In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra. By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

This article describes how to use capacity of the theory of formal concept analysis (AFP) to create a mathematical model of information resource focused on people with disabilities. Describes methods of constructing formal concept grids using Galois operators in relation to data socially-oriented geographic information systems (GIS).

The aim of the work is to develop a computer model of the dynamics of the city, using combined technologies: agent approach, methods of system dynamics and GIS technologies. The computer model can be used as a tool to support decision-making in the management of urban systems. The conceptual model of the city dynamics and the technology of its implementation in AnyLogic environment are presented, as well as the study of the impact of changes in the level of comfort of areas on the dynamics of the city population.

The paper describes the practical results received from digital images topographic characteristics calculating in Matlab, such as length and curvature of contour lines, density of lengths and curvature, as well as irregularities of contour lines of the first and second order. Topological characteristics contain complete information about the shape and contours of a digital image, which allows them to be effectively used in solving the problems of remote sensing data processing, analysis of biomedical images, classification and pattern recognition problems.