Vol 60, No 1 (2019)

We obtain an efficient lower bound of complexity for n-ary functions over a finite field of arbitrary order in the class of polarized polynomials. The complexity of a function is defined as the minimal possible number of nonzero terms in a polarized polynomial realizing the function.

We prove that linear orderings are primitively recursively categorical over a class of structures K_Σ if and only if they contain only finitely many successivities.

An arithmetic graph function is a mapping associating to a finite group G the graph whose vertices are the divisors of |G|. We formulate and study the problem of recognizing hereditary saturated formations by arithmetic graph functions, and solve it for some arithmetic graph functions.

We prove the negativity of separable enumerations of division rings and establish that the effective embeddability of a commutative integral domain in a separably enumerated field is equivalent to its negativity.

We prove the next result. If two isometric regular surfaces with regular boundaries, of an arbitrary finite genus, and positive Gaussian curvature in the three-dimensional Euclidean space, consist of two congruent arcs corresponding under the isometry (lying on the boundaries of these surfaces or inside these surfaces) then these surfaces are congruent.

We construct new radially symmetric exact solutions of the multidimensional nonlinear diffusion equation, which can be expressed in terms of elementary functions, Bessel functions, Jacobi elliptic functions, Lambert W-function, and the exponential integral. We find new self-similar solutions of a spatially one-dimensional parabolic equation similar to the nonlinear heat equation. Our exact solutions can help verify difference schemes and numerical calculations used in the mathematical modeling of processes and phenomena described by these equations.

Necessary and sufficient conditions are found under which the sum of N order bounded disjointness preserving operators is n-disjoint with n and N naturals. It is shown that the decomposition of an order bounded n-disjoint operator into a sum of disjointness preserving operators is unique up to “Boolean permutation,” the meaning of which is clarified in the course of the presentation.

We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in some Banach space. The equation is furnished with the Cauchy data, and the overdetermination condition is the value of some operator at a solution. The existence and uniqueness theorems local in time are established.

This article lays foundations for the theory of vector conjugation boundary value problems on a compact Riemann surface of arbitrary positive genus. The main constructions of the classical theory of vector boundary value problems on the plane are carried over to Riemann surfaces: reduction of the problem to a system of integral equations on a contour, the concepts of companion and adjoint problems, as well as their connection with the original problem, the construction of a meromorphic matrix solution. We show that each vector conjugation boundary value problem reduces to a problem with a triangular coefficient matrix, which in fact reduces the problem to a succession of one-dimensional problems. This reduction to the well-understood one-dimensional problems opens up a path towards a complete construction of the general solution of vector boundary value problems on Riemann surfaces.

We prove the local finiteness of some periodic groups generated by odd transpositions. As a consequence of our results we will show that the Suzuki simple groups Sz(2^2m+1) are recognizable by their spectrum in the class of periodic groups without subgroups isomorphic to D₈, the dihedral group of order 8.