Authors:M. B. Karmanova Pages: 1 - 4 Abstract: Abstract For classes of Hölder mappings of Carnot groups and the corresponding graph mappings, a method for constructing intrinsic bases is described, which makes it possible to push forward the Hausdorff dimension of preimages to images. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010033 Issue No:Vol. 95, No. 1 (2017)

Authors:D. D. Cherkashin; A. M. Raigorodskii Pages: 5 - 6 Abstract: Abstract New lower bounds are found for the minimum number of colors needed to color all points of a Euclidean space in such a way that any two points at a distance of 1 have different colors. PubDate: 2017-01-01 DOI: 10.1134/s106456241701001x Issue No:Vol. 95, No. 1 (2017)

Authors:B. N. Chetverushkin; M. V. Yakobovskiy Pages: 7 - 11 Abstract: Abstract A new method is discussed which provides the possibility of long-term continuous calculations on a computing systems consisting of millions of operating devices, some of which may suffer failures in the course of calculation. The method relies on the properties of hyperbolized systems of partial differential equations, for which the domain of influence on the solution is localized in space. As a result, the necessary part of the solution can be rapidly recalculated without restarting the whole calculation process. The number of additional processors required for executing the recalculation is estimated. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010021 Issue No:Vol. 95, No. 1 (2017)

Authors:A. M. Raigorodskii; A. A. Sagdeev Pages: 15 - 16 Abstract: Abstract Explicit exponential lower bounds for the chromatic numbers of spaces with forbidden monochromatic regular simplexes are found for the first time. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010082 Issue No:Vol. 95, No. 1 (2017)

Authors:P. E. Ryabov Pages: 17 - 20 Abstract: Abstract For the general integrability case of M. Adler and P. van Moerbeke, invariant relations are obtained in which the rank of the momentum map is 1. Thereby, special periodic solutions generating the edges of the bifurcation diagram are defined. All phase variables are expressed in terms of a set of constants and one auxiliary variable, for which a differential equation integrable in elliptic functions time is given. An explicit expression for the characteristic exponent determining the type of special periodic solutions is presented, which makes it possible to study the character of stability of the obtained solution. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010094 Issue No:Vol. 95, No. 1 (2017)

Authors:V. V. Arestov; M. V. Deikalova Pages: 21 - 25 Abstract: Abstract The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [−1, 1] between the uniform norm and the norm of the space L q (α,β) , 1 ≤ q < ∞, with Jacobi weight ϕ(α,β)(x) = (1 − x)α(1 + x)β α ≥ β > −1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L q (α,β) , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\) , is attained. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010100 Issue No:Vol. 95, No. 1 (2017)

Authors:J. Gough; T. S. Ratiu; O. G. Smolyanov Pages: 26 - 30 Abstract: Abstract A quantum anomaly is the breaking of symmetry with respect to some transformations after the quantization of a classical Hamiltonian or Lagrangian system. It is shown that both the Noether theorems (including their infinite-dimensional versions) and the explanation of the origin of quantum anomalies can be obtained by using similar formulas for derivatives of functions whose values are measure in the former case and pseudomeasures in the latter. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010112 Issue No:Vol. 95, No. 1 (2017)

Authors:E. R. Smol’yakov Pages: 37 - 42 Abstract: Abstract A generalized equilibrium for conflict problems with partially overlapping game sets of participants is proposed; this equilibrium turns out to be nonempty even in those cases where all known equilibria are empty. PubDate: 2017-01-01 DOI: 10.1134/s1064562416060259 Issue No:Vol. 95, No. 1 (2017)

Authors:E. I. Khukhro; P. Shumyatsky Pages: 43 - 45 Abstract: Abstract Let α be an automorphism of a finite group G. For a positive integer n, let E G,n (α) be the subgroup generated by all commutators [...[[x,α],α],…,α] in the semidirect product G 〈α〉 over x ∈ G, where α is repeated n times. By Baer’s theorem, if E G,n (α)=1, then the commutator subgroup [G,α] is nilpotent. We generalize this theorem in terms of certain length parameters of E G,n (α). For soluble G we prove that if, for some n, the Fitting height of E G,n (α) is equal to k, then the Fitting height of [G,α] is at most k + 1. For nonsoluble G the results are in terms of the nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F h* (H) = H, where F 0* (H) = 1, and F i+1* (H) is the inverse image of the generalized Fitting subgroup F*(H/F i *(H)). Let m be the number of prime factors of the order α counting multiplicities. It is proved that if, for some n, the generalized Fitting height E G,n (α) of is equal to k, then the generalized Fitting height of [G,α] is bounded in terms of k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λE G,n (α)= k, then the nonsoluble length of [G,α] is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010124 Issue No:Vol. 95, No. 1 (2017)

Authors:A. V. Arutyunov Pages: 46 - 49 Abstract: Abstract Nonlinear equations in Banach spaces are considered. Solvability conditions are obtained for them. These results are a generalization of the Hadamard diffeomorphism theorem. Additionally, conditions for the existence of coincidence points of two mappings acting on metric spaces are obtained. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010148 Issue No:Vol. 95, No. 1 (2017)

Authors:A. G. Chechkin; A. S. Shamaev Pages: 55 - 59 Abstract: Abstract The Fokker–Planck–Kolmogorov parabolic second-order differential operator is considered, for which its fundamental solution is derived in explicit form. Such operators arise in numerous applications, including signal filtering, portfolio control in financial mathematics, plasma physics, and problems involving linear-quadratic regulators. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010161 Issue No:Vol. 95, No. 1 (2017)

Authors:M. E. Zhukovskii; A. B. Kupavskii Pages: 60 - 61 Abstract: Abstract A random graph G(n, p) is said to obey the (monadic) zero–one k-law if, for any monadic formula of quantifier depth k, the probability that it is true for the random graph tends to either zero or one. In this paper, following J. Spencer and S. Shelah, we consider the case p = n −α. It is proved that the least k for which there are infinitely many α such that a random graph does not obey the zero–one k-law is equal to 4. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010227 Issue No:Vol. 95, No. 1 (2017)

Authors:P. N. Klepikov; E. D. Rodionov Pages: 62 - 64 Abstract: Abstract Algebraic Ricci solitons on Lie groups with left-invariant (pseudo)Riemannian metric and zero Schouten–Weyl tensor are studied. The absence of nontrivial algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor and diagonalizable Ricci operator is proved. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010197 Issue No:Vol. 95, No. 1 (2017)

Authors:B. N. Chetverushkin; N. D’Ascenzo; A. V. Saveliev; V. I. Saveliev Pages: 68 - 71 Abstract: Abstract A solution algorithm developed for three-dimensional magnetogasdynamic problems is used to simulate the accretion of interstellar matter onto a massive astronomical object with the formation of collinear jets. A kinetically consistent algorithm is well adapted to the architecture of high-performance computer systems with massive parallelism and has improved conditions for time discretization. The three-dimensional accretion of interstellar matter is computed in detail on a spatial grid consisting of 1 billion nodes, and the possibility of studying the formation of collinear jets is demonstrated. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010185 Issue No:Vol. 95, No. 1 (2017)

Authors:A. V. Akhmetzyanov; A. G. Kushner; V. V. Lychagin Pages: 72 - 75 Abstract: Abstract A constructive method is proposed for finding finite-dimensional submanifolds in the space of smooth functions that are invariant with respect to flows defined by evolutionary partial differential equations. Conditions for the stability of these submanifolds are obtained. Such submanifolds are constructed for generalized Rapoport–Leas equations that arise in the theory of porous media flows. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010239 Issue No:Vol. 95, No. 1 (2017)

Authors:A. I. Tyulenev; S. K. Vodop’yanov Pages: 79 - 83 Abstract: Abstract Given a closed weakly regular d-thick subset S of ℝ n , we prove the existence of a bounded linear extension operator Ext: Tr S W p 1 (ℝ n , γ) → W p 1 (ℝ n , γ) for p ∈ (1, ∞), 0 ≤ d ≤ n, r ∈ (max{1, n − d}, p), l ∈ ℕ, and \(\gamma \in {A_{\frac{p}{r}}}\) (ℝ n ). In particular, we prove that a linear bounded trace space exists in the case where S is the closure of an arbitrary domain in ℝ n , γ ≡ 1, and p > n − 1. The obtained results supplement those of previous studies, in which a similar problem was considered either in the case of p ∈ (n, ∞) without constraints on the set S or in the case of p ∈ (1, ∞) under stronger constraints on the set S. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010276 Issue No:Vol. 95, No. 1 (2017)

Authors:E. G. Shifrin Pages: 84 - 86 Abstract: Abstract The Schwarz alternating method makes it possible to construct a solution of the Dirichlet problem for the two-dimensional Laplace equation in a finite union of overlapping domains, provided that this problem has a solution in each domain. The existing proof of the method convergence and estimation of the convergence rate use the condition that the normals to the boundaries of the domains at the intersection points are different. In the paper, it is proved that this constraint can be removed for domains with Hölder continuous normals. Removing the constraint does not affect the rate of convergence. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010264 Issue No:Vol. 95, No. 1 (2017)

Authors:A. V. Lotov Pages: 95 - 98 Abstract: Abstract A numerical method is proposed for constructing an external polyhedral estimate for the trajectory tube of a nonlinear dynamic system described by a differential inclusion. The method is based on the approximation of cross sections of the trajectory tube (reachable sets) for an auxiliary system described by the convex hull of the graph of the differential inclusion. It produces polyhedral estimates suitable for the direct study of tubes via computer visualization and for the solution of more general problems. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010045 Issue No:Vol. 95, No. 1 (2017)

Authors:M. J. Mardanov; R. A. Teymurov Pages: 99 - 102 Abstract: Abstract A variational method for the optimal control of moving sources governed by a parabolic equation with nonlocal integral conditions is considered. For this problem, an existence and uniqueness theorem is proved, necessary optimality conditions in the form of pointwise and integral maximum principles are obtained, sufficient conditions for the Fréchet differentiability of the cost functional are found, and an expression for its gradient is derived. PubDate: 2017-01-01 DOI: 10.1134/s1064562417010070 Issue No:Vol. 95, No. 1 (2017)