Cube

Abstract: The continuous symmetry measures approach, designed to assess quantitatively the degree of any symmetry within any structure, is extended to the important class of the polyhedra. For this purpose, we developed a general methodology and a general computational tool, which identify the minimal distance of a given structure to a desired general shape with the same number of vertexes. Specifically, we employ this tool to evaluate quantitatively the degree of polyhedricity within distorted polyhedra, taking as examples the most central and abundant polyhedral structures in chemistry in general and in coordination chemistry in particular, namely the tetrahedron, the bipyramid, the octahedron, the cube, the icosahedron, and the dodecahedron. After describing the properties of the symmetry measurement tool, we show its application and versatility in a number of cases where the deviation from exact symmetry has been an issue, including z-axis Jahn−Teller type polyhedral distortions, tantalum hydride complexes, pen...

Abstract: A simple transformation is introduced which facilitates the evaluation of integrals over certain square based pyramids or cubes in cases where the integrand has a singularity at a vertex.

Abstract: We introduce the Iceberg-CUBE problem as a reformulation of the datacube (CUBE) problem. The Iceberg-CUBE problem is to compute only those group-by partitions with an aggregate value (e.g., count) above some minimum support threshold. The result of Iceberg-CUBE can be used (1) to answer group-by queries with a clause such as HAVING COUNT(*) >= X, where X is greater than the threshold, (2) for mining multidimensional association rules, and (3) to complement existing strategies for identifying interesting subsets of the CUBE for precomputation.We present a new algorithm (BUC) for Iceberg-CUBE computation. BUC builds the CUBE bottom-up; i.e., it builds the CUBE by starting from a group-by on a single attribute, then a group-by on a pair of attributes, then a group-by on three attributes, and so on. This is the opposite of all techniques proposed earlier for computing the CUBE, and has an important practical advantage: BUC avoids computing the larger group-bys that do not meet minimum support. The pruning in BUC is similar to the pruning in the Apriori algorithm for association rules, except that BUC trades some pruning for locality of reference and reduced memory requirements. BUC uses the same pruning strategy when computing sparse, complete CUBEs.We present a thorough performance evaluation over a broad range of workloads. Our evaluation demonstrates that (in contrast to earlier assumptions) minimizing the aggregations or the number of sorts is not the most important aspect of the sparse CUBE problem. The pruning in BUC, combined with an efficient sort method, enables BUC to outperform all previous algorithms for sparse CUBEs, even for computing entire CUBEs, and to dramatically improve Iceberg-CUBE computation.

Abstract: The traditional approach for parametrizing a surface involves cutting it into charts and mapping these piecewise onto a planar domain. We introduce a robust technique for directly parametrizing a genus-zero surface onto a spherical domain. A key ingredient for making such a parametrization practical is the minimization of a stretch-based measure, to reduce scale-distortion and thereby prevent undersampling. Our second contribution is a scheme for sampling the spherical domain using uniformly subdivided polyhedral domains, namely the tetrahedron, octahedron, and cube. We show that these particular semi-regular samplings can be conveniently represented as completely regular 2D grids, i.e. geometry images. Moreover, these images have simple boundary extension rules that aid many processing operations. Applications include geometry remeshing, level-of-detail, morphing, compression, and smooth surface subdivision.