Effective partitioning method for computing weighted Moore-Penrose inverse

We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizations of the partitioning method for computing the Moore-Penrose inverse of rational and polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004) 137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82 (2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are implemented in the symbolic computational package MATHEMATICA

Late Utilitarian Moral Theory and Its Development: Sidgwick, Moore

Henry Sidgwick taught G.E. Moore as an undergraduate at the University of Cambridge. Moore found Sidgwick’s personality less than attractive and his lectures “rather dull”. Still, philosophically speaking, Moore absorbed a great deal from Sidgwick. In the Preface to the Trinity College Prize Fellowship dissertation that he submitted in 1898, just two years after graduation, he wrote “For my ethical views it will be obvious how much I owe to Prof. Sidgwick.” Later, in Principia Ethica, Moore credited Sidgwick with having “first clearly exposed the [naturalistic] fallacy” – a fallacy putatively committed when one defines naturalistically or super-naturalistically “good” – which was one of the book’s main ambitions (PE 39; also 17, 59). It is therefore unsurprising that Moore remarks in the intellectual autobiography he wrote years later that “From…[Sidgwick’s] published works…I have gained a good deal, and his clarity and his belief in Common Sense were very sympathetic to me.” This influence did not, however, prevent Moore from registering disagreements with Sidgwick, the sharpest of which concern the viability of egoism and the nature of the good. The disagreements between Sidgwick and Moore speak to many important moral theoretical issues arising both within and without the utilitarian tradition in ethical thinking. Because the two share much in common, a critical comparison of them on a range of moral philosophical questions proves instructive. It will tell us in particular something about the general direction of ethical thinking in the utilitarian tradition at the dawn of the twentieth century. This chapter has four parts. Part I compares the versions of utilitarianism to which Sidgwick and Moore subscribed. Part II examines the arguments each provides for the view. Part III discusses their conflicting theories of value. Part IV sums things up