{"id":"https://openalex.org/W1840515818","doi":"https://doi.org/10.2422/2036-2145.2011.3.08","title":"Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities","display_name":"Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities","publication_year":2011,"publication_date":"2011-10-05","ids":{"openalex":"https://openalex.org/W1840515818","doi":"https://doi.org/10.2422/2036-2145.2011.3.08","mag":"1840515818"},"language":"en","primary_location":{"id":"doi:10.2422/2036-2145.2011.3.08","doi":"https://doi.org/10.2422/2036-2145.2011.3.08","is_oa":true,"landing_page_url":"https://doi.org/10.2422/2036-2145.2011.3.08","pdf_url":"https://journals.sns.it/index.php/annaliscienze/article/download/248/241","source":{"id":"https://openalex.org/S4210213170","display_name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","issn_l":"0391-173X","issn":["0391-173X","2036-2145"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-1"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://journals.sns.it/index.php/annaliscienze/article/download/248/241","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5038797648","display_name":"Scott N. Armstrong","orcid":"https://orcid.org/0000-0002-1395-081X","observed_orcids":["https://orcid.org/0000-0002-1395-081X"]},"institutions":[{"id":"https://openalex.org/I40434647","display_name":"Universit\u00e9 Paris Nanterre","ror":"https://ror.org/013bkhk48","country_code":"FR","type":"education","lineage":["https://openalex.org/I40434647"]},{"id":"https://openalex.org/I4210100624","display_name":"Centre d'Analyse et de Math\u00e9matique Sociales","ror":"https://ror.org/00vb8fc53","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210100624","https://openalex.org/I4210150854","https://openalex.org/I90669466"]},{"id":"https://openalex.org/I90669466","display_name":"\u00c9cole des hautes \u00e9tudes en sciences sociales","ror":"https://ror.org/02d9dg697","country_code":"FR","type":"facility","lineage":["https://openalex.org/I90669466"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Scott N. Armstrong","raw_affiliation_strings":["UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex , France","EHESS 54 bd Raspail 75270 Paris Cedex 06, France","UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex, France and CAMS,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex , France","institution_ids":["https://openalex.org/I40434647"]},{"raw_affiliation_string":"EHESS 54 bd Raspail 75270 Paris Cedex 06, France","institution_ids":["https://openalex.org/I90669466"]},{"raw_affiliation_string":"UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex, France and CAMS,","institution_ids":["https://openalex.org/I40434647","https://openalex.org/I4210100624"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024241586","display_name":"Boyan Sirakov","orcid":"https://orcid.org/0000-0003-4601-6350","observed_orcids":["https://orcid.org/0000-0003-4601-6350"]},"institutions":[{"id":"https://openalex.org/I40434647","display_name":"Universit\u00e9 Paris Nanterre","ror":"https://ror.org/013bkhk48","country_code":"FR","type":"education","lineage":["https://openalex.org/I40434647"]},{"id":"https://openalex.org/I4210100624","display_name":"Centre d'Analyse et de Math\u00e9matique Sociales","ror":"https://ror.org/00vb8fc53","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I4210100624","https://openalex.org/I4210150854","https://openalex.org/I90669466"]},{"id":"https://openalex.org/I90669466","display_name":"\u00c9cole des hautes \u00e9tudes en sciences sociales","ror":"https://ror.org/02d9dg697","country_code":"FR","type":"facility","lineage":["https://openalex.org/I90669466"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Boyan Sirakov","raw_affiliation_strings":["UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex , France","and CAMS , EHESS 54 bd Raspail 75270 Paris Cedex 06 , France","EHESS 54 bd Raspail 75270 Paris Cedex 06, France","UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex, France and CAMS,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex , France","institution_ids":["https://openalex.org/I40434647"]},{"raw_affiliation_string":"and CAMS , EHESS 54 bd Raspail 75270 Paris Cedex 06 , France","institution_ids":["https://openalex.org/I4210100624","https://openalex.org/I90669466"]},{"raw_affiliation_string":"EHESS 54 bd Raspail 75270 Paris Cedex 06, France","institution_ids":["https://openalex.org/I90669466"]},{"raw_affiliation_string":"UFR SEGMI Universit\u00e9 Paris 10 92001 Nanterre Cedex, France and CAMS,","institution_ids":["https://openalex.org/I40434647","https://openalex.org/I4210100624"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5024241586"],"corresponding_institution_ids":["https://openalex.org/I40434647","https://openalex.org/I4210100624","https://openalex.org/I90669466"],"apc_list":null,"apc_paid":null,"fwci":4.0789,"has_fulltext":true,"cited_by_count":37,"citation_normalized_percentile":{"value":0.93878234,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":88,"max":98},"biblio":{"volume":"10","issue":null,"first_page":"711","last_page":"728"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9990000128746033,"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"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.0003000000142492354,"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"}},{"id":"https://openalex.org/T10541","display_name":"Nonlinear Differential Equations Analysis","score":0.0,"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/fully-nonlinear-elliptic-equations","display_name":"fully nonlinear elliptic equations","score":0.996999979019165},{"id":"https://openalex.org/keywords/critical-exponents","display_name":"critical exponents","score":0.7310000061988831},{"id":"https://openalex.org/keywords/exterior-domains","display_name":"exterior domains","score":0.8429999947547913},{"id":"https://openalex.org/keywords/scaling-behavior","display_name":"scaling behavior","score":0.6420000195503235},{"id":"https://openalex.org/keywords/maximum-principle","display_name":"maximum principle","score":0.9700000286102295}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6693152189254761},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.6537823677062988},{"id":"https://openalex.org/C142259097","wikidata":"https://www.wikidata.org/wiki/Q5891314","display_name":"Homogeneity (statistics)","level":2,"score":0.6406542658805847},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6105604767799377},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5976932048797607},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.5760466456413269},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5352894067764282},{"id":"https://openalex.org/C164154869","wikidata":"https://www.wikidata.org/wiki/Q901927","display_name":"Critical exponent","level":3,"score":0.5339798927307129},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.516242265701294},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.48771533370018005},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.48460468649864197},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.44950827956199646},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.4422636032104492},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.43491944670677185},{"id":"https://openalex.org/C70615421","wikidata":"https://www.wikidata.org/wiki/Q339444","display_name":"Laplace's equation","level":3,"score":0.4338729977607727},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36640220880508423},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2092040479183197},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.17585501074790955},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.1097000241279602},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08879917860031128},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06439313292503357},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.05025911331176758},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.2422/2036-2145.2011.3.08","doi":"https://doi.org/10.2422/2036-2145.2011.3.08","is_oa":true,"landing_page_url":"https://doi.org/10.2422/2036-2145.2011.3.08","pdf_url":"https://journals.sns.it/index.php/annaliscienze/article/download/248/241","source":{"id":"https://openalex.org/S4210213170","display_name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","issn_l":"0391-173X","issn":["0391-173X","2036-2145"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-1"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1001.4489","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1001.4489","pdf_url":"https://arxiv.org/pdf/1001.4489","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":"pmh:oai:CiteSeerX.psu:10.1.1.393.2566","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.393.2566","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://hal.archives-ouvertes.fr/docs/00/45/61/02/PDF/Liouv_Power.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.753.9133","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.753.9133","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1001.4489.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.988.2634","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.988.2634","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"https://hal.archives-ouvertes.fr/file/index/docid/450368/filename/liouvpow.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-00450368v2","doi":null,"is_oa":true,"landing_page_url":"https://hal.science/hal-00450368v2/document","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405252976","host_organization_name":"Centre pour la Communication Scientifique Directe","host_organization_lineage":["https://openalex.org/I4405252976"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2011, 10, pp.711-728","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:numdam.org:ASNSP_2011_5_10_3_729_0","doi":null,"is_oa":false,"landing_page_url":"https://www.numdam.org/item/ASNSP_2011_5_10_3_729_0/","pdf_url":null,"source":{"id":"https://openalex.org/S4306402264","display_name":"French digital mathematics library (Numdam)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I4405278365","host_organization_name":"MathDoc","host_organization_lineage":["https://openalex.org/I4405278365"],"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.1001.4489","doi":"https://doi.org/10.48550/arxiv.1001.4489","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1001.4489","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:10.2422/2036-2145.2011.3.08","doi":"https://doi.org/10.2422/2036-2145.2011.3.08","is_oa":true,"landing_page_url":"https://doi.org/10.2422/2036-2145.2011.3.08","pdf_url":"https://journals.sns.it/index.php/annaliscienze/article/download/248/241","source":{"id":"https://openalex.org/S4210213170","display_name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","issn_l":"0391-173X","issn":["0391-173X","2036-2145"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-1"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1840515818.pdf","grobid_xml":"https://content.openalex.org/works/W1840515818.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W131752915","https://openalex.org/W400586990","https://openalex.org/W1559997346","https://openalex.org/W1584606662","https://openalex.org/W1967864386","https://openalex.org/W1968466383","https://openalex.org/W1969826220","https://openalex.org/W1984242990","https://openalex.org/W1996204566","https://openalex.org/W1999136561","https://openalex.org/W2006875973","https://openalex.org/W2009555731","https://openalex.org/W2032316144","https://openalex.org/W2056878197","https://openalex.org/W2058762375","https://openalex.org/W2061688023","https://openalex.org/W2066105192","https://openalex.org/W2075709627","https://openalex.org/W2094247338","https://openalex.org/W2129112030","https://openalex.org/W2158030819","https://openalex.org/W2163940086","https://openalex.org/W2325928540","https://openalex.org/W2964228928","https://openalex.org/W4210791600","https://openalex.org/W4301074654","https://openalex.org/W6736753013"],"related_works":["https://openalex.org/W7172256357","https://openalex.org/W2963776844","https://openalex.org/W2987591679","https://openalex.org/W1965159361","https://openalex.org/W7165615869","https://openalex.org/W2513117957","https://openalex.org/W2884924150","https://openalex.org/W2101182244","https://openalex.org/W2158030819","https://openalex.org/W7169514194"],"abstract_inverted_index":{"We":[0],"study":[1],"fully":[2],"nonlinear":[3],"elliptic":[4],"equations":[5],"such":[6],"asor":[7],"in":[8,54,77,84],"exterior":[9,56],"domains,":[10],"where":[11],"F":[12],"is":[13,87],"any":[14,55],"uniformly":[15],"elliptic,":[16],"positively":[17],"homogeneous":[18],"operator.We":[19],"show":[20],"that":[21,37],"there":[22,48],"exists":[23],"a":[24],"critical":[25,73],"exponent,":[26],"depending":[27],"on":[28,102],"the":[29,32,40,78,103,107,110],"homogeneity":[30],"of":[31,35,42,61,106],"fundamental":[33],"solution":[34],"F,":[36],"sharply":[38],"characterizes":[39],"range":[41],"p":[43],">":[44],"1":[45],"for":[46,75],"which":[47],"exist":[49],"positive":[50],"supersolutions":[51,76],"or":[52],"solutions":[53],"domain.Our":[57],"result":[58],"generalizes":[59],"theorems":[60],"Bidaut-V\u00e9ron":[62],"[6]":[63],"as":[64,66],"well":[65],"Cutri":[67],"and":[68,90,99,109],"Leoni":[69],"[11],":[70],"who":[71],"found":[72],"exponents":[74],"whole":[79],"space":[80],"R":[81],"n":[82],",":[83],"case":[85],"-F":[86],"Laplace's":[88],"operator":[89],"Pucci's":[91],"operator,":[92],"respectively.The":[93],"arguments":[94],"we":[95],"present":[96],"are":[97],"new":[98],"rely":[100],"only":[101],"scaling":[104],"properties":[105],"equation":[108],"maximum":[111],"principle.":[112]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":4}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}