Vol 208 (2022)
Статьи
On two-dimensional systems of Volterra integral equations of the first kind
Abstract
In this paper, we consider two-dimensional systems of Volterra integral equations of the first kind. The case where a system of integral equations of the second kind is obtained by differentiating the equations is well studied. We examine the case where this approach leads to a system of integral equations with an degenerate matrix of the principal part. We formulate sufficient conditions for the existence of a unique smooth solution in terms of matrix pencils.
3-10
Two combinatorial identities related to enumeration of graphs
Abstract
From the explicit formula for the number of labeled, series-parallel, 2-connected graphs with a given number of vertices obtained by the author, two combinatorial identities are derived. Also, proofs of these identities independent of the enumeration of graphs are given.
11-14
On spectral properties of one difference operator with involution
Abstract
We consider a difference operator with involution acting in the complex Hilbert space l₂(ℤ). Using the method of similar operators, we reduce it to the diagonal (block diagonal) form, which allows one to obtain various spectral characteristics of the original operator and to construct biinvariant subspaces for it.
15-23
Flows in networks with barrier reachability
Abstract
The problem of flows in networks with barrier-type reachability restrictions is considered. We introduce new definitions that allow one to describe a flow in a network with reachability constraints, in particular, a representation of a flow as a vector-valued function. Conditions for preserving the flow and restricting the maximum flow along an arc are formulated in terms of vector-valued functions. This allows one to consider flow problems without passing to an unfolding, which is a graph with connected arcs.
24-28
Asymptotic problem of restoring the high-frequency right-hand side of the telegraph equation
Abstract
In this paper, we consider the Cauchy problem for the telegraph equation. The lower coefficient and the right-hand side of the equation oscillate in time with a high frequency, the amplitude of the lower coefficient is small, namely, is inversely proportional to the frequency, and the right-hand side is unknown. We examine the problem on the recovery of the right-hand side from the three-term asymptotics of the solution given at some point in space. For this purpose, we use a nonclassical algorithm for solving inverse coefficient problems with rapidly oscillating data.
29-36
Qualitative properties of solutions to fourth-order differential equations on graphs
Abstract
In this paper, we examine properties of solutions to fourth-order differential equations on geometric graphs (positivity, oscillatory behavior, distribution of zeros, etc.). We prove theorems on alternation of zeros of solutions and develop the theory of nonoscillation. The definition of nonoscillation for fourth-order equations on graphs is based on the concept of a double constancy zone introduced in the paper. The new approach allows one to generalize the basic principles of the theory of nonoscillation of second-order equations on a graph to fourth-order equations.
37-48
Inverse problem for the Sturm–Liouville operator with a frozen argument on the time scale
Abstract
In this paper, we consider the problem of constructing the potential of the Sturm–Liouville equation with a frozen argument on the time scale by the spectrum of the Dirichlet boundary-value problem, where the time scale consists of two segments and the argument is frozen at the end of the first segment. We obtain the uniqueness theorem and construct an algorithm for solving the inverse problem together with necessary and sufficient conditions for its solvability. The case considered substantially differs from the case of the classical Sturm– Liouville operator with a frozen argument.
49-62
The Lagrange principle and the Pontryagin maximum principle in ill-posed optimal control problems
Abstract
We consider the regularization of the classical optimality conditions—the Lagrange principle and the Pontryagin maximum principle—in a convex optimal control problem for a parabolic equation with distributed and boundary controls, and also with a finite number functional equality constraints given by “point’ functionals nondifferentiable in the Fréchet sense, which are the values of the solution of the third initial-boundary-value problem for the specified equation at preselected fixed (possibly boundary) points of the cylindrical domain of the independent variables.
63-78
Linkedness of families of sets, supercompactness, and some generalizations
Abstract
We examine a construction that has the meaning of an abstract analog of a superextension of a topological space and new types of supercompact topological spaces. In addition, we study relations between ultrafilters and maximal linked systems on measurable spaces.
79-90
Systems with dissipation with five degrees of freedom: analysis and integrability. I. Primordial problem from dynamics of a multidimensional rigid body in a nonconservative field of forces
Abstract
This paper is the first part of a survey on the integrability of systems with five degrees of freedom. The review consists of three parts. In this first part, the primordial problem from the dynamics of a multidimensional rigid body placed in a nonconservative force field is described in detail. In the second and third parts, which will be published in the next issue, we consider more general dynamical systems on tangent bundles to the five-dimensional sphere and other smooth manifolds of a sufficiently wide class. Theorems on sufficient conditions for the integrability of the considered dynamical systems in the class of transcendental functions are proved.
91-121
One of the statistical solutions of the problem of differential diagnostics
Abstract
In this paper, we show that in the case of trajectory measurements with noise, which is a normal white-noise random process with zero mean value and a limited spectrum, diagnostics is feasible using the diagnostic algorithms developed in the author’s previous works. The diagnostic functional was obtained, which was introduced a priori in the previous works of the author.
122-127