The von Mises-Fisher distribution of the first exit point from the hypersphere of the drifted Brownian motion and the density of the first exit time

A characterization is provided for the von Mises–Fisher random variable, in terms of first exit point from the unit hypersphere of the drifted Wiener process. Laplace transform formulae for the first exit time from the unit hypersphere of the drifted Wiener process are provided. Post representations in terms of Bell polynomials are provided for the densities of the first exit times from the circle and from the sphere.

A generalisation of the fractional Brownian field based on non-Euclidean norms

We explore a generalisation of the L´evy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial pth mean body and the polar projection transforms.

Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features

We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task, although the minimax rates for pointwise estimation are very slow.

Optimal Confidence Bands for Shape-Restricted Curves

Let Y be a stochastic process on [0,1] satisfying dY(t)=n 1/2 f(t)dt+dW(t) , where n≥1 is a given scale parameter (`sample size'), W is standard Brownian motion and f is an unknown function. Utilizing suitable multiscale tests, we construct confidence bands for f with guaranteed given coverage probability, assuming that f is isotonic or convex. These confidence bands are computationally feasible and shown to be asymptotically sharp optimal in an appropriate sense.

Use of echocardiography reveals reestablishment of ventricular pumping efficiency and partial ventricular wall motion recovery upon ventricular cryoinjury in the zebrafish

AIMS While zebrafish embryos are amenable to in vivo imaging, allowing the study of morphogenetic processes during development, intravital imaging of adults is hampered by their small size and loss of transparency. The use of adult zebrafish as a vertebrate model of cardiac disease and regeneration is increasing at high speed. It is therefore of great importance to establish appropriate and robust methods to measure cardiac function parameters. METHODS AND RESULTS Here we describe the use of 2D-echocardiography to study the fractional volume shortening and segmental wall motion of the ventricle. Our data show that 2D-echocardiography can be used to evaluate cardiac injury and also to study recovery of cardiac function. Interestingly, our results show that while global systolic function recovered following cardiac cryoinjury, ventricular wall motion was only partially restored. CONCLUSION Cryoinjury leads to long-lasting impairment of cardiac contraction, partially mimicking the consequences of myocardial infarction in humans. Functional assessment of heart regeneration by echocardiography allows a deeper understanding of the mechanisms of cardiac regeneration and has the advantage of being easily transferable to other cardiovascular zebrafish disease models.