On the Multiplicity of Orthogonal Geodesics in Riemannian Manifold With Concave Boundary. Applications to Brake Orbits and Homoclinics

Let (M, g) be a complete Riemannian Manifold, Omega subset of M an open subset whose closure is diffeomorphic to an annulus. If partial derivative Omega is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in (Omega) over bar = Omega boolean OR partial derivative Omega starting orthogonally to one connected component of partial derivative Omega and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a. class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

Ectopic mineralization of articular cartilage in the bullfrog Rana catesbeiana and its possible involvement in bone closure

Mineralization of the articular cartilage is a pathological condition associated with age and certain joint diseases in humans and other mammals. In this work, we describe a physiological process of articular cartilage mineralization in bullfrogs. Articular cartilage of the proximal and distal ends of the femur and of the proximal end of the tibia-fibula was studied in animals of different ages. Mineralization of the articular cartilage was detected in animals at 1 month post-transformation. This mineralization, which appeared before the hypertrophic cartilage showed any calcium deposition, began at a restricted site in the lateral expansion of the cartilage and then progressed to other areas of the epiphyseal cartilage. Mineralized structures were identified by von Kossa's staining and by in vivo incorporation of calcein green. Element analysis showed that calcium crystals consisted of poorly crystalline hydroxyapatite. Mineralized matrix was initially spherical structures that generally coalesced after a certain size to occupy larger areas of the cartilage. Alkaline phosphatase activity was detected at the plasma membrane of nearby chondrocytes and in extracellular matrix. Apoptosis was detected by the TUNEL (TDT-mediated dUTP-biotin nick end-labeling) reaction in some articular chondrocytes from mineralized areas. The area occupied by calcium crystals increased significantly in older animals, especially in areas under compression. Ultrastructural analyses showed clusters of needle-like crystals in the extracellular matrix around the chondrocytes and large blocks of mineralized matrix. In 4-year-old animals, some lamellar bone (containing bone marrow) occurred in the same area as articular cartilage mineralization. These results show that the articular cartilage of R. catesbeiana undergoes precocious and progressive mineralization that is apparently stimulated by compressive forces. We suggest that this mineralization is involved in the closure of bone extremities, since mineralization appears to precede the formation of a rudimentary secondary center of ossification in older animals.

Dynamics of a spacecraft and normalization around Lagrangian points in the Neptune-Triton system

The problem of a spacecraft orbiting the Neptune-Triton system is presented. The new ingredients in this restricted three body problem are the Neptune oblateness and the high inclined and retrograde motion of Triton. First we present some interesting simulations showing the role played by the oblateness on a Neptune's satellite, disturbed by Triton. We also give an extensive numerical exploration in the case when the spacecraft orbits Triton, considering Sun, Neptune and its planetary oblateness as disturbers. In the plane a x I (a = semi-major axis, I = inclination), we give a plot of the stable regions where the massless body can survive for thousand of years. Retrograde and direct orbits were considered and as usual, the region of stability is much more significant for the case of direct orbit of the spacecraft (Triton's orbit is retrograde). Next we explore the dynamics in a vicinity of the Lagrangian points. The Birkhoff normalization is constructed around L-2, followed by its reduction to the center manifold. In this reduced dynamics, a convenient Poincare section shows the interplay of the Lyapunov and halo periodic orbits, Lissajous and quasi-halo tori as well as the stable and unstable manifolds of the planar Lyapunov orbit. To show the effect of the oblateness, the planar Lyapunov family emanating from the Lagrangian points and three-dimensional halo orbits are obtained by the numerical continuation method. Published by Elsevier Ltd. on behalf of COSPAR.

Dynamics in dumbbell domains I. Continuity of the set of equilibria

We analyze the dynamics of a reaction-diffusion equation with homogeneous Neumann boundary conditions in a dumbbell domain. We provide an appropriate functional setting to treat this problem and, as a first step, we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds. (c) 2006 Elsevier B.V. All rights reserved.

On the existence of infinite heteroclinic cycles in polynomial systems and its dynamic consequences

In this work we consider the dynamic consequences of the existence of infinite heteroclinic cycle in planar polynomial vector fields, which is a trajectory connecting two saddle points at infinity. It is stated that, although the saddles which form the cycle belong to infinity, for certain types of nonautonomous perturbations the perturbed system may present a complex dynamic behavior of the solutions in a finite part of the phase plane, due to the existence of tangencies and transversal intersections of their stable and unstable manifolds. This phenomenon might be called the chaos arising from infinity. The global study at infinity is made via the Poincare Compactification and the argument used to prove the statement is the Birkhoff-Smale Theorem. (c) 2004 WILEY-NCH Verlag GmbH & Co. KGaA, Weinheim.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Randomized clinical trial comparing percutaneous closure of patent foramen ovale (PFO) using the Amplatzer PFO Occluder with medical treatment in patients with cryptogenic embolism (PC-Trial): rationale and design

Background Several studies have shown an association of cryptogenic stroke and embolism with patent foramen ovale (PFO), but the question how to prevent further events in such patients is unresolved. Options include antithrombotic treatment with warfarin or antiplatelet agents or surgical or endovascular closure of the PFO. The PC-Trial was set up to compare endovascular closure and best medical treatment for prevention of recurrent events. Methods The PC-Trial is a randomized clinical trial comparing the efficacy of percutaneous closure of the PFO using the Amplatzer PFO occluder with best medical treatment in patients with cryptogenic embolism, i.e. mostly cryptogenic stroke. Warfarin for 6 months followed by antiplatelet agents is recommended as medical treatment. Randomization is stratified according to patients age (<45 versus ≥45 years), presence of atrial septal aneurysm (ASA yes or no) and number of embolic events before randomization (one versus more than one event). Primary endpoints are death, nonfatal stroke and peripheral embolism. Discussion patients were randomized in 29 centers of Europe, Canada, and Australia. Randomization started February 2000. Enrollment of 414 patients was completed in February 2009. All patients will be followed-up longitudinally. Follow-up is maintained until the last enrolled patient is beyond 2.5 years of follow-up (expected in 2011).

Surgery for complications of trans-catheter closure of atrial septal defects: a multi-institutional study from the European Congenital Heart Surgeons Association

This study aims to analyse the collective experience of participating European Congenital Heart Surgeons Association centres in the surgical management of complications resulting from trans-catheter closure of atrial septal defects (ASDs).

Accidental closure of the left upper pulmonary vein with an Amplatzer atrial septal defect occluder

We report the clinical outcome of a 46-year-old man referred for percutaneous closure of an atrial septal defect under transthoracic echocardiographic and fluoroscopic guidance, whose upper left pulmonary vein was erroneously obliterated using an Amplatzer atrial septal defect occluder. Various medical conditions have been associated with pulmonary vein stenosis including dyspnea on exertion or at rest, cough, and hemoptysis. However, there were no short- or long-term symptoms in this patient.