Complementary pivoting and the Hopf Degree Theorem

We describe a combinatorial process for the extension of triangulations and integer labeling functions. Together with a discussion of topological degree via completely labeled simplices our result yields an elementary proof of the Hopf Degree Theorem. A further application of our techniques will provide a regularization result, similar to theorems of H. Hopf and B. O'Neill.     

On spherical spline interpolation and approximation

Spherical spline functions are introduced by use of Green's surface functions with respect to the (Laplace-)Beltrami operator of the (unit) sphere. Natural (spherical) spline functions are used to interpolate data discretely given on the sphere. A method is presented that allows the smoothing of irregularities in measured values or experimental data. Extensions of Peano's theorem and Sard's theory of best approximation to the spherical case are given by integral formulas. Schoenberg's theorem is transcribed into spherical nomenclature.     

Euclidean Green's functions for Jaffe fields

We extend the axioms for Euclidean Green's functions recently proposed by Osterwalder and Schrader to Jaffe fields.     

A Simple Approximate Formula for the Aspect Ratio of Oblate Particles

A simple approximative formula is derived, which can be used to quantify the shape of oblate particles or an average shape of the corresponding particle system, when the results of sedimentation analysis (Stokes equivalent diameters) are known and results from either microscopic image analysis (assuming stable orientation, i.e., with the plane perpendicular to the direction of observation) or laser diffraction (assuming random orientation) are available for the same sample. In the latter case Cauchy's stereological theorem is applied to account for random orientation. Furthermore, it is shown that for sufficiently large aspect ratios, this formula is very close to the well‐known Jennings‐Parslow relation and can replace this more complicated expression in many practical cases, e.g., in the routine characterization of ceramic raw materials (kaolins and oxide or non‐oxide platelet powders).     

The Telluraboranes closo-TeB5Cl5 and closo-TeB11Cl11 with Exceptionally Long Body Diagonals: Synthetic and Bonding Motifs for Innovative Octahedral and Icosahedral Geometries

Six-vertex closo-TeB₅Cl₅ ( 1 ) and twelve-vertex closo-TeB₁₁Cl₁₁ ( 2 ) telluraboranes have been prepared via co-pyrolysis of B₂Cl₄ with TeCl₄ in vacuo at temperatures between 360 °C and 400 °C. Both compounds are sublimable, off-white solids, and they have been characterized by one- and two-dimensional ¹¹B NMR and high-resolution mass spectroscopy. Both ab initio/GIAO/NMR and DFT/ZORA/NMR computations support octahedral and icosahedral geometries for 1 and 2 , respectively, as expected due to their closo-electron counts. The octahedral structure of 1 has been confirmed by single-crystal X-ray diffraction on an incommensurately modulated crystal. The corresponding bonding properties have been analyzed in terms of the intrinsic bond orbital (IBO) approach. 1 is the first example of a polyhedral telluraborane with a cluster size smaller than 10 vertices.