{"id":"https://openalex.org/W4416062342","doi":"https://doi.org/10.48550/arxiv.2507.06158","title":"Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices","display_name":"Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices","publication_year":2025,"publication_date":"2025-07-08","ids":{"openalex":"https://openalex.org/W4416062342","doi":"https://doi.org/10.48550/arxiv.2507.06158"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2507.06158","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2507.06158","pdf_url":"https://arxiv.org/pdf/2507.06158","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":null,"raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2507.06158","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Cruz, Anjelo Gabriel R.","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cruz, Anjelo Gabriel R.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029715004","display_name":"Manuel Joseph C. Loquias","orcid":"https://orcid.org/0000-0002-1427-1069"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Loquias, Manuel Joseph C.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5020243260","display_name":"J\u00f6rg \u039c. Thuswaldner","orcid":"https://orcid.org/0000-0001-5308-762X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thuswaldner, J\u00f6rg M.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"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/T11567","display_name":"semigroups and automata theory","score":0.3474000096321106,"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":0.3474000096321106,"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/T11435","display_name":"Polynomial and algebraic computation","score":0.155799999833107,"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/T12162","display_name":"Cellular Automata and Applications","score":0.1103999987244606,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/numerical-digit","display_name":"Numerical digit","score":0.6557000279426575},{"id":"https://openalex.org/keywords/attractor","display_name":"Attractor","score":0.5554999709129333},{"id":"https://openalex.org/keywords/automaton","display_name":"Automaton","score":0.5515000224113464},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4625000059604645},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.4577000141143799},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.359499990940094},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.3474000096321106}],"concepts":[{"id":"https://openalex.org/C94957134","wikidata":"https://www.wikidata.org/wiki/Q82990","display_name":"Numerical digit","level":2,"score":0.6557000279426575},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6485999822616577},{"id":"https://openalex.org/C164380108","wikidata":"https://www.wikidata.org/wiki/Q507187","display_name":"Attractor","level":2,"score":0.5554999709129333},{"id":"https://openalex.org/C112505250","wikidata":"https://www.wikidata.org/wiki/Q787116","display_name":"Automaton","level":2,"score":0.5515000224113464},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4659000039100647},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4625000059604645},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.4577000141143799},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.359499990940094},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.3474000096321106},{"id":"https://openalex.org/C116248031","wikidata":"https://www.wikidata.org/wiki/Q214526","display_name":"Automata theory","level":3,"score":0.3264999985694885},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3125},{"id":"https://openalex.org/C174327141","wikidata":"https://www.wikidata.org/wiki/Q176837","display_name":"Quantum finite automata","level":4,"score":0.3086000084877014},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.30630001425743103},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2768000066280365},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2766999900341034},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.26989999413490295},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26440000534057617},{"id":"https://openalex.org/C198931094","wikidata":"https://www.wikidata.org/wiki/Q2610096","display_name":"Hurwitz matrix","level":3,"score":0.2615000009536743},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.25690001249313354}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2507.06158","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2507.06158","pdf_url":"https://arxiv.org/pdf/2507.06158","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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2507.06158","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2507.06158","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2507.06158","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2507.06158","pdf_url":"https://arxiv.org/pdf/2507.06158","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":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Let":[0,21],"$A$":[1,49],"be":[2,14,23,66,113,116],"an":[3],"expanding":[4],"$2":[5],"\\times":[6],"2$":[7],"matrix":[8],"with":[9,46,85,157],"rational":[10],"entries":[11],"and":[12,50,92,144],"$\\mathbb{Z}^2[A]$":[13,28],"the":[15,70,103,123,128,138,141,172,180],"smallest":[16],"$A$-invariant":[17],"$\\mathbb{Z}$-module":[18],"containing":[19],"$\\mathbb{Z}^2$.":[20],"$\\mathcal{D}$":[22],"a":[24,31,42,96,147,153],"finite":[25,97],"subset":[26],"of":[27,35,99,107,140,182],"which":[29,136],"is":[30,40,56],"complete":[32],"residue":[33],"system":[34,155],"$\\mathbb{Z}^2[A]/A\\mathbb{Z}^2[A]$.":[36],"The":[37],"pair":[38],"$(A,\\mathcal{D})$":[39,121,156,169],"called":[41],"{\\em":[43,47,51,100,105,124],"digit":[44,52,159,184],"system}":[45],"base}":[48],"set}":[53],"$\\mathcal{D}$.":[54],"It":[55],"well":[57],"known":[58],"that":[59,120,170],"every":[60],"vector":[61,149],"$x":[62],"\\in":[63,90],"\\mathbb{Z}^2[A]$":[64,151],"can":[65,111],"written":[67],"uniquely":[68],"in":[69,152],"form":[71],"\\[":[72],"x":[73],"=":[74],"d_0":[75],"+":[76,78,80,82],"Ad_1":[77],"\\cdots":[79],"A^kd_k":[81],"A^{k+1}p,":[83],"\\]":[84],"$k\\in":[86],"\\mathbb{N}$":[87],"minimal,":[88],"$d_0,\\dots,d_k":[89],"\\mathcal{D}$,":[91],"$p$":[93,110],"taken":[94],"from":[95],"set":[98],"periodic":[101],"elements},":[102],"so-called":[104],"attractor}":[106],"$(A,\\mathcal{D})$.":[108],"If":[109],"always":[112],"chosen":[114],"to":[115,146,165,178],"$0$":[117],"we":[118,131],"say":[119],"has":[122],"finiteness":[125,173],"property}.":[126],"In":[127],"present":[129],"paper":[130],"introduce":[132],"finite-state":[133],"transducer":[134],"automata":[135,162],"realize":[137],"addition":[139],"vectors":[142],"$\\pm(1,0)^\\top$":[143],"$\\pm(0,1)^\\top$":[145],"given":[148],"$x\\in":[150],"number":[154],"collinear":[158],"set.":[160],"These":[161],"are":[163],"applied":[164],"characterize":[166,179],"all":[167],"pairs":[168],"have":[171],"property":[174],"and,":[175],"more":[176],"generally,":[177],"attractors":[181],"these":[183],"systems.":[185]},"counts_by_year":[],"updated_date":"2026-07-03T08:13:44.112507","created_date":"2025-10-10T00:00:00"}