Computable bounds for the reach and rr-convexity of subsets of Rd\mathbb{R}^d

The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the point cloud becomes dense in the set, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the $\beta$-reach, a generalization of the reach that excludes small-scale features of size less than a parameter $\beta\in[0,\infty)$. Numerical studies suggest how the $\beta$-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds.Comment: 37 pages, 23 figure

Decentralized collaborative transport of fabrics using micro-UAVs

Small unmanned aerial vehicles (UAVs) have generally little capacity to carry payloads. Through collaboration, the UAVs can increase their joint payload capacity and carry more significant loads. For maximum flexibility to dynamic and unstructured environments and task demands, we propose a fully decentralized control infrastructure based on a swarm-specific scripting language, Buzz. In this paper, we describe the control infrastructure and use it to compare two algorithms for collaborative transport: field potentials and spring-damper. We test the performance of our approach with a fleet of micro-UAVs, demonstrating the potential of decentralized control for collaborative transport.Comment: Submitted to 2019 International Conference on Robotics and Automation (ICRA). 6 page

A local statistic for the spatial extent of extreme threshold exceedances

We introduce the extremal range, a local statistic for studying the spatial extent of extreme events in random fields on $\mathbb{R}^2$. Conditioned on exceedance of a high threshold at a location $s$, the extremal range at $s$ is the random variable defined as the smallest distance from $s$ to a location where there is a non-exceedance. We leverage tools from excursion-set theory to study distributional properties of the extremal range, propose parametric models and predict the median extremal range at extreme threshold levels. The extremal range captures the rate at which the spatial extent of conditional extreme events scales for increasingly high thresholds, and we relate its distributional properties with the bivariate tail dependence coefficient and the extremal index of time series in classical Extreme-Value Theory. Consistent estimation of the distribution function of the extremal range for stationary random fields is proven. For non-stationary random fields, we implement generalized additive median regression to predict extremal-range maps at very high threshold levels. An application to two large daily temperature datasets, namely reanalyses and climate-model simulations for France, highlights decreasing extremal dependence for increasing threshold levels and reveals strong differences in joint tail decay rates between reanalyses and simulations.Comment: 32 pages, 5 figure

Indirect chiral magnetic exchange through Dzyaloshinskii–Moriya-enhanced RKKY interactions in manganese oxide chains on Ir(100)

Localized electron spins can couple magnetically via the Ruderman–Kittel–Kasuya–Yosida interaction even if their wave functions lack direct overlap. Theory predicts that spin–orbit scattering leads to a Dzyaloshinskii–Moriya type enhancement of this indirect exchange interaction, giving rise to chiral exchange terms. Here we present a combined spin-polarized scanning tunneling microscopy, angle-resolved photoemission, and density functional theory study of MnO_2 chains on Ir(100). Whereas we find antiferromagnetic Mn–Mn coupling along the chain, the inter-chain coupling across the non-magnetic Ir substrate turns out to be chiral with a 120° rotation between adjacent MnO_2 chains. Calculations reveal that the Dzyaloshinskii–Moriya interaction results in spin spirals with a periodicity in agreement with experiment. Our findings confirm the existence of indirect chiral magnetic exchange, potentially giving rise to exotic phenomena, such as chiral spin-liquid states in spin ice systems or the emergence of new quasiparticles

Isotropisation of Bianchi class A models with curvature for a minimally coupled scalar tensor theory

We look for necessary isotropisation conditions of Bianchi class $A$ models with curvature in presence of a massive and minimally coupled scalar field when a function $\ell$ of the scalar field tends to a constant, diverges monotonically or with sufficiently small oscillations. Isotropisation leads the metric functions to tend to a power or exponential law of the proper time $t$ and the potential respectively to vanish as $t^{-2}$ or to a constant. Moreover, isotropisation always requires late time accelerated expansion and flatness of the Universe.Comment: 20 page

Indirect chiral magnetic exchange through Dzyaloshinskii–Moriya-enhanced RKKY interactions in manganese oxide chains on Ir(100)

Localized electron spins can couple magnetically via the Ruderman–Kittel–Kasuya–Yosida interaction even if their wave functions lack direct overlap. Theory predicts that spin–orbit scattering leads to a Dzyaloshinskii–Moriya type enhancement of this indirect exchange interaction, giving rise to chiral exchange terms. Here we present a combined spin-polarized scanning tunneling microscopy, angle-resolved photoemission, and density functional theory study of MnO_2 chains on Ir(100). Whereas we find antiferromagnetic Mn–Mn coupling along the chain, the inter-chain coupling across the non-magnetic Ir substrate turns out to be chiral with a 120° rotation between adjacent MnO_2 chains. Calculations reveal that the Dzyaloshinskii–Moriya interaction results in spin spirals with a periodicity in agreement with experiment. Our findings confirm the existence of indirect chiral magnetic exchange, potentially giving rise to exotic phenomena, such as chiral spin-liquid states in spin ice systems or the emergence of new quasiparticles

Numerical Approaches to Spacetime Singularities

This Living Review updates a previous version which its itself an update of a review article. Numerical exploration of the properties of singularities could, in principle, yield detailed understanding of their nature in physically realistic cases. Examples of numerical investigations into the formation of naked singularities, critical behavior in collapse, passage through the Cauchy horizon, chaos of the Mixmaster singularity, and singularities in spatially inhomogeneous cosmologies are discussed.Comment: 51 pages, 6 figures may be found in online version: Living Rev. Relativity 2002-1 at www.livingreviews.or

Surface area and volume of excursion sets observed on point cloud based polytopic tessellations

The excursion set of a C2 smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension d. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in Rd , including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices

On the perimeter estimation of pixelated excursion sets of 2D anisotropic random fields

We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets.To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb R^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb R^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting

On the perimeter estimation of pixelated excursion sets of 2D anisotropic random fields

We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets.To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb R^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb R^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting