Scalar curvature via local extent

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.Comment: 22 pages. A new rigidity result has been added (see Proposition 17). Some typos have been correcte

Extent of the slowdown

Economic indicators ; Business cycles

Ulcerative colitis: let's talk about extent

Ulcerative colitis (UC) is a chronic inflammatory disease in which clinical course varies substantially between patients. The extent of the disease is usually pointed out as one of the factors responsible for this variation. With this study, we pretended to evaluate the differences in natural history and pharmacological therapy prescription between left-sided and extended UC

Extent of stacking disorder in diamond

Hexagonal diamond has been predicted computationally to display extraordinary physical properties including a hardness that exceeds cubic diamond. However, a recent electron microscopy study has shown that so-called hexagonal diamond samples are in fact not discrete materials but faulted and twinned cubic diamond. We now provide a quantitative analysis of cubic and hexagonal stacking in diamond samples by analysing X-ray diffraction data with the DIFFaX software package. The highest fractions of hexagonal stacking we find in materials which were previously referred to as hexagonal diamond are below 60%. The remainder of the stacking sequences are cubic. We show that the cubic and hexagonal sequences are interlaced in a complex way and that naturally occurring Lonsdaleite is not a simple phase mixture of cubic and hexagonal diamond. Instead, it is structurally best described as stacking disordered diamond. The future experimental challenge will be to prepare diamond samples beyond 60% hexagonality and towards the so far elusive 'perfect' hexagonal diamond

Spatial Extent of Random Laser Modes

We have experimentally studied the distribution of the spatial extent of modes and the crossover from essentially single-mode to distinctly multimode behavior inside a porous gallium phosphide random laser. This system serves as a paragon for random lasers due to its exemplary high index contrast. In the multimode regime, we observed mode competition. We have measured the distribution of spectral mode spacings in our emission spectra and found level repulsion that is well described by the Gaussian orthogonal ensemble of random-matrix theory