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Let G be a connected subgroup of the group Diff(M ) of diffeomorphisms of a manifold M . It is well known that every element φ ∈ π 1 (G, id) defines an endomorphism ∂ φ : H * (M, Q) → H * +1 (M, Q) as follows. Choose a loop {φ t }, t ∈ S 1 , of diffeomorphisms from G representing φ and a cycle C in M . Then the homology classSuppose now that (M, ω) is a closed symplectic manifold, and take G to be its group Ham(M, ω) of Hamiltonian diffeomorphisms. In this paper, we discuss the following statement. As we will see in section 2 it has a number of applications to the geometry and topology of the group of symplectomorphisms. Theorem 1.A. Let φ be a loop in the group Ham(M, ω) of Hamiltonian diffeomorphisms. Then ∂ φ vanishes identically for all φ ∈ π 1 (Ham(M, ω), id).Below we give the proof of this statement when M is 4-dimensional as well as for some higher dimensional symplectic manifolds -the so-called spherically *

We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball $B^4(c) \subset \R^4$ into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form $M_{\lambda}= (S^2 \times S^2, \mu \omega_0 \oplus \omega_0)$ where $\omega_0$ is the area form on the sphere with total area 1 and $\mu$ belongs to the interval $[1,2]$. We show that, when $\mu$ is 1, this space retracts to the space of symplectic frames, for any value of $c$. However, for any given $1 < \mu < 2$, the rational homotopy type of that space changes as $c$ crosses the critical parameter $c_{crit} = \mu - 1$, which is the difference of areas between the two $S^2$ factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of $c$ remain either below or above that critical value.Comment: Typos corrected, 2 minor corrections in the text. Numbering consistant with the published versio

Heart rate variability (HRV) corresponds to the adaptation of the heart to any stimulus. In fact, among the pathologies affecting HRV the most, there are the cardiovascular diseases and depressive disorders, which are associated with high medical cost in Western societies. Consequently, HRV is now widely used as an index of health.In order to better understand how this adaptation takes place, it is necessary to examine which factors directly influence HRV, whether they have a physiological or environmental origin. The primary objective of this research is therefore to conduct a literature review in order to get a comprehensive overview of the subject.The system of these factors affecting HRV can be divided into the following five categories: physiological and pathological factors, environmental factors, lifestyle factors, non-modifiable factors and effects. The direct interrelationships between these factors and HRV can be regrouped into an influence diagram. This diagram can therefore serve as a basis to improve daily clinical practice as well as help design even more precise research protocols.

Abstract. Let M be an oriented S 2 -bundle over a compact Riemann surface Σ. We show that up to diffeomorphism there is at most one symplectic form on M in each cohomology class. Since the possible cohomology classes of symplectic forms on M are known, this completes the classification of symplectic forms on these manifolds. Our proof relies on a simplification of our previous arguments and on the equivalence between Gromov and Seiberg-Witten invariants that we apply twice.