{"id":"https://openalex.org/W7214020154","doi":"https://doi.org/10.48550/arxiv.2609.24886","title":"Minimal Products via Weighted Spheres","display_name":"Minimal Products via Weighted Spheres","publication_year":2026,"publication_date":"2026-09-21","ids":{"openalex":"https://openalex.org/W7214020154","doi":"https://doi.org/10.48550/arxiv.2609.24886"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.24886","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24886","pdf_url":"https://arxiv.org/pdf/2609.24886","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/2609.24886","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5152489844","display_name":"Yongsheng Zhang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhang, Yongsheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5152489844"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9854000210762024,"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"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9854000210762024,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.012500000186264515,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/composition-principles","display_name":"compositional principles","score":0.7519999742507935},{"id":"https://openalex.org/keywords/self-shrinkers","display_name":"self-shrinkers","score":0.8610000014305115},{"id":"https://openalex.org/keywords/lusternik-schnirelmann-category","display_name":"Lusternik-Schnirelmann category","score":0.9890000224113464}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.777400016784668},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.6351000070571899},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.6284000277519226},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.4957999885082245},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.45680001378059387},{"id":"https://openalex.org/C2776175482","wikidata":"https://www.wikidata.org/wiki/Q1195816","display_name":"Transfer (computing)","level":2,"score":0.44909998774528503},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.42570000886917114},{"id":"https://openalex.org/C4530962","wikidata":"https://www.wikidata.org/wiki/Q777407","display_name":"Closed-form expression","level":2,"score":0.42149999737739563},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41749998927116394},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.3495999872684479},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34790000319480896},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.33079999685287476},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.32739999890327454},{"id":"https://openalex.org/C202887219","wikidata":"https://www.wikidata.org/wiki/Q3895221","display_name":"Parametrization (atmospheric modeling)","level":3,"score":0.30320000648498535},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.2980000078678131},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.29319998621940613},{"id":"https://openalex.org/C109786386","wikidata":"https://www.wikidata.org/wiki/Q861804","display_name":"Bilinear transform","level":4,"score":0.290800005197525},{"id":"https://openalex.org/C139505817","wikidata":"https://www.wikidata.org/wiki/Q5527834","display_name":"Gaussian binomial coefficient","level":4,"score":0.28349998593330383},{"id":"https://openalex.org/C168167062","wikidata":"https://www.wikidata.org/wiki/Q1117970","display_name":"Component (thermodynamics)","level":2,"score":0.26579999923706055}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.24886","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24886","pdf_url":"https://arxiv.org/pdf/2609.24886","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.2609.24886","doi":"https://doi.org/10.48550/arxiv.2609.24886","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.24886","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.24886","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.24886","pdf_url":"https://arxiv.org/pdf/2609.24886","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214020154.pdf","grobid_xml":"https://content.openalex.org/works/W7214020154.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W7154390575","https://openalex.org/W7172404526"],"related_works":["https://openalex.org/W2033334413","https://openalex.org/W2034208378","https://openalex.org/W1567571844","https://openalex.org/W7010143100","https://openalex.org/W2045804816","https://openalex.org/W2151391197","https://openalex.org/W1573415141","https://openalex.org/W2049355082","https://openalex.org/W7099900676","https://openalex.org/W2031085082"],"abstract_inverted_index":{"We":[0],"establish":[1],"an":[2],"exact":[3],"composition":[4,34],"principle":[5],"for":[6,50],"spherical":[7,22],"immersions":[8],"coupled":[9],"through":[10],"coefficient":[11,95],"profiles":[12,59,96,111],"and":[13,30,38,45,72],"norm-preserving":[14],"bilinear":[15],"maps.":[16],"Under":[17],"explicit":[18],"mixed":[19],"orthogonality,":[20],"the":[21,46,51,81,116,120],"mean":[23],"curvature":[24],"vector":[25],"splits":[26],"into":[27],"orthogonal":[28],"input":[29,42],"profile":[31,47],"contributions.":[32],"The":[33,106],"is":[35,43,48,112],"minimal":[36,44],"if":[37,40],"only":[39],"every":[41],"$W$-minimal":[49,78],"induced":[52],"monomial":[53],"weight.":[54],"Tensor":[55],"iteration":[56],"allows":[57],"weighted":[58],"to":[60,85],"serve":[61],"as":[62],"inputs":[63],"in":[64,87],"further":[65],"compositions.":[66],"At":[67],"fixed":[68],"weights,":[69],"Hsiang--Lawson":[70],"reduction":[71],"Kapouleas--McGrath":[73],"doubling":[74],"yield":[75],"closed":[76,98],"embedded":[77,99],"profiles.":[79],"When":[80],"effective":[82],"exponents":[83],"tend":[84],"infinity":[86],"positive":[88],"limiting":[89],"proportions,":[90],"Gaussian":[91],"transfer":[92],"produces":[93],"interior":[94],"from":[97],"self-shrinkers":[100],"that":[101],"are":[102],"nondegenerate":[103],"modulo":[104],"rotations.":[105],"number":[107],"of":[108,119],"distinct":[109],"corresponding":[110],"bounded":[113],"below":[114],"by":[115],"Lusternik--Schnirelmann":[117],"category":[118],"rotation":[121],"orbit.":[122]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-09-23T00:00:00"}