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- ItemON NON-OSCILLATION FOR TWO DIMENSIONAL SYSTEMS OF NON-LINEAR ORDINARY DIFFERENTIAL EQUATIONS(UNIV MISKOLC INST MATH, 2024-11-28) Opluštil, ZdeněkThe 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.
- ItemOn moduli and arguments of roots of complex trinomials(Mathematical Sciences Publishers, 2024-11-20) Čermák, Jan; Fedorková, Lucie; Jánský, JiříRoot 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.
- ItemDynamic 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, IvanNowadays, 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.
- ItemSYMMETRIC 2 x 2 MATRIX FUNCTIONS WITH ORDER PRESERVING PROPERTY(AGH University Press, 2024-10-10) Štoudková Růžičková, VieraIt is known that the discrete matrix Riccati equation has the order preserving property under some assumptions. In this paper we formulate and prove the converse statement for the case when the dimensions of the matrices are 2 x 2 and the order preserving property holds for all such symmetric matrices.
- ItemQuantization of two- and three-player cooperative games based on QRA(IOP Publishing, 2024-10-01) Eryganov, Ivan; Hrdina, Jaroslav; Návrat, AlešIn this paper, a novel quantization scheme for cooperative games is proposed. The circuit is inspired by the Eisert-Wilkens-Lewenstein protocol, which was modified to represent cooperation between players and extended to 3-qubit states. The framework of Clifford algebra is used to perform necessary computations. In particular, we use a direct analogy between Dirac formalism and Quantum Register Algebra (QRA) to represent circuits. This analogy enables us to perform automated proofs of the circuit equivalence in a simple fashion. The expected value of the Shapley value concerning quantum probabilities is employed to distribute players' payoffs after the measurement. We study how entanglement, representing the level of pre-agreement between players, affects the final utility distribution. The paper also demonstrates how the QRA and GAALOP software can automate all necessary calculations.