Ordinary Percolation with Discontinuous Transitions

Percolation on a one-dimensional lattice and fractals such as the Sierpinski gasket is typically considered to be trivial because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a nontrivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here, we provide a simple example of this transition in form of a small-world network consisting of a one-dimensional lattice combined with a hierarchy of long-range bonds that reveals many features of the transition in a mathematically rigorous manner.Comment: RevTex, 5 pages, 4 eps-figs, and Mathematica Notebook as Supplement included. Final version, with several corrections and improvements. For related work, see http://www.physics.emory.edu/faculty/boettcher

On Fano schemes of complete intersections

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete intersection is irregular of dimension at least 2, and for the Fano surfaces we deduce formulas for their holomorphic Euler characteristic.Comment: Added lacking references, corrected acknowledgments, minor editorial change

Protocol for a pilot randomised controlled clinical trial to compare the effectiveness of a graduated three layer straight tubular bandaging system when compared to a standard short stretch compression bandaging system in the management of people with venous ulceration: 3VSS2008

<p>Abstract</p> <p>Background</p> <p>The incidence of venous ulceration is rising with the increasing age of the general population. Venous ulceration represents the most prevalent form of difficult to heal wounds and these problematic wounds require a significant amount of health care resources for treatment. Based on current knowledge multi-layer high compression system is described as the gold standard for treating venous ulcers. However, to date, despite our advances in venous ulcer therapy, no convincing low cost compression therapy studies have been conducted and there are no clear differences in the effectiveness of different types of high compression.</p> <p>Methods/Design</p> <p>The trial is designed as a pilot multicentre open label parallel group randomised trial. Male and female participants aged greater than 18 years with a venous ulcer confirmed by clinical assessment will be randomised to either the intervention compression bandage which consists of graduated lengths of 3 layers of elastic tubular compression bandage or to the short stretch inelastic compression bandage (control). The primary objective is to assess the percentage wound reduction from baseline compared to week 12 following randomisation. Randomisation will be allocated via a web based central independent randomisation service (nQuery v7) and stratified by study centre and wound size ≤ 10 cm²or >10 cm². Neither participants nor study staff will be blinded to treatment. Outcome assessments will be undertaken by an assessor who is blinded to the randomisation process.</p> <p>Discussion</p> <p>The aim of this study is to evaluate the efficacy and safety of two compression bandages; graduated three layer straight tubular bandaging (3L) when compared to standard short stretch (SS) compression bandaging in healing venous ulcers in patients with chronic venous ulceration. The trial investigates the differences in clinical outcomes of two currently accepted ways of treating people with venous ulcers. This study will help answer the question whether the 3L compression system or the SS compression system is associated with better outcomes.</p> <p>Trial Registration</p> <p>ACTRN12608000599370</p