{"id":"https://openalex.org/W2131359258","doi":"https://doi.org/10.1142/s0217595915400059","title":"A Computable Characterization of the Extrinsic Mean of Reflection Shapes and Its Asymptotic Properties","display_name":"A Computable Characterization of the Extrinsic Mean of Reflection Shapes and Its Asymptotic Properties","publication_year":2015,"publication_date":"2015-02-01","ids":{"openalex":"https://openalex.org/W2131359258","doi":"https://doi.org/10.1142/s0217595915400059","mag":"2131359258"},"language":"en","primary_location":{"id":"doi:10.1142/s0217595915400059","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0217595915400059","pdf_url":null,"source":{"id":"https://openalex.org/S1254427","display_name":"Asia Pacific Journal of Operational Research","issn_l":"0217-5959","issn":["0217-5959","1793-7019"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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":"Asia-Pacific Journal of Operational Research","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064906166","display_name":"Chao Ding","orcid":"https://orcid.org/0000-0001-5721-9650"},"institutions":[{"id":"https://openalex.org/I43439940","display_name":"University of Southampton","ror":"https://ror.org/01ryk1543","country_code":"GB","type":"education","lineage":["https://openalex.org/I43439940"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Chao Ding","raw_affiliation_strings":["School of Mathematics, University of Southampton, National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Southampton SO17 1BJ, United Kingdom"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Southampton, National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Southampton SO17 1BJ, United Kingdom","institution_ids":["https://openalex.org/I43439940"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5112621269","display_name":"Houduo Qi","orcid":"https://orcid.org/0000-0003-3481-4814"},"institutions":[{"id":"https://openalex.org/I43439940","display_name":"University of Southampton","ror":"https://ror.org/01ryk1543","country_code":"GB","type":"education","lineage":["https://openalex.org/I43439940"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Hou-Duo Qi","raw_affiliation_strings":["School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom","School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom","institution_ids":["https://openalex.org/I43439940"]},{"raw_affiliation_string":"School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom","institution_ids":["https://openalex.org/I43439940"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5064906166"],"corresponding_institution_ids":["https://openalex.org/I43439940"],"apc_list":null,"apc_paid":null,"fwci":0.5288,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.67526983,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"32","issue":"01","first_page":"1540005","last_page":"1540005"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12417","display_name":"Morphological variations and asymmetry","score":0.9993000030517578,"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/T12417","display_name":"Morphological variations and asymmetry","score":0.9993000030517578,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9502000212669373,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T13487","display_name":"Statistical and numerical algorithms","score":0.9384999871253967,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/reflection","display_name":"Reflection (computer programming)","score":0.76108717918396},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.651924192905426},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.5846498608589172},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.4959660470485687},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.49551355838775635},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.48787403106689453},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.45518845319747925},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.4418550431728363},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42806458473205566},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.41794946789741516},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3520854115486145},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.25463348627090454},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.1963190734386444},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13834092020988464},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10457953810691833},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.06970778107643127}],"concepts":[{"id":"https://openalex.org/C65682993","wikidata":"https://www.wikidata.org/wiki/Q1056451","display_name":"Reflection (computer programming)","level":2,"score":0.76108717918396},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.651924192905426},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5846498608589172},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.4959660470485687},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.49551355838775635},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.48787403106689453},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.45518845319747925},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.4418550431728363},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42806458473205566},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.41794946789741516},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3520854115486145},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.25463348627090454},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.1963190734386444},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13834092020988464},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10457953810691833},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.06970778107643127},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s0217595915400059","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0217595915400059","pdf_url":null,"source":{"id":"https://openalex.org/S1254427","display_name":"Asia Pacific Journal of Operational Research","issn_l":"0217-5959","issn":["0217-5959","1793-7019"],"is_oa":false,"is_in_doaj":false,"is_core":true,"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":"Asia-Pacific Journal of Operational Research","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:wsi:apjorx:v:32:y:2015:i:01:n:s0217595915400059","is_oa":false,"landing_page_url":"http://www.worldscientific.com/doi/abs/10.1142/S0217595915400059","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"},{"id":"pmh:oai:RePEc:wsi:apjorx:v:32:y:2015:i:01:p:1540005-1-1540005-16","is_oa":false,"landing_page_url":"http://www.worldscinet.com/cgi-bin/details.cgi?type=html&id=pii:S0217595915400059","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.5,"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities"}],"awards":[{"id":"https://openalex.org/G1261773691","display_name":null,"funder_award_id":"EP/K007645/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1572610412","https://openalex.org/W1996440619","https://openalex.org/W2008154674","https://openalex.org/W2026231217","https://openalex.org/W2086940601","https://openalex.org/W2087219114","https://openalex.org/W2091804476","https://openalex.org/W2114261252","https://openalex.org/W2120176508","https://openalex.org/W2146932984","https://openalex.org/W2169628220","https://openalex.org/W2217701558","https://openalex.org/W2500756573","https://openalex.org/W2798707604","https://openalex.org/W2949519581","https://openalex.org/W2951656971","https://openalex.org/W4205548518","https://openalex.org/W4252097634","https://openalex.org/W4255333822","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2952384777","https://openalex.org/W4238233526","https://openalex.org/W2903675318","https://openalex.org/W4294294347","https://openalex.org/W2279239548","https://openalex.org/W115379796","https://openalex.org/W2404658402","https://openalex.org/W2610298398","https://openalex.org/W4281992692","https://openalex.org/W4233426741"],"abstract_inverted_index":{"The":[0],"reflection":[1,25,36,69,84,108,120],"shapes":[2,70,85,121],"of":[3,12,22,43,63,67,78,82,92,101,106,118],"configurations":[4],"in":[5,71],"\u211c":[6],"m":[7],"with":[8],"k":[9],"landmarks":[10],"consist":[11],"all":[13,44],"the":[14,29,35,40,64,68,75,79,83,90,102,107,114,119],"geometric":[15],"information":[16],"that":[17],"is":[18,86,122],"invariant":[19],"under":[20],"compositions":[21],"similarity":[23],"and":[24],"transformations.":[26],"By":[27,88],"considering":[28],"corresponding":[30],"Schoenberg":[31],"embedding,":[32],"we":[33,58,95],"embed":[34],"shape":[37],"space":[38,42],"into":[39],"Euclidean":[41],"(k":[45,49],"-":[46,50],"1)":[47,51],"by":[48],"real":[52],"symmetric":[53],"matrices.":[54],"In":[55],"this":[56],"paper,":[57],"provide":[59],"a":[60,97,111],"computable":[61],"formula":[62],"extrinsic":[65,80,104],"mean":[66,81,105],"arbitrary":[72],"dimensions.":[73],"Moreover,":[74],"asymptotic":[76],"analysis":[77],"studied.":[87],"using":[89],"differentiability":[91],"spectral":[93],"operators,":[94],"obtain":[96],"central":[98],"limit":[99],"theorem":[100],"sample":[103],"shapes.":[109],"As":[110],"direct":[112],"application,":[113],"two-example":[115],"hypothesis":[116],"test":[117],"also":[123],"derived.":[124]},"counts_by_year":[{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}