We consider a nonlinear stationary Fokker-Planck-Kolmogorov equation with a parameter. We obtained sufficient conditions for existence of a selection (a map that assigns to every parameter value one of solutions of the equation with this parameter value) with either Souslin or Borel dependence with respect to the parameter.

Using the method of monotone (in the sense of Browder-Minty) operators, global theorems on the existence, uniqueness, and estimates of the solution for various classes of inhomogeneous nonlinear integral equations are proved, in which the operators of fractional (in the sense
of Riemann-Liouville) integration enter linearly or nonlinearly, or these operators contain nonlinearity under the sign of the integral (Gummerstein-type equations). In the latter case, the existence and uniqueness of the solution are established without the coercivity condition on the nonlinearity. In all cases, the conditions found in the work play an important role, under which the fractional integration operators act continuously from the real Lebesgue space 𝐿𝑝(𝑎, 𝑏) to the
spaces conjugate to them and are strictly positive. The theorems proved within the framework of the space 𝐿2(𝑎, 𝑏) cover the corresponding linear equations with integrals of fractional order.
From the obtained estimates, in particular, it directly follows that under the conditions of the proved theorems, the corresponding homogeneous linear and nonlinear integral equations have only a trivial (zero) solution.

This paper considers a new formulation of the interpolation problem, where the interpolating function coincides with the interpolated function on the direct product of an interval and a finite generalized parallelepipedal grid. A new object of study is the interpolation trigonometric polynomial with variable coefficients.

The article is devoted to the development of S. A. Lomov’s regularization method for singularly perturbed problems in the presence of spectral singularities in the limit operator. In particular, a regularized asymptotic solution to the singularly perturbed Cauchy problem for the Schr¨odinger equation is constructed on time intervals containing focal points. Based on the ideas of asymptotic integration of problems with an unstable spectrum, it is indicated how regularizing functions should be introduced, the formalism of the regularization method for the specified type of singularity is described in detail, the justification of this algorithm is carried out and an asymptotic solution of any order with respect to a small parameter is constructed.

This paper investigates the Cauchy problem for a system of parabolic-type equations with constant coefficients and an upper-triangular interaction matrix. A theorem on the structure of the solution is proved, and explicit formulas for the components via the matrix fundamental
solution are obtained.
In the general case of an arbitrary (non-upper-triangular) interaction matrix, an analytical solution of the system cannot be obtained in an explicit closed form analogous to the Poisson formula for the heat equation. The standard method of separation of variables and
representation of the solution via eigenfunctions becomes inapplicable due to the non-selfadjointness of the operator for an arbitrary interaction matrix. This is because the diffusion and interaction operators do not commute in general, necessitating the use of the Dyson series to obtain the solution. The main result consists in establishing necessary and sufficient conditions for the termination of the Dyson series, which makes it possible to obtain an exact solution in
the form of a finite sum.

An asymptotic formula is obtained for the number of representations of a sufficiently large natural number 𝑁 in the form

under the conditions

where 𝑏1, 𝑏2, 𝑏3 and 𝑁 are pairwise coprime natural numbers, and 𝐵𝑖 are arbitrary fixed positive constants.
Previously, a similar asymptotic formula was established under the condition

This paper focuses on the Faber–Schauder system, which serves as a basis in the space of continuous functions on an interval and as a representation system in Lebesgue spaces 𝐿𝑝[0; 1]. We consider constructing function representations through series expansions with
respect to the Faber–Schauder system using orthorecursive expansion. The work presents proofs of convergence theorems for orthorecursive expansions in various functional spaces. Specifically, we show that for any continuous function, its orthorecursive expansion with respect to the Faber–Schauder system converges pointwise to the decomposed function everywhere except possibly at a countable set of points. For the broader class of Lebesgue integrable functions, convergence almost everywhere of orthorecursive expansions is proved. Furthermore, results on convergence in 𝐿𝑝-norm in Lebesgue spaces are obtained. The paper also examines a modified version of the Faber–Schauder system constructed using semitranslates of basis functions. This modification allows to obtain uniform convergence of orthorecursive expansions for continuous functions. A comparison is provided of convergence theorems for orthorecursive expansions with respect to both the standard Faber–Schauder system and its modified version.

Let 𝑇1, . . . , 𝑇𝑚 be pairwise coprime natural numbers. Consider the sequences of integers {𝑎_𝑛}^(𝑘,𝑗)_𝑛 , 𝑗 = 1, . . . ,𝑚, 𝑘 = 1, . . . , 𝑇𝑗 that satisfy the conditions

We assume that the following conditions are met:
1) For each 𝑗 = 1, . . . ,𝑚 let 𝑎(1,𝑗)𝑛 = 1 for all 𝑛.
2) For each 𝑗 = 1, . . . ,𝑚 let the vectors be (︁ 𝑎(𝑘,𝑗) 0 , . . . , 𝑎(𝑘,𝑗) 𝑇𝑗−1 )︁
, 𝑘 = 2, . . . , 𝑇𝑗 are linearly independent with the vector (︁ 𝑎(1,𝑗) 0 , . . . , 𝑎(1,𝑗) 𝑇𝑗−1 )︁ = (1, . . . , 1).
Denote

The experiments conducted demonstrated good statistical properties of the binary digit sequences of these function values. The experimental results are presented in the article.

A. O. Gelfond proved that for coprime 𝑏 − 1 and 𝑑, the sums of digits of 𝑏-ary expansions of natural numbers are uniformly distributed over arithmetic progressions with difference 𝑑. This result was later generalized to expansions over linear recurrent bases and to Ostrowsky expansions.
Eminyan solved the problem of the joint distribution modulo 2 of the sums of digits of binary expansions for pairs of consecutive natural numbers. Eminyan’s theorem can also be easily derived from one old result of Mahler. Later, a similar result was obtained for Zeckendorf
expansions over Fibonacci numbers and, further, for expansions over a wide class of linear recurrent bases.
In the present paper, we consider an analogue of the Mahler-Eminyan problem for Ostrowsky expansions associated with an arbitrary irrational 𝛼. Asymptotic formulas are obtained for the number of numbers not exceeding 𝑋, with a given distribution of the Ostrowsky expansions sums of digits of 𝑛 and 𝑛 + 1 modulo 2. The constants in the main term of the asymptotics are expressed as the sum of some infinite series, whose summands are expressed in terms of the
continued fraction expansion of 𝛼. These constants are explicitly calculated for 𝛼 = (√5−1)/2 .
The proof is based on the geometrization theorem previously obtained by the author and A. A. Zhukova, which describes sets of numbers with a fixed ending of the Ostrowsky expansion, as well as the theory of uniform distribution for the sequence {𝑛𝛼}.

We numerically compare two methods for solving the periodic Poisson equation −Δ𝑢 = 𝑓 on the two-dimensional torus [0, 1)2: the classical two-dimensional FFT on a uniform 𝑁 × 𝑁 grid, and A. V. Rodionov’s algorithm on the parallelepipedal grid 𝑀(𝑎, 𝑝), which reduces the
computation of Fourier coefficients of the solution to a one-imensional DFT of length 𝑝 over the values of the right-hand side at the grid nodes. The comparison of methods is carried out on two model problems with smooth and piecewise-smooth right-hand sides.

We study the applicability of Рунге’s rule for estimating the uniform interpolation error ‖𝑓 −𝑆𝑀(Λ),𝑀*(Λ),𝑓 ‖ for two-dimensional trigonometric interpolation on Korobov parallelepipedal grids. As a indicator we propose comparing the interpolant against its values on the shifted parallelepipedal grid 𝑀(Λ)2𝑁: the indicator reduces to a single inverse FFT of length 2𝑁 and has the same cost as the interpolation. On test functions from the Korobov classes 𝐸22 and 𝐸∞2 , the ratio of the exact error to the Рунге indicator is found to be stable in 𝑁..

In the paper we prove that the topological Baer radical of a right bounded left duo ring coincides with its set of the all topologically nilpotent elements. After that we define topologically nil-Armendariz rings. We prove that every right bounded left duo ring is a topologically nil-Armendariz ring.

The paper considers a new formulation of the problem of studying Korobov lattices.

Pythagorean triangles, or Pythagorean triples, have been the subject of such extensive popular science literature, including articles from Kvant and Kvantik, that it makes no sense to list all the sources in a bibliography; simply searching for the relevant terms in an online search
engine is sufficient. You can find websites that explicitly list all Pythagorean triangles in which the hypotenuse does not exceed a certain large number, such as 1,000,000. Each publication, along with well-known facts for centuries, highlights new developments and intriguing numerical details.
Below, we propose one possible algorithm for enumerating all irreducible Pythagorean triangles, which boils down to dividing them into series, each with its own invariant.

Singularly perturbed parabolic equations in domains with corner points of the boundary are considered. The method of corner boundary functions is used to construct the asymptotics of the solution. The purpose of this work is to identify the features of parabolic equations for which the method of corner boundary functions can be applied.

This paper studies best simultaneous approximations of the second kind for rationally dependent irrationalities. It is shown that, starting from a certain point, they are determined by the convergents of the common measure of these irrationalities, which are multiples of these irrationalities.

Natural numbers representable as the sum of a prime and a power of two with a natural exponent are called Romanov numbers. The study of such numbers was initiated by L. Euler, C. Goldbach [3] and A. de Polignac [1, Th´eor`eme 2, IV]. In his 1934 work [10], N. Romanov proved
that the set of Romanov numbers has positive lower density, i.e., for some constant 𝛼 > 0 every interval [1, 𝑥] for 𝑥 ≥ 4 contains at least 𝛼𝑥 numbers representable as 𝑝 + 2𝑛. This result also generalizes to the case of powers of an arbitrary natural number 𝑎: 𝑝 + 𝑎𝑛, with a constant
depending on 𝑎. The main method of proof consists in combining the Cauchy–Bunyakovsky–Schwarz inequality with sieve methods. Romanov’s theorem admits many generalizations: for instance, G. Rieger [8] proved its analogue for number fields, I. Shparlinski and A. Weingartner [11] established the same result for polynomials over finite fields, and A. Radomskii [7] obtained a series of results on the number of representations of natural numbers as 𝑎 + 𝑏, where 𝑎 is an element of a sifted set (e.g., a prime or a sum of two squares of integers) and 𝑏 is taken from a more intricately structured set. In particular, the cited work contains results on sums 𝑝 + #𝐸(Fℓ), where 𝑝 and ℓ are primes, and #𝐸(Fℓ) is the number of points of a fixed elliptic curve 𝐸 over the field Fℓ.
Concerning odd numbers that are not Romanov numbers, P. Erd˝os [2] established in 1950 that the upper density of Romanov numbers does not exceed 1/2− 1/(2^(241)·3·5·7·13·17·241). The proof uses
covering systems of congruences to construct an explicit arithmetic progression with difference 2^241 ·3·5·7·13·17·241 that contains no Romanov numbers. In the same work, he formulated the conjecture on the unboundedness of the smallest modulus in a covering system, which received a negative answer only 63 years later in [5]. Erd˝os’s estimate was later lowered to 0.490491 by L. Habsieger and X.-F. Roblot [4].
The present work is also devoted to results about the complement of the set of Romanov numbers. Namely, lower bounds are proved for the length of the largest subinterval of [1,𝑋] containing no Romanov numbers. Theorem 2 provides a general method for obtaining such
lower bounds, depending on an arbitrary set 𝒫 consisting of prime numbers. The main result — Theorem 1 — is proved in two different ways: an elementary unconditional approach uses primitive prime divisors of the numbers 2𝑚−1, while the second approach relies on the Extended Riemann Hypothesis for the zeta functions of a certain family of number fields. The estimates obtained by these two approaches coincide.

As is well known, fuzzy sets are classical objects that generalize classical sets, where the membership of an element is characterized by a function taking values from the interval [0, 1].
Fuzzy sets are often used in various applied research. Fuzzy topology, as an area in mathematics, arose from the synthesis of ideas and concepts of fuzzy set theory and fuzzy logic with the subject of general topology and allows, in particular, to take a new look at many facts of
general topology.
In turn, in the papers of a number of mathematicians fuzzy topology methods are applied to different algebraic systems (groups, vector spaces, etc).
In this note we investigate fuzzy topologies on unary algebras, i. e. universal algebras whose signature operations all have arity 1. Such algebras can be interpreted as acts over semigroups or as algebraic automata without output.
We prove that fuzzy topologies on an arbitrary unary algebra form a complete lattice under inclusion. This lattice is a sublattice of the lattice of all fuzzy topologies on the underlying set of the given algebra. Problems related to the characterization of the properties and structure
of fuzzy topologies lattices on unary algebras are currently open.

In this paper, using Yu. V. Linnik’s discrete ergodic method, we investigate the asymptotic behavior of integer points on a two-sheeted hyperboloid corresponding to second-order matrices with a given even trace 2𝑡 and a growing determinant 𝑚 → ∞. The special case of a matrix with a zero trace relates to previously conducted studies (see [1]–[4]).

Besides the standard canonical representation of a polyadic number

one can consider series of some other type, representing polyadic numbers. Let 𝑝𝑛 be the 𝑛−th prime number. Let

Then any polyadic number 𝛼 can be represented as

This representation allows to obtain examples of globally algebraically independent polyadic numbers simply enough.

In this paper the solvability of a multidimensional additive system of congruences prime modulo for polynomials of several variables is proved. A.A.Karatsuba completely studied the one-dimensional case, getting the right order of the number of summands for solvability of the considering system of congruences.

This article presents a method and a numerical algorithm for estimating the effective nonlinear elastic properties of a porous material under finite strain. The approach, described for the 2D case, is based on the numerical solution of boundary value problems from the nonlinear
theory of elasticity on a representative surface element or a unit cell of a porous material. The fundamental relations of the theory of repeated superposition of large deformations are employed. The boundary value problems are solved using the nonlinear computational kernel of the Fidesys commercial computer-aided engineering (CAE) system. For modeling a material with a regular structure, periodic boundary conditions are applied to its rectangular unit cell in the form of constraints on the displacements of opposite sides. The effective properties are represented by a quadratic dependence of the generalized stress tensor on the strain tensor. Two calculation series for effective properties are performed: for porous copper and porous rubber. The influence of pore pressure on the first- and second-order effective moduli is investigated. It is shown that the second-order elastic moduli are significantly dependent on the applied pressure.
It is concluded that internal pressure must be taken into account when modeling the nonlinear deformation of a porous material.

Due to the practical value of the results of studying sound scattering on bodies of various structures and configurations, checking their consistency is important and relevant. The existing methods for monitoring accuracy of solutions to diffraction problems don’t allow assessing correctness of the data obtained. To achieve this goal, it is proposed to use an energy criterion, which consists in checking fulfilment of the law of conservation of acoustic energy. The energy
characteristic whose balance must be investigated according to this criterion should be a timeaveraged energy flux of a sound wave. The expression for the integral energy criterion represents the equality of an average energy flux in a scattered wave and an average energy flux of the interaction of incident and scattered fields. The problem of diffraction of a plane sound wave on an absolutely rigid stationary sphere served as a model problem to demonstrate the application
of the energy criterion. The energy balance study was carried out for different values of a wave number as well as a radius of the body and the spherical surface surrounding the obstacle.
Based on the analysis of the results obtained, it was found that a high degree of accuracy in fulfilling the energy criterion indicates the correctness of the solution to the posed diffraction problem. It is shown that the use of the energy criterion makes it possible to detect errors in the values of the sought coefficients. The inaccuracy of the found solution was revealed graphically by analyzing the dependence of a discrepancy between the energy indicators on the numerical
parameter characterizing a magnitude of the introduced error. The angular distribution of the period-average energy flux in the scattered wave was studied, and it was determined that a wrong calculation of the coefficients leads to its modification.

In this paper, the problem of diffraction of a cylindrical sound wave on an elastic multilayer cylinder located in a homogeneous ideal compressible liquid is considered.
The primary disturbance field is a cylindrical harmonic sound wave emitted by a linear source. It is assumed that the axes of the source and the diffuser are not parallel and do not lie in the same plane.
The analytical solution of the problem is obtained.