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1
Content available Symmetric polynomials in the 3D Fourier equation
Biernat G., Lara-Dziembek S., Pawlak E.
Journal of Applied Mathematics and Computational Mechanics
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2015
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Vol. 14, nr 1
5--12
EN
The work is a continuation of the method of calculating the determinant of the block matrix in the three-dimensional case. In this article the symmetric polynomials are used.
2
Content available Symmetric polynomials in the 2D Fourier equation
Biernat G., Lara-Dziembek S., Pawlak E.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 4
5--12
EN
The work is a continuation of the method of calculating the determinant of the block matrix in the two-dimensional case. In this paper we use the Finite Differences Method and the symmetric polynomials.
3
Content available remote Araucaria Trees: Construction and Grafting Theorems
Schmitt D., Spehner J-C.
Fundamenta Informaticae
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2013
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Vol. 123, nr 4
417--445
EN
Araucarias have been introduced by Schott and Spehner as trees which appear in the minimal automaton of the shuffle of words. We give here a new definition of araucarias which is more constructive and we prove that our definition of araucarias is equivalent to the original one. From the new definition we derive an optimal algorithm for the construction of araucarias and a new method for calculating their size. Moreover we characterize araucarias by properties of their maximal paths, by associating a capacity to every edge. We then show that every araucaria can be obtained by grafting and merging smaller araucarias. We prove also that every directed tree can be embedded in an araucaria. Moreover we define a capacity for every vertex of an araucaria, which leads to different new enumeration formulas for araucarias.
4
Content available Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases
Mesquita T. A., Rocha Z. Da
Opuscula Mathematica
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2012
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Vol. 32, no. 4
675-687
EN
We deal with a symbolic approach to the cubic decomposition (CD) of polynomial sequences - presented in a previous article referenced herein - which allows us to compute explicitly the first elements of the nine component sequences of a CD. Properties are investigated and several experimental results are discussed, related to the CD of some widely known orthogonal sequences. Results concerning the symmetric character of the component sequences are established.
5
Content available About a certain relation between two polynomials of the same degree
Piwowarczyk T.
Archives of Control Sciences
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2011
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Vol. 21, no. 1
85-103
EN
This article presents several different methods for solving the problem of how to find a certain relation defined in chapter 2. The first method deals with the identities known in the theory of symmetric polynomials as the elements of a certain vector space. The second method is designed around the matrix transformations between symmetric polynomials. The third method is designed around the property of a linear operator and its characteristic polynomial. The fourth method is designed in the area of complex numbers, and introduces the multiplication group of 'complex roots of one'. Significant improvement in the third and fourth method is made by introducing so called 'block method'. It facilitates all calculations by making them much shorter. The article ends with an example showing symmetry and regularity of all procedures. Finally, the article shows how to solve the problem for any degree n of the polynomial, and for any degree k. At the end of the paper solutions for n < 5 and k < 5 are tabulated.
6
Content available remote Program for displaying and counting all possible symmetrical monospectrals at the degree N
Monvoisin S.
Czasopismo Techniczne. Elektrotechnika
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2000
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R. 97, z. 4-E
99--105
PL
Teoria wielomianów symetrycznych wielu zmiennych jest potężnym narzędziem matematycznym, które może być z powodzeniem stosowane w inżynierskich obliczeniach. Zawdzięcza to specjalnemu zapisowi zwanemu notacją spektralną. Celem niniejszego artykułu jest przedstawienie programu, który realizuje wszystkie możliwe elementarne wielomiany symetryczne w notacji spektralnej dla dowolnego stopnia. Dodatkowo program oblicza liczbę elementarnych wielomianów. Program jest napisany w C++. Użytkownik tego programu po wprowadzeniu stopnia wielomianu otrzymuje na ekranie monitora wszystkie możliwe elementarne wielomiany oraz ich całkowitą liczbę. Opisano wszystkie podane procedury; wszystkie obliczenia rozpoczynają się od formuł Vieta, a następnie krok po kroku są tworzone inne symbole spektralne. Zaproponowany program uzupełnia procedury tworzenia symboli spektralnych i w przyszłości wraz z innymi programami ma stworzyć bazę do komputerowego wykorzystania wielomianów symetrycznych wielu zmiennych w aplikacjach inżynierskich.
7
Content available remote Podstawy mnogościowe wielomianów symetrycznych wielu zmiennych w notacji spektralnej : zarys teorii
Piwowarczyk T.
Czasopismo Techniczne. Elektrotechnika
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2000
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R. 97, z. 4-E
38--60
EN
All symmetric polynomials of multiple variables, with the variables belonging to the fixed, final set, are denoted by means of some abstract symbols. They are known as spectral, or multipower symbols. This set is analysed from a point of view of a set theory. The article contains the definitions of those relations which are used for computing elements in numerous subsets, such as equivalence relation, ordering relation, quotient sets, combinatorics formulas. In other words, some set theory model is proposed for symmetric polynomials of multiple variables. Such a model provides a solid basis for the further study of symmetric polynomials, and first of all, for the study of their numerous vector subspaces. This study will be developed in a theoretically unlimited set of algebraic identities which are particularly useful when it comes to engineering applications
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