In [16] and [17], Tan looks at the principal blocks of the symmetric groups [Sfr ]9, [Sfr ]10, [Sfr ]11, [Sfr ]12 over a field of characteristic three, and highlights the contrast with blocks of small abelian defect. Here we continue by determining the Ext-quivers of the blocks of [Sfr ]13 over the same field; this in particular provides examples of a [4:1]-pair and a [3:2]-pair with non-abelian defect.

Let G be a group, and let 〈A[mid ]R〉 be a finite group presentation for G with [mid ]R[mid ][ges ][mid ]A[mid ]. Then there exists a finite semigroup presentation 〈B[mid ]Q〉 for G such that [mid ]Q[mid ]- [mid ]B[mid ] = [mid ]R[mid ]- [mid ]A[mid ]. Moreover, B is either the same generating set or else it contains one additional generator.

Let A be an Artin algebra. Then there are finitely many non-isomorphic simple A-modules. Suppose S1, S2, …, Sn form a complete list of all non-isomorphic simple A-modules and we fix this ordering of simple modules. Let Pi and Qi be the projective cover and the injective envelope of Si respectively. With this order of simple modules we define for each i the standard module Δ(i) to be the maximal quotient of Pi with composition factors Sj with j [les ] i. Let Δ be the set of all these standard modules Δ(i). We denote by [Fscr ](Δ) the subcategory of A-mod whose objects are the modules M which have a Δ-filtration, namely there is a finite chain

0 = M0 ⊂ M1 ⊂ M2 ⊂ … ⊂ Mt = M

of submodules of M such that Mi/Mi−1 is isomorphic to a module in Δ for all i. The modules in [Fscr ](Δ) are called Δ-good modules. Dually, we define the costandard module ∇(i) to be the maximal submodule of Qi with composition factors Sj with j [les ] i and denote by ∇ the collection of all costandard modules. In this way, we have also the subcategory [Fscr ](∇) of A-mod whose objects are these modules which have a ∇-filtration. Of course, we have the notion of ∇-good modules. Note that Δ(n) is always projective and ∇(n) is always injective.

We investigate a generalization of the method introduced by Kouchnirenko to compute the codimension (colength) of an ideal under a certain non-degeneracy condition on a given system of generators of I. We also discuss Newton non-degenerate ideals and give characterizations using the notion of reductions and Newton polyhedra of ideals.

We introduce order conserving embeddings as a more general form of order preserving embeddings between finite dimensional nest algebras. The structure of these embeddings is determined, in terms of order indecomposable decompositions, and they are shown to be determined up to inner conjugacy by their induced maps on K0. Classifications of direct systems and limit algebras are obtained in terms of dimension distribution groups.

Let X be a finite CW-complex. We show that the image of the homotopy groups of X under suspension have an exponent at every prime. As a corollary we recover Long's result that finite H-spaces have exponents at all primes. We show that the stable homotopy groups of X have an exponent at p if and only if X is rationally equivalent to a point. This allows us to construct many examples of spaces with infinite dimensional rational homotopy groups and without an exponent at any prime. These examples further support Moore's conjecture.

The Kuhn–Schwartz non-realizability theorem states the following: if the mod p cohomology of a topological space is finitely generated as a module over the Steenrod algebra [Ascr ] then it is finite. We generalize this result to the category of G-spaces, where G is a compact Lie group, by considering the equivariant cohomology of a G-space as an object in the category of all [Ascr ]-modules with a compatible H*(BG; [ ]p)-module structure.

We consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these results we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.

We establish convergence results involving the nth Cesaro mean σβn(θ) of order β of f(eiθ) = [sum ]aneinθ where the coefficients (an) satisfy a Dirichlet-type condition [sum ]nα[mid ]an[mid ]2 < ∞ for a fixed α in (0, 1]. When α = 1, for example, we prove that [sum ][mid ]σβn(θ)−f(eiθ) [mid ]2 < ∞ for every β > −½ at almost all points θ in [−π,π]. We also derive weighted convergence results of this type outside exceptional sets of capacity zero, and show by examples that these results are, in a certain sense, sharp with respect to the size of the exceptional sets.

Matsumoto and Yor have recently discovered an interesting invariance property of a product of the generalized inverse Gaussian and gamma distributions. In this paper we obtain: (1) a complete regression version of its converse; (2) a converse to the matrix variate Matsumoto–Yor property which extends an earlier result. Of independent interest is a functional equation for matrix valued functions, which has been solved in the course of investigation of the second problem.

In this paper we study properties of the fundamental domain [Fscr ]β of number systems, which are defined in rings of integers of number fields. First we construct addition automata for these number systems. Since [Fscr ]β defines a tiling of the n-dimensional vector space, we ask, which tiles of this tiling ‘touch’ [Fscr ]β. It turns out that the set of these tiles can be described with help of an automaton, which can be constructed via an easy algorithm which starts with the above-mentioned addition automaton. The addition automaton is also useful in order to determine the box counting dimension of the boundary of [Fscr ]β. Since this boundary is a so-called graph-directed self-affine set, it is not possible to apply the general theory for the calculation of the box counting dimension of self similar sets. Thus we have to use direct methods.

Let E be a Banach function lattice such that L∞[0, 1] [rarrhk ] E [rarrhk ] L1[0, 1]. We characterize the strict singularity of the embedding L∞[0, 1] [rarrhk ] E and the strict cosingularity of E [rarrhk ] L1[0, 1] in terms of functionals defined by using characteristic functions.