In English-language literature, Beatty sequence means a sequence of the form [αn] and, more generally, [αn + β], where α is a positive irrational number, β is a real number (if β = 0, then the sequence is called homogeneous, otherwise it is called non-homogeneous). In Russian literature, such sequences are usually referred to as greatest-integer sequences of a special form, or as generalized arithmetic progressions. The properties of these sequences have been under extensive study ever since late 19th century and up to nowadays. This paper contains a review of main directions in Beatty sequences research, and points out some key results. 

The investigation of the distribution of prime numbers in Beatty sequences, once started in 1970s, was continued in 2000s, when due to application of new methods it became possible to improve estimates of remainder terms in asymptotic formulas. A wide range of tasks deal with sums of the values of arithmetical functions over Beatty sequences. Various authors obtained asymptotic formulas for sums of the values of divisor function τ(n) and multidimensional divisor function τk(n), of divisor-summing function σ(n), of Euler function ϕ(n), of Dirichlet characters, of prime divisor counting function ω(n). Besides that, there appeared various results concerning quadratic residues and nonresidues in Beatty sequences. Since 1990s additive tasks associated with Beatty sequences became a topical direction of study. Someanaloguesof classical Goldbachtype problems, where primes belong to Beatty sequences, are under research, along with tasks of representation of integers as a sum, a part of summands of which are members of such a sequence.

The paper presents newly obtained upper and lower bounds for the number of zeros for functions of a special type, as well as an estimate for the measure of the set where these functions attain small values. Let f1 (x), ..., fn (x) be functions differentiable on the interval I, n+1 times and Wronskian from derivatives almost everywhere on I is different from 0. Such functions are called nondegenerate. The problem of the distribution of the zeros of the function F (x) = anfn (x) + ... + a1f1 (x) + a0, aj ∈ Z, 1 ≤ j ≤ n is important in the metric theory of Diophantine approximations.

Let Q > 1 be a sufficiently large integer, and the interval I has length Q−γ, 0 ≤ γ < 1. We obtain upper and lower bounds for the number of zeros of the function F (x) on the interval I, with |aj| ≤ Q, 0 ≤ γ < 1. For γ = 0 such estimates were obtained by A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevitch, N. V. Budarina.

The distribution of algebraic numbers is quite complicated. Probably this is why they are rarely used as dense sets. Nevertheless, A. Baker and W. Schmidt proved in 1970 that the distribution of algebraic numbers still have some kind of uniformity on long intervals, which they called regularity. Recently many works have appeared addressing the problems concerning the lengths of the intervals on which real algebraic numbers have regularity property. It was discovered that for any integer Q > 1 there are intervals of length 0.5Q⁻¹, which don’t contain algebraic numbers of any degree n and of height H(α) ≤ Q. At the same time it’s possible to find such c0 = c0(n) that for any c > c0 algebraic numbers on any interval of length exceeding cQ⁻¹ have regularity property. Such "friendly" to algebraic numbers intervals are intervals free of rational numbers with small denominators and algebraic numbers of small degree and height. In order to find algebraic numbers we build integral polynomials with small values on an interval and large height using Minkowski’s linear forms theorem. It turns out that for "most" points x of an interval these polynomials have similar and nice characteristics (degree, height, module of polynomial value at point x). These characteristics are sufficient for building algebraic numbers on an interval. In this paper we prove existence of algebraic numbers of high degree on “very short“ intervals.

Now it is well known that dynamical systems can be categorized into systems with selfexcited attractors and systems with hidden attractors. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. Hidden attractors play the important role in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing. In addition, complex behaviors of chaotic systems have been applied in various areas from image watermarking, audio encryption scheme, asymmetric color pathological image encryption, chaotic masking communication to random number generator. Recently so-called chameleons systems have been found out by researchers. These systems were so are named for the reason, that they shows self-excited or hidden oscillations depending on the value of parameters entering into them. In the present work the simple algorithm of synthesizing of oneparametrical chameleons systems is offered. Evolution Lyapunov exponents and Kaplan-Yorke dimension of such systems at change of parameter is traced.

The paper is devoted to a survey of the main results obtained in the solution of the Tura´n and Fejer extremal problems on the torus; the Tur´an, Delsarte, Bohmann, and Logan extremal problems on the Euclidean space, half-line, and hyperboloid. We also give results obtained when solving a similar problem on the optimal argument in the module of continuity in the sharp Jackson inequality in the space L² on the Euclidean space and half-line. Most of the results were obtained by the authors of the review. The survey is based on a talk made by V.I. Ivanov at the conference «6th Workshop on Fourier Analysis and Related Fields, Pecs, Hungary, 2431 August 2017». We solve also the problem of the optimal argument on the hyperboloid. As the basic apparatus for solving extremal problems on the half-line, we use the Gauss and Markov quadrature formulae on the half-line with respect to the zeros of the eigenfunctions of the Sturm–Liouville problem. For multidimensional extremal problems we apply a reduction to one-dimensional problems by means of averaging of admissible functions over the Euclidean sphere. Extremal function is unique in all cases.

In this paper we consider a new class of Dirichlet series, the zeta functions of monoids of natural numbers. The inverse Dirichlet series for the zeta function of monoids of natural numbers are studied. It is shown that the existence of an Euler product for the zeta function of a monoid is related to the uniqueness of the factorization into prime factors in this monoid.

The notion of coprime sets of natural numbers is introduced and it is shown that for such sets the multiplicativity of minimal monoids and corresponding zeta-functions of monoids takes place.

It is shown that if all prime elements of a monoid are prime numbers, then the characteristic function of the monoid is a multiplicative function and in this case the zeta function of the monoid is a generalized L-function.

Various examples of monoids and corresponding zeta functions of monoids are considered. The relation between the inversion of the zeta function of a monoid and the generalized Mo¨bius function on a monoid as a partially ordered set is studied by means of the divisibility of natural numbers. A number of properties of the zeta functions of monoids of natural numbers with a unique decomposition into prime factors are obtained.

The paper deals with taking the logarithm of an Eulerian product as a function of a complex argument. It is shown that a continuous function that determines the value of the logarithm of an Euler product runs through all branches of the infinite-valued function of the logarithm near its pole. The corollaries on the value of a complex-valued function of a special form near a singular point are obtained. These properties imply statements about the values of the Riemann zeta function near the boundary of the region of absolute convergence.

Using Bertrand’s postulate, infinite exponential sequences of prime numbers are introduced. It is shown that corresponding zeta-functions of monoids of natural numbers converge absolutely in the whole half-plane with a positive real part. Since such zeta-functions of monoids of natural numbers can be decomposed into an Euler product in the whole region of absolute convergence, they do not have zeros in the entire half-plane with a positive real part.

In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

In the paper we introduce generalized Mumford polynomials describing additive law on generalized Jacobian of singular hyperelliptic curve over the field K of characteristics different from 2, and smooth at infinity and defined in the affine chart by the equation y² = φ(x)²f(x), where f is a square-free polynomial. We describe the relation between the continued fraction expansion of the quadratic irrationalities in the hyperelliptic function field K(x,√ f(x))  and the generalized Mumford polynomials describing the additive law in the divisor class group of the singular hyperelliptic curve. This correspondence between the continued fraction expansion of the quadratic irrationalities and the generalized Mumford polynomials allow us to prove the theorem on equivalence of two conditions: the condition (i) of quasi-periodicity of continued fraction expansion (related with valuation of a point of degree 1 on the normalization of the curve) of a quadratic irrationality of the special type and the condition (ii) of the finiteness of the order of the class, related to the point of degree 1 on the normalization of the curve. By means of this correspondence we also obtain the results on the symmetry of quasi-period and we give estimates for its length, generalizing results obtained before by the author and collaborators. 

Problem of separation for differential operators was first investigated by W. N. Everitt and M. Giertz in the beginning of seventieth of the last century. They mainly have investigated, in their works, separation of Sturm–Liouville operator operator and its powers. Later, this problem was investigated by K. Kh. Boimatov, M. Otelbaev, F. V. Atcinson, W. D. Evans, A. Zettl and others. The main part of papers published in this direction concerns with the case of linear operators(both ordinary differential operators and partial differential operators). Separation of nonlinear differential operators was mainly investigated in case when operator under consideration was a weak perturbation of linear one. The case when operator under consideration is not a weak perturbation of linear one was investigated only in some works. Results of this paper also concerns with this poorly studied case. The paper is devoted to studying coercive properties of nonlinear differential operator of the form

                        L[u(x)] = −u^{V I}(x) + V (x,u(x))u(x)

in Hilbert space L2(R) and separation theorem for this operator is proved. The investigated operator L[u(x)] is strictly nonlinear, in the sense that in the general case it cannot be represented as a weak perturbation of a linear operator.

The authors carried out the basic parameters calculation of compacting powders of metallic systems (metals, steels, non-ferrous alloys of different alloying systems) during plastic compression deformation. The paper considers variants of calculating the pressures and densities distribution in powder compacts of various cross sections in different coordinate systems. The authors found that the highest density of the rectangular cross-section powder compact after pressing is observed near a die wall under the pressing punch, at the middle of the short side of the rectangle the density is higher than in the middle of the long side; the lowest density is observed at the die wall above the fixed punch, and in the middle of the short side of the rectangle the density is lower than in the middle of the long side; the lowest density under the pressing punch and the highest density over the fixed punch are observed at the centers of the corresponding rectangular sections. It is shown that in elliptical section compacts of the powders of metal systems after pressing at positive value ζ (ζ > 0) the pressure and density at the ends of the ellipse major axis are higher than at the ends of the minor axis and, conversely, when the ζ < 0 pressure and density at the ends of the major axis are less than at the ends of the minor axis. The obtained results can be used to develop low-waste and resource-saving processes and technologies for processing industrial materials under various conditions and states.

One well-known approach to the problem of analytic continuation of Dirichlet series is analysis of properties of a sequence of primitive integrals, which arise in iterations of a summatory function of the coefficients of these series. With this approach it was possible to obtain an analytic continuation of the Riemann zeta function and Dirichlet L-functions. In 1975 N. G. Chudakov presented necessary and sufficient conditions for an analytic continuation of Dirichlet series as meromorphic functions with a finite Lindelo¨f function, expressed through behavior of primitive integrals.

In this paper we formulate necessary and sufficient conditions of analytic continuation of Dirichlet series with finite-valued coefficients to an entire function. These conditions are expressed in terms of behavior of Cesa`ro means of coefficients of a Dirichlet series. Unlike the result of N. G. Chudakov, where conditions of analytic continuation are expressed as an existence theorem, in this paper we obtain an explicit form of the asymptotics of Cesa`ro means. This result is based on the approximation approach developed earlier by V. N. Kuznetsov and the author, which made it possible to establish a connection between the solution of this problem and a possibility to approximate entire functions defined by Dirichlet series by Dirichlet polynomials in the critical strip.

In this paper we study the analytic properties of Dirichlet L -functions in the critical strip, characteristic for almost periodic functions. The research is based on Approximation approach, consisting in the construction of Dirichlet polynomials, which are almost periodic functions, "rapidly convergent"in the critical strip to Dirichlet L -functions.

On this path, for any rectangle lying in the critical strip, the existence of ε -almost period for the Dirichlet L-function, we obtain the estimate constants of uniform continuity. Issues related to studying other properties of Dirichlet L -functions are discussed.

Almost nilpotent varieties of nonassociative algebras over a field of zero characteristic in the class of all algebras satisfying identical relation x(yz) ≡ 0 are studied. Earlier in this class of algebras for each natural number m ≥ 2 the algebra Am generating the almost nilpotent variety var(A_m) of exponential growth with exponent of m was defined. In the paper numerical characteristics of varieties var(A_m) are studied.

To this end in the relatively free algebras of the varieties var(A_m) the spaces of multilinear elements corresponding to left normed polynomials with fixed variable on the first position are considered.

Each space is considered as completely reducible module of the symmetric group and multiplicities in the decomposition of the corresponding cocharacter into sum of irreducible characters are calculated. The multiplicities corresponding to the multilinear parts of relatively free algebras of the variety var(A_m) are defined by the calculated values. Colengths of the varieties var(A_m), m≥ 2 are obtained using this method. For each m ≥ 2 the set of identical relations that defines the variety var(A_m) is obtained.

The paper presents brief information about the untimely death of Yaroslav Andreevich, Vagramenko