Measuring magnetic fields in plasmas by means of light scattering

The theory of light scattering in plasmas containing a magnetic field yields the special case of modulated scattering spectra. The modulation frequency is governed by the field in the plasma and is equal to the electron cyclotron frequency. In this investigation magnetic fields in a plasma were determined by a laser scattering experiment. The experimental data were: electron densityn _e=10¹⁶cm^?3, electron temperatureT _e=3.2 eV, scattering angle θ=90 °, scattering parameter α=0.6, and a maximum field in the plasma of 125 kG. The spectrum measured at the maximum magnetic field was modulated with 3.6 × 10¹¹ Hz. In scattering experiments with a field reduced by about 20% the observed modulation frequency was 2.8 × 10¹¹ Hz. A thermal spectrum with a smooth profile was found when no field was present in the plasma. Applying the theory of cyclotron modulated spectra one obtains from the scattering experiment magnetic fields of 128, 100, and 0 kG. Within the experimental accuracy these values agree well with the fields determined by means of magnetic probes. Other possible interpretations of the measured deviations from thermal spectra (modulation with the plasma frequency or additional cold electron components in the plasma) are discussed, but they afford no explanation. This experiment has domonstrated that magnetic fields in plasmas can be measured locally and almost without disturbance by means of light scattering.     

Dynamic rating of European football teams

^** Formerly Knorr-Held. Email: held{at}stat.uni-muenchen.de In this paper we apply and extend recently proposed methodsfor the dynamic analysis of pairwise comparison data to Europeanfootball teams. Our statistical model is based on the cumulativelogistic link model with time-changing parameters for the strengthof each team. We jointly analyse the results from the five topEuropean leagues from 1996 to 2001 and all international matchesbetween teams from these leagues. We introduce weights for internationalmatches and also allow for a different size of the home teamadvantage in the different leagues. We suggest that the resultsfrom such an analysis may be taken as an alternative to theUEFA coefficient, which is currently used to determine the numberof teams from each league to take part in the European footballcontests.     

Holomorphic Extensions of Whitney Jets

Let φ be a Whitney jet on a closed set F ? ?. By Whitney’s extension theorem φ can be extended to an infinitely differentiable function f on ? which is real analytic on ? F. The main purpose of this article is to show that f can be chosen in such a way that f¦_?F has a holomorphic continuation to the open set (? F) × i? ? ?. In the special case that F is a compact interval or a single point we can even achieve that f¦_?F has a holomorphic continuation to all of $\hat {\rm C}\setminus F$ . In particular, this implies an improvement of the well-known theorem of E. Borel. We also investigate the question when such extensions are given by a so-called extension operator.     

Frequent hypercyclicity,chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. In contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.     

Extension Operators with Analytic Values

If F ? ? is a closed set such that the space of all Whitney jets on F admits an extension operator then there exists such an extension operator whose values are holomorphic in ?F if and only if ?F is compact. In the case F is a compact set, there is even an extension operatorfor which the extensions are holomorphic in (? ∪ {∞})F.     

Tame linear extension operators for smooth Whitney functions

For a compact set K⊆Rd we present a rather easy construction of a linear extension operator E:E(K)→C^∞(Rd) for the space of Whitney jets E(K) which satisfies linear tame continuity estimates , where ‖⋅s_‖ denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milman holds for every s>r on K. In particular, E(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s?1.     

Numerical Investigation of Particulate Flow over a Backward‐Facing Step

Flow separation and recirculation caused by a sudden expansion in the channel geometry in the form of a backwardfacing step (BFS) appear in numerous practical applications. Additionally, BFS flow has been used as a generic test case to study fundamental flow properties, such as separation or re‐attachment. In the present work, BFS flow laden with dispersed particles is investigated by numerical simulations using a spectral element method [1]. The motion of the dispersed particles is computed by Lagrangian particle tracking. In a first step, only the influence of the flow on the particles is accounted for, while possible effects of the particle motion on the flow are neglected. Spatial distribution of the particles is investigated, and effects of different wall‐particle interaction models on the computational results are examined.     

Subharmonic transition to turbulence in channel flow

A direct numerical simulation of subharmonic transition to turbulence in channel flow has been performed. The stages of primary and secondary instability have been identified in the results leading to a staggered pattern of A-shaped vortices. The associated staggered symmetry has been found to persist up to rather late stages of the breakdown process. This symmetry rapidly disappears in the final stage of transition and a developed turbulent flow is attained. A pronounced difference in the development between the two channel halves is observed which is consistent with the predictions of secondary instability theory.