Coupling of the GB set property for ergodic averages

Let (Y,,,T) be an ergodic dynamical system. LetA be an nonempty subset ofL ²() such that , whereA=sup{||sȒt||₂ ,s, tA} andN(A, u) is the smallest number ofL ²()-open balls of radiusu, centered inA, enough to coverA. Let . We prove as a consequence of a more general result, thatC(A) is aGB subset ofL ²().     

How to improve transmissive optics in the very far infrared?

Conclusion We have presented the possibility to make optical coatings on quartz in the FIR and we have got a transmission coefficient as high as 97% at room temperature. It will rise probably close to 100% at liquid helium temperature where the quartz two-phonon difference processes are frozen. The use of polyethylene or TPX is also possible to make plates and lenses. The reflectivity losses are smaller because the refractive index is much smaller, but they cannot be further reduced by any coating because there is no transparent material with n1.2 in the FIR. These lenses are very useful in the FIR (except around 60 cm^–1 if they are made of polyethylene). TPX is quite transparent over the whole FIR range, and also in the visible with nearly the same refractive index as in the FIR. Alignments can be made with visible light, and still work in the FIR.     

Modular curves and Poncelet polygons

Parts of the results and the essential techniques of this note are taken from the Erlangen thesis (1991) of the second author. They were circulated as Nr. 122 of Schriftenreihe Komplexe Mannigfaltigkeiten. Our research was supported by DFG grant Ba 423/3-3 and the European Science Project Geometry of Algebraic Varieties SCI-0398-C(A)