Sharp multidimensional numerical integration for strongly convex functions on convex polytopes

This paper introduces and studies a new class of multidimensional numerical integration, which we call ?strongly positive definite cubature formulas?. We establish, among others, a characterization theorem providing necessary and sufficient conditions for the approximation error (based on such cubature formulas) to be bounded by the approximation error of the quadratic function. This result is derived as a consequence of two characterization results, which are of independent interest, for linear functionals obtained in a more general seeting. Thus, this paper extends some result previously reported in [2, 3] when convexity in the classical sense is only assumed. We also show that the centroidal Voronoi Tesselations provide an efficient way for constructing a class of optimal of cubature formulas. Numerical results for the two-dimensional test functions are given to illustrate the efficiency of our resulting cubature formulas.

A Multiscale Nonhydrostatic Atmospheric Model Using Centroidal Voronoi Tesselations and C-Grid Staggering

The formulation of a fully compressible nonhydrostatic atmospheric model called the Model for Prediction Across Scales–Atmosphere (MPAS-A) is described. The solver is discretized using centroidal Voronoi meshes and a C-grid staggering of the prognostic variables, and it incorporates a split-explicit time-integration technique used in many existing nonhydrostatic meso- and cloud-scale models. MPAS can be applied to the globe, over limited areas of the globe, and on Cartesian planes. The Voronoi meshes are unstructured grids that permit variable horizontal resolution. These meshes allow for applications beyond uniform-resolution NWP and climate prediction, in particular allowing embedded high-resolution regions to be used for regional NWP and regional climate applications. The rationales for aspects of this formulation are discussed, and results from tests for nonhydrostatic flows on Cartesian planes and for large-scale flow on the sphere are presented. The results indicate that the solver is as accurate as existing nonhydrostatic solvers for nonhydrostatic-scale flows, and has accuracy comparable to existing global models using icosahedral (hexagonal) meshes for large-scale flows in idealized tests. Preliminary full-physics forecast results indicate that the solver formulation is robust and that the variable-resolution-mesh solutions are well resolved and exhibit no obvious problems in the mesh-transition zones.

Spatial Correlation of Gene Expression Measures in Tissue Microarray Core Analysis

Tissue microarrays (TMAs) make possible the screening of hundreds of different tumour samples for the expression of a specific protein. Automatic features extraction procedures lead to a series of covariates corresponding to the averaged stained scores. In this article, we model the random geometry of TMA cores using voronoi tesselations. This formalism enables the computation of indices of spatial correlation of stained scores using both classical and novel approaches. The potential of these spatial statistics to correctly discriminate between diseased and non-diseased cases is evaluated through the analysis of a TMA containing samples of breast carcinoma data. The results indicate a significant improvement in the breast cancer prognosis.

Sorting, texture and structure

SynopsisSands are sorted by the actions of air and water. A mathematical–physical model for the ensuing spatial and temporal variations in the size distributions of the sand grains is outlined. A crucial element in aeolian sorting and transport of sand is the process that takes place when a saltating grain impinges on the sand surface. This raises the problem of describing the texture of sand surfaces. Some initial empirical and mathematical findings concerning this problem are discussed. A closely connected question is that of how the grains in a sand deposit are packed and how this relates to the size distribution. Voronoi tesselations in combination with random point processes seem to offer one useful approach to the question.

Distortion of certain Voronoi tesselations when one particle moves

Some properties of the Voronoi polygon associated with a moving particle in a regular tessellation are studied. In particular, the area of the variable polygon has played a relevant role in previous studies on plant competition, animal predation, adjustment of the sex ratio in bisexual populations and the gregarious behaviour of animal species.