7 problems

Let $V$ be a quadratic stochastic operator on the simplex $S^{m-1}$. It is called ergodic if, for every $\boldsymbol{x}\in S^{m-1}$, the limit … exists. Ulam's ergodic theorem conj…

Ulam's ergodicity conjecture. Any quadratic stochastic operator is ergodic.

Let $E$ be the finite set underlying the quadratic stochastic operator, let $F\subset E$, and let ${\mathbf P}$ be a matrix whose elements satisfy (7), thereby determining an $F$-Q…

Let $V\in\tilde{\mathcal V}^-$ and let $\frak{f}(V)=(a_{ki})$. For ${\mathbf{x}}\in S$, write $\operatorname{supp}({\mathbf{x}})$ for its support, let ${\mathbf e}_j$ denote the…

Let ${\mathcal E}_1$ and ${\mathcal E}_2$ be two $n$-dimensional isomorphic Volterra evolution algebras, and let the corresponding Volterra quadratic stochastic operators be the op…

Equivalence conjecture. For any ${\mathbf{x}},{\mathbf{y}}\in S^{n-1}$, the corresponding Markov measures are equivalent:

Let $S^2$ be the two-dimensional simplex, and let $W_\alpha:S^2\to S^2$ be the quadratic stochastic operator defined in the source by equation $W_\alpha$. Assume…