Wyniki wyszukiwania - Biblioteka Nauki

1

Combinatorial minors for matrix functions and their applications

100%

Shevelev V.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

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2014

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tom z. 4

5--16

EN

As well known, permanent of a square (0,1)-matrix A of order n enumerates the permutations β of 1, 2, ..., n with the incidence matrices B ≤ A. To obtain enumerative information on even and odd permutations with condition B ≤ A, we should calculate two-fold vector (ɑ1, ɑ2) with ɑ1 + ɑ2 = per A. More general, the introduced ω-permanent, where ω = e2πi/m, we calculate as m-fold vector. For these and other matrix functions we generalize the Laplace theorem of their expansion over elements of the first row, using the defined so-called “combinatorial minors”. In particular, in this way, we calculate the cycle index of permutations with condition B ≤ A.

PL

Jak wiadomo, permanent (0, 1)-macierzy kwadratowej A stopnia n podaje liczbę permutacji β liczb 1, 2, ..., n, mających macierz incydencji B ≤ A. Aby otrzymać informację o liczbie parzystych i nieparzystych permutacji z warunkiem B ≤ A, należy obliczyć dwuskładowy wektor (ɑ1, ɑ2), gdzie ɑ1 + ɑ2 = per A. Ogólniej wprowadzamy pojęcie ω-permanentu, gdzie ω = e2πi/m, który obliczamy jako odpowiedni m-składowy wektor. Dla takich i innych funkcji macierzowych uogólniamy twierdzenie Laplace’a o ich rozwinięciu względem elementów pierwszego wiersza, wykorzystując zdefiniowane w tym celu tak zwane minory kombinatoryczne. W szczególności obliczamy w ten sposób indeks cyklowy permutacji spełniających warunek B ≤ A.

2

A Note on the Permanental Roots of Bipartite Graphs

94%

Zhang H. , Liu S. , Li W.

Discussiones Mathematicae Graph Theory

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2014

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tom 34

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nr 1

49-56

EN

It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imaginary axes. Furthermore, we prove that any graph has no negative real permanental root, and any graph containing at least one edge has complex permanental roots.

3

Per-Spectral Characterizations Of Some Bipartite Graphs

94%

Wu T. , Zhang H.

Discussiones Mathematicae Graph Theory

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2017

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tom 37

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nr 4

935-951

EN

A graph is said to be characterized by its permanental spectrum if there is no other non-isomorphic graph with the same permanental spectrum. In this paper, we investigate when a complete bipartite graph Kp,p with some edges deleted is determined by its permanental spectrum. We first prove that a graph obtained from Kp,p by deleting all edges of a star K1,l, provided l < p, is determined by its permanental spectrum. Furthermore, we show that all graphs with a perfect matching obtained from Kp,p by removing five or fewer edges are determined by their permanental spectra.

4

Immanant Conversion on Symmetric Matrices

71%

Purificação Coelho M. , Duffner M. A. , Guterman A. E.

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nr 1

EN

Letr Σn(C) denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C) -> Σn (C) satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB) = dχ·(Φ(Α ) + αΦ(Β)) for all matrices A,В ε Σ„(С) and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С).

5

On some conjectures regarding tridiagonal matrices

71%

Fonseca C. M.

Journal of Applied Mathematics and Computational Mechanics

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2018

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tom Vol. 17, nr 4

13--17

EN

We discuss several conjectures proposed recently by A.Z. Küçük and M. Düz on the permanent of certain type of tridiagonal matrices. We recall some less known results on tridiagonal matrices and, at the same time, bring other results together to a common framework.

6

On a recurrence for permanents of a sequence of 3-tridiagonal matrices

59%

Trojovský P. , Zvoníková I.

Journal of Applied Mathematics and Computational Mechanics

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2019

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tom Vol. 18, nr 4

95--100

EN

This is a corrigendum of the paper: Küçük, A. Z. & Düz, M. (2017). Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz and the Chebyshev polynomials. Journal of Applied Mathematics and Computational Mechanics, 16, 75-86. We will show that Remark 9, on page 84, does not hold, what is the consequence of the incorrect proof, which authors formulated there.

7

An update on a few permanent conjectures

48%

Zhang F.

Special Matrices

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2016

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tom 4

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nr 1

305-316

EN

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise a few new questions for future study. (†conjectures have been recently settled negatively.)