{"id":"https://openalex.org/W7225566802","doi":"https://doi.org/10.5281/zenodo.24051852","title":"Symmetric relative equilibria with asymmetric positive masses in the Newtonian 1+N coorbital problem, for N = 10 and every even N \u2265 4","display_name":"Symmetric relative equilibria with asymmetric positive masses in the Newtonian 1+N coorbital problem, for N = 10 and every even N \u2265 4","publication_year":2026,"publication_date":"2026-11-08","ids":{"openalex":"https://openalex.org/W7225566802","doi":"https://doi.org/10.5281/zenodo.24051852"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24051852","doi":"https://doi.org/10.5281/zenodo.24051852","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051852","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24051852","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5017497939","display_name":"William Huanshan Chuang","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Chuang, William Huanshan","raw_affiliation_strings":["Logarcheon Inc"],"raw_orcid":"https://orcid.org/0009-0008-0796-9478","affiliations":[{"raw_affiliation_string":"Logarcheon Inc","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5017497939"],"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.4099000096321106,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.4099000096321106,"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/T11082","display_name":"Spacecraft Dynamics and Control","score":0.33739998936653137,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/T11394","display_name":"Dynamics and Control of Mechanical Systems","score":0.06509999930858612,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/relative-equilibria","display_name":"relative equilibria","score":0.8450000286102295},{"id":"https://openalex.org/keywords/newtonian-potential","display_name":"Newtonian potential","score":0.7509999871253967},{"id":"https://openalex.org/keywords/symmetric-configurations","display_name":"symmetric configurations","score":0.7250000238418579},{"id":"https://openalex.org/keywords/axis-of-symmetry","display_name":"axis of symmetry","score":0.8199999928474426},{"id":"https://openalex.org/keywords/reflection-symmetry","display_name":"reflection symmetry","score":0.7889999747276306},{"id":"https://openalex.org/keywords/positive-mass","display_name":"positive mass","score":0.6499999761581421},{"id":"https://openalex.org/keywords/regular-n-gon","display_name":"regular n-gon","score":0.7390000224113464},{"id":"https://openalex.org/keywords/trigonometric-sums","display_name":"trigonometric sums","score":0.8619999885559082},{"id":"https://openalex.org/keywords/lean-4-formalization","display_name":"Lean 4 formalization","score":0.8500000238418579},{"id":"https://openalex.org/keywords/celestial-mechanics","display_name":"celestial mechanics","score":0.8820000290870667}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7504000067710876},{"id":"https://openalex.org/C114410712","wikidata":"https://www.wikidata.org/wiki/Q973321","display_name":"Hypersurface","level":2,"score":0.5929999947547913},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5913000106811523},{"id":"https://openalex.org/C2775956649","wikidata":"https://www.wikidata.org/wiki/Q1189744","display_name":"Pfaffian","level":2,"score":0.558899998664856},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.42340001463890076},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41929998993873596},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.3767000138759613},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.367900013923645},{"id":"https://openalex.org/C2779886137","wikidata":"https://www.wikidata.org/wiki/Q21030012","display_name":"Symmetry (geometry)","level":2,"score":0.3278999924659729},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3100000023841858},{"id":"https://openalex.org/C151300846","wikidata":"https://www.wikidata.org/wiki/Q125539","display_name":"Submanifold","level":2,"score":0.30149999260902405},{"id":"https://openalex.org/C150397156","wikidata":"https://www.wikidata.org/wiki/Q591310","display_name":"General position","level":2,"score":0.28769999742507935},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.28369998931884766},{"id":"https://openalex.org/C52153879","wikidata":"https://www.wikidata.org/wiki/Q930499","display_name":"Symmetric polynomial","level":4,"score":0.2833000123500824},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.2799000144004822},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.27309998869895935},{"id":"https://openalex.org/C294558","wikidata":"https://www.wikidata.org/wiki/Q366968","display_name":"Newtonian fluid","level":2,"score":0.2727999985218048},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.2700999975204468},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.2619999945163727},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.2572999894618988},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.2572999894618988}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24051852","doi":"https://doi.org/10.5281/zenodo.24051852","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051852","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},{"id":"doi:10.5281/zenodo.24051853","doi":"https://doi.org/10.5281/zenodo.24051853","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051853","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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":"doi:10.5281/zenodo.24051852","doi":"https://doi.org/10.5281/zenodo.24051852","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24051852","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4226376361","https://openalex.org/W3023086428","https://openalex.org/W7223386151","https://openalex.org/W2014832380","https://openalex.org/W2884129947","https://openalex.org/W7223377064","https://openalex.org/W7163048278","https://openalex.org/W7096022399","https://openalex.org/W3102726320","https://openalex.org/W1948036342"],"abstract_inverted_index":{"In":[0],"2022":[1],"Deng,":[2,150],"Hampton":[3,151],"and":[4,45,112,152,264,280,297,309],"Wang":[5],"proved":[6],"that":[7,37,71,91,180,226],"the":[8,24,38,46,55,60,104,113,141,146,155,166,193,199,224,277,281,299,311],"Newtonian":[9,56],"1+N":[10],"coorbital":[11],"problem":[12],"has":[13,18,81],"relative":[14,118,127,258],"equilibria":[15],"whose":[16],"configuration":[17,90],"an":[19,256],"axis":[20,267],"of":[21,99,133,138,149,172,177,215,268],"symmetry":[22],"while":[23],"positive":[25,121,134,182,218,262],"masses":[26,123,263],"do":[27],"not,":[28],"for":[29,41,62,129,210,232,237,245],"N":[30,51,65,74,235,238,246,250,282],"=":[31,52,75,251,283],"4,":[32,67,76],"6,":[33,77],"8.":[34,78],"They":[35],"conjectured":[36],"same":[39],"holds":[40,209],"larger":[42],"even":[43,64,234],"N,":[44],"first":[47,85,227],"open":[48],"case":[49],"is":[50,86,92,124,163,230],"10.":[53],"For":[54,249],"potential":[57],"we":[58,253],"prove":[59],"conjecture":[61],"every":[63,117,211,233],"\u2265":[66,239],"by":[68,219,241,271],"one":[69,190,206],"argument":[70,222],"also":[72,125,144,254],"recovers":[73],"The":[79,84,160,195,274,302],"proof":[80],"two":[82],"parts.":[83],"algebraic.":[87],"Consider":[88],"a":[89,95,109,126,130,170,213,220,265],"symmetric":[93,122,178,183],"under":[94],"reflection":[96],"fixing":[97],"none":[98],"its":[100],"points.":[101],"On":[102],"it,":[103],"mass":[105],"matrix":[106],"splits":[107],"into":[108],"block":[110,114],"A":[111],"\u2212A\u1d40.":[115],"Hence":[116],"equilibrium":[119,128,259],"with":[120,260,292,294],"two-dimensional":[131],"cone":[132],"masses,":[135],"almost":[136],"all":[137],"which":[139,208,229],"lack":[140],"symmetry.":[142],"This":[143],"explains":[145],"Pfaffian":[147,156],"computations":[148],"Wang,":[153],"since":[154],"equals":[157],"\u00b1det":[158],"A.":[159],"second":[161],"part":[162],"local.":[164],"Through":[165],"regular":[167,200],"N-gon":[168,201],"passes":[169],"hypersurface":[171],"dimension":[173],"N/2":[174],"\u2212":[175],"1":[176],"configurations":[179],"admit":[181],"masses.":[184],"Its":[185],"generic":[186],"points":[187],"have":[188],"exactly":[189],"nontrivial":[191],"symmetry,":[192,269],"reflection.":[194],"nondegeneracy":[196],"needed":[197],"at":[198],"reduces":[202],"to":[203],"three":[204],"conditions:":[205],"sign,":[207],"N;":[212],"family":[214],"trigonometric":[216],"sums,":[217],"concavity":[221,278],"except":[223],"first;":[225],"sum,":[228],"nonzero":[231],"(analytically":[236],"16,":[240],"certified":[242,270],"ball":[243,272],"arithmetic":[244],"\u2264":[247],"14).":[248],"10":[252,284],"give":[255],"explicit":[257],"ten":[261],"single":[266],"arithmetic.":[273],"algebraic":[275],"part,":[276],"lemma":[279],"sign":[285],"inequality":[286],"are":[287],"verified":[288],"in":[289],"Lean":[290],"4":[291],"Mathlib,":[293],"no":[295],"sorry":[296],"only":[298],"standard":[300],"axioms.":[301],"paper":[303],"names":[304],"each":[305],"step":[306],"not":[307],"formalised":[308],"gives":[310],"reason.":[312]},"counts_by_year":[],"updated_date":"2026-11-10T11:39:43.021295","created_date":"2026-11-09T00:00:00"}
