Vector network equilibrium problems with elastic demands

We consider vector models for complex systems with spatially distributed elements which arise in communication and transportation networks. In order to describe the flow distribution within such a network, we utilize the equilibrium approach, which extends the shortest path one. Being based on this approach, we investigate several networking control problems, with taking into account many factors. As a result, general vector equilibrium problems models with complex behavior of elements are suggested. In particular, they involve elastic demand functions. Due to the presence of many factors, we utilize multicriteria models with respect to different preference relations. The corresponding problems admit efficient solution methods within optimization and equilibrium frameworks.     

Vector equilibrium problems with elastic demands and capacity constraints

In this paper, a (weak) vector equilibrium principle for vector network problems with capacity constraints and elastic demands is introduced. A sufficient condition for a (weak) vector equilibrium flow to be a solution for a system of (weak) vector quasi-variational inequalities is obtained. By virtue of Gerstewitz’s nonconvex separation functional ξ, a (weak) ξ-equilibrium flow is introduced. Relations between a weak vector equilibrium flow and a (weak) ξ-equilibrium flow is investigated. Relations between weak vector equilibrium flows and two classes of variational inequalities are also studied.     

Torsional stability of a glass-reinforced plastic cylindrical shell with an elastic core

The problem of the stability of a glass-reinforced plastic cylindrical shell with an elastic core subjected to twisting moments applied to the edges of the shell is considered. As in various other studies [4–6], the glass-reinforced plastic is treated as an elastically orthotropic material. The core is treated as an isotropic elastic cylinder, whose outer surface is bonded to the shell. Expressions for the critical stresses are obtained for an infinitely long shell and a shell of finite length.Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1082–1086, November–December, 1970.     

Three-dimensional contact problems for an elastic wedge with a coating

Three-dimensional contact problems for an elastic wedge, one face of which is reinforced with a Winkler-type coating with different boundary conditions on the other face of the wedge, are investigated. A power-law dependence of the normal displacement of the coating on the pressure is assumed. The contact area, the pressure in this region, and the relation between the force and the indentation of a punch are determined using the method of non-linear boundary integral equations and the method of successive approximations. The results of calculations are analysed for different values of the aperture angle of the wedge, the relative distance of the punch from the edge of the wedge, the ratio of the radii of curvature of the punch (an elliptic paraboloid), and the non-linearity factors of the coating. The results obtained are compared with the solutions of similar problems for a wedge without a coating.     

Three-dimensional contact problems with friction for a composite elastic wedge

Contact problems for a composite elastic wedge in the form of two joined wedge-shaped layers with different aperture angles joined by a sliding clamp, where the layer under the punch is incompressible, are studied in a three-dimensional formulation. Conditions for a sliding or rigid clamp or the absence of stresses are set up on one face of the composite wedge. The integral equations of the problems are derived taking account of the friction forces perpendicular to the edge of the wedge. The method of non-linear boundary integral equations of the Hammerstein type is used when the contact area is unknown. A regular asymptotic solution is constructed for an elliptic contact area. By virtue of the incompressibility of the material of the layer in contact with the punch, this solution retains the well known root singularity in the boundary of the contact area when account is taken of friction.     

Some optimization problems for elastic bodies

Two optimization problems for finite deformations of an elastic body are formulated. In the first problem, the motion that gives minimum value (maximum value) to a certain functional is chosen from a single family of controllable motions. The necessary conditions for optimality are stated for one specific problem of this type. In the second problem, the minimizing (maximizing) motion is chosen from the class of all possible motions connecting two configurations of the body. The necessary conditions for the optimality of singular solutions are obtained for one specific example of this type.This work was supported by the Pokrajinska Zajednica za Naucni Rad, Novi Sad, Yugoslavia.     

Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles

In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of −_g1(x)$-g_1(x)$ for extra production of one unit at location x   and an extra cost of _g2(y)$g_2(y)$ for creating one unit of demand at y  . The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞$p\to \infty$ to a double obstacle problem (with obstacles _g1$g_1$, _g2$g_2$) for the p  -Laplacian. In fact, under a certain natural constraint on the extra costs (that is equivalent to impose that the total optimal cost is bounded) we prove that this limit gives the extra material and extra demand needed for optimality and a Kantorovich potential for the mass transport problem involved. We also show that this problem can be interpreted as an optimal mass transport problem in which one can make the transport directly (paying a cost given by the Euclidean distance) or may hire a courier that cost _g2(y)−_g1(x)$g_2(y)-g_1(x)$ to pick up a unit of mass at y and deliver it to x. For this different interpretation we provide examples and a decomposition of the optimal transport plan that shows when we have to use the courier.     

Compact mixed methods for convection/diffusion type problems

This paper presents a class of fourth-order compact finite difference technique for solving two-dimensional convection diffusion equation. The equation is recasted as a first-order mixed system, introducing a conservation and flux equations. Since flux appears explicitly in the mixed formulation, we search a fourth-order compact approximation of the primary solution field and flux. Based on Taylor series expansion, the proposed compact mixed formulation generalizes the work of Carey and Spotz [G.F. Carey, W.F. Spotz, Higher-order compact mixed methods, Commun. Numer. Meth. Eng. 13 (1997)]. We show that their fourth-order formulation corresponds to a particular case of our presented scheme, and we extend their work to variable diffusion and convection coefficients. Some numerical experiments are performed to demonstrate the fourth-order effective convergence rate.     

Hele-Shaw type problems with dynamical boundary conditions

In this paper, we study the nonlinear evolution equation of Hele-Shaw type with dynamical boundary conditions. That is, the equation ut=Δw+f where u∈H(w) and H is the Heaviside function, with boundary condition μ(x,w)t_∂w+k∇w⋅ν=g, where ν denotes the outward normal vector of the fixed boundary of the domain. We prove existence, uniqueness and some qualitative properties of the solution.     

PLANE ELASTIC PROBLEMS OF DIFFERENT MEDIA WITH CRACKS ON THE INTERFACE

The equllibrium problem for the infinite elastic plane consisting of two different media with many cracks on the interface is discussed.It is transferred to a boundary value problem for analytic functions and then further reduced to a singular integral equation,the unique solvability and an effective method of solution for which are eatablished.A practical example in applications is illustrated,the solution of which is obtained in closed form.     

Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.     

Variational problems connected with Monge-Ampere type operators

In this note we describe a geometric problem leading to Monge-Ampere type operators. Some variational problems and theorems on their solvability are formulated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 186–189, 1988.     

Boundary-value problems for shallow elastic membrane caps

Consider the general nonlinear boundary-value problem (p(t)y' (t))' = p(t)q(t) f (t, y(t), y' (t)), t 1, g(y(1), y' (1))= 0, where the function f may be singular at the point y(1)= 0 and p(1) 0. We obtain conditions which guarantee existenceof positive and bounded solutions of the above problem. As anapplication we prove existence and uniqueness of rotationallysymmetric solutions to a nonlinear boundary-value problem, representingthe elastic deformation of a spherical cap.     

Stability of glass-reinforced plastic plates resting on an elastic foundation

The stability of oriented glass-reinforced plastic plates resting on an elastic Winkler foundation is investigated using the theories of Timoshenko [2, 6] and Ambartsumyan [7], which permit the shear strains to be taken into account. The solution obtained is compared with the corresponding solution based on the classical Kirchhoff theory [8].L'vov Polytechnic Institute. Institute of Polymer Mechanics, AS LatSSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1082–1088, November–December, 1968.     

Mixed interface problems for anisotropic elastic bodies

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where, on one part of the interface, rigid contact conditions are give (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

Romanov type problems

Romanov proved that the proportion of positive integers which can be represented as a sum of a prime and a power of 2 is positive. We establish similar results for integers of the form \(n=p+2^{2^k}+m!\) and \(n=p+2^{2^k}+2^q\) where \(m,k \in \mathbb {N}\) and p, q are primes. In the opposite direction, Erd?s constructed a full arithmetic progression of odd integers none of which is the sum of a prime and a power of two. While we also exhibit in both cases full arithmetic progressions which do not contain any integers of the two forms, respectively, we prove a much better result for the proportion of integers not of these forms: (1) The proportion of positive integers not of the form \(p+2^{2^k}+m!\) is larger than \(\frac{3}{4}\). (2) The proportion of positive integers not of the form \(p+2^{2^k}+2^q\) is at least \(\frac{2}{3}\).     

Completion problems and scattering problems for Dirac type differential equations with singularities

A completion problem to recover a rational matrix function which is j-unitary on the line is treated. A Dirac type system with singularities on the semi-axis is recovered explicitly from its left reflection coefficient. The close relation between these two problems is discussed.     

Variational problems of least area type with constraints

Summary We establish a new regularity result for boundaries of sets minimizing the area functional perturbed by a curvature term, and subject to a mass constraint of the type , with a fixed, strictly positive integrable functiong(x). The existence of Lagrange parameters is also considered. A preliminary version of the paper was presented at the meeting on “Partial Differential Equations”, Ferrara (Italy), October 13–16, 1986. Work partially supported by the Italian M.P.I.