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1
Content available 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
Content available remote 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
Content available remote The Karlin-McGregor formula for paths connected with a clique
Obata N.
Probability and Mathematical Statistics
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2013
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Vol. 33, Fasc. 2
451--466
EN
The Karlin-McGregor formula, a well-known integral expression of the m-step transition probability for a nearest-neighbor random walk on the non-negative integers (an infinite path graph), is reformulated in terms of one-mode interacting Fock spaces. A truncated direct sum of onemode interacting Fock spaces is newly introduced and an integral expression for the m-th moment of the associated operator is derived. This integral expression gives rise to an extension of the Karlin-McGregor formula to the graph of paths connected with a clique.
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 remote Recurrent Construction of MacWilliams and Chebyshev Matrices
Gogin N., Hirvensalo M.
Fundamenta Informaticae
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2012
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Vol. 116, nr 1-4
93-110
EN
We give two recursive expressions for both MacWilliams and Chebyshev matrices. The expressions give rise to simple recursive algorithms for constructing the matrices. In order to derive the second recursion for the Chebyshev matrices we find out the Krawtchouk coefficients of the discrete Chebyshev polynomials, a task interesting on its own.
6
Content available remote 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
Content available Two steps piecewise affine identification of nonlinear systems
Stevek J., Szucs A., Kvasnica M., Fikar M., Kozak S.
Archives of Control Sciences
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2012
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Vol. 22, no. 4
371-388
EN
Given a set of input-output measurements, the paper proposes a method for approximation of a nonlinear system by a piecewise affine model (PWA). First step of the two-stage procedure is identification from input-output data, in order to obtain an appropriate nonlinear function in analytic form. The analytic expression of the model can be represented either by a static nonlinear function or by a dynamic system and can be obtained using a basis function expansion modeling approach. Subsequently we employ nonlinear programming to derive optimal PWA approximation of the identified model such that the approximation error is minimized. Moreover, we show that approximation of multivariate systems can be transformed into a series of one-dimensional approximations, which can be solved efficiently using standard optimization techniques.
8
Content available remote Ultraspherical type generating functions for orthogonal polynomials
Demni N.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 2
281--296
EN
We characterize, under some technical assumptions and up to a conjecture, probability distributions of finite all order moments with ultraspherical type generating functions for orthogonal polynomials. Our method is based on differential equations and the obtained measures are particular beta distributions. We actually recover the free Meixner family of probability distributions so that our method gives a new approach to the characterization of free Meixner distributions.
9
Content available remote Image encryption using edge diffusion and selective permutations in orthogonal polynomials based transformation domain
Krishnamoorthi R., Sheba Kezia Malarchelvi P. D.
Image Processing & Communications
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2009
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Vol. 14, no 4
13-27
EN
This paper proposes a new image encryption technique in orthogonal polynomials based transformation domain (OPT) with edge diffusion and selective combinational permutations for secure transmission of images. In this technique, we propose the use of the polynomials based edge detection algorithm to decompose the image blocks into edge blocks and non-edge blocks. The edge blocks are first encrypted with a cryptographic algorithm and then a selective combinational permutation is applied over the low frequency coefficients of the edge as well as non-edge blocks in order to reduce the number of bits to be permuted. Then a combinational shuffling of bits, coefficients and blocks are carried out in addition to sign bit encryption. A symmetric key based cryptographically secure pseudo random process controls the entire encryption process. Experimental results reveal that the proposed encryption scheme provides very low encryption PSNRs and security analyses prove that the proposed technique offers effective encryption.
10
Content available remote 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.
11
Content available remote Holomorphic extension of locally holder functions
Rusev P.
Demonstratio Mathematica
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2003
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Vol. 36, nr 2
335--342
EN
Let γ be a smooth Jordan curve in the extended complex plane passing trough the point at infinity. In the paper are given sufficiently conditions under which a complex function defined on γ admits a holomorphic extensions into a region complementary to γ.
12
Content available remote The infinite divisibility and orthogonal polynomials with a constant recursion formula in free probability theory
Saitoh N., Yoshida H.
Probability and Mathematical Statistics
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2001
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Vol. 21, Fasc. 1
159--170
EN
We calculate Voiculescu’s R-transform of the compactly supported probability measure on Rinduced from the orthogonal polynomials with a constant recursion formula, and investigate its infinite divisibility with respect to the additive free convolution. In the case of infinite divisibility, we give the Lévy-Hinčin measure explicitly for the integral representation of the R-transform of the free analogue of the Lévy-Hinčin formula.
13
Integrals of legendre polynomials and solution of some partial differential equations
Belinsky R.
Journal of Applied Analysis
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2000
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Vol. 6, nr 2
259-282
EN
We show a connection between the polynomials whose inflection points coincide with their interior roots (let us write shorter PIPCIR), Legendre polynomials, and Jacobi polynomials, and study some properties of PIPCIRs (Part I). In addition, we give new formulas for some classical orthogonal polynomials. Then we use PIPCIRs to solve some partial differential equations (Part II).
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