Phylogeography of a marine insular endemic in the Atlantic Macaronesia : the Azorean barnacle, Megabalanus azoricus (Pilsbry,1916)

[EN]The Azorean barnacle, Megabalanus azoricus (Pilsbry, 1916), is a Macaronesian endemic whose obscure taxonomy and the unknown relationships among forms inhabiting isolated Northern Atlantic oceanic islands is investigated by means of molecular analysis herein. Mitochondrial data from the 16S rRNA and COX1 genes support its current species status, tropical ancestry, and the taxonomic homogeneity throughout its distribution range. In contrast, at the intraspecific level and based on control region sequences, we detected an overall low level of genetic diversity and three divergent lineages. The haplogroups α and γ were sampled in the Azores, Madeira, Canary, and Cabo Verde archipelagos; whereas haplogroup β was absent from Cabo Verde

A generalization of the Moore-Penrose inverse related to matrix subspaces of C(nxm)

[EN]A natural generalization of the classical Moore-Penrose inverse is presented. The so-called S-Moore-Penrose inverse of a m x n complex matrix A, denoted by As, is defined for any linear subspace S of the matrix vector space Cnxm. The S-Moore-Penrose inverse As is characterized using either the singular value decomposition or (for the nonsingular square case) the orthogonal complements with respect to the Frobenius inner product. These results are applied to the preconditioning of linear systems based on Frobenius norm minimization and to the linearly constrained linear least squares problem.