We introduce the concept of a pseudoconvex set in an odd-dimensional Euclidean space. The inversion formula is obtained for the Radon transform in the case where the integrand is a piecewise continuous function defined on a pseudoconvex set. The result achieved is a generalization of a previously known property proved for smooth functions.
The problem of threshold stability for a bilevel problem with a median type of facility location and discriminatory pricing is considered. When solving such a problem, it is necessary to find the threshold stability radius and a semifeasible solution of the original bilevel problem such that the leader’s revenue is not less than a predetermined value (threshold) for any deviation of budgets that does not exceed the threshold stability radius and which preserves its semifeasibility. Thus, the threshold stability radius determines the limit of disturbances of consumer budgets with which these conditions are satisfied.
Two approximate algorithms for solving the threshold stability problem based on the heuristic of descent with alternating neighborhoods are developed. These algorithms are based on finding a good approximate location of facilities as well as on calculating the optimal set of prices for the found location of facilities. The algorithms differ in the way they compare various locations of facilities; this ultimately leads to different estimates of threshold stability radius. A numerical experiment has shown the efficiency of the chosen approach both in terms of the running time of the algorithms and the quality of the solutions obtained.
A submatrix hypergraph ( G_{ntimes m} ) is a hypergraph whose vertices are entries of an ( ntimes m ) matrix and hyperedges are submatrices of order ( 2 ). In this paper, we consider perfect colorings of submatrix hypergraphs and study their parameters. We provide several constructions of perfect colorings of ( G_{ntimes m} ) and prove that the incidence matrices of ( 2 )-designs are perfect colorings of the submatrix hypergraph. Moreover, we describe all perfect 2-colorings of hypergraphs ( G_{2times m} ) and ( G_{3times m} ).
The article presents the statement of inverse problems of finding the right-hand side of the heat equation from an additional integral condition and justifies their Hadamard well-posedness in the class of regular solutions. The uniqueness of solutions of the problems is proved on the basis of integral identities. The solutions of the problems are constructed explicitly using separation of variables and the integral equation method.
The dynamics of axisymmetric two-dimensional strains in a linear elastic half-space bounded by a smooth surface of revolution with positive Gaussian curvature is considered. An approximate solution of the initial–boundary value problem is constructed on the basis of ray series with expansion in a time-like variable. The limited number of terms of the ray series is used for near-front domains of curvilinear waves of strong discontinuities. The coefficients of this series are the discontinuities of the derivatives of displacements with respect to time (starting from the first derivative). It is shown that it is necessary to take into account the ray series components up to the ( (k + 1) )st order inclusive at the ( k )th step of the ray method for a two-dimensional type of the deformation process.
The boundary control method is applied to the solution of one-dimensional discrete inverse problems. The discrete counterparts of the operators used in the method (the control, response, and connecting operators) are defined. The relations between the operators corresponding to the discrete wave equation and the discrete heat equation are established.
Lattice-based cryptosystems are one of the main post-quantum alternatives to asymmetric cryptography currently in use. Most attacks on these cryptosystems can be reduced to the shortest vector problem (SVP) in a lattice. Previously, the authors proposed a quantum oracle model from Grover’s algorithm to implement a hybrid quantum-classical algorithm based on the GaussSieve algorithm and solving SVP. In this paper, a new model of a quantum oracle is proposed and analyzed. Two implementations of the new quantum oracle model are proposed and estimated. The complexity of implementing the new quantum oracle model to attack post-quantum lattice-based cryptosystems that are finalists of the NIST post-quantum cryptography competition is analyzed. Comparison of obtained results for new and existing models of quantum oracle is given.
The article is devoted to the development of a numerical algorithm for finding an approximate solution of an optimal control problem with terminal constraints and control constraints. The algorithm is based on the reduction of the original optimal control problem to a finite-dimensional problem and the use of the penalty method and the differential evolution method to solve the latter. A feature of the proposed approach is that the solution found is independent of the choice of the initial approximation. The operation of the algorithm is illustrated by its application to applied optimal control problems. The results of computational experiments are consistent with the results of calculations based on other methods.
The three-dimensional flow of a system of a viscous heat-conducting fluid and a binary mixture with a common interface in a layer bounded by solid walls is studied. A radial time-varying temperature distribution is specified on the lower substrate; the upper wall is assumed to be thermally insulated. Assuming a small Marangoni number, the structure of a steady-state flow is described depending on the layer thickness ratio and taking into account the influence of mass forces. The solution of the nonstationary problem is determined in Laplace transforms by quadratures. It is shown that if the given temperature on the lower substrate stabilizes over time, then with increasing time the solution reaches the resulting steady-state mode only under certain conditions on the initial distribution of concentrations in the mixture.
If a system of differential equations admits a continuous transformation group, then, in some cases, the system can be represented as a combination of two systems of differential equations. These systems, as a rule, are of smaller order than the original one. This information pertains to the linear equations of elasticity theory. The first system is automorphic and is characterized by the fact that all of its solutions are obtained from a single solution using transformations in this group. The second system is resolving, with its solutions passing into themselves under the group action. The resolving system carries basic information about the original system. The present paper studies the automorphic and resolving systems of two- and three-dimensional time-invariant elasticity equations, which are systems of first-order differential equations. We have constructed infinite series of conservation laws for the resolving systems and automorphic systems. There exist infinitely many such laws, since the systems of elasticity equations under consideration are linear. Infinite series of linear conservation laws with respect to the first derivatives are constructed in this article. It is these laws that permit solving the first boundary value problem for the equations of elasticity theory in the two- and three-dimensional cases. The solutions are constructed by quadratures, which are calculated along the boundary of the studied domains.
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