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1
Content available remote Unfoldings and Coverings of Weighted Graphs
Courcelle Bruno
Fundamenta Informaticae
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2022
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Vol. 189, nr 1.
1--47
EN
Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings of finite graphs. We prove similar results for unfoldings of finite directed graphs. Moreover, we generalize coverings and similarly, unfoldings to graphs and digraphs equipped with finite or infinite weights attached to edges of the covered or unfolded graphs. This generalization yields a canonical "factorization" of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing ω as an infinite weight provides us with finite descriptions of regular trees having nodes of countably infinite degree. Regular trees (trees having finitely many subtrees up to isomorphism) play an important role in the extension of Formal Language Theory to infinite structures described in finitary ways. Our weighted graphs offer effective descriptions of the above mentioned regular trees and yield decidability results. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.
2
Content available The Stability Interval of the Set of Linear System
Mazakov Talgat, Wójcik Waldemar, Jomartova Sholpan, Karymsakova Nurgul, Ziyatbekova Gulzat, Tursynbai Aisulu
International Journal of Electronics and Telecommunications
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2021
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Vol. 67, No. 2
155--161
EN
The article considers the problem of stability of interval-defined linear systems based on the Hurwitz and LienardShipar interval criteria. Krylov, Leverier, and Leverier-Danilevsky algorithms are implemented for automated construction and analysis of the interval characteristic polynomial. The interval mathematics library was used while developing the software. The stability of the dynamic system described by linear ordinary differential equations is determined and based on the properties of the eigenvalues of the interval characteristic polynomial. On the basis of numerical calculations, the authors compare several methods of constructing the characteristic polynomial. The developed software that implements the introduced interval arithmetic operations can be used in the study of dynamic properties of automatic control systems, energy, economic and other non-linear systems.
3
Content available Digraph-building method for finding a set of minimal realizations of 2-D dynamic systems
Markowski K. A., Hryniów K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 5
589--597
EN
This paper presents a digraph-building method designed to find the determination of realization of two-dimensional dynamic system. The main differences between the method proposed and other state-of-the-art solutions used include finding a set of realizations (belonging to a defined class) instead of only one realization, and the fact that obtained realizations have minimal size of state matrices. In the article, the proposed method is described, compared to state-of-the-art methods and illustrated with numerical examples. To the best of authors’ knowledge, the method shown in the paper is superior to all other state-of-the-art solutions both in terms of number of solutions and their matrix size. Additionally, MATLAB function for determination of realization based on the set of state matrices is included.
4
Content available Parallel Digraphs-building Computer Algorithm for Finding a Set of Characteristic Polynomial Realisations of Dynamic System
Hryniów K., Markowski K. A.
Journal of Automation Mobile Robotics and Intelligent Systems
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2016
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Vol. 10, No. 3
38--51
EN
This paper presents in-depth the parallel computer algorithm for the determination of characteristic polynomial realisations of dynamic system. The main differences between the depicted method and other state of- the-art solutions include finding not few realisations, but a whole set, and the fact that the found realisations are always minimal among all possible. As digraphsbuilding methods used in the algorithm are NP-complete or NP-hard problems, the algorithm is paralleled and GPGPU (General-Purpose computing on Graphics Processor Units) computation is proposed as the only feasible solution. The article describes in detail the proposed method, discusses it’s complexity, presents optimisation solutions and still open problems. The working algorithm is illustrated with a numerical example and compared to results of other known methods.
5
Content available remote An algorithm for the calculation of the minimal polynomial
Białas S., Białas M.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2008
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Vol. 56, nr 4
391-393
EN
This paper gives the simple algorithm for calculation of the degree and coefficients of the minimal polynomial for the complex matrix A = [aij ]n×n.
6
Content available remote Wyznaczanie macierzy stanu dodatniego układu 2D na podstawie wielomianu charakterystycznego
Markowski K.
Przegląd Elektrotechniczny
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2007
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R. 83, nr 2
109-112
PL
W artykule przedstawiono metodę wyznaczanie elementów macierzy stanu dodatniego układu dwuwymiarowego (2D) opisanego za pomocą modelu ogólnego używając teorii dwuwymiarowych grafów skierowanych oddziaływań. Metodę zilustrowano przykładem numerycznym.
EN
In this paper a method for determination elements of the state matrices of two-dimensional (2D) systems described by general model using digraphs theory is presented. The method is illustrated by numerical examples.
7
On linear dependence of iterates
Matkowski J., Zhang W.
Journal of Applied Analysis
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2000
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Vol. 6, nr 1
149-157
EN
A functional equation related to a problem of linear dependence of iterates is considered.
8
A note on eigenvalues of interval matrix
Skrzypczyk J., Witek H.
Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska
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1999
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z. 10
273-278
EN
In the paper, we are concerned with interval - oriented methodology to model uncertainties of eigenvalues of an nxn interval real matrix. We investigate methods of calculation for characteristic polynomials of interval matrices. Presented methodology is probably the simplest way to model and to approximate vibration properties of systems with uncertain parameters.
PL
W pracy przedstawiono nowe pojęcia i wyniki dotyczące metodologii analizy zagadnień związanych z wartościami własnymi i wyznaczaniem współczynników wielomianu charakterystycznego macierzy rzeczywistych o współczynnikach interwałowych. Przedstawione ujęcie jest prawdopodobnie najprostszym sposobem aproksymacji w modelowaniu drgań i ich własności w systemach o parametrach niepewnych.
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