{"id":"https://openalex.org/W4400078525","doi":"https://doi.org/10.3390/axioms13070435","title":"The Existence and Representation of the Solutions to the System of Operator Equations AiXBi + CiYDi + EiZFi = Gi(i = 1, 2)","display_name":"The Existence and Representation of the Solutions to the System of Operator Equations AiXBi + CiYDi + EiZFi = Gi(i = 1, 2)","publication_year":2024,"publication_date":"2024-06-27","ids":{"openalex":"https://openalex.org/W4400078525","doi":"https://doi.org/10.3390/axioms13070435"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13070435","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13070435","pdf_url":"https://www.mdpi.com/2075-1680/13/7/435/pdf?version=1721033801","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/13/7/435/pdf?version=1721033801","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055740937","display_name":"Gen Che","orcid":"https://orcid.org/0000-0003-3853-1176"},"institutions":[{"id":"https://openalex.org/I120379545","display_name":"Inner Mongolia Agricultural University","ror":"https://ror.org/015d0jq83","country_code":"CN","type":"education","lineage":["https://openalex.org/I120379545"]},{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Gen Che","raw_affiliation_strings":["College of Science, Inner Mongolia Agricultural University, Hohhot 010018, China","School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Science, Inner Mongolia Agricultural University, Hohhot 010018, China","institution_ids":["https://openalex.org/I120379545"]},{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101018234","display_name":"Guojun Hai","orcid":null},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guojun Hai","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109752057","display_name":"Jiarui Mei","orcid":null},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jiarui Mei","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5114183056","display_name":"Xiang Cao","orcid":null},"institutions":[{"id":"https://openalex.org/I2722730","display_name":"Inner Mongolia University","ror":"https://ror.org/0106qb496","country_code":"CN","type":"education","lineage":["https://openalex.org/I2722730"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiang Cao","raw_affiliation_strings":["School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China","institution_ids":["https://openalex.org/I2722730"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5101018234"],"corresponding_institution_ids":["https://openalex.org/I2722730"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.11183268,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"13","issue":"7","first_page":"435","last_page":"435"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9987000226974487,"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":0.9987000226974487,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9983999729156494,"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.9975000023841858,"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/representation","display_name":"Representation (politics)","score":0.7512763738632202},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6560168266296387},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.6466482877731323},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5143371820449829},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.44668444991111755},{"id":"https://openalex.org/keywords/matrix-representation","display_name":"Matrix representation","score":0.43410632014274597},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4236748218536377},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4109213352203369},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3815404176712036},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3538905382156372},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.15620958805084229},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1318420171737671},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.08344599604606628}],"concepts":[{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.7512763738632202},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6560168266296387},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.6466482877731323},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5143371820449829},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.44668444991111755},{"id":"https://openalex.org/C103275481","wikidata":"https://www.wikidata.org/wiki/Q6787889","display_name":"Matrix representation","level":3,"score":0.43410632014274597},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4236748218536377},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4109213352203369},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3815404176712036},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3538905382156372},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.15620958805084229},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1318420171737671},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.08344599604606628},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","level":1,"score":0.0},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms13070435","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13070435","pdf_url":"https://www.mdpi.com/2075-1680/13/7/435/pdf?version=1721033801","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:3230038e4f3643d5b1b2bda00f2734f3","is_oa":false,"landing_page_url":"https://doaj.org/article/3230038e4f3643d5b1b2bda00f2734f3","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":"Axioms, Vol 13, Iss 7, p 435 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms13070435","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13070435","pdf_url":"https://www.mdpi.com/2075-1680/13/7/435/pdf?version=1721033801","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"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":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4400078525.pdf"},"referenced_works_count":42,"referenced_works":["https://openalex.org/W74403139","https://openalex.org/W1430720837","https://openalex.org/W1599279268","https://openalex.org/W1966592410","https://openalex.org/W1968328641","https://openalex.org/W1974511160","https://openalex.org/W1980549954","https://openalex.org/W1984736553","https://openalex.org/W1999674602","https://openalex.org/W2010333970","https://openalex.org/W2011990554","https://openalex.org/W2013698113","https://openalex.org/W2020339875","https://openalex.org/W2021752762","https://openalex.org/W2025256645","https://openalex.org/W2027056747","https://openalex.org/W2039408173","https://openalex.org/W2039926115","https://openalex.org/W2041705422","https://openalex.org/W2051812350","https://openalex.org/W2060946158","https://openalex.org/W2061502163","https://openalex.org/W2067429836","https://openalex.org/W2076449511","https://openalex.org/W2083842595","https://openalex.org/W2087405824","https://openalex.org/W2097720652","https://openalex.org/W2103256163","https://openalex.org/W2115576014","https://openalex.org/W2327368185","https://openalex.org/W2747157424","https://openalex.org/W2912091068","https://openalex.org/W2963273768","https://openalex.org/W2979951043","https://openalex.org/W3133949164","https://openalex.org/W3135438476","https://openalex.org/W3162467412","https://openalex.org/W3217074047","https://openalex.org/W4220866850","https://openalex.org/W4283799571","https://openalex.org/W6603054850","https://openalex.org/W6791670325"],"related_works":["https://openalex.org/W2006221579","https://openalex.org/W2509138561","https://openalex.org/W2157322189","https://openalex.org/W2783489612","https://openalex.org/W3081697272","https://openalex.org/W2137256705","https://openalex.org/W2349259413","https://openalex.org/W2915764004","https://openalex.org/W1588620092","https://openalex.org/W2483953803"],"abstract_inverted_index":{"In":[0,29],"this":[1],"paper,":[2],"we":[3,31],"give":[4],"the":[5,11,22,33,39,45,49,53,57,60],"necessary":[6],"and":[7,18,43,52],"sufficient":[8],"conditions":[9],"for":[10],"existence":[12],"of":[13,24,35,41,48,59],"general":[14,36],"solutions,":[15,17],"self-adjoint":[16,50],"positive":[19,54],"solutions":[20,37,51,55],"to":[21,38],"system":[23,40],"AiXBi+CiYDi+EiZFi=Gi(i=1,2)":[25],"under":[26],"additional":[27],"conditions.":[28],"addition,":[30],"derive":[32],"representation":[34,47],"AiXBi+CiYDi+EiZFi=Gi(i=1,2),":[42],"provide":[44],"matrix":[46],"in":[56],"sense":[58],"star":[61],"order.":[62]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}