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- Densities of maximal embedding dimension numerical semigroupsPublication . Iglésias, Laura; Neto, Ana MargaridaWe define the density of a numerical semigroup and study the densities of all the maximal embedding dimension numerical semigroups with a fixed Frobenius number, as well as the possible Frobenius number for a fixed density. We also prove that for a given possible density, in the sense of Wilf's conjecture, one can find a maximal embedding dimension numerical semigroup with that density.
- Weakly spectrally complete pair of matricesPublication . Iglésias, Laura; Silva, Fernando C.Let and be matrices over an algebraically closed field. Let be elements of such that and . We give necessary and sufficient condition for the existence of matrices and similar to and, respectively, such that has eigenvalues.
- Unified min-max and interlacing theorems for linear operatorsPublication . Iglésias, Laura; Santa-Clara, Catarina; Silva, Fernando C.There exist striking analogies in the behaviour of eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely diagonalization theorems, interlacing inequalities and Courant-Fischer type formulae. Carlson and Sa [D. Carlson and E.M. Sa, Generalized minimax and interlacing inequalities, Linear Multilinear Algebra 15 (1984) pp. 77-103.] introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite-dimensional case. We show that this unification can be done using modular lattices with Goldie dimension, which have a natural structure of s-space in the finite-dimensional case, and extend the unification to the countable-dimensional case.
- Spectral and weak polynomial completeness for the product of nonsingular matricesPublication . Iglésias, Laura; Silva, Fernando C.Let F be a field with at least four elements. In this paper, we identify all the pairs (A, B) of n x n nonsingular matrices over F, satisfying the following property: for every monic polynomial f (x) = x(n) + a(n-1)x(n-1) +... + a(1)x + a(0) over F, with a root in F and a(0) = (-1)(n) det(AB), there are nonsingular matrices X, Y is an element of F-nxn such that XAX(-1)Y BY-1 has characteristic polynomial f (x).