Abstract The problem of optimal trajectory planning for a moving object with a nonuniform radiation pattern is considered and analytically solved as a variational problem. The object tries to evade detection by a search system consisting of a single sensor. Necessary and sufficient conditions for trajectory optimality are obtained. Analytical expressions for optimal trajectories, the velocity law,

Abstract We consider the conditions for a finite set with a given system of operations (a finite algebra) to be subject to a probability limit theorem, i.e., arbitrary computations with mutually independent random variables have value distributions that tend to a certain limit (limit law) as the number of random variables used in the computation grows. Such behavior may be seen as a generalization

Abstract The problem of entropy estimation of probability density functions with allowance for real data is posed (the maximum entropy estimation (MEE) problem). Global existence conditions for the implicit dependence of Lagrange multipliers on data collection are obtained. The asymptotic efficiency of maximum entropy estimates is proved.

Abstract Let A be a nonsingular complex (n × n) matrix. The congruence centralizer of A is the collection \(\mathcal{L}\) of matrices X satisfying the relation \(X{\kern 1pt} {\text{*}}AX = A\). The dimension of \(\mathcal{L}\)as a real variety in the matrix space \({{M}_{n}}({\mathbf{C}})\) is shown to be equal to the difference of the real dimensions of the following two sets: the conventional centralizer

Abstract We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories

Abstract We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter \(f = {{f}_{q}}(r,u)\), the local density \(\rho = \rho (r)\), and the Newtonian potential \(U = U(r)\), where \(r: = {\text{|}}x{\text{|}}\), \(u: = {\text{|}}v{\text{|}}\) (\((x,v) \in {{\mathbb{R}}^{3}} \times {{\mathbb{R}}^{3}}\) are the space–velocity

Abstract A mathematical model for viscous gas flow simulation with time scales bounded by the characteristic time between molecular collisions is considered. This approach is a variant of the quasi-gasdynamic system of equations, which is based on the relationship between the kinetic and macroscopic descriptions of continuous medium motion. Based on the presented model, a numerical algorithm is constructed

Abstract For the problem of an autonomous object moving under hostile observation, the observer’s positions are characterized in which the object following any route can choose a speed mode that allows observation evasion, and positions guaranteeing that the observer is able to track the object on the initial part of the trajectory and only on it are described.

Abstract— Terminal invariance sufficient conditions for nonlinear dynamical stochastic controllable systems of diffusion-jump type are proposed. These conditions have no analogues in the world literature. Both perturbation invariance conditions (for a fixed initial point) and absolute invariance conditions (ensuring that the terminal criterion takes a constant value for any initial data) are formulated

Abstract A local version of A.T. Fomenko’s conjecture on modeling of integrable systems by billiards is formulated. It is proved that billiard systems realize arbitrary numerical marks of Fomenko–Zieschang invariants. Thus, numerical marks are not a priori a topological obstacle to the realization of the Liouville foliation of integrable systems by billiards.

Abstract An operation generalizing convolution is introduced using the Young transform and Fenchel’s duality theorem. Based on this operation, an aggregation procedure for a nonlinear input–output model with concave positively homogeneous production functions is proposed.

Abstract Let Δn be the discriminant of a general polynomial of degree n and \(\mathcal{N}\) be the Newton polytope of Δn. We give a geometric proof of the fact that the truncations of Δn to faces of \(\mathcal{N}\) are equal to products of discriminants of lesser n degrees. The proof is based on the blow-up property of the logarithmic Gauss map for the zero set of Δn.

Abstract Interference-based logic gates serve as a basis for photonic computers intended (as distinct from quantum computers) for the same class of problems as electronic computers and making it possible to increase performance irrespective of the limitations inherent in electronic technologies. The proposed interference-based logic gates represent a complete basic functional set. They meet the requirement

Abstract A singularly perturbed initial-boundary value problem for a parabolic equation, which is called in applications an equation of Burgers type, is considered. Existence conditions are obtained, and an asymptotic approximation of a new class of solutions with a moving front is constructed. The results are applied to problems with quadratic and modular nonlinearity and nonlinear amplification.

Abstract We solve an optimal control problem for thermodynamic processes in an ideal gas. The thermodynamic state is given by a Legendrian manifold in a contact space. Pontryagin’s maximum principle is used to find an optimal trajectory (thermodynamic process) on this manifold that maximizes the work of the gas. In the case of ideal gases, it is shown that the corresponding Hamiltonian system is completely

Abstract A theorem on the zero existence preservation for a parametric family of multivalued (α, β)-search functionals on an open subset of a metric space is proved. Several corollaries on the existence preservation for preimages of a closed subspace, for coincidence points, and for common fixed points under the action of a parametric family (a number of families) of mappings are obtained. The notion

Abstract The problems of artificial intelligence, as well as control and decision-making with incomplete or inaccurate information, cover a wide class of problems of abductive explanation, including tasks in terms of cause–effect. This paper is devoted to the logical formation of hypotheses that explain observed effects. Means of representing knowledge and hypothesizing are proposed. A language is

Abstract An important stage in the design of electronic circuits of various classes and purposes is the calculation of the zero-level instability of a developed device under the influence exerted on the parameters of its components by external disturbances (variations in temperature, humidity, pressure, radiation, etc.), as well as by technological variations in these parameters in the course of serial

Abstract We obtain a complete description of fields \(\mathbb{K}\) that are quadratic extensions of \(\mathbb{Q}\) and of cubic polynomials \(f \in \mathbb{K}[x]\) for which a continued fraction expansion of \(\sqrt f \) in the field of formal power series \(\mathbb{K}((x))\) is periodic. We also prove a finiteness theorem for cubic polynomials \(f \in \mathbb{K}[x]\) with a periodic expansion of \(\sqrt

Abstract In this paper, for the first time, we provide a quasi-polynomial time approximation scheme for the well-known capacitated vehicle routing problem formulated in metric spaces of an arbitrary fixed doubling dimension.

Abstract A quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For a simplified formulation, a criterion for the existence of a global-in-time smooth solution is obtained. For the original system, a sufficient condition for singularity formation is found, and a sufficient condition for the smoothness of the

Abstract Spectral problems for the Dirac operator specified on a finite interval with regular, but not strongly regular boundary conditions and a complex-valued integrable potential are studied. This work is aimed at finding the conditions under which the root function system forms a common Riesz basis rather than a Riesz basis with parentheses.

Abstract Two results concerning the number \(P(2,n)\) of threshold functions and the singularity probability \({{\mathbb{P}}_{n}}\) of random (\(n \times n\)) \({\text{\{ }} \pm 1{\text{\} }}\)-matrices are established. The following asymptotics are obtained: \(P(2,n)\sim 2\left( {\begin{array}{*{20}{c}} {{{2}^{n}} - 1} \\ n \end{array}} \right)\quad {\text{and}}\quad {{P}_{n}}\sim {{n}^{2}} \times

Abstract A new approach to the problems of estimating the global maximum point and the integral of a continuous function on a compact set is proposed. The approach combines a simple Monte Carlo method and the ideas of the Lebesgue theory of measure and integration. Quality estimates for the proposed methods are given.

Abstract In this paper we extend Y. Eliashberg’s theorem on the maps with fold type singularities to arbitrary Thom-Boardman singularities. Namely, we state a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity ΣI at each point. In dimensions 2 and 3 we rephrase this condition

Abstract An antipodal nonbipartite distance-regular graph Γ of diameter 3 has an intersection array \(\{ k,(r - 1){{c}_{2}},\;1;\;1,\;{{c}_{2}},\;k\} \) (\({{c}_{2}} < k - 1\)) and eigenvalues k, n, –1, and –m, where n and –m are the roots of the quadratic equation \({{x}^{2}} - ({{a}_{1}} - {{c}_{2}})x - k = 0\). The Krein bound \(q_{{33}}^{3} \geqslant 0\) gives \(m \leqslant {{n}^{2}}\) if \(r \ne

Abstract For two-step free nilpotent Lie algebras, we describe symplectic foliations and Casimir functions. A left-invariant time-optimal problem is considered in which the set of admissible controls is given by a strictly convex compact set in the first layer of the Lie algebra that contains the origin in its interior. We describe integrals for the vertical subsystem of the Hamiltonian system of the

Abstract The classical Pólya–Schur inequality for the logarithmic energy of a point charge distributed on a circle is generalized to the Green energy with respect to the concentric circular ring.

Abstract Let 0 < α, σ < 1 be arbitrary fixed constants, let \({{q}_{1}} < {{q}_{2}} < \ldots < {{q}_{n}} < {{q}_{{n + 1}}}\) < … be the set of primes satisfying the condition \(\{ q_{n}^{\alpha }\} < \sigma \) and indexed in ascending order, and let \(m \geqslant 1\) be any fixed integer. Using an analogue of the Bombieri–Vinogradov theorem for the above set of primes, upper bounds are obtained for

Abstract In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite mean energy density and study the convergence of distributions of the solutions to a limiting Gaussian measure for large times. We derive the formulas for the limiting energy current density (in mean) and find a new class

Abstract The problem of seismic wave propagation from a source located in a well is considered. Acoustic equations are used to describe the dynamic behavior of the fluid. The damaged zone is described as a porous fluid-saturated medium by applying the Dorovsky model. The elastic approximation is used to describe the dynamic behavior of the surrounding rock. A unified algorithm based on the grid-characteristic

Abstract The minimax-maximin relations for vector-valued functionals over the real field are studied. An increase in the dimensionality of criteria may result in a violation of some basic relations, for example, in an inequality between maximin and minimax that is always true for classic problems. Accordingly, the conditions for its correctness or violation need to be established. This paper introduces

Abstract In layered media, the solution of Maxwell’s equations suffers a strong or weak discontinuity at the layer boundaries. Finite-difference schemes providing convergence on strong discontinuities have been proposed for the first time. These are conservative bicompact two-point schemes with mesh nodes lying on the layer boundaries. A fundamentally new technique for taking into account the medium

Abstract We propose a new way of justifying the accelerated gradient sliding of G. Lan, which allows one to extend the sliding technique to a combination of an accelerated gradient method with an accelerated variance reduction method. New optimal estimates for the solution of the problem of minimizing a sum of smooth strongly convex functions with a smooth regularizer are obtained.

Abstract The concept of stability of continuous extension of mappings with respect to a Nemytskii superposition operator is studied. Sufficient conditions of such stability with respect to a Nemytskii superposition operator are obtained. The essentiality of the corresponding assumptions is illustrated by examples.

Abstract An approach is proposed to consider heuristic metrics introduced and used in data analysis problems. In the approach, the entire information on pairwise distances expressed by numerical values is reduced to information on a metric belonging as a point of a semi-metric cone to corresponding subcones, which are elements of factor sets for proposed relations of kernel equivalences for mappings

Abstract The central-difference Nessyahu–Tadmor (NT) scheme is considered, which is built using second-order MUSCL reconstruction of fluxes. The accuracy of the NT scheme is studied as applied to calculating shock waves propagating with a variable velocity. It is shown that this scheme has the first order of integral convergence on intervals with one of the boundaries lying in the region of influence

Abstract According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrödinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized

Abstract The goal of this paper is spectral analysis of an evolutionary semigroup generator describing the dynamics of a rarefied two-component plasma subjected to a self-consistent electromagnetic field. For the problem in question, the spectrum is given in terms of the dispersion relationship and an effective approach to the calculation of the instability index is developed.

Abstract We study an explicit two-level symmetric (in space) finite-difference scheme for the multidimensional barotropic gas dynamics system of equations with quasi-gasdynamic regularization linearized at a constant solution (with an arbitrary velocity). A criterion and both necessary and sufficient conditions for the L2-dissipativity of the solutions to the Cauchy problem for the scheme are derived

AbstractAn infinite finitely presented nilsemigroup with identity x9 = 0 is constructed. This construction answers the question of L.N. Shevrin and M.V. Sapir. The proof is based on the construction of a sequence of geometric complexes, each obtained by gluing several simple 4-cycles (squares). These complexes have certain geometric and combinatorial properties. Actually, the semigroup is the set of

AbstractA strange term arising in the homogenization of elliptic (and parabolic) equations with dynamic boundary conditions given on some boundary parts of critical size is considered. A problem with dynamic boundary conditions given on the union of some boundary subsets of critical size arranged ε-periodically along the boundary and with homogeneous Neumann conditions given on the rest of the boundary

AbstractIn this paper we discuss a number of rigorous results in the stochastic model for wave turbulence due to Zakharov–L’vov. Namely, we consider the damped/driven (modified) cubic nonlinear Schrödinger equation on a large torus and decompose its solutions to formal series in the amplitude. We show that when the amplitude goes to zero and the torus’ size goes to infinity the energy spectrum of the

AbstractFor problems related to radiative heat conduction, an algorithm is proposed that is well adapted to the architecture of systems with extramassive parallelism. According to the underlying method, a term with a small parameter multiplying the second time derivative is included in the model describing the process. Examples of numerical results obtained using this model on detailed spatial meshes

AbstractFor an autonomous model of the motion of a nonlinear viscous fluid, we study the limiting behavior of its weak solutions as time tends to infinity. Namely, the existence of weak solutions on the positive half-axis is established, the trajectory space corresponding to the solutions of this model is determined, and the existence of a minimum trajectory attractor and, then, a global attractor

AbstractInverse problems of fracture exploration seismology are solved using machine learning methods. A single fracture of fixed size and subvertical orientation is considered in the two-dimensional case. The spatial position and the inclination angle of the fracture are determined using a neural network. The training set consists of solutions of direct problems produced by the grid-characteristic

AbstractA one-parameter family of global solutions of a two-dimensional hyperbolic differential-difference equation with an operator acting with respect to a space variable is constructed. A theorem is proved stating that the resulting solutions are classical for all parameter values if the symbol of the difference operator of the equation has a positive real part. Classes of equations for which this

AbstractA theorem about the behavior of Cauchy-type integrals at the endpoints of the integration contour and at discontinuity points of the density is stated, and its application to boundary value problems for 2n-order parabolic equations with a changing direction of time are described. The theory of singular equations, along with the smoothness of the initial data, makes it possible to specify necessary

AbstractThe existence of a weak solution for the Navier–Stokes-alpha model with temperature-dependent viscosity is proved using the topological approximation method for the study of hydrodynamic problems.

AbstractFor the numerical solution of the Cauchy problem with multiple poles, we propose a reciprocal function method. In the case of first-order poles, it makes it possible to continue the solution through the poles and to determine the solution and the pole positions with good accuracy. The method allows one to employ conventional explicit and implicit schemes, for example, explicit Runge–Kutta schemes

AbstractWe consider C1-functions defined on two-step Carnot groups with a sub-Lorentzian structure defined by one horizontal direction with a negative squared length along it, and prove a nonholonomic coarea formula. A result of interest in itself concerns the correctness of the problem statement, namely, the level sets have to be spacelike.

AbstractWe study a two-dimensional vector logarithmic-potential equilibrium problem with the Nikishin matrix of interaction. A constructive method for finding the supports of a vector equilibrium measure is given. The densities of the components of the equilibrium measure are expressed in terms of an algebraic function that is explicitly written out. The problem is motivated by the study of the convergence

AbstractGiven a sum of a finite number of independent random variables, the asymptotic behavior of its distributions and densities at infinity is investigated in the case when the densities or tails of these distributions decrease faster than the densities or tails of gamma distributions.

AbstractResult on the uniform, over the entire real line, equiconvergence of spectral expansions related to the self-adjoint extension of a general differential operation of any even order with coefficients from the one-dimensional Kato class, with the Fourier integral expansion is presented. The statement is based on the obtained uniform estimates for the spectral function of this operator.

AbstractFollowing V.I. Arnold, we define the stochasticity parameter S(U) of a set \(U \subseteq {{\mathbb{Z}}_{M}}\) to be the sum of squares of consecutive distances between the elements of U. The stochasticity parameter of the set RM of quadratic residues modulo M is studied. We compare \(S({{R}_{M}})\) with the average value \(s(k) = s(k,M)\) of S(U) over all subsets of \(U \subseteq {{\mathbb{Z}}_{M}}\)

AbstractThe evolution of a system of mutually gravitating particles is considered taking into account the energy loss in collisions. Collisions can be described in various ways. One can use the theory of inelastic impact of solids with Newton’s recovery coefficient for the relative velocity of bouncing particles. In numerical implementation, the main difficulty of this approach is to track and refine

AbstractTwo approaches are suggested for constructing a probabilistic approximation of the evolution operator e–itH, where \(H = \tfrac{{{{{( - 1)}}^{m}}}}{{(2m)!}}\tfrac{{{{d}^{{2m}}}}}{{d{{x}^{{2m}}}}}\), in the strong operator topology. In the first approach, the approximating operators have the form of expectations of functionals of a certain Poisson point field, while, in the second approach,

AbstractThe paper presents a plane regular fractal Peano curve with a Euclidean square-to-line ratio (L2-locality) of \(5\frac{{43}}{{73}}\), which is minimal among all known curves of this class. The presented curve has a fractal genus of 25. Performed calculations allow us to state that all the other regular curves with a fractal genus not exceeding 36 have a strictly greater square-to-line ratio

AbstractThis paper summarizes the results of studies of the laminar–turbulent transition in some fluid and gas dynamics problems obtained by applying numerical methods and methods of chaotic dynamics. The following problems are analyzed: 2D and 3D Kolmogorov problems in a periodic domain, 3D Rayleigh–Benard convection in rectangular domains, 3D backward-facing step flow, and development of 3D Rayleigh–Taylor

AbstractThis paper shows the integrability of certain classes of odd-order dynamical systems that are homogeneous with respect to some of the variables and in which a system on the tangent bundle of smooth manifolds is distinguished. In this case, the force fields have dissipation of different signs and generalize previously considered cases.