{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,14]],"date-time":"2025-11-14T07:26:43Z","timestamp":1763105203983,"version":"build-2065373602"},"reference-count":7,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2008,12,22]],"date-time":"2008-12-22T00:00:00Z","timestamp":1229904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We construct non-linear information inequalities from Mat\u00b4u\u02c7s\u2019 infinite series of linear information inequalities. Each single non-linear inequality is sufficiently strong to prove that the closure of the set of all entropy functions is not polyhedral for four or more random variables, a fact that was already established using the series of linear inequalities. To the best of our knowledge, they are the first non-trivial examples of non-linear information inequalities.<\/jats:p>","DOI":"10.3390\/e10040765","type":"journal-article","created":{"date-parts":[[2008,12,17]],"date-time":"2008-12-17T05:55:18Z","timestamp":1229493318000},"page":"765-775","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Non-linear Information Inequalities"],"prefix":"10.3390","volume":"10","author":[{"given":"Terence","family":"Chan","sequence":"first","affiliation":[{"name":"Institute for Telecommunications Research, University of South Australia, Australia"}]},{"given":"Alex","family":"Grant","sequence":"additional","affiliation":[{"name":"Institute for Telecommunications Research, University of South Australia, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2008,12,22]]},"reference":[{"key":"ref_1","unstructured":"Mat\u00fa\u0161, F. (2007). IEEE Int. Symp. Inform. Theory."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1440","DOI":"10.1109\/18.681320","article-title":"On the characterization of entropy function via information inequalities","volume":"44","author":"Zhang","year":"1998","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1992","DOI":"10.1109\/TIT.2002.1013138","article-title":"On a relation between information inequalities and group theory","volume":"48","author":"Chan","year":"2002","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_4","unstructured":"Dougherty, R., Freiling, C., and Zeger, K. (2006). IEEE Int. Symp. Inform. Theory."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4470","DOI":"10.1109\/TIT.2008.928963","article-title":"Dualities between entropy functions and network codes","volume":"54","author":"Chan","year":"2008","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1924","DOI":"10.1109\/18.641556","article-title":"A framework for linear information inequalities","volume":"43","author":"Yeung","year":"1997","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3129","DOI":"10.1109\/TIT.2003.820015","article-title":"Zero-error network coding for acyclic networks","volume":"49","author":"Song","year":"2003","journal-title":"IEEE Trans. Inform. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/4\/765\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:21:16Z","timestamp":1760221276000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/4\/765"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12,22]]},"references-count":7,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2008,12]]}},"alternative-id":["e10040765"],"URL":"https:\/\/doi.org\/10.3390\/e10040765","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2008,12,22]]}}}