{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:17:10Z","timestamp":1760235430938,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,21]],"date-time":"2021-08-21T00:00:00Z","timestamp":1629504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-20-1-0348"],"award-info":[{"award-number":["FA9550-20-1-0348"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+\u22ef+a22y+a11y+a00y for some n,a0,\u2026,an\u2208N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir\u2192Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R\u2a7e0\u00d7R\u2a7e0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.<\/jats:p>","DOI":"10.3390\/e23081085","type":"journal-article","created":{"date-parts":[[2021,8,22]],"date-time":"2021-08-22T21:47:52Z","timestamp":1629668872000},"page":"1085","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dirichlet Polynomials and Entropy"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9326-5328","authenticated-orcid":false,"given":"David I.","family":"Spivak","sequence":"first","affiliation":[{"name":"Topos Institute, Berkeley, CA 94704, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4580-0741","authenticated-orcid":false,"given":"Timothy","family":"Hosgood","sequence":"additional","affiliation":[{"name":"Topos Institute, Berkeley, CA 94704, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,21]]},"reference":[{"key":"ref_1","unstructured":"Spivak, D.I., and Myers, D.J. (2020). Dirichlet Polynomials form a Topos. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Leinster, T. (2014). Basic Category Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781107360068"},{"key":"ref_3","first-page":"170","article-title":"A probability monad as the colimit of spaces of finite samples","volume":"34","author":"Fritz","year":"2019","journal-title":"Theory Appl. Categ."},{"key":"ref_4","first-page":"489","article-title":"Higher-Dimensional Algebra VII: Groupoidification","volume":"24","author":"Baez","year":"2010","journal-title":"Theory Appl. Categ."},{"key":"ref_5","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 Syst. Tech. J."},{"key":"ref_6","unstructured":"KidzSearch Wiki (2021, August 18). Rectangle Facts for Kids. Available online: https:\/\/wiki.kidzsearch.com\/wiki\/Rectangle."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1945","DOI":"10.3390\/e13111945","article-title":"A Characterization of Entropy in Terms of Information Loss","volume":"13","author":"Baez","year":"2011","journal-title":"Entropy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1109\/JRPROC.1952.273898","article-title":"A method for the construction of minimum-redundancy codes","volume":"40","author":"Huffman","year":"1952","journal-title":"Proc. IRE"},{"key":"ref_9","first-page":"421","article-title":"A Bayesian characterization of relative entropy","volume":"29","author":"Baez","year":"2014","journal-title":"Theory Appl. Categ."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1085\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:48:30Z","timestamp":1760165310000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1085"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,21]]},"references-count":9,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081085"],"URL":"https:\/\/doi.org\/10.3390\/e23081085","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,8,21]]}}}