{"id":"https://openalex.org/W2997721922","doi":"https://doi.org/10.3390/sym12010086","title":"Generation of Julia and Mandelbrot Sets via Fixed Points","display_name":"Generation of Julia and Mandelbrot Sets via Fixed Points","publication_year":2020,"publication_date":"2020-01-02","ids":{"openalex":"https://openalex.org/W2997721922","doi":"https://doi.org/10.3390/sym12010086","mag":"2997721922"},"language":"en","primary_location":{"id":"doi:10.3390/sym12010086","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12010086","pdf_url":"https://www.mdpi.com/2073-8994/12/1/86/pdf?version=1604817829","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/12/1/86/pdf?version=1604817829","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054055136","display_name":"Mujahid Abbas","orcid":"https://orcid.org/0000-0001-5528-1207"},"institutions":[{"id":"https://openalex.org/I184693016","display_name":"China Medical University","ror":"https://ror.org/00v408z34","country_code":"TW","type":"education","lineage":["https://openalex.org/I184693016"]},{"id":"https://openalex.org/I43879129","display_name":"Government College University, Lahore","ror":"https://ror.org/040gec961","country_code":"PK","type":"education","lineage":["https://openalex.org/I43879129"]}],"countries":["PK","TW"],"is_corresponding":false,"raw_author_name":"Mujahid Abbas","raw_affiliation_strings":["Department of Mathematics, Government College University, Lahore 54000, Pakistan","Department of Medical Research, China Medical University No. 91, Hsueh-Shih Road, Taichung 400, Taiwan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Government College University, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I43879129"]},{"raw_affiliation_string":"Department of Medical Research, China Medical University No. 91, Hsueh-Shih Road, Taichung 400, Taiwan","institution_ids":["https://openalex.org/I184693016"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5007292994","display_name":"Hira Iqbal","orcid":"https://orcid.org/0000-0002-1286-4880"},"institutions":[{"id":"https://openalex.org/I201384688","display_name":"National University of Computer and Emerging Sciences","ror":"https://ror.org/003eyb898","country_code":"PK","type":"education","lineage":["https://openalex.org/I201384688"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Hira Iqbal","raw_affiliation_strings":["Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Lahore 54000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I201384688"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042326806","display_name":"Manuel De la Sen","orcid":"https://orcid.org/0000-0001-9320-9433"},"institutions":[{"id":"https://openalex.org/I169108374","display_name":"University of the Basque Country","ror":"https://ror.org/000xsnr85","country_code":"ES","type":"education","lineage":["https://openalex.org/I169108374"]}],"countries":["ES"],"is_corresponding":true,"raw_author_name":"Manuel De la Sen","raw_affiliation_strings":["Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain"],"raw_orcid":"https://orcid.org/0000-0001-9320-9433","affiliations":[{"raw_affiliation_string":"Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, Barrio Sarriena, 48940 Leioa, Spain","institution_ids":["https://openalex.org/I169108374"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5042326806"],"corresponding_institution_ids":["https://openalex.org/I169108374"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":5.4881,"has_fulltext":false,"cited_by_count":34,"citation_normalized_percentile":{"value":0.96829179,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"12","issue":"1","first_page":"86","last_page":"86"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.9725000262260437,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12812","display_name":"Meromorphic and Entire Functions","score":0.9423999786376953,"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/mandelbrot-set","display_name":"Mandelbrot set","score":0.9955118298530579},{"id":"https://openalex.org/keywords/julia-set","display_name":"Julia set","score":0.9470323324203491},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.7421759366989136},{"id":"https://openalex.org/keywords/complex-quadratic-polynomial","display_name":"Complex quadratic polynomial","score":0.7130356431007385},{"id":"https://openalex.org/keywords/newton-fractal","display_name":"Newton fractal","score":0.663520336151123},{"id":"https://openalex.org/keywords/fixed-point","display_name":"Fixed point","score":0.6611917614936829},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5957881808280945},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4279683828353882},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3563729524612427},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.19448107481002808},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.1784912645816803},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.15091180801391602},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.10519939661026001}],"concepts":[{"id":"https://openalex.org/C97519476","wikidata":"https://www.wikidata.org/wiki/Q257","display_name":"Mandelbrot set","level":3,"score":0.9955118298530579},{"id":"https://openalex.org/C187619975","wikidata":"https://www.wikidata.org/wiki/Q848427","display_name":"Julia set","level":2,"score":0.9470323324203491},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.7421759366989136},{"id":"https://openalex.org/C6077466","wikidata":"https://www.wikidata.org/wiki/Q5156597","display_name":"Complex quadratic polynomial","level":3,"score":0.7130356431007385},{"id":"https://openalex.org/C194342247","wikidata":"https://www.wikidata.org/wiki/Q1151001","display_name":"Newton fractal","level":4,"score":0.663520336151123},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.6611917614936829},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5957881808280945},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4279683828353882},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3563729524612427},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.19448107481002808},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.1784912645816803},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.15091180801391602},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.10519939661026001},{"id":"https://openalex.org/C117898588","wikidata":"https://www.wikidata.org/wiki/Q6664310","display_name":"Local convergence","level":3,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym12010086","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12010086","pdf_url":"https://www.mdpi.com/2073-8994/12/1/86/pdf?version=1604817829","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:addi.ehu.es:10810/48965","is_oa":true,"landing_page_url":"http://hdl.handle.net/10810/48965","pdf_url":null,"source":{"id":"https://openalex.org/S4306401964","display_name":"Communities in ADDI (University of the Basque Country)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I169108374","host_organization_name":"University of the Basque Country","host_organization_lineage":["https://openalex.org/I169108374"],"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":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:2b92f68f62fb448fbe1de97fbaf643cb","is_oa":true,"landing_page_url":"https://doaj.org/article/2b92f68f62fb448fbe1de97fbaf643cb","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 12, Iss 1, p 86 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym12010086","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12010086","pdf_url":"https://www.mdpi.com/2073-8994/12/1/86/pdf?version=1604817829","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6857220102","display_name":null,"funder_award_id":"IT 1207-19","funder_id":"https://openalex.org/F4320321705","funder_display_name":"Eusko Jaurlaritza"}],"funders":[{"id":"https://openalex.org/F4320321705","display_name":"Eusko Jaurlaritza","ror":"https://ror.org/00pz2fp31"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W832618888","https://openalex.org/W1579586326","https://openalex.org/W1856667174","https://openalex.org/W1950518118","https://openalex.org/W2004061703","https://openalex.org/W2041118851","https://openalex.org/W2041221205","https://openalex.org/W2058442100","https://openalex.org/W2059214908","https://openalex.org/W2073556564","https://openalex.org/W2078206416","https://openalex.org/W2094347864","https://openalex.org/W2105056989","https://openalex.org/W2132822938","https://openalex.org/W2149375763","https://openalex.org/W2261545930","https://openalex.org/W2319365926","https://openalex.org/W2352978371","https://openalex.org/W2390700995","https://openalex.org/W2536868324","https://openalex.org/W2890428587","https://openalex.org/W2922108273","https://openalex.org/W2964335031","https://openalex.org/W4238652859","https://openalex.org/W4246447829","https://openalex.org/W4250883246","https://openalex.org/W6675775263","https://openalex.org/W6682278902","https://openalex.org/W6705513857","https://openalex.org/W6711739752"],"related_works":["https://openalex.org/W4249675078","https://openalex.org/W2370909541","https://openalex.org/W2081216589","https://openalex.org/W1544032841","https://openalex.org/W2997721922","https://openalex.org/W1972311046","https://openalex.org/W109862761","https://openalex.org/W4239801010","https://openalex.org/W2063835118","https://openalex.org/W3008531049"],"abstract_inverted_index":{"The":[0],"aim":[1],"of":[2,10,18,29,59,81,95,118,126],"this":[3],"paper":[4],"is":[5,105,111],"to":[6],"present":[7],"an":[8],"application":[9],"a":[11,87],"fixed":[12],"point":[13],"iterative":[14,91],"process":[15],"in":[16],"generation":[17,80],"fractals":[19],"namely":[20],"Julia":[21,82,97],"and":[22,50,83,98,107,124],"Mandelbrot":[23,84,99],"sets":[24,85,100],"for":[25,78,101],"the":[26,30,57,79,96,116,121],"complex":[27,103],"polynomials":[28,104],"form":[31],"T":[32],"(":[33],"x":[34,37,41],")":[35],"=":[36],"n":[38,51],"+":[39,42],"m":[40,45],"r":[43,47],"where":[44],",":[46],"\u2208":[48],"C":[49],"\u2265":[52],"2":[53],".":[54],"Fractals":[55],"represent":[56],"phenomena":[58],"expanding":[60],"or":[61],"unfolding":[62],"symmetries":[63],"which":[64],"exhibit":[65],"similar":[66],"patterns":[67],"displayed":[68],"at":[69],"every":[70],"scale.":[71],"We":[72,113],"prove":[73],"some":[74],"escape":[75],"time":[76],"results":[77],"using":[86],"Picard":[88],"Ishikawa":[89],"type":[90],"process.":[92],"A":[93],"visualization":[94],"certain":[102],"presented":[106],"their":[108],"graphical":[109],"behaviour":[110],"examined.":[112],"also":[114],"discuss":[115],"effects":[117],"parameters":[119],"on":[120],"color":[122],"variation":[123],"shape":[125],"fractals.":[127]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":3}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2020-01-10T00:00:00"}
