{"id":"https://openalex.org/W4393162623","doi":"https://doi.org/10.3390/axioms13040216","title":"On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities","display_name":"On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities","publication_year":2024,"publication_date":"2024-03-25","ids":{"openalex":"https://openalex.org/W4393162623","doi":"https://doi.org/10.3390/axioms13040216"},"language":"en","primary_location":{"id":"doi:10.3390/axioms13040216","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13040216","pdf_url":"https://www.mdpi.com/2075-1680/13/4/216/pdf?version=1711373886","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/2075-1680/13/4/216/pdf?version=1711373886","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064553367","display_name":"Naveed Hussain","orcid":"https://orcid.org/0000-0002-0269-3803"},"institutions":[{"id":"https://openalex.org/I61623833","display_name":"University of Agriculture Faisalabad","ror":"https://ror.org/054d77k59","country_code":"PK","type":"education","lineage":["https://openalex.org/I61623833"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Naveed Hussain","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan","institution_ids":["https://openalex.org/I61623833"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009981956","display_name":"Ahmad N. Al\u2010Kenani","orcid":null},"institutions":[{"id":"https://openalex.org/I185163786","display_name":"King Abdulaziz University","ror":"https://ror.org/02ma4wv74","country_code":"SA","type":"education","lineage":["https://openalex.org/I185163786"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ahmad N. Al-Kenani","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia","Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;"],"raw_orcid":"https://orcid.org/0000-0001-5559-5951","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia","institution_ids":["https://openalex.org/I185163786"]},{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia;","institution_ids":["https://openalex.org/I185163786"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101921717","display_name":"M. Asif","orcid":"https://orcid.org/0000-0001-8669-942X"},"institutions":[{"id":"https://openalex.org/I16076960","display_name":"COMSATS University Islamabad","ror":"https://ror.org/00nqqvk19","country_code":"PK","type":"education","lineage":["https://openalex.org/I16076960"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Muhammad Asif","raw_affiliation_strings":["Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan","Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan;"],"raw_orcid":"https://orcid.org/0000-0001-8669-942X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I16076960"]},{"raw_affiliation_string":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan;","institution_ids":["https://openalex.org/I16076960"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0799173,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"13","issue":"4","first_page":"216","last_page":"216"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":1.0,"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9998999834060669,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/gravitational-singularity","display_name":"Gravitational singularity","score":0.756409764289856},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.7399541735649109},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6157128214836121},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.5227739214897156},{"id":"https://openalex.org/keywords/lie-algebra","display_name":"Lie algebra","score":0.5038730502128601},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.49550116062164307},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.17386522889137268}],"concepts":[{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.756409764289856},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.7399541735649109},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6157128214836121},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5227739214897156},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.5038730502128601},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.49550116062164307},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.17386522889137268}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms13040216","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13040216","pdf_url":"https://www.mdpi.com/2075-1680/13/4/216/pdf?version=1711373886","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:38787ca4d68c48c99d3ebd753215baf3","is_oa":true,"landing_page_url":"https://doaj.org/article/38787ca4d68c48c99d3ebd753215baf3","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 13, Iss 4, p 216 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/13/4/216/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms13040216","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":"Axioms","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms13040216","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms13040216","pdf_url":"https://www.mdpi.com/2075-1680/13/4/216/pdf?version=1711373886","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4393162623.pdf","grobid_xml":"https://content.openalex.org/works/W4393162623.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W1972469603","https://openalex.org/W1989428971","https://openalex.org/W2046043855","https://openalex.org/W2051697815","https://openalex.org/W2058299490","https://openalex.org/W2061465222","https://openalex.org/W2089860881","https://openalex.org/W2090587397","https://openalex.org/W2104370788","https://openalex.org/W2304480337","https://openalex.org/W2328519477","https://openalex.org/W2332083703","https://openalex.org/W2567044466","https://openalex.org/W2588222758","https://openalex.org/W2921542787","https://openalex.org/W2922185274","https://openalex.org/W3026358768","https://openalex.org/W3048397446","https://openalex.org/W3102976169","https://openalex.org/W3181491506","https://openalex.org/W3211773328","https://openalex.org/W4288081285","https://openalex.org/W4382753768","https://openalex.org/W6731035463","https://openalex.org/W6819088363"],"related_works":["https://openalex.org/W4245687327","https://openalex.org/W2072289333","https://openalex.org/W2112549839","https://openalex.org/W4254637135","https://openalex.org/W2905483137","https://openalex.org/W4288586675","https://openalex.org/W2964155584","https://openalex.org/W2248023582","https://openalex.org/W2754149351","https://openalex.org/W2490469197"],"abstract_inverted_index":{"Lie":[0,40,78,131],"algebra":[1,41,63,79,132],"plays":[2],"an":[3],"important":[4],"role":[5],"in":[6,30],"the":[7,31,58,108,128,139,151,169],"study":[8,51,170],"of":[9,15,33,36,52,60,77,120,130,143,171],"singularity":[10,45,172],"theory":[11,35],"and":[12,70,85,136,162],"other":[13],"fields":[14],"sciences.":[16],"Finding":[17],"numerous":[18],"invariants":[19],"linked":[20],"with":[21],"isolated":[22,37,145],"singularities":[23],"has":[24],"always":[25],"been":[26],"a":[27,47,65,74,87,117,165],"primary":[28],"interest":[29],"field":[32],"classification":[34],"singularities.":[38,146],"Any":[39],"that":[42],"characterizes":[43],"simple":[44],"produces":[46],"natural":[48],"question.":[49],"The":[50],"properties":[53],"such":[54],"as":[55,164],"to":[56,103,123,168],"find":[57],"dimensions":[59],"newly":[61],"defined":[62],"is":[64,111,114],"remarkable":[66],"work.":[67],"Hussain,":[68],"Yau":[69],"Zuo":[71],"have":[72],"found":[73],"new":[75,166],"class":[76],"Lk(V),":[80],"k\u22651,":[81],"i.e.,":[82],"Der":[83],"(Mk(V),Mk(V))":[84],"proposed":[86],"conjecture":[88,142,153],"over":[89],"its":[90],"dimension":[91,129],"\u03b4k(V)":[92],"for":[93,100,116,134,155],"k\u22650.":[94],"Later,":[95],"they":[96],"proved":[97],"it":[98,113],"true":[99,115],"k":[101],"up":[102],"k=1,2,3,4,5.":[104],"In":[105],"this":[106],"work,":[107],"main":[109],"concern":[110],"whether":[112],"higher":[118],"value":[119],"k.":[121],"According":[122],"this,":[124],"we":[125,148],"first":[126],"calculate":[127],"Lk(V)":[133],"k=6":[135],"then":[137],"compute":[138],"upper":[140],"estimate":[141],"fewnomial":[144],"Additionally,":[147],"also":[149],"justify":[150],"inequality":[152],"\u03b4k+1(V)<\u03b4k(V)":[154],"k=6.":[156],"Our":[157],"calculated":[158],"results":[159],"are":[160],"innovative":[161],"serve":[163],"addition":[167],"theory.":[173]},"counts_by_year":[],"updated_date":"2026-06-15T08:34:33.830935","created_date":"2025-10-10T00:00:00"}