Vol 59, No (2016)

At designing structures and devices interacting with the ow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The de nition of stability of an elastic body, taken in the article, corresponds to the Lyapunov’s concept of stability of dynamical system. On the base of a proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic ow of gas or liquid (in an ideal model of a incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensi cation of technological processes, for example, the process of mixing is considered. The action of these devices is based on the oscillations of elastic elements at the owing around a of gas or liquid ow. The dynamic stability of the elastic element, located on one wall of the ow channel with the subsonic ow of gas or liquid (in an ideal model of a compressible environment) is investigated. The both models is described by coupled nonlinear system of di erential equations for the unknown functions - the potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functionals, the su cient conditions of the stability, impose restrictions on the free-stream velocity of the gas, the exural sti ness of the elastic element, and other parameters of the mechanical system is obtained. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.

We consider boundary-value problems for di erential-di erence operators with perturbations in shifts of the argument. We prove that the family of di erential-di erence operators is positive de nite uniformly with respect to the shifts of the argument. Solutions of such problems depend continuously on these shifts. We consider the coercivity problem for di erential-di erence operators с with incommensurable shifts of the argument and study the approximation of such operators by rational operators.

We develop the theory of generalized Jordan chains of multiparameter operator functions A(λ) : E1 → E2, λ ∈ Λ, dimΛ = k, dimE1 = dimE2 = n, where A0 = A(0) is a noninvertible operator. To simplify the notation, in Secs. 1-3 the geometric multiplicity λ0 is set to 1, i. e. dimN(A0) = 1, N(A0) = span{ϕ}, dimN∗(A∗0) = 1, N∗(A∗0) = span{ψ}, and the operator function A(λ) is supposed to be linear with respect to λ. For the polynomial dependence of A(λ), in Sec. 4 we consider a linearization. However, the bifurcation existence theorems hold in the case of several Jordan chains as well. We consider applications to degenerate differential equations of the form [A0 + R(·, x)]x*= Bx.

We study elliptic operators on manifolds with singularities such that a discrete group G acts on the manifold. Following the standard elliptic theory approach, we de ne the Fredholm property of an operator by its principal symbol. For this problem, we prove that the symbol is a pair consisting of the symbol on the principal stratum (the inner symbol) and the symbol at the conical point (the conormal symbol). We establish the Fredholm property of elliptic elements.