{"id":"https://openalex.org/W1551773547","doi":"https://doi.org/10.1137/15m1022653","title":"Remarks on the Symmetric Rank of Symmetric Tensors","display_name":"Remarks on the Symmetric Rank of Symmetric Tensors","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W1551773547","doi":"https://doi.org/10.1137/15m1022653","mag":"1551773547"},"language":"en","primary_location":{"id":"doi:10.1137/15m1022653","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1022653","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://figshare.com/articles/journal_contribution/Remarks_On_The_Symmetric_Rank_Of_Symmetric_Tensors/10772993","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002855648","display_name":"Shmuel Friedland","orcid":"https://orcid.org/0000-0002-0423-5005"},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Shmuel Friedland","raw_affiliation_strings":["Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045","institution_ids":["https://openalex.org/I39422238"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5002855648"],"corresponding_institution_ids":["https://openalex.org/I39422238"],"apc_list":null,"apc_paid":null,"fwci":4.1741,"has_fulltext":false,"cited_by_count":50,"citation_normalized_percentile":{"value":0.95052474,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"37","issue":"1","first_page":"320","last_page":"337"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":1.0,"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/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9696999788284302,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9487000107765198,"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/rank","display_name":"Rank (graph theory)","score":0.8651454448699951},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7677037715911865},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5978115200996399},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.5267536044120789},{"id":"https://openalex.org/keywords/symmetric-tensor","display_name":"Symmetric tensor","score":0.5098134875297546},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4246135354042053},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12322542071342468},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1177072525024414},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.05475282669067383}],"concepts":[{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.8651454448699951},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7677037715911865},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5978115200996399},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.5267536044120789},{"id":"https://openalex.org/C20178491","wikidata":"https://www.wikidata.org/wiki/Q2204117","display_name":"Symmetric tensor","level":3,"score":0.5098134875297546},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4246135354042053},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12322542071342468},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1177072525024414},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.05475282669067383},{"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/15m1022653","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1022653","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:figshare.com:article/10772993","is_oa":true,"landing_page_url":"https://figshare.com/articles/journal_contribution/Remarks_On_The_Symmetric_Rank_Of_Symmetric_Tensors/10772993","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"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":"","raw_type":"Text"}],"best_oa_location":{"id":"pmh:oai:figshare.com:article/10772993","is_oa":true,"landing_page_url":"https://figshare.com/articles/journal_contribution/Remarks_On_The_Symmetric_Rank_Of_Symmetric_Tensors/10772993","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"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":"","raw_type":"Text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1361267407","display_name":"Computational problems in tensors and quantum information theory","funder_award_id":"1216393","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G1872689415","display_name":null,"funder_award_id":"DMS-1216393","funder_id":"https://openalex.org/F4320332172","funder_display_name":"Directorate for Mathematical and Physical Sciences"},{"id":"https://openalex.org/G7604122832","display_name":null,"funder_award_id":"DMS-1216393","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332172","display_name":"Directorate for Mathematical and Physical Sciences","ror":"https://ror.org/029b7h395"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W802720919","https://openalex.org/W1548129254","https://openalex.org/W1599023517","https://openalex.org/W1871447361","https://openalex.org/W1992223193","https://openalex.org/W2000067655","https://openalex.org/W2020062309","https://openalex.org/W2025120643","https://openalex.org/W2039748980","https://openalex.org/W2057503509","https://openalex.org/W2070249148","https://openalex.org/W2114613785","https://openalex.org/W2118065075","https://openalex.org/W2127585680","https://openalex.org/W2258781388","https://openalex.org/W2264065105","https://openalex.org/W2499634235","https://openalex.org/W2787233150","https://openalex.org/W2962827512","https://openalex.org/W2963262648","https://openalex.org/W2964227090","https://openalex.org/W3098420274","https://openalex.org/W3098477899","https://openalex.org/W3099166182","https://openalex.org/W3104141160","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W3172166127","https://openalex.org/W2170114491","https://openalex.org/W2242624680","https://openalex.org/W2950186459","https://openalex.org/W2781510240","https://openalex.org/W2897298721","https://openalex.org/W2569661359","https://openalex.org/W2216309014","https://openalex.org/W2952474002","https://openalex.org/W4301967441"],"abstract_inverted_index":{"We":[0,61],"give":[1],"sufficient":[2],"conditions":[3],"on":[4],"a":[5,45],"symmetric":[6,15,76],"tensor":[7],"$\\mathcal{S}\\in\\mathrm{S}^d\\mathbb{F}^n$":[8],"to":[9,24],"satisfy":[10],"the":[11,14,25,37,40,58,63,66],"following":[12],"equality:":[13],"rank":[16,26,38,71],"of":[17,27,39,65,75],"$\\mathcal{S}$,":[18,28],"denoted":[19,29],"as":[20,30,44],"$\\mathrm{srank\\;}\\mathcal{S}$,":[21],"is":[22,33,50],"equal":[23],"$\\mathrm{rank\\;}\\mathcal{S}$.":[31],"This":[32],"done":[34],"by":[35],"considering":[36],"unfolded":[41],"$\\mathcal{S}$":[42],"viewed":[43],"matrix":[46],"$A(\\mathcal{S})$.":[47],"The":[48],"condition":[49],"$\\mathrm{rank\\;}\\mathcal{S}\\in\\{\\mathrm{rank\\;}A(\\mathcal{S}),\\mathrm{rank\\;}A(\\mathcal{S})+1\\}$.":[51],"In":[52],"particular,":[53],"$\\mathrm{srank\\;}\\mathcal{S}=\\mathrm{rank\\;}\\mathcal{S}$":[54],"for":[55,57,69],"$\\mathcal{S}\\in\\mathrm{S}^d\\mathbb{C}^n$":[56],"cases":[59],"$(d,n)\\in\\{(3,2),(4,2),(3,3)\\}$.":[60],"discuss":[62],"analogues":[64],"above":[67],"results":[68],"border":[70],"and":[72],"best":[73],"approximations":[74],"tensors.":[77]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":11},{"year":2017,"cited_by_count":10},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
