Knuth-Bendix algorithm and the conjugacy problems in monoids

We present an algorithmic approach to the conjugacy problems in monoids, using rewriting systems. We extend the classical theory of rewriting developed by Knuth and Bendix to a rewriting that takes into account the cyclic conjugates.Comment: This is a new version of the paper 'The conjugacy problems in monoids and semigroups'. This version will appear in the journal 'Semigroup forum

Geodesic rewriting systems and pregroups

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at our main object of study which we call geodesically perfect rewriting systems. We show that these are well-behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations. We describe a Knuth-Bendix completion process to construct such systems, show how they may be found with the help of Stallings' pregroups and conversely may be used to construct such pregroups.Comment: 44 pages, to appear in "Combinatorial and Geometric Group Theory, Dortmund and Carleton Conferences". Series: Trends in Mathematics. Bogopolski, O.; Bumagin, I.; Kharlampovich, O.; Ventura, E. (Eds.) 2009, Approx. 350 p., Hardcover. ISBN: 978-3-7643-9910-8 Birkhause

Polymer-based paclitaxel-eluting stents reduce in-stent neointimal tissue proliferation A serial volumetric intravascular ultrasound analysis from the TAXUS-IV trial

ObjectivesThe aim of this study was to use serial volumetric intravascular ultrasound (IVUS) to evaluate the effects of polymer-based, paclitaxel-eluting stents on in-stent neointima formation and late incomplete stent apposition.BackgroundThe TAXUS-IV trial demonstrated that the slow-release, polymer-based, paclitaxel-eluting stent reduces angiographic restenosis and the need for repeat revascularization procedures. Serial IVUS studies reveal details of the pattern of vascular responses provoked by stent implantation that provide insight into device safety and efficacy.MethodsIn the TAXUS-IV trial, patients were randomized to the slow-release, polymer-based, paclitaxel-eluting TAXUS stent or a bare-metal EXPRESS stent (Boston Scientific Corp., Natick, Massachusetts). As part of a formal substudy, complete volumetric IVUS data were available in 170 patients, including 88 TAXUS patients and 82 controls, at implantation and at nine-month follow-up.ResultsNo baseline differences were present in the clinical characteristics or IVUS parameters between the control and TAXUS groups. At nine-month follow-up, IVUS lumen volumes were larger in the TAXUS group (123 ± 43 mm3vs. 104 ± 44 mm3, p = 0.005), due to a reduction in neointimal volume (18 ± 18 mm3vs. 41 ± 23 mm3, p < 0.001). Millimeter-by-millimeter analysis within the stent demonstrated uniform suppression of neointimal growth along the entire stent length. Late lumen loss was similar at the proximal edge of the stent between the two groups, and reduced with the TAXUS stent at the distal edge (p = 0.004). Incomplete stent apposition at nine months was observed in only 3.0% of control and 4.0% of TAXUS stents (p = 0.12).ConclusionsPolymer-based, paclitaxel-eluting TAXUS stents are effective in inhibiting neointimal tissue proliferation, and do not result in late incomplete stent apposition

Insights into the pathogenesis of vein graft disease: lessons from intravascular ultrasound

The success of coronary artery bypass grafting (CABG) is limited by poor long-term graft patency. Saphenous vein is used in the vast majority of CABG operations, although 15% are occluded at one year with as many as 50% occluded at 10 years due to progressive graft atherosclerosis. Intravascular ultrasound (IVUS) has greatly increased our understanding of this process. IVUS studies have shown that early wall thickening and adaptive remodeling of vein grafts occurs within the first few weeks post implantation, with these changes stabilising in angiographically normal vein grafts after six months. Early changes predispose to later atherosclerosis with occlusive plaque detectable in vein grafts within the first year. Both expansive and constrictive remodelling is present in diseased vein grafts, where the latter contributes significantly to occlusive disease. These findings correlate closely with experimental and clinicopathological studies and help define the windows for prevention, intervention or plaque stabilisation strategies. IVUS is also the natural tool for evaluating the effectiveness of pharmacological and other treatments that may prevent or slow the progression of vein graft disease in clinical trials

Revascularization for coronary artery disease in diabetes mellitus: Angioplasty, stents and coronary artery bypass grafting

Author Manuscript: 2011 April 14Patients with diabetes mellitus (DM) are prone to a diffuse and rapidly progressive form of atherosclerosis, which increases their likelihood of requiring revascularization. However, the unique pathophysiology of atherosclerosis in patients with DM modifies the response to arterial injury, with profound clinical consequences for patients undergoing percutaneous coronary intervention (PCI). Multiple studies have shown that DM is a strong risk factor for restenosis following successful balloon angioplasty or coronary stenting, with greater need for repeat revascularization and inferior clinical outcomes. Early data suggest that drug eluting stents reduce restenosis rates and the need for repeat revascularization irrespective of the diabetic state and with no significant reduction in hard clinical endpoints such as myocardial infarction and mortality. For many patients with 1- or 2-vessel coronary artery disease, there is little prognostic benefit from any intervention over optimal medical therapy. PCI with drug-eluting or bare metal stents is appropriate for patients who remain symptomatic with medical therapy. However, selection of the optimal myocardial revascularization strategy for patients with DM and multivessel coronary artery disease is crucial. Randomized trials comparing multivessel PCI with balloon angioplasty or bare metal stents to coronary artery bypass grafting (CABG) consistently demonstrated the superiority of CABG in patients with treated DM. In the setting of diabetes CABG had greater survival, fewer recurrent infarctions or need for re-intervention. Limited data suggests that CABG is superior to multivessel PCI even when drug-eluting stents are used. Several ongoing randomized trials are evaluating the long-term comparative efficacy of PCI with drug-eluting stents and CABG in patients with DM. Only further study will continue to unravel the mechanisms at play and optimal therapy in the face of the profoundly virulent atherosclerotic potential that accompanies diabetes mellitus.National Institutes of Health (U.S.) (GM 49039

Evacetrapib and Cardiovascular Outcomes in High-Risk Vascular Disease

BACKGROUND: The cholesteryl ester transfer protein inhibitor evacetrapib substantially raises the high-density lipoprotein (HDL) cholesterol level, reduces the low-density lipoprotein (LDL) cholesterol level, and enhances cellular cholesterol efflux capacity. We sought to determine the effect of evacetrapib on major adverse cardiovascular outcomes in patients with high-risk vascular disease. METHODS: In a multicenter, randomized, double-blind, placebo-controlled phase 3 trial, we enrolled 12,092 patients who had at least one of the following conditions: an acute coronary syndrome within the previous 30 to 365 days, cerebrovascular atherosclerotic disease, peripheral vascular arterial disease, or diabetes mellitus with coronary artery disease. Patients were randomly assigned to receive either evacetrapib at a dose of 130 mg or matching placebo, administered daily, in addition to standard medical therapy. The primary efficacy end point was the first occurrence of any component of the composite of death from cardiovascular causes, myocardial infarction, stroke, coronary revascularization, or hospitalization for unstable angina. RESULTS: At 3 months, a 31.1% decrease in the mean LDL cholesterol level was observed with evacetrapib versus a 6.0% increase with placebo, and a 133.2% increase in the mean HDL cholesterol level was seen with evacetrapib versus a 1.6% increase with placebo. After 1363 of the planned 1670 primary end-point events had occurred, the data and safety monitoring board recommended that the trial be terminated early because of a lack of efficacy. After a median of 26 months of evacetrapib or placebo, a primary end-point event occurred in 12.9% of the patients in the evacetrapib group and in 12.8% of those in the placebo group (hazard ratio, 1.01; 95% confidence interval, 0.91 to 1.11; P=0.91). CONCLUSIONS: Although the cholesteryl ester transfer protein inhibitor evacetrapib had favorable effects on established lipid biomarkers, treatment with evacetrapib did not result in a lower rate of cardiovascular events than placebo among patients with high-risk vascular disease. (Funded by Eli Lilly; ACCELERATE ClinicalTrials.gov number, NCT01687998 .)