We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang–Baxter maps, which are set-theoretical solutions to the quantum Yang–Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang–Baxter and entwining Yang–Baxter maps are also presented. Using the obtained general results, we construct new examples of (parametric) Yang–Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and Müller-Hoissen (2019 Lett. Math. Phys. 109 799–827) in a study of soliton solutions of vector Kadomtsev–Petviashvili equations. Also, we present invariants for this nonlinear tetrahedron map.

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The concept of coupled reactors could be of considerable interest for transmutation of radioactive wastes on the basis of a system consisting of a heavy-particle accelerator, a neutron target, and a blanket in the form of a subcritical assembly [1]. The theory of such systems was first developed by R. Avery at the Argonne National Laboratory in the USA and was reported in 1958 at the 2nd Geneva Conference on peaceful uses of atomic energy [2].Coupled reactors are under study by many specialists. Considerable attention is often devoted to reactor safety with breeding of fuel. Usually, a reactor in which fission in the active zone is produced only by fast neutrons and fission in the blanket reflector, which is the breeding zone, is produced by both fast and thermal neutrons. This is a combined system consisting of a fast reactor and a thermal subcritical assembly, which is safer than standard fast reactors with respect to positive-reactivity accidents.Multiple-section coupled systems in the form of a combination of a high-power reactor and serially positioned subcritical assemblies with neutron valve coupling between sections have been discussed [3, 4]. Layers of 235U, cadmium, and a moderator, placed at the boundaries between sections, have been proposed for use as valves. It has been observed that practical implementation of a system of this type provides the opportunity to increase substantially the degree of burnup of the fissioning substances, increase by several factors, due to the higher dilution of fuel in the subcritical assembly, the neutron flux density in the active zone, and guarantee safer conditions of power production because the valves prevent the accident from spreading from the subcritical assembly to the reactor. Two-section reactor systems are also under intensive discussion in the context of aperiodic pulsed reactors, in application to which they open up the possibility of decreasing the duration of the radiation pulse or increasing the volume of experimentally accessible zones with a high radiation level [5, 6]. Devices of the latter type have been realized in practice [7, 8] or are incorporated in important projects, for example, high-power nuclear-pumped lasers [9].It is shown in [10] that the creation of neutron valve coupling between sections is the most effective means for decreasing pulse duration in coupled reactors or subcritical assemblies. An effective method of valve coupling based on the use of a threshold fissioning substance in one of the sections is also examined in [10]. In the two-section BR-K booster reactor, developed at VNIII~F, using 237Np as the material in one of the sections the radiation pulse duration is almost ten times shorter than that obtained with the standard single-section arrangement [11].Returning to transmutation and a system consisting of an accelerator, a target, and a blanket, developed as part of the ABC/ATW program, it can be conjectured that coupled subcritical assemblies will be useful for achieving deeper subcriticality of the blanket in the workin...

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