{"id":"https://openalex.org/W7214710616","doi":"https://doi.org/10.48550/arxiv.2609.35558","title":"Entrywise nonnegativity of the inverse relative gain array in dimension six","display_name":"Entrywise nonnegativity of the inverse relative gain array in dimension six","publication_year":2026,"publication_date":"2026-09-28","ids":{"openalex":"https://openalex.org/W7214710616","doi":"https://doi.org/10.48550/arxiv.2609.35558"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.35558","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35558","pdf_url":"https://arxiv.org/pdf/2609.35558","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.35558","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5123352847","display_name":"Jeffrey Uhlmann","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Uhlmann, Jeffrey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5159279843","display_name":"Jie Wang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Jie","raw_affiliation_strings":["State Key Laboratory of Mathematical Sciences , Academy of Mathematics and Systems Science , Chinese Academy of Sciences , Beijing , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"State Key Laboratory of Mathematical Sciences , Academy of Mathematics and Systems Science , Chinese Academy of Sciences , Beijing , China","institution_ids":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.6484000086784363,"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":0.6484000086784363,"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.3379000127315521,"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"}},{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.0038999998942017555,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/symmetric-positive-definite-matrices","display_name":"symmetric positive definite matrices","score":0.8230000138282776},{"id":"https://openalex.org/keywords/matrix-inverse","display_name":"matrix inverse","score":0.4580000042915344},{"id":"https://openalex.org/keywords/doubly-stochastic-matrices","display_name":"doubly stochastic matrices","score":0.7149999737739563}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8683000206947327},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.7422999739646912},{"id":"https://openalex.org/C2777818330","wikidata":"https://www.wikidata.org/wiki/Q1994695","display_name":"Majorization","level":2,"score":0.6873000264167786},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6567999720573425},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.6333000063896179},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5436000227928162},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5267000198364258},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5189999938011169},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.459199994802475},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.42410001158714294},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4066999852657318},{"id":"https://openalex.org/C134567657","wikidata":"https://www.wikidata.org/wiki/Q193794","display_name":"Identity matrix","level":3,"score":0.3944000005722046},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.37709999084472656},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.34299999475479126},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33320000767707825},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.32249999046325684},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.32019999623298645},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.3084999918937683},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.30379998683929443},{"id":"https://openalex.org/C139018669","wikidata":"https://www.wikidata.org/wiki/Q6961560","display_name":"Nonnegative matrix","level":4,"score":0.29910001158714294},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2957000136375427},{"id":"https://openalex.org/C39757861","wikidata":"https://www.wikidata.org/wiki/Q1395814","display_name":"Fano plane","level":2,"score":0.28369998931884766},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.2831999957561493},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.27639999985694885},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.2754000127315521}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.35558","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35558","pdf_url":"https://arxiv.org/pdf/2609.35558","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.35558","doi":"https://doi.org/10.48550/arxiv.2609.35558","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.35558","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.35558","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35558","pdf_url":"https://arxiv.org/pdf/2609.35558","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3356097721","display_name":null,"funder_award_id":"12171324","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7864953252","display_name":null,"funder_award_id":"12201618","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214710616.pdf","grobid_xml":"https://content.openalex.org/works/W7214710616.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2947399339","https://openalex.org/W4400840611"],"related_works":["https://openalex.org/W2054014428","https://openalex.org/W1540662194","https://openalex.org/W6977887123","https://openalex.org/W1985015255","https://openalex.org/W2117986650","https://openalex.org/W2072259116","https://openalex.org/W7095470859","https://openalex.org/W4362721336","https://openalex.org/W2065309551","https://openalex.org/W2566999989"],"abstract_inverted_index":{"We":[0,46],"prove":[1],"that":[2],"$(G\\circ":[3],"G^{-1})^{-1}$":[4],"is":[5],"entrywise":[6],"nonnegative":[7],"for":[8],"every":[9,131],"real":[10],"symmetric":[11],"positive":[12,25,62,92],"definite":[13],"matrix":[14],"$G$":[15],"of":[16,31,35,41,61,136],"order":[17,42],"at":[18],"most":[19],"six.":[20],"Its":[21],"entries":[22],"are":[23,142],"strictly":[24],"exactly":[26],"within":[27],"the":[28,32,115,125,137],"connected":[29],"components":[30],"nonzero-entry":[33],"graph":[34],"$G$.":[36],"An":[37],"explicit":[38,80],"rational":[39],"example":[40],"seven":[43],"establishes":[44,72],"sharpness.":[45],"give":[47],"two":[48],"complementary":[49],"structural":[50],"proofs.":[51],"The":[52,69],"analytic":[53],"proof":[54,71],"combines":[55],"a":[56,66,73,119],"trace":[57],"obstruction":[58],"with":[59,114,124],"continuation":[60],"column":[63,90],"forms":[64],"and":[65,96,118],"bordering":[67],"identity.":[68],"algebraic":[70],"stronger":[74],"four-dimensional":[75],"polarized":[76],"inequality":[77,123],"through":[78],"an":[79],"ten-square":[81],"polynomial":[82,111],"certificate.":[83],"A":[84],"common":[85],"framework":[86],"relates":[87],"biquadratic":[88],"positivity,":[89],"forms,":[91],"semidefinite":[93],"precision":[94],"loads,":[95],"inverse":[97,138],"relative":[98,139],"gain":[99,140],"array":[100,141],"nonnegativity":[101],"in":[102],"consecutive":[103],"dimensions.":[104],"Further":[105],"consequences":[106],"include":[107],"exact":[108],"load-deficit":[109],"identities,":[110],"sum-of-squares":[112],"certificates":[113],"covariance":[116],"fixed,":[117],"reverse":[120],"Schur--Horn":[121],"majorization":[122],"same":[126],"sharp":[127],"dimension":[128],"bound.":[129],"In":[130],"dimension,":[132],"sufficiently":[133],"large":[134],"powers":[135],"doubly":[143],"stochastic.":[144]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-30T00:00:00"}
