We consider the class N of groups in which the normaliser of every subgroup is normal, and the class C of groups in which the commutator subgroup normalises every subgroup. It is clear that C ⊆ N, and it is known that groups in the class N are nilpotent of class at most 3. We show that every finite p-group in N is also in C, provided that p ≥ 5, and we give an example showing that this is not true for p = 2.

Zagier showed that the number of integer solutions to the Markoff equation with components bounded by T grows asymptotically like C(log T)2, where C is explicity computable. Rosenberger showed that there are only a finite number of equations ax2 + by2 + cz2 = dxyz with a, b, and c dividing d, and for which the equation admits an infinite number of integer solutions. In this paper, we generalise Zagier's techniques so that they may be applied to the Rosenberger equations. We also apply these techniques to the equations ax2 + by2 + cz2 = dxyz + 1.

We consider the space Sn of all nonempty bounded closed normal subsets of the cone where is the set of all vectors x ∈ Rn with nonnegative coordinates. We equip the space Sn with the Hausdorff metric and show that most elements of Sn are, in fact, strictly normal. More precisely, we show that the complement of the collection of all stricly normal elements of Sn is a σ-porous subset of Sn.

We study the tensor norm defined by a sequence space λ and its minimal and maximal operator ideals associated in the sense of Defant and Floret. Our results extend the classical theory related to the tensor norms of Saphar [16]. They show the key role played by the finite dimensional structure of the ultrapowers of λ in this kind of problems.

A new reverse of Bessel's inequality for orthornormal families in real or complex inner product space is obtained. Applications for some Grüss type results are also provided.

In this paper we shall prove that for any 0 < d ≤ 2, holds for n ≥ 1.

As an application, we shall then show that the following recursively defined sequence

satisfies

The difference equation above originates from a heat conduction problem studied by Myshkis (J. Difference Equ. Appl. 3(1997), 89–91).

Δ–finitistic dimensions of standardly stratified algebras are defined similarly to properly stratified algebras. It is proved that the finitistic dimension for any standardly stratified algebra is bounded by the sum of the Δ–finitistic dimension and the good filtration dimension. Finally, the –good filtration dimension of standardly stratified algebras is equal to the –good filtration dimension of their Ringel duals.

We prove that, given a quasi-primitive complex representation D for a finite group G, the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G. More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G.

Due to a printing error, a significant graph on page 12 was omitted from the paper [1].