In this note we show by means of a simple example that, if the maximin problem with (nonlinear) concave increasing utility
functions is solved by inspecting the extreme points of the (generalized) Voronoi diagram (as usually proposed), one may have
to inspect an infinite number of candidate points.
The research of the second and third authors is partially supported by Grant PB96-1416-C02-02 of Ministerio de Educación y
Cultura, Spain 相似文献
This paper describes a new approach to discretizing first- and second-order partial differential equations. It combines the advantages of finite elements and finite differences in having both unstructured (triangular/tetrahedral) meshes and low-order physically intuitive schemes. In this ‘co-volume’ framework, the discretized gradient, divergence, curl, (scalar) Laplacian, and vector Laplacian operators satisfy relationships found in standard vector field theory, such as a Helmholtz decomposition. This article focuses on the vorticity–velocity formulation for planar incompressible flows. The algorithm is described and some supporting numerical evidence is provided. 相似文献
We strengthen the well-known Oxtoby theorem for strictly ergodic transformations by replacing the standard Cesaro convergence by the weaker Riesz or Voronoi convergence with monotonically increasing or decreasing weight coefficients. This general result allows, in particular, to strengthen the classical Weyl theorem on the uniform distribution of fractional parts of polynomials with irrational coefficients. 相似文献
We tackle the problem of computing the Voronoi diagram of a 3-D polyhedron whose faces are planar. The main difficulty with the computation is that the diagram's edges and vertices are of relatively high algebraic degrees. As a result, previous approaches to the problem have been non-robust, difficult to implement, or not provenly correct.
We introduce three new proximity skeletons related to the Voronoi diagram: (1) the Voronoi graph (VG), which contains the complete symbolic information of the Voronoi diagram without containing any geometry; (2) the approximate Voronoi graph (AVG), which deals with degenerate diagrams by collapsing sub-graphs of the VG into single nodes; and (3) the proximity structure diagram (PSD), which enhances the VG with a geometric approximation of Voronoi elements to any desired accuracy. The new skeletons are important for both theoretical and practical reasons. Many applications that extract the proximity information of the object from its Voronoi diagram can use the Voronoi graphs or the proximity structure diagram instead. In addition, the skeletons can be used as initial structures for a robust and efficient global or local computation of the Voronoi diagram.
We present a space subdivision algorithm to construct the new skeletons, having three main advantages. First, it solves at most uni-variate quartic polynomials. This stands in sharp contrast to previous approaches, which require the solution of a non-linear tri-variate system of equations. Second, the algorithm enables purely local computation of the skeletons in any limited region of interest. Third, the algorithm is simple to implement. 相似文献
We present an algorithm to reconstruct smooth surfaces of arbitrary topology from unorganised sample points and normals. The method uses natural neighbour interpolation, works in any dimension and accommodates non-uniform samples. The reconstructed surface interpolates the data points and is implicitly represented as the zero set of some pseudo-distance function. It can be meshed so as to satisfy a user-defined error bound, which makes the method especially relevant for small point sets. Experimental results are presented for surfaces in
. 相似文献
We present a novel method for the computation of well-defined optimized atomic partial charges and radii from the total electron density. Our method is based on a two-step radical Voronoi tessellation of the (possibly periodic) system and subsequent integration of the total electron density within each Voronoi cell. First, the total electron density is partitioned into the contributions of each molecule, and subsequently the electron density within each molecule is assigned to the individual atoms using a second set of atomic radii for the radical Voronoi tessellation. The radii are optimized on-the-fly to minimize the fluctuation (variance) of molecular and atomic charges. Therefore, our method is completely free of empirical parameters. As a by-product, two sets of optimized atomic radii are produced in each run, which take into account many specific properties of the system investigated. The application of an on-the-fly interpolation scheme reduces discretization noise in the Voronoi integration. The approach is particularly well suited for the calculation of partial charges in periodic bulk phase systems. We apply the method to five exemplary liquid phase simulations and show how the optimized charges can help to understand the interactions in the systems. Well-known effects such as reduced ion charges below unity in ionic liquid systems are correctly predicted without any tuning, empiricism, or rescaling. We show that the basis set dependence of our method is very small. Only the total electron density is evaluated, and thus, the approach can be combined with any electronic structure method that provides volumetric total electron densities—it is not limited to Hartree–Fock or density functional theory (DFT). We have implemented the method into our open-source software tool TRAVIS. 相似文献