arXiv subjects

Ravil Bildanov

Publications and source records attributed to Ravil Bildanov.

3 recordsLinked to original sources

arXiv2024

On $3$-generated axial algebras of Jordan type $\frac{1}{2}$

Axial algebras of Jordan type $η$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-η)$, where $η$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field.

math.RA↗

arXiv2020

On WL-rank and WL-dimension of some Deza circulant graphs

The WL-rank of a digraph $Γ$ is defined to be the rank of the coherent configuration of $Γ$. The WL-dimension of $Γ$ is defined to be the smallest positive integer $m$ for which $Γ$ is identified by the $m$-dimensional Weisfeiler-Leman algorithm. We classify the Deza circulant graphs of WL-rank $4$. In additional, it is proved that each of these graphs has WL-dimension at most $3$. Finally, we establish that some families of Deza circulant graphs have WL-rank $5$ or $6$ and WL-dimension at most $3$.

math.CO↗

arXiv2020

Factoring nonabelian finite groups into two subsets

A group $G$ is said to be factorized into subsets $A_1, A_2, \ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following conjecture: for every finite group $G$ and every factorization $n=ab$ of its order, there is a factorization $G=AB$ with $|A|=a$ and $|B|=b$. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than $10\,000$.

math.GR↗