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.

The article proposes a technology for controlling access through a Web application. In the first part, the implementation of the electromechanical part of the ACS, the key feature of which is the preservation of the last state when the power is turned off and the presence of an anti-panic mode. This feature is obtained by installing a geared motor on the lock cylinder. This solution allows you to abandon the need for certification of the lock, use both electronic control, for example, by means of a telephone or proximity cards, and classical control using a regular standard key, as well as finding the lock in any state for an arbitrarily long time, regardless of the presence of power. In the second part, the implementation of network interaction between the user and the directly controlled object is proposed.

This article discusses the problem of determining traffic flows at an intersection by identifying vehicles using neural networks. A model of a neural network used to detect vehicles in the image is described. The paper describes an algorithm for identifying cars at an intersection. The article also describes an algorithm for obtaining the trajectory of the vehicle.

The article discusses the structure of the hardware and internetworking of automated room heat loss identification system on example of the Ugra State University classroom. The temperature controller has been developed taking into account the specifics of the room. A heat meter with telemetry capability has been selected.