{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T16:18:00Z","timestamp":1764173880677,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2012,8,17]],"date-time":"2012-08-17T00:00:00Z","timestamp":1345161600000},"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>The discrete Shannon entropy H was formulated only to measure indeterminacy effected through a set of probabilities, but the indeterminacy in a real-valued discrete variable depends on both the allowed outcomes x and the corresponding probabilities \u00de. A fundamental measure that is sensitive to both x and p is derived here from the total differential entropy of a continuous real variable and its conjugate in the discrete limit, where the conjugate is universally eliminated. The asymptotic differential entropy recovers H plus the new measure, named \u2261, which provides a novel probe of intrinsic organization in sequences of real numbers.<\/jats:p>","DOI":"10.3390\/e14081522","type":"journal-article","created":{"date-parts":[[2012,8,17]],"date-time":"2012-08-17T11:02:57Z","timestamp":1345201377000},"page":"1522-1538","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["The Entropy of a Discrete Real Variable"],"prefix":"10.3390","volume":"14","author":[{"given":"Scott","family":"Funkhouser","sequence":"first","affiliation":[{"name":"SPAWAR Systems Center Atlantic, Joint Base Charleston, North Charleston, SC 29406, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,8,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell. 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Math., submitted for publication."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"012311","DOI":"10.1103\/PhysRevA.84.012311","article-title":"Informational derivation of quantum theory","volume":"84","author":"Chiribella","year":"2011","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/8\/1522\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:51:51Z","timestamp":1760219511000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/14\/8\/1522"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,8,17]]},"references-count":13,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2012,8]]}},"alternative-id":["e14081522"],"URL":"https:\/\/doi.org\/10.3390\/e14081522","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2012,8,17]]}}}