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991.
Christian Bär 《Communications in Mathematical Physics》1997,188(3):709-721
We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension
2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that
the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface
and a singular set of lower dimension. We also see that the nodal set of a Δ-harmonic differential form on a closed manifold
has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal.
Received: 28 October 1996 / Accepted: 3 March 1997 相似文献
992.
Christian Richter 《Elemente der Mathematik》1998,53(4):149-158
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Christian Steglich 《Results in Mathematics》1995,27(1-2):188-193
While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M n → An+1, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry. 相似文献
996.
Ingrid Lukkari Jaromir Ruzicka Gary D. Christian 《Fresenius' Journal of Analytical Chemistry》1993,346(6-9):813-818
Summary Automated methods for the determination of ammonium and ammonia are reviewed, and techniques based on gas diffusion using a semi-permeable membrane were selected for the determination of both total ammonium-nitrogen and free ammonia in fermentation samples. A simple and robust instrument based on sequential injection analysis (SIA) consisting of a piston pump and two selector valves was used. Two different methods of ammonia detection, the Berthelot method and detection using pH-indicators, have been evaluated and compared. The indicator method showed higher reproducibility and the range of determination could easily be adjusted to match the concentrations of the samples. The range of determination could be adjusted from 0.05 mmol/l to 350 mmol/l, depending on choice of acceptor solution. This method has been evaluated with fermentation medium samples and tested on-line in a yeast fermentation process.Dedicated to Professor Dr. Wilhelm Fresenius on the occasion of his 80th birthday 相似文献
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