Natural number
27 (twenty-seven) is the natural number following26 and preceding28.
Including the null-motif, there are 27 distincthypergraph motifs.[1]
TheClebsch surface, with 27 straight linesThere are exactlytwenty-seven straight lines on a smoothcubic surface,[2] which give a basis of thefundamental representation ofLie algebra
.[3][4]
The unique simple formally realJordan algebra, the exceptional Jordan algebra of self-adjoint3 by 3 matrices ofquaternions, is 27-dimensional;[5] itsautomorphism group is the 52-dimensionalexceptional Lie algebra
[6]
There are twenty-sevensporadic groups, if thenon-strict group of Lie type
(with anirreducible representation that is twice that of
in 104 dimensions)[7] is included.[8]
InRobin's theorem for theRiemann hypothesis, twenty-seven integers fail to hold
for values
where
is theEuler–Mascheroni constant; this hypothesis is trueif and only if this inequality holds for every larger
[9][10][11]
TheClebsch surface has 27exceptional lines can be defined over the real numbers.
It is possible to arrange 27 vertices and connect them with edges to create theHolt graph.
27 is 33, and therefore, it is the secondtetration of 3 (23).
- ^Lee, Geon; Ko, Jihoon; Shin, Kijung (2020). "Hypergraph Motifs: Concepts, Algorithms, and Discoveries". In Balazinska, Magdalena; Zhou, Xiaofang (eds.).46th International Conference on Very Large Data Bases. Proceedings of the VLDB Endowment. Vol. 13.ACM Digital Library. pp. 2256–2269.arXiv:2003.01853.doi:10.14778/3407790.3407823.ISBN 9781713816126.OCLC 1246551346.S2CID 221779386.
- ^Baez, John Carlos (February 15, 2016)."27 Lines on a Cubic Surface".AMS Blogs.American Mathematical Society. RetrievedOctober 31, 2023.
- ^Aschbacher, Michael (1987). "The 27-dimensional module for E6. I".Inventiones Mathematicae.89. Heidelberg, DE:Springer:166–172.Bibcode:1987InMat..89..159A.doi:10.1007/BF01404676.MR 0892190.S2CID 121262085.Zbl 0629.20018.
- ^Sloane, N. J. A. (ed.)."Sequence A121737 (Dimensions of the irreducible representations of the simple Lie algebra of type E6 over the complex numbers, listed in increasing order.)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. RetrievedOctober 31, 2023.
- ^Kac, Victor Grigorievich (1977). "Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras".Communications in Algebra.5 (13).Taylor & Francis: 1380.doi:10.1080/00927877708822224.MR 0498755.S2CID 122274196.Zbl 0367.17007.
- ^Baez, John Carlos (2002)."The Octonions".Bulletin of the American Mathematical Society.39 (2). Providence, RI:American Mathematical Society:189–191.doi:10.1090/S0273-0979-01-00934-X.MR 1886087.S2CID 586512.Zbl 1026.17001.
- ^Lubeck, Frank (2001)."Smallest degrees of representations of exceptional groups of Lie type".Communications in Algebra.29 (5). Philadelphia, PA:Taylor & Francis: 2151.doi:10.1081/AGB-100002175.MR 1837968.S2CID 122060727.Zbl 1004.20003.
- ^Hartley, Michael I.; Hulpke, Alexander (2010)."Polytopes Derived from Sporadic Simple Groups".Contributions to Discrete Mathematics.5 (2). Alberta, CA:University of Calgary Department of Mathematics and Statistics: 27.doi:10.11575/cdm.v5i2.61945.ISSN 1715-0868.MR 2791293.S2CID 40845205.Zbl 1320.51021.
- ^Axler, Christian (2023)."On Robin's inequality".The Ramanujan Journal.61 (3). Heidelberg, GE:Springer:909–919.arXiv:2110.13478.Bibcode:2021arXiv211013478A.doi:10.1007/s11139-022-00683-0.S2CID 239885788.Zbl 1532.11010.
- ^Robin, Guy (1984)."Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann"(PDF).Journal de Mathématiques Pures et Appliquées. Neuvième Série (in French).63 (2):187–213.ISSN 0021-7824.MR 0774171.Zbl 0516.10036.
- ^Sloane, N. J. A. (ed.)."Sequence A067698 (Positive integers such that sigma(n) is greater than or equal to exp(gamma) * n * log(log(n)).)".TheOn-Line Encyclopedia of Integer Sequences. OEIS Foundation. RetrievedOctober 31, 2023.
Wells, D.The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987), p. 106.
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