{"id":"https://openalex.org/W2777081909","doi":"https://doi.org/10.1080/00029890.2018.1392737","title":"Preimages of the Numerical Range","display_name":"Preimages of the Numerical Range","publication_year":2017,"publication_date":"2017-12-21","ids":{"openalex":"https://openalex.org/W2777081909","doi":"https://doi.org/10.1080/00029890.2018.1392737","mag":"2777081909"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2018.1392737","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1392737","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5042837801","display_name":"John H. Clifford","orcid":null},"institutions":[{"id":"https://openalex.org/I4210130704","display_name":"University of Michigan\u2013Dearborn","ror":"https://ror.org/035wtm547","country_code":"US","type":"education","lineage":["https://openalex.org/I4210130704"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"John Clifford","raw_affiliation_strings":["Department of Mathematics &amp; Statistics, University of Michigan \u2013 Dearborn, Dearborn, MI, 48128"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics &amp; Statistics, University of Michigan \u2013 Dearborn, Dearborn, MI, 48128","institution_ids":["https://openalex.org/I4210130704"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057220826","display_name":"Kelly Jabbusch","orcid":null},"institutions":[{"id":"https://openalex.org/I4210130704","display_name":"University of Michigan\u2013Dearborn","ror":"https://ror.org/035wtm547","country_code":"US","type":"education","lineage":["https://openalex.org/I4210130704"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kelly Jabbusch","raw_affiliation_strings":["Department of Mathematics &amp; Statistics, University of Michigan \u2013 Dearborn, Dearborn, MI, 48128"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics &amp; Statistics, University of Michigan \u2013 Dearborn, Dearborn, MI, 48128","institution_ids":["https://openalex.org/I4210130704"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043259417","display_name":"Micha\u00ebl Lachance","orcid":null},"institutions":[{"id":"https://openalex.org/I4210130704","display_name":"University of Michigan\u2013Dearborn","ror":"https://ror.org/035wtm547","country_code":"US","type":"education","lineage":["https://openalex.org/I4210130704"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael Lachance","raw_affiliation_strings":["Department of Mathematics &amp; Statistics, University of Michigan \u2013 Dearborn, Dearborn, MI, 48128"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics &amp; Statistics, University of Michigan \u2013 Dearborn, Dearborn, MI, 48128","institution_ids":["https://openalex.org/I4210130704"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210130704"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.21766608,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"125","issue":"1","first_page":"69","last_page":"74"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9995999932289124,"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":0.9995999932289124,"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/T11673","display_name":"Advanced Topics in Algebra","score":0.9983000159263611,"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/T10884","display_name":"Holomorphic and Operator Theory","score":0.998199999332428,"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/bijection","display_name":"Bijection","score":0.8785379528999329},{"id":"https://openalex.org/keywords/unit-sphere","display_name":"Unit sphere","score":0.738743245601654},{"id":"https://openalex.org/keywords/numerical-range","display_name":"Numerical range","score":0.6708942651748657},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6551809310913086},{"id":"https://openalex.org/keywords/unit","display_name":"Unit (ring theory)","score":0.6346753239631653},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.592503547668457},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5193882584571838},{"id":"https://openalex.org/keywords/disjoint-union","display_name":"Disjoint union (topology)","score":0.49445438385009766},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49323076009750366},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.43719664216041565},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3756451904773712},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3454573452472687},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.07629457116127014},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.07275885343551636}],"concepts":[{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.8785379528999329},{"id":"https://openalex.org/C191948623","wikidata":"https://www.wikidata.org/wiki/Q1307800","display_name":"Unit sphere","level":2,"score":0.738743245601654},{"id":"https://openalex.org/C193060811","wikidata":"https://www.wikidata.org/wiki/Q1579053","display_name":"Numerical range","level":2,"score":0.6708942651748657},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6551809310913086},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.6346753239631653},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.592503547668457},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5193882584571838},{"id":"https://openalex.org/C7655956","wikidata":"https://www.wikidata.org/wiki/Q5282259","display_name":"Disjoint union (topology)","level":2,"score":0.49445438385009766},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49323076009750366},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.43719664216041565},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3756451904773712},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3454573452472687},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.07629457116127014},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.07275885343551636},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.0},{"id":"https://openalex.org/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00029890.2018.1392737","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1392737","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547","https://openalex.org/P4310320449"],"host_organization_lineage_names":["Taylor & Francis","Informa"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W141863670","https://openalex.org/W1488435683","https://openalex.org/W1980557251","https://openalex.org/W2328205117","https://openalex.org/W2330597313","https://openalex.org/W4205158050"],"related_works":["https://openalex.org/W55187247","https://openalex.org/W2058834578","https://openalex.org/W1891636518","https://openalex.org/W264408731","https://openalex.org/W2073363580","https://openalex.org/W411819533","https://openalex.org/W2067430117","https://openalex.org/W4312202183","https://openalex.org/W2004734763","https://openalex.org/W1972516737"],"abstract_inverted_index":{"For":[0],"a":[1,23],"nonnormal":[2],"two-by-two":[3],"matrix":[4],"B,":[5],"there":[6],"exist":[7],"two":[8,42],"open":[9],"disjoint":[10],"subsets":[11],"of":[12,28,32,37,40],"the":[13,19,26,29,35,38,48],"complex":[14],"unit":[15,50],"sphere":[16],"from":[17],"which":[18],"mapping":[20],"z*Bz":[21],"is":[22],"bijection":[24],"onto":[25],"interior":[27],"numerical":[30],"range":[31],"B.":[33],"Moreover,":[34],"union":[36],"closures":[39],"these":[41],"subsets,":[43],"up":[44],"to":[45],"rotation,":[46],"yields":[47],"entire":[49],"sphere.":[51]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-08-15T07:11:24.734988","created_date":"2025-10-10T00:00:00"}
