A metric ρ_I that defines a metrization of the functor I of idempotent measures is introduced on the space I(X) of idempotent measures (Maslov measures) given on a compact metric space X . For each measure μ∈ I(X) , the upper and lower quantization dimensions of this measure can be defined with respect to the metric ρ_I ; they do not exceed the corresponding box dimensions of the support of the measure μ . The following theorem on intermediate values of quantization dimensions of idempotent measures is proved: on every compact metric space of box dimension a<∞ , for all two reals b and c related by the inequalities 0≤ b≤ c≤ a , there exists an idempotent measure whose lower and upper quantization dimensions are equal to b and c , respectively. An analogous theorem on intermediate values was previously proved by the author for quantization dimensions of probability measures, whose idempotent analogs are Maslov measures.
We describe derivations of finite-dimensional simple Novikov algebras and pre-Lie Witt doubles over an algebraically closed field of positive characteristic in terms of special derivations of the underlying associative commutative differentially simple algebras.
We prove that, under certain circumstances, a quasivariety contains continuum many Q -universal subquasivarieties for which the finite membership problem and the quasiequational theory are undecidable.
We consider a class of systems of nonlinear difference equations with variable delay and periodic coefficients in the linear terms. We study the asymptotic stability of the zero solution, obtain an estimate for the domain of attraction of the zero solution, and establish estimates characterizing the stabilization rates of solutions to the systems at infinity. To obtain the results, we use a Lyapunov–Krasovskii functional of a special form.
Variational inequalities with a constraint on the solution for abstract hyperbolic equations are considered. An existence theorem for a solution is proved by the penalty method. Sufficient geometric conditions on the constraint set are given that guarantee the convergence of penalized solutions to a solution to the problem. Under some additional conditions, the strong convergence of approximate solutions is proved. As a result, a solution is obtained for which the law of conservation of energy holds, which corresponds to a perfectly elastic impact.
We study the Laplace transform of generalized functions whose supports are contained in a half-space, its properties, an inversion theorem, and its application to the construction of weak generalized solutions to the Cauchy problem for linear Sobolev-type equations with constant coefficients in a class of functions that grow at infinity.
In this paper, we focus on a topological aspect, namely, the genera of nonnilpotent graphs associated with nonweakly nilpotent groups. We determine the genera of the nonnilpotent graphs of some classes of finite nonnilpotent groups. We classify all nonweakly nilpotent groups whose nonnilpotent graphs are planar, toroidal, double-toroidal, or triple-toroidal. We also show that A_4 is the only nonweakly nilpotent group whose nonnilpotent graph is either quadruple-toroidal or pentuple-toroidal.
In the low-frequency range of the spectrum of an isotropic waveguide in the shape of a thick layer of thin-walled honeycombs with completely clamped surface, we find a family of open wide gaps between narrow spectral segments (the wave-stopping and wave-passing zones, respectively). Elastic waves concentrate near and oscillate along the edges of the honeycomb cells. The results are obtained by constructing asymptotics for eigenpairs of the model problem on the periodicity cell depending on the Floquet parameter. The main role is played by the boundary layer phenomenon, described by solutions to two elasticity problems—the plane vector problem and the out-of-plane scalar problem—in a symmetric two-dimensional tripod composed of unit half-strips. The decisive observation is as follows: the unique eigenvalue in the discrete spectrum of the scalar problem lies strictly below the spectrum of the vector problem. The asymptotics is justified by means of the classical lemma on “almost eigenvalues” and “almost eigenvectors,” as well as by checking the convergence of the attributes of vector-valued eigenfunctions.
We consider a problem closely related to a mathematical model for temperature control. It is based on a one-dimensional non-self-adjoint parabolic equation with variable coefficients. We study qualitative properties of an optimal control defined as a minimizer of a general functional. Controllability is proved for a certain class of control functions.
We consider the Vlasov–Poisson system with an external homogeneous magnetic field describing the kinetics of a two-component rarefied plasma in the three-dimensional case. For all initial density distribution functions with compact supports, we obtain sufficient conditions on the external magnetic field that provide the global existence of density distribution functions whose supports have arbitrarily small growth with respect to the space variables.
Let (𝒦_𝒮,⟨·,·⟩) denote a reproducing kernel Hilbert space defined over a nonempty set 𝒮 . For a given positive bounded linear operator A on 𝒦_𝒮 , the A -Berezin number of an A -bounded linear operator T acting on 𝒦_𝒮 is defined by 𝐛𝐞𝐫_A(T)=sup_μ∈𝒮|⟨ T v_μ,v_μ⟩_A| , while the A -Berezin norm of T is defined by T_𝐛𝐞𝐫_A=sup_μ,ν∈𝒮|⟨ Tv_μ,v_ν⟩_A| . Here, v_μ and v_ν are normalized reproducing kernels in 𝒦_𝒮 , and ⟨ξ_1,ξ_2⟩_A=⟨ Aξ_1,ξ_2⟩ for all ξ_1,ξ_2∈𝒦_𝒮 . The main objective of this paper is to establish various inequalities involving the A -Berezin number 𝐛𝐞𝐫_A(·) and the A -Berezin norm ·_𝐛𝐞𝐫_A . In addition, we revisit and correct certain results recently published by Huban in Turkish J. Math., vol. 46, no. 1, 189–206 (2022). These corrections contribute to a more complete understanding of weighted reproducing kernel Hilbert spaces and highlight the symmetry inherent in the behavior of A -bounded linear operators.
We prove that every automorphism of a Chevalley group of type 𝐆_2 over a commutative local ring without 1/3 is standard, i.e., is a composition of a ring automorphism, a graph automorphism, and an inner automorphism.
A deterministic longest-prefix rewriting system is a string-rewriting system such that there are no rewriting rules X→ Y , X→ Z with Y ≠ Z , and only the longest prefixes of words are subject to rewriting. For such a system, analogs of some concepts related to object-oriented data systems are defined and studied: inheritance of classes and objects, instances of classes, class and instance attributes, conceptual dependence and consistency, conceptual scheme, types and subtypes, etc. Special attention is paid to the effective verification of various properties of the rewriting systems under consideration.
For a class of models 𝒦 , we study the Weihrauch complexity of the following model-theoretic problem: finding an elementary embedding of an arbitrary model ℳ∈𝒦 into a countable saturated model 𝒩 such that 𝒩∈𝒦 and 𝒩 is elementarily equivalent to ℳ . We prove that for the classes of linear orders, Boolean algebras, and abelian groups, this problem is strongly Weihrauch equivalent to the problem lim (that is, the problem of computing the limit in Baire space). We isolate some natural classes 𝒦 consisting of equivalence structures such that the corresponding problem for 𝒦 has Weihrauch degree strictly less than the degree of lim .
We study well-posedness issues for inverse problems of recovering a heat transfer coefficient using a set of solution values at fixed points on the boundary of the domain. Diffraction-type conditions are used on the interface between the media. The boundary conditions are nonlinear, and the heat transfer coefficient is represented as a finite segment of a series with unknown coefficients depending on time. Under certain conditions on the data, we prove that there exists a unique solution to the problem locally in time and this solution depends continuously on the data of the problem. The proof relies on a priori estimates and the contraction mapping principle.
Finite groups are isospectral if they have the same sets of element orders. In this paper, we complete the description of finite groups isospectral to the simple groups PSL_n(q) or PSU_n(q) , where n≥ 11 . We also obtain a substantial restriction on the structure of finite groups isospectral to orthogonal and symplectic groups.
We consider three-dimensional hyperbolic polyhedra of finite volume with finitely many vertices. The normalized volume of a polyhedron is the ratio of its volume to the number of vertices. Given some set of hyperbolic polyhedra, we can associate with it the set of normalized volumes of the polyhedra belonging to it. We call this set the spectrum of normalized volumes of the set under consideration. We focus on right-angled hyperbolic polyhedra. For the subset of ideal polyhedra, we find bounds for the spectrum of normalized volumes and prove that they are sharp. Moreover, we show that the spectrum splits into discrete and dense parts. For the subset of compact polyhedra, we obtain estimates for the spectrum of normalized volumes, prove that the upper bound is sharp, and also establish numerical intervals on which the spectrum is discrete and dense. The endpoints of the indicated numerical intervals are expressed in terms of the volume of the regular ideal hyperbolic tetrahedron and the volume of the regular ideal hyperbolic octahedron.
