{"id":"https://openalex.org/W4392950814","doi":"https://doi.org/10.3390/sym16030372","title":"On Generalized t-Transformation of Free Convolution","display_name":"On Generalized t-Transformation of Free Convolution","publication_year":2024,"publication_date":"2024-03-19","ids":{"openalex":"https://openalex.org/W4392950814","doi":"https://doi.org/10.3390/sym16030372"},"language":"en","primary_location":{"id":"doi:10.3390/sym16030372","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16030372","pdf_url":"https://www.mdpi.com/2073-8994/16/3/372/pdf?version=1710842138","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/2073-8994/16/3/372/pdf?version=1710842138","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000311601","display_name":"Ayed R. A. Alanzi","orcid":"https://orcid.org/0000-0002-5337-2276"},"institutions":[{"id":"https://openalex.org/I199702508","display_name":"Jouf University","ror":"https://ror.org/02zsyt821","country_code":"SA","type":"education","lineage":["https://openalex.org/I199702508"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Ayed. R. A. Alanzi","raw_affiliation_strings":["Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia","Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0002-5337-2276","affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia","institution_ids":["https://openalex.org/I199702508"]},{"raw_affiliation_string":"Department of Mathematics, College of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia","institution_ids":["https://openalex.org/I199702508"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089048863","display_name":"Raouf Fakhfakh","orcid":"https://orcid.org/0000-0003-2233-4899"},"institutions":[{"id":"https://openalex.org/I142899784","display_name":"University of Sfax","ror":"https://ror.org/04d4sd432","country_code":"TN","type":"education","lineage":["https://openalex.org/I142899784"]},{"id":"https://openalex.org/I199702508","display_name":"Jouf University","ror":"https://ror.org/02zsyt821","country_code":"SA","type":"education","lineage":["https://openalex.org/I199702508"]}],"countries":["SA","TN"],"is_corresponding":true,"raw_author_name":"Raouf Fakhfakh","raw_affiliation_strings":["Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia","Laboratory of Probability and Statistics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3000, Tunisia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Science and Arts in Gurayat, Jouf University, Gurayat 77454, Saudi Arabia","institution_ids":["https://openalex.org/I199702508"]},{"raw_affiliation_string":"Laboratory of Probability and Statistics, Faculty of Sciences of Sfax, University of Sfax, Sfax 3000, Tunisia","institution_ids":["https://openalex.org/I142899784"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041542650","display_name":"Fatimah Alshahrani","orcid":"https://orcid.org/0000-0003-1306-7688"},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Fatimah Alshahrani","raw_affiliation_strings":["Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0003-1306-7688","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5089048863"],"corresponding_institution_ids":["https://openalex.org/I142899784","https://openalex.org/I199702508"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.3244,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.78036621,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"16","issue":"3","first_page":"372","last_page":"372"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9975000023841858,"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"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9975000023841858,"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/T11191","display_name":"Mathematical functions and polynomials","score":0.9973000288009644,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9950000047683716,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/convolution","display_name":"Convolution (computer science)","score":0.7556039094924927},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.632281482219696},{"id":"https://openalex.org/keywords/convolution-power","display_name":"Convolution power","score":0.6132001280784607},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.555131196975708},{"id":"https://openalex.org/keywords/free-probability","display_name":"Free probability","score":0.5268251299858093},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.5143495798110962},{"id":"https://openalex.org/keywords/cauchy-distribution","display_name":"Cauchy distribution","score":0.4585854709148407},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4477241635322571},{"id":"https://openalex.org/keywords/transformation","display_name":"Transformation (genetics)","score":0.43962353467941284},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4362432360649109},{"id":"https://openalex.org/keywords/convolution-theorem","display_name":"Convolution theorem","score":0.4272342920303345},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.312049925327301},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.16612473130226135},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1021716296672821}],"concepts":[{"id":"https://openalex.org/C45347329","wikidata":"https://www.wikidata.org/wiki/Q5166604","display_name":"Convolution (computer science)","level":3,"score":0.7556039094924927},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.632281482219696},{"id":"https://openalex.org/C86254553","wikidata":"https://www.wikidata.org/wiki/Q5166605","display_name":"Convolution power","level":5,"score":0.6132001280784607},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.555131196975708},{"id":"https://openalex.org/C70120467","wikidata":"https://www.wikidata.org/wiki/Q515893","display_name":"Free probability","level":2,"score":0.5268251299858093},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5143495798110962},{"id":"https://openalex.org/C49344536","wikidata":"https://www.wikidata.org/wiki/Q726441","display_name":"Cauchy distribution","level":2,"score":0.4585854709148407},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4477241635322571},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.43962353467941284},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4362432360649109},{"id":"https://openalex.org/C79587385","wikidata":"https://www.wikidata.org/wiki/Q2638931","display_name":"Convolution theorem","level":5,"score":0.4272342920303345},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.312049925327301},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.16612473130226135},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1021716296672821},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","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/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.0},{"id":"https://openalex.org/C76563020","wikidata":"https://www.wikidata.org/wiki/Q4817582","display_name":"Fractional Fourier transform","level":4,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym16030372","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16030372","pdf_url":"https://www.mdpi.com/2073-8994/16/3/372/pdf?version=1710842138","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:99bfe79e253d4df3a91d557492fc2fbc","is_oa":false,"landing_page_url":"https://doaj.org/article/99bfe79e253d4df3a91d557492fc2fbc","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 16, Iss 3, p 372 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/16/3/372/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym16030372","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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym16030372","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16030372","pdf_url":"https://www.mdpi.com/2073-8994/16/3/372/pdf?version=1710842138","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4392950814.pdf"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W1570229043","https://openalex.org/W1594682445","https://openalex.org/W1629277479","https://openalex.org/W1988344476","https://openalex.org/W2015349482","https://openalex.org/W2037382166","https://openalex.org/W2319822297","https://openalex.org/W2517649823","https://openalex.org/W2963009812","https://openalex.org/W3011059581","https://openalex.org/W3016046145","https://openalex.org/W3098442735","https://openalex.org/W4226359107","https://openalex.org/W4253567193","https://openalex.org/W4321446955","https://openalex.org/W6726050288","https://openalex.org/W6774775550"],"related_works":["https://openalex.org/W4363675452","https://openalex.org/W52748724","https://openalex.org/W2293685972","https://openalex.org/W4392383417","https://openalex.org/W2899125270","https://openalex.org/W2267589039","https://openalex.org/W2165363605","https://openalex.org/W2033203427","https://openalex.org/W4299871285","https://openalex.org/W2950284538"],"abstract_inverted_index":{"A":[0],"notion":[1],"of":[2,6,25,34,67,77,86],"generalized":[3],"(two-parameterized)":[4],"t-transformation":[5],"free":[7,12,27,59,80,89,98,111],"convolution,":[8,13,100],"also":[9],"called":[10],"(t=(a,b))-deformed":[11],"is":[14,42,51,73,105],"introduced":[15],"for":[16,53,75],"a\u2208R":[17],"and":[18,84],"b&gt;0.":[19],"In":[20,62],"this":[21],"article,":[22],"some":[23],"results":[24],"t-deformed":[26,58,79,88,110],"convolution":[28,60],"are":[29],"given":[30],"within":[31],"the":[32,54,65,68,78,87,97],"theory":[33],"Cauchy-Stieltjes":[35],"Kernel":[36],"(CSK)":[37],"families.":[38,48],"The":[39],"variance":[40,55,69],"function":[41,56],"a":[43,101],"fundamental":[44],"concept":[45],"in":[46],"CSK":[47,82,91],"An":[49],"expression":[50],"provided":[52,74,106],"under":[57],"power.":[61],"addition,":[63],"through":[64],"use":[66],"function,":[70],"an":[71],"approximation":[72],"members":[76,85],"Gaussian":[81],"family":[83,92],"Poisson":[90],"respectively.":[93],"Furthermore,":[94],"by":[95],"involving":[96],"multiplicative":[99],"new":[102],"limit":[103],"theorem":[104],"with":[107],"respect":[108],"to":[109],"convolution.":[112]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2024-03-20T00:00:00"}
