Journal of Applied and Industrial Mathematics

Under consideration is the identification problem for a time-dependent source in the wave equation. The Dirichlet conditions are used as the boundary conditions, whereas the weighted integral of the conormal derivative of the solution over the boundary of the spatial domain serves as the overdetermination condition. Using the Duhamel method, the problem is reduced to the Volterra integral equation of the first and then the second kind. These results are applied to studying nonlinear coefficient problems. The existence and uniqueness of a local solution is proved by the contraction mapping principle.

We consider some piecewise linear 4-dimensional dynamical system that models a gene network regulated by one negative feedback and three positive feedbacks. Glass and Pasternack described the conditions for the existence of a stable cycle in this model. We construct an invariant piecewise linear surface with nontrivial link with the Glass-Pasternack cycle outside the attraction domain of this stable cycle in the phase portrait of this system.

Under study is the problem of reconstructing the memory function of a. viscoelastic medium from the system of viscoelastic equations for a. homogeneous anisotropic medium. As additional information, the Fourier image of the displacement vector with respect to the spatial variables for the values V₀# 0 of the transformation parameter is given. It is demonstrated that, provided the data of the problem satisfy some conditions of agreement and smoothness, the solution of the problem is uniquely determined in the class of continuous functions and depends continuously on the given functions.

Under study is the process of producing irreversible deformations in a rotating cylinder of a material with elastic, viscous, and plastic properties. It is assumed that, before the prescribed maximum angular velocity is reached, the cylinder with a rigid inclusion rotates with acceleration and after that, with deceleration. As the inertial forces change, the irreversible deformations initially grow as creep deformations, and, as the angular velocity increases and the stress states reach the yield surface, some plastic flow region originates and develops. Thereafter, an elastoplastic unloading boundary emerges: The plastic flow region decreases as this boundary surface moves across the volume. The elastoplastic boundaries turn out to be the place where the mechanism is turned on (or off) of fast and intensive production of irreversible deformations (the plastic flow). The results of simulation of the time-varying deformations and stresses, including the residual stresses and their relaxation, are presented and discussed.

Under study is the mathematical model describing the inverse temperature hysteresis as well as the self-sustained oscillations in the CO oxidation over a palladium catalyst in an chemical stirred tank reactor (CSTR). We consider the reaction dynamics under temperature-programmed conditions: At first, the temperature T of the CSTR monotonically increases (due to outside heating) and then it decreases to the initial value. As the temperature goes up, on the surface and in the bulk of the catalyst two palladium oxide forms appear and then, while the temperature decreases, the catalyst reduces to its original state. The mathematical model of nonstatinary processes in such a CSTR is the piecewise continuous system of nonlinear ordinary differential equations (ODE), i.e, a discrete-continuous system. Using the theory of dynamical systems and bifurcation theory as well as numerical methods, we study the structure of the maximal families of the steady states and periodic solutions in dependence on temperature. For the system under study some sufficient conditions are given under which an inverse hysteresis is observed on the dependence of the conversion of the main reagent versus T. Moreover, as temperature decreases, there are self-oscillations of the reaction rate and CO conversion on the lower back branch of the hysteresis. The parameters of the model are found such that the experimental data are qualitatively described.

Under study is the equilibrium problem of a two-dimensional elastic body with a crack crossing a thin rigid inclusion at some point. Nonpenetration conditions in the form of inequalities are put on the crack faces and at the intersection point of the crack with the rigid inclusion. The equilibrium problem of an elastic body with a crack crossing a thin elastic inclusion is also considered. The theorems of unique solvability of these problems are proved, and some complete systems of boundary conditions are obtained. The equivalence of the two formulations, variational and differential, is examined. We establish that the limit transition with respect to the rigidity parameter in the problems on the equilibrium of an elastic body with an elastic inclusion leads to the equilibrium problem of an elastic body with a rigid inclusion.

We consider the cyclic job shop problem with no-wait constraints which consists in minimizing the cycle time. We assume that a single product is produced on a few machines. A job is processed by performing a given set of operations in a predetermined sequence. Each operation can be performed on exactly one machine. We consider the problem of minimization the cycle time with no-wait constraints between some pairs of sequential operations and investigate the complexity of the problem and some of its subproblems. In general, the problem is proved to be strongly NP-hard. In the case when the job is processed without downtime between operations, polynomial solvability is proved and the two algorithms are proposed. Also we develop an algorithm for the general case which is pseudopolynomial if the number of admissible downtime is fixed. The case of a single no-wait constraint is polynomially solvable. The problem with two no-wait constraints becomes NP-hard. We found effectively solvable cases and propose the corresponding algorithms.

Point symmetries allowed by plasticity equations in the dynamical case are used to construct solutions for the dynamical equations of ideal plasticity. These symmetries make it possible to convert the exact solutions of stationary dynamical equations to nonstationary solutions. The so-constructed solutions include arbitrary functions of time. The solutions allow us to describe the plastic flow between the plates changing their shape under the action of dynamical loads. Some new spatial self-similar solution is also presented.

We study the relationship between homogeneous bent functions and some intersection graphs of a special type that are called Nagy graphs and denoted by Γ_(n,k). The graph Γ_(n,k) is the graph whose vertices correspond to (n_k) unordered subsets of size k of the set 1,..., n. Two vertices of Γ(_n,k) are joined by an edge whenever the corresponding k-sets have exactly one common element. Those n and k for which the cliques of size k + 1 are maximal in Γ_(n,k) are identified. We obtain a formula for the number of cliques of size k + 1 in Γ(_n,k) for n = (k + 1)k/2. We prove that homogeneous Boolean functions of 10 and 28 variables obtained by taking the complement to the cliques of maximal size in Γ(₁₀,4) and Γ_(28,7) respectively are not bent functions.

We consider a model of economy with two markets for each product: one—state and the other—competitive. Moreover, both markets coexist in the same economic space allowing free movement of goods and means of payment. In particular, it is assumed that the surplus of products purchased at fixed state prices can be sold at free prices of the competitive market. The important feature of the model is that the manufacturing activity is taken into account both in the state and in the competitive market. While most literature on mixed economies is devoted to the issues of existence and Pareto optimality of equilibria, the focus of the present paper is on analyzing their coalition stability. We continue studying the fuzzy cores of mixed economic models of Arrow—Debreu type which was started earlier for the case of high free market prices. New conditions are established for the coincidence of the sets of undominated and equilibrium allocations, covering the cases of low equilibrium prices for some of the products.