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 Âûïóñê 80
- Gubiy E., Zorkaltsev V., Perzhabinsky S. Chebyshev and euclidean projections of point on linear manifold
Results of research of properties and interrelations of Chebyshev and Euclidean projections of the origin on linear manifold are considered in the article. Many problems of applied mathematics can be presented in the such view. They are problems of linear approximations, problems of search solutions of balance models closed to the given infeasible solutions, search of pseudosolutions of the models with inconsistent conditions. Euclidean projections are corresponded to application of the least square method. Chebyshev projections are corresponded to minimization of a maximal deviation. We developed and theoretical justified algorithm of searching of Chebyshev projections. The algorithm gives single-valued result and allows to dispense without the difficult verified and sometimes violated Haar condition. The algorithm is based on using of lexicographic optimization. The relative interior point of set of optimal solutions is found on each stage of lexicographic optimization. The property of producing of relative interior points is the main property of algorithms of interior point method. The sets of Chebyshev and Euclidean projections of the origin on linear manifold are formed by way of varying of positive coefficients corresponding to components of vectors in Chebyshev and Euclidean norms. We justified that closure of these sets are equal with the set of vectors of the linear manifold with Pareto-efficient absolute meanings of the components. Consequently, any Chebyshev and Euclidean projection can be get with any required accuracy through choosing the weight coefficients. It was also proved any Euclidean projection with any set of positive weight coefficients in Euclidian norm can be get for the account of choosing the weight coefficients in the form of Chebyshev projection.
- Sobolev V. One queueing system and Fibonacci numbers
This paper deals with a queuing system with Poisson arrivals, exponential service times, single service channel and infinite number of waiting positions, customers are serviced in the order of their arrival. The requests arrives in groups and the number of requests in a group is one or two. For this queueing system be found in algebraic form the steady-state probabilities for the number of customers in the system. A probability mass function of this distribution can be defined by polynomials like polynomials Fibonacci. The geometric distribution is a special case of this distribution. Fibonacci numbers can be expressed in terms of the polynomials like polynomials Fibonacci. Consequently our distribution expressed in terms of this polynomials under certain conditions can be written in terms of Fibonacci numbers. Using the Binet formula is shown that in some cases the found distribution is asymptotically geometric distribution. In this paper it is shown that the Bernoulli numbers can be expressed as an elementary double sum of binomial coefficients. Changing the order in that double sum and summing one of them get a formula for Fibonacci numbers which Catalan developed or Lucas formula for Fibonacci numbers.
- Agasandyan G. Continuous VaR-criterion and investor's optimal portfolio
The work continues author’s investigations connected with correctness conditions ascertained previously for families of risk-preferences functions (r.p.f.) that might be used in financial markets in problems of optimization on continuous VaR-criterion (CC-VaR). These conditions were used in analyzing an example of families deduced from the super-family of piecewise-linear functions by pure analytical means. Numerical methods of checking the correctness of r.p.f.-families that are useful when difficulties arise in analytical investigations are suggested. These methods are based on discrete algorithms of optimization under CC-VaR for scenario markets and solve correctness problems with quite high-degree approximation. Methods are tested on the former super-family and applied to the super-family of the generalized circles. Results demonstrate adequacy and generality of methodology.
- Kuznetsov E. Network interactions structure analysis: context-sensitive centrality measures
Classical centrality indexes for the analysis of system of interrelating and interacting objects and widespread methods of key elements detection by ranging nodes on the value of common network centrality index not always consider intensity of intra-group interactions in the system. We offer context-dependent measures of centrality – based on intra-group interaction of elements in the network. Namely the proposed centrality measure of each element depends on the subset of elements for which is considered. For general integrated characteristic of importance, influence, etc. of some group of nodes we offer to use minimum or maximum, but not an average, value of the centrality index of nodes in this group. For definition of subset of key nodes in the network, we offer to use algorithms of special cluster analysis – algorithms for monotone system kernel detection. It not only provides determination of global extremum of functional in the corresponding optimization task, but also allows to analyze structure of network in more detail. As an example of application of the offered approach, the network of export links of the member countries of the European Union is considered. The special front boundary role of United Kingdom in the core of the European Union is revealed.
- Ivanov N. Algorithms of resource management in generalized stochastic networks
The modeling of the control process in real time by a computing system with a limited number of computing resources is considered. Networking is used as a modeling tool. Limiting the number of resources can lead to the formation of queues ready to work. In this article, the work is associated with the corresponding arc of the network. Methods are proposed for managing the resources of a computing system for various disciplines of performing work that are in the buffer. In particular, the FIFO and LIFO disciplines are considered, as well as one of the heuristic disciplines. The basis for the analysis of these methods is the concept of network state. The components of the state are considered to be a set of executed works and a set of works that are ready for execution and are in the buffer. Network graphics states form a finite state tree, which serves as the basis for building simulation programs. Using these programs, the user can make a comparative analysis of the considered resource management algorithms of the computing system. Using the proposed simulation algorithms also allows the user to solve the problem of the optimal choice of the number of performers. The solution to this problem allows us to find an acceptable compromise between this number and the program execution time in real time.
- Brokarev I. Using artificial neural networks for solving the problem of analysis of the composition of gas mixtures
The problem formulation of natural gas composition analysis is described. The statistical method is proposed to use for solving the problem of gas composition analysis. The main stages of the statistical model development for the natural gas composition analysis are described. The results of the correlation analysis for the selection of input and output parameters for the statistical model are presented. The main statistical models that used for solving of the problem of gas mixtures composition analysis are shown. The artificial neural networks and Levenberg – Marquardt algorithm are used within the study. The description of the Levenberg – Marquardt training algorithm is given taking into account possible modifications of the algorithm. The architecture of the proposed neural network is described. The gas mixtures ranges that used in the training and test samples are given. The accuracy characteristics of the proposed model are given. It was concluded that the chosen neural network model architecture is adequate, based on the calculated accuracy characteristics of the model. The results of predicting the gas mixtures composition by measurements of gas physical parameters are shown. Further research directions in the development of the proposed method for analyzing the natural gas composition are given.
- Ogorodnikov K. Position accuracy analysis in terrain-aided navigation systems
The influence of information support uncertainties and operating conditions on corrections provided by the recurrent-research algorithm of the terrain-aided navigation system has been studied. As is known, uncertainties of the position coordinates depends on many factors, including the inertial system errors, reference information errors, a measurer of the geophysical field errors, the information value of the terrain area, etc. The results of the analysis show that the uncertainty caused by the error in speed rises with an increase of the number of the geophysical field measurements. The component caused by uncertainties of information support, on the contrary, decreases. Also, in this article expressions were obtained that can be used when planning the flight route and choosing terrain areas for performing correction. In particular, an equation has been synthesized that establishes a relationship between the standard deviation of the heights gradient, standard deviation of the heights, and the height field correlation radius. An expression has also been formed that allows us to calculate the optimal measurements number necessary for correction, under the conditions of the measurers errors and the movement trajectory.
- Shevlyakov A. Cube-robot climbing a wall
Mobile robotics is a constantly evolving field, synergetic with artificial intelligence. Collective and cooperative autonomous robots are a short-term goal at the moment, with new breakthroughs reported yearly. While aerial robots get the most attention, other environments are equally important and pose challenging problems in motion control, sensing and engineering. Cube robots are interesting in this regard, as their shape allows groups of such robots to form stable structures. Additionally, it grants the robot an ability to jump. In this article we propose a control for a cube robot to climb a wall, assuming a symmetric wall facing it is available. We consider two-dimensional motion in vertical plane. The proposed control allows to optimize the rebound trajectory after a collision to reach maximum flight distance. We also present a software to model the movement of a cube robot and try various control strategies, based on Box2D and ImGUI libraries.
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