{"id":"https://openalex.org/W4387107249","doi":"https://doi.org/10.2478/forma-2023-0002","title":"Differentiation on Interval","display_name":"Differentiation on Interval","publication_year":2023,"publication_date":"2023-09-01","ids":{"openalex":"https://openalex.org/W4387107249","doi":"https://doi.org/10.2478/forma-2023-0002"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2023-0002","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0002","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0002","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://sciendo.com/pdf/10.2478/forma-2023-0002","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015450295","display_name":"Noboru Endou","orcid":"https://orcid.org/0000-0002-5922-2332"},"institutions":[{"id":"https://openalex.org/I4210113483","display_name":"National Institute of Technology, Gifu College","ror":"https://ror.org/025b2hx26","country_code":"JP","type":"education","lineage":["https://openalex.org/I4210113483","https://openalex.org/I4210120810"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Noboru Endou","raw_affiliation_strings":["National Institute of Technology, Gifu College , 2236-2 Kamimakuwa, Motosu, Gifu , Japan"],"raw_orcid":"https://orcid.org/0000-0002-5922-2332","affiliations":[{"raw_affiliation_string":"National Institute of Technology, Gifu College , 2236-2 Kamimakuwa, Motosu, Gifu , Japan","institution_ids":["https://openalex.org/I4210113483"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5015450295"],"corresponding_institution_ids":["https://openalex.org/I4210113483"],"apc_list":null,"apc_paid":null,"fwci":0.4773,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":{"value":0.69349819,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"31","issue":"1","first_page":"9","last_page":"21"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.9962999820709229,"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/interval","display_name":"Interval (graph theory)","score":0.7284271121025085},{"id":"https://openalex.org/keywords/automatic-differentiation","display_name":"Automatic differentiation","score":0.7058071494102478},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.6961800456047058},{"id":"https://openalex.org/keywords/interval-arithmetic","display_name":"Interval arithmetic","score":0.5853870511054993},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5734645128250122},{"id":"https://openalex.org/keywords/hol","display_name":"HOL","score":0.5718310475349426},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.5320794582366943},{"id":"https://openalex.org/keywords/chain-rule","display_name":"Chain rule (probability)","score":0.5255205631256104},{"id":"https://openalex.org/keywords/relation","display_name":"Relation (database)","score":0.45324885845184326},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4244352877140045},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3602949380874634},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.34112581610679626},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3351893424987793},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2922975420951843},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.26038339734077454},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.17014774680137634},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10750120878219604},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.07357382774353027}],"concepts":[{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.7284271121025085},{"id":"https://openalex.org/C133512626","wikidata":"https://www.wikidata.org/wiki/Q787371","display_name":"Automatic differentiation","level":3,"score":0.7058071494102478},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.6961800456047058},{"id":"https://openalex.org/C191252586","wikidata":"https://www.wikidata.org/wiki/Q1671453","display_name":"Interval arithmetic","level":3,"score":0.5853870511054993},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5734645128250122},{"id":"https://openalex.org/C17435882","wikidata":"https://www.wikidata.org/wiki/Q17030435","display_name":"HOL","level":2,"score":0.5718310475349426},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.5320794582366943},{"id":"https://openalex.org/C33825631","wikidata":"https://www.wikidata.org/wiki/Q17004731","display_name":"Chain rule (probability)","level":5,"score":0.5255205631256104},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.45324885845184326},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4244352877140045},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3602949380874634},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.34112581610679626},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3351893424987793},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2922975420951843},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26038339734077454},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.17014774680137634},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10750120878219604},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.07357382774353027},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C107673813","wikidata":"https://www.wikidata.org/wiki/Q812534","display_name":"Bayesian probability","level":2,"score":0.0},{"id":"https://openalex.org/C57830394","wikidata":"https://www.wikidata.org/wiki/Q278079","display_name":"Posterior probability","level":3,"score":0.0},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C49698424","wikidata":"https://www.wikidata.org/wiki/Q1989192","display_name":"Law of total probability","level":4,"score":0.0},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2478/forma-2023-0002","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0002","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0002","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/15401","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/15401","pdf_url":"https://repozytorium.uwb.edu.pl/jspui/bitstream/11320/15401/1/10.2478_forma-2023-0002.pdf","source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2023-0002","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0002","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0002","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.4300000071525574,"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10"}],"awards":[{"id":"https://openalex.org/G5043624120","display_name":"Expansion of the Reasoning Library of Analysis for Formal Verification","funder_award_id":"23K11242","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4387107249.pdf"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W36540068","https://openalex.org/W112168304","https://openalex.org/W1591577758","https://openalex.org/W2143235494","https://openalex.org/W2184838578","https://openalex.org/W2295728669","https://openalex.org/W2770430340","https://openalex.org/W4285324967","https://openalex.org/W4285325472","https://openalex.org/W4285326779"],"related_works":["https://openalex.org/W2097846334","https://openalex.org/W166200282","https://openalex.org/W2009634354","https://openalex.org/W4205764719","https://openalex.org/W4318903784","https://openalex.org/W2958554379","https://openalex.org/W4389139242","https://openalex.org/W2969957441","https://openalex.org/W3149592096","https://openalex.org/W1528742518"],"abstract_inverted_index":{"Summary":[0],"This":[1],"article":[2],"generalizes":[3],"the":[4,10,50,61,99,112,115,146,160,181],"differential":[5],"method":[6],"on":[7,71,87,95,133,148],"intervals,":[8],"using":[9],"Mizar":[11,24],"system":[12],"[2],":[13],"[3],":[14],"[12].":[15],"Differentiation":[16],"of":[17,32,34,117,184],"real":[18,35,141],"one-variable":[19,142],"functions":[20],"is":[21,53,68,138,164],"introduced":[22],"in":[23,37,129,168,180],"[13],":[25],"along":[26],"standard":[27],"lines":[28],"(for":[29],"interesting":[30],"survey":[31],"formalizations":[33],"analysis":[36],"various":[38,155],"proof-assistants":[39],"like":[40],"ACL2":[41],"[11],":[42],"Isabelle/HOL":[43],"[10],":[44],"Coq":[45],"[4],":[46],"see":[47],"[5]),":[48],"but":[49,109],"differentiable":[51],"interval":[52,90,119],"restricted":[54],"to":[55,81,84,140,170],"open":[56],"intervals.":[57,127],"However,":[58],"when":[59],"considering":[60],"relationship":[62],"with":[63],"integration":[64,67],"[9],":[65],"since":[66],"an":[69,118,165],"operation":[70],"a":[72,88,96,124],"closed":[73,89,97],"interval,":[74,98],"it":[75,137],"would":[76],"be":[77,82],"convenient":[78],"for":[79],"differentiation":[80,125,147,171],"able":[83],"handle":[85],"derivates":[86],"as":[91,123,154],"well.":[92],"Regarding":[93],"differentiability":[94,103],"right":[100],"and":[101,120,151,172],"left":[102],"have":[104],"already":[105],"been":[106],"formalized":[107,176],"[6],":[108],"they":[110],"are":[111],"derivatives":[113],"at":[114],"endpoints":[116],"not":[121],"demonstrated":[122],"over":[126],"Therefore,":[128],"this":[130],"paper,":[131],"based":[132],"these":[134],"results,":[135],"although":[136],"limited":[139],"functions,":[143],"we":[144],"formalize":[145],"arbitrary":[149],"intervals":[150],"summarize":[152],"them":[153],"basic":[156],"propositions.":[157],"In":[158],"particular,":[159],"chain":[161],"rule":[162],"[1]":[163],"important":[166],"formula":[167],"relation":[169],"integration,":[173],"extending":[174],"recent":[175],"results":[177],"[7],":[178],"[8]":[179],"latter":[182],"field":[183],"research.":[185]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1}],"updated_date":"2026-07-09T07:52:08.696243","created_date":"2025-10-10T00:00:00"}