Authors:A. N. Agadzhanov Pages: 109 - 112 Abstract: Properties of fractal functions which are not differentiable in the classical sense but have continuous Weil-type derivatives of variable order at each point are studied. It is shown that the Weierstrass, Takagi, and Besicovitch classical fractal functions have such derivatives. An example of an oscillatory system controlling which requires constructing a fractal control function having a Weil-type derivative of variable order at each point is considered. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020016 Issue No:Vol. 95, No. 2 (2017)

Authors:V. I. Bogachev; A. N. Doledenok; S. V. Shaposhnikov Pages: 113 - 117 Abstract: In this paper we study properties of weighted Zolotarev metrics and compare them with the Kantorovich metric. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020028 Issue No:Vol. 95, No. 2 (2017)

Authors:M. B. Karmanova Pages: 118 - 121 Abstract: Area formulas for classes of Hölder mappings of Carnot groups and the corresponding graph mappings are obtained. The calculation of a nonintrinsic measure is exemplified. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020041 Issue No:Vol. 95, No. 2 (2017)

Authors:A. G. Chechkin; A. S. Shamaev Pages: 122 - 124 Abstract: A second-order Schrödinger differential operator of parabolic type is considered, for which an explicit form of a fundamental solution is derived. Such operators arise in many problems of physics, and the fundamental solution plays the role of the Feynman propagation function. PubDate: 2017-03-01 DOI: 10.1134/s106456241702003x Issue No:Vol. 95, No. 2 (2017)

Authors:A. E. Dyusembaev Pages: 125 - 128 Abstract: Conditions are determined under which, for pattern recognition problems with standard information (Ω-regular problems), a correct algorithm and a six-level spatial neural network reproducing the calculations performed by the correct algorithm can be constructed. The proposed approach to constructing the neural network is not related to the traditional approach based on minimizing a functional. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020053 Issue No:Vol. 95, No. 2 (2017)

Authors:A. A. Zlotnik; I. A. Zlotnik Pages: 129 - 135 Abstract: Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020089 Issue No:Vol. 95, No. 2 (2017)

Authors:M. D. Bragin; B. V. Rogov Pages: 140 - 143 Abstract: An iterative method for solving equations of multidimensional bicompact schemes based on an approximate factorization of their difference operators is proposed for the first time. Its algorithm is described as applied to a system of two-dimensional nonhomogeneous quasilinear hyperbolic equations. The convergence of the iterative method is proved in the case of the two-dimensional homogeneous linear advection equation. The performance of the method is demonstrated on two numerical examples. It is shown that the method preserves a high (greater than the second) order of accuracy in time and performs 3–4 times faster than Newton’s method. Moreover, the method can be efficiently parallelized. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020107 Issue No:Vol. 95, No. 2 (2017)

Authors:N. N. Kozlov; T. M. Eneev Pages: 144 - 146 Abstract: This paper continues the authors’ long-time studies concerning biomathematics (see [1]). One direction of research was related to the development of a new area of modern biomathematics, namely, a mathematical theory of genetic codes. The foundations of this theory are described below. To the author’s knowledge, the results presented have not been found in other researchers’ publications. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020120 Issue No:Vol. 95, No. 2 (2017)

Authors:G. M. Feldman Pages: 147 - 150 Abstract: According to the well-known Heyde theorem Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in n independent random variables given another. For n = 2 we prove analogs of this theorem in the case when random variables take values in a locally compact Abelian group X, and coefficients of the linear forms are topological automorphisms of the group X. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020119 Issue No:Vol. 95, No. 2 (2017)

Authors:J. I. Díaz; D. Gómez-Castro; A. V. Podolskiy; T. A. Shaposhnikova Pages: 151 - 156 Abstract: The asymptotic behavior, as ε → 0, of the solution uε to a variational inequality with nonlinear constraints for the p-Laplacian in an ε-periodically perforated domain when p ∈ (1, 2) is studied. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020132 Issue No:Vol. 95, No. 2 (2017)

Authors:N. N. Kalitkin; S. A. Kolganov Pages: 157 - 160 Abstract: A class of functions for which the trapezoidal rule has superpower convergence is described: these are infinitely differentiable functions all of whose odd derivatives take equal values at the left and right endpoints of the integration interval. An heuristic law is revealed; namely, the convergence exponentially depends on the number of nodes, and the exponent equals the ratio of the length of integration interval to the distance from this interval to the nearest pole of the integrand. On the basis of the obtained formulas, a method for calculating the Fermi–Dirac integrals of half-integer order is proposed, which is substantially more economical than all known computational methods. As a byproduct, an asymptotics of the Bernoulli numbers is found. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020156 Issue No:Vol. 95, No. 2 (2017)

Authors:N. N. Kozlov; E. I. Kugushev; T. M. Eneev Pages: 161 - 163 Abstract: Mathematical analysis of large genomes is a problem of current interest motivated by the development of DNA sequencing methods. To date, human and some other genomes have been sequenced. Certain characteristics of a genetic code shared by all these genomes are numerically analyzed in this paper. Relying on the results, a new property of the genetic code for overlaps of six and three genes in a single DNA strand is formulated. The choice of three termination codons out of 64 possible genetic code triplets does not influence the cardinality of the sets of nucleotide chains admitting sextuple and triple overlaps of genes. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020168 Issue No:Vol. 95, No. 2 (2017)

Authors:P. P. Zabreiko; A. V. Lebedev Pages: 164 - 167 Abstract: Banach geometric objects imitating a phenomenon of the type of the absence of arbitrage in financial markets models are analyzed. The role played in this field by reflexive subspaces (which replace classically considered finite-dimensional subspaces) and by plasterable cones is revealed. A series of new geometric criteria for the absence of arbitrage are proved. An alternative description of the existence of a martingale measure is given, which does not use dual objects. PubDate: 2017-03-01 DOI: 10.1134/s106456241702020x Issue No:Vol. 95, No. 2 (2017)

Authors:A. P. Mashtakov; Remco Duits; Yu. L. Sachkov; Erik Bekkers; I. Yu. Beschastnyi Pages: 168 - 171 Abstract: In order to detect vessel locations in spherical images of retina we consider the problem of minimizing the functional \(\int\limits_0^l {\mathfrak{C}\left( {\gamma \left( s \right)} \right)\sqrt {{\xi ^2} + k_g^2\left( s \right)} ds}\) for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k g denotes the geodesic curvature of γ. Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and propose numerical solution to this problem with consequent comparison to exact solution in the case C = 1. An experiment of vessel tracking in a spherical image of the retina shows a benefit of using SO(3) geodesics. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020181 Issue No:Vol. 95, No. 2 (2017)

Authors:A. A. Duyunova; V. V. Lychagin; S. N. Tychkov Pages: 172 - 175 Abstract: This paper presents a classification of equations of state for viscous fluids (or gases) whose motion is governed by the Navier–Stokes equations. The classification is based on an analysis of admissible symmetries. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020211 Issue No:Vol. 95, No. 2 (2017)

Authors:S. T. Sadetov Pages: 178 - 180 Abstract: Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D K → D L , defining the tensor product L ⊗ K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2]. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020235 Issue No:Vol. 95, No. 2 (2017)

Authors:P. S. Petrov; S. A. Sergeev; A. A. Tolchennikov Pages: 181 - 184 Abstract: An asymptotic solution to the problem of sound pulse propagation in deep sea is derived using the Maslov canonical operator. An example of a waveguide with the Munk sound speed profile and a point source is considered, and an asymptotic expression for the pulse signal time series at the receiver is obtained. The asymptotic solution is compared with the solution computed using the normal mode theory. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020065 Issue No:Vol. 95, No. 2 (2017)

Authors:E. V. Biryal’tsev; M. R. Galimov; A. M. Elizarov Pages: 185 - 189 Abstract: An experience of designing integrated hardware and software solutions for high-performance computing in solving modern geophysical problems on the basis of full-wave inversion is described. Problems of designing mass high-performance software systems intended for extensive use in industry are discussed. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020090 Issue No:Vol. 95, No. 2 (2017)

Authors:A. V. Il’in; E. I. Atamas’; V. V. Fomichev Pages: 190 - 193 Abstract: An inversion problem for a linear time-invariant MIMO system with possibly unstable zero dynamics is considered. Sufficient conditions for the invertibility of such systems are given, and an algorithm for invertor synthesis is proposed. The results are extended to time-delay systems with commensurable delays. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020144 Issue No:Vol. 95, No. 2 (2017)

Authors:A. I. Ovseevich; A. K. Fedorov Pages: 194 - 197 Abstract: We study the problem of the minimum-time damping of a closed string under a bounded load, applied at a single fixed point. A constructive feedback control law is designed, which allows bringing the system to a bounded neighbourhood of the terminal manifold. The law has the form of the dry friction at the point, where the load is applied. The motion under the control is governed by a nonlinear wave equation. The existence and uniqueness of solution of the Cauchy problem for this equation are proved. The main result is the asymptotic optimality of the suggested control law. PubDate: 2017-03-01 DOI: 10.1134/s1064562417020193 Issue No:Vol. 95, No. 2 (2017)