Ab initio study on structure and phase transition of A- and B-type rare-earth sesquioxides Ln2O3 (Ln=La-Lu, Y, and Sc) based on density function theory

Ab initio energetic calculations based on the density functional theory (DFT) and projector augmented wave (PAW) pseudo-potentials method were performanced to determine the crystal structural parameters and phase transition data of the polymorphic rare-earth sesquioxides Ln₂O₃ (where Ln=La-Lu, Y, and Sc) with A-type (hexagonal) and B-type (monoclinic) configurations at ground state. The calculated results agree well with the limited experimental data and the critically assessed results. A set of systematic and self-consistent crystal structural parameters, energies and pressures of the phase transition were established for the whole series of the A- and B-type rare-earth sesquioxides Ln₂O₃. With the increase of the atomic number, the ionic radii of rare-earth elements Ln and the volumes of the sesquioxides Ln₂O₃ reflect the so-called “lanthanide contraction”. With the increase of the Ln³⁺-cation radius, the bulk modulus of Ln₂O₃ decreases and the polymorphic structures show a degenerative tendency.     

A localization argument for characters of reductive Lie groups

This article provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed by Schmid (in: Deformation Theory and Symplectic Geometry, Mathematical Physics Studies, Vol. 20, Kluwer Academic Publishers, Dordrecht, 1997, pp. 259-270).A corresponding problem in the compact group setting was solved by Berline et al. (Heat Kernels and Dirac Operators, Springer, Berlin, 1992) by an application of the theory of equivariant forms and particularly the fixed point integral localization formula. This article (besides its representation-theoretical significance) provides a whole family of examples where it is possible to localize integrals to fixed points with respect to an action of a noncompact group. Moreover, a localization argument given here is not specific to the particular setting considered in this article and can be extended to a more general situation.There is a broadly accessible article (Libine, A Localization Argument for Characters of Reductive Lie Groups: An Introduction and Examples, 2002, math.RT/0208024) which explains how the argument works in the case, where the key ideas are not obstructed by technical details and where it becomes clear how it extends to the general case.     

Integrals of equivariant forms and a Gauss-Bonnet theorem for constructible sheaves

The Berline-Vergne integral localization formula for equivariantly closed forms ([N. Berline, M. Vergne, Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris 295 (1982) 539-541], Theorem 7.11 in [N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, 1992]) is well-known and requires the acting Lie group to be compact. In this article, we extend this result to real reductive Lie groups G_R.As an application of this generalization, we prove an analogue of the Gauss-Bonnet theorem for constructible sheaves. If F is a G_R-equivariant sheaf on a complex projective manifold M, then the Euler characteristic of M with respect to F

     

Thermodynamic and NMR studies of mixed-ligand complex formation of cadmium ethylenediaminetetraacetate with diamines in an aqueous solution

The mixed-ligand complex formation in the systems Cd2 + Edta4–(CH₂) _n (NH₂)₂, n = 2 (En), 6 (L) has been NMR and calorimetrically studied in aqueous solution at 298.15 K and the ionic strength of I = 0.5 (KNO₃). The thermodynamic parameters of formation of the CdEdtaL²⁻, CdEdtaHL⁻, (CdEdta)₂L⁴⁻, and (CdEdta)₂En⁴⁻ complexes have been determined. The most probable coordination mode for the complexone and the diamine ligand in the mixed-ligand complexes was discussed.     

Phase Equilibria in the Ternary System Fe-Gd-Mo at 600 °C

 Phase equilibria in the ternary system Fe-Gd-Mo at 600 °C were determined. The phase composition for different element concentrations were quantitatively determined from the diffraction patterns by means of the multi-phase Rietveld-refinement as well as through evaluation of the microstructure images obtained by the scanning electron microscopy. Both methods are compared with each other with respect to their precision and limitations. The complete isothermal section at 600 °C includes one ternary phase τ (ThMn₁₂-type of structure), four pseudo-binary phases (Fe,Mo)₁₇Gd₂, (Fe,Mo)₂₃Gd₆, (Fe,Mo)₃Gd and (Fe,Mo)₂Gd and binary phases Fe₂Mo and μ-(Fe,Mo). The ternary phase τ forms the tie-lines with the solid solutions α-Fe, (Mo)-, Fe₂Mo and μ-(Fe,Mo) phases as well as with the pseudo-binary Fe-Gd compounds. Three phases (Fe,Mo)₂Gd, (Mo) and Gd coexist in a wide concentrations range. The homogeneity region of the ternary phase τ as well as the solubilities of the third element in the binary phases were determined.     

A fixed point localization formula for the Fourier transform of regular semisimple coadjoint orbits

Let be a Lie group acting on an oriented manifold M, and let ω be an equivariantly closed form on M. If both and M are compact, then the integral is given by the fixed point integral localization formula (Theorem 7.11 in Berline et al. Heat Kernels and Dirac Operators, Springer, Berlin, 1992). Unfortunately, this formula fails when the acting Lie group is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of in such a way that all fixed points are accounted for.Let be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly closed forms is the symplectic volume form dβ of a coadjoint orbit Ω. Even if Ω is not compact, the integral exists as a distribution on the Lie algebra . This distribution is called the Fourier transform of the coadjoint orbit.In this article, we will apply the localization results described in [L1,L2] to get a geometric derivation of Harish-Chandra's formula (9) for the Fourier transforms of regular semisimple coadjoint orbits. Then, we will make an explicit computation for the coadjoint orbits of elements of which are dual to regular semisimple elements lying in a maximally split Cartan subalgebra of .     

Republication of: Quantum theory of weak gravitational fields

This is an English translation of a paper by Matvei Bronstein, first published in German in 1936 in a long-extinct Soviet journal, in which he presented the first attempt at quantizing a weak (linearized) gravitational field, rather modern in its approach. The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by Stanley Deser and Alexei Starobinsky, and Bronstein’s brief biography written by Stanley Deser.