arXiv subjects

Petr Akhmet'ev

Publications and source records attributed to Petr Akhmet'ev.

3 recordsLinked to original sources

arXiv2024

Combinatorial formula for the $M$ invariant of magnetic lines

To solve MHD problems within the framework of the theory of two-scale mean fields, it is important to study the invariants of magnetic lines. Such invariants are constructed on the basis of invariants of classical links, which must satisfy the asymptotic property. We choose the simplest asymptotic invariant $M_3$ of three-component links, which is not expressed in terms of the pairwise linking coefficients of the components. We check the asymptotic property based only on the combinatorial definition of the invariant and do not use the analytic integral. For simple examples, the proven formula is verified by calculation.

math.GT↗

arXiv2024

Invariants of magnetic lines for Yang-Mills solutions

We construct a new Yang-Mills 3D-solution on the space of negative scalar curvarure. We discuss a problem of non-abelian gauge symmetry is broken with the assumption that a scalar curvature of the domain is a negative small parameter. In this case we use the following fact: a geometrical scale related with Vassiliev's discriminant of magnetic lines coincids with a physical Kolmogorov scale. This gives an estimation of $α$-effect by the dispersion of the asymptotic ergodic Hopf invariant in the limite with a negative scalar curvature parameter.

math.AT↗

arXiv2016

Kairvaire Problems in Stable Homotopy Theory

The Kervaire Problem is an unsolved problem in Stable Homotopy Theory. The first interesting example is in dimension $30$. There exists a closed stably-parallelizable manifold $\tilde{M}^{30}$ with Arf-Kervaire invariant 1. It is unknown, if such a manifold exists in dimension $126$? The goal of the paper is to recall the construction of the manifold $M^{30}$ and investigate its properties.

math.AT↗