{"id":"https://openalex.org/W2884239031","doi":"https://doi.org/10.1007/s00454-025-00816-4","title":"Fixed-Parameter Extrapolation and Aperiodic Order","display_name":"Fixed-Parameter Extrapolation and Aperiodic Order","publication_year":2018,"publication_date":"2018-10-24","ids":{"openalex":"https://openalex.org/W2884239031","doi":"https://doi.org/10.1007/s00454-025-00816-4","mag":"2884239031"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-025-00816-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-025-00816-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00816-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00816-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111848148","display_name":"Stephen Fenner","orcid":null},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Stephen Fenner","raw_affiliation_strings":["Computer Science and Engineering Department, University of South Carolina, Columbia, SC, 29208, USA","UNIVERSITY OF SOUTH CAROLINA"],"raw_orcid":"https://orcid.org/0000-0002-6028-9323","affiliations":[{"raw_affiliation_string":"Computer Science and Engineering Department, University of South Carolina, Columbia, SC, 29208, USA","institution_ids":["https://openalex.org/I155781252"]},{"raw_affiliation_string":"UNIVERSITY OF SOUTH CAROLINA","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028130995","display_name":"Frederic Green","orcid":null},"institutions":[{"id":"https://openalex.org/I130785548","display_name":"Clark University","ror":"https://ror.org/04123ky43","country_code":"US","type":"education","lineage":["https://openalex.org/I130785548"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Frederic Green","raw_affiliation_strings":["Departments of Mathematics and Computer Science, Clark University, Worcester, MA, 01610, USA","Clark Univ"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departments of Mathematics and Computer Science, Clark University, Worcester, MA, 01610, USA","institution_ids":["https://openalex.org/I130785548"]},{"raw_affiliation_string":"Clark Univ","institution_ids":["https://openalex.org/I130785548"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5012415648","display_name":"Steven Homer","orcid":"https://orcid.org/0000-0002-1515-1093"},"institutions":[{"id":"https://openalex.org/I111088046","display_name":"Boston University","ror":"https://ror.org/05qwgg493","country_code":"US","type":"education","lineage":["https://openalex.org/I111088046"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Steven Homer","raw_affiliation_strings":["Computer Science Department, Boston University, Boston, MA, 02215, USA","Boston Univ"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Computer Science Department, Boston University, Boston, MA, 02215, USA","institution_ids":["https://openalex.org/I111088046"]},{"raw_affiliation_string":"Boston Univ","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5111848148"],"corresponding_institution_ids":["https://openalex.org/I155781252"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.12120928,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"49","issue":"1","first_page":"1","last_page":"47"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9977999925613403,"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/T11567","display_name":"semigroups and automata theory","score":0.9977999925613403,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9944999814033508,"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"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9940999746322632,"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/lambda","display_name":"Lambda","score":0.7716686129570007},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.664762020111084},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5926995277404785},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5335543751716614},{"id":"https://openalex.org/keywords/aperiodic-graph","display_name":"Aperiodic graph","score":0.505418062210083},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.49172404408454895},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.47644928097724915},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.4493017792701721},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.2094046175479889},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2057364284992218},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13715553283691406}],"concepts":[{"id":"https://openalex.org/C2778113609","wikidata":"https://www.wikidata.org/wiki/Q10897","display_name":"Lambda","level":2,"score":0.7716686129570007},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.664762020111084},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5926995277404785},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5335543751716614},{"id":"https://openalex.org/C104247578","wikidata":"https://www.wikidata.org/wiki/Q4779368","display_name":"Aperiodic graph","level":2,"score":0.505418062210083},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.49172404408454895},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.47644928097724915},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.4493017792701721},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.2094046175479889},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2057364284992218},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13715553283691406},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s00454-025-00816-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-025-00816-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00816-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},{"id":"doi:10.1145/3289137.3289145","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3289137.3289145","pdf_url":null,"source":{"id":"https://openalex.org/S4210206003","display_name":"ACM SIGACT News","issn_l":"0163-5700","issn":["0163-5700","1943-5827"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM SIGACT News","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1212.2889","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1212.2889","pdf_url":"https://arxiv.org/pdf/1212.2889","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"","raw_type":"text"},{"id":"mag:2884239031","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/1212.2889","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:commons.clarku.edu:mathematics-1056","is_oa":true,"landing_page_url":"https://commons.clarku.edu/mathematics/55","pdf_url":null,"source":{"id":"https://openalex.org/S4377196688","display_name":"Clark Digital Commons (Clark University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I130785548","host_organization_name":"Clark University","host_organization_lineage":["https://openalex.org/I130785548"],"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":"Mathematics","raw_type":"text"},{"id":"doi:10.48550/arxiv.1212.2889","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1212.2889","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.1007/s00454-025-00816-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-025-00816-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-025-00816-4.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete &amp; Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3700370057","display_name":null,"funder_award_id":"CCF-1533663","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6145246782","display_name":"XPS: FULL: CCA: Collaborative Research: Automatically Scalable Computation","funder_award_id":"1533663","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7575781653","display_name":"AF: Small: Computational Complexity from Physical Constraints","funder_award_id":"0915948","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8065122670","display_name":null,"funder_award_id":"CCF-0915948","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2884239031.pdf","grobid_xml":"https://content.openalex.org/works/W2884239031.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W56268668","https://openalex.org/W88162231","https://openalex.org/W203799894","https://openalex.org/W1482997031","https://openalex.org/W1489341787","https://openalex.org/W1552419556","https://openalex.org/W1976063591","https://openalex.org/W1977183350","https://openalex.org/W1977852626","https://openalex.org/W1994584977","https://openalex.org/W2030691099","https://openalex.org/W2032148870","https://openalex.org/W2088007210","https://openalex.org/W2089164015","https://openalex.org/W2103012681","https://openalex.org/W2115103988","https://openalex.org/W2118775355","https://openalex.org/W2129727166","https://openalex.org/W2133847090","https://openalex.org/W2160114101","https://openalex.org/W2161122290","https://openalex.org/W2280238320","https://openalex.org/W2608463343","https://openalex.org/W3021045375","https://openalex.org/W3118657136","https://openalex.org/W4232040968","https://openalex.org/W4241746776","https://openalex.org/W6602268012","https://openalex.org/W6603631806"],"related_works":["https://openalex.org/W2731769302","https://openalex.org/W2992775237","https://openalex.org/W3009252519","https://openalex.org/W1488192226","https://openalex.org/W3102599922","https://openalex.org/W2962775804","https://openalex.org/W3099751044","https://openalex.org/W2277695700","https://openalex.org/W2145581050","https://openalex.org/W3087298269","https://openalex.org/W3164698335","https://openalex.org/W2262252406","https://openalex.org/W2810921360","https://openalex.org/W3196542841","https://openalex.org/W2087071857","https://openalex.org/W3099732631","https://openalex.org/W2957530950","https://openalex.org/W2141906371","https://openalex.org/W2936395102","https://openalex.org/W3042953477"],"abstract_inverted_index":{"Abstract":[0],"Fix":[1],"any":[2],"$$\\lambda":[3,33,78,118,152,172,229,242,277],"\\in":[4],"\\mathbb":[5,22,102],"{C}$$":[6,23,103],"<mml:math":[7,24,35,54,80,104,120,128,154,161,174,231,244,279],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[8,25,36,55,81,105,121,129,155,162,175,232,245,280],"<mml:mrow>":[9,26,56,106,130],"<mml:mi>\u03bb</mml:mi>":[10,37,60,64,82,109,122,156,165,176,233,246,281],"<mml:mo>\u2208</mml:mo>":[11],"<mml:mi>C</mml:mi>":[12,29,112],"</mml:mrow>":[13,30,66,113,136],"</mml:math>":[14,31,38,67,83,114,123,137,157,167,177,234,247,282],".":[15,72,138,192],"We":[16,73],"say":[17],"that":[18,337],"a":[19,42,93,148,188,198,264],"set":[20],"$$S\\subseteq":[21],"<mml:mi>S</mml:mi>":[27],"<mml:mo>\u2286</mml:mo>":[28,111],"is":[32,68,216,346],"$$":[34,79,119,153,160,173,230,243,278],"-convex":[39,84,124,178,235,248,283],"if,":[40],"whenever":[41],"and":[43,86,90,168,202,237,250,299,310,314,320,329],"b":[44],"are":[45],"in":[46,70,217,292],"S":[47,71],",":[48],"the":[49,75,116,185,209,239,258,339],"point":[50],"$$(1-\\lambda":[51],")a+\\lambda":[52],"b$$":[53],"<mml:mo>(</mml:mo>":[57],"<mml:mn>1</mml:mn>":[58,134],"<mml:mo>-</mml:mo>":[59],"<mml:mo>)</mml:mo>":[61],"<mml:mi>a</mml:mi>":[62],"<mml:mo>+</mml:mo>":[63],"<mml:mi>b</mml:mi>":[65],"also":[69,273],"investigate":[74],"properties":[76],"of":[77,95,126,141,187,200,212,260,266,324],"sets":[85,179,205,236,249],"their":[87],"(topological)":[88],"closures,":[89],"we":[91,146,272],"prove":[92],"number":[94,191,199,265],"facts":[96],"about":[97,228],"them.":[98],"Let":[99],"$$Q_\\lambda":[100,159],"\\subseteq":[101],"<mml:msub>":[107,163],"<mml:mi>Q</mml:mi>":[108,164],"</mml:msub>":[110,166],"be":[115,181],"least":[117],"superset":[125],"$$\\{0,1\\}$$":[127],"<mml:mo>{</mml:mo>":[131],"<mml:mn>0</mml:mn>":[132],"<mml:mo>,</mml:mo>":[133],"<mml:mo>}</mml:mo>":[135],"Generalizing":[139],"results":[140,227,259,318,340],"R.":[142],"G.":[143],"E.":[144],"Pinch,":[145,325],"give":[147,195],"sufficient":[149],"condition":[150],"on":[151],"for":[158,302],"some":[169],"other":[170,300],"related":[171],"to":[180,197,263],"discrete":[182],"by":[183],"introducing":[184],"notion":[186],"strong":[189],"PV":[190],"These":[193],"conditions":[194],"rise":[196],"periodic":[201],"aperiodic":[203,276],"Meyer":[204],"(often":[206],"regarded":[207],"as":[208],"mathematical":[210],"counterpart":[211],"\u201cquasicrystals\u201d).":[213],"This":[214],"paper":[215,336],"two":[218],"parts:":[219],"Part":[220,252,261,270,293],"I":[221,262],"(Theory,":[222],"Sections":[223,255],"1\u20136)":[224],"gives":[225],"general":[226],"explores":[238],"connections":[240],"between":[241],"quasicrystals;":[251],"II":[253,294],"(Applications,":[254],"7\u201311)":[256],"applies":[257],"special":[267],"cases.":[268],"In":[269],"II,":[271],"display":[274],"several":[275,286],"sets,":[284],"including":[285],"with":[287],"dihedral":[288],"symmetry.":[289],"Section":[290],"11":[291],"contains":[295],"conjectures,":[296],"open":[297],"problems,":[298],"suggestions":[301],"further":[303],"research.":[304],"Our":[305,317],"work":[306,323],"combines":[307],"elementary":[308],"concepts":[309],"techniques":[311],"from":[312],"algebra":[313],"plane":[315],"geometry.":[316],"extend":[319],"generalize":[321],"previous":[322],"Berman":[326],"&amp;":[327,332],"Moody,":[328],"Mas\u00e1kov\u00e1,":[330],"Patera,":[331],"Pelantov\u00e1.":[333],"An":[334],"extended":[335],"includes":[338],"given":[341],"here":[342],"among":[343],"many":[344],"others":[345],"available":[347],"online.":[348]},"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2018-08-03T00:00:00"}
