Vol 24, No 1 (2022)

In this paper, the question of uniqueness of the solution for one class of Volterra-Stieltjes linear integral equations of the third kind is investigated. The notion of derivative with respect to an increasing function was introduced by A. Asanov in 2001 and plays special role in the study. This notion is a generalization of the usual concept of a derivative function and is an inverse operator for one class of the Stieltjes integral. Basing on idea of such derivative, using the method of integral transformations and the method of non-negative quadratic forms, the uniqueness theorems for the solution of the considered class of integral equations are proved. Examples satisfying the conditions of uniqueness theorems are also constructed in the paper. It becomes clear from these examples that it is difficult to study Volterra-Stieltjes linear integral equations of the first and third kind without using the notion of derivative with respect to increasing function.

The work solves the classification problem for structurally stable flows, which goes back to the classical works of Andronov, Pontryagin, Leontovich and Mayer. One of important examples of such flows is so-called Morse-Smale flow, whose non-wandering set consists of a finite number of fixed points and periodic trajectories. To date, there are exhaustive classification results for Morse-Smale flows given on manifolds whose dimension does not exceed three, and a very small number of results for higher dimensions. This is explained by increasing complexity of the topological problems that arise while describing the structure of the partition of a multidimensional phase space into trajectories. In this paper authors investigate the class G(Mⁿ) of Morse-Smale flows on a closed connected orientable manifold Mⁿ whose non-wandering set consists of exactly four points: a source, a sink, and two saddles. For the case when the dimension of the supporting manifold is greater or equal than four, it is additionally assumed that one of the invariant manifolds for each saddle equilibrium state is one-dimensional. For flows from this class, authors describe the topology of the supporting manifold, estimate minimum number of heteroclinic curves, and obtain necessary and sufficient conditions of topological equivalence. Authors also describe an algorithm that constructs standard representative in each class of topological equivalence. One of the surprising results of this paper is that while for n=3 there is a countable set of manifolds that admit flows from class G(M³)  there is only one supporting manifold (up to homeomorphism) for dimension  n>3

In this paper, we study anti-endomorphisms of some finite groupoids. Previously, special groupoids S(k,q) of order k(k+1) with a generating set of k elements were introduced. Previously, the element-by-element description of the monoid of all endomorphisms (in particular, automorphisms) of a given groupoid was studied. It was shown that every finite monoid is isomorphically embeddable in the monoid of all endomorphisms of a suitable groupoid  S(k,q). In recent article, we give an element-by-element description for the set of all anti-endomorphisms of the groupoid S(k,q). We establish that, depending on the groupoid S(k,q), the set of all its anti-endomorphisms may be closed or not closed under the composition of mappings. For an element-by-element description of anti-endomorphisms, we study the action of an arbitrary anti-endomorphism on generating elements of a groupoid. With this approach, the anti-endomorphism will fall into one of three classes. Anti-endomorphisms from the two classes obtained will be endomorphisms of given groupoid. The remaining class of anti-endomorphisms, depending on the particular groupoid  S(k,q), may either consist or not consist of endomorphisms. In this paper, we study endomorphisms of some finite groupoids G whose order satisfies some inequality. We construct some endomorphisms of such groupoids and show that every finite monoid is isomorphically embedded in the monoid of all endomorphisms of a suitable groupoid G. To prove this result, we essentially use a generalization of Cayley's theorem to the case of monoids (semigroups with identity).