Vladikavkaz Institute of Management

A linear functional on a real vector space is determined up to a scalar by its zero hyperplane. A linear operator is recovered from its kernel up to a simple multiplier in rather special cases. Nevertheless, some operator analog of the functional case is valid: each operator with target a Kantorovich space, a Dedekind complete vector lattice, is determined up to some orthomorphism by the kernels of its strata. We call this idea Kutateladze’s stratification principle. Two approaches are outlined: a standard approach, based on the concept of a local Hamel basis, and a nonstandard approach, involving Boolean valued analysis. It is clarified under what conditions the principle works. Some new applications to simultaneous inequalities with linear and multilinear operators are also given.

This work investigates a problem for a fractional diffusion equation with nonclassical boundary conditions. A family of weighted difference schemes is studied for the considered problem. An algorithm for finding a numerical solution is provided. Using the maximum principle for the difference problem, an a priori estimate is derived, which implies the stability of the difference schemes and the convergence of the numerical solution to the exact solution in the C-norm.

Для Кавказа, «исключительно сложной лаборатории лингвистической мысли» (А.С. Чикобава), проблема языковых контактов всегда актуальна. Древние контакты – это постоянное взаимное сближение, обмен, взаимообогащение через общение: торговое, культурное, экономическое, через иные формы сотрудничества. Древние контакты давали импульс развитию культуры, отпечатывались в языке, устном народном творчестве, в обрядовой жизни и материальной культуре носителей контактирующих языков. Иранизмы в языках ареала являются также результатом древних торгово-экономических связей с Ираном, о чем свидетельствует преобладающее количество названий, связанных с торговлей, наименованиями материй, одежды, предметов обихода, различных продуктов, блюд, пряностей, растений; это также общественно-политические и религиозные термины, части мужских и женских имен. Иранизмы, как и заимствования из других языков, представляют собой общий для заимствующих языков словарный фонд и способствуют их сближению. Статья посвящена древним историко-культурным и этнолингвистическим взаимосвязям и взаимовлияниям носителей иберийско-кавказских, тюркских и индоевропейских языков на примере термина пехуымпар/ пахуымпарв различных его значениях. Предпринята попытка установить время, место и вектор заимствования; выявить особенности развития семантики; определить исторические и этнокультурные факторы, способствовавшие его появлению. В результате исследования выявлено, что этноязыковые контакты населявших Кавказ сложились еще в доисламское время; языки ареала подвергались процессам дивергенции, конвергентным явлениям, независимо от генетического родства; заимствовались иранизмы как без участия языка-посредника, так и с его участием. Заимствования в течение долгой жизни, приспосабливаясь, претерпевали в тюркских и кавказских языках освоение – фонетическое, морфологическое и лексико-семантическое. For the Caucasus, "an exceptionally complex laboratory of linguistic thought" (A.S. Chikobava), the problem of linguistic contacts is always topical. Ancient contacts are a constant mutual rapprochement, exchange, and mutual enrichment through communication – trade, cultural, economic, and other forms of co-operation. Ancient contacts gave impetus to the development of culture, imprinted in the language, oral folk art, ritual life and material culture of the speakers of the contacting languages. Iranisms in the languages of the area are also the result of ancient trade and economic relations with Iran, as evidenced by the predominant number of trade-related names, names of fabrics, clothes, household items, various products, dishes, spices, plants; these are also socio-political and religious terms, parts of male and female names. Iranisms, like borrowings from other languages, represent a common vocabulary for the borrowing languages and contribute to their rapprochement. The article is devoted to ancient historical, cultural and ethnolinguistic relationships and mutual influences of speakers of Iberian-Caucasian, Turkic and Indo-European languages using the example of the term pehuympar/pahuymparin its various meanings. An attempt has been made to establish the time, place and vector of borrowing; identify features of the development of semantics; determine the historical and ethnocultural factors that contributed to its emergence. As a result of the study, it was revealed that ethno-linguistic contacts of those inhabiting the Caucasus developed in pre-Islamic times; the languages of the area were subject to processes of divergence and convergent phenomena, regardless of genetic relatedness; Iranianisms were borrowed both without the participation of an intermediary language and with its participation. Over the course of a long life, borrowings, adapting, underwent development in the Turkic and Caucasian languages – phonetic, morphological and lexical-semantic.

We implement the Boolean valued analysis of Banach spaces. The realizations of Banach spaces in a Boolean valued universe are lattice normed spaces. We present the basic techniques of studying these objects as well as the Boolean valued approach to injective Banach lattices.

We present the inverse problem of successive determination of the two unknowns that are a coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of an integral operator describing the memory of the medium. The direct initial-boundary value problem involves the zero data and the Neumann boundary condition. The trace of the Fourier image of a solution to the direct problem on the boundary of the medium serves as additional information. Studying inverse problems, we assume that the unknown coefficient is expanded into an asymptotic series in powers of a small parameter. Also, we construct some method for finding the coefficient that accounts for the memory of the environment to within an error of order $ O(\varepsilon^{2}) $ . At the first stage, we determine a solution to the direct problem in the zero approximation and the kernel of the integral operator, while the inverse problem reduces to an equivalent problem of solving the system of nonlinear Volterra integral equations of the second kind. At the second stage, we consider the kernel given and recover a solution to the direct problem in the first approximation and the unknown coefficient. In this case, the solution to the equivalent inverse problem agrees with a solution to the linear system of Volterra integral equations of the second kind. We prove some theorems on the unique local solvability of the inverse problems and present the results of numerical calculations of the kernel and the coefficient.

A linear operator $ T $ in a locally solid vector lattice $ (E,\tau) $ is a Lebesgue operator, if $ Tx_{\alpha}\overset{\tau}{\rightarrow}0 $ for every net in $ E $ satisfying $ x_{\alpha}\downarrow 0 $ ; a $ KB $ -operator, if for every $ \tau $ -bounded increasing net $ x_{\alpha} $ in $ E_{+} $ there exists $ x\in E $ with $ Tx_{\alpha}\overset{\tau}{\rightarrow}Tx $ ; a quasi $ KB $ -operator, if $ T $ takes $ \tau $ -bounded increasing nets in $ E_{+} $ to $ \tau $ -Cauchy nets; a Levi operator, if for every $ \tau $ -bounded increasing net $ x_{\alpha} $ in $ E_{+} $ there exists $ x\in E $ such that $ Tx_{\alpha}\overset{o}{\rightarrow}Tx $ ; and a quasi Levi operator, if $ T $ takes $ \tau $ -bounded increasing nets in $ E_{+} $ to $ o $ -Cauchy ones. We address the domination problem for the quasi $ KB $ -operators and quasi Levi operators in locally solid vector lattices. Moreover, under study are some properties of Lebesgue, Levi, and $ KB $ -operators. In particular, we prove that the vector space of regularly Lebesgue operators is a subalgebra of the algebra of all regular operators.