Establishing robustness of a spatial dataset in a tolerance‐based vector model

Spatial data are usually described through a vector model in which geometries are represented by a set of coordinates embedded into an Euclidean space. The use of a finite representation, instead of the real numbers theoretically required, causes many robustness problems which are well known in the literature. Such problems are made even worse in a distributed context, where data is exchanged between different systems and several perturbations can be introduced in the data representation. In order to discuss the robustness of a spatial dataset, two implementation models have to be distinguished: the identity and the tolerance model. The robustness of a dataset in the identity model has been widely discussed in the literature and some algorithms of the Snap Rounding (SR) family can be successfully applied in such contexts. Conversely, this problem has been less explored in the tolerance model. The aim of this article is to propose an algorithm inspired by those of the SR family for establishing or restoring the robustness of a vector dataset in the tolerance model. The main ideas are to introduce an additional operation which spreads instead of snapping geometries, in order to preserve the original relation between them, and to use a tolerance region for such an operation instead of a single snapping location. Finally, some experiments on real‐world datasets are presented, confirming how the proposed algorithm can establish the robustness of a dataset.     

Ab initio quantum-mechanical modeling of pyrophyllite [Al2Si4O10(OH)2] and talc [Mg3Si4O10(OH)2] surfaces

Bulk and slab geometry optimizations and calculations of the electrostatic potential at the surface of both pyrophyllite [Al₂Si₄O₁₀(OH)₂] and talc [Mg₃Si₄O₁₀(OH)₂] were performed at Hartree–Fock and DFT level. In both pyrophyllite and talc cases, a modest (001) surface relaxation was observed, and the surface preserves the structural features of the crystal: in the case of pyrophyllite the tetrahedral and octahedral sheets are strongly distorted with respect to the ideal hexagonal symmetry (and basal oxygen are located at different heights along the direction normal to the basal plane), whereas the structure of talc deviates slightly from the ideal hexagonal symmetry (almost co-planar basal oxygen). The calculated distortions are fully consistent with those experimentally observed. Although the potentials at the surface of pyrophyllite and talc are of the same order of magnitude, large topological differences were observed, which could possibly be ascribed to the differences between the surface structures of the two minerals. Negative values of the potential are located above the basal oxygen and at the center of the tetrahedral ring; above silicon the potential is always positive. The value of the potential minimum above the center of the tetrahedral ring of pyrophyllite is ?0.05 V (at 2 Å from the surface), whereas in the case of talc the minimum is ?0.01 V, at 2.7 Å. In the case of pyrophyllite the minimum of potential above the higher basal oxygen is located at 1.1 Å and it has a value of ?1.25 V, whereas above the lower oxygen the value of the potential at the minimum is ?0.2 V, at 1.25 Å; the talc exhibits a minimum of ?0.75 V at 1.2 Å, above the basal oxygen.