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1

Meshless local Petrov-Galerkin method for rotatingRayleigh beam using Chebyshev and Legendre polynomials

Panchore Vijay

Archive of Mechanical Engineering

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2022

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LXIX, nr 2

301-–318

EN

The numerical solutions are obtained for rotating beams; the inclusion of centrifugal force term makes it difficult to get the analytical solutions. In this paper, we solve the free vibration problem of rotating Rayleigh beam using Chebyshev and Legendre polynomials where weak form of meshless local Petrov-Galerkin method is used. The equations which are derived for rotating beams result in stiffness matrices and the mass matrix. The orthogonal polynomials are used and results obtained with Chebyshev polynomials and Legendre polynomials are exactly the same. The results are compared with the literature and the conventional finite element method where only first seven terms of both the polynomials are considered. The first five natural frequencies and respective mode shapes are calculated. The results are accurate when compared to literature.

2

Moments of the weighted Cantor measures

Harding Steven N., Riasanovsky Alexander W. N.

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

256--273

EN

Based on the seminal work of Hutchinson, we investigate properties of α-weighted Cantor measures whose support is a fractal contained in the unit interval. Here, α is a vector of nonnegative weights summing to 1, and the corresponding weighted Cantor measure μα is the unique Borel probability measure on [0, 1] satisfying [wzór] where φn : x ↦ (x+n)/N. In Sections 1 and 2 we examine several general properties of the measure μα and the associated Legendre polynomials in L2μα[0,1]. In Section 3, we (1) compute the Laplacian and moment generating function of μα, (2) characterize precisely when the moments Im = ∫[0,1]xmdμα exhibit either polynomial or exponential decay, and (3) describe an algorithm which estimates the firstmmoments within uniform error ε in O((loglog(1/ε))·m log m). We also state analogous results in the natural case where α is palindromic for the measure να attained by shifting μα to [−1/2,1/2].

3

Empirical tests of performance of some M – estimators

Banaś M., Ligas M.

Geodesy and Cartography

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2014

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Vol. 63, no. 2

127--146

EN

The paper presents an empirical comparison of performance of three well known M – estimators (i.e. Huber, Tukey and Hampel’s M – estimators) and also some new ones. The new M – estimators were motivated by weighting functions applied in orthogonal polynomials theory, kernel density estimation as well as one derived from Wigner semicircle probability distribution. M – estimators were used to detect outlying observations in contaminated datasets. Calculations were performed using iteratively reweighted least-squares (IRLS). Since the residual variance (used in covariance matrices construction) is not a robust measure of scale the tests employed also robust measures i.e. interquartile range and normalized median absolute deviation. The methods were tested on a simple leveling network in a large number of variants showing bad and good sides of M – estimation. The new M – estimators have been equipped with theoretical tubing constants to obtain 95% efficiency with respect to the standard normal distribution. The need for data – dependent tuning constants rather than those established theoretically is also pointed out.

PL

W artykule przedstawiono empiryczne porównanie trzech dobrze znanych M – estymatorów (Huber’a, Tukey’a oraz Hampel’a) jak również kilku nowych. Nowe estymatory motywowane były funkcjami wagowymi wykorzystywanymi w teorii wielomianów ortogonalnych, estymacji jądrowej oraz jeden motywowany przez funkcję gęstości „półokręgu” Wigner’a. Każdy z estymatorów został użyty do wykrywania obserwacji odstających w skażonych zbiorach danych. Obliczenia wykonano za pomocą „reważonej” metody najmniejszych kwadratów. Ze względu na fakt, iż wariancja resztowa (używana w konstrukcji macierzy kowariancyjnych) nie jest odpornym estymatorem skali, w testach wykorzystano również odporne miary takie jak: rozstęp ćwiartkowy oraz znormalizowane odchylenie medianowe. Testy wykonano na prostej sieci niwelacyjnej w dużej ilości wariantów ukazujących dobre i złe strony M – estymacji. Nowe estymatory zostały wyposażone w teoretyczne stałe odcinania zapewniające 95% efektywność względem standaryzowanego rozkładu normalnego. Kwestia rozwijania metod bazujących na stałych odcinania pochodzących z danych została również pokrótce poruszona.

4

Orthogonal polynomials on ellipses and their recurrence relations

Lauric V.

Demonstratio Mathematica

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2014

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Vol. 47, nr 1

137--145

EN

In this note we study the connection between orthogonal polynomials on an ellipse and orthogonal Laurent polynomials on the unit circle relative to some multiplicative measures and then establish the recurrence relations for orthogonal polynomials on an ellipse. The matrix representation of the operator of multiplication by coordinate function is obtained.

5

Modeling of antennas for teleinformation system working in Wi-Fi standard

Syroka Z., Łabarzewski C.

Diagnostyka

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2012

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nr 4(64)

45-48

EN

The article is describing model projects, and simulation analysis of the antennas intended for the cooperation with wireless network cards working in the Wi-Fi standard, reported as applications for acquiring the protection right to the utility models. The projects assumed designing the omnidirectional antenna, and the directional one, providing the maximum possible value of amplifying the signal, at the simultaneous keeping the smallnesses of the size, so that the antennas could work in room conditions. Models, and simulations were performed with the help of a non-commercial program 4nec2, based on the method of moments. Moreover, for an every model, there was drawn up a model of the system fitting the impedance of the antenna to the signal line about the impedance of 50Ω.

PL

W pracy zostały przedstawione założenia bezprzewodowego systemu diagnostycznego będącego przedmiotem zgłoszenia patentowego [5]. System ten wykorzystuje w nadajniku metodę rozwijania sygnałów w szereg Fouriera.

6

Jednoczesne metody znajdowania wartości szczególnych oraz zer wielomianów ortogonalnych

Wróbel I.

Biuletyn Naukowy Wrocławskiej Wyższej Szkoły Informatyki Stosowanej. Informatyka

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2012

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vol. 2

10--15

PL

Rozważamy zastosowania pewnych metod wyznaczania miejsc zerowych w problemie obliczania wartości szczególnych macierzy dwudiagonalnych. Proponujemy algorytmy będące modyfikacjami metod klasycznych: Weierstrassa, Abertha i Bairstowa obliczania wszystkich pierwiastków wielomianu. Wykorzystywane są własności rozpatrywanych macierzy zarówno w konstrukcji samego algorytmu jak i odpowiednim doborze wartości początkowych oraz w wyborze warunku zakończenia obliczeń. Rozważane zmodyfikowane metody mogą być również stosowane do wyznaczania pierwiastków wielomianów ortogonalnych.

EN

We consider applications of certain rootfinding methods for the bidiagonal singular value problem. The problem of computing singular values of a bidiagonal n-by-n matrix is equivalent to computing eigenvalues of a symmetric tridiagonal n2-by-n2 matrix. The algorithms we propose are modifications of the classical Weierstrass, Aberth and Bairstow methods for computing all roots of a polynomial. We make use of the properties of the matrix, both in algorithms themselves and in the choice of the initial approximation and the stopping criterion. We also apply these modified methods to finding roots of orthogonal polynomials.

7

Zastosowanie wielomianów ortogonalnych do aproksymacji odwzorowań kartograficznych

Balcerzak J., Pędzich P.

Roczniki Geomatyki

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2006

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T. 4, z. 3

33-40

EN

In the paper, an approximation method is presented, which uses orthogonal polynomials. Coefficients of polynomials are determined by recurrence formulas on discrete sets of points. Utilization of orthogonal polynomials allows to avoid creation and solving of normal equation systems. In the paper application of the method to map projection approximation function in .1992. system is also presented. The method may be used in geodesy and cartography for obtaining map projection approximation functions, and calculating elementary scale, convergence and geodetic reductions.

8

Asymptotic of Lp extremal polynomials off the unit circle

Khaldi R.

Demonstratio Mathematica

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2005

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Vol. 38, nr 3

623--632

EN

Let sigma be a positive measure whose support is the unit circle gamma plus a denumerable set of mass points, which accumulate at gamma and satisfy Blaschke^s condition. We study the asymptotics of Lp extremal polynomials (0 < p < oo) in the region exterior to r under Szego's condition.