{"id":"https://openalex.org/W1966864455","doi":"https://doi.org/10.1142/s0218127406016008","title":"THE \u0141OJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2","display_name":"THE \u0141OJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2","publication_year":2006,"publication_date":"2006-08-01","ids":{"openalex":"https://openalex.org/W1966864455","doi":"https://doi.org/10.1142/s0218127406016008","mag":"1966864455"},"language":"en","primary_location":{"id":"doi:10.1142/s0218127406016008","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127406016008","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"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":"International Journal of Bifurcation and Chaos","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/A5069809502","display_name":"Mauro Forti","orcid":"https://orcid.org/0000-0002-3970-7201"},"institutions":[{"id":"https://openalex.org/I102064193","display_name":"University of Siena","ror":"https://ror.org/01tevnk56","country_code":"IT","type":"education","lineage":["https://openalex.org/I102064193"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"MAURO FORTI","raw_affiliation_strings":["Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Siena, v. Roma 56, 53100 Siena, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Ingegneria dell'Informazione, Universit\u00e0 di Siena, v. Roma 56, 53100 Siena, Italy","institution_ids":["https://openalex.org/I102064193"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5074942219","display_name":"A. Tesi","orcid":"https://orcid.org/0000-0002-0234-5999"},"institutions":[{"id":"https://openalex.org/I45084792","display_name":"University of Florence","ror":"https://ror.org/04jr1s763","country_code":"IT","type":"education","lineage":["https://openalex.org/I45084792"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"ALBERTO TESI","raw_affiliation_strings":["Dipartimento di Sistemi e Informatica, Universit\u00e0 di Firenze, v. S. Marta 3, 50139 Firenze, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Sistemi e Informatica, Universit\u00e0 di Firenze, v. S. Marta 3, 50139 Firenze, Italy","institution_ids":["https://openalex.org/I45084792"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5069809502"],"corresponding_institution_ids":["https://openalex.org/I102064193"],"apc_list":null,"apc_paid":null,"fwci":0.9053,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.76270653,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"16","issue":"08","first_page":"2191","last_page":"2205"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11347","display_name":"Neural Networks Stability and Synchronization","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11347","display_name":"Neural Networks Stability and Synchronization","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11476","display_name":"Graph theory and applications","score":0.9721999764442444,"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/T10320","display_name":"Neural Networks and Applications","score":0.9681000113487244,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7042195200920105},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.6242483258247375},{"id":"https://openalex.org/keywords/hypercube","display_name":"Hypercube","score":0.5327544212341309},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4574202597141266},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.45072320103645325},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.27165040373802185},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11522132158279419}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7042195200920105},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6242483258247375},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.5327544212341309},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4574202597141266},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.45072320103645325},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27165040373802185},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11522132158279419},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s0218127406016008","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218127406016008","pdf_url":null,"source":{"id":"https://openalex.org/S81011612","display_name":"International Journal of Bifurcation and Chaos","issn_l":"0218-1274","issn":["0218-1274","1793-6551"],"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":"International Journal of Bifurcation and Chaos","raw_type":"journal-article"},{"id":"pmh:oai:flore.unifi.it:2158/224909","is_oa":false,"landing_page_url":"http://hdl.handle.net/2158/224909","pdf_url":null,"source":{"id":"https://openalex.org/S4306402033","display_name":"Florence Research (University of Florence)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I45084792","host_organization_name":"University of Florence","host_organization_lineage":["https://openalex.org/I45084792"],"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":"info:eu-repo/semantics/article"},{"id":"pmh:oai:usiena-air.unisi.it:11365/8662","is_oa":false,"landing_page_url":"http://hdl.handle.net/11365/8662","pdf_url":null,"source":{"id":"https://openalex.org/S4377196319","display_name":"Use Siena air (University of Siena)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I102064193","host_organization_name":"University of Siena","host_organization_lineage":["https://openalex.org/I102064193"],"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":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","score":0.44999998807907104,"id":"https://metadata.un.org/sdg/10"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W1514578360","https://openalex.org/W1551360398","https://openalex.org/W2010141219","https://openalex.org/W2066263844","https://openalex.org/W2067292421","https://openalex.org/W2157312889","https://openalex.org/W2157976960","https://openalex.org/W2160121923","https://openalex.org/W4210605524"],"related_works":["https://openalex.org/W4286907761","https://openalex.org/W2916710807","https://openalex.org/W2413588304","https://openalex.org/W2009411567","https://openalex.org/W3205447645","https://openalex.org/W2097769911","https://openalex.org/W2371629703","https://openalex.org/W2774705104","https://openalex.org/W72835634","https://openalex.org/W2122785676"],"abstract_inverted_index":{"In":[0,61,253],"the":[1,13,25,28,51,58,85,92,95,102,105,125,151,154,172,202,240,245,255,264],"sixties,":[2],"\u0141ojasiewicz":[3,59,109,126,174,203],"proved":[4],"a":[5,21,31,34,42,64,73,88,132,144,158,165,184,192,210],"fundamental":[6],"inequality":[7,40,110],"for":[8,111],"vector":[9,67,82,115,133,147],"fields":[10,68,116,142,148],"defined":[11,71],"by":[12],"gradient":[14,29,86],"of":[15,27,33,66,75,87,94,104,114,146,153,157,183,209,266,269],"an":[16,197,231],"analytic":[17],"function,":[18],"which":[19,48,69,149],"gives":[20],"lower":[22],"bound":[23],"on":[24,50,72,101],"norm":[26],"in":[30,91,239],"neighborhood":[32],"(possibly)":[35],"non-isolated":[36],"critical":[37,52],"point.":[38,199],"The":[39,140],"involves":[41],"number":[43],"belonging":[44],"to":[45,138,196,215],"(0,":[46],"1),":[47],"depends":[49],"point,":[53,233],"and":[54,119,190,223,234],"is":[55,79,84,99,117,121,136,177,213,227,236],"known":[56],"as":[57,191],"exponent.":[60],"this":[62,112,235],"paper,":[63],"class":[65,113,145],"are":[70],"hypercube":[74],"\u211d":[76],"n":[77],",":[78],"considered.":[80],"Each":[81],"field":[83,134],"quadratic":[89],"function":[90],"interior":[93],"hypercube,":[96],"however":[97],"it":[98,120,176,194,217],"discontinuous":[100],"boundary":[103],"hypercube.":[106],"An":[107],"extended":[108,173],"proved,":[118],"also":[122],"shown":[123,178],"that":[124,179,219,259],"exponent":[127,204],"at":[128,205],"each":[129,180,206,220,224],"point":[130,208],"where":[131,244],"vanishes":[135],"equal":[137,214],"1/2.":[139],"considered":[141],"include":[143],"describe":[150],"dynamics":[152],"output":[155,181,225],"trajectories":[156],"standard":[159,260],"Cellular":[160],"Neural":[161],"Network":[162],"(CNN)":[163],"with":[164],"symmetric":[166,185,211,261],"neuron":[167],"interconnection":[168],"matrix.":[169],"By":[170],"applying":[171],"inequality,":[175],"trajectory":[182],"CNN":[186,212,246],"has":[187],"finite":[188],"length,":[189],"consequence":[193],"converges":[195],"equilibrium":[198,207,232,251],"Furthermore,":[200],"since":[201],"1/2,":[216],"follows":[218],"(state)":[221],"trajectory,":[222,226],"exponentially":[228],"convergent":[229],"toward":[230],"true":[237],"even":[238],"most":[241],"general":[242],"case":[243],"possesses":[247],"infinitely":[248],"many":[249],"nonisolated":[250],"points.":[252],"essence,":[254],"obtained":[256],"results":[257],"mean":[258],"CNNs":[262],"enjoy":[263],"property":[265],"absolute":[267],"stability":[268],"exponential":[270],"convergence.":[271]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}