It is well known that the ring of integers Z is an ????-ring, therefore it is possible to define unique (up to isomorphism) structure of a ring with identity on the additive group Z. A natural question arises about the uniqueness of the ring structure with identity constructed on a multiplicative monoid Z. It is shown in this paper that this question is solved negatively. Moreover, a method of construction new various ring structures on the multiplicative monoid Z by dint of multiplicative automorphisms was developed and described. The concept of basis was introduced for the multiplicative monoid Z, and it was shown that there are no bases (up to sign) that are differ to a basis consists of all prime numbers, and bases that are obtain of that basis by a permutations of its elements. The example of construction a new ring structure on the set Z for fixed standart multiplication is given in the end of this paper. The new addition on the multiplicative monoid Z is obtained by a permutation of prime numbers (it is 2 ↦→ 3 ↦→ 5 ↦→ 2 permutation in the detailed example). From the results obtained in the paper it follows in particular, that the ring Z is not an unique addition ring (UA-ring).

It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J. C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems.

The Tur´an, Fej´er, Delsarte, Bohman, and Logan extremal problems for positive definite functions in Euclidean space or for functions with nonnegative Fourier transform have many applications in the theory of functions, approximation theory, probability theory, and metric geometry. Since the extremal functions in them are radial, by means of averaging over the Euclidean sphere they admit a reduction to analogous problems for the Hankel transform on the half-line. For the solution of these problems we can use the Gauss and Markov quadrature formulae on the half-line at zeros of the Bessel function, constructed by Frappier and Olivier.

The normalized Bessel function, as the kernel of the Hankel transform, is the solution of the Sturm–Liouville problem with power weight. Another important example is the Jacobi transform, the kernel of which is the solution of the Sturm–Liouville problem with hyperbolic weight. The authors of the paper recently constructed the Gauss and Markov quadrature formulae on the half-line at zeros of the eigenfunctions of the Sturm–Liouville problem under natural conditions on the weight function, which, in particular, are satisfied for power and hyperbolic weights.

Under these conditions on the weight function, the Tur´an, Fej´er, Delsarte, Bohman, and Logan extremal problems for the Fourier transform over eigenfunctions of the Sturm–Liouville problem are solved. Extremal functions are constructed. For the Tur´an, Fej´er, Bohman, and Logan problems their uniqueness is proved. 

The work builds on the algebraic theory of polynomials Tue. The theory is based on the study of submodules of Z[????]-module Z[????] 2 . Considers submodules that are defined by one defining relation and one defining relation ????-th order. More complex submodule is the submodule given by one polynomial relation. Sub par Tue ????-th order are directly connected with polynomials Tue ????-th order. Using the algebraic theory of pairs of submodules of Tue ????-th order managed to obtain a new proof of the theorem of M. N. Dobrowolski (senior) that for each ???? there are two fundamental polynomial Tue ????-th order, which are expressed through others. Basic polynomials are determined with an accuracy of unimodular polynomial matrices over the ring of integer polynomials.

In the work introduced linear-fractional conversion of TDP-forms. It is shown that the transition from TDP-forms associated with an algebraic number ???? to TDP-the form associated with the residual fraction to algebraic number ????, TDP-form is converted under the law, similar to the transformation of minimal polynomials and the numerators and denominators of the respective pairs of Tue is converted using the linear-fractional transformations of the second kind. 

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions.

It is shown that for purely real algebraic irrationalities ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of given algebraic irrationalities.

The definition of the generalized number of Piso, which differs from the definition of numbers he’s also the lack of any requirement of integrality.

It is shown that for arbitrary real algebraic irrationals ???? of degree ???? > 2, starting from some number ????0 = ????0(????), the sequence of residual fractions ???????? is a sequence of generalized numbers Piso.

Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction ???????? are concentrated about fractions − ????????−2 ????????−1 is either in the interval of radius ???? (︁ 1 ????2 ????−1 )︁ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number.

It is established that, starting from some number ????0 = ????0(????), fair recurrent formula for incomplete private ???????? expansions of real algebraic irrationals ????, Express ???????? using the values of the minimal polynomial ????????−1(????) for residual fractions ????????−1 and its derivative at the point ????????−1.

Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations. Composition this fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of ???? in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction − ????????−2 ????????−1 .

It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly.

In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number ???? and its limit points. 

The article provides an overview of the system GeoGebra is designed to perform various transformations of objects on the plane using built-in tools and commands. Transformations that can be implemented with tools and commands – a mirror image of the object relative to the direct (axial symmetry), reflection of an object about a point (сentral symmetry), inversion relative to the circle, rotate around a point, parallel to the migration on a vector, homothety relative to the point. In addition, with commands, Shear and Stretch, not having tools analogues, you can make variety shifts of objects along and across of directions.

Induced transformation of almost Hermitian structure for linear extension of the manifold with almost contact metric structure was considered in this paper. We got formulas for induced transformation of almost Hermitian structure for linear extension of the smooth manifold with almost contact metric structure. There exist four equations for the Gray–Hervella’s classification of the smooth manifolds with almost Hermitian structures. In this paper we studied invariance of these equations. One equation is invariant. The conditions of invariance for three other equations were got in this paper. These equations defined sixteen classes of the smooth manifolds with almost Hermitian structure. In this paper we studied invariance for these classes. One class is invariant. Six classes are invariant if and only if exterior differential of function of induced transformation is contained in the second fundamental distribution. Other classes are invariant if and only if the function of induced transformation is constant.

The paper contains a classification of congruence-coherent Rees algebras and algebras with an operator. The concept of coherence was introduced by D.Geiger. An algebra A is called coherent if each of its subalgebras containing a class of some congruence on A is a union of such classes.

In Section 3 conditions for the absence of congruence-coherence property for algebras having proper subalgebras are found. Necessary condition of congruence-coherence for Rees algebras are obtained. Sufficient condition of congruence-coherence for algebras with an operator are obtained. In this section we give a complete classification of congruence-coherent unars.

In Section 4 some modification of the congruence-coherent is considered. The concept of weak and locally coherence was introduced by I.Chajda. An algebra A with a nullary operation 0 is called weakly coherent if each of its subalgebras including the kernel of some congruence on A is a union of classes of this congruence. An algebra A with a nullary operation 0 is called locally coherent if each of its subalgebras including a class of some congruence on A also includes a class the kernel of this congruence. Section 4 is devoted to proving sufficient conditions for algebras with an operator being weakly and locally coherent.

In Section 5 deals with algebras ⟨A,d,f⟩ with one ternary operation d(x,y,z) and one unary operation f acting as endomorphism with respect to the operation d(x,y,z). Ternary operation d(x,y,z) was defined according to the approach offered by V.K. Kartashov. Necessary and sufficient conditions of congruence-coherent for algebras ⟨A,d,f⟩ are obtained. Also, necessary and sufficient conditions of weakly and locally coherent for algebras ⟨A,d,f,0⟩ with nullary operation 0 for which f(0) = 0 are obtained.

In this paper we consider such generalizations of conformal transformations for contact metric manifolds as generalized conformal transformations, ????-transformations, generalized ????- transformations. Components of tensor fields for almost contact metric structure are given. These components are found in A-frame. Components for the tensor of affine deformation by Riemannian connection are calculated in this paper.We study six structure tensors of almost contact metric manifold.They are not invariant under generalized conformal transformations.We consider a particular case of the generalized conformal transformation, i.e. ????- transformation, third, fifth structure tensors are invariant under this transformation. Conditions of invariance for other structure tensors are received. The invariance of six structure tensors under generalized ????-transformations is studied. The second structure tensor is invariant under the generalized ????-transformation. Vanishing of third and fifth structure tensors is invariant under this transformation.We got the conditions of invariance under these transformations for first structured tensor under generalized ????-transformation.

In paper we studied almost Hermitian structures of total space of principal fiber T¹bundle with flat connection over some classes of almost contact metric manifolds, such as contact, K−contact, Sasakian, normal, cosymplectic, nearly cosymplectic, exactly cosymplectic and weakly cosymplectic manifolds. Over contact and K−contact manifolds almost Hermitian structure belongs to the W₂⊕W₄ class. Lee’s form is different from the form of the flat connection by constant factor, equal to −2. Moreover, dual Lee’s vector field is different from some vector field from vertical distribution by the same constant factor. Also, this almost Hermitian structure is local conformal almost Kahlerian. Over Sasakian manifolds almost Hermitian structure belongs to the W₄ class. Lee’s form is different from the form of the flat connection by constant factor, equal to 2. Moreover, dual Lee’s vector field also is different from some vector field from vertical distribution by the same constant factor. Over weakly cosymplectic manifolds almost Hermitian structure is semiKahlerian. Lee’s form and dual Lee’s vector field are identically zero. Over cosymplectic manifolds almost Hermitian structure is Kahlerian. Also, Lee’s form and dual Lee’s vector field are identically zero. Over normal manifolds almost Hermitian structure is Hermitian. Over exactly cosymplectic manifolds almost Hermitian structure is G₁ almost Hermitian structure, and over nearly cosymplectic manifolds almost Hermitian structure is G₂ almost Hermitian structure.

One way to study the properties of rings, algebras, Lie algebras and their ideals presupposes their description via the properties of modules over these rings, algebras, Lie algebras. This article deals with the study of radicals of Lie algebras. We discuss the possibility of homological descriptions of the Jacobson radical of Lie algebras and nilpotent radical of the special Lie algebra.

The first section introduces the concepts of radicals of Lie algebras.

The second section is devoted to the Jacobson radical of Lie algebras. It is proved that the intersection of all annihilators of irreducible modules over an arbitrary Lie algebra L coincides with the intersection of the Lie algebras L and the Jacobson radical of the universal enveloping algebra. This section contains examples that prove this fact. This examples allows to prove the equality of the nilpotent radical of PI-irreducible represented radical of finite-dimensional Lie algebra over a field of characteristic zero. We find the correlation between the locally nilpotent radical and others radicals of Lie algebras such that the irreducible represented radical, the PI-irreducible represented radical and the finitely irreducible represented radical.

In the third section it is shown that the locally nilpotent radical is included in the PI-irreducible represented radical for an arbitrary special Lie algebra L over a field F of characteristics zero. We have proved that the prime radical is not included in the PI-irreducible represented radical. The reverse inclusion for these radicals does not hold. The PI-irreducible represented radical is not locally solvable in the general case. Shows an example of a special Lie algebra L over a field F with the locally nilpotent radical, which has is equal to zero. 

In this paper we present new two-sided estimates of gamma-function Γ(x + 1) on the real semiaxis x > 0. Based on this result, we improve well-known estimates for the factorial n!, which hold for all n ≥ 1. Some of obtained estimates of gamma-function Γ(x+1) hold only for x ≥ 1/2 and some only for x ≥ 1. The main estimates are connected to the notion of alternation round of a function by asymptotic series in the strong sense. However such a strong alternation is proved only for several partial sums. We have a conjecture that the asymptotic series alternates round a logarithm of gamma-function in strong sense. Similary we propose new inequalities for the number of n-combination from 2n. These considerations indicate that next investigation is promissing and give a method for construction of new two-sided estimates for functions having alternating asymptotic series.

When modeling an extensive class of technical systems, the mathematical apparatus of queuing systems (QMS) is widely used. An example of such a system is the computer network, where computer applications are generated and executed. Applications are generated usually not regularly, but by accident, forming the so-called random order of applications (requirements). Service requests, it also continues some random time. One of the central issues in the organization of mass-service systems is the elucidation of the regularities that subordinate the moments when system requirements for service are submitted.

The article explores the flow of events in technical systems of various purposes. On the basis of the fact that under the Poisson character of the flow mathematical modeling of the systems is greatly simplified, the problem of obtaining a simple criterion for determining the degree of approximation of the flow of events to a Poisson one is posed. Pearson’s criterion, regression, correlation and parametric criteria were investigated. A criterion based on the calculation of the waiting function was obtained again. On the example of the study of the system with "competitions"it is shown that the flow of events generated by the system tends to Poisson with an infinite increase in the number of "competing"subjects. 

An associative ring R is called an E-ring if all endomorphisms of its additive group R⁺ are left multiplications, that is, for any α ∈ EndR⁺ there is r ∈ R such that α(x) = x · r for all x ∈ R. E-rings were introduced in 1973 by P. Schultz. A lot of articles are devoted to E-rings. But most of them are considered torsion free E-rings. In this work we consider E-rings (including mixed rings) whose ranks do not exceed 2. It is well known that an E-ring of rank 0 is exactly a ring classes of residues. It is proved that E-rings of rank 1 coincide with infinite T-ring (with rings R_χ). The main result of the paper is the description of E-rings of rank 2. Namely, it is proved that an E-ring R of rank 2 or decomposes into a direct sum of E-rings of rank 1, or R = Zm ⊕ J, where J is an m-divisible torsion free E-ring, or ring R is S-pure embedded in the ring ^∏︀t_p(R). In addition, we obtain some results about nilradical of a mixed

p∈S E-ring.

By the quasi-endomorphism ring ℰ(G) of a torsion-free Abelian group G of finite rank we mean divisible hull of the endomorphism ring of the group. The elements of ℰ(G) is called quasiendomorphisms of G. Thus the quasi-endomorphisms of the group G is normal endomorphisms, which formally divided by non-zero integers.

In the paper it is considered quasi-endomorphism rings of class of strongly indecomposable torsion-free Abelian groups of rank 4 with one τ-adic relation, whose pseudo-socles have rank 1. Let τ = [(m_p)] be a fixed type, where m_p is a non-negative integer or the symbol ∞, indexed by elemets of P, the set of primes numbers. Denote by K_p = Zpm_p the residue class ring modulo p^mp in the case m_p < ∞ and ring of p-adic integers if m_p = ∞. We use the description of the groups from the above class up to quasi-isomorphism in terms of four-dimension over the field of rational numbers Q subspaces of algebra Q(τ) = Q⊗^∏︀_p∈pK_p. The existing relationship between the quasi-endomorphisms of a group G of this class and endomorphisms of the corresponding of this group subspace U of the algebra Q(τ) allows us to represent the quasi-endomorphisms of the group G in the form of a matrices of order 4 over the field of rational numbers.

In this paper, a classification of the quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 4 with one τ-adic relation, whose pseudosocles have rank 1, is obtained. It is proved that, up to isomorphism, there exist two algebras and one infinite series of algebras with rational parameter, which are realized as quasi-endomorphism rings of groups of this class.

In this paper we study mixed modules, with the following property: every homogeneous function of several variables of a module is additive. By a homogeneous function we mean any mapping of the direct sum of a finite number of copies of a module into the module itself that commutes with the endomorphisms of the given module. In the universal algebra, the algebraic structure is said to be endoprimal if all its term-functions commute with endomorphisms. It is well-known that each endodualizable finite algebra is endoprimal. Some authors have studied endoprimal algebras in varieties of vector spaces, semilattices, Boolean algebras, Stone algebras, Heyting algebras, and Abelian groups. In this article, the links between endoprimality and the properties of the multiplicative semigroup of the endomorphism ring of a module, which the author started earlier. Classes of mixed non-reduced splitting modules and reduced modules over commutative Dedekind ring have been investigated. Links between this problem and the property of unique additivity has been shown.

We study character sums over shifted powers modulo a prime p. Such sums can be viewed as generalizations of character sums over shifted multiplicative subgroups. We obtain some new results on upper estimates for absolute value of these sums. The case when the cardinality of subgroup is less than √p, it is a question of non-trivial upper bounds for such sums that remains open and is unsolved today. It was proposed by J. Burgain and M. Ch. Chang in the review of 2010. Nevertheless, some intermediate results were achieved by Professor K. Gong, who established non-trivial estimates of such sums in the case when the subgroup is much larger than √p. In this paper, we obtain some new results on the upper bound for the absolute value of the generalization of such sums, which are incomplete sums of character sums over shifted subgroups. Two proofs of the main result are given. The first one is based on reduction of this sum to the well-known estimate of A. Weil and the method of smoothing such sums. The method of estimating the incomplete sum through the full one is also applied. One result of M. Z. Garaev is also used. The second proof is based on the original idea of I. M. Vinogradov. This approach was proposed to refine the known inequality of Poya-Vinogradov and uses in its essence some geometric and combinatorial ideas. The second proof is not fully presented. We only prove a key statement, and for the rest of the calculations we refer the reader to the initial work of I. M. Vinogradov.

This article considers activities of an outstanding expert in the field of algorithmic issues of algebra and mathematical logic Professor, Doctor of Physical and Mathematical Sciences, head of the Chair of Computer Security and Mathematical Methods of Information Processing of the P.G. Demidov Yaroslavl State University Valeriy Georgievich Durnev. It provides a brief overview of his main scientific results, touches upon achievements in pedagogical and organizational activities, and includes biographical information.

This article is dedicated to the 80th anniversary of a prominent specialist in universal algebra — Vladimir Konstantinovich Kartashov.

An overview of the main stages of professional formation and growth V. K. Kartashov.

Analyzed the main directions of fundamental and applied mathematical research.

The list of the main scientific publications of V. K. Kartashov and topics of theses, which V. K. Kartashov was the supervisor. 

This article is dedicated to the 70th anniversary of a prominent specialist in the analytic theory of numbers — Valentin Nikolaevich Kuznetsov. An overview of the main stages of professional formation and growth V. N. Kuznetsova. Analyzed the main directions of fundamental and applied mathematical research. The list of the main scientific publications of V. N. Kuznetsova and topics of theses, for which V. N. Kuznetsov was the supervisor.

Рассматривается задача, относящаяся к общей проблеме построения последовательности псевдослучайных чисел. Одним из важных свойств псевдослучайных последовательностей хорошего качества является их непериодичность. Но бесконечная непериодическая последовательность может иметь начальные отрезки, вид которых далёк от желаемого. Например, отрезок десятичного разложения лиувиллева числа

∑︁∞ ????=0 10−????!

имеет лишь небольшое количество единиц, а подавляющее большинство остальных цифр равны нулю.

При рассмотрении конечных отрезков разложений чисел возникает, таким образом, необходимость определения понятий периодичности и достаточной непериодичности конечной последовательности чисел, что и сделано в работе.

Рассматриваются разложения действительных чисел и исследуется вопрос о связи арифметических свойств разлагаемого числа с достаточной непериодичностью отрезков его разложения.

Обсуждаются способы построения чисел, имеющих последовательности достаточно непериодических разложений. Описаны некоторые результаты в этом направлении и их возможное развитие.

Вкратце изложены задачи, связанные с представлениями полиадических чисел. Эти представления удобны тем, что в них не используется операция деления чисел, что значительно упрощает процесс получения искомого разложения. Описаны полученные результаты и сформулированы задачи. 

This paper is devoted to the 90th anniversary of the founder Vladimir’s school of number theory, Doctor of Physical and Mathematical Sciences, Professor Boris Veniaminovich Levin. It contains some biographical information and brief analysis of his scientific works.

This paper is devoted to the 90th anniversary of the founder Vladimir’s school of number theory, Doctor of Physical and Mathematical Sciences, Professor Boris Veniaminovich Levin. It contains some biographical information and brief analysis of his scientific works.