A functional system is a set of functions endowed with a set of operations on these functions.
The operations allow one to obtain new functions from the existing ones.
Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.
The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.
In our paper we consider the functional system of rational functions with rational coefficients endowed with the superposition operation. We investigate the problem of bases of complete systems, namely:
Does every complete system have a (finite) basis? ;
For any positive integer n, is there a basis of a complete system consisting of n functions?
a number of examples of basis consisting of n functions are presented explicitly
(𝑛 = 1, 2, 3, ...).

The answers to all these questions are positive, which is the main result of this article.

Let 𝑋− be a sufficiently large real number, 𝑏1, 𝑏2-integers with 1 ⩽ 𝑏1, 𝑏2 ⩽ 𝑋, 𝑎𝑖𝑗 ,(𝑖 = 1, 2; 𝑗 = 1, 4)− positive integers, 𝑝1,. . ., 𝑝4−prime numbers.Let 𝐵 = max {3 |𝑎𝑖𝑗|} , (𝑖 = 1, 2; 𝑗 = 1, 4), ¯𝑏 = (𝑏1, 𝑏2), 𝐾 = 9√2𝐵3⃒ ⃒¯𝑏⃒⃒,𝐸2,4(𝑋) ={︀𝑏𝑖⃒⃒1 ≤ 𝑏𝑖 ≤ 𝑋, 𝑏𝑖 ̸= 𝑎𝑖1𝑝1 + · · · + 𝑎𝑖4𝑝4, 𝑖 = 1, 2}︀.
The paper studies the solvability of a system of linear equations 𝑏𝑖 = 𝑎𝑖1𝑝1+· · ·+𝑎𝑖4𝑝4, 𝑖 = 1, 2,
in primes 𝑝1, . . . , 𝑝4 and for the first time a power estimate for the exceptional set 𝐸2,4(𝑋) and a lower estimate for 𝑅(¯𝑏)− the number of solutions of the system under consideration in prime numbers, are obtained, namely, that if 𝑋 is sufficiently large and 𝛿(0 < 𝛿 < 1) is sufficiently small real numbers, then: there exists a sufficiently large number 𝐴, such that for 𝑋 > 𝐵𝐴 estimate is fair 𝐸2,4(𝑋) < 𝑋2−𝛿; and for 𝑅(¯ 𝑏) given ¯𝑏= (𝑏1, 𝑏2), 1 ⩽ 𝑏1, 𝑏2 ⩽ 𝑋 fair estimate 𝑅(¯ 𝑏) ⩾ 𝐾2−𝛿(ln𝐾)−4, for all ¯𝑏= (𝑏1, 𝑏2) except for at most 𝑋2−𝛿 pairs of them.

In this paper, the rate of convergence of Ces`aro means of the double Fourier series of a 2𝜋-periodic function in each variable and of generalized bounded variation, is estimated. The result obtained is a generalization of a result of S. M. Mazhar for a single Fourier series and of our earlier result for a function of two variables.

A classical property of a non-constant 2𝑟-periodic function on the real axis is that it has no period incommensurable with 𝑟. One of the multidimensional analogues of this statement is the following well-known theorem of L. Zalcman on two radii: for the existence of a nonzero locally summable function 𝑓 : R𝑛 → C with nonzero integrals over all balls of radii 𝑟1 and 𝑟2 in R𝑛 it is necessary and sufficient that 𝑟1/𝑟2 ∈ 𝐸𝑛, where 𝐸𝑛 is the set of all possible ratios of positive zeros of the Bessel function 𝐽𝑛/2. The condition 𝑟1/𝑟2 /∈ 𝐸𝑛is equivalent to the equality 𝒵+(︀̃︀𝜒𝑟1)︀∩ 𝒵+(︀̃︀𝜒𝑟2)︀= ∅, where 𝜒𝑟 is the indicator of the ball 𝐵𝑟 = {𝑥 ∈ R𝑛 : |𝑥| < 𝑟}, ̃︀𝜒𝑟 is the spherical transform (Fourier-Bessel transform) of the indicator 𝜒𝑟, 𝒵+(̃︀𝜒𝑟) is the set of all positive zeros of even entire function ̃︀𝜒𝑟. In terms of convolutions, L. Zalcman’s theorem means that the operator 

$$𝒫𝑓 = (𝑓 * 𝜒𝑟1 , 𝑓 * 𝜒𝑟2 ), 𝑓 ∈ 𝐿^(1,loc) (R𝑛)$$ 

is injective if and only if 𝑟1/𝑟2 /∈ 𝐸𝑛. In this paper, a new formula for the inversion of the operator 𝒫 is found under the condition 𝑟1/𝑟2 /∈ 𝐸𝑛. The result obtained significantly simplifies the previously known procedures for recovering a function 𝑓 from given ball means values 𝑓 *𝜒𝑟1 и 𝑓 * 𝜒𝑟2 . The proofs use the methods of harmonic analysis, as well as the theory of entire and special functions.

The Fermat–Steiner problem is to find all points of the metric space 𝑌 such that the sum of the distances from each of them to points from some fixed finite subset 𝐴 = {𝐴1, . . . ,𝐴𝑛} of the space 𝑌 is minimal. In this paper, this problem is considered in the case when 𝑌 = ℋ(𝑋) is
the space of non-empty compact subsets of a finite-dimensional normed space 𝑋 endowed with the Hausdorff metric, i.e. ℋ(𝑋) is a hyperspace over 𝑋. The set 𝐴 is called boundary, all 𝐴𝑖 are called boundary sets, and the compact sets that realize the minimum of the sum of distances to 𝐴𝑖 are called Steiner compacts.
In this paper, we study the question of stability in the Fermat–Steiner problem when passing from a boundary consisting of finite compact sets 𝐴𝑖 to a boundary consisting of their convex hulls Conv(𝐴𝑖). By stability here we mean that the minimum of the sum of distances 𝑆𝐴 does not change when passing to convex hulls of boundary compact sets.
The paper continued the study of geometric objects, namely, hook sets that arise in the Fermat–Steiner problem. Also three different sufficient conditions for the instability of the boundary from ℋ(𝑋) were derived, two of which are based on the constructed theory of such
sets. For the case of an unstable boundary 𝐴 = {𝐴1, . . . ,𝐴𝑛}, a method was developed to search for deformations of some element from ℋ(𝑋), which lead to compact sets that give a smaller value of the sum of distances to Conv(𝐴𝑖) than 𝑆𝐴.
The theory constructed within the framework of this study was applied to one of the wellknown from recent works boundary 𝐴 ⊂ ℋ(R2), namely, its instability was proved and compact sets were found realizing the sum of distances to Conv(𝐴𝑖), less than 𝑆𝐴.

The question of integrability of the Fourier transform and other integral transformations ℱ(𝑓) on classes of functions in weighted spaces 𝐿𝑝(R𝑑) is a fundamental problem of harmonic analysis. The classical Hausdorff–Young result says that if a function 𝑓 from 𝐿𝑝(R𝑑) with 𝑝 ∈ [1, 2], then its Fourier transform ℱ(𝑓) ∈ 𝐿𝑝′ (R𝑑). For 𝑝 > 2 the Fourier transform in the general situation will be a generalized function. The Fourier transform can be defined as an usual function for 𝑝 > 2 by considering the weighted spaces 𝐿𝑝(R𝑑). In particular, the classical Pitt inequality implies that if 𝑝, 𝑞 ∈ (1,∞), 𝛿 = 𝑑( 1/𝑞 − 1/𝑝′ ), 𝛾 ∈ [(𝛿)+, 𝑑/𝑞 ) and function 𝑓 is integrable in 𝐿𝑝(R𝑑) with power weight |𝑥|𝑝(𝛾−𝛿), then its Fourier transform ℱ(𝑓) belongs to the space 𝐿𝑞(R𝑑) with weight |𝑥|−𝑞𝛾. The case 𝑝 = 𝑞 corresponds to the well-known Hardy–Littlewood inequality.
The question arises of extending the conditions for the integrability of the Fourier transform under additional conditions on the functions. In the one-dimensional case, G. Hardy and J. Littlewood proved that if 𝑓 is an even nonincreasing function tending to zero and 𝑓 ∈ 𝐿𝑝(R)
for 𝑝 ∈ (1,∞), then ℱ(𝑓) belongs to 𝐿𝑝(R) with weight |𝑥|𝑝−2. R. Boas (1972) suggested that for a monotone function 𝑓 the membership | · |𝛾−𝛿𝑓 ∈ 𝐿𝑝(R) is equivalent to | · |−𝛾ℱ(𝑓) ∈ 𝐿𝑝(R) if and only if 𝛾 ∈ (−1/𝑝′ , 1/𝑝 ). The one-dimensional Boas conjecture was proved by Y. Sagher (1976).
D. Gorbachev, E. Liflyand and S. Tikhonov (2011) proved the multidimensional Boas conjecture for radial functions, moreover, on a wider class of general monotone non-negative radial functions 𝑓: ‖| · |−𝛾ℱ(𝑓)‖𝑝 ≍ ‖| · |𝛾−𝛿𝑓‖𝑝 if and only if 𝛾 ∈ ( 𝑑/𝑝 − (𝑑+1)/2, 𝑑/𝑝 ), where
𝛿 = 𝑑( 1/𝑝 − 1/𝑝′ ). For radial functions, the Fourier transform is expressed in terms of the Bessel transform of half-integer order, which reduces to the classical Hankel transform and includes the cosine and sine Fourier transforms. For the latter, the Boas conjecture was proved by E. Liflyand and S. Tikhonov (2008). For the Bessel–Hankel transform with an arbitrary order, the Boas conjecture was proved by L. De Carli, D. Gorbachev and S. Tikhonov (2013). D. Gorbachev, V. Ivanov and S. Tikhonov (2016) generalized these results to the case of (𝜅, 𝑎)-generalized Fourier transform. A. Debernardi (2019) studied the case of the Hankel transform and general monotone alternating functions.
So far, the Boas conjecture has been considered for functions on the semiaxis. In this paper, it is studied on the entire axis. To do this, we consider the integral Dunkl transform, which for even functions reduces to the Bessel–Hankel transform. It is also shown that the Boas conjecture remains valid for the (𝜅, 𝑎)-generalized Fourier transform, which gives the Dunkl transform for 𝑎 = 2. As a result, we have 

$$‖| · |−𝛾ℱ𝜅,𝑎(𝑓)‖𝑝,𝜅,𝑎 ≍ ‖| · |𝛾−𝛿𝑓‖𝑝,𝜅,𝑎,$$ 

where 𝛾 ∈ ( 𝑑_𝜅,𝑎/𝑝 − )𝑑_𝜅,𝑎+𝑎/2)/2 , (𝑑_𝜅,𝑎)/𝑝 ), 𝛿 = 𝑑𝜅,𝑎( 1/𝑝 − 1/𝑝′ ), 𝑑𝜅,𝑎 = 2𝜅 + 𝑎 − 1.

The paper investigates the question of the domain of absolute convergence of the zeta series for some monoids of natural numbers. Two main cases are considered: monoids with 𝐶 power𝜃- density and monoids with 𝐶-logarithmic 𝜃-power density. A new concept is introduced — strong ⃗𝐶= (𝐶1, . . . ,𝐶𝑛) power ⃗𝜃 is the density. For the zeta function of a sequence of natural numbers 𝐴 with a strong ⃗𝐶
= (𝐶1, . . . ,𝐶𝑛)power ⃗𝜃-density proved the theorem according to which the zeta function 𝜁(𝐴|𝛼) is an analytical function of the variable 𝛼, regular at 𝜎 > 0, having 𝑛 poles of the first order, and deductions are found in these poles.
For the case of 𝐶 logarithmic 𝜃-power density, a fundamentally different result is proved: if the monoid 𝑀 has a 𝐶 logarithmic𝜃-power density with 0 < 𝜃 < 1, then the zeta function of the monoid 𝑀 has a holomorphic half-plane 𝜎 > 0 and the imaginary axis is the singularity
line.
In the third section, the question of the analytical continuation of the zeta function of the monoid of natural numbers in three cases is considered: for a monoid of 𝑘-th powers of natural numbers, for a set of natural numbers free of 𝑘-th powers, and for the union of two monoids of 𝑘-th powers of natural numbers when the exponents of the degrees are mutually prime numbers.
In all three cases, it is shown that the analytic continuation exists on the entire complex plane. Functional equations are found for each of the three cases. They all have a fundamentally different look. In addition, new properties of the zeta function that are missing from the Riemann zeta function are found for each analytic continuation in the critical band.
In conclusion, promising, relevant topics for further research are listed.

We consider a Cauchy problem for a system of two quasilinear first order partial differential equations with continuous and bounded free terms. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. The sufficient conditions for the existence and uniqueness of a local solution of the Cauchy problem in the initial coordinates at which the solution has the same smoothness with respect to 𝑥 as the initial
functions of the Cauchy problem are determined. The sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem in the initial coordinates (for a given finite interval 𝑡 ∈ [0, 𝑇]) are determined. Local existence and uniqueness theorem of the solution of the Cauchy problem for a system of quasilinear first order partial differential equations with continuous and bounded free terms is proved with the method of an additional argument. The
investigation of a nonlocal solvability of the Cauchy problem is based on the method of an additional argument. The proof of the nonlocal solvability of the Cauchy problem for a system of quasilinear first order partial differential equations with continuous and bounded free terms
relies on global estimates.

This survey article is devoted to 𝑝-extensions of complete discrete valuation fields of mixed characteristic where 𝑝 is the characteristic of the residue field. It is known that any totally ramified Galois extension with a non-maximal ramification jump can be determined by an
Artin-Schreier equation, and the upper bound for the ramification jump corresponds to the lower bound of the valuation in the right-hand side of the equation. The problem of construction of extensions with arbitrary Galois groups is not solved.

The work is devoted to establishing coercive estimates and proofs of separability theorems for a nonlinear elliptic differential operator of non-divergence form in a weighted space. On the basis of the obtained coercive estimates, the coercive solvability of a nonlinear elliptic differential second-order operator in the space 𝐿2,𝜌(𝑅𝑛) is investigated. The problem of "separability of differential expressions"was first studied by mathematicians V.N.Everitt and M. Girtz. They studied in detail the separability of the Sturm-Liouville operator. Further development of this theory belongs to K.H.Boymatov, M. Otelbaev and their students. Most of the published works on this theory relate to linear operators. There are only some papers that consider nonlinear differential operators, which are weak nonlinear perturbations of linear operators. The case when the operator under study is nonlinear, i.e. it cannot be represented as a weak perturbation of a linear operator, is considered only in some separate papers. The results obtained here also relate to this little-studied case. In this work, the coercive properties of a non-divergence nonlinear elliptic differential operator are studied 

$$𝐿[𝑢] = −Σ︁𝑛𝑖,𝑗=1 𝑎𝑖𝑗(𝑥)𝜕^2𝑢/𝜕𝑥_𝑖𝜕𝑥_𝑗+ 𝑉 (𝑥, 𝑢)𝑢(𝑥),$$ 

in the weight space 𝐿2,𝜌(𝑅𝑛) and on the basis of coercive estimates, its separability in this space is proved. Based on the separability of the considered elliptic operator of nondivergent form, we study the coercive solvability of a nonlinear elliptic differential equation in a weighted Hilbert space 𝐿2,𝜌(𝑅𝑛).

This paper is devoted to obtaining estimates of the type of Bykovsky estimates for the deviation of a generalized parallelepipedal grid. It continues the studies similar to those that we previously performed to assess the quality measure and the quantitative measure of the
parallelepipedal grid.
The main idea used in this paper goes back to the work of V. A. Bykovsky (2002) on estimating the error of approximate integration over parallelepipedal grids and its generalization in the work of O. A. Gorkusha and N. M. Dobrovolsky (2005) for the case of a hyperbolic zeta function of an arbitrary lattice. The central place in these works is played by the Bykovsky set, consisting of local minima of the second kind, and sums over these sets.
As in the work "On Bykovsky estimates for a measure of the quality of optimal coefficients the effect was found that a multiplier with a logarithmic order of growth appears in the deviation estimates, which began to include the definition of the modified Bykovsky sum.
The method of work is to combine the approaches from the work "Estimates of deviations of generalized parallelepipedal grids"(1984) with the approaches of 2005.
Further ways to obtain clarification of the received estimates are outlined.

The representation problem of a natural number 𝑛 in the diagonal quadratic form with four variables 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑧2 + 𝑑𝑡2, where 𝑎, 𝑏, 𝑐, 𝑑 are given positive integers, is considered in this paper. The question is posed to define under what conditions on the coefficients 𝑎, 𝑏, 𝑐, 𝑑 such representation does not exist for a given 𝑛. These conditions, which obtained based on the theory of congruences or without proof, are given in the Kloosterman’s work (1926).
Kloosterman also has obtained an asymptotic formula for the number of solutions to the equation 𝑛 = 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑧2 + 𝑑𝑡2. The main term of this formula is a series +Σ︀∞𝑞=1Φ(𝑞) of a multiplicative function Φ(𝑞) containing the one-dimensional Gaussian sums with coefficients 𝑎,
𝑏, 𝑐, 𝑑. Our work is related to the study of the representation of this special series as a product over primes Π︀ 𝑝|𝑞 (1 + Φ(𝑝) + Φ(𝑝2) + · · · ).
Previously, the authors have been considered the case when 𝑝 ̸= 2. Conditions for the coefficients 𝑎, 𝑏, 𝑐, 𝑑, 𝑛 under which the equation 𝑛 = 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑧2 + 𝑑𝑡2 has no solutions have been proved with using exact formulas for the one-dimensional Gaussian sums, Ramanujan sum and the generalized Ramanujan sum from the power of a prime.
The case for 𝑝 = 2 and 𝑛 odd is considering in this paper. Taking into account formulas for the one-dimensional Gaussian sums from the power of two, the some not previously studied sums that are close to the Kloosterman sum, are appeared. For such sums from the power of
two, we obtained the exact values. This allowed us to give a complete proof of the conditions on the coefficients 𝑎, 𝑏, 𝑐, 𝑑, at least two of which are even. Under these conditions an odd natural number cannot be represented by a diagonal quadratic form with four variables. Note
that some of these conditions are new and are not mentioned in Kloosterman’s work.

This paper covers the student biography period of the outstanding Soviet mathematician, twice Hero of Socialist Labour academician Ivan Matveevich Vinogradov. In particulary, the personal matter of the student of the Saint Petersburg Emperor University I.M. Vinogradov from St. Petersburg Historical archive allows one to reveal the list of his addresses in St. Petersburg during 1910-1914. This paper is a slight modification of the report presented by the authors at the local history conference «The history of Velikiye Luki in faces» dedicated to the life and work of outstanding citizens of Velikiye Luki and to the Year of science and technologies. This conference took place in Velikiye Luki at November 18, 2021 under the support of the Cultural commitee of Velikiye Luki Administration, Public Council on Historical and Cultural Heritage, the Local history society of Velikiye Luki and Central City Library of M.I. Semevsky.
The text of the report (with the different set of photos) was published after the submission to «Chebyshevskii Sbornik» in annual local history almanac «Velikiye Luki Bulletin» and is posted here with kindly permission of D.A. Belyukov, the chief editor of the bulletin.

In this paper theorems on the expression of real numbers on Fibonacci sequence. It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations.
We note that unifiing of an expression of a real number over inverse values of a multiplicaticative system permits to get the estimation of the form 

$$𝑒 −Σ︁𝑛𝑘=0 1/𝑘!=(𝑥_𝑛)/𝑛!,1/(𝑛 + 1)≤ 𝑥𝑛 <1/𝑛.$$

Expressions of numbers over the sequence of inverse of Fibonacci numbers essentially uses these representation throw powers of “the gold section”  𝜙 = (1+√5)/2 .

The article is devoted to the Ramsey — Kass — Koopmans economic growth model. We investigated the monotonicity of the functions 𝐶(𝑡) and 𝐾(𝑡) under a special initial condition.
Our results are obtained using an auxiliary system of differential equations, which is similar to the original system of differential equations arising in the case of constancy of the stationary rate of savings.

In this paper, we consider the KdV equation of negative order with a free term in the class of periodic functions. It is shown that the KdV equation of negative order with a free term in the class of periodic functions can be integrated by the method of the inverse spectral problem.
The evolution of the spectral data of the Sturm-Liouville operator with a periodic potential associated with the solution of a negative-order KdV equation with a free term in the class of periodic functions is determined. The results obtained make it possible to apply the inverse
problem method to the solution of the KdV equation of negative order with a free term in the class of periodic functions.

The ring of polyadic numbers can be defined in several ways. One can introduce a metrizable topology on the ring of integers by counting the set of ideals (𝑚) by a complete system of neighborhoods of zero. The complete system of neighborhoods in the ring of integers is a collection of sets of the form 𝑎 + (𝑚). The operations of addition and multiplication are continuous in this topology and the ring of integers with this topology is a topological ring.
Completion of the resulting topological ring of integers - this is the ring of polyadic numbers.
An equivalent definition is the inverse (projective) limit 

$$ lim ←−𝑚Z/𝑚!Z.$$ 

Let’s recall that the canonical decomposition of the polyadic number 𝜆 has the form

$$𝜆 =∞Σ︁𝑛=0𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z, 0 ≤ 𝑎𝑛 ≤ 𝑛.$$

This series converges in any field of 𝑝− adic numbers Q𝑝 .Denoting the sum of this series in the field Q𝑝 with the symbol 𝜆(𝑝), we get that any polyadic number 𝜆 can be considered as an element of the direct product of rings of integer 𝑝− adic numbers Z𝑝 for all primes 𝑝. The
converse statement is also true, meaning that the ring of polyadic integers coincides with this direct product. However, evidence of the latter claim could not be found. The purpose of this note is to fill this gap. In addition, some applications of polyadic numbers are described.