A Centre-Stable Manifold for the Focussing Cubic NLS in R1+3R^{1+3}

Consider the focussing cubic nonlinear Schr\"odinger equation in $R^3$: $$i\psi_t+\Delta\psi = -|\psi|^2 \psi.$$ It admits special solutions of the form $e^{it\alpha}\phi$, where $\phi$ is a Schwartz function and a positive ($\phi>0$) solution of $$-\Delta \phi + \alpha\phi = \phi^3.$$ The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form $e^{i(v \cdot + \Gamma)} \phi(\cdot - y, \alpha)$. We prove that any solution starting sufficiently close to a standing wave in the $\Sigma = W^{1, 2}(R^3) \cap |x|^{-1}L^2(R^3)$ norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \mc N is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones. The proof is based on the modulation method introduced by Soffer and Weinstein for the $L^2$-subcritical case and adapted by Schlag to the $L^2$-supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in $R^3$ for the nonselfadjoint Schr\"odinger operator obtained by linearizing around a standing wave solution.Comment: 56 page

A randomised, blinded, trial of clopidogrel versus aspirin in patients at risk of ischaemic events (CAPRIE). CAPRIE Steering Committee

Many clinical trials have evaluated the benefit of long-term use of antiplatelet drugs in reducing the risk of clinical thrombotic events. Aspirin and ticlopidine have been shown to be effective, but both have potentially serious adverse effects. Clopidogrel, a new thienopyridine derivative similar to ticlopidine, is an inhibitor of platelet aggregation induced by adenosine diphosphate. METHODS: CAPRIE was a randomised, blinded, international trial designed to assess the relative efficacy of clopidogrel (75 mg once daily) and aspirin (325 mg once daily) in reducing the risk of a composite outcome cluster of ischaemic stroke, myocardial infarction, or vascular death; their relative safety was also assessed. The population studied comprised subgroups of patients with atherosclerotic vascular disease manifested as either recent ischaemic stroke, recent myocardial infarction, or symptomatic peripheral arterial disease. Patients were followed for 1 to 3 years. FINDINGS: 19,185 patients, with more than 6300 in each of the clinical subgroups, were recruited over 3 years, with a mean follow-up of 1.91 years. There were 1960 first events included in the outcome cluster on which an intention-to-treat analysis showed that patients treated with clopidogrel had an annual 5.32% risk of ischaemic stroke, myocardial infarction, or vascular death compared with 5.83% with aspirin. These rates reflect a statistically significant (p = 0.043) relative-risk reduction of 8.7% in favour of clopidogrel (95% Cl 0.3-16.5). Corresponding on-treatment analysis yielded a relative-risk reduction of 9.4%. There were no major differences in terms of safety. Reported adverse experiences in the clopidogrel and aspirin groups judged to be severe included rash (0.26% vs 0.10%), diarrhoea (0.23% vs 0.11%), upper gastrointestinal discomfort (0.97% vs 1.22%), intracranial haemorrhage (0.33% vs 0.47%), and gastrointestinal haemorrhage (0.52% vs 0.72%), respectively. There were ten (0.10%) patients in the clopidogrel group with significant reductions in neutrophils (< 1.2 x 10(9)/L) and 16 (0.17%) in the aspirin group. INTERPRETATION: Long-term administration of clopidogrel to patients with atherosclerotic vascular disease is more effective than aspirin in reducing the combined risk of ischaemic stroke, myocardial infarction, or vascular death. The overall safety profile of clopidogrel is at least as good as that of medium-dose aspirin