Coordination polyhedra of hydration shells of Na+ and K+ cations in aqueous solutions

Simulation of the hydration of Na⁺ and K⁺ cations in dilute solution was performed by the Monte Carlo method. A novel approach to structural analysis of hydration shells of ions was developed. Specific sets of coordination polyhedra formed by water molecules of the first coordination sphere were found. Structural and energy characteristics of hydration were calculated. The effect of Na⁺ and K⁺ cations on the structure of the network of H-bonds and mobility of water molecules in hydration shells was studied. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 852–857, May, 1999.     

The Moduli Space of n+1 Points in Complex Hyperbolic n-Space

The structure of the space of orbits of PU(n, 1) acting on (n+1)-tuples of points in complex hyperbolic space is characterized in terms of side lengths and angular invariants. The more general situation in which some of the points lie on the boundary of complex hyperbolic space is described in terms of other basic invariants.     

Helical Structures: The Geometry of Protein Helices and Nanotubes

In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant. A helical structure is thus generated by the repeated action of a screw transformation acting on a subunit. A plane hexagonal lattice wrapped round a cylinder provides a useful starting point for describing the helical conformations of protein molecules, for investigating the geometrical properties of carbon nanotubes, and for certain types of dense packings of equal spheres.     

Comparison of various generalizations of continued fractions

We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectorsL related to two Davenport cubic formsg ₁ andg ₂. The Klein polyhedra ofg ₁ andg ₂ were calculated in another paper. Here the integer convergentsP _k given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 339–348, March, 1997. Translated by V. E. Nazaikinskii     

Multicomponent self-assembly of a nested co(24) @co(48) metal-organic polyhedral framework

A tale of two polyhedra: Two nested Archimedean metal-organic polyhedra, a rhombicuboctahedron (Co(48) cage) and a cuboctahedron (Co(24) cage), have been assembled from two types of cobalt dimers and two complementary ligands. Within the 3D covalent cubic array of outer Co(48) cages and framework lie encapsulated inner Co(24) cages that are linked into a separate "hidden" 3D framework.     

Crystal and electronic structure study of AgPd3Se

The AgPd₃Se compound was synthesised from individual elements by solid-state chemical reactions and structurally characterized by powder X-ray diffraction data. AgPd₃Se displays cubic symmetry, space group Pa3?, unit cell parameter a=8.6289(1) Å and Z=8. Double-Friauf polyhedra (DFP), defined by Pd and Se atoms, form the basic structural building block of the AgPd₃Se crystal structure. The Ag atoms occupy the centres of DFPs forming an Ag–Ag dimer (2.792(2) Å). The packing of DFPs forms two kinds of interpenetrating networks that show similar features as three-dimensional Penrose tiles. AgPd₃Se is isostructural with CaAu₃Ga. The electric resistivity as well as the electronic structure calculation suggests metallic behaviour.     

Mean width and caliper characteristics of network polyhedra

The kinetic modelling of space-filling grain networks has been approached traditionally by representing the grains as spheres of equivalent volume. A spherical approximation used to describe polyhedral grains, unfortunately, relinquishes most geometrical and all topological details of the grain structure. Techniques developed by Hilgenfeldt et al., and by Glicksman, describe network structures comprised of space-filling irregular polyhedra and their kinetics with regular polyhedra, which act as ‘proxies’ that preserve both local topology and network constraints. Analytical formulas based on regular polyhedra and Surface Evolver simulations are used in this study to calculate the average caliper width and mean width for extended sets of polyhedra that vary systematically from convex to concave objects. Of importance, caliper width and mean width measures allow estimation of the growth rates of grains. Comparison of these calculations and simulations, however, reveal a weak dependence between average caliper width and mean width measures and the detailed shapes of polyhedra, especially their face curvatures. This finding might, in fact, affect the application and use of linear measures as kinetic tools in quantitative microstructure measurements.     

Approximate indexing of icosahedral polyhedra with fibonacci numbers

In general, indexing faces of icosahedral face-forms requires irrational numbers. However, for many practical purposes an approximate indexing based on triplets of integer numbers can be used. Two possible approaches called, respectively, “Fibonacci Matrix Methods” (FMM) and the “Linear Combination Method” (LCM) are described. FMM relies on the use of “auxiliary” matrices F_n, F² _n, F³ _n and F⁴ _n which have Fibonacci numbers as their elements. These matrices allow good approximation of the results usually obtained using the standard five-fold rotation matrices which are typical of icosahedral symmetry. LCM is based on the use of a classical crystallographic rule i.e. the so-called “Goldschmidt Complication Law” which is just a particular case of linear combination of triplets of face indices, with integers as coefficients. The occurrence of large integer indices is remarked.     

An overview on distance and pseudo distance functions and their applications

The fast and accurate collision detection and distance calculation between geometric models is fundamental in robotics, manufacturing and computer-simulated environments. This paper surveys the state of the art in the theory and algorithms of distance and pseudo distance functions between convex sets. An overview on the existing results,including fast algorithms for distance calculation, definitions and properties of various pseudo distance functions (the J-function, the growth distance, and the pseudo minimum translational distance), is presented. The distance and pseudo distance functions are versatile in a wide range of areas.Their applications in robot motion planning, grasp analysis/synthesis, form error evaluation, and engineering optimization is demonstrated.