12 problems

Let $X$ be a finite-dimensional non-degenerate evolution ${\mathbb K}$-algebra, and write its optimal direct-sum decomposition into indecomposable evolution ideals as … For an evol…

Let $\mathcal{E}$ be a complex evolution algebra. An element $u\in\mathcal{E}$ is an idempotent if $u^2=u$. If $\mathcal{E}$ has natural basis $\{e_1,\dots,e_n\}$ and structure mat…

Nonzero-solution conjecture. This system admits a solution $(x_1,x_2,\dots,x_n)$ such that $x_i\neq0$ for all $i$. The paper proves a relaxed version of this conjecture, rather tha…

Let $n\in\mathbb{N}$. A field $\mathbb{K}$ is an $n$-FSEANI if the parametric family of systems of equations given by (EE) has a non-zero solution in $\mathbb{K}$ for every choice…

Let $A$ be a two-dimensional evolution algebra over a field $K$, and let $\operatorname{Aut}(A)$ denote its group of automorphisms. An associative representation $\mathcal U_p$ is…

Let $G$ be a graph, and let $\mathcal{A}_{RW}(G)$ and $\mathcal{A}(G)$ denote its random-walk and graph-associated evolution algebras, respectively. Suppose that…

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…

Graph evolution-algebra isomorphism conjecture.

Attractiveness conjecture for $Q_2$. The point $Q_2$ is an attracting point for any $a,b,c,d$ satisfying this inequality.

Attractiveness conjecture for $P_0$. Based on numerical computations, $P_0$ is attractive for any parameters $a,b,c,d$ satisfying these conditions.

Let $E$ be an $n$-dimensional non-nilpotent evolution algebra satisfying condition $P$. Let $ES_1$ denote the one-dimensional evolution algebra used in the source, let…

Let $A=(a_{i,j})_{1\leq i,j\leq n}$ be a complex invertible matrix, and let $x_1,\dots,x_n$ be complex variables. Nonvanishing solution conjecture. The system … has a solution…