We consider a model of simplest circular gene network regulated by one negative and two positive feedbacks. The model is represented in the form of 3-dimensional dynamical system with piecewise linear threshold righthand sides. In the phase portrait of this system, we describe a hidden attractor. Conditions of existence and uniqueness of a cycle of this system are established.

A nonlinear mathematical model of the equilibrium of a plate contacting with two obstacles is investigated. The first non-deformable obstacle is defined by inclined generatrices, and the second one restricts the plate displacements on the side face. In this case, the plate can contact both along the side edge and at the points of the curve corresponding to the intersection of the front surface of the plate and the side cylindrical surface of the plate. These circumstances lead to the fact that boundary conditions are imposed in the form of three inequalities fulfilled on the same curve. Along with the model of a homogeneous plate, the case of a nonhomogeneous plate in which a rigid inclusion is located near the contact boundary is also considered. The unique solvability of the problems for both models is proven. Under the condition of additional smoothness of the solutions to these problems, optimality conditions are found in the form of boundary conditions, as well as the corresponding equivalent differential formulations.

We examine the solvability questions in Sobolev spaces of parabolic inverse coefficient problems in stratified media with transmission conditions of the imperfect contact type. A solution has all generalized derivatives involved in the equation summable to some power. The overdetermination conditions are the values of a solution at some points lying in the domain. The proof relies a priori estimates and the fixed-point theorem.

We study two-dimensional nonlinear partial differential equations of the second order with variable coefficients. The left-hand side of these equations is a homogeneous polynomial of the second degree on unknown function and its derivatives. We consider a set of linear multiplicative transformations of the unknown function which keep a form of initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of this transformation. Expressions for the Laplace invariants are obtained through the coefficients of the equation and their first derivatives. For the equations under consideration, equivalent systems of first-order equations are found, containing the Laplace invariants. It is shown that if one of the Laplace invariants is equal to zero, then the corresponding system is reduced to a single first-order equation. Also in this case, if certain additional conditions on the coefficients are met, a solution to the original equation in quadratures can be obtained. The studies were carried out for a hyperbolic equation with a mixed derivative and for a nonlinear second-order equation of general form with a homogeneous polynomial of the second degree in the unknown function and its derivatives. In these cases, expressions for the Laplace invariants are obtained and the corresponding equivalent systems are given.

We obtain a result by combining three prevalent trends of the fixed point theory, namely (i) replacement of the Lipschitz constants in contraction inequality by functions, (ii) considerations of functions without continuity assumption and (iii) use of binary relations in the space. Specifically, we define a Mizoguchi–Takahashi–Kannan type contraction, which is shown to have fixed points in a metric space with an appropriate binary relation. The issue of the uniqueness of fixed point is separately considered. There are two illustrative examples, in one of which the discontinuity of the function occurs at a fixed point. We discuss Hyers–Ulam–Rassias stability of the fixed point problem and also establish a data dependence result.

We consider a boundary control problem for a fourth-order parabolic equation in a bounded one-dimensional domain. At a part of the boundary, a value of the solution is given and it is required to find control to get the average value of solution. By the method of separation of variables, the problem is reduced to the Volterra integral equation of the first kind. The existence of the control function was proven by the Laplace transform method and an estimate on the minimum time to reach the given average temperature in the rod was found.

The authors present a method of indicator random processes, applicable to constructing models of jump processes associated with the diffusion process. Indicator random processes are processes that take only two values: 1 and 0, in accordance with some probabilistic laws. It is shown that the indicator random process is invariant when reduced to an arbitrary positive degree. Equations with random coefficients used in modeling dynamic systems, when applying the method of indicator random processes, can take into account the possibility of adaptation to external changes, including random ones, in order to preserve indicators important for the existence of the system, which can be continuous or discrete. In the case of indicator random processes, defined as functions of the Poisson process, equations for dynamic processes in a media with abruptly changing properties are constructed and studied. To study the capabilities of the proposed method, dynamic models of the diffusion process in media were studied with delay centers and diffusion processes during transitions by switching from one subspace to another. For these models, equations for characteristic functions are constructed. Using the method of indicator random processes, a characteristic function for the Kac model was constructed. It is shown that in the case of dependence of the indicator random process on the Poisson process, the equation for the characteristic function corresponds to the telegraph equation. This result coincides with the result of Kac.

The paper presents a new formulation of the production-distribution problem in networks with a complex structure of manufacturing the end product. The paper highlights different features of the problem, including the sequence of procedures for product manufacturing and delivery, taking into account various types of products at one stage, the distinction between a step and a stage. The concepts of "fictitious" and "real" parts (volumes) of product supply from real suppliers were introduced, which made it possible to avoid the use of heuristic methods in solving the problem. The paper presents a technique based on the simplex method for optimizing the production and supply of different types of products at each stage of the production chain, allowing the problem to be solved in one optimization procedure. The results of the study can be useful for planning in complex and multidisciplinary ("network") companies to make economically sound decisions in the field of production chain management.