A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simpl...

The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maxim...

In this paper we deal with C*-algebras of real rank zero that can be represented as inductive limits A = lim −→(An, νn+1,n) of direct sums of homogeneous C*-algebras of the form An = ⊕ i=1 Pn,iM[n,i](C(Xn,i))Pn,i, where Xn,i are finite CW complexes and Pn,i are selfadjoint projections. Following [3] we called these C*-algebras approximately homogeneous. Our main result asserts that any approxim...

we derive the formulas of the maximal andminimal ranks of four real matrices $x_{1},x_{2},x_{3}$ and $x_{4}$in common solution $x=x_{1}+x_{2}i+x_{3}j+x_{4}k$ to quaternionmatrix equations $a_{1}x=c_{1},xb_{2}=c_{2},a_{3}xb_{3}=c_{3}$. asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. we give the exp...

We classify all essential extensions of the form 0 → B → D → C(X) → 0 where B is a nonunital simple separable finite real rank zero Z-stable C*algebra with continuous scale, and where X is a finite CW complex. In fact, we prove that there is a group isomorphism Ext(C(X),B) → KK(C(X),M(B)/B).