Volume 13, Nº 1 (2019)

This is theoretical study of the ill-posed problem on localization (determination of position) of discontinuities of the first kind of a function of one variable. The exact function x is smooth but has finitely many discontinuities of the first kind. Given some approximate function x^δ, ||x^δ − x|| L₂(ℝ) ≤ δ, and the error level δ, it is required to determine the number of discontinuities and approximate their location with an estimate of the approximation accuracy. Regular localization methods are constructed on the basis of averages that are scaled by the regularization parameter. The investigation of these methods consists in carrying out estimates for their three main characteristics on the classes of correctness: accuracy of localization, separability, and observability. Under consideration is the general formulation of the problem that generalizes the previously obtained results. The necessary conditions are obtained that must be satisfied by the accuracy of localization, separability, and observability. Also, the sufficient conditions close to the necessary are found, under which a localization method is constructed with the specified accuracy, observability, and separability. The concept of optimality of the localization methods is introduced in terms of the order of accuracy, separability, and observability (in the whole) and the methods are constructed that are optimal in order in the whole.

Under consideration are the multicriteria integer linear programming problems with finitely many feasible solutions. The problem itself consists in finding a set of extremal solutions. We derive some lower and upper bounds for the T₁-stability radius under assumption that arbitrary Hölder norms are given in the solution and criteria spaces. A class of the problems with an infinitely large stability radius is specified. We also consider the case of the multicriteria linear Boolean problem.

An algorithm is described for determining the optimal and the entire set of suboptimal trajectories of development of technical and economic systems. The dynamics of the possible development of a system is considered as a directed graph whose nodes characterize the possible system states in the future time intervals, while the arcs represent all possible transitions from one state to another during given time intervals. The algorithm is based on the dynamic programming principles. It is applied in the software package “Dynamics” that realizes the methods of combinatorial modeling to study the long-term options for the development of energy systems.

Under study is a three-level pricing problem formulated as a Stackelberg game in which the two companies, the Leader and the Follower, compete with each other for customers demand by setting prices for homogeneous products on their facilities. The first decision is made by the Leader. Then, having full information about the Leader’s choice, the Follower makes his own decision.After that each customer chooses the facilitywithminimal service costs to be serviced from. The Leader and the Follower use different pricing strategies: uniform and mill pricing respectively. We study the behavior of company revenues depending on the number of facilities. For this, an exact decomposition type algorithm is proposed. Moreover, we developed a hybrid approximation algorithm that is based on the variable neighborhood descent and coordinate descent.

Under study is some two-dimensional model describing equilibriumof a composite solid with a thin rigid inclusion and a crack. A boundary condition of Signorini’s type is prescribed on the crack curve. For a family of corresponding variational problems, the dependence is analyzed of their solutions on the parameter characterizing the location of the rigid inclusion. The existence of solution of the optimal control problem is proved. For this problem, the quality functional is defined with the help of an arbitrary continuous functional on the solution space, while the location of the inclusion is chosen as the control parameter.

Under consideration is a multiple well platform where the metal pipes of various diameters that are located above and below ground act as interferences together with boreholes with metal tubing. The sought-for objects of study are the thawing zones located near the boreholes. Numerical calculations of direct problems are performed by the nodal finite element method implemented in the author’s software package for solving the problems of electrical resistivity tomography.

It is established that we can implement almost every Boolean function on n variables by a logic network in the basis {x&y, x ∨ y, x ⨁ y, 1}, allowing a complete fault detection test with length at most 4 under arbitrary stuck-at faults at outputs of gates. The following assertions are also proved:We can implement each Boolean function on n variables by a logic network in the basis {x&y, x ∨ y, x ⨁ y, 1} (in the basis {x&y, x ∨ y, x ∨ y, x ⨁ y}) containing at most one dummy variable and allowing a complete fault detection test of length at most 5 (at most 4, respectively) under faults of the same type.

We study the problem of finding a fixed point for a special class of piecewise-constant mappings of a simplex into itself which arise in connection with the search for equilibrium prices in the classical exchange model and its various versions. The consideration is based on the polyhedral complementarity which is a natural generalization of linear complementarity. Here we study the mappings arising from models with fixed budgets. Mappings of this class possess a special property of monotonicity (logarithmic monotonicity), which makes it possible to prove that they are potential. We show that the problem of finding fixed points of these mappings is reducible to optimization problems for which it is possible to propose finite suboptimization algorithms.We give description of two algorithms.

The second Riddell relation relates the generating functions for the number of labeled connected graphs and the number of labeled blocks. We consider the conditions under which this relation is true for a subclass of connected graphs.Under these conditions, the formulas are valid that express the number of graphs from a subclass of labeled connected graphs trough the generating function of their blocks. By way of application, we obtain expressions for the numbers of labeled connected and 2-connected series-parallel graphs.