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We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution of mean curvature flow in ℝ3 with genus 0. More generally, we show that the only properly embedded self-similar shrinkers in ℝ3 with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.

In this paper, sufficient conditions for the parallelism of the normal curvature tensor of submanifolds in spaces of constant curvature are obtained. Keywords and phrases: submanifold, space of constant curvature, second fundamental form, normal curvature tensor, normal curvature vector, normal torsion vector. AMS Subject Classification: 53B25, 53C40

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space ℍ⁴ with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, a nowhere vanishing second fundamental form and a scalar curvature bounded from below.

High-dimension low-sample size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, the popular support vector machine suffers from \"data piling\" at the margin, which can diminish generalizability. This leads naturally to the development of distance-weighted discrimination, which is based on second-order cone programming, a modern computationally intensive optimization method.

We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails.

We propose a novel approach for the parametrically robust design of dynamic systems. The approach can be applied to system models with parameters that are uncertain in the sense that values for these parameters are not known precisely, but only within certain bounds. The novel approach is guaranteed to find an optimal steady state that is stable for each parameter combination within these bounds. Our approach combines the use of a standard solver for constrained optimization problems with the rigorous solution of nonlinear systems. The constraints for the optimization problems are based on the concept of parameter space normal vectors that measure the distance of a tentative optimum to the nearest known critical point, i.e., a point where stability may be lost. Such normal vectors are derived using methods from nonlinear dynamics. After the optimization, the rigorous solver is used to provide a guarantee that no critical points exist in the vicinity of the optimum, or to detect such points. In the latter case, the optimization is resumed, taking the newly found critical points into account. This optimize-and-verify procedure is repeated until the rigorous nonlinear solver can guarantee that the vicinity of the optimum is free from critical points and therefore the optimum is parametrically robust. In contrast to existing design methodologies, our approach can be automated and does not rely on the experience of the designing engineer. A simple model of a fermenter is used to illustrate the concepts and the order of activities arising in a typical design process.

Due to the complexity of surrounding environments, lidar point cloud data (PCD) are often degraded by plane noise. In order to eliminate noise, this paper proposes a filtering scheme based on the grid principal component analysis (PCA) technique and the ground splicing method. The 3D PCD is first projected onto a desired 2D plane, within which the ground and wall data are well separated from the PCD via a prescribed index based on the statistics of points in all 2D mesh grids. Then, a KD-tree is constructed for the ground data, and rough segmentation in an unsupervised method is conducted to obtain the true ground data by using the normal vector as a distinctive feature. To improve the performance of noise removal, we propose an elaborate K nearest neighbor (KNN)-based segmentation method via an optimization strategy. Finally, the denoised data of the wall and ground are spliced for further 3D reconstruction. The experimental results show that the proposed method is efficient at noise removal and is superior to several traditional methods in terms of both denoising performance and run speed.

It is very challenging for robots to perform grinding and polishing tasks on surfaces with unknown geometry. Most existing methods solve this problem by modeling the relationship between the force sensing information and surface normal vectors by analyzing the forces on special end tools such as spherical tools and cylindrical tools and simplified friction model. In this paper, we propose a normal vectors learning method to simultaneously control end-effector force and direction on unknown surfaces. First, the relation that mapping the force sensing information to the surface normal vectors is learned from the demonstrated data on the known plane using locally weighted regression. Next, the learned relation is used to estimate surface normal vectors on the unknown surface. To improve the force control precision on the unknown geometry surface, the adaptive force control is developed. To improve the direction control precision due to friction, the iterative learning control is developed. The proposed method is verified by comparative simulations and experiments using the Franka robot. Results show that the end-effector can be controlled perpendicular to the surface with a certain force.