{"id":"https://openalex.org/W4386282967","doi":"https://doi.org/10.3390/axioms12090845","title":"Derivative Formulas and Gradient of Functions with Non-Independent Variables","display_name":"Derivative Formulas and Gradient of Functions with Non-Independent Variables","publication_year":2023,"publication_date":"2023-08-30","ids":{"openalex":"https://openalex.org/W4386282967","doi":"https://doi.org/10.3390/axioms12090845"},"language":"en","primary_location":{"id":"doi:10.3390/axioms12090845","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090845","pdf_url":"https://www.mdpi.com/2075-1680/12/9/845/pdf?version=1693393386","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/12/9/845/pdf?version=1693393386","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112639749","display_name":"Matieyendou Lamboni","orcid":null},"institutions":[{"id":"https://openalex.org/I19894307","display_name":"Universit\u00e9 de Montpellier","ror":"https://ror.org/051escj72","country_code":"FR","type":"education","lineage":["https://openalex.org/I19894307"]},{"id":"https://openalex.org/I4210166444","display_name":"Institut de Recherche pour le D\u00e9veloppement","ror":"https://ror.org/05q3vnk25","country_code":"FR","type":"funder","lineage":["https://openalex.org/I2802818602","https://openalex.org/I4210090127","https://openalex.org/I4210131494","https://openalex.org/I4210166444"]},{"id":"https://openalex.org/I2800447973","display_name":"University of French Guiana","ror":"https://ror.org/00nb39k71","country_code":"GF","type":"education","lineage":["https://openalex.org/I2800447973"]}],"countries":["FR","GF"],"is_corresponding":true,"raw_author_name":"Matieyendou Lamboni","raw_affiliation_strings":["228-UMR Espace-Dev, University of Guyane, University of R\u00e9union, IRD, University of Montpellier, 34090 Montpellier, France","Department DFR-ST, University of Guyane, 97346 Cayenne, France","UMR Espace-Dev Guyane (Campus de Troubiran 97 300 Cayenne - French Guiana)"],"affiliations":[{"raw_affiliation_string":"228-UMR Espace-Dev, University of Guyane, University of R\u00e9union, IRD, University of Montpellier, 34090 Montpellier, France","institution_ids":["https://openalex.org/I19894307","https://openalex.org/I4210166444"]},{"raw_affiliation_string":"Department DFR-ST, University of Guyane, 97346 Cayenne, France","institution_ids":[]},{"raw_affiliation_string":"UMR Espace-Dev Guyane (Campus de Troubiran 97 300 Cayenne - French Guiana)","institution_ids":["https://openalex.org/I2800447973"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5112639749"],"corresponding_institution_ids":["https://openalex.org/I19894307","https://openalex.org/I2800447973","https://openalex.org/I4210166444"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":1.3242,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.82575408,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":99},"biblio":{"volume":"12","issue":"9","first_page":"845","last_page":"845"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9886999726295471,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9886999726295471,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T13487","display_name":"Statistical and numerical algorithms","score":0.9872999787330627,"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/T11186","display_name":"Hydrology and Drought Analysis","score":0.9693999886512756,"subfield":{"id":"https://openalex.org/subfields/2306","display_name":"Global and Planetary Change"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.8123225569725037},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.777121365070343},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.717106282711029},{"id":"https://openalex.org/keywords/variables","display_name":"Variables","score":0.6303574442863464},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.6203804016113281},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.5817480087280273},{"id":"https://openalex.org/keywords/copula","display_name":"Copula (linguistics)","score":0.5590293407440186},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.528312623500824},{"id":"https://openalex.org/keywords/change-of-variables","display_name":"Change of variables (PDE)","score":0.44203120470046997},{"id":"https://openalex.org/keywords/separation-of-variables","display_name":"Separation of variables","score":0.42343148589134216},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.3025967478752136},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.25232791900634766},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.15600994229316711},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.12975230813026428},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.12338036298751831},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.11968079209327698}],"concepts":[{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.8123225569725037},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.777121365070343},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.717106282711029},{"id":"https://openalex.org/C27574286","wikidata":"https://www.wikidata.org/wiki/Q320723","display_name":"Variables","level":2,"score":0.6303574442863464},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.6203804016113281},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.5817480087280273},{"id":"https://openalex.org/C17618745","wikidata":"https://www.wikidata.org/wiki/Q207509","display_name":"Copula (linguistics)","level":2,"score":0.5590293407440186},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.528312623500824},{"id":"https://openalex.org/C107393986","wikidata":"https://www.wikidata.org/wiki/Q11912046","display_name":"Change of variables (PDE)","level":5,"score":0.44203120470046997},{"id":"https://openalex.org/C159722891","wikidata":"https://www.wikidata.org/wiki/Q913323","display_name":"Separation of variables","level":3,"score":0.42343148589134216},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3025967478752136},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25232791900634766},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.15600994229316711},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.12975230813026428},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.12338036298751831},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.11968079209327698},{"id":"https://openalex.org/C186219872","wikidata":"https://www.wikidata.org/wiki/Q955889","display_name":"Differential algebraic equation","level":4,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/axioms12090845","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090845","pdf_url":"https://www.mdpi.com/2075-1680/12/9/845/pdf?version=1693393386","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-04621210v1","is_oa":true,"landing_page_url":"https://hal.science/hal-04621210","pdf_url":"https://hal.science/hal-04621210/document","source":{"id":"https://openalex.org/S4406922466","display_name":"SPIRE - Sciences Po Institutional REpository","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Axioms, 2023, 12 (9), pp.845. &#x27E8;10.3390/axioms12090845&#x27E9;","raw_type":"Journal articles"},{"id":"pmh:oai:doaj.org/article:c29b12f16b9d424e92ab0fa21c535b8b","is_oa":true,"landing_page_url":"https://doaj.org/article/c29b12f16b9d424e92ab0fa21c535b8b","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 12, Iss 9, p 845 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/12/9/845/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms12090845","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Axioms","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/axioms12090845","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms12090845","pdf_url":"https://www.mdpi.com/2075-1680/12/9/845/pdf?version=1693393386","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4386282967.pdf"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1457220277","https://openalex.org/W1974511160","https://openalex.org/W1979917364","https://openalex.org/W1994669566","https://openalex.org/W1995565517","https://openalex.org/W2014268383","https://openalex.org/W2053469438","https://openalex.org/W2088997392","https://openalex.org/W2118020555","https://openalex.org/W2137295153","https://openalex.org/W2157671726","https://openalex.org/W2544972693","https://openalex.org/W2740980173","https://openalex.org/W2972277001","https://openalex.org/W3134200530","https://openalex.org/W4200166684","https://openalex.org/W4200304176","https://openalex.org/W4224881919","https://openalex.org/W4234061158","https://openalex.org/W4253652099","https://openalex.org/W4292082790","https://openalex.org/W4296645165","https://openalex.org/W4360979025"],"related_works":["https://openalex.org/W2040131087","https://openalex.org/W1503814607","https://openalex.org/W1531734355","https://openalex.org/W4293862336","https://openalex.org/W1984857247","https://openalex.org/W4386282967","https://openalex.org/W2800475502","https://openalex.org/W4365456661","https://openalex.org/W1927924852","https://openalex.org/W2154123151"],"abstract_inverted_index":{"Stochastic":[0],"characterizations":[1],"of":[2,23,36,39,53,59,70,96],"functions":[3,12,40,60,97],"subject":[4],"to":[5,77,88],"constraints":[6],"result":[7],"in":[8,51],"treating":[9],"them":[10],"as":[11],"with":[13,29,41,61,98],"non-independent":[14,42,62,71,99],"variables.":[15,100],"By":[16],"using":[17],"the":[18,24,54,66,90],"distribution":[19],"function":[20],"or":[21],"copula":[22],"input":[25],"variables":[26,43,63],"that":[27],"comply":[28],"such":[30],"constraints,":[31],"we":[32],"derive":[33,89],"two":[34],"types":[35],"partial":[37,57],"derivatives":[38,58],"(i.e.,":[44],"actual":[45],"and":[46,49,93],"dependent":[47,67],"derivatives)":[48],"argue":[50],"favor":[52],"latter.":[55],"Dependent":[56],"rely":[64],"on":[65],"Jacobian":[68],"matrix":[69],"variables,":[72],"which":[73],"is":[74],"also":[75],"used":[76],"define":[78],"a":[79],"tensor":[80],"metric.":[81],"The":[82],"differential":[83],"geometric":[84],"framework":[85],"allows":[86],"us":[87],"gradient,":[91],"Hessian,":[92],"Taylor-type":[94],"expansions":[95]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3}],"updated_date":"2026-03-14T08:43:22.919905","created_date":"2025-10-10T00:00:00"}
