Vol 11, No 4 (2017)
- Year: 2017
- Articles: 19
- URL: https://journals.rcsi.science/1990-4789/issue/view/13240
Article
A discrete algorithm for localizing the discontinuity lines of a function of two variables
Abstract
We consider an ill-posed problem of localizing the discontinuity lines of a function of two variables. It is assumed that, instead of a precisely given function f, the values are available of the averages on the square of the perturbed function fδ at the points of a uniform grid as well as the error level δ so that \({\left\| {f - {f^\delta }} \right\|_{{L_2}}}{(_\mathbb{R}}^2)\) ≤ δ. An algorithm is constructed for localizing the discontinuity lines, its convergence is proved with the estimates of the approximation accuracy, which coincide in the order of magnitude with the estimates obtained earlier by the authors for the case when, instead of the average values of the function fδ, the function itself is given. Also, we substantiate the estimates for an important characteristic of localization methods, i.e. separability threshold.
An upper bound for the competitive location and capacity choice problem with multiple demand scenarios
Abstract
A new mathematical model is considered related to competitive location problems where two competing parties, the Leader and the Follower, successively open their facilities and try to win customers. In the model, we consider a situation of several alternative demand scenarios which differ by the composition of customers and their preferences.We assume that the costs of opening a facility depend on its capacity; therefore, the Leader, making decisions on the placement of facilities, must determine their capacities taking into account all possible demand scenarios and the response of the Follower. For the bilevel model suggested, a problem of finding an optimistic optimal solution is formulated. We show that this problem can be represented as a problem of maximizing a pseudo- Boolean function with the number of variables equal to the number of possible locations of the Leader’s facilities.We propose a novel systemof estimating the subsets that allows us to supplement the estimating problems, used to calculate the upper bounds for the constructed pseudo-Boolean function, with additional constraints which improve the upper bounds.
Lattice complete graphs
Abstract
We study the properties of graphs that can be placed in a rectangular lattice so that all vertices located in the same (horizontal or vertical) row be adjacent. Some criterion is formulated for an arbitrary graph to be in the specified class.
Numerical study of a two-dimensional mathematical model with variable heat exchange coefficient which arises in cryosurgery
Abstract
We formulate and numerically solve a two-dimensional boundary value problem of Stefan type with nonlinear heat sources of a special kind and a variable heat exchange coefficient. The model under study arises in cryosurgery in the process of freezing some living biological tissue by a cryoinstrument of cylindrical shape placed on the surface of the tissue. The model takes into account the actually observed effect of spatial localization of heat. Some results of the computer simulation are presented.
On computational complexity of the electric power flow optimization problem in market environment
Abstract
Under consideration is the electric power flow optimization problem for an electric power system which typically arises in calculation of electrical power auctions in the “day-ahead” and balancing markets. It was established that the problem of finding a feasible flow in the balancing market is NP-hard in the strong sense even in case of one generator. The problem of finding an optimal flow in the day-ahead market is proved to be NP-hard even with one generator and without controlled cuts.
Determination of heat transfer properties of media with a single-needle probe
Abstract
Needle probes with a line heater inside are often used in studying the heat transfer properties of loose rocks. The key problem of contact methods of measuring thermal properties of various media consists in finding thermal contact resistance at the probe/medium interface which must be taken into account in determining the thermal diffusivity of the medium. We describe a mathematical model of heating of a long needle probe in the medium under study, taking into account dimensions and thermal properties of the needle source and assuming that thermal contact between the source and the medium is not ideal. Based on the proposed model, we formulate and solve the inverse problem of finding the thermal diffusivity coefficient of the medium and the heat exchange coefficient at the probe/medium interface. The purpose of the article is to create methodology for determining thermal properties of various media in the field.
On (1,l)-coloring of incidentors of multigraphs
Abstract
It is proved that if l is at least Δ/2 − 1 then (1, l)-chromatic number of an arbitrary multigraph of maximum degree Δ is at most Δ+1. Moreover, it is proved that the incidentors of every directed prism can be colored in four colors so that every two adjacent incidentors are colored distinctly and the difference between the colors of the final and initial incidentors of each arc is 1.
Invariant operators and separation of residual stresses
Abstract
We consider the equations of linear theory of elasticity in stresses for the threedimensional space. Solutions are decomposed into sums of stationary solutions not satisfying the compatibility condition (residual stresses) and nonstationary solutions satisfying the compatibility condition and hence represented through the displacements. The construction of this decomposition is reduced to solving a series of Poisson equations.
Determination of the possibility of rock burst in a coal seam
Abstract
The article presents an algorithm for computing a quantity that serves as a criterion for the possibility of rock burst in a coal seam.We propose to seek this quantity in the two steps: At the first step, an inverse problem is solved to find the necessary quantities; while at the second step, we solve a boundary value problem for the biharmonic equation. The inverse problem can be solved by minimizing an objective functional that is shown to be strongly convex.
Using Lagrange multipliers in the triangular element of a nonshallow shell under variable interpolation of displacements
Abstract
We present an algorithmfor calculation of arbitrary thin shells on the basis of a triangular element of discretization with corrective Lagrange multipliers. The stiffness matrix of this element is formed using some ways of approximating displacements as scalar or vector quantities.
On the operations of bounded suffix summation and multiplication
Abstract
The operations of bounded suffix summation and bounded suffix multiplication are introduced. Using these operations, we define the class BSSM of polynomially computable functions. It is proved that the class BSSMcontains the class BPC defined by the operation of bounded prefix concatenation and has finite basis under superposition.
The network equilibrium problem with mixed demand
Abstract
We formulate the network equilibrium problem with mixed demand which generalizes the problems of network equilibrium with fixed and elastic demand. We prove the equilibrium conditions for this problem and propose some conditions of existence of a solution that are based on the coercivity property.We establish a connection between the problem of network equilibrium with mixed demand and the problem of auction equilibrium. The results of test calculations are presented for a model example.
On facet-inducing inequalities for combinatorial polytopes
Abstract
One of the central questions of polyhedral combinatorics is the question of the algorithmic relationship between the vertex and facet descriptions of convex polytopes. From the standpoint of combinatorial optimization, the main reason for the actuality of this question is the possibility of applying the methods of convex analysis to solving the extremal combinatorial problems. In this paper, we consider the combinatorial polytopes of a sufficiently general form. We obtain a few of necessary conditions and a sufficient condition for a supporting inequality of a polytope to be a facet inequality and give an illustration of the use of the developed technology to the polytope of some graph approximation problem.
Auto-oscillations during deceleration of a rigid body in a resisting medium
Abstract
Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.
An exact algorithm for finding a vector subset with the longest sum
Abstract
We consider the problem: Given a set of n vectors in the d-dimensional Euclidean space, find a subsetmaximizing the length of the sum vector.We propose an algorithm that finds an optimal solution to this problem in time O(nd−1(d + logn)). In particular, if the input vectors lie in a plane then the problem is solvable in almost linear time.
Geometric support in the optimization problem for the surface of the hydroturbine draft tube based on numerical simulation of the flow
Abstract
Some methods are developed for generating a set of surfaces of the smooth hydroturbine draft tube depending on the nine geometric parameters, and also for constructing meshes in the region of the draft tube for the subsequent optimization of its shape based on numerical simulation of flows.
Vibrations of a conductive string in a nonstationary magnetic field under presence of two nonlinear factors
Abstract
We consider vibrations of a conductive string with fixed ends in a magnetic field. Induction of the magnetic field is a preassigned function of time. Two nonlinear factors are taken into account simultaneously: the variation of string tension with displacement and the magnetostrictive effect. It is shown that, in the case of a periodic magnetic field, the nonlinear factors can compensate each other and the problem can be reduced to studying the linearized parametric vibrations.
Stability of the cylindrical flame front in an annular combustion chamber
Abstract
Using small perturbations, within the framework of phenomenological theory of mixture combustion we study stability of the cylindrical front of deflagration combustion in an annular combustion chamber. The flame front is described as a discontinuity of gasdynamic parameters. It is discovered that the flame front is unstable for some types of small perturbations of the mainstream flow of the fuel mixture and the flame front. The mechanics of instability is examined using both numerical and analytical methods. The cases are presented of evolution of the instabilities rotating in the annular channel.