Über asymmetrische diophantische approximationen

For irrational numbers θ define α(θ) = lim sup{1/(q(p ? qθ))|p ∈ $\text{Z}$, q ∈ $\text{N}$, p ? qθ > 0} and α(θ) = 0 for rationals. Put $\text{α}\text{(θ) = max{α(θ), α(?0)}}$. Then $\text{U}$ = α($\text{R}$β$\text{Q}$) is an asymmetric analogue to the Lagrange spectrum $\text{U}\text{=}\text{α}\text{(RβQ)}$. Our results concerning $\text{U}$ partly contrast the known properties of $\text{U}$. In fact, $\text{U}$ is a perfect set, each element of which is a condensation point of the spectrum and has continuously many preimages. $\text{U}$ is the closure of its rational elements and of its elements of the form p√m (p ∈ $\text{Q}$), as well. The arbitrarily well approximable numbers form a G_δ-set of 2. category. One has, roughly speaking, $\text{α}\text{→ ∞}$ for α → 1. Finally, the well-known Markov sequence which constitutes the lower Lagrange and Markov spectrum is proved to be a (small) subset of $\text{U}$?[√5,3).     

Unicity of best one-sidedL 1-approximations

Summary This paper deals with the problem of uniqueness in one-sidedL ₁-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL ₁-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.     

Einfluss funktioneller Gruppen auf die Massenspektren von 17β-Hydroxysteroiden

The different influence of functional groups on fragmentation reactions is demonstrated by the example of 17β-hydroxy-5α-androstanes. In 17β-hydroxy-5α-androstane the tendency of steroids to be cleaved by rupture of the C-13/C-17 bond under production of a charged centre at C-13 is only enhanced by the hydroxyl group. This influence of the hydroxyl group only predominates as long as substituents are absent which need less energy for ionization than is necessary to ionize the C-13/C-17 bond. Even the presence of a double bond may change the fragmentation pattern completely. If, however, an additional alkyl group is located at C-17, ionization occurs with high probability by elimination of an electron from the OH group connected to C-17. Therefore, other main degradation reactions occur than in molecular ions, having the charge located at C-13. In such compounds even the dominant influence of a conjugated system on the fragmentation reactions is considerably reduced. By etherfication of the hydroxyl group this effect is still very much increased. In the course of this investigation some not yet known degradation reactions, rather typical for the presence of certain groups in the steroid skeleton, were found. These are discussed briefly.     

Semiconductor-laser amplifier as an optically controlled switch

The application of a semiconductor-laser amplifier with reduced facet reflectivity as all-optical switch is described. An optical data signal is switched on and off with an extinction ratio of 10 dB when an optical control signal is turned on and off. The two optical signals are polarized perpendicularly to each other. The device was operated up to 100 MHz.Nanjing Institute of Posts and Telecommunication Nanjing, PR China     

Solid-State Photochemistry—A Method of Generating Unusual Valence States

High-energy radiation can give rise to pairs of complementary defects in nonmetallic solids by the transfer of electrons between various types of atoms. These “color centers” which are generally paramagnetic, can usually be described as unusual valence states of an element. They are destroyed by heating and in most cases regenerated by renewed irradiation. In a heteropolar solid the formation of color centers usually leads to cancellation of point charges due to foreign ions of other valence or to vacancies. This is shown by the examples of kunzite, brazilianite, smoky quartz, and citrine; the most important methods for the structural elucidation of color centers are also described. Application of the principle of charge balance opens up possibilities for the production of unusual valence states, e.g. Al²⁺, F^2?, Fe⁴⁺, and O^?. Moreover, the type of the color center often permits far-reaching conclusions to be drawn about the defect structure of real crystals, which could hardly be clarified in other ways.     

Fast-n abelsche Gruppen

Ohne Zusammenfassung