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We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points
A detailed structural analysis of a Langmuir-Blodgett (LB) multilayer composed of a polyelectrolyte-amphiphile complex (PAC) is presented. The PAC is self-assembled from metal ions, ditopic bis-terpyridines, and amphiphiles. The vertical structure of the LB multilayer is investigated by X-ray reflectometry. The multilayer has a periodicity of 57 A, which corresponds to an architecture of flat lying metallo-supramolecular coordination polyelectrolyte (MEPE) rods and upright-standing amphiphiles (dihexadecyl phosphate, DHP). In-plane diffraction reveals hexagonal packing of the DHP molecules. Using extended X-ray absorption fine structure (EXAFS) experiments, we prove that the central metal ion is coordinated to the terpyridine moieties in a pseudo-octahedral coordination environment. The Fe-N bond distances are 1.82 and 2.0 angstrom, respectively. Temperature resolved measurements indicate a reversible phase transition in a temperature range up to 55 degrees C. EXAFS measurements indicate a lengthening of the average Fe-N bond distance from 1.91 to 1.95 angstrom. The widening of the coordination cage upon heating is expected to lower the ligand field stabilization, thus giving rise to spin transitions in these composite materials