If a i .erential one-map between normed spaces is continuous on a star-like open set an is i .erentiable possibly except of one point then it has a primitive if an only if its erivative as a bilinear form is symmetric at every point where it exists.A generalization of the theorem on the existence of the potential operator is also obtained.The proof is based on the idea of the Goursat Lemma.
Authors:B. Handke, T. Ślęzak, M. Kubik and J. Korecki
Thin magnetite films grown on cleaved MgO (100) substrates were studied by in-situ conversion electron Mössbauer spectroscopy (CEMS) in the temperature range of 80 K–300 K. The film growth was monitored by low energy electron diffraction revealing a perfect epitaxy in the whole thickness range from 430 nm down to 3 nm. CEMS measurements proved an ideal magnetite stoichiometry of the 430 nm thick film reflected both in the hyperfine pattern and in the Verwey transition. The thinner films showed strong deviation from the bulk properties, which was attributed to the modification of composition at the MgO/Fe3O4 interface, at which a magnesium rich phase is supposed based on CEM spectra. This phase, which was not observed previously, could be detected thanks to highly sensitive low-temperature measurements.