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H., private communication."},{"key":"BF02414066_CR2","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/978-1-4612-0963-8","volume-title":"Unsolved Problems in Geometry","author":"H. T. Croft","year":"1991","unstructured":"Croft, H. T., Fowler, K. J. and Guy, R. K.,Unsolved Problems in Geometry, Springer-Verlag, New York, 1991, pp. 171\u2013172."},{"key":"BF02414066_CR3","first-page":"811","volume":"84","author":"R. K. Guy","year":"1977","unstructured":"Guy, R. K., \u2018Research problems\u2019,Amer. Math. Monthly 84 (1977), 811.","journal-title":"Amer. Math. Monthly"},{"key":"BF02414066_CR4","first-page":"66","volume":"4","author":"J. M. Hammersley","year":"1968","unstructured":"Hammersley, J. M., \u2018On the enfeeblement of mathematical skills by ldModern Mathematicsrd and by similar soft intellectural trash in schools and universities\u2019,Bull. Inst. Math. Appl. 4 (1968), 66\u201385.","journal-title":"Bull. Inst. Math. 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Ilm. Mat. Sains dan Teknol."],"abstract":"<jats:p>The Moving Sofa Problem concerns the planar shape of maximum area that can be moved around a right-angled corner in a two-dimensional hallway of unit width. The objectives of this study are (1) to validate the upper bound estimate of the sofa area obtained from the intersection of the straight corridor and the right-angled corridor models, (2) to analyze the movement mechanism of the Hammersley sofa through rotation paths and contact paths, and (3) to provide details of the Gerver sofa area calculation for numerical validation. The methods used include analysis of the function A(u, \u03b8), which is the upper bound value function of the sofa area obtained from the area of the intersection of the straight corridor and the right-angled corridor, the application of the concepts of rotation paths, contact points, and contact paths to prove that the Hammersley sofa construction can pass through the right-angled corridor, and the calculation of the area using Green\u2019s Theorem to validate the Gerver sofa area. The main results show that (1) the minimum upper bound of the function A(u, \u03b8) reaches two times the square root of two under certain conditions, (2) the rotation path used proves that the Hammersley sofa satisfies the definition of a shape that can pass through a right-angled corridor, and (3) calculations using Green\u2019s Theorem yield an area of approximately 2.2195 area units. The findings of this study clarify the geometric construction elements of the Hammersley and Gerver sofas, and provide validation details that have rarely been fully described before.<\/jats:p>","DOI":"10.37905\/euler.v13i3.33457","type":"journal-article","created":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T05:38:06Z","timestamp":1766468286000},"page":"446-454","source":"Crossref","is-referenced-by-count":0,"title":["Validasi Estimasi Batas Atas Luas Sofa dan Perhitungan Luas Sofa Gerver serta Analisis Mekanisme Pergerakan Sofa Hammersley"],"prefix":"10.37905","volume":"13","author":[{"given":"Muhammad Imam","family":"Akbar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aji","family":"Gumilang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Denny Ivanal","family":"Hakim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"24930","published-online":{"date-parts":[[2025,12,16]]},"container-title":["Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi"],"link":[{"URL":"https:\/\/ejurnal.ung.ac.id\/index.php\/Euler\/article\/viewFile\/33457\/12150","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/ejurnal.ung.ac.id\/index.php\/Euler\/article\/viewFile\/33457\/12150","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T05:38:07Z","timestamp":1766468287000},"score":26.88855,"resource":{"primary":{"URL":"https:\/\/ejurnal.ung.ac.id\/index.php\/Euler\/article\/view\/33457"}},"issued":{"date-parts":[[2025,12,16]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,11,27]]}},"URL":"https:\/\/doi.org\/10.37905\/euler.v13i3.33457","ISSN":["2776-3706","2087-9393"],"issn-type":[{"value":"2776-3706","type":"electronic"},{"value":"2087-9393","type":"print"}],"published":{"date-parts":[[2025,12,16]]}},{"indexed":{"date-parts":[[2024,9,26]],"date-time":"2024-09-26T04:07:16Z","timestamp":1727323636115},"posted":{"date-parts":[[2024]]},"group-title":"SSRN","reference-count":0,"publisher":"Elsevier BV","content-domain":{"domain":[],"crossmark-restriction":false},"DOI":"10.2139\/ssrn.4967224","type":"posted-content","created":{"date-parts":[[2024,9,25]],"date-time":"2024-09-25T08:54:13Z","timestamp":1727254453000},"source":"Crossref","is-referenced-by-count":0,"title":["Deep Learning Evidence for Global Optimality of Gerver's Sofa"],"prefix":"10.2139","author":[{"given":"Kuangdai","family":"Leng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jia","family":"Bi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jaehoon","family":"Cha","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Samuel","family":"Pinilla","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeyan","family":"Thiyagalingam","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","deposited":{"date-parts":[[2024,9,25]],"date-time":"2024-09-25T08:54:13Z","timestamp":1727254453000},"score":23.870977,"resource":{"primary":{"URL":"https:\/\/www.ssrn.com\/abstract=4967224"}},"issued":{"date-parts":[[2024]]},"references-count":0,"URL":"https:\/\/doi.org\/10.2139\/ssrn.4967224","published":{"date-parts":[[2024]]},"subtype":"preprint"},{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:30:36Z","timestamp":1760146236928,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T00:00:00Z","timestamp":1729209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"EPSRC grant","doi-asserted-by":"publisher","award":["EP\/X019918\/1"],"award-info":[{"award-number":["EP\/X019918\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver\u2019s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks\u2019 approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver\u2019s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our \u201cwaterfall\u201d algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver\u2019s solution. The second approach applies discrete optimization to the Kallus\u2013Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver\u2019s area from above, indicating that no larger sofa exists.<\/jats:p>","DOI":"10.3390\/sym16101388","type":"journal-article","created":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T06:46:52Z","timestamp":1729234012000},"page":"1388","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Deep Learning Evidence for Global Optimality of Gerver\u2019s Sofa"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3007-1825","authenticated-orcid":false,"given":"Kuangdai","family":"Leng","sequence":"first","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jia","family":"Bi","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jaehoon","family":"Cha","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Samuel","family":"Pinilla","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2167-1343","authenticated-orcid":false,"given":"Jeyan","family":"Thiyagalingam","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/1008074","article-title":"Problem 66-11, Moving furniture through a hallway","volume":"8","author":"Moser","year":"1966","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF00426226","article-title":"On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities","volume":"1","author":"Hammersley","year":"1968","journal-title":"Educ. 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This is why we developed an alternative method based on digital photography for taking these measurements which is faster and easier to use, to make anthropometry more user-friendly and approachable to paediatricians.<\/jats:p><jats:p>We conducted a cross-sectional study in 54 children between 2 and 18 years of age. We compared manual measurements with photogrammetric measurements to validate our method.<\/jats:p><jats:p>Inter-observer correlations of all measurements are \u22650.96 and mean differences are 0.3\u20130.9 cm, except for arm span. Comparison of manual to photogrammetric measurements shows mean differences of 0.6\u20131.3 cm, with correlations \u22650.92, except for sitting height and arm span. 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