Authors:V. A. Bykovskii; A. V. Ustinov Pages: 611 - 614 Abstract: Abstract A general formula for elements of double Somos-4 sequences is obtained. A sufficient integrality condition for such sequences is presented. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060016 Issue No:Vol. 94, No. 3 (2016)

Authors:A. A. Kudryavtsev; O. V. Shestakov Pages: 615 - 619 Abstract: Abstract The problem of nonparametric estimation of a signal function by thresholding the coefficients of its wavelet decomposition is considered. In models with various noise distributions, asymptotically optimal thresholds and orders of the loss functions are calculated on the basis of probabilities of errors in the calculation of wavelet coefficients. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060028 Issue No:Vol. 94, No. 3 (2016)

Authors:D. A. Podoprikhin; T. N. Fomenko Pages: 620 - 622 Abstract: Abstract New results on fixed points and coincidences of families of set-valued mappings of partially ordered sets obtained without commutativity assumptions are presented. These results develop theorems on fixed points of an isotone self-mapping of an ordered set (for families of set-valued mappings) and theorems about coincidences of two set-valued mappings one of which is isotone and the other is covering (for finite families of set-valued mappings). PubDate: 2016-11-01 DOI: 10.1134/s106456241606003x Issue No:Vol. 94, No. 3 (2016)

Authors:L. A. Prokhorenkova; A. V. Krot Pages: 623 - 626 Abstract: Abstract The local clustering coefficients of preferential attachment models are analyzed. Previously, a general approach to preferential attachment was proposed (the PA-class was introduced); it was shown that the degree distribution in all models of the PA-class obeys a power law. The global clustering coefficient was also analyzed, and a lower bound for the mean local clustering coefficient was found. In the paper, new results are obtained by analyzing the local clustering coefficients of models of the PA-class. Namely, the behavior of the mean value C 2(n, d) of local clustering over vertices of degree d is studied. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060041 Issue No:Vol. 94, No. 3 (2016)

Authors:M. L. Goldman Pages: 627 - 631 Abstract: Abstract A monotone operator P mapping the Orlicz–Lorentz class to an ideal space is considered. The Orlicz–Lorentz class is the cone of measurable functions on R + =(0, ∞) whose decreasing rearrangements with respect to the Lebesgue measure on R + belong to the weighted Orlicz space L Φ,ν. Reduction theorems are proved, which make it possible to reduce estimates of the norm of the operator P: ΛΦ,ν →Y to those of the norm of its restriction to the cone of nonnegative step functions in L Φ,ν. The application of these results to the identity operator from ΛΦ,ν to the weighted Lebesgue space Y = L 1(R +; g) gives exact descriptions of associated norms for ΛΦ,ν. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060065 Issue No:Vol. 94, No. 3 (2016)

Authors:A. V. Kel’manov; A. V. Pyatkin Pages: 635 - 638 Abstract: Abstract Problems of partitioning a finite set of Euclidean points (vectors) into clusters are considered. The criterion is to minimize the sum, over all clusters, of (1) squared norms of the sums of cluster elements normalized by the cardinality, (2) squared norms of the sums of cluster elements, and (3) norms of the sum of cluster elements. It is proved that all these problems are strongly NP-hard if the number of clusters is a part of the input and are NP-hard in the ordinary sense if the number of clusters is not a part of the input (is fixed). Moreover, the problems are NP-hard even in the case of dimension 1 (on a line). PubDate: 2016-11-01 DOI: 10.1134/s1064562416060089 Issue No:Vol. 94, No. 3 (2016)

Authors:E. A. Lukashev; N. N. Yakovlev; E. V. Radkevich; O. A. Vasil’yeva Pages: 649 - 653 Abstract: Abstract The initial stage of the laminar–turbulent transition is reconstructed. Its mechanism is based on spinodal decomposition (diffusion separation). PubDate: 2016-11-01 DOI: 10.1134/s1064562416060119 Issue No:Vol. 94, No. 3 (2016)

Authors:V. G. Sakbaev; O. G. Smolyanov Pages: 654 - 658 Abstract: Abstract Representations of Schrödinger semigroups and groups by Feynman iterations are studied. The compactness, rather than convergence, of the sequence of Feynman iterations is considered. Approximations of solutions of the Cauchy problem for the Schrödinger equation by Feynman iterations are investigated. The Cauchy problem for the Schrödinger equation under consideration is ill-posed. From the point of view of the approach of the paper, this means that the problem has no solution in the sense of integral identity for some initial data. The well-posedness of the Cauchy problem can be recovered by extending the operator to a selfadjoint one; however, there exists continuum many such extensions. Feynman iterations whose partial limits are the solutions of all Cauchy problems obtained for various self-adjoint extensions are studied. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060132 Issue No:Vol. 94, No. 3 (2016)

Authors:A. D. Baev; R. A. Kovalevskii; P. A. Kobylinskii Pages: 659 - 662 Abstract: Abstract Problems for high-order degenerate elliptic equations in a half-space are studied. Coercive a priori estimates and existence theorems for solutions of such problems in special weighted Sobolev-type spaces are obtained. The norms in these spaces are defined with the help of a special integral transform. Pseudodifferential operators with degeneration constructed using a special integral transform are studied. Pseudodifferential operators with degeneration are used to factorize the symbol of a high-order degenerate elliptic operator and to construct a separating operator of the Leray–Sakamoto type. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060168 Issue No:Vol. 94, No. 3 (2016)

Authors:M. B. Karmanova Pages: 663 - 666 Abstract: Abstract The polynomial sub-Riemannian differentiability of classes of mappings of Carnot groups and graphs is proved. Examples of polynomial sub-Riemannian differentials preserving Hausdorff dimension are given. PubDate: 2016-11-01 DOI: 10.1134/s106456241606017x Issue No:Vol. 94, No. 3 (2016)

Authors:E. A. Baderko; M. F. Cherepova Pages: 670 - 672 Abstract: Abstract The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060211 Issue No:Vol. 94, No. 3 (2016)

Authors:V. V. Konev Pages: 676 - 680 Abstract: Abstract A transformation of a discrete-time martingale with conditionally Gaussian increments into a sequence of i.i.d. standard Gaussian random variables is proposed as based on a sequence of stopping times constructed using the quadratic variation. It is shown that sequential estimators for the parameters in AR(1) and generalized first-order autoregressive models have a nonasymptotic normal distribution. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060235 Issue No:Vol. 94, No. 3 (2016)

Authors:A. L. Skubachevskii; Y. Tsuzuki Pages: 681 - 683 Abstract: Abstract The Vlasov–Poisson equations for a two-component high-temperature plasma with an external magnetic field in a half-space are considered. The electric field potential satisfies the Dirichlet condition on the boundary, and the initial density distributions of charged particles satisfy the Cauchy conditions. Sufficient conditions for the induction of the external magnetic field and the initial charged-particle density distributions are obtained that guarantee the existence of a classical solution for which the supports of the charged-particle density distributions are located at some distance from the boundary. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060247 Issue No:Vol. 94, No. 3 (2016)

Authors:M. L. Blank Pages: 688 - 691 Abstract: Abstract A novel approach to the fair division problem is proposed, which is based on the concept of a priori estimates and ideas of dynamical systems theory. For several problems on the division of a resource with discrete components, this approach leads to explicit constructive solutions in cases for which even the existence of solutions has not been previously known. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060272 Issue No:Vol. 94, No. 3 (2016)

Authors:V. P. Platonov; V. S. Zhgoon; G. V. Fedorov Pages: 692 - 696 Abstract: Abstract A relationship between the continued fraction expansion of the quadratic irrationalities of hyperelliptic fields and the Mumford polynomials determining addition in the group of divisor classes on a hyperelliptic curve is described. A theorem on the equivalence of the quasi-periodicity of a quadratic irrationality and the existence of a point of finite order is proved; results on the symmetry of the quasi-period and estimates of its length are obtained. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060284 Issue No:Vol. 94, No. 3 (2016)

Authors:V. D. Stepanov; G. E. Shambilova Pages: 697 - 702 Abstract: Abstract Necessary and sufficient conditions for the weighted boundedness of a class of positive quasilinear integral two-kernel operators of iterative type on the real half-line are given. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060302 Issue No:Vol. 94, No. 3 (2016)

Authors:O. V. Rudenko Pages: 703 - 707 Abstract: Abstract A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types—modular, quadratic, and quadratically cubic—which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060053 Issue No:Vol. 94, No. 3 (2016)

Authors:O. V. Rudenko Pages: 708 - 711 Abstract: Abstract Solutions to a partial differential equation of the third order containing the modular nonlinearity are studied. The model describes, in particular, elastic waves in media with weak high-frequency dispersion and with different response to tensile and compressive stresses. This equation is linear for solutions preserving their sign. Nonlinear phenomena only manifest themselves to alternating solutions. Stationary solutions in the form of solitary waves or solitons are found. It is shown how the linear periodic wave becomes nonlinear after exceeding a certain critical value of the amplitude, and how it transforms into a soliton with further increase in the amplitude. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060296 Issue No:Vol. 94, No. 3 (2016)

Authors:E. M. Solnechnyi Pages: 712 - 714 Abstract: Abstract Stability and roughness conditions for a distributed plant control system with a two-link controller close to a degenerated system (Shchipanov’s controller) are considered. A mathematical model of the plant and the controller is specified by operator relations between the output, control, and disturbance. For a linear plant, conditions for the nearly invariant behavior of the closed system are obtained. For a nonlinear plant that is linear with respect to control, conditions for the stability and nearly invariant behavior of the closed system are obtained. PubDate: 2016-11-01 DOI: 10.1134/s1064562416060090 Issue No:Vol. 94, No. 3 (2016)