Vol 59, No 4 (2018)

We show that each computably enumerable Turing degree is a degree of autostability relative to strong constructivizations for a decidable directed graph. We construct a decidable undirected graph whose autostability spectrum relative to strong constructivizations is equal to the set of all PA-degrees.

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a₁,..., a_n) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.

The spectrum of a finite group is the set of the orders of its elements. We consider the problem that arises within the framework of recognition of finite simple groups by spectrum: Determine all finite almost simple groups having the same spectrum as its socle. This problem was solved for all almost simple groups with exception of the case that the socle is a simple even-dimensional orthogonal group over a field of odd characteristic. Here we address this remaining case and determine the almost simple groups in question.

Also we prove that there are infinitely many pairwise nonisomorphic finite groups having the same spectrum as the simple 8-dimensional symplectic group over a field of characteristic other than 7.

We introduce the notion of A-numbering which generalizes the classical notion of numbering. All main attributes of classical numberings are carried over to the objects considered here. The problem is investigated of the existence of positive and decidable computable A-numberings for the natural families of sets e-reducible to a fixed set. We prove that, for every computable A-family containing an inclusion-greatest set, there also exists a positive computable A-numbering. Furthermore, for certain families we construct a decidable (and even single-valued) computable total A-numbering when A is a low set; we also consider a relativization containing all cases of total sets (this in fact corresponds to computability with a usual oracle).

We establish some algebraic, metric, and spectral properties of convolution integral operators in L₂(ℝ^N).

A regularizing algorithm is constructed for solving the problem of extension of a wave field from the boundary to the interior of a half-plane. The cases are considered of the wave equation and hyperbolic equations with variable lower order coefficients.

We study multiagent logics and use temporal relational models with multivaluations. The key distinction from the standard relational models is the introduction of a particular valuation for each agent and the computation of the global valuation using all agents’ valuations. We discuss this approach, illustrate it with examples, and demonstrate that this is not a mechanical combination of standard models, but a much more subtle and sophisticated modeling of the computation of truth values in multiagent environments. To express the properties of these models we define a logical language with temporal formulas and introduce the logics based at classes of such models. The main mathematical problem under study is the satisfiability problem. We solve it and find deciding algorithms. Also we discuss some interesting open problems and trends of possible further investigations.

For a real solution (u, p) to the Euler stationary equations for an ideal fluid, we derive an infinite series of the orthogonality relations that equate some linear combinations of mth degree integral momenta of the functions u_iu_j and p to zero (m = 0, 1,... ). In particular, the zeroth degree orthogonality relations state that the components ui of the velocity field are L²-orthogonal to each other and have coincident L²-norms. Orthogonality relations of degree m are valid for a solution belonging to a weighted Sobolev space with the weight depending on m.