{"id":"https://openalex.org/W4214583387","doi":"https://doi.org/10.3390/axioms11030099","title":"The Smallest Singular Value Anomaly and the Condition Number Anomaly","display_name":"The Smallest Singular Value Anomaly and the Condition Number Anomaly","publication_year":2022,"publication_date":"2022-02-25","ids":{"openalex":"https://openalex.org/W4214583387","doi":"https://doi.org/10.3390/axioms11030099"},"language":"en","primary_location":{"id":"doi:10.3390/axioms11030099","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11030099","pdf_url":"https://www.mdpi.com/2075-1680/11/3/99/pdf?version=1645780548","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/2075-1680/11/3/99/pdf?version=1645780548","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5009299181","display_name":"Achiya Dax","orcid":"https://orcid.org/0000-0002-6403-1020"},"institutions":[{"id":"https://openalex.org/I4210100416","display_name":"Israel Water Authority","ror":"https://ror.org/013zszd91","country_code":"IL","type":"government","lineage":["https://openalex.org/I4210100416"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"Achiya Dax","raw_affiliation_strings":["Hydrological Service, P.O. Box 36118, Jerusalem 91360, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hydrological Service, P.O. Box 36118, Jerusalem 91360, Israel","institution_ids":["https://openalex.org/I4210100416"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5009299181"],"corresponding_institution_ids":["https://openalex.org/I4210100416"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":1.2273,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.78831162,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"11","issue":"3","first_page":"99","last_page":"99"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11716","display_name":"Random Matrices and Applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994000196456909,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/row","display_name":"Row","score":0.7733509540557861},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7277477979660034},{"id":"https://openalex.org/keywords/anomaly","display_name":"Anomaly (physics)","score":0.6856119632720947},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.6750794649124146},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.613736093044281},{"id":"https://openalex.org/keywords/row-and-column-spaces","display_name":"Row and column spaces","score":0.5925632119178772},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5769868493080139},{"id":"https://openalex.org/keywords/condition-number","display_name":"Condition number","score":0.5306868553161621},{"id":"https://openalex.org/keywords/random-matrix","display_name":"Random matrix","score":0.5046952962875366},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4936750829219818},{"id":"https://openalex.org/keywords/expected-value","display_name":"Expected value","score":0.4618024230003357},{"id":"https://openalex.org/keywords/value","display_name":"Value (mathematics)","score":0.4383416175842285},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.12912076711654663},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11718598008155823},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08145153522491455}],"concepts":[{"id":"https://openalex.org/C135598885","wikidata":"https://www.wikidata.org/wiki/Q1366302","display_name":"Row","level":2,"score":0.7733509540557861},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7277477979660034},{"id":"https://openalex.org/C12997251","wikidata":"https://www.wikidata.org/wiki/Q567560","display_name":"Anomaly (physics)","level":2,"score":0.6856119632720947},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.6750794649124146},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.613736093044281},{"id":"https://openalex.org/C104140500","wikidata":"https://www.wikidata.org/wiki/Q2088159","display_name":"Row and column spaces","level":3,"score":0.5925632119178772},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5769868493080139},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.5306868553161621},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.5046952962875366},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4936750829219818},{"id":"https://openalex.org/C141042865","wikidata":"https://www.wikidata.org/wiki/Q200125","display_name":"Expected value","level":2,"score":0.4618024230003357},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.4383416175842285},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.12912076711654663},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11718598008155823},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08145153522491455},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C26873012","wikidata":"https://www.wikidata.org/wiki/Q214781","display_name":"Condensed matter physics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/axioms11030099","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11030099","pdf_url":"https://www.mdpi.com/2075-1680/11/3/99/pdf?version=1645780548","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:3f21bbaa9e6244c88f18806240eedc19","is_oa":false,"landing_page_url":"https://doaj.org/article/3f21bbaa9e6244c88f18806240eedc19","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 11, Iss 3, p 99 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/11/3/99/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms11030099","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; Volume 11; Issue 3; Pages: 99","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms11030099","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11030099","pdf_url":"https://www.mdpi.com/2075-1680/11/3/99/pdf?version=1645780548","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":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4214583387.pdf","grobid_xml":"https://content.openalex.org/works/W4214583387.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W391578156","https://openalex.org/W1488435683","https://openalex.org/W1506342804","https://openalex.org/W1974755392","https://openalex.org/W1976576957","https://openalex.org/W1979857262","https://openalex.org/W1981573335","https://openalex.org/W2000748685","https://openalex.org/W2018233872","https://openalex.org/W2024978582","https://openalex.org/W2038118478","https://openalex.org/W2060581589","https://openalex.org/W2087284858","https://openalex.org/W2102988735","https://openalex.org/W2130076109","https://openalex.org/W2168494960","https://openalex.org/W2261576411","https://openalex.org/W2963750720","https://openalex.org/W4301491118","https://openalex.org/W4312258136","https://openalex.org/W4320800818"],"related_works":["https://openalex.org/W4321517655","https://openalex.org/W4287867775","https://openalex.org/W3005625810","https://openalex.org/W4249386048","https://openalex.org/W2127210545","https://openalex.org/W2036965437","https://openalex.org/W3017172632","https://openalex.org/W4287815178","https://openalex.org/W4214583387","https://openalex.org/W3133613038"],"abstract_inverted_index":{"Let":[0,22,35],"A":[1,163],"be":[2,26,125],"an":[3],"arbitrary":[4],"matrix":[5],"in":[6,182,189,210,234,258],"which":[7,123,144],"the":[8,17,23,29,38,48,58,61,69,87,105,134,160,168,172,176,179,211,226,259,271],"number":[9,18,50,130,232,244,251,260],"of":[10,19,28,33,42,51,60,68,80,104,115,142,162,178,193,201,206,213,238,261],"rows,":[11],"m,":[12],"is":[13,74,84,91,95,145,151,155],"considerably":[14,146],"larger":[15],"than":[16,148],"columns,":[20],"n.":[21],"submatrix":[24],"Ai,i=1,\u2026,m,":[25],"composed":[27],"first":[30,78,113],"i":[31],"rows":[32,161,169,180],"A.":[34],"\u03b2i":[36],"denote":[37,47],"smallest":[39,70,98],"singular":[40,71,99],"value":[41,100,141],"Ai,":[43],"and":[44,64,175],"let":[45],"ki":[46],"condition":[49,106,129,231,243,250],"Ai.":[52],"In":[53],"this":[54,81,116,154],"paper,":[55],"we":[56],"examine":[57],"behavior":[59,67],"sequences":[62],"\u03b21,\u2026,\u03b2m,":[63],"k1,\u2026,km.":[65],"The":[66,77,102,112,128,203,263],"values":[72],"sequence":[73,103],"somewhat":[75],"surprising.":[76],"part":[79,114],"sequence,":[82,117],"\u03b21,\u2026,\u03b2n,":[83],"descending,":[85],"while":[86,240],"second":[88,135],"part,":[89,136],"\u03b2n,\u2026,\u03b2m,":[90],"ascending.":[92],"This":[93],"phenomenon":[94],"called":[96],"\u201cthe":[97],"anomaly\u201d.":[101],"numbers":[107],"has":[108],"a":[109,140,190,229,235,241,249,255],"different":[110],"character.":[111],"k1,\u2026,kn,":[118],"always":[119],"ascends":[120],"toward":[121,139],"kn,":[122],"can":[124],"very":[126],"large.":[127],"anomaly":[131,252,257],"occurs":[132],"when":[133],"kn,\u2026,km,":[137],"descends":[138],"km,":[143],"smaller":[147],"kn.":[149],"It":[150],"shown":[152],"that":[153,228,269],"likely":[156],"to":[157,254],"happen":[158],"whenever":[159],"satisfy":[164],"two":[165],"conditions:":[166],"all":[167],"are":[170],"about":[171],"same":[173],"size,":[174],"directions":[177],"scatter":[181],"some":[183],"random":[184,194],"way.":[185],"These":[186],"conditions":[187],"hold":[188],"wide":[191],"range":[192],"matrices,":[195],"as":[196,198],"well":[197],"other":[199],"types":[200],"matrices.":[202],"practical":[204],"importance":[205],"these":[207],"phenomena":[208],"lies":[209],"use":[212],"iterative":[214,223],"methods":[215],"for":[216],"solving":[217],"large":[218,230],"linear":[219],"systems,":[220],"since":[221],"several":[222],"solvers":[224],"have":[225],"property":[227],"results":[233],"slow":[236],"rate":[237],"convergence,":[239],"small":[242],"yields":[245],"fast":[246],"convergence.":[247],"Consequently,":[248],"leads":[253],"similar":[256],"iterations.":[262],"paper":[264],"ends":[265],"with":[266],"numerical":[267],"experiments":[268],"illustrate":[270],"above":[272],"phenomena.":[273]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
