Random line tessellations of the plane: statistical properties of many-sided cells

We consider a family of random line tessellations of the Euclidean plane introduced in a much more formal context by Hug and Schneider [Geom. Funct. Anal. 17, 156 (2007)] and described by a parameter \alpha\geq 1. For \alpha=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for \alpha=2 it coincides with the typical Poisson-Voronoi cell. Let p_n(\alpha) be the probability for the zero-cell to have n sides. By the methods of statistical mechanics we construct the asymptotic expansion of \log p_n(\alpha) up to terms that vanish as n\to\infty. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when \alpha>1, but gets delocalized for the Crofton cell, \alpha=1, which is a singular point of the parameter range. The large-n expansion of \log p_n(1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the {\it typical} n-sided cell of a Poisson line tessellation.Comment: 26 pages, 3 figure

A digital twin of bridges for structural health monitoring

© International Workshop on Structural Health Monitoring. All rights reserved. Bridges are critical infrastructure systems connecting different regions and providing widespread social and economic benefits. It is therefore essential that they are designed, constructed and maintained properly to adapt to changing conditions of use and climate-driven events. With the rapid development in capability of collecting bridge monitoring data, a data challenge emerges due to insufficient capability in managing, processing and interpreting large monitoring datasets to extract useful information which is of practical value to the industry. One emerging area of research which focuses on addressing this challenge is the creation of 'digital twins' for bridges. A digital twin serves as a virtual representation of the physical infrastructure (i.e. the physical twin), which can be updated in near real time as new data is collected, provide feedback into the physical twin and perform 'what-if scenarios for assessing asset risks and predicting asset performance. This paper presents and broadly discusses two years of exploratory study towards creating a digital twin of bridges for structural health monitoring purposes. In particular, it has involved an interdisciplinary collaboration between civil engineers at the Cambridge Centre for Smart Infrastructure and Construction (CSIC) and statisticians at the Alan Turing Institute (ATI), using two monitored railway bridges in Staffordshire, UK as a case study. Four areas of research were investigated: (i) real-time data management using BIM, (ii) physics-based approaches, (iii) data-driven approaches, and (iv) data-centric engineering approaches (i.e. synthesis of physics-based and data-driven approaches). A framework for creating a digital twin of bridges, particularly for structural health monitoring purposes, is proposed and briefly discussed

Structure and magnetic properties of Bi5Ti3FeO15 ceramics prepared by sintering, mechanical activation and EDAMM process. A comparative study

Three different methods were used to obtain Bi5Ti3FeO15 ceramics, i.e. solid-state sintering, mechanical activation (MA) with subsequent thermal treatment, and electrical discharge assisted mechanical milling (EDAMM). The structure and magnetic properties of produced Bi5Ti3FeO15 samples were characterized using X-ray diffraction and Mössbauer spectroscopy. The purest Bi5Ti3FeO15 ceramics was obtained by standard solid-state sintering method. Mechanical milling methods are attractive because the Bi5Ti3FeO15 compound may be formed at lower temperature or without subsequent thermal treatment. In the case of EDAMM process also the time of processing is significantly shorter in comparison with solid-state sintering method. As revealed by Mössbauer spectroscopy, at room temperature the Bi5Ti3FeO15 ceramics produced by various methods is in paramagnetic state

Geometric conductive filament confinement by nanotips for resistive switching of HfO2-RRAM devices with high performance

Filament-type HfO2-based RRAM has been considered as one of the most promising candidates for future non-volatile memories. Further improvement of the stability, particularly at the "OFF" state, of such devices is mainly hindered by resistance variation induced by the uncontrolled oxygen vacancies distribution and filament growth in HfO2 films. We report highly stable endurance of TiN/Ti/HfO2/Si-tip RRAM devices using a CMOS compatible nanotip method. Simulations indicate that the nanotip bottom electrode provides a local confinement for the electrical field and ionic current density; thus a nano-confinement for the oxygen vacancy distribution and nano-filament location is created by this approach. Conductive atomic force microscopy measurements confirm that the filaments form only on the nanotip region. Resistance switching by using pulses shows highly stable endurance for both ON and OFF modes, thanks to the geometric confinement of the conductive path and filament only above the nanotip. This nano-engineering approach opens a new pathway to realize forming-free RRAM devices with improved stability and reliability

Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes

We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces

Abel Symposia

Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics

Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting