Hassane Abbas

Lebanese University · Faculty of Science-1, Department of Mathematics

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eige...

We study the inverse eigenvalue problem for finding doubly stochastic matrices with specified eigenvalues. By making use of a combination of Dykstra's algorithm and an alternating projection process onto a non-convex set, we derive hybrid algorithms for finding doubly stochastic matrices and symmetric doubly stochastic matrices with prescribed eige...

CITATIONS 0 READS 8 4 authors, including: Some of the authors of this publication are also working on these related projects: Lin's conjecture about determinantial inequalities View project Lin's conjecture about determinantial inequalities Project View project Bassam Mourad Lebanese University Abstract An n-list λ := (r; λ 2 ,. .. , λ n) of comple...

In this paper, we prove that each of the following functions is convex on R : f (t) = w N (A t XA 1−t ± A 1−t XA t), (t) = w N (A t XA 1−t), and h(t) = w N (A t XA t) where A > 0, X ∈ M n and N(.) is a unitarily invariant norm on M n. Consequently, we answer positively the question concerning the convexity of the function t → w(A t XA t) proposed b...

Let A be an n×n matrix and let ∨kA be its k-th symmetric tensor power. We express the normalized trace of ∨kA as an integral of the k-th powers of the numerical values of A over the unit sphere Sn of Cn with respect to the rotation-invariant probability measure. Equivalently, this expression in turn can be interpreted as an integral representation...

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by Lemos and Soares in 2018. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by Bh...

Let \(\vee ^k A\) be the k-th symmetric tensor power of \(A\in M_n(\mathbb {C})\). In 2021, we have expressed the normalized trace of \(\vee ^kA\) as an integral of the k-th powers of the numerical values of A over the unit sphere \(\mathbb {S}^{n}\) of \(\mathbb {C}^{n}\) with respect to the normalized Euclidean surface measure \(\sigma\). In this...

Let $A$ be an $n\times n$ matrix and let $\vee^k A$ be its $k$-th symmetric tensor product. We express the normalized trace of $\vee^k A$ as an integral of the $k$-th powers of the numerical values of $A$ over the unit sphere $\mathbb{S}^{n}$ of $\mathbb{C}^{n}$ with respect to the normalized Euclidean surface measure. Equivalently, this expression...

An n-list λ := (r; λ 2 ,. .. , λ n) of complex numbers with r > 0, is said to be realizable if λ is the spectrum of n × n nonnegative matrix A and in this case A is said to be a nonnegative realization of λ. If, in addition, each row and column sum of A equals r, then λ is said to be doubly stochastically realizable and in such case A is said to be...

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by R. Lemos and G. Soares in \cite{lemos}. In addition, we present a complement of a unitarily invariant norm inequality which was con...

Let A, B be n×n matrices such that A is positive semi-definite and B is Her-mitian. In this note, it is shown, among other inequalities, the following determinantal inequality det(A k + (AB) 2) ≥ det(A k + A 2 B 2) for all k ∈ [1, ∞[ .

In this note, we consider the problem of characterizing the conditions under which the positive semi-definite pth root of a positive semi-definite doubly stochastic matrix is doubly stochastic. First, we obtain new sufficient conditions for this problem that improve the existing ones for the case p = 2. In addition, if we let K n denote the set of...

The main goal of this paper is to prove the following determinantal inequality:det⁡(Ak+|Bs2As2|2ts)≤det⁡(Ak+AtBt)≤det⁡(Ak+|As2Bs2|2ts) for any positive semi-definite matrices A and B, and for all 0≤t≤s≤k. It generalizes several known determinantal inequalities, and one main consequence of it confirms Lin's conjecture which states that for positive...

In the present paper, we provide several inequalities for the generalized numerical radius of operator matrices as introduced by A. Abu-omar and F. Kittaneh in [3]. We generalize the convexity and the log-convexity results obtained by M. Sababheh in [12] for the case of the numerical radius to the case of the generalized numerical radius. We illust...

In 2016, Lin conjectured that if ABB∗C∈M2(Mn) is a positive semi-definite matrix then sj(Φ(B))≤sj(Φ(A)♯Φ(C)), j=1,2,…,n, where Φ(X)=X+Tr(X)In and sj(.) means the jth largest singular value. In this note, we confirm this conjecture when AB = BA and prove the more general result sj(Ψf(B))≤sj(Ψf(A)♯Ψf(C))andsj(Ψf(|B|))≤sj(Ψf(A♯C)),j=1,2,…,n where Ψf(X...

In 2017, M. Lin formulated two conjectures concerning determinantal inequalities for positive semi-definite matrices A and B, and which can be stated as followsdet⁡(A2+|AB|p)≥det⁡(A2+|BA|p) for p≥0 anddet⁡(A2+|AB|p)≥det⁡(A2+ApBp) for 0≤p≤2. The main goal of this paper is to confirm the first conjecture in a slightly more general setting namely in t...

The real (resp. symmetric) doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real n-tuple λ=(1,λ2,…,λn) to be the spectrum of an n×n (resp. symmetric) doubly stochastic matrix. If λi≤0 for all i=2,…,n and the sum of all the entries in λ is nonnegative, then we refer to such λ as a nor...

The notion of similarity is a fundamental concept in different scientific fields. Similarity measures aim at quantifying the extent to which objects resemble each other. This paper is concerned with the analysis of the properties of similarity matrices. More specifically, we focus on their positive semi-definite property, which is important to deri...

The inverse eigenvalue problems play an important role in broad application areas such as system identification, Hopfield neural networks, control design, mass–spring system and molecular spectroscopy. This paper proposes an algorithm that yields a new method to efficiently and accurately compute the partially bisymmetric solutions (M,C,K) under pr...

In this article, we present an algorithm to compute the asymptotic representations of solutions of singularly-perturbed linear differential systems, in a neighborhood of a turning point. Our algorithm is based on an analysis by a Newton polygon and is implemented in the computer algebra system Maple.

The symmetric doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real -tuple to be the spectrum of an nxn symmetric doubly stochastic matrix. For n>3 , this problem remains open though many partial results are known. In this note, we present a new family of necessary conditions for thi...

We introduce and investigate the notion of an operator P-class function. We show that every nonnegative operator convex function is of operator P-class, but the converse is not true in general. We present some Jensen type operator inequalities involving P-class functions and some Hermite-Hadamard inequalities for operator P-class functions. MSC: 4...

For any unitarily invariant norm vertical bar vertical bar vertical bar . vertical bar vertical bar vertical bar, the Heinz inequalities for operators assert that 2 vertical bar vertical bar vertical bar A(1/2)XB(1/2)vertical bar vertical bar vertical bar <= vertical bar vertical bar vertical bar A(nu)XB(1-nu) + A(1-nu)XB(nu) vertical bar vertical...

A symmetric doubly stochastic matrix is said to be determined by its spectrum (DS) if the only symmetric doubly stochastic matrices that are similar to are of the form for some permutation matrix The problem of characterizing such matrices is considered here. An ‘almost’ the same but a more difficult problem is the following: ‘Characterize all the...

In this article, we recover singularly-perturbed linear differential systems from their turning points and reduce the rank of the singularity in the parameter to its minimal integer value. Our treatment is Moser-based; that is to say it is based on the reduction criterion introduced for linear singular differential systems by Moser. Such algorithms...

In this paper, we present an algorithm which computes a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in two variables, based on (Barkatou, 1997). A first step was set in (Barkatou-LeRoux, 2006) where the problem of rank reduction was tackled via the approach of (Levelt, 1991). We give instea...

In this note, we present an algorithm that yields a new method for constructing doubly stochastic and symmetric doubly stochastic matrices for the inverse eigenvalue problem. In addition, we introduce new open problems in this area that lay the ground for future work.

In this note a simple extension of the complex algebra to higher dimension is proposed. Using the postulated algebra a two dimensional Dirac equation is formulated and its solution is calculated. It is found that there is a sub-algebra where the associative nature can be recovered.

In this paper, we develop a double scale numerical method which may be of great significance in computing discontinuous solutions of certain systems of PDE's arising in application. It is of basic importance, especially in industrial applications, to study the solutions which represent shock waves. Then there appear "multiplications of distribution...

In this paper, we developed a Godunov scheme for solving nonconservative systems. The main idea of this method is a new type of projection which illustrated the essential role of the numerical viscosity to determine the solution with shocks for system in a nonconservative form. We apply our study to a system modeling elasticity and we observe a com...