Rosário Fernandes

The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles

In 1979, Ferguson characterized the periodic Jacobi matrices with given eigenvalues and showed how to use the Lanzcos Algorithm to construct each such matrix. This article provides general characterizations and constructions for the complex analogue of periodic Jacobi matrices. As a consequence of the main procedure, we prove that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2.

Diabetic mastopathy: a case report

Diabetic mastopathy (DMP) is an uncommon collection of clinical, radiological, and histological features, classically described in premenopausal women with long-term insulin-dependent diabetes mellitus. This entity can mimic breast carcinoma, but, in the appropriate clinical and imaging setting, the diagnosis can be made by core biopsy, avoiding unnecessary surgeries. We report the case of a 34-year-old female, with a 12-year history of type 1 diabetes, who presented with bilateral breast lumps. Mammography, ultrasonography, and magnetic resonance imaging could not exclude the suspicion of malignancy, and a core biopsy was performed showing the typical histologic features of DMP. The literature is briefly reviewed.

On the inverse eigenvalue problems: the case of superstars

Let T be a tree and let x0 be a vertex of T. T is called a superstar with central vertex x0 if Tx0 is a union of paths. The General Inverse Eigenvalue Problem for certain trees is partially answered. Using this description, some superstars are presented for which the problem of ordered multiplicity lists and the Inverse Eigenvalue Problem are not equivalent.

An extension of Brualdi’s algorithm for the construction of (0,1)-matrices with prescribed row and column sum vectors

Abstract Given partitions R and S with the same weight and S ⪯ R ∗ , the Robinson–Schensted–Knuth correspondence establishes a bijection between the class A ( R , S ) of ( 0 , 1 ) -matrices with row-sum R and column-sum S , and pairs of Young tableaux with conjugate shape λ and λ ∗ , with S ⪯ λ ⪯ R ∗ . We give a canonical construction for matrices in A ( R , S ) whose insertion tableau has a prescribed shape λ , with S ⪯ λ ⪯ R ∗ . This algorithm generalizes some recent constructions due to R. Brualdi for the extremal cases λ = S and λ = R ∗ (using a Ryser-like algorithm), and due to C.M. da Fonseca and R. Mamede for particular cases of λ .

Decomposable λ-critical tensors

Let λ=(λ1,…,λs) be a partition of m and let V be a finite dimensional vector space over C. We also denote by λ the irreducible character of Sm associated with the partition λ and by Vλ we denote the symmetry class of tensors associated with λ and V. Let j∈{1,…,λ1} and z∈Vλ. The concept of j-reach of z is defined. Using this concept we introduce the concept of λ-critical element of Vλ. Generalized Plucker polynomials are constructed in a way that the set of their common roots contains the set of the families of components of decomposable λ-critical elements of Vλ. The concepts and results are generalizations of those defined and proved in [4].

Small perturbations and pairs of matrices that have the same immanent

We study the behaviour of the covering number of an element of a family of vectors under small perturbations. We apply this study to obtain results on the pairs of matrices that have the same immanent.

Location of the eigenvalues of weighted graphs with a cut edge

We establish some identities for the characteristic polynomial of Hermitian matrices whose graph is a cycle. We use the paper C.M da Fonesca: Interlacing properties for Hermitian matrices whose graph is a given tree, SIAM J. Matrix Anal. Appl. 27 (2005) pp. 130–141 and we extend some interlacing results obtained in this article to graphs with a cut edge. For some cases, we give a graphical interpretation of the results.

Rank partitions and covering numbers under small perturbations of an element

We investigate how the rank partitions and the covering number of the elements can change with arbitrarily small perturbations of a fixed element.

Semilinear preservers of the immanants in the set of the doubly stochastic matrices

Let

On the spectra of some graphs like weighted rooted trees

S_n