Minimum Distance Between the Faces of Two Convex Polyhedra: A Sufficient Condition

The problem of finding the Euclidean distance between two convex polyhedra can be reduced to the combinatorial optimization problem of finding the minimum distance between their faces. This paper presents a global optimality criterion for this problem. An algorithm (QLDPA) for the fast computation of the distance between convex and bounded polyhedra is proposed as an application of it. Computer experiments show its fast performance, especially when the total number of vertices is large.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

Near-perfect matrices

A 0, 1 matrixA isnear-perfect if the integer hull of the polyhedron {x0: Ax } can be obtained by adding one extra (rank) constraint. We show that in general, such matrices arise as the cliquenode incidence matrices of graphs. We give a colouring-like characterization of the corresponding class of near-perfect graphs which shows that one need only check integrality of a certain linear program for each 0, 1, 2-valued objective function. This in contrast with perfect matrices where it is sufficient to check 0, 1-valued objective functions. We also make the following conjecture: a graph is near-perfect if and only if sequentially lifting any rank inequality associated with a minimally imperfect graph results in the rank inequality for the whole graph. We show that the conjecture is implied by the Strong Perfect Graph Conjecture. (It is also shown to hold for graphs with no stable set of size eleven.) Our results are used to strengthen (and give a new proof of) a theorem of Padberg. This results in a new characterization of minimally imperfect graphs: a graph is minimally imperfect if and only if both the graph and its complement are near-perfect.The research has partially been done when the author visited Mathematic Centrum, CWI, Amsterdam, The Netherlands.     

Effective atom volumes for implicit solvent models: comparison between Voronoi volumes and minimum fluctuation volumes

An essential element of implicit solvent models, such as the generalized Born method, is a knowledge of the volume associated with the individual atoms of the solute. Two approaches for determining atomic volumes for the generalized Born model are described; one is based on Voronoi polyhedra and the other, on minimizing the fluctuations in the overall volume of the solute. Volumes to be used with various parameter sets for protein and nucleic acids in the CHARMM force field are determined from a large set of known structures. The volumes resulting from the two different approaches are compared with respect to various parameters, including the size and solvent accessibility of the structures from which they are determined. The question of whether to include hydrogens in the atomic representation of the solute volume is examined. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1857-1879, 2001     

Mackay,Anti-Mackay,Double-Mackay,Pseudo-Mackay,and Related Icosahedral Shell Clusters

Mackay introduced two important crystallographic concepts in a short paper published 40 years ago. One is the icosahedral shell structure (iss) consisting of concentric icosahedra displaying fivefold rotational symmetry. The number of atoms contained within these icosahedral shells and subshells agrees well with the magic numbers in rare gas clusters, (C₆₀) _N molecules, and some metal clusters determined by mass spectroscopy or simulated on energy considerations. The cluster of 55 atoms within the second icosahedral shell occurs frequently and has been called Mackay icosahedron, or simply MI, which occurs not only in various clusters, but also in intermetallic compounds and quasicrystals. The second concept is the hierarchic icosahedral structures caused by the presence of a stacking fault in the fcc packing of the successive triangular faces in the iss. For instance, a fault occurs after the ABC layers resulting an ABCB packing. This is, in fact, a hierarchic icosahedral structure of a core icosahedron connected to 12 outer icosahedra by vertex sharing, or an icosahedron of icosahedra (double MI. Contrary to Mackay's iss, a faulted hierarchic icosahedral shell is, in fact, a twinlike face capping of the underlying triangles; it is, therefore, called an anti-Mackay cluster. The hierarchic icosahedral structure in an Al-Mn-Pd icosahedral quasicrystal has a core of body-centered cube rather than an icosahedron and, therefore, is called a pseudo-Mackay cluster. The hierarchic icosahedral structures have been studied separately in the past in the fields of clusters, nanoparticles, intermetallic compounds, and quasicrystals, but the underlying geometry should be the same. In the following a unified geometrical analysis is presented.     

Ligand design for alkali-metal-templated self-assembly of unique high-nuclearity CuII aggregates with diverse coordination cage units: crystal structures and properties

The construction of two unique, high-nuclearity Cu(II) supramolecular aggregates with tetrahedral or octahedral cage units, [(mu(3)-Cl)[Li subset Cu(4)(mu-L(1))(3)](3)](ClO(4))(8)(H(2)O)(4.5) (1) and [[Na(2) subset Cu(12)(mu-L(2))(8)(mu-Cl)(4)](ClO(4))(8)(H(2)O)(10)(H(3)O(+))(2)](infinity) (2) by alkali-metal-templated (Li(+) or Na(+)) self-assembly, was achieved by the use of two newly designed carboxylic-functionalized diazamesocyclic ligands, N,N'-bis(3-propionyloxy)-1,4-diazacycloheptane (H(2)L(1)) or 1,5-diazacyclooctane-N,N'-diacetate acid (H(2)L(2)). Complex 1 crystallizes in the trigonal R3c space group (a = b = 20.866(3), c = 126.26(4) A and Z = 12), and 2 in the triclinic P1 space group (a = 13.632(4), b = 14.754(4), c = 19.517(6) A, alpha = 99.836(6), beta = 95.793(5), gamma = 116.124(5) degrees and Z = 1). By subtle variation of the ligand structures and the alkali-metal templates, different polymeric motifs were obtained: a dodecanuclear architecture 1 consisting of three Cu(4) tetrahedral cage units with a Li(+) template, and a supramolecular chain 2 consisting of two crystallographically nonequivalent octahedral Cu(6) polyhedra with a Na(+) template. The effects of ligand functionality and alkali metal template ions on the self-assembly processes of both coordination supramolecular aggregates, and their magnetic behaviors are discussed in detail.     

Toward a universal quantum QSPR operator

This article presents a discussion about the formalism, which might be associated to a general Quantum quantitative structure–properties relations operator, appearing in a Boltzmann‐like exponential form, which is based in turn on the definition of the concept of thermal voltage, applied to thermally scaled electronic density functions. Three practical numerical examples are presented, corresponding to the calculation of the polarization angle in assorted chiral molecules, the estimation of fish toxicity for perchlorobenzene within the set of chlorobenzenes and a typical quantum QSAR study on the popular Cramer steroid set.     

Azobenzene‐Functionalized Metal–Organic Polyhedra for the Optically Responsive Capture and Release of Guest Molecules

Stimuli‐responsive metal–organic polyhedra (srMOPs) functionalized with azobenzene showed UV‐irradiation‐induced isomerization from the insoluble trans‐srMOP to the soluble cis‐srMOP, whereas irradiation with blue light reversed this process. Guest molecules were trapped and released upon cis‐to‐trans and trans‐to‐cis isomerization of the srMOPs, respectively. This study provides a new direction in the ever‐diversifying field of MOPs, while laying the groundwork for a new class of optically responsive materials.     

A General Strategy for Hollow Metal-Phytate Coordination Complex Micropolyhedra Enabled by Cation Exchange

The ability to incorporate functional metal ions (Mⁿ⁺) into metal–organic coordination complexes adds remarkable flexibility in the synthesis of multifunctional organic–inorganic hybrid materials with tailorable electronic, optical, and magnetic properties. We report the cation-exchanged synthesis of a diverse range of hollow Mⁿ⁺-phytate (PA) micropolyhedra via the use of hollow Co²⁺-PA polyhedral networks as templates at room temperature. The attributes of the incoming Mⁿ⁺, namely Lewis acidity and ionic radius, control the exchange of the parent Co²⁺ ions and the degree of morphological deformation of the resulting hollow micropolyhedra. New functions can be obtained for both completely and partially exchanged products, as supported by the observation of Ln³⁺ (Ln³⁺=Tb³⁺, Eu³⁺, and Sm³⁺) luminescence from as-prepared hollow Ln³⁺-PA micropolyhedra after surface modification with dipicolinic acid as an antenna. Moreover, Fe³⁺- and Mn²⁺-PA polyhedral complexes were employed as magnetic contrast agents.     

A Phosphine Functionalized β-Diketimine Ligand for the Synthesis of Manifold Metal Complexes

A bis(diphenyl)-phosphine functionalized β-diketimine (PNac-H) was synthesized as a flexible ligand for transition metal complexes. The newly designed ligand features symmetrically placed phosphine moieties around a β-diketimine unit, forming a PNNP-type pocket. Due to the hard and soft donor atoms (N vs. P) the ligand can stabilize various coordination polyhedra. A complete series ranging from coordination numbers 2 to 6 was realized. Linear, trigonal planar, square planar, tetrahedral, square pyramidal, and octahedral coordination arrangements containing the PNac-ligand around the metal center were observed by using suitable metal sources. Hereby, PNac-H or its anion PNac⁻ acts as mono-, bi- and tetradendate ligand. Such a broad flexibility is unusual for a rigid tetradentate system. The structural motifs were realized by treatment of PNac-H with a series of late transition metal precursors, for example, silver, gold, nickel, copper, platinum, and rhodium. The new complexes have been fully characterized by single crystal X-ray diffraction, NMR, IR, UV/Vis spectroscopy, mass spectrometry as well as elemental analysis. Additionally, selected complexes were investigated regarding their photophysical properties. Thus, PNac-H proved to be an ideal ligand platform for the selective coordination and stabilization of various metal ions in diverse polyhedra and oxidation states.