In this paper, we obtain estimates from above and from below the number of zeros of functions of a special kind, as well as an estimate of the measure of the set of points in which such functions take small values. Let f₁ (x), ..., f_n (x) function defined on an interval I, n + 1 times differentiable and Wronskian of derivatives almost everywhere (in the sense of Lebesgue measure) on I different from 0. Such functions are called nondegenerate. The problem of distributing zeros of F (x) = a_nf_n (x) + ... + a₁f₁ (x) + a₀, a_j ∈ Z, 1 ≤ j ≤ n is a generalization of many problems about the distribution of zeros of polynomials is important in the metric theory of Diophantine approximations. An interesting fact is that there is a lot in common in the distribution of roots of the function F (x) and the distribution of zeros of polynomials. For example, the number of zeros of F (x) on a fixed interval does not exceed n, as well as for polynomials — the number of zeros does not exceed the polynomial degree.

Three theorems were proved: on the evaluation of the number of zeros from above, on the evaluation of the number of zeros from below, as well as an auxiliary metric theorem, which is necessary to obtain estimates from below. While obtaining lower bounds method was used for major and minor fields, who introduced V. G. Sprindzuk.

Let Q > 1 be a sufficiently large integer, and the interval I has the length Q^−γ, 0 ≤ γ < 1. Produced estimates on the top and bottom for the number of zeros of the function F (x) on the interval I, with |a_j| ≤ Q, 0 ≤ γ < 1, and also indicate the dependence of this quantity from the interval I. When γ = 0 similar results are available from A. S. Pyartli, V. G. Sprindzhuk, V. I. Bernik, V. V. Beresnevich, N. V. Budarina.

An estimate of the irrationality measure of various transcendental numbers is one of the directions in the theory of Diophantine approximations foundations. Nowadays there is a range of methods which make possible to obtain similar estimates for the values of analytic functions. The most effective method is the adding of various integral constructions; one of the first early constructions is the classical intuitive representation of the Gauss hypergeometric function.

Lower estimates of the irrationality measure of rational numbers logarithms were considered by many foreign authors: А. Baker and G. Wüstholz [4], А. Heimonen, Т. Matala-aho, К. Vaänanen [5], Q. Wu [6], G. Rhin and P. Toffin [7]. In their works they used various integral constructions, giving small linear forms from logarithms and other numbers, calculated asymptotic of integrals and coefficients of the linear forms using the saddle point method, Laplace theorem, evaluated the denominator coefficients of the linear forms using various schemes "reduction of prime numbers". Review of some methods from the theory of diophantine approximation of rational numbers logarithms at that time was introduced in 2004 by V. Zudilin [8].

Then V. Kh. Salikhov in [3] considerably improved estimate of the irrationality measure of ln3, based on the same asymptotic methods, but used a new type of integral construction, which has property of summetry. Subsequently, V. Kh. Salikhov due to usage of already complex symmetrized integral improved estimate of the irrationality measures of π [15]. In the future, this method (as applied to diophantine approximation of logarithms of rational numbers) was developed by his pupils: E. S. Zolotuhina [10, 11], M. Yu. Luchin [12, 13], Е. B. Tomashevskaya [14]. It led to improvement of the irrationality measure estimates for the following numbers:

µ(log(5/3)) ≤ 5.512... [14], µ(log(8/5)) < 5.9897 [12], µ(log(7/5)) ≤ 4.865... [14], µ(log(9/7)) ≤6 3.6455... [10], µ(log(7/4)) < 8.1004 [13].

In this paper due to usage the symmetrized real integral we obtain a new estimate of the irrationality measure of ln3. The previous irrationality measure estimate of ln3 was received in 2014 by Q. Wu and L. Wang [1].

The estimate improvement had resulted from the addition of a special square symmetrized polynomial to the symmetrized polynomials used in the integral construction of K. Wu and L. Wang.

A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. Any such algebra can be considered as partially ordered by the relation of set-theoretic inclusion. For a given set Ω of operations on relations, we denote by V  ar{Ω} [V ar{Ω,⊂}] the variety generated by the algebras [respectively ordered algebras] of relations with operations from Ω. Operations on relations, as a rule, are given by formulas of the first order predicate calculus. Such operations are called logical operations. An important class of logical operations is the class of Diophantine operations. An operation on relations is called Diophantine if it can be defined by a formula containing in its prenex normal form only existential quantifiers and conjunctions. We study algebras of relations with one binary Diophantine operation, i.e., groupoids of relations. As the operation being considered, the Diophantine operation * that is defined in the following way: ρ*σ = {( x,у) ∈ X × X : (∃z)( x, z) ∈ ρ∧( x, z) ∈ σ}. The relation ρ*σ is the result of the cylindrification of the intersection ρ∩σ of the binary relations ρ and σ. In the paper, the finite bases of identities for varieties V ar{*}  and V ar{*,⊂}. are found. The groupoid (A,·) belongs to the variety V ar {*} if and only if it satisfies the identities: xy = уx (1), (xу)2 = xу (2), (xу)у = xу (3), x 2у 2 = x 2у (4), (x 2у 2)z = x 2(у 2z) (5). The partially ordered groupoid (A,·,≤) belongs to the variety V ar{*,⊂}if and only if it satisfies the identities (1) - (5) and the identities: х ≤ x2 (6), xу ≤ x 2 (7). As a consequence, we also obtain a finite basis of identities for the variety V ar{*,∪}.

The problem of modeling the behavior of reinforced concrete cylindrical shell under conditions of radiation exposure is considered. Solving equations for the shell are obtained, taking into account the joint action of the load and radiation exposure. It is believed that the mechanical characteristics of concrete and reinforcement depend on the radiation dose. The work of concrete on tension and compression is also taken into account. In the output, a method is used to replace the discrete reinforcement in each direction with an equivalent layer. To solve the equations, it was suggested to use a time-stepping technique with finding at each step the law of distribution of the irradiation dose over the body of the shell, the swelling deformation, and according to them the laws of distribution of the mechanical characteristics of concrete and reinforcement, taking into account the effect of irradiation and the nature of the stress state.

The article is dedicated to the memory of Oleg Nikolaevich Vvedenskii (1937-1981). O.N. Vvedenskii was a student of the academician I.R. Shafarevich. O.N. Vvedenskii’s research and the results obtained are related to duality in elliptic curves and with the corresponding Galois cohomology over local fields, with Shafarevich-Tate pairing and with other pairings, with local and quasi-local of class fields theory of elliptic curves, with the theory of Abelian varieties of dimension greater than 1, with the theory of commutative formal groups over local fields. The paper presents both the results obtained by O.N. Vvedenskii, and new selected results, developing research in the directions of the fundamental groups of schemes, the principal homogeneous spaces (torsors), and duality. The first part of the article presented here is an introduction both to the results obtained by O.N. Vvedenskii in the direction of duality of Abelian varieties and formal groups, and in new selected results, developing research in the directions of the fundamental groups of schemes, the principal homogeneous spaces (torsors), and duality. The Introduction gives preliminary information and presents the content of the article. In the first section we give a brief survey of selected results on the theory of algebraic, quasialgebraic and proalgebraic groups and group schemes. Further, in Section 2 we present selected results on fundamental groups of algebraic varieties, on fundamental groups of schemes, and in Section 3 - selected results on principal homogeneous spaces (torsors), developing research by O.N. Vvedenskii and other authors. In Section 4 we give information on duality, and in Section 5 the paper presents the results by O.N. Vvedenskii on the arithmetic theory of formal groups and their development. The results of this section, represented over local and quasi-local fields K, over their rings of integers, and over their residue fields k, are connected (1) with the formal structure of Abelian varieties, (2) with commutative formal groups, (3) with corresponding homomorphisms. In the article, algebraic varieties, Abelian schemes, and commutative formal group schemes are defined, as a rule, over local and quasi-local fields, over their rings of integers, and over their residue fields. But these objects are also briey considered over global fields, since O.N. was interested in the subject of algebraic varieties over global fields and he carried out corresponding studies. It is assumed that the characteristic of the residue fields is more than 3, unless otherwise specified.

I am grateful to V.N. Chubarikov for offering to publish the article in Chebyshevskii Sbornik.

Special thanks to N.M. Dobrovolsky for help and support in the process of preparing the article for publication.

For the cosine Fourier transform on the half-line two extremal problems were posed and solved by B. Logan in 1983. In the first problem it was necessary to find a minimal neighborhood of zero outside of which a nontrivial integrable even entire function of exponential type at most τ, having a nonnegative Fourier transform, is nonpositive. In the second problem it was necessary to find a minimal neighborhood of zero outside of which a nontrivial integrable even entire function of exponential type at most τ, having a nonnegative Fourier transform and a zero mean value, is nonnegative. The first Logan problem got the greatest development, because it turned out to be connected with the problem of the optimal argument in the modulus of continuity in the sharp Jackson inequality in the space L² between the value of the best approximation of function by entire functions of exponential type and its modulus of continuity. It was solved for the Fourier transform on Euclidean space and for the Dunkl transform as its generalization, for the Fourier transform over eigenfunctions of the Sturm-Liouville problem on the half-line, and the Fourier transform on the hyperboloid.

The second Logan problem was solved only for the Fourier transform on Euclidean space. In the present paper, it is solved for the Fourier transform over eigenfunctions of the SturmLiouville problem on the half-line, in particular, for the Hankel and Jacobi transforms. As a consequence of these results, using the averaging method of functions over the Euclidean sphere, we obtain a solution of the second Logan problem for the Dunkl transform and the Fourier transform on the hyperboloid. General estimates are obtained using the Gauss quadrature formula over the zeros of the eigenfunctions of the Sturm-Liouville problem on the half-line, which was recently proved by the authors of the paper. In all cases, extremal functions are constructed. Their uniqueness is proved.

The paper continues research on a new class of Dirichlet series — zeta functions of monoids of natural numbers. The inverse Dirichlet series for zeta functions of monoids of natural numbers with unique factorization into prime elements and for zeta-functions of sets of prime elements of monoids with unique factorization into prime elements are studied.

For any β > 1 examples of Dirichlet series with an abscissa of absolute convergence σ =  are constructed. For any natural β > 1 examples of a pair of zeta functions ζ(B|α) and ζ(A_B,β|α) with the equality σ_AB,β = σB/ β  are constructed.

Various examples of monoids and corresponding zeta functions of monoids are considered. A number of properties of the zeta functions of monoids of natural numbers with unique factorization into prime factors are obtained. An explicit form of the inverse series to the zeta-function of the set of primes supplemented by one is found.

An explicit form of the ratio of the Riemann zeta-function to the zeta-function of the set of primes supplemented by one is found.

Nested sequences of monoids generated by primes are considered. For the zeta-functions of these monoids the nesting principle is formulated, which allows to transfer the results about the coefficients of one zeta-functions to the coefficients of other zeta-functions.

In this paper the general form of all monoids of natural numbers with unique factorization into prime factors was described for the first time.

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

The work continues the study of a new class of Dirichlet series — Zeta function of monoids of natural numbers. First of all, we study in detail the Zeta function ζ(M(q)|α) of geometric progress M(q) with the first term equal to 1 and an arbitrary natural denominator q > 1, which is the simplest monoid of natural numbers with a unique decomposition into simple elements of the monoid. For a meromorphic function ζ(M(q)|α) = qα /qα−1 with many poles S(M(q)) ={︂2πik lnq⃒ ⃒ ⃒ ⃒k ∈Z}︂

representations are received:

ζ(M(q)|α) =qα 2 αlnq∞ ∏︁ n=1(︂1 + α2 ln2 q 4π2n2 )︂−1 = 1 2+1 αlnq

+∞ ∑︁ n=12αlnq α2 ln2 q + 4n2π2==qα 2 αlnq 4π2

Γ(︂αilnq 2π )︂Γ(︂−αilnq 2π )︂.

For the Zeta function ζ(M(p~)|α) of the monoid M(p~) with a finite number of primesp~ = (p1,...,pn) the decomposition into an infinite product is obtained

ζ(M(p~)|α) =P(p~)α 2 αnQ(p~)n ∏︁ ν=1∞ ∏︁ m=1(︂1 + α2 ln2 pν 4π2m2 )︂−1 ,

where P(p~) = p1 ...pn, Q(p~) = lnp1 ...lnpn, and a functional equation is found ζ(M(p~)|−α) = (−1)n ζ(M(p~)|α) P(p~)α .

For the monoid of positive integers M*(p~) = N·M−1(p~) with a unique Prime factorization consisting of positive integers n mutually Prime with P(p~) = p1 ...pn, and for the Euler product P(M*(p~)/ alpha), consisting of factors for all primes other than p1,...,pn, a functional equation is found

ζ(M*(p~)|α) = M(p~,α)ζ(M*(p~)|1−α),

where

M(p~,α) = M(α)· M1(p~,α) M1(p~,1−α) , M1(p~,α) = n ∏︁ ν=1(︂1− 1 pα ν)︂.

It is proved that for any infinite set of Prime P1 there is no analytic function equal to

lim /n→∞ζ(M(p~n)|α)

on the whole complex plane.

The protective series conjecture is formulated for any exponential set of PE primes.

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

The paper considers the class of Dirichlet series with multiplicative coefficients defining Functions regular in the right half-plane of the complex plane and admitting Approximation by Dirichlet polynomials in the critical strip. It is shown that the regularity condition on the imaginary axis allows one to analytically continue such series as entire functions on the complex plane.

The proof of this fact is based on the properties of approximation Dirichlet polynomials and the Riemann-Schwartz ideas, embedded in the symmetry principle of analytic continuation functions of a complex variable. The class of Dirichlet series for which Analyticity analysis on the imaginary axis.

It should be noted that the result obtained in the work has a direct relation to the solution of the well-known problem of generalized characters posed by Y. V. Linnik and N. G. Chudakov in the 1950s.

The approach indicated in the paper in the problem of analytic continuation of Dirichlet series with numerical properties admits a generalization to Dirichlet series with characters of numeric fields. This encourages credit continuation without using the functional equation of the Dirichlet L-functions of numeric fields on the complex plane.

We also note that the class of Dirichlet series studied in this paper belongs to the Dirichlet series whose coefficients are determined by non-principal generalized characters. It can be shown that for these series the condition of analytic continuation. As far back as 1984, V. N. Kuznetsov showed that in the case of an analytic continuation of such series in an integral way onto the complex plane determined by the order of growth of the module, then Chudakov’s hypothesis that the generalized character is a Dirichlet character will take place. But the final solution of the problem of generalized characters, put in 1950 by Y. V. Linnik and N. G. Chudakov, will be given in the following papers of the authors.

After Voronin’s work of 1975, it is known that some of zeta and L-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Two cases of shifts, continuous and discrete, are considered.

The present paper is devoted to the universality of Lerch zeta-functions L(λ,α,s), s = σ+it, which are defined, for σ > 1, by the Dirichlet series with terms e^2πiλm(m+α)^−s with parameters λ ∈ R and α, 0 < α ≤ 1, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions f₁(s),...,f_r(s) simultaneously is approximated by shifts L(λ₁,α₁,s+ikh),...,L(λ_r,α_r,s+ikh), k = 0,1,2,..., where h > 0 is a fixed number. For this, the linear independence over the field of rational numbers for the set {(log(m + α_j) : m ∈N₀, j = 1,...,r), }is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.

The paper provides the review of calculation method and basic parameters of moulding processes in dilatant materials which are typical representatives of powder metal systems of different chemical compositions. They are based on mathematical models that use not only qualitative explanation, but also quantitative description of the dilatancy effect. The work shows the complete system of basic plasticity theory equations of the rigid-plastic isotropic dilatant media. It considers an example of the steady-state plastic flow calculation under conditions of axisymmetric deformation. It is shown that for axisymmetric deformation the equations relative to velocity vector projection on the characteristic directions are similar to the equations for planar deformation. It is established that the current yield conditions with varying degrees of accuracy describe the types of dilatancy (loosening and compaction). Therefore, for a more precise solution of some problems, it is necessary to refine the mathematical models of the yield condition. For some processes of plastic shaping when solving the system of equations of dilatant media, it is expedient to represent the flow conditions in the form of separate regions: hyperbolic, parabolic and elliptic.

In paper is considered the development of the concept of "artinian"for Lie algebras. The concept of artinian was introduced for associative rings with the minimality condition. At the same time, it extended to modules and subalgebras. A little later they began to consider Artinian Jordan algebras. For such algebras the role of a one-sided ideal is played by a quadratic ideal or, as N. Djecobson called it, the inner ideal. Artinian for Lie algebras through ideals determined Yu.A. Bakhturin, S.A. Pikhtilkov and V.М. Polyakov. They considered special Artinian Lie algebras. S.A. Pikhtilkov applied Artinian Lie algebras to construct the structural theory of special Lie algebras. Georgia Benkart defined the artinian for Lie algebras through inner ideals. F. Lopez, E. Garcia, G. Lozano explored the concept of the inner ideal applied to artinian with the help of Jordan pairs. The definition of artinian for Lie algebras in this paper is presented in three senses: via subalgebras, ideals, and inner ideals. The relationship established between these definitions is established by the authors earlier. Examples of Artinian Lie algebras are considered. The application of Artinian Lie algebras to the solution of the Mikhalev problem is described: the primary radical of the Artinian Lie algebra is solvable.

According to A.I. Maltsev, a set of linear algebras in which a fixed set of identities is called a variety. Using the language of the theory of Lie algebras, we say that the algebra is metabelian if it satisfies the identity (xy)(zt) ≡ 0. A variety is called Specht if it is such a variety and any of its subvariety has a finite basis of identities. Codimension growth is determined by sequence of dimensions multilinear parts of a relatively free algebra of a variety. This sequence is called a sequence codimensions, referring to the multilinear spaces of the ideal identities of the variety. This article presents the results related to the problem of fractional polynomial growth. The review gives new examples of such varieties, and also give a new example of a variety with an infinite basis of identities.

Sequence classification problems often arise in such areas as bioinformatics and natural language processing. In the last few year best results in this field were achieved by the deep learning methods, especially by architectures based on recurrent neural networks (RNN). However, the common problem of such models is a lack of interpretability, i.e., extraction of key features from data that affect the most the model’s decision. Meanwhile, using of less complicated neural network leads to decreasing predictive performance thus limiting usage of state-of-art machine learning methods in many subject areas. In this work we propose a novel interpretable deep learning architecture based on extraction of principal sets of short substrings — sequence motifs. The presence of extracted motif in the input sequence is a marker for a certain class. The key component of proposed solution is differential alignment algorithm developed by us, which provides a smooth analog of classical string comparison methods such as Levenshtein edit distance, and Smith–Waterman local alignment. Unlike previous works devoted to the motif based classification, which used CNN for shift-invariant searching, ours model provide a way to shift and gap invariant extraction of motifs.

The paper is devoted to the study of trigonometric sums of algebraic grids with weights, which play a Central role in the modification of K. K. Frolov’s method proposed by N. M. Dobrovolsky in 1984. The trigonometric sum of the algebraic grid with weights for the vector ⃗m = ⃗0 is naturally called the weighted number of points of the algebraic grid.

In the introduction of this paper, the justification of the relevance of the research topic is proposed, the necessary definitions and facts from the modern theory of K. K. Frolov’s method are given, an important theorem on the decomposition of the trigonometric sum of an algebraic grid with weights in a row by points of an algebraic grid is proved. In the section "Auxiliary lemmas"the necessary facts from the theory of weight functions of a special kind which play a principal role in modification of H. M. Dobrovolsky are given without proof. method K. K. Frolov.

Using a theorem on the decomposition of the trigonometric sum of an algebraic grid with weights in a row by points of an algebraic grid and a Lemma on the value of a trigonometric integral of the weight function, we derive an asymptotic formula for the weighted number of points of an algebraic grid with a special weight function of order 2, which States that such a number tends to unity.

Similarly, it is shown that when the determinant of an algebraic lattice grows for any vector ⃗m ≠ ⃗0, the trigonometric sum of algebraic grids with weights given by the special weight function tends to 0.

For simplicity, only the case of the simplest weight function of order 2 is considered in the main text of the article.

In conclusion, we formulate without proof similar statements about the values of trigonometric sums of algebraic grids with special weight functions of the order r + 1 for any natural R.

Namely, it is argued that for the weighted number of points of algebraic nets with a special weight function r is true desire-to-1 with the residual member of the order s−1 of the logarithm of the determinant is an algebraic lattice, divided by r + 1 the degree of the determinant is an algebraic lattice. A similar statement is true about the tendency to zero the trigonometric sum of an algebraic grid with weights given by a special weight function of the order r + 1.

The solution of the diffraction problem for a plane sound wave on an elastic ellipsoid E with an outer inhomogeneous layer is presented. The ellipsoid is in a half-space filled with an ideal fluid. The boundary of a half-space Π is an acoustically rigid or acoustically soft surface.

To obtain a solution, the area occupied by the liquid is expanded to full space. An additional scattering obstacle is introduced. This obstacle is a copy of E, located mirror-wise with respect to the plane Π. A second incident plane wave is also added. This wave ensures the fulfillment of that condition at the points of the plane Π, which corresponds to the type of the half-space boundary in the initial formulation of the problem. Thus, the problem is transformed into the problem of scattering of two plane sound waves on two ellipsoids in unbounded space.

The solution is based on the linear theory of elasticity and the model of propagation of small vibrations in an ideal fluid. In the outer part of the environment, the solution is sought analytically in the form of an expansion in spherical harmonics and Bessel functions. In the spherical region, which includes two ellipsoids and an adjacent layer of liquid, the finite element method (FEM) is used. The results of the calculation of the directivity patterns of the scattered sound field in the far zone are presented.

These dependences show the influence of the geometric and material parameters of the ellipsoid on the diffraction of sound.

In paper the analytical solution of a problem about diffraction of a plane sound wave on two uniform elastic cylinders with radially non-uniform coatings is received. Analytic expressions are obtained which describe the wave fields in the containing medium and the homogeneous elastic bodies. The boundary-value problem for the system of ordinary differential equations of the second order is constructed for determination of the displacement fields in non-uniform coatings.

By means of an continuous-non-uniform elastic coatings it is possible to change effectively scattering performances of bodies in determinate directions if to pick up corresponding the inhomogeneity laws for mechanical parametres of a coating. The problem is of interest for analysis of sound diffraction on a lattice of cylindrical bodies and also serves as a necessary element of solution by a method of imaginary sources of a problem about sound diffraction on the single homogeneous elastic cylinder with the non-uniform covering which is close to acoustically soft or absolutely rigid flat surfaces.