Authors:M. B. Karmanova Pages: 549 - 552 Abstract: A new idea of constructing the parallel translation map on a Carnot–Carathéodory space is proposed, the basic properties of this map are described, and the polynomials approximating graph mappings are qualitatively described. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060035 Issue No:Vol. 96, No. 3 (2017)

Authors:E. G. Bakhtigareeva; M. L. Gol’dman Pages: 553 - 557 Abstract: Modular inequalities and inequalities for the norms of Hardy-type operators on the cone Ω of positive functions and on the cone of positive decreasing functions with common weight and common Young function in a weighted Orlicz space are considered. A reduction theorem for the norm of the Hardy operator on the cone Ω is obtained. It is shown that this norm is equivalent to the norm of a modified operator on the cone of all positive functions in the space under consideration. It is proved that the modified operator is a generalized Hardy-type operator. The equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all positive functions in the Orlicz space is shown. A criterion for the validity of such inequalities in general Orlicz spaces is obtained and refined for weighted Lebesgue spaces. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060059 Issue No:Vol. 96, No. 3 (2017)

Authors:F. Götze; A. A. Naumov; A. N. Tikhomirov Pages: 558 - 560 Abstract: The product of m ∈ N independent random square matrices whose elements are independent identically distributed random variables with zero mean and unit variance is considered. It is known that, as the size of the matrices increases to infinity, the empirical spectral measure of the normalized eigenvalues of the product converges with probability 1 to the distribution of the mth power of the random variable uniformly distributed on the unit disk of the complex plane. In particular, in the case of m = 1, the circular law holds. The purpose of this paper is to prove the validity of the local circular law (as well as its generalization to the case of any fixed m) in the case where the distribution of the matrix elements has finite absolute moment of order 4 + δ,δ > 0,. Recent results of Bourgade, Yau, and Yin, of Tao and Vu, and of Nemish are generalized. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060072 Issue No:Vol. 96, No. 3 (2017)

Authors:A. N. Zarubin Pages: 561 - 567 Abstract: Boundary value problems for equations of mixed type with multiple functional lead and lag are studied. General solutions of such equations are constructed. The problems are uniquely solvable. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060096 Issue No:Vol. 96, No. 3 (2017)

Authors:I. M. Petrushko Pages: 568 - 570 Abstract: Properties of the solutions of a parabolic equation in the case of a Keldysh-type degeneracy on the boundary of the domain are investigated. The unique solvability of the first mixed problem for this equation is studied. Necessary and sufficient conditions for the existence of limits of the solution on the lateral surface of a cylindrical domain and on its lower base in L2-type spaces are found. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060084 Issue No:Vol. 96, No. 3 (2017)

Authors:N. N. Kozlov; E. I. Kugushev; T. M. Eneeva Pages: 571 - 573 Abstract: A mathematical analysis was performed, which shows that the maximum length of DNA in which translation is possible for at least one of three reading frames for any DNA strand is 10 nucleotides. In the case of any nonstandard stop codon triplet, this number can only decrease. Moreover, for any DNA consisting of at most 11 nucleotides, two and more genes overlaps in plus and minus strands are possible. The probability that a stop codon triplet chosen at random belongs to the group of stop codon triplets with these properties is less than 0.06. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060102 Issue No:Vol. 96, No. 3 (2017)

Authors:Yu. N. Orlov; V. Zh. Sakbaev; O. G. Smolyanov Pages: 574 - 577 Abstract: Transformations of measures, generalized measures, and functions generated by evolution differential equations on a Hilbert space E are studied. In particular, by using Feynman formulas, a procedure for averaging nonlinear random flows is described and an analogue of the law of large number for such flows is established (see [1, 2]). PubDate: 2017-11-01 DOI: 10.1134/s1064562417060126 Issue No:Vol. 96, No. 3 (2017)

Authors:Sh. Tetunashvili Pages: 578 - 579 Abstract: Series with respect to systems Φ{φn(x)}n=1∞ of measurable and almost everywhere finite functions are discussed. A necessary and sufficient condition for representing any series with respect to a system Φ as a sum of two universal series is formulated. A consequence of the condition is that any series with respect to an arbitrary complete and orthonormal system Φ is a sum of two universal series. PubDate: 2017-11-01 DOI: 10.1134/s106456241706014x Issue No:Vol. 96, No. 3 (2017)

Authors:S. V. Selivanova; V. L. Selivanov Pages: 580 - 582 Abstract: In this paper we find a connection between constructive number fields and computable reals. This connection is applied to prove the computability in the rigorous sense of computable analysis) of solutions of some important systems of partial differential equations, by means of algorithms which are really used in numerical analysis. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060138 Issue No:Vol. 96, No. 3 (2017)

Authors:V. I. Bogachev; S. V. Shaposhnikov Pages: 583 - 586 Abstract: New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060175 Issue No:Vol. 96, No. 3 (2017)

Authors:V. N. Ushakov; G. V. Parshikov Pages: 594 - 597 Abstract: A differential inclusion (d.i.) with a parameter α ∈ ℒ, where ℒ is a compact set in a finite-dimensional Euclidean space is considered. The problem under study is one of guiding a d.i. to a given compact set M in the phase space of the d.i. at a fixed moment of time. Issues concerning the unification of the d.i. with a parameter are discussed. A specific feature of the present unification is that it is performed for a parameterdependent d.i. that cannot be initially structured by a vector function similar to those determining the dynamics of systems in control and conflict control problems. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060205 Issue No:Vol. 96, No. 3 (2017)

Authors:M. E. Zhukovskii; M. G. Sánchez Pages: 598 - 600 Abstract: We consider existential monadic second-order sentences ∃X φ(X) about undirected graphs, where ∃X is a finite sequence of monadic quantifiers and φ(X) ∈ +∞ωω is an infinite first-order formula. We prove that there exists a sentence (in the considered logic) with two monadic variables and two first-order variables such that the probability that it is true on G(n, p) does not converge. Moreover, such an example is also obtained for one monadic variable and three first-order variables. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060242 Issue No:Vol. 96, No. 3 (2017)

Authors:S. N. Tumanov; A. A. Shkalikov Pages: 607 - 611 Abstract: The dynamics of the eigenvalues of a family of Sturm–Liouville operators with complex integrable PT-symmetric potential on a finite interval is studied. In the model case of the complex Airy operator, a criterion for the similarity of operators in the family to self-adjoint and normal operators is stated and the exceptional parameter values corresponding to multiple eigenvalues are analytically calculated. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060230 Issue No:Vol. 96, No. 3 (2017)

Authors:I. A. Finogenko Pages: 612 - 615 Abstract: For nonautonomous functional-differential equations with piecewise continuous right-hand sides and solutions understood in the sense of A.F. Filippov, the method of limit differential inclusions is proposed to study the asymptotic properties of solutions. Properties of the type of the invariance of -limit sets and analogues of LaSalle’s invariance principle are proved by applying invariantly differentiable Lyapunov functionals with derivatives of constant sign. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060217 Issue No:Vol. 96, No. 3 (2017)

Authors:V. V. Vlasov; N. A. Rautian Pages: 620 - 624 Abstract: Functional-differential and integro-differential equations with the principal part being an abstract hyperbolic equation perturbed by terms with unbounded variable operator coefficients multiplying variable delays are studied. Additionally, Volterra integral operators are considered. For the equations under study, the well-posedness of initial value problems in Sobolev spaces of vector functions is proved. In the autonomous case, spectral analysis of the operator functions that are the symbols of the indicated equations is performed. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060291 Issue No:Vol. 96, No. 3 (2017)

Authors:V. V. Kozlov Pages: 625 - 627 Abstract: A linear transformation with an invariant being a nondegenerate quadratic form is symplectic. The geometric properties of such transformations are discussed. A complete set of quadratic invariants which are pairwise in involution is explicitly specified. The structure of the isotropic cone on which all these integrals simultaneously vanish is investigated. Applications of the general results to the problem on the stability of a fixed point of a linear transformation with a quadratic invariant are discussed. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060254 Issue No:Vol. 96, No. 3 (2017)

Authors:V. D. Stepanov; G. E. Shambilova Pages: 631 - 635 Abstract: Necessary and sufficient conditions for the weighted boundedness in Lebesgue spaces of a certain class of bilinear weighted inequalities with integration operators on the cone of nondecreasing functions on a real half-axis are given. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060278 Issue No:Vol. 96, No. 3 (2017)

Authors:O. A. Vasiljeva; E. A. Lapshin; O. V. Rudenko Pages: 646 - 649 Abstract: An equation is obtained that describes the nonlinear diffraction of a focused wave in a half-space x > 0 starting from the wave source, then through the focus region up to the far zone, where the wave becomes spherically divergent. In contrast to the Khokhlov–Zabolotskaya equation (KZ), which contains two spatial variables, the calculation of the field on the beam axis is reduced to a simpler one-dimensional problem. Integral relations that are useful for numerical calculation are indicated. The profiles of a periodic wave harmonic at the input to the medium are constructed. A comparison with the results of a numerical solution of problems based on KZ is made, which revealed a good accuracy of the approximate model. The passage of a wave through the focus region, accompanied by the formation of shock fronts, diffraction phase shifts and asymmetric distortion of regions of different polarity, is traced. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060151 Issue No:Vol. 96, No. 3 (2017)

Authors:V. E. Krivonozhko; A. V. Lychev Pages: 650 - 653 Abstract: Free Disposal Hull (FDH) model was proposed in the end of 20-th century for performance measurement and management of complex units. Production possibility set of FDH model is a nonconvex set. For this reason, nonlinear integer programming methods and enumeration methods have been used for computations of various characteristics of units’ performance. However, as Cesaroni, Kerstens and Van de Woestyne noted, there are no methods in the world scientific literature for frontier visualization in nonconvex FDH models. In this paper, an approach is proposed for frontier visualization with the use of methods of purposeful enumeration. Computational experiments, conducted with the use of real-life data sets, confirm reliability and effectiveness of proposed algorithms. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060047 Issue No:Vol. 96, No. 3 (2017)

Authors:N. Yu. Lukoyanov; M. I. Gomoyunov; A. R. Plaksin Pages: 654 - 657 Abstract: The relation between a differential game for neutral-type systems and a Hamilton–Jacobi functional equation with coinvariant derivatives is established. The value functional of the game is proved to coincide with the minimax solution of this equation. Optimal strategies for the players are described. PubDate: 2017-11-01 DOI: 10.1134/s1064562417060114 Issue No:Vol. 96, No. 3 (2017)