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We find the limit distributions for a spectrum of a system of n particles governed by a k-body interaction. The hamiltonian of this system is modelled by a Gaussian random matrix. We show that the limit distribution is a q-deformed Gaussian distribution with the deformation parameter q depending on the fraction k/√n. The family of q-deformed Gaussian distributions include the Gaussian distribution and the semicircular law; therefore our result is a generalization of the results of Wigner [Wig1,...

Limit points of eigenvalues of (di)graphs

Fu Ji Zhang, Zhibo Chen (2006)

Czechoslovak Mathematical Journal

The study on limit points of eigenvalues of undirected graphs was initiated by A. J. Hoffman in 1972. Now we extend the study to digraphs. We prove: 1. Every real number is a limit point of eigenvalues of graphs. Every complex number is a limit point of eigenvalues of digraphs. 2. For a digraph $D$ , the set of limit points of eigenvalues of iterated subdivision digraphs of $D$ is the unit circle in the complex plane if and only if $D$ has a directed cycle. 3. Every limit point of eigenvalues of a set...

Limiting spectral distribution of circulant type matrices with dependent inputs.

Bose, Arup, Hazra, Rajat Subhra, Saha, Koushik (2009)

Electronic Journal of Probability [electronic only]

Limiting spectral distribution of XX' matrices

Arup Bose, Sreela Gangopadhyay, Arnab Sen (2010)

Annales de l'I.H.P. Probabilités et statistiques

The methods to establish the limiting spectral distribution (LSD) of large dimensional random matrices includes the well-known moment method which invokes the trace formula. Its success has been demonstrated in several types of matrices such as the Wigner matrix and the sample covariance matrix. In a recent article Bryc, Dembo and Jiang [Ann. Probab.34 (2006) 1–38] establish the LSD for random Toeplitz and Hankel matrices using the moment method. They perform the necessary counting of terms in the...

Linear discrepancy of basic totally unimodular matrices.

Doerr, Benjamin (2000)

The Electronic Journal of Combinatorics [electronic only]

Linear maps that strongly preserve regular matrices over the Boolean algebra

Kyung-Tae Kang, Seok-Zun Song (2011)

Czechoslovak Mathematical Journal

The set of all $m \times n$ Boolean matrices is denoted by $𝕄_{m, n}$ . We call a matrix $A \in 𝕄_{m, n}$ regular if there is a matrix $G \in 𝕄_{n, m}$ such that $A G A = A$ . In this paper, we study the problem of characterizing linear operators on $𝕄_{m, n}$ that strongly preserve regular matrices. Consequently, we obtain that if $min {m, n} \leq 2$ , then all operators on $𝕄_{m, n}$ strongly preserve regular matrices, and if $min {m, n} \geq 3$ , then an operator $T$ on $𝕄_{m, n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T (X) = U X V$ for all $X \in 𝕄_{m, n}$ , or $m = n$ and $T (X) = U X^{T} V$ for all $X \in 𝕄_{n}$ .

Linear maps which preserve or strongly preserve weak majorization.

Hasani, Ahmad Mohammad, Vali, Mohammad Ali (2007)

Journal of Inequalities and Applications [electronic only]

Linear operators preserving maximal column ranks of nonbinary boolean matrices

Seok-Zun Song, Sung-Dae Yang, Sung-Min Hong, Young-Bae Jun, Seon-Jeong Kim (2000)

Discussiones Mathematicae - General Algebra and Applications

The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.

Linear operators that preserve Boolean rank of Boolean matrices

LeRoy B. Beasley, Seok-Zun Song (2013)

Czechoslovak Mathematical Journal

The Boolean rank of a nonzero $m \times n$ Boolean matrix $A$ is the minimum number $k$ such that there exist an $m \times k$ Boolean matrix $B$ and a $k \times n$ Boolean matrix $C$ such that $A = B C$ . In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks $1$ and $2$ . In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank...

Linear operators that preserve graphical properties of matrices: isolation numbers

LeRoy B. Beasley, Seok-Zun Song, Young Bae Jun (2014)

Czechoslovak Mathematical Journal

Let $A$ be a Boolean ${0, 1}$ matrix. The isolation number of $A$ is the maximum number of ones in $A$ such that no two are in any row or any column (that is they are independent), and no two are in a $2 \times 2$ submatrix of all ones. The isolation number of $A$ is a lower bound on the Boolean rank of $A$ . A linear operator on the set of $m \times n$ Boolean matrices is a mapping which is additive and maps the zero matrix, $O$ , to itself. A mapping strongly preserves a set, $S$ , if it maps the set $S$ into the set $S$ and the complement of...

Linear preservers of left matrix majorization.

Khalooei, Fatemeh, Radjabalipour, Mehdi, Torabian, Parisa (2008)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Linear preservers of regular matrices over general Boolean algebras.

Kang, Kyung-Tae, Song, Seok-Zun, Heo, Seong-Hee, Jun, Young-Bae (2011)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

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