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Andrey Vasil'ev

Publications and source records attributed to Andrey Vasil'ev.

11 recordsLinked to original sources
arXiv2021

Effects of spatial quantization and Rabi-shifted resonances in single and double excitation of quantum wells and wires induced by few-photon optical field

We develop a fully quantum theoretical approach which describes the dynamics of Frenkel excitons and bi-excitons induced by few photon quantum light in a quantum well or wire (atomic chain) of finite size. The eigenenergies and eigenfunctions of the coupled exciton-photon states in a multiatomic system are found and the role of spatial confinement as well as the energy quantization effects in 1D and 2D cases is analyzed. Due to the spatial quantization, the excitation process is found to consist in the Rabi-like oscillations between the collective symmetric states characterized by discrete energy levels and arising in the picture of the ladder bosonic operators. At the same time, the enhanced excitation of additional states with energy close to the upper polariton branch is revealed. The found new effect is referred to as the formation of Rabi-shifted resonances and is analyzed in details. Such states are shown to influence dramatically on the dynamics of excitation especially in the limit of large times.

quant-ph↗
arXiv2020

A characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers

Recent classification of $\frac{3}{2}$-transitive permutation groups leaves us with three infinite families of groups which are neither $2$-transitive, nor Frobenius, nor one-dimensional affine. The groups of the first two families correspond to special actions of ${\mathrm{PSL}}(2,q)$ and ${\mathrm{PΓL}}(2,q),$ whereas those of the third family are the affine solvable subgroups of ${\mathrm{AGL}}(2,q)$ found by D. Passman in 1967. The association schemes of the groups in each of these families are known to be pseudocyclic. It is proved that apart from three particular cases, each of these exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its $3$-dimensional intersection numbers.

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↗
arXiv2019

Two-closure of supersolvable permutation group in polynomial time

The $2$-closure $\overline{G}$ of a permutation group $G$ on $Ω$ is defined to be the largest permutation group on $Ω$, having the same orbits on $Ω\timesΩ$ as $G$. It is proved that if $G$ is supersolvable, then $\overline{G}$ can be found in polynomial time in $|Ω|$. As a byproduct of our technique, it is shown that the composition factors of $\overline{G}$ are cyclic or alternating of prime degree.

math.GR↗
arXiv2019

The proper definition and Wielandt-Hartley's theorem for submaximal $\mathfrak{X}$-subgroups

A nonempty class $\mathfrak{X}$ of finite groups is called complete if it is closed under taking subgroups, homomorphic images and extensions. We deal with a classical problem of determining $\mathfrak{X}$-maximal subgroups. We consider two definitions of submaximal $\mathfrak{X}$-subgroups suggested by Wielandt and discuss which one better suits our task. We prove that these definitions are not equivalent yet Wielandt-Hartley's theorem holds true for either definition of $\mathfrak{X}$-submaximality. We also give some applications of the strong version of Wielandt-Hartley's theorem.

math.GR↗
arXiv2017

Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog.

math.GR↗
arXiv2016

Cartan coherent configurations

The Cartan scheme $\cal X$ of a finite group $G$ with a $(B,N)$-pair is defined to be the coherent configuration associated with the action of $G$ on the right cosets of the Cartan subgroup $B\cap N$ by the right multiplications. It is proved that if $G$ is a simple group of Lie type, then asymptotically, the coherent configuration $\cal X$ is 2-separable, i.e., the array of 2-dimensional intersection numbers determines $\cal X$ up to isomorphism. It is also proved that in this case, the base number of $\cal X$ equals 2. This enables us to construct a polynomial-time algorithm for recognizing the Cartan schemes when the rank of $G$ and order of the underlying field are sufficiently large. One of the key points in the proof of the main results is a new sufficient condition for an arbitrary homogeneous coherent configuration to be 2-separable.

math.CO↗
arXiv2014

On finite groups isospectral to simple classical groups

The spectrum $ω(G)$ of a finite group $G$ is the set of element orders of $G$. Finite groups $G$ and $H$ are isospectral if their spectra coincide. Suppose that $L$ is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic $p$. It is proved that a finite group $G$ isospectral to $L$ cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from $p$. Together with a series of previous results this implies that every finite group $G$ isospectral to $L$ is `close' to $L$. Namely, if $L$ is a linear or unitary group, then $L\leqslant G\leqslant\operatorname{Aut}(L)$, in particular, there are only finitely many such groups $G$ for given $L$. If $L$ is a symplectic or orthogonal group, then $G$ has a unique nonabelian composition factor $S$ and, for given $L$, there are at most 3 variants for $S$ (including $S\simeq L$).

math.GR↗
arXiv2014

On non-abelian Schur groups

A finite group G is called Schur, if every Schur ring over G is associated in a natural way with a regular subgroup of Sym(G) that is isomorphic to G. We prove that any nonabelian Schur group G is metabelian and the number of distinct prime divisors of the order of G does not exceed 7.

math.CO↗