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 $ \mathbf R ^ {n} $ whose coordinates satisfy $ x _ {i} \geq 0 $( $ i = 1 \dots n $) and $ \sum _ {i = 1 } ^ {n} x _ {i} = 1 $. The set $ S $ of all states is a simplex in $ \mathbf R ^ {n} $ and the vertices $ e _ {i} $( $ i = 1 \dots n $) of $ S $ are the different types of individuals in the population. If $ \gamma _ {ijk } $ is the probability that an individual $ e _ {k} $ appears in the next generation from parents of types $ e _ {i} $ and $ e _ {j} $, then $ \sum _ {k = 1 } ^ {n} \gamma _ {ijk } = 1 $( $ i,j = 1 \dots n $) and $ \gamma _ {ijk } = \gamma _ {jik } $( $ i,j,k = 1 \dots n $). 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 $ x _ {k} ^ \prime = \sum _ {i,j = 1 } ^ {n} \gamma _ {ijk } x _ {i} x _ {j} $( $ k = 1 \dots n $). These relations define a quadratic operator $ V : S \rightarrow S $ called the evolutionary quadratic operator. The Bernstein stationarity principle says that $ V ^ {2} = V $ and the Bernstein problem aims at describing all quadratic operators satisfying this principle. Bernstein solved his problem for $ n = 3 $ 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 $ \mathbf R ^ {n} $ an algebra structure can be defined via the operator $ V $ by

$$ xy = { \frac{1}{2} } ( V ( x + y ) - V ( x ) - V ( y ) ) $$

for all $ x, y \in \mathbf R ^ {n} $, and if $ \omega : {\mathbf R ^ {n} } \rightarrow \mathbf R $ is the mapping defined by $ x = ( x _ {1} \dots x _ {n} ) \mapsto \sum _ {i = 1 } ^ {n} x _ {i} $, then $ V ^ {2} = V $ if and only if $ ( x ^ {2} ) ^ {2} = \omega ( x ) ^ {2} x ^ {2} $ for all $ x \in \mathbf R ^ {n} $. Moreover, $ \omega ( xy ) = \omega ( x ) \omega ( y ) $ for all $ x, y \in \mathbf R ^ {n} $. Of course, to define this multiplication over the whole space $ \mathbf R ^ {n} $ starting from the simplex $ S $, one has to make convenient extensions of this multiplication by bilinearity. Now, in general, if $ K $ is a (commutative) field and $ A $ is a commutative $ K $- algebra, then a weighted algebra $ ( A, \omega ) $ over $ K $ is said to be a Bernstein algebra if $ ( x ^ {2} ) ^ {2} = \omega ( x ) ^ {2} x ^ {2} $ for all $ x \in A $( 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 $ x \in A $, where $ ( A, \omega ) $ is a weighted algebra, the plenary powers $ x ^ {[ m ] } $ of $ x $ are defined by $ x ^ {[ 1 ] } = x $ and $ x ^ {[ m -1 ] } x ^ {[ m -1 ] } = x ^ {[ m ] } $ for all integer $ m \geq 2 $. The plenary powers can be interpreted by saying that they represent random mating between discrete non-overlapping generations. $ ( A, \omega ) $ is called an $ n $ th order Bernstein algebra if $ x ^ {[ n + 2 ] } = \omega ( x ) ^ {2 ^ {n} } x ^ {[ n + 1 ] } $ for all $ x \in A $, where $ n \geq 1 $ 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 $ x ^ {[ n + 2 ] } = x ^ {[ n + 1 ] } $( $ x \in A $ such that $ \omega ( x ) = 1 $) is that equilibrium in the population is reached after exactly $ n $ 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 $5$" 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 $(n-2,2)$" 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