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41.
42.
Ulrich Brehm 《Geometriae Dedicata》1990,33(1):59-76
We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces P
n
and P
n
(n2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces H
n
and H
n
(n2). 相似文献
43.
Ulrich Koschorke 《manuscripta mathematica》1988,61(4):383-415
The geometry of two types of link homotopy invariants of a link map f:SpSqSm is discussed. The first one is the -invariant which greatly generalizes the classical notion of linking number. The second one, the -invariant, is closely related to the linking behaviour of f|sp with only the double point set of f|Sq, and therefore measures (to some extend) the obstruction to embedding Sq. These invariants are related by a Hopf invariant homomorphism. In many cases link maps are classified up to link homotopy here, and a setting is provided e.g. for future injectivity results for . Also the image of is studied, yielding an interesting double filtration of stable homotopy groups of spheres. 相似文献
44.
45.
Hlushkou D Seidel-Morgenstern A Tallarek U 《Langmuir : the ACS journal of surfaces and colloids》2005,21(13):6097-6112
We present a numerical scheme for analyzing steady-state isothermal electroosmotic flow (EOF) in three-dimensional random porous media, involving solution of the coupled Poisson, Nernst-Planck, and Navier-Stokes equations. While traditional finite-difference methods were used to resolve the Poisson-Nernst-Planck problem, the (electro)hydrodynamics has been addressed with high efficiency using the lattice-Boltzmann method. The developed model allows simulation of electrokinetic transport under most general conditions, including arbitrary value and distribution of electrokinetic potential at the solid-liquid interface, electrolyte composition, and pore space morphology. The approach provides quantitative information on a spatial distribution of simulated velocities. This feature was utilized to characterize EOF fields in regular and random, confined and bulk packings of hard (i.e., impermeable, nonconducting) spheres. Important aspects of pore space morphology (sphere size distribution), surface heterogeneity (mismatch in electrokinetic potentials at confining wall and sphere surface), and fluid phase properties (electrical double layer thickness) were investigated with respect to their influence on the EOF dynamics over microscopic and macroscopic spatial domains. Most important is the observation of a generally nonuniform pore-level EOF velocity profile in the sphere packings (even in the thin double layer limit) which is caused by pore space morphology and which is in contrast to the pluglike velocity distribution in a single, straight capillary under the same conditions. 相似文献
46.
47.
VNCl2(Pyridine)2; Synthesis, I. R. Spectrum, and Crystal Structure VNCl2(Pyridine)2 is formed from the cyclothiazeno-vanadium(IV) complex [VCl(N3S2)(Pyridine)2]2 · 2 CH2Cl2 in boiling toluene in form of brown-red, hydrolysis sensitive crystal. It was characterized by its I.R. spectrum and an X-ray crystal structure determination. VNCl2(Pyridine)2crystallizes in the orthorhombic space group Pccn with four formula units per unit cell (668 observed independent reflexions, R = 0.055). Lattice constants: a = 1550, b = 924, c = 832 pm. Monomer molecules are situated on twofold rotation axes. They are stacked along the V?N axis to form columns V?N…V. The short VN bond of 160 pm corresponds to a triple bond, whereas the V…N distance of 256 pm indicates a weak interaction. The V? N(Pyridine) and V? Cl bond lengths are 213.0 and 233.5 pm, respectively. 相似文献
48.
Häussermann U Amerioun S Eriksson L Lee CS Miller GJ 《Journal of the American Chemical Society》2002,124(16):4371-4383
This work presents a detailed, combined experimental and theoretical study on the structural stability of s-p bonded compounds with the BaAl4 structure type (space group I4/mmm, Z = 2) as part of a broad program to investigate the complex questions of structure formation and atomic arrangements in polar intermetallics. From ab initio calculations employing pseudopotentials and a plane wave basis set, we extracted optimized structural parameters, binding energies, and the electronic structure of the systems AeX(III)4, AeX(II)2X(IV)2, AeX(II)2X(III)2 (Ae = Ca, Sr, Ba; X(II) = Mg, Zn; X(III) = Al, Ga; X(IV) = Si, Ge). For all systems we found a pronounced pseudo-gap in the density of states separating network X42- bonding from antibonding electronic states that coincides with the Fermi level for an electron count of 14 electrons per formula unit, the optimum value for stable BaAl4-type polar intermetallics. However, the synthesis and structural characterization (from X-ray single crystal and powder diffraction data) of the new compounds AeZn2-Al2+, AeZn2-deltaGa2+delta (Ae = Ca, Sr, Ba; delta = 0-0.2) and AeMg0.9Al3.1, AeMg1.7Ga2.3 (Ae = Sr, Ba) manifested that electron deficiency is rather frequent for BaAl4-type polar intermetallics. The site preference for different "X" elements in the ternary systems was quantified by calculating "coloring energies", which, for some systems, was strongly dependent on the size of the electropositive Ae component. The Ae2+ cations decisively influence the nearest neighbor distances in the encapsulating polyanionic networks X4(2-) and the structures of these networks are surprisingly flexible to the size of the Ae component without changing the overall bonding picture. A monoclinically distorted variant of the BaAl4 structure occurs when the cations become too small for matching the size of encapsulating X4(2-) cages. An even larger size mismatch leads to the formation of the EuIn4 structure type. 相似文献
49.
50.