An estimate of the attraction domain with a specified exponential stability index in a wheeled robot control problem

The problem of synthesizing a law for the control of the plane motion of a wheeled robot is investigated. The rear wheels are the drive wheels and the front wheels are responsible for the turning of the platform. The aim of the control is to steer a target point to a specified trajectory and to stabilize the motion along it. The trajectory is assumed to be specified by a smooth curve. The actual curvature of the trajectory of the target point, which is related to the angle of rotation of the front wheels by a simple algebraic relation, is considered as the control. The control is subjected to bilateral constraints by virtue of the fact that the angle of rotation of the front wheels is limited. The attraction domain in the distance to trajectory - orientation space, is investigated for the proposed control law. Arrival at a trajectory with a specified exponential stability index is guaranteed in the case of initial conditions belonging to the given domain. An estimate of the attraction domain in the form of an ellipse is given.     

On Two-Phase Flow in a Rotating Boundary Layer

The two-phase flow induced by a rotating disk in a stationary unbounded mixture is considered. The generalized similarity assumption of von Karman reduces the averaged equations of motion with a linear drag between the phases to a system of ordinary differential equations. These are investigated by asymptotic and numerical techniques. The equations display a nontrivial behavior in a sublayer near the boundary, whose thickness is of the order of the particle size. The volume fraction of the dispersed phase is singular unless a small suction is applied on the disk or a small diffusion term is added to the continuity equations. Outside this sublayer, the velocity field is quite similar to a rescaled classical von Karman flow. Good agreement between asymptotic and numerical solution is obtained, although there is considerable stiffness in the equations. The motion of a solid particle in a von Karman flow is also discussed, but the present investigation is restricted to small radii because the shear-lift force is neglected.     

Decision-making in a fuzzy environment with fuzzy termination time

The multistage control of a deterministic and stochastic system in a fuzzy environment is considered. The fuzzy environment is meant as fuzzy constraints imposed on subsequent controls and a fuzzy goal to be attained. The fuzzy decision is assumed to be the intersection of fuzzy constraints and a fuzzy goal. The problem is to find a maximizing decision. The termination time is given as a specified fuzzy set in the space of control stages. For solving the problem, the dynamic programming is applied.     

Workforce-constrained maintenance scheduling for military aircraft fleet: a case study

The problem is related to a fleet of military aircraft with a certain flying program in which the availability of the aircraft sufficient to meet the flying program is a challenging issue. During the pre- or after-flight inspections, some component failures of the aircraft may be found. In such cases, the aircraft are sent to the repair shop to be scheduled for maintenance jobs, consisting of failure repairs or preventive maintenance tasks. The objective is to schedule the jobs in such a way that sufficient number of aircrafts is available for the next flight programs. The main resource, as well as the main constraint, in the shop is skilled-workforce. The problem is formulated as a mixed-integer mathematical programming model in which the network flow structure is used to simulate the flow of aircraft between missions, hanger and repair shop. The proposed model is solved using the classical Branch-and-Bound method and its performance is verified and analyzed in terms of a number of test problems adopted from the real data. The results empirically supported practical utility of the proposed model.     

Optimal pricing and quantity of products with two offerings

This paper analyzes the decision of a firm offering two versions of a product, a deluxe and a regular. While both products satisfy the same market, the deluxe version is sold at a high price relative to its cost and is aimed at the high end of the demand curve. The regular version is sold at a low price relative to its cost and is targeted to customers at the low end of the demand curve. This two-offering strategy is especially popular with book publishers where a paperback book is introduced some time after the hardbound version is introduced. The time between the introduction of the two versions of the product is accompanied by a downward shift in the demand curve due to customers losing interest in the product or satisfying their demand from a secondary used market. We solve a profit maximization model for a firm using a two-offering strategy. The model is solved for linear and exponential deterioration in demand, which is assumed to be deterministic. Also, a model with linear deterioration in demand, which is assumed to be stochastic, is solved. The results indicate that substantial improvements in profit can be obtained by using the two-offering strategy. Numerical sensitivity analysis and examples are used to illustrate the results.     

A group invariant solution for a pre-existing fluid-driven fracture in permeable rock

The propagation of a two-dimensional pre-existing fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is derived. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The partial differential equation admits four Lie point symmetries provided the leak-off velocity satisfies a first order linear partial differential equation. The solution of this equation yields the leak-off velocity as a function of the distance along the fracture and time. The group invariant solution is derived by considering a linear combination of the Lie point symmetries. The boundary value problem is reformulated as a pair of initial value problems. The model in which the leak-off velocity is proportional to the fracture half-width is considered. The working condition of constant pressure at the fracture entry is analysed in detail.     

The stress state of an elastic composite cone with centre of rotation at the apex

The torsion of a composite cone that has a centre of rotation at its apex is investigated in a spherical system of coordinates. A composite cone is a cone with one shear modulus, inserted into a conical funnel having another shear modulus and with ideal mechanical contact between its surface and the inner surface of the conical funnel. The auxiliary problem of a composite cone with its apex truncated by a spherical surface is considered first. The outer surface of such a conical body is not loaded, but a load that reduces to a torque is applied to its spherical surface. The auxiliary problem is reduced to a one-dimensional discontinuous boundary-value problem using a specially constructed integral transformation. The exact solution of this boundary-value problem is constructed. The limit is then taken in the solution obtained as the radius of the spherical surface tends to zero for the purpose of obtaining an exact solution of the problem of the torsion of a composite cone that has a centre of rotation at the apex.     

Ohmic heating of foods during aspetic processing

The problem of heating food by applying a large electrical currentacross it is considered. Various aspects of the problem areconsidered, with emphasis on the electrical and thermal behaviourdue to variations in electrical conductivity of the food. Suchvariations are due to variations in temperature within the foodand differences in material properties of the food. The stabilityof ohmic heating as a method used for processing is consideredfor the simple case of a homogeneous food. The heating of ahomogeneous food in a shear layer close to a wall is analysed.The case of a single spherical solid food particle with temperature-dependentconductivity surrounded by a liquid sauce is considered to modelthe behaviour of a multiphase food.     

The problem of the contact between a linear elastic body and elastic and rigid bodies (a variational approach)

The problem of the contact between a linear elastic body and a rigid body is formulated as a one-sided problem. The solution is determined from the variational inequality, equivalent to the problem of minimizing the energy functional in a set of allowable displacements. The regularity of the solution is established down to internal points of the contact boundary. A measure is constructed in the subsets of the contact boundary that enables the effect of a stamp on an elastic body to be characterized. The absolute continuity of this measure is proved at the internal point. The problem of the contact of two elastic bodies is examined in a similar formulation. The regularity of the solution is established and the nature of the effect of one body on the other is clarified.     

Perspectives sur les recherches en didactique des mathématiques

The paper is a review of chosen approaches to research in mathematics education in several countries: Germany, France, United States, Russia, Poland, Canada. The review is done in the literary form of a satire, in which a character is taken on a voyage to a variety of “islands” respresenting different research interests and methodologies in mathematics education. The story is a parody of Homer’sOdyssee, and the main character is called Odysseus. Odysseus’ role is played by the famous arithmetic problem about a team of an unknown number of scythers who are given the task of scything two meadows one of which is double the size of the other. As the problem travels from one “island” to another, mathematics educators do different things to and with the problem and it is solved is a variety of ways. The main text of the paper reads as a story and there are no explicit references and names of authors, whose work is only alluded to. However, the solution to all allusions, i.e. explicit references, can be found in the footnotes.     

The out-of-kilter algorithm applied to traffic congestion

The out-of-kilter algorithm finds a minimum cost assignment of flows to a network defined in terms of one-way arcs, each with upper and lower bounds on flow, and a cost. It is a mathematical programming algorithm which exploits the network structure of the data. The objective function, being the sum taken over all the arcs of the products, cost×flow, is linear. The algorithm is applied in a new way to minimise a series of linear functions in a heuristic method to reduce the value of a non-convex quadratic function which is a measure of traffic congestion. The coefficients in these linear functions are chosen in a way which avoids one of the pitfalls occurring when Beale's method is applied to such a quadratic function. The method does not guarantee optimality but has produced optimal results with networks small enough for an integer linear programming method to be feasible.     

A parametric method of constructing Poincaré mappings in hydrodynamic systems

The plane-parallel motion of the particles of an incompressible medium reduces to an investigation of a Hamilton system. The stream function is a Hamilton function. A Hamilton function, which depends periodically on time and corresponds to the agitation of an incompressible medium in a domain which varies periodically with time, is considered. This agitation of the medium is due to dynamic chaos. The transition to dynamic chaos is described by investigating the location of the Lagrangian particles over time intervals which are multiples of the period (Poincaré points (PP)). The set of PP can be obtained using a Poincaré mapping in the phase flow. The method which has been developed is used to investigate the plane-parallel motion of the particles in an incompressible fluid in a thin layer, bounded by a flat bottom, rectilinear side walls and an upper boundary which is deformed according to a specified periodic law. The motion of the particles is determined from Hamilton's system of equations. The Hamiltonian (the stream function) is found in the thin-layer approximation and depends on two dimensionless parameters: the amplitude of deformation and the tangential velocity in the deforming boundary. The characteristic boundary, which separates the domain of the chaotic motion of the PP from the domain of ordered motion, is determined numerically in the domain of the two parameters. The topological structure of the phase trajectories up to the transition to chaotic conditions is analysed using the Poincaré mapping, found with an accuracy up to the third order with respect to the amplitude. The phase trajectories of the PP, found analytically, turn out to be close to the trajectories of the initial Hamilton system, determined numerically. The mapping found in the domain of the two dimensionless parameters enables one to determine, qualitatively, the boundary of the transition to chaos.     

A neural network system for solving an assortment problem in the steel industry

A neural network model for solving an assortment problem found in the iron and steel industry is discussed in this paper. The problem arises in the yard where steel plate is cut into rectangular pieces. The neural network model can be categorized as a Hopfield model, but the model is expanded to handle inequality constraints. The idea of a penalty function is used. A large penalty is applied to the network if a constraint is not satisfied. The weights are updated based on the penalty values. A special term is added to the energy function of the network to guarantee the convergence of the neural network which has this feature. The performance of the neural network was evaluated by comparison with an existing expert system. The results showed that the neural network has the potential to identify in a short time near-optimal solutions to the assortment problem. The neural network is used as the core of a system for dealing with the assortment problem. In building the neural networks system for practical use, there were many implementation issues. Some of them are presented here, and the fundamental ideas are explained. The performance of the neural network system is compared to that of the expert system and evaluated from the practical viewpoint. The results show that the neural network system is useful in handling the assortment problem.     

Generalized Latin squares on the torus

The notion of a Latin square is generalized. The natural object on which to define this extension is the torus. A theorem is proved which shows that the existence of a Latin square implies the existence of a linear Latin square, a Latin square with special form. The theorems in the paper are used to provide alternate proofs of results due to Pólya and Chandra (in relation to a problem of Moser). The inability to extend the results to orthogonal Latin squares is noted.     

Diffraction of a plane wave by a narrow cone

The problem of diffraction of a plane scalar wave by a narrow cone is considered. The shape of the cone is arbitrary. The boundary condition is the Dirichlet or Neumann one. The wave scattered by the cone vertex arises as a result of the diffraction process. The subject of this paper is to calculate the wave amplitude. If the cone is narrow, it is possible to obtain simpler approximate formulas in comparison with Smyshlayev's one. The exactness of the approximate formulas is checked numerically. The etalon is a solution in explicit form in the axially symmetric case. The calculation shows that our formula is more exact in the case of the Dirichlet boundary condition than Felsen's formula. The approximate formula is a generalization of Felsen's one for circular cone to an arbitrary narrow cone in the case of the Neumann boundary condition. Bibliography: 6 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 67–74. Translated by D. B. Dement'ev.     

Inverse coefficient problem for a wave equation in a bounded domain

The nonlinear inverse problem for a wave equation is investigated in a three-dimensional bounded domain subject to the Dirichlet boundary condition. Given a family of solutions to the equation defined on a closed surface within the original domain, it is required to reconstruct the coefficient determining the velocity of sound in the medium. The solutions used for this purpose correspond to the acoustic medium perturbations localized in the neighborhood of a certain closed surface. The inverse problem is reduced to a linear integral equation of the first kind, and the uniqueness of the solution to this equation is established. Numerical results are presented.     

Evolution to a steady state for a rarefied gas flowing from a tank into a vacuum through a plane channel

A kinetic equation (S-model) is used to solve the nonstationary problem of a monatomic rarefied gas flowing from a tank of infinite capacity into a vacuum through a long plane channel. Initially, the gas is at rest and is separated from the vacuum by a barrier. The temperature of the channel walls is kept constant. The flow is found to evolve to a steady state. The time required for reaching a steady state is examined depending on the channel length and the degree of gas rarefaction. The kinetic equation is solved numerically by applying a conservative explicit finite-difference scheme that is firstorder accurate in time and second-order accurate in space. An approximate law is proposed for the asymptotic behavior of the solution at long times when the evolution to a steady state becomes a diffusion process.     

Split fields and direction of propagation for the solution to first-order systems of equations

The standard wave-splitting approach for the wave equation in inhomogeneous media is first reexamined. Next, by analogy with the theory of wave propagation through singular surfaces, a characterization is given for a function in space-time to represent a wave propagating in a direction. The condition is applied in connection with a simple example and found to be quite restrictive. The same problem is then considered in the Fourier-transform domain where the unknown function is an n-tuple satisfying a system of ordinary differential equations. The condition for propagation in a direction is established for the Fourier components. Next, some physical problems are considered which are expressed by partial differential equations or by integro-differential equations. The associated first-order system of equations is examined in terms of the eigenvalues of a matrix. This shows that, for any eigenvalue, the direction of propagation may change with the frequency and that arguments about the dominance of the principal part of the operator may cease to hold.     

A constraint generation scheme to probabilistic linear problems with an application to power system expansion planning

In this paper, we first describe a constraint generation scheme for probabilistic mixed integer programming problems. Next, we present a decomposition approach to the peak capacity expansion planning of interconnected hydrothermal generating systems, with bounds on the transmission capacity between the regions. The objective is to minimize investments in generating units and interconnection links, subject to constraints on supply reliability. The problem is formulated as a stochastic integer program. The constraint generation scheme, which is similar to Benders decomposition, is applied in the solution of the peak capacity expansion problem. The master problem in this decomposition scheme is an integer program, solved by implicit enumeration. The operating subproblem corresponds to a stochastic network flow problem, and is solved by a maximum flow algorithm and Monte Carlo simulation. The approach is illustrated through a case study involving the expansion of the system of the Brazilian Southeastern region.