Ústav matematiky

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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.
SYMMETRIC 2 x 2 MATRIX FUNCTIONS WITH ORDER PRESERVING PROPERTY
(AGH University Press, 2024-10-10) Štoudková Růžičková, Viera
It 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.
Quantization of two- and three-player cooperative games based on QRA
(IOP Publishing, 2024-10-01) Eryganov, Ivan; Hrdina, Jaroslav; Návrat, Aleš
In 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.
Higher Order Geometric Algebras and Their Implementations Using Bott Periodicity
(Springer Nature, 2024-08-31) Stodola, Marek; Hrdina, Jaroslav
Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC) as a matrix over the Compass ruler algebra (CRA).
A Novel Technique for the Extraction of Dynamic Events in Extreme Ultraviolet Solar Images
(The American Astronomical Society, 2024-11-07) Kalenská, Petra; Rajmic, Pavel; Gebrtová, Karolína; Druckmüller, Miloslav
High-spatial-resolution images of the solar corona acquired in the extreme ultraviolet (EUV), most notably with the Atmospheric Imaging Assembly (AIA) instrument on the Solar Dynamics Observatory (SDO) reveal the abundance of dynamic events which range from flaring bright points and jets to erupting prominences and coronal mass ejections (CMEs). In this work we present novel techniques to extract such dynamic events from the more steady background corona using 17.1 nm SDO-AIA images. The techniques presented here treat any time series of coronal images as a matrix that can be decomposed into two matrices representing the background and the dynamic component, respectively. The latter has the properties of a so-called sparse matrix, and the proposed methods are classified as methods based on sparse representations. The proposed methods are the median-filter method, the principal component pursuit, and the dynamic-mode decomposition, all of which include data pre-processing using the noise-adaptive fuzzy equalization method. The study reveals that the median-filter method and the dynamic-mode decomposition enhance all motions in the time series and produce similar results. On the other hand, the principal component pursuit enables the clear differentiation of CMEs from the background corona, thus providing a valuable tool for the characterization of their acceleration profiles in the low corona as seen in the EUV.

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