Writing certain commutators as products of cubes in free groups

Let Cu(γ) be the minimal number of cubes required to express an element γ of a free group F. We establish a method for showing that certain equations do not have solutions in free groups. Using it, we find Cu(γ) for certain elements of the derived subgroup of F. If is the wreath product of F by the infinite cyclic group, we also show that every element of W′ is a product of at most one commutator and three cubes in W.     

Remarks on the frame envelope of a σ-frame

The familiar equivalence between σ-frames and σ-coherent frames, given by the frame envelopes of σ-frames, is shown to induce an equivalence between stably continuous σ-frames and stably continuous frames. Similarly, the analogue of the former for σ-biframes is proved to provide an equivalence between compact regular σ-biframes and compact regular biframes. As an application we obtain the equivalence between stably continuous σ-frames and compact regular σ-biframes due to Matutu as an easy consequence of its frame counterpart established earlier by Banaschewski and Brümmer. This provides an affirmative answer to a question posed by Dana Scott.     

An obstruction to the positivity of relative Yamabe invariants

For a supergroup , we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact n-dimensional -manifold W with nonempty boundary M, , in terms of the bordism class [M] in the Stolz obstruction group associated to [St2]. In par ticular, when W is a 5-dimensional spin manifold and the -invariant of a connected component of M is nonzero, we prove that W does not admit a positive scalar curvature metric with minimal boundary condition. Received: 4 July 2001; in final form: 5 February 2002 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 11640070.     

Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte polynomial. Furthermore, the cardinalities of the sets of matroids in a given family grow exponentially as a function of rank, and there are many such families.In Memory of Gian-Carlo Rota     

A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .