Structure of Bernstein populations of type \((3,n-3)\)
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 the last years. The second section of this paper is devoted to exposing one of these: the case stochastic nonexceptional nonregular nonnuclear Bernstein algebras of type \((n-2,2)\). As a consequence of this survey, the Bernstein problem in the degenerate case can be solved. The last theorem of the work is a reformulation of the main result for stationary evolutionary operators, which is a new support to the Lyubich conjecture.
scientific article; zbMATH DE number 1100759
scientific article