Volume 237, Nº 3 (2019)

Statistical models of a single open string avoiding self-intersections in the d-dimensional Euclidean space ℝ^d, 2 ≤ d < 4, and the ensemble of strings are considered. The presentation of these models is based on the Darwin–Fowler method, used in statistical mechanics to derive the canonical ensemble. The configuration of the string in space ℝ^d is described by its contour length L and the spatial distance R between its ends. We establish an integral equation for a transformed probability density W(R, L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations. This allows us using the renormalization group method to investigate the asymptotic behavior of this density in the case where R → ∞ and L → ∞. For the model of an ensemble of M open strings with the mean string contour length over the ensemble given by \( \overline{L} \), we obtain the most probable distribution of strings over their lengths in the limit as M → ∞. Averaging the probability density W(R, L) over the canonical ensemble eventually gives the sought density 〈W(R, L)〉.

We construct new bases of real functions from L²(B_r) and from L²(ℚ_p). These functions are eigenfunctions of the p-adic pseudo-differential Vladimirov operator, which is defined on a compact set B_r ⊂ ℚ_p of the field of p-adic numbers ℚ_p or, respectively, on the entire field ℚ_p. A relation between the basis of functions from L²(ℚ_p) and the basis of p-adic wavelets from L²(ℚ_p) is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator that is diagonal in the basis constructed.

In this paper, we study universal equivalence of general and special linear groups over fields. We give the following criterion for this relation to hold: two groups G_n(K) and G_m(L) (G = GL, SL, K and L are infinite fields) are universally equivalent if and only if n = m and the fields K and L are universally equivalent.

We extend Haver’s theorem on the characterization of absolute extensors in the class of countable-dimensional spaces to the category of metric filtered spaces.

Universal hypergraphical automata are universally attracting objects in the category of automata for which the set of states and the set of output symbols are equipped with structures of hypergraphs. It was proved earlier that a wide class of such sort of automata are determined up to isomorphism by their semigroups of input symbols. We investigate the connection between isomorphisms of universal hypergraphical automata and isomorphisms of their components: semigroups of input symbols and hypergraphs of states and output symbols.

This paper studies the spectrum of the Hénon map and the spectrum of the baker’s map. The character of fixed points of the Hénon map and randomness of the baker’s map are analyzed. Attractors of the modified Hénon map and the modified baker’s map are considered; cases where attractors are fractal sets are selected.

The paper analyzes the product of m regular permutation groups G₁· . . . · G_m, where m ≥ 2 is a natural number. Each of the regular permutation groups is a subgroup of the symmetric permutation group S(Ω) of degree |Ω| for the set Ω. M. M. Glukhov proved that for k = 2 and m = 2, 2-transitivity of the product G₁· G₂ is equivalent to the absence of zeros in the corresponding square matrix with the number of rows and columns equal to |Ω| − 1. Also M. M. Glukhov has given necessary conditions of 2-transitivity of such a product of regular permutation groups.

In this paper, we consider the general case for any natural m and k such that m ≥ 2 and k ≥ 2. It is proved that k-transitivity of the product of regular permutation groups G₁· . . . · G_m is equivalent to the absence of zeros in the square matrix with the number of rows and columns equal to (|Ω| − 1)!/(|Ω| − k)!. We obtain correlation between the number of arcs corresponding to this matrix and a natural number l such that the product (PsQt)^l is 2-transitive, where P,Q ⊆ S(Ω) are some regular permutation groups and the permutation st is an (|Ω| − 1)-cycle. We provide an example of the building of AES ciphers such that their round transformations are k-transitive on a number of rounds.