{"id":"https://openalex.org/W2964236115","doi":"https://doi.org/10.1007/s00211-017-0904-8","title":"Discrete maximal regularity of time-stepping schemes for fractional evolution equations","display_name":"Discrete maximal regularity of time-stepping schemes for fractional evolution equations","publication_year":2017,"publication_date":"2017-07-22","ids":{"openalex":"https://openalex.org/W2964236115","doi":"https://doi.org/10.1007/s00211-017-0904-8","mag":"2964236115","pmid":"https://pubmed.ncbi.nlm.nih.gov/29375159"},"language":"en","primary_location":{"id":"doi:10.1007/s00211-017-0904-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00211-017-0904-8","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs00211-017-0904-8.pdf","source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007%2Fs00211-017-0904-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039605463","display_name":"Bangti Jin","orcid":"https://orcid.org/0000-0002-3775-9155"},"institutions":[{"id":"https://openalex.org/I45129253","display_name":"University College London","ror":"https://ror.org/02jx3x895","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I45129253"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Bangti Jin","raw_affiliation_strings":["1Department of Computer Science, University College London, Gower Street, London, WC1E 6BT UK","Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"1Department of Computer Science, University College London, Gower Street, London, WC1E 6BT UK","institution_ids":[]},{"raw_affiliation_string":"Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, UK","institution_ids":["https://openalex.org/I45129253"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039197834","display_name":"Buyang Li","orcid":"https://orcid.org/0000-0001-7566-3464"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Buyang Li","raw_affiliation_strings":["2Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong","Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"2Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong","institution_ids":[]},{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5002445122","display_name":"Zhi Zhou","orcid":"https://orcid.org/0000-0002-3439-0313"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Zhi Zhou","raw_affiliation_strings":["2Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong","Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"2Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong","institution_ids":[]},{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong","institution_ids":["https://openalex.org/I14243506"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5039605463"],"corresponding_institution_ids":["https://openalex.org/I45129253"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":14.8793,"has_fulltext":true,"cited_by_count":86,"citation_normalized_percentile":{"value":0.99498807,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"138","issue":"1","first_page":"101","last_page":"131"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12727","display_name":"Differential Equations and Numerical Methods","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10288","display_name":"Fractional Differential Equations Solutions","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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":0.9988999962806702,"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/mathematics","display_name":"Mathematics","score":0.9090225696563721},{"id":"https://openalex.org/keywords/multiplier","display_name":"Multiplier (economics)","score":0.6665809154510498},{"id":"https://openalex.org/keywords/backward-euler-method","display_name":"Backward Euler method","score":0.6152251362800598},{"id":"https://openalex.org/keywords/fractional-calculus","display_name":"Fractional calculus","score":0.565247654914856},{"id":"https://openalex.org/keywords/eulers-formula","display_name":"Euler's formula","score":0.4931792616844177},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.47951534390449524},{"id":"https://openalex.org/keywords/convolution","display_name":"Convolution (computer science)","score":0.4663611650466919},{"id":"https://openalex.org/keywords/time-derivative","display_name":"Time derivative","score":0.4571724534034729},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4489606022834778},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.4381408989429474},{"id":"https://openalex.org/keywords/fourier-analysis","display_name":"Fourier analysis","score":0.43189936876296997},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4208250641822815},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.319112092256546},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.2979205846786499}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9090225696563721},{"id":"https://openalex.org/C124584101","wikidata":"https://www.wikidata.org/wiki/Q1053266","display_name":"Multiplier (economics)","level":2,"score":0.6665809154510498},{"id":"https://openalex.org/C768646","wikidata":"https://www.wikidata.org/wiki/Q2736820","display_name":"Backward Euler method","level":3,"score":0.6152251362800598},{"id":"https://openalex.org/C154249771","wikidata":"https://www.wikidata.org/wiki/Q1339058","display_name":"Fractional calculus","level":2,"score":0.565247654914856},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.4931792616844177},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47951534390449524},{"id":"https://openalex.org/C45347329","wikidata":"https://www.wikidata.org/wiki/Q5166604","display_name":"Convolution (computer science)","level":3,"score":0.4663611650466919},{"id":"https://openalex.org/C17685861","wikidata":"https://www.wikidata.org/wiki/Q6960532","display_name":"Time derivative","level":2,"score":0.4571724534034729},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4489606022834778},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4381408989429474},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.43189936876296997},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4208250641822815},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.319112092256546},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.2979205846786499},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"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/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","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/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C139719470","wikidata":"https://www.wikidata.org/wiki/Q39680","display_name":"Macroeconomics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s00211-017-0904-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00211-017-0904-8","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs00211-017-0904-8.pdf","source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},{"id":"pmid:29375159","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/29375159","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische mathematik","raw_type":null},{"id":"pmh:oai:ira.lib.polyu.edu.hk:10397/74469","is_oa":true,"landing_page_url":"http://hdl.handle.net/10397/74469","pdf_url":null,"source":{"id":"https://openalex.org/S4306400205","display_name":"PolyU Institutional Research Archive (Hong Kong Polytechnic University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I14243506","host_organization_name":"Hong Kong Polytechnic University","host_organization_lineage":["https://openalex.org/I14243506"],"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":null,"raw_type":"Journal/Magazine Article"},{"id":"pmh:oai:pubmedcentral.nih.gov:5762870","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/5762870","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Numer Math (Heidelb)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.1007/s00211-017-0904-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00211-017-0904-8","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs00211-017-0904-8.pdf","source":{"id":"https://openalex.org/S4210205683","display_name":"Numerische Mathematik","issn_l":"0029-599X","issn":["0029-599X","0945-3245"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Numerische Mathematik","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2000473417","display_name":null,"funder_award_id":"ARO MURI Grant W911NF-15-1-0562","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G2208209013","display_name":null,"funder_award_id":"MURI Grant W911NF-15-1-0562","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G3552121337","display_name":null,"funder_award_id":"W911NF-15-1-0562","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"},{"id":"https://openalex.org/G3722276629","display_name":null,"funder_award_id":"EP/M025160/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G5385673730","display_name":null,"funder_award_id":"W911NF-15-1-","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G7452299184","display_name":null,"funder_award_id":"W911NF","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G7640431553","display_name":"Sparsity Regularization for Inverse Problems -- Theory, Algorithm and Application","funder_award_id":"EP/M025160/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G7913808730","display_name":null,"funder_award_id":"W911NF-15-1-0562","funder_id":"https://openalex.org/F4320333591","funder_display_name":"Multidisciplinary University Research Initiative"},{"id":"https://openalex.org/G8944192383","display_name":null,"funder_award_id":"W911NF-15-1-0562","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"}],"funders":[{"id":"https://openalex.org/F4320320286","display_name":"University College London","ror":"https://ror.org/02jx3x895"},{"id":"https://openalex.org/F4320333591","display_name":"Multidisciplinary University Research Initiative","ror":null},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"},{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2964236115.pdf","grobid_xml":"https://content.openalex.org/works/W2964236115.grobid-xml"},"referenced_works_count":55,"referenced_works":["https://openalex.org/W560409768","https://openalex.org/W628383724","https://openalex.org/W1480637530","https://openalex.org/W1510033248","https://openalex.org/W1510266502","https://openalex.org/W1527296550","https://openalex.org/W1560185428","https://openalex.org/W1568437275","https://openalex.org/W1584611818","https://openalex.org/W1901372027","https://openalex.org/W1967436950","https://openalex.org/W1969312973","https://openalex.org/W1977696410","https://openalex.org/W1982386299","https://openalex.org/W1984507032","https://openalex.org/W2007203088","https://openalex.org/W2031029580","https://openalex.org/W2042322027","https://openalex.org/W2048909372","https://openalex.org/W2049030310","https://openalex.org/W2053695265","https://openalex.org/W2054872369","https://openalex.org/W2073659542","https://openalex.org/W2079024344","https://openalex.org/W2080951455","https://openalex.org/W2081214876","https://openalex.org/W2081714730","https://openalex.org/W2083565006","https://openalex.org/W2111271983","https://openalex.org/W2164094632","https://openalex.org/W2165076033","https://openalex.org/W2166518045","https://openalex.org/W2187871078","https://openalex.org/W2243872394","https://openalex.org/W2274070693","https://openalex.org/W2280020194","https://openalex.org/W2302793323","https://openalex.org/W2435960546","https://openalex.org/W2437878828","https://openalex.org/W2479918603","https://openalex.org/W2594191637","https://openalex.org/W2612471483","https://openalex.org/W2787959293","https://openalex.org/W2962904825","https://openalex.org/W2963253432","https://openalex.org/W2963353751","https://openalex.org/W2963582054","https://openalex.org/W2963914435","https://openalex.org/W2964032975","https://openalex.org/W2964083141","https://openalex.org/W4205158050","https://openalex.org/W4230923404","https://openalex.org/W4247635576","https://openalex.org/W4253840677","https://openalex.org/W4300857049"],"related_works":["https://openalex.org/W3162423291","https://openalex.org/W2911367846","https://openalex.org/W1697484951","https://openalex.org/W4297084911","https://openalex.org/W2021837978","https://openalex.org/W3005204046","https://openalex.org/W2037834696","https://openalex.org/W2104831383","https://openalex.org/W2525825098","https://openalex.org/W2754120606"],"abstract_inverted_index":{"In":[0],"this":[1],"work,":[2],"we":[3],"establish":[4],"the":[5,52,63,70,104],"maximal":[6],"$$\\ell":[7],"^p$$":[8],"-regularity":[9],"for":[10,15,69,107],"several":[11],"time":[12],"stepping":[13],"schemes":[14,38],"a":[16,22,59],"fractional":[17,23,60],"evolution":[18],"model,":[19],"which":[20],"involves":[21],"derivative":[24],"of":[25,62],"order":[26],"$$\\alpha":[27,31],"\\in":[28],"(0,2)$$":[29],",":[30,34],"\\ne":[32],"1$$":[33],"in":[35],"time.":[36],"These":[37,101],"include":[39,72],"convolution":[40],"quadratures":[41],"generated":[42],"by":[43],"backward":[44,49],"Euler":[45,56],"method":[46,57],"and":[47,58,87],"second-order":[48],"difference":[50],"formula,":[51],"L1":[53],"scheme,":[54],"explicit":[55],"variant":[61],"Crank\u2013Nicolson":[64],"method.":[65],"The":[66],"main":[67],"tools":[68],"analysis":[71],"operator-valued":[73],"Fourier":[74],"multiplier":[75],"theorem":[76],"due":[77,91],"to":[78,92],"Weis":[79],"(Math":[80],"Ann":[81],"319:735\u2013758,":[82],"2001.":[83,97],"doi:":[84,98],"10.1007/PL00004457":[85],")":[86],"its":[88],"discrete":[89],"analogue":[90],"Blunck":[93],"(Stud":[94],"Math":[95],"146:157\u2013176,":[96],"10.4064/sm146-2-3":[99],").":[100],"results":[102,106],"generalize":[103],"corresponding":[105],"parabolic":[108],"problems.":[109]},"counts_by_year":[{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":10},{"year":2021,"cited_by_count":15},{"year":2020,"cited_by_count":16},{"year":2019,"cited_by_count":16},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":4}],"updated_date":"2026-05-16T08:24:45.110214","created_date":"2025-10-10T00:00:00"}
