{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:10:18Z","timestamp":1760238618497,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,8,28]],"date-time":"2020-08-28T00:00:00Z","timestamp":1598572800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["No. 2020R1A2B5B01001864","No. 2020R1F1A1A01065568"],"award-info":[{"award-number":["No. 2020R1A2B5B01001864","No. 2020R1F1A1A01065568"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider mixture experiments in which the proportions of the components must be non-negative and their sum must equal one. Thus, the experimental region for a mixture of components is a simplex. Li and Zhang (2017) made the conjecture that the pseudo component transformation of the lattice points in the simplex has a special property. In this paper, we show that this conjecture is not true in general. Furthermore, we refine this conjecture and prove the refined conjecture.<\/jats:p>","DOI":"10.3390\/sym12091427","type":"journal-article","created":{"date-parts":[[2020,8,28]],"date-time":"2020-08-28T10:23:08Z","timestamp":1598610188000},"page":"1427","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Refinement of the Conjecture on the Pseudo Component Transformation of the Lattice Points in the Simplex"],"prefix":"10.3390","volume":"12","author":[{"given":"Bara","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, Korea"}]},{"given":"Jeongsim","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju, Chungbuk 28644, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cornell, J.A. (2011). A Primer on Experiments with Mixtures, John Wiley & Sons Inc.","DOI":"10.1002\/9780470907443"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Raghavarao, D., Wiley, J.B., and Chitturi, P. (2011). Choice-Based Conjoint Analysis: Models and Designs, Taylor & Francis Group, LLC.","DOI":"10.1201\/9781420099973"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2172","DOI":"10.1080\/03610926.2013.875576","article-title":"Construction of uniform designs for mixture experiments with complex constraints","volume":"45","author":"Liu","year":"2016","journal-title":"Commun. Stat.-Theor. Methods"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1111\/j.2517-6161.1958.tb00299.x","article-title":"Experiments with mixtures","volume":"20","year":"1958","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.spl.2017.07.017","article-title":"The pseudo component transformation design for experiment with mixture","volume":"131","author":"Li","year":"2017","journal-title":"Stat. Probabil. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1080\/00401706.1962.10490034","article-title":"Simplex lattice designs for multicomponent systems","volume":"4","author":"Gorman","year":"1962","journal-title":"Technometrics"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Fang, K.-T., and Wang, Y. (1994). Number Theoretic Methods in Statistics, Chapman and Hall.","DOI":"10.1007\/978-1-4899-3095-8"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1080\/00224065.2009.11917758","article-title":"Uniform designs for highly constrained mixture experiments","volume":"41","author":"Borkowski","year":"2009","journal-title":"J. Qual. Technol."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1427\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:04:13Z","timestamp":1760177053000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1427"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,28]]},"references-count":8,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091427"],"URL":"https:\/\/doi.org\/10.3390\/sym12091427","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,8,28]]}}}