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Equilateral Triangle (equilateral + triangle)
Selected AbstractsA lattice Boltzmann-BGK algorithm for a diffusion equation with Robin boundary condition,application to NMR relaxationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2009A. Hiorth Abstract We present a lattice Boltzmann-BGK (LBGK) algorithm for a diffusion equation together with a Robin boundary condition, which we apply in the case of nuclear magnetic resonance relaxation. The boundary condition we employ is independent of the direction of the wall. This makes the algorithm very suitable for complicated geometries, such as porous media. We discuss the effect of lattice topology by using, respectively, an eight-speed and a four-speed lattice. The numerical algorithm is compared with analytical results for a square and an equilateral triangle. The eight-speed lattice performs well in both cases. The four-speed lattice performs well for the square, but fails in the case of an equilateral triangle. Comparison with a random walk algorithm is also included. The LBGK algorithm presented here can also be used for a convective diffusion problem if the speed of the fluid can be neglected close to the boundary. Copyright © 2008 John Wiley & Sons, Ltd. [source] 5-Amino-2,4,6-tribromoisophthalic acid: the MAD triangle for experimental phasingACTA CRYSTALLOGRAPHICA SECTION C, Issue 5 2009Tobias Beck The title compound, C8H4Br3NO4, shows an extensive hydrogen-bond network. In the crystal structure, molecules are linked into chains by COO,H...O bonds, and pairs of chains are connected by additional COO,H...O bonds. This chain bundle shows stacking interactions and weak N,H...O hydrogen bonds with adjacent chain bundles. The three Br atoms present in the molecule form an equilateral triangle. This can be easily identified in the heavy-atom substructure when this compound is used as a heavy-atom derivative for experimental phasing of macromolecules. The title compound crystallizes as a nonmerohedral twin. [source] The magic triangle goes MAD: experimental phasing with a bromine derivativeACTA CRYSTALLOGRAPHICA SECTION D, Issue 4 2010Tobias Beck Experimental phasing is an essential technique for the solution of macromolecular structures. Since many heavy-atom ion soaks suffer from nonspecific binding, a novel class of compounds has been developed that combines heavy atoms with functional groups for binding to proteins. The phasing tool 5-amino-2,4,6-tribromoisophthalic acid (B3C) contains three functional groups (two carboxylate groups and one amino group) that interact with proteins via hydrogen bonds. Three Br atoms suitable for anomalous dispersion phasing are arranged in an equilateral triangle and are thus readily identified in the heavy-atom substructure. B3C was incorporated into proteinase K and a multiwavelength anomalous dispersion (MAD) experiment at the Br,K edge was successfully carried out. Radiation damage to the bromine,carbon bond was investigated. A comparison with the phasing tool I3C that contains three I atoms for single-wavelength anomalous dispersion (SAD) phasing was also carried out. [source] Unprecedented Stereoselective Synthesis of Catalytically Active Chiral Mo3CuS4 ClustersCHEMISTRY - A EUROPEAN JOURNAL, Issue 5 2006Marta Feliz Dr. Abstract Cluster excision of polymeric {Mo3S7Cl4}n phases with chiral phosphane (+)-1,2-bis[(2R,5R)-2,5-(dimethylphospholan-1-yl)]ethane ((R,R)-Me-BPE) or with its enantiomer ((S,S)-Me-BPE) yields the stereoselective formation of the trinuclear cluster complexes [Mo3S4{(R,R)-Me-BPE}3Cl3]+ ([(P)- 1]+) and [Mo3S4{(S,S)-Me-BPE}3Cl3]+ ([(M)- 1]+), respectively. These complexes posses an incomplete cuboidal structure with the metal atoms defining an equilateral triangle and one capping and three bridging sulfur atoms. The P and M symbols refer to the rotation of the chlorine atoms around the C3 axis, with the capping sulphur atom pointing towards the viewer. Incorporation of copper into these trinuclear complexes affords heterodimetallic cubane-type compounds of formula [Mo3CuS4{(R,R)-Me-BPE}3Cl4]+ ([(P)- 2]+) or [Mo3CuS4{(S,S)-Me-BPE}3Cl4]+ ([(M)- 2]+), respectively, for which the chirality of the trinuclear precursor is preserved in the final product. Cationic complexes [(P)- 1]+, [(M)- 1]+, [(P)- 2]+, and [(M)- 2]+ combine the chirality of the metal cluster framework with that of the optically active diphosphane ligands. The known racemic [Mo3CuS4(dmpe)3Cl4]+ cluster (dmpe=1,2-bis(dimethylphosphanyl)ethane) as well as the new enantiomerically pure Mo3CuS4 [(P)- 2]+ and [(M)- 2]+ complexes are efficient catalysts for the intramolecular cyclopropanation of 1-diazo-5-hexen-2-one (3) and for the intermolecular cyclopropanation of alkenes, such as styrene and 2-phenylpropene, with ethyl diazoacetate. In all cases, the cyclopropanation products were obtained in high yields. The diastereoselectivity in the intermolecular cyclopropanation of the alkenes and the enantioselectivity in the inter- or intramolecular processes are only moderate. La reacción de escisión de la fase polimérica {Mo3S7Cl4}ncon la fosfina quiral (+)-1,2-bis[(2R,5R)-2,5-(dimetilfosfolan-1-il)]etano, (R,R)-Me-BPE, o con su enantiómero, (S,S)-Me-BPE, conduce a la formación estereoselectiva de los complejos clúster trinucleares [Mo3S4(R,R -Me-BPE)3Cl3]+([(P)- 1]+) y [Mo3S4(S,S -Me-BPE)3Cl3]+([(M)- 1]+), respectivamente. Estos complejos poseen una estructura de cubo incompleto, dónde los átomos metálicos definen un triángulo equilátero, con un azufre unido a tres átomos de molibdeno y tres azufres puente. Los símbolos P y M hacen referencia a la rotación de los átomos de cloro alrededor del eje C3, con el azufre apuntado dirigido hacia el observador. La incorporación de cobre a estos complejos trinucleares conduce a compuestos heterodimetálicos con estructura tipo cubano de fórmula [Mo3CuS4(R,R -Me-BPE)3Cl4]+([(P)- 2]+) y [Mo3CuS4(S,S -Me-BPE)3Cl4]+([(M)- 2]+) donde la quiralidad del precursor trinuclear se mantiene en el producto final. Los complejos catiónicos [(P)- 1]+, [(M)- 1]+, [(P)- 2]+y [(M)- 2]+combinan la quiralidad del esqueleto clúster con la de los ligandos difosfina. El clúster racémico [Mo3CuS4(dmpe)3Cl4]+(dmpe=1,2-bis(dimetilfosfino)etano), así como los complejos Mo3CuS4 enantioméricamente puros [(P)- 2]+o [(M)- 2]+son catalizadores eficaces para la reacción de ciclopropanación intramolecular de 1-diazo-5-hexen-2-ona (3) y para la ciclopropanación intermolecular de alquenos, estireno y 2-fenilpropeno, con etil diazoacetato. En todos los casos los productos de ciclopropanación se obtienen con rendimientos elevados. La diastereoselectividad en la ciclopropanación intermolecular de alquenos y la enantioselectividad en los procesos tanto inter- como intramoleculares son únicamente moderadas. [source] Adaptive Implicit Surface Polygonization Using Marching TrianglesCOMPUTER GRAPHICS FORUM, Issue 2 2001Samir Akkouche This paper presents several improvements to the marching triangles algorithm for general implicit surfaces. The original method generates equilateral triangles of constant size almost everywhere on the surface. We present several modifications to adapt the size of the triangles to the curvature of the surface. As cracks may arise in the resulting polygonization, we propose a specific crack-closing method invoked at the end of the mesh growing step. Eventually, we show that the marching triangles can be used as an incremental meshing technique in an interactive modeling environment. In contrast to existing incremental techniques based on spatial subdvision, no extra data-structure is needed to incrementally edit skeletal implicit surfaces, which saves both memory and computation time. [source] Di-2-pyridyl Ketone Oxime in Zinc Chemistry: Inverse 12-Metallacrown-4 Complexes and Cationic Pentanuclear ClustersEUROPEAN JOURNAL OF INORGANIC CHEMISTRY, Issue 10 2005Maria Alexiou Abstract The use of di-2-pyridyl ketone oxime (Hpko)/X, "blends" (X, = PhCO2,, N3,, NCO,, acac,, NCS,) in zinc chemistry yields neutral tetranuclear and cationic pentanuclear clusters. Various synthetic procedures have led to the synthesis of compounds [Zn4(OH)2(O2CPh)2(pko)4]·3MeCN (1·3MeCN), [Zn4(OH)2(N3)2(pko)4]·4DMF (2·4DMF), [Zn4(OH)2(NCO)2(pko)4]·3DMF·H2O (3·3DMF·H2O), [Zn4(OH)2(acac)2(pko)4]·4CH2Cl2 (4·4CH2Cl2), [Zn5Cl2(pko)6][ZnCl(NCS)3]·2.5H2O·1.5MeOH (5·2.5H2O·1.5MeOH) and [Zn5(NCS)2(pko)6(MeOH)][Zn(NCS)4]·2.5H2O·MeOH (6·2.5H2O·MeOH). The structures of the six complexes have been determined by single-crystal X-ray crystallography. The tetranuclear molecules of 1,4 lie on a crystallographic inversion centre and have an inverse 12-metallacrown-4 topology. Two triply bridging hydroxides are accommodated in the centre of the metallacrown ring. The pko, ligands form a propeller configuration that imposes absolute stereoisomerism with , and , chirality. Two metal ions are in distorted O2N4 octahedral environments, whereas the rest are in severely distorted tetrahedral or trigonal bipyramidal environments. The five Zn ions of the cations of 5 and 6 are held together by six pko, ligands which adopt three different coordination modes; the chloro (5) and isothiocyanato (6) ligands are terminal. The five Zn ions define two nearly equilateral triangles sharing a common apex, and the novel Zn5 topology can be described as two "collapsed" 9-metallacrown-3 structures sharing a common Zn apex. Besides the pentanuclear cations, the structures of 5 and 6 contain slightly distorted tetrahedral [ZnCl(NCS)3]2, and [Zn(NCS)4]2, ions, respectively, with the isothiocyanato ligands binding the metal ion in a virtually linear fashion. (© Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005) [source] Dense quasicrystalline tilings by squares and equilateral trianglesACTA CRYSTALLOGRAPHICA SECTION A, Issue 1 2010Michael O'Keeffe Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses. [source] | ||||||||||