We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky’s theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally we con...

Abstract We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami [BH2], essentially continues to hold for a general real symmetric Wigner matrix. To obtain this result, we ad...

In this paper, we extend and generalize some of the well known spectral properties of a nonnegative matrix by establishing analogous properties for the zeros of derivatives of the characteristic polynomial of such a matrix.

One special subalgebra of the algebra of fuzzy truth values is the set of those elements that take only 0 and 1 as values. This subalgebra is in one-to-one correspondence with the subsets of the unit interval, but the operations do not correspond to ordinary union and intersection. The automorphisms of this subalgebra are studied here, resulting in showing that the subobjects corresponding to n...

For the polynomial planar vector fields with a hyperbolic or nilpotent critical point at the origin, the monodromy problem has been solved, but for the strongly degenerate critical points this problem is still open. When the critical point is monodromic, the stability problem or the center- focus problem is an open problem too. In this paper we will consider the polynomial planar vector fields ...