A note on the graded K-theory of certain graded rings

Following ideas of Quillen we prove that the graded K-theory of a Z-multi-graded ring with support contained in a pointed cone is entirely determined by the K-theory of the sub-ring of elements of degree 0.

Finite domination and Novikov homology over strongly Z-graded rings

Let L be a unital Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent to a bounded complex of finitely generated projective L0-modules, generalising known results for twisted Laurent polynomial rings. The crucial hypothesis is that L is a strongly graded ring. 

On Quillen's calculation of graded K-theory

We adapt Quillen’s calculation of graded K-groups of Z-graded rings with support in N to graded K-theory, allowing gradings in a product Z×G with G an arbitrary group. This in turn allows us to use induction and calculate graded K-theory of Z -multigraded rings.

SK1 of graded division algebras

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.

Unitary SK1 of unitary and graded division algebras

The reduced unitary Whitehead group $\SK$ of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary $\SK$ of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary $\SK$ of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary $\SK$ for much wider classes of division algebras over henselian fields.

A review of behaviour change theories and techniques used in group based self-management programmes for chronic low back pain and arthritis

Background: Medical Research Council (MRC) guidelines recommend applying theory within complex interventions to explain how behaviour change occurs. Guidelines endorse self-management of chronic low back pain (CLBP) and osteoarthritis (OA), but evidence for its effectiveness is weak. Objective: This literature review aimed to determine the use of behaviour change theory and techniques within randomised controlled trials of group-based self-management programmes for chronic musculoskeletal pain, specifically CLBP and OA. Methods: A two-phase search strategy of electronic databases was used to identify systematic reviews and studies relevant to this area. Articles were coded for their use of behaviour change theory, and the number of behaviour change techniques (BCTs) was identified using a 93-item taxonomy, Taxonomy (v1). Results: 25 articles of 22 studies met the inclusion criteria, of which only three reported having based their intervention on theory, and all used Social Cognitive Theory. A total of 33 BCTs were coded across all articles with the most commonly identified techniques being '. instruction on how to perform the behaviour', '. demonstration of the behaviour', '. behavioural practice', '. credible source', '. graded tasks' and '. body changes'. Conclusion: Results demonstrate that theoretically driven research within group based self-management programmes for chronic musculoskeletal pain is lacking, or is poorly reported. Future research that follows recommended guidelines regarding the use of theory in study design and reporting is warranted.