Vol 215, No 11 (2024)

The theory of $n$-valued groups and its applications is developed by going over from groups defined axiomatically to combinatorial groups defined by generators and relations. A wide class of cyclic $n$-valued groups is introduced on the basis of cyclically presented groups. The best-known cyclically presented groups are the Fibonacci groups introduced by Conway. The problem of the existence of the orbit space of $n$-valued groups is related to the problem of the integrability of $n$-valued dynamics. Conditions for the existence of such spaces are presented. Actions of cyclic $n$-valued groups on $\mathbb R^3$ with orbit space homeomorphic to $S^3$ are constructed. The projections $\mathbb R^3 \to S^3$ onto the orbit space are shown to be connected, by means of commutative diagrams, with coverings of the sphere $S^3$ by three-dimensional compact hyperbolic manifolds which are cyclically branched along a hyperbolic knot. Bibliography: 54 titles.

The topological equivalence of low-dimensional Morse–Smale flows without fixed point (NMS-flows) under assumptions of various generality is the subject of a number of publications. Starting from dimension 4, there are only few results on classification. However, it is known that there exists nonsingular flows with wildly embedded invariant saddle manifolds. In this paper the class of nonsingular Morse–Smale flows on closed orientable 4-manifolds with a unique saddle orbit which is, moreover, nontwisted, is considered. It is shown that the equivalence class of a certain knot embedded in $\mathbb S^2\times\mathbb S^1$ is a complete invariant of such a flow. Given a knot in $\mathbb S^2\times\mathbb S^1$, a standard representative in the class of flows under consideration is constructed. The supporting manifold of all such flows is shown to be the manifold $\mathbb S^3\times\mathbb S^1$.Bibliography: 24 titles.

It is proved that the decrease of quantum relative entropy under the action of a quantum operation is a lower semicontinuous function of the pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under the action of quantum operations. It also implies the lower semicontinuity of the modulus of the joint convexity of quantum relative entropy (as a function of ensembles of quantum states).Various corollaries and applications of these results are considered.Bibliography: 42 titles.