In this paper we prove that any irreducible Coxeter group of infinite order is directly indecomposable as an abstract group, without the finite rank assumption. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W , the centralizers in W of the normal subgroups of W that are generated by involutions. As a consequence, we show that the problem of deciding ...

An element A of the n× n copositive cone C is called irreducible with respect to the nonnegative cone N if it cannot be written as a nontrivial sum A = C + N of a copositive matrix C and an elementwise nonnegative matrix N . This property was studied by Baumert [2] who gave a characterisation of irreducible matrices. We demonstrate here that Baumert’s characterisation is incorrect and give a co...

We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree n polynomials of Hamming distance one from a randomly chosen set of b2n/nc o...

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface intersects its image non-trivially. We show that a three-manifold admits only finitely many surface bundle structures with strongly irreducible monodromy.

This handout, which accompanies the course on analogies between Z and F [T ], discusses some properties of polynomials in F [T ]. The results in Sections 1 and 2 work with any F , but the results in Section 3 and 4 are (somewhat) special to the field F = Fp. The main result in these notes in Theorem 3.7. The notation F [T ]h(T ), for the ring of polynomials in F [T ] considered modulo h(T ), is...

In this paper L is a lattice and l is an element of L. The scheme NonUniqExD1 deals with a non empty relational structure A , a subset B of A , a non empty subset C of A , and a binary predicate P , and states that: There exists a function f from B into C such that for every element e of A if e ∈ B, then there exists an element u of A such that u ∈ C and u = f (e) and P [e,u] provided the param...

Michael Behe’s concept of irreducible complexity, and in particular his use of this concept to critique Darwinism, continues to come under heavy fire from the biological community. The problem with Behe, so Darwinists inform us, is that he has created a problem where there is no problem. Far from constituting an obstacle to the Darwinian mechanism of random variation and natural selection, irre...

We begin a systematic study of those finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups S that generate join irreducible pseudovari-eties are characterized as follows: whenever S divides a direct product A × B of finite semigroups, then S divides either A ...

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset Ω̃. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield...