# Bernstein algebra

Around 1900, S.N. Bernstein (cf. [a3], [a4], [a5]) worked on an important problem concerning the laws of formal genetics. This problem is known today as the Bernstein problem. Following Yu.I. Lyubich (cf. [a10]), this problem can be expressed as follows. The state of a population in a given generation is described by a vector in whose coordinates satisfy () and . The set of all states is a simplex in and the vertices () of are the different types of individuals in the population. If is the probability that an individual appears in the next generation from parents of types and , then () and (). In absence of selection and under random hypothesis, the state of the population in the next generation can be written, in terms of coordinates, as (). These relations define a quadratic operator called the evolutionary quadratic operator. The Bernstein stationarity principle says that and the Bernstein problem aims at describing all quadratic operators satisfying this principle. Bernstein solved his problem for and much progress was achieved recently (cf. [a6], [a8]) in this direction. The Bernstein problem can be translated in terms of algebra structure. In fact, over an algebra structure can be defined via the operator by

for all , and if is the mapping defined by , then if and only if for all . Moreover, for all . Of course, to define this multiplication over the whole space starting from the simplex , one has to make convenient extensions of this multiplication by bilinearity. Now, in general, if is a (commutative) field and is a commutative -algebra, then a weighted algebra over is said to be a Bernstein algebra if for all (cf. [a2]). In recent years (1990s), the theory of Bernstein algebras has been substantially improved. V.M. Abraham (cf. [a1]) suggests the construction of a generalized Bernstein algebra. In this perspective, for an element , where is a weighted algebra, the plenary powers of are defined by and for all integer . The plenary powers can be interpreted by saying that they represent random mating between discrete non-overlapping generations. is called an th order Bernstein algebra if for all , where is the smallest such integer (cf. [a11]). Second-order Bernstein algebras are simply called Bernstein algebras and first-order Bernstein algebras are also called gametic diploid algebras. The interpretation of the equation ( such that ) is that equilibrium in the population is reached after exactly generations of intermixing. For genetic properties of Bernstein algebras, see [a7] and [a12].

See also Genetic algebra; Baric algebra.

#### References

[a1] | V.M. Abraham, "Linearising quadratic transformations in genetic algebras" Thesis, Univ. London (1975) |

[a2] | M.T. Alcalde, C. Burgueno, A. Labra, A. Micali, "Sur les algèbres de Bernstein" Proc. London Math. Soc. (3) , 58 (1989) pp. 51–68 |

[a3] | S.N. Bernstein, "Principe de stationarité et généralisation de la loi de Mendel" C.R. Acad. Sci. Paris , 177 (1923) pp. 528–531 |

[a4] | S.N. Bernstein, "Démonstration mathématique de la loi d'hérédité de Mendel" C.R. Acad. Sci. Paris , 177 (1923) pp. 581–584 |

[a5] | S.N. Bernstein, "Solution of a mathematical problem connected with the theory of heredity" Ann. Math. Stat. , 13 (1942) pp. 53–61 |

[a6] | S. González, J.C. Gutiérrez, C. Martínez, "The Bernstein problem in dimension " J. Algebra , 177 (1995) pp. 676–697 |

[a7] | A.N. Griskhov, "On the genetic property of Bernstein algebras" Soviet Math. Dokl. , 35 (1987) pp. 489–492 (In Russian) |

[a8] | J.C. Gutiérrez, "The Bernstein problem for type " J. Algebra , 181 (1996) pp. 613–627 |

[a9] | P. Holgate, "Genetic algebras satisfying Bernstein's stationarity principle" J. London Math. Soc. (2) , 9 (1975) pp. 613–623 |

[a10] | Yu.I. Lyubich, "Mathematical structures in population genetics" Biomathematics , 22 (1992) |

[a11] | C. Mallol, A. Micali, M. Ouattara, "Sur les algèbres de Bernstein IV" Linear Alg. & Its Appl. , 158 (1991) pp. 1–26 |

[a12] | A. Micali, M. Ouattara, "Structure des algèbres de Bernstein" Linear Alg. & Its Appl. , 218 (1995) pp. 77–88 |

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Bernstein algebra.

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