Ellipsoidal space-times,sources for the Kerr metric

The paper develops a systematic derivation of the Kerr metric and its possible sources in a clear geometric manner. It starts with a concise account of previous attempts at constructing an interior Kerr solution. Then a treatment of stationary-axisymmetric spacetimes, specially fitted to the needs of the following analysis, is presented. A new notion of an ellipsoidal space-time is introduced: it is a space-time in which local rest 3-spaces of some observers split naturally into congruences of concentric and coaxial ellipsoids. It is shown that these 3-spaces are natural spaces to consider the ellipsoidal figures of equilibrium. The investigation is carried out in detail for axially symmetric oblate confocal ellipsoids, but possible generalizations are indicated. The Kerr metric is found to be an ellipsoidal space-time of this special kind. Some remarks concerning an (unfound) explicit interior Kerr solution conclude the paper.     

Refinement of the crystal structure of acetylacetonatodicarbonylrhodium(I)

The structure of acetylacetonatodicarbonylrhodium(I), Rh(acac)(CO)₂, has been refined from three-dimensional X-ray diffractometer data. The complex crystallises in the triclinic space groupP¯1 with two molecules in a unit cell of dimensionsa = 6.5189(5),b = 7.7614(8),c = 9.2049(12)Å, = 106.04(1), = 91.15(1), = 100.21(1) °. Full-matrix least-squares refinement, using 1456 independent reflections, has reachedR = 0.038.The rhodium atom has a square-planar coordination with two Rh-O(acac) distances of 2.040 and 2.044Å, and two Rh-C(carbonyl) distances both equal to 1.831Å, the O-Rh-O angle is 90.8 ° and the C-Rh-C angle is 88.9 °. All twelve non-hydrogen atoms are closely planar, with an average deviation of 0.003Å, and a maximum deviation of 0.006Å, from the least-squares plane through the molecule. A second non-crystallographic plane of symmetry bisects the molecule, which therefore has essentiallymm2 (C _2v ) point symmetry. The molecules stack in such a way that the rhodium atoms of neighbouring molecules occupy the two remaining pseudo-octahedral positions, with Rh...Rh distances of 3.253 and 3.271Å.     

Gaussian geminal basis set optimization with crude SCF reference state

Gaussian geminal basis functions for second-order correlation energy calculations according to the Sinanogˇlu method are optimized with reference to rather crude SCF functions. The optimized geminal basis set is then used in a one-step calculation of the correlation energy with respect to the near-Hartree-Fock reference State. The numerical results for the beryllium atom indicate the usefulness of the proposed technique.     

The Lamplighter Group as a Group Generated by a 2-state Automaton, and its Spectrum

We realize the lamplighter group /2 as a group defined by a 2-state automaton. We study the corresponding action of this group on a binary tree and on its boundary. The final goal is the computation for a special system of generators of the spectrum of the Markov (or the random walk) operator which is [–1,1] in this case and of the spectral measure which is a discrete measure concentrated on a dense countable set of points in [–1,1] (a new effect unseen before for Markovian operators on groups which leads to a counterexample to the Strong Atiyah Conjecture). This is done by the computation of spectra of finite-dimensional approximations of the operator and uses an idea of fractalness in a similar way it was used by Bartholdi and Grigorchuk for the computation of the spectra of some branch groups. We also obtain the asymptotic of type e^–1/1–x of the spectral measure in the neighborhood of 1 and show that Følner sets grow exponentially.     

The yellow cake

In this paper we consider the following property:

For every function there are functions

(for ) such that

We show that, despite some expectation suggested by S. Shelah (1997), does not imply . Next, we introduce cardinal characteristics of the continuum responsible for the failure of .