{"id":"https://openalex.org/W4401960797","doi":"https://doi.org/10.3390/sym16091117","title":"Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices","display_name":"Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices","publication_year":2024,"publication_date":"2024-08-28","ids":{"openalex":"https://openalex.org/W4401960797","doi":"https://doi.org/10.3390/sym16091117"},"language":"en","primary_location":{"id":"doi:10.3390/sym16091117","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16091117","pdf_url":"https://www.mdpi.com/2073-8994/16/9/1117/pdf?version=1724833163","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/16/9/1117/pdf?version=1724833163","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5106831675","display_name":"Ruyu Tao","orcid":null},"institutions":[{"id":"https://openalex.org/I196934937","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709","country_code":"CN","type":"education","lineage":["https://openalex.org/I196934937"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ruyu Tao","raw_affiliation_strings":["Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"],"affiliations":[{"raw_affiliation_string":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China","institution_ids":["https://openalex.org/I196934937"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111305004","display_name":"Ying Li","orcid":null},"institutions":[{"id":"https://openalex.org/I196934937","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709","country_code":"CN","type":"education","lineage":["https://openalex.org/I196934937"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Ying Li","raw_affiliation_strings":["Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"],"affiliations":[{"raw_affiliation_string":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China","institution_ids":["https://openalex.org/I196934937"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087097238","display_name":"Mingcui Zhang","orcid":"https://orcid.org/0000-0001-8087-766X"},"institutions":[{"id":"https://openalex.org/I196934937","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709","country_code":"CN","type":"education","lineage":["https://openalex.org/I196934937"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Mingcui Zhang","raw_affiliation_strings":["Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"],"affiliations":[{"raw_affiliation_string":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China","institution_ids":["https://openalex.org/I196934937"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100430486","display_name":"Xiaochen Liu","orcid":"https://orcid.org/0000-0001-9824-094X"},"institutions":[{"id":"https://openalex.org/I196934937","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709","country_code":"CN","type":"education","lineage":["https://openalex.org/I196934937"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiaochen Liu","raw_affiliation_strings":["Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China"],"affiliations":[{"raw_affiliation_string":"Research Center of Semi-Tensor Product of Matrices: Theory and Applications, College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China","institution_ids":["https://openalex.org/I196934937"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5106381930","display_name":"Musheng Wei","orcid":null},"institutions":[{"id":"https://openalex.org/I21945476","display_name":"Shanghai Normal University","ror":"https://ror.org/01cxqmw89","country_code":"CN","type":"education","lineage":["https://openalex.org/I21945476"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Musheng Wei","raw_affiliation_strings":["College of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China"],"affiliations":[{"raw_affiliation_string":"College of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China","institution_ids":["https://openalex.org/I21945476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5111305004"],"corresponding_institution_ids":["https://openalex.org/I196934937"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":2.2253,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.89094304,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":98,"max":99},"biblio":{"volume":"16","issue":"9","first_page":"1117","last_page":"1117"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9937999844551086,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9937999844551086,"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/T12303","display_name":"Tensor decomposition and applications","score":0.9785000085830688,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.9743000268936157,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/quaternion","display_name":"Quaternion","score":0.9196934700012207},{"id":"https://openalex.org/keywords/dual-quaternion","display_name":"Dual quaternion","score":0.66801518201828},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6550600528717041},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5791893601417542},{"id":"https://openalex.org/keywords/dual","display_name":"Dual (grammatical number)","score":0.5699110627174377},{"id":"https://openalex.org/keywords/hermitian-matrix","display_name":"Hermitian matrix","score":0.5556909441947937},{"id":"https://openalex.org/keywords/quaternion-algebra","display_name":"Quaternion algebra","score":0.5223292708396912},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4953223764896393},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.35382145643234253},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.280185341835022},{"id":"https://openalex.org/keywords/algebra-representation","display_name":"Algebra representation","score":0.07532253861427307},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07267129421234131}],"concepts":[{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.9196934700012207},{"id":"https://openalex.org/C145941777","wikidata":"https://www.wikidata.org/wiki/Q5310237","display_name":"Dual quaternion","level":3,"score":0.66801518201828},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6550600528717041},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5791893601417542},{"id":"https://openalex.org/C2780980858","wikidata":"https://www.wikidata.org/wiki/Q110022","display_name":"Dual (grammatical number)","level":2,"score":0.5699110627174377},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.5556909441947937},{"id":"https://openalex.org/C151640129","wikidata":"https://www.wikidata.org/wiki/Q2835967","display_name":"Quaternion algebra","level":5,"score":0.5223292708396912},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4953223764896393},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35382145643234253},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.280185341835022},{"id":"https://openalex.org/C14394260","wikidata":"https://www.wikidata.org/wiki/Q4723980","display_name":"Algebra representation","level":3,"score":0.07532253861427307},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07267129421234131},{"id":"https://openalex.org/C124952713","wikidata":"https://www.wikidata.org/wiki/Q8242","display_name":"Literature","level":1,"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/C100899422","wikidata":"https://www.wikidata.org/wiki/Q5058347","display_name":"Cellular algebra","level":4,"score":0.0},{"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym16091117","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16091117","pdf_url":"https://www.mdpi.com/2073-8994/16/9/1117/pdf?version=1724833163","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:4620ed350a8b4d09981fc9c22ec7a0d3","is_oa":true,"landing_page_url":"https://doaj.org/article/4620ed350a8b4d09981fc9c22ec7a0d3","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 16, Iss 9, p 1117 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/16/9/1117/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym16091117","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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym16091117","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16091117","pdf_url":"https://www.mdpi.com/2073-8994/16/9/1117/pdf?version=1724833163","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1231421488","display_name":null,"funder_award_id":"under","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G1425722638","display_name":null,"funder_award_id":"319462208","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G144932472","display_name":null,"funder_award_id":"62176112","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2954701993","display_name":null,"funder_award_id":"ZR2022","funder_id":"https://openalex.org/F4320324174","funder_display_name":"Natural Science Foundation of Shandong Province"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4861881532","display_name":null,"funder_award_id":"ZR2022MA030","funder_id":"https://openalex.org/F4320324174","funder_display_name":"Natural Science Foundation of Shandong Province"},{"id":"https://openalex.org/G5625434284","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320324155","funder_display_name":"Liaocheng University"},{"id":"https://openalex.org/G5655394322","display_name":null,"funder_award_id":"319462208","funder_id":"https://openalex.org/F4320324155","funder_display_name":"Liaocheng University"},{"id":"https://openalex.org/G5939423041","display_name":null,"funder_award_id":"Technology","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320314997","display_name":"Strong","ror":"https://ror.org/041vyzr56"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320324155","display_name":"Liaocheng University","ror":"https://ror.org/03yh0n709"},{"id":"https://openalex.org/F4320324174","display_name":"Natural Science Foundation of Shandong Province","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4401960797.pdf"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W567151721","https://openalex.org/W1139739466","https://openalex.org/W1843791525","https://openalex.org/W1979296307","https://openalex.org/W1983473126","https://openalex.org/W2002056129","https://openalex.org/W2028691689","https://openalex.org/W2036202791","https://openalex.org/W2055495357","https://openalex.org/W2078838908","https://openalex.org/W2130276958","https://openalex.org/W2133702739","https://openalex.org/W2170553290","https://openalex.org/W3010997318","https://openalex.org/W3018632141","https://openalex.org/W3102718792","https://openalex.org/W3209990524","https://openalex.org/W3214066271","https://openalex.org/W3217433320","https://openalex.org/W4311061965","https://openalex.org/W4378697069","https://openalex.org/W4380487384","https://openalex.org/W4386890919","https://openalex.org/W4388002703","https://openalex.org/W4388634706","https://openalex.org/W4391611382","https://openalex.org/W4399330529","https://openalex.org/W6670684666","https://openalex.org/W6805156164"],"related_works":["https://openalex.org/W2156842304","https://openalex.org/W2128405419","https://openalex.org/W4213061425","https://openalex.org/W4321633639","https://openalex.org/W2528648357","https://openalex.org/W4235237238","https://openalex.org/W2749494690","https://openalex.org/W1499641851","https://openalex.org/W4387870045","https://openalex.org/W4387596512"],"abstract_inverted_index":{"Dual":[0],"algebra":[1],"plays":[2],"an":[3,26],"important":[4],"role":[5],"in":[6],"kinematic":[7],"synthesis":[8],"and":[9,64,92,121],"dynamic":[10],"analysis,":[11],"but":[12],"there":[13],"are":[14,46,125],"still":[15],"few":[16],"studies":[17],"on":[18],"dual":[19,38,48,61,79,87,90,97,112],"quaternion":[20,39,49,62,80,91,98,113],"matrix":[21,40,81,114],"theory.":[22],"This":[23],"paper":[24],"provides":[25],"efficient":[27],"method":[28],"for":[29],"solving":[30],"the":[31,37,74,78,86,93,102,111,122,129,132],"QLY":[32,105],"least":[33,106],"squares":[34,107],"problem":[35],"of":[36,60,77,89,95,104,110,131],"equation":[41,115],"AXB+CYD\u2248E,":[42],"where":[43],"X,":[44],"Y":[45],"unknown":[47],"matrices":[50,63],"with":[51],"special":[52,96],"structures.":[53],"First,":[54],"we":[55,100],"define":[56],"a":[57],"semi-tensor":[58],"product":[59],"study":[65,101],"its":[66],"properties,":[67],"which":[68],"can":[69],"be":[70],"used":[71],"to":[72,127],"achieve":[73],"equivalent":[75],"form":[76],"equation.":[82],"Then,":[83],"by":[84],"using":[85],"representation":[88],"GH-representation":[94],"matrices,":[99],"expression":[103],"Hermitian":[108],"solution":[109],"AXB+CYD\u2248E.":[116],"The":[117],"algorithm":[118],"is":[119],"given":[120],"numerical":[123],"examples":[124],"provided":[126],"illustrate":[128],"efficiency":[130],"method.":[133]},"counts_by_year":[{"year":2025,"cited_by_count":6}],"updated_date":"2026-03-18T14:38:29.013473","created_date":"2025-10-10T00:00:00"}