Vol 208 (2022)

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.

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.

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.

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.

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.