Superregular matrices and applications to convolutional codes

The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matrix is superregular if all of its minors that are not trivially zero are nonzero. Given a a×b, a ≥ b, superregular matrix over a field, we show that if all of its rows are nonzero then any linear combination of its columns, with nonzero coefficients, has at least a−b + 1 nonzero entries. Secondly, we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code, namely, the rate and the Forney indices. These results answer some open questions on distances and constructions of convolutional codes posted in the literature.

Bacterial collagenases: a review

Bacterial collagenases are metalloproteinases involved in the degradation of the extracellular matrices of animal cells, due to their ability to digest native collagen. These enzymes are important virulence factors in a variety of pathogenic bacteria. Nonetheless, there is a lack of scientific consensus for a proper and well-defined classification of these enzymes and a vast controversy regarding the correct identification of collagenases. Clostridial collagenases were the first ones to be identified and characterized and are the reference enzymes for comparison of newly discovered collagenolytic enzymes. In this review we present the most recent data regarding bacterial collagenases and overview the functional and structural diversity of bacterial collagenases. An overall picture of the molecular diversity and distribution of these proteins in nature will also be given. Particular aspects of the different proteolytic activities will be contextualized within relevant areas of application, mainly biotechnological processes and therapeutic uses. At last, we will present a new classification guide for bacterial collagenases that will allow the correct and straightforward classification of these enzymes.

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph.