Dirac—Fock one-centre calculations. The molecules BH,AlH, GaH,InH and TlH

Relativistic and non-relativistic Hartree—Fock calculations including np_{$\text{1}\text{2}$} and np_{$\text{3}\text{2}$} orbitals in the valence MO are reported for the molecules BH, AlH, GaH, InH and TlH. The relativistic contraction of the bond length reaches 7.0% for TlH. Relativistic effects make the chemical bond stronger for AlH, GaH and InH and weaker for TlH. The calculated force constants and dissociation energies are in reasonable agreement with experiment. A change from npσ towards np_{$\text{1}\text{2}$} bonding takes place along the series. For Tl the MO is only 39% p_{$\text{3}\text{2}$} as compared with the non-relativistic value of 67%. This probably explains the dominant monovalency of thallium.     

Ab initio study of bonding trends for f0 actinide oxyfluoride species

 Fully relativistic, four-component Dirac–Fock calculations and quasirelativistic pseudopotential calculations at different ab initio levels are used to study the bonding trends among the naked, triatomic [OAnO] ^q+ groups or the oxyfluorides [AnO _n F _m ] ^q with f ⁰ configurations. The triatomic f ⁰ series is suggested to range from the bent ThO₂ via the linear OPaO⁺ to at least NpO₂ ³⁺, a possible new gas-phase species. The neutral oxyfluoride molecules include the experimentally unknown NpO₂F₃ and PuO₂F₄. The latter is a candidate for the so far unknown oxidation state Pu(VIII), which is found to lie considerably above Pu(VI), but to be locally stable. Their all-oxygen isoelectronic analogues are NpO₅ ³⁻, known in the solid state, and the unknown PuO₆ ⁴⁻. Further possible candidates for Pu(VIII) are PuO₄(D _4h ) and the cube-shaped PuF₈(O _h ). Isoelectronic UF₈ ²⁻ is calculated to be D _4d , in agreement with experiment. Received: 18 May 2001 / Accepted: 21 June 2001 / Published online: 11 October 2001     

Sobolev embeddings, extensions and measure density condition

There are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Ω, in any of all the possible cases, then Ω satisfies the measure density condition. The second main result, Theorem 5, provides several characterizations of the Wm,p-extension domains for 1<p<∞. As a corollary we prove that the property of being a W1,p-extension domain, 1<p?∞, is invariant under bi-Lipschitz mappings, Theorem 8.     

Energy‐dispersive diffraction with synchrotron radiation and a germanium detector

The response of an intrinsic Ge detector in energy‐dispersive diffraction measurements with synchrotron radiation is studied with model calculations and diffraction from perfect Si single‐crystal samples. The high intensity and time‐structure of the synchrotron radiation beam leads to pile‐up of the output pulses, and the energy distribution of the pile‐up pulses is characteristic of the fill pattern of the storage ring. The pile‐up distribution has a single peak and long tail when the interval of the radiation bunches is small, as in the uniform fill pattern, but there are many pile‐up peaks when the bunch distance is a sizable fraction of the length of the shaping amplifier output pulse. A model for the detecting chain response is used to resolve the diffraction spectrum from a perfect Si crystal wafer in the symmetrical Laue case. In the 16‐bunch fill pattern of the ESRF storage ring the spectrum includes a large number of `extra reflections' owing to pile‐up, and the model parameters are refined by a fit to the observed energy spectrum. The model is used to correct for the effects of pile‐up in a measurement with the 1/3 fill pattern of the storage ring. Si reflections (2h,2h,0) are resolved up to h = 7. The pile‐up corrections are very large, but a perfect agreement with the integrated intensities calculated from dynamical diffraction theory is achieved after the corrections. The result also demonstrates the convergence of kinematical and dynamical theories at the limit where the extinction length is much larger than the effective thickness of the perfect crystal. The model is applied to powder diffraction using different fill patterns in simulations of the diffraction pattern, and it is demonstrated that the regularly spaced pile‐up peaks might be misinterpreted to arise from superlattices or phase transitions. The use of energy‐dispersive diffraction in strain mapping in polycrystalline materials is discussed, and it is shown that low count rates but still good statistical accuracy are needed for reliable results.     

A big Picard theorem for quasiregular mappings into manifolds with many ends

We study quasiregular mappings from a punctured Euclidean ball into -manifolds with many ends and prove, by using Harnack's inequality, a version of the big Picard theorem.

     

Quantum theory of measurement and the polar decomposition of an interaction

The measurement-theoretic content of the polar decomposition of an interaction is analyzed. It is shown that the polar decomposition arises exactly from the strong correlation premeasurement of a discrete physical quantity.     

Unitary and Antiunitary Operators Mapping Vectors to Parallel or to Orthogonal Ones, with Applications to Symmetry Transformations

Unitary and antiunitary operators which map vectors either to parallel or to orthogonal vectors are characterized. As an application the results are used to show that symmetry transformations induced by real functions are identity transformations.