Showing papers in "Doklady Mathematics in 2016"
TL;DR: In this paper, a second-order partial differential equation admitting exact linearization is discussed, which contains terms with nonlinearities of three types (modular, quadratic and quadratically cubic) which can be present jointly or separately.
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.
37 citations
TL;DR: An approach to the implementation of a recommender system based on ontologies of mathematical knowledge is presented, and various adaptations of the system to user scenarios aimed at the preparation of personalized recommendations are discussed.
Abstract: An approach to the implementation of a recommender system based on ontologies of mathematical knowledge is presented. On the basis of a document browsed by a user, the system forms on line a list of recommendations, which include similar documents, key words, and definitions of these words from ontology and other terminological sources. The method of recommendations yields a vector representation of documents, taking into account the position of terms in the logical structure of the document and their ontological connections. On the basis of the cosine measure, a measure of proximity between documents is calculated. The order of documents in the list of recommendations is determined by values of the proximity measure. Various adaptations of the system to user scenarios aimed at the preparation of personalized recommendations are discussed.
26 citations
TL;DR: New first-order methods are introduced for solving convex optimization problems from a fairly broad class and a stochastic intermediate gradient method is proposed that allows using an arbitrary norm in the space of variables and a prox-function.
Abstract: New first-order methods are introduced for solving convex optimization problems from a fairly broad class. For composite optimization problems with an inexact stochastic oracle, a stochastic intermediate gradient method is proposed that allows using an arbitrary norm in the space of variables and a prox-function. The mean rate of convergence of this method and the probability of large deviations from this rate are estimated. For problems with a strongly convex objective function, a modification of this method is proposed and its rate of convergence is estimated. The resulting estimates coincide, up to a multiplicative constant, with lower complexity bounds for the class of composite optimization problems with an inexact stochastic oracle and for all usually considered subclasses of this class.
22 citations
TL;DR: It is proved that the splitting error is zero for a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law).
Abstract: A dimensional splitting scheme is applied to a multidimensional scalar homogeneous quasilinear hyperbolic equation (conservation law). It is proved that the splitting error is zero. The proof is presented for the above partial differential equation in an arbitrary number of dimensions. A numerical example is given that illustrates the proved accuracy of the splitting scheme. In the example, the grid convergence of split (locally one-dimensional) compact and bicompact difference schemes and unsplit bicompact schemes combined with high-order accurate time-stepping schemes (namely, Runge–Kutta methods of order 3, 4, and 5) is analyzed. The errors of the numerical solutions produced by these schemes are compared. It is shown that the orders of convergence of the split schemes remain high, which agrees with the conclusion that the splitting error is zero.
19 citations
TL;DR: In this paper, the existence of a coincidence point of two mappings acting between (q1, q2)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition is investigated.
Abstract: We introduce (q1, q2)-quasimetric spaces and examine their properties. Covering mappings between (q1, q2)-quasimetric spaces are investigated. Sufficient conditions for the existence of a coincidence point of two mappings acting between (q1, q2)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition are obtained.
18 citations
TL;DR: A face recognition method based on a matching algorithm with recursive calculation of oriented gradient histograms for several circular sliding windows and a pyramidal image decomposition is proposed, which produces good results for geometrically distorted and scaled images.
Abstract: A face recognition method based on a matching algorithm with recursive calculation of oriented gradient histograms for several circular sliding windows and a pyramidal image decomposition is proposed. The algorithm produces good results for geometrically distorted and scaled images.
17 citations
TL;DR: In this article, the dominance of general word maps with constants has been shown for a simple algebraic group G of rank m, where w is a nontrivial word in the free group Fm and wΣ = w1σ1w2 ··· wrσrwr + 1, w1, wr + 1 ∈ Fm, w2, wr ≠ 1, σr | σi ∈ GZ(G)}.
Abstract: In the present paper, we consider word maps w: Gm → G and word maps with constants wΣ: Gm → G of a simple algebraic group G, where w is a nontrivial word in the free group Fm of rank m, wΣ = w1σ1w2 ··· wrσrwr + 1, w1, …, wr + 1 ∈ Fm, w2, …, wr ≠ 1, Σ = {σ1, …, σr | σi ∈ GZ(G)}. We present results on the images of such maps, in particular, we prove a theorem on the dominance of “general” word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the representation variety R(Γw, G) of the group Γw = Fm/ .
16 citations
TL;DR: In this article, a series of results on the stability of the independence number of random subgraphs of distance graphs were obtained, which are natural generalizations of the classical Kneser graphs.
Abstract: A series of results are obtained on the stability of the independence number of random subgraphs of distance graphs, which are natural generalizations of the classical Kneser graphs.
16 citations
TL;DR: In this paper, the asymptotic stability and global stability of equilibria in autonomous systems of differential equations are analyzed in terms of compact invariant sets and positively invariants.
Abstract: The asymptotic stability and global asymptotic stability of equilibria in autonomous systems of differential equations are analyzed. Conditions for asymptotic stability and global asymptotic stability in terms of compact invariant sets and positively invariant sets are proved. The functional method of localization of compact invariant sets is proposed for verifying the fulfillment of these conditions. Illustrative examples are given.
16 citations
TL;DR: In this article, the Mumford polynomials determining addition in the group of divisor classes on a hyperelliptic curve were studied and a theorem on the equivalence of the quasi-periodicity of a quadratic irrationality and the existence of a point of finite order was proved.
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.
14 citations
TL;DR: In this paper, a quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained, and a more accurate criterion is found for key elements.
Abstract: Given a polynomial f of odd degree, the nontrivial S-units can be effectively related to the continued fraction expansions of the elements associated with \(\sqrt f \) only in the case where S contains an infinite valuation and a finite valuation determined by first-degree polynomial. A quasi-periodicity criterion for any element of the field of formal power series in a first-degree polynomial is obtained. For key elements, a more accurate criterion is found. The criterion is used to show that, for S specified above, in the presence of a nontrivial S-unit, the expansion of \(\sqrt f \) can be both nonperiodic and periodic. Estimates relating the quasi-period to the degree of the fundamental S-unit are obtained. Examples in which the bounds of these estimates are attained are given.
TL;DR: In this article, the structure of the Jacobian group for circulant graphs is studied and an efficient algorithm for calculating it is proposed for the simplest graphs in this family, and in the general case, and an effective algorithm for computing it is also proposed.
Abstract: The Jacobian of a graph is defined as the maximal Abelian group generated by flows obeying two Kirchhoff’s laws. This notion, also known as the Picard group, sandpile group, or critical group, has been extensively studied by many authors in the past decade. This is an important algebraic invariant of a finite graph. At the same time, the structure of the Jacobian is known only in particular cases. The paper is devoted to the study of the structure of the Jacobian group for circulant graphs. For the simplest graphs in this family, the Jacobian group is explicitly described, and in the general case, and effective algorithm for calculating it is proposed.
TL;DR: In this article, the monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux is analyzed, assuming that the propagation velocity of characteristics of the approximated conservative equation is positive.
Abstract: The monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux is analyzed. Monotonicity conditions for this scheme are obtained assuming that the propagation velocity of characteristics of the approximated conservative equation is positive. Test computations are presented that illustrate these properties of the CABARET scheme.
TL;DR: In this paper, conditions on the real kernel K(x, t) of an integral operator under which this operator satisfies a well-defined boundary condition for the corresponding differential equation are found.
Abstract: Integral operators of the form \(L_K^{ - 1} f(x) = \int\limits_\Omega {K(x,t)f(t)dt}\) for the case of a finite domain Ω ⊂ Rn with smooth boundary ∂Ω are considered. Conditions on the real kernel K(x, t) of an integral operator under which this operator satisfies a well-defined boundary condition for the corresponding differential equation are found. The application of the results is demonstrated on the example of a Sturm–Liouville equation, for which the derivation of the general form of well-posed boundary value problems is presented.
TL;DR: A priori accuracy estimates for low-rank approximations using a small number of rows and columns of the initial matrix are proposed, which are substantially more accurate than those known previously.
Abstract: A priori accuracy estimates for low-rank approximations using a small number of rows and columns of the initial matrix are proposed. Unlike in the existing methods of pseudoskeleton approximation, this number is larger than the rank of approximation, but the estimates are substantially more accurate than those known previously.
TL;DR: In this paper, the authors consider nonlinear boundary conditions involving some maximal monotone graphs which may correspond to discontinuous or non-Lipschitz functions arising in some catalysis problems.
Abstract: We extend previous papers in the literature concerning the homogenization of Robin type boundary conditions for quasilinear equations, in the case of microscopic obstacles of critical size: here we consider nonlinear boundary conditions involving some maximal monotone graphs which may correspond to discontinuous or non-Lipschitz functions arising in some catalysis problems.
TL;DR: In this paper, a family of distance graphs whose structure is close to that of Kneser graphs is studied, and lower and upper bounds for the chromatic numbers of such graphs are obtained, and relations between these numbers are considered.
Abstract: A family of distance graphs whose structure is close to that of Kneser graphs is studied. New lower and upper bounds for the chromatic numbers of such graphs are obtained, and relations between these numbers are considered. The structure of certain important independent sets of the family of graphs under consideration is described, and the cardinalities of these sets are explicitly calculated.
TL;DR: In this article, new inequalities relating Sobolev and Kantorovich norms for functions on Riemannian manifolds satisfying certain curvature conditions are obtained, which are related to our inequalities.
Abstract: New inequalities relating Sobolev and Kantorovich norms for functions on Riemannian manifolds satisfying certain curvature conditions are obtained.
TL;DR: It is proved that localizing sets separate simple and complex dynamics of nonlinear systems, namely, in the complement of any localizing set, the trajectory behavior of a system admits a standard description in the form of several scenarios, while, in localizer sets and their intersections, the trajectories of the system can be very complex.
Abstract: The problem is considered of finding domains in the phase space in which the trajectories of a system have a fairly simple behavior determined by a typical scenario. The problem is solved by applying the method of localization of compact invariant sets of the system. It is proved that localizing sets separate simple and complex dynamics of nonlinear systems, namely, in the complement of any localizing set, the trajectory behavior of a system admits a standard description in the form of several scenarios, while, in localizing sets and their intersections, the trajectory behavior of the system can be very complex. Specifically, it is shown that the α- and ω-limit sets of any trajectory are contained in localizing sets.
TL;DR: In this article, a complete system of formulas for the analytic continuation of the Lauricella generalized hypergeometric function FD(N) with any N beyond the boundary of the unit polydisk is proposed.
Abstract: An approach for constructing a complete system of formulas for the analytic continuation of the Lauricella generalized hypergeometric function FD(N) with any N beyond the boundary of the unit polydisk is proposed. The approach is exposed in detail for the continuation of the function under consideration in neighborhoods of points whose all N components equal 1 or ∞. For the Lauricella function, differential relations being analogues of Jacobi’s formula for the Gaussian hypergeometric function are also presented. The results can be applied to solve the crowding problem for the Schwarz–Christoffel integral and to the theory of the Riemann–Hilbert problem.
TL;DR: It is shown that the high-order accurate bicompact schemes can be efficiently parallelized on multicore and multiprocessor computers.
Abstract: The method of lines is used to obtain semidiscrete equations for a bicompact scheme in operator form for the inhomogeneous linear transport equation in two and three dimensions. In each spatial direction, the scheme has a two-point stencil, on which the spatial derivatives are approximated to fourth-order accuracy due to expanding the list of unknown grid functions. This order of accuracy is preserved on an arbitrary nonuniform grid. The equations of the method of lines are integrated in time using diagonally implicit multistage Runge–Kutta methods of the third up fifth orders of accuracy. Test computations on refined meshes are presented. It is shown that the high-order accurate bicompact schemes can be efficiently parallelized on multicore and multiprocessor computers.
TL;DR: In this article, a recurrence method is developed for synthesizing an output control for the finite spectrum of a descriptor dynamical system defined in a state space, which can be used to construct a parametrization of the solution set ensuring the same finite spectrum.
Abstract: A recurrence method is developed for synthesizing an output control for the finite spectrum of a descriptor dynamical system defined in a state space. The method is new and generalizing. It produces analytical solutions of the synthesis problem and can be used to construct a parametrization of the solution set ensuring the same finite spectrum.
TL;DR: In this article, self-similar solutions for a quadratically cubic second-order partial differential equation governing the behavior of nonlinear waves in various distributed systems, for example, in some metamaterials, are compared with selfsimilar solutions of the Burgers equation.
Abstract: Self-similar solutions are found for a quadratically cubic second-order partial differential equation governing the behavior of nonlinear waves in various distributed systems, for example, in some metamaterials. They are compared with self-similar solutions of the Burgers equation. One of them describing a single unipolar pulse is shown to satisfy both equations. The other self-similar solutions of the quadratically cubic equation behave differently from the solutions of the Burgers equation. They are constructed by matching the positive and negative branches of the solution, so that the function itself and its first derivative are continuous. One of these solutions corresponds to an asymmetric solitary N-wave of the sonic shock type. Self-similar solutions of a quadratically cubic equation describing the propagation of cylindrically symmetric waves are also found.
TL;DR: In this article, the construction of the Maslov canonical operator adapted to an arbitrary coordinate system on the corresponding Lagrangian manifold is presented, which does not require any additional choice of the phase function.
Abstract: We present the construction of the Maslov canonical operator adapted to an arbitrary coordinate system on the corresponding Lagrangian manifold. The construction does not require any additional choice of the phase function.
TL;DR: In this article, a numerical analytic algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems was proposed. But the resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients and the conditions for the stability of the closed-loop system were established, and a stability result was stated.
Abstract: A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.
TL;DR: The problem of estimating the signal function from noisy observations by thresholding the coefficients of its wavelet decomposition by minimizing the average probability of error in calculating the wavelet coefficients is considered.
Abstract: The problem of estimating the signal function from noisy observations by thresholding the coefficients of its wavelet decomposition is considered. The asymptotic orders of the threshold and risk are calculated by minimizing the average probability of error in calculating the wavelet coefficients.
TL;DR: In this article, the authors developed theorems on fixed points of an isotone self-mapping of an ordered set (for families of set-valued mappings) and about coincidences of two set-value mappings one of which is isotone and the other is covering.
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).
TL;DR: In this article, a multiscale approach to computing real gas flows in engineering microchannels on high-performance computer systems in a wide range of Knudsen numbers is described.
Abstract: A multiscale approach to computing real gas flows in engineering microchannels on high-performance computer systems in a wide range of Knudsen numbers is described. The numerical implementation of the approach combines the solution of quasigasdynamic equations and the molecular dynamics method. Following the approach, the parameters of the real gas equation of state are found at the molecular level, the kinetic gas properties are calculated, and the form of boundary conditions on the microchannel walls are determined. The technique is verified by computing several test problems. The results agree well with available theoretical and experimental data.
TL;DR: In this paper, a system of equations describing the motion of viscous fluids with polymer additives is considered, which involves a temperature-dependent viscosity and a velocity-dependent external force, which is used as a control.
Abstract: A system of equations describing the motion of viscous fluids with polymer additives is considered. The system involves a temperature-dependent viscosity and a velocity-dependent external force, which is used as a control. The existence of an optimal feedback control for the system is proved.
TL;DR: In this paper, an approximation based on the double period method designed for smooth aperiodic functions is proposed for cross sections of nuclear reactions important for controlled thermonuclear fusion, which makes it possible to solve broad classes of problems in a unified manner.
Abstract: In physical and engineering applications, an important task is the processing of experimental curves measured with considerable errors. Such problems are solved by applying the regularization method, in which success relies heavily on the researcher’s intuition. We propose using an approximation based on the double period method designed for smooth aperiodic functions. Regularization makes use of a Tikhonov stabilizer with the second derivative squared. As a result, the spurious oscillations are suppressed and the shape of an experimental curve is well approximated. This approach makes it possible to solve broad classes of problems in a unified manner. The method is demonstrated as applied to the approximation of cross sections of nuclear reactions important for controlled thermonuclear fusion. Tables recommended as reference data are obtained.