The author suggests the famous description of cubic field units of negative discriminant with recurrent sequences which are analogous to Fibonacci numbers. It runs to derived algebraic field and it is interpreted as applied to Diophantine equations.

 

The article is devoted to investigation of an variety N generated by twoelement commutative multiplicatively idempotent semirings. Two classical theorems of Birkhoff (about the characterization of varieties of algebraic structures, and subdirect reducibility) are initial in the studying of semiring varieties. In 1971 J. A. Kalman proved that there exist up to isomorphism three subdirectly irreducible commutative idempotent semirings satisfying the dual distributive law x + yz = (x + y)(x + z), namely a two-element field, a twoelement mono-semiring, and the some three-element semiring. In 1999 S. Ghosh showed that any commutative multiplicatively idempotent semiring with identity x + 2xy = x is the subdirect product of a Boolean ring and a distributive lattice. In 1992 F. Guzman got a similar result for the variety of all multiplicatively idempotent semirings with zero and unit, satisfying the identity 1 + 2x = 1. It was proved that every such semiring is commutative. This one is the subdirect product of two-element fields and two-element chains and it may be generated by a single three-element semiring. We obtained the following results in the work. We proved some necessary conditions for subdirect irreducibility of semirings from the variety M of all the semirings with commutative indempotent multiplication. It was shown that an arbitrary semiring from M is subdirect product of two commutative multiplicatively idempotent semirings, one of which has the identity 3x = x, and the other has the identity 3x = 2x. We found all the subdirectly irreducible semirings in N and discribed varieties in N. It was obtained that in the class M the variety N is defined by the single identity x+ 2xy +yz = x+ 2xz +yz. We proved that the lattice of all the subvarieties of the variety N is a 16-element Boolean lattice.

In 2008–2011, we solved several well–known additive problems such that Ternary Goldbach’s Problem, Hua Loo Keng’s Problem, Lagrange’s Problem with restriction on the set of variables. Asymptotic formulas were obtained for these problems. The main terms of our formulas differ from ones of the corresponding classical problems. In the main terms the series of the form 2πim(ηN−0,5k(a+b)) sink πm(b − a) σk(N, a, b) = e . πkmk |m|<∞ appear. These series were investigated by the authors. Suppose that k > 2 and n > 1 are naturals. Consider the equation n n n x1 + x2 + . . . + xk = N (1)in natural numbers x1, x2, . . . , xk. The question on the number of solutions of the equation (1) is Waring’s problem. Let η be the irrational algebraic number, n > 3, 2n + 1, if 3 6 n 6 10, k > k0 = 2[n2(2 log n + log log n + 5)], if n > 10. In this report we represent the variant of Waring’s Problem involving natural numbers such that a 6 {ηxn} < b, where a and b are arbitrary real numbers i of the interval [0, 1). Let J(N) be the number of solutions of (1) in natural numbers of a special type, and I(N) be the number of solutions of (1) in arbitrary natural numbers. Then the equality holds J(N) ∼ I(N)σk(N, a, b). The series σk(N, a, b) is presented in the main term of the asymptotic formula in this problem as well as in Goldbach’s Problem, Hua Loo Keng’s Problem.

 

The paper discusses the generalization of the method embodiments N. M. Korobov approximate solution of the Dirichlet problem for equations of the form ∂ ∂ Q ∂x1 , . . . , ∂xs u(x) = f(x), where the functions u(x), f(x), ϕ(x) belongs to the class of functions Eα s in case of using generalized Parallelepipedal nets M(Λ) integral lattices Λ. Particular attention is paid to the class of differential operators, consisting ∂ ∂ of operators Q , . . . , with zero kernel. The importance of this class of ∂x1 ∂xs operators due to the fact that up to a constant solution of differential equations with partial derivatives for these operators is uniquely determined. An example of such an operator is the Laplace operator. In the work, an approximate solution of the Dirichlet problem for partial differential equations using arbitrary generalized parallelepiped mesh M(Λ) integer lattice Λ for a certain class of periodic functions and shown that by using an infinite sequence of nested grids is generalized parallelepipedal nets sufficiently fast convergence of the approximate solutions to the function u(x).

 

By means of additive arithmetic functions on a sequence of the shifted prime numbers the processes with realizations from a space of functions without ruptures of the second sort are based. In this space with a topology of Skorokhod and σ-algebra of the borelean multitudes a sequence of the measures corresponding to constructed arithmetic processes is entered. Exactly, the relative frequency of prime numbers is accepted to a measure of the borelean multitudes. These numbers don’t surpass natural number of n to which there a correspond realization of the constructed processes getting to this multitude. Necessary and sufficient conditions of weak convergence of sequence of the entered measures to the measure corresponding to a process are found. Thus process with the independent increments, which distributions are not expressed, is limited. Necessary and sufficient conditions represent two limit ratios the first of which is an infinite of a sequence of the set sums. The proofs of need of performance of this ratio for weak convergence of sequence of measures are the main part of all proofs of the theorem. This proof is carried out by consideration of distributions of increments of arithmetic processes on the intervals close to a unit and a transition to characteristic functions, corresponding to these distributions. Further, using an independence of increments of a limit process and a weak compactness of a sequence of measures (taken from Yu. Prokhorov’s known theorem of weak convergence of probability measures), by an asymptotic formula for average values of multiplicative functions on sequence of the shifted prime numbers of N. Timofeev, we receive the first condition of the theorem. At the proof of sufficiency of both conditions for weak convergence of sequence of measures characteristic functions are applied again. It allows, in a particular, to use early the limit theorems received by the author in functional spaces for additive functions on "rare" multitudes. The sequence {p+1} is included in a class of the sequences considered in these theorems. However, in them the condition similar to the first condition considered here, isn’t necessary, but is sufficient. It allows, applying the specified theorems to a considered case to receive a weak convergence of sequence of measures. A representation for a characteristic function of a limit process is also received.

In this work is given the description of Hamiltonian algebras in some subclasses of class of algebras with operators having one ternary basic operation and one operator. Universal algebra A is a Hamiltonian algebra if every subuniverse of A is the block of some congruence of the algebra A. Algebra with operators is an universal algebra with additional system of the unary operations acting as endomorphisms with respect to basic operations. These operations are called permutable with basic operations. An algebra with operators is ternary if it has exactly one basic operation and this operation is ternary. It is obtained the sufficient condition of Hamiltonity for arbitrary universal algebras with operators. It is described Hamiltonian algebras in classes of ternary algebras with one operator and with basic operation that is either Pixley operation, or minority function, or majority function of special view. Let V be a variety of algebras with operators and V has signature Ω1 ∪Ω2, where Ω1 is an arbitrary signature containing near-unanimity function and Ω2 is a set of operators. It is proved that V not contains nontrivial Abelian algebras.

The so-called Bergman representation formula (reproducing formula) and various estimates for Bergman projection with positive reproducing kernel and sharp Forelli-Rudin type estimates of Bergman kernel are playing a crucial role in certain new extremal problems related to so-called distance function in analytic function spaces in various domains in Cn . In this paper based on some recent embedding theorems for analytic spaces in bounded domains with C2 boundary and admissible domains new results for Bergman-type analytic function spaces related with this extremal problem will be provided. Some(not sharp) assertions for BMOA and Nevanlinna spaces, for analytic Besov spaces in any D bounded domain with C2 boundary or admissible domains in Cn will be also provided. We remark for readers in addition the problems related to regularity of Bergman projection which we use always in proof in various types of domains with various types of boundaries (or properties of boundaries) are currently and in the past already are under intensive attention. Many estimates for reproducing operators and there kernels and Lp boundedness of Bergman projections have been also the object of considerable interest for more than 40 years. These tools serve as the core of all our proofs. When the boundary of the domain D is sufficiently smooth decisive results were obtained in various settings. Our intention in this paper is the same as a previous our papers on this topic, namely we collect some facts from earlier investigation concerning Bergman projection and use them for our purposes in estimates of distY (f, X ) function (distance function). Based on our previous work and recent results on embeddings in classical analytic spaces in domains of various type in Cn we provide several new general assertions for distance function in products of strictly pseudoconvex domains with smooth boundary, general bounded domains with C2 boundary and in admissible domains in various spaces of analytic functions of several complex variables.These are first results of this type for bounded domains with C2 boundary and admissible domains. In addition to our results we add some new sharp results for special kind of domains so called products of strictly pseudoconex domains with smooth boundary in Cn extendind our sharp results in strictly pseudoconex domains.

Recently in publications for preparation of high-quality images are increasingly used freely distributed graphical TikZ package [3], which became an integral part of the scientific and technical publishing system TEX/LATEX. It’s library turtle implemented one of the versions elegant programming language high level Logo, developed in 1967 by Seymour Papert and Idit Harel [1] and is based on recursion. The version of the Logo, provided by the package turtle, this is just one of about 250 different implementations of the language. But, with a minimum of resources, this version is very easy to use and allows you to programmatically from the comfort of LATEX, easily supports simple technical drawing. About the version of the Logo provided by the library tirtle package TikZ, , and is discussed in this article. Here is described not only all means of this version, but also demonstrated their use on illustrative examples. Note that the Logo this is not just a language, but a bunch of language and integrated development environment, allowing to give the simplicity and clarity to the process of teaching programming. In the standard version of the Logo has executor of the commands that can be moved point (turtle) around the screen forward on a specified number of steps and change the direction of its motion. This turtle when moving can leave a mark on the screen in the form of line segments describing these or other shapes, or move, without leaving a trace. A set of commands that are understood by the executor turtle, and is the basis of language Logo, which had in its simplicity a revolutionary impact on the start learning programming. To understand Logo in the state of children of primary school and even pre-school age. According to the conclusions of dissertational work Idit Harel, in developing the design of the program, thinking on how best to structure and present information, the student is much deeper begins to understand the content of the task that lies at the base of the program. Note two important circumstances. In tirtle-implementation Logo change of direction turtle can occur at any angle to the current direction of its movement, and to move the turtle besides the standard commands can be used and the typical structure of a cycle \foreach.