{"id":"https://openalex.org/W2953367536","doi":"https://doi.org/10.3390/sym11060825","title":"Directional Thermodynamic Formalism","display_name":"Directional Thermodynamic Formalism","publication_year":2019,"publication_date":"2019-06-21","ids":{"openalex":"https://openalex.org/W2953367536","doi":"https://doi.org/10.3390/sym11060825","mag":"2953367536"},"language":"en","primary_location":{"id":"doi:10.3390/sym11060825","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11060825","pdf_url":"https://www.mdpi.com/2073-8994/11/6/825/pdf?version=1561521714","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/2073-8994/11/6/825/pdf?version=1561521714","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5083856807","display_name":"Mourad Ben Slimane","orcid":"https://orcid.org/0000-0003-3543-9251"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":true,"raw_author_name":"Mourad Ben Slimane","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5061812859","display_name":"Moez Ben Abid","orcid":"https://orcid.org/0000-0002-2922-0184"},"institutions":[{"id":"https://openalex.org/I8636806","display_name":"University of Sousse","ror":"https://ror.org/00dmpgj58","country_code":"TN","type":"education","lineage":["https://openalex.org/I8636806"]}],"countries":["TN"],"is_corresponding":false,"raw_author_name":"Moez Ben Abid","raw_affiliation_strings":["Ecole Sup\u00e9rieure des Sciences et de la Technologie de Hammam Sousse, Universit\u00e9 de Sousse, Sousse 4011, Tunisia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ecole Sup\u00e9rieure des Sciences et de la Technologie de Hammam Sousse, Universit\u00e9 de Sousse, Sousse 4011, Tunisia","institution_ids":["https://openalex.org/I8636806"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5023592054","display_name":"Ines Ben Omrane","orcid":"https://orcid.org/0000-0002-7911-7556"},"institutions":[{"id":"https://openalex.org/I240666556","display_name":"Imam Mohammad ibn Saud Islamic University","ror":"https://ror.org/05gxjyb39","country_code":"SA","type":"education","lineage":["https://openalex.org/I240666556"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ines Ben Omrane","raw_affiliation_strings":["Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, Riyadh 11623, Saudi Arabia","institution_ids":["https://openalex.org/I240666556"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041111512","display_name":"Borhen Halouani","orcid":"https://orcid.org/0000-0001-9566-7235"},"institutions":[{"id":"https://openalex.org/I28022161","display_name":"King Saud University","ror":"https://ror.org/02f81g417","country_code":"SA","type":"education","lineage":["https://openalex.org/I28022161"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Borhen Halouani","raw_affiliation_strings":["Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia","institution_ids":["https://openalex.org/I28022161"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5083856807"],"corresponding_institution_ids":["https://openalex.org/I28022161"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.05213733,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"11","issue":"6","first_page":"825","last_page":"825"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10688","display_name":"Image and Signal Denoising Methods","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9778000116348267,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.953499972820282,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.7488579154014587},{"id":"https://openalex.org/keywords/isotropy","display_name":"Isotropy","score":0.6682744026184082},{"id":"https://openalex.org/keywords/multifractal-system","display_name":"Multifractal system","score":0.6302821040153503},{"id":"https://openalex.org/keywords/formalism","display_name":"Formalism (music)","score":0.5783951282501221},{"id":"https://openalex.org/keywords/anisotropy","display_name":"Anisotropy","score":0.5347093343734741},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.5217729210853577},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5066397786140442},{"id":"https://openalex.org/keywords/wavelet","display_name":"Wavelet","score":0.4798690378665924},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4667496979236603},{"id":"https://openalex.org/keywords/directional-derivative","display_name":"Directional derivative","score":0.4547696113586426},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.45178350806236267},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.4385886490345001},{"id":"https://openalex.org/keywords/besov-space","display_name":"Besov space","score":0.4159679114818573},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.36492210626602173},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.20737284421920776},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14482232928276062},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.11557644605636597}],"concepts":[{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.7488579154014587},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.6682744026184082},{"id":"https://openalex.org/C133905733","wikidata":"https://www.wikidata.org/wiki/Q2629238","display_name":"Multifractal system","level":3,"score":0.6302821040153503},{"id":"https://openalex.org/C73301696","wikidata":"https://www.wikidata.org/wiki/Q5469984","display_name":"Formalism (music)","level":3,"score":0.5783951282501221},{"id":"https://openalex.org/C85725439","wikidata":"https://www.wikidata.org/wiki/Q466686","display_name":"Anisotropy","level":2,"score":0.5347093343734741},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.5217729210853577},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5066397786140442},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.4798690378665924},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4667496979236603},{"id":"https://openalex.org/C2011187","wikidata":"https://www.wikidata.org/wiki/Q383851","display_name":"Directional derivative","level":2,"score":0.4547696113586426},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.45178350806236267},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.4385886490345001},{"id":"https://openalex.org/C129698243","wikidata":"https://www.wikidata.org/wiki/Q526261","display_name":"Besov space","level":5,"score":0.4159679114818573},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.36492210626602173},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.20737284421920776},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14482232928276062},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.11557644605636597},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C6183050","wikidata":"https://www.wikidata.org/wiki/Q3058180","display_name":"Interpolation space","level":4,"score":0.0},{"id":"https://openalex.org/C55112680","wikidata":"https://www.wikidata.org/wiki/Q190549","display_name":"Functional analysis","level":3,"score":0.0},{"id":"https://openalex.org/C558565934","wikidata":"https://www.wikidata.org/wiki/Q2743","display_name":"Musical","level":2,"score":0.0},{"id":"https://openalex.org/C153349607","wikidata":"https://www.wikidata.org/wiki/Q36649","display_name":"Visual arts","level":1,"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/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym11060825","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11060825","pdf_url":"https://www.mdpi.com/2073-8994/11/6/825/pdf?version=1561521714","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:doaj.org/article:e6ed4135879f431691e43173b50b1492","is_oa":true,"landing_page_url":"https://doaj.org/article/e6ed4135879f431691e43173b50b1492","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 11, Iss 6, p 825 (2019)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym11060825","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11060825","pdf_url":"https://www.mdpi.com/2073-8994/11/6/825/pdf?version=1561521714","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":[{"display_name":"Sustainable cities and communities","score":0.4699999988079071,"id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":96,"referenced_works":["https://openalex.org/W162854683","https://openalex.org/W578051105","https://openalex.org/W658985630","https://openalex.org/W954928309","https://openalex.org/W1522305672","https://openalex.org/W1524073616","https://openalex.org/W1527515332","https://openalex.org/W1529603256","https://openalex.org/W1553381056","https://openalex.org/W1599594355","https://openalex.org/W1710462594","https://openalex.org/W1885000897","https://openalex.org/W1964767713","https://openalex.org/W1966215731","https://openalex.org/W1967924750","https://openalex.org/W1980118756","https://openalex.org/W1986333957","https://openalex.org/W1987596796","https://openalex.org/W1991218660","https://openalex.org/W1992797860","https://openalex.org/W1993444341","https://openalex.org/W1994143674","https://openalex.org/W1998045092","https://openalex.org/W1998727969","https://openalex.org/W1999143616","https://openalex.org/W2001745801","https://openalex.org/W2003010749","https://openalex.org/W2007912280","https://openalex.org/W2009043930","https://openalex.org/W2013987111","https://openalex.org/W2021537669","https://openalex.org/W2022140185","https://openalex.org/W2023110771","https://openalex.org/W2031638306","https://openalex.org/W2032121576","https://openalex.org/W2032456190","https://openalex.org/W2033750748","https://openalex.org/W2038675637","https://openalex.org/W2041213761","https://openalex.org/W2044897742","https://openalex.org/W2053276035","https://openalex.org/W2056994213","https://openalex.org/W2059069690","https://openalex.org/W2060567785","https://openalex.org/W2062099911","https://openalex.org/W2063413320","https://openalex.org/W2066462711","https://openalex.org/W2067393519","https://openalex.org/W2068295468","https://openalex.org/W2072802027","https://openalex.org/W2073377943","https://openalex.org/W2074145530","https://openalex.org/W2077998334","https://openalex.org/W2078206416","https://openalex.org/W2079081125","https://openalex.org/W2081195621","https://openalex.org/W2086151797","https://openalex.org/W2088555890","https://openalex.org/W2089062302","https://openalex.org/W2090541396","https://openalex.org/W2090769263","https://openalex.org/W2093728332","https://openalex.org/W2098869675","https://openalex.org/W2098914003","https://openalex.org/W2099211525","https://openalex.org/W2102901030","https://openalex.org/W2103264301","https://openalex.org/W2109504624","https://openalex.org/W2118203293","https://openalex.org/W2125564887","https://openalex.org/W2138865513","https://openalex.org/W2146424137","https://openalex.org/W2149195003","https://openalex.org/W2151184871","https://openalex.org/W2156158657","https://openalex.org/W2161798353","https://openalex.org/W2164550108","https://openalex.org/W2199571618","https://openalex.org/W2249778981","https://openalex.org/W2315631835","https://openalex.org/W2504381165","https://openalex.org/W2559304291","https://openalex.org/W2771941706","https://openalex.org/W2795902834","https://openalex.org/W2963162152","https://openalex.org/W2964197071","https://openalex.org/W2998250468","https://openalex.org/W3098420962","https://openalex.org/W3132971798","https://openalex.org/W3148918439","https://openalex.org/W4249775755","https://openalex.org/W4300794971","https://openalex.org/W6641563507","https://openalex.org/W6669047953","https://openalex.org/W6675635722","https://openalex.org/W6729878695"],"related_works":["https://openalex.org/W1639315488","https://openalex.org/W2056486084","https://openalex.org/W4382488079","https://openalex.org/W2010006040","https://openalex.org/W2014428041","https://openalex.org/W4367684936","https://openalex.org/W2940189448","https://openalex.org/W2027961474","https://openalex.org/W2124361952","https://openalex.org/W2054265775"],"abstract_inverted_index":{"The":[0,39],"usual":[1],"thermodynamic":[2],"formalism":[3],"is":[4,12,24],"uniform":[5],"in":[6,31,60,134],"all":[7],"directions":[8],"and,":[9],"therefore,":[10],"it":[11],"not":[13],"adapted":[14],"to":[15],"study":[16],"multi-dimensional":[17],"functions":[18,138],"with":[19],"various":[20],"directional":[21,56,63,72,80,92,105],"behaviors.":[22],"It":[23],"based":[25],"on":[26,90],"a":[27,53],"scaling":[28,57,93,97],"function":[29,58,94],"characterized":[30],"terms":[32,61],"of":[33,41,62,117,119],"isotropic":[34],"Sobolev":[35,64],"or":[36,65,78,85,95],"Besov-type":[37],"norms.":[38],"purpose":[40],"the":[42,71],"present":[43],"paper":[44],"was":[45],"twofold.":[46],"Firstly,":[47],"we":[48,69,76],"proved":[49],"wavelet":[50],"criteria":[51],"for":[52,104,113,126],"natural":[54],"extended":[55],"expressed":[59],"Besov":[66],"spaces.":[67],"Secondly,":[68],"performed":[70],"multifractal":[73],"formalism,":[74],"i.e.,":[75],"computed":[77],"estimated":[79],"H\u00f6lder":[81,106],"spectra,":[82],"either":[83,91],"directly":[84],"via":[86],"some":[87],"Legendre":[88],"transforms":[89],"anisotropic":[96,123],"functions.":[98],"We":[99,108],"obtained":[100],"general":[101],"upper":[102],"bounds":[103],"spectra.":[107],"also":[109],"showed":[110],"optimal":[111],"results":[112],"two":[114],"large":[115],"classes":[116],"examples":[118],"deterministic":[120],"and":[121,132,139],"random":[122],"self-similar":[124],"tools":[125],"possible":[127],"modeling":[128],"turbulence":[129],"(or":[130],"cascades)":[131],"textures":[133],"images:":[135],"Sierpinski":[136],"cascade":[137],"fractional":[140],"Brownian":[141],"sheets.":[142]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
