{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:07:55Z","timestamp":1760238475866,"version":"build-2065373602"},"reference-count":5,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,10]],"date-time":"2020-08-10T00:00:00Z","timestamp":1597017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["19K03536"],"award-info":[{"award-number":["19K03536"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.<\/jats:p>","DOI":"10.3390\/sym12081333","type":"journal-article","created":{"date-parts":[[2020,8,10]],"date-time":"2020-08-10T07:25:03Z","timestamp":1597044303000},"page":"1333","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Gyrogeometric Mean in the Einstein Gyrogroup"],"prefix":"10.3390","volume":"12","author":[{"given":"Takuro","family":"Honma","sequence":"first","affiliation":[{"name":"Niigata Pref. Takada Senior High School, Yasuduka Branch School, Yasuduka 942-0411, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4338-1355","authenticated-orcid":false,"given":"Osamu","family":"Hatori","sequence":"additional","affiliation":[{"name":"Institute of Science and Technology, Niigata University, Niigata 950-2181, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ungar, A.A. (2008). Analytic Hyperbolic Geometry and Albert Einstein\u2019s Special Theory of Relativity, World Scientific Publishing Co. Pte. Ltd.. 981-277-229-4.","DOI":"10.1142\/9789812772305"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/S0034-4877(15)30019-7","article-title":"Einstein gyrogroup as a B-loop","volume":"76","author":"Suksumran","year":"2015","journal-title":"Rep. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1016\/j.laa.2005.08.025","article-title":"Riemannian geometry and matrix geometric means","volume":"413","author":"Bhatia","year":"2006","journal-title":"Linear Algebra Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.laa.2003.11.019","article-title":"Geometric means","volume":"385","author":"Ando","year":"2004","journal-title":"Linear Algebra Appl."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ungar, A.A. (2009). A gyrovector space approach to hyperbolic geometry. Synthesis Lectures on Mathematics and Statistics, Morgan & Claypool Publishers.","DOI":"10.1007\/978-3-031-02396-5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1333\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:58:43Z","timestamp":1760176723000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1333"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,10]]},"references-count":5,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081333"],"URL":"https:\/\/doi.org\/10.3390\/sym12081333","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,8,10]]}}}