Saddle-point systems appear in many scientific and engineering applications. The systems can be sparse, symmetric or nonsymmetric, and possibly singular. In many of these applications, the number of constraints is relatively small compared to the number of unknowns. The traditional null-space method is inefficient for these systems, because it is expensive to find the null space explicitly. Som...

An n × n matrix is called a principally nonsingular matrix (NSmatrix) if all its principal minors are different from zero and it is called a totally nonsingular matrix (TNS-matrix) if all its minors are different from zero. In this paper, we are interested in the NS-matrix (TNSmatrix) completion problem: whether a partial NS-matrix (TNS-matrix) has a NS-matrix (TNS-matrix) completion. Here, we ...

a toroidalization of a dominant morphism $varphi: xto y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $x$ and $y$. we give a proof of toroidalization of locally toroidal morphisms of 3-folds.