Vol 12, No 4 (2018)

Under study is the stability of the inverted pendulum motion whose suspension point vibrates according to a sinusoidal law along a straight line having a small angle with the vertical. Formulating and using the contracting mapping principle and the criterion of asymptotic stability in terms of solvability of a special boundary value problem for the Lyapunov differential equation, we prove that the pendulum performs stable periodic movements under sufficiently small amplitude of oscillations of the suspension point and sufficiently high frequency of oscillations.

The transmission of a vertex v in a graph is the sum of the distances from v to all other vertices of the graph. In a transmission irregular graph, the transmissions of all vertices are pairwise distinct. It is known that almost all graphs are not transmission irregular. Some infinite family of transmission irregular trees was constructed by Alizadeh and Klavžar [Appl.Math. Comput. 328, 113–118 (2018)] and the following problemwas formulated: Is there an infinite family of 2-connected graphs with the property? In this article, we construct an infinite family of 2-connected transmission irregular graphs.

Under study is a bilevel stochastic linear programming problem with quantile criterion. Bilevel programming problems can be considered as formalization of the process of interaction between two parties. The first party is a Leader making a decision first; the second is a Follower making a decision knowing the Leader’s strategy and the realization of the random parameters. It is assumed that the Follower’s problem is linear if the realization of the random parameters and the Leader’s strategy are given. The aim of the Leader is the minimization of the quantile function of a loss function that depends on his own strategy and the optimal Follower’s strategy. It is shown that the Follower’s problem has a unique solution with probability 1 if the distribution of the random parameters is absolutely continuous. The lower-semicontinuity of the loss function is proved and some conditions are obtained of the solvability of the problem under consideration. Some example shows that the continuity of the quantile function cannot be provided. The sample average approximation of the problem is formulated. The conditions are given to provide that, as the sample size increases, the sample average approximation converges to the original problem with respect to the strategy and the objective value. It is shown that the convergence conditions hold for almost all values of the reliability level. A model example is given of determining the tax rate, and the numerical experiments are executed for this example.

The positive closure operator is defined on using the logical formulas containing the logical connectives ∨, & and the quantifier ∃. Extensions of the positive closure operator are considered by using arbitrary (and not necessarily binary) logical connectives. It is proved that each proper extension of the positive closure operator by using local connectives gives either an operator with a full systemof logical connectives or an implication closure operator (extension by using logical implication). For the implication closure operator, the description of all closed classes is found in terms of endomorphism semigroups.

We consider binomial functions over a finite field of order 2ⁿ. Some necessary condition is found for such a binomial function to be a permutation. It is proved that there are no permutation binomial functions in the case that 2ⁿ − 1 is prime. Permutation binomial functions are constructed in the case when n is composite and found for n ≥ 8.

Some regularization algorithm is proposed related to the problem of continuation of the wave field from the planar boundary into the half-plane. We consider a hyperbolic equation whose main part coincideswith the wave operator, whereas the lowest term contains a coefficient depending on the two spatial variables. The regularization algorithm is based on the quasi-reversibility method proposed by Lattes and Lions. We consider the solution of an auxiliary regularizing equation with a small parameter; the existence, the uniqueness, and the stability of the solution in the Cauchy data are proved. The convergence is substantiated of this solution to the exact solution as the small parameter vanishes. A solution of an auxiliary problem is constructed with the Cauchy data having some error. It is proved that, for a suitable choice of a small parameter, the approximate solution converges to the exact solution.

Under study is the wave propagation process in the half-space y₃ = 0 with the Cartesian coordinates y₁, y₂, and y₃ which is filled with an elastic medium. The parameters of the medium are discontinuous and depend only on the coordinate y₃. The wave process is induced by an external perturbation source that generates a plane wave moving from the domain y₃ > h > 0. It is proved that the direct dynamic problem is uniquely solvable in the corresponding function space, and a special presentation is found for the solution. The problem of determination of the acoustic impedance of the medium from the wave field measurements on the surface is investigated by the spectral methods of the theory of differential equations.

The 3-coloring problem for a given graph consists in verifying whether it is possible to divide the vertex set of the graph into three subsets of pairwise nonadjacent vertices. A complete complexity classification is known for this problem for the hereditary classes defined by triples of forbidden induced subgraphs, each on at most 5 vertices. In this article, the quadruples of forbidden induced subgraphs is under consideration, each on atmost 5 vertices. For all but three corresponding hereditary classes, the computational status of the 3-coloring problem is determined. Considering two of the remaining three classes, we prove their polynomial equivalence and polynomial reducibility to the third class.

Under consideration are the numerical methods for simulation of a fluid flow in fractured porous media. The fractures are taken into account explicitly by using a discrete fracture model. The formulated single-phase filtering problem is approximated by an implicit finite element method on unstructured grids that resolve fractures at the grid level. The systems of linear algebraic equations (SLAE) are solved by the iterative methods of domain decomposition in the Krylov subspaces using the KRYLOVlibrary of parallel algorithms. The results of solving some model problem are presented. A study is conducted of the efficiency of the computational implementation for various values of contrast coefficients which significantly affect the condition number and the number of iterations required for convergence of the method.