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DISCRETE MATHEMATICS
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DISCRETE APPLIED MATHEMATICS
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Elements of Discrete Mathematics 2023
Kursy OnlineThis course helps with learning Elements of Discrete Mathematics.
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Elements of Discrete Mathematics 2022
Kursy OnlineThis course helps with learning Elements of Discrete Mathematics.
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Elements of Discrete Mathematics 2024
Kursy OnlineThis course helps with learning Elements of Discrete Mathematics.
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The modelling method of discrete-continuous systems
PublikacjaThe paper introduces a method of discrete-continuous systems modelling. In the proposed method a three-dimensional system is divided into finite elements in only two directions, with the third direction remaining continuous. The thus obtained discrete-continuous model is described by a set of partial differential equations. General difference equations of discrete system are obtained using the rigid finite element method. The limit...
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On the Characteristic Graph of a Discrete Symmetric Channel
PublikacjaWe present some characterizations of characteristic graphs of row and/or column symmetric channels. We also give a polynomial-time algorithm that decides whether there exists a discrete symmetric channel whose characteristic graph is equal to a given input graph. In addition, we show several applications of our results.
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Acceleration of the discrete Green's function computations
PublikacjaResults of the acceleration of the 3-D discrete Green's function (DGF) computations on the multicore processor are presented. The code was developed in the multiple precision arithmetic with use of the OpenMP parallel programming interface. As a result, the speedup factor of three orders of magnitude compared to the previous implementation was obtained thus applicability of the DGF in FDTD simulations was significantly improved.
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A DISCRETE-CONTINUOUS METHOD OF MECHANICAL SYSTEM MODELLING
PublikacjaThe paper describes a discrete-continuous method of dynamic system modelling. The presented approach is hybrid in its nature, as it combines the advantages of spatial discretization methods with those of continuous system modelling methods. In the proposed method, a three-dimensional system is discretised in two directions only, with the third direction remaining continuous. The thus obtained discrete-continuous model is described...
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Accuracy of the discrete Green's function computations
PublikacjaThis paper discusses the accuracy of the discrete Green's function (DGF) computations. Recently closed-form expression of the DGF and its efficient numerical implementation were presented which facilitate the DGF applications in FDTD simulations of radiation and scattering problems. By carefully comparing the DGF results to those of the FDTD simulation, one can make conclusions about the range of the applicability of the DGF for...
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Discrete convolution based on polynomial residue representation
PublikacjaThis paper presents the study of fast discrete convolution calculation with use of the Polynomial Residue Number System (PRNS). Convolution can be based the algorithm similar to polynomial multiplication. The residue arithmetic allows for fast realization of multiplication and addition, which are the most important arithmetic operations in the implementation of convolution. The practical aspects of hardware realization of PRNS...
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Discrete and continuous fractional persistence problems – the positivity property and applications
PublikacjaIn this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical (α=1) differential equation in the fractional case (here using fractional Caputo’s derivatives) and the numerical scheme which are associated (here with discrete Grünwald–Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability...
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Discrete Modelling of Micro-structural Phenomena in Granular Shear Zones
PublikacjaThe micro-structure evolution in shear zones in cohesionless sand for quasi-static problems was analyzed with a discrete element method (DEM). The passive sand failure for a very rought retaining wall undergoing horizontal translation towards the sand backfill was discussed. To simulate the behaviour of sand, the spherical discrete element model was used with elements in the form of rigid spheres with contacts moments.
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Numerical Test for Stability Evaluation of Discrete-Time Systems
PublikacjaIn this paper, a new numerical test for stability evaluation of discrete-time systems is presented. It is based on modern root-finding techniques at the complex plane employing the Delaunay triangulation and Cauchy's Argument Principle. The method evaluates if a system is stable and returns possible values and multiplicities of unstable zeros of the characteristic equation. For state-space discrete-time models, the developed test...
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Windowing of the Discrete Green's Function for Accurate FDTD Computations
PublikacjaThe paper presents systematic evaluation of the applicability of parametric and nonparametric window functions for truncation of the discrete Green's function (DGF). This function is directly derived from the FDTD update equations, thus the FDTD method and its integral discrete formulation can be perfectly coupled using DGF. Unfortunately, the DGF computations require processor time, hence DGF has to be truncated with appropriate...
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Discrete-time estimation of nonlinear continuous-time stochastic systems
PublikacjaIn this paper we consider the problem of state estimation of a dynamic system whose evolution is described by a nonlinear continuous-time stochastic model. We also assume that the system is observed by a sensor in discrete-time moments. To perform state estimation using uncertain discrete-time data, the system model needs to be discretized. We compare two methods of discretization. The first method uses the classical forward Euler...
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Discrete-time estimation of nonlinear continuous-time stochastic systems
PublikacjaIn this paper we consider the problem of state estimation of a dynamic system whose evolution is described by a nonlinear continuous-time stochastic model. We also assume that the system is observed by a sensor in discrete-time moments. To perform state estimation using uncertain discrete-time data, the system model needs to be discretized. We compare two methods of discretization. The first method uses the classical forward Euler...
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Periodic Properties of 1D FE Discrete Models in High Frequency Dynamics
PublikacjaFinite element discrete models of various engineering 1D structures may be considered as structures of certain periodic characteristics. The source of this periodicity comes from the discontinuity of stress/strain field between the elements. This behaviour remains unnoticeable, when low frequency dynamics of these structures is investigated. At high frequency regimes, however, its influence may be strong enough to dominate calculated...
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Connection matrix theory for discrete dynamical systems
PublikacjaIn [C] and [F1] the connection matrix theory for Morse decomposition is developedin the case of continuous dynamical systems. Our purpose is to study the case of discrete timedynamical systems.
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Index filtrations and Morse decomposition for discrete dynamical systems
PublikacjaOn a Morse decomposition of an isolated invariant set of a homeomorphism(discrete dynamical system) there are partial orderings defined by the homeomorphism.These are called admissible orderings of the...