arXiv subjects

Rachid Marsli

Publications and source records attributed to Rachid Marsli.

3 recordsLinked to original sources

arXiv2020

Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair

Let (λ, v) be a known real eigenpair of a square real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than λ. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any square real matrix.

math.CO↗

arXiv2019

Some properties of ergodicity coefficients with applications in spectral graph theory

The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm based ergodicity coefficients Tp. If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using Tp. In the last section, these two results are applied to bound the spectral radius of the Laplacian matrix as well as the algebraic connectivity of its associated graph. Many other results are obtained. In particular, Theorem 2.15 is a convergence theorem for Tp and Theorem 4.7 compares some ergodicity coefficients to each other.

math.CO↗

arXiv2019

On the prime decomposition of integers of the form $(z^n-y^n)/(z-y)$

In this work, the author shows a sufficient and necessary condition for an integer of the form $(zn-y^n)/(z-y)$ to be divisible by some perfect $mth$ power $p$, where $p$ is an odd prime and $m$ is a positive integer. A constructive method of this type of integers is explained with details and examples. Links between the main result and known ideas such as Termat's last theorem, Goormaghtigh conjecture and Mersenne numbers are discussed. other related ideas, examples and applications are provides.

math.GM↗