In the theory of polyadic groups plays an important role groups A∗ and A0, appearing in Post’s Coset Theorem [2], asserts that for every n-ary groups ⟨A, [ ]⟩ exists a group of A∗ , in which there is normal subgroup A0 such that the factor group A∗/A0 — cyclic group of order n−1. Generator xA0 this cyclic group is the n-ary group with n-ary operation derived from operation in the group A∗ , wherein n-ary groups ⟨A, [ ]⟩ and ⟨xA0, [ ]⟩ isomorphic. Group A∗ is called the Post’s universal covering group, and the group A0 — appropriate group. The article begins with a generalization of the Post’s Coset Theorem: for every n-ary groups ⟨A, [ ]⟩, n = k(m − 1) + 1, the Post’s universal covering group A∗ has a normal subgroup mA such that the factor group A∗/ mA — cyclic group of order m − 1. Moreover, A0 ⊆ mA ⊆ A∗ and mA/A0 - cyclic group of order k. In this paper we study the permutability of elements in n-ary group. In particular, we study the m-semi-commutativity in n-ary groups, which is a generalization of of the well-known concepts of commutativity and semicommutativity. Recall that the n-ary group ⟨A, [ ]⟩ is called abelian if it contains any substitution σ of the set {1, 2, . . . , n} true identity [a1a2 . . . an] = [aσ(1)aσ(2) . . . aσ(n) ], and n-ary group ⟨A, [ ]⟩ is called a semi-abelian if it true identity [aa1 . . . an−2b] = [ba1 . . . an−2a] Summarizing these two definitions, E. Post called n-ary group ⟨A, [ ]⟩ m-semiabelian if m − 1 divides n − 1 and (aa1 . . . am−2b, ba1 . . . am−2a) ∈ θA for any a, a1, . . . , am−2, b ∈ A. We have established a new criterion of m-semi-commutativity of n-ary group, formulated by a subgroup mA of the Post’s universal covering group: n-ary group ⟨A, [ ]⟩ is m-semi-abelian if and only if the group mA is abelian. For n = k(m−1) + 1 by fixed elements c1, . . . , cm−2 ∈ A on n-ary group of ⟨A, [ ]⟩ construct (k + 1)-ary group ⟨A, [ ]k+1,c1...cm−2 ⟩. On the coset A(m−1) in generalized Post’s Coset Theorem construct (k+ 1)-ary group ⟨A(m−1) , [ ]k+1⟩. Proved isomorphism of constructed (k + 1)-ary groups. This isomorphism allows us to prove another criterion m-semi-commutativity n-ary group: nary group ⟨A, [ ]⟩ is m-semi-abelian if and only if for some c1, . . . , cm−2 ∈ A (k + 1)-ary group ⟨A, [ ]k+1,c1...cm−2 ⟩ is abelian.

We study the evolution inclusion of hyperbolic type with a linear damping, which describes a class of piezoelectric controlled fields with non-monotonic potential. Discontinuous on the phase variable interaction function can be represented as the difference of subdifferentials of convex functionals. This system describes a wide class of controlled Continuum Mechanics processes, in particular, the piezoelectric controlled processes with a multivalued "reaction-displacement" law. The representation of "reaction-displacement" law as the difference of subdifferentials of convex functionals allows more flexible control for piezoelectric system. In such processes, the properties of operator presented in the model play the key role. Therefore, we impose conditions on parameters of the problem such that allow investigated model with acceptable accuracy to describe real physical process and, at the same time, provide an opportunity to use existing mathematical apparatus for it. In this paper, using the methods of the theory of global and trajectory attractors for multivalued operator semigroups the finitedimensioness of weak solutions of the model is substantiated up to a small parameter. Furthermore, the results are applied to a piezoelectric problem.

In the number theory additive problems is very important. One of them is the Ingam binary additive divisor problem on the representation of natural number as the difference of product of numbers. Many mathematician like T. Esterman, D. I. Ismoilov, D. R. Heath-Brown, G. I. Arkhipov and V. N. Chubarikov, J.-M. Deshouillers and H. Iwaniec improved the remainder term in the asymptotic formula of the number of solution of this diophantine equation. In present paper one problem with quadratic forms is considered. This problem is analog of the Ingam binary additive divisor problem. Let d — negative square-free number, F = Q( √ d) — imaginary quadratic field, δF — discriminant of field F, Q1(m), Q2(k) — binary positive defined primitive quadratic forms with matrixes A1, A2, det A1 = det A2 = −δF , ε > 0 — arbitrarily small number; n ∈ N, h ∈ N. The asymptotical formula of the number of solution of diophantine equation Q1(m) − Q2(k) = h with weight coefficient exp ( −(Q1(m) + Q2(k))/n) is received. In this asymptotical formula discriminant of field δF is fixed and the remainder term is estimating as O(h εn 3/4+ε ), which not depend of δF . Moreover the parameter h grow as O(n) with growing on the main parameter n. Proof of the asymptotical formula based on circular method when sum, which is solution of diophantine equation, may be representing as integral. Interval of integration divided by numbers of Farey series. The taking weight coefficient allow to use the functional equation of the theta-function. Moreover the estimation of one sum with Gauss sums is important. Using the evident formula of some product of Gauss sums of the number which coprimes of discriminant of field this sum represented of Kloosterman’s sum which estimate by A. Weil.

 

In the work of the positive solution of the conjugation of words in HNN-extension with the system of entrance letters. The base HNN-extensions is a wood product of the infinite cyclic groups with cyclic subgroups. The result is a generalization of the conjugacy problem in HNN-extension of a wood product of cyclic groups associated cyclic subgroups with one entrance letter. The conjugacy problem for words is of interest in free designs groups. The problem was solved in free groups with cyclic subgroups by S.Lipshutz, in the HNNextension of a free group by an associate of cyclic subgroups by A. Friedman, in HNN-extension of a tree product with the association cyclic groups associated with cyclic subgroups by author with V.N. Bezverkhny. In this paper a positive solution of the conjugation problem for words in HNNextension with the system of entrance letters. The base HNN-extensions is a tree product of the infinite cyclic groups with cyclic subgroups. The result is a generalization of the conjugacy problem in HNN-extension of a wood product of cyclic groups associated cyclic subgroups with one entrance letters. Assertion is proved for any number of entrance letters using the method of mathematical induction. In the proof of the main theorem the author proved self result assertion : - algorithmic solvability of intersection of finitely generated subgroup of the core group with an associated sub-group; - algorithmic solvability of intersection of the related class of finitely generated subgroup of the core group with an associated sub-group.

 

A class of algebraic systems which is closed under homomorphic images and finite subdirect products is called a formation. Formations was widely used in group theory. Particularly, the saturated formations of groups is one of the most studied formations. A formation of finite groups is said to be a saturated formation if G/Φ(G) ∈ F implies G ∈ F for an arbitrary finite group G and it’s Frattini subgroup Φ(G). A generalization of these definitions is as follows. A congruence θ on the algebraic system A is called a Frattini congruence if the union of all θ-classes generated by the elements of B differs from A for each proper subsystem B of the algebraic system A. A class X is saturated in the class Y, if A ∈ Y and A/θ ∈ X for some Frattini congruence θ on A implies A ∈ X. We consider finite formations of monounary algebras in this paper. An element a of a monounary algebra ⟨A, f⟩ is cyclic if f n (a) = a for some positive integer n. A monounary algebra is cyclic if all of it’s elements are cyclic. First we give a condition for a congruence of finite monounary algebra to be a Frattini congruence. Then we prove that the empty formation, the formation of all finite cyclic monounary algebras and the formation of all finite monounary algebras are the only saturated formations in the class of all finite monounary algebras.

The new estimate for the sum of the values of a primitive Dirichlet character modulo an integer q has been obtained over the sequence of shifted primes p − l, (l, q) = 1, p 6 x. This estimate is nontrivial for x > q 5 6 +ε and refines the estimate obtained by J. B. Friedlander, K. Gong, I. E. Shparlinskii. Their estimate holds provided that x > q 8 9 +ε .

 

The notion of n-ary group is a generalization of the binary group so many of the results from the theory of groups have n-ary analogue in theory of n-ary groups. But there are significant differences in these theories. For example, multiplier of the direct product of n-ary groups does not always have isomorphic copy in this product (in paper there is an example). It is proved that the direct product ∏ i∈I ⟨Ai , fi⟩ n-ary groups has n-ary subgroup isomorphic to ⟨Aj , fj ⟩ (j ∈ I), then and only when there is a homomorphism of ⟨Aj , fj ⟩ in ∏ i∈I,i̸=j ⟨Ai , fi⟩. Were found necessary and sufficient conditions for in direct product of n-ary groups, each of the direct factors had isomorphic copy in this product and the intersection of these copies singleton (as well as in groups) – each direct factor has a idempotent. For every n-ary group, can define a binary group which helps to study the n-ary group, that is true Gluskin-Hossu theorem: for every n-ary group of ⟨G, f⟩ for an element e ∈ G can define a binary group ⟨G, ·⟩, in which there will be an automorphism φ(x) = f(e, x, cn−2 1 ) and an element d = f( (n) e ) such that the following conditions are satisfied: f(x n 1 ) = x1 · φ(x2) · . . . · φ n−1 (xn) · d, x1, x2, . . . , xn ∈ G; (4) φ(d) = d; (5) φ n−1 (x) = d · x · d −1 , x ∈ G. (6) Group ⟨G, ·⟩, which occurs in Gluskin-Hossu theorem called retract n-ary groups ⟨G, f⟩. Converse Gluskin-Hossu theorem is also true: in any group ⟨G, ·⟩ for selected automorphism φ and element d with the terms (5) and (6), given n-ary group ⟨G, f⟩, where f defined by the rule (4). A n-ary group called (φ, d)- defined on group ⟨G, ·⟩ and denote derφ,d⟨G, ·⟩. Was found connections between n-ary group, (φ, d)-derived from the direct product of groups and n-ary groups that (φi , di)-derived on multipliers of this product: let ∏ i∈I ⟨Ai , ·i⟩ – direct product groups and φi , di – automorphism and an element in group ⟨Ai , ·i⟩ with the terms of (5) and (6) for any i ∈ I. Then derφ,d ∏ i∈I ⟨Ai , ·i⟩ = ∏ i∈I derφi,di ⟨Ai , ·i⟩, where φ – automorphism of direct product of groups ∏ i inI ⟨Ai , ·i⟩, componentwise given by the rule: for every a ∈ ∏ i∈I Ai , φ(a)(i) = φi(a(i)) (called diagonal automorphism), and d(i) = di for any i ∈ I. In the theory of n-ary groups indecomposable n-ary groups are finite primary and infinite semicyclic n-ary groups (built by Gluskin-Hossu theorem on cyclic groups). We observe n-ary analogue indecomposability cyclic groups. However, unlike groups, finitely generated semi-abelian n-ary group is not always decomposable into a direct product of a finite number of indecomposable semicyclic n-ary groups. It is proved that any finitely generated semiabelian n-ary group is isomorphic to the direct product finite number of indecomposable semicyclic n-ary groups (infinite or finite primary) if and only if in retract this n-ary group automorphism φ from Gluskin-Hossu theorem conjugate to some diagonal automorphism.

Measure Theory plays an important role in many questions of Mathematics. The notion of a measure being introduced as a generalization of a notion of the size of a segment made many of limiting processes be a formal procedure, and by this reason stood very productive in the questions of Harmonic analysis. Discovery of Haar measure was a valuable event for the harmonic analysis in topological groups. It stood clear that many of measures, particularly, the product of Lebesgue measure in finite dimensional cube [0, 1]n could be considered as a Haar measure. The product measure has many important properties concerning projections (see [1,3]). The theorems of Fubini and Tonelly made it very useful in applications. In this work we show that the coinsidence of considered measures, observed in finite dimensional case, impossible for infinite dimensional case, despite that such a representation was in use without proof. Considering infinite dimensional unite cube Ω = [0, 1] × [0, 1] × · · · , we define in this cube the Tichonoff metric by a special way despite that it induces the same topology. This makes possible to introduce a regular measure eliminating difficulties connected with concentration of a measure, with the progress of a dimension, around the bound. We use the metric to define a set function in the algebra of open balls defining their measure as a volume of open balls. By this way we introduce a new measure in infinite dimensional unite cube different from the Haar and product measures and discuss some differences between introduced measure and the product measure. Main difference between the introduced measure and Haar measure consisted in non invariance of the first. The difference between the new measure and product measure connected with the property: let we are given with a infinite family of open balls every of which does not contain any other with total finite measure; then they have an empty intersection. Consequently, every point contained in by a finite number of considered balls only. This property does not satisfied by cylindrical set. For example, let D1 = I1 × I × I × · · · , D2 = I2 × I1 × I × I × · · · ,. . . I = [0, 1], Ik = [ 0, k k + 1] , k = 1, 2, .... It clear that every of these cylindrical sets does contain any other, but their intersection is not empty (contains zero). This makes two measures currently different.

 

Recently graphics system TikZ has become an integral part of the scientific text editor TEX/LATEX and turned it into a vector graphics editor. The article discusses the means TikZ to create compact, attractive and informative conceptual schemes. We are talking about tools, located in the library of extensions of mindmap (\usetikzlibrary{mindmap}) [2, c. 635]. Conceptual schemes are useful in training and are also called maps of lectures, activity, etc. In fact, they are representations of multi-level lists in the form of a special graph. Therefore, to control the appearance of conceptual schemes using standard tikz-tools trees, supplemented with several special options. The article gives a brief description of these special options, and shows their usage when forming the concrete scheme "Computer devices".