Elastic state of a plate with a circular plug and a rectilinear thin elastic inclusion

There is considered the problem of the state of stress of an infinite elastic plane with a bonded circular plug and an arbitrarily located thin elastic inclusion under biaxial tension. Conditions of ideal mechanical contact are satisfied on the line separating the materials. By using the complex Kolosov — Muskhelishvili potentials, the problem is reduced to a system of integro-differential equations which is solved numerically by utilization of a mechanical quadrature method. A numerical analysis is given for the solution of the problem of the elastic equilibrium of a plane with a circular hole and an arbitrarily located thin inclusion.     

A multidimensional tropical optimization problem with a non-linear objective function and linear constraints

We examine a multidimensional optimization problem in the tropical mathematics setting. The problem involves the minimization of a non-linear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear inequality constraints. We start with an overview of known tropical optimization problems with linear and non-linear objective functions. A short introduction to tropical algebra is provided to offer a formal framework for solving the problem under study. As a preliminary result, a solution to a linear inequality with an arbitrary matrix is presented. We describe an example optimization problem drawn from project scheduling and then offer a general representation of the problem. To solve the problem, we introduce an additional variable and reduce the problem to the solving of a linear inequality, in which the variable plays the role of a parameter. A necessary and sufficient condition for the inequality to hold is used to evaluate the parameter, whereas the solution to the inequality is considered a solution to the problem. Based on this approach, a complete direct solution in a compact vector form is derived for the optimization problem under fairly general conditions. Numerical and graphical examples for two-dimensional problems are given to illustrate the obtained results.     

Nonlocal diffusion,a Mittag-Leffler function and a two-dimensional Volterra integral equation

In this paper we consider a particular class of two-dimensional singular Volterra integral equations. Firstly we show that these integral equations can indeed arise in practice by considering a diffusion problem with an output flux which is nonlocal in time; this problem is shown to admit an analytic solution in the form of an integral. More crucially, the problem can be re-characterized as an integral equation of this particular class. This example then provides motivation for a more general study: an analytic solution is obtained for the case when the kernel and the forcing function are both unity. This analytic solution, in the form of a series solution, is a variant of the Mittag-Leffler function. As a consequence it is an entire function. A Gronwall lemma is obtained. This then permits a general existence and uniqueness theorem to be proved.     

An Optimization Problem with a Separable Non-Convex Objective Function and a Linear Constraint

For a class of global optimization (maximization) problems, with a separable non-concave objective function and a linear constraint a computationally efficient heuristic has been developed.The concave relaxation of a global optimization problem is introduced. An algorithm for solving this problem to optimality is presented. The optimal solution of the relaxation problem is shown to provide an upper bound for the optimal value of the objective function of the original global optimization problem. An easily checked sufficient optimality condition is formulated under which the optimal solution of concave relaxation problem is optimal for the corresponding non-concave problem. An heuristic algorithm for solving the considered global optimization problem is developed.The considered global optimization problem models a wide class of optimal distribution of a unidimensional resource over subsystems to provide maximum total output in a multicomponent systems.In the presented computational experiments the developed heuristic algorithm generated solutions, which either met optimality conditions or had objective function values with a negligible deviation from optimality (less than 1/10 of a percent over entire range of problems tested).     

A recognition problem for a vector alphabet generating a sequence with a quasiperiodic structure

An off-line recognition problem is analyzed for a vector alphabet generating sequences with quasiperiodic vector fragments that coincide with alphabet vectors. It is shown that the solution of this problem reduces to solving a special optimization problem. It is proved that the problem considered is solvable in polynomial time, and an algorithm for its exact solution is justified. The algorithm ensures the maximum likelihood recognition of a vector alphabet for the case of additive noise which is a Gaussian sequence of independent random values having an identical distribution.     

Modeling and solving a real-life assignment problem at universities

A real-life problem of assigning students to exams during an examination period is modeled as an optimization problem over the set of maximal cliques of a specially structured weighted graph. The problem is solved using a combination of special-purpose heuristic and Tabu search. Numerical results of experiments with real-life data from Belgrade Law School are reported.     

The interaction between a penny-shaped crack and a spherical inclusion under torsion

The axisymmetric interaction problem of an elastic spherical inclusion with a penny-shaped crack in an elastic space under torsion is considered. The superposition and reflection methods [3]-[4] are used to solve the mixed boundary value problem in question. With the help of the dual integral equations technique and appropriate re-expansion of the eigenfunction, the problem is reduced to an infinite system of linear algebraic equations of the second kind. The matrix elements of that system decrease exponentially along the rows and the columns. Its unique solution is proved to exist in a proper class of sequences and is shown to be represented by a convergent, in the vicinity of the origin, power series in a geometric parameter, equal to the ratio of the radius of the inclusion to its distance from the crack. This procedure provides an efficient formula for the stress intensity factor.     

Joint deformation of a circular inclusion and a matrix

An elastic infinite plane containing a circular inclusion with given jumps of tractions and displacements along the interface and nonzero conditions at infinity is considered. Explicit expressions are derived for the Goursat-Kolosov complex potentials of this problem. The solution constructed can be used to examine various circular interfacial defects, including interfacial cracks and rigid parts of the interface. The problem under consideration is fundamental for the superposition method, which solves many problems in which a circular region is an element of a polyphase elastic medium. In such cases, the well-posedness of the problem, which depends on the interrelation between the jumps of tractions and displacements, follows from the very superposition method. The application techniques of this method are demonstrated for singular problems on the action of a point force and an edge dislocation located inside an inclusion or in the matrix. Computational results for the tractions arising at the interface under the action of a point force concentrated in the inclusion are given.     

Strong controllability and optimal control of the heat equation with a thermal source

In this paper we consider an optimal control system described byn-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

Location of a discrete resource and its allocation according to a fractional objective

The problem considered is that of the location of a discrete resource and its allocation to activities with concave return functions in such a way as to maximize the ratio of ‘return’ to ‘cost’, the total cost being the sum of a fixed cost and linearly variable costs. It is assumed that each resource has an effectiveness of 0 or 1 against each activity. It is demonstrated that an optimal solution can be determined by the rounding to integers of the solution of an associated problem in continuous variables. Solutions with objective values arbitrarily close to the optimal value can be generated by resource-wise optimizations. An upper bound of the number of non-zero integer allocations in an optimal solution is derived.     

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile

In this paper we consider a general mathematical model for the collision between the free-fall hammer of a pile-driver and an elastic pile whose ends are furnished with a bearing. When the free-fall hammer collides with the pile, the displacement of a cross-sectional area of the pile is the weak solution of an initial-boundary value problem involving a linear wave equation with memory boundary conditions. We generalize this problem into a nonlinear one with more general boundary conditions. Then we obtain the unique solvability and the regularity of the weak solution of this nonlinear problem. The unique solvability is shortly discussed in regard to the Galerkin method. The regularity result is obtained by a combination of a fixed-point technique and an energy method, and the convenience of this procedure is also pointed out.     

Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument

We consider a problem for a quasilinear hyperbolic equation with a nonlocal condition that contains a retarded argument. By reducing this problem to a nonlinear integrofunctional equation, we prove the existence and uniqueness theorem for its solution. We pose an inverse problem of finding a solution-dependent coefficient of the equation on the basis of additional information on the solution; the information is given at a fixed point in space and is a function of time. We prove the uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation and analysis of an integro-functional equation for the difference of two solutions of the inverse problem.     

Existence and comparison results for a singular semilinear elliptic equation with a lower order term

This paper deals with the homogeneous Dirichlet problem for a singular semilinear elliptic equation with a first order term. When the datum is bounded we prove an existence result and we show that any solution can be compared with the solution to a suitable symmetrized problem.     

Job selection and sequencing on a single machine in a random environment

We examine a single machine scheduling problem with random processing times and deadline. Given a set of independent jobs having specified initiation costs and terminal revenues, the objective is to select a subset of the jobs and sequence the selected jobs such that the expected profit is maximized. The job selection aspect considered by us marks a clear departure from the pure sequencing focus found in the traditional scheduling literature. In this paper, we assume an exponentially distributed deadline and do not allow preemption. Even under these conditions, the selection and sequencing problem remains quite difficult (unlike its pure sequencing counterpart); we in fact conjecture that the problem is NP-hard. However, we show that the problem can be efficiently solved as long as the cost parameter is agreeable or an approximate solution is acceptable. To this end, we describe several solution properties, present dynamic programming algorithms (one of which exhibits a pseudo-polynomial time worst-case complexity), and propose a fully-polynomial time approximation scheme. In addition, we study a number of special cases which can be solved in polynomial time. Finally, we summarize our work and discuss an extension where the jobs are precedence related.     

On the Relation Between an Inverse Problem for a System of Ordinary Differential Equations and an Initial Boundary Value Problem for a Hyperbolic System

This paper deals with a coefficient inverse problem of a system of ODEs whose coefficient matrix is the so-called generalized negative definite matrix. To solve the problem, an initial-boundary value problem of a hyperbolic system of PDEs is constructed. The existence and uniqueness of its solution and its asymptotic convergence with respect to one of the variables to the original inverse problem are proved. As a result, the solution of the inverse problem is reduced to the solution of the direct problem. A few numerical examples were solved to show the effectiveness of the method.     

A Transmission Problem Involving a Parameter and a Weight

In the course of developing a spectral theory for non-selfadjoint elliptic boundary problems involving an indefinite weight function, there arises a transmission problem which has not hitherto been dealt with in a L_p setting. By educing our problem to one for ordinary differential equations with the aid of the Fourier transformation, we are able to resolve the transmission problem,that is to say, we are able to establish L_p estimates for its solutions which are supported in a neighbourhood of the origin, and this is p ecisely what is required for the furthe development of the spectral theory for the boundary problems cited above.     

Homogeneous solutions and Saint-Venant problems for a helical spring

The three-dimensional problem of the theory of elasticity for a spring with a stress-free side surface is investigated. In [1] the problem was reduced to an eigenvalue problem on a section, which enables a complete system of homogeneous elementary solutions to be constructed, and a group of 12 elementary solutions were distinguished, on the basis of which the construction of a Saint-Venant solution was reduced to two types of two-dimensional problems and an algebraic system of equations in the coefficients of the expansion. A variational formulation of these problems is given and the results of an asymptotic and numerical investigation of all solutions and of the stiffness matrix are presented.     

Impact of a cone on a thin elastic membrane

The problem of computation of parameters of motion of a thin elastic membrane under the Impact of a rigid body, was considered by various authors (see [1] together with bibliography) without, however, yielding a rational solution. This paper presents a full qualitative analysis of solution of this problem for the case of normal impact of a circular cone moving with constant velocity on an infinite elastic membrane of constant thickness. Although this is the simplest case, it is important, insofar as it brings to light the characteristic features of the problem. In the “membrane” approximation the thickness of the layer is found to be an unessential parameter, therefore the problem, as postulated by us, is self-similar and its solution is reducible to the problem for ordinary differential equations.     

Simultaneously optimizing link tolls and signal settings in a road network

The problem of determining link tolls to reduce traffic congestion is often referred as a toll design problem. In this paper, optimal tolls are determined for signal-controlled junctions in urban traffic road networks where the rerouting traffic is properly taken into account. This problem can be formulated as a mathematical program with equilibrium constraints (MPEC) where the user equilibrium is expressed as a variational inequality problem. Due to the non-differentiability of the equilibrium problem, an efficient convergent solution scheme is established. Numerical calculations are conducted on a variety of example road networks and comparisons are made with earlier methods.