Geodesic
Two-level Bernsteinian populations
Sbornik. Mathematics, Tome 24 (1974) no. 4, pp. 593-615 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this article we give a partial solution to Bernstein's problem concerning quadratic mappings of a simplex into itself: the description of all those mappings $V$ satisfying the condition $V^2=V$. The formulation of this question is connected with mathematical genetics. In this article, the selected class of mappings is characterized by the existence of linear preservation laws sufficient for the parametrization of a mapping. In genetic terms, this means the existence of genes (but, generally speaking, with a more complex behavior than in classical genetics). This article describes all these mappings. Bibliography: 8 titles. 
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Yu. I. Lyubich. Two-level Bernsteinian populations. Sbornik. Mathematics, Tome 24 (1974) no. 4, pp. 593-615. http://geodesic.mathdoc.fr/item/SM_1974_24_4_a7/

[1] S. N. Bernshtein, “O prilozhenii matematiki k biologii”, Nauka na Ukraine, 1922, no. 1, 14–19

[2] S. N. Bernstein, “Principe de stationarite et generalisation de la loi de Mendel”, C. r. Acad. Sci. (Paris), 177 (1923), 528–531 | Zbl

[3] S. N. Bernshtein, “Reshenie odnoi matematicheskoi problemy, svyazannoi s teoriei nasledstvennosti”, Uchenye zapiski N.-i. kaf. Ukrainy, otd. matem., 1924, no. 1, 83–115

[4] Yu. I. Lyubich, “Osnovnye ponyatiya i teoremy evolyutsionnoi genetiki svobodnykh populyatsii”, Uspekhi matem. nauk, XXVI:5 (161) (1971), 51–116

[5] Yu. I. Lyubich, “K matematicheskoi teorii nasledstvennosti”, DAN SSSR, 209:5 (1973), 1028–1030 | Zbl

[6] Yu. I. Lyubich, “Ob analogakh zakona Khardi-Vainberga”, Genetika, IX:10 (1973), 139–144

[7] Yu. I. Lyubich, “Stroenie bernshteinovskikh populyatsii tipa $(2,n-2)$”, Uspekhi matem. nauk, XXX:1(181) (1975), 247–248 | MR | Zbl

[8] Yu. I. Lyubich, “Stroenie bernshteinovskikh populyatsii tipa $(n-1, 1)$”, Uspekhi matem. nauk, XXVIII:5 (173) (1973), 247–248