2167846685","https://openalex.org/W2167862310","https://openalex.org/W2181825792","https://openalex.org/W4245192867","https://openalex.org/W4253086862","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W3106133691","https://openalex.org/W4249580765","https://openalex.org/W2096753949","https://openalex.org/W1985218657","https://openalex.org/W1557945163","https://openalex.org/W2742285599","https://openalex.org/W2963341196","https://openalex.org/W2023661790","https://openalex.org/W3124205579","https://openalex.org/W2073994398"],"abstract_inverted_index":{"Summary":[0],"In":[1],"this":[2,87],"article,":[3],"we":[4,14],"formalize":[5,15],"a":[6,16,19,24,28,41],"matrix":[7,17,29,51],"of":[8,18,22,30,48,55,57,101],"\u2124-module":[9,44,60],"and":[10,27,69,74,80,97],"its":[11,49,58],"properties.":[12],"Specially,":[13],"linear":[20],"transformation":[21],"\u2124-module,":[23],"bilinear":[25,32],"form":[26,33],"the":[31],"(Gramian":[34],"matrix).":[35],"We":[36],"formally":[37],"prove":[38],"that":[39],"for":[40,63],"finite-rank":[42],"free":[43],"V":[45],",":[46],"determinant":[47],"Gramian":[50],"is":[52,61],"constant":[53],"regardless":[54],"selection":[56],"basis.":[59],"necessary":[62],"lattice":[64],"problems,":[65],"LLL":[66],"(Lenstra,":[67],"Lenstra":[68],"Lov\u00e1sz)":[70],"base":[71],"reduction":[72],"algorithm":[73],"cryptographic":[75],"systems":[76],"with":[77],"lattices":[78],"[22]":[79],"coding":[81],"theory":[82],"[14].":[83],"Some":[84],"theorems":[85,93,100],"in":[86,94],"article":[88],"are":[89],"described":[90],"by":[91],"translating":[92],"[24],":[95],"[26]":[96],"[19]":[98],"into":[99],"\u2124-module.":[102]},"counts_by_year":[{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2025-10-10T00:00:00"}