Distributional properties of the typical cell of stationary iterated tessellations

Distributional properties are considered of the typical cell of stationary iterated tessellations (SIT), which are generated by stationary Poisson-Voronoi tessellations (SPVT) and stationary Poisson line tessellations (SPLT), respectively. Using Neveus exchange formula, the typical cell of SIT can be represented by those cells of its component tessellation hitting the typical cell of its initial tessellation. This provides a simulation algorithm without consideration of limits in space. It has been applied in order to estimate the probability densities of geometric characteristics of the typical cell of SIT generated by SPVT and SPLT. In particular, the probability densities of the number of vertices, the perimeter, and the area of the typical cell of such SIT have been determined.Acknowledgement. This work was supported by France Telecom R&D through research grant no. 001B130.     

Model Based Estimation of Geometric Characteristics of Open Foams

Open cell foams are a class of modern materials which is interesting for a wide variety of applications and which is not accessible to classical materialography based on 2d images. 3d imaging by micro computed tomography is a practicable alternative. Analysis of the resulting volume images is either based on a simple binarisation of the image or on so-called cell reconstruction by image processing. The first approach allows to estimate mean characteristics like the mean cell volume using the typical cell of a random spatial tessellation as model for the cell shape. The cell reconstruction allows estimation of empirical distributions of cell characteristics. This paper summarises the theoretical background for the first method, in particular estimation of the intrinsic volumes and their densities from discretized data and models for random spatial tessellations. The accuracy of the estimation method is assessed using the dilated edge systems of simulated random spatial tessellations.     

Joint distribution of direction and length of typical I-segment in a homogeneous random planar tessellation stable under iteration

The paper deals with homogeneous random planar tessellations stable under iteration (random STIT tessellations). The length distribution of the typical I-segment is already known in the isotropic case [8]. In the present paper, the anisotropic case is treated. Then also the direction of the typical I-segment is of interest. The joint distribution of direction and length of the typical I-segment is evaluated. As a first step, the corresponding joint distribution for the so-called typical remaining I-segment is derived. Dedicated to the 80th birthday of Klaus Krickeberg     

The Combinatorial Structure of Spatial STIT Tessellations

Spatially homogeneous random tessellations that are stable under iteration (nesting) in the $3$ 3 -dimensional Euclidean space are considered, so-called STIT tessellations. They arise as outcome of a space-time process of subsequent cell division and, consequently, they are not facet-to-facet. The intent of this paper is to develop a detailed analysis of the combinatorial structure of such tessellations and to determine a number of new geometric mean values, for example for the neighbourhood of the typical vertex. The heart of the results is a fine classification of tessellation edges based on the type of their endpoints or on the equality relationship with other types of line segments. In the background of the proofs are delicate distributional properties of spatial STIT tessellations.     

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.     

Do students confuse dimensionality and “directionality”?

The aim of this research is to understand the way in which students struggle with the distinction between dimensionality and “directionality” and if this type of potential confusion could be a factor affecting students’ tendency toward improper linear reasoning in the context of the relations between length and area of geometrical figures. 131 9th grade students were confronted with a multiple-choice test consisting of six problems related to the perimeter or the area of an enlarged geometrical figure, then some interviews were carried out to obtain qualitative data in relation to students’ reasoning. Results indicate that more than one fifth of the students’ answers could be characterized as based on directional thinking, suggesting that students struggled with the distinction between dimensionality and “directionality”. A single arrow showing one direction (image provided to the students) seemed to strengthen the tendency toward improper linear reasoning for the area problems. Two arrows showing two directions helped students to see a quadratic relation for the area problems.     

Shape-driven nested Markov tessellations

A new and rather broad class of stationary random tessellations of the d-dimensional Euclidean space is introduced, which we call shape-driven nested Markov tessellations. Locally, these tessellations are constructed by means of a spatio-temporal random recursive split dynamics governed by a family of Markovian split kernel, generalizing thereby the – by now classical – construction of iteration stable random tessellations. By providing an explicit global construction of the tessellations, it is shown that under suitable assumptions on the split kernels (shape-driven), there exists a unique time-consistent whole-space tessellation-valued Markov process of stationary random tessellations compatible with the given split kernels. Beside the existence and uniqueness result, the typical cell and some aspects of the first-order geometry of these tessellations are in the focus of our discussion.     

Simulation of typical Cox–Voronoi cells with a special regard to implementation tests

We consider stationary Poisson line processes in the Euclidean plane and analyze properties of Voronoi tessellations induced by Poisson point processes on these lines. In particular, we describe and test an algorithm for the simulation of typical cells of this class of Cox–Voronoi tessellations. Using random testing, we validate our algorithm by comparing theoretical values of functionals of the zero cell to simulated values obtained by our algorithm. Finally, we analyze geometric properties of the typical Cox–Voronoi cell and compare them to properties of the typical cell of other well-known classes of tessellations, especially Poisson–Voronoi tessellations. Our results can be applied to stochastic–geometric modelling of networks in telecommunication and life sciences, for example. The lines can then represent roads in urban road systems, blood arteries or filament structures in biological tissues or cells, while the points can be locations of telecommunication equipment or vesicles, respectively.     

Length distributions of edges in planar stationary and isotropic STIT tessellations

Stationary and isotropic random tessellations of the euclidean plane are studied which have the characteristic property to be stable with respect to iteration (or nesting), STIT for short. Since their cells are not in a face-to-face position, three different types of linear segments appear. For all the types the distribution of the length of the typical segment is given. The text was submitted by the authors in English.     

Asymptotic shape of small cells

A stationary Poisson line tessellation is considered whose directional distribution is concentrated on two different atoms with some positive weights. The shape of the typical cell of such a tessellation is studied when its area or its perimeter tends to zero. In contrast to known results where the area or the perimeter tends to infinity, it is shown that the asymptotic shape of cells having small area is degenerate. Again in contrast to the case of large cells, the asymptotic shape of cells with small perimeter is not uniquely determined. The results are accompanied by a large scale simulation study.     

Second-Order Characteristics of the Edge System of Random Tessellations and the PPI Value of Foams

The mean number of pores per inch (PPI) is widely used as a pore size characteristic for foams. Nevertheless, there is still a lack of fast and reliable methods for estimating this quantity. We propose a method for estimating the PPI value based on the Bartlett spectrum of a dark field image of the material. To this end, second-order properties of the edge systems of random tessellations are investigated in detail. In particular, we study the spectral density of the random length measure of the edges. It turns out that the location of its first local maximum is proportional to the PPI value. To determine the factor of proportionality, several random tessellation models as well as examples of real foams are investigated. To mimic the image acquisition process, 2D sections and projections of 3D tessellations are considered.     

Phase-space growth rates,local Lyapunov spectra,and symmetry breaking for time-reversible dissipative oscillators

We investigate and discuss the time-reversible nature of phase-space instabilities for several flows, $\dot{x}=f(x)$. The flows describe thermostated oscillator systems in from two through eight phase-space dimensions. We determine the local extremal phase-space growth rates, which bound the instantaneous comoving Lyapunov exponents. The extremal rates are point functions which vary continuously in phase space. The extremal rates can best be determined with a “singular-value decomposition” algorithm. In contrast to these precisely time-reversible local “point function” values, a time-reversibility analysis of the comoving Lyapunov spectra is more complex. The latter analysis is nonlocal and requires the additional storing and playback of relatively long (billion-step) trajectories.All the oscillator models studied here show the same time reversibility symmetry linking their time-reversed and time-averaged “global” Lyapunov spectra. Averaged over a long-time-reversed trajectory, each of the long-time-averaged Lyapunov exponents simply changes signs. The negative/positive sign of the summed-up and long-time-averaged spectra in the forward/backward time directions is the microscopic analog of the Second Law of Thermodynamics. This sign changing of the individual global exponents contrasts with typical more-complex instantaneous “local” behavior, where there is no simple relation between the forward and backward exponents other than the local (instantaneous) dissipative constraint on their sum. As the extremal rates are point functions, they too always satisfy the sum rule.     

Mixed Random Mosaics

Cowan [2] has defined random mosaics processes RMP in R² and has given some basic properties of them. In particular Cowan introduces the fundamental parameters α, β_k, γ_k of the process and, in terms of them, he computes the mean values of the area α, perimeter h, number of ares w and number of vertices v of a typical polygon of the RMP. Our purpose is to consider the RMP obtained by superposition of two independent random mosaics. Then, the characteristics a₁₂, h₁₂, w₁₂, v₁₂ of the resulting process are computed in terms of the characteristics a_i, h_i, w_i, v_i, of each process. The case of non random tessellations mixed with random mosaics is also considered.     

Wrapping spheres with flat paper

We study wrappings of smooth (convex) surfaces by a flat piece of paper or foil. Such wrappings differ from standard mathematical origami because they require infinitely many infinitesimally small folds (“crumpling”) in order to transform the flat sheet into a surface of nonzero curvature. Our goal is to find shapes that wrap a given surface, have small area and small perimeter (for efficient material usage), and tile the plane (for efficient mass production). Our results focus on the case of wrapping a sphere. We characterize the smallest square that wraps the unit sphere, show that a 0.1% smaller equilateral triangle suffices, and find a 20% smaller shape contained in the equilateral triangle that still tiles the plane and has small perimeter.     

Geometric Measures for Random Curved Mosaics of Rd

This paper deals with stationary random mosaics of R^d with general cell shapes. As geometric measures concentrated on the i-skeleton (i = 0, 1,…,d) the i-dimensional surface area (volume) measure and (i — 1) different curvature measures are chosen. The corresponding densities are calculated as well as for the mosaics and their superpositions in terms of mean cell parameters and mean cell numbers. This leads to various relations between the characteristic which are applied, in particular, to two- and three-dimensional tessellations. A comparison with known formulas for mosaics with convex cells in R² and R³ is given.     

On barotropic trapped wave solutions with no-slip boundary conditions

Barotropic trapped wave solutions of a linearized system of the ocean dynamics equations are described for a semi-infinite, f-plane model basin of constant depth bordering a straight, vertical coast, for some “typical” values of the model parameters. No-slip boundary conditions are considered. When the wave length is shorter than the Rossby deformation radius, the main features of the wave solutions are as follows: the Kelvin wave exponential offshore decay scale essentially decreases as the wave length decreases, and an additional wave solution propagating in the opposite direction appears.     

Topological Mean Value Relations for Random Cell Complexes

There are investigated stationary random q-dimensional topological cell complexes in ?^d, in particular, random tessellations. General relationships between the mean values of topological characteristics are derived. Then they are specified for the cases d = 2, 3, 4.     

On stochastic calculus with respect to q-Brownian motion

Following the approach and the terminology introduced in Deya and Schott (2013) [6], we construct a product Lévy area above the q-Brownian motion (for $q\in [0,1)$) and use this object to study differential equations driven by the process.We also provide a detailed comparison between the resulting “rough” integral and the stochastic “Itô” integral exhibited by Donati-Martin (2003) [7].     

A minimal partition problem with trace constraint in the Grushin plane

We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the least amount of perimeter, under an additional “one-dimensional” constraint on the intersections of their boundaries. We prove existence of regular solutions for this problem, and we characterize them in terms of isoperimetric sets, showing differences with the Euclidean case. The problem arises from the study of quantitative isoperimetric inequalities and has connections with the theory of minimal clusters.