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Institute
Stability of control systems is one of the central subjects in control theory. The classical asymptotic stability theorem states that the norm of the residual between the state trajectory and the equilibrium is zero in limit. Unfortunately, it does not in general allow computing a concrete rate of convergence particularly due to algorithmic uncertainty which is related to numerical imperfections of floating-point arithmetic. This work proposes to revisit the asymptotic stability theory with the aim of computation of convergence rates using constructive analysis which is a mathematical tool that realizes equivalence between certain theorems and computation algorithms. Consequently, it also offers a framework which allows controlling numerical imperfections in a coherent and formal way. The overall goal of the current study also matches with the trend of introducing formal verification tools into the control theory. Besides existing approaches, constructive analysis, suggested within this work, can also be considered for formal verification of control systems. A computational example is provided that demonstrates extraction of a convergence certificate for example dynamical systems.
It is possible to obtain a common updating rule for k-means and Neural Gas algorithms by using a generalized Expectation Maximization method. This result is used to derive two variants of these methods. The use of a similarity measure, specifically the gaussian function, provides another clustering alternative to the before mentioned methods. The main benefit of using the gaussian function is that it inherently looks for a common cluster center for similar data points (depending on the value of the parameter s ). In different experiments we report similar behaviour of batch and proposed variants. Also we show some useful results for the “alternative” similarity method, specifically when there is no clue about the number of clusters in the data sets.
This thesis was written in order to prove the expediency of startup ecosystem support and to develop practical recommendations for Belarusian government based on the analysis of successful practices in the U.S. and Lithuania.
It covers the essence of a “startup company” and a “startup ecosystem” as well as provides the analysis of socioeconomic impact of startup companies with particular focus on job creation. It sheds light on the best startup support policies in the U.S., where most prominent startup ecosystems are operating, and Lithuania as a country with similar to Belarusian preconditions and a rapidly
developing ecosystem. Furthermore, this paper deals with Belarus‘s peculiarities regarding fostering startup ecosystem growth. It assesses recent economic development of Belarusian IT sector and gives an insight into its competitive advantages and challenges.
The subsequent paper is based on internet research using articles, presentations, reports and studies, websites and official legal documents.