{"id":"https://openalex.org/W7225792576","doi":"https://doi.org/10.5281/zenodo.24135749","title":"Three Regimes in the Sharp Quadratic Stability Problem for Dittert's Functional","display_name":"Three Regimes in the Sharp Quadratic Stability Problem for Dittert's Functional","publication_year":2026,"publication_date":"2026-11-11","ids":{"openalex":"https://openalex.org/W7225792576","doi":"https://doi.org/10.5281/zenodo.24135749"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24135749","doi":"https://doi.org/10.5281/zenodo.24135749","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24135749","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24135749","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058111521","display_name":"Jerzy Kaczorowski","orcid":"https://orcid.org/0009-0000-7461-1179","observed_orcids":["https://orcid.org/0009-0000-7461-1179"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kaczorowski, Jaros\u0142aw","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0000-7461-1179","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5058111521"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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":0.836899995803833,"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":0.836899995803833,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.07090000063180923,"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"}},{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.051600001752376556,"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":[],"concepts":[{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.8646000027656555},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8076000213623047},{"id":"https://openalex.org/C62438384","wikidata":"https://www.wikidata.org/wiki/Q331350","display_name":"Simplex","level":2,"score":0.6783000230789185},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5533000230789185},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5291000008583069},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.48339998722076416},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.48260000348091125},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.47850000858306885},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4034000039100647},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3984000086784363},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.3813000023365021},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3709000051021576},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3630000054836273},{"id":"https://openalex.org/C144521790","wikidata":"https://www.wikidata.org/wiki/Q134164","display_name":"Simplex algorithm","level":3,"score":0.36149999499320984},{"id":"https://openalex.org/C85817219","wikidata":"https://www.wikidata.org/wiki/Q884772","display_name":"Block matrix","level":3,"score":0.3449000120162964},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.33390000462532043},{"id":"https://openalex.org/C2983524802","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic model","level":3,"score":0.33219999074935913},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.325300008058548},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32409998774528503},{"id":"https://openalex.org/C73729460","wikidata":"https://www.wikidata.org/wiki/Q17023339","display_name":"Hessian equation","level":4,"score":0.2939000129699707},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.2824000120162964},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2818000018596649},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.263700008392334},{"id":"https://openalex.org/C81845259","wikidata":"https://www.wikidata.org/wiki/Q290117","display_name":"Quadratic programming","level":2,"score":0.26170000433921814},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2554999887943268},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.2517000138759613}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24135749","doi":"https://doi.org/10.5281/zenodo.24135749","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24135749","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.24135750","doi":"https://doi.org/10.5281/zenodo.24135750","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24135750","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.24135749","doi":"https://doi.org/10.5281/zenodo.24135749","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24135749","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"We":[0,120],"study":[1],"the":[2,12,43,52,68,91,107,115,122,130,135,167],"sharp":[3],"global":[4],"quadratic":[5],"stability":[6],"constant":[7],"for":[8,87,149],"Dittert's":[9],"functional":[10],"on":[11],"simplex":[13],"of":[14,51,134,157],"nonnegative":[15],"(n\\times":[16],"n)":[17],"matrices":[18,50],"with":[19,77],"total":[20],"entry":[21],"sum":[22],"(n).":[23],"Writing":[24],"(U_n=J_n/n),":[25],"we":[26],"consider":[27],"[\\kappa_n^*=\\inf_{A\\ne":[28],"U_n}\\frac{\\Phi(U_n)-\\Phi(A)}{|A-U_n|_F^2}.]":[29],"Exact":[30],"witnesses":[31,47],"and":[32,128,152,155],"reproducible":[33],"numerical":[34],"searches":[35],"reveal":[36],"three":[37],"distinct":[38],"low-dimensional":[39],"regimes.":[40],"For":[41,55],"(n=3,4,5),":[42],"best":[44,69],"observed":[45,70],"finite-distance":[46],"are":[48,58,85],"cycle":[49],"form":[53],"((P+Q)/2).":[54],"(n=6,7),":[56],"they":[57],"replaced":[59],"by":[60],"highly":[61],"symmetric":[62],"block":[63],"minimizers.":[64],"From":[65],"(n=8)":[66],"onward,":[67],"branch":[71,137],"is":[72,96,114],"a":[73,78],"nearly":[74],"uniform":[75],"matrix":[76],"single":[79],"zero":[80],"entry.":[81],"Two":[82],"general":[83],"results":[84],"proved":[86],"every":[88],"(n\\ge3).":[89],"First,":[90],"local":[92],"Hessian":[93,153],"scale":[94],"(L_n)":[95],"never":[97],"sharp:":[98],"(\\kappa_n^<L_n).":[99],"Second,":[100],"an":[101],"explicit":[102],"one-dimensional":[103],"one-zero":[104,118],"family":[105],"yields":[106],"strict":[108],"upper":[109,147],"bound[\\kappa_n^\\le":[110],"\\widetilde":[111],"Z_n<Z_n,]where":[112],"(Z_n)":[113],"doubly":[116],"stochastic":[117],"value.":[119],"derive":[121],"asymptotic":[123],"expansion":[124],"(Z_n/\\gamma_n\\to1/2)":[125],"from":[126,138],"below":[127],"quantify":[129],"exponentially":[131],"small":[132],"departure":[133],"optimized":[136],"double":[139],"stochasticity.":[140],"The":[141],"computational":[142],"archive":[143],"provides":[144],"exact":[145],"rational":[146],"bounds":[148],"(n=6,\\ldots,9),":[150],"KKT":[151],"diagnostics,":[154],"comparisons":[156],"seven":[158],"prescribed-zero":[159],"boundary":[160],"patterns":[161],"through":[162],"(n=12).":[163],"Global":[164],"optimality":[165],"beyond":[166],"rigorously":[168],"established":[169],"(n=3)":[170],"case":[171],"remains":[172],"open.":[173]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-11-12T00:00:00"}