{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T04:07:47Z","timestamp":1746158867881,"version":"3.40.4"},"reference-count":0,"publisher":"New York Business Global LLC","issue":"2","license":[{"start":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T00:00:00Z","timestamp":1746057600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Eur. J. Pure Appl. Math."],"abstract":"<jats:p>After the fundamental work of Livschitz in \\cite{L1,L2}, various research directions emerged, among which the following stand out: (i) the study of cocycles with values in groups and semigroups beyond $\\mathbb{R}$, as well as the investigation of corresponding regularity results;(ii) the analysis of how a certain degree of regularity ($C^{k}$ for $k=1,2,\\ldots,\\infty ,\\omega$) of the cocycle can confer corresponding regularity to the solution of the cohomological equation; and (iii) the study of higher-dimensional cohomology naturally associated with the action of groups such as $\\mathbb{Z}^{k}$ or $\\mathbb{R}^{k}$.The aim of this article is to present, as self-contained as possible, a review of the natural generalizations of the notions of cocycles and cochains, as well as their corresponding results, in the study of cohomological equations.<\/jats:p>","DOI":"10.29020\/nybg.ejpam.v18i2.5830","type":"journal-article","created":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T12:12:36Z","timestamp":1746101556000},"page":"5830","source":"Crossref","is-referenced-by-count":0,"title":["Cocycles in Lie Groups, Cochains and Regularity Problem"],"prefix":"10.29020","volume":"18","author":[{"given":"Ros\u00e1rio D.","family":"Laureano","sequence":"first","affiliation":[]}],"member":"11552","published-online":{"date-parts":[[2025,5,1]]},"container-title":["European Journal of Pure and Applied Mathematics"],"original-title":[],"link":[{"URL":"https:\/\/www.ejpam.com\/index.php\/ejpam\/article\/download\/5830\/2163","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.ejpam.com\/index.php\/ejpam\/article\/download\/5830\/2163","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,1]],"date-time":"2025-05-01T12:12:46Z","timestamp":1746101566000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ejpam.com\/index.php\/ejpam\/article\/view\/5830"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,1]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,5,1]]}},"URL":"https:\/\/doi.org\/10.29020\/nybg.ejpam.v18i2.5830","relation":{},"ISSN":["1307-5543"],"issn-type":[{"value":"1307-5543","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,1]]}}}