Ústav matematiky

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Periodic solutions in a linear delay difference system
(Bolyai Institute, University of Szeged, 2025-04-05) Čermák, Jan; Fedorková, Lucie; Nechvátal, Luděk
The paper investigates periodicity properties of a linear autonomous difference system with two delayed terms. Assuming that the system matrices are simultaneously triangularizable, we formulate necessary and sufficient conditions guaranteeing the existence of a nonzero periodic solution (with an a priori given period) of the studied system. The analytical form of such conditions is shown to generalize the existing results on this topic. Moreover, it is supported by a geometric reformulation, offering a better understanding of the derived periodicity conditions. Information on the form of the searched periodic solution (including its prime period) is also provided.
Damage behaviour of quasi-brittle composites: mathematical and computational aspects
(MDPI, 2025-04-11) Vala, Jiří; Tomáš, Jiří
In the present paper, an evaluation of the damage behaviour of quasi-brittle composites exposed to mechanical, thermal, and other loads is studied by means of viscoelastic and/or viscoplastic material models, applying some non-local regularisation techniques to the initiation and development of damages. The methods above are presented as a strong tool for a deeper understanding of material structures in miscellaneous engineering disciplines like civil, mechanical, and many others. Nevertheless, all of the software packages reflect certain compromises between the need for effective computational tools, with parameters obtained from inexpensive experiments, within the possibilities and the complexity of both physical and geometrical descriptions of structure deformation within processes. The article is devoted to the mathematical aspects regarding a considerably wide class of computational modelling problems, emphasising the following ones: (i) the existence and the uniqueness of solutions of engineering problems formulated in terms of the deterministic initial and boundary value problems of partial differential equations theory; (ii) the problems of convergence of computational algorithms applied to (i). Both aspects have numerous references to possible generalisations and investigations connected with open problems.
ON NON-OSCILLATION FOR TWO DIMENSIONAL SYSTEMS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS
(UNIV MISKOLC INST MATH, 2024-11-28) Opluštil, Zdeněk
The paper studies the non-oscillatory properties of two-dimensional systems of non-linear differential equations u ' = g(t)|v|1/alpha sgn v, v ' = -p(t)|u|(alpha)sgn u, where the functions g: [0, +infinity[-> [0, +infinity[, p: [0, +infinity[-> & Ropf; are locally integrable and alpha > 0. We are especially interested in the case of integral(+infinity)g(s) ds < +infinity. In the paper, new non-oscillation criteria are established. Among others, they generalize well-known results for linear systems as well as second order linear and also half-linear differential equations. The criteria presented complement the results of Hartman-Wintner's type for the system in question.
On moduli and arguments of roots of complex trinomials
(Mathematical Sciences Publishers, 2024-11-20) Čermák, Jan; Fedorková, Lucie; Jánský, Jiří
Root properties of a general complex trinomial have been explored in numerous papers. Two questions have attracted a significant attention: the relationships between the moduli of these roots and the trinomial’s entries, and the location of the roots in the complex plane. We consider several particular problems connected with these topics, and provide new insights into them. As two main results, we describe the set of all trinomials having a root with a given modulus, and derive explicit formula for calculations of the arguments of such roots. In this fashion, we obtain a comprehensive characterization of these roots. In addition, we develop a procedure enabling us to compute moduli and arguments of all roots of a general complex trinomial with arbitrary precision. This procedure is based on the derivation of a family of real transcendental equations for the roots’ moduli, and it is supported by the formula for their arguments. All our findings are compared with the existing results.
Dynamic Reduction of Network Flow Optimization Problem: Case of Waste-to-Energy Infrastructure Planning in Czech Republic
(Elsevier, 2024-10-01) Pluskal, Jaroslav; Šomplák, Radovan; Kůdela, Jakub; Eryganov, Ivan
Nowadays, many sophisticated tools based on various mathematical approaches are used to support planning and strategic decision-making. In the field of waste management, allocation and location problems based mainly on the structure network flow problem are used with respect to infrastructure planning. Modern formulations of the problem allow the inclusion of integer and nonlinear constraints that reflect real-world operations. However, despite the advanced computational technology, such real-world problems are difficult to solve in adequate detail due to the large scale of the problem. Thus, the links in the system are simplified, but most often a transport network is aggregated. The individual nodes in the system may then represent areas with tens or hundreds of thousands of inhabitants, which does not provide sufficient insight for location tasks. This paper presents an approach to dynamically reduce the network with respect to selected points of interest. The selected areas are modeled in greater detail, while with increasing distance the entities are more aggregated into larger units. The approach is based on a transformation of the original network and subsequent cluster analysis, preferably using existing transport infrastructure. The presented approach provides the possibility of practical application of complex tools that are currently mostly theoretical due to high computational demands. The methodology is applied to a case study of Waste-to-Energy infrastructure planning, which needs to model a large area to fill a large capacity facility.

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