卷 13, 编号 3 (2019)

The study is presented of the aircraft landing gear deformations in the axial direction of the aircraft (gear-walk) under the load from a runway. Some beam-based model is proposed to describe the elastic properties of landing gear and the oleo strut behavior. The primary verification of the model is carried out: The characteristic frequencies of oscillations are found, and the behavior of the system under harmonic load is investigated.

—We say that two edges in the hypercube are close if their endpoints form a 2-dimensional subcube. We consider the problem of constructing a 2-factor not containing close edges in the hypercube graph. For solving this problem,we use the new construction for building 2-factors which generalizes the previously known stream construction for Hamiltonian cycles in a hypercube.Owing to this construction, we create a family of 2-factors without close edges in cubes of all dimensions starting from 10, where the length of the cycles in the obtained 2-factors grows together with the dimension.

The system of equations of electrodynamics is considered for a nonmagnetic nonconducting medium. For this system, the problem is under study of determining the permittivity ε from the given modulus of the electric intensity vector of the electromagnetic field that is the result of the interference of the two fields generated by some point sources of an extraneous current. It is assumed that permittivity is different from a given positive constant ε₀ only inside a compact domain Ω₀ ⊂ ℝ³, whereas the modulus of the electric field intensity vector is given for all frequencies starting from a fixed frequency ω₀ on the boundary S of some domain Ω that includes Ω₀. It is shown that this information allows us to reduce the original problem to the well-known inverse kinematic problem of determining the refraction index inside Ω by the travel time of the electromagnetic wave between the points of the boundary of this domain. The algorithm for numerical solution of the inverse problem is constructed, and the test calculations on the simulated data are presented.

We study the problem of the linear stability of stationary plane-parallel shear flows of an inviscid stratified incompressible fluid in the gravity field between two fixed impermeable solid parallel infinite plates with respect to plane perturbations in the Boussinesq approximation and without it. For both cases, we construct some analytical examples of steady plane-parallel shear flows of an ideal density-heterogeneous incompressible fluid and small plane perturbations in the form of normal waves imposed on them, whose asymptotic behavior proves that these perturbations grow in time regardless of whether the well-known result of spectral stability theory (the Miles Theorem) is valid or not.

Under study is the polynomial orthogonal basis system of vector fields in the ball which corresponds to the Helmholtz decomposition and is divided into the three parts: potential, harmonic, and solenoidal. It is shown that the decomposition of a solenoidal vector field with respect to this basis is a poloidal-toroidal decomposition (the Mie representation). In this case, the toroidal potentials are Zernike polynomials, whereas the poloidal potentials are generalized Zernike polynomials. The polynomial system of toroidal and poloidal vector fields in a ball can be used for solving practical problems, in particular, to represent the geomagnetic field in the Earth’s core.

We consider the class of linear systems of delay differential equations with periodic coefficients. Using a special class of Lyapunov–Krasovskiĭ functionals, we establish conditions for the exponential stability of the zero solution and obtain estimates characterizing the exponential decay rate of solutions at infinity.

We consider predicates on finite sets. The predicates invariant under some (m + 1)-ary near-unanimity function are called m-junctive. We propose to represent the predicates on a finite set in generalized conjunctive normal forms (GCNFs). The properties for GCNFs of m-junctive predicates are obtained. We prove that each m-junctive predicate can be represented by a strongly consistent GCNF in which every conjunct contains at most m variables. This representation of an m-junctive predicate is called reduced. Some fast algorithm is proposed for finding a reduced representation for an m-junctive predicate. It is shown how the obtained properties of GCNFs of m-junctive predicates can be applied for constructing a fast algorithm for the generalized S-satisfiability problem in the case that S contains only the predicates invariant under a common near unanimity function.

A numerical study is carried out of the branch and cut method adapted for solving the clique partitioning problem (CPP). The problem is to find a family of pairwise disjoint cliques with minimum total weight in a complete edge-weighted graph. The two particular cases of the CPP are considered: The first is known as the aggregating binary relations problem (ABRP), and the second is the graph approximation problem (GAP). For the previously known class of facet inequalities of the polytope of the problem, the cutting-plane algorithm is developed. This algorithm includes the two new basic elements: finding a solution with given guaranteed accuracy and a local search procedure to solve the problem of inequality identification. The proposed cutting-plane algorithm is used to construct lower bounds in the branch and cut method. Some special heuristics are used to search upper bounds for the exact solution. We perform a numerical experiment on randomly generated graphs. Our method makes it possible to find an optimal solution for the previously studied cases of the ABRP and for new problems of large dimension. The GAP turns out to be a more complicated case of the CPP in the computational aspect. Moreover, some simple and difficult classes of the GAPs are identified for our algorithm.

We consider one class of systems of nonautonomous linear differential equations of neutral type with distributed delay. We obtain sufficient conditions for the exponential stability of the zero solution and conditions on perturbations of the coefficients under which the exponential stability of the zero solution is preserved. Using a Lyapunov—Kjasovskiĭ functional of a special kind, we prove some estimates that characterize the exponential decay of solutions at infinity.