Given a rational matrix G with complex coefficients and a domain Γ in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over Γ, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the pol...

Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. We say that F is a degreesequence-forcing set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in C. A degreesequence-forcing set is minimal if no proper subset is degree-sequenceforcing. We characterize the non-minimal degree-sequ...

In this paper we construct minimal cubature formulae of trigonometric degree: we obtain explicit formulae for low dimensions of arbitrary degree and for low degrees in all dimensions. A useful tool is a closed form expression for the reproducing kernels in two dimensions.

We show that there exists a Turing degree which is minimal and fixed point free.

We write H → G if every 2-coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G-minimal if H → G, but for every proper subgraph H ′ of H, H ′ 6→ G. We define s(G) to be the minimum s such that there exists a G-minimal graph with a vertex of degree s. We prove that s(Kk) = (k − 1) and s(Ka,b) = 2 min(a, b) − 1. We also pose several related open problems. c © 2...

In this paper we compute the minimal degree of a faithful representation by partial transformations finite semigroup admitting completely reducible matrix over field complex numbers. This includes all inverse semigroups, and hence our results generalize earlier Easdown Schein on an semigroup. It also well-studied monoids like full fields monoid binary relations (i.e., matrices Boolean semiring)...