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Vol 214, No 1 (2023)
Sharp Bernstein-type inequalities for Fourier-Dunkl multipliers
Abstract
A method for the proof of analogues of the classical Bernstein, Riesz and Boas inequalities for differentiation and difference operators defined by means of multipliers in terms of the Fourier-Dunkl transform is developed. This method is based on Civin-type interpolation formulae. Some of the inequalities obtained are sharp in the uniform norm. Bibliography: 42 titles.
Matematicheskii Sbornik. 2023;214(1):3-30
3-30
Jordan property for groups of bimeromorphic automorphisms of compact Kähler threefolds
Abstract
Let $X$ be a nonuniruled compact Kähler space of dimension $3$. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.Bibliography: 29 titles.
Matematicheskii Sbornik. 2023;214(1):31-42
31-42
Structure of the spectrum of a nonselfadjoint Dirac operator
Abstract
For the Dirac operator with two-point boundary conditions and an arbitrary complex-valued $L_2$-integrable potential $V(x)$ the spectral problem is considered. Necessary and sufficient conditions on an entire function to be the characteristic function of such a boundary value problem are obtained. Necessary and sufficient conditions on the spectrum of the above operator are established in the case when the boundary conditions are regular. Bibliography: 16 titles.
Matematicheskii Sbornik. 2023;214(1):43-60
43-60
‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain
Abstract
A formally selfadjoint system of second-order differential equations is considered in a three-dimensional domain on small parts of whose boundary an analogue of Steklov spectral conditions is set, while the Neumann boundary conditions are set on the rest of the boundary. Under certain algebraic and geometric conditions an asymptotic expression for the eigenvalues of this problem is presented and a limiting problem is put together, which produces the leading asymptotic terms and involves systems of integro-differential equations in half-spaces, interconnected by means of certain integral characteristics of vector-valued eigenfunctions. One example of a concrete problem in mathematical physics describes surface waves in several ice holes made in the ice cover of a water basin, and the asymptotic formula for eigenfrequencies shows that the local wave processes interact independently of the distance between the holes. Another series of applied problems relates to elastic fixings of bodies along small pieces of their surfaces. Possible generalizations are discussed; a number of related open questions are stated. Bibliography: 41 titles.
Matematicheskii Sbornik. 2023;214(1):61-112
61-112
On the sharp Baer-Suzuki theorem for the $\pi$-radical of a finite group
Abstract
Let $\pi$ be a proper subset of the set of prime numbers. Denote by $r$ the least prime not contained in $\pi$ and set $m=r$ for $r=2$ and $3$ and $m=r-1$ for $r\ge5$. The conjecture under consideration claims that a conjugacy class $D$ of a finite group $G$ generates a $\pi$-subgroup of $G$ (equivalently, is contained in the $\pi$-radical) if and only if any $m$ elements of $D$ generate a $\pi$-group. It is shown that this conjecture holds if every non-Abelian composition factor of $G$ is isomorphic to a sporadic, an alternating, a linear, or a unitary simple group. Bibliography: 49 titles.
Matematicheskii Sbornik. 2023;214(1):113-154
113-154