Vol 219, No 1 (2016)

We propose a mathematical model of a system of wing profiles of tandem type with elastic ailerons in subsonic flow of an ideal gas (liquid). The aerohydroelasticity problem is reduced to the study of a system of integro-differential equations. The solution is constructed by the Bubnov–Galerkin method. Numerical experiments are presented in the case of two profiles with elastic ailerons. We establish the dynamic stability of the aileron and obtain the stability conditions if only one profile has an elastic aileron.

We consider a dynamical system simulating a switching device that changes its state finitely many times during operation. Changes of state (switchings) are described by a recurrent inclusion that corresponds to a representation of the system as a dynamic automaton with memory. The switching times and the number of switchings are not a priori known and are found by minimizing the functional. We find sufficient and necessary optimality conditions for such systems. We obtain an equation for synthesis of optimal trajectories. The optimality condition is illustrated by examples.

We study a new class of homogeneous operators in L₂(\( {\mathbb{R}}^2 \)) that, after foliation of \( {\mathbb{R}}^2 \) into concentric circles, are represented in fibres as singular integral operators with measurable essentially bounded coefficients. We find necessary and sufficient conditions for the invertibility of such operators and construct the operator-valued symbolic calculus for the C^∗–algebra generated by such operators and operators of multiplication by multiplicatively weakly oscillating functions. We obtain a criterion for the generalized Fredholm property of operators and find effectively verifiable functional necessary conditions for the classical Fredholm property.

We obtain exact formulas describing the nonwandering set of a C¹-smooth skew product of interval maps with Ω-stable quotient map of type ≻ 2^∞.

We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M. We prove that for any foliation (M,F) there exists an open, not necessarily connected, saturated, and everywhere dense subset M₀ of M and a manifold L₀ such that the induced foliation (M₀, F_M0) is formed by the fibers of a locally trivial fibration with the standard fiber L₀ over (possibly, non-Hausdorff) smooth q-dimensional manifold. In the case of codimension 1, the induced foliation on each connected component of the manifold M₀ is formed by the fibers of a locally trivial fibration over a circle or over a line.

We establish the existence and uniqueness of continuous solutions to systems of linear and nonlinear integral equations with partial integrals, with partial integrals and potential type kernels, and fractional partial integrals. We obtain conditions guaranteeing that the solutions have continuous partial derivatives and are continuously differentiable.

We analyze the method of localization of invariant compact sets in the case where the right-hand side of an autonomous system includes an uncertainty. We consider two cases depending on the current state of an autonomous system: the uncertainty on the right-hand side of the system is expressed by a constant parameter or a parameter varying in time. It is shown that a localizing set constructed in the first case is also a localizing set in the second case.

We establish the uniqueness of a solution to the Dirichlet problem for a class of anisotropic second order elliptic equations with lower-order terms and nonpower nonlinearities. We construct an example demonstrating that the class of equations under consideration is larger than the class of equations with power nonlinearites.