Vol 59, No 1 (2018)
- Year: 2018
- Articles: 19
- URL: https://journals.rcsi.science/0037-4466/issue/view/10455
Article
Recovering Linear Operators and Lagrange Function Minimality Condition
Abstract
This article concerns the recovery of the operators by noisy information in the case that their norms are defined by integrals over infinite intervals. We study the conditions under which the dual extremal problem (often nonconvex) can be solved using the Lagrange function minimality condition.
On Dark Computably Enumerable Equivalence Relations
Abstract
We study computably enumerable (c.e.) relations on the set of naturals. A binary relation R on ω is computably reducible to a relation S (which is denoted by R ≤cS) if there exists a computable function f(x) such that the conditions (xRy) and (f(x)Sf(y)) are equivalent for all x and y. An equivalence relation E is called dark if it is incomparable with respect to ≤c with the identity equivalence relation. We prove that, for every dark c.e. equivalence relation E there exists a weakly precomplete dark c.e. relation F such that E ≤cF. As a consequence of this result, we construct an infinite increasing ≤c-chain of weakly precomplete dark c.e. equivalence relations. We also show the existence of a universal c.e. linear order with respect to ≤c.
Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups
Abstract
Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S3 = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S3 of all geodesics on (G, d) with respect to p.
Describing Neighborhoods of 5-Vertices in a Class of 3-Polytopes with Minimum Degree 5
Abstract
Lebesgue proved in 1940 that each 3-polytope with minimum degree 5 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences
(6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11)
(5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17)
(5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6,∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11)
(5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13).
We prove that each 3-polytope with minimum degree 5 without vertices of degree from 7 to 10 contains a 5-vertex whose set of degrees of its neighbors is majorized by one of the following sequences: (5, 6, 6, 5, ∞), (5, 6, 6, 6, 15), and (6, 6, 6, 6, 6), where all parameters are tight.
Finite Groups with Three Given Subgroups
Abstract
Given a hereditary saturated formation F of soluble groups, we study finite groups with three F-subgroups of coprime indices. We obtain the new criteria for these groups to lie in the Shemetkov formations, the formations of all supersoluble groups, the formations of all groups with nilpotent commutator subgroup, and other formations.
Interpolation Problems for Entire Functions Induced by Regular Hexagons
Abstract
We consider linear equations for analytic functions in the plane with cuts along a “half” of the boundary of a hexagon. We propose a regularization method, reducing them to an equation with difference kernel. Applications are given to the moment problem for entire functions of exponential type.
Unconditional Convergence of Fourier Series for Functions of Bounded Variation
Abstract
This article concerns the unconditional convergence a.e. of Fourier series with respect to general orthonormal systems. We find certain conditions to be satisfied by the functions in the orthonormal system so that the Fourier series of each function of finite variation unconditionally converge a.e. The results are best possible.
Limit Automorphisms of the C*-Algebras Generated by Isometric Representations for Semigroups of Rationals
Abstract
We consider inductive sequences of Toeplitz algebras whose connecting homomorphisms are defined by collections of primes. The inductive limits of these sequences are C*-algebras generated by representations for semigroups of rationals. We study the limit endomorphisms of these C*-algebras induced by morphisms between copies of the same inductive sequences of Toeplitz algebras. We establish necessary and sufficient conditions for these endomorphisms to be automorphisms of the algebras.
Transmission of Waves Through a Small Aperture in the Cross-Wall in an Acoustic Waveguide
Abstract
We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter ε > 0. We describe the effects of almost complete reflection or transmission of waves related to the classical Vainstein anomaly and the presence of almost standing waves for the threshold value Λk of the spectral parameter λ in continuous spectrum. The greatest attention is paid to analyzing the range λε = Λk + ε2μ2 of the spectral parameter with μ ≤ μ0, which generates scattering coefficients depending on μ > 0 and presents the greatest difficulties in constructing and justifying the asymptotics. Almost complete reflection and transmission correspond to the cases of going away from the threshold (as μ → +∞) and approaching it (as μ → +0) characterized by simpler asymptotics.
Alternative Proof of Mironov’s Results on Commuting Self-Adjoint Operators of Rank 2
Abstract
We give an alternative proof of Mironov’s results on commuting self-adjoint operators of rank 2. Mironov’s proof is based on Krichever’s complicated theory of the existence of a high-rank Baker–Akhiezer function. In contrast to Mironov’s proof, our proof is simpler but the results are slightly weaker. Note that the method of this article can be extended to matrix operators. Using the method, we can construct the first explicit examples of matrix commuting differential operators of rank 2 and arbitrary genus.
On the Number of Vedernikov–Ein Irreducible Components of the Moduli Space of Stable Rank 2 Bundles on the Projective Space
Abstract
We propose a method for finding the exact number of Vedernikov–Ein irreducible components of the first and second types in the moduli space M(0, n) of stable rank 2 bundles on the projective space P3 with Chern classes c1 = 0 and c2 = n ≥ 1. We give formulas for the number of Vedernikov–Ein components and find a criterion for their existence for arbitrary n ≥ 1.
Global Solvability and Estimates of Solutions to the Cauchy Problem for the Retarded Functional Differential Equations That Are Used to Model Living Systems
Abstract
We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.
Sobolev Embedding Theorems and Generalizations for Functions on a Metric Measure Space
Abstract
Considering the metric case, we define an analog of the Sobolev space of functions with generalized derivatives of order greater than 1. The space of functions with fractional generalized derivatives is also treated. We prove generalizations of the Sobolev embedding theorems and Gagliardo–Nirenberg interpolation inequalities to the metric case.
Degenerate Linear Evolution Equations with the Riemann–Liouville Fractional Derivative
Abstract
We study the unique solvability of the Cauchy and Schowalter–Sidorov type problems in a Banach space for an evolution equation with a degenerate operator at the fractional derivative under the assumption that the operator acting on the unknown function in the equation is p-bounded with respect to the operator at the fractional derivative. The conditions are found ensuring existence of a unique solution representable by means of the Mittag-Leffler type functions. Some abstract results are illustrated by an example of a finite-dimensional degenerate system of equations of a fractional order and employed in the study of unique solvability of an initial-boundary value problem for the linearized Scott-Blair system of dynamics of a medium.
Weakly Periodic Gibbs Measures for HC-Models on Cayley Trees
Abstract
We study hard-core (HC) models on Cayley trees. Given a 2-state HC-model, we prove that exactly two weakly periodic (aperiodic) Gibbs measures exist under certain conditions on the parameters. Moreover, we consider fertile 4-state HC-models with the activity parameter λ > 0. The three types of these models are known to exist. For one of the models we show that the translationinvariant Gibbs measure is not unique.
Finite Groups with Given Weakly σ-Permutable Subgroups
Abstract
Let G be a finite group and let σ = {σi | i ∈ I} be a partition of the set of all primes P. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σi-subgroup of G and ℋ has exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HAx = AxH for all A ∈ ℋ and all x ∈ G. A subgroup H of G is said to be weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ HσG, where HσG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G. We study the structure of G under the condition that some given subgroups of G are weakly σ-permutable in G. In particular, we give the conditions under which a normal subgroup of G is hypercyclically embedded. Some available results are generalized.
Controllability of Differential-Algebraic Equations in the Class of Impulse Effects
Abstract
Considering a control linear system of differential-algebraic equations with infinitely differentiable coefficients we establish the existence of solutions in the class of Sobolev–Schwartz distributions. The solution is expressed as the sum of a regular generalized function and a singular generalized function. We study controllability with a jump of a regular component and a singular component of the solution.
Recognizability of All WIP-Minimal Logics
Abstract
We consider extensions of Johansson’s minimal logic J. It was proved in [1] that the weak interpolation property (WIP) is decidable over the minimal logic. Moreover, all logics with WIP are divided into eight pairwise disjoint intervals. The notion of recognizable logic was introduced in [2]. The recognizability over J of five of the eight WIP-minimal logics, i.e. of the lower ends of intervals with WIP, was proved earlier in [2, 3]. We prove the recognizability over J of the remaining three WIP-minimal logics.