arXiv subjects

M. V. Volkov

Publications and source records attributed to M. V. Volkov.

10 recordsLinked to original sources

arXiv2026

The lattice of varieties of semigroups with completely regular square. II

We solve (modulo groups) the word problem for free semigroup satisfying $xy = (xy)^n$. This enables us to show that the variety ${\cal {SCR}}_n$ given by this identity is not equal to the join of the varieties ${\cal{CR}}_n$ and ${\cal{SI}}$ defined by the identities $x = x^n$ and $xy = (xy)^2$ respectively. Therefore the result of (M.V.Volkov, T.A.Ershova, The lattice of varieties of semigroup with completely regular square, Monash Conference on Semigroup Theory in Honour of G. B. Preston, World Scientific, Singapore, 1991, 306-322) that the lattice $L({\cal {SCR}}_n)$ is modular does not follow from the earlier results concerning $L({\cal{CR}}_n)$ and $L({\cal{SI}})$. However the variety ${\cal{SO}}_n$ consisting of all semigroups from ${\cal {SCR}}_n$ whose idempotents form a subsemigroup turns out to be equal to the join of the varieties ${\cal O}_n$ of all orthogroups and ${\cal{SI}}$. We deduce from this a description of the lattice $L({\cal{SO}}_n)$ modulo groups.

math.GR↗

arXiv2023

Identities in twisted Brauer monoids

We show that it is co-NP-hard to check whether a given semigroup identity holds in the twisted Brauer monoid $\mathcal{B}^τ_n$ with $n\ge5$.

math.GR↗

arXiv2021

Cross-connection structure of locally inverse semigroups

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterise the categories that correspond to locally inverse semigroups in the realm of Nambooripad's cross-connection theory. Further, we specialise our cross-connection description of locally inverse semigroups to inverse semigroups and completely 0-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann-Schein-Nambooripad Theorem; for completely 0-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.

math.GR↗

arXiv2020

Identities of the Kauffman Monoid $\mathcal{K}_3$

We give a transparent combinatorial characterization of the identities satisfied by the Kauffman monoid $\mathcal{K}_3$. Our characterization leads to a polynomial time algorithm to check whether a given identity holds in $\mathcal{K}_3$.

math.GR↗

arXiv2019

Identities of the Kauffman Monoid $\mathcal{K}_4$ and of the Jones monoid $\mathcal{J}_4$

Kauffman monoids $\mathcal{K}_n$ and Jones monoids $\mathcal{J}_n$, $n=2,3,\dots$, are two families of monoids relevant in knot theory. We prove a somewhat counterintuitive result that the Kauffman monoids $\mathcal{K}_3$ and $\mathcal{K}_4$ satisfy exactly the same identities. This leads to a polynomial time algorithm to check whether a given identity holds in $\mathcal{K}_4$. As a byproduct, we also find a polynomial time algorithm for checking identities in the Jones monoid $\mathcal{J}_4$.

math.GR↗

arXiv2018

Inductive groupoids and cross-connections of regular semigroups

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann--Schein--Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

math.GR↗

arXiv2015

The Finite Basis Problem for Kiselman Monoids

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke--Kiselman monoids including the Kiselman monoids $\mathcal{K}_n$. As a consequence, we conclude that the identities of $\mathcal{K}_n$ are nonfinitely based for every $n\ge 4$ and exhibit a finite identity basis for the identities of each of the monoids $\mathcal{K}_2$ and $\mathcal{K}_3$. In the third version a question left open in the initial submission has beed answered.

math.GR↗

arXiv2011

Potential splitting approach to multichannel Coulomb scattering: the driven Schrödinger equation formulation

In this paper we suggest a new approach for the multichannel Coulomb scattering problem. The Schrödinger equation for the problem is reformulated in the form of a set of inhomogeneous equations with a finite-range driving term. The boundary conditions at infinity for this set of equations have been proven to be purely outgoing waves. The formulation {presented here} is based on splitting the interaction potential into a finite range core part and a long range tail part. The conventional matching procedure coupled with the integral Lippmann-Schwinger equations technique are used in the formal theoretical basis of this approach. The reformulated scattering problem is suitable for application in the exterior complex scaling technique: the practical advantage is that after the complex scaling the problem is reduced to a boundary problem with zero boundary conditions. The Coulomb wave functions are used only at a single point: if this point is chosen to be at a sufficiently large distance, on using the asymptotic expansion of Coulomb functions, one may completely avoid the Coulomb functions in the calculations. The theoretical results are illustrated with numerical calculations for two models.

physics.atom-ph↗

arXiv2011

Solving the Coulomb scattering problem using the complex scaling method

Based on the work of Nuttall and Cohen [Phys. Rev. {\bf 188} (1969) 1542] and Resigno et al{} [Phys. Rev. A {\bf 55} (1997) 4253] we present a rigorous formalism for solving the scattering problem for long-range interactions without using exact asymptotic boundary conditions. The long-range interaction may contain both Coulomb and short-range potentials. The exterior complex scaling method, applied to a specially constructed inhomogeneous Schrödinger equation, transforms the scattering problem into a boundary problem with zero boundary conditions. The local and integral representations for the scattering amplitudes have been derived. The formalism is illustrated with numerical examples.

physics.atom-ph↗

arXiv2006

Exact results for state-to-state transition probabilities in the multistate Landau-Zener model by non-stationary perturbation theory

Multistate generalizations of Landau-Zener model are studied by summing entire series of perturbation theory. A new technique for analysis of the series is developed. Analytical expressions for probabilities of survival at the diabatic potential curves with extreme slope are proved. Degenerate situations are considered when there are several potential curves with extreme slope. New expressions for some state-to-state transition probabilities are derived in degenerate cases.

cond-mat.other↗