Computing regular equivalence: practical and theoretical issues

Social network analysts have tried to capture the idea of social role explicitly by proposing a framework that precisely gives conditions under which group actors are playing equivalent roles. They term these methods positional analysis techniques. The most general definition is regular equivalence which captures the idea that equivalent actors are related in a similar way to equivalent alters. Regular equivalence gives rise to a whole class of partitions on a network. Given a network we have two different computational problems. The first is how to find a particular regular equivalence. An algorithm exists to find the largest regular partition but there are not efficient algorithms to test whether there is a regular k-partition. That is a partition in k groups that is regular. In addition, when dealing with real data, it is unlikely that any regular partitions exist. To overcome this problem relaxations of regular equivalence have been proposed along with optimisation techniques to find nearly regular partitions. In this paper we review the algorithms that have developed to find particular regular equivalences and look at some of the recent theoretical results which give an insight into the complexity of finding regular partitions.

HDL composition and HDL antioxidant capacity in patients on regular haemodialysis

Recent evidence suggests that HDL can directly inhibit LDL oxidation, a key early stage in atherogenesis. Patients with chronic renal failure are at increased cardiovascular risk, have reduced HDL levels and altered HDL composition. We have therefore investigated whether compositional changes in HDL lead to decreased HDL antioxidant capacity in these patients. In comparison to control subject HDL, patient HDL contained less total cholesterol, cholesterol esters, phospholipids and alpha-tocopherol. LDL, HDL and LDL + HDL were standardised for protein and oxidised in the presence of Cu2+. The rate of propagation during HDL oxidation was reduced in the patient group (3.28 +/- 0.65 x 10(-5) vs. 4.60 +/- 0.97 x 10(-5) abs. U/min, P

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U s = Spec A s . A quasi-coherent sheaf on X gives rise, by taking sections over the U s , to a diagram of modules over the coordinate rings A s , indexed by the intersection poset S of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Sop-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan S. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U s agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.