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1
Content available Maximum packings of the a-fold complete 3-uniform hypergraph with loose 3-cycles
Bunge Ryan C., Collins Dontez, Conko-Camel Daryl, El-Zanati Saad I., Liebrecht Rachel, Vasquez Alexander
Opuscula Mathematica
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2020
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Vol. 40, no. 2
209--225
EN
It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order v if and only if v ≡0, 1, or 2 (mod 9). For all positive integers λand v, we find a maximum packing with loose 3-cycles of the λ-fold complete 3-uniform hypergraph of order v. We show that, if v ≥6, such a packing has a leave of two or fewer edges.
2
Content available Asymptotic expansion of large eigenvalues for a class of unbounded Jacobi matrices
Harrat Ayoub, Zerouali El Hassan, Zielinski Lech
Opuscula Mathematica
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2020
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Vol. 40, no. 2
241--270
EN
We investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly.
3
Content available Asymptotics of the discrete spectrum for complex Jacobi matrices
Malejki M.
Opuscula Mathematica
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2014
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Vol. 34, no. 1
139--160
EN
The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l2(N).
4
Content available Relation between determinants of tridiagonal and certain pentadiagonal matrices
Borowska J., Łacińska L.
Journal of Applied Mathematics and Computational Mechanics
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2014
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Vol. 13, nr 3
21--28
EN
In this paper we consider a pentadiagonal matrix which consists of only three non-zero bands. We prove that the determinant of such a matrix can be represented by a product of two determinants of corresponding tridiagonal matrices. It is shown that such an approach gives greatly shorter time of computer calculations.
5
Content available Efektywne programowanie w Matlabie : odwracanie macierzy trójprzekątniowych metodą eliminacji Gaussa
Keller P., Wróbel I.
Biuletyn Naukowy Wrocławskiej Wyższej Szkoły Informatyki Stosowanej. Informatyka
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2014
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vol. 4
8--13
PL
Celem cyklu artykułów Efektywna programowanie, w Matlabie. jest prezentacja sposobów pisania bardzo wydajnych algorytmów w języku Matlab, rozwiązujących wybrane problemy obliczeniowe. W niniejszym artykule przedstawiamy efektywną implementację metody eliminacji Gaussa zastosowanej do wyznaczania odwrotności macierzy trój przekątniowych. Zaimplementowane zostały warianty eliminacji zarówno bez, jak i z wyborem elementów głównych. Wysoka efektywność stworzonych funkcji potwierdzona jest wykonanymi testami obliczeniowymi.
EN
The scries Effective programming in Mallab is meant to present very fast implementations of al- gorithms for solving various computational problems in the Matlab programming language. In this paper, we present a very efficient implementation of the Gaussian elimination algorithm applied to computing the inverse of a tridiagonal matrix. Two variants of the elimination, without and with pivoting, are considered. The high efficiency of the presented solutions is supported by computational examples.
6
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.
7
Content available On determinant of certain pentadiagonal matrix
Borowska J., Łacińska L., Rychlewska J.
Journal of Applied Mathematics and Computational Mechanics
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2013
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Vol. 12, nr 3
21--26
EN
In this paper, using the LU factorization, the relation between the determinant of a certain pentadiagonal matrix and the determinant of a corresponding tridiagonal matrix will be derived. Moreover, it will be shown that determinant of this special pentadiagonal matrix can be calculated by applying the fourth order homogeneous linear difference equation.
8
Content available Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices
Malejki M.
Opuscula Mathematica
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2010
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Vol. 30, no. 3
311-330
EN
The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from 1 to N, for a Jacobi matrix J by the eigenvalues of the finite submatrix J(n) of order pn x pn, where N = max{k ∈ N : k ≤ rpn} and r ∈ (0, 1) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of J in the case p = 3.
9
Content available Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvaules of finite submatrices
Malejki M.
Opuscula Mathematica
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2007
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Vol. 27, no. 1
37-49
EN
We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space l2(N) by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the first n eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order n x n. The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [7]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.
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