{"id":"https://openalex.org/W7220954070","doi":"https://doi.org/10.48550/arxiv.2610.08470","title":"A symmetric counterexample to Strang's conjecture for bivariate $C^1$ cubic splines on triangulations","display_name":"A symmetric counterexample to Strang's conjecture for bivariate $C^1$ cubic splines on triangulations","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220954070","doi":"https://doi.org/10.48550/arxiv.2610.08470"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08470","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08470","pdf_url":"https://arxiv.org/pdf/2610.08470","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/2610.08470","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5115458777","display_name":"Pratyush Potu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I27804330","display_name":"Brown University","ror":"https://ror.org/05gq02987","country_code":"US","type":"education","lineage":["https://openalex.org/I27804330"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Potu, Pratyush","raw_affiliation_strings":["Division of Applied Mathematics , Brown University , Providence , RI 02912 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Division of Applied Mathematics , Brown University , Providence , RI 02912 , USA","institution_ids":["https://openalex.org/I27804330"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5115458777"],"corresponding_institution_ids":["https://openalex.org/I27804330"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.38749998807907104,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.38749998807907104,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.18379999697208405,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.12680000066757202,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/c1-splines","display_name":"C1 splines","score":0.7590000033378601},{"id":"https://openalex.org/keywords/spline-spaces","display_name":"spline spaces","score":0.5130000114440918},{"id":"https://openalex.org/keywords/dimension-formula","display_name":"dimension formula","score":0.6759999990463257},{"id":"https://openalex.org/keywords/counterexample","display_name":"counterexample","score":0.9319999814033508},{"id":"https://openalex.org/keywords/triangulation","display_name":"triangulation","score":0.8759999871253967},{"id":"https://openalex.org/keywords/equilateral-triangle","display_name":"equilateral triangle","score":0.9860000014305115},{"id":"https://openalex.org/keywords/d3-symmetry","display_name":"D3 symmetry","score":0.6959999799728394},{"id":"https://openalex.org/keywords/isometry-groups","display_name":"isometry groups","score":0.5479999780654907},{"id":"https://openalex.org/keywords/multicollinearity","display_name":"multicollinearity","score":0.5289999842643738}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.842199981212616},{"id":"https://openalex.org/C146786541","wikidata":"https://www.wikidata.org/wiki/Q157002","display_name":"Equilateral triangle","level":2,"score":0.7139000296592712},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6735000014305115},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.6330000162124634},{"id":"https://openalex.org/C135981907","wikidata":"https://www.wikidata.org/wiki/Q188056","display_name":"Triangulation","level":2,"score":0.5860999822616577},{"id":"https://openalex.org/C178609930","wikidata":"https://www.wikidata.org/wiki/Q3538712","display_name":"Pitteway triangulation","level":3,"score":0.5544000267982483},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5200999975204468},{"id":"https://openalex.org/C68010082","wikidata":"https://www.wikidata.org/wiki/Q192445","display_name":"Delaunay triangulation","level":2,"score":0.46709999442100525},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.45649999380111694},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4408999979496002},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.41839998960494995},{"id":"https://openalex.org/C64341305","wikidata":"https://www.wikidata.org/wiki/Q4919225","display_name":"Bivariate analysis","level":2,"score":0.41179999709129333},{"id":"https://openalex.org/C167117609","wikidata":"https://www.wikidata.org/wiki/Q17078161","display_name":"Point set triangulation","level":3,"score":0.383899986743927},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.3637999892234802},{"id":"https://openalex.org/C84160026","wikidata":"https://www.wikidata.org/wiki/Q2211166","display_name":"Isometry group","level":2,"score":0.3472999930381775},{"id":"https://openalex.org/C206352148","wikidata":"https://www.wikidata.org/wiki/Q209306","display_name":"Hyperbolic geometry","level":3,"score":0.33160001039505005},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3246999979019165},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.3043999969959259},{"id":"https://openalex.org/C105313907","wikidata":"https://www.wikidata.org/wiki/Q1053857","display_name":"Dual polyhedron","level":2,"score":0.29829999804496765},{"id":"https://openalex.org/C98653994","wikidata":"https://www.wikidata.org/wiki/Q875937","display_name":"Isosceles triangle","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29660001397132874},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.29190000891685486},{"id":"https://openalex.org/C143669375","wikidata":"https://www.wikidata.org/wiki/Q7874246","display_name":"Coxeter group","level":2,"score":0.27390000224113464},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.26499998569488525},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.2515000104904175},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.250900000333786},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.2508000135421753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08470","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08470","pdf_url":"https://arxiv.org/pdf/2610.08470","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.2610.08470","doi":"https://doi.org/10.48550/arxiv.2610.08470","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08470","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:2610.08470","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08470","pdf_url":"https://arxiv.org/pdf/2610.08470","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/W7220954070.pdf","grobid_xml":"https://content.openalex.org/works/W7220954070.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W7204254521","https://openalex.org/W7207574047"],"related_works":["https://openalex.org/W88062295","https://openalex.org/W7095394030","https://openalex.org/W2624684107","https://openalex.org/W7212074696","https://openalex.org/W7207574047","https://openalex.org/W2074922035","https://openalex.org/W1986614163","https://openalex.org/W2043907736","https://openalex.org/W2023659726","https://openalex.org/W2386193515"],"abstract_inverted_index":{"We":[0],"exhibit":[1],"a":[2,37],"triangulation":[3,29,43],"of":[4,12,49,54],"an":[5],"equilateral":[6,56],"triangle":[7,57],"for":[8],"which":[9,58],"the":[10,21,28,42,47,50,55],"space":[11],"bivariate":[13],"$C^1$":[14],"cubic":[15],"splines":[16],"has":[17],"dimension":[18,22],"larger":[19],"than":[20],"formula":[23],"conjectured":[24],"by":[25],"Strang.":[26],"Notably,":[27],"is":[30,44,59],"such":[31],"that":[32],"no":[33],"two":[34],"edges":[35],"sharing":[36],"vertex":[38],"are":[39],"collinear,":[40],"and":[41],"invariant":[45],"under":[46],"action":[48],"isometry":[51],"group":[52],"($D_3$)":[53],"triangulated.":[60]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}
