Authors:M. B. Karmanova Pages: 199 - 202 Abstract: Abstract The general nature of approximation of certain Hölder mappings of Carnot–Carathéodory spaces is described. The local existence of a basis in the image which preserves sub-Riemannian Hausdorff dimension is also proved. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030012 Issue No:Vol. 95, No. 3 (2017)

Authors:S. B. Kolonitskii Pages: 203 - 206 Abstract: Abstract The Dirichlet problem for the generalized Ginzburg–Landau system is considered. The existence of positive vector solutions is proved in the following three cases: (1) the cross term has weak growth; (2) the interaction constant is large enough; and (3) the cross term has strong growth and the interaction constant is positive and close to zero. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030024 Issue No:Vol. 95, No. 3 (2017)

Authors:L. V. Lokutsievskii; Yu. L. Sachkov Pages: 211 - 213 Abstract: Abstract One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030048 Issue No:Vol. 95, No. 3 (2017)

Authors:M. L. Goldman Pages: 214 - 217 Abstract: Abstract Modular and norm inequalities are considered for positively homogeneous operators on the cone of all nonnegative functions and on the cone Ω of nonnegative decreasing functions from the weighted Orlicz space with a general weight and a general Young function. A reduction theorem is obtained for the norm of an operator on Ω. This norm is shown to be equivalent to the norm of a modified operator on the cone of all nonnegative functions in the above Orlicz space. A similar theorem is obtained for modular inequalities. The results are based on the application of the principle of duality, which gives a description of the associated Orlicz norm for Ω. We also establish the equivalence of modular inequalities on the cone Ω and modified modular inequalities on the cone of all nonnegative functions in Orlicz space. In the general situation, the forms of these answers are substantially different from the descriptions obtained earlier by P. Drabek, A. Kufner, and H. Heinig under the assumption that the Young function and its complementary function satisfy Δ2-conditions. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030061 Issue No:Vol. 95, No. 3 (2017)

Authors:M. B. Karmanova Pages: 218 - 221 Abstract: Abstract Two-step sub-Lorentzian structures of arbitrary dimension with negative squared length in several directions are considered. An area formula for space-like graph surfaces is derived. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030085 Issue No:Vol. 95, No. 3 (2017)

Authors:M. M. Petrunin Pages: 222 - 225 Abstract: Abstract For a polynomial f of odd degree, nontrivial S-units can be effectively related to the continued fraction expansion of elements involving the square root of the polynomial f only in the case where S consists of an infinite valuation and a finite valuation determined by a first-degree polynomial h. In the paper, the proof that the quasi-periodicity of the continued fraction expansion of an element of the form \(\frac{{\sqrt f }}{{{h^s}}}\) implies periodicity is completed. In particular, it is proved that the continued fraction expansion of \(\sqrt f \) for f of any degree is quasi-periodic in k((h)) it and only if it is periodic. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030097 Issue No:Vol. 95, No. 3 (2017)

Authors:V. G. Romanov Pages: 230 - 234 Abstract: Abstract The stationary system of Maxwell equations for a unmagnetized nonconducting medium is considered. For this system, the problem of determining the permittivity ε from given electric or magnetic fields is studied. It is assumed that the electromagnetic field is induced by a plane wave coming from infinity in the direction ν. It is also assumed that the permittivity is different from a given positive constant ε0 only inside a compact domain Ω ⊂ R 3 with a smooth boundary S. To find ε inside Ω, the solution of the corresponding direct problem for the system of electrodynamic equations on the shadow portion of the boundary of Ω is specified for all frequencies starting at some fixed ω0 and for all ν. The high-frequency asymptotics of the solution to the direct problem is studied. It is shown that the information specified makes it possible to reduce the original problem to the well-known inverse kinematic problem of determining the refraction coefficient inside Ω from the traveling times of an electromagnetic wave. This leads to a uniqueness theorem for the solution of the problem under consideration and opens up the opportunity of its constructive solution. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030164 Issue No:Vol. 95, No. 3 (2017)

Authors:M. I. Imanaliev; A. Asanov; R. A. Asanov Pages: 235 - 239 Abstract: Abstract A new approach is used to show that the solution for one class of systems of linear Fredholm integral equations of the third kind with multipoint singularities is equivalent to the solution of systems of linear Fredholm integral equations of the second kind with additional conditions. The existence, nonexistence, uniqueness, and nonuniqueness of solutions to systems of linear Fredholm integral equations of the third kind with multipoint singularities are analyzed. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030140 Issue No:Vol. 95, No. 3 (2017)

Authors:A. S. Romanov Pages: 243 - 246 Abstract: Abstract Functions from the Sobolev spaces W p 1(Q) are considered on a unit cube Q ⊂ R n , and the properties of their traces on Lipschitz surfaces are examined. The relation is found between the Hölder exponent α and the Hausdorff dimension of the family of poor k-dimensional planes Γ on which the traces do not belong to C α(Γ). For the corresponding families of poor k-dimensional Lipschitz surfaces, estimates in terms of p-modules are obtained. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030127 Issue No:Vol. 95, No. 3 (2017)

Authors:A. V. Zvyagin; V. G. Zvyagin Pages: 247 - 249 Abstract: Abstract The qualitative dynamics of weak solutions to a nonautonomous model of polymer solution motion (with a rheological relation satisfying the objectivity principle) is studied using the theory of pullback attractors of trajectory spaces. For this purpose, the existence of weak solutions is proved for the model under study, a family of trajectory spaces is defined, trajectory and minimal pullback attractors are introduced, and their existence is proved. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030218 Issue No:Vol. 95, No. 3 (2017)

Authors:L. Mattner; I. G. Shevtsova Pages: 250 - 253 Abstract: Abstract We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ3-metric measuring the difference between expectations of sufficiently smooth functions, like · 3, of a sum of independent random variables X 1,..., X n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X 1,..., X n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009). PubDate: 2017-05-01 DOI: 10.1134/s1064562417030188 Issue No:Vol. 95, No. 3 (2017)

Authors:V. P. Platonov; G. V. Fedorov Pages: 254 - 258 Abstract: Abstract On the basis of a given criterion for the quasi-periodicity of continued fractions for elements of the hyperelliptic field L = K(x)( \(\sqrt f \) ), where K is an arbitrary field of characteristic different from 2 and f ∈ K[x] is a square-free polynomial, new polynomials f ∈ Q[x] of odd degree for which the elements of \(\sqrt f \) have periodic continued fraction expansion are found. PubDate: 2017-05-01 DOI: 10.1134/s106456241703019x Issue No:Vol. 95, No. 3 (2017)

Authors:E. R. Smol’yakov Pages: 259 - 263 Abstract: Abstract A strengthened equilibrium for conflict problems with partially overlapping game sets of participants is proposed, which is of substantial help in determining fair division in cooperative games and makes it possible to refine the hierarchy of all known equilibria. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030206 Issue No:Vol. 95, No. 3 (2017)

Authors:T. N. Fomenko Pages: 264 - 266 Abstract: Abstract The paper is devoted to the problem of the existence of common fixed points and coincidence points of a family of set-valued maps of ordered sets. Fixed-point and coincidence theorems for families of set-values maps are presented, which generalize some of the known results. The presented theorems, unlike previous ones, do not assume the maps to be isotone or coverable. They require only the existence of special chains having lower bounds with certain properties in the ordered set. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030176 Issue No:Vol. 95, No. 3 (2017)

Authors:A. I. Kozhanov; S. V. Potapova Pages: 267 - 269 Abstract: Abstract The solvability of a boundary value problem for the differential equation \(h\left( x \right){u_t} + {\left( { - 1} \right)^m}\frac{{{\partial ^{2m + 1}}u}}{{\partial {a^{2m + 1}}}} - {u_{xx}} = f\left( {x,t,a} \right)\) is studied in the case where h(x) has a jump discontinuity and reverses its sign on passing through the discontinuity point. Existence and uniqueness theorems are proved in the case of solutions having all Sobolev generalized derivatives involved in the equation. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030231 Issue No:Vol. 95, No. 3 (2017)

Authors:V. V. Napalkov; V. V. Napalkov Pages: 270 - 272 Abstract: Abstract Reproducing kernel Hilbert spaces having orthogonally similar decomposition systems are considered. Conditions under which such spaces coincide or are isomorphic are found. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030243 Issue No:Vol. 95, No. 3 (2017)

Authors:A. P. Khromov Pages: 273 - 275 Abstract: Abstract The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L 2[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem. PubDate: 2017-05-01 DOI: 10.1134/s106456241703022x Issue No:Vol. 95, No. 3 (2017)

Authors:A. V. Favorskaya; I. B. Petrov Pages: 287 - 290 Abstract: Abstract Spatial dynamic wave effects occurring in rocks with ravines and caverns were studied. The influence exerted by the explosion type and the cavern-to-ravine distance on the formation of spatial dynamic wave patterns and seismograms was analyzed in the case of horizontal and vertical reception lines. The gridcharacteristic method and the full wave joint numerical modeling of elastic and acoustic waves were used. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030139 Issue No:Vol. 95, No. 3 (2017)

Authors:O. V. Rudenko Pages: 291 - 294 Abstract: Abstract Solutions to an inhomogeneous partial differential equation of the second-order like Burgers equation are derived. Instead of the common quadratically nonlinear term, this equation contains the term with modular nonlinearity. This model describes the excitation of elastic waves in dissipative media differently reacting to tensile and compressive stresses. The equation is linear for the functions, preserving the sign. Nonlinear effects are manifested only to alternating functions. The solution for the periodic signal is found. The processes of generation of fundamental and higher harmonics are studied. The stationary wave profile is constructed. For one special kind of right-hand-side of the “modular” equation the solution in the form of S-wave is pointed out which is a bipolar single pulse. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030255 Issue No:Vol. 95, No. 3 (2017)

Authors:S. N. Vassilyev; A. A. Galyaev Pages: 299 - 304 Abstract: Abstract Problems of automatic action planning and optimal control of unmanned vehicles in an adversarial environment are considered. An approach based on nonmonotonic logic in the language of positively formed formulas for automatically planning a route to a chosen target area is developed. The choice of this area, as well as classification and determination of priority targets, is implemented by means of logical inference with classical semantics. The optimal pursuit problem for the example of a group of three targets is solved. PubDate: 2017-05-01 DOI: 10.1134/s1064562417030267 Issue No:Vol. 95, No. 3 (2017)