The Title Catalog: A Third Dimension

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A toral diffeomorphism with a non-polygonal rotation set

We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon

The Stratified Structure of Spaces of Smooth Orbifold Mappings

We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between O and P with the C^r topology carries the structure of a smooth C^\infty Banach (r finite)/Frechet (r=infty) manifold. For the notion of complete reduced orbifold map, the corresponding set of C^r maps between O and P with the C^r topology carries the structure of a smooth C^\infty Banach (r finite)/Frechet (r=infty) orbifold. The remaining two notions carry a stratified structure: The C^r orbifold maps between O and P is locally a stratified space with strata modeled on smooth C^\infty Banach (r finite)/Frechet (r=infty) manifolds while the set of C^r reduced orbifold maps between O and P locally has the structure of a stratified space with strata modeled on smooth C^\infty Banach (r finite)/Frechet (r=infty) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show they inherit the structure of C^\infty Banach (r finite)/Frechet (r=infty) manifolds. In fact, for r finite they are topological groups, and for r=infty they are convenient Frechet Lie groups.Comment: 31 pages, 2 figures; corrected and expande

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

The purpose of this paper is to enhance a correspondence between the dynamics of the differential equations $\dot y(t)=g(y(t))$ on $\mathbb{R}^d$ and those of the parabolic equations $\dot u=\Delta u +f(x,u,\nabla u)$ on a bounded domain $\Omega$. We give details on the similarities of these dynamics in the cases $d=1$, $d=2$ and $d\geq 3$ and in the corresponding cases $\Omega=(0,1)$, $\Omega=\mathbb{T}^1$ and dim($\Omega$)$\geq 2$ respectively. In addition to the beauty of such a correspondence, this could serve as a guideline for future research on the dynamics of parabolic equations

Status Update and Interim Results from the Asymptomatic Carotid Surgery Trial-2 (ACST-2)

Objectives: ACST-2 is currently the largest trial ever conducted to compare carotid artery stenting (CAS) with carotid endarterectomy (CEA) in patients with severe asymptomatic carotid stenosis requiring revascularization. Methods: Patients are entered into ACST-2 when revascularization is felt to be clearly indicated, when CEA and CAS are both possible, but where there is substantial uncertainty as to which is most appropriate. Trial surgeons and interventionalists are expected to use their usual techniques and CE-approved devices. We report baseline characteristics and blinded combined interim results for 30-day mortality and major morbidity for 986 patients in the ongoing trial up to September 2012. Results: A total of 986 patients (687 men, 299 women), mean age 68.7 years (SD ± 8.1) were randomized equally to CEA or CAS. Most (96%) had ipsilateral stenosis of 70-99% (median 80%) with contralateral stenoses of 50-99% in 30% and contralateral occlusion in 8%. Patients were on appropriate medical treatment. For 691 patients undergoing intervention with at least 1-month follow-up and Rankin scoring at 6 months for any stroke, the overall serious cardiovascular event rate of periprocedural (within 30 days) disabling stroke, fatal myocardial infarction, and death at 30 days was 1.0%. Conclusions: Early ACST-2 results suggest contemporary carotid intervention for asymptomatic stenosis has a low risk of serious morbidity and mortality, on par with other recent trials. The trial continues to recruit, to monitor periprocedural events and all types of stroke, aiming to randomize up to 5,000 patients to determine any differential outcomes between interventions. Clinical trial: ISRCTN21144362. © 2013 European Society for Vascular Surgery. Published by Elsevier Ltd. All rights reserved