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- ItemA precise asymptotic description of half-linear differential equations(WILEY-V C H VERLAG GMBH, 2024-04-08) Řehák, PavelWe study asymptotic behavior of solutions of nonoscillatory second-order half-linear differential equations. We give (in some sense optimal) conditions that guarantee generalized regular variation of all solutions, where no sign condition on the potential is assumed. For all of these solutions, we establish precise asymptotic formulas, where positive as well as negative potential is considered. We examine, as consequences, also equations with regularly varying coefficients, or with the coefficients viewed as perturbations of exponentials, or the equations under certain critical (double roots) settings. We make also asymptotic analysis of Poincare-Perron solutions. Many of our results are new even in the linear case.
- ItemVanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point(De Gruyter, 2024-02-05) Diblík, Josef; Růžičková, MiroslavaA singular nonlinear differential equation z(sigma) dw/dz = aw + zwf(z , w), where sigma > 1, is considered in a neighbourhood of the point z = 0 z=0 located either in the complex plane C if sigma is a natural number, in a Riemann surface of a rational function if sigma is a rational number, or in the Riemann surface of logarithmic function if sigma is an irrational number. It is assumed that w = w ( z ) w=w\left(z) , a is an element of C { 0 } a, and that the function f f is analytic in a neighbourhood of the origin in C x C . Considering sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w = w (z ) w=w(z) in a domain that is part of a neighbourhood of the point z = 0 z=0 in C or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z -> 0 w (z) = 0 is proved and an asymptotic behaviour of w (z) s established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.
- ItemOptimal control of combined heat and power station operation(Springer Nature, 2023-09-13) Kůdela, Jakub; Suja, Jerguš; Šomplák, Radovan; Pluskal, Jaroslav; Hrabec, DušanCombined heat and power stations have become one of the most utilized units of district heating systems. These stations usually contain several boilers for burning fossil fuels and renewable resources used for heating up steam, which can be used either for residential and commercial heating or electricity generation. To ensure efficiency, a boiler should either run continuously (for at least a given period) on a power output higher than a given threshold or switch off. The optimal control of the plant operations should combine an efficient setup for the turbine and boilers in operation, reflecting the demand for steam and the price of electricity, and a schedule that describes which boilers should be in operation at a given time. This paper proposes a method for optimal control of combined heat and power station operation for a given time horizon. The method is based on a two-level approach. The lower-level models correspond to finding the optimal setup of the combined heat and power station parameters for an hourly demand for different kinds of steam. The upper-level model corresponds to the optimal schedule of the operations of the individual boilers, which is planned for the entire time horizon. The lower-level model is modeled as a mixed-integer linear programming problem and is solved using parametric programming. A dynamic programming algorithm solves the upper-level model with a rolling horizon. The validity of the proposed method and its computational complexity for different granularity of the time horizon, different ranges of the parameters, varying demand for various kinds of steam, and varying electricity prices are investigated in a case study. The presented approach can be readily applied to other control problems with a similar structure.
- ItemA digital Jordan surface theorem with respect to a graph connectedness(De Gruyter, 2023-12-31) Šlapal, JosefAfter introducing a graph connectedness induced by a given set of paths of the same length, we focus on the 2-adjacency graph on the digital line Z with a certain set of paths of length n for every positive integer n . The connectedness in the strong product of three copies of the graph is used to define digital Jordan surfaces. These are obtained as polyhedral surfaces bounding the polyhedra that can be face-to-face tiled with digital tetrahedra.
- ItemMachine Learning Method for Changepoint Detection in Short Time Series Data(MDPI, 2023-10-05) Smejkalová, Veronika; Šomplák, Radovan; Rosecký, Martin; Šramková, KristínaAnalysis of data is crucial in waste management to improve effective planning from both short- and long-term perspectives. Real-world data often presents anomalies, but in the waste management sector, anomaly detection is seldom performed. The main goal and contribution of this paper is a proposal of a complex machine learning framework for changepoint detection in a large number of short time series from waste management. In such a case, it is not possible to use only an expert-based approach due to the time-consuming nature of this process and subjectivity. The proposed framework consists of two steps: (1) outlier detection via outlier test for trend-adjusted data, and (2) changepoints are identified via comparison of linear model parameters. In order to use the proposed method, it is necessary to have a sufficient number of experts’ assessments of the presence of anomalies in time series. The proposed framework is demonstrated on waste management data from the Czech Republic. It is observed that certain waste categories in specific regions frequently exhibit changepoints. On the micro-regional level, approximately 31.1% of time series contain at least one outlier and 16.4% exhibit changepoints. Certain groups of waste are more prone to the occurrence of anomalies. The results indicate that even in the case of aggregated data, anomalies are not rare, and their presence should always be checked.