Blind fluorescence structured illumination microscopy: A new reconstruction strategy

In this communication, a fast reconstruction algorithm is proposed for fluorescence \textit{blind} structured illumination microscopy (SIM) under the sample positivity constraint. This new algorithm is by far simpler and faster than existing solutions, paving the way to 3D and/or real-time 2D reconstruction.Comment: submitted to IEEE ICIP 201

A Metric for Gradient RG Flow of the Worldsheet Sigma Model Beyond First Order

Tseytlin has recently proposed that an action functional exists whose gradient generates to all orders in perturbation theory the Renormalization Group (RG) flow of the target space metric in the worldsheet sigma model. The gradient is defined with respect to a metric on the space of coupling constants which is explicitly known only to leading order in perturbation theory, but at that order is positive semi-definite, as follows from Perelman's work on the Ricci flow. This gives rise to a monotonicity formula for the flow which is expected to fail only if the beta function perturbation series fails to converge, which can happen if curvatures or their derivatives grow large. We test the validity of the monotonicity formula at next-to-leading order in perturbation theory by explicitly computing the second-order terms in the metric on the space of coupling constants. At this order, this metric is found not to be positive semi-definite. In situations where this might spoil monotonicity, derivatives of curvature become large enough for higher order perturbative corrections to be significant.Comment: 15 pages; Erroneous sentence in footnote 14 removed; this version therefore supersedes the published version (our thanks to Dezhong Chen for the correction

A characterization of Dirac morphisms

Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.Comment: 18 pages; restricted to the even-dimensional cas

Surgery and the Spectrum of the Dirac Operator

We show that for generic Riemannian metrics on a simply-connected closed spin manifold of dimension at least 5 the dimension of the space of harmonic spinors is no larger than it must be by the index theorem. The same result holds for periodic fundamental groups of odd order. The proof is based on a surgery theorem for the Dirac spectrum which says that if one performs surgery of codimension at least 3 on a closed Riemannian spin manifold, then the Dirac spectrum changes arbitrarily little provided the metric on the manifold after surgery is chosen properly.Comment: 23 pages, 4 figures, to appear in J. Reine Angew. Mat

Mu rhythm: State of the art with special focus on cerebral palsy

Various specific early rehabilitation strategies are proposed to decrease functional disabilities in patients with cerebral palsy (CP). These strategies are thought to favour the mechanisms of brain plasticity that take place after brain injury. However, the level of evidence is low. Markers of brain plasticity would favour validation of these rehabilitation programs. In this paper, we consider the study of mu rhythm for this goal by describing the characteristics of mu rhythm in adults and children with typical development, then review the current literature on mu rhythm in CP. Mu rhythm is composed of brain oscillations recorded by electroencephalography (EEG) or magnetoencephalography (MEG) over the sensorimotor areas. The oscillations are characterized by their frequency, topography and modulation. Frequency ranges within the alpha band (∼10Hz, mu alpha) or beta band (∼20Hz, mu beta). Source location analyses suggest that mu alpha reflects somatosensory functions, whereas mu beta reflects motor functions. Event-related desynchronisation (ERD) followed by event-related (re-)synchronisation (ERS) of mu rhythm occur in association with a movement or somatosensory input. Even if the functional role of the different mu rhythm components remains incompletely understood, their maturational trajectory is well described. Increasing age from infancy to adolescence is associated with increasing ERD as well as increasing ERS. A few studies characterised mu rhythm in adolescents with spastic CP and showed atypical patterns of modulation in most of them. The most frequent findings in patients with unilateral CP are decreased ERD and decreased ERS over the central electrodes, but atypical topography may also be found. The patterns of modulations are more variable in bilateral CP. Data in infants and young children with CP are lacking and studies did not address the questions of intra-individual reliability of mu rhythm modulations in patients with CP nor their modification after motor learning. Better characterization of mu rhythm in CP, especially in infants and young children, is warranted before considering this rhythm as a potential neurophysiological marker of brain plasticity

A Reilly formula and eigenvalue estimates for differential forms

We derive a Reilly-type formula for differential p-forms on a compact manifold with boundary and apply it to give a sharp lower bound of the spectrum of the Hodge Laplacian acting on differential forms of an embedded hypersurface of a Riemannian manifold. The equality case of our inequality gives rise to a number of rigidity results, when the geometry of the boundary has special properties and the domain is non-negatively curved. Finally we also obtain, as a by-product of our calculations, an upper bound of the first eigenvalue of the Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page

Generic metrics and the mass endomorphism on spin three-manifolds

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $p\in M$ is called the mass endomorphism in $p$ associated to the metric $g$ due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.Comment: 8 page

The Dirac operator on generalized Taub-NUT spaces

We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact K\"ahler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author.Comment: Final version, 16 page