The Bernstein problem for type \((n-2,2)\)
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 mathematical expressions 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 \(A=\text{Re}\oplus U_e\oplus V_e\), where \(\text{ker} (\omega)= U_e\oplus V_e\), \(U_e=\{x\in A:ex={1\over2}x\}\) and \(V_e=\{x\in A:ex=0\}\). It is a well-known fact that \(\dim U_e\) (and then \(\dim V_e\)), \(\dim U^2_e\) and \(\dim(u_eV_e+ V^2_e)\) do not depend on the choice of the idempotent element, so the type \(A=(m,\delta)\) can be defined where \(m-1=\dim U_e\) and \(\delta= \dim V_e\) (so \(n=\dim A=m+\delta)\). Following definitions introduced by Lyubich, if \(U_eV_e+ V^2_e=0\), the Bernstein algebra \(A\) is called regular, and if \(U^2_e=0\) is 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 in the last years. The second section of this paper is devoted to exposing one of these: the case of stochastic nonexceptional nonregular nonnuclear 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 paper 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 883778