On the L2-Poincaré duality for incomplete Riemannian manifolds: A general construction with applications

Let [Formula: see text] be an open, oriented and incomplete Riemannian manifold of dimension [Formula: see text]. Under some general conditions we show the existence of a Hilbert complex [Formula: see text] such that its cohomology groups, labeled with [Formula: see text], satisfy the following properties: [Formula: see text] [Formula: see text] (Poincaré duality holds) There exists a well-defined and nondegenerate pairing: [Formula: see text] If [Formula: see text] is a Fredholm complex, then every closed extension of the de Rham complex [Formula: see text] is a Fredholm complex and, for each [Formula: see text], the quotient [Formula: see text] is a finite dimensional vector space.

Some Sampling Properties of Common Phase Estimators

The instantaneous phase of neural rhythms is important to many neuroscience-related studies. In this letter, we show that the statistical sampling properties of three instantaneous phase estimators commonly employed to analyze neuroscience data share common features, allowing an analytical investigation into their behavior. These three phase estimators—the Hilbert, complex Morlet, and discrete Fourier transform—are each shown to maximize the likelihood of the data, assuming the observation of different neural signals. This connection, explored with the use of a geometric argument, is used to describe the bias and variance properties of each of the phase estimators, their temporal dependence, and the effect of model misspecification. This analysis suggests how prior knowledge about a rhythmic signal can be used to improve the accuracy of phase estimates.