On C1-robust transitivity of volume-preserving flows

We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincare flow associated to it admits a dominated splitting over M

Removing zero Lyapunov exponents in volume-preserving flows

Baraviera and Bonatti proved that it is possible to perturb, in the c^1 topology, a volume-preserving and partial hyperbolic diffeomorphism in order to obtain a non-zero sum of all the Lyapunov exponents in the central direction. In this article we obtain the analogous result for volume-preserving flows.Comment: 10 page

Eigenvalue Estimates for submanifolds of N×RN \times \mathbb{R} with locally bounded mean curvature

We give lower bounds for the fundamental tone of open sets in submanifolds with locally bounded mean curvature in $N \times \mathbb{R}$, where $N$ is an $n$-dimensional complete Riemannian manifold with radial sectional curvature $K_{N} \leq \kappa$. When the immersion is minimal our estimates are sharp. We also show that cylindrically bounded minimal surfaces has positive fundamental tone.Comment: 9 page

Generic dynamics of 4-dimensional C2 Hamiltonian systems

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which every regular energy surface is either Anosov or it is in the closure of energy surfaces with zero Lyapunov exponents a.e. This is in the spirit of the Bochi-Mane dichotomy for area-preserving diffeomorphisms on compact surfaces and its continuous-time version for 3-dimensional volume-preserving flows

There are no proper topological hyperbolic homoclinic classes for area-preserving maps

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class \Lambda associated to an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then \Lambda=M and f is an Anosov homeomorphism.MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the ‘Fundação para a Ciência e a Tecnologia’, through the Project UID/MAT/00013/2013

Foliations and Chern-Heinz inequalities

We extend the Chern-Heinz inequalities about mean curvature and scalar curvature of graphs of $C^{2}$-functions to leaves of transversally oriented codimension one $C^{2}$-foliations of Riemannian manifolds. That extends partially Salavessa's work on mean curvature of graphs and generalize results of Barbosa-Kenmotsu-Oshikiri \cite{barbosa-kenmotsu-Oshikiri} and Barbosa-Gomes-Silveira \cite{barbosa-gomes-silveira} about foliations of 3-dimensional Riemannian manifolds by constant mean curvature surfaces. These Chern-Heinz inequalities for foliations can be applied to prove Haymann-Makai-Osserman inequality (lower bounds of the fundamental tones of bounded open subsets $\Omega \subset \mathbb{R}^{2}$ in terms of its inradius) for embedded tubular neighborhoods of simple curves of $\mathbb{R}^{n}$.Comment: This paper is an improvment of an earlier paper titled On Chern-Heinz Inequalities. 8 Pages, Late

Perceived stress in obsessive-compulsive disorder is related with obsessive but not cmpulsive symptoms

Obsessive–compulsive disorder (OCD) is achronic psychiatric disorder characterized by recurrent intrusive thoughts and/or repetitive compulsory behaviors. This psychiatric disorder is known to be stress responsive, as symptoms increase during periods of stress but also because stressful events may precede the onset of OCD. However, only a few and inconsistent reports have been published about the stress perception and the stress-response in these patients. Herein, we have characterized the correlations of OCD symptoms with basal serum cortisol levels and scores in a stress perceived questionnaire (PSS-10). The present data reveals that cortisol levels and the stress scores in the PSS-10 were significantly higher in OCD patients that in controls. Moreover, stress levels self-reported by patients using the PSS-10 correlated positively with OCD severity in the Yale–Brown Obsessive–Compulsive Scale (Y–BOCS). Interestingly, PSS-10 scores correlated with the obsessive component, but not with the compulsive component, of Y–BOCS. These results confirm that stress is relevant in the context of OCD, particularly for the obsessive symptomatology.Pedro Morgado is supported by a fellowship “SFRH/SINTD/60129/2009” funded by FCT – Foundation for Science and Technology. Supported by FEDER funds through Operational program for competitive factors – COMPETE and by national funds through FCT –Foundation for Science and Technology to project “PTDC/SAU-NSC/111814/2009.