Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second version mainly resolves minor LaTeX problems.

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Increased cancer risk in heavy drinkers with the alcohol dehydrogenase 1C*1 allele, possibly due to salivary acetaldehyde

Background: Chronic ethanol consumption is associated with an increased risk of upper aerodigestive tract cancer. As acetaldehyde seems to be a carcinogenic factor associated with chronic alcohol consumption, alcoholics with the alcohol dehydrogenase (ADH) 1C*1 allele seem to be particularly at risk as this allele encodes for a rapidly ethanol metabolising enzyme leading to increased acetaldehyde levels. Recent epidemiological studies resulted in contradictory results and therefore we have investigated ADH1C genotypes in heavy alcohol consumers only. Methods: We analysed the ADH1C genotype in 107 heavy drinkers with upper aerodigestive tract cancer and in 103 age matched alcoholic controls without cancer who consumed similar amounts of alcohol. Genotyping of the ADH1C locus was performed using polymerase chain reaction based on restriction fragment length polymorphism methods on leucocyte DNA. In addition, ethanol was administered orally (0.3 g/kg body weight) to 21 healthy volunteers with the ADH1C*1,1, ADH1C*1,2, and ADH1C*2,2 genotypes, and 12 volunteers with various ADH genotypes consumed ethanol ad libitum (mean 211 (29) g). Subsequently, salivary acetaldehyde concentrations were measured by gas chromatography or high performance liquid chromatography. Results: The allele frequency of the ADH1C*1 allele was found to be significantly increased in heavy drinkers with upper aerodigestive tract cancer compared with age matched alcoholic controls without cancer (61.7% v 49.0%; p = 0.011). The unadjusted and adjusted odds ratios for all cancer cases versus all alcoholic controls were 1.67 and 1.69, respectively. Healthy volunteers homozygous for the ADH1C*1 allele had higher salivary acetaldehyde concentrations following alcohol ingestion than volunteers heterozygous for ADH1C (p = 0.056) or homozygous for ADH1C*2 (p = 0.011). Conclusions: These data demonstrate that heavy drinkers homozygous for the ADH1C*1 allele have a predisposition to develop upper aerodigestive tract cancer, possibly due to elevated salivary acetaldehyde levels following alcohol consumption

Poisson point process convergence and extreme values in stochastic geometry

Let η t be a Poisson point process with intensity measure tμ , t>0 , over a Borel space X , where μ is a fixed measure. Another point process ξ t on the real line is constructed by applying a symmetric function f to every k -tuple of distinct points of η t . It is shown that ξ t behaves after appropriate rescaling like a Poisson point process, as t→∞ , under suitable conditions on η t and f . This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints and non-intersecting k -flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process

Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values