Vol 10, No 3 (2016)

The results are presented of computer simulation of the operation of a three-layer perceptron trained for solving inverse problems of anomalous diffusion theory. Several types of inverse problems are considered, including the problem of determining the Hurst exponent of a selfsimilar medium.

Under study is an equilibrium problem for a plate under the influence of external forces. The plate is assumed to have a thin rigid inclusion that reaches the boundary at the zero angle and partially contacts a rigid body. On the inclusion face, there is a delamination. We consider the complete Kirchhoff–Love model, where the unknown functions are the vertical and horizontal displacements of the middle surface points of the plate. We present differential and variational formulations of the problem and prove the existence and uniqueness of a solution.

We consider the strongly NP-hard problem of partitioning a set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sum of the squared intracluster distances from the elements of the clusters to their centers. The weights of sums are the sizes of the clusters. The center of one cluster is given as input, while the center of the other cluster is unknown and determined as the average value over all points in the cluster (as the geometric center). Two variants of the problems are analyzed in which the cluster sizes are either given or unknown. We present and prove some exact pseudopolynomial algorithms in the case of integer components of the input points and fixed space dimension.

This paper deals with the following question: Can combinatorial properties of polytopes help in finding an estimate for the complexity of the corresponding optimization problem? Sometimes, these key characteristics of complexity were the number of hyperfaces of the polytope, diameter and clique number of the graph of the polytope, the rectangle covering number of the vertex-facet incidence matrix, and some other characteristics. In this paper, we provide several families of polytopes for which the above-mentioned characteristics differ significantly from the real computational complexity of the corresponding optimization problems. In particular, we give two examples of discrete optimization problem whose polytopes are combinatorially equivalent and they have the same lengths of the binary representation of the coordinates of the polytope vertices. Nevertheless, the first problem is solvable in polynomial time, while the second problem has exponential complexity.

The inverse problems are considered for the equation of longitudinal wave propagation under the overdetermination conditions of the final and integral types. The main purpose of the study is to prove the existence of regular solutions of inverse problems of recovering the unknown external sources in addition to the solution. One of the suggested approaches bases on reducing the inverse problem to an integrodifferential equation.

Under consideration is the problem of contact of a viscoelastic plate with an elastic beam. To characterize the viscoelastic deformation of the plate, the hereditary integrals are used. The differential formulation of the problem with the conditions in the form of a system of equalities and inequalities in the domain of possible contact is presented, and its equivalence to a variational inequality is proved. The unique solvability of the problem is proved as well as the existence of the time derivative of the solution. A limit problem is also considered as the bending rigidity of the plate tends to infinity.

The article deals with a flow in the tube of rectangular cross-section with an inner circular cylindrical element. We use the numerical method basing on R-functions. This method is meshfree and therefore more efficient than the finite element method that requires remeshing when the geometry of problem is changed. The dependence of the flow on the diameter of the central cylinder and its position in the duct is investigated under constant pressure gradient. It was found that the resistance decreases if the inner element is moved from the center of the duct.

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.