{"id":"https://openalex.org/W3181136734","doi":"https://doi.org/10.1142/s0218216521400071","title":"Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths","display_name":"Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths","publication_year":2021,"publication_date":"2021-09-01","ids":{"openalex":"https://openalex.org/W3181136734","doi":"https://doi.org/10.1142/s0218216521400071","mag":"3181136734"},"language":"en","primary_location":{"id":"doi:10.1142/s0218216521400071","doi":"https://doi.org/10.1142/s0218216521400071","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218216521400071","pdf_url":null,"source":{"id":"https://openalex.org/S55337115","display_name":"Journal of Knot Theory and Its Ramifications","issn_l":"0218-2165","issn":["0218-2165","1793-6527"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-1","norway-1","poland-70","russia-white-list-4"],"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Knot Theory and Its Ramifications","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2107.03004","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5055495202","display_name":"Nikolay Abrosimov","orcid":"https://orcid.org/0000-0001-5429-6589","observed_orcids":["https://orcid.org/0000-0001-5429-6589"]},"institutions":[{"id":"https://openalex.org/I188973947","display_name":"Novosibirsk State University","ror":"https://ror.org/04t2ss102","country_code":"RU","type":"education","lineage":["https://openalex.org/I188973947"]},{"id":"https://openalex.org/I196355604","display_name":"National Research Tomsk State University","ror":"https://ror.org/02he2nc27","country_code":"RU","type":"education","lineage":["https://openalex.org/I196355604"]},{"id":"https://openalex.org/I4210096862","display_name":"Sobolev Institute of Mathematics","ror":"https://ror.org/00shc0s02","country_code":"RU","type":"facility","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210096862","https://openalex.org/I4210124601","https://openalex.org/I4210127387"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Nikolay Abrosimov","raw_affiliation_strings":["Novosibirsk State University, Novosibirsk 630090, Russia","Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk 634050, Russia","Sobolev Institute of Mathematics, Novosibirsk 630090, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Novosibirsk State University, Novosibirsk 630090, Russia","institution_ids":["https://openalex.org/I188973947"]},{"raw_affiliation_string":"Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk 634050, Russia","institution_ids":["https://openalex.org/I196355604"]},{"raw_affiliation_string":"Sobolev Institute of Mathematics, Novosibirsk 630090, Russia","institution_ids":["https://openalex.org/I4210096862"]}]},{"author_position":"last","author":{"id":null,"display_name":"Bao Vuong","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I188973947","display_name":"Novosibirsk State University","ror":"https://ror.org/04t2ss102","country_code":"RU","type":"education","lineage":["https://openalex.org/I188973947"]},{"id":"https://openalex.org/I196355604","display_name":"National Research Tomsk State University","ror":"https://ror.org/02he2nc27","country_code":"RU","type":"education","lineage":["https://openalex.org/I196355604"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"Bao Vuong","raw_affiliation_strings":["Novosibirsk State University, Novosibirsk 630090, Russia","Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk 634050, Russia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Novosibirsk State University, Novosibirsk 630090, Russia","institution_ids":["https://openalex.org/I188973947"]},{"raw_affiliation_string":"Regional Scientific and Educational Mathematical Center, Tomsk State University, Tomsk 634050, Russia","institution_ids":["https://openalex.org/I196355604"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3085,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.50042058,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":93},"biblio":{"volume":"30","issue":"10","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.6967999935150146,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.6967999935150146,"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":0.27489998936653137,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.005499999970197678,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hyperbolic-space","display_name":"hyperbolic space","score":0.8240000009536743},{"id":"https://openalex.org/keywords/volume-equations","display_name":"volume equations","score":0.9169999957084656},{"id":"https://openalex.org/keywords/edge-length","display_name":"edge length","score":0.9679999947547913},{"id":"https://openalex.org/keywords/hyperbolic-cosine","display_name":"hyperbolic cosine","score":0.8100000023841858},{"id":"https://openalex.org/keywords/dihedral-angle","display_name":"dihedral angle","score":0.9570000171661377},{"id":"https://openalex.org/keywords/cosine-rule","display_name":"cosine rule","score":0.8569999933242798},{"id":"https://openalex.org/keywords/gram-matrix","display_name":"Gram matrix","score":0.8769999742507935},{"id":"https://openalex.org/keywords/existence-conditions","display_name":"existence conditions","score":0.7400000095367432},{"id":"https://openalex.org/keywords/convex-hull","display_name":"convex hull","score":0.6359999775886536}],"concepts":[{"id":"https://openalex.org/C105239961","wikidata":"https://www.wikidata.org/wiki/Q160003","display_name":"Tetrahedron","level":2,"score":0.9159514904022217},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8336758613586426},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.5002193450927734},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.4734192490577698},{"id":"https://openalex.org/C83677898","wikidata":"https://www.wikidata.org/wiki/Q1878538","display_name":"Hyperbolic space","level":2,"score":0.4550924301147461},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.45395299792289734},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.44862252473831177},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.4416659474372864},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36575907468795776},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s0218216521400071","doi":"https://doi.org/10.1142/s0218216521400071","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218216521400071","pdf_url":null,"source":{"id":"https://openalex.org/S55337115","display_name":"Journal of Knot Theory and Its Ramifications","issn_l":"0218-2165","issn":["0218-2165","1793-6527"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-c","jufo-1","norway-1","poland-70","russia-white-list-4"],"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Knot Theory and Its Ramifications","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2107.03004","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2107.03004","pdf_url":"https://arxiv.org/pdf/2107.03004","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2107.03004","doi":"https://doi.org/10.48550/arxiv.2107.03004","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2107.03004","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:2107.03004","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2107.03004","pdf_url":"https://arxiv.org/pdf/2107.03004","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3181136734.pdf","grobid_xml":"https://content.openalex.org/works/W3181136734.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W164083653","https://openalex.org/W1509033957","https://openalex.org/W1560025104","https://openalex.org/W1631732175","https://openalex.org/W1995697411","https://openalex.org/W1997626431","https://openalex.org/W2013110893","https://openalex.org/W2029163643","https://openalex.org/W2129480560","https://openalex.org/W2764278474","https://openalex.org/W2773591465"],"related_works":["https://openalex.org/W2296243042","https://openalex.org/W2106618694","https://openalex.org/W1995697411","https://openalex.org/W4295942364","https://openalex.org/W3009286427","https://openalex.org/W2013110893","https://openalex.org/W1997626431","https://openalex.org/W4394658608","https://openalex.org/W7097198737","https://openalex.org/W1631732175"],"abstract_inverted_index":{"We":[0,62],"consider":[1],"a":[2,7,12,72,93,106,123,134],"compact":[3],"hyperbolic":[4,22,56,107],"tetrahedron":[5,53,73,108,135],"of":[6,15,34,48,51,58,71,92,105,111,126,133,139,144],"general":[8],"type.":[9],"It":[10,27],"is":[11],"convex":[13],"hull":[14],"four":[16],"points":[17],"called":[18],"vertices":[19],"in":[20,74,89,109,137],"the":[21,32,46,52,69,90,103,112,131,140,145],"space":[23],"[Formula:":[24,75],"see":[25,76],"text].":[26,77],"can":[28,119],"be":[29,120],"determined":[30],"by":[31,55],"set":[33],"six":[35],"edge":[36,49,60,87,113,141],"lengths":[37,88],"up":[38],"to":[39],"isometry.":[40],"For":[41],"further":[42],"considerations,":[43],"we":[44,79,97],"use":[45],"notion":[47],"matrix":[50,142],"formed":[54],"cosines":[57],"its":[59],"lengths.":[61,114],"establish":[63],"necessary":[64],"and":[65,86],"sufficient":[66],"conditions":[67],"for":[68,130],"existence":[70],"Then":[78],"find":[80],"relations":[81],"between":[82],"their":[83],"dihedral":[84],"angles":[85],"form":[91],"cosine":[94],"rule.":[95],"Finally,":[96],"obtain":[98],"exact":[99],"integral":[100],"formula":[101,118,129],"expressing":[102],"volume":[104,117,132],"terms":[110,138],"The":[115],"latter":[116],"regarded":[121],"as":[122],"new":[124],"version":[125],"classical":[127],"Sforza\u2019s":[128],"but":[136],"instead":[143],"Gram":[146],"matrix.":[147]},"counts_by_year":[{"year":2022,"cited_by_count":1}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2021-07-19T00:00:00"}
