{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T01:23:04Z","timestamp":1768612984828,"version":"3.49.0"},"reference-count":27,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T00:00:00Z","timestamp":1707177600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia, I.P.","award":["UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/00144\/2020"]}]},{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia, I.P.","award":["UIDP\/00144\/2020"],"award-info":[{"award-number":["UIDP\/00144\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>In this short communication, following a brief introduction, we undertake a comprehensive analytical study of the weighted Simpson index. Our primary emphasis concerns the precise determination of the optimal point (minimizer) coordinates and of the minimum value of the index, a differentiable convex function, which is related to the harmonic mean concept. Furthermore, we address and solve the inversion problem and show the tight connection between both approaches. Last, we give some insights and final remarks on this subject.<\/jats:p>","DOI":"10.3390\/math12040507","type":"journal-article","created":{"date-parts":[[2024,2,6]],"date-time":"2024-02-06T12:38:24Z","timestamp":1707223104000},"page":"507","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Optimal Point of the Weighted Simpson Index"],"prefix":"10.3390","volume":"12","author":[{"given":"Jos\u00e9 Pinto","family":"Casquilho","sequence":"first","affiliation":[{"name":"Postgraduate and Research Program, Universidade Nacional Timor Lorosa\u2019e (UNTL), D\u00edli, Timor-Leste"}]},{"given":"Helena","family":"Mena-Matos","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica da Universidade do Porto (CMUP), Faculdade de Ci\u00eancias da Universidade do Porto (FCUP), 4169-007 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1038\/163688a0","article-title":"Measurement of diversity","volume":"163","author":"Simpson","year":"1949","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1007\/s11538-013-9817-6","article-title":"Mathematical models of memory CD8+ T-cell repertoire dynamics in response to viral infections","volume":"75","author":"Davis","year":"2013","journal-title":"Bull. 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