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  1. 3 Open Mathematics
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  5. 1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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1
Content available On some properties of Chebyshev polynomials
100%
Belbachir H. , Bencherif F.
Discussiones Mathematicae - General Algebra and Applications
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2008
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tom 28
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nr 1
121-133
EN
Letting $T_{n}$ (resp. $U_{n}$) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences $(X^{k}T_{n-k})_{k}$ and $(X^{k}U_{n-k})_{k}$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space $𝔼_{n}[X]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_{n}$ and $U_{n}$ admit remarkableness integer coordinates on each of the two basis.
2
Content available A geometric property of the roots of Chebyshev polynomials
80%
Wituła R. , Hetmaniok E. , Słota D.
|
2014
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tom Vol. 13, nr 1
143--148
EN
In this text a new property of geometric nature of the Chebyshev polynomials is given. It is proven that the lengths of diagonals of a regular n-gon with the side of length equal to one are the sums of positive roots of the respective renormalized Chebyshev polynomials of one from among four types. Some new special decompositions of differences of values of the Chebyshev polynomials are also presented.
3
Content available remote Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions
80%
Prodinger H.
Open Mathematics
|
2017
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tom 15
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nr 1
1156-1160
EN
A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula. Similar formulae are derived for scaled Fibonacci numbers.
4
Content available Application of modified chebyshev polynomials in asymmetric cryptography
80%
Lawnik M. , Kapczyński A.
Computer Science
|
2019
|
tom Vol. 20 (3)
289--303
EN
Based on Chebyshov polynomials, one can create an asymmetric cryptosystem that allows for secure communication. Such a cryptosystem is based on the fact that these polynomials form a semi-group due to the composition operation. This article presents two new cryptosystems based on modifications of Chebyshev's polynomials. The presented analysis shows that their security is the same as in the case of algorithms associated with the problem of discrete logarithms. The article also shows methods that allow for the faster calculation of Chebyshev polynomials.
5
Content available remote On the multivariate transfinite diameter
70%
Bloom T. , Calvi J.-P.
|
1999
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tom 72
|
nr 3
285-305
EN
We prove several new results on the multivariate transfinite diameter and its connection with pluripotential theory: a formula for the transfinite diameter of a general product set, a comparison theorem and a new expression involving Robin's functions. We also study the transfinite diameter of the pre-image under certain proper polynomial mappings.
6
Content available remote A note on the rate of convergence for Chebyshev-Lobatto and Radau systems
70%
Berriochoa E. , Cachafeiro A. , Díaz J. , Martínez E.
Open Mathematics
|
2016
|
tom 14
|
nr 1
156-166
EN
This paper is devoted to Hermite interpolation with Chebyshev-Lobatto and Chebyshev-Radau nodal points. The aim of this piece of work is to establish the rate of convergence for some types of smooth functions. Although the rate of convergence is similar to that of Lagrange interpolation, taking into account the asymptotic constants that we obtain, the use of this method is justified and it is very suitable when we dispose of the appropriate information.
7
Content available remote Application of Chebyshev Collocation Method to Solving the Heat Equation
60%
Povstenko J.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
|
2000/2001
|
tom Vol. 8
57--66
8
Content available remote Aproksymacja jednostajna odwzorowań kartograficznych
60%
Pędzich P.
Roczniki Geomatyki
|
2009
|
tom T. 7, z. 4
67-76
PL
W artykule zaprezentowane zostaną własności wielomianów Czebyszewa oraz ich zastosowanie do aproksymacji odwzorowań kartograficznych. Ponadto przedstawione zostanie porównanie wyników aproksymacji jednostajnej oraz aproksymacji średniokwadratowej odwzorowań kartograficznych.
EN
Usually the least square method is used for approximation of map projection. Determination of polynomial coefficients requires solution of a complicated system of equations. It is possible to avoid such a problem using orthogonal Chebyshev polynomials. This is a completely different method of approximation, where the maximum difference between the value of the function and the value calculated from polynomial is minimized. In the paper, properties of Chebyshev polynomials approximations are presented as wall as their application to map projection approximation and comparison with other methods of map projection.
9
Content available remote Symbolic description of the polynomial roots and their numerical implementation - better than in Mathematica software?
60%
Hetmaniok E. , Słota D. , Pleszczyński M. , Wituła R. , Różański M. , Szczygieł M.
Annals of Computer Science and Information Systems
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2019
|
tom Vol. 20
41--45
EN
This paper is a continuation of the discussion undertaken in one of our previous papers. We present in the current paper the corrected, and also given in a slightly changed form, Vandermonde formulae for the roots of some quintic polynomials considered in J.P. Tignol's monograph. The proofs of selected trigonometric identities from our previous paper are given and some new identities have been generated by the occasion, which also can be used for testing our Langrange algorithm for the case of cubic polynomials. Moreover, we present here the decomposition of polynomials belonging to some two-parameter family of polynomials related to the Chebyshev polynomials of the first kind.
10
Content available remote Parametric excitation of pipes through fluid flow
60%
Szabó Z.
Computer Assisted Mechanics and Engineering Sciences
|
1999
|
tom Vol. 6, No. 3-4
487-494
EN
In this paper the dynamic behaviour of a continuum inextensible pipe containing fluid flow is investigated. The fluid is considered to be incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The equations of motion are derived via Lagrangian equations and Hamilton's principle. The system is non-conservative, and the amount of energy carried in and out by the flow appears in the model. It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic. The method which constructs the state transition matrix used in Floquet theory in terms of Chebyshev polynomials is especially effective for stability analysis of systems with multi-degree-of-freedom. The stability charts are created w.r.t. the forcing frequency omega, the perturbation amplitude nu and the average flow velocity U.
11
Content available An extension of typically-real functions and associated orthogonal polynomials
60%
Naraniecka I. , Szynal J. , Tatarczak A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2011
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tom 65
|
nr 2
EN
Two-parameters extension of the family of typically-real functions is studied. The definition is obtained by the Stjeltjes integral formula. The kernel function in this definition serves as a generating function for some family of orthogonal polynomials generalizing Chebyshev polynomials of the second kind. The results of this paper concern the exact region of local univalence, bounds for the radius of univalence, the coefficient problems within the considered family as well as the basic properties of obtained orthogonal polynomials.
12
Content available remote Numerical stability of the Richardson second order method
51%
Smoktunowicz A. , Wróbel I.
|
2005
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tom Vol. 38, nr 1
255--263
EN
In this paper we study numerical properties of the Richardson second order method (RS) for solving a linear system Ax = b, where A € Rnxn is infinitysymmetric and positive definite. We consider the standard model of floating point arithmetic (cf. [6], [7], [11]). We prove that the RS-algorithm is numerically stable. This means that the algorithm computes approximations xk to the exact solution x* = A-1b such that the error limfk||xk - x*ll2 ls of order eMcond(A), where eM is the machine precision and cond(A) = ||A || 2 ||A-1|| denotes the condition number of the matrix A.
13
Content available remote Metoda kolokacji rozwiązywania równań różniczkowych typu parabolicznego
51%
Povstenko Y.
Studia i Materiały / Europejska Uczelnia Informatyczno-Ekonomiczna w Warszawie
|
2011
|
tom Nr 1(1)
43--50
PL
W artykule omówiono metodę spektralną rozwiązywania równań typu parabolicznego. Opisano procedurę numeryczną i przedstawiono wyniki eksperymentu numerycznego.
EN
In this paper the spectral method for solving equations of parabolic type is discussed. Numerical procedure is described, and results of numerical experiment are presented.
14
Content available Irregularity of post mining deformations as indicator revealing effects of processes of unknown origin in area of Bochnia
51%
Szczerbowski Z.
|
2020
|
tom T. 19
81--93
EN
The presented work deals with the problem of terrain surface and rock mass deformation in the area of the Bochnia Salt Mine. The deformations are related to natural causes (mainly the tectonic stress of the Carpathian orogen) as well as anthropogenic ones related to the past mining activity conducted directly under the buildings of the town of Bochnia. The discussed characteristics of land surface deformation are important from the point of view of threats to surface features and contribute to spatial development. Particularly anomalous zones of observed subsidence basins are examined as places of second order deformation effects. The author presents a method of determinations of these anomalous areas and he discusses their origins.
PL
Przestawiona praca dotyczy problematyki deformacji powierzchni i górotworu w rejonie Kopalni Soli Bochnia. Deformacje te związane są zarówno z przyczynami naturalnymi (głównie nacisk tektoniczny orogenu karpackiego), jak i antropogenicznymi związanymi z minioną aktywnością górniczą prowadzoną bezpośrednio pod zabudową miejską Bochni. Omówione charakterystyki deformacji powierzchni terenu są istotne z punktu widzenia zagrożeń dla obiektów powierzchni oraz są elementem planowania przestrzennego. Szczególnie przeanalizowane zostały strefy anomalne niecki obniżeniowej jako efekty drugiego rzędu deformacji. Autor przedstawił sposób ich wyznaczania oraz omówił genezę ich powstania.
15
Content available Numerical solution of two-dimensional Fredholm integro-differential equations by Chebyshev integral operational matrix method
51%
Ishola C. Y. , Taiwo O. A. , Adebisi A. F. , Peter O. J.
Journal of Applied Mathematics and Computational Mechanics
|
2022
|
tom Vol. 21, nr 1
29--40
EN
This paper presents the Chebyshev Integral Operational Matrix Method (CIOMM) for the numerical solution of two-dimensional Fredholm Integro-Differential Equations (2D-FIDEs). The process of the method is obtaining the operational matrix of integration by evaluating a 2D integral of 2D Chebyshev polynomial basis functions and assuming approximate solutions of the 2D-FIDEs as a truncated 2D Chebyshev series. This leads to a system of linear algebraic equations which are solved to obtain the values of the unknown constants using Maple 18. Some numerical problems are solved to illustrate the practicability of the method.
16
Content available remote Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences
51%
Wituła R. , Słota D.
|
|
tom 4
|
nr 3
531-546
EN
In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x 2n + y 2n , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.
17
Content available remote High-speed modular structures for parallel computing in the space of orthogonal projections
51%
Selianinau M.
|
|
tom T. 5
89-98
EN
The numerical-analytical method of digital signal processing based on the modular model of space of orthogonal projections are presented in the article. This gives new possibilities for the high-performance processing of discrete signals on numerical-analytical level at the realization not only arithmetic but also more complicated operations such as convolution, correlation, algorithms of spectral analysis and others.
PL
W artykule przedstawiono liczbowo-analityczną metodę cyfrowego przetwarzania sygnałów, która oparta jest na wykorzystaniu modułowego modelu przestrzeni rzutów ortogonalnych. Daje ona nowe możliwości dla wysokowydajnego przetwarzania sygnałów dyskretnych na liczbowo-analitycznym poziomie przy realizacji nie tylko arytmetycznych, ale również bardziej skomplikowanych operacji takich, w szczególności, jak splot, korelacja, algorytmy widmowej analizy i inne.
18
Content available remote On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials
41%
Szabłowski P. J.
|
|
tom Vol. 46, nr 4
679--708
EN
We review properties of the q-Hermite polynomials and indicate their links with the Chebyshev, Rogers–Szegö, Al-Salam–Chihara, continuous q-utraspherical polynomials. In particular, we recall the connection coefficients between these families of polynomials. We also present some useful and important finite and infinite expansions involving polynomials of these families including symmetric and non-symmetric kernels. In the paper, we collect scattered throughout literature useful but not widely known facts concerning these polynomials. It is based on 43 positions of predominantly recent literature.
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