Lie Algebras and Growth in Branch Groups

We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov dimension $\log3/\log(1+\sqrt2)$. We then draw a general result relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call "parabolic space", namely the quotient of the group by the stabilizer of an infinite ray. The growth of the group is bounded from below by the growth of its graded group ring, which connects to the growth of the Lie algebra by a product-sum formula, and the growth of the parabolic space is bounded from below by the growth of the Lie algebra. Finally we use this information to explicitly describe the normal subgroups of the "Grigorchuk group". All normal subgroups are characteristic, and the number of normal subgroups of index $2^n$ is odd and is asymptotically $n^{\log_2(3)}$

Complex morphology and functional dynamics of vital murine intestinal mucosa revealed by autofluorescence 2-photon microscopy

The mucosa of the gastrointestinal tract is a dynamic tissue composed of numerous cell types with complex cellular functions. Study of the vital intestinal mucosa has been hampered by lack of suitable model systems. We here present a novel animal model that enables highly resolved three-dimensional imaging of the vital murine intestine in anaesthetized mice. Using intravital autofluorescence 2-photon (A2P) microscopy we studied the choreographed interactions of enterocytes, goblet cells, enteroendocrine cells and brush cells with other cellular constituents of the small intestinal mucosa over several hours at a subcellular resolution and in three dimensions. Vigorously moving lymphoid cells and their interaction with constituent parts of the lamina propria were examined and quantitatively analyzed. Nuclear and lectin staining permitted simultaneous characterization of autofluorescence and admitted dyes and yielded additional spectral information that is crucial to the interpretation of the complex intestinal mucosa. This novel intravital approach provides detailed insights into the physiology of the small intestine and especially opens a new window for investigating cellular dynamics under nearly physiological conditions

Optische Spektroskopie an hochgeladenen Bismut-Ionen und Konstruktion eines hochauflösenden VUV-Gitterspektrometers

Due to the presence of 4p and 4d open shells, the electronic structure of highly charged bismuth ions is particularly di_cult to calculate. Since practically no experimental data for these ions are available, the optical spectrum of Bi ions in charge states q = 36 to 55 was systematically investigated. The ions were produced with the Heidelberg Electron Beam Ion Trap, and the spectra observed by means of a Czerny-Turner spectrometer. A total of 25 previously unknown spectral lines were found, and their wavelengths determined with a relative accuracy of 170 ppm and better.Furthermore, in the second part of this work a spectrometer for the VUV range was designed and built. The instrument was subsequently used in two beamtimes at the FLASH. The aim of the experiments was to study the laser activity of atomic transitions in xenon and krypton after inner-shell excitation with FEL pulses. The instrument performed excellently, achieving a very high resolution which will enable better determination of the properties of both the FEL and the atomic lasers under study

On the normal basis theorem

The normal basis theorem is a fundamental result in Galois theory. For infinite fields, textbooks and monographs usually refer to a proof given by Artin in 1948. For finite fields, a completely different argument is commonly used. We give two short proofs of the normal basis theorem which work without this distinction.They build on Dedekind’s theorem on the linear independence of Galois automorphisms, and on the Krull–Schmidt theorem. The rest is elementary linear algebra. Both proofs are inspired by but simpler than the one given by Deuring in 1932

Fine-structure investigations in highly charged ions using spectroscopy in the vacuum ultraviolet regime

Spectroscopy of the fine structure of lithium-like ions allows for some of the best precision tests of quantum electrodynamics (QED) calculations. In this work, the fine-structure doublet of lithium-like nitrogen, oxygen, neon, and argon ions were studied. For neon, the results are of comparable accuracy to the best laboratory data found in literature. The here reported values for argon are the first laboratory results and more accurate than those of previous solar observations, and in good agreement with recent ab initio full-scale QED calculations. A related subject is the search for a possible variation of the fine-structure constant α. Two highly forbidden transitions in neodymium-like iridium are among the best-suited ones for this search. In a systematic study, for the final determination of the upper and lower levels, 68 previously unknown emission lines were found and assigned to this species, almost doubling the number of known transitions. For these purposes, a normal-incidence spectrometer was developed. A high-performance electron beam ion trap (and source) was co-developed and commissioned for TRIUMF (Vancouver); while record-breaking specifications were achieved with it, the reported vacuum ultraviolet (VUV) work was carried out using a similar system at MPIK Heidelberg