We nd innnitely many pairs of coprime integers, a and q, such that the least prime a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators.

The purpose of this paper is to give new necessary and sufficient conditions for internal stabilizability and existence of left/right/doubly coprime factorizations for linear MIMO systems. In particular, we generalize the YoulaKučera parametrization of all stabilizing controllers for every internally stabilizable MIMO plant which does not necessarily admit doubly coprime factorizations.

The Barát-Thomassen conjecture, recently proved in [3], asserts that for every tree T , there is a constant cT such that every cT -edge connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T -decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For i...

A well-established research area in finite group theory consists in exploring the interplay between the structure of a group G and certain sets of positive integers, which are naturally associated to G. One of those sets, denoted by cs(G), is the set of conjugacy class sizes of G. In order to have a better understanding of the arithmetical structure of cs(G), it is useful to introduce two kind ...

In this section, R denotes a ring with identity I. We study conditions under which I - Aλ and I - Bλ are left coprime or right coprime, where A, B ∈ R. As applications, we get sufficient conditions under which the Kneading determinant of a finite rank pair of operators on an infinite dimensional space is rational.

We study the problem of existence of weak right or left or strong coprime factorizations in H-infinity over the right half-plane of an analytic function defined in some subset of the right half-plane. We give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous-time axis. In particular, we...

This paper studies the antiwindup synthesis problems for plants with coprime factor uncertainties. The proposed technique utilizes an equivalent framework of compensated systems to investigate the robustness of the antiwindup compensators. The penalty operators of the input saturation are studied originally in general cases. Robustness criteria are proposed for an antiwindup compensated system ...