Pseudographs and Lax-Oleinik semi-group: a geometric and dynamical interpretation

Let H be a Tonelli Hamiltonian defined on the cotangent bundle of a compact and connected manifold and let u be a semi-concave function defined on M. If E (u) is the set of all the super-differentials of u and (\phi t) the Hamiltonian flow of H, we prove that for t > 0 small enough, \phi-t (E (u)) is an exact Lagrangian Lipschitz graph. This provides a geometric interpretation/explanation of a regularization tool that was introduced by P.~Bernard to prove the existence of C 1,1 subsolutions

On iterated translated points for contactomorphisms of R^{2n+1} and R^{2n} x S^1

A point q in a contact manifold is called a translated point for a contactomorphism \phi, with respect to some fixed contact form, if \phi (q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points is related to the chord conjecture and to the problem of leafwise coisotropic intersections. In the case of a compactly supported contactomorphism of R^{2n+1} or R^{2n} x S^1 contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if \phi is positive then there are infinitely many non-trivial geometrically distinct iterated translated points, i.e. translated points of some iteration \phi^k. This result can be seen as a (partial) contact analogue of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of R^{2n}, and is obtained with generating functions techniques in the setting of arXiv:0901.3112.Comment: 10 pages, revised version. I removed the discussion on linear growth of iterated translated points, because it contained a mistake. To appear in the International Journal of Mathematic

Finite reduction and Morse index estimates for mechanical systems

A simple version of exact finite dimensional reduction for the variational setting of mechanical systems is presented. It is worked out by means of a thorough global version of the implicit function theorem for monotone operators. Moreover, the Hessian of the reduced function preserves all the relevant information of the original one, by Schur's complement, which spontaneously appears in this context. Finally, the results are straightforwardly extended to the case of a Dirichlet problem on a bounded domain.Comment: 13 pages; v2: minor changes, to appear in Nonlinear Differential Equations and Application

A variational principle for actions on symmetric symplectic spaces

We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such ``extended hamiltonians'' in terms of Hamilton-Jacobi-type equations. Simplest spaces are treated explicitly.Comment: 28 pages. Edited english translation of first author's PhD thesis (2000

Large normally hyperbolic cylinders in a priori stable Hamiltonian systems

We prove the existence of normally hyperbolic invariant cylinders in nearly integrable hamiltonian systems

Entrepreneurship in island contexts: a systematic review of the tourism and hospitality literature

Entrepreneurship is vital to the success of tourism and hospitality and the sector makes an important contribution to many island economies. Despite this, far too little attention has been paid by researchers to tourism and hospitality entrepreneurship in islands (THEI). This research helps to address this gap through a systematic review of the literature, conducted to provide a platform for further research and to help investigators set their research priorities and thereby advance understanding of this important field. Using the Scopus database and the PRISMA technique, a total of 132 articles were included in bibliometric and thematic content analyses. The review revealed that, although there has been an increase in THEI research, this has tended to focus on the Asia-Pacific region rather than the European and North American contexts. It was also found that, hitherto, the generalizability of much THEI published research is limited. It is therefore suggested that researchers consider redressing this geographical bias and conduct more quantitative and comparative THEI studies. Further opportunities exist for scholars to investigate the characteristics and behaviors of tourism and hospitality island entrepreneurs as well as the impacts of the industrial and spatial aspects of THEI. For professionals working in island economic development and business support, this research identifies many of the challenges and opportunities associated with supporting THEI

Analysis of the UK Government's 2011 tourism policy

This review considers the UK Government's 2011 tourism policy document. The policy was produced during a period of public sector restructuring in the UK and also during the global economic crisis, which began in 2008. The policy sets out a number of reforms to the governance of tourism at the national and local levels, which aim to increase the level of private sector involvement in leading and developing the tourism sector and to reduce the sector's dependence on public funding. During a period of economic slowdown in the UK, the tourism industry can make a significant contribution to growth, but it is not yet clear whether these proposed reforms will support or impede the future development of the tourism industry in the UK

Legendrian links, causality, and the Low conjecture

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Nat\'ario and Tod says that two events x,y\in\ss are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic $(X^{m+1}, g)$ such that a cover of its Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$Comment: Version 3 - minor improvements, references added 11 pages, 1 figur

Behavioral Characterization of the Novel GABA B Receptor- Positive Modulator GS39783 (N,NЈ-Dicyclopentyl-2- methylsulfanyl-5-nitro-pyrimidine-4,6-diamine): Anxiolytic-Like Activity without Side Effects Associated with Baclofen or Benzodiazepines

ABSTRACT The role of GABA B receptors in various behavioral processes has been largely defined using the prototypical GABA B receptor agonist baclofen. However, baclofen induces sedation, hypothermia and muscle relaxation, which may interfere with its use in behavioral paradigms. Although there is much evidence for a role of the inhibitory neurotransmitter GABA in the pathophysiology of anxiety, the role of GABA B receptors in these disorders is largely unclear. We recently identified GS39783 (N,NЈ-dicyclopentyl-2-methylsulfanyl-5-nitro-pyrimidine-4,6-diamine) as a selective allosteric positive modulator at GABA B receptors. The aim of the present study was to broadly characterize the effects of GS39783 in well-validated rodent models for motor activity, cognition, and anxiety. The following tests were included: locomotor activity in rats and mice, rotarod and traction tests (including determinations of core temperature) in mice, passive avoidance in mice and rats, elevated plus maze in rats, elevated zero maze in mice and rats, stress-induced hyperthermia in mice, and pentobarbital-and ethanol-induced sleep in mice. Unlike baclofen and/or the benzodiazepine chlordiazepoxide, GS39783 had no effect in any of the tests for locomotion, cognition, temperature, or narcosis. Most interestingly, GS39783 had anxiolytic-like effects in all the tests used. Overall, the data obtained here suggest that positive modulation of GABA B receptors may serve as a novel therapeutic strategy for the development of anxiolytics, with a superior side effect profile to both baclofen and benzodiazepines