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Abstract: Abstract We construct some class of selfadjoint operators in the Krein spaces consisting of functions on the straight line \( \{\operatorname{Re}s=\frac{1}{2}\} \) . Each of these operators is a rank-one perturbation of a selfadjoint operator in the corresponding Hilbert space and has eigenvalues complex numbers of the form \( \frac{1}{s(1-s)} \) , where \( s \) ranges over the set of nontrivial zeros of the Riemann zeta-function. PubDate: 2024-01-01

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Abstract: Abstract We implement the Boolean valued analysis of Banach spaces. The realizations of Banach spaces in a Boolean valued universe are lattice normed spaces. We present the basic techniques of studying these objects as well as the Boolean valued approach to injective Banach lattices. PubDate: 2024-01-01

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Abstract: Abstract We show that the Levi class of the quasivariety of right-orderable groups strictly includes this quasivariety. PubDate: 2024-01-01

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Abstract: Abstract We obtain two-sided estimates for Alexandrov’s \( n \) -width of the compact set of infinitely smooth functions boundedly embedded into the space of continuous functions on a finite segment. PubDate: 2024-01-01

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Abstract: Abstract We construct an additive basis for a relatively free alternative algebra of Lie-nilpotent degree 5, describe the associative center and core of this algebra, and find the T-generators of the full center. Also, we give some asymptotic estimate for the codimension of the T-ideal generated by a commutator of degree 5 in a free alternative algebra, and find a finite-dimensional superalgebra that generates the variety of alternative algebras with the Lie-nilpotency of the selfadjoint operator of degree 5. PubDate: 2024-01-01

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Abstract: Abstract We prove that every mapping with finite distortion on a Carnot group is open and discrete provided that it is quasilight and the distortion coefficient is integrable. Also, we estimate the Hausdorff dimension of the preimages of points for mappings on a Carnot group with a bounded multiplicity function and summable distortion coefficient. Furthermore, we give some example showing that the obtained estimates cannot be improved. PubDate: 2024-01-01

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Abstract: Abstract We found the geodesics, shortest arcs, cut loci, and injectivity radius of any oblate ellipsoid of revolution in three-dimensional Euclidean space. PubDate: 2024-01-01

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Abstract: Abstract Considering the approximation of a function \( f \) from a Sobolev space by the partial sums of Fourier series in a system of Sobolev orthogonal polynomials generated by classical Laguerre polynomials, we obtain an estimate for the convergence rate of the partial sums to \( f \) . PubDate: 2024-01-01

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Abstract: Abstract We study admissible rules for the extensions of the modal logics S4 and GL with the weak co-covering property and describe some explicit independent basis for the admissible rules of these logics. The resulting basis consists of an infinite sequence of rules in compact and simple form. PubDate: 2024-01-01

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Abstract: Abstract We construct two new infinite series of irreducible components of the moduli space of semistable nonlocally free reflexive rank 2 sheaves on the three-dimensional complex projective space. In the first series the sheaves have an even first Chern class, and in the second series they have an odd one, while the second and third Chern classes can be expressed as polynomials of a special form in three integer variables. We prove the uniqueness of components in these series for the Chern classes given by those polynomials. PubDate: 2024-01-01

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Abstract: Abstract We consider Kolmogorov operators with constant diffusion matrices and linear drifts, i.e., Ornstein–Uhlenbeck operators, and show that all solutions to the corresponding stationary Fokker–Planck–Kolmogorov equations (including signed solutions) are invariant measures for the generated semigroups. This also gives a relatively explicit description of all solutions. PubDate: 2024-01-01

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Abstract: Abstract We study unification and admissibility for an infinite class of modal logics. Conditions superimposed to these logics are to be decidable, Kripke complete, and generated by the classes of rooted frames possessing the greatest clusters of states (in particular, these logics extend modal logic S4.2). Given such logic \( L \) and each formula \( \alpha \) unifiable in \( L \) , we construct a unifier \( \sigma \) for \( \alpha \) in \( L \) , where \( \sigma \) verifies admissibility in \( L \) of arbitrary inference rules \( \alpha/\beta \) with a switched-modality conclusions \( \beta \) (i.e., \( \sigma \) solves the admissibility problem for such rules). PubDate: 2024-01-01

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Abstract: Abstract A topological fibered space is a Birman–Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We present a series of sufficient conditions for a fiber bundle over the circle to be a Birman–Hilden space. PubDate: 2024-01-01

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Abstract: Abstract Consider the fundamental group \( {\mathfrak{G}} \) of an arbitrary graph of groups and some root class \( {\mathcal{C}} \) of groups, i.e., a class containing a nontrivial group and closed under subgroups, extensions, and unrestricted direct products of the form \( \prod_{y\in Y}X_{y} \) , where \( X,Y\in{\mathcal{C}} \) and \( X_{y} \) is an isomorphic copy of \( X \) for each \( y\in Y \) . We provide some criterion for the separability by \( {\mathcal{C}} \) of a finitely generated abelian subgroup of \( {\mathfrak{G}} \) valid when the group satisfies an analog of the Baumslag filtration condition. This enables us to describe the \( {\mathcal{C}} \) -separable finitely generated abelian subgroups for the fundamental groups of some graphs of groups with central edge subgroups. PubDate: 2024-01-01

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Abstract: Abstract Studying the approximation properties of a certain Riesz–Zygmund sum of Fourier–Chebyshev rational integral operators with constraints on the number of geometrically distinct poles, we obtain an integral expression of the operators. We find upper bounds for pointwise and uniform approximations to the function \( x ^{s} \) with \( s\in(0,2) \) on the segment \( [-1,1] \) , an asymptotic expression for the majorant of uniform approximations, and the optimal values of the parameter of the approximant providing the greatest decrease rate of the majorant. We separately study the approximation properties of the Riesz–Zygmund sums for Fourier–Chebyshev polynomial series, establish an asymptotic expression for the Lebesgue constants, and estimate approximations to \( f\in H^{(\gamma)}[-1,1] \) and \( \gamma\in(0,1] \) as well as pointwise and uniform approximations to the function \( x ^{s} \) with \( s\in(0,2) \) . PubDate: 2024-01-01

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Abstract: Abstract We fully study the oriented rotatability exponents of solutions to homogeneous autonomous linear differential systems and establish that the strong and weak oriented rotatability exponents coincide for each solution to an autonomous system of differential equations. We also show that the spectrum of this exponent (i.e., the set of values of nonzero solutions) is naturally determined by the number-theoretic properties of the set of imaginary parts of the eigenvalues of the matrix of a system. This set (in contrast to the oscillation and wandering exponents) can contain other than zero values and the imaginary parts of the eigenvalues of the system matrix; moreover, the power of this spectrum can be exponentially large in comparison with the dimension of the space. In demonstration we use the basics of ergodic theory, in particular, Weyl’s Theorem. We prove that the spectra of all oriented rotatability exponents of autonomous systems with a symmetrical matrix consist of a single zero value. We also establish relationships between the main values of the exponents on a set of autonomous systems. The obtained results allow us to conclude that the exponents of oriented rotatability, despite their simple and natural definitions, are not analogs of the Lyapunov exponent in oscillation theory. PubDate: 2024-01-01

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Abstract: Abstract We prove the equivalence of the power-law convergence rate in the \( L_{2} \) -norm of ergodic averages for \( {ð•‘}^{d} \) and \( {ð•‰}^{d} \) actions and the same power-law estimate for the spectral measure of symmetric \( d \) -dimensional parallelepipeds: for the degrees that are roots of some special symmetric polynomial in \( d \) variables. Particularly, all possible range of power-law rates is covered for \( d=1 \) . PubDate: 2024-01-01

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Abstract: Abstract We present the inverse problem of successive determination of the two unknowns that are a coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of an integral operator describing the memory of the medium. The direct initial-boundary value problem involves the zero data and the Neumann boundary condition. The trace of the Fourier image of a solution to the direct problem on the boundary of the medium serves as additional information. Studying inverse problems, we assume that the unknown coefficient is expanded into an asymptotic series in powers of a small parameter. Also, we construct some method for finding the coefficient that accounts for the memory of the environment to within an error of order \( O(\varepsilon^{2}) \) . At the first stage, we determine a solution to the direct problem in the zero approximation and the kernel of the integral operator, while the inverse problem reduces to an equivalent problem of solving the system of nonlinear Volterra integral equations of the second kind. At the second stage, we consider the kernel given and recover a solution to the direct problem in the first approximation and the unknown coefficient. In this case, the solution to the equivalent inverse problem agrees with a solution to the linear system of Volterra integral equations of the second kind. We prove some theorems on the unique local solvability of the inverse problems and present the results of numerical calculations of the kernel and the coefficient. PubDate: 2023-11-01

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Abstract: Abstract We obtain some sufficient conditions for potency and virtual potency for automorphism groups and the split extensions of some groups. In particular, considering a finitely generated group \( G \) residually \( p \) -finite for every prime \( p \) , we prove that each split extension of \( G \) by a torsion-free potent group is a potent group, and if the abelianization rank of \( G \) is at most 2 then the automorphism group of \( G \) is virtually potent. As a corollary, we derive the necessary and sufficient conditions of virtual potency for certain generalized free products and HNN-extensions. PubDate: 2023-11-01

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Abstract: Abstract Under study are the homeomorphisms that induce the bounded composition operators of \( BV \) -functions on Carnot groups. We characterize continuous \( BV_{\operatorname{loc}} \) -mappings on Carnot groups in terms of the variation on integral lines and estimate the variation of the \( BV \) -derivative of the composition of a \( C^{1} \) -function and a continuous \( BV_{\operatorname{loc}} \) -mapping. PubDate: 2023-11-01