Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Transplantation harmonique,transplantation par modules,et théorèmes isopérimétriques

Summary The “harmonic transplantation” allows to extend some isoperimetric theorems, so far proved by conformal mapping, to higher connectivity and to higher dimensions; for the first eigenvalue λ₁ of a membrane, it again can give only upper bounds.—The “transplantation by moduli” is much more flexible; for example, it leads to a simple one-dimensional interpretation of the Rayleigh-Faber-Krahn theorem.      

Scattering theory of Kondo mirages and observation of single Kondo atom phase shift

We explain the origin of the Kondo mirage seen in recent quantum corral scanning tunneling microscope experiments with a scattering theory of electrons on the surfaces of metals. Our theory, combined with experimental data, provides a direct observation of a single Kondo atom phase shift. The Kondo mirage observed at the empty focus of an elliptical quantum corral is shown to arise from multiple electron bounces off the corral wall adatoms. We demonstrate our theory with direct quantitive comparison to experimental data.     

Isoperimetrische Ungleichungen für elastisch gebundene Membranen auf Dreiseiten und auf Vierseiten

Summary Some isoperimetric inequalities for vibrating membranes are extended to an elastically supported boundary, using conformal transplantation. A further inequality for fixed-free boundaries is given in section 3.