On the outer automorphism groups of triangular alternation limit algebras

Let $A$ denote the alternation limit algebra, studied by Hopenwasser and Power, and by Poon, which is the closed direct limit of upper triangular matrix algebras determined by refinement embeddings of multiplicity $r_k$ and standard embeddings of multiplicity $s_k$. It is shown that the quotient of the isometric automorphism group by the approximately inner automorphisms is the abelian group \ZZ ^d where $d$ is the number of primes that are divisors of infinitely many terms of each of the sequences $(r_k)$ and $(s_k)$. This group is also the group of automorphisms of the fundamental relation of $A$.Comment: 12 pages, Late

Sex differences in the associations between birthweight and lipid levels in middle-age: findings from the 1958 British birth cohort

Objective To examine sex differences in birthweight–lipid associations. Methods and results Using prospectively collected data on birthweight and non-fasting lipid levels at age 44–45 y from the 1958 British birth cohort (3603 men and 3583 women), sex differences in birthweight–lipid associations were examined. There were inverse associations between birthweight and total and low-density-lipoprotein (LDL)-cholesterol among women (a 1 kg increase in birthweight was associated with a 0.13 mmol/L reduction in total cholesterol (p < 0.001) and a 0.07 mmol/L reduction in LDL-cholesterol (p = 0.02)) but no associations among men (p = 0.005 and p = 0.01, respectively, for birthweight × sex interactions). There was an inverse association between birthweight and triglycerides of a similar magnitude in both sexes (a 1 kg increase in birthweight was associated with a 7% reduction in triglyceride levels in sex-adjusted models (p < 0.001)). There was no association between birthweight and high-density-lipoprotein-cholesterol. Associations were largely unaltered after adjustment for covariates. Of birthweight, current height and BMI, the latter was the strongest predictor of lipid levels. Conclusions The finding of an inverse association between birthweight and triglycerides in both sexes and of inverse associations between birthweight and total and LDL-cholesterol only in women suggests that the mechanisms underlying the associations with birthweight may vary for different lipids

Frameworks, Symmetry and Rigidity

Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.Comment: 5 Figures. Replaces Dec. 2008 version. To appear in IJCG

Performance and strategy:simultaneous equations analysis of long-lived firms

A simultaneous equations model of performance, strategy and size is tested using fieldwork evidence on long-lived firms in Scotland. Estimation is by I3SLS, with correction for sample selection bias. The contributions of this paper are that it: (a) grounds estimation on fieldwork evidence; (b) calibrates performance and competitive strategy; (c) tests and models endogeneity; and (d) computes robust trade-off elasticities between firm size and performance. It shows how this trade-off provides the entrepreneur with two strong incentives: (i) to seek greater efficiency typically by an increase in the human capital of the ‘core’ workforce; (ii) to achieve higher levels of performance by adopting more diverse competitive strategies

The rigidity of infinite graphs

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in d-dimensional Euclidean space is generalised to the non-Euclidean l^p norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit of an inclusion tower of finite graphs for which the inclusions satisfy a relative rigidity property. For d>2 a countable graph which is rigid for generic placements in R^d may fail the stronger property of sequential rigidity, while for d=2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.Comment: 51 page