Computation of generalized inverses by using the LDL∗ decomposition

AbstractAn efficient algorithm, based on the LDL∗ factorization, for computing {1,2,3} and {1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix A, is developed. We consider matrix products A∗A and AA∗ and corresponding LDL∗ factorizations in order to compute the generalized inverse of A. By considering the matrix products (R∗A)†R∗ and T∗(AT∗)†, where R and T are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3} and A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse

Symbolic computation of AT,S(2)-inverses using QDR factorization

AbstractEfficient evaluation of the full-rank QDR decomposition is established. A method and algorithm for efficient symbolic computation of AT,S(2) inverses of a given rational matrix A is defined using the full-rank QDR decomposition of an appropriate rational matrix W. The algorithm is implemented using MATHEMATICA’s ability to deal with symbolic expressions as well as with numbers. Examples including polynomial and rational matrices are presented. Some comparisons with well-known methods for symbolic evaluation of generalized inverses are obtained

Environmental Effects on Superoxide Dismutase and Catalase Activity and Expression in Honey Bee

Understanding the cellular stress response in honey bees will significantly contribute to their conservation. The aim of this study was to analyze the response of the antioxidative enzymes superoxide dismutase and catalase in honey bees related to the presence of toxic metals in different habitats. Three locations were selected: (i) Tunovo on the mountain Golija, as control area, without industry and large human impact, (ii) Belgrade as urban area, and (iii) Zajaca, as mining and industrial zone. Our results showed that the concentrations of lead (Pb) in whole body of bees vary according to habitat, but there was very significant increase of Pb in bees from investigated industrial area. Bees from urban and industrial area had increased expression of both Sod1 and Cat genes, suggesting adaptation to increased oxidative stress. However, in spite increased gene expression, the enzyme activity of catalase was lower in bees from industrial area suggesting inhibitory effect of Pb on catalase