{"id":"https://openalex.org/W7204753417","doi":"https://doi.org/10.20944/preprints202608.2304.v1","title":"Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers","display_name":"Arbitrary Stage Spectra for Raikov Remainders Along Quotient Towers","publication_year":2026,"publication_date":"2026-08-31","ids":{"openalex":"https://openalex.org/W7204753417","doi":"https://doi.org/10.20944/preprints202608.2304.v1"},"language":"en","primary_location":{"id":"doi:10.20944/preprints202608.2304.v1","doi":"https://doi.org/10.20944/preprints202608.2304.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202608.2304.v1","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.20944/preprints202608.2304.v1","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5047506639","display_name":"Xing-Yu Hu","orcid":"https://orcid.org/0009-0007-9953-9362","observed_orcids":["https://orcid.org/0009-0007-9953-9362","https://orcid.org/0009-0004-4351-413X"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xing-Yu Hu","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0007-9953-9362","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5047506639"],"corresponding_institution_ids":[],"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9599999785423279,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9599999785423279,"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/T10899","display_name":"Advanced Operator Algebra Research","score":0.014999999664723873,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11545","display_name":"Advanced Banach Space Theory","score":0.004000000189989805,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/hausdorff-topological-groups","display_name":"Hausdorff topological groups","score":0.8270000219345093},{"id":"https://openalex.org/keywords/closed-normal-subgroups","display_name":"closed normal subgroups","score":0.9390000104904175},{"id":"https://openalex.org/keywords/realization-theorem","display_name":"realization theorem","score":0.4320000112056732},{"id":"https://openalex.org/keywords/boolean-algebra","display_name":"Boolean algebra","score":0.3709999918937683}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8158000111579895},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.7979000210762024},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.6460000276565552},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5620999932289124},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.5529999732971191},{"id":"https://openalex.org/C2777831296","wikidata":"https://www.wikidata.org/wiki/Q12518","display_name":"Tower","level":2,"score":0.5289000272750854},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4700999855995178},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.45989999175071716},{"id":"https://openalex.org/C105515060","wikidata":"https://www.wikidata.org/wiki/Q743179","display_name":"Normal subgroup","level":3,"score":0.45509999990463257},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.453900009393692},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.41429999470710754},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.4142000079154968},{"id":"https://openalex.org/C141898687","wikidata":"https://www.wikidata.org/wiki/Q1501997","display_name":"Hausdorff distance","level":2,"score":0.4050000011920929},{"id":"https://openalex.org/C40753290","wikidata":"https://www.wikidata.org/wiki/Q1139111","display_name":"Quotient space (topology)","level":3,"score":0.37439998984336853},{"id":"https://openalex.org/C134565946","wikidata":"https://www.wikidata.org/wiki/Q906015","display_name":"Direct limit","level":2,"score":0.373199999332428},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.35359999537467957},{"id":"https://openalex.org/C16310669","wikidata":"https://www.wikidata.org/wiki/Q1347208","display_name":"Epimorphism","level":2,"score":0.3474000096321106},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.33390000462532043},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.32420000433921814},{"id":"https://openalex.org/C95627345","wikidata":"https://www.wikidata.org/wiki/Q1138961","display_name":"Quotient group","level":4,"score":0.31869998574256897},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.30379998683929443},{"id":"https://openalex.org/C51287006","wikidata":"https://www.wikidata.org/wiki/Q63565120","display_name":"Quotient algebra","level":4,"score":0.2935999929904938},{"id":"https://openalex.org/C2779483109","wikidata":"https://www.wikidata.org/wiki/Q4681347","display_name":"Additive group","level":3,"score":0.2890999913215637},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.2797999978065491},{"id":"https://openalex.org/C130133044","wikidata":"https://www.wikidata.org/wiki/Q1887083","display_name":"Compact group","level":3,"score":0.2676999866962433},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26489999890327454},{"id":"https://openalex.org/C96386209","wikidata":"https://www.wikidata.org/wiki/Q1046291","display_name":"Topological group","level":3,"score":0.25119999051094055}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.20944/preprints202608.2304.v1","doi":"https://doi.org/10.20944/preprints202608.2304.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202608.2304.v1","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},{"id":"pmh:oai:arXiv.org:2610.02212","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.02212","pdf_url":"https://arxiv.org/pdf/2610.02212","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.2610.02212","doi":"https://doi.org/10.48550/arxiv.2610.02212","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.02212","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":"doi:10.20944/preprints202608.2304.v1","doi":"https://doi.org/10.20944/preprints202608.2304.v1","is_oa":true,"landing_page_url":"https://doi.org/10.20944/preprints202608.2304.v1","pdf_url":null,"source":{"id":"https://openalex.org/S6309402219","display_name":"Preprints.org","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204753417.pdf","grobid_xml":"https://content.openalex.org/works/W7204753417.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W73691789","https://openalex.org/W1580492876","https://openalex.org/W1977080006","https://openalex.org/W1977946233","https://openalex.org/W1993947583","https://openalex.org/W1995706090","https://openalex.org/W2218738517","https://openalex.org/W2493073921","https://openalex.org/W2547676428","https://openalex.org/W2570737369","https://openalex.org/W2794766999","https://openalex.org/W2952020424","https://openalex.org/W3154238838","https://openalex.org/W4405076463","https://openalex.org/W4414083003","https://openalex.org/W4415310130","https://openalex.org/W4417028089","https://openalex.org/W4417304128","https://openalex.org/W7160999582"],"related_works":["https://openalex.org/W7171741030","https://openalex.org/W2085344907","https://openalex.org/W1580492876","https://openalex.org/W1800228","https://openalex.org/W2064026906","https://openalex.org/W2884802740","https://openalex.org/W2098458600","https://openalex.org/W2528886949","https://openalex.org/W2044480287","https://openalex.org/W2054144799"],"abstract_inverted_index":{"Let":[0],"\\(":[1,9,26,30,34,43,48,53,67,103,109,146,158,166,172,179,221,225,229,240,248,258,262,271,286,292,302],"\\theta":[2,104],"\\)":[3,11,37,51,69,105,168,174,223,233,252,260,283,296],"be":[4,12,93,154],"an":[5,207,254],"ordinal":[6,102],"and":[7,46,52,59,106,136,163,238,297,314,332,350,357],"let":[8,228],"(N_\\alpha)_{\\alpha\\leq\\theta}":[10],"a":[13,22,114,120,184,188,193,217,278,299,311],"continuous":[14,121],"increasing":[15],"tower":[16,122,133,185],"of":[17,21,42,64,77,87,123,131,210,213,224,301,329],"closed":[18,124,218],"normal":[19,125,219],"subgroups":[20,126],"Hausdorff":[23,115],"topological":[24],"group":[25,118],"G":[27,36,44,226,251,295],"\\),":[28,45,111,162,227,265,275],"with":[29,89,119],"N_0=\\{e\\}":[31],"\\).":[32,55,148,182,244,305],"Write":[33],"\\rho":[35,249,272,293,303],"for":[38,164,277],"the":[39,60,65,71,78,132,137,143,169,211,235,268,284,318,322,330,341,346],"Raikov":[40,79,269],"completion":[41,80],"put":[47],"K_\\alpha=\\overline{N_\\alpha}^{\\rho":[49],"G}":[50,243],"H_\\alpha=GK_\\alpha":[54],"The":[56,85,149,203,245],"successive":[57,170,199],"differences":[58,63],"corresponding":[61],"limit":[62,194],"groups":[66],"H_\\alpha":[68],"mark":[70],"first":[72],"stages":[73,88,144],"at":[74,142,192],"which":[75],"points":[76],"become":[81],"old":[82],"quotient":[83,151,214,236,256,333,348,359],"points.":[84],"set":[86,108],"nonempty":[90,138,190],"strata":[91,139],"can":[92,152],"prescribed":[94],"arbitrarily":[95],"even":[96,196],"under":[97],"pseudocompactness.":[98],"For":[99,216,306,325,337],"every":[100,107,129,165,198],"nonzero":[101],"S\\subseteq(0,\\theta]":[110],"there":[112],"is":[113,134,175,201,261,298,335],"pseudocompact":[116,135,356],"Boolean":[117],"such":[127],"that":[128],"member":[130],"occur":[140],"exactly":[141,177],"in":[145,247,291],"S":[147,181],"terminal":[150],"simultaneously":[153],"chosen":[155],"isomorphic":[156],"to":[157,267],"\\mathbb":[159],"Z/2\\mathbb":[160],"Z":[161],"\\delta<\\theta":[167],"factor":[171,200],"N_{\\delta+1}/N_\\delta":[173],"compact":[176,326],"when":[178,197],"\\delta+1\\notin":[180],"Thus":[183],"may":[186],"have":[187],"single":[189],"stratum":[191],"stage":[195],"compact.":[202],"realization":[204],"theorem":[205],"uses":[206],"exact":[208],"description":[209],"fibers":[212],"extensions.":[215],"subgroup":[220],"N":[222,274,304],"\\widehat":[230,287],"q:\\rho":[231],"G\\to\\rho(G/N)":[232],"extend":[234],"homomorphism":[237],"write":[239],"K=\\overline":[241],"N^{\\rho":[242],"fiber":[246],"G\\setminus":[250,294],"over":[253],"embedded":[255],"point":[257],"gN":[259],"g(K\\setminus":[263],"N)":[264],"homeomorphic":[266],"remainder":[270,343],"N\\setminus":[273],"while":[276],"point\\(":[279],"y\\in\\widehat":[280],"q(\\rho":[281],"G)\\setminus(G/N)":[282],"whole":[285],"q":[288],"\\)-fiber":[289],"lies":[290],"copy":[300],"precompact":[307,339],"kernels":[308],"this":[309],"gives":[310],"canonical":[312],"decomposition,":[313],"nested":[315],"quotients":[316],"give":[317],"two-stage":[319],"identity":[320],"underlying":[321],"transfinite":[323],"filtration.":[324],"kernels,":[327,340],"pseudocompactness":[328],"source":[331,342],"remainders":[334],"equivalent.":[336],"noncompact":[338],"maps":[344],"onto":[345],"entire":[347],"completion,":[349],"countable":[351],"Abelian":[352],"examples":[353],"exhibit":[354],"both":[355],"non-pseudocompact":[358],"remainders.":[360]},"counts_by_year":[],"updated_date":"2026-11-12T12:28:57.059296","created_date":"2026-09-01T00:00:00"}