{"id":"https://openalex.org/W3021122852","doi":"https://doi.org/10.1137/20m1338265","title":"Growth in Some Finite Three-Dimensional Matrix Groups","display_name":"Growth in Some Finite Three-Dimensional Matrix Groups","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3021122852","doi":"https://doi.org/10.1137/20m1338265","mag":"3021122852"},"language":"en","primary_location":{"id":"doi:10.1137/20m1338265","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1338265","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2005.05077","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Brendan Murphy","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Brendan Murphy","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5103059249","display_name":"James Thomas Wheeler","orcid":"https://orcid.org/0000-0002-3644-4629"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"James Wheeler","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0002-3644-4629","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.17947872,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"34","issue":"3","first_page":"1984","last_page":"1998"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11476","display_name":"Graph theory and applications","score":0.9883999824523926,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9847000241279602,"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/multiplicative-function","display_name":"Multiplicative function","score":0.7822585701942444},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6465334296226501},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.6044821739196777},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.5892730355262756},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5801699161529541},{"id":"https://openalex.org/keywords/incidence-matrix","display_name":"Incidence matrix","score":0.5230008959770203},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.501788854598999},{"id":"https://openalex.org/keywords/triangular-matrix","display_name":"Triangular matrix","score":0.4988410472869873},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.47821328043937683},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.47165337204933167},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4716345965862274},{"id":"https://openalex.org/keywords/generating-set-of-a-group","display_name":"Generating set of a group","score":0.45284783840179443},{"id":"https://openalex.org/keywords/matrix-multiplication","display_name":"Matrix multiplication","score":0.4300524592399597},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.41222789883613586},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3107066750526428},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2948017120361328},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22573116421699524},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16079002618789673},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14356699585914612},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08602729439735413},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07622620463371277}],"concepts":[{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.7822585701942444},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6465334296226501},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.6044821739196777},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.5892730355262756},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5801699161529541},{"id":"https://openalex.org/C85136909","wikidata":"https://www.wikidata.org/wiki/Q939272","display_name":"Incidence matrix","level":3,"score":0.5230008959770203},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.501788854598999},{"id":"https://openalex.org/C104528550","wikidata":"https://www.wikidata.org/wiki/Q506265","display_name":"Triangular matrix","level":3,"score":0.4988410472869873},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.47821328043937683},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.47165337204933167},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4716345965862274},{"id":"https://openalex.org/C18116961","wikidata":"https://www.wikidata.org/wiki/Q734209","display_name":"Generating set of a group","level":2,"score":0.45284783840179443},{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.4300524592399597},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.41222789883613586},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3107066750526428},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2948017120361328},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22573116421699524},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16079002618789673},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14356699585914612},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08602729439735413},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07622620463371277},{"id":"https://openalex.org/C62611344","wikidata":"https://www.wikidata.org/wiki/Q1062658","display_name":"Node (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C96442724","wikidata":"https://www.wikidata.org/wiki/Q242188","display_name":"Invertible matrix","level":2,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"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/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"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":4,"locations":[{"id":"doi:10.1137/20m1338265","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m1338265","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2005.05077","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2005.05077","pdf_url":"https://arxiv.org/pdf/2005.05077","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,"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":"mag:3021122852","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/2005.05077.pdf","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,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.2005.05077","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2005.05077","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,"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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2005.05077","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2005.05077","pdf_url":"https://arxiv.org/pdf/2005.05077","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,"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":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.6700000166893005}],"awards":[{"id":"https://openalex.org/G666958914","display_name":null,"funder_award_id":"RPG\u20102017\u2013371","funder_id":"https://openalex.org/F4320319993","funder_display_name":"Leverhulme Trust"}],"funders":[{"id":"https://openalex.org/F4320319993","display_name":"Leverhulme Trust","ror":"https://ror.org/012mzw131"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W1235945937","https://openalex.org/W1513029095","https://openalex.org/W1555768633","https://openalex.org/W1595225713","https://openalex.org/W1822888486","https://openalex.org/W1964692317","https://openalex.org/W2000713788","https://openalex.org/W2016920904","https://openalex.org/W2129684959","https://openalex.org/W2137285381","https://openalex.org/W2171162943","https://openalex.org/W2902266326","https://openalex.org/W2942579974","https://openalex.org/W2962855781","https://openalex.org/W2963200630","https://openalex.org/W2963470755","https://openalex.org/W2963907663","https://openalex.org/W2963992709","https://openalex.org/W2973699483","https://openalex.org/W2982852936","https://openalex.org/W2986162662"],"related_works":["https://openalex.org/W2080059175","https://openalex.org/W2779271145","https://openalex.org/W2019009914","https://openalex.org/W3104525903","https://openalex.org/W3098715261","https://openalex.org/W2735244160","https://openalex.org/W1802110487","https://openalex.org/W132969035","https://openalex.org/W3165456727","https://openalex.org/W1501757166","https://openalex.org/W3146621223","https://openalex.org/W2131439180","https://openalex.org/W3105371085","https://openalex.org/W32983157","https://openalex.org/W3041014133","https://openalex.org/W2009003139","https://openalex.org/W3103049123","https://openalex.org/W3101156383","https://openalex.org/W3098233318","https://openalex.org/W3100248043"],"abstract_inverted_index":{"We":[0,61],"study":[1],"the":[2,20,68],"growth":[3],"of":[4,22],"product":[5,33],"sets":[6],"in":[7],"some":[8],"finite":[9,30],"three-dimensional":[10],"matrix":[11],"groups.":[12],"In":[13],"particular,":[14],"we":[15],"prove":[16,63],"two":[17],"results":[18],"about":[19],"group":[21],"$2\\times":[23],"2$":[24],"upper":[25],"triangular":[26],"matrices":[27],"over":[28],"arbitrary":[29],"fields:":[31],"a":[32],"set":[34],"estimate":[35,44],"using":[36,45],"techniques":[37],"from":[38],"multiplicative":[39],"combinatorics,":[40],"and":[41],"an":[42,64],"energy":[43,49,65],"incidence":[46],"geometry.":[47],"The":[48],"method":[50],"gives":[51],"better":[52],"quantitative":[53],"results,":[54],"but":[55],"only":[56],"applies":[57],"to":[58],"small":[59],"sets.":[60],"also":[62],"result":[66],"for":[67],"Heisenberg":[69],"group.":[70]},"counts_by_year":[],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2020-05-13T00:00:00"}
