Structure of Bernstein populations of type \((3,n-3)\)
type: Item
instance of: scholarly article
title: Structure of Bernstein populations of type \((3,n-3)\)
zbMATH Open document ID: 0908.17024
DOI: 10.1016/S0024-3795(97)81512-1
author: Juan C. Gutierrez Fernandez
published in: Linear Algebra and its Applications
publication date: 1998-05-18T00:00:00Z
review text: In this paper, a full description of stochastic nonregular nonexceptional nonnuclear Bernstein algebras of type \((n-2,2)\) is obtained. So, if the Lyubich conjecture is proved to be true, the above result would completely solve the Bernstein problem for the type \((n-2,2)\). The first section is devoted to introducing the Bernstein problem, concerning the mathematical expression of fundamental laws of biological heredity, and the basic related concepts. We recall that a Bernstein algebra \((A,\omega)\) is a (commutative) real algebra \(A\) endowed with a nonzero algebra homomorphism \(\omega :A\to{\mathbb R}\) satisfying \((x^2)^2=\omega (x)^2x^2\) for any \(x\in A\). It is said that the Bernstein algebra \(A\) is a stochastic Bernstein algebra if it has a finite basis (named stochastic basis) such that the simplex \(\Delta\) spanned by this basis is invariant with respect to the multiplication (\(\Delta\cdot\Delta\subset\Delta\)). Such an algebra has a nonzero idempotent \(e\) which gives a Peirce decomposition \(A={\mathbb R}e\oplus U_e\oplus V_e\), where \(\ker (\omega)=U_e\oplus V_e\), \(U_e=\{ x\in A : ex={1\over 2}x\}\) and \(V_e=\{ x\in A : ex=0\}\). It is a well known fact that \(\dim U_e\) (and then \(\dim V_e\)), \(\dim U_e^2\) and \(\dim (u_eV_e+V_e^2)\) does not depend on the choice of the idempotent element, so one can define the type \(A=(m,\delta)\) where \(m-1=\dim U_e\) and \(\delta =\dim V_e\) (so \(n=\dim A=m+\delta\)). Following definitions introduced by Lyubich, if \(U_eV_e+V_e^2=0\), the Bernstein algebra \(A\) is called regular, and if \(U_e^2=0\), exceptional. Every Bernstein algebra of type \((m,\delta)\) with \(m-1\) or \(\delta\) less than or equal to \(1\) is regular or exceptional. An algebra \(A\) is called nuclear if \(A^2=A\). The Bernstein problem is reduced to describing all stochastic Bernstein algebras and the multiplication with respect to a stochastic basis. Several partial solutions to this problem have been obtained during th…
Mathematics Subject Classification ID: 17D92
zbMATH DE Number: 1100759
zbMATH Keywords: genetic algebras
zbMATH Keywords: Bernstein algebras
zbMATH Keywords: Bernstein problem
zbMATH Keywords: stochastic Bernstein algebra
reviewed by: A. Castellón Serrano
MaRDI profile type: Publication
cites work: Idempotent Elements in a Bernstein Algebra
cites work: Classification of bernstein algebras of type (3,n- 3)
cites work: The Bernstein problem in dimension 5
cites work: On regular Bernstein algebras
cites work: The Bernstein problem for type \((n-2,2)\)
cites work: The Bernstein problem in dimension 6
cites work: Genetic Algebras Satisfying Bernstein's Stationarity Principle
cites work: TWO-LEVEL BERNSTEINIAN POPULATIONS
cites work: Q4079379
cites work: Q3869128
cites work: Q4182313
cites work: Q4283475
cites work: Q3724143
full work available at URL: s0024-3795(97)81512-1
OpenAlex ID: W2044114988
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description: scientific article; zbMATH DE number 1100759