{"id":"https://openalex.org/W7223920180","doi":"https://doi.org/10.48550/arxiv.2610.33084","title":"Space curves in Hahn calculus: rotation structure, a fundamental theorem, and $q$-deformed invariants at the fixed point","display_name":"Space curves in Hahn calculus: rotation structure, a fundamental theorem, and $q$-deformed invariants at the fixed point","publication_year":2026,"publication_date":"2026-10-21","ids":{"openalex":"https://openalex.org/W7223920180","doi":"https://doi.org/10.48550/arxiv.2610.33084"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.33084","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.33084","pdf_url":"https://arxiv.org/pdf/2610.33084","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/2610.33084","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125743526","display_name":"Umut Selvi","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I3129222432","display_name":"Ankara Hac\u0131 Bayram Veli University","ror":"https://ror.org/05mskc574","country_code":"TR","type":"education","lineage":["https://openalex.org/I3129222432"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Selvi, Umut","raw_affiliation_strings":["Department of Mathematics , Ankara Hac\u0131 Bayram Veli University , Ankara , 06900 , T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Ankara Hac\u0131 Bayram Veli University , Ankara , 06900 , T\u00fcrkiye","institution_ids":["https://openalex.org/I3129222432"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5016312344","display_name":"Derya Sa\u011flam","orcid":"https://orcid.org/0000-0002-7034-2183","observed_orcids":["https://orcid.org/0000-0002-7034-2183"]},"institutions":[{"id":"https://openalex.org/I3129222432","display_name":"Ankara Hac\u0131 Bayram Veli University","ror":"https://ror.org/05mskc574","country_code":"TR","type":"education","lineage":["https://openalex.org/I3129222432"]}],"countries":["TR"],"is_corresponding":false,"raw_author_name":"Sa\u011flam, Derya","raw_affiliation_strings":["Department of Mathematics , Ankara Hac\u0131 Bayram Veli University , Ankara , 06900 , T\u00fcrkiye"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Ankara Hac\u0131 Bayram Veli University , Ankara , 06900 , T\u00fcrkiye","institution_ids":["https://openalex.org/I3129222432"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I3129222432"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.9473000168800354,"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"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9473000168800354,"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/T12504","display_name":"Mathematics and Applications","score":0.012000000104308128,"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.00570000009611249,"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":[{"id":"https://openalex.org/keywords/q-calculus","display_name":"q-calculus","score":0.6949999928474426},{"id":"https://openalex.org/keywords/space-curves","display_name":"space curves","score":0.8799999952316284},{"id":"https://openalex.org/keywords/fixed-points","display_name":"fixed points","score":0.48899999260902405},{"id":"https://openalex.org/keywords/inscribed-polygons","display_name":"inscribed polygons","score":0.492000013589859},{"id":"https://openalex.org/keywords/sampling-bias","display_name":"sampling bias","score":0.3869999945163727}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7713000178337097},{"id":"https://openalex.org/C162546045","wikidata":"https://www.wikidata.org/wiki/Q947922","display_name":"Frenet\u2013Serret formulas","level":3,"score":0.7627000212669373},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.7170000076293945},{"id":"https://openalex.org/C84985374","wikidata":"https://www.wikidata.org/wiki/Q1892644","display_name":"Torsion of a curve","level":5,"score":0.7062000036239624},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.6169999837875366},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5983999967575073},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.576200008392334},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5446000099182129},{"id":"https://openalex.org/C41779539","wikidata":"https://www.wikidata.org/wiki/Q2543717","display_name":"Equiangular polygon","level":3,"score":0.4912000000476837},{"id":"https://openalex.org/C135007989","wikidata":"https://www.wikidata.org/wiki/Q643156","display_name":"Rotation number","level":3,"score":0.44369998574256897},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.4296000003814697},{"id":"https://openalex.org/C44074806","wikidata":"https://www.wikidata.org/wiki/Q1066790","display_name":"Inscribed figure","level":2,"score":0.4250999987125397},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.4165000021457672},{"id":"https://openalex.org/C2776196297","wikidata":"https://www.wikidata.org/wiki/Q17138781","display_name":"Twist","level":2,"score":0.3644999861717224},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3499000072479248},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.34450000524520874},{"id":"https://openalex.org/C89025888","wikidata":"https://www.wikidata.org/wiki/Q911907","display_name":"Dihedral angle","level":4,"score":0.2752000093460083},{"id":"https://openalex.org/C68260962","wikidata":"https://www.wikidata.org/wiki/Q7200967","display_name":"Planarity testing","level":2,"score":0.27309998869895935},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.272599995136261},{"id":"https://openalex.org/C161223928","wikidata":"https://www.wikidata.org/wiki/Q3775868","display_name":"Center of curvature","level":4,"score":0.26269999146461487},{"id":"https://openalex.org/C163415756","wikidata":"https://www.wikidata.org/wiki/Q126473","display_name":"Contraction (grammar)","level":2,"score":0.25780001282691956},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.2574999928474426},{"id":"https://openalex.org/C16063025","wikidata":"https://www.wikidata.org/wiki/Q4370338","display_name":"Total curvature","level":5,"score":0.25600001215934753},{"id":"https://openalex.org/C51999183","wikidata":"https://www.wikidata.org/wiki/Q3822160","display_name":"Torsion constant","level":3,"score":0.25600001215934753}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.33084","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.33084","pdf_url":"https://arxiv.org/pdf/2610.33084","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.2610.33084","doi":"https://doi.org/10.48550/arxiv.2610.33084","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.33084","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":"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":"pmh:oai:arXiv.org:2610.33084","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.33084","pdf_url":"https://arxiv.org/pdf/2610.33084","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223920180.pdf","grobid_xml":"https://content.openalex.org/works/W7223920180.grobid-xml"},"referenced_works_count":26,"referenced_works":["https://openalex.org/W271810198","https://openalex.org/W603006721","https://openalex.org/W1595749660","https://openalex.org/W1658634785","https://openalex.org/W1969373001","https://openalex.org/W1993671632","https://openalex.org/W1995863087","https://openalex.org/W2003108457","https://openalex.org/W2033429912","https://openalex.org/W2068864491","https://openalex.org/W2090107137","https://openalex.org/W2108665243","https://openalex.org/W2115941077","https://openalex.org/W2128949314","https://openalex.org/W2168050137","https://openalex.org/W2321807852","https://openalex.org/W2343401604","https://openalex.org/W2766512039","https://openalex.org/W2962827375","https://openalex.org/W3082949096","https://openalex.org/W4232115556","https://openalex.org/W4240644230","https://openalex.org/W4405419361","https://openalex.org/W4416379092","https://openalex.org/W7149763957","https://openalex.org/W7206208545"],"related_works":["https://openalex.org/W6885021328","https://openalex.org/W2511285111","https://openalex.org/W7160005082","https://openalex.org/W7163711990","https://openalex.org/W2063216546","https://openalex.org/W7133550843","https://openalex.org/W4309744081","https://openalex.org/W3168181882","https://openalex.org/W2486347144","https://openalex.org/W4386947502"],"abstract_inverted_index":{"We":[0,78,178],"develop":[1],"the":[2,13,16,20,49,54,75,95,109,115,138,172,202,206],"differential":[3],"geometry":[4],"of":[5,36,53,92,125,141,146,212],"space":[6],"curves":[7,220],"in":[8,103,137,150],"Hahn's":[9],"quantum":[10],"calculus,":[11],"where":[12,199],"derivative":[14],"is":[15,31,63,197,201],"difference":[17],"quotient":[18,135],"along":[19,33],"contraction":[21],"$\u03c3(t)=qt+\u03c9$":[22],"with":[23,48,154,162,221,236],"fixed":[24],"point":[25],"$\u03c9_0=\u03c9/(1-q)$.":[26],"The":[27],"Hahn":[28,61,110,181,229],"Frenet":[29,51],"frame":[30,52],"transported":[32],"every":[34,147],"orbit":[35,56],"$\u03c3$":[37],"by":[38,71,183],"rotations":[39,93],"through":[40],"turning":[41,144,188],"and":[42,74,84,94,112,118,143,158,189,214,224,245],"torsion":[43,113,142,190,213,225],"angles,":[44],"which":[45],"identifies":[46],"it":[47],"discrete":[50],"inscribed":[55,149],"polygon.":[57],"An":[58],"embedded":[59],"regular":[60],"curve":[62,153],"determined,":[64],"up":[65,242],"to":[66,243],"linear":[67,246],"reparametrization":[68],"about":[69],"$\u03c9_0$,":[70,233],"its":[72],"trace":[73],"induced":[76],"contraction.":[77],"prove":[79],"a":[80,85,122,151,169,184,209,237],"sharp":[81],"planarity":[82],"criterion":[83],"fundamental":[86],"theorem":[87],"via":[88],"an":[89],"infinite":[90],"product":[91],"Jackson--N\u00f6rlund":[96],"integral.":[97],"For":[98],"smooth":[99,152],"curves,":[100],"expansions":[101],"uniform":[102],"$q$":[104],"show":[105],"that":[106],"at":[107,195],"$\u03c9_0$":[108,196],"curvature":[111,215,223],"carry":[114],"factors":[116,173],"$[2]_q/2$":[117],"$[3]_q/3$.":[119],"These":[120],"factors,":[121],"sampling":[123],"effect":[124],"geometrically":[126],"decaying":[127,161],"steps,":[128],"disappear":[129],"under":[130],"edge-length":[131],"normalization,":[132],"but":[133],"their":[134,192],"persists":[136],"limiting":[139],"ratio":[140,194,211,239],"angles":[145],"polygon":[148],"monotonically":[155],"converging":[156],"vertices":[157],"edge":[159],"lengths":[160],"ratio~$q$.":[163],"In":[164],"uniformly":[165],"sampled":[166],"relaxation":[167],"towards":[168],"stable":[170],"node,":[171],"can":[174],"become":[175],"measurable":[176],"biases.":[177],"characterize":[179,218],"generalized":[180,228],"helices":[182,235],"two-step":[185],"relation":[186],"between":[187],"angles;":[191],"Lancret":[193],"$\\pm([2]_q/2)\\cot\\vartheta$,":[198],"$\\vartheta$":[200],"helix":[203],"angle.":[204],"Unlike":[205],"classical":[207],"case,":[208],"constant":[210,222],"does":[216],"not":[217],"helices:":[219],"are":[226,240],"never":[227],"helices,":[230],"and,":[231],"near":[232],"unit-speed":[234],"given":[238],"unique":[241],"similarities":[244],"reparametrization.":[247]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-28T00:00:00"}
