If k = 1, then X is called the group inverse of A, and is denoted by X = Ag. The Drazin inverse is very useful in various applications (see, e.g. [1]–[4]; applications in singular differential and difference equations, Markov chains and iterative methods). In 1980, Cline and Greville [5] extended the Drazin inverse of square matrix to rectangular matrix, which can be generalized to the quaterni...

Abstract In the present work, firstly, we use a minimax equality to prove existence of solution certain system varitional equations providing numerical approximation such solution. Then, propose method solve collage-type inverse problem associated with corresponding system, and illustrate behaviour example.

Consider any nilpotent group G of finite odd order. We ask if we can always find a galois extension K of Q such that Gal(K/Q) ∼= G. This is the famous Inverse Galois Problem applied to nilpotent groups of finite odd order. By solving the Group Extension Problem and the Embedding Problem, two problems that are related to the Inverse Galois Problem, we show that such a K always exists. A major re...